The Catalogue of Computational Material Models: Basic Geometrically Linear Models in 1D 3030636836, 9783030636838

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Paul Steinmann Kenneth Runesson

The Catalogue of Computational Material Models Basic Geometrically Linear Models in 1D

The Catalogue of Computational Material Models

Paul Steinmann Kenneth Runesson •

The Catalogue of Computational Material Models Basic Geometrically Linear Models in 1D

123

Paul Steinmann Department of Mechanical Engineering Friedrich-Alexander-Universität Erlangen-Nürnberg Erlangen, Germany

Kenneth Runesson Department of Industrial and Material Sciences Chalmers University of Technology Göteborg, Sweden

ISBN 978-3-030-63683-8 ISBN 978-3-030-63684-5 https://doi.org/10.1007/978-3-030-63684-5

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The present first part of the Catalogue of Computational Material Models reflects our (indeed very) long-standing ambition to compile a rigorous approach to the continuum modelling of materials and its corresponding computational treatment in a comprehensive, unifying and (hopefully) didactic treatise. However, as it often goes with human ambitions, our energy to pursue this encyclopedically scheduled endeavour came and went in waves. During an initial high-tide phase almost two decades ago we attempted to cover the topic in the widest possible fashion in what we called among us—unhumbly and somewhat pompously—The Book. At the end of that phase we encountered the typical frustration of authors when the own high-flying plans fail in front of seemingly insurmountable obstacles piling up in front. On the one hand, the envisioned all-encompassing, yet unified treatment became rather cumbersome in view of the ever increasing topical issues that we wanted to include. On the other hand, daily routine and a variety of professional and private distractions took their toll (that we gladly paid). As a consequence, in the sequel we endured a quite long low-tide phase for the project with some smaller superposed waves of new attempts here and there. Nevertheless, it seemed our endeavour was more or less doomed. Only recently, with a quite healthy distance to our earlier attempts, we re-viewed and re-scheduled our initial plans. Unexpectedly, having not pushed the project too desperately during the low-tide phase was rather fortunate. In hindsight the long maturation time was indeed beneficial. It turned out that the way we liked to attempt continuum modelling had somewhat evolved to a more stringent approach based on concepts from convex analysis. This allowed us to attack the various material models entirely afresh. Likewise, with the calm (to not call it wisdom) of age it proved much easier to decide on substantial re-arrangements of the overwhelming material with an initial restriction to only Basic Geometrically Linear One-Dimensional Models. These already showcase all archetypical material behaviours and computational approaches that are likewise displayed by more sophisticated models to be tackled in separate parts later, however without being overshadowed by too much of complexity.

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Preface

With our energy re-loaded and the firm determination to assemble in a first step an introduction to computational material modelling, we eventually rode on a high-tide surf again. This time we succeeded to cope with our more realistic ambitions, the result of which is the present first part of the Catalogue of Computational Material Models. We wish to believe that the Catalogue of Computational Material Models with its focus on Basic Geometrically Linear One-Dimensional Models is of great help to post-graduate students, researchers, course lectures and professionals in computational engineering and neighbouring disciplines alike. Erlangen, Germany Gothenburg, Sweden January 2021

Paul Steinmann Kenneth Runesson

Disclaimer

The Catalogue of Computational Material Models is NEITHER a textbook NOR a monograph, however, it is a Catalogue in its true encyclopaedic sense. After starting out with elasticity as a reference that the reader is expected to be familiar with, it outlines the theoretical formulation and algorithmic treatment of 15 different basic variants (5  each of visco-elasticity, plasticity, visco-plasticity, respectively) of paradigmatic material models. The presentation for each of these basic material models is a stand-alone account and intentionally follows exactly the same structure. Thereby, the central idea is that the reader can separately consult the Catalogue for any of the computational material models and find a self-contained exposition of the corresponding theoretical formulation and algorithmic treatment without the need to cross-refer to other basic material models. Nevertheless, this concept also allows easy comparison of the theoretical formulation and resulting algorithmic treatment for different basic material models and, thereby, to uncover in detail similarities and differences. It is, of course, obvious that one-dimensional material models are not immediately suited for real world analyses without extension to 3D. Yet, however, we are firmly convinced that basic material models in a one-dimensional, geometrically linear setting are indeed of utmost relevance and value, since they already highlight the main characteristics and differences of paradigmatic material behaviour without requiring mastering the intricacies of tensor calculus in multiple dimensions. To facilitate comparison, the Catalogue analyses the response of each basic material model for identical histories of prescribed strain and stress. This seemingly repetitious approach allows clearly showcasing and contrasting the characteristics of various modelling options. Finally, it is clear that it is often necessary to adopt geometrically nonlinear material models, e.g. when modelling metal forming or soft matter response. However, the kinematical intricacies of a geometrically nonlinear approach, which

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Disclaimer

are rooted in concepts from differential geometry, easily overshadow the main characteristics of paradigmatic material models. Thus, the Catalogue focusses entirely on the geometrically linear setting, although without sacrificing generality. Indeed, all of the models presented in the Catalogue are of utmost relevance and routinely used in a geometrically linear setting.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Compositions of Rheological Models . . . . . . . . . . . . . . . . 1.2.1 Serial Arrangement of ERMs . . . . . . . . . . . . . . . 1.2.2 Parallel Arrangement of ERMs . . . . . . . . . . . . . . 1.2.3 Arrangement of Serial-CRMs and Parallel-CRMs . 1.2.4 Serial Arrangement of ERMs and Parallel-CRMs . 1.2.5 Parallel Arrangement of ERMs and Serial-CRMs . 1.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Modelling Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Continuum Mechanics . . . . . . . . . . . . . . . 2.1.2 Dissipation Consistent Material Modelling . 2.2 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Constitutive Integrator . . . . . . . . . . . . . . . 2.2.2 Finite Element Method . . . . . . . . . . . . . . . 2.3 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Heaviside Function and Causal Signals . . . 2.3.2 Laplace Transformation . . . . . . . . . . . . . . 2.3.3 Complex Representations . . . . . . . . . . . . . 2.3.4 Legendre Transformation . . . . . . . . . . . . . 2.3.5 Constrained Optimization . . . . . . . . . . . . .

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3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hooke Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Specific Hooke Model: Formulation . . . . . . 3.1.2 Specific Hooke Model: Algorithmic Update .

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3.1.3 3.1.4

Specific Hooke Model: Response Analysis . . . . . . . . . . . Generic Hooke Model: Formulation . . . . . . . . . . . . . . . .

4 Visco-Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Newton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Specific Newton Model: Formulation . . . . . . . . . . . 4.1.2 Specific Newton Model: Algorithmic Update . . . . . . 4.1.3 Specific Newton Model: Response Analysis . . . . . . 4.1.4 Generic Newton Model: Formulation . . . . . . . . . . . 4.2 Kelvin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Specific Kelvin Model: Formulation . . . . . . . . . . . . 4.2.2 Specific Kelvin Model: Algorithmic Update . . . . . . . 4.2.3 Specific Kelvin Model: Response Analysis . . . . . . . 4.2.4 Generic Kelvin Model: Formulation . . . . . . . . . . . . 4.3 Generalized-Kelvin Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Standard-Linear-Solid Kelvin Model: Formulation . . 4.3.2 Standard-Linear-Solid Kelvin Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Standard-Linear-Solid Kelvin Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Generic Generalized-Kelvin Model: Formulation . . . 4.4 Maxwell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Specific Maxwell Model: Formulation . . . . . . . . . . . 4.4.2 Specific Maxwell Model: Algorithmic Update . . . . . 4.4.3 Specific Maxwell Model: Response Analysis . . . . . . 4.4.4 Generic Maxwell Model: Formulation . . . . . . . . . . . 4.5 Generalized-Maxwell Model . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Standard-Linear-Solid Maxwell Model: Formulation 4.5.2 Standard-Linear-Solid Maxwell Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Standard-Linear-Solid Maxwell Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Generic Generalized-Maxwell Model: Formulation . .

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5 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 St. Venant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Specific St. Venant Model: Formulation . . . . . . 5.1.2 Specific St. Venant Model: Algorithmic Update . 5.1.3 Specific St. Venant Model: Response Analysis . 5.1.4 Generic St. Venant Model: Formulation . . . . . . 5.2 Prandtl Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Specific Prandtl Model: Formulation . . . . . . . . . 5.2.2 Specific Prandtl Model: Algorithmic Update . . .

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5.2.3 5.2.4 5.3 Prandtl 5.3.1

Specific Prandtl Model: Response Analysis . . . . . . . . . Generic Prandtl Model: Formulation . . . . . . . . . . . . . . Hardening Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Prandtl Isotropic Hardening Model: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Specific Prandtl Isotropic Hardening Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Specific Prandtl Isotropic Hardening Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Specific Prandtl Kinematic Hardening Model: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Specific Prandtl Kinematic Hardening Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Specific Prandtl Kinematic Hardening Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.7 Specific Prandtl Mixed Hardening Model: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.8 Specific Prandtl Mixed Hardening Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.9 Specific Prandtl Mixed Hardening Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.10 Generic Prandtl Hardening Model: Formulation . . . . . .

6 Visco-Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bingham Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Specific Bingham Model: Formulation . . . . . . 6.1.2 Specific Bingham Model: Algorithmic Update . 6.1.3 Specific Bingham Model: Response Analysis . . 6.1.4 Generic Bingham Model: Formulation . . . . . . . 6.2 Perzyna Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Specific Perzyna Model: Formulation . . . . . . . 6.2.2 Specific Perzyna Model: Algorithmic Update . . 6.2.3 Specific Perzyna Model: Response Analysis . . 6.2.4 Generic Perzyna Model: Formulation . . . . . . . 6.3 Perzyna Hardening Model . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Specific Perzyna Isotropic Hardening Model: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Specific Perzyna Isotropic Hardening Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . 6.3.3 Specific Perzyna Isotropic Hardening Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . 6.3.4 Specific Perzyna Kinematic Hardening Model: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Specific Perzyna Kinematic Hardening Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . .

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6.3.6

Specific Perzyna Kinematic Hardening Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.7 Specific Perzyna Mixed Hardening Model: Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.8 Specific Perzyna Mixed Hardening Model: Algorithmic Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.9 Specific Perzyna Mixed Hardening Model: Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.10 Generic Perzyna Hardening Model: Formulation . . . . .

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Chapter 1

Introduction

What you have in your head, put down on paper. The head is a fragile vessel. — Dmitri Shostakovich, 1906–1975 —

1.1 Motivation All materials respond in one way or other to external stimuli, e.g. mechanical, thermal, chemical, electro-magnetic (or other) loading. The material’s response may typically be observed in terms of state variables—deformation, temperature, concentration and electro-magnetic potentials—and/or in terms of their spatial gradients— strain, temperature gradient, concentration gradient, electric field and magnetic flux -, together with the corresponding state functions—stress, heat and mass flux, electric flux and magnetic field. However, one should recall that from a philosophical point of view these observations already rely in some way on the underlying phenomenological concepts of the above listed state variables and state functions (collectively forming the state quantities). Thus, even the most careful and sophisticated observations at smallest and largest time and length scales are necessarily only phenomenological in nature. Any model (including quantum and cosmological mechanics) based on necessarily phenomenological observations is consequently also only describing phenomena, since true knowledge or understanding of the underlying nature is intrinsically unavailable (recall also Plato’s cave allegory). Of course, this is not at all a desperate state of affairs, mastering nature based on a mere phenomenological understanding—to date ranging from the quantum to the cosmological level—proves to be extremely powerful!

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Steinmann and K. Runesson, The Catalogue of Computational Material Models, https://doi.org/10.1007/978-3-030-63684-5_1

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1 Introduction

Table 1.1 Role and composition of a material model

Observation of Phenomena ↓ Relevance of Phenomena ↓ Material Model Physical Model Mathematical Model (Concepts) (Equations) ↓ Algorithm ↓ Prediction of Phenomena

Moreover, the common believe is that the more observed phenomena a model captures, the closer are its predictions to the true response, however usually at the expense of more sophisticated modelling. This allows the modelling effort to be adapted to the detail of phenomena that shall be captured. It is in this regard that a material model aims in phenomenologically describing only the relevant response of a material to external stimuli. Thereby, a material model consists of a physical model and a mathematical model, see Table 1.1. A physical model assembles concepts regarding the underlying physical structures and mechanisms, take for example the physical model of a regular arrangement of atoms in a crystalline lattice and the movement of atomic defects through this lattice. Likewise, all of the above listed phenomenological concepts of state quantities are part of the physical model. A mathematical model consist of (sets of) equations of either algebraic and/or differential (and/or integral) format that relate the state variables and state functions. Thereby, for modelling economy a mathematical model shall only capture what are deemed to be the most relevant phenomena. Relevance is of course a subjective perception that needs to be defined on a case by case basis with the concrete application in mind (“relevance is in the eye of the beholder” or “relevance follows (the) application”). For example, whereas the details of the intricate electronic structure of matter as phenomenologically described by quantum mechanics may be irrelevant for modelling the mechanical response of a material at an engineering scale, they may be of utmost relevance when describing the material’s micro electronic properties. Oftentimes the underlying equations of a material model are either cumbersome or even impossible to solve analytically. Here computational discretisation comes to a help by transforming the (time and space) continuous setting of the mathematical part of the material model into a corresponding algorithm. The resulting, so-called computational material model then allows for approximate response predictions with controllable accuracy that can be compared to the analytical solution of the mathematical part of the material model (at least in principle). Thereby an algorithm shall be convergent (stable and consistent) so that the accuracy of the response prediction

1.1 Motivation

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Table 1.2 Classification of material’s basic mechanical phenomena into basic categories and resulting paradigmatic material models Mechanical phenomena Rate-independent Rate-dependent Quasi-statically reversible Quasi-statically irreversible

Elasticity Plasticity

Visco-elasticity Visco-plasticity

monotonically scales with the computational effort and eventually converges to the analytical solution in the limit of an infinitely refined discretisation. In this treatise restriction is purposely made to capture only mechanical phenomena, essentially in terms of stress and strain histories at the engineering scale for archetypical material response behaviour. Thereby, since both observation and modelling are restricted to the engineering scale, typically non-observable, so-called internal variables that model hidden sub-scale processes, i.e. processes at underlying non-observed/non-observable scales, need additionally to be included into the list of state variables. Basic mechanical phenomena observed at the engineering scale may be classified into basic categories. Therein the mechanical response of a material to external mechanical stimuli is characterised, on the one hand, as being either rate-independent or rate-dependent and, on the other hand, as being either quasi-statically reversible or irreversible. Paradigms for the possible combinations of these categories are material models for elasticity, visco-elasticity, plasticity and visco-plasticity, see the matrix arrangement in Table 1.2. By-passing the challenges of tensor calculus in multiple dimensions, the main characteristics of these paradigmatic material models are best highlighted in a onedimensional, geometrically linear context. This is the motivation for the present Catalogue of Computational Material Models to exclusively focus on Basic Geometrically Linear One-Dimensional Models that are captured by arbitrarily sophisticated combinations of elementary rheological models, see the remainder of Chap. 1. To set the stage, some preliminaries regarding necessary modelling, computational, and mathematical tools are assembled in Chap. 2. Thereafter, the remaining Chaps. 3–6 are concerned with the actual Catalogue of Computational Material Models. To this end, after starting out with elasticity as a reference, further 15 different basic variants (5×each of {visco-elasticity, plasticity, visco-plasticity}, respectively) of the paradigmatic material models in Table 1.2 are systematically explored. The presentation for each of these basic material models is a stand-alone account and follows in each case exactly the same generic structure as outlined in Table 1.3. On the one hand, this allows, in the true sense of a catalogue, to consult each of the basic material models separately without the need to refer to other basic material models. On the other hand, even though this somewhat repetitious concept may seem tedious, it allows to easily compare the formulation and resulting algorithmic setting of the various basic material models and thereby to uncover, in detail, similarities and differences. In particular, the response of each basic material model is analysed

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1 Introduction

Table 1.3 Generic structure for the presentation of basic material models Specific model: formulation Specific model: algorithmic update Specific model: response analysis Generic model: formulation

for the identical histories (Zig-Zag, Sine, Ramp) of prescribed strain and stress so as to clearly showcase and to contrast to each other the characteristics of the various modelling options.

1.2 Compositions of Rheological Models Phenomenologically, the mechanical behavior of generic materials (either solid or fluid) can be captured in a one-dimensional setting by arbitrarily sophisticated combined rheological models (CRM). These can be composed by serial and parallel arrangements (and arbitrarily sophisticated combinations thereof) of only a few elementary rheological models (ERM) such as elastic springs, viscous dashpots, and frictional sliders, which are denoted the Hooke model, the Newton model, and the St. Venant model, respectively, see Fig. 1.1 (more exotic ERMs capturing, e.g., curing or ageing are ignored here for the sake of conciseness). In the following a few of the infinitely many possibilities for the systematic arrangements of these ERMs are explored. Thereby only those resulting CRMs that carry the name of a scientist in bold font are of relevance for the computational material models considered in this treatise (especially if solid materials are concerned), the remaining resulting CRMs either carry a name of a scientist in sans serif font or, if no established name is available, are denoted by simply stringing together their

Fig. 1.1 Elementary rheological models (ERM): elastic spring, viscous dashpot, and frictional slider. These are denoted the Hooke, the Newton, and the St. Venant model, respectively

σ

σ

σ

σ

σ

σ

1.2 Compositions of Rheological Models

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constituting ERMs in sans serif font1 (the named and un-named CRMs in sans serif font that are mainly of relevance to model fluid materials will be considered separately elsewhere). Generically, the ERMs from Fig. 1.1 provide input/output relations between the stress σ and the strain  (or the strain rate ˙ ) σ (σ

ERM ↔  ↔ ˙ )

(1.1)

1.2.1 Serial Arrangement of ERMs For a serial arrangement of two ERMs, say ERM1 and ERM2 , the strain  of the resulting serial-CRM is the summation of the strains 1 and 2 of the two ERMs, whereas their stresses coincide  = 1 + 2 and

σ = σ1 = σ2 .

(1.2)

Possible serial arrangements of any two of the three ERMs from Fig. 1.1 are showcased in Fig. 1.2. Out of these, the catalogue contains accounts on the Maxwell model and the Prandtl model, respectively.

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.2 Serial arrangements of ERMs. These are denoted the Maxwell, the dashpot-slider, and the Prandtl model, respectively

1 Like,

e.g., ERM1 -[ERM2 +ERM3 ] for a CRM consisting of a serial arrangement of ERM1 with a parallel arrangement of ERM2 and ERM3 .

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1 Introduction σ

σ

1

2

3

Fig. 1.3 Serial arrangement of all ERMs denoted the spring-dashpot-slider model

Likewise, for the serial arrangement of all three ERMs, say ERM1 to ERM3 , the strain  of the resulting serial-CRM is the summation of the strains 1 to 3 of the three ERMs, whereas their stresses coincide  = 1 + 2 + 3 and

σ = σ1 = σ2 = σ3 .

(1.3)

The serial arrangement of all three ERMs from Fig. 1.1 is showcased for completeness in Fig. 1.3.

1.2.2 Parallel Arrangement of ERMs For a parallel arrangement of two ERMs, say ERM1 and ERM2 , the stress σ of the resulting parallel-CRM is the summation of the stresses σ1 and σ2 of the two ERMs, whereas their strains coincide σ = σ1 + σ2 and

 = 1 =  2 .

(1.4)

Possible parallel arrangements of any two of the three ERMs from Fig. 1.1 are showcased in Fig. 1.4. Out of these, the catalogue contains accounts on the Kelvin model and the Bingham model, respectively. Likewise, for the parallel arrangement of all three ERMs, say ERM1 to ERM3 , the stress σ of the resulting parallel-CRM is the summation of the stresses σ1 to σ3 of the three ERMs, whereas their strains coincide σ = σ1 + σ2 + σ3 and

 = 1 =  2 =  3 .

(1.5)

The parallel arrangement of all three ERMs from Fig. 1.1 is showcased for completeness in Fig. 1.5.

1.2 Compositions of Rheological Models

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Fig. 1.4 Parallel arrangements of ERMs. These are denoted the Kelvin, the Bingham, and the St. Venant hardening model, respectively

σ

σ

σ

σ

σ

σ

Fig. 1.5 Parallel arrangement of all ERMs denoted the Bingham Hardening model σ

σ

1.2.3 Arrangement of Serial-CRMs and Parallel-CRMs The three serial-CRMs from Fig. 1.2 and the three parallel-CRMs from Fig. 1.4 may be further combined into more complex-CRMs in various ways. Here only one of the many possible permutations, the Burgers model, shall be considered: For a serial arrangement of a serial-CRM, say CRM1 , that consists of two serial secondary ERMs, say ERM1.1 and ERM1.2 , with a parallel-CRM, say CRM2 , that

8

1 Introduction

σ

σ

1.1

1.2 1

2

Fig. 1.6 Serial arrangement of serial- and parallel-CRMs. The particular choice is denoted the Burgers model

consists of two parallel secondary ERMs, say ERM2.1 and ERM2.2 , the strain  of the resulting complex-CRM is a summation of the resulting strain 1 of the serial-CRM (the summation of the strains 1.1 and 1.2 in the two serial secondary ERMs) and the strain 2 of the parallel-CRM (the strains in the two parallel secondary ERM coincide 2.1 = 2.2 ), whereas the stress σ of the resulting complex-CRM coincides with the stress σ1 in the serial-CRM (the stresses in the two serial secondary ERM coincide σ1.1 = σ1.2 ) and the resulting stress σ2 in the parallel-CRM (the summation of the stresses σ2.1 and σ2.2 in the two parallel secondary ERMs)  = 1 + 2 and σ = σ1 = σ2 with 1 = 1.1 + 1.2 and σ1 = σ1.1 = σ1.2 as well as σ2 = σ2.1 + σ2.2 and 2 = 2.1 = 2.2 .

(1.6)

The serial arrangement of the Maxwell model, a particular serial-CRM from Fig. 1.2, and the Kelvin model, a particular parallel-CRM from Fig. 1.4, into the Burgers model is showcased in Fig. 1.6.

1.2.4 Serial Arrangement of ERMs and Parallel-CRMs For a serial arrangement of a primary ERM, say ERM1 , with a parallel-CRM, say CRM2 , that consists of two parallel secondary ERMs, say ERM2.1 and ERM2.2 , the strain  of the resulting serial-parallel-CRM is a summation of the strain 1 of the primary ERM and the strain 2 of the parallel-CRM (the strains in the two parallel secondary ERM coincide 2.1 = 2.2 ), whereas the stress σ of the resulting serialparallel-CRM coincides with the stress σ1 in the primary ERM and the resulting stress σ2 in the parallel-CRM (the summation of the stresses σ2.1 and σ2.2 in the two parallel secondary ERMs)

1.2 Compositions of Rheological Models

9

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.7 Serial-parallel arrangements of ERMs. These are denoted the Generalized Kelvin SLS (Zener-K), the Perzyna (Bingham–Hooke), and the Prandtl hardening model, respectively

 = 1 + 2 and σ = σ1 = σ2 with σ2 = σ2.1 + σ2.2 and 2 = 2.1 = 2.2 .

(1.7)

Possible serial-parallel combined arrangements of the three ERMs from Fig. 1.1 are showcased in Figs. 1.7, 1.8, and 1.9. Out of these, the catalogue contains accounts on the Generalized Kelvin SLS (standard linear solid) model, the Perzyna model and the Prandtl hardening model, respectively. For a serial arrangement of a primary ERM, say ERM1 , with a parallel-CRM, say CRM2 , that consists of three parallel secondary ERMs, say ERM2.1 to ERM2.3 , the strain  of the resulting serial-parallel-CRM is a summation of the strain 1 of the primary ERM and the strain 2 of the parallel-CRM (the strains in the three parallel secondary ERM coincide 2.1 = 2.2 = 2.3 ), whereas the stress σ of the resulting serial-parallel-CRM coincides with the stress σ1 in the primary ERM and the resulting stress σ2 in the parallel-CRM (the summation of the stresses σ2.1 to σ2.3 in the three parallel secondary ERMs)

10

1 Introduction

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.8 Serial-parallel arrangements of ERMs. These are denoted the Generalized Kelvin SLF (Lethersich), the dashpot-[dashpot+slider], and the dashpot-[slider+spring] model, respectively

 = 1 + 2 and σ = σ1 = σ2 with σ2 = σ2.1 + σ2.2 + σ2.3 and 2 = 2.1 = 2.2 = 2.2 .

(1.8)

Possible serial-parallel combined arrangements of the three ERMs from Fig. 1.1 are showcased in Fig. 1.10. Out of these, the catalogue contains an account on the Perzyna hardening model.

1.2.5 Parallel Arrangement of ERMs and Serial-CRMs For a parallel arrangement of a primary ERM, say ERM1 , with a serial-CRM, say CRM2 , that consists of two serial secondary ERMs, say ERM2.1 and ERM2.2 , the stress σ of the resulting parallel-serial-CRM is a summation of the stress σ1 of the

1.2 Compositions of Rheological Models

11

σ

σ

σ σ

σ σ

1

2

Fig. 1.9 Serial-parallel arrangements of ERMs. These are denoted the slider-[spring+dashpot], the slider-[dashpot+slider], and the slider-[slider+spring] model, respectively

primary ERM and the stress σ2 of the serial-CRM (the stresses in the two serial secondary ERM coincide σ2.1 = σ2.2 ), whereas the strain  of the resulting parallelserial-CRM coincides with the strain 1 in the primary ERM and the resulting strain 2 in the serial-CRM (the summation of the strains 2.1 and 2.2 in the two serial secondary ERMs) σ = σ1 + σ2 and  = 1 = 2 with 2 = 2.1 + 2.2 and σ2 = σ2.1 = σ2.2 .

(1.9)

Possible parallel-serial combined arrangements of the three ERMs from Fig. 1.1 are showcased in Figs. 1.11, 1.12, and 1.13. Out of these, the catalogue contains an account on the Generalized Maxwell SLS model. For a parallel arrangement of a primary ERM, say ERM1 , with a serial-CRM, say CRM2 , that consists of three serial secondary ERMs, say ERM2.1 to ERM2.3 , the stress σ of the resulting parallel-serial-CRM is a summation of the stress σ1 of the primary ERM and the stress σ2 of the serial-CRM (the stresses in the three serial

12

1 Introduction

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.10 Serial-parallel arrangements of ERMs. These are denoted the Perzyna hardening, the dashpot-[spring+dashpot+slider], and the slider-[spring+dashpot+slider] model, respectively

1.2 Compositions of Rheological Models

13

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.11 Parallel-serial arrangements of ERMs. These are denoted the Generalized Maxwell SLS (Zener-M), the spring+[dashpot-slider], and the Generalized Prandtl SLS model, respectively

secondary ERM coincide σ2.1 = σ2.2 = σ2.3 ), whereas the strain  of the resulting parallel-serial-CRM coincides with the strain 1 in the primary ERM and the resulting strain 2 in the serial-CRM (the summation of the strains 2.1 to 2.3 in the three serial secondary ERMs) σ = σ1 + σ2 and  = 1 = 2 with 2 = 2.1 + 2.2 + 2.3 and σ2 = σ2.1 = σ2.2 = σ2.2 .

(1.10)

Possible parallel-serial combined arrangements of the three ERMs from Fig. 1.1 are showcased for completeness in Fig. 1.14.

14

1 Introduction

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.12 Parallel-serial arrangements of ERMs. These are denoted the Generalized Maxwell SLF (Jeffreys), the dashpot+[dashpot-slider], and the dashpot+[slider-spring] model, respectively

1.3 Further Reading The Catalogue of Computational Material Models focuses on Basic Geometrically Linear Models in 1D. These display already all characteristic features of arbitrarily advanced material models without overshadowing the affairs by too much of sophistication and complexity. However, oftentimes advanced geometrically non-linear models in 3D, with possible coupling to non-mechanical fields, are required for reallife applications. For extensions to such more general scenarios, the following further reading is recommended: The roots of present-days Continuum Mechanics trace back to the epochal publications by Truesdell and Toupin [1], that for the first time develops a rigorously unified theory of continuum physics, Truesdell and Noll [2], that lays the foundation of rational continuum mechanics, and Truesdell [3], that gives a more accessible introduction to rational continuum mechanics. Alongside these contributions, Flügge [4] provides a classical guide to tensor analysis in continuum mechanics. Classical references on continuum mechanics, that perhaps require less obstacles to overcome

1.3 Further Reading

15

σ

σ

σ

σ

σ

σ

1

2

Fig. 1.13 Parallel-serial arrangements of ERMs. These are denoted the slider+[spring-dashpot], the slider+[dashpot-slider], and the slider+[slider-spring] model, respectively

by the reader, are those of Prager [5], Chadwick [6], and Spencer [7]. As an early precursor to today’s numerical methods, Washizu [8] pioneered variational methods in elasticity and plasticity. The comprehensive overview by Malvern [9] is undoubtedly the all-times classic reference on continuum mechanics. Succeeding the advent of rational continuum mechanics are the monographs by Gurtin [10], that delivers a precise introduction to continuum mechanics, Eringen [11, 12], that assemble scholarly overviews on continuum mechanics, and, more recently, Liu [13], that contributes a concise account on rational continuum thermodynamics. The today’s take on continuum mechanics is dominated by the expositions by Holzapfel [14], that represents a stage-setting textbook on non-linear solid mechanics, Maugin [15], that dwells on the fundamentals of continuum thermodynamics based on internal variables, Gurtin, Fried and Anand [16], that demonstrates an expert outline of continuum mechanics and thermodynamics, Silhavy [17] that elaborates a mathematically stringent presentation of continuum mechanics and thermodynamics, and Tadmor, Miller and Elliot [18], that outlines the fundamentals of continuum mechanics and thermodynamics. Murdoch [19] elucidates the intricate relation between molecular mechanics and con-

16

1 Introduction

σ

σ

σ

σ

σ

σ

1

2

3

Fig. 1.14 Parallel-serial arrangements of ERMs. These are denoted the spring+[spring-dashpotslider], the dashpot+[spring-dashpot-slider], and the slider+[spring-dashpot-slider] model, respectively

tinuum mechanics. Further noteworthy texts on continuum mechanics are, e.g., by Basar and Weichert [20], that systematically presents its mathematical fundamentals and physical concepts, Lai et al. [21], that accounts for an introductory exposition, and Reddy [22] that aims for a more advanced textbook. Besides the generic kinematic and balance equations, the material-specific constitutive relations that derive from Material Modelling represent a corner stone of continuum mechanics. In this regard the book by Lemaitre and Chaboche [23] is a milestone representative of the French school of mechanics. A rigorous account on finite constitutive modelling of generic materials is contributed by Haupt [24]. Ottosen and Ristinmaa [25] assemble a comprehensive overview on a broad bandwidth of constitutive modelling, whereas Tadmor and Miller [26] elaborate on the atomistic foundations of the continuum constitutive modelling approach. A self-consistent introduction specifically devoted to non-linear elasticity and visco-elasticity is due to Drozdov [27]. Likewise, Bertram [28] presents a careful introduction to modern non-linear elasticity and plasticity, while Bertram and Glüge [29] focus on a concise introduction to generic material continuum modelling.

1.3 Further Reading

17

Elasticity characterising rate-independent, reversible material response is perhaps the best studied material model in continuum mechanics. Some of the classical references on theoretical and mathematical elasticity are the books by Green and Zerna [30], Love [31], and Muskhelishvili [32]. A masterly introduction to rational nonlinear elasticity is given by Wang and Truesdell [33]. Emphasizing its geometrical underpinnings, Marsden and Hughes [34] elaborate a quite challenging, yet frequently cited, tract on the mathematical foundations of elasticity. The monograph by Ogden [35] is acknowledgedly a must-have introduction to non-linear elastic deformations. A comprehensive exposition of demanding non-linear problems in elasticity is due to Antmann [36]. The tracts by Ciarlet [37, 38] dwell on analysis-based presentations of non-linear elasticity. Based on quasi-convexity, Pedregal [39] analyses the non/existence of solutions to variational methods in non-linear elasticity. Lurie [40] provides a detailed examination of theoretical elasticity. Linear elasticity is covered, e.g., in the rational treatise by Gurtin [41], the scholarly essay by Podio-Guidugli [42], and the more recent treatment by Slaughter [43]. Rate-dependent, reversible material response is captured by models from the realm of Visco-Elasticity. The early monograph by Flügge [44] is a reference survey on the formulation of visco-elasticity. Likewise, Bland [45] represents a classical reference on linear visco-elasticity. Drozdov [46] comprehensively reviews various visco-elastic response behaviours and the corresponding continuum modelling. Cho [47] and Phan-Thien and Mai-Duy [48], e.g., provide introductions to viscoelasticity that emphasize the importance of polymer rheology. A classical account on visco-elasticity is given in the introductory text by Christensen [49]. Fractional calculus and waves are considered by Mainardi [50] for mathematical models of linear visco-elasticity. With an interest in the analytical properties of the solutions to initial-boundary-value problems, Fabrizio and Morro [51] analyse linear viscoelasticity mathematically. A more engineering-oriented approach to visco-elasticity is presented by Gutierrez-Lemini [52]. As a framework for rate-independent, irreversible material response, Plasticity poses severe mathematical challenges to the modeler. In this regard the treatise by Hill [53] is the key reference on mathematical plasticity. The fundamentals of plasticity are outlined, e.g., in the classical account by Kachanov [54]. Chakrabarty [55] offers a comprehensive exposition of classical plasticity. The focus of the tract by Maugin [56] is on the mathematical setting of the thermodynamics of plasticity. Lubliner [57] presents a benchmark compilation of paradigmatic problems in plasticity. An advanced treatment of finite plasticity is given by Lubarda [58]. Likewise with a view on finite plasticity, Nemat-Nasser [59] treats the case of heterogeneous materials. Khan and Huang [60] contribute a lucid exposition of continuum plasticity. An integrated coverage of continuum mechanics and plasticity is due to Wu [61]. An up-to-date review of finite plasticity is represented in the monograph by Hashiguchi and Yamakawa [62]. The convex analysis setting of plasticity relevant to the present Catalogue of Computational Material Models is covered by Han and Reddy [63]. Visco-Plasticity combines rate-dependent and irreversible material response and is typically modelled as an extension to an underlying plasticity formulation. The foundational formulation of overstress-type visco-plasticity is due to Perzyna [64].

18

1 Introduction

The alternative formulation of visco-plasticity, based on convex analysis in the spirit of the ‘closest-point-projection’, is advocated by Duvant and Lions [65]. In the realm of fluid mechanics, Huilgol [66] considers fluids exhibiting a yield stress. Only few problems of continuum mechanics are amenable to analytical solutions, thus emphatically asking for Computational Continuum Mechanics. Bonet and Wood [67] present a bespoke combination of non-linear continuum mechanics and the finite element method. An examination of non-linear continuum mechanics in concert with the finite element method is offered by Ibrahimbegovic [68]. The early treatise by Oden [69] develops an amazing integration of non-linear continuum mechanics and the finite element method. Algorithmic illustrations of computational inelasticity are the topic of the contribution by Kojic and Bathe [70]. Shabana [71] attempts a state-of-the-art coverage of computational continuum mechanics. Le Tallec [72] provides an in-depth compilation of non-linear computational elasticity. A computational guide to visco-elasticity is given by Marques and Creus [73]. Computational elasticity and plasticity characterise the book by Anandarajah [74]. Borja [75] gives a narrative coverage of theoretical and computational plasticity. Dunne and Petrinic [76] strive for an engineering tutorial to computational plasticity. Advanced computational methods in plasticity are contained in De Souza Neto, Peric and Owen [77]. The comprehensive monograph by Hashiguchi [78] outlines theoretical and computational plasticity. Probably the key references to computational plasticity are the expositions by Simo and Hughes [79], that contains most influential computational approaches towards inelasticity, and by Simo [80], that focusses on the algorithmically-driven analysis and computation of plasticity. Continuum mechanics and material modelling are also underlying Structural Modelling. One example is the monograph by Jirasek and Bazant [81] that gives a systematic presentation of inelastic structural analysis. Chen and Han [82] provide an application-driven account on engineering plasticity, and Krenk [83] focusses on the combination of solid and structural mechanics from an engineering viewpoint. Furthermore, many aspects of Biomechanics are attacked by tools from non-linear continuum mechanics and the corresponding material modelling. A few examples are the book by Epstein [84], that takes a differential geometry view on continuum biomechanics, and the comprehensive exposition on mathematical and mechanical formulations of biological growth by Goriely [85]. Oftentimes, combinations of continuum mechanics with other areas of continuum physics, so-called Coupled Problems, are of considerable interest. A non-exhaustive list comprise the account by Coussy [86] on the mechanics of porous solids, the landmark tracts on continuum electrodynamics by Eringen and Maugin [87, 88], and the state-of-the-art treatment of electro- and magneto-elasticity by Dorfmann and Ogden [89]. Finally, various problems involving defects such as inclusions, vacancies and interfaces as well as singularities at crack tips, wedges and in dislocation cores can be approached within the unifying framework of Configurational Mechanics. Here the main protagonists are the milestone publications on material forces by Maugin [90, 91], the engineering presentation of mechanics in material space by Kienzler and Hermann [92], and the views on configurational forces expressed by Gurtin [93].

1.3 Further Reading

19

Miniaturisation requires continuum modelling to capture length-scale effects as predicted by Generalised Continua. A small selection are the authoritative treatises by Eringen [94–96] on micro-continuum and non-local field theories, the unifying account by Iesan [97] on thermo-elasticity of generalised continua, the comprehensive overview by Madenci and Oterkus [98] on peridynamics modelling, and the differential-geometry-driven treatise by Steinmann [99] on higher-order elasticity and plasticity. The above set of topics from the arena of continuum mechanics, material modelling and their periphery together with the corresponding references are by no means exhaustive. However, these are representative and shall serve as a good starting point if more detailed information is needed.

References 1. C. Truesdell, R. Toupin. The classical field theories, in Principles of classical Mechanics and Field Theory (Springer, Berlin, 1960), pp. 226–858 2. C. Truesdell, W. Noll, The Non-linear Field Theories of Mechanics (Springer, Berlin, 2004) 3. C. Truesdell, A First Course in Rational Continuum Mechanics (Academic, Cambridge, 1992) 4. W. Flügge, Tensor Analysis and Continuum Mechanics (Springer, Berlin, 1972) 5. W. Prager, Introduction to Mechanics of Continua (Dover, Illinois, 2004) 6. P. Chadwick, Continuum Mechanics: Concise Theory and Problems (Dover, Illinois, 2012) 7. A.J.M. Spencer, Continuum Mechanics (Dover, Illinois, 2004) 8. K. Washizu, Variational Methods in Elasticity and Plasticity (Pergamon Press, Oxford, 1975) 9. L.E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice Hall, Upper Saddle River, 1969) 10. M.E. Gurtin, An Introduction to Continuum Mechanics (Academic, Cambridge, 1982) 11. A.C. Eringen, Mechanics of Continua (R.E. Krieger Publishing Co., 1980) 12. A.C. Eringen, Continuum Mechanics of Single-substance Bodies (Academic, Cambridge, 2016) 13. I-S. Liu, Continuum Mechanics (Springer, Berlin, 2013) 14. G.A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering (Wiley, Hoboken, 2000) 15. G.A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction (World Scientific, Singapore, 1999) 16. M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua (Cambridge University Press, Cambridge, 2010) 17. M. Silhavy, The Mechanics and Thermodynamics of Continuous Media (Springer, Berlin, 2013) 18. E.B. Tadmor, R.E. Miller, R.S. Elliott, Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing Equations (Cambridge University Press, Cambridge, 2012) 19. A.I. Murdoch, Physical Foundations of Continuum Mechanics (Cambridge University Press, Cambridge, 2012) 20. Y. Basar, D. Weichert, Nonlinear Continuum Mechanics of Solids: Fundamental Mathematical and Physical Concepts (Springer, Berlin, 2013) 21. W.M. Lai, D.H. Rubin, E. Krempl, D. Rubin, Introduction to Continuum Mechanics (Butterworth-Heinemann, Oxford, 2009) 22. J.N. Reddy, An Introduction to Continuum Mechanics (Cambridge University Press, Cambridge, 2007) 23. J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials (Cambridge University Press, Cambridge, 1994)

20

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24. P. Haupt, Continuum Mechanics and Theory of Materials (Springer, Berlin, 2013) 25. N.S. Ottosen, M. Ristinmaa, The Mechanics of Constitutive Modeling (Elsevier, Amsterdam, 2005) 26. E.B. Tadmor, R.E. Miller, Modeling Materials: Continuum, Atomistic and Multiscale Techniques (Cambridge University Press, Cambridge, 2011) 27. A.D. Drozdov, Finite Elasticity and Viscoelasticity: A Course in the Nonlinear Mechanics of Solids (World Scientific, Singapore, 1996) 28. A. Bertram, Elasticity and Plasticity of Large Deformations (Springer, Berlin, 2012) 29. A. Bertram, R. Glüge, Solid Mechanics (Springer, Berlin, 2015) 30. A.E. Green, W. Zerna, Theoretical Elasticity (Dover, Illinois, 1992) 31. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge University Press, Cambridge, 2013) 32. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (Springer, Berlin, 2013) 33. C.-C. Wang, C. Truesdell, Introduction to Rational Elasticity (Springer, Berlin, 1973) 34. J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity (Dover, Illinois, 1994) 35. R.W. Ogden, Non-linear Elastic Deformations (Dover, Illinois, 1997) 36. S.S. Antman, Nonlinear Problems of Elasticity (Springer, Berlin, 2005) 37. P.G. Ciarlet, Mathematical Elasticity: Volume I: Three-dimensional Elasticity (North-Holland, 1988) 38. P.G. Ciarlet, Mathematical Elasticity: Volume II: Theory of Plates (Elsevier, Amsterdam, 1997) 39. P. Pedregal, Variational Methods in Nonlinear Elasticity (Siam, 2000) 40. A.I. Lurie, Non-linear Theory of Elasticity (Elsevier, Amsterdam, 2012) 41. M.E. Gurtin, The linear theory of elasticity, in Linear Theories of Elasticity and Thermoelasticity (Springer, Berlin, 1973), pp. 1–295 42. P. Podio-Guidugli, A Primer in Elasticity (Springer, Berlin, 2000) 43. W.S. Slaughter, The Linearized Theory of Elasticity (Springer, Berlin, 2012) 44. W. Flügge, Viscoelasticity (Springer, Berlin, 1975) 45. D.R. Bland, The Theory of Linear Viscoelasticity (Dover, Illinois, 2016) 46. A.D. Drozdov, Mechanics of Viscoelastic Solids (Wiley, Hoboken, 1998) 47. K.S. Cho, Viscoelasticity of Polymers: Theory and Numerical Algorithms (Springer, Berlin, 2016) 48. N. Phan-Thien, N. Mai-Duy, Understanding Viscoelasticity: An Introduction to Rheology (Springer, Berlin, 2017) 49. R. Christensen, Theory of Viscoelasticity: An Introduction (Elsevier, Amsterdam, 2012) 50. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010) 51. M. Fabrizio, A. Morro, Mathematical Problems in Linear Viscoelasticity (Siam, 1992) 52. D. Gutierrez-Lemini, Engineering Viscoelasticity (Springer, Berlin, 2014) 53. R. Hill, The Mathematical Theory of Plasticity (Oxford University Press, Oxford, 1998) 54. L.M. Kachanov, Fundamentals of the Theory of Plasticity (Dover, Illinois, 2004) 55. J. Chakrabarty, Theory of Plasticity (Elsevier, Amsterdam, 2012) 56. G.A. Maugin, The Thermomechanics of Plasticity and Fracture (Cambridge University Press, Cambridge, 1992) 57. J. Lubliner, Plasticity Theory (Dover, Illinois, 2008) 58. V.A. Lubarda, Elastoplasticity Theory (CRC Press, Boca Raton, 2001) 59. S. Nemat-Nasser, Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials (Cambridge University Press, Cambridge, 2004) 60. A.S. Khan, S. Huang, Continuum Theory of Plasticity (Wiley, Hoboken, 1995) 61. H.-C. Wu, Continuum Mechanics and Plasticity (Chapman and Hall/CRC, Boca Raton, 2004) 62. K. Hashiguchi, Y. Yamakawa, Introduction to Finite Strain Theory for Continuum Elastoplasticity (Wiley, Hoboken, 2012) 63. W. Han, B.D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis (Springer, Berlin, 2012)

References

21

64. P. Perzyna, Fundamental problems in viscoplasticity, in Advances in Applied Mechanics (Elsevier, Amsterdam, 1966), pp. 243–377 65. G. Duvant, J.L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 2012) 66. R.R. Huilgol, Fluid Mechanics of Viscoplasticity (Springer, Berlin, 2015) 67. J. Bonet, R.D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis (Cambridge University Press, Cambridge, 1997) 68. A. Ibrahimbegovic, Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods (Springer, Berlin, 2009) 69. J.T. Oden, Finite Elements of Nonlinear Continua (Dover, Illinois, 2006) 70. M. Kojic, K.-J. Bathe, Inelastic Analysis of Solids and Structures (Springer, Berlin, 2005) 71. A.A. Shabana, Computational Continuum Mechanics (Wiley, Hoboken, 2018) 72. P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis (Elsevier, Illinois, 1994), pp. 465–622 73. S.P.C. Marques, G.J. Creus, Computational Viscoelasticity (Springer, Berlin, 2012) 74. A. Anandarajah, Computational Methods in Elasticity and plasticity: Solids and Porous Media (Springer, Berlin, 2011) 75. R.I. Borja, Plasticity: Modelling and Computation (Springer, Berlin, 2013) 76. F. Dunne, N. Petrinic, Introduction to Computational Plasticity (Oxford University Press, Oxford, 2005) 77. E.A. de Souza Neto, D. Peric, D.R.J. Owen, Computational Methods for Plasticity: Theory and Applications (Wiley, Hoboken, 2011) 78. K. Hashiguchi, Elastoplasticity Theory (Springer, Berlin, 2014) 79. J.C. Simo, T.J.R. Hughes, Computational Inelasticity (Springer, Berlin, 2006) 80. J.C. Simo, Numerical analysis and simulation of plasticity, in Handbook of Numerical Analysis (Elsevier, Amsterdam, 1998), pp. 183–499 81. M. Jirasek, Z.P. Bazant, Inelastic Analysis of Structures (Wiley, Hoboken, 2001) 82. W.-F. Chen, D.-J. Han, Plasticity for Structural Engineers (J. Ross Publishing, 2007) 83. S. Krenk, Non-linear Modeling and Analysis of Solids and Structures (Cambridge University Press, Cambridge, 2009) 84. M. Epstein, The Elements of Continuum Biomechanics (Wiley, Hoboken, 2012) 85. A. Goriely, The Mathematics and Mechanics of Biological Growth (Springer, Berlin, 2017) 86. O. Coussy, Mechanics and Physics of Porous Solids (Wiley, Hoboken, 2011) 87. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua I: Foundations and Solid Media (Springer, Berlin, 2012) 88. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua II: Fluids and Complex Media (Springer, Berlin, 2012) 89. L. Dorfmann, R.W. Ogden, Nonlinear Theory of Electroelastic and Magnetoelastic Interactions (Springer, Berlin, 2014) 90. G.A. Maugin, Material Inhomogeneities in Elasticity (Chapman and Hall/CRC, Boca Raton, 1993) 91. G.A. Maugin, Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics (Chapman and Hall/CRC, Boca Raton, 2010) 92. R. Kienzler, G. Herrmann, Mechanics in Material Space: With Applications to Defect and Fracture Mechanics (Springer, Berlin, 2000) 93. M.E. Gurtin, Configurational Forces as Basic Concepts of Continuum Physics (Springer, Berlin, 2000) 94. A.C. Eringen, Microcontinuum Field Theories: I. Foundations and Solids (Springer, Berlin, 2012) 95. A.C. Eringen, Microcontinuum Field Theories: II. Fluent Media (Springer, Berlin, 2001) 96. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, Berlin, 2002) 97. D. Iesan, Thermoelastic Models of Continua (Springer, Berlin, 2013) 98. E. Madenci, E. Oterkus, Peridynamic Theory and Its Applications (Springer, Berlin, 2013) 99. P. Steinmann, Geometrical Foundations of Continuum Mechanics (Springer, Berlin, 2015)

Chapter 2

Preliminaries

The material of music is sound and silence. Integrating these is composing. — John Cage, 1912–1992 —

For the formulation, the algorithmic treatment, and the analysis of the computational material models to be considered in this treatise, a number of modelling, computational, and mathematical tools are needed. Modelling tools are concerned with continuum mechanics in one dimension and dissipation consistent material modelling based on concepts from convex analysis. Computational tools are concerned with constitutive integrators for either strain or stress control and—in order to give some context—the finite element method in one dimension. Mathematical tools are concerned with Laplace transformation, complex representations, Legendre transformation, and constrained optimization. All these tools will be outlined only briefly as preliminaries in the sequel of this chapter.

2.1 Modelling Tools 2.1.1 Continuum Mechanics Let the one-dimensional Euclidean space E1 be parameterized by the coordinate x. A one-dimensional continuum body B consists of a continuous set of physical points p, which are embedded into the Euclidean space by a mapping p → x ∈ E1 (Fig. 2.1). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Steinmann and K. Runesson, The Catalogue of Computational Material Models, https://doi.org/10.1007/978-3-030-63684-5_2

23

24

2 Preliminaries

Fig. 2.1 Embedding of continuum body B consisting of physical points p into the Euclidean space E1 with coordinates x. In E1 the continuum body occupies the configuration B with boundary ∂ B = ∂ Bu ∪ ∂ Bσ . The unknown in B is the displacement function u = u(x) as a response to external loading by distributed body forces b = b(x) in B and prescribed boundary stress σ = σ¯ at ∂ Bσ . The displacement u = u¯ is prescribed at ∂ Bu

Then the continuous set of coordinates x occupied by the one-dimensional continuum body B is denoted its configuration B, its boundary is correspondingly denoted as ∂B (more precisely the closure of B is denoted as B¯ := B ∪ ∂B, vice versa the interior of B is denoted as B := B¯ \ ∂B). The coordinates occupying the boundary ∂B are denoted as ∂ x := x|∂B . The differential domain element in B is denoted by dx, the (signed) differential boundary element on ∂B is denoted by d dx (sgn( d dx) = ±1 for x leaving/entering the domain B). The total boundary decomposes into two disjunct parts ∂B = ∂B u ∪ ∂B σ with ∂B u ∩ ∂B σ = ∅. Derivatives of functions f = f (x) defined over B with respect to the one-dimensional coordinate x are denoted by f  = f  (x). The one-dimensional configuration B is the solution domain for a continuum mechanical problem that seeks to determine as its primary solution the onedimensional displacement function u = u(x) (together with other secondary functions derived therefrom as, e.g., the one-dimensional geometrically linear strain  = u  and its work conjugate one-dimensional stress σ ) as response to given external data within B (the distributed body forces b = b(x) per unit length) and given external data at ∂B (prescribed displacements u = u¯ at ∂B u and prescribed boundary stresses σ = σ¯ at ∂B σ ). The displacement function u(x) assigns new positions u(x) : x → x + u(x) to the physical points p occupying the positions x ∈ B before application of external data. The one-dimensional global statement for the equilibrium of forces requires the resultant of the distributed body forces and the boundary stresses (as a reaction at ∂B u and as prescribed at ∂B σ ) to vanish    0= b dx + σ d dx + σ¯ d dx. (2.1) B

∂Bu

∂Bσ

Incorporating the boundary condition σ = σ¯ at ∂B σ and assuming sufficient smoothness for σ = σ (x), the resulting boundary integral (expanding over the two endpoints

2.1 Modelling Tools

25

∂B of the one-dimensional domain B) may be transformed to a domain integral by the one-dimensional format of the Gauss theorem (which follows in a one-dimensional context simply from the fundamental lemma of calculus)     σ d dx = σ dx =⇒ 0 = [b + σ  ] dx. (2.2) ∂B

B

B

Requiring finally that the above global statement holds as well for arbitrary subsets V ⊂ B of the configuration, a process called localization, renders the corresponding local statement for the equilibrium of forces1 − σ  = b in B and

σ = σ¯ on ∂B σ .

(2.3)

The generic continuum mechanics problem is summarized in Table 2.1. The local statement is also denoted as the strong form of the equilibrium of forces due to the differentiability requirements posed by the presence of the σ  term. A corresponding weak form, with reduced differentiability requirements for σ is most useful. The standard2 solution space is then taken as U = {u ∈ H1 (B) in B and

u = u¯ on ∂B u },

(2.4)

whereas the space of test functions, which may be interpreted as virtual displacements, is defined as U0 = {u ∈ H1 (B) in B and

u = 0 on ∂B u }.

(2.5)

Recall that H1 (B) denotes the space of functions δu with square-integrable derivatives δu  := δ on B. Then multiplying the strong form by δu, integrating over B and ∂B σ , applying integration by parts and the one-dimensional format of the Gauss theorem renders eventually the weak form as follows: Find u ∈ U that solves    δ σ dx = δu b dx + δu σ¯ d dx ∀ δu ∈ U0 . (2.6) B

B

∂Bσ

Note the differentiation has here been shifted from the stress σ to the test function δu  . The weak form is also denoted as the Principle of Virtual Work: If equilibrium of forces holds, the internal virtual work performed by the stresses along the virtual strains equals the virtual work performed by the external forces along the virtual displacements. An approximate solution to the weak form is typically determined via the Finite Element Method.

1 In

one dimension the equilibrium statement  x is statically determinate. Thus the stress may here be computed directly from σ (x) = σ (x0 ) − x0 b(x) ¯ dx. ¯ Note however, that in two and three dimensions the equilibrium statement is obviously statically indeterminate. 2 The regularity that is implicit in this choice of solution and test spaces is considered as the appropriate one for a very large class of problems. However, it would exclude, say, slip line solutions in rigid plasticity that involve displacement discontinuities.

26

2 Preliminaries

Table 2.1 Summary of generic continuum mechanics problem

2.1.2 Dissipation Consistent Material Modelling It remains to establish (dissipation consistent) material models for the stress σ as a function of the strain , the strain rate ˙ and, possibly, a set of internal state variables ˙ α and their rates α. Dissipation consistent material modelling denotes a continuum thermodynamics modelling framework that automatically guarantees non-violation of the dissipation inequality as a manifestation of the second law of thermodynamics. Under the assumptions of (i) isothermal conditions, (ii) geometrical linearity, and (iii) a purely one-dimensional setting, the dissipation inequality3 reads d := σ ˙ − ψ˙ ≥ 0,

(2.7)

where d is the dissipation power density and ψ is the free energy density here characterizing the density of stored energy. For a generic case, we assume that the free energy density ψ = ψ(, α)

(2.8)

is parameterized in the total strain  and the set of internal state variables4 α (in short the set of internal variables) that collects the scalar-valued internal state variables αi (in short the internal variables), i.e. α := {α1 , α2 , . . .} (with appropriate scalar prod3 In

continuum thermodynamics the dissipation inequality is a consequence of the balances of (i) mass, (ii) linear momentum, (iii) angular momentum, (iv) energy, and (v) entropy. In integral format the dissipation inequality states that the working of external forces (that coincides with the working of stress under the condition of mechanical equilibrium) exerted on a continuous body can never be smaller than the energy storage within the continuum body. 4 Collectively, state functions and state variables denote the state quantities that determine the state of a system.

2.1 Modelling Tools

27

uct ◦, norm |α|2 := α ◦ α and time derivative defined as α˙ := {α˙ 1 , α˙ 2 , . . .}). Without loss of generality, it is assumed that ψ is smooth, i.e. continuously differentiable; hence we may rewrite Eq. 2.7 as  ∂ψ  ◦ α˙ =: [σ − σ  ] ˙ + [−a ] ◦ α˙ ≥ 0 d = [σ − ∂ ψ] ˙ + − ∂α

(2.9)

where we introduced the energetic stress σ  = σ  (, α) and the set of energetic driving forces a = a (, α), defined as σ  :=

∂ψ ∂ψ and a := . ∂ ∂α

(2.10)

Next, upon introducing the dissipative stress σ and the set of dissipative driving forces a , respectively, defined by the identities σ := σ − σ  and

a := −a (=: a)

(2.11)

we may eventually rewrite Eq. 2.9 as d = σ ˙ + a ◦ α˙ ≥ 0.

(2.12)

Constitutive relations are next introduced for σ and a in terms of the dissipation potential π. For a generic case, the dissipation potential ˙ π = π(˙ , α).

(2.13)

is parameterized in the rates of the strain and the set of internal state variables ˙ and ˙ respectively. Then, for dissipation consistent material modelling, the dissipation α, potential π is required to be: (i) (ii) (iii) (iv)

˙ ≥ 0, positive, i.e. π(˙ , α) convex,5 zero at the origin, i.e. π(0, 0) = 0; moreover it is: positive homogeneous6 of degree ≥ 1,

function f = f (x) is convex if f (β x1 + [1 − β] x2 ) ≤ β f (x1 ) + [1 − β] f (x2 ) holds for β ∈ [0, 1]. In particular for differentiable functions f = f (x) convexity implies f (x2 ) ≥ f (x1 ) + f  (x1 ) [x2 − x1 ]. For the special case that f (0) = 0 the inequality f  (x) x ≥ f (x) follows, which implies f  (x) x ≥ 0 for f (x) ≥ 0. 6 A function f = f (x) is homogeneous of degree δ ∈ R+ if f (β x) = β δ f (x) holds for all β ∈ R. Furthermore a function f = f (x) is positive homogeneous of degree δ ∈ R if f (β x) = β δ f (x) holds for all β ∈ R+ . For the latter the Euler homogeneous function theorem reads x f  (x) = δ f (x). 5A

28

2 Preliminaries

(v) either smooth or non-smooth.7 This choice characterizes a Standard Dissipative Material. To simplify the notation we henceforth assume smoothness in this overview. The more general non-smooth situation in which partial derivatives have to be substituted by sub-differentials is left to the chapters on Plasticity and Visco-Plasticity.8 For the dissipative stress and the set of dissipative driving forces we then propose σ =

∂π and ∂ ˙

a =

∂π ∂ α˙

(2.14)

whereby the dissipation inequality in Eq. 2.12 becomes d=

∂π ∂π ˙ + ◦ α˙ ≥ 0 ∂ ˙ ∂ α˙

(2.15)

and thus d ≥ 0 is trivially satisfied due to the (i) positivity and (ii) convexity of π together with (iii) its ‘zero-at-zero’ property π(0, 0) = 0. Summarizing, we have the constitutive relations ˙ σ = σ  (, α) + σ (˙ , α),

(2.16a)

˙ 0 = a (, α) + a (˙ , α).

(2.16b)

Equation 2.16b is commonly denoted Biot’s equation. When subjected to the initial condition α(t = 0) = 0, the constitutive evolution equations can then be solved for two principally different (loading) scenarios: Strain Control: (t) is prescribed, whereby α(t) is solved from Eq. 2.16b. σ (t) is then computed from Eq. 2.16a in a post-processing step. Stress Control: σ (t) is prescribed, whereby (t) and α(t) are solved from the coupled Eqs. 2.16a and 2.16b. Sometimes it is convenient to express the constitutive relations in terms of the dual dissipation potential Note that π being either smooth or non-smooth is a property that has profound consequences for the resulting model characteristics. The model classes that are discussed in the subsequent chapters of this treatise possess the following characteristics for π:

7

Elasticity : Visco-Elasticity : Plasticity : Visco-Plasticity :

π ≡ 0, π is smooth and positive homogeneous of degree > 1, π is non-smooth at (0, 0) and positive homogeneous of degree = 1, π is non-smooth at (0, 0) and positive homogeneous of degree > 1.

˙ , α) since π can, indeed, depend on the state. be more careful, we should write π = π(˙ , α; An example is the Prandtl hardening model. However, since , α merely play the role of parameters of π in its role as dissipation potential, we prefer to suppress them as arguments henceforth.

8 To

2.1 Modelling Tools

29

π∗ = π∗ (σ , a )

(2.17)

that is parameterized in terms of the dissipative stress σ and the set of dissipative driving forces a , respectively, and that is also (i) positive, i.e. π∗ (σ , a ) ≥ 0, (ii) convex, and (iii) zero at the origin, i.e. π∗ (0, 0) = 0. The relations between π and π∗ are given via the Legendre transformations π (˙ , α˙ ) = max{σ ˙ + a ◦ α˙ − π∗ (σ , a )},

(2.18a)

π∗ (σ , a ) = max {σ ˙ + a ◦ α˙ − π (˙ , α˙ )}.

(2.18b)

σ ,a

˙ ,α˙

The stationarity conditions corresponding to Eqs. 2.18a and 2.18b are, respectively, ˙ =

∂π∗ and ∂σ

α˙ =

∂π∗ , ∂ a

(2.19a)

∂π and ∂ ˙

a =

∂π . ∂ α˙

(2.19b)

σ =

As a consequence from the Legendre transformation it also holds that ˙ + π∗ (σ , a ) ≥ 0. d = π(˙ , α)

(2.20)

When the formulation is based on π∗ (rather than on π), we may summarize the constitutive relations as follows σ = σ  (, α) + σ ,

(2.21a)

0 = a (, α) + a

(2.21b)

˙ =

∂π∗ (σ , a ) , ∂σ

(2.22a)

α˙ =

∂π∗ (σ , a ) . ∂ a

(2.22b)

and

Upon expressing σ and a in terms of  and α from Eq. 2.21, we may then rephrase the representation in Eq. 2.22 as   ∂π∗ σ − σ  (, α), −a (, α) , (2.23a) ˙ = ˙ (, α, σ ) = ∂σ   ∂π∗ σ − σ  (, α), −a (, α) ˙ α, σ ) = . (2.23b) α˙ = α(, ∂ a

30

2 Preliminaries

Once again, we identify the two (loading) scenarios: Strain Control: (t) is prescribed, whereby α(t) and σ (t) are solved from the coupled Eqs. 2.23a and 2.23b. Stress Control: σ (t) is prescribed, whereby α(t) and (t) are solved from the coupled Eqs. 2.23a and 2.23b. Example: Consider as examples from the area of plasticity and visco-plasticity the following families π(α; ˙ γ ) of positive and convex dissipation potentials that are zero at the origin 1 |α| ˙ 1+γ , 1+γ  π2 (α; ˙ γ ) = α˙ 2 + γ 2 − γ , α˙ π3 (α; ˙ γ ) = γ ln cosh . γ ˙ γ) = π1 (α;

(2.24)

All of these are valid alternatives that are regularized by the family parameter γ ≥ 0. For γ > 0 all π(α; ˙ γ ) are smooth, however for γ → 0 all π(α; ˙ γ ) converge to the non-smooth π(α) ˙ = |α|. ˙ The corresponding driving forces a(α; ˙ γ ) then compute as ˙ γ ) = |α| ˙ γ a1 (α; ˙ γ) =  a2 (α;

α˙ , |α| ˙ α˙

α˙ 2 + γ 2 α˙ ˙ γ ) = γ tanh . a3 (α; γ

(2.25) ,

Observe that only a2 and a3 asymptotically approach the value 1 with increasing α, ˙ in the more classical case of a1 an overshoot can be observed for α˙ > 1. The above relations may be inverted to render the evolution of the corresponding internal variables α(a; ˙ γ ) as α˙ 1 (a; γ ) = |a|1/γ α˙ 2 (a; γ ) = γ √

a , |a| a

1 − a2 α˙ 3 (a; γ ) = γ artanh a.

(2.26) ,

With these results at hand the dual potentials π∗ (a; γ ) are finally determined from a Legendre transformation as 1+γ γ |a| γ , 1+γ  π∗2 (a; γ ) = γ [1 − 1 − a 2 ],

π∗1 (a; γ ) =

(2.27)

2.1 Modelling Tools

31

π∗3 (a; γ ) = γ [a artanh a + ln

 1 − a 2 ].

Observe that π∗1 , π∗2 and π∗3 approach finite values for a → 1. For γ > 0 all π∗ (a; γ ) are smooth, however for γ → 0 all π∗ (a; γ ) converge to the non-smooth indicator function I{|a|−1} of |a| ≤ 1, i.e. ⎧ |a| ≤ 1 ⎨ 0 for π∗ (a; γ → 0) = I{|a|−1} := (2.28) ⎩ ∞ |a| > 1

π1 (α; ˙ γ)

1 0.5

0.5

γ α˙

0 1 |α| ˙ 1+γ 1+γ

0.5

˙ γ) a1 (α;

1

γ α˙

0 −0.5

−1

−1 −1

−0.5

0

0.5

−1

1

π2 (α; ˙ γ)

1 0.5

α˙

0.5

+

−γ

0.5

1

a2 (α; ˙ γ)

0.5

0 γ2

0

1 γ

α˙ 2

−0.5

γ α˙

0 −0.5 −1

−1 −1

−0.5

0

0.5

−1

1

˙ γ) π3 (α;

1 0.5

0

0.5 α˙

0 α˙ γ ln cosh γ

0.5

1

˙ γ) a3 (α;

1 γ

0.5

−0.5

γ α˙

0 −0.5 −1

−1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

Fig. 2.2 Alternative families of dissipation potentials π(α; ˙ γ ) together with resulting subdifferentials a = a(α; ˙ γ ) for regularization parameter γ ∈ {0, 0.1, 0.2, 0.3}. For γ → 0 all π(α; ˙ γ) converge to the non-smooth π(α) ˙ = |α| ˙

32

2 Preliminaries

The families of dissipation potentials π(α; ˙ γ ) and dual dissipation potentials π∗ (a; γ ) are depicted together with their sub-differentials a = a(α; ˙ γ ) and α˙ = α(a; ˙ γ ) in Figs. 2.2 and 2.3. ∞

∞

π1∗ (a; γ)

0.2

γ

γ

0.5 a

0 1+γ γ |a| γ 1+γ

0.2

α˙ 1 (a; γ)

1

a

0 −0.5 −1

−1

−0.5

0

0.5

−1

1

π2∗ (a; γ) 0.2

a

0.5

1

α˙ 2 (a; γ) γ

0.5

0 √

0

1 γ

γ [1 −

−0.5

a

0 −0.5

1 − a2 ]

0.2

−1 −1

−0.5

0

0.5

−1

1

π3∗ (a; γ) 0.2

−0.5

0

γ

0.5 a

0 √ γ [a artanh a + ln 1 − a2 ]

1

α˙ 3 (a; γ)

1 γ

0.5

a

0 −0.5

0.2

−1 −1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

Fig. 2.3 Alternative families of dual dissipation potentials π∗ (a; γ ) together with resulting sub-differentials α˙ = α(a; ˙ γ ) for regularization parameter γ ∈ {0, 0.1, 0.2, 0.3}. For γ → 0 all π∗ (a; γ ) converge to the non-smooth indicator function I{|a|−1} of |a| ≤ 1

2.2 Computational Tools

33

2.2 Computational Tools 2.2.1 Constitutive Integrator 2.2.1.1

General Setting

Material models are typically formulated as functional and/or differential relations between the state functions, here the free energy ψ = ψ(, α) and the stress σ = σ (, α), and the (external and internal) state variables, here the strain  and the set of internal (state) variables α. For a discrete representation of these state quantities the time arrow within the time interval of interest T = [0, T ] is discretized, see Fig. 2.4, into discrete time instants t n with n = 0, 1, . . . n mx , whereby the starting and the terminating time instants are t 0 = 0 and t n mx = T , resulting in n mx time steps with time step size t n := t n − t n−1 , i.e. T=

n mx

[t n − t n−1 ] =

n=1

n mx

t n .

(2.29)

n=1

Then, for known (external and internal) state variables at the discrete time instant t n−1 , i.e. for known  n−1 and α n−1 and thus for known state functions at the discrete time instant t n−1 , i.e. for known ψ n−1 = ψ( n−1 , α n−1 ) and σ n−1 = σ ( n−1 , α n−1 ), either  n or σ n are prescribed at the discrete time instant t n and the remaining state quantities are sought to be updated algorithmically by a constitutive integrator. Moreover, the derivative of σ n with respect to  n , i.e. the algorithmic tangent E an , is often needed. Algorithmic Strain Control The algorithm for strain control is denoted the basic constitutive integrator: For known  n−1 , α n−1 (and σ n−1 ) and prescribed strain  n it computes the corresponding σ n , α n along with the algorithmic tangent stiffness E an , see Fig. 2.5.

= [0 , T ]

t ∈

t0 = 0

···

t1

Δt 1

Δt 2

···

···

tn

t n−1

Δt n

···

t nmx

=T

Δt nmx

T

Fig. 2.4 Discretization of time interval T = [0, T ] into discrete time instants t n with n = 0, 1, . . . n mx , whereby t 0 = 0 and t n mx = T , resulting in n mx time steps t n := t n − t n−1

34

2 Preliminaries n = n+ 1

{

n−1

, αn−1 ;

n

}

{σ n , αn , Ean }

Integrator

Fig. 2.5 Algorithmic strain control: Basic constitutive integrator to determine, for given  n , the remaining discrete state quantities σ n , α n along with the algorithmic tangent stiffness E an , all at time instant t n . The counter for the load/time steps is denoted by n

Algorithmic Stress Control The algorithm for stress control is denoted the extended constitutive integrator, it is based on the basic constitutive integrator: . For known σ n−1 , α n−1 (and  n−1 ) and prescribed stress σ n = σ¯ n it computes the n n corresponding  , α along with the algorithmic tangent stiffness E an , see Fig. 2.6. . Thereby the strain  n is determined form the condition σ n = σ¯ n within a Newton iteration loop (external to the basic constitutive integrator) based on the algorithmic tangent stiffness E an , thus the iterative update for  n reads  n ⇐  n + [σ¯ n − σ n ]/E an

(2.30)

The quadratic convergence of the Newton iteration for |σ¯ n − σ n | → 0 (in the vicinity of the solution) depends crucially on the correct determination of E an (and the smoothness of the stress-strain relation). Observe that no stress control is possible as soon as the algorithmic tangent stiffness degenerates with E an ≤ 0.

{

n−1

(⇒

n

n

{

n−1

n=n+1

), αn−1 , Ean−1 (⇒ Ean ); σ ¯ n−1 (⇒ σ n ), σ ¯n}

, αn−1 ;

n

⇐

}

n

+ [¯ σ n − σ n ]/Ean

Integrator

{σ n , αn , Ean }

?

σn = σ ¯n no yes

{ n , αn , Ean } . Fig. 2.6 Algorithmic stress control: Extended constitutive integrator to determine, for given σ n = σ¯ n , the remaining discrete state quantities  n , α n along with the algorithmic tangent stiffness E an , all at time instant t n . The counter for the load/time steps is denoted by n

2.2 Computational Tools

35

Algorithmic Time Integration For a generic (autonomous) evolution problem the corresponding first-order evolution equation in time for the evolving variable x = x(t) reads together with appropriate initial conditions at time t = 0 as   x(t) ˙ = f x(t) for t ∈ T = [0, T ] with x(0) = x 0 .

(2.31)

Then, in principle, the solution trajectory x(t) within the time interval of interest T = [0, T ] follows analytically from integrating the evolution equation  t   f x(t) dt for t ∈ [0, T ]. (2.32) x(t) − x 0 = 0

In many cases analytical integration of the evolution equation is however not possible or at least cumbersome. Thus approximations x n to the discrete instants x(t n ) of the solution trajectory x(t) are sought, i.e. x n ≈ x(t n ) for n ∈ [0, n mx ].

(2.33)

Then the approximations x n to the discrete instants x(t n ) of the solution trajectory x(t) are typically obtained by a one-step (and one-stage) integration algorithm, i.e. an integration algorithm that involves only the discrete solutions x n−1 and x n at the start and end of one time step. Such an integration algorithm is generically represented in terms of the time step size t n and an integration rule   x n − x n−1 = t n RULE{ f x(t) , x n , x n−1 }.

(2.34)

  Thereby the integration rule depends on the format f x(t) of the evolution equation, possibly the unknown approximation x n to x(t n ), and also possibly the known approximation x n−1 to x(t n−1 ). Integration algorithms depending only on the known approximation x n−1 to x(t n−1 ) are denoted as explicit, whereas integration algorithms depending in one way or other on the unknown approximation x n to x(t n ) are termed implicit. Further important criteria to discriminate between different integration algorithms are their conditional or unconditional stability and their order of accuracy. Typical representatives of one-step integration algorithms as showcased in Fig. 2.7 are for example: The Euler Forward Integrator x n − x n−1 = t n f (x n−1 ). Explicit, conditionally stable, and first-order accurate.

(2.35)

36

2 Preliminaries

x(t)

x(t)

xn

xn

xn−1

xn−1 tn−1

a)

c)

tn

t

tn−1

b)

(t)

x(t)

n

xn xn−0.5

n−1

xn−1 tn−1

tn

t

d)

tn−1

tn−0.5

tn

t

tn

t



   Fig. 2.7 Integrators for x˙ = f x(t) (note that f x(t) denotes the slope of x = x(t): a Eulerforward (explicit, conditionally stable, first-order accurate), b Euler-backward (implicit, unconditionally stable, first-order accurate), c Trapezoidal (implicit, unconditionally stable, second-order accurate), d Midpoint (implicit, unconditionally stable, second-order accurate)

The Euler Backward Integrator x n − x n−1 = t n f (x n ).

(2.36)

Implicit, unconditionally stable, and first-order accurate. The Heun Trapezoidal Integrator

x n − x n−1 = t n

1 [ f (x p ) + f (x n−1 )] with x p − x n−1 = t n f (x n−1 ). 2

(2.37)

Explicit, conditionally stable, and second-order accurate. The Heun Midpoint Integrator

x n − x n−1 = t n f

1 p [x + x n−1 ] 2

 with x p − x n−1 = t n f (x n−1 ). (2.38)

Explicit, conditionally stable, and second-order accurate.

2.2 Computational Tools

37

The Trapezoidal Integrator 1 x n − x n−1 = t n [ f (x n ) + f (x n−1 )]. 2

(2.39)

Implicit, unconditionally stable, and second-order accurate. The Midpoint Integrator

x n − x n−1 = t n f

 1 n [x + x n−1 ] . 2

(2.40)

Implicit, unconditionally stable, and second-order accurate. Computational Example The various algorithmic time integration methods listed in the above are compared for the example     x(t) ˙ = f x(t) with f x(t) = 1 − x(t) and

x 0 = 0.

(2.41)

  Note that the right-hand-side f x(t) is linear, thus the explicit (Heun) and the implicit, respectively, trapezoidal and midpoint integrators here coincide identically. Then discrete algorithmic approximations x n to x(t n ) are determined in the time interval T = [0, T ] with T = 10. Furthermore x h (t) denotes the approximate solution as the piece-wise linear interpolation of the time-ordered sequence {x n }. The analytical solution of the above evolution equation is the saturation function x(t) = 1 − exp(−t) with x(0) = 0 and

x(t → ∞) → 1.

(2.42)

The error is defined as the difference between the approximate and the analytical solution (2.43) ε h (t) := x h (t) − x(t). Over the time interval T, the error is best evaluated in terms of its L 2 -norm, which, for the sake of simplicity, is here also evaluated only approximately (in an Euler Backward fashion) 



ε h (t; { t n })2 := 0

T

  n mx  h 2 |ε (t)| dt ≈  |x n − x(tn )|2 t n .

(2.44)

1

For constant t n =: t the L 2 -norm of the error is a function of the constant time step size and is abbreviated as eh = eh ( t) := ε h (t; t)2 . The computational results obtained with the explicit Euler Forward and the implicit Euler Backward integrators together with those obtained with the explicit Heun Trapezoidal/Midpoint and the implicit Trapezoidal/Midpoint integrators are

38

2 Preliminaries

displayed in Fig. 2.8. The analytical time history x = x(t) together with its approximations x h = x h (t; t) are indicated in the bold continuous coordinate axes, whereas h h := eh ( t)/emax (normalized to its maximum value) is the L 2 -norm of the error erel indicated in the bold dashed coordinate axes. Thereby, the L 2 -norm of the errors of the explicit Euler Forward and the implicit Euler Backward integrators—displayed for the range t ∈ [1/3, 1/2] corresponding to maximum time step numbers in the range n mx ∈ [30, 20]—show the typical linear convergence rate with constant time step sizes, i.e. eh ( t) ∝ [ t]1 . Correspondingly, the L 2 -norm of the errors of the explicit Heun Trapezoidal/Midpoint and the implicit Trapezoidal/Midpoint integrators—displayed for the range t ∈ [1/2, 1] corresponding to maximum time step numbers in the range n mx ∈ [20, 10]—show the typical quadratic convergence rate with constant time step sizes, i.e. eh ( t) ∝ [ t]2 . Observe the different error levels of the first-order and second-order accurate integrators for the explicit and the implicit cases, respectively, despite otherwise identical convergence rates:

ehrel ∈ [0, 1]

x ∈ [0, 1]

ehmax = 0.14946

ehrel ∈ [0, 1]

x ∈ [0, 1]

Euler Forward

Euler Backward

ehmax = 0.10990

t ∈ [0, 10] Δt ∈ [0.¯ 3, 0.5]

ehrel ∈ [0, 1]

x ∈ [0, 1]

ehrel ∈ [0, 1]

x ∈ [0, 1] Midpoint/Trapezoidal

Heun

ehmax = 0.19753

t ∈ [0, 10] Δt ∈ [0.¯3, 0.5]

t ∈ [0, 10] Δt ∈ [0.5, 1.0]

ehmax = 0.04415

t ∈ [0, 10] Δt ∈ [0.5, 1.0]

    Fig. 2.8 Algorithmic time integration of x(t) ˙ = f x(t) with right-hand-side f x(t) = 1 − x(t) and initial value x0 = 0 over the time interval T = [0, 10] into a sequence of approximations x n ≈ x(t n ) together with the L 2 -norm of the error eh ( t) := x h (t; t) − x(t)2 compared to the analytical solution x(t) = 1 − exp(−t) (the relative L 2 -norm of the error is here defined as h ( t) := e h ( t)/e h ) erel max

2.2 Computational Tools

39

eh ≈ [1, 0.62] × 0.15 ≈ [0.15, 0.09] Euler Forward Integrator versus eh ≈ [1, 0.69] × 0.11 ≈ [0.11, 0.07] Euler Backward Integrator and eh ≈ [1, 0.15] × 0.20 ≈ [0.20, 0.03] Heun Trapezoidal/Midpoint Integrator versus eh ≈ [1, 0.21] × 0.04 ≈ [0.04, 0.01] Trapezoidal/Midpoint Integrator. The time step sizes ( t ∈ [1/3, 1/2] versus t ∈ [1/2, 1]) for the first-order accurate integrators and the second-order accurate integrators, respectively, are here merely chosen differently for the sake of graphical representability of the L 2 -norm of the error. Stability of Time Integrators The stability of the time integrators can be assessed based on the evolution problem of the previous computational example. This is most easily achieved by the substitution y(t) := 1 − x(t) resulting in y˙ (t) = −y(t) with y 0 := y(0) = 1,

(2.45)

so that the analytical solution reads as y(t) = exp(−t)

(2.46)

y(t → ∞) → 0.

(2.47)

with the decay property When the algorithmic update with constant time step t applied to y˙ (t) = −y(t) is re-formulated in terms of the amplification function a( t) (denoted the stability function when analysing Runge–Kutta methods) as y n = a( t)y n−1 = [a( t)]n y 0 ,

(2.48)

the two following notions of stability are introduced: A-Stability: L-Stability:

y n decays to zero for n → ∞, thus |a( t)| < 1, y n decays to zero for t → ∞, thus a( t → ∞) → 0.

The absolute values of the amplification functions for the time integrators used in the previous computational example result in: Euler Forward : Euler Backward : Heun Trapezoidal/Midpoint : Trapezoidal/Midpoint :

|a( t)| = |1 − t| , |a( t)| = |1 + t|−1 , |a( t)| = |1 − t + 21 [ t]2 | , |a( t)| = |1 − 21 t||1 + 21 t|−1 .

40

2 Preliminaries

The absolute values of the amplification functions are displayed as function of the time step size in Fig. 2.9. From these the stability properties of the various time integrators are easily assessed: conditionally stable for t < 2, A-stable and L-stable, conditionally stable for t < 2, A-stable .

Euler Forward : Euler Backward : Heun Trapezoidal/Midpoint : Trapezoidal/Midpoint :

The sequence y n only converges monotonically to zero if the amplification function satisfies 1 > a( t) > 0, otherwise, i.e. if the amplification function satisfies 0 > a( t) > −1, the sequence y n converges in an oscillatory manner.

2.2.2 Finite Element Method Although not explicitly used in the sequel, some basics of the one-dimensional finite element method shall here be outlined briefly in order to give some context for the computational material models considered in this treatise. Thereby, it is particularly illuminating to showcase how the algorithmic update of the stress and its linearization, as outlined in the sequel of the Catalogue in much detail for each computational

|a(Δt)| Euler Forward 1

|a(Δt)| Euler Backward 1

Conditionally Stable |1 − Δt|

.5

0.5

Δt

0 0

0.5

1

A- and L-Stable

1.5

|1 + Δt|−1

2

0

50

1

1

Conditionally Stable 0

0.5

1

150

200

1.5

|1 − 12 Δt||1 + 12 Δt|−1

0.5

|1 − Δt + 12 Δt2 |

0

100

|a(Δt)| Midpoint/Trapezoidal

|a(Δt)| Heun

.5

Δt

0

Δt 2

A-Stable

0 0

50

100

Δt 150

200

Fig. 2.9 Modulus of the amplification function |a( t)| for the algorithmic time integrators applied to y˙ (t) = −y. Time integrators are A-stable for |a( t)| < 1 and, in addition, L-stable for |a( t → ∞)| → 0

2.2 Computational Tools

41

material model, contribute to an archetypical structural computation. Obviously, the one-dimensional, geometrically linear setting is normally not sufficient for real world analyses without extension to multiple dimensions; however, it already contains all the main ingredients characterizing structural analyses without requiring mastering the algebraically challenging multi-dimensional setting. The key concept underlying the finite element method is the discretization of the solution domain B ≈ Bh into n el elements Be , compare Fig. 2.10 B ≈ Bh =

n el 

Be .

(2.49)

e=1

Accordingly, due to the additivity of integrals, the weak form (or rather the principle of virtual work) takes the expression    n el  n el   δ σ dx = δu b dx + δu σ¯ d∂ x ∀ δu. (2.50) e=1

Be

∂Beσ

Be

e=1

Then the trial and test functions will be approximated element-wise (locally) on the given discretization in terms of shape functions Nk (ξ ) associated with n en element nodes u ≈ u h with u h (ξ )| L e =

n en 

u|k Nk (ξ )

(2.51)

k=1

and δu ≈ δu h with δu h (ξ )| L e =

n en 

δu|k Nk (ξ ).

(2.52)

k=1

Here the coordinate ξ takes values in the so-called isoparametric domain, i.e. ξ ∈ [−1, +1]. Within the isoparametric concept, the physical coordinates are approximated element-wise in an identical fashion as the trial and test functions. Thus the isoparametric map reads

x ∈ Bh

1

2 B1

3 B2

4 B3

5 B4

6 B5

7 B6

Bh

Fig. 2.10 Discretization of the solution domain B ≈ Bh into elements Be with e = 1, . . . n el

42

2 Preliminaries n en 

x ≈ x h with x h (ξ )| L e =

x|k Nk (ξ ).

(2.53)

k=1

An example for the (local) element-wise node and dof numbering for an element with quadratic shape functions is depicted in Fig. 2.11. Accordingly, the Jacobian of the isoparametric map computes element-wise Je (ξ ) := ∂ξ x h (ξ )| L e =

n en 

x|k ∂ξ Nk (ξ ).

(2.54)

k=1

The Jacobian allows in particular to determine the total differential of the physical coordinate dx h | L e = Je dξ.

(2.55)

Thus, the gradient of the shape functions expands element-wise into Nk = ∂ξ Nk Je−1 .

(2.56)

Consequently, the gradients of the trial and test functions compute as  ≈ h with h (ξ )| L e =

n en 

u|k Nk (ξ )

(2.57)

k=1

and δ ≈ δh with δh (ξ )| L e =

n en 

δu|k Nk (ξ ).

(2.58)

k=1

Accordingly, the discretized weak form expands eventually into n el  n en  e=1 k=1

 δu|k

Be

Nk σ dx =

Fig. 2.11 Local node and dof numbering for an element with quadratic shape functions. The isoparametric coordinate is ξ ∈ [−1, +1]

n el  n en  e=1 k=1

 δu|k



 Be

Nk b dx +

∂Beσ

Nk σ¯ d∂ x . (2.59)

u |1

u |2

u |3

x |1

x |2

x |3

ξ −1

+1

2.2 Computational Tools

43

Next, for a more compact representation, the shape functions and their gradients are arranged into element-wise (local) row-matrices     Ne := N1 , . . . , Nk , . . . , Nn en , Be := N1 , . . . , Nk , . . . , Nn en .

(2.60)

Likewise, the local dofs of the trial and test functions are assembled into element-wise (local) column-matrices   t t ue := u|1 , . . . , u|k , . . . , u|n en , δue := δu|1 , . . . , δu|k , . . . , δu|n en .

(2.61)

Then the trial and test functions along with their gradients read in compact matrix notation as u h (ξ )| L e = Ne ue and h (ξ )| L e = Be ue

(2.62)

δu h (ξ )| L e = Ne δue and δh (ξ )| L e = Be δue .

(2.63)

and

Consequently, the discretized weak form takes the compact representation n el 

 δute

e=1

Be

Bte

σh dx =

n el 

 δute

Be

e=1



 Nte

b dx +

∂Beσ

Nte

σ¯ d∂ x

(2.64)

that may be further abbreviated by introducing element-wise (local) column-matrices of internal and external forces n el 

δute se =

e=1

n el 

δute fe .

(2.65)

e=1

Here, the element-wise (local) column-matrices of internal and external forces are defined as  se :=

 Be

Bte σh dx and

fe :=

 Be

Nte b dx +

∂Beσ

Nte σ¯ d∂ x.

(2.66)

Note that σh is determined from the algorithmic stress update as outlined in the sequel of the Catalogue in much detail for each computational material model. The element-wise (local) dofs are formally assigned to n np global dofs by Boolean matrices (containing mostly zeros and only a few ones)

44

2 Preliminaries

ue = ae u with dim ae = n en × n np .

(2.67)

Thus, the discretized weak form reads eventually in terms of global column-matrices of internal and external forces δut s = δut f ∀δu

(2.68)

that follow from the assembly of the corresponding (local) element-wise columnmatrices of internal and external forces s :=

n el 

n el

se =: A se and f :=

ate

e=1

e=1

n el 

n el

ate fe =: A fe .

e=1

e=1

(2.69)

For non-linear material models (and conservative external loads) also the linearization of the internal part of the discretized weak form is needed, i.e. n el  e=1

 δute

el  ∂σh Be dx due =: δute k e due . ∂h e=1

n

Be

Bte

(2.70)

Here the element-wise (local) tangent stiffness matrix is introduced in terms of the algorithmic tangent stiffness   ∂σh k e := Bte Be dx =: Bte E a Be dx. (2.71) ∂h Be Be Observe that ∂σh /∂h =: E a denotes the algorithmic tangent that follows from linearizing the algorithmic stress update as outlined in the sequel of the Catalogue for each computational material model. Consequently, the global tangent stiffness follows from assembly as k :=

n el 

n el

ate k e ae =: A k e .

e=1

e=1

(2.72)

Finally, in terms of the previously introduced global column-matrices of internal and external forces, the global residual for a (either linear or non-linear) equilibrium problem is defined as . r := f − s = 0.

(2.73)

Within a global Newton iteration loop the linearisation of the residual reads dr = −k du.

(2.74)

2.2 Computational Tools

45

Thus an iterative update for the global unknowns collected in u follows as the solution of a linear algebraic equation system in terms of the global residual and the global tangent stiffness matrix u ⇐ u + du with du = k −1 r.

(2.75)

Subsequently, the residual has to be re-evaluated with the updated estimate for u and the iteration proceeds until convergence with |r| → 0 is achieved. The quadratic convergence of the Newton iteration (in the vicinity of the solution) depends crucially on the correct determination of k (and the smoothness of the global force-displacement relation). Observe that no force control is possible as soon as the global tangent stiffness matrix degenerates with wt k w ≤ 0 for some w = 0.

2.3 Mathematical Tools 2.3.1 Heaviside Function and Causal Signals The Heaviside function is denoted by H(t); as depicted in Fig. 2.12 (left) it is defined as ⎧ ⎫ t < 0⎬ ⎨0 for H(t) := . (2.76) ⎩ ⎭ 1 t ≥0 Then, based on the definition of the Heaviside function, a causal signal is any timedependent function with property

2

2

H(t)

H0 (t)

1

1

t

0

t

0

−1

−1 −2

−1

0

1

2

−2

−1

0

1

2

Fig. 2.12 (Left) Heaviside function H(t): the paradigm of a causal signal. (Right) Modified Heaviside function H0 (t)

46

2 Preliminaries

x(t) ≡ H(t) x(t) =⇒ x(t) =

⎧ ⎨ 0 ⎩

⎫ t < 0⎬ for

x(t)

t ≥0

⎭

.

(2.77)

Moreover, it sometimes proves convenient to also define the modified Heaviside function H0 (t), see Fig. 2.12 (right), as ⎧ ⎫ t ≤ 0⎬ ⎨0 for H0 (t) := . (2.78) ⎩ ⎭ 1 t >0 The modified Heaviside function is particularly helpful in the context of representing the algorithmic tangent of computational material models.

2.3.2 Laplace Transformation The complex-valued Laplace transformed signal X (s) in complex frequency domain of a real-valued causal signal x(t) in time domain is obtained by Laplace transformation L{x(t)} defined as  ∞ x(t) exp(−s t) dt. (2.79) X (s) = L{x(t)} := −0

Thereby t and s denote the real-valued time variable and the complex-valued frequency or rather Laplace transformation variable, respectively. Likewise the realvalued inversely Laplace transformed causal signal x(t) of a complex-valued signal X (s) is obtained by inverse Laplace transformation L−1 {X (s)} defined by 1 x(t) = L {X (s)} := 2π i −1



(s)+i ∞

(s)−i ∞

X (s) exp(s t) ds.

(2.80)

From these definitions the Laplace transformation inherits a number of useful properties, of which only a few that are relevant for the discussion in the sequel are mentioned: • The Laplace transformation of a linear combination a1 x1 (t) + a2 x2 (t) of two real-valued causal signals x1 (t) and x2 (t) renders the same linear combination of Laplace transformed signals X 1 (s) and X 2 (s), i.e. L{a1 x1 (t) + a2 x2 (t)} = a1 X 1 (s) + a2 X 2 (s).

(2.81)

• The Laplace transformation of the rate x(t) ˙ of a real-valued causal signal x(t) with initial condition x(0) = 0 computes as

2.3 Mathematical Tools

47

L{x(t)} ˙ = s X (s).

(2.82)

• If the convolution integral x1 (t)  x2 (t) of two real-valued causal signals x1 (t) and x2 (t) is defined as  t  t x1 (t − t  ) x2 (t  ) dt  = x1 (t  ) x2 (t − t  ) dt  (2.83) x1 (t)  x2 (t) := 0

0

the corresponding Laplace transformation of the convolution integral computes as the product of the corresponding Laplace transformed signals X 1 (s) and X 2 (s), i.e. L{x1 (t)  x2 (t)} = X 1 (s) X 2 (s).

(2.84)

With these properties at hand the inversion x(t), a real-valued causal signal with initial condition x(0) = 0, of the convolution integral z(t) = y(t)  x(t) ˙ of a realvalued causal signal y(t) and the rate x(t), ˙ a typical problem in what follows, is obtained in complex frequency domain by the Laplace transformed signals Z (s) and Y (s) as X (s) = Z (s)/[s Y (s)] with a subsequent inverse Laplace transformation of X (s) to the time domain, see the sketch in Fig. 2.13. A brief list of real-valued causal signals x(t) in time domain and corresponding Laplace transformed complex-valued signals X (s) in complex frequency domain that are relevant in the sequel is given in Table 2.2.

2.3.3 Complex Representations Complex Numbers and Harmonically Oscillating Signals Complex numbers allow the condensed representation of twice the information that is contained in an ordinary real number. Thereby a complex number x ∈ C allows the following alternative representations

Laplace Transformation z(t) = y(t) x(t) ˙

Z(s) = L{z(t)} = s Y (s) X(s) Inversion

x(t) = L−1 {X(s)}

X(s) = Z(s)/[s Y (s)]

Inverse Laplace Transformation Fig. 2.13 Inversion x(t) of a frequently occurring, typical convolution integral z(t) = y(t)  x(t) ˙

48

2 Preliminaries

Table 2.2 Real-valued causal signals x(t) in time domain and corresponding Laplace transformed complex-valued signals X (s) in complex frequency domain

⎧ ⎨

x  + i x  , x = x0 [cos α + i sin α], ⎩ x 0 ei α .

(2.85)

Here i denotes the imaginary unit, and x  := x0 cos α and x  := x0 sin α are the real and imaginary parts of x. The length x0 and angle α in the complex plane thus compute from [x0 ]2 := [x  ]2 + [x  ]2 and tan α := x  /x  . If the angle α varies linear in time as α = ω t, the complex representation ⎧ x  (t) + i x  (t), ⎨ x(t) = x0 [cos(ω t) + i sin(ω t)], ⎩ x 0 ei ω t .

(2.86)

(2.87)

of a harmonically oscillating signal x(t) = x0 cos(ω t)

(2.88)

2.3 Mathematical Tools

49

and its rate x(t) ˙ = x0 ω sin(ω t)

(2.89)

is obtained in the complex plane, see Fig. 2.14. Here ω :=

2π and T T

(2.90)

x(t) ˙ = sin(ωt) ω +1

eiωt ωt 0

t

−1

−1

0

+1 x(t) = cos(ωt)

t

Fig. 2.14 Complex representation x(t) of a harmonically oscillating signal x(t) = cos(ω t), whereby ω := 2 π/T and T denote and the period duration, respectively.  the  angular  frequency  The real and imaginary parts  x(t) and  x(t) relate to the signal x(t) and its scaled rate x(t)/ω, ˙ respectively

50

2 Preliminaries

denote the angular frequency and the period duration, respectively. The real and imaginary parts     x  (t) =  x(t) = x0 cos(ω t) and x  (t) =  x(t) = x0 sin(ω t)

(2.91)

obviously relate to the signal x(t) and its scaled rate x(t)/ω, ˙ respectively. Harmonically Oscillating Stress and Strain: Generic Case A stress signal oscillating harmonically with angular frequency ω σ (t) = σ0 cos(ω t + δσ )

(2.92)

and stress phase shift angle δσ has complex representation σ (t; ω) := σ ∗ exp(i ω t) =: σ  (t; ω) + i σ  (t; ω),

(2.93)

whereby (•; ω) denotes parametrization in the angular frequency. Here σ ∗ denotes the complex amplitude comprising the amplitude σ0 and the phase shift angle δσ of the stress signal as σ ∗ := σ0 exp(i δσ ),

(2.94)

whereas σ  (t; ω) and σ  (t; ω) are the real and imaginary parts of the complex stress signal. Accordingly the rate of the harmonically oscillating stress signal has complex representation σ˙ (t; ω) = i ω σ ∗ exp(i ω t) =: σ˙  (t; ω) + i σ˙  (t; ω).

(2.95)

Likewise a strain signal oscillating harmonically with angular frequency ω (t) = 0 cos(ω t + δ )

(2.96)

and strain phase shift angle δ has complex representation (t; ω) :=  ∗ exp(i ω t) =:   (t; ω) + i   (t; ω).

(2.97)

Here  ∗ denotes the complex amplitude comprising the amplitude 0 and the phase shift angle δ of the strain signal as  ∗ := 0 exp(i δ ),

(2.98)

2.3 Mathematical Tools

51

whereas   (t; ω) and   (t; ω) are the real and imaginary parts of the complex strain signal. Accordingly the rate of the harmonically oscillating strain signal has complex representation ˙ (t; ω) = i ω  ∗ exp(i ω t) =: ˙  (t; ω) + i ˙  (t; ω).

(2.99)

Then for linear response and steady state conditions the complex representation of a harmonically oscillating stress-strain relation reads σ (t; ω) = E ∗ (ω) (t; ω) or ˙ (t; ω) = C ∗ (ω) σ˙ (t; ω)

(2.100)

with the angular frequency dependent complex stiffness and compliance modulus E ∗ (ω) and C ∗ (ω) both consisting of real and imaginary parts E ∗ (ω) := E  (ω) + i E  (ω) and C ∗ (ω) := C  (ω) − i C  (ω).

(2.101)

Thus for linear response the complex representation of a harmonically oscillating steady state stress-strain relation expands alternatively as σ (t; ω) = E  (ω) (t; ω) + E  (ω)/ω ˙ (t; ω),

(2.102)

˙ (t; ω) = C  (ω) σ˙ (t; ω) + C  (ω) ω σ (t; ω). Here ˙ (t; ω) = i ω (t; ω) and σ (t; ω) = −i σ˙ (t; ω)/ω have been used. Then the real part of the harmonically oscillating steady state stress-strain relation is expressed as σ  (t; ω) = E  (ω)   (t; ω) + E  (ω)/ω ˙  (t; ω),

(2.103)

˙  (t; ω) = C  (ω) σ˙  (t; ω) + C  (ω) ω σ  (t; ω). Consequently the real-valued stress power can be formulated in terms of the real and imaginary parts of either the complex stiffness modulus or the complex compliance modulus

σ  (t; ω) ˙  (t; ω) =

1  ˙ E (ω) |   (t; ω)|2 + E  (ω)/ω | ˙  (t; ω)|2 , 2

=

1  ˙ C (ω) |σ  (t; ω)|2 + C  (ω) ω |σ  (t; ω)|2 . 2

(2.104)

Integrating the stress power over time with either   (0) = 0 for strain control or σ  (0) = 0 for stress control, respectively, and with integration variable s ∈ [0, t], results eventually in the expended work as

52

2 Preliminaries

W (t; ω) =

1  E (ω) |   (t; ω)|2 + E  (ω)/ω 2

1 = C  (ω) |σ  (t; ω)|2 + C  (ω) ω 2

 

t

| ˙  (s; ω)|2 ds,

(2.105)

0 t

|σ  (s; ω)|2 ds.

0

It is thus obvious that the real parts of the complex stiffness modulus E  (ω) and the complex compliance modulus C  (ω) relate to energy storage, whereas the imaginary parts of the complex stiffness modulus E  (ω) and the complex compliance modulus C  (ω) connect to energy loss (dissipation). Thus E  (ω) and E  (ω) are commonly denoted as storage and loss stiffness modulus, and C  (ω) and C  (ω) are commonly denoted as storage and loss compliance modulus, respectively. Harmonically Oscillating Stress and Strain: Strain Control Specializing the above discussion to the case of strain control, i.e. the strain is prescribed as a sinusoidal signal, the phase angles δ and δσ of the strain and the stress signal, that are related by the phase angle δ (see Fig. 2.15), are given by δ = −

π π and δσ = δ − . 2 2

(2.106)

Accordingly the complex strain and stress amplitudes compute as  ∗ = −i 0 and σ ∗ = σ0 sin δ − i σ0 cos δ.

(2.107)

Moreover the complex strain and stress signals and their rates follow as

1.5

1.5 δ/2π

1

δ/2π

1

0.5

0.5 ωt/2π

ωt/2π

0

0

0.5

−0.5

−1 0

0.2

sin(ω tˆ) sin(ωt) 0.4 0.6 0.8

−1 1

0

0.2

0.4

sin(ωt) sin(ω tˇ) 0.6 0.8 1

Fig. 2.15 Sinusoidal signals sin(ω t), sin(ω tˇ) := sin(ω t + δ), sin(ω tˆ) := sin(ω t − δ) without and with phase angles δ (here displayed for π/2). Left) Positive phase angle δ shifts the original signal sin(ω t) to the left into sin(ω tˆ); Right) Negative phase angle δ shifts the original signal sin(ω t) to the right into sin(ω tˇ)

2.3 Mathematical Tools

53

(t; ω) =

0 [sin(ω t) − i cos(ω t)],

(2.108)

˙ (t; ω) = ω 0 [cos(ω t) + i sin(ω t)], σ (t; ω) = σ0 [sin(ω tˆ) − i cos(ω tˆ)], σ˙ (t; ω) = ω σ0 [cos(ω tˆ) + i sin(ω tˆ)], whereby the shifted time tˆ has been defined so as to include the positive phase shift angle tˆ := t + δ/ω.

(2.109)

Finally the complex stress-strain relations expand as σ  (t; ω) = σ  (t; ω) =

0 [E  (ω) sin(ω t) + E  (ω) cos(ω t)], 0 [E  (ω) sin(ω t) − E  (ω) cos(ω t)], ˙  (t; ω) = ω σ0 [C  (ω) sin(ω tˆ) + C  (ω) cos(ω tˆ)], ˙  (t; ω) = ω σ0 [C  (ω) sin(ω tˆ) − C  (ω) cos(ω tˆ)].

(2.110)

Then with the corresponding real parts of the stress and the strain rate σ  (t; ω) = σ0 sin(ω tˆ) and ˙  (t; ω) = ω 0 cos(ω t)

(2.111)

the stress power computes for the case of strain control as σ  (t; ω) ˙  (t; ω) = ω 02 [E  (ω) sin(ω t) cos(ω t) + E  (ω) cos2 (ω t)] (2.112) = ω σ02 [C  (ω) sin(ω tˆ) cos(ω tˆ) + C  (ω) sin2 (ω tˆ)]. Consequently the work expended in either a complete strain or stress cycle (with integration over either t or tˆ,9 respectively) is entirely transformed into dissipation 

2π ω

σ  (t; ω) ˙  (t; ω) dt

0

-cycle

= π E  (ω) 02 ,

σ -cycle

= π C  (ω) σ02 ,

9

.



t

1 sin2 ( ω t) 2 ω 0  t 1 cos(ω s) cos(ω s) ds = sin (2 ω t) + 4ω 0  t 1 sin(ω s) sin(ω s) ds = − sin (2 ω t) + 4ω 0 sin(ω s) cos(ω s) ds =

t 2 t 2

(2.113)

54

2 Preliminaries

whereas the work expended in either a quarter strain or stress cycle consists of energy storage and dissipation 

π

2ω

σ  (t; ω) ˙  (t; ω) dt

-cycle

=

0 σ -cycle

=

1  1 E (ω) 02 + π E  (ω) 02 , 2 4 1  1 C (ω) σ02 + π C  (ω) σ02 . 2 4

(2.114)

In summary these detailed results again clearly justify the terminology storage and loss stiffness or compliance modulus for E  (ω) and E  (ω) or C  (ω) and C  (ω), respectively. Harmonically Oscillating Stress and Strain: Stress Control Specializing the above discussion to the case of stress control, i.e. the stress is prescribed as a sinusoidal signal, the phase angles δσ and δ of the stress and the strain signal, that are related by the phase angle δ, are given by δσ = −

π π and δ = −δ − . 2 2

(2.115)

Accordingly the complex stress and strain amplitudes compute as σ ∗ = −i σ0 and ε∗ = −ε0 sin δ − i ε0 cos δ.

(2.116)

Moreover the complex strain and stress signals and their rates follow as 0 [sin(ω tˇ) − i cos(ω tˇ)], ˙ (t; ω) = ω 0 [cos(ω tˇ) + i sin(ω tˇ)], (t; ω) =

(2.117)

σ (t; ω) = σ0 [sin(ω t) − i cos(ω t)], σ˙ (t; ω) = ω σ0 [cos(ω t) + i sin(ω t)], whereby the shifted time tˇ has been defined so as to include the negative phase shift angle tˇ := t − δ/ω.

(2.118)

Finally the complex stress-strain relations expand as σ  (t; ω) = σ  (t; ω) =

0 [E  (ω) sin(ω tˇ) + E  (ω) cos(ω tˇ)], 0 [E  (ω) sin(ω tˇ) − E  (ω) cos(ω tˇ)],

˙  (t; ω) = ω σ0 [C  (ω) sin(ω t) + C  (ω) cos(ω t)], ˙  (t; ω) = ω σ0 [C  (ω) sin(ω t) − C  (ω) cos(ω t)].

(2.119)

2.3 Mathematical Tools

55

Then with the corresponding real parts of the strain rate and the stress ˙  (t; ω) = ω 0 cos(ω tˇ) and σ  (t; ω) = σ0 sin(ω t)

(2.120)

the stress power computes for the case of stress control as σ  (t¯; ω) ˙  (t¯; ω) = ω 02 [E  (ω) sin(ω tˇ) cos(ω tˇ) + E  (ω) cos2 (ω tˇ)], (2.121) = ω σ02 [C  (ω) sin(ω t) cos(ω t) + C  (ω) sin2 (ω t)]. Consequently the work expended in either a complete stress or strain cycle (with integration over either t or tˇ, respectively) is entirely transformed into dissipation 

2π ω

σ  (t; ω) ˙  (t; ω) dt

0

σ -cycle

= π E  (ω) 02 ,

(2.122)

-cycle

= π C  (ω) σ02 ,

whereas the work expended in a quarter stress or strain cycle consists of energy storage and dissipation 

π

2ω

σ  (t; ω) ˙  (t; ω) dt

σ -cycle

=

0 -cycle

=

1  1 E (ω) 02 + π E  (ω) 02 , 2 4 1  1 C (ω) σ02 + π C  (ω) σ02 . 2 4

(2.123)

In summary these detailed results again clearly justify the terminology storage and loss stiffness or compliance modulus for E  (ω) and E  (ω) or C  (ω) and C  (ω), respectively.

2.3.4 Legendre Transformation The Legendre transformation of a convex (l.s.c),10 possibly non-smooth function f (x) into its convex (l.s.c), possibly non-smooth dual function f ∗ (y) is defined as f (x) := max{y x − f ∗ (y)} and y

f ∗ (y) := max{y x − f (x)}. x

(2.124)

Observe the different parametrization of f = f (x) and f ∗ = f ∗ (y). A prominent example is the Legendre transformation of the non-smooth function f (x) = |x| as displayed in Fig. 2.17. The following properties hold for convex (l.s.c), possibly 10 For a lower semicontinuous function

if n → ∞.

f (x) ≤ limn→∞ inf f (xn ) is satisfied for a sequence xn → x

56

2 Preliminaries

x (y)

y

y (x) ∗

f (y)

f (x)

x Fig. 2.16 The smooth function f (x) and its smooth dual function f ∗ (y) are related via Legendre transformation. As a consequence y(x) = ∂x f (x) and x(y) = ∂ y f ∗ (y) hold with y x = f (x) + f ∗ (y)

f (x) = |x|

1 0.5

0.5

0 0.5

x 0.5 x

0.5 x − |x|

−1

0

max{0.5 x − |x|} = 0 −0.5 0 0.5

1.0 x −1

1

f (x) = |x|

1

1.0 x − |x|

x

−0.5 −1

−1

max{1.0 x − |x|} = 0 −0.5 0 0.5

1

f ∗ (y) = I{|y|−1} ∞

∞

1 0.5

0.5 1.5 x − |x| x

0

0

−1

−0.5

0.5 −1

f (x) = |x|

1

1.5 x −1

−1 max{1.5 x − |x|} = ∞ −0.5 0 0.5

1

I{|y|−1} := −1

−0.5

+1 ⎧ ⎨ 0 ⎩

|y| ≤ 1 for

∞ 0

y

.

|y| > 1 0.5

1

Fig. 2.17 Graphical Legendre transformation of f (x) = |x|: For any fixed y (here y = {0.5, 1.0, 1.5}) the line y x is drawn (dotted) together with the function y x − |x| (bold dotted) in the same diagram. The supremum of the latter function (here y = {0, 0, ∞}) is then drawn as the value of f ∗ (y) = I{|y|−1} , the indicator function of the set |y| − 1 ≤ 0 in the dual diagram

2.3 Mathematical Tools

57

non-smooth functions related by a Legendre transformation y(x) = dx f (x) and

x(y) = d y f ∗ (y) and

y x = f (x) + f ∗ (y).

(2.125)

Here, for non-smooth functions, dx f (x) denotes the set of sub-derivatives, i.e. the sub-differential11 of f (x) with respect to x, and d y f ∗ (y) denotes the set of subderivatives, i.e. the sub-differential of f ∗ (y) with respect to y. As consequences of the above properties (i) dx f (x) and d y f ∗ (y) are inverses of one another, and (ii) applying Legendre transformation twice renders f (x) = f ∗∗ (x). For a motivation consider the smooth y versus x diagram in Fig. 2.16 with the dependence between y and x either  x given by y = y(x) or by x = x(y). Then f (x) follows as the integral f (x) := 0 y(x  ) dx  , i.e. the area under the y = y(x) curve, rendering ∂x f = y(x), whereas the dual f ∗ (y) follows as the integral f ∗ (y) = thus y   ∗ 0 x(y ) dy , i.e. the area under the x = x(y) curve, thus rendering ∂ y f = x(y). ∗ Obviously f (x) and f (y) sum up to y x.

2.3.5 Constrained Optimization General Setting Generically, a constrained optimization problem consists in seeking the variable (or function) x0 that minimizes a function (or a functional) F(x), i.e. x0 = arg{min F(x)} x

(2.126)

typically under some equality and inequality constraints h(x) = 0 and g(x) ≤ 0

(2.127)

(box constraints and other types of constraints shall not be considered here). Since in the sequel this treatise is essentially concerned with problems restricted by inequality constraints, the equality constraint h(x) = 0 shall not be considered further. Finally, based on the McCauley brackets 2 g(x) := g(x) + |g(x)| ≥ 0

(2.128)

that extract the positive part of a function, the remaining inequality constraint g(x) ≤ 0 may alternatively be expressed as (non-smooth) equality constraint

f = f (x) with x ∈ X is defined as the set of sub-differentials dx f |x0 := {z| f (x) − f (x0 ) − z [x − x0 ] ≥ 0 ∀x ∈ X}. For smooth functions the sub-differential degenerates to the ordinary derivative z = f  .

11 The sub-differential of a non-smooth convex function

58

2 Preliminaries F, g

100

x0 = arg{g(x) = 0} F (x)

80

60

40

40 20

g(x) x

0 0

5

10

15

20

F (x)

80

60

20

F, g x0 = max arg g(x) = 0}

100

g(x) x

0

25

0

5

10

15

20

25

Fig. 2.18 One-dimensional minimization of function F(x) restricted by inequality constraint g(x) ≤ 0 where x ≤ x0 denotes the admissible domain. Left) Original formulation in terms of inequality constraint with g(x0 ) = 0; Right) Equivalent formulation in terms of alternative equality constraint with g(x ≤ x0 ) = 0

g(x) = 0.

(2.129)

A one-dimensional optimization problem restricted by either the inequality constraint g(x) ≤ 0 or the corresponding alternative equality constraint g(x) = 0 is displayed for the sake of demonstration in Fig. 2.18. Lagrange Multiplier Method For the Lagrange multiplier method the function (functional) F(x) to be minimized (under the restriction of the negative inequality constraint g(x) ≤ 0) is amended by the product of the positive Lagrange multiplier λ ≥ 0 and the negative inequality constraint g(x) ≤ 0 to render the Lagrange function (functional) (x, λ) (x, λ) := F(x) + λ g(x).

(2.130)

Then, the Lagrange function (functional) is required to take a saddle point at the optimal solution characterized by the so-called Karush–Kuhn–Tucker (KKT) complementary conditions. With  denoting the derivative with respect to x the corresponding optimality condition reads F  (x) + λ g  (x) = 0

(2.131)

which has to be solved jointly with the Karush–Kuhn–Tucker (KKT) complementary conditions g(x) ≤ 0, λ ≥ 0, λ g(x) = 0.

(2.132)

2.3 Mathematical Tools

59

The last equality determines that λ = 0 whenever g(x) < 0, i.e. when the constrained is not active, and g(x) = 0 when λ > 0, i.e. when the constraint is active. The special (neutral) case that λ = 0 and g(x) = 0 hold simultaneously is obtained when F(x) attains its minimum for the same x0 that also denotes the root of g(x) = 0. Alternatively, reformulating the inequality constraint into a corresponding (nonsmooth) equality constraint g(x) = 0, the Lagrange multiplier method is expressed ˆ λ) in terms of a modified Lagrange function (functional) (x, ˆ λ) := F(x) + λ g(x). (x,

(2.133)

Consequently, the optimality condition then reads formally F  (x) + λ g(x)  0

(2.134)

which has to be solved jointly with the constraint condition g(x) = 0.

(2.135)

Of course, the non-smoothness of the constraint g(x) may pose difficulties upon computing its derivative (indeed its sub-differential, i.e. the set of sub-derivatives) with respect to x at x = x0 . Nevertheless, formally, the use of g(x) as an equality constraint alleviates the representation of the following variants of the Lagrange multiplier method. Perturbed Lagrange Multiplier Method For the perturbed Lagrange multiplier method the function (functional) F(x) to be minimized (under the restriction of the alternative equality constraint g(x) = 0) is amended by the product of the positive Lagrange multiplier λ ≥ 0 and the alternative equality constraint g(x) = 0 together with the weighted square −1/[2 ε] λ2 ≤ 0 of the Lagrange multiplier to render the perturbed Lagrange function (functional) ˆε (x, λ)   1 λ . (2.136) ˆε (x, λ) := F(x) + λ g(x) − 2ε Correspondingly, the optimality condition then reads F  (x) + λ g(x)  0

(2.137)

which has to be solved jointly with the constraint condition λ = ε g(x) ≥ 0.

(2.138)

60

2 Preliminaries

Thus, the Lagrange multiplier is proportional to the constraint violation, whereby the proportionality factor ε is denoted the perturbation or rather the penalty parameter. Introducing the explicit expression for the Lagrange multiplier into the perturbed Lagrange function (functional) ˆε (x, λ) renders the corresponding penalty method that minimizes a penalized Lagrange function (functional) ˜ε (x) as 1 ˜ε (x) := F(x) + ε g(x)2 → min . x 2

(2.139)

Correspondingly, the unconstrained optimality condition for ˜ε (x) then reads F  (x) + ε g(x) g(x)  0.

(2.140)

When comparing ˜ε (x)  0 with its corresponding (modified) Lagrange multiplier pendant ˆ (x, λ)  0 it becomes obvious that the Lagrange multiplier λ as computed in the latter is explicitly modeled in the former as λ := ε g(x).

(2.141)

It is moreover obvious from the structure of the perturbed Lagrange function (functional) ˆε (x, λ) that the solution to the penalty method ˜ε (x) → min x converges to ˆ λ) the solution based on the original (modified) Lagrange function (functional) (x, as the penalty parameter approaches infinity ε → ∞. The benefit of the penalty method is the reduced number of unknowns (there is no Lagrange multiplier that needs to be determined), its drawback, however, is an increasing bad-conditioning of the resulting equation ˜ε (x)  0 for ε → ∞. Augmented Lagrange Multiplier Method For the augmented Lagrange multiplier method the function (functional) F(x) to be minimized (under the restriction of the alternative equality constraint g(x) = 0) is amended by the product of the positive Lagrange multiplier λ ≥ 0 and the alternative equality constraint g(x) = 0 together with the weighted square ε/2 g(x)2 ≥ 0 of the alternative equality constraint to render the augmented Lagrange function (functional) ˇε (x, λ) 1 ˇε (x, λ) := F(x) + λ g(x) + ε g(x)2 . 2

(2.142)

Correspondingly, the optimality condition then reads F  (x) + [λ + ε g(x)] g(x)  0 which has to be solved jointly with the constraint condition

(2.143)

2.3 Mathematical Tools

61

g(x) = 0.

(2.144)

Typically, the optimality condition is satisfied iteratively, whereby at the current iteration a solution x new is sought at fixed Lagrange multiplier λold . Subsequently, the Lagrange multiplier is recomputed for the next iteration in terms of an Usawa-type update that takes into account the constraint violation after the current iteration λnew = λold + ε g(x new )

(2.145)

Thereby, as a benefit of the method, regardless of the penalty parameter’s value, the constraint may be satisfied with arbitrary precision, however at the expense of additional iterations in an (external) Usawa-type update loop.

Chapter 3

Elasticity

There is still plenty of good music to be written in C major. Arnold Schoenberg, 1874–1951

Elasticity is certainly the most intensely studied and best understood paradigm model for elementary material behavior. Thereby, due to his systematic studies of elastic springs in his book De potentia restitutiva from 1678, Robert Hooke [1635–1702] may indeed be called the founder of elasticity. The corresponding illustration (to the right) is a detail from the frontispiece of this eminent publication. Interestingly, maybe out of fear of his colleagues, Hooke published his finding on the behavior of linear springs, that was later denoted as Hooke’s law in his honor, in the form of an anagram ceiiinosssttuv, which, when disentangled reads in Latin ut tensio sic vis. The Hooke model is a template for three-dimensional linear and nonlinear models of elasticity, and, moreover an essential contribution to most of the other, more sophisticated, material models discussed in subsequent chapters. Due to the simplicity of the Hooke model, the techniques relevant only later for visco-elasticity (convolution integral representation, Laplace transformation representation, complex harmonic oscillation representation) shall here first be exercised for linear elasticity.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Steinmann and K. Runesson, The Catalogue of Computational Material Models, https://doi.org/10.1007/978-3-030-63684-5_3

63

64

3 Elasticity

3.1 Hooke Model The Hooke model of an elastic solid (in short the Hooke model) consists of an elastic spring (see the sketch of the specific Hooke model in Fig. 3.1). The basic kinematic assumption of the Hooke model is the equality of the total strain  and the elastic strain e (representing the elongation of the elastic spring), i.e. (3.1)  ≡ e . Note that the set of internal variables is empty for the Hooke model, i.e. α = ∅. Robert Hooke [b. 18.7.1635, Isle of Wight, England, d. 3.3.1703, London, England] was polymath, Curator of Experiments of the Royal Society, Professor of Geometry at the Gresham College in London, Surveyor of the City of London and architect. He largely contributed to mechanics, microscopy, astronomy, meteorology and geology. Due to his tract “De potentia restitutiva” on the elasticity of springs from 1678, the Hooke model of linear elastic solids is named in his honour.

3.1.1 Specific Hooke Model: Formulation The specific Hooke model, displayed in Fig. 3.1, consists of a linear elastic spring with stiffness E (the elastic modulus). Direct Representation For the specific Hooke model the free energy density ψ is expressed as a quadratic (and thus convex) function of  (the total strain) E

Fig. 3.1 Specific Hooke model σ

σ

e

3.1 Hooke Model

65

ψ() =

1 2 E . 2

(3.2)

Then the (energetic =) ˆ total stress σ, which is conjugated to the total strain , follows as σ() = ∂ ψ() = E .

(3.3)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ. The specific Hooke model is entirely energetic, thus the dissipation potential π = 0 is zero (as is the corresponding dual dissipation potential π ∗ = 0). Consequently, the dissipative stress σ  ≡ 0 vanishes identically and no distinction is made between the total stress σ and the energetic stress σ  . The corresponding free enthalpy density ψ ∗ , as determined from a Legendre transformation   1 (3.4) ψ ∗ (σ) = max σ  − E ||2  2 then reads, with C := 1/E denoting the compliance, ψ ∗ (σ) =

1 C σ2 . 2

(3.5)

Accordingly, the respective constitutive law for the total strain follows as (σ) = ∂σ ψ ∗ (σ) = C σ.

(3.6)

Obviously, the expressions in Eqs. 3.3 and 3.6 are inverse relations. The free energy and free enthalpy densities ψ = ψ() and ψ ∗ = ψ ∗ (σ) that are quadratic in  and σ, respectively, together with the resulting constitutive relations σ = σ() and  = (σ) that are linear in  and σ, respectively, are displayed in Fig. 3.2. The specific Hooke model is summarized in Table 3.1.

66

3 Elasticity

ψ( )

1

1 2

0.5

2

0.5

0

0

0.5

−0.5

−1

−1 −1

−0.5

0

0.5

−1

1

ψ ∗ (σ)

1

σ

−0.5

−1

−1 0

0.5

0.5

1 σ/E

(σ)

σ

0

0.5

−0.5

0

0.5

0

−1

−0.5

1 1 2 σ /E 2

0.5

σ( )

1

1

−1

−0.5

0

0.5

1

Fig. 3.2 Specific Hooke model: Quadratic free energy density ψ() together with resulting linear σ = σ() and quadratic free enthalpy density ψ ∗ (σ) together with resulting linear  = (σ) Table 3.1 Summary of the specific Hooke model

3.1 Hooke Model

3

67

3

E(t)/E = H(t)

2

C(t) E = H(t)

2 (t)

0

= H(t)

1

1 σ(t)/σ0 = H(t)

t

0

0

1

2

t

0

3

0

1

2

3

Fig. 3.3 Specific Hooke model: Normalized relaxation function E (t)/E (left) and normalized creep function C (t) E (right)

Convolution Integral Representation The constitutive relation for the total stress may be considered a degenerated differential (i.e. an algebraic) equation relating the total stress and strain as σ(t) = E (t).

(3.7)

Imposing a constant strain step (t) = 0 H(t)1 renders also a Heaviside-type solution for the stress history σ(t) = E H(t) 0 =⇒ σ(t) =: E(t) 0 .

(3.8)

Here E(t), i.e. the normalized stress history as response to an imposed constant unit strain step (t) = H(t), has been introduced as the Heaviside-type “relaxation” function that is illustrated in Fig. 3.3 (left) E(t) := E H(t).

(3.9)

Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as response to an arbitrary strain history (t) ≡ H(t) (t) is formally retrieved from the convolution integral  t E(t − t  ) ˙(t  ) dt  =: E(t)  ˙(t). (3.10) σ(t) = E (t) = 0

Thereby the convolution of the “relaxation” function with the strain rate history is abbreviated symbolically as E(t)  ˙(t).

1 With

H(t) the Heaviside function, (t) is a causal signal: (t < 0) = 0 and (t ≥ 0) = 0 .

68

3 Elasticity

Imposing, alternatively, a constant stress step σ(t) = σ0 H(t) renders also a Heaviside-type solution for the strain history E (t) = H(t) σ0 =⇒ (t) =: C(t) σ0 .

(3.11)

Here C(t), i.e. the normalized strain history as response to an imposed constant unit stress step σ(t) = H(t), has been introduced as the Heaviside-type “creep” function that is illustrated in Fig. 3.3 (right) C(t) := C H(t).

(3.12)

Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as response to an arbitrary stress history σ(t) ≡ H(t) σ(t) is formally retrieved from the convolution integral  t C(t − t  ) σ(t ˙  ) dt  =: C(t)  σ(t). ˙ (3.13) (t) = C σ(t) = 0

Thereby the convolution of the “creep” function with the stress rate history is abbreviated symbolically as C(t)  σ(t). ˙ Laplace Transformation Representation Upon Laplace transformation, the convolution integral of the “relaxation” function E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0, results in L{σ(t)} = L{E(t)  ˙(t)} = s L{E(t)} L{(t)} = E L{(t)}.

(3.14)

Choosing, as a particular example, a causal harmonic strain history with (t) = H(t) a sin(ω t) and thus L{(t)} = a ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,2 also a causal in-phase harmonic signal for the resulting stress history, whereby the initial condition σ(0) = E (0) is captured. σ(t) = H(t) E  sin(ω t) a .

(3.15)

Thereby E  := E is here formally introduced as abbreviation, however as will become transparent in the sequel, it denotes the so-called storage stiffness modulus. The stress history resulting from a sinusoidal strain history is shown in Fig. 3.4 (left).

2 The

inverse Laplace transformation for the stress history follows from the following step by step computation: L{σ(t)}

a

= E

ω ω2 + s 2

= E  L{H(t) sin(ω t)}.

3.1 Hooke Model

69

σ(t)/E

10

(t) E

10 5

5 t

0 −5

−5

−10

−10 0

2

4

6

8

10

t

0

0

2

4

6

8

10

Fig. 3.4 Specific Hooke model with E = 1.0: Normalized stress history σ(t)/E resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history (t) E resulting from sinusoidal stress history with σa = 5 and ω = 2 π/4 (right)

Upon Laplace transformation, the convolution integral of the “creep” function C(t) with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0, results in L{(t)} = L{C(t)  σ(t)} ˙ = s L{C(t)} L{σ(t)} = C L{σ(t)}.

(3.16)

Choosing, as a particular example, a causal harmonic stress history with σ(t) = H(t) σa sin(ω t) and thus L{σ(t)} = σa ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,3 also a causal in-phase harmonic signal for the resulting strain history, whereby the initial condition (0) = C σ(0) is captured (t) = H(t) C  sin(ω t) σa .

(3.17)

Thereby C  := C = 1/E is here formally introduced as abbreviation, in terminological accordance to E  it is denoted the storage compliance modulus. The strain history resulting from a sinusoidal stress history is shown in Fig. 3.4 (right). It is interesting to note that the “relaxation” function and the “creep” function are related via their Laplace transformations as s 2 L{E(t)} L{C(t)} = 1.

3 The

(3.18)

inverse Laplace transformation for the strain history follows from the following step by step computation: L{(t)}

σa

=C

ω ω2 + s 2

= C  L{H(t) sin(ω t)}.

70

3 Elasticity

Observe, furthermore, that direct application of the Laplace transformation to the degenerated differential (i.e. algebraic) equation relating the total stress and strain renders immediately a relation L{σ(t)} = E L{(t)}

(3.19)

that is entirely conforming with the convolution integral representation. Complex Harmonic Oscillation Representation The degenerated differential (i.e. algebraic) equation relating the total stress and strain reads in complex representation as σ(t) = E (t).

(3.20)

Then, for a stationary harmonic oscillation of the total stress and strain with σ(t) = σ ∗ ei ω t and (t) = ∗ ei ω t the relation between the corresponding complex amplitudes ∗ = a ei δ and σ ∗ = σa ei δσ (where δ = −π/2 for sinusoidal strain control and δσ = −π/2 for sinusoidal stress control) follows as σ ∗ = E ∗ =: E ∗ ∗ .

(3.21)

Thereby the quantity relating the complex amplitudes of the total strain and stress is denoted the complex stiffness modulus, its inverse is the complex compliance modulus (so that E ∗ C ∗ = 1) E ∗ (ω) := E  := E and C ∗ (ω) := C  := C.

(3.22)

Note that for the specific Hooke model, trivially, the complex moduli E ∗ and C ∗ have only real parts. Here, E  and C  denote the so-called storage stiffness modulus and storage compliance modulus, respectively. The storage stiffness modulus and the storage compliance modulus are plotted against the angular frequency ω for various elasticity moduli in Fig. 3.5. Finally, the (real) amplitudes E a and Ca of the complex stiffness modulus and the complex compliance modulus are defined via E ∗ (ω) =: E  ei0 =: E a ei0 and C ∗ (ω) =: C  ei0 =: Ca ei0

(3.23)

so that σa = E a a (or a = Ca σa ) with δσ = δ . Specifically, the angular frequency independent amplitudes E a = E a (ω) and Ca = Ca (ω) follow as E a := E  = E and Ca := C  = C.

(3.24)

The amplitudes of the complex stiffness modulus and the complex compliance modulus, when plotted against the angular frequency ω for various elasticity moduli, are

3.1 Hooke Model

E

71

E = 102 , 101 , 100 , 10−1 , 10−2

C

102

102 101

101

100

100

10−1

10−1

10−2

E = 102 , 101 , 100 , 10−1 , 10−2

E 10−3 10−2 10−1 100 101 102 103

ω

C := 1/E

10−2

10−3 10−2 10−1 100 101 102 103

ω

Fig. 3.5 Specific Hooke model: Storage stiffness modulus and amplitude E  (ω) ≡ E a (ω) (left) and storage compliance modulus and amplitude C  (ω) ≡ Ca (ω) (right) plotted against the angular frequency ω for five decades of elasticity moduli E Table 3.2 Algorithmic update for the specific Hooke model

identical to the storage moduli that are displayed in Fig. 3.5. The phase shift angle between the harmonically oscillating total stress and strain and its tangent are also denoted the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the loss angle are identically zero for all angular frequencies.

3.1.2 Specific Hooke Model: Algorithmic Update The trivial (algorithmic) update for the specific Hooke model is summarized in Table 3.2.

3.1.3 Specific Hooke Model: Response Analysis The following elementary strain and stress histories (Zig-Zag, Sine, Ramp) will also be used throughout for the response analysis of the other material models to be

72

3 Elasticity

discussed in subsequent chapters. Thereby, either the strain or the stress history will be prescribed and the stress or strain history resulting thereof computed and analyzed. Strain and Stress History: Zig-Zag For unit stiffness E = 1 the coinciding and non-harmonic cyclic Zig-Zag strain and stress history reads (t) E=1 σ(t) = := 4 a 4 σa ⎧ 2 ⎪ ⎪ −k+ ⎪ ⎪ 2 ⎪ ⎪ ⎨ 1 k− − ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 −k+ 2

t T t T t T

⎧ ⎧ ⎫ 4⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k − k− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ t 3 for ≤ ≤ k− k− ⎪ ⎪ ⎪ T 4⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩k − ⎭ ⎭ k− 4 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎫ 3⎪ ⎪ ⎪ 4⎪ ⎪ ⎪ ⎬ 1 4⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ ⎭ 4

(3.25)

Thereby k = 1, 2, . . . , n cy counts the cycles with n cy the maximum cycle number, T denotes the period length of one cycle, whereas a and σa are the strain and stress amplitude, respectively. The corresponding Zig-Zag strain and stress histories for E = 1, amplitudes a = σa = 5 and period T = 4 are showcased for the time interval t ∈ [0, tmax = 10 = 2.5 × T ] in Fig. 3.6a, b. Strain and Stress History: Sine For unit stiffness E = 1 the coinciding and harmonic cyclic Sine strain and stress history reads (t) E=1 σ(t) = := sin(2 π [t/T ]). a σa

(3.26)

Thereby, T denotes the period length of one cycle, and a and σa are the strain and stress amplitude, respectively. The corresponding Sine strain and stress histories for E = 1, amplitudes a = σa = 5 and period T = 4 are showcased for the time interval t ∈ [0, tmax = 10 = 2.5 × T ] in Fig. 3.6c, d. Strain and Stress History: Ramp For unit stiffness E = 1 the coinciding Ramp strain and stress history reads

3.1 Hooke Model

73 σ

t

t

a)

(tmax = 100 × 0.1

max,min

b)

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

σ

t

t

c)

(tmax = 100 × 0.1

max,min

d)

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

σ

t

t

e)

(tmax = 100 × 0.1

max,min

= ± 5.0)

f)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

Fig. 3.6 Response analysis of Hooke model with material data: E = 1.0. a, b Zig-Zag, c, d Sine, e, f Ramp strain and stress history with data: a = σa = 5.0

74

3 Elasticity

⎧ t − t0 ⎪ ⎪ ⎪t −t ⎪ ⎪ 1 0 ⎪ ⎨ (t) E=1 σ(t) = := 1 ⎪ a σa ⎪ ⎪ ⎪ t −t ⎪ ⎪ ⎩ 3 t3 − t2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎧ ⎫ t0 ≤ t ≤ t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ t1 ≤ t ≤ t2 for ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎭ t2 ≤ t ≤ t3

(3.27)

Thereby t0 , t1 , t2 , and t3 denote the starting time, the time after ramp-up, the time before ramp-down, and the terminating time, respectively, and a and σa are the strain and stress amplitude, respectively. The corresponding Ramp strain and stress histories for E = 1, amplitudes a = σa = 5, and time instants t0 = 0.0, t1 = 1.0, t2 = 9.0, and t3 = 10.0 are showcased for the time interval t ∈ [0, tmax = 10] in Fig. 3.6e, f.

3.1.4 Generic Hooke Model: Formulation A generic formulation of the Hooke model can be obtained from generalizing the specific Hooke model in Fig. 3.1 by assuming the elastic spring as nonlinear. For the generic Hooke model the free energy density ψ is expressed as a nonquadratic, yet convex, function of  (the total strain) ψ = ψ().

(3.28)

Then the (energetic =) ˆ total stress σ follows as σ() = ∂ ψ().

(3.29)

Recall that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides here identically with the energetic stress σ  ≡ σ. The generic Hooke model is entirely energetic, thus the dissipation potential π = 0 is zero (as is the corresponding dual dissipation potential π ∗ = 0). Consequently, the dissipative stress σ  ≡ 0 vanishes identically and no distinction is made between the total stress σ and the energetic stress σ  . Furthermore, for the generic Hooke model it is possible to introduce the nonquadratic, convex free energy and free enthalpy densities as ψ = ψ() and ψ ∗ = ψ ∗ (σ), respectively, which are related via corresponding Legendre transformations4

4 Remark

on Legendre Transformation of Free Energy/Enthalpy Densities:

3.1 Hooke Model

75

ψ ( ) = max{σ  − ψ ∗ (σ)},

(3.30a)

ψ ∗ (σ) = max{σ  − ψ ( )}.

(3.30b)

σ



The stationarity conditions corresponding to Eqs. 3.30a and 3.30b are the constitutive relations (3.31a) (σ) = ∂σ ψ ∗ (σ), σ( ) = ∂  ψ ( ).

(3.31b)

Obviously, the relations in Eqs. 3.31a and 3.31b determine entirely the energetic behaviour of the generic Hooke model, thus the formulation is completed at this stage. It is finally emphasized that due to the entirely energetic character of the generic Hooke model the dissipation inequality d ≤ 0 degenerates to the strict equality d = 0. The generic Hooke model is summarized in Table 3.3.

Table 3.3 Summary of the generic Hooke model

Consider the σ versus  diagram (to the right), whereby the dependence between σ and  is either given by σ = σ σ() or by   = (σ). Then ψ() follows as the integral ψ() := 0 σ( ) d , i.e. the area under the σ = σ() curve, thus rendering ∂ ψ = σ(), whereas ψ ∗ (σ) folσ lows as the integral ψ ∗ (σ) = 0 (σ  ) dσ  , i.e. the area under the  = (σ) curve, thus rendering ∂σ ψ ∗ = (σ). Obviously, ψ() and ψ ∗ (σ) sum up to σ .

(σ) σ( ) ψ ∗ (σ) ψ( )

Chapter 4

Visco-Elasticity

I’d say the differences are more interesting than the similarities at this point. — Philip Glass, b. 1973 —

Visco-elasticity is the paradigm for rate-dependent, either (asymptotically) reversible (in the case of solids) or non-reversible (in the case of fluids) material behavior. Thereby experimental evidence for various classes of materials, in particular for polymers (typically above the glass transition temperature) and fluids, suggests to introduce the total stress or parts of it (as the response to an external load) as being rate-dependent. Rate-dependence is rooted in sub-scale time-dependent relaxation processes in the material, take as an example chain disentanglements in polymeric materials with a chain network sub-scale structure. From a convex analysis point of view, rate-dependence is engraved in the smoothness of the convex dissipation potential and its dual. The elementary rheological model to capture viscous, i.e. ratedependent material behavior in terms of a one-to-one relation between the stress and the rate of strain is the viscous dashpot. The rheological model for a viscous fluid consisting of a viscous dashpot only is denoted the Newton model. Parallel and serial arrangements of a viscous dashpot with an elastic spring render the Kelvin model for visco-elastic solids and the Maxwell model for visco-elastic fluids, respectively. Further, serial or parallel arrangements of either several Kelvin models or several Maxwell models are established as the Generalized-Kelvin model and the Generalized-Maxwell model, respectively. Particular three parameter sub-cases of the corresponding specific Generalized-Kelvin or Generalized-Maxwell model are denoted as the Standard-Linear-Solid (SLS) Kelvin or Maxwell model, respectively, and the Standard-Linear-Fluid (SLF) Kelvin or Maxwell model (which shall not be considered here), respectively. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Steinmann and K. Runesson, The Catalogue of Computational Material Models, https://doi.org/10.1007/978-3-030-63684-5_4

77

78

4 Visco-Elasticity

4.1 Newton Model The Newton model of a viscous fluid (in short the Newton model) consists of a viscous dashpot (see the sketch of the specific Newton model in Fig. 4.1). The basic kinematic assumption of the Newton model is the equality of the total strain  and the viscous strain v (representing the elongation of the viscous dashpot), i.e. (4.1)  ≡ v . Note that the set of internal variables is empty for the Newton model, i.e. α = ∅. Sir Isaac Newton [b. 4.1.1643, Woolsthorpe, England, d. 31.3.1727, London, England] was Lucasian Professor of Mathematics at the Trinity College in Cambridge. He laid the foundations of classical mechanics in his landmark treatise “Philosophiae Naturalis Principia Mathematica” from 1687. Likewise he is the co-founder of differential and integral calculus (as the opponent to Leibniz) and greatly contributed to optics. The Newton model of viscous fluids is named after him.

4.1.1 Specific Newton Model: Formulation The specific Newton model, displayed in Fig. 4.1, consists of a linear viscous dashpot with viscosity η. Direct Representation Since there is no energy storage for the specific Newton model the free energy density ψ vanishes identically ψ() ≡ 0. (4.2) η

Fig. 4.1 Specific Newton model

σ

σ

v

4.1 Newton Model

79

Thus the energetic stress σ  , which is conjugated to the total strain , vanishes identically as well (4.3) σ  () ≡ 0. Furthermore, for the specific Newton model the convex and smooth (quadratic) dissipation potential π is chosen as π(˙) =

1 η |˙|2 . 2

(4.4)

Observe that (i) π does depend on ˙, thus the dissipative stress σ  = 0 for ˙ = 0, and that (ii) π is positively homogenous of degree two in ˙ and obviously smooth at the origin ˙ = 0. Consequently, the dissipative stress σ  computes as partial derivative of the dissipation potential with respect to its conjugated variable σ  (˙) = ∂˙ π(˙) = η ˙.

(4.5)

Recall that the total stress σ applied to the rheological model (that enters the equilibrium condition) and the energetic and dissipative stresses are constitutively related by σ = σ  + σ  , thus (since here σ  ≡ 0) the value σ ≡ σ

(4.6)

will exclusively be used in the sequel for convenience of exposition. The corresponding dual dissipation potential π ∗ , as determined from a Legendre transformation   1 (4.7) π ∗ (σ) = max σ ˙ − η |˙|2 ˙ 2 then reads π ∗ (σ) =

1 1 2 |σ| . 2 η

(4.8)

The evolution law for the total strain then follows as partial derivative of the dual dissipation potential with respect to its conjugated variable ˙(σ) = ∂σ π ∗ (σ) =

1 σ. η

(4.9)

Obviously the expressions in Eqs. 4.5 and 4.9 are inverse relations. The smooth dissipation and dual dissipation potentials π = π(˙) and π ∗ = π ∗ (σ) together with the resulting smooth constitutive relations σ = σ(˙) and ˙ = ˙(σ) are displayed in Fig. 4.2. The specific Newton model is summarized in Table 4.1.

80

4 Visco-Elasticity

π( ˙)

1

1 η ˙2 2

0.5

0.5

0

0

˙

0.5

−0.5

−1

−1 −1

−0.5

0

0.5

1 2 σ /η 2

σ

−0.5

−1

−1 0.5

0.5

1 σ/η

˙(σ)

σ

0

0.5

0

0

0.5

0

−0.5

−0.5

1

0.5

−1

˙

−1

1

π ∗ (σ)

1

η˙

σ( ˙)

1

1

−1

−0.5

0

0.5

1

Fig. 4.2 Specific Newton model: Smooth dissipation potential π(˙) together with resulting smooth σ = σ(˙) and smooth dual dissipation potential π ∗ (σ) together with resulting smooth ˙ = ˙ (σ) Table 4.1 Summary of the specific Newton model (1) Strain



≡ v

(2) Potential π = (3) Stress

1 2

η |˙|2

σ = η ˙

≡

or (2) Potential π ∗ =

1 1 |σ|2 2 η

(3) Evolution ˙

1 σ η

=

σ

4.1 Newton Model

81

Convolution Integral Representation The constitutive relation for the total stress may be considered a differential equation relating the total stress and strain as σ(t) = η ˙(t).

(4.10)

Imposing a constant strain step (t) = 0 H(t) (and thus ˙(t) = 0 δ(t)) renders a Dirac-delta-type solution for the stress history σ(t) = η δ(t) 0 =⇒ σ(t) =: E(t) 0 .

(4.11)

Here E(t), i.e. the normalized stress history as response to an imposed constant unit strain step (t) = H(t), has been introduced as the Dirac-delta-type relaxation function that is illustrated in Fig. 4.3 (left) E(t) := η δ(t).

(4.12)

Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution integral  t

σ(t) =

E(t − t  ) ˙(t  ) dt  =: E(t)  ˙(t).

(4.13)

0

Thereby, the convolution of the relaxation function with the strain rate history is abbreviated symbolically as E(t)  ˙(t). ˙ = Imposing, alternatively, a constant stress step σ(t) = σ0 H(t) (and thus σ(t) σ0 δ(t)) renders linearly increasing creep strain in time

3

∞ E(t)/η = δ(t)

3

2

C(t) η = H(t) t

2 (t)

0 = H(t)

1

1 σ(t)/σ0 = H(t)

t

0

0

1

2

3

t

0

0

1

2

3

Fig. 4.3 Specific Newton model: Normalized relaxation function E (t)/η (left) and normalized creep function C (t) η (right). The dotted lines depict the normalized step functions for the prescribed strain and stress, respectively

82

4 Visco-Elasticity

η ˙(t) = H(t) σ0 =⇒ (t) =: C(t) σ0 .

(4.14)

Here C(t), i.e. the normalized strain history as response to an imposed constant unit stress step σ(t) = H(t), has been introduced as the linearly increasing creep function that is illustrated in Fig. 4.3 (right) C(t) := H(t) t/η.

(4.15)

Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as response to an arbitrary stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral  t

(t) =

C(t − t  ) σ(t ˙  ) dt  =: C(t)  σ(t). ˙

(4.16)

0

Thereby the convolution of the creep function with the stress rate history is abbreviated symbolically as C(t)  σ(t). ˙ Laplace Transformation Representation Upon Laplace transformation, the convolution integral of the relaxation function E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0, results in L{σ(t)} = L{E(t)  ˙(t)} = s L{E(t)} L{(t)} = η s L{(t)}.

(4.17)

Choosing, as a particular example, a causal harmonic strain history with (t) = H(t) a sin(ω t) and thus L{(t)} = a ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,1 a causal harmonic signal (shifted by π/2) for the resulting stress history, whereby the initial condition σ(0) = η ˙(0) is captured. σ(t) = H(t) E  cos(ω t) a .

(4.18)

Thereby E  := η ω is here formally introduced as abbreviation, however as will become transparent in the sequel, it denotes the so-called loss stiffness modulus. The stress history resulting from a sinusoidal strain history is shown in Fig. 4.4 (left). Upon Laplace transformation, the convolution integral of the creep function C(t) with a prescribed stress (rate) history, supposed a causal signal σ(t) = H(t) σ(t) with σ(0) = 0, results in

1 The

inverse Laplace transformation for the stress history follows from the following step by step computation: L{σ(t)}

a

= ηs

ω s = ηω 2 = E  L{H(t) cos(ω t)}. [ω 2 + s 2 ] [ω + s 2 ]

4.1 Newton Model

83

σ(t)/η

10

(t) η

10

t

−5

−10

−10 2

4

6

8

10

cos(ωt)

−5

0

t

0

˜ −C

0

˜ C

5

5

0

2

4

6

8

10

Fig. 4.4 Specific Newton model with η = 1.0: Normalized stress history σ(t)/η resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history (t) η resulting from sinusoidal stress history with σa = 5 and ω = 2 π/4 (right)

L{(t)} = L{C(t)  σ(t)} ˙ = s L{C(t)} L{σ(t)} =

1 L{σ(t)}. ηs

(4.19)

Choosing, as a particular example, a causal harmonic stress history with σ(t) = H(t) σa sin(ω t) and thus L{σ(t)} = σa ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,2 a causal harmonic signal (shifted by −π/2) for the resulting strain history superposed by a constant signal that is needed to enforce the initial condition (0) = 0   (4.20) (t) = H(t) −C  cos(ω t) + C  σa . Thereby C  := 1/[η ω] is here formally introduced as abbreviation, in terminological accordance to E  it is denoted the loss compliance modulus. The strain history resulting from a sinusoidal stress history is shown in Fig. 4.4 (right). 2 The

inverse Laplace transformation for the strain history follows from the following step by step computation: L{(t)}

σa

=−

−ω 1 η s [ω 2 + s 2 ]

=−

−ω 2 1 η ω s [ω 2 + s 2 ]

1 s 2 − [ω 2 + s 2 ] η ω s [ω 2 + s 2 ]   1 1 s =− − η ω ω2 + s 2 s s   1 = −C +C ω2 + s 2 s =−

= −C  L{H(t) cos(ω t)} + C  L{H(t)}.

84

4 Visco-Elasticity

It is interesting to note that the relaxation function and the creep function are related via their Laplace transformations as s 2 L{E(t)} L{C(t)} = 1.

(4.21)

Observe, furthermore, that direct application of the Laplace transformation to the differential equation relating the total stress and strain renders immediately a relation L{σ(t)} = η s L{(t)}

(4.22)

that is entirely conforming with the convolution integral representation. Complex Harmonic Oscillation Representation The differential equation relating the total stress and strain reads in complex representation as (4.23) σ(t) = η ˙(t). Then, for a stationary harmonic oscillation of the total stress and strain with σ(t) = σ ∗ ei ω t and (t) = ∗ ei ω t the relation between the corresponding complex amplitudes ∗ = a ei δ and σ ∗ = σa ei δσ (where δ = −π/2 for sinusoidal strain control and δσ = −π/2 for sinusoidal stress control) follows as σ ∗ = i η ω ∗ =: E ∗ ∗ .

(4.24)

Thereby the quantity relating the complex amplitudes of the total strain and stress is denoted the complex stiffness modulus, its inverse is the complex compliance modulus (so that E ∗ C ∗ = 1) E ∗ (ω) = i η ω =: i E  and C ∗ (ω) = −i

1 =: −i C  . ηω

(4.25)

Note that for the specific Newton model, trivially, the complex moduli E ∗ and C have only imaginary parts. Here, E  and C  denote the so-called loss stiffness modulus and loss compliance modulus, respectively, that are defined as ∗

E  := η ω and C  :=

1 . ηω

(4.26)

The loss stiffness modulus and the loss compliance modulus are plotted against the angular frequency ω for various viscosities in Fig. 4.5. Finally, the (real) amplitudes E a and Ca of the complex stiffness modulus and the complex compliance modulus are defined via

4.1 Newton Model

E

6

85

η = 102 , 101 , 100 , 10−1 , 10−2

10

C

6

10

103

103

100

100

10−3 10−6

η = 102 , 101 , 100 , 10−1 , 10−2

10−3

ωη 10−3 10−2 10−1 100 101 102 103

10−6

ω

1 ωη 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.5 Specific Newton model: Loss stiffness modulus and amplitude E  (ω) ≡ E a (ω) (left) and loss compliance modulus and amplitude C  (ω) ≡ Ca (ω) (right) plotted against the angular frequency ω for five decades of viscosities η

E ∗ (ω) =: E  ei π/2 =: E a ei π/2 and C ∗ (ω) =: C  e−i π/2 =: Ca e−i π/2

(4.27)

so that σa = E a a (or a = Ca σa ) with δσ = δ + π/2. Specifically, the angular frequency dependent amplitudes E a (ω) and Ca (ω) follow as E a := E  = η ω and Ca := C  =

1 . ηω

(4.28)

The amplitudes of the complex stiffness modulus and the complex compliance modulus, when plotted against the angular frequency ω for various viscosities, are identical to the loss moduli that are displayed in Fig. 4.5. The phase shift angle between the harmonically oscillating total stress and strain and its tangent are also denoted the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the loss angle are constant for all angular frequencies.

4.1.2 Specific Newton Model: Algorithmic Update For the specific Newton model the evolution law for the total strain  is integrated by the implicit Euler backwards method to render n := n − n−1 =

t n n σ . η

Consequently, the total stress σ is updated at the end of the time step by

(4.29)

86

4 Visco-Elasticity

Table 4.2 Algorithmic update for the specific Newton model Input

n n−1

Trial Strain

 = n − n−1 η   t n η E an = t n

Update Stress σ n = Tangent Output

σ n E an

σn =

η  . t n

(4.30)

The trial strain  is computable exclusively from known quantities at the beginning and at the end of the time step and follows as  := n .

(4.31)

The sensitivity of σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is straightforwardly computed as ∂ σ n =

η . t n

(4.32)

Note that, consequently, the algorithmic tangent degenerates to E a → ∞ for t n → 0, i.e. for very fast processes (as compared to the viscosity) the response is rigid. The algorithmic step-by-step update for the specific Newton model is summarized in Table 4.2.

4.1.3 Specific Newton Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Newton model to a prescribed Zig-Zag strain history is displayed in Fig. 4.6a–f. Figure 4.6a, c depict the prescribed Zig-Zag (viscous) strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed.

4.1 Newton Model

87 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.6 Response analysis of the specific Newton model with material data: η = 1.0. Prescribed Zig-Zag strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 40.0; t = 1.0, N = 100, f T = 400.0; t = 10.0, N = 100

88

4 Visco-Elasticity

Figure 4.6b showcases the resulting stress history σ(t) that displays a periodic block signal with σ(t) = ±η ˙(t) = ±5 since ˙(t) = ±5. The resulting σ = σ() diagram is highlighted in Fig. 4.6d. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = ±5 obviously tend to ∞ when t → 0. Finally, Fig. 4.6e, f depict the resulting σ = σ() diagrams for a 10 and 100 times longer period T corresponding to a 10 and 100 times lower strain rate |˙(t)|, respectively. They clearly demonstrate a viscous fluid-like behaviour with vanishing stress for |˙(t)| → 0. Prescribed Strain History: Sine The response of the specific Newton model to a prescribed Sine strain history is displayed in Fig. 4.7a–f. Figure 4.7a, c depict the prescribed Sine (viscous) strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.7b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.18, a cosine signal with σ(t) = η ˙(t) = η ω a cos(ω t) and amplitude η ω a = 2.5π ≈ 7.85. The resulting, slightly distorted σ = σ() diagram is highlighted in Fig. 4.7d. Due to the finite sized time step t and corresponding finite sized strain increment  the expected slope of the σ = σ() diagram at  = 0 is only approximately captured; however, it obviously tends to ∞ when t → 0. The numerical integration error due to the finite sized time step t explains likewise the slight distortion of the σ = σ() ellipsoidal path. Finally, Fig. 4.7e, f depict the resulting σ = σ() diagrams for a 10 and 100 times longer period T corresponding to lower strain rates |˙(t)|, respectively. They clearly demonstrate a viscous fluid-like behaviour with vanishing stress for |˙(t)| → 0. Prescribed Strain History: Ramp The response of the specific Newton model to a prescribed Ramp strain history is displayed in Fig. 4.8a–f. Figure 4.8a, c depict the prescribed Ramp (viscous) strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.8b showcases the resulting stress history σ(t) that displays a block-type signal with σ(t) = η ˙(t) = ±5 when ˙(t) = ±5 in the loading and the unloading phases and σ(t) = η ˙(t) = 0 when ˙(t) = 0 during the holding phase, and |σ(t)| ∈ [0, 5].

4.1 Newton Model

89 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.7 Response analysis of the specific Newton model with material data: η = 1.0. Prescribed Sine strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 40.0; t = 1.0, N = 100, f T = 400.0; t = 10.0, N = 100

90

4 Visco-Elasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.8 Response analysis of the specific Newton model with material data: η = 1.0. Prescribed Ramp strain history with data: a = 5.0, a)-d) t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e) t0 = 0.0, t1 = 10.0, t2 = 90.0, t3 = 100.0; t = 1.0, N = 100, f) t0 = 0.0, t1 = 100.0, t2 = 900.0, t3 = 1000.0; t = 10.0, N = 100

4.1 Newton Model

91

The resulting σ = σ() diagram is highlighted in Fig. 4.8d. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = 5 obviously tend to ∞ when t → 0. Finally, Fig. 4.8e, f depict the resulting σ = σ() diagrams for 10 and 100 times larger t1 , t2 , t3 corresponding to lower strain rates |˙(t)|, respectively. They clearly demonstrate a viscous fluid-like behaviour with vanishing stress for |˙(t)| → 0. Prescribed Stress History: Zig-Zag The response of the specific Newton model to a prescribed Zig-Zag stress history is displayed in Fig. 4.9a–f. Figure 4.9a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.9b, d showcase the resulting (viscous) strain history (t) that displays a periodic signal with ˙(t) = σ(t)/η ∝ ±5t, thus leading to piecewise quadratic (t) with |(t)| ∈ [0, 5]. The resulting, slightly tilted, σ = σ() diagram is highlighted in Fig. 4.9c. The numerical integration error due to the finite sized time step t explains the slight tilting of the σ = σ() path. Finally, Fig. 4.9e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to a 10 and 100 times higher stress rate |σ(t)|, ˙ respectively. They clearly demonstrate a rigid behaviour with vanishing strain for → ∞. |σ(t)| ˙ Prescribed Stress History: Sine The response of the specific Newton model to a prescribed Sine stress history is displayed in Fig. 4.10a–f. Figure 4.10a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.10b, d showcase the resulting (viscous) strain history (t) that displays, in accordance with the analytical solution in Eq. 4.20, a (negative and upwards shifted) cosine signal with ˙(t) = σ(t)/η = σa /η sin(ω t) oscillating about the mean value a = σa /η/ω = 10/π ≈ 3.18 with amplitude a ≈ 3.18. The resulting, slightly tilted σ = σ() diagram is highlighted in Fig. 4.10c. The numerical integration error due to the finite sized time step t explains the slight tilting of the σ = σ() ellipsoidal path.

92

4 Visco-Elasticity σ

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

t

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 10.0 σmax,min = ± 5.0)

= ± 10.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.9 Response analysis of the specific Newton model with material data: η = 1.0. Prescribed Zig-Zag stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

4.1 Newton Model

93

σ

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

t

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 10.0 σmax,min = ± 5.0)

= ± 10.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.10 Response analysis of the specific Newton model with material data: η = 1.0. Prescribed Sine stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

94

4 Visco-Elasticity

Finally, Fig. 4.10e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to higher stress rates |σ(t)|, ˙ respectively. They clearly demonstrate a rigid behaviour with vanishing strain for |σ(t)| ˙ → ∞. Prescribed Stress History: Ramp The response of the specific Newton model to a prescribed Ramp stress history is documented in Fig. 4.11a–f. Figure 4.11a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.11b, d showcase the resulting (viscous) strain history (t) that displays a monotonic signal with ˙(t) = σ(t)/η ∝ ±5t when σ(t) ∝ ±5t in the loading and the unloading phases, thus leading to piecewise quadratic (t), and ˙(t) = σ(t)/η = 5 when σ(t) = 5 during the holding phase, thus leading to piecewise linear (t). Taken together, (t3 ) = 2 × 2.5 + 5 × 8 = 45. The resulting σ = σ() diagram of nonlinear parallelogram format is highlighted in Fig. 4.11c. Finally, Fig. 4.11e, f depict the resulting σ = σ() diagrams for 10 and 100 times ˙ respectively. They clearly smaller t1 , t2 , t3 corresponding to higher stress rates |σ(t)|, demonstrate a rigid behaviour with vanishing strain for |σ(t)| ˙ → ∞.

4.1.4 Generic Newton Model: Formulation A generic formulation of the Newton can be obtained from generalizing the specific Newton model in Fig. 4.1 by assuming the viscous dashpot as nonlinear. For the generic Newton model the free energy density ψ vanishes identically ψ() ≡ 0.

(4.33)

Thus the energetic stress σ  vanishes identically as well σ  () ≡ 0.

(4.34)

Recall that the energetic and the dissipative stresses are constitutively related to the total stress σ (that enters the equilibrium condition) by σ = σ  + σ  , thus (with σ  ≡ 0) the total stress σ ≡ σ  will exclusively be used in the sequel. Furthermore, for the generic Newton model it is possible to introduce the convex and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙) and π ∗ = π ∗ (σ), respectively, which are related via corresponding Legendre trans-

4.1 Newton Model

95

σ

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

t

(tmax = 100 × 0.1

b)

σ

max,min

= ± 50.0)

v

t

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 50.0 σmax,min = ± 5.0)

p,max,min

= ± 50.0)

σ

f)

(

max,min

= ± 50.0 σmax,min = ± 5.0)

Fig. 4.11 Response analysis of the specific Newton model with material data: η = 1.0. Prescribed Ramp stress history with data: σa = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.1, t2 = 0.9, t3 = 1.0; t = 0.01, N = 100, f t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100

96

4 Visco-Elasticity

formations3

π ( ˙) = max{σ ˙ − π ∗ (σ)},

(4.35a)

π ∗ (σ) = max{σ ˙ − π ( ˙)}.

(4.35b)

σ

˙

The stationarity conditions corresponding to Eqs. 4.35a and 4.35b are the constitutive relations ˙(σ) = ∂σ π ∗ (σ), (4.36a) σ( ˙) = ∂ ˙ π ( ˙).

(4.36b)

Obviously, the relations in Eqs. 4.36a and 4.36b determine entirely the dissipative behavior of the generic Newton model, thus the formulation is completed at this stage. Finally, as a further interesting aspect, the dissipation d = σ ˙ is alternatively expressed from Eqs. 4.35a and 4.35b in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙) + π ∗ (σ) ≥ 0.

(4.37)

The generic Newton model is summarized in Table 4.3. Table 4.3 Summary of the generic Newton model (1) Strain



≡ v

(2) Potential π = π(˙) (3) Stress

σ = ∂˙ π

≡

σ

or (2) Potential π ∗ = π ∗ (σ) (3) Evolution ˙

= ∂σ π ∗

3 Remark

on Legendre Transformation of Dissipation Potentials: Consider the σ versus ˙ diagram (to the right), whereby the dependence between σ and ˙ is either given by σ = σ(˙)σ or by ˙ = ˙ (σ). Then π(˙) follows as the integral π(˙) := ˙ π ∗ (σ)   0 σ(˙ ) d˙ , i.e. the area under the σ = σ(˙) curve, thus ∗ rendering ∂˙ π = σ(˙), whereas π (σ) follows as the inteσ gral π ∗ (σ) = 0 ˙ (σ  ) dσ  , i.e. the area under the ˙ = ˙ (σ) curve, thus rendering ∂σ π ∗ = ˙ (σ). Obviously, π(˙) and π ∗ (σ) sum up to σ ˙ .

˙(σ) σ( ˙)

. π( ˙) ˙

4.2 Kelvin Model

97

4.2 Kelvin Model The Kelvin model of a visco-elastic solid (in short the Kelvin model) consists of a parallel arrangement of (1) an elastic spring and (2) a viscous dashpot (see the sketc.h of the specific Kelvin model in Fig. 4.12). The basic kinematic assumption of the Kelvin model is the equality of the total strain , the elastic strain e (representing the elongation of the elastic spring), and the viscous strain v (representing the elongation of the viscous dashpot), i.e.  ≡ e ≡ v .

(4.38)

Note that the set of internal variables is empty for the Kelvin model, i.e. α = ∅. Sir William Thomson (Lord Kelvin) [b. 26.6.1824, Belfast, Ireland, d. 17.12.1907, Netherhall, Scotland] was Professor of Theoretical Physics at the University of Glasgow. He worked mainly on thermo- and electrodynamics for which he developed ingenious (measuring) devices. The so-called Kelvin scale is based on his correct determination of the absolute zero temperature. 1892 he became Baron Kelvin in recognition of his achievements in thermodynamics. The Kelvin model for viscoelastic solids is named after him.

E

Fig. 4.12 Specific Kelvin model

σ

σ

η

≡

e

≡

v

98

4 Visco-Elasticity

4.2.1 Specific Kelvin Model: Formulation The specific Kelvin model, displayed in Fig. 4.12, consists of a parallel arrangement of (1) a linear elastic spring with stiffness E and (2) a linear viscous dashpot with viscosity η. Direct Representation For the specific Kelvin model the free energy density ψ is expressed as a quadratic (and thus convex) function of  (the total strain) ψ() =

1 2 E . 2

(4.39)

Then the energetic stress σ  , which is conjugated to the total strain , follows as σ  () = ∂ ψ() = E .

(4.40)

Furthermore, for the specific Kelvin model the convex and smooth (quadratic) dissipation potential π is chosen as π(˙) =

1 η |˙|2 . 2

(4.41)

Observe that (i) π does depend on ˙, thus the dissipative stress σ  = 0 for ˙ = 0, and that (ii) π is positively homogenous of degree two in ˙ and obviously smooth at the origin ˙ = 0. Consequently, the dissipative stress σ  computes as partial derivative of the dissipation potential with respect to its conjugated variable σ  (˙) = ∂˙ π(˙) = η ˙.

(4.42)

Recall that the total stress σ applied to the rheological model (that enters the equilibrium condition), the energetic stress σ  =: σe (the stress in the elastic spring), and the dissipative stress σ  =: σv (the stress in the viscous dashpot) are constitutively related by σ = σ  + σ  =: σe + σv with σe := σ  and

σv := σ  .

(4.43)

The corresponding dual dissipation potential π ∗ , as determined from a Legendre transformation 1 (4.44) π ∗ (σv ) = max{σv ˙ − η |˙|2 } ˙ 2 then reads

4.2 Kelvin Model

99

Table 4.4 Summary of the specific Kelvin model (1) Strain



≡ e

(2) Energy

ψ =

(3) Stress

σe = E 

(4) Potential π = (5) Stress

≡

v

1 2 2E

1 2

=

σ − σv

≡

σ

σ − σe

≡

σ

η |˙|2

σv = η ˙

=

or (4) Potential π ∗ =

1 1 |σv |2 2 η

(5) Evolution ˙

1 σv η

=

π ∗ (σv ) =

1 1 |σv |2 . 2 η

(4.45)

The evolution law for the total strain then follows as partial derivative of the dual dissipation potential with respect to its conjugated variable ˙(σv ) = ∂σv π ∗ (σv ) =

1 σv . η

(4.46)

Obviously the expressions in Eqs. 4.42 and 4.46 are inverse relations. The smooth dissipation and dual dissipation potentials π = π(˙) and π ∗ = π ∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙) and ˙ = ˙(σv ) are similar to those displayed in Fig. 4.2. The specific Kelvin model is summarized in Table 4.4. Convolution Integral Representation The constitutive relations σe = E  = σ − σv for the elastic stress and σv = η ˙ = σ − σe for the viscous stress may be arranged in a differential equation relating the total stress and strain as σ(t) = E (t) + η ˙(t). (4.47) Accordingly, at  = 0 and σ = σ0 the rheological element satisfies instantaneously σ0 /E = τ ˙, thus τ := η/E has been introduced as the relaxation time of the Kelvin model.

100

3

4 Visco-Elasticity

∞ E(t)/η = δ(t) + H(t)/τ

3

2

C(t)/C = H(t) [1 − e−t/τ ]

2 (t)

0

= H(t)

σ(t)/σ0 = H(t)

1

1

t

0

0

1

2

3

τ t

0

0

1

2

3

Fig. 4.13 Specific Kelvin model: Normalized relaxation function E (t)/η (left) and normalized creep function C (t)/C (right) for a relaxation time τ = 2. Observe the jump in the relaxation function at t = 0. The dotted lines depict the normalized step functions for the prescribed strain and stress, respectively. The dashed line illustrates the meaning of the relaxation time

Imposing a constant strain step (t) = 0 H(t) (and thus ˙(t) = 0 δ(t)) renders the superposition of a Dirac-delta-type and a constant solution for the stress history σ(t) = [E H(t) + η δ(t)] 0 =⇒ σ(t) =: E(t) 0 .

(4.48)

Here E(t), i.e. the normalized stress history as response to an imposed constant unit strain step (t) = H(t), has been introduced as the constant relaxation function that is illustrated in Fig. 4.13 (left) E(t) := E [H(t) + τ δ(t)].

(4.49)

Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution integral  t

σ(t) =

E(t − t  ) ˙(t  ) dt  =: E(t)  ˙(t).

(4.50)

0

Thereby the convolution of the relaxation function with the strain rate history is abbreviated symbolically as E(t)  ˙(t). Imposing, alternatively, a constant stress step σ(t) = σ0 H(t) (and thus σ(t) ˙ = σ0 δ(t)) renders an exponentially saturating creep strain in time E (t) + η ˙(t) = H(t) σ0 =⇒ (t) =: C(t) σ0 .

(4.51)

Here C(t), i.e. the normalized strain history as response to an imposed constant unit stress step σ(t) = H(t), has been introduced as the exponentially saturating creep function that is illustrated in Fig. 4.13 (right)

4.2 Kelvin Model

101

C(t) := C H(t) [1 − e−t/τ ],

(4.52)

where C := E −1 denotes the inverse of the elastic stiffness, i.e. the elastic compliance. Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as response to an arbitrary stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral 

t

(t) =

C(t − t  ) σ(t ˙  ) dt  =: C(t)  σ(t). ˙

(4.53)

0

Thereby the convolution of the creep function with the stress rate history is abbreviated symbolically as C(t)  σ(t). ˙ Laplace Transformation Representation Upon Laplace transformation, the convolution integral of the relaxation function E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0, results in L{σ(t)} = L{E(t)  ˙(t)} = s L{E(t)} L{(t)} = E [1 + τ s] L{(t)}.

(4.54)

Choosing, as a particular example, a causal harmonic strain history with (t) = H(t) a sin(ω t) and thus L{(t)} = a ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,4 a phase shifted causal harmonic signal for the resulting stress history, whereby the initial condition σ(0) = η ˙(0) is captured.   σ(t) = H(t) E  sin(ω t) + E  cos(ω t) a .

(4.55)

Thereby E  := E and E  := E τ ω are here formally introduced as abbreviations, however as will become transparent in the sequel, they denote the so-called storage and loss stiffness moduli (note that E  /E  = τ ω). The stress history resulting from a sinusoidal strain history is shown in Fig. 4.14 (left). Upon Laplace transformation, the convolution integral of the creep function C(t) with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0, results in

4

The inverse Laplace transformation for the stress history follows from the following step by step computation: L{σ(t)}

a

= E [1 + τ s]

ω [ω + τ ω s] = E [ω 2 + s 2 ] [ω 2 + s 2 ]

= E  L{H(t) sin(ω t)} + E  L{H(t) cos(ω t)} .

102

4 Visco-Elasticity

σ(t)/E

10

(t)/C 5

0

˜ ˜ sin(ωt) − C C

−5 −5

−10 0

2

4

6

8

10

t

cos(ωt)

t

0

˜ e−t/τ C

5

0

2

4

6

8

10

Fig. 4.14 Specific Kelvin model with τ = 1.0: Normalized stress history σ(t)/E resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history (t)/C resulting from sinusoidal stress history with σa = 5 and ω = 2 π/4 (right)

L{(t)} = L{C(t)  σ(t)} ˙ = s L{C(t)} L{σ(t)} = C

1 L{σ(t)}. 1+τs

(4.56)

Choosing, as a particular example, a causal harmonic stress history with σ(t) = H(t) σa sin(ω t) and thus L{σ(t)} = σa ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,5 a phase shifted causal harmonic signal for the resulting strain history superposed by an exponentially decaying signal that is needed to enforce the initial condition (0) = 0   (t) = H(t) C  sin(ω t) − C  cos(ω t) + C  e−t/τ σa .

(4.57)

Thereby C  := C/[1 + τ 2 ω 2 ] and C  := C τ ω/[1 + τ 2 ω 2 ] are here formally introduced as abbreviations, in terminological accordance to E  and E  they are 5 The inverse Laplace transformation for the strain history follows from the following step by step computation:

L{(t)}

σa

=C

1 ω [1 + τ s] [ω 2 + s 2 ]

=C

1 ω [1 + τ 2 ω 2 ] 2 2 [1 + τ ω ] [1 + τ s] [ω 2 + s 2 ]

1 [ω − τ ω s] [1 + τ s] + τ ω τ [ω 2 + s 2 ] 2 2 [1 + τ ω ] [1 + τ s] [ω 2 + s 2 ]   ω−τ ωs τ ωτ 1 + =C 1+τs [1 + τ 2 ω 2 ] ω 2 + s 2 τ ω s   + C  =C 2 −C ω2 + s 2 1+τs ω + s2 =C

= C  L{H(t) sin(ω t)} − C  L{H(t) cos(ω t)} + C  L{H(t) e−t/τ } .

4.2 Kelvin Model

103

denote the storage and the loss compliance moduli (note that C  /C  = τ ω). The strain history resulting from a sinusoidal stress history is shown in Fig. 4.14 (right). It is interesting to note that the relaxation function and the creep function are related via their Laplace transformations as s 2 L{E(t)} L{C(t)} = 1.

(4.58)

Observe, furthermore, that direct application of the Laplace transformation to the differential equation relating the total stress and strain L{σ(t)} = η s L{(t)} + E L{(t)}

(4.59)

renders immediately the relation L{σ(t)} = E [1 + τ s] L{(t)},

(4.60)

that is entirely conforming with the convolution integral representation. Complex Harmonic Oscillation Representation The differential equation relating the total stress and strain reads in complex representation as σ(t) = E (t) + η ˙(t).

(4.61)

Then, for a stationary harmonic oscillation of the total stress and strain with σ(t) = σ ∗ ei ω t and (t) = ∗ ei ω t the relation between the corresponding complex amplitudes ∗ = a ei δ and σ ∗ = σa ei δσ (where δ = −π/2 for sinusoidal strain control and δσ = −π/2 for sinusoidal stress control) follows as σ ∗ = E [1 + i τ ω] ∗ =: E ∗ ∗ .

(4.62)

Thereby the quantity relating the complex amplitudes of the total strain and stress is denoted the complex stiffness modulus E ∗ (ω) = E [1 + i τ ω] =: E  + i E  , its inverse is the complex compliance modulus (so that E ∗ C ∗ = 1)   1 τω =: C  − i C  . − i C ∗ (ω) = C 1 + τ 2 ω2 1 + τ 2 ω2

(4.63)

(4.64)

104

4 Visco-Elasticity

Note that for the specific Kelvin model the complex moduli E ∗ and C ∗ have indeed real and imaginary parts. Here, E  and E  denote the so-called storage and loss stiffness moduli, respectively, that are defined as E  := E and E  := E τ ω,

(4.65)

whereas C  and C  denote the so-called storage and loss compliance moduli, respectively, that are defined as C  := C

1 τω and C  := C . 2 2 1+τ ω 1 + τ 2 ω2

(4.66)

The storage and loss stiffness and compliance moduli are plotted against the angular frequency ω for various relaxation times in Fig. 4.15.

E 0.4 E 10

τ = 102 , 101 , 100 , 10−1 , 10−2

E E

106

100.2

103

100

100

10−0.2

10−3

1 −0.4

10

10−6

10−3 10−2 10−1 100 101 102 103

ω

C C 100

2

1

0

−1

−2

τ = 10 , 10 , 10 , 10 , 10

10−3 10−2 10−1 100 101 102 103

ω

C C

2

τ = 10 , 10 , 100 , 10−1 , 10−2

1

100

10−4

10−2

10−6

10−3

10−10

ωτ

10−1

10−2

10−8

τ = 102 , 101 , 100 , 10−1 , 10−2

1 1 + ω2τ 2 10−3 10−2 10−1 100 101 102 103

ω

10−4 10−5

ωτ 1 + ω2τ 2 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.15 Specific Kelvin model: Normalized storage stiffness modulus E  (ω)/E (top left) and normalized loss stiffness modulus E  (ω)/E (top right) together with normalized storage compliance modulus C  (ω)/C (bottom left) and normalized loss compliance modulus C  (ω)/C (bottom right) plotted against the angular frequency ω for five decades of relaxation times τ

4.2 Kelvin Model

Ea E 105

105

τ = 102 , 101 , 100 , 10−1 , 10−2 √

104

E E

1 + ω2τ 2

6

10

103

103

100

102

10−3

101 100

10−6

10−3 10−2 10−1 100 101 102 103

ω

Ca C 100

τ = 102 , 101 , 100 , 10−1 , 10−2

10−2

10−3 10−2 10−1 100 101 102 103

ω

C C

106

τ = 102 , 101 , 100 , 10−1 , 10−2

100

10−3

10−5

ωτ

103

10−1

10−4

τ = 102 , 101 , 100 , 10−1 , 10−2

√

1 1 + ω2τ 2

10−3 10−2 10−1 100 101 102 103

ω

10−3 10−6

ωτ 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.16 Specific Kelvin model: Normalized amplitude E a (ω)/E (top left) and tangent of phase shift angle tan δ(ω) = E  (ω)/E  (ω) (top right) together with normalized amplitude Ca (ω)/C (bottom left) and tangent of phase shift angle tan δ(ω) = C  (ω)/C  (ω) (bottom right) plotted against the angular frequency ω for five decades of relaxation times τ

Finally, the (real) amplitude E a and the phase shift angle δ of the complex stiffness modulus are defined as

−1   (4.67) E ∗ (ω) =: [E  ]2 + [E  ]2 ei tan (E /E ) =: E a ei δ , likewise the (real) amplitude Ca and the phase shift angle δ of the complex compliance modulus are defined as

−1   (4.68) C ∗ (ω) =: [C  ]2 + [C  ]2 e−i tan (C /C ) =: Ca e−i δ , so that σa = E a a (or a = Ca σa ) with δσ = δ + δ. Specifically, the angular frequency dependent amplitude E a (ω) and phase shift angle δ(ω) follow from

106

4 Visco-Elasticity

E a :=



E  [E  ]2 + [E  ]2 = E 1 + τ 2 ω 2 and tan δ :=  = τ ω, E

(4.69)

correspondingly, the angular frequency dependent amplitude Ca (ω) and phase shift angle δ(ω) follow from

Ca :=

[C  ]2 + [C  ]2 = C √

1 1 + τ 2 ω2

and tan δ :=

C  = τ ω. C

(4.70)

The amplitudes and (the tangent of) the phase shift angle of the complex stiffness modulus and the complex compliance modulus are plotted against the angular frequency ω for various relaxation times in Fig. 4.16. The phase shift angle δ between the harmonically oscillating total stress and strain and its tangent tan δ are also denoted the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the loss angle tend to infinity for large angular frequencies, since in this limit the dissipation of the viscous dashpot is dominating over the energy storage in the elastic spring.

4.2.2 Specific Kelvin Model: Algorithmic Update For the specific Kelvin model the evolution law for the total strain  is integrated by the implicit Euler backwards method to render n := n − n−1 =

t n n σ . η v

(4.71)

Consequently, the viscous stress σv is updated at the end of the time step by σvn =

η  . t n

(4.72)

The trial strain  is computable exclusively from known quantities at the beginning and at the end of the time step and follows as  := n .

(4.73)

Finally the total stress reads at the end of the time step σ n = σvn + σen =

η   + E n . t n

(4.74)

The sensitivity of σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is straightforwardly computed as

4.2 Kelvin Model

107

Table 4.5 Algorithmic update for the specific Kelvin model Input

n n−1

Trial Strain

 = n − n−1 η   + E n t n η E an = +E t n

Update Stress σ n = Tangent Output

σ n E an

∂ σ n =

η + E. t n

(4.75)

Here the definition for the relaxation time τ := η/E has been incorporated. Note that, consequently, the algorithmic tangent degenerates to E a → ∞ for t n → 0, i.e. for very fast processes (as compared to the relaxation time) the response is rigid. Likewise, for vanishing elastic stiffness E → 0 the algorithmic tangent degenerates to the case of the Newton model. Finally, for vanishing viscosity η → 0 or for t n → ∞, i.e. for very slow processes (as compared to the relaxation time) the algorithmic tangent degenerates to E a → E, i.e. the response is elastic. The algorithmic step-by-step update for the specific Kelvin model is summarized in Table 4.5.

4.2.3 Specific Kelvin Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Kelvin model to a prescribed Zig-Zag strain history is documented in Fig. 4.17a–f. Figure 4.17a, c depicts the prescribed Zig-Zag (viscous) strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.17b showcases the resulting stress history σ(t) that displays a periodic, distorted block signal with σ(t) ˙ = E ˙(t) = ±5 since ˙(t) = ±5 and |σ(t)| ∈ [0, 10]. The resulting σ = σ() diagram is highlighted in Fig. 4.17d. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram format of the σ = σ() diagram with vertical slopes at  = 0 and  = ±5 is only approximately captured, however the slopes at  = 0 and  = ±5 obviously tend to ∞ when t → 0. Finally, Fig. 4.17e, f depict the resulting σ = σ() diagrams for a 10 and 100 times longer period T corresponding to a 10 and 100 times lower strain rate |˙(t)|,

108

4 Visco-Elasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.17 Response analysis of the specific Kelvin model with material data: η = 1.0, E = 1.0. Prescribed Zig-Zag strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 40.0; t = 1.0, N = 100, f T = 400.0; t = 10.0, N = 100

4.2 Kelvin Model

109

respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙(t)| → 0. Prescribed Strain History: Sine The response of the specific Kelvin model to a prescribed Sine strain history is documented in Fig. 4.18a–f. Figure 4.18a, c depicts the prescribed Sine (viscous) strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.18b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.55, a harmonic signal with σ(t) √ = E (t) + η ˙(t) = E a sin(ω t) + η ω a cos(ω t) and amplitude σa = E 1 + τ 2 ω 2 a ≈ 9.31. The resulting σ = σ() diagram is highlighted in Fig. 4.18d. Due to the finite sized time step t and corresponding finite sized strain increment  the expected slope of the σ = σ() diagram at  = 0 is only approximately captured, however it obviously tends to ∞ when t → 0. Finally, Fig. 4.18e, f depict the resulting σ = σ() diagrams for a 10 and 100 times longer period T corresponding to lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙(t)| → 0. Prescribed Strain History: Ramp The response of the specific Kelvin model to a prescribed Ramp strain history is documented in Fig. 4.19a–f. Figure 4.19a, c depicts the prescribed Ramp (viscous) strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.19b showcases the resulting stress history σ(t) that displays a distorted block-type signal with σ(t) ˙ = E ˙(t) = ±5 when ˙(t) = ±5 in the loading and the unloading phases and σ(t) = E (t) = 5 when ˙(t) = 0 during the holding phase, and |σ(t)| ∈ [0, 10]. The resulting σ = σ() diagram is highlighted in Fig. 4.19d. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram format of the σ = σ() diagram with vertical slopes at  = 0 and  = ±5 is only approximately captured, however the slopes at  = 0 and  = 5 obviously tend to ∞ with t → 0. Finally, Fig. 4.19e, f depict the resulting σ = σ() diagrams for 10 and 100 times larger t1 , t2 , t3 corresponding to lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙(t)| → 0.

110

4 Visco-Elasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.18 Response analysis of the specific Kelvin model with material data: η = 1.0, E = 1.0. Prescribed Sine strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 40.0; t = 1.0, N = 100, f T = 400.0; t = 10.0, N = 100

4.2 Kelvin Model

111 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.19 Response analysis of the specific Kelvin model with material data: η = 1.0, E = 1.0. Prescribed Ramp strain history with data: a = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 10.0, t2 = 90.0, t3 = 100.0; t = 1.0, N = 100, f t0 = 0.0, t1 = 100.0, t2 = 900.0, t3 = 1000.0; t = 10.0, N = 100

112

4 Visco-Elasticity

Prescribed Stress History: Zig-Zag The response of the specific Kelvin model to a prescribed Zig-Zag stress history is documented in Fig. 4.20a–f. Figure 4.20a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.20b, d showcases the resulting (viscous) strain history (t) that displays a periodic nonlinear signal after an initial transient phase. The resulting, slightly tilted σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 4.20c. Finally, Fig. 4.20e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to a 10 and 100 times higher stress rate |σ(t)|, ˙ respectively. They clearly demonstrate a rigid behaviour with vanishing strain for |σ(t)| ˙ → ∞. Prescribed Stress History: Sine The response of the specific Kelvin model to a prescribed Sine stress history is documented in Fig. 4.21a–f Figure 4.21a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.21b, d showcases the resulting (viscous) strain history (t) that displays, in accordance with √ the analytical solution in Eq. 4.57, a harmonic signal with amplitude a = σa /E/ 1 + τ 2 ω 2 ≈ 2.68 after an initial transient phase. The resulting, slightly tilted σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 4.21c. Finally, Fig. 4.21e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to higher stress rates |σ(t)|, ˙ respectively. They clearly demonstrate a rigid behaviour with vanishing strain for |σ(t)| ˙ → ∞. Prescribed Stress History: Ramp The response of the specific Kelvin model to a prescribed Ramp stress history is documented in Fig. 4.22a–f. Figure 4.22a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.22b, d showcases the resulting (viscous) strain history (t) that displays a monotonic signal until the end of the holding phase with (t2 ) → 5 when the creep

4.2 Kelvin Model

113

σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 5.0)

v

t

c)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

p,max,min

= ± 5.0)

σ

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.20 Response analysis of the specific Kelvin model with material data: η = 1.0, E = 1.0. Prescribed Zig-Zag stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

114

4 Visco-Elasticity σ

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

t

(tmax = 100 × 0.1

b)

σ

max,min

= ± 5.0)

v

t

c)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

p,max,min

= ± 5.0)

σ

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.21 Response analysis of the specific Kelvin model with material data: η = 1.0, E = 1.0. Prescribed Sine stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

4.2 Kelvin Model

115

process has ended. The unloading phase activates again visco-elasticity and terminates with E (t3 ) + η ˙(t3 ) = 0 with (t3 ) = −˙(t3 ) ≈ 3 (from visual inspection). The resulting σ = σ() diagram of nonlinear parallelogram format is highlighted in Fig. 4.22c. Finally, Fig. 4.22e, f depict the resulting σ = σ() diagrams for 10 and 100 times ˙ respectively. They clearly smaller t1 , t2 , t3 corresponding to higher stress rates |σ(t)|, demonstrate a rigid behaviour with vanishing strain for |σ(t)| ˙ → ∞.

4.2.4 Generic Kelvin Model: Formulation A generic formulation of the Kelvin model can be obtained from generalizing the specific Kelvin model in Fig. 4.12 by assuming the elastic spring or/and the viscous dashpot as nonlinear. For the generic Kelvin model the free energy density ψ is expressed as a nonquadratic, yet convex, function of  (the total strain) ψ = ψ().

(4.76)

Then the energetic stress σ  follows as σ  () = ∂ ψ().

(4.77)

Recall that the energetic and the dissipative stresses are constitutively related to the total stress σ (that enters the equilibrium condition) by σ = σ  + σ  . Moreover the definitions σe := σ  (in the elastic spring) and σv := σ  (in the viscous dashpot) are introduced. Furthermore, for the generic Kelvin model it is possible to introduce the convex and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙) and π ∗ = π ∗ (σv ), respectively, which are related via corresponding Legendre transformations (4.78a) π ( ˙ ) = max{σv ˙ − π ∗ (σv )}, σv

π ∗ (σv ) = max{σv ˙ − π ( ˙ )}. ˙

(4.78b)

The stationarity conditions corresponding to Eqs. 4.78a and 4.78b are the constitutive relations (4.79a) ˙ (σv ) = ∂σv π ∗ (σv ), σv ( ˙ ) = ∂ ˙ π ( ˙ ).

(4.79b)

Obviously, the relations in Eqs. 4.79a and 4.79b determine entirely the dissipative behavior of the generic Kelvin model, thus the formulation is completed at this stage.

116

4 Visco-Elasticity σ

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

t

(tmax = 100 × 0.1

b)

σ

max,min

= ± 5.0)

v

t

c)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

p,max,min

= ± 5.0)

σ

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.22 Response analysis of the specific Kelvin model with material data: η = 1.0, E = 1.0. Prescribed Ramp stress history with data: σa = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.1, t2 = 0.9, t3 = 1.0; t = 0.01, N = 100, f t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100

4.2 Kelvin Model

117

Table 4.6 Summary of the generic Kelvin model (1) Strain



(2) Energy

ψ = ψ()

Stress

σ e = ∂ ψ

(3)

≡ e

≡

v

=:

σ − σv

≡

σ

=:

σ − σe

≡

σ

(4) Potential π = π(˙) (5) Stress

σv = ∂˙ π

or (4) Potential π ∗ = π ∗ (σv ) (5) Evolution ˙

= ∂σ v π ∗

Finally, as a further interesting aspect, the dissipation d = σv ˙ is alternatively expressed from Eqs. 4.78a and 4.78b in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙) + π ∗ (σv ) ≥ 0.

(4.80)

The generic Kelvin model is summarized in Table 4.6.

4.3 Generalized-Kelvin Model The Generalized-Kelvin model of a visco-elastic solid/fluid (in short the Generalized-Kelvin model) consists of a serial arrangement of K Kelvin elements each (k = 1, . . . , K ) consisting of a parallel arrangement of (i) an elastic spring and (ii) a viscous dashpot (see the sketc.h of the specific Generalized-Kelvin model in Fig. 4.23). The basic kinematic assumption of the kth Kelvin element is the equality of the total strain contribution k , the elastic strain contribution ek (representing the elongation of the kth elastic spring), and the viscous strain contribution vk (representing the elongation of the kth viscous dashpot), i.e. k ≡ ek ≡ vk , whereby the overall total strain adds up from all corresponding contributions

(4.81)

118

4 Visco-Elasticity

σ

1

Ek1

EkK

ηk1

ηkK

≡

e1

≡

v1

K

≡

eK

σ

≡

vK

Fig. 4.23 Specific Generalized-Kelvin model

=

K  k=1

k =

K  k=1

ek =

K 

vk .

(4.82)

k=1

Note that the set of viscous strain contributions {v2 , . . . , vk , . . . , v K } denote the K − 1 elements contained in the set of internal variables α = {v2 , . . . , vk , . . . , v K } for the Generalized-Kelvin model.

4.3.1 Standard-Linear-Solid Kelvin Model: Formulation The specific Generalized-Kelvin model, displayed in Fig. 4.23, consists of a serial arrangement of K specific Kelvin elements each (k = 1, . . . , K ) consisting of a parallel arrangement of (i) a linear elastic spring with stiffness E kk and (ii) a linear viscous dashpot with viscosity ηkk . As a particular three parameter sub-case of the specific Generalized-Kelvin model the Standard-Linear-Solid Kelvin model, displayed in Fig. 4.24, consists of a serial arrangement of (1) a linear elastic spring with stiffness E 0 (a Hooke element representing the elastic instantaneous response) and (2) a specific Kelvin element consisting of a parallel arrangement of (i) a linear elastic spring with stiffness E k and (ii) a linear viscous dashpot with viscosity ηk . Direct Representation For the Standard-Linear-Solid Kelvin model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − v (the elastic strain contribution e ) and v (the viscous strain contribution)

4.3 Generalized-Kelvin Model

119

Ek E0 σ

σ

ηk

1

≡

e

2

≡

v

Fig. 4.24 Standard-Linear-Solid Kelvin model

ψ(, v ) =

1 1 E 0 [ − v ]2 + E k 2v . 2 2

(4.83)

Then the energetic stress σ  , which is conjugated to the total strain , and the energetic viscous stress σv , which is conjugated to the viscous strain v , follow as σ  (, v ) = ∂ ψ(, v ) = σv (, v )

E 0 [ − v ]

,

(4.84a)

= ∂v ψ(, v ) = −E 0 [ − v ] + E k v .

(4.84b)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic (or rather instantaneous) stress, σ  := σ0 ≡ σ, and, due to the serial arrangement of the (elastic instantaneous) Hooke element and the Kelvin element, also with the difference of the elastic stress in the Kelvin element, σe := E k v , and the energetic viscous stress, σe − σv = σ0 ≡ σ. Furthermore, for the Standard-Linear-Solid Kelvin model the convex and smooth (quadratic) dissipation potential π is chosen as π(˙v ) =

1 ηk |˙v |2 . 2

(4.85)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically (however the dissipative viscous stress σv = 0 for ˙v = 0), and that (ii) π is positively homogenous of degree two in ˙v and obviously smooth at the origin ˙v = 0. Consequently, the dissipative viscous stress σv computes as partial derivative of the dissipation potential with respect to its conjugated variable σv (˙v ) = ∂˙v π(˙v ) = ηk ˙v .

(4.86)

120

4 Visco-Elasticity

Recall that the energetic and the dissipative viscous stresses (in the viscous dashpot of the Kelvin element) are constitutively related by σv + σv = 0, thus the notion of viscous stress defined as the value σv := σv = −σv with σ0 = σe + σv

(4.87)

will exclusively be used in the sequel for convenience of exposition. The corresponding dual dissipation potential π ∗ , as determined from a Legendre transformation 1 (4.88) π ∗ (σv ) = max{σv ˙v − ηk |˙v |2 } ˙ v 2 then reads π ∗ (σv ) =

1 1 |σv |2 . 2 ηk

(4.89)

The evolution law for the viscous strain contribution of the Kelvin element follows as partial derivative of the dual dissipation potential with respect to its conjugated variable 1 σv . (4.90) ˙v (σv ) = ∂σv π ∗ (σv ) = ηk Obviously the expressions in Eqs. 4.86 and 4.90 are inverse relations. The smooth dissipation and dual dissipation potentials π(˙v ) and π ∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙v ) and ˙v = ˙v (σv ) are similar to those displayed in Fig. 4.2. The Standard-Linear-Solid Kelvin model is summarized in Table 4.7.

Table 4.7 Summary of the Standard-Linear-Solid Kelvin model =  e + v

(1) Strain



(2) Energy

ψ =

(3) Stress

σ = E 0 [ − v ]

=

σv + σe

≡

σ

(4) Stress

σv = E 0 [ − v ] − E k v

=

σ − σe

≡

−σv

≡

σv

(5) Potential π = (6) Stress

1 2 E0

1 2

[ − v ]2 + 21 E k 2v 

ηk |˙v |2

σv = ηk ˙ v

or (5) Potential π ∗ =

1 1 |σv |2 2 ηk

(6) Evolution ˙ v =

1 σv ηk



4.3 Generalized-Kelvin Model

121

Convolution Integral Representation The constitutive relations v =  − σ/E 0 for the total stress in the Hooke element (and its rate form) and σv = ηk ˙v = σ − σe for the viscous stress in the Kelvin element together with the constitutive relation σe = E k v = σ − σv for the elastic stress in the Kelvin element may be arranged in a differential equation relating the total stress and strain as ˙ = E ∞ (t) + E ∞ τk ˙(t). σ(t) + c τk σ(t)

(4.91)

Here E ∞ := E 0 E k /[E 0 + E k ] and τk := ηk /E k denote the equilibrium elastic stiffness of the Standard-Linear-Solid Kelvin model and the relaxation time of the Kelvin element. Moreover the stiffness ratio e := E 0 /E ∞ and the compliance ratio −1 = Ck + C0 and C0 := E 0−1 . c := C0 /C∞ are introduced with C∞ := E ∞ Imposing a constant strain step (t) = 0 H(t) (and thus ˙(t) = 0 δ(t)) renders an exponential stress relaxation in time ˙ = [E ∞ H(t) + E ∞ τk δ(t)] 0 =⇒ σ(t) =: E(t) 0 . σ(t) + c τk σ(t)

(4.92)

Here E(t), i.e. the normalized stress history as response to an imposed constant unit strain step (t) = H(t), has been introduced as the exponentially decaying relaxation function that is illustrated in Fig. 4.25 (left)   E(t) := E ∞ H(t) 1 + [e − 1] e−e t/τk .

(4.93)

Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution integral  t

σ(t) =

E(t − t  ) ˙(t  ) dt  =: E(t)  ˙(t).

(4.94)

0

Thereby the convolution of the relaxation function with the strain rate history is abbreviated symbolically as E(t)  ˙(t). ˙ = Imposing, alternatively, a constant stress step σ(t) = σ0 H(t) (and thus σ(t) σ0 δ(t)) renders an exponentially saturating creep strain in time E ∞ (t) + E ∞ τk ˙(t) = [H(t) + c τk δ(t)] σ0 =⇒ (t) =: C(t) σ0 .

(4.95)

Here C(t), i.e. the normalized strain history as response to an imposed constant unit stress step σ(t) = H(t), has been introduced as the exponentially saturating creep function that is illustrated in Fig. 4.25 (right)     C(t) := C∞ H(t) 1 + [c − 1] e−t/τk = Ck H(t) 1 − e−t/τk + C0 H(t). (4.96) The latter expansion clearly highlights the serial arrangement of a Kelvin and a Hooke element in the Standard-Linear-Solid Kelvin model. Based on the Boltzmann

122

4 Visco-Elasticity

3

E(t)/E∞ = H(t) [1 + ¯e e−et/τ ]

3

2

2

1

1

C(t)/C∞ = H(t) [1 + ¯c e−t/τ ]

σ(t)/σ0 = clH(t) (t)

0

0

= H(t)

t

cτ 0

1

2

3

t

0

0

1

2

3

Fig. 4.25 Standard-Linear-Solid Kelvin model: Normalized relaxation function E (t)/E ∞ (left) and normalized creep function C (t)/C∞ (right) for a relaxation time τk = 2 and E 0 = E k (thus E ∞ = 1/2 E 0 , e = 2, e¯ := e − 1 = 1, c = 1/2 and c¯ := c − 1 = −1/2). Observe the jumps in both functions at t = 0. The dotted lines depict the normalized step functions for the prescribed strain and stress, respectively. The dashed line illustrates the meaning of the modified relaxation time c τk

superposition process, the strain history (t) for t ≥ 0 as response to an arbitrary stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral 

t

(t) =

C(t − t  ) σ(t ˙  ) dt  =: C(t)  σ(t). ˙

(4.97)

0

Thereby the convolution of the creep function with the stress rate history is abbreviated symbolically as C(t)  σ(t). ˙ Laplace Transformation Representation Upon Laplace transformation, the convolution integral of the relaxation function E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0, results in L{σ(t)} = L{E(t)  ˙(t)} = s L{E(t)} L{(t)} 1 + τk s = E∞ L{(t)}. 1 + c τk s

(4.98)

Choosing, as a particular example, a causal harmonic strain history with (t) = H(t) a sin(ω t) and thus L{(t)} = a ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,6 a phase shifted causal harmonic signal for the resulting stress history superposed by an exponentially decaying signal that is needed to enforce the initial condition σ(0) = 0 6 The inverse Laplace transformation for the stress history follows from the following step by step computation:

4.3 Generalized-Kelvin Model

  σ(t) = H(t) E  sin(ω t) + E  cos(ω t) − E  e−e t/τk a .

123

(4.99)

Thereby E  := E ∞ [1 + c τk2 ω 2 ]/[1 + c2 τk2 ω 2 ] and E  := E ∞ [1 − c] τk ω/[1 + c τk2 ω 2 ] are here formally introduced as abbreviations, however as will become transparent in the sequel, they denote the so-called storage and loss stiffness moduli (note that E  /E  = [1 − c] τk ω/[1 + c τk2 ω 2 ]). The stress history resulting from a sinusoidal stress history is shown in Fig. 4.26 (left). Upon Laplace transformation, the convolution integral of the creep function C(t) with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0, results in 2

L{(t)} = L{C(t)  σ(t)} ˙ = s L{C(t)} L{σ(t)} 1 + c τk s = C∞ L{σ(t)}. 1 + τk s

(4.100)

Note that the nominator C∞ [1 + c τk s] in the above expands as Ck + C0 [1 + τk s] thus highlighting again the serial arrangement of a Kelvin and a Hooke element in the Standard-Linear-Solid Kelvin model   1 (4.101) + C0 L{σ(t)}. L{(t)} = Ck 1 + τk s Choosing, as a particular example, a causal harmonic stress history with σ(t) = H(t) σa sin(ω t) and thus L{σ(t)} = σa ω/[ω 2 + s 2 ] renders, after inverse Laplace

L{σ(t)}

a

= E∞ = E∞ = E∞ = E∞

[1 + τk s] ω [1 + c τk s] [ω 2 + s 2 ] [1 + c τk s] ω − [c − 1] τk ω s [1 + c τk s] [ω 2 + s 2 ] [1 + c2 τk2 ω 2 ] s ω + E  ω2 + s 2 [1 + c τk s] [ω 2 + s 2 ] ω2

ω s − c τk ω 2 ω + E  c τk ω 2 + E  2 2 +s ω +s [1 + c τk s] [ω 2 + s 2 ]

ω [1 + c τk s] s c τk [ω 2 + s 2 ]   + E − E ω2 + s 2 [1 + c τk s] [ω 2 + s 2 ] [1 + c τk s] [ω 2 + s 2 ] ω s c τk    +E −E = E 2 ω + s2 ω2 + s 2 1 + c τk s

= E

= E  L{H(t) sin(ω t)} + E  L{H(t) cos(ω t)} − E  L{H(t) e−t/[c τk ] } .

124

4 Visco-Elasticity

transformation7, ,8 a phase shifted causal harmonic signal for the resulting strain history superposed by an exponentially decaying signal that is needed to enforce the initial condition (0) = 0   (t) = H(t) C  sin(ω t) − C  cos(ω t) + C  e−t/τk σa .

(4.102)

Thereby C  := C∞ [1 + c τk2 ω 2 ]/[1 + τk2 ω 2 ] = Ck /[1 + τk2 ω 2 ] + C0 and C  := C∞ [1 − c] τk ω/[1 + τk2 ω 2 ] = Ck τk ω/[1 + τk2 ω 2 ] are here formally introduced as abbreviations, in terminological accordance to E  and E  they are denote the storage and the loss compliance moduli (note that C  /C  = [1 − c] τk ω/[1 + c τk2 ω 2 ]). The strain history resulting from a sinusoidal stress history is shown in Fig. 4.26 (right). 7 The inverse Laplace transformation for the strain history follows from the following step by step computation:

L{(t)}

σa

= C∞ = C∞ = C∞ = C∞

ω [1 + c τk s] [1 + τk s] [ω 2 + s 2 ] [1 + τk s] ω − [1 − c] τk ω s [1 + τk s] [ω 2 + s 2 ] ω2

[1 + τk2 ω 2 ] s ω − C  2 +s [1 + τk s] [ω 2 + s 2 ]

ω s − τk ω 2 ω − C  τk ω 2 − C  ω2 + s 2 ω + s2 [1 + τk s] [ω 2 + s 2 ]

ω [1 + τk s] s τk [ω 2 + s 2 ] − C  + C  2 2 2 +s [1 + τk s] [ω + s ] [1 + τk s] [ω 2 + s 2 ] ω s τ k − C  2 + C  = C 2 ω + s2 ω + s2 1 + τk s = C

ω2

= C  L{H(t) sin(ω t)} − C  L{H(t) cos(ω t)} + C  L{H(t) e−t/τk } . 8

Alternatively a more direct derivation that highlights the serial arrangement of a Kelvin and a Hooke element reads: L{(t)}

σa

= C0 = C0

ω 1 ω + Ck [ω 2 + s 2 ] [1 + τk s] [ω 2 + s 2 ] [ω 2

ω [1 + τk2 ω 2 ] ω 1 + Ck 2 2 2 +s ] [1 + τk ω ] [1 + τk s] [ω 2 + s 2 ]

[ω − τk ω s] [1 + τk s] + τk ω τk [ω 2 + s 2 ] ω 1 + Ck 2 2 [ω 2 + s 2 ] [1 + τk s] [ω 2 + s 2 ] [1 + τk ω ]   τk ω τk ω 1 ω − τk ω s = C0 2 + + C k [ω + s 2 ] ω2 + s 2 1 + τk s [1 + τk2 ω 2 ] ω s τk    =C 2 −C +C ω + s2 ω2 + s 2 1 + τk s = C0

= C  L{H(t) sin(ω t)} − C  L{H(t) cos(ω t)} + C  L{H(t) e−t/τk } .

4.3 Generalized-Kelvin Model

σ(t)/E

4

˜ sin( E

ω t)

˜ +E

cos(

125

ω t)

(t)/C

10 5

2

τ

t ˜ e−t/[cτ ] −E

0 −2 −4 0

˜ C

0

t/ −

e

t

−5 ˜ ˜ sin(ωt) − C C

cos(ωt)

−10 2

4

6

8

10

0

2

4

6

8

10

Fig. 4.26 Standard-Linear-Solid Kelvin model with τk = 1.0 and E 0 = E k : Normalized stress history σ(t)/E k resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history (t)/Ck resulting from sinusoidal stress history with σa = 5 and ω = 2 π/4 (right)

It is interesting to note that the relaxation function and the creep function are related via their Laplace transformations as s 2 L{E(t)} L{C(t)} = 1.

(4.103)

Observe, furthermore, that direct application of the Laplace transformation to the differential equation relating the total stress and strain L{σ(t)} + c τk s L{σ(t)} = E ∞ L{(t)} + E ∞ τk s L{(t)}

(4.104)

renders immediately the relation L{σ(t)} = E ∞

1 + τk s L{(t)}, 1 + c τk s

(4.105)

with inverse L{(t)} = C∞

  1 + c τk s 1 L{σ(t)} = Ck + C0 L{σ(t)}, 1 + τk s 1 + τk s

(4.106)

that are entirely conforming with the convolution integral representations. Complex Harmonic Oscillation Representation The differential equation relating the total stress and strain reads in complex representation as σ(t) + c τk σ(t) ˙ = E ∞ (t) + E ∞ τk ˙(t). (4.107)

126

4 Visco-Elasticity

Then, for a stationary harmonic oscillation of the total stress and strain with σ(t) = σ ∗ ei ω t and (t) = ∗ ei ω t the relation between the corresponding complex amplitudes ∗ = a ei δ and σ ∗ = σa ei δσ (where δ = −π/2 for sinusoidal strain control and δσ = −π/2 for sinusoidal stress control) follows as σ∗ = E∞

1 + i τk ω ∗  =: E ∗ ∗ . 1 + i c τk ω

(4.108)

Thereby the quantity relating the complex amplitudes of the total strain and stress is denoted the complex stiffness modulus E ∗ (ω) = E ∞

1 + i τk ω =: E  + i E  , 1 + i c τk ω

(4.109)

its inverse is the complex compliance modulus (so that E ∗ C ∗ = 1)9 C ∗ (ω) = C∞

1 + i c τk ω =: C  − i C  . 1 + i τk ω

(4.110)

Note that for the Standard-Linear-Solid Kelvin model the complex moduli E ∗ and C ∗ have indeed real and imaginary parts. Here, E  and E  denote the so-called storage and loss stiffness moduli, respectively, that are defined as E  := E ∞

1 + c τk2 ω 2 [1 − c] τk ω and E  := E ∞ , 2 2 2 1 + c τk ω 1 + c2 τk2 ω 2

(4.111)

whereas C  and C  denote the so-called storage and loss compliance moduli, respectively, that are defined as10 C  := C∞

1 + c τk2 ω 2 [1 − c] τk ω and C  := C∞ . 2 2 1 + τk ω 1 + τk2 ω 2

(4.112)

The storage and loss stiffness and compliance moduli are plotted against the angular frequency ω for various relaxation times in Fig. 4.27.

9 Observe

that the complex compliance modulus may alternatively be expressed as C ∗ (ω) = Ck

1 + C0 . 1 + i τk ω

. 10 Observe

that the storage and loss compliance moduli may alternatively be expressed as C  := Ck

.

1 τk ω + C0 and C  := Ck . 1 + τm2 ω 2 1 + τk2 ω 2

4.3 Generalized-Kelvin Model

E E∞100.3

τ = 102 , 101 , 100 , 10−1 , 10−2

127

E 0 E∞ 10

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 100.2

0.1

10

10−2

1 + c ω2τ 2 1 + c2 ω 2 τ 2

10−3 10−4 10−5

0

10

10−3 10−2 10−1 100 101 102 103

10−3 10−2 10−1 100 101 102 103

ω

C C∞ 100 10−0.1

10−0.2

τ = 102 , 101 , 100 , 10−1 , 10−2

[1 − c] ωτ 1 + c2 ω 2 τ 2 ω

C C∞

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 10−2

1 + c ω2τ 2 1 + ω2τ 2

10−3 10−4 10−5

−0.3

10

10−3 10−2 10−1 100 101 102 103

ω

[1 − c] ωτ 1 + ω2τ 2 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.27 Standard-Linear-Solid Kelvin model: Normalized storage stiffness modulus E  (ω)/E ∞ (top left) and normalized loss stiffness modulus E  (ω)/E ∞ (top right) together with normalized storage compliance modulus C  (ω)/C∞ (bottom left) and normalized loss compliance modulus C  (ω)/C∞ (bottom right) plotted against the angular frequency ω for five decades of relaxation times τk and E 0 = E k

Finally, the (real) amplitude E a and the phase shift angle δ of the complex stiffness modulus are defined as

−1   (4.113) E ∗ (ω) =: [E  ]2 + [E  ]2 ei tan (E /E ) =: E a ei δ , likewise the (real) amplitude Ca and the phase shift angle δ of the complex compliance modulus are defined as

−1   (4.114) C ∗ (ω) =: [C  ]2 + [C  ]2 e−i tan (C /C ) =: Ca e−i δ , so that σa = E a a (or a = Ca σa ) and δσ = δ + δ. Specifically, the angular frequency dependent amplitude E a (ω) follows as

128

4 Visco-Elasticity

E a :=



[E  ]2 + [E  ]2 = E ∞

1 + τk2 ω 2 , 1 + c2 τk2 ω 2

(4.115)

correspondingly, the angular frequency dependent amplitude Ca (ω) follows as Ca :=



[C  ]2 + [C  ]2 = C∞

1 + c2 τk2 ω 2 . 1 + τk2 ω 2

(4.116)

Finally the phase shift angle δ(ω) is expressed as tan δ :=

E  C  [1 − c] τk ω = = .   E C 1 + c τk2 ω 2

(4.117)

The amplitudes and (the tangent of) the phase shift angle of the complex stiffness modulus and the complex compliance modulus are plotted against the angular frequency ω for various relaxation times in Fig. 4.28. The phase shift angle δ between the harmonically oscillating total stress and strain and its tangent tan δ are also denoted the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the loss angle tend to zero for large angular frequencies, since in this limit the viscous dashpot is too inert to react. Equivalence to Standard-Linear-Solid Maxwell Model The Standard-Linear-Solid Kelvin model is characterized by differential equations for (i) the global response relating total stress and total strain as well as for (ii) the local response relating viscous strain and total strain as σ + c τk σ˙ = E ∞  + E ∞ τk ˙ and v + c τk ˙v = [1 − c] .

(4.118)

By re-parametrization of the relaxation time as τm := c τk the global response of the Standard-Linear-Solid Kelvin model and the Standard-Linear-Solid Maxwell model (as sketc.hed in Fig. 4.29) coincides, however the local response or rather the viscous strain predicted by of the two models obviously still differs σ + τm σ˙ = E ∞  + E 0 τm ˙ and ˜v + τm ˙˜ v = .

(4.119)

Here the viscous strain ˜v as predicted by the equivalent Standard-Linear-Solid Maxwell model is related to the corresponding viscous strain v as predicted by the underlying Standard-Linear-Solid Kelvin model via ˜v =

v . 1−c

(4.120)

If furthermore the instantaneous elastic stiffness E 0 and the equilibrium elastic stiffness E ∞ of the Standard-Linear-Solid Maxwell model and the Standard-Linear-

4.3 Generalized-Kelvin Model

τ = 102 , 101 , 100 , 10−1 , 10−2

Ea E∞100.3

129

E E

100

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 10−2

100.2

τ 2 ω2

1+ 1 + c2 τ 2 ω 2

0.1

10

10−3 10−4 10−5

0

10

10−3 10−2 10−1 100 101 102 103

10−3 10−2 10−1 100 101 102 103

ω

τ = 102 , 101 , 100 , 10−1 , 10−2

Ca C∞ 100

[1 − c] ωτ 1 + c τ 2ω2 ω

C C

0

τ = 102 , 101 , 100 , 10−1 , 10−2

10

10−1 10−2

10−0.1

c2 τ 2 ω 2

1+ 1 + ω2τ 2

−0.2

10

10−3 10−4 10−5

−0.3

10

10−3 10−2 10−1 100 101 102 103

ω

[1 − c] ωτ 1 + c τ 2ω2 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.28 Standard-Linear-Solid Kelvin model: Normalized amplitude E a (ω)/E ∞ (top left) and tangent of phase shift angle tan δ(ω) = E  (ω)/E  (ω) (top right) together with normalized amplitude Ca (ω)/C∞ (bottom left) and tangent of phase shift angle tan δ(ω) = C  (ω)/C  (ω) (bottom right) plotted against the angular frequency ω for five decades of relaxation times τk and E 0 = E k

Solid Kelvin model are required to coincide, the three parameter sets {E ∞ , E m , ηm } and {E 0 , E k , ηk } representing both models are related via E∞ =

E 02 E0 Ek E 02 and E m = and ηm := ηk . E0 + Ek E0 + Ek [E 0 + E k ]2

(4.121)

Here the resulting stiffness E ∞ of a serial arrangement of elastic springs with stiffness E 0 and E k together with the relations E m = E 0 − E ∞ and ηm,k = E m,k τm,k have been used.

130

4 Visco-Elasticity

E∞ := E0 Ek /[E0 + Ek ]

σ

Em := E02 /[E0 + Ek ] ηm := ηk E02 /[E0 + Ek ]2

˜e

σ

˜v

Fig. 4.29 Standard-Linear-Solid Maxwell model equivalent to Standard-Linear-Solid Kelvin model: Parallel arrangement of (1) a linear elastic spring with stiffness E ∞ := E 0 E k /[E 0 + E k ] and (2) a specific Maxwell element consisting of a serial arrangement of (i) a linear elastic spring with stiffness E m := E 02 /[E 0 + E k ] and (ii) a linear viscous dashpot with viscosity ηm := ηk E 02 /[E 0 + E k ]2 . The total strain  is decomposed additively into an elastic and a viscous part  = ˜ e + ˜ v

4.3.2 Standard-Linear-Solid Kelvin Model: Algorithmic Update For the Standard-Linear-Solid Kelvin model the evolution law for the viscous strain v is integrated by the implicit Euler backwards method to render nv := nv − n−1 = v

t n n σ . ηk v

(4.122)

Thus, on the one hand, by incorporating the discretized evolution law for the viscous strain, the total stress σ is updated at the end of the time step by σ n = E k nv +

ηk ηk + t n E k n   =: E  + nv . k v v t n t n

(4.123)

Here the trial viscous strain v is trivially defined as its known value at the beginning of the time step (4.124) v := n−1 v . On the other hand the total stress σ is updated at the end of the time step σ n = E 0 [n − nv ] =: E 0 e − E 0 nv .

(4.125)

The trial elastic strain e is computable exclusively from known quantities at the beginning and at the end of the time step and follows as

4.3 Generalized-Kelvin Model

131

e := n − n−1 v .

(4.126)

Eliminating the viscous strain increment nv from the two expressions for the updated total stress σ n then renders σ n = E 0 e − E 0

 n  t n σ − E k v . n ηk + t E k

(4.127)

The above relation is regrouped in order to separate the unknowns at the end of the time step from the known strain and trial strain τ0 + t˜n n t˜n n τ0 + t˜n n σ = e +  = [1 − c] ˜e +  . ηk τ∞ τ∞

(4.128)

Here the definitions for the relaxation times τ0 := ηk /E 0 and τ∞ := ηk /E ∞ as well as for the scaled time step t˜n := t n E k /E ∞ = t n /[1 − c] and the modified elastic trail strain ˜e := [e − c n ] E k /E ∞ = [e − c n ]/[1 − c] = n − v /[1 − c] with c := τ0 /τ∞ have been introduced. Thus the total stress and the increment of the viscous strain read at the end of the time step σn =

ηk t˜n  n n [1 − c]  ˜ + E  and  = [1 − c] e . ∞ e v τ0 + t˜n τ0 + t˜n

(4.129)

The sensitivity of σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is straightforwardly computed as ∂ σ n =

ηk [1 − c] + E ∞ . τ0 + t˜n

(4.130)

Note that, consequently, the algorithmic tangent degenerates to E a → E ∞ + E 0 [1 − c] = E 0 for t˜n → 0, i.e. for very fast processes (as compared to the relaxation time) the response is (stiff) elastic. Likewise, for a rigid (spontaneous elastic) spring with E 0 → ∞ and thus for a vanishing relaxation time τ0 → 0 and t˜n → t n the algorithmic tangent degenerates to the case of the Kelvin model. Finally, for vanishing viscosity ηk → 0 or for t˜n → ∞, i.e. for very slow processes (as compared to the relaxation time) the algorithmic tangent degenerates to E a → E ∞ , i.e. the response is (soft) elastic. The algorithmic step-by-step update for the Standard-Linear-Solid Kelvin model is summarized in Table 4.8.

132

4 Visco-Elasticity

Table 4.8 Algorithmic update for the Standard-Linear-Solid Kelvin model Input

n n−1 v

Trial Strain

˜ e = n − n−1 v /[1 − c]

t˜n [1 − c] ˜ e + n−1 v τ0 + t˜n ηk Update Stress σ n = [1 − c] ˜ e + E ∞ n τ0 + t˜n Update Strain nv =

ηk [1 − c] + E ∞ τ0 + t˜n

Tangent

E an =

Output

σ n nv E an

4.3.3 Standard-Linear-Solid Kelvin Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the Standard-Linear-Solid Kelvin model to a prescribed Zig-Zag strain history is documented in Fig. 4.30a–f. Figure 4.30a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.30b showcases the resulting stress history σ(t) that displays a periodic, distorted zig-zag or rather sawtooth-type signal after a slight initial transient phase. The viscous strain v (t), which—after a slight initial transient phase—is also a periodic signal, is demonstrated in Fig. 4.30c. The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that also displays the slight initial transient phase is highlighted in Fig. 4.30d. Finally, Fig. 4.30e, f depict the resulting σ = σ() diagrams for a 100 times shorter and a 100 times longer period T corresponding to a 100 times higher and a 100 times lower strain rate |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙(t)| → ∞ or E ∞ = 0.5 for |˙(t)| → 0. Prescribed Strain History: Sine The response of the Standard-Linear-Solid Kelvin model to a prescribed Sine strain history is documented in Fig. 4.31a–f. Figure 4.31a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T

4.3 Generalized-Kelvin Model

133 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

b)

(tmax = 100 × 0.1 σmax,min = ± 5.0) σ

v

t

c)

(tmax = 100 × 0.1

v,max,min

= ± 5.0)

d)

(

max,min

σ

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0 σmax,min = ± 5.0)

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.30 Response analysis of Standard-Linear-Solid Kelvin model with material data: ηk = 1.0, E k = 1.0 (τk = 1.0), E 0 = 1.0 (E ∞ = E m = 0.5, ηm = 0.25, τm = 0.5). Prescribed Zig-Zag strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.04; t = 0.001, N = 100, f T = 400.0; t = 10.0, N = 100

134

4 Visco-Elasticity

in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.31b showcases the resulting stress history σ(t) that displays, in accordance with

the analytical solution in Eq. 4.99, a harmonic signal with amplitude σa = E ∞ [1 + τk2 ω 2 ]/[1 + c2 τk2 ω 2 ] a ≈ 3.66 after an initial transient phase. The viscous strain v (t), which—after a slight initial transient phase—is also a (phase shifted) harmonic signal is demonstrated in Fig. 4.31c. The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that also displays the slight initial transient phase is highlighted in Fig. 4.31d. Finally, Fig. 4.31e, f depict the resulting σ = σ() diagrams for a 100 times shorter and a 100 times longer period T corresponding to higher and lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙(t)| → ∞ or E ∞ = 0.5 for |˙(t)| → 0. Prescribed Strain History: Ramp The response of the Standard-Linear-Solid Kelvin model to a prescribed Ramp strain history is documented in Fig. 4.32a–f. Figure 4.32a depicts the prescribed Ramp (viscous) strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.32b showcases the resulting stress history σ(t) that especially displays relaxation to the equilibrium response σ(t) → E ∞ a = 2.5 during the holding phase and nonlinear stress response during the un/loading phases approaching σ(t) ≈ 3.5 (σ(t) ≈ −1) at the end of the loading (unloading) phase. The viscous strain v (t), which approaches v (t) → 2.5 during the holding phase and v (t) ≈ 1.5 (v (t) ≈ 1) at the end of the loading (unloading) phase is demonstrated in Fig. 4.32c. The resulting σ = σ() diagram is highlighted in Fig. 4.32d. Finally, Fig. 4.32e, f depict the resulting σ = σ() diagrams for 100 times smaller and 100 times larger t1 , t2 , t3 corresponding to higher and lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙(t)| → ∞ or E ∞ = 0.5 for |˙(t)| → 0. Prescribed Stress History: Zig-Zag The response of the Standard-Linear-Solid Kelvin model to a prescribed Zig-Zag stress history is documented in Fig. 4.33a–f. Figure 4.33a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed.

4.3 Generalized-Kelvin Model

135 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

b)

(tmax = 100 × 0.1 σmax,min = ± 5.0) σ

v

t

c)

(tmax = 100 × 0.1

v,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0 σmax,min = ± 5.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.31 Response analysis of Standard-Linear-Solid Kelvin model with material data: ηk = 1.0, E k = 1.0 (τk = 1.0), E 0 = 1.0 (E ∞ = E m = 0.5, ηm = 0.25, τm = 0.5). Prescribed Sine strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.04; t = 0.001, N = 100, f T = 400.0; t = 10.0, N = 100

136

4 Visco-Elasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 5.0) σ

v

t

c)

(tmax = 100 × 0.1

v,max,min

= ± 5.0)

d)

(

max,min

σ

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0 σmax,min = ± 5.0)

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.32 Response analysis of Standard-Linear-Solid Kelvin model with material data: ηk = 1.0, E k = 1.0 (τk = 1.0), E 0 = 1.0 (E ∞ = E m = 0.5, ηm = 0.25, τm = 0.5). Prescribed Ramp strain history with data: a = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100, f t0 = 0.0, t1 = 100.0, t2 = 900.0, t3 = 1000.0; t = 10.0, N = 100

4.3 Generalized-Kelvin Model

137

Figure 4.33b showcases the resulting strain history (t) that displays a periodic, distorted zigzag or rather sawtooth-like signal after an initial transient phase. The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.33c. The viscous strain v (t), which—after an initial transient phase— is also a (phase shifted) periodic signal, is demonstrated in Fig. 4.33d. Finally, Fig. 4.33e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to a 10 and 100 times higher stress rate |σ(t)|, ˙ respectively. They clearly demonstrate an elastic solid-type behaviour with linear ˙ → ∞. σ = σ() relation and stiffness approaching E 0 = 1 for |σ(t)| Prescribed Stress History: Sine The response of the Standard-Linear-Solid Kelvin model to a prescribed Sine stress history is documented in Fig. 4.34a–f. Figure 4.34a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.34b showcases the resulting strain history (t) that displays, in accordance with

the analytical solution in Eq. 4.102, a harmonic signal with amplitude

a = σa [1 + c2 τk2 ω 2 ]/[1 + τk2 ω 2 ]/E ∞ ≈ 6.83 after an initial transient phase. The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.34c. The viscous strain v (t), which—after an initial transient phase—is also a (phase shifted) harmonic signal, is demonstrated in Fig. 4.34d. Finally, Fig. 4.34e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to higher stress rates |σ(t)|, ˙ respectively. They clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation and ˙ → ∞. stiffness approaching E 0 = 1 for |σ(t)| Prescribed Stress History: Ramp

The response of the Standard-Linear-Solid Kelvin model to a prescribed Ramp stress history is documented in Fig. 4.35a–f. Figure 4.35a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.35b showcases the resulting strain history (t) that especially displays nonlinear creep to the equilibrium response (t) → σa /E ∞ = 10 during the holding phase and nonlinear strain response during the un/loading phases. The resulting σ = σ() diagram is highlighted in Fig. 4.35c. The viscous strain v (t), which approaches v (t) → 5 during the holding phase, is demonstrated in Fig. 4.35d.

138

4 Visco-Elasticity σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

v,max,min

= ± 10.0)

σ

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.33 Response analysis of Standard-Linear-Solid Kelvin model with material data: ηk = 1.0, E k = 1.0 (τk = 1.0), E 0 = 1.0 (E ∞ = E m = 0.5, ηm = 0.25, τm = 0.5). Prescribed Zig-Zag stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

4.3 Generalized-Kelvin Model

139

σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

v,max,min

= ± 10.0)

σ

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.34 Response analysis of Standard-Linear-Solid Kelvin model with material data: ηk = 1.0, E k = 1.0 (τk = 1.0), E 0 = 1.0 (E ∞ = E m = 0.5, ηm = 0.25, τm = 0.5). Prescribed Sine stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

140

4 Visco-Elasticity

Finally, Fig. 4.35e, f depict the resulting σ = σ() diagrams for 10 and 100 times ˙ respectively. They clearly smaller t1 , t2 , t3 corresponding to higher stress rates |σ(t)|, demonstrate an elastic solid-type behaviour with linear σ = σ() relation and stiff˙ → ∞. ness approaching E 0 = 1 for |σ(t)| Equivalence to Standard-Linear-Solid Maxwell Model The global response behaviour of the Standard-Linear-Solid Kelvin model with parameters ηk = 1.0, E k = 1.0 (thus τk = 1.0), and E 0 = 1.0 in terms of the total strain and the total stress, as discussed in the above, is equivalent to the global response behaviour of the Standard-Linear-Solid Maxwell model with parameters ηm = 0.25, E m = 0.5 (thus τm = 0.5), and E ∞ = 0.5. The local response behaviour of the Standard-Linear-Solid Maxwell model with these parameters in terms of the viscous strain, however, is related to that of the Standard-Linear-Solid Kelvin model by a factor 1/[1 − c] = 2.0.

4.3.4 Generic Generalized-Kelvin Model: Formulation A generic formulation of the Generalized-Kelvin model can be obtained from generalizing the specific Generalized-Kelvin model in Fig. 4.23 by assuming the K elastic springs or/and the K viscous dashpots as nonlinear. For the generic Generalized-Kelvin model the free energy density ψ follows as the sum K  ψk (vk ), (4.131) ψ(, {v j }\1 ) = ψ1 (, v ) + k=2

with abbreviations {v j }\1 := {v2 , . . . , vk , . . . , v K } for the set of viscous strains K and v := k=2 vk for the total viscous strain. Thereby, the free energy densities ψ1 and ψk for the generic Kelvin elements (1 and k = 2, . . . , K ) are expressed as non-quadratic, yet convex, functions of  − v and vk (the elastic strain contribution e1 and the (k = 2, . . . , K ) viscous contributions vk ) ψ1 (, v ) = ψ1 ( − v ) and

ψk = ψk (vk ).

(4.132)

Note that ψ1 (, v ) and ψ1 ( − v ) are different functions that return, however, the same function value for the same values of  and v . Then the energetic stress σ  and the energetic viscous stresses σvk for the generic Kelvin elements (k = 2, . . . , K ) follow as σ  (, v ) = ∂ ψ(, {v j }\1 ) = ∂ ψ1 ( − v ) σvk (

,

(4.133a)

vk ) = ∂vk ψ(, {v j }\1 ) = ∂vk ψ1 ( − v ) + ∂vk ψk (vk ).

(4.133b)

4.3 Generalized-Kelvin Model

141

σ

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

t

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 10.0 σmax,min = ± 5.0)

= ± 10.0)

σ

σ

e)

v,max,min

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.35 Response analysis of Standard-Linear-Solid Kelvin model with material data: ηk = 1.0, E k = 1.0 (τk = 1.0), E 0 = 1.0 (E ∞ = E m = 0.5, ηm = 0.25, τm = 0.5). Prescribed Ramp stress history with data: σa = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.1, t2 = 0.9, t3 = 1.0; t = 0.01, N = 100, f t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100

142

4 Visco-Elasticity

Note that the energetic stress σ  coincides here with the elastic stress σe1 := σ  in the first generic Kelvin element and that, due ∂ ψ1 = −∂vk ψ1 , the elastic stresses σek in the remaining (k = 2, . . . , K ) generic Kelvin elements follow as σek := σ  + σvk . In analogy the definitions σv1 := σ  and σvk := σ  + σvk (k = 2, . . . , K ) are introduced for the viscous stresses in the generic Kelvin elements in terms of the dissipative stress σ  and dissipative viscous stresses σvk . Recall finally that the energetic and the dissipative stress are constitutively related to the total stress σ (that enters the equilibrium condition) by σ = σ  + σ  which is likewise engraved in the relation between the energetic and the dissipative viscous stresses σvk = −σvk resulting in σe1 + σv1 = σek + σvk ≡ σ. Thus the sum of the elastic stress contribution σek and the viscous stress contribution σvk for each generic Kelvin element (k = 2, . . . , K ) equals likewise the total stress σ (the equilibrium stress). Furthermore, for the generic Generalized-Kelvin model the convex and smooth (non-quadratic) dissipation and dual dissipation potentials follow as the sums π ( ˙ , { ˙v j }\1 ) = π¯ ( ˙v1 , { ˙v j }\1 ) = π¯ 1 ( ˙v1 ) +

K 

π¯ k ( ˙vk ),

(4.134a)

π¯ k∗ (σvk ),

(4.134b)

k=2

π ∗ (σ  , {σv j }\1 ) = π¯ ∗ (σv1 , {σv j }\1 ) = π¯ 1∗ (σv1 ) +

K  k=2

with abbreviations {˙v j }\1 := {˙v2 , . . . , ˙vk , . . . , ˙v K } for the set of viscous strain rates, ˙v1 := ˙ − ˙v for the rate of the total viscous strain and {σv j }\1 := {σv2 , . . . , σvk , . . . , σv K } for the set of dissipative viscous stresses, {σv j }\1 := {σv2 , . . . , σvk , . . . , σv K } for the set of viscous stresses (with σv1 := σ  and σvk := σ  + σvk ). Thereby, the convex and smooth (non-quadratic) dissipation and dual dissipation potentials for the generic Generalized-Kelvin model introduced as π = π(˙, {˙v j }\1 ) and π ∗ = π ∗ (σ  , {σv j }\1 ), respectively, are related via corresponding Legendre transK K σvk ˙vk = σv1 ˙v1 + k=2 σvk ˙vk ) formations (note that σ  ˙ + k=2 π ( ˙ , { ˙v j }\1 ) = max {σ  ˙ +

K 



σ  ,{σv j }\1

π ∗ (σ  , {σv j }\1 ) = max {σ  ˙ + ˙ ,{ ˙ v j }\1

σvk ˙vk − π ∗ (σ  , {σv j }\1 )},

(4.135a)

σvk ˙vk − π ( ˙ , { ˙v j }\1 )}.

(4.135b)

k=2 K  k=2

The stationarity conditions corresponding to Eqs. 4.135a and 4.135b are the constitutive relations ˙ (•) = ∂σ π ∗ (•) and

˙vk (•) = ∂σv π ∗ (•) (• = σ  , {σv j }\1 ), k

(4.136a)

4.3 Generalized-Kelvin Model

143

σ  (•) = ∂ ˙ π (•) and

σvk (•) = ∂ ˙vk π (•) (• = ˙ , { ˙v j }\1 ).

(4.136b)

Alternatively, after re-parametrization (˙v1 = ˙ − ˙v and σvk = σ  + σvk ), the stationarity conditions corresponding to Eqs. 4.135a and 4.135b read more conventiently ˙v1 (σv1 ) = ∂σv1 π¯ 1∗ (σv1 ) and

˙vk (σvk ) = ∂σvk π¯ k∗ (σvk ),

(4.137a)

σv1 ( ˙v1 ) = ∂ ˙v1 π¯ 1 ( ˙v1 ) and

σvk ( ˙vk ) = ∂ ˙vk π¯ k ( ˙vk ).

(4.137b)

Obviously, the relations in Eqs. 4.136a and 4.136b (or likewise in Eqs. 4.137a and 4.137b) determine entirely the dissipative behavior of the generic Generalized-Kelvin model, thus the formulation is completed at this stage. Finally, as a further interesting aspect, the dissipation for the generic GeneralizedKelvin model follows as the sum d(σ  , {σv j }\1 , ˙, { ˙v j }\1 ) = d1 (σ  , ˙) +

K 

dk (σvk , ˙vk ) ≥ 0.

(4.138)

k=2

Alternatively, after re-parametrization (˙v1 = ˙ − ˙v and σvk = σ  + σvk ), the dissipation reads d(σ  , {σv j }\1 , ˙, { ˙v j }\1 ) = d¯1 (σv1 , ˙v1 ) +

K 

d¯k (σvk , ˙vk ) ≥ 0.

(4.139)

k=2

Thereby, the re-parameterized dissipation d¯1 = σv1 ˙v1 and d¯k = σvk ˙vk (k = 2, . . . , K ) for each generic Kelvin element is alternatively expressed from Eqs. 4.135a and 4.135b in terms of the dissipation potentials π¯ 1 , π¯ k and the dual dissipation potentials π¯ 1∗ , π¯ k∗ as

Table 4.9 Summary of the generic Generalized-Kelvin model 1) Strain 2) Energy

= ψk

=

K k=1 k

ψk (

=

σek

=

∂ ek ψk

4) Potential

π ¯k

=

π ¯k ( ˙vk )

π

=

σvk

=

∂ ˙vk π ¯k π ¯k∗ (σvk )

5) Evolution

π ¯k∗

=

π∗

=

˙vk

=

˙

=

=:

σ − σvk

=:

σ − σek

K ¯k k=1 π

or 4) Potential

K k=1 ek

K k=1 ψk

ψ 3) Stress

5) Stress

=

ek )

K ¯k∗ k=1 π

∂σvk π ¯k∗ K k=1 ˙vk

=

K k=1 vk

144

4 Visco-Elasticity

d¯1 = π¯ 1 (˙v1 ) + π¯ 1∗ (σv1 ) ≥ 0 and

d¯k = π¯ k (˙vk ) + π¯ k∗ (σvk ) ≥ 0.

(4.140)

The generic Generalized-Kelvin model is summarized in Table 4.9.

4.4 Maxwell Model The Maxwell model of a visco-elastic fluid (in short the Maxwell model) consists of a serial arrangement of (1) an elastic spring and (2) a viscous dashpot (see the sketc.h of the specific Maxwell model in Fig. 4.36). The basic kinematic assumption of the Maxwell model is the additive decomposition of the total strain  into the elastic strain e (representing the elongation of the elastic spring) and the viscous strain v (representing the elongation of the viscous dashpot), i.e. (4.141)  = e + v . Note that the viscous strain v denotes the only element contained in the set of internal variables α = {v } for the Maxwell model. E

η

σ

σ

e

Fig. 4.36 Specific Maxwell model

v

4.4 Maxwell Model

145

James Clerk Maxwell [b. 13.6.1831 Edinburgh, Scotland, d. 5.11.1879 Cambridge, England] was Professor of Physics at various British Universities. He is best known for his unification of electricity, magnetism and light in the socalled Maxwell equations of electromagnetism that imply i.a. the finite speed of light, see his “A Treatise on Electricity and Magnetism” from 1873. As part of his occupation with the dynamical theory of gases he proposed in 1867 what is now called the Maxwell model for visco-elastic fluids.

4.4.1 Specific Maxwell Model: Formulation The specific Maxwell model, displayed in Fig. 4.36, consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear viscous dashpot with viscosity η. Direct Representation For the specific Maxwell model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − v (the elastic strain e ) ψ(, v ) =

1 E [ − v ]2 . 2

(4.142)

Then the energetic stress σ  , which is conjugated to the total strain , and the energetic viscous stress σv , which is conjugated to the viscous strain v , follow as σ  (, v ) = ∂ ψ(, v )

=

E [ − v ],

(4.143a)

σv (, v )

= −E [ − v ].

(4.143b)

= ∂v ψ(, v )

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the viscous dashpot, also with the negative of the energetic viscous stress, −σv ≡ σ. Furthermore, for the specific Maxwell model the convex and smooth (quadratic) dissipation potential π is chosen as

146

4 Visco-Elasticity

π(˙v ) =

1 η |˙v |2 . 2

(4.144)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) π is positively homogenous of degree two in ˙v and obviously smooth at the origin ˙v = 0. Consequently, the dissipative viscous stress σv computes as partial derivative of the dissipation potential with respect to its conjugated variable (4.145) σv (˙v ) = ∂˙v π(˙v ) = η ˙v . Recall that the energetic and the dissipative viscous stresses are constitutively related by σv + σv = 0, thus the notion of viscous stress defined as the value σv := σv = −σv

(4.146)

will exclusively be used in the sequel for convenience of exposition. The corresponding dual dissipation potential π ∗ , as determined from a Legendre transformation 1 (4.147) π ∗ (σv ) = max{σv ˙v − η |˙v |2 } ˙ v 2 then reads π ∗ (σv ) =

1 1 |σv |2 . 2 η

(4.148)

The evolution law for the viscous strain then follows as partial derivative of the dual dissipation potential with respect to its conjugated variable ˙v (σv ) = ∂σv π ∗ (σv ) =

Table 4.10 Summary of the specific Maxwell model (1) Strain



=  e + v

(2) Energy

ψ =

(3) Stress

σ = E [ − v ]

(4) Potential π = (5) Stress

1 2E

1 2

[ − v ]2 ≡

η |˙v |2

σv = η ˙ v

≡

or (4) Potential π ∗ =

1 1 |σv |2 2 η

(5) Evolution ˙ v =

1 σv η



σv

σ

≡



−σv

1 σv . η

(4.149)

4.4 Maxwell Model

147

Obviously the expressions in Eqs. 4.145 and 4.149 are inverse relations. The smooth dissipation and dual dissipation potentials π = π(˙v ) and π ∗ = π ∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙v ) and ˙v = ˙v (σv ) are similar to those displayed in Fig. 4.2. The specific Maxwell model is summarized in Table 4.10. Convolution Integral Representation The rate form of the constitutive relation σ˙ v = E [˙ − ˙v ] = σ˙ for the viscous stress (as well as for the total stress) together with the evolution equation for the viscous strain ˙v = σv /η = σ/η may be arranged in a differential equation relating the total stress and strain as σ(t) + τ σ(t) ˙ = η ˙(t). (4.150) Accordingly, for ˙ = 0 the stress satisfies σ = −τ σ˙ at any t, thus τ := η/E has been introduced as the relaxation time of the Maxwell model. Imposing a constant strain step (t) = 0 H(t) (and thus ˙(t) = 0 δ(t)) renders an exponential stress relaxation in time σ(t) + τ σ(t) ˙ = η δ(t) 0 =⇒ σ(t) =: E(t) 0 .

(4.151)

Here E(t), i.e. the normalized stress history as response to an imposed constant unit strain step (t) = H(t), has been introduced as the exponentially decaying relaxation function that is illustrated in Fig. 4.37 (left) E(t) := E H(t) e−t/τ .

(4.152)

Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution integral  t

σ(t) =

E(t − t  ) ˙(t  ) dt  =: E(t)  ˙(t).

(4.153)

0

Thereby the convolution of the relaxation function with the strain rate history is abbreviated symbolically as E(t)  ˙(t). ˙ = Imposing, alternatively, a constant stress step σ(t) = σ0 H(t) (and thus σ(t) σ0 δ(t)) renders a linearly increasing creep strain in time η ˙(t) = [H(t) + τ δ(t)] σ0 =⇒ (t) =: C(t) σ0 .

(4.154)

Here C(t), i.e. the normalized strain history as response to an imposed constant unit stress step σ(t) = H(t), has been introduced as the linearly increasing creep function that is illustrated in Fig. 4.37 (right)

148

4 Visco-Elasticity

3

3

E(t)/E = H(t) e−t/τ

2

C(t)/C = H(t) [1 + t/τ ]

2 (t)

0

= H(t)

1

1 σ(t)/σ0 = H(t)

0

t

τ 0

1

2

3

t

0

0

1

2

3

Fig. 4.37 Specific Maxwell model: Normalized relaxation function E (t)/E (left) and normalized creep function C (t)/C (right) for a relaxation time τ = 2. Observe the jumps in both functions at t = 0. The dotted lines depict the normalized step functions for the prescribed strain and stress, respectively. The dashed line illustrates the meaning of the relaxation time

C(t) := C H(t) [1 + t/τ ],

(4.155)

where C := E −1 denotes the inverse of the elastic stiffness, i.e. the elastic compliance. Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as response to an arbitrary stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral 

t

(t) =

C(t − t  ) σ(t ˙  ) dt  =: C(t)  σ(t). ˙

(4.156)

0

Thereby the convolution of the creep function with the stress rate history is abbreviated symbolically as C(t)  σ(t). ˙ Laplace Transformation Representation Upon Laplace transformation, the convolution integral of the relaxation function E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0, results in L{σ(t)} = L{E(t)  ˙(t)} = s L{E(t)} L{(t)} = E

τs L{(t)}. 1+τs

(4.157)

Choosing, as a particular example, a causal harmonic strain history with (t) = H(t) a sin(ω t) and thus L{(t)} = a ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,11 a phase shifted causal harmonic signal for the resulting stress his11

The inverse Laplace transformation for the stress history follows from the following step by step computation:

4.4 Maxwell Model

149

σ(t)/E

5

˜ ˜ sin(ωt) + E E

(t)/C

10 cos(ωt)

5 t

0

˜ C

t

˜ e−t/τ −E

0 −5

−5 ˜ ˜ sin(ωt) − C C

0

2

4

6

8

10

0

2

cos(ωt)

4

6

8

10

Fig. 4.38 Specific Maxwell model with τ = 1.0: Normalized stress history σ(t)/E resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history (t)/C resulting from sinusoidal stress history with σa = 5 and ω = 2 π/4 (right)

tory superposed by an exponentially decaying signal that is needed to enforce the initial condition σ(0) = 0   σ(t) = H(t) E  sin(ω t) + E  cos(ω t) − E  e−t/τ a .

(4.158)

Thereby E  := E τ 2 ω 2 /[1 + τ 2 ω 2 ] and E  := E τ ω/[1 + τ 2 ω 2 ] are here formally introduced as abbreviations, however as will become transparent in the sequel, they denote the so-called storage and loss stiffness moduli (note that E  /E  = 1/[τ ω]). The stress history resulting from a sinusoidal strain history is shown in Fig. 4.38 (left). Upon Laplace transformation, the convolution integral of the creep function C(t) with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0, results in

L{σ(t)}

a

= E

τs ω [1 + τ s] [ω 2 + s 2 ]

= E

s [1 + τ 2 ω 2 ] τω 2 2 [1 + τ ω ] [1 + τ s] [ω 2 + s 2 ]

[τ ω 2 + s] [1 + τ s] − τ [ω 2 + s 2 ] τω 2 2 [1 + τ ω ] [1 + τ s] [ω 2 + s 2 ]   2 τω +s τω τ = E − 1+τs [1 + τ 2 ω 2 ] ω 2 + s 2 τ ω s   − E  = E 2 +E ω2 + s 2 1+τs ω + s2

= E

= E  L{H(t) sin(ω t)} + E  L{H(t) cos(ω t)} − E  L{H(t) e−t/τ } .

150

4 Visco-Elasticity

L{(t)} = L{C(t)  σ(t)} ˙ = s L{C(t)} L{σ(t)} = C

1+τs L{σ(t)}. τs

(4.159)

Choosing, as a particular example, a causal harmonic stress history with σ(t) = H(t) σa sin(ω t) and thus L{σ(t)} = σa ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,12 a phase shifted causal harmonic signal for the resulting strain history superposed by a constant signal that is needed to enforce the initial condition (0) = 0   (4.160) (t) = H(t) C  sin(ω t) − C  cos(ω t) + C  σa . Thereby C  := C and C  := C/[τ ω] are here formally introduced as abbreviations, in terminological accordance to E  and E  they are denote the storage and the loss compliance moduli (note that C  /C  = 1/[τ ω]). The strain history resulting from a sinusoidal stress history is shown in Fig. 4.38 (right). It is interesting to note that the relaxation function and the creep function are related via their Laplace transformations as s 2 L{E(t)} L{C(t)} = 1.

(4.161)

Observe, furthermore, that direct application of the Laplace transformation to the differential equation relating the total stress and strain L{σ(t)} + τ s L{σ(t)} = η s L{(t)}

(4.162)

renders immediately the relation L{σ(t)} = E

τs L{(t)}, 1+τs

(4.163)

that is entirely conforming with the convolution integral representation. 12

The inverse Laplace transformation for the strain history follows from the following step by step computation: L{(t)}

σa

=C

ω [1 + τ s] τs [ω 2 + s 2 ]

=C

1 [1 + τ s] ω 2 τ ω s [ω 2 + s 2 ]

1 s [τ ω 2 − s] + [ω 2 + s 2 ] τω s [ω 2 + s 2 ]   τ ω2 − s 1 1 + =C τ ω ω2 + s 2 s ω s 1 = C 2 − C  2 + C  ω + s2 ω + s2 s

=C

= C  L{H(t) sin(ω t)} − C  L{H(t) cos(ω t)} + C  L{H(t)} .

4.4 Maxwell Model

151

Complex Harmonic Oscillation Representation The differential equation relating the total stress and strain reads in complex representation as σ(t) + τ σ(t) ˙ = η ˙(t). (4.164) Then, for a stationary harmonic oscillation of the total stress and strain with σ(t) = σ ∗ ei ω t and (t) = ∗ ei ω t the relation between the corresponding complex amplitudes ∗ = a ei δ and σ ∗ = σa ei δσ (where δ = −π/2 for sinusoidal strain control and δσ = −π/2 for sinusoidal stress control) follows as σ∗ = E

iτω ∗ =: E ∗ ∗ . 1+iτ ω

(4.165)

Thereby the quantity relating the complex amplitudes of the total strain and stress is denoted the complex stiffness modulus ∗



E (ω) = E

 τ 2 ω2 τω +i =: E  + i E  , 1 + τ 2 ω2 1 + τ 2 ω2

(4.166)

its inverse is the complex compliance modulus (so that E ∗ C ∗ = 1)  1 =: C  − i C  . C (ω) = C 1 − i τω ∗



(4.167)

Note that for the specific Maxwell model the complex moduli E ∗ and C ∗ have indeed real and imaginary parts. Here, E  and E  denote the so-called storage and loss stiffness moduli, respectively, that are defined as E  := E

τ 2 ω2 τω and E  := E , 2 2 1+τ ω 1 + τ 2 ω2

(4.168)

whereas C  and C  denote the so-called storage and loss compliance moduli, respectively, that are defined as C  := C and C  := C

1 . τω

(4.169)

The storage and loss stiffness and compliance moduli are plotted against the angular frequency ω for various relaxation times in Fig. 4.39. Finally, the (real) amplitude E a and the phase shift angle δ of the complex stiffness modulus are defined as

−1   (4.170) E ∗ (ω) =: [E  ]2 + [E  ]2 ei tan (E /E ) =: E a ei δ ,

152

4 Visco-Elasticity

τ = 102 , 101 , 100 , 10−1 , 10−2

E 101 E

E E

τ = 102 , 101 , 100 , 10−1 , 10−2 100 10−1

10−2

10−2 −5

10

10−3

10−11

10−4

ω2τ 2 1 + ω2τ 2

10−8

10−5

10−3 10−2 10−1 100 101 102 103

10−3 10−2 10−1 100 101 102 103

ω

ω

τ = 102 , 101 , 100 , 10−1 , 10−2

C 0.4 C 10

C C

6

103

100

100

1 10−0.4 10−3 10−2 10−1 100 101 102 103

ω

τ = 102 , 101 , 100 , 10−1 , 10−2

10

100.2

10−0.2

ωτ 1 + ω2τ 2

10−3 10−6

1 ωτ 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.39 Specific Maxwell model: Normalized storage stiffness modulus E  (ω)/E (top left) and normalized loss stiffness modulus E  (ω)/E (top right) together with normalized storage compliance modulus C  (ω)/C (bottom left) and normalized loss compliance modulus C  (ω)/C (bottom right) plotted against the angular frequency ω for five decades of relaxation times τ

likewise the (real) amplitude Ca and the phase shift angle δ of the complex compliance modulus are defined as

−1   (4.171) C ∗ (ω) =: [C  ]2 + [C  ]2 e−i tan (C /C ) =: Ca e−i δ , so that σa = E a a (or a = Ca σa ) and δσ = δ + δ. Specifically, the angular frequency dependent amplitude E a (ω) and phase shift angle δ(ω) follow from E a :=



τω E  1 , [E  ]2 + [E  ]2 = E √ and tan δ :=  = 2 2 E τω 1+τ ω

(4.172)

correspondingly, the angular frequency dependent amplitude Ca (ω) and phase shift angle δ(ω) follow from

4.4 Maxwell Model

153

τ = 102 , 101 , 100 , 10−1 , 10−2

Ea E 100

E E

10−1

6

τ = 102 , 101 , 100 , 10−1 , 10−2

10

103

10−2

100

10−3 10−4

√

−5

10

ωτ 1 + ω2τ 2

10−3 10−6

10−3 10−2 10−1 100 101 102 103

ω

τ = 102 , 101 , 100 , 10−1 , 10−2

Ca C 105

√

4

10

1 + ω2τ 2 ωτ

1 ωτ 10−3 10−2 10−1 100 101 102 103

ω

C C

106

τ = 102 , 101 , 100 , 10−1 , 10−2

103

103

100

102 101

10−3

100 10−3 10−2 10−1 100 101 102 103

ω

10−6

1 ωτ 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.40 Specific Maxwell model: Normalized amplitude E a (ω)/E (top left) and tangent of phase shift angle tan δ(ω) = E  (ω)/E  (ω) (top right) together with normalized amplitude Ca (ω)/C (bottom left) and tangent of phase shift angle tan δ(ω) = C  (ω)/C  (ω) (bottom right) plotted against the angular frequency ω for five decades of relaxation times τ

Ca :=



[C  ]2 + [C  ]2 = C

√ C  1 + τ 2 ω2 1 and tan δ :=  = . τω C τω

(4.173)

The amplitudes and (the tangent of) the phase shift angle of the complex stiffness modulus and the complex compliance modulus are plotted against the angular frequency ω for various relaxation times in Fig. 4.40. The phase shift angle δ between the harmonically oscillating total stress and strain and its tangent tan δ are also denoted the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the loss angle tend to zero for large angular frequencies, since in this limit the viscous dashpot is too inert to react.

154

4 Visco-Elasticity

4.4.2 Specific Maxwell Model: Algorithmic Update For the specific Maxwell model the evolution law for the viscous strain v is integrated by the implicit Euler backwards method to render nv := nv − n−1 = v

t n n σ . η v

(4.174)

Moreover, the viscous stress σv is updated at the end of the time step by σvn = −E [nv − n ] =: E e − E nv .

(4.175)

The trial elastic strain e is computable exclusively from known quantities at the beginning and at the end of the time step and follows as e := n − n−1 v .

(4.176)

Incorporating the discretized evolution law for the viscous strain then renders σvn = E e − E

t n n σ . η v

(4.177)

The above relation is regrouped in order to separate the unknowns at the end of the time step from the known trial strain τ + t n n σv = e . η

(4.178)

Here the definition for the relaxation time τ := η/E has been incorporated. Thus the viscous stress and the increment of the viscous strain read at the end of the time step η t n σvn = e and nv =  . (4.179) n τ + t τ + t n e The sensitivity of σvn = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is straightforwardly computed as ∂ σvn =

η . τ + t n

(4.180)

Note that, consequently, the algorithmic tangent degenerates to E a → E for t n → 0, i.e. for very fast processes (as compared to the relaxation time) the response is elastic. Likewise, for a rigid (elastic) spring with E → ∞ and thus for a vanishing relaxation time τ → 0 the algorithmic tangent degenerates to the case of the Newton model. Finally, for vanishing viscosity η → 0 or for t n → ∞, i.e. for very slow

4.4 Maxwell Model

155

Table 4.11 Algorithmic update for the specific Maxwell model Input

n n−1 v

Trial Strain

e = n − n−1 v

t n  + n−1 v τ + t n e η Update Stress σvn =  = σ n τ + t n e η Tangent E an = τ + t n Update Strain nv =

Output

σ n nv E an

processes (as compared to the relaxation time) the algorithmic tangent degenerates to E a → 0, i.e. the response is infinitely soft. The algorithmic step-by-step update for the specific Maxwell model is summarized in Table 4.11.

4.4.3 Specific Maxwell Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Maxwell model to a prescribed Zig-Zag strain history is documented in Fig. 4.41a–f. Figure 4.41a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.41b showcases the resulting stress history σ(t) that displays a periodic, distorted zig-zag or rather sawtooth-type signal after an initial transient phase. The viscous strain v (t) = (t) − σ(t)/E with ˙v (t) = σ(t)/η, which—after an initial transient phase—is also a periodic signal (phase shifted with respect to she stress signal), is demonstrated in Fig. 4.41c. The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.41d. Finally, Fig. 4.41e, f depict the resulting σ = σ() diagrams for a 100 times shorter and a 100 times longer period T corresponding to a 100 times higher and a 100 times lower strain rate |˙(t)|, respectively. They clearly demonstrate an elastic solid-like

156

4 Visco-Elasticity

behaviour with linear σ = σ() relation for |˙(t)| → ∞ and a viscous fluid-like behaviour with vanishing stress for |˙(t)| → 0. Prescribed Strain History: Sine The response of the specific Maxwell model to a prescribed Sine strain history is documented in Fig. 4.42a–f. Figure 4.42a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.42b showcases the resulting stress history σ(t) that displays, in accordance with the √ analytical solution in Eq. 4.158, a harmonic signal with amplitude σa = E τ ω/ 1 + τ 2 ω 2 a ≈ 4.21 after an initial transient phase. The viscous strain v (t) = (t) − σ(t)/E with ˙v (t) = σ(t)/η, which—after an initial transient phase—is also a harmonic signal (phase shifted by π/2 with respect to she stress signal) with amplitude σa /η/ω ≈ 2.68, is demonstrated in Fig. 4.42c. The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.42d. Finally, Fig. 4.42e, f depict the resulting σ = σ() diagrams for a 100 times shorter and a 100 times longer period T corresponding to higher and lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙(t)| → ∞ and a viscous fluid-like behaviour with vanishing stress for |˙(t)| → 0. Prescribed Strain History: Ramp The response of the specific Maxwell model to a prescribed Ramp strain history is documented in Fig. 4.43a–f. Figure 4.43a depicts the prescribed Ramp (viscous) strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.43b showcases the resulting stress history σ(t) that especially displays complete relaxation during the holding phase and nonlinear stress response during the un/loading phases approaching |σ(t)| → 3. The viscous strain v (t) = (t) − σ(t)/E with ˙v (t) = σ(t)/η, which approaches v (t) → 5 during the holding phase and v (t) → 2 (v (t) → 3) at the end of the loading (unloading) phase is demonstrated in Fig. 4.43c. The resulting σ = σ() diagram is highlighted in Fig. 4.43d. Finally, Fig. 4.43e, f depict the resulting σ = σ() diagrams for 100 times smaller and 100 times larger t1 , t2 , t3 corresponding to higher and lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙(t)| → ∞ and a viscous fluid-like behaviour with vanishing stress for |˙(t)| → 0.

4.4 Maxwell Model

157 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

b)

(tmax = 100 × 0.1 σmax,min = ± 5.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0 σmax,min = ± 5.0)

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.41 Response analysis of the specific Maxwell model with material data: η = 1.0, E = 1.0 (τ = 1.0). Prescribed Zig-Zag strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.04; t = 0.001, N = 100, f T = 400.0; t = 10.0, N = 100

158

4 Visco-Elasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 5.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0 σmax,min = ± 5.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.42 Response analysis of the specific Maxwell model with material data: η = 1.0, E = 1.0 (τ = 1.0). Prescribed Sine strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.04; t = 0.001, N = 100, f T = 400.0; t = 10.0, N = 100

4.4 Maxwell Model

159 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 5.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0 σmax,min = ± 5.0)

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.43 Response analysis of the specific Maxwell model with material data: η = 1.0, E = 1.0 (τ = 1.0). Prescribed Ramp strain history with data: a = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100, f t0 = 0.0, t1 = 100.0, t2 = 900.0, t3 = 1000.0; t = 10.0, N = 100

160

4 Visco-Elasticity

Prescribed Stress History: Zig-Zag The response of the specific Maxwell model to a prescribed Zig-Zag stress history is documented in Fig. 4.44a–f. Figure 4.44a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.44b showcases the resulting strain history (t) that displays a periodic, distorted zigzag or rather sawtooth-like signal. The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 4.44c. The viscous strain v (t) = (t) − σ(t)/E with ˙v (t) = σ(t)/η, which is also a periodic signal (phase shifted by π/2 with respect to she stress signal) with v (t) ∈ [0, 5], is demonstrated in Fig. 4.44d. Finally, Fig. 4.44e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to a 10 and 100 times higher stress rate |σ(t)|, ˙ respectively. They clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation for |σ(t)| ˙ → ∞. Prescribed Stress History: Sine The response of the specific Maxwell model to a prescribed Sine stress history is documented in Fig. 4.45a–f. Figure 4.45a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.45b showcases the resulting strain history (t) that displays, in accordance with the analytical solution in Eq.√4.160, a harmonic signal oscillating about σa /τ /ω ≈ 3.18 with amplitude a = σa 1 + τ 2 ω 2 /τ /ω/E ≈ 5.93 thus rendering max(min)(t) ≈ 9.11(−2.74). The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 4.45c. The viscous strain v (t) = (t) − σ(t)/E with ˙v (t) = σ(t)/η, which is also a harmonic signal (phase shifted by π/2 with respect to she stress signal) that oscillates about σa /η/ω ≈ 3.18 with amplitude σa /η/ω ≈ 3.18 thus rendering max v (t) ≈ 6.37, is demonstrated in Fig. 4.45d. Finally, Fig. 4.45e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to higher stress rates |σ(t)|, ˙ respectively. They clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation for |σ(t)| ˙ → ∞. Prescribed Stress History: Ramp The response of the specific Maxwell model to a prescribed Ramp stress history is documented in Fig. 4.46a–f.

4.4 Maxwell Model

161

σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 10.0 σmax,min = ± 5.0)

= ± 10.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.44 Response analysis of the specific Maxwell model with material data: η = 1.0, E = 1.0 (τ = 1.0). Prescribed Zig-Zag stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

162

4 Visco-Elasticity σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 10.0)

v

t

c)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

σ

e)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

p,max,min

= ± 10.0)

σ

f)

(

max,min

= ± 10.0 σmax,min = ± 5.0)

Fig. 4.45 Response analysis of the specific Maxwell model with material data: η = 1.0, E = 1.0 (τ = 1.0). Prescribed Sine stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

4.4 Maxwell Model

163

σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 50.0)

v

t

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 50.0 σmax,min = ± 5.0)

= ± 50.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 50.0 σmax,min = ± 5.0)

Fig. 4.46 Response analysis of the specific Maxwell model with material data: η = 1.0, E = 1.0 (τ = 1.0). Prescribed Ramp stress history with data: σa = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.1, t2 = 0.9, t3 = 1.0; t = 0.01, N = 100, f t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100

164

4 Visco-Elasticity

Figure 4.46a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.46b showcases the resulting strain history (t) that especially displays linear creep during the holding phase and nonlinear strain response during the un/loading phases. The resulting σ = σ() diagram is highlighted in Fig. 4.46c. The viscous strain v (t) = (t) − σ(t)/E with ˙v (t) = σ(t)/η, which is also a linear signal during the holding phase, is demonstrated in Fig. 4.46d. Finally, Fig. 4.46e, f depict the resulting σ = σ() diagrams for 10 and 100 ˙ respectively. They times smaller t1 , t2 , t3 corresponding to higher stress rates |σ(t)|, clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation for |σ(t)| ˙ → ∞.

4.4.4 Generic Maxwell Model: Formulation A generic formulation of the Maxwell model can be obtained from generalizing the specific Maxwell model in Fig. 4.36 by assuming the elastic spring or/and the viscous dashpot as nonlinear. For the generic Maxwell model the free energy density ψ is expressed as a nonquadratic, yet convex, function of  − v (the elastic strain e ) ψ(, v ) = ψ( − v ).

(4.181)

Note that ψ(, v ) and ψ( − v ) are different functions that return, however, the same function value for the same values of  and v . Then the energetic stress σ  and the energetic viscous stress σv follow as σ  (, v ) = ∂ ψ(, v ) = ∂ ψ( − v ),

(4.182a)

σv (, v )

(4.182b)

= ∂v ψ(, v ) = ∂v ψ( − v ).

Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ  ≡ σ and the negative of the energetic viscous stress −σv ≡ σ. Moreover the energetic and the dissipative viscous stresses are constitutively related by σv + σv = 0, thus the notion of viscous stress defined as σv := σv = −σv will exclusively be used in the sequel. Furthermore, for the generic Maxwell model it is possible to introduce the convex and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙v ) and π ∗ = π ∗ (σv ), respectively, are related via corresponding Legendre transformations (4.183a) π ( ˙v ) = max{σv ˙v − π ∗ (σv )}, σv

4.4 Maxwell Model

165

Table 4.12 Summary of the generic Maxwell model (1) Strain



(2) Energy

ψ = ψ( − v )

(3)

Stress

=  e + v

σ = ∂ ψ

≡

σ

≡

σv

≡



−σv

(4) Potential π = π(˙v ) (5) Stress

σv = ∂˙v π



or (4) Potential π ∗ = π ∗ (σv ) (5) Evolution ˙ v = ∂σv π ∗

π ∗ (σv ) = max{σv ˙v − π ( ˙v )}. ˙ v

(4.183b)

The stationarity conditions corresponding to Eqs. 4.183a and 4.183b are the constitutive relations ˙v (σv ) = ∂σv π ∗ (σv ), (4.184a) σv ( ˙v ) = ∂ ˙v π ( ˙v ).

(4.184b)

Obviously, the relations in Eqs. 4.184a and 4.184b determine entirely the dissipative behavior of the generic Maxwell model, thus the formulation is completed at this stage. Finally, as a further interesting aspect, the dissipation d = σv ˙v is alternatively expressed from Eqs. 4.183a and 4.183b in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙v ) + π ∗ (σv ) ≥ 0.

(4.185)

The generic Maxwell model is summarized in Table 4.12.

4.5 Generalized-Maxwell Model The Generalized-Maxwell model of a visco-elastic fluid/solid (in short the Generalized-Maxwell model) consists of a parallel arrangement of M Maxwell elements each (m = 1, . . . , M) consisting of a serial arrangement of (i) an elastic spring

166

4 Visco-Elasticity

Em1

ηm1

Emm

ηmm

σ

σ

EmM

em

ηmM

vm

Fig. 4.47 Specific Generalized-Maxwell model

and (ii) a viscous dashpot (see the sketc.h of the specific Generalized-Maxwell model in Fig. 4.47). The basic kinematic assumption of the mth Maxwell element is the additive decomposition of the total strain  into the elastic strain em (representing the elongation of the mth elastic spring) and the viscous strain vm (representing the elongation of the mth viscous dashpot), i.e.  = em + vm for m = 1, . . . , M.

(4.186)

Note that the set of viscous strains {v1 , . . . , vm , . . . , v M } denote the M elements contained in the set of internal variables α = {v1 , . . . , vm , . . . , v M } for the Generalized-Maxwell model.

4.5.1 Standard-Linear-Solid Maxwell Model: Formulation The specific Generalized-Maxwell model, displayed in Fig. 4.47, consists of a parallel arrangement of M specific Maxwell elements each (m = 1, . . . , M) consisting of a serial arrangement of (i) a linear elastic spring with stiffness E mm and (ii) a linear viscous dashpot with viscosity ηmm .

4.5 Generalized-Maxwell Model

167

E∞

σ

σ

ηm

Em

e

v

Fig. 4.48 Standard-Linear-Solid Maxwell model

As a particular three parameter sub-case of the specific Generalized-Maxwell model the Standard-Linear-Solid Maxwell model, displayed in Fig. 4.48, consists of a parallel arrangement of (1) a linear elastic spring with stiffness E ∞ (a Hooke element representing the elastic equilibrium response) and (2) a specific Maxwell element consisting of a serial arrangement of (i) a linear elastic spring with stiffness E m and (ii) a linear viscous dashpot with viscosity ηm . Direct Representation For the Standard-Linear-Solid Maxwell model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − v (the elastic strain e ) and  (the total strain) 1 1 ψ(, v ) = E m [ − v ]2 + E ∞ 2 . (4.187) 2 2 Then the energetic stress σ  , which is conjugated to the total strain , and the energetic viscous stress σv , which is conjugated to the viscous strain v , follow as σ  (, v ) = ∂ ψ(, v ) = σv (, v )

E m [ − v ] + E ∞ ,

= ∂v ψ(, v ) = −E m [ − v ]

.

(4.188a) (4.188b)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the parallel arrangement of the (elastic equilibrium) Hooke element and the Maxwell element, also with the difference of the elastic equilibrium stress, σ∞ := E ∞ , and the energetic viscous stress, σ∞ − σv ≡ σ.

168

4 Visco-Elasticity

Furthermore, for the Standard-Linear-Solid Maxwell model the convex and smooth (quadratic) dissipation potential π is chosen as π(˙v ) =

1 ηm |˙v |2 . 2

(4.189)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) π is positively homogenous of degree two in ˙v and obviously smooth at the origin ˙v = 0. Consequently, the dissipative viscous stress σv computes as partial derivative of the dissipation potential with respect to its conjugated variable (4.190) σv (˙v ) = ∂˙v π(˙v ) = ηm ˙v . Recall that the energetic and the dissipative viscous stresses (in the elastic spring and the viscous dashpot of the Maxwell element) are constitutively related by σv + σv = 0, thus the notion of viscous stress defined as the value σv := σv = −σv

(4.191)

will exclusively be used in the sequel for convenience of exposition. The corresponding dual dissipation potential π ∗ , as determined from a Legendre transformation 1 (4.192) π ∗ (σv ) = max{σv ˙v − ηm |˙v |2 } ˙ v 2 then reads

Table 4.13 Summary of the Standard-Linear-Solid Maxwell model (1) Strain



=  e + v

(2) Energy

ψ =

(3) Stress

σ = E m [ − v ] + E ∞  =

σv + σ∞

≡

σ

(4) Stress

σv = E m [ − v ]

σ − σ∞

≡

−σv

≡

σv

(5) Potential π = (6) Stress

1 2 Em

1 2

[ − v ]2 + 21 E ∞ 2

=



ηm |˙v |2

σv = ηm ˙ v

or (5) Potential π ∗ =

1 1 |σv |2 2 ηm

(6) Evolution ˙ v =

1 σv ηm



4.5 Generalized-Maxwell Model

169

π ∗ (σv ) =

1 1 |σv |2 . 2 ηm

(4.193)

The evolution law for the viscous strain of the Maxwell element then follows as partial derivative of the dual dissipation potential with respect to its conjugated variable 1 σv . (4.194) ˙v (σv ) = ∂σv π ∗ (σv ) = ηm Obviously the expressions in Eqs. 4.190 and 4.194 are inverse relations. The smooth dissipation and dual dissipation potentials π = π(˙v ) and π ∗ = π ∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙v ) and ˙v = ˙v (σv ) are similar to those displayed in Fig. 4.2. The Standard-Linear-Solid Maxwell model is summarized in Table 4.13. Convolution Integral Representation The rate form of the constitutive relation σ˙ v = E m [˙ − ˙v ] for the viscous stress and the evolution equation for the viscous strain ˙v = σv /ηm in the Maxwell element together with the rate form of the constitutive relation σ˙ ∞ = E ∞ ˙ in the Hooke element may be arranged in a differential equation relating the total stress and strain as ˙ = E ∞ (t) + E 0 τm ˙(t). (4.195) σ(t) + τm σ(t) Here E 0 := E m + E ∞ and τm := ηm /E m denote the instantaneous elastic stiffness of the Standard-Linear-Solid Maxwell model and the relaxation time of the Maxwell element. Moreover the stiffness ratio e := E 0 /E ∞ and the compliance ratio c := −1 . C0 /C∞ are introduced with C0 := E 0−1 = C∞ Cm /[C∞ + Cm ] and C∞ := E ∞ Imposing a constant strain step (t) = 0 H(t) (and thus ˙(t) = 0 δ(t)) renders an exponential stress relaxation in time ˙ = [E ∞ H(t) + E 0 τm δ(t)] 0 =⇒ σ(t) =: E(t) 0 . σ(t) + τm σ(t)

(4.196)

Here E(t), i.e. the normalized stress history as response to an imposed constant unit strain step (t) = H(t), has been introduced as the exponentially decaying relaxation function that is illustrated in Fig. 4.49 (left)   E(t) := E ∞ H(t) 1 + [e − 1] e−t/τm = E ∞ H(t) + E m H(t) e−t/τm .

(4.197)

The latter expansion clearly highlights the parallel arrangement of a Hooke and a Maxwell element in the Standard-Linear-Solid Maxwell model. Based on the Boltzmann superposition process, the stress history σ(t) for t ≥ 0 as response to an arbitrary strain history (t) ≡ H(t) (t) follows from the convolution integral

170

4 Visco-Elasticity

3

E(t)/E∞ = H(t) [1 + ¯e e−t/τ ]

3

2

2

1

1

C(t)/C∞ = H(t) [1 + ¯c e−ct/τ ]

σ(t)/σ0 = H(t) (t)

0

= H(t)

0

t

τ 0

1

2

3

t

0

0

1

2

3

Fig. 4.49 Standard-Linear-Solid Maxwell model: Normalized relaxation function E (t)/E ∞ (left) and normalized creep function C (t)/C∞ (right) for a relaxation time τm = 2 and E ∞ = E m (thus e = 2, e¯ := e − 1 = 1, c = 1/2 and c¯ := c − 1 = −1/2). Observe the jumps in both functions at t = 0. The dotted lines depict the normalized step functions for the prescribed strain and stress, respectively. The dashed line illustrates the meaning of the relaxation time τm



t

σ(t) =

E(t − t  ) ˙(t  ) dt  =: E(t)  ˙(t).

(4.198)

0

Thereby the convolution of the relaxation function with the strain rate history is abbreviated symbolically as E(t)  ˙(t). ˙ = Imposing, alternatively, a constant stress step σ(t) = σ0 H(t) (and thus σ(t) σ0 δ(t)) renders an exponentially saturating creep strain in time E ∞ (t) + E 0 τm ˙(t) = [H(t) + τm δ(t)] σ0 =⇒ (t) =: C(t) σ0 .

(4.199)

Here C(t), i.e. the normalized strain history as response to an imposed constant unit stress step σ(t) = H(t), has been introduced as the exponentially saturating creep function that is illustrated in Fig. 4.49 (right)   C(t) := C∞ H(t) 1 + [c − 1] e−c t/τm .

(4.200)

Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as response to an arbitrary stress history σ(t) ≡ H(t) σ(t) follows from the convolution integral  t

(t) = 0

C(t − t  ) σ(t ˙  ) dt  =: C(t)  σ(t). ˙

(4.201)

4.5 Generalized-Maxwell Model

171

Thereby the convolution of the creep function with the stress rate history is abbreviated symbolically as C(t)  σ(t). ˙ Laplace Transformation Representation Upon Laplace transformation, the convolution integral of the relaxation function E(t) with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0, results in L{σ(t)} = L{E(t)  ˙(t)} = s L{E(t)} L{(t)} 1 + e τm s = E∞ L{(t)}. 1 + τm s

(4.202)

Note that the nominator E ∞ [1 + e τm s] in the above expands as E ∞ [1 + τm s] + E m τm s thus highlighting again the parallel arrangement of a Hooke and a Maxwell element   τm s L{σ(t)} = E ∞ + E m L{(t)}. (4.203) 1 + τm s Choosing, as a particular example, a causal harmonic strain history with (t) = H(t) a sin(ω t) and thus L{(t)} = a ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,13 a causal phase shifted harmonic signal for the resulting stress history superposed by an exponentially decaying signal that is needed to enforce the initial condition σ(0) = 0   σ(t) = H(t) E  sin(ω t) + E  cos(ω t) − E  e−t/τm a .

(4.204)

13

The inverse Laplace transformation for the stress history follows from the following step by step computation: L{σ(t)}

a

= E∞ = E∞ = E∞ = E∞

[1 + e τm s] ω [1 + τm s] [ω 2 + s 2 ] [1 + τm s] ω − [1 − e] τm ω s [1 + τm s] [ω 2 + s 2 ] ω2

ω [1 + τm2 ω 2 ] s + E  2 +s [1 + τm s] [ω 2 + s 2 ]

ω2

ω ω s − τm ω 2 + E  τm ω 2 + E  2 2 +s ω +s [1 + τm s] [ω 2 + s 2 ]

ω [1 + τm s] s τm [ω 2 + s 2 ]   + E − E ω2 + s 2 [1 + τm s] [ω 2 + s 2 ] [1 + τm s] [ω 2 + s 2 ] ω s τm    +E −E = E 2 ω + s2 ω2 + s 2 1 + τm s = E

= E  L{H(t) sin(ω t)} + E  L{H(t) cos(ω t)} − E  L{H(t) e−t/τm } Alternatively a more direct derivation that highlights the parallel arrangement of a Hooke and a Maxwell element reads:

172

4 Visco-Elasticity

σ(t)/E

10

˜ sin( E

+ ω t)

˜ E

cos(

ω t)

(t)/C

4

5

2 t ˜ e−t/τ −E

0 −5

˜ C

0

−

e

t/

τ] [e

t

−2 ˜ ˜ sin(ωt) − C C

cos(ωt)

−4

−10 0

2

4

6

8

10

0

2

4

6

8

10

Fig. 4.50 Standard-Linear-Solid Maxwell model with τm = 1.0 and E ∞ = E m : Normalized stress history σ(t)/E m resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history (t)/Cm resulting from sinusoidal stress history with σa = 5 and ω = 2 π/4 (right)

Thereby E  := E ∞ [1 + e τm2 ω 2 ]/[1 + τm2 ω 2 ] = E ∞ + E m τm2 ω 2 /[1 + τm2 ω 2 ] and E  := E ∞ [e − 1] τm ω/[1 + τm2 ω 2 ] = E τm ω/[1 + τm2 ω 2 ] are here formally introduced as abbreviations, however as will become transparent in the sequel, they denote the so-called storage and loss stiffness moduli (note that E  /E  = [e − 1] τm ω/[1 + e τm2 ω 2 ]). The stress history resulting from a sinusoidal strain history is shown in Fig. 4.50 (left). Upon Laplace transformation, the convolution integral of the creep function C(t) with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0, results in L{(t)} = L{C(t)  σ(t)} ˙ = s L{C(t)} L{σ(t)} 1 + τm s = C∞ L{σ(t)}. 1 + e τm s L{σ(t)}

a

= E∞

ω τm s ω + Em [1 + τm s] [ω 2 + s 2 ] [ω 2 + s 2 ]

= E∞

ω τm ω s [1 + τm2 ω 2 ] + Em [ω 2 + s 2 ] [1 + τm2 ω 2 ] [1 + τm s] [ω 2 + s 2 ]

(4.205)

[τm ω 2 + s] [1 + τm s] − τm [ω 2 + s 2 ] ω τm ω + Em 2 2 2 +s ] [1 + τm ω ] [1 + τm s] [ω 2 + s 2 ]   2 τm ω τm ω τm ω + s − = E∞ 2 + E m ω2 + s 2 1 + τm s [ω + s 2 ] [1 + τm2 ω 2 ] τm ω s    = E 2 −E +E ω2 + s 2 1 + τm s ω + s2 = E∞

[ω 2

= E  L{H(t) sin(ω t)} + E  L{H(t) cos(ω t)} − E  L{H(t) e−t/τm } .

4.5 Generalized-Maxwell Model

173

Choosing, as a particular example, a causal harmonic stress history with σ(t) = H(t) σa sin(ω t) and thus L{σ(t)} = σa ω/[ω 2 + s 2 ] renders, after inverse Laplace transformation,14 a causal phase shifted harmonic signal for the resulting strain history superposed by an exponentially decaying signal that is needed to enforce the initial condition (0) = 0   (t) = H(t) C  sin(ω t) − C  cos(ω t) + C  e−c t/τm σa .

(4.206)

Thereby C  := C∞ [1 + e τ 2 ω 2 ]/[1 + e2 τ 2 ω 2 ] and C  := C∞ [e − 1] τ ω/[1 + e τ 2 ω 2 ] are here formally introduced as abbreviations, in terminological accordance to E  and E  they are denoted the storage and the loss compliance moduli (note that C  /C  = [e − 1] τ ω/[1 + e τ 2 ω 2 ]). The strain history resulting from a sinusoidal stress history is shown in Fig. 4.50 (right). 2

It is interesting to note that the relaxation function and the creep function are related via their Laplace transformations as s 2 L{E(t)} L{C(t)} = 1.

(4.207)

Observe, furthermore, that direct application of the Laplace transformation to the differential equation relating the total stress and strain L{σ(t)} + τm s L{σ(t)} = E ∞ L{(t)} + E 0 τm s L{(t)}

(4.208)

renders immediately the relation

14 The inverse Laplace transformation for the strain history follows from the following step by step computation:

L{(t)}

σa

= C∞ = C∞ = C∞ = C∞

ω [1 + τm s] [1 + e τm s] [ω 2 + s 2 ] [1 + e τm s] ω − [e − 1] τm ω s [1 + e τm s] [ω 2 + s 2 ] ω [1 + e2 τm2 ω 2 ] s − C  ω2 + s 2 [1 + e τm s] [ω 2 + s 2 ] ω2

ω s − e τm ω 2 ω − C  e τm ω 2 − C  2 2 +s ω +s [1 + e τm s] [ω 2 + s 2 ]

ω [1 + e τm s] s e τm [ω 2 + s 2 ]   − C + C ω2 + s 2 [1 + e τm s] [ω 2 + s 2 ] [1 + e τm s] [ω 2 + s 2 ] ω s e τm    −C +C =C 2 ω + s2 ω2 + s 2 1 + e τm s = C

= C  L{H(t) sin(ω t)} − C  L{H(t) cos(ω t)} + C  L{H(t) e−t/[e τm ] } .

174

4 Visco-Elasticity

L{σ(t)} = E ∞

  1 + e τm s τm s L{(t)}, L{(t)} = E ∞ + E m 1 + τm s 1 + τm s

with inverse L{(t)} = C∞

1 + τm s L{σ(t)}, 1 + e τm s

(4.209)

(4.210)

that are entirely conforming with the convolution integral representations. Complex Harmonic Oscillation Representation The differential equation relating the total stress and strain reads in complex representation as σ(t) + τm σ(t) ˙ = E ∞ (t) + E 0 τm ˙(t). (4.211) Then, for a stationary harmonic oscillation of the total stress and strain with σ(t) = σ ∗ ei ω t and (t) = ∗ ei ω t the relation between the corresponding complex amplitudes ∗ = a ei δ and σ ∗ = σa ei δσ (where δ = −π/2 for sinusoidal strain control and δσ = −π/2 for sinusoidal stress control) follows as σ∗ = E∞

1 + i e τm ω ∗  =: E ∗ ∗ . 1 + i τm ω

(4.212)

Thereby the quantity relating the complex amplitudes of the total strain and stress is denoted the complex stiffness modulus15 E ∗ (ω) = E ∞

1 + i e τm ω =: E  + i E  , 1 + i τm ω

(4.213)

its inverse is the complex compliance modulus (so that E ∗ C ∗ = 1) C ∗ (ω) = C∞

1 + i τm ω =: C  − i C  . 1 + i e τm ω

(4.214)

Note that for the Standard-Linear-Solid Maxwell model the complex moduli E ∗ and C ∗ have indeed real and imaginary parts. Here, E  and E  denote the so-called storage and loss stiffness moduli, respectively, that are defined as16 15 Observe

that the complex stiffness modulus may alternatively be expressed as E ∗ (ω) = E ∞ + E m

i τm ω . 1 + i τm ω

. 16 Observe

that the storage and loss stiffness moduli may alternatively be expressed as E  := E ∞ + E m

τm2 ω 2 τm ω and E  := E m . 1 + τm2 ω 2 1 + τm2 ω 2

4.5 Generalized-Maxwell Model

175

τ = 102 , 101 , 100 , 10−1 , 10−2

E E∞100.3

E 0 E∞ 10

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 100.2

10−2

1 + e ω2τ 2 1 + ω2τ 2

100.1

10−3 10−4 −5

0

10

10 −3

10

C C∞ 100 10−0.1

10−0.2

−2

10

−1

10

0

10

ω

1

10

2

10

3

10−3 10−2 10−1 100 101 102 103

10

τ = 102 , 101 , 100 , 10−1 , 10−2

[e − 1] ωτ 1 + ω2τ 2 ω

C C∞

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 10−2

1 + e ω2τ 2 1 + e2 ω 2 τ 2

10−3 10−4 10−5

−0.3

10

10−3 10−2 10−1 100 101 102 103

ω

10−6

[e − 1] ωτ 1 + e2 ω 2 τ 2 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.51 Standard-Linear-Solid Maxwell model: Normalized storage stiffness modulus E  (ω)/E ∞ (top left) and normalized loss stiffness modulus E  (ω)/E ∞ (top right) together with normalized storage compliance modulus C  (ω)/C∞ (bottom left) and normalized loss compliance modulus C  (ω)/C∞ (bottom right) plotted against the angular frequency ω for five decades of relaxation times τm and E ∞ = E m

E  := E ∞

1 + e τm2 ω 2 [e − 1] τm ω and E  := E ∞ , 2 2 1 + τm ω 1 + τm2 ω 2

(4.215)

whereas C  and C  denote the so-called storage and loss compliance moduli, respectively, that are defined as C  := C∞

1 + e τm2 ω 2 [e − 1] τm ω and C  := C∞ . 2 2 2 1 + e τm ω 1 + e2 τm2 ω 2

(4.216)

The storage and loss stiffness and compliance moduli are plotted against the angular frequency ω for various relaxation times in Fig. 4.51. .

176

4 Visco-Elasticity

Finally, the (real) amplitude E a and the phase shift angle δ of the complex stiffness modulus are defined as

−1   (4.217) E ∗ (ω) =: [E  ]2 + [E  ]2 ei tan (E /E ) =: E a ei δ , likewise the (real) amplitude Ca and the phase shift angle δ of the complex compliance modulus are defined as

−1   (4.218) C ∗ (ω) =: [C  ]2 + [C  ]2 e−i tan (C /C ) =: Ca e−i δ , so that σa = E a a (or a = Ca σa ) and δσ = δ + δ. Specifically, the angular frequency dependent amplitude E a (ω) follows as

1 + e2 τm2 ω 2  2  2 E a := [E ] + [E ] = E ∞ , (4.219) 1 + τm2 ω 2

Ea E∞100.3

τ = 102 , 101 , 100 , 10−1 , 10−2

E E

100

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 100.2

0.1

10

10−2

1 + e2 τ 2 ω 2 1 + ω2τ 2

10−3 10−4 10−5

100

10−3 10−2 10−1 100 101 102 103

10−3 10−2 10−1 100 101 102 103

ω

Ca C∞ 100

τ = 102 , 101 , 100 , 10−1 , 10−2

[e − 1] ωτ 1 + e τ 2ω2 ω

C C

100

τ = 102 , 101 , 100 , 10−1 , 10−2

10−1 10−2

10−0.1

−0.2

10

τ 2 ω2

1+ 1 + e2 τ 2 ω 2

10−3 10−4 10−5

−0.3

10

10−3 10−2 10−1 100 101 102 103

ω

[e − 1] ωτ 1 + e τ 2ω2 10−3 10−2 10−1 100 101 102 103

ω

Fig. 4.52 Standard-Linear-Solid Maxwell model: Normalized amplitude E a (ω)/E ∞ (top left) and tangent of phase shift angle tan δ(ω) = E  (ω)/E  (ω) (top right) together with normalized amplitude Ca (ω)/C∞ (bottom left) and tangent of phase shift angle tan δ(ω) = C  (ω)/C  (ω) (bottom right) plotted against the angular frequency ω for five decades of relaxation times τm and E ∞ = E m

4.5 Generalized-Maxwell Model

177

correspondingly, the angular frequency dependent amplitude Ca (ω) follows as

Ca := [C  ]2 + [C  ]2 = C∞



1 + τm2 ω 2 . 1 + e2 τm2 ω 2

(4.220)

Finally, the phase shift angle δ(ω) is expressed as tan δ :=

E  C  [e − 1] τm ω = = .   E C 1 + e τm2 ω 2

(4.221)

The amplitudes and (the tangent of) the phase shift angle of the complex stiffness modulus and the complex compliance modulus are plotted against the angular frequency ω for various relaxation times in Fig. 4.52. The phase shift angle δ between the harmonically oscillating total stress and strain and its tangent tan δ are also denoted the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the loss angle tend to zero for large angular frequencies, since in this limit the viscous dashpot is too inert to react. Equivalence to Standard-Linear-Solid Kelvin Model The Standard-Linear-Solid Maxwell model is characterized by differential equations for (i) the global response relating total stress and total strain as well as for (ii) the local response relating viscous strain and total strain as σ + τm σ˙ = E ∞  + E 0 τm ˙ and v + τm ˙v = .

(4.222)

By re-parametrization of the relaxation time as τk := e τm the global response of the Standard-Linear-Solid Maxwell model and the Standard-Linear-Solid Kelvin model (as sketc.hed in Fig. 4.53) coincides, however the local response or rather the viscous strain predicted by the two models obviously still differs σ + c τk σ˙ = E ∞  + E ∞ τk ˙ and ˜v + c τk ˙˜ v = [1 − c] .

(4.223)

Here the viscous strain ˜v as predicted by the equivalent Standard-Linear-Solid Kelvin model is related to the corresponding viscous strain v as predicted by the underlying Standard-Linear-Solid Maxwell model via ˜v = [1 − c] v .

(4.224)

If furthermore the instantaneous elastic stiffness E 0 and the equilibrium elastic stiffness E ∞ of the Standard-Linear-Solid Kelvin model and the Standard-LinearSolid Maxwell model are required to coincide, the three parameter sets {E 0 , E k , ηk } and {E ∞ , E m , ηm } representing both models are related via

178

4 Visco-Elasticity

Ek := E∞ [Em + E∞ ]/Em E0 := Em + E∞

σ

σ

2 ηk := ηm [Em + E∞ ]2 /Em

˜e

˜v

Fig. 4.53 Standard-Linear-Solid Kelvin model equivalent to Standard-Linear-Solid Maxwell model: Serial arrangement of (1) a linear elastic spring with stiffness E 0 := E m + E ∞ and (2) a specific Kelvin element consisting of a parallel arrangement of (i) a linear elastic spring with stiffness E k := E ∞ [E m + E ∞ ]/E m and (ii) a linear viscous dashpot with viscosity ηk := 2 . The total strain  is decomposed additively into an elastic and a viscous part ηm [E m + E ∞ ]2 /E m  = ˜ e + ˜ v

E 0 := E m + E ∞ and E k := E ∞

E0 E2 and ηk := ηm 20 . Em Em

(4.225)

Here the resulting stiffness E 0 of a parallel arrangement of elastic springs with stiffness E m and E ∞ together with the relations Ck = C∞ − C0 and ηm,k = E m,k τm,k have been used.

4.5.2 Standard-Linear-Solid Maxwell Model: Algorithmic Update For the Standard-Linear-Solid Maxwell model the evolution law for the viscous strain v is integrated by the implicit Euler backwards method to render nv := nv − n−1 = v

t n n σ . ηm v

(4.226)

Likewise, the viscous stress σv is updated at the end of the time step by σvn = −E m [nv − n ] =: E m e − E m nv .

(4.227)

The trial elastic strain e is computable exclusively from known quantities at the beginning and at the end of the time step and follows as e := n − n−1 v .

(4.228)

4.5 Generalized-Maxwell Model

179

Incorporating the discretized evolution law for the viscous strain then renders σvn = E m e − E m

t n n σ . ηm v

(4.229)

The above relation is regrouped in order to separate the unknowns at the end of the time step from the known trial strain τm + t n n σv = e . ηm

(4.230)

Here the definition for the relaxation time τm := ηm /E m has been incorporated. Thus the total and the viscous stress and the increment of the viscous strain read at the end of the time step σ n = σvn + E ∞ n with σvn =

ηm t n  n  and  =  . (4.231) e v τm + t n τm + t n e

The sensitivity of σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is straightforwardly computed as ∂ σ n =

ηm + E∞. τm + t n

(4.232)

Note that, consequently, the algorithmic tangent degenerates to E a → E ∞ + E m for t n → 0, i.e. for very fast processes (as compared to the relaxation time) the response is (stiff) elastic. Likewise, for a rigid (elastic) spring with E m → ∞ and thus for a vanishing relaxation time τm → 0 the algorithmic tangent degenerates to the case of the Kelvin model. Finally, for vanishing viscosity ηm → 0 or for t n → ∞, i.e. for very slow processes (as compared to the relaxation time) the algorithmic tangent degenerates to E a → E ∞ , i.e. the response is (soft) elastic. The algorithmic step-by-step update for the Standard-Linear-Solid Maxwell model is summarized in Table 4.14.

Table 4.14 Algorithmic update for the Standard-Linear-Solid Maxwell model Input

n n−1 v

Trial Strain

e = n − n−1 v

t n  + n−1 v τm + t n e ηm Update Stress σ n =  + E ∞ n τm + t n e ηm Tangent E an = + E∞ τm + t n Update Strain nv =

Output

σ n nv E an

180

4 Visco-Elasticity

4.5.3 Standard-Linear-Solid Maxwell Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the Standard-Linear-Solid Maxwell model to a prescribed Zig-Zag strain history is documented in Fig. 4.54a–f. Figure 4.54a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.54b showcases the resulting stress history σ(t) that displays a periodic, distorted zig-zag or rather sawtooth-type signal after an initial transient phase. The viscous strain v (t), which—after an initial transient phase—is also a periodic signal, is demonstrated in Fig. 4.54c. The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.54d. Finally, Fig. 4.54e, f depict the resulting σ = σ() diagrams for a 100 times shorter and a 100 times longer period T corresponding to a 100 times higher and a 100 times lower strain rate |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 2 for |˙(t)| → ∞ or E ∞ = 1 for |˙(t)| → 0. Prescribed Strain History: Sine The response of the Standard-Linear-Solid Maxwell model to a prescribed Sine strain history is documented in Fig. 4.55a–f. Figure 4.55a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.55b showcases the resulting stress history σ(t) that displays, in accordance with

the analytical solution in Eq. 4.204, a harmonic signal with amplitude σa = E ∞ [1 + e2 τm2 ω 2 ]/[1 + τm2 ω 2 ] a ≈ 8.85 after an initial transient phase. The viscous strain v (t), which—after an initial transient phase—is also a (phase shifted) harmonic signal is demonstrated in Fig. 4.55c. The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.55d. Finally, Fig. 4.55e, f depict the resulting σ = σ() diagrams for a 100 times shorter and a 100 times longer period T corresponding to higher and lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 2 for |˙(t)| → ∞ or E ∞ = 1 for |˙(t)| → 0.

4.5 Generalized-Maxwell Model

181 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0)

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.54 Response analysis of Standard-Linear-Solid Maxwell model with material data: ηm = 1.0, E m = 1.0 (τm = 1.0), E ∞ = 1.0 (E 0 = E k = 2.0, ηk = 4.0, τk = 2.0). Prescribed Zig-Zag strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.04; t = 0.001, N = 100, f T = 400.0; t = 10.0, N = 100

182

4 Visco-Elasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.55 Response analysis of Standard-Linear-Solid Maxwell model with material data: ηm = 1.0, E m = 1.0 (τm = 1.0), E ∞ = 1.0 (E 0 = E k = 2.0, ηk = 4.0, τk = 2.0). Prescribed Sine strain history with data: a = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.04; t = 0.001, N = 100, f T = 400.0; t = 10.0, N = 100

4.5 Generalized-Maxwell Model

183

Prescribed Strain History: Ramp The response of the Standard-Linear-Solid Maxwell model to a prescribed Ramp strain history is documented in Fig. 4.56a–f. Figure 4.56a depicts the prescribed Ramp (viscous) strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.56b showcases the resulting stress history σ(t) that especially displays relaxation to the equilibrium response σ(t) → E ∞ a = 5 during the holding phase and nonlinear stress response during the un/loading phases approaching σ(t) ≈ 8 (σ(t) ≈ −3) at the end of the loading (unloading) phase. The viscous strain v (t), which approaches v (t) → 5 during the holding phase and v (t) ≈ 2 (v (t) ≈ 3) at the end of the loading (unloading) phase is demonstrated in Fig. 4.56c. The resulting σ = σ() diagram is highlighted in Fig. 4.56d. Finally, Fig. 4.56e, f depict the resulting σ = σ() diagrams for 100 times smaller and 100 times larger t1 , t2 , t3 corresponding to higher and lower strain rates |˙(t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 2 for |˙(t)| → ∞ or E ∞ = 1 for |˙(t)| → 0. Prescribed Stress History: Zig-Zag The response of the Standard-Linear-Solid Maxwell model to a prescribed Zig-Zag stress history is documented in Fig. 4.57a–f. Figure 4.57a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.57b showcases the resulting strain history (t) that displays a periodic, distorted zigzag or rather sawtooth-like signal after an initial transient phase. The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.57c. The viscous strain v (t), which—after an initial transient phase— is also a (phase shifted) periodic signal, is demonstrated in Fig. 4.57d. Finally, Fig. 4.57e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to a 10 and 100 times higher stress rate |σ(t)|, ˙ respectively. They clearly demonstrate an elastic solid-type behaviour with linear ˙ → ∞. σ = σ() relation and stiffness approaching E 0 = 2 for |σ(t)|

184

4 Visco-Elasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

b)

(tmax = 100 × 0.1 σmax,min = ± 10.0) σ

v

t

c)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

d)

(

max,min

σ

e)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

= ± 5.0 σmax,min = ± 10.0) σ

f)

(

max,min

= ± 5.0 σmax,min = ± 10.0)

Fig. 4.56 Response analysis of Standard-Linear-Solid Maxwell model with material data: ηm = 1.0, E m = 1.0 (τm = 1.0), E ∞ = 1.0 (E 0 = E k = 2.0, ηk = 4.0, τk = 2.0). Prescribed Ramp strain history with data: a = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100, f t0 = 0.0, t1 = 100.0, t2 = 900.0, t3 = 1000.0; t = 10.0, N = 100

4.5 Generalized-Maxwell Model

185

σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 5.0)

v

t

c)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.57 Response analysis of Standard-Linear-Solid Maxwell model with material data: ηm = 1.0, E m = 1.0 (τm = 1.0), E ∞ = 1.0 (E 0 = E k = 2.0, ηk = 4.0, τk = 2.0). Prescribed Zig-Zag stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

186

4 Visco-Elasticity

Prescribed Stress History: Sine The response of the Standard-Linear-Solid Maxwell model to a prescribed Sine stress history is documented in Fig. 4.58a–f. Figure 4.58a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.58b showcases the resulting strain history (t) that displays, in accordance with

the analytical solution in Eq. 4.206, a harmonic signal with amplitude a = σa [1 + τm2 ω 2 ]/[1 + e2 τm2 ω 2 ]/E ∞ ≈ 2.82 after an initial transient phase. The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that also displays the initial transient phase is highlighted in Fig. 4.58c. The viscous strain v (t), which—after an initial transient phase—is also a (phase shifted) harmonic signal, is demonstrated in Fig. 4.58d. Finally, Fig. 4.58e, f depict the resulting σ = σ() diagrams for a 10 and 100 times shorter period T corresponding to higher stress rates |σ(t)|, ˙ respectively. They clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation and ˙ → ∞. stiffness approaching E 0 = 2 for |σ(t)| Prescribed Stress History: Ramp The response of the Standard-Linear-Solid Maxwell model to a prescribed Ramp stress history is documented in Fig. 4.59a–f. Figure 4.59a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 4.59b showcases the resulting strain history (t) that especially displays nonlinear creep to the equilibrium response (t) → σa /E ∞ = 5 during the holding phase and nonlinear strain response during the un/loading phases. The resulting σ = σ() diagram is highlighted in Fig. 4.59c. The viscous strain v (t), which approaches v (t) → 5 during the holding phase, is demonstrated in Fig. 4.59d. Finally, Fig. 4.59e, f depict the resulting σ = σ() diagrams for 10 and 100 times ˙ respectively. They clearly smaller t1 , t2 , t3 corresponding to higher stress rates |σ(t)|, demonstrate an elastic solid-type behaviour with linear σ = σ() relation and stiff˙ → ∞. ness approaching E 0 = 2 for |σ(t)| Equivalence to Standard-Linear-Solid Kelvin Model The global response behaviour of the Standard-Linear-Solid Maxwell model with parameters ηm = 1.0, E m = 1.0 (thus τm = 1.0), and E ∞ = 1.0 in terms of the total strain and the total stress, as discussed in the above, is equivalent to the global

4.5 Generalized-Maxwell Model

187

σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 5.0)

v

t

c)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.58 Response analysis of Standard-Linear-Solid Maxwell model with material data: ηm = 1.0, E m = 1.0 (τm = 1.0), E ∞ = 1.0 (E 0 = E k = 2.0, ηk = 4.0, τk = 2.0). Prescribed Sine stress history with data: σa = 5.0, a–d T = 4.0; t = 0.1, N = 100, e T = 0.4; t = 0.01, N = 100, f T = 0.04; t = 0.001, N = 100

188

4 Visco-Elasticity σ

t

t

a)

(tmax = 100 × 0.1 σmax,min = ± 5.0)

(tmax = 100 × 0.1

b)

σ

max,min

= ± 5.0)

v

t

c)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

d)

(tmax = 100 × 0.1

(

max,min

= ± 5.0 σmax,min = ± 5.0)

= ± 5.0)

σ

σ

e)

p,max,min

f)

(

max,min

= ± 5.0 σmax,min = ± 5.0)

Fig. 4.59 Response analysis of Standard-Linear-Solid Maxwell model with material data: ηm = 1.0, E m = 1.0 (τm = 1.0), E ∞ = 1.0 (E 0 = E k = 2.0, ηk = 4.0, τk = 2.0). Prescribed Ramp stress history with data: σa = 5.0, a–d t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100, e t0 = 0.0, t1 = 0.1, t2 = 0.9, t3 = 1.0; t = 0.01, N = 100, f t0 = 0.0, t1 = 0.01, t2 = 0.09, t3 = 0.1; t = 0.001, N = 100

4.5 Generalized-Maxwell Model

189

response behaviour of the Standard-Linear-Solid Kelvin model with parameters ηk = 4.0, E k = 2.0 (thus τk = 2.0), and E 0 = 2.0. The local response behaviour of the Standard-Linear-Solid Kelvin model with these parameters in terms of the viscous strain, however, is related to that of the Standard-Linear-Solid Maxwell model by a factor 1 − c = 0.5.

4.5.4 Generic Generalized-Maxwell Model: Formulation A generic formulation of the Generalized-Maxwell model can be obtained from generalizing the specific Generalized-Maxwell model in Fig. 4.47 by assuming the M elastic springs or/and the M viscous dashpots as nonlinear. For the generic Generalized-Maxwell model the free energy density ψ follows as the sum M  ψm (, vm ), (4.233) ψ(, {v j }) = m=1

with abbreviation {v j } := {v1 , . . . , vm , . . . , v M } for the set of viscous strains. Thereby, the free energy density ψm for each generic Maxwell element (m = 1, . . . , M) is expressed as a non-quadratic, yet convex, function of  − vm (the elastic strain em ) (4.234) ψm (, vm ) = ψm ( − vm ). Note that ψ(, vm ) and ψ( − vm ) are different functions that return, however, the  same function value for the same values of  and vm . Then the energetic stress σm and  the energetic viscous stress σvm for each generic Maxwell element (m = 1, . . . , M) follow as σm (, vm ) = ∂ ψm (, vm )

= ∂ ψm ( − vm ),

(4.235a)

σvm (, vm )

= ∂vm ψm ( − vm ).

(4.235b)

= ∂vm ψm (, vm )

Consequently the energetic stress σ  and the energetic viscous stress σv for the generic Generalized-Maxwell model follow as the sums σ  (, {v j }) =

M 

σm (, vm ),

(4.236a)

σvm (, vm ).

(4.236b)

m=1

σv (, {v j }) =

M  m=1

Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ  ≡ σ and the negative of the energetic vis-

190

4 Visco-Elasticity

cous stress −σv ≡ σ. Moreover the energetic and the dissipative viscous stresses are constitutively related by σv + σv = 0, thus the notion of viscous stress defined as σv := σv = −σv will exclusively be used in the sequel. Analogous statements hold for the stresses related to each generic Maxwell element (m = 1, . . . , M). Furthermore, for the generic Generalized-Maxwell model the convex and smooth (non-quadratic) dissipation and dual dissipation potentials follow as the sums π ({ ˙v j }) =

M 

πm ( ˙vm ),

(4.237a)

πm∗ (σvm ),

(4.237b)

m=1

∗

π ({σv j }) =

M  m=1

with abbreviations {˙v j } := {˙v1 , . . . , ˙vm , . . . , ˙v M } for the set of viscous strain rates and {σv j } := {σv1 , . . . , σvm , . . . , σv M } for the set of viscous stresses. Thereby, the convex and smooth (non-quadratic) dissipation and dual dissipation potentials for each generic Maxwell element (m = 1, . . . , M) introduced as πm = πm (˙vm ) and π ∗ = π ∗ (σvm ), respectively, are related via corresponding Legendre transformations πm ( ˙vm ) = max{σvm ˙vm − πm∗ (σvm )},

(4.238a)

πm∗ (σvm ) = max{σvm ˙vm − πm ( ˙vm )}.

(4.238b)

σ vm

˙ vm

The stationarity conditions corresponding to Eqs. 4.238a and 4.238b are the constitutive relations (4.239a) ˙vm (σvm ) = ∂σvm π ∗ (σvm ), σvm ( ˙vm ) = ∂ ˙vm π ( ˙vm ).

(4.239b)

Obviously, the relations in Eqs. 4.239a and 4.239b determine entirely the dissipative behavior of the generic Generalized-Maxwell model, thus the formulation is completed at this stage. Finally, as a further interesting aspect, the dissipation for the generic Generalized-Maxwell model follows as the sum d({σv j }, {˙v j }) =

M 

dm (σvm , ˙vm ) ≥ 0.

(4.240)

m=1

Thereby, the dissipation dm = σvm ˙vm (m = 1, . . . , M) for each generic Maxwell element is alternatively expressed from Eqs. 4.238a and 4.238b in terms of the dissipation potential πm and the dual dissipation potential πm∗ as

4.5 Generalized-Maxwell Model

191

dm = πm (˙vm ) + πm∗ (σvm ) ≥ 0. The generic Generalized-Maxwell model is summarized in Table 4.15. Table 4.15 Summary of the generic Generalized-Maxwell model (1) Strain



= em + vm

(2) Energy

ψm = ψm ( − vm ) M ψ = m=1 ψm

(3) Stress

σ m = ∂ ψm ≡ M σ = m=1 σm



σm

≡

(4) Potential πm = πm (˙vm ) M π = m=1 πm (5) Stress

σvm = ∂˙vm πm ≡ M σv = m=1 σ vm

or ∗ = π ∗ (σ ) (4) Potential πm m vm M ∗ ∗ π = m=1 πm ∗ (5) Evolution ˙ vm = ∂σvm πm



σ vm



−σvm

(4.241)

Chapter 5

Plasticity

I have discovered that it is enough when a single note is beautifully played. — Arvo Pärt, b. 1935 —

Plasticity is the paradigm for non-reversible, rate-independent material behavior. Experimental evidence for various classes of materials, in particular for ductile metals, suggests to decompose the total strain into an elastic, stress producing part and a plastic, irreversible part. Irreversibility is rooted in sub-scale mechanisms in the material, e.g. dislocation motion in ductile metallic materials with crystalline subscale structure. Conceptually, the plastic part of the strain remains after removing the external load, however, in general together with reversible residual strains. These only vanish when the material is allowed to locally relax into a non-strained state (which usually requires to sacrifice global compatibility). The onset and evolution of irreversibility are defined by a yield condition and a flow rule, which are fundamental concepts at the core of plasticity. They are intimately related to the non-smoothness of the convex dissipation potential and its dual. The elementary rheological model to capture plastic, i.e. non-reversible, rateindependent material behavior is the frictional slider. The rheological model for a perfect rigid-plastic solid consisting of a frictional slider only is denoted the St. Venant model. Parallel and serial arrangements of a frictional slider with an elastic spring render the St. Venant hardening model for a hardening rigid-plastic solid (not considered here) and the Prandtl model for a perfect elasto-plastic solid, respectively. Further, the serial arrangement of an elastic spring and the St. Venant hardening model is established as the Prandtl model for a hardening elasto-plastic solid.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Steinmann and K. Runesson, The Catalogue of Computational Material Models, https://doi.org/10.1007/978-3-030-63684-5_5

193

194

5 Plasticity

5.1 St. Venant Model The St. Venant model of a perfect rigid-plastic solid (in short the St. Venant model) consists of a frictional slider (see the sketch of the specific St. Venant model in Fig. 5.1). The basic kinematic assumption of the St. Venant model (that shall be exploited in the algorithmic setting) is the equality of the total strain  and the plastic strain p (representing the elongation of the frictional slider), i.e.  ≡ p .

(5.1)

Note that the set of internal variables is empty for the St. Venant model, i.e. α = ∅. Adhémar Jean Claude Barré de SaintVenant [b. 23.8.1797, Villiers-en-Biére, Seineet-Marne, France, d. 6.1.1886, Saint-Ouen, Loiret-Cher, France] worked initially as Engineer and later became Professor of Mathematics at the École des Ponts et Chaussées in Paris. He contributed largely to mechanics, elasticity, plasticity, hydrostatics and hydrodynamics. He was first to present a correct derivation of the Navier– Stokes equations in 1843. The St. Venant model for rigid-plastic solids is named after him.

5.1.1 Specific St. Venant Model: Formulation The specific St. Venant model, displayed in Fig. 5.1, consists of a linear frictional slider with threshold σy . Since there is no energy storage for the specific St. Venant model the free energy density ψ vanishes identically ψ() ≡ 0.

Fig. 5.1 Specific St. Venant model

(5.2)

σy

σ

σ ≡

p

5.1 St. Venant Model

195

Thus the energetic stress σ  conjugated to the total strain  vanishes identically as well σ  () ≡ 0.

(5.3)

Furthermore, for the specific St. Venant model the convex but non-smooth dissipation potential π is chosen as π(˙) = σy |˙|.

(5.4)

Observe that (i) π does depend on ˙, thus the dissipative stress σ  = 0 for ˙ = 0, and that (ii) π is positively homogenous of degree one in ˙ and obviously non-smooth at the origin ˙ = 0. Consequently, the dissipative stress σ  computes as some subderivative of the dissipation potential with respect to its conjugated variable ⎧ ⎫ +σy ˙ > 0 ⎬ ⎨ σ  (˙) ∈ d˙ π(˙) = [ − σy , +σy ] for ˙ = 0 , (5.5) ⎩ ⎭ ˙ < 0 −σy whereby d˙ π denotes the set of sub-derivatives, i.e. the sub-differential of π with respect to ˙. Observe that σ  is constitutively not determined for ˙ = 0 (thus it can at best be computed from equilibrium considerations). Recall that the total stress σ applied to the rheological model (that enters the equilibrium condition) and the energetic and the dissipative stresses are constitutively related by σ = σ  + σ  , thus the value (since here σ  ≡ 0) σ ≡ σ

(5.6)

will exclusively be used in the sequel for convenience of exposition. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving force, i.e. in the σ-space, is next introduced as the union of the rigid domain and the yield surface, compare the representation in Fig. 5.2. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙ π in Eq. 5.5, or, alternatively, from evaluating the formal definition of the sub-differential  d˙ π(˙) = {σ | σ [˙ − ˙] ≤ σy |˙ | − |˙| ∀˙ },

(5.7)

196

5 Plasticity σ

Fig. 5.2 Specific St. Venant model: The rigid domain for σ defined by |σ| − σy < 0 in the σ space. The two points |σ| − σy = 0 define the yield surface. The union of the rigid domain and the yield surface renders the admissible domain |σ| − σy ≤ 0

σ = +σy +σy

0

−σy σ = −σy

whereby ˙ denotes any admissible strain rate. Then at ˙ = 0 it holds for any admissible ˙ that σ ˙ ≤ σy |˙ | and, with max˙ {σ ˙ /|˙ |} = |σ|, the admissible domain follows as |σ| ≤ σy . The rigid domain is defined as the interior of the admissible domain, i.e. int A := {σ | |σ| − σy < 0},

(5.8)

whereas the yield surface, which in the present one-dimensional case collapses to the two end points σ = ±σy , is defined as the boundary of the admissible domain, i.e. ∂ A := {σ | |σ| − σy = 0}.

(5.9)

Collectively, the admissible domain in the σ-space is characterized by the yield condition |σ| − σy ≤ 0.

(5.10)

States in the interior int A of the admissible domain with |σ| < σy are rigid, whereas states on the boundary ∂ A of the admissible domain with |σ| = σy are plastic. Recall that σ is constitutively not determined in the rigid domain. The corresponding dual dissipation potential π ∗ , as determined from the Legendre transformation π ∗ (σ) = max{σ ˙ − σy |˙|} ˙

then reads

(5.11)

5.1 St. Venant Model

197

π ∗ (σ) = IA (σ) :=

⎧ ⎨ 0 ⎩

⎫ |σ| ≤ σy ⎬ for

∞

|σ| > σy

⎭

,

(5.12)

where IA denotes the indicator function of the admissible domain A in the σ-space. The evolution law (the associated flow rule) for the total strain then follows as some sub-derivative of the dual dissipation potential with respect to its conjugated variable ⎧ ⎫ |σ| < σy ⎪ ⎪ ⎨ 0 ⎬ for , (5.13) ˙(σ) ∈ dσ π ∗ (σ) = dσ IA (σ) = σ ⎪ ⎩λ ⎭ |σ| = σy ⎪ |σ| whereby dσ π ∗ denotes the set of sub-derivatives, i.e. the sub-differential of π ∗ with respect to σ and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 5.5 and 5.13 are inverse relations. The nonsmooth dissipation and dual dissipation potentials π(˙) and π ∗ (σ) together with the resulting non-smooth constitutive relations σ = σ(˙) and ˙ = ˙(σ) are displayed in Fig. 5.3. Interestingly, the result in Eq. 5.13 can be rephrased in terms of the postulate of maximum dissipation (due to perfect plasticity) that follows from the reverse Legendre transformation π(˙) = max{d(σ; ˙p ) − IA (σ)} = max{d(σ; ˙)}, σ

σ∈ A

(5.14)

whereby d(σ; ˙) := σ ˙ denotes the dissipation power density. The postulate of maximum dissipation can, alternatively, be recast as a variational inequality: For given ˙, find σ ∈ A as the solution of d(σ; ˙) ≥ d(σ  ; ˙) ∀σ  ∈ A,

(5.15)

whereby σ  denotes any admissible stress. As yet another alternative, the postulate of maximum dissipation may be reformulated as constrained optimization problem with  a Lagrange functional incorporating the admissibility constraint |σ| ≤ σy by the Lagrange multiplier λ ≥ 0  (σ, λ; ˙) := −d(σ; ˙) + λ |σ| − σy .

(5.16)

In accordance with Eq. 5.13 the stationarity condition of this constrained optimization problem then reads ˙ = λ

σ , |σ|

(5.17)

198

5 Plasticity

π( ˙)

1

σy | ˙|

σ( ˙)

1

0.5

+σy

0.5

0

˙

0

0.5

−0.5

−1

−1 −1

−0.5

0 π ∗ (σ)

∞

1

0.5

˙

−σy −1

1

∞

−0.5

0

0.5 ˙(σ)

1

1

| ˙|

0.5

0.5 0

−σy

σ

+σy

0.5

−0.5

−1

−1 −1

−0.5

0

0.5

1

−σy

0

σ

+σy

| ˙| −1

−0.5

0

0.5

1

Fig. 5.3 Specific St. Venant model: Non-smooth dissipation potential π(˙) together with resulting non-smooth σ = σ(˙) and non-smooth dual dissipation potential π ∗ (σ) together with resulting non-smooth ˙ = ˙ (σ)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format λ ≥ 0, |σ| ≤ σy , λ |σ| = λ σy .

(5.18)

Note that it follows immediately from Eq. 5.17 that |˙| = λ. Finally, the strain arclength, denoted κ, may conveniently be introduced as a measure of the accumulated deformation, i.e.  κ = κ˙ dt with κ˙ := |˙| = λ ≥ 0. (5.19) The specific St. Venant model is summarized in Table 5.1.

5.1 St. Venant Model

199

Table 5.1 Summary of the specific St. Venant model (1) Strain

 ≡ p

(2) Potential π = σy |˙| (3) Stress

σ = σy

˙ |˙|

≡

σ  for ˙ p = 0

or (2) Yield

0 ≥ |σ| − σy

(3) Evolution ˙ = λ (4) KKT

λ ≥ 0,

σ |σ| |σ|

≤

σy ,

λ |σ|

=

λ σy

Determination of Total Stress The basic kinematic assumption of the St. Venant model may be re-formulated as a constraint . e :=  − p = 0.

(5.20)

Clearly e = 0 compares to a vanishing elastic strain (as present in the Prandtl model discussed in the sequel). Then regarding the rate format of the kinematic constraint ˙e = ˙ − ˙p = 0 the postulate of maximum (plastic) dissipation may be stated in three alternative formats: (1) The Lagrange multiplier format ˆ λ, ˙p , σp ; ˙) := (σ, λ, ˙p ) − σp [˙ − ˙p ], (σ,

(5.21)

whereby ˆ is the Lagrange functional incorporating the rate format of the kinematic constraint ˙e = ˙ − ˙p = 0 by the Lagrange multiplier σp (i.e. the plastic stress). Accordingly, the evolution law for the plastic strain, and the total stress versus plastic stress relation follow from the constrained optimization problem as ˙p = λ

σ (+KKT) and σ = σp , |σ|

(5.22)

whereas the corresponding optimality condition regarding the kinematic constraint reads (5.23) ˙e = ˙ − ˙p = 0.

200

5 Plasticity

(2) The perturbed Lagrange multiplier format  1 ˆE (σ, λ, ˙p , σp ; ˙) := (σ, λ, ˙p ) − σp ˙ − ˙p − σp , 2E

(5.24)

whereby ˆE is the perturbed Lagrange functional with E > 0 the perturbation parameter of dimension stress × time (obviously for E → ∞ the original Lagrange multiplier format is retrieved). Accordingly, the evolution law for the plastic strain, and the total stress versus plastic stress relation are unchanged, whereas the optimality condition regarding the kinematic constraint now reads σp = E ˙e = E [˙ − ˙p ].

(5.25)

Back-substitution of this result for the plastic stress σp into the perturbed Lagrange multiplier format renders the penalty format as 1 ˜E (σ, λ, ˙p ; ˙) := (σ, λ, ˙p ) − E [˙ − ˙p ]2 , 2

(5.26)

whereby ˜E is the penalty functional with E > 0 the penalty parameter. Accordingly, the evolution law for the plastic strain is unchanged, whereas the total stress versus kinematic constraint violation relation follows from the penalized constrained optimization problem as σ = E ˙e = E [˙ − ˙p ].

(5.27)

(3) The augmented Lagrange multiplier format 1 ˇE (σ, λ, ˙p , σp ; ˙) := (σ, λ, ˙p ) − σp [˙ − ˙p ] − E [˙ − ˙p ]2 , 2

(5.28)

whereby ˇE is the augmented Lagrange functional. Accordingly, the evolution law for the plastic strain is unchanged, and the total stress versus plastic stress and kinematic constraint violation relation follows from the augmented constrained optimization problem as σ = σp + E ˙e = σp + E [˙ − ˙p ],

(5.29)

whereas the corresponding optimality condition regarding the kinematic constraint reads ˙e = ˙ − ˙p = 0.

(5.30)

The augmented Lagrange multiplier format suggests an iterative determination of the Lagrange multiplier σp , i.e. the plastic stress is obtained from an Usawa update scheme upon substituting σp by

5.1 St. Venant Model

201

σp ⇐ σp + E ˙e = σp + E [˙ − ˙p ].

(5.31)

Note that once the kinematic constraint ˙e = ˙ − ˙p = 0 is satisfied, the total and the plastic stress coincide σ = σp .

5.1.2 Specific St. Venant Model: Algorithmic Update The integration algorithm for the specific St. Venant model is based on the augmented Lagrange multiplier format of the postulate of maximum (plastic) dissipation that allows to incorporate the kinematic constraint  = p (in rate form). Thereby the evolution law for the plastic strain p is integrated by the implicit Euler backward method to render np := np − n−1 = λ p

σn , |σ n |

(5.32)

whereby λ = t n λn . Consequently, and based on the assumption that the kine− n−1 = 0, the matic constraint is satisfied at the end of the previous time step n−1 p total stress σ (versus plastic stress σp and kinematic constraint violation e relation) reads after Euler backward integration at the end of the current time step as σ n = σpk−1 − E [np − n ] =: σ  − E np .

(5.33)

Here σpk−1 denotes the plastic stress, i.e. the Lagrange multiplier enforcing the kinematic constraint within an Usawa iteration. Moreover E := E/t is a penalty parameter of dimension stress. In contrast to the somewhat naive penalty format, the augmented Lagrange multiplier format allows for arbitrary small penalty parameters that do not compromise the condition number of the equation (system) to be solved. The trial total stress σ  is computable exclusively from known quantities at the beginning of the time step and the previous Usawa update, it follows as − n ]. σ  := σpk−1 − E [n−1 p

(5.34)

Incorporating the discretized evolution law for the plastic strain then renders σ n = σ  − E λ

σn . |σ n |

(5.35)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress

σn |σ n | + E λ = σ . |σ n |

(5.36)

202

5 Plasticity

As an immediate consequence the equivalent stress and its trial value are related via |σ n | = |σ  | − E λ.

(5.37)

A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously σ σn ≡ . |σ n | |σ  |

(5.38)

Eventually, the yield function at the end of the time step is expressed as φn := |σ n | − σy = φ − E λ.

(5.39)

Here the trial value of the yield function φ has been defined as φ := |σ  | − σy .

(5.40)

Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is computed in closed form from λ =

φ  ≥ 0. E

(5.41)

Once λ is computed all other variables may be updated. In particular, the elastic strain (that represents the kinematic constraint) at the end of the time step reads + λ ne = n − np with np = n−1 p

σ . |σ  |

(5.42)

Then, if ne exceeds a given tolerance, the plastic stress is reset according to an Usawa update scheme as σpk = σpk−1 − E [np − n ]

(5.43)

and the Usawa iteration is continued upon incrementing k and re-computing the trial total stress σ  . Otherwise, if ne falls below the given tolerance, the total stress is updated as σ n = σpk−1 .

(5.44)

The algorithmic step-by-step update for the specific St. Venant model is summarized in Table 5.2.

5.1 St. Venant Model

203

Table 5.2 Algorithmic update for the specific St. Venant model Input

n n−1 σpk−1 p

Trial Strain

p = n−1 p

Trial Stress

σp = σpk−1 σ  = σp − E [p − n ]

Trial Yield

φ = |σ  | − σy

Loading Check

IF φ < 0 THEN λ = 0 ELSE λ =

φ E

ENDIF Update Strain

np = p + λ

σ |σ  |

Constraint Check IF |n − np | ≥ tol THEN Update Multiplier σpk = σp + E [n − np ] k =k+1 goto Trial Stress ELSE Update Stress σ n = σp ENDIF Output

σ n np σpk−1

5.1.3 Specific St. Venant Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the specific St. Venant model to a prescribed Zig-Zag strain history is documented in Fig. 5.4a, b, c, d, e.

204

5 Plasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.4 Response analysis of the specific St. Venant model with material data: E = 1.0, σy = 1.0. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

5.1 St. Venant Model

205

Figure 5.4a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 5.4b showcases the resulting stress history σ(t) that displays a block signal with σ(t) = ±σy = ±1 whenever ˙(t) = ±5. The resulting σ = σ() diagram is highlighted in Fig. 5.4c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = ±5 obviously tend to ∞ with t → 0. Figure 5.4d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint p (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 5.4e follows linear in time from integrating κ(t) ˙ = |˙(t)| = 5 over two and a half periods, thus κmax = 50. Prescribed Strain History: Sine The response of the specific St. Venant model to a prescribed Sine strain history is documented in Fig. 5.5a, b, c, d, e. Figure 5.5a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 5.5b showcases the resulting stress history σ(t) that displays a block signal with σ(t) = ±1 whenever ˙(t) = a ω cos(ω t) = 0. The resulting σ = σ() diagram is highlighted in Fig. 5.5c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = ±5 obviously tend to ∞ with t → 0. Figure 5.5d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint p (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 5.5e follows as a sequence of (positive and negative) sine quarter-waves from integrating κ(t) ˙ = |˙(t)| = a ω | cos(ω t)| over two and a half periods, thus κmax = 50. Prescribed Strain History: Ramp The response of the specific St. Venant model to a prescribed Ramp strain history is documented in Fig. 5.6a, b, c, d, e. Figure 5.6a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 5.6b showcases the resulting stress history σ(t) that displays a block signal with σ(t) = ±1 whenever ˙(t) = ±5 in the loading and the unloading phases. In particular during the holding phase σ(t) = σp = +1 results as a reaction to the kinematic constraint p (t) ≡ (t).

206

5 Plasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.5 Response analysis of the specific St. Venant model with material data: E = 1.0, σy = 1.0. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

5.1 St. Venant Model

207 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 5.6 Response analysis of the specific St. Venant model with material data: E = 1.0, σy = 1.0. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

208

5 Plasticity

The resulting σ = σ() diagram is highlighted in Fig. 5.6c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = 5 obviously tend to ∞ with t → 0. Figure 5.6d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint p (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 5.6e follows linear-constant-linear in time from integrating κ(t) ˙ = |˙(t)| = {5, 0, 5} over the time interval t ∈ [0, tmax = 10], thus κmax = 10.

5.1.4 Generic St. Venant Model: Formulation A generic formulation of the St. Venant model can be obtained from generalizing the specific St. Venant model in Fig. 5.1 by assuming the frictional slider as nonlinear. For the generic St. Venant model the free energy density ψ vanishes identically ψ() ≡ 0.

(5.45)

Thus the energetic stress σ  vanishes identically as well σ  () ≡ 0.

(5.46)

Recall that the energetic and the dissipative stresses are constitutively related to the total stress σ (that enters the equilibrium condition) by σ = σ  + σ  , thus (with σ  ≡ 0) the total stress σ ≡ σ  will exclusively be used in the sequel. Furthermore, for the generic St. Venant model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙) and π ∗ = π ∗ (σ), respectively, are related via corresponding Legendre transformations π ( ˙) = max{σ ˙ − π ∗ (σ)},

(5.47a)

π ∗ (σ) = max{σ ˙ − π ( ˙)}.

(5.47b)

σ

˙

Then the stationarity conditions corresponding to Eqs. 5.47a and 5.47b are the constitutive relations ˙(σ) ∈ dσ π ∗ (σ), (5.48a) σ( ˙) ∈ d ˙ π ( ˙).

(5.48b)

5.1 St. Venant Model

209

Obviously the relations in Eqs. 5.48a and 5.48b determine entirely the dissipative behavior of the generic St. Venant model, thus the formulation would be completed at this stage. To be more explicit, however, alternatively to Eq. 5.48b the closed and convex admissible domain A in the σ-space is introduced. It is characterized by the convex yield condition φ = φ(σ) := ϕ(σ) − σy ≤ 0.

(5.49)

Here φ = φ(σ) is the yield function and ϕ(σ) denotes the equivalent stress that is compared to the yield limit σy , a material property. Then the evolution law for the (total) strain (i.e. the associated flow rule) follows alternatively to Eq. 5.48a from the postulate of maximum dissipation (due to plasticity) with  a Lagrange functional incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0 (σ, λ; ˙) := −d(σ; ˙) + λ φ(σ).

(5.50)

Consequently, the stationarity condition of this constrained optimization problem reads ˙ = λ ∂σ φ,

(5.51)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form λ ≥ 0, φ ≤ 0, λ φ = 0.

(5.52)

It shall be noted that collectively Eqs. 5.49, 5.51 and 5.52 are entirely equivalent statements to Eqs. 5.48a and 5.48b. As a further interesting aspect the dissipation d = σ ˙ shall next be examined more closely. From Eqs. 5.47a and 5.47b the dissipation d is alternatively expressed in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙) + π ∗ (σ) ≥ 0.

(5.53)

However, based on the above introduction of the yield condition φ ≤ 0 the dual dissipation potential is identified as the indicator function IA of the admissible domain A ⎧ φ(σ) ≤ 0 ⎨ 0 for π ∗ (σ) = IA (σ) := . (5.54) ⎩ ∞ φ(σ) > 0 Thus for the generic St. Venant model the dual dissipation potential equals zero in the admissible domain A. Consequently, provided the stress is admissible, the dissipation is indeed expressed in terms of the dissipation potential only

210

5 Plasticity

Table 5.3 Summary of the generic St. Venant model (1) Strain

 ≡ p

(2) Potential π = π(˙) (3) Stress

σ ∈ d˙ π

σ



or (2) Yield

φ = φ(σ) ≤ 0

(3) Evolution ˙ = λ ∂σ φ (4) KKT

λ ≥ 0, φ

≤

0, λ φ

=

0

d = π(˙) ≥ 0.

(5.55)

Finally for an equivalent stress that is homogeneous of degree one in the stress (thus σ ∂σ ϕ = ϕ), the dissipation d = σ ˙ is exclusively given in terms of the Lagrange multiplier λ and the yield limit σy , since then d = λ σ ∂σ ϕ = λ ϕ = λ σ y .

(5.56)

The generic St. Venant model is summarized in Table 5.3.

5.2 Prandtl Model The Prandtl model of a perfect elasto-plastic solid (in short the Prandtl model) consists of a serial arrangement of (1) an elastic spring and (2) a frictional slider (see the sketch of the specific Prandtl model in Fig. 5.7).

E

σy

e

p

σ

Fig. 5.7 Specific Prandtl model

σ

5.2 Prandtl Model

211

The basic kinematic assumption of the Prandtl model is the additive decomposition of the total strain  into the elastic strain e (representing the elongation of the elastic spring) and the plastic strain p (representing the elongation of the frictional slider), i.e. (5.57)  = e + p . Note that the plastic strain p denotes the only element contained in the set of internal variables α = {p } for the Prandtl model. Ludwig Prandtl [b. 4.2.1875, Freising, Germany, d. 15.8.1953, Göttingen, Germany] was Professor of Fluid Mechanics at the University Göttingen and the attached Kaiser Wilhelm Institute for Flow Research. He is the founding father of the boundary layer theory in fluid dynamics. Although mainly known for his research in aerodynamics, he also worked on problems of plasticity. His contribution to dry friction from 1928 gave an atomistic explanation of the static friction force. The Prandtl model thus describes solids displaying a yield stress after an initial elastic phase.

5.2.1 Specific Prandtl Model: Formulation The specific Prandtl model, displayed in Fig. 5.7, consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear frictional slider with threshold σy . For the specific Prandtl model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − p (the elastic strain e ) ψ(, p ) =

1 E [ − p ]2 . 2

(5.58)

Then the energetic stress σ  conjugated to the total strain  and the energetic plastic stress σp conjugated to the plastic strain p follow as σ  (, p ) = ∂ ψ(, p ) = σp (, p )

E [ − p ],

(5.59a)

= ∂p ψ(, p ) = −E [ − p ].

(5.59b)

212

5 Plasticity

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the frictional slider, also with the negative of the energetic plastic stress, −σp ≡ σ. Furthermore, for the specific Prandtl model the convex but non-smooth dissipation potential π is chosen as π(˙p ) = σy |˙p |.

(5.60)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) π is positively homogenous of degree one in ˙p and obviously non-smooth at the origin ˙p = 0. Consequently, the dissipative plastic stress σp computes as some sub-derivative of the dissipation potential with respect to its conjugated variable ⎧ ⎫ +σy ˙p > 0 ⎬ ⎨ σp (˙p ) ∈ d˙p π(˙p ) = [ − σy , +σy ] for ˙p = 0 , (5.61) ⎩ ⎭ ˙p < 0 −σy whereby d˙p π denotes the set of sub-derivatives, i.e. the sub-differential of π with respect to ˙p . Recall that the energetic and the dissipative plastic stresses are constitutively related by σp + σp = 0, thus the notion of plastic stress defined as the value σp := σp = −σp

(5.62)

will exclusively be used in the sequel for convenience of exposition. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving force, i.e. in the σp -space, is next introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.8. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙p π in Eq. 5.61, or, alternatively, from evaluating the formal definition of the sub-differential  d˙p π(˙p ) = {σp | σp [˙p − ˙p ] ≤ σy |˙p | − |˙p | ∀˙p },

(5.63)

whereby ˙p denotes any admissible plastic strain rate. Then at ˙p = 0 it holds for any admissible ˙p that σp ˙p ≤ σy |˙p | and, with max˙p {σp ˙p /|˙p |} = |σp |, the admissible domain follows as |σp | ≤ σy .

5.2 Prandtl Model

213 σp

Fig. 5.8 Specific Prandtl model: The elastic domain for σp defined by |σp | − σy < 0 in the σp -space. The two points |σp | − σy = 0 define the yield surface. The union of the elastic domain and the yield surface renders the admissible domain |σp | − σy ≤ 0

σp = +σy +σy

0

−σy σp = −σy

The elastic domain is defined as the interior of the admissible domain, i.e. int A := {σp | |σp | − σy < 0},

(5.64)

whereas the yield surface, which in the present one-dimensional case collapses to the two end points σp = ±σy , is defined as the boundary of the admissible domain, i.e. ∂ A := {σp | |σp | − σy = 0}.

(5.65)

Collectively, the admissible domain in the σp -space is characterized by the yield condition |σp | − σy ≤ 0.

(5.66)

States in the interior int A of the admissible domain with |σp | < σy are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp | = σy are plastic. The corresponding dual dissipation potential π ∗ , as determined from the Legendre transformation π ∗ (σp ) = max{σp ˙p − σy |˙p |}

(5.67)

˙ p

then reads π ∗ (σp ) = IA (σp ) :=

⎧ ⎨ 0 ⎩

⎫ |σp | ≤ σy ⎬ for

∞

|σp | > σy

⎭

,

(5.68)

214

5 Plasticity

where IA denotes the indicator function of the admissible domain A in the σp -space. The evolution law (the associated flow rule) for the plastic strain then follows as some sub-derivative of the dual dissipation potential with respect to its conjugated variable ⎧ ⎫ 0 |σp | < σy ⎪ ⎪ ⎨ ⎬ for , (5.69) ˙p (σp ) ∈ dσp π ∗ (σp ) = dσp IA (σp ) = σ ⎪ ⎩λ p ⎭ |σp | = σy ⎪ |σp | whereby dσp π ∗ denotes the set of sub-derivatives, i.e. the sub-differential of π ∗ with respect to σp and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 5.61 and 5.69 are inverse relations. The non-smooth dissipation and dual dissipation potentials π = π(˙p ) and π ∗ = π ∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙p ) and ˙p = ˙p (σp ) are similar to those displayed in Fig. 5.3. Interestingly, the result in Eq. 5.69 can be rephrased in terms of the postulate of maximum dissipation (due to perfect plasticity) that follows from the reverse Legendre transformation π(˙p ) = max{d(σp ; ˙p ) − IA (σp )} = max{d(σp ; ˙p )}, σp

σp ∈ A

(5.70)

whereby d(σp ; ˙p ) := σp ˙p denotes the dissipation power density. The postulate of maximum dissipation can, alternatively, be recast as a variational inequality: For given ˙p , find σp ∈ A as the solution of d(σp ; ˙p ) ≥ d(σp ; ˙p ) ∀σp ∈ A,

(5.71)

whereby σp denotes any admissible plastic stress. As yet another alternative, the postulate of maximum dissipation may be reformulated as constrained optimization problem with  a Lagrange functional incorporating the admissibility constraint |σp | ≤ σy by the Lagrange multiplier λ ≥ 0  (σp , λ; ˙p ) := −d(σp ; ˙p ) + λ |σp | − σy .

(5.72)

In accordance with Eq. 5.69 the stationarity condition of this constrained optimization problem then reads ˙p = λ

σp , |σp |

(5.73)

5.2 Prandtl Model

215

Table 5.4 Summary of the specific Prandtl model (1) Strain

 =  e + p

(2) Energy

ψ =

(3) Stress

σ = E [ − p ]

1 2E

[ − p ]2 ≡

σ

≡



−σp

(4) Potential π = σy |˙p | (5) Stress

σp = σy

˙ p |˙p |

≡

σp for ˙ p = 0

or (4) Yield

0 ≥ |σp | − σy

(5) Evolution ˙ p = λ (6) KKT

λ ≥ 0,

σp |σp | |σp |

≤

σy ,

λ |σp |

=

λ σy

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format λ ≥ 0, |σp | ≤ σy , λ |σp | = λ σy .

(5.74)

Note that it follows immediately from Eq. 5.73 that |˙p | = λ. Finally, the plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated plastic deformation, i.e.  κ = κ˙ dt with κ˙ := |˙p | = λ ≥ 0. (5.75) The specific Prandtl model is summarized in Table 5.4.

5.2.2 Specific Prandtl Model: Algorithmic Update For the specific Prandtl model the evolution law for the plastic strain p is integrated by the implicit Euler backwards method to render np := np − n−1 = λ p

σpn |σpn |

,

(5.76)

whereby λ = t n λn . Consequently, the plastic stress σp is updated at the end of the time step by

216

5 Plasticity

σpn = −E [np − n ] =: σp − E np .

(5.77)

Here the trial plastic stress σp is computable exclusively from known quantities at the beginning of the time step and follows as − n ]. σp := −E [n−1 p

(5.78)

Incorporating the discretized evolution law for the plastic strain then renders σpn = σp − E λ

σpn |σpn |

.

(5.79)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress

σn p = σp . |σpn | + E λ |σpn |

(5.80)

As an immediate consequence the equivalent stress and its trial value are related via |σpn | = |σp | − E λ.

(5.81)

A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously σpn |σpn |

≡

σp |σp |

.

(5.82)

Eventually, the yield function at the end of the time step is expressed as φn := |σpn | − σy = φ − E λ.

(5.83)

Here the trial value of the yield function φ has been defined as φ := |σp | − σy .

(5.84)

Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is computed in closed form from λ =

φ  ≥ 0. E

(5.85)

Once λ is computed all other variables may be updated. In particular, the plastic stress at the end of the time step reads

5.2 Prandtl Model

217

σpn = σp − E λ

σp |σp |

.

(5.86)

The sensitivity of σpn = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n   σp σp n ∂ (λ). ∂ σp = E − E λ ∂ (5.87) − E |σp | |σp | The first derivative term on the right-hand-side computes to zero since   σp σp σp 1 E − E ≡ 0. ∂ = |σp | |σp | |σp |2 |σp |

(5.88)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-hand-side computes from requiring satisfaction of (the yield condition) ∂ φn = 0 for ongoing plastic flow at the end of the time step, i.e.

Table 5.5 Algorithmic update for the specific Prandtl model Input

n n−1 p

Trial Strain

p = n−1 p

Trial Stress

σp = −E [p − n ]

Trial Yield

φ = |σp | − σy

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E

ENDIF Update Strain np = p + λ

σp |σp |

Update Stress

σ n = E [n − np ]

Tangent

E an = E − H0 (λ) E

Output

σ n np E an

218

5 Plasticity

∂ φ − E ∂ (λ) =

σp |σp |

. E − E ∂ (λ) = 0.

(5.89)

As a conclusion the algorithmic tangent E a is thus finally expressed as E an = E − H0 (λ) E.

(5.90)

Note that, consequently, the algorithmic tangent degenerates to E a = 0 for λ > 0. In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Prandtl model is summarized in Table 5.5.

5.2.3 Specific Prandtl Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the specific Prandtl model to a prescribed Zig-Zag strain history is documented in Fig. 5.9a, b, c, d, e. (These shall be compared to the corresponding response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.4a, b, c, d, e.) Figure 5.9a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.9b showcases the resulting stress history σ(t) that displays a trapezoidal signal with σ(t) ˙ = E ˙(t) in the elastic phases where |σ(t)| < σy = 1 (E = 1, thus the slopes in the elastic phases in Fig. 5.9a, b coincide), and σ(t) ˙ = 0 in the plastic phases where |σ(t)| = σy = 1. The resulting σ = σ() diagram is highlighted in Fig. 5.9c. The expected parallelogram-type format of the σ = σ() diagram is captured exactly, whereby the slopes at  = 0 and  = ±5 obviously coincide with the elastic modulus E = 1. Figure 5.9d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| = 5, whereas p (t) stays constant with |p (t)| = 4 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.9e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 5 during the plastic phases and constant in time during the elastic phases, thus κmax = [0.5 + 4 + 0.375] × 8 = 39 (for 4.875 plastic phases of plastic arc-length 8 each).

5.2 Prandtl Model

219 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.9 Response analysis of the specific Prandtl model with material data: E = 1.0, σy = 1.0. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

220

5 Plasticity

Prescribed Strain History: Sine The response of the specific Prandtl model to a prescribed Sine strain history is documented in Fig. 5.10a, b, c, d, e. (These shall be compared to the corresponding response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.5a, b, c, d, e.) Figure 5.10a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.10b showcases the resulting stress history σ(t) that displays a periodic signal with σ(t) ˙ = ˙(t) in the elastic phases where |σ(t)| < 1 (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.10a, b are affine), and σ(t) ˙ = 0 in the plastic phases where |σ(t)| = 1. The resulting σ = σ() diagram is highlighted in Fig. 5.10c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at  = 0 and  = ±5 obviously tend to the elastic modulus E = 1 for t → 0. Figure 5.10d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)|, whereas p (t) stays constant with |p (t)| = 4 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.10e follows as a sequence of sine waves segments from integrating κ(t) ˙ = |˙p (t)| during the plastic phases and constant in time during the elastic phases, thus κmax = [0.5 + 4 + 0.375] × 8 = 39 (for 4.875 plastic phases of plastic arc-length 8 each). Prescribed Strain History: Ramp The response of the specific Prandtl model to a prescribed Ramp strain history is documented in Fig. 5.11a, b, c, d, e. (These shall be compared to the corresponding response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.6a, b, c, d, e.) Figure 5.11a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.11b showcases the resulting stress history σ(t) that displays a trapezoidal signal with σ(t) ˙ = ˙(t) in the two elastic phases where |σ(t)| < 1 (thus the slopes in the two elastic phases in Fig. 5.11a, b coincide), and σ(t) ˙ = 0 in the two plastic phases where |σ(t)| = 1. In particular during the holding phase σ(t) = 1 results as the response to the elastic strain e (t) = (t) − p (t) = 5 − 4 = 1.

5.2 Prandtl Model

221 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.10 Response analysis of the specific Prandtl model with material data: E = 1.0, σy = 1.0. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

222

5 Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 5.11 Response analysis of the specific Prandtl model with material data: E = 1.0, σy = 1.0. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

5.2 Prandtl Model

223

The resulting σ = σ() diagram is highlighted in Fig. 5.11c. The expected parallelogram-type format of the σ = σ() diagram is captured exactly, whereby the slopes at  = 0 and  = 5 obviously coincide with the elastic modulus E = 1. Figure 5.11d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| = 5 (or ˙p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 4 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.11e follows constant-linear-constantlinear in time from integrating κ(t) ˙ = |˙(t)| = {0, 5, 0, 5} over the time interval t ∈ [0, tmax = 10], thus κmax = 4 + 3 = 7.

5.2.4 Generic Prandtl Model: Formulation A generic formulation of the Prandtl model can be obtained from generalizing the specific Prandtl model in Fig. 5.7 by assuming the elastic spring or/and the frictional slider as nonlinear. For the generic Prandtl model the free energy density ψ is expressed as a nonquadratic but convex function of  − p (the elastic strain e ) ψ(, p ) = ψ( − p ).

(5.91)

Note that ψ(, p ) and ψ( − p ) are different functions that return, however, the same function value for the same values of  and p . Then the energetic stress σ  and the energetic plastic stress σp follow as σ  (, p ) = ∂ ψ(, p ) = ∂ ψ( − p ),

(5.92a)

σp (, p )

(5.92b)

= ∂p ψ(, p ) = ∂p ψ( − p ).

Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ  ≡ σ and the negative of the energetic plastic stress −σp ≡ σ. Moreover the energetic and the dissipative plastic stresses are constitutively related by σp + σp = 0, thus the notion of plastic stress defined as σp := σp = −σp will exclusively be used in the sequel. Furthermore, for the generic Prandtl model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙p ) and π ∗ = π ∗ (σp ), respectively, are related via corresponding Legendre transformations π ( ˙p ) = max{σp ˙p − π ∗ (σp )},

(5.93a)

π ∗ (σp ) = max{σp ˙p − π ( ˙p )}.

(5.93b)

σp

˙ p

224

5 Plasticity

Then the stationarity conditions corresponding to Eqs. 5.93a and 5.93b are the constitutive relations (5.94a) ˙p (σp ) ∈ dσp π ∗ (σp ), σp ( ˙p ) ∈ d ˙p π ( ˙p ).

(5.94b)

Obviously the relations in Eqs. 5.94a and 5.94b determine entirely the dissipative behavior of the generic Prandtl model, thus the formulation would be completed at this stage. To be more explicit, however, alternatively to Eq. 5.94b the closed and convex admissible domain A in the σp -space is introduced. It is characterized by the convex yield condition φ = φ(σp ) := ϕ(σp ) − σy ≤ 0.

(5.95)

Here φ = φ(σp ) is the yield function and ϕ(σp ) denotes the equivalent (plastic) stress that is compared to the yield limit σy , a material property. Then the evolution law for the plastic strain (i.e. the associated flow rule) follows alternatively to Eq. 5.94a from the postulate of maximum dissipation (due to plasticity) with  a Lagrange functional incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0 (σp , λ; ˙p ) := −d(σp ; ˙p ) + λ φ(σp ).

(5.96)

Consequently, the stationarity condition of this constrained optimization problem reads ˙p = λ ∂σp φ,

(5.97)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form λ ≥ 0, φ ≤ 0, λ φ = 0.

(5.98)

It shall be noted that collectively Eqs. 5.95, 5.97 and 5.98 are entirely equivalent statements to Eqs. 5.94a and 5.94b. As a further interesting aspect the dissipation d = σp ˙p shall next be examined more closely. From Eqs. 5.93a and 5.93b the dissipation d is alternatively expressed in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙p ) + π ∗ (σp ) ≥ 0.

(5.99)

However, based on the above introduction of the yield condition φ ≤ 0 the dual dissipation potential is identified as the indicator function IA of the admissible domain A

5.2 Prandtl Model

225

Table 5.6 Summary of the generic Prandtl model (1) Strain



(2) Energy

ψ = ψ( − p )

(3)

Stress

=  e + p

σ = ∂ ψ

≡

σ



σp

≡



−σp

(4) Potential π = π(˙p ) (5) Stress

σp ∈ d˙p π



or (4) Yield

φ = φ(σp ) ≤ 0

(5) Evolution ˙ p = λ ∂σp φ (6) KKT

λ ≥ 0, φ

≤

0, λ φ

=

π ∗ (σp ) = IA (σp ) :=

0

⎧ ⎨ 0 ⎩

φ(σp ) ≤ 0 for

∞

φ(σp ) > 0

.

(5.100)

Thus for the generic Prandtl model the dual dissipation potential equals zero in the admissible domain A. Consequently, provided the plastic stress is admissible, the dissipation is indeed expressed in terms of the dissipation potential only d = π(˙p ) ≥ 0.

(5.101)

Finally for an equivalent (plastic) stress that is homogeneous of degree one in the plastic stress (thus σp ∂σp ϕ = ϕ), the dissipation d = σp ˙p is exclusively given in terms of the Lagrange multiplier λ and the yield limit σy , since then d = λ σ p ∂σ p ϕ = λ ϕ = λ σ y .

(5.102)

The generic Prandtl model is summarized in Table 5.6.

5.3 Prandtl Hardening Model The Prandtl model of a hardening elasto-plastic solid (in short the Prandtl hardening model) consists of a serial arrangement of (1) an elastic spring and (2) a hardening frictional slider consisting of a parallel arrangement of (i) a frictional slider and (ii) a hardening spring (see the sketch of the specific Prandtl hardening model in Fig. 5.12).

226

5 Plasticity εh H E σ

σ

σy

p

e

Fig. 5.12 Specific Prandtl hardening model

The basic kinematic assumption of the Prandtl hardening model is the additive decomposition of the total strain  into the elastic strain e (representing the elongation of the elastic spring) and the plastic strain p (representing the elongation of the hardening frictional slider), i.e.  = e + p .

(5.103)

Note that the plastic strain p together with the hardening strain εh measuring the elongation of the hardening spring denote the only elements contained in the set of internal variables α = {p , εh } for the Prandtl hardening model.

5.3.1 Specific Prandtl Isotropic Hardening Model: Formulation The specific Prandtl isotropic hardening model, similar to that displayed in Fig. 5.12 (however with the hardening modulus H and the hardening strain εh coinciding here with the isotropic-hardening modulus H and the isotropic-hardening strain hi , respectively), consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear isotropic-hardening frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy and (ii) a linear isotropichardening spring with stiffness H (the isotropic-hardening modulus). For the specific Prandtl isotropic hardening model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − p (the elastic strain e ) and hi (the isotropic-hardening strain) ψ(, p , hi ) =

1 1 E [ − p ]2 + H 2hi . 2 2

(5.104)

5.3 Prandtl Hardening Model

227

Then the energetic stress σ  conjugated to the total strain  and the energetic plastic stress σp conjugated to the plastic strain p together with the isotropic-hardening  stress σhi conjugated to the isotropic-hardening strain hi follow as σ  (, p

) = ∂ ψ(, p , hi ) =

E [ − p ],

(5.105a)

σp (, p  σhi (

) = ∂p ψ(, p , hi ) = −E [ − p ],

(5.105b)

hi ) = ∂hi ψ(, p , hi ) =

H hi

.

(5.105c)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the isotropic-hardening frictional slider, also with the negative of the energetic plastic stress, −σp ≡ σ. Furthermore, for the specific Prandtl isotropic hardening model the convex but non-smooth dissipation potential π is chosen as π(˙p , ˙hi ) = [σy + H hi ] |˙p | − H hi ˙hi .

(5.106)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) π is positively homogenous of degree one in {˙p , ˙hi } and is obviously non-smooth at the origin {˙p , ˙hi } = {0, 0}. Consequently,  the dissipative plastic stress σp and the dissipative isotropic-hardening stress σhi compute as some sub-derivatives of the dissipation potential with respect to their conjugated variables σp (˙p , ˙hi ) ∈ d˙p π(˙p , ˙hi ),  σhi (˙p , ˙hi ) ∈ d˙hi π(˙p , ˙hi ),

(5.107)

with ⎧ ⎫ +[σy + H hi ] ˙p > 0 ⎬ ⎨ d˙p π(˙p , ˙hi ) = −[σy + H hi ], +[σy + H hi ] for ˙p = 0 , ⎩ ⎭ ˙p < 0 −[σy + H hi ] −H hi , d˙hi π(˙p , ˙hi ) =

(5.108)

whereby d˙p π and d˙hi π denote the sets of sub-derivatives, i.e. the sub-differentials of π with respect to ˙p and ˙hi , respectively. Recall that the energetic and the dissipative plastic as well as isotropic-hardening   + σhi = 0, respectively, stresses are constitutively related by σp + σp = 0 and σhi thus the notions of plastic stress and isotropic-hardening stress defined as the values σp := σp = −σp , σhi :=

 σhi

=

 −σhi ,

(5.109a) (5.109b)

228

5 Plasticity σp

+σy

H

σp = +σy +

hi

hi

0

−σy H

σp = −σy −

hi

Fig. 5.13 Specific Prandtl isotropic hardening model: The elastic domain for σp defined by |σp | − [σy + H hi ] < 0 in the {σp , hi }-space expands uniformly with the isotropic hardening strain hi ∈ [0, ∞). The slopes of the two lines |σp | − [σy + H hi ] = 0 defining the yield surface denote the isotropic hardening modulus H . The union of the elastic domain and the yield surface renders the admissible domain |σp | − [σy + H hi ] ≤ 0

will exclusively be used in the sequel for convenience of exposition. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving forces, i.e. in the {σp , σhi }-space, is next introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.13. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙p π in Eq. 5.108, or, alternatively, from evaluating the formal definition of the sub-differential

{σp |

σp [˙p

d˙p π(˙p , ˙hi ) =  − ˙p ] ≤ [σy + H hi ] |˙p | − |˙p | ∀˙p },

(5.110)

whereby ˙p denotes any admissible plastic strain rate. Then at ˙p = 0 it holds for any admissible ˙p that σp ˙p ≤ [σy + H hi ] |˙p | and, with max˙p {σp ˙p /|˙p |} = |σp |, the admissible domain follows as |σp | ≤ σy + H hi . Moreover, the sub-differential d˙hi π reduces to the partial derivative ∂˙hi π and renders σhi = −H hi Thus the admissible domain is eventually expressed as |σp | ≤ σy − σhi . The elastic domain is defined as the interior of the admissible domain, i.e.   int A := {σp , σhi } | |σp | − [σy − σhi ] < 0 ,

(5.111)

whereas the yield surface, which in the present one-dimensional case collapses to the two lines σp = ±[σy − σhi ], is defined as the boundary of the admissible domain, i.e.

5.3 Prandtl Hardening Model

229

  ∂ A := {σp , σhi } | |σp | − [σy − σhi ] = 0 .

(5.112)

Collectively, the admissible domain in the {σp , σhi }-space is characterized by the yield condition |σp | − [σy − σhi ] ≤ 0.

(5.113)

States in the interior int A of the admissible domain with |σp | < σy − σhi are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp | = σy − σhi are plastic. The corresponding dual dissipation potential π ∗ , as determined from the Legendre transformation π ∗ (σp , σhi ) = max{σp ˙p + σhi ˙hi − [σy + H hi ] |˙p | + H hi ˙hi } ˙ p ,˙hi

(5.114)

then reads with the stationarity condition σhi = −H hi (note the minus sign) ⎧ ⎫ |σp | ≤ σy − σhi ⎬ ⎨ 0 for π ∗ (σp , σhi ) = IA (σp , σhi ) := , (5.115) ⎩ ⎭ ∞ |σp | > σy − σhi where IA denotes the indicator function of the admissible domain A in the {σp , σhi }space. The evolution laws (the associated flow rules) for the plastic and the isotropichardening strains then follow as some sub-derivatives of the dual dissipation potential with respect to their conjugated variables ˙p (σp , σhi ) ∈ dσp π ∗ (σp , σhi ) = dσp IA (σp , σhi ), ˙hi (σp , σhi ) ∈ dσhi π ∗ (σp , σhi ) = dσhi IA (σp , σhi ), with

⎧ ⎪ ⎨

(5.116)

⎫ |σp | < σy − σhi ⎪ ⎬

0

dσp π ∗ (σp , σhi ) for = dσp IA (σp , σhi ) = ⎩ ⎪ λ σp ⎭ |σp | = σy − σhi ⎪ |σp |

(5.117a)

and ∗

⎧ ⎨0

dσhi π (σp , σhi ) = dσhi IA (σp , σhi ) = ⎩

⎫ |σp | < σy − σhi ⎬ for

λ

|σp | = σy − σhi

⎭

,

(5.117b)

whereby dσp π ∗ and dσhi π ∗ denote the sets of sub-derivatives, i.e. the sub-differentials of π ∗ with respect to σp and σhi , respectively, and λ is a positive Lagrange (or rather plastic) multiplier.

230

5 Plasticity

Obviously, the expressions in Eqs. 5.107 and 5.116 are inverse relations. Identifying ˙hi with |˙p | and setting σhi = 0, the remaining non-smooth dissipation and dual dissipation potentials π = π(˙p ) and π ∗ = π ∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙p ) and ˙p = ˙p (σp ) are similar to those displayed in Fig. 5.3. Interestingly, the result in Eq. 5.116 can be rephrased in terms of the postulate of maximum dissipation (due to isotropic-hardening plasticity) that follows from the reverse Legendre transformation π(˙p , ˙hi )) = =

max

{d(σp , σhi ; ˙p , ˙hi ) − IA (σp , σhi )}

max

{d(σp , σhi ; ˙p , ˙hi )},

σp ,σhi

{σp ,σhi }∈ A

(5.118)

whereby d(σp , σhi ; ˙p , ˙hi ) := σp ˙p + σhi ˙hi denotes the dissipation power density. The postulate of maximum dissipation can, alternatively, be recast as a variational inequality: For given {˙p , ˙hi }, find {σp , σhi } ∈ A as the solution of   ; ˙p , ˙hi ) ∀{σp , σhi } ∈ A, d(σp , σhi ; ˙p , ˙hi ) ≥ d(σp , σhi

(5.119)

 } denote any admissible plastic and isotropic-hardening stress. As whereby {σp , σhi yet another alternative, the postulate of maximum dissipation may be reformulated as constrained optimization problem with  a Lagrange functional incorporating the admissibility constraint |σp | ≤ [σy − σhi ] by the Lagrange multiplier λ ≥ 0

 (σp , σhi , λ; ˙p , ˙hi ) := −d(σp , σhi ; ˙p , ˙hi ) + λ |σp | − [σy − σhi ] .

(5.120)

In accordance with Eq. 5.116 the stationarity conditions of this constrained optimization problem then read ˙p = λ

σp and ˙hi = λ, |σp |

(5.121)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format λ ≥ 0, |σp | ≤ [σy − σhi ], λ |σp | = λ [σy − σhi ].

(5.122)

Note that it follows immediately from Eq. 5.121 that |˙p | = ˙hi = λ. Finally, the plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated plastic deformation, i.e.  κ = κ˙ dt with κ˙ := |˙p | = ˙hi = λ ≥ 0. (5.123)

5.3 Prandtl Hardening Model

231

Table 5.7 Summary of the specific Prandtl isotropic hardening model (1) Strain



(2) Energy

ψ =

(3) Stress

σ

(4) Stress

σhi = −H hi

(5) Potential π

=  e + p 1 2E

[ − p ]2 + 21 H 2hi

= E [ − p ]

σ

≡

=



−σp

= σyhi |˙p | − H hi ˙ hi with σyhi := σy + H hi ˙ p |˙p |

(6) Stress

σp = σyhi

(7) Stress

σhi = −H hi

σp for ˙ p = 0

≡

or (5) Yield

0

≥ |σp | − σyhi

(6) Evolution ˙ p = λ

σp |σp |

(7) Evolution ˙ hi = λ (8) KKT

λ

≥ 0,

|σp |

≤

σyhi ,

λ |σp |

=

λ σyhi

The specific Prandtl isotropic hardening model is summarized in Table 5.7.

5.3.2 Specific Prandtl Isotropic Hardening Model: Algorithmic Update For the specific Prandtl isotropic hardening model the evolution laws for the plastic strain p and the isotropic-hardening strain hi are integrated by the implicit Euler backwards method to render

np := np − n−1 = λ p

σpn |σpn |

and nhi := nhi − n−1 = λ, hi

(5.124)

whereby λ = t n λn . Consequently, the plastic stress σp and the isotropichardening stress σhi are updated at the end of the time step by

232

5 Plasticity

σpn = −E [np − n ] =: σp − E np , n σhi

= −H

nhi

=:

 σhi

−H

(5.125)

nhi .

 are comHere the trial plastic stress σp and the trial isotropic-hardening stress σhi putable exclusively from known quantities at the beginning of the time step and follow as

σp := −E [n−1 − n ], p  σhi

:= −H

n−1 hi

(5.126)

.

Incorporating the discretized evolution law for the plastic strain then renders σpn = σp − E λ

σpn |σpn |

.

(5.127)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stresses

σn p = σp . |σpn | + E λ |σpn |

(5.128)

As an immediate consequence the equivalent stress and its trial value are related via |σpn | = |σp | − E λ.

(5.129)

A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously σpn |σpn |

≡

σp |σp |

.

(5.130)

Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore n  = σhi − H λ. σhi

(5.131)

Consequently, the yield function at the end of the time step is expressed as n = φ − [E + H ] λ. φn := |σpn | − σy + σhi

(5.132)

Here the trial value of the yield function φ has been defined as  . φ := |σp | − σy + σhi

(5.133)

5.3 Prandtl Hardening Model

233

Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is computed in closed form from λ =

φ  ≥ 0. E+H

(5.134)

Once λ is computed all other variables may be updated. In particular, the plastic stress at the end of the time step reads σpn = σp − E λ

σp |σp |

.

(5.135)

The sensitivity of σpn = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n   σp σp n ∂ (λ). ∂ σp = E − E λ ∂ (5.136) − E |σp | |σp | The first derivative term on the right-hand-side computes to zero since   σp σp σp 1 E − E ≡0 ∂ = |σp | |σp | |σp |2 |σp |

(5.137)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-hand-side computes from requiring satisfaction of (the yield condition) ∂ φn = 0 for ongoing plastic flow at the end of the time step, i.e. ∂ φ − [E + H ] ∂ (λ) =

σp |σp |

. E − [E + H ] ∂ (λ) = 0.

(5.138)

As a conclusion the algorithmic tangent E a is thus finally expressed as E an = E − H0 (λ)

E2 . E+H

(5.139)

In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Prandtl hardening model capturing isotropic hardening is summarized in Table 5.8.

234

5 Plasticity

Table 5.8 Algorithmic update for the specific Prandtl isotropic hardening model Input

n n−1 n−1 p hi

Trial Strain

p = n−1 p hi = n−1 hi

Trial Stress

σp = −E [p − n ]  = −H  σhi hi

Trial Yield

φ



 = |σp | − σy + σhi

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E+H

ENDIF Update Strain np = p + λ hi

σp |σp |

nhi

=

Update Stress

σn

= E [n − np ]

Tangent

E an = E − H0 (λ)

Output

σ n np nhi E an

+ λ

E2 E+H

5.3.3 Specific Prandtl Isotropic Hardening Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Prandtl isotropic hardening model to a prescribed ZigZag strain history is documented in Fig. 5.14a, b, c, d, e. Figure 5.14a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.14b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) ˙ = E ˙(t) in the elastic phases where |σ(t)| < σy + H κ(t) = 1 + 0.1 κ (E = 1, thus the slopes in the elastic phases in Fig. 5.14a, b

5.3 Prandtl Hardening Model

235 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.14 Response analysis of the specific Prandtl isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

236

5 Plasticity

coincide), and σ(t) ˙ = [E − E 2 /[E + H ]] ˙(t) = ˙(t)/11 in the plastic phases where |σ(t)| = σy + H κ(t) = 1 + 0.1 κ(t). The resulting σ = σ() diagram is highlighted in Fig. 5.14c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format (contracting in the  direction and isotropically expanding in the σ direction) of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at  = 0 and  = ±5 obviously tend to the elastic modulus E = 1 for t → 0. It is tedious but easy to verify that the stress varies between [0, 1], ±15/11, ∓245/112 , ±3415/113 , ∓44045/114 and [542815/115 , −262440/115 ] in the elastic phases. Figure 5.14d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + H ] = 5 × 10/11, whereas p (t) stays constant with |p (t)| = {0, 40/11, 360/112 , 3240/113 , 29160/114 , 262440/115 } during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.14e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 5 × 10/11 during the plastic phases and constant in time during the elastic phases, thus κmax = 2 × [40/11 + 360/112 + 3240/113 + 29160/114 ] + 262440/115 = 3817640/115 ≈ 24. Prescribed Strain History: Sine The response of the specific Prandtl isotropic hardening model to a prescribed Sine strain history is documented in Fig. 5.15a, b, c, d, e. Figure 5.15a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.15b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) ˙ = ˙(t) in the elastic phases where |σ(t)| < 1 + 0.1 κ(t) (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.15a, b are affine), and σ(t) ˙ = ˙(t)/11 in the plastic phases where |σ(t)| = 1 + 0.1 κ(t). The resulting σ = σ() diagram is highlighted in Fig. 5.15c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format (contracting in the  direction and isotropically expanding in the σ direction) of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at  = 0 and  = ±5 obviously tend to the elastic modulus E = 1 for t → 0. It is tedious but easy to verify that the stress varies between [0, 1], ±15/11, ∓245/112 , ±3415/113 , ∓44045/114 and [542815/115 , −262440/115 ] in the elastic phases. Figure 5.15d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + H ] = |˙(t)| ×

5.3 Prandtl Hardening Model

237 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.15 Response analysis of the specific Prandtl isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

238

5 Plasticity

10/11, whereas p (t) stays constant with |p (t)| = {0, 40/11, 360/112 , 3240/113 , 29160/114 , 262440/115 } during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.15e follows as a sequence of sine waves segments from integrating κ(t) ˙ = |˙p (t)| = |˙(t)| × 10/11 during the plastic phases and constant in time during the elastic phases, thus κmax = 2 × [40/11 + 360/112 + 3240/113 + 29160/114 ] + 262440/115 = 3817640/115 ≈ 24. Prescribed Strain History: Ramp The response of the specific Prandtl isotropic hardening model to a prescribed Ramp strain history is documented in Fig. 5.16a, b, c, d, e. Figure 5.16a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.16b showcases the resulting stress history σ(t) with σ(t) ˙ = ˙(t) in the two elastic phases where |σ(t)| < 1 + 0.1 κ(t) (thus the slopes in the two elastic phases in Fig. 5.16a, b coincide), and σ(t) ˙ = ˙(t)/11 in the two plastic phases where |σ(t)| = 1 + 0.1 κ(t). In particular during the holding phase σ(t) = 15/11 results as the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/11 = 15/11. The resulting σ = σ() diagram is highlighted in Fig. 5.16c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format (isotropically expanding in the σ direction) of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at  = 0 and  = 5 obviously tend to the elastic modulus E = 1 for t → 0. Figure 5.16d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + H ] = 50/11 (or ˙p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 0.8 × 50/11 = 40/11 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.16e follows constant-linear-constantlinear in time from integrating κ(t) ˙ = |˙p (t)| = {0, 50/11, 0, 50/11} over the time interval t ∈ [0, tmax = 10], thus κmax = 40/11 + 250/112 = 690/112 ≈ 5.7. Prescribed Stress History: Zig-Zag The response of the specific Prandtl isotropic hardening model to a prescribed ZigZag stress history is documented in Fig. 5.17a, b, c, d, e. Figure 5.17a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles.

5.3 Prandtl Hardening Model

239 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 5.16 Response analysis of the specific Prandtl isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

240

5 Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 5.17 Response analysis of the specific Prandtl isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

241

Figure 5.17b showcases the resulting strain history (t) that displays a periodic signal with ˙(t) = σ(t)/E ˙ = σ(t)/1 ˙ (i.e. a purely elastic phase) after the initial elastic and the subsequent plastic phase with ˙(t) = σ(t)/E ˙ = σ(t)/1 ˙ and ˙(t) = ˙ × 11, respectively. Thus (1) = σy /1 + [σa − σy ] σ(t)/[E ˙ − E 2 /[E + H ]] = σ(t) × 11 = 1/1 + 4 × 11 = 45 at the end of the plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 5.17c. Once the plastic phase is completed the cyclic σ = σ() behavior in the remaining elastic phase is linear elastic with slope E = 1. It is easy to verify that the strain varies between [45, 35] in the remaining elastic phase. Figure 5.17d demonstrates the plastic strain history p (t): during the plastic phase ˙ = 55 − 5/1 = 50, whereas p (t) stays p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E| constant with p (t) = 40 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.17e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 50 during the plastic phase and constant in time during the elastic phases, thus κmax = 40. Prescribed Stress History: Sine The response of the specific Prandtl isotropic hardening model to a prescribed Sine stress history is documented in Fig. 5.18a, b, c, d, e. Figure 5.18a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.18b showcases the resulting strain history (t) that displays a periodic signal with ˙(t) = σ(t) ˙ (i.e. a purely elastic phase) after the initial elastic and the subsequent plastic phase with ˙(t) = σ(t) ˙ and ˙(t) = σ(t) ˙ × 11, respectively. Thus (1) = 1/1 + 4 × 11 = 45 at the end of the plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 5.18c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected σ = σ() behavior is only approximately captured in the elastic-plastic transition. Once the plastic phase is completed the cyclic σ = σ() behavior in the remaining elastic phase is linear elastic with slope E = 1. It is easy to verify that the strain varies between [45, 35] in the remaining elastic phase. Figure 5.18d demonstrates the plastic strain history p (t): during the plastic ˙ whereas p (t) stays constant with phase p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E|, p (t) = 40 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.18e follows as sinusoidal in time from integrating κ(t) ˙ = |˙p (t)| during the plastic phase and constant in time during the elastic phases, thus κmax = 40.

242

5 Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 5.18 Response analysis of the specific Prandtl isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

243

Prescribed Stress History: Ramp The response of the specific Prandtl isotropic hardening model to a prescribed Ramp stress history is documented in Fig. 5.19a, b, c, d, e. Figure 5.19a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.19b showcases the resulting strain history (t) that displays an initial elastic phase with ˙(t) = σ(t), ˙ a subsequent plastic phase with ˙(t) = σ(t) ˙ × 11 (thus in particular during the holding phase (t) = 1/1 + 4 × 11 = 45), and a final elastic phase with ˙(t) = σ(t), ˙ respectively. The resulting σ = σ() diagram is highlighted in Fig. 5.19c. Once the holding phase is completed the σ = σ() behavior in the unloading phase is linear elastic with slope E = 1. It is easy to verify that the strain varies between [45, 40] in the final elastic phase. Figure 5.19d demonstrates the plastic strain history p (t): during the plastic phase ˙ = 55 − 5/1 = 50, whereas p (t) stays p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E| constant with p (t) = 40 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.19e follows constant-linear-constant in time from integrating κ(t) ˙ = |˙p (t)| = {0, 50, 0} over the time interval t ∈ [0, tmax = 10], thus κmax = 40.

5.3.4 Specific Prandtl Kinematic Hardening Model: Formulation The specific Prandtl kinematic hardening model, similar to that displayed in Fig. 5.12 (however with the hardening modulus H and the hardening strain εh coinciding here with the kinematic-hardening modulus K and the kinematic-hardening strain hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear kinematic-hardening frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy and (ii) a linear kinematic-hardening spring with stiffness K (the kinematic-hardening modulus). For the specific Prandtl kinematic hardening model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − p (i.e. the elastic strain e ) and hk (the kinematic-hardening strain) ψ(, p , hk ) =

1 1 E [ − p ]2 + K 2hk . 2 2

(5.140)

Then the energetic stress σ  conjugated to the total strain  and the energetic plastic stress σp conjugated to the plastic strain p together with the kinematic-hardening

244

5 Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 80.0)

Fig. 5.19 Response analysis of the specific Prandtl isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

245





stress σhk conjugated to the kinematic-hardening strain hk follow as σ  (, p

) = ∂ ψ(, p , hk ) =

E [ − p ],

(5.141a)

σp (, p  σhk (

) = ∂p ψ(, p , hk ) = −E [ − p ],

(5.141b)

hk ) = ∂hk ψ(, p , hk ) =

K hk

.

(5.141c)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the kinematic-hardening frictional slider, also with the negative of the energetic plastic stress, −σp ≡ σ. Furthermore, for the specific Prandtl kinematic hardening model the convex but non-smooth dissipation potential π is chosen as π(˙p , ˙hk ) = σy |˙p | + K hk [˙p − ˙hk ].

(5.142)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) π is positively homogenous of degree one in {˙p , ˙hk } and is obviously non-smooth at the origin {˙p , ˙hk } = {0, 0}. Consequently,  the dissipative plastic stress σp and the dissipative kinematic-hardening stress σhk compute as some sub-derivatives of the dissipation potential with respect to their conjugated variables σp (˙p , ˙hk ) ∈ d˙p π(˙p , ˙hk ),  σhk (˙p , ˙hk ) ∈ d˙hk π(˙p , ˙hk ),

(5.143)

with ⎧ ⎫ +[σy + K hk ] ˙p > 0 ⎬ ⎨ d˙p π(˙p , ˙hk ) = −[σy − K hk ], +[σy + K hk ] for ˙p = 0 , ⎩ ⎭ ˙p < 0 −[σy − K hk ] −K hk , d˙hk π(˙p , ˙hk ) =

(5.144)

whereby d˙p π and d˙hk π denote the sets of sub-derivatives, i.e. the sub-differentials of π with respect to ˙p and ˙hk , respectively. Recall that the energetic and the dissipative plastic as well as kinematic-hardening   stresses are constitutively related by σp + σp = 0 and σhk + σhk = 0, respectively, thus the notions of plastic stress and isotropic-hardening stress defined as the values σp := σp = −σp , σhk :=

 σhk

=

 −σhk ,

will exclusively be used in the sequel for convenience of exposition.

(5.145a) (5.145b)

246

5 Plasticity σp

+σy

0

−σy

K σp = +σy +

hk

hk

K σp = −σy +

hk

Fig. 5.20 Specific Prandtl kinematic hardening model: The elastic domain for σp defined by |σp − K hk | − σy < 0 in the {σp , hk }-space shifts with the kinematic hardening strain hk ∈ (−∞, +∞). The slope of the two lines |σp − K hk | − σy = 0 defining the yield surface denotes the kinematic hardening modulus K . The union of the elastic domain and the yield surface renders the admissible domain |σp − K hk | − σy ≤ 0

The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving forces, i.e. in the {σp , σhk }-space, is introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.20. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙p π in Eq. 5.144, or, alternatively, from evaluating the formal definition of the sub-differential

{σp |

σp [˙p

d˙p π(˙p , ˙hk ) =  − ˙p ] ≤ σy |˙p | − |˙p | + K hk [˙p − ˙p ] ∀˙p },

(5.146)

whereby ˙p denotes any admissible plastic strain rate. Then at ˙p = 0 it holds for any admissible ˙p that σp ˙p ≤ σy |˙p | + K hk ˙p and, with max˙p {[σp − K hk ] ˙p /|˙p |} = |σp − K hk |, the admissible domain follows as |σp − K hk | ≤ σy . Moreover, the sub-differential d˙hk π reduces to the partial derivative ∂˙hk π and renders σhk = −K hk . Thus the admissible domain is eventually expressed as |σp + σhk | ≤ σy . The elastic domain is defined as the interior of the admissible domain, i.e.   int A := {σp , σhk } | |σp + σhk | − σy < 0 ,

(5.147)

whereas the yield surface, which in the present one-dimensional case collapses to the two lines σp + σhk = ±σy , is defined as the boundary of the admissible domain, i.e.

5.3 Prandtl Hardening Model

247

  ∂ A := {σp , σhk } | |σp + σhk | − σy = 0 ,

(5.148)

Collectively, the admissible domain in the {σp , σhk }-space is characterized by the yield condition |σp + σhk | − σy ≤ 0.

(5.149)

States in the interior int A of the admissible domain with |σp + σhk | < σy are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp + σhk | = σy are plastic. The corresponding dual dissipation potential π ∗ , as determined from the Legendre transformation π ∗ (σp , σhk ) = max{σp ˙p + σhk ˙hk − σy |˙p | − K hk [˙p − ˙hk ]} ˙ p ,˙hk

(5.150)

then reads with the stationarity condition σhk = −K hk (note the minus sign) ⎧ ⎫ |σp + σhk | ≤ σy ⎬ ⎨ 0 for π ∗ (σp , σhk ) = IA (σp , σhk ) := , (5.151) ⎩ ⎭ ∞ |σp + σhk | > σy where IA denotes the indicator function of the admissible domain A in the {σp , σhk }space. The evolution laws (the associated flow rules) for the plastic and the kinematichardening strains then follow as some sub-derivatives of the dual dissipation potential with respect to their conjugated variables ˙p (σp , σhk ) ∈ dσp π ∗ (σp , σhk ) = dσp IA (σp , σhk ), ˙hk (σp , σhk ) ∈ dσhk π ∗ (σp , σhk ) = dσhk IA (σp , σhi ),

(5.152)

with ⎧ ⎪ ⎪ ⎨

0

⎫ |σp + σhk | < σy ⎪ ⎪ ⎬

dσp π ∗ (σp , σhk ) for σp + σhk = dσp IA (σp , σhk ) = ⎪ ⎪ ⎪ |σp + σhk | = σy ⎪ ⎭ ⎩λ |σp + σhk |

(5.153a)

and ⎧ ⎪ ⎪ ⎨

0

⎫ |σp + σhk | < σy ⎪ ⎪ ⎬

dσhk π ∗ (σp , σhk ) for , σp + σhk = dσhk IA (σp , σhh ) = ⎪ ⎪ ⎪ ⎪ |σ + σ | = σ λ p hk y⎭ ⎩ |σp + σhk |

(5.153b)

248

5 Plasticity

whereby dσp π ∗ and dσhk π ∗ denote the sets of sub-derivatives, i.e. the sub-differentials of π ∗ with respect to σp and σhk , respectively, and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 5.143 and 5.152 are inverse relations. Identifying ˙hk with ˙p and setting σhk = 0, the remaining non-smooth dissipation and dual dissipation potentials π = π(˙p ) and π ∗ = π ∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙p ) and ˙p = ˙p (σp ) are similar to those displayed in Fig. 5.3. Interestingly, the result in Eq. 5.152 can be rephrased in terms of the postulate of maximum dissipation (due to kinematic-hardening plasticity) that follows from the reverse Legendre transformation π(˙p , ˙hk )) = =

max

{d(σp , σhk ; ˙p , ˙hk ) − IA (σp , σhk )}

max

{d(σp , σhk ; ˙p , ˙hk )},

σp ,σhk (σp ,σhk )∈ A

(5.154)

whereby d(σp , σhk ; ˙p , ˙hk ) := σp ˙p + σhk ˙hk denotes the dissipation power density. The postulate of maximum dissipation can, alternatively, be recast as a variational inequality: For given {˙p , ˙hk }, find {σp , σhk } ∈ A as the solution of   ; ˙p , ˙hk ) ∀{σp , σhk } ∈ A, d(σp , σhk ; ˙p , ˙hk ) ≥ d(σp , σhk

(5.155)

 } denote any admissible plastic and kinematic-hardening stress. As whereby {σp , σhk yet another alternative, the postulate of maximum dissipation may be reformulated as constrained optimization problem with  a Lagrange functional incorporating the admissibility constraint |σp + σhk | ≤ σy by the Lagrange multiplier λ ≥ 0

 (σp , σhk , λ; ˙p , ˙hk ) := −d(σp , σhk ; ˙p , ˙hk ) + λ |σp + σhk | − σy .

(5.156)

In accordance with Eq. 5.152 the stationarity conditions of this constrained optimization problem then read ˙p = λ

σp + σhk and |σp + σhk |

˙hk = λ

σp + σhk , |σp + σhk |

(5.157)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format λ ≥ 0, |σp + σhk | ≤ σy , λ |σp + σhk | = λ σy .

(5.158)

Note that it follows immediately from Eq. 5.157 that |˙p | = |˙hk | = λ. Finally, the plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated plastic deformation, i.e.

5.3 Prandtl Hardening Model

249

Table 5.9 Summary of the specific Prandtl kinematic hardening model (1) Strain



=  e + p

(2) Energy

ψ

=

(3) Stress

σ

= E [ − p ]

(4) Stress

σhk = −K hk

(5) Potential π

1 2E

[ − p ]2 + 21 K 2hk σ

≡

≡



−σp

= σy |˙p | + K hk [˙p − ˙ hk ] ˙ p + K hk |˙p |

(6) Stress

σp = σy

(7) Stress

σhk = −K hk

≡

σp for ˙ p = 0

or (5) Yield

0

≥ |σphk | − σy with σphk := σp − K hk σphk

(6) Evolution ˙ p = λ

|σphk | σphk

(7) Evolution ˙ hk = λ (8) KKT

λ

|σphk |

≥ 0,

|σphk |

≤

σy ,

λ |σphk |

=

λ σy

 κ=

κ˙ dt with κ˙ := |˙p | = |˙hk | = λ ≥ 0.

(5.159)

The specific Prandtl kinematic hardening model is summarized in Table 5.9.

5.3.5 Specific Prandtl Kinematic Hardening Model: Algorithmic Update For the specific Prandtl kinematic hardening model the evolution laws for the plastic strain p and the kinematic-hardening strain hk are integrated by the implicit Euler backwards method to render = λ np := np − n−1 p

n σpn + σhk n |σpn + σhk |

n = nhk − n−1 hk =: hk ,

(5.160)

whereby λ = t n λn . Consequently, the plastic stress σp and the kinematichardening stress σhk are updated at the end of the time step by

250

5 Plasticity

σpn = −E [np − n ] =: σp − E np , n σhk

= −K

nhk

=:

 σhk

−K

(5.161)

nhk .

 are comHere the trial plastic stress σp and the trial kinematic-hardening stress σhk putable exclusively from known quantities at the beginning of the time step and follow as

− n ], σp := −E [n−1 p  σhk

:= −K

n−1 hk

(5.162)

.

Combining the plastic stress and the kinematic-hardening stress at the end of the time step and incorporating the discretized evolution laws for the plastic strain and the kinematic-hardening strain then renders n  = σp + σhk − [E + K ] λ σpn + σhk

n σpn + σhk n |σpn + σhk |

.

(5.163)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stresses

σn + σn p hk n  = σp + σhk | + [E + K ] λ . |σpn + σhk n |σpn + σhk |

(5.164)

As an immediate consequence the equivalent stress and its trial value are related via n  | = |σp + σhk | − [E + K ] λ. |σpn + σhk

(5.165)

A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously n σpn + σhk n |σpn + σhk |

≡

 σp + σhk  |σp + σhk |

.

(5.166)

Eventually, the yield function at the end of the time step is expressed as n | − σy = φ − [E + K ] λ. φn := |σpn + σhk

(5.167)

Here the trial value of the yield function φ has been defined as  | − σy . φ := |σp + σhk

(5.168)

Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is computed in closed form from

5.3 Prandtl Hardening Model

251

λ =

φ  ≥ 0. E+K

(5.169)

Once λ is computed all other variables may be updated. In particular, the plastic stress at the end of the time step reads σpn = σp − E λ

 σp + σhk  |σp + σhk |

.

(5.170)

The sensitivity of σpn = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n  ∂ σpn = E − E λ ∂

 σp + σhk

|σp +



 σhk |

−E

 σp + σhk  |σp + σhk |

∂ (λ).

(5.171)

The first derivative term on the right-hand-side computes to zero since  ∂

 σp + σhk  |σp + σhk |

 =

  σp + σhk σp + σhk 1 E −   2  E ≡0 |σp + σhk | |σp + σhk | |σp + σhk |

(5.172)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-hand-side computes from requiring satisfaction of (the yield condition) ∂ φn = 0 for ongoing plastic flow at the end of the time step, i.e. ∂ φ − [E + K ] ∂ (λ) =

 σp + σhk  |σp + σhk |

. E − [E + K ] ∂ (λ) = 0.

(5.173)

As a conclusion the algorithmic tangent E a is thus finally expressed as E an = E − H0 (λ)

E2 . E+K

(5.174)

In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Prandtl hardening model capturing kinematic hardening is summarized in Table 5.10.

252

5 Plasticity

Table 5.10 Algorithmic update for the specific Prandtl kinematic hardening model Input

n n−1 n−1 p hk

Trial Strain

p = n−1 p hk = n−1 hk

Trial Stress

σp = −E [p − n ]  = −K  σhk hk

Trial Yield

φ



 |−σ = |σp + σhk y

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E+K

ENDIF Update Strain np = p + λ

 σp + σhk

 | |σp + σhk

nhk = hk + λ

 σp + σhk

 | |σp + σhk

Update Stress

σ n = E [n − np ]

Tangent

E an = E − H0 (λ)

Output

σ n np nhk E an

E2 E+K

5.3.6 Specific Prandtl Kinematic Hardening Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Prandtl kinematic hardening model to a prescribed ZigZag strain history is documented in Fig. 5.21a, b, c, d, e. Figure 5.21a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles.

5.3 Prandtl Hardening Model

253 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.21 Response analysis of the specific Prandtl kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

254

5 Plasticity

Figure 5.21b showcases the resulting stress history σ(t) that displays a periodic signal with σ(t) ˙ = E ˙(t) in the elastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| < σy = 1 (E = 1, thus the slopes in the elastic phases in Fig. 5.21a, b coincide), and σ(t) ˙ = [E − E 2 /[E + K ]] ˙(t) = ˙(t)/11 in the plastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = σy = 1. The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in Fig. 5.21c. It is easy to verify that the stress varies between [0, 1] , ±15/11 and ∓7/11 in the elastic phases. Figure 5.21d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + K ] = 5 × 10/11, whereas p (t) stays constant with |p (t)| = 40/11 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.21e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 5 × 10/11 during the plastic phases and constant in time during the elastic phases, thus κmax = [0.5 + 4 + 0.375] × 80/11 = 390/11 (for 4.875 plastic phases of plastic arc-length 80/11 each). Prescribed Strain History: Sine The response of the specific Prandtl kinematic hardening model to a prescribed Sine strain history is documented in Fig. 5.22a, b, c, d, e. Figure 5.22a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.22b showcases the resulting stress history σ(t) that displays a periodic signal with σ(t) ˙ = ˙(t) in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.22a, b are affine), and σ(t) ˙ = ˙(t)/11 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1. The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in Fig. 5.22c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected σ = σ() behavior is only approximately captured in the elastic-plastic transition. It is easy to verify that the stress varies between [0, 1], ±15/11 and ∓7/11 in the elastic phases. Figure 5.22d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + K ] = |˙(t)| × 10/11, whereas p (t) stays constant with |p (t)| = 40/11 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.22e follows as a sequence of sine waves segments from integrating κ(t) ˙ = |˙p (t)| = |˙(t)| × 10/11 during the plastic phases and constant in time during the elastic phases, thus κmax = [0.5 + 4 + 0.375] × 80/11 = 390/11 (for 4.875 plastic phases of plastic arc-length 80/11 each).

5.3 Prandtl Hardening Model

255 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.22 Response analysis of the specific Prandtl kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

256

5 Plasticity

Prescribed Strain History: Ramp The response of the specific Prandtl kinematic hardening model to a prescribed Ramp strain history is documented in Fig. 5.23a, b, c, d, e. Figure 5.23a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.23b showcases the resulting stress history σ(t) with σ(t) ˙ = ˙(t) in the two elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the slopes in the two elastic phases in Fig. 5.23a, b coincide), and σ(t) ˙ = ˙(t)/11 in the two plastic phases where |σ(t) − 0.1 κ(t)| = 1. In particular during the holding phase σ(t) = 15/11 results as the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/11 = 15/11. The resulting parallelogram-type σ = σ() diagram is highlighted in Fig. 5.23c. It is easy to verify that the stress varies between 15/11 and −7/11 in the second elastic phase. Figure 5.23d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + K ] = 50/11 (or ˙p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 40/11 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.23e follows constant-linear-constantlinear in time from integrating κ(t) ˙ = |˙p (t)| = {0, 50/11, 0, 50/11} over the time interval t ∈ [0, tmax = 10], thus κmax = 40/11 + 30/11 = 70/11. Prescribed Stress History: Zig-Zag The response of the specific Prandtl kinematic hardening model to a prescribed ZigZag stress history is documented in Fig. 5.24a, b, c, d, e. Figure 5.24a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.24b showcases the resulting strain history (t) that displays a periodic signal with ˙(t) = σ(t)/E ˙ = σ(t)/1 ˙ in the elastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| < σy = 1 (E = 1, thus the slopes in the elastic phases in Fig. 5.24a, ˙ × 11 in the plastic phases b coincide), and ˙(t) = σ(t)/[E ˙ − E 2 /[E + K ]] = σ(t) where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = σy = 1. Thus a = σy /1 + [σa − σy ] × 11 = 1/1 + 4 × 11 = 45 denotes the corresponding strain amplitude. The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in Fig. 5.24c. It is easy to verify that the strain varies between [0, 1], and ±[45, 43] in the elastic phases.

5.3 Prandtl Hardening Model

257 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 5.23 Response analysis of the specific Prandtl kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

258

5 Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 5.24 Response analysis of the specific Prandtl kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

259

Figure 5.24d demonstrates the plastic strain history p (t): during the plastic phase ˙ = 55 − 5/1 = 50, whereas p (t) stays p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E| constant with |p (t)| = 40 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.24e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 50 during the plastic phases and constant in time during the elastic phases, thus κmax = [0.5 + 4 + 0.375] × 80 = 390 (for 4.875 plastic phases of plastic arc-length 80 each). Prescribed Stress History: Sine The response of the specific Prandtl kinematic hardening model to a prescribed Sine stress history is documented in Fig. 5.25a, b, c, d, e. Figure 5.25a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.25b showcases the resulting strain history (t) that displays a periodic signal with ˙(t) = σ(t) ˙ in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.25a, b are affine), and ˙(t) = σ(t) ˙ × 11 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1. Thus a = 45 denotes the corresponding strain amplitude. The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in Fig. 5.25c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected σ = σ() behavior is only approximately captured in the elastic-plastic transition. It is easy to verify that the strain varies between [0, 1], and ±[45, 43] in the elastic phases. Figure 5.25d demonstrates the plastic strain history p (t): during the plastic ˙ whereas p (t) stays constant with phase p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E|, |p (t)| = 40 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.25e follows as sinusoidal in time from integrating κ(t) ˙ = |˙p (t)| during the plastic phases and constant in time during the elastic phases, thus κmax = [0.5 + 4 + 0.375] × 80 = 390 (for 4.875 plastic phases of plastic arc-length 80 each). Prescribed Stress History: Ramp The response of the specific Prandtl kinematic hardening model to a prescribed Ramp stress history is documented in Fig. 5.26a, b, c, d, e. Figure 5.26a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps

260

5 Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 5.25 Response analysis of the specific Prandtl kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

261

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 80.0)

Fig. 5.26 Response analysis of the specific Prandtl kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

262

5 Plasticity

with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.26b showcases the resulting strain history (t) that displays an initial elastic phase with ˙(t) = σ(t), ˙ a subsequent plastic phase with ˙(t) = σ(t) ˙ × 11 (thus in particular during the holding phase (t) = 1/1 + 4 × 11 = 45), a second elastic phase with ˙(t) = σ(t), ˙ and a final plastic phase with ˙(t) = σ(t) ˙ × 11, respectively. The resulting parallelogram-type σ = σ() diagram is highlighted in Fig. 5.26c. It is easy to verify that the strain varies between 45 and 43 in the second elastic phase (and between 0 and 1 in the initial elastic phase). Figure 5.26d demonstrates the plastic strain history p (t): during the plastic phase ˙ = 55 − 5/1 = 50, whereas p (t) stays p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E| constant with p (t) = 40 (or as initial value p (t) = 0) during the second elastic phase. Finally, the plastic arc-length κ(t) in Fig. 5.26e follows constant-linear-constantlinear in time from integrating κ(t) ˙ = |˙p (t)| = {0, 50, 0, 50} over the time interval t ∈ [0, tmax = 10], thus κmax = 40 + 30 = 70.

5.3.7 Specific Prandtl Mixed Hardening Model: Formulation The specific Prandtl (isotropic and kinematic) mixed hardening model, similar to that displayed in Fig. 5.12 (however with the hardening modulus H and the hardening strain εh coinciding here with the isotropic- and kinematic-hardening moduli H and K , respectively, and the isotropic- and kinematic-hardening strains hi and hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear mixed-hardening frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy and (ii) linear mixedhardening springs with stiffnesses H and K (the isotropic- and kinematic-hardening moduli). For the specific Prandtl mixed hardening model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − p (i.e. the elastic strain e ), hi (the isotropic-hardening strain) and hk (the kinematic-hardening strain) ψ(, p , hi , hk ) =

1 1 1 E [ − p ]2 + H 2hi + K 2hk . 2 2 2

(5.175)

Then the energetic stress σ  conjugated to the total strain  and the energetic plastic stress σp conjugated to the plastic strain p together with the isotropic-hardening stress   σhi conjugated to the isotropic-hardening strain hi and the kinematic-hardening stress   σhk conjugated to the kinematic-hardening strain hk follow as

5.3 Prandtl Hardening Model

263

σ  (, p

) = ∂ ψ(, p , hi , hk ) =

E [ − p ],

(5.176a)

σp (, p  σhi (  σhk (

) = ∂p ψ(, p , hi , hk ) = −E [ − p ],

(5.176b)

) = ∂hi ψ(, p , hi , hk ) =

H hi

,

(5.176c)

hk ) = ∂hk ψ(, p , hi , hk ) =

K hk

.

(5.176d)

hi

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the mixed-hardening frictional slider, also with the negative of the energetic plastic stress, −σp ≡ σ. Furthermore, for the specific Prandtl mixed hardening model the convex but nonsmooth dissipation potential π is chosen as π(˙p , ˙hi , ˙hk ) = [σy + H hi ] |˙p | − H hi ˙hi + K hk [˙p − ˙hk ].

(5.177)

Observe that (i) π does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) π is positively homogenous of degree one in {˙p , ˙hi , ˙hk } and is obviously non-smooth at the origin {˙p , ˙hi , ˙hk } = {0, 0, 0}. Consequently, the dissipative plastic stress σp and the dissipative isotropic- and kinematic  hardening stresses σhi and σhk compute as some sub-derivatives of the dissipation potential with respect to their conjugated variables σp (˙p , ˙hi , ˙hk ) ∈ d˙p π(˙p , ˙hi , ˙hk ),  σhi (˙p , ˙hi , ˙hk ) ∈ d˙hi π(˙p , ˙hi , ˙hk ),  σhk (˙p , ˙hi , ˙hk ) ∈ d˙hk π(˙p , ˙hi , ˙hk ),

(5.178)

with

d˙p π(˙p , ˙hi , ˙hk ) =

d˙hi π(˙p , ˙hi , ˙hk ) = d˙hk π(˙p , ˙hi , ˙hk ) =

⎧  + [σy + H hi ] + K hk ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎨ − [σy + H hi ] − K hk , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎫ ˙p > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ = 0 for  ˙ , p ⎪  ⎪ ⎪ + [σy + H hi ] + K hk ⎪ ⎪ ⎪  ⎭ − [σy + H hi ] − K hk ˙p < 0 −H hi , −K hk .

(5.179)

whereby d˙p π, d˙hi π and d˙hk π denote the sets of sub-derivatives, i.e. the subdifferentials of π with respect to ˙p , ˙hi and ˙hk , respectively. Recall that the energetic and the dissipative plastic as well as isotropic- and   kinematic-hardening stresses are constitutively related by σp + σp = 0, σhi + σhi =   0 and σhk + σhk = 0, respectively, thus the notions of plastic stress and isotropicand kinematic-hardening stresses defined as the values

264

5 Plasticity σp

σp H

+σy

σp = +σy +

hk

+

hi

+σy

0

hi

0

−σy

−σy H

K

σp = −σy +

hk

−

hk

K

hi

Fig. 5.27 Specific Prandtl (isotropic and kinematic) mixed hardening model: The elastic domain for σp defined by |σp − K hk | − [σy + H hi ] < 0 in the {σp , hi , hk }-space expands uniformly with the isotropic hardening strain hi ∈ [0, ∞) and shifts with the kinematic hardening strain hk ∈ (−∞, +∞). The slopes of the two lines |σp | − [σy + H hi ] = 0 defining the yield surface for hk = 0 denote the isotropic hardening modulus H . The slope of the two lines |σp − K hk | − σy = 0 defining the yield surface for hi = 0 denotes the kinematic hardening modulus K . The union of the elastic domain and the yield surface renders the admissible domain |σp − K hk | − [σy + H hi ] ≤ 0

σp := σp = −σp , 



σhi := σhi = −σhi , σhk :=

 σhk

=

 −σhk ,

(5.180a) (5.180b) (5.180c)

will exclusively be used in the sequel for convenience of exposition. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving forces, i.e. in the {σp , σhi , σhk }-space, is introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.27. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙p π in Eq. 5.179, or, alternatively, from evaluating the formal definition of the sub-differential

{σp |

σp [˙p

(5.181) d˙p π(˙p , ˙hi , ˙hk ) =     − ˙p ] ≤ [σy + H hi ] |˙p | − |˙p | + K hk [˙p − ˙p ] ∀˙p },

whereby ˙p denotes any admissible plastic strain rate. Then at ˙p = 0 it holds for any admissible ˙p that σp ˙p ≤ [σy + H hi ] |˙p | + K hk ˙p and, with max˙p {[σp − K hk ] ˙p /|˙p |} = |σp − K hk |, the admissible domain follows as |σp − K hk | ≤ σy + H hi . Moreover, the sub-differentials d˙hi π and d˙hk π reduce to the partial derivatives ∂˙hi π and ∂˙hk π, respectively, and render σhi = −H hi and σhk = −K hk . Thus the admissible domain is eventually expressed as |σp + σhk | ≤ σy − σhi . The elastic domain is defined as the interior of the admissible domain, i.e.

5.3 Prandtl Hardening Model

265

  int A := {σp , σhk } | |σp + σhk | − [σy − σhi ] < 0 ,

(5.182)

whereas the yield surface, which in the present one-dimensional case collapses to the two planes σp + σhk = ±[σy − σhi ], is defined as the boundary of the admissible domain, i.e.   ∂ A := {σp , σhi } | |σp + σhk | − [σy − σhi ] = 0 ,

(5.183)

Collectively, the admissible domain in the {σp , σhi , σhk }-space is characterized by the yield condition |σp + σhk | − [σy − σhi ] ≤ 0.

(5.184)

States in the interior int A of the admissible domain with |σp + σhk | < σy − σhi are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp + σhk | = σy − σhi are plastic. The corresponding dual dissipation potential π ∗ , as determined from a Legendre transformation π ∗ (σp , σhi , σhk ) = max

˙ p ,˙hi ,˙hk

(5.185)

{σp ˙p + σhi ˙hi + σhk ˙hk − [σy − H hi ] |˙p | + H hi ˙hi − K hk [˙p − ˙hk ]} then reads with the stationarity conditions σhi = −H hi and σhk = −K hk (note the minus signs) π ∗ (σp , σhi , σhk ) = IA (σp , σhi , σhk ) := ⎧ ⎫ |σp + σhk | ≤ σy − σhi ⎬ ⎨ 0 for , ⎩ ⎭ ∞ |σp + σhk | > σy − σhi

(5.186)

where IA denotes the indicator function of the admissible domain A in the {σp , σhi , σhk }-space. The evolution laws (the associated flow rules) for the plastic and the isotropic- and kinematic-hardening strains then follow as some sub-derivatives of the dual dissipation potential with respect to their conjugated variables ˙p (σp , σhi , σhk ) ∈ dσp π ∗ (σp , σhi , σhk ) = dσp IA (σp , σhi , σhk ), ˙hi (σp , σhi , σhk ) ∈ dσhi π ∗ (σp , σhi , σhk ) = dσhi IA (σp , σhi , σhk ), ˙hk (σp , σhi , σhk ) ∈ dσhk π ∗ (σp , σhi , σhk ) = dσhk IA (σp , σhi , σhk ),

(5.187)

266

5 Plasticity

with dσp π ∗ (σp , σhi , σhk ) = dσp IA (σp , σhi , σhk ) = ⎫ ⎧ 0 |σp + σhk | < σy − σhi ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ for σp + σhk ⎪ ⎪ ⎪ |σp + σhk | = σy − σhi ⎪ ⎭ ⎩λ |σp + σhk |

(5.188a)

dσhi π ∗ (σp , σhi , σhk ) = dσhi IA (σp , σhi , σhk ) = ⎧ ⎫ |σp + σhk | < σy − σhi ⎬ ⎨0 for ⎩ ⎭ λ |σp + σhk | = σy − σhi

(5.188b)

dσ π ∗ (σp , σhi , σhk ) = dσhk IA (σp , σhi , σhk ) = ⎧ hk ⎫ 0 |σp + σhk | < σy − σhi ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ for , σp + σhk ⎪ ⎪ ⎪ |σp + σhk | = σy − σhi ⎪ ⎩λ ⎭ |σp + σhk |

(5.188c)

and

and

whereby dσp π ∗ , dσhi π ∗ and dσhk π ∗ denote the sets of sub-derivatives, i.e. the subdifferentials of π ∗ with respect to σp , σhi and σhk , respectively, and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 5.178 and 5.187 are inverse relations. Identifying ˙hi with |˙p | and ˙hk with ˙p , respectively, and setting σhi = 0 and σhk = 0, the remaining non-smooth dissipation and dual dissipation potentials π(˙p ) and π ∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙p ) and ˙p = ˙p (σp ) are similar to those displayed in Fig. 5.3. Interestingly, the result in Eq. 5.187 can be rephrased in terms of the postulate of maximum dissipation (due to kinematic-hardening plasticity) that follows from the reverse Legendre transformation π(˙p , ˙hi , ˙hk )) = =

max

{d(σp , σhi , σhk ; ˙p , ˙hi , ˙hk ) − IA (σp , σhi , σhk )}

max

{d(σp , σhi , σhk ; ˙p , ˙hi , ˙hk )},

σp ,σhi ,σhk (σp ,σhi ,σhk )∈ A

(5.189)

whereby d(σp , σhi , σhk ; ˙p , ˙hi , ˙hk ) := σp ˙p + σhi ˙hi + σhk ˙hk denotes the dissipation power density. The postulate of maximum dissipation can, alternatively, be recast as a variational inequality: For given {˙p , ˙hi , ˙hk }, find {σp , σhi , σhk } ∈ A as the solution of

5.3 Prandtl Hardening Model

267

d(σp , σhi , σhk ; ˙p , ˙hi , ˙hk ) ≥      d(σp , σhi , σhk ; ˙p , ˙hi , ˙hk ) ∀{σp , σhi , σhk }∈

(5.190) A,

  , σhk } denote any admissible plastic, isotropic- and kinematicwhereby {σp , σhi hardening stress. As yet another alternative, the postulate of maximum dissipation may be reformulated as constrained optimization problem with  a Lagrange functional incorporating the admissibility constraint |σp + σhk | ≤ [σy − σhi ] by the Lagrange multiplier λ ≥ 0

(σp , σhi , σhk , λ; ˙p , ˙hi , ˙hk ) :=  −d(σp , σhi , σhk ; ˙p , ˙hi , ˙hk ) + λ |σp + σhk | − [σy − σhi ] .

(5.191)

In accordance with Eq. 5.187 the stationarity conditions of this constrained optimization problem then read ˙p = λ

σp + σhk and |σp + σhk |

˙hi = λ and

˙hk = λ

σp + σhk , |σp + σhk |

(5.192)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format λ ≥ 0, |σp + σhk | ≤ [σy − σhi ], λ |σp + σhk | = λ [σy − σhi ].

(5.193)

Note that it follows immediately from Eq. 5.192 that |˙p | = ˙hi = |˙hk | = λ. Finally, the plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated plastic deformation, i.e.  κ = κ˙ dt with κ˙ := |˙p | = ˙hi = |˙hk | = λ ≥ 0. (5.194) The specific Prandtl mixed hardening model is summarized in Table 5.11.

5.3.8 Specific Prandtl Mixed Hardening Model: Algorithmic Update For the specific Prandtl mixed (isotropic and kinematic) hardening model the evolution laws for the plastic strain p , the kinematic-hardening strain hk and the isotropichardening strain hi are integrated by the implicit Euler backwards method to render = λ np := np − n−1 p

n σpn + σhk n |σpn + σhk |

n = nhk − n−1 hk =: hk

(5.195)

268

5 Plasticity

Table 5.11 Summary of the specific Prandtl mixed hardening model (1) Strain



=  e + p

(2) Energy

ψ

=

(3) Stress

σ

= E [ − p ]

(4) Stress

σhi = −H hi

Stress

σhk = −K hk

(5)

(6) Potential π

1 2E

[ − p ]2 + 21 H 2hi + 21 K 2hk σ

≡



σp

= [σy + H hi ] |˙p | − H hi ˙ hi + K hk [˙p − ˙ hk ]

(7) Stress

σp = [σy + H hi ]

(8) Stress

σhi = −H hi

Stress

σhk = −K hk

(9)

≡

˙ p + K hk ≡ σp for ˙ p = 0 |˙p |

or (6) Yield

0

≥ |σphk | − σyhi with σyhi := [σy + H hi ]

(7) Evolution ˙ p = λ

σphk |σphk |

with σphk := [σp − K hk ]

(8) Evolution ˙ hi = λ (9) Evolution ˙ hk = λ (10) KKT

λ

σphk |σphk |

≥ 0,

|σphk |

≤

σyhi ,

λ |σphk |

=

λ σyhi

and = λ, nhi := nhi − n−1 hi

(5.196)

whereby λ = t n λn . Consequently, the plastic stress σp , the kinematic-hardening stress σhk and the isotropic-hardening stress σhi are updated at the end of the time step by σpn = −E [np − n ] =: σp − E np , n σhk n σhi

= −K = −H

nhk nhi

 =: σhk  =: σhi

− −

K nhk , H nhi .

(5.197)

5.3 Prandtl Hardening Model

269

 Here the trial plastic stress σp , the trial kinematic-hardening stress σhk and the trial  isotropic-hardening stress σhi are computable exclusively from known quantities at the beginning of the time step and follow as

σp := −E [n−1 − n ], p  σhk  σhi

:= −K := −H

n−1 hk n−1 hi

(5.198)

, .

Combining the plastic stress and the kinematic-hardening stress at the end of the time step and incorporating the discretized evolution laws for the plastic strain and the kinematic-hardening strain then renders n  = σp + σhk − [E + K ] λ σpn + σhk

n σpn + σhk n |σpn + σhk |

.

(5.199)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stresses

σn + σn p hk n  |σpn + σhk = σp + σhk | + [E + K ] λ . n n |σp + σhk |

(5.200)

As an immediate consequence the equivalent stress and its trial value are related via n  | = |σp + σhk | − [E + K ] λ. |σpn + σhk

(5.201)

A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously n σpn + σhk

|σpn +

n σhk |

≡

 σp + σhk  |σp + σhk |

.

(5.202)

Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore n  = σhi − H λ. σhi

(5.203)

Consequently, the yield function at the end of the time step is expressed as n n | − σy + σhi = φ − [E + H + K ] λ. φn := |σpn + σhk

(5.204)

Here the trial value of the yield function φ has been defined as   | − σy + σhi . φ := |σp + σhk

(5.205)

270

5 Plasticity

Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is computed in closed form from λ =

φ  ≥ 0. E+H+K

(5.206)

Once λ is computed all other variables may be updated. In particular, the plastic stress at the end of the time step reads σpn = σp − E λ

 σp + σhk  |σp + σhk |

.

(5.207)

The sensitivity of σpn = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n     σp + σhk σp + σhk n ∂ σp = E − E λ ∂ (5.208) − E   ∂ (λ). |σp + σhk | |σp + σhk | The first derivative term on the right-hand-side computes to zero since  ∂

 σp + σhk  |σp + σhk |

 =

  σp + σhk σp + σhk 1 E −   2  E ≡0 |σp + σhk | |σp + σhk | |σp + σhk |

(5.209)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-hand-side computes from requiring satisfaction of (the yield condition) ∂ φn = 0 for ongoing plastic flow at the end of the time step, i.e.

∂ φ − [E + H + K ] ∂ (λ) =

 σp + σhk

|σp

+

 σhk |

. E − [E + H + K ] ∂ (λ) = 0. (5.210)

As a conclusion the algorithmic tangent E a is thus finally expressed as E an = E − H0 (λ)

E2 . E+H+K

(5.211)

In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Prandtl hardening model capturing mixed hardening is summarized in Table 5.12.

5.3 Prandtl Hardening Model

271

Table 5.12 Algorithmic update for the specific Prandtl mixed (isotropic and kinematic) hardening model Input

n n−1 n−1 n−1 p hi hk

Trial Strain

p = n−1 p hi = n−1 hi hk = n−1 hk

Trial Stress

σp = −E [p − n ]  = −H  σhi hi  = −K  σhk hk

Trial Yield

φ



 | − σ + σ = |σp + σhk y hi

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E+H+K

ENDIF Update Strain np = p + λ nhi

=

hi

+ λ

nhk = hk + λ

 σp + σhk

 | |σp + σhk  σp + σhk

 | |σp + σhk

Update Stress

σ n = E [n − np ]

Tangent

E an = E − H0 (λ)

Output

σ n np nhi nhk E an

E2 E+H+K

5.3.9 Specific Prandtl Mixed Hardening Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the specific Prandtl mixed (isotropic and kinematic) hardening model to a prescribed Zig-Zag strain history is documented in Fig. 5.28a, b, c, d, e. Figure 5.28a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100

272

5 Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.28 Response analysis of the specific Prandtl mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

273

time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.28b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) ˙ = E ˙(t) in the elastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ = σy + H κ(t) (E = 1, thus the slopes in the elastic phases in Fig. 5.28a, b coincide), and σ(t) ˙ = [E − E 2 /[E + H + K ]] ˙(t) = ˙(t)/6 in the plastic phases where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t) = σy + H κ(t). The resulting σ = σ() diagram is highlighted in Fig. 5.28c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format (contracting in the  direction and isotropically expanding, kinematically shifting in the σ direction) of the σ = σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at  = 0 and  = ±5 obviously tend to the elastic modulus E = 1 for t → 0. It is tedious but easy to verify that the stress varies between [0, 1], [5/3, −1], [ − 20/9, 15/9], [145/54, −120/54], [ − 995/324, 870/324] and [6595/1944, −3125/1944] in the elastic phases. Figure 5.28d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + H + K ] = 5 × 10/12, whereas p (t) stays constant with |p (t)| = {0, 10/3, 25/9, 125/54, 625/324, 3125/1944} during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.28e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 5 × 10/12 during the plastic phases and constant in time during the elastic phases, thus κmax = 2 × [10/3 + 25/9 + 125/54 + 625/324] + 3125/1944 = 43385/1944 ≈ 22.3. Prescribed Strain History: Sine The response of the specific Prandtl mixed (isotropic and kinematic) hardening model to a prescribed Sine strain history is documented in Fig. 5.29a, b, c, d, e. Figure 5.29a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.29b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) ˙ = ˙(t) in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ(t) (thus the corresponding curve segments representing elastic loading/unloading in Fig. 5.29a, b are affine), and σ(t) ˙ = ˙(t)/6 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t). The resulting σ = σ() diagram is highlighted in Fig. 5.29c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format (contracting in the  direction and isotropically expanding, kinematically shifting in the σ direction) of the σ = σ() diagram is only approx-

274

5 Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 5.29 Response analysis of the specific Prandtl mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

275

imately captured in the elastic-plastic transition, however the slopes at  = 0 and  = ±5 obviously tend to the elastic modulus E = 1 for t → 0. It is tedious but easy to verify that the stress varies between [0, 1], [5/3, −1], [ − 20/9, 15/9], [145/54, −120/54], [ − 995/324, 870/324] and [6595/1944, −3125/1944] in the elastic phases. Figure 5.29d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + H + K ] = |˙(t)| × 10/12, whereas p (t) stays constant with |p (t)| = {0, 10/3, 25/9, 125/54, 625/324, 3125/1944} during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.29e follows as a sequence of sine waves segments from integrating κ(t) ˙ = |˙p (t)| = |˙(t)| × 10/12 during the plastic phases and constant in time during the elastic phases, thus κmax = 2 × [10/3 + 25/9 + 125/54 + 625/324] + 3125/1944 = 43385/1944 ≈ 22.3. Prescribed Strain History: Ramp The response of the specific Prandtl mixed (isotropic and kinematic) hardening model to a prescribed Ramp strain history is documented in Fig. 5.30a, b, c, d, e. Figure 5.30a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.30b showcases the resulting stress history σ(t) with σ(t) ˙ = ˙(t) in the two elastic phases where |σ(t) − 0.1 κ(t)| < 1 + 0.1 κ(t) (thus the slopes in the two elastic phases in Fig. 5.30a, b coincide), and σ(t) ˙ = ˙(t)/6 in the two plastic phases where |σ(t) − 0.1 κ(t)| = 1 + 0.1 κ(t). In particular during the holding phase σ(t) = 5/3 results as the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/12 = 5/3. The resulting σ = σ() diagram is highlighted in Fig. 5.30c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected parallelogram-type format (isotropically expanding, kinematically shifting in the σ direction) of the σ = σ() diagram is only approximately captured in the elasticplastic transition, however the slopes at  = 0 and  = 5 obviously tend to the elastic modulus E = 1 for t → 0. Figure 5.30d demonstrates the plastic strain history p (t): during the plastic phases p (t) evolves in parallel to the total strain with |˙p (t)| = |˙(t)| E/[E + H + K ] = 50/12 (or ˙p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 0.8 × 50/12 = 10/3 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.30e follows constant-linear-constantlinear in time from integrating κ(t) ˙ = |˙p (t)| = {0, 50/12, 0, 50/12} over the time interval t ∈ [0, tmax = 10], thus κmax = 10/3 + 35/18 = 95/18 ≈ 5.3.

276

5 Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 5.30 Response analysis of the specific Prandtl mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

277

Prescribed Stress History: Zig-Zag The response of the specific Prandtl mixed (isotropic and kinematic) hardening model to a prescribed Zig-Zag stress history is documented in Fig. 5.31a, b, c, d, e. Figure 5.31a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.31b showcases the resulting strain history (t) that displays a periodic signal with ˙(t) = σ(t)/E ˙ = σ(t)/1 ˙ (i.e. a purely elastic phase) after an initial series of elastic and plastic phases with ˙(t) = σ(t)/E ˙ = σ(t)/1 ˙ and ˙(t) = σ(t)/[E ˙ − ˙ × 6, respectively. Thus (1) = σy /1 + [σa − σy ] × 6 = E 2 /[E + H + K ]] = σ(t) 1/1 + 4 × 6 = 25 at the end of the first plastic phase and (3) = (1) − [σa + 1]/1 − [σa − 1] × 6 = 25 − 6 − 4 × 6 = −5 at the end of the second plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 5.31c. Once the second plastic phase is completed the cyclic σ = σ() behavior in the remaining elastic phase is linear elastic with slope E = 1. It is easy to verify that the strain varies between ∓5 in the remaining elastic phase. Figure 5.31d demonstrates the plastic strain history p (t): during the two plastic ˙ = 30 − 5/1 = 25, whereas p (t) phases p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E| stays constant with p (t) = 20 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.31e follows linear in time from integrating κ(t) ˙ = |˙p (t)| = 25 during the plastic phase and constant in time during the elastic phases, thus κmax = 40. Prescribed Stress History: Sine The response of the specific Prandtl mixed (isotropic and kinematic) hardening model to a prescribed Sine stress history is documented in Fig. 5.32a, b, c, d, e. Figure 5.32a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.32b showcases the resulting strain history (t) that displays a periodic signal with ˙(t) = σ(t) ˙ (i.e. a purely elastic phase) after an initial series of elastic and plastic phases with ˙(t) = σ(t) ˙ and ˙(t) = σ(t) ˙ × 6, respectively. Thus (1) = 1/1 + 4 × 6 = 25 at the end of the first plastic phase and (3) = 25 − 6 − 4 × 6 = −5 at the end of the second plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 5.32c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected σ = σ() behavior is only approximately captured in the elastic-plastic transition. Once the second plastic phase is completed the cyclic σ = σ() behavior in the

278

5 Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 5.31 Response analysis of the specific Prandtl mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

5.3 Prandtl Hardening Model

279

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 5.32 Response analysis of the specific Prandtl mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

280

5 Plasticity

remaining elastic phase is linear elastic with slope E = 1. It is easy to verify that the strain varies between ∓5 in the remaining elastic phase. Figure 5.32d demonstrates the plastic strain history p (t): during the two plastic phases p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E|, ˙ whereas p (t) stays constant with p (t) = 20 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.32e follows as sinusoidal in time from integrating κ(t) ˙ = |˙p (t)| during the plastic phase and constant in time during the elastic phases, thus κmax = 40. Prescribed Stress History: Ramp The response of the specific Prandtl mixed (isotropic and kinematic) hardening model to a prescribed Ramp stress history is documented in Fig. 5.33a, b, c, d, e. Figure 5.33a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 5.33b showcases the resulting strain history (t) that displays an initial elastic phase with ˙(t) = σ(t), ˙ a subsequent plastic phase with ˙(t) = σ(t) ˙ × 6 (thus in particular during the holding phase (t) = 1/1 + 4 × 6 = 25), and a final elastic phase with ˙(t) = σ(t), ˙ respectively. The resulting σ = σ() diagram is highlighted in Fig. 5.33c. Once the holding phase is completed the σ = σ() behavior in the unloading phase is linear elastic with slope E = 1. It is easy to verify that the strain varies between [25, 20] in the final elastic phase. Figure 5.33d demonstrates the plastic strain history p (t): during the plastic phase ˙ = 30 − 5/1 = 25, whereas p (t) stays p (t) evolves with |˙p (t)| = |˙(t) − σ(t)/E| constant with p (t) = 20 (or as initial value p (t) = 0) during the elastic phases. Finally, the plastic arc-length κ(t) in Fig. 5.33e follows constant-linear-constant in time from integrating κ(t) ˙ = |˙p (t)| = {0, 25, 0} over the time interval t ∈ [0, tmax = 10], thus κmax = 20.

5.3.10 Generic Prandtl Hardening Model: Formulation A generic formulation of the Prandtl hardening model can be obtained from generalizing of the specific Prandtl hardening model in Fig. 5.12 by assuming the elastic spring or/and the hardening spring or/and the frictional slider as nonlinear. For the generic Prandtl hardening model the free energy density ψ is expressed as a non-quadratic but convex function of  − p (the elastic strain e ) and εh (the hardening strain)

5.3 Prandtl Hardening Model

281

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

p

t

t

d)

(tmax = 100 × 0.1

p,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 80.0)

Fig. 5.33 Response analysis of the specific Prandtl mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

282

5 Plasticity

ψ(, p , εh ) = ψ( − p , εh ).

(5.212)

Note that ψ(, p , εh ) and ψ( − p , εh ) are different functions that return, however, the same function value for the same values of , p and εh . Then the energetic stress  σ  and the energetic plastic stress σp together with the energetic hardening stress σh follow as σ  (, p , εh ) = ∂ ψ(, p , εh ) = ∂ ψ( − p , εh ),

(5.213a)

σp (, p , εh )  σh (, p , εh )

= ∂p ψ(, p , εh ) = ∂p ψ( − p , εh ),

(5.213b)

= ∂εh ψ(, p , εh ) = ∂εh ψ( − p , εh ).

(5.213c)

Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ  ≡ σ and the negative of the energetic plastic stress −σp ≡ σ. Moreover the energetic and the dissipative plastic and hardening   stresses are constitutively related by σp + σp = 0 and σh + σh = 0, respectively, thus the notions of plastic stress and hardening stress defined as σp := σp = −σp and σh := σh = −σh will exclusively be used in the sequel. Furthermore, for the generic Prandtl hardening model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙p , εh ) and π ∗ = π ∗ (σp , σh ), respectively, are related via corresponding Legendre transformations π ( ˙p , ε˙h ) = max{σp ˙p + σh ε˙h − π ∗ (σp , σh )},

(5.214a)

π ∗ (σp , σh ) = max{σp ˙p + σh ε˙h − π ( ˙p , ε˙h )}.

(5.214b)

σp ,σh

˙ p ,˙εh

Then the stationarity conditions corresponding to Eqs. 5.214a and 5.214b are the constitutive relations ˙p (σp , σh ) ∈ dσp π ∗ (σp , σh ) and

ε˙h (σp , σh ) ∈ dσh π ∗ (σp , σh ),

(5.215a)

σp ( ˙p , ε˙h ) ∈ d ˙p π ( ˙p , ε˙h ) and

σh ( ˙p , ε˙h ) ∈ d ε˙ h π ( ˙p , ε˙h ).

(5.215b)

Obviously the relations in Eqs. 5.215a and 5.215b determine entirely the dissipative behavior of the generic Prandtl hardening model, thus the formulation would be completed at this stage. To be more explicit, however, alternatively to Eq. 5.215b the closed and convex admissible domain A in the {σp , σh }-space is introduced. It is characterized by the convex yield condition φ = φ(σp , σh ) = ϕh (σp , σh ) − σy ≤ 0.

(5.216)

5.3 Prandtl Hardening Model

283

Here φ = φ(σp , σh ) is the yield function and ϕh (σp , σh ) denotes the equivalent (combined plastic and hardening) stress that is compared to the initial yield limit σy , a material property. Then the evolution laws for the plastic and hardening strains (i.e. the associated flow rules) follow alternatively to Eq. 5.215a from the postulate of maximum dissipation (due to hardening plasticity) with  a Lagrange functional incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0 (σp , σh , λ; ˙p , ε˙h ) := −d(σp , σh ; ˙p , ε˙h ) + λ φ(σp , σh ).

(5.217)

Consequently, the stationarity conditions of this constrained optimization problem read ˙p = λ ∂σp φ and ε˙h = λ ∂σh φ,

(5.218)

subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form λ ≥ 0, φ ≤ 0, λ φ = 0.

(5.219)

It shall be noted that collectively Eqs. 5.216, 5.218 and 5.219 are entirely equivalent statements to Eqs. 5.215a and 5.215b. As a further interesting aspect the dissipation d = σp ˙p + σh ε˙h shall next be examined more closely. From Eqs. 5.214a and 5.214b the dissipation d is alternatively expressed in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙p , ε˙h ) + π ∗ (σp , σh ) ≥ 0.

(5.220)

However, based on the above introduction of the yield condition φ ≤ 0 the dual dissipation potential is identified as the indicator function IA of the admissible domain A ⎧ φ(σp , σh ) ≤ 0 ⎨ 0 for π ∗ (σp , σh ) = IA (σp , σh ) := . (5.221) ⎩ ∞ φ(σp , σh ) > 0 Thus for the generic Prandtl hardening model the dual dissipation potential equals zero in the admissible domain A. Consequently, provided the plastic and hardening stresses are admissible, the dissipation is indeed expressed in terms of the dissipation potential only d = π(˙p , ε˙h ) ≥ 0.

(5.222)

Finally for an equivalent (combined plastic and hardening) stress that is homogeneous of degree one in the plastic and hardening stresses (thus σp ∂σp ϕh + σh ∂σh ϕh =

284

5 Plasticity

Table 5.13 Summary of the generic Prandtl hardening model (1) Strain

 =  e + p

(2) Energy

ψ = ψ( − p , εh )

Stress

σ = ∂ ψ

(4) Stress

σh = ∂εh ψ

(3)

≡

σ

≡



−σp

(5) Potential π = π(˙p , ε˙ h ) 

(6) Stress

σp ∈ d˙p π



σp

(7) Stress

σh ∈ dε˙ h π



σh



or (5) Yield

φ = φ(σp , σh ) ≤ 0

(6) Evolution ˙ p = λ ∂σp φ (7) Evolution ε˙ h = λ ∂σh φ (8) KKT

λ ≥ 0, φ

≤

0, λ φ

=

0

ϕh ), the dissipation d = σp ˙p + σh ε˙h is exclusively given in terms of the Lagrange multiplier λ and the initial yield limit σy , since then  d = λ σ p ∂ σ p ϕh + σ h ∂ σ h ϕh = λ ϕ h = λ σ y . The generic Prandtl hardening model is summarized in Table 5.13.

(5.223)

Chapter 6

Visco-Plasticity

In serial music, the series itself is seldom audible. —Steve Reich, b. 1936

Visco-plasticity is the quintessential characteristics of both non-reversible and ratedependent material behavior. Thereby experimental evidence for various classes of materials, including thermoplastic polymers and ductile metals, suggests that a part of the total strain is irreversible with its evolution being rate-dependent. Irreversibility thus motivates the decomposition of the total strain into an elastic, stress producing part and a visco-plastic, irreversible part. The onset and evolution of irreversibility are then captured by a yield condition and a rate-dependent flow rule, concepts that are at the core of any overstress-based visco-plasticity formulation. From a convex analysis point of view, non-reversibility and rate-dependence are intimately related to the non-smoothness and non-linearity of the convex dissipation potential and its dual. The Bingham model is the basic combined rheological model for a perfect rigidvisco-plastic solid that displays both non-reversible and rate-dependent material behavior. It consists of a parallel arrangement of a frictional slider and a viscous dashpot. Parallel and serial arrangements of a Bingham model with an elastic spring render the Bingham hardening model for a hardening rigid-visco-plastic solid (not considered here) and the Perzyna model for a perfect elasto-visco-plastic solid, respectively. Further, the serial arrangement of an elastic spring and the Bingham hardening model is established as the Perzyna model for a hardening elasto-visco-plastic solid.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 P. Steinmann and K. Runesson, The Catalogue of Computational Material Models, https://doi.org/10.1007/978-3-030-63684-5_6

285

286

6 Visco-Plasticity η

Fig. 6.1 Specific Bingham model σ

σ

σy

≡

vp

6.1 Bingham Model The Bingham model of a perfect rigid-visco-plastic fluid (in short the Bingham model) consists of a parallel arrangement of (i) a frictional slider and (ii) a viscous dashpot (see the sketch of the specific Bingham model in Fig. 6.1). The basic kinematic assumption of the Bingham model (that shall be exploited in the algorithmic setting) is the equality of the total strain  and the visco-plastic strain vp (representing the elongation of the viscous frictional slider), i.e.  ≡ vp .

(6.1)

Note that the set of internal variables is empty for the Bingham model, i.e. α = ∅. Eugen Cook Bingham [b. 8.12.1878, Cornwall, Vermont, USA, d. 6.11.1945, Easton, Pennsylvania, USA] was Professor of Chemistry at Lafayette College in Easton. He is the founding father (together with Markus Reiner) of the notion and science of rheology. Following a number of publications dealing with viscosity and plastic flow, he published his most popular treatise “Fluidity and Plasticity” in 1922. Fluids displaying a yield stress are named after him as Bingham fluids.

6.1.1 Specific Bingham Model: Formulation The specific Bingham model, displayed in Fig. 6.1, consists of a parallel arrangement of (i) a linear frictional slider with threshold σy and (ii) a linear viscous dashpot with viscosity η.

6.1 Bingham Model

287

Since there is no energy storage for the specific Bingham model the free energy density ψ vanishes identically ψ() ≡ 0. (6.2) Thus the energetic stress σ  conjugated to the total strain  vanishes identically as well (6.3) σ  () ≡ 0. Furthermore, for the specific Bingham model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous dissipation potential πv , together with a convex and non-smooth plastic contribution, the plastic dissipation potential πp , i.e. (6.4) π(˙) = πv (˙) + πp (˙). Thereby the viscous and plastic contributions πv and πp to the (total) dissipation potential π are chosen as πv (˙) =

1 η |˙|2 and 2

πp (˙) = σy |˙|.

(6.5)

Observe that (i) π = πv + πp does depend on ˙, thus the dissipative stress σ  = 0 for ˙ = 0, and that (ii) πv is positively homogenous of degree two in ˙ and obviously smooth at the origin ˙ = 0, and that (iii) πp is positively homogenous of degree one in ˙ and obviously non-smooth at the origin ˙ = 0. As a consequence of the additive structure of the (total) dissipation potential the dissipative stress σ  consists of a viscous contribution, the dissipative viscous overstress σv , and a plastic contribution, the dissipative plastic stress σp , i.e. σ  (˙) = σv (˙) + σp (˙).

(6.6)

Thereby the dissipative viscous overstress σv computes as partial derivative of the viscous dissipation potential with respect to its conjugated variable σv (˙) = ∂˙ πv (˙) = η ˙,

(6.7)

whereas the dissipative plastic stress σp computes as some sub-derivative of the plastic dissipation potential with respect to its conjugated flux ⎧ ⎨

⎫ +σy ˙ > 0 ⎬ σp (˙) ∈ d˙ πp (˙) = [−σy , +σy ] for ˙ = 0 , ⎩ ⎭ −σy ˙ < 0

(6.8)

whereby d˙ πp denotes the set of sub-derivatives, i.e. the sub-differential of πp with respect to ˙. Observe that σp (and thus σ  ) is constitutively not determined for ˙ = 0 (thus it can at best be computed from equilibrium considerations).

288

6 Visco-Plasticity

Recall that the total stress applied to the rheological model (that enters the equilibrium condition) and the energetic and the dissipative stresses are constitutively related by σ = σ  + σ  , thus (since here σ  ≡ 0) σ ≡ σ ,

(6.9)

moreover the notions of viscous overstress and plastic stress defined as the values σ := σv + σp with σv := σv and

σp := σp

(6.10)

will exclusively be used in the sequel for convenience of exposition. Separate Viscous Overstress and Plastic Stress The Bingham model may be formulated further by considering the viscous overstress σv and the plastic stress σp separately. Thereby, due to the non-smooth plastic dissipation potential, the plastic stress is constrained to reside in an admissible domain. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the (plastic) dissipative driving force, i.e. in the σp -space, is next introduced as the union of the rigid domain and the yield surface, compare the representation in Fig. 5.2. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙ πp in Eq. 6.8, or, alternatively, from evaluating the formal definition of the sub-differential  d˙ πp (˙) = {σp | σp [˙ − ˙] ≤ σy |˙ | − |˙| ∀˙ },

(6.11)

whereby ˙ denotes any admissible strain rate. Then at ˙ = 0 it holds for any admissible ˙ that σp ˙ ≤ σy |˙ | and, with max˙ {σp ˙ /|˙ |} = |σp |, the admissible domain follows as |σp | ≤ σy . The rigid domain is defined as the interior of the admissible domain, i.e. int A := {σp | |σp | − σy < 0},

(6.12)

whereas the yield surface, which in the present one-dimensional case collapses to the two end points σp = ±σy , is defined as the boundary of the admissible domain, i.e. (6.13) ∂ A := {σp | |σp | − σy = 0}. Collectively, the admissible domain in the σp -space is characterized by the yield condition (6.14) |σp | − σy ≤ 0.

6.1 Bingham Model

289

States in the interior int A of the admissible domain with |σp | < σy are rigid, whereas states on the boundary ∂ A of the admissible domain with |σp | = σy are visco-plastic. Recall that σp is constitutively not determined in the rigid domain. The corresponding dual viscous and plastic dissipation potentials πv∗ and πp∗ , as determined from the Legendre transformations

 1 2 = max σv ˙ − η |˙| ˙ 2 ∗ πp (σp ) = max{σp ˙ − σy |˙| }

πv∗ (σv )

(6.15a) (6.15b)

˙

then read πv∗ (σv ) =

1 1 |σv |2 2 η

πp∗ (σp ) = IA (σp ) :=

(6.16a) ⎧ ⎨ 0 ⎩

⎫ |σp | ≤ σy ⎬ for

∞

|σp | > σy

⎭

,

(6.16b)

where IA denotes the indicator function of the admissible domain A in the σp -space. The evolution law (the associated flow rule) for the total strain then follows either as partial derivative of the dual viscous dissipation potential or likewise as some sub-derivative of the dual plastic dissipation potential, in either case with respect to its conjugated variable 1 σv η ⎧ ⎫ 0 |σp | < σy ⎪ ⎪ ⎨ ⎬ for , ˙(σp ) ∈ dσp πp∗ (σp ) = dσp IA (σp ) = σ ⎪ ⎩λ p ⎭ |σp | = σy ⎪ |σp |

˙(σv ) = ∂σv πv∗ (σv )

=

(6.17a)

(6.17b)

whereby dσp πp∗ denotes the set of sub-derivatives, i.e. the sub-differential of πp∗ with respect to σp and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 6.7, 6.8 and 6.17a, 6.17b are inverse relations. The smooth viscous dissipation and dual viscous dissipation potentials πv = πv (˙) and πv∗ = πv∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙) and ˙ = ˙(σv ) are similar to those displayed in Fig. 4.2. The non-smooth plastic dissipation and dual plastic dissipation potentials πp = πp (˙) and πp∗ = πp∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙) and ˙ = ˙(σp ) are similar to those displayed in Fig. 5.3.

290

6 Visco-Plasticity

Total Stress Alternatively, the Bingham model may be formulated further by considering the total stress. To this end the viscous and the plastic stress need to be related to the total stress. Remarkably, since at yield the plastic stress satisfies |σp | = σy , the viscous overstress σv = σ − σp allows representation in terms of the yield condition that is, however, evaluated in terms of the total stress σ ⎫ ⎧ +|σ| − σy σ > +σy ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 0 else . (6.18) if σv = ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ −|σ| + σy σ < −σy The reasoning for the representation in Eq. 6.18 is highlighted in Fig. 6.2 and follows as: • For ˙ = 0 the overstress in the viscous damper is identically zero, i.e. σv ≡ 0 and thus the stress σp in the frictional slider coincides identically with the total stress σp ≡ σ. Consequently, and again since ˙ = 0, the total stress satisfies |σ| ≤ σy . • For ˙ > 0 (with σ > +σy ) the overstress in the viscous damper results in σv = σ − σy ≡ |σ| − σy and thus the stress σp in the frictional slider coincides identically with the (positive) yield stress σp ≡ +σy . σv

σp

+σy −σy +σy

σ

σ −σy

Fig. 6.2 Specific Bingham model: The total stress σ = σv + σp is the sum of the viscous overstress σv and the plastic stress σp . The viscous damper is only activated once the load carrying capacity of the frictional slider is exceeded. Accordingly the viscous overstress is identically zero σv ≡ 0 for |σ| − σy ≤ 0 (left), while the plastic stress remains constant with |σp | = σy for |σ| − σy > 0 (right)

6.1 Bingham Model

291

• For ˙ < 0 (with σ < −σy ) the overstress in the viscous damper results in σv = σ + σy ≡ −|σ| + σy and thus the stress σp in the frictional slider coincides identically with the (negative) yield stress σp ≡ −σy . Finally the above relations may conveniently be summarized as

σv =

⎧ ⎪ ⎨

0

σ  ⎪ ⎩ |σ| − σy |σ|

for

⎫ |σ| < σy ⎪ ⎬ ⎭ |σ| ≥ σy ⎪

.

(6.19)

Then, based on the representation for the viscous stress in terms of the total stress in Eq. 6.19, the two variants of the associated evolution law for the total strain in Eqs. 6.17a, 6.17b are alternatively expressed in terms of the total stress

˙(σ) =

⎧ ⎪ ⎪ ⎨

0

|σ| − σy σ ⎪ ⎪ ⎩ η |σ|

for

⎫ |σ| < σy ⎪ ⎪ ⎬ ⎪ |σ| ≥ σy ⎪ ⎭

.

(6.20)

Obviously, the expressions in Eqs. 6.7, 6.8 and 6.20 are inverse relations. With the representation for the evolution of the total strain in Eq. 6.20, the corresponding (total) dual dissipation potential π ∗ , as determined from the Legendre transformation 

1 π ∗ (σ) = max σ ˙ − η |˙|2 − σy |˙| ˙ 2 then reads1

1 The

expression for the evolution of the total strain rate in Eq. 6.20 results in

σ ˙ (σ) = |σ|

and

⎧ ⎪ ⎨

0

|σ| − σy ⎪ ⎩ η

for

⎫ |σ| < σy ⎪ ⎬ ⎪ |σ| ≥ σy ⎭

⎫ |σ| < σy ⎪ ⎪ ⎬ for 1  2 η |˙(σ)|2 = ⎪ ⎪ 1 |σ| − σy 2 ⎪ ⎭ ⎩ |σ| ≥ σy ⎪ 2 η ⎧ ⎪ ⎪ ⎨

and σy |˙(σ)| = σy

0

⎧ ⎪ ⎨

0

|σ| − σy ⎪ ⎩ η

for

⎫ |σ| < σy ⎪ ⎬ ⎪ |σ| ≥ σy ⎭

Taken together, the (total) dual dissipation potential π ∗ (σ) follows.

.

(6.21)

292

6 Visco-Plasticity

π( ˙)

2 1

+ σy | ˙|

0

0

˙

1

−1

2

−2 −1

0

1

0

−σy

σ| − σy

+σy

2

1

σ

0

1

−1

2

−2 −2

−1

0

−1

0

1

1 η

2

σ| − σy sgn(σ)

−σy

σ

+σy

−2

2

1 ˙(σ)

2

11 2η

1

˙

−2

2

π ∗ (σ)

2

η ˙ + sgn( ˙)σy

1

1 η ˙2 2

−2

σ( ˙)

2

−1

0

1

2

Fig. 6.3 Specific Bingham model: Non-smooth dissipation potential π(˙) together with resulting non-smooth σ = σ(˙) and non-smooth dual dissipation potential π ∗ (σ) together with resulting non-smooth ˙ = ˙ (σ)

π ∗ (σ) =

⎧ ⎪ ⎪ ⎨

0

2  ⎪ 1 |σ| − σy ⎪ ⎩ 2 η

⎫ |σ| < σy ⎪ ⎪ ⎬ for ⎪ ⎭ |σ| ≥ σy ⎪

.

(6.22)

The above relations may conveniently be condensed by the help of the Macaulay bracket • := 21 [• + | • |], e.g. the dual dissipation potential is expressed as π ∗ (σ) =

1 |σ| − σy 2 . 2 η

(6.23)

The non-smooth (total) dissipation and dual (total) dissipation potentials π = π(˙) and π ∗ = π ∗ (σ) together with the resulting non-smooth constitutive relations σ = σ(˙) and ˙ = ˙(σ) are displayed in Fig. 6.3. The result in Eq. 6.20 for the evolution of the total strain thus follows directly from the reverse Legendre transformation

6.1 Bingham Model

293

 1 |σ| − σy 2 , π(˙) = max d(σ; ˙) − σ 2 η

(6.24)

whereby d(σ; ˙) := σ ˙ denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.24 embodies the unconstrained optimization problem 1 |σ| − σy 2 ˜1/η (σ; ˙) := −d(σ; ˙) + (6.25) → min, σ 2 η whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint |σ| ≤ σy penalized by the penalty parameter 1/η. In accordance with Eq. 6.20 the stationarity condition of this unconstrained optimization problem then reads ˙(σ) =

|σ| − σy σ . η |σ|

(6.26)

Finally, the total strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated total deformation, i.e.

|σ| − σy κ = κ˙ dt with κ˙ := |˙| = ≥ 0. (6.27) η The specific Bingham model is summarized in Table 6.1. Table 6.1 Summary of the specific Bingham model (1) Strain



= vp

(2) Potential πv =

1 2

η |˙|2

(3) Potential πp = σy |˙| (4) Stress

σ = η ˙ + σy

˙ |˙|

≡

σ  for ˙ = 0

or (2) Potential πv∗ = (3) Yield

|σv |2 /η

0 ≥ |σp | − σy

(4) Evolution ˙ (5) KKT

1 2

σp |σp |

= λ

λ ≥ 0,

|σp |

σv η

= ≤

or (2) Potential π ∗ =

1 2

(3) Evolution ˙

|σ| − σy σ η |σ|

=

|σ| − σy 2 /η

σy ,

λ |σp |

=

λ σy

294

6 Visco-Plasticity

Determination of Total Stress The basic kinematic assumption of the Bingham model may be re-formulated as a constraint . (6.28) e :=  − vp = 0. Clearly e = 0 compares to a vanishing elastic strain (as present in the Perzyna model discussed in the sequel). Then regarding the rate format of the kinematic constraint ˙e = ˙ − ˙vp = 0 the postulate of maximum (visco-plastic) dissipation may be stated in three alternative formats: (1) The Lagrange multiplier format ˆ v , σp , λ, ˙vp , σvp ; ˙) := (σ −d(σv , σp , ˙vp ) +

πv∗ (σv )

(6.29)

+ λ φ(σp ) − σvp [˙ − ˙vp ],

(with φ := |σp | − σy and πv∗ = 21 |σv |2 /η) whereby ˆ is the Lagrange functional incorporating the rate format of the kinematic constraint ˙e = ˙ − ˙vp = 0 by the Lagrange multiplier σvp (i.e. the visco-plastic stress). Accordingly, the evolution law for the visco-plastic strain, and the total stress versus visco-plastic stress relation follow from the constrained optimization problem as ˙vp = ∂σv πv∗ = λ ∂σp φ

(+KKT) and σ := σv + σp = σvp ,

(6.30)

whereas the corresponding optimality condition regarding the kinematic constraint reads (6.31) ˙e = ˙ − ˙vp = 0. (2) The perturbed Lagrange multiplier format ˆE (σv , σp , λ, ˙vp , σvp ; ˙) :=   1 ∗ σvp , −d(σv , σp , ˙vp ) + πv (σv ) + λ φ(σp ) − σvp ˙ − ˙vp − 2E

(6.32)

whereby ˆE is the perturbed Lagrange functional with E > 0 the perturbation parameter of dimension stress × time (obviously for E → ∞ the original Lagrange multiplier format is retrieved). Accordingly, the evolution law for the visco-plastic strain, and the total stress versus visco-plastic stress relation are unchanged, whereas the optimality condition regarding the kinematic constraint now reads

6.1 Bingham Model

295

σvp = E ˙e = E [˙ − ˙vp ].

(6.33)

Back-substitution of this result for the visco-plastic stress σvp into the perturbed Lagrange multiplier format renders the penalty format as ˜E (σv , σp , λ, ˙vp ; ˙) :=

(6.34)

1 −d(σv , σp , ˙vp ) + πv∗ (σv ) + λ φ(σp ) − E [˙ − ˙vp ]2 , 2 whereby ˜E is the penalty functional with E > 0 the penalty parameter. Accordingly, the evolution law for the visco-plastic strain is unchanged, whereas the total stress versus kinematic constraint violation relation follows from the penalized constrained optimization problem as σ := σv + σp = E ˙e = E [˙ − ˙vp ].

(6.35)

(3) The augmented Lagrange multiplier format ˇE (σv , σp , λ, ˙vp , σvp ; ˙) :=

(6.36)

1 −d(σv , σp , ˙vp ) + πv∗ (σv ) + λ φ(σp ) − σvp [˙ − ˙vp ] − E [˙ − ˙vp ]2 , 2 whereby ˇE is the augmented Lagrange functional. Accordingly, the evolution law for the visco-plastic strain is unchanged, and the total stress versus visco-plastic stress and kinematic constraint violation relation follows from the augmented constrained optimization problem as σ := σv + σp = σvp + E ˙e = σvp + E [˙ − ˙vp ],

(6.37)

whereas the corresponding optimality condition regarding the kinematic constraint reads (6.38) ˙e = ˙ − ˙vp = 0. The augmented Lagrange multiplier format suggests an iterative determination of the Lagrange multiplier σvp , i.e. the visco-plastic stress is obtained from an Usawa update scheme upon substituting σvp by σvp ⇐ σvp + E ˙e = σvp + E [˙ − ˙vp ].

(6.39)

Note that once the kinematic constraint ˙e = ˙ − ˙vp = 0 is satisfied, the total and the visco-plastic stress coincide σ = σvp .

296

6 Visco-Plasticity

6.1.2 Specific Bingham Model: Algorithmic Update The integration algorithm for the specific Bingham model is based on the augmented Lagrange multiplier format of the postulate of maximum (visco-plastic) dissipation that allows to incorporate the kinematic constraint  = vp (in rate form). Thereby the evolution law for the visco-plastic strain vp is integrated by the implicit Euler backwards method to render nvp := nvp − n−1 vp = λ

σn , |σ n |

(6.40)

whereby the incremental visco-plastic multiplier λ is defined as λ := t n λn := t n

|σ n | − σy ≥ 0. η

(6.41)

Consequently, and based on the assumption that the kinematic constraint is satn−1 = 0, the total stress σ (versus isfied at the end of the previous time step n−1 vp −  visco-plastic stress σvp and kinematic constraint violation e relation) is updated at the end of the time step by k−1 − E [nvp − n ] =: σ  − E nvp . σ n = σvp

(6.42)

k−1 Here σvp denotes the visco-plastic stress, i.e. the Lagrange multiplier enforcing the kinematic constraint within an Usawa iteration. Moreover E := E/t is a penalty parameter of dimension stress. In contrast to the somewhat naive penalty format, the augmented Lagrange multiplier format allows for arbitrary small penalty parameters that do not compromise the condition number of the equation (system) to be solved. The trial total stress σ  is computable exclusively from known quantities at the beginning of the time step and the previous Usawa update, it follows as k−1 n − E [n−1 σ  := σvp vp −  ].

(6.43)

Incorporating the discretized evolution law for the visco-plastic strain then renders σ n = σ  − E λ

σn . |σ n |

(6.44)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress   σn |σ n | + E λ = σ . |σ n |

(6.45)

6.1 Bingham Model

297

As an immediate consequence the equivalent stress and its trial value are related via (6.46) |σ n | = |σ  | − E λ. A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously σ σn ≡  . n |σ | |σ |

(6.47)

Eventually, the yield function at the end of the time step is expressed as φn := |σ n | − σy = φ − E λ.

(6.48)

Here the trial value of the yield function φ has been defined as φ := |σ  | − σy .

(6.49)

Next for visco-plastic loading with λ > 0 the definition for the incremental visco-plastic multiplier is regrouped to render φn − λ

η = 0. t n

(6.50)

Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form from φ ≥ 0. (6.51) λ = E + η/t n Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t n → 0. Once λ is computed all other variables may be updated. In particular, the elastic strain (that represents the kinematic constraint) at the end of the time step reads as ne = n − nvp with nvp = n−1 vp + λ

σ . |σ  |

(6.52)

Then, if ne exceeds a given tolerance, the visco-plastic stress is reset according to an Usawa update scheme as k k−1 = σvp − E [nvp − n ] σvp

(6.53)

298

6 Visco-Plasticity

and the Usawa iteration is continued2 upon incrementing k and re-computing the trial total stress σ  . Otherwise, if ne falls below the given tolerance, the total stress is updated as k−1 . (6.54) σ n = σvp The algorithmic step-by-step update for the specific Bingham model is summarized in Table 6.2.

6.1.3 Specific Bingham Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Bingham model to a prescribed Zig-Zag strain history is documented in Fig. 6.4a–e. (These shall be compared to the corresponding response of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models in Figs. 4.6a–f and 5.4a–e, respectively.) Figure 6.4a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 6.4b showcases the resulting stress history σ(t) that displays a block signal with σ(t) = ±[σy + η |˙(t)|] = ±1.375 whenever ˙(t) = ±5. The resulting σ = σ() diagram is highlighted in Fig. 6.4c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = ±5 obviously tend to ∞ with t → 0. Figure 6.4d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint vp (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 6.4e follows linear in time from integrating κ(t) ˙ = |˙(t)| = 5 over two and a half periods, thus κmax = 50. Prescribed Strain History: Sine The response of the specific Bingham model to a prescribed Sine strain history is documented in Fig. 6.5a–e. (These shall be compared to the corresponding response of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models in Figs. 4.7a–f and 5.5a–e, respectively.) . = σ¯ n necessitates a special treatment: After resetting the visco-plastic stress the total strain needs first to be updated according to  k k − E [nvp − n ] σ¯ n − σvp σ¯ n − σvp σ¯ n − σ n n n n  ← + = + = nvp + E E E

2 The case of stress control with prescribed σ n

before re-computing the trial total stress σ  . Thereby it is clearly understood that the penalty parameter E serves as the algorithmic tangent stiffness within the combined augmented Lagrange/Newton iterations for stress control.

6.1 Bingham Model

299

Table 6.2 Algorithmic update for the specific Bingham model. (Note the different input and output arguments for either strain or stress control!) Input

k−1 n (or σ¯ n ) n−1 σvp vp

Trial Strain

vp = n−1 vp

Trial Stress

 = σ k−1 σvp vp  − E [ − n ] σ  = σvp vp

Trial Yield

φ = |σ  | − σy

Loading Check

IF φ < 0 THEN λ = 0 ELSE λ =

φ E + η/t n

ENDIF Update Strain

nvp = vp + λ

σ |σ  |

Constraint Check IF |n − nvp | ≥ tol THEN Update Multiplier k = σ  + E [n − n ] σvp vp vp k ]/E if stress control n = nvp + [σ¯ n − σvp

k =k+1 goto Trial Stress ELSE Update Stress  σ n = σvp

ENDIF Output

k−1 σ n (or n ) nvp σvp

300

6 Visco-Plasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.4 Response analysis of the specific Bingham model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

6.1 Bingham Model

301

Figure 6.5a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 6.5b showcases the resulting stress history σ(t) that displays an alternating signal with σ(t) = ±[σy + η |˙(t)|] = ±[1 + 0.59 | cos(ω t)|] with σmax/min = ±1.59 whenever ˙(t) = 0. The resulting σ = σ() diagram is highlighted in Fig. 6.5c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected vertical slopes of the σ = σ() diagram at  = 0 and  = ±5 are only approximately captured, however they obviously tend to ∞ with t → 0. Figure 6.5d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint vp (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 6.5e follows as a sequence of (positive and negative) sine quarter-waves from integrating κ(t) ˙ = |˙(t)| = a ω | cos(ω t)| over two and a half periods, thus κmax = 50. Prescribed Strain History: Ramp The response of the specific Bingham model to a prescribed Ramp strain history is documented in Fig. 6.6a–e. (These shall be compared to the corresponding response of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models in Figs. 4.8a–f and 5.6a–e, respectively.) Figure 6.6a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Figure 6.6b showcases the resulting stress history σ(t) that displays a stepwise signal with σ(t) = ±1.375 whenever ˙(t) = ±5 in the loading and the unloading phases. During the holding phase the relaxed σ(t) = σvp = +1 results as a reaction to the kinematic constraint vp (t) ≡ (t) with ˙(t) = 0. The resulting σ = σ() diagram is highlighted in Fig. 6.6c. Due to the finite sized time step t and corresponding finite sized strain increment  the expected rectangular format of the σ = σ() diagram is only approximately captured, however the slopes at  = 0 and  = 5 obviously tend to ∞ with t → 0. Figure 6.6d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint vp (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 6.6e follows linear-constant-linear in time from integrating κ(t) ˙ = |˙(t)| = {5, 0, 5} over the time interval t ∈ [0, tmax = 10], thus κmax = 10. Prescribed Stress History: Zig-Zag The response of the specific Bingham model to a prescribed Zig-Zag stress history is documented in Fig. 6.7a–e.

302

6 Visco-Plasticity σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.5 Response analysis of the specific Bingham model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

6.1 Bingham Model

303 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 6.6 Response analysis of the specific Bingham model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

304

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 75.0)

σ

c)

(

max,min

= ± 75.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 75.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.7 Response analysis of the specific Bingham model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.1 Bingham Model

305

Figure 6.7a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas rigid time steps are indicated by smaller filled circles. Figure 6.7b showcases the resulting strain history (t) that displays a periodic signal with non-constant (t) for the visco-plastic phases with |σ(t)| ∈ [1, 5] and with either (t) = 0 or (t) = 42.6¯ for the rigid phases with |σ(t)| ∈ [0, 1]. The resulting (lens-shaped) σ = σ() diagram is highlighted in Fig. 6.7c. Figure 6.7d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint vp (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 6.7e follows from integrating κ(t) ˙ = |˙(t)| during the visco-plastic phases and constant in time during the rigid phases, ¯ thus κmax = 5 × 42.6¯ = 213.3. Prescribed Stress History: Sine The response of the specific Bingham model to a prescribed Sine stress history is documented in Fig. 6.8a–e. Figure 6.8a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas rigid time steps are indicated by smaller filled circles. Figure 6.8b showcases the resulting strain history (t) that displays a periodic signal with non-constant (t) for the visco-plastic phases with |σ(t)| ∈ [1, 5] and with either (t) = 0 or (t) ≈ 59.9 for the rigid phases with |σ(t)| ∈ [0, 1]. The resulting (lentil-shaped) σ = σ() diagram is highlighted in Fig. 6.8c. Figure 6.8d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint vp (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 6.8e follows from integrating κ(t) ˙ = |˙(t)| during the visco-plastic phases and constant in time during the rigid phases, thus κmax ≈ 5 × 59.9 ≈ 299.6. Prescribed Stress History: Ramp The response of the specific Bingham model to a prescribed Ramp stress history is documented in Fig. 6.9a–e. Figure 6.9a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.9b showcases the resulting strain history (t) that displays an initial rigid phase with (t) = 0 for σ(t) ∈ [0, 1], subsequent visco-plastic loading, creep and visco-plastic unloading phases for σ(t) ∈ [1, 5], σ(t) = 5 and σ(t) ∈ [5, 1] with

306

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 75.0)

σ

c)

(

max,min

= ± 75.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 75.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.8 Response analysis of the specific Bingham model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.1 Bingham Model

307

σ

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

t

(tmax = 100 × 0.1

b)

max,min

= ± 500.0)

σ

c)

(

max,min

= ± 500.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1 vp,max,min = ± 500.0)

e)

(tmax = 100 × 0.1

κmax = 500.0)

Fig. 6.9 Response analysis of the specific Bingham model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

308

6 Visco-Plasticity

(t1 ) = 21.3¯ and (t2 ) = 448, respectively, and a final rigid phase for σ(t) ∈ [1, 0] ¯ with (t) = 469.3. The resulting σ = σ() diagram is highlighted in Fig. 6.9c. Figure 6.9d clearly demonstrates that the augmented Lagrange multiplier format effectively enforces the constraint vp (t) ≡ (t). Finally, the strain arc-length κ(t) in Fig. 6.9e follows from integrating κ(t) ˙ = ¯ |˙(t)| over the time interval t ∈ [0, tmax = 10], thus κmax = 469.3.

6.1.4 Generic Bingham Model: Formulation A generic formulation of the Bingham model can be obtained from generalizing the specific Bingham model in Fig. 6.1 by assuming the viscous dashpot or/and the frictional slider as nonlinear. For the generic Bingham model the free energy density ψ vanishes identically ψ() ≡ 0.

(6.55)

Thus the energetic stress σ  vanishes identically as well σ  () ≡ 0.

(6.56)

Recall that the energetic and the dissipative stresses are constitutively related to the total stress σ (that enters the equilibrium condition) by σ = σ  + σ  , thus (with σ  ≡ 0) the total stress σ ≡ σ  will exclusively be used in the sequel. Furthermore, for the generic Bingham model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙) and π ∗ = π ∗ (σ), respectively, are related via corresponding Legendre transformations π ( ˙) = max{σ ˙ − π ∗ (σ)},

(6.57a)

π ∗ (σ) = max{σ ˙ − π ( ˙)}.

(6.57b)

σ

˙

Then the stationarity conditions corresponding to Eqs. 6.57a and 6.57b are the constitutive relations ˙(σ) ∈ dσ π ∗ (σ), (6.58a) σ( ˙) ∈ d ˙ π ( ˙).

(6.58b)

Obviously the relations in Eqs. 6.58a and 6.58b determine entirely the dissipative behavior of the generic Bingham model, thus the formulation would be completed at this stage. To be more explicit, however, alternatively to Eq. 6.58b the closed and convex admissible domain A in the σ-space is introduced. It is characterized by the convex

6.1 Bingham Model

309

yield condition φ = φ(σ) := ϕ(σ) − σy ≤ 0.

(6.59)

Here φ = φ(σ) is the overstress function and ϕ(σ) denotes the equivalent stress that is compared to the yield limit σy , a material property. Then the evolution law for the (total) strain (i.e. the associated flow rule) follows alternatively to Eq. 6.58a from the postulate of maximum dissipation (due to visco-plasticity) 1 φ(σ) 2 → min, ˜1/η (σ; ˙) := −d(σ; ˙) + 2 η

(6.60)

whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint φ ≤ 0 penalized by the penalty parameter 1/η. Consequently, the stationarity condition of this unconstrained optimization problem reads ˙ = λ ∂σ φ with λ := φ(σ) /η ≥ 0.

(6.61)

It shall be noted that collectively Eqs. 6.59 and 6.61 are entirely equivalent statements to Eqs. 6.58a and 6.58b. As a further interesting aspect the dissipation d = σ ˙ shall next be examined more closely. From Eqs. 6.57a and 6.57b the dissipation d is alternatively expressed in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙) + π ∗ (σ) ≥ 0.

(6.62)

Thereby, based on the above introduction of the overstress function φ (and in view of Eqs. 6.57a, 6.58a, 6.60 and 6.61) the dual dissipation potential is identified as π ∗ (σ) =

⎧ ⎨

⎫ φ(σ) ≤ 0 ⎬

0

⎩1

φ(σ)2 /η 2

for φ(σ) > 0

⎭

=

1 φ(σ) 2 . 2 η

(6.63)

Finally for an equivalent stress that is homogeneous of degree one in the stress (thus σ ∂σ ϕ = ϕ), the dissipation d = σ ˙ is exclusively given in terms of the overstress function φ (with abbreviation λ := φ /η ≥ 0 for the visco-plastic multiplier and equivalent stress ϕ = φ + σy ≥ 0), since then d = λ σ ∂σ ϕ = λ ϕ = φ [φ + σy ]/η. The generic Bingham model is summarized in Table 6.3.

(6.64)

310

6 Visco-Plasticity

Table 6.3 Summary of the generic Bingham model (1) Strain



= vp

(2) Potential π = π(˙) (3) Stress

σ ∈ d˙ π



σ

or (2) Potential π ∗ = (3) Evolution ˙

1 2

φ(σ) 2 /η

= λ ∂σ φ with λ : =

φ(σ) /η

6.2 Perzyna Model The Perzyna model of a perfect elasto-visco-plastic solid (in short the Perzyna model) consists of a serial arrangement of (1) an elastic spring and (2) a viscous frictional slider consisting of a parallel arrangement of (i) a frictional slider and (ii) a viscous dashpot (see the sketch of the specific Perzyna model in Fig. 6.10). The basic kinematic assumption of the Perzyna model is the additive decomposition of the total strain  into the elastic strain e (representing the elongation of the elastic spring) and the visco-plastic strain vp (representing the elongation of the viscous frictional slider), i.e. (6.65)  = e + vp . Note that the visco-plastic strain vp denotes the only element contained in the set of internal variables α = {vp } for the Perzyna model. Piotr Perzyna [b. 1.8.1931, Nied´zwiada, Poland, d. 22.6.2013, Warsaw, Poland] was Professor of Solid Mechanics at the Institute of Fundamental Technological Research, Polish Academy of Sciences (IPPT PAN) in Warsaw. He is the founding father of overstress-type viscoplasticity that is oftentimes named after him as Perzyna visco-plasticity. Since the foundation for Perzyna visco-plasticity was laid in his milestone publication from 1966, he continued to work on inelastic material models with internal variables.

6.2 Perzyna Model

311

6.2.1 Specific Perzyna Model: Formulation The specific Perzyna model, displayed in Fig. 6.10, consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear viscous frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy and (ii) a linear viscous dashpot with viscosity η. For the specific Perzyna model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − vp (the elastic strain e ) ψ(, vp ) =

1 E [ − vp ]2 . 2

(6.66)

Then the energetic stress σ  conjugated to the total strain  and the energetic  conjugated to the visco-plastic strain vp follow as visco-plastic stress σvp σ  (, vp ) = ∂ ψ(, vp ) =  σvp (, vp )

E [ − vp ],

(6.67a)

= ∂vp ψ(, vp ) = −E [ − vp ].

(6.67b)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the viscous frictional slider, also  ≡ σ. with the negative of the energetic visco-plastic stress, −σvp Furthermore, for the specific Perzyna model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous dissipation potential πv , together with a convex and non-smooth plastic contribution, the plastic dissipation potential πp , i.e. (6.68) π(˙vp ) = πv (˙vp ) + πp (˙vp ). η E σ

σ

σy

e

Fig. 6.10 Specific Perzyna model

vp

312

6 Visco-Plasticity

Thereby the viscous and plastic contributions πv and πp to the (total) dissipation potential π are chosen as πv (˙vp ) =

1 η |˙vp |2 and 2

πp (˙vp ) = σy |˙vp |.

(6.69)

Observe that (i) π = πv + πp does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) πv is positively homogenous of degree two in ˙vp and obviously smooth at the origin ˙vp = 0, and that (iii) πp is positively homogenous of degree one in ˙vp and obviously non-smooth at the origin ˙vp = 0. As a consequence of the additive structure of the (total) dissipation potential the  consists of a viscous contribution, the dissipative dissipative visco-plastic stress σvp  viscous overstress σv , and a plastic contribution, the dissipative plastic stress σp , i.e.  σvp (˙vp ) = σv (˙vp ) + σp (˙vp ).

(6.70)

Thereby the dissipative viscous overstress σv computes as partial derivative of the viscous dissipation potential with respect to its conjugated variable σv (˙vp ) = ∂˙vp πv (˙vp ) = η ˙vp ,

(6.71)

whereas the dissipative plastic stress σp computes as some sub-derivative of the plastic dissipation potential with respect to its conjugated variable ⎧ ⎨

⎫ ˙vp > 0 ⎬ +σy σp (˙vp ) ∈ d˙vp πp (˙vp ) = [−σy , +σy ] for ˙vp = 0 , ⎩ ⎭ ˙vp < 0 −σy

(6.72)

whereby d˙vp πp denotes the set of sub-derivatives, i.e. the sub-differential of πp with respect to ˙vp . Recall that the energetic and the dissipative visco-plastic stresses are constitutively   + σvp = 0, thus the notion of visco-plastic stress together with the related by σvp notions of viscous overstress and plastic stress defined as the values   = −σvp with σv := σv and σvp = σv + σp := σvp

σp := σp

(6.73)

will exclusively be used in the sequel for convenience of exposition. Separate Viscous Overstress and Plastic Stress The Perzyna model may be formulated further by considering the viscous overstress σv and the plastic stress σp separately. Thereby, due to the non-smooth plastic dissipation potential, the plastic stress is constrained to reside in an admissible domain. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the (plastic) dissipative driving force, i.e. in the σp -space, is next introduced as the

6.2 Perzyna Model

313

union of the elastic domain and the yield surface, compare the representation in Fig. 5.8. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙vp πp in Eq. 6.72, or, alternatively, from evaluating the formal definition of the sub-differential  d˙vp πp (˙vp ) = {σp | σp [˙vp − ˙vp ] ≤ σy |˙vp | − |˙vp | ∀˙vp },

(6.74)

whereby ˙vp denotes any admissible visco-plastic strain rate. Then at ˙vp = 0 it holds for any admissible ˙vp that σp ˙vp ≤ σy |˙vp | and, with max˙vp {σp ˙vp /|˙vp |} = |σp |, the admissible domain follows as |σp | ≤ σy . The elastic domain is defined as the interior of the admissible domain, i.e. int A := {σp | |σp | − σy < 0},

(6.75)

whereas the yield surface, which in the present one-dimensional case collapses to the two end points σp = ±σy , is defined as the boundary of the admissible domain, i.e. (6.76) ∂ A := {σp | |σp | − σy = 0}. Collectively, the admissible domain in the σp -space is characterized by the yield condition (6.77) |σp | − σy ≤ 0. States in the interior int A of the admissible domain with |σp | < σy are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp | = σy are visco-plastic. The corresponding dual viscous and plastic dissipation potentials πv∗ and πp∗ , as determined from the Legendre transformations

 1 πv∗ (σv ) = max σv ˙vp − η |˙vp |2 ˙ vp 2 ∗ πp (σp ) = max{σp ˙vp − σy |˙vp | }

(6.78a) (6.78b)

˙ vp

then read πv∗ (σv ) =

1 1 |σv |2 2 η

πp∗ (σp ) = IA (σp ) :=

(6.79a) ⎧ ⎨ 0 ⎩

⎫ |σp | ≤ σy ⎬ for

∞

|σp | > σy

⎭

,

(6.79b)

where IA denotes the indicator function of the admissible domain A in the σp -space. The evolution law (the associated flow rule) for the visco-plastic strain then follows either as partial derivative of the dual viscous dissipation potential or likewise as some

314

6 Visco-Plasticity

sub-derivative of the dual plastic dissipation potential, in either case with respect to its conjugated variable 1 σv η ⎫ ⎧ 0 |σp | < σy ⎪ ⎪ ⎬ ⎨ for ˙vp (σp ) ∈ dσp πp∗ (σp ) = dσp IA (σp ) = , σ ⎪ ⎭ ⎩λ p |σp | = σy ⎪ |σp |

˙vp (σv ) = ∂σv πv∗ (σv )

=

(6.80a)

(6.80b)

whereby dσp πp∗ denotes the set of sub-derivatives, i.e. the sub-differential of πp∗ with respect to σp and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 6.71, 6.72 and 6.80a, 6.80b are inverse relations. The smooth viscous dissipation and dual viscous dissipation potentials πv = πv (˙vp ) and πv∗ = πv∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙vp ) and ˙vp = ˙vp (σv ) are similar to those displayed in Fig. 4.2. The nonsmooth plastic dissipation and dual plastic dissipation potentials πp = πp (˙vp ) and πp∗ = πp∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙vp ) and ˙vp = ˙vp (σp ) are similar to those displayed in Fig. 5.3. Visco-Plastic Stress Alternatively, the Perzyna model may be formulated further by considering the viscoplastic stress. To this end the viscous and the plastic stress need to be related to the visco-plastic stress. σp

σv

+σy −σy +σy

σvp

σvp −σy

Fig. 6.11 Specific Perzyna model: The visco-plastic stress σvp = σv + σp is the sum of the viscous overstress σv and the plastic stress σp . The viscous damper is only activated once the load carrying capacity of the frictional slider is exceeded. Accordingly the viscous overstress is identically zero σv ≡ 0 for |σvp | − σy ≤ 0 (left), while the plastic stress remains constant with |σp | = σy for |σvp | − σy > 0 (right)

6.2 Perzyna Model

315

Remarkably, since at yield the plastic stress satisfies |σp | = σy , the viscous overstress σv = σvp − σp allows representation in terms of the yield condition that is, however, evaluated in terms of the visco-plastic stress σvp ⎧ +|σvp | − σy ⎪ ⎪ ⎨ 0 σv = ⎪ ⎪ ⎩ −|σvp | + σy

⎫ σvp > +σy ⎪ ⎪ ⎬ else . ⎪ ⎪ ⎭ σvp < −σy

if

(6.81)

The reasoning for the representation in Eq. 6.81 is highlighted in Fig. 6.11 and follows as: • For ˙vp = 0 the overstress in the viscous damper is identically zero, i.e. σv ≡ 0 and thus the stress σp in the frictional slider coincides identically with the viscoplastic stress σp ≡ σvp . Consequently, and again since ˙vp = 0, the visco-plastic stress satisfies |σvp | ≤ σy . • For ˙vp > 0 (with σvp > +σy ) the overstress in the viscous damper results in σv = σvp − σy ≡ | + σvp | − σy and thus the stress σp in the frictional slider coincides identically with the (positive) yield stress σp ≡ +σy . • For ˙vp < 0 (with σvp < −σy ) the overstress in the viscous damper results in σv = σvp + σy ≡ −|σvp | + σy and thus the stress σp in the frictional slider coincides identically with the (negative) yield stress σp ≡ −σy . Finally the above relations may conveniently be summarized as

σv =

⎧ ⎪ ⎨

0

 σvp ⎪ ⎩ |σvp | − σy |σvp |

for

⎫ |σvp | < σy ⎪ ⎬ ⎭ |σvp | ≥ σy ⎪

.

(6.82)

Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.82, the two variants of the associated evolution law for the visco-plastic strain in Eqs. 6.80a, 6.80b are alternatively expressed in terms of the visco-plastic stress

˙vp (σvp ) =

⎧ ⎪ ⎪ ⎨

0

|σvp | − σy σvp ⎪ ⎪ ⎩ η |σvp |

for

⎫ |σvp | < σy ⎪ ⎪ ⎬ ⎪ |σvp | ≥ σy ⎪ ⎭

.

(6.83)

Obviously, the expressions in Eqs. 6.71, 6.72 and 6.83 are inverse relations. With the representation for the evolution of the visco-plastic strain in Eq. 6.83, the corresponding (total) dual dissipation potential π ∗ , as determined from the Legendre transformation

 1 ∗ 2 (6.84) π (σvp ) = max σvp ˙vp − η |˙vp | − σy |˙vp | ˙ vp 2

316

6 Visco-Plasticity

then reads3

⎫ |σvp | < σy ⎪ ⎪ ⎬ for 2  . π ∗ (σvp ) = ⎪ ⎪ 1 |σvp | − σy ⎪ ⎪ ⎩ |σvp | ≥ σy ⎭ 2 η ⎧ ⎪ ⎪ ⎨

0

(6.85)

The above relations may conveniently be condensed by the help of the Macaulay bracket • := 21 [• + | • |], e.g. the dual dissipation potential is expressed as π ∗ (σvp ) =

1 |σvp | − σy 2 . 2 η

(6.86)

The non-smooth (total) dissipation and dual (total) dissipation potentials π = π(˙vp ) and π ∗ = π ∗ (σvp ) together with the resulting non-smooth constitutive relations σvp = σvp (˙vp ) and ˙vp = ˙vp (σvp ) are similar to those displayed in Fig. 6.3. The result in Eq. 6.83 for the evolution of the visco-plastic strain thus follows directly from the reverse Legendre transformation

 1 |σvp | − σy 2 π(˙vp ) = max d(σvp ; ˙vp ) − , σvp 2 η

(6.87)

whereby d(σvp ; ˙vp ) := σvp ˙vp denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.87 embodies the unconstrained optimization problem 1 |σvp | − σy 2 → min, ˜1/η (σvp ; ˙vp ) := −d(σvp ; ˙vp ) + σvp 2 η

3 The

expression for the evolution of the visco-plastic strain in Eq. 6.83 results in

σvp ˙ vp (σvp ) = |σvp |

and

⎧ ⎪ ⎨

0

|σ | − σy ⎪ ⎩ vp η

for

⎫ |σvp | < σy ⎪ ⎬ ⎪ |σvp | ≥ σy ⎭

⎫ |σvp | < σy ⎪ ⎪ ⎬ for 1  2 η |˙vp (σvp )|2 = ⎪ ⎪ 1 |σvp | − σy 2 ⎪ ⎭ ⎩ |σvp | ≥ σy ⎪ 2 η ⎧ ⎪ ⎪ ⎨

and σy |˙vp (σvp )| = σy

0

⎧ ⎪ ⎨

0

|σ | − σy ⎪ ⎩ vp η

for

⎫ |σvp | < σy ⎪ ⎬ ⎪ |σvp | ≥ σy ⎭

Taken together, the (total) dual dissipation potential π ∗ (σvp ) follows.

.

(6.88)

6.2 Perzyna Model

317

whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint |σvp | ≤ σy penalized by the penalty parameter 1/η. In accordance with Eq. 6.83 the stationarity condition of this unconstrained optimization problem then reads |σvp | − σy σvp . (6.89) ˙vp (σvp ) = η |σvp | Finally, the visco-plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated visco-plastic deformation, i.e.

κ=

κ˙ dt with κ˙ := |˙vp | =

|σvp | − σy ≥ 0. η

The specific Perzyna model is summarized in Table 6.4.

Table 6.4 Summary of the specific Perzyna model (1) Strain



= e + vp

(2) Energy

ψ

=

(3) Stress

σ

= E [ − vp ]

(4) Potential πv =

1 2E

1 2

[ − vp ]2 

≡

σ

≡

˙ vp |˙vp |

≡

 for ˙ vp = 0 σvp

−σvp

η |˙vp |2

(5) Potential πp = σy |˙vp | (6) Stress

σvp = η ˙ vp + σy

or (4) Potential πv∗ = (5) Yield

0

1 2

|σv |2 /η

≥ |σp | − σy σp |σp |

(6) Evolution ˙ vp = λ (7) KKT

λ

≥ 0,

|σp |

σv η

= ≤

σy ,

or (4) Potential π ∗ =

1 2

(5) Evolution ˙ vp =

|σvp | − σy σvp η |σvp |

|σvp | − σy 2 /η

λ |σp |

=

λ σy

(6.90)

318

6 Visco-Plasticity

6.2.2 Specific Perzyna Model: Algorithmic Update For the specific Perzyna model the evolution law for the visco-plastic strain vp is integrated by the implicit Euler backwards method to render nvp := nvp − n−1 vp = λ

n σvp n | |σvp

,

(6.91)

whereby the incremental visco-plastic multiplier λ is defined as λ := t n λn := t n

n |σvp )| − σy

η

≥ 0.

(6.92)

Consequently, the visco-plastic stress σvp is updated at the end of the time step by n  = −E [nvp − n ] =: σvp − E nvp . σvp

(6.93)

 is computable exclusively from known quanHere the trial visco-plastic stress σvp tities at the beginning of the time step and follows as  n := −E [n−1 σvp vp −  ].

(6.94)

Incorporating the discretized evolution law for the visco-plastic strain then renders n  σvp = σvp − E λ

n σvp n | |σvp

.

(6.95)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress  σn  vp n  = σvp | + E λ . |σvp n | |σvp

(6.96)

As an immediate consequence the equivalent stress and its trial value are related via n  | = |σvp | − E λ. (6.97) |σvp A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously n σvp n | |σvp

≡

 σvp  | |σvp

.

(6.98)

Eventually, the overstress function at the end of the time step is expressed as

6.2 Perzyna Model

319 n φn := |σvp | − σy = φ − E λ.

(6.99)

Here the trial value of the overstress function φ has been defined as  | − σy . φ := |σvp

(6.100)

Next for visco-plastic loading with λ > 0 the definition for the incremental visco-plastic multiplier is regrouped to render φn − λ

η = 0. t n

(6.101)

Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form from φ ≥ 0. (6.102) λ = E + η/t n Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t n → 0. Once λ is computed all other variables may be updated. In particular, the visco-plastic stress at the end of the time step reads  σvp n  (6.103) = σvp − E λ  . σvp |σvp | n The sensitivity of σvp = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n

 n ∂ σvp

= E − E λ ∂

 σvp  | |σvp

 −E

 σvp  | |σvp

∂ (λ).

(6.104)

The first derivative term on the right-hand-side computes to zero since  ∂

 σvp  | |σvp

 =

  σvp σvp 1 E − E ≡ 0.  |  |2 |σ  | |σvp |σvp vp

(6.105)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-handside computes from requiring satisfaction of ∂ [φn − λ η/t n ] = 0 for ongoing visco-plastic flow at the end of the time step, i.e.    σvp η  η  . ∂ ∂ (λ) = 0. (λ) = ∂ φ  − E + E − E +   | t n |σvp t n

(6.106)

320

6 Visco-Plasticity

As a conclusion the algorithmic tangent E a is thus finally expressed as E an = E − H0 (λ)

E2 . E + η/t n

(6.107)

Note that, consequently, the algorithmic tangent degenerates to E a = 0 for η → 0 and λ > 0, likewise it degenerates to E a = E for t n → 0. In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Perzyna model is summarized in Table 6.5.

Table 6.5 Algorithmic update for the specific Perzyna model Input

n n−1 vp

Trial Strain

vp = n−1 vp

Trial Stress

 = −E [ − n ] σvp vp

Trial Yield

 |−σ φ = |σvp y

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E + η/t n

ENDIF Update Strain nvp = vp + λ

 σvp  | |σvp

Update Stress

σ n = E [n − nvp ]

Tangent

E an = E − H0 (λ)

Output

σ n nvp E an

E2 E + η/t n

6.2 Perzyna Model

321

6.2.3 Specific Perzyna Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the specific Perzyna model to a prescribed Zig-Zag strain history is documented in Fig. 6.12a–e. (These shall be compared to the corresponding response of the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell, Bingham and Prandtl models in Figs. 4.41a–f, 6.4a–e and 5.9a–e, respectively.) Figure 6.12a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.12b showcases the resulting stress history σ(t) that displays a periodic signal with |σ(t)| → σy + η |˙(t)| = 1.375 (since ˙(t) = ±5) at the Zig-Zag turning points. The resulting σ = σ() diagram is highlighted in Fig. 6.12c. The initial elastic slope (E = 1) and the elastic slope upon strain reversal at |σ(t)| = 1.375 are easy to verify. Figure 6.12d demonstrates the corresponding visco-plastic strain history vp (t) with |vp (t)| → 5 − 1.375/1 = 3.625. Finally, the strain arc-length κ(t) in Fig. 6.12e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 35 (from visual inspection). Prescribed Strain History: Sine The response of the specific Perzyna model to a prescribed Sine strain history is documented in Fig. 6.13a–e. (These shall be compared to the corresponding response of the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell, Bingham and Prandtl models in Figs. 4.42a–f, 6.5a–e and 5.10a–e, respectively.) Figure 6.13a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.13b showcases the resulting stress history σ(t) that displays—after an initial phase—a (slightly upwards shifted) periodic signal. The resulting σ = σ() diagram is highlighted in Fig. 6.13c. The initial elastic slope (E = 1) and the elastic slope upon strain reversal at |(t)| = 5 are easy to verify.

322

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.12 Response analysis of the specific Perzyna model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

6.2 Perzyna Model

323 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.13 Response analysis of the specific Perzyna model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

324

6 Visco-Plasticity

Figure 6.13d demonstrates the corresponding visco-plastic strain history vp (t) with |vp (t)| → 4 (from visual inspection). Finally, the strain arc-length κ(t) in Fig. 6.13e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 37.5 (from visual inspection). Prescribed Strain History: Ramp The response of the specific Perzyna model to a prescribed Ramp strain history is documented in Fig. 6.14a–e. (These shall be compared to the corresponding response of the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell, Bingham and Prandtl models in Figs. 4.43a–f, 6.6a–e and 5.11a–e, respectively.) Figure 6.14a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.14b showcases the resulting stress history σ(t) that displays an increasing signal with |σ(t)| → σy + η |˙(t)| = 1.375 whenever ˙(t) = ±5 in the loading and the unloading phases. During the holding phase with ˙(t) = 0 the stress relaxes to σ(t) → σy = 1. The resulting σ = σ() diagram is highlighted in Fig. 6.14c. The elastic slope (E = 1) in the loading and unloading phase are easy to verify. Likewise the stress relaxation from σ = 1.375 to σ = 1 during the holding phase is clearly visible at  = 5. Figure 6.14d demonstrates the corresponding visco-plastic strain history vp (t) with |vp (t)| → 4 and |vp (t)| → 1.3 in the loading and unloading phase, respectively (from visual inspection). Finally, the strain arc-length κ(t) in Fig. 6.14e follows from integrating κ(t) ˙ = |˙vp (t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax ≈ 6.7 (from visual inspection). Prescribed Stress History: Zig-Zag The response of the specific Perzyna model to a prescribed Zig-Zag stress history is documented in Fig. 6.15a–e. (These shall be compared to the corresponding response of the underlying, visco-elastic and rigid-visco-plastic, specific Maxwell and Bingham models in Figs. 4.44a–f and 6.7a–e, respectively.) Figure 6.15a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.15b showcases the resulting strain history (t) that displays a periodic signal with (t) ∈ [0, 45] (from visual inspection).

6.2 Perzyna Model

325 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

(

c)

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 6.14 Response analysis of the specific Perzyna model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

326

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 75.0)

σ

c)

(

max,min

= ± 75.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 75.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.15 Response analysis of the specific Perzyna model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.2 Perzyna Model

327

The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 6.15c. Figure 6.15d demonstrates the corresponding visco-plastic strain history vp (t), ¯ which is also a periodic signal with vp (t) ∈ [0, 42.6]. Finally, the strain arc-length κ(t) in Fig. 6.15e follows from integrating κ(t) ˙ = ¯ |˙vp (t)| over two and a half periods and approaches κmax ≈ 213.3. Figure 6.15d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.7d and e since they are both exposed to the identical stress history. Prescribed Stress History: Sine The response of the specific Perzyna model to a prescribed Sine stress history is documented in Fig. 6.16a–e. (These shall be compared to the corresponding response of the underlying, visco-elastic and rigid-visco-plastic, specific Maxwell and Bingham models in Figs. 4.45a–f and 6.8a–e, respectively.) Figure 6.16a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.16b showcases the resulting strain history (t) that displays a periodic signal with (t) ∈ [0, 47] (from visual inspection). The resulting (lentil-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 6.16c. Figure 6.16d demonstrates the corresponding visco-plastic strain history vp (t), which is also a periodic signal with vp (t) ∈ [0, 59.9]. Finally, the strain arc-length κ(t) in Fig. 6.16e follows from integrating κ(t) ˙ = |˙(t)| over two and a half periods and approaches κmax = 299.6. Figure 6.16d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.8d and e since they are both exposed to the identical stress history. Prescribed Stress History: Ramp The response of the specific Perzyna model to a prescribed Ramp stress history is documented in Fig. 6.17a–e. Figure 6.17a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles.

328

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 75.0)

σ

c)

(

max,min

= ± 75.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 75.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.16 Response analysis of the specific Perzyna model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.2 Perzyna Model

329

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 500.0)

σ

c)

(

max,min

= ± 500.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1 vp,max,min = ± 500.0)

e)

(tmax = 100 × 0.1

κmax = 500.0)

Fig. 6.17 Response analysis of the specific Perzyna model with material data: E = 1.0, σy = 1.0, η = 0.075. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

330

6 Visco-Plasticity

Figure 6.17b showcases the resulting strain history (t) that displays an increas¯ The holding phase with σ(t) ing signal with (t) → 469.3. ˙ = 0 is characterised by ¯ (Here the elastic contribution to extensive creep with ˙(t) = [5 − 1]/0.075 = 53.3. the strain with at most σa /E = 5 is negligible in the eye-ball-norm.) The resulting σ = σ() diagram is highlighted in Fig. 6.17c. The creep towards  ≈ 450 (from visual inspection) during the holding phase is clearly visible at σ = 5. Figure 6.17d demonstrates the corresponding visco-plastic strain history vp (t) ¯ (In the eye-ball-norm the difference to the strain signal is with |vp (t)| → 469.3. negligible due to the smallness of the elastic strain e ≤ 5.) Finally, the strain arc-length κ(t) in Fig. 6.17e follows from integrating κ(t) ˙ = ¯ |˙(t)| over two and a half periods and approaches κmax = 469.3. Figure 6.17d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.9d and e since they are both exposed to the identical stress history.

6.2.4 Generic Perzyna Model: Formulation A generic formulation of the Perzyna model can be obtained from generalizing the specific Perzyna model in Fig. 6.10 by assuming the elastic spring or/and the viscous dashpot or/and the frictional slider as nonlinear. For the generic Perzyna model the free energy density ψ is expressed as a nonquadratic but convex function of  − vp (the elastic strain e ) ψ(, vp = ψ( − vp ).

(6.108)

Note that ψ(, vp ) and ψ( − vp ) are different functions that return, however, the same function value for the same values of  and vp . Then the energetic stress σ   and the energetic visco-plastic stress σvp follow as σ  (, vp ) = ∂ ψ(, vp ) = ∂ ψ( − vp ),

(6.109a)

 σvp (, vp )

(6.109b)

= ∂vp ψ(, vp ) = ∂vp ψ( − vp ).

Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ  ≡ σ and the negative of the energetic visco ≡ σ. Moreover the energetic and the dissipative visco-plastic plastic stress −σvp   stresses are constitutively related by σvp + σvp = 0, thus the notion of visco-plastic   stress defined as σvp := σvp = −σvp will exclusively be used in the sequel. Note moreover that the visco-plastic stress σvp in the viscous frictional slider decomposes additively into the viscous stress σv in the viscous dashpot and the plastic stress σp in the frictional slider. Furthermore, for the generic Perzyna model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙vp ) and π ∗ = π ∗ (σvp ), respectively, are related via corresponding Legendre transformations

6.2 Perzyna Model

331

π ( ˙vp ) = max{σvp ˙vp − π ∗ (σvp )},

(6.110a)

π ∗ (σvp ) = max{σvp ˙vp − π ( ˙vp )}.

(6.110b)

σvp

˙ vp

Then the stationarity conditions corresponding to Eqs. 6.110a and 6.110b are the constitutive relations (6.111a) ˙vp (σvp ) ∈ dσvp π ∗ (σvp ), σvp ( ˙vp ) ∈ d ˙vp π ( ˙vp ).

(6.111b)

Obviously the relations in Eqs. 6.111a and 6.111b determine entirely the dissipative behavior of the generic Perzyna model, thus the formulation would be completed at this stage. To be more explicit, however, alternatively to Eq. 6.111b the closed and convex admissible domain A in the σvp -space is introduced. It is characterized by the convex yield condition (6.112) φ = φ(σvp ) := ϕ(σvp ) − σy ≤ 0. Here φ = φ(σvp ) is the overstress function and ϕ(σvp ) denotes the equivalent (visco-plastic) stress that is compared to the yield limit σy , a material property. Then the evolution law for the visco-plastic strain (i.e. the associated flow rule) follows alternatively to Eq. 6.111a from the postulate of maximum dissipation (due to viscoplasticity) 1 φ(σvp ) 2 → min, ˜1/η (σvp ; ˙vp ) := −d(σvp ; ˙vp ) + 2 η

(6.113)

whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint φ ≤ 0 penalized by the penalty parameter 1/η. Consequently, the stationarity condition of this unconstrained optimization problem reads ˙vp = λ ∂σvp φ with λ := φ(σvp ) /η ≥ 0.

(6.114)

It shall be noted that collectively Eqs. 6.112 and 6.114 are entirely equivalent statements to Eqs. 6.111a and 6.111b. As a further interesting aspect the dissipation d = σvp ˙vp shall next be examined more closely. From Eqs. 6.110a and 6.110b the dissipation d is alternatively expressed in terms of the dissipation potential π and the dual dissipation potential π ∗ as d = π(˙vp ) + π ∗ (σvp ) ≥ 0.

(6.115)

Thereby, based on the above introduction of the overstress function φ (and in view of Eqs. 6.110a, 6.111a, 6.113 and 6.114) the dual dissipation potential is identified

332

6 Visco-Plasticity

Table 6.6 Summary of the generic Perzyna model (1) Strain



= e + vp

(2) Energy

ψ

= ψ( − vp )

Stress

σ

= ∂ ψ

(3)

(4) Potential π (5)

Stress

≡

σ

≡



−σvp

= π(˙vp )

σvp ∈ d˙vp π





σvp

or (4) Potential π ∗ =

1 2

φ(σvp ) 2 /η

(5) Evolution ˙ vp = λ ∂σvp φ with λ : =

as π ∗ (σvp ) =

⎧ ⎨

⎫ φ(σvp ) ≤ 0 ⎬

0

⎩1

φ(σvp )2 /η 2

φ(σvp ) /η

for φ(σvp ) > 0

⎭

=

1 φ(σvp ) 2 . 2 η

(6.116)

Finally for an equivalent (visco-plastic) stress that is homogeneous of degree one in the visco-plastic stress (thus σvp ∂σvp ϕ = ϕ), the dissipation d = σvp ˙vp is exclusively given in terms of the overstress function φ (with abbreviation λ := φ /η ≥ 0 for the visco-plastic multiplier and equivalent stress ϕ = φ + σy ≥ 0), since then d = λ σvp ∂σvp ϕ = λ ϕ = φ [φ + σy ]/η.

(6.117)

The generic Perzyna model is summarized in Table 6.6.

6.3 Perzyna Hardening Model The Perzyna model of a hardening elasto-visco-plastic solid (in short the Perzyna hardening model) consists of a serial arrangement of (1) an elastic spring and (2) a hardening viscous frictional slider consisting of a parallel arrangement of (i) a frictional slider, (ii) a viscous dashpot and (iii) a hardening spring (see the sketch of the specific Perzyna hardening model in Fig. 6.18). The basic kinematic assumption of the Perzyna hardening model is the additive decomposition of the total strain  into the elastic strain e (representing the elongation of the elastic spring) and the visco-plastic strain vp (representing the elongation of the hardening viscous frictional slider), i.e.  = e + vp .

(6.118)

6.3 Perzyna Hardening Model

333

Note that the visco-plastic strain vp together with the hardening strain εh measuring the elongation of the hardening spring denote the only elements contained in the set of internal variables α = {vp , εh } for the Perzyna hardening model.

6.3.1 Specific Perzyna Isotropic Hardening Model: Formulation The specific Perzyna isotropic hardening model, similar to that displayed in Fig. 6.18 (however with the hardening modulus H and the hardening strain εh coinciding here with the isotropic-hardening modulus H and the isotropic-hardening strain hi , respectively), consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear isotropic-hardening viscous frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy , (ii) a linear viscous dashpot with viscosity η, and (iii) a linear isotropic-hardening spring with stiffness H (the isotropic-hardening modulus). For the specific Perzyna isotropic hardening model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − vp (the elastic strain e ) and hi (the isotropic-hardening strain) ψ(, vp , hi ) =

1 1 E [ − vp ]2 + H 2hi . 2 2

(6.119)

εh H

E

η

σ

σ σy

e

Fig. 6.18 Specific Perzyna hardening model

vp

334

6 Visco-Plasticity

Then the energetic stress σ  conjugated to the total strain  and the energetic  conjugated to the visco-plastic strain vp together with the visco-plastic stress σvp  isotropic-hardening stress σhi conjugated to the isotropic-hardening strain hi follow as σ  (, vp

) = ∂ ψ(, vp , hi ) =

E [ − vp ],

(6.120a)

 σvp (, vp  σhi (

) = ∂vp ψ(, vp , hi ) = −E [ − vp ],

(6.120b)

hi ) = ∂hi ψ(, vp , hi ) =

H hi

.

(6.120c)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the isotropic-hardening viscous  ≡ σ. frictional slider, also with the negative of the energetic visco-plastic stress, −σvp Furthermore, for the specific Perzyna isotropic hardening model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous dissipation potential πv , together with a convex and non-smooth plastic contribution, the plastic dissipation potential πp , i.e. π(˙vp , ˙hi ) = πv (˙vp ) + πp (˙vp , ˙hi ).

(6.121)

Thereby the viscous and plastic contributions πv and πp to the (total) dissipation potential π are chosen as πv (˙vp ) =

1 η |˙vp |2 and 2

πp (˙vp , ˙hi ) = [σy + H hi ] |˙vp | − H hi ˙hi . (6.122)

Observe that (i) π = πv + πp does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) πv is positively homogenous of degree two in ˙vp and obviously smooth at the origin ˙vp = 0, and that (iii) πp is positively homogenous of degree one in {˙p , ˙hi } and obviously non-smooth at the origin {˙p , ˙hi } = {0, 0}. As a consequence of the additive structure of the (total)  consists of a viscous dissipation potential the dissipative visco-plastic stress σvp  contribution, the dissipative viscous overstress σv , and a plastic contribution, the dissipative plastic stress σp , i.e.  σvp (˙vp , ˙hi ) = σv (˙vp ) + σp (˙vp , ˙hi ).

(6.123)

Thereby the dissipative viscous overstress σv computes as partial derivative of the viscous dissipation potential with respect to its conjugated variable σv (˙vp ) = ∂˙vp πv (˙vp ) = η ˙vp ,

(6.124)

6.3 Perzyna Hardening Model

335

whereas the dissipative plastic stress σp and the dissipative isotropic-hardening stress  σhi compute as some sub-derivatives of the plastic dissipation potential with respect to their conjugated variables σp (˙vp , ˙hi ) ∈ d˙vp π(˙vp , ˙hi ),

(6.125)



σhi (˙vp , ˙hi ) ∈ d˙hi π(˙vp , ˙hi ), with ⎧ ⎫ +[σy + H hi ]  ˙ > 0 ⎨ ⎬ vp d˙vp πp (˙vp , ˙hi ) = −[σy + H hi ], +[σy + H hi ] for ˙vp = 0 , ⎩ ⎭ ˙vp < 0 −[σy + H hi ] d˙hi πp (˙vp , ˙hi ) =

(6.126)

−H hi ,

whereby d˙vp πp and d˙hi πp denote the sets of sub-derivatives, i.e. the sub-differentials of πp with respect to ˙vp and ˙hi , respectively. Recall that the energetic and the dissipative visco-plastic as well a the isotropic    + σvp = 0 and σhi + σhi = 0, hardening stresses are constitutively related by σvp respectively, thus the notions of visco-plastic stress (together with the notions of viscous overstress and plastic stress) as well as of isotropic-hardening stress defined as the values   = −σvp with σv := σv and σvp = σv + σp := σvp

:=

σhi

 σhi

=

σp := σp ,

 −σhi ,

(6.127a) (6.127b)

will exclusively be used in the sequel for convenience of exposition. Separate Viscous Overstress and Plastic Stress The Perzyna isotropic hardening model may be formulated further by considering the viscous overstress σv and the plastic stress σp separately. Thereby, due to the non-smooth plastic dissipation potential, the plastic stress is constrained to reside in an admissible domain. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving forces, i.e. in the {σvp , σhi }-space, is next introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.13. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙vp πp in Eq. 6.126, or, alternatively, from evaluating the formal definition of the sub-differential

{σp |

σp [˙vp

d˙vp πp (˙vp , ˙hi ) =  − ˙vp ] ≤ [σy + H hi ] |˙vp | − |˙vp | ∀˙vp },

(6.128)

336

6 Visco-Plasticity

whereby ˙vp denotes any admissible visco-plastic strain rate. Then at ˙vp = 0 it holds for any admissible ˙vp that σp ˙vp ≤ [σy + H hi ] |˙vp | and, with max˙vp {σp ˙vp /|˙vp |} = |σp |, the admissible domain follows as |σp | ≤ σy + H hi . Moreover, the subdifferential d˙hi πp reduces to the partial derivative ∂˙hi πp and renders σhi = −H hi . Thus the admissible domain is eventually expressed as |σp | ≤ σy − σhi . The elastic domain is defined as the interior of the admissible domain, i.e.   int A := {σp , σhi } | |σp | − [σy − σhi ] < 0 ,

(6.129)

whereas the yield surface, which in the present one-dimensional case collapses to the two lines σp = ±[σy − σhi ], is defined as the boundary of the admissible domain, i.e.   (6.130) ∂ A := {σp , σhi } | |σp | − [σy − σhi ] = 0 . Collectively, the admissible domain in the {σp , σhi }-space is characterized by the yield condition (6.131) |σp | − [σy − σhi ] ≤ 0. States in the interior int A of the admissible domain with |σp | < σy − σhi are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp | = σy − σhi are visco-plastic. The corresponding dual viscous and plastic dissipation potentials πv∗ and πp∗ , as determined from the Legendre transformations

 1 ) = max σv ˙vp − η |˙vp |2 ˙ vp 2 ∗ πp (σp , σhi ) = max {σp ˙vp + σhi ˙hi − [σy + H hi ] |˙vp | + H hi ˙hi }

πv∗ (σv

˙ vp ,˙hi

(6.132a) (6.132b)

then read with the stationarity condition σhi = −H hi (note the minus sign) πv∗ (σv

)=

1 1 |σv |2 2 η

πp∗ (σp , σhi ) = IA (σp , σhi ) :=

(6.133a) ⎧ ⎨ 0 ⎩

⎫ |σp | ≤ σy − σhi ⎬ for

∞

|σp | > σy − σhi

⎭

,

(6.133b)

where IA denotes the indicator function of the admissible domain A in the {σp , σhi }space. The evolution laws (the associated flow rules) for the visco-plastic and the isotropic-hardening strains then follow either as the partial derivative of the dual viscous dissipation potential or likewise as some sub-derivatives of the dual plastic dissipation potential, in either case with respect to their conjugated variables

6.3 Perzyna Hardening Model

˙vp (σv

337

) = ∂σv πv∗ (σv

˙vp (σp , σhi ) ∈

),

dσp πp∗ (σp , σhi )

= dσp IA (σp , σhi ),

(6.134)

˙hi (σp , σhi ) ∈ dσhi πp∗ (σp , σhi ) = dσhi IA (σp , σhi ), with

1 σv η

∂σv πv∗ (σv ) = and dσp πp∗ (σp , σhi )

⎧ ⎪ ⎨

0

σ = dσp IA (σp , σhi ) = ⎪ ⎩λ p |σp | and dσhi πp∗ (σp , σhi )

for

⎫ |σp | < σy − σhi ⎪ ⎬

⎫ |σp | < σy − σhi ⎬ for

λ

(6.135b)

⎭ |σp | = σy − σhi ⎪

⎧ ⎨0

= dσhi IA (σp , σhi ) = ⎩

(6.135a)

|σp | = σy − σhi

⎭

,

(6.135c)

whereby dσp πp∗ and dσhi πp∗ denote the sets of sub-derivatives, i.e. the sub-differentials of πp∗ with respect to σp and σhi , respectively, and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 6.124, 6.125 and 6.134 are inverse relations. The smooth viscous dissipation and dual viscous dissipation potentials πv = πv (˙vp ) and πv∗ = πv∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙vp ) and ˙vp = ˙vp (σv ) are similar to those displayed in Fig. 4.2. Identifying ˙hi with |˙vp | and setting σhi = 0, the remaining non-smooth plastic dissipation and dual plastic dissipation potentials πp = πp (˙vp ) and πp∗ = πp∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙vp ) and ˙vp = ˙vp (σp ) are similar to those displayed in Fig. 5.3. Visco-Plastic Stress Alternatively, the Perzyna isotropic hardening model may be formulated further by considering the visco-plastic stress. To this end the viscous and the plastic stress need to be related to the visco-plastic stress. Remarkably, since at yield the plastic stress satisfies |σp | = σy − σhi , the viscous overstress σv = σvp − σp allows representation in terms of the yield condition that is, however, evaluated in terms of the visco-plastic stress σvp ⎧ +|σvp | − σy + σhi ⎪ ⎪ ⎨ 0 σv = ⎪ ⎪ ⎩ −|σvp | + σy − σhi

if

⎫ σvp > +σy − σhi ⎪ ⎪ ⎬ else . ⎪ ⎪ ⎭ σvp < −σy + σhi

(6.136)

338

6 Visco-Plasticity

The reasoning for the representation in Eq. 6.136 is highlighted in Fig. 6.19 and follows as: • For ˙vp = 0 the overstress in the viscous damper is identically zero, i.e. σv ≡ 0 and thus the stress σp in the isotropic-hardening frictional slider coincides identically with the visco-plastic stress σp ≡ σvp . Consequently, and again since ˙vp = 0, the visco-plastic stress satisfies |σvp | ≤ σy − σhi . • For ˙vp > 0 (with σvp > +[σy − σhi ]) the overstress in the viscous damper results in σv = σvp − [σy − σhi ] ≡ +|σvp | − σy + σhi and thus the stress σp in the isotropic-hardening frictional slider coincides identically with the (positive) current yield stress σp ≡ +[σy − σhi ]. • For ˙vp < 0 (with σvp < −[σy − σhi ]) the overstress in the viscous damper results in σv = σvp + [σy − σhi ] ≡ −|σvp | + σy − σhi and thus the stress σp in the isotropic-hardening frictional slider coincides identically with the (negative) current yield stress σp ≡ −[σy − σhi ]. Finally the above relations may conveniently be summarized as

σv =

⎧ ⎪ ⎨

0

 σvp ⎪ ⎩ |σvp | − [σy − σhi ] |σvp |

for

⎫ |σvp | < σy − σhi ⎪ ⎬ ⎭ |σvp | ≥ σy − σhi ⎪

.

(6.137)

Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.137, the two variants of the associated evolution law for the visco-plastic strain in Eqs. 6.135a, 6.135b are alternatively expressed in terms of the visco-plastic stress

˙vp (σvp , σhi ) =

⎧ ⎪ ⎪ ⎨

0

|σvp | − [σy − σhi ] σvp ⎪ ⎪ ⎩ η |σvp |

for

⎫ |σvp | < σy − σhi ⎪ ⎪ ⎬ ⎪ |σvp | ≥ σy − σhi ⎪ ⎭

.

(6.138)

Consequently, based on Eq. 6.135c rendering ˙hi = |˙vp |, the associated evolution law for the isotropic-hardening strain follows as

˙hi (σvp , σhi ) =

⎧ ⎪ ⎪ ⎨

0

|σvp | − [σy − σhi ] ⎪ ⎪ ⎩ η

for

⎫ |σvp | < σy − σhi ⎪ ⎪ ⎬ ⎪ |σvp | ≥ σy − σhi ⎪ ⎭

.

(6.139)

Obviously, the expressions in Eqs. 6.124, 6.125 and 6.138, 6.139 are inverse relations. With the representation for the evolution of the visco-plastic and isotropichardening strains in Eqs. 6.138 and 6.139, the corresponding (total) dual dissipation potential π ∗ , as determined from the Legendre transformation

6.3 Perzyna Hardening Model

339 σp

σv

+[σy − σhi ] −[σy − σhi ] +[σy − σhi ]

σvp

σvp −[σy − σhi ]

Fig. 6.19 Specific Perzyna isotropic hardening model: The visco-plastic stress σvp = σv + σp is the sum of the viscous overstress σv and the plastic stress σp . The viscous damper is only activated once the load carrying capacity of the isotropic-hardening frictional slider is exceeded. Accordingly the viscous overstress is identically zero σv ≡ 0 for |σvp | − [σy − σhi ] ≤ 0 (left), while the plastic stress remains constant (at a particular σhi that may be considered to expand along a third dimension perpendicular to the plane displayed in the above) with |σp | = σy − σhi for |σvp | − [σy − σhi ] > 0 (right)



∗

π (σvp , σhi ) = max σvp ˙vp + σhi ˙hi

(6.140)

˙ vp ,˙hi

1 − η |˙vp |2 − [σy + H hi ] |˙vp | + H hi ˙hi 2



then reads4

4 The

expressions for the evolution of the visco-plastic and the isotropic-hardening strains in Eqs. 6.138, 6.139 result in σvp ˙ vp (σvp , σhi ) + σhi ˙ hi (σvp , σhi ) = ⎧ ⎫ 0 |σvp | < σy − σhi ⎪ ⎪ ⎬ ⎨  for |σvp | + σhi |σ | − [σy − σhi ] ⎪ ⎪ ⎩ vp |σvp | ≥ σy − σhi ⎭ η and ⎫ ⎧ 0 |σvp | < σy − σhi ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ for 1 2  η |˙vp (σvp , σhi )|2 = | − [σ − σ ] |σ ⎪ ⎪ 1 2 vp y hi ⎪ ⎭ ⎩ |σvp | ≥ σy − σhi ⎪ 2 η and [σy + H hi (σhi )] |˙vp (σvp , σhi )| − H hi (σhi ) ˙ hi (σvp , σhi ) =

340

6 Visco-Plasticity

⎫ |σvp | < σy − σhi ⎪ ⎪ ⎬ for ∗  . π (σvp , σhi ) = 2 ⎪ ⎪ 1 |σvp | − [σy − σhi ] ⎪ ⎭ ⎩ |σvp | ≥ σy − σhi ⎪ 2 η ⎧ ⎪ ⎪ ⎨

0

(6.141)

The above relations may conveniently be condensed by the help of the Macaulay bracket • := 21 [• + | • |], e.g. the dual dissipation potential is expressed as π ∗ (σvp , σhi ) =

1 |σvp | − [σy − σhi ] 2 . 2 η

(6.142)

Identifying ˙hi with |˙vp | and setting σhi = 0, the remaining non-smooth (total) dissipation and dual (total) dissipation potentials π = π(˙vp ) and π ∗ = π ∗ (σvp ) together with the resulting non-smooth constitutive relations σvp = σvp (˙vp ) and ˙vp = ˙vp (σvp ) are similar to those displayed in Fig. 6.3. The result in Eqs. 6.138, 6.139 for the evolution of the visco-plastic and the isotropic-hardening strains thus follows directly from the reverse Legendre transformation 

1 |σvp | − [σy − σhi ] 2 , (6.143) π(˙vp , ˙hi ) = max d(σvp , σhi ; ˙vp , ˙hi ) − σvp ,σhi 2 η whereby d(σvp , σhi ; ˙vp , ˙hi ) := σvp ˙vp + σhi ˙hi denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.143 embodies the unconstrained optimization problem ˜1/η (σvp , σhi ; ˙vp , ˙hi ) := −d(σvp , σhi ; ˙vp , ˙hi ) +

(6.144)

1 |σvp | − [σy − σhi ] 2 → min , σvp ,σhi 2 η

whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint |σvp | ≤ σy − σhi penalized by the penalty parameter 1/η. In accordance with Eqs. 6.138, 6.139 the stationarity conditions of this unconstrained optimization problem then read

σy

⎧ ⎪ ⎨

0

|σ | − [σy − σhi ] ⎪ ⎩ vp η

for

⎫ |σvp | < σy − σhi ⎪ ⎬ ⎪ |σvp | ≥ σy − σhi ⎭

Taken together, the (total) dual dissipation potential π ∗ (σvp , σhi ) follows.

.

6.3 Perzyna Hardening Model

341

|σvp | − [σy − σhi ] σvp , η |σvp | |σvp | − [σy − σhi ] . ˙hi (σvp , σhi ) = η ˙vp (σvp , σhi ) =

(6.145a) (6.145b)

Finally, the visco-plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated visco-plastic deformation, i.e.

Table 6.7 Summary of the specific Perzyna isotropic hardening model (1) Strain



= e + vp

(2) Energy

ψ

=

(3) Stress

σ

= E [ − vp ]

(4) Stress

σhi = −H hi

(5) Potential πv =

1 2E

1 2

[ − vp ]2 + 21 H 2hi σ

≡

≡



−σvp

η |˙vp |2

(6) Potential πp = σyhi |˙vp | − H hi ˙ hi with σyhi := σy + H hi (7) Stress

σvp = η ˙ vp + σyhi

(8) Stress

σhi = −H hi

˙ vp |˙vp |

≡

 for ˙ vp = 0 σvp

or (5) Potential πv∗ = (6) Yield

0

1 2

|σv |2 /η

≥ |σp | − σyhi σp |σp |

(7) Evolution ˙ vp = λ

σv η

=

(8) Evolution ˙ hi = λ (9) KKT

λ

≥ 0,

|σp |

≤

σyhi ,

or (5) Potential π ∗ =

1 2

(6) Evolution ˙ vp =

|σvp | − σyhi σvp η |σvp |

(7) Evolution ˙ hi =

|σvp | − σyhi 2 /η

|σvp | − σyhi η

λ |σp |

=

λ σyhi

342

6 Visco-Plasticity

κ=

κ˙ dt with κ˙ := |˙vp | = ˙hi =

|σvp | − [σy − σhi ] ≥ 0. η

(6.146)

The specific Perzyna isotropic hardening model is summarized in Table 6.7.

6.3.2 Specific Perzyna Isotropic Hardening Model: Algorithmic Update For the specific Perzyna isotropic hardening model the evolution laws for the viscoplastic strain vp and the isotropic-hardening strain hi are integrated by the implicit Euler backwards method to render nvp := nvp − n−1 vp = λ

n σvp n | |σvp

and nhi := nhi − n−1 = λ, hi

(6.147)

whereby the incremental visco-plastic multiplier λ is defined as λ := t n λn := t n

n n |σvp | − [σy − σhi ]

≥ 0.

η

(6.148)

Consequently, the visco-plastic stress σvp and the isotropic-hardening stress σhi are updated at the end of the time step by n  = −E [nvp − n ] =: σvp − E nvp , σvp n σhi

= −H

nhi

=:

 σhi

−H

nhi

(6.149)

.

  and the trial isotropic-hardening stress σhi Here the trial visco-plastic stress σvp are computable exclusively from known quantities at the beginning of the time step and follow as

 n σvp := −E [n−1 vp −  ],  σhi

:= −H

n−1 hi

(6.150)

.

Incorporating the discretized evolution law for the visco-plastic strain then renders n  σvp = σvp − E λ

n σvp n | |σvp

.

(6.151)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress

6.3 Perzyna Hardening Model

343

  σn vp n  = σvp |σvp | + E λ . n | |σvp

(6.152)

As an immediate consequence the equivalent stress and its trial value are related via n  | = |σvp | − E λ. (6.153) |σvp A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously n σvp n | |σvp

≡

 σvp  | |σvp

.

(6.154)

Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore n  = σhi − H λ. (6.155) σhi Consequently, the yield function at the end of the time step is expressed as n n | − σy + σhi = φ − [E + H ] λ. φn := |σvp

(6.156)

Here the trial value of the yield function φ has been defined as   | − σy + σhi . φ := |σvp

(6.157)

Next for visco-plastic loading with λ > 0 the definition for the incremental visco-plastic multiplier is regrouped to render φn − λ

η = 0. t n

(6.158)

Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form from φ ≥ 0. (6.159) λ = E + H + η/t n Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t n → 0. Once λ is computed all other variables may be updated. In particular, the visco-plastic stress at the end of the time step reads  σvp n  (6.160) = σvp − E λ  . σvp |σvp | n The sensitivity of σvp = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ

344

6 Visco-Plasticity

depends implicitly on n  n ∂ σvp

= E − E λ ∂

 σvp  | |σvp

 −E

 σvp  | |σvp

∂ (λ).

(6.161)

The first derivative term on the right-hand-side computes to zero since  ∂

 σvp



 | |σvp

=

  σvp σvp 1 E − E ≡ 0.  |  |2 |σ  | |σvp |σvp vp

(6.162)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-handside computes from requiring satisfaction of ∂ [φn − λ η/t n ] = 0 for ongoing visco-plastic flow at the end of the time step, i.e.

Table 6.8 Algorithmic update for the specific Perzyna isotropic hardening model Input

n n−1 n−1 vp hi

Trial Strain

vp = n−1 vp hi = n−1 hi

Trial Stress

Trial Yield

 = −E [ − n ] σvp vp  = −H  σhi hi  | − σ + σ φ = |σvp y hi

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E + H + η/t n

ENDIF Update Strain nvp = vp + λ nhi

=

hi

 σvp  | |σvp

+ λ

Update Stress

σ n = E [n − nvp ]

Tangent

E an = E − H0 (λ)

Output

σ n nvp nhi E an

E2 E + H + η/t n

6.3 Perzyna Hardening Model

345

 η  ∂ (λ) = ∂ φ − E + H + t n   σvp η  . E − E + H + ∂ (λ) = 0.  n |σvp | t

(6.163)

As a conclusion the algorithmic tangent E a is thus finally expressed as E an = E − H0 (λ)

E2 . E + H + η/t n

(6.164)

Note that, consequently, the algorithmic tangent degenerates to the plastic case for η → 0 and λ > 0, likewise it degenerates to E a = E for t n → 0. In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Perzyna hardening model capturing isotropic hardening is summarized in Table 6.8.

6.3.3 Specific Perzyna Isotropic Hardening Model: Response Analysis Prescribed Strain History: Zig-Zag The response of the specific Perzyna isotropic hardening model to a prescribed ZigZag strain history is documented in Fig. 6.20a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl isotropic hardening and Perzyna models in Figs. 5.14a–e and 6.12a–e, respectively.) Figure 6.20a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.20b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) = σy + η ˙(t) + H κ(t) = 1.375 + 0.1 κ(t) whenever σ(t) > 1 + 0.1 κ(t) and with ˙(t) = 5, thus σmax ≈ 3.5 (from visual inspection). The resulting σ = σ() diagram is highlighted in Fig. 6.20c. Due to isotropic hardening its resulting rounded parallelogram-type format has constant amplitude in the  direction and is isotropically expanding in the σ direction.

346

6 Visco-Plasticity

Figure 6.20d demonstrates the corresponding visco-plastic strain history vp (t): during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas vp (t) stays constant during the elastic phases with decreasing amplitude after each half-period (and eventually vp (t) → 1.5). Finally, the strain arc-length κ(t) in Fig. 6.20e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax = 21.25 (from visual inspection). Prescribed Strain History: Sine The response of the specific Perzyna isotropic hardening model to a prescribed Sine strain history is documented in Fig. 6.21a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl isotropic hardening and Perzyna models in Figs. 5.15a–e and 6.13a–e, respectively.) Figure 6.21a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.21b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal. The resulting σ = σ() diagram is highlighted in Fig. 6.21c. Due to isotropic hardening its resulting rounded parallelogram-type format has constant amplitude in the  direction and is isotropically expanding in the σ direction. Figure 6.21d demonstrates the corresponding visco-plastic strain history vp (t): during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas vp (t) stays constant during the elastic phases with decreasing amplitude after each half-period. Finally, the strain arc-length κ(t) in Fig. 6.21e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax = 23 (from visual inspection). Prescribed Strain History: Ramp The response of the specific Perzyna isotropic hardening model to a prescribed Ramp strain history is documented in Fig. 6.22a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl isotropic hardening and Perzyna models in Figs. 5.16a–e and 6.14a–e, respectively.) Figure 6.22a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.22b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙(t) = ±5 in the loading and the unloading phases.

6.3 Perzyna Hardening Model

347 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.20 Response analysis of the specific Perzyna isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, η = 0.075. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

348

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.21 Response analysis of the specific Perzyna isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, η = 0.075. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

349 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 6.22 Response analysis of the specific Perzyna isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, η = 0.075. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

350

6 Visco-Plasticity

During the holding phase with ˙(t) = 0 the stress relaxes to σ(t) → σy + 0.1 κ(t) ≈ 1.375 (from visual inspection). The resulting σ = σ() diagram is highlighted in Fig. 6.22c. The elastic slope (E = 1) in the loading and unloading phase are easy to verify. Likewise the stress relaxation to σ = 1.375 during the holding phase is clearly visible at  = 5. Figure 6.22d demonstrates the corresponding visco-plastic strain history vp (t) with vp (t) → 3.75 and vp (t) → 1.925 in the loading and unloading phase, respectively (from visual inspection). Finally, the strain arc-length κ(t) in Fig. 6.22e follows from integrating κ(t) ˙ = |˙vp (t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax ≈ 5.5 (from visual inspection). Prescribed Stress History: Zig-Zag The response of the specific Perzyna isotropic hardening model to a prescribed ZigZag stress history is documented in Fig. 6.23a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl isotropic hardening and Perzyna models in Figs. 5.17a–e and 6.15a–e, respectively.) Figure 6.23a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.23b showcases the resulting strain history (t) that displays a nearly periodic signal after the initial elastic and visco-plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 6.23c. Once the initial elastic and visco-plastic phase is completed the σ = σ() behavior displays less and less hysteresis in the remaining cycles and approaches a purely elastic response. Figure 6.23d demonstrates the corresponding visco-plastic strain history vp (t), which is also a nearly periodic signal with decreasing amplitude after the initial elastic and visco-plastic phase. Finally, the strain arc-length κ(t) in Fig. 6.23e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 35 (from visual inspection). Prescribed Stress History: Sine The response of the specific Perzyna isotropic hardening model to a prescribed Sine stress history is documented in Fig. 6.24a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl isotropic hardening and Perzyna models in Figs. 5.18a–e and 6.16a–e, respectively.) Figure 6.24a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are

6.3 Perzyna Hardening Model

351

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.23 Response analysis of the specific Perzyna isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, η = 0.075. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

352

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.24 Response analysis of the specific Perzyna isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, η = 0.075. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

353

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 80.0)

Fig. 6.25 Response analysis of the specific Perzyna isotropic hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, η = 0.075. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

354

6 Visco-Plasticity

computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.24b showcases the resulting strain history (t) that displays a nearly periodic signal after the initial elastic and visco-plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 6.24c. Once the initial elastic and visco-plastic phase is completed the σ = σ() behavior displays less and less hysteresis in the remaining cycles and approaches a purely elastic response. Figure 6.24d demonstrates the corresponding visco-plastic strain history vp (t), which is also a nearly periodic signal with decreasing amplitude after the initial elastic and visco-plastic phase. Finally, the strain arc-length κ(t) in Fig. 6.24e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 40 (from visual inspection). Prescribed Stress History: Ramp The response of the specific Perzyna isotropic hardening model to a prescribed Ramp stress history is documented in Fig. 6.25a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl isotropic hardening and Perzyna models in Figs. 5.19a–e and 6.17a–e, respectively.) Figure 6.25a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.25b showcases the resulting strain history (t) that displays a smoothly increasing signal in the loading and holding phases with (t) → 45 and a purely elastic behaviour in the unloading phase with (t) → 40. The nonlinear creep during the holding phase saturates due to the isotropic hardening. The resulting σ = σ() diagram is highlighted in Fig. 6.25c. Once the holding phase is completed the σ = σ() behavior in the unloading phase is purely elastic with σ(t) ∈ [5, 0] and slope E = 1, whereby the strain approaches (t) → 40. Likewise the creep towards  = 45 during the holding phase is clearly visible at σ = 5. Figure 6.25d demonstrates the corresponding smoothly and monotonically increasing visco-plastic strain history vp (t) with vp (t) → 40. Finally, the strain arc-length κ(t) in Fig. 6.25e follows from integrating κ(t) ˙ = |˙(t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax = 40.

6.3.4 Specific Perzyna Kinematic Hardening Model: Formulation The specific Perzyna kinematic hardening model, similar to that displayed in Fig. 6.18 (however with the hardening modulus H and the hardening strain εh coinciding

6.3 Perzyna Hardening Model

355

here with the kinematic-hardening modulus K and the kinematic-hardening strain hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear kinematic-hardening viscous frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy , (ii) a linear viscous dashpot with viscosity η, and (iii) a linear kinematic-hardening spring with stiffness K (the kinematic-hardening modulus). For the specific Perzyna kinematic hardening model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − vp (the elastic strain e ) and hk (the kinematic-hardening strain) ψ(, vp , hk ) =

1 1 E [ − vp ]2 + K 2hk . 2 2

(6.165)

Then the energetic stress σ  conjugated to the total strain  and the energetic visco conjugated to the visco-plastic strain vp together with the kinematicplastic stress σvp  hardening stress σhk conjugated to the kinematic-hardening strain hk follow as σ  (, vp

) = ∂ ψ(, vp , hk ) =

E [ − vp ],

(6.166a)

 σvp (, vp  σhk (

) = ∂vp ψ(, vp , hk ) = −E [ − vp ],

(6.166b)

hk ) = ∂hk ψ(, vp , hk ) =

K hk

.

(6.166c)

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the kinematic-hardening viscous  ≡ σ. frictional slider, also with the negative of the energetic visco-plastic stress, −σvp Furthermore, for the specific Perzyna kinematic hardening model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous dissipation potential πv , together with a convex and non-smooth plastic contribution, the plastic dissipation potential πp , i.e. π(˙vp , ˙hk ) = πv (˙vp ) + πp (˙vp , ˙hk ).

(6.167)

Thereby the viscous and plastic contributions πv and πp to the (total) dissipation potential π are chosen as πv (˙vp ) =

1 η |˙vp |2 and 2

πp (˙vp , ˙hk ) = σy |˙vp | + K hk [˙vp − ˙hk ].

(6.168)

Observe that (i) π = πv + πp does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) πv is positively homogenous of degree two in ˙vp and obviously smooth at the origin ˙vp = 0, and that (iii) πp is positively homogenous of degree one in {˙p , ˙hk } and obviously non-smooth at the origin {˙p , ˙hk } = {0, 0}. As a consequence of the additive structure of the (total)  consists of a viscous dissipation potential the dissipative visco-plastic stress σvp

356

6 Visco-Plasticity

contribution, the dissipative viscous overstress σv , and a plastic contribution, the dissipative plastic stress σp , i.e.  σvp (˙vp , ˙hk ) = σv (˙vp ) + σp (˙vp , ˙hk ).

(6.169)

Thereby the dissipative viscous overstress σv computes as partial derivative of the viscous dissipation potential with respect to its conjugated variable σv (˙vp ) = ∂˙vp πv (˙vp ) = η ˙vp ,

(6.170)

whereas the dissipative plastic stress σp and the dissipative kinematic-hardening  stress σhk compute as some sub-derivatives of the plastic dissipation potential with respect to their conjugated variables σp (˙vp , ˙hk ) ∈ d˙vp π(˙vp , ˙hk ),

(6.171)



σhi (˙vp , ˙hk ) ∈ d˙hk π(˙vp , ˙hk ), with ⎧ ⎫ +[σy + K hk ] ˙vp > 0 ⎬ ⎨ d˙vp πp (˙vp , ˙hk ) = −[σy − K hk ], +[σy + K hi ] for ˙vp = 0 , ⎩ ⎭ ˙vp < 0 −[σy − K hi ] d˙hk πp (˙vp , ˙hk ) =

(6.172)

−K hk ,

whereby d˙vp πp and d˙hk πp denote the sets of sub-derivatives, i.e. the sub-differentials of πp with respect to ˙vp and ˙hk , respectively. Recall that the energetic and the dissipative visco-plastic as well a the kinematic    + σvp = 0 and σhk + σhk = 0, hardening stresses are constitutively related by σvp respectively, thus the notions of visco-plastic stress (together with the notions of viscous overstress and plastic stress) as well as of kinematic-hardening stress defined as the values   = −σvp with σv := σv and σvp = σv + σp := σvp

σhk

:=

 σhk

=

σp := σp ,

 −σhk ,

(6.173a) (6.173b)

will exclusively be used in the sequel for convenience of exposition. Separate Viscous Overstress and Plastic Stress The Perzyna kinematic hardening model may be formulated further by considering the viscous overstress σv and the plastic stress σp separately. Thereby, due to the non-smooth plastic dissipation potential, the plastic stress is constrained to reside in an admissible domain.

6.3 Perzyna Hardening Model

357

The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving forces, i.e. in the {σvp , σhk }-space, is next introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.20. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙vp πp in Eq. 6.172, or, alternatively, from evaluating the formal definition of the sub-differential

{σp |

σp [˙vp

(6.174) d˙vp πp (˙vp , ˙hk ) =     − ˙vp ] ≤ σy |˙vp | − |˙vp | + K hk [˙vp − ˙vp ] ∀˙vp },

whereby ˙vp denotes any admissible visco-plastic strain rate. Then at ˙vp = 0 it holds for any admissible ˙vp that σp ˙vp ≤ σy |˙vp | + K hk ˙vp and, with max˙vp {[σp − K hk ] ˙vp /|˙vp |} = |σp − K hk |, the admissible domain follows as |σp − K hk | ≤ σy . Moreover, the sub-differential d˙hk πp reduces to the partial derivative ∂˙hk πp and renders σhk = −K hk . Thus the admissible domain is eventually expressed as |σp + σhk | ≤ σy . The elastic domain is defined as the interior of the admissible domain, i.e.   int A := {σp , σhk } | |σp + σhk | − σy < 0 ,

(6.175)

whereas the yield surface, which in the present one-dimensional case collapses to the two lines σp + σhk = ±σy , is defined as the boundary of the admissible domain, i.e.   (6.176) ∂ A := {σp , σhk } | |σp + σhk | − σy = 0 . Collectively, the admissible domain in the {σp , σhk }-space is characterized by the yield condition (6.177) |σp + σhk | − σy ≤ 0. States in the interior int A of the admissible domain with |σp + σhk | < σy are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp + σhk | = σy are visco-plastic. The corresponding dual viscous and plastic dissipation potentials πv∗ and πp∗ , as determined from the Legendre transformations

 1 ) = max σv ˙vp − η |˙vp |2 ˙ vp 2 πp∗ (σp , σhk ) = max {σp ˙vp + σhk ˙hk − σy |˙vp | − K hk [˙vp − ˙hk ]} πv∗ (σv

˙ vp ,˙hk

(6.178a) (6.178b)

then read with the stationarity condition σhk = −K hk (note the minus sign)

358

6 Visco-Plasticity

πv∗ (σv

)=

1 1 |σv |2 2 η

(6.179a)

πp∗ (σp , σhk ) = IA (σp , σhk ) :=

⎧ ⎨ 0 ⎩

⎫ |σp + σhk | ≤ σy ⎬ for |σp + σhk | > σy

∞

⎭

,

(6.179b)

where IA denotes the indicator function of the admissible domain A in the {σp , σhk }space. The evolution laws (the associated flow rules) for the visco-plastic and the kinematic-hardening strains then follow either as the partial derivative of the dual viscous dissipation potential or likewise as some sub-derivatives of the dual plastic dissipation potential, in either case with respect to their conjugated variables ˙vp (σv

) = ∂σv πv∗ (σv

),

˙vp (σp , σhk ) ∈ dσp πp∗ (σp , σhk ) = dσp IA (σp , σhk ), ˙hk (σp , σhk ) ∈

dσhk πp∗ (σp , σhk )

with ∂σv πv∗ (σv ) = and

⎧ ⎪ ⎪ ⎨

0

(6.180)

= dσhk IA (σp , σhk ),

1 σv η

(6.181a)

⎫ |σp + σhk | < σy ⎪ ⎪ ⎬

dσp πp∗ (σp , σhk ) for σp + σhk = dσp IA (σp , σhk ) = ⎪ ⎪ ⎪ |σp + σhk | = σy ⎪ ⎭ ⎩λ |σp + σhk |

(6.181b)

and ⎧ ⎪ ⎪ ⎨

0

⎫ |σp + σhk | < σy ⎪ ⎪ ⎬

dσhk πp∗ (σp , σhk ) for , σp + σhk = dσhk IA (σp , σhk ) = ⎪ ⎪ ⎪ |σp + σhk | = σy ⎪ ⎭ ⎩λ |σp + σhk |

(6.181c)

whereby dσp πp∗ and dσhk πp∗ denote the sets of sub-derivatives, i.e. the sub-differentials of πp∗ with respect to σp and σhk , respectively, and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 6.170, 6.171 and 6.180 are inverse relations. The smooth viscous dissipation and dual viscous dissipation potentials πv = πv (˙vp ) and πv∗ = πv∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙vp ) and ˙vp = ˙vp (σv ) are similar to those displayed in Fig. 4.2. Identifying ˙hk with ˙vp and setting σhk = 0, the remaining non-smooth plastic dissipation and dual plastic dissipation potentials πp = πp (˙vp ) and πp∗ = πp∗ (σp ) together with the result-

6.3 Perzyna Hardening Model

359

ing non-smooth constitutive relations σp = σp (˙vp ) and ˙vp = ˙vp (σp ) are similar to those displayed in Fig. 5.3. Visco-Plastic Stress Alternatively, the Perzyna kinematic hardening model may be formulated further by considering the visco-plastic stress. To this end the viscous and the plastic stress need to be related to the visco-plastic stress. Remarkably, since at yield the plastic stress satisfies |σp + σhk | = σy , the viscous overstress σv = σvp − σp allows representation in terms of the yield condition that is, however, evaluated in terms of the visco-plastic stress σvp ⎧ +|σvp + σhk | − σy ⎪ ⎪ ⎨ 0 σv = ⎪ ⎪ ⎩ −|σvp + σhk | + σy

if

⎫ σvp + σhk > +σy ⎪ ⎪ ⎬ else . ⎪ ⎪ ⎭ σvp + σhk < −σy

(6.182)

The reasoning for the representation in Eq. 6.182 is highlighted in Fig. 6.26 and follows as: • For ˙vp = 0 the overstress in the viscous damper is identically zero, i.e. σv ≡ 0 and thus the stress σp in the kinematic-hardening frictional slider coincides identically with the visco-plastic stress σp ≡ σvp . Consequently, and again since ˙vp = 0, the visco-plastic stress satisfies |σvp + σhk | ≤ σy . σv

σp + σhk

+σy −σy +σy

σvp + σhk

σvp + σhk −σy

Fig. 6.26 Specific Perzyna kinematic hardening model: The visco-plastic stress σvp = σv + σp is the sum of the viscous overstress σv and the plastic stress σp . The viscous damper is only activated once the load carrying capacity of the frictional slider is exceeded. Accordingly the viscous overstress is identically zero σv ≡ 0 for |σvp + σhk | − σy ≤ 0 (left), while the plastic stress, shifted by the kinematic-hardening stress σhk , remains constant with |σp + σhk | = σy for |σvp + σhk | − σy > 0 (right)

360

6 Visco-Plasticity

• For ˙vp > 0 (with σvp + σhk > +σy ) the overstress in the viscous damper results in σv = [σvp + σhk ] − σy ≡ +|σvp + σhk | − σy and thus the stress σp + σhk in the frictional slider coincides identically with the (positive) yield stress σp + σhk ≡ +σy . • For ˙vp < 0 (with σvp + σhk < −σy ) the overstress in the viscous damper results in σv = [σvp + σhk ] + σy ≡ −|σvp + σhk | + σy and thus the stress σp + σhk in the frictional slider coincides identically with the (negative) yield stress σp + σhk ≡ −σy . Finally the above relations may conveniently be summarized as

σv =

⎧ ⎪ ⎪ ⎨

0

σvp + σhk  ⎪ ⎪ ⎩ |σvp + σhk | − σy ] |σvp + σhk |

for

⎫ |σvp + σhk | < σy ⎪ ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy ⎪ ⎭

.

(6.183)

Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.183, the two variants of the associated evolution law for the visco-plastic strain in Eqs. 6.181a, 6.181b are alternatively expressed in terms of the visco-plastic stress

⎧ ⎪ ⎪ ⎨

˙vp (σvp , σhk ) = 0

|σvp + σhk | − σy σvp + σhk ⎪ ⎪ ⎩ η |σvp + σhk |

for

(6.184) ⎫ |σvp + σhk | < σy ⎪ ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy ⎪ ⎭

.

Consequently, based on Eq. 6.181c rendering ˙hk = ˙vp , the associated evolution law for the kinematic-hardening strain follows as

⎧ ⎪ ⎪ ⎨

˙hk (σvp , σhk ) = 0

|σvp + σhk | − σy σvp + σhk ⎪ ⎪ ⎩ η |σvp + σhk |

for

(6.185) ⎫ |σvp + σhk | < σy ⎪ ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy ⎪ ⎭

.

Obviously, the expressions in Eqs. 6.170, 6.171 and 6.184, 6.185 are inverse relations. With the representation for the evolution of the visco-plastic and kinematichardening strains in Eqs. 6.184 and 6.185, the corresponding (total) dual dissipation potential π ∗ , as determined from the Legendre transformation

6.3 Perzyna Hardening Model

361

π ∗ (σvp , σhk ) = max σvp ˙vp + σhk ˙hk

(6.186)

˙ vp ,˙hk

1 − η |˙vp |2 − σy |˙vp | − K hk [˙vp − ˙hk ] 2



then reads5 ⎫ |σvp + σhk | < σy ⎪ ⎪ ⎬ for ∗  . π (σvp , σhk ) = 2 ⎪ ⎪ 1 |σvp + σhk | − σy ⎪ ⎭ ⎩ |σvp + σhk | ≥ σy ⎪ 2 η ⎧ ⎪ ⎪ ⎨

0

(6.187)

The above relations may conveniently be condensed by the help of the Macaulay bracket • := 21 [• + | • |], e.g. the dual dissipation potential is expressed as π ∗ (σvp , σhk ) =

1 |σvp + σhk | − σy 2 . 2 η

(6.188)

Identifying ˙hk with ˙vp and setting σhk = 0, the remaining non-smooth (total) dissipation and dual (total) dissipation potentials π = π(˙vp ) and π ∗ = π ∗ (σvp ) together with the resulting non-smooth constitutive relations σvp = σvp (˙vp ) and ˙vp = ˙vp (σvp ) are similar to those displayed in Fig. 6.3. The result in Eqs. 6.184, 6.185 for the evolution of the visco-plastic and the kinematic-hardening strains thus follows directly from the reverse Legendre trans5 The

expressions for the evolution of the visco-plastic and the kinematic-hardening strains in Eqs. 6.184, 6.185 result in σvp ˙ vp (σvp , σhk ) + σhk ˙ hk (σvp , σhk ) =

|σvp + σhk | and

⎧ ⎪ ⎨

0

|σ + σhk | − σy ⎪ ⎩ vp η ⎧ ⎪ ⎪ ⎨

for

⎫ |σvp + σhk | < σy ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy ⎭

0

1 2  η |˙vp (σvp , σhk )|2 = ⎪ 1 |σvp + σhk | − σy 2 ⎪ ⎩ 2 η and

for

⎫ |σvp + σhk | < σy ⎪ ⎪ ⎬ ⎪ ⎭ |σvp + σhk | ≥ σy ⎪

 σy |˙vp (σvp , σhk )| + K hk (σhk ) ˙ vp (σvp , σhk ) − ˙ hk (σvp , σhk ) = ⎧ ⎫ 0 |σvp + σhk | < σy ⎪ ⎪ ⎨ ⎬ for σy . |σ + σ | − σ hk y ⎪ ⎪ ⎩ vp |σvp + σhk | ≥ σy ⎭ η

Taken together, the (total) dual dissipation potential π ∗ (σvp , σhk ) follows.

362

6 Visco-Plasticity

formation 

1 |σvp + σhk | − σy 2 , d(σvp , σhk ; ˙vp , ˙hk ) − σvp ,σhk 2 η

π(˙vp , ˙hk ) = max

Table 6.9 Summary of the specific Perzyna kinematic hardening model (1) Strain



= e + vp

(2) Energy

ψ

=

(3) Stress

σ

= E [ − vp ]

(4) Stress

σhk = −K hk

(5) Potential πv =

1 2E

1 2

[ − vp ]2 + 21 K 2hk σ

≡



≡

−σvp

η |˙vp |2

(6) Potential πp = σy |˙vp | + K hk [˙vp − ˙ hk ] (7) Stress

σvp = η ˙ vp + σy

˙ vp + K hk |˙vp |

(8) Stress

σhk = −K hk

↑for ˙ vp = 0

≡

 σvp

or (5) Potential πv∗ = (6) Yield

0

1 2

|σv |2 /η

≥ |σphk | − σy with σphk := σp − K hk σphk

(7) Evolution ˙ vp = λ

|σphk | σphk

(8) Evolution ˙ hk = λ (9) KKT

λ

|σphk |

≥ 0,

|σphk |

=

σv η

=

σv η ≤

σy ,

λ |σphk |

=

λ σy

or (5) Potential π ∗ = (6) Evolution ˙ vp = (7) Evolution ˙ hk =

1 2

hk | − σ 2 /η with σ hk := σ − K  |σvp y vp hk vp

hk | − σ σ hk |σvp y vp

η

hk | |σvp

hk | − σ σ hk |σvp y vp

η

hk | |σvp

(6.189)

6.3 Perzyna Hardening Model

363

whereby d(σvp , σhk ; ˙vp , ˙hk ) := σvp ˙vp + σhk ˙hk denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.189 embodies the unconstrained optimization problem ˜1/η (σvp , σhk ; ˙vp , ˙hk ) := −d(σvp , σhk ; ˙vp , ˙hk ) +

(6.190)

1 |σvp + σhk | − σy 2 → min , σvp ,σhk 2 η

whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint |σvp + σhk | ≤ σy penalized by the penalty parameter 1/η. In accordance with Eqs. 6.184, 6.185 the stationarity conditions of this unconstrained optimization problem then read |σvp + σhk | − σy η |σvp + σhk | − σy ˙hk (σvp , σhk ) = η ˙vp (σvp , σhk ) =

σvp + σhk , |σvp + σhk | σvp + σhk . |σvp + σhk |

(6.191a) (6.191b)

Finally, the visco-plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated visco-plastic deformation, i.e.

κ=

κ˙ dt with κ˙ := |˙vp | = |˙hk | =

|σvp + σhk | − σy ≥ 0. η

(6.192)

The specific Perzyna kinematic hardening model is summarized in Table 6.9.

6.3.5 Specific Perzyna Kinematic Hardening Model: Algorithmic Update For the specific Perzyna kinematic hardening model the evolution laws for the viscoplastic strain vp and the kinematic-hardening strain hk are integrated by the implicit Euler backwards method to render nvp := nvp − n−1 vp = λ

n n + σhk σvp n + σn | |σvp hk

n = nhk − n−1 hk =: hk ,

(6.193)

whereby the incremental visco-plastic multiplier λ is defined as λ := t n λn := t n

n n |σvp + σhk | − σy

η

≥ 0.

(6.194)

364

6 Visco-Plasticity

Consequently, the visco-plastic stress σvp and the kinematic-hardening stress σhk are updated at the end of the time step by n  σvp = −E [nvp − n ] =: σvp − E nvp , n  σhk = −K nhk =: σhk − K nhk .

(6.195)

  and the trial kinematic-hardening stress σhk Here the trial visco-plastic stress σvp are computable exclusively from known quantities at the beginning of the time step and follow as

 n σvp := −E [n−1 vp −  ],  σhk

:= −K

n−1 hk

(6.196)

.

Combining the visco-plastic stress and the kinematic-hardening stress at the end of the time step and incorporating the discretized evolution laws for the visco-plastic strain and the kinematic-hardening strain then renders n n   + σhk = σvp + σhk − [E + K ] λ σvp

n n + σhk σvp n + σn | |σvp hk

.

(6.197)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress  σn + σn  vp hk n n   = σvp + σhk | + [E + K ] λ + σhk . |σvp n + σn | |σvp hk

(6.198)

As an immediate consequence the equivalent stress and its trial value are related via n n   + σhk | = |σvp + σhk | − [E + K ] λ. (6.199) |σvp A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously n n + σhk σvp n + σn | |σvp hk

≡

  σvp + σhk  + σ | |σvp hk

.

(6.200)

Consequently, the yield function at the end of the time step is expressed as n n + σhk | − σy = φ − [E + K ] λ. φn := |σvp

Here the trial value of the yield function φ has been defined as

(6.201)

6.3 Perzyna Hardening Model

365   φ := |σvp + σhk | − σy .

(6.202)

Next for visco-plastic loading with λ > 0 the definition for the incremental visco-plastic multiplier is regrouped to render φn − λ

η = 0. t n

(6.203)

Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form from φ ≥ 0. (6.204) λ = E + K + η/t n Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t n → 0. Once λ is computed all other variables may be updated. In particular, the visco-plastic stress at the end of the time step reads   + σhk σvp n  = σvp − E λ  σvp (6.205)  . |σvp + σhk | n = σ n with respect to n is denoted the algorithmic tangent The sensitivity of σvp E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n

 n ∂ σvp

= E − E λ ∂

  + σhk σvp  + σ | |σvp hk

 −E

  + σhk σvp  + σ | |σvp hk

∂ (λ).

(6.206)

The first derivative term on the right-hand-side computes to zero since  ∂

  + σhk σvp  + σ | |σvp hk

 =

    σvp + σhk σvp + σhk 1 E − E ≡ 0.  + σ |  + σ  |2 |σ  + σ  | |σvp |σvp vp hk hk hk

(6.207)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-handside computes from requiring satisfaction of ∂ [φn − λ η/t n ] = 0 for ongoing visco-plastic flow at the end of the time step, i.e.  η  ∂ φ  − E + K + ∂ (λ) = t n   σ¯ vp η  . E − E + K + ∂ (λ) = 0  | |σ¯ vp t n

(6.208)

   with abbreviation σ¯ vp := σvp + σhk . As a conclusion the algorithmic tangent E a is thus finally expressed as

366

6 Visco-Plasticity

E an = E − H0 (λ)

E2 . E + K + η/t n

(6.209)

Note that, consequently, the algorithmic tangent degenerates to the plastic case for η → 0 and λ > 0, likewise it degenerates to E a = E for t n → 0. In one dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Perzyna hardening model capturing kinematic hardening is summarized in Table 6.10.

Table 6.10 Algorithmic update for the specific Perzyna kinematic hardening model Input

n n−1 n−1 vp hk

Trial Strain

vp = n−1 vp hk = n−1 hk

Trial Stress

Trial Yield

 = −E [ − n ] σvp vp  = −K  σhk hk  + σ | − σ φ = |σvp y hk

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E + K + η/t n

ENDIF Update Strain nvp = vp + λ nhk = hk + λ

 + σ σvp hk

 + σ | |σvp hk  + σ σvp hk

 + σ | |σvp hk

Update Stress

σ n = E [n − nvp ]

Tangent

E an = E − H0 (λ)

Output

σ n nvp nhk E an

E2 E + K + η/t n

6.3 Perzyna Hardening Model

367

6.3.6 Specific Perzyna Kinematic Hardening Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the specific Perzyna kinematic hardening model to a prescribed Zig-Zag strain history is documented in Fig. 6.27a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.21a–e and 6.12a– e, respectively.) Figure 6.27a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.27b showcases the resulting stress history σ(t) that displays a periodic, saw-tooth-type signal with σ(t) = σy + η ˙(t) + K vp (t) = 1.375 + 0.1 vp (t) whenever σ(t) > 1 + 0.1 vp (t) and with ˙(t) = 5, thus σmax ≈ 1.715 (from visual inspection). The resulting σ = σ() diagram is highlighted in Fig. 6.27c. Due to kinematic hardening its resulting rounded parallelogram-type format has constant amplitude in both the  and σ direction. Figure 6.27d demonstrates the corresponding visco-plastic strain history vp (t): during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas vp (t) stays constant with |vp (t)| ≈ 3.4 (from visual inspection) during the elastic phases. Finally, the strain arc-length κ(t) in Fig. 6.27e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 65 (from visual inspection). Prescribed Strain History: Sine The response of the specific Perzyna kinematic hardening model to a prescribed Sine strain history is documented in Fig. 6.28a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.22a–e and 6.13a–e, respectively.) Figure 6.28a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.28b showcases the resulting stress history σ(t) that displays a periodic signal.

368

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.27 Response analysis of the specific Perzyna kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1, η = 0.075. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

369 σ

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

t

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.28 Response analysis of the specific Perzyna kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1, η = 0.075. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

370

6 Visco-Plasticity

The resulting σ = σ() diagram is highlighted in Fig. 6.28c. Due to kinematic hardening its resulting rounded parallelogram-type format has constant amplitude in both the  and σ direction. Figure 6.28d demonstrates the corresponding visco-plastic strain history vp (t): during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas vp (t) stays constant with |vp (t)| ≈ 3.5 (from visual inspection) during the elastic phases. Finally, the strain arc-length κ(t) in Fig. 6.28e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax = 68 (from visual inspection). Prescribed Strain History: Ramp The response of the specific Perzyna kinematic hardening model to a prescribed Ramp strain history is documented in Fig. 6.29a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.23a–e and 6.14a–e, respectively.) Figure 6.29a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.29b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙(t) = ±5 in the loading and the unloading phases. During the holding phase with ˙(t) = 0 the stress relaxes to σ(t) → σy + 0.1 κ(t) ≈ 1.375 (from visual inspection). The resulting σ = σ() diagram is highlighted in Fig. 6.29c. The elastic slope (E = 1) in the loading and unloading phase are easy to verify. Likewise the stress relaxation to σ = 1.375 during the holding phase is clearly visible at  = 5. Figure 6.29d demonstrates the corresponding visco-plastic strain history vp (t) with vp (t) → 3.75 and vp (t) → 1.45 in the loading and unloading phase, respectively (from visual inspection). Finally, the strain arc-length κ(t) in Fig. 6.29e follows from integrating κ(t) ˙ = |˙vp (t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax ≈ 6.05 (from visual inspection). Prescribed Stress History: Zig-Zag The response of the specific Perzyna kinematic hardening model to a prescribed Zig-Zag stress history is documented in Fig. 6.30a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.24a–e and 6.15a– e, respectively.)

6.3 Perzyna Hardening Model

371 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 6.29 Response analysis of the specific Perzyna kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1, η = 0.075. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

372

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.30 Response analysis of the specific Perzyna kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1, η = 0.075. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

373

Figure 6.30a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.30b showcases the resulting strain history (t) that displays a periodic signal after an initial transient phase. The resulting (distorted lens-shaped) σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 6.30c. Figure 6.30d demonstrates the corresponding visco-plastic strain history vp (t), which is also a periodic signal after the initial transient phase. Finally, the strain arc-length κ(t) in Fig. 6.30e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 162 (from visual inspection). Prescribed Stress History: Sine The response of the specific Perzyna kinematic hardening model to a prescribed Sine stress history is documented in Fig. 6.31a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.25a–e and 6.16a–e, respectively.) Figure 6.31a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.31b showcases the resulting strain history (t) that displays a periodic signal after the initial transient phase. The resulting (distorted lentil-shaped) σ = σ() diagram is highlighted in Fig. 6.31c. Figure 6.31d demonstrates the corresponding visco-plastic strain history vp (t), which is also a periodic signal after the initial transient phase. Finally, the strain arc-length κ(t) in Fig. 6.31e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 218 (from visual inspection). Prescribed Stress History: Ramp The response of the specific Perzyna kinematic hardening model to a prescribed Ramp stress history is documented in Fig. 6.32a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.26a–e and 6.17a–e, respectively.) Figure 6.32a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps

374

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.31 Response analysis of the specific Perzyna kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1, η = 0.075. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

375

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 80.0)

Fig. 6.32 Response analysis of the specific Perzyna kinematic hardening model with material data: E = 1.0, σy = 1.0, K = 0.1, η = 0.075. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

376

6 Visco-Plasticity

with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.32b showcases the resulting strain history (t) that displays a smoothly increasing signal in the loading and holding phases with (t) → 45 and a smoothly decreasing signal in the unloading phase with (t) → 30 (from visual inspection). The nonlinear creep during the holding phase saturates due to the kinematic hardening. The resulting σ = σ() diagram is highlighted in Fig. 6.32c. Once the holding phase is completed the σ = σ() behavior in the unloading phase is initially purely elastic with σ(t) ∈ [5, 3] and slope E = 1, and subsequently visco-plastic whereby the strain approaches (t) → 30. Likewise the creep towards  = 45 during the holding phase is clearly visible at σ = 5. Figure 6.32d demonstrates the corresponding smoothly and monotonically in/decreasing visco-plastic strain history vp (t) with vp (t) → 40 during the holding phase and vp (t) → 30 in the unloading phase. Finally, the strain arc-length κ(t) in Fig. 6.32e follows from integrating κ(t) ˙ = |˙(t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax ≈ 50 (from visual inspection).

6.3.7 Specific Perzyna Mixed Hardening Model: Formulation The specific Perzyna (isotropic and kinematic) mixed hardening model, similar to that displayed in Fig. 6.18 (however with the hardening modulus H and the hardening strain εh coinciding here with the isotropic- and kinematic-hardening moduli H and K , respectively, and the isotropic- and kinematic-hardening strains hi and hk , respectively), consists of a serial arrangement of (1) a linear elastic spring with stiffness E and (2) a linear mixed-hardening viscous frictional slider consisting of a parallel arrangement of (i) a linear frictional slider with threshold σy , (ii) a linear viscous dashpot with viscosity η, and (iii) linear mixed-hardening springs with stiffnesses H and K (the isotropic- and kinematic-hardening moduli). For the specific Perzyna mixed hardening model the free energy density ψ is expressed as a quadratic (and thus convex) function of  − vp (the elastic strain e ), hi (the isotropic-hardening strain) and hk (the kinematic-hardening strain) ψ(, vp , hi , hk ) =

1 1 1 E [ − vp ]2 + H 2hi + K 2hk . 2 2 2

(6.210)

Then the energetic stress σ  conjugated to the total strain  and the energetic  conjugated to the visco-plastic strain vp together with the visco-plastic stress σvp   isotropic-hardening stress σhi conjugated to the isotropic-hardening strain hi and   the kinematic-hardening stress σhk conjugated to the kinematic-hardening strain hk follow as

6.3 Perzyna Hardening Model

377

σ  (, vp

) = ∂ ψ(, vp , hi , hk ) =

E [ − vp ],

(6.211a)

 σvp (, vp  σhi (  σhk (

) = ∂vp ψ(, vp , hi , hk ) = −E [ − vp ],

(6.211b)

) = ∂hi ψ(, vp , hi , hk ) =

H hi

,

(6.211c)

hk ) = ∂hk ψ(, vp , hi , hk ) =

K hk

.

(6.211d)

hi

Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic stress, σ  ≡ σ, and, due to the serial arrangement of the elastic spring and the mixed-hardening viscous fric ≡ σ. tional slider, also with the negative of the energetic visco-plastic stress, −σvp Furthermore, for the specific Perzyna mixed hardening model the (total) dissipation potential π consists of a convex and smooth viscous contribution, the viscous dissipation potential πv , together with a convex and non-smooth plastic contribution, the plastic dissipation potential πp , i.e. π(˙vp , ˙hi , ˙hk ) = πv (˙vp ) + πp (˙vp , ˙hi , ˙hk ).

(6.212)

Thereby the viscous and plastic contributions πv and πp to the (total) dissipation potential π are chosen as πv (˙vp ) =

1 η |˙vp |2 2

(6.213)

πp (˙vp , ˙hi , ˙hk ) = [σy + H hi ] |˙vp | − H hi ˙hi + K hk [˙vp − ˙hk ]. Observe that (i) π = πv + πp does not depend on ˙, thus the dissipative stress σ  = σ − σ  ≡ 0 vanishes identically, and that (ii) πv is positively homogenous of degree two in ˙vp and obviously smooth at the origin ˙vp = 0, and that (iii) πp is positively homogenous of degree one in {˙p , ˙hi , ˙hk } and obviously non-smooth at the origin {˙p , ˙hi , ˙hk } = {0, 0, 0}. As a consequence of the additive structure of  consists of a the (total) dissipation potential the dissipative visco-plastic stress σvp  viscous contribution, the dissipative viscous overstress σv , and a plastic contribution, the dissipative plastic stress σp , i.e.  σvp (˙vp , ˙hi , ˙hk ) = σv (˙vp ) + σp (˙vp , ˙hi , ˙hk ).

(6.214)

Thereby the dissipative viscous overstress σv computes as partial derivative of the viscous dissipation potential with respect to its conjugated variable σv (˙vp ) = ∂˙vp πv (˙vp ) = η ˙vp ,

(6.215)

whereas the dissipative plastic stress σp and the dissipative isotropic- and kinematic  hardening stresses σhi and σhk compute as some sub-derivatives of the plastic dissipation potential with respect to their conjugated variables

378

6 Visco-Plasticity

σp (˙vp , ˙hi , ˙hk ) ∈ d˙vp π(˙vp , ˙hi , ˙hk ), 

σhi (˙vp , ˙hi , ˙hk ) ∈ d˙hi π(˙vp , ˙hi , ˙hk ),  (˙vp , ˙hi , ˙hk ) σhi

with

(6.216)

∈ d˙hk π(˙vp , ˙hi , ˙hk ),

⎧ ⎫  + [σy + H hi ] + K hk ˙vp > 0 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ − [σ + H  ] − K  , ⎪ ⎪ y hi hk ⎨ ⎬ = 0 for  ˙ , d˙vp πp (˙vp , ˙hi , ˙hk ) = vp ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ + [σy + H hi ] + K hk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎩ ⎭ − [σy + H hi ] − K hk ˙vp < 0 d˙hi πp (˙vp , ˙hi , ˙hk ) =

−H hi ,

d˙hk πp (˙vp , ˙hi , ˙hk ) =

−K hk .

(6.217)

whereby d˙vp πp , d˙hi πp and d˙hk πp denote the sets of sub-derivatives, i.e. the subdifferentials of πp with respect to ˙vp , ˙hi and ˙hk , respectively. Recall that the energetic and the dissipative visco-plastic as well a the isotro  + σvp = 0, pic- and kinematic-hardening stresses are constitutively related by σvp     σhi + σhi = 0 and σhk + σhk = 0, respectively, thus the notions of visco-plastic stress (together with the notions of viscous overstress and plastic stress) as well as of isotropic- and kinematic-hardening stresses defined as the values   = −σvp with σv := σv and σvp = σv + σp := σvp

σhi

:=

σhk

:=

 σhi  σhk

= =

σp := σp ,

 −σhi ,  −σhk ,

(6.218a) (6.218b) (6.218c)

will exclusively be used in the sequel for convenience of exposition. Separate Viscous Overstress and Plastic Stress The Perzyna mixed hardening model may be formulated further by considering the viscous overstress σv and the plastic stress σp separately. Thereby, due to the nonsmooth plastic dissipation potential, the plastic stress is constrained to reside in an admissible domain. The closed and convex admissible domain A = int A ∪ ∂ A in the space of the dissipative driving forces, i.e. in the {σvp , σhi , σhk }-space, is next introduced as the union of the elastic domain and the yield surface, compare the representation in Fig. 5.27. Thereby, the admissible domain may either be determined directly from the expression of the sub-differential d˙vp πp in Eq. 6.217, or, alternatively, from evaluating the formal definition of the sub-differential

6.3 Perzyna Hardening Model

{σp |

σp [˙vp

379

d˙vp πp (˙vp , ˙hi , ˙hk ) = (6.219)     − ˙vp ] ≤ [σy + H hi ] |˙vp | − |˙vp | + K hk [˙vp − ˙vp ] ∀˙vp },

whereby ˙vp denotes any admissible visco-plastic strain rate. Then at ˙vp = 0 it holds for any admissible ˙vp that σp ˙vp ≤ [σy + H hi ] |˙vp | + K hk ˙vp and, with max˙vp {[σp − K hk ] ˙vp /|˙vp |} = |σp − K hk |, the admissible domain follows as |σp − K hk | ≤ σy + H hi . Moreover, the sub-differential d˙hi πp and d˙hk πp reduce to the partial derivative ∂˙hi πp and ∂˙hk πp , respectively, and render σhi = −H hi and σhk = −K hk . Thus the admissible domain is eventually expressed as |σp + σhk | ≤ σy − σhi . The elastic domain is defined as the interior of the admissible domain, i.e.   int A := {σp , σhi , σhk } | |σp + σhk | − [σy − σhi ] < 0 ,

(6.220)

whereas the yield surface, which in the present one-dimensional case collapses to the two planes σp + σhk = ±[σy − σhi ], is defined as the boundary of the admissible domain, i.e.   ∂ A := {σp , σhi , σhk } | |σp + σhk | − [σy − σhi ] = 0 .

(6.221)

Collectively, the admissible domain in the {σp , σhi , σhk }-space is characterized by the yield condition (6.222) |σp + σhk | − [σy − σhi ] ≤ 0. States in the interior int A of the admissible domain with |σp + σhk | < σy − σhi are elastic, whereas states on the boundary ∂ A of the admissible domain with |σp + σhk | = σy − σhi are visco-plastic. The corresponding dual viscous and plastic dissipation potentials πv∗ and πp∗ , as determined from the Legendre transformations

 1 πv∗ (σv ) = max σv ˙vp − η |˙vp |2 ˙ vp 2 ∗ πp (σp , σhi , σhk ) = max ˙ vp ,˙hi ,˙hk

(6.223a) (6.223b)

{σp ˙vp + σhi ˙hi + σhk ˙hk − [σy − H hi ] |˙vp | + H hi ˙hi − K hk [˙vp − ˙hk ]} then read with the stationarity conditions σhi = −H hi and σhk = −K hk (note the minus signs)

380

6 Visco-Plasticity

1 1 |σv |2 2 η πp∗ (σp , σhi , σhk ) = IA (σp , σhi , σhk ) := ⎫ ⎧ |σp + σhk | ≤ σy − σhi ⎬ ⎨ 0 for , ⎭ ⎩ ∞ |σp + σhk | > σy − σhi πv∗ (σv ) =

(6.224a) (6.224b)

where IA denotes the indicator function of the admissible domain A in the {σp , σhi , σhk }-space. The evolution laws (the associated flow rules) for the viscoplastic and the isotropic- and kinematic-hardening strains then follow either as the partial derivative of the dual viscous dissipation potential or likewise as some subderivatives of the dual plastic dissipation potential, in either case with respect to their conjugated variables ˙vp (σv

) = ∂σv πv∗ (σv

˙vp (σp , σhi , σhk ) ∈

),

dσp πp∗ (σp , σhi , σhk )

= dσp IA (σp , σhi , σhk ),

˙hi (σp , σhi , σhk ) ∈ dσhi πp∗ (σp , σhi , σhk ) = dσhi IA (σp , σhi , σhk ),

(6.225)

˙hk (σp , σhi , σhk ) ∈ dσhk πp∗ (σp , σhk , σhk ) = dσhk IA (σp , σhk , σhk ), with ∂σv πv∗ (σv ) =

1 σv η

(6.226a)

and dσp πp∗ (σp , σhi , σhk ) = dσp IA (σp , σhi , σhk ) = ⎫ ⎧ 0 |σp + σhk | < σy − σhi ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ for σ + σ hk ⎪ ⎪ ⎪λ p |σp + σhk | = σy − σhi ⎪ ⎭ ⎩ |σp + σhk |

(6.226b)

dσhi πp∗ (σp , σhi , σhk ) = dσhi IA (σp , σhi , σhk ) = ⎫ ⎧ |σp + σhk | < σy − σhi ⎬ ⎨0 for ⎭ ⎩ λ |σp + σhk | = σy − σhi

(6.226c)

and

and

6.3 Perzyna Hardening Model

381

dσhk πp∗ (σp , σhi , σhk ) = dσhk IA (σp , σhk , σhk ) = ⎫ ⎧ 0 |σp + σhk | < σy − σhi ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ for , σp + σhk ⎪ ⎪ ⎪ |σp + σhk | = σy − σhi ⎪ ⎭ ⎩λ |σp + σhk |

(6.226d)

whereby dσp πp∗ , dσhi πp∗ and dσhk πp∗ denote the sets of sub-derivatives, i.e. the subdifferentials of πp∗ with respect to σp , σhi and σhk , respectively, and λ is a positive Lagrange (or rather plastic) multiplier. Obviously, the expressions in Eqs. 6.215, 6.216 and 6.225 are inverse relations. The smooth viscous dissipation and dual viscous dissipation potentials πv = πv (˙vp ) and πv∗ = πv∗ (σv ) together with the resulting smooth constitutive relations σv = σv (˙vp ) and ˙vp = ˙vp (σv ) are similar to those displayed in Fig. 4.2. Identifying ˙hi with |˙vp | and ˙hk with ˙vp and setting σhi = 0 and σhk = 0, the remaining non-smooth plastic dissipation and dual plastic dissipation potentials πp = πp (˙vp ) and πp∗ = πp∗ (σp ) together with the resulting non-smooth constitutive relations σp = σp (˙vp ) and ˙vp = ˙vp (σp ) are similar to those displayed in Fig. 5.3. Visco-Plastic Stress Alternatively, the Perzyna mixed hardening model may be formulated further by considering the visco-plastic stress. To this end the viscous and the plastic stress need to be related to the visco-plastic stress.

σv

σp + σhk

+[σy − σhi ] −[σy − σhi ] +[σy − σhi ] σvp + σhk

σvp + σhk −[σy − σhi ]

Fig. 6.33 Specific Perzyna mixed hardening model: The visco-plastic stress σvp = σv + σp is the sum of the viscous overstress σv and the plastic stress σp . The viscous damper is only activated once the load carrying capacity of the frictional slider is exceeded. Accordingly the viscous overstress is identically zero σv ≡ 0 for |σvp + σhk | − [σy − σhi ] ≤ 0 (left), while the plastic stress, shifted by the kinematic-hardening stress σhk , remains constant (at a particular σhi that may be considered to expand along a third dimension perpendicular to the plane displayed in the above) with |σp + σhk | = σy − σhi for |σvp + σhk | − [σy − σhi ] > 0 (right)

382

6 Visco-Plasticity

Remarkably, since at yield the plastic stress satisfies |σp + σhk | = σy − σhi , the viscous overstress σv = σvp − σp allows representation in terms of the yield condition that is, however, evaluated in terms of the visco-plastic stress σvp ⎧ +|σvp + σhk | − σy + σhi ⎪ ⎪ ⎨ 0 σv = ⎪ ⎪ ⎩ −|σvp + σhk | + σy − σhi

if

⎫ σvp + σhk > +σy − σhi ⎪ ⎪ ⎬ else . ⎪ ⎪ ⎭ σvp + σhk < −σy + σhi

(6.227)

The reasoning for the representation in Eq. 6.227 is highlighted in Fig. 6.33 and follows as: • For ˙vp = 0 the overstress in the viscous damper is identically zero, i.e. σv ≡ 0 and thus the stress σp in the kinematic-hardening frictional slider coincides identically with the visco-plastic stress σp ≡ σvp . Consequently, and again since ˙vp = 0, the visco-plastic stress satisfies |σvp + σhk | ≤ σy − σhi . • For ˙vp > 0 (with σvp + σhk > +[σy − σhi ]) the overstress in the viscous damper results in σv = [σvp + σhk ] − [σy − σhi ] ≡ +|σvp + σhk | − σy + σhi and thus the stress σp + σhk in the isotropic-hardening frictional slider coincides identically with the (positive) current yield stress σp + σhk ≡ +[σy − σhi ]. • For ˙vp < 0 (with σvp + σhk < −[σy − σhi ]) the overstress in the viscous damper results in σv = [σvp + σhk ] + [σy − σhi ] ≡ −|σvp + σhk | + σy − σhi and thus the stress σp + σhk in the isotropic-hardening frictional slider coincides identically with the (negative) current yield stress σp + σhk ≡ −[σy − σhi ]. Finally the above relations may conveniently be summarized as ⎧ ⎪ ⎪ ⎨

σv = 0

σvp + σhk  ⎪ ⎪ ⎩ |σvp + σhk | − [σy − σhi ] |σvp + σhk |

(6.228) ⎫ |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ for . ⎪ |σvp + σhk | ≥ σy − σhi ⎪ ⎭

Then, based on the representation for the viscous stress in terms of the viscoplastic stress in Eq. 6.228, the two variants of the associated evolution law for the visco-plastic strain in Eqs. 6.226a, 6.226b are alternatively expressed in terms of the visco-plastic stress

⎧ ⎪ ⎪ ⎨

˙vp (σvp , σhi , σhk ) = 0

|σvp + σhk | − [σy − σhi ] σvp + σhk ⎪ ⎪ ⎩ η |σvp + σhk |

for

⎫ |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy − σhi ⎪ ⎭

(6.229)

.

6.3 Perzyna Hardening Model

383

Consequently, based on Eq. 6.226c rendering ˙hi = |˙vp |, the associated evolution law for the isotropic-hardening strain follows as ˙hi (σvp , σhi , σhk ) =

⎧ ⎪ ⎪ ⎨

0

|σvp + σhk | − [σy − σhi ] ⎪ ⎪ ⎩ η

for

⎫ |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy − σhi ⎪ ⎭

(6.230)

.

Likewise, based on Eq. 6.226d rendering ˙hk = ˙vp , the associated evolution law for the kinematic-hardening strain follows as ˙hk (σvp , σhi , σhk ) =

⎧ ⎪ ⎪ ⎨

0

|σvp + σhk | − [σy − σhi ] σvp + σhk ⎪ ⎪ ⎩ η |σvp + σhk |

for

(6.231)

⎫ |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ ⎪ |σvp + σhk | ≥ σy − σhi ⎪ ⎭

.

Obviously, the expressions in Eqs. 6.215, 6.216 and 6.229, 6.230, 6.231 are inverse relations. With the representation for the evolution of the visco-plastic and kinematichardening strains in Eqs. 6.229 and 6.230, 6.231, the corresponding (total) dual dissipation potential π ∗ , as determined from the Legendre transformation

π (σvp , σhi , σhk ) = max σvp ˙vp + σhi ˙hi + σhk ˙hk ∗

˙ vp ,˙hi ,˙hk

1 − η |˙vp |2 − [σy + H hi ] |˙vp | + H hi ˙hi − K hk [˙vp − ˙hk ] 2

(6.232) 

then reads6 6 The

expressions for the evolution of the visco-plastic and the isotropic- and kinematic-hardening strains in Eqs. 6.229, 6.230, 6.231, result in σvp ˙ vp (σvp , σhi , σhk ) + σhi ˙ hi (σvp , σhi , σhk ) + σhk ˙ hk (σvp , σhi , σhk ) = 

⎧ ⎫ 0 |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ ⎨ for |σvp + σhk | + σhi |σ + σhk | − [σy − σhi ] ⎪ ⎪ ⎩ vp |σvp + σhk | ≥ σy − σhi ⎭ η

and ⎫ |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ for 1 2  η |˙vp (σvp , σhi , σhk )|2 = + σ | − [σ − σ ] |σ ⎪ ⎪ 1 2 vp hk y hi ⎪ ⎭ ⎩ |σvp + σhk | ≥ σy − σhi ⎪ 2 η ⎧ ⎪ ⎪ ⎨

0

384

6 Visco-Plasticity

⎧ ⎪ ⎪ ⎨

π ∗ (σvp , σhi , σhk ) =

⎫ 0 |σvp + σhk | < σy − σhi ⎪ ⎪ ⎬ for  2 . ⎪ ⎪ 1 |σvp + σhk | − [σy − σhi ] ⎪ ⎩ ⎭ |σvp + σhk | ≥ σy − σhi ⎪ 2 η

(6.233)

The above relations may conveniently be condensed by the help of the Macaulay bracket • := 21 [• + | • |], e.g. the dual dissipation potential is expressed as π ∗ (σvp , σhi , σhk ) =

1 |σvp + σhk | − [σy − σhi ] 2 . 2 η

(6.234)

Identifying ˙hi with |˙vp | and ˙hk with ˙vp and setting σhi = 0 and σhk = 0, the remaining non-smooth (total) dissipation and dual (total) dissipation potentials π = π(˙vp ) and π ∗ = π ∗ (σvp ) together with the resulting non-smooth constitutive relations σvp = σvp (˙vp ) and ˙vp = ˙vp (σvp ) are similar to those displayed in Fig. 6.3. The result in Eqs. 6.229, 6.230, 6.231 for the evolution of the visco-plastic and the isotropic- and kinematic-hardening strains thus follows directly from the reverse Legendre transformation (6.235) π(˙vp , ˙hi , ˙hk ) = max σvp ,σhi ,σhk 

1 |σvp + σhk | − [σy − σhi ] 2 , d(σvp , σhi , σhk ; ˙vp , ˙hi , ˙hk ) − 2 η whereby d(σvp , σhi , σhk ; ˙vp , ˙hi , ˙hk ) := σvp ˙vp + σhi ˙hi + σhk ˙hk denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.235 embodies the unconstrained optimization problem ˜1/η (σvp , σhi , σhk ; ˙vp , ˙hi , ˙hk ) := −d(σvp , σhi , σhk ; ˙vp , ˙hi , ˙hk ) +

(6.236)

1 |σvp + σhk | − [σy − σhi ] 2 → min , σvp ,σhi ,σhk 2 η

whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint |σvp + σhk | ≤ σy − σhi penalized by the penalty parameter 1/η. In accorand

 σy + H hi (σhi ) |˙vp (σvp , σhi , σhk )| − H hi (σhi ) ˙ hi (σvp , σhi , σhk )+  K hk (σhk ) ˙ vp (σvp , σhi , σhk ) − ˙ hk (σvp , σhi , σhk ) = ⎧ ⎫ 0 |σvp + σhk | < σy − σhi ⎪ ⎪ ⎨ ⎬ for σy . |σ + σhk | − [σy − σhi ] ⎪ ⎪ ⎩ vp |σvp + σhk | ≥ σy − σhi ⎭ η

Taken together, the (total) dual dissipation potential π ∗ (σvp , σhi , σhk ) follows.

6.3 Perzyna Hardening Model

385

Table 6.11 Summary of the specific Perzyna mixed hardening model (1) Strain



= e + vp

(2) Energy

ψ

=

(3) Stress

σ

= E [ − vp ]

(4) Stress

σhi = −H hi

Stress

σhk = −K hk

(5)

(6) Potential πv =

1 2E

1 2

[ − vp ]2 + 21 H 2hi + 21 K 2hk σ

≡

η |˙vp |2

≡



−σvp

↓ σyhi := σy + H hi

(7) Potential πp = σyhi |˙vp | − H hi ˙ hi + K hk [˙vp − ˙ hk ] (8) Stress

σvp = η ˙ vp + σyhi

˙ vp + K hk |˙vp |

(9) Stress

σhi = −H hi

↑for ˙ vp = 0

(10) Stress

σhk = −K hk

≡

 σvp

or (6) Potential πv∗ = (7) Yield

0

1 2

|σv |2 /η

≥ |σphk | − σyhi with σphk := σp − K hk σphk

(8) Evolution ˙ vp = λ

|σphk |

=

σv η

=

σv η

(9) Evolution ˙ hi = λ σphk

(10) Evolution ˙ hk = λ (11) KKT

λ

|σphk |

≥ 0,

|σphk |

≤

σyhi ,

λ |σphk |

=

λ σyhi

or (6) Potential π ∗ = (7) Evolution ˙ vp = (8) Evolution ˙ hi = (9) Evolution ˙ hk =

1 2

hk | − σ hi 2 /η with σ hk := σ − K  |σvp vp hk y vp

hk | − σ hi σ hk |σvp y vp

η

hk | |σvp

hk | − σ hi |σvp y

η hk | − σ hi σ hk |σvp y vp

η

hk | |σvp

386

6 Visco-Plasticity

dance with Eqs. 6.229, 6.230, 6.231 the stationarity conditions of this unconstrained optimization problem then read |σvp + σhk | − [σy − σhi ] η |σvp + σhk | − [σy − σhi ] ˙hi (σvp , σhi , σhk ) = η |σvp + σhk | − [σy − σhi ] ˙hk (σvp , σhi , σhk ) = η ˙vp (σvp , σhi , σhk ) =

σvp + σhk , |σvp + σhk | , σvp + σhk . |σvp + σhk |

(6.237a) (6.237b) (6.237c)

Finally, the visco-plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated visco-plastic deformation, i.e.

κ= with κ˙ := |˙vp | = ˙hi = |˙hk | =

κ˙ dt

(6.238)

|σvp + σhk | − [σy − σhi ] ≥ 0. η

The specific Perzyna mixed hardening model is summarized in Table 6.11.

6.3.8 Specific Perzyna Mixed Hardening Model: Algorithmic Update For the specific Perzyna mixed (isotropic and kinematic) hardening model the evolution laws for the visco-plastic strain vp , the kinematic-hardening strain hk and the isotropic-hardening strain hi are integrated by the implicit Euler backwards method to render nvp := nvp − n−1 vp = λ

n n + σhk σvp n + σn | |σvp hk

n = nhk − n−1 hk =: hk ,

(6.239)

and = λ, nhi := nhi − n−1 hi

(6.240)

whereby the incremental visco-plastic multiplier λ is defined as λ := t n λn := t n

n n n |σvp + σhk | − [σy − σhi ]

η

≥ 0.

(6.241)

Consequently, the visco-plastic stress σvp , the kinematic-hardening stress σhk and the isotropic-hardening stress σhi are updated at the end of the time step by

6.3 Perzyna Hardening Model

387

n  σvp = −E [nvp − n ] =: σvp − E nvp , n σhk n σhi

= −K = −H

nhk nhi

=: =:

 σhk  σhi

−K −H

(6.242)

nhk , nhi .

  , the trial kinematic-hardening stress σhk Here the trial visco-plastic stress σvp  and the trial isotropic-hardening stress σhi are computable exclusively from known quantities at the beginning of the time step and follow as

 n := −E [n−1 σvp vp −  ],  σhk  σhi

:= −K := −H

n−1 hk n−1 hi

(6.243)

, .

Combining the visco-plastic stress and the kinematic-hardening stress at the end of the time step and incorporating the discretized evolution laws for the visco-plastic strain and the kinematic-hardening strain then renders n n   + σhk = σvp + σhk − [E + K ] λ σvp

n n + σhk σvp n + σn | |σvp hk

.

(6.244)

This relation is regrouped in order to separate the unknowns at the end of the time step from the known trial stress  σn + σn  vp hk n n   + σhk | + [E + K ] λ + σhk . |σvp = σvp n + σn | |σvp hk

(6.245)

As an immediate consequence the equivalent stress and its trial value are related via n n   + σhk | = |σvp + σhk | − [E + K ] λ. (6.246) |σvp A direct further consequence that alleviates the computation of the flow direction at the end of the time step in terms of trial values is then obviously n n + σhk σvp n + |σvp

n σhk |

≡

  σvp + σhk  + σ | |σvp hk

.

(6.247)

Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore n  = σhi − H λ. (6.248) σhi Consequently, the yield function at the end of the time step is expressed as n n n + σhk | − σy + σhi = φ − [E + H + K ] λ. φn := |σvp

(6.249)

388

6 Visco-Plasticity

Here the trial value of the yield function φ has been defined as    + σhk | − σy + σhi . φ := |σvp

(6.250)

Next for visco-plastic loading with λ > 0 the definition for the incremental visco-plastic multiplier is regrouped to render φn − λ

η = 0. t n

(6.251)

Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form from φ ≥ 0. (6.252) λ = E + H + K + η/t n Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t n → 0. Once λ is computed all other variables may be updated. In particular, the visco-plastic stress at the end of the time step reads   + σhk σvp n  (6.253) = σvp − E λ  σvp  . |σvp + σhk | n The sensitivity of σvp = σ n with respect to n is denoted the algorithmic tangent E a (thus dσ = E a d) and is computed from the product rule while noting that λ depends implicitly on n

 n ∂ σvp

= E − E λ ∂

  + σhk σvp  + σ | |σvp hk

 −E

  + σhk σvp  + σ | |σvp hk

∂ (λ).

(6.254)

The first derivative term on the right-hand-side computes to zero since  ∂

  + σhk σvp  + σ | |σvp hk

 =

    σvp + σhk σvp + σhk 1 E − E ≡ 0.  + σ |  + σ  |2 |σ  + σ  | |σvp |σvp vp hk hk hk

(6.255)

It shall be noted that the corresponding tangent modulus (tensor) in more than one dimension is different from zero. The second derivative term on the right-handside computes from requiring satisfaction of ∂ [φn − λ η/t n ] = 0 for ongoing visco-plastic flow at the end of the time step, i.e.  η  ∂ (λ) = ∂ φ − E¯ + H + t n    σ¯ vp . ¯ + H + η ∂ (λ) = E − E 0  | |σ¯ vp t n

(6.256)

6.3 Perzyna Hardening Model

389

   with abbreviations σ¯ vp := σvp + σhk and E¯ := E + K . As a conclusion the algorithmic tangent E a is thus finally expressed as

E an = E − H0 (λ)

E2 . E + H + K + η/t n

(6.257)

Note that, consequently, the algorithmic tangent degenerates to the plastic case for η → 0 and λ > 0, likewise it degenerates to E a = E for t n → 0. In one

Table 6.12 Algorithmic update for the specific Perzyna mixed (isotropic and kinematic) hardening model Input

n n−1 n−1 n−1 vp hi hk

Trial Strain

vp = n−1 vp hi = n−1 hi hk = n−1 hk

Trial Stress

Trial Yield

 = −E [ − n ] σvp vp  = −H  σhi hi  = −K  σhk hk  + σ | − σ + σ φ = |σvp y hk hi

Loading Check IF φ < 0 THEN λ = 0 ELSE λ =

φ E + H + K + η/t n

ENDIF Update Strain nvp = vp + λ

 + σ σvp hk

 + σ | |σvp hk

nhi = hi + λ nhk = hk + λ

 + σ σvp hk

 + σ | |σvp hk

Update Stress

σ n = E [n − nvp ]

Tangent

E an = E − H0 (λ)

Output

σ n nvp nhi nhk E an

E2 E + H + K + η/t n

390

6 Visco-Plasticity

dimension the algorithmic tangent trivially coincides with its continuous counterpart. It shall be noted, however, that this is at variance with the corresponding result in two and three dimensions. The algorithmic step-by-step update for the specific Perzyna hardening model capturing mixed hardening is summarized in Table 6.12.

6.3.9 Specific Perzyna Mixed Hardening Model: Response Analysis

Prescribed Strain History: Zig-Zag The response of the specific Perzyna mixed (isotropic and kinematic) hardening model to a prescribed Zig-Zag strain history is documented in Fig. 6.34a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and Perzyna models in Figs. 5.28a–e and 6.12a–e, respectively.) Figure 6.34a depicts the prescribed Zig-Zag strain history (t) with amplitude a = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.34b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) = σy + η ˙(t) + H κ(t) + K vp (t) = 1.375 + 0.1 κ(t) + 0.1 vp (t) whenever σ(t) > 1 + 0.1 κ(t) + 0.1 vp (t) and with ˙(t) = 5, thus σmax ≈ 3.55 (from visual inspection). The resulting σ = σ() diagram is highlighted in Fig. 6.34c. Due to mixed (isotropic and kinematic) hardening its resulting rounded parallelogram-type format has constant amplitude in the  direction and is isotropically expanding and kinematically shifting back and forth in the σ direction. Figure 6.34d demonstrates the corresponding visco-plastic strain history vp (t): during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas vp (t) stays constant during the elastic phases with decreasing amplitude after each half-period (and eventually vp (t) → 1.5). Finally, the strain arc-length κ(t) in Fig. 6.34e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax = 20.25 (from visual inspection). Prescribed Strain History: Sine The response of the specific Perzyna mixed (isotropic and kinematic) hardening model to a prescribed Sine strain history is documented in Fig. 6.35a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic

6.3 Perzyna Hardening Model

391 σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.34 Response analysis of the specific Perzyna mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1, η = 0.075. Prescribed Zig-Zag strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

392

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 50.0)

Fig. 6.35 Response analysis of the specific Perzyna mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1, η = 0.075. Prescribed Sine strain history with data: a = 5.0, T = 4.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

393

and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and Perzyna models in Figs. 5.29a–e and 6.13a–e, respectively.) Figure 6.35a depicts the prescribed Sine strain history (t) = a sin(ω t) with amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.35b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal. The resulting σ = σ() diagram is highlighted in Fig. 6.35c. Due to mixed (isotropic and kinematic) hardening its resulting rounded parallelogram-type format has constant amplitude in the  direction and is isotropically expanding and kinematically shifting back and forth in the σ direction. Figure 6.35d demonstrates the corresponding visco-plastic strain history vp (t): during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas vp (t) stays constant during the elastic phases with decreasing amplitude after each half-period (and eventually vp (t) → 1.6). Finally, the strain arc-length κ(t) in Fig. 6.35e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax = 24 (from visual inspection). Prescribed Strain History: Ramp The response of the specific Perzyna mixed (isotropic and kinematic) hardening model to a prescribed Ramp strain history is documented in Fig. 6.36a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and Perzyna models in Figs. 5.30a–e and 6.14a–e, respectively.) Figure 6.36a depicts the prescribed Ramp strain history (t) with maximum a = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.36b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙(t) = ±5 in the loading and the unloading phases. During the holding phase with ˙(t) = 0 the stress relaxes to σ(t) → σy + 0.1 κ(t) + 0.1 vp (t) ≈ 1.66 (from visual inspection). The resulting σ = σ() diagram is highlighted in Fig. 6.36c. The elastic slope (E = 1) in the loading and unloading phase are easy to verify. Likewise the stress relaxation to σ = 1.66 during the holding phase is clearly visible at  = 5. Figure 6.36d demonstrates the corresponding visco-plastic strain history vp (t) with vp (t) → 3.3 and vp (t) → 1.6 in the loading and unloading phase, respectively (from visual inspection).

394

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1

a)

max,min

= ± 5.0)

(tmax = 100 × 0.1 σmax,min = ± 3.5)

b) σ

c)

(

max,min

= ± 5.0 σmax,min = ± 3.5) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 5.0)

e)

(tmax = 100 × 0.1

κmax = 10.0)

Fig. 6.36 Response analysis of the specific Perzyna mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1, η = 0.075. Prescribed Ramp strain history with data: a = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

395

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.37 Response analysis of the specific Perzyna mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1, η = 0.075. Prescribed Zig-Zag stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

396

6 Visco-Plasticity

Finally, the strain arc-length κ(t) in Fig. 6.36e follows from integrating κ(t) ˙ = |˙vp (t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax ≈ 5 (from visual inspection). Prescribed Stress History: Zig-Zag The response of the specific Perzyna mixed (isotropic and kinematic) hardening model to a prescribed Zig-Zag stress history is documented in Fig. 6.37a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and Perzyna models in Figs. 5.31a–e and 6.15a–e, respectively.) Figure 6.37a depicts the prescribed Zig-Zag stress history σ(t) with amplitude σa = 5 and period T = 4 in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.37b showcases the resulting strain history (t) that displays a nearly periodic signal with decreasing amplitude after the initial elastic and visco-plastic phase. The resulting σ = σ() diagram is highlighted in Fig. 6.37c. Once the initial elastic and visco-plastic phase is completed the σ = σ() behavior displays less and less hysteresis in the remaining cycles and approaches a purely elastic response. Figure 6.37d demonstrates the corresponding visco-plastic strain history vp (t), which is also a nearly periodic signal with decreasing amplitude after the initial elastic and visco-plastic phase. Finally, the strain arc-length κ(t) in Fig. 6.37e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 35 (from visual inspection). Prescribed Stress History: Sine The response of the specific Perzyna mixed (isotropic and kinematic) hardening model to a prescribed Sine stress history is documented in Fig. 6.38a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and Perzyna models in Figs. 5.32a–e and 6.16a–e, respectively.) Figure 6.38a depicts the prescribed Sine stress history σ(t) = σa sin(ω t) with amplitude σa = 5, period T = 4 and corresponding angular frequency ω = 2π/T in the time interval t ∈ [0, tmax = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.38b showcases the resulting strain history (t) that displays a nearly periodic signal with decreasing amplitude after the initial elastic and visco-plastic phase.

6.3 Perzyna Hardening Model

397

σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 400.0)

Fig. 6.38 Response analysis of the specific Perzyna mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1, η = 0.075. Prescribed Sine stress history with data: σa = 5.0, T = 4.0; t = 0.1, N = 100

398

6 Visco-Plasticity σ

t

t

(tmax = 100 × 0.1 σmax,min = ± 5.0)

a)

(tmax = 100 × 0.1

b)

max,min

= ± 50.0)

σ

c)

(

max,min

= ± 50.0 σmax,min = ± 5.0) κ

vp

t

t

d)

(tmax = 100 × 0.1

vp,max,min

= ± 50.0)

e)

(tmax = 100 × 0.1

κmax = 80.0)

Fig. 6.39 Response analysis of the specific Perzyna mixed (isotropic and kinematic) hardening model with material data: E = 1.0, σy = 1.0, H = 0.1, K = 0.1, η = 0.075. Prescribed Ramp stress history with data: σa = 5.0, t0 = 0.0, t1 = 1.0, t2 = 9.0, t3 = 10.0; t = 0.1, N = 100

6.3 Perzyna Hardening Model

399

The resulting σ = σ() diagram is highlighted in Fig. 6.38c. Once the initial elastic and visco-plastic phase is completed the σ = σ() behavior displays less and less hysteresis in the remaining cycles and approaches a purely elastic response. Figure 6.38d demonstrates the corresponding visco-plastic strain history vp (t), which is also a nearly periodic signal with decreasing amplitude after the initial elastic and visco-plastic phase. Finally, the strain arc-length κ(t) in Fig. 6.38e follows from integrating κ(t) ˙ = |˙vp (t)| over two and a half periods and approaches κmax ≈ 40 (from visual inspection). Prescribed Stress History: Ramp The response of the specific Perzyna mixed (isotropic and kinematic) hardening model to a prescribed Ramp stress history is documented in Fig. 6.39a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and Perzyna models in Figs. 5.33a–e and 6.17a–e, respectively.) Figure 6.39a depicts the prescribed Ramp stress history σ(t) with maximum σa = 5, loading phase during t ∈ [t0 = 0, t1 = 1), holding phase during t ∈ [t1 = 1, t2 = 9], and unloading phase during t ∈ (t2 = 9, t3 = 10], whereby N = 100 time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas elastic time steps are indicated by smaller filled circles. Figure 6.39b showcases the resulting strain history (t) that displays a smoothly increasing signal in the loading and holding phases with (t) → 25 and a purely elastic behaviour in the unloading phase with (t) → 20. The nonlinear creep during the holding phase saturates due to the mixed (isotropic and kinematic) hardening. The resulting σ = σ() diagram is highlighted in Fig. 6.39c. Once the holding phase is completed the σ = σ() behavior in the unloading phase is purely elastic with σ(t) ∈ [5, 0] and slope E = 1, whereby the strain approaches (t) → 20. Likewise the creep towards  = 25 during the holding phase is clearly visible at σ = 5. Figure 6.39d demonstrates the corresponding smoothly and monotonically increasing visco-plastic strain history vp (t) with vp (t) → 20. Finally, the strain arc-length κ(t) in Fig. 6.39e follows from integrating κ(t) ˙ = |˙(t)| over the time interval t ∈ [0, tmax = 10] and approaches κmax = 20.

6.3.10 Generic Perzyna Hardening Model: Formulation A generic formulation of the Perzyna hardening model can be obtained from generalizing the specific Perzyna hardening model in Fig. 6.18 by assuming the elastic spring or/and the hardening spring or/and the viscous dashpot or/and the frictional slider as nonlinear. For the generic Perzyna hardening model the free energy density ψ is expressed as a non-quadratic but convex function of  − vp (the elastic strain e ) and εh (the

400

6 Visco-Plasticity

hardening strain) ψ(, vp , εh ) = ψ( − vp , εh ).

(6.258)

Note that ψ(, vp , εh ) and ψ( − vp , εh ) are different functions that return, however, the same function value for the same values of , vp and εh . Then the energetic  together with the energetic hardstress σ  and the energetic visco-plastic stress σvp  ening stress σh follow as σ  (, vp , εh ) = ∂ ψ(, vp , εh ) = ∂ ψ( − vp , εh ),  σvp (, vp , εh )  σh (, vp , εh )

(6.259a)

= ∂vp ψ(, vp , εh ) = ∂vp ψ( − vp , εh ),

(6.259b)

= ∂εh ψ(, vp , εh ) = ∂εh ψ( − vp , εh ).

(6.259c)

Recall that the total stress σ (that enters the equilibrium condition) coincides identically with the energetic stress σ  ≡ σ and the negative of the energetic visco ≡ σ. Moreover the energetic and the dissipative visco-plastic plastic stress −σvp     and hardening stresses are constitutively related by σvp + σvp = 0 and σh + σh = 0, respectively, thus the notions of visco-plastic stress and hardening stress defined     = −σvp and σh := σh = −σh will exclusively be used in the sequel. as σvp := σvp Note moreover that the visco-plastic stress σvp in the hardening viscous frictional slider decomposes additively into the viscous stress σv in the viscous dashpot and the hardening plastic stress σp in the frictional slider. Furthermore, for the generic Perzyna hardening model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙vp , εh ) and π ∗ = π ∗ (σvp , σh ), respectively, are related via corresponding Legendre transformations π ( ˙vp , ε˙h ) = max {σvp ˙vp + σh ε˙h − π ∗ (σvp , σh )},

(6.260a)

π ∗ (σvp , σh ) = max{σvp ˙vp + σh ε˙h − π ( ˙vp , ε˙h )}.

(6.260b)

σvp ,σh

˙ vp ,˙εh

Then the stationarity conditions corresponding to Eqs. 6.260a and 6.260b are the constitutive relations ˙vp (σvp , σh ) ∈ dσvp π ∗ (σvp , σh ) and

ε˙h (σvp , σh ) ∈ dσh π ∗ (σvp , σh ),

(6.261a)

σvp ( ˙vp , ε˙h ) ∈ d ˙vp π ( ˙vp , ε˙h ) and σh ( ˙vp , ε˙h ) ∈ d ε˙ h π ( ˙vp , ε˙h ).

(6.261b)

Obviously the relations in Eqs. 6.261a and 6.261b determine entirely the dissipative behavior of the generic Perzyna hardening model, thus the formulation would be completed at this stage. To be more explicit, however, alternatively to Eq. 6.261b the closed and convex admissible domain A in the {σvp , σh }-space is introduced. It is characterized by the convex yield condition

6.3 Perzyna Hardening Model

401

φ = φ(σvp , σh ) := ϕh (σvp , σh ) − σy ≤ 0.

(6.262)

Here φ = φ(σvp , σh ) is the overstress function and ϕh (σvp , σh ) denotes the equivalent (combined visco-plastic and hardening) stress that is compared to the initial yield limit σy , a material property. Then the evolution law for the visco-plastic and hardening strains (i.e. the associated flow rules) follow alternatively to Eq. 6.261a from the postulate of maximum dissipation (due to hardening visco-plasticity) 1 φ(σvp , σh ) 2 ˜1/η (σvp , σh ; ˙p , ε˙h ) := −d(σvp , σh ; ˙vp , ε˙h ) + → min, (6.263) 2 η whereby ˜1/η is a penalized Lagrange functional incorporating the admissibility constraint φ ≤ 0 penalized by the penalty parameter 1/η. Consequently, the stationarity condition of this unconstrained optimization problem reads ˙vp = λ ∂σvp φ and

ε˙h = λ ∂σh φ with λ := φ(σvp , σh ) /η ≥ 0.

(6.264)

It shall be noted that collectively Eqs. 6.262 and 6.264 are entirely equivalent statements to Eqs. 6.261a and 6.261b. As a further interesting aspect the dissipation d = σvp ˙vp + σh ε˙h shall next be examined more closely. From Eqs. 6.260a and 6.260b the dissipation d is alternatively expressed in terms of the dissipation potential π and the dual dissipation potential π ∗ as (6.265) d = π(˙vp , ε˙h ) + π ∗ (σvp , σh ) ≥ 0. Thereby, based on the above introduction of the overstress function φ (and in view of Eqs. 6.260a, 6.261a, 6.263 and 6.264) the dual dissipation potential is identified as ⎧ ⎫ 0 φ(σvp , σh ) ≤ 0 ⎬ ⎨ for (6.266) π ∗ (σvp , σh ) = ⎩1 ⎭ 2 φ(σ , σ ) /η φ(σ , σ ) > 0 vp h vp h 2 =

1 φ(σvp , σh ) 2 . 2 η

Finally for an equivalent (combined visco-plastic and hardening) stress that is homogeneous of degree one in the visco-plastic and hardening stresses (thus σvp ∂σvp ϕh + σh ∂σh ϕh = ϕh ), the dissipation d = σvp ˙vp + σh ε˙h is exclusively given in terms of the overstress function φ (with abbreviation λ := φ /η ≥ 0 for the viscoplastic multiplier and equivalent stress ϕh = φ + σy ≥ 0), since then  d = λ σvp ∂σvp ϕh + σh ∂σh ϕh = λ ϕh = φ [φ + σy ]/η. The generic Perzyna hardening model is summarized in Table 6.13.

(6.267)

402

6 Visco-Plasticity

Table 6.13 Summary of the generic Perzyna hardening model (1) Strain



= e + vp

(2) Energy

ψ

= ψ( − vp , εh )

Stress

σ

= ∂ ψ

(3)

(4) Stress

≡

σ

≡



−σvp

σh = ∂εh ψ

(5) Potential π

= π(˙vp , ε˙ h ) 

(6) Stress

σvp ∈ d˙vp π



σvp

(7) Stress

σh ∈ dε˙ h π



σh



or (5) Potential π ∗ =

1 2

φ(σvp , σh ) 2 /η

(6) Evolution ˙ vp = λ ∂σvp φ (7) Evolution ε˙ h = λ ∂σh φ with λ : = φ(σvp , σh ) /η