Metal Forming: Formability, Simulation, and Tool Design [1 ed.] 0323852556, 9780323852555

Citation preview

METAL FORMING

METAL FORMING Formability, Simulation, and Tool Design CHRIS V. NIELSEN Technical University of Denmark, Lyngby, Denmark

PAULO A.F. MARTINS IDMEC, Instituto Superior Técnico, University of Lisbon, Portugal

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-323-85255-5 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Charlotte Rowley Production Project Manager: Poulouse Joseph Cover Designer: Christian Bilbow Typeset by SPi Global, India

To our families.

Contributors Brian N. Legarth Technical University of Denmark, Lyngby, Denmark (Chapter 5) Luis M. Alves Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal (Chapter 6) Maria Beatriz Silva Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal (Chapter 2) Niels Bay Technical University of Denmark, Lyngby, Denmark (Chapter 6)

xi

Preface The present book is intended as an advanced textbook on metal forming, which aims at bridging the gap between standard BSc and MSc teaching and advanced research topics focusing on three main topics: (i) formability, (ii) finite element modelling and (iii) tool design. The objectives are as follows: firstly, to provide essential knowledge on formability and finite element topics, which students and engineers use in their daily activities, sometimes without fully understanding the theoretical background; secondly, to give readers, who have attended basic metal forming courses to be able to understand some of the topics that are currently dealt with in advanced research on metal forming; and finally, to provide young engineers taking the first steps in advanced tool design and sizing, a gateway to the industry. The introductory chapter (Chapter 1) gives a broader description of the scope of the book and ends with a classification of metal forming processes based on their load/deformation pattern according to the German DIN standard. Chapter 2 is dedicated to the mechanisms of fracture and plastic instability. The most commonly applied phenomenological models of ductile damage and fracture based on continuum mechanics are presented. The second part of Chapter 2 is dedicated to plastic instability presenting the classic theory by Swift as well as Marciniak-Kuczynski theory for instability in sheet forming. A final part describes typical material flow defects in bulk and sheet metal forming. Chapter 3 presents a user’s perspective on finite element modelling of metal forming processes providing information about accuracy, reliability and validity. A number of simple examples are given, which illustrates the pros and cons of the different formulations (flow formulation, solid formulation and dynamic formulation). The chapter is very useful to users of commercial codes with little knowledge about finite element modelling. Chapter 4 provides deeper knowledge on finite element analysis of metal forming exemplified by the flow formulation, in which the two authors have long-term expertise. Throughout the chapter, some very useful, simple examples illustrate the numerical issues and their solutions. Chapter 5 provides introductory knowledge on the finite element solid formulation applied to metal forming by discretisation and implementation

xiii

xiv

Preface

of the finite element equations for an elastic-viscoplastic material model. It opens the way for readers, who want to progress to more sophisticated implementations of the solid formulation. Chapter 6 provides a comprehensive description of tool design. The chapter gives examples on tools, design guidelines including dimensioning as well as information on tool materials and lubricants. There are several books covering the fundamentals on metal forming at BSc and MSc levels and a significant number of books covering specific research topics of metal forming. The present book aims at bridging the gap between these two types of books. Niels Bay Emeritus Professor Technical University of Denmark, Lyngby, Denmark

Acknowledgements This book would never have been written without the collaboration and support of several people, companies and institutions. We start by thanking our coauthors Beatriz Silva, Brian Legarth, Luis Alves and Niels Bay, who generously contributed with their knowledge to several chapters of this book. Niels Bay is further acknowledged for his long-time scientific and technical mentoring of the authors. Tony Akins, who died in 2018, is also remembered for his inspirational vision on the interaction between formability and fracture mechanics. Erman Tekkaya has been sharing high-quality teaching material and knowledge with Paulo Martins for a long time. This was an important and inspiring source of information for some of the chapters included in this book. The support of Klaus Schreiner from Hatebur, Viktor Lazorkin from Lazorkin Engineering and Motonobu Furuya from NICHIDAI Corporation, and of their companies to the tool design chapter is greatly acknowledged. Similar acknowledgements go to the permissions given by the editorial company Casteilla. The English language revision of the chapters in this book by Tom and Jan Goodwin is also greatly acknowledged by the authors. The authors also want to thank Dennis McGonagle for believing and supporting the authors’ book proposal and Poulouse Joseph for the great help provided during the preparation of this book. The support of the Technical University of Denmark, Instituto Superior T
ecnico at the University of Lisbon, Fundac¸a˜o para a Ci^encia e a Tecnologia of Portugal and IDMEC under LAETA-UIDB/50022/2020 and PTDC/ EME-EME/0949/2020 is also acknowledged. Finally, we would like to thank our families for the support and encouragement they gave us during the writing of this book. Chris V. Nielsen Paulo A.F. Martins

xv

CHAPTER 1

Introduction* Students attending metal forming courses are exposed to a wide range of topics covering plasticity theory, analytical and numerical methods of analysis, material science, formability and fracture, friction and lubrication, processes and machine tools, and tool design. However, the type of exposure varies from university to university. In general, it can be said, with due exceptions, that the higher the intensity level in plasticity, methods of analysis, and formability and fracture, the lower the intensity level and depth in processes, machine tools, and tool design. The vice versa is equally true. Elective advanced courses in metal forming are more difficult to characterise because they are often designed in the image of the professors who are responsible for them, namely on their research expertise. Although this situation may in some cases solve the low-intensity levels of exposure that students may have had to some topics of metal forming in their mandatory BSc or MSc courses, it may in other cases further deepen the differences between the levels of exposure to different topics. This book does not aim to solve all these problems nor to provide readers with a uniform level of knowledge in all the aforementioned topics of metal forming. However, it picks three major key topics in metal forming, which are often not taught with the same level of intensity: (i) formability, (ii) finite element modelling, and (iii) tool design, in order to provide readers with an integrated, high level of exposure to theoretical, experimental, and designrelated subjects. Taking these objectives into account, this book starts with a chapter on formability in metal forming (see Chapter 2). In contrast to other textbooks of metal forming, the chapter starts with fracture and ends with plastic instability and necking because the latter is not critical in all metal forming processes. The presentation draws from nucleation, growth, and coalescence of voids, to the characterisation of different crack opening modes in terms of plastic flow, microstructural damage and fractography. The role played by * Chris V. Nielsen (Technical University of Denmark, Lyngby, Denmark) and Paulo A.F. Martins (IDMEC, Instituto Superior T
ecnico, University of Lisbon, Portugal). Metal Forming https://doi.org/10.1016/B978-0-323-85255-5.00010-8

© 2021 Elsevier Inc. All rights reserved.

1

2

Metal forming

stress triaxiality and shear (through the Lode coefficient) is considered and several uncoupled ductile damage criteria are introduced. The chapter ends with an introduction to both micro- and macromechanics-based damage analyses using Gurson-Tvergaard and Lemaitre-based approaches. Chapters 3–5 are focused on finite elements in metal forming, but they have different objectives. Chapter 3 provides a user’s perspective on accuracy, reliability and validity of finite element modelling, and aims at answering questions such as follows: Which finite element formulations are available for solving metal forming processes? What are the basic aspects of modelling and computation that need to be considered when selecting a specific formulation for solving a metal forming process? How accurate and reliable are the numerical estimates provided by finite element computer programs? Which methodologies can be utilised for validating the numerical estimates? Which mesh generation technologies are available to create finite element models? Refreshment of large deformation kinematics and solution techniques to solve nonlinear systems of equations is also provided in Chapter 3 to help reduce the gap between developers and users of finite element computer programs. Chapter 4 is designed for those readers who want to obtain a deeper knowledge in finite elements, namely in the understanding of how finite element formulations, iterative solution methods, friction and contact between objects and many other scientific and numerical ingredients available in literature can be integrated and merged in the development of an electrothermo-mechanical finite element computer program for metal forming. The option for building a chapter upon the quasi-static flow formulation is explained by the experience of the authors in using this formulation to develop computer programs for metal forming and resistance welding processes. A second reason for this option is the possibility offered by the flow formulation of merging solid mechanics, energy and fluid mechanics views of material deformation into a single set of finite element equations. Chapter 5 performs a brief introduction to the quasi-static solid formulation. The presentation draws from large deformation kinematics and from the principle of virtual power that had previously been introduced in Chapters 3 and 4, to the discretisation and implementation of the finite element equations for elasto-viscoplasticity. The additive decomposition of the elastic and plastic rates of deformation (hypo-elastic-plastic model) is utilised, and an equilibrium correction term is applied to reduce the drift of the solution away from the true equilibrium path. Together with the use of an explicit time integration scheme, this simplifies the presentation and opens

Introduction

3

the way for readers who want to progress to more sophisticated readings and implementations of the solid formulation involving implicit integration schemes for checking equilibrium at each increment of time by means of iterative procedures meant to minimise the residual force vector. Chapter 6 is focused on tool design, because the main objective of finite element modelling is the support of dimensioning and tool design in metal forming. Writing this last chapter was a challenge because the classification and presentation of all different types of tools for metal forming is a cyclopean task, impossible to accomplish in a single chapter of a book with the aforementioned objectives. In order to cover a wide variety of tools in cold, warm, and hot forming processes, the authors decided to merge the classification of metal forming processes as shown in Table 1.1 with the process mechanisms as shown in Table 1.2 to distinguish between forming tools in which the shape of the workpiece is obtained by (i) plasticity and friction, (ii) predominant plasticity and friction, and (iii) plasticity, friction, and fracture. This classification is also linked to formability in the sense that fracture is an undesirable phenomenon in the first two groups of tools, but a necessary phenomenon, responsible for the formation of new surfaces in forming by shearing. A similar difference exists in finite element modelling because the use of damage/separation criteria at the edges of forming by shearing tools is necessary for the tool movement during the press stroke. Chapter 6 provides examples of tools, design guidelines and information related to tool materials and lubricants and finishes with a glimpse of tool

Table 1.1 Classification of metal forming processes. Process group

Process subgroup

Compressive forming

Open die forming Closed die forming Extrusion Rolling

Combined tensile and compressive forming

Drawing Deep drawing Bending

Tensile forming

Stretch forming

Forming by shearing

Punching and blanking

4

Metal forming

Table 1.2 Process mechanisms and classification of metal forming processes and tools. Process mechanism

Process group

Process subgroup

Tool group

Plasticity and friction

Compressive forming

Open die forming

Compressive forming

Closed die forming Extrusion Rolling

Predominant plasticity and friction

Plasticity, friction, and fracture

Drawing

Combined tensile and compressive forming

Deep drawing

Tensile forming

Stretch forming

Forming by shearing

Punching and blanking

Bending

Tensile forming, combined tensile and compression forming, and bending Shearing

failure. This is aimed to provide a background for readers who are giving the first steps in industrial tool design and dimensioning. Before finishing this introductory chapter, the authors would like to include a list of recommended publications that cover some of the topics dealt with in the chapters in more detail. The list is intended for those who aim to learn beyond the contents of this book.

Further reading Formability and fracture

Atkins, A.G., Mai, Y.-W., 1985. Elastic and Plastic Fracture. Wiley, Chichester. Col, A., 2002. Emboutissage des t^ oles: importance des modes de d
eformation. In: Techniques de l’Ing
enieur—trait
e genie m
ecanique, BM7510. (in French). Lemaitre, J., 1996. A Course on Damage Mechanics. Springer, Berlin. Montheillet, F., Briottet, L., 1998. Endommagement et ductilit
e en mise en forme. In: Techniques de l’Ing
enieur—trait
e mat
eriaux m
etalliques, M601. (in French). Pokorny, A., Pokorny, J., 2002. Fractographie: Morphologies des cassures. In: Techniques de l’Ing
enieur—trait
e Mat
eriaux m
etalliques, M4121. (in French). Tekkaya, A.E., Bouchard, P.-O., Bruschi, S., Tasan, C.C., 2020. Damage in metal forming. CIRP Ann. Manuf. Technol. 69, 600–623. Volk, W., Groche, P., Brosius, A., Ghiotti, A., Kinsey, B.L., Liewald, M., Madej, L., Min, J., Yanagimoto, J., 2019. Models and modelling for process limits in metal forming. CIRP Ann. Manuf. Technol. 68, 775–798.

Introduction

5

Finite element method

Dunne, F., Petrinic, N., 2005. Introduction to Computational Plasticity. Oxford, New York. Kobayashi, S., Altan, T., Oh, S.-I., 1989. Metal Forming and the Finite Element Method. Oxford, New York. Mattiasson, K., 2010. FE-models of the sheet metal forming processes. In: Banabic, D. (Ed.), Sheet Metal Forming Processes: Constitutive Modelling and Numerical Simulation. Springer, Berlin. Nielsen, C.V., Zhang, W., Alves, L.M., Bay, N., Martins, P.A.F., 2013. Finite element formulations. Modelling of Thermo-Electro-Mechanical Manufacturing Processes With Applications in Metal Forming and Resistance Welding. Springer, London. https:// doi.org/10.1007/978-1-4471-4643-8_2. Tekkaya, A.E., 2000. Simulation of metal forming processes. In: Banabic, D. (Ed.), Formability of Metallic Materials. Springer, Berlin. Zienkiewicz, O.C., Taylor, R.L., 2000. The Finite Element Method. ButterworthHeinemann, Oxford.

Tool design

Altan, T., Ngaile, G., Shen, G., 2004. Cold and Hot Forging: Fundamentals and Applications. ASM International, Materials Park. Altan, T., Tekkaya, A.E., 2012a. Sheet Metal Forming. Fundamentals. ASM International, Materials Park. Altan, T., Tekkaya, A.E., 2012b. Sheet Metal Forming. Processes and applications. ASM International, Materials Park. Boljanovic, V., 2004. Sheet Metal Forming Processes and Die Design. Industrial Press, New York. Bostbarge, G., Faure, H., Gobard, Y., 2001. Forgeage a` froid de l’acier: choix de l’acier et proceeds. In: Techniques de l’Ing
enieur—trait
e mat
eriaux m
etalliques, M3085. (in French). Bostbarge, G., Faure, H., Gobard, Y., 2002. Forgeage a` froid de l’acier: gamme de forgeage et pie`ces extrudes. In: Techniques de l’Ing
enieur—trait
e mat
eriaux m
etalliques, M3086. (in French). Col, A., 2010. L’emboutissage des aciers. Dunot, Paris (in French). Corbet, C., 2005. Proc
edes de mise en forme des mat
eriaux. Casteilla, Paris (in French). ICFG, 1992. International Cold Forging Group 1967-1982 Objectives, History, Published Documents. Bamberg, Meisenbach. Lange, K., 1985. Handbook of Metal Forming. SME-Society of Manufacturing Engineers, Dearborn. Lange, K., Kammerer, M., P€ ohlandt, K., Sch€ ock, J., 2008. Fließpressen—Wirtschaftliche Fertigung metallischer Pr€azisionswerkst€ ucke. Springer, Berlin (in German). Montheillet, F., 2002. M
etallurgie en mise en forme. In: Techniques de l’Ing
enieur—trait
e mat
eriaux m
etalliques, M600. Schuler, 1998. In: Altan, T. (Ed.), Metal Forming Handbook. Springer, Berlin. Thomas, A., 1998. Forging Methods. Materials Forming Technology, Sheffield. Tschaetsch, H., 2006. Metal Forming Practice. Springer, Berlin. Waller, J.A., 1978. Press Tools and Presswork. Portcullis Press, London.

CHAPTER 2

Formability* 2.1 Introduction Formability is defined as the degree of deformation that can be achieved in a metal forming process without creating an undesirable condition, such as cracking, necking, buckling or the formation of folds and flaws. This chapter is focused on the physics behind formability and discusses the origins of the major defects and cracks that are commonly found in metal forming processes. Some defects, such as wrinkling and necking, are explained in terms of continuum mechanics whilst others, like orange peel, grain coarsening and mechanical fibering, arise because real materials are not continua. The explanation of cracking requires consideration of continuum mechanics, fracture mechanics and metallurgy, and its occurrence is determined by critical levels of damage within regions that are highly strained due to extensive material flow. In most metal forming processes (e.g. forging, extrusion, rolling and deep drawing), the occurrence of surface or interior cracks sets the formability limits and, therefore, crack nucleation must be prevented. However, there are other processes such as blanking, shearing and machining that concern separation of parts and formation of new surfaces. In these other processes, controlled crack nucleation, initiation and propagation is desirable because it is an inherent part of the deformation mechanics. To conclude this introduction, it is worth explaining the usage of the term ‘formability’ throughout the chapter. The term will be utilised for both sheet, tube and bulk metal forming processes instead of using it exclusively for sheet and tube forming, in which no appreciable, intentional changes in the thickness of the parts are usually made. This means that the term ‘workability’ often used in bulk forming, in which intentional, significant changes are made in the thickness of the parts, will be considered as a synonym of the term ‘formability’. * Chris V. Nielsen (Technical University of Denmark, Lyngby, Denmark), Maria Beatriz Silva (Instituto Superior T
ecnico, University of Lisbon, Lisbon, Portugal), and Paulo A.F. Martins (IDMEC, Instituto Superior T
ecnico, University of Lisbon, Portugal). Metal Forming https://doi.org/10.1016/B978-0-323-85255-5.00006-6

© 2021 Elsevier Inc. All rights reserved.

7

8

Metal forming

2.2 Deformation-zone geometry The major parameters controlling formability are invariably processdependent or material-dependent. The shape of the deformation zone was one of the first process-dependent parameters controlling formability to be investigated because it exerts a strong influence upon homogeneity of material flow, internal porosity, and tendency to open cracks during material processing. Caddell and Atkins (1968) characterised the influence of the deformation-zone geometry by means of a parameter Δ, defined as the ratio of the mean thickness (or diameter) h of the workpiece to the contact length l between the tool and the workpiece (Fig. 2.1): h (2.1) l The theoretical foundations of the deformation-zone geometry parameter (hereafter designated as the Δ-parameter) derive from the slip-line field analysis of isotropic, nonstrain hardening slabs compressed by opposing, frictionless, flat punches, for different ratios of the slab thickness h to punch width l (Green, 1951; Hill, 1950, Fig. 2.2). In fact, the slip-line fields for Δ ¼ 1 and Δ > 8.74 (semiinfinite slab) allow concluding that the corresponding Mohr’s stress circles are under compression ( Johnson and Mellor, 1973, Fig. 2.3) and, therefore, crack initiation and propagation is avoided. However, the slip-line fields for intermediate geometries (e.g. Δ ¼ 5.43) show that the hydrostatic (mean) stress σ m is tensile at the centre (refer to the rightmost Mohr’s stress circle corresponding to point 1,I in Fig. 2.4) and that its value increases with the slab thickness h. Hydrostatic tension is responsible Δ¼

45º

h

l

Fig. 2.1 Deformation-zone geometry parameter Δ ¼ h/l.

45º

l

h

Formability

9

h

h/l = 0.5

h/l = 1

l h/l =5.43

h/l > 8.74

Fig. 2.2 Slip-line fields for the compression of slabs by opposing, frictionless, flat punches in a rigid-perfectly plastic material for different thickness to punch width ratios (Δ ¼ h/l).

for the accumulation of internal damage in the form of void formation and subsequent crack initiation and propagation. Patterns of changing deformation fields like those depicted in Fig. 2.4 are found with opposed wedge indentation: when the fields intersect at sufficiently deep penetration, plastic flow that accommodates vertical motion of the wedges occurs along the axis of the plate, putting the centre of the bar into tension and leading to eventual fracture. This is how cutting by pliers takes place and explains why, in other cases, inhomogeneous plastic flow is dangerous for void formation and crack initiation and propagation. Centre voids are also found in rotary piercing and are used as a start-up of seamless tube production in the Mannesmann process. The aforementioned evolution of slip-line fields and Mohr’s stress circle diagrams with the Δ-parameter (Figs 2.3 and 2.4) allows concluding that material flow evolves from homogeneous when Δ ¼ 1, to inhomogeneous extending throughout the slab thickness when 1 < Δ < 8.74 and, finally, to inhomogeneous localised under the punch when the slab thickness h becomes very large (Δ > 8.74). The corresponding evolution of the normalised values of the vertical and horizontal stresses in the material placed under the tools is shown in Fig. 2.5 (Hill, 1950). Despite the relevance of slip-line field analysis and the Δ-parameter to understand the physics behind the occurrence of cracking in open-

10

Metal forming

(A)

(B) Fig. 2.3 Slip-line fields for the compression of slabs with (A) Δ ¼ 1 and (B) Δ > 8.74 by opposing, frictionless, flat punches in a rigid-perfectly plastic material.

die forging, practical knowledge about how to avoid defects was applied before the publication of the slip-line fields for the compression of slabs by Hill (1950a). 100 years before, Nasmyth (1850) introduced V-anvil forging (Fig. 2.6) as the solution to combat the tendency for crack formation. V-anvil forging is still used nowadays to fabricate shafts utilised

Formability

11

Fig. 2.4 Slip-line field for the compression of slabs with Δ ¼ 5.43 by opposing, frictionless, flat punches in a rigid-perfectly plastic material.

in ships, power plants and wind turbines from large cast ingots above the recrystallisation temperature. This is because it is the most effective solution to prevent internal porosities to grow into cracks during open-die forging. The porosities originate from improper feeding, slag inclusions due to impurities in the melt and segregations or coarse

12

Metal forming

Fig. 2.5 Influence of the deformation-zone geometry parameter Δ on the normalised values of the vertical and horizontal stresses. Typical values for contact stresses in the deformation zone of metal forming processes are included.

Dead zone

Fig. 2.6 V-anvil forging of large cast ingots and the corresponding slip-line field explaining the avoidance of tensile hydrostatic stresses at the centre region.

microstructure due to the long cooling time during the production process. The explanation of the effectiveness of V-anvil forging in preventing crack initiation and propagation in open-die forging is based on the avoidance of tensile hydrostatic stresses σ m > 0 at the centre due to the formation of a dead metal zone.

2.3 Voids and void-growth mechanisms The development of hydrostatic tension for Δ-parameters in the range 1 < Δ < 8.74 often leads to internal damage in the form of microscopic

Formability

13

porosity (voids) at the centre that will eventually give rise to internal cracks. This was originally confirmed by Coffin and Rogers (1967) who showed a significant decrease in density near the centre after drawing slabs with high values of the Δ-parameter. However, depending on the metallic alloy and manufacturing process of the semifinished part, material can already present some inclusions and voids before applying plastic deformation associated with the metal forming processes. Plastic deformation, at the grain boundaries, in the neighbourhood of inclusions or around hard secondary-phase particles, can nucleate new voids and cause them to grow progressively and simultaneously. In fact, after a void nucleates, it will become a location for stress concentration and, therefore, it will experience higher deformation rates than the remaining material. Continuation of plastic deformation will result in void shape changes and void coalescence by linking up to form a microscopic crack that will eventually give rise to a macroscopic crack (Fig. 2.7). Fig. 2.7 illustrates the nucleation, growth and coalescence of voids in normal and shear stress fields, respectively. McClintock et al. (1966, 1968) developed models for the growth of these two types of voids considering a spacing-to-size ratio l/d between the initial intervoid distance l and diameter d and accounting for the influence of the normal or shear stresses and material strain hardening. The work of McClintock (1968) in the mechanics of void growth in tension (Fig. 2.7A) enabled Atkins and Mai (1985) to propose the following relation between the spacing-to-size l/d ratio and stress triaxiality

s τ l d

τ Inclusion

(A)

s

Void

(B)

Fig. 2.7 Schematic illustration of the nucleation, growth and coalescence of voids formed under (A) tension and (B) shear at inclusions (or hard secondary-phase particles). Coalescence by linking up of voids originates microscopic cracks.

14

Metal forming

η ¼ σm =σ (defined as the ratio η between hydrostatic stress σ m and effective stress σ):
 ðε l σm ln dε  σ d

(2.2)

0

Christiansen et al. (2019) followed a similar approach to that of Atkins and Mai (1985) in order to model growth and coalescence of voids subjected to in-plane shear stresses τ (Fig. 2.7B), and proposed the following relation between the spacing-to-size ratio l/d, shear strain γ and stress triaxiality η:
 ðγ ðγ l 1 σm ln dγ  dγ + 2τ d 3 0

(2.3)

0

The equations mentioned earlier allow concluding that modelling the growth and coalescence of voids subjected to general stress states involving normal and shear stresses must consider both the dilatational effects related to stress triaxiality and the distortional effects related to shear stresses.

2.4 Fractography and fracture Void growth and distortion due to the accumulation of damage result in coalescence and formation of microscopic cracks that will eventually give rise to a final visible macroscopic crack. The surface morphology of a visible crack contains the history of its initiation and propagation and is very often utilised to identify and understand the origins of failure. This type of analysis is designated as fractography and is commonly performed by means of scanning electron microscopy (SEM). In the case of ductile fracture, which is the type of fracture that is relevant to metal forming, surface morphology is made of dimples and may be classified into two different groups: (a) Circular dimple-based structures typical of fracture caused by normal stresses originated in remote loading orthogonal to the fracture surface (Fig. 2.8A); (b) Parabolic dimple-based structures with their open ends pointing towards (or rotated against) the shearing direction that are typical of fracture caused by sliding under shear stresses (Fig. 2.8B).

Formability

15

Fig. 2.8 Scanning electron microscopy of the fracture surface of cracks in aluminium alloy AA1050-H111 caused by (A) normal stresses and (B) shear stresses. (Adapted from Magrinho, J.P., Silva, M.B., Reis, L., Martins, P.A.F., 2019. Formability limits, fractography and fracture toughness in sheet metal forming. Materials 12, 1493.)

Mode I

Mode II

Mode III

Fig. 2.9 The three crack opening modes: Mode I—tensile stresses, Mode II—in-plane shear stresses and Mode III—out-of-plane shear stresses with schematic representation of the corresponding surface morphology.

Fig. 2.9 associates the different types of dimpled structures and loading conditions to the three fundamental crack opening modes of fracture mechanics: (a) Mode I—opening by tensile stresses—circular dimples; (b) Mode II—opening by in-plane shear stresses—parabolic elongated dimples that point in the same direction as the crack propagation; (c) Mode III—opening by out-of-plane shear stresses—parabolic elongated dimples that are rotated with respect to crack propagation direction.

16

Metal forming

Besides these three fundamental crack opening modes, there are also mixed modes resulting from the simultaneous application of normal and shear stresses. Mixed modes are characterised by intermediate morphologies with dimple-based structures that may be considered a combination of circle and parabolic shapes.

2.5 Uncoupled ductile damage criteria and fracture The utilisation of ductile damage mechanics to predict the onset of cracking in metal forming has a long research pedigree and is usually carried out by means of three different procedures: (a) Uncoupled procedures based on the utilisation of simple ductile damage criteria that are weighted integrations of the effective plastic strain (Atkins, 1996): ðεf Dcrit ¼ gdε

(2.4)

0

where the nondimensional term g is a weighting function that corrects the accumulated value of effective strain at fracture εf as a function of the loading path; (b) Coupled procedures based on microbased continuum damage mechanics (Gurson, 1977; Tvergaard and Needleman, 1984); (c) Coupled procedures based on macrobased continuum damage mechanics (Lemaitre, 1985). The reason why the uncoupled ductile damage criteria are focused on correcting the effective strain at fracture is due to the phenomenological link with the critical size of voids at the instance of coalescence. To _ understand this link, let us consider the coalescence rate ℜ=ℜ of spherical voids with radius ℜ subjected to tension obtained by Budiansky et al. (1982):  1=m _ ℜ 1 3 (2.5) ¼ mη + ð1  mÞð1 + 0:43mÞ ε_ ℜ 2 2 _ is the time derivative of the radius, ε_ is the strain rate, and m is the where ℜ strain rate sensitivity of the flow stress given by σ ¼ C ε_ m where C is a strength constant that depends upon strain, temperature and material. Then, performing the integrating on time, one obtains the desired link between the effective strain at fracture εf and the critical value of the ratio ℜ/ℜ0 between

Formability

17

the actual ℜ and original ℜ0 radius of the voids at the instant when void coalescence occurs:  1=m
 3 ℜ εf ¼ 2 mη + ð1  mÞð1 + 0:43mÞ ln (2.6) 2 ℜ 0 crit Uncoupled ductile damage criteria generally make use of the knowledge on void mechanisms formed under tension (Eq. 2.2) and shear (Eq. 2.3) to predict the onset of cracking. One of these criteria takes the nondimensional weighting function g as the stress triaxiality g ¼ η in order to setup an uncoupled ductile damage criterion Ayada et al. (1987) that is directly related to the original work of McClintock (1968): ðεf Dcrit ¼ 0

σm dε σ

(2.7)

The right-hand side of Eq. (2.7) is the accumulated ductile damage D resulting from dilatation changes in voids subjected to normal stresses (Eq. 2.2) and the onset of cracking is given by the condition of the accumulated ductile damage D being equal to a critical value Dcrit. Other wellknown uncoupled ductile damage criteria take different approaches for setting up the nondimensional weighting function g, as shown in Table 2.1. The criterion proposed by Cockcroft and Latham (1968) is purely empirical and recognises the dependence of fracture upon the largest principal stress σ 1. Brozzo et al. (1972) modified the Cockcroft and Latham (1968) criterion to incorporate an explicit dependence on the hydrostatic stress. This criterion like the one proposed by Norris (1978) is suitable for predicting the formability limit of metal sheets under high levels of triaxiality. In Norris’ criterion, the symbol A is a material parameter to be identified. The criterion proposed by Rice and Tracey (1969) is based on an alternative coalescence approach to that of Budiansky et al. (1982) to model the growth of a spherical void in an infinite perfectly plastic material subjected to remote normal stresses:
 _ _ ℜ ℜ 3 σm V ¼ 0:28exp ¼3 (2.8) ε_ ℜ V ℜ 2 σf 



where V is the volume of the spherical void, V is its variation in time, ε_ is the strain rate, and σ f is the flow stress resulting from the application of a threedimensional stress state or the yield stress σ Y in the case of rigid-perfectly plastic materials.

18

Metal forming

Table 2.1 Uncoupled ductile damage criteria for metal forming.

Ayada et al. (1987) (after McClintock, 1968)

Dcrit ¼

Cockcroft and Latham (normalised) (1968)

Dcrit ¼

Brozzo et al. (1972)

Dcrit ¼

Norris et al. (1978)

Dcrit ¼

Rice and Tracey (1969)

Dcrit ¼

Oyane (1972) Christiansen et al. (2019)

Ðεf σm 0

σ

Ðεf σ1 0

Ðεf 0

Ðεf 0

Ðεf 0

σ

dε dε

2σ 1 3ðσ 1 σ m Þ dε

1 ð1A σ m Þ dε

exp

3 σm  2 σ

dε

Dcrit ¼

 Ðεf  1 + B σσm dε

Dcrit ¼

Ðγf τ

Ðγf 3 σ m

0

0

0

σ dγ +

2 σ

dγ


 Ðεf  τ A h1 + 3σ m =σ i B Dcrit ¼ dε 2 0 σ

Lou et al. (2012)

The criterion proposed by Oyane (1972) is built upon the theory of plasticity for porous materials taking into consideration that density decreases with void growth in tension. The symbol B is a material parameter to be identified. The main idea behind the uncoupled ductile damage criteria discussed previously is to account for the influence of the largest principal stress and/or the hydrostatic stress on crack opening. This makes them suitable for describing failure under tension (high triaxiality) but unsuitable to describe failure in low-stress triaxiality conditions where the effect of shear is important. A recent attempt to overcome this problem was due to Christiansen et al. (2019) who proposed a criterion that derives explicitly from void growth and coalescence mechanisms in shear (Eq. 2.3). This was accomplished by multiplying the right-hand side of Eq. (2.3) by the normalised shear stress ratio τ=σ, in order to obtain the following criterion: ðγ f Dcrit ¼ 0

τ dγ + σ

ðγf 0

3 σm dγ 2σ

(2.9)

Formability

19

The first right-hand side term of Eq. (2.9) is related to the accumulation of damage caused by distortion and is expressed as a function of the normalised plastic shear work per unit of volume τdγ. The second right-hand side term accounts for the dilatational effects due to stress triaxiality η. Christiansen et al. (2019) successfully applied this criterion to model initiation and propagation of cracks in side-pressing of cylindrical specimens under plane strain deformation conditions. The criterion due to Lou et al. (2012) multiplies the stress ratiorelated distortional and dilatational void effects to obtain the following expression:  ðεf  A
τ h1 + 3σ m =σ i B Dcrit ¼ dε 2 σ

(2.10)

0

In Eq. (2.10), ‘h i’ are the Macaulay brackets that are used to prevent the accumulation of damage when the stress triaxiality η <  1/3, and the symbols A and B are material parameters to be identified. Moreover, the criterion considers that cracking does not occur when the stress triaxiality η <  1/3. The procedure for determining the critical values Dcrit of damage for the criteria listed in Table 2.1 requires experimental testing to obtain the strains at fracture. It is also worth noting that all the criteria listed in Table 2.1 should not account for the accumulation of negative damage due to dilatational void changes when σ m < 0, because the closing up of voids under hydrostatic compression does not always ensure material healing and recovery of strength. This means that Macaulay brackets are implicitly considered for all the criteria listed in Table 2.1, which makes them conservative towards void closure. Another conclusion derived from practical use of uncoupled ductile damage criteria in real metal forming processes is that some criteria work well for predicting the onset of cracking by tension, whilst others work well for predicting the onset of cracking by in-plane or out-of-plane shearing. As a result of this, the utilisation of uncoupled damage criteria may often seem to be a little erratic, not to say chaotic, due to inadequate selection of the appropriate criterion for a specific metal forming application. Another key point missing in what has been said so far is the link between ductile damage and fracture mechanics. This link will be discussed in the following section.

20

Metal forming

2.5.1 Link between uncoupled ductile damage and fracture mechanics Martins et al. (2014) proposed an analytical framework for uncoupled ductile damage that solves the lack of integration mentioned earlier between damage criteria and fracture mechanics. The framework establishes the connection between the strains and stresses derived from plane stress plasticity theory under anisotropic plastic flow, the fundamentals of ductile damage mechanics and the three crack separation modes (opening by tensile stresses, by in-plane shear stresses and by out-of-plane shear stresses) (Fig. 2.9). Mode I—Tensile fracture Experiments in sheet metal forming show that, irrespective of the initial loading history, tensile fracture occurs approximately at a constant throughthickness true strain ε3f ¼ Const. The value of ε3f corresponds to a constant percentage of reduction in thickness at fracture Rf given by (t0  tf)/t0 where t0 is the initial thickness of the sheet and tf is the thickness at fracture. Rf and ε3f are related by ε3f ¼ ln(1  Rf). Owing to constancy of volume during plastic material flow, it follows that ε1f ¼  ε2f  ε3f. This result means that in principal strain space, the formability limit associated with crack opening by tension (hereafter designated as the fracture forming limit—FFL) is a straight line falling from left to right with slope ‘1’ (refer to Fig. 2.10 where lines of constant thickness reduction R are shown). Fig. 2.10 also presents a schematic evolution of Mohr’s circles of strain for two proportional loading paths (0C and 0F) corresponding to uniaxial tension and balanced biaxial stretching. For simplifying the presentation, both loading paths are taken as linear up to the onset of fracture. As shown in Fig. 2.10, in the case of uniaxial tension, the diameters of Mohr’s strain circles increase as deformation progresses from A to C but this increase is not concentric due to the necessity of ensuring the ratios between the principal strains (ε1 : ε2 : ε3 ¼ 1 :  0.5 :  0.5). Fracture occurs at point C. In the case of balanced biaxial stretching, the diameter of the circles increases as deformation progresses from D to F, but this increase is also not concentric due to the necessity of ensuring the ratios between principal strains (ε1 : ε2 : ε3 ¼ 0.5 : 0.5 :  1.0). The overall levels of strain are smaller than in the case of uniaxial tension and fracture occurs at point F on the FFL. Fracture is sometimes said to take place at constant effective strain. Fig. 2.10 shows ellipses of constant effective strain ε ¼ K1 and ε ¼ K2 where

Fig. 2.10 Schematic representation of the fracture forming limit (FFL) line in principal strain space together with Mohr’s circles of strains and strain loading paths corresponding to different points A, B before fracture at C and D, E before fracture at F (Martins et al., 2014).

22

Metal forming

failure strain pairs should lie, if cracking occurred at a critical value of effective strain. However, experiments show that a criterion based on a constant fracture strain is incorrect. This is the reason for modifying the criterion by a nondimensional weighting function g that corrects the accumulated value of the effective strain until fracture εf as a function of the loading path (refer to Eq. 2.4). Now, taking the weighting function g as the normalised stress triaxiality g ¼ η, Eq. (2.4) results in the ductile damage criterion due to Ayada et al. (1987) that is directly related to the pioneering work of McClintock (1968). The integrand σ m =σ and the variable of integration dε in the criterion may be expressed as the product of three partial ratios involving the increment of strain dε1 and the stress σ 1. Then, using the constitutive equations associated with Hill’48 anisotropic yield criterion (Hill, 1948), and assuming rotational symmetric anisotropy rα ¼ r ¼ r, where r is the anisotropy coefficient (also known as the Lankford coefficient) defined as the ratio r between the width εw and thickness εt strains during a uniaxial tensile test, and r is the normal anisotropy defined as the weighted average ratio r of strains along the directions parallel r0, diagonal r45 and perpendicular r90 to the rolling direction: r¼

εw εt

r¼

r0 + 2r45 + r90 4

(2.11)

it is possible to rewrite Eq. (2.7) as follows: ðεf

σm Dcrit ¼ dε ¼ σ 0

εð1f

0

σ m σ 1 dε dε1 ¼ σ 1 σ dε1

εð1f

0

ð1 + r Þ ð1 + βÞdε1 3

(2.12)

where β ¼ dε2/dε1 is the slope of a general proportional strain path and the three partial ratios σ m/σ 1, σ 1 =σ and dε=dε1 are given by (where α ¼ σ 2/σ 1 is the principal stress ratio): σ m 1 + α ð1 + 2r Þð1 + βÞ ¼ ¼ σ1 3 3½ð1 + r Þ + rβ

σ1 1 ½ð1 + r Þ + rβ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 1 + 2r 2r β + β2 1+ (2.13) 1 + r rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dε 1+r 2r ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 + β + β2 dε1 1+r 1 + 2r

Appendix A provides the details of the algebraic procedure that was utilised for obtaining the identity corresponding to σ m/σ 1 and for building-up

Formability

23

the two other partial ratios σ 1 =σ and dε=dε1 of Eq. (2.13). It is worth noting that although other anisotropic yield criteria could have been chosen for deriving the aforementioned equations, they would generally suffer from limited applicability due to difficulties of mathematical treatment. The integrand in Eq. (2.12) has the form (A + B β), implying that the damage function for a constant strain ratio β is independent of the loading path history. Path-independence of damage functions based on integrand terms in the form (A + B β), where A and B are constants is comprehensively discussed by Atkins and Mai (1985) and provide additional justification for the reason why strain loading paths in Fig. 2.10 were assumed as linear. Rewriting Eq. (2.12) as a function of the major and minor in-plane strains (ε1f, ε2f) at the onset of fracture, Dcrit ¼

 ð1 + r Þ  ε1f + ε2f 3

(2.14)

it follows that the critical value of the stress triaxiality σ m =σ based damage criterion due to Ayada et al. (1987) is also a straight line with slope ‘1’ falling from left to right in agreement with the FFL and the condition of critical thickness reduction (Fig. 2.10). If the lower limit of the integral in Eq. (2.12) is ε0 rather than zero, corresponding to situations where there is a threshold strain ε0 below which damage is not accumulated, Eq. (2.14) becomes the following: Dcrit ¼

ð1 + r Þ  ε1f + ε2f  ð1 + βÞ ε0 3

(2.15)

The existence of a threshold strain ε0 below which damage is not accumulated leads to an ‘upward curvature’ tail of the FFL as it is schematically plotted in Fig. 2.11. The explanation is simple and starts by understanding that for a specific value of ε2f there has to be an increase in ε1f when ε0 > 0 because ductile damage is a material property that must remain independent from strain loading paths. This means that the FFL moves upwardly when a threshold strain ε0 exists. However, because β ¼ dε2/dε1 increases from the second quadrant (where it takes negative values) to biaxial strain in the first quadrant (where it takes the highest positive value equal to ‘+1’), the FFL will not only move upwardly as it will also present an upward tail. In other words, the threshold strain ε0 is responsible for the upward tail of the FFL depicted in Fig. 2.11.

24

Metal forming

ε1

FFL R3 R2

1 –1

εo > 0

R1

εo = 0 ε = K2 ε = K1

1 –1

1 1

0

ε2

Fig. 2.11 Schematic representation of the deviation from linearity of the FFL (Martins et al., 2014).

Muscat-Fenech et al. (1996) made the connection between the FFL and fracture toughness in mode I. Taking their observation in conjunction with the conclusions mentioned earlier regarding the dual condition of the critical thickness reduction Rf and the critical ductile damage Dcrit being constant and independent from the material deformation history up to fracture, it is possible to see that the FFL is a material property in contrast to the forming limit curve (FLC) that will be analysed later in this chapter, which is not a material property independent of the strain path. Mode II—In-plane shear fracture The formulation of the analytical conditions under which fracture occurs by in-plane shear stresses will be informed by Mohr’s circle of strains, from which it may be concluded that straight lines γ 1, γ 2, and γ 3 rising from left to right and corresponding to maximum values of the in-plane distortion γ have slope ‘+1’ and are perpendicular to the FFL (Fig. 2.12). In-plane distortions γ are caused by in-plane shear stresses τ and, therefore, it is likely that the in-plane shear fracture forming limit (SFFL) will coincide with a straight line of slope equal to ‘+1’, in which the major

Fig. 2.12 Schematic representation of the in-plane shear fracture forming limit (SFFL) in the principal strain space together with Mohr’s circles of strain and a strain loading path corresponding to point A before fracture at B (Martins et al., 2014).

26

Metal forming

and minor in-plane strains and distortions take critical values at fracture (refer to γ ¼ γ f in Fig. 2.12): ε1f  ε2f ¼ γ f

(2.16)

Fig. 2.12 also presents a schematic evolution of Mohr’s circles of strain for a loading path 0AB consisting of pure shear deformation. As shown, the diameters of the circles increase concentrically as deformation progresses due to the necessity of ensuring the ratios between the principal strains (ε1 : ε2 : ε3 ¼ 1 :  1 : 0). Fracture occurs at point B in mode II. Damage mechanics also predicts a slope ‘+1’ for the SFFL when the weighting function g in Eq. (2.4) is taken as the in-plane shear stress ratio g ¼ τ=σ: Dscrit ¼

ðεf 0

τ dε: σ

(2.17)

The weighting function g in Eq. (2.17) may be thought of as a correction of the accumulated values of the effective strain until fracture at εf by means of the maximum in-plane shear stress τ corresponding to different strain loading paths. In other words, the new damage function combines the old fracture criterion of maximum shear stress associated with void shape changes by distortion and the effective plastic strain at fracture. By expressing the integrand τ=σ and the variable of integration dε as the product of two partial ratios involving the increment of strain dε1, the in-plane shear stress τ and using the constitutive equations associated with Hill’48 anisotropic yield criterion (Hill, 1948), it is possible to rewrite Eq. (2.17) as follows, Dscrit

ðεf

τ ¼ dε ¼ σ 0

εð1f

0

τ dε dε1 ¼ σ dε1

εð1f

0

1 ð1 + r Þ ð1  βÞ dε1 2 ð1 + 2r Þ

(2.18)

where the ratio dε=dε1 was previously defined in Eq. (2.13) and the in-plane shear stress τ and the shear stress ratio τ=σ are given by τ ¼ τ12 ¼

σ1  σ2 1  α ¼ σ1 2 2

τ 1 1 1β ¼ pffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 2 1 + 2r 2r 1+ β + β2 1+r

(2.19)

When linear strain paths β in conjunction with the integrand form (A + B β) are employed, the critical value of damage Dscrit in Eq. (2.18) is

Formability

27

strain path-independent and may be expressed as a function of the major and minor in-plane strains at the limiting locus of in-plane shear fracture: Dscrit ¼

 1 ð1 + r Þ  ε1f  ε2f 2 ð1 + 2r Þ

(2.20)

This result shows that critical values of damage by in-plane shear are located along a straight line rising from left to right with a slope equal to ‘+1’ in agreement with the condition of critical distortion γ f along the shear fracture forming limit (SFFL). The location of the SFFL in principal strain space is determined by experimentation, but in strict theoretical terms and disregarding the development of mixed-crack separation modes at the transition between tensile and in-plane shear fracture modes, the SFFL only requires the experimental values of one type of shear test (for instance, the torsion shear test) because of the known slope. However, good experimental practice recommends the utilisation of other tests such as in-plane shear tests instead of assuming the perpendicular condition from theory. Mode III—Out-of-plane shear fracture Another important formability limit is that arising from material flow conditions where cracks develop not due to in-plane (ε1, ε2) distortion γ but due to out-of-plane (ε1, ε3) distortion γ 13. In other words, where cracks are triggered by critical values of the out-of-plane shear stresses τ13 in mode III, instead of in-plane shear stresses τ in mode II. The symbols γ 13 and τ13 are utilised to denote the maximum distortion γ max and the maximum shear stress τmax in the largest Mohr’s circles of strains and stresses corresponding to the plane defined by principal directions 1 and 3. Fracture mechanics employs the term ‘through-thickness shear’ as a synonymous for ‘out-of-plane shear’ but from a metal forming point of view such designation is only appropriate in the case of thick sheets, not in the case of bulk forming workpieces. The out-of-plane shear fracture locus is of key importance to bulk forming due to three-dimensional nature of material flow conditions at the free surfaces of the workpieces where cracks are triggered. This is exactly what makes bulk forming material flow conditions different from those of sheet forming and, therefore, the out-of-plane shear stresses are expected to play a key role in formulating the conditions for out-of-plane shear fracture forming limit (OSFFL). It is worth noting that only surface cracks will be considered in this section of the chapter due to the overall plane stress assumption (σ 3 ¼ 0).

28

Metal forming

Fig. 2.13 Cracking by out-of-plane shear stresses. (A) Surface cracks in upsetting of cylinders (plane stress conditions σ 3 ¼ 0 when initiated); (B) internal (hidden) chevron cracks in forward rod extrusion.

Internal cracks such as those found in open-die forging of cast ingots or the chevron cracks found in extrusion will not be addressed here (Fig. 2.13). The classical analytical treatment of these cracks makes use of the deformation-zone geometry parameter Δ ¼ h/l (refer to Section 2.2) to investigate the tendency to open internal cracks during bulk forming. Another possibility is, for example, to make use of some of the uncoupled ductile damage criteria that are listed in Table 2.1. Following the analytical procedures used in previous sections, the weighting function g in Eq. (2.4) equals the ratio of the out-of-plane shear stress to the effective stress g ¼ τ13 =σ. Eq. (2.4) can then be rewritten as follows: Dtscrit

ðεf ¼ 0

τ13 dε σ

(2.21)

Now, by expressing the integrand τ13 =σ and the variable of integration dε in Eq. (2.21) as the product of two partial ratios involving the increment of strain dε1 and the out-of-plane shear stress τ13, Dtscrit

ðεf

τ13 dε 1 ¼ dε1 ¼ σ dε1 2 0

εð1f

0

  ð1 + r Þ2 rβ 1+ dε1 ð1 + 2r Þ ð1 + r Þ

(2.22)

and, subsequently, expressing the maximum out-of-plane shear stresses τ13 as follows: σ1  σ3 σ1 ¼ (2.23) τ13 ¼ 2 2

Formability

29

where σ 3 ¼ 0 is the stress perpendicular to the surface where cracking occurs, it is possible to obtain the normalised ratio τ13 =σ that will be utilised as the weighting function g in Eq. (2.21): τ13 1 1 ½ð1 + r Þ + rβ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2.24) σ 2 1 + 2r 2r 2 β+β 1+ 1+r By replacing Eq. (2.24) into Eq. (2.22), one obtains the following result after some algebraic treatment:   1 ð1 + r Þ2 r ts Dcrit ¼ (2.25) ε1f + ε2f 2 ð1 + 2r Þ ð1 + r Þ This result demonstrates that critical values of damage at the limiting locus of out-of-plane shear fracture are located along a straight line falling from left to right with a slope equal to ‘r/(1 + r)’ (Fig. 2.14). In Fig. 2.14, it is worth noting that the slope of the OSFFL depends upon the anisotropy factor r, whereas the slopes of the FFL and SFFL are independent of r and equal to ‘1’ and ‘+1’, respectively (Eqs. 2.14, 2.20), even when using Hill’s 48 anisotropic yield criterion. So, in case r ¼ 1, Eq. (2.25) predicts that separation in mode III is characterised by a limiting locus of out-of-plane shear fracture consisting of a straight line with a slope equal to ‘1/2’. Fig. 2.14 also presents a schematic evolution of Mohr’s circles of strain for a strain loading path 0C consisting of uniaxial compression 0A followed by balanced biaxial stretching AB and final uniaxial tension BC without necking. As seen, Mohr’s circle is first translated from A to B without changing its diameter but ensuring the following ratios between the principal A A B B B strains (εA 1 : ε2 : ε3 ¼ 0.5 : 0.5 :  1.0), (ε1 : ε2 : ε3 ¼ 1.0 :  0.5 :  0.5). From B to C, Mohr’s circle expands in a nonconcentric way to ensure the ratios between the principal strains that are typical of uniaxial tension. From a fracture mechanics point of view, failure at C is caused by the out-of-plane shear stresses giving rise to crack separation in mode III. Because of the aforementioned path-independence of damage functions based on integrand terms in the form (A + B β) , the accumulated damages obtained from a complex strain path 0ABC and from a linear strain path 0 BC are identical. Fig. 2.14 also presents an isoline corresponding to a constant value of the plastic work per unit of volume w ¼ w1, in the case of a material having a stress-strain curve approximated by Ludwik-Hollomon’s equation σ ¼ kεn . ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )n + 1 ð k 1+r 2r ¼ w1 (2.26) w ¼ σdε ¼ 1+ β + β2 dε1 n + 1 1 + 2r 1+r

Fig. 2.14 Schematic representation of the out-of-plane shear fracture forming limit (OFFL) in the principal strain space together with Mohr’s circles of strains and a strain loading path corresponding to points A and B before fracture at C for an isotropic material. The concentric ellipses are isolines of effective strain and plastic work per unit of volume (Martins et al., 2014).

Formability

31

The shape of the isoplastic work per unit of volume w is concentric around the corresponding isoeffective strain ε ¼ K1 and expands as the strain hardening n increases. Its shape explains why in some special conditions and specific loading paths (e.g. between uniaxial tension and plane strain conditions), the plastic work per unit of volume may eventually provide good results as a ductile damage indicator. However, as shown in Fig. 2.14, there are many other loading paths where the difference between w and the OSFFL is extremely large, for example, the loading paths between uniaxial tension 0 BC and uniaxial compression 0A. In fact, this is the reason why the plastic work per unit of volume and related energy-based criteria should not be used to analyse formability in metal forming. Another interesting conclusion arises from comparing the formability locus by out-of-plane shear Dtscrit in Eq. (2.25) against the ductile damage criterion proposed by Cockcroft and Latham (1968): ð i σ1 ð1 + r Þ2 h r DCL ¼ (2.27) DCL ¼ + dε ε ε 1f 2f crit σ ð1 + 2r Þ 1+r The result of such comparison is Dtscrit ¼ DCL crit /2, meaning that the Cockcroft-Latham damage criterion is based on a ‘hidden’ out-of-plane shear-based condition and, therefore, should not be seen as a principal stress damage-based model. This explains why the Cockcroft-Latham damage criterion works well in bulk forming, like in case of the bulk formability test specimens that fail by vertical or inclined cracks along the outer surface (Fig. 2.13), because these cracks do not run radially and, therefore, must be triggered by shear. Graphical representation of the analytical framework for ductile damage Depending on the loading path, experiments show that two types of cracks are found in all sheet metal forming processes. They are (i) tensile cracks (Fig. 2.15A), as found in traditional and incremental sheet metal forming processes and described by the FFL (mode I), and also (ii) in-plane shear cracks (Fig. 2.15B) that had previously been noted in incremental sheet metal forming by Soeiro et al. (2015). The latter are not characterised by the FFL but the SFFL (mode II), in the tension-compression quadrant (Fig. 2.15C). Cracks opening by tension at the lobes show similarity to what is commonly found in deep drawing, whereas cracks opening by in-plane shear at the root of the valleys are induced in the material by the tool travelling along the sharp concave corners as a result of the meridional cross sections along the valleys being artificially held stiff by the ribs.

Metal forming

32

(A)

(B)

2.0 1.8 1.6 1.4 1.2

e1

FFL

1.0 0.8

SFFL

FFL

SFFL

0.6 0.4 0.2

FLC

FLC SFFL FFL

0.0 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2

e2

0.0

0.2

0.4

FLC

FLC SFFL FFL 0.6 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2

e2

0.0

0.2

0.4

0.6

(C) Fig. 2.15 Failure by cracking in single-point incremental forming of a truncated lobe conical shape with varying drawing angles (Soeiro et al., 2015). (A) Crack opening by tension in a convex region; (B) crack opening by in-plane shear in a sharp concave region; (C) representation of the strain loading paths corresponding to (A) and (B) in principal strain space.

Experimental work by Kudo and Aoi (1967) and Kuhn et al. (1973) showed that the limiting fracture strain pairs on the outside surfaces of upset test specimens subjected to conditions typical of bulk deformation, fall on a straight line of slope ‘1/2’ in principal strain space (i.e. parallel to the loading path in uniaxial compression, refer to leftmost line in Fig. 2.16). Since both vertical and inclined surface cracks that are found in the specimens do not run radially, they must be produced by shear as previously

Formability

33

ε1

OSFFL 1

-2

B

A

FFL 1 -1

Homogeneous compression 1

-2

C

1 -1 1 -1/2 1 1

ε2

0

Fig. 2.16 Bilinear fracture locus in bulk metal forming obtained from upset formability tests performed in cylindrical, tapered and flanged specimens (Martins et al., 2014).

noticed by Kobayashi (1970) and Atkins and Mai (1985). Inclined surface cracks are generally found in specimens with high friction and low aspect ratios, whilst vertical cracks occur in specimens with low friction and larger aspect ratios. Subsequent investigations by Erman et al. (1983) disclosed new experimental results showing the existence of a bilinear fracture locus in bulk metal forming resulting from the combination of the previously mentioned straight line of slope ‘1/2’ and a straight line of slope ‘1’, parallel to the loading path in pure shear. The bilinear fracture locus of bulk forming is schematically illustrated in Fig. 2.16. Vertical cracks are usually found in the upset formability tests corresponding to the fracture locus given by the rightmost line. Taking into consideration Fig. 2.16 and what was mentioned in previous sections of this chapter, it follows that failure in the upset bulk test specimens is characterised by the OSFFL (mode III) and the FFL (mode I). Under these circumstances, Martins et al. (2014) presented a plot of the analytical framework for ductile damage in principal strain space and in the space of effective strain at fracture vs stress triaxiality (Vujovic and Shabaik, 1986) (Fig. 2.17).

(A)

(B) Fig. 2.17 Schematic representation of the fracture loci in (A) sheet and (B) bulk forming that result from the analytical framework in principal strain space (left) and in the space of the effective strain at fracture vs stress triaxiality (right). The threshold strain ε0 responsible for the upward tail of the FFL and SFFL was not considered (Martins et al., 2014).

Formability

35

The plots consider that the differences in plastic flow resulting from the plane-stress conditions of sheet metal forming (Fig. 2.17A) and the threedimensional stress conditions of bulk metal forming (Fig. 2.17B) that are commonly used as a rationale to classify metal forming processes into two different groups should also be employed to distinguish the circumstances under which different processes fail by fracture.

Example 2.1

Consider the flanged upset compression test specimen made from steel AISI 1040 annealed that is shown in Table 2.2. The specimen fails by cracking in opening mode I (refer to the intersection between the loading path ‘C’ and the FFL in Fig. 2.16) when the strains at the outer flange surface are (ε1f, ε2f) ¼ (4/9,  2/9 ). To determine the critical damage according to Cockcroft and Latham, and Rice and Tracey criteria (Table 2.1), we start by determining the strain at fracture in the radial direction (principal direction 3) ε3f and the corresponding value of effective strain εf : 

ε3f ¼  ε1f + ε2f



2 ¼  εf ¼ 9

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2 ε1f + ε22f + ε23f ¼ 3 9

Table 2.2 Flanged upset compression test specimen. (mm)

h

18

d

16

di

12

t

8

36

Metal forming

The critical damage at fracture Dcrit according to both criteria are given by DCL crit

DRT crit ¼

εðf


exp

0



3 σm dε ¼ 2σ

εðf 0

εðf

¼ 0

εðf σ1 4 dε ¼ dε ¼ ¼ 0:444 σ 9 0




 3 σ1 1 1 4 exp dε ¼ exp  εf ¼ exp  ¼ 0:733 2 3σ 1 2 2 9

Calculations took into consideration that cracking at the outer flange surface occurs by uniaxial tension.

2.5.2 Uncoupled ductile damage and the Lode parameter Although the analytical framework for ductile damage in metal forming proposed by Martins et al. (2014) provides a unique link between uncoupled ductile damage criteria and fracture mechanics, it results from Fig. 2.17 that stress triaxiality η was the only factor utilised for distinguishing between the different stress states. Moreover, the observation of the schematic strain loading paths in upset formability tests (Fig. 2.16) allows concluding that compression of cylindrical rotated test specimens experience in-plane shear ε1 ¼  ε2 strain loading paths similar to those obtained in torsion and shear tests. The necessity of distinguishing between different stress states in metal forming stimulated researchers to introduce an additional stress ratio designated as the Lode coefficient. The Lode coefficient μ is a measure of the effect of the intermediate principal stress σ 2 and was originally proposed to analyse the influence of σ 2 on yielding according to Tresca and von Mises’ yield criteria (Fig. 2.18): 1 2σ 2  σ 1  σ 3 σ 2  2 ðσ 1 + σ 3 Þ σ 2  σ 13 ¼  1  μ  1 (2.28) μ¼ ¼ 1 σ1  σ3 τ13 ðσ 1  σ 3 Þ 2 In the case of the von Mises yield criteria, the combined use of the stress triaxiality η and the Lode coefficient μ allows expressing the principal stresses σ 1, σ 2, σ 3 as follows (Mendelson, 1986):

Formability

37

τ τ13 σ3 σ 13

σ2

σ1

σ

σ 2 − σ13

Fig. 2.18 Notation utilised in the definition of the Lode coefficient μ.

! ð3  μÞσ ð3  μÞ σ 1 ¼ σ m + pffiffiffiffiffiffiffiffiffiffiffiffi ¼ η + pffiffiffiffiffiffiffiffiffiffiffiffi σ 3 μ2 + 3 3 μ2 + 3 ! 2μσ 2μ σ 2 ¼ σ m + pffiffiffiffiffiffiffiffiffiffiffiffi ¼ η + pffiffiffiffiffiffiffiffiffiffiffiffi σ 3 μ2 + 3 3 μ2 + 3 ! ð3 + μÞσ ð3 + μÞ σ 3 ¼ σ m  pffiffiffiffiffiffiffiffiffiffiffiffi ¼ η  pffiffiffiffiffiffiffiffiffiffiffiffi σ 3 μ2 + 3 3 μ2 + 3

(2.29)

Eq. (2.29) associate the Lode coefficient μ to the deviatoric part of the stress tensor, which is responsible for controlling the distortional effects in material deformation. Another way of expressing the Lode coefficient μ is by means of the Lode angle parameter ξ, which is obtained from the Lode angle θ (Figs 2.19 and 2.20). For this purpose, let us consider a stress state given by point P(σ 1, σ 2, σ 3) in principal stress space (Fig. 2.19) and write its location as P(ρ, s, θ), where ρ is the magnitude of the hydrostatic stress vector: pffiffiffi 1 1 1 I1 (2.30) ρ ¼ jON j ¼ pffiffiffi σ 1 + pffiffiffi σ 2 + pffiffiffi σ 3 ¼ 3σ m ¼ pffiffiffi 3 3 3 3 s is the magnitude of the deviatoric stress vector: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ jPN j ¼ jOP j2  jON j2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðσ 21 + σ 22 + σ 23 Þ  3σ 2m ¼ ðσ 1  σ m Þ2 + ðσ 2  σ m Þ2 + ðσ 3  σ m Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ¼ σ 0 21 + σ 0 22 + σ 0 23 ¼ 2J2 (2.31)

38

Metal forming

σ

3

σ3

e lan

cos-11/ 3

πP

P S

O

N

cos-11/ 3

σ2

σ1

σ2

cos-11/ 3

σ1 Fig. 2.19 Representation of a stress state in principal stress space, also designated as the Westergaard (1920) stress space.

and θ is the positive Lode angle measured from the projection of point P lying on the yield surface to the projection of the maximum principal stress axis σ 1 on the π-plane (Fig. 2.20). In the aforementioned equations, I1 and J2 are the first invariant of the stress tensor and the second invariant of the deviatoric stress tensor, respectively. The Lode angle θ is related to the Lode coefficient μ as follows (Mendelson, 1986): pffiffiffi 3ðμ + 1Þ (2.32) tanθ ¼ 3μ In case the stress state P is on the yield locus (Fig. 2.20), itsplocation may ffiffiffi alternatively be given by means of the distance ρ ¼ jON j ¼ 3 σ m and the Lode angle θ. The Lode angle parameter ξ is defined as pffiffiffi 27 J3 3 3 J3 ξ ¼ cos3θ ¼ ¼ (2.33) 2 σ3 2 J23=2 and, in the case of plane stress conditions σ 3 ¼ 0, Wierzbicki and Xue (2005) obtained the following relation with the stress triaxiality η:
 27 1 2 ξ¼ η η  (2.34) 2 3 The relation given by Eq. (2.34) is plotted in Fig. 2.21.

Formability

σ2

σ = −σ

π Plane

2

39

3

Yield locus

−σ

−σ

1

3

(compression)

30°

P σ1 = −σ 3

O

(shear)

θ 30°

σ3

σ1 (tension)

−σ

2

120°

Fig. 2.20 Projection of the stress state given by point P on the π-plane.

η 1 Plane strain tension

Equal biaxial tension

0.5

Uniaxial tension Torsion or shear

0 Uniaxial compression

-0.5 Equal biaxial compression

Plane strain compression

-1 -1

-0.5

0

0.5

1

ξ

Fig. 2.21 Typical values of triaxiality and Lode angle parameter for different formability tests. (Adapted from Bai, Y., Wierzbicki, T., 2008. A new model of metal plasticity and fracture with pressure and Lode dependence. Int. J. Plast. 24, 1071–1096.)

40

Metal forming

2τmax σ

1.155

1

-0.33

0

0.33

0.67

η

Fig. 2.22 Relation between the normalised maximum shear stress τmax =σ and the stress triaxiality η. (Adapted from Isik, K., Doig, M., Richter, H., Clausmeyer, T., Tekkaya, A.E., 2015. Enhancement of Lemaitre model to predict cracks at low and negative triaxialities in sheet metal forming. Key Eng. Mater. 639, 427–434.)

Another important result obtained by Lou et al. (2012) is the relation between the Lode angle θ and the normalised maximum shear stress τmax =σ: 
 τmax σ 1  σ 3 1 1 4 p ffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ ¼ cosθ  cos π  θ (2.35) σ 2σ 3 μ2 + 3 3 because it allows understanding the importance of shear for different levels of stress triaxiality η. The relation between the normalised maximum shear stress τmax =σ and the stress triaxiality η is plotted in Fig. 2.22. Revisiting Table 2.1, it is possible to conclude that the uncoupled ductile damage criteria due to Christiansen et al. (2019) and Lou et al. (2012) account for the influence of both stress triaxiality η (explicitly) and Lode angle parameter ξ (θ) (implicitly). Christiansen et al. (2019), for example, consider the summation of both terms, whereas Lou et al. (2012) consider the multiplication: ðγf

ðγf τ 3 σm dγ + dγ ! DðξÞ + DðηÞ Dcrit ¼ σ 2σ 0

0

 ðεf  A
τ 1 + 3σ m =σ B dε ! DðξÞ  DðηÞ Dcrit ¼ σ 2 0

(2.36)

Formability

41

Nevertheless, both strategies result in a limited range of applicability. For instance, the criterion due to Christiansen et al. (2019) is only valid for analysing the onset of cracking under in-plane shearing or under combined in-plane shearing and tension because it experiences difficulties in situations where cracks open exclusively by tension. This type of limitations was recently addressed by Isik (2018) and will be described in what follows. Uncoupled ductile damage criterion due to Isik (2018) The previous sections of this chapter allow drawing the following conclusions: (a) The hydrostatic part of the stress tensor controls dilatational growth of voids by means of stress triaxiality; (b) The deviatoric part of the stress tensor controls void shape changes by distortion by means of the Lode angle parameter; (c) Fracture locus has a trend change near the uniaxial tensile condition (Fig. 2.17); (d) There is no possibility of having void growth in materials subjected to compressive states of stress under η <  1/3. This set of conclusions was the motivation behind the development of the uncoupled ductile damage criterion of Isik (2018), who proposed a nondimensional g function that corrects the accumulated value of effective strain at fracture εf in Eq. (2.4) by a weighted combination of stress triaxiality η (explicitly) and Lode angle parameter ξ (θ) (implicitly): ðεf  Dcrit ¼ 0


 2τmax  1 dε A h1 + 3ηi + B σ

(2.37)

In Eq. (2.37), A and B are material parameters to be identified. They scale the effect of the dilatational and distortional effects in the growth and coalescence of voids. It is also clear from Eq. (2.35) and Fig. 2.22 that there is no accumulation of damage due to Lode angle parameter ξ in case η ¼  1/3 (uniaxial compression), η ¼ 1/3 (uniaxial tension), and η ¼ 2/3 (equal biaxial stretching) because 2τmax =σ ¼ 1 and, therefore, the second right-hand term of Eq. (2.37) is equal to zero. Similar to what was done by Martins et al. (2014), the critical damage Dcrit can also be plotted in the space of effective strain at fracture vs stress

42

Metal forming

triaxiality as a fracture locus. For this purpose and assuming proportional strain loading paths, Eq. (2.37) can be rewritten as εf ¼

Dcrit
 2τmax 1 A h1 + 3ηi + B σ

(2.38)

The plot is shown in Fig. 2.23 after normalising the material parameters A and B by the critical damage Dcrit εf ¼

1
 2τmax A h1 + 3ηi + B 1 σ

(2.39)

The main difference of Eq. (2.39) and Fig. 2.23 to the analytical framework due to Martins et al. (2014) is the absence of distinction between sheet and bulk metal forming. In the case of plane stress loading conditions, and similar to Martins et al. (2014), the fracture locus is asymptotic for uniaxial compression (η ¼  1/3). However, the ratio between material parameter A and B influences the overall shape of the fracture locus. In particular, as the ratio B/A increases, the shape evolves from monotonic to ‘double valley’ with a local peak value at uniaxial tension (η ¼ 1/3). In such cases of ‘double valley’-shaped curves, there are two local minima of effective strain at failure for triaxiality values equal to η ¼ 0 (pure shear) and

εf

εf 5

5

A = 0.2

B = 1.0 4

4 A = 0.2

B = 1.0

3

3

B = 2.0 2

2

A = 0.4

1

1

B = 4.0

A = 1.0 0 –0.33

(A)

0

0.33

0.66

η

0 –0.33

0

0.33

0.66

η

(B)

Fig. 2.23 Schematic representation of the fracture loci due to Isik et al. (2018) in the space of the effective strain at fracture vs stress triaxiality. The influence of material parameters (A) A and (B) B of Eq. (2.39) is included. (Adapted from Isik, K., 2018. Modelling and Characterization of Damage and Fracture in Sheet-Bulk Metal Forming (Ph.D. Thesis). Technical University of Dortmund, Dortmund.)

Formability

43

Table 2.3 A and B parameters for the uncoupled ductile damage criteria of Isik et al. (2018). Material

A

B

Steel DP600

0.35

0.10

Steel DC04

0.14

0.43

pffiffiffi η ¼ 1= 3 (plane strain). Table 2.3 presents the values of A and B parameters of Eq. (2.37) for a dual-phase steel DP600 and a mild steel DP04 (Isik, 2018). Uncoupled ductile damage criterion due to Bao and Wierzbicki (2004) The ‘double valley’ shape (also known as the ‘butterfly’ shape) of the fracture locus disclosed in Fig. 2.23 had been originally reported by Bao and Wierzbicki (2004) after carrying out formability tests for negative stress triaxialities (1/3  η < 0)—using compression tests and notched compression tests; for low-stress triaxialities (0  η < 1/3)—using shear tests, combined shear and tension tests and tension in central hole specimens; and for high-stress triaxialities (η 1/3)—using conventional tensile tests and tensile tests in notched specimens. The shape of the fracture locus obtained from all these tests is schematically depicted in Fig. 2.24. The main difference between the analytical framework for uncoupled ductile damage (Martins et al., 2014) and the empirical uncoupled ductile damage criterion that Bao and Wierzbicki (2004) built upon their εf

1

1

-2

1 -1/2 1 1

0

(A)

ε2

σm / σ -1/3

0

1/3

1/ 3 2/3

(B)

Fig. 2.24 Fracture loci derived from experiments utilised to set up the uncoupled ductile damage criterion of Bao and Wierzbicki (2004). (A) Schematic representation in principal strain space; (B) schematic representation in the space of the effective strain at fracture vs stress triaxiality.

44

Metal forming

experimental results is that sheet and bulk forming are treated independently in the analytical framework. As a result of this, the fracture locus for sheet forming derived from the analytical framework (Fig. 2.17A) presents some differences against that obtained by Bao and Wierzbicki (2004) (Fig. 2.24). In fact, neglecting the deviations from linearity of the FFL that were previously mentioned in Section 2.5.1 (Fig. 2.17A (left)), which can be properly modelled by adjusting the g function by permitting ε0 to exist, and assuming the absence of mixed-mode cracking, the main differences between the two approaches are found in the second quadrant of principal strain space. The analytical framework (Martins et al., 2014) considers the limiting locus of in-plane shear fracture located in the tension-compression domain to be defined by means of a straight line (SFFL), whereas Bao and Wierzbicki (2004) consider the fracture locus in this region of principal strain space to be defined by means of a curved line that is asymptotic towards strain loading in pure compression. The difference between the two fracture loci is attributed to the fact that, in contrast to the methods and procedures utilised by Bao and Wierzbicki (2004), the analytical framework does not make use of bulk formability tests to determine the fracture strains in sheet forming. In other words, there is an independent treatment of sheet and bulk metal forming conditions. The differences in the space of the effective strain at fracture vs stress triaxiality (Fig. 2.24B vs Fig. 2.17A (right)), especially the absence of the asymptote at σ m = σ¼ 1=3, result from the aforementioned differences in principal strain space. The elimination of the peak at σ m =σ¼ 2=3 (Fig. 2.17A) is dependent on the existence of the aforementioned nonlinearity of the FFL and, therefore, cannot be claimed to be a difference between the two damage criteria. In bulk forming, the differences between the fracture locus derived from the analytical framework (Fig. 2.17B) and from Bao and Wierzbicki (2004) are more significant. In fact, whilst the analytical framework is capable of replicating the onset of fracture that is commonly found in the bulk formability tests performed by Kudo and Aoi (1967), Kuhn et al. (1973), and Erman et al. (1983), amongst others, the criterion of Bao and Wierzbicki (2004) seems not to agree with the slope ‘1/2’. In connection to this, it is worth noting that crack formation by means of out-of-plane shear is characteristic of the fracture locus that is commonly found in bulk formability tests (Fig. 2.16). The criterion of Bao and Wierzbicki (2004) also indicated positive dependence of plastic fracture strain on stress triaxiality, which conflicted

Formability

45

with the generally accepted concept that high-stress triaxiality accelerates void growth whilst reducing the plastic fracture strain of metals (Hung et al., 2020). This was attributed to the fact that the influence of the Lode angle parameters had been neglected and stimulated the development of a new damage criterion by Bai and Wierzbicki (2008) that considers the effective strain at fracture εf as a function of both stress triaxiality η and Lode angle parameter ξ:    1 2 d2 η d6 η d4 η εf ðη ξÞ ¼  d3 exp + d5 exp θ d1 exp 2 +

 1 d1 expd2 η  d5 expd6 η θ + d3 expd4 η 2

2 θ ¼ 1  arccosξ π

(2.40)

This new criterion, where the six material parameters di need to be identified, allowed Bai and Wierzbicki (2008) to construct a three-dimensional   fracture locus in the space εf , η, ξ that defines the strain at fracture as a function of the stress triaxiality η and the Lode angle parameter ξ. The result is schematically plotted in Fig. 2.25. Later, Bai and Wierzbicki (2010) modified the Mohr-Coulomb failure criteria (Coulomb, 1776; Mohr and Beyer, 1928) that is widely used in rock and solid mechanics, as well as in brittle materials, into a stress triaxiality and Lode angle parameter-dependent criterion, and successfully applied it to predict a fracture in metal forming. This new criterion also gives rise to a three-dimensional fracture locus in the space εf , η, ξ : K εf ðη, ξÞ ¼ c2

"rffiffiffiffiffiffiffiffiffiffiffi


#!1=n 1 + c12 θπ 1 θπ  εps cos + c1 η + sin 3 6 3 6

θπ π 1 π ¼  arccos ξ ¼  θ 6 6 3 6

(2.41)

The material parameters ci need to be identified and K, n, εps are taken from the isotropic hardening law of Swift (Swift, 1952):  n σ ¼ K εps + ε (2.42)

46

Metal forming

1.5

Plane stress 1.0

εf 0.5 1 0.5 0 -1

0 -0.5 -0.5

0

η

0.5

1

θ

-1

Fig. 2.25 Schematic representation of the fracture loci associated with the uncoupled ductile damage criterion proposed by Bai and Wierzbicki (2008) in an extended threedimensional version of the space of the effective strain at fracture   vs stress triaxiality that also includes the influence of the Lode angle parameter ξ θ .

2.6 Coupled ductile damage criteria and fracture Uncoupled ductile damage criteria are based on weighted integrations of plastic strain increments and the weighting functions are generally taken as nondimensional and stress state-dependent (Eq. 2.4). The uncoupled ductile damage approaches are simple to implement in computer programs because they do not affect material behaviour (the influence of damage is calculated at the postprocessing stage) and because the critical values of damage and other parameters to be identified are relatively easy to be determined. In contrast, coupled ductile damage approaches include softening effects due to damage in material deformation and will eventually influence the final strain values at the onset of fracture. The dependence of material deformation on damage complicates the implementation in finite element computer programs as well as the determination of the unknown parameters to be identified. The following two sections will make a brief introduction to coupled ductile damage using two different continuum damage mechanics approaches: (a) A microbased approach built upon the theory of plasticity of porous materials (Gurson, 1977) that make direct use of the void volume fraction fv (or, porosity) as an internal variable to represent damage and its softening effect on material strength;

Formability

47

(b) A macrobased approach in which a continuum damage variable built upon a thermodynamic dissipation potential is introduced for accomplishing the same objective (Kachanov, 1958; Lemaitre, 1985).

2.6.1 Micromechanics-based damage criteria The presence of voids in damaged materials makes their deformation different from that of the parent undamaged (virgin) materials. One of the main differences results from the fact that localised shear stresses can be created by hydrostatic loading to produce plastic deformation. For this reason, the yield surfaces coupled to micromechanics’ ductile damage must include both deviatoric and hydrostatic stresses: AJ2 + BI12 ¼ C σ 2f

(2.43)

In Eq. (2.43), J2 ¼ σ 2 =3 is the second invariant of the deviatoric stress tensor; I1 ¼ 3 σ m is the first invariant of the stress tensor; σ f is the flow stress of the undamaged (virgin) material; and A, B, and C are coefficients that are functions of the porosity inside the material. Assuming the distribution of voids to be uniform, the material can be considered as a homogeneous medium with an average value of porosity characterised by the volumetric fraction fv of voids. The first approach in the field was carried out by Shima and Oyane (1976), who proposed the following empirical modification of the von Mises yield criterion: σ 2   2 m 2 σ + ¼ βσ f (2.44) α where α and β are two parameters related to the volumetric void fraction fv. These two parameters were experimentally determined by Shima and Oyane (1976) through mechanical testing of samples made of sintered copper: α¼

ð1 + fv Þ0:514 2:49fv 0:514

β¼

1 ð1 + fv Þ2:5

(2.45)

The criterion proposed by Gurson (1977) was built upon the analysis of the mechanical behaviour of a rigid-perfectly plastic material containing a spherical void and is given as follows:
2
  σ 3σ m 2  (2.46) + 2fv cosh  1 + fv 2 ¼ 0 2σ f σf

48

Metal forming

As seen, the criterion becomes equal to the von Mises yield criterion in case fv ¼ 0 because there is no porosity inside the material. Tvergaard (1981) modified the original Gurson (1977) criterion by introducing three additional parameters qi to account for the local interactions between voids:
2
  σ 3q2 σ m 2  (2.47) + 2q1 fv cosh  1 + q3 fv 2 ¼ 0 2σ f σf and the values of these parameters were estimated by means of finite element modelling q1 ¼ 1:5 q2 ¼ 1 q3 ¼ ðq1 Þ2

(2.48)

which in fact reduces to a single independent parameter q1. Later, Tvergaard and Needleman (1984) modified the Tvergaard (1981) criterion by introducing an ‘equivalent’ volumetric void fraction f∗v that increases faster than the volumetric void fraction fv in order to account for the accelerating coalescence effects resulting from the local interactions between adjacent voids: 8 fv  fc < fv ∗  fc ∗ f (2.49) fv ¼ f + u ðfv  fc Þ fv > fc :c ff  fc In Eq. (2.49), fc is critical volumetric void fraction at incipient coalescence and ff is the volumetric void fraction at fracture. fu∗ ¼ 1/q1 is the maximal value of the fv∗ at which the stress-carrying capacity vanishes and corresponds to a triggering for element deletion and fracture. It is also worth mentioning that the previous criteria based on the original work of Gurson (1977) predict no increase in damage and, therefore, no cracking in situations of zero triaxiality η ¼ σ m/σ f ¼ 0. A solution to this problem was later proposed by Nahshon and Hutchinson (2008) who modified the criterion to include a shear term that is related to void growth in pure shear. Fig. 2.26 provides a comparison between the yield surfaces provided by the three aforementioned criteria for a small volumetric void fraction fv ffi 0.05. The constitutive equations associated with these criteria can be obtained by applying the principle of normality, and in the case of the criterion due to Shima and Oyane (1976), this allows expressing the volumetric strain rate as follows: η ε_ v ¼ ε_ 1 + ε_ 2 + ε_ 3 ¼ 2 1:6 ε_ (2.50) αβ

Formability

σ σf

49

Shima and Oyane Tvergaard

Gurson

1 0.8 0.6 0.4 0.2

σm σf

0 0

0.5

1

1.5

2

2.5

3

Fig. 2.26 Comparison of the yield locus given by the criteria of Shima and Oyane (1976), Gurson (1977), and Tvergaard (1981) for a volumetric void fraction fv ffi 0.05.

where ε_ is the effective strain rate of the porous material. Assuming the overall level of porosity to be low fv ≪ 1, the following relation can be established for the volumetric strain rate ε_ v : 

V ε_ v ¼ fv V

(2.51)



where V is the volume and V is the time derivative of the volume. To conclude, it is worth saying that coupled ductile damage criteria based on micromechanics approaches allow the application of direct material characterisation techniques using scanning electron microscopy to obtain the model parameters and their changes during deformation. However, the overall procedure is challenging and tedious and sometimes needs to be complemented with inverse parameter identification by means of finite element analysis.

2.6.2 Macromechanics-based approaches Macromechanics approaches are based on the definition of an internal scalar damage variable D for characterising damage in a material subjected to plastic deformation. Fig. 2.27 shows a representative volume element (RVE) of such a material that contains traces of defects (microcracks and microvoids, shown in black).

50

Metal forming

n

δSD

n

δS

–1

Fig. 2.27 Schematic representation of a representative volume element with microcracks ! and microvoids after intersection by a plane with normal n .

The density of microcracks and intersections of microvoids on a surface ! δS resulting from the intersection of the RVE by a plane with a normal n is given as follows: δSD (2.52) δS where Dn is designated as the local damage variable, δSD is the part of the surface where microcracks and intersections of microvoids exist, and δSe is the remaining part of the surface that effectively resists to deformation: Dn ¼

δSe ¼ δS  δSD

(2.53)

If the material is subjected to uniaxial loading in the direction of the normal, the local damage variable Dn allows establishing its physical meaning as follows: • Dn ¼ 0 undamaged material; • 1 < Dn < 0 damaged material; • Dn ¼ 1 cracked material. By assuming a homogeneous distribution of defects and the hypothesis of strain equivalence that states that ‘the strain behaviour of a damaged material is represented by the constitutive equations of the undamaged (virgin) material in the potential of which the stress is simply replaced by the effective stress’ (Lemaitre, 1985), the effective stress tensor e σ may be written as follows: σ ij e σ ij ¼ (2.54) ð1  DÞ The effective stress tensor e σ in Eq. (2.54) must be understood as the stress tensor that effectively resists to deformation, in order to distinguish its meaning from that of the effective (equivalent) stress σ associated with yield

Formability

51

σ

F, n

SD S ~ E1

~ E2

E

ε1

ε2

ε

Fig. 2.28 Damage in tensile testing and its influence on the elasticity modulus.

criteria. The damage variable D ¼ Dn results from assuming that damage does not depend on the normal and will assume values between 0 (undamaged) and 1 (cracked) inside the material. Let us now consider the uniaxial tensile test shown in Fig. 2.28 to understand the physical meaning of Eq. (2.54). For this purpose, we start by writing the uniaxial ‘effective’ stress e σ in the specimen as follows: F F σ e σ¼ ¼ ¼ Se S  SD 1  D

(2.55)

From the hypothesis of strain equivalence stated before and using the constitutive equations of elasticity, the following relation can be obtained: e σ σ σ εe ¼ ¼ ¼ E Eð1  DÞ Ee

(2.56)

where E is the elasticity modulus of the undamaged (virgin) material and Ee ¼ Eð1  DÞ is the actual elasticity modulus of the damaged material. This result allows understanding damage as a measure of the loss of stiffness (i.e. a measure of softening): Ee (2.57) E Under these circumstances, the evolution of damage throughout the tensile test may be determined by measuring the changes in the elasticity D¼1

52

Metal forming

modulus or, alternatively, by determining the changes in the elastic energy release per unit of volume during unloading: 1 e 1 σ2 σε ¼ Y¼ e 2 2E ð1  DÞ2

(2.58)

The generalisation of Eq. (2.58) for three-dimensional stress conditions was originally performed by Lemaitre and Desmorat (2005) and is given by the following:  σ 2  1 σ2 2 m (2.59) ð1 + νÞ + 3 ð1  2νÞ Y¼ 2 σ 2E ð1  DÞ 3 Coupling material deformation with the elastic energy release rate in order to include the softening effects due to damage is accomplished by considering a dissipation potential F(σ ij) ¼ FY + FD that includes not only the plastic potential from plasticity theory (i.e. the yield function) FY but also the damage dissipative potential FD given by (Lemaitre, 1986):
s + 1 S Y (2.60) FD ¼ ðs + 1Þð1  DÞ S where S (MPa) and s are two material parameters to be identified, which may depend on the temperature. This procedure allows writing the damage evolution associated with the Lemaitre criterion as follows:
s
s λ_ ∂FD Y p Y D_ ¼ λ_ ¼ ε_ (2.61) ¼ ∂Y 1  D S S p where λ_ ¼ ε_ ð1  DÞ is retrieved from the plastic multiplier of the LevyMises constitutive equations modified to include damage softening (these equations were obtained after generalisation to plasticity of what was previously made in elasticity): p ε_ ij ¼

p λ_ 3 ε_ 0 3

0 σ

σ ij ¼

2 σ 2 σ ð1  DÞ ij

(2.62)

As a result of this, the evolution of ductile damage according to Lemaitre’s ductile damage criterion is given as follows: 8
s < _p Y p _ (2.63) D ¼ ε S , ε  ε0 : 0, εp < ε0

Formability

53

where ε0 is a threshold value of effective strain below which damage is not accumulated. Three final notes must be given regarding Lemaitre’s coupled ductile damage criterion. Firstly, it does not predict damage for values of stress triaxiality η <  1/3 because it does not include a dependency on the shear stresses (Lode angle parameter). Secondly, the original version of the criterion (Lemaitre, 1985) assumes that microvoids and microcracks play the same role in both tension and compression. This was later admitted not to be realistic and gave rise to modifications of the original criterion to account for crack closure or partial crack closure (Lemaitre, 1996). Thirdly, the material parameters are commonly identified by inverse analysis combining results from experiments and finite element modelling.

2.7 Experimental determination of the fracture loci The methods and procedures utilised for determining the fracture loci in sheet and bulk forming involve the use of experimental tests capable of covering a wide range of strain loading paths.

2.7.1 Sheet metal forming The fracture loci (fracture forming limits) in sheet metal forming are associated with crack opening by tension (FFL) and by in-plane shear (SFFL). Table 2.4 summarises the experimental sheet formability tests that are commonly used to determine these two fracture loci with an indication of the corresponding states of stress and strain in the case of an isotropic material. The FFL can be determined by combination of tensile, bulge (circular and elliptical) and Nakajima tests (with different radius r0) to cover strain loading paths from uniaxial tension to balanced biaxial stretching. Double-notched tensile tests (DNTT) can be used to obtain additional strain loading paths near plane strain deformation conditions. The SFFL can be determined by means of shear tests with different ligament sizes l0. The mixed-mode fracture region located in-between the FFL and the SFFL can be characterised by means of staggered DNTT’s tests with different ligament sizes l0 and inclination angles α. The strain loading paths may be determined by means of circle grid analysis (CGA). CGA is an experimental technique that consists of etching a grid of circles with diameter d0 on a blank before deformation and measuring the major l and minor w axes of the ellipses resulting from the circles after deformation (Keeler, 1968). The major ε1 and minor ε2 surface strains applied to

Table 2.4 Schematic representation of sheet formability tests. Experimental test

Tensile

Geometry lc lo

State of stress (initial)

State of strain (initial)

σ1 > 0 σ2 ¼ σ3 ¼ 0

ε1 > 0 ε2 ¼ ε3 ¼ ε1 =2 < 0

σ1  σ2 > 0 σ3 ¼ 0

ε 1  ε2 > 0 ε3 < 0

σ1 > σ2  0 σ3 ¼ 0

ε1  0 ε1 =2 < ε2 < ε1 ε3 < 0

wo

Hydraulic bulge (circular and elliptical) do

Nakajima do

ro

Double-notched tension (DNTT)

L

w

σ1 > 0 σ2 < 0 σ3 ¼ 0

ε1 > 0 ε2 ¼ 0 ε3 < 0

σ 1 > σ 2 σ3 ¼ 0

ε1 >  ε2 ε3 < 0

σ 1 ¼ σ 2 σ3 ¼ 0

ε 1 ¼  ε2 ε3 ¼ 0

lo

Staggered DNTT

L lo

w

α

Shear

L

w lloo Adapted from Magrinho, J.P., Silva, M.B., Reis, L., Martins, P.A.F., 2019. Formability limits, fractography and fracture toughness in sheet metal forming. Materials 12, 1493.

Metal forming

56

Crack Digital camera

w

2 1

1 d0

l

2

wf

t f1 t f2 t fn

Detail A

(A)

Detail B

(B)

Grips

1.19 1.02 0.85 0.68 0.51 0.34 0.17 0.00

Specimen

1.19 1.02 0.85 0.68 0.51 0.34 0.17 0.00

Camera

(C) Fig. 2.29 Determination of surface strains in sheet test specimens. (A) Circle grid analysis for measuring the surface strains in a circular bulge test; (B) procedure for measuring the thickness strain ε3f and the minor strain ε2f at fracture in a circular bulge test; (C) schematic illustration of a digital image correlation system (DIC) for measuring the surface strains ε1 and ε2 with experimental plots of the major strain ε1 in aluminium AA1050-H111 (left) and copper (right) staggered DNTT test specimens with α ¼ 80° (Table 2.4) at the onset of fracture. (Adapted from Cristino, V.A., Silva, M.B., Wong, P.K., Martins, P.A.F., 2017. Determining the fracture forming limits in sheet metal forming: a technical note. J. Strain Anal. Eng. Des. 52, 467–471; Magrinho, J.P., Silva, M.B., Reis, L., Martins, P.A.F., 2019. Formability limits, fractography and fracture toughness in sheet metal forming. Materials 12, 1493.)

an ellipse at a specific instant of time (i.e. at a specific point of the strain loading path) are obtained as follows (Fig. 2.29A): ε1 ¼ ln

l d0

ε2 ¼ ln

w d0

(2.64)

The fracture strain pairs defining the FFL and the SFFL cannot be directly determined by CGA because the application of grids, even with very small

Formability

57

circles, to obtain the surface strains in the necked region after it forms and, therefore, close to fracture, will always provide values that cannot be considered the fracture strains. In fact, grids with very small circles are difficult to measure and the values are dependent on the initial size of the circles in the neighbourhood of the cracks due to inhomogeneous plastic deformation. As a result of this, the methodology to obtain the fracture strain pairs involves measuring the thickness of the specimens before and after deformation in order to obtain the ‘gauge length’ strains (Fig. 2.29B). The procedure involves the determination of the fracture strain ε3f in the thickness direction by measuring the sheet thickness before t0 and after tf failure by cracking (refer to detail A in Fig. 2.29B). The sheet thickness tf in the cracked region is obtained from the average of a number of thickness measurements tfi, where nm is the number of measurements: tf ¼

nm X

tfi =nm

(2.65)

i¼1

The fracture strain ε3f is obtained from ε3f ¼ ln

tf t0

(2.66)

The fracture strain ε2f in the sheet surface (minor fracture strain) is determined by measuring the minor axis wf of the ellipse that resulted from plastic deformation and fracture of a circle (refer to detail B in Fig. 2.29B): wf (2.67) ε2f ¼ ln d0 Alternatively, the minor fracture strain ε2f can be obtained by means of a digital image correlation (DIC) system that uses a 3D noncontact optical technique for measuring the evolution of the strains with time (Fig. 2.29C). In the case of using a DIC system, the surface of the specimens has to be painted with a nonuniform speckle pattern before deformation (Magrinho et al., 2019). The major surface fracture strain ε1f ¼  (ε2f + ε3f) is determined by incompressibility. Fig. 2.30 presents the fracture loci (FFL and SFFL) for aluminium AA1050-H111 sheets with 1 mm of thickness obtained by means of the sheet formability tests that are listed in Table 2.4. The fracture strain pairs (ε1f, ε2f) were determined by the application of the procedure that was previously described, and the strain loading paths ε1( ε2) of the DNTT, staggered DNTT and shear tests were obtained by using a DIC system.

58

Metal forming

Fig. 2.30 Fracture loci (FFL and SFFL) of aluminium AA1050-H111 sheets with 1 mm thickness in (A) principal strain space and (B) in the space of effective strain vs stress triaxiality. The solid markers correspond to strains at fracture. Note: The physical meaning of the open markers and of the associated curve labelled as FLC will be given later in this chapter (Sections 2.8.4 and 2.9). (Adapted from Magrinho, J.P., 2019. Sheet-Bulk Forming (Ph.D. Thesis). University of Lisbon, Lisbon.)

A fractography analysis performed in the DNTT test specimens (Fig. 2.8A) revealed a circular dimple-based structure typical of fracture by tension (mode I), which is consistent with the fracture strains of DNTT test specimens being located on top of the FFL. The same analysis performed in shear test specimens (Fig. 2.8B) revealed elongated, parabolic dimple-based structures typical of fracture caused by shear (mode II) that is also consistent with the fracture strains of shear test specimens being located on top of the SFFL. The curve labelled as ‘FLC’ in Fig. 2.30 is the forming limit curve, which will be later described in this chapter (refer to Section 2.8.4). The FFL and SFFL may be approximated by straight lines as follows: FFL : ε1 ¼ 0:68 ε2 + 1:34 SFFL : ε1 ¼ 1:38 ε2 + 2:14

(2.68)

The FFL and SFFL of aluminium AA1050-H111 sheets with 1 mm thickness have slopes of 0.68 and +1.38, respectively. These slopes are different from the theoretical estimates of ‘1’ and ‘+1’ because

Formability

59

the experimental testing conditions deviate from the simplifying assumptions used in the theoretical model presented in Section 2.5.1. Despite these differences, the perpendicularity between the FFL and the SFFL maintains. The representation of the FFL and SFFL in the space of effective strain vs stress triaxiality considered Hill’48 anisotropic yield criterion (Hill, 1948), and the stress triaxiality and effective strain were calculated by means of the equations that are included in Appendix A.

2.7.2 Bulk metal forming The fracture loci in bulk metal forming are associated with crack opening by tension (FFL) and by out-of-plane shear stresses (OSFFL). Table 2.5 summarises the experimental bulk formability tests that are commonly used to determine these two fracture loci with an indication of the corresponding states of stress and strain in the case of an isotropic material at the surface locations marked with black squares. The surface strains εθf and εzf at the onset of fracture (fracture strains) are determined by means of a technique similar to that utilised in sheet metal forming (Gouveia et al., 1996). The technique consists of imprinting a square with side length l0 ¼ w0 before deformation at the surface locations marked with black squares in Table 2.5, and measuring the major l and minor w side lengths of the rectangles resulting from the squares after deformation and cracking (Fig. 2.31A). εθf ¼ ln

w l εzf ¼ ln w0 l0

(2.69)

Alternatively, the fracture strains can be obtained by digital image correlation (DIC), which allows measuring the evolution of the surface strains with time (Fig. 2.31C, Magrinho et al., 2018). In this other case, the surface of the bulk formability test specimens needs to be painted with a nonuniform speckle pattern (Fig. 2.31B). The evolution of the strains with time obtained from a DIC system combined with the evolution of the force with displacement provided from a testing machine allows identifying the instant of time when the cracks are triggered. Fig. 2.32 shows the result of this correlation for hemispherical and tapered test specimens made of AISI 1045 carbon steel. In the case of the hemispherical test specimen (Fig. 2.32A), results show a

Table 2.5 Schematic representation of bulk formability tests. Experimental test

Geometry

Cylindrical

D

State of stress (initial)

State of strain (initial)

σ1 ¼ σ2 ¼ 0 σ3 < 0

ε1 ¼ ε2 ¼ ε3 =2 > 0 ε3 < 0

σ 1 ¼ σ 3 > 0 σ2 ¼ 0

ε1 ¼ ε3 > 0 ε2 ¼ 0

σ1  0  σ3 σ2 ¼ 0

ε1 > 0 ε1  ε2  ε3 ε3 < 0

H

Shear

L w2 h2 H h1 w1 W

Tapered

D d

H h1

Flanged

D d

σ1 > 0 σ2 ¼ σ3 ¼ 0

ε1 > 0 ε2 ¼ ε3 ¼ ε1 =2 < 0

σ1 > 0 σ2 ¼ σ3 ¼ 0

ε1 > 0 ε2 ¼ ε3 ¼ ε1 =2 < 0

H h1

Hemispherical

d

H

D

Adapted from Gouveia, B.P.P.A., Rodrigues, J.M.C., Martins, P.A.F., 1996. Fracture predicting in bulk metal forming. Int. J. Mech. Sci. 38, 361–372; Magrinho, J.P., Silva, M.B., Alves, L.M., Atkins, A.G., Martins, P.A.F., 2018. New methodology for the characterization of failure by fracture in bulk forming. J. Strain Anal. Eng. Des. 53, 242–247.

62

Metal forming

l0 w0

l w

(A)

(B)

Measuring region

Non-uniform speckle pattern

40º ~ 60º

Camera

(C) Fig. 2.31 Methodology to determine the fracture strains in bulk formability test specimens. (A) Square grid analysis for measuring the surface strains in a cylindrical test specimen; (B) photograph of a cylindrical test specimen painted with a nonuniform speckle pattern at three different stages of deformation; (C) schematic illustration of a digital image correlation system (DIC) for measuring the surface strains in a cylindrical test specimen similar to that shown in (B).

monotonic force grown with the displacement up to an absolute maximum after which the force drops as a result of crack propagation (refer to the dashed vertical line). In the case of the tapered test specimen (Fig. 2.32B), the force grows up to a local maximum (refer to the dashed vertical line) after which drops and rises again due to strain hardening and to the increase in the top and the bottom surface areas of the specimen, as compression progresses. Fig. 2.33 presents the strain loading paths, the fracture loci and the corresponding fracture strains for the entire set of bulk formability tests listed in Table 2.5 in principal strain space (Fig. 2.33A), and in the space of effective strain vs stress triaxiality (Fig. 2.33B). The latter was obtained by considering the von Mises yield criterion. The strain loading paths in Fig. 2.33 were obtained by means of the DIC system and the fracture strain pairs were determined as previously described.

Formability

63

Fig. 2.32 Experimental force-displacement and strain-displacement evolutions for (A) hemispherical and (B) tapered test specimens of Table 2.5 made from AISI 1045 carbon steel with photographs of the cracks. (Adapted from Magrinho, J.P., Silva, M.B., Alves, L.M., Atkins, A.G., Martins, P.A.F., 2018. New methodology for the characterization of failure by fracture in bulk forming. J. Strain Anal. Eng. Des. 53, 242–247.)

The FFL is approximated by a straight line with a slope of ‘1.15’, in close agreement with the theoretical value of ‘1’ that was presented for crack opening by tension (mode I, refer to Section 2.5.1). The OSFFL is approximated by a straight line with a slope of ‘0.5’ that is identical to the theoretical slope associated with crack opening by through-thickness shear (mode III, refer to Section 2.5.1).

Fig. 2.33 Fracture loci (FFL and OSFFL) of AISI 1045 carbon steel in (A) principal strain space and (B) in the space of effective strain vs stress triaxiality. (Adapted from Magrinho, J.P., Silva, M.B., Alves, L.M., Atkins, A.G., Martins, P.A.F., 2018. New methodology for the characterization of failure by fracture in bulk forming. J. Strain Anal. Eng. Des. 53, 242–247; Magrinho, J.P., 2019. Sheet-Bulk Forming (Ph.D. Thesis). University of Lisbon, Lisbon.)

Formability

65

2.8 Plastic instability 2.8.1 Diffuse necking in uniaxial tension Plastic deformation in simple tension is controlled by two competing mechanisms: (i) strain hardening and (ii) geometric softening. Strain hardening prevails on the left-hand side of the nominal stress vs nominal strain curve (engineering stress vs strain curve) of the material up to the maximum force from which the ultimate tensile strength σ UTS is obtained (Fig. 2.34). In the left-hand side of the nominal stress-strain curve, the specimen becomes longer and its cross-sectional area decreases uniformly through the gage length when increasing the tension force F. The increase in strength due to strain hardening compensates the geometric softening due to uniform reduction of the cross-sectional area. Plastic deformation is stable (tension force F increases with deformation) and is homogeneous with an equal distribution of stress in the specimen. When the strain hardening does not compensate geometric softening, the reduction of the cross-sectional area ceases to be uniform and the plastic deformation becomes inhomogeneous due to concentration within a region that extends over an appreciable length of the specimen. Necking is triggered (the type of necking shown in Fig. 2.34 is designated as ‘diffuse necking’) and the force-carrying capacity of the specimen will decrease up to F

σnom Maximum force

σUTS

σr = σ θ

σz

σY

σr

R a

σθ

0

eY

e inst

e F

Fig. 2.34 Nominal stress vs nominal strain curve and detail of the necked portion of a round tensile specimen showing the triaxial state of stress.

66

Metal forming

fracture in a relatively uncontrolled manner and the plastic deformation becomes unstable. The peak in force F corresponds to the onset of necking (refer to the ultimate tensile strength σ UTS in Fig. 2.34) and to the instant when the uniaxial state of stress is replaced by a triaxial state of stress (Bridgman, 1944). Application of the condition of maximum applied force dF ¼ 0 in the true stress vs true strain curve of the material (commonly designated as the stress vs strain curve) gives dF ¼ d ðσ AÞ ¼ σ dA + Adσ ¼ 0

(2.70)

where A is the actual cross-sectional area of the specimen and σ is the uniaxial tensile stress. Because plastic deformation is essentially a volume constancy-based mechanism dV ¼ dA l + dl A ¼ 0 and Eq. (2.70) becomes dσ dA dl ¼  ¼ ¼ dε (2.71) σ A l From Eq. (2.71), we may derive the condition of plastic instability corresponding to diffuse necking in tension as follows: dσ ¼σ (2.72) dε Fig. 2.35 associates the plastic instability point to the location where the magnitude of the slope dσ/dε of the stress vs strain curve is equal to σ. σ

σ σ, d⎯

d σ ⎯ dε

dε

ε inst

σ inst

Instability point

d σ = σR ⎯ dε

α

1

(A)

0

ε inst

ε

0

ε inst

ε

(B)

Fig. 2.35 Graphical construction of diffuse necking in (A) a true stress vs true strain curve and (B) through combination of the diagrams (σ, ε) and (dσ/dε, ε).

Formability

67

2.8.2 Localised necking in uniaxial tension Fracture in sheet specimens subjected to tension is also preceded by plastic instability and necking. However, two types of necking may occur in these types of specimens: diffuse and localised necking (Fig. 2.36). Diffuse necking (Fig. 2.36A) is similar to what happens in the round test specimen subjected to tension and is accompanied by contraction strains in both the width and thickness directions (dεx ¼ dεz ¼  dεy/2, in case the material is isotropic). The necked area of the specimen is bound by two circular arcs, symmetrical in relation to the loading axis, and is considerably greater than the cross-sectional area of the specimen. Localised necking occurs inside the region of diffuse necking, along a very narrow width w inclined at an angle θ to the loading direction (refer to the grey zone in Fig. 2.36B). Since the neck is very narrow (w has the same magnitude as the sheet thickness t), its length l remains unchanged and plane strain conditions prevail, dεx0 ¼ 0. The strain εx0 along the neck is obtained from the strain tensor εij by using the transformation law of Cartesian tensors from the axis x, y, z to the axis, x0 , y0 , z where the direction x0 is aligned with the neck:

F

Fdn< F

t y' y

y

θ

x'

x

x

w

z

l

Fdn< F (A)

F (B)

Fig. 2.36 (A) Diffuse and (B) localised necking in a sheet test specimen subjected to uniaxial tension. Note: The applied force for diffuse necking Fdn is smaller than that for localised necking F.

68

Metal forming

2

0 0

xyz

εij

3 sinθ cosθ 0 R ¼ 4 cos θ sinθ 0 5 0 0 1

xyz

¼ R εij RT

(2.73)

This allows expressing the increment of strain dεx0 along the neck as follows: dεx0 ¼ dεy cos 2 θ + dεx sin 2 θ ¼ 0

(2.74)

In the aforementioned equations, x and y are principal directions (γ xy ¼ 0). Then, using the constitutive equations associated with Hill’48 anisotropic yield criterion (Hill, 1948), and assuming plane stress conditions σ z ¼ 0 and rotational symmetric anisotropy, where r is the normal anisotropy, it is possible to rewrite the increments of strain along the x and y directions, as follows: dεx ¼

d ε 1  d ε 1  ðr + 1Þσ x  rσ y dεy ¼ ðr + 1Þσ y  rσ x σ ð1 + r Þ σ ð1 + r Þ (2.75)

Replacing Eq. (2.75) into Eq. (2.74) gives tan 2 θ ¼ 

dεy ðr + 1Þσ y  rσ x ¼ dεx rσ y  ðr + 1Þσ x

(2.76)

This result allows concluding that for an isotropic material (r ¼ 1) under uniaxial tension (σ x ¼ 0), the localised neck forms at angle θ ¼ 54.7° to the loading axis (Fig. 2.36B). Now, taking into consideration that the cross-sectional area of the neck A0 ¼ lt, the area A perpendicular to the loading y-direction may be written as follows: A ¼ A0 sin θ

(2.77)

Since the length l of the neck does not vary, the following identities are obtained: dA dA0 dt (2.78) ¼ 0 ¼ ¼ dεz A A t The criterion for the formation of localised necking requires the force F ¼ σ yA to fall under the plane strain deformation constraint dεx0 , dF ¼ σ y dA + Adσ y ¼ 0

(2.79)

Formability

69

Substituting Eq. (2.78) into Eq. (2.79) gives dσ y dA ¼  ¼ dεz σy A

(2.80)

Taking into consideration Eq. (2.75), the definition of normal anisotropy and the volume incompressibility condition, it is possible to obtain the following relation between the increments of strain: dεz ¼

dε 1  σ y  σ x σ ð1 + r Þ

dεz ¼ 

dεy ð r + 1Þ

(2.81)

Eqs. (2.80), (2.81) allow writing the condition of localised necking in uniaxial tension (y is the principal direction ‘1’) as follows: dσ y σy dσ 1 σ1 ¼ ¼ ) dεy ð1 + r Þ dε1 ð1 + r Þ

(2.82)

In case, for example, of a sheet test specimen made from an isotropic material (r ¼ 1), the localised neck forms when dσ/dε ¼ σ/2. Fig. 2.37 shows the graphical representation of this result and a comparison with the previously derived condition of diffuse necking dσ/dε ¼ σ that precedes localised necking in sheet specimens.

σ

αl Localized

αd

d σ ⎯ σ ⎯ dε = 2

d σ ⎯ σ ⎯ dε = 1

Diffuse

1

0

ε

2

Fig. 2.37 Graphical construction of the diffuse and localised necking points in a true stress vs true strain curve for an isotropic material.

70

Metal forming

2.8.3 Localised necking in biaxial tension and the forming limit curve The first step towards the representation of the localised necking locus in principal strain space is given by determining the conditions for localised necking under a general state of biaxial tension (Fig. 2.38). For this purpose, we start by assuming proportional loading under a constant strain ratio β ¼ dεx/dεy ¼ εx/εy so that the increment of strain dεx0 along the neck (Eq. 2.74) may be written as follows: dεx0 ¼ dεy cos 2 θ + βdεy sin 2 θ ¼ 0

(2.83)

This gives tan 2 θ ¼ 

1 β

(2.84)

and allows concluding that localised necking can only be triggered for negative strain ratios β < 0. In fact, according to Eq. (2.84), the limiting condition corresponding to β ¼ 0 should correspond to a local neck perpendicular to the loading direction (θ ¼ 90º). In practical terms, the result obtained in Eq. (2.84) implies that localised necking cannot occur under stretching conditions where εx > 0 and, therefore, cracking at the fracture forming limit (FFL) (refer to Fig. 2.10 and Section 2.5.1) should not be preceded by localised necking in the tension-tension (right-hand side) quadrant of principal strain space.

Fy t y'

y

θ

Fx

x' x

w

Fx

l

Fy

Fig. 2.38 Localised necking in a sheet test specimen subjected to biaxial tension.

Formability

71

In order to get a better understanding of the formability limit by localised necking associated with Eq. (2.84), we will consider the condition of maximum applied force Fy ¼ σ yAy (with Fy > Fx), where Ay is the cross-sectional area perpendicular to the y-loading direction (Fig. 2.38) dF y ¼ σ y dAy + Ay dσ y ¼ 0

(2.85)

and obtain the following identity dσ y dAy ¼ ¼ dεz σy Ay

(2.86)

dσ y ¼ ð1 + βÞdεy σy

(2.87)

or

after replacing the volume constancy condition dεz ¼  (1 + β) dεy for a proportional strain loading path in Eq. (2.86). Eq. (2.87) allows writing the condition for localised necking in biaxial tension (y is the principal direction ‘1’) as follows: dσ y dσ 1 ¼ ð1 + βÞσ y ) ¼ ð1 + βÞσ 1 dεy dε1

(2.88)

Considering the relations between the maximum principal stress σ 1 and the maximum increment of principal strain dε1 with the corresponding effective values associated with Hill’48 anisotropic yield criterion (Hill, 1948) that were provided in Eq. (2.13), rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 2r 2 Bpffiffiffiffiffiffiffiffiffiffiffiffi 1 + 1 + r β + β C C σ¼B @ 1 + 2r ½ð1 + r Þ + rβ A σ 1 ¼ f ðr, βÞσ 1 (2.89) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1+r 2r ε ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 + β + β2 ε1 ¼ gðr, βÞε1 1+r 1 + 2r it is possible to write the following identity for a material with a stress-strain curve approximated by Ludwik-Hollomon’s equation σ ¼ kεn, under a proportional strain loading path: σ1 ¼ K

gn ðr, βÞ n ε ¼ K 0 ðr, βÞεn1 f ðr, βÞ 1

(2.90)

72

Metal forming

Replacing Eq. (2.90) into Eq. (2.88), the following expressions due to Hill (1952) can be utilised for plotting the locus of strain pairs at which localised necking occurs in principal strain space (Fig. 2.39): n inst εinst (2.91) εinst 1 ¼ 2 ¼ βε1 1+β Necking is an undesirable surface blemish in components made from sheet metals, so limits in sheet metal forming are most often controlled by localised necking rather than by fracture because once a neck appears (point A in Fig. 2.39) and spreads sideways under subsequent deformation, thinning will progress very fast under decreasing forces until the sheet cracks (point B in Fig. 2.39). The localised necking locus is designated as the forming limit curve (FLC), which for the moment will be assumed as a straight line given by Eq. (2.91) in the tension-compression domain of principal strain space (refer also to the result obtained from Eq. (2.78)). As shown in Fig. 2.39, the FLC may also be regarded as the locus of all the in-plane strains where sharp

Fig. 2.39 Locus for localised necking according to Eq. (2.85) and fracture forming limit (FFL) due to cracking by tension (mode I).

Formability

73

changes in loading path occur since all prior loading paths become plane strain (dε2 ¼ 0) after necking (refer to the strain loading path 0AB in Fig. 2.39). Example 2.2

This example is focused on the diffuse necking condition in a sheet test specimen subjected to balanced biaxial stretching (Fig. 2.40) and is solved upon the symmetry condition that allows using either the force Fx or the force Fy to characterise the onset of plastic instability. Fy

y

Fx

Fx ly

x z

lx

Fy

Fig. 2.40 Sheet test specimen under balanced biaxial stretching with Fx ¼ Fy.

Using, for example, the condition of maximum applied force in the y direction, dFy ¼ 0 may be written as   dF y ¼ d σ y Ay ¼ σ y dAy + Ay dσ y ¼ 0 where Ay (Ax ¼ Ay) is the actual cross-sectional area of the specimen perpendicular to the loading direction y. Because the volume of the specimen remains constant during plastic deformation, dV¼ dAy ly + dly Ay ¼ 0, the previous equation becomes dσ y dAy dl y ¼ ¼ ¼ dεy σy Ay ly Considering the effective stress and the increment of effective strain associated with Hill’48 anisotropic yield criterion (Hill, 1948), assuming plane stress conditions σ z ¼ 0 and rotational symmetric anisotropy, and taking into account that the axes x, y, z coincide with the principal directions (refer to Appendix A)

74

Metal forming

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi 2r 2 σ ¼ σ 2x + σ 2y  σx σy ¼ σy ð1 + r Þ 1+r sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1+r 2r dε ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dε2x + dε2y + dεx dεy ¼ 2 ð1 + r Þ dεy ð1 + r Þ ð1 + 2r Þ The condition of maximum applied force results as follows: dσ 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ dε 2 ð1 + r Þ The previously mentioned equation is the condition for plastic instability in balanced biaxial stretching and allows concluding that for a sheet specimen of an isotropic material (r ¼ 1) subjected to balanced biaxial loading, diffuse necking forms when dσ=dε ¼ σ=2. The right-hand side of the equation is half the value predicted for diffuse necking in uniaxial tension, meaning that in the case of a material having a stress-strain curve approximated by means of Ludwik-Hollomon’s equation σ ¼ kεn , the effective strain at the onset of diffuse necking in balanced biaxial stretching εbbs inst ¼ 2n is twice that predicted for uniaxial tension εut inst ¼ n.

2.8.4 Representation of the forming limit curve in principal strain space As shown in the previous section, in the tension-compression (left-hand side) quadrant of principal strain space, there is a theory to predict the in-plane strain pairs at which localised necks occur and the angles with respect to the major loading axis at which localised necks form (Hill, 1952). Localised necking is preceded by diffuse necking, and Swift (1952) proposed a theory for obtaining its locus for a material with a stress-strain curve approximated by means of Ludwik-Hollomon’s equation σ ¼ kεn, under proportional strain loading paths. With Fig. 2.41 showing a representative part of the sheet material loaded by in-plane forces F1 and F2 in the principal stress and strain directions, the condition for diffuse instability is dF 1 ¼ dðσ 1 A1 Þ ¼ A1 dσ 1 + σ 1 dA1 ¼ 0 dF 2 ¼ dðσ 2 A2 Þ ¼ A2 dσ 2 + σ 2 dA2 ¼ 0

(2.92)

Formability

F2

75

A2

2

F1

F1

l2

1 3

A1

l1

F2

Fig. 2.41 Sheet test specimen under biaxial loading with F1 6¼ F2.

where further loading occurs without a corresponding increase of the forces. With dA/A ¼  dl/l ¼  dε due to volume constancy, the condition can be written as follows: dσ 1 ¼ σ 1 dε1 dσ 2 ¼ σ 2 dε2

(2.93)

The FLC based on the condition in Eq. (2.93) will be established in the following for a rotational symmetric material with normal anisotropy r under plane stress conditions (σ 3 ¼ 0). Under these conditions, Hill’s quadratic anisotropic yield criterion can be written as (refer to Appendix A) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r 2r σ2 2 2 σ ¼ σ1  and σ 1 > 0 σ1σ2 + σ2 ¼ σ1 1  α + α2 for α ¼ 1+r 1+r σ1 (2.94) and the anisotropic flow rule takes the following form: dε dε1 dε2 dε3 ¼ ¼ ¼ ð1 + r Þσ ð1 + r Þσ 1  rσ 2 ð1 + r Þσ 2  rσ 1 σ 1 + σ 2

(2.95)

At this point, it becomes convenient to introduce the subtangent at diffuse instability Zd according to Fig. 2.42. This subtangent becomes a calculation parameter linking the stress-strain curve with the onset of diffuse instability depending on the anisotropy and loading condition. Figs. 2.35 and 2.37 already showed a special case corresponding to uniaxial tension where Zd ¼ 1. A similar concept exists for local instability, in which

76

Metal forming

σ

σ dif. inst

Zd

0

ε dif. inst.

ε

Fig. 2.42 Ludwik-Hollomon’s stress-strain curve with the definition of the subtangent at diffuse instability Zd.

Fig. 2.37 includes the subtangent for local instability Zl ¼ 2 for the special case of uniaxial tension. From Fig. 2.42, the subtangent is defined as

σ dif :inst dσ

σdε

¼ , Zd ¼ (2.96) Zd dε dif :inst dσ dif :inst and by inserting Ludwik-Hollomon’s equation σ ¼ kεn in Eq. (2.96), it can be determined that the effective strain at diffuse instability is given through the subtangent and the strain hardening exponent as follows: εdif :inst ¼ Zd n

(2.97)

In order to set up the equation for Zd, it is necessary to express strain and stress increments. From the flow rule in Eq. (2.95), it follows that the increment in effective strain can be expressed through as dε ¼

1+r σdε1 1 + r  rα σ 1

(2.98)

Formability

77

where α ¼ σ 2/σ 1 is the principal stress ratio. The increment of effective stress is expressed through differentiation of the yield criterion in Eq. (2.94):   r r dσ dσ 2 σ  + σ  σ σ 1 2 1 2 1 ∂σ ∂σ 1+r 1+r dσ ¼ (2.99) dσ 1 + dσ 2 ¼ σ ∂σ 1 ∂σ 2 Rewriting Eq. (2.99) by using the principal stress ratio α and writing the stress increments according to the condition for diffuse instability (2.93) gives dσ ¼

ð1 + r  rαÞσ 1 dε1 + ðð1 + r Þα  r Þσ 2 dε2 σ1 ð1 + r Þσ

(2.100)

Using α once again and also utilising the principal strain ratio β¼

ε2 ð1 + r Þα  r ¼ ε1  ε2 ε1 1 + r  rα

(2.101)

it is possible to rewrite Eq. (2.100) as follows: dσ ¼

ð1 + r  rαÞ2 + ðð1 + r Þα  r Þ2 α 2 σ 1 dε1 ð1 + r Þð1 + r  rαÞσ

(2.102)

Finally, inserting Eqs. (2.98), (2.102) into Eq. (2.96) and expressing σ=σ 1 through the yield criterion (2.94), the subtangent becomes pffiffiffiffiffiffiffiffiffi 3 1 + r ð1 + r  2rα + ð1 + r Þα2 Þ2 (2.103) Zd ¼ ð1 + r Þ2  ð2 + r Þrα  ð2 + r Þrα2 + ð1 + r Þ2 α3 For any normal anisotropy r and principal stress ratio α, the effective strain at diffuse instability is now given by Eqs. (2.97), (2.103). The special case of anisotropy and balanced biaxial stretching in Example 2.2 is contained in the general equations with r ¼ 1 and α ¼ 1. In terms of principal strains, it appears from the flow rule under proportional loading, the yield criterion and (2.101) that 1 + r  rα ε1 ¼ pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε 1 + r 1 + r  2rα + ð1 + r Þα2

ε2 ¼ βε1

(2.104)

and hence with Eqs. (2.97), (2.103) that the principal strains at diffuse instability are dif :inst

ε1

ð1 + r  rαÞZd n ¼ pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + r 1 + r  2rα + ð1 + r Þα2

dif :inst

ε2

dif :inst

¼ βε1

(2.105)

Metal forming

78

or, dif :inst ε1 dif :inst

ε2

¼

 ð1 + r Þ2  3r ð1 + r Þα + ð1 + 2r + 3r 2 Þα2  r ð1 + r Þα3 n ð1 + r Þ2  ð2 + r Þrα  ð2 + r Þrα2 + ð1 + r Þ2 α3 dif :inst

¼ βε1

(2.106)

It may be convenient also to express the principal strains at diffuse instability in terms of the principal strain ratio β instead of the principal stress ratio α. Replacement in Eq. (2.106) through the relation in Eq. (2.101) gives   1 + r + 2rβ + ð1 + r Þβ2 n dif :inst   ε2dif :inst ¼ βε1dif :inst ¼ (2.107) ε1 2 ð1 + βÞ 1 + r  β + ð1 + r Þβ In the special case of an isotropic material with r ¼ 1, Eqs. (2.106), (2.107) reduce to the following: 2ð2  αÞð1  α + α2 Þn dif :inst dif :inst ε2 ¼ βε1 for r ¼ 1 2 3 4  3α  3α + 4α   2 1 + β + β2 n dif :inst :inst   ε2dif :inst ¼ βεdif ¼ for r ¼ 1 ε1 1 ð1 + βÞ 2  β + 2β2 dif :inst

ε1

¼

(2.108) (2.109)

The FLC corresponding to diffuse instability is included in Fig. 2.43 (refer to the grey curve) for the special case of an isotropic material. In contrast to the locus for localised necking due to Hill (1952), the locus for diffuse instability due to Swift (1952) is applicable to both quadrants of principal strain space. There is no continuum theory explaining the occurrence of localised necks in the right quadrant of the principal strain space (stretching), but experiments reveal their existence, usually occurring perpendicular to the greatest tensile strain. This led Marciniak and Kuckzynski (1967) to postulate the existence of locally thinned regions in the sheet workpieces at which necks initiate, and their theory provides loci for localised necking in both strain quadrants. A simplified illustration of the influence of a thinned region due to material and geometry imperfections is given here as it is done by other authors (Mielnik, 1991; Hosford and Caddell, 2011) for the special case of uniaxial tension. Consider the sheet material shown in Fig. 2.44 with a thickness imperfection giving rise to an initial cross-sectional area defined as f¼

A a, 0 A b, 0

(2.110)

Formability

79

ε1

FFL A

1 -1

Localized necking (Hill)

A Diffuse necking (Swift)

1 -1

n

A

1 -1/2 1

1 1

-1

0

n

ε2

Fig. 2.43 Locus for diffuse and localised necking together with the fracture forming limit (FFL) due to cracking by tension (mode I) for an isotropic material.

When the imperfection parameter f is equal to one, the imperfection vanishes, and decreasing values of f correspond to increasing imperfection. If the sheet material is loaded in uniaxial tension perpendicular to the groove like imperfection, force equilibrium implies: F ¼ σ a Aa ¼ σ b Ab

(2.111) n

Considering Ludwik-Hollomon’s equation σ ¼ kε applicable for uniaxial tension and expressing a cross-sectional area through the longitudinal strain, A ¼ A0exp(ε1), Eq. (2.111) becomes kεn1a Aa, 0 expðε1a Þ ¼ kεn1b Ab, 0 expðε1b Þ

(2.112)

and by using the imperfection parameter in Eq. (2.110), f εn1a expðε1a Þ ¼ εn1b expðε1b Þ

(2.113)

Numerical solution of Eq. (2.113) for n ¼ 0.25 and various values of f results in the dependency between the longitudinal strains ε1a and ε1b as shown in Fig. 2.45A (Hosford and Caddell, 2011). Without an imperfection, f ¼ 1, the longitudinal strains ε1a and ε1b are identical up until necking, which occurs when ε1a ¼ ε1b ¼ n ¼ 0.25.

80

Metal forming

Ab

F

Aa

Ab

F

Fig. 2.44 Uniaxial tension perpendicular to a groove like imperfection resulting in a local smaller cross-sectional area, Aa < Ab.

With an imperfection, f < 1, the strain ε1b in the vicinity of the imperfection lacks behind the strain ε1a in the imperfection and saturates at a lower strain level, ε1b < n ¼ 0.25, when necking localises in the imperfection and ε1a increases alone. Fig. 2.45B relates the localisation in longitudinal strain to the principal strain space in a uniaxial tensile test, where the slope of the strain path is 2 until the transition into local instability, where the strain path changes into that of plane strain. Similar, although more complicated, analyses are made in stretching by Marciniak and Kuckzynski (1967), who were able to explain the localisation of strain in biaxial stretching although there is no continuum theory similar to the local instability by Hill (1952).

2.8.5 Factors influencing the forming limit curve A number of factors influence the forming limit curve. Two of the factors, the strain hardening exponent n and the normal anisotropy factor r, are already included in the aforementioned derivations leading to Eq. (2.107) for diffuse instability, and Eq. (2.91) for local instability, as plotted in Fig. 2.46. The forming limit curves are proportional to the strain-hardening exponent, in accordance with Eq. (2.97). The normal anisotropy has little effect in stretching (right side of Fig. 2.46) and a negative effect on the formability in the tension-compression (left side of Fig. 2.46), i.e. smaller limiting strains when the values of normal anisotropy are higher. This conclusion should not be confused with the fact that normal anisotropy increases the limiting drawing ratio in deep drawing, where the influence of anisotropy also alters the stress distribution by a relative increase of flow stress in the critical zone between cup bottom and cup wall (tension-tension) as compared to the flow stress in the flange (tension-compression). The FLD in Fig. 2.46 does not include the effect of strain rate, which may influence the position of the FLCs. This is because the local strain rates are higher in the necking zone than in adjacent zones. Strain rate hardening,

Formability

ε1b

81

ε1

0.30

f 1.0 0.995 0.990 0.980

0.20

0.950

0.10

ε 1a

0.900

ε1b

-1

ε1a 0

0.10

0.20

2

ε2 0

0.30

ε1a,b (A)

(B)

Fig. 2.45 Localisation of necking seen (A) by the relationship between longitudinal strains ε1a and ε1b for a strain hardening exponent n ¼ 0.25 and various values of imperfection parameters f according to Eq. (2.113) and (B) Veerman’s graphical method (Veerman et al., 1971). ((A) Adapted from Hosford, W.F., Caddell, R.M., 2011. Metal Forming Mechanics and Metallurgy. Cambridge University Press, New York.)

Fig. 2.46 Instability curves for diffuse and local instability for a material following Ludwik-Hollomon’s flow stress curve σ ¼ kεn shown for different values of the strain hardening exponent n and normal anisotropy r. Variation of r is only shown for n ¼ 0.1 and n ¼ 0.4 to avoid too many overlapping curves.

82

Metal forming

therefore, postpones instability by hardening the necking zone relatively more than the adjacent material (Mielnik, 1991). Fig. 2.47 shows schematically how FLCs are influenced by the sheet thickness. Thin sheets will experience instability earlier than thicker sheets because imperfections have a relatively higher impact. If a certain size of an inclusion, groove or another imperfection is considered, the imperfection parameter in Eq. (2.110) becomes smaller (corresponding to a relatively larger imperfection) for smaller sheet thickness, and the theory by Marciniak and Kuckzynski (1967) as illustrated in Fig. 2.45 then explains the quicker localisation in the thin sheets. As the sheet thickness increases, the effect decreases and above a certain threshold thickness, the formability becomes almost independent from the sheet thickness (the values of thickness in Fig. 2.47 are just examples). Imperfections at the edges also influence the formability whenever the critical zones are near of close to the sheet edge, e.g., in collar drawing or even in a simple tensile test of a sheet material (hence the requirement of surface quality in tensile test standards). The mechanisms are again due to imperfections giving rise to strain localisation and potential necking and edge cracks.

ε1

> 2 mm 1.5 mm 1 mm 0.5 mm

Failure area

1

Safe area 1

0

ε

2

Fig. 2.47 Schematic influence of sheet thickness on the instability curve.

Formability

83

Until this point, the loading path has been assumed proportional, i.e., with linear strain path corresponding to a constant principal strain ratio β. Experiments show, however, a strain path dependence, and M€ uschenborn and Sonne (1975) proposed the following procedure to estimate formability under bilinear strain paths. Fig. 2.48 shows an example of the application of M€ uschenborn and Sonne’s theory to establish the new formability curves after a first linear deformation step. Fig. 2.48A shows a diffuse instability curve based on Eq. (2.107) with r ¼ 1 and n ¼ 0.1 together with part of an ellipse corresponding to constant effective strain ε ¼ 0:05 in the principal strain space. Four grey lines exemplify linear strain paths until diffuse instability. Let us now consider deformation until the drawn curve of constant effective strain. Then, M€ uschenborn and Sonne’s theory is that the formability represented by the dotted part of a grey line inside the ellipse is already ‘spent’, and the solid part of the grey line is the ‘remaining’ formability in the specific loading direction. Fig. 2.48B shows how the remaining formability lines can be used after a first loading step to establish the new

Fig. 2.48 Influence of nonlinear strain paths after M€ uschenborn and Sonne’s theory for an isotropic material (r ¼ 1) with a strain hardening exponent n ¼ 0.1. (A) Diffuse instability curve under linear strain paths and identification of the remaining formability after a certain effective strain (here ε ¼ 0:05); (B) diffuse instability curves after bilinear strain paths exemplified by first loading path being either uniaxial tension (β ¼  1/2) or balanced biaxial stretching (β ¼ 1).

84

Metal forming

formability curve. The example includes uniaxial tension and balanced biaxial stretching as the first deformation step, but the same theory can easily be adapted to other strain ratios. The example in Fig. 2.48B shows that the formability is higher during nonlinear strain paths if the loading path bends clockwise and less if it bends anticlockwise. If possible, this should be considered when designing sheet forming processes.

2.8.6 Representation of the forming limit curve in space of effective strain vs stress triaxiality The forming limit curves are sometimes presented in the space of effective strain vs stress triaxiality (see Fig. 2.17). It is therefore also convenient to be able to represent the instability curves in this space. For doing that, the principal strains at instability (see Eq. (2.107) for diffuse instability and Eq. (2.91) for local instability) are converted into the effective strain at instability through sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1+r 2r inst inst 2 2 (2.114) εinst ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðεinst ε1 ε2 + ðεinst 1 Þ + 2 Þ ð1 + r Þ ð1 + 2r Þ The stress triaxiality under plane stress (σ 3 ¼ 0) is given by σ m =σ ¼ ðσ 1 + σ 2 Þ=ð3σ Þ. With the effective stress in Eq. (2.94) and the relation between the principal stress ratio α and the principal strain ratio β in Eq. (2.101), the stress triaxiality can be expressed through the principal strain ratio as follows: pffiffiffiffiffiffiffiffiffiffiffiffi 1 + 2r σm 1+β rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ (2.115) σ 3 2r 2 1+ β+β 1+r Based on Eqs. (2.114), (2.115), the instability curves that were presented in the principal strain space in Fig. 2.46 are converted into the space of effective strain vs stress triaxiality in Fig. 2.49. Only two strain hardening exponents are included in order to avoid too many overlapping curves. The instability curve (FLC) and the fracture forming limit (FFL) are plotted together in Fig. 2.50, where also experimental values are included. The data are from Martins (2014) and based on a sheet material with normal anisotropy r ¼ 0.84.

Formability

85

Fig. 2.49 Diffuse and local instability curves presented in the space of effective strain vs stress triaxiality.

Balanced biaxial stretching

Plate strain

sm = 0.64 s

Uniaxial tension

1

sm = 0.33 s

Effective strain e

1.5

sm = 0.55 s

Normal anisotropy r=0.84

2

FFL FLC Tensile test Circular bulge test Elliptical bulge test Nakajima test Hemispherical dome test

0.5

0 0.2

0.3

0.4 0.5 s Stress triaxiality m

0.6

0.7

s

Fig. 2.50 Fracture forming limit (FFL) and forming limit curve (FLC) together with corresponding experiments in the space of effective strain vs stress triaxiality for a sheet material with normal anisotropy r ¼ 0.84. Note: Data are from Martins (2014).

86

Metal forming

2.9 Experimental determination of the forming limit curve The experimental determination of the forming limit curve (FLC) (refer to the grey line in Fig. 2.50) by means of the sheet formability tests that are listed in Table 2.4 can be performed in three different ways: (i) positiondependent, (ii) time-dependent, and (iii) time-position-dependent. The position-dependent method determines the onset of necking by analysing the principal strain distribution on the sheet surface of the specimens at a fixed and unique instant of time, immediately before cracking. The method requires etching or imprinting of a grid of circles on the blank before forming. The ellipses resulting from plastic deformation of the circles along a direction s perpendicular to the crack (Fig. 2.51A) are measured in accordance with the Z€ urich no. 5 procedure proposed by Rossard (1976), which later evolved to the ISO standard 12004-2 (2008). The measurements can be made with a digital camera (Fig. 2.51A) and the maximum strain at the onset of necking is obtained by mathematical interpolation of the strains along the direction s by means of a parabolic ‘bell-shaped curve’ (Fig. 2.51B). The use of digital image correlation (DIC) systems for measuring the evolution of strains on the sheet surface during testing (i.e. along time) allowed the development of time-dependent methods for determining the FLC in both principal strain space and space of effective strain vs stress triaxiality (Fig. 2.30). In time-dependent methods, necking is estimated by analysing the evolution of strains over time in the fracture area. The time-dependent method proposed by Martı´nez-Donaire et al. (2014) makes use of a temporal analysis of the major strain ε1 distribution and of its first derivative (_ε 1 , major strain rate) for a series of points of the sheet surface placed along a cross section perpendicular to the crack. Its application involves the following steps: (a) Definition of the necking boundary corresponding to point ‘B’ in Fig. 2.51C where the strain rate is gradually reduced until it ceases, and where the strain reaches a constant level (or even undergoes some elastic unloading) before fracture, (b) Determination of the time instant, tnecking, for the onset of necking (refer to the dashed vertical line in Fig. 2.51C). This point corresponds to the instant of time when the strain rate of point B reaches a local maximum after which it begins to decrease, (c) Identification of the fracture point. The point inside the necking area with the greatest strain corresponds to the fracture point (refer to point ‘A’ in Fig. 2.51C),

Formability

87

Digital camera Necking strain

Strain

Crack

s Position (s)

(A)

(B)

Major Strain

Point B Point B - Strain Rate Necking Strain

B A

Major Strain Rate

Point A

Time

(C) Fig. 2.51 Methods for determining the necking strains in sheet formability tests. (A) Schematic representation of a circular bulge test specimen with the identification of the ellipses located along a direction s perpendicular to the crack; (B) determination of the necking strain along s by means of a parabolic ‘bell-shaped curve’—position-dependent method; (C) determination of the necking strain using a time-dependent method. (Adapted from Martínez-Donaire, A.J., García-Lomas, F.J., Vallellano, C., 2014. New approaches to detect the onset of localised necking in sheets under through-thickness strain gradients. Mater. Des. 57, 135–45; Magrinho, J.P., 2019. Sheet-Bulk Forming (Ph.D. Thesis). University of Lisbon, Lisbon.)

(d) The major and minor strains at necking correspond to the strains ε1 and ε2 of point ‘A’ at the instant of time tnecking. The time-position-dependent method proposed by Martı´nez-Donaire et al. (2014) also known as ‘the flat-valley method’ is based on the observation and analysis of the displacements of the outer surface of the sheet test specimens in order to identify the instant of time corresponding to the onset

88

Metal forming

Fig. 2.52 Time-position-dependent method to determine the necking strains in sheet test specimens. (A) Evolution of the z-displacement with x-position at different instants of time; (B) evolution of the derivative of the z-displacement with x-position at different instants of time. (Adapted from Martínez-Donaire, A.J., García-Lomas, F.J., Vallellano, C., 2014. New approaches to detect the onset of localised necking in sheets under throughthickness strain gradients. Mater. Des. 57, 135–45; Magrinho, J.P., 2019. Sheet-Bulk Forming (Ph.D. Thesis). University of Lisbon, Lisbon.)

of necking. Fig. 2.52A shows a schematic evolution of the z-displacement (perpendicular to the surface of the nondeformed test specimen) vs the x-position along a section perpendicular to the necking for a Nakajima formability test. Each z-displacement vs x-position evolution corresponds to a different instant of time. As seen, for an instant of time t1 far from the onset of necking, the surface of the sheet follows the curvature imposed by the tool (hemispherical punch in the case of the Nakajima formability test). However, at the onset of necking (time t2), the curve begins to flatten because the thickness of the central region reduces more rapidly than the adjacent points. When the profile becomes flat, the sheet simultaneously begins to deform locally and independently of the curvature imposed by the punch. This event physically corresponds to the onset of necking. After the onset of necking (time t3), the curve presents a valley in the sheet surface that can be clearly seen and progressively deepens until fracture. Alternatively, the identification of the onset of necking can be made by considering the first spatial derivative of the z-displacement (Fig. 2.52B), and looking at the instant of time t2 for which the slope remains locally constant as a result of the sheet developing a flat profile. Other relevant methods for the experimental determination of the onset of necking are the timedependent methods proposed by Volk and Hora (2011) and Merklein et al. (2015), and the time-position-dependent method proposed by Wang et al. (2014) and Min et al. (2017), amongst others.

Formability

89

2.10 Process defects A number of typical defects can be identified for each metal forming process as summarised by Johnson and Mamalis (1976), who provide the overview reported in Table 2.6 in their survey. Defects include the formation of voids and cracks, plastic instability, and metallurgical defects as presented earlier in this chapter. Another group of defects are due to material flow defects, and a number of examples will be provided in this section. For a description of all defects listed in Table 2.6, the reader is referred to Johnson and Mamalis (1976) and their references for detailed descriptions. The rest of this section provides examples of material flow defects related to the following processes: (a) Rolling, (b) Forging (exemplified by closed-die forging, injection forging and extrusion), (c) Deep drawing, (d) Tube bending. Table 2.6 Process defects summarised by Johnson and Mamalis (1976).

Rolling Flat and section rolling Edge cracking Transverse-fire cracking Alligatoring (crocodiling) Fishtail Folds, laps Flash, fins Laminations Ridges-spouty material Ribbing Sinusoidal fracture Cross, transverse, and helical rolling Central cavity (axial or annular fissure) Overheated ball bearing Roll mark Folding (or seeming), laps, fringes Squaring

Extrusion-piercing Christmas tree (fir tree) Hot-, cold shortness Radial- and circumferential cracking Internal cracking Central bursts (chevrons) Piping (cavity formation) Sucking-in Corner lifting Skin inclusions (side and bottom of the billet) Longitudinal streaks Laps Laminated fractures Mottled appearance Extrusion-defect Impact extrusion Multiple tensile ‘necks’ Thermal break-off Continued

90

Metal forming

Table 2.6 Process defects summarised by Johnson and Mamalis (1976)—cont’d

Necking Triangulation and triangular fishtail Ring rolling Cavities Fishtail Edge cracking Straight-sided forms Forging Open- and closed-die forging, upsetting, indentation Longitudinal cracking Hot tears and tears Edge cracking Central cavity Centre bursts Cracks due to t.v.ds and thermal cracks Folds, laps Flash, fins Laminations Orange peel Shearing fracture Piping Rotary forging Mushrooming Central fracture Flaking High-energy-rate forging Piping Dead metal region Laps Turbulent metal flow

Drawing of rod, sheet, wipe, and tube Internal bursts (cup and con

chevron) Transverse surface cracking Chips of metal Poor surface finish Folding and buckling Fins, laps Deep drawing Wrinkling Puckering Tearing (necking) Edge cracking Orange peel Stretcher-strains (L€ uders lines) Earing Bending and contour forming Cracking Wrinkling Hole flanging Lip formation Petal formation Plug formation Blanking and cropping Distortion of the part Cracking Martensitic lines Eyes, ears, warts, beards, tongues

2.10.1 Rolling In rolling, the ideal parallel gap is altered by elastic deflection of the rollers under load, resulting in uneven thickness of the rolled sheet. When cylindrical rollers elastically bend, the sheet becomes thicker at the centre as shown in Fig. 2.53A. Rollers can be cambered (made barrel-shaped) to compensate for the bending, such that the roll gap in the loaded condition results in an even sheet thickness. It is difficult to obtain a stable parallel roll gap, and if the rollers are overcambered, the sheet thickness will again

Formability

(A)

(A1)

(A2)

(A3)

(A4)

(B)

(B1)

(B2)

(B3)

(B4)

91

Fig. 2.53 Defects in rolling due to (A) elastic deflection of rollers (or undercambering), and (B) overcambering resulting in overcompensation of elastic deflection. Potential defects include (A) uneven sheet thickness (thick centre), (A1) longitudinal tensile stress along the centre and compression along edges, (A2) transverse cracking, (A3) warping, (A4) wavy edges, (B) uneven sheet thickness (thin centre), (B1) longitudinal compressive stress along the centre and tension along edges, (B2) edges cracks, (B3) axial centre crack, (B4) wavy sheet along the centre.

be uneven, namely thinner at the centre as illustrated in Fig. 2.53B. Undercambering, on the contrary, goes in the direction as shown in Fig. 2.53A. When the centre of the sheet is the thickest after rolling (Fig. 2.53A), there is a longitudinal competition resulting in tensile stress along the centre and compressive stress along the edges (Fig. 2.53A1). The tensile stress may cause transverse cracking (Fig. 2.53A2), the coexisting tension and compression may cause warping of the sheet (Fig. 2.53A3), and the compressive stress may cause wavy edges due to buckling (Fig. 2.53A4). The opposite types of defects are typical of the case where the rolled sheet is thinner along the centre (Fig. 2.53B), i.e. a compressive stress develops along the centre and a tensile stress along the edges (Fig. 2.53B1). The tensile stress may lead to edge cracks (Fig. 2.53B2) or axial centre cracking (Fig. 2.53B3), whilst the compressive stress may cause the sheet centre to be wavy due to buckling (Fig. 2.53B4). Cambering may also occur due to temperature changes and resulting thermal expansion. Thermal camber takes time to develop, so that shape is a problem in the start-up of mills. Lubrication, cooling and roll camber all interact in this complicated problem. The front and back tensions between mills often hide shape defects, which do not become evident until the stresses are relaxed upon cooling. Shape meters indicate flatness problems whilst the mill is operating. Corrections for shape may be attempted by differential cooling of the rollers, by roll-bending jacks acting on the roll

92

Metal forming

bearings or by using four high-rolling mill arrangements with smalldiameter active rollers subjected to lower forces being supported by large-diameter rollers.

2.10.2 Forging In closed-die forging, defects are mainly associated with the development of folds and flaws as shown in Fig. 2.54. Folds (also known as ‘laps’, Fig. 2.54A) may be caused by buckling of the web during forging. Folding of material back on itself when fillet radii are very small may also lead to the occurrence of voids (Fig. 2.54B). Flaws (also known as ‘cold shuts’, Fig. 2.54C) are highly sheared regions associated with dead metal zones that develop when the billet is oversized. Folds and flaws are also likely to occur in other forging processes. Fig. 2.55A, for example, shows a pulley produced by injection forging in commercially pure aluminium with two possible cross sections (Colla et al., 1997). The cross section in Fig. 2.55B is typical of a sound pulley

Folds

(A)

Voids

(B)

Flaws

(C) Fig. 2.54 Typical defects in closed-die forging: (A) Folds formed by buckling of the web, (B) voids caused by small fillet radii, and (C) flaws due to oversized billet.

Formability

93

Fig. 2.55 (A) Injection forging of a pulley component made from commercially pure aluminium showing (B) a sound cross section or (C) a cross section with a fold. (Adapted from Colla, D., Petersen, S.B., Balendra, R., Martins, P.A.F., 1997. Injection forging of industrial components from thick-walled tubes. Trans. ASME J. Eng. Ind. 119, 537–541.)

produced by injection forging with preforming, whilst the cross section in Fig. 2.55C discloses an inadmissible pulley with a fold produced by injection forging without preforming. Folds would not matter if the metal healed fully when they were pressed back into the workpiece (as in the rehealing which occurs in extrusion with porthole and bridge dies), but unhealed pressed back folds are weak interfaces which become potential cracks. Such defects can open up during subsequent forming operations; even if they do not, they can be troublesome in service causing fatigue failures and/or lead to other problems such as corrosion and wear. Of course, the folds themselves may crack before being folded over, leading to more defects, such as flash cracks in steel forgings. An example of a dead zone is provided in Fig. 2.56, where flow lines during extrusion are visualised in Fig. 2.56B as compared to a reference grid marked on the specimen in Fig. 2.56A. Dead metal zones result from large die angles and/or high friction and are generally not desirable as a dead zone results in a poorer surface of the extruded surface, and a crack inside the component if the nonextruded workpiece material is part of the final component. Only in a few cases, the dead zone is desirable as it results in a smoother surface of the extruded part for some materials when extruded hot.

94

Metal forming

Fig. 2.56 Cross sections of round specimens with grids showing (A) a regular reference grid and (B) deformed grid after extrusion through a die with 90 degrees semidie angle and resulting dead-zone formation.

2.10.3 Wrinkling in deep drawing One of the typical defects in deep drawing is wrinkling. Wrinkling in the flange (Fig. 2.57A), together with bottom fracture in the cup, is often defining the process window. Wrinkling in the flange is due to compressive circumferential stress σ θ occurring when the flange is drawn to a smaller radius (Fig. 2.57B) as it leads to buckling. The tendency for wrinkling increases with the increase of unsupported/unconstrained surface area of the sheet metal in compression, decreasing thickness, and nonuniformity in sheet thickness. Wrinkles may also be initiated by trapped lubricants distributed nonuniformly between the surface of the sheet and the dies. For a limited range of products such as thin aluminium food trays, the wrinkles are left in place, whilst they are unacceptable in most situations. A blank holder is often used to suppress the wrinkles, by either a controlled blank holder force or a fixed gap between the deep drawing die and the blank holder. If the blank holder pressure is too high, the deep drawing force increases (because of friction) and may lead to tearing of the cup wall. Conversely, if the blank holder pressure is too low, wrinkling will occur in the flange. Deep drawing may be carried out without a blank holder

Formability

95

σθ σθ σr (A)

(B)

Fig. 2.57 Wrinkling in deep drawing due to compressive hoop stress and insufficient blank holder pressure. (A) Deep drawn cup with wrinkles in the flange and (B) schematic illustration of compressive hoop stress.

provided that the sheet metal is relatively thick to prevent wrinkling, or specialised dies based on the tractrix curve are used.

2.10.4 Tube bending Wrinkling is also seen in other sheet forming processes where compressive stresses occur. An example is tube bending, where compressive stresses arise on the inside of the bend. Fig. 2.58A shows schematically bow tube bending, and Fig. 2.58B shows three examples of bent tubes. The bending operation induces compressive stresses on the inside of the bend, which may cause wrinkling, and tensile stresses on the outside of the bend, which may cause cracks. Fig. 2.58C shows the result of bending without a mandrel or other support on the inside of the tube. The compressive stresses lead to buckling, which is seen as wrinkles. In other cases, the tube cross section will collapse during the bending. Many variants of the internal support during bending exist. Fig. 2.58D and E shows the improvement obtained by filling the tube by sand. In the former case, there is still a bit of wrinkling, whilst the latter is completely free of wrinkles, which is obtained by annealing the tube before bending.

2.11 Metallurgy Plastic deformation and formability are strongly influenced by the chemical composition of the materials and by other metallurgical factors such as the

96

Metal forming

Radius block Roller

Ram

(A)

(C)

(D)

(E) (B) Fig. 2.58 Tube bending shown (A) schematically, (B) by examples of tubes with details of the bend resulting from bending (C) of an empty tube, (D) a tube supported on the inside by sand and (E) an annealed tube supported on the inside by sand.

crystalline structure, the defects (imperfections) in the crystalline structure, the homogeneity of the material matrix, the grain size, and the surface quality.

2.11.1 Crystalline structure Metals present three basic types of geometric arrangement of the atoms in a crystal, at room temperature (Fig. 2.59): • BCC—body-centred cubic (e.g. chromium, iron, carbon steel); • FCC—face-centred cubic (e.g. aluminium, copper, brass, bronze, austenitic stainless steel); • HCP—hexagonal close-packed (e.g. zinc, magnesium, titanium). Plastic deformation of the crystalline structures takes place by slipping and twinning under shear stresses. In the case of slipping (Fig. 2.60A), one plane of atoms slides over an adjacent plane (called the slip plane). The critical shear

Formability

97

Fig. 2.59 Crystalline structures (unit cells) of metals with indication of a typical slip plane (in grey colour).

stress τcr to initiate slip (Fig. 2.61A) and dislocation motion (Fig. 2.61B) combined with the total number of slip systems (slip planes multiplied by slip directions) available in a crystal allows understanding the influence of the crystalline structure on material strength and formability (Table 2.7). The critical shear stress τcr to cause slip to occur is given by (Felbeck and Atkins, 1996) Gb (2.116) 2π a where a and b are defined in Fig. 2.61A and G ¼ τ/γ is the shear modulus. In the case of twinning (Fig. 2.60B), a portion of a crystal abruptly forms a mirror image of itself across a plane. Twinning is an alternative plastic deformation mechanism that is observed when dislocations are suppressed in both BCC and HCP crystalline structures. The changes in orientation caused by twinning often place new slip systems into favourable orientations with respect to the shear stress and, therefore, enable additional slipping to τcr ¼

Metal forming

98

Slip plane

Twinning plane

Slip direction

(A)

(B)

Fig. 2.60 Plastic deformation mechanisms. (A) Slipping and (B) twinning.

τcr Dislocations ( )

b x

γ Grain

a

τ cr (A)

(B)

Fig. 2.61 (A) Critical shear stress to initiate slip and (B) schematic planar representation of crystalline structures with dislocations in adjacent grains. Table 2.7 Mechanical behaviour as a function of the crystalline structure, at room temperature. Structure

BCC

FCC

HCP

Critical shear stress

High

Low

Low

Number of slip systems

12

12

3

Atomic packing factor

0.68

0.74

0.74

Mechanical behaviour

Good strength and moderate formability

Moderate strength and good formability

Moderate strength and low formability (brittle)

Note: The atomic packing factor is defined as the ratio between the total volume of atoms in a unit cell and the total volume of the unit cell.

Formability

99

occur. In other words, twinning indirectly contributes to slipping and dislocation motion, and to the increase in formability.

2.11.2 Defects The crystalline structures are not perfect. Plastic deformation and formability are greatly influenced by the nature of the defects and by their distribution and concentration in the material. The defects in crystalline structures can be classified into five different categories: • Point defects, such as vacancies (missing atoms), interstitials (extra atoms) or impurities (foreign atoms); • One-dimensional (linear) defects (called dislocations), which form in a region where a plane of atoms terminates abruptly in the lattice; • Two-dimensional defects (called grain boundaries), which correspond to the interfaces formed between grains during solidification; • Three-dimensional defects, such as voids, cracks or inclusions. Although the presence of dislocations lowers the critical shear stress required to slip, they often become entangled and their motion constrained by interstitials, impurities and grain boundaries. Entanglement and motion constraints raise the critical shear stress required to slip and, therefore, increases the strength and strain hardening of the materials. Inclusions are nonmetallic elements such as oxides, nitrides, sulphides or carbonitrides that form in the metal at the end of solidification. They are known to decrease formability—voids form around inclusions—and to produce effects not predicted by continuum mechanics analysis. Plastic deformation in steel rolling, for example, tends to elongate and align MnS inclusions in the longitudinal direction and to reduce fracture toughness and ductility in the width direction (Hosford and Caddell, 2011). The anisotropy produced by these inclusions is called mechanical fibering and is a prerequisite for alligatoring (crocodiling) defects in rolling (Table 2.5). Mechanical fibering also affects sheet bending, because formability along and across the sheet is different—cracks may occur when the sheet is bent one way (when the outer tensile stresses are perpendicular to the stingers resulting from the elongated inclusions), but not the other (Fig. 2.62). Mechanical fibering is also known to stimulate the occurrence of centreline fractures in wire drawing performed with high-deformation-zone geometry parameters Δ ¼ h/l. It is worth noticing that the anisotropy caused by mechanical fibering is different from crystallographic anisotropy (also known as ‘texture’ and

100

Metal forming

Cracking of the outer fibers along the elongated inclusions

Rolling direction No cracks

Rolling direction

Elongated inclusions

Fig. 2.62 Influence of mechanical fibering on the formability in sheet bending. r

r

1.5

1.5

Δ r≅0

Δ r >0 45º

1.0

0º

135º 90º

45º

1.0

0º

180º

135º 90º

180º

r-

0.5

0.5

0º

45º

0º

90º

45º

90º

Fig. 2.63 Influence of anisotropy on the formation of ears in a deep drawn cup.

commonly referred to as ‘anisotropy’) caused by the grains in a material not being randomly oriented, but have one or more preferred orientations. This other type of anisotropy is commonly introduced by deformation processing in well-known processes such as cold rolling, wire drawing and extrusion. The degree of anisotropy in a sheet material is commonly defined by the anisotropy coefficient r and by the normal anisotropy r previously defined in Eq. (2.11). The anisotropy within the plane of the sheet material is refereed as planar anisotropy Δr and given as r0  2r45 + r90 (2.117) 2 One of the effects of anisotropy in deep drawing is the formation of ‘ears’ in the directions in which the sheet is softer and easier to draw due to different flow stresses along different directions. Planar anisotropy Δr is a very useful quantity for determining the earing tendency of the material as shown in Fig. 2.63. Δr ¼

Formability

101

2.11.3 Material matrix Dispersing a high density of fine precipitate particles over the material matrix is a well-known technique to improve the strength and hardness of certain metallic alloys, because the precipitates create obstacles to slipping and dislocation motion. The process is called precipitation hardening and generally involves heat treatment but it should not be confused with quenching of a steel to obtain martensite, which is a completely different process. Precipitation hardening is commonly used in high-strength aluminium alloys, nickel-based superalloys, and high-strength steels. In case precipitation hardening is performed before forming (e.g. in the case of high-strength steels), the obstacles to slipping and dislocation motion initiate at the beginning of deformation. Precipitation hardening increases material strength and diminishes formability. There are cases in which precipitation hardening is only carried out after forming in order to ensure a good level of formability throughout the manufacturing process. This is, for example, the case of bake hardening steels used in automotive doors, which experience precipitation after forming and during the application of temperature for drying the paint applied on the doors. There are other materials, like, for example, aluminium-copper alloys AA-2XXX (also known as duralumin) that experience age hardening, which is a form of precipitation hardening that develops with time, at room temperature. In the case of these materials, forming must be performed before ageing or after solubilisation to dissolve the precipitates and ensure good formability.

2.11.4 Grain size Grain size influences plastic deformation. Materials with small grain sizes have larger grain boundary areas to constrain the movement of dislocations due to crystallographic misorientation and to discontinuity of the slip planes between adjacent grains. As a result of this, small grain sizes are commonly associated with higher material strengths. The grain size influence on the yield strength σ Y of a monolithic crystal is described by the Hall (1951)-Petch (1953) equation: σ Y ¼ σ i + kd 1=2

(2.118)

where σ i is a stress that opposes slipping and dislocation motion, k is a material constant and d is the average grain diameter.

102

Metal forming

Combining Eq. (2.118) with the following relation between the strain hardening exponent n and the average grain diameter d due to Morrison (1966) c1 n¼ (2.119) c2 + d1=2 where c1 and c2 are material-dependent parameters (taking values c1 ¼ 5 and c2 ¼ 10 in the case of low-carbon steels), it may be concluded that larger grain sizes give rise to lower material strengths and to higher values of the strain hardening exponent n. Because the strain hardening exponent n indicates the capability of a material to harden sufficiently in critical areas and better distribute the plastic deformation over surrounding areas (i.e. the capability to prevent the local build-up of strains and the tendency to form voids in critical areas), it may be concluded that formability increases with the grain size. In general, high-strength materials have lower values of n than lowstrength materials because the hardening mechanisms interact with the motion of dislocations, and, therefore, with the overall formability. However, readers must be aware that the relation between grain size and formability is not straightforward because very large grains may encourage cracking along the grain boundaries, forcing materials to experience a decrease in formability when compared to a similar material with smaller grain sizes.

2.11.5 Surface quality Surface quality also influences the formability of metals. The roughness of the surface (amplitude and texture) plays an important role in sheet metal forming and must be properly controlled during the final stage of cold rolling (called the ‘skin-pass rolling’). The skin-pass rolling must also prevent L€ uders lines formation (also known as ‘stretcher-strains’) in the areas of the sheet blanks that will be slightly deformed in subsequent forming processes because the yield point plateau only disappears in regions subjected to larger amounts of plastic deformations. Skin-pass rolling is commonly performed with parameters that are different from those utilised in the other rolling stages, namely a small reduction (approximately 1%), a large contact length and a large roll radius compared to the change in thickness. Furthermore, skin-pass rolling is usually performed under dry friction conditions or with a detergent as lubricant and with roughened roll surfaces, which leads to high friction between the

Formability

103

material and the work rolls causing severe inhomogeneous deformation (Kijima and Bay, 2009). Other ways of preventing L€ uders lines to appear in the forming processes are either by prestraining the sheet prior to forming (beyond the L€ uders band region of the uniaxial tensile test) or by alloying the material in such a way as to eliminate the yield drop and plateau from the stress-strain curve (Fig. 2.64). Besides the aforementioned defect related to L€ uders lines formation, there are two other surface defects (scratches and orange peel) that should also be taken into consideration because they compromise the overall formability and surface quality of the metal-formed parts. Scratches can be originated in material production (typically aligned in the rolling direction, in the case of sheets), in subsequent material handling or in the metal forming process. They lead to reductions in formability and, therefore, it is important to identify their source and to eliminate its cause. Orange peel in sheet metal forming occurs when a limited number of large grains deform independently of each other throughout the sheet thickness. The differences in the amount of deformation between neighbouring

σnom Maximum force

σ UTS σ upY σ loY

A B

A

Yield plateau

B

Elastic region Plastic region (Lüders band)

0

e

Fig. 2.64 Development of L€ uders band in a tensile test sample of a material exhibiting a well-defined yield point. Note: σ upY and σ loY denote the upper and lower yield stresses.

104

Metal forming

grains and within each grain give rise to a rough surface appearance (similar to the skin of an orange) perfectly visible to the naked eye. HCP metals are known to suffer this type of defect because they have a limited number of slip systems. In case the grains are smaller, there is less variation in the amount of deformation between neighbouring grains and the surface imperfections are too small for the eye to detect.

References Atkins, A.G., 1996. Fracture in forming. J. Mater. Process. Technol. 56, 609–618. Atkins, A.G., Mai, Y.-W., 1985. Elastic and Plastic Fracture. Wiley, New York. Ayada, M., Higashino, T., Mori, K., 1987. Central bursting in extrusion of inhomogeneous materials. In: Proceedings of ICTP 1987—2nd International Conference on Technology of Plasticity, Stuttgart, vol. 1, pp. 553–558. Bai, Y., Wierzbicki, T., 2008. A new model of metal plasticity and fracture with pressure and Lode dependence. Int. J. Plast. 24, 1071–1096. Bai, Y., Wierzbicki, T., 2010. Application of extended Mohr–Coulomb criterion to ductile fracture. Int. J. Fract. 161, 1–20. Bao, Y., Wierzbicki, T., 2004. On fracture locus in the equivalent strain and stress triaxiality space. Int. J. Mech. Sci. 46, 81–98. Bridgman, P.W., 1944. The stress distribution at the neck of a tension specimen. Trans. ASM 32, 553–574. Brozzo, P., Deluca, B., Rendina, R., 1972. A new method for the prediction of formability limits in metal sheets, sheet metal forming and formability. In: 7th Biennial Conference of the International Deep Drawing Research Group, Amsterdam. Budiansky, B., Hutchinson, J.W., Slutsky, S., 1982. Void growth and collapse in viscous solids. In: Hopkins, H.G., Sewell, M.J. (Eds.), The Rodney Hill 60th Anniversary Volume. Pergamon Press, Oxford, pp. 13–44. Caddell, R.M., Atkins, A.G., 1968. The influence of redundant work when drawing rods through conical dies. Trans. ASME J. Eng. Ind. 90, 411–419. Christiansen, P., Nielsen, C.V., Bay, N., Martins, P.A.F., 2019. Internal shear cracking in bulk metal forming. J. Mater. Des. Appl. 233, 603–614. Cockcroft, M.G., Latham, D.J., 1968. Ductility and the workability of metals. J. Inst. Met. 96, 33–39. Coffin, L.F., Rogers, H.C., 1967. Influence of pressure on the structural damage in metal forming processes. Trans. ASM 60, 672–686. Colla, D., Petersen, S.B., Balendra, R., Martins, P.A.F., 1997. Injection forging of industrial components from thick-walled tubes. Trans. ASME J. Eng. Ind. 119, 537–541. Coulomb, C.A., 1776. Essai sur une application des re`gles de maximis & minimis a´ quelques proble`mes de statique, relatifs a´ l’architecture. De l’Imprimerie Royale, Paris. Erman, E., Kuhn, H.A., Fitzsimons, G., 1983. Novel test specimens for workability testing. In: Chait, R., Papirno, R. (Eds.), Compression Testing of Homogeneous Materials and Composites (ASTM STP 808). ASTM, p. 279. Felbeck, D.K., Atkins, A.G., 1996. Strength and Fracture of Engineering Solids. Prentice Hall, New Jersey. Gouveia, B.P.P.A., Rodrigues, J.M.C., Martins, P.A.F., 1996. Fracture predicting in bulk metal forming. Int. J. Mech. Sci. 38, 361–372.

Formability

105

Green, A.P., 1951. A theoretical investigation of the compression of a ductile material between smooth flat dies. Lond. Edinb. Dublin Philos. Mag. J. Sci. 42, 900–918. Gurson, A., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rules for porous ductile media. Trans. ASME J. Eng. Mater. Technol. 99, 2–15. Hall, E.O., 1951. The deformation and ageing of mild steel: Ill. Discussion of results. Proc. Phys. Soc. B 64, 747–753. Hill, R., 1948. A theory of yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. Ser. A 193, 281–297. Hill, R., 1950a. On the inhomogeneous deformation of a plastic lamina in compression test. Lond. Edinb. Dublin Philos. Mag. J. Sci. 41, 733–744. Hill, R., 1950. Plasticity. Clarendon Press, Oxford. Hill, R., 1952. On discontinuous plastic states with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1, 19–30. Hosford, W.F., Caddell, R.M., 2011. Metal Forming Mechanics and Metallurgy. Cambridge University Press, New York. Hung, Q., Kim, J.J., Nguyen, D.T., Kim, Y.S., 2020. Uncoupled ductile fracture criterion considering secondary void band behaviours for failure prediction in sheet metal forming. Int. J. Mech. Sci. 169, 105297. International Organisation for Standardization, 2008. International Standard ISO 120042:2008 Metallic Materials-Sheet and Strip-Determination of Forming Limit Curves, Part 2: Determination of Forming Limit Curves in the Laboratory. ISO, Geneva. Isik, K., 2018. Modelling and Characterization of Damage and Fracture in Sheet-Bulk Metal Forming (Ph.D. Thesis). Technical University of Dortmund, Dortmund. Johnson, W., Mamalis, A.G., 1976. A survey of some physical defects arising in metal working processes. In: Tobias, S.A. (Ed.), Proceedings of the Seventeenths International Machine Tool Design and Research Conference, Birmingham, pp. 607–621. Johnson, W., Mellor, P.B., 1973. Engineering Plasticity. Van Nostrand Reinhold, London. Kachanov, L., 1958. On creep rupture time. Proc. Acad. Sci. USSR Div. Eng. Sci. 8, 2631. Keeler, S.P., 1968. Circular Grid System—A Valuable Aid for Evaluating Sheet Metal Formability (SAE Technical Paper 680092). Kijima, H., Bay, N., 2009. Influence of tool roughness and lubrication on contact conditions in skin-pass rolling. J. Mater. Process. Technol. 209, 4835–4841. Kobayashi, S., 1970. Deformation characteristics and ductile fracture of 1040 steel in simple upsetting of solid cylinders and rings. Trans. ASME J. Eng. Ind. 92, 391–398. Kudo, H., Aoi, K., 1967. Effect of compression test condition upon fracturing of a medium carbon steel. J. Jpn. Soc. Technol. Plast. 8, 17–27. Kuhn, H.A., Lee, P.W., Erturk, T., 1973. A fracture criterion for cold forming. Trans. ASME J. Eng. Mater. Technol. 95, 213–218. Lemaitre, J., 1985. Continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107, 83–89. Lemaitre, J., 1986. Local approach of fracture. Eng. Fract. Mech. 25, 523–537. Lemaitre, J., 1996. A Course on Damage Mechanics. Springer, Berlin. Lemaitre, J., Desmorat, R., 2005. Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer, Berlin. Lou, Y., Huh, H., Lim, S., Pack, K., 2012. New ductile fracture criterion for prediction of fracture forming limit diagrams of sheet metals. Int. J. Solids Struct. 49, 3605–3615. Magrinho, J.P., Silva, M.B., Alves, L.M., Atkins, A.G., Martins, P.A.F., 2018. New methodology for the characterization of failure by fracture in bulk forming. J. Strain Anal. Eng. Des. 53, 242–247. Magrinho, J.P., Silva, M.B., Reis, L., Martins, P.A.F., 2019. Formability limits, fractography and fracture toughness in sheet metal forming. Materials 12, 1493.

106

Metal forming

Marciniak, Z., Kuckzynski, K., 1967. Limit strains in the process of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609–620. Martı´nez-Donaire, A.J., Garcı´a-Lomas, F.J., Vallellano, C., 2014. New approaches to detect the onset of localised necking in sheets under through-thickness strain gradients. Mater. Des. 57, 135–145. Martins, R.J.P., 2014. Numerical Simulation of Sheet Metal Formability Tests (M.Sc. Thesis). University of Lisbon, Lisbon. Martins, P.A.F., Bay, N., Tekkaya, A.E., Atkins, A.G., 2014. Characterization of fracture loci in metal forming. Int. J. Mech. Sci. 83, 112–123. McClintock, F.A., 1968. A criterion for ductile fracture by the growth of holes. Trans. ASME J. Appl. Mech. 35, 363–371. McClintock, F.A., Kaplan, S.M., Berg, C.A., 1966. Ductile fracture by hole growth in shear bands. Int. J. Fract. Mech. 2, 614–627. Mendelson, A., 1986. Plasticity, Theory and Application. McMillan, New York. Merklein, M., Maier, A., Kinnst€atter, D., Jaremenko, C., Affronti, E., 2015. A new approach to the evaluation of forming limits in sheet metal forming. Key Eng. Mater. 639, 333–338. Mielnik, E.M., 1991. Metalworking Science and Engineering. McGraw-Hill, New York. Min, J., Stoughton, T.B., Carsley, J.E., Lin, J., 2017. A method of detecting the onset of localized necking based on surface geometry measurements. Exp. Mech. 57, 521–535. Mohr, C.O., Beyer, K., 1928. Abhandlungen aus dem gebiete der technischen mechanik 3. erweiterte Aufl. zur Jahrhundertfeier der Technischen Hochschule Dresden. W. Ernst & Sohn, Berlin. Morrison, W.B., 1966. The effect of grain size on the stress-strain relationship in low-carbon steel. Trans. ASM 59, 824–846. Muscat-Fenech, C.M., Arndt, S., Atkins, A.G., 1996. The determination of fracture forming limit diagrams from fracture toughness. In: Kals, H.J.J., Shirvani, B., Singh, U.P., Geiger, M. (Eds.), Proceedings Sheet Metal 1996—4th International Conference on Sheet Metal, Enschede, vol. 1, pp. 249–260. M€ uschenborn, W., Sonne, H.-M., 1975. Einfluss de form€anderungsweges auf die grenzform€anderungen des feinblechs. Arch. Das Eisenh€ uttenwesen 46, 597–602. Nahshon, K., Hutchinson, J., 2008. Modification of the Gurson model for shear failure. Eur. J. Mech. A Solids 27, 1–17. Nasmyth, J., 1850. Improvements in forging iron. J. Frankl. Inst. 50, 404–408. Norris, D.M., 1978. A plastic-strain, mean stress criterion for ductile fracture. Trans. ASME J. Eng. Mater. Technol. 100, 279–286. Norris, D.M., Reaugh, J.E., Moran, B., Quinones, D.F., 1978. A plastic-strain, mean-stress criterion for ductile fracture. J. Eng. Mater. Technol. 100, 279–286. Oyane, M., 1972. Criteria of ductile fracture strain. Bull. JSME 15, 1507–1513. Petch, N.J., 1953. The cleavage strength of polycrystals. J. Iron Steel Inst. 174, 25–28. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 210–217. Rossard, C., 1976. Mise en forme des m
etaux et alliages. Centre National de la Recherche Scientifique—CNRS, Paris. Shima, S., Oyane, M., 1976. Plasticity theory for porous metals. Int. J. Mech. Sci. 18, 285–291. Soeiro, J.M.C., Silva, C.M.A., Silva, M.B., Martins, P.A.F., 2015. Revisiting the formability limits by fracture in sheet metal forming. J. Mater. Process. Technol. 217, 184–192. Swift, H., 1952. Plastic instability under plane stress. J. Mech. Phys. Solids 1, 1–18. Tvergaard, V., 1981. Influence of voids on shear bands instabilities under plane strain conditions. Int. J. Fract. 17, 389–407.

Formability

107

Tvergaard, V., Needleman, A., 1984. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, 157–169. Veerman, C.C., Hartman, L., Peels, J.J., Neve, P.F., 1971. Determination of appearing and admissible strains in cold-reduced sheets. Sheet Met. Ind. 48, 678–694. Volk, W., Hora, P., 2011. New algorithm for a robust user-independent evaluation of beginning instability for the experimental FLC determination. Int. J. Mater. Form. 4, 339–346. Vujovic, V., Shabaik, A.H., 1986. A new workability criterion for ductile metals. Trans. ASME J. Eng. Mater. Technol. 108, 245–249. Wang, K., Carsley, J.E., He, B., Li, J., Zhang, L., 2014. Measuring forming limit strains with digital image correlation analysis. J. Mater. Process. Technol. 214, 1120–1130. Westergaard, H.M., 1920. On the resistance of ductile materials to combined stresses. J. Frankl. Inst. 189, 627–640. Wierzbicki, T., Xue, L., 2005. On the effect of the third invariant of the stress deviator on fracture (Technical Report). Impact and Crashworthiness Laboratory, Massachusetts Institute of Technology, Cambridge.

CHAPTER 3

Finite element simulation: A user’s perspective* 3.1 Introduction Taking a general view of the present state of the art in terms of modelling and computation of metal forming processes, it may be concluded that the finite element method is the most widespread numerical method for the analysis of complex industrial processes. The finite element method is capable of providing very efficient computer programs that can easily take into account the geometric and material nonlinearities as well as the contact changes that are typical of metal forming processes to produce accurate predictions of displacements, strain rates, strains, stresses, damage, temperature and microstructure evolution. The widespread utilisation of finite elements in metal forming by industry, research and education institutions is intended to: (i) optimise existing products and processes by cost and quality, (ii) develop new products and processes in shorter time, (iii) increase process know-how and compensate the gap of technological experience, and (iv) assist training and marketing efficiently. However, the widespread utilisation of finite element modelling in metal forming is also being accomplished with a growing evidence that most of the users are currently utilising commercial programs as ‘black-boxes’. Nowadays, users are mainly being trained on specific issues and procedures related to the utilisation of a particular computer program instead of being educated or refreshed with fundamental and detailed knowledge on material science, theory of continuum mechanics, and finite element methodologies. This is giving rise to a large group of users that are unable to recognise the basic pitfalls of the commercial programs and its methodologies and to a new generation of students and engineers that has little or no experience at all in the development and implementation of finite element computer programs. * Chris V. Nielsen (Technical University of Denmark, Lyngby, Denmark) and Paulo A.F. Martins (IDMEC, Instituto Superior T
ecnico, University of Lisbon, Portugal). Metal Forming https://doi.org/10.1016/B978-0-323-85255-5.00011-X

© 2021 Elsevier Inc. All rights reserved.

109

110

Metal forming

If the present situation needs to be changed, it is important to understand what has led to it. During the early developments of finite elements in metal forming, research groups working in the field together with leading industrial companies played a key strategic role by pushing forwards the application of finite elements particularly through writing in-house computer programs and sharing details of its implementation in the open literature. Those were the times users and developers could not be distinguished from each other and computer programs could easily be identified with individuals and individual research groups. In contrast to the active role performed by the metal forming research groups and industrial companies during the 1980s and early 1990s, current practice seems to indicate a total or near-total engagement of the majority of these groups and companies on applications rather than on developments. A critical gap is now being formed between the developers of the finite element computer programs and the users having the know-how on metal forming. The present relationship between developers and users seems more like a forced marriage below one’s station. As a result of this, students and engineers are increasingly being confronted with important key questions that need to be properly addressed. Which finite element formulations are available for solving metal forming processes? What are the basic aspects of modelling and computation behind selecting a specific formulation for solving a particular metal forming process? How accurate and reliable are the numerical estimates provided by finite element computer programs? Which methodologies can be utilised for validating the numerical estimates? These are critical questions namely in the case of industrial users because they may involve decisions on the acquisition of a new press, worth several hundred thousand Euros, or on the manufacturing of a new set of tools, worth tens of thousands of Euros as well as on the estimate of the set-up and production lead times. This chapter is concerned with the aforementioned gap between developers and users and it is written with the objective of providing students and engineers a basic overview that will enable them to recognise the main differences between the existing finite element formulations and mesh generation strategies, to identify the possible sources of errors and to understand the routes for validating their numerical results. The objective is to diminish the gap between developers of the computer programs and users having the know-how on the metal forming technology.

Finite element simulation: A user's perspective

111

The overall structure and contents of this chapter draws from a publication by Tekkaya and Martins (2009) and from the authors’ experience in lecturing this topic to students and specialists of metal forming in several countries around the world.

3.2 The finite element environment To run a finite element analysis, a mesh needs first to be generated, containing the appropriate number of elements that make up the overall shape of the workpieces and tools. This is the way of transcribing a metal forming process into a series of elements and nodes (with several degrees of freedom—d.o.f.) to be analysed. The density of the mesh can be altered based upon how complex or simple a simulation should be. This first stage of the analysis is called ‘preprocessing’ and consists of supplying a set of information (input data) indispensable for running the finite element computer programs: (a) Geometry of workpiece and tools—discretisation of the workpiece and tools by finite elements, (b) Boundary conditions—identification of process time, moving and stationary tool elements, symmetry conditions, applied pressure and friction, and thermal and electrical data along the contact interfaces, (c) Analysis type—decision on the type of simulation to be performed: static vs dynamic and on the type of coupling; mechanical, thermomechanical, magneto-mechanical or electro-thermomechanical, (d) Material data—selection of material behaviour; elastic, rigid-plastic/ viscoplastic, elastic-plastic/viscoplastic and supply of mechanical, thermal and electrical data (flow curve, thermal and electrical parameters), (e) Numerical parameters—choice of solution procedures, time stepping, convergence conditions and major output requirements. Finite element calculations are based on every single element or node of the mesh and assembled into a global system to be solved for the overall result of the entire metal forming system. Since there are several sources of nonlinearity in a metal forming simulation, the calculations are carried out incrementally (through subdivision of the process time) and sometimes also iteratively within a time increment. Because the calculations are done in a mesh, rather than in the entire physical workpiece and tools, the output data generally requires interpolation between the nodes to produce the final output data. Output data are handled in ‘postprocessing’ and consist of force vs displacement curves,

112

Metal forming

PRE-PROCESSING

FE COMPUTER PROGRAM

Geometry Boundary conditions Analysis type Material data Numerical parameters

POST-PROCESSING

Force vs. displacement Displacements Velocities Field variables

Fig. 3.1 Finite element environment in metal forming.

displacements, velocities, strain rates, strains, stresses, and other field variables such as temperature and ductile damage. Fig. 3.1 summarises the three steps: preprocessing, finite element calculations and postprocessing.

3.3 Mesh generation Mesh generation can be defined as the process of breaking up (or discretising) a physical domain (workpiece and tools) into smaller subdomains (elements) in order to simplify the numerical solution of the partial differential equations that govern the physical behaviour of a metal forming system. The principal application of mesh generation is the finite element method, but it can also be found in other applications such as computer graphics for animation films and games.

3.3.1 Structured vs unstructured meshes In mesh generation, the term ‘valence’ V designates the number of elements that are shared by a node (Fig. 3.2). Depending on the valence, meshes can be divided into two groups, structured and unstructured. A structured mesh is characterised by all the interior nodes having an equal number of adjacent elements (i.e. equal valence V). The meshes are typically all-quadrilateral or all-hexahedral and when nontrivial boundaries are required, ‘block-structured’ techniques may have to be employed to break the domain up into topological blocks. The algorithms for generating structured meshes also involve iterative smoothing techniques to align elements with boundaries or physical domains. Unstructured meshes, on the contrary, relax the node valence requirement, allowing any number of elements to meet at a single node. The

Finite element simulation: A user's perspective

113

V3

V4 V4

V5

Structured mesh

Unstructured mesh

Fig. 3.2 Valence of a node and structured vs unstructured meshes.

meshes are typically made from triangular and tetrahedral elements, although quadrilateral and hexahedral elements may also be utilised to generate unstructured meshes. The valence is also related to the quality of meshes. In two-dimensional meshes like those shown in Fig. 3.2, the valence V ¼ 4 is the best for quadrilateral meshes because the resulting elements have good quality with corner angles of approximately 90 degrees. The choice between structured vs unstructured meshes as well as between triangular vs quadrilateral (or tetrahedral vs hexahedral) elements is a recurring discussion in mesh generation forums. Considering, for example, three-dimensional applications the following conclusions can be drawn: (a) Mesh generation with tetrahedral elements is simple and robust, (b) Tetrahedral elements are overly stiff, have poor ability to withstand severe distortions (they may be viewed as degenerated hexahedral elements) and are very much sensitive to mesh orientation, (c) Tetrahedral elements frequently require up to an order of magnitude more elements to achieve the same level of accuracy as alternative hexahedral elements. This results in higher central processing unit (CPU) times, (d) Hexahedral elements are able to handle much severer distortions than tetrahedral elements making them attractive for finite element modelling of metal forming processes, (e) The utilisation of hexahedral elements is often limited by robustness of the meshing algorithms and automatic mesh generation difficulties. In fact, the arbitrary generation of hexahedral elements is state-of-the-art technology with ongoing developments that are not always available in commercial preprocessors.

114

Metal forming

3.3.2 Mesh generation techniques Taking a general view to the techniques currently available for the automatic generation of structured and unstructured meshes of hexahedral elements in arbitrary domains, it is possible to identify five different categories (other techniques like Voronoi tessellation that are used to generate tetrahedral elements will not be discussed here): (a) Mapped meshing, (b) Sweeping, (c) Plastering, (d) Indirect meshing, (e) Grid-based meshing. Mapped meshing This technique involves decomposition of the volume into blocks (labelled as ‘A’ to ‘D’ in Fig. 3.3) by the application of hexahedral elements (also named ‘superelements’). The (x, y, z) coordinates of these blocks are then mapped into a parametric coordinate system (η, ζ, ξ) for subsequent division into smaller hexahedral elements by indicating the number of divisions and grading along each coordinate axis (Zienkiewicz and Phillips, 1971; Martins and Marques, 1992).

A

B D

C 8

z

5

ζ

7 6

7

8 3

η

6

5

4 1

x

2

y

Fig. 3.3 Mapped meshing technique.

ξ

4 1

3 2

Finite element simulation: A user's perspective

115

In some cases, users can also utilise semiautomatic procedures that are capable of subdividing relatively easy geometries into block-structured regions. One of the main advantages of mapped meshing is that the parametric coordinate system intrinsically satisfies the topology and quality constraints for setting up a hexahedral mesh. Therefore, the only truly difficult task associated with mapped meshing is to ensure that opposite surfaces of the volume to be meshed have equal number of divisions. This can often be impossible for arbitrary geometric configurations or can involve considerable user interaction to decompose the geometry into mapped meshable blocks. Sweeping Sweeping (Fig. 3.4) is a 2.5D hexahedral mapped meshing technique based on a quadrilateral mesh that can be picked up through space along a curve (Knupp, 1998; Staten et al., 1998). Regular layers of hexahedra are formed at specified intervals using the same topology as the quadrilateral source mesh. Provided the source and target surfaces have similar topology and that they are connected by a set of mapped meshable surfaces, the quadrilateral elements of the source area can be swept through the volume to generate good-quality hexahedral elements. Plastering Mesh generation by plastering (Fig. 3.5) is an extension into 3D of the advancing front algorithm commonly utilised in 2D (Blacker and Meyers, 1993). The procedure starts with a quadrilateral (source) mesh, and hexahedra are then added individually onto the quadrilaterals in an advancing front manner. The technique may experience difficulties in filling complex arbitrary

Target (3D Mesh) Sweep

Source (2D Mesh)

Fig. 3.4 Sweeping meshing technique. (Adapted from Owen, S.J., 1993. Non-Simplical Unstructured Mesh Generation (Ph.D. Thesis). Carnegie Mellon University, Pittsburgh.)

116

Metal forming

Advancing front Source

Fig. 3.5 Plastering technique. (Adapted from Owen, S.J., 1993. Non-Simplical Unstructured Mesh Generation (Ph.D. Thesis). Carnegie Mellon University, Pittsburgh.)

domains, namely to solve interior voids where no hexahedra could be placed to close the volume. Indirect meshing Unstructured mesh generation of arbitrary-shaped volumes with tetrahedral elements is fast, flexible, and simple to automate (major preprocessors have this possibility). In contrast, unstructured mesh generation with hexahedral elements is generally an ill-defined problem that is difficult to automate. Indirect meshing (Fig. 3.6) takes advantage of the tetrahedral mesh generators’ robustness and performs the automatic decomposition of each tetrahedron into four hexahedra. The major disadvantage of this technique is related to the fact that hexahedral elements obtained from unstructured tetrahedral meshes have poor aspect ratios and nodes with high valence, which artificially increase the overall stiffness of the finite element models (Alves et al., 2003a). Grid-based meshing Grid-based meshing was proposed by Schneiders and B€ unten (1995) for the automatic generation of good-quality hexahedral elements in arbitrary domains. The technique involves the construction of a structured threedimensional mesh of hexahedra in the interior of the volume (core mesh) followed by subsequent generation of an extra layer of elements for linking the core with its projection on the boundary.

Finite element simulation: A user's perspective

117

Hexahedron

Fig. 3.6 Indirect meshing technique.

The following example retrieved from Nielsen et al. (2013) shows the result of grid-based procedures using a cylindrical core mesh and a cuboid core mesh (Fig. 3.7). Taking into account the pros and cons of the different meshing techniques in conjunction with the research efforts that are currently being made in the development of three-dimensional automatic meshing based on hexahedral elements, it can be stated that the grid-based approach is the best available option for discretising complex three-dimensional geometries with hexahedral elements for metal forming applications.

Fig. 3.7 Influence of the core mesh in the grid-based meshing technique. (A) Triangular surface mesh of the volume to be discretised, (B) cylindrical core mesh and final hexahedral mesh, and (C) cuboid core mesh and final hexahedral mesh.

118

Metal forming

3.4 Nonlinearity in finite element modelling of metal forming There are three major sources of nonlinearity in finite element simulation of metal forming processes: (a) Material nonlinearity, (b) Changing of static and kinematic boundary conditions, (c) Geometric nonlinearity. Material nonlinearity is associated with the inelastic behaviour of deformation beyond the yield stress of the material since the flow stress (stress vs strain curve) in the plastic regime is essentially nonlinear (Fig. 3.8A). The source of nonlinearity arising from the changes in static and kinematic boundary conditions is related to the interaction between workpieces and tools or between different workpieces, for example, contact with friction along the material-tool interfaces. Geometric nonlinearity results from the differences of the final (current) geometry with respect to its original (initial) geometry caused by deformation. Deformation of a vector (or line) embedded in the material generally involves stretching, rigid-body rotation and translation but only stretching contributes to the accumulation of strain and changes in stress state. Rigidbody rotation and translation contribute neither to shape (or size) changes nor to accumulation of strain and changes in stress state. Shearing results from deformation of adjacent vectors (or lines) embedded in the material. σ

Y,y

Linear

(mm)

2 Non-linear

1

A 0

E

–1

B

Original C

dX

P(X,Y,Z) E

X 1

x

B'

A' 3

2

u

4

5

X,x

dx

(mm)

Final (Current)

P'(x,y,z)

ε (A)

–2

D'

C'

(B)

Fig. 3.8 (A) Material nonlinearity and (B) rigid-body rotation associated with geometric nonlinearity.

Finite element simulation: A user's perspective

Example 3.1

Let us consider the rigid body motion of the two-dimensional workpiece shown in Fig. 3.8B from original ABCD to final A0 B0 C0 D0 configurations (adapted from Tekkaya, 2007). The position of an arbitrary point P in the original configuration is given by vector X ¼ (X, Y, Z) and its final configuration P0 after rigid-body translation and rotation is given by vector x ¼ (x, y, z). The displacement vector u ¼ (u, v, w) between P and P0 is given by 0 1 0 1 u 4X +Y u ¼ @ v A ¼ @ X  Y A w 0 so that,

0 1 0 1 x X + ð4  X + Y Þ x ¼ X + u ¼ @ y A ¼ @ Y + ðX  Y Þ A z Z

The displacement gradient eij can be obtained from 2

∂u 6 ∂X 6 ∂u 6 6 ∂v eij ¼ ¼6 ∂X 6 ∂X 6 4 ∂w ∂X

∂u ∂Y ∂v ∂Y ∂w ∂Y

3 ∂u 3 ∂Z 7 7 2 1 1 0 7 ∂v 7 4 7 ¼ 1 1 0 5 ∂Z 7 0 0 0 7 ∂w 5 ∂Z

and allow writing the infinitesimal strain tensor εij as    3 1 ∂u ∂v 1 ∂u ∂w + + 6 2 ∂y ∂x 2 ∂z ∂x 7 7 2 3 6 6   7 εx εxy εxz 6 1 ∂u ∂v  ∂v 1 ∂v ∂w 7 6 7 ε ¼ 4 εyx εy εyz 5 ¼ 6 + + 7 6 2 ∂y ∂x 7 ∂y 2 ∂z ∂y εzx εzy εz 6 7 6  7    4 1 ∂u ∂w 1 ∂v ∂w 5 ∂w + + 2 ∂z ∂x 2 ∂z ∂y ∂z 2 3 1 0 0 ¼ 4 0 1 0 5 0 0 0 2

∂u ∂x

and the infinitesimal spin tensor ωij as

119

120

Metal forming

3 ∂u ∂w  6 ∂z ∂x 7 7 2 3 6 6    7 ωx ωxy ωxz 6 1 ∂v ∂u 7 1 ∂v ∂w 7 0  ω ¼ 4 ωyx ωy ωyz 5 ¼ 6 6 2 ∂x  ∂y 7 2 ∂z ∂y 6 7 ωzx ωzy ωz 6  7    4 1 ∂w ∂u 1 ∂w ∂v 5   0 2 ∂x ∂z 2 ∂y ∂z 2 3 0 1 0 ¼ 4 1 0 0 5 0 0 0 2

0

1 2



∂u ∂v  ∂y ∂x



1 2



The conclusion from the results given by the infinitesimal strain and spin tensors is that although there is no shape and size changes caused by the rigid-body rotation, their components are different from zero εx ¼ 1 εy ¼ 1 ωxy ¼ 1 ffi 57:3º This result is obviously wrong and allows concluding that geometric nonlinearity cannot be described by infinitesimal strain measures. Appropriate measures of strain need to be used in continuum mechanics of large deformations. Translation will not be considered in what follows because it has not influence in strain.

3.5 Kinematics of large deformations 3.5.1 Measures of strain Measures of strain for large deformations are based on the deformation gradient F, which is a nonsymmetric tensor describing the deformation between the final configuration dx and the original configuration dX of a vector (or line) embedded in the material 2 3 ∂x ∂x ∂x 0 1 6 70 1 dx 6 ∂X ∂Y ∂Z 7 dX 7 B C 6 C ∂y ∂y ∂y 7 B C¼6 B dY C dx ¼ FdXB dy 6 @ A 6 ∂X ∂Y ∂Z 7 @ A 7 6 7 dz 4 ∂z ∂z ∂z 5 dZ ∂X ∂x 6 ∂X ∂x 6 6 ∂y ¼6 F¼ ∂X 6 ∂X 4 ∂z ∂X 2

∂Y ∂x ∂Y ∂y ∂Y ∂z ∂Y

∂Z 3 ∂x ∂Z 7 7 ∂xi ∂y 7 7 Fij ¼ ∂Xj ∂Z 7 ∂z 5 ∂Z

(3.1)

Finite element simulation: A user's perspective

121

The deformation gradient F cannot be directly used as a strain measure for large deformations because it includes stretching and rigid-body rotation, and the latter does not contribute to shape or size changes, nor to internal stresses. To obtain a strain measure for large deformations that is free from rigidbody rotation, we first define the (left) Cauchy-Green tensor C that can be understood as the ratio dl2/dL2 between the square of the final dl and initial dL lengths of the infinitesimal vector embedded in the material, dl2 ¼ dxdx + dydy ¼ dx  dx ¼ dxT dx ¼ dXT FT FdX ¼ dXT CdX C ¼ FT F

(3.2)

where the symbol ‘’ in Eq. (3.2) is the dot (scalar) product. The (left) Cauchy-Green tensor C is symmetric and is used as a measure of stretching independent from rigid-body rotation because it is exclusively related to changes in length. However, because C ¼ I, where I is the identity tensor, in case there is no material deformation, it is preferable to use an alternative strain measure, known as the Green-Lagrange finite strain tensor E, which is equal to zero when there is only rigid-body rotation:   1 T 1 F F  I Eij ¼ Fki Fkj  δij 2 2  T   !     !  1 1 T ∂x ∂x 1 ∂ðu + XÞ T ∂ðu + XÞ E¼ F FI ¼ I ¼ I 2 2 ∂X ∂X 2 ∂X ∂X

E¼

1 ¼ 2



∂u +I ∂X

 !  T  T ! ∂u 1 ∂u ∂u ∂u ∂u + +I I ¼ + ∂X 2 ∂X ∂X ∂X ∂X

T 

(3.3)

Typical components of E obtained from Eq. (3.3) are written as  2  2  2 ! ∂u 1 ∂u ∂v ∂w + + Ex ¼ + ∂X 2 ∂X ∂X ∂X (3.4)     1 ∂u ∂v 1 ∂u ∂u ∂v ∂v ∂w ∂w Exy ¼ + + + + 2 ∂Y ∂X 2 ∂X ∂Y ∂X ∂Y ∂X ∂Y and allow concluding that in case of infinitesimal deformations (where X ffi x), the second-order terms in Eq. (3.4) can be neglected, so that the normal and shear strains become similar to those of the infinitesimal strain tensor (refer to Example 3.1)

Metal forming

122

 T  T !  T !  1 T 1 ∂u ∂u ∂u ∂u 1 ∂u ∂u E¼ F FI ¼ + + ffi + 2 2 ∂X ∂X ∂X ∂X 2 ∂X ∂X ffiε (3.5) Another strain measure that is commonly utilised in large deformations is the Almansi strain tensor e, obtained from the (right) Cauchy-Green tensor B:  1 (3.6) I  B1 B ¼ FFT 2 In the aforementioned equations, B may be understood as the ratio dL2/dl2 between the square of the initial dL and final dl lengths of the infinitesimal vector embedded in the material. A final measure of strain for large deformations that can also be derived from the (right) Cauchy-Green tensor B is the logarithmic (or true) strain ε defined as e¼

 1  ε ¼  ln B1 2

(3.7)

Example 3.2

In order to check the adequacy of E as a strain measure for large deformations, let us revisit Example 3.1. This time, we will start by calculating the deformation gradient F by applying Eq. (3.1) to the relationship x ¼ X + u between final x and original X configurations: 3 2 ∂x ∂x ∂x 6 ∂X ∂Y ∂Z 7 2 3 7 6 0 1 0 7 6 ∂x 6 ∂y ∂y ∂y 7 4 ¼6 F¼ 7 ¼ 1 0 0 5 ∂X 6 ∂X ∂Y ∂Z 7 0 0 1 7 6 4 ∂z ∂z ∂z 5 ∂X ∂Y ∂Z Then, replace the result just found in Eq. (3.3) to obtain

 1 1 E ¼ ðC  I Þ ¼ F T F  I ¼ 0 2 2 This result is physically correct because rigid-body rotation cannot produce strains. Thus, the Green-Lagrange strain tensor E is a correct strain measure for large deformations and its nonlinear characteristics will be responsible for introducing nonlinearity in finite element computations.

Finite element simulation: A user's perspective

123

Example 3.3

In order to obtain the physical meaning of the different strain measures, let us consider a rod with initial length L and final length l subjected to stretching and rigid-body rotation (Fig. 3.9). Y,y

l

Final

L Original

0

X,x

Fig. 3.9 Rod subjected to stretching and rigid-body rotation.

Table 3.1 provides the equations derived from different measures of strain that can be utilised in both infinitesimal and large deformation analysis of the rod as shown in Fig. 3.9. Fig. 3.10 provides a comparison of the results obtained from these equations. Table 3.1 Comparison of different strain measures for the rod shown in Fig. 3.9. n

Strain measure

Equation

2

Green-Lagrange

1 l2 2 L2  1

1

Engineering (linear infinitesimal deformations)

0

Logarithmic

1

Swainger (nonlinear approximation of the Almansi)

2

Almansi

n

General relation (for uniaxial loading conditions)





lL  L

ln

l L

lL  l

1 2 1 n



2

1  Ll2



  n  L 1 l

124

Metal forming

Fig. 3.10 Plot of the different strain measures listed in Table 3.1.

3.5.2 Rate of deformation Some finite element formulations are developed in terms of rate (timedependent) quantities. Often, even rate-independent plasticity models (such as rigid-plastic models) are written in terms of velocities for implementation into finite element computer programs. This is easy to understand given that plasticity is an incremental process: rather than deal with increments of plastic strain, it is more convenient to work with the plastic strain rate. Thus, it is important to understand how the tensors and strain measures previously described can be written in rate form. For this purpose, we will describe the increment dv in velocity v occurring over an incremental change in position dx of a point P0 of the final (current) configuration as follows (Fig. 3.11): dv ¼

∂v dx ∂x

L¼

∂v ∂x

Lij ¼

where L is named the velocity gradient tensor.

∂vi ∂xj

(3.8)

Finite element simulation: A user's perspective

125

Y,y (mm)

dv

2

1

B

Original C

dX

0 –1

x

P(X,Y,Z) E

X A 1

x

B'

A' 3

2

v+dv

v

u

4

5

v

dx

Final

P'(x,y,z) X,x (mm)

dx Final

P'(x,y,z) –2

D'

C'

Detail

Fig. 3.11 Spatially varying velocity in a material point P0 of the final (current) configuration.

The velocity gradient L is related to the deformation gradient F through     _F ¼ ∂ ∂x ¼ ∂ ∂x ¼ ∂v ¼ ∂v ∂x ¼ LF L ¼ FF _ 1 (3.9) ∂t ∂X ∂X ∂t ∂X ∂x ∂X and may be physically understood as a tensor that maps the deformation gra_ dient F onto the rate of change of the deformation gradient F. The velocity gradient L can be decomposed into symmetric (stretch related) and antisymmetric (rotation-related) parts,  1  1 (3.10) L + LT + L  LT ¼ D + W 2 2 The symmetric part D is named as the rate of deformation tensor    T !  1 ∂v 1 ∂v 1 ∂vi ∂vj T D¼ L+L ¼ ¼ (3.11) + + 2 2 ∂x ∂x 2 ∂xj ∂xi L ¼ symðLÞ + asymðLÞ ¼

and the antisymmetric part W is named the continuum spin tensor    T !  1 ∂v 1 ∂v 1 ∂vi ∂vj T ¼ (3.12) W¼ LL ¼   2 2 ∂x ∂x 2 ∂xj ∂xi The rate of equivalent strain (hereafter named the effective strain rate) may be defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ε_ ¼ ðD + D22 + D23 Þ (3.13) ð D : DÞ ¼ Dij Dij ¼ 3 3 3 1

126

Metal forming

where the symbol ‘:’ in Eq. (3.13) is the double contraction (double dot product) of two second-order tensors, and D1, D2, D3 are the eigenvalues of the rate of deformation tensor D. A final rate of deformation measure that can also be obtained from the velocity gradient tensor L is the time rate of the Green-Lagrange strain tensor E_  

1     dE d 1  T 1 T E_ ¼ ¼ F F  I ¼ FT F_ + F_ F ¼ FT ðL FÞ + FT LT FÞ dt dt 2 2 2  T   1 L+L ¼ FT LF + FT LT F ¼ FT F ¼ FT DF 2 2 (3.14) where D is the symmetric part of L as defined in Eq. (3.10). Example 3.4

Let us calculate the rate of deformation of the rod subjected to uniaxial tension in y-direction shown in Fig. 3.12 (adapted from Dunne and Petrinic, 2005). We will start by writing the stretch ratios as  1=2 r l l λy ¼ λx ¼ ¼ R L L

 1=2 r l λz ¼ ¼ R L

The deformation gradient tensor F can be written by taking into account the following relation between the final x and original X configurations: x ¼ λ x X y ¼ λy Y z ¼ λz Z and can be written as follows: 2 3 2  ∂x ∂x ∂x l 1=2 6 ∂X ∂Y ∂Z 7 6 6 7 6 L 7 ∂x 6 6 ∂y ∂y ∂y 7 6 ¼6 F¼ 7¼6 0 ∂X 6 ∂X ∂Y ∂Z 7 6 6 7 6 4 ∂z ∂z ∂z 5 4 0 ∂X ∂Y ∂Z

3 0

0

7 7 7 l 7 0 7 L   7 1=2 5 l 0 L

The velocity gradient L is obtained from Eq. (3.9)

Finite element simulation: A user's perspective

  3 1 dl=dt  0 0 6 2 7 l 6 7 6 7 dl=dt 6 7 _ 1 ¼ 6 L ¼ FF 7 0 0 6 7 l  7 6 4 1 dl=dt 5 0 0  2 l 2

and finally, the rate of deformation tensor D is obtained by applying Eq. (3.11):   2 3 1 dl=dt 0 0 62 7 l 6 7 6 7  6 1 dl=dt 7 T D¼ L+L ¼6 7 0 0 6 7 2 l  7 6 4 1 dl=dt 5 0 0  2 l This result allows concluding that in the case of a rod subjected to uniaxial tension, the rate of deformation tensor D is identical to the true strain rate. This provides a reasonable understanding of the physical meaning of the rate of deformation.

Y, y

Final (deformed after stretching)

Initial (undeformed) l L

Z, z

r R

Fig. 3.12 Rod subjected to stretching in y-direction.

X, x

127

128

Metal forming

3.5.3 Measures of stress The true stress or Cauchy stress tensor σ is a second-order symmetric tensor that refers to the stress vector t resulting from an infinitesimal force dT applied on the infinitesimal surface dS with normal n (vector of direction cosines) of the final (current) configuration (Fig. 3.13)   dT t ¼ lim ¼ σn (3.15) dS!0 dS The infinitesimal force dT can be written in two different ways. One way results from multiplying both terms of Eq. (3.15) by the surface dS dT ¼ σn dS

(3.16)

and the other way consists of using the area dS0 and its normal n0 of the original (initial) configuration. This other way allows rewriting Eq. (3.16) as dT ¼ pn0 dS0

(3.17)

where p is named the (first) Piola-Kirchhoff stress tensor. The relationship between the first Piola-Kirchhoff and the Cauchy stress tensors is obtained making Eqs. (3.16), (3.17) equal and considering the mass conservation m ¼ ρ ndxdS ¼ ρ0 n0dXdS0, p¼

dS ρ ∂X σ¼ 0 σ ρ ∂x dS0

(3.18)

Eq. (3.18) allows concluding that the first Piola-Kirchhoff stress p tensor is nonsymmetric and resembles the nominal (engineering) stress because it is Q

Z,z

Original t 0

dX P

X

d T0 x

dS0 n0 dT n

dS dx

P'

Q'

Y,y

Final t0 + dt

O X,x

Fig. 3.13 Surface dS with normal n subjected to a resultant infinitesimal force dT.

Finite element simulation: A user's perspective

129

defined in terms of the infinitesimal force dT applied on the current infinitesimal surface dS per unit of the original infinitesimal surface dS0. Another quantity that it is worth defining is the equivalent Cauchy stress (hereafter designated as the effective Cauchy stress) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3 0 0 3 0 0 3  02 (3.19) σ¼ ðσ : σ Þ ¼ σ ij σ ij ¼ σ 1 + σ 0 22 + σ 0 23 2 2 2 where the symbol σ0 denotes the deviatoric Cauchy stress tensor given by 1 σ0 ¼ σ  TrðσÞ I 3

(3.20)

The effective strain rate given by Eq. (3.13) and the effective Cauchy stress given by Eq. (3.19) can be used to calculate the work rate (power) required for material deformation Z Z Z _ (3.21) W ¼ w_ dV ¼ σ : DdV ¼ σ ε_ dV V

V

V

and, from this relation, it is possible to derive the second Piola-Kirchhoff S as a work rate conjugate stress measure (energetically conjugate pair) of the Green-Lagrange strain rate E_ given in Eq. (3.14) Z Z σ : D dV ¼ J σ : D dV 0 (3.22) V  T 1V0 1 T ¼ J ðF σF Þ E_ J σD ¼ J σ F E_ F The second Piola-Kirchhoff S is obtained from Eq. (3.22) as a symmetric stress tensor given by 

1

S¼J F σF

T



  ρ0 ∂X ∂X T ¼ ¼ pFT σ ρ ∂x ∂x

(3.23)

In the aforementioned equations, J ¼ dV/dV0 ¼ det F is the Jacobian of the transformation. A final measure of stress that can also be obtained from the Cauchy stress tensor σ and that is also work conjugate of the rate of deformation tensor D is the Kirchhoff stress τ tensor τ¼Jσ¼

ρ0 σ ρ

(3.24)

130

Metal forming

The Kirchhoff stress tensor is identical to the Cauchy stress tensor corrected by a scale factor J. This means that the Kirchhoff and Cauchy stress tensors are identical whenever material deformation is incompressible like, for example, in the plastic deformation of metallic materials. Further information on the kinematics of large deformations may be obtained in Dunne and Petrinic (2005).

3.6 Finite element formulations Having finished introducing the sources of nonlinearity and the kinematics of large deformation, the presentation will now move to the fundamentals of the quasi-static and dynamic formulations that are commonly utilised in the numerical modelling of metal forming processes. The aim is to provide users with the background that will enable them to select a specific formulation for solving a particular metal forming process. This requires understanding of available finite element formulations and the basic aspects of modelling and computation.

3.6.1 Quasi-static finite element formulations The quasi-static finite element formulations are based on the static governing equilibrium equations in current (deformed) configuration in the absence of body forces ∂σ x ∂τyx ∂τzx + + ¼0 ∂x ∂y ∂z ∂τxy ∂σ y ∂τzy + + ¼0 ∂x ∂y ∂z

∂σ ij ¼0 ∂xj

(3.25)

∂τxz ∂τyz ∂σ z + + ¼0 ∂x ∂y ∂z By employing the Galerkin form of the weighted residual method, it is possible to write an integral form of Eq. (3.25), which instead of satisfying the equilibrium requirements exactly (i.e. point-wise) will only fulfil them in an average sense over the entire domain (volume V), Z ∂σ ij δui dV ¼ 0 (3.26) ∂xj V

Finite element simulation: A user's perspective

131

The symbol δui denotes an arbitrary variation in the displacement (or, velocity) ui. The duality between displacement and velocity is due to the fact that finite element formulations for metal forming can be set up in terms of displacements or velocities. Applying integration by parts in Eq. (3.26), followed by the divergence theorem, and considering the natural and essential boundary conditions, it is possible to rewrite Eq. (3.26) as follows: Z Z ∂ðδui Þ σ ij dV  ti δui dS ¼ 0 (3.27) ∂xj V

St

where ti ¼ σ ijnj denotes the tractions applied on the boundary St with prescribed traction having the unit normal vector nj. Eq. (3.27) is named the ‘weak form’ of Eq. (3.25) because the governing quasi-static equilibrium equations are now only satisfied in weaker continuity requirements. In other words, Eq. (3.27) requires the stress field to be continuous but not anymore its derivative. Eq. (3.27) together with appropriate constitutive equations for elasticplastic/viscoplastic or rigid-plastic/viscoplastic materials enables quasi-static finite element formulations to be given by the following matrix set of nonlinear equations: Ku ¼ F

(3.28)

The aforementioned equation expresses the quasi-static equilibrium condition in the current configuration (at the instant of time ‘t’). The symbol K denotes the stiffness matrix and F is the generalised force vector resulting from the forces, pressure and friction stresses applied on the boundary. The basic feature of K is its nonlinearity with u. The quasi-static finite element formulations are commonly implemented in conjunction with implicit schemes. The main advantage of implicit schemes over alternative solutions based on explicit schemes is that equilibrium is to be checked at each increment of time by means of iterative procedures meant to minimise the residual force vector R within a specified tolerance. This ensures a very good level of accuracy in terms of geometry and field variables. RðuÞ ¼ F  Ku

(3.29)

The nonlinear set of Eq. (3.29) derived from the quasi-static finite element formulations with implicit schemes can be solved by different

132

Metal forming

numerical techniques such as the direct iterative method (also known as the successive replacement method) and the Newton-Raphson-based methods. In connection to this, it is worth mentioning that the term ‘implicit’ is here utilised in the sense that the solution is made incrementally (typically hundreds of increments) and that an iterative procedure is employed in each step in order to fulfil the static equilibrium conditions.

K(u 0 ) - fi rst it erati on K(u 1) sec ond K( iter u) atio 2 n - th ird ite rat ion

Direct iterative method The direct iteration method (also named ‘successive replacement method’) evaluates the stiffness matrix based on the displacements of the previous iteration to reduce Eq. (3.29) into a linear set of equations. In practical terms, the method consists of solving the set of equations, F ¼ Kn1un where the symbol ‘n’ denotes the iteration number. Alternatively, the method can be formulated by considering the residual force vector R at iteration n and proceed as it is graphically illustrated in Fig. 3.14

Force

u K(u).

Given force

Solution R1

Δ u2 Displacement u0

u1

u2

u3

Fig. 3.14 Basic principle of the direct iteration method.

un

Finite element simulation: A user's perspective

133

Incremental Process

.... R nt = Ft – Knt –1 utn –1

R1t + Dt = Ft+Dt – K0t+Dt u0t+Dt .... t+Dt Dunt+Dt = Rnt+Dt Kn–1 t +Dt +Dt unt +D t = utn–1 + Dun

....

Iterative Process

+Dt Rtn+1 = Knt+Dt unt+Dt – Ft + D t

....

Fig. 3.15 Implementation of the direct iteration method in quasi-static finite element formulations.

Rn ðun1 Þ ¼ F  Kn1 un1 Rn ðun1 Þ ¼ Kn1 un  Kn1 un1 ¼ Kn1 Δun Δun ¼ Rn ðun1 Þ=Kn1 un ¼ un1 + Δun

(3.30)

The implementation of the direct iterative method in a computer program built upon a quasi-static finite element formulation is illustrated in Fig. 3.15. Example 3.5

Let us consider the equation x2  x  6 ¼ 0, which has the solutions x ¼  2 and x ¼ 3, and obtain the first of these two solutions (x ¼  2) by means of the direct iteration method. We start by writing the equation in a similar way to that of Eq. (3.28) K x ¼ F ) ðx  1Þx ¼ 6 The residual R and the increment of displacement Δu ¼ Δx are given by Eq. (3.30): R ¼ 6  ðxn1  1Þxn1 Δxn ¼

Rn ðxn1 Þ 6  ðxn1  1Þxn1 ¼ xn1  1 Kn1 xn ¼ xn1 + Δxn

134

Metal forming

The results of the iterative calculation process illustrated in Fig. 3.15 are given (starting from an initial guess x ¼ 0): Iteration x

Δx

Iteration x

6

Δx

0

0

1

6

2

0.85714 2.373626374 10

1.94286 0.095980508

3

3.23077

2.03884

4

1.41818 1.063021189 12

5

2.4812

6

1.72354 0.479471365 14

1.98861 0.019016076

7

2.20301

2.00762

5.142857143

8

1.87324 0.215001879

9

2.08824

1.812587413 11 0.757660891 13 0.329777532 15

0.14538228 0.064395879

1.97444 0.042746132 2.01719

0.028578584 0.012693452

and allow concluding that the direct iteration method converges unconditionally towards the solution, but convergence becomes slower as the exact solution is approached.

Newton-Raphson iterative methods The Newton-Raphson iterative methods are based on a linear Taylor series expansion of the residual R near the actual velocity estimate as follows: 

∂R R ðun Þ ffi Rn ¼ Rn1 + ∂u

 n1

Δun ¼ 0 un ¼ un1 + Δun

(3.31)

The term Kt ¼  ∂ R/∂ u is named the tangent stiffness matrix and accounts for the sensitivities of the internal forces to perturbations in the nodal displacement degrees of freedom of the system. Depending on updating or not updating Kt during the iterative process, two main procedures can be used: (a) Tangent stiffness procedure, in which the tangent stiffness matrix Ktn1 is updated after each iteration by using the values obtained in previous iteration; (b) Initial stiffness procedure, in which the tangent stiffness matrix Kt0 is not updated and takes the values of the first iteration. In some cases,

Finite element simulation: A user's perspective

135

the procedure may be modified to include intermediate updates of the tangent stiffness matrix. The Newton-Raphson iterative method based on a tangent stiffness procedure can be applied by means of the following set of equations, where the symbol ‘n’ denotes the iteration number (Fig. 3.16): F ¼ Kn1 un Rn ðun1 Þ ¼ F  Kn1 un1   ∂R R ðun Þ ffi Rn ¼ Rn1 + Δun ¼ 0 Rn1  Ktn1 Δun ¼ 0 (3.32) ∂u n1 Δun ¼ Rn1 =Ktn1 un ¼ un1 + Δun The Newton-Raphson iterative method based on a tangent stiffness approach is only conditionally convergent due to its dependency on the initial displacement guess u0. The method converges quadratically at the vicinity of the exact solution, but its convergence radius is often limited. The convergence of the Newton-Raphson iterative method can be improved by relaxation un ¼ un1 + βΔun

(3.33)

where β is the relaxation factor to be determined by the analysis of the convergence behaviour of previous iterations and/or by the application of line search algorithms. The application of the Newton-Raphson iterative method with an initial stiffness approach requires replacing Ktn1 by Kt0 in the overall set of Eq. (3.32).

Example 3.6

Let us consider the equation x2  4 ¼ 0, whose solutions are x ¼  2 and x ¼ 2 and obtain the second of these solutions (x ¼ 2) by means of the Newton-Raphson iterative method. We start by writing the equation in a similar way to that of Eq. (3.28), K x ¼ F ) xx ¼ 4 The residual R and the increment of displacement Δu ¼ Δx are given by Eq. (3.32)

Fig. 3.16 Newton-Raphson method based on (A) tangent and (B) initial stiffness procedures.

Finite element simulation: A user's perspective

137

R ¼ 4  xn1 xn1   ∂R t Kn1 ¼ ¼ 2xn1 ∂x n1 Δxn ¼

Rn1 ðxn1 Þ 4  xn1 xn1 ¼ t 2xn1 Kn1

xn ¼ xn1 + Δxn

The results of the iterative calculation process are provided in the following table for an initial guess x0 ¼ 3: Iteration

x

Δx

0

3

0.83333

1

2.166667

0.16026

2

2.00641

0.0064

3

2.00001

-1E  05

In case the initial stiffness procedure were utilised instead of the tangential stiffness procedure, the set of equations mentioned earlier has to be written as R ¼ 4  xn1 xn1   ∂R t K0 ¼  ¼ 2x0 ∂x 0 Δxn ¼

Rn1 ðxn1 Þ 4  xn1 xn1 ¼ 2x0 K0t

xn ¼ xn1 + Δxn

and the results of the iterative calculation process with initial guess x0 ¼ 3 are given as follows: Iteration

x

Δx

Iteration

x

Δx

0

3

0.83333

5

2.001818

0.00121

1

2.166667

0.11574

6

2.000605

0.0004

2

2.050926

0.03438

7

2.000202

0.00013

3

2.016543

0.01107

8

2.000067

4.5E  05

4

2.005469

0.00365

9

2.000022

1.5E  05

As shown, the Newton-Raphson iterative method based on an initial stiffness approach has a lower convergence rate than the NewtonRaphson iterative method based on a tangent stiffness approach.

138

Metal forming

Convergence criteria The direct iterative method and the Newton-Raphson iterative-based methods are designed to minimise the residual force vector R within a specified tolerance and their control and assessment needs to be performed by means of a convergence criterion. The following convergence criteria are commonly utilised: (a) Relative displacement norm: kun  un1 k  ζu kun1 k

(3.34)

where the symbol ‘k k’ denotes the Euclidean norm and ζ u is a number typically ranging from 102 to 105 depending on the accuracy requirements for the solution. (b) Relative residual norm: kR n k  ζF kFk

(3.35)

where ζ F is a number typically ranging 101 to 104 (one order of magnitude less than ζ u) because stresses and hence forces are usually one order of magnitude less accurate than displacements. Sometimes a maximum number of iterations may be used in situations when the programs are not able to reach convergence of the relative displacement and residual norms within the present number of iterations. Three different actions may be taken in such cases: force the program to stop, reduce the increment of time and retry the iterative procedure, or allow advancing to the next increment of time with the information of lack of convergence. Further information on convergence criteria is given in Tekkaya (2000). Finite element equations As mentioned earlier, the matrix set of nonlinear equations Ku ¼ F (Eq. 3.28) can be built from constitutive equations for rigid-plastic/ viscoplastic or elastic-plastic/viscoplastic material behaviour. In case of rigid-plastic/viscoplastic material behaviour, the procedure is based on the flow formulation (Cornfield and Johnson, 1973; Lee and Kobayashi, 1973; Zienkiewicz and Godbole, 1974) and the matrix set of nonlinear equations may be derived from the weak form of the quasi-static equilibrium equations by replacing the displacement ui by the velocity vi in Eq. (3.27)

Finite element simulation: A user's perspective

Z σ ij V

or Z V

∂ðδvi Þ dV  ∂xj

139

Z ti δvi dS ¼ 0

(3.36)

St

   Z 1 ∂ðδvi Þ ∂ δvj dV  ti δvi dS ¼ 0 σ ij + 2 ∂xj ∂xi

(3.37)

St

Considering that δD ¼ (1/2)[δL + δLT], the weak variational form of the stress equilibrium equations can be written as follows: Z Z σ : δDdV  t δvdS ¼ 0 (3.38) V

St

where the rate of deformation tensor D corresponds to the plastic rate of deformation tensor D ¼ Dp since elastic effects are not taken into account by the flow formulation. Since the finite element equations (3.38) are written in the current (deformed) configuration at time t and make use of a control volume approach with velocities v as the primary unknowns, the overall computer implementation is similar to that of an incompressible viscous fluid (and explains the designation ‘flow formulation’). In fact, the flow formulation uses a kind of ‘updated Eulerian’ formulation (Mattiasson, 2010). The geometry is fixed and velocities at given points in space remain constant in each time step t, whilst equilibrium is checked by means of iterative procedures meant to minimise the residual force vector R to within a specified tolerance. The geometry is then updated based on the calculated velocities. One of the main advantages of the flow formulation is that implementation is simpler, robustness is higher, and calculations are generally faster than those of alternative solutions based on solid formulations for elasticplastic or elastic-viscoplastic material behaviour. An obvious disadvantage of the flow formulation is of course that computer programs are not able to properly handle phenomena related to elasticity, such as spring back, unless it happens after proportional loading such that spring back can be simulated by one elastic solution. The pseudoelastic constitutive law that is commonly utilised for modelling the rigid zones may also lead to convergence difficulties whenever the size of the plastically deforming region is very limited as well as to difficulties

140

Metal forming

in calculating the normal pressure at the workpiece-tool contact interfaces, where the workpiece behaves quasi-rigidly. The latter may originate difficulties in estimating the frictional forces (Tekkaya, 2000). Details on the finite element flow formulation and on its computer implementation are given in another chapter of this book. In case of elastic-plastic or elastic-viscoplastic material behaviour, the procedure to build the matrix set of nonlinear equations K u ¼ F (Eq. 3.28) is based on the solid formulation (Hibbitt et al., 1970; McMeeking and Rice, 1975). The solid formulation is built upon the weak form of the quasi-static equilibrium equations (3.25) in the configuration at time t0, taken as the initial stress-free configuration (total Lagrangian approach) or as the last calculated configuration (updated Lagrangian approach). Taking into consideration the work rate required for material deformation given by Eq. (3.21) and the work rate conjugate relation of Eq. (3.22), it is possible to write the weak form of the quasi-static equilibrium equations (3.25) with reference to the last calculated configuration at time t0 as follows: Z Z S : δ E_ dV 0  t0 δvdS0 ¼ 0 (3.39) V0

St0

Solid formulations generally make use of the following alternative expression derived after replacing the second by the first Piola-Kirchhoff stress tensor p ¼ SF (3.23) and taking Eq. (3.14) into account, Z Z p : δ F_ dV 0  t0 δvdS0 ¼ 0 (3.40) V0

St0

In case solid formulations are to be implemented in rate form, the starting point is the stress rate equilibrium equations ∂p_ ij ∂Xj

¼0

(3.41)

Then, using a procedure like that described previously, it is possible to obtain the following weak form of the quasi-static rate equilibrium equations: Z Z p_ : δ F_ dV 0  t_ 0 δvdS0 ¼ 0 (3.42) V0

where t_ is the traction rate.

St0

Finite element simulation: A user's perspective

141

The need to write the weak form given by Eq. (3.42) in terms of the first Piola-Kirchhoff stress rate p_ instead of the Cauchy stress rate σ_ is because the latter is not an objective measure of stress rate, even though the Cauchy stress σ itself is an objective measure of stress because it is invariant (or indifferent) against rigid-body rotation. In other words, the time derivative of the stress equilibrium equations given by Eq. (3.25) cannot serve as starting point to derive the finite element equations of the solid formulation in the rate form _ due to inadequacy of the Cauchy stress rate σ. It can be proved that including the Cauchy stress tensor in the weak form of the quasi-static rate equations requires modifying Eq. (3.42) as follows (Tekkaya, 1986): Z Z  T 

r σ : δD  2 ðDσÞ : δD + σ : δL L dV 0  t_ 0 δvd S0 ¼ 0 (3.43) V0

St0

This equation makes use of the Jaumann rate of the Cauchy stress tensor, σ ¼ σ_ + σW  Wσ which can be proved to be an objective measure of stress rate. The finite element computer programs based on the solid formulation do not have the disadvantages of those based on the flow formulation regarding the calculation of residual stresses and elastic spring back. However, they are more difficult to implement, require higher computational resources and are generally less robust and much slower to perform calculations than those build upon the flow formulation. Further details on the computer implementation of solid and flow formulations are given in Tekkaya (2000) and Mattiasson (2010). r

Pros and cons of quasi-static finite element formulations Basic aspects of modelling and computation behind selecting a specific finite element formulation require users to have a full understanding on the advantages and disadvantages of each existing formulation. In the case of the quasistatic finite element formulations, the main advantages may be summarised as follows: (a) Equilibrium is checked at each iteration within the step (time or load increment) to minimise the residual force R to a specified tolerance, (b) Accuracy is very good in both geometry and distribution of major field variables. The main drawbacks are as follows: (a) The solution of linear systems of equations in each iteration is required, (b) High computation times and high memory requirements are needed,

142

Metal forming

(c) Computation time depends quadratically on the number of degrees of freedom (d.o.f.) if a direct solver is utilised, (d) Stiffness matrix is often ill-conditioned which can turn the solution procedure unstable and deteriorate the performance of iterative solvers, (e) Difficulties in dealing with complex nonlinear geometric, material and tribological boundary conditions. Numerical implementation of the formulations can often present convergence problems, (f ) Not able to properly handle phenomena related to elasticity in the case of the flow formulation. Example 3.7

A system of equations is classified as ill-conditioned if a small change in the inputs (independent coefficients or the right-hand side term) produces a large change in the outputs (answer). This gives rise to computational problems and to difficulties in obtaining the correct solution of the physical phenomenon. Let us consider the following system of equations (adapted from Tekkaya, 2007),      4 u1 2004 1 1 u1 ¼ ! ¼ u2 2 2000 1 1:001 u2 Now let us a consider a small change in one of the coefficients,      1 1 u1 4 u1 1004 ¼ ! ¼ 1 1:002 u2 u2 2 1000 As shown, the result changes drastically due to the ill-conditioning of the system of equations. In fact, the eigenvalues λmax, λmin of the original system of equations give rise to a very large condition number κ ¼ j λmax j/j λmin j:   jλmax j λmax 2:000500125 ffi 4002≫1 ¼ !κ¼ λmin 0:000499875 jλmin j

3.6.2 Dynamic finite element formulations The dynamic finite element formulations for metal forming are based on the dynamic equilibrium equations in the absence of body forces (Honecker and Mattiasson, 1989) ∂σ ij  ρ u€i ¼ 0 ∂xj

(3.44)

Finite element simulation: A user's perspective

143

where ρ denotes the density and u€ is the acceleration of the mass at the current instant of time t. The most significant difference between dynamic and quasi-static equilibrium equations is that the former takes inertial effects into account whereas the latter does not take inertia into consideration (refer also to Eq. (3.25). Applying a mathematical procedure similar to that described in the previous section, Eq. (3.44) results in the following weak form Z Z Z ∂ðδui Þ ρ€ ui δui dV + σij dV  ti δui dS ¼ 0 (3.45) ∂xj V

V

St

that may be transformed into the following matrix set of nonlinear equations at the instant of time t: ext Mu € t + Fint t ¼ Ft

(3.46)

In the aforementioned equation, the symbol M denotes the mass matrix, F ¼ Ku is the vector of internal forces resulting from the stiffness of the metal forming system and Fext is the external force vector. The nonlinear set of equations (3.46), associated with dynamic finite element formulations is commonly solved by means of an explicit time integration scheme that do not check equilibrium at each increment of time and, therefore, do not minimise the residual force vector. As a result of this, computer programs do not experience convergence problems, but their overall level of accuracy can be poor in terms of both geometry and distribution of the major field variables. An important topic related to dynamic formulations is the stability of the explicit time integration scheme that sets the maximum increment of time that is possible to be utilised in the time-stepping procedure of an incremental process. The following example addresses this topic by considering the analogy with a typical undamped mass-spring system. int

Example 3.8

The stability of the explicit time integration scheme of the nonlinear set of equations (3.46) is determined through the analysis of the undamped massspring system as shown in Fig. 3.17.

144

Metal forming

Applying Newton’s law to the mass m results in the following differential equation of motion for the mass-spring system: m x€ + kx ¼ 0 Selecting a cosine-based tentative solution, since the second derivative of a cosine is proportional to the negative of the same cosine, xðt Þ ¼ A cos ðωt Þ and putting this solution into the differential equation of motion:   md2 ðAcos ðωt ÞÞ + kA cos ðωt Þ ¼ mω2 + k ðA cos ðωt ÞÞ ¼ 0 2 dt The solution of the above equation is rffiffiffiffi   k 2 mω + k ¼ 0 ) ω ¼ m pffiffiffiffiffiffiffiffi meaning that the system will vibrate with a frequency (in rad/s) ω ¼ k=m, which we understand as the oscillation frequency of a single d.o.f. massspring system. The analogy with the rod subjected to elastic deformation that is included in Fig. 3.17 is made by writing the mass m as m ¼ ρπ R2 L and by relating the force vs displacement F ¼ k x evolution with the stress vs strain σ ¼ E ε evolution, F ¼ kx ,

F L x ¼k 2 πR2 πR L

or L ε πR2 The relationship discussed earlier allows expressing the spring constant k as a function of the elasticity modulus E and of the radius R and length L of the rod as σ ¼ Eε ¼ k

EπR2 L Substitution of the spring constant in the equation of the frequency gives rffiffiffiffi pffiffiffiffiffiffiffiffi E=ρ k L ω¼ ¼ ) t ¼ pffiffiffiffiffiffiffiffi m L E=ρ k¼

Finite element simulation: A user's perspective

145

pffiffiffiffiffiffiffiffi where t is the period. Since E=ρ ¼ ce where ce is the dilatational wave speed (speed of sound) in the rod material, the previous equation can be written as follows: t¼

L ce

In finite element applications, L ¼ Le is the characteristic element length, which can be taken as the minimum distance between any two nodes of an element, and the maximum time increment of the explicit time integration scheme is obtained from Δt  Le/ce. This result is known as the Courant stability condition and allows concluding that explicit dynamic formulations require very small increments of time Δt (e.g. μ-seconds). Since metal forming processes take several seconds to be finished, it follows that a typical finite simulation would require millions of increments to be finished if no modifications in Eq. (3.46) are made.

F

x

m

A

x

k R

m

x 0

σ

-kx

L

m

k E

ε 0

Fig. 3.17 Mass-spring system and analogy with a rod subjected to elastic deformation.

Example 3.9

Let us consider the deep drawing operation of a steel sheet. The blank is discretised by means of quadrilateral shell elements with 1 mm side length and the modelling conditions are summarised in Table 3.2 (adapted from Tekkaya, 2000).

146

Metal forming

The overall process time is given by T¼

H 300  103 m ¼ ¼ 0:6 s vp 0:5 m=s

The maximum time increment of the explicit time integration scheme obtained from the Courant stability condition is obtained from Δt ¼

Le 1 mm ¼ 2  107 s ¼ ce 5  106 mm=s

Dividing the total process time by the maximum time increment of the explicit time integration scheme gives an estimate of the total number of increments N performed by the explicit dynamic finite element computer program N¼

T 0:6s ¼ 3  106 ¼ Δt 2  107 s

This is a huge number of increments and justifies the modification of Eq. (3.46) to make it more efficient. After this example, readers will be introduced to mass scaling, which is a procedure commonly used to increase the maximum time increment Δt in order to decrease the total number of increments N. A small note is left warning readers about the fact that mass scaling of unstructured meshes with elements having different pside ffiffiffiffiffiffiffiffilengths Le requires that for a specified increment of time Δt ¼ Le = E=ρ, there will be a different value of density per element due to the nonlinear relation between the element side length Le and density ρ.

Table 3.2 Modelling conditions of Example 3.9.

Element side length Le (mm)

1

Punch velocity vp (m/s)

0.5

Cup height H (mm)

300

Speed of sound ce (m/s)

5000

Finite element simulation: A user's perspective

147

Explicit integration using the central difference method There are two main procedures to increase the maximum time increment Δt of the explicit time integration scheme and to reduce the overall CPU time: (a) Mass scaling, generally accomplished by increasing the density of the material ρ and artificially reducing the speed ce of the longitudinal wave, (b) Load factoring, generally accomplished by changing the rate of loading through an artificial increase in the velocity of the tooling as compared to the real forming velocity. This second approach is not allowed whenever the material is strain-rate sensitive or thermomechanical phenomena are involved. Because these two procedures can artificially add undesirable inertia effects, it is usual to add an extra damping term C in Eq. (3.46) and transform the differential equation of motion into that of a mass-spring-damped system: ext Mu € t + C u_ t + Fint t ¼ Ft

(3.47)

Integration of the dynamic equilibrium equations resulting from Eq. (3.47) is carried out by an explicit time integration scheme based on central difference approximations derived from the Taylor series expansions of the displacements u: f 0 ðaÞ f 00 ðaÞ (3.48) ðx  aÞ + ðx  aÞ2 + ⋯ 1 2 The following time discretisation scheme can be defined from Eq. (3.48) (refer also to Fig. 3.18): f ðxÞ ¼ f ðaÞ +

Δt 2 +⋯ 2 Δt2 uðt  Δt Þ ¼ utΔt ¼ ut  u_ t Δt + u €t +⋯ 2 ut + Δt  utΔt ut + Δt  2ut + utΔt u €t ¼ u_ t ¼ 2Δt ðΔtÞ2 €t uðt + ΔtÞ ¼ ut + Δt ¼ ut + u_ t Δt + u

(3.49)

Substituting the velocity and acceleration given in Eq. (3.49) into Eq. (3.47), the following solution is obtained for calculating the displacements u at time t + Δt, " # 1 1 1 M½2ut  utΔt  C ut + Δt ¼ Fext  Fint + 2M+ 2Δt ðΔtÞ ðΔt Þ2 1 + (3.50) CutΔt 2Δt

148

Metal forming

Displacement t +Δ t

u

t

u

t - Δt

Slope

u

Δt

.t

u

Δt Time t

t - Δt

t

t +Δ t

Fig. 3.18 Graphical illustration of the central difference approximation scheme.

The designation ‘explicit time integration’ is now easy to understand because the solution for time t + Δt is only dependent on the displacements of the previous known states of equilibrium at times t and t  Δt. A typical computer implementation is illustrated in Fig. 3.19). In connection to this, it is important to distinguish between the use of the terms ‘implicit or explicit’ in the sense that an iterative procedure is employed or not in each simulation increment to fulfil the residual of the equilibrium equations from the use of the same terms in the context of time integration. In fact, the use of the terms ‘implicit or explicit’ in the context of time integration simply means that the solution for time t + Δt depends on the displacements at time t + Δt or is solely dependent on the displacements at previous time steps t and t  Δt. To conclude it is worth mentioning that if the mass M and damping C matrices in Eq. (3.50) are diagonalised (‘lumped’), its inversion is straightforward. The system of differential equations uncouples, and the overall solution can be performed independently and very fast for each degree of freedom. The utilisation of reduced integration schemes that often are even applied to the deviatoric parts of the stiffness matrix and the resort to numerical actions related to mass scaling and load factoring also account for strong reductions of the computation time. Further details on the computer implementation of explicit dynamic formulations are given in Tekkaya (2000).

Finite element simulation: A user's perspective

149

Incremental Process

.... –1

ut + Dt =

1

(Dt)2

M+

1

2Dt

C

Fext – Fint +

1

(Dt)2

M[2ut – ut – Dt ] +

1

2Dt

Cut – Dt

int = Fint (u Ft+Dt t+Dt ) ext = Fext (u Ft+Dt t+Dt )

....

Fig. 3.19 Implementation of dynamic explicit formulations.

Pros and cons of explicit dynamic formulations The two main advantages of dynamic explicit formulations are as follows: (a) The computer programs are robust and do not present convergence problems, (b) The computation time depends linearly on the number of degrees of freedom whilst in alternative quasi-static implicit formulations, it depends more than linearly (in case of iterative solvers) and up to quadratically (in the case of direct solvers). (a) (b) (c) (d)

(e)

The main drawbacks can be listed as follows: The utilisation of very small increments of time per step is required, The equilibrium after each increment of time is not checked, The assignment of the system damping is rather arbitrary, Experienced users for adequately designing the mesh and choosing the scaling parameters for mass, velocity, and damping are needed. Otherwise, it may lead to inaccurate solutions for the deformation, prediction of forming defects and distribution of the major field variables within the workpiece, Spring back calculations are very time-consuming and may lead to errors. This specific problem is frequently overtaken by combining dynamic explicit with quasi-static implicit analysis.

Table 3.3 presents a summary of the main characteristics of the finite element formulations commonly utilised in the numerical simulation of industrial metal forming processes.

Table 3.3 Finite element formulations and computer implementations utilised in metal forming. Quasi-static formulations Flow formulation

Solid formulation

Dynamic formulation

Equilibrium equation

Quasi-static

Quasi-static

Dynamic

Constitutive equations

Rigid-plastic/ viscoplastic

Elastic-plastic/viscoplastic

Elastic-plastic/viscoplastic

Main structure

Stiffness matrix and force vector

Stiffness matrix and force vector

Mass and damping matrices plus internal and external force vectors

Minimisation of the residual force at each incremental step

Yes

Yes

No

Size of the incremental step

Medium

Medium to large

Very small

CPU time per incremental step

Medium

Medium to long

Very short

Explicit

Implicit

Explicit

Accuracy of the results (stress and strain distributions)

Medium to high

High

Medium

Spring back and residual stresses

No

Yes

Yes/No

Abaqus (implicit), Marc, Superform, Autoform

Abaqus (explicit), Dyna3D, PamStamp

Time integration scheme

a

Commercial FEM computer programs a

b

b

Forge , Deform , QForm, Eesy-Form

In the sense that algorithms do not need values of the next time step to compute the solution. Also have solid formulation options. Adapted from Tekkaya, A.E., Martins, P.A.F., 2009. Accuracy, reliability and validity of finite element analysis in metal forming: a user’s perspective. Eng. Comput. 26, 1026–1055. b

Finite element simulation: A user's perspective

151

3.7 Errors in finite element analysis Basic sources of errors in the finite element analysis of metal forming processes can be classified into three different groups: (a) Modelling errors, (b) Numerical errors, (c) Other errors. Awareness and understanding of these errors is the key for users to improve the overall accuracy and reliability of their simulation estimates obtained by the utilisation of commercial finite element programs.

3.7.1 Modelling errors Modelling errors are directly related to the physics of the processes, and therefore, avoiding or minimising these errors requires a deep insight into the deformation mechanics of the metal forming processes. In what follows, modelling errors will be subdivided into different classes and comprehensively analysed. Analysis type A critical source of errors in the numerical simulation of metal forming processes by means of commercial finite element programs is the wrong selection of the type of analysis. Simulations must be performed using the large displacement/large strain available options because not using them will lead to overly stiff material behaviour and, therefore, to unexpectedly high stresses at given strains, or equivalently, unexpectedly low strains at given forces. Another typical source of numerical errors is related to not activating the coupled thermomechanical option. Not using this option will lead to a constant temperature field within the workpiece as can be easily recognised by the users during postprocessing. Material models One of the first choices after selecting the type of analysis is related to the selection of the material model. Models utilised in the numerical simulation of metal forming processes can be rigid-plastic, rigid-viscoplastic, elasticplastic or elastic-viscoplastic. Computer programs based on rigid-plastic/viscoplastic material models (refer to the ‘flow formulation’ in Table 3.3) simplify the analysis by neglecting the elastic deformation of the workpiece. Therefore, they are not able to properly compute the residual stresses as well as the elastic spring back

152

Metal forming

(although some computer programs claim being capable of performing the elastic unloading of the workpiece at the end of rigid-plastic or rigidviscoplastic calculations, which is only strictly correct if loading was proportional). They may also experience difficulties in calculating the strains and stresses in zones of workpieces that behave quasi-rigidly as a result of using a pseudoelastic constitutive law. The computer programs based on elastic-plastic/viscoplastic material models (refer to the ‘solid formulation’ and ‘dynamic formulation’ in Table 3.3) do not have the early mentioned disadvantages. However, computer programs based on the solid formulation are slower than those built upon the flow formulation. Computer programs based on the dynamic formulation are faster and more robust at the price of their results being less accurate and also having difficulties in calculating spring back and residual stresses. There are two different approaches for implementing elastic-plastic material models: hypo- and hyperelastic plastic. For small, isotropic, strains, the results of these two approaches do not differ significantly, but for large strains and for modelling kinematic hardening or general anisotropic hardening behaviour, even for small strains, the approach is not strictly correct because the additive decomposition of the rate of deformation tensor D into elastic and plastic parts is not valid (refer to Appendix B) D 6¼ De + Dp

(3.51)

Hyperelastic plastic approaches are thermodynamically consistent and are based on the multiplicative decomposition of the deformation gradient F into elastic and plastic parts, F ¼ Fe Fp

(3.52)

Kinematic hardening, or more generally, induced anisotropy, has gained attention in the last few years in the simulation of metal forming processes. One fundamental reason is that most of the metal forming processes are multistage processes with different loading paths in each stage. This brings to light the question of using the correct material model to predict material flow, forming load and final properties such as residual stresses and hardness correctly. Fig. 3.20 shows the computed and experimental evolution of the force vs displacement for the cogging of a bar with 90 degrees rotation after the first blow (corresponding to 25% height reduction). As shown, there are significant deviations between calculations and experiments originated by the intermediate 90 degrees turning of the workpiece. This problem, which

Finite element simulation: A user's perspective

153

187.5 170.7 154.0 137.3 120.5 103.8 87.1 70.3 53.6 36.9

(A)

(B)

120 100

1st Blow 2nd Blow

Force (kN)

80 60 40 20

Experimental FEM

0 0

2

4

6

8

10

12

Displacement (mm)

(C) Fig. 3.20 Cogging of a technically pure (99.95%) aluminium bar with an intermediate 90 degrees turning. (A) Geometry and effective stress (MPa) at the end of the first and second blows; (B) experimental parts corresponding to (A); (C) computed and experimental evolution of the force vs displacement.

is frequently ignored by users of finite element computer programs, cannot be understood in terms of plastic behaviour under conditions of unidirectional loading (e.g. tensile or compression testing), but can be well explained in terms of the influence of the strain path on material behaviour (Alves et al., 2003b). Fundamental physical phenomenon Another key decision is related to the dimensions of the physical problem. The possibilities span from two-dimensional (plane stress, plane strain or axisymmetric) to three-dimensional models and decision must bring into consideration savings in computation time against possible modelling errors derived from simplifications of the physical phenomenon. Fig. 3.21 shows the computed and experimental evolution of the force vs displacement for the cylindrical side pressing of a rod made of technically pure aluminium (99.95%). The numerical predictions were obtained by the

154

Metal forming

1200

Force (kN)

1000 800 600 400 FEM 2D (Plane Strain)

200

FEM 3D Experimental

0 0

2

4

6

8

10

12

14

16

18

20

Displacement (mm)

Fig. 3.21 Computed and experimental evolution of the force vs displacement for the cylindrical side pressing of a technically pure aluminium rod. The computed results were obtained by means of two- and three-dimensional finite element models.

utilisation of three-dimensional and simplified two-dimensional plane strain finite element models and put evidence into the influence of the model in the overall accuracy of the results. The reduction in computation time resulting from the utilisation of a simplified two-dimensional model is up to 94%. Good practice suggests that plane strain models can be utilised whenever very long preforms are to be considered, such as in the case of thin plate rolling, whereas plane stress models can be utilised whenever thin preforms without through thickness stresses are to be considered, such as in a tensile test of a flat sheet for instance. The implementation of axisymmetric models requires rotational symmetry conditions in the geometry, boundary conditions, applied loading and material properties. The last requirement makes axisymmetric models unsuitable for the numerical simulation of sheet metal forming processes in which planar anisotropy effects will need to be taken into consideration. The assignment of symmetry conditions also plays a key role in settingup finite element models because their application reduces the overall size of the models and significantly improves the numerical stability of the computations. Fig. 3.22A shows an example resulting from the assignment of a 90 degree angle symmetry condition in the case of the forging of a gear that leads to much faster computation than that resulting from the utilisation of a full model. Alternatively, the assignment of an 11.25 degrees angle symmetry could also have been applied. However, users must be aware of the typical

Finite element simulation: A user's perspective

155

1.71 1.52 1.33 1.14 0.95 0.76 0.57 0.38 0.19 0

(A)

(B) Fig. 3.22 Assignment of symmetry conditions in finite element models. (A) Successful and effective assignment of symmetry conditions in the forging of a gear; (B) unsuccessful assignment of an axisymmetric symmetry condition in the heading of a slender rod. (Adapted from Alves, M.L., Rodrigues, J.M.C., Martins, P.A.F., 2003b. Simulation of three-dimensional bulk forming processes by the finite element flow formulation. Model. Simul. Mater. Sci. Eng. 11, 803–821.)

sources of errors that are commonly derived from the utilisation of finite element models equipped with symmetry reductions: (a) Computation of the total force directly from the reduced symmetric model, (b) Application of a symmetry condition that is violated due to the material model (e.g. anisotropy), applied loading or instability (Fig. 3.22B). Another important decision of a finite element user has to do with the selection between an isothermal and a coupled thermomechanical analysis. This choice can affect simulation conditions drastically in terms of both the computation time and the need for input material data. The simulation of

156

Metal forming

hot or warm forming is usually performed as thermomechanically coupled although, for certain hot forming processes, an isothermal analysis at a constant elevated temperature may be sufficient. The simulation of cold forming is commonly performed as isothermal, but it is worth to notice that cold forming may heat up materials by several hundred degree Celsius affecting mechanical and tribological properties of the material and tooling. In what concerns the selection between an isothermal or a coupled thermomechanical analysis, it is important to understand that the exchange of heat between workpiece and tooling is greatly influenced by the contact time under pressure. In the case of a hydraulic press, the contact time under pressure can be up to 40 times longer than in a drop hammer and, as a result of this, the account of the exchange of heat is much more relevant if the forming process is to be performed in an hydraulic press instead of a drop hammer. The interaction between workpiece and tooling requires users to select the most appropriate procedure for modelling the tools during the settingup of finite element simulations. Tools deform elastically but are commonly modelled as rigid because their deformations are negligible when compared to the plastic deformations of the workpieces. Although this is plausible in many applications, it can also be responsible for supplying wrong results. An example of the latter is provided in Fig. 3.23 where only by modelling the elastic deformation of the dies, it is possible for finite element estimates to match the experimental force-displacement evolution during the cold forward extrusion of a phosphated and soap-coated 20MnB4 steel (annealed). The utilisation of rigid tools is seen as responsible for deviations up to 20% against experimental measurements. Limitations in terms of the overall size and complexity of the finite element models together with increasing computation requirements due to the need of constantly updating the contact interface between the workpiece and tools often lead users to leave the elastic analysis of tooling out of the simulation procedure and only to be performed at the postprocessing stage. This procedure is known as ‘decoupled (or noncoupled) elastic analysis of tooling’ or ‘loose coupled elastic analysis of tooling’. However, it is important to notice that the decoupled elastic analysis of tooling is mainly suitable for providing estimates of the deformations and applied pressures for those metal forming problems in which the elastic deformation of the tool has a trivial influence on the overall size of the deformed workpieces. The exact procedure for performing the elastic analysis of tooling requires numerical coupling between finite element modelling of the workpiece and elastic deformation of the tools.

Finite element simulation: A user's perspective

157

120 FE rigid die

Force (kN)

FE elastic die 80 Experimental

40

0

4

8 12 Displacement (mm)

16

20

Fig. 3.23 Modelling the elastic deformation of a die in order to properly account for the experimental force-displacement evolution curve of a cold forward extrusion process. (Adapted from Tekkaya, A.E., Martins, P.A.F., 2009. Accuracy, reliability and validity of finite element analysis in metal forming: a user’s perspective. Eng. Comput. 26, 1026–1055.)

A final issue in setting-up finite element models is related to the choice between quasi-static and dynamic formulations. This subject was already analysed, and readers are advised to refer to Table 3.3 for information on basic guidelines. Still, it is worth noticing that typical velocities of metal forming processes allow inertial forces to be neglected and numerical analysis to be performed statically. Even in metal forming processes performed in drop hammers, where there is a large impact of the tool, inertia forces can be ignored. Only special processes such as shot peening, explosive and electromagnetic forming need to include inertia forces into consideration for a more accurate analysis. Material data The second aspect directly related to material models is the set of material data to be inserted into finite element computer programs. From a metal forming point of view, the most important material data is the flow curve because it characterises the strain-hardening behaviour. In the case of cold forming processes, the flow curve at room temperature is usually sufficient; it should be available up to an equivalent plastic strain of about 3, in the case of bulk forming; and up to 1, in the case of sheet metal forming. However, in most of the cases, characterisation of the flow curve for bulk forming is

158

Metal forming

performed by means of tensile tests up to a level of equivalent strain equal to 0.2–0.3, or by means of compression tests up to a level of equivalent strain equal to 0.8–1.0. In the case of sheet metal forming, upsetting is not feasible (unless being performed by means of stack compression specimens), and basically, the hydraulic bulge test is utilised delivering flow curves up to 0.6 equivalent plastic strain. However, it should be noticed that if the material exhibits normal anisotropy considerably different from 1.0, replacement of tensile testing with bulge testing will lead to errors since the bulge test in reality tests the material in the thickness direction. As a result of this, there is a generalised practice (and important source of modelling errors) of extrapolating the remaining part of the flow curve. Fig. 3.24 shows the deviation between the Bridgman’s corrected flow curve after necking and the extrapolation of the remaining part of the flow curve based on a Ludwik-Hollomon approximation of the experimental stressstrain data before necking. The material is stainless steel X5CrNiMo 17-12-2 (DIN), and the tensile test was performed at room temperature (Koc¸aker, 2003). For warm and hot forming processes, the flow curve is a function of not only the strain but also the strain rate and temperature. As a result of this, flow curves are much more difficult to obtain and may sometimes present considerable deviations (up to 40%) when provided from different sources (Arfman, 2004). The need to perform material characterisation for higher values of strain and for levels of strain rate above those currently attained by means of conventional universal (tensile and compression) testing machines requires the utilisation of torsion testing machines, drop weight testing machines, and Hopkinson bars (Fig. 3.25). Besides the flow curve, the yield locus for sheet metal forming represents other crucial data. Especially for aluminium alloys and high strength steels, the quadratic yield loci are not appropriate, so that complicated tests must be conducted to determine the nonquadratic yield locus of the material. Ductile damage Currently available models for ductile damage are expected to successfully predict the onset of cracking for different stress and strain states (internal vs surface cracks). There are two main approaches for analysing ductile damage in metal forming. The first approach commonly referred to as the

Finite element simulation: A user's perspective

159

σ 1600 Corrected flow curve after necking (Bridgman) 1200

Extrapolated from experimental data

800

400

0

Experimental data

0.4

0.8

1.2

1.6

ε

Fig. 3.24 Comparison between corrected and extrapolated stress-strain curves (MPa) illustrating the deviations after necking. (Adapted from Koc¸aker, B., 2003. Product Properties Prediction After Forming Process Sequence (M.Sc. Thesis). The Middle East Technical University, Ankara.)

Fig. 3.25 Conditions of similitude between mechanical testing and mechanical processing of materials. (Adapted from Chastel, Y., 1999. Approaches scientifiques des proc
ed
es de mise en forme des m
etaux—Rh
eologie des m
etaux et essais rh
eologiques. CEMEF, Sophia Antipolis (in French); Silva, C.M.A., Rosa, P.A.R., Martins, P.A.F., 2016. Innovative testing machines and methodologies for the mechanical characterization of materials. Exp. Tech. 40, 569–581.)

160

Metal forming

‘uncoupled damage approach’ assumes material to be continuous and isotropic throughout the deformation and makes use of the conventional constitutive equations for elastic-plastic or rigid-plastic behaviour. Ductile damage criteria based on uncoupled damage are built upon weighted integrations of plastic strain increments and the weighting functions are generally taken as nondimensional and stress-state-dependent. They are simple to implement in computer programs because they neglect the influence of damage in the yield surface (the influence of damage is calculated at the postprocessing stage) and critical values of damage and other parameters to be identified are relatively easy to be determined. The second approach is named ‘coupled damage approach’ and includes softening effects due to damage in material deformation that will eventually influence the final strain values at the onset of fracture. The dependence of material deformation on damage complicates the implementation in finite element computer programs as well as the determination of the unknown parameters to be identified. The ductile damage criteria built upon coupled damage approaches can be derived using two different continuum damage mechanic routes: (a) A microbased route, built upon the theory of plasticity of porous materials (Gurson, 1977; Tvergaard, 1981; Tvergaard and Needleman, 1984) that make direct use of the void volume fraction fv (or porosity) as an internal variable to represent damage and its softening effect on material strength, (b) A macrobased route, in which a continuum damage variable built upon a thermodynamic dissipation potential is introduced for accomplishing the same objective (Kachanov, 1958; Lemaitre, 1985). Fig. 3.26 shows finite element estimates of the load-displacement evolution provided by a numerical model that does not take ductile damage into account and by means of an alternative model based on a coupled damage mechanics approach (Soyarslan et al., 2008). As expected, the differences between the two curves are only significant after necking because the latter is capable of taking into account the opening, growth and coalescence of voids to form a crack near the centre of the neck of the tensile bar whilst the former is only capable of taking into account the geometrical effect derived from the reduction of area in the region of the neck. The chapter on formability provides detailed information regarding ductile damage and fracture.

Finite element simulation: A user's perspective

161

100

Force (kN)

80

No damage

60 Damage coupled 40

20

0

2

4 6 Displacement (mm)

8

Fig. 3.26 Modelling load-displacement evolution during a tensile test using a coupled damage approach and comparison with the estimate provided by means of finite element analysis without damage. (Adapted from Soyarslan, C., Tekkaya, A.E., Aky€ uz, U., 2008. Application of continuum damage mechanics in crack propagation problems: forward extrusion chevron predictions. J. Appl. Math. Mech. 88, 436–453.)

Friction and heat transfer Knowledge about friction and lubrication is still limited in metal forming. The friction model usually adopted in finite element modelling is either Amontons’ law (also known as Coloumb’s law, although Amontons has suggested it much earlier): τ ¼ μq

(3.53)

where τ is the frictional shear stress, q is the interfacial normal pressure and μ is the coefficient of friction or the law of constant friction: τ ¼ mk

(3.54)

where k is the shear flow stress and m is the friction factor. However, the aforementioned friction models are not satisfactory compared to the advanced and detailed studies that are possible to be carried out by currently available finite element computer programs. In the simulation of bulk metal forming processes, the use of Amontons’ law gives occasion for an overestimation of the friction stresses at the tool-workpiece contact interface. When the coefficients of friction are large or the contact pressures are

162

Metal forming

extreme, the friction stresses can easily become greater than the yield stress of the material in pure shear. In case of sheet metal forming processes, the law of constant friction generally overestimates the friction stresses at low normal pressures because the friction does not depend on the current state of stress at the tool-workpiece contact interface, but simply on a material property. The utilisation of the Bay and Wanheim’s (1976) general friction model, when available, allows much better modelling conditions because it assumes friction to be proportional to the normal stress at low normal pressures q/σ Y < 1.5, but going towards a constant value at high normal pressures q/σ Y > 3, the two ranges being combined via an intermediate transition region. A great number of tests have been suggested in the literature for modelling the tribological conditions at the tool-workpiece interface (Schey, 1983), but the determination of the friction coefficient μ or the friction factor m is rather complicated, since they depend on the contact pressure, the surface area expansion, the relative sliding velocity and the interface temperature, amongst other factors (Bay, 2002). One of the well-known standard tests to determine the friction data is the ring compression test. However, this test supplies only friction data for the given contact pressure and surface expansion valid for this test. Since modelling and quantification of friction by means of simple models such as Amontons’ law and the law of constant friction is questionable, friction coefficients or factors are in many cases tuned by the users during the numerical simulation in order to provide good estimates of the forming loads and of the deformed shape of the workpiece. A similar problem exists with the heat transfer coefficient, which is also depending on the contact pressure and interface temperature, amongst other factors. Description of tooling One further source of error in finite element modelling of metal forming processes is the mathematical description of the tool surfaces (Fig. 3.27). In most of the commercial finite element computer programs, the surfaces are described by means of surface meshes (e.g. triangular elements). The utilisation of a grid of triangular elements instead of alternative approaches based on clouds of points, analytical functions or parametric surfaces is due to the fact that the former always guarantees the successful discretisation of the surfaces whilst the other techniques often face difficulties whenever complex shapes and/or small geometrical details are to be

Finite element simulation: A user's perspective

163

Fig. 3.27 Schematic representation of the different methodologies that can be utilised in the discretisation of tooling in metal forming (Santos and Makinouchi, 1995). (A) Analytical functions or parametric surfaces; (B) surface meshes; and (C) clouds of points.

discretised. However, triangular elements fail to ensure smoothness and, therefore, introduce artificial roughness on the surface of tooling. This can bring in geometrical errors in case of small fillet radii, for instance. Another common source of error in the description of tooling arises from the mismatches between the engineering drawings of tools and the real dimensions of these. Machine tools The final source of errors to be presented in this section is related to the modelling of the kinematics of the tools. For drop hammers, energy-based

Metal forming

164

2500

2000

Force (kN)

2nd

1500

1000 1st

500

0

10

20

30

40

50

100

300

600

900

1200

Accumulated contact time (ms)

Fig. 3.28 Interaction between process and machine tools during finite element analysis of the upsetting of an aluminium AA1100 billet using (i) a gravity drop hammer with a ram mass of 5000 kg and a useful energy per blow (2 blows) of approximately 26 kJ and (ii) a hydraulic press with a force rating of 2000 kN.

kinematics of the tool motion is necessary, whereas for mechanical presses, a displacement-dependent force characteristic must be used. The simplest forming machines to model are the hydraulic presses. Fig. 3.28 shows the computed evolution of the forming load with time for the upsetting of an AA1100 cylindrical billet obtained from a finite element simulation coupled with an appropriate kinematic description of a drop hammer and of a hydraulic press. Two blows are necessary for deforming the cylindrical billet when using the drop hammer. At each blow load builds up and deformation proceeds until the useful available energy of the machine is totally dissipated. Since the required deformation of the cylindrical billet is attained after 24 ms, the remaining energy is to be transmitted to the tools, the foundations and the machine structure. This phenomenon raises the issue of modelling the whole machine instead of modelling just the tool because the press may distort leading to the generation of moments. For precision forging or generally net-shaped forming, this kind of distortion can affect the processes considerably. The resulting contact time under pressure ranges from less than 30 ms for the gravity drop hammer (the shortest time) to around 1000 ms for the hydraulic press. This information is relevant for modelling warm and hot forming processes, for which the strain rates and cooling times have a significant influence on the flow behaviour of the material. In fact, this knowledge is fundamental for understanding the heat transfer mechanisms between the workpiece and the tools as well as to select the most adequate lubrication system.

Finite element simulation: A user's perspective

165

3.7.2 Numerical errors The numerical errors arise from the mathematical nature of the simulation algorithms. Avoiding or minimising these errors requires an insight into the underlying finite element theories and computer implementation strategies. Selection of elements Discussion around the selection of elements for performing two- and threedimensional modelling of metal forming processes is an open field of research, discussion, and controversy. In general terms, the most important factors that need to be taken into consideration by the users during the selection of an element type are the following: (a) Performance in large deformation analysis, (b) Availability of reduced or selectively reduced integration techniques, (c) Availability of meshing and remeshing automatic generators, (d) Number, distribution and overall quality of the resulting finite element mesh. In what concerns three-dimensional modelling of bulk metal forming processes, the choice of elements is often to be made between tetrahedral and hexahedral solid elements. Tetrahedral elements are geometrically more versatile and easier to handle in automatic meshing and remeshing of complex forming shapes than hexahedral elements. However, standard tetrahedral elements cannot perform accurately in the numerical simulation of metal forming processes, because the metal incompressibility constraint results in volumetric locking. Although the utilisation of second-order tetrahedral elements may seem a good option, due to the geometrical complexity of bulk metal forming systems (workpiece and tooling), it is also not recommended. In fact, second-order tetrahedral elements perform poorly in the tool-workpiece contact interfaces because of the manner distributed forces and surface pressures are transmitted through its nodes. This problem often leads to stability problems in the contact algorithms. As a result of this, special tetrahedral elements with interior nodes and equipped with reduced or selectively reduced integration techniques that prevent locking at high levels of plastic strains are commonly made available to the users. However, these elements still suffer from some of the typical drawbacks of tetrahedral elements: (a) They are overly stiff and very sensitive to mesh orientation (Fig. 3.29), (b) They frequently require up to an order of magnitude more elements to achieve the same level of accuracy as alternative hexahedral elements

166

Metal forming

Fig. 3.29 Discretisation of a cylindrical billet by means of 1800 hexahedral and tetrahedral elements. The irregular pattern of the tetrahedral mesh is expected to provide nonsymmetrical distributions of the field variables.

since a tetrahedral element can always be seen as a degenerated hexahedral element. The aforementioned drawbacks result in higher computation times and turn hexahedra into a preferred choice for the discretisation of threedimensional bulk metal forming parts. There are, however, two important limitations related to the selection of hexahedral elements: (a) Hexahedral elements do not conform to arbitrary volumes and prescribed surfaces as good as tetrahedral elements. Its widespread utilisation has been, so far, limited by robustness and automatisation difficulties, (b) Automatic unstructured meshing and remeshing of arbitrarily volumes with all well-shaped hexahedral elements is still an open field of research. Some commercial packages do not yet have the facility for this type of preprocessing although developments are being reported in the open literature since the mid 1990s. A similar discussion can now be performed regarding the selection of elements for modelling sheet metal forming processes. The choice is usually between membrane and shell elements that are the so-called structural elements. Structural elements bear the property that certain field variables are integrated over the smallest dimension, here the thickness of the sheets. Basic guidelines can be summarised as follows: (a) Membrane elements are simple to implement and to utilise. However, they are limited in terms of applications because bending effects are not considered. Their main advantage results from the fact that they

Finite element simulation: A user's perspective

167

frequently require 5–20 times less computation time to achieve the same level of accuracy as shell elements; (b) Shell elements take bending effects into consideration but are often considered the ‘prima donna’ of finite elements because they can be very sensitive to small changes in geometry and in the boundary conditions. Thick shell elements account for the variations of some field variables through integration points within the thickness of the forming parts whereas thin shell elements do not. Because membrane and shell elements do not allow variation of shear stresses across the thickness and present difficulties in modelling contact with friction on the die and on the blank holder independently, developers and users are showing growing interest in the utilisation of solid elements (tetrahedral and hexahedral). Solid elements not only account the variation of shear stresses across the thickness as they can easily handle both frictional contact interfaces separately. Unfortunately, the availability of computer programs offering solid elements for sheet metal forming applications is still limited. A compromise between solid and shell elements for linear and nonlinear analyses is the utilisation of ‘solid-shell’ elements (Hauptmann and Schweizerhof, 1998). Discretisation and mesh convergence After having selected the correct type of element, it is crucial to analyse the number, distribution and overall quality of the elements that were utilised in the discretisation of the workpiece. This must be performed considering the geometry of the workpiece in conjunction with the field variables and process parameters that are to be investigated in a particular metal forming analysis. Good practice suggests that users must avoid finite element models containing acute and obtuse angles and distorted elements because these are unconformities that considerably affect the overall accuracy of the simulation results. Small elements are to be applied near small geometrical features or in the regions where abrupt changes in the solution are expected whilst larger elements are sufficient in the uninteresting parts of domain. In finite element modelling, a finer mesh typically results in a more accurate solution. However, as a mesh is made finer, the computation time increases. This leads users being confronted with the fundamental question of how to specify a mesh that satisfactorily balances accuracy and computing resources. The answer to this question can easily be found in mesh convergence studies. Users are advised to start their analysis by creating a mesh using the fewest, reasonable number of elements, analyse the process and subsequently

168

Metal forming

Fig. 3.30 Mesh convergence in the finite element compression of a cylinder with friction. Results show the hydrostatic stress (p ¼  σ m) at the midpoint of the centreline and the relative CPU time as a function of the number of elements.

plot the field variable (strain, stress, etc.) or the parameters (force, deflection, etc.) under investigation against the number of elements (or against the number of nodes). They should then recreate the mesh with a denser element distribution, reanalyse the process and compare results with those of the previous mesh (Fig. 3.30). By keeping increasing the mesh density and reanalysing the model, users should be able to notice when results start converging. This will enable obtaining an accurate solution with a mesh that is sufficiently dense and not overly demanding of computing resources. Unfortunately, mesh convergence studies are rarely conducted in industrial applications due to users being short of time. Nevertheless, it is worth to notice that although material flow patterns can be little affected by mesh discretisation, residual stresses are highly influenced by element sizes and therefore, no residual stress analysis should be trusted unless a convergence study is performed. Time step and convergence criteria Discretisation in time is a multiple-aspect ranking source of errors where the goal is to utilise the largest possible increment of time Δt per step whilst preserving the overall accuracy of the simulation results. Despite some

Finite element simulation: A user's perspective

169

commercial programs offer the so-called automatic procedures for setting up the increment of time, it is essential for users to understand that the appropriate procedure must take into consideration a wide range of aspects such as the type of finite element formulation, contact conditions, volume accuracy and hardening behaviour of the material. In general terms, the quasi-static formulations can handle much larger Δt than explicit dynamic formulations. Nevertheless, current practice shows that the size of the increment of time in case of quasi-static formulations is usually below theoretical expectancy due to material and geometrical nonlinearities related to the flow rule and contact. In what concerns the influence of the increment of time Δt on volume constancy (Fig. 3.31), it is recommended that users undertake assessment by means of convergence studies with respect to changing time steps. It is worth to notice that this problem only affects Eulerian time integration schemes such as that utilised in the flow formulation. The solid formulations do not suffer from this problem. Convergence criteria are other important sources of numerical errors because they are responsible for defining the maximum error admissible in the solution of the nonlinear system of equations that forms the basis of the computational implementation of the finite element method. As

Fig. 3.31 Volume loss vs reduction in height in the finite element compression of a bar using the flow formulation with different values Δh/h of the increment size per step.

170

Metal forming

shown in Section 3.6.1, the criteria can be set up in terms of velocity and residual norms but appropriate setting-up of its values for achieving good finite element results has need of convergence studies regarding these limits.

Force

Other factors Other important sources of numerical errors can be attributed to the selection of penalty factors for variational contact algorithms or incompressibility constraints, time increments for direct (also known as linear) contact algorithms, procedures for discretisation of tooling, convergence criteria for iterative solvers, friction algorithms, remeshing procedures and output parameters. Fig. 3.32 schematically illustrates how poor contact boundary conditions resulting from an inappropriate selection of a penalty factor may lead to wrong estimates of the forming load. It is worth to notice that a similar result could be attained if a too large time step had been utilised in conjunction with a direct contact algorithm.

Mesh B

Mesh B Mesh A

Mesh A

Displacement

Fig. 3.32 Schematic representation of a wrong estimate of the forming load due to numerical errors related with the selection of penalty factors for contact algorithms.

3.7.3 Other errors There are other sources of errors besides the modelling and numerical errors. The following list provides several examples: (a) Wrong utilisation of commercial finite element programs mainly due to misunderstanding the manuals and/or misunderstanding the scope of the underlying theories, (b) Hazards of commercial finite element programs related to bugs or theoretical/numerical mistakes imbedded in the source code,

Finite element simulation: A user's perspective

171

(c) Wrong interpretation of finite element results due to erroneous representation in the postprocessor, due to inadequate exchange of data with a third-party postprocessor software or due to lack of knowledge and experience of the user, (d) Hardware errors (e.g. input/output operations on storage media).

3.8 Validation of finite element procedures Validation of finite element procedures is crucial for understanding how accurate and reliable are the computer estimates provided by the numerical simulation of a specific metal forming process. In general terms, validation is concerned with the numerical and physical appraisal of the conceptualised finite element models and with the discharge of users and/or applied computer programs. The methodologies to be utilised for this purpose can be divided into two different categories, process-independent and processdependent, and both will be comprehensively discussed in what follows.

3.8.1 Process-independent validation procedures Process-independent validation procedures comprise checking material flow and distribution of stresses in the regions where boundary or symmetry conditions are prescribed and performing convergence studies of selected field and process variables with respect to changing numerical and physical simulation parameters. Material flow The fulfilment of the prescribed boundary conditions is a trivial, but very important, validation of the material flow computed by means of finite element procedures. In fact, after finishing the computations (i.e. during postprocessing) all the nodal points with prescribed zero displacements or velocities in the conceptualised finite element model must remain at their individual initial locations. This sort of validation also applies for the nodes in contact with rigid bodies (such as tools) because they are not allowed to penetrate these bodies. Symmetry entities (such as axes or planes) also provide helpful validation routes because structured meshes with grids perpendicular to these entities will remain perpendicular as the deformation occurs. Fig. 3.33 provides an example derived from the finite element analysis of a forward rod extrusion. In the case of employing unstructured mesh models or making use of remeshing procedures, the aforementioned validation can only be

172

Metal forming

Fig. 3.33 Finite element estimate of the material flow pattern in forward rod extrusion. (Adapted from Tekkaya, A.E., Martins, P.A.F., 2009. Accuracy, reliability and validity of finite element analysis in metal forming: a user’s perspective. Eng. Comput. 26, 1026–1055.)

performed with the help of postprocessing packages having the ability to depict material flow patterns on a virtual uniform grid (much like physically etching a pattern on a cross section of a workpiece and having the ability to view it after being deformed). Stress field In what concerns the numerical validation of the stress field, users can check the vanishing of the normal and shear stresses on the free surfaces where no traction is specified as well as the absence of shear stresses at the symmetry entities of the conceptualised finite element model. It is worth noticing that the aforementioned conditions are not always simple to check because when nodal or elemental stresses are the output, the condition of zero stresses is not necessarily satisfied. A more accurate way to check this condition is to analyse the stresses at the centre of the elements (the most accurate stress locations of the elements) from the interior to the surface (or to the symmetry) of the workpiece. Extrapolating these stresses to the surface (or to the symmetry) should yield the expected zero values. The distribution of residual stresses can also be utilised for validating finite element procedures because they must be self-equilibrated (i.e. they must fulfil static equilibrium). Taking an extruded or drawn rod as an example, it follows that the axial residual stresses σz in the portion of the rod where the properties are steady state and, hence, no shear stresses are present, must fulfil the following condition: ZR σz r dr ¼ 0 0

where r is the radial coordinate and R is the outer radius of the rod.

(3.55)

Finite element simulation: A user's perspective

173

On the other hand, the tangential stresses σ θ must fulfil the condition: ZR σθ dr ¼ 0

(3.56)

0

Distribution of forces First checking of the computed distribution of forces can be performed quite straight forwardly if their sense of direction is compared against the physics of the metal forming process. In addition, users are advised to verify self-equilibrium of the computed distribution of forces in the absence of friction and applied externals forces or to check the value of the resultant force against forces and pressure applied on the boundaries of the workpieces.

Convergence studies Convergence tests help performing the numerical validation of the finite element procedures. In general terms, the convergence of the required field or process variable towards a constant value must be investigated by changing various numerical and physical simulation parameters. The parameters that must be controlled are the mesh size and topology, geometrical modelling of the tools, contact and friction modelling parameters, remeshing schemes and factors, increment of time per step, volume constancy, convergence criteria and limits and solver parameters, amongst others. In case of iterative solvers, parameters can greatly influence solution stability if high precision is required in the analysis of complex contact problems or in the computation of residual stresses. In general terms, it can be stated that the most important parameters to check by means of convergence studies are the mesh size and topology, convergence criteria and limits and the increment of time per step. However, there are other parameters not so commonly available to the users that can also significantly influence the overall quality of the results. One of these hidden parameters can be found in the mathematical treatment of the friction model and is due to stability problems caused by the abrupt changes in the friction shear stress at the vicinity of the neutral points. In fact, the friction laws are commonly implemented by means of alternative mathematical forms that ensure a smooth transition of the frictional stresses near the neutral points. This is performed by replacing the step function

174

Metal forming

τ N

ur N

τ N - Neutral Point

τ

τ

dτ dur

dτ dur

Fig. 3.34 Schematic comparison between the physical friction law and its numerical implementation in finite element computer programs.

(typical of friction laws near the neutral points) by an arc tangent as close to the step as desired (refer to Fig. 3.34). In the case of the law of constant friction, this is achieved as shown in the following:    ur 2 |ur | ur τf ¼ mk ffi mk (3.57) arctan |ur | |ur | π a where ur is the relative sliding velocity at the contact interface between material and tooling and ‘a’ is an arbitrary constant much smaller than the relative velocity. The critical hidden parameter is ‘a’ (regulating the slope of the curve at zero relative velocity ur ¼ 0) and may have a significant influence in the overall frictional behaviour. In what regards volume constancy, users must be aware that this requirement is only applicable for the plastic part of the deformation meaning that, in case of finite element formulations using elastic-plastic constitutive equations this validation route will not be truly accurate from a pure theoretical point of view. However, because the elastic part of deformation is very small compared

Finite element simulation: A user's perspective

175

with the large deformations that are typical of metal forming processes, the validation procedure will always make sense in current modelling practice. Reasons for the loss of volume are usually derived from numerical problems due to incorrect choices of the increment of time per step (in the case of applications making use of the flow formulation), to difficulties related to the contact algorithms, to inappropriate penalty factors used for the incompressibility constraint or to an excessive number of low performance remeshing procedures. Some of the aforementioned issues were already considered in the previous sections of this chapter and no further discussion will be made.

3.8.2 Process-dependent validation procedures Process-dependent validation procedures comprise checking finite element results against theoretical solutions and technical data available in literature or against experimental and industrial observations and measurements. Both topics will be discussed in what follows. Validation against theoretical solutions and technical data available in literature In most cases, users are not able to verify their finite element estimates against experimental or industrial data. However, it is quite often possible to check their overall numerical estimates against similar or related solutions available in the literature. The aim should be to check whether the magnitude of the forming load and the distribution of the main field variables (stress, strain, temperature, etc.) make sense. Fig. 3.35 shows the computed and experimental evolution of the force vs displacement for the cold forward extrusion of a rod made from a steel St32-3 (DIN) annealed. The approximate value of the maximum extrusion force can be estimated by means of the ideal work method as follows: Fmax ¼ pe A0 ffi 207 kN

(3.58)

where pe is the extrusion pressure and A0 is the initial cross sectional area of the rod. The comparison between finite element predictions and ideal work method calculations account for an error of approximately 30% but may be enough for verification purposes. And, if users would like to have more enhanced comparisons, they could always replace the ideal work method by more sophisticated analytical methods such as the slab or the upper-bound methods. The overall trend of the computed evolution of the force vs displacement can also be checked against the typical force-displacement characteristics of the

176

Metal forming

180 160

Load (kN)

140 120 100 80 60 40

Experimental

20

Finite Element Method

0 0

5

10

15

20

25

30

35

40

Displacement (mm)

Fig. 3.35 Force vs displacement evolution during cold forward extrusion of a cylindrical rod with 15 mm diameter for 60% reduction obtained from experimentation and calculated by the finite element method.

metal forming processes under investigation. Typical force-displacement curves for standard metal forming processes can be found in the specialised literature. In what concerns the distribution of temperature, a simple and efficient method of validation consists of comparing finite element estimates with the average maximum increase of temperature provided by means of an analytical adiabatic heat balance calculation ΔT ¼

ασavg εf ρc

(3.59)

where σavg is the average value of the flow stress over the strain interval 0 to εf , ρ is the density, c is the mass heat capacity and α is the fraction of energy stored (0.8–0.95). The strain field can also be roughly checked by comparing finite element estimates of the effective strain against the maximum analytical strain value derived from a resembling ideal deformation process. In case of the forward rod extrusion depicted in Fig. 3.35, this can be achieved by calculating   εf ¼ ln A0 =Af , which is the exact strain value at the centreline of the extruded rod in the steady-state region. Checking process limits can also be utilised to validate finite element procedures. For instance, interface contact pressures larger than 2000–2500 MPa are not plausible due to the limiting strength of materials in tool construction. Large contact pressures may therefore indicate either a wrong modelling

Finite element simulation: A user's perspective

177

procedure or a correct modelling procedure that will lead to a die failure in practice. Limiting values of strength depend on the materials, heat and surface treatments, and design and information can be found in the specialised literature. Another useful check is provided by means of existing formability data, as, for example, the forming limit curve in sheet metal forming processes, which can be utilised for evaluating the simulation results and for assessing how critical the results are for the overall feasibility of the process. Validation against experimental and industrial observations and measurements This type of validation can only be performed after manufacturing the metal forming parts. However, it is strongly recommended to be followed by the developers of finite element programs and by the users in a company or in a research centre to verify if an existing program is reliable and correctly being applied. It can also be utilised for analysing the relative performance of different finite element computer programs and to help deciding between renewing a software licence or taking the offer of another software vendor. Several experimental techniques can be utilised to accomplish this type of validation and comprise the utilisation of the following: (a) Load cells, piezoelectric force transducers, pressure transducers and displacements transducers to register the evolution of the force with the displacement, (b) Strain measurements on dies and punches give indications of the loads on the tools and hence the forming forces that can be compared with the computed ones, (c) Coordinate measuring machines, shadowgraphs and related metrology equipment to obtain the geometry of the formed parts that will be compared against finite element results, (d) Visioplasticity or physical modelling techniques using soft model materials to obtain the experimental distribution of the velocity and strains, (e) Grid analysis (circle or rectangular) techniques to obtain the experimental distribution of strains, (f ) Microhardness equipment to get the experimental distribution of hardness and indirectly obtain the distribution of effective strain and stress, (g) Thermocouples and thermographic equipment to register the experimental distribution of temperature. Finally, validation can be performed by comparing finite element estimates against the so-called benchmark processes. Typical benchmark

178

Metal forming

processes are provided by IDDRG (The International Deep Drawing Research Group) for sheet metal forming applications and by the Process Simulation Subgroup of ICFG (The International Cold Forging Group) for bulk metal forming applications. One typical benchmark process suggested by ICFG is the production of the pipe screw shown in Fig. 3.36 (Tekkaya, 2003). The flow curve and the CAD data are provided by the ICFG and the experimental evolution of the force vs displacement is given in Fig. 3.36.

1.OP

2.OP

3.OP

4.OP

5.OP

Material: Q St 32-3 (DIN 1.0303) Initial Volume: ~ 905 mm3

(A) Forming stroke

FDC

58 kN 4. OP

130 kN 3. OP 31 kN 2. OP 33 kN 1. OP

0.025 s

(B)

Courtesy Hatebur, Reinach

Fig. 3.36 Benchmark test case suggested by the Process Simulation Subgroup of the ICFG. (A) Forming sequence and (B) force-displacement curve.

Finite element simulation: A user's perspective

179

References Alves, M.L., Fernandes, J.L.M., Rodrigues, J.M.C., Martins, P.A.F., 2003a. Finite element remeshing in metal forming using hexahedral elements. J. Mater. Process. Technol. 141, 395–403. Alves, M.L., Rodrigues, J.M.C., Martins, P.A.F., 2003b. Simulation of three-dimensional bulk forming processes by the finite element flow formulation. Model. Simul. Mater. Sci. Eng. 11, 803–821. Arfman, G., 2004. The variability of material property measurements. In: The 4th ICFGWorkshop on Process Simulation in Metal Forming Industry, Shangai. Bay, N., 2002. Modelling of friction in cold metal forming processes. In: 2nd Workshop on Process Simulation in Metal Forming Industry, Padova. Bay, N., Wanheim, T., 1976. Real area of contact and friction stress at high pressure sliding contact. Wear 38, 201–209. Blacker, T.D., Meyers, R.J., 1993. Seams and wedges in plastering: a 3D hexahedral mesh generation algorithm. Eng. Comput. 2, 83–93. Cornfield, G.C., Johnson, R.H., 1973. Theoretical prediction of plastic flow in hot rolling including the effect of various temperature distributions. J. Iron Steel Inst. 211, 567–573. Dunne, F., Petrinic, N., 2005. Introduction to Computational Plasticity. Oxford, New York. Gurson, A., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rules for porous ductile media. Trans. ASME J. Eng. Mater. Technol. 99, 2–15. Hauptmann, R., Schweizerhof, K., 1998. A systematic development of ’solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom. Int. J. Numer. Methods Eng. 42, 49–69. Hibbitt, H.D., Marcal, P.V., Rice, J.R., 1970. A finite element formulation for problems of large strain and large displacement. Int. J. Solids Struct. 6, 1069–1086. Honecker, A., Mattiasson, K., 1989. Finite element procedures for 3D sheet forming simulation. In: NUMIFORM’89-3rd International Conference on Numerical Methods in Industrial Forming Processes, Fort Collins. Kachanov, L., 1958. On creep rupture time. Proc. Acad. Sci. USSR Div. Eng. Sci. 8, 2631. Knupp, P., 1998. Next-generation sweep tool: a method for generating all-hex meshes on two-and-one-half dimensional geometries. In: 7th International Meshing Roundtable, Dearborn, pp. 505–513. Koc¸aker, B., 2003. Product Properties Prediction After Forming Process Sequence (M.Sc. Thesis). The Middle East Technical University, Ankara. Lee, C.H., Kobayashi, S., 1973. New solutions to rigid-plastic deformation problems using a matrix method. Trans. ASME J. Eng. Ind. 95, 865–873. Lemaitre, J., 1985. Continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107, 83–89. Martins, P.A.F., Marques, M.J.M.B., 1992. Model 3—a three-dimensional mesh generator. Comput. Struct. 42, 511–529. Mattiasson, K., 2010. FE-models of the sheet metal forming processes. In: Banabic, D. (Ed.), Sheet Metal Forming Processes: Constitutive Modelling and Numerical Simulation. Springer, Berlin. McMeeking, R.M., Rice, J.R., 1975. Finite-element formulations for problems of large elastic-plastic deformation. Int. J. Solids Struct. 10, 601–616. Nielsen, C.V., Fernandes, J.L.M., Martins, P.A.F., 2013. All-hexahedral meshing and remeshing for multi-object manufacturing applications. Comput. Aided Des. 45, 911–922. Santos, A., Makinouchi, A., 1995. Contact strategies to deal with different tool descriptions in static explicit FEM for 3-D sheet metal forming simulations. J. Mater. Process. Technol. 50, 277–291.

180

Metal forming

Schey, J.A., 1983. Tribology in Metalworking: Friction, Lubrication and Wear. ASM International, Materials Park. Schneiders, R., B€ unten, R., 1995. Automatic generation of hexahedral finite element meshes. Comput. Aided Geom. Des. 12, 693–707. Soyarslan, C., Tekkaya, A.E., Aky€ uz, U., 2008. Application of continuum damage mechanics in crack propagation problems: forward extrusion chevron predictions. J. Appl. Math. Mech. 88, 436–453. Staten, M.L., Canann, S.A., Owen, S.J., 1998. BMSWEEP: locating interior nodes during sweeping. In: 7th International Meshing Roundtable, Dearborn, pp. 7–18. Tekkaya, A.E., 1986. Determination of Residual stresses in Cold Bulk Metal Forming. Berichte aus dem Institut f€ ur Umformtechnik, University of Stuttgart, Springer. Tekkaya, A.E., 2000. Simulation of metal forming processes. In: Banabic, D. (Ed.), Formability of Metallic Materials. Springer, Berlin. Tekkaya, A.E., 2003. Process simulation in forging: a report from the ICFG subgroup on simulation. In: ICFG/NACFG International Cold Forging Conference, Columbus. Tekkaya, A.E., 2007. Master of Engineering in Applied Computational Mechanics (ESoCAET). Atilim University, Ankara. Tekkaya, A.E., Martins, P.A.F., 2009. Accuracy, reliability and validity of finite element analysis in metal forming: a user’s perspective. Eng. Comput. 26, 1026–1055. Tvergaard, V., 1981. Influence of voids on shear bands instabilities under plane strain conditions. Int. J. Fract. 17, 389–407. Tvergaard, V., Needleman, A., 1984. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, 157–169. Zienkiewicz, O.C., Godbole, P.N., 1974. Flow of plastic and viscoplastic solids with special reference to extrusion and forming processes. Int. J. Numer. Methods Eng. 8, 3–16. Zienkiewicz, O.C., Phillips, D.V., 1971. An automatic mesh generation scheme for plane and curved surfaces by isoparametric coordinates. Int. J. Numer. Methods Eng. 3, 519–528.

CHAPTER 4

Finite element flow formulation* 4.1 Introduction The finite element flow formulation was originally developed by Lee and Kobayashi (1973), Cornfield and Johnson (1973), and Zienkiewicz and Godbole (1974) with the aim of simulating metal forming processes. During the 1980s, the flow formulation was primarily set up for modelling two-dimensional bulk forming processes and such efforts gave rise to the development of a first generation of computer programmes with applicability limited to plane strain and axisymmetric conditions. Even so, authors such as Altan and Knoerr (1992) were able to report case studies in which the two-dimensional constraint was ingeniously stretched out to obtain useful information regarding three-dimensional metal forming applications. In order to extend applicability of the flow formulation to modelling conditions involving more than the mechanical behaviour alone, a thermal model was introduced to simulate thermo-mechanical metal forming processes. The first attempt to handle a coupled thermo-mechanical metal forming process was made by Zienkiewicz et al. (1978a) who used a finite element iterative procedure to solve the material flow for a given distribution of temperature, in conjunction with heat transfer, during plane strain extrusion. Later, Zienkiewicz et al. (1978b, 1981) modified the procedure to obtain the temperature distribution within the workpiece simultaneously with the solution of the velocity field. The approach, commonly known as ‘strong coupled thermo-mechanical’, was applied to solve steady-state extrusion and rolling. Strong coupled thermo-mechanical finite element algorithms were further developed by Rebelo and Kobayashi (1980a,b) to allow the numerical simulation of nonsteady-state metal forming processes. The technique was applied to solid cylinder and ring compression testing. Developments in computers and reduction in the associated computational costs over several decades have stimulated the extension of availability * Chris V. Nielsen (Technical University of Denmark, Lyngby, Denmark) and Paulo A.F. Martins (IDMEC, Instituto Superior T
ecnico, University of Lisbon, Portugal). Metal Forming https://doi.org/10.1016/B978-0-323-85255-5.00008-X

© 2021 Elsevier Inc. All rights reserved.

181

182

Metal forming

and effectiveness of finite element computer programmes based on the flow formulation to simulate three-dimensional metal forming processes. As a result of these efforts, complex metal forming processes are now being simulated with high accuracy without the need to use geometrical and material flow simplifications. Most of the scientific and numerical ingredients that are necessary to develop a finite element computer programme based on the flow formulation are available in the literature but there is a need to understand how this knowledge can be integrated to develop a computer programme. This chapter is focused on this need and is aimed at eliminating the gap between theory and numerical implementation.

4.2 Theoretical fundamentals The chapter starts by exposing the readers to three different views of the flow formulation: (i) the solid mechanics view based on the governing quasi-static equilibrium equations, (ii) the energy (variational principle) view based on the second extremum principle of plasticity and (iii) the fluid mechanics view based on the momentum balance equation. Then, readers are introduced to the discretisation procedures, solution techniques, contact algorithms and thermal and electrical coupling, amongst other topics that are crucial for developing a finite element computer programme based on the finite element flow formulation.

4.2.1 Quasi-static equilibrium—A solid mechanics view Looking at the finite element flow formulation from a solid mechanics point of view, the starting point is the governing quasi-static equilibrium equations in current configuration (in the absence of body forces), ∂σ ij ¼0 ∂xj

∂σ ¼ rσ ¼ 0 ∂x

(4.1)

where r ¼ ∂/∂ x is the gradient operator. There are two important notes that should now be transmitted to the readers. Firstly, in several sections of this chapter, we will make use of both matrix (bold symbols) and tensor (index) notations for pedagogical reasons. Secondly, the velocity will be denoted by the symbol u (or u) in order to be able to utilise the symbol v (or v) later in the presentation for the velocity at elemental level. This choice of symbols is different from that utilised in the

Finite element flow formulation

183

previous chapter where the symbol u (or u) was exclusively utilised for displacements. The Galerkin form of the weighted residual method allows writing an integral form of equation (4.1), which instead of satisfying the quasi-static equilibrium requirements exactly (that is, pointwise) will only fulfil them in an average sense over the entire domain (volume V, Fig. 4.1), ð ð ∂σ ij ∂σ δui dV ¼ 0 δudV ¼ 0 (4.2) ∂xj ∂x V

V

The weighting function δui (or δu) denotes an arbitrary variation in the velocity because the flow formulation is set up in terms of velocities. Applying the chain rule of derivatives to the integral in Eq. (4.2) results in ð ð ð ð  ∂ðδui Þ ∂  ∂ðδuÞ ∂ σ ij dV  σ ij δui dV ¼ 0 σ: dV  ðσδuÞdV ¼ 0 ∂xj ∂xj ∂x ∂x V

V

V

V

(4.3)

which can be further modified by the application of the divergence theorem to convert the second volume integral into a surface integral over the essential boundary conditions along the surface St with outward normal nj (or n)

Z

u

S

velocities

V Y

X

t

tractions

Fig. 4.1 Notation used in the static equilibrium of a body with a volume V and total surface area S.

Metal forming

184

ð V

ð ∂ðδui Þ σ ij dV  σ ij nj δui dS ¼ 0 ∂xj

ð V

St

ð ∂ðδuÞ σ: dV  σnδudS ¼ 0 ∂x St

(4.4) Alternatively, Eq. (4.4) can be written as    ð ð 1 ∂ðδui Þ ∂ δuj dV  ti δui dS ¼ 0 σ ij + 2 ∂xj ∂xi V St   ! ð ð 1 ∂ðδuÞ ∂ðδuÞ T dV  tδudS ¼ 0 σ: + 2 ∂x ∂x V

(4.5)

St

The symbol ti ¼ σ ijnj (or t ¼ σ n) denotes the tractions applied on the boundary St with prescribed tractions and a normal with a vector of direction cosines given by nj (or n). Eq. (4.5) is named the ‘weak form’ of equation (4.1) because the governing quasi-static equilibrium equations are now only satisfied in weaker continuity requirements because only the stress field is required to be continuous and not anymore its derivative. Considering the rate of deformation tensor D ¼ (1/2) (L + LT), where the velocity gradient tensor L ¼ ∂ u/∂ x (Lij ¼ ∂ui/∂ xj), the weak form of the quasi-static equilibrium equations can be expressed as follows: ð ð ð ð σ ij δDij dV  ti δui dS ¼ 0 σ : δDdV  tδudS ¼ 0 (4.6) V

St

V

St

where the rate of deformation tensor D ¼ Dp is equal to the plastic rate of deformation tensor because the flow formulation does not account for elastic deformation (hereafter the symbol D will be exclusively used for the plastic rate of deformation tensor, which will also be simply designated as the rate of deformation tensor). Using the decomposition of the Cauchy stress tensor σ into a deviatoric tensor σ0 related to shape change and a hydrostatic tensor σH related to volume change, Eq. (4.6) becomes  ð ð 1 0 σ ij + δij σ kk δDij dV  ti δui dS ¼ 0 3 V St  ð ð (4.7) 1 0 σ + TrðσÞI : δDdV  tδudS ¼ 0 3 V

St

Finite element flow formulation

185

where the hydrostatic tensor is given by σH ¼ δijσ kk/3 ¼ (1/3) (Tr(σ)) I, and the symbol δij denotes the Kronecker delta (and I denotes the identity matrix), ð ð ð 1 0 σ ij δDij dV + δij σ kk δDij dV  ti δui dS ¼ 0 3 V Vð ð ð St (4.8) 0 σ : δDdV + σH : δDdV  tδudS ¼ 0 V

V

St

An alternative expression for the weak form of the governing equilibrium equations (4.8) in tensor notation is given by ð ð ð 0 σ ij δDij dV + σ m δDv dV  ti δui dS ¼ 0 (4.9) V

V

St

where Dv ¼ δijDij ¼ Dii is the volumetric rate of deformation and σ m ¼ σ kk/3 is the hydrostatic (or mean) stress. Replacing ε_ v ¼ Dv in the aforementioned equation and taking into consideration that the increment of plastic work rate (plastic power) per unit of volume δw_ p is given by δw_ p ¼ σ 0ij δDij ¼ σ δε_

(4.10)

because the flow formulation neglects elastic effects, Eq. (4.9) becomes ð ð ð _ σ δεdV + σ m δ_ε v dV  ti δui dS ¼ 0 (4.11) V

V

St

The second term in Eq. (4.11) is equal to zero because of volume constancy. To conclude, we will add a second zero term to Eq. (4.11) related to the variation of the hydrostatic stress σ m for a reason that will be understood later in the presentation ð ð ð ð _ σ δεdV + σ m δ_ε v dV + δσ m ε_ v dV  ti δui dS ¼ 0 (4.12) V

V

V

St

There are three important facts that need to be clarified concerning the (plastic) rate of deformation tensor D: (a) Firstly, it is important to understand that the usual limitations of small strains and/or small rotations that complicate most strain calculations do not apply to the rate of deformation tensor. The rate of deformation

186

Metal forming

tensor is not limited by any of these issues because, as an operator on velocity rather than displacement, a rate of deformation is a calculation over an infinitesimal time step. During this time step, the workpiece orientation and level of strain only changes by an infinitesimal amount, and since this amount of change is not finite, it does not complicate matters in any way. So the rate of deformation tensor is always applicable regardless of the level of strain or rotation (McGinty, 2020). (b) Secondly, it is important to remember that the rate of deformation tensor D is an objective (frame indifferent) measure of rate of deformation because like the Cauchy stress tensor it rotates according to the general transformation rule D0 ¼ R D RT of second order tensors, where R is the rotation matrix. (c) Finally, although the rate of deformation tensor cannot be additively decomposed in elastic and plastic tensors D 6¼ De + Dp due to the usual limitations of small strains and/or small rotations, it can be proved that if the elastic strains are neglected D ¼ Dp (Dunne and Petrinic, 2005) (refer to Appendix A of Chapter 3). These three facts facilitate the introduction of the rate form of the LevyMises constitutive equations in the weak form of the governing quasi-static equilibrium equations, as required by the finite element flow formulation.

4.2.2 Second extremum principle—An energy view Rates of energy The first law of thermodynamics imposes the conservation of energy and it does so in the rate form by stating that the net sum of the rates at which energy is transferred amongst different forms is zero. Table 4.1 lists all the Table 4.1 The relevant rates of energy. Forms of energy

Rates of energy (J/s 5 W)

Internal energy

d dt V

Kinetic energy

d 1 2 dt V 2 ρu dV

Internal forces

d dt V

Surface tractions Heat generation Heat flux

Ð Ð Ð

ρw dV ¼

Ð

V

¼ Ð

Ð

ρ w_ dV w_ ¼ dw dt V

ρaudV a ¼ du dt

f sdV ¼ V f udV u ¼ ds dt Ð Ð d ds dt S tsdS ¼ S tudS u ¼ dt  Ð Ð Ð d _ _ V Q dV dt ¼ V Q dV dt  Ð Ð Ð d S qndS dt ¼ S qndS dt

Finite element flow formulation

187

relevant rates of energy by differentiating the corresponding forms of energy with respect to time (McGinty, 2020). In Table 4.1, the symbol ρ denotes the density (a scalar), w is the internal energy per unit of volume (a scalar), s is the displacement vector, u is the velocity vector, a is the acceleration vector, f is the body force vector, t is the traction vector, Q_ is the heat generation rate per unit of volume (a scalar), q is the heat flux vector, n is unit normal vector to the surface of the volume, dV is the differential volume element, dS is the differential surface element of the volume and dt is the differential time increment. The principle of conservation of energy states that “the time rate of change of the kinetic plus the internal energy is equal to the sum of the rate of work of all the other energies supplied to, or removed from the volume per unit of time.” The supplied energies may include thermal energy, chemical energy or electromagnetic energy but in the following equation, which takes the form of the well-known first law of thermodynamics, only the mechanical and thermal energies are considered: ð ð ð ð ð ð ρaudV + ρ w_ dV ¼ f udV + tudS + Q_ dV  qndS (4.13) V

V

V

S

V

S

The first term in the left-hand side of Eq. (4.13) is the rate form of kinetic energy and the second term is the rate form of internal energy. The righthand-side terms of Eq. (4.13) are the rate form of energy due to body forces, surface forces, heat generation and heat flux. The last term in Eq. (4.13) has a negative sign because the energy is flowing out of the volume. Applying the divergence theorem to Eq. (4.13) and taking into account that t ¼ σ n, we obtain ð ð ð ð ð ð ρaudV + ρ w_ dV ¼ f udV + r  ðuσ ÞdV + Q_ dV  rqdV V

V

V

V

V

V

(4.14) Now applying the chain rule of derivatives to the right-hand-side term derived from surface tractions, ð ð ð r  ðuσÞdV ¼ u  ðrσÞdV + ru : σdV (4.15) V

V

V

and manipulating the second term of the right-hand side of Eq. (4.13),

188

Metal forming

ð

ð

ð

ð

ð

ru : σdV ¼ L : σdV ¼ σ : LdV ¼ σ : ðD + WÞdV ¼ σ : DdV V

V

V

V

V

(4.16)

Ð

because V σ : WdV ¼ 0, due to the fact that the Cauchy stress tensor is symmetric and the continuum spin tensor is antisymmetric, the following result is obtained: ð

ð

ρaudV + V

V

ð ð ð ð ð ρ w_ dV ¼ f udV + σ : DdV + ðrσÞudV + Q_ dV  rqdV V

ð

)



V

V

V ð V  ρ w_  σ : D  Q_ + rq dV ¼ ðrσ + f  ρaÞudV

V

V

(4.17)

This expression allows concluding that the right-hand side of Eq. (4.17) corresponds to the integral form of the linear momentum principle. The linear momentum principle states that “the time rate of change of an arbitrary portion of a volume is equal to the resultant force acting upon the considered portion” and allows concluding that in static equilibrium conditions, the right-hand side of Eq. (4.17) is equal to zero. This means that the internal energy increases as mechanical work is performed and heat is generated within the volume, but decreases as heat flows out. ρ w_  σ : D  Q_ + rq ¼ 0 ) ρ w_ ¼ σ : D + Q_  rq

(4.18)

The application of what was said earlier to the static equilibrium of a body with a volume V and total surface area S subjected to plastic deformation due to prescribed external tractions, and in the absence of heat generation, heat flux, body forces and inertia effects, allows modifying Eq. (4.13) by using ρ w_ ¼ σ : D and cancelling out aforementioned terms as ð ð σ : DdV ¼ tudS (4.19) V

S

Extremum principles The extremum principles consist of two theorems applicable to rigid-plastic materials that are commonly utilised to derive the lower and upper bound methods utilised in the analytical modelling of metal forming processes. The extremum principles are comprehensively described in Johnson and Mellor

Finite element flow formulation

189

(1973) and consider external power involving tractions applied on the surface S to be split into two parts: ð ð ð tu dS ¼ tu dS + tu dS (4.20) S

Su

St

where the first part is related to the surface portion Su with prescribed velocities and the second part is related to the surface portion St where tractions are applied (Fig. 4.2). Combining Eqs (4.19), (4.20), it follows that (Fig. 4.2) ð ð ð tudS ¼ σ : DdV  tudS (4.21) V

Su

St

Eq. (4.21) is the starting point for the first extremum principle which states that “when a body is yielding and small velocities are undergone, the rate of power done by the actual forces or surface tractions on Su is greater than, or equal to, that done by the surface tractions t∗ of any other statically admissible stress field σ∗,” ð ð tudS  t∗ udS (4.22) Su

Sv ∗

A statically admissible stress field σ is defined as one that fulfils the equilibrium equations and all the boundary conditions. The first extremum principle is the starting point for setting up the lower bound method.

Z

u velocities

Su

V St

X

Y

t = t*

prescribed tractions

Fig. 4.2 Notation used in the first extremum principle.

190

Metal forming

The second extremum principle states that “for any kinematically admissible velocity field u∗, the power done is always larger than the actual one, so that the forces computed are always larger than the actual ones” (Fig. 4.3). ð ð ð ð ∗ ∗ ∗ ∗ σ : D dV + kjΔu jdS  tu dS  tu dS (4.23) V

Sd

St

Su

In the aforementioned equation, k is the shear yield stress and j Δu∗j is the velocity discontinuity (a scalar) in the tangential component of the kinematically admissible velocity field u∗ along the discontinuity surface Sd. A kinematically admissible velocity field u∗ is defined as one that ensures volume constancy and fulfils all the boundary conditions. The upper bound method makes use of the second extremum principle to set up the following procedure to calculate the force applied by a tool Ftool to plastically deform a body: 0 1 ð ð ð 1 @ ∗ ∗ Ftool ¼ σ : D dV + kjΔu∗ jdS  tu∗ dSA (4.24) utool V

Sd

St

where utool is the tool velocity. Often, the upper bound method considers the body as rigid and deformation to occur by shear along several velocity discontinuities placed within the volume. The second extremum principle (4.23) can be also utilised to set up the finite element flow formulation. For this purpose, the volume is approximated by a summation of a finite number of elements, and velocity discontinuities are eliminated because the continuity requirement of the velocity field is inherent to the finite element method. By doing this and taking into account equation (4.10), Eq. (4.23) may be transformed into the following rate of energy variational statement (which Z

u = u*

Su

prescribed velocities

V Δu

St

X

t

Sd

Y

tractions

Fig. 4.3 Notation utilised in the second extremum principle.

Finite element flow formulation

191

has to be minimised) expressed in terms of an assumed kinematically admissible velocity field (Markov, 1947) (hereafter the superscripts ‘*’ will not be utilised): ð ð _ ΠðuÞ ¼ σ εdV  ti ui dS ) minimum (4.25) V

St

However, it is important to understand that Markov’s rate of energy variational statement (4.25) is based on a kinematically admissible velocity field that needs to additionally satisfy the incompressibility condition derived from the continuity equation when density ρ does not change with time: ∂ρ (4.26) + ðρui Þ, i ¼ 0 ! ui, i ¼ ε_ v ¼ 0 ∂t In fact, the requirement of a geometrically self-consistent velocity field that ensures the incompressibility condition reduces the family of possible velocity field guesses drastically. From a computational point of view, this is done through relaxation of the incompressibility condition on the velocity field and implies subjecting the rate of energy variational statement (4.25) to an additional constraint ε_ v ¼ 0 by means of a Lagrange multiplier λ: ð ð ð _ Πðu, λÞ ¼ σ εdV + λ ε_ v dV  ti ui dS ) minimum (4.27) V

V

St

Or, alternatively, by means of a penalty factor K (a large positive number): ð ð ð 1 2 _ ΠðuÞ ¼ σ εdV + K ε_ v dV  ti ui dS ) minimum (4.28) 2 V

V

St

Applying the stationary condition δ Π(u,λ) ¼ (∂ Π/∂u) δu + (∂ Π/∂ λ) δλ ¼ 0 to Eq. (4.27) the first variation of the functional Π(u,λ) becomes, ð ð ð ð _ σ δεdV + λδ_ε v dV + δλ ε_ v dV  ti δui dS ¼ 0 (4.29) V

V

V

St

A similar procedure can be done for the variation δ Π(u) ¼ 0 of the functional given by Eq. (4.28):

192

Metal forming

ð

ð ð _ σ δεdV + K ε_ v δ_ε v dV  ti δui dS

V

V

(4.30)

St

The main difference between the two stationary forms of Markov’s rate of energy variational statement just mentioned is that the Lagrange multiplier form increases the number of unknown variables of the finite element model whereas the penalty form keeps the original number of unknown variables on the expense of increasing ill-conditioning of the system of equations. Finally, it is important to understand that Markov’s rate of energy variational statement (4.25) could, alternatively, be rewritten as ð ð ð _ Πðu, σ m Þ ¼ σ εdV + σ m ε_ v dV  ti ui dS ) minimum (4.31) V

V

St

because the second term in the right-hand side is equal to zero. Observation of the variational statement in Eq. (4.31) leads to the following conclusions: (a) The comparison of Eqs (4.27), (4.28) with Eq. (4.31) provides the physical meaning of the Lagrange multiplier and penalty function λ¼

K ε_ v ¼ σ m 2

(4.32)

(b) Applying the condition δ Π(u) ¼ 0 to Eq. (4.31) provides a result identical to that of Eq. (4.12) ð ð ð ð _ δΠ ¼ σ δεdV + σ m δ_ε v dV + δσ m ε_ v dV  ti δui dS ¼ 0 (4.33) V

V

V

St

and allows concluding that the weak form of the governing quasi-static equilibrium equations is identical to the minimisation of the rate of energy variational statement. (c) Eqs (4.27), (4.31) involve simultaneous solution of velocity and Lagrange multiplier (hydrostatic stress or pressure) and is the link with the fluid dynamics view of the finite element flow formulation that will be considered in the next section.

4.2.3 Momentum balance equation—A fluid dynamics view Viscous fluids From the fluid mechanics point of view, all matter consists of basically two states, fluid and solid. The difference between these two states of matter is related to the reaction to applied shear (or tangential) stresses. A solid can

Finite element flow formulation

193

resist shear stresses by static deformation whereas a fluid cannot. Any shear stress applied to a fluid, no matter how small, will always result in motion (Fig. 4.4A). In fact, a fluid moves and deforms continuously as long as the shear stress is applied and only stays at rest in a state of zero shear stress. Constitutive equations for fluids generally assume that the shear stress is proportional to the velocity gradient, or in general that the deviatoric stress is proportional to the velocity gradient τ¼μ

dθ du ∂ui ¼ μ or σ 0ij ¼ μ ∂xj dt dy

(4.34)

where μ is the (shear) coefficient of viscosity. As shown in Fig. 4.4B, the relation between the shear stress and the velocity gradient may be linear (Newtonian fluids) or nonlinear (nonNewtonian fluids). In metal forming, we are particularly interested in a special type of non-Newtonian fluids named ‘viscoplastic fluids’ that require a finite yield stress σ Y before they begin to flow and have a decreasing resistance to shear with increasing values of stress, σ Y + γ ðT , pÞ ε_ m (4.35) 3 ε_ In this equation, ε_ is the strain rate, m is the strain rate sensitivity and γ is a material parameter to be determined that is dependent on temperature T and hydrostatic pressure p. The deviatoric stress can be written as follows: μ¼

(A)

(B)

Fig. 4.4 Fluids. (A) Deformation of a fluid caused by shear stresses and (B) behaviour of three different types of viscous materials.

194

Metal forming

σ 0ij ¼ Cijpq D0pq

(4.36)

0 is the deviatoric rate of deformation tensor and Cijpq is a matrix where Dpq containing the viscosity coefficients. The equation previously mentioned resembles the Levy-Mises constitutive equation of metal plasticity that relates the deviatoric Cauchy stress tensor σ ij0 with the rate of deformation tensor Dpq.

Differential momentum balance equation The basic differential momentum balance equation for an infinitesimal element in a fluid flow is given by Du (4.37) Dt where ρ is the density, g is the acceleration of gravity vector, σ0 is the deviatoric Cauchy stress tensor and p is the hydrostatic pressure. The symbol Du/Dt in the right-hand side of Eq. (4.37) is the material derivative of the velocity u (i.e. the acceleration a) in an Eulerian framework and is calculated by rσ0  rp + ρg ¼ ρ

  Du du ∂u ¼ +u ¼ u_ + uðruÞ Dt dt ∂x

(4.38)

From a physical point of view, Eq. (4.37) means that the viscous force per unit of volume, plus the force from the hydrostatic pressure per unit of volume, plus the body force per unit of volume, is equal to inertia force (density times the acceleration) per unit of volume. Note that the gradient r u in the material derivative is performed with respect to x (actual configuration) and not to X (previous configuration). This copes with an Eulerian description of motion that concerns with a particular region of the space occupied by the volume and not to the motion of individual particles of the volume from their original position X to their actual position x. In the case of an incompressible flow, the material derivative becomes identical to the time derivative and Eq. (4.37) becomes the well-known Navier-Stokes equation: du rσ0  rp + ρg ¼ ρ (4.39) ) rσ0  rp + ρg ¼ ρa dt The Navier-Stokes equation allows understanding the viscoplastic flow of metals as analogous to that of incompressible fluids with a

Finite element flow formulation

195

very high viscosity in which the viscous-pressure (velocity–pressure) phenomena are linked and, therefore, need to be solved simultaneously. In case body and inertia forces are absent, flow is performed under quasistatic equilibrium conditions and equation (4.39) becomes identical to the governing quasi-static equilibrium equation (4.1) that were taken as the starting point of the solid mechanics view of the finite element flow formulation, rσ0  rp ¼ 0 ) rðσ0  pÞ ¼ 0 ) rðσ0 + σH Þ ¼ 0 ) rσ ¼ 0 (4.40) The fluid dynamics view allows understating the flow formulation as a mixed velocity–pressure (u-p) formulation with an inherent Eulerian description of motion. Some authors prefer using the term ‘updated Eulerian’ instead of ‘Eulerian’ (Mattiasson, 2010) due to the ability of the flow formulation to handle unsteady-state metal forming processes through the use of an explicit time integration scheme in which the new configuration at time t + Δt is obtained from the velocity at the known configuration at time t as follows: t +ðΔt

xt + Δt ¼ xt +

udt ffi xt + ut Δt

(4.41)

t

After obtaining the weak form of the governing quasi-static equilibrium equations (4.1), understanding the need to relax the incompressibility condition of the velocity field by adding an additional constraint ε_ v ¼ 0 through Lagrange multiplier or penalty forms, and after understanding the need to solve the velocity and the hydrostatic stress simultaneously under an Eulerian description of motion, it is time to proceed towards the discretisation of the finite element equations.

Example 4.1

In order to understand how variational principle problems subjected to constraints are used, let us consider the problem of determining the minimum of the function Π(x, y) ¼ 2 x2  2 xy + y2 + 18 x + 6 y subject to constraint x ¼ y on the variables (adapted from Zienkiewicz (1983)). Continued

196

Metal forming

Example 4.1—cont’d

The obvious algebraic approach to solve this problem is to replace x ¼ y in Π(x, y) Π ¼ 2x2  2x2 + x2 + 18x + 6x ¼ x2 + 24x from which we can easily find the minimum ∂Π ¼ 2x + 24 ¼ 0 ) x ¼ y ¼ 12 ∂x Unfortunately, this procedure cannot be implemented in finite element computer programmes and one alternative is to use the method of Lagrange multipliers. This is done by extending the function Π with an extra variable λ (named the Lagrange multiplier) by means of the following additional term (equal to zero) Πλ ¼ λ (x  y),

Π ¼ Π + Πλ ¼ 2x2  2xy + y2 + 18x + 6y + λðx  yÞ The minimum is now obtained by solving the following system of equations: 8 > ∂Π > > ¼ 4x  2y + 18 + λ ¼ 0 > > ∂x > > > > < ∂Π ¼ 2x + 2y + 6  λ ¼ 0 ) x ¼ y ¼ 12, λ ¼ 6 > ∂y > > > > > > > ∂Π > : ¼xy¼0 ∂λ Another alternative consists of penalising the constraint x  y ¼ 0 by extending the function Π(x, y) by means of an additional penalty term ΠΚ ¼ K (x  y)2, where K is a large positive number with the same meaning of K/2 in Eq. (4.28): Π ¼ Π + ΠΚ ¼ 2x2  2xy + y2 + 18x + 6y + K ðx  yÞ2 The advantage of this approach against the Lagrange multiplier method is that no extra variables are introduced. The disadvantage is that the solution becomes dependent on the value chosen for the penalty K (refer to the Table in the following): 8 ∂Π > > > ¼ 4x  2y + 18 + 2K ðx  yÞ ¼ 0 < ∂x > ∂Π > > : ¼ 2x + 2y + 6  2K ðx  yÞ ¼ 0 ∂y

Finite element flow formulation

K

x

y

0

12

15.0000000

1

12

13.5000000

10

12

12.2727273

100

12

12.0297030

1000

12

12.0029970

10,000

12

12.0003000

100,000

12

12.0000300

1,000,000

12

12.0000030

197

As shown, the penalty method is effective and less costly than the Lagrange multiplier method but it may provide a trivial result x ¼ y ¼ 0 (i.e. the solution ‘locks’) if the value of K becomes very large. In fact, when K is very large, the relative weight of the penalty term ΠK in the overall extended function Π becomes excessive and a trivial result is obtained: Π ¼ Π + ΠK ¼ 2x2  2xy + y2 + 18x + 6y + K ðx  yÞ2 ffi K ðx  yÞ2 8 > ∂Π > > > < ∂x ffi 2K ðx  yÞ ¼ 0 ) x ¼ y ¼ 0 for K ! ∞ > > ∂Π > > : ffi 2K ðx  yÞ ¼ 0 ∂y

4.3 Discretisation by finite elements The previous section showed that the finite element flow formulation is based on a set of integral equations having the velocity u and the hydrostatic stress σ m ¼ λ as primary unknowns ð ð ð ð _ σδεdV + λδ_ε v dV + δλ ε_ v dV  ti δui dS ¼ 0 (4.42) V

V

V

St

Or, having just the velocity u as primary unknown in case of using a penalty form for relaxing the incompressibility constraint ε_ v ¼ 0 of the velocity field, ð ð ð _ σδεdV + K ε_ v δ_ε v dV  ti δui dS (4.43) V

V

St

198

Metal forming

From a computer implementation point of view, the integrals in the system of Eq. (4.42) or (4.43) need to be approximated by a summation over a finite number of elements discretising the volume ð M X f ðx, uÞdV ffi f ðxn , vÞΔV m (4.44) m¼1

V

In the equation mentioned earlier, the symbols x and u denote two vectors containing the position and velocity in an arbitrary position whereas xn and v are two vectors containing the coordinates of the nodes and the corresponding velocities in an element ‘m’. Fig. 4.5 illustrates a three-dimensional domain subdivided (discretised) by hexahedral elements with eight nodes. This type of element will be hereafter utilised to provide the details regarding the implementation of the finite element integral equations at elemental level. In hexahedral elements, any arbitrary position x ¼ {x, y, z}T may be described in terms of the coordinates of the nodes xn ¼ {x1, y1, z1, .…, x8, y8, z8}T by means of an interpolation procedure that uses the shape functions Ni at the corresponding arbitrary local position in an element, x¼

8 X

Ni xi

y¼

i¼1

8 X i¼1

Ni yi

z¼

8 X

Ni zi

i¼1

1 Ni ¼ ð1 + ξi ξÞð1 + ηi ηÞð1 + ςi ςÞ 8 (4.45) 8 5

ζ 7

ξ z

η

4

6

F

Node n u zn

y

uyn

1

3 2

Node n

x

uxn

Fig. 4.5 Illustration of a three-dimensional finite element model composed of hexahedral elements with eight nodes. Each node has three degrees of freedom for the representation of velocity field v at elemental level and one degree of freedom for representation of scalar fields (e.g. temperature and electric potential).

Finite element flow formulation

199

where ξ, η, ς are the natural (local) coordinates of the hexahedral element and ξi, ηi, ςi take values +1 or 1 according to the local position of node i in Fig. 4.5. The hexahedral element provides three degrees of freedom in each node for the velocity components in a mechanical finite element model and one degree of freedom for the scalar fields of temperature and electric potential in thermal and electrical finite element models, respectively. Considering the element as isoparametric, the same shape functions Ni are used for both coordinates and velocities. This makes life much easier and allow describing the velocity u ¼ {ux, uy, uz}T in an arbitrary position within the element by interpolation from its nodal values v ¼ {ux1, uy1, uz1, …, ux8, uy8, uz8}T 8 8 8 X X X ux ¼ Ni uxi uy ¼ Ni uyi uz ¼ Ni uzi i¼1

i¼1

i¼1

8 9 ux1 > > > > > > > > > > > uy1 > > > > > > > 8 9 2 > 3> > > > > 0 0 ⋯ N 0 0 u u N > > > x 1 8 z1 > > > > < > = 6 < = 7 6 7 u ¼ uy ¼ 4 0 N1 0 ⋯ 0 N8 0 5 ⋯ ¼ NT v > > > > > > > : > ; > > > 0 0 N 1 ⋯ 0 0 N8 > ux8 > uz > > > > > > > > > > > > u y8 > > > > > > : > ; uz8

(4.46)

where N is the matrix containing the shape functions Ni in natural coordinates ξ, η, ς. The rate of deformation tensor D within the element is given by  T !  1 ∂u 1 ∂u T D¼ L+L ¼ + 2 2 ∂x ∂x

(4.47)

and can be related to the nodal velocity vector v by means of the rate of deformation matrix B

200

Metal forming

2∂ 6 ∂x 6 8 9 6 60 Dx > > 6 > > > > 6 > > > > 6 > > Dy > > 6 > > > > > > 6 > < Dz > = 60 6 ¼6 > > ∂ 2Dxy > > > 6 > 6 > > > > 6 > > 2Dyz > > 6 ∂y > > > > 6 > > > > : ; 6 60 2Dxz 6 6 4 ∂ ∂z

0 ∂ ∂y 0 ∂ ∂x ∂ ∂z 0

fDg ¼ Bv ¼ LNT v 3 0 7 8 9 7 ux1 > 7 > > > > > 07 > > 7 > > > u 7 > > y1 > > > 72 > > > 3> > > ∂7 > u > 7 N1 0 0 ⋯ N8 0 0 > z1 > > > < = 7 7 ∂z 7 6 6 7 7 4 0 N 1 0 ⋯ 0 N8 0 5 ⋯ > > 7 > > 07 0 0 N ⋯ 0 0 N > > ux8 > > > 7 1 8 > > > > > 7 > > > > 7 > > > > u ∂7 y8 > > > > 7 > > : ; 7 ∂y 7 uz8 5 ∂ ∂x (4.48)

In the preceding equation, the symbol {D} denotes a vector form of the rate of deformation tensor D according to the Voigt notation for symmetric tensors, and L is a matrix operator built from spatial partial derivatives. Users and developers of the finite element flow formulation commonly designate the rate of deformation matrix B as the ‘strain rate matrix’. Because of this popular use of terminology, the components of the vector form of rate of deforn oT mation tensor {D} are sometimes written as ε_ ij ¼ ε_ x , ε_ y , ε_ z , γ_ xy , γ_ yz , γ_ xz . However, readers must be aware that there are differences between the rate of deformation tensor D and the time rate of the logarithmic (or true) strain tensor ε_ (refer to Section 3.5 of Chapter 3), because only the normal components of the rate of deformation tensor are identical to the time rate of change of the logarithmic strain. The incompressibility constraint ε_ v ¼ 0 of the velocity field can be related to the rate of deformation matrix B as follows: ε_ v ¼ Dv ¼ CT fDg ¼ CT Bv ¼ CT LNT v CT ¼ ½ 1 1 1 0 0 0 

(4.49)

where C is the matrix form of the Kronecker delta δij. The Levy-Mises constitutive equations for isotropic metals are written in vectorial form as follows:

Finite element flow formulation

2

2 63 6 60 6 6 6 60 6 d¼6 60 6 6 6 60 6 4 0

0 0 0 0 2 0 0 0 3 2 0 0 0 3 1 0 0 0 3 1 0 0 0 3 0 0 0 0

0

201

3

7 7 07 7 7 7 07 σ 2 σ 7 0 0 Dij fσ g ¼ dfDg (4.50) σ ij ¼ 7 _ε 3 ε_ 07 7 7 7 07 7 15 3 0 where {σ } is the Voigt notation of the deviatoric Cauchy stress tensor σ0 . _ the effective Cauchy stress Under these circumstances, with σ 0ij Dij ¼ σ ε, σ in Eq. (4.50) can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 0 3 0 0 σ¼ ðσ : σ Þ ¼ fσ0 gT d1 fσ0 g σ ij σ ij ¼ (4.51) 2 2 Similarly, the effective strain rate ε_ in Eq. (4.50) can be written as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ε_ ¼ (4.52) Dij Dij ¼ ðD : DÞ ¼ fDgT dfDg 3 3 or by replacing Eq. (4.48) into Eq. (4.52): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε_ ¼ vT BT d B v

(4.53)

Now, inserting the equations mentioned earlier into Eqs (4.42), (4.43), it is possible to rewrite the finite element equations as an assembly of integral equations at elemental level. Starting, for example, with the Lagrange multiplier form in Eq. (4.42), and introducing T as the vector form of the tractions ti the following system of equations is obtained: 8 ð M ð ð = T m T m + δλm C Bv dV  δv NTdS ¼0 > ; m m V

St

(4.54)

Metal forming

202

Due to the necessity of treating the hydrostatic stress σ m ¼ λ as an independent variable in mixed velocity–pressure formulations, the following system of equations is used: 8 9 8 > > ð ð ð > M < = > X > σ > m T m m > + λ B C dV  NTdS k v dV ¼0 > m > > < m¼1 > : m ε_ ; m m V (4.55) 8 V 9 St > ð > M < = X > > > > CT Bv dV m ¼ 0 > : : ; m¼1

Vm

T

where k ¼ B d B. The preceding system of equations may be written in a condensed form as follows: 

M  X F σP Q v ¼ T 0 Q 0 λ m m¼1 ð ð ð (4.56) 1 m T m P¼ Q ¼ B C dV F ¼ NTdSm k dV ε_ m m m V

V

St

In case of using the penalty form (4.43), the following system of equations is obtained (σ m ¼ K ε_ v =2): 8 9 > ð ð

X T T m m T T T m T m δv B d B vdV + K δv C BvC BdV  δv NTdS > _ > ; m¼1 : m ε m m V

V

St

¼0 (4.57) T

where k ¼ B d B. This system of equations may also be written in a condensed form M X

f ½σP + K m Q fvg ¼ fFgg m¼1 ð ð ð 1 m T T m Q ¼ C BC B dV F ¼ NTdSm P¼ k dV _ε m m m V

V

(4.58)

St

Because the most significant part of the central processing unit (CPU) time of a metal forming simulation is consumed in solving the main system of equations resulting from Eq. (4.56) or (4.58), it is of paramount importance to utilise solvers that can effectively reduce the overall CPU time. This

Finite element flow formulation

203

is commonly done through the decomposition of the finite element models into subdomains that are simultaneously solved by means of parallel computation. Nielsen and Martins (2017) discuss the parallelisation of solvers in finite element computer programmes and provide a source code of a parallel direct solver, which can also be partially utilised as a preconditioner of conjugate gradient iterative solvers recommended for large finite element models.

4.4 Iterative solution methods The solution of the discretised finite element equations involves the utilisation of iterative methods that were explained in Chapter 3. The application of these methods is meant to minimise the residual force vector R to within a specified tolerance. In what follows, the presentation will be exclusively based on the penalty form of the finite element flow formulation (also named the ‘irreducible flow formulation’) but readers can easily extend its contents to mixed velocity–pressure forms based on the use of Lagrange multipliers. Additional information may also be obtained in Tekkaya (2000).

4.4.1 Direct iterative method Application of the direct iterative method consists of using the velocity field from previous iteration n  1 to determine the velocity field at the current iteration n according to Eq. (4.58): 82 9 3 > M > M < = X X

6 m 7 σ P + K Q 5 fvgn ¼ fFg Kn1 fvgn ¼ fFg ) 4|{z} > > |fflffl{zfflffl} : ; m¼1

KD

KH

n1

m¼1

(4.59) where F is the nodal force vector and K is the stiffness matrix composed of a deviatoric stiffness matrix KD ¼ σP and a hydrostatic stiffness matrix KH ¼ KQ. Alternatively, as it was explained in Section 3.6.1 of Chapter 3, the direct iterative method consists of writing the residual force vector R as Rn ¼

M X m¼1

fFg  ½σ P + K m Qn1 fvgn1

(4.60)

204

Metal forming

and replacing the nodal force vector F from Eq. (4.59) into Eq. (4.60) to obtain M X

½σ P + K m Qn1 fvgn  ½σ P + K m Qn1 fvgn1 Rn ¼ m¼1 82 9 3 (4.61) M < = X m 5 4 σP+K Q Rn ¼ fΔvgn : |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ; m¼1 K

n1

Then, by solving (4.61) in order to find {Δv}n, one obtains the new velocity vn at the current iteration n by means of fΔvgn ¼ K1 n1 R n and vn ¼ vn1 + Δvn

(4.62)

The iterative procedure is carried out until fulfilling some convergence criteria such as the relative residual force norm kR n k  ζF kFk

(4.63)

where the specified tolerance ζF is a number typically ranging from 101 to 104. Other convergence criteria like the relative velocity norm are also used: kvn  vn1 k  ζv kvn1 k

(4.64)

where ζ v is a number typically ranging from 102 to 105 depending on the accuracy requirements for the solution.

4.4.2 Newton-Raphson iterative method The Newton-Raphson iterative method is particularly suitable at the vicinity of the solution because it converges faster than the direct iterative method. However, the method is conditionally convergent and very sensitive to the quality of the initial velocity guess. For this reason, it is sometimes utilised in conjunction with the direct iterative method because the latter is insensitive with respect to the initial guess and capable of providing good velocity guesses for the Newton-Raphson iterative method that help circumventing its convergence problems. The Newton-Raphson method is explained in Section 3.6.1 of Chapter 3 and its application to the finite element flow formulation involves

Finite element flow formulation

205

expanding the residual force vector R near the velocity estimate at current iteration n as follows:   ∂R Rn ¼ Rn1 + fΔvgn ¼ 0 ∂v n1 M X

Rn ¼ fFg  ½σ P + K m Qn1 fvgn (4.65) m¼1

Rn1 ¼

M X m¼1

fFg  ½σ P + K m Qn1 fvgn1

From the chain rule, Eq. (4.53) and introducing b, the following expressions can be stated: ∂σ ∂σ ∂ε_ ¼ ∂v ∂ε_ ∂v

∂ε_ kv ¼ ∂v ε_

b ¼ kv

They can be used to define a matrix H ¼ ð∂ðσPÞ=∂vÞv as # ð" ∂σ 1 σ 1  2 b bT dV m Hn1 ¼ _ n1 ε_ n1 ε_ _ n1 n1 n1 ∂ ε ε n1 m

(4.66)

(4.67)

V

which together with P in Eq. (4.58) gives the following derivative of the residual force vector R   M X

∂R (4.68) ¼ ½σ P + H + K m Qn1 ∂v n1 m¼1 This result allows rewriting the expansion of the residual force vector R near the velocity estimate at current iteration n as follows:   M X ∂R Rn1 + fΔvgn ¼ fF  σ Pv  K m Qvgn1 ∂v n1 m¼1 8 9 > > M < = X m  ½σ P + H + K Qn1 fΔvgn > > ; m¼1 :|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} t ¼0

Kn1

(4.69)

where Kt ¼  ∂R/∂ v is known as the tangent stiffness matrix. Eq. (4.69) allows determining {Δv}n ¼ (Ktn1)1Rn1 and obtaining the new velocity vn at the current iteration n as vn ¼ vn1 + Δvn

(4.70)

206

Metal forming

The iterative process is carried out until fulfilling convergence criteria similar to those previously described for the direct iterative method.

4.4.3 Line search algorithms As mentioned in Section 3.6.1 of Chapter 3, the iterative velocity scheme of Eqs (4.62), (4.70) can be improved by means of a relaxation factor β that is aimed at diminishing the total number of iterations, vn ¼ vn1 + βΔvn

(4.71)

This is particularly important in situations where the straight application of Eqs (4.62), (4.70) is insufficient and additional control on the iterative solution is needed. The relaxation factor β can be set up by means of empirical procedures based on the analysis of convergence between successive iterations or by means of line search algorithms. A line search algorithm that can easily be implemented in the finite element flow formulation is based on the observation that the residual force vector R is orthogonal to the velocity force v. For this purpose, Eq. (4.57) is rewritten as follows: 8 9 > > > 8 9> > > > > > > > > ð ð

< = M > X BT d B vdV m + K m CT BvCT BdV m  NTdSm ¼0 fδvgT  > > > : m ε_ ;> > m¼1 > > > m m > > V V S t > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> : ; R

(4.72)

This observation can be translated into a residual function (a scalar) R (β) defined as RðβÞ ¼ fΔvgTn Rðvn1 + βΔvn Þ ¼ 0

(4.73)

Due to the extreme nonlinearity of the residual function R (β), the condition in Eq. (4.73) is too stringent, and in practice, it is sufficient to obtain a value of the relaxation factor β such that j R ðβÞj < j R ð0Þj

(4.74)

Under normal conditions, the value β ¼ 1 corresponds to the standard relaxation procedure satisfying equation (4.74) and, therefore, no extra operations are needed. However, in complex deformation processes, this is not the case and a more suitable value of β must be obtained. This is done

Finite element flow formulation

207

by approximating R(β) as a quadratic function built upon the values of R(0), R(1) and of the derivative dR(β)/dβ at β ¼ 0 (Bonet and Wood, 1997): R ð0Þ ¼ fΔvgTn  Rðvn1 Þ T  R ð1Þ ¼ fΔvgn  Rðvn1 + Δvn Þ dR ∂R ∂v ¼ fΔvgTn ¼ fΔvgTn Ktn1  fΔvgn ¼ fΔvgTn Rn1 ¼ Rð0Þ  dβ ∂v n1 ∂β (4.75) The result of the approximation of the residual R (β) by a quadratic function satisfying the two end points β ¼ 0 and β ¼ 1, and the derivative in β ¼ 0 is given by (Fig. 4.6) R ðβÞ ¼ ΔvTn  Rðvn1 + βΔvn Þ  ð1  βÞR ð0Þ + R ð1Þβ2 ¼ 0

(4.76)

Hence, a good estimation of the relaxation factor β is given by r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α α2 Rð0Þ α α¼ (4.77) β1 ¼ 2 2 Rð1Þ Eq. (4.77) allows concluding that if α < 0, there is a well-defined value for the relaxation factor β (Fig. 4.6A). However, if α > 0, the square root of Eq. (4.77) provides a complex number (with real and imaginary parts) and

R

R R(1)

β β1

0

β1

β 1

0

1

R(1) R(0)

R(0)

(A)

(B)

Fig. 4.6 Illustration of the line search algorithm utilised to determine the value of the relaxation factor β of the iterative solution scheme in case the residual function R(β) ¼ 0 provides a solution in (A) the real axis ℝ or in (B) the complex plane ℂ.

208

Metal forming

the relaxation factor β should be taken as the value that minimises the quadratic function R(β) (Fig. 4.6B):  dR d  1 Rð0Þ α ¼ ð1  βÞRð0Þ + Rð1Þβ2 ¼ 0 ) β1 ¼ ¼ dβ dβ 2 Rð1Þ 2

(4.78)

4.5 Explicit solution scheme Iterative solution methods to minimise the residual force vector R should not be confused with the overall explicit solution utilised by the finite element flow formulation. In fact, considering the condense matrix form of the nonlinear set of equations given by Eq. (4.58), it is necessary to understand that all calculations are performed with respect to the actual geometry x, which relates to a particular region of the space occupied by the volume at time t. Hence, Eq. (4.58) may be written as M X m¼1

½σ P + K m Qtn1 fvgtn ¼ fFgt

)

M X m¼1

Ktn1 fvgtn ¼ fFgt

(4.79) and the new geometry at time t + Δt is obtained by means of the following explicit time integration scheme based on an ‘updated Eulerian’ description of motion at elemental level: x

t + Δt

t

t +ðΔt

¼x +

vdt ffi xt + vtn Δt

(4.80)

t

Another interesting characteristic of the finite element flow formulation is that there is no strain measure because all the equations are built upon the velocity vector v and the rate of deformation tensor D. Hence, the strain tensor and the effective strain should be obtained from the rate of deformation tensor by means of an explicit time integration scheme like that utilised for the geometry. The following equation shows the procedure utilised for determining the effective strain ε: t + Δt

ε

t

t +ðΔt

¼ ε + t

_ ffi ε εdt

t

t + ε_ n Δt

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ε_ ¼ ð D : DÞ 3

(4.81)

Finite element flow formulation

209

Finally, the Cauchy stress tensor σ at time t + Δt is directly obtained from the rate of deformation tensor D at time t: σ K (4.82) fσgt + Δt ¼ dfDgt + CT fDgt 2 ε_ where the first term is the deviatoric stress from Eq. (4.50) and the second term is the hydrostatic stress obtained from Eqs (4.32), (4.49). This means that the stress field is directly obtained from the geometry at time t without reference to previous configurations.

4.6 Numerical integration Computer implementation of the finite element flow formulation requires the integrals of the finite element equations to be numerically evaluated by means of Gauss-Legendre quadrature rules. The quadrature rules in three dimensions are of the form ð1 ð1 ð1 f ðξ, η, ζÞdξdηdζ ffi

n X n X n X

  wi wj wk f ξi , ηj , ζ k

(4.83)

i¼1 j¼1 k¼1

1 1 1

where wi, wj, wk are the weighting coefficients, ξi, ηj, ζ k are the Gauss points (coordinate positions within the elements) and n is the total number of Gauss points along a local axis within the hexahedral elements utilised in the finite element discretisation (Table 4.2). The total number of integration points nip ¼ n3. The question that must now be asked is the number of integration points to be utilised in the numerical integration of the stiffness matrix K of Eq. (4.58) or (4.59). The answer to this question is found in the work of Malkus and Hughes (1978) on mixed velocity–pressure and penalty forms utilised in nonlinear incompressible and near incompressible continuum problems in solid and fluid mechanics. According to this work, penalty forms are only applicable with reduced integration and are equivalent to Table 4.2 Coordinates and weights in quadrature rules of first and second orders. n (order)

nip

ξi, ηj, ζ k

wi, wj, wk

1

1

0

2

8

p1ffiffi3

1

2

210

Metal forming

mixed velocity–pressure forms in which the pressure distribution is discontinuous between the elements. Taking also into consideration that the use of penalty forms in incompressible fluid flows must satisfy the Babusˇka-Brezzi condition, which requires the number of velocity degrees of freedom (d.o.f.) to be larger than the number of d.o.f. in pressure to prevent overstiff responses (locking) and spurious oscillations in pressure, it is concluded that the deviatoric stiffness matrix KD ¼ σP will be integrated using a full 2 2 2 scheme and the hydrostatic stiffness matrix KH ¼ K Q will be integrated using a reduced 1 1 1 Gauss scheme. Example 4.2

In order to understand the application of numerical integration, let us consider the solution of the following integral (Fig. 4.7) using one-point (n ¼ 1) and two-point (n ¼ 2) Gauss quadrature rules, ð1 

ð1 f ðξÞdξ ¼

I¼ 1

1

 1 3 exp ðξÞ + ξ + dξ ξ+2 2

Application of the one-dimensional version of the quadrature rule given by Eq. (4.83) to the previous integral becomes

f(ξ ) 10

8

Area = 8.8165

6

4

2

–1

(A)

–0.577

0

0.577

1

ξ

(B)

Fig. 4.7 Illustration of the use of quadrature rules to calculate the grey area below the function.

Finite element flow formulation

ð1 n¼1 ) I¼

 f ðξÞdξ ffi w1 f ð0Þ ¼ 2 3 exp ð0Þ + 02 + 0

1

ð1 n¼2 ) I¼

f ðξÞdξ ffi 1

2 X i¼1

 1 ¼7 0+2

  B 1 pffiffi wi f ðξi Þ ¼ 1 B = 3 @3 exp

1  2 C 1 1 C +  pffiffiffi + A 1 3  pffiffiffi + 2 3 0 1    2 B 1 1 1 C C ¼ 8:78566 +1 B @3exp pffiffi3ffi + pffiffi3ffi + 1 A pffiffiffi + 2 3 The exact solution is 8.8165 and better estimates could be obtained by increasing the number of Gauss points.

Example 4.3

The Babusˇka-Brezzi condition can be illustrated by means of the following two systems of equations: 2 3 1 2 3 8 9 879 v < v1 = < = 1 ¼ 1 6 7 v ¼ 3 4 2 1 1 5 v2 ¼ 9 : ; : ; 2 0 p p¼4 3 1 0 2 3 1 2 3 8 9 8 9 v¼0 6 7< v = < 14 = v ¼ 0 v ¼ 0 6 2 0 0 7 p1 ¼ 0 p ¼ 7 ¼ 4 ¼ 2 …, etc p p 1 1 1 4 5: ; : ; 0 p2 p2 ¼ 0 p2 ¼ 2 p 2 ¼ 6 3 0 0 These two systems of equations are compared with the mixed velocity– pressure form of the finite element flow formulation (Eq. 4.56), 

M  X v F σP Q ¼ T 0 λ 0 Q m m¼1 It is possible to conclude that the Babusˇka-Brezzi condition prevents locking due to overconstraint. In fact, when the number of pressure unknowns p ¼ σ m ¼ λ is larger than the number of velocity unknowns, the solution becomes overstiff (locking occurs) v ¼ 0, and the system of equations provides infinite (spurious) solutions for the pressure distribution.

211

212

Metal forming

4.7 Treatment of rigid zones Rigid zones are characterised by very small values of the effective strain rate ε_ in comparison with those occurring in the remaining plastically deforming regions. This creates numerical difficulties in the solution of the system of equations because the deviatoric stiffness matrix KD ¼ σP becomes very ill-conditioned: 0 1 ð 1 ε_  0 ) @P ¼ _ k dV m A ! ∞ ε Vm

82 9 3 M < = X 4 σ P + K m Q5 fvg ¼ fFg : |{z} ; m¼1

(4.84)

KD !∞

Strictly speaking, the flow formulation is only applicable to plastically deforming regions. Therefore, rigid zones should be removed from calculations and the elements located in these regions should be excluded from the overall stiffness matrix. The limiting threshold value of strain rate below which material is considered rigid is named the ‘cut-off strain rate’ ε_ 0 and may be obtained from the average strain rate values in the overall volume as follows: ε_ 0  ξ ε_ avg

ð _εavg ¼ 1 εdV _ V

(4.85)

V

In the preceding equation, ξ is a value usually taken in the range 102 to 104. A too large value of ξ will model rigid zones poorly, and a too small value of ξ may lead to numerical inaccuracies. The technique to handle these rigid regions without excluding the elements from the overall stiffness matrix is by assuming a Newtonian fluid-like behaviour and modify the Levy-Mises constitutive equations as follows (refer to Eqs (4.35), (4.36)): μ¼

2 σ0 2 σ0 ! σ 0ij ¼ Dij _ 3 ε0 3 ε_ 0

fσ0 g ¼

σ0 dfDg ε_ 0

(4.86)

The procedure is graphically illustrated in Fig. 4.8. An improvement of this approach is to avoid excessive strain accumulation in rigid regions. The strain should only be accumulated if the effective strain rate is increasing or if

Finite element flow formulation

213

σ Plastic

σ0

Newtonian

μ ε0

ε

Fig. 4.8 Illustration of the Newtonian fluid-like behaviour of the material for effective strain rates below the limiting threshold ε_ 0 .

the effective strain has already exceeded a certain strain level meaning that the region should not be treated as rigid (Nielsen et al., 2013). Example 4.4

The danger of dividing by zero in Eq. (4.84) can be further understood by means of this very simple analytical example. Let us consider two variables a ¼ b. So, we can write a2 ¼ ab ) a2  b2 ¼ ab  b2 ) ða + bÞða  bÞ ¼ bða  bÞ Now comes the illegal division of the previous equation by zero, ða + bÞ ¼ b leading to the following wrong result, 2b ¼ b ) 2 ¼ 1 ! This justifies the necessity of taking care of Eq. (4.84) when the effective strain rate is zero.

4.8 Treatment of friction Friction between workpiece and tools (and between adjacent workpieces) is influenced by the prevailing local state of stress, the relative sliding velocity vr at the contact interface and the changing boundary conditions. The

214

Metal forming

numerical treatment of friction is usually done with the classical friction laws that resort to constant model parameters, such as Coulomb-Amontons’ friction law, that express the ratio τf/σ n ¼ μ of the friction shear stress τf to the normal contact pressure σ n being equal to a constant friction coefficient μ. Other friction laws also used are the constant friction law, which expresses the ratio τf/k ¼ m of the friction shear stress τf to the shear yield stress k being equal to a friction factor m, and the Bay and Wanheim (1976) friction law that assumes friction to be proportional to the normal contact pressure at low normal pressure, but progressively going towards a constant value at high normal pressure. Computer implementation of these friction laws is performed by considering friction as a traction boundary condition and neglecting most of the phenomenological parameters that influence the interaction between lubricant, workpiece and tool during plastic deformation. Alternative procedures to account for the influence of some of these parameters in the lubricant response can be grouped into two different categories: (a) By discretising the interface lubricant layer between workpiece and tools by means of solid elements with a fictitious small stiffness (Hartley et al., 1979) and (b) By modelling the lubricant as a viscous fluid through the combination of fluid and solid mechanics (Carretta et al., 2015). However, the explored alternatives generally require the use of numerical procedures to prevent excessive distortion of the interface layer elements and often suffer from numerical difficulties caused by the differences in stiffness between metals and fluids, which may be up to 9 or 10 orders of mag€ unyagiz et al., 2017). For these reasons, they will not be nitude (Ust€ considered in this section. Treatment of friction as a traction boundary condition requires dedicated numerical implementation by including an extra term in Eq. (4.58) instead of using the force vector F resulting from the other tractions acting on surface St, 9 8 > > > > > > ð ð ð ð = < M ∂τf X σ T T T T T m T T m m m m δv C BvC B dV + δvr N δv B d B v dV + K dS  δv N TdS > > ∂vr > ε_ > > m¼1 > ; :V m Vm Sm Sm f

t

¼0 (4.87)

In the preceding equation, the symbols τf and vr denote the friction shear stress vector and the relative velocity vector between the workpiece and

Finite element flow formulation

215

vr N

τ

τf

τf

τf

τf

vr

vr

N τ

N - Neutral point

Singularity

(A)

(B)

Fig. 4.9 (A) Friction shear stress and its derivative in case of the law of constant friction and (B) approximation by the arc tangent function given by Eq. (4.88).

tools, respectively. Because the derivative ∂ τf/∂vr is discontinuous at the neutral points N where vr ¼ 0 (Fig. 4.9), it is necessary to modify the friction laws in order to prevent the occurrence of singularities. This is usually done by approximating the friction shear stress τf at the vicinity of the neutral points by means of an arc tangent function (Chen and Kobayashi, 1978):  

vr 2 vr jvr j τ ¼ mk ffi mk arctan (4.88) a π jvr j jvr j where a is a relatively small number. Eq. (4.88) is valid for the law of constant friction but similar expressions can be written for the other friction laws. To conclude, it is worth mentioning that the surface integrals of the dedicated friction term of Eq. (4.87) must be numerically integrated in (at least) five Gauss points along each local direction in order to correctly model the evolution of the friction shear stresses near the neutral points.

4.9 Contact between objects Contact between objects can be classified into two different categories (Fig. 4.10): (a) Contact between deformable and rigid objects (Fig. 4.10A) and (b) Contact between deformable objects (Fig. 4.10B).

Metal forming

216

Deformable object Finite element discretization

t

t

V N

z

Contact surface S

Γ

V

y x Rigid object (tool)

Contact surface SΓ

Time t + Δ t

Time t

(A)

t

Deformable object 1

V1

Deformable object 2

t

V1

z

y

V2

x

Contact surface S

Time t

V2 Time t + Δ t

(B) Fig. 4.10 Contact (A) between a deformable and a rigid object and (B) between two deformable objects.

The fundamental problem in contact between objects consists of determining the new contact surface after a time increment Δt whilst satisfying the mechanical, thermal and electrical governing equations and preventing overlapping between the objects. In what follows, we will start by addressing the contact between deformable and rigid objects and consider frictionless conditions along the contact surfaces SΓ in order to simplify the presentation. The unitary contact condition for an arbitrary nodal point N with coordinate xt at time t is given by (Tekkaya, 2007) σ n ðvN  vtool ÞT  n ¼ 0

(4.89)

Finite element flow formulation

217

Deformable object

n

N

xt

xt

vN

xt vN

Δ vN Δ x t+ t

vtool

SΓ

x t+Δ t

vtool

vtool

Rigid object (tool)

(A)

(B)

(C)

Fig. 4.11 Contact between deformable and rigid objects at nodal point N. (A) Notation, (B) sliding, (C) separation.

where σ n, vN and n are the normal stress, the velocity vector and the normal vector of direction cosines to SΓ at point N, respectively. The symbol vtool denotes the tool velocity vector (Fig. 4.11A). The unitary contact condition (4.89) includes the two possible contact situations between objects: (a) Contact with or without sliding, during which (vN  vtool)T  n ¼ 0 and σ n < 0 (Fig. 4.11B) and (b) Separation, during which (vN  vtool)T  n > 0 and σ n ¼ 0 (Fig. 4.11C). A contact algorithm must be capable of handling the unitary contact condition given by Eq. (4.89) for all the nodes in contact with surface SΓ and of monitoring and resolve all the new contacts arising from nodes that will reach SΓ during the increment of time Δt (Fig. 4.12A). The following Deformable object Rigid object (tool)

x t vN

SΓ

(A)

x

n v x t N t+Δ t x

x t vN

t+Δ t

x t+ t

vtool

vtool

vtool

(B)

(C)

Fig. 4.12 Contact between deformable and rigid objects at nodal point N. (A) Explicit time integration scheme, (B) linear contact algorithm showing correction by projection onto the contact surface, (C) notation for the penalty-based contact algorithms.

218

Metal forming

sections will present three different types of contact algorithms that can be implemented in finite element computer programmes: (a) Linear contact algorithms, (b) Penalty-based contact algorithms, (c) Lagrange multiplier-based contact algorithms.

4.9.1 Linear contact algorithms Linear contact algorithms are based on the explicit time integration scheme that is utilised to obtain the new geometry of an arbitrary point N at time t +Δt (refer to Eq. (4.80)). Because the new position of N generally results in overlapping between the two objects, it is mandatory to correct the final position of point N through an additional projection onto the surface SΓ, as shown in Fig. 4.12B. Linear contact algorithms are robust and easy to implement but they give rise to loss of volume due to the multiple corrections that are carried out during a simulation. However, the loss of volume can be minimised by using small increments of time Δt or by implementing procedures that will predict the minimum increment of time Δtmin for an arbitrary nodal point N to get into contact with surface SΓ, so that the increment of time Δt can be split into two (or more) partial increments (Δtmin and Δt  Δtmin).

4.9.2 Penalty-based contact algorithms In penalty-based contact algorithms, the normal gap velocity g is penalised by including an additional term in the original rate of energy variational statement (4.25) at elemental level, ð ð ð ΠðvÞ ¼ σ εdV _  tvdS + 1 Kc g2 dS ) minimum 2 V St

SΓ

g ¼ ðvN  vtool Þ  n

(4.90)

where vN is the velocity of node nodal point N, vtool is the velocity of the tool and n is the outward normal unit vector to the tool at the contact point. The goal is to force the normal gap velocity of nodal point N to be zero at the contact surface SΓ by means of a penalty factor for contact Kc. The method allows a certain amount of penetration and the larger the value of the penalty, the better the constraints will be respected. The main drawback of this type of contact algorithm is that high values of the penalty factor Kc may lead to ill-conditioned systems of equations.

Finite element flow formulation

219

4.9.3 Lagrange multiplier-based contact algorithms In Lagrange multiplier-based contact algorithms, the original rate of energy variational statement (4.25) is modified to include an additional term that contains a function f describing the contact surface. ð ð ð   _ Πðv, λc Þ ¼ σ εdV  tvdS + λc f xt + Δt , t + Δt dS ) minimum V

St

SΓ

(4.91) This method provides the exact solution but at cost of increasing the total number of unknowns because the Lagrange multipliers λc are extra unknowns that are introduced in the overall system of equations. Example 4.5

A penalty-based contact algorithm will now be illustrated by means of a simple uniaxial case consisting of a nodal point N with free velocity in the x-direction. Nodal point N moves towards the rigid object and the corresponding normal gap velocity at time t is given by (Fig. 4.13A) g ¼ ðvN  vtool Þ  n ¼ vNx  vtoolx Once nodal point N enters the contact bandwidth (at time t + Δt), the contact penalty term in the extended rate of energy variational statement given by Eq. (4.90) (Fig. 4.13B) is activated: ð ð ð 1 _ Kc g2 dS ) minimum ΠðvÞ ¼ σ εdV  tv dS + 2 V

St

SΓ

Continued

Rigid object

Rigid object

x

vN

xt

vtool

x

vN

Δ x t+ t

vtool

Kc Deformable object

Deformable object Contact

(A)

Contact

(B)

Fig. 4.13 Application of the penalty-based contact algorithm in a uniaxial contact case. (A) No contact at time t and (B) contact at time t + Δt.

Metal forming

220

Example 4.5—cont’d

This penalty term may be regarded as the rate of strain energy of a spring with stiffness Kc connecting the nodal point N to the rigid object (Fig. 4.13B): 1 1 1 U_ ¼ Kc g2 ¼ Kc ½ðvN  vtool Þ  n2 ¼ Kc ðvNx  vtoolx Þ2 2 2 2 Minimising the extended rate of energy variational statement ∂ Π(v)/∂ v ¼ 0

8 9 > ð ð ð

X σ T T δv B d B vdV m + K m δvT CT BvCT BdV m  δvT NTdSm + Kcm gδg dS > > _ : ε ;

m¼1

Vm

Vm

Stm

SΓ

¼0

the following system of equations is obtained, 9 2 3 2 38 ⋯ ⋯ ⋯ ⋯ ⋯ > ⋯> > > 6 ⋯ ðKN ,N + Kc Þ KN ,N + 1 ⋯ 7< vNx = 6 FNx + Kc vtoolx 7 7 6 7 ¼6 4 5 4⋯ vNy > ⋯ ⋯ ⋯ ⋯ 5> > > ; : ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ The foregoing system of equations allows concluding that large values of Kc (large spring stiffness) prevent penetration between the two objects because the horizontal velocity of the nodal point N and that of the rigid object are forced to be identical vNx ¼ vtool. In practical terms, because values of Kc cannot be very large in order to avoid locking, it follows that vNx ffi vtool. The consequence of this is that prevention of penetration of nodal point N into the rigid object is ensured within a region where nodal point N can be slightly away from the rigid object or slightly overlapped with the rigid object as a result of the spring stiffness Kc not being large enough. Finally, the reaction force exerted by the ‘spring’ on the deformable body at nodal point N is given by RNx ¼ FNx ¼ Kc ðvN  vtool Þ  n ¼ Kc ðvNx  vtoolx Þ

4.9.4 Contact between deformable objects The contact described earlier is valid for a deformable object against a rigid object. The rigid object acts as a master, whilst the node from the deformable object is the slave. Between deformable objects, there is no natural master– slave relation, and the contact interface has to find its velocity based on an

Finite element flow formulation

221

Object 1

L

N1

NP v

P

v2

n

N1

(1- β)L

n

βL

v1

N2

N2 Object 2

Detail of object 2

Fig. 4.14 Contact pair between two deformable objects. The node NP of object 1 is in contact with edge segment N1  N2 of object 2. The parameter β  [0; 1] identifies the location where NP contacts the edge segment.

overall minimisation of the energy (or power) of the system, whilst obeying the condition in (4.89). Fig. 4.14 shows two objects in contact, one of the identified contact pairs consisting of a node NP in contact with an edge segment N1  N2 from the other object. Self-contact of one object against itself, e.g. upon folding, is handled in the same way. The contact search algorithm is based on a two-pass strategy when both sides of the contact can deform. As a result of this, the nodes in the edge segment are likely to be part of other contact pairs, where they take the role similar to that of NP. Considering here only the normal contact based on the penalty formulation, Eq. (4.90) is valid also for deformable objects in contact when the normal gap velocity is expressed as g ¼ ðvP  βv1  ð1  βÞv2 Þ  n

(4.92)

where v is the velocity vector of the nodes indicated by the subscripts. The parameter β  [0; 1] identifies the contact position along the edge segment N1  N2 as illustrated in detail in Fig. 4.14 and is utilised to weight the velocities of each node in the edge segment. The penalty terms arising from Eq. (4.90) with Eq. (4.92) to account for two deformable objects in contact are distributed as shown in Fig. 4.15 in the global stiffness matrix for a two-dimensional problem. The order of the diagonal positions related to nodes NP, N1, and N2 depends on the global node numbering. In the two-dimensional example, each of the blocks consists of four positions in the stiffness matrix. The off-diagonal blocks relating nodes to each other will potentially increase the necessary bandwidth or the skyline heights and increase the memory for storing the stiffness matrix and increase the computational time to solve the equation system. Node numbering optimisation is performed to minimise the distance between

222

Metal forming

PP Diagonal related to Np

P1

P2

Each block contains four positions:

11

12

xx xy yx yy

Diagonal related to N1

22

Diagonal related to N2

Fig. 4.15 Positions of the contact penalty terms in the global stiffness matrix. The order and position of the diagonals related to the three nodes in a contact pair depends on the global node numbering.

connected nodes in the stiffness matrix, and when contact between deformable objects occur, it should be considered to include contact relations in this optimisation, as well. The node-to-segment contact pairs may result in locking or an overly stiff response of the contact. Segment-to-segment contact pairs may therefore be considered as they obey the contact in an average sense instead of pointwise. The reader is referred to specialised literature such as Zavarise and Wriggers (1998) and Puso and Laursen (2004) for further details of different contact algorithms and to Silva et al. (2015) for a segment-to-segment contact implementation in the flow formulation.

4.10 Thermo-mechanical analysis The importance of temperature calculations during finite element simulation of warm and hot metal-forming processes has been recognised for a long time. In the past, several approximate techniques were developed and applied to the analysis of such processes. Most of these techniques assumed that the temperature of the deforming material would remain approximately constant during the entire forming operation. This procedure, commonly known as ‘isothermal modelling’, neglects the heat generated during plastic deformation, as well as the heat exchanged between the workpiece, the tools, and the surrounding environment. Hence, the corresponding finite element models only consider the flow stress of the deforming material at an average temperature of the forming process. In terms of metal forming practice, it is important to notice that isothermal modelling considers no significant variation of the material properties with temperature and requires the tools to be heated in the range of temperatures of the deforming material. In general, isothermal modelling is not appropriate to the analysis of warm and hot metal forming processes and

Finite element flow formulation

223

when plastic deformation must be solved in conjunction with the calculation of temperature changes in both workpiece and tools. This requires the utilisation of coupled thermo-mechanical analysis in which the computed temperature field at time t is utilised to update the flow stress in the constitutive equations from which the stiffness matrix is formed and the velocity at time t is calculated.

4.10.1 Heat transfer equation The heat transfer equation in a continuous isotropic and homogeneous body with a volume V bound by a closed surface S can be written as ∂T + Q_ ¼ 0 (4.93) ∂t where k is the thermal conductivity, ρ c is the specific heat capacity and Q_ is the heat generation rate. In thermo-mechanical analysis of metal forming processes, the heat generation rate is obtained from the latent heat produced during plastic deformation, kr2 T  ρc

Q_ ¼ β σ ε_

β ¼ 0:85  0:95

(4.94)

where β is the irreversible factor that gives the fraction of plastic work converted to heat. The Galerkin form of the weighted residual method allows writing an integral form of Eq. (4.93), which instead of satisfying the heat transfer equation exactly (i.e. pointwise) will only fulfil it in an average sense over the entire domain. ð ð ð dT kr2 T δT dV  ρc (4.95) δT dV + βσ ij Dij δT dV ¼ 0 dt V

V

v

The symbol δT denotes an arbitrary variation in the temperature. Applying the chain rule of derivatives r(rT δT) ¼ r2T δT + r T r (δT) to the first integral of Eq. (4.95) results in the following expression ð V

ð ð ð dT krT rðδT ÞdV  krðrT δT ÞdV + ρc δT dV  βσ ij Dij δT dV dt ¼0

V

V

V

(4.96) which can be further modified by the application of the divergence theorem to convert the second volume integral into a surface integral:

Metal forming

224

ð V

ð ð ð dT krT rðδT ÞdV  qn δT dSq + ρc δT dV  βσ ij Dij δT dV ¼ 0 dt Sq

V

v

(4.97) where qn is the heat flux along the surface Sq (belonging to the closed surface S) qn ¼ krT  n ¼ ðkrT ÞT n

(4.98)

where the symbol n is the vector of direction cosines of the normal to the surface Sq.

4.10.2 Finite element discretisation Following a procedure like that employed in the discretisation of the governing quasi-static equilibrium equations of the mechanical model, it is possible to write the heat transfer equation in matrix form at elemental level as follows: 8

X

> m¼1 :

Vm

k MMT TdV m +

ð

ð

m _ ρc NNT TdV 

Vm

  β σ ε_ NdV m 

Vm

ð Sqm

qn NdSm q

9 > = > ;

¼0 (4.99)

The temperature T ¼ NTT is interpolated using identical shape functions as those utilised in the velocity field of the mechanical model (Eq. 4.45) because the hexahedral elements are assumed as isoparametric, and M ¼ r N is given by 2 ∂N ∂N ∂N 3 1

1

6 ∂x ∂y ⋯ ⋯ M¼6 4 ∂N 8 ∂N8 ∂x ∂y

1

∂z 7 ⋯ 7 ∂N8 5 ∂z

(4.100)

The equation system (4.99) can be expressed in the abbreviated form as _ ¼Q KT + C T

(4.101)

Finite element flow formulation

225

where K is the heat conduction matrix, C is the heat capacity matrix and Q is the heat flux vector. The heat flux vector Q has several components and is given by ð

Q¼

ð

  β σ ε_ N dV +

Vm

ð

 4  σ sb εn Tenv  T 4 N dS +

Srm

+

ð

hlub ðTtool  T Þ N dS + m Stool

ð

henv ðTenv  T Þ N dS + Scm

qf N dS Sfm

(4.102)

In the aforementioned equation, Tenv and Ttool are environment and tool temperatures, respectively. The first integral is the contribution of the net heat generated by plastic deformation inside the workpiece. The second integral is the dissipation by heat radiation, where σ sb is the StefanBoltzmann constant and εn is the emissivity number. The third integral accounts for the heat convection between the surface of the workpiece and the environment, where henv is the heat convection coefficient. The fourth integral describes the heat transferred between the workpiece and the die through their contact interface (usually a thin layer of lubricant characterised by a heat transfer coefficient hlub). The last integral is the contribution of the heat generated by friction along the tool-workpiece interface, where qf is the surface heat generation rate due to friction qf ¼ τf jvr j

(4.103)

where τf is the friction shear stress at the tool-workpiece interface and vr is the relative velocity between the two objects. The integration of Eq. (4.101) is performed by means of the following time integration scheme: h i _ t +θT _ t + Δt (4.104) Tt + Δt ¼ Tt + Δt ð1  θÞ T where t and Δt are time and its increment and θ is a parameter varying between 0 and 1. By replacing Eq. (4.104) into Eq. (4.101), the following procedure is obtained: ðθΔt K + CÞTt + Δt ¼ ðð1  θÞΔt K + CÞTt + ð1  θÞΔt Qt + θΔt Qt + Δt (4.105)

226

Metal forming

Fig. 4.16 Thermo-mechanical coupling.

Typical values for θ are 0 (forward difference), 1/2 (Crank-Nicholson), 2/3 (Galerkin) and 1 (backward difference). The value θ ¼ 2/3 is commonly used because it may be proved to be unconditionally stable. Fig. 4.16 provides a schematic flow chart of a typical thermo-mechanical coupling. As shown, the coupling is strong because the new temperature field and resulting changes in mechanical and thermal material behaviour are converged with the mechanical response including the heat generation at the end of each step.

4.11 Electro-thermo-mechanical analysis Electrical analysis is utilised to calculate the distribution of the electric potential Φ (a scalar) which, after differentiation and multiplication by the conductivity, provides the current density j in the workpiece and tools (usually electrodes). Electro-thermo-mechanical analysis based on the finite element formulation is commonly utilised to simulate resistance welding processes (Nielsen et al., 2015), amongst other processes. Electrical analysis is based on the electric potential equation in a continuous body with an arbitrary volume V bound by a closed surface S under steady-state conditions, given by r2 Φ + Q ¼ 0

(4.106)

where Q is the electric potential source (or sink). Applying once more the Galerkin form of the weighted residual method to Eq. (4.106), the following integral form of equation is obtained:

Finite element flow formulation

ð

ð r2 ΦδΦdV + QδΦdV ¼ 0

V

227

(4.107)

V

The symbol δΦ denotes an arbitrary variation in the electric potential. Applying the chain rule of derivatives r(r Φ δΦ) ¼ r2Φ δΦ + r Φ r (δΦ) to the first integral of Eq. (4.107) results in the following expression ð ð ð rΦrðδΦÞdV  rðrΦδΦÞdV  QδΦdV ¼ 0 (4.108) V

V

V

which can be further modified by the application of the divergence theorem to convert the second volume integral into a surface integral: ð ð ð rΦrðδΦÞdV  Φn δΦdS  QδΦdV ¼ 0 (4.109) V

V

S

where Φn is the normal gradient of the electric potential Φ to the free surfaces, Φn ¼ rΦ  n ¼ ðrΦÞT n

(4.110)

Because Φn ¼ 0 and in the absence of sources or sinks (Q ¼ 0), Eq. (4.109) becomes ð rΦrðδΦÞdV ¼ 0 (4.111) V

Following a procedure like that employed in the discretisation of the heat transfer equation, it is possible to write Eq. (4.111) in matrix form at elemental level as follows: 8 9 M ð M >

> : ε_ R ; m¼1

Vm

231

(4.122)

Stm

In the equation mentioned earlier, k ¼ BTd B, and d results from the constitutive Eq. (4.115) that relates the rate of deformation directly with the Cauchy stress instead of relating with the deviatoric Cauchy stress as in the case of dense metals: 2 σR σR Dij fσg ¼ dfDg 3 ε_ R ε_ R 2 2 3 3 4 2 2 0 0 0 1 1 1 0 0 0 6 6 4 2 0 0 0 7 1 1 0 0 07 6 6 7 7 6 6 7 1 1 4 0 0 07 1 0 0 07 6 7 d¼ 6 + 6 3 0 07 0 0 07 3A 6 6 7 3ð3  AÞ 6 7 4 4 3 05 0 05 sym: 3 sym: 0 (4.123) σ ij ¼

To conclude, it must be noted that void opening by tension is inherent to this type of formulation due to the physical relation with the decreases in relative density. Hence, the extension of the flow formulation to porous metals may also be used in coupled damage analysis of dense materials based on microbased continuum damage mechanics (refer to Section 2.5 of Chapter 2 for additional information on ductile damage).

4.12.2 Ductile polymers Ductile polymers such as thermoplastics undergo plastic deformation but unlike the yield strength of metals that of polymers depends on the hydrostatic stress. In particular, the yield strength in compression σ C is greater than in tension σ T by approximately 10%–20%. This difference is named the ‘strength differential effect’. Cold forming of ductile polymers has been comprehensively investigated since the mid-1960s. Whitney and Andrews (1967) proposed a pressuremodified Tresca yield criterion and Sternstein and Ongchin (1969) developed a first version of the pressure-modified von Mises yield criterion that was later enhanced by Raghava and Caddell (1973) and Caddell et al. (1974) in order to explicitly account for the differences between the compressive

232

Metal forming

σj von Mises

σT

−σ C

σT

σi Crazing

σ= 0 k

−σC Raghava-Caddell-Atkins

Fig. 4.18 Comparison between the von Mises yield function used in metals and the Raghava-Caddell-Atkins yield function for ductile polymers. The envelope defining crazing under biaxial stress is also included.

and tensile flow stresses of a polymeric material at a given strain. The yield function F (σ ij) associated with the Caddell et al. (1974) yield criterion is given by (Fig. 4.18) rffiffiffiffiffiffiffiffiffiffiffiffi   3 0 0 2 (4.124) σ ¼ F σ ij ¼ σ  σ C  σ T + ðσ C  σ T Þσ kk ¼ 0 σ σ 2 ij ij where σ is the effective stress and σ kk ¼ δij σ ij ¼ 3 σ m is related to the hydrostatic stress. In case σ T ¼ σ C, the yield function F (σ ij) becomes identical to that of von Mises commonly used in isotropic metals. In what concerns incompressibility, several researchers proved, after measuring volume variation of different polymers, that there is negligible volume variation after yielding and concluded that the normality rule, typical of associated plasticity, does not hold in the case of the pressuredependent yield surfaces of ductile polymers (Whitney and Andrews, 1967). This conclusion was confirmed by Spitzig and Richmond (1979) who showed that the associated flow rule based on a pressure-sensitive yield surface leads to predictions of volume variation at least one order of magnitude larger than observed in ductile polymers. As a result of this, the incompressibility of polymers in the plastic deformation regime can be taken into account by setting up constitutive

Finite element flow formulation

233

equations built upon the following nonassociated flow rule (Alves and Martins, 2009):   ∂Q σ ij p _ ε_ ij ¼ λ (4.125) ∂σ ij where λ_ is the viscoplastic multiplier and Q ¼ J2 is the plastic potential of conventional von Mises isotropic plasticity of metals instead of the yield criterion (4.124). The considerations described allow extending the finite element formulation currently utilised for isotropic metals to the special case of ductile polymers subjected to cold plastic deformation. However, it is worth mentioning that some polymers (PVC, for example) show a progressive change of colour when subjected to tensile states of stress (Fig. 4.18). This is attributed to the phenomenon of crazing that involves the formation of microvoids and stretched polymer chains. The occurrence of crazing mechanisms can be predicted by means of the normalised version of the stress-bias criterion (Sternstein and Ongchin, 1969; Bucknall, 2007): σ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi (4.126) ¼ 1  Rc + Rc σ0 where Rc ¼ σ 2/σ 0, σ 0 is the craze opening stress and σ 1 and σ 2 are the first and second principal stresses at craze initiation.

4.12.3 Viscous flow The fluid mechanics view of the flow formulation (Section 3.2.3) revealed its adequacy for solving problems of both metal forming and viscous fluid flow (e.g. polymer and glass forming). Considering a unified notation for both types of matters, the utilisation of the finite element flow formulation for viscous fluids requires the constitutive equations of metals to be replaced by those of viscous fluids (4.36): σ 0ij ¼ 2μD0ij

(4.127)

where μ is the shear viscosity coefficient. In the case of non-Newtonian fluids, μ assumes the form m1

μ ¼ μ0 ε_

m < 1, μ0 ¼ μ0 ðε, T , pÞ

(4.128)

For the particular case of viscoplastic fluids, the shear viscosity coefficient μ takes the form of Eq. (4.35):

234

Metal forming

m μ ¼ σ Y + γ ε_ (4.129) 3 ε_ For pure plasticity without strain rate sensitivity, the shear viscosity coefficient μ takes the special form of Eq. (4.35),

σ (4.130) 3 ε_ which correspond to the well-known Levy-Mises equations utilised for isotropic metals. In fact, the main difference between the constitutive equations of metals and fluids stems from the separation of the deviatoric and volumetric responses of the viscous fluid that is inherent to the momentum balance equation. In practical terms, this means that the deviatoric response of a viscous fluid requires the following changes in Eq. (4.50): μ¼

2

σ 0ij ¼ 2μD0ij

fσ0 g ¼ dμ fD0 g

2μ 60 6 60 dμ ¼ 6 60 6 40 0

0 2μ 0 0 0 0

0 0 2μ 0 0 0

0 0 0 μ 0 0

0 0 0 0 μ 0

3 0 07 7 07 7 (4.131) 07 7 05 μ

The volumetric response is obtained directly from the bulk viscosity coefficient μv, which takes the role of the penalty factor K in case of metals σ m ¼ μv Dv ¼ μv ε_ v

μv ¼ f ðσ m Þ  K

(4.132)

because bulk viscosity is generally a large number.

4.12.4 Anisotropic metals The implementation of Hill’s quadratic anisotropic yield criterion, after Hill (1948), will be presented here with derivations compared to those of the isotropic formulation presented earlier in this chapter. The reference coordinate system will be 1,2,3 instead of x,y,z to make the presentation more natural in tensor notation. The 1,2,3 notation in this section shall not be confused with the typical meaning of principal stress or strain directions. The isotropic von Mises yield criterion is

Finite element flow formulation

235

    2f σ ij ¼ ðσ 11  σ 22 Þ2 + ðσ 22  σ 33 Þ2 + ðσ 33  σ 11 Þ2 + 6 σ 212 + σ 223 + σ 231 ¼ 2σ 2 (4.133)

where σ is the effective stress being equivalent to the yield stress in uniaxial tension (in any direction since Eq. (4.133) is isotropic), which can also be written in terms of the deviatoric Cauchy stress components as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 σ¼ ðσ 11  σ 22 Þ2 + ðσ 22  σ 33 Þ2 + ðσ 33  σ 11 Þ2 + 6ðσ 212 + σ 223 + σ 231 Þ r2ffiffiffiffiffiffiffiffiffiffiffiffi 3 0 0 ¼ σ σ 2 ij ij (4.134) The anisotropic yield criterion when following Hill (1948) is   2f σ ij ¼ F ðσ 22  σ 33 Þ2 + Gðσ 33  σ 11 Þ2 + H ðσ 11  σ 22 Þ2 + 2Lσ 223 + 2Mσ 231 + 2N σ 212 ¼ 2e σ2 (4.135) where F, G, H, L, M, and N are constants to be determined by material testing. In the original yield criterion by Hill (1948), the most right-hand side was 1, but here it is chosen to introduce the reference yield stress e σ in order to make the constants dimensionless and to recover the units of the yield function f(σ ij) to be similar to the units of (4.133). It can already be noticed here that if F ¼ G ¼ H¼ 1 and L ¼ M ¼ N ¼ 3, the yield function (4.135) reduces to the von Mises yield criterion and the reference yield stress will simply be the yield stress in a tensile test in any direction, i.e. e σ ¼ σ. Whenever, the constants differ from this special case, the yield function (4.135) represents an anisotropic material with symmetry in material properties in three mutual orthogonal symmetry planes coinciding with the coordinate system. This type of anisotropy is close to that of rolled sheet material. If the uniaxial yield stress is tested in the 1, 2, and 3 directions to be X, Y, and Z, and if the yield shear stresses in the 23, 31, and 13 directions are R, S, and T, it follows from (4.135) that 2

ðG + H ÞX 2 ¼ 2σ

2

2LR2 ¼ 2σ

2

2MS2 ¼ 2σ

2

2NT 2 ¼ 2σ

ðH + F ÞY 2 ¼ 2σ ðF + GÞZ 2 ¼ 2σ

2

2

(4.136)

236

Metal forming

Isolation of the constants gives   1 1 1 2 +  F ¼σ Y 2 Z2 X 2   1 1 1 2 G¼σ +  Z2 X 2 Y 2   1 1 1 2 H ¼σ +  X 2 Y 2 Z2

2

σ L¼ 2 R 2

σ M¼ 2 S

2

σ N¼ 2 T

(4.137)

The reference yield stress e σ can with advantage be chosen as one of the yield stresses X, Y or Z. Returning to the special case of an isotropic pffiffiffi material, pffiffiffi σ = 3 ¼ σ= 3, it will follow that X ¼ Y ¼ Z ¼ e σ ¼ σ, and R ¼ S ¼ T ¼ e and (4.137) yields F ¼ G ¼ H ¼ 1 and L ¼ M ¼ N ¼ 3, as also stated earlier. If the constants in the anisotropic yield criterion are arranged as 2 3 P1111 P1122 P1133 P1112 P1123 P1131 6 7 6 P2211 P2222 P2233 P2212 P2223 P2231 7 6 7 6 7 6 P3311 P3322 P3333 P3312 P3323 P3331 7 6 7 ¼ Pijkl 6P 7 6 1211 P1222 P1233 P1212 P1223 P1231 7 6 7 6P 7 4 2311 P2322 P2333 P2312 P2323 P2331 5 2 P3111 P3122 P3133 P3112 P3123 P31313 G + H H G 0 0 0 6 7 6 H F + H F 0 0 0 7 6 7 6 7   F F + G 0 0 0 7 6 G PUL 0 6 7 ¼6 ¼P¼ 0 PLR 0 0 2N 0 0 7 6 0 7 6 7 6 0 0 0 0 2L 0 7 4 5 0

0

0

0

0 2M (4.138)

it is possible to write (4.135) as follows:   1 1 (4.139) f σ ij ¼ σ ij Pijkl σ kl ¼ σ 0ij Pijkl σ 0kl 2 2 where the last equality is realised by inserting σ ij ¼ σ ij0 + δijσ m. The identification of four submatrices of size 3 3 in (4.138) is for later use.

Finite element flow formulation

237

qffiffiffiffiffiffiffiffiffiffiffi   An effective stress for the anisotropic material is defined as σ ¼ f σ ij as in the isotropic case and hence follows also σ ¼ e σ . The effective stress related to Hill’s quadratic yield criterion becomes (refer to Eqs (4.135), (4.138))



1 1 1 1 0 σ ij Pijkl σ kl 2 ¼ σ ij Pijkl σ 0kl 2 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 F ðσ 22  σ 33 Þ2 + Gðσ 33  σ 11 Þ2 + H ðσ 11  σ 22 Þ2 + 2Lσ 223 + 2Mσ 231 + 2Nσ 212 ¼ 2 (4.140)

σ¼

which is the equivalent to Eq. (4.134) for the von Mises effective stress. The relationship between the Cauchy stress tensor σ ij and the rate of deformation tensor Dij is established through the principle of normality for associated plasticity (also known as ‘flow rule’)   ∂f σ ij _ Dij ¼ λ (4.141) ∂σ ij where λ_ is a proportionality factor (also known as the ‘plastic multiplier’) to be determined later. In order to have the analogy with the isotropic case, the derivatives of the yield function f(σ ij) with respect to the stress components are first determined for the von Mises yield criterion in (4.133) (notice here that, e.g. 6σ 12 ¼ 3σ 12 + 3σ 21) 9 ∂f > ¼ 2σ 11  σ 22  σ 33 ¼ 3σ 011 > > > ∂σ 11 > > > > > ∂f 0 > ¼ 2σ 22  σ 11  σ 33 ¼ 3σ 22 > > = ∂σ 22 ∂f ¼ 3σ 0ij ) ∂f > ∂σ ij 0 > ¼ 2σ 33  σ 11  σ 22 ¼ 3σ 33 > > > ∂σ 33 > > > > > ∂f > 0 > ¼ 3σ ij ¼ 3σ ij i 6¼ j ; ∂σ ij

(4.142)

Inserting (4.142) into (4.141) gives for the isotropic material Dij ¼ 3σ 0ij λ_ , σ 0ij ¼

1 Dij 3λ_

(4.143)

238

Metal forming

which are Levy-Mises constitutive equations stating proportionality between the rate of deformation tensor and the deviatoric Cauchy stress tensor. Now for the anisotropic material, the partial derivatives of the yield function (4.135) are 9 ∂f > ¼ ðG + H Þσ 11  Hσ 22  Gσ 33 > > > ∂σ 11 > > > > > ∂f > ¼ Hσ 11 + ðF + H Þσ 22  Fσ 33 > > > ∂σ 22 > > > > > ∂f > > ¼ Gσ 11  Fσ 22 + ðF + GÞσ 33 > = ∂σ 33 ∂f ¼ Pijkl σ kl ¼ Pijkl σ 0kl (4.144) ) > ∂σ ∂f ∂f ij > > ¼ ¼ N σ 12 > > ∂σ 12 ∂σ 21 > > > > > ∂f ∂f > > > ¼ ¼ Lσ 23 > > ∂σ 23 ∂σ 32 > > > > > ∂f ∂f > ; ¼ ¼ Mσ 31 ∂σ 31 ∂σ 13 where the introduction of Pijkl is directly seen from (4.138) and the last equality is realised by inserting σ ij ¼ σ ij0 + δijσ m.ched file. Considering the principle of normality for associated plasticity (4.141) and taking into consideration the partial derivatives of the yield function given by (4.144), it is possible to obtain the constitutive equations for the anisotropic material as follows: _ ijkl σ 0 Dij ¼ λP kl

(4.145)

Hence, the deviatoric Cauchy stress tensor can be related to the rate of deformation tensor as follows: 1 (4.146) σ 0ij ¼ Mijkl Dkl λ_ where Mijkl has the effect of the inversion of Pijkl. However, because Pijkl is singular, it cannot be inverted. Mijkl is therefore introduced to replace the nonexisting P1 ijkl . The structure of Mijkl is   MUL 0 (4.147) Mijkl ¼ M ¼ 0 MLR due to the structure of Pijkl. All the four submatrices have dimensions 3 3. The lower right submatrix is directly determined as MLR ¼ P1 LR ¼

Finite element flow formulation

239

1 1 1 diag 2N , 2L , 2M whilst the upper left submatrix PUL is the one giving rise to the singularity. The submatrix MUL is therefore set up in a different way, by fulfilling the equation, 8 0 9 2 38 9 8 9 < σ 11 = 1 2 1 1 < σ 011 = < σ 011 = MUL PUL σ 022 ¼ 4 1 2 1 5 σ 022 ¼ σ 022 (4.148) : 0 ; 3 : 0 ; : 0 ; σ 33 σ 33 σ 33 1 1 2 where the effect of MUL is the same as if it was the inversion of PUL. The last 0 equality in (4.148) is shown by recognising that σ kk ¼ 0. The matrix MUL satisfying Eq. (4.148) is inserted in Eq. (4.147) together with the diagonal inversion MLR ¼ P1 LR. The full Mijkl with positions as in (4.138) becomes 2

Fk ðF + GÞk ðF + H Þk 0 0 6 ðF + GÞk Gk ðG + H Þk 0 0 6 6 6 ðF + H Þk ðG + H Þk Hk 0 0 6 1 6 Mijkl ¼ 6 0 0 0 0 6 2N 6 1 6 0 0 0 0 6 2L 4 0 0 0 0 0

0 0 0

3

7 7 7 7 7 1 7 0 7, k ¼ 7 3ðFG + FH + GH Þ 7 0 7 7 5 1 2M (4.149)

It is now possible to define the effective strain rate such that σ ε_ ¼ σ ij Dij ¼ σ 0ij Dij

(4.150)

where the last equality is shown when using δijDij ¼ Dkk ¼ 0 due to volume constancy. The effective strain rate related to the von Mises yield criterion will be established first by inserting von Mises effective stress (4.134) and then by using Levy-Mises constitutive equations (4.143): σ ε_ ¼ σ 0ij Dij ) ε_ ¼

σ 0ij Dij σ

1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D D σ 0ij Dij 2 _ ij ij 3 λ _ Dij Dij ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ε ¼ 3 3 1 3 0 0 D D σ σ   ij ij 2 3λ_ 2 2 ij ij (4.151)

Following a similar approach, the effective strain rate for the anisotropic material can be determined from Eq. (4.150) when inserting the effective stress defined in Eq. (4.140) and using Hill’s constitutive equations in Eqs (4.145), (4.146):

Metal forming

240

1 DM D σ 0ij Dij pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ ij ijkl kl λ _ε ¼ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ε_ ¼ 2Dij Mijkl Dkl σ 11 1 0 Dij Mijkl Dkl Pijkl σ ij Pijkl σ 0kl 2 λ_ 2 2 (4.152) σ 0ij Dij

Written out, the effective strain rate is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi u " 2 2 2 u F D2 + GD2 + H D2 ð2D12 Þ ð2D23 Þ ð2D31 Þ 11 22 33 ε_ ¼ t2 + + + FG + FH + GH 2N 2L 2M (4.153) An equation similar to Eq. (4.52) can be written by introducing a different d matrix as follows: 2

ε_ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fDgT dfDg

F 6 k’ 6 60 6 6 6 60 6 d¼6 60 6 6 6 60 6 4 0

0 0 G 0 k’ H 0 k’ 0 0

0

0

0

0

0

0

0

0

0 1 2L 0

0

0 1 2N 0 0 0 0 0

0

0 1 2M

3

7 7 7 7 7 7 7 7 7, k’ ¼ FG + FH + GH 7 7 7 7 7 7 5 (4.154)

To complete the formulation, it is necessary to determine the proportionality factor of the principle of normality for associated plasticity (4.141). This is done first for the isotropic material isolating the proportionality factor in LevyMises flow rule (4.143), applying (4.150) and the von Mises effective stress, Dij σ 0ij Dij σ ε_ ε_ λ_ ¼ 0 ¼ 0 0 ¼ 2 ¼ 3σ ij 3σ ij σ ij 2σ 2σ

(4.155)

which completes Levy-Mises constitutive equations: Dij ¼ σ 0ij

3ε_ 2σ , σ 0ij ¼ Dij 2σ 3ε_

(4.156)

Similarly, the anisotropic proportionality factor is found by isolation in the constitutive equations (4.145), application of Eq. (4.150) and the effective stress (4.140),

Finite element flow formulation

σ 0ij Dij Dij ε_ ε_ λ_ ¼ ¼ ¼σ 2¼ 0 0 0 Pijkl σ kl σ ij Pijkl σ kl 2σ 2σ

241

(4.157)

The proportionality factors are seen to be identical for the isotropic and anisotropic materials. Hence, the complete anisotropic constitutive equations are given by Dij ¼

ε_ Pijkl σ 0kl 2σ

(4.158)

or 2σ (4.159) Mijkl Dkl ε_ The modifications in the implementation to include the anisotropic formulation involves the replacement of the d matrix as defined in (4.154) when building the stiffness matrix and replacement of the calculation of deviatoric Cauchy stresses according to (4.159) with Mijkl defined in (4.149). In matrix notation, Eq. (4.159) can be written as follows: σ 0ij ¼

2σ MfDg (4.160) ε_ An additional complication is that the anisotropic yield criterion is dependent on the orientation relative to the coordinate system, which was not the case for the von Mises yield criterion. Therefore, when the workpiece deforms and elements start to rotate in the global coordinate system, there is a need for rotation of the anisotropic yield criterion to be aligned with the local element axes. Rotation between global and local axes is presented in the continuation. fσ0 g ¼

Rotation between global axes and material axes The increment of rigid body rotation in each element is described by the incremental rotation matrix ΔR ¼ I + WΔt

(4.161)

which is a summation of the identity matrix I and the product of the time increment and the continuum spin tensor W defined as follows:

242

Metal forming



W¼

1 ∂vi ∂vj  2 ∂xj ∂xi



2

3 0 W12 W13 ¼ 4 W12 0 W23 5 W13 W23 0

(4.162)

The continuum spin tensor is calculated at each time step in each element through the shape function derivatives in the centre of the element (in natural coordinates). The total rigid body rotation of an element can be accumulated according to Ri ¼ Ri1 + ΔRi  I

(4.163)

where i is the step number. The rotation matrices in Eqs (4.161), (4.163) are capable of rotating 3 3 matrices. In the anisotropic formulation, it is necessary to rotate the d matrix in each time increment to update it to the current material axes. The diagonal matrix is therefore stored as a 3 3 matrix 2 F 1 1 3

F G H 1 1 1 d ¼ diag , , , , , k’ k’ k’ 2N 2L 2M k0 ¼ FG + FH + GH



6 k0 2N 2M 7 6 1 G 1 7 7 ) d¼6 6 2N k0 2L 7 4 5 1 1 H 2M 2L k’ (4.164)

which follows the same storage format as used for stress components 2 3 σ 11 σ 12 σ 13 fσg ¼ fσ 11 , σ 22 , σ 33 , σ 12 , σ 23 , σ 31 gT ) σ ¼ 4 σ 12 σ 22 σ 23 5 (4.165) σ 13 σ 23 σ 33 With the 3 3 format in (4.164), the d matrix is rotated into the new material axes in each time increment i according to di ¼ ΔRdi1 ΔRT

(4.166)

and transferred back into the diagonal matrix storage when building the element stiffness matrices. In the calculation of deviatoric stress components (4.160), the M matrix is defined in the material axes. The rate of deformation tensor is therefore temporarily rotated from the global axes to the material axes by applying the total rotation matrix Dmat ¼ RDRT

(4.167)

Finite element flow formulation

243

Calculation of the deviatoric stresses with respect to material axes is now performed as fσ0 gmat ¼ 2σε_ MfDgmat according to (4.160), and the deviatoric stress components with reference to the global axes are recovered by reverse rotation: σ0 ¼ RT σ0 mat R

(4.168)

As the Cauchy stress tensor is objective, there are no complications related to rotation.

4.12.5 Elastic effects The flow formulation is based on rigid-plastic/viscoplastic constitutive modelling and does not consider elastic deformation, which is typically a valid approximation when deformations are large. However, there are metal forming processes such as deep drawing and bending, amongst others, where elastic deformation and spring back play a significant role. Mori et al. (1996) presented the inclusion of elastic deformation in the flow formulation, and the implementation described in this section follows a similar procedure. All elements initially start in the elastic regime. When the effective stress of an element comes to the vicinity of the initial yield stress σ Y, its deformation regime is changed into elastic–plastic, and the elastic constitutive equations are replaced by the elastic–plastic constitutive equations (also known as the Prandtl-Reuss constitutive equations). In practice, this requires comparing the effective strain ε against εY ¼ σ Y/E (where E is the elasticity modulus). When the effective strain ε has exceeded this threshold, the element’s deformation regime is changed once more time, but this time into rigidplastic in order to account for large deformations, whilst ignoring further elastic deformation. With reference to Fig. 4.19, the choice of the deformation regime for each element is decided based on 8 elastic, ε fl σ Y =E < deformation regime ¼ elastic  plastic, fl σ Y =E < ε< fu σ Y =E : rigid  plastic=viscoplastic, ε fu σ Y =E (4.169) where fl and fu are parameters defining the range of strains near the initial yielding, where elastic–plastic deformation shall be taken into account. Typical values around fl ¼ 0.95 and fu ¼ 1.05 can be chosen, but must be decided in connection with the applied time step for a given problem.

244

Metal forming

Fig. 4.19 Limiting strain ranges defining the elastic, elastic–plastic and rigid-plastic/ viscoplastic deformation regimes.

A stress situation in the vicinity of yielding is illustrated in Fig. 4.20, where a stress path is exceeding the initial yield stress of the material causing strain hardening. The present stress state (point P) is elastic, i.e. σ P < σ Y . The assumed load increment will cause a stress path through initial yielding (point Q) followed by strain hardening to a stress state in point R with effective stress, σ R ¼ σ P + Δσ, equal to the new flow stress. With reference to

Initial yield locus

R S

Y 3 2

σ2

Q P

σ

Δσ

O 3 σ 2 3

3 2

σ1

Stress path of the element

Fig. 4.20 Schematic representation in the π-plane of the three-dimensional principal stress space of the simplified procedure to determine the increment ratio r that is necessary for an elastic element to yield. (Adapted from Yamada, Y., Yoshimura, N., Sakurai, T., 1968. Plastic stress strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Int. J. Mech. Sci. 10, 343–354.)

Finite element flow formulation

245

Fig. 4.20, a ratio r of the elastic part of the stress increment to the total stress increment is defined and approximated as r¼

PQ σ Y  σ P  PR σR  σ P

(4.170)

The right-hand side of the foregoing equation is an approximation of the increment ratio proposed by Yamada et al. (1968) for easier implementation. Full derivation of the original increment ratio is included in Appendix D. The ratio r in (4.170) was originally used to scale the force increment according to the elastic element closest to yielding to achieve a situation where it just reaches the yield stress. Hereafter, the element will be considered plastic. Another approach is to avoid splitting the time step or the force increment. The ratio of the elastic contribution to the stress increment is instead used to scale the amount of the stress–strain matrix stemming from both the elastic and the elastic–plastic constitutive equations according to Δσ ¼ Eðrde + ð1  r Þdep ÞΔε

(4.171)

The elastic de and elastic–plastic dep partial stiffness coefficient matrices were obtained numerically by Marcal (1965) after inversion of the elastic and elastic–plastic constitutive equations, respectively. Inversion of the elastic constitutive equations results in 2 3 1ν SYM 6 1  2ν 7 6 ν 7 1  ν 6 7 6 1  2ν 1  2ν 7 6 7 6 ν 7 ν 1ν 6 7 1 6 1  2ν 1  2ν 1  2ν 7 de ¼ (4.172) 6 7 1 7 1 + ν6 0 0 0 6 7 2 6 7 6 7 1 6 0 7 0 0 0 6 7 2 4 1 5 0 0 0 0 0 2 Yamada et al. (1968) derived explicitly the corresponding matrix for the elastic–plastic deformation regime by inversion of the elastic–plastic constitutive equations (also known as the Prandtl-Reuss equations)

246

Metal forming

3 2 1  ν σ0 x  SYM 7 6 7 6 1  2ν S 7 6 2 0 0 0 7 6 ν σx σy 1  ν σ y 7 6  7 6 1  2ν  S 1  2ν S 7 6 0 σ0 7 6 ν 0 σ0 02 σ σ σ ν 1  ν y z 7 6 x z z    7 6 1 6 1  2ν S 1  2ν S 1  2ν S 7 dep ¼ 7 6 0 2 0 0 1 + ν6 σ y τxy σ x τxy σ z τxy 7 1 τxy 7 6     7 6 2 S S S S 7 6 0 2 0 0 7 6 σ τ τ σ τ σ τ τ τ yz 1 y yz yz yz xy yz z 7 6  x     7 6 2 S S S S S 7 6 0 0 0 2 4 σ y τzx τxy τzx τyz τzx 1 τzx 5 σ x τzx σ z τzx       2 S S S S S S (4.173) 2

where S is defined as follows with H being the slope of the stress–strain curve and G being the shear modulus:   2 2 H dσ E S ¼ σ 1 + H¼ G¼ (4.174) 3 3G dε 2ð1 + νÞ The elastic–plastic solution in (4.171) requires the stress to be incremented in each step, which is not the case in the flow formulation, where the stress is given solely by the accumulated effective strain and the strain rate of the current step. In the flow formulation, the stress is therefore not necessarily saved between steps. On the contrary, in the solid formulations, the stress field of the previous step is of great importance as the new step consists of solving a stress increment. The stress of the previous step enters the equations as an initial stress, and at the end of the step, it is incremented by the solution obtained in (4.171). In general, the deformation will include rigid body rotation between simulation steps. It is therefore necessary at each step to rotate the stress from the previous step into the new configuration, both for the role of initial stress and for the incremental update in the end of the step. With the incremental rotation matrix defined in (4.161), the total stress in step i is calculated as σi ¼ Δσi + ΔRσi1 ΔRT

(4.175)

where the stress increment in step i is calculated from Eq. (4.171). The last term in Eq. (4.175) is the stress field of step i  1 rotated into the new configuration. The last term is also applied as the initial stress to step i. Elastic unloading at the end of a simulation for calculation of the residual stresses and spring back is performed by changing all elements to follow the

Finite element flow formulation

247

elastic constitutive equations and performing one more simulation step with the actual stress field as the initial stress. This calculation is valid when all loading was proportional such that no dynamic unloading occurred before the end of the process. Simulation involving dynamic unloading requires the use of a solid formulation as presented in Chapter 5.

References Altan, T., Knoerr, M., 1992. Application of 2D finite element method to simulation of cold forging process. J. Mater. Process. Technol. 35, 275–302. Alves, L.M., Martins, P.A.F., 2009. Nosing of thin-walled PVC tubes into hollow spheres using a die. Int. J. Adv. Manuf. Technol. 44, 26–37. Bay, N., Wanheim, T., 1976. Real area of contact and friction stress at high pressure sliding contact. Wear 38, 201–209. Bonet, J., Wood, R.D., 1997. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge. Bucknall, C.B., 2007. New criterion for craze initiation. Polymer 48, 1030–1041. Caddell, R.M., Raghava, R.S., Atkins, A.G., 1974. Pressure dependent yield criteria for polymers. Mater. Sci. Eng. 13, 113–120. Carretta, Y., Boman, R., Legrand, N., Laugier, M., Ponthot, J.P., 2015. Multiscale modelling of the micro-plasto-hydrodynamic lubrication—a crucial mechanism for friction in metal forming. In: 6th International Conference on Coupled Problems in Science and Engineering, Venice. Chen, C.C., Kobayashi, S., 1978. Rigid-plastic finite element analysis of ring compression. In: Application of Numerical Methods to Forming Processes. Applied Mechanics Division, vol. 28. ASME, p. 163. Cornfield, G.C., Johnson, R.H., 1973. Theoretical prediction of plastic flow in hot rolling including the effect of various temperature distributions. J. Iron Steel Inst. 211, 567–573. Doraivelu, S.M., Gegel, H.L., Gunasekera, J.S., Malas, J.C., Morgan, J.T., Thomas, J.F., 1984. A new yield function for compressible P/M materials. Int. J. Mech. Sci. 26, 527–535. Dunne, F., Petrinic, N., 2005. Introduction to Computational Plasticity. Oxford University Press, New York. Green, R.J., 1972. A plasticity theory for porous solids. Int. J. Mech. Sci. 14, 215–224. Hartley, P., Sturgess, C.E.N., Rowe, G.W., 1979. Friction in finite-element analyses of metal forming processes. Int. J. Mech. Sci. 21, 301–311. Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A Math. Phys. Sci. 193 (1033), 281–297. Johnson, W., Mellor, P.B., 1973. Engineering Plasticity. van Nostrand Reinhold, London. Kuhn, H.A., Downey, C.L., 1971. Deformation characteristics and plasticity theory of sintered powder materials. Int. J. Powder Metall. 7, 15–25. Lee, D.N., Kim, H.S., 1992. Plasticity yield behaviour of porous metals. Powder Metall. 35, 275–279. Lee, C.H., Kobayashi, S., 1973. New solutions to rigid plastic deformation problems using a matrix method. J. Eng. Ind. 95, 865–873. Malkus, D.S., Hughes, T.J.R., 1978. Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81. Marcal, P.V., 1965. A stiffness method for elastic-plastic problems. Int. J. Mech. Sci. 7, 229–238.

248

Metal forming

Markov, A.A., 1947. On variational principles in theory of plasticity. Prikladnaia Matematika i Mekhanika II, 339–350. Martins, P.A.F., Barata Marques, M.J.M., 1991. A general three-dimensional finite element approach for porous and dense metal-forming processes. J. Eng. Manuf. 205, 257–263. Mattiasson, K., 2010. FE-models of the sheet metal forming processes. In: Banabic, D. (Ed.), Sheet Metal Forming Processes: Constitutive Modelling and Numerical Simulation. Springer, Berlin. McGinty, R., 2020. Continuum Mechanics Notes. www.continuummechanics.org. Mori, K., Wang, C.C., Osakada, K., 1996. Inclusion of elastic deformation in rigid-plastic finite element analysis. Int. J. Mech. Sci. 38, 621–631. Nielsen, C.V., Martins, P.A.F., 2017. Parallel direct solver for finite element modelling of manufacturing processes. In: Davim, J.P. (Ed.), Computational Methods and Production Engineering. Woodhead Publishing, Cambridge. Nielsen, C.V., Zhang, W., Alves, L.M., Bay, N., Martins, P.A.F., 2013. Modelling of Thermo-Electro-Mechanical Manufacturing Processes With Applications in Metal Forming and Resistance Welding. Springer, London. Nielsen, C.V., Zhang, W., Perret, W., Martins, P.A.F., Bay, N., 2015. Three-dimensional simulations of resistance spot welding. J. Automob. Eng. 229, 885–897. Puso, M.A., Laursen, T.A., 2004. A mortar segment-to-segment contact method for large deformation solid mechanics. Comput. Methods Appl. Mech. Eng. 193, 601–629. Raghava, R.S., Caddell, R.M., 1973. A macroscopic yield criterion for crystalline polymers. Int. J. Mech. Sci. 15, 967–974. Rebelo, N., Kobayashi, S., 1980a. A coupled analysis of viscoplastic deformation and heat transfer—I. Theoretical considerations. Int. J. Mech. Sci. 22, 699–706. Rebelo, N., Kobayashi, S., 1980b. A coupled analysis of viscoplastic deformation and heat transfer—II. Applications. Int. J. Mech. Sci. 22, 707–718. Shima, S., Oyane, M., 1976. Plasticity theory for porous metals. Int. J. Mech. Sci. 18, 285–291. Silva, C.M.A., Nielsen, C.V., Alves, L.M., Martins, P.A.F., 2015. Environmentally friendly joining of tubes by their ends. J. Clean. Prod. 87, 777–786. Spitzig, W.A., Richmond, O., 1979. Effect of hydrostatic pressure on the deformation behaviour of polyethylene and polycarbonate in tension and in compression. Polym. Eng. Sci. 19, 1129–1139. Sternstein, S.S., Ongchin, L., 1969. Yield criteria for plastic deformation of glassy high polymers in general stress fields. Polymer Prepr. 10, 1117–1124. Tekkaya, A.E., 2000. Simulation of metal forming processes. In: Banabic, D. (Ed.), Formability of Metallic Materials. Springer, Berlin. Tekkaya, A.E., 2007. Master of Engineering in Applied Computational Mechanics (ESoCAET). Atilim University, Ankara. € unyagiz, E., Christiansen, P., Nielsen, C.V., Bay, N., Martins, P.A.F., 2017. Revisiting Ust€ liquid lubrication methods by means of a fully coupled approach combining plastic deformation and liquid lubrication. J. Eng. Tribol. 231, 1425–1433. Whitney, W., Andrews, R.D., 1967. Yielding of glassy polymers: volume effects. J. Polym. Sci. Part C 16, 2981–2990. Yamada, Y., Yoshimura, N., Sakurai, T., 1968. Plastic stress strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Int. J. Mech. Sci. 10, 343–354. Zavarise, G., Wriggers, P., 1998. A segment-to-segment contact strategy. Math. Comput. Model. 28, 497–515. Zienkiewicz, O.C., 1983. The Finite Element Method. McGraw-Hill, New York. Zienkiewicz, O.C., Godbole, P.N., 1974. Flow of plastic and viscoplastic solids with special reference to extrusion and forming processes. Int. J. Numer. Methods Eng. 8, 3–16.

Finite element flow formulation

249

Zienkiewicz, O.C., Jain, P.C., On˜ate, E., 1978a. Flow of solids during forming and extrusion. Some aspects of numerical solutions. Int. J. Solids Struct. 12, 15–38. Zienkiewicz, O.C., On˜ate, E., Heinrich, J.C., 1978b. Plastic flow in metal forming—I. Coupled thermal behaviour in extrusion—II. Thin sheet forming. In: Proceedings of Winter Annual Meeting of ASME on Application of Numerical Methods to Forming Processes. AMD, vol. 28. ASME, p. 107. Zienkiewicz, O.C., On˜ate, E., Heinrich, J.C., 1981. A general formulation for coupled thermo flow of metals using finite elements. Int. J. Numer. Methods Eng. 17, 1497–1514.

CHAPTER 5

Introduction to the finite element solid formulation* 5.1 Introduction In the mid-1960s, the finite element method was already well established for the analysis of elastic structures subjected to small deformations (Zienkiewicz, 1967). Therefore, it is not surprising that the earliest applications of the finite element method in plasticity were built upon the existing computer programs after breaking the total applied force into increments, and considering both linear elastic and nonlinear elastic-plastic constitutive equations. Marcal (1965) was among the research pioneers in the field and introduced the concept of partial stiffness coefficients to relate the change in stress {dσ} with the strain increment {dε} as follows: fdσg ¼ dfdεg ) dσ i ¼

6 X ∂σ i j¼1

∂εj

dεj dij ¼

∂σ i ∂εj

(5.1)

Partial stiffness coefficients were applied in both elastic and elastic-plastic regimes, and mean partial stiffness coefficients resulting from the weighted contributions of elastic and elastic-plastic regimes were considered for those regions in the material that would yield during an increment:    ∂σ i  ∂σ i  ∂σ i  ¼ r  + ð1  r Þ  (5.2) ∂εj mean ∂εj e ∂εj ep The parameter r in the equation mentioned earlier was obtained from the ratio between the strain to cause yield and the total strain calculated from the increment of force in case deformation was elastic. The partial stiffness coefficients (5.1) in both elastic and elastic-plastic regimes were later designated as the elastic de and elastic-plastic dep stressstrain matrices and were obtained numerically by Marcal (1965) after * Brian N. Legarth (Technical University of Denmark, Lyngby, Denmark), Chris V. Nielsen (Technical University of Denmark, Lyngby, Denmark), and Paulo A.F. Martins (IDMEC, Instituto Superior T
ecnico, University of Lisbon, Portugal). Metal Forming https://doi.org/10.1016/B978-0-323-85255-5.00007-8

© 2021 Elsevier Inc. All rights reserved.

251

252

Metal forming

inversion of the Hooke and Prandtl-Reuss stress-strain constitutive equations, respectively. The first applications of the aforementioned methodology consisted of the analysis of tubes under internal pressure (Marcal, 1965) and notched test specimens (Marcal and King, 1967), but its use in metal forming only began after Yamada et al. (1968) derived the explicit form of the elastic-plastic stiffness coefficients, i.e. the explicit form of the elastic-plastic stress-strain matrix as follows: fdσg ¼ Edep fdεg ¼ 2ð1 + νÞGdep fdεg

(5.3)

where ν is the Poisson ratio, E is the elasticity modulus, G is the shear elasn oT ticity modulus, dε ¼ εx , εy , εz , γ xy , γ yz , γ zx , and dep is the elastic-plastic stress-strain matrix as given in Eq. (4.173) of Chapter 4. The methodology proposed by Yamada et al. (1968) was straightforwardly applied to metal forming processes and consisted of replacing the elastic stress-strain matrix de (see Eq. (4.172)) relating the stress and strain increments in elasticity by the elastic-plastic stress-strain matrix dep in the yielded elements along the successive increments of force. The transition of the elements from elastic to elastic-plastic regimes was controlled by means of an explicit integration scheme that consisted of determining a factor designated as rmin (Appendix D with reference to Fig. D.1) for estimating the force increment that was necessary for the elastic element having the largest effective stress to yield. Lee and Kobayashi (1970) considered the weak form of the governing quasi-static equilibrium equations in rate form (in current configuration) ∂σ_ ij =∂xj ¼ 0 and utilised an explicit integration scheme similar to that of Yamada et al. (1968) to analyse the indentation of a solid specimen by a flat punch. The finite element computer programs for small deformations were a breakthrough in metal forming due to their capability of removing many of the restrictions imposed by other methods of analysis regarding the utilisation of complex geometries, realistic frictional conditions and actual material properties. However, despite all these advantages, the use of small deformations soon revealed two major drawbacks: (a) The geometric nonlinearity resulting from the differences of the final geometry with respect to its original geometry cannot be described by small deformations due to the amount of rigid-body rotation and translation involved in metal forming processes (Example 3.1 of Chapter 3),

Introduction to the finite element solid formulation

253

(b) The use of explicit integration schemes without control of the residual force vector gives rise to a steady increase in the nonequilibrated forces (i.e. to differences between the external and internal forces) along the incremental solution process. As a result of these drawbacks, the 1970s saw the appearance of two major approaches to successfully deal with large plastic deformations: firstly, the flow formulation (Chapter 4), which neglects elastic deformation but is a valuable approach for the numerical modelling of mechanical, thermo-mechanical and electro-thermo-mechanical metal forming processes. The other approach is the solid formulation based on large deformations that was introduced by Hibbitt et al. (1970). In fact, because the original development by Hibbitt et al. (1970) made use of a total Lagrangian formulation, in which the reference state is the original undeformed configuration, and because this reference state is not convenient for metal forming due to the large amount of deformation involved, the application of the solid formulation to metal forming only started a couple of years later when Nagtegaal et al. (1974) introduced the updated Lagrangian approach that uses the current state under consideration as the reference state. The appeal of the updated Lagrangian approach to metal forming was obvious because it allowed upgrading the existing small deformation finite element computer programs to large deformations by including stress rate variables that could effectively handle the nonlinear geometrical effects such as the rate of deformation tensor and the Jaumann rate of the Cauchy stress tensor (refer to Chapter 3). The following sections present an introduction to the theoretical foundations of the quasi-static solid formulation for large deformations using an updated Lagrangian approach. Several choices were made to keep the overall presentation as simple as possible, to avoid repetition of concepts and to keep the notation as close as possible to that utilised in the previous finite element chapters. The first choice was to present the quasi-static finite element equations in the rate form so that comparisons with the finite element flow formulation (Chapter 4) are easier. The second choice was to employ a hypoelastic plastic formulation (refer to Appendix B) based on the additive decomposition of the rate of deformation tensor D into elastic De and plastic Dp parts: D ¼ De + Dp

(5.4)

254

Metal forming

This is based on the assumption that elastic strains are small and allows determining the stress rate by using De in combination with the rate form of the elastic constitutive equations (also known as Hooke’s law) and Dp in combination with the rate form of the plastic constitutive equations. Once the stress rate is known, integration over time allows obtaining the stress state. The third choice was to avoid repeating the presentation of implicit integration schemes for checking equilibrium at each increment of time by means of iterative procedures meant to minimise the residual force vector R (see Eq. (3.29)), although it violates the general approach outlined in Table 3.3 of Chapter 3. The implicit integration schemes are widely used in computer programs based on the quasi-static solid formulation (e.g. Menezes and Teodosiu, 2000) and are based on the utilisation of different numerical techniques such as the direct iterative and the Newton-Raphsonbased methods (Section 3.6 of Chapter 3 and Section 4.4 of Chapter 4). Alternatively, the authors decided to utilise an explicit integration scheme that minimises the drift of the solution away from the true equilibrium path based on the utilisation of equilibrium correction terms. The fourth choice was to simplify the stress-state update by avoiding iterative procedures on the time integration of the constitutive equation. In conclusion, the chapter is not aimed at providing knowledge on the quasi-static solid formulation as profound as that provided on the quasi-static flow formulation of Chapter 4, but to provide the basis for those readers who want to progress into more complex readings and computer implementations that make use of implicit time integration schemes for the minimisation of the residual force and the time integration of the constitutive equations.

5.2 Theoretical fundamentals The velocity gradient L is a nonsymmetric tensor L 6¼ LT that can be decomposed into symmetric D ¼ DT and antisymmetric parts W ¼  WT as follows (3.10): L¼D+W

(5.5)

The symmetric part of the velocity gradient L is the rate of deformation tensor D as given in Eq. (3.11): D¼

 1 L + LT 2

(5.6)

Introduction to the finite element solid formulation

255

The antisymmetric part of the velocity gradient L is the continuum spin tensor W as given in Eq. (3.12):  1 (5.7) L  LT 2 In the solid formulation based on hypoelastic plastic implementations, both small elastic rates of deformation (small elastic strain rates) DE and finite plastic rates of deformation DP are considered, such that W¼

D ¼ D E + DP

(5.8)

The weak variational form of the stress equilibrium equations can be written as follows: ð ð ð σ : δDdV  bδvdV  tδvdS ¼ 0 (5.9) V

V

St

where all integrations are carried out in the current deformed configuration, i.e. V and S. Here, σ ¼ σT is the symmetric Cauchy stress tensor, b are the body forces, v is the velocity field and t are surface tractions referring to the current configuration. Neglecting body forces, the weak variational form of the stress equilibrium equations (5.9) can be written as (see also Eq. (3.38)): ð ð σ : δDdV  tδvdS ¼ 0 (5.10) V

St

Example 5.1 Inadequacy of the Cauchy stress rate

The kinematics of large deformations is not consistent with the weak variational form of the stress rate equilibrium equations being directly obtained from Eq. (5.10) as follows: ð ð σ_ : δDdV  t_ δvdS ¼ 0 V

St

For understanding the reason why this is not possible, let us consider a rod with cross-sectional area A subjected to a uniaxial force F and undergoing pure rotation in the XY original reference frame (Fig. 5.1). We will first demonstrate that the aforementioned inconsistency is not related to the rate of deformation tensor D, which is independent of rigidbody rotation (Section 3.5 of Chapter 3). For this purpose, let us consider the deformation gradient F (3.1) of the rod undergoing rigid-body rotation:

256

Metal forming

Y

Y F

θ

F

y

A

y x

o

o

x

z

z

F

O

F

O X

Z

X

Z

(A)

(B) Y

x

F

F

y o z

O Z

X

(C) Fig. 5.1 Rotation of a rod of cross-sectional area A subjected to uniaxial loading at the (A) initial, (B) intermediate, and (C) final positions.

2

3 ∂x ∂x ∂x 6 ∂X ∂Y ∂Z 7 2 3 6 7 cos θ sin θ 0 6 7 ∂x 6 ∂y ∂y ∂y 7 4 F¼ ¼6 7 ¼  sin θ cos θ 0 5 ∂X 6 ∂X ∂Y ∂Z 7 0 0 1 6 7 4 ∂z ∂z ∂z 5 ∂X ∂Y ∂Z As expected, the deformation gradient F is equal to the rotation matrix R corresponding to the transformation between the XY original coordinate system and the xy corotational coordinate system. The rate of change of the deformation gradient F_ is given by

Introduction to the finite element solid formulation

2

3  sin θ cosθ 0 F_ ¼ θ_ 4  cos θ  sin θ 0 5 0 0 0 and the velocity gradient L (3.9) 2  sin θ cos θ _ 1 ¼ θ_ 4  cos θ  sin θ L ¼ FF 0 0

is equal to 32 3 2 3 0 cos θ  sin θ 0 0 1 0 0 54 sin θ cos θ 0 5 ¼ θ_ 4 1 0 0 5 0 0 0 1 0 0 0

This allows concluding that the rate of deformation tensor D (3.11) is equal to zero, meaning that it is independent of rigid-body rotation: 0 2 3 2 31 0 1 0 0 1 0  1 1 D ¼ L + LT ¼ @θ_ 4 1 0 0 5 + θ_ 4 1 0 0 5A ¼ 0 2 2 0 0 0 0 0 0 Thus, the problem of inconsistency must be due to the fact that the stress rate equilibrium equations cannot be expressed in terms of the Cauchy stress rate σ_ tensor as follows:         ∂σ_ ij ∂ ∂σ ij ∂ ∂σ ∂σ_ ¼ ¼0 ¼ ¼0 ∂Xj ∂t ∂Xj ∂t ∂X ∂X For this purpose, let us consider the stress state at time t (Fig. 5.1A): 2 3 20 0 03 σ XX σ XY σ XZ 6 F 7 05 σt ¼ 4 σ YX σ YY σ YZ 5 ¼ 4 0 A σ ZX σ ZY σ ZZ 0 0 0 and the stress state at time t + Δt (Fig. 5.1C), i.e. after completing the 90 degrees rotation: 2 3 2 3 F σ XX σ XY σ XZ 0 0 6A 7 7 σt + Δt ¼ 4 σ YX σ YY σ YZ 5 ¼ 6 4 0 0 05 σ ZX σ ZY σ ZZ 0 0 0 It may be concluded that the stresses change with rotation with respect to the original coordinate system XY, and therefore, the Cauchy stress rate tensor σ_ 6¼ 0. This result explains the reason why the time derivative of the stress equilibrium equations expressed in terms of the Cauchy stress rate tensor σ_ cannot serve as starting point to derive the finite element equations of the solid formulation in the rate form. However, it is worth noticing that although σ_ is dependent upon rigidbody rotation, the Cauchy stress tensor σ itself obeys the transformation law for second-order tensors:

257

258

Metal forming

σ∗ ¼ R σ R T where R is rotation matrix and σ∗ is the corotational stress tensor. These results mean that although rigid-body rotation changes the Cauchy stress _ it does not change the Cauchy stress state tensor σ with rate tensor σ, respect to the xy corotational coordinate system. In other words, the Cauchy stress tensor σ is an objective (or frame indifferent) measure of stress.

Example 5.2 Adequacy of the Jaumann rate of the Cauchy and Kirchhoff stress tensors r

The Jaumann rate of the Cauchy stress tensor σ was introduced in Section 3.6 of Chapter 3 as follows: r

σ ¼ σ_ + σW  Wσ where W is the continuum spin tensor (3.12). The objective of this example r is to confirm that σ is independent of rigid-body rotation in a rod subjected to uniaxial loading and rigid-body rotation (refer to Fig. 5.1). For this purpose, let us rearrange the previous equation as follows: r

σ_ ¼ σ σW + Wσ We start by determining the continuum spin tensor W: 0 2 3 2 31 2 3 0 1 0 0 1 0 0 1 0   1 1 W ¼ L  LT ¼ @θ_ 4 1 0 0 5  θ_ 4 1 0 0 5A ¼ θ_ 4 1 0 0 5 2 2 0 0 0 0 0 0 0 0 0 Now, let us calculate  σW + Wσ for the final position of the rod corresponding to 90 degrees rotation (θ ¼ π/2, Fig. 5.1C): 2

σ XX σ XY σ XZ

32

0

1 0

0

0 0 3

3

2

0

1 0

32

σ XX σ XY σ XZ

3

6 76 7 6 76 7 76 7 _6 76 7 σW + Wσ ¼ θ_ 6 4 σ YX σ YY σ YZ 54 1 0 0 5 + θ 4 1 0 0 5 4 σ YX σ YY σ YZ 5 0 0 σ ZX σ ZY σ ZZ F 3 0 0  2σ XY σ YY  σ XX σ yz A 7 6 7 π6 6 F 7 _ θ 6 7 _ 2σ XY σ zx 5 ¼ 6 ¼ θ 4 σ YY  σ XX 7 θ¼ 0 05 2t 4  t A σ yz σ zx 0 0 0 0 σ ZX σ ZY σ ZZ

2

2

0

Taking into consideration that the Cauchy stress tensor σ is frame indifferent, the transformation law for second-order tensors allows writing the stresses σ ∗ij in the final configuration (after 90 degrees rotation) as a function of the stresses σ ij in the XY original reference frame as follows:

Introduction to the finite element solid formulation

2 3 0 cos θ sin θ 0 6 σ∗ ¼ R σ RT ¼ 4  sin θ cos θ 0 56 40 0 0 1 0 2

259

3 0 0 2 cos θ  sin θ 0 3 F 7 0 74 sin θ cos θ 0 5 A 5 0 0 1 0 0

2 3 2 sin θ cos θ 0 F 4 sin θ ¼ sin θ cos θ cos 2 θ 0 5 A 0 0 1 Thus, the Cauchy stress rate tensor σ_ after 90 degrees rotation is given by 2 3 F 3 2 2 2 0  0 2 sin θ cosθ cos θ  sin θ 0 6 A 7 7 dσ∗ _ F 6 7 π6 2 2 6 7 F ¼ θ 4 cos θ  sin θ 2 cosθ sin θ 0 5 ¼ 6 σ_ ¼ 0 07 dt A 2t 4  5 A 0 0 0 0 0 0 ¼ σW + Wσ r

Remembering that σ_ ¼ σ σW + Wσ and taking into consideration that σ_ ¼ σW + Wσ during the rigid-body rotation of a bar subjected to uniaxial loading, we were able to verify the spin invariance (or corotational characteristics) of the Jaumann rate of the Cauchy stress r tensor (i.e. σ ¼ 0) for this particular example. It is worth mentioning that a similar result is obtained for the Jaumann r rate of the Kirchhoff stress tensor τ ¼ 0 because the Kirchhoff stress tensor τ T is a symmetric tensor (τ ¼ τ ) identical to the Cauchy stress tensor σ whenever material deformation is incompressible like, for example, in the plastic deformation of metallic materials. r The Jaumann rate of the Kirchhoff stress tensor τ will be utilised throughout this chapter. :

Introducing the superposed dot, i.e. ðÞ ¼ ∂ðÞ=∂t, as the intrinsic time rate as well as the nonsymmetric first Piola-Kirchhoff stress tensor, hereafter named as p 6¼ pT, and its work-rate conjugate measure, the nonsymmetric deformation gradient F 6¼ FT, we can write the weak form of the quasi-static rate equilibrium equations (i.e. principle of virtual work rate) (McMeeking and Rice, 1975) (see also Eq. (3.42)) as follows: ð ð _ p_ : δFdV 0 ¼ (5.11) t_ 0 δvdS0 V0

St0

where all integrations are carried out in the initial reference configuration, i.e. V0 and S0, such that the surface tractions t0 are referring to the initial configuration as well.

Metal forming

260

Aiming at a formulation that leads to a symmetric set of equations to be solved in the finite element implementation, it is convenient to relate the time rate of the nonsymmetric first Piola-Kirchhoff stress tensor p_ with r

r

the Jaumann rate ðÞ of the symmetric Kirchhoff stress tensor τ as follows (McMeeking and Rice, 1975): r

p_ ¼ τ σD  Dσ + σLT r

p_ ij ¼ τij  σ jk Dki  Djk σ ki + σ ik vj, k

(5.12)

where the last three terms account for the geometrical nonlinearities (see Section 3.4). It is worth mentioning that several equations of this chapter (e.g. Eqs. (5.12), (5.13)) will be presented in both matrix (bold symbols) and tensor (index) notations for pedagogical reasons. Then, the principle of virtual work rate becomes (see similarity with Eq. (3.43))

"ð # ð ð  T o 1 τ : δD  2ðDσÞ : δD + σ : δL L dV ¼ t_ 0 δvdS  σ : δDdV  t0 δvdS Δt V V St St "ð # ð ð ð ( 1 r τ ij δDij  2σ ij Dik δDkj + σ ij Lkj δLki gdV ¼ t_i0 δvi dS  σ δD dV  ti0 δvi dS Δt V ij ij V St St ð n

r

(5.13)

where all the integrations are carried out in the current deformed configuration, i.e. V and S. Obviously, the square bracket on the right-hand side is zero according to Eq. (3.38) if the current state is in equilibrium. However, during time integration with time increments Δt, the solution may drift away from the true equilibrium path. Including the bracket term on the right-hand side of Eq. (5.13) prevents such drifting and is therefore called the equilibrium correction term. For isotropic elasticity, the elastic rate of deformation DE is given by the incremental generalised Hooke’s law through Young’s modulus E and Poisson’s ratio ν. The elastic rate of deformation is arranged in a 6  1 vector: 8 E9 D11 > > > > > > > E > > > 2D > 12 > > > > > > < DE > = 22 DE ¼ (5.14) > 2DE23 > > > > > > > > > 2DE > > > > 31 > > > : E> ; D33

Introduction to the finite element solid formulation

261

n oT which for small deformations is identical to ε_ ¼ ε_ x , γ_ xy , ε_ y , γ_ yz , γ_ zx , ε_ z . The incremental constitutive relation is given by r

τ ¼ E : DE

r τij

where

2

6 6 6 6 6 E 6 E¼ ð1 + νÞð1  2νÞ 6 6 6 6 4

1ν 0 ν

(5.15)

¼ Eijkl DEkl

0 ν 1  2ν 0 2 0 1ν

0

0

0

0 0

0

0

0

0 1  2ν 2

0

0

0

0

ν

0

ν

0

0

ν

3

0 7 7 7 ν 7 7 7 (5.16) 0 7 7 7 7 0 5

1  2ν 2 0 1ν

If the incremental constitutive relation was to be based on the symmetric Cauchy stress tensor σ ¼ σT, an asymmetric set of equations were to be solved as discussed by Yamada and Sasaki (1995). The Jaumann rate of the Kirchhoff stress and the intrinsic rate of the Cauchy stress are related as follows (Yamada and Sasaki, 1995): r

r

σ_ ¼ τ + Wσ  σW  σTrðDÞ  τ + Wσ  σW r

r

(5.17)

σ_ ij ¼ τij + σ jk Wki + σ ik Wkj  σ ij Dkk  τij + σ jk Wki + σ ik Wkj where volume changes are often neglected, i.e. Tr(D)  Tr(DP) ¼ 0, due to plastic incompressibility. The updated Kirchhoff stress and the Cauchy stress are related as τ ¼ J σ (3.24), where J is the ratio of the volume in the reference configuration to the volume in the current configuration, so that J ¼ 1 instantaneously (McMeeking and Rice, 1975). In the updated Lagrangian formulation, the current configuration is always taken as the instantaneous one, which means that τ ¼ σ. The plastic part DP of the rate of deformation tensor D may be implemented by means of a rate (or time)-independent or rate-dependent incremental plasticity approach. In this chapter, we restrict ourselves to the rate-dependent (also known as ‘viscoplastic’) implementation, and write the Jaumann rate of the Kirchhoff stress as follows:   r τ ¼ E : D  DP (5.18)   r τij ¼ Eijkl Dkl  DPkl

262

Metal forming

Assuming the principle of normality and denoting the yield surface normal by nP, it is possible to write DP as P

DP ¼ ε_ nP P

DPij ¼ ε_ nPij

(5.19)

In the case of the isotropic von Mises yield surface, the normal nP is given by (Eq. (4.50)) 3σ0 (5.20) 2σ Considering the analogy between metal forming and viscous flow of non-Newtonian fluids (Sections 4.2.3 and 4.12.3 of Chapter 4), the effective P plastic strain rate ε_ may be written as follows:    1  m P n  P σ ε m P P σ ¼ C ε_ ! ε_ ¼ ε_ 0 with g ε ¼ σ 0 1 + (5.21) ε0 gðεP Þ np ¼

where ε_ 0 is a reference loading rate, m is the rate-sensitivity parameter and gðεP Þ is a hardening function, which depends on the stress value σ 0 ¼ σ Y, the hardening exponent n and the fitting parameter ε0. In Eq. (5.21), the strength coefficient C may be seen as a quantity related to the shear viscosity of the metallic materials (see Eq. (4.128) in Chapter 4), which neglecting temperature is mainly dependent on plastic strain, like the hardening function gðεP Þ. The elastic perfect-plastic behaviour is represented by n ¼ 0, and the larger the value of n is, the more the material hardens. r Hence, the plastic strain rates do not depend on the stress rates τ , but on the effective stress σ itself. It should be noted that the explicit hardening gðεP Þ presented here is taken more out of convenience than by matching a typical uniaxial elastoplastic stress-strain curve usually of the form 8σ > σ  σ0 < E ε ¼ σ 0  σ n (5.22) > : σ > σ0 E σ0 From this, the hardening function gðεP Þ can implicitly be obtained from  n σ 0 gðεP Þ gðεP Þ P ε ¼  (5.23) E σ0 E which generally requires solving for gðεP Þ by some nonlinear approach.

Introduction to the finite element solid formulation

263

Example 5.3 Analysis of the rate-sensitivity parameter

For εP ¼ 0, the hardening function is g(0) ¼ σ 0. However, from Eq. (5.21), it P is evident that the effective plastic strain rate ε_ is zero when—and only when—the effective stress σ is zero as well. In other words, plastic deformation occurs even for stresses below the ‘initial’ yield stress σ 0. Hence, no distinct yield condition can be established, but plasticity gradually evolves if the effective stress σ is nonzero. To illustrate this, the effect of the rate-sensitivity parameter m, on the P P effective plastic strain rate ε_ , is shown in Fig. 5.2 where ε_ =_ε 0 as P function of σ=gðε Þ as obtained from Eq. (5.21) is depicted for different values of the rate-sensitivity parameter m. From Fig. 5.2, it is clearly shown that the smaller the rate-sensitivity parameter m is, the more abrupt the effective plastic strain rate evolves. For m ¼ 0.005, an almost infinite large effective plastic strain rate is obtained when σ  gðεP Þ, whereas large values of m make the evolution of the effective plastic strain rate more smooth. For m ¼ 0.03 in Fig. 5.2B, the effective plastic strain rate is practically zero for σ  1:4gðεP Þ, but for σ > 1:4gðεP Þ, a nonvanishing plastic strain starts to evolve. The extreme (unrealistic) case of m ¼ 1, where the effective plastic strain rate evolves linearly with stress, is often referred to as linear viscosity and is typical of Newtonian fluids, like water. Realistic values of

4 5 ×10

1.5 m=0.005 m=0.01 m=0.03 m=1

−⋅ P ε ⋅ ε0

m=0.005 m=0.01 m=0.03 m=1

−⋅ P 4.5 ε ⋅ ε0 4

3.5

1

3 2.5 2 0.5 1.5 1 0.5 0

0 0

0.5

1

(A)

− σ g −ε P

1.5

0

0.5

1

1.5

− σ

2

g −ε P

(B)

Fig. 5.2 Effect of the rate-sensitivity parameter m on the plastic response at (A) very low P strain rates (e.g. up to ε_ ¼ 0:003=s when ε_ 0 ¼ 0:002=s) and (B) high strain rates (e.g. up P to ε_ ¼ 100=s when ε_ 0 ¼ 0:002=s).

264

Metal forming

the rate-sensitivity parameter is m  [0.005; 0.03], such that m < 0.005 can be considered as rate-independent behaviour (see Section 5.5 for the special case of m ! 0). It is also clear from Eq. (5.21) that loading by a strain rate P identical to the reference strain rate, i.e. ε_ ¼ ε_ 0 , yields gðεP Þ ¼ σ for all values of the rate-sensitivity parameter, m. A typical value of the reference strain rate is ε_ 0 ¼ 0:002=s.

5.3 Discretisation by finite elements On matrix-vector form, the intrinsic rate of the first Piola-Kirchhoff stress tensor (resembling the nominal stress) p_ can be written as (Yamada and Hirakawa, 1978) 8 9 r > 8 9 > 8 9 τ11 > > > > p L11 > > > > > > 11 > > > > > > > > > > > > > > > > > > > > > > > r > > > > > > > τ > > > > > > 12 > > > > > p L 12 > 12 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > r > > > > > > > > > > > > τ 13 p L > > > > > 13 > 13 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > r > > > > > > > > > > > > p τ L 21 21 > 21 > > > > > > > > > > > > > > > > >

< < < = = = r r r r p_ ¼ τ + EG L , p22 ¼ τ22 + EG L22 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > r > > > > > > p L 23 23 > > > > > > τ > > > > 23 > > > > > > > > > > > > > > > > > > > > > > > > > > > p31 > > >r > > > > > L > > 31 > > > > > > > τ > > > > > > 31 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > p L 32 32 > > > > > r > > > > > τ32 > > > > > > > > > > > > > > > > > > : > : > ; > ; > > > > p33 L > 33 :r ; τ33 T r r with the symmetric matrix EG ¼ EG given as follows:

(5.24)

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 r 6 EG ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

σ 11

σ 12 σ 11  σ 22 2

σ 13

0

0

0

0

0

σ 23 σ 11 + σ 22   0 2 2 σ 11  σ 33 σ 23  0 2 2 σ 22  σ 11 σ 12 2

σ 13  2 σ 12 2 σ 13  2

σ 23  2 σ 11 + σ 33  2 σ 23 2

σ 13 2 σ 12  2 σ 13  2

σ 22

σ 23

0

0

σ 22  σ 33 2

symm:

σ 12 σ 22 + σ 33  2 2 σ 33  σ 11 σ 12  2 2 σ 33  σ 22 2 

0

3

7 7 0 7 7 7 7 7 0 7 7 7 7 0 7 7 7 7 0 7 7 7 7 0 7 7 7 7 σ 13 7 7 7 7 7 σ 23 7 5 σ 33

(5.25)

266

Metal forming

The finite element discretisation now follows the procedure previously used for the flow formulation in Chapter 4. The velocities (displacement rates) v within an element are expressed in terms of nodal velocities ui of the i ¼ 1, 2, 3, …, ne nodes and the ne shape functions, Ni. Following the example in Section 4.3 for an eight-node (ne ¼ 8) hexahedral element (see Eq. (4.46)), the velocities within an element are given as follows: 8 9 vx > > > < > = (5.26) v ¼ vy ¼ NT u > > > : > ; vz where N is the 24  3 shape function matrix and u is a 24  1 vector containing the nodal velocities of the element. Arranging the rate of deformation tensor as in Eq. (5.14) leads to D ¼ Bu

(5.27)

B ¼ ½∂NT

(5.28)

where [∂] is a differentiation operator of size 6  3 2 3 ∂ 0 0 6 ∂x 7 6 7 6 7 6∂ ∂ 7 6 7 6 ∂y ∂x 0 7 6 7 6 7 6 7 6 0 ∂ 07 6 7 ∂y 6 7 6 7 ½∂ ¼ 6 7 6 ∂ ∂7 6 0 7 6 ∂z ∂y 7 6 7 6 7 6∂ ∂7 6 7 0 6 7 ∂x 7 6 ∂z 6 7 4 ∂5 0 0 ∂z

(5.29)

such that B is the standard 6  24 rate of deformation matrix (refer to the resemblance with Eq. (4.48)). Similarly, arranging the velocity gradient tensor in a 9  1 vector,

Introduction to the finite element solid formulation

8 9 L11 > > > > > > L12 > > > > > > > > > L > 13 > > > > > > > < L21 > = L ¼ L22 > > > L23 > > > > > > > > > L31 > > > > > > > > > > L 32 > > : ; L33

267

(5.30)

the velocity gradient vector L is obtained as follows: L ¼ BL u

(5.31)

BL ¼ ½∂L NT

(5.32)

where BL is a 9  24 matrix and [∂]L is another differentiation operator of size 9  3 given by the following: 2 3 ∂ 0 0 6 ∂x 7 6 7 6 7 ∂ 60 07 6 7 ∂x 6 7 6 7 ∂7 6 60 0 7 6 ∂x 7 6 7 6∂ 7 6 7 6 ∂y 0 0 7 6 7 6 7 6 7 ∂ 6 07 ½∂L ¼ 6 0 (5.33) 7 ∂y 6 7 6 7 ∂7 6 60 0 7 6 ∂y 7 6 7 6∂ 7 6 7 0 07 6 6 ∂z 7 6 7 6 7 60 ∂ 07 6 7 ∂z 6 7 4 ∂5 0 0 ∂z The finite element equations are obtained by writing the principle of virtual work rate (Eq. (5.13)) as a summation of integrations over each element m. With M being the total number of elements utilised in the discretisation, this gives

Metal forming

268

8 M ð ð ð = 16 7 m m m ¼ t_ 0 δvdS  4 σ : δDdV  t0 δvdS 5 > Δt ; m m m V

St

(5.34)

St

Adopting the notation in Eq. (5.24) after Yamada and Hirakawa (1978) instead of Eq. (5.12) to express p, _ the finite element equation for the principle of virtual work (5.34) can be written as 8 2 2 39 > ð ð ð =

X r 16 6r 7 4 τ : δD + δLT EG LdV m ¼ t_ 0 δvdSm  4 σ : δDdV m  t0 δvdSm 5 > > Δt ; m¼1 : m m m m V

V

St

St

(5.35) r

The Jaumann rate of the Kirchhoff stress tensor τ is expressed by the conr

stitutive equation τ ¼ E : DE (5.18). The finite element computation of the   r first left-hand term in Eq. (5.35), τ : δD ¼ E : D  DP : δD, indirectly r

handles the increment of the Kirchhoff stress tensor Δτ ffi τ Δt from the configuration at time t to the configuration at time t + Δt as follows: t +ðΔt

Δτ ¼

r

t +ðΔt

τ dt ¼

t

E

t +ðΔt

E : D dt ¼ t

  E : D  DP dt ¼

t

¼

E : Δε |fflffl{zfflffl}

Elastic predictor increment

t +ðΔt

  E : D  E : DP dt

t

 E : ΔεP |fflfflffl{zfflfflffl}

Plastic corrector

(5.36)

In this equation, Δτ is the spin-invariant (or corotational) increment of the Kirchhoff stress tensor and Δε and ΔεP are the total and plastic finite strain increments. The elastic predictor increment in Eq. (5.36) gives rise to a trial stress e τt + Δt placed outside the yield surface. The plastic corrector is directed towards the centre of the yield surface and is obtained from the plastic finite strain loaded elastically to bring the stress τt+Δt back onto the yield surface at time t + Δt (Fig. 5.3). With the Jaumann rate of the Kirchhoff stress defined in Eq. (5.18) and the previously defined matrices in Eqs. (5.26), (5.27) and (5.31), it is

Introduction to the finite element solid formulation

269

Plastic corrector Yield surface at time t+ Δ t

~

T t+

Δt

Elastic predictor increment T

t+ Δt

Δt 2 σ t+ 3

σ2

Stress path Tt

O

σ3

σ1

2 3

σt Yield surface at time t

Fig. 5.3 Schematic representation of the radial return method in the π-plane of the three-dimensional principal stress space (Wilkins, 1964).

possible to rewrite Eq. (5.35) as follows when the arbitrary nodal velocity δu cancels out: 8

M ð ð ð = 16 T m T m T m7 _ ¼ Nt 0 dS  4 B σdV  Nt0 dS 5 (5.37) > Δt ; m m m V

St

St

It is now shown that the term with the plastic deformation rate can be moved to the right-hand side, which leaves the following form of the global system of equations: M  X

 kmS + kmG fug ¼ fF + P + Cg

(5.38)

m¼1

where the contributions to the stiffness matrix are ð m kS ¼ BT EBdV m Vm

(5.39)

270

Metal forming

kmG

ð

r

BTL EG BL dVm

¼

(5.40)

Vm

and the contributions to the force vector are ð F ¼ Nt_ T0 dSm ð P¼ 2 C¼

16 4 Δt

BT EDP dV m

Vm

ð Vm

(5.41)

Stm

BT σdV m +

ð

(5.42) 3

7 NtT0 dSm 5

(5.43)

Stm

It is shown that if a viscoplastic formulation is considered, the small deformation element stiffness matrix km S is identical to the stiffness matrix for a standard small elastic deformation formulation and that km G is a contribution arising from the finite deformations to handle geometric nonlinearities. All plastic contributions enter through the right-hand side P, which is called the viscoplastic force (or stress) vector. It is emphasised that for a time-independent formulation, the local stiffness matrix km S will depend on the instantaneous tangent modulus (see Section 5.5). As discussed, C is the equilibrium correction vector, which can be included in order to avoid the numerical solution to drift away from its true equilibrium path. It is noted that C should contain zeros for all unloaded nodes, while nonzero forces in equilibrium with the boundary conditions appear in all the loaded nodes. In summary, these are the steps to be performed to upgrade a small deformation finite element computer program to finite deformations: 1. Add the additional element stiffness km G arising from finite deformations to the standard small deformation elastic stiffness matrix km S. 2. Obtain the rates of deformation and stress rates from D ¼ Bu (5.27) and   r τ ¼ E : D  DP (Eq. (5.18)) where DP is known from the previous increment (initially, DP ≡ 0). r 3. Update the Cauchy stress rate according to σ_ ¼ τ + Wσ  σW  σTrðDÞ (5.17).

Introduction to the finite element solid formulation

271

4. Update the Cauchy stress according to the following explicit time integration scheme t + Δt

σ

t

t +ðΔt

¼σ + t

_ _ ffi σt + σΔt σdt |{z}

(5.44)

Δσ

which is then used to determine the plastic rate of deformation DP for the next increment. 5. Update the geometry based on velocity u and increment of time Δt.

5.4 Implicit vs explicit integration procedures The procedure outlined is a simple forward Euler procedure, without any iterations or check of residuals. However, including the equilibrium correction term C in Eq. (5.38) prevents numerical drifting, which essentially corresponds to a single iteration. It is possible to further improve the accuracy at a given load step by solving the system of equations multiple times while adding the updated C without changing the external load vector F. It is clear from Fig. 5.2 that for small values of the rate-sensitivity parameter m, numerical instabilities may occur due to the sudden evolution of the effective plastic strain rate. To increase the stable time step Δt, Peirce et al. (1984) proposed the tangent modulus method. The method is still a forward Euler method without iterations, but based on a Taylor expansion of the effective plastic strain rate P ε_ in the interval between two simulation steps, the stable time step can be increased by almost a factor of 10. However, doing so the small strain element stiffness matrix km S will no longer be constant, but needs to be updated at every simulation step, as the elastic stiffness matrix E in Eq. (5.16) is affected. Similarly, the viscoplastic stress vector P in Eq. (5.42) should also be corrected (Tvergaard, 2001). An unconditionally stable algorithm for viscoplasticity has also been presented by Hughes and Taylor (1978). Example 5.4 Elastoplastic necking in uniaxial tension

To illustrate the elastoplastic large deformation finite element formulation, uniaxial tension of a bar is considered as also in Chapter 2. The initial length is L0 ¼ 20 mm, the initial square cross-sectional area is A0 ¼ 4 mm2 and the applied force is denoted as F. Due to symmetry, only one-eighth of the bar is discretised by means of 20-node hexahedral elements.

Z

F

X

Y

Y

Z

εy = 0.2

2 σ0

1.5

0.36 0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.15 0.1 0.05 0

F

σynom εy = 0.35

εy = 0.05

1 εy = 0.0015

0.5

0

0

0.1

εy = n = 0.2

0.2

0.3

εy

0.4

εy = 0.0015

X

0.05

0.20

0.35

Fig. 5.4 Tensile testing of a bar with simulated nominal stress normalised by the initial yield stress as a function of the true strain. The contour plots show the effective plastic strain at selected elongations.

Introduction to the finite element solid formulation

273

The material parameters used in the example are σ 0/E ¼ 0.003, ν ¼ 1/3, n ¼ 0.2, ε0 ¼ 0.004, ε_ 0 ¼ 0:005=s and m ¼ 0.001. Such a low value of the rate-sensitivity parameter m means that viscous effects can be ignored. Defining the strain as εy ¼ ln

L L0

where L is the current length and the stress as σ nom Y ¼

F , A0

the stress-strain response is shown in Fig. 5.4. The stress is normalised by σ 0, such that initial yielding occurs at σ y/σ 0 ¼ 1 (at the strain εy ¼ E/σ 0 ¼ 0.003). The maximum nominal load carried by the bar is at εy ¼ n ¼ 0.2 in agreement with Fig. 2.43, which includes instability (by diffuse necking) in uniaxial tension. After this point, the deformation becomes inhomogeneous as the deformation localises in the necking region. This is shown in Fig. 5.4 by the contours of the effective plastic strain at εy ¼ 0.35. The elastic part can be calculated from the stress at εy ¼ 0.35 to be εEy  1.6σ 0/E ¼ 0.005 ≪ 0.35.

5.5 Rate-independent limit By inverting the effective plastic strain rate in Eq. (5.21), it follows that: ! ! P m P m σ ε_ σ ε_ ¼ ) P ¼ ¼ 1 for m ¼ 0 (5.45) ε_ 0 ε_ 0 gðεP Þ g ðε Þ P

Because this result means that ε_ ¼ ε_ 0 at all times for m ¼ 0, it is necessary to derive the effective strain rate directly from the rate form of Eq. (5.23):  P n1 P   P n  P P g_ ðε Þ g_ ðεP Þ _εP ¼ ∂ε ¼ ∂ σ 0 gðε Þ  gðε Þ ¼ σ 0 n gðε Þ  ∂t ∂t E σ0 E E σ0 σ0 E (5.46) _ we obtain Considering Eq. (5.45) with m ¼ 0, gðεP Þ ¼ σ and g_ ðεP Þ ¼ σ, !  n1 n σ 1 P ε_ ¼  σ_ (5.47) E σ0 E

274

Metal forming

Fig. 5.5 Elastic and elastic-plastic (tangent) modulus in elastic linearly plastic stressstrain material behaviour.

Now, taking into consideration the elastic linearly plastic stress-strain curve shown in Fig. 5.5, in which the linearly plastic response is a local approximation of a general nonlinear strain hardening plastic response, the following relations are obtained for a hypoelastic plastic approach: dεt ¼ dεe + dεp )

dσ dσ dσ 1 1 1 ¼ + , ¼  Et E H H Et E

(5.48)

In the above equation, H ¼ dσ/dεp is the plastic modulus and Et ¼ dσ/dε is the tangent modulus (also known as the elastic-plastic tangent modulus) in the plastic deformation regime (σ > σ 0). Combining Eq. (5.47) over a time increment dt and Eq. (5.48) and remembering that uniaxial and effective values of stress and strain are identical, we obtain   1 n σ n1 ¼ (5.49) Et E σ 0 This result allows concluding that rate (or time)-independent plasticity can be represented through the rate-dependent plasticity model in the limiting condition of m ! 0. In practical terms, small values m < 0.005 are already sufficient to model rate-independent plasticity, while m ¼ 0 cannot be used because of numerical issues. Extrapolating the elastic-plastic tangent modulus Et to three dimensions, it follows that an alternative to Eq. (5.18) and to the finite element equations of the principal of virtual work rate Eqs. (5.34), (5.35) in rate-independent

Introduction to the finite element solid formulation

275

plasticity is to replace the elastic constitutive tensor E by the elastoplastic constitutive tensor Cep (also known as the elastoplastic tangent operator) obtained from the Prandtl-Reuss constitutive equations: r

τ ¼ Cep : D:

(5.50)

However, as shown by Simo and Taylor (1985), the utilisation of an elastoplastic tangent operator in combination with the iterative solution schemes employed in the solution of the nonlinear system of equations requires replacing the elastic-plastic constitutive tensor Cep by the so-called elastoplastic consistent constitutive tensor. This is the approach followed by commonly used elastic-plastic finite element computer programs, which requires changing the constitutive Eq. (5.18) by Eq. (5.50) and applying implicit iterative procedures for solving the resulting nonlinear set of equations (Section 3.6). Details are given in Tekkaya (2000) and Dunne and Petrinic (2005), among others.

References Dunne, F., Petrinic, N., 2005. Introduction to Computational Plasticity. Oxford, New York. Hibbitt, H.D., Marcal, P.V., Rice, J.R., 1970. A finite element formulation for problems of large strain and large displacement. Int. J. Solids Struct. 6, 1069–1086. Hughes, T.J.R., Taylor, R.L., 1978. Unconditionally stable algorithms for quasi-static elasto/visco-plastic finite element analysis. Comput. Struct. 8, 169–173. Lee, C.H., Kobayashi, S., 1970. Elastoplastic analysis of plane-strain and axisymmetric flat punch indentation by the finite element method. Int. J. Mech. Sci. 12, 349–370. Marcal, P.V., 1965. A stiffness method for elastic-plastic problems. Int. J. Mech. Sci. 7, 229–238. Marcal, P.V., King, I.P., 1967. Elastic plastic analysis of two dimensional stress systems by the FEM. Int. J. Mech. Sci. 9, 143–155. McMeeking, R.M., Rice, J.R., 1975. Finite element formulation for problems of large elastic-plastic deformation. Int. J. Solids Struct. 11, 601–616. Menezes, L.F., Teodosiu, C., 2000. Three-dimensional numerical simulation of the deepdrawing process using solid finite elements. J. Mater. Process. Technol. 97, 100–106. Nagtegaal, J.C., Parks, D.M., Rice, J.R., 1974. On the numerically accurate finite element solutions in the fully plastic range. Comput. Methods Appl. Mech. Eng. 4, 153–177. Peirce, D., Shih, C.F., Needleman, A., 1984. A tangent modulus method for rate dependent solids. Comput. Struct. 18, 875–887. Simo, J.C., Taylor, R.L., 1985. Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Eng. 48, 101–118. Tekkaya, A.E., 2000. Simulation of metal forming processes. In: Banabic, D. (Ed.), Formability of Metallic Materials. Springer, Berlin. Tvergaard, V., 2001. Plasticity and Creep in Structural Materials. Department of Mechanical Engineering—Solid Mechanics, Technical University of Denmark, Lyngby.

276

Metal forming

Wilkins, M.L., 1964. Calculation of elastic-plastic flow. In: Alder, B., Fernbach, S., Rotenberg, M. (Eds.), Methods of Computational Physics, Volume 3—Fundamental Methods in Hydrodynamics. Academic Press, New York. Yamada, Y., Hirakawa, H., 1978. Large deformation and instability analysis in metal forming process. Appl. Numer. Methods Form. Process. 28, 27–38. Yamada, Y., Sasaki, M., 1995. Elastic-plastic large deformation analysis program and lamina compression test. Int. J. Mech. Sci. 37, 691–707. Yamada, Y., Yoshimura, N., Sakurai, T., 1968. Plastic stress strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Int. J. Mech. Sci. 10, 343–354. Zienkiewicz, O.C., 1967. The Finite Element Method in Structural and Continuum Mechanics. McGraw-Hill, New York.

CHAPTER 6

Tool design* 6.1 Introduction Tools for metal forming are usually developed for specific parts. Design and construction costs are significant and may easily reach several tens of thousands of Euros before the first part is produced. There are many factors to be considered when planning a production process and designing the tools: (a) Batch size and production rate, (b) Complexity of the part to be formed and number of intermediate forming stages (preforms), (c) Critical functional surfaces and required level of geometrical and dimensional accuracy, (d) Forming temperature, (e) Equipment (presses, drop hammers) to be used, (f ) Tool materials, heat and surface treatments, (g) Preheating die temperature and procedure, (h) Lubrication system, (i) Force and distribution of main field variables (e.g. stress, strain, and temperature) applied in punches and dies. In addition to this, tools are usually designed in accordance with the characteristics of the different types and combinations of processes that are utilised for producing the parts. This opens the door for multiple ways of classifying and categorising tools. In this chapter, we will adopt a classification that is based on the classification of metal forming processes that were introduced in Chapter 1. Three groups of tools will be considered: (a) Tools for compressive forming, (b) Tools for tensile forming, combined tensile and compressive forming and bending, (c) Tools for forming by shearing. * Chris V. Nielsen (Technical University of Denmark, Lyngby, Denmark), Luis M. Alves (Instituto Superior T
ecnico, University of Lisbon, Lisbon, Portugal), Niels Bay (Technical University of Denmark, Lyngby, Denmark), and Paulo A.F. Martins (IDMEC, Instituto Superior T
ecnico, University of Lisbon, Portugal). Metal Forming https://doi.org/10.1016/B978-0-323-85255-5.00009-1

© 2021 Elsevier Inc. All rights reserved.

277

278

Metal forming

Due to this multiplicity of factors, it is impossible to write a chapter containing all the information and accumulated knowledge and experience of different toolmakers. In fact, some of the information is scattered across different books, other information is confidential and only available for internal use of toolmakers, and yet other information is personal and gets lost when a designer retires or dies. The purpose of this chapter is not to present the accumulated skills and experience of different designers, nor to replace some of the existing books, standards and internal procedures or guidelines used by designers and toolmakers around the world. The goal is to provide a set of structured information about tools, materials utilised in tool construction, procedures used in the design of punches and dies, and topics related to temperature, failure and lubrication that are inseparable from tool design. The chapter is written to give readers the first steps in tool design and the references include a number of publications where additional comprehensive information is available. For this reason, this chapter will not cover all the factors that were previously identified as being important in the design of tools. It is also worth noticing that some of these factors, like, for example, the determination of force, pressure and distribution of field variables, were already presented to readers in this book, or in courses previously attended by the readers, where the different theoretical methods of analysis of metal forming processes were presented (Fig. 6.1). We recommend the following books to get information and refresh knowledge on the theoretical methods of analysis that were not covered in the previous chapters of this book: Mielnik (1991), Rodrigues and Martins (2005), Valberg (2010), and Hosford and Caddell (2011). Theoretical Methods of Analysis Analytical Methods Static Equilibrium Methods

Num erical Methods Finite Element Method

Slab Method

Boundary Element Method

Energy Methods

Finite Volume Method

Ideal Work Method

Meshless Method

Lower Bound Method Upper Bound Method Static and Kinematic Equilibrium Methods Slip-Line Field Method

Fig. 6.1 Theoretical methods of analysis of metal forming processes.

Tool design

279

6.2 Tools for compressive forming 6.2.1 Open-die tools Open-die tools are utilised in compressive forming (forging) processes, in which there is no major lateral constraint except for friction and, consequently, no important three-dimensional confinement. Barrelling of the workpieces is caused by frictional forces that oppose outward material flow at the die interfaces and, in case of hot forming, by the utilisation of cold dies, which will rapidly lower the temperature of the billet at the die interfaces and will increase its resistance to deformation. Appropriate lubrication systems and preheating of the dies for hot forming can minimise both sources of barrelling. Typical open-die tools are simple (Fig. 6.2) and their use requires manipulation of the workpiece by hand or by a mechanical manipulator in a predetermined sequence to obtain a geometry that in most of the cases will be subsequently machined or formed with closed-die tools to produce the final required part. Recent developments in compressive forming have seen the design and construction of new open-die tools for large-scale forging of long axial components, in which a billet is reduced in vertical and horizontal directions simultaneously (Lazorkin and Melnykov, 2011; Lazorkin and Lazorkin, 2019). These new open-die tools with four acting dies (named as fourdie forging device, Fig. 6.3A) explore the concept of multidirectional dies, which will be discussed in more detail in Section 6.2.3. The lower die remains stationary in the bolster that is attached to the press table, the upper die travels with the bolster that is attached to the press ram, and two lateral dies mounted in sliding wedge holders redirect the vertical force and Flat die

Edging die

Fullering die

1

2

3

5

4

Sequence V-anvil die

Double V-anvil die

Fig. 6.2 Typical open-die tools with an example of a fabrication sequence.

280

Metal forming

Upper die Tensile stress

Dead metal zone

Bolster Billet Sliding die holder Guide

Lower die

Bolster

(A)

(B)

(C)

(D) Fig. 6.3 Schematic representation of a four-die-based tool system for compressive forging. (A) Open (left) and working (right) position of the tool during compressive forming of a billet with initial circular cross section; (B) slip-line field utilised to explain the possible occurrence of tensile stresses in the centre of a round billet compressed by conventional flat dies; (C) slip-line field utilised to explain the possible formation of a dead metal zone in the centre of a round billet compressed in a radial forging machine. (D) Photographs of a four-die-based tool system for compressive forging in the open and closed positions. (Courtesy of Lazorkin-Engineering, LLC. Adapted from Lazorkin ,V., Lazorkin, D., Kuralekh, S., 2018. New four-die forging devices (FDFD) design solutions and open-die forging technologies. Adv. Mater. 7, 1–8.)

movement of the press into the horizontal direction to simultaneously compress the billet from both sides. The advantage of reducing the cross section of the billets from four different directions simultaneously instead of using conventional open-die tools (Fig. 6.2) installed in upper and lower bolsters is the diminishing of lateral spread because material flows mainly in the longitudinal direction. This ensures better control of plastic deformation, avoids possible development of tensile stresses in the centre, which may lead to failure by cracking (Fig. 6.3B), and increases the overall productivity.

Tool design

281

The use of four-die-based tool systems is also advantageous in relation to alternative solutions based on the utilisation of radial forging machines due to the following reasons: (a) The design and construction of a four-die-based tool is much less expensive than the acquisition of a special-purpose radial forging machine, (b) Four-die-based tools can be installed in conventional presses, (c) In contrast to radial forging machines that only apply radial normal stresses on the billets (Fig. 6.3C), the four-die-based tools apply both normal and shear stresses as a result of friction and of material flow in between the dies. This allows plastic deformation to extend to the entire cross section and prevents the formation of a dead metal zone with different mechanical and metallurgical properties. The four-die-based tool concept with sliding die holders is also flexible because the flat die shown in the figure can be easily replaced by an edging or fullering die shape, if necessary. Like in conventional open-die tools, the working area of the four-die tools is usually smaller than the overall size of the workpiece subjected to compressive forming sequences (Fig. 6.3D). This allows processing large billets without requiring very large forces or heavy equipment. The component can be subsequently moved as in cogging to be forged in a different axial position.

6.2.2 Impression-die tools (closed-die tools) Impression-die tools, also named closed-die tools, enable improved control of workpiece shape in the lateral direction by the die walls and in the axial direction by bringing the die faces to a predetermined position (Fig. 6.4). Compressive forming processes performed in impression-dies are usually carried out at elevated temperatures with extra material that overfills the required final geometry extruded through a restricted narrow gap and appearing as a flash around the part. Flash is removed in secondary operations by piercing, trimming and/or machining and accounts for 5% up to 30% of the total volume of the billets. Design of impression-dies for hot compressive forming with flash sacrifices accuracy and net-shaped capability to obtain high formability, due to low material flow stresses and absence of strain hardening. As a result of this, most forming sequences involve a preforming die (if necessary) and only one, two or three impression-dies (named as ‘blocker(s)’ and ‘finish’ dies).

282

Metal forming

Upper die block

Dowel Die insert Die notch

Anvil bolster

Blocker

Blocker impression

Billet (preform)

Finish impression Dowel slot Flash Die insert

Die wing Anvil bolster key Die shank Die key Lower die block

Lower die block

(A)

(B)

Fig. 6.4 Impression-die tool to install in a hammer. (A) Billet (preform) and blocker in the open and closed positions of the impression-die. (B) Die attachment to the anvil bolster in a drop hammer. (Adapted from Finkl, 1994. Finkl Die Design Handbook 1–3. A Finkl & Sons, Chicago.)

The impression-dies can be designed and assembled as individual dies (Fig. 6.4A) or placed side by side in a single die block (Fig. 6.4B). Preforming is intended to reduce material requirements, reduce die wear, minimise the potential for defect formation, and to displace and modify the shape of the billet so that it closely matches the geometry of the impression-dies. Preforming dies will not be discussed here but they are generally flat, fullering or edging dies as shown in Fig. 6.2 or bending-shaped dies. Preforms may also be obtained by forge rolling of bars. The blocker (impression-die) imparts a general, but not exact, shape to the preforms (or billets), omitting details of the final workpiece that inhibit the natural flow of the material. It may be considered a streamline replica of the finish impression. The finish (impression-die) imparts the final size and shape to the workpiece. Impression-die tools are often installed in hammers and screw presses. Fig. 6.4A shows a typical two halve-die design, in which the attachment of the lower die block to the anvil bolster in a hammer is made by means of a dovetail-type arrangement (Fig. 6.4B). The male shank of the die block is fitted in the female notch of the anvil bolster and a tapered key is driven tight, providing a prestressed arrangement.

Tool design

283

Die insert Bolster Flash land

Flash gutter

Support plate t

Billet (preform) Blocker

w

Support plate Die insert Bolster

Flash land

(A)

t

w

Blocker Final part Finish forging

(B)

Final piercing and trimming not shown in the figure

Fig. 6.5 Detail of (A) blocker and (B) finish impression-dies to be installed in a press with details of the flash land and flash gutter. (Adapted from Thomas, A., 1998. Forging Methods. Materials Forming Technology, Sheffield.)

Fig. 6.5 shows typical blocker and finish impression-dies to be installed in a press. The die blocks are now replaced by support plates and bolsters. The latter are needed to distribute the forces over a large area and to prevent the risk of damaging the press tables, which are difficult to repair and may compromise the installation and accuracy of other tools. Bolts (not shown in the figure) are utilised to attach the tools to the ram and to the press table. The following design rules apply to the design of blocker and finish impression-dies for hot compressive forming. (a) The finish should be made slightly larger than the final workpiece to allow for shrinkage as materials cool and to account for the amount of material to be removed by machining (Fig. 6.6 and Table 6.1), (b) The batch size determines the design of tools. Expensive tooling with several preforming stages and reduced amount of material to be

Metal forming

284

Removed by machining

Low degree of replicability

Cost tm Machining Production Material Tooling

Large batch

Small batch

Final workpiece

Fig. 6.6 Machining allowance tm, tool design and overall profitability. Table 6.1 Basic guidelines for workpiece design according to notation. Height h (mm)

Fillet radius r0 (mm)

Round-off radius r1 (mm)

15

5

2.5

25

8

4

40

12

4.5

50

15

5

65

18

5.5

75

20

6

Machining allowance tm (mm)a

Inner draft angle αi (º)

Outer draft angle αo (º)

7–10 5–8 (close)

5–7 3–5 (close)

1.0

8–12 5–9 (close)

5–10 3–7 (close)

1.5 2.0

a Does not include compensation for shrinkage during cooling. Adapted from IS 3469-1, 2004.

machined from the workpiece to obtain the final geometry are easier to justify for large batch sizes (Fig. 6.6), (c) The finish and blocker are surrounded by a flash that must be designed to generate sufficient cavity pressure to fill the most difficult regions. This is because rapid cooling of the flash (because it is thin) will increase

Tool design

285

its resistance to further deformation and will help to build up pressure to force material into the remaining unfilled regions of the die cavity. The flash land thickness t (mm) of the finish dies (Fig. 6.5) can be dimensioned as follows: pffiffiffiffiffiffi (6.1) t ¼ 0:015 Ap where Ap is the projected surface area of the workpiece without flash (Tschaetsch, 2006). Typical flash land thicknesses vary in the range 1 < t < 10 mm and the flash land width w is usually dimensioned according to w=t ¼ 2  5

(6.2)

(d) The blocker should be narrower than the finish to help the operator in placing the blocked workpiece in the finish cavity. Hence, it should also be made deeper to maintain proper volume, (e) The blocker flash thickness should be made larger than that of the finish, (f ) The radii of the blocker should be at least twice as large as in the finish. It is worth noticing that small fillet radii are more difficult to fill than larger ones and that fillet radii located at extremities or distant from the centre of gravity are generally more difficult to fill. (Refer to Table 6.1 for information about typical radii and draft angles in impression-dies.) Hammers and screw presses are machines controlled by energy (Fig. 6.7). In hammers, the nominal energy capacity EN is determined by the ram mass

Fig. 6.7 Hammers vs screw presses.

286

Metal forming

m and drop height H, and by extra energy provided by air, steam or oil pressure p in the case of double-acting hammers, EN ¼ η ðmg + pAÞH

(6.3)

where η is the efficiency factor and A is the piston area. In screw presses, the nominal energy capacity EN is mainly stored in the flywheel and depends upon its angular velocity ωF and mass moment of inertia IF (Fig. 6.7) r ω2 vθF ¼ vθD ) ωF ¼ ωD EN ¼ η F IF (6.4) 2 R where vθ is the circumferential velocity, ωD is the angular velocity of the driving disc, R is the radius of the flywheel, and r is the radial position of the driving disc in contact with the flywheel (Rodrigues and Martins, 2005). Both hammers and screw presses are impact machines that are suitable for multiple strokes, as it is commonly used in hot compressive forming with impression-dies. In both types of machines, the ram velocity increases up to the contact between the upper impression-die half and the billet, being approximately 10 times higher in case of hammers. The force, on the contrary, cannot be directly controlled and depends on workpiece material and geometry, and required deformation distance or energy. Because of the smaller impact velocity of screw presses, the deformation resistance during hot forming is lower than with hammers. As a guideline in the choice of hammers, it may additionally be said that drop hammers are mainly applied for small to medium-sized impression-die tools whereas double-acting hammers are utilised for medium-to-large-sized impressiondie tools.

6.2.3 Precision tools Precision tools are utilised in net-shape and near-net-shape compressive forming processes in which a billet with carefully controlled volume is deformed (hot, warm, or cold) by a punch in order to fill a die cavity without any loss of material (flashless forming). The punches and dies may be made of one or several pieces, and the production processes usually involve a forming sequence that must be designed to eliminate flash and reduce or eliminate secondary finishing operations. Fig. 6.8 shows two forming sequences used in precision tools for cold and hot compressive forming. The basic problem in precision forming is how to eliminate the excess material resulting from variations in the permissible tolerances (0.5%–1%

Tool design

287

Fig. 6.8 Forming sequence in (A) cold and (B) hot compressive forming processes. (Courtesy of Hatebur Umformmaschinen AG.)

in volume) and temperature (30°C) of the billets, and from the progressive abrasive wear of the internal die cavities. In impression-die tools (standard closed-die tools) for compressive forming, the excess of material is accumulated in the flash or in machining allowances and is removed in secondary operations. In precision forming, the design strategy is to look for spaces in the die cavity to accommodate excess material. This is obviously very much dependent on the geometry of the part to be formed but still  three main solutions can be applied (Cermak and Gra´f, 2003): (a) Some regions of the parts, usually the peripheral edge fillet radii, may remain slightly underfilled within the approved dimensional tolerances (Fig. 6.8A). This solution is more complicated than it seems because the ratio of cavity volume to the edge fillet radii volume is very large and sometimes requires exploiting the elastic deformation of both the machine and the tool to accommodate the excess material, (b) Utilisation of internal flashes (Fig. 6.8B). This solution is commonly used when forging axisymmetric parts with a central hole and is based on the accumulation of the excess material in the central hole region of the part that will later be removed by punching. This is commonly done when producing rings, wheels and gears. If there is no central hole, some other regions of the internal webs that will be punched later can also be used for this purpose,

288

Metal forming

(c) Some regions of the parts can increase its dimensions. This solution transfers the excess material into the final part height (either the total part height or a local part height—e.g. a gear rim or a flange height) and requires special tool design involving floating and split dies. Precision tools for cold compressive forming processes (hereafter designated as ‘cold forging’) may involve upsetting, heading, extrusion, ironing, coining or a combination of these processes (refer to Appendix E). Their design sacrifices material formability and low material flow stresses of hot forming, to obtain significant reductions in material waste and elimination of secondary finishing operations, and significant gains in hardness and strength due to material strain hardening. Cold forging is also capable of providing good surface finish and close tolerances, as compared to those of conventional machining and hot forging (Table 6.2). For this reason, it is sometimes also used as a final calibration stage after warm or hot forging processes. Material formability in cold forging is lower than in warm or hot forging because plastic deformation occurs at a temperature appreciably below the recrystallisation temperature (below 0.3 times the absolute melting temperature in Kelvin of the material). As a result of this, the applied pressures and forces on the punches and dies are higher and the overall complexity and cost of the tools are also higher than in warm or hot forging. Precision tools for cold forging may be classified into three different groups: (a) Conventional precision tools, (b) Precision tools with floating dies, (c) Precision tools with multidirectional dies. Table 6.2 Accuracy of different manufacturing processes. Manufacturing process

ISO IT tolerance grade (ISO 286) 2

3

Hot forging (impression-die) Hot forging and sizing (near net-shape forming) Warm forging Cold forging (precision forming) Turning Milling Grinding Honing Lapping Additive manufacturing Standard range

Under favourable conditions

4

5

6

7

8

9

10

11

12

13

14

15

16

Tool design

289

Example 6.1

Determine the tolerance corresponding to IT7 (ISO 286) for a nominal hole size in the range 18–30 mm. First, we must introduce the ISO standard expression that relates the tolerance T in micrometres (μm) with the geometric mean dimension D in millimetres (mm) and the IT grade   pffiffiffiffi T ¼ 100:2 ðITG1Þ 0:45 3 D + 0:001D where ITG is the IT grade (a positive number). Then, we calculate the geometric mean dimension D as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ 18  30 ¼ 23:24 mm and replace this value, and ITG ¼ 7 into the ISO standard expression to obtain   pffiffiffiffiffiffiffiffiffiffiffi T ¼ 100:2 ð71Þ 0:45 3 23:24 + 0:001  23:24 ¼ 20:72 ffi 21 μm This value corresponds to a tolerance H7 (0 to +21 μm) for a hole or h7 (21 μm to 0) for a shaft.

Conventional precision tools Fig. 6.9 shows typical tool designs for forward rod extrusion, backward can extrusion, heading and forward tube extrusion. The tools consist of a number of components that can be divided into three groups: (a) Active tool elements that come into direct contact with the workpiece during operation, (b) Passive tool elements that surround the active elements and support them directly or indirectly, (c) Structural elements that accommodate active and passive tool elements. Referring to Fig. 6.9, the following elements are included in conventional precision tool systems: (a) Punch (b) Mandrel

Bolster

Bolster

Stripper plate

Support plate

Support plate Stress ring Punch

Workpiece

Workpiece

Pillar Stress ring

Punch Die insert

Counter punch

Support plate

Bolster

Die insert

Support plate Bolster Ejector

Ejector

(B)

(A)

Bolster Support plate Bush

Bolster Support plate

Punch Stress ring

Punch

Stress ring Workpiece

Pillar

Mandrel

Support plate Die insert

Support plate

Die insert

Counter punch Workpiece

Support plate

Bolster Bolster

Ejector

(C)

Ejector

(D)

Fig. 6.9 Cold forging tools for: (A) forward rod extrusion, (B) backward can extrusion, (C) heading and (D) forward tube extrusion. (Adapted from ICFG, 1992. International Cold Forging Group 1967–1982 Objectives, History, Published Documents. Meisenbach, Bamberg.)

Tool design

(c) (d) (e) (f ) (g) (h) (i) (j)

291

Die insert Stress rings Counterpunch Support plates Ejector Stripper plate Guide pillar Bolster.

The elements (a)–(c) are active elements. Active elements are in direct contact with the part during the forming process and are the most severely stressed tool elements. Their design should therefore be considered carefully to minimise wear and to prevent breakage. Depending on the wear, they may have to be replaced or maintained regularly. The elements (d)–(g) are passive elements. Their main function is to distribute the heavy process loads towards the press frame without exceeding the yield stress of any element. They are very important because the tool life can be influenced negatively if incorrect dimensioning or form inaccuracies of the passive elements lead to overloading of the active elements. The elements (h)–(j) are structural elements, which besides accommodating the other elements, also provide guidance and centring between the upper and lower tool halves and enable the installation in the press. They vary in size and design according to the geometry of the parts and required forming forces. Bolsters, for example, are generally made from large thick plates of low hardness, mild steel to distribute the forces over a large area of the press table or ram. Passive elements that are part-independent (e.g. support plates) and structural elements (die sets) consisting of top and bottom bolsters, guide pillars and stripper plates, are reusable and commonly used in other tool set-ups. In forward rod extrusion Fig. 6.9A, mutual guiding of the upper and lower tool parts lies in the guiding of the punch in the container. In backward can extrusion, external guiding by the subpress pillars shown in Fig. 6.9B is essential for the attainment of good concentricity. By utilising the facilities offered by a hydraulic press with triple cylinder, it is possible to guide the upper and lower tool parts together, as in the can extrusion shown in Fig. 6.10A. In this way, poor alignment can be avoided. The tool represents the first step in the manufacturing of a slider axle, where the two subsequent steps are carried out in the toolset shown in Fig. 6.10B (Grønbæk, 1981b). The subsequent steps shown in Fig. 6.10B consist of ironing and forward can extrusion to form an internal spline. Close tolerances and coaxiality of

Metal forming

292

Bolster Die insert

Support plate

Support plate Punch Stress ring Punch Stress ring Die insert

Counter punch Ejector

Bolster Support plate Counter punch

(A)

Bolster

(B)

(C)

Fig. 6.10 Cold forging tools for the three-stage cold forging of a slider axle in 17Cr3: (A) forward can extrusion, (B) ironing and forward can extrusion, (C) initial billet, intermediate workpiece obtained in (A) and final workpiece obtained in (B). (Adapted from Grønbæk, J.,1981b. Koldflydepresning. Temarapport: massivformgivning og pulverteknologi. Jernet (in Danish).)

the upper and lower cup are essential in these components. The tool design ensures these requirements, and the initial billet, intermediate workpiece and final workpiece are schematically shown in Fig. 6.10C. From what was said earlier, the main task of a designer is to dimension the active tool elements in order to ensure a long service life with an economic manufacturing cost, because tool costs play a significant role in precision forming. Service life is limited by excessive wear or by overload or fatigue cracking, amongst other types of failure. For this reason, selection of tool materials and of lubrication and cooling systems, if necessary, are of paramount importance and need to be considered by the designers. In some applications of cold compressive forming, the raw material is conveniently supplied in the form of tubes instead of rods. Fig. 6.11A shows a precision tool for producing liners for composite overwrapped pressure vessels (COPV’s) and Fig. 6.11B illustrates the tool at the open and closed positions and provides photographs of three liners after being formed. The tool consists of structural, passive and active elements. The structural elements comprise several individual components such as the pillars, the bolsters, the ram holder, the support plate and the ram, which are independent of the material and geometry of the liners to be fabricated. The stress ring,

Tool design

293

Upper press table Pillar

Ram holder

Upper die

Ram

Bush

Upper die

Stress ring Support plate

Die insert

COPV liner Tube with mandrel Support ring

Clamping ring Lower bolster

Lower die

Mandrel

Press table

(A)

(B) Fig. 6.11 Shaping commercial tubes into seamless liners for composite overwrapped pressure vessels (COPV’s). (A) Schematic representation of the cold forming precision tool. (B) Illustration of the open, closed and extraction positions of the tool with a picture showing three liners made from commercial aluminium AA6063-T0 tubes. (Adapted from Alves, L.M., Santana, P., Moreira, H., Martins, P.A.F., 2013. Fabrication of metallic liners for composite overwrapped pressure vessels by tube forming. Int. J. Press. Vessel. Pip. 111–112, 36–43; Alves, L.M., Martins, P.A.F., Pardal, T.C., Almeida, P.J., Valverde, N.M., 2015. Plastic Deformation Technological Process for Production of ThinWall Revolution Shells from Tubular Billets (Patent EP2265396 (B1)).)

294

Metal forming

the support ring and the clamping ring are the passive elements and the die insert and the upper and lower, semiellipsoidal-shaped dies are the active elements that are dependent on the geometry of the liner to be fabricated. The upper die is characterised by its unconventional very sharp edge and it is protected against collapse by circumferential tensile stresses through the inner container, which acts as a shrink-fit tool part, and by plastic flow that inhibits the tube from contacting the sharp edge during forming. The clamping ring fixes the stress ring to the support ring and lower bolster. This part is removed after forming, to enable the stress ring and die insert to move up with the ram holder in order to extract the liner (Fig. 6.11B). Dimensioning of punches, dies and inserts will not be addressed in this section because they are inherent to other types of tools. They will be considered later in Section 6.6. Instead, we will address the problem of selecting a suitable press for installing the tool, because presses have different geometric and operating characteristics. The relevant geometric characteristics to consider are the table and ram surface area, the installation height, the slide stroke and the ejector stroke, if available, and the convenience of the existing tool change systems. In what regards the operating characteristics of a press, the key factors are the nominal force FN, the energy capacity and the stroke-time and forcestroke curves (Fig. 6.12) of the press. This is because precision forming tools utilised in cold forging require substantially long strokes and operate under high forces with powerful ejector systems. Generally, mechanical presses are recommended for large production batch sizes, but their slide stroke limit and force-stroke curve often constrain the maximum part lengths to be produced. As shown in Fig. 6.12, mechanical presses are controlled by the ram path between the upper dead centre (UDC) and the bottom dead centre (BDC), which comes from the crank radius R of the eccentric drive mechanism. In the case of eccentric presses, the nominal force FN is available at a crank angle α ¼ 30º before the BDC and the vertical force FV is related to the crank angle as follows (Rodrigues and Martins, 2005): FV ¼

FN 1 2 sin α

(6.5)

The plot FN/FV given by Eq. (6.5) is illustrated in Fig. 6.12. Hydraulic presses are controlled by force and operate at substantially lower speeds than mechanical presses. They have longer ram and ejector

Tool design

295

UDC

R

Oil

BDC

UDC

L

H=2R F

h BDC

FV

FV

FH

T-slot Ram Ram path Tool Hydraulic press

Mechanical press (eccentric drive)

UDC BDC

Overload safety Stroke

FV FN

UDC

1.5

1.0

Forming process

Mechanical press (knuckle-joint drive)

0.5

BDC Time

1.0

0.5

UDC

h/H

0

BDC

Fig. 6.12 Mechanical vs hydraulic presses with stroke-time and force-stroke schematic curves.

strokes as well as the capability of maintaining a constant force throughout the ram stroke (Fig. 6.12). Recent years have seen the development of servo presses that replace the classic electric motor, clutch, and flywheel of mechanical presses by one or more servomotors in direct contact with the mechanism giving the reciprocating movement. These new type of presses allow adjusting the ram speed and obtaining stroke-time evolutions where the ram is maintained at the bottom dead centre (BDC) for a longer amount of time than in mechanical presses with eccentric or knuckle-joint drives. The main disadvantage of servo presses is the installed power, which must be significantly higher than that of conventional mechanical press because they have to be capable of delivering large instantaneous amounts of energy in certain instants of the stroke-time evolution.

296

Metal forming

Precision tools with floating dies Precision tools with floating dies explore the possibility of different tool elements supported by mechanical, elastomer and gas springs, or by hydraulic cylinders being moved at different rates to close the dies at different times. In practical terms, these types of tools replicate the relative movements that could be alternatively obtained in multiple-action presses with smaller efforts on the tool elements and higher flexibility, at a much lower cost (Fig. 6.13). As shown in the figure, the billet (or preform) is first loaded and positioned on the punch/ejector of the lower die. Then, the press ram moves the upper die down to close the die cavity, without deforming the billet. By continuing the press movement, the upper and lower punches start to deform the billet and to fill the cavity symmetrically around the horizontal symmetry plane whilst the springs ensure the closing of the dies. Once the die cavity is filled, the excess of material flows through the radial flash gap to compensate inaccuracies in volume. When the part is formed, the ram moves upwards, and the part is kept in the lower die half until the upper punch is retracted. Once the dies are opened and the upper punch reaches its initial upward position, the formed part is ejected from the lower die. The overall principle of floating dies relies on a free, self-adjusting, movement of the closed-die halves, which can be tuned by acting directly on the design of the mechanical, elastomer and gas springs, or on the working conditions of the hydraulic power units. Typical springs used in precision tools with floating dies consist of high-performance compression or disk springs made from chromium-vanadium spring steel (e.g. 50CrV4 steel) with high fatigue and impact resistance in the heat-treated condition.

Spring

Upper die Punch

Billet

Final part Lower die Punch/Ejector Loading billet

Start forming

End forming

Unloading the part

Fig. 6.13 Basic working principle of a precision tool with floating dies.

Tool design

297

Elastomers may also be used as alternative rubber-elastic spring elements and are usually made from polyurethane with shore hardness ratings of 80, 90, or 95. Gas springs use nitrogen as a pressure medium and operate under working pressures in the range of 25–150 bar. Alternative set-ups based on conventional (nonfloating) precision dies require independent movements of the two counteracting punches plus an extra movement to close the upper and lower dies in order to form a symmetric part similar to that obtained in Fig. 6.13. Nonfloating dies with a single punch movement would inevitably lead to the production of asymmetric parts (Fig. 6.14). Alternative designs of precision tools with floating dies utilise pressurised hydraulic cushions to support the dies and replicate the two independent movements of a typical double-action press in a single-action press. As an example, Fig. 6.15 shows a solution by the Japanese company NICHIDAI Corporation (1992), where a pantograph link is applied to control the speed of the upper and lower tool parts. The upper hinge of the pantograph is mounted on the upper bolster and moves with the same velocity as the press ram, whilst the lower hinge is fixed to the lower bolster and does not move. The upper punch moves with the upper bolster, the lower punch remains stationary in the lower bolster and the divided die is resting on a middle plate supported by the middle hinges of the pantograph. If the upper and lower links of the pantograph have the same length, the upper punch moves downwards (refer to ‘y’) twice as fast as the upper die (refer to ‘y/2’), implying that the lower punch will penetrate the die with an upward speed equal to and opposed to the upper punch.

Asymmetric shape Billet

Loading billet

Lock dies

Start forming

End forming

Unloading the part

Fig. 6.14 Alternative precision tool without floating dies and a single punch movement.

298

Metal forming

Punch Bolster Die insert

Hydraulic pressure y

Pantograph Middle plate

y /2

Stress ring Workpiece Support plate Bolster

Hydraulic pressure

(A)

(B) Fig. 6.15 Precision tool with floating dies supported by pressurised pillows. (A) Schematic representation of a tool that utilises a pantograph to obtain a balanced upward and downward material flow on a single-acting press. (B) Photograph of a tool with floating dies supported by pressurised pillows together with a bevel gear and a crossjoint fabricated by this tool. ((A) Adapted from Yoshimura, H., Wang, C.C., 1997. Manufacturing of dies for precision forging. In: 1st JSTP International Seminar on Precision Forging, Osaka, pp. 15-1–15-7; Nakamura, T., Osakada, K., 2005. Research and development of precision forging in Japan. In: Proceedings 8th ICTP, Verona, pp. 113–114. (B) Courtesy of Nichidai Corporation.)

In this way, the forging will be exactly symmetrical. By changing the length ratio between the upper and lower links, the ratio between the two punch speeds can be controlled to forge asymmetrical components. Precision tools with multidirectional dies Precision tools with multidirectional dies offer the possibility of forming a billet from several spatial directions in a single press stroke. This can be done by modifying conventional tools to include gas springs or hydraulic cylinders

Tool design

299

to obtain the movements of the active tool elements in the required directions, or by including wedges and levers in the tool design. In contrast to sheet metal forming, in which the design and utilisation of tools with multidirectional dies is relatively widespread, the design of compressive forming tools with multidirectional dies is still scarce and remains an open topic of applied research. Applications in compressive forming started to be exclusively focused on the production of crankshafts (White, 1914; Rut, 1971) until the work performed by Doege and Broß (1997), and others (Doege and Behrens, 2010). Fig. 6.16 shows an example of the use of die wedge actuators to redirect the vertical force and movement of a press into the horizontal direction to form a billet that is enclosed in a die cavity. The design of precision tools with multidirectional dies must take the different directions and kinematics of the individual punches and sliding die elements into account to assure die closing with adequate filling, to prevent collisions between the different moving elements and to enable die opening and removal of the formed parts. Basic kinematics of die wedge actuators enables writing the following relations in terms of position and velocities (Fig. 6.16) dx ¼ vx ¼ vy tan α dt

x ¼ y tanα

(6.6)

Typical wedge working angles α are in the range of 35–45 degrees.

Die wedge actuator

Punch

Spring

Bolster Workpiece

Billet Sliding die

α

y

Spring Guide

x Bolster

Loading the billet

Locking the sliding dies

Start forming

End forming

Fig. 6.16 Basic working principle of a precision tool with multidirectional dies. (Adapted from Behrens, B.-A., 2000–2011. Komplexe umformung in einem pressenhub durch schmieden mit mehrdirektionalen werkzeugen. White Paper of the Project SFB 489: Prozesskette zur herstellung pra€zisionsgeschmiedeter hochleistungsbauteile. DFG (in German).)

300

Metal forming

Outer tapered ring

Upper inner punch Upper outer punch Spring

Workpiece

Sliding die element Billet

Fig. 6.17 Precision tool with multidirectional dies. (Adapted from Vazquez, V., Altan, T., 2000. Die design for flashless forging of complex parts. J. Mater. Process. Technol. 98, 81–89.)

Fig. 6.17 shows a detail of a precision tool with multidirectional dies to produce an inner race for a constant velocity joint (Vazquez and Altan, 2000). The inner and outer counteracting punches take care of vertical forming whilst the six individual sliding die elements (segments) that are driven by the outer tapered ring take care of the horizontal forming and of the closing of the die. In this case, the outer tapered ring acts as a series of wedges placed along the circumferential direction. From a kinematical point of view, there are three possible combinations of movements in a multidirectional tool with elements acting in the vertical and horizontal directions (Behrens, 2000–2011): (a) First horizontal then vertical forming, (b) First vertical then horizontal forming, (c) Simultaneous vertical and horizontal forming. Tool design must ensure that proper wedge angles are chosen, and springs are utilised to move the dies back to the initial position after forming, amongst other requirements. Although the technical feasibility of precision tools with multidirectional dies has been demonstrated in numerous applications, there are still open questions regarding its industrial utilisation (Behrens, 2000–2011): (a) Service life of the tool elements—for example, to what extent will the individual tool elements fail by wear? (b) Economic viability—for example, to what extent is it valid to assume that the amount of investment in such complex tools is compensated by the savings in production time and material consumption?

Tool design

301

6.3 Tools for tensile, combined tensile and compressive forming, and bending Tools for tensile, combined tensile and compressive forming and bending include those used in deep drawing, ironing, stretch forming and bending processes. This is a vast domain for tool design due to the variety of processes to be covered, to the different ways in which raw material is supplied (e.g. rods, tubes or sheets), to the number of operations that are required in each die (single-operation or multioperation) and to the number of stations involved (single-station or multistation (progressive) dies). In view of the earlier, the authors decided to focus on raw materials supplied in the form of sheets and to organise the presentation in the following three groups of tools, which result from merging the different ways of classifying tools according to the manufacturing processes, the number of operations in each die and the number of stations: (a) Conventional tools with single-station dies, (b) Multidirectional tools with single-station dies, (c) Progressive tools with multistation dies.

6.3.1 Conventional tools with single-station dies Conventional tools with single-station dies are designed to accomplish one of the following two objectives: (a) Single-operation dies, in which only one operation (e.g. deep drawing or stretching) is performed per die during a press stroke, (b) Multioperation dies, in which more than one operation (e.g. two deep drawing operations) are performed in a die during a press stroke. Conventional tools with single-station, multioperation dies may sometimes involve both forming (e.g. deep drawing, bending or hole flanging) and shearing operations (e.g. blanking or punching). These types of tools are usually named as ‘combination tools’ or ‘compound tools’ and will be addressed in the following section. Fig. 6.18 shows a conventional tool with a single-station, multioperation die to produce a cylindrical cup by means of deep drawing. The deep drawing and subsequent reverse redrawing are carried out within the same tool. The term ‘reverse’ is used because in the tool shown in Fig. 6.18, redrawing forms the intermediate drawn cup in a direction opposite to that of the first drawing operation. Some designers also name reverse drawing as ‘inside-out redrawing’ because the outside walls of the intermediate cup are on the inside of the final cup.

Metal forming

302

Punch Air hole

Bolster Die

Intermediate cup

Die Final cup

Blankholder Punch

Bolster

Pin Air hole Ejector

Air cylinder

(A)

(B)

Fig. 6.18 Conventional tool with a single-station, multioperation die that performs deep drawing and reverse deep drawing. (Adapted from Waller, J.A., 1978. Press Tools and Presswork. Portcullis Press, London.)

As shown in Fig. 6.18, the hollow tool element mounted on the lower bolster works as the punch in the first drawing operation and as the die in the subsequent reverse redrawing operation. The die mounted on the upper bolster descends over the initial blank, which is drawn over the hollow bottom punch to produce the first cup, whereas the blankholder retreats under counterload delivered by the cushion system of the press. After this, the upper punch is moved downwards to redraw the cup into the inside of the hollow bottom punch (refer to the rightmost picture of Fig. 6.18). Redrawing operations are needed when producing cups with a greater depth than that permitted by the limiting draw ratio (LDR) of the material, LDR ¼

d0 dp

(6.7)

In the aforementioned equation, d0 is the diameter of the initial disk blank and dp is the diameter of the punch. Typical values of LDR for some common materials are given in Table 6.3.

Tool design

303

Table 6.3 Maximum limiting draw ratio (LDR) values for several materials. Material

Drawing

Redrawing

Carbon steel (% Mn 0:2 t0 2q  q2

(6.8)

where q is the reduction in area (decimal value) at fracture in uniaxial tension. (b) Elastic spring back When the bend punch is retracted, the elastic components of stress will cause spring back and both the angle and the radius of the bent parts are increased. This may be avoided by calculating the amount of spring back using one of the theoretical methods of analysis that are listed in Fig. 6.1 and compensate by over bending. Alternatively, the tool can be designed as shown in Fig. 6.21 with radius a R¼ (6.9) 2 sinα and the bottom will be flattened in a subsequent finishing operation. a Punch Sheet

Die

α

R

Springback

α

(A)

(B)

Fig. 6.21 (A) Evolution of the sheet during U-bending and (B) tool design to compensate spring back. (Adapted from Corbet, C., 2005. Proc
edes de mise en forme des mat
eriaux. Casteilla, Paris (in French).)

Tool design

307

Multidirectional tools with single-station dies to pierce or punch holes from different angles through cups or shells require the transformation of the vertical force and movement of the press into various three-dimensional directions by using die wedge actuators. The tools are generally more complex, but the working principle is similar to that shown in Fig. 6.20.

6.3.3 Progressive tools with multistation dies In progressive tools, several successive and different operations can be carried out on a strip being fed in exact increments of displacement, typically directly from a coil. Multiple punches and dies corresponding to different stations that perform deep drawing, punching and blanking operations are generally used and attached to the common die plates. Fig. 6.22A shows a progressive tool to produce a hemispherical shell that comprises several initial lancing and planish operations, two intermediate drawing stages and a final drawing and trimming stage. The initial lancing and planish operations are critical for the tool to form and transport the strip correctly from one station to another without operator handling. In fact,

1st Drawing counter punch Die Bolster

Holder

Support plate Strip Pad Support plate

Bolster

Planish

2nd Drawing Tab

(B)

1st Lancing punch

1st Drawing punch

(A)

Final drawing and triming

1st Drawing

Bridge

Planish

2nd Lancing

Pilot hole

1st Lancing

Fig. 6.22 (A) Progressive tool to fabricate a hemispherical cup and (B) schematic detail of the strip showing the work of the different stations. (Adapted from Waller, J.A., 1978. Press Tools and Presswork. Portcullis Press, London.)

308

Metal forming

drawing and final trimming operations are only possible if the strip is able to pass through the tool without distortion and this is only feasible if the blank is ‘almost separated’ from the rest of the strip before performing the drawing operations. As shown in Fig. 6.22B, the initial lancing operations perform partial cuts concentric to the part to be obtained in order to only leave a small portion of metal holding the initial drawing blank to the strip. These cuts open as ‘tongues’ during the drawing operations but keep the three tabs which hold the intermediate drawn cup to the strip. An amount of material named as ‘bridge’ is left between each blank to minimise the deformation of the strip. Final trimming of the drawn part from the strip is performed by a circular punch. Small circular holes (named ‘pilot holes’) placed at the edge of the strip are used by pilot pins (not shown in Fig. 6.22) to position (index) the strip accurately in each station. Fig. 6.23 shows a picture of an industrial progressive tool with multistation dies to produce an automotive suspension support. This type of tool is expensive, but the investment is rapidly amortised by the high productivity resulting from its large production speeds when compared to other types of tools. To conclude, it is worth mentioning that the availability of transfer systems for transporting a blank and moving it from one station to the other avoids the necessity of designing the tool to account for the strip to transport the blank throughout the stations. In such cases, the tools are designated as ‘transfer tools’ instead of ‘progressive tools’ because the transfer of parts from one station to the next is carried out by means of sidebars, gripper rails or crossbar suction mechanical systems, amongst others.

Fig. 6.23 Progressive tools with multistation dies to fabricate an automotive suspension support.

Tool design

309

6.4 Tools for forming by shearing In progressive tools with multistation dies, there are shearing operations working side by side with deep drawing, hole-flanging and bending operations. However, there are tools that are mainly designed to perform forming by shearing. These tools are usually grouped in the following three different categories: (a) Combination tools (b) Compound tools (c) Fine blanking tools.

6.4.1 Combination tools Combination tools perform two shearing operations or one-shearing and another nonshearing operation in a single stroke on a strip or precut blank. There are several types of combinations such as blanking and punching, blanking, drawing, and drawing and trimming, amongst other examples. Fig. 6.24 presents a typical combination tool for blanking the periphery and punching the two small circular holes simultaneously. The design is different from that of a progressive tool in which the two circular holes have to be first punched so that the strip is fed to another station, where another punch and die set will cut the periphery to obtain the required part. A typical feature of these tools is that there is always an active tool element like that shown in Fig. 6.24 that has a dual function of being a punch for the outer shearing perimeter and a die for the inner shearing perimeter. Punch Punch plate

Ejector Die Punch and die

Ejector Bolster Punch backing plate

Bush

Workpiece Strip

Pillar Pin Blankholder

Die plate

Bolster

Spring

Fig. 6.24 Tool with combined die for producing a component where both speed and concentricity of the diameters are essential.

310

Metal forming

The upper die half carries the die for the outer perimeter, the punches for the inner holes and the ejector. The ejector makes use of the second acting movement of the press, but it could also be designed to be actuated by springs. Usually, the main difference between utilising one or the other type of ejector is related to the removal of the part after being cut. Second acting movements allows the part to be removed and blown from the strip by compressed air whereas spring-loaded ejectors tend to reinsert the part back into the strip requiring additional procedures to remove the part. From a design point of view, it is important to understand that the cut surface is not perfectly perpendicular to the sheet surface and its quality is greatly influenced by the clearance between the punch and die. Excessive clearance allows extensive plastic deformation, separation is delayed, and a long burr is pulled out at the upper surface. This type of cut surface is also obtained when the punch and die are used beyond their tool life. With a very tight clearance, the cracks—originating from the tool edges—miss each other and the cut is then completed by a secondary tearing process, producing a jagged edge roughly midway in the sheet thickness. This type of clearance leads to more rapid tool wear and to shorter tool life. Values of punch-die clearance for several materials are given in Table 6.5. Fig. 6.25 shows a tool that performs two shearing and one deep drawing operations, but only two of these operations are performed simultaneously. The strip is fed through the material guide and the first stage trims a blank from the strip using a blanking punch and a die. The blank falls on the punch over which it is drawn (second stage) by the action of the deep drawing die. The ejector/blankholder acts as a pressure pad. Once the workpiece approaches its final shape, a hole in the top surface is simultaneously punched (third stage) by the upper cylindrical punch. The counterpunch acts as a blankholder during the punching operation and the falling scrap is led away from the tool by including a central opening in the lower bolster. Stripping the drawn workpiece (cup) from the punch or die is accomplished by the action of the upper and lower ejectors.

6.4.2 Compound tools Compound tools perform two or more shearing operations together with other sheet forming operations simultaneously on a strip or precut blank. The other sheet forming operations typically consist of drawing, hole flanging and bending, with or without die wedge or cam-based actuators to redirect the vertical force and movement of the press into the horizontal or inclined movement of the sliding die elements.

Table 6.5 Punch-die clearance in forming by shearing as a function of the morphology of the cut surface for several materials.

Material

(Punch-die clearance j as a percentage (%) of sheet thickness)

High-carbon steel (0.6%–1.0% C)

23

15

–

Carbon steel

21

9

2

Stainless steel

23

10

1.5

Hard

25

4

1.25

Soft

26

6

0.75

25

11

2.5

Hard

24

7

0.8

Soft

21

6

1

Hard

20

10

1

Soft

17

7

1

Magnesium

16

4

0.75

Lead

22

7

2.5

Copper Phosphor bronze Brass Aluminium

Adapted from Rodrigues, J.M.C., Martins, P.A.F., 2005. Tecnologia Mec^ anica—Tecnologia da Deformac¸a˜o Pla´stica I–II. Escolar Editora, Lisboa (in Portuguese).

312

Metal forming

Fixing ring Plate Deep drawing die Blanking punch Support plate Strip

Scrap

Material guide Counter punch / Ejector Blanking die Punch

Air hole

Final workpiece

Falling blank

Falling scrap Ejector / Blankholder

Die

Deep drawing punch

Pin

Fig. 6.25 Combination tool consisting of one initial blanking operation followed by a deep drawing and punching operations performed simultaneously. (Adapted from Corbet, C., 2005. Proc
edes de mise en forme des mat
eriaux. Casteilla, Paris (in French).)

Because compound tools make use of almost all the fundamental principles that were described in the previous sections, their construction is complex and requires much ingenuity and experience from the designers. Fig. 6.26 presents a compound tool for producing a mild steel part, which involve blanking, punching, drawing and hole flanging. Regarding hole flanging, it is worth mentioning the following practical design guidelines illustrated in Fig. 6.27A. r ¼ 2d1 0:15t0 r1 ¼ q

(6.10)

r2 ¼ 0:3d1 where q is the reduction in area (decimal value) at fracture in uniaxial tension. Other punch geometries like those shown in Fig. 6.27B can alternatively be used. Conical (‘B’), hemispherical (‘C’) and cylindrical (‘D’) punches are

Tool design

313

Piercing punch Pin

Support plate Bolster

Bush Counter punch Pillar

Strip

Blanking punch / Drawing die

Blanking die

Stripper

Blankholder

Falling scrap Bolster Drawing punch Hole flanging punch / Piercing die Pin

Fig. 6.26 Compound tool for producing a mild steel part from flat strip. (Adapted from Waller JA: Press Tools and Presswork, London, 1978, Portcullis Press.)

t0 d 1 d2

r1

A

C

B

D

d1 j 9 r r2 d 2 H7

(A)

(B)

Fig. 6.27 Hole flanging of sheets. (A) Design guidelines for tractrix punches. (B) Alternative punch geometries. ((A) Adapted from Corbet, C., 2005. Proc
edes de mise en forme des mat
eriaux. Casteilla, Paris (in French). (B) Adapted from Wilken, R., 1958. Das biegen von innenborden mit stempeln. In: Mitteilungen der forschungsgesellschaft blechverarbeitung (in German), pp. 56–63.)

Metal forming

314

easier to manufacture than tractrix (‘A’) punches but require higher forces. The cylindrical punch (‘D’) is the one requiring the greatest force but also the lowest stroke to produce a collar.

6.4.3 Fine blanking tools Fine blanking tools are used for producing parts and holes with very cleancut edges, perpendicular to the sheet surface and with a surface finish sufficiently smooth (comparable to that attained in machining), to allow immediate use of the parts. The mechanics of the process is based on the delay of fracture caused by the superposition of a high hydrostatic state-of-stress in the clearance between the punch and the die. The clearance is also considerably smaller than in conventional blanking (0.5 to 1% of the sheet thickness against 5 to 10% of the sheet thickness in the case of conventional forming by shearing). The main active elements of a fine blanking tool are the following (Fig. 6.28):

Cutting force

Punch

Blank holder and V-ring force

Blankholder

Scrap

V-ring Die

Counter punch

Final part

Counter force

(A)

A

H

R

(mm) 2.5

(mm) 0.6

(mm) 0.6

3.3-3.7

2.5

0.7

0.7

3.8-4.5

2.8

0.8

0.8

R

R

t0

4 5°

H

4 5°

A

(mm) 2.8-3.2

(B) Fig. 6.28 Fine blanking. (A) The main active elements and working principle of fine blanking. (B) Geometry and dimensions of the V-ring. (Adapted from Schuler, Altan, T. (Ed.), 1998. Metal Forming Handbook. Springer, Berlin.)

Tool design

(a) (b) (c) (d)

315

Punch Die Blankholder with a V-impingement ring Counterpunch (lower pressure cushion).

Fine blanking is preferentially performed on triple-action hydraulic presses where the pressure and movement of the punch, blankholder and die can be controlled independently during the cutting cycle. Fig. 6.29 presents a detail of an industrial fine blanking tool utilised for producing a bicycle spur gear with teeth placed along the outside edge. The advantage of fine blanking for producing this type of component is that it reduces the nine working operations required in conventional production, including blanking on individual presses and machining through deburring and grinding, to only three operations. In fact, after obtaining the part by fine blanking and bevel turning the tooth tips to one side of the gear, it is only necessary to perform deburring (Schuler, 1998). Pressure unit Punch/ejector Support ring

Ram

Upper bolster Pin Pillar

Punch Blankholder with V-ring Final workpiece Die

Lower Bolster

Counter punch Punch

Press table Lower pressure unit Thrust ring

Fig. 6.29 Detail of a fine blanking tool to produce a bicycle spur gear. (Adapted from Waller, J.A., 1978. Press Tools and Presswork. Portcullis Press, London.)

6.5 Tool materials 6.5.1 Selection of tool materials Selection of tool materials requires detailed information on the forming operation conditions. The workpiece material to be formed, the operating

316

Metal forming

temperature, the type of machine tool to be used and the predicted distribution of strains and stresses in the active tool elements need to be known in order to select the appropriate tool materials. In general, the following guidelines must be taken into consideration during the selection of tool materials: (a) Strength and hardness Tool materials must provide high strength and hardness. In cold forming, for example, it is not unusual to reach pressures above 2000 MPa creating serious difficulties when values in the range of 2800 MPa are reached. The theoretical methods listed in Fig. 6.1, and in particular the finite element method, are important to determine the average values of stresses applied on tools and the localised critical peaks that may lead to failure. Very high tool pressures justify the choice of warm or hot forming instead of cold forming. (b) Temperature resistance The choice of warm and hot forming is not without its problems because new requirements related to the maintenance of strength and hardness at elevated temperature need to be considered. In particular, the preheating and operating temperature of the tools must not exceed its tempering temperature. The dwell time, which is the time that the dies are actually in contact, under pressure, with the workpiece is also critical in hot forming, because long dwell times will increase the temperature in the tools and the risk of softening. High-speed forming processes with high strain rates, such as those performed on hammers result in shorter dwell times than lowspeed forming processes performed on hydraulic presses (Table 6.6). In some cases, cooling of the tool must be considered in order not to exceed the tempering temperature of the tool materials during forming. Table 6.6 Typical contact time and acting speed of various forming machine tools. Forming machine tool

Dwell time (ms)

Acting speed (m/s)

High-speed hammers

0.5–5

16

Drop hammers

1–10

6

Mechanical presses (eccentric drive)

20–100

1

Screw presses

50–150

0.5

Hydraulic presses

250–500

0.1

Adapted from Tekkaya, A.E., 2007. Master of Engineering in Applied Computational Mechanics (ESoCAET). Atilim University, Ankara.

Tool design

317

(c) Toughness and fatigue resistance Tool materials must also provide good toughness and fatigue resistance and be able to maintain these performances at elevated temperatures, if necessary. The force vs displacement evolution is very important because it describes the transfer of energy to the workpiece being formed. A force vs displacement evolution resulting from high-speed forming processes (Table 6.6) produces high peak forces on the tools and requires the selection of tool materials with an emphasis on fracture toughness. Low-speed forming processes, such as those performed on hydraulic presses, produce lower forces over a longer period of time for the same amount of energy expended and, therefore, result in higher dwell times. This may compromise toughness resistance and yield strength at elevated temperatures and shows that each force vs displacement profile has its advantages and disadvantages. Cyclic loading of the forming tools inherent to production magnifies these problems and may lead to progressive and localised tool damage and to the progressive growth of cracks that will eventually lead to complete fracture of the tool when reaching a critical size. (d) Wear resistance The problem of choosing tool materials is that there is no rule of thumb nor a single type of tool steel capable of ensuring the best of all the required properties. For example, carbon contents up to 2.5% are important to improve the strength, hardness and wear resistance of tool steels at elevated temperature, but it adversely affects fracture toughness. In fact, from a metallurgical point of view, the factors that improve the resistance of a tool material to softening and wear at high temperatures are the same factors that reduce its fracture toughness. In the case of tool steels, the balance of strength, hardness and wear resistance vs fracture toughness is regulated by controlling the alloy composition and the heat treatment. This allows the required factors to be coordinated to provide the best choice for a particular application. (e) Surface treatments, coatings and lubrication Another aspect that should not be forgotten is the temperature resistance of the tool material in the case of applying surface treatments and coatings involving high temperature. Lubrication may also influence the choice of tool materials, but this topic will be analysed later in the presentation.

318

Metal forming

(f ) Machinability Machining dominates the costs in tool production (typically, machining 65%, material 20%, heat treatment 5%, assembly and adjustment 10% of total costs). Hence, the choice of tool materials with good machinability will reduce the cost of machining through less cutting tool consumption, power consumption and operation time. Machinability of tool steels is influenced by many factors such as chemical composition, microstructure, inclusions and thermomechanical properties. In general, the high-carbon and alloy contents that are typical of tool steels make them more difficult to machine than the lower carbon and the low-alloy steels. Several of the alloying elements used in tool steels, especially chromium, molybdenum and vanadium, form carbides that have adverse effects on machining. These effects are markedly influenced by the size, shape and distribution of the carbide particles in the steel matrix. To permit higher removal rates and easier machinability tool steels can be heat treated by annealing to reduce the hardness to a minimum. Table 6.7 provides a resume of the principal types of tool steels for cold, warm and hot forming. However, readers are advised to contact tool steel suppliers for technical guidance on the selection of the most appropriate materials for their tool designs. The commercial tool steel names included in the table are based on Uddeholm (voestalpine HPM Denmark A/S), but other tool suppliers provide similar tool steels with other commercial designations. The active tool parts, which are in contact with the workpiece during forming, are subjected to high normal pressures and require high compressive strength. They are made from cold work tool steels, high-speed steels, powder metallurgical tool steels or cemented carbides. The latter may be applied for punches and die inserts in the case of very high pressures and/ or large-quantity production requiring large wear resistance, but they are more expensive and delivery time is often long since the tools normally have to be custom made except for simple cylindrical geometries. They are, therefore, mainly used for mass production tools, e.g. for bolt manufacturing. Table 6.8 lists possible tool steels and recommended hardness ranges for active tool parts. AISI W2 is a low-alloyed high-carbon steel, which does not harden through. It is sometimes applied for monoblocks in cold forging of softer materials like aluminium. Stress rings must take up substantial tensile stresses during forming and are made in more ductile warm working tool steels. Table 6.9 lists suggested tool materials and hardness ranges.

Table 6.7 Selection of tool steels for metal forming. Type

Group

AISI

DIN no.

Cold work steel

A.1 Unalloyed and low alloyed

W1

A.2 Medium and high alloyed

Warm work tool steel

UHB name

C

Si

Mn

Co

Cr

Mo

Ni

V

1.1545

1.05

0.20

0.20

W2

1.2833

1.00

0.20

0.20

(A 2)

1.2363

Rigor

1.00

0.30

0.55

5.00

(O 1)

1.2510

Arne

1.00

0.25

1.10

0.60

(S 1)

1.2542

Regin 3

0.45

1.00

0.30

1.05

6F7

1.2767

0.45

0.20

0.40

1.35

1.2718

0.55

0.20

0.45

0.60

Caldie

0.7

0.2

0.5

5.0

2.3

0.5

Sleipner

0.9

0.9

0.5

7.8

2.5

0.5

0.70

1.00

W

0.10

A.2.1 12% Cr-steels

(D 2)

1.2379

Sverker 21

1.55

0.40

0.40

12.0

(D 3)

1.2080

Sverker 3

2.10

0.30

0.30

11.5

Prehardened mould steel

(P20)

Impax

0.37

0.3

1.4

2.0

1.05

0.25

0.20 0.10

0.60

0.20

2.00

4.00

0.50

2.75

0.2

1.0

Continued

Table 6.7 Selection of tool steels for metal forming—cont’d Type

Hot work tool steel

Group

B.1 Medium alloyed B.2 High alloyed

Highspeed steel and PM steel

C

AISI

6F2

DIN no. x

H 11

UHB name

C

Si

Mn

Cr

Mo

Ni

V

0.55

0.25

0.65

0.70

0.30

1.65

0.10

Dievar

0.35

0.2

0.5

5.0

2.3

0.6

Vidar Sup.

0.36

0.3

0.3

5.0

1.3

0.5

Unimax

0.5

0.2

0.5

5.0

2.3

0.5

Orvar Sup.

0.40

1.05

0.40

5.25

1.35

1.00

1.2713

1.2343

Co

H13

1.2344

H 10

1.2365

0.32

0.30

0.30

3.00

2.80

0.55

1.2367

0.40

0.40

0.45

5.00

3.00

0.90

H 19

1.2889

0.45

0.40

0.40

4.50

3.00

2.00

(M 2)

1.3343

UHB 29

0.90

0.45

0.40

4.15

5.00

1.85

Vanadis 4E

1.4

0.4

0.4

4.7

3.5

3.7

Vanadis 6

2.1

1.0

0.4

6.8

1.5

5.4

Vanadis 8

2.3

0.4

0.4

Vanadis 23

1.28

(M 3:2)

1.3395

T 15

1.3202

1.40

0.45

0.40

T 42

1.3207

1.30

0.45

0.40

4.50

W

6.35

4.8

3.6

8.0

4.2

5.00

3.1

6.4

4.50

4.15

0.85

3.75

12.0

10.5

4.15

3.75

3.25

10.25

Adapted and updated from ICFG, 1992. International Cold Forging Group 1967–1982 Objectives, History, Published Documents. Meisenbach, Bamberg.

Tool design

321

Table 6.8 Tool steels for high compressive loads typical for active tool parts. AISI

Uddeholm name

Recommended hardness, HRC

W2

–

58–62

Caldie

60

Sleipner

61–63

Vanadis 4 (M3:2)

62–64

a

62–66

a

62–65

Vanadis 6 Vanadis 8

a

In case of excessive tool wear.

Table 6.9 Tool steels for die cores and stress rings. AISI

AISI

Uddeholm name

Recommended hardness, HRC

Die cores

D2

Sverker 21

58–60

6F7

56–58

M2

UHB 29

59–61

(M 3:2)

Vanadis 23

59–61

T 42 Stress rings

63–65

(4340)

Impax

360–390 HB ( 40 HRC)

H11

Vidar Sup.

46–48

(H13)

Orvar Sup.

44–52

Dievar

48–54

Unimax a

a

57

Only for intermediate stress rings.

Tool materials for less loaded tool parts, e.g. support plates distributing the load behind the punch and the die, are listed in Table 6.10. The thickness of these plates is important for an effective distribution of the load and to avoid bending. The materials should have a high compressive strength and a plane parallelism within 0.02 mm is required. Table 6.10 Tool steels for supporting plates. AISI

Uddeholm name

Nimax (H13) (D2) a

a

350–400 HB ( 40 HRC)

Orvar Sup.

50–54

Unimaxb

57

Sverker 21

58–62

Tough hardened, machinable. For supporting plates with many thread holes.

b

Recommended hardness, HRC

322

Metal forming

Tables 6.11 and 6.12 give mechanical properties of tool steels hardened to various degrees. Table 6.13 provides the machinability index Mi of some tool materials defined as Mi ¼

cutting speed of material for 20 min tool life  100%: (6.11) cutting speed of free  cutting steel for 20 min tool life

Table 6.11 Tool steels for high compressive loading and their properties. Hardness HRC

Elasticity modulus (GPa)

Compression test σ 0.2 (MPa)

Bending test σ 0.1 (MPa)

AISI

UHB-name

A2

Rigour

62 60 58

206

2150 1950 1650

2350 2550 2450

D2

Sverker 21

62 60 58

209

2150 1950 1650

2350 2550 2450

M2

UHB 29

65 64 63 62

220

2450 2350 2250 2150

 3000  3000  3000  3000

M3:2

Vanadis 6

66 65 64 63 62

220

2450 2350 2250 2150

 3000  3000  3000  3000

Vanadis 23

65 61 57

230

3000 2500 2000

 4000

O1

Arne

61 60 58 54 50 44 40

206

2050 1950 1650 1500 1300 1200 1100

2450 2550 2450 2150 1850 1550 1350

S1

Regin 3

58 56

(206)

1650 1550

64 60

(201)

W2

Adapted and updated from Dahlin, A˚., 1974. Material f€ or kallsmideverktyg. Mekan resultata 74003 (in Swedish).

Table 6.12 Tool steels for high-tensile loading and their properties. Hardness AISI

UHB-name

HRC

H 13

Orvar 2 M

56 54 52 50

4340

Impax

Brinell

Tensile test Elasticity modulus (GPa)

σ0.2 (MPa)

Elongt. 5 × d (%)

Compression test σ0.2 (MPa)

Bending test σ0.1 (MPa)

212

1550 1500 1450 1350 1500 1450 1350 1200 1100 900

6 7 7 7 7 7 8 8 10 12

1550 1500 1450 1300 1500 1450 1350 1200 1100 900

2250 2150 2050 1850 2050 1950 1750 1650 1450

206

1100 900 700

10 12 15

530 510 470 440 390 330 390 330 270

Adapted from Dahlin, A˚., 1974. Material f€ or kallsmideverktyg. Mekan resultata 74003 (in Swedish).

324

Metal forming

Table 6.13 Machinability index of some tool materials. AISI

Machinability index

A2

42

A6

33

D2

27

D3

27

M2

39

O1

42

O2

42

Hard powder metallurgy metals produced by sintering can alternatively be used for some special applications involving very high compressive stresses. Tungsten carbide is a hard metal consisting of hard tungsten carbide (WC) particles in a softer metal bond of cobalt (Co). Depending on the composition (percentage of WC and Co) and grain size, it may provide compressive strengths up to 4000 MPa and hardness up to 1000 DPN. The elasticity modulus of hard metals is greater than those of tool steels making them also advantageous for applications requiring very low elastic tool deflections (e.g. the outside collar of the coining tools utilised in coin minting). To conclude, it is worth noting that simple passive and structural tool elements such as bolsters, top and bottom plates, punch holders, die holders, guide plates, jigs and fixtures are usually made from low- or medium-carbon steels (e.g. DIN 10050 or 1.1730), or from chromium-molybdenum-alloyed steels (e.g. DIN 1.2312) supplied in the hardened and tempered condition. These steels are primarily intended to be used in the ‘as delivered’ condition.

6.5.2 Hardening and tempering Fig. 6.30 shows a typical example of the different steps in heat treatment of a high-alloyed tool steel. After premachining, the tool material may contain residual stresses, which can be removed by stress annealing to 600–650°C. It is subsequently finish machined. Due to limited thermal conductivity and the effect of varying thicknesses, significant thermal stresses may appear during hardening, which may lead to geometric distortions and crack formation. Preheating is therefore absolutely essential. It is performed by slow heating to the required temperature and maintaining this temperature until the tool is heated thoroughly. Heating to the next level can then take place. After preheating,

Temperature

Tool design

Stress-Relief Annealing

Pre-machining

Finish-machining

Pre-heating

600-650ºC

Austenitizing

Quenching

Tempering

Hardening Temperature 3rd Pre-heating Stage 1 min/mm (900ºC) Hot bath 500-600ºC

Air/Oil

2nd Pre-heating Stage 1/2 min/mm (650ºC) Slow Furnace Cooling

325

1st Pre-heating Stage 1/2 min/mm (400ºC)

1st Tempering

2nd Tempering

1h/20 mm

1h/20 mm

Air

Air

Equalizing Temperature 1h/100 mm

Time

Fig. 6.30 Course of heat treatment for a high-alloyed tool steel. (Adapted from ICFG, 1992. International Cold Forging Group 1967–1982 Objectives, History, Published Documents. Meisenbach, Bamberg.)

the component is heated to the hardening temperature at which temperature it should be held until the desired phase transformation has taken place. Depending on the steel quality and the tool geometry, sudden cooling can take place in water, oil, salt bath or air (e.g. AISI types ‘W’, ‘O’, ‘D’, and ‘A’). In quenching to room temperature, there is a risk of cracking due to thermal stresses. Therefore, cooling to approximately 80°C is recommended, after which the component can be transferred to an oven of 100–150°C. Especially for thick items, it is important to hold the temperature to ensure phase transformation and temperature balance throughout the component. Immediately after these operations, the component should be tempered to avoid crack formation. The component is heated slowly to the required temperature, which is established according to the specifications of the tool material in question and the hardness desired. The holding time is 2 h for every 20 mm thickness, and at least 2 h. The tool material is then air-cooled, and the hardness checked. Especially for high-alloyed steel at least two tempering operations are needed, and three are often advisable. The hardness should be checked after each tempering operation.

6.5.3 Surface treatments and coatings Tool life is often limited by surface deterioration due to wear, corrosion and fatigue. Wear, for example, is influenced by the abrasivity of the workpiece

326

Metal forming

materials (e.g. in forming of titanium) and by the geometry of the workpiece. Complicated geometries create more wear than simple geometries because of the higher pressures applied on the tools. To counteract this, various surface treatments and coatings can be used, which may furthermore reduce the tendency of workpiece material pick-up on the tool surface. They can be applied by means of two different groups of techniques: (i) deposition and (ii) diffusion techniques. Deposition techniques (Table 6.14) are characterised by transporting hard layers of ceramic coatings such as titanium nitride (TiN) or titanium carbide (TiC) from a source and depositing it onto the surface of a tool component. Two main processes are used: physical vapour deposition (PVD) and chemical vapour deposition (CVD). In PVD, the tool steel to be coated must have a high resistance towards undesirable tempering during coating. Deposition is performed after hardening and tempering of the tool steel and the heat treatment and coating operations should not produce significant distortion or loss of hardness of the tool component. CVD is carried out at temperatures of approximately 800–1100°C and requires the tools to be hardened (preferentially secondary hardened) and tempered after coating. The process enables deposition of thicker layers of hard ceramic coatings than PVD but, due to the elevated temperatures involved, there is a risk of dimensional changes. For this reason, CVD is not recommended for tools with narrow-dimensional tolerances. A way to circumvent this problem is to carry out the hardening an extra time before the final hardening and CVD coating and finish grinding the tool before the second hardening and coating. Deposition of hard layers of ceramic coatings by PVD and CVD is carried out by specialised companies, but designers must take into account the need to avoid grooves and to specify mean surface roughness’s Rz < 1 on the surfaces to be coated to ensure good results (Note: Rz/Ra ¼ 4 7, where Ra is the average surface roughness). PVD also presents difficulties in coating internal holes. A typical limit of a maximum length of the internal hole to be coated equals to the diameter of the hole is often used (Schuler, 1998). However, as a general recommendation, designers should contact the companies specialised in PVD and CVD to obtain information and guidance on the active tool components to be coated. Diffusion techniques (Table 6.15) are characterised by the input of an element, such as carbon, nitrogen, boron and oxygen, into the surface of a tool steel component by the application of the appropriate amount of heat, time and surface catalytic reaction.

Table 6.14 Treatment of forming tools by deposition of hard ceramic coatings. TiN

TiC

TiCN

TiAlN

CrN

Synthetic diamond

Colour

Golden

Grey

Violet/brown

Black

Silver

Black

Method

PVD/CVD

CVD

PVD

PVD

PVD

Plasma-CVD

Treatment temperature (°C)

800

950–1100