Damage Modeling of Composite Structures: Strength, Fracture, and Finite Element Analysis [1 ed.] 0128209631, 9780128209639

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DAMAGE MODELING OF COMPOSITE STRUCTURES

DAMAGE MODELING OF COMPOSITE STRUCTURES Strength, Fracture, and Finite Element Analysis PENGFEI LIU Associate Professor, Ocean college, Zhejiang University, Zhoushan, China

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 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: 978-0-12-820963-9 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

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Preface In 2019, I moved to a famous east island city, Zhoushan, Zhejiang province in China, to pursue my new teaching and research career, where Ocean College of Zhejiang University creates an excellent work platform with new office, original labs, as well as fresh air and seafood. Everything in each-day life at Zhoushan campus is valuable and inspiring, although it is also quite challenging. Again, I met many enthusiastic collaborative partners in the new discipline. Particularly, my specialty changes from mechanical engineering, solid mechanics, chemical machinery to ocean and ship engineering, which creates a multidisciplinary opportunity for me. Of course, one invariable topic is just about my research direction in the area of advanced composite materials and structures, which obviously shows strong adaptive capacity for various engineering application. Because my graduate study was on fiber-reinforced metal matrix composites at School of Aeronautics and Astronautics, Zhejiang University, I started to love magic composite materials which has so many advantages because I always believe in the philosophy that one plus one is greater than two, which are bound to innovate future industry and people’s life. Now, my research pays more attention to thermoset and thermoplastic fiber-reinforced resin matrix composite and structures, and I am building my research group with postdoctoral students, doctoral students, graduate students, and undergraduates, who concentrate on some practical composite engineering structures including composite wind turbine blade, and composite pressure vessel. Of course, lightweight design and manufacturing application of composite materials in ship and ocean structures deserves favorable expectation, although serve ocean environments are complicated. After all, so long east coastline in China provides a potential to develop lightweight composite ocean structures and techniques. We believe composites will make an important breakthrough in the field of ship and ocean engineering. In recent years, my group addressed theoretical modeling, numerical analysis, and nondestructive test of laminated and adhesive composite structures. Now, an important task is to apply these methods and techniques to lightweight design, manufacturing, and in-serve performance analysis of practical fiber-reinforced resin matrix composite structures. In 2019, I received the book invitation from Dr. Dennis Mcgonagle on Mechanics of Materials, Elsevier Press. I just attempt to make a summary on our recent fundamental work although it is so tiny. Currently,

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Preface

laminated and adhesive processing for composite materials shows advantages that brings about high strength and stiffness as well as low density in the areas of manufacturing, assembly, and repair of composite structures. However, failure mechanisms of composites are complicated, which includes intralaminar damage and interlaminar delamination as well as the failure of adhesive joints. Various damage models are introduced to provide theoretical and technique supports for damage tolerance design and lifetime prediction of composite structures, with the purpose of reducing the weight and cost. Of course, these models can be developed or are being improved in order to reach high-efficiency and high-fidelity simulation as well as optimal design of composite structures and their components. Before this work, there have been many publications on damage models of composite laminates. However, little work integrates both the intralaminar damage and interlaminar delamination as well as adhesive failure of composite structures systematically under different loads and environments, including intralaminar damage constitutive models, failure criteria, and damage evolution laws as well as virtual crack closure technique, cohesive models, and extended finite element method for predicting delamination and adhesive failure. In addition, numerical techniques for implementing progressive failure analysis using implicit and explicit finite element analysis as well as some important numerical issues including numerical convergence and mesh sensitivity are also significantly required to be deeply discussed. Furthermore, multiscale damage behaviors of composite structures also require more powerful theories and numerical technique. It is therefore necessary and timely to publish this book that outlines the links between various damage models, numerical algorithms, and failure mechanisms of composites. Although it is absolutely not comprehensive, the book would be partly a collection of publications of some world-leading experts and authors in relevant research areas. Therefore, it will cover advanced knowledge on composite mechanics, making it informative reading for the experts, and, concurrently, it can serve as a fruitful reference source for peers intending to work in this area. I would appreciate all my graduate students, undergraduate students, and collaborative colleagues who are/were so zealous in spending beautiful time on academic discussion and exchange as well as collaboration. This stimulates my research motivation and confidence, and enhances my thinking ability progressively. I would also appreciate the strong support from my family—my parents, my wife, and two daughters. Finally, I thank the invitation by Elsevier Press. No retreat but forward for my work and life. Pengfei Liu Ocean College, Zhejiang University, Zhoushan, China

C H A P T E R

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Cohesive models and implicit finite element analysis for delamination analysis of composite laminates with different fracture modes 1.1 Introduction Currently, carbon/glass fiber reinforced polymer composites have been increasingly used in areas such as aerocraft, ocean, and new energy vehicles due to a series of favorable advantages including high stiffnessto-weight ratios and designability. Two important applications in ocean engineering are composite wind turbine blade and composite pressure vessel, in which the composite skin in the blade is laminated structure and the composite pressure vessel belongs to laminated winding structure. They are generally designed and manufactured by deploying fiber layups in different orientations for different layers. These stacked angleply layers are expected to resist loads that come from different directions. However, laminated properties of composites bring about complicated failure mechanisms. In most cases, interlaminar delamination of composites caused by weak shear strength of polymer matrix leads to severe degradation of physical and mechanical performances of composites. Besides, delamination is usually coupled with intralaminar transverse matrix cracking [1]. The delamination issue of composites is essentially represented by the crack initiation and propagation. The energy dissipation near the crack tip due to crack propagation is the main reason for catastrophic material failure, and the mechanical response of composites is largely affected by

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00014-6

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Copyright © 2021 Elsevier Inc. All rights reserved.

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1. Cohesive models and implicit finite element analysis

interlaminar fracture behaviors. To analyze interlaminar fracture properties, some advanced fracture theories have been developed. For example, the virtual crack closure technique (VCCT) [2–4] and the cohesive theory, which are increasingly applied to delamination analysis of composites. While VCCT is efficient in calculating the energy release rate for crack propagation within the linear elastic fracture mechanics, it cannot predict the crack initiation. By comparison, Dugdale and Barenblatt [5, 6] originally proposed the cohesive process zone conception near the crack tip that describes discrete fracture as the material separation across the surface. In the cohesive model, the nonlinear fracture behaviors are controlled by an assumed traction-displacement jump (or called separation) law, which can be essentially considered as a projection of continuous bulk constitutive model into the discontinuous interface, according to the variational stress boundary/initial value problem with an interface discontinuity. One of the major advantages for the cohesive model is that it can predict both the crack initiation and propagation, effectively. Besides, it can avoid the consideration for the complicated stress singularity at the crack tip. If the width of the fracture zone in this discrete representation is approximated to be zero, the length scale associated with failure is infinitely smaller than that in the structure [7]. Now, many popular cohesive models were developed to predict the delamination crack initiation and propagation of composite laminates from the phenomenological perspective. Particularly, numerical application of cohesive models in delamination analysis of composites was reviewed [8–12], which were expected to represent true governed traction-displacement jump constitutive relationships and energy dissipation mechanisms due to crack propagation to large extents that are generally identified by experimental approaches. On one hand, many triangle cohesive models were developed for mixed-mode fracture delamination of composites by many researchers [13–22]. Specially, a discrete triangle cohesive model for delamination analysis was developed [23]. Although the triangle cohesive models have achieved success in predicting the delamination growth of composites, they cannot represent accurately true nonlinear shape of cohesive zone such as the fiber bridging mechanisms in composites. On the other hand, the potential-based exponential cohesive models for mixed-mode fracture were developed [24–26]. Currently, the exponential cohesive models [24–26] and the polynomial cohesive models [27] have been used to represent nonlinear cohesive behaviors. The advantage of the exponential cohesive models is that the exponential function is continuous for the traction-displacement jump cohesive relationship and their derivatives. It could be the reason for the exponential cohesive model to become the most popular for the mathematical derivation and finite element implementation. In order to consider further the damage effect, we develop a damage-type exponential

1.2 Stress boundary value problem with an interface discontinuity

3

cohesive model to predict mixed-mode delamination of composite laminates, which introduces damage variables to describe the irreversible damage evolution behaviors during the delamination of composites. Numerical convergence using finite element analysis (FEA) is solved by introducing the viscous effect and the mesh size effect is solved by following the crack band model [28]. The four typical delamination cases: (a) double cantilever beam (DCB), (b) end notch flexure (ENF), (c) mixed-mode bending fracture (MMB), and (d) fixed ratio mixed mode (FRMM) are used to validate the cohesive model. For the MMB and FRMM cases, the damage evolution locus between individual failure mode is further discussed.

1.2 Stress boundary value problem with an interface discontinuity Let us consider a bounded domain Ω
ℝ3 occupied by an initially linear elastic body, as shown in Fig. 1.1. The boundary ∂ Ω of the domain Ω with the outward normal nΩ is subdivided into either Neumann or Dirichlet boundary conditions defined on ∂ Ωt and ∂ Ωu, respectively, such that ∂ Ω ¼ ∂ Ωu [ ∂ Ωt and ∂ Ωu \ ∂ Ωt ¼ ∅. Further, Ω is divided into Ω+ and Ω by an internal discontinuity Γd such that Ω ¼ Ω+ [ Ω in the reference configuration. The displacement field u across Γd is discontinuous but continuous on Ω+ and Ω. Neglecting the inertia and body force, the strong form is given by the balance of linear momentum and the boundary conditions 8 —  σ ¼ 0, in Ω + [ Ω > > < u ¼ ub , on ∂Ωu (1.1) σ  nΩ ¼ T b , on ∂Ωt > > : σ  N ¼ t ð½½uÞ, on Γd

Tb

∂Ωt –

Γd

–

Γd Ω– Ω+

+

Γd

Point X– ∂Ωu

nΩ N

FIG. 1.1

Point X+

Boundary value problem with an interface discontinuity.

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1. Cohesive models and implicit finite element analysis

where ub : ∂ Ωu ! ℝ3 is the field of prescribed displacement on the Dirichlet boundary and Tb : ∂ Ωt ! ℝ3 is the field of prescribed traction on the Neumann boundary. The Cauchy stress tensor is denoted by σ. The Cauchy cohesive traction t which acts on the internal boundary Γd depends on the displacement jump ½½u at the interface Γd. The reference normal vector N points from Ω to Ω+. The virtual work equation for the body with volume V containing the cohesive interface Γd is given by the following weak form ð ð ð —X δu : σdV + t  δ½½udA ¼ δu  T b dA (1.2) Ω + [Ω

Γd

∂Ωt

1.3 Cohesive modeling for mixed-mode delamination of composite laminates The exponential cohesive traction-displacement jump relationship for interface failure is used, as shown in Fig. 1.2. The expression for each failure mode is given by. ti ¼ etci =½½ui c ½½ui exp ð½½ui =½½ui c Þ, ði ¼ 1, 2, 3Þ

(1.3)

where i ¼ 1, 2, and 3 represent the normal and two tangential directions, respectively. σ max ¼ tc3 and τmax ¼ tc1 ¼ tc2 are the maximum tractions in the normal and tangential directions, respectively. The fracture toughness for each fracture mode is equal to the area under the traction-displacement jump curve ð∞ t1 d½½u1  ¼ eσ max ½½u1 c ¼ Gc1 (1.4) ð∞ 0

t2 d½½u2  ¼

ð∞ 0

0

t3 d½½u3  ¼ eτmax ½½u2 c ¼ eτmax ½½u3 c ¼ Gc2 ¼ Gc3

(1.5)

where Gc1, Gc2, and Gc3 are the fracture toughness for fracture mode-I, II, and III, respectively.

FIG. 1.2 Exponential traction-displacement jump law.

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1.3 Cohesive modeling for mixed-mode delamination of composite laminates

1.3.1 Single-mode delamination model based on continuum damage mechanics Based on Eq. (1.3), the damaged cohesive traction-displacement jump relationship is developed   ½½ui  1 ti ¼ etci =½½ui c ½½ui exp  , ði ¼ 1, 2, 3Þ (1.6) ½½ui c 1  di where di are damage variables. The Helmholtz free energy ψ is written as 1 ψ ¼ ψ e ðt, ½½u, d, TÞ + ψ d ðκ, TÞ , ψ d ðκÞ ¼ aκ  κ 2

(1.7)

where ψ e and ψ d are the free energy representing the elastic deformation and damage evolvement of interface, respectively. T is the temperature and κ is a vector of internal variable. a is a material parameter. According to the thermodynamic theory, the relationships between the thermodynamic conjugate forces (Y, B) and internal variables (d, κ) are given by t¼

∂ψ ∂ψ ∂ψ , Y ¼ , B¼ ¼ aκ ∂½½u ∂d ∂κ

(1.8)

The Clausius-Duhem dissipation inequality imposed by the continuum thermodynamics on the evolution laws is [29]. 1 γ ¼ t  ½½u_  + Y  d_  B  κ_  —T  q  0 T

(1.9)

where γ is the power of dissipation due to damage. q is the heat flux. If the dissipation γ reaches the maximum, the damage evolution law is given by d ∂γ ∂Fd ∂γ d ∂F ¼ 0 ) d_ ¼ λ_ d ¼ 0 ) κ_ ¼ λ_ , ∂Y ∂Y ∂B ∂B

(1.10)

where Fd(Y, d, B) is the damage potential function and λd is the damage consistency parameter. They are assumed to obey the Kuhn-Tucker loading/uploading conditions [29]. d d λ_  0, Fd  0, λ_ Fd ¼ 0

(1.11)

Here, a damage potential function Fd(Y, d, B) is proposed for each failure mode if the temperature effect is neglected Fd ðY, d, BÞ ¼ Y  Y c  ðB  Bc Þ

(1.12)

d From Eqs. (1.11), (1.12), the damage criterion is not satisfied and λ_ 50 d holds at Fd < 0 while the further damage appears and λ_ >0 holds at Fd ¼ 0.

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1. Cohesive models and implicit finite element analysis d

Thus, the consistency condition F_ ¼ 0 for the damage evolution by using Eq. (1.12) derives the consistency parameter λd.  d  ∂Fd : ∂F ∂B Y: d Y= (1.13) d_ ¼ λ_ Ι5  ¼ ∂Y ∂B ∂κ a where Ι is the unit vector. Here, an exponential function for the conjugate force Y is proposed when the displacement jump-based failure criterion is assumed ! κ  ½½uc 1  exp  f ½½u  ½½uc (1.14) Y=a ¼ 1  exp ð1Þ where ½½uc and ½½uf are the initial and critical failure displacement jump, respectively. By combining Eqs. (1.11)–(1.14), the damage vector d is found to be equal to the right-hand formula of Eq. (1.14). The progression of damage is possible if the actual displacement jump ½½u is equal to the evolving internal variable κ 0 and increases for different delamination modes. The damage remains constant if ½½u is smaller than κ or ½½u is not increasing. Thus, the damage evolution conditions for the load function g(½½u, κ) ¼ ½½ u  κ are given by  _ d>0, if ½½u ¼ κ and ½½u_  > 0, Damage evolution (1.15) _d50, if ½½u < κ or ½½u_   0 , No damage evolution The critical displacement jump ½½uf in Eq. (1.14) is numerically solved by Eq. (1.16) ð ½½u f et c =½½uc ½½u exp ð½½u=½½uc Þd½½u ¼ (1.16) 0     c c f f c c c c et ½½u  et ½½u + ½½u exp ½½u =½½u ¼ μG where μ is a constant approximating unity. For example, ½½uf ¼ 11.76½½uc is obtained at μ ¼ 0.9999 by combining Eqs. (1.4) or (1.5) and (1.16).

1.3.2 Mixed-mode delamination model based on continuum damage mechanics The traction-displacement jump relationship for mixed-mode delamination is assumed by   ½½u 1 c c t ¼ et =½½u ½½u exp  c (1.17) ½½u 1  d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 c c 2 2 where t ¼ 2τmax + σ max and ½½u ¼ ð½½u1 c Þ + ð½½u2 c Þ + ð½½u3 c Þ .

1.3 Cohesive modeling for mixed-mode delamination of composite laminates

7

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½½u ¼ ðh½½u1 iÞ2 + ð½½u2 Þ2 + ð½½u3 Þ2 and hi denotes the MacAuley bracket. For mixed-mode delamination, the anisotropic damage variable d in Eq. (1.6) is replaced by the scalar damage variable d in Eq. (1.18) ! κ  ½½uc 1  exp  f ½½u  ½½uc d¼ (1.18) 1  exp ð1Þ The damage evolution conditions for mixed-mode delamination are given by.  _ d>0, if ½½u ¼ κ and ½½u_  > 0 , Damage evolution (1.19) _d50, if ½½u < κ or ½½u_   0, No damage evolution The mixed-mode interlaminar fracture toughness Gc is assumed to obey the B-K law [30]  

 Gshear η (1.20) Gc ¼ Gc1 + Gc2  Gc1 GT where Gshear ¼ G2 + G3 and GT ¼ G1 + G2 + G3. η is a constant depending on the mode mixity. The mixed-mode critical failure displacement jump ½½uf can be numerically solved by combining Eqs. (1.20), (1.21). ð ½½u f etc =½½uc ½½uexp ð½½u=½½uc Þd½½u (1.21) 0     c c f f c c c c ¼ et ½½u  et ½½u + ½½u exp ½½u =½½u ¼ G The tangent stiffness Dtan ij for mixed-mode interface fracture is derived from Eqs. (1.6), (1.18) by considering the interpenetration problem between the delaminated surface. ∂t h ii Dtan ij ¼ ∂ ½uj  8 h D h i E h i i h i h i c ic c h > gij 1  f g1j  ½uj  = ½uj  1  ½uj = ½uj  tcj e1½½uj  =½½uj  = ½uj  , 0 < ½½u < ½½u c > > > 0 1 > h i > > > h D h i E h i i tc > ½u  > j 1 C j B > > gij 1  f g1j  ½uj  = ½uj  h i c exp @1  h i c A > > < 1d ½uj  ½uj  ¼ 8 39 h i2 h i > > > > > < = ½uj  6 ½uj  > ∂d 1 > 7 > > , ½½u c  ½½u < ½½u f 1  h i c 41  d + h i h i c 5 > 2 > > > > : ; ð 1  d Þ >  ∂ ½u   ½u ½u j > j j > > : ½½u  0 or ½½u  ½½u f 0

(1.22)

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1. Cohesive models and implicit finite element analysis

where ∂d ∂d ∂κ ∂½½u ¼ , ði ¼ 1, 2, 3Þ ∂ ½uj  ∂κ ∂½½u ∂ ½uj 

∂d 1 ¼ f ∂κ ½½u  ½½uc

exp 

κ  ½½uc

!

½½uf  ½½uc 1  exp ð1Þ

  ∂½½u ½½ui  h½½ui i 1 + g1i ¼ ∂½½ui  ½½u ½½ui  and f is a penalty factor and gij is the Kronecker dalta. age loading and

∂κ ¼ 0 is for damage uploading. ∂½½u

(1.23)

(1.24)

(1.25) ∂κ ¼ 1 is for dam∂½½u

1.3.3 Numerical considerations for cohesive models using FEA The implicit finite element formulation for implementing the cohesive model for quasistatic problem sees Segurado and LLorca [31], where the middle-plane numerical interpolation technique is used. FEA of developed cohesive model is performed using ABAQUS-UEL (user element subroutine). The zero-thickness interface element is used and the middle-plane Γd is used for numerical interpolation. In the following, the mesh size effect and numerical convergence using FEA are addressed, respectively. 1.3.3.1 Solution for the mesh size effect Yang and Cox [12] pointed out a necessary condition for meshindependent numerical results is that the length of a cohesive zone should be larger than the size of the solid element on either side of crack. For the triangular cohesive model, Turon et al. [32] explored the size effect by introducing the cohesive characteristic length lcz and adjusting the cohesive strength. Yet, the problem for the nonlinear cohesive model is more complicated due to variable interface stiffness. Here, the thought of crack band model [28] is introduced into the exponential cohesive model to solve the size effect. The objective response by FEA is ensured by regularizing the fracture toughness Gc1 in Eq. (1.4) and Gc2 in Eq. (1.5) gc ¼ Gc =lcz where gc is the energy dissipation per unit volume.

(1.26)

1.4 VCCT for predicting the delamination growth of composite laminates

9

The cohesive length lcz was evaluated that is defined as one over which stress increases up to the maximum interfacial strength some distance ahead of the crack tip [5, 6, 33–38]. For different fracture modes, the generalized formula is given by lcz ¼ λ

EGc ðt c Þ2

(1.27)

where E is Young’s modulus and λ is a constant depending on different cohesive models and failure cases. For practical delamination cases, the cohesive lengths are evaluated by adjusting the values for tc and λ. Here, two points should be noted: ① the cohesive length for mode-II is larger than that for mode-I since the delaminated fracture toughness for mode-II is generally larger than that for mode-I; ② for the mixed-mode delamination, the cohesive length for mode-I and mode-II is taken respectively. 1.3.3.2 Solution for convergence issue The convergence problem arises from the softening effect of cohesive model. Essentially, the negative eigenvalues in the stiffness matrix lead to the ill-conditioned finite element equations and unstability of numerical solution. In order to solve the convergence problem, Camanho et al. [15], and Alfano and Crisfield [17] used the local-control arc-length and line searches, Hamitouche et al. [39] introduced the damping effect into the cohesive model. Specially, Gao and Bower [40] introduced the viscous effect into the Xu and Needleman’s cohesive model [24]. Recently, Liu et al. [41] gave a review on the convergence problem and found the viscous damping technique shows higher calculation efficiency than the arc-length algorithm since the latter imposes additional constraint equations between the displacement increment and the arc-length radius. Here, the viscous damping method is used for stabilizing the numerical solution.

1.4 VCCT for predicting the delamination growth of composite laminates Fig. 1.3 shows crack propagation from the crack tip node i to j with the increment crack length Δa. Node i is separated into two nodes i1 and i2 after crack propagation. For node i, the relative displacements in three directions (x, y, z) are Δuix, Δuiy, and Δuiz after propagation and the node forces before propagation are Fix, Fiy, and Fiz. The energy release rate due to crack propagation is given by [2, 3]

10

1. Cohesive models and implicit finite element analysis

j

Δa

Y Z X

i

Δuiy

Δuix

FIG. 1.3 Schematic representation of the VCCT with crack propagation from i and j.

G ¼ GI + GII + GIII ¼ 1 lim Δa!0 2S

ð Δa 0

Fix ðaÞΔuix ðaÞda +

ð Δa 0

Fiy ðaÞΔuiy ðaÞda +

ð Δa

 Fiz ðaÞΔuiz ðaÞda

0

(1.28) where GI, GII and GIII are energy release rate for the mode-I, mode-II, and mode-III, and S is the new area generated due to the crack propagation length Δa. Assume a two-step method is adopted by using the node force before crack propagation and node relative displacement after crack propagation, the G due to crack propagation on the crack closure surface is calculated as [2, 3]

1 G¼ Fix Δuix + Fiy Δuiy + Fiz Δuiz (1.29) 2S Often, the two-step method can be approximately substituted by the one-step method if the mesh sizes on the crack closure surface are sufficiently small. The relative displacement after crack propagation can be substituted by the relative displacements between nearest node pairs before crack propagation [42]. For example, the relative displacements in three directions for the node pair i1 and i2 can be approximately substituted by those for the node pair e1 and e2. In this analysis, the one-step method is used for the calculation of energy release rate. The master-slave node technique is used to simulate crack propagation in FEA. The technique divides the node pair at the same position into two nodes by releasing the coupling freedom degree if the critical energy release rate G is larger than the equivalent one Gc according to the B-K law in Eq. (1.20). The finite element implementation of the VCCT has been integrated into ABAQUS finite element software as a module. In this chapter, the delamination analysis of composite laminates using the VCCT adopts this module.

1.5 Numerical examples and discussion

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1.5 Numerical examples and discussion The DCB, ENF, MMB, and FRMM cases are used to testify the cohesive model, in which the DCB case belongs to the mode-I delamination type, the ENF case belongs to the mode-II delamination type, and the MMB and FRMM cases belong to the mixed-mode delamination type. In order to study the mesh size effect on the load-displacement response using the cohesive model and VCCT, four mesh sizes with different element side length along the direction of crack propagation h ¼ 0.015, 0.03, 0.1, and 0.5 mm are applied. The number of bulk elements is 2E5, 1E5, 3E4, and 6E3, respectively. The Newton-Cotes integration is used [43] and two integration points are set at each direction for each cohesive element, and the displacement, traction, and damage variable as the status variables that depend on the solution are monitored. The contact pairs between the delaminated surface are initially set to prevent the normal interpenetration. The Newton-Raphson algorithm is used and the time increment is properly controlled. λ in Eq. (1.27) is taken 0.5 and 0.1 respectively for the modeI and mode-II delamination.

1.5.1 Numerical results for the DCB delamination case Fig. 1.4 shows the DCB case with boundary and loads. Two loads are symmetrically and oppositely applied to the upper lamina and low lamina of specimen. The material and size parameters are listed in Table 1.1 [23]. Fig. 1.5 shows three mesh models for the DCB case and the number of cohesive interface elements is 4668, 700, and 140 for h ¼ 0.015, 0.1, and 0.5 mm mesh sizes, respectively. Fig. 1.6 shows the crack length a-crack opening displacement 2w curves for the DCB case for three mesh models at σ max ¼ 60 MPa. The ratio a/2w changes from about 12.5 to 4.5. The delamination crack propagates quickly at the beginning, but then enters the unstable delamination stage until the end of the specimen. Fig. 1.7 shows the global loaddisplacement curves by considering the effects of the cohesive

FIG. 1.4

The DCB delamination model with boundary conditions and loads.

12

1. Cohesive models and implicit finite element analysis

TABLE 1.1 Material and geometry parameters for the DCB case [23]. E1 (GPa)

E2 (GPa)

G12 (GPa)

135.9

9

5.2

v12

Gc1 (N/mm)

L (mm)

a0 (mm)

2h (mm)

B (mm)

0.24

0.28

100

30

3

10

FIG. 1.5 Three mesh models for the DCB case.

strengths and mesh sizes. The load first increases and then decreases with the displacement. It can be seen the curves at low cohesive strengths using cohesive model deviate from the true results, but the curve at σ max ¼ 60 MPa is in good agreement with those by the VCCT and discrete cohesive model at 5.7 MPa cohesive strength by [23]. Yet, the peak load using the VCCT is largest. The error may arise from the evaluation of energy release rate as the failure criterion for the VCCT. Spurious numerical oscillation appears using both the cohesive model and VCCT at the softening stage for coarse mesh size, but coarse and fine mesh models lead to consistent results, showing the

1.5 Numerical examples and discussion

13

FIG. 1.6 The crack length-crack opening displacement curves for the DCB case for three mesh models at σ max ¼ 60 MPa.

developed cohesive model is insensitive to the mesh size. The main reason for numerical oscillation arises from the sudden release of strain energy in the intralaminar materials which leads to instantaneous failure of the element, and thus spurious solution jump appears in the unstable snap-back delamination process. Since the practical cohesive model must reflect the stress distribution associated with damage mechanisms occurring ahead of the physical crack tip [44], the numerical oscillation shows the coarse mesh size cannot accurately capture the cohesive zone stress distribution. However, the curve becomes smooth at the decreasing stage when the mesh size h is equal to the cohesive length lcz 1 . Xie and Waas [23] also explored the effect of the bias mesh on the curve and found it can more easily lead to oscillation using coarse and bias mesh sizes. According to the present numerical results, only the element side length h along the direction of crack propagation shows large influence on the numerical results. Besides, the initial crack propagation is shown to appear much earlier than the peak load. This conclusion is also consistent with that obtained by Chen et al. [14]. For the mode-I delamination, no convergence problem occurs without the viscous technique.

14

1. Cohesive models and implicit finite element analysis

FIG. 1.7 The load-displacement curves for the DCB case: (A) effects of the cohesive

strengths using coarse mesh size h ¼ 33lcz 1 ¼ 0.5 mm and (B) effects of the mesh sizes at σ max ¼ 60 MPa. The curves using proposed cohesive model are also compared with the results obtained using the VCCT and existing cohesive models.

15

1.5 Numerical examples and discussion

1.5.2 Numerical results for the ENF delamination case Fig. 1.8 shows the ENF case with boundary and load. Three-point bending with an interlaminar initial crack is generally used to test the mode-II fracture toughness. The contact pair is introduced into the delaminated surface. The material and size parameters are listed in Table 1.2 [23]. The number of cohesive interface elements is 2334, 700, and 140 for h ¼ 0.03, 0.1, and 0.5 mm mesh sizes, respectively. Fig. 1.9 shows the global load-displacement curves. By using low cohesive strengths and coarse mesh sizes, the load-displacement curves by the cohesive model differ from the results using the VCCT and the discrete cohesive model at 57 MPa cohesive strength by Xie and Waas [23]. The load-displacement curve experiences three stages: ① the load first increases with displacement; ② a large “snap” phenomenon appears indicating the unstable crack propagation; ③ and then the load increases rapidly again. Goyal et al. [25] classified this case into the unstable delamination growth at a0 < 0.35L for the ENF specimen. It can also be seen the curves are basically consistent using three mesh models. Different from the DCB case, the convergence becomes more difficult using the Newton-Raphson algorithm for the ENF case which requires the viscous regularization and linear search algorithm as the cohesive strength increases to 500 MPa. The convergence problem due to the increase of the cohesive strength and interface stiffness is also found by Chen et al. [14] and Turon et al. [19]. In addition, Feih [45] found the restriction on the element size can be improved by choosing a larger number of integration points instead of reducing the element sizes. Yet, this still requires sufficiently fine mesh sizes since the combination of a large number of integration points and

FIG. 1.8

The ENF delamination model with boundary and load.

TABLE 1.2 Material and geometry parameters for the ENF case [23]. E1 (GPa)

E2 (GPa)

70

70

v12

Gc2 (N/mm)

L (mm)

a0 (mm)

2h (mm)

B (mm)

0.24

1.45

100

30

3

10

16

1. Cohesive models and implicit finite element analysis

FIG. 1.9 The load-displacement curves for the ENF case: (A) effects of the cohesive

strengths using coarse mesh size h ¼ 33lcz 2 ¼ 0.5 mm and (B) effects of the mesh sizes at τmax ¼ 500 MPa. The curves using proposed cohesive model are also compared with the results obtained using the VCCT and existing cohesive models.

17

1.5 Numerical examples and discussion

coarse mesh sizes may also lead to numerical oscillation. The oscillatory performance of interface elements in terms of the selection of integration points is specially studied by Schellekens and de Borst [43]. They showed Gaussian integration can easily lead to the oscillation of traction when large traction gradients are present over an element. Thus, the NewtonCotes integration is used here.

1.5.3 Numerical results for the MMB delamination case Fig. 1.10 shows the MMB case with boundary and loads. The experimental equipment of the MMB is initially proposed by Reeder and Crews [46] to predict the mixed-mode I/II delamination fracture toughness. By loading with a lever, a single applied load produces bending loads on the specimen. Different mode I/II ratios can be obtained by adjusting the length e of the lever. The material and size parameters are listed in Table 1.3 [13, 23]. The number of cohesive interface elements is 700 and 140 for h ¼ 0.1 and 0.5 mm mesh sizes, respectively. Fig. 1.11 shows the load-displacement curves for the MMB case. Similar to the DCB and ENF cases, the curve shows small numerical oscillation at the softening stage for coarse mesh sizes using the cohesive model and VCCT. As the cohesive strength σ max(or τmax) increases, the initial stiffness etc/½½uc of cohesive interface increases rapidly and differs for normal and shear failure. The initial normal interface stiffness reaches about 5E4 N/mm at σ max ¼ 500 MPa. As the cohesive strength increases to 500 MPa, the curve is close to that obtained by the VCCT. Same to the ENF case, the curve for the MMB case also exhibits three distinct stages, in which the sudden

FIG. 1.10

The MMB delamination model with boundary and loads.

TABLE 1.3 Material and geometry parameters for the MMB case [13, 23]. E1 (GPa)

E2 (GPa)

G12 (GPa)

v12

Gc1 5 Gc2 (N/mm)

L (mm)

e (mm)

a0 (mm)

2h (mm)

B (mm)

135.9

9

5.2

0.24

4

100

50

30

3

10

18

1. Cohesive models and implicit finite element analysis

FIG. 1.11 The load-displacement curves for the MMB case: (A) effects of the cohesive

strengths using coarse mesh size h ¼ 5lcz 1 ¼ 0.5 mm and (B) effects of the mesh sizes at σ max ¼ τmax ¼ 500 MPa. The curves using proposed cohesive model are also compared with the results obtained using the VCCT and existing cohesive models.

“snap” effect shows unstable delamination properties. The delamination appears just at the beginning of loading, much earlier than the moment at which the peak load reaches. The numerical results by the developed cohesive model are also compared with the results using the interface

1.5 Numerical examples and discussion

19

delamination model and cohesive model [7, 13, 23]. In general, these curves show good agreement except small errors at the last increasing stage. Compared with the B-K law used here, the power law is given by [13, 22, 23, 25, 27]  β  β G1 G2 + ¼1 (1.30) Gc1 Gc2 where β ¼ 1 and β ¼ 2 denote the linear and quadratic failure criteria, respectively. The power law in Eq. (1.30) fails to accurately capture the dependence of the mixed-mode fracture toughness on the mode ratio [15]. By contrast, the difficulty using the B-K law is to determine the coefficient η in Eq. (1.20) which yet depends on the mode-mixity and the phase angle between the tangent and normal displacement jumps [30]. Turon et al. [18] explored the effect of different mode ratios on the load-displacement curves and got the valueη ¼ 2.284 in their cohesive models by applying a least-square fit to the experimental data for AS4/PEEK carbon fiber composites. Here, the determination for η is to calculate ½½uf/½½uc in Eq. (1.18). We found ½½uf/½½uc varies little as η changes. For example, ½½uf/½½uc changes from 2.1 to 2.3 as η varies from 1.5 to 2.5 at σ max ¼ τmax ¼ 500 MPa. Thus, η ¼ 2.0 is used here. Fig. 1.12A shows the damage variable d-mixed mode displacement jump ½½u curves for three cohesive strengths and Fig. 1.12B shows the damage evolution locuses at σ max ¼ τmax ¼ 500 MPa. As the cohesive strength increases, the ability for the interface to resist the delamination becomes stronger. The approximate elliptic damage evolution locus according to the present cohesive model is similar to that obtained by Turon et al. [18]. The only difference is that the present cohesive model requires higher cohesive strengths than that by Turon et al. [18] in order to get accurate numerical results since a nonlinear exponential cohesive law instead of a simpler triangle relationship is used. Similar to the developed cohesive model, the conclusion for the requirement for the cohesive strength to increase to high values such as 500 MPa in order to lead to accurate results is also obtained by the potential-based exponential cohesive model [26]. Besides, the numerical convergence is also a large challenge for both the triangle and exponential cohesive models for softening problems. For the exponential cohesive models, high cohesive strength adds the difficulty of convergence. Yet, the convergence problem for the triangle cohesive models lies in the sharp and discontinuous transition at the peak traction. Although the artificial viscous method is an ideal one for improving the convergence, the viscous coefficients should be properly chosen, similar to the analysis by Gao and Bower [40].

20

1. Cohesive models and implicit finite element analysis

1.00

Damage variable, d

0.75

0.50 smax = tmax = 100 MPa 0.25

smax = tmax = 300 MPa smax =tmax = 500 MPa

0.00

(A)

0.05

0.10 0.15 0.20 0.25 0.30 Displacement jump [[u]] (mm)

0.35

0.06 d=0 0.05

d = 0.05 d = 0.856

[[u2]] (mm)

0.04

d = 1.0

0.03 0.02 0.01 0.00

(B)

0.01

0.02

0.03

0.04

0.05

0.06

[[u1]] (mm)

FIG. 1.12 Damage evolution of cohesive elements for the MMB case: (A) damage variable d-mixed mode displacement jump ½½u curve for three cohesive strengths and (B) damage evolution locus at σ max ¼ τmax ¼ 500 MPa.

1.5.4 Numerical results for the FRMM delamination case Fig. 1.13 shows the FRMM case with boundary and load. The material and size parameters are listed in Table 1.4 [14]. Different from the MMB case, the mode-I and II delamination fracture toughness for the FRMM case is different. The number of cohesive interface elements is

21

1.5 Numerical examples and discussion

FIG. 1.13

The FRMM delamination model with boundary and load.

TABLE 1.4 Material and geometry parameters for the FRMM case [14]. E1 (GPa)

E2 (GPa)

G12 (GPa)

v12

Gc1 (N/mm)

Gc2 (N/mm)

L (mm)

a0 (mm)

2h (mm)

B (mm)

176

8

6

0.27

0.257

0.856

105

45

3.1

24

1834, 550 and 110 for h ¼ 0.03, 0.1, and 0.5 mm mesh sizes, respectively. Fig. 1.14 shows the load-displacement curves for the FRMM case. Fig. 1.15 shows the damage evolution locuses. Same to the DCB, ENF and MMB cases, the delamination starts much earlier than the peak load. Similar to the ENF and MMB cases, three stages appear in the process of delamination, accompanied by the sudden softening effect. It can be seen all curves for the DCB, ENF, MMB, and FRMM cases are close to the true results as the cohesive strength increases. This conclusion is also obtained by Xie and Waas [23]. At σ max ¼ τmax ¼ 500 MPa, the curve using the developed cohesive model is basically consistent with the results obtained by the experiments, the VCCT and Chen et al. [14]. Also, they studied the effect of parameter β in Eq. (1.30) and found the results using the quadratic failure criterion β ¼ 2 are more close to the experimental results. In addition, Chandra et al. [8] gave an overview on the selection of cohesive strength (or displacement jump) for current cohesive models and applied cases. The traction-displacement cohesive properties between bi-material interface can be determined by inverse optimization computation by measuring the crack-tip opening displacements directly [47, 48]. Besides, such as the infra-red crack opening interferometry measurements [49] and laser treatment technique [50] are also used to determine the cohesive properties and to improve the bond toughness of interface. In terms of the sensitivity analysis, the effects of cohesive parameters including the cohesive fracture energy, cohesive shape, and cohesive strengths on the mixed-mode delamination mechanisms using cohesive models are studied. The cohesive models as a phenomenological method is expected to expound the true bonding properties for various bi-material interface by adjusting the cohesive parameters.

22

1. Cohesive models and implicit finite element analysis

FIG. 1.14 The load-displacement curves for the FRMM case: (A) effects of the cohesive

strengths using coarse mesh h ¼ 12.5lcz 1 ¼ 0.5 mm and (B) effects of the mesh sizes at σ max ¼ τmax ¼ 500 MPa. The curves using proposed cohesive model are also compared with the results using the VCCT, existing cohesive model and experiment.

23

1.6 Concluding remarks

Damage variable, d

1.00

(A)

0.75 smax = tmax = 100 MPa smax = tmax = 300 MPa

0.50

smax =tmax = 500 MPa

0.25

0.00 0.00

0.01 0.02 0.03 0.04 0.05 Displacement jump [[u]] (mm)

0.06

0.009 d=0 d = 0.58 d = 0.94 d = 1.0

0.008

[[u2]] (mm)

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000

(B)

0.004

0.005

0.006 0.007 [[u1]] (mm)

0.008

0.009

FIG. 1.15 Damage evolution of cohesive elements for the FRMM case: (A) damage variable d-mixed mode displacement jump ½½u curve for three cohesive strengths and (B) damage evolution locus at σ max ¼ τmax ¼ 500 MPa.

1.6 Concluding remarks Cohesive models have been increasingly used for delamination analysis of composite laminates. Various cohesive models were developed to predict the delamination growth. In this chapter, the exponential and bilinear cohesive models are used to predict mixed-mode delamination of composites. By performing numerical simulation on the DCB, ENF, MMB, and FRMM delamination cases, the global load-displacement

24

1. Cohesive models and implicit finite element analysis

curves using developed cohesive model are shown to be in good agreement with the results obtained by using VCCT and existing cohesive models after the mesh size effect and convergence problems are solved effectively. The developed cohesive model can be also applied to adhesive failure analysis.

References [1] Ladeveze P, Lubineau G, Marsal D. Towards a bridge between the micro- and the mesomechanics of delamination for laminated composites. Compos Sci Technol 2006;66:698–712. [2] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 1977;9:931–8. [3] Krueger R. Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 2004;57:109–43. [4] Liu PF, Hou SJ, Chu JK, et al. Finite element analysis of postbuckling and delamination of composite laminates using virtual crack closure technique. Compos Struct 2011;93:1549–60. [5] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8:100–4. [6] Barenblatt I. Mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 1962;7:55–129. [7] Mosler J, Scheider I. A thermodynamically and variationally consistent class of damagetype cohesive models. J Mech Phys Solids 2011;59:1647–66. [8] Chandra N, Li H, Shet C, Ghonem H. Some issues in the application of cohesive zone models for metal-ceramic interfaces. Inter J Solids Struct 2002;39:2827–55. [9] Elices M, Guinea GV, Go´mez J, Planas J. The cohesive zone model advantages, limitations and challenge. Eng Fract Mech 2002;69:137–63. [10] Cornec A, Scheider I, Schwalbe KH. On the practical application of the cohesive model. Eng Fract Mech 2003;70:1963–87. [11] de Borst R. Numerical aspects of cohesive-zone models. Eng Fract Mech 2003;70:1743–57. [12] Yang QD, Cox BN. Cohesive models for damage evolution in laminated composites. Int J Fract 2005;133:107–37. [13] Mi Y, Crisfield MA, Davies GA. Progressive delamination using interface elements. J Compos Mater 1998;32:1246–72. [14] Chen J, Crisfield M, Kinloch AJ, et al. Predicting progressive delamination of composite material specimens via interface elements. Mech Compos Mater Struct 1999;6:301–17. [15] Camanho PP, Davila CG, De Moura MF. Numerical simulation of mixed-mode progressive delamination in the composite materials. J Compos Mater 2003;37:1415–38. [16] Alfano G, Crisfield MA. Finite element interface models for the delamination analysis of laminated composites: mechanics and computational issues. Int J Numer Methods Eng 2001;50:1701–36. [17] Alfano G, Crisfield MA. Solution strategies for the delamination analysis based on a combination of local-control arc-length and line searches. Int J Numer Methods Eng 2003;58:999–1048. [18] Turon A, Camanho PP, Costaa J, Da´vila CG. A damage model for the simulation of delamination in advanced composites under variable-mode loading. Mech Mater 2006;38:1072–89. [19] Turon A, Camanho PP, Costa J, Renart J. Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: definition of interlaminar strengths and elastic stiffness. Compos Struct 2010;92:1857–64.

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[20] Swaminathan S, Pagano NJ, Ghosh S. Analysis of interfacial debonding in threedimensional composite microstructures. J Eng Mater Technol 2006;128:96–106. [21] Bui VQ, Iannucci L, Robinson P, Pinho ST. A coupled mixed-mode delamination model for laminated composites. J Compos Mater 2011;45:1717–29. [22] Jiang WG, Hallett SR, Green BG, Wisnom MR. A concise interface constitutive law for analysis of delamination and splitting in composite materials and its application to scaled notched tensile specimens. Int J Numer Methods Eng 2007;69:1982–95. [23] Xie D, Waas AM. Discrete cohesive zone model for mixed-mode fracture using finite element analysis. Eng Fract Mech 2006;73:1783–96. [24] Xu XP, Needleman A. Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 1994;42:1397–434. [25] Goyal VK, Johnson ER, Davila CG. Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities. Compos Struct 2004;65:289–305. [26] Park K, Paulino GH, Roesler JR. A unified potential-based cohesive model of mixedmode fracture. J Mech Phys Solids 2009;57:891–908. [27] El-Sayed S, Sridharan S. Predicting and tracking interlaminar crack growth in composites using a cohesive layer model. Compos Part B Eng 2001;32:545–53. [28] Bazˇant ZP, Oh B. Crack band theory for fracture of concrete. Mater Struct 1983;16:155–77. [29] Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge: Cambridge University Press; 1990. [30] Benzeggagh ML, Kenane M. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos Sci Technol 1996;56:439–49. [31] Segurado J, LLorca J. A new three-dimensional interface finite element to simulate fracture in composites. Inter J Solids Struct 2004;41:2977–93. [32] Turon A, Da´vila CG, Camanho PP, Costa J. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng Fract Mech 2007;74:1665–82. [33] Irwin GR. Plastic zone near a crack and fracture toughness, In: Proceedings of the seventh Sagamore Ordnance materials conference, vol. IVNew York: Syracuse University; 1960. p. 63–78. [34] Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6:773–82. [35] Rice JR. The mechanics of earthquake rupture. In: Dziewonski AM, Boschhi E, editors. Physics of the Earth’s interior. Proceedings of the international school of physics “Enrico Fermi”, Course 78, 1979. Amsterdam: Italian Physical Society/North-Holland; 1980. p. 555–649. [36] Bao G, Suo Z. Remarks on crack-bridging concepts. Appl Mech Rev 1992;24:355–66. [37] Cox BN, Marshall DB. Concepts for bridged cracks in fracture and fatigue. Acta Metall Mater 1994;42:341–63. [38] Harper PW, Hallett SR. Cohesive zone length in numerical simulations of composite delamination. Eng Fract Mech 2008;75:4774–92. [39] Hamitouche L, Tarfaoui M, Vautrin A. An interface debonding law subject to viscous regularization for avoiding instability: application to the delamination problems. Eng Fract Mech 2008;75:3084–100. [40] Gao YF, Bower AF. A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces. Model Simul Mater Sci Eng 2004;12(3):453–63. [41] Liu PF, Zheng JY. Recent developments on damage modeling and finite element analysis for composite laminates: a review. Mater Des 2010;31:3825–34.

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[42] Xie D, Biggers SB. Progressive crack growth analysis using interface element based on the virtual crack closure technique. Finite Elem Anal Des 2006;42:977–84. [43] Schellekens JCJ, de Borst R. On the numerical integration of interface elements. Int J Numer Methods En 1993;36:43–66. [44] Schellekens JCJ, de Borst R. A nonlinear finite-element approach for the analysis of mode-I free edge delamination in composites. Int J Solids Struct 1993;30:1239–53. [45] Feih S. Development of a user element in ABAQUS for modelling of cohesive laws in composite structures. Roskilde, Denmark: Risø National Laboratory; 2005. Risø-R-1501. [46] Reeder JR, Crews Jr. JR. Mixed-mode bending method for delamination testing. AIAA J 1990;28:1270–6. [47] Jacobsen TK, Sorensen BF. Mode I intra-laminar crack growth in composites-modelling of R-curves from measured bridging laws. Compos Part A 2001;32:1–11. [48] Shen B, Stanciulescu I, Paulino GH. Inverse computation of cohesive fracture properties from displacement fields. Inverse Prob Sci Eng 2012;18:1103–28. [49] Gowrishankar S, Haixia M, Liechti KM, Huang R. A comparison of direct and iterative methods for determining traction-separation relations. Int J Fract 2012;177:109–28. [50] Alfano M, Lubineau G, Furgiuele F, Paulino GH. On the enhancement of bond toughness for Al-epoxy T-peel joints with laser surface treated surfaces. Int J Fract 2011;171:139–50.

C H A P T E R

2

Implicit finite element analysis of failure behaviors of adhesive composite joints using exponential and bilinear cohesive models 2.1 Introduction Adhesive jointing is considered as one of the most favorable techniques employed in composite structures because they can reduce detrimental stress concentrations and guarantee structural integrity compared with traditional fasteners such as riveting and bolting, in which fiber cutting and interruption resulting in severely nonuniform stress distributions are often needed [1]. However, interface failure between the adherend and finite-thickness adhesive layer due to low bonding strength of the adhesive layer often leads to the loss of stiffness and strength of structures. Deep insight into the adhesive failure mechanism requires advanced numerical methods. Because of very thin thickness (e.g., 0.1 mm) and complicated mechanical properties of adhesive layers, modeling and simulation on the adhesive failure are a tough task. Currently, the cohesive theory [2, 3] has been demonstrated to be the most popular approach for the delamination/ adhesive failure analysis of composites. The cohesive zone model (CZM) assumes a traction-displacement jump relationship, which characterizes progressive separation of the bonded interface, as shown in Fig. 2.1. Now, there are a lot of cohesive models in terms of the shape of tractiondisplacement jump curves, such as the bilinear cohesive models [4–8], the trapezoidal and polynomial cohesive models, [9, 10] and the exponential

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00008-0

27

Copyright © 2021 Elsevier Inc. All rights reserved.

28

2. Failure behaviors of adhesive composite joints

FIG. 2.1 Fracture process zone and cohesive traction-displacement jump relationships.

cohesive models [11–14]. These phenomenal cohesive models have achieved success in predicting the delamination/adhesive failure of composites to a large extent when they are superimposed to represent the true fracture process zone. However, numerical convergence due to “snapback” instability as a fundamental issue using implicit finite element analysis (FEA) poses a large challenge to robust and accurate prediction of failure mechanisms. This issue arises essentially from cohesive softening behavior with nonpositive definite stiffness matrix, leading to the divergence of solutions from the equilibrium path during Newton iterations. The convergence issue is typically circumvented by introducing some advanced numerical techniques. One famous numerical method is the cylindrical arc-length scheme (or called Riks scheme) that attempts to trace an equilibrium path using the arc-length search, avoiding the singular point in the load responses encountered in the Newton-Raphson algorithm [4, 5, 15]. However, the Riks scheme often requires high computational cost because it has to solve an additional constraint equation between the arc-length radius and the displacement increment in each iteration. By comparison, another feasible scheme is to introduce an artificial viscous damping into the nonlinear stiffness equation. Introduced viscous damping or energy is proven to regularize the instability problem with zero or negative eigenvalues in the stiffness matrix well. By adopting this idea, Chaboche et al. [16] introduced the viscous effect into Tvergaard’s polynomial cohesive model to improve the robustness of numerical results, e.g., the elimination of “displacement jump” in the load curves, and Hamitouche et al. [17] introduced the damage rate dependency to limit the cohesive softening problem. Gao and Bower [18] introduced a fictitious small viscosity into Xu and Needleman’s exponential cohesive model [11], and Hu et al. [19] proposed a move-limit method by building up a rigid wall to remove the instability. Although viscous regularization scheme shows some advantages in improving convergence over the arc-length algorithm, an unsolved issue occurs about how to select appropriate viscous parameters which should be large enough to regularize cohesive softening behavior but small enough not to affect numerical accuracy. Furthermore, it is not yet clear how some important

2.2 Bilinear cohesive model

29

cohesive parameters including the cohesive strength and cohesive shape affect the numerical convergence and computational efficiency based on viscous regularization for simulating failure of adhesive composite joints. In this chapter, we first introduce the bilinear cohesive model developed by Camanho et al. [6] and the exponential cohesive model developed by Liu and Islam [14]. Then, the implicit finite element formulation for cohesive models is presented based on the middle-plane interpolation idea by using ABAQUS-UEL subroutine (user element subroutine) and the artificial viscous effect is introduced into the stiffness equation in implicit FEA to improve numerical convergence. Furthermore, the purpose of this chapter is to study the effects of different cohesive law parameters including the cohesive strength and cohesive shape on convergence in simulating the failure of the single lap composite joint and the skin/ stiffener composite panel under tension by using two mesh models. In addition, the effect of different viscous parameters on the load responses of two composite structures is also studied. From the convergence and accuracy perspective, this chapter presents some suggestions for appropriately selecting cohesive strengths and viscous parameters for failure analysis of composite adhesive joints using cohesive models.

2.2 Bilinear cohesive model Camanho et al. [6] proposed a bilinear cohesive model, as shown in Fig. 2.2. The single-mode traction-displacement jump relationships are given by. 8 Δmax , Δmax  Δc > < ti ¼ K
i s  i max ci f ti ¼ 1  di KΔi , Δi  Δmax < Δi , i > f : t ¼ 0, Δmax  Δi i i  max
max  Δi ¼ max
Δi , jΔij , i ¼ 1, 2, mode II and III (2.1) Δmax ¼ max Δmax , Δ3 with Δmax  0, mode I 3 3 3

FIG. 2.2

Bilinear cohesive model.

30 where dsi ¼

2. Failure behaviors of adhesive composite joints Δi ðΔmax Δci Þ
i f

Δmax ðΔi Δci Þ i f

 0  dsi  1, i ¼ 1, 2, 3 are damage variables. Δ ¼ ½½u

is the displacement jump vector and u is the displacement vector. Δf3 ¼ 2Gc3/σ max, Δf1, 2 ¼ 2Gc1, 2/τmax. σ max ¼ tc3 and τmax ¼ tc1 ¼ tc2 are the maximum tractions in the normal and tangential directions, respectively. Δc1 ¼ Δc2 ¼ τmax/K, Δc3 ¼ σ max/K. Gci (i ¼ 1, 2, 3) are the modes-III, II, and I fracture toughness for adhesive failure, respectively. Δmax ¼ max(Δmax , Δi) i i are the maximum displacement jumps in the load history indicating the irreversibility. In order to avoid the interpenetration of the completely failed interface under compression, the penalty stiffness is introduced in the normal-3 direction t3 ¼ KΔ3 (Δ3  0). The mixed-mode traction-displacement jump relationships are written as 8 max c > < ti ¼ KΔi , Δi  Δi , f c s ti ¼ ð1  d ÞKΔi , Δi  Δmax < Δi , (2.2) i > f : t ¼ 0, max Δi  Δi i where the scalar mixed-mode damage variable ds(0  ds  1) is written as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f
c
max  Δm Δmax m  Δm s   , Δm ¼ Δ21 + Δ22 + hΔ3 i2 , Δmax ¼ max Δ , Δ d ¼ , m m m f c Δmax Δ  Δ m m m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 > < Δc Δc
1 +
β  , Δ3 > 0 3 1 c 2 c 2 , Δcm ¼ Δ + βΔ 1 3 > :
c 2
c 2 Δ1 + Δ2 , Δ3  0 " 8  2 η # > < 2 Gc +
Gc  Gc  β , Δ3 > 0 2 1 f (2.3) Δm ¼ KΔcm 1 1 + β2 >
c 2
c 2 : Δ1 + Δ2 , Δ3  0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2  ffi where hi denotes the MacAuley bracket. β ¼ Δ1 + Δ22 =Δ23 ðΔ3 > 0Þ is the mode mixity ratio and η is the parameter defined in the B-K law [20].

2.3 Exponential cohesive model The exponential cohesive model in last chapter developed by us [14] is used, as shown in Fig. 2.3. For single-mode, the damaged tractiondisplacement jump cohesive laws ti  Δmax (Δmax ¼ max(Δmax , Δi)) are writi i i ten as


 Δmax s 1 i ti ¼ etci =Δci Δmax exp  1  d , ði ¼ 1, 2, 3Þ (2.4) i i Δci

2.3 Exponential cohesive model

FIG. 2.3

31

Exponential cohesive model.

where dsi (i ¼ 1, 2, 3) are damage variables in the normal and tangential directions, respectively. ! Δmax  Δci i 1  exp  f Δi  Δci s , ði ¼ 1, 2, 3Þ (2.5) di ¼ 1  exp ð1Þ where Δci (i ¼ 1, 2, 3) and Δfi(i ¼ 1, 2, 3) are the initial and critical failure displacement jumps, respectively. The critical displacement jumps Δfi(i ¼ 1, 2, 3) in Eq. (2.5) can be numerically solved by the following equation: ð Δf i
 etci =Δci Δmax exp Δmax =Δci dΔi ¼ i i (2.6) 0     f f c c c c c c eti Δi  eti Δi + Δi exp Δi =Δi ¼ μGi , ð i ¼ 1, 2, 3Þ where μ is a constant approximating unity and Δfi ¼ 11.76Δci (i ¼ 1, 2, 3) is obtained at μ ¼ 0.9999. The traction-displacement jump relationship t  Δmax(Δmax ¼ max(Δmax, Δ)) for mixed-mode fracture is assumed by  Δmax 1 c c max t ¼ et =Δ Δ exp  c (2.7) Δ 1  ds where ds is the scalar interfacial damage variable for mixed-mode fracture. q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
2 Δc ¼ Δc1 + Δc2 + Δc3 and tc ¼ 2τ2max + σ 2max are the equivalent critical displacement jump and critical traction, respectively. The variable qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi displacement jump Δ ¼ ðΔ1 Þ2 + ðΔ2 Þ2 + ðΔ3 Þ2 is assumed. For the mixed-mode fracture, the damage variable ds is given by  Δmax  Δc 1  exp  f Δ  Δc (2.8) ds ¼ 1  exp ð1Þ The mixed-mode critical failure displacement jump Δf can be numerically solved by

32

2. Failure behaviors of adhesive composite joints

ð Δf 0

etc =Δc Δmax exp ðΔmax =Δc ÞdΔ

 ¼ etc Δc  etc Δc + Δf exp Δf =Δc ¼ Gc

(2.9)

where the mixed-mode fracture toughness Gc is assumed to obey the B-K law [20]. The second-order tangent stiffness tensor Dtan for mixed-mode fracture is derived from Eqs. (2.4), (2.8) by considering the interpenetration between the delaminated interface ∂ti ∂Δmax D E i  8j h Δmax =Δcj > =Δcj etcj e j =Δcj , 0 < Δmax < Δc , =Δmax 1  Δmax δij 1  ξδ1j Δmax > > j j j > ! > > h D E i tc Δmax > > 1 j j > max =Δmax >  exp 1  > c j < δij 1  ξ δ1j Δj Δj Δcj 1  ds 8 2 39 ¼ > < = Δmax s Δmax > > 1 j j > 41  ds + ∂d 5 , > 1  Δc  Δmax < Δf , > > : Δcj ∂Δmax Δcj ð1  ds Þ2 ; > > j > > : 0, Δmax  0 or Δmax  Δf

Dtan ij ¼

(2.10)

where ∂ds ∂ds ∂Δmax ¼ ∂Δmax ∂Δmax ∂Δmax j j  ∂ds 1 1 Δmax  Δc ¼ exp  f ∂Δmax 1  exp ð1Þ Δf  Δc Δ  Δc max  Δi ∂Δmax Δmax i ¼ 1 + δ 1i max Δmax ∂Δmax Δ i i

(2.11)

(2.12) (2.13)

and ξ is a penalty factor and δij is the Kronecker delta.

2.4 Implicit finite element formulation for implementing cohesive models Segurado and LLorca [21] developed a three-dimensional (3D) finite element formulation for implementing cohesive models by adopting the middle-plane numerical technique, as shown in Fig. 2.4. Similar to the standard isoparametric elements, the local coordinate system (ξ, η, ζ) and the global coordinate system (x, y, z) are defined. The displacement jump in the local coordinate system ½½u for the cohesive element is calculated as

2.4 Implicit finite element formulation for implementing cohesive models

FIG. 2.4

33

Middle-plane interpolation technique for implementing 3D cohesive element

using FEA.


 (2.14) Δ ¼ ½½u ¼ φT ½½uðξ, ηÞ, φ ¼ φ n, et 1 , et 2
 where φ n, et 1 , et 2 is a 3  3 transformation matrix of the local coordinate system to the global one and three orthotropic directions are given by  R ∂x ∂xR ∂xR  ∂ξ ∂η ∂ξ  , et 1 ¼  R  , et 2 ¼ et 1  n n¼  R (2.15) R ∂x   ∂x   ∂x    ∂ξ  ∂ξ  ∂η  where the coordinate of middle-plane point xR is written as  1 xR ¼ H ðξ, ηÞð I 1212 j I 1212 Þ x + u 2 ~

(2.16)

where x and u are the 24  1 node coordinate matrix and the 24  1 node displacement~matrix for 3D eight-node elements, respectively. H(ξ, η) is a 3  12 matrix for 3D eight-node elements including the shape function and I1212 is the identity matrix. The relative displacement at the point (ξ, η) between element sides is interpolated as a function of the relative displacement between the paired nodes 8 9 < ½½u1 ðξ, ηÞ = ½½uðξ, ηÞ ¼ ½½u2 ðξ, ηÞ ¼ H ðξ, ηÞ½½uN  ¼ H ðξ, ηÞϕu ¼ Lu (2.17) : ; ~ ~ ½½u3 ðξ, ηÞ where L is a 3  24 matrix and 1  ξ, η  1 are local coordinates of elements. ϕ ¼ ð I 1212 j I 1212 Þ is a 12  24 matrix. Finally, the node force vector Fc(24  1 matrix) and the stiffness matrix Kc(24  24 matrix) are written as

34

2. Failure behaviors of adhesive composite joints

ðð Fc ¼

LT φT tdA ¼

ðð

e T tdA ¼ M

XX

ðð

ðð

i

e T t jJ j, ω i ωj M

j

∂F c e T ∂t ∂½½u dA ¼ e T ∂t dA ¼ M ¼ M ∂u ∂½½u ∂u ∂u ~ ~ ~ ðð ðð T ∂t ∂½½u T ∂t e e e dA ¼ M MdA ¼ ¼ M ∂½½u ∂u ∂½½u ~ XX e T Dtan M e jJ j ¼ ω i ωj M

Kc ¼

i

(2.18)

j

where j Jj is the Jacobian for isoparametric transformation and ω is the e weight. M5φL is a 3  24 shape function matrix for 3D cohesive element.

2.5 Cohesive zone length Cohesive models introduce a length scale (generally called cohesive zone length) due to cohesive softening behavior. The cohesive zone length is defined as the distance from the crack tip to the position where the maximum cohesive traction is attained [22, 23]. If the cohesive length scale is not considered in the analysis, then the dissipation of adhesive fracture energy cannot be accurately captured which will lead to mesh sensitivity problem. Thus, the cohesive zone length must be properly evaluated. Turon et al. [22] evaluated the cohesive zone length by comparing existing theoretical formulas based on the crack-tip stress and energy fields and gave an expression for the cohesive length Lcz Lcz ¼

9 Gc πE 32 ðtc Þ2

(2.19)

where E is the Young’s modulus of the adhesive material that is approximately equal to the transverse modulus, Gc is the critical energy release rate. Turon et al. [22] suggested three elements in the cohesive zone are sufficient to predict the crack growth, which is adopted here.

2.6 Numerical results and discussion Cohesive models are implemented using ABAQUS-UEL (user implicit element subroutine), where the main work in each iteration is to update the node residual force ΔFc(24  1 matrix) and the element stiffness matrix Kc(24  24 matrix) in Eq. (2.18) for 3D eight-node zero-thickness interface

2.6 Numerical results and discussion

35

element. Due to cohesive softening behavior, the artificial viscous force Fv is introduced to improve numerical convergence [24] ( F ext  F c  F v ¼ R, (2.20) F v ¼ cM∗ v, v ¼ Δu=Δt ~ where M∗ is an artificial mass matrix calculated with unity density, v is the node velocity, Δu is the node displacement increment, R is the tolerance, c ~ is a constant damping factor, and Δt is the pseudo time increment during nonlinear iterations. In the following, two cases on the single lap joint and the skin/stiffener composite panel under tension are used to study the numerical convergence. Because the mode-II shear failure governs the crack growth process for these two cases, the effect of normal cohesive strength σ max on convergence and accuracy is very small and σ max ¼ 40 MPa is used. The tangential cohesive strength τmax changes for comparison. The viscous constant c ¼ 5e-3 in Eq. (2.20) is generally used. Small time increment is required to guarantee numerical precision (the initial, minimum, and maximum time increments are △t ¼ 0.0001s, 1e-10s, and 0.001s, respectively). Volokh [25] showed the convergence rate for four cohesive models including the bilinear, parabolic, sinusoidal, and exponential cohesive models are similar, but the exponential cohesive model with a smooth transition at the peak point is most attractive. Alfano et al. [26] pointed out convergence for the trapezoidal cohesive model is the worst because of severely discontinuous transitions at the peak point, leading to more iterations although it is often superimposed to predict ductile failure of adhesive joints well [27]. By comparison, convergence using the exponential cohesive model is best, which shows a large advantage over the trapezoidal cohesive model especially for the coarse model, and the bilinear cohesive model shows a compromise between numerical convergence and accuracy. In the following, the exponential and bilinear cohesive models are adopted for numerical analysis.

2.6.1 Numerical results of single adhesive lap joint under tension In this section, 3D FEA on the failure of the adhesively bonded single lap joint (SLJ) is performed. The geometry model and material parameters are taken from Panigrahi [28], as listed in Fig. 2.5 and Table 2.1, respectively. η ¼ 2.0 in Eq. (2.20) is taken from Liu et al. [14]. Campilho et al. [29], Krueger and O’Brien [30], Dattaguru et al., [31], and Pradhan and Panda [32] pointed out sufficiently fine mesh is strongly required to guarantee numerical convergence and accuracy. Fig. 2.6 shows two mesh models. Fig. 2.7 shows the load-displacement curves using two mesh models at the cohesive strength τmax ¼ 40 MPa and Fig. 2.8 shows the

36

2. Failure behaviors of adhesive composite joints

FIG. 2.5 Adhesively bonded single lap joint (SLJ).

TABLE 2.1 Material parameters for single-lap joint [28]. Ply longitudinal modulus

E1

181 GPa

Ply transverse modulus

E2

10.3 GPa

Out-of-plane modulus

E3

10.3 GPa

Inplane shear modulus

G12

7.17 GPa

Out-of-plane shear modulus

G13

7.17 GPa

G23

4 GPa

v12

0.28

v13

0.28

v23

0.3

Mode-II delamination fracture toughness

Gc1

0.66 N/mm

Mode-III delamination fracture toughness

Gc2

0.66 N/mm

Mode-I delamination fracture toughness

Gc3

0.23 N/mm

Poisson’s ratio

FIG. 2.6 Two mesh models for the single lap joint (SLJ).

2.6 Numerical results and discussion

37

FIG. 2.7 Load-displacement curves for the SLJ using two cohesive models at τmax ¼ 40 MPa and two mesh models.

FIG. 2.8 Load-displacement curves for the SLJ using two cohesive models at different cohesive strengths and coarse mesh model.

38

2. Failure behaviors of adhesive composite joints

load-displacement curves using the coarse mesh model and two cohesive models at the cohesive strength τmax ¼ 20, 40, and 60 MPa, respectively. For the bilinear cohesive model, Camanho et al. [6] suggested an accurate value of the cohesive stiffness for carbon/epoxy composites is K ¼ 106 N/mm3 for the bilinear cohesive model. From numerical analysis, numerical convergence and computational efficiency are better by using K ¼ 105 N/mm3 than those by using K ¼ 106 N/mm3 although they lead to almost the same load responses, which is consistent with the conclusion obtained by Turon et al. [22]. From Figs. 2.7 and 2.8, the adhesive failure occurs suddenly without distinct damage evolution, similar to the conclusions obtained by Campilho et al. [29] and Hu and Soutis [33]. Under the same cohesive strength and fracture toughness, the exponential cohesive model leads to larger debond load than that by the bilinear cohesive model, which was explained by longer fracture process zone for the exponential cohesive model than that for the bilinear cohesive model by Campilho et al. [29]. Convergence using the exponential cohesive model is better than the bilinear cohesive model because a sharp transition at the peak point for the bilinear cohesive model requires more iterations to reach equilibrium. In addition, Fig. 2.7 also shows the effect of mesh sizes on convergence is small which require almost same increments and iterations. This shows the coarse mesh model is fine enough to guarantee convergence and accuracy. From Fig. 2.8, the debond load increases, but the convergence difficulty adds when the cohesive strength τmax increases, which was also found by Alfano and Crisfield [5], Turon et al., [22] and Liu and Islam [14]. From numerical analysis, adhesive failure will not occur at τmax > 200 MPa for the exponential cohesive model and at τmax > 150 MPa for the bilinear cohesive model. Turon et al. [22] pointed out low cohesive strength represents the softening properties of the fracture process zone more accurately although the stress distribution may be altered. In terms of convergence, τmax > 80 MPa is not generally suggested according to the present analysis. Fig. 2.9 shows the effect of different viscous constants c on the load responses using two cohesive models at τmax ¼ 40 MPa. Computational CPU time at different c using the exponential cohesive model is almost the same, but decreases by 20% at c ¼ 0.01 compared with that at c ¼ 0.0001 using the bilinear cohesive model. In addition, c has to increase to get convergent solutions when the cohesive strength τmax is larger than 60 MPa for the bilinear cohesive model. Fig. 2.10 shows the predicted debond loads at the crack initiation for the SLJ case using two mesh models and two cohesive models at τmax ¼ 20  80 MPa. Kim et al. [34] and Liljedahl et al. [35] studied the failure of adhesive composite joints with different adhesive materials by experimental and numerical approaches, in which the debond loads show difference between ductile

2.6 Numerical results and discussion

FIG. 2.9

39

Viscous effect on the load responses for the SLJ using coarse mesh model.

FIG. 2.10 Predicted debond loads for the SLJ using two cohesive models and coarse mesh model.

40

2. Failure behaviors of adhesive composite joints

FIG. 2.11 Crack growth processes for the SLJ at the tensile displacements (A) 0.584 mm (initial crack) and (B) 0.621 mm (complete damage), respectively.

and brittle adhesive failure. From Fig. 2.10, the debond load increases largely when the cohesive strengths increase from 20 to 80 MPa, but increase few when the cohesive strength continues to add. Thus, τmax ¼ 40  80 MPa are good choices for achieving excellent convergence and accuracy for some kind of adhesive material. Fig. 2.11 shows the adhesive failure results, in which there is a short time interval from the crack initiation at 0.584 mm displacement to the complete damage at 0.621 mm displacement.

2.6.2 Numerical results of skin/stiffener adhesive composite panel under tension Fig. 2.12 shows a skin/stiffener composite panel in airplane structures. Material parameters and geometry sizes are taken from Krueger and O’Brien [30] and Turon et al. [7], as listed in Table 2.2. The layup patterns

FIG. 2.12

Skin/stiffener composite panel under tensile load.

41

2.6 Numerical results and discussion

TABLE 2.2 Material parameters for IM6/3501-6 composites [7, 30]. Ply longitudinal modulus

E1

144.7 GPa

Ply transverse modulus

E2

9.65 GPa

Out-of-plane modulus

E3

9.65 GPa

Inplane shear modulus

G12

5.2 GPa

Out-of-plane shear modulus

G13

5.2 GPa

G23

3.14 GPa

v12

0.3

v13

0.3

v23

0.45

Mode-II delamination fracture toughness

Gc1

0.547 N/mm

Mode-III delamination fracture toughness

Gc2

0.547 N/mm

Mode-I delamination fracture toughness

Gc3

0.075 N/mm

Poisson’s ratio

for the skin and stiffener are [0°/45°/90°/45°/+45°/45°/0°]s and [45°/ 90°/45°/0°/90°]s, respectively. η ¼ 1.45 in Eq. (2.20) is taken from Turon et al. [7]. Fig. 2.13 shows two mesh models. Fig. 2.14 shows the loaddisplacement curves using two mesh models and two cohesive models at τmax ¼ 40 MPa and Figs. 2.15 and 2.16 further compare the effect of cohesive strengths using two cohesive models. Fig. 2.17 shows the effect of different viscous constants c. Fig. 2.18 compares the debond loads using cohesive models with the experimental results. From Fig. 2.14, the load curves are close using two mesh models for the exponential cohesive model, but are slightly different from the results using two mesh models for the bilinear cohesive model. For the exponential cohesive model, convergence is the same using two mesh models. However, the difference of convergence becomes large for two mesh

FIG. 2.13

Two mesh models for the skin/stiffener composite panel.

42

2. Failure behaviors of adhesive composite joints

FIG. 2.14

Load-extensometer displacement curves of the skin/stiffener composite panel using two cohesive models at τmax ¼ 40 MPa and two mesh models.

FIG. 2.15

Load-extensometer displacement curves of the skin/stiffener composite panel using the exponential cohesive model at different cohesive strengths τmax and two mesh models.

2.6 Numerical results and discussion

43

FIG. 2.16

Load-extensometer displacement curves of the skin/stiffener composite panel using the bilinear cohesive models at different cohesive strengths τmax and two mesh models.

FIG. 2.17

Viscous effect on the load responses for the skin/stiffener composite panel.

44

FIG. 2.18

2. Failure behaviors of adhesive composite joints

Predicted debond loads for the skin/stiffener composite panel using two cohe-

sive models.

models using the bilinear cohesive model when the cohesive strength τmax increases. For example, there is a difference of 20% for the computational CPU time at τmax < 40 MPa, but a difference of about 40% at τmax ¼ 40  80 MPa. The effect of mesh sizes on convergence is more distinct for the bilinear cohesive model than for the exponential cohesive model, and the computational CPU time using the fine mesh model is twice that using the coarse mesh model. From Figs. 2.15 and 2.16, large cohesive strength τmax shows a stronger interfacial bonding ability, which requires more energy to drive the crack growth. However, convergence difficulty also adds when cohesive strengths increase, which is reflected by more increments and iterations. For the exponential cohesive model, the effect of cohesive strength τmax on convergence is small at τmax < 40 MPa, but becomes distinct at τmax > 40 MPa. For the bilinear cohesive model, convergence becomes poor when the cohesive strength τmax increases at τmax < 40 MPa, but becomes much more difficult at τmax > 40 MPa. At τmax ¼ 60 MPa, the computational CPU time by the bilinear cohesive model is four times that by the exponential cohesive model. In addition, it is noted too large or small cohesive strength τmax cannot be acceptable. Large strengths lead to the convergence difficulty while small strengths lead to inaccurate results, e.g., at τmax ¼ 20 MPa in Figs. 2.14 and 2.15. Fig. 2.17 shows the effect of different viscous constants c on the load-displacement curves using two cohesive

2.6 Numerical results and discussion

45

models. Load curves keep consistent when c changes from 0.0001 to 0.01. Because convergence for this case is hard especially at the stage of crack initiation which requires many iterations and time cuts in the nonlinear FEA, the effect of viscous regularization for improving convergence is distinct. In this case, c ¼ 0.0005 can meet the requirement of convergence and accuracy. For this case using the bilinear cohesive model, Turon et al. [7] adopted the normal strength σ max ¼ 61 MPa and the tangential strengths τmax ¼ 68 MPa directly. However, they failed to study numerical convergence and computational efficiency. Here, we further compare the effect of cohesive strengths τmax on the predicted debond loads, as shown in Figs. 2.18. Fig. 2.19 shows the crack growth process from the crack initiation at 0.128 mm displacement to complete damage at 0.184 mm displacement, different from the SLJ case without distinct damage evolution. It is shown τmax ¼ 68 MPa lead to accurate debond loads by comparing the experimental results using the bilinear cohesive model indeed. However, the effect of shape of cohesive models on the load responses is not neglectable. Chandra et al. [36] performed parameter sensitivity analysis using different cohesive models by push-out test and pointed out the results using the bilinear cohesive model are closer to the experimental results than the nonlinear cohesive model. However, they neglected the effect of cohesive strengths on the load curves. Later, Volokh [25] pointed out the shape of different cohesive models will affect the fracture process by rigid block-peel test at the same fracture toughness and cohesive strength. Alfano [26] studied the modes-I and II fracture mechanisms using four cohesive models (bilinear, linear-parabolic, exponential, and trapezoidal models), which were found to depend on the ratio of the stiffness of the lamina material to the adhesive layer material. For the ductile interface fracture here, Campilho et al. [29] showed the length of the process zone for the exponential cohesive model is longer than that for the

FIG. 2.19

Crack growth processes for the skin/stiffener composite panel at the displacements (A) 0.128 mm (initial crack), (B) 0.140 mm, and (C) 0.184 mm (complete damage) at τmax ¼ 40 MPa.

46

2. Failure behaviors of adhesive composite joints

bilinear cohesive model. Thus, the predicted debond load using the exponential cohesive model is larger than that using the bilinear cohesive model at the same cohesive strength, which is also consistent with that in the SLJ case. In this case, the cohesive strengths τmax ¼ 60  80 MPa for the bilinear cohesive model and τmax ¼ 40  60 MPa for the exponential cohesive model are good choices for achieving an excellent combination of convergence and accuracy.

2.7 Concluding remarks Cohesive models have been widely used for delamination/adhesive failure analysis of composite structures. Unfortunately, there is still no unified cohesive model that can represent the true fracture process zone under different load conditions. Furthermore, numerical convergence using cohesive models emerges in the implicit nonlinear FEA as a challenging issue in simulating the crack initiation and growth of composite adhesive joints. This chapter first develops 3D finite element codes to implement the bilinear and exponential cohesive models for failure analysis of composite adhesive joints using ABAQUS-UEL. Then, this chapter aims to study the influence of the cohesive shape and cohesive strength on the numerical convergence of the single lap joint and the skin/stiffener composite panel under tension. From 3D FEA, four main conclusions are obtained: (1) In general, the bilinear cohesive model shows poorer convergence than the exponential cohesive model at the same cohesive strength, fracture toughness and mesh size. (2) Fine mesh size is beneficial to improving convergence, especially by the bilinear cohesive model. (3) In general, relatively low cohesive strengths, e.g., τmax ¼ 40  60 MPa for the exponential cohesive model and τmax ¼ 60  80 MPa for the bilinear cohesive model for the SLJ and skin/stiffener panel cases are good choices to achieve an excellent combination of convergence and accuracy while higher cohesive strengths will add the numerical difficulty remarkably, especially for the bilinear cohesive model. (4) Viscous regularization is shown to be an excellent method to improve convergence. For the SLJ and skin/stiffener panel cases, convergent solutions cannot be obtained without viscous effects. From numerical analysis, c ¼ 0.0001–0.01 for the SLJ case and c ¼ 0.0005–0.01 for the skin/stiffener panel case are reasonable ranges.

References [1] Stapleton SE, Pineda EJ, Gries T, Waas AM. Adaptive shape functions and internal mesh adaptation for modeling progressive failure in adhesively bonded joints. Int J Solids Struct 2014;51(18):3252–64.

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C H A P T E R

3

Implicit finite element analysis of progressive failure and strain localization of notched carbon fiber/epoxy composite laminates under tension 3.1 Introduction Carbon fiber-reinforced polymer matrix composites have been increasingly used in aircraft, wind turbine blades, and pressure vessels for energy storage, in which laminates are important structural types that aim to achieve high stiffness and strength in different directions from the design perspective [1]. Yet, the difficulty for the design of composite laminates lies in their complicated failure mechanisms and damage evolution behaviors, especially the interaction between the intralaminar damage and interlaminar delamination, upon which deep insight into the structural strength and damage tolerance of composites are based [2]. Currently, a large amount of theoretical and numerical research on progressive failure analysis of composite laminates has been performed. For intralaminar damage of composites, the phenomenological research at the macroscopic scale includes the failure criteria, damage constitutive modeling, damage evolution modeling, and finite element implementation [3–5]. For the delamination of composites, the cohesive model and virtual crack closure technique (VCCT) are often adopted [6–14]. Especially, numerical analysis on the intralaminar damage and interlaminar delamination are performed simultaneously on the composite panels [15–17].

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00006-7

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Copyright © 2021 Elsevier Inc. All rights reserved.

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3. Implicit finite element analysis of progressive failure

From the fracture mechanics perspective, Ling et al. [18] developed an augmented finite element model based on the extended finite element (XFEM) to perform high-fidelity analysis at various length scales. Iarve et al. [19] used the XFEM to study the interactions between matrix cracking and delamination. van der Meer et al. [20, 21] developed a phantom node formulation with mixed mode cohesive law in the framework of XFEM to study various failure mechanisms of laminates. It should be widely recognized that progressive failure mechanisms of composite laminates are multiscale, which relates the macroscopic homogenized damage to the microscopic discrete fracture. An important issue in the macroscopic progressive failure analysis for predicting the strength of structures lies in the strain localization, which widely occurs in composite structures, especially for those with geometry discontinuities [22, 23]. The strain localization means intense deformation, the energy dissipation and the subsequent damage appear at the narrow regions with high-gradient strain fields. In fact, composite structures will ultimately experience the localized damage stage and even the softening stage since they start failing initially. The multiscale nature of the progressive failure of composite structures creates an inherent length scale into the governing equations because the sizes of the localization area are much smaller than those of macroscopic structure. Yet, the classic continuum mechanics fails to provide an objective description of the strain localization due to the absence of an internal length scale. This leads to the loss of ellipticity of the governing equations for quasi-static problems and the spurious mesh dependency because the volume and energy dissipation are wrongly calculated to zero within the localized area. Thus, numerical results using implicit finite element analysis (FEA) on the interactions between the intralaminar and interlaminar damage become meaningless. It is significant to introduce nonlocal theories to predict objective progressive failure properties of composites. Yet, there is still little research that focuses on this area using nonlocal FEA. In this chapter, three-dimensional (3D) nonlocal FEA on intralaminar damage and interlaminar delamination of carbon fiber/epoxy composite laminates with a central hole is performed. The strain-based failure criteria and damage evolution models are presented, and the nonlinear cohesive model is adopted for delamination analysis. The well-known nonlocal integral theory introduces a length scale to the stress governing equation for solving the strain localization issue by assuming long-range interactions among different material points. Numerical algorithms are implemented by successfully combining ABAQUS-UEL (user element subroutine), UMAT (user material subroutine), USDFLD (user defined field subroutine), and UEXTERNALDB (user external data utility). The strain localization issue of composites is shown to be well-regularized

3.2 Damage models of composite laminates

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by eliminating mesh sensitivity. Then, the effects of the hole radius and layup patterns on the load responses of composites are also studied.

3.2 Damage models of composite laminates The damaged stress-strain constitutive relationship is written as σ ¼ Cd : ε

(3.1)

where Cd(1  di(i ¼ 1, 2, 3), C) is the damaged four-order elastic tensor. C is the undamaged four-order elastic tensor. di(i ¼ 1, 2, 3) represent the microscopic fiber breakage, matrix cracking, and fiber/matrix interface debonding, respectively. A typical damage configuration is given by Lapczyk and Hurtado [24]. 2 3 ð1  d1 ÞE1 ð1  d1 Þð1  d2 Þv21 E1 0 14 5 d3 ð1  d1 Þð1  d2 Þv12 E2 ð1  d2 ÞE2 0 Cd ¼ D 0 0 Dð1  d3 ÞG12 ¼ 1  ð1  d1 Þð1  d2 Þ, D ¼ 1  ð1  d1 Þð1  d2 Þv12 v21 , (3.2) where E1 and E2 are the longitudinal and transverse Young’s moduli, and v12 and v21 are Poisson’s ratio, and G12 is the shear modulus. The damage variables for fiber breakage and matrix cracking are given by ! 8 f eq > ε1 ε1  εc1 > > , fiber breakage > < d1 ¼ 1  exp λ1 εeq f c 1 ε1  ε1 ! (3.3) f eq c > ε ε  ε > 2 2 2 > > , matrix cracking : d2 ¼ 1  exp λ2 εeq f c 2 ε2  ε2 eq where λ1 and λ2 are materials parameters. The equivalent strains εeq 1 and ε2 are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8  c 2 > > eq > ε ¼ ðε11 Þ2 + ε1 ðε12 Þ2 > εc , fiber breakage > < 1 1 εc12 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3.4)  c 2 > > ε eq > 2 2 2 c > ε2 ¼ ðε22 Þ + ðε12 Þ > ε2 , matrix cracking : εc12

where εc1 ¼ XT/C11 is the initial failure strain for fiber breakage and εc2 ¼ YT/C22 is that for matrix cracking, and εc12 ¼ SLT/C44 is that for fiber/matrix interface failure. εf1 and εf2are the critical fiber and matrix failure strains, respectively. XT and XC are the longitudinal tensile and compressive strengths, YT and YC are the transverse tensile and compressive

52

3. Implicit finite element analysis of progressive failure

strengths, and SLT is the longitudinal shear strength. Eq. (3.4) is applicable to both tension and compression. The consistent tangent modulus D for the intralaminar damage is derived as ∂σ ∂Cd D ¼ ¼ Cd + : ε, a ∂ε ∂ε eq 8 eq ∂C ∂C ∂d1 ∂ε ∂C ∂d2 ∂ε > > > d ¼ d eq 1 + d eq 2 , > > ∂ε ∂d1 ∂ε1 ∂ε ∂d2 ∂ε2 ∂ε > > ! > f f eq > c < ∂d1 ε1 ε1 ε1  εc1 ε1 ,  eq 2 exp λ1 eq f eq ¼ λ1 f ∂ε1 ε1 ε1  εc1 > ε1  εc1 ε1 > ! > > f f eq > > ε2 ε2 ε2  εc2 ∂d2 εc2 > > ¼ λ exp λ , > 2 2   eq eq : ∂ε f eq 2 ε2 εf2  εc2 ε2  εc2 ε2 2

8 eq eq > ∂ε ∂ε2 ε11 ε22 > > 1 ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi, , ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > !2 ! u > ∂ε11 u ∂ε 2 > c c 2 u u > ε ε > tðε Þ2 + t ðε Þ2 + 1 2 > > ðε12 Þ2 ðε12 Þ2 11 22 > c c > ε ε > 12 12 > > > !2 !2 < eq eq ∂ε1 ∂ε2 εc1 εc2 ε12 ε12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi c ffi c , ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , > ∂ε ¼ v ! ! > ε12 ∂ε12 u ε12 12 u > c 2 c 2 u u > ε ε > tðε Þ2 + tðε Þ2 + 1 2 > > ðε12 Þ2 ðε12 Þ2 11 22 > > εc12 εc12 > > > > eq eq eq eq eq eq eq > > ∂εeq ∂ε1 ∂ε1 ∂ε1 ∂ε2 ∂ε2 ∂ε2 ∂ε2 > > : 1 ¼ 0, ¼ 0, ¼ 0, ¼ 0, ¼ 0, ¼ 0, ¼ 0, ¼0 ∂ε22 ∂ε33 ∂ε13 ∂ε23 ∂ε11 ∂ε33 ∂ε23 ∂ε13 (3.5)

where the second term in Eq. (3.5) includes two parts for fiber and matrix. The exponential cohesive model [11] and zero-thickness interface elements [25] are used for delamination analysis.

3.3 Nonlocal integral model Strain localization means a high strain gradient appears within the narrow area. Across this localization area, there exists a discontinuity in the displacement gradient (or velocity gradient) field, which is often referred to as weak discontinuity. The strain localization for the quasistatic problem is represented by the singularity of the acoustic tensor. eq To solve the localization problem, the nonlocal equivalent strains ε1 eq and ε2 at the material point X are introduced as the weighted averaging of all material points around this point within some distance for fiber and matrix [26].

3.4 Numerical results and discussion

8 ð 1 eq eq > > α0 ðX, Y Þε1 ðY Þ dY, fiber < ε1 ð X Þ ¼ α ðX Þ ð Ω eq > > εeq ðX Þ ¼ 1 α0 ðX, Y Þε2 ðY Þ dY, matrix : 2 α ðX Þ Ω

53

(3.6)

where Y is the neighboring material point around the point X. α0(X, Y) is the nonlocal weight function, which is given by the Gaussian-type function " # 1 kX  Y k2 (3.7) α0 ðX, Y Þ ¼ exp  2l2 ð2π Þ3=2 l3 where kkdenotes the norm. l is the length scale and different values for l eq can be chosen for fiber and matrix. The nonlocal equivalent strains ε1 and eq eq eq ε2 are reduced to the local equivalent strains ε1 and ε2 at l ! 0, respectively. The normalization factor α(X)in Eq. (3.6) is given by ð (3.8) αðX Þ ¼ α0 ðX, Y Þ dY Ω

3.4 Numerical results and discussion Fig. 3.1 shows the finite element algorithm for implementing the nonlocal progressive failure models by combining ABAQUS-UEL (user element subroutine), UMAT (user material subroutine), USDFLD (user defined field subroutine), and UEXTERNALDB (user external data utility subroutine). The nonlocal damage model is implemented by combining ABAQUS-UMAT, USDFLD, and UEXTERNALDB; the cohesive model is implemented by ABAQUS-UEL. Similar to the isoparametric element, the displacement jump of the Gaussian integration point in the global coordinate system (x, y, z) is transformed into that in the local coordinate system (ξ, η, ζ). A cohesive length is introduced to regularize the cohesive softening. Numerical codes by ABAQUS are developed to predict 3D nonlocal progressive failure properties of composites. Material parameters are defined by the array “PROPS” for preprocessing and the status variables about the damage variables and stress and strain levels are defined by the array “STATEV,” and “SVARS” for postprocessing in ABAQUS-UMAT and UEL. Once the failure of fiber or matrix is identified, the damage variables d1 or d2 are updated and the stiffness degradations are performed. At this time, the nonlocal averaging on the equivalent strains for fiber and matrix elements are performed using ABAQUS-UEXTERNALDB utility

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3. Implicit finite element analysis of progressive failure

FIG. 3.1

Flow chart of the nonlocal intralaminar damage and interlaminar delamination of composite laminates using implicit FEA.

subroutine. This is implemented by placing the local strain and the local strain increment in a defined COMMON BLOCK data module for both ABAQUS-UMAT and ABAQUS-UEXTERNALDB at the end of each increment by setting LOP ¼ 2. Besides, nonlocal averaging in ABAQUSUEXTERNALDB also requires the coordinates and volume for each Gauss integration point, which are initially obtained by the ABAQUS-USDFLD subroutine, in which the command “CALL GETVRM” acquires the coordinates and volume of the integration points, and places them into the field variables that are also stored in the COMMON BLOCK data module. After performing nonlocal averaging, the nonlocal strain increment in ABAQUS-UEXTERNALDB is returned to ABAQUS-UMAT by COMMON BLOCK. For each load increment, the stiffnesses of the bulk elements and cohesive elements are assembled and the unbalanced forces

3.4 Numerical results and discussion

FIG. 3.2

55

Two mesh models of composite laminates with a central hole.

are calculated unitedly. Implicit FEA includes two steps: (1) the NewtonRaphson method is used before the global collapse of structures, and (2) the viscous effect is activated during the collapse stage to solve the “snap” convergence issue. Fig. 3.2 shows two mesh models of T700/8911 composite laminates with a central hole including 1080 and 540 bulk elements, and 540 and 270 cohesive interface elements, respectively. Only one-quarter of the geometry model is used for the finite element model. The mesh models are established by using the ABAQUS command language in the inp main file. Two layup patterns [0°/90°]4s and [0°/45°]4s are used. Material parameters listed in Table 3.1 are obtained from experiments by Liu et al. [27], and the tensile test is shown in Fig. 3.3. The damage parameters λ1 ¼ 0.08 and λ2 ¼ 0.8 in Eq. (3.3) are chosen. Accurate calibration of length scale requires special experimental approaches. From experimental analysis, Geers et al. [28] evaluated the length scale by the available strain fields. Dontsov et al. [29] evaluated the length scale by measuring the dispersion of elastic waves. From Eq. (3.7), the nonlocal radius R, which is related to the length scale regularizes the localization problem. Here, we compare the length scale, which affects the localization area to the hole radius. By comparison, two parameters l ¼ 1.0 and 2.0 mm in Eq. (3.7) are used. Because the nonlocal stress and strain solutions require data exchange among different Gauss integration points, parallel calculations are necessary to improve computational efficiency. Figs. 3.4–3.14 shows the damage evolution processes for fiber and matrix for the [0°/90°]4s and [0°/45°]4s laminates with a 10 mm-diameter hole. In the numerical subroutine, the status variables SDV7 and SDV8 represent the damage variables d1 and d2 in Eq. (3.3), respectively. The initial failure of fiber and matrix as well as delamination appears near the hole due to the stress concentration. For the [0°/90°]4s laminate, matrix cracking first appears in the 90° lamina at the strain 0.23% (the tensile

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3. Implicit finite element analysis of progressive failure

TABLE 3.1 Material parameters for T700/8911 composites [27]. Ply longitudinal modulus

E1

135 GPa

Ply transverse modulus

E2

11.41 GPa

Out-of-plane modulus

E3

11.41 GPa

Inplane shear modulus

G12

7.92 GPa

Out-of-plane shear modulus

G13

3.792 GPa

G23

7.92 GPa

v12

0.33

v13

0.33

v23

0.49

Longitudinal tensile strength

XT

2600 MPa

Longitudinal compressive strength

XC

1422 MPa

Transverse tensile strength

YT

603 MPa

Transverse compressive strength

YC

241 MPa

Longitudinal shear strength

SLT

94 MPa

Interlaminar mode-I fracture toughness

GIC

0.252 N/mm

Interlaminar mode-II fracture toughness

GIIC

0.665 N/mm

Interlaminar mode-III fracture toughness

GIIIC

0.665 N/mm

Initial fiber breakage strain

εc1 ¼ XT/C11

0.019

Initial matrix cracking strain

εc2 ¼ YT/C22

0.005

Initial interface shear strain

εc12 ¼ SLT/C44

0.012

Critical fiber breakage strain

εf1

0.021

Critical matrix cracking strain

εf2

0.035

Poisson’s ratio

stress level is 117 MPa), and fiber and matrix start to damage in the 0° lamina at the strain 0.52% (the tensile stress level is 268 MPa). Yet, fiber starts to damage in the 90° lamina at the strain 0.56% (the tensile stress level is 290 MPa). It is shown the fiber strength governs the load-bearing ability of the 0 º lamina while the load-bearing ability of the 90° lamina is determined by matrix. When the strain reaches 0.65% (the tensile stress level is 334 MPa), the delamination starts to appear, which is marked by the distinct debonding of paired nodes. The delamination phenomenon is largely attributed to the mismatch of the load-bearing abilities between the 0° lamina and 90° lamina, which leads to high interlaminar shear stresses,

3.4 Numerical results and discussion

57

FIG. 3.3

Tensile test for T700/8911 composite laminate with a central hole.

FIG. 3.4 Damage evolution processes for fiber in 0° ply for the [0°/90°]4s laminate with a 10 mm-diameter hole at the strains (A) 0.672%, (B) 1.001%, and (C) 1.622%, respectively.

FIG. 3.5 Damage evolution processes for matrix in 0° ply for the [0°/90°]4s laminate with a 10 mm-diameter hole at the strains (A) 0.672%, (B) 1.001%, and (C) 1.622%, respectively.

FIG. 3.6 Damage evolution processes for fiber in 90° ply for the [0°/90°]4s laminate with a 10 mm-diameter hole at the strains (A) 0.672%, (B) 1.001%, and (C) 1.622%, respectively.

59

3.4 Numerical results and discussion

FIG. 3.7 Damage evolution processes for matrix in 90° ply for the [0°/90°]4s laminate with a 10 mm-diameter hole at the strains (A) 0.672%, (B) 1.001%, and (C) 1.622%, respectively.

800

Initial fiber damage

Stress (MPa)

600

400

Initial matrix cracking

200

0 0.0

Test result [29] Numerical result by fine mesh Numerical result by coarse mesh 0.5

1.0

Strain (%)

1.5

FIG. 3.8 Stress-strain curves for two mesh models for the [0°/90°]4s laminate using nonlocal model at l ¼ 1 mm.

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3. Implicit finite element analysis of progressive failure

800

Initial fiber damage

Stress (MPa)

600

400

Initial matrix cracking

200

Test result [29] Numerical results by l =1mm Numerical results by l =2mm

0 0.0

0.5

1.0

Strain (%)

1.5

FIG. 3.9 Stress-strain curves for fine models for the [0°/90°]4s laminate using nonlocal model at l ¼ 1 mm and 2 mm.

Initial fiber damage

Stress (MPa)

600

400

Initial matrix cracking

200

0 0.0

Test result [29] Local result by fine mesh Local result by coarse mesh 0.5

1.0

1.5

Strain (%) FIG. 3.10 Stress-strain curves for two mesh models for the [0°/90°]4s laminate using local model at l ¼ 0 mm.

and the delamination further evolves with the progressive failure of fiber and matrix, which results in the strain-softening at the location near the hole. From Figs. 3.8–3.12, the strain localization problem due to the progressive failure of composites is well-regularized by achieving relatively consistent load-displacement curves for two mesh models when the

3.4 Numerical results and discussion

61

FIG. 3.11 Damage evolution processes for fiber in 0° ply for the [0°/45°]4s laminate with a 10 mm-diameter hole at the strains (A) 0.529%, (B) 0.643%, and (C) 1.622%, respectively.

FIG. 3.12 Damage evolution processes for matrix in 0° ply for the [0°/45°]4s laminate with a 10 mm-diameter hole at the strains (a) 0.529%, (b) 0.643% and (c) 1.622%, respectively.

FIG. 3.13 Damage evolution processes for fiber in 45° ply for the [0°/45°]4s laminate with a 10 mm-diameter hole at the strains (A) 0.529%, (B) 0.643%, and (C) 1.622%, respectively.

FIG. 3.14 Damage evolution processes for matrix in 45° ply for the [0°/45°]4s laminate with a 10 mm- diameter hole at the strains (A) 0.529%, (B) 0.643%, and (C) 1.622%, respectively.

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3.4 Numerical results and discussion

length scale parameter is properly chosen. This is actually achieved only when energy dissipation due to the damage of fiber, matrix, and interface occurs at a larger area than the localized hole area. By comparison, the decreasing curves for the load-displacement responses using local models show a large difference using two mesh models. If only intralaminar damage or interlaminar delamination is considered, the stress levels are slightly smaller than those by considering the intralaminar damage and interlaminar delamination simultaneously. This shows the complicated interactions between them weaken the load-bearing ability of composites. On one hand, matrix cracking and intralaminar stiffness degradation can trigger the delamination, and the delamination also aggravates the intralaminar damage, especially for the [0°/90°]4s with larger interlaminar shear stress. On the other hand, the energy required by the delamination is little because more energy dissipation appears for the damage of fiber and matrix. Thus, the delamination is not very distinct for the laminate with a central hole, which makes no large difference in the load-displacement curves when the cohesive strength changes from 50 MPa to 100 MPa. Figs. 3.15–3.17 show the load-displacement curves by comparing the effects of the hole diameter and layup patterns. It is shown the load-bearing ability for the [0°/90°]4s laminate is slightly higher than that for the [0°/45°]4s laminate. Fig. 3.18 shows the delamination results.

800

Stress (MPa)

600

400

Initial matrix cracking

Initial fiber damage

Numerical results for the composite laminates with 5mm hole diameter 10mm hole diameter

200

0 0.0

0.5

1.0

Strain (%)

1.5

FIG. 3.15 Stress-strain curves of the [0°/90°]4s laminates with different hole sizes using nonlocal model.

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3. Implicit finite element analysis of progressive failure

800

Stress (MPa)

600

400

Initial matrix cracking

Initial fiber damage

Numerical results for the composite laminates with 5mm hole diameter 10mm hole diameter

200

0 0.0

0.5

1.0

1.5

FIG. 3.16 Stress-strain curves of the [0°/45°]4s laminates with different hole sizes using nonlocal model.

Stress (MPa)

800

600

400

200

0 0.0

FIG. 3.17

Initial fiber damage Initial matrix cracking

Numerical results for [0°/45°]4s composite laminate [ 0°/90°]4s composite laminate 0.5

1.0

Strain (%)

1.5

Stress-strain curves of different laminates with two layups using nonlocal

model.

3.5 Concluding remarks As a real physical phenomenon. Strain localization occurs widely in composite structures. It is always a challenging research topic when the localization problem is involved. For the quasi-brittle carbon fiber/epoxy composites, the localization problems have not been solved well, which prohibits accurate prediction of progressive failure properties and

References

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FIG. 3.18 Deformation results near the hole by considering the (A) only intralaminar damage, (B) only interlaminar delamination, and (C) intralaminar damage and interlaminar delamination.

strengths of composites. This chapter performs 3D implicit FEA on nonlocal progressive failure of composite laminates by combining the intralaminar damage model, the cohesive model, and the nonlocal integral theory. The ABAQUS-UEL, UMAT, USDFLD, and UEXTERNALDB subroutines in ABAQUS are commonly used to implement the nonlocal models. An important aspect lies in the development of the finite element algorithm and numerical codes using ABAQUS. For composite laminates with a central hole, the interaction mechanisms between intralaminar damage and interlaminar delamination, and the progressive failure process and load-displacement curves of composites are well-predicted by discussing the effects of the hole radius and layup angle. It is shown the introduced length scale for multiscale progressive failure issues of composites, which essentially origins from the fiber/matrix microscopic structures of composites, plays an important role in regularizing the strain localization problem of composite structures. The developed finite element algorithm and numerical codes can be further applied to a generic composite structure including adhesive composite joints, which promotes insight into the complicated failure mechanisms and damage evolution behaviors of structures.

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C H A P T E R

4

Localized damage models and implicit finite element analysis of notched carbon fiber/epoxy composite laminates under tension 4.1 Introduction Beyond the initial failure of composites, after the stress levels exceed the material strengths, mechanical properties of composites become highly nonlinear, represented by progressive failure and continuous stiffness degradation. It is important to understand how the damage evolves after the initial failure as most of the energy dissipation takes place there. In general, failure modes of composites include fiber pullout and breakage or kinking, matrix cracking, fiber/matrix shear failure, and delamination. Degradation mechanisms and damage evolution behaviors depend on the ply’s orientation, the stacking sequence, the geometry size, as well as the load and environment. Currently, progressive failure analysis (PFA) based on continuum damage mechanics using finite element analysis (FEA) has grown up to be a popular method to predict stiffness degradation behaviors and load-bearing abilities of composite structures under various loads [1–13]. Associative intralaminar damage and interlaminar delamination of notched composite laminates by FEA are studied, where an important mechanism lies in the interaction between matrix cracking and delamination [14–19]. However, progressive failure of composites inevitably meets

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00011-0

69

Copyright © 2021 Elsevier Inc. All rights reserved.

70

4. Localized damage models and implicit finite element analysis

the strain localization problem for geometrically discontinuous composite structures with holes, notches, and cut-outs [9, 10]. Localization means smoothly varying deformation field suddenly gives rise to one with a highly localized field where high-gradient strain field, severe damage, and energy dissipation concentrate on the local region, leading to ill-posed boundary value problems with the loss of ellipticity of the governing equation for quasi-static problems. No matter by experimental or numerical approaches, more efforts are needed to address localized deformation and damage evolution behaviors of composites with instabilities. Geers et al. [20] revealed the process zone where damage is accumulated by monitoring the strain localization behaviors of composites. Ibnabdeljalil and Curtin [21] and Martı´nez-Esnaola et al. [22] studied the localized stress concentration and stress redistribution around the fiber breakage of composites by micromechanical damage modeling. Kelly and Hallstr€ om [23] found that dominating localized matrix cracking under transverse load induces delamination by experimental and numerical methods. Hallett et al. [24] divided the damage evolution process of [45°/90°/
45°/0°]4s composite laminates with a central hole under tension into four stages by some experimental approaches (such as X-ray and C-scanning): (1) isolated matrix cracking at the hole and free edge and (2) interconnected damage at the hole (inner delamination regions) and localized damage at the free edge from matrix cracking and (3) damage across the width of the specimen in an “influence zone” of the hole and (4) final catastrophic failure. The second stage shows that the strain localization with softening behavior starts to appear after initial matrix cracking. Mechraoui et al. [25] showed by acoustic emission test that the localization signals come from different failure mechanisms of composites and the extent of localization is represented by a real wave velocity. Bazˇant and Jira´sek [26] showed the model needs to be enriched so as to capture the real process more adequately if the deformation field is expected to have important components with wavelengths below the resolution level of the material model, i.e., the minimum size of the region into which the strain can localize. This shows there exists an intrinsic physical characteristic length scale that transmits the deformation information from the continuum body to the microstructure when the localization occurs. Unfortunately, FEA based on continuum mechanics lacks a length scale to describe the deformation field of microstructure, leading to spurious mesh sensitivity. In fact, zero characteristic length scale is approached when the mesh is gradually refined within the localized region. In this case, the ellipticity of quasi-static governing equation transforms into a hyperbolic system even though the boundary conditions are still prescribed in an elliptical system. It is beyond doubt that the reproduction of the localized damage zone of composites in their entirety, e.g., the delamination cracks and various types of transverse macrocracks is an

4.1 Introduction

71

important work. For example, Tay et al. [27] studied the localized damage of composites under transverse loads by FEA. Flatscher et al. [28] explored the intralaminar plasticity and localized damage mechanisms of openhole composites by FEA. Moure et al. [29] investigated the damage localization around material defects in composites using a discrete damage model. Crivelli et al. [30] monitored the localization behaviors for composite panels with fatigue matrix cracks and delamination using acoustic emission. However, these workers did not attempt to introduce any length scale to solve the localization problem. In recent decades, efforts to introduce a length scale as the localization limiter to regularize ill-posed boundary value problems have led to a resurgence of research. The length scale is generally considered as the mean spacing of neighboring representative microstructures. A common approach to deal with spurious mesh dependency by FEA is to refer to the smeared crack band model proposed by Bazˇant and Oh [31] who imposed a minimum element size on the fracture toughness, which was later adopted by such as Lapczyk and Hurtado [9] and Maimı´ et al. [10] in intralaminar damage models for composites as built-in ABAQUS modules. However, Jira´sek and Zimmermann [32] showed FEA by the crack band model is still suffered from stress locking with spurious stress transfer across widely opening cracks, meshinduced directional bias, and possible instability at late stages of the loading process. By comparison, a fundamental and general way to deal with the localization problem is to introduce the nonlocal integral theory developed mainly by Eingern and Edelen [33] and Pijaudier-Cabot and Bazˇant [34], in which the nonlocal internal variable is assumed as a spatial convolution of the local corresponding internal variable and the kernel function with a length scale. Pijaudier-Cabot and Bazˇant [34] pointed out that a key idea of the nonlocal integral theory is to perform nonlocal averaging on those physically motivated internal variables that control strain softening and to treat the strain, stress, and other variables as local definitions. Bazˇant and Jira´sek [26] further pointed out the motivation of nonlocal averaging is that the damage model and FEA with strain softening lead to spurious localization of damage into a zone of zero volume, which causes unobjective numerical solutions with a vanishing energy dissipation during structural failure. This showed nonlocal averaging plays an important role in solving the localization problem with strain softening. Although they demonstrated only a onedimensional (1D) problem, Bazˇant and Jira´sek [26] concluded objective numerical results can also be achieved in multiple spatial dimensions. Recently, Li et al. [35] proposed a nonlocal fracture model for composite laminates and performed numerical calculations using Fast Fourier Transforms (FFTs). Patel and Gupta [36] proposed a nonlocal continuum damage model for composite laminated panels, but numerical implementation using FEA was not introduced. Forghani et al. [37] and Liu et al. [38] performed nonlocal FEA on the localized intralaminar damage and interlaminar delamination of

72

4. Localized damage models and implicit finite element analysis

composites using LS-DYNA and ABAQUS respectively. Germain et al. [39], Patel and Gupta [36], and Wu et al. [40] used the implicit gradient theory as an approximation to the nonlocal integral theory to solve the localization problem of composites. It is worth pointing out nonlocal averaging by implicit FEA should ultimately contribute to stiffness equations instead of only local internal variables in order to achieve quadratic convergence. However, there are few nonlocal finite element models including the nonlocal consistent tangent stiffness for composites. This chapter addresses the localization problem of composites by introducing the nonlocal integral theory developed by Pijaudier-Cabot and Bazˇant [34] into the intralaminar damage model of composites. An important work is to derive the nonlocal implicit stiffness equation including the nonlocal intralaminar consistent tangent stiffness and the cohesive interface stiffness based on the variational form of the stress boundary value problem with an interface discontinuity. Two length scales are introduced into the damage evolution behaviors of fiber and matrix, indicating longrange interactions among different material points. Different from the work of Forghani et al. [37] and Liu et al. [38] who performed nonlocal averaging on the local equivalent strains for fiber and matrix, a remarkable feature here is that the derivatives of the equivalent strains with respect to the strain components are taken as the nonlocal variables, which are included in the nonlocal consistent tangent stiffness that is assembled with direct integration contribution from neighboring elements, leading to a larger bandwidth in the stiffness matrix. Therefore, the advantage of the developed nonlocal finite element model over the work by Forghani et al. [37] and Liu et al. [38] is to guarantee quadratic convergence of nonlinear implicit solutions. The developed nonlocal model is implemented by combining ABAQUS-UEL, UMAT, and USDFLD subroutines and UEXTERNALDB utility subroutine. The update of nonlocal data is semi-implicit because nonlocal averaging is performed during each load increment in UEXTERNALDB. 3D implicit FEA on the intralaminar damage and interlaminar delamination of two [0°/90°]4s T700/8911 composite specimens with different hole diameters is performed using the developed numerical technique to predict the nonlocal progressive failure properties and structural strengths.

4.2 Nonlocal progressive failure model of composite laminates 4.2.1 Intralaminar damage model The intralaminar damaged stress-strain relationship is written as σ ¼ Cd : ε

(4.1)

where Cd(1
di(i ¼ 1, 2, 3), C) is the damaged four-order elastic tensor. C is the undamaged four-order elastic tensor. di(i ¼ 1, 2, 3) represent the fiber

4.2 Nonlocal progressive failure model of composite laminates

73

breakage, matrix cracking, and fiber/matrix interface debonding, respectively. The damage configuration in the fiber principal coordinate system proposed by Wei et al. [41] is given by 1 1
v12 v21 2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi ð1
d1 ÞE1 0 1
d1 1
d2 v21 E1 6 pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 7 4 1
d1 1
d2 v12 E2 5, ð1
d2 ÞE2 0 ð1
d3 Þð1
v12 v21 ÞG12 0 0 2 3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 E1 v21 E1 0 1
d1 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 6 7 6 7 E2 0 C5 4v E 5, M ¼ 4 5, 1
d2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1
v12 v21 12 2 ð1
v12 v21 ÞG12 0 0 1
d3 ( d1t , ε11 > 0 , and d2 d3 ¼ 1
ð1
d1t Þð1
d1c Þð1
d2t Þð1
d2c Þ, d1 ¼ d1c , ε11 < 0 ( d2t , ε22 + ε33 > 0 (4.2) ¼ d2c , ε22 + ε33 < 0 Cd ¼ MCM T ¼

where E1 and E2 are the longitudinal and transverse Young’s moduli, and v12 and v21 are the Poisson’s ratios, and G12 is the shear modulus. All damage variables change from zero (no damage) to one (complete damage). The matrix Cd is symmetric due to v21E1 ¼ v12E2. d1t, d1c, d2t, and d2c are damage variables for fiber tension, fiber compression, matrix tension, and matrix compression, respectively. εij(i, j ¼ 1, 2, 3) are the strain components. Hashin [42] proposed the stress-based failure criteria for fiber and matrix which can identify detailed failure modes under tensile and compressive states. Huang and Lee [43] showed by experiments that strain may be a more favorable approach to assess the failure of composites because the strain is more continuous and smoother than the stress during the damage evolution of composites. Later, Linde et al. [44] proposed the strain-based failure criteria, which are yet limited to plane cases. Recently, Forghani et al. [37] and Liu et al. [38] proposed simple expressions for the equivalent strains as the failure criteria of fiber and matrix. However, tensile and compressive failure cannot be differentiated. We adopt the modified 3D strain-based Hashin failure criteria for fiber and matrix proposed by Huang and Lee [43], but the forms are changed to those proposed by eq Linde et al. [44] who introduced the equivalent strains εeq 1 and ε2 for the damage evolution laws. The failure criteria for fiber and matrix under tensile and compressive states are written as (a) Fiber tensile failure ε11 > 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 2  0 2ffi ε ε eq ε1 ¼ ðε11 Þ2 + ðε12 Þ2 01t + ðε13 Þ2 01t  ε01t ε12 ε13

(4.3)

74

4. Localized damage models and implicit finite element analysis

(b) Fiber compressive failure ε11 < 0 eq

ε1 ¼ jε11 j  ε01c

(4.4)

(c) Matrix tensile failure ε22 + ε33 > 0

eq ε2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 2    0 2  0 2 ε ε22 ε33 ε ε ¼ ðε22 + ε33 Þ2 + 02t ε223
2 E2 E3 + ðε12 Þ2 02t + ðε13 Þ2 02t G23 ε23 ε12 ε13  ε02t (4.5)

(d) Matrix compressive failure in the through-thickness direction ε22 + ε33 < 0 eq ε2

 " # 2    E2 ε02c ε22 E2 + ε33 E3  0 ¼ ðε22 + ε33 Þ
1 + ε 2c  2G12 ε012 2G12 ε012    2  2  ε02c E 2 E3 ε12 ε13  2 0 0 +  2 ε23
2 ε22 ε33 + ε2c 0 + ε2c 0   ε02c 0 G23 ε12 ε13  ε

(4.6)

23

where ε01t ¼ XT/E1 and ε01c ¼ XC/E1 are the fiber tensile and compressive strains, respectively. ε02t ¼ YT/E2 and ε02c ¼ YC/E2 are the matrix tensile and compressive strains, respectively. XT, XC, YT, and YC are the tensile and compressive strengths along and perpendicular to the fiber principal direction, respectively. ε012 ¼ S12/(2G12), ε013 ¼ S13/(2G13), and ε023 ¼ S23/(2G23) are the shear strain strengths in the 1–2, 1–3, and 2–3 plane, respectively. S12, S13, and S23 are the shear strengths. E1 and E2 are the elastic moduli. G12, G23, and G13 are the shear moduli. Similar to the work of Lapczyk and Hurtado [9], Maimı´ et al. [10], Forghani et al. [37], and Wei et al. [41], the damage variables for fiber breakage and matrix cracking are given by  8 f > ε1t,1c κ 1
ε01t,1c > > > , fiber tensile or compressive failure, > d1t,1c ¼  f > > < κ1 ε1t,1c
ε01t,1c  (4.7) f 0 > > ε κ
ε 2 2t,2c > 2t,2c > > , matrix tensile or compressive failure d2t,2c ¼  > > f :
ε0 κ2 ε 2t,2c

2t,2c

4.2 Nonlocal progressive failure model of composite laminates

75

where the damage load functions F1,2 ¼ εeq 1,2
κ 1,2 and the Kuhn-Tucker loading-unloading conditions F1,2  0, κ_ 1,2  0, κ_ 1,2 F1,2 ¼ 0 are introf f and ε2t,2c are the final fiber duced. κ 1 and κ2 are the internal variables. ε1t,1c breakage and matrix cracking strains which are determined by experif 0 f 0 > ε1t,1c and ε2t,2c > ε2t,2c are required. ments. ε1t,1c

4.2.2 Bilinear cohesive models for predicting delamination Camanho et al. [45] and Turon et al. [46] proposed bilinear cohesive models. The single-mode traction-displacement jump relationships are given by. 8 > Δmax  Δ0i < ti ¼ KΔi ,  i f ti ¼ 1
dsi KΔi , Δ0i  Δmax < Δi , i > : f ti ¼ 0, Δmax  Δi i (   Δmax ¼ max Δmax , jΔi j , i ¼ 1,2, mode II and III i i  max  (4.8) max  0, mode I Δ3 ¼ max Δ3 , Δ3 with Δmax 3 where dsi ¼

Δi ðΔmax
Δ0i Þ i f

Δmax ðΔi
Δ0i Þ i f



 0  dsi  1, i ¼ 1, 2, 3 are damage variables. u is the

displacement and Δ ¼ ½½u is the displacement jump. Δf3 ¼ 2Gc1/σ max, Δf1,2 ¼ 2Gc2,3/τmax. σ max and τmax are the maximum tractions in the normal and tangential directions, respectively. Δ01 ¼ Δ02 ¼ τmax/K, Δ03 ¼ σ max/ K. Gci (i ¼ 1, 2, 3) are mode-I, II, and III interlaminar fracture toughnesses, (i ¼ 1, 2, 3) are the maximum displacement jumps in the respectively. Δmax i load history indicating the irreversibility. In order to avoid the interpenetration of the completely failed interface under compression, the penalty stiffness T3 ¼ KΔ3 (Δ3  0) is introduced in the normal direction, where the penalty stiffness K ¼ 1E6 N/mm3 was suggested by Camanho et al. [45] and Turon et al. [46]. The mixed-mode traction-displacement jump relationships are written as 8 > Δmax  Δ0i , < ti ¼ KΔi , i f 0 (4.9) < Δi , ti ¼ ð1
ds ÞKΔi , Δi  Δmax i > : f max Δi  Δi ti ¼ 0, where the scalar mixed-mode damage variable ds (0  ds  1) is written as  f  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  max  Δm Δmax
Δ0m m , Δm ¼ Δ21 + Δ22 + hΔ3 i2 , Δmax m ¼ max Δm , Δm , f 0 max Δm Δm
Δ m 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 u " !η # > 2 u >  β2 > > > 2 > Δ0 Δ0 u 1 + β , Δ3 > 0 c + G c
Gc > > > > t , Δ3 > 0 G 3 1 2 2 < < 2 1 f 1 + β2 KΔ0m 1 Δ01 + βΔ03 Δ0m ¼ , Δm ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >  2  2ffi > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > : >  0 2  0 2 Δ01 + Δ02 , Δ3  0 > : Δ 1 + Δ2 , Δ3  0

ds ¼

(4.10)

76 where

4. Localized damage models and implicit finite element analysis

hi

denotes the MacAuley bracket.

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 β¼ Δ1 + Δ2 =Δ3 ðΔ3 > 0Þ

is the

mode mixity ratio and η is the parameter defined in the B-K law [47]. The cohesive model introduces a length scale parameter that is generally called cohesive zone length due to cohesive softening behavior, which is defined as the distance from the crack tip to the position where the maximum cohesive traction is attained [48, 49]. Turon et al. [48] suggested three elements in the cohesive zone are sufficient to predict the delamination growth.

4.2.3 Nonlocal integral model Strain localization as a real physical phenomenon appears widely in composites. Forghani et al. [37] showed the microscopic crack propagation of composites is essentially a structural problem instead of a material problem, which means the stress and strain at a material point are nonlocal. In order to solve the localization problem, a length scale is required to transmit the deformation information between the localized region and the entire continuum solid. Eringen [33] and Pijaudier-Cabot and Bazˇant [34] developed the nonlocal integral theory which assumes the motion of one point depends not only on this point itself, but also on all points around this point. This actually introduces the nonlocal concept with long-range interactions among different material points. Pijaudier-Cabot and Bazˇant [34] showed only those internal variables that control the strain-softening can be taken as the localization limiter. Forghani et al. [37] further showed the damage variable is not a good choice for nonlocal averaging due to the locking problem. Because nonlocal averaging directly contributes to the consistent tangent stiffness here, the derivatives eq of the equivalent strains εeq 1 and ε2 for fiber and matrix in Eqs. (4.3)–(4.6) with respect to the strain ε are used for nonlocal averaging, which are included in the nonlocal tangent stiffness as shown in Appendix A   ð eq  eq   ∂ε1,2 X q ∂ε1,2 X p 1 (4.11) ¼   α0 X p , X q dX q ∂ε ∂ε α Xp Ω where Xq is the neighboring material point around the point Xp. The weight function α0(Xp, Xq) and the normalization factor α(Xp) are given by "

# ð

X p
X q 2       1 ¼ exp
α0 X p , X q dX q , α X α0 X p , X q ¼ p 2 3=2 3 2l Ω ð2πÞ l (4.12) where k  k denotes the norm. Here, the length scale l is l(f ) and l(m) for fiber eq ∂ε ðX p Þ and matrix respectively. The nonlocal variables 1, 2∂ε are reduced to the eq ∂ε ðX p Þ local variables 1, 2∂ε at l(f ) ! 0 and l(m) ! 0, respectively.

77

4.2 Nonlocal progressive failure model of composite laminates

Bazˇant and Jira´sek [26] showed there is a subtle difference between the notions of the intrinsic material length and the length scale l. The latter is a special case of the former because nonlocal averaging is only one of the possible enrichments that introduce a length parameter into the continuum theory. This demonstrates the length scale l is introduced from not only physical but also mathematical considerations. Bazˇant and Jira´sek [26] also showed the incremental nonlocal variables are separated into the average part and interactive part and the latter represents the statistical continuum smearing of discrete cracks. This indicates the motivation of nonlocal averaging due to microcrack interactions contains the smeared crack band concept [31]. A controversial issue using the nonlocal integral theory is the treatment of the boundary. The reason is that the regions which contribute to the nonlocal average near the boundary are smaller than those far from the boundary. Thus, damage evolution near the boundary is delayed and these regions are stronger than the internal parts of the body. Jira´sek Ð et al. [50] showed the kernel function ( Ωα0(Xp, Xq)dXq)/α(Xp) ¼ 1 in Eq. (4.11) is a feasible strategy to eliminate the boundary effect during nonlocal averaging.

4.2.4 Unified finite element model of nonlocal intralaminar damage and interlaminar delamination of composite laminates The virtual work equation for the body Ω containing the discontinuous interface Γd is given by the following weak form ð ð ð —X δu : σdV + δΔ  t dA ¼ δu  t b dA (4.13) Ω + [Ω


Γd

∂Ωt

where δu is the virtual displacement. ub is the Dirichlet boundary condition and tb is the Neumann boundary condition. It is shown the virtual work done by the external load is divided into the virtual strain energy (including the elastic deformation energy and damage dissipation energy) and the virtual cohesive interface energy. After performing finite element discretization and linearization on Eq. (4.13), the nonlocal implicit stiffness equation is written as nbulk + ncoh

A

e¼1

½ðK L + K c ÞΔu ¼ F ext ~

(4.14)

where A denotes the assembly of the bulk and interface element stiffnesses. nbulk and ncoh represent the number of bulk elements and cohesive elements. Δu is the increment of the node displacement. The derivations of the nonlocal~ element stiffness KL and node force FL for bulk elements see Appendix A. The derivations of the stiffness Kc and node force Fc for cohesive elements see Appendix B. The external node force Fext is taken as the incremental form according to the Newton-Raphson algorithm.

78

4. Localized damage models and implicit finite element analysis

Numerical calculation ends until the residual node force Fext
FL
Fc reduces to the tolerance R. Implicit FEA often meets numerical convergence problem due to softening behavior. Same as Chapter 2, artificial viscous effect is introduced to improve convergence [51] ( F ext
F L
F c
F v ¼ R, (4.15) F v ¼ cM∗ v, v ¼ Δu=Δt ~ where M∗ is the mass matrix calculated with unity density, v is the node velocity, Δu is the node displacement increment, Fv is the artificial viscous force, c is a~constant damping factor, and Δt is the time increment during nonlinear iterations. It is noted introduced viscous forces should be large enough to regularize cohesive softening behaviors but small enough not to affect numerical accuracy.

4.3 Numerical results and discussion FEA is used to implement the developed nonlocal model. Although Pijaudier-Cabot and Bazˇant [34] and Jira´sek et al. [50] demonstrated the role of the length scale by nonlocal averaging for 1D and 2D localization problems, little work is done for 3D problems. Nonlocal averaging contributes to the consistent tangent stiffness according to Appendix A, difeq ferent from that performed only on the equivalent strains εeq 1 and ε2 in Eq. (4.11) and the damage variables d1 and d2 in Eq. (4.6) according to Forghani et al. [37] and Liu et al. [38] in last chapter. Fig. 4.1 shows the flow chart of nonlocal FEA, in which ABAQUS-UEL (User element subroutine), UMAT (User material subroutine), USDFLD (User defined field subroutine) and UEXTERNALDB (User external data utility subroutine) are used. The nonlocal intralaminar damage model is implemented by combining ABAQUS-UMAT, USDFLD and UEXTERNALDB, and the cohesive model for delamination is implemented by ABAQUS-UEL, where zerothickness 3D eight-node cohesive elements are established. After the initial failure of fiber or matrix elements are detected, the nonlocal bulk element stiffness KL and cohesive element stiffness Kc are assembled and the node forces FL and Fc are calculated at each iteration according to Eq. (4.14). Nonlocal averaging includes two key points: (1) the coordinate and volume at each integration point are needed in ABAQUSUEXTERNALDB, which are initially obtained by ABAQUS-USDFLD subroutine and assigned to those field variables stored in the COMMON BLOCK data module, which are shared by both ABAQUSUEXTERNALDB and ABAQUS-USDFLD and (2) the derivatives of the local equivalent strains with respect to the strain components are also

4.3 Numerical results and discussion

FIG. 4.1

79

Flow chart of the nonlocal FEA of progressive failure of composite laminates.

placed in a defined COMMON BLOCK data module shared by both ABAQUS-UMAT and ABAQUS-UEXTERNALDB. Nonlocal averaging is performed on the derivatives of the equivalent strains with respect to the strain components by ABAQUS-UEXTERNALDB, which are returned to ABAQUS-UMAT by COMMON BLOCK. It is noted the update of nonlocal data is semi-implicit because nonlocal averaging is performed during each load increment instead of each iteration. After performing nonlinear calculations on the stiffness equation, all variables including the strains, equivalent strains, damage variables, and stresses become nonlocal. Because nonlocal FEA requires data exchanges among different integration points, parallel calculations are needed to improve computational efficiency. Besides, the numerical technique by dynamically allocating and releasing arrays is also developed in UEXTERNALDB in order to increase computational capacity. Jira´sek et al. [50] pointed out the convergence and equilibrium iteration at the global level are usually more regular for the nonlocal model than for the local one. The reason is that the transition of strain fields from a diffuse damage mode to a localized

80

4. Localized damage models and implicit finite element analysis

one is relatively smooth, similar to the conclusion by Huang and Lee [43]. Therefore, required iterations that achieve equilibrium are reduced, which partly compensates increased computational cost for nonlocal search for neighboring integration points. The geometry structure of two [0°/90°]4s composite specimens is shown in Fig. 4.2. The thickness of each ply is 0.1125 mm. Liu et al. [38] studied the effect of different layup patterns on the damage evolution behaviors of composites by nonlocal FEA. Here, we further study the effects of two hole diameters: 5 and 10 mm. Fig. 4.3 shows two symmetric

FIG. 4.2 Geometry structures and sizes for two composite specimens with a central hole.

FIG. 4.3 Two mesh models for two composite specimens.

81

4.3 Numerical results and discussion

mesh models including 1080 and 1440 bulk elements and 540 and 720 cohesive interface elements, respectively. Material parameters for T700/8911 composites are listed in Table 4.1 [52]. Turon et al. [46] obtained η ¼ 2.284 for AS4/PEEK composite by experiments. As η in Eq. (4.10) changes from 1.5 to 2.5 for T700/8911 composite, delamination results are not affected distinctly and η ¼ 2.0 is adopted here. The length scale controls the spread of the kernel function, but cannot be directly measured. In general, these parameters are explained as the mean half spacing between the representative microstructures [26]. However, the fiber/matrix microstructures of composites often take on random TABLE 4.1 Mechanical parameters of T700/8911 composites [52]. Ply longitudinal modulus

E1

135 GPa

Ply transverse modulus

E2

11.41 GPa

Out-of-plane modulus

E3

11.41 GPa

Inplane shear modulus

G12

7.92 GPa

Out-of-plane shear modulus

G13

7.92 GPa

G23

3.79 GPa

v12

0.33

v13

0.33

v23

0.49

Longitudinal tensile strength

XT

2600 MPa

Longitudinal compressive strength

XC

1422 MPa

Transverse tensile strength

YT

603 MPa

Transverse compressive strength

YC

241 MPa

Shear strength

S12

94 Mpa

Shear strength

S13

94 Mpa

Shear strength

S23

40 Mpa

Interlaminar mode-I fracture toughness

Gc1

0.252 N/mm

Interlaminar mode-II/III fracture toughness

Gc2 ¼ Gc3

0.665 N/mm

Initial fiber tensile breakage strain

ε01t ¼ XT/E1

0.019

Initial matrix tensile cracking strain

ε02t ¼ YT/E2

0.005

Critical fiber tensile breakage strain

ε f1t

0.021

Critical matrix tensile cracking strain

ε f2t

0.035

Poisson’s ratio

82

4. Localized damage models and implicit finite element analysis

arrangements. Bazˇant and Jira´sek [26] pointed out there are two length scale parameters: one of them characterizing the length of process zone and the other describing the width of the process zone, and any identification procedure through experimental data extracts from only a single length scale parameter when the fixed ratio of these two length scale parameters are assumed. However, this assumption can hardly be considered as a general rule and a favorable method is to extract two length scale parameters. In the developed model, two length scales l(f ) and l(m) for longitudinal fiber breakage and transverse matrix cracking are introduced. Currently, there are some advanced experimental approaches for the identification of the length scale of composite structure, such as the strain field method proposed by Geers et al. [53], the elastic wave dispersion method proposed by Dontsov et al. [54] and the microcantilever test method proposed by Dehrouyeh-Semnani and Nikkhah-Bahrami [55]. Yet, it is almost impossible to determine the length scales rigorously for complex structures [26]. It is more believed the length scale is introduced from not only the physical concept but also the numerical consideration. Here, l(f ) and l(m) are assumed as adjustable parameters comparable to the minimum element size suggested by Rousseller et al. [56]. Ahad et al. [57] further showed the length scale must be greater than (or at least equal to) the minimum element size in the regions where the damage may occur so that the regularization by nonlocal averaging is variable. In last chapter, same values for l(f ) ¼ l(m) ¼ 1 mm and 2 mm are adopted for simplification. Yet, Forghani et al. [37] showed employing a relatively small transverse nonlocal radius by orthotropic averaging is helpful to capture a narrow matrix cracking pattern, and took l(f ) ¼ 2 mm and l(m) ¼ 0.5 mm, 1 mm, and 2 mm respectively. Here, we attempt to study the effect of different length scales l(f ), l(m) ¼ 0.5 mm, 1 mm, and 2 mm on the load-displacement curves for two mesh models. Furthermore, the length scales are approximately determined by the load stress-strain curves, similar to the work by Ahad et al. [57] and Korsunsky and Kim [58] who determined the length scale parameter for ductile metal materials by notched tensile tests. The tensile stress-strain curves for two composite specimens using different length scales and two mesh models are shown in Fig. 4.4. Table 4.2 lists the tensile strengths and fracture strains by FEA. After the initial failure of fiber or matrix, the stiffness of the structure degrades which leads to nonlinear load response. Intralaminar matrix cracking and delamination are the main failure modes before fiber breakage, where matrix cracking induces a little delamination growth near the hole. From the nonlocal FEA, the decreasing parts of tensile curves tend to be consistent for two mesh models, which show objective description of the damage evolution and energy dissipation after performing nonlocal averaging. By comparison, the local FEA that lacks a length scale in the governing equation, leads to some mesh dependence. When the length scales l(f ) and l(m) vary from

83

4.3 Numerical results and discussion

800

800 750 700

600

Stress (MPa)

650

Fine mesh and l(f) = l(m) = 0.5mm

1.5

1.6

1.7

Fine mesh and l(f) = l(m) = 1.0mm Fine mesh and l(f) = l(m) = 2.0mm

Initial fiber damage

Fine mesh and l(f) = 2.0mm, l(m) = 0.5mm

400

Fine mesh and l(f) = 2.0mm, l(m) = 1.0mm Coarse mesh and l(f) = l(m) = 0.5mm

Initial matrix cracking

Coarse mesh and l(f) = l(m) = 1.0mm Coarse mesh and l(f) = l(m) = 2.0mm Coarse mesh and l(f) = 2.0mm, l(m) = 0.5mm

200

Coarse mesh and l(f) = 2.0mm, l(m) = 1.0mm Fine mesh and l(f) = l(m) = 0mm Coarse mesh and l(f) = l(m) = 0mm Test result [49]

0 0.0

0.4

0.8

(A)

1.6

0.2

2.0

Strain (%)

800

700

600

650

Fine mesh and l(f) = l(m) = 0.5mm

Stress (MPa)

1.4

Fine mesh and l(f) = l(m) = 1.0mm

1.5

Fine mesh and l(f) = l(m) = 2.0mm

Initial fiber damage

Fine mesh and l(f) = 2.0mm, l(m) = 0.5mm

400

Fine mesh and l(f) = 2.0mm, l(m) = 1.0mm

Initial matrix cracking

Coarse mesh and l(f) = l(m) = 0.5mm Coarse mesh and l(f) = l(m) = 1.0mm Coarse mesh and l(f) = l(m) = 2.0mm Coarse mesh and l(f) = 2.0mm, l(m) = 0.5mm

200

Coarse mesh and l(f) = 2.0mm, l(m) = 1.0mm Fine mesh and l(f) = l(m) = 0mm Coarse mesh and l(f) = l(m) = 0mm Test result [49]

0 0.0

(B)

0.4

0.8

1.2

1.6

2.0

Strain (%)

FIG. 4.4 Stress-strain curves for [0°/90°]4s specimens with (A) 5 mm-diameter hole and (B) a 10 mm-diameter hole using different length scales and mesh models.

84

4. Localized damage models and implicit finite element analysis

TABLE 4.2 Tensile strengths and fracture strains using nonlocal and local FEA for two mesh models.

Composite laminate with a 5 mm-diameter hole

Composite laminate with a 10 mm-diameter hole

Tensile strengths (MPa)

Fracture strains (%)

(1) Test results

809

1.560

(2) Coarse mesh and nonlocal model at l( f ) ¼ l(m) ¼ 0.5 mm

790

1.575

(3) Coarse mesh and nonlocal model at l( f ) ¼ l(m) ¼ 1.0 mm

795

1.585

(4) Coarse mesh and nonlocal model at l( f ) ¼ l(m) ¼ 2.0 mm

800

1.595

(5) Coarse mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 0.5 mm

791

1.577

(6) Coarse mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 1.0 mm

793

1.580

(7) Fine mesh and nonlocal model at l( f ) ¼ l(m) ¼ 0.5 mm

781

1.557

(8) Fine mesh and nonlocal model at l( f ) ¼ l(m) ¼ 1.0 mm

785

1.566

(9) Fine mesh and nonlocal model at l( f ) ¼ l(m) ¼ 2.0 mm

784

1.564

(10) Fine mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 0.5 mm

782

1.559

(11) Fine mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 1.0 mm

784

1.564

(12) Coarse mesh and local model

756

1.503

(13) Fine mesh and local model

776

1.549

(1) Test results

714

1.524

(2) Coarse mesh and nonlocal model at l( f ) ¼ l(m) ¼ 0.5 mm

723

1.480

(3) Coarse mesh and nonlocal model at l( f ) ¼ l(m) ¼ 1.0 mm

726

1.489

(4) Coarse mesh and nonlocal model at l( f ) ¼ l(m) ¼ 2.0 mm

729

1.493

85

4.3 Numerical results and discussion

TABLE 4.2 Tensile strengths and fracture strains using nonlocal and local FEA for two mesh models—cont’d Tensile strengths (MPa)

Fracture strains (%)

(5) Coarse mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 0.5 mm

722

1.479

(6) Coarse mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 1.0 mm

725

1.486

(7) Fine mesh and nonlocal model at l( f ) ¼ l(m) ¼ 0.5 mm

706

1.445

(8) Fine mesh and nonlocal model at l( f ) ¼ l(m) ¼ 1.0 mm

705

1.443

(9) Fine mesh and nonlocal model at l( f ) ¼ l(m) ¼ 2.0 mm

713

1.460

(10) Fine mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 0.5 mm

710

1.453

(11) Fine mesh and nonlocal model at l( f ) ¼ 2.0 mm and l(m) ¼ 1.0 mm

711

1.456

(12) Coarse mesh and local model

677

1.379

(13) Fine mesh and local model

711

1.456

0.5 to 2.0 mm, the load curves do not change too much. By referring to the suggestion by Rousseller et al. [56] that the length scale should be larger than the minimum element side length in Fig. 4.3, we take l(f ) ¼ l(m) ¼ 0.5 mm in the following analysis. eq eq The contours of the nonlocal equivalent strains ε1 and ε2 the damage variables d1 and d2 for fiber and matrix for the specimen with a 5 mmdiameter hole by nonlocal FEA are shown in Figs. 4.5–4.8. The corresponding results for the specimen with a 10 mm-diameter hole by nonlocal FEA are shown in Figs. 4.9–4.12. The predicted damage evolution behaviors are close using two mesh models for two specimens. The initial failure for both fiber and matrix appears near the hole. For the specimen with a 5 mm-diameter hole, the matrix starts to fail at the strain 0.342%, and the equivalent matrix strain 0.85%, and fiber fails initially at the strain 0.702% and equivalent fiber strain 2.654%. For the specimen with a 10 mm-diameter hole, the matrix starts to fail at the strain 0.229%, and

(1) Equivalent strain e1eq +3.104e-02 +2.854e-02 +2.605e-02 +2.355e-02 +2.106e-02 +1.857e-02 +1.607e-02 +1.358e-02 +1.108e-02 +8.590e-03 +6.096e-03 +3.602e-03 +1.108e-03

(2) Damage variable d1 +3.563e-01 +3.226e-01 +2.889e-01 +2.553e-01 +2.216e-01 +1.879e-01 +1.543e-01 +1.206e-01 +8.693e-02 +5.327e-02 +1.960e-02 -1.406e-02 -4.773e-02

(A)

+6.505e-02 +5.963e-02 +5.421e-02 +4.879e-02 +4.337e-02 +3.795e-02 +3.253e-02 +2.711e-02 +2.169e-02 +1.627e-02 +1.085e-02 +5.429e-03 +9.590e-06

+5.119e-01 +4.575e-01 +4.032e-01 +3.488e-01 +2.945e-01 +2.401e-01 +1.858e-01 +1.314e-01 +7.710e-02 +2.276e-02 -3.159e-02 -8.593e-02 -1.403e-01

(B) +1.143e-01 +1.045e-01 +9.472e-02 +8.491e-02 +7.510e-02 +6.529e-02 +5.548e-02 +4.567e-02 +3.587e-02 +2.606e-02 +1.625e-02 +6.440e-03 -3.368e-03

+5.564e-01 +4.995e-01 +4.426e-01 +3.857e-01 +3.288e-01 +2.720e-01 +2.151e-01 +1.582e-01 +1.013e-01 +4.444e-02 -1.244e-02 -6.932e-02 -1.262e-01

(C) FIG. 4.5 Contours of the equivalent strain εeq 1 and damage variable d1 of the fiber for the [0°/90°]4s laminate.

(1) Equivalent strain e2eq +3.104e-02 +2.854e-02 +2.605e-02 +2.355e-02 +2.106e-02 +1.857e-02 +1.607e-02 +1.358e-02 +1.108e-02 +8.590e-03 +6.096e-03 +3.602e-03 +1.108e-03

(2) Damage variable d2 +3.563e-01 +3.226e-01 +2.889e-01 +2.553e-01 +2.216e-01 +1.879e-01 +1.543e-01 +1.206e-01 +8.693e-02 +5.327e-02 +1.960e-02 -1.406e-02 -4.773e-02

(A) +6.505e-02 +5.963e-02 +5.421e-02 +4.879e-02 +4.337e-02 +3.795e-02 +3.253e-02 +2.711e-02 +2.169e-02 +1.627e-02 +1.085e-02 +5.429e-03 +9.590e-06

+5.119e-01 +4.575e-01 +4.032e-01 +3.488e-01 +2.945e-01 +2.401e-01 +1.858e-01 +1.314e-01 +7.710e-02 +2.276e-02 -3.159e-02 -8.593e-02 -1.403e-01

+1.143e-01 +1.045e-01 +9.472e-02 +8.491e-02 +7.510e-02 +6.529e-02 +5.548e-02 +4.567e-02 +3.587e-02 +2.606e-02 +1.625e-02 +6.440e-03 -3.368e-03

+5.564e-01 +4.995e-01 +4.426e-01 +3.857e-01 +3.288e-01 +2.720e-01 +2.151e-01 +1.582e-01 +1.013e-01 +4.444e-02 -1.244e-02 -6.932e-02 -1.262e-01

(B)

(C) Contours of the equivalent strain εeq 2 and damage variable d2 of the matrix for the [0°/90°]4s laminate with a 5 mm-diameter hole using fine mesh model at the strains (A) 0.342%, (B) 1.195%, and (C) 1.557%, respectively.

FIG. 4.6

(1) Equivalent strain e1eq

(2) Damage variable d1

+3.577e-02 +3.295e-02 +3.013e-02 +2.732e-02 +2.450e-02 +2.168e-02 +1.886e-02 +1.605e-02 +1.323e-02 +1.041e-03 +7.594e-03 +4.776e-03 +1.959e-03

+4.049e-01 +3.677e-01 +3.305e-01 +2.933e-01 +2.560e-01 +2.188e-01 +1.816e-01 +1.444e-01 +1.071e-01 +6.991e-02 +3.269e-02 -4.537e-03 -4.176e-02

+6.736e-02 +6.194e-02 +5.652e-02 +5.110e-02 +4.569e-02 +4.027e-02 +3.485e-02 +2.943e-02 +2.401e-02 +1.860e-02 +1.318e-02 +7.759e-03 +2.341e-03

+5.318e-01 +4.796e-01 +4.274e-01 +3.751e-01 +3.229e-01 +2.707e-01 +2.185e-01 +1.663e-01 +1.141e-01 +6.186e-02 +9.648e-03 -4.257e-02 -9.478e-02

+1.018e-01 +9.359e-02 +8.535e-02 +7.711e-02 +6.886e-02 +6.062e-02 +5.237e-02 +4.413e-02 +3.588e-02 +2.764e-02 +1.940e-02 +1.115e-02 +2.909e-03

+5.826e-01 +5.226e-01 +4.626e-01 +4.026e-01 +3.426e-01 +2.826e-01 +2.226e-01 +1.626e-01 +1.026e-01 +4.260e-02 -1.740e-02 -7.740e-02 -1.374e-01

(A)

(B)

(C) FIG. 4.7 Contours of the equivalent strain εeq 1 and damage variable d1 of the fiber for the [0°/90°]4s laminate with a 5 mm-diameter hole using coarse mesh model at the strains (A) 0.794%, (B) 1.244%, and (C) 1.557%, respectively.

(1) Equivalent strain e2eq +9.000e-03 +8.285e-03 +7.571e-03 +6.850e-03 +6.141e-03 +5.427e-03 +4.712e-03 +3.997e-03 +3.283e-03 +2.568e-03 +1.853e-03 +1.138e-03 +4.238e-04

(2) Damage variable d2 +1.962e-01 +1.777e-01 +1.592e-01 +1.406e-01 +1.221e-01 +1.035e-01 +8.497e-02 +6.642e-02 +4.789e-02 +2.934e-02 +1.080e-02 -7.747e-03 -2.629e-02

(A) +2.900e-02 +2.681e-02 +2.463e-02 +2.244e-02 +2.025e-02 +1.807e-02 +1.588e-02 +1.370e-02 +1.151e-02 +9.323e-03 +7.137e-03 +4.950e-03 +2.764e-03

+5.283e-01 +4.800e-01 +4.316e-01 +3.833e-01 +3.349e-01 +2.866e-01 +2.382e-01 +1.899e-01 +1.415e-01 +9.315e-02 +4.480e-02 -3.350e-03 -5.190e-02

+3.677e-02 +3.477e-02 +3.229e-02 +2.981e-02 +2.733e-02 +2.485e-02 +2.237e-02 +1.990e-02 +1.742e-02 +1.494e-02 +1.246e-03 +9.978e-03 +7.499e-03 +5.020e-03

+5.425e-01 +4.992e-01 +4.560e-01 +4.128e-01 +3.696e-01 +3.264e-01 +2.831e-01 +2.399e-01 +1.967e-01 +1.535e-02 +1.103e-02 +6.704e-02 +2.382e-02

(B)

(C) FIG. 4.8 Contours of the equivalent strain εeq 2 and damage variable d2 of the matrix for the [0°/90°]4s laminate with a 5 mm-diameter hole using coarse mesh model at the strains (A) 0.342%, (B) 1.244%, and (C) 1.557%, respectively.

(1) Equivalent strain e1eq

(2) Damage variable d1

+2.253e-02 +2.067e-02 +1.880e-02 +1.694e-02 +1.507e-02 +1.320e-02 +1.134e-02 +9.471e-03 +7.605e-03 +5.738e-03 +3.872e-03 +2.006e-03 +1.401e-04

+1.954e-01 +1.778e-01 +1.602e-01 +1.425e-01 +1.249e-01 +1.073e-01 +8.962e-02 +7.198e-02 +5.434e-02 +3.671e-02 +1.907e-02 +1.431e-03 -1.621e-02

+8.465e-02 +7.725e-02 +6.985e-02 +6.244e-02 +5.504e-02 +4.764e-02 +4.024e-02 +3.283e-02 +2.543e-02 +1.803e-02 +1.062e-02 +3.220e-03 -4.183e-03

+5.073e-01 +4.548e-01 +4.023e-01 +3.499e-01 +2.974e-01 +2.449e-01 +1.924e-01 +1.400e-01 +8.748e-02 +3.500e-02 -1.748e-02 -6.995e-02 -1.224e-01

+1.335e-01 +1.220e-01 +1.105e-01 +9.895e-02 +8.743e-02 +7.592e-02 +6.440e-02 +5.288e-02 +4.136e-02 +2.984e-02 +1.832e-02 +6.803e-03 -4.716e-03

+5.414e-01 +4.852e-01 +4.290e-01 +3.729e-01 +3.167e-01 +2.605e-01 +2.043e-01 +1.482e-01 +9.200e-02 +3.582e-02 -2.035e-02 -7.653e-02 -1.327e-01

(A)

(B)

(C) FIG. 4.9 Contours of the equivalent strain εeq 1 and damage variable d1 of the fiber for the [0°/90°]4s laminate with a 10 mm-diameter hole using fine mesh model at the strains (A) 0.472%, (B) 1.092%, and (C) 1.445%, respectively.

(1) Equivalent strain e2eq

(2) Damage variable d2

+8.000e-03 +7.334e-03 +6.668e-03 +6.002e-03 +5.335e-03 +4.669e-03 +4.003e-03 +3.337e-03 +2.671e-03 +2.005e-03 +1.338e-03 +6.723e-04 +6.125e-06

+1.163e-01 +1.053e-01 +9.434e-02 +8.335e-02 +7.236e-02 +6.136e-02 +5.037e-02 +3.938e-02 +2.839e-02 +1.739e-02 +6.401e-03 -4.592e-02 -1.559e-02

+3.000e-02 +2.748e-02 +2.495e-02 +2.243e-02 +1.990e-02 +1.738e-02 +1.485e-02 +1.233e-02 +9.803e-03 +7.279e-03 +4.754e-03 +2.230e-03 -2.951e-04

+5.173e-01 +4.663e-01 +4.153e-01 +3.642e-01 +3.132e-01 +2.622e-01 +2.112e-01 +1.602e-01 +1.092e-01 +5.815e-02 +7.140e-03 +4.388e-02 -9.489e-02

(A)

(B)

+4.121e-02 +3.774e-02 +3.427e-02 +3.080e-02 +2.732e-02 +2.385e-02 +2.038e-02 +1.691e-02 +1.344e-02 +9.970e-03 +6.499e-03 +3.028e-03 -4.430e-04

+5.393e-01 +4.913e-01 +4.432e-01 +3.952e-01 +3.471e-01 +2.991e-01 +2.510e-01 +2.029e-01 +1.549e-01 +1.068e-01 +5.876e-02 +1.071e-02 -3.735e-02

(C) Contours of the equivalent strain εeq 2 and damage variable d2 of the matrix for the [0°/90°]4s laminate with a 10 mm-diameter hole using fine mesh model at the strains (A) 0.229%, (B) 1.092%, and (C) 1.445%, respectively.

FIG. 4.10

(1) Equivalent strain e1eq +3.179e-02 +2.920e-02 +2.661e-02 +2.402e-02 +2.143e-02 +1.884e-02 +1.625e-02 +1.366e-02 +1.107e-02 +8.479e-03 +5.889e-03 +3.299e-03 +7.096e-04

(2) Damage variable d1 +2.466e-01 +2.233e-01 +2.000e-01 +1.767e-01 +1.534e-01 +1.301e-01 +1.068e-01 +8.347e-02 +6.017e-02 +3.687e-02 +1.357e-02 -9.734e-03 -3.303e-02

(A)

+6.165e-02 +5.641e-02 +5.117e-02 +4.592e-02 +4.068e-02 +3.544e-02 +3.020e-02 +2.496e-02 +1.972e-02 +1.448e-02 +9.235e-03 +3.994e-03 -1.248e-03

+4.650e-01 +4.172e-01 +3.694e-01 +3.216e-01 +2.738e-01 +2.260e-01 +1.782e-01 +1.304e-01 +8.257e-02 +3.477e-02 -1.303e-02 -6.083e-02 -1.086e-01

(B)

+1.092e-01 +9.965e-02 +9.011e-02 +8.057e-02 +7.102e-02 +6.148e-02 +5.194e-02 +4.240e-02 +3.286e-02 +2.332e-02 +1.378e-02 +4.238e-03 -5.303e-03

+5.182e-01 +4.647e-01 +4.112e-01 +3.578e-01 +3.043e-01 +2.508e-01 +1.973e-01 +1.438e-01 +9.036e-02 +3.688e-02 -1.660e-02 -7.008e-02 -1.236e-01

(C) FIG. 4.11 Contours of the equivalent strain εeq 1 and damage variable d1 of the fiber for the [0°/90°]4s laminate with a 10 mm-diameter hole using coarse mesh model at the strains (A) 0.635%, (B) 1.053%, and (C) 1.445%, respectively.

(1) Equivalent strain e2eq +8.191e-03 +7.517e-03 +6.843e-03 +6.168e-03 +5.494e-03 +4.820e-03 +4.146e-03 +3.472e-03 +2.797e-03 +2.123e-03 +1.449e-03 +7.747e-04 +1.005e-04

(2) Damage variable d2 +2.668e-01 +2.416e-01 +2.164e-01 +1.912e-01 +1.660e-01 +1.408e-01 +1.155e-01 +9.033e-02 +6.511e-02 +3.990e-02 +1.468e-02 -1.053e-02 -3.575e-02

(A)

+3.043e-02 +2.791e-02 +2.540e-02 +2.289e-02 +2.037e-02 +1.786e-02 +1.534e-02 +1.283e-02 +1.032e-02 +7.802e-03 +5.288e-03 +2.774e-03 +2.598e-04

+5.303e-01 +4.753e-01 +4.203e-01 +3.652e-01 +3.102e-01 +2.552e-01 +2.002e-01 +1.452e-01 +9.019e-02 +3.518e-02 -1.984e-02 -7.485e-02 -1.299e-01

+4.014e-02 +3.679e-02 +3.344e-02 +3.010e-02 +2.675e-02 +2.340e-02 +2.006e-02 +1.671e-02 +1.336e-02 +1.002e-02 +6.671e-03 +3.324e-03 -2.207e-05

+5.451e-01 +5.950e-01 +4.449e-01 +3.948e-01 +3.447e-01 +2.946e-01 +2.445e-01 +1.944e-01 +1.443e-01 +9.421e-02 +4.411e-02 -5.998e-03 -5.610e-02

(B)

(C) Contours of the equivalent strain εeq 2 and damage variable d2 of the matrix for the [0°/90°]4s laminate with a 10 mm-diameter hole using coarse mesh model at the strains (A) 0.310%, (B) 1.053%, and (C) 1.445%, respectively.

FIG. 4.12

94

4. Localized damage models and implicit finite element analysis

the equivalent matrix strain 0.637%, and fiber fails initially at the strain 0.472% and equivalent fiber strain 2.367%. The load-bearing ability decreases when the diameter of the hole adds. After the initial failure, matrix cracks propagate to the whole specimen, but fiber breakage propagates at about 30° angle to the loading direction, which is consistent with the results obtained by Lee et al. [18] who adopted the Puck’s failure criteria [59] and damage mechanics method to study local failure of fiber and matrix of [(0°/90°)6]s composite laminates under a particular stress combination. Chang and Chang [1] proposed a damage model to study in-plane damage mechanisms of [0°/45°/90°]2s composites although the delamination was not considered. They showed the damage initiates through matrix cracking in the 90° plies before propagating across the 45° plies, and fiber breakage follows as the cracks reach the 0° plies. Kortschot and Beaumont [60] performed experimental research on the observed subcritical notch tip cracking patterns in [0°n/90°n]ns carbon fiber/epoxy laminates and established a qualitative relationship between the terminal damage and notched strength. They divided the visible failure modes into the transverse cracks in the 90° plies, the axial splitting in the 0° plies, and the triangular delamination zone at the 0°/90° interface. Kongshavn and Poursartip [61] developed a strain-softening approach to predict the failure of [
45°/90°/45°/0°/]2s notched composites and showed the damage zone propagates from the notch-tip in a stable, self-similar manner which includes matrix cracks and delamination in the off-axis plies. Green et al. [62] performed a series of experiments on open-hole [45°m/90°m/
45°m/ 0°m]ns composite laminates by employing three different scaling methods: in-plane scaling, sublaminate-scaling, and ply-scaling. Failure modes for the sublaminate-scaling specimens (m ¼ 1, n ¼ 1, 2, 4, 8) are the fiber pullout or brittle failure, and load-drop is caused by fiber breakage in the 0° plies. By comparison, ply-scaled specimens (m ¼ 1,2,4,8, n ¼ 1) all fail in the delamination mode, and load-drop is caused by delamination at the
45°/0° interface prior to fiber breakage in the 0° plies. Hallett et al. [24] divided the damage evolution of sublaminate-scaling [45°/ 90°/
45°/0°]4s laminates into four stages mentioned above. For the sublaminate-scaling [0°/90°]4s laminates here, transverse matrix cracking appears at the 90° plies and then induces a little delamination near the hole. With progressive delamination, a little splitting (longitudinal matrix cracking) appears in the 0° plies, accompanied by a little fiber breakage. Suddenly much fiber breakage appears indicating the final collapse of structures. Either fiber breakage or matrix cracking occurs across the laminate where the weakest sections of these plies are located. Figs. 4.13–4.16 compare the results by local and nonlocal FEA for two specimens using a fine mesh model. After performing nonlocal averaging, the damage evolution of fiber and matrix is alleviated to some extent, which is represented

(1) Local model Equivalent strain e1eq +1.150e-01 +1.050e-01 +9.500e-02 +8.500e-02 +7.500e-02 +6.500e-02 +5.500e-02 +4.500e-02 +3.500e-02 +2.500e-02 +1.500e-02 +5.500e-03 -5.000e-03

Damage variable d1 +5.600e-01 +5.008e-01 +4.417e-01 +3.825e-01 +3.233e-01 +2.642e-01 +2.050e-01 +1.458e-01 +8.667e-02 +2.750e-02 -3.167e-02 -9.083e-02 -1.500e-01

(A)

(2) Nonlocal model Equivalent strain e1eq +1.150e-01 +1.050e-01 +9.500e-02 +8.500e-02 +7.500e-02 +6.500e-02 +5.500e-02 +4.500e-02 +3.500e-02 +2.500e-02 +1.500e-02 +5.500e-03 -5.000e-03

Damage variable d1 +5.600e-01 +5.008e-01 +4.417e-01 +3.825e-01 +3.233e-01 +2.642e-01 +2.050e-01 +1.458e-01 +8.667e-02 +2.750e-02 -3.167e-02 -9.083e-02 -1.500e-01

(B) FIG. 4.13 Contours of the equivalent strain εeq 1 and damage variable d1 of the fiber for the [0°/90°]4s laminate with a 5 mm-diameter hole using fine mesh model at the strain 1.557%.

96

4. Localized damage models and implicit finite element analysis

(1) Local model Equivalent strain e1eq

Damage variable d1 +5.600e-01 +5.008e-01 +4.417e-01 +3.825e-01 +3.233e-01 +2.642e-01 +2.050e-01 +1.458e-01 +8.667e-02 +2.750e-02 -3.167e-02 -9.083e-02 -1.500e-01

+1.150e-01 +1.050e-01 +9.500e-02 +8.500e-02 +7.500e-02 +6.500e-02 +5.500e-02 +4.500e-02 +3.500e-02 +2.500e-02 +1.500e-02 +5.500e-03 -5.000e-03

(2) Nonlocal model Equivalent strain e1eq

Damage variable d1 +5.600e-01 +5.008e-01 +4.417e-01 +3.825e-01 +3.233e-01 +2.642e-01 +2.050e-01 +1.458e-01 +8.667e-02 +2.750e-02 -3.167e-02 -9.083e-02 -1.500e-01

+1.150e-01 +1.050e-01 +9.500e-02 +8.500e-02 +7.500e-02 +6.500e-02 +5.500e-02 +4.500e-02 +3.500e-02 +2.500e-02 +1.500e-02 +5.500e-03 -5.000e-03

(1) Local model Equivalent strain e2eq

Damage variable d2

+4.400e-02 +4.037e-02 +3.673e-02 +3.310e-02 +2.947e-02 +2.583e-02 +2.220e-02 +1.857e-02 +1.493e-02 +1.130e-02 +7.667e-03 +4.033e-03 +4.000e-04

+5.600e-01 +5.008e-01 +4.417e-01 +3.825e-01 +3.233e-01 +2.642e-01 +2.050e-01 +1.458e-01 +8.667e-02 +2.750e-02 -3.167e-02 -9.083e-02 -1.500e-01

(2) Nonlocal model Equivalent strain e2eq

Damage variable d2 +5.600e-01 +5.008e-01 +4.417e-01 +3.825e-01 +3.233e-01 +2.642e-01 +2.050e-01 +1.458e-01 +8.667e-02 +2.750e-02 -3.167e-02 -9.083e-02 -1.500e-01

+4.400e-02 +4.037e-02 +3.673e-02 +3.310e-02 +2.947e-02 +2.583e-02 +2.220e-02 +1.857e-02 +1.493e-02 +1.130e-02 +7.667e-03 +4.033e-03 +4.000e-04

(A)

(B) εeq 2

FIG. 4.14 Contour of the equivalent strain and damage variable d2 of the matrix for the [0°/90°]4s laminate with a 5 mm-diameter hole using fine mesh model at the strain 1.557%.

(1) Local model Equivalent strain e1eq +1.700e-02 +1.553e-02 +1.405e-02 +1.258e-02 +1.110e-02 +9.625e-03 +8.150e-03 +6.675e-03 +5.200e-03 +3.725e-03 +2.250e-03 +7.750e-04 -7.000e-04

(A)

Damage variable d1 +5.500e-01 +4.933e-01 +4.367e-01 +3.800e-01 +3.233e-01 +2.667e-01 +2.100e-01 +1.533e-01 +9.667e-02 +4.000e-02 -1.667e-02 -7.333e-02 -1.300e-01

(2) Nonlocal model Equivalent strain e1eq +1.700e-02 +1.553e-02 +1.405e-02 +1.258e-02 +1.110e-02 +9.625e-03 +8.150e-03 +6.675e-03 +5.200e-03 +3.725e-03 +2.250e-03 +7.750e-04 -7.000e-04

Damage variable d1 +5.500e-01 +4.933e-01 +4.367e-01 +3.800e-01 +3.233e-01 +2.667e-01 +2.100e-01 +1.533e-01 +9.667e-02 +4.000e-02 -1.667e-02 -7.333e-02 -1.300e-01 -1.327e-01

(A) FIG. 4.15 Contours of the equivalent strain εeq 1 and damage variable d1 of the fiber for the [0°/90°]4s laminate with a 10 mm-diameter hole using fine mesh model at the strain 1.445%.

4.3 Numerical results and discussion

(1) Local model Equivalent strain e2eq +6.500e-03 +5.954e-03 +5.408e-03 +4.862e-03 +4.317e-03 +3.771e-03 +3.225e-03 +2.679e-03 +2.133e-03 +1.587e-03 +1.042e-03 +4.958e-04 -5.000e-05

97

Damage variable d2 +5.500e-01 +4.933e-01 +4.367e-01 +3.800e-01 +3.233e-01 +2.667e-01 +2.100e-01 +1.533e-01 +9.667e-02 +4.000e-02 -1.667e-02 -7.333e-02 -1.300e-01

(A) (2) Nonlocal model Equivalent strain e2eq +6.500e-03 +5.954e-03 +5.408e-03 +4.862e-03 +4.317e-03 +3.771e-03 +3.225e-03 +2.679e-03 +2.133e-03 +1.587e-03 +1.042e-03 +4.958e-04 -5.000e-05 -5.918e-05

Damage variable d2 +5.500e-01 +4.933e-01 +4.367e-01 +3.800e-01 +3.233e-01 +2.667e-01 +2.100e-01 +1.533e-01 +9.667e-02 +4.000e-02 -1.667e-02 -7.333e-02 -1.300e-01

(A) FIG. 4.16

Contours of the equivalent strain and damage variable of the matrix for the [0°/90°]4s laminate with a 10 mm-diameter hole using fine mesh model at the strain 1.445%.

by the reduction in the intensity of damage within the localized area. In addition, selected nonlocal variables, i.e., equivalent strains for fiber and matrix which control the damage evolution and strain softening are also demonstrated to be suitable for nonlocal averaging. The effect of different viscous constants c in Eq. (4.15) on the load responses for two specimens is shown in Fig. 4.17. In general, c ¼ (1E
3)
(5E
3) is acceptable for achieving excellent convergence and numerical accuracy. Figs. 4.18 and 4.19 show the delamination results at the final failure strain using different cohesive strengths for two specimens. Here, delamination along the load direction is detected by the dislocation of pair nodes at the same position. The delamination initiation and growth concentrate mainly on the area near the hole. Besides, predicted delamination using the cohesive model becomes more difficult when the normal and tangential cohesive strengths σ max and τmax increase from 5 to 20 MPa. When σ max and τmax exceed 20 MPa, no delamination appears. In addition, it is found the load-displacement curves are the same for the same specimen and mesh model when σ max and τmax exceed 1 MPa. This shows the developed nonlocal model can eliminate the mesh dependency without considering delamination. The convergence difficulty arises when σ max and τmax increase, similar to the conclusions obtained by Turon et al. [48] and Liu et al. [38]. Compared with the specimen with a 10 mm-diameter hole, the specimen with a 5 mm-diameter hole leads to more distinct delamination because of more severe deformation mismatch between the 0° plies and 90° plies near the hole, similar to the conclusion obtained by Forghani et al. [37].

98

4. Localized damage models and implicit finite element analysis

800

800 750 700

600 650

Stress (MPa)

1.4

1.5

1.6

1.7

Results for the specimens with a (1) 5mm-diameter hole Test result (49) Viscous constant c=0.01 Viscous constant c=0.005 Viscous constant c=0.001 (2) 10mm-diameter hole Test result (49)

400

200

Viscous constant c=0.01 Viscous constant c=0.005 Viscous constant c=0.001 0 0.0

0.4

0.8

1.2

1.6

2.0

Strain (%)

FIG. 4.17 Stress-strain curves of two [0°/90°]4s specimens using different viscous constants using fine mesh model.

FIG. 4.18 Delamination of the [0°/90°]4s laminate with a 5 mm-diameter hole using (1) coarse mesh model and (2) fine-mesh model at the strain 1.557% and cohesive strengths (A) 5 MPa, (B) 10 MPa, and (C) 20 MPa, respectively.

4.4 Concluding remarks

99

FIG. 4.19 Delamination of the [0°/90°]4s laminate with a 10 mm-diameter hole using (1) coarse mesh model and (2) fine-mesh model at the strain 1.445% and cohesive strengths (A) 5 MPa, (B) 10 MPa, and (C) 20 MPa, respectively.

In general, fiber breakage plays a dominating role in governing the tensile strength of open-hole composite laminates although matrix cracking and delamination decrease the stiffness and integrity of structures largely [63].

4.4 Concluding remarks Strain localization is commonly considered as a challenging issue in composite structures. Local FEA cannot predict the damage evolution behaviors and load-bearing abilities of composites accurately due to the lack of a length scale. This chapter introduces the nonlocal integral theory developed by Pijaudier-Cabot and Bazˇant [34] into the damage model to solve the localization problem of composites. Based on the variational form of the stress boundary value problem with an interface discontinuity, this chapter derives a finite element model on the nonlocal intralaminar damage and interlaminar delamination of composite laminates, where two length scales for fiber and matrix are introduced. It should be emphasized that nonlocal averaging on the derivatives of equivalent strains with respect to the strain components for fiber and matrix here ultimately contributes to the implicit nonlocal consistent tangent stiffness. The developed nonlocal finite element model is implemented by combining ABAQUS-UMAT, UEL, USDFLD subroutines, and UEXTERNALD utility subroutine. For two [0°/90°]4s T700/8911 composite specimens with 5 mm- and 10 mm-diameter central holes respectively, the effects of different length scales on the load responses are comparatively studied. By

100

4. Localized damage models and implicit finite element analysis

comparing the local and nonlocal results in terms of load-displacement curves and damage evolution behaviors, introduced length scales play an important role in predicting localized damage evolution behaviors of composites accurately after eliminating spurious mesh dependence. Of course, there may be some disadvantages for the predictions of damage evolution distributions for fiber and matrix for discontinuous composite structures since numerical results are obtained from only phenomenological description. The developed finite element model and numerical codes by ABAQUS as robust and reliable computational technique will be further applied to a series of composite structures with a variety of geometry discontinuities including holes, notches, and cut-outs. The localization problem for composite structures that is expected to be fundamentally solved or discussed using nonlocal implicit FEA would have to resort to high-performance computation, except for the requirement for fine treatment of boundary conditions.

A Appendix: Nonlocal consistent tangent stiffness tensor KL for bulk elements Jira´sek and Patza´k [64] derived an isotropic consistent tangent stiffness for damaged concrete materials. In the following, we further derive an anisotropic nonlocal consistent tangent stiffness for damaged composites. The discrete incremental form of stress tensor Δσ is given by X X   X   Δσ X p ¼ Cd Δε5 CdðpÞ B X p Δu (A.1) ~ p p p where Cd( p) ¼ Cd(C, 1
di( p)) (i ¼ 1, 2) is defined and the strain matrix B(Xp) is simplified as Bp. ωp is the weight of the integration point Xp. The element stiffness tensor KL is derived by 0 1 ð nbulk ∂C @BT Cd B + BT d BΔuAdV KL ¼ A e¼1 Ω + [Ω
∂Δu ~ ~ 2 3 (A.2) X nbulk X ∂C d ¼ A4 Bp Δu5 ωp BTp CdðpÞ Bp + ωp BTp e¼1 ∂Δu ~ p p ~ where it is noted the last term in Eq. (A.2) includes two parts for fiber and matrix. From Eq. (4.2), differentiating the damaged elastic tensor Cd with respect to the node displacement increment Δu leads to ~

101

A Appendix: Nonlocal consistent tangent stiffness tensor KL for bulk elements

8 eq eq ∂Cd ∂Cd ∂d1 ∂κ1 ∂ε1 ∂Cd ∂d2 ∂κ 2 ∂ε2 > > ¼ + , > > ∂Δu ∂d1 ∂κ 1 ∂εeq ∂Δu ∂d2 ∂κ2 ∂εeq ∂Δu > > 1 2 > ~ ~ ~ > > > f > c > ε ∂d1t ε1t > > ¼ f 1t , Fiber tensile failure ε11 > 0, > > 2 0 > ∂κ 1 ð κ > ε1t
ε1t 1 Þ > > > > f > ε1c > ∂d1c εc1c > > ¼ , Fiber compressive failure ε11 < 0, > > ∂κ 1 εf
ε0 ðκ 1 Þ2 > > 1c 1c > > > f < ∂d ε εc2t 2t ¼ f 2t , Matrix tensile failure ε22 + ε33 > 0, > ∂κ2 ε
ε0 ðκ 2 Þ2 > 2t 2t > > > > f > ε2c ∂d2c εc2c > > > ¼ , Matrix compressive failure ε22 + ε33 < 0, > f > ∂κ 2 ε
ε0 ðκ 2 Þ2 > > 2c 2c > > eq  eq   eq  > > ∂εq X ∂ε1 X ∂ε1 ∂ε1 > > > ¼ ¼ ω α ω α Bq , q pq q pq > > ∂Δ u ∂ε ∂Δ u ∂ε q > q q q > > ~ ~ > >  eq   eq  eq > X X > ∂ε ∂ε ∂ε ∂ε2 q > 2 2 > ¼ ωq αpq ωq αpq Bq , > : ∂Δu ¼ ∂ε ∂Δ u ∂ε q q q q ~ ~ where

∂κ 1 eq ∂ε1

¼ 1 and

∂κ 2 eq ∂ε2

¼ 1 represent loading, and

∂κ 1 eq ∂ε1

¼ 0 and

∂κ2 eq ∂ε2

(A.3)

¼ 0 repre-

sent unloading. The nonlocal derivatives of the local equivalent strains eq εeq 1 and ε2 with respect to the strain ε are derived by Eq. (4.11) 8 eq eq eq eq ∂ε1   X ∂ε1   ∂ε2   X ∂ε2   > > X X X Xq , ω α ω α ¼ , ¼ > p q pq q p q pq > ∂ε ∂ε ∂ε > < ∂ε q q

 

(A.4) α0 X p
X q

> > X



¼ α   > pq > > wr α0 X p
X r

: r

Based on Eqs. (4.3)–(4.6), the differentiation of the equivalent strains εeq 1 and εeq 2 with respect to the components of strain tensor ε leads to the following expressions (a) Fiber tensile failure ε11 > 0 "  0 2  0 2 #
12 eq ∂ε1 2 2 ε1t 2 ε1t ¼ ε11 ðε11 Þ + ðε12 Þ + ðε13 Þ ∂ε11 ε012 ε013  0 2 "  0 2  0 2 #
12 eq ∂ε1 ε1t ε ε ¼ ε12 0 ðε11 Þ2 + ðε12 Þ2 01t + ðε13 Þ2 01t ∂ε12 ε12 ε12 ε13

(A.5)

(A.6)

102

4. Localized damage models and implicit finite element analysis

 0 2 "  0 2  0 2 #
12 eq ∂ε1 ε1t 2 2 ε1t 2 ε1t ¼ ε13 0 ðε11 Þ + ðε12 Þ + ðε13 Þ ∂ε13 ε13 ε012 ε013

(A.7)

(b) Fiber compressive failure ε11 < 0  eq   ∂ε1    ∂ε  ¼ 1 11 (c) Matrix tensile failure ε22 + ε33 > 0  0 2 "  0 2   eq ∂ε2 ε ε ε22 ε33 ¼ ε12 02t ðε22 + ε33 Þ2 + 02t ε223
2 E2 E3 ∂ε12 G23 ε12 ε23 1 #  0 2  0 2
2 2 ε2t 2 ε2t + ðε13 Þ +ðε12 Þ ε012 ε013  0 2 "  0 2   eq ∂ε2 ε ε ε22 ε33 ¼ ε13 02t ðε22 + ε33 Þ2 + 02t ε223
2 E2 E3 ∂ε13 G23 ε13 ε23 1 #  0 2  0 2
2 ε ε +ðε12 Þ2 02t + ðε13 Þ2 02t ε12 ε13

(A.8)

(A.9)

(A.10)

8 2 !2 3 9 2 !2 ! eq = 0 ∂ε2 < ε ε02t E E ε22 ε33 2 3 2 2 2t 4 4 5 ε33  ðε22 + ε33 Þ + 0 ¼ ε22 + 1
2 ε23
2 E2 E3 ; ∂ε22 : G23 ε023 G23 ε23 !2 !2 3
12 0 0 2 ε2t 2 ε2t 5 +ðε12 Þ + ðε13 Þ ε012 ε013 (A.11) !2 2 !2 ! eq ∂ε2 ε02t 4 ε02t ε22 ε33 2 2 ¼ ε23 0 ðε22 + ε33 Þ + 0 ε23
2 E2 E3 ∂ε23 G23 ε23 ε23 1 !2 !2 3
2 0 0 2 ε2t 2 ε2t 5 +ðε12 Þ + ðε13 Þ ε012 ε013

(A.12)

8 2 ! 3 9 2 !2 ! eq = 0 2 ∂ε2 < ε ε02t E E ε22 ε33 2 3 2 2 2t 4 4 5 ε22  ðε22 + ε33 Þ + 0 ¼ ε + 1
2 ε23
2 E2 E3 ; ∂ε33 : 33 G23 ε023 G23 ε23 + ðε12 Þ2

ε02t ε012

!2 + ðε13 Þ2

ε02t ε013

!2 3
12 5

(A.13)

B Appendix: Stiffness matrix Kc and internal force Fc for cohesive elements

(d) Matrix compressive failure in the through-thickness direction ε22 + ε33 < 0   eq   0  ∂ε2  2ε2c ε12    ∂ε  ¼   0 2  12 ε12    eq   0  ∂ε2  2ε2c ε13    ∂ε  ¼   0 2  13 ε13    eq   0  2  0  ∂ε2  ε2c E2 ðε22 E2 + ε33 E3 Þ ε02c ε33 E2 E3 E ε  2 2c  ¼
+
1      ∂ε   0 0 2 0 2  2G ε 22 12 12 2 G12 ε12 G23 ε23   eq   0  ∂ε2  2ε2c ε23    ∂ε  ¼   0 2  23 ε23    eq   0  2  0  ∂ε2  ε2c E3 ðε22 E2 + ε33 E3 Þ ε02c ε22 E2 E3 E ε  2 2c  ¼
+
1      ∂ε   0 0 2 0 2  2G ε 33 12 2 G12 ε12 G23 ε23 12

103

(A.14)

(A.15)

(A.16)

(A.17)

(A.18)

Substituting Eq. (A.3) into Eq. (A.2) leads to the weighted nonlocal consistent tangent stiffness KL 8

!    eq ! ∂ε1 ∂Cd ∂d1 ∂κ1 KL ¼ A Bq Δ u ∂d1 p ∂κ1 p ∂εeq ∂ε e¼1 : p ~ p, q q 1 p 9 !     eq ! = X ∂ε2 ∂Cd ∂d2 ∂κ2 + ωp ωq αpq BTp Bp Bq Δ u eq ∂d ∂κ ~; 2 p 2 p ∂ε2 p ∂ε q p, q nbulk = < ½½u1 ðξ, ηÞ > (B.4) ½½uðξ, ηÞ ¼ ½½u2 ðξ, ηÞ ¼ H ðξ, ηÞ½½uN  ¼ H ðξ, ηÞϕu ¼ Lu > > ~ ~ ; : ½½u3 ðξ, ηÞ where L is a 3  24 matrix and
1  ξ, η  1 are the local coordinates of elements. ϕ ¼ (I1212 j I1212) is a 12  24 matrix. The node force vector Fc(24  1 matrix) and the stiffness matrix Kc(24  24 matrix) are written as ðð ðð XX T T F c ¼ L φ TdA ¼ M T TdA ¼ ωi ωj M T T jJ j, ðð

ðð

i

j

ðð ∂F c ∂T ∂½½u T ∂T T ∂T ∂½½u Kc ¼ ¼ M dA ¼ M dA ¼ M T dA ∂u ∂u ∂½½u ∂u ∂½½u ∂u ~ ~ ~ ~ ðð XX ∂T MdA ¼ ωi ωj M T KM jJ j ¼ MT ∂½½u i j

(B.5)

where jJj is the Jacobian of isoparametric transformation and ω is the weight. M is a 3  24 shape function matrix for the cohesive element and K is a 3  3 cohesive stiffness matrix in which diagonal values are K in Eq. (4.8).

References

105

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C H A P T E R

5

Implicit finite element analysis of postbuckling and delamination of symmetric and unsymmetric composite laminates under compression using virtual crack closure technique 5.1 Introduction When composites are applied to such as stiffened laminates [1], they are often subjected to compressive loads, where complicated interactions between buckling, postbuckling, and delamination growth affect the mechanical properties of structures largely. The stability issue has become an important topic for the analysis and design of composite laminates [2–4]. Conventional stability-based design criteria are simply based on the linear-elastic theory, i.e., the initial unstability behavior due to the limitation of immature design and calculation tools. However, experiments [5] showed composite structures still exhibit high load-bearing abilities when they enter into the postbuckling unstability stage, which enables it insufficient to consider only the buckling and delamination behaviors, without taking full use of the relatively long postbucking load-bearing stage. With the development of numerical methods and computational ability, the nonlinear analysis including the buckling, postbuckling, and

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00016-X

109

Copyright © 2021 Elsevier Inc. All rights reserved.

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5. Implicit finite element analysis of postbuckling

delamination failure is increasingly becoming an indispensable means to gaining insight into the stability mechanisms that help to improve the design efficiency of composite structures to a large extent. Currently, there is already a lot of theoretical and experimental work which focuses on the buckling, postbuckling, and delamination mechanisms of composite structures. Generally, the unstability modes under compressive loads can be divided into two categories: local buckling and global buckling [6]. A sublaminate may locally buckling and impose the additional bending stresses on neighboring sublaminates, and the unstable delamination is followed by global buckling for composite laminates. Chai et al. [7, 8] established the 1D and 2D delamination buckling models by evaluating the energy release rate (ERR) for the delamination crack propagation. Kardomateas and Schmueser [9], Kutlu and Chang [10], and Whitcomb [11] and studied the effects of delamination growth on the stability of composite laminates by developing 2D and 3D analytical models. Bolotin [12] and Singh and Kumar [13] explored the delamination buckling mechanisms by considering the interactions between the local buckling, damage accumulation, crack growth, and global buckling. For the through-the-width multiple delamination, Hu [14], Hwang and Liu [15], Cappello and Tumino [16], Karihaloo and Stang [17], Suemasu et al. [18], Wang and Zhang [19], Wang et al. [20], Aslan and Sahin [21], and Liu et al. [22, 23] studied the effects of nonlinear deformations on the buckling and postbuckling behaviors. Besides, Han et al. [24] studied the combination of different load types on the postbuckling behavior of composite laminates. However, most of these work was concentrated on the symmetric structures. The widely used unsymmetric composite structures inevitably add the complexity of the delamination unstability mechanisms and the corresponding design. It can be imagined such as the distributions of geometry sizes can easily lead to the unsymmetry of structure. Furthermore, it is important to predict the delamination mechanisms of symmetric and unsymmetric composite laminates under compressive loads by analyzing the evolvement of ERR for crack propagation in order to study the interactions between the buckling, postbuckling, and multiple delamination. This chapter designs a generic 3D laminated composite specimens with different initial delamination sizes and their distributions in order to study the effects of the unsymmetry on the load response, and delamination and postbuckling behavior. The virtual crack closure technique (VCCT) is used to predict the delamination growth and to calculate the ERR. Compared with symmetric composite laminates, unsymmetric laminates show different unstable delamination behavior from symmetric laminates when the initial delamination sizes are small.

111

5.2 VCCT for predicting the delamination growth

5.2 VCCT for predicting the delamination growth Rybicki and Kanninen [25] proposed a crack closure integral-VCCT to calculate the ERR. The only assumption of VCCT is the energy needed for the crack propagation length Δa is equal to that for closing two separate crack surfaces with the crack length Δa. Krueger [26] gave a comprehensive review on the VCCT and pointed out there are three main advantages using the VCCT: ① it is applicable to 3D structures, ② the requirement for the mesh is low, and ③ the consideration for the crack-tip singularity can be avoided. Xie and Biggers [27] developed an interface element to implement VCCT. Fig. 5.1 shows crack propagation from the crack tip node i to j with the incremental crack length Δa. Node i is separated into two nodes i1 and i2 after crack propagation. For node i, the relative displacements in three directions (x, y, z) are Δuix, Δuiy, and uiz after propagation and the node forces before propagation are Fix, Fiy, and Fiz. The ERR due to crack propagation is given by [26] G¼

3 X

GI

I¼1

¼ lim

Δa!0

1 2S

ð Δa 0

Fix ðaÞΔuix ðaÞda +

ð Δa 0

Fiy ðaÞΔuiy ðaÞda +

ð Δa 0

 Fiz ðaÞΔuiz ðaÞda (5.1)

FIG. 5.1 2D Schematic representation of crack propagation between two layers using virtual crack closure technique.

112

5. Implicit finite element analysis of postbuckling

where GI(I ¼ 1, 2, 3) are the ERR for three failure modes, and S is the new area generated due to the crack propagation length Δa. Assume a two-step method is used by using the node force before crack propagation and node relative displacement after crack propagation, the G due to crack propagation on the crack closure surface is calculated as  1 G¼ Fix Δuix + Fiy Δuiy + Fiz Δuiz (5.2) 2S For VCCT, the two-step method can be approximately substituted by the one-step method if the mesh sizes on the crack closure surface are sufficiently small. The relative displacement after crack propagation can be substituted by the relative displacements between nearest node pairs before crack propagation. For example, the relative displacements in three directions for the node pair i1 and i2 can be approximately substituted by those for the node pair e1 and e2. In this analysis, the one-step method is used for the calculation of ERR. The delamination crack propagates if the following two delamination criteria are satisfied. ① B-K law [28]  η GII + GIII (5.3) G  GC ¼ GIC + ðGIIC  GIC Þ GI + GII + GIII ② Power-law



GI GIC

α     GII α GIII α + + 1 GIIC GIIIC

(5.4)

where GIC, GIIC, and GIIIC are critical ERR for different fracture modes. G and GC are the equivalent and critical ERR, respectively. η and α are constants.

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates 5.3.1 Composite specimens The through-the-width delamination of composite flat laminates is considered. Material parameters for T300/976 composites are listed in Table 5.1 [29]. η ¼ 2 in Eq. (5.3) and α ¼ 1 in Eq. (5.4) are used. As shown in Fig. 5.2, unsymmetric composite specimens are designed based on Kutlu and Chang’s experiments [29], including two angle-ply configurations 04//( θ)6//04 (θ ¼ 0° and 45°, and “//” denotes the delaminated interface). Case-a, c, e, and g belong to symmetric structures and Caseb, d, f, and h are unsymmetric structures. Initial delamination sizes and their distributions for eight cases are listed in Table 5.2. The number of finite elements is 40,000.

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates

113

TABLE 5.1 Material parameters [29]. Ply longitudinal modulus

E1

139.3 GPa

Ply transverse modulus

E2

9.72 GPa

Out-of-plane modulus

E3

9.72 GPa

Inplane shear modulus

G12

5.58 GPa

Out-of-plane shear modulus

G13

5.58 GPa

G23

3.45 GPa

v12

0.29

v13

0.29

v23

0.40

Interlaminar fracture toughness for mode-I

GIC

0.0876 N/mm

Interlaminar fracture toughness for mode-II

GIIC

0.3152 N/mm

Interlaminar fracture toughness for mode-III

GIIIC

0.3152 N/mm

Poisson’s ratio

y

Z

Clamped end

Clamped

X end

h c3 c1

a2 a1

c4

B c2

A FIG. 5.2

Unsymmetric multiple delamination of composite laminates under compressive

strain.

5.3.2 Numerical technique for buckling and delamination analysis Fig. 5.3 shows 3D finite element model with boundary conditions. The FEA of delamination buckling and postbuckling uses ABAQUS software, including the initial buckling analysis and subsequent postbucking analysis. In the buckling analysis, the multipoint constraint is used to tie the nodes at the same position on delaminated surface. Fig. 5.4 shows the

114

5. Implicit finite element analysis of postbuckling

TABLE 5.2 Geometry sizes (h ¼ 2.54 mm, A ¼ 50.8 mm and B ¼ 5.08 mm are used for all ages). Case

layup

a1 (mm)

a2 (mm)

c1 (mm)

c2 (mm)

c3 (mm)

c4 (mm)

a

04//012//04

38.1

19.05

6.35

6.35

15.875

15.875

b

04//012//04

38.1

19.05

6.35

6.35

19.05

12.7

c

04// (45)6//04

38.1

19.05

6.35

6.35

15.875

15.875

d

04// (45)6//04

38.1

19.05

6.35

6.35

19.05

12.7

e

04//012//04

25.4

12.7

12.7

19.05

19.05

f

04//012//04

25.4

12.7

19.05

25.4

12.7

g

04// (45)6//04

25.4

12.7

12.7

19.05

19.05

h

04// (45)6//04

25.4

12.7

19.05

25.4

12.7

12.7 6.35 12.7 6.35

(a) Symmetric laminates

y

Delaminated surface

z x

Axial displacement (x direction) Constraints (y and z direction)

Axial displacement (x direction) Constraints (y and z direction)

(b) Unsymmetric laminates

y

Delaminated surface

z x

Axial displacement (x direction) Constraints (y and z direction)

Axial displacement (x direction) Constraints (y and z direction)

FIG. 5.3 Parametric finite element model of (A) symmetric and (B) unsymmetric composite laminates with boundary conditions (element side length ¼ 50.8/100 mm).

buckling modals for symmetric and unsymmetric laminates. The buckling analysis is to get the modals (eigenvalues) using the perturbation method when the stiffness matrix is singular, and introduces an initial imperfection for the postbuckling analysis by multiplying a very small value on the first-order buckling modal. The postbuckling analysis employs the master-slave node technique, in which the slave nodes on the slave surface are to be debonded from those on the master surface. The contact relationships are set for the delaminated surface to prevent the interpenetration.

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates

FIG. 5.4

115

Buckling modes for (A) symmetric and (B) unsymmetric composite laminates.

The delamination starts if the delamination criterion above is satisfied within a given tolerance. The geometrically nonlinear finite element equations for the postbuckling analysis using implicit Newton-Raphson algorithm are given by KT u ¼ ΔF K T ¼ K 0 + KL + K σ ,

(5.5)

where the stiffness matrix KT includes the elastic stiffness K0, geometry stiffness KL, and large displacement stiffness Kσ . ΔF is the residual force vector. The viscous damping technique is used to improve the numerical convergence and the line search strategies are used to accelerate the convergence. Very small time incrementation is used to ensure the numerical stability and accuracy (the initial, maximum, and minimum time increments are 0.001 s, 1e 6 s, and 0.005 s, respectively).

5.3.3 Numerical results for a1 5 38.1 mm and a2 5 19.05 mm laminates Fig. 5.5 shows the axial compressive load-strain curves and Fig. 5.6 shows the compressive load-central deflection curves. It can be seen two delamination criteria lead to basically consistent results. The power-law fails to accurately capture the dependence of the mixed-mode fracture toughness on the mode ratio. In contrast, the difficulty using the B-K law is to determine the coefficient η in Eq. (5.4) which yet depends on the mode-mixing and the phase angle for different ERR. Turon et al. [30] explored the effect of different mode ratios on the load-displacement curves and got η ¼ 2.284 in Eq. (5.3) for AS4/PEEK composites. We find the load response does not change largely when as η varies from 1.5 to 2.5. Thus, η ¼ 1.75 is used.

116

5. Implicit finite element analysis of postbuckling

FIG. 5.5 Axial compressive load-strain curves for composite laminates with a1 ¼ 38.1 mm and a2 ¼ 19.05 mm (local buckling, initial delamination, unstable delamination, and global buckling are highlighted).

For the symmetric Case-a laminate, the calculated local buckling strain is about 5.1  104, is in good agreement with the result 5.6  104 by Wang and Zhang [19]. When the strain increases to 2.2  103, the upper sublaminate starts to debond. When the strain increases to 3.1  103, the delamination crack length for the upper sublaminate adds from the initial length a1 ¼ 38.1 mm to about 42 mm, and at this time the bottom sublaminate starts to debond, and unstable snap-back delamination appears which arises from the complicated interactions between the delamination and buckling. After that, the bottom sublaminate delaminates in a higher crack growth rate than the upper sublaminate. For the unsymmetric Case-b laminate, the structure show a slightly different response at the unstable delamination stage. Figs. 5.7 and 5.8 show the delamination growth process for the Case-c and d laminates, respectively. The angle-ply Case-c and Case-d laminates show weaker load-bearing abilities than the unidirectional Case-a and b laminates. Local buckling for the Case-c and d laminates appears at the strain about 5.07  104, slightly smaller than 5.1  104 for the Case-a laminate. For the Case-d laminate, at about 2.0  103 strain, the upper sublaminate starts to debond and then unstable delamination appears, earlier than the about 2.2  103 strain for the Case-a and b laminates. Then,

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates

117

FIG. 5.6 Compressive load-central deflection curves for (A) Case-a and c and (B) Case-b and d laminates.

118

5. Implicit finite element analysis of postbuckling

FIG. 5.7 Delamination growth process of Case-c composite laminate at the compressive strain (A) 2.05  103, (B) 2.2  103, (C) 2.35  103, and (D) 7.87  103 using B-K law.

FIG. 5.8 Delamination growth process of Case-d composite laminate at the compressive strain (A) 2.0  103, (B) 2.1  103, (C) 2.2  103, and (D) 7.87  103 using B-K law.

unstable delamination appears before the delamination initiation of bottom sublaminate at about 2.2  103 strain. At this time, the crack length for the upper sublaminate increases to about 42 mm. Consistent response appears for the Case-c and d laminates which are almost not affected by the unsymmetry. From about 2.0  103 to 7.87  103 strain, the load response experiences very long unstable stage before the global buckling

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates

119

TABLE 5.3 Local buckling load, initial delamination load and global buckling load for eight cases. Case

Local buckling load (N)

Initial delamination load (N)

Global buckling load (N)

a

493

1971

1659

b

462

1835

1650

c

451

945

463

d

400

935

462

e

900

1750

1539

f

884

1628

1536

g

810

1140

462

h

750

1092

458

because of the layup effect, distinctly different from the Case-a and b laminates. This shows more complicated unstable delamination behavior for the angle-ply Case-c and d laminates than the unidirectional Case-a and b laminates. From Table 5.3, the initial delamination load, and local and global buckling loads for the Case-c and d laminates are smaller than those for the Case-a and b laminates. The global buckling load 1659 N for the Case-a laminate is in good agreement with 1656 N obtained by Wang and Zhang using the finite strip method [19]. The delamination growth process can also be described by the ERR at the crack-tip. For the Case-d laminate, at the 2.0  103 strain, the delamination crack on the slave surface for the upper sublaminate starts to propagate, and the mode-I and mode-II ERR reaches 0.19 N/mm and 4.5  102 N/mm respectively. At the 2.2  103 strain, the increasing ERR drives the bottom sublaminate for the Case-d laminate to debond. At this time, the mode-I and mode-II ERR at the crack-tip for the upper sublaminate increases to 0.2 and 5.0  102 N/mm, and reaches 9  102 N/mm and 2.2  102 N/mm for the bottom sublaminate. By comparing with the interlaminar fracture toughness, the mode-I failure is the main form at the delamination initiation stage for both the upper and bottom sublaminates. During the second unstable delamination stage at the strain 5.8  102 strain, the mode-I, and mode-II ERR at the crack-tip for the upper sublaminate increases to 0.23 and 0.83 N/mm, and reaches 0.24 and 0.29 N/mm for the bottom sublaminate. At the 7.87  103 strain for the global bucking stage, the mode-I and mode-II ERR is 0.23 and 0.84 N/mm, and 0.24 and 0.36 N/mm for the upper and bottom sublaminates, respectively. This shows the dominating mode-I fracture mode develops gradually to mixed-mode fracture modes for the delamination

120

5. Implicit finite element analysis of postbuckling

growth, and the mode-II ERR gradually governs the delamination growth with the increase of strain. By comparison, the upper and bottom sublaminates for the Case-c laminate almost start to propagate simultaneously at the 2.05  103 strain, at which the mode-I and mode-II ERR is 0.2 and 4.2  102, 8.8  102 and 1.6  102 N/mm for the upper and bottom sublaminates, respectively. From the 5.8  103 to 7.87  103 strain, the mode-I and mode-II ERR is 0.22 and 0.93 N/mm, and 0.24 and 0.3 N/mm for the upper and bottom sublaminates, respectively. Besides, the mixed-mode delamination failure always exists from the delamination initiation of the unidirectional Case-a and b laminates because the mode-II ERR is always larger than the mode-I ERR for the delamination growth of the upper and bottom sublaminates.

5.3.4 Numerical results for a1 5 25.4 mm and a2 5 12.7 mm laminates Fig. 5.9 shows the axial compressive load-strain curves and Fig. 5.10 shows the compressive load-central deflection curves using B-K law. Compared with the unidirectional laminates, the angle-ply laminates show weaker load-bearing abilities and more unstable delamination behavior. Local buckling strain 1.1  103 for the unsymmetric Case-f and h laminates is slightly larger than 1.0  103 for the symmetric Case-e and g laminates. The initial delamination for the upper and bottom sublaminates at the 1.5  103 and 3.5  103 strains for the unsymmetric Case-h laminate is slightly earlier than those at the 1.7  103 and 3.8  103 strains for the symmetric Case-f laminate. Similar to the a1 ¼ 38.1 mm and a2 ¼ 19.05 mm laminates, the immediate unstable delamination growth appears between the delamination initiation of upper and bottom sublaminates. Furthermore, the angle-ply layup aggravates the delamination unstability behavior. However, the a1 ¼ 25.4 mm and a2 ¼ 12.7 mm laminates show stronger load-bearing abilities than a1 ¼ 38.1 mm and a2 ¼ 19.05 mm laminates, which can be explained as the requirement for more energy to drive the crack propagation. Compared with the symmetric laminates, the unsymmetric laminates show slightly weaker load-bearing abilities after initial delamination when the initial delamination sizes are small. This arises mainly from the smaller delamination resistance for the unsymmetric laminates than that for the symmetric laminates. For the unsymmetric Case-f and h laminates, the delamination crack propagates for both the upper and bottom sublaminates more quickly towards the negative x direction than towards the positive x direction. As listed in Table 5.3, the a1 ¼ 25.4 mm and a2 ¼ 12.7 mm cases show similar relationships between the symmetric and unsymmetric laminates for the initial buckling loads, initial delamination loads, and

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates

121

FIG. 5.9 Axial compressive load–strain curves for composite laminates with a1 ¼ 25.4 mm and a2 ¼ 12.7 mm using B-K law (local buckling, initial delamination, unstable delamination, and global buckling are highlighted).

global buckling loads. Local buckling loads and initial delamination loads for the unsymmetric laminates are slightly smaller than those for the symmetric laminates, but the global buckling loads are almost the same. Figs. 5.11 and 5.12 show the delamination growth process for the Case-g and h laminates. The delamination growth process is also explained by the ERR. For the unsymmetric Case-h laminate, at the 1.5  103 strain, the delamination crack on the slave surface for the upper sublaminate starts to propagate, and the mode-I and mode-II ERR reaches 0.1 N/mm and 3.8  102 N/mm, respectively, close to the corresponding values 9.4  102 and 2.5  102 N/mm at the 1.7  103 strain for the symmetric Case-g laminate. At the 3.1  103 strain for unstable delamination, the mode-I and II ERR increases to 0.24 and 8.0  102 N/mm, and 2.1  102 and 4.5  102 N/mm for the upper and bottom sublaminates respectively for the Case-h laminate, and 0.24 and 7.95  102 N/mm for the upper sublaminate and 1.08  102 and 2.36  103 N/mm for the bottom sublaminate for the Case-g laminate. At the 3.5  103 and 3.8  103 strains respectively, the increasing ERR drives the bottom sublaminate for the Case-h and g laminates to debond. At this time, the mode-I and modeII ERR at the crack-tip for the upper sublaminate increases to 0.24 and 0.12 N/mm, and reaches 0.13 and 0.8 N/mm for the bottom sublaminate for the Case-h laminate. In contrast, the mode-I and mode-II ERR for

122

5. Implicit finite element analysis of postbuckling

FIG. 5.10 Compressive load-central deflection curves for (A) Case-e and g and (B) Case-f and h laminates.

the upper and bottom sublaminates for the Case-g laminate is 0.24 and 0.1 N/mm, and 0.24 and 0.29 N/mm, respectively. This shows the same conclusion to the Case-d laminate that the delamination evolves gradually from the dominating mode-I failure to mixed-mode failure. By comparison, the mode-I and II ERR changes from 0.24 and 0.27 N/mm at the initial

5.3 Implicit FEA of buckling, postbuckling, and delamination growth of composite laminates

123

FIG. 5.11 Delamination growth process of symmetric Case-g composite laminate at the compressive strain (A) 1.7  103, (B) 1.9  103, (C) 3.8  103, and (D) 7.87  103 using B-K law.

FIG. 5.12 Delamination growth process of unsymmetric Case-h composite laminate at the compressive strain (A) 1.5  103, (B) 1.9  103, (C) 3.5  103, and (D) 4.4  103 using B-K law.

delamination of the upper sublaminate to 0.24 and 0.53 N/mm, and 0.16 and 0.19 N/mm at the initial delamination of the bottom sublaminate, and to 0.24 and 0.69 N/mm, and to 0.25 and 0.68 N/mm at the global buckling stage for the Case-f laminate. In general, more energy is needed to drive the mixed-mode delamination crack propagation for the unidirectional laminates. Finally, the global buckling strain 4.4  103 for the

124

5. Implicit finite element analysis of postbuckling

unsymmetric Case-h laminate is smaller than 7.87  103 for the symmetric Case-g laminate.

5.4 Concluding remarks 3D implicit FEA of buckling, postbuckling, and through-the-width multiple delamination of composite laminates are performed to study the delamination mechanisms and load-strain response. VCCT is used to predict the delamination crack propagation by calculating the ERR for different fracture modes. Eight composite specimens are designed to study the mechanical properties of the unidirectional and angle-ply laminates, and the symmetric and unsymmetric laminates. From numerical results, the following conclusions are obtained: (1) The unsymmetry affects the delamination and buckling behavior of composite laminates largely only when the initial multiple delamination sizes are relatively small. (2) Compared with the symmetric laminates, the unsymmetry arising from the distributions of initial multiple delamination cracks decreases the local buckling load and initial delamination load slightly, but does not affect the global buckling load. (3) The angle-ply laminates show more complicated unstable delamination behavior than unidirectional laminates. In general, the unstable delamination stage that is largely affected by the postbuckling behavior appears between the delamination initiation of upper and bottom sublaminates. (4) For angle-ply laminates, the dominating mode-I failure model at the initial delamination stage evolves gradually to the mixed-mode failure from the unstable delamination stage to the global buckling stage, in contrast with the dominating mixed-mode failure from the beginning of delamination initiation for unidirectional laminates.

References [1] G€ urdal Z, Tatting BF, Wu CK. Variable stiffness composite panels: effects of stiffness variation on the in-plane and buckling response. Compos Part A 2008;39(5):911–22. [2] Craven R, Iannucci L, Olsson R. Delamination buckling: a finite element study with realistic delamination shapes, multiple delaminations and fibre fracture cracks. Compos Part A 2010;41(5):684–92. [3] Kim HJ, Hong CS. Buckling and postbuckling behavior of composite laminates with a delamination. Compos Sci Technol 1997;57(5):557–64. [4] Singh SB, Kumar A. Postbuckling response and strength of laminates under combined in-plane loads. Compos Sci Technol 1999;59(6):727–36.

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[5] Gu HZ, Chattopadhyay A. An experimental investigation of delamination buckling and postbuckling of composite laminate. Compos Sci Technol 1999;59(6):903–10. [6] Kim H, Kedward KT. A method for modeling the local and global buckling of delaminated composite plates. Compos Struct 1999;44(1):43–53. [7] Chai H, Babcock C, Knauss W. One dimensional modelling of failure in laminated plates by delamination buckling. Int J Solids Struct 1981;17(11):1069–83. [8] Chai H, Babcock C. Two-dimensional modelling of compressive failure in delaminated laminates. J Compos Mater 1985;19(1):67–98. [9] Kardomateas GA, Schmueser DW. Buckling and postbuckling of delaminated composites under compressive loads including transverse shear effects. AIAA J 1988;26 (3):337–43. [10] Kutlu Z, Chang FK. Modeling compression failure of laminated composites containing multiple through-the-width delamination. J Compos Mater 1992;26(3):350–86. [11] Whitcomb JD. Analysis of a laminate with a postbuckled embedded delamination, including contact effects. J Compos Mater 1992;26(10):1523–35. [12] Bolotin VV. Delamination in composite structures: its origin, buckling, growth and stability. Compos Part B Eng 1996;27(2):129–45. [13] Singh SB, Kumar A. Postbuckling response and failure of symmetric laminates under in-plane shear. Compos Sci Technol 1998;58(12):1949–60. [14] Hu HT. Influence of In-plane shear nonlinearity on buckling and postbuckling responses of composite plates and shells. J Compos Mater 1993;27(2):138–51. [15] Hwang SF, Liu GH. Buckling behavior of composite laminates with multiple delaminations under uniaxial compression. Compos Struct 2001;53(2):235–43. [16] Cappello F, Tumino D. Numerical analysis of composite plates with multiple delamination subjected to uniaxial buckling load. Compos Sci Technol 2006;66(2):264–72. [17] Karihaloo BL, Stang H. Buckling-driven delamination growth in composite laminates: guidelines for assessing the threat posed by interlaminar matrix delamination. Compos Part B Eng 2008;39(2):386–95. [18] Suemasu H, Irie T, Ishikawa T. Buckling and post-buckling behavior of composite plates containing multiple delamination. J Compos Mater 2009;43(2):191–202. [19] Wang S, Zhang Y. Buckling, post-buckling and delamination propagation in debonded composite laminates part: 2: numerical applications. Compos Struct 2009;88(1):131–46. [20] Wang XW, Pont-Lezica I, Harris JM, Guild FJ, Pavier MJ. Compressive failure of composite laminates containing multiple delaminations. Compos Sci Technol 2005;65 (2):191–200. [21] Aslan Z, Sahin M. Buckling behavior and compressive failure of composite laminates containing multiple large delaminations. Compos Sci Technol 2009;89(3):382–90. [22] Liu PF, Hou SJ, Chu JK, et al. Finite element analysis of postbuckling and delamination of composite laminates using virtual crack closure technique. Compos Struct 2011;93 (6):1549–60. [23] Liu PF, Zheng JY. On the through-the-width multiple delamination, and buckling and postbuckling behaviors of symmetric and unsymmetric composite laminates. Appl Compos Mater 2013;20(6):1147–60. [24] Han SC, Lee SY, Rus G. Postbuckling analysis of laminated composite plates subjected to the combination of in-plane shear, compression and lateral loading. Int J Solids Struct 2006;43(18–19):5713–35. [25] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 1977;9(4):931–8. [26] Krueger R. Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 2004;57(2):109–43. [27] Xie D, Biggers SB. Progressive crack growth analysis using interface element based on the virtual crack closure technique. Finite Elem Anal Des 2006;42:977–84.

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[28] Benzeggagh ML, Kenane M. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos Sci Technol 1996;56(4):439–49. [29] Kutlu Z, Chang FK. Composite panels containing multiple through-the-width delamination and subjected to compression part II: experiments and verification. Compos Struct 1995;31(4):297–314. [30] Turon A, Camanho PP, Costaa J, D’avila CG. A damage model for the simulation of delamination in advanced composites under variable-mode loading. Mech Mater 2006;38(11):1072–89.

C H A P T E R

6

Implicit finite element analysis of the influence of cohesive law parameters for delamination analysis of composite laminates under compression 6.1 Introduction In the last chapter, we introduce VCCT to predict the delamination and buckling behaviors of composite laminates. Compared with VCCT, CZM shows large advantages as it can predict both crack initiation and propagation for delamination and adhesive failure efficiently. Currently, there are a lot of CZMs in terms of the shape of traction-displacement jump curves. Unfortunately, there is still no conclusion on what kind of models the true interface is subjected to although various CZMs are increasingly used to predict the interface separation. Thus, an important issue emerges on how the shape and strength of CZM govern the progressive failure of the interface. Rots [1] showed that the shape of the strain-softening branch of the constitutive model is an important input parameter, which has a significant influence. Volokh [2] compared the bilinear, parabolic, sinusoidal, and exponential CZMs in simulating a rigid block-peel test and showed the shape of models has a significant influence on the numerical results. Chandra et al. [3] made a comprehensive review on a series of issues on the models and pointed out that the shape of interface models has a big influence on the load responses of composites. Alfano [4] compared the effects of the bilinear, linear-parabolic, exponential, and trapezoidal cohesive law shapes on the load-responses of a mode-I double cantilever beam. His conclusion was that whether or not the shape of the softening

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00010-9

127

Copyright © 2021 Elsevier Inc. All rights reserved.

128

6. Element analysis of the influence of cohesive law parameters

curve is an important parameter may depend on the boundary value problem, in particular on the ratio between the interface toughness and the stiffness of the bulk material. Song et al. [5] showed the influence of cohesive softening shape becomes significant as the relative size of the fracture process zone compared to the structure size increases for quasibrittle materials, while the cohesive parameters including material strength and cohesive fracture energy are considered more important than the cohesive softening shape. Turon et al. [6] and Harper et al. [7] used the bilinear CZM, and Goyal et al. [8] and Liu et al. [9] used the exponential CZM to study the effect of cohesive strength on the mixed-mode delamination of composite laminates in terms of the double cantilever beam (DCB), the end-notched flexure (ENF) test, and mixed-mode bending (MMB) specimens. Freed and Banks-Sills [10] studied the effect of cohesive strength on the load responses of composites by using the polynomial CZM proposed by Needleman [11]. The general conclusion is that the cohesive strength affects the softening part of the load-displacement curve represented by the unstable delamination behavior to a large extent. Campilho et al. [12] also showed that the composite joint with ductile adhesive is highly affected by different cohesive shapes including the triangular, exponential, and trapezoidal shapes. As a particular delamination case of composite laminates under compression, complicated interactions between the postbuckling and delamination affect the stiffness, stability, and strength largely. For the stability and delamination problems, the evolving profile of the true ductile/brittle fracture process zone is interrupted by the geometrically nonlinear deformation of structures. From the mathematical point of view, some cohesive laws in front of the crack tip governing the interface failure were measured well by experiments [13–15] provided that these CZMs with different shapes were superimposed representing the profile of the true physical process zone. However, it may be a more challenging task for measuring mixed-mode CZMs for the delamination growth under compression. Thus, a favorable and important work is to model the delamination growth under the interruption of stability factors by evaluating the effects of different modes including the cohesive shape and strength on the load responses of composites. Recently, Alfano et al. [16, 17] demonstrated by a global sensitivity analysis that few cohesive law parameters including the cohesive shape, fracture energy, and cohesive strength are important under beam loading. However, it is not clear about the effects of cohesive law parameters for the compressive case because the delamination problems for composites under compression using 3D models are almost not studied. This chapter further adopts the bilinear and exponential CZMs in Chapter 2 to perform the delamination analysis of composite laminates under compression [18]. The effects of cohesive law parameters including the cohesive strengths, the shape of CZMs, and cohesive element thickness on the postbuckling and delamination behaviors are discussed.

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates

129

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates 6.2.1 Numerical examples Fig. 6.1 shows the through-the-width delamination specimens of composite flat laminates under compression. Fig. 6.2 shows the finite element model for symmetric and unsymmetric laminates after delamination. Material parameters for T300/976 composites are listed in the last Chapter [19] and Table 6.1 lists the geometry parameters including the initial multiple delamination cracks. In FEA, symmetric models are used for Case-A, B, and C, and the whole model is used for Case-D. The number of bulk elements is 10,000, 10,000, 10,000, and 20,000, respectively for 4 cases and the number of cohesive elements is 450, 450, 630, and 1260, respectively.

y

Z

Clamped end

X

h c3 c1

Clamped end

Screenshot(Alt + A)

a2 a1

c4

B c2

A FIG. 6.1

Composite laminates with multiple delaminations under compression.

FIG. 6.2 Finite element models of (A) symmetric and (B) unsymmetric composite laminates with boundary conditions (element side length ¼ 50.8/100 mm).

130

6. Element analysis of the influence of cohesive law parameters

TABLE 6.1 Geometry sizes of composite specimens (h ¼ 2.54 mm, A ¼ 50.8 mm and B ¼ 5.08 mm are used for four cases). Case

Layup

a1 (mm)

a2 (mm)

c1 (mm)

c2 (mm)

c3(mm)

c4(mm)

A

04//012//04

38.1

19.05

6.35

6.35

15.875

15.875

B

04// (45)6//04

38.1

19.05

6.35

6.35

15.875

15.875

C

04// (45)6//04

25.4

12.7

12.7

19.05

19.05

D

04// (45)6//04

25.4

12.7

19.05

25.4

12.7

12.7 6.35

The postbuckling analysis is based on the finite deformation theory. The weak form for the postbuckling and delamination problems of composites at finite deformation is given by ð ð ð δE  S dV + δ½½u  T dA ¼ δu  T b dA (6.1) Ω + [Ω

Γd

∂Ωt

where E and S are the Green-Lagrange strain tensor and the second PiolaKirchoff stress tensor, respectively. T is taken as the cohesive Piola’s traction vector at finite deformation (we use t at small deformation in Chapter 2). Tb is the prescribed outer traction vector. After performing linearization and discretization on Eq. (6.1) using the total Lagrange finite element formulation [20], the nonlinear global stiffness equation is assembled as     nbulk + ncoh nbulk + ncoh Α KΔu ¼ Α ðKT + Kc ÞΔu ¼ F ext  F s  F c , (6.2) e¼1 e¼1 ~ ~ KT ¼ K0 + KL + Kσ where KT is the stiffness matrix of bulk elements including the elastic stiffness K0, the geometry stiffness KL, and the large displacement stiffness Kσ . Δu is the node displacement increment in nonlinear FEA. Fext is the exter~ force vector and F is the node force vector for the bulk elements. K nal s c and Fc are the stiffness tensor and node force vector for cohesive elements in Chapter 2. nbulk and ncoh are the number of bulk elements and cohesive elements, respectively. After the normal cohesive zone fails completely for delamination, the normal contact behavior is activated by defining the master and slave contact surfaces. FEA includes the initial buckling analysis and subsequent postbuckling analysis. The buckling analysis is performed using the perturbation method, which introduces an initial imperfection for the postbuckling analysis by multiplying a very small value on the first-order buckling mode.

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates

131

6.2.2 Numerical results and discussion The effects of cohesive shape, cohesive strength, and cohesive element thickness on the load responses of composite structures are studied. Numerical results at the same normal and tangential cohesive strengths by using two CZMs are compared. Figs. 6.3–6.6 show the compressive load-strain curves by considering the effects of cohesive strengths for two CZMs and four cases. Figs. 6.7–6.10 show the compressive loadcentral deflection curves for four cases using the exponential CZM. First, the load curves are slightly different at the unstable delamination stage and the peak strength increases a little when the cohesive strength increases. The influence of cohesive strengths becomes larger for the angle-ply laminates than for the unidirectional laminates. Second, the exponential CZM leads to a stronger peak strength than the bilinear CZM at the same cohesive strengths σ max ¼ τmax ¼ 20 MPa, 40 MPa, and 60 MPa, respectively, but the global buckling loads by two CZMs with different cohesive strengths differ very little. Turon et al. [6] pointed out that low cohesive strength can represent the softening properties of the fracture process zone accurately in front of the crack tip well. Here, the difference among load responses lies mainly in the unstable delamination stage using different cohesive strengths, affected by the postbuckling behavior. Besides, the unstable delamination stage requires longer computational

FIG. 6.3 Compressive load-strain curves for the Case-A laminate by two CZMs at different cohesive strengths.

132

6. Element analysis of the influence of cohesive law parameters

FIG. 6.4

Compressive load-strain curves for the Case-B laminate by two CZMs at different cohesive strengths.

FIG. 6.5 Compressive load-strain curves for the Case-C laminate by two CZMs at different cohesive strengths.

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates

133

FIG. 6.6 Compressive load-strain curves for the Case-D laminate by two CZMs at different cohesive strengths.

FIG. 6.7 Compressive load-central deflection curves for the Case-A laminate by two CZMs at different cohesive strengths.

134

6. Element analysis of the influence of cohesive law parameters

FIG. 6.8 Compressive load-central deflection curves for the Case-B laminate by two CZMs at the cohesive strengths σ max ¼ τmax ¼ 40 MPa.

FIG. 6.9 Compressive load-central deflection curves for the Case-C laminate by two CZMs at the cohesive strengths σ max ¼ τmax ¼ 40 MPa.

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates

135

FIG. 6.10 Compressive load-central deflection curves for the Case-D laminate by two CZMs at the cohesive strengths σ max ¼ τmax ¼ 40 MPa.

time using the bilinear CZM than using the exponential CZM. Third, the angle-ply laminate-B shows much weaker compressive load-bearing ability than the unidirectional laminate-A, and the unsymmetric laminate-D shows much weaker load-bearing ability than the symmetric laminateC. Further, more complicated delamination processes for the top sublaminate and bottom sublaminate interacting with the postbuckling appear for unsymmetric laminates. Fourth, the geometrically nonlinear postbuckling accelerates the delamination growth, affecting the evolvement of the fracture process zone. Fifth, from the cohesive shape, the nonlinear exponential CZM may be a better choice to represent the process zone for predicting the ductile delamination than the bilinear CZM. Finally, the increase of cohesive strength adds the convergence difficulty, which was also found by Turon et al. [6] and Liu and Islam [9], and Alfano and Crisfield [21]. Liu and Islam [9] found coarse mesh sizes lead to the spurious numerical oscillation at the decreasing stage of load-displacement curves although coarse and fine mesh sizes lead to consistent curves. The main reason arises from the sudden release of strain energy in the intralaminar materials, which leads to instantaneous failure of elements, and thus spurious solution jump appears for the unstable snap-back delamination. Numerical oscillation shows that a coarse mesh size cannot accurately

136

6. Element analysis of the influence of cohesive law parameters

capture the stress distribution in front of the physical crack tip within the fracture process zone, which was first found by Schellekens and de Borst [22]. However, one has to find the equilibrium between numerical robustness and computational efficiency in the practical simulation by CZMs. From Figs. 6.3–6.6, adopted mesh sizes are shown to be fine enough to avoid this problem. Feih [23] showed a large number of integration points can relax the restriction on the element sizes. Yet, this still requires sufficiently fine mesh sizes as the combination of a large number of integration points and coarse mesh sizes may also lead to numerical oscillation. In addition, Turon et al. [6] also showed that large cohesive stiffness can cause numerical oscillation of tractions although the cohesive contribution to the global compliance before crack propagation should be small enough to avoid the introduction of fictitious compliance to the model. Zou et al. [24] suggested semiempirical values for the stiffness between 104 and 107 times the value of the interfacial strength per unit length. Camanho et al. [25] suggested an accurate value of the cohesive stiffness for carbon/ epoxy composite specimens is K ¼ 106 N/mm3 for the bilinear CZM. From numerical calculations, the convergence and computational efficiency are slightly better by using K ¼ 105 N/mm3 than those by using K ¼ 106 N/ mm3 although they lead to almost the same load responses, which were consistent with the conclusion by Turon et al. [6]. Li and Sridharan [26] modeled the finite-thickness and zero-thickness cohesive elements, where ABAQUS-UEL (User element subroutine) is for zero-thickness elements and ABAQUS-UMAT (User material subroutine) is for finite-thickness elements. They showed that the finite-thickness cohesive element cannot predict the crack propagation accurately. Figs. 6.11–6.14 show the effects of cohesive element thickness on the load-strain curves and the load-deflection curves for the Case-A and C laminates by the exponential CZM at the thickness 0, 0.05, and 0.1 mm, respectively. The finite-thickness and zero-thickness cohesive elements lead to basically consistent results. However, the finite-thickness element adds convergence difficulty, which is represented by the larger numerical oscillation of tractions at the unstable delamination stage. Thus, the zero-thickness cohesive element is a better choice for modeling the delamination growth although there are some other numerical problems, e.g., zero mass for modeling the dynamic delamination, which needs the introduction of artificial mass for time integration in the finite difference algorithm. Figs. 6.15 and 6.14 show the delamination growth processes accompanied by postbuckling behaviors for the Case-C and D laminates by the exponential CZM. For the symmetric Case-C laminate, local buckling occurs at the strain 1.04  103 and load 826 N. When the strain increases to 1.83  103, the top sublaminate starts to debond, as shown in Fig. 6.15A. From the strain 1.83  103 to 2.48  103, the delamination crack for the top sublaminate propagates quickly from 25.4 to 34.5 mm. At the

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates

137

FIG. 6.11 Compressive load-strain curves for the Case-A laminate by the exponential CZM at the cohesive strengths σ max ¼ τmax ¼ 40 MPa and different cohesive element thickness.

FIG. 6.12 Compressive load-strain curves for the Case-C laminate by the exponential CZM at the cohesive strengths σ max ¼ τmax ¼ 40 MPa and different cohesive element thickness.

138

6. Element analysis of the influence of cohesive law parameters

FIG. 6.13

Compressive load-central deflection curves for the Case-A laminate by the exponential CZM at the cohesive strengths σ max ¼ τmax ¼ 40 MPa and different cohesive element thickness.

FIG. 6.14

Compressive load-central deflection curves for the Case-C laminate by the exponential CZM at the cohesive strengths σ max ¼ τmax ¼ 40 MPa and different cohesive element thickness.

6.2 Implicit FEA of buckling, postbuckling, and multiple delaminations of composite laminates

139

strain 3.45  103, the bottom sublaminate starts to debond and propagates up to the length of the top sublaminate crack at the strain 3.75  103, as shown in Fig. 6.15B and C. From the strain 3.75  103, unstable delamination appears and delamination and buckling interact in a complicated way, represented by the oscillation of traction. The delamination mode for the upper sublaminate is mixed-mode, but mode-I fracture dominates the bottom sublaminate. At the strain 4.57  103, the unstable delamination ends and the global buckling stage appears, as shown in Fig. 6.15D. For the unsymmetric Case-D laminate, local buckling occurs at the strain 1.02  103 and load 815 N. The top sublaminate starts to debond at the strain 1.97  103, as shown in Fig. 6.16A. The delamination crack propagates quickly from the crack length 25.4 to 39.1 mm until the strain 2.95  103. At the strain 2.98  103, the bottom sublaminate starts to debond and propagates quickly until the unstable

FIG. 6.15 Delamination growth processes for the Case-C laminate at the compressive strains (A) 1.83  103, (B) 3.45  103, (C) 3.75  103, and (D) 5.26  103 by the exponential CZM.

140

6. Element analysis of the influence of cohesive law parameters

FIG. 6.16 Delamination growth processes for the Case-D laminate at the compressive strains (A) 1.97  103, (B) 2.98  103, (C) 3.63  103, and (D) 4.42  103 by the exponential CZM.

delamination stage at the strain 3.63  103, where the delamination and buckling occur simultaneously, as shown in Fig. 6.16B and C. At the strain 4.42  103, the global buckling stage appears. In general, the unsymmetry of laminate affects the unstable delamination process and delamination rate but does not affect the local and global buckling loads. Table 6.2 lists the loads at the local buckling, the delamination initiation for the top sublaminate and bottom sublaminate and the global bucking. The compressive experiments were performed by Kutlu and Chang [19]. In general, numerical results by two CZMs are close to the experimental results [19]. Differently, the initial delamination for the top and bottom sublaminates by the bilinear CZM is earlier than that by the exponential CZM. It is noted the shape of CZM in front of the crack tip representing the evolving process zone is much affected by the postbuckling behavior although we use two CZMs from the mathematical point of view. By considering the complexity of the true process zone, the main difference for the two CZMs resides on the numerical convergence and computational efficiency by implicit FEA. This is the reason why different shapes of CZMs can coexist for simulating various interface failure problems. In addition, because the cohesive strength as the compressive traction on the finite-size surface in front of the crack tip does not affect the load responses largely except the softening stage of load curves, small cohesive strength is a favorable selection from computational considerations. This

141

6.3 Concluding remarks

TABLE 6.2 Local buckling loads, initial delamination loads and global buckling loads for four cases.

Local buckling load (N)

Initial delamination load for top sublaminate (N)

Initial delamination load for bottom sublaminate (N)

Global buckling load (N)

Exponential CZM

482

1984

2165

1636

Bilinear CZM

481

1934

2136

1611

Case A

Experiment [18] B

1620

Exponential CZM

432

604

951

462

Bilinear CZM

426

591

912

460

Experiment [18] C

481

Exponential CZM

826

1159

1264

487

Bilinear CZM

821

1105

1234

463

Experiment [18] D

488

Exponential CZM

815

1237

1289

531

Bilinear CZM

826

1246

1303

546

conclusion is consistent with the results by Turon et al. [6]. Finally, more energy dissipation within the fracture process zone is needed to drive the mixed-mode delamination, especially at the unstable delamination stage.

6.3 Concluding remarks This chapter studies the influence of cohesive law parameters including the cohesive shape, cohesive strength, and cohesive element thickness on the load responses of carbon fiber/epoxy composite laminates with initial multiple delaminations under compression. The exponential and bilinear

142

6. Element analysis of the influence of cohesive law parameters

CZMs are comparatively used for predicting delamination growth. From FEA, three conclusions are obtained: First, the cohesive shapes have not large effects on the load responses, expect that the bilinear CZM shows poorer convergence than the exponential CZM. Second, the cohesive strengths mainly affect the unstable delamination stage for the laminates under compression, but do not affect the local and global buckling loads largely. This shows low cohesive strength is a good choice because high cohesive strength adds the convergence difficulty. Third, zero-thickness cohesive element is a best modeling strategy by considering both the computational efficiency and numerical convergence, but the zero-mass problem should also be reconsidered for dynamic problems. Finally, the delamination prediction of composites under compression using CZMs, which represent the evolving fracture process zone was affected by the geometrically nonlinear deformation largely at the unstable delamination stage between the initial buckling and global buckling. The evolvement of the process zone is governed by the selected cohesive shape and cohesive strength to some extent, which is demonstrated by slightly different delamination behaviors for the top and bottom sublaminates. Although it is very hard to accurately determine the shape and strength of the fracture process zone for the delamination problem of laminates under compression, we attempt to superimpose and compare the role of different CZMs and expect to solve some critical issues of CZMs by FEA including the numerical robustness and computational efficiency in modeling the delamination growth. By considering these problems, there should be a weight between the reasonable selection and use of various cohesive law parameters under complicated loads and environments.

References [1] Rots JG. Strain-softening analysis of concrete fracture specimens. In: Wittmann FH, editor. Fracture toughness and fracture energy of concrete. Amsterdam: Elsevier; 1986. p. 137–48. [2] Volokh KY. Comparison between cohesive zone models. Commun Numer Methods Eng 2004;20(11):845–56. [3] Chandra N, Li H, Shet C, Ghonemb H. Some issues in the application of cohesive zone models for metal-ceramic interfaces. Int J Solids Struct 2002;39(10):2827–55. [4] Alfano G. On the influence of the shape of the interface law on the application of cohesive-zone models. Compos Sci Technol 2006;66(6):723–30. [5] Song SH, Paulino GH, Buttlar WG. Influence of the cohesive zone model shape parameter on asphalt concrete fracture behavior, In: Multiscale and functionally graded materials conference; 2006. [6] Turon A, Camanho PP, Costa J, Renarta J. Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: definition of interlaminar strengths and elastic stiffness. Compos Struct 2010;92(8):1857–64.

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[7] Harper PW, Sun L, Hallett SR. A study on the influence of cohesive zone interface element strength parameters on mixed mode behavior. Compos A: Appl Sci Manuf 2012;43 (4):722–34. [8] Goyal VK, Johnson ER, Davila CG. Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities. Compos Struct 2004;65 (3–4):289–305. [9] Liu PF, Islam MM. A nonlinear cohesive model for mixed-mode delamination of composite laminates. Compos Struct 2013;106:47–56. [10] Freed Y, Banks-Sills L. A new cohesive zone model for mixed mode interface fracture in bimaterials. Eng Fract Mech 2008;75(15):4583–93. [11] Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl Mech 1987;54(3):525–31. [12] Campilho RDSG, Banea MD, Neto JABP, da Silva LFM. Modelling adhesive joints with cohesive zone models effect of the cohesive law shape of the adhesive layer. Int J Adhes 2013;44:48–56. [13] Ferracin T, Landis CM, Delannay F, Pardoen T. On the determination of the cohesive zone properties of an adhesive layer from the analysis of the wedge-peel test. Int J Solids Struct 2003;40(11):2889–904. [14] Sørensen BF, Jacobsen TK. Characterising delamination of fibre composites by mixedmode cohesive laws. Compos Sci Technol 2009;69(3–4):445–56. [15] Blaysat B, Hoefnagels J, Lubineau G, Alfano M, Geers M. Interfacial debonding quantification by image correlation integrated with double cantilever beam kinematics. Int J Solids Struct 2014;55:79–91. [16] Alfano M, Lubineau G, Furgiuele F, Paulino GH. On the enhancement of bond toughness for Al/epoxy T-peel joints with laser surface treated surfaces. Int J Fract 2011;171 (2):139–50. [17] Alfano M, Lubineau G, Paulino GH. Global sensitivity analysis in the identification of cohesive models using full-field kinematic data. Int J Solids Struct 2015;55:66–78. [18] Liu PF, Gu ZP, Peng XQ, et al. Finite element analysis of the influence of cohesive law parameters on the multiple delamination behaviors of composites under compression. Compos Struct 2015;131:975–86. [19] Kutlu Z, Chang FK. Composite panels containing multiple through-the-width delamination and subjected to compression. Part II: experiments and verification. Compos Struct 1995;31:297–314. [20] Hibbitt HD, Marcal PV, Rice JR. A finite element formulation for problems of large strain and large displacement. Int J Solids Struct 1970;6(8):1069–86. [21] Alfano G, Crisfield MA. Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. Int J Numer Methods Eng 2001;50(7):1701–36. [22] Schellekens JCJ, de Borst R. A nonlinear finite-element approach for the analysis of mode-I free edge delamination in composites. Int J Solids Struct 1993;30(9):1239–53. [23] Feih S. Development of a user element in ABAQUS for modelling of cohesive laws in composite structures. Risø National Laboratory Roskilde, Denmark, Risø-R-1501 2005. [24] Zou Z, Reid SR, Li S, Soden PD. Modelling interlaminar and intralaminar damage in filament wound pipes under quasi-static indention. J Compos Mater 2002;36(4):477–99. [25] Camanho PP, Davila CG, de Moura MF. Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 2003;37(16):1415–38. [26] Li YP, Sridharan S. Performance of two distinct cohesive layer models for tracking composite delamination. Int J Fract 2005;136(1–4):99–131.

C H A P T E R

7

Cohesive/friction coupled model and implicit finite element analysis for delamination analysis of composite laminates under three-point bending 7.1 Introduction In Chapters 2 and 6, we introduce CZM to predict delamination of composites by combining implicit FEA, after some critical numerical issues are solved. However, the degradation process of cohesive zones is often accompanied by gradually increased friction from the physical perspective because the delamination interface is generally very thin. Wisnom et al. [1] highlighted the frictional effect in the mode-II shear delamination of composites by experiments. Fan et al. [2] developed a unified approach to quantify the frictional effect using the mode-II delamination tests including the end-notched flexure (ENF), end-loaded-split (ELS), fourpoint bending ENF (4ENF), and over-notched-flexure (ONF). Their conclusions showed the friction plays an important role in affecting the delamination resistance. Therefore, a favorable strategy is to couple the cohesive model and friction in the delamination analysis. Raous et al. [3] developed a micromechanical adhesion/friction coupled model by considering the contact zone as a material boundary. When the interface is decomposed into a damaged part with the friction and an undamaged part, Alfano and Sacco [4] developed a bilinear cohesive/friction coupled model. Later, Parrinello et al. [5, 6], Guiamatsia and Nguyen [7] and Serpieri et al. [8] extended the work of Alfano and Sacco [4] by further considering the plastic interface and mixed-mode interface failure under

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00005-5

145

Copyright © 2021 Elsevier Inc. All rights reserved.

146

7. Cohesive/friction coupled model

different load conditions. As pointed out previously, the exponential CZM with a continuous transition at the peak point is more suitable for modeling the delamination with the friction than the bilinear CZM from the mathematical perspective. This chapter developed a nonlinear cohesive/ friction coupled model for the mode-II shear delamination of composites under three-point bending based on a modified Xu and Needleman’s exponential cohesive model [9–11], which includes a nonassociative frictional slip criterion and a slip potential as well as a return mapping algorithm for the plastic frictional contact problem. The developed model is implemented using zero-thickness interface element by ABAQUS-UEL, which is limited to small frictional sliding state. Effects of the friction coefficient and cohesive strength on the load responses and delamination mechanisms of composites using FEA. Numerical results for the mode-II delamination of the [0°]6, [
30°]5, [
45°]5 angle-ply end-notched flexure composites demonstrate the developed model by comparing the analytical solutions.

7.2 Cohesive/friction coupled model for mode-II delamination of composites Mode-II shear delamination for an end-notched flexure (ENF) composite specimen with the frictional contact is shown in Fig. 7.1. The normal and tangential displacement jumps ½½u1 and ½½u2 are written as

 ½½u1 ¼ x1  x2  n, ½½u2 ¼ x1  x2  t (7.1) where x1 and x2 are material points at the discontinuous interface, and n and t denote the normal and tangential directions. For the contacted Delaminated and contacted surface

Zoom in n x2 x1

t

FIG. 7.1 Mode-II shear delamination for the ENF composite specimen with the frictional contact (x1 and x2 are separated material points at the discontinuous interface).

7.2 Cohesive/friction coupled model for mode-II delamination of composites

147

interface, the nonpenetration condition requires the inequality constraint ½½u1 0 holds. Xu and Needleman [9] proposed an interface potential-based cohesive model for predicting the interface failure.

0 19 8 !8 ! #  "     > ½½u 2 = > ½½u 1 < ½½u 1 ½½u 1 1  q r  q > 2 A , > > 1r+ exp @  q+ ϕ ½½u 1 , ½½u 2 ¼ ϕ1 + ϕ1 exp  > > δ1 : δ1 r1 r  1 δ1 > δ2 ; > > 2 > > > 0 1 2 0 13 > !8 !9 >   >  > ½½u 2 ½½u 2 ϕ1 ½½u 1  1q 4 ∂ϕ > 2 2 @ A @ A 5 > , exp  exp  ¼ + r 1  exp  < t1 ½½u 1 , ½½u 2 ¼ δ1 : δ1 δ1 ; r1 ∂½u ½  1 δ1 δ2 δ2 2 2 > > 0 1 > "  # ! >  >   > ½½u 2 > ϕ1 ½½u 2 ½½u 1 r  q ½½u 1 ∂ϕ > 2 @ A, > > t ½½u 1 , ½½u 2 ¼ q+ exp  exp  ¼2 > > 2 2 δ2 δ2 δ1 r  1 δ1 ∂½u ½ 2 > δ > > 2 > > > h i > > : δ ¼ ϕ =½σ 0:5 exp ð 1 Þ , δ ¼ ϕ = τ ð 0:5 exp ð 1 Þ Þ 1 1 max 2 2 max (7.2)

where ϕ is the interface potential including the normal and tangential parts ϕ1 and ϕ2. t1 and t2 are the normal and tangential cohesive tractions. σ max is the maximum normal traction without the tangential displacement jump and τmax is the maximum tangential traction without the normal displacement jump. The normal traction t1 reaches the maximum value σ max at the normal displacement jump t2, and the tangential traction t2 reaches pffiffiffi the maximum value τmax at the tangential displacement jump δ2 = 2. Specially, Gao and Bower [12] considered the interface potential as the fracture toughness. In the following, GcI ¼ ϕ1 and GcII ¼ ϕ2 are taken as the delamination fracture toughness in the normal and tangential directions, respectively. q ¼ ϕ2/ϕ1 and r ¼ ½½u∗11/δ1 are coupling constants between the normal and tangential directions. ½½u∗11 is the value of ½½u11 after complete shear separation with t1 ¼ 0. Abdul-Baqi et al. [13] and Hattiangadi and Siegmund [14] suggested q  0.43 so that the maximum tractions have the same values t1 ¼ t2 at δ1 ¼ δ2. However, ϕ1 ¼ ϕ2 holds at q ¼ 1, which is yet inconsistent with the experimental delamination results. In order to allow different values for ϕ1 and ϕ2, van den Bosch et al. [10] proposed a modified version of Xu and Needleman’s cohesive model by taking q ¼ 1 in Eq. (7.2) and replacing ϕ1 by ϕ2 in the expression of the traction t2. " !# 8    
 > ½½u1 ½½u1 ½½u22 > > ϕ ½½u1 , ½½u2 ¼ ϕ1 1  1 + exp  exp  2 , > > δ1 δ1 δ2 > > > ! >   <
 ϕ1 ½½u1 ½½u1 ½½u2 (7.3) exp  exp  2 2 , t1 ½½u1 , ½½u2 ¼ > δ1 δ1 δ1 δ2 > > ! >     > >
 > ϕ2 ½½u2 ½½u1 ½½u1 ½½u22 > > ½ ½u  , ½ ½u  1 + exp  exp  ¼ 2 t : 2 1 2 δ2 δ2 δ1 δ1 δ22

148

7. Cohesive/friction coupled model

where the work done during mixed-mode separation is path-dependent at ϕ1 ¼ ϕ2. Further, McGarry et al. [11] pointed out that unphysical behavior appears for mixed-mode overclosure using Eq. (7.3): the work of tangential separation reduces with increasing normal over-closure, and negative tangential work appears for large normal overclosure ½½u1/δ1 <  1, resulting in repulsive tangential traction. In order to correct this problem, McGarry et al. [11] modified Eq. (7.3) to the following nonpotential form by removing the term 1 + ½½u1/δ1. ! 8   2
 > ½ ½u  ½ ½u  ½ ½u  > > t1 ½½u1 , ½½u2 ¼ σ max exp ð1Þ 1 exp  1 exp  2 2 , > > δ1 δ1 > δ2 > < !   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½½u1
 ½½u1 ½½u22 (7.4) exp  exp  2 , t2 ½½u1 , ½½u2 ¼ τmax 2exp ð1Þ > > > δ δ δ 1 1 > 2 > h i > > 0:5 : δ ¼ ϕ =½σ exp ð1Þ, δ ¼ ϕ = τ ð0:5 exp ð1ÞÞ 1

1

max

2

2

max

In the following, a cohesive/friction coupled model based on Eq. (7.4) for the mode-II shear delamination of composites is developed. After the tangential cohesive softening appears, nonassociative plasticity interface with the frictional contact is assumed, where both the normal and tangential displacement jumps ½½u1 and ½½u2 are divided into the elastic and plastic parts ½½u1 ¼ ½½u1(e) + ½½u1( p) and ½½u2 ¼ ½½u2(e) + ½½u2( p). If the tangential traction-elastic displacement jump curve obeys the exponential distribution, Eq. (7.4) is further adjusted to the following form for the mode-II shear delamination with the normal contact.  h i 8  > t1 ½½u1ðeÞ ¼ K1 ½½u1ðeÞ , δ2 ¼ ϕ2 = τmax ð0:5 exp ð1ÞÞ0:5 , > > ! > > >   ½½u22ðeÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½½u2ðeÞ > < exp  2 , t2 ½½u2ðeÞ ¼ τmax 2 exp ð1Þ δ2 δ2 > ! ! > 2 2 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ½ ½u  2 ½ ½u  > ∂t τ 2 ð e Þ 2 ð e Þ 2 max >K ¼ > 2exp ð1Þ exp  2 ¼ 1 : 2 ½½u2ðeÞ δ2 δ2 δ22

(7.5)

where the normal traction t1 is adjusted by introducing the normal penalty stiffness K1 based on the Kuhn-Tucker contact relationships ½½u1(e) 0, t1  0, ½½u1(e)t1 ¼ 0. K p2ffiffiffiis the tangential stiffness which becomes negative at ½½u2ðeÞ  ½½u02ðeÞ ¼ 2=2δ2 .Theoretically, infinite penalty parameter leads to accurate enforcement of constraints. Here, the penalty method is used, but the penalty stiffness K1 must be appropriately chosen so that illconditioned problems can be avoided. After the initial tangential cohesive softening appears, μ jt1 j (μis the variable friction coefficient) is superimposed onto the tangential traction t2 to describe continuous transition from the cohesive state to the friction state

7.2 Cohesive/friction coupled model for mode-II delamination of composites

149

explicitly. Thus, the tangential traction becomes t2 + μ j t1 j. μ jt1 jincreases exponentially from zero to the smooth value μs jt1 j (μs ¼ tan ϕ is the smooth friction coefficient and ϕ is the friction angle) where pure frictional contact appears, as shown in Fig. 7.2. The variable friction coefficient μ is written as.   c 0 ½ ½u  ½ ½u   ½ ½u  2ðeÞ 2ðeÞ 2ðeÞ exp ðdÞ  1   , d¼ μ ¼ μs (7.6) c exp ð1Þ  1 ½½u ½½u  ½½u0 2ðeÞ

2ðeÞ

2ðeÞ

where d is the damage variable. ½½u02(e) and ½½u02(e) are the initial and critical displacement jumps, respectively. Based on Eq. (7.5), ½½uc2(e) is solved by assuming the area under the tangential traction-elastic displacement jump curve reaches the maximum. ! Z ½½u c2ðeÞ ½½u2ðeÞ ½½u22ðeÞ exp  2 d½½u2ðeÞ δt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 0 c !  ln ð1  eÞ ¼ e ! 1 ) ½ ½u  ¼ δ 2 2 ð e Þ Z ∞ ½½u2ðeÞ ½½u22ðeÞ exp  2 d½½u2ðeÞ δ2 δ2 0 (7.7) ½½uc2(e)  3δ2

holds at e ¼ 0.9999. where The finite critical separation ½½uc2(e) is assumed, which is solved by assuming the accumulative cohesive fracture energy reaches the maximum, similar to the work by Goyal et al. [15] and Liu et al. [16]. The plastic deformation of adhesive materials in shear is nonassociative. After the initial cohesive softening appears, the frictional contact problem resorts to the classic nonassociative plasticity theory. Based on the work of Raous et al. [3], Parrinello et al. [5], and Weyler et al. [17], the frictional slip criterion F and the slip potential G according to the nonassociative MohrCoulomb frictional contact law are written as.

FIG. 7.2 Superimposed tangential friction increases exponentially from zero at the beginning of tangential cohesive softening to the smooth friction (friction coefficient is μs) at the complete tangential cohesive failure where pure frictional contact appears.

150

7. Cohesive/friction coupled model

F ¼ t2 + μjt1 j  μs jt1 j, G ¼ t2 + βs jt1 j, ∂G _ ∂G ½½u_ 2ðpÞ ¼ λ_ ¼ λ, ½½u_ 1ðpÞ ¼ λ_ ¼ βs λ_ ¼ βs ½½u_ 2ðpÞ ∂t2 ∂jt1 j

(7.8)

where λ_ is the consistency factor. F obeys the slip/stick Kuhn-Tucker conditions F  0, λ_  0 , and λ_ F ¼ 0. βs ¼ tan(φ) is the dilatancy and φ(0  φ  ϕ) is the dilatant angle. In the Mohr-Coulomb plasticity model, the friction angle ϕ defines the ratio of the shear stress to the normal stress, and the dilatant angle φ denotes the ratio of the volumetric strain rate to the shear strain rate, and dilatancy βs is the volume expansion in shear. The limit friction surface F ¼ 0 of the Coulomb law is a cone in the space of contact traction, as shown in Fig. 7.3. From Eq. (7.8), the normal and tangential plastic displacement jumps ½½u1( p) and ½½u2( p) take on the same increasing tendency. Parrinello et al. [5] developed a bilinear cohesive/friction coupled interface model, where a nonassociative frictional interface is considered. Here, the tangential traction t2 + μ jt1 j by replacing t2 in their model is used in the slip criterion in Eq. (7.8). After introducing the return mapping algorithm in Appendix A, the updated normal  traction t2 + μ jt1 j, the updated tangential traction t2 ¼ μs jt1 j ½½u1ðeÞ and the tangential stiffness K2 ¼ K2(½½u2(e)) are obtained.

7.3 Numerical algorithm for the cohesive/friction coupled model In last several chapter, we introduce the finite element formulation for cohesive models. Wriggers [18] pointed out that the isoparametric element within the frictional contact area does not allow large relative displacement.

FIG. 7.3 Limit friction surfaces of the Coulomb law as a cone in the space of contact traction.

7.4 Numerical results and discussion

151

Thus, the developed cohesive/friction model is limited to small deformation. These cohesive/frictional contact elements are considered as intermediate regions connecting the contacted master/slave surfaces in each discrete contact domain of the deformed body [17]. Numerical algorithm using implicit FEA is shown in Fig. 7.4. After the tangential cohesive softening appears, the tangential trial traction is calculated. If the frictional slip appears, Newton iterations are performed to calculate the updated normal and tangential tractions and stiffnesses. Then, the node residual force and element stiffness are calculated for solving the node displacement increment.

7.4 Numerical results and discussion The end-notched flexure (ENF) glass/polyester composite specimens with an initial middle-plane delamination crack are shown in Fig. 7.5. Three types of layup specimens [0°]6, [
30°]5, and [
45°]5 are used. These specimens were first studied by Ozdil et al. [19] using analytical and numerical approaches and later studied by Theotokoglou and Vrettos [20] using FEA. Geometry parameters are listed in Table 7.1 and material parameters are listed in Table 7.2 [19], in which the delamination fracture toughness is calculated using four methods: (1) Compliance theory [21], (2) Beam theory [22], (3) Modified beam theory [23], and (4) Associated beam theory and first-order shear deformation theory [19]. The formulas for four methods are attached in Appendix B. It is worth pointing out that the cohesive/friction model developed by Parrinello et al. [5, 6] allows only the same delamination fracture toughness for the mode-I and II delamination. In fact, the mode-I fracture toughness is not necessary for the mode-II shear delamination. Although the friction affects the mode-II delamination fracture toughness [23], the purpose is to explore the effects of the cohesive strength and friction on the delamination mechanisms and load responses of composites at a constant mode-II delamination fracture toughness, similar to the work by Parrinello et al. [5, 6]. Thus, the average value of the upper and lower limits of the delamination fracture toughness using four methods is used. Three finite element models with different mesh sizes are shown in Fig. 7.6, where the number of bulk and cohesive elements for three mesh models and three layups is listed in Table 7.3. First, the effect of viscous constant c in Chapter 2 on the loaddisplacement curves for Case-A laminate using two mesh models is shown in Figs. 7.7 and 7.8. Numerical results are also compared with the analytical results without considering the frictional effect using the theoretical formulas derived by Mi et al. [24], as attached in Appendix C. Second, coarse mesh size leads to numerical oscillation at the decreasing stage of loaddisplacement curves, similar to the work by Liu et al. [16]. The main reason arises from the sudden release of strain energy and the instantaneous

152

FIG. 7.4 using FEA.

7. Cohesive/friction coupled model

Numerical algorithm for implementing the cohesive/friction coupled model

153

7.4 Numerical results and discussion

FIG. 7.5

Geometry of the ENF composite specimen.

failure of elements in the intralaminar materials, leading to nonphysical limit points in the form of a snap-through or a snap-back situation in the load-displacement curves. However, oscillation does not represent large loss of accuracy for load responses. By comparing Figs. 7.7 with 7.8, fine mesh sizes along the direction of crack propagation helps to alleviate the oscillation largely by capturing the sudden release of strain energy during decreasing stage more precisely. It is believed finer mesh sizes can eliminate the oscillation completely. Besides, the viscous parameter c produces very small impact on the oscillation, compared with large sensitivity for mesh sizes. However, it is noted too large value for c will lead to inaccurate results indeed due to additionally dissipated viscous energy, but too small value

TABLE 7.1 Mechanical properties of quasiunidirectional glass/polyester composites [19]. Ply longitudinal modulus

E1

34.7 GPa

Ply transverse modulus

E2

8.50 GPa

Out-of-plane modulus

E3

8.50 GPa

Inplane shear modulus

G12

4.34 GPa

Out-of-plane shear modulus

G13

4.34 GPa

G23

3.27 GPa

v12

0.27

v13

0.27

v23

0.30

GcII

0.496
0.135 N/mm ([0°]6) 0.976
0.071 N/mm ([
30°]5) 1.485
0.158 N/mm ([
45°]5)

Poisson’s ratio

Mode-II delamination fracture toughness

154

7. Cohesive/friction coupled model

TABLE 7.2 Geometry sizes of glass/polyester composite specimens [19]. Case

Layup

2 L (mm)

b (mm)

2 h (mm)

a0 (mm)

A

[0°]6

100

20

4.38

25

B

[
30°]5

100

20

7.30

25

C

[
45°]5

100

20

7.30

25

for c will add the convergence difficulty. From Figs. 7.7 and 7.8, c ¼ 1e-4 is a good choice for achieving favorable combination of numerical convergence and accuracy, which is used in the following analysis. In summary, robust and accurate results can be obtained once mesh sizes and viscous constant are properly selected. Third, the effect of normal stiffness K1 on the load-displacement curves using two mesh models for Case-A laminate is shown in Fig. 7.9. It is found when K1 is larger than 9000 N/mm3 or smaller than 3000N/mm3, convergence becomes difficult at c ¼ 1e-4. By comparison, K1 ¼ 3000 9000N/mm3 leads to good convergence at c ¼ 1e-4. Thus, K1 ¼ 6000N/mm3 is adopted in the following analysis. The effects of the friction coefficient μs, cohesive strength τmax and mesh size on the load-displacement curves for three cases are shown in Figs. 7.10–7.18. Numerical results by the proposed model are also compared with the analytical results. For angle-ply laminates, equivalent moduli [20] are used for analytical solutions. First, Sch€ on [25] measured the friction coefficient for the epoxy with frictional sliding against metal, where the friction is related to the shear deformation or fracture of the polymer

FIG. 7.6 Three finite element models for the ENF laminate with loads and boundary conditions.

7.4 Numerical results and discussion

155

TABLE 7.3 Number of the bulk and cohesive elements for three mesh models and three layups.

Coarse mesh

Middle mesh

Fine mesh

FIG. 7.7

Layup

Number of bulk elements

Number of cohesive elements

Sizes of bulk elements: length × width × height (mm × mm × mm)

Case-A

2688

336

1.78 2.5 0.73

Case-B and C

4480

Case-A

4800

600

1.0 2.5 0.73

Case-B and C

8000

Case-A

6720

840

0.71 2.5 0.73

Case-B and C

11,200

Effect of the viscous constant c on the load-displacement curves for Case-A laminate at ϕ ¼ 0° and τmax ¼ 30 MPa using middle mesh model.

156

7. Cohesive/friction coupled model

FIG. 7.8 Effect of the viscous constant c on the load-displacement curves for Case-A laminate at ϕ ¼ 0° and τmax ¼ 30 MPa using fine mesh model.

FIG. 7.9 Effect of the normal stiffness on the load-displacement curves for Case-A laminate at ϕ ¼ 0° and τmax ¼ 30 MPa using fine and middle mesh models.

7.4 Numerical results and discussion

157

FIG. 7.10 Load-displacement curves for Case-A laminate using three friction angles and middle mesh model.

surface. They obtained the friction coefficients 0.2–0.7 in the rubber region. Here, we take three friction angles ϕ ¼ 0 ° , 15 ° , and 30° and three friction coefficients are calculated as μs ¼ tan ϕ ¼ 0, 0.27, and 0.54, respectively. From Figs. 7.10, 7.13, and 7.16, the frictional effect appears at the initial unstable delamination stage a ¼ L and large friction coefficient leads to large peak load and slight oscillating load response. After the stable delamination stage is reached, the difference between the load curves using different friction coefficients becomes large, represented by large contact area and frictional energy dissipation. Besides, the frictional effects are larger for angleply Case-B and C laminates than unidirectional Case-A laminate. Second, take Case-A laminate for example, the delamination starts and the crack-tip energy dissipates as the displacement Δ increases to about 3.1 mm. As the crack propagates from a ¼ 25–35 mm, the load response starts to change suddenly. From 4 ¼ 3.1 to 5.5 mm displacements, the unstable delamination is represented by oscillating load response. After Δ ¼ 5.5 mm displacement, the stable delamination stage appears, and the frictional effect increases. Third, the stiffness of specimens increases slightly when τmax adds. In addition, τmax also affects the peak load for three cases, as shown in Figs. 7.11, 7.14, and 7.17. However, large value τmax ¼ 50 MPa will lead to convergence difficulty and oscillating load responses during the unstable delamination. Fourth, from Figs. 7.12, 7.15, and 7.18, the load curves using

158

FIG. 7.11

7. Cohesive/friction coupled model

Load-displacement curves for Case-A laminate using three cohesive strengths and (A) middle mesh model and (B) fine mesh model.

7.4 Numerical results and discussion

FIG. 7.12

159

Load-displacement curves for Case-A laminate using three mesh models.

FIG. 7.13 Load-displacement curves for Case-B laminate using three friction angles and middle mesh model.

160

7. Cohesive/friction coupled model

FIG. 7.14 Load-displacement curves for Case-B laminate using three cohesive strengths and (A) middle mesh model and (B) fine mesh model.

7.4 Numerical results and discussion

FIG. 7.15

161

Load-displacement curves for Case-B laminate using three mesh models.

FIG. 7.16 Load-displacement curves for Case-C laminate using three friction angles and middle mesh model.

162

7. Cohesive/friction coupled model

FIG. 7.17 Load-displacement curves for Case-C laminate using three cohesive strengths and (A) middle mesh model and (B) fine mesh model.

7.4 Numerical results and discussion

FIG. 7.18

163

Load-displacement curves for Case-C laminate using three mesh models.

three mesh models (element side sizes along the crack propagation direction are 1, 1.8, and 2.5 mm, respectively) are consistent. Although fine mesh model helps to eliminate the numerical oscillation, computational efficiency should also be considered, especially at large τmax. Delamination growth processes for Case-A laminate are shown in Fig. 7.19. The crack length-load displacement curves by discussing the effects of the friction coefficient and layup are shown in Figs. 7.20 and 7.21. The contours of shear stress τ23 for three cases at different stages are shown in Figs. 7.22–7.24. The crack tip and the crack propagation length are determined by detecting the dislocated paired nodes for cohesive elements in Fig. 7.19 and the high-gradient area for the shear stress τ23 in Fig. 7.22. First, we approximately divide the whole delamination process into the fast delamination stage from a ¼ 25 mm at 4 ¼ 3.1mm to about a ¼ 45 mm at 4 ¼ 3.6 mm, and the unstable delamination stage from a ¼ 45 to 55 mm at 4 ¼ 4.0 mm, and the stable delamination stage at a > 55 mm for Case-A laminate. Second, the frictional effect is not distinct for the unidirectional Case-A laminate, which is represented by a small change of the shear stress τ23 when the friction coefficient increases. However, the frictional effect becomes distinct for angleply laminates as shown in Figs. 7.10, 7.13, and 7.16. Table 7.4 lists the bending strengths by comparing the effects of the layup, friction coefficient, and cohesive strength. It is shown that large friction angle and cohesive strength lead to large strength, especially for angle-ply laminates. Third, the crack

164

7. Cohesive/friction coupled model

FIG. 7.19

Delamination growth processes for Case-A laminate at the displacement (A) 3.11 mm, (B) 4.34 mm, and (C) 8.79 mm respectively at τmax ¼ 30 MPa and ϕ ¼ 45° using middle mesh model.

propagation rate is faster for Case-B laminate than that for Case-C laminate because the displacement Δ increases from 2.3 to 3.2 mm for Case-B laminate and from 3.7 to 4.9 mm for Case-C laminate as the crack propagates from a ¼ 25 to 50 mm. The effect of the superimposed traction μ j t1 j in Eq. (7.8) on the loaddisplacement curves are shown in Fig. 7.25. For unidirectional Case-A laminate, μ jt1 j has almost no effect on the load-displacement curves, but becomes a little more distinct for angle-ply Case-C laminate. By comparison, the introduction of μ jt1 j decreases the load-bearing ability slightly after the laminate enters into the unstable delamination stage for Case-C laminate.

7.4 Numerical results and discussion

FIG. 7.20

165

Layup effect on the crack propagation rate at τmax ¼ 30 MPa using middle mesh

model.

FIG. 7.21 mesh model.

Frictional effect on the crack propagation rate at τmax ¼ 30 MPa using middle

166

7. Cohesive/friction coupled model

(A)

S, S13 +1.628e+01 +1.360e+01 +1.091e+01 +8.224e+00 +5.539e+00 +2.854e+00 +1.680e-01 –2.517e+00 –5.203e+00 –7.888e+00 –1.057e+01 –1.326e+01 –1.594e+01

(B)

S, S13 +1.234e+01 +1.019e+01 +8.037e+00 +5.884e+00 +3.731e+00 +1.577e+00 –5.759e-01 –2.729e+00 –4.882e+00 –7.036e+00 –9.189e+00 –1.134e+01 –1.350e+01

(C)

S, S13 +1.184e+01 +9.903e+00 +7.969e+00 +6.034e+00 +4.100e+00 +2.165e+00 +2.310e-01 –1.703e+00 –3.638e+00 –5.572e+00 –7.507e+00 –9.441e+00 –1.138e+01

(D)

S, S13 +1.929e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.883e+01

(A) (B)

(C)

(D) 3 1

FIG. 7.22 Contours of the tangential stress τ23 at the displacement (A) 3.12 mm, (B) 3.65 mm, (C) 4.85 mm, and (D) 8.95 mm respectively for Case-A laminate at ϕ ¼ 0° and τmax ¼ 30 MPa using middle mesh model.

(A)

+1.651e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.617e+01

(B)

+1.429e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01

(C)

+1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.415e+01

(D)

+1.940e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.925e+01

(A)

(B)

(C)

(D) FIG. 7.23 Contours of the shear stress τ23 at the displacement (A) 3.12 mm, (B) 3.65 mm, (C) 4.85 mm, and (D) 8.95 mm respectively for Case-A laminate at ϕ ¼ 15° and τmax ¼ 30 MPa using middle mesh model.

167

7.4 Numerical results and discussion S, S13

(A)

S, S13

+1.651e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.617e+01

(B)

+1.429e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01

S, S13

(C)

S, S13

+1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.415e+01

(D)

+1.940e+01 +1.271e+01 +1.065e+01 +8.585e+00 +6.522e+00 +4.460e+00 +2.398e+00 +3.362e-01 –1.726e+00 –3.788e+00 –5.850e+00 –7.912e+00 –9.974e+00 –1.204e+01 –1.925e+01

(A) (B)

(C)

(D) 3 1

FIG. 7.24 Contours of the shear stress τ23 at the displacement (A) 3.12 mm, (B) 3.65 mm, (C) 4.85 mm, and (D) 8.95 mm, respectively for Case-A laminate at ϕ ¼ 30° and τmax ¼ 30 MPa using middle mesh model.

TABLE 7.4 Bending strengths predicted using proposed model (Unit: N). Middle mesh size, ϕ 5 0°, τmax

Layup

10 MPa

τmax 5 30 MPa, middle mesh size

τmax 5 30 MPa, ϕ 5 0°

30 MPa

50 MPa

ϕ 5 15°

ϕ 5 30°

Coarse mesh size

Fine mesh size

[0°]6

399.9

505.9

569.9

511.5

519.4

484.9

481.6

[
30°]5

814.8

1028.4

1181.1

1077.1

1124.1

1048.9

1044.9

[
45°]5

808.3

1049.0

1136.8

1085.8

1127.9

1034.4

1046.4

168

7. Cohesive/friction coupled model

FIG. 7.25 Numerical results by considering the effect of the superimposed tangential traction μjt1 j for Case-C laminate at τmax ¼ 30 MPa using middle mesh model t1.

7.5 Concluding remarks This chapter develops a cohesive/friction coupled model for the modeII shear delamination of angle-ply adhesive composite joint based on a modified Xu and Needleman’s exponential cohesive model. The tangential friction is assumed to appear after the initial tangential cohesive failure, and the exponentially increased tangential friction is superimposed onto the tangential cohesive traction to describe continuous transition from the cohesive state to the frictional state explicitly. Then, a frictional slip criterion and a slip potential function as well as a numerical algorithm by FEA are proposed to solve the updated normal and tangential tractions and stiffnesses. Some important numerical issues for the proposed model are discussed in FEA including the numerical interpolation technique for cohesive element, the numerical convergence and the frictional contact algorithm based on the discrete contact domain. Numerical results on the [0°]6, [
30°]5, and [
45°]5 angle-ply end-notched flexure (ENF) composite specimens demonstrate the proposed model by comparing the analytical results. The main purpose of numerical calculations is to discuss the effects of the friction coefficient, tangential cohesive strength and normal contact stiffness on the load responses and delamination mechanisms of composites. It is shown the frictional effect becomes distinct after the unstable delamination for angle-ply laminates, leading to slightly stronger load-bearing ability.

Appendix A: Cohesive/friction contact algorithm using implicit FEA

169

Appendix A: Cohesive/friction contact algorithm using implicit FEA By referring to the return mapping algorithm, the updated tangential traction t2 and the consistent tangential stiffness K2 are solved by the following Newton iterations: (a) After the tangential cohesive softening appears, the trial tangential traction is calculated as.   trði + 1Þ ð i + 1Þ trði + 1Þ ði + 1Þ ðiÞ t2 ¼ t2 ½½u2ðeÞ ¼ ½½u2  ½½u2ðpÞ   trði + 1Þ ði + 1Þ ðiÞ + μjt1 jði + 1Þ ½½u1ðeÞ ¼ ½½u1  ½½u1ðpÞ (A.1) where ½½u2 > 0 and ½½u1 < 0 are the normal and tangential displacement (i) jumps at the beginning of new (i+ 1)th increment, and ½½u(i) 1( p) and ½½u2( p) are the normal and tangential plastic displacement jumps at the end of ith increment. (b) If ttr(i+1)  μs j t1 j(i+1), the tangential stiffness K2(½½u(i+1) ¼ ½½utr(i+1) 2 2 2(e) ) and the tangential traction t2 ¼ ttr(i+1) ¼ t2(½½u(i+1) ¼ ½½utr(i+1) 2 2 2(e) ) in Eq. (7.5) are calculated. There is no plastic loading and slip for the contacted interface. (c) If ttr(i+1) > μs jt1 j(i+1), the plastic loading and frictional slip appear. After 2 the initial value for Δλ ¼ 0 is given, the following Newton iterations are performed to solve the updated elastic displacement jump ½½u(i+1) 2(e) and the plastic displacement jump ½½u(i+1) . 2( p) 8 9 2  2 3 ði + 1Þ > > ð i + 1Þ > > ½½u > > ½½u 7 6 > > > > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2ðeÞ 2ðeÞ > > 7 6 ð i + 1Þ > > > > 7 > 0, exp 6 t ¼ τmax 2 exp ð1Þ > > > > 5 4 2 2 δ > > δ 2 > > > > 2 > > > > > > > > > > > >   > > > > ði + 1Þ > > ð i + 1 Þ > > > > > 0, ¼ K ½ ½u  t > > 1 1 > > 1 ð e Þ > > > > > > > > > > ð i + 1Þ ði + 1Þ > > > > > > ði + 1Þ ði + 1Þ ð i + 1 Þ ð i + 1 Þ > > t1 > > F¼t +μ  μs ½½u , G¼t + β s t1 , > > > > 2 2 > > > > > > > > > > ∂F ði + 1Þ > > ð i + 1 Þ ð i Þ ð i + 1 Þ > > > 0, dΔλ + ⋯ ¼ 0 ) dΔλ ¼ F=d > 0 ) Δλ ¼ Δλ  F =d F + > > 2 > > 2 > > ∂Δλ > > > > > >   > > > > > > ð i + 1 Þ   ð i + 1 Þ > > exp d  1 > > ∂F ð i + 1 Þ ð i + 1 Þ > > ði + 1Þ  μ K < 0, μði + 1Þ ¼ μ

> > > 2  2 30  2 1 > > > > > > > > ði + 1Þ ði + 1Þ > > > > ½½u 2 ½½u 7 C 6 B q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > > 2 ð e Þ 2 ð e Þ τ 7 C 6 B ð i + 1 Þ max > > >K > 7 C 6 B 1  < 0, ¼ 2exp ð 1 Þ exp  > > > > 5 A 4 @ 2 2 2 > > δ > > δ δ 2 > > > > 2 2 > > > > > > > > > > > > > > > ði + 1Þ > ði + 1Þ ði + 1Þ ði + 1Þ i ð i + 1 Þ > > > > ½½u ¼ ½½u + Δλ > 0, ½½u ¼ ½½u  ½½u > 0, > > > > 2 2 ð p Þ 2 ð p Þ 2 ð e Þ 2 ð p Þ > > > > > > > > > > > > ð i Þ ð i + 1 Þ ð i + 1 Þ ð i + 1 Þ ð i + 1 Þ > > ð i + 1 Þ > > + β Δλ > 0, ¼ ½ ½u  ½ ½u  < 0,  ½ ½u  ¼ ½ ½u  ½ ½u  > > s > > 1 > 1ðpÞ > 1ðpÞ 1ðpÞ 1ðeÞ > > > > > > 0 1 > > > > > > > > 0 c > > ½ ½u  ½ ½u  > > B C > > 2 ð e Þ 2 ð e Þ > > ði + 1Þ ¼ > > B1  C > > d > > @ A > > c 0 ð i + 1 Þ > > ½½u  ½½u : ; ½½u 2ðeÞ 2ðeÞ 2ðeÞ

(A.2)

170

7. Cohesive/friction coupled model

where Δ() denotes the increment. It is shown the return mapping algorithm ensures that the frictional slip criterion meets the KuhnTucker load conditions λ_ > 0 and F ¼ 0. During numerical iterations in Eq. (A.2), the tangential elastic displacement jump ½½u2(e) increases and the tangential traction t2 decreases, but the normal elastic displacement jump j½½ui+1 1(e) j increases and the normal traction jt1 j increases, which guarantees the convergence of the proposed algorithm at jF j < tol (tol is the tolerance). In addition, it is noted the introduction of μ jt1 j helps to accelerate the convergence of the proposed algorithm in Eq. (A.2) because jd(i+1) j 2 decreases and Δλ increases and j t1 j(i+1)increases. Finally, the normal traction t1 ¼ t1(½½u1(e)), the tangential traction t2 ¼ μs jt1 j and the tangential stiffness K2 ¼ K2½½u2(e) are updated after convergence.

Appendix B: Mode-II shear delamination fracture toughness of composites Four methods for calculating the mode-II energy release rate (ERR) are: (1) Compliance theory [21], (2) Beam theory [22], (3) Modified beam theory [23] and (4) Associative beam theory and first-order shear deformation theory [19]. The mode-II delamination fracture toughness GcII is determined using the initial crack length a ¼ a0 and the experimental load value P [20]. (a) Compliance theory [21] Load-line compliance C ¼ Δ/P is defined, where Δ is the displacement at the central loading point and P is the applied load. The ERR GII takes the form. GII ¼

P2 ∂C 2b ∂a

(B.1)

where a is the actual crack length and b is the width of the specimen. (b) Beam theory [22] The load-line compliance CBT and the mode-II ERR GBT II are given by the beam theory CBT ¼ GBT II ¼

2L3 + 3a3 8E1 bh3 9a2 P2 16E1 b2 h3

where E1 is the longitudinal elastic modulus.

(B.2) (B.3)

Appendix C: Analytical formulas for the load-displacement curves

171

(c) Modified beam theory [23] Eqs. (C.2), (C.3) were modified to include the effect of transverse shear deformation. 1:2L + 0:9a 4bG13 h "   2 # E1 h BT GSH II ¼ GII 1 + 0:2 a G13 CSH ¼ CBT +

(B.4)

(B.5)

where G13 is the shear modulus. (d) Associated beam theory and first-order shear deformation theory [19] By combining the laminate beam theory and the first order shear deformation theory, the compliance and the mode-II ERR are given by



 a ða55 ÞAB  ða55 ÞBC L3 ðd11 ÞBC Lða55 ÞBC a3 ðd11 ÞAB  ðd11 ÞBC + + + CSBT ¼ 12b 4bk 6b 2bk (B.6)

 ða55 ÞAB  ða55 ÞBC P2 2 + GSBT ¼ a ð d Þ  ð d Þ (B.7) 11 AB 11 BC II 8b2 k where k ¼ 5/6 is the shear correction factor, AB is the delaminated section, BC is a part of the intact section of the ENF specimen in Fig. 7.5, and (d11)AB, (d11)BC, (a55)AB, and (a55)BC are the effective bending and shear compliances, respectively. To calculate CSBT and GSBT for the ENF specimen, the effective II bending and shear compliances are calculated according to the method proposed by Ozdil et al. [19].

Appendix C: Analytical formulas for the load-displacement curves for mode-II delamination of ENF composites According to the analytical formulas derived by Mi et al. [24], the loaddisplacement curve of the ENF composite specimen is divided into three parts:
 P 2L3 + 3a30 (C.1) Curve  OB : Δ ¼ 96E1 I "
3=2 # c 64G bE I P 1 II p ffiffiffi 2L3 + Curve  ABCða < LÞ : Δ ¼ (C.2) 96E1 I 3 P3

172

7. Cohesive/friction coupled model

"
3=2 # 64GcII bE1 I P 3 pffiffiffi 2L  Curve  DEða > LÞ : Δ ¼ 24E1 I 4 3 P3

(C.3)

where I is the second-order inertia moment and a0 is the initial crack length.

References [1] Wisnom MR, Petrossian ZJ, Jones MI. Interlaminar failure of unidirectional glass/epoxy due to combined through thickness shear and tension. Compos A: Appl Sci Manuf 1996;27(10):921–9. [2] Fan C, Ben Jar PY, Roger Cheng JJ. A unified approach to quantify the role of friction in beam-type specimens for the measurement of mode II delamination resistance of fibrereinforced polymers. Compos Sci Technol 2007;67:989–95. [3] Raous M, Cangemi L, Cocu M. A consistent model coupling adhesion, friction, and unilateral contact. Comput Method Appl Mech Eng 1999;177:383–99. [4] Alfano G, Sacco E. Combining interface damage and friction in a cohesive zone model. Int J Numer Methods Eng 2006;68:542–82. [5] Parrinello F, Failla B, Borino G. Cohesive-frictional interface constitutive model. Int J Solids Struct 2009;46:2680–92. [6] Parrinello F, Marannano G, Borino G, Pasta A. Frictional effect in mode II delamination: experimental test and numerical simulation. Eng Fract Mech 2013;110:258–69. [7] Guiamatsia I, Nguyen GD. A thermodynamics-based cohesive model for interface debonding and friction. Int J Solids Struct 2014;51:647–59. [8] Serpieri R, Sacco E, Alfano G. A thermodynamically consistent derivation of a frictionaldamage cohesive-zone model with different mode I and mode II fracture energies. Eur J Mech-A/Solids 2015;49:13–25. [9] Xu XP, Needleman A. Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 1994;42(9):1397–434. [10] Van den Bosch M, Schreurs P, Geers M. An improved description of the exponential Xu and Needleman cohesive zone law for mixed-mode decohesion. Eng Fract Mech 2006;73:1220–34. ´ Ma´irtı´n E,  Parry G, Beltz GE. Potential-based and non-potential-based [11] McGarry JP, O cohesive zone formulations under mixed-mode separation and over-closure. Part I: theoretical analysis. J Mech Phys Solids 2014;63:336–62. [12] Gao YF, Bower AF. A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces. Model Simul Mater Sci Eng 2004;12:453–63. [13] Abdul-Baqi A, Van der Giessen E. Numerical analysis of indentation-induced cracking of brittle coatings on ductile substrates. Int J Solids Struct 2002;39:1427–42. [14] Hattiangadi A, Siegmund T. An analysis of the delamination of an environmental protection coating under cyclic heat loads. Eur J Mech-A/Solids 2005;24:361–70. [15] Goyal VK, Johnson ER, Davila CG. Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities. Compos Struct 2004;65:289–305. [16] Liu PF, Islam MM. A nonlinear cohesive model for mixed-mode delamination of composite laminates. Compos Struct 2013;106:47–56. [17] Weyler R, Oliver J, Sain T, Cante J. On the contact domain method: a comparison of penalty and Lagrange multiplier implementations. Comput Methods Appl Mech Eng 2012;205:68–82. [18] Wriggers P. Computational contact mechanics. 2nd ed. Berlin, New York: Springer; 2006.

References

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[19] Ozdil F, Carlsson L, Davies P. Beam analysis of angle-ply laminate end-notched flexure specimens. Compos Sci Technol 1998;58:1929–38. [20] Theotokoglou E, Vrettos C. A finite element analysis of angle-ply laminate end-notched flexure specimens. Compos Struct 2006;73:370–9. [21] Broek D. Elementary engineering fracture mechanics. 3rd ed. The Hague: Martinus Nijhoff Publishers; 1984. [22] Russell A, Street K. Factors affecting the interlaminar fracture energy of graphite/epoxy laminates. In: Progress in science and engineering of composites. Tokyo, Japan: ICCMIV; 1982. p. 279–86. [23] Carlsson L, Gillespie J, Pipes R. On the analysis and design of the end notched flexure (ENF) specimen for mode II testing. J Compos Mater 1986;20:594–604. [24] Mi Y, Crisfield MA, Davies GA, Hellweg HB. Progressive delamination using interface elements. J Compos Mater 1998;32:1246–72. [25] Sch€ on J. Coefficient of friction of composite delamination surfaces. Wear 2000;237:77–89.

C H A P T E R

8

Viscoelastic bilinear cohesive model and parameter identification for failure analysis of adhesive composite joints using explicit finite element analysis 8.1 Introduction In Chapter 2, we introduce the bilinear and exponential cohesive models to predict the quasi-static failure adhesive joints. By introducing a phenomenological traction-separation relationship within a finite length process zone ahead of the crack tip, cohesive models have achieved success in predicting failure behaviors of adhesive composite joints with complex geometry, material nonlinearity, and large deformation. Various cohesive models in terms of the shape of cohesive laws were developed a priori for adhesive joints with different adhesive resins and curing parameters [1–6]. Since one main task is to link the fracture toughness for adhesive joints with various fracture modes to the cohesive law parameters, the bilinear cohesive model proposed mainly by Turon et al. [7] shows some advantage in deriving mixed-mode cohesive laws, although the peak in the curve is discontinuous. From Campilho et al. [4], the bilinear cohesive model is shown to be more applicable to brittle failure of adhesive joints. As a numerical approximation to the true finite-thickness

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00004-3

175

Copyright © 2021 Elsevier Inc. All rights reserved.

176

8. Viscoelastic bilinear cohesive model

adhesive layer, the zero-thickness middle-plane interface element technique shows favorable numerical robustness for implicit FEA although it requires to address the mesh size effect [8] and the numerical convergence problem [9] for both the bilinear and exponential cohesive models [7, 10]. Bazˇant and Li [11] showed that time dependence for interface fracture includes two cases: one is involved in the rate effect of solid materials and another denotes the rate effect of interface separation. The rate dependence of interface failure [12, 13] in the first case can be essentially taken as a projection of the rate effect of solid materials on the discontinuous interface from the variational principle of rate-dependent stress governing equation. For adhesive joints, it is natural to imagine that the rate effect of interface failure [14, 15] does not need to resort to the rate effect of solid materials, which then falls into the framework of the second case. In this case, cohesive models can be extended to further consider the rate effect of failure behaviors of adhesive joints due to creep or relaxation behaviors of adhesive resin, which exhibit a time delay relationship physically between the cohesive traction and the separation. Currently, there are some viscoelastic cohesive models to predict the rate effect of interface fracture. Rahul Kumar et al. [16] extended Xu and Needleman’s exponential cohesive model [17] to predict the rate-dependent interface fracture toughness as a function of the cohesive law parameters and loading velocity for peel testing of polymers. Xu et al. [18] developed a rate-dependent model to predict the failure behaviors of adhesive joints by adding the contribution of a Maxwell element with the combination of springs and dashpots. By using the modified model developed by Van Den Bosch et al. [19] on the Xu and Needleman’s exponential cohesive model [17], Zreid et al. [20] developed a thermodynamically consistent viscoelastic cohesive model, in which the rate-dependent cohesive traction is represented by a Maxwell element. Allen and Searcy [21] developed a microscopicmacroscopic homogenization viscoelastic cohesive model to yield a damage dependent integral-type cohesive traction-separation law, but the relaxation moduli are directly given to analyze time-dependent interface fracture behaviors. Although there was several work on the identification of rate-independent cohesive law parameters [22–26], little work was focused on the parameter identification of viscoelastic cohesive models except for the work of Upadhyaya et al. [27]. Based on the Schapery’s linear viscoelastic theory [28], this chapter extends the bilinear cohesive model [7] to the viscoelastic case, in which the additional integral-type viscous cohesive traction is superimposed. The relaxation moduli in the viscous cohesive traction are further expanded by the Prony series, which can be equivalent to the spring and dashpot models above [18–20]. The finite-thickness middle-plane interface element technique [5, 6] is adopted to implement the regular

8.2 Viscoelastic bilinear cohesive model

177

bilinear cohesive model. It should be recognized that the phenomenological viscoelastic cohesive models [18–20] and the homogenized viscoelastic cohesive models [21, 27] were developed from different perspectives. Different from the work by Upadhyaya et al. [27] who used the macroscopic J-integral data to extract the microscopic cohesive tractionseparation behaviors according to the homogenized viscoelastic cohesive model [21], the originality is to develop an inverse method to identify the macroscopic viscoelastic cohesive law parameters from the phenomenological perspective by combining explicit FEA and experiments. Numerical analysis by ABAQUS-Python script language and MATLAB software is performed to explore the effects of cohesive strengths and load rates on the load responses for single-lap adhesive joints with brittle adhesive materials under low strain rate tension. Form FEA, the developed inverse method is shown to find optimal viscoelastic cohesive parameter, i.e., the Prony series parameter configurations efficiently.

8.2 Viscoelastic bilinear cohesive model The viscoelastic cohesive traction-separation behaviors are similar to the relaxation tests of viscoelastic materials: the separations Δi are loaded at the start of the test and then kept constant. The instantaneous tractionsseparations are written as ðt ti ðtÞ ¼ Ki ðt  sÞΔ_ i ðsÞds ¼ Ki ðtÞΔi (8.1) 0

where Ki(t)(i ¼ 1, 2, 3) and Δi (i ¼ 1, 2, 3) are the relaxation moduli and separations in the tangential and normal directions, respectively. i ¼ 1 denotes the normal direction and i ¼ 2, 3 are the tangential direction. Ki(t) can be further written as the dimensionless forms ki ðtÞ ¼ Ki ðtÞ=Ki 0 where Ki ¼ Ki ð0Þ are the instantaneous moduli. Thus, the traction-separation relationships are rewritten as ðt ti ðtÞ ¼ Ki 0 ki ðt  sÞΔ_ i ðsÞds

(8.2)

0

(8.3)

0

By performing integration by part, Eq. (8.3) is transformed to   ðt ti ðtÞ ¼ Ki 0 Δi + k_ i ðsÞΔ_ i ðt  sÞds

(8.4)

0

Finally, the tractions are expressed as the summation of the regular instantaneous tractions ti 0 ðtÞ and the rate-dependent tractions at time t

178

8. Viscoelastic bilinear cohesive model

t i ðt Þ ¼ t i 0 ðt Þ +

ðt

k_ i ðsÞti 0 ðt  sÞds

(8.5)

0

where the regular term ti 0 ðtÞ in the bilinear cohesive model [7] are given in Chapter 2. k_ i is the rate of ki. It is noted that the viscoelastic effect will disappear completely when the normal and tangential separations Δi (i ¼ 1, 2, 3) exceed the corresponding critical values Δfi (i ¼ 1, 2, 3). Thus, the last term in Eq. (8.5) is set to zero after that time. The dimensionless relaxation moduli ki can be expanded by the Prony series N   X P t=τi ki ¼ 1  kij 1  e j (8.6) j¼1

P kij

where (j ¼ 1, 2, 3, …, N) is the jth dimensionless coefficient of the Prony series and τij (j ¼ 1, 2, 3, …, N) is the jth reciprocal of retardation time (unit: P s). As they represent the viscoelastic effect, the parameters kij , and τij in Eq. (8.6) will be determined inversely by combining numerical analysis and experiments below. Then, Eq. (8.5) can be further expressed as 8 9 8 9 8 t0 9 > tj1 > > > = < > < t1 = < 1 = X N > N X 0 j ¼ t0  t ¼ t2 ¼ t2  tj (8.7) t 2> > > : ; > > ; > : j¼1 j¼1 t3 ; : 0 j t3 t3 8 Pð > > k1j t s=τ1 0 > j > t ¼ e j t1 ðt  sÞds, > > 1 > τ1j 0 > > > > > > Pð < k2j t s=τ2 0 j (8.8) t2 ¼ 2 e j t2 ðt  sÞds, > τj 0 > > > > > > Pð > > k3j t s=τ3 0 > j > j t ðt  sÞds, > > 3 : t3 ¼ τ 3 e 0 j

8.3 Explicit FEA for predicting rate-dependent cohesive failure The recursive algorithm [29] is used to solve the viscous cohesive traction. There are three reasons for selecting explicit FEA: (1) The timestepping algorithm is appropriate for dealing with time relevance of failure. (2) Implicit FEA for implementing cohesive models often meets severe numerical convergence problems. (3) The consistent tangent stiffness for integral-type viscoelastic model cannot be derived. Dynamic equilibrium of the weak form of rate-dependent cohesive interface Γc is given by

8.4 Numerical analysis for predicting rate-dependent failure of adhesive joints

ð

ð Γc

ρc δu_  u€ tc dΓ d +

Γc

δΔ_  t dΓ d ¼

179

ð Γ

δu_  f dΓ

(8.9)

where δu_ is the virtual velocity, u€ is the acceleration, ρc is the density of cohesive interface and f is the external force on the cohesive interface. In Eq. (8.9), the left first term denotes the virtual dynamic energy for cohesive interface, the left second term represents the virtual internal energy and the right term denotes the virtual external work. In general, a finite-thickness cohesive interface tc 6¼ 0 is used for explicit FEA. After performing discretization and linearization on Eq. (8.9), the nonlinear finite element equation for the cohesive interface is given by   nc Α M c u€ + F coh ¼ F ext , e¼1 ð~ ð (8.10) e TM e tc dΓ, F coh ¼ M e T t dΓ M c ¼ ρc M Γc

Γc

where Α denotes the assembly of cohesive element stiffnesses. nc is the number of cohesive elements. u€ is the node acceleration and Mc is the ~ elements. F and F are the cohesive lumped mass matrix for cohesive coh ext e is node force and the external node force, respectively. The strain matrix M shown in Chapter 2. The central difference algorithm [30] is used to solve Eq. (8.10), where sufficiently small time increment is required to guarantees numerical stability. The total cohesive traction t is split additively into a regular cohesive traction and a viscous cohesive traction, in which the viscous traction is related to the regular traction according to Eq. (8.5). Because the total traction is used for explicit FEA in Eq. (8.10), the calculated separation according to Eq. (8.10) is affected by the viscoelastic effect. Because the viscoelastic effect applies to the cohesive traction-separation law indirectly, the traction-separation law still shows progressive damage behaviors.

8.4 Numerical analysis for predicting rate-dependent failure of adhesive joints 8.4.1 Inverse parameter identification of the Prony series in the relaxation moduli Fig. 8.1 shows the SLJ with geometry sizes and boundary conditions for numerical analysis and experiments. Carbon fiber reinforced bismaleimide resin (T700/QY8911) composite laminates with [0°/0°]4s layup pattern are used for adherends and J-116 epoxy resin is used for adhesive layer. We assume mechanical parameters of adherends are

180 25 mm

8. Viscoelastic bilinear cohesive model

25 mm 2 mm

tc = 0.2 mm

2 mm U 25 mm

70 mm 120 mm

FIG. 8.1 Single-lap joint (SLJ) with geometry sizes and boundary conditions. TABLE 8.1 Material parameters for T700/8911 composites. E11 (GPa)

E22 5 E33 (GPa)

G12 5 G13 (GPa)

G23 (GPa)

ν12 5 ν13

ν23

135

9.25

5.65

3.55

0.31

0.299

rate-independent and their elastic parameters are listed in Table 8.1 by performing longitudinal and transverse tensile and shear experiments on unidirectional composites. By considering the short time before final tensile fracture of specimens, the tensile tests of SLJ with three low strain rates 1, 5, and 50 mm/min, respectively, are performed by using the electronic universal testing, as shown in Fig. 8.2. During tests, the loaddisplacement/time curves are recorded until complete failure of joints. The location of failure of joints concentrates almost on the adhesive layer. The finite element software ABAQUS is used, in which the reduced integration element CPE4R in the plane strain condition is used for solids and the cohesive element COH2D4 is used for adhesive joints. For explicit FEA, the densities of solid elements and cohesive elements are ρ ¼ 2000 kg/m3 and ρc ¼ 1200 kg/m3, respectively. Here, the density of cohesive elements ρc is equal to that of adhesive material J-116 resin. It deserves pointing out the inertia effect in explicit FEA should be reduced largely that requires to select ρc properly. Mesh model is shown in Fig. 8.3, where the sizes of solid elements and cohesive elements for adhesive joint

FIG. 8.2 Tensile experiments of SLJ and fractured specimens.

8.4 Numerical analysis for predicting rate-dependent failure of adhesive joints

FIG. 8.3

181

Mesh model for SLJ with solid elements and cohesive elements.

area are 0.5 mm  0.25 mm and 0.25 mm  0.2 mm, respectively. The displacement load is applied to the right end of specimens. In general, the loading rate 1 mm/min is used and the total calculation time is 20 s in explicit FEA. In ABAQUS explicit module, the stable time will be automatically evaluated.  Here, the 1st dimensionless coefficient of the Prony series P

P

P

k11 , k21 ¼ k31 and the 1st reciprocal of retardation time (τ11 ¼ τ12 ¼ τ13) are

considered. The input viscoelastic cohesive law parameters in explicit FEA include the cohesive strengths (σ max, τmax), the fracture toughnesses c c c of  adhesive  joints (G1, G2 ¼ G3), and the parameters in the Prony series P

P

k11 , k21 , τ11 . For rate-independent cohesive models, the cohesive

strengths were evaluated using some advanced experimental approaches [22–25]. Here, we do not attempt to identify the cohesive strengths, but to compare their effects on the load responses and failure behaviors of adhesive joints, similar to the work of Campilho et al. [4] and Liu et al. [5, 6]. In general, the cohesive strengths σ max ¼ 45 MPa and τmax ¼ 25 MPa are generally used, which are equal to the tensile and shear strengths of epoxy materials. After they are assumed constant under low strain rate loads, the normal and tangential fracture toughnesses of SLJ are Gc1 ¼ 0.40 N/mm and Gc2 ¼ Gc3 ¼ 0.61 N/mm, respectively. η ¼2.0 in Eq. (A.3) is used.  In order to determine the parameters

P

P

k11 , k21 , τ11

in the relaxation

moduli, an inverse method is developed that aims to reach an optimal parameter configuration when the numerical load curve approximates the experimental load curve with a least-square error after all curves are discretized into a series of data points (p ¼ 1,2, …, n). The objective function from the mathematical perspective is defined as n   P P  X  RFexp  RFnum =RFexp (8.11) F k11 , k21 , τ11 ¼ P P P P¼1

where RFPexp and RFnum are numerical and experimental forces at discrete P data points, respectively. The inverse parameter identification method is implemented by combining ABAQUS-Python script language and MATLAB software, as shown in Fig. 8.4. The inverse algorithm is divided into

182

8. Viscoelastic bilinear cohesive model

FIG. 8.4 Parameter identification algorithm by combining MATLAB and ABAQUS software.

three steps: (1) MATLAB calls ABAQUS to get the load-time curve. (2) ABAQUS-Python script language reads the ABAQUS-ODB result file and transfers data to MATLAB. (3) MATLAB calculates the objective function and performs optimization search using the pattern recognition   algorithm. Finally, the optimal parameter configuration P

P

k11 , k21 , τ11 is obtained.

183

8.4 Numerical analysis for predicting rate-dependent failure of adhesive joints

Here, we first use one experimental/numerical load under the  P curve  P 1 load rate 1 mm/min to determine the parameters k11 , k21 , τ1 by the  P P  inverse method and then apply the determined parameters k11 , k21 , τ11 to different load rate cases for validation by comparing numerical with experimental load curves under the load rates 5 mm/min and 50 mm/min.  Figs. 8.5and 8.6 show the iterative processes of the inverse parameters P

P

P

P

k11 , k21 , τ11 and the objective function F at the initial values k11 ¼ k21 ¼ 0:4 P

P

and τ11 ¼ 0.4 and their ranges 0:001 < k11 ¼ k21 < 1 and 0.001 < τ11 < 1. The P

P

optimal results are k11 ¼ k21 ¼ 0:3 and τ11 ¼ 0.21 after 60 iterations when the objective function F is below 1%. In order to check the sensitivity of P

P

initial parameters, two group of parameters k11 ¼ k21 ¼ 0:01, τ11 ¼ 0.01, P

P

and k11 ¼ k21 ¼ 0:9 and τ11 ¼ 0.9 are further selected. Figs. 8.7–8.10 show the corresponding results. It is shown that the optimal results are almost the same for different initial parameter configurations. However, when the difference between the initial and optimal parameter values is large, P

P

more iterations are required to reach stability. In the following, k11 ¼ k21 ¼ 0:3 and τ11 ¼ 0.21 are used.

FIG. 8.5

P

P

Parameter identification process at the initial values k11 ¼ k21 ¼ 0:4 and τ11 ¼ 0.4.

184

8. Viscoelastic bilinear cohesive model

P

P

FIG. 8.6 Objective function at the initial values k11 ¼ k21 ¼ 0:4 and τ11 ¼ 0.4.

FIG. 8.7

P

P

Parameter identification process at the initial values k11 ¼ k21 ¼ 0:01 and τ11 ¼ 0.01.

8.4 Numerical analysis for predicting rate-dependent failure of adhesive joints

P

185

P

FIG. 8.8

Objective function at the initial values k11 ¼ k21 ¼ 0:01 and τ11 ¼ 0.01.

FIG. 8.9

Parameter identification process at the initial values k11 ¼ k21 ¼ 0:9 and τ11 ¼ 0.9.

P

P

186

FIG. 8.10

8. Viscoelastic bilinear cohesive model

P

P

Objective function at the initial values k11 ¼ k21 ¼ 0:9 and τ11 ¼ 0.9.

8.4.2 Rate-dependent failure analysis of adhesive joints By comparison, two mesh models with the sizes of cohesive elements 0.5 mm  0.2 mm and 0.25 mm  0.2 mm are used. Because explicit FEA is used, no convergence problem appears. Fig. 8.11 shows the crack initiation and propagation behaviors at the cohesive strengths σ max ¼ 45 MPa and τmax ¼ 25 MPa. Crack initiates and propagates both two ends of the cohesive interface at the displacement 0.276 mm. The adhesive specimens fail completely in the form of the dominating shear fracture at the displacement 0.306 mm. From the tensile test of SLJ, the experienced time

FIG. 8.11

Cohesive crack initiation and propagation at the displacement 0.276 mm.

8.4 Numerical analysis for predicting rate-dependent failure of adhesive joints

187

from crack initiation to complete failure, i.e., the decreasing part of load curves is very short. For rate-independent interface failure, the shape of cohesive model and the cohesive strengths are two important factors that affect the failure behaviors and strengths of adhesive joints [3]. Campilho et al. [4] studied the effects of cohesive law parameters for brittle and ductile adhesives, and found the bilinear cohesive model is more applicable to brittle epoxy adhesive. Here, we extend the bilinear cohesive model to the viscoelastic case and then compare the effects of different cohesive strengths below. Fig. 8.12 shows the load-time curves at three groups of cohesive strengths: σ max ¼ 30 MPa and τmax ¼ 15 MPa, and σ max ¼ 45 MPa and τmax ¼ 25 MPa, and σ max ¼ 60 MPa and τmax ¼ 40 MPa. The numerical curve is consistent with the experimental curve at σ max ¼ 45 MPa and τmax ¼ 25 MPa, similar to the results of Campilho et al. [4]. The increase of cohesive strengths leads to larger strengths of adhesive joints, similar to the conclusion by Liao and Huang [31]. In addition, two mesh sizes lead to almost consistent load curves. Liu et al. [6] and Turon et al. [7] show that computational efficiency by implicit FEA will decrease with the increase of cohesive strengths, but will not be affected by explicit FEA.

FIG. 8.12

Load-time curves at different cohesive strengths using two mesh models.

188

FIG. 8.13

8. Viscoelastic bilinear cohesive model

Numerical and experimental load-displacement curves at three load rates.

Fig. 8.13 shows the effects of three load rates on the load-displacement curves. Here, we assume the mode-I and II fracture toughnesses Gc1 and Gc2 of SLJ keep constant under low load rates. It is concluded that the decreasing part in the load curve shows some difference for different load rates and the ultimate failure load adds with the increase of load rate, similar to the conclusion obtained by Musto and Alfano [15] and Zreid et al. [20]. In addition, numerical results are in relatively good agreement with experimental results at three load rates, which shows the developed parameter identification method is acceptable.

8.5 Concluding remarks This chapter develops an inverse parameter identification method to identify viscoelastic cohesive law parameters for failure analysis of adhesive joints under low strain rate loading. From explicit FEA and experiments of SLJ, the developed method is shown to be robust and efficient. The increase of load rates and cohesive strengths increases the loadbearing abilities of joints. The numerical method can be applied to joints with various adhesive materials with rate effects.

References

189

References [1] Sarrado C, Turon A, Costa J, Renart J. An experimental analysis of the fracture behavior of composite bonded joints in terms of cohesive laws. Compos Part A Appl Sci Manuf 2016;90:234–42. [2] Belnoue JPH, Hallett SR. Cohesive/adhesive failure interaction in ductile adhesive joints part I: a smeared-crack model for cohesive failure. Int J Adhes Adhes 2016;68:359–68. [3] Gustafson PA, Waas AM. The influence of adhesive constitutive parameters in cohesive zone finite element models of adhesively bonded joints. Int J Solids Struct 2009;46:2201–15. [4] Campilho RDSG, Banea MD, Neto JABP, da Silva LFM. Modelling adhesive joints with cohesive zone models effect of the cohesive law shape of the adhesive layer. Int J Adhes Adhes 2013;44:48–56. [5] Liu PF, Gu ZP, Peng XQ. A nonlinear cohesive/friction coupled model for shear induced delamination of adhesive composite joint. Int J Fract 2016;199:135–56. [6] Liu PF, Gu ZP, Peng XQ. Finite element analysis of the influence of cohesive law parameters on the multiple delamination behaviors of composites under compression. Compos Struct 2015;131:975–86. [7] Turon A, Camanho PP, Costaa J, Da´vila CG. A damage model for the simulation of delamination in advanced composites under variable-mode loading. Mech Mater 2006;38(11):1072–89. [8] Harper PW, Hallett SR. Cohesive zone length in numerical simulations of composite delamination. Eng Fract Mech 2008;75(16):4774–92. [9] Liu PF, Islam MM. A nonlinear cohesive model for mixed-mode delamination of composite laminates. Compos Struct 2013;106:47–56. [10] Park K, Paulino GH, Roesler JR. A unified potential-based cohesive model of mixedmode fracture. J Mech Phys Solids 2009;57(6):891–908. [11] Bazˇant ZP, Li YN. Cohesive crack with rate-dependent opening and viscoelasticity: I. Mathematical model and scaling. Int J Fract 1997;86(3):247–65. [12] Van Der Bosch MJ, Schreurs PJG, Geers MGD. On the development of a 3D cohesive zone element in the presence of large deformations. Comput Mech 2006;42:171–80. [13] Nguyen TD, Govindjee S. Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth. Int J Fract 2006;141(102):255–68. [14] Lißner M, Alabort E, Cui H, Pellegrino A, Petrinic N. On the rate dependent behaviour of epoxy adhesive joints: experimental characterisation and modelling of mode I failure. Compos Struct 2018;189(1):286–303. [15] Musto M, Alfano G. A novel rate-dependent cohesive-zone model combining damage and visco-elasticity. Comput Struct 2013;118(6):126–33. [16] Rahulkumar P, Jagota A, Bennison SJ, et al. Cohesive element modeling of viscoelastic fracture: application to peel testing of polymers. Int J Solids Struct 2000;37(13):1873–97. [17] Xu XP, Needleman A. Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 1994;42(9):1397–434. [18] Xu CC, Siegmund T, Ramani K. Rate-dependent crack growth in adhesives: I. Modeling approach. Inter J Adhes Adhes 2003;23(1):9–13. [19] Van Den Bosch M, Schreurs P, Geers M. An improved description of the exponential Xu and Needleman cohesive zone law for mixed-mode decohesion. Eng Fract Mech 2006;73:1220–34. [20] Zreid I, Fleischhauer R, Kaliske M. A thermomechanically coupled viscoelastic cohesive zone model at large deformation. Int J Solids Struct 2013;50(25–26):4279–91. [21] Allen DH, Searcy CR. A micromechanical model for a viscoelastic cohesive zone. Int J Fract 2001;107:159–76. [22] Ferracin T, Landis CM, Delannay F, Pardoen T. On the determination of the cohesive zone properties of an adhesive layer from the analysis of the wedge-peel test. Int J Solids Struct 2003;40(11):2889–904.

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[23] Sørensen BF, Jacobsen TK. Characterising delamination of fibre composites by mixedmode cohesive laws. Compos Sci Technol 2009;69(3–4):445–56. [24] Azevedo JCS, Campilho RDSG, da Silva FJG, Faneco TMS, Lopes RM. Cohesive law estimation of adhesive joints in mode II condition. Theor Appl Fract Mech 2015;80:143–54. [25] Alfano M, Lubineau G, Paulino GH. Global sensitivity analysis in the identification of cohesive models using full-field kinematic data. Int J Solids Struct 2015;55(1):66–78. [26] Cui H. Simulation of ductile adhesive failure with experimentally determined cohesive law. Compos Part B Eng 2016;92(1):193–201. [27] Upadhyaya P, Roy S, Haque MH, Lu HB. A novel numerical-experimental approach for predicting delamination in high temperature polymer matrix composites. Compos Struct 2013;118–24. [28] Schapery RA. Nonlinear viscoelastic solids. Int J Solids Struct 2000;37(1–2):359–66. [29] Haj-Ali RM, Muliana AH. Numerical finite element formulation of the Schapery nonlinear viscoelastic material model. Int J Numer Methods Eng 2004;59:25–45. [30] Hughes TJR. The finite element method: linear static and dynamic finite element analysis. NY: Dover Publications; 2000. [31] Liao LJ, Huang CG. Numerical analysis of effects of adhesive type and geometry on mixed-mode failure of adhesive joint. Int J Adhes Adhes 2016;68:389–96.

C H A P T E R

9

Viscoelastic cohesive/friction coupled model and explicit finite element analysis for delamination analysis of composite laminates under dynamic three-point bending 9.1 Introduction The rate effect shows significance for the delamination growth of composites. Van Der Bosch et al. [1] captured the rate sensitivity for interface debonding by performing peel tests of PET material, and Nguyen and Govindjee [2] demonstrated how the total fracture energy that is dissipated during crack propagation increases with the crack speed for an infinitely long strip of viscoelastic materials. Bazˇant and Li [3] pointed out that time dependence of fracture comes from two sources: the viscoelasticity of bulk materials and the rate process of the breakage of bonds, leading to rate-dependent softening law for crack opening. Theoretically, the variational weak form for the rate-dependent initial value problem also demonstrates the first argument. The second argument shows the interface failure may also be rate-dependent regardless of the bulk material. Thus, rate-dependent cohesive models are necessary for delamination analysis. The rate effect for the cohesive model means the cohesive traction depends on not only the separation, but also the variation of separation. From the energy perspective, the viscous dissipation disturbs the crack

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00007-9

191

Copyright © 2021 Elsevier Inc. All rights reserved.

192

9. Viscoelastic cohesive/friction coupled model

propagation behaviors within the fracture process zone. Kubair et al. [4] developed a numerical technique for the underlying Riemann-Hilbert problem for the chosen rate and damage-dependent cohesive law, and their results showed that the energy required for crack propagation increases with the crack propagation rate. Zhang et al. [5] studied the mode-I steady-state dynamic crack growth in rate-dependent solids using a cohesive model, in which the traction in the damage region is characterized by a softening power-law and the crack growth is controlled by the critical crack opening displacement at the physical crack-tip. Zhou et al. [6] adopted an initially rigid, linear-decaying cohesive law that relates the traction to the effective displacement and its rate of change, to simulate dynamic separation. Musto and Alfano [7] developed a novel ratedependent cohesive-zone model based on two assumptions including the rate-independence of fracture energy and the irrelevance of damage with fracture energy. Zreid et al. [8] developed a thermomechanically coupled differential-type viscoelastic cohesive model. Salih et al. [9] developed a differential-type viscoelastic cohesive model by considering the configuration of different springs and dashpots. Allen and Searcy [10] adopted a homogenization procedure to yield a damage dependent viscoelastic cohesive traction-separation law. Corigliano and Ricci [11] developed two time-dependent models for softening interface and the timedependent elastic damage interface. Giambanco and Scimemi [12] developed an interface law with softening response by considering the rate dependency of the fracture process zone. For dynamic delamination of L-shape composite laminates, Gozluklu et al. [13, 14] developed a modified rate-dependent bilinear cohesive model to predict the intersonic crack propagation. Another important problem is the cohesive fracture with the frictional contact. The degradation process of cohesive zone is often accompanied by the friction from the physical perspective because the thickness of adhesive layers is very thin. Yu et al. [15] used the linearly softening cohesive model developed by Camacho and Ortiz [16] to account for dynamic crack nucleation, closure and frictional sliding of composite plates, where the rate effect is assumed to come from the inertia. From the theoretical perspective, there is a natural transition from the cohesive state to the frictional contact state. However, it is very hard to identify which state is cohesive or frictional state from the physical perspective. Therefore, a favorable strategy is to couple the tangential shear traction and the friction contact stress. Raous et al. [17] developed a micromechanical adhesion/ friction coupled model by considering the contact zone as a material boundary. When the interface is decomposed into a damaged part with the friction and an undamaged part, Serpieri et al. [18] developed a bilinear cohesive/friction coupled model, and Parrinello et al. [19–21], Guiamatsia and Nguyen [22] and Serpieri et al. [18] further considered the

9.2 Viscoelastic cohesive model and numerical algorithm

193

mixed-mode interface failure under different load conditions. Further, Snozzi and Molinari [23] developed a linear cohesive/friction contact coupled model and Liu et al. [24] developed an exponential cohesive/ friction contact coupled model in Chapter 7. However, there is still no rate-dependent cohesive/friction coupled model to account for viscous interface failure that is mixed with the frictional effect. This chapter extends the cohesive model in Chapter 7 to the viscoelastic case by developing a viscoelastic cohesive/friction coupled model and numerical algorithm for delamination analysis of composite laminates under strain rate loading, where one important task to describe a continuous transition from the cohesive state to the frictional state by superimposing a variable tangential friction onto the regular tangential cohesive traction in the frictional sliding criterion. The regular cohesive traction comes from a modified Xu and Needleman’s exponential cohesive model with continuous mathematical description at the peak cohesive traction and the viscous cohesive traction is taken as the hereditary integral representation of linear viscoelasticity based on the Schapery viscoelastic theory [25]. The reason why we use the integral-type viscoelastic model is to facilitate theoretical derivation and algorithm implementation. By means of the stress relaxation concept, the viscous traction becomes a convolution of the dimensionless integral kernel function and the regular traction, in which the kernel function is expanded by the Prony series. After the frictional sliding criterion is satisfied, the numerical algorithm for solving the tangential cohesive traction is developed. Then, the recursive algorithm for solving the incremental stress and strain of viscoelastic solids developed by Haj-Ali and Muliana [26] is extended to the cohesive interface. Finally, the explicit finite element analysis is adopted to solve the traction and separation. Numerical results for the single-leg bending (SLB) and the end notched flexure (ENF) composite specimens using ABAQUS-VUEL subroutine are given.

9.2 Viscoelastic cohesive model and numerical algorithm for delamination analysis of composite laminates The modified exponential cohesive model by McGarry et al. [27] is given by ! 8  
 > ½½u1 ½½u1 ½½u22 > > t1 ½½u1 , ½½u2 ¼ σ max exp ð1Þ exp  exp  2 , > > δ1 δ1 > δ2 > < !   p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
 ½½u1 ½½u1 ½½u22 (9.1) exp  exp  2 , t2 ½½u1 , ½½u2 ¼ τmax 2 exp ð1Þ > > > δ δ δ 1 1 > 2 > h i > > 0:5 : δ ¼ ϕ =½σ exp ð1Þ, δ ¼ ϕ = τ ð0:5 exp ð1ÞÞ 1

1

max

2

2

max

194

9. Viscoelastic cohesive/friction coupled model

where t1 and t2 are the regular normal and tangential cohesive tractions, respectively. σ max and τmax are the normal and tangential cohesive strengths, respectively. ½½u1 and ½½u2 are the normal and tangential separations, respectively. δ1 and δ2 are the characteristic normal and tangential separations, respectively. GcI ¼ ϕ1 and GcII ¼ ϕ2 are the normal and tangential delamination fracture toughnesses, respectively. It is noted that the exponential cohesive tractions t1 and t2 hardly reach zero except at infinite separations ½½u1 and ½½u2. However, the actual critical separations ½½u1 and ½½u2 cannot reach infinity. Therefore, finite critical separations ½½uc1 and ½½uc2 are solved by assuming the accumulative cohesive fracture energy for the normal and tangential delamination respectively reaches the maximum 8ð c   ½½u 1 > ½½u1 ½½u1 > >   d½½u1 exp  > >
c  ½½uc1 > δ1 δ1 0 > >   ð ¼ δ1 ð1  λn Þ, ¼ λ ! 1 ) ½ ½u  + δ exp  1 1 > ∞ ½½u 1 > ½½u1 δ1 > 1 > exp  d ½ ½u  > 1 > > δ1 < 0 δ1 ! ð ½½u c2 > ½½u2 ½½u22 > > d½½u2 exp  > > 0 δ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > δ22 > c > ! > ¼ λ ! 1 ) ½ ½u  ¼ δ  ln ð1  λ2 Þ 2 2 > ð 2 > ∞ > ½½u2 ½½u22 > > exp  2 d½½u2 > : δ2 0 δ2 (9.2)

½½uc1  11.76δ1

½½uc2  3.035δ2

where at λ1 ¼ 0.9999 and at λ2 ¼ 0.9999. In the following, a viscoelastic cohesive model is originally developed by extending the Schapery-type viscoelastic theory for solids [25] to the interface. The total normal and tangential tractions t1,total and t2,total are divided into the regular parts t1 and t2 above, and the viscous parts t1,v and t2,v below, respectively. The normal and tangential viscous tractions t1,v and t2,v are written as the integral-type functions 8 ðt > > > < t1,v ðtÞ ¼ G1 ðt  γ Þ½½u_ 1 ðγ Þdγ, o (9.3) ðt > > > : t ðtÞ ¼ G ðt  γ Þ½½u_  ðγ Þdγ 2,v

o

2

2

where t denotes time.G1 and G2 are the normal and tangential relaxation stiffnesses, respectively. By considering a relaxation test for interface in which the separations are assumed to be applied to the delamination specimen suddenly and then held for a long time, the normal and tangential viscous tractions can be further written as

9.2 Viscoelastic cohesive model and numerical algorithm

195

8 ðt
 > > t ð t Þ ¼ G1 ðt  γ Þ½½u_ 1 ðγ Þdγ ¼ G1 ðtÞ½½u1 , ½½u_ 1 ¼ 0 for t > 0 > < 1,v o

ðt >
 > > : t2,v ðtÞ ¼ G2 ðt  γ Þ½½u_ 2 ðγ Þdγ ¼ G2 ðtÞ½½u2 , ½½u_ 2 ¼ 0 for t > 0

(9.4)

o

The viscoelastic interface is “long-term elastic” which means the response settles down to the constant tractions, i.e., G1(t) ! G1,∞ and G2(t) ! G2,∞ at t ! ∞ after having been subjected to a displacement jump for a relatively long time. In this case, the normal and tangential relaxation stiffnesses are written as the dimensionless forms g1 ðtÞ ¼ G1 ðtÞ=G1,0 , g2 ðtÞ ¼ G2 ðtÞ=G2, 0

(9.5)

where G1,0 ¼ G1(t ¼ 0) and G2,0 ¼ G2(t ¼ 0) are the instantaneous stiffnesses. By combining Eqs. (9.4), (9.5), the expressions for the viscous tractions can be further written as 8 ðt ðt > > _ > < t1,v ðtÞ ¼ G1,0 g1 ðt  γ Þ½½u_ 1 ðγ Þdγ ¼ g1 ðt  γ Þt 1 ðγ Þdγ, o o (9.6) ðt ðt > > > t ðt Þ ¼ G _ : g2 ðt  γ Þ½½u_  ðγ Þdγ ¼ g2 ðt  γ Þt 2 ðγ Þdγ 2,v 2,0 o

2

o

where g1(t ¼ 0) ¼ 1 and g2(t ¼ 0) ¼ 1, andg1(t ¼ ∞) ¼ G1(t ¼ ∞)/G1,0 and g2(t ¼ ∞) ¼ G2(t ¼ ∞)/G2,0. It is noted that the viscoelastic effect will disappear when the critical normal and tangential separations ½½uc1 and ½½uc2 are reached according to Eq. (9.4), and then the viscoelastic tractions t1,v and t2,v are set to zero after that time. Remarks (1) Eq. (9.6) shows that the viscous tractions are the convolution of the dimensionless stiffness and the rate of regular tractions, and the viscous tractions appear as the delayed relaxation responses of the regular tractions. (2) we do not attempt to divide the normal and tangential separations ½½u1 and ½½u2 into the elastic and viscoelastic parts in the present viscoelastic cohesive model since we take directly t1 and t2 as the functions of ½½u1 and ½½u2 in Eq. (9.1). (3) The total normal or tangential traction-the normal or tangential separation curves can be directly described no longer by a simple mathematical function. (4) It is noted that the fracture toughnesses GcI ¼ ϕ1 and GcII ¼ ϕ2 are related to the maximum values σ max and τmax for the regular tractions t1 and t2. Since the viscous traction t1,v(t) and t2,v(t) are also related to the regular tractions t1 and t2, which shows the viscous tractions affect the interface fracture toughness indirectly. Thus, we still approximately adopt the fracture toughnesses GcI and GcII in the viscoelastic cohesive model although the viscous effect may affect them. (5) Similarly, the critical separations in the viscoelastic cohesive model are also explicitly obtained no longer. According to the relaxation concept, the total normal and tangential cohesive tractions will decrease at

196

9. Viscoelastic cohesive/friction coupled model

constant normal and tangential separations. Thus, we still adopt the critical separations ½½uc1 and ½½uc2 in Eq. (9.2) in the viscoelastic cohesive model. By considering the temperature effect, the total normal and tangential tractions t1, total and t2, total are further written as 8 ðt > > < t1,total ðtÞ ¼ t1 ðtÞ + g1 ðψ ðtÞ  ψ ðγ ÞÞt_ 1 ðγ Þdγ ðot (9.7) > > : t2,total ðtÞ ¼ t2 ðtÞ + g2 ðψ ðtÞ  ψ ðγ ÞÞt_ 2 ðγ Þdγ o

where Eq. (9.7) shows the total tractions are considered as the addition of the regular tractions and the delayed viscoelastic tractions from both the physical and mathematical perspectives. The reduced time is Ðt ψ ðtÞ ¼ 0 Aðds θðsÞÞ, and A(θ(s)) is the shift function at time t that is generally given by the Arrhenius form or Williams-Landel-Ferry (WLF) form [3].   8 E0 1 1 > >  , Arrhenius form, < ln ðAÞ ¼ R θ  θ z θ 0  θz (9.8) C1 ðθ  θ0 Þ > > : log 10 ðAÞ ¼  , WLF form C 2 + ðθ  θ 0 Þ where θ is the actual temperature, θ0 is the reference temperature, θz is the absolute temperature, E0 is the activation energy, R is the universal gas constant and C1 and C2 are constants. The relaxation stiffnesses g1 and g2 in Eq. (9.7) are defined by the Prony series expansion 8 N X > > > g ð ψ ð t Þ Þ ¼ gi ½1  exp ðαi ψ ðtÞÞ 1 > < i¼1 (9.9) N h  i X > > > g2 ðψ ðtÞÞ ¼ g 1  exp β ψ ð t Þ > j j : j¼1

where N is the number of terms, gi and gj are the ith coefficient of the Prony series, and αi and βj are the ith reciprocal of retardation time with the unit s1. Substituting Eqs. (9.9) into (9.7) yields 8 N N X X > > > > t ð t Þ ¼ t ð t Þ + g t ð t Þ  gi q1 ðtÞ, 1 i 1 1,total > > > i¼1 i¼1 > > ðt > > > > q1 ðtÞ ¼ exp ½αi ðψ ðtÞ  ψ ðγ ÞÞt_ 1 ðγ Þdγ, < o (9.10) N N X X > > > > gi t 2 ð t Þ  gi q2 ðtÞ, t2,total ðtÞ ¼ t2 ðtÞ + > > > i¼1 i¼1 > > ðt > > > > : q2 ðtÞ ¼ exp ½βi ðψ ðtÞ  ψ ðγ ÞÞt_ 2 ðγ Þdγ o

9.3 Viscoelastic cohesive/friction coupled model and numerical algorithm

197

The recursive algorithm for solving the total cohesive tractions t2,total(t) and t1,total(t) sees Appendix-A. It is noted that the derivation above does not relate to the frictional problem.

9.3 Viscoelastic cohesive/friction coupled model and numerical algorithm for delamination analysis of composite laminates For pure mode-I delamination, no contact and friction after delamination appear. For mode-II and mixed-mode I/II delamination, frictional contact will generally appear. The contact mechanics requires no penetration between two neighboring solids. Fig. 9.1 shows the mode-II delamination mixed with friction. The normal relative displacement ½½u1 ¼ (x1  x2)  n < 0 is used to detect the contact and the penalty function method is generally adopted to calculate the normal traction t1 ¼  k1½½u1 based on the Kuhn-Tucker normal contact relationship ½½u1 0, t1 < 0 and ½½u1t1 ¼ 0. The normal stiffness k1 should be chosen appropriately to avoid ill-conditioned numerical problems. In the following, a viscoelastic cohesive/friction coupled model is developed, in which the basic thought is to couple the regular tangential traction and the friction. In this way, the total tangential traction as the addition of the regular and viscous tractions is also actually coupled with the friction. Liu et al. [24] developed a cohesive/friction contact coupled model, where the basic idea is to superimpose the friction onto the tangential traction to describe continuous transition from the cohesive state to the frictional state explicitly after the initial tangential softening, and the tangential separation ½½u2 is divided into the elastic part ½½ue2, and the inelastic part ½½ui2 due to friction,

FIG. 9.1

Delamination with the frictional contact (x1 and x2 are separated material points at the discontinuous interface).

198

9. Viscoelastic cohesive/friction coupled model

FIG. 9.2 Superimposed tangential friction increases exponentially from zero at the beginning of tangential cohesive softening to the smooth friction (friction coefficient μs) at the complete tangential cohesive failure where pure frictional contact appears.

and the friction coefficient μ increases gradually from zero to the smooth value μs. In addition, the tangential traction t2-the elastic tangential separation ½½ue2 is assumed to obey the curve in Fig. 9.2. The variable friction coefficient μ is assumed as pffiffiffi ! 2 e,c e δ2 ½½u2 ½½u2  2 exp ðdÞ  1 , d¼ μ ¼ μs (9.11) pffiffiffi ! exp ð1Þ  1 2 e e,c δ2 ½½u2 ½½u2  2 where d is the damage-like variable which represents the intensity of adhee sion. The critical value ½½ue,c 2  3.035δ2 of ½½u2 is still adopted in the cohesive/friction coupled model that is calculated similar to Eq. (9.4). Thus, the tangential delamination fracture toughness GcII becomes the area under the tangential traction-the elastic separation curve. In this work, we continue to adopt this idea above. From Liu et al. [24], it is not necessary to divide the normal separation ½½u1 into the elastic and inelastic parts for the contact surface. Remarks The cohesive/friction coupled model developed by Liu et al. [24] is only applicable to the pure mode-II delamination which means the normal contact always holds. However, it may not be this case for I/II mixed-mode delamination because we do not know when the normal contact or the initial tangential softening first appears. The developed cohesive/friction coupled model is assumed to become valid only when both these two conditions hold simultaneously. If the normal contact appears ahead of the initial tangential softening, there is no problem with the assumption in Fig. 9.2B. If the normal contact appears later than the initial tangential softening, we can still assume the relationship in Fig. 9.2B, which shows the increasing friction stress μ jt1 j is still calculated using the exponential function in Eq. (9.11). Differently, we assume

9.3 Viscoelastic cohesive/friction coupled model and numerical algorithm

199

pffiffiffi nonzero start of the friction stress at ½½ue2 > 2=2δ2 . If the normal contact does not appear after the critical separation ½½ue2 > ½½ue,c 2 , the cohesive/ friction coupled model will become unnecessary. The frictional slip criterion F is written as F ¼ t2 + μjt1 j  μ2 jt1 j > 0

(9.12)

where the tangential traction actually becomes t2 + μ jt1 j. Here, we adopt a new expression for calculating the inelastic tangential separation rate ½½u_ i2 in the cohesive/friction coupled model after Eq. (9.12) is satisfied

8 ϕðFÞ n > i > ½½u_  ¼ δ2 , > > < 2 t2, max ( (9.13) > 0, F  0 > > > : ϕðFÞ ¼ F=η, F > 0 where η and n are positive constants. It is noted that ½½ui2 increases but ½½u_ i2 decreases when the frictional contact criterion F > 0 holds during the tangential softening stage. Remarks The large-deformation theory is generally applicable to the cohesive model by introducing the middle-plane numerical interpolation technique. However, a point in a contact pair in one cohesive element cannot jump out to establish the contact relationship with another point in other cohesive elements in the cohesive/friction coupled model. Thus, small deformation is reduced from the initial tangential failure to the critical tangential failure in this work. After the critical cohesive failure, pure frictional contact status appears and large deformation contact becomes applicable again. After Eq. (9.12) is satisfied, the cohesive/friction contact algorithm for solving the elastic tangential separation ½½ue2 and the regular tangential traction t2(½½ue2) sees Appendix-B. The numerical algorithm for the viscoelastic cohesive/friction coupled model is summarized as follows. After the normal contact and the tangential cohesive softening appear, we judge whether the frictional slip criterion in Eq. (9.12) holds, where the tangential coupled cohesive/friction traction is assumed to become t2 + μ jt1 j by superimposing a variable tangential friction μ jt1 j onto the regular tangential cohesive traction t2. If Eq. (9.12) is satisfied, the tangential separation ½½u2 is divided into the elastic part ½½ue2 and the inelastic part ½½ui2, and the relationship between the tangential cohesive traction t2(½½ue2)-the elastic tangential separation ½½ue2 is assumed in Fig. 9.2, and the superimposed μ jt1 j is used to describe a continuous transition from the cohesive state to the frictional state explicitly. After Eq. (9.12) is satisfied, the elastic tangential separation ½½ue2 and the

200

9. Viscoelastic cohesive/friction coupled model

regular tangential traction t2(½½ue2) is solved according to the developed algorithm in Appendix-B by combining Eqs. (9.11)–(9.13). Finally, the total viscoelastic cohesive/friction traction t2, total is solved using Eq. (9.10) and Appendix-A. Similar to Chapter 8, explicit FEA is performed for dynamic delamination analysis.

9.4 Numerical results and discussion The developed theoretical model is implemented using ABAUS-VUEL (user dynamic element subroutine). The flowchart for numerical algorithm using VUEL is shown in Fig. 9.3, where the main task is to update the element mass, the cohesive node internal force, and the residual node force as well as the external force. In VUEL, the main material parameters transferred by the main imp file are: the normal contact stiffness, the cohesive strengths, the normal and tangential fracture toughnesses, the firstorder Prony series parameter and the first-order reciprocal of retardation time by “props” array, and several important history variables including the viscoelastic cohesive traction, the total cohesive traction, the inelastic tangential separation, the variable friction coefficient are defined by “svars” array. Also, a small value for the variable “stabletime” is required to guarantee stability. In the following, numerical results in terms of the single-leg bending (SLB) composite specimens with frictionless mixedmode delamination and the end notched flexure (ENF) composite specimens with frictional shear delamination under low strain rate loading are given. Because explicit FEA using cohesive elements with softening effect can easily lead to numerical oscillation of load curve, the numerical filtering technique is often needed to eliminate it largely. In addition, the “mass scaling method” in ABAQUS also helps to reduce the oscillation, but large mass will lead to inaccurate results. Thus, an appropriate mass is also important to get robust and accurate results. Of course, the mass scaling also helps to reduce computational time.

9.4.1 Single-leg bending (SLB) composite specimens Fig. 9.4 shows the unidirectional SLB composite specimens with frictionless mixed-mode delamination. T700/8911 composite specimens are prepared according to the Chinese standard- ASTM D6671/D6671M-06 and a Teflon film is inserted at the middle plane of specimens to make an initial delamination crack. The temperature effect is not considered in this analysis. The delamination analysis of [0°16//0°16] (“//” denotes the initial delaminated interface) unidirectional composite laminates

201

9.4 Numerical results and discussion

Give initial strain increment

Calculate cohesive element mass

Calculate global and local node separations

Normal separation

> 9P h > c > 1 + 0:22 : GII ¼ a 16b2 h3 E11

! ! ! !#  2    2 E11 1 E11 1 E11 1 E11 h h h 4 + 0:71 2 + 0:32 2 + 0:1 , a a a E33 E33 E33 G13 ! !#  2 E11 1 E11 h 2 + 0:048 a G13 G13

(9.14)

where E11 and E33 are the elastic moduli and G13 is the shear modulus. The plane strain problem is considered. The mesh model with 1 mm  0.5 mm solid element and 1 mm  0.001 mm cohesive element is generally used. The density for solid elements is 2000 kg/m and the TABLE 9.2 Material properties for T700/8911 composites and SLB specimens. Ply longitudinal modulus

E1

135 GPa

Ply transverse modulus

E2

11.41 GPa

Out-of-plane modulus

E3

11.41 GPa

Inplane shear modulus

G12

7.92 GPa

Out-of-plane shear modulus

G13

7.92 GPa

G23

3.79 GPa

v12

0.33

v13

0.33

v23

0.49

Mode-I delamination fracture toughness

GcI

0.1092 N/mm (a ¼ 41 mm)

Mode-II delamination fracture toughness

GcII

Poisson’s ratio

0.2699 N/mm (a ¼ 66 mm) 0.0754 N/mm (a ¼ 41 mm) 0.1922 N/mm (a ¼ 66 mm)

204

9. Viscoelastic cohesive/friction coupled model



density of cohesive elements 1000 kg/m3 is chosen to be equal to that of epoxy. In general, the low load rate 100 mm/min is considered, the displacement load is 10 mm and the total load time is 6 s. In order to determine the first-order parameters (g1, α1 ¼ β1) in the relaxation moduli in Eq. (9.10), the inverse method is adopted that aims to reach an optimal parameter configuration when the numerical load-time curve approximates the experimental curve with a least-squares error, where numerical and experimental curves are discretized into a series data points (p ¼ 1, 2, …, n). The objective function is defined as    X n   P P G num F k1 , k3 , τ i ¼ RFexp RFexp (9.15) p  RFp p p¼1

num where RFexp are experimental and numerical forces at discrete p and RFp data points, respectively. The inverse parameter identification method is implemented by combining ABAQUS software, Python script language, and MATLAB software. The whole algorithm is divided into three steps: (1) MATLAB calls ABAQUS to get the load-time curve. (2) ABAQUS-Python script language reads the ABAQUS-ODB result file and transfers data to MATLAB. (3) MATLAB calculates the objective function and performs optimization search using the pattern recognition algorithm. Finally, the optimal parameter configuration (g1, α1 ¼ β1) is obtained. It is noted that different strain rate loading cases can be applied when the optimal parameters (g1, α1 ¼ β1) are determined by the inverse method using one strain rate loading case. Figs. 9.6 and 9.7 show the load-displacement curves at the optimal parameters g1 ¼ 0.5 and α1 ¼ β1 ¼ 5.0 s1, and two cohesive strengths σ max ¼ τmax ¼ 20 MPa and 50 MPa, respectively. Fig. 9.8 shows the delamination growth process at different displacements. Numerical results at about σ max ¼ τmax ¼ 30 MPa are in agreement with experimental results. The specimen-A experiences the initial delamination at about 1.6 mm displacement, the unstable delamination at about 2 mm displacement, and then the stable delamination stages, but for specimen-B the initial delamination appears at about 2.6 mm displacement and no distinct unstable delamination stage appears. Unstable delamination for specimen-A means that the softening property of a cohesive element leads to the sudden release of strain energy in the lamina which leads to instantaneous failure of the element. In such a case, a snap-through situation arises in the load curve. By comparing two specimens, larger stiffness is obtained before the delamination for specimen-A than that for specimen-B because of shorter initial crack length for specimen-A than that for specimen-B. From the delamination initiation to stable delamination, the delamination rate is faster for specimen-A than for specimen-B. Before collapse of

9.4 Numerical results and discussion

205

FIG. 9.6 Load-displacement curves for SLB specimen-A with 50 mm initial crack length.

FIG. 9.7 Load–displacement curves for SLB specimen-B with 75 mm initial crack length.

206

9. Viscoelastic cohesive/friction coupled model

FIG. 9.8 Delamination process for SLB specimen-A at the displacement (A) 4 mm, (B) 5 mm, (C)8 mm, and (D) 10 mm.

specimens, the crack propagation length is 61 mm for specimen-A and 29 mm for specimen-B. In addition, the load-bearing ability for specimen-A is higher than that for specimen-B. We assume a continuous transition from the cohesive state to the pure frictional state once both the normal contact and the tangential softening conditions are met. It is noted that large tangential frictional sliding may appear that leads to severe distortion and premature complete failure of cohesive elements, and thus the frictional contact of delaminated surface will disappear suddenly and energy dissipation due to friction is underestimated, if the tangential frictional sliding and the cohesive element failure are not coupled. Snozzi and Molinari [23] introduced material damping to reduce the oscillation during frictional sliding and the amount of energy injected in the system in implicit FEA. This also offsets the underestimated energy dissipation due to friction to some extent. However, only small frictional sliding is allowed to appear when they are coupled according to the present cohesive/friction coupled framework, because a point in a contact pair in one cohesive element cannot jump out to establish the contact relationship with another point in other cohesive elements. Thus, severe cohesive element distortion problem that kills the job can be avoided in numerical analysis. After the cohesive element failure, pure frictional sliding is reduced that can still render point-topoint contact affiliated to the failed cohesive element surface.

9.4 Numerical results and discussion

207

9.4.2 End notched flexure (ENF) composite specimens Fig. 9.9 shows the unidirectional ENF composite specimens with shear/ frictional delamination. Elastic parameters for bulk elements and viscoelastic parameters for cohesive elements remain unchanged. The load rate is 100 mm/min, the displacement load is 10 mm and the total load time is 6 s. The parameters η ¼ 100 and n ¼ 1 in Eq. (9.12) are used. The mode-II delamination fracture toughnesses GcII ¼ 0.25 N/mm and 0.5625 N/mm for two group of initial delamination sizes s ¼ 50 mm and 75 mm are obtained from the modified beam theory [28]. "   2 # 9a2 P2 E1 h c GII ¼ 1 + 0:2 (9.16) a 16E1 b2 h3 G13 Figs. 9.10 and 9.11 show the load-displacement curves at two cohesive strengths τmax ¼ 20 MPa and 50 MPa, respectively. Fig. 9.12 shows the delamination growth process at different displacements. Numerical results are in agreement with experimental results at the cohesive strength about τmax ¼ 30 MPa. The ENF specimens show similar load responses for two groups of sizes, where the model with short initial crack length leads to distinct unstable delamination growth stage and that with length initial crack length results in distinct unstable delamination growth stage. Sch€ on [29] measured the friction coefficient for the epoxy with frictional sliding against metal, where the friction is related to the shear deformation or fracture of the polymer surface. They obtained the friction coefficients 0.2–0.7 in the rubber region. By comparison, the frictional effect appears at the initial unstable delamination stage and leads to slightly higher load-bearing

FIG. 9.9

Geometry of the ENF composite specimen.

208

FIG. 9.10

9. Viscoelastic cohesive/friction coupled model

Load-displacement curves for ENF specimen-A with 50 mm initial crack length.

FIG. 9.11 Load-displacement curves for ENF specimen-B with 75 mm initial crack length.

9.4 Numerical results and discussion

209

FIG. 9.12 Delamination growth process for ENF specimen-A at the displacement (A) 4 mm, (B) 5 mm, (C) 8 mm, and (D) 10 mm.

ability from the unstable delamination stage to the stable delamination stage. Figs. 9.13 and 9.14 show the distributions of the interlaminar shear stress τ12 at different loads. It is shown that the friction slightly increases the shear stress after specimens enter the unstable delamination stage. In addition, the maximum shear stress can also mark the crack tip location and the delamination growth process accurately. By comparison, Fig. 9.15 shows the load-displacement curve for ENF specimen A using a finer mesh model with the cohesive element size 0.25 mm  0.001 mm. It is shown that coarse and fine mesh models lead to basically consistent results. If the mode-II fracture toughness for ENF specimens under low strain rates keeps unchanged, Figs. 9.16 and 9.17 compare the loaddisplacement curves for specimen-A and B at 100, 200, and 300 mm/ min load rates at the cohesive strengths σ max ¼ τmax ¼ 30 MPa. The displacement load is also 10 mm, the total load time is 3 and 2 s for the load rate 200 and 300 mm/min, respectively. It is shown that the critical load increases slightly with the increase of strain rate, which is consistent with the conclusion obtained by Yasaee et al. [30]. For mode-II dynamic delamination, the shear-dominated crack growth along the fibers of unidirectional composites has been reported by Coker and Rosakis [31] who showed that shear fracture was featured by two discernible shock waves

210

9. Viscoelastic cohesive/friction coupled model

FIG. 9.13 Distribution of the interlaminar shear stress for ENF specimen-A at the displacement (A) 4 mm, (B) 5 mm, (C) 8 mm, and (D) 10 mm as well as the frictional coefficient μs ¼ 0.

FIG. 9.14 Distribution of the interlaminar shear stress for ENF specimen-A at the displacement (A) 4 mm, (B) 5 mm, (C) 8 mm, and (D) 10 mm as well as the frictional coefficient μs ¼ 0.3.

9.5 Concluding remarks

211

FIG. 9.15 (A) Fine mesh model and (B) load-displacement curves for ENF specimen-A with 50 mm initial crack length.

emanating from the crack tip. In addition, their experiments also furnished the evidence of large-scale frictional contact in the form of double-shock wave structures emitted from the crack tip and the end of the traveling contact zone.

9.5 Concluding remarks This chapter develops a viscoelastic cohesive/friction coupled model for delamination analysis of composite laminates. First, based on the modified Xu and Needleman’s exponential cohesive model and the Schapery viscoelastic theory, the viscoelastic traction is introduced as an addition of

212

9. Viscoelastic cohesive/friction coupled model

FIG. 9.16 Load-displacement curves for ENF specimen-A with 50 mm initial crack length

at the cohesive strengths σ max ¼ τmax ¼ 30 MPa under three load rates 100, 200, and 300 mm/ min, respectively.

FIG. 9.17 Load-displacement curves for ENF specimen-B with 75 mm initial crack length at the cohesive strengths σ max ¼ τmax ¼ 30 MPa under three load rates 100, 200, and 300 mm/ min, respectively.

Appendix-A. The recursive algorithm

213

the regular traction, which is the convolution between the dimensionless cohesive stiffness and the regular traction. The recursive algorithm is developed to solve the viscoelastic traction. Then, the viscoelastic cohesive/friction coupled model is developed, in which the tangential friction is assumed to appear after the initial tangential cohesive failure, and the exponentially increased tangential friction is superimposed onto the tangential cohesive traction in the frictional sliding criterion to describe a continuous transition from the cohesive state to the frictional state explicitly. Numerical algorithm is also developed to solve the cohesive/friction traction. Finally, two numerical examples including the SLB and ENF using explicit FEA are employed to explore the effects of cohesive strengths, friction and load rates on the load-displacement responses and the delamination growth behaviors. The developed cohesive model and numerical algorithm can be applied to the impact induced delamination research of composite laminates.

Appendix-A. The recursive algorithm for solving the tractions in the developed viscoelastic cohesive model Haj-Ali and Muliana [26] presented the recursive algorithm to solve the stress and strain in the Schapery-type nonlinear viscoelastic solid materials. Here, we extend this algorithm to the cohesive interface by taking the tangential direction for example. The integration expression q2(t) in Eq. (9.10) is divided into two parts for two-time ranges ð tΔt q 2 ðt Þ ¼ exp ½αi ðψ ðtÞ  ψ ðγ ÞÞt_ 2 ðγ Þdγ oð (A.1) t _ exp ½αi ðψ ðtÞ  ψ ðγ ÞÞt 2 ðγ Þdγ + tΔt

A reduced time increment is defined by ΔψðtÞ ¼ ψ ðtÞ  ψ ðt  ΔtÞ

(A.2)

The first term in Eq. (A.1) can be expressed by the integral at the time t  Δt and the current reduced time ð tΔt exp ½αi ðψ ðtÞ  ψ ðγ ÞÞt_ 2 ðγ Þdγ ¼ exp ½αi ΔψðtÞq2 ðt  ΔtÞ (A.3) o

where q2(t  Δt) is a hereditary integral at the end of previous time t  Δt that can be taken as the history variables updated and stored at the end of each time increment.

214

9. Viscoelastic cohesive/friction coupled model

If t2(γ) is considered to be linear over the current time step increment Δt and the shift parameter is not directly a function of time, the second term in Eq. (A.1) can be performed by parts ðt exp ½αi ðψ ðtÞ  ψ ðγ ÞÞt_ 2 ðγ Þdγ tΔt (A.4) 1  exp ½αi ΔψðtÞ ½t2 ðtÞ  t2 ðt  ΔtÞ ¼ αi ΔψðtÞ Substituting Eqs. (A.3), (A.4) into Eq. (A.1), the following expression can be further written as q2 ðtÞ ¼ exp ½αi ΔψðtÞq2 ðt  ΔtÞ +

1  exp ½αi ΔψðtÞ ½t2 ðtÞ  t2 ðt  ΔtÞ αi ΔψðtÞ (A.5)

The current total tangential traction t2,total(t) can be further written as t2,total ðtÞ ¼ t2 ðtÞ +

N N X X gi t2 ðtÞ  gi exp ½αi ΔψðtÞq2 ðt  ΔtÞ i¼1

N X

i¼1

1  exp ½αi ΔψðtÞ ½t2 ðtÞ  t2 ðt  ΔtÞ αi ΔψðtÞ i¼1 ! N N X X 1  exp ½αi ΔψðtÞ ¼ 1+ gi  gi t2 ðtÞ αi ΔψðtÞ i¼1 i¼1





gi

 N X 1  exp ½αi ΔψðtÞ t2 ðt  ΔtÞ gi exp ½αi ΔψðtÞq2 ðt  ΔtÞ  αi ΔψðtÞ i¼1 (A.6)

Substituting Eq. (A.5) in Eq. (A.6) yields 8  N X > 1  exp ½αi ΔψðtÞ > > t t ð t Þ ¼ D ð t Þt ð t Þ  g q ð t Þ  ð t Þ , > 2,total 2 i 2 2 > αi ΔψðtÞ < i¼1 ! > N N X X > 1  exp ½ α Δψ ð t Þ  > i > D ðt Þ ¼ 1 + > gi  gi : αi ΔψðtÞ i¼1 i¼1

(A.7)

The equation above allows for the incremental calculation of the traction for a time increment Δt, which accumulates to that from the previous time step t  Δt. By using Eq. (A.5), the incremental form of the second term in Eq. (A.7) is written as

Appendix-B. The cohesive/frictional contact algorithm

215

 N X 1  exp ½αi ΔψðtÞ t2 ðtÞ gi q2 ðtÞ  αi ΔψðtÞ i¼1  N X 1  exp ½αi Δψðt  ΔtÞ  t2 ðt  ΔtÞ gi q2 ðt  ΔtÞ  αi Δψðt  ΔtÞ i¼1 ( N X ¼ gi ð exp ½αi ΔψðtÞ  1Þq2 ðt  ΔtÞ + t2 ðt  ΔtÞ i¼1



) 1  exp ½αi Δψðt  ΔtÞ 1  exp ½αi ΔψðtÞ  αi Δψðt  ΔtÞ αi ΔψðtÞ (A.8)

Finally, the incremental total traction Δt2,total(t) is obtained by

 Δt2,total ðtÞ ¼ t2,total ðtÞ  t2,total ðt  ΔtÞ ¼ DðtÞt2 ðtÞ  Dðt  ΔtÞT2 ðt  ΔtÞ N X  gi ð exp ½αi ΔψðtÞ  1Þq2 ðt  ΔtÞ i¼1

 N X 1  exp ½αi Δψðt  ΔtÞ 1  exp ½αi ΔψðtÞ   gi t2 ðt  ΔtÞ αi Δψðt  ΔtÞ αi ΔψðtÞ i¼1 (A.9) By combining Eqs. (A.5), (A.9), Δt2,total(t) is recursively solved. Then, the total tangential traction Δt2,total(t) is updated. Similarly, the total normal traction t1,total(t) is solved.

Appendix-B. The cohesive/frictional contact algorithm for solving the elastic tangential separation ½½ue2 and the regular tangential traction t2 in the developed viscoelastic cohesive/friction coupled model Let us define the initial load time t ¼ t0 and the time t ¼ tc at which both the normal contact and the tangential softening start to appear. After t ¼ tc, the trial tangential traction is calculated as   e,tr i ttr ð t + Δt Þ ¼ t ð t + Δt Þ ½ ½u  ð t + Δt Þ ¼ ½ ½u  ð t + Δt Þ  ½ ½u  ð t Þ c 2 c c c c 2 2 2 2 (B.1)
 + μjt1 jðtc + ΔtÞ ½½u1 ðtc + ΔtÞ where t2 is the function of time tc and (tc + Δt), which is calculated using Eq. (9.1). ½½u2(tc + Δt) > 0 and ½½u1(tc + Δt) < 0 are the tangential and normal separations at the beginning of the new (tc + Δt) time step, and ½½ui2(tc) is the inelastic tangential separation at the end of tc time step. 4 t is the time increment.

216

9. Viscoelastic cohesive/friction coupled model

(a) If ttr 2 (tc + Δt)  μs jt1 j(tc + Δt), the inelastic tangential separation is ½½ui2(tc + Δt) ¼ 0 and the tangential traction t2(tc + Δt) ¼ e, tr ttr 2 (tc + Δt) ¼ t2(½½u2 (tc + Δt) ¼ ½½u2(tc + Δt))is calculated. There is no frictional slip for the contacted surface. (b) If t tr 2 (t c + Δt) > μs jt 1 j(t c + Δt), the frictional slip appears. After the initial value for ½½ui2 (t ¼ t c ) ¼ 0 is given, numerical iterations are performed to solve the updated elastic tangential separation ½½ue2 (t c + Δt) and the inelastic tangential separation ½½ui2 (t c + Δt) 8 0  2 1 ! > > ½½u e2 ðtc + ΔtÞ C > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½½u e2 ðtc + ΔtÞ ½½u 1 ðtc + ΔtÞ > B > > A @ exp  2exp ð 1 Þ t ð t + Δt Þ ¼ τ exp  > 0, max > 2 c > δ2 δ1 > δ22 > > > pffiffiffi  >  > > > e, c ½½u e ðt + ΔtÞ  2 δ > > ½ ½u  c >  2 > 2 2 < t ðt + ΔtÞ ¼ k ½½u ðt + ΔtÞ > 0, dðt + ΔtÞ ¼ 2 pffiffiffi   0,  c 1 c 1 1 c 2 > δ ½½u e2 ðtc + ΔtÞ ½½u 2e, c  > > > 2 2 > >     > exp ðdðtc + ΔtÞÞ  1 > > > , Fðtc + ΔtÞ ¼ t2 ðtc + ΔtÞ + μðtc + ΔtÞt1 ðtc + ΔtÞ  μ2 t1 ðtc + ΔtÞ, > μðtc + ΔtÞ ¼ μs > exp ð 1 Þ  1 > > !n > > > > Fðtc + ΔtÞ > i i > > 0, ½½u e2 ðtc + ΔtÞ ¼ ½½u 2 ðtc + ΔtÞ  ½½u i2 ðtc + ΔtÞ > 0 : ½½u 2 ðtc + ΔtÞ ¼ ½½u 2 ðtc Þ + δ2 ηt2, max (B.2)

where it is noted that the direction for the inelastic tangential separation ½½ui2(tc + Δt) and the total tangential separation ½½u2(tc + Δt) is opposite because of cohesive softening. Thus, both ½½ui2(tc + Δt) and ½½ue2(tc + Δt) increase during iterations above. Because t2(tc + Δt) decreases with increasing ½½ue2(tc + Δt) and the slip criterion F(tc + Δt) decreases up to the tolerance, which guarantees the convergence of the developed algorithm. Although it increases with d(tc + Δt) and ½½ue2(tc + Δt), the introduced term μ(tc + Δt)jt1 j(tc + Δt) is much smaller than the decrease of t2(tc + Δt). Then, the regular tangential traction t2(tc + Δt) is updated after convergence for each time increment 4t. Finally, the regular tangential cohesive traction t2 in Appendix-A is substituted by the tangential cohesive/friction traction t2 in Appendix-B aftert tc. It should be pointed out there is no derivation for the consistent tangential stiffness for the regular traction-separation relationship because the explicit time-stepping algorithm is used. After all, it is almost impossible to derive the viscoelastic cohesive stiffness for the Schapery-type model. It is also noted that explicit FEA also helps to avoid numerical convergence due to cohesive softening, similar to the work by Elmarakbi et al. [32].

References

217

References [1] Van Der Bosch MJ, Schreurs PJG, Geers MGD. On the development of a 3D cohesive zone element in the presence of large deformations. Comput Mech 2006;42:171–80. [2] Nguyen TD, Govindjee S. Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth. Int J Fract 2006;141(1–2):255–68. [3] Bazˇant ZP, Li YN. Cohesive crack with rate-dependent opening and viscoelasticity: I. Mathematical model and scaling. Int J Fract 1997;86:247–65. [4] Kubair DV, Geubelle PH, Huang YG. Analysis of a rate-dependent cohesive model for dynamic crack propagation. Eng Fract Mech 2003;70:685–704. [5] Zhang X, Mai YM, Jeffrey RG. A cohesive plastic and damage zone model for dynamic crack growth in rate-dependent materials. Int J Solids Struct 2003;40:5819–37. [6] Zhou FH, Molinari JF, Shioya T. A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials. Eng Fract Mech 2005;72:1383–410. [7] Musto M, Alfano G. A novel rate-dependent cohesive-zone model combining damage and visco-elasticity. Comput Struct 2013;118:126–33. [8] Zreid I, Fleischhauer R, Kaliske M. A thermomechanically coupled viscoelastic cohesive zone model at large deformation. Int J Solids Struct 2013;50:4279–91. [9] Salih S, Davey K, Zou Z. Rate-dependent elastic and elasto-plastic cohesive zone models for dynamic crack propagation. Int J Solids Struct 2016;90:95–115. [10] Allen DH, Searcy CR. A micromechanical model for a viscoelastic cohesive zone. Int J Fract 2001;107:159–76. [11] Corigliano A, Ricci M. Rate-dependent interface models: formulation and numerical applications. Int J Solids Struct 2001;38:547–76. [12] Giambanco G, Fileccia Scimemi G. Mixed mode failure analysis of bonded joints with rate-dependent interface models. Int J Numer Methods Eng 2006;67:1160–92. [13] Gozluklu B, Uyar I, Coker D. Intersonic delamination in curved thick composite laminates under quasi-static loading. Mech Mater 2015;80:163–82. [14] Gozluklu B, Coker D. Modeling of dynamic crack propagation using rate dependent interface model. Theor Appl Fract Mech 2016;85:191–206. [15] Yu C, Pandolfi A, Ortiz M, Coker D, Rosakis AJ. Three-dimensional modeling of intersonic shear-crack growth in asymmetrically loaded unidirectional composite plates. Int J Solids Struct 2002;39(25):6135–57. [16] Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. Int J Solids Struct 1996;33(20 22):2899–938. [17] Raous M, Cangemi L, Cocu M. A consistent model coupling adhesion, friction, and unilateral contact. Comput Methods Appl Mech Eng 1999;177:383–99. [18] Serpieri R, Sacco E, Alfano G. A thermodynamically consistent derivation of a frictionaldamage cohesive-zone model with different mode I and mode II fracture energies. Eur J Mech A Solids 2015;49:13–25. [19] Parrinello F, Failla B, Borino G. Cohesive-frictional interface constitutive model. Int J Solids Struct 2009;46:2680–92. [20] Parrinello F, Marannano G, Borino G, Pasta A. Frictional effect in mode II delamination: experimental test and numerical simulation. Eng Fract Mech 2013;110:258–69. [21] Parrinello F, Marannano G, Borino G, Pasta A. A thermodynamically consistent cohesive- frictional interface model for mixed mode delamination. Eng Fract Mech 2016;153:61–79. [22] Guiamatsia I, Nguyen GD. A thermodynamics-based cohesive model for interface debonding and friction. Int J Solids Struct 2014;51:647–59. [23] Snozzi L, Molinari JF. A cohesive element model for mixed mode loading with frictional contact capability. Int J Numer Methods Eng 2013;93:510–26.

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[24] Liu PF, Gu ZP, Peng XQ. A nonlinear cohesive/friction coupled model for shear induced delamination of adhesive composite joint. Int J Fract 2016;199:135–56. [25] Schapery RA. Nonlinear viscoelastic solids. Int J Solids Struct 2000;37(1–2):359–66. [26] Haj-Ali RM, Muliana AH. Numerical finite element formulation of the Schapery nonlinear viscoelastic material model. Int J Numer Methods Eng 2004;59:25–45. ´ , Parry G, Beltz GE. Potential-based and non-potential-based O [27] McGarry JP, Ma´irtı´n E cohesive zone formulations under mixed-mode separation and over-closure. Part I: theoretical analysis. J Mech Phys Solids 2014;63:336–62. [28] Carlsson L, Gillespie J, Pipes R. On the analysis and design of the end notched flexure (ENF) specimen for mode II testing. J Compos Mater 1986;20:594–604. [29] Sch€ on J. Coefficient of friction of composite delamination surfaces. Wear 2000;237:77–89. [30] Yasaee M, Mohamed G, Pellegrino A, Petrinic N, Hallett SR. Strain rate dependence of mode II delamination resistance in through thickness reinforced laminated composites. Int J Impact Eng 2017;107:1–11. [31] Coker D, Rosakis AJ. Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composites plates. Philos Mag 2001;81:571–95. [32] Elmarakbi AM, Hu N, Fukunaga H. Finite element implementation of delamination growth in composite materials using LS-DYNA. Compos Sci Technol 2009;69:2383–91.

C H A P T E R

10

Damage models and explicit finite element analysis of thermoset composite laminates under low-velocity impact 10.1 Introduction Composite structures are often subjected to dynamic impact loads, which often result in progressive damage up to fracture of structures. Research on the damage behaviors of composites under impact loads is an important task to improve the impact resistance and residual lifetime of structures. The impact problems of composites are generally divided into low-velocity impact and high-velocity impact, leading to different damage mechanisms. To promote the lightweight design of composite structures, a transverse low-velocity impact is typically used to evaluate the damage tolerance of composites. The main failure mechanisms for laminated composites under impact load include longitudinal fiber breakage, transverse matrix cracking, and delamination as well as their interactions [1]. In order to predict the dynamic progressive failure properties of composites efficiently in the design of composite structures, developing a robust numerical technique using finite element analysis (FEA) is significant. As a challenging work, integrated impact simulation by FEA includes a series of complicated numerical issues represented mainly by three nonlinear problems: geometry, material, and boundary nonlinearities, which generally relate to the damage constitutive model, the failure criteria, the damage evolution law, the frictional contact algorithm and the time-stepping algorithm for impact analysis.

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00013-4

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Copyright © 2021 Elsevier Inc. All rights reserved.

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10. Damage models and explicit finite element analysis

The main purpose of this chapter is to compare the effects of different failure criteria including Hashin criteria [2, 3], Chang-Chang criteria [4], and Puck criteria [5] on the prediction of damage behaviors of composite laminates under transverse low-velocity impact using FEA. During the past decades, a large number of failure criteria have been developed for fiber-reinforced polymer composites. Initially, Tsai and Wu [6] proposed quadratic failure criteria in the stress space, but detailed failure modes cannot be identified. Afterward, Hashin [2, 3] proposed failure criteria for fiber and matrix respectively based on the Mohr-Coulomb failure theory. Similarly, Chang-Chang [4] proposed stress-based longitudinal and transverse tension and compression criteria for fiber and matrix. However, Hashin [2, 3] didn’t continue to determine the orientation of an inclined fracture plane although he pointed out the failure was caused by stresses acting on that plane. As a theory that ranks high in the world wide failure exercise [7–10], the failure criteria proposed by Puck and Sch€ urmann [5] are based on Hashin’s criteria [2, 3] further distinguish between the fiber fracture and inter-fiber fracture where the fracture plane is considered in detail. Currently, much work performed numerical analysis by FEA on the mechanical responses and damage mechanisms of composite laminates under low-velocity impact. In particular, Zhao and Cho [11] used TsaiWu criteria, Riccio et al. [12] and Li et al. [13] used Hashin criteria [2, 3], Meo et al. [14] used Chang-Chang criteria [4], and Donadon et al. [15], Faggiani and Falzon [16], Shi et al. [17], and Feng and Aymerich [18] used Puck criteria [5] in their analysis. It deserves pointing out they are not really demonstrated and only used a priori for dynamic loads although all failure criteria above are validated in quasi-static loads to a large extent. On one hand, accurate determination of the fracture plane angle and related parameters is difficult [19–21] although Puck criteria are considered as a large improvement over other criteria. On the other hand, it is not clear how the fracture plane affects the dynamic progressive failure processes because it is not easy to validate them by experiments. A favorable strategy is to compare the effects of different failure criteria on the impact responses of composite structures. After the strong and weak forms for the dynamic initial value problem for the laminate with an interface discontinuity (delamination) are given, this chapter studies the intralaminar damage and interlaminar delamination of composite laminates under low-velocity impact, where important work is to compare the effects of Hashin, Chang-Chang, and Puck criteria on the dynamic progressive failure properties of laminates. The impact modeling platform is established by ABAQUS-PYTHON scripting language and numerical technique based on the central difference algorithm for the elastic damage model with three failure criteria is developed by ABAQUS-VUMAT. The delamination is simulated by the bilinear cohesive model in Chapter 2 [22]. By comparison, high-efficiency and high-

10.2 Strong and weak forms of the initial value problem for composite laminates

221

fidelity simulation of dynamic progressive failure properties are expected to be achieved by appropriately selecting failure criteria and related numerical parameters for a generic aerospace composite structure.

10.2 Strong and weak forms of the initial value problem for composite laminates under finite deformation under low-velocity drop-weight impact 10.2Let us consider a composite laminate as a bounded domain Ω
ℝ3 with an interface discontinuity (delamination) subjected to drop-weight impact, as shown in Fig. 10.1. The boundary ∂ Ω of the domain Ω with the outward normal N is subdivided into either Neumann or Dirichlet boundary conditions defined on ∂ Ωc and ∂ Ωu respectively, such that ∂ Ω ¼ ∂ Ωu [ ∂ Ωc and ∂ Ωu \ ∂ Ωc ¼ Ø. The reference normal vector N points from Ω to Ω+. The displacement field u across Γ d is discontinuous but continuous in Ω+ and Ω. Further, the normal vector n acts on the middle + plane between two discontinuous interfaces Γ  d and Γ d . The strong form in the reference configuration for the laminate is given by the balance of momentum and the boundary conditions under finite deformation 8 € in Ω —  S + ρ0 b ¼ ρ0 u, > > > > > uðX, t ¼ 0Þ ¼ 0, in Ω > > > > > on ∂Ωu > < uðX, tÞ ¼ 0 , (10.1) u_ ðX, t ¼ 0Þ ¼ 0 , in Ω=∂Ωc > > > u_ ðX, t ¼ 0Þ ¼ u_ 0 , on ∂Ωc > > > > > > SðX, tÞ  N ¼ Τ b ðX, tÞ, on ∂Ωc > > : SðX, tÞ  n + 5SðX, tÞ  n ¼ T ð½½uÞ, on Γ d

FIG. 10.1 Dynamic frictional contact between the impactor and laminate with a discontinuous interface.

222

10. Damage models and explicit finite element analysis

where t ¼ 0 denotes the time when the impactor and laminate start to contact and u_ 0 is the initial velocity on the contact surface at time t ¼ 0. S is the second Piola-Kirchhoff stress tensor and b is the body force per unit mass. τ : ∂ Ωc ! ℝ3 is the field of the dynamic contact traction on ∂ Ωc. T : Γ d ! ℝ3 is the Piola’s cohesive traction at the delamination interface Γ d which depends on the displacement jump ½½u at the interface Γ d. X is the material point in the reference configuration. Let us define two contact points x1 ¼ X1 + w and x2 ¼ X2 + u on the contact surface ∂ Ωc under finite deformation. The contact constraint that prevents the interpenetration of two normal contact surfaces is gN 5 (x1  x2)  N 0, where gN is the normal gap and gN < 0 denotes thepenetration condition.  The relative tangential displacement is gt 5 x1  x2  t and t is the tangential direction on the contact surface. The penalty method that introduces a normal positive contact parameter εN is used to deal with the dynamic contact and the normal contact/separation relationship obeys the KuhnTucker conditions gN 0, τN  0, and gNτN ¼ 0. τN ¼ τ  N 5 εNgN < 0 is the normal contact traction. The tangential contact traction τt ¼ τ  t is assumed to obey the Mohr-Coulomb frictional contact law, where the contact potential and contact criterion are defined to describe the slip/stick state [23]. Q Q By performing variations on Eq. (10.1), the potential energies Lam and Imp of the laminate and impactor in the reference configuration is derived by using the divergence theorem ð ð ð 8 Q > > € u dΩ + ¼ δu  ρ δE : S dΩ + δ½½u  T dA > 0 Lam > > Ω + [Ω Ω + [Ω Γd > > > ð ð > > > > δu  τ dA  ρ0 δu  b dΩ, For deformed laminate,  > < ∂Ωc Ω + [Ω ð ð ð > Q > > € w dΩ + ¼ δw  ρ δw  τ dA  ρ1 δw  b dΩ, For rigid impactor, > 1 Imp > > Ω ∂Ωc Ω > > > ð > > Q Q Q > > ¼ + + ðτN δgN + τt δgt Þ dA : Tot

Lam

Imp

∂Ωc

(10.2) where the dynamic contact problem is equivalent to the constraint Q optimization problem which requires that the total potential energy Tot reaches the minimum under the constraint condition gN 0. δu and δw are virtual displacements. Ε 5 (FTF  I)/2 is the Green-Lagrange strain tensor in the reference configuration. FTF is the right Gauss-Green deformation tensor and F ¼ —X x 5 I + u  — is the deformation gradient tensor in the reference configuration. I is a second-order identity tensor. ρ0 and ρ1 are the densities of the deformed laminate and the rigid impactor, respectively. It is shown the work by the impact force and body force on the laminate is divided into the kinetic energy, the strain energy, and the dissipated energy.

10.3 Intralaminar constitutive model, failure criteria, and damage evolution laws of composites

223

10.3 Intralaminar constitutive model, failure criteria, and damage evolution laws of composites 10.3.1 Damaged stress-strain relationships of composites All analysis is limited to the elastic deformation and the strain rate effect is neglected. Finite deformation is considered. The effective stress tensor S and the nominal stress tensor S for damaged composites are written as respectively S ¼ C : E, S ¼ Cd : E 2

E1 ð1  ν23 ν32 Þ E2 ðν12  ν13 ν32 Þ E3 ðν13  ν12 ν23 Þ 6 E1 ðν21  ν31 ν23 Þ E2 ð1  ν13 ν31 Þ E3 ðν23  ν21 ν13 Þ 6 1 6 E1 ðν31  ν21 ν32 Þ E2 ðν32  ν12 ν31 Þ E3 ð1  ν12 ν21 Þ C5 6 ΩG12 Ω6 6 4 ΩG23

(10.3) 3 7 7 7 7 7 7 5 ΩG31 (10.4)

with Ω ¼ 1  ν12 ν21  ν23 ν32  ν31 ν13  2ν12 ν23 ν31 The relationship between two stresses in Eq. (10.3) was given by Donadon et al. [15] 8 9 2 0 1=ð1  d11 Þ S11 > > > > > > 6 > > > > 0 1=ð1  d22 Þ S > > 22 6 > > < = 6 0 0 S33 6 56 > S12 > > 6 0 0 > > > 6 > > > > 4 0 0 S > > 23 > > : ; 0 0 S31

9 38 0 0 0 0 S11 > > > > > > 7> > 0 0 0 0 S22 > > > 7> > = 7< S > 1 0 0 0 7 33 7 7> 0 0 S > 0 1=ð1  d12 Þ > 12 > > 7> > > 5> 0 S23 > 0 0 1=ð1  d23 Þ > > > > : ; S31 0 0 0 1=ð1  d31 Þ (10.5)

     8 T C d12 ¼ 1  1  dT11 1  dC > 11 1  smt d22 1  smc d22 > <      d23 ¼ 1  1  dT11 1  dC 1  smt dT22 1  smc dC 11 22 > >      : T C d31 ¼ 1  1  dT11 1  dC 11 1  smt d22 1  smc d22

(10.6)

where C is the undamaged elastic tensor and Cd is the damaged elastic tensor. The superscripts “T” and “C” in the damage variables dT11, dC11, dT22, and dC22 denote the tension and compression, respectively. smt and smc are introduced coefficients to control the loss of shear stiffness due to matrix tension and compression [24]. E1, E2, and E3 are Young’s moduli and G12, G23, and G31 are shear moduli. Eij(i, j ¼ 1, 2, 3) are the components of Green-Lagrange strain tensor E.

224

10. Damage models and explicit finite element analysis

10.3.2 Failure criteria and damage evolution laws of composites 1The failure criteria of fiber and matrix may be defined by the stress or strain. Hashin [2, 3] proposed stress-based criteria for fiber and matrix. Huang and Lee [25] showed by experiments strain may be a better choice for defining the failure because the strain is more continuous compared with stress during the damage evolution of composites. Currently, Hashin criteria [2, 3], Chang-Chang criteria [4], and Puck criteria [5] have been applied to the impact analysis of composites. Compared with Hashin and Chang-Chang criteria, Puck criteria further consider a fracture plane due to matrix compression. Donadon et al. [15] initially adopted the strainbased Puck criteria in the impact analysis. By comparing with three criteria, the formulas for fiber tension, fiber compression, and matrix tension are the same except matrix compression. In this following, four failure modes for three criteria are given respectively. (1) Fiber tension and compression !2 !2 E11 E11 T C F11 ¼  1  0, F11 ¼ 10 ET0,1 EC0,1

(10.7)

where the corresponding initial failure strains ET0,1 and EC0,1 are given by. ET0,1 ¼

XT C XC , E0,1 ¼ E1 E1

(10.8)

where XT and XC are the fiber tensile and compressive strengths, respectively. The damage evolution laws for fiber tension and compression are assumed by ! T ðCÞ T ð CÞ Ef , 1 E0,1 T ðCÞ d11 ¼ TðCÞ 1 (10.9) T ðCÞ E11 E E f ,1

0,1

where ETf,1 and ECf,1 are the critical failure strains when the damage variables reach one. In order to eliminate the mesh sensitivity problem, a characteristic length l is introduced to solve the critical failure strain by regularizing T ðCÞ the fiber fracture energy Γ 11 [26]. The stress-strain curve is shown Fig. 10.2. The expression is given by T ðCÞ

Ef , 1 ¼

T ðCÞ

2Γ 11 XTðCÞ l

(10.10)

where it is noted fiber compression is induced by microbuckling mechanism, e.g., kink band. However, it does not result in the global collapse although it decreases the load-bearing abilities. The residual

10.3 Intralaminar constitutive model, failure criteria, and damage evolution laws of composites

225

S11 XT

ECf,1

E1(1– d11)

EC0,1 Xres

ET0,1

ETf,1

E11

XC

FIG. 10.2

Intralaminar fiber tension and compression.

strength, i.e., fiber crushing strength X res is equal to the matrix compressive strength YC [16]. Therefore, the damage mechanisms for fiber tension and compression are generally considered as the same. (2) Matrix tension !2 E22 YT T F22 ¼  1  0, ET0,2 ¼ (10.11) T E2 E0,2 where ET0, 2 is the initial failure strain for matrix tension. YT is the matrix tensile strength and the tensile stress-strain curve is shown in Fig. 10.3. The damage evolution law for matrix tension is written as ! ETf, 2 ET0, 2 T d22 ¼ T 1 (10.12) E22 Ef , 2  ET0, 2 where ETf, 2 is the critical failure strain for matrix tension which is given by [16, 17].

FIG. 10.3

Intralaminar matrix tension.

226

10. Damage models and explicit finite element analysis

ETf, 2 ¼

2Γ T22 YT l

(10.13)

where Γ T22 is the fracture energy for matrix tension. (3) Matrix compression ① Chang-Chang failure criteria [4, 14] 3 8 !2 !2  2   C > E > E E E12 2 22 22 0, 2 > 4 5 > + 1 +  1, < 2E0, 12 2E0,12 E0,12 EC0,2 > > ^ > YC S > : E0,12 ¼ 12 , EC0,2 ¼ 2G12 E2

(10.14)

where E0,12 is the initial shear failure strain and EC0,2 is the initial failure ^12 is the in-plane shear strength and YC strain for matrix compression. S is the matrix compressive strength. ② Hashin failure criteria [2, 3, 13]

8 !2 !2 !2 ! C 2 > > E13 E23 > ðE22 + E33 Þ + E22 + E33 E0, 2  1  E22 E33 + E12 > + +  1, >  2 < EC EC 2E0, 12 E0, 12 E0, 13 E0, 23 EC 0, 2 0, 3 0, 2 E0, 23 > > ^ ^ > ZC S S > > : E0, 13 ¼ 13 , E0, 23 ¼ 23 , Ec0, 3 ¼ 2G13 2G23 E3 (10.15)

where E0, 12, E0, 23, and E0, 13 are the initial shear failure strains. EC0, 3 is ^13 and S ^23 are the initial failure strain along the thickness direction. S the out-of-plane shear strengths and ZC is the strength along the thickness direction. For Hashin and Chang-Chang criteria, the damage evolution laws are written as ! ECf, 2 EC0,2 2Γ C22 C C d22 ¼ C (10.16) 1  ¼ , E f ,2 E22 YC l Ef ,2  EC0, 2 where ECf,2 is the critical failure strain for matrix compression. Γ C22 is the fracture energy for matrix compression. ③ Puck failure criteria [5, 15–17, 26] Experimental research by Puck and Sch€ urmann [5] showed the unidirectional composite laminate under transverse compression fails by shear with a resultant fracture plane oriented at an angle θf ¼ 53 2° with respect to the through-thickness direction, as shown in Fig. 10.4. The fracture angle can be a combination of two or even three stress transformations. They suggested it is much more appropriate to use the Mohr-Coulomb theory-based failure criteria that are shown in Fig. 10.5. Puck failure criteria are written as [5, 15–17, 26]

10.3 Intralaminar constitutive model, failure criteria, and damage evolution laws of composites

227

3

SNT

θf

SNT SNN

2

1

FIG. 10.4

Fracture plane after stress rotation for matrix compression.

SNT A

SNT = S23 +

NTSNN

(SNN, SNT) A

S 23

θf |Y C|

FIG. 10.5

SNN

Mohr-Coulomb behavior for matrix compression.

8 !2 !2 > S S > TN LN > FðSNN , SNT , SNL Þ ¼ +  1, > > ^12 + μ SNN > ^A + μTN SNN S > S LN > 23 > > >   > T > > SLTN ¼ T ðαÞS123 TðαÞ , μTN ¼ tan ðϕÞ ¼ tan 2θf  90° , > > > < 2 3 1 0 0      7 A YC 1  sin ðϕÞ 6 > > ^ ¼ 7, S > TðαÞ ¼ 6 0 cos θf sin θ , > f 4 5 23 > > cos ðϕÞ 2     > > > 0  sin θf cos θf > > > > > μ μ > > > ϕ ¼ 2θf  90°, TN ¼ NL : A ^12 ^ S S 23

(10.17)

228

10. Damage models and explicit finite element analysis

Sr =

S2NT + S2NL

C

F22 (SNT , SNL) = 1

Sr = max

r

FIG. 10.6

0 (1– d C 22)Sr

f r

r

=

2

NT

+

2

NL

Intralaminar damage model for matrix compression.

where N is the normal direction, and T and L are the tangential directions with respect to the fracture plane direction. YC is the transverse compressive strength. The rotation of the Piola-Kirchhoff stresses Sij(i, j ¼ 1, 2, 3) to the fracture plane by using a transformation ^A is the transverse matrix T(α) leads to the stresses Sij(i, j ¼ L, T, N). S 23 shear strength in the fracture plane which is determined by the transverse compressive strength YC and the material friction angle ϕ. After introducing the bilinear model in the fracture plane as shown in Fig. 10.6, the damage evolution law is expressed as ! 8 max f γ > γ γ > C > 1 , > f < d22 ðγ NT , γ NL Þ ¼ max γγ γγ  γγ (10.18) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2Γ C22 > max 2 2 2 2 > : γ γ ¼ γ NT + γ NL , γ γ ¼ 0 , Sr ¼ STN + SNL Sr l where γ fr and γ max are the initial and critical combined shear strains, r respectively. The strains γ NT and γ NL are related to the stresses SNT and SNL. S0r is the shear stress at the initial failure.

10.4 Time integration algorithm for calculating the impact responses of composites After performing discretization and linearization on Eq. (10.3), the nonlinear finite element equations for the deformed laminate and the rigid impactor respectively are derived as.

10.4 Time integration algorithm for calculating the impact responses of composites

ðaÞ Laminate ðbÞ Impactor     nimp nlam + ncoh € + F imp ¼ 0, M u€ + F int + F coh ¼ F imp , Α M w Α e¼1 e¼1 ~ ~ ð ð T ρ0 N T N dΩ, M ¼ ρ1 N N dΩ, M¼ +  Ω [Ω Ω ð ð T T B S dΩ, F imp ¼ N τ dA F int ¼ Ω + [Ω ∂Ωc ð e T T dA F coh ¼ M

229

(10.19)

Γd

where Α denotes the assembly of the bulk and interface element stiffnesses. The body force is neglected here. nlam, ncoh and nimp are the number of the laminate elements, delamination interface elements, and impactor elements, respectively. M and M are 24 24 lumped mass matrices for 3D eight-node solid elements for the laminate and impactor, respectively. e is a N and N are 3 24 shape function matrices for 3D solid elements. M 3 24 shape function matrix for 3D cohesive element in Chapter 2. Corresponding, Fcoh(24 1 matrix), Fint(24 1 matrix), and Fimp(24 1 matrix) are the cohesive force vector, the internal force vector, and the impact € are 24 1 node acceleration matricontact force vector, respectively. u€ and w ~ M ~ e refers to Liu et al. [27, 28]. The ces for 3D solid elements. The expression strain matrix B(6 24 matrix) for 3D solid elements is written as 9 8 9 8 E11 > > E11 > > > > > > > > > > > > > > > > E22 > E22 > > > > > > > > > > > > =

33 33 ¼ δE ¼ > > γ 23 > > > > > 2E12 > > > > > > > > > > > > > > > > > γ 2E > > > > 23 13 > : > > > ; ; : γ 12 2 2E13 3 F11 NI,1 F21 NI,1 F31 NI, 1 7 6 F12 NI,2 F22 NI,2 F32 NI, 2 7 6 7 6 7 6 F13 NI,3 F23 NI,3 F33 NI, 3 7 ¼ Bδu,B56 7 6 ~ 6 F11 NI,2 + F12 NI,1 F21 NI,2 + F22 NI,1 F31 NI,2 + F32 NI, 1 7 7 6 4 F12 NI,3 + F13 NI,2 F22 NI,3 + F23 NI,2 F32 NI,3 + F33 NI, 2 5 F11 NI,3 + F13 NI,1 F21 NI,3 + F23 NI,1 F31 NI,3 + F33 NI, 1 (10.20) where δ denotes the variation in FEA. Fij(i, j ¼ 1, 2, 3) are the components of the deformation gradient tensor F. I ¼ 1, 2, ⋯, 7, 8 is the number of nodes in a 3D eight-node element. According to total Lagrange finite element formulation for finite deformation, the strain matrix B can be divided into a linear strain matrix and two nonlinear strain matrices.

230

10. Damage models and explicit finite element analysis

The Newmark time integration algorithm [29] is used to update the node displacement u and the node velocity u_ in Eq. (10.19) for the lami~ is divided into two steps: ~ nate. The explicit FEA (a) The node forces Fint, Fcoh, and Fimp at time t + Δt are calculated. The derivation of Fcoh sees Liu et al. [27, 28]. Fimp is calculated by the penalty method. In order to calculate Fint, the nominal stress S at time t + Δt must be obtained in advance. The incremental constitutive equations are written as Et + Δt ¼ Et + ΔEt + Δt , St + Δt ¼ C : Εt + Δt , St + Δt ¼ Cd : Εt + Δt

(10.21)

where the damaged elastic matrix Cd depends on the damage variables dij(i, j ¼ 1, 2, 3) in Eq. (10.4) which are updated according to different failure modes. (b) When the initial node displacement uðt ¼ 0Þ ¼ 0 and the velocity ~ u_ ðt ¼ 0Þ ¼ u_ are given, the initial acceleration u€ðt ¼ 0Þ is calculated as ~ ~ ~0   (10.22) u€ðt ¼ 0Þ ¼ M 1 F imp  F int  F coh ~ Then, the time-stepping algorithm is given by   8 €u ðt + ΔtÞ ¼ M 1 F imp  F int  F coh , > > > ~  > > < Δt2 ð1  βÞ€u ðtÞ + β€u ðt + ΔtÞ , uðt + ΔtÞ  uðtÞ + Δt_u ðtÞ + ~ ~ ~ ~ ~  2 > > > > > : _u ðt + ΔtÞ  _u ðtÞ + Δt_u ðtÞ ð1  αÞ€u ðtÞ + α€u ðt + ΔtÞ ~ ~ ~ ~ ~

(10.23)

where α and β are two adjusted parameters determining the implicit or explicit solutions. As a particular case of Newmark time integration algorithm [29], the explicit central difference algorithm at α ¼ 0.5 and β ¼ 0 in ABAQUS is used. The Courant conditional stability for the finite difference algorithm introduces a constraint condition on the time increment Δt pffiffiffiffiffiffiffiffiffiffiffiffi Δt < 2=ω ¼ Lmin = E1 =ρ0 (10.24) where ω is generally considered as the largest natural frequency of the smallest element and Lmin is the minimum side size in a solid element.

10.5 Numerical results and discussion 10.5.1 Finite element modeling of impact problems of composites The drop-weight impact of composite laminates is adopted and the finite element model including laminate and impactor is shown in Fig. 10.7. The layup parameters for the laminate, the contact parameters

10.5 Numerical results and discussion

FIG. 10.7

231

Mesh model for impact analysis.

between the laminate and impactor are set. The reduced integration element C3D8R and the cohesive interface element are used for the laminate and the finite-thickness (0.001 mm) delamination interface, respectively. The supporting plate below the laminate is fixed in all directions to represent experimental constraint conditions. The intralaminar damage models with three failure criteria respectively are implemented using ABAQUS-VUMAT (User dynamic material subroutine module). Finite deformation effect in ABAQUS is activated. The delamination analysis is performed using the bilinear cohesive model in ABAQUS. smt ¼ 0.9 and smc ¼ 0.5 in Eq. (10.6) are used [24]. The dynamic penalty method is used to deal with the contact between the laminate and impactor. The friction contact coefficient is generally set to 0.3 [16, 26] and effects of different friction coefficients on the numerical results are discussed below. In the following, numerical results for two specimens with different materials, geometry sizes, and impact energies are discussed.

10.5.2 Numerical results and analysis for case-A specimens T700GC/M21 composite is adopted for Case-A specimen and material parameters are listed in Table 10.1 [26, 30]. Case-A specimens include two layup patterns: [02°/452°/902°/  452°]s (Layup-A) and [902°/02°/  452°/  452°]s (Layup-B). Geometry sizes for the laminate are TABLE 10.1

Material parameters for T700GC/M21 composite laminates [26, 30]

Layer properties

E1 ¼ 130 GPa, E2 ¼ E3 ¼ 7.7 GPa, G23 ¼ 3.8 GPa, G12 ¼ G13 ¼ 4.8 GPa, v12 ¼ v13 ¼ 0.33, v23 ¼ 0.35 XT ¼ 2080 MPa, XC ¼ 1250 MPa, YT ¼ 60M Pa, YC ¼ 140 MPa ^12 ¼ S ^23 ¼ S ^13 ¼ 110MPa ZC ¼ YC ¼ 290MPa, S C T C ΓT ¼ 133N=mm,Γ ¼ 40N=mm,Γ 11 11 22 ¼ 0:6N=mm,Γ 22 ¼ 2:1N=mm

Interface properties

Gc1 ¼ 0.6 N/mm, Gc2 ¼ Gc3 ¼ 2.1 N/mm, η ¼ 1.45, σ max ¼ τmax ¼ 30 MPa

232

10. Damage models and explicit finite element analysis

100mm 150 mm 4 mm. The thickness of each ply is 0.25mm. The diameter and mass of the impactor are 16mm and 2 kg, respectively. The impact energy is 25J [30]. The number of all elements is 60,352. Hongkarnjanakul et al. [30] got the impact contact time about 3.5ms by experiments and thus the total impact time period in the explicit FEA for Case-A specimen is set to 4.0 ms. Numerical results using FEA are compared with the experimental data [30]. Fig. 10.8 shows the impact force-time curves using three failure criteria. Three aspects of issues are discussed here. (1) Some numerical oscillation appears before the peak impact force which indicates severe damage. In general, numerical oscillations are caused by two reasons: dynamic responses and numerical effects. Numerical filtering is used and the output time step for sampling of numerical results is set to 1e-5s. (2) The first damage load Fcrit is found to be caused by delamination represented by large fluctuation in the impact force-time curve, which is consistent with the conclusions by Schoeppner and Abrate [31], Gao and Zhang [32], Yang and Cantwell [33], and Gonza´lez et al. [34]. The values for Fcrit by FEA are obtained as the damage variable for delamination ds > 0 in Chapter 2 is detected, and are experimentally determined by large fluctuation in the impact force-time curve. By comparison, the values for Fcrit by FEA and experiments [30] are about 3200 and 3600 N respectively for Layup-A laminate, and about 4500 and 5000 N respectively for Layup-B laminate. (3) After the peak force, the impactor starts to rebound at time about t ¼ 1.5 ms, where the impact curves become smooth and three failure criteria lead to consistent results by comparing with the experimental data [30]. For Layup-A laminate, Chang-Chang criteria slightly overestimate the peak force, but Hashin criteria and Puck criteria lead to accurate results. For Layup-B laminate, both Chang-Chang and Hashin criteria overestimate the peak force. The possible reason is that Chang-Chang criteria do not consider the out-of-plane shear deformation as well as their interaction with matrix damage. Fig. 10.9 shows the impact force-displacement curves for two layup specimens using three failure criteria. The maximum central displacements using three failure criteria are almost the same, which yet shows some difference from experimental results. In addition, irreversible energy dissipation due to matrix damage and delamination leads to the final indentation depth at the end of impact, whose values using three criteria are slightly larger than those by experiments. Fig. 10.10 shows the dissipated energy of the laminate, which comes from the damage of composites and the frictional contact. When the impactor starts to contact with the laminate, the energy dissipation appears and the laminate deforms by absorbing the kinetic energy of the impactor. When the impactor reaches the lowest point at about t ¼ 1.5 ms, all the kinetic energy has been transformed partly into the elastic deformation energy and partly into the dissipated energy. After this

10.5 Numerical results and discussion

FIG. 10.8

233

Impact force-time curves using Puck, Hashin, and Chang-Chang failure criteria for Case-A specimens: (A) Layup-A laminate and (B) Layup-B laminate.

234

FIG. 10.9

10. Damage models and explicit finite element analysis

Impact force-displacement curves using Puck, Hashin, and Chang-Chang failure criteria for Case-A specimens: (A) Layup-A laminate and (B) Layup-B laminate.

10.5 Numerical results and discussion

235

FIG. 10.10 Dissipated energy after impact at about t ¼ 3.5 ms using Puck, Hashin, and Chang-Chang failure criteria for Case-A specimens: for (A) Layup-A laminate and (B) Layup-B laminate.

time, the impactor rebounds and elastic unloading occurs, where the elastic strain energy of the laminate transformed partly into the kinetic energy of the impactor and partly into the dissipated energy again. The dissipated energy is calculated as the difference between the initial and final kinetic energy of the impactor. By comparison, three failure criteria lead to basically consistent results. The errors for the dissipated energy using Puck, Hashin, and Chang-Chang criteria by comparing with the experimental data [30] are 1.7%, 4.2%, and 18.9% respectively for Layup-A laminate, and 1.5%, 10.4%, and 20.1% respectively for Layup-B laminate. Therefore, Puck criteria lead to higher precision because more detailed consideration for 3D nonlinear failure mechanisms, particularly the fracture plane due to transverse matrix compression that increases the shear strength. This was also pointed out in the world-wide failure exercise by Hinton et al. [7, 8], Soden et al. [9], and Kaddour et al. [10]. Figs. 10.11 and 10.12 show the matrix tension damage, matrix compression damage and delamination interface of the laminate at about t ¼ 3.5 ms at complete separation between the laminate and impactor for Layup-A and B laminates, respectively. Two damage variables (or called status variables) in ABAQUS-VUMAT are defined for matrix tension and compression, respectively. There are totally 16 plies and 15 delamination interfaces for each layup. In 0 ° /0 ° /45° /45° /90 ° /90 ° /  45 ° /  45° /  45° /  45° /90° / 90 ° /45 ° /45 ° /0 ° /0° Layup-A laminate and 90 ° /90 ° /0 ° /0 ° /  45 ° /  45 ° /45 ° /45 ° /45 ° /45 ° /  45 ° /  45 ° /0 ° /0 ° /90 ° /90° Layup-B laminate, 8 plies (colored number) are selected for analysis. The selected 7

236

10. Damage models and explicit finite element analysis

0°

45°

90°

–45°

–45°

90°

45°

0°

Matrix tension

Matrix compression 0°

45°

90°

–45°

–45°

90°

45°

0°

+1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

0° / 45°

45° / 90° 90° / –45° 45° / –45° –45° / –90° 90° / 45° 45° / 0°

Delamination

(A) 0°

45°

90°

–45°

–45°

90°

45°

90°

–45°

–45°

90°

45°

0°

Matrix tension Z X Y

Matrix compression 0°

45°

0°

+1.000e+00 +9.167e-01a +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

Delamination

0° / 45°

45° / 90°

90° / –45°

45° / –45°

–45° / 90°

90° / 45°

45° / 0°

(B) FIG. 10.11 Matrix tension, matrix compression, and delamination interface at about t ¼ 3.5 ms for Case-A specimen: Layup-A laminate using (A) Puck criteria, (B) Hashin criteria, and (Continued)

237

10.5 Numerical results and discussion

0°

45°

90°

–45°

–45°

90°

45°

0°

0°

45°

90°

–45°

–45°

90°

45°

0°

Matrix tension Y

Z X

Matrix compression +1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

Delamination

0° / 45°

45° / 90° 90° / –45° –45° / –45° –45° / 90° 90° / 45° 45° / 0°

(C) Fig. 10.11, cont’d (C) Chang-Chang criteria.

delamination interfaces for Layup-A and B laminates respectively are 0° /45° , 45 ° /90° , 90 ° /  45 ° ,  45 ° /  45 ° ,  45° /90 ° , 90° /45° , 45 ° /0°, and 90 ° / 0° , 0° /  45° , 45° /45° , 45 ° /45 ° , 45° / 45° , 45° /0 ° , 0 ° /90°. The damage evolution of composites can be divided into four stages at about t¼ 0.35ms, 0.65 ms, 1.5 ms, and 3.5ms, respectively. At about t¼ 0.35 ms, matrix tension damage appears initially at the bottom of the laminate due to large bending stress, similar to the conclusion obtained by Johnson et al. [35]. In addition, the extent of matrix tension damage using three failure criteria is almost the same. By comparison, matrix compression damage appears initially at the top of laminate, and the extent of damage using Puck criteria is larger than those using Hashin and Chang-Chang criteria. The delamination crack propagates along the fiber direction in the ply below the delamination interface, which is consistent with the conclusions by Shi et al. [17] and Richardson and Wisheart [36] by using the ultrasonic c-scan technique. For example, the delamination crack at the 45 °/0° interface for Layup-A Laminate propagates along the 0° ply, and propagates along the 45° ply at the 90 ° /  45° interface. Similar phenomena are also found at the 90° /0° interface and 0° /  45° interface for Layup-B laminate. At about t¼ 0.65 ms, the stiffness

238

10. Damage models and explicit finite element analysis

90°

0°

–45°

45°

45°

–45°

0°

90°

45°

–45°

0°

90°

Matrix tension Z X

Y

Matrix compression

90°

0°

–45°

45°

+1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

Delamination

90° / 0° 0° / –45° –45° / 45° 45° / 45° 45° / –45° –45° / 0° 0° / 90°

(A) 90°

0°

–45°

45°

45°

–45°

0°

90°

90°

0°

–45°

45°

45°

–45°

0°

90°

Matrix tension Z X Y

Matrix compression +1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

Delamination

90° / 0°

0° / –45° –45° / 45°

45° / 45°

45° / –45° –45° / 0° 0° / 90°

(B) FIG. 10.12 Matrix tension, matrix compression, and delamination interface at about t ¼ 3.5 ms for Case-A specimen: Layup-B laminate using (A) Puck criteria, (B) Hashin criteria, and (Continued)

239

10.5 Numerical results and discussion

90°

0°

–45°

45°

45°

–45°

0°

–45°

0°

90°

Matrix tension Z X Y

Matrix compression

90°

0°

–45°

45°

45°

90°

+1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

Delamination

90° / 0°

0° / –45° –45° / 45°

45° / 45°

45° / –45° –45° / 0° 0° / 90°

(C) Fig. 10.12, cont’d (C) Chang-Chang criteria.

of elements degrades remarkably at the central region of laminate. Similar to the conclusions obtained by Faggiani and Falzon [16], Tan et al. [26], and Kim et al. [37], matrix tension damage for two layup specimens becomes distinct along the fiber direction, which evolves from the bottom to the top of laminate. By comparison, the damage area for matrix tension at the bottom of laminate predicted using three criteria is basically consistent. By using Puck criteria, matrix compression damage evolves from the top to the bottom of laminate and most of the damage appears at several top plies. By comparison, matrix compression damage predicted using Hashin and Chang-Chang criteria is more severe than that using Puck criteria for several top plies. Hinton et al. [7], Soden et al. [9], and Dong et al. [38] showed there may be some allowance when Puck criteria are used to predict the initial failure stresses, especially in the presence of thermal stresses. Delamination cracks propagate along the fiber direction below the interface and the delamination areas using three criteria are approximately the same. In addition, progressive delamination interacts with matrix cracks in a complicated way, which was demonstrated by Hongkarnjanakul et al. [30]. At about t¼ 1.5 ms, the velocity and kinetic energy of the impactor become zero as it reaches the very bottom. Matrix tension damage still evolves along the fiber direction which concentrates on the central

240

10. Damage models and explicit finite element analysis

region. In general, matrix tension damage using Puck criteria is relatively large, but small using Hashin criteria, especially at the middle and bottom of laminate. In addition, the damage evolution direction for matrix tension using Puck criteria is more distinct than those using Hashin and Chang-Chang criteria. Matrix compression damage continues to evolve and the damaged area using Puck criteria is still the minimum. Similar to matrix tension damage, matrix compression damage also continues to evolve and becomes distinct along the fiber direction, which are mainly represented by the 0°, 45°, and 45° plies for Layup-A laminate and the 90° and 45° plies for Layup-B laminate. Besides, delamination crack also continues to propagate among the internal plies of laminates, which may arise from large transverse shear forces over there, consistent with the experimental phenomena observed by Johnson et al. [35]. By comparison, delamination is most severe at the 45 ° /90° interface and the +45° /  45°interface at the middle and low parts of Layup-A and B laminates, respectively. By comparing with the second stage at about t¼ 1.5ms, matrix tension damage and delamination at about t¼ 3.5 ms do not show large difference, in which the impactor rebounds and starts to separate from the laminate, and the impact force becomes zero, but matrix compression damage continues to evolve during the rebound stage, which may be caused by the compressive stress during rebound at local areas. During four stages above, matrix tension and delamination play an important role in affecting the damage tolerance and residual strength of the laminate. By comparison, four cohesive strengths σ max ¼ τmax ¼ 20 MPa, 30 MPa, 40 MPa, and 50 MPa are used to discuss the effect of interface strengths on the delamination damage. Fig. 10.13 shows the +45 ° /90°, 90 ° /  45°, and 45 ° /90° delamination interface for Layup-A laminate using Puck criteria. It is shown the delamination damage area predicted using σ max ¼ τmax ¼ 20MPa is the largest, larger than the experimental results +1.000e+00 +9.167e-01 +8.333e-01 +7.500e-01 +6.667e-01 +5.833e-01 +5.000e-01 +4.167e-01 +3.333e-01 +2.500e-01 +1.667e-01 +8.333e-02 +0.000e+00

σmax = τmax = 20MPa

σmax = τmax = 30MPa

σmax = τmax = 40MPa

σmax = τmax = 50MPa

+45°/90°

90°/–45°

–45°/90°

FIG. 10.13 Delamination interface using four cohesive strengths at about t ¼ 3.5 ms for Case-A specimen: Layup-A laminate.

10.5 Numerical results and discussion

241

obtained by Hongkarnjanakul et al. [30]. By comparison, σ max ¼ τmax ¼ 30MPa lead to consistent damage area and σ max ¼ τmax ¼ 40 MPa and 50MPa underestimate the damaged area. Thus, σ max ¼ τmax ¼ 30MPa are generally used. Fig. 10.14 shows the effects of different frictional coefficients on the impact-time curve and the impact-displacement curve. There is no large difference in terms of the impact force-time and impact forcedisplacement by using three friction coefficients 0.2, 0.3, and 0.4. In addition, we use the reduced integration element C3D8R for the laminate, which may lead to the hourglass effect with spurious energy dissipation. We generally use the enhanced hourglass method in ABAQUS to improve the numerical robustness. Fig. 10.15 shows the artificial strain energy (ALLAE)-time curve and the internal energy (ALLIE)-time curve. It is shown ALLAE accounts for less than 5% of ALLIE, which shows hourglass model is well controlled.

10.5.3 Numerical results and analysis for case-B specimens In order to compare the effects of three failure criteria on the impact responses, HS300/ET223 Case-B composite specimen is further used and material parameters are listed in Table 10.2. Geometry sizes for the laminate are 65 mm 87.5 mm 3.2 mm. The thickness of each ply is 0.32 mm. The diameter and mass of the impactor are 12.5 mm and 2.34 kg, respectively. The number of all elements is 41,210. The layup pattern is [0°3/ + 45 ° /  45°]s and three impact energies: 2, 4, and 8 J are used [18]. Feng and Aymerich [18] got the impact contact time about 4.2 ms by experiments and thus the total impact time period in the explicit FEA for Case-B specimen is set to 5.0 ms. Numerical results using FEA are compared with the experimental data [18]. Figs. 10.16–10.18 show the impact force-time curve, the impact forcedisplacement curve, and the energy dissipation diagram. Five aspects of issues are discussed here. (1) From Fig. 10.16, some numerical oscillation for Case-B specimen also appears before the peak impact force and initial large fluctuation for the impact force is caused by the delamination initiation, similar to Case-A specimens. (2) By comparison, the values Fcrit obtained by FEA and experiments are about 1400 and 1200 N at 2 J impact energy, about 1500 and 1500 N at 4 J impact energy, and about 1800 and 1600 N at 8 J impact energy. It is shown Fcrit is not affected by the impact energy largely, which is consistent with the conclusion obtained by Gonza´lez et al. [34]. The impact force reaches the maximum at about t ¼ 2 ms when the impactor reaches the lowest location, and then the impactor rebounds until complete separation between the impactor and laminate at about t ¼ 4.2 ms. (3) From Figs. 10.16 and 10.17, Puck criteria lead to

242

10. Damage models and explicit finite element analysis

FIG. 10.14 Effects of different friction coefficients on numerical results: (A) impact forcetime curves and (B) impact force-displacement curves using Puck criteria for Case-A specimen: Layup-A laminate.

10.5 Numerical results and discussion

243

FIG. 10.15

Hourglass effects on the dissipated energy using Puck criteria for Case-A specimens: (A) Layup-A laminate and (B) Layup-B laminate.

consistent results with the experimental results. By comparison, Hashin and Chang-Chang criteria lead to slightly larger errors. In addition, Puck criteria capture sudden drop at the peak force 4.5 kN in Fig. 10.16, but Hashin and Chang-Chang criteria fail to capture it. (4) From Fig. 10.18,

TABLE 10.2 Material parameters for HS300/ET223 composite laminates [18]. Layer properties

Interface properties

E1 ¼ 122 GPa, E2 ¼ E3 ¼ 6.2 GPa, G23 ¼ 3.7 GPa G12 ¼ G13 ¼ 4.4 GPa, v12 ¼ v13 ¼ 0.35, v23 ¼ 0.5 XT ¼ 1850 MPa, XC ¼ 1470 MPa, YT ¼ 29 MPa, YC ¼ 290 MPa ^12 ¼ S ^23 ¼ S ^13 ¼ 65MPa ZC ¼ YC ¼ 140MPa, S C T C ¼ 92N=mm,Γ ¼ 80N=mm,Γ ΓT 11 11 22 ¼ 0:52N=mm,Γ 22 ¼ 1:61N=mm Gc1 ¼ 0.52 N/mm, Gc2 ¼ Gc3 ¼ 0.92 N/mm, η ¼ 1.45 σ max ¼ 30 MPa, τmax ¼ 80 MPa

FIG. 10.16 Impact force-time curves using Puck, Hashin, and Chang-Chang failure criteria under different impact energies: (A) 2 J, (B) 4 J, (Continued)

10.5 Numerical results and discussion

245

Fig. 10.16, cont’d and (C) 8 J for Case-B specimen.

the dissipated energy increases rapidly from 2 and 4 J to 8 J impact energy. Besides, Puck criteria lead to a closer value to the averaged experimental value. (5) The damage features for Case-B specimens are similar to those for Case-A specimens: matrix tension damage appears initially at the bottom 0° plies due to large bending stress and then evolves into neighboring upper +45° plies, but matrix compression damage appears initially at the top 45° plies. All three failure criteria can capture delamination growth at the 0 ° / + 45°, +45 ° /  45°, 45 ° / + 45°, and +45 ° /0° interfaces. The delamination areas at the 45 ° / + 45° and +45 ° /0° interfaces are the largest. From FEA on Case-A and B specimens above, numerical results using Puck criteria are generally more close to the experimental results obtained by Hongkarnjanakul et al. [30] and Feng and Aymerich [18] since they are considered as an improvement over the other two criteria. However, Puck criteria also show some disadvantages. (1) Accurate determination of the ^A fracture plane angle and related parameters such as the strength value S 23 needs some special algorithms. For example, Puck et al. [19] proposed pragmatic solutions to these parameters and Schirmaier et al. [20] and Wiegand et al. [21] proposed search algorithms respectively to determine the fracture plane angle. (2) The effects of the ply thickness and stacking sequence in the failure analysis of laminate have not been properly

246

10. Damage models and explicit finite element analysis

addressed in the world wide failure exercise by Soden et al. [9]. (3) The transverse strength and the shear strength are simply modified by some correction factors to include in situ effects in Puck criteria, which yet cannot reflect the influence of the ply thickness and neighboring layup angle,

FIG. 10.17 Impact force-displacement curves using Puck, Hashin, and Chang-Chang failure criteria under different impact energies: (A) 2 J, (B) 4 J, (Continued)

10.5 Numerical results and discussion

247

Fig. 10.17, cont’d and (C) 8 J for Case-B specimen.

FIG. 10.18 Dissipated energy using Puck, Hashin, and Chang-Chang failure criteria under different impact energies: (A) 2 J, (B) 4 J, and (C) 8 J for Case-B specimen.

248

10. Damage models and explicit finite element analysis

as pointed out by Dong et al. [38]. By comparison, Chang-Chang criteria are still used in some finite element software including LS-DYNA and MSC. Dytran [39], and Hashin criteria have been integrated into ABAQUS software [40] due to their simpler formulas and numerical implementation by FEA. Thus, it is crucial to select appropriate failure criteria to achieve high-efficiency and high-fidelity simulation in practical impact analysis.

10.6 Concluding remarks This chapter studies the effects of different failure criteria including Puck, Hashin, and Chang-Chang criteria on the mechanical responses of carbon fiber composite laminates under low-velocity impact. A unified finite element formulation on the dynamic intralaminar damage and interlaminar delamination of laminates is developed. The intralaminar damage models using three failure criteria are implemented using ABAQUS-VUMAT and the delamination is simulated by the bilinear cohesive model. From numerical analysis on composite specimens with different materials, geometry sizes, and impact energies, three main conclusions are obtained: (a) In general, the errors between numerical and averaged experimental results using Puck criteria are the smallest, but the largest using Chang-Chang criteria. The main reason may lie that Puck criteria consider the strengthening effect of transverse compression on the nonlinear shear deformation on the fracture plane, and Chang-Chang criteria neglect the out-of-plane shear deformation effects. (b) Matrix tension and delamination are the main failure modes. Matrix tension damage initiates at the bottom and evolves along the fiber direction, but matrix compression damage initiates at the top and also evolves along the fiber direction. In general, matrix tension damage using Puck criteria at the middle and low plies of laminate using Puck criteria is most severe, but matrix compression damage at the middle and upper plies of laminate using Hashin and Chang-Chang criteria is most severe. In addition, delamination cracks propagate along the fiber direction in the ply below the delamination interface. (c) The first damage is caused by delamination which is represented by initial large fluctuation in the impact force-time curves. The first damage load is not affected by the impact energy largely. Dissipated energy due to the damage of composites and the dynamic friction contact adds largely with the increase of the impact velocity or impact energy.

References

249

Although Puck criteria theoretically show some advantages over other two criteria, there are still some intractable problems such as the accurate determination of the fracture plane angle and related strength parameters. Because of simpler formulas and numerical implementation by FEA, we also see Hashin and Chang-Chang criteria are widely applied to the impact analysis of composites as an integrated module in some commercial finite element software including ABAQUS, LS-DYNA, and MSC. Dytran. By weighting numerical precision and cost, there should be an appropriate selection and use for these failure criteria in the impact analysis of composites. After all, these failure criteria as macroscopically phenomenological theories should commit to solving some practical impact problems of composite structures accurately and efficiently.

References [1] de Moura MFSF, Gonc¸alves JPM. Modelling the interaction between matrix cracking and delamination in carbon-epoxy laminates under low velocity impact. Compos Sci Technol 2004;64:1021–7. [2] Hashin Z, Rotem A. A fatigue failure criterion for fiber reinforced materials. J Compos Mater 1973;7:448–64. [3] Hashin Z. Fatigue failure criteria for unidirectional fiber composites. J Appl Mech 1980;48:846–52. [4] Chang FK, Chang KY. A progressive damage model for laminated composites containing stress concentrations. J Compos Mater 1987;21:834–55. [5] Puck A, Sch€ urmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos Sci Technol 2002;62:1633–62. [6] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Compos Mater 1971;5:58–80. [7] Hinton MJ, Kaddour AS, Soden PD. Evaluation of failure prediction in composite laminates: background to ’part C’ of the exercise. Compos Sci Technol 2004;64:321–7. [8] Hinton MJ, Kaddour AS, Soden PD. A further assessment of the predictive capabilities of current failure theories for composite laminates: comparison with experimental evidence. Compos Sci Technol 2004;64:549–88. [9] Soden PD, Kaddour AS, Hinton MJ. Recommendations for designers and researchers resulting from the world-wide failure exercise. Compos Sci Technol 2004;64 (3–4):589–604. [10] Kaddour AS, Hinton MJ, Soden PD. A comparison of the predictive capabilities of current failure theories for composite laminates: additional contributions. Compos Sci Technol 2004;64:449–76. [11] Zhao GP, Cho CD. Damage initiation and propagation in composite shells subjected to impact. Compos Struct 2007;78:91–100. [12] Riccio A, De Luca A, Di Felice G, Caputo F. Modelling the simulation of impact induced damage onset and evolution in composites. Compos Part B 2014;66:340–7. [13] Li DH, Liu Y, Zhang X. Low-velocity impact responses of the stiffened composite laminated plates based on the progressive failure model and the layerwise/solid-elements method. Compos Struct 2014;110:249–75. [14] Meo M, Morris AJ, Vignjevic R, Marengo G. Numerical simulations of low-velocity impact on an aircraft sandwich panel. Compos Struct 2003;62:353–60.

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[15] Donadon MV, Iannucci L, Falzon BG, Hodgkinson JM, de Almeida SFM. A progressive failure model for composite laminates subjected to low velocity impact damage. Comput Struct 2008;86:1232–52. [16] Faggiani A, Falzon BG. Predicting low-velocity impact damage on a stiffened composite panel. Compos Part A 2010;41:737–49. [17] Shi Y, Swait T, Soutis C. Modelling damage evolution in composite laminates subjected to low velocity impact. Compos Struct 2012;94:2902–13. [18] Feng D, Aymerich F. Finite element modelling of damage induced by low-velocity impact on composite laminates. Compos Struct 2014;108:161–71. [19] Puck A, Kopp J, Knops M. Guidelines for the determination of the parameters in Puck’s action plane strength criterion. Compos Sci Technol 2002;62:371–8. [20] Schirmaier FJ, Weiland J, K€arger L, Henning F. A new efficient and reliable algorithm to determine the fracture angle for Puck’s 3D matrix failure criterion for UD composites. Compos Sci Technol 2014;100:19–25. [21] Wiegand J, Petrinic N, Elliott B. An algorithm for determination of the fracture angle for the three-dimensional Puck matrix failure criterion for UD composites. Compos Sci Technol 2008;68:2511–7. [22] Camanho PP, Davila CG, Moura MFD. Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 2003;37:1415–38. [23] Wriggers P. Computational contact mechanics. Berlin Heidelberg: Springer; 2006. [24] Pederson J. Finite element analysis of carbon fiber composite ripping using ABAQUS. Dissertations & Theses-Gradworks; 2008. [25] Huang CH, Lee YJ. Experiments and simulation of the static contact crush of composite laminated plates. Compos Struct 2003;61:265–70. [26] Tan W, Falzon BG, Chiu LNS, Price M. Predicting low velocity impact damage and compression-after-impact (CAI) behaviour of composite laminates. Compos Part A 2015;71:212–26. [27] Liu PF, Yang YH, Gu ZP, Zheng JY. Finite element analysis of progressive failure and strain localization of carbon fiber/epoxy composite laminates by ABAQUS. Appl Compos Mater 2015;22(6):711–31. [28] Liu PF, Gu ZP, Peng XQ, Zheng JY. Finite element analysis of the influence of cohesive law parameters on the multiple delamination behaviors of composites under compression. Compos Struct 2015;131:975–86. [29] Hughes TJR. The finite element method: Linear static and dynamic finite element analysis. Dover Publications; 2000. [30] Hongkarnjanakul N, Bouvet C, Rivallant S. Validation of low velocity impact modelling on different stacking sequences of CFRP laminates and influence of fibre failure. Compos Struct 2013;106:549–59. [31] Schoeppner GA, Abrate S. Delamination threshold loads for low velocity impact on composite laminates. Compos Part A 2000;31:903–15. [32] Gao D, Zhang X. Impact damage prediction in carbon composite structures. Int J Impact Eng 1995;16:149–70. [33] Yang FJ, Cantwell WJ. Impact damage initiation in composite materials. Compos Sci Technol 2010;70:336–42. [34] Gonza´lez EV, Maimı´ P, Camanho PP, Lopes CS, Blanco N. Effects of ply clustering in laminated composite plates under low-velocity impact loading. Compos Sci Technol 2011;71:805–17. [35] Johnson HE, Louca LA, Mouring S, Fallah AS. Modelling impact damage in marine composite panels. Int J Impact Eng 2009;36:25–39. [36] Richardson MOW, Wisheart MJ. Review of low-velocity impact properties of composite materials. Compos Part A 1996;27:1123–31.

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[37] Kim EH, Rim MS, Lee I, Hwang TK. Composite damage model based on continuum damage mechanics and low velocity impact analysis of composite plates. Compos Struct 2013;95:123–34. [38] Dong H, Wang J, Karihaloo BL. An improved Puck’s failure theory for fibre-reinforced composite laminates including the in situ strength effect. Compos Sci Technol 2014;98:86–92. [39] Nguyen MQ. A review of explicit finite element software for composite impact analysis. J Compos Mater 2005;39:375–86. [40] Syste`mes D. Abaqus analysis user’s manual. RI, USA: Simulia Corp Providence; 2007.

C H A P T E R

11

Explicit finite element analysis of plastic composite laminates under low-velocity impact

11.1 Introduction This chapter explores impact-induced damage mechanisms of plastic composite laminates. Pierron and Vautrin [1], Vogler and Kyriakides [2], Wang et al. [3], and Vogler et al. [4] showed by experiments some composites such as T300/914, AS4/PEEK, and IM7-8552 show distinct plastic deformation and nonlinear shear effect. From the thermodynamic perspective, Ladeve`ze and Lubineau [5, 6] developed damage models for plastic composite laminates, which show complexity for parameter identification. Later, Espinosa et al. [7] performed FEA on dynamic delamination of composite laminates, but did not introduce failure criteria and damage evolution laws for fiber and matrix. Based on Sun and Chen’s flow rule [8], Chen et al. [9] developed an elastoplastic damage model for AS4/PEEK composites that leads to a more accurate prediction than the models by Maa and Cheng [10] and Lapczyk and Hurtado [11]. However, the work by Chen et al. [9] was limited to quasi-static load. In terms of impact load, He et al. [12] used the maximum stress failure criteria to study the impact responses of plastic composites, but did not study the damage evolution mechanisms for fiber and matrix. Based on continuum damage mechanics, Kim et al. [13] developed a 3D inelastic model to predict the plastic deformation of composites under impact, which yet failed to define the plastic flow rule and hardening law. Singh and Mahajan [14] implemented Sun and Chen’s flow rule [8] using the forward Euler algorithm, which yet shows errors by explicit FEA.

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00012-2

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Copyright © 2021 Elsevier Inc. All rights reserved.

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11. Explicit finite element analysis of plastic composite laminates

This chapter performs explicit FEA for plastic composites under impact loads by further using the Puck’s failure criteria [15], Sun and Chen’s flow rule [8] as well as the bilinear cohesive model [16] for delamination analysis. Based on the strain equivalence principle proposed by Simo and Ju [17], an uncoupled numerical scheme for predicting intralaminar plastic deformation and damage evolution is developed, in which the effective strains and stresses are first solved using the backward Euler algorithm and then the nominal stress and strains, as well as damage variables, are updated. Explicit FEA is performed by developing numerical codes using ABAQUS-VUMAT (Explicit user material subroutine). Finally, two composite specimens under low-velocity impact are used to demonstrate the proposed numerical scheme.

11.2 Theoretical model for composites under impact 11.2.1 Damaged stress-strain relationships of composites For transverse impact of composite laminates, large strain theory is generally needed. The constitutive models are written as S5C : Εe , S5Cd : Εe

(11.1)

where the matrices C and S are written as 2 E1 ð1  ν23 ν32 Þ E2 ðν12  ν13 ν32 Þ E3 ðν13  ν12 ν23 Þ 6 E 6 1 ðν21  ν31 ν23 Þ E2 ð1  ν13 ν31 Þ E3 ðν23  ν21 ν13 Þ 6 16 6 E1 ðν31  ν21 ν32 Þ E2 ðν32  ν12 ν31 Þ E3 ð1  ν12 ν21 Þ C¼ 6 Ω6 ΩG12 6 6 ΩG23 4

3 7 7 7 7 7 7 7 7 7 5 ΩG31

with Ω ¼ 1  ν12 ν21  ν23 ν32  ν31 ν13  2ν12 ν23 ν31 (11.2)

and

8 9 S11 > 2 > > > > > 0 1=ð1  d11 Þ > > > > > > > > S22 > > 6 0 1=ð1  d22 Þ > > 6 >

= 6 0 0 33 6 ¼6 > S12 > > 6 0 0 > > > 6 > > > > 4 0 0 > > > S23 > > > > > > > 0 0 : ; S31

9 38 S11 > 0 0 0 0 > > > > > 7> > 0 0 0 0 S22 > > > 7> > = 7< S > 1 0 0 0 7 33 7 7> 0 0 S > 0 1=ð1  d12 Þ > 12 > > 7> > > 5> 0 0 1=ð1  d23 Þ 0 S > > > > : 23 > ; 0 0 0 1=ð1  d31 Þ S31 (11.3)

255

11.2 Theoretical model for composites under impact

8      > > 1  smt dT22 1  smc dC d12 ¼ 1  1  dT11 1  dC > 11 22 > > <      1  smt dT22 1  smc dC d23 ¼ 1  1  dT11 1  dC 11 22 > >   > >    > T C : d31 ¼ 1  1  dT11 1  dC 1  s d d 1  s mc mt 22 11 22

(11.4)

where C is the undamaged 4th-order elastic tensor and Cd is the damaged 4th-order elastic tensor. Εe ¼ Ε  Εp and Εp are the elastic and plastic parts of the Green-Lagrange strain tensor Ε, respectively. S and S are the effective and nominal second Piola-Kirchoff stress tensors, respectively. E1, E2, and E3 are the Young’s moduli, where “1” denotes the fiber direction and “2” and “3” are the transverse directions.G12, G23, and G13 are the shear moduli and vij(i, j ¼ 1, 2, 3) are the Poisson’s ratios. “T” and “C” in the damage variables dT11, dC11, dT22, and dC22 denote the tension and compression, respectively. Similar to the last chapter, Smt ¼ 0.9 and Smc ¼ 0.5 are used to control the loss of shear stiffness due to matrix tension and compression [18].

11.2.2 Plastic flow rule The simple plastic flow rule for composites is used [8].   qffiffiffiffiffiffiffiffiffiffiffi  p  ffi e p ¼ 3f S  R Ε e ¼ 0, F S, Ε   1 1 1 2 2 2 f Sij ¼ a11 S11 + a22 S22 + a33 S33 + a12 S11 S12 + a13 S11 S33 + a23 S22 S33 2 2 2 2 2 2 + a44 S23 + a55 S13 + a66 S12 (11.5) where aij are constants which are determined by experiments. F is the yielding function and f is the plastic potential function. R is the plastic e p is the equivalent plastic strain. In general, the hardening stress and Ε  p n e power-law function R ¼ β Ε is assumed, and β and n are constants that fit the experimental curve. Here, the plastic deformation is assumed to occur in the undamaged area of damaged materials, and the plastic flow and hardening law are expressed in the effective stress space S. For composite laminates under out-of-plane impact, Eq. (11.5) is modified to the following one-parameter form [19].  2 i   1 h 2 2 2 S22  S33 + 4S23 + 2a66 S13 + S12 (11.6) f Sij ¼ 2 where the constant a66 is determined by the off-axis tensile test. p e_ is calculated as The equivalent plastic strain rate Ε p e_ ¼ λ_ ∂F=∂R ¼ λ_ Ε

where λ_ is the plastic consistency factor.

(11.7)

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11. Explicit finite element analysis of plastic composite laminates

From the equivalence of the rate of plastic work, the equivalent stress Ris calculated as p pffiffiffiffiffi p e_ ) R ¼ 3f S : Ε_ ¼ RΕ (11.8) p where the plastic strain rate Ε_ is calculated as 9 8 0 > > > >   > > > > > S  S 3 22 33 > > > > > p ffiffiffiffi ffi ∂F=∂S ¼ > > 22 > > 2 8 p 9 > > 3f > > > > _ > > > >   Ε > > > > 11 > > > > S  S > > > 3 33 22 > p > > > > > > > > _ p ffiffiffiffi ffi ∂F=∂S ¼ Ε 33 > > > > 22 > > > > 2 > > > > 3f > > > = = < < Ε_ p > p 33 _ S ¼ λ Ε_ ¼ λ_ ∂F=∂S ) 23 p ∂F=∂S23 ¼ 6 pffiffiffiffiffi > > > Ε_ 23 > > > > > > > > > 3f > > > > > > > p > > > > > _ > > > > Ε > > > 13 > > > > S13 > > > > > > :_p ; ffi > > > > > ∂F=∂S13 ¼ 3a66 pffiffiffiffi Ε 12 > > 3f > > > > > > > > > S12 > > > > > > > p ffiffiffiffi ffi ¼ 3a ∂F=∂S 12 66 ; : 3f

(11.9)

11.3 Numerical algorithm using explicit FEA The strain rate effect is neglected. Essentially, the plastic deformation and damage evolution of composites under impact are coupled. Based on the strain equivalency principle proposed mainly by Simo and Ju [17], we adopt an uncoupled scheme that first solves the effective stresses and strains and then the damage variables and nominal stresses. This simplification shows that plastic deformation and damage evolution are independent. For the update of the plastic stresses and strains, the backward Euler algorithm using Newton iteration is most robust, which requires all stresses at the end of the new time increment must return to the updated yielding surface simultaneously. By comparison, the forward Euler algorithm used by Singh and Mahajan [14] shows large errors because the trial stresses do not return to the updated yielding surface and numerical accuracy is guaranteed by reducing the time increment. In the following, the backward Euler algorithm is introduced to update the effective stresses and strains, nominal stresses, and strains as well as damage variables. The effective stress-strain relationship in Eq. (11.1), the yield function in Eq. (11.5), and the hardening law represent the plastic constitutive model without considering damage. Iterative solutions are discretized into a sequence of time steps [tn, tn+1](n ¼ 0, 1, 2, 3, …) and each step refers to

11.3 Numerical algorithm using explicit FEA

257

the (n + 1)th increment. Driven by ΔEn+1, the o backn the strain increment p ep , Sn + 1 , Rn + 1 , dij, n + 1 oare ward Euler algorithm requires En + 1 , En + 1 ,nE n+1 p ep at updated at the end of (n + 1)th step after En , En , E n , Sn , Rn , dij, n the nth step are given. The updated stresses and strains are stored at the end of (n + 1)th increment, which are taken as the initial values at the start of next increment. The incremental plastic constitutive model is written as 8 En + 1 ¼ En + ΔEn + 1 , > > > p p > En + 1 ¼ En + Δλn + 1 ∂S Fn + 1 , > > > < p ep e E (11.10) n + 1 ¼ En + Δλn + 1 ,   > p > > , S ¼ C : Ε  Ε n+1 n+1 > n+1 >   > > p :F e ¼F S ,E 0 n+1

n+1

n+1

In order to solve the plastic stresses and strains, the elastic trial stresses are used and the “frozen” plastic response is assumed. At the start of new (n + 1)th increment, the constitutive equations are written as 8 e, tr En + 1 ¼ Een + ΔEe,n +tr1 ¼ Een + ΔEn + 1 , > > > > p, tr p > > E n + 1 ¼ En , > > > > >

n, > > n+1   > > tr p, tr > > Sn + 1 ¼ C : En + 1  En + 1 ¼ Sn + C : ΔEn + 1 , > > > > : dtr ¼ d , ij ¼ 11,22, 12,13,23 ij, n + 1

ij, n

where the superscript “tr” denotes the trial concept. Then, the stresses and strains are updated by judging whether the elastic or plastic deformation appears.  tr  e p, tr  0 and all trial (a) Only the elastic stage appears if Fn + 1 ¼ F Sn + 1 , E n+1 stresses and arethe updated stresses and strains.  strains tr e p, tr > 0, the following Newton iterations (b) If Fn + 1 ¼ F Sn + 1 , E n+1 according to the backward Euler algorithm are required to perform the closest projection of the trial stresses on the yielding surface, which guarantees all stresses persist on the updated yielding surface. At this time,the Kuhn-Tucker loading condition ep ðΔλn + 1 Þ ¼ 0 is used to solve the plastic Fn + 1 Sn + 1 ðΔλn + 1 Þ, E n+1

consistency factor increment Δλn+1 > 0 by performing the following numerical iterations. During iterations, the total strain En+1 is constant. The initial conditions for iterations are set as ð0Þ tr p, ð0Þ p,tr e p, ð0Þ ¼ E e p, tr κ ¼ 0, Δλ0n + 1 ¼ 0, Sn + 1 ¼ Sn + 1 , En + 1 ¼ En + 1 , E n+1 n+1

(11.12)

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11. Explicit finite element analysis of plastic composite laminates

The Newton iterations are written as 8 ðκ + 1Þ ðκ Þ ðκ + 1Þ > Δλn + 1 ¼ Δλn + 1 + δΔλn + 1 , > > > > ðκ + 1Þ p, ðκÞ p, ðκ + 1Þ > < Εp, ¼ Εn + 1 + δΕn + 1 , n+1 e p, ðκ + 1Þ ¼ E ep, ðκÞ + δE ep, ðκ + 1Þ , > E > n+1 n+1 n+1 > >   > > : Sðκ + 1Þ ¼ C : Εn + 1  Εp, ðκ + 1Þ n+1 n+1

(11.13)

where δΔλ(κ+1) n+1 is given by

  ðκ Þ F Δλn + 1 ðκ + 1Þ  δΔλn + 1 ¼   ðκ Þ F0 Δλn + 1

(11.14)

where F0 is the partial derivative of yielding function with respect to Δλ. iterations above end until the yielding criterion  Numerical  ðκ + 1Þ ep, ðκ + 1Þ  tol (tol is tolerance) is met. Then, the effective stress F Sn + 1 , E n+1 e p , and the Sn + 1 , the elastic strain Εen+1, the equivalent plastic strain E n+1 plastic strain Εpn+1 are obtained 8 p p, ðκ + 1Þ > Εn + 1 ¼ Εn + 1 , > > > > < ep e p, ðκ + 1Þ , En + 1 ¼ E n+1 > p, ðκ + 1Þ e > ¼ Ε , Ε > n + 1  Εn + 1 n+1 > > : e Sn + 1 ¼ C : Ε n + 1

(11.15)

Then, the plastic state keeps constant during the update of damage state. Puck criteria above are used to judge whether there is fiber or matrix failure, and damage variables dij are updated if fiber or matrix fails. Finally, the nominal stress S is updated according to Eq. (11.1), which is then used in the update of the governing equation in the explicit FEA M u€ + F int ¼ F ext , ð F int ¼ BT SdΩ, ð Ω M ¼ ρN T NdΩ

(11.16)

Ω

where ρ is the density of laminate. N is the shape function and B is the large-deformation strain matrix according to the total Lagrange finite element formulation [20]. M is the lumped mass matrix instead of the consistent mass matrix in order to improve calculation efficiency [21]. Fint and Fext are internal and external node force vectors, respectively. The explicit central difference algorithm is used to solve the node acceleration,

259

11.4 Results and discussion

velocity, and displacement, in which time increment △ t must be small enough to avoid the Courant stability problem pffiffiffiffiffiffiffiffiffiffi (11.17) Δt < 2=ω ¼ Lmin = E1 =ρ where ω is generally considered as the largest natural frequency of the smallest element and Lmin is the minimum side size in a solid element.

11.4 Results and discussion Two graphite/epoxy composite specimens are used to validate the developed algorithm. The geometry model includes the laminate, fixed frame, and impactor. The reduced integration element C3D8R is used for the laminate and impactor, and the finite-thickness (0.001 mm) bilinear cohesive element COH3D8 is used for the delamination interface.

11.4.1 Case-A specimen For Case-A laminate, the layup pattern and geometry size are [0°2/90°2/ and 100 mm  100 mm  2.1 mm, respectively. The mass and diameter of the impactor are 6.5 kg and 1.27 cm, respectively. From impact experiments by Kim et al. [13], the impact contact time is about 11 ms and thus the total calculation time 12 ms is set in the explicit FEA. In order to prevent the interpenetration between the delaminated interface of any two neighboring plies, the penalty contact algorithm is employed and the friction coefficient is set to 0.3 [22]. Material parameters including elastic constants, strength parameters, and fracture energy are listed in Table 11.1 and plastic parameters are listed in Table 11.2.

0°2/90°2/0°2]s

TABLE 11.1 Material parameters for graphite/epoxy composites for Case-A specimen [4]. E1 ¼ 143.4 GPa, E2 ¼ E3 ¼ 9.27 GPa, G23 ¼ 3.2 GPa, G12 ¼ G13 ¼ 3.8GPa, v12 ¼ v13 ¼ 0.31, v23 ¼ 0.52

Ply properties

XT ¼ 2945MPa, XC ¼ 1650MPa, YT ¼ 54MPa, YC ¼ 240MPa, C ^12 ¼ S ^23 ¼ S ^13 ¼ 89:9MPa, GT S 11 ¼ 91.6 N/mm, G ¼ 79:9N=mm, 11

GT22 ¼ 0.22 N/mm, GC 22 ¼ 0.76 N/mm

Interface properties

Gc1 ¼ 0.33N/mm, Gc2 ¼ Gc3 ¼ 0.8N/mm, η ¼ 1.45 T01 ¼ 33MPa, T02 ¼ T03 ¼ 54MPa

TABLE 11.2 Material parameters for plastic graphite/epoxy composites [8]. a66 β n

1.25 567.9 MPa 0.272

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11. Explicit finite element analysis of plastic composite laminates

Fig. 11.1 shows three mesh models including 15,675, 72,928, and 121,360 bulk elements with the element sizes 2.0 mm  2.0 mm, 0.94 mm  0.94 mm, and 0.5 mm  0.5 mm at the contact area, respectively. Figs. 11.2 and 11.3 show the impact force-time curves and the artificial strain energy (ALLAE)-time curves at 5J impact energy, respectively. From Fig. 11.2, coarse mesh model overestimates the peak stress, but middle and fine mesh models lead to consistent results with experimental results. From Fig. 11.3, the artificial strain energy (ALLAE) using coarse mesh model accounts for about 10% of the total impact energy, but only about 5% using middle and fine mesh models, which shows the hourglass mode is better controlled using the fine mesh model. In the following, the middle mesh model is used by considering computational efficiency and accuracy. Three impact energies 5J, 10J, and 15J are used in FEA and numerical results are compared with the experimental results and numerical results

FIG. 11.1 Finite element model for the impact analysis of the laminate with three mesh sizes.

11.4 Results and discussion

FIG. 11.2

FIG. 11.3

261

The impact force-time curve using three mesh models at 5J impact energy.

The artificial strain energy (ALLAE)-time curve using three mesh models at 5J impact energy.

262

11. Explicit finite element analysis of plastic composite laminates

in the last chapter using the elastic damage model and Singh and Mahajan [14] using forward Euler algorithm in terms of the impact force-time curves, central displacement-time curves as well as damage evolution behaviors of fiber, matrix, and delamination interface. Fig. 11.4 shows the impact force-time curves at three impact energies. Some numerical oscillation near the peak force in the curves appears, which arises mainly from two reasons: dynamic response and numerical effect. Here, a numerical filtering technique is used and the output sampling time is 0.01 ms. At about 5.5 ms time, the impactor reaches the lowest end and then starts to rebound until complete separation between the impactor and laminate at about 11 ms time. Relatively good agreement is obtained between our numerical results and experimental results [13] in terms of the impact force-time curves. By comparison, Singh and Mahajan [14] underestimated the peak impact force and cannot get the sudden drop in the impact force-time curves at 15J impact energy. At 5J and 10J impact energies, elastic damage model, and plastic damage model lead to basically consistent results. However, elastic damage model shows large errors compared with experimental results at 15J impact energy. By comparing with experimental results, numerical results using the plastic damage model show higher precision than those using the elastic damage model as the impact energy and plastic deformation become relatively large. By considering the sudden drop in the impact force-time curves from numerical and experimental results [13], the impact energy can be divided into two levels. The first energy level is lower than 15J and the second energy level is over 15J. At 5J and 10J impact energies, smooth impact force-time curve is obtained indicating no severe damage. However, unstable drop after the peak value shows relatively severe damage at 15J impact energy. Figs. 11.5 and 11.6 show the maximum central displacement and the permanent central displacement respectively, corresponding to the lowest end of the impactor at about 5.5 ms time and the zero impact force at the end of impact at about 11 ms time, respectively. Relatively good agreement is also obtained between our numerical results and experimental results [13]. By comparison, elastic damage model and plastic damage model lead to the same results, but Singh and Mahajan [14] overestimated the maximum central displacement. Fig. 11.7 shows fiber damage at about 11 ms time. Three aspects of conclusions are obtained by FEA. (1) It appears only at the second energy level with the impact energy higher than 15J. (2) Most of fiber damage appears both at the upper plies of the laminate because large compressive stress arising from the plastic deformation at the upper 0°2 plies, similar to the conclusion obtained by Johnson et al. [23]. Also, it also appears at the bottom 0°2 plies and 90°2 plies due to tensile stress [13]. (3) Fiber damage propagates along the transverse of the fiber principal direction, consistent with the conclusions obtained by Kim et al. [13] and Faggiani and Falzon [22].

11.4 Results and discussion

263

FIG. 11.4 Numerical results for Case-A specimen: (A) the impact force-time curve at 5J impact energy, (B) the central displacement-time curve at 5J impact energy, (Continued)

264

11. Explicit finite element analysis of plastic composite laminates

FIG. 11.4, CONT’D (C) the impact force-time curve at 10J impact energy, (D) the central displacement-time curve at 10J impact energy, (Continued)

11.4 Results and discussion

265

FIG. 11.4,CONT’D (E) the impact force-time curve at 15J impact energy, and (F) the central displacement-time curve at 15J impact energy.

266

FIG. 11.5

11. Explicit finite element analysis of plastic composite laminates

Maximum central displacement for Case-A specimen at different impact

energies.

FIG. 11.6 energies.

Permanent central displacement for Case-A specimen at different impact

11.4 Results and discussion

267

FIG. 11.7 Fiber damage evolution for Case-A specimen at the impact energies (A) 5J and 10J and (B) 15J impact energy.

Fig. 11.8 shows matrix tension damage, matrix compression damage and delamination of the laminate at about 11 ms time. Three damage variables (or called status variables) in ABAQUS-VUMAT are defined for fiber damage, matrix tension and compression damage, respectively. The delamination is shown by the damage variable ds in last chapter. Several aspects of conclusions are obtained here. (1) Matrix tension damage appears initially at the bottom of the laminate and propagates upward. By comparison, matrix compression damage appears mainly at the upper and bottom plies. (2) In general, matrix tension damage is dominating compared with matrix compression damage. In addition, matrix tension and compression damage become more severe with the increase of impact energy, but evolves always along the fiber principal direction, similar to the conclusions obtained by Kim et al. [13] and Faggiani and Falzon [14]. (3) Delamination appears mainly at the upper plies of the laminate around the impact areas at the first energy level and delamination at the bottom plies of the laminate also occurs at the second energy level, accompanied by severe fiber and matrix damage, consistent with the experimental observation by using ultrasonic C-scan technique [13].

11.4.2 Case-B specimen In order to validate the developed algorithm, Case-B specimen is further used and material parameters are listed in Table 11.3. The layup pattern and geometry size are [0°3/90°3]s and 65 mm  87.5 mm  2 mm, respectively. The mass and diameter of the impactor are 2.28 kg and 1.25 cm, respectively. The mesh model for Case-B specimen is similar to that for Case-A specimen. The number of bulk elements is 42,510.

268

FIG. 11.8

11. Explicit finite element analysis of plastic composite laminates

Matrix tension, matrix compression, and delamination damage evolution for Case-A specimen at the impact energies (A) 5J, (B) 10J, (Continued)

11.4 Results and discussion

269

FIG. 11.8, CONT’D and (C) 15J, respectively.

TABLE 11.3 Material parameters for graphite/epoxy composites for Case-B specimen [4]. Ply properties

E1 ¼ 93.7 GPa, E2 ¼ E3 ¼ 7.45 GPa G12 ¼ G13 ¼ G23 ¼ 3.97 GPa, v12 ¼ v13 ¼ v23 ¼ 0.261 XT ¼ 2945 MPa, XC ¼ 1650 MPa, YT ¼ 54 MPa, YC ¼ 240 MPa C ^12 ¼ S ^23 ¼ S ^13 ¼ 89:9MPa, GT S 11 ¼ 91.6 N/mm, G11 ¼ 79.9 N/mm ΓT22 ¼ 0.22 N/mm, ΓC22 ¼ 0.76 N/mm

Interface properties

Gc1 ¼ 0.52 N/mm, Gc2 ¼ Gc3 ¼ 0.97 N/mm, η ¼ 1.45, T01 ¼ 30 MPa, T02 ¼ T03 ¼ 80 MPa

Three impact energies 2J, 5J, and 9.2J are used [24]. Aymerich et al. [24] got the impact contact time of about 5.5 ms for Case-B specimen by experiments and thus the total impact time period in the explicit FEA is set to 6 ms. Numerical results using FEA are compared with the experimental data [24–26]. Figs. 11.9 and 11.10 show the impact force-time curves and the impact force-displacement curves. From Fig. 11.9, some numerical oscillation for Case-B specimen also appears before the peak force and numerical

270

FIG. 11.9

11. Explicit finite element analysis of plastic composite laminates

Impact force-time curve for Case-B specimen at the impact energies (A) 2J,

(B) 5J, (Continued)

11.4 Results and discussion

271

FIG. 11.9, CONT’D and (C) 9.2J, respectively.

filtering technique is also used, similar to Case-A specimen. The impact force reaches the maximum at about 2.5 ms time when the impactor reaches the lowest end and then the impactor rebounds until complete separation between the impactor and laminate at about 5.5 ms time. Numerical results are shown to be basically consistent with the experimental data [24]. Similar to Case-A specimen, two energy levels are divided for Case-B specimen: the first level is lower than 8J and the second level is higher than 8J. There is no large damage for the laminate at 2J and 5J impact energies. However, a sudden drop at the peak force 4.5 kN in Fig. 11.9 indicates large plastic deformation and localized damage at the impact contact area at 9.2J impact energy. Similarly, experimental results show the impact force-time curves capture sudden force drop and subsequent high-amplitude oscillation at the impact energy higher than 8J [24–26]. From Fig. 11.10, the maximum central displacement and the permanent central displacement by FEA and experiments [24–26] are basically the same except slightly high peak force. From the impact force-time curves and the impact force-displacement curves, elastic damage model and plastic damage model lead to consistent results at 2J and 5J impact energies, but elastic damage model cannot capture a sudden drop in the impact force-time curves at 9.2J impact energy.

272

FIG. 11.10

11. Explicit finite element analysis of plastic composite laminates

Impact force-displacement curve for Case-B specimen at the impact energies

(A) 2J, (B) 5J, (Continued)

11.4 Results and discussion

273

FIG. 11.10, CONT’D and (C) 9.2J, respectively.

Fig. 11.11 shows the energy dissipation of the laminate at different impact energies, which comes from the plastic deformation, damage evolution, and frictional contact. When the impactor starts to contact with the laminate, the energy dissipation starts to appear and the laminate deforms by absorbing the kinetic energy of the impactor. When the impactor reaches the lowest end at about 2.5 ms time, all kinetic energy has been transformed partly into the strain energy and partly into the dissipated energy. After this time, the impactor rebounds and elastic unloading occurs, and the strain energy of the laminate transformed partly into the kinetic energy of the impactor and partly into the dissipated energy again. The dissipated energy can be calculated as the difference between the initial and final kinetic energy of the impactor. In addition, the energy dissipation is relatively small when the impact energy is lower than 8J, but increases largely at 9.2J impact energy, consistent with the experimental results [24]. The possible reason lies in more fiber damage indicating large energy dissipation at higher impact energies. However, elastic damage model underestimates the energy dissipation at 9.2J impact energy because it cannot capture sudden drop in the impact force-time curves. Damage features for Case-B specimen are similar to those for Case-A specimen. Matrix tension damage appears initially at the bottom 0° plies

274

FIG. 11.11

11. Explicit finite element analysis of plastic composite laminates

Dissipated energy for Case-B specimen at different impact energies.

and then evolves upward. By comparison, matrix compression damage appears at the upper 0° plies. With the increase of impact energy, the 90 ° /0° interface delamination evolves along the 0° direction. Fiber damage evolves at the upper 0° plies, represented mainly by a sudden drop at the peak force.

11.5 Concluding remarks This chapter performs explicit FEA on plastic composite laminates under low-velocity impact. First, a plastic damage model for polymer matrix composites is presented which considers various failure mechanisms and damage evolution behaviors including fiber, matrix, and shear failure. Second, a numerical scheme is originally proposed to predict the plastic deformation and damage evolution behaviors, in which the backward Euler algorithm is developed to solve the effective and nominal stresses and strains as well as damage variables. Finally, based on the nominal stress, the explicit time-stepping algorithm is used to solve the node displacement, velocity, and acceleration. For two composite specimens with different materials, geometry sizes, and impact energy, relatively good agreement is obtained between numerical and experimental

References

275

results in terms of the impact force-time/displacement curves and the dissipated energy. In addition, matrix tension damage initiates at the bottom plies and propagates upwards, but matrix compression and fiber damage appear mainly at the upper and bottom plies. Finally, distinct plastic deformation and fiber damage appear only at high impact energy, and the damage features of elastic composite laminates can be approximated using the present plastic model at low impact energy.

References [1] Pierron F, Vautrin A. Accurate comparative determination of the in-plane shear modulus of T300/914 by the iosipescu and 45° off-axis tests. Compos Sci Technol 1994;52: 61–72. [2] Vogler TJ, Kyriakides S. On the axial propagation of kink bands in fiber composites: part I experiments. Int J Solids Struct 1999;36:557–74. [3] Wang J, Callus PJ, Bannister MK. Experimental and numerical investigation of the tension and compression strength of un-notched and notched quasi-isotropic laminates. Compos Struct 2004;64:297–306. [4] Vogler M, Rolfes R, Camanho PP. Modeling the inelastic deformation and fracture of polymer composites-Part I: plasticity model. Mech Mater 2013;59:50–64. [5] Ladeve`ze P, Lubineau G. On a damage mesomodel for laminates : micro-meso relationships, possibilities and limits. Compos Sci Technol 2001;61:2149–58. [6] Lade`veze P, Lubineau G. An enhanced mesomodel for laminates based on micromechanics. Compos Sci Technol 2002;62:533–41. [7] Espinosa H, Dwivedi S, Lu HC. Modeling impact induced delamination of woven fiber reinforced composites with contact/cohesive laws. Comput Methods Appl Mech Eng 2000;183:259–90. [8] Sun CT, Chen JL. A simple flow rule for characterizing nonlinear behavior of fiber composites. J Compos Mater 1989;23:1009–20. [9] Chen JF, Morozov EV, Shankar K. A combined elastoplastic damage model for progressive failure analysis of composite materials and structures. Compos Struct 2012;94: 3478–89. [10] Maa RH, Cheng JH. A CDM-based failure model for predicting strength of notched composite laminates. Compos Part B 2002;33:479–89. [11] Lapczyk I, Hurtado JA. Progressive damage modeling in fiber-reinforced materials. Compos Part A 2007;38:2333–41. [12] He W, Guan ZD, Li X, Liu DB. Prediction of permanent indentation due to impact on laminated composites based on an elasto-plastic model incorporating fiber failure. Compos Struct 2013;96:232–42. [13] Kim EH, Rim MS, Lee I, Hwang TK. Composite damage model based on continuum damage mechanics and low velocity impact analysis of composite plates. Compos Struct 2013;95:123–34. [14] Singh H, Mahajan P. Modeling damage induced plasticity for low velocity impact simulation of three dimensional fiber reinforced composite. Compos Struct 2015;131: 290–303. [15] Puck A, Sch€ urmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos Sci Technol 2002;62:1633–62. [16] Camanho PP, Davila CG, Moura MFD. Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 2003;37:1415–38.

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[17] Simo JC, Ju JW. Strain-based and stress-based continuum damage models. I. Formulation. Int J Solids Struct 1987;23:821–40. [18] Pederson J. Finite element analysis of carbon fiber composite ripping using ABAQUS [Master thesis]. Clemson University; 2008. [19] Thiruppukuzhi SV, Sun CT. Testing and modeling high strain rate behavior of polymeric composites. Compos Part B 1998;29:535–46. [20] Hibbitt HD, Marcal PV, Rice JR. A finite element formulation for problems of large strain and large displacement. Int J Solids Struct 1970;6:1069–86. [21] Hughes TJR. The finite element method: linear static and dynamic finite element analysis. Dover Publications; 2000. [22] Faggiani A, Falzon BG. Predicting low-velocity impact damage on a stiffened composite panel. Compos Part A 2010;41:737–49. [23] Johnson HE, Louca LA, Mouring S, Fallah AS. Modelling impact damage in marine composite panels. Int J Impact Eng 2009;36:25–39. [24] Aymerich F, Pani C, Priolo P. Damage response of stitched cross-ply laminates under impact loadings. Eng Fract Mech 2007;74:500–14. [25] Aymerich F, Pani C, Priolo P. Effect of stitching on the low-velocity impact response of [03/903]s graphite/epoxy laminates. Compos Part A 2007;38:1174–82. [26] Aymerich F, Dore F, Priolo P. Prediction of impact-induced delamination in cross-ply composite laminates using cohesive interface elements. Compos Sci Technol 2008;68:2383–90.

C H A P T E R

12

Explicit finite element analysis of GLARE hybrid composite laminates under low-velocity impact 12.1 Introduction As hybrid composite laminates, glass fiber composite/aluminum metal (GLARE) laminates which were first proposed at Delft University [1], have been used to manufacture components in Airbus A380 airplane including flat skin, cargo bay lining floor, firewall, and fuselage. By layup and size design, favorable mechanical properties of structures can be reached. Since airplanes often meet dynamic loads, an important work in the lightweight design of GLARE laminates is to improve impact resistance. There are some fundamental experimental and numerical research on impact responses, damage mechanisms, and energy absorption abilities of GLARE laminates under low-velocity impact [2–5]. However, they still show insufficiently accurate and mature predictions of impact responses of structures due to two reasons: (1) The damage evolution laws were not introduced; (2) The energy dissipation mechanisms with respect to different failure modes including intralaminar fiber and matrix damage, delamination, and plastic deformation in the aluminum layers were not discussed. The energy dissipation mechanisms of GLARE laminates are closely related to the damage evolution and plastic deformation. This chapter further explains the damage evolution behaviors and energy dissipation mechanisms of GLARE laminate corresponding to different failure modes including fiber breakage, matrix cracking, shear failure, and delamination. Numerical technique using the finite difference algorithm for composite layers is developed by ABAQUS-VUMAT (user

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00015-8

277

Copyright © 2021 Elsevier Inc. All rights reserved.

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12. Explicit finite element analysis of GLARE hybrid composite

dynamic material subroutine). In terms of the impact force-time/ displacement curves and the energy dissipation, numerical results with different metal thickness and impact energy by explicit FEA are compared with experimental results.

12.2 Johnson-Cook model for aluminum layer The Johnson-Cook model without considering temperature change is used to predict the dynamic mechanical properties of aluminum layers [6]. " # 
n  ε_ pl σ ¼ A + B εpl 1 + C ln (12.1) ε_ 0 where A, B, and C are material parameters, εpl is the equivalent plastic strain, n is material constants, ε_ pl =_ε 0 is the dimensionless plastic strain rate. To simulate ductile damage in aluminum layers, a damage parameter ψ is introduced [7]. 0 1 X Δεpl @ A (12.2) ψ¼ pl εf pl

where Δεpl is the equivalent plastic strain increment and εf is the ratedependent failure strain which is given by [7]. " # ε_ pl pl εf ¼ ½d1 + d2 exp ðd3 p=qÞ 1 + d4 ln (12.3) ε_ 0 where p/q is the hydrostatic pressure/deviatoric stress ratio and di(i ¼ 1  4) are material parameters.

12.3 Finite element model for impact analysis of GLARE laminates The drop-weight impact of GLARE laminates is adopted. Each Glare 5-3/ 2 laminate includes three 2024-T3 aluminum layers and eight FM94/S2 glass fiber composite layers. The layup pattern is, where Ali(i ¼ 1,2, 3) denotes three aluminum layers. The thickness of each composite layer is 0.127 mm [8]. Here, thickness 0.4 and 0.5mm respectively for each aluminum layer [Al1/0 ° /90 ° /90° /0 ° /Al2/0 ° /90 ° /90° /0 ° /Al3] in two specimens (GL 5-1 and GL 5-2 specimens) are selected. Thus, the total thickness for the two specimens is 2.23 and 2.52mm, respectively. Material parameters for aluminum and composite layers are listed in Tables 12.1 and 12.2, respectively. Similar to the last two chapters, Intralaminar damage models with Puck failure criteria are implemented using ABAQUS-VUMAT (user dynamic

12.3 Finite element model for impact analysis of GLARE laminates

TABLE 12.1

279

Material parameters for 2024-T3 aluminum.

Composite layer Young’s moduli E1 ¼ 50.6 GPa, E2 ¼ E3 ¼ 9.9 GPa Shear moduli G12 ¼ G13 ¼ 3.7 GPa, G23 ¼ 1.65 GPa Poisson’s ratios v12 ¼ v13 ¼ 0.32, v23 ¼ 0.063 Strength parameters XT ¼ 2500 MPa, XC ¼ 2000 MPa, YT ¼ 50 MPa, YC ¼ 150 MPa ^12 ¼ S ^13 ¼ 75MPa, S ^23 ¼ 50MPa [6] S

Intralaminar fracture toughness GT11 ¼ 81.5 N/mm, GC 11 ¼ 106.3 N/mm, GT22 ¼ 14.955 N/mm, GC 22 ¼ 14.955 N/mm Interlaminar fracture toughness Gc1 ¼ 0.5 N/mm, Gc2 ¼ Gc3 ¼ 0.9 N/mm Strength parameters T01 ¼ 60 MPa, T02 ¼ T03 ¼ 90 MPa

Interface

TABLE 12.2

Material parameters for FM94/S2 composite layers.

Elastic parameters

ρ ¼ 2770 kg/m3, E ¼ 72.4 GPa, ν ¼ 0.34

Plastic parameters

A ¼ 369 MPa, B ¼ 689 MPa, C ¼ 0.083 MPa, n ¼ 0.34

Fracture parameters

d1 ¼ 0:13,d2 ¼ 0:13, d3 ¼ 1:5, d4 ¼ 0:011, ε_ 0 ¼ 1:0s1

material subroutine). Finite deformation effect in ABAQUS is activated. Reduced integration element C3D8R, enhanced hourglass control, and distortion control are used. Bilinear cohesive model is used to predict the delamination and finite-thickness (0.001 mm) cohesive element is established for delamination analysis between two layers. The dynamic penalty method is used to deal with the contact between the laminate and impactor. The friction contact coefficient is set to 0.3. From impact experiments by P€ arn€ anen et al. [8], the total impact contact time is about 3.5 ms, and the total calculation time 4 ms is set in FEA. The diameter and mass of the impactor are 15.9 mm and 2.76 kg, respectively. Two levels of impact energy 20 and 30 J are used. In general, impact experiments by Sadighi et al. [9] and Seo et al. [10] adopted a square GLARE laminate, clamped by two square steel supporting frames with a central hole. By simplification, they substituted the real structure with the circular laminate and support frame by assuming the gripped area is rigid. However, this may lead to some errors between experimental and numerical results. By comparison, we establish two finite element models including the real model and simplified model [9, 10], as shown in Fig. 12.1A and B respectively. Based on experiments by P€arn€anen et al. [8], the sizes of square laminates are 110  110 mm and the hole diameter of supporting frames is 80 mm. Fine mesh sizes with 0.87  0.87 mm are used at the impact contact area and the number of bulk elements is 32,594 and 52,089 for two models respectively. Impact simulation is performed at 20 and 30 J impact energy for GL 5-1 specimen. By comparing with experimental data [8], the impact of force-time/displacement curves by FEA are shown in Fig. 12.2. In the impact force-time curves, the real model leads to

280

12. Explicit finite element analysis of GLARE hybrid composite

FIG. 12.1 Mesh models for GL 5-1 specimen under low-velocity impact: (A) simplified model and (B) real model.

FIG. 12.2 Numerical results for GL 5-1 specimen using real model and simplified model: (A) the impact force-time curve at 20 J impact energy, (Continued)

12.3 Finite element model for impact analysis of GLARE laminates

281

FIG. 12.2, CONT’D (B) the impact force-displacement curve at 20 J impact energy, (C) the impact force-time curve at 30 J impact energy (Continued)

282

12. Explicit finite element analysis of GLARE hybrid composite

FIG. 12.2, CONT’D (D) the impact force-displacement curve at 30 J impact energy.

accurate results, but the reduced model slightly overestimates the peak force because the reduced model is stiffer than the real model. In the impact force-displacement curves, both two models underestimate the permanent central displacement, but smaller value obtained by the reduced model validates the over-constrained condition. In general, the real model leads to more accurate results than the simplified model [9, 10]. In the following, the real model is used.

12.4 Numerical results and analysis Fig. 12.3 shows the impact force-time/displacement curves for GL 5-1 and GL 5-2 specimens at 20 and 30 J energy. Some numerical oscillation appears near the peak force in the curves, which arises mainly from two reasons: dynamic response and numerical effect. Here, numerical filtering technique is used and the output sampling time is 0.01 ms. At about 2.2 and 2 ms time for GL 5-1 and GL 5-2 specimens respectively, the impactor reaches the lowest end and then starts to rebound until complete separation between the impactor and laminate at about 3.7 and 3.5 ms time, respectively. Relatively good agreement is obtained between numerical and experimental results in terms of the impact of force-time/displacement curves. GL 5-2 specimen shows higher peak force and less contact time than GL 5-1 specimen for both numerical and experimental results,

12.4 Numerical results and analysis

283

FIG. 12.3 Numerical results for GL 5-1 and GL 5-2 specimens: (A) the impact force-time curve at 20 J impact energy, (B) the impact force-displacement curve at 20 J impact energy, (Continued)

284

12. Explicit finite element analysis of GLARE hybrid composite

FIG. 12.3, CONT’D (C) the impact force-time curve at 30 J impact energy (D) the impact force-displacement curve at 30 J impact energy.

12.4 Numerical results and analysis

285

which arises from higher stiffness and impact resistance for GL 5-2 specimen with thicker aluminum layers. From Fig. 12.3B and D, the maximum central displacement and permanent central displacement correspond to the lowest end of the impactor and the zero impact force at the end of impact respectively. From both numerical and experimental results, GL 5-2 specimen leads to smaller maximum/permanent central displacements than GL 5-1 specimen because of smaller local deformation. Fig. 12.4 shows matrix tension damage, matrix compression damage, and delamination for composite layers and the equivalent plastic strain for aluminum layers for GL 5-1 specimen at about 4 ms time. Two damage variables (or called status variables) in ABAQUS-VUMAT are defined for matrix tension and compression, respectively. The delamination is represented by the damage variable ds. Three issues are discussed here. (1) Delamination and matrix tension damage are found to be the dominant damage modes and delamination is more severe around the impact contact area; (2) Experimental results by ultrasonic C-scans [8] showed that nonimpacted metal layers are first cracked and separated from the composite layer at the impact area, and delamination appears near the crack and tends to grow around the damaged areas of metal layers. Besides, Ultrasonic tests [8] also showed that more severe damage and plastic deformation lead to larger delamination areas. From numerical results, dominating delamination takes place mainly at the interface between the middle and bottom aluminum layers and the composite layers under both 20 and 30 J impact energy, which are also consistent with the experimental observation by C-scans [8]. This may be explained as smaller bonding strength and fracture toughness between the aluminum layer and the composite layer than between composite layers due to plastic deformation and damage evolution of metal layers. (3) From Fig. 12.4, matrix tension damage is dominating compared with matrix compression damage. Most of the matrix damage appears at the bottom layers of laminates. This can be explained by the fact that damage is suppressed by energy absorption by the middle aluminum layer. Similar to the last chapter, matrix damage always evolves along the fiber principal direction. Fig. 12.5 shows the energy dissipation of laminates, which comes from the plastic deformation of aluminum layers, the damage evolution of composite layers, the delamination between different layers, and the frictional contact. When the impactor starts to contact with the laminate, the energy dissipation starts to appear and the laminate deforms by absorbing the kinetic energy of the impactor. When the impactor reaches the lowest end, all kinetic energy has been transformed partly into the strain energy and partly into the dissipated energy. After this time, the impactor rebounds and elastic unloading occurs, and the strain energy of the laminate transforms partly into the kinetic energy of the impactor and partly into the dissipated energy again. In general, the total dissipated energy of the system can be calculated by two equivalent methods: (1) The difference between the kinetic energy of

286

12. Explicit finite element analysis of GLARE hybrid composite Al1

0°

90°

90°

0°

Al2

0°

90°

90°

0°

Al3

Al1

0°

90°

90°

0°

Al2

0°

90°

90°

0°

Al3

Matrix tension X Z Y

Matrix compression +1.000e+00 +9.167e–01 +8.333e–01 +7.500e–01 +6.667e–01 +5.833e–01 +5.000e–01 +4.167e–01 +3.333e–01 +2.500e–01 +1.667e–01 +8.333e–02 +0.000e–00

Al1/0° 0° / 90° 90° / 90° 90° / 0° 0° / Al2 Al2 / 0° 0° / 90° 90° / 90° 90° / 0° 0° / Al3

Delamination

(A)

Al1

0°

90°

90°

0°

Al2

0°

90°

Al1

0°

90°

90°

0°

Al2

0°

90°

90°

0°

Al3

Matrix tension X Z Y

90°

0°

Al3

Matrix compression +1.000e+00 +9.167e–01 +8.333e–01 +7.500e–01 +6.667e–01 +5.833e–01 +5.000e–01 +4.167e–01 +3.333e–01 +2.500e–01 +1.667e–01 +8.333e–02 +0.000e–00

Al1/0° 0° / 90° 90° / 90° 90° / 0° 0° / Al2 Al2 / 0° 0° / 90° 90° / 90° 90° / 0° 0° / Al3

Delamination

(B) FIG. 12.4 Matrix tension, matrix compression and delamination for composite layers and

equivalent plastic strain for aluminum layers at about t ¼ 4 ms for GL 5-1 specimen at impact energy (A) 20 J and (B) 30 J.

the impactor before and after impact and (2) The area under the impact forcedisplacement curve. By using the second method, the final value of the history output variable ALLIE in ABAQUS is acquired to get the total dissipated energy. By comparison, errors for the total dissipated energy obtained by FEA and experiments [8] at 20 and 30 J impact energy are 12.7% and 14%

12.4 Numerical results and analysis

287

FIG. 12.5 Total dissipated energy at 20 and 30 J impact energy after impact at about t ¼ 4 ms for (A) GL 5-1 specimen and (B) GL 5-2 specimen.

288

12. Explicit finite element analysis of GLARE hybrid composite

respectively for GL 5-1 specimen, and 8.5% and 8.0% respectively for GL 5-2 specimen. Besides, the ratios of the total dissipated energy to the initial kinetic energy 20 and 30 J of the impactor are 57% and 60% respectively for GL 5-1 specimen, and 61.5% and 62.2% respectively for GL 5-2 specimen. This means more energy is dissipated with the increase of impact energy. In order to study dynamic progressive failure properties of laminates, energy dissipation corresponding to different failure modes is further discusses. Fig. 12.6 shows the dissipated energy due to the intralaminar damage of composite layers, the plasticity deformation of aluminum layers and the delamination failure, respectively. In the history output module in ABAQUS, the energy dissipation due to intralaminar damage is calculated as the difference (ALLIE-ALLSE) between the total energy dissipation (the final value of the history output variable ALLIE) and the recoverable elastic strain energy ALLSE. The dissipated energy due to delamination is acquired by the output variable ALLDMD in ABAQUS. From Fig. 12.6, three issues are discussed here. (1) About 60% of the total dissipated energy is absorbed by the plastic deformation of aluminum layers. By comparison, dissipated energy due to intralaminar damage and delamination accounts for only about 3.5% and 9% of the total dissipated energy, respectively. At the same time, delamination is more severe than intralaminar damage, as shown in Fig. 12.5. (2) The plastic dissipation of aluminum layers increases up to the maximum value when the impactor starts to rebound and then basically remains unchanged. The ratios of the plastic dissipation energy to the total dissipated energy at 20 and 30 J impact energy are 63% and 66% respectively for GL 5-1 specimen, and 57% and 64% respectively for GL 5-2 specimen, which shows the plastic deformation of aluminum layers absorbs more energy with the increase of impact energy. (3) The plastic energy dissipation will decrease and the energy dissipation due to damage and delamination will increase when the aluminum layers become thicker. For example, the plastic dissipation of aluminum layers for GL 5-1 and GL 5-2 specimens accounts for 63% and 57% of the total dissipated energy respectively at 20 J impact energy, and 66% and 64% respectively at 30 J impact energy. Fig. 12.7 shows the dissipated energy by each aluminum layer and the bottom four composite layers. The stable value of strain energy at the end of impact is the dissipated energy of each layer. In general, the aluminum layers absorb most of the impact energy, the bottom aluminum layer dissipates the maximum energy but the middle aluminum layer absorbs the minimum energy among three aluminum layers. The main reason lies that the dominant shear stress at the middle plane is smaller than the tensile and compressive stresses on the top and bottom surface. Finally, the dissipated energy of lower layers is relatively higher than that of the upper layers, which demonstrates larger tensile stress due to bending in lower layers than compressive stress in upper layers causes more damage.

12.4 Numerical results and analysis

289

FIG. 12.6 Total dissipated energy and energy dissipation including intralaminar damage, delamination, and plasticity for GL 5-1 and GL 5-2 specimens at impact energy (A) 20 J and (B) 30 J.

290

12. Explicit finite element analysis of GLARE hybrid composite

FIG. 12.7 Strain energy for three aluminum layer and the bottom four composite layers (A) for GL 5-1 specimen at 20 J impact energy, (B) for GL 5-1 specimen at 30 J impact energy, (Continued)

12.4 Numerical results and analysis

291

FIG. 12.7, CONT’D (C) for GL 5-2 specimen at 20 J impact energy, and (D) for GL 5-2 specimen at 30 J impact energy.

292

12. Explicit finite element analysis of GLARE hybrid composite

In order to compare the impact resistance and energy dissipation between fiber composite laminates and GLARE hybrid laminates, mechanical analysis of glass fiber composite laminates (CL) without aluminum layers with the same layup patterns and impact conditions to GL 5-1 laminates is further performed. Fig. 12.8 shows the impact force-time curves and the total energy dissipation for GL 5-1 and CL specimens

FIG. 12.8

Numerical results for GL 5-1 and CL specimens at 5 and 10 J impact energy: (A) the impact force-time curves and (B) the total energy dissipation.

12.4 Numerical results and analysis

293

respectively at 5 and 10 J impact energy. From Fig. 12.8A, GL 5-1 specimen shows a higher impact force and less contact time than the CL specimen due to high impact resistance. From Fig. 12.8B, the ratios of the total dissipated energy to the initial kinetic energy for GL 5-1 and CL specimens are 51.2% and 21.8% respectively at 5 J impact energy, and 52.4% and 17% respectively at 10 J impact energy. From Fig. 12.9, at 5 and 10 J impact energy, about 55% total energy is dissipated by the plastic deformation of

FIG. 12.9

Detailed energy dissipation including the total energy dissipation, the plasticity dissipation, and the intralaminar damage and interlaminar delamination dissipation at (A) 5 J and (B) 10 J impact energy.

294

12. Explicit finite element analysis of GLARE hybrid composite

aluminum layers and less energy is dissipated by intralaminar damage and interlaminar delamination for composite layers of GL 5-1 than CL, which shows composite layers for GL 5-1 have smaller damage. This means GLARE laminates generally show stronger energy dissipation ability and smaller damage for composite layers than fiber composite laminates.

12.5 Concluding remarks This chapter explores the dynamic mechanical properties of glass fiber/aluminum (GLARE) hybrid laminates under low-velocity impact by explicit FEA that includes dynamic intralaminar damage for composite layers, plastic deformation for aluminum layers, interlaminar delamination, and frictional contact. From numerical analysis on GLARE laminates with different thickness and impact energy, three main conclusions are obtained: (a) GL 5-2 specimen with a thicker aluminum layer shows a higher peak impact force, less contact time, and smaller maximum/permanent central displacements than GL 5-1 specimen due to higher stiffer and impact resistance. (b) Plastic deformation of aluminum layers accounts for most dissipated energy in the whole laminate that increases with the impact energy. Among the three aluminum layers, the bottom aluminum layer dissipates the maximum energy, but the middle aluminum layer dissipates the minimum energy. In addition, laminates with thicker aluminum layers dissipate less energy. (c) Delamination and matrix tension are dominant damage modes. Delamination concentrates mainly on the interface between aluminum layers and composite layers. Matrix damage always evolves along the fiber principal direction. Most plastic deformation for aluminum layers, matrix damage, and delamination for composite layers take place at the bottom part of laminate due to larger tensile stress, compared with smaller compressive stress at the upper part.

References [1] Vlot A, Gunnink JW. Fiber metal laminates-an introduction. Kluwer Academic Publishers; 2001. [2] Chai GB, Manikandan P. Low velocity impact response of fiber-metal laminates–a review. Compos Struct 2014;107(3):363–81. [3] Morinie`re FD, Alderliesten RC, Benedictus R. Modelling of impact damage and dynamics in fiber-metal laminates-a review. Int J Impact Eng 2014;67(5):27–38. [4] Kakati S, Chakraborty D. Delamination in GLARE laminates under low velocity impact. Compos Struct 2020;240, 112083.

References

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[5] Wang ZW, Zhao JP. Low velocity impact response of GLARE laminates based on a new efficient implementation of Puck’s criterion. Thin-Walled Struct 2019;144, 106321. [6] Soutis C, Mohamed G, Hodzic A. Modelling the structural response of GLARE panels to blast load. Compos Struct 2011;94(1):267–76. [7] Sitnikova E, Guan ZW, Schleyer GK, Cantwell WJ. Modelling of perforation failure in fiber metal laminates subjected to high impulsive blast loading. Int J Solids Struct 2014;51(18):3135–46. [8] P€ arn€ anen T, Alderliesten R, Rans C, Brander T, Saarela O. Applicability of AZ31B-H24 magnesium in fiber metal laminates–an experimental impact research. Compos Part A 2012;43(9):1578–86. [9] Sadighi M, P€ arn€anen T, Alderliesten RC, Sayeaftabi M, Benedictus R. Experimental and numerical investigation of metal type and thickness effects on the impact resistance of fiber metal laminates. Appl Compos Mater 2012;19(3–4):545–59. [10] Seo H, Hundley J, Hahn HT, Yang JM. Numerical simulation of glass-fiber-reinforced aluminium laminates with diverse impact damage. AIAA J 2010;26(48):676–87.

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13

Explicit finite element analysis of impact-induced damage of composite wind turbine blade under typhoon by considering fluid/solid interaction 3.1 Introduction Currently, wind power has been widely considered as one of the most dominating clean, environmental, and renewable energy, especially offshore wind energy. Chinese government has invested increased amount of wind power fields in east coast areas including Zhejiang Province, Jiangsu Province and Shandong Province, Fujian Province and Guangdong Province. Wind turbine blade is representative of lightweight structural component in wind turbine system, which are designed and manufactured by high-strength/stiffness and low density carbon fiber or glass fiber resin matrix composites. Through the advanced aerodynamic design, the composite blade is generally constructed to be a slim, thin, and flexible airfoil with its section changing with the edgewise direction. By leveraging the prepreg laminate, vacuum resin transfer and adhesive curing technology, the blade includes mainly the laminated skin and sandwich web. Due to the difference between the upper and low surfaces of the blade as wind flows across the airfoil section, the blade moves circularly within the plane that the wind turbine rotates around, transforming wind energy to the dynamic energy which is then amplified by gear. Under low wind velocity, the blade runs at a stable rotating state when the interaction between wind and blade reaches equilibrium. However, the blade runs in a series of complex environments, especially for typhoon

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00003-1

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298

13. Element analysis of composite wind turbine blade

cases with such as 50 m/s wind velocity in Zhoushan city area, Zhejang Province. The failure mode of blade includes the progressive damage and delamination of skin, the adhesive failure between the skin and web. By considering the fluid-solid interaction, there is some research on high-velocity impact of blade [1–3]. However, these works failed to consider complicated failure mechanisms above. Similar to the work above, the prevailing design criteria used in China [4] still suffers from two important disadvantages: (1) it considers equivalent aerodynamic loads for strength analysis instead of fluid-solid coupling and (2) Progressive damage analysis of skin and the adhesive failure analysis between the skin and web are severely insufficient, although it mentions detailed failure modes such as the cohesive and adhesive failure. Based on the work of Chapters 10-12, this chapter further performed explicit FEA on the blade under extreme high-velocity wind impact. The highlighted features of the present work are represented by two aspects: (1) The dynamic fluid-solid interaction is considered; (2) The adhesive failure between the web and skin is simulated using the bilinear cohesive model presented in Chapter 10. Explicit FEA is carried out to explore the effects of different wind velocity and thickness of adhesive layer on the impact damage mechanisms of T-shape adhesive piecewise blade.

3.2 Numerical algorithm and fluid/solid model Under high-velocity wind pressure, the blade produces large deformation that in turn affects the wind fluid field largely. Thus, the movement of wind fluid and the deformation of blade is coupled and interactive. It is significant to consider the fluid-solid interaction during lightweight design and performance analysis of blade. The well-known k-ε turbulence model is adopted for wind fluid [5]. The arbitrary Euler-Lagrange (ALE) movement form and finite volume algorithm are adopted to discretize the fluid domain. Here, the pressure load is considered and viscous boundary layer effect with shear load is neglected. The finite difference algorithm is used for solving the movement of blade. In this work, the single-way coupling is adopted to improve computational efficiency, which shows the pressure and velocity of wind fluid and the deformation and displacement of blade is sequentially solved instead of strong coupling with simultaneous solution at the end of each small time increment. The local geometry model of T-shape piecewise blade includes skin, adhesive and web. T700/epoxy composites are used for skin and the layup pattern is [0°, 0°, 45°, 45°, 90°, 90°, 45°, 45°, 45°, 45°, 90°, 90°, 45°, 45°, 0°, 0°]. The density and dynamic viscosity of wind fluid are 1.3 kg/m3 and 1.79e-6 Pa s, respectively. The densities of composite layers and adhesive layers are 2000 and 1600 kg/m3, respectively. The cohesive

3.3 Numerical results and discussion

FIG. 13.1

299

Geometry model of piecewise blade.

strength is set 50 MPa. The geometry model of the blade is shown in Fig. 13.1 and the mesh models for wind and T-shape piecewise blade are shown in Fig. 13.2. The single-way coupling between the wind and blade is implemented by Tie contact. The boundary conditions of fluid and solid meshes are that velocity and displacement are zero. Steady fluid for wind is assumed and the wind velocity is set for pressure load.

3.3 Numerical results and discussion 3.3.1 Damage analysis of blade under different wind velocity Three wind velocity is considered: 20, 50, and 100 m/s. Thickness of adhesive layer is 0.02 mm. Figs. 13.3 and 13.4 show the maximum Mises stress, the maximum damage for cohesive layer, respectively. Fig. 13.5

FIG. 13.2

Mesh model of piecewise blade.

300

13. Element analysis of composite wind turbine blade

22

Maximum stress (MPa)

20 18 16 14 Numerical results using 20m/s 50m/s 100m/s

12 10 0

2

4

6

8

10

12

14

16

18

Number of layers FIG. 13.3

Maximum Mises stress for each layer under different wind velocity.

shows the distributions of Mises stress. The increase of wind velocity aggravates the maximum stress, displacement, and damage. Results also show the absence of fluid-solid coupling analysis will increase the numerical errors, especially for high wind velocity. Thus, numerical analysis is significant to improve design and damage prediction of the blade as a reliable supplement of criteria [4].

Maximum damage rate

1.00

0.95

0.90

0.85 Numerical results using 20m/s 50m/s 100m/s

0.80

0

2

4

6

8

10

12

14

16

Number of layers FIG. 13.4

Maximum damage for each cohesive layer under different wind velocity.

3.3 Numerical results and discussion

301

S, Mises (Avg: 75%) +5.491e+07 +5.034e+07 +4.576e+07 +4.118e+07 +3.661e+07 +3.203e+07 +2.746e+07 +2.288e+07 +1.830e+07 +1.373e+07 +9.152e+06 +4.576e+06 +4.713e+06

(A)

S, Mises (Avg: 75%) +5.848e+07 +5.361e+07 +4.874e+07 +4.386e+07 +3.899e+07 +3.412e+07 +2.924e+07 +2.437e+07 +1.950e+07 +1.462e+07 +9.748e+06 +4.874e+06 +6.808e+02

(B) S, Mises (Avg: 75%)

+6.888e+07 +6.314e+07 +5.740e+07 +5.166e+07 +4.592e+07 +4.018e+07 +3.444e+07 +2.870e+07 +2.296e+07 +1.722e+07 +1.148e+07 +5.741e+06 +1.274e+03

(C) FIG. 13.5 Distributions of Maximum Mises stress for each layer under wind velocity (A) 20 m/s, (B) 50 m/s, and (C) 100 m/s, respectively.

302

13. Element analysis of composite wind turbine blade

3.3.2 Damage analysis of blade with different thickness of adhesive layer The wind velocity is 50 m/s. Three thickness of adhesive layer is considered: 0.1, 0.15, and 0.2 mm. Figs. 13.6 and 13.7 show the maximum Mises stress, the maximum damage for cohesive layer, respectively. 26

Maximum stress (MPa)

24 22 20 18 16 14

Numerical results using 0.1mm 0.15mm 0.2mm

12 10 0

2

4

6

8

10

12

14

16

18

Number of layers FIG. 13.6 Maximum Mises stress for each layer under different thickness of adhesive layer.

1.02 1.00

Maximum damage rate

0.98 0.96 0.94 0.92 0.90 Numerical results using 0.1mm 0.15mm 0.2mm

0.88 0.86 0.84 0

2

4

6

8

10

12

14

16

Number of layers FIG. 13.7

Maximum damage for each layer under different thickness of adhesive layer.

References

303

The maximum stress, displacement, and damage decrease with the increase of thickness of adhesive layer. However, the amount for the decrease is not distinct from 0.1 to 0.15 mm. Thus, the increase of thickness can improve the stress and damage level well within some range for thickness.

References [1] Wang L, Quant R, Kolios A. Fluid structure interaction modelling of horizontal-axis wind turbine blades based on CFD and FEA. J Wind Eng Ind Aerodyn 2016;158:11–25. [2] Santo G, Peeters M, Van Paepegem W, Degroote J. Dynamic load and stress analysis of a large horizontal axis wind turbine using full scale fluid-structure interaction simulation. Renew Energy 2019;140:212–26. [3] Miao WP, Li C, Wang YB, Xiang B, Deng YH. Study of adaptive blades in extreme environment using fluid-structure interaction method. J Fluids Struct 2019;91:102734. [4] DNVGL-ST-0376. Rotor blades for wind turbines. 2015. [5] Juretic F, Kozmar H. Computational modeling of the neutrally stratified atmospheric boundary layer flow using the standard k–ε turbulence model. J Wind Eng Ind Aerodyn 2013;115:112–20.

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Explicit finite element analysis of micromechanical failure behaviors of thermoplastic composites under transverse tension and shear 14.1 Introduction It is well-known the mechanical properties of composites are multiscale. Now, multiscale features of composites have attracted much research on the representative volume element (RVE) and microscopic damage mechanics, which provide an efficient approach to predict overall behaviors of composites through bottom-to-up homogenization by performing analytical and numerical analysis on periodic RVEs [1]. Due to much weaker load-bearing ability of fiber in the transverse direction than in the longitudinal direction, many scholars have developed micromechanical damage models and performed FEA on thermoplastic composites under transverse loads. Transverse damage and fracture behaviors of composites depend on a series of factors including the constituent properties, the interfacial bonding strength, the fiber distribution, and other residual effects. From micromechanical analysis on RVE, Gonza´lez and LLorca [2], Totry et al. [3] and Canal et al. [4] studied the failure locus and damage evolution behaviors under transverse tension/compression and out-of-plane shear, Wongsto and Li [5] and Okabe et al. [6] investigated the effects of fiber arrangement on the damage behaviors and tensile strength of composites, Vaughan and McCarthy [7] studied the effects on the thermal residual stress and interface failure on transverse fracture behaviors of HTA/6376 thermoplastic composites, Vajari [8] explored the effects of the microvoids and interfacial fracture toughness on the Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00001-8

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14. Explicit finite element analysis of micromechanical failure

macroscopic mechanical behaviors of thermoplastic composites under tension and shear. These works above are limited to implicit FEA. Different from thermoset epoxy, some thermoplastic resins such as polycarbonate (PC) and polymethylmethacrylate (PMMA) resin often show first softening and then rehardening plastic deformation behaviors near the glassy transition temperature, which are closely related to complicated molecular chain orienting and stretching, where two competitive damage mechanisms exist: shear yielding dominates until molecular chain scission under compression, and crazing whose surfaces are bridged by numerous fine fibrils dominates due to the nucleation, growth, and coalescence of microvoids under tension [9]. Furthermore, these damage mechanisms depend on the hydrostatic pressure, strain rate, and temperature largely. The strain-softening phenomenon of thermoplastic resin with plastic flow instability accelerates the appearance of localization bands and the evolution of microvoids between cavitated particles, where severe plastic deformation and energy dissipation concentrate on a very narrow area. Although the physical origin for the plastic rehardening is commonly ascribed to the orientation of the macromolecular chains, there is still no consistent conclusion for explaining the initial plastic softening that was partly believed to depend on the void evolution within the high strain gradient zones. It is noted the cohesive softening for interface and the plastic softening for matrix inevitably bring about severe numerical difficulty for implicit FEA. By comparison, a favorable approach is to resort to explicit finite difference method to perform microscopic progressive failure analysis of thermoplastic composites. This chapter performed micromechanical damage modeling and explicit finite element analysis on the damage evolution behaviors and failure surface of glass fiber/PC resin thermoplastic composites under quasistatic transverse tension and shear loads using explicit FEA. Microscopic RVE is established using ABAQUS-PYTHON script language. Plane strain state is assumed for long fiber composites. The viscoplastic Gurson model [10] is used to predict the plastic deformation and void growth of the matrix. The fiber/matrix interface failure is simulated by the bilinear cohesive model, as described in Chapter 2. The effects of the load mode and the cohesive interface strength on the load responses, the void growth, the strain concentration or localization behaviors, and the failure surface of composites are studied.

14.2 Gurson’s type model for thermoplastic resin Since the rate of Cauchy stress σ is not objective, the rate constitutive equation is written as σ r ¼ C : de ¼ C : ðd  dp Þ

(14.1)

14.2 Gurson’s type model for thermoplastic resin

307

where σ r is the Jaumann stress rate. d and dp are the strain rate and plastic strain rate, respectively. C is the four-order elastic tensor. The purpose of this work is to study the void evolution and failure surface of glassy polymer such as PC resin. Damage is ascribed to the distributed growth of void volume and crazing during plastic deformation. The void model was first proposed by Gurson [11] and later mainly modified by Tvergaard and Needleman [12], which assumes that the deformation mode surrounding a void is homogenous and the softening behaviors result from the growth of voids. Similar to the work of Holopainen [13], we assume the voids are sphere in shape and uniformly distributed, and thus the damage evolution of material is assumed to be isotropic that is represented by a scalar void volume fraction f ¼ dVv/dV, where Vv is the volume of voids and dV is the total volume. The plastic yielding function Φ is expressed as
2
   q 3q2 p Φðq, p, σ 0 , f Þ ¼ + 2fq1 cosh (14.2)  1 + q1 f 2 σ0 2 σ0 where σ is the Cauchy p stress spatial ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi configuration. p ¼  tr(σ)/3 is the ffiffiffiffiffiffi inpthe hydrostatic stress. q ¼ 3J2 ¼ 3s : s=2 is the equivalent stress. J2 ¼ 12 s : s is the second stress invariant and s ¼ σ  p1 is the deviatoric stress tensor. Because constant values for q1 ¼ 1.5 and q2 ¼ 1.0 suggested by Tvergaard and Needleman [12] cannot satisfactorily predict the void growth and the change of void shape for amorphous glassy polymer, Zaı¨ri et al. [10] suggested they should be taken as the functions of the equivalent plastic strain εp q1 ¼ q10 ð1 + cεp ÞNv , q2 ¼ q20 ð1 + cεp ÞNv

(14.3)

where q10, q20, c and Nv are constants. Based on the  work  of Canal et al. [4], the plastic hardening/softening p stress σ 0 ¼ σ 0 εp , ε_ for glassy polymer is fitted as "
!# N
p N1
#" p  α εp ε εp ε_ σ0 ¼ 1 + σy +1 + C1 log C2 1 + C3 log 3 εy εy εy ε_ 0 (14.4) p

where σ y ¼ Eεy is the initial yield stress, ε is the equivalent plastic strain p and ε_ is the equivalent plastic strain rate. α, C1, C2, C3, N, N1, and ε_ 0 are constants. The rate of void volume fraction f_ is divided into the growth and nucleation parts 8 f_ ¼ f_ growth + f_ nucleation , > > > > p < f_ growth ¼ ð1  f Þd : 1, "
(14.5) 2 # p > f 1 ε  ε >_ p N N > > : f nucleation ¼ Aε_ , A ¼ pffiffiffiffiffi exp  2 sN sN 2π

308

14. Explicit finite element analysis of micromechanical failure

where fN is the amplitude, sN and εN are the standard deviation and the mean value of the normal distribution function, respectively. p The equivalent plastic strain rate ε_ is assumed to change according to the equivalent plastic work and the plastic rate of deformation is given by the associated flow rule ∂Φ p ð1  f Þσ 0 ε_ ¼ σ : dp ,dp ¼ λ_ ∂σ

(14.6)

where λ_ is the plastic consistency factor. The Kuhn-Tucker loadingunloading conditions and the consistency condition for Φ and λ_ are met. The plastic rate of deformation dp can be divided into the deviatoric and hydrostatic parts
 1 ∂Φ ∂Φ 3s 1+ n , n¼ (14.7) dp ¼ λ_  3 ∂p ∂q 2q where n is the unit vector in the deviatoric stress space.

14.3 Explicit FEA on microscopic RVEs of thermoplastic composites 14.3.1 Finite element modeling by ABAQUS A square RVE with 8 randomly distributed fibers is established for glass fiber/PC resin composites by developing numerical codes using ABAQUS-PYTHON script language, as shown in Fig. 14.1. The planestrain state for long fiber composites is assumed. The side length of RVE is 46 μm, the fiber radius is 5 μm and the fiber volume fraction is 30%. The reduced integration element CPE4R in ABAQUS software is

FIG. 14.1

Geometry model of RVE with displacement loads and boundary conditions.

14.3 Explicit FEA on microscopic RVEs of thermoplastic composites

309

used for fiber and matrix. For glass fiber, the Young’s modulus is Ef ¼ 70 GPa, the Poisson’s ratio νf ¼ 0.22 and the density ρf ¼ 2500 kg/m3. For PC resin, the elastic and plastic parameters are listed in Table 14.1 and the void parameters are listed in Table 14.2. The backward Euler algorithm for the finite-deformation Gurson model is given by Liu and Li [14], which is implemented by developing numerical codes using ABAQUSVUMAT subroutine. COH2D4 interface element in ABAQUS for the bilinear cohesive model is used. The density and thickness for the interface are assumed as ρc ¼ 1200 kg/m3 and 0.2 μm, respectively. The cohesive strengths 60 MPa are used. Periodic boundary conditions are applied to the edges of the RVE to ensure displacement and traction continuities between neighboring RVEs (which deform like a jigsaw puzzle), which can be expressed as the relative displacements u1 and u2 between opposite edges uð0, yÞ  uðL0 , yÞ ¼ u1 , (14.8) uðx, 0Þ  uðx, L0 Þ ¼ u2 In Fig. 14.1, displacements on point N1 in the x and y directions are set to zero, and displacement on point N2 in the y direction is set to zero. Combined uniaxial normal displacement along the y axis and shear displacement u2 ¼ (δt, δs) are imposed on point N3, where δt and δs are the tensile and shear displacements, respectively. In u1 ¼ (ux, uy), the normal part ux is computed from the condition that the average normal stresses perpendicular to the x axis should be zero, and the shear part uy is determined by the equilibrium that the total moment induced by the shear stresses acting on the side should be zero. The tensile and shear strains are calculated by ln(1 + δt/L0) and arctan[δs/(L0 + δt)], and the tensile and shear stresses are calculated by the total normal and shear forces on the side divided by the actual cross-section. Thus, the tensile and shear strain rates are calculated by ln(1 + δt/L0)/t and arctan[δs/(L0 + δt)]/t (t is time), respectively. In general, a low tensile strain rate 104 s1 for quasistatic loading is used. The TABLE 14.1

Elastic and plastic parameters for PC resin. ρm(kg/ m3)

Em (MPa)

vm

2300

0.33 1200

α

C1

C2

C3

ε_ 0

εy

σ y(MPa) N1 ⋯N

0.225 0.0124 0.0386 0.125 104 s1 0.024 55.4

TABLE 14.2

Void parameters for PC resin.

f0

εN

sN

fN

q1

q2

q3

0

0.03

0. 15

0.15

0.9

1.2

0.9

1.3 ⋯0.1

310

14. Explicit finite element analysis of micromechanical failure

time integration solution is performed using ABAQUS-EXPLICIT module. The tensile displacement δt ¼ 5 μm is generally chosen and the shear displacement δs changes according to the ratio δt/δs in the following analysis. The stable time increment 1.5e-6s is used in explicit FEA. The linear bulk viscosity parameter in ABAQUS-EXPLICIT is set to 0.06.

14.3.2 Numerical results and discussion First, the RVE with the first random fiber distribution (case-1) and three mesh sizes under transverse tension is evaluated. As shown in Fig. 14.2, the number of elements for three mesh sizes is 545 (coarse mesh), 913 (middle mesh) and 1249 (fine mesh), respectively. The number of elements for fiber, matrix and interface using three mesh sizes is listed in Table 14.3. Fig. 14.3 shows the stress-strain curves by explicit FEA and experiment. Transverse tensile test for glassy fiber/PC resin composite specimens is carried out using the UTM5000 electronic test machine, as shown in Fig. 14.4. The stress decreases obviously with the increase of mesh precision during the softening stage. Mesh sensitivity is eliminated when the mesh size becomes fine enough and the middle mesh size is acceptable. However, the curve using the coarse mesh size leads to large error. Fig. 14.5 shows the damage dissipation energy and the total strain energy of one RVE using middle mesh, which indicates the stable time increment leads to the smooth energy output without numerical oscillation. In the

FIG. 14.2

Three mesh models for the RVE with case-1 fiber distribution.

TABLE 14.3 Number of elements for three mesh models. Coarse mesh

Middle mesh

Fine mesh

Number of fiber elements

177

278

293

Number of cohesive elements

92

132

155

Number of matrix elements

272

503

801

FIG. 14.3

Quasistatic tensile stress-strain curves for three mesh models.

FIG. 14.4 Quasistatic tensile experiment on glass fiber/PC composite specimen.

312

14. Explicit finite element analysis of micromechanical failure

FIG. 14.5 The damage dissipation energy and the total strain energy-time curves of one RVE using middle mesh model.

following, middle mesh size (around 900 elements) will be used by considering both computational accuracy and efficiency. In order to verify the RVE is large enough to represent macroscopic mechanical responses of composites accurately, three models with different fiber distributions under tension are analyzed and the stress-strain curves are shown in Fig. 14.6. The results for three fiber distributions show small errors for the load responses of composites, indicating that the size of the RVE is large enough to assure the representativeness of results. In addition, numerical oscillation during the softening stage appears in Figs. 14.3 and 14.6, which arises from the sudden release of strain energy due to plastic softening that leads to nonphysical limit points in the form of a snap-back situation using not very fine mesh size. Because a huge amount of computation is needed in explicit FEA, the number of fibers will not increase although there is still small difference among these curves. Figs. 14.7 and 14.8 show the distributions of the equivalent plastic strain (PEEQ) and the void volume fraction (VVF) in the deformed configuration for three RVEs with different fiber distributions at the strain 3.3%. Within the high-strain area, the PEEQ exceeds the initial yielding strain εy ¼ 0.024, which shows the plastic deformation appears at several narrow bands. However, most areas are still under the elastic deformation stage. With different fiber distributions of the RVE, the position and orientation of a strain concentration band or a localization band will change and there

14.3 Explicit FEA on microscopic RVEs of thermoplastic composites

FIG. 14.6

313

Tensile stress-strain curves for RVEs with three fiber distributions.

is also some difference between the maximum values for the PEEQ and VVF. This indicates that fiber distribution affects the macroscopic load responses slightly, but the microscopic deformation and damage largely. In addition, the contours of the PEEQ and VVF are consistent since they are coupled. Thus, only the contour of VVF is discussed here. The damage evolution and strain concentration band or localization band of matrix under transverse tension are discussed using case-1 with middle mesh size. Fig. 14.9A shows the elastic stage without damage at the strain 1.4%. As shown in Fig. 14.9B, matrix damage zones appear at an about 45 degrees angle with the fiber. As shown in Fig. 14.9C, the stress concentration at the interface crack tip induces the formation of very short high-strain bands in the matrix that link up interface cracks in neighboring fibers, where the maximum strength is attained. This leads to a reduction of the matrix flow stress which is responsible for the drop in the tensile strength after yielding and accelerates the localization in the matrix between fibers. Matrix damage increases largely and interface debonding at different locations are linked by matrix damage, which induces the main strain concentration throughout the RVE model, as shown in Fig. 14.9D. Progressive interface debonding using cohesive elements is shown in Fig. 14.10, in which SDEG represents the scalar stiffness degradation of cohesive element, and SDEG ¼ 0 represents perfect interface and SDEG ¼ 1 means completely debonded interface.

FIG. 14.7 Contours of the equivalent plastic strain for RVEs with three fiber distributions at the tensile strain 3.3%.

FIG. 14.8 Contours of the void volume fraction for RVEs with three fiber distributions at the tensile strain 3.3%.

FIG. 14.9 Contours of the void volume fraction at the tensile strains 1.4%, 1.9%, 2.2%, and 3.3% respectively.

FIG. 14.10

Contours of the scalar stiffness degradation (SDEG) at the tensile strains 1.4%, 1.9%, 2.2%, and 3.3% respectively.

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14. Explicit finite element analysis of micromechanical failure

The influence of cohesive interface strengths on the damage mechanisms of RVE model under tension is discussed using case-1. The experimentally observed dominating damage mechanisms and strain localization of the matrix and the interface debonding are shown to be affected by the interface strength. Fig. 14.11 shows the stress-strain curves at 40 MPa (low), 60 MPa (middle) and 100 MPa (high) cohesive interface strengths, respectively. The numerical result with 60 MPa cohesive strength is closest to the experimental result. Low interface strength significantly reduces the macroscopic strength of composites and leads to a distinct nonlinearity in the stress-strain curves at a relatively low strain level. This is due to the fact that the strength of composites is mainly controlled by interfacial debonding in the case of weak interface, where the load on the matrix cannot be effectively transferred into the fibers. Fig. 14.12 shows the distributions of the VVF at 40, 60, and 100 MPa cohesive strengths, respectively. In Fig. 14.12A, the void growth of the matrix is always accompanied by the nucleation and growth of interface cracks. On one hand, stress concentration at the interface crack tip promotes the plastic deformation of the matrix and the damage localization along the weakest path, dictated by the spatial distribution of interface cracks perpendicular to the tensile loads. On the other hand, since the remaining perfect interface can carry more stress, interface debonding extends along the interface and finally leads to the complete failure of the interface. In

FIG. 14.11 100 Mpa.

Tensile stress-strain curves using the cohesive interface strengths 40, 60, and

FIG. 14.12

Contours of the void volume fraction using the cohesive interface strengths 40, 60, and 100 MPa at the tensile strain 3.3%.

320

14. Explicit finite element analysis of micromechanical failure

contrast, the appearance of severe matrix damage without obvious interface debonding indicates that the dominant failure mechanism is still the matrix plastic deformation instead of interface debonding when the cohesive strength reaches 100 MPa in Fig. 14.12C. The matrix between neighboring fibers rather than the interface is the origin of the damage, and the failure processes for strongly bonded interface can be divided into three steps: the incipient development of strain concentration bands in the matrix channels between the fibers, the nucleation of interface cracks and the final deformation localization in the matrix in a dominant band. It should be noted that this usually does not match the real case as the interfacial strength is commonly lower than the matrix strength for glassy fiber/PC resin composites. The degree of failure with 60 MPa cohesive strength falls in between strong and weak interfaces. Besides, the maximum value of VVF increases with the increase of interface strength. The failure surface is obtained for three cohesive interface strengths under the tensile load and different shear loads. The evolution of the shear stress with the tensile stress for different δt/δs ratios (from 1/2 to 5/1) is shown in Figs. 14.13–14.15, where δt and δs stand for the imposed tensile and shear displacements, respectively. As we can see, the normal and tangential stresses increase proportionally during the elastic stage, but this proportionality disappears with the onset of the plastic deformation and interface debonding. Numerical results can be used to get the failure

FIG. 14.13

Tensile stress-shear stress curves under biaxial loading with different δt/δs ratios (transverse tension and shear) and 40 MPa interface strength.

14.3 Explicit FEA on microscopic RVEs of thermoplastic composites

321

FIG. 14.14

Tensile stress-shear stress curves under biaxial loading with different δt/δs ratios (transverse tension and shear) and 60 MPa interface strength.

FIG. 14.15

Tensile stress-shear stress curves under biaxial loading with different δt/δs ratios (transverse tension and shear) and 100 MPa interface strength.

322

14. Explicit finite element analysis of micromechanical failure

surface of composites under transverse tension and shear by linking the failure points, where a sharp change of slope occurs in the tensile stress-shear stress curves, as shown in Figs. 14.13–14.15. Fig. 14.16 shows the failure surface for different interface strengths. The difference between the failure surface for weak and middle interfaces is larger than that for middle and strong interfaces, indicating the weak interface significantly reduces the macroscopic strength of composites again. Besides, the failure surface becomes larger with the increase of the ratio δt/δs, indicating the interface strength affects the transverse tensile strength more largely than the traverse shear strength of composites. This may be caused by the fact that interface debonding is more sensitive to tensile loads rather than shear loads. Finally, numerical results for the failure surface using the present method are compared with two classical failure criteria for composites developed by Hashin [15] and Puck et al. [16]. The Hashin failure criteria distinguish between the fiber-dominated and the matrix-dominated fracture, which can be further subdivided into the tensile and compressive modes. Furthermore, the Hashin failure criteria assume the failure comes from the normal and tangential stresses acting on the fracture plane, which is parallel to the fibers in the case of the matrix-dominated failure under tension. By considering the composite lamina is isotropic in the x-y plane, the Hashin failure criteria [15] can be expressed as

FIG. 14.16

Prediction of the failure surface using cohesive interface strengths 40, 60, and 100 MPa under transverse tension and shear.

14.3 Explicit FEA on microscopic RVEs of thermoplastic composites



σ2 YT

2 2 τ12 + ¼1 ST

323

(14.9)

where both the tension perpendicular to the fibers and the shear in the x-y plane are considered. YT and ST are the transverse tensile and out-of-plane shear strengths, respectively. σ 2 and τ12 are the tensile stresses along the y axis and the shear stress in the x-y plane, respectively. However, the Hashin criteria cannot capture the increase of the shear strength in the presence of compressive stress since they cannot determine the actual orientation of the fracture plane but assume a quadratic interaction between the stress invariants. Among many modified criteria, the Puck failure criteria are consistent with the experimental results under tension, compression, and in-plane shear. Puck et al. [16] assume that failure is caused by the normal and tangential stresses acting on a fracture plane for composites under compression, which forms an angle θf with the direction perpendicular to the tensile stresses, where θf can be explicitly determined for each combination of the normal stress σ 2 and shear stress τ12 acting on composites. The Puck failure criteria are given by [16] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

   2   2 τt θf 1 cos ð2θc Þ cos ð2θc Þ    + + σ n θf ¼ 1 σ n θf 4 2 YT 2Yc cos θc 2Yc cos 4 θc Yc cos θc (14.10) with



  σ n θf ¼ σ 2 cos 2 θf + 2τ12 sin θf cos θf ,  τt θf ¼ σ 2 sin θf cosθf + τ12 cos 2 θf  sin 2 θf

(14.11)

where Yc and θc stand for the failure stress and the orientation of the fracture plane under transverse compression, respectively. θc ¼ 47 degrees is used here in reference to the advice by Canal et al. [4]. The predictions using both criteria are based on mechanical tests, which provide the transverse tensile strength YT, the transverse shear strength ST, the transverse compressive strength YC and the corresponding fracture angle θf. As mechanical tests for composites under multiaxial loads are complicated due to some problems such as the buckling, the free edge effect and failure at the end constraints, these parameters are obtained from the present numerical model, as listed in Table 14.4. The predictions of the failure surface for different interface strengths under transverse tension and out-of-plane shear using the Hashin and Puck criteria are compared with the present numerical results, as shown in Figs. 14.17–14.19. Numerical and analytical results are basically consistent. However, the Puck criteria, which are based on the tensile and compressive strengths of composites, in good agreements with the present numerical results at the tension-dominated region, but underestimate

324

14. Explicit finite element analysis of micromechanical failure

TABLE 14.4 Failure strengths under uniaxial loads for different cohesive strengths. Cohesive strengths (MPa)

Yc (MPa)

ST (MPa)

YT (MPa)

40

55.3

27.9

37.0

60

68.5

33.5

44.6

100

74.3

35.6

48.3

FIG. 14.17 Comparison of numerical and theoretical results using 40 MPa cohesive interface strength under transverse tension and shear.

the strength of composites in shear. The Puck failure criteria assume that the failure surface is determined by the interaction between two main failure mechanisms: the brittle fracture of the matrix under transverse tension/compression and the yielding of the matrix in shear. The latter mechanism is still dominant when the matrix is thermoplastic, but failure under transverse tension is not brittle in this case and does not follow a path totally perpendicular to the tensile stress, as shown in Fig. 14.9. Thus, the Puck criteria cannot provide accurate results in shear, if it is expected to be calibrated by YT to provide an accurate estimation for the transverse tensile strength. Furthermore, the difference of shear strengths between numerical and theoretical results increases slightly with the decrease of interface strength. This may be caused by the fact that the Puck criteria

14.3 Explicit FEA on microscopic RVEs of thermoplastic composites

325

FIG. 14.18 Comparison of numerical and theoretical results using 60 MPa interface strength under transverse tension and shear.

FIG. 14.19

Comparison of numerical and theoretical results using 100 MPa interface strength under transverse tension and shear.

326

14. Explicit finite element analysis of micromechanical failure

assume that the main failure mechanism is the shear yielding of the matrix, while the failure is controlled by the interface debonding for low interface strength. As for the Hashin criteria, quadratic interaction between the normal and shear stresses for the description of the failure surface overestimates the strength of composites under biaxial loading to some extent, although the difference is not distinct because the prediction is based on the numerically obtained values of YT and ST. However, the prediction is based on the shear strength ST of composites, which is generally very difficult to measure experimentally.

14.4 Conclusions This chapter studies failure behaviors of thermoplastic composites under transverse tension or combined tension-shear loads using micromechanical approach. The RVE of composites is established that can predict the void growth, the plastic softening of matrix and interface debonding. Effects of the random fiber distribution and the cohesive strength on the void growth and failure surface of the RVE using different mesh models are studied using explicit FEA. Three main conclusions are obtained: (1) Random fiber distributions show small effects on the macroscopic loadbearing ability of composites, but large effects on the microscopic void growth and its distributions as well as the strain concentration or localization behaviors of composites. (2) There is a distinct competition between the interface debonding and the void growth of the matrix. The interface debonding is the dominant failure mechanism if the interface strength is low, while the void growth of matrix is the dominant failure mechanism if the interface strength is high. In addition, interface debonding changes the propagation directions and distributions of strain concentration bands in the matrix. (3) Comparisons between numerical results and those using the Hashin and Puck failure criteria show the limitation of the phenomenological failure criteria to accurately predict the failure surface of thermoplastic composites because they do not consider the yielding of the thermoplastic polymer and interface debonding. Thus, compared with the analytical approach suitable for thermoset composites, a numerical tool by micromechanical analysis provides a more efficient approach to evaluate multiscale failure mechanisms and damage evolution behaviors of thermoplastic composites.

References

327

References [1] LLorca J, Gonza´lez C, Molina-Aldareguia JM, Segurado J, Seltzer R, Sket F, Rodriguez M, Sa´daba S, Mun˜oz R, Canal LP. Multiscale modeling of composite materials: a roadmap towards virtual testing. Adv Mater 2011;23:5130–47. [2] Gonza´lez C, LLorca J. Mechanical behavior of unidirectional fiber-reinforced polymers under transverse compression: microscopic mechanisms and modeling. Compos Sci Technol 2007;67:2795–806. [3] Totry E, Gonza´lez C, LLorca J. Influence of the loading path on the strength of fiber- reinforced composites subjected to transverse compression and shear. Int J Solids Struct 2008;45:1663–75. [4] Canal LP, Segurado J, LLorca J. Failure surface of epoxy-modified fiber-reinforced composites under transverse tension and out-of-plane shear. Int J Solids Struct 2009;46:2265–74. [5] Wongsto A, Li S. Micromechanical FE analysis of UD fibre-reinforced composites with fibres distributed at random over the transverse cross-section. Compos Part A 2005;36:1246–66. [6] Okabe T, Nishikawa M, Toyoshima H. A periodic unit-cell simulation of fiber arrangement dependence on the transverse tensile failure in unidirectional carbon fiber reinforced composites. Int J Solids Struct 2011;48:2948–59. [7] Vaughan TJ, McCarthy CT. Micromechanical modelling of the transverse damage behaviour in fibre reinforced composites. Compos Sci Technol 2011;71:388–96. [8] Ashouri Vajari D. A micromechanical study of porous composites under longitudinal shear and transverse normal loading. Compos Struct 2015;125:266–76. [9] Mulliken AD, Boyce MC. Mechanics of the rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates. Int J Solids Struct 2006;43(5):1331–56. [10] Zaı¨ri F, Nait-Abdelaziz M, Gloaguen JM, Lefebvre JM. Modelling of the elastoviscoplastic damage behaviour of glassy polymers. Int J Plast 2008;24:945–65. [11] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth. 1. Yield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 1977;99(1):2–15. [12] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69. [13] Holopainen S. Influence of damage on inhomogeneous deformation behavior of amorphous glassy polymers. Modeling and algorithmic implementation in a finite element setting. Eng Fract Mech 2014;117:28–50. [14] Liu PF, Li XK. Dynamic void growth and localization behaviors of glassy polymer using nonlocal explicit finite element analysis. J Peri Nonlocal Model 2019;1(1):3–13. [15] Hashin Z. Failure criteria for unidirectional fiber composites. ASME J Appl Mech 1980;47:329–34. [16] Puck A, Sch€ urmann H. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos Sci Technol 2002;62:1633–62.

C H A P T E R

15

Numerical simulation of micromechanical crack initiation and propagation of thermoplastic composites using extended finite element analysis with embedded cohesive model 15.1 Introduction It is well-known failure mechanisms and damage evolution behaviors of fiber-reinforced composites are multiscale, which ultimately depends on the microscopic mechanical fiber/matrix properties. To promote lightweight design and application of composite structures, it is significant to predict the strength and damage tolerance of them by developing robust theoretical and numerical methods from both macroscopic and microscopic scales. Research on evolutive microscopic crack behaviors plays an important role in evaluating the macroscopic damage tolerance of composite structures. From the multiscale perspective, macroscopic damage evolution of composites is represented by continuous degradation and residual load-bearing abilities of the microscopic fiber, matrix, and fiber/matrix interface, which are tightly related to the microscopic fracture of different phases. As an important phase that transfers loads in composites, the toughness of epoxy matrix is much improved by adding the second-phase polymer particles such as rubbers and thermoplastics, which is represented by the shear yielding of ductile matrix enhanced by the cavitation

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00009-2

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Copyright © 2021 Elsevier Inc. All rights reserved.

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15. Numerical simulation of micromechanical crack initiation

of elastomeric particles [1, 2]. However, this facilitates the evolution of microvoids and the formation of discrete shear bands between cavitated particles, particularly resulting from the intrinsic softening upon yielding followed by the hardening behaviors for glassy polymer. Localization band (or called shear band) means severe plastic deformation and energy dissipation concentrate on a very narrow area. Because the thickness of shear bands is generally thin, a small length scale in the framework of weak discontinuity is naturally created once the localization appears, which demonstrates the multiscale essence of localization phenomena [3]. Nowadays, it should be recognized that shear bands for ductile materials are represented by the initiation of localized damage and microcracks, characterized by plastic flow instability. A crack forms by the nucleation, growth, and coalescence of microvoids for porous ductile materials. From the continuum mechanics perspective, localized damage accumulates to produce a crack, driven by irreversible energy dissipation. For example, a crack is assumed to appear a priori when the porosity reaches a limit value according to the Gurson-Tvergaard-Needleman (GTN) model that predicts the damage evolution of ductile materials phenomenologically [4, 5]. Because of the absence of a length scale, the classic continuum mechanics fail to predict accurately the damage evolution and energy dissipation within the localized band. This often leads to the mesh sensitivity problem using finite element analysis (FEA), without fully transmitting deformation information from those localized elements to the whole body [6, 7]. From the mathematical perspective, localization means the loss of the ellipticity (or hyperbolicity) of the stress governing equation and the singularity of acoustic tensor even if boundary conditions are prescribed in an elliptic (or hyperbolic) state. Furthermore, spurious energy dissipation by continuum models due to strain softening often contradicts with the dissipated energy needed to drive a crack according to the fracture mechanics theory [8]. Within the framework of weak discontinuity, the localization problem with the strain discontinuity was typically solved by introducing a length scale such as in the nonlocal integral models [9, 10] and nonlocal gradient models [11, 12] to regularize the stress boundary value (or initial value) problem with strain softening. Although physical meanings for these methods are not different, similar numerical effects with respect to accurate and robust prediction of finite energy dissipation are reached. For example, Mazars and Pijaudier-Cabot [13] linked damage mechanics to fracture mechanics effectively based on the nonlocal integral theory and thermodynamic theory provided that a damage zone develops to form an equivalent crack. Moe¨s et al. [14] proposed a thick level set based model to study the damage growth and fracture based on the equivalence of the fracture energy, in which nonlocal treatment on the damage variable over the thickness as an explicit function of the level set is used to describe the transition from the critical damage to

15.1 Introduction

331

fracture. Broumand et al. [15] and Cr
et
e et al. [16] associated the nonlocal damage/plasticity model, the rate-dependent GTN model respectively with the extended finite element method (XFEM) [17] to predict the crack initiation and propagation. Wang and Waisman [18] developed a coupled continuous/discontinuous model to capture the transition from nonlocal diffusion damage to a sharp crack with an energy equivalence description, and placed it into the framework of XFEM. However, one of the highlighted challenges we still have to confront is accurate calibration of the length scale that is closely relevant to the thickness of the band, which perhaps has to resort to some special experiments [19]. According to the linear elastic fracture mechanics, relatively mature numerical analysis in predicting the crack propagation of brittle materials assumes the motion of a crack is led by the crack tip provided that an initial flaw size exists, where the crack propagation direction is determined by the nonlocal stress/strain, the stress intensity factor or J-integral near the crack-tip, particularly using the XFEM. However, this is no longer feasible when a strongly nonlinear deformation occurs at the crack tip for ductile materials. Carpinteri and Colombo [20] showed most nonlinear fracture processes appear around the crack tip, called as the fracture process zone that governs the crack behaviors. Based on the elastic-plastic fracture mechanics, the cohesive concept which was proposed by Dugdale [21] and Barenblatt [22] to describe discrete fracture as material separation across the interface, predicts the nucleation of microscopic voids and cavitations well by assuming a traction-displacement jump relationship. Areias and Belyschko [23] pointed out a small length scale from the continuum mechanics perspective allows us to consider a representative surface dissipation mechanism where the orientation effect is explicit, e.t., the so-called Barenblatt model. This shows there exists a physical cohesive softening law that dominates the nonlinear fracture behaviors for the damage evolution of the localization band. Later, Benvenuti [24], Ja
skowiec and van der Meer [25], Pike and Oskay [26], and Wu and Li [27] embedded the cohesive model into the XFEM to predict the crack propagation efficiently, but they are not really related to the localized damage mechanisms. An alternative strategy that links the localized damage to the fracture is to assume the thickness of the shear band is zero, where the strong discontinuity concept with displacement discontinuity is recovered as a limit case of the weak discontinuity [28–31]. In the strong discontinuity analysis, displacement is assumed to possess a Heaviside jump at the localization interface and strain becomes a bounded measure including a Dirac-delta function, but the traction must be continuous at the localization interface. The concept of strong discontinuity is introduced that ensures the classic plasticity theory is compatible with the discontinuous displacement field arising from the localization. In this case, finite

332

15. Numerical simulation of micromechanical crack initiation

dissipation energy due to plastic and damage in the weak discontinuity analysis translates fully to that due to damage evolution and crack propagation in the strong discontinuity analysis. The localization bifurcation condition and the propagation direction of zero-thickness interface in the strong discontinuity analysis were determined by discontinuous bifurcation analysis instead of the crack-tip fracture parameters for rateindependent solids. Oliver et al. [30] also showed strong discontinuity analysis provides an effective way to link damage mechanics to fracture mechanics. However, two aspects of work deserves further study: (1) Most of them used J2 plasticity model and Drucker-Prager model for strong discontinuity analysis, but the effects of damage evolution behaviors of bulk materials on the localization initiation condition and propagation direction were not studied. (2) Most of them adopted the assumed strain method to implement the strong discontinuity model, which yet shows some disadvantages compared with the XFEM, e.g., the interpolation issue of slip continuously across element boundaries [31]. This chapter introduces strong discontinuity analysis to the research on the localized damage and fracture of fiber-reinforced ductile polymer composites. An important originality is to combine the strong discontinuity analysis and the GTN model to study the fracture behaviors of the ductile polymer matrix. First, the theoretical work in this research is to derive explicit expressions for the localization bifurcation condition and propagation direction for the GTN model. By comparison, Mansouri et al. [32] combined the GTN model and the discontinuous bifurcation theory [33] to predict the ductility limit of sheet metals, and Li and Karr [34] performed bifurcation, localization, and imperfection analysis by the GTN model to study the ductile fracture initiation, which were yet limited to weak discontinuity analysis. Second, once the localization condition is met, the XFEM with the embedded cohesive model is used to predict the crack initiation and propagation of the matrix. Finally, two numerical examples are given to validate the developed model and the numerical technique. This research provides a link between the localized damage and fracture behaviors of fiber-reinforced ductile polymer matrix composites.

15.2 Strong discontinuity analysis on the strain localization for the GTN model Strain localization is generally related to plastic flow instabilities, where irreversible energy dissipation localized into a narrow area. The classic localization criteria can trackback to the positiveness of second-order work theory for the stability and bifurcation perspectives proposed by Hill [35], which involves a bifurcation condition for the loss of the uniqueness of solutions for the boundary value problem. Later, the pioneering work

15.2 Strong discontinuity analysis on the strain localization for the GTN model

333

by Rudnicki and Rice [33] is the derivation of the general bifurcation condition for rate-independent plastic materials based on the singularity of acoustic tensor. Yet, their analysis is limited to the framework of weak discontinuity. Simo et al. [28] proposed the strong discontinuity concept by assuming zero-thickness localization interface with displacement discontinuity. In the following, explicit expressions for the localization bifurcation condition and propagation direction by strong discontinuity analysis for the GTN model are derived. According to the strong discontinuity analysis, the displacement is composed of the regular part and discontinuous part. The displacement rate u_ and the strain rate ε_ are given by respectively _ u_ + ½½u_ Hs , ε_ ¼ —s u_ ¼ ε_ + ð½½u_ nÞδs u5

(15.1)

where Hs is the Heaviside function and δs is the Dirac-delta function. ½½u_  ¼ ς_ m is the displacement jump rate in a given area, m is the motion direction of the localization band, and ς_ is the magnitude of the displacement jump rate. n is the normal direction of localization interface u_ is the regular displacement rate and ε_ ¼ —s u_ is the strain rate. Here, ½½u is assumed constant across the discontinuous interface. The continuum elastoplastic stress-strain model is written as p _ _ _ ε_ ¼ λ∂G=∂σ _ 0 ¼ λΗ σ_ ¼ C : ðε_  ε_ p Þ, Fðσ, σ 0 Þ  0, ε_ p ¼ λ∂G=∂σ, 0, σ

(15.2)

where associative plastic model G ¼ F is assumed. Η is the hardening/
 softening modulus. C ¼ 2μ I  13 11 + K11 is the four-order elastic tensor, K ¼ λ + 23 μ is the bulk modulus, λ and μ are Lam
e constants, (1)ij 5 δij is the Kronecker delta and Iijkl ¼ (δijδkl + δilδjk)/2 is the four-order identity tensor. The plastic factor λ_ which obeys the Kuhn-Tucker loading/unloading conditions and the consistency requirement is given by λ_ ¼

∂σ F : C : ε_ , H ¼ Hð∂σ0 F Þð ∂σ0 GÞ ∂σ F : C : ∂σ G + H

By using Eqs. (15.2), (15.3), the stress rate σ_ is expressed as   _ σ_ ¼ C : ðε_ 2_ε p Þ ¼ σ_ + C : ð½½u_ nÞs δs  λ∂G=∂σ

(15.3)

(15.4)

where σ_ ¼ C : ε_ is the regular part of the stress rate σ_ . n is the normal vector of the band. Because the stress rate σ_ in Eq. (15.4) must be regular, the only requirement is that the plastic consistency parameter λ_ ¼ λ_ δ δs is a singular distribution function [28]. From Eq. (15.4), the following equations are obtained _ ð½½u_ nÞs ¼ λ_ δ ∂G=∂σ σ_ ¼ σ_ ¼ C : ε,

(15.5)

By combining Eqs. (15.1), (15.4), the traction rate τ_ across the discontinuity interface can be written as the following form

334

15. Numerical simulation of micromechanical crack initiation

  τ_ 5C : ε_ + ð½½u_ nÞs δs 2λ_ δ δs ∂σ G  n

(15.6)

where the jump of the traction rate τ_ across the discontinuity must be zero ½½_τ  ¼ 0 for equilibrium to be satisfied on the discontinuity. The plastic loading consistency condition F(σ, σ 0) ¼ 0 is extended as   F_ ðσ, σ 0 Þ ¼ ∂σ F : C : ε_ + ð½½u_ nÞs δs 2λ_ δ δs ð∂σ GÞ  λ_ δ δs Hð∂σ 0 FÞð∂σ0 GÞ ¼ 0 (15.7) where σ_ 0 ¼ 2λ_ δ δs Hð∂σ0 GÞ is derived from Eq. (15.2). By considering the consistency condition in Eq. (15.7), both the bounded and unbounded terms must sum to zero respectively. The bounded term in Eq. (15.7) leads to the first expression for _ ∂σ F : C : ε_  λ_ δ δs H ¼ 0 ) λ_ δ ¼ H1 δ ∂σ F : C : ε

(15.8)

where the scalar softening modulus H1 ¼ Hδ 1 δs must also be a singular distribution [28]. The unbounded term in Eq. (15.7) leads to the second expression forλ_ δ ∂σ F : C : ð½½u_ nÞ ∂σ F : C : ð½½u_ nÞs δs 2λ_ δ δs ∂σ F : C : ∂σ G ¼ 0 ) λ_ δ ¼ ∂σ F : C : ∂σ G

s

(15.9)

Substituting Eq. (15.9) into Eq. (15.6) leads to the initiation condition of the strain localization by the strong discontinuity analysis [28] A  ½½u_  ¼ n  Ctan  n  ½½u_  ¼ 0, Ctan ¼ C 

C : ∂σ G∂σ F : C ∂σ F : C : ∂σ G

(15.10)

where Ctan is the elastic perfectly plastic tangent modulus tensor. The acoustic tensor A must be singular, i.e., the determinant det(A) of the acoustic tensor A is zero for a nontrivial solution ½½u_  6¼ 0. It should be pointed out Eq. (15.10) is a function of actual stress state although it does not contain the plastic modulus explicitly. In the last chapter, we introduced the GTN model. By combining Eq. (15.10) with the GTN model with some tensor operations, the explicit expression for Ctan is obtained   1 Ctan ¼K11 + 2μ I  11 3 2

9K ω2 11 + 36μ2 =σ 40 ss + 18Kμω=σ 20 1s + 18Kμω=σ 20 s1 , χ   3 q2 ω ¼q1 q2 f sinh σ m =σ 0 , χ ¼ 18μs : s=σ 40 + 9Kω2 2 σ0 (15.11) 

15.2 Strong discontinuity analysis on the strain localization for the GTN model

335

Eq. (15.10) leads to Eq. (15.12) naturally m  A  m ¼ ðmnÞ : Ctan : ðmnÞ ¼ 0

(15.12)

By combining Eqs. (15.5), (15.10), (15.12) with some tensor operations, the expression for η_ ¼ λ_ δ =_ς is obtained rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 μ > _ > _ , > η ¼ λ δ =_ς ¼ < λ2 + λ3 !λ1 2 2 2 2 > > λ ¼ K + μ  9K ω 9ω2 , λ ¼ 36μ  9 ðs : sÞ2 , λ ¼ 18Kμω  18 s : s > 2 3 : 1 4 4 3 χ χ σ0 χ σ0 σ 40 (15.13) where it is shown η_ is a function of the actual stress levels. The normal direction n, the tangential direction t, and the motion direction m of the localization band in the plane strain state are shown in Fig. 15.1, where two angles θi(i ¼ 1, 2) are defined. By taking the trace with respect to Eq. (15.5), the angle θ1 is obtained sinθ1 ¼ tr½ðmnÞs 5m  n ¼ η_ trð∂G=∂σ Þ ¼ η_ ð∂σ G : 1Þ ¼ 3ωη_

(15.14)

By performing some tensor operations on Eq. (15.5), the relationship between the normal and tangential displacement jump rates is obtained sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 s : s=σ 40 + ω2 ∂σ G : ∂σ G  1 ¼  1 (15.15) jς_ t j ¼ βς_ n , jς_ t j5_ς cos θ1 , β ¼ 3ω2 ½trð∂σ GÞ2 where β is a large number because the actual stress levels within the localized band are high ∂σ G ¼ 3s/σ 20 + ω1.

FIG. 15.1

Motion direction and normal direction of the localization band.

336

15. Numerical simulation of micromechanical crack initiation

The displacement jump rate ½½u_  can be decomposed into two parts including the tangential and normal directions ½½u_  ¼ ς_ m ¼ ς_ n n + jς_ t jt. By using Eqs (15.15), (15.5) is further written as the following form  1 η_ η_
ðnnÞs + signðς_ t ÞðtnÞs ¼ ∂σ G ¼ 3s=σ 20 + ω1 (15.16) β cos θ1 cos θ1 where it is noted there is a transition from global to local displacement jumps, which was studied by a consistent iterative scheme proposed by Ja
skowiec and van der Meer [25]. Substituting n 5 [cosθ2, sinθ2]T and t 5 [sin θ2, cosθ2]T into Eq. (15.16) leads to the following matrix equations under the σ 1  σ 2 principal stress plane 21 3s 1 ð cos θ2 Þ2  signðς_ t Þ sin θ2 cosθ2 sin θ2 cosθ2  signðς_ t Þð sin θ2 Þ2 6β 7 β 6 7 41 5 2 1 2 sin θ2 cos θ2 + signðς_ t Þð cosθ2 Þ ð sin θ2 Þ + signðς_ t Þsin θ2 cos θ2 β β " # 3s1 =σ 20 + ω 0 η_ ¼ cos θ1 0 3s2 =σ 20 + ω (15.17) where s1 and s2 are the maximum and minimum principal deviatoric stress components for the plane strain problem. Eq. (15.17) can be extended to three equations 8  1 η_
> > ð cosθ2 Þ2  signðς_ t Þsin θ2 cos θ2 ¼ 3s1 =σ 20 + ω , > > > β cos θ1 > >

β cos θ 1 > > > h i > 2 > 2 2 > : sin θ2 cos θ2 + signðς_ t Þ ð cosθ2 Þ  ð sin θ2 Þ ¼ 0 β From Eq. (15.18-3), the angle tan ð2θ2 Þ ¼  signðς_ t Þβ is obtained. By compffiffiffiffiffiffiffiffiffiffiffiffi bining cos θ1 ¼ m  t ¼ β= 1 + β2 , another two expressions for the angle θ2 are obtained cos θ1 ¼ sin ð2θ2 Þ ¼ cos ðπ=2  2θ2 Þ ) θ2 ¼ ðπ=4  θ1 =2Þ or (15.19) cos θ1 ¼  sin ð2θ2 Þ ¼ cos ðπ  π=2 + 2θ2 Þ ) θ2 ¼ ðπ=4  θ1 =2Þ Combining Eq. (15.18-1), (15.18-2) with some tensor operations leads to the localization bifurcation (or initiation) condition for plane strain problems 3_η ðσ 1  σ 2 Þ ¼ σ 20

(15.20)

where σ i(i ¼ 1, 2) are the in-plane principal stresses. The localization condition in Eq. (15.20) is shown to be related to the actual stress levels.

15.3 Finite element formulation by XFEM with an embedded cohesive model

337

For the plane strain problem, the σ 1  σ 2 stress plane is the major-minor principal stress plane.

15.3 Finite element formulation by XFEM with an embedded cohesive model The boundary value problem with a discontinuous interface is shown in Fig. 15.2. The boundary ∂ Ω of the domain Ω with the outward normal N is subdivided into either Neumann or Dirichlet boundary conditions defined on ∂ Ωt and ∂ Ωu respectively such that ∂ Ω ¼ ∂ Ωu [ ∂ Ωt and ∂ Ωu \ ∂ Ωt ¼ Ø. Further, Ω is divided into Ω+ and Ω by an internal discontinuity Γ d such that Ω ¼ Ω+ [ Ω. The displacement field u across Γ d is discontinuous but continuous within Ω+ and Ω. The strong form is given by the balance of the linear momentum and boundary conditions 8 —  σ + ρb ¼ 0, in Ω + [ Ω > > < u ¼ ub , on ∂Ωu (15.21) σ  N ¼ t , on ∂Ωt > b > : +  σ  n ¼ σ  n ¼ τ ð½½uÞ, on Γ d where ub is the prescribed displacement on the Dirichlet boundary, tb is the prescribed traction on the Neumann boundary, and ρb is the body force. The traction vector τ which acts on the localization interface Γ d depends on the displacement jump ½½u ¼ u+  u at Γ d. The normal vector + + n directs from the point x on Γ  d to the point x on Γ d . By introducing the virtual displacement variation, the weak form of the boundary value problem with an interface discontinuity is written as the virtual work equation Tb

∂Ωt Γd–

Γd+

– Γd Ω+

Point x–

∂Ωu

∂Ωu n

FIG. 15.2

Ω–

Point x+

N

Stress boundary value problem with an interface discontinuity.

338

15. Numerical simulation of micromechanical crack initiation

ð

ð Ω + [Ω

ð—x δuÞ : σ dΩ +

Γd

ð

ð δ½½u  τ dΓ5

∂Ωt

δu  t b dΓ +

Ω

δu  ðρbÞ dΩ (15.22)

According to the extended finite element method (XFEM), the displacement field u(x) is split into the regular continuous part and the discontinuous part [14] 8 X X e J ΦJ ðxÞaJ , > NI dI + N < uðxÞ5 I J enriched (15.23) > : Φ ðxÞ ¼ H ðxÞ  H
x , I,J51, nnode J s s J where nnode denotes the number of nodes in one element. The set of rege J are the partition ular nodes is I and the set of enriched nodes is J. NI and N of unity interpolation shape functions for the continuum elements and enriched elements, respectively. Here, the shifted Heaviside function for Hs(x) is used which is zero on the negative side and one on the positive side of the discontinuous interface. Hs(xJ) is the value of Heaviside function at the node J in the enriched elements. dI and aJ are the standard and enriched node displacements to be solved. The enrichment of the crack tip is neglected and the enrichment applies only to those elements which are intersected by the crack. After combining Eqs. (15.22), (15.23) with some variational operations, the following finite element equations by the XFEM are established to solve the standard node displacement dI and the enriched node displacement aJ " #  " ext # f K dd K da dI ¼ Iext (15.24) ad dd aJ fJ K K where K is the assembled stiffness matrix and its four components are written as ð 8 int dd > ¼ ∂f =∂d ¼ BI T DBI dΩ, K > I > I > > > > Ω +ð [Ω > > > > int da > e J dΩ, > BI T DB > K ¼ ∂f I =∂aJ ¼ > < Ω +ð [Ω (15.25) > int ad > e T DBI dΩ, ¼ ∂f =∂d ¼ B K > I J J > > > > > Ω + [Ω > ð ð >

> > int coh dd > e J T Ds N e J dΓ e J dΩ + N e T DB K ¼ ∂ f + f ¼ B =∂a > J J > J J : Γd Ω + [Ω

where D is the tangent modulus of bulk elements and Ds is the stiffness of interface elements. The localization problem is equivalent to the crack

15.4 Numerical results and discussion

339

FIG. 15.3 Linearly softening cohesive model for governing the progressive failure of localization band after localization bifurcation condition is met.

initiation and propagation using XFEM with embedded cohesive model. After initial localization is detected, the cohesive model is introduced. Currently, there are many cohesive models in terms of the shape of cohesive traction-displacement jump curves. Since the bifurcation condition in Eq. (15.20) is given, the extrinsic linearly decreasing cohesive model proposed by Camacho and Ortiz [36] is used in this research, as shown Fig. 15.3. The tractions τn and τt are written as the functions of ςn and j ςt j respectively   τcn
c τc
ςn  ςn , τt ¼ tc ςct  jςt j (15.26) c ςn ςt pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where ς ¼ ς2n + ς2t ¼ 1 + β2 ςn and τ ¼ τ2n + τ2t ¼ 1 + β2 τn are obtained when ςct ¼ βςcn and τct ¼ βτcn are assumed. The surface energy (or called fracture energy) Gs which is calculated as the active area under the curve τ  ς is written as   ð ςc ð ð ςcn ð ςc t t 1 Gs ¼ τ  d ½½u ¼ τn dςn + τt dςt ¼ 1 + 2 τt dςt β 0 0 0 (15.27)   1 1 1 ¼ τct τct 1 + 2  τct τct 2 2 β τn ¼

where τct and ςct are determined by Gs. Thus, Eqs. (15.26), (15.27) relate the surface energy to the cohesive model. It is noted the cohesive model in Eq. (15.27) does not consider coupling between the normal and tangential directions because of the assumed same cohesive stiffness in Ds.

15.4 Numerical results and discussion 15.4.1 Numerical issues for XFEM and cohesive model As shown in Fig. 15.4, the cohesive segment technique proposed by Remmers et al. [37] and the phantom node method proposed by Song et al. [38] are used in FEA. The crack is assumed to start at the center of

340

15. Numerical simulation of micromechanical crack initiation

FIG. 15.4 XFEM with embedded cohesive model that simulates the crack initiation and propagation.

an enriched element where the initiation condition of the localization is met. Then, the crack propagates to intersect with two edges of enriched elements. Because the crack-tip enrichment is not needed, the crack propagates across an entire element at one time. In addition, two signed functions ψ 1(x) and ψ 2(x) are used to determine the geometry locations of the interface and crack tip, and also provide a local coordinate to determine the enriched Heaviside functions Hs(x) and Hs(xJ) in Eq. (15.23) 8  !  ! > AB  AC ðxB  xA ÞðyC  yA Þ  ðxC  xA ÞðyB  yA Þ > > > ψ ð x Þ ¼ ¼ > < 1 ðxA  xB Þ2 + ðyA  yB Þ2 kABk2 (15.28)  !  ! > > > AB  AC ðxB  xA ÞðyB  yA Þ + ðxC  xA ÞðyC  yA Þ > > ¼ : ψ 2 ðxÞ ¼ ð xA  xB Þ 2 + ð yA  yB Þ 2 kABk2 The crack tip is represented by the intersection of zero values of ψ 1(x) and ψ 2(x). No explicit representation of the interface is needed because the interface is completely characterized by node data. After the crack passes across an enriched element, the element is divided into two elements, each of which contains real or phantom nodes at the intersected edges. The initiation condition of localization in Eq. (15.20) is rewritten as 1:0  3η_ ðσ 1  σ 2 Þ=σ 20  ð1:0 + tolÞ

(15.29)

where the tolerance tol ¼ 1e  3 is chosen. Numerical convergence due to cohesive softening is solved by properly introducing viscous effect into Eq. (15.25)

15.4 Numerical results and discussion

8 ext int v > > < f I  f I  f I ¼ R, int coh v f ext J  f J  f J  f J ¼ R, > > : f v 5f v ¼ cM∗ v, v ¼ Δu=Δt I

341

(15.30)

J

∗

where M is an artificial mass matrix calculated by unity density, v is the node velocity, c is a damping factor, Δu is the node displacement increment, Δt is the time increment during Newton iterations and R is the tolerance.

15.4.2 Finite element model and numerical results for single-edge notched bending specimen The single-edge notched bending (SENB) carboxy-terminated butadiene-acrylonitrile rubber (CTBN)-modified epoxy specimen is used. The experiment was performed by Imanaka et al. [39] and the geometry model and sizes are shown in Fig. 15.5. From Aravas [40], the implicit backward Euler algorithm for implementing the GTN model by FEA is given in Appendix-A and numerical codes are developed by using ABAQUS-UMAT (User material subroutine). The plane strain problem is assumed. The elastic modulus and Poisson’s ratio are 2058 MPa and 0.38, respectively. The void parameters in the GTN model are q1 ¼ 1.9, q2 ¼ 1, q3 ¼ 1.9, σ N ¼ 19.6 MPa, and s ¼ 14.7 MPa/23.6 MPa ¼ 0.62 [41]. The initial and nucleated void volume fractions are 1% and 12%. The hardening/softening stress σ 0 is fitted by a function of the equivalent plastic strain εp [41]

FIG. 15.5

Single-edge notch bending (SENB) rubber-modified epoxy specimen.

342

15. Numerical simulation of micromechanical crack initiation

" N  p N1  # α εp ε εp σ0 ¼ 1 + σy +1 + C1 log C2 3 εy εy εy

(15.31)

where α ¼ 0.225, C1 ¼ 0.03, C2 ¼ 0.05, εy ¼ σ y/Em, N1 ¼ 0.1 and N ¼ 1.3 [41] and σ y ¼ 23.6 MPa [39]. The surface energy Gs ¼ 4N/mm is used [39] and the corresponding critical traction and displacement jump are τct ¼ 8 MPa and ςct ¼ 1.0 mm according to the linear cohesive model. Numerical codes for implementing the localization (or crack) initiation condition and propagation direction for matrix are developed using the subroutine module ABAQUS-UDMGINI (user damage initiation). The XFEM function in the main input file in ABAQUS is activated to simulate the crack propagation along the specified path by performing enrichment on matrix elements. Besides, the utility subroutine GETVARM is called to acquire the void volume fraction. Very small time increment is required to guarantee numerical precision (the initial, minimum, and maximum time increments are Δt ¼ 1e  4s, 1e  10s, and 1e  3s, respectively). The viscous constant c ¼ 1e  5 in Eq. (15.32) is generally used. Two mesh models including 1204 and 636 elements are established to study the mesh sensitivity. Two support points at the bottom are constrained and 1mm displacement is exerted on the upper middle part. Fig. 15.6 shows the crack propagation path using two mesh models. One of two symmetric crack paths with an angle to the x axis may appear. By comparison, the crack propagates along the y axis according to the direction perpendicular to the maximum principal stress by using the built-in XFEM module in ABAQUS. The difference between the two methods can also be found in Fig. 15.1. In addition, two mesh models lead to consistent crack paths, which demonstrate the XFEM leads to results independent of mesh sizes. Fig. 15.7 shows the load-displacement curves by comparing the numerical results with the experimental results. It is shown two mesh sizes lead to basically consistent load responses for strong or weak discontinuity analysis. However, mesh sizes must be fine enough to get accurate results. The weak discontinuity analysis without crack leads to consistent results with experiments [41]. However, loadbearing ability by strong discontinuity analysis with crack is much weaker than that by weak discontinuity analysis because of the loss of integrity of the structure with crack. Fig. 15.8 shows the void volume fraction with crack propagation at four stages. Crack starts to propagate at about 0.1 mm load. From about 0.1 to 0.3 mm loads, crack propagates slowly, but more quickly from about 0.3 to 0.35 mm loads. Finally, load curves after numerical convergence do not change when the viscous constant c change from 1e-4 to 1e-6.

15.4 Numerical results and discussion

343

FIG. 15.6

Crack contours with one of two possible propagation directions for SENB specimen using (A) coarse mesh size and (B) fine mesh size.

15.4.3 Finite element models and numerical results for random composite microstructures under transverse tension Carbon fiber/rubber-modified epoxy composites are used. The elastic moduli and Poisson’ ratios for fiber and matrix are 40 GPa, 2.35 GPa, 0.25, and 0428, respectively [41]. The hardening/softening stress for rubbermodified epoxy also refers to Eq. (15.31) and σ y ¼ 75 MPa [41]. The void

344

15. Numerical simulation of micromechanical crack initiation

FIG. 15.7

Load-displacement curves by comparing numerical and experimental results using two mesh models, strong and weak discontinuity analysis for SENB specimen.

FIG. 15.8

Distributions of void volume fractions with crack propagation for SENB specimen at (A) 0.1 mm load, (B) 0.2 mm load, (C) 0.3 mm load, and (D) 0.35 mm load, respectively.

15.4 Numerical results and discussion

345

parameters in the GTN moduli are q1 ¼ 1.35, q2 ¼ 0.95, q3 ¼ 1.35, σ N ¼ 69.76 MPa, and s ¼ 0.39 [41]. The initial and nucleated void volume fractions are 1% and 14% [41]. Since the fiber/matrix interface debonding is an important factor affecting the strength, stiffness, and the localized damage properties of composites, Canal et al. [41] and Vaughan and McCarthy [42] compared the effects of different interface strengths and fracture toughnesses in terms of strong and weak interface on the damage and failure mechanisms of composites under transverse tension and compression. In this research, a perfectly bonded fiber/matrix interface is assumed because our focus is placed on the localization and crack behaviors of the matrix. The surface energy Gs ¼ 4N/mm is also used [41]. Square 2D plane strain finite element model for representative composite microstructures with the sizes L0  L0 ¼ 40 μm  40 μm including 10 randomly distributed fibers is generated by using ABAQUS-PYTHON language, as shown in Fig. 15.9, in which two fiber distributions are used by comparison. The fiber radius and the fiber volume fraction are 5 μm and 50%, respectively. Four-node plane strain element CPE4 is adopted in FEA and the number of all elements is 1089 and 1118, respectively. The structure is subjected to the periodic displacement boundary condition uðL0 , yÞ  uð0, yÞ ¼ u0

(15.32)

where u0 is the exerted displacement. The tensile strain is calculated by ln(1 + u0/L0) and the tensile stress σ 2 is computed by dividing the total node forces at the load edges by the length L0. Constraint equations between corresponding nodes on the opposite edges along the y axis are set to implement the periodic boundary conditions.

FIG. 15.9 Two composite microstructures with 10 randomly distributed fibers: (A) first fiber distribution and (B) second fiber distribution.

346

15. Numerical simulation of micromechanical crack initiation

For composite microstructures with multiple fibers, localization can appear simultaneously in many areas near the fiber/matrix interface. Therefore, it is believed multiple crack initiation and propagation are generated theoretically for multifiber structures. In this research, we consider only one crack for the element that first meets the initiation condition. In fact, there is a primary matrix crack before the final damage for composites under transverse loading according to the experimental observation by Parı´s et al. [43] and Hobbiebrunken et al. [44]. Fig. 15.10 shows the crack initiation and propagation paths by acquiring the status variable STATUSXFEM in ABAQUS and Fig. 15.11 shows the stress-strain curves for two microstructures with 10 fibers. Nine issues are discussed here. (1) Two possible propagation directions exist and the first localized element appears at one edge. In this research, we do not attempt to determine the direction exactly, but give two possible directions because of random composite microstructures. In addition, the first localized element is shown to be the same for two crack propagating paths. (2) Only matrix crack is considered and no interface crack appears.

FIG. 15.10 Crack contours with two possible propagation directions for two composite microstructures with 10 fibers: (A) first fiber distribution and (B) second fiber distribution.

15.4 Numerical results and discussion

347

According to Fig. 15.10, we do not attempt to continue to perform calculations on further crack propagation after the matrix crack propagates and reaches the interface in terms of the crack propagation direction-1. The interaction between two kinds of cracks is beyond the present research. (3) Nonlinear properties appear at the initial stage of the stress-strain

FIG. 15.11 Stress-strain curves for two composite microstructures with 10 fibers: (A) first fiber distribution and (B) second fiber distribution using strong and weak discontinuity analysis.

348

15. Numerical simulation of micromechanical crack initiation

curve in Fig. 15.11 due to matrix softening and localization. In fact, once the plastic deformation for the matrix is found, the consistent tangent modulus becomes negative and the acoustic tensor becomes singular. For the GTN model that considers the coupling between the plastic deformation and microvoid damage evolution, the physical softening phenomenon of rubber-modified epoxy matrix after initial yielding accelerates the initiation of localization. (4) The work done by the external load is partly dissipated by both the damage evolution of the matrix and the crack propagation, and partly becomes the deformation energy. (5) The stress-strain curve does not show large fluctuation when the cohesive surface energy Gs changes from 2 to 6 N/mm, which shows the structural integrity is not largely affected due to one short crack propagation in multifiber composite microstructures. (6) Fig. 15.12 shows the crack propagation path using finer mesh models, which also demonstrates the strong discontinuity analysis is almost independent of mesh sizes. (7) Fig. 15.13 shows distributions of the equivalent strain, the Mises stress, and the void volume fraction with crack contours for the first fiber distribution and two crack propagation directions. The localization appears near the fiber/matrix interface, indicating severe plastic deformation. (8) The average strengths for two fiber distributions and crack propagation directions in Fig. 15.11 are about 104, 101, 91, and 101 MPa, respectively. By comparison, slightly larger strengths 116 and 119 MPa for two fiber distributions are obtained by weak discontinuity analysis without crack. It should be pointed out that we select 10 fibers instead of 30 fibers adopted by Canal et al. [41] because we attempt to get a longer crack propagation length. In addition, computational cost for strong discontinuity analysis is also higher than that for weak discontinuity analysis. (9) Since crack propagates straightly across

FIG. 15.12 Crack contours using finer mesh sizes for composite microstructure with 10 fibers: (A) first fiber distribution and (B) second fiber distribution.

15.4 Numerical results and discussion

349

Equivalent strain PE, Max. Principal (Avg: 75%) +6.443e-01 +5.906e-01 +5.369e-01 +4.832e-01 +4.295e-01 +3.758e-01 +3.221e-01 +2.684e-01 +2.148e-01 +1.611e-01 +1.074e-01 +5.369e-02 +0.000e-00

PE, Max. Principal (Avg: 75%) +5.084e-01 +4.661e-01 +4.237e-01 +3.813e-01 +3.390e-01 +2.966e-01 +2.542e-01 +2.119e-01 +1.695e-01 +1.271e-01 +8.474e-02 +4.237e-02 +0.000e+00

(A) Mises stress S, Mises (Avg: 75%) +6.501e+02 +5.960e+02 +5.419e+02 +4.878e+02 +4.337e+02 +3.797e+02 +3.256e+02 +2.715e+02 +2.174e+02 +1.633e+02 +1.093e+02 +5.517e+01 +1.088e+00

S, Mises (Avg: 75%) +7.834e+02 +7.186e+02 +6.539e+02 +5.891e+02 +5.243e+02 +4.595e+02 +3.948e+02 +3.300e+02 +2.652e+02 +2.005e+02 +1.357e+02 +7.091e+01 +6.140e+00

(B) Void volume fraction VVF (Avg: 75%) +4.908e-01 +4.590e-01 +4.283e-01 +3.971e-01 +3.659e-01 +3.346e-01 +3.034e-01 +2.721e-01 +2.409e-01 +2.097e-01 +1.784e-01 +1.472e-01 +1.160e-01

VVF (Avg: 75%) +4.908e-01 +4.597e-01 +4.286e-01 +3.975e-01 +3.663e-01 +3.352e-01 +3.041e-01 +2.730e-01 +2.419e-01 +2.108e-01 +1.796e-01 +1.485e-01 +1.174e-01

(C) FIG. 15.13 Distributions of (A) the equivalent strain, (B) the Mises stress, and (C) the void volume fraction with crack contours for the composite microstructures with 10 randomly distributed fibers, the first fiber distribution and two crack propagation directions.

350

15. Numerical simulation of micromechanical crack initiation

an entire element when the crack-tip enrichment is neglected, small-time increment and mesh size are needed in order to eliminate mesh dependence for crack initiation and propagation largely.

15.5 Concluding remarks This chapter introduces the strong discontinuity analysis into the research on the strain localization and crack behaviors of rateindependent fiber-reinforced ductile polymer matrix composites. The appearance and evolution of the localization band are equivalent to the crack initiation and propagation. An important work is to derive the localization bifurcation initiation and propagation direction for the GTN model based on the singularity of the acoustic tensor. The strong discontinuity analysis typically falls into the framework of XFEM with an embedded cohesive model. The complicated crack-tip enrichment in the XFEM is not needed because we do not use those crack-tip fracture parameters such as the J-integral and the energy release rate to determine the crack propagation direction. It should be emphasized that the particular softening phenomenon of the rubber-modified epoxy matrix after initial yielding accelerates the evolution of localization band (or crack) no matter from a weak or strong discontinuity perspective. Based on the present work, a link between the localized damage and fracture is established, which also provides a new thread to study the damage tolerance of composites from microscopic scale apart from the phenomenological damage mechanics approach.

Appendix-A: Implicit backward Euler algorithm for implementing the GTN model by FEA [40] The strain increment Δε is decomposed into the elastic and plastic parts Δε 5 Δεe + Δεp. The stress σ is divided into the hydrostatic stress p and the deviatoric stress S: σ5S  p15 23 qe n  p1. The yielding function F(p, q, Hα), the plastic potential function G(p, q, Hα) and the internal variable increment function dHα ¼ hα(dεp, σ, Hβ) are defined. α and β define the number of internal variables. e n denotes the normal direction of yielding surface. The plastic strain increment Δεp is written as 1 ∂G ∂G 3S n¼ n, Δεp ¼ Δλ , Δεq ¼ Δλ , e Δεp ¼ Δεp 1 + Δεq e 3 ∂p ∂q 2q

(A.1)

where Δεp and Δεq are the hydrostatic and equivalent strain increments, respectively. Δλ is the plastic factor increment.

Appendix-A: Implicit backward Euler algorithm

351

The stress update algorithm addresses the solution of the stress and strain after a new strain increment Δε is given. In the following, fully implicit backward Euler algorithm is used which guarantees the incremental stress reaches the new yielding surface and all variables are updated simultaneously at the end of each increment. The trial stress concept is introduced which ultimately returns onto the updated yielding surface. The stress σ n+1 at a new strain increment n + 1 is calculated as trðeÞ

p

trðeÞ

nn + 1 , σ n + 1 ¼ σ n + 1  C : Δεn ¼ σ n + 1  KΔεp 1  2μΔεq e 3 S n + 1 trðeÞ σ n + 1 ¼ σ n + C : Δεen + 1 , e nn + 1 ¼ 2 qn + 1

(A.2)

where σ tr(e) n+1 is the elastic trial stress at the start of a new strain increment. The yielding function and internal variable increment function at n +1 increment are written as


 F pn + 1 , qn + 1 , Hnα + 1 ¼ 0, ΔHα ¼ hα Δεp , σ n + 1 , Hnβ + 1 (A.3) pΔε + qΔε

where ΔHα,β¼1 ¼ Δεp ¼ ð1fp Þσ0 q and ΔHα,β¼2 ¼ Δf ¼ ð1  f ÞΔεp + AΔεp are calculated for the GTN model in terms of two internal variables εp andf. Elimination of Δλ from Eq. (A.1) leads to     ∂G ∂G Δεp + Δεq ¼0 (A.4) ∂q n + 1 ∂p n + 1 n leads to the Decomposition of the stress σ n+1 in the directions 1 and e hydrostatic stress pn+1 and the equivalent plastic stress qn+1 nn + 1 ) qn + 1 ¼ qe  3μ Δεq pn + 1 ¼ pe  KΔεp , Sn + 1 ¼ Se  2μ Δεq e

(A.5)

where pe, qe and Se are the trial stresses. In summary, the implicit algorithm solves Δεp, Δεq, pn+1, qn+1, and ΔHα by combining the following equations 8     ∂G ∂G > > > Δεp + Δεq ¼ 0, > > >
∂q n + 1 α  ∂p n + 1 > > < F pn + 1 , qn + 1 , Hn + 1 ¼ 0, (A.6) pn + 1 ¼ pe + KΔεp , > > > e > qn + 1 ¼ q  > 3μ Δεq ,

> > > : ΔHα ¼ hα Δεp , Δεq , pn + 1 , qn + 1 , Hnβ + 1 When Δεp and Δεq are selected as the primary unknown, Eq. (A.6-1), (A.6-2) are considered as the basic equations in which the iterative variables cp and cq are solved using the Newton’s method A11 cp + A12 cq ¼ b1 , (A.7) A21 cp + A22 cq ¼ b2 where the constants Aij(i, j ¼ 1, 2) and bi(i ¼ 1, 2) are given by

352

15. Numerical simulation of micromechanical crack initiation

! ! 8 2 2 > ∂G ∂2 G X ∂2 G ∂Hα ∂2 G X ∂2 G ∂Hα > > + Δεp K + + Δεq K 2 + , A11 ¼ > > ∂q ∂p∂q α¼1 ∂q∂Hα ∂Δεp ∂p∂Hα ∂Δεp > ∂p > α¼1 > > ! ! > > 2 2 > > ∂G ∂2 G X ∂2 G ∂Hα ∂2 G X ∂2 G ∂Hα > > + Δεp 3μ 2 + + A12 ¼  Δεq 3μ , > > ∂p ∂q∂Hα ∂Δεp ∂p∂q α¼1 ∂p∂Hα ∂Δεp > ∂q > α¼1 > > > > 2 2 < ∂F X ∂F ∂Hα ∂F X ∂F ∂Hα A21 ¼ K + , A22 ¼ 3μ + , α ∂p α¼1 ∂H ∂Δεp ∂q α¼1 ∂Hα ∂Δεq > > > > ! > > 2 > ∂G ∂G ∂Hα X ∂hβ ∂hβ > > > , b1 ¼ Δεp  Δεq , b2 ¼ F, ¼ Cαβ +K > > ∂Δεp β¼1 ∂Δεp ∂p ∂q ∂p > > > > ! >   > 2 > α β β X > ∂H ∂h ∂h ∂hα 1 > > , C ¼ C  3μ ¼ δ  > αβ αβ αβ : ∂Δεq ∂Δεq ∂q ∂Hβ β¼1 (A.8)

Thus, the plastic strain increments Δεp and Δεq are updated Δεp ¼ Δεp + cp , Δεq ¼ Δεq + cq

(A.9)

Then, all other variables including pn+1, qn+1, and ΔHα are also updated according to Eq. (A.6-3)–(A.6-5). Numerical iteration continues until the yielding function is met. In the following, the consistent modulus D ¼ (∂ σ/∂ ε)n+1 is derived. The elastic equation (A.2) is rewritten as   1 p nn + 1 ) σ n + 1 ¼ C : εn + 1  εn  Δεp 1  Δεq e 3   1 ∂e nn + 1 nn + 1  Δεq : ∂e nn + 1 , ∂σ n + 1 ¼ C : ∂εn + 1  ∂Δεp 1  ∂Δεq e (A.10) 3 ∂σ n + 1   ∂e nn + 1 1 3 1 I  11  e nn + 1 e ¼ nn + 1 2 ∂σ n + 1 qn + 1 2 The variations ∂ Δεq and ∂ Δεq with respect to ∂ σ are given by " #  2 X ∂G ∂2 G ∂p ∂2 G ∂q ∂2 G + Δεp + ∂Δεp ∂Hα + : ∂σ + ∂q ∂p∂q ∂σ ∂q2 ∂σ ∂q∂Hα α¼1 " #  2 X ∂G ∂2 G ∂p ∂2 G ∂q ∂2 G α + ∂H ¼ 0 : ∂σ + ∂Δεq  Δεq ∂p ∂p2 ∂σ ∂p∂q ∂σ ∂p∂Hα α¼1 The yielding function also gives   2 X ∂F ∂p ∂F ∂q ∂F + ∂Hα ¼ 0 : ∂σ + α ∂p ∂σ ∂q ∂σ ∂H α¼1

(A.11)

(A.12)

Appendix-A: Implicit backward Euler algorithm

The variations of internal variables Hα are calculated as

β  β   2 X ∂h ∂hβ ∂h ∂p ∂hβ ∂q α + ∂H ¼ Cαβ ∂Δεp + ∂Δεq + : ∂σ ∂Δεp ∂Δεq ∂p ∂σ ∂q ∂σ β¼1 Combination of Eqs. (A.11), (A.12) leads to (
 nn + 1 : ∂σ n + 1 , A11 ∂Δεp + A12 ∂Δεq ¼ B11 1 + B12 e
 nn + 1 : ∂σ n + 1 A21 ∂Δεp + A22 ∂Δεq ¼ B21 1 + B22 e where the constants Aij ði, j ¼ 1, 2Þ and Bij ði, j ¼ 1, 2Þ are given by 8  2 X 2  > ∂G X ∂2 G ∂2 G ∂hβ > > + A ¼ Δε + Δε , C > 11 p q αβ > > ∂q α¼1 β¼1 ∂q∂Hα ∂p∂Hα ∂Δεp > > > > > >  2 X 2  > > ∂G X ∂2 G ∂2 G ∂hβ > > + A ¼  Δε + Δε , Cαβ > 12 p q > α α > ∂p α¼1 β¼1 ∂q∂H ∂p∂H ∂Δεq > > > > >   > 2 X 2  2 X 2  > X X > ∂F ∂hβ ∂F ∂hβ > > A21 ¼ Cαβ Cαβ , A22 ¼ , > > > ∂Hα ∂Hα ∂Δεp ∂Δεq > α¼1 β¼1 α¼1 β¼1 > > > " > # > 2 X 2  2 > 2 β X > 1 ∂ G ∂ G ∂h > > B11 ¼  Δεp + Cαβ > > > 3 ∂p∂q α¼1 β¼1 ∂q∂Hα ∂p > > > > " > # > 2 X 2  2 < 1 ∂2 G X ∂ G ∂hβ + Cαβ + Δεq , 3 ∂p2 α¼1 β¼1 ∂p∂Hα ∂p > > > > > " > # > 2 X 2  2 > > ∂2 G X ∂ G ∂hβ > > B12 ¼ Δεp + Cαβ > > ∂q2 α¼1 β¼1 ∂q∂Hα ∂q > > > > > " > # > 2 X 2  2 > > ∂2 G X ∂ G ∂hβ > > + Cαβ + Δεq , > > ∂p∂q α¼1 β¼1 ∂p∂Hα ∂q > > > > > " > # > 2 X 2  > > 1 ∂F X ∂F ∂hβ > > B21 ¼  + Cαβ , > > 3 ∂p α¼1 β¼1 ∂Hα ∂p > > > > > " > # > 2 X 2  > > ∂F X ∂F ∂hβ > > Cαβ > : B22 ¼  ∂q + ∂Hα ∂q α¼1 β¼1 Eq. (A.14) is transformed to another form

353

(A.13)

(A.14)

(A.15)

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15. Numerical simulation of micromechanical crack initiation

8

> < ∂Δεp ¼ mp1 1 + mpn n n + 1 : ∂σ n + 1 ,

> : ∂Δεq ¼ mq1 1 + mqn n n + 1 : ∂σ n + 1

(A.16)

Substituting Δεp and Δεq in Eq. (A.14) into Eq. (A.10) leads to ðI + C : M Þ : ∂σ n + 1 ¼ C : ∂εn + 1 , 1 1 M ¼ mp1 11 + mpn 1e nn + 1 1 nn + 1 + mq1 e 3 3 ∂e nn + 1 nn + 1 e nn + 1 + Δεq + mqn e ∂σ n + 1

(A.17)

From Eq. (A.14), the consistent tangent modulus D ¼ (∂ σ/∂ ε)n+1 is calculated as
1 (A.18) D ¼ ð∂σ=∂εÞn + 1 ¼ ðI + C : M Þ1 : C ¼ C1 + M where D is symmetric only at 13 mpn ¼ mq1 . For associative plasticity model, D is approximately symmetric.

References [1] Boyce MC, Parks DM, Argon AS. Large inelastic deformation of glassy polymers part I: rate dependent constitutive model. Mech Mater 1988;7:15–33. [2] Gearing BP, Anand I. On modeling the deformation and fracture response of glassy polymers due to shear-yielding and crazing. Inter J Solids Struct 2004;41:3125–53. [3] Coenen EWC, Kouznetsova VG, Geers MGD. Multi-scale continuous-discontinuous framework for computational-homogenization-localization. J Mech Phys Solids 2012;60:1486–507. [4] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 1977;99:2–15. [5] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [6] Bazˇant ZP, Belytschko T, Chang TP. Continuum theory for strain-softening. ASCE J Eng Mech 1984;110:1666–92. [7] Ortiz M, Leroy Y, Needleman A. A finite element method for localized failure analysis. Comput Meth Appl Mech Eng 1987;61:189–214. [8] Bazˇant ZP. Instability, ductility and size effect in strain-softening concrete. ASCE J Eng Mech 1976;102:331–44. [9] Eringen AC. On nonlocal plasticity. Inter J Eng Sci 1987;19:1461–74. [10] Pijaudier-Cabot G, Bazˇant ZP. Nonlocal damage theory. ASCE J Eng Mech 1987;113:1512–33. [11] Triantafyllidis N, Aifantis EC. A gradient approach to localization of deformation-I hyperelastic materials. J Elast 1986;16:225–38. [12] Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Strain gradient plasticity: theory and experiment. Acta Metall et Mater 1994;42:475–87.

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e JP, Longe`re P, Cadou JM. Numerical modelling of crack propagation in ductile materials combining the GTN model and X-FEM. Comput Meth Appl Mech Eng 2014;275:204–33. [17] Moe¨s N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Inter J Numer Meth Eng 1999;46:131–50. [18] Wang YX, Waisman H. From diffuse damage to sharp cohesive cracks: a coupled XFEM framework for failure analysis of quasi-brittle materials. Comput Meth Appl Mech Eng 2016;299:57–89. [19] Abu Al-Rub RK, Voyiadjis GZ. Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nanoindentation experiments. Inter J Plast 2004;20:1139–82. [20] Carpinteri A, Colombo G. Numerical analysis of catastrophic softening behaviour (snap-back instability). Comput Struct 1989;31:607–36. [21] Dugdale DS. Yielding of steel sheets containing slit. J Mech Phy Solids 1960;8:100–4. [22] Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 1962;7:55–129. [23] Areias PM, Belyschko T. Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Inter J Numer Meth Eng 2005;63:760–88. [24] Benvenuti E. A regularized XFEM framework for embedded cohesive interfaces. Comput Method Appl Mech Eng 2008;197:4367–78. [25] Ja
skowiec J, van der Meer FP. A consistent iterative scheme for 2D and 3D cohesive crack analysis in XFEM. Comput Struct 2014;136:98–107. [26] Pike MG, Oskay C. XFEM modeling of short microfiber reinforced composites with cohesive interfaces. Finite Elem Anal Des 2015;106:16–31. [27] Wu JY, Li FB. An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks. Comput Method Appl Mech Eng 2015;295:77–107. [28] Simo JC, Oliver J, Armero F. An analysis of strong discontinuities induced by strain- softening in rate-independent inelastic solids. Comput Mech 1993;12:277–96. [29] Armero F, Garikipati K. An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Inter J Solids Struct 1996;33:2863–85. [30] Oliver J, Huespe AE, Pulido MDG, Chaves E. From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 2002;69:113–36. [31] Borja RI, Regueiro RA. Strain localization in frictional materials exhibiting displacement jumps. Comput Method Appl Mech Eng 2001;190:2555–80. [32] Mansouri LZ, Chalal H, Abed-Meraim F. Ductility limit prediction using a GTN damage model coupled with localization bifurcation analysis. Mech Mater 2014;76:64–92. [33] Rudnicki JW, Rice JR. Conditions for the localization of deformation in pressuresensitive dilatant materials. J Mech Phys Solids 1975;23:371–94. [34] Li YN, Karr DG. Prediction of ductile fracture in tension by bifurcation, localization, and imperfection analyses. Inter J Plast 2009;25:1128–53. [35] Hill R. A general theory of uniqueness and stability in elastic-plastic solids. J Mech Phys Solids 1958;6:236–49. [36] Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. Inter J Solids Struct 1996;33:2899–938.

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[37] Remmers JJC, de Borst R, Needleman A. A cohesive segments method for the simulation of crack growth. Comput Mech 2003;31:69–77. [38] Song JH, Areias PMA, Belytschko T. A method for dynamic crack and shear band propagation with phantom nodes. Inter J Numer Meth Eng 2006;67:868–93. [39] Imanaka M, Motohashi S, Nishi K, Nakamura Y, Kimoto M. Crack-growth behavior of epoxy adhesives modified with liquid rubber and cross-linked rubber particles under mode I loading. Inter J Adhes Adhes 2009;29:45–55. [40] Aravas N. On the numerical integration of a class of pressure-dependent plasticity models. Int J Numer Meth Eng 1987;24:1395–416. [41] Canal LP, Segurado J, LLorca J. Failure surface of epoxy-modified fiber-reinforced composites under transverse tension and out-of-plane shear. Inter J Solids Struct 2009;46:2265–74. [42] Vaughan TJ, McCarthy CT. Micromechanical modelling of the transverse damage behaviour in fibre reinforced composites. Compos Sci Technol 2011;71:388–96. [43] Parı´s F, Correa E, Can˜as J. Micromechanical view of failure of the matrix in fibrous composite materials. Compos Sci Technol 2003;63:1041–52. [44] Hobbiebrunken T, Hojo M, Adachi T, De Jong C, Fiedler B. Evaluation of interfacial strength in CF/epoxies using FEM and in-situ experiments. Compos Part A 2006;37:2248–56.

C H A P T E R

16

Micromechanical damage modeling and multiscale progressive failure analysis of composite pressure vessel 16.1 Introduction Progressive damage and failure properties of lightweight composites are essentially multiscale ranging from microscopic to macroscopic scales, which are involved in such as fiber breakage, matrix cracking, and interface failure. Moreover, there complicated micromechanical failure mechanisms evolve and interact, leading to macroscopic performance degradation and ultimate failure of composites. In order to predict damage tolerance and load-bearing ability of composite structures, an ultimate strategy is to perform a multiscale analysis by developing advanced theory and numerical technique [1]. In general, comprehensive research on the damage and failure of composites should consider the fiber/matrix/interface phases thoroughly. An initial matrix crack due to defects nucleates, propagates, and reaches the interface, leading to the fiber bridging effect for fiber-reinforced composites. Once the fiber bridging stress exceeds the fiber strength, quasi-brittle fiber breakage results in the load redistribution and excessive loadbearing on neighboring fiber and matrix near the broken fiber. On a small scale, the conventional shear-lag theory [2] analyzes the interface stress transfer and interface fracture properties well by establishing the relationship between the interface frictional stress and the fiber tensile stress. Later, many scholars [3–9] further studied the single fiber and multiple

Damage Modeling of Composite Structures https://doi.org/10.1016/B978-0-12-820963-9.00002-X

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Copyright © 2021 Elsevier Inc. All rights reserved.

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16. Micromechanical damage modeling

fiber/matrix interface stress transfer with elastic and elastic-plastic matrix using the shear-lag theory although it neglected some factors such as the Poisson’s effect, fiber and matrix shear, and radial and hoop stresses. No matter from theory and experiment, fiber pull-out becomes an effective approach to test the interface bonding properties of composites [10–12], where the stress-based and energy release rate-based interface fracture criteria are widely adopted. As an extension over the shear-lag theory, the Green’s function-based method [13, 14] was also proposed to predict the damage evolution and tensile strength of large-scale composite models with multiple fibers. A central feature for these advanced theories is to emphasize the effect of multiple fiber breakage on the stress concentration, interface stress transfer, and statistical tensile strength of composites. For nonhomogenized composites, the stress concentration and strain localization phenomena due to broken fibers inevitably aggravate the damage and failure of different phases [15, 16]. Currently, two types of load-sharing models based on the shear-lag theory were developed to describe the stress redistributions after fiber breakage: the global loadsharing (GLS) model [17] and the local load-sharing (LLS) model [18]. Landis et al. [19] showed the failure of composites can fall between the extremes of GLS and nearest neighboring LLS. GLS is associated with the material where fiber breakage does not cause distinct stress concentrations in neighboring fiber and the stress along a broken fiber recovers to the applied stress linearly from the break. In contrast, LLS assumes that intact fiber experiences a stress concentration in the presence of a fiber break. Although these models achieved large success at different scales to a large extent, there are still some defects for LLS as it does not accurately describe the extent of load redistribution and the influence of some micromechanical parameters. With the increase of computer ability, finite element analysis (FEA) has become a powerful tool to predict multiscale damage and failure behaviors of large-scale composite structures. Xia et al. [20] showed it is insufficient to focus only on the single fiber breakage problem and myriad detail associated with it. In a large-scale analysis of the representative volume element (RVE), all details from the local micromechanical fiber/ matrix/interface failure to the macroscopic damage evolution of composite structures are expected to be evaluated appropriately. Micromechanical modeling and FEA have been performed to predict progressive failure properties of composites [21–24], but little work concentrates on the longitudinal progressive failure analysis of large-scale composite structures by 3D FEA. In FEA by Xia et al. [13, 20], zero interface bonding strength and completed debonded frictional interface were assumed by considering thermal residual stress for SiC fiber/metal matrix composites, but may

16.2 Micromechanical damage modeling and FEA on RVE

359

not be this case for carbon fiber/epoxy composites with much lower processing temperature. Essentially, it is beyond doubt complete failure characteristics of different phases should be considered in multiscale FEA, particularly in the presence of thermal residual stress. Liu et al. [25] developed a set of theoretical models and numerical techniques to predict micromechanical damage and failure behaviors of small-scale composite models with square fiber distribution. This chapter further employs these methods to predict damage and failure behaviors of large-scale composite structures, in which RVE with hexagonal fiber distribution is established for FEA. It is noted the hexagonal cell model describes more precise load transfer and stress concentration behaviors than the square cell model. Fiber breakage with random fiber strength is predicted using Monte Carlo simulation, progressive matrix cracking is predicted by developing a damage model, and interface failure is simulated using the exponential cohesive model developed by Xu and Needleman [26]. Numerical results are obtained in terms of the stress concentration, progressive damage behaviors, and tensile strengths of a composite model. The developed numerical technique helps to gain deeper insight into the multiscale failure mechanisms and the strength prediction of composite pressure vessels and other composite structures.

16.2 Micromechanical damage modeling and FEA on RVE 16.2.1 Finite element model of RVE According to the microscopic observation of composites, two relatively regular fiber distributions in the matrix for T700/epoxy composites exist square and hexagonal distributions [27]. In this work, the hexagonal RVE for large-scale micromechanical model is established for the failure analysis. The fiber radius is Rf ¼ 3.5E  3 mm, the fiber volume fraction is 62%, and the axial length is L ¼ 80E 3 mm. FEA of the RVE is performed with ANSYS-APDL (ANSYS PARAMETRIC DESIGN LANGUAGE). 3D eightnode Solid45 element is used to mesh the fiber and matrix. Fiber breakage is simulated using the Weibull distribution, the zero-thickness Inter205 element using the Xu and Needleman’s interface cohesive model [26] is employed to simulate the crack initiation and propagation of the fiber/ matrix interface. In order to capture the stress concentration accurately, the mesh is refined at the initially broken fiber. The finite element model is shown in Fig. 16.1. The axial length Δz ¼ L/20 ¼ 4E  3 mm of each element is selected. The number of fiber and matrix is 11,360 and 4480, respectively.

360

FIG. 16.1

16. Micromechanical damage modeling

The finite element model for the hexagonal RVE.

16.2.2 Failure criteria and damage models of the RVE 16.2.2.1 Failure criterion and strength distributions for carbon fiber Fiber is considered as transversely isotropic and quasi-brittle. The maximum stress failure criterion for fiber is assumed as


σ fz
< Xf (16.1) where Xf is the statistical tensile strength obeying the Weibull distributions. Similar to Li [28], the strength distribution of each fiber element is written by the two-parameter Weibull function       Δz σ fz m F σ f ¼ 1  exp  (16.2) L0 σ 0 where F is the failure probability of fiber element at a stress level equal to or smaller than the axial fiber stress σ f, m is the shape parameter that characterizes the flaw size distributions, and σ 0 is the scale parameter at a gauge length L0 that describes the characteristic strength of fibers. After a random number η within the interval [0,1] is selected, the tensile strength Xf of each fiber element is obtained  1=m L0 ln ð1  ηÞ (16.3) Xf ¼ σ 0  Δz where m ¼ 12.06 is Weibull modulus, L0 ¼ 8 mm, and σ 0 ¼ 4.41e3 GPa for T700 fiber based on the fiber bundles theory and statistical theory [29].

16.2.2.2 Failure criterion and damage modeling for epoxy matrix In the conventional shear-lag model [3–9], the epoxy matrix is considered to be linear elastic without damage and failure. In this work, the initial damage and progressive failure are introduced for the epoxy matrix.

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16.2 Micromechanical damage modeling and FEA on RVE

By considering the hydrostatic effect, Ha et al. [30] proposed a modified Mises failure criterion !2 σ eq J1 + cr ¼ 1 (16.4) σ cr J eq 1 qffiffiffiffiffiffiffiffiffiffiffi where σ eq ¼ 32 S : S is the Mises effective stress, J1 ¼ tr(σ) is the first stress invariant and S ¼ σ  tr(σ)/3I2 is the deviatoric stress tensor, and I2 is the pffiffiffiffiffiffiffiffiffiffiffiffi cr two-order unit tensor. σ cr eq ¼ Tm Cm and J1 ¼ CmTm/(Cm  Tm) are critical values. Cm ¼ 120 MPa and Tm ¼ 80 MPa are the compressive and tensile strengths of the epoxy matrix, respectively. Liu et al. [25] developed a continuum damage mechanics-based model for isotropic epoxy. The relationship between the effective elastic tensor C and the nominal tensor C is given by C ¼ C=ð1  dÞ, C ¼ λI2 I2 + 2μI4

(16.5)

where d is the damage variable, I4 is the four-order unit tensor. λ and μ are Lame constants. The damage variable d is written as [25]. d ¼ 1  exp ðY=Yc Þ

(16.6)

where Yc is the critical value of the thermal conjugate force Y that is given by Lemaitre and Desmorat [31].    σ eq 2 trðσ Þ Y¼ ð1 + νm Þ + 3ð1  2νm Þ (16.7) 3σ eq 2Em 3 where Em and νm are the Young’s modulus and Poisson’s ratio of epoxy. From Eqs. (16.6), (16.7), the damage variable d is obtained. 16.2.2.3 Exponential cohesive model for interface debonding Xu and Needleman’s exponential cohesive model [26] is used, as shown in Fig. 16.2A. The surface potential function ϕ(δ) is given by Nodes

8 7

t smax

4 5

z

3

h x

6

Middle plane D

c

(A)

FIG. 16.2

D

f

D (B)

1

3 2 1

2

Exponential cohesive model implemented by middle-plane interpolation.

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16. Micromechanical damage modeling

8 h i 2 > ϕðδÞ ¼ eσ max δn 1  ð1 + Δn ÞeΔn eΔt > > > > > > 2 > > Tn ¼ eσ max Δn eΔn eΔt > < 2 δn > Tt ¼ 2eσ max Δt ð1 + Δn ÞeΔn eΔt > > δt > > > > > δ δt > n > : Δn ¼ , Δt ¼ δn δt

(16.8)

where Tn and Tt are the normal and tangential tractions, respectively. Δn and Δt are the normalized normal and tangential opening displacements, respectively. σ max and δ are the maximum traction and the corresponding displacement jump at that point, respectively. The cohesive model is implemented using a middle-plane interpolation technique, as shown in Fig. 16.2B. After interface debonding, the debonded interface may contact with friction. From micromechanical damage and failure analysis of metal matrix composites, Xia et al. [13, 20] showed the composite tensile strength is rather insensitive to the friction coefficient. Thus, the frictional contact of the debonded interface is not considered in this work.

16.2.3 Mori-Tanaka homogenization method for composite with interface debonding Hori and Nemat-Nasser [32] pointed out that the fundamental assumption of the mean-field homogenization method is that the strain and stress are piecewise uniform in each phase in an RVE. The volume average of the strain ε and stress σ for fiber and matrix is given by respectively ð ð 1 1 (16.9) hεip ¼ p εdV, hσ ip ¼ p σdV, ðp ¼ f , mÞ V V Vp

Vp

where V is the volume. The conventional Mori-Tanaka method does not consider the effect of interface debonding. Hori and Nemat-Nasser [32] showed the macroscopic strain must introduce an additional strain due to interface failure. For composites with interface debonding, the average macroscopic stress hσi and strain hεi are written as [33]. 8 > hσ i ¼ f hσ if + ð1  f Þhσ im , > > > < hεi ¼ f hεif + ð1  f Þhεim + f hεiint , (16.10) ð > > 1 > int > ð½u  n + n  ½uÞdA, ½u ¼ um  uf : hεi ¼ f 2V sint where f is the fiber volume fraction. hεiint is the superimposed strain due to interface debonding. sint denotes the discontinuous interface. n is the unit

16.2 Micromechanical damage modeling and FEA on RVE

363

normal vector on the interface pointing to the matrix. um and u f are the displacements on the interface from the matrix side to the fiber side, respectively. Vf is the fiber volume in the RVE. The strain concentration factors Bε and Aε are defined as follows h i (16.11) hεif ¼ Bε : hεim ¼ Aε : hεi ¼ Aε : f hεif + ð1  f Þhεim + f hεiint Liu et al. [25] performed a multiscale analysis of composite structures using the asymptotic homogenization method that involves the interaction between the macroscopic and microscopic data. However, this requires a huge amount of computational resources for large composite structures. By comparison, the Mori-Tanaka mean-field method is easier to implement by FEA and requires several orders of magnitude smaller computational time.

16.2.4 Material parameters and finite element analysis The Young’s modulus Ef and Poisson’s ratio νf for T700 fiber are 227 GPa and 0.3, respectively. The longitudinal and transverse thermal expansion coefficients αf for T700 fiber are 0.52  1E  6/°C and 10  1E  6/°C, respectively [25]. Yc ¼ 1000 MPa in Eq. (16.6) is used. The cohesive interface parameters in Eq. (16.8) are taken as σ max ¼ 75 MPa and δn ¼ δt ¼ 2E  4mm, respectively. The Poisson’s ratio νm of epoxy is 0.4. Experiments showed that Young’s modulus and thermal expansion coefficient of epoxy depend on temperature largely. The Young’s modulus Em(T) of epoxy can be written by three expressions [34].   8

 T  Tg + ΔT1 > > , Tg  ΔT1  T  Tg + ΔT2 , ðaÞ Em ðT Þ ¼ Em Tg  ΔT1 exp k2 > > ΔT1 + ΔT2 > > > > ! < T  Tr , T  Tg  ΔT1 , ðbÞ Em ðT Þ ¼ Em ðTr Þ exp k1 > > Tg  ΔT1 + Tr > > > > > > : ðcÞ Em ðT Þ ¼ 0:01  Em ðTr Þ, T  Tg + ΔT2

(16.12)

The thermal expansion coefficient αm of epoxy is assumed to vary linearly with temperature and the slope is written as [34]. k¼

α1  αm ðTr Þ T g  Tr

(16.13)

where the glassy transition temperature Tg ¼ 110 ° C, Tr ¼ 23 ° C, ΔT1 ¼ ΔT2 ¼ 35 ° C, Em(Tr) ¼ 3.35 GPa, Em(Tr  ΔT1) ¼ 0.7Em(Tr), Em(Tr + ΔT2) ¼ 0.01Em(Tr), k1 ¼ 0.35667, k2 ¼ 4.2485, αm(Tr) ¼ 58  1E  6/ ° C, α1 ¼ 139  1E  6/ ° C. Progressive failure analysis of the RVE is implemented using ANSYSAPDL (ANSYS Parametric Design Language). FEA is divided into two

364

16. Micromechanical damage modeling

steps: (1) Periodic boundary conditions [35] are exerted on all opposite sides of the RVE for calculating the thermal residual stress from curing temperature of epoxy 149°C to room temperature 20°C, in which discrete thermal expansion coefficients with temperature are used. In addition, all constraints are exerted on anyone selected node to eliminate the rigid displacement. (2) Periodic boundary conditions for the broken fiber surface are removed, and the axial tensile load is applied on the opposite surfaces at room temperature. The restart analysis is used to update the degraded material parameters of fiber and matrix after each load increment and the viscous stabilization method is used to solve the numerical convergence problem at the strength limit of the RVE.

16.3 Numerical results and discussion Fig. 16.3 shows the distributions of thermal residual stress after curing. As the temperature-dependent thermal expansion coefficients of epoxy are higher than those of fiber, thermal residual stress occurs mainly at the epoxy. However, the thermal stress does not lead to the initial failure of the matrix. Fig. 16.4 shows the progressive damage evolution process for fiber and matrix elements. Fig. 16.5 shows the interface debonding

FIG. 16.3 The distributions of thermal residual stress (Mises stress) of RVE after curing.

16.3 Numerical results and discussion

365

FIG. 16.4 The progressive damage evolution processes for fiber and matrix elements at the tensile strains (A) 1%, (B) 1.125%, (C) 1.25%, (D) 1.75%, and (E) 2.125%, respectively.

process. At about the strain 0.5%, the RVE starts to fail in the form of matrix cracking with 6 failed matrix elements, which appears near the initially broken fiber. The fiber/matrix interface starts to debond near the broken fiber at about the strain 0.625%, accompanied by 16 failed matrix elements. At the strain 1.125%, there are 1137 failed matrix elements. Fiber breakage starts to appear at the strain 1.25% and at this time all matrix elements (4480) have failed and more severe interface debonding appears. At the strain 2.125%, 26 fiber elements and 4480 matrix elements fail, indicating complete damage of the RVE. It is noted the initially broken fiber induces the accumulation of failed fiber, matrix and interface elements, which accelerates the localized damage of RVE. The progressive failure properties of the RVE can be compared with the acoustic emission (AE) test results on T700 fiber/epoxy composites under tension [25]. AE is mainly represented by the amplitude parameters reflecting the signal strength of different failure modes. The damage evolution process of composites is divided into three stages: (a) At the early stage (0–200 s), the matrix cracking with the 50–60 dB amplitude other than fiber fracture appears. (b) At the middle damage stage (200–1000 s), the dominating failure modes are still matrix cracking, accompanied by very little fiber breakage with beyond 80 dB amplitude. (c) At the stage of fracture (1000–1200 s), the number of fiber fracture adds, indicating the ultimate collapse of the specimen. Experimental results are consistent with numerical analysis. The initially broken fiber introduces the stress redistribution and localized damage on the neighboring fibers. Fig. 16.6 shows the axial stresses

366

16. Micromechanical damage modeling

FIG. 16.5 The interface debonding process at the tensile strains (A) 0.25%, (B) 0.625%, (C) 1.25%, and (D) 1.875%.

on fiber and matrix elements at different strains. It is noted severe stress concentration appears the broken fiber and neighboring matrix, and gradually extends to neighboring fibers. As the cohesive model belongs to a phenomenological model, true interface shear and normal stresses are not given here. Fig. 16.7 shows the axial stress distributions on the broken fiber from the fracture surface (normalized by the far-field fiber stress σ ∞ and averaged over the fiber cross-section) at the tensile strains 0.625% and 1.25%, respectively. The axial fiber stress first recovers gradually from zero at the fracture surface to the smooth far-field fiber stress and then remains constant. Xia et al. [13, 20] showed the slip length ls for the initially

16.3 Numerical results and discussion

367

FIG. 16.6

The axial stresses on fiber and matrix elements at the tensile strains (A) 0.25%, (B) 0.625%, (C) 1.25%, and (D) 1.875%, respectively.

debonded interface (or called as the stress recovery length or inefficient load transfer length for initially bonded interface) can be evaluated by the formula ls ¼ rfσ ∞/2τ using the simple shear-lag model and constant interface friction stress τ. If the frictional stress is replaced by the cohesive traction, the analytical ls is about 0.02 mm at the tensile strain 0.625%, close to the numerical solution in Fig. 16.7. However, the numerical result for the slip length is larger than the analytical solution when the strain continues to increase. Fig. 16.8 shows the axial stress concentration factor k(z) (SCF, the local axial fiber stress normalized by the far-field fiber stress σ f∞, and averaged over the fiber cross-section) of neighboring fibers around the broken fiber.

368

16. Micromechanical damage modeling

FIG. 16.7 The axial fiber stress distributions on the broken fiber from the fracture surface at the tensile strains 0.625% and 1.25%, respectively.

FIG. 16.8 The axial stress concentration factor k(z) of neighboring fibers around the initial broken fiber with the increase of tensile strain.

The SCF adds slightly with the increase of tensile strain. More stresses are redistributed on the first nearest-neighboring fibers than those on the second and third nearest-neighboring fibers. By comparison, the analytical solutions using the global load-sharing (GLS) model [17] and the local load-sharing (LLS) model [18] are also provided. The GLS model assumed

369

16.3 Numerical results and discussion

all intact fibers share the applied loads equally and, thus, equally carry the loads lost from broken fibers, where sufficiently low interfacial shear stress is assumed. In the LLS model, only the immediate unbroken neighboring fibers carry these lost loads, thereby causing more severe overloads than those that appear by the GLS model. The LLS model is considered a fiber composite consisting of infinite hexagonally aligned fibers embedded in the elastic matrix and is more accurate than the GLS model because it accounts for the effects of the elastic deformations of the fiber and matrix on the redistribution of tensile stress based on the shear-lag theory. Further, the modified shear-lag model [13, 14] considered the geometry of finite periodic patches based on the shear-lag model and applied an influence function technique to solve a 3D stress field around the broken fiber. Table 16.1 compares the SCFs obtained by FEA and analytical models. The SCFs obtained by the GLS model are the smallest among all models. The GLS model decreases the stress concentration around the broken fiber and underestimates the sharing portions of the loads borne by the neighboring intact fibers. For the second-neighboring fibers, the SCF obtained by the RVE is basically consistent with that by the LLS, but a large difference appears for the first nearest-neighboring fibers. This can be interpreted as follows: the LLS model neglects the matrix tensile stress and assumes the matrix to be in a state of only pure elastic shear. Therefore, the first nearest-neighboring fibers bear more stress in the LLS. In addition, Blassiau et al. [27] compared the SCF using the square and hexagonal cell models for carbon fiber/epoxy composites with initially bonded and debonded interface. They considered the elastic matrix without damage evolution and used the interface node bonding technique to simulate the interface debonding. From the present FEA, the hexagonal cell model leads to slightly smaller SCF for the same neighboring fibers, which can be explained by the fact that more fibers in the hexagonal cell model share the redistributed loads, consistent with the conclusion with Blassiau et al. [27]. However, it deserves pointing out the progressive failure of matrix also accelerates the localized fiber damage to some extent, which was not considered in the work of Blassiau et al. [27]. Besides, the cohesive interface model for predicting the interface failure is more advantageous than the interface bonding technique. Thus, modeling the transition from the initially bonded interface to debonded interface is significant to describe TABLE 16.1

Stress concentration factor (SCF) at the tensile strain 1.25%.

Square RVE [25]

Hexagonal RVE

GLS model [17]

LLS model [18]

1

1.085

1.035

1.0011

1.1046

2

1.055

1.002

1.0011

1.0100

where “1” and “2” represent the first and second nearest-neighboring fibers around the broken fiber.

370

16. Micromechanical damage modeling

the load transfer and damage evolution of composites in this work, different from the work of Xia et al. [13, 20]. Fig. 16.9 shows the tensile experiments and fracture specimens of carbon fiber/epoxy composites and Fig. 16.10 shows the tensile stress-strain curves of composites by FEA and experiments for four composite specimens. Based on the shear-lag model and Weibull distribution, the expression for the ultimate tensile strength (UTS) σ u is given as [13, 14, 17, 18, 20].  1=ðm + 1Þ    m  2 m+1 σ τL0 σu ¼ f σc + ð1  f Þσ m , σ c ¼ 0 (16.14) m+2 m+2 rf where τ is the frictional shear stress at the debonded interface. The UTS of composites by analytical solutions, FEA, and experiments are listed in Table 16.2. By comparison, the UTS obtained by FEA is smaller

FIG. 16.9 The tensile experiments of unidirectional carbon fiber/epoxy composite specimens.

FIG. 16.10

The tensile stress-strain curves of composites by FEA and experiments.

371

16.4 Concluding remarks

TABLE 16.2

Composite tensile strengths, as predicted and measured.

Hexagonal RVE UTS (MPa)

2475

Curtin’s model [13, 14, 17, 18, 20] τ 5 20–120 MPa

Experiments (average value)

2256–2586

2585

than the experimental results. Besides, the UTS predicted using the analytical model approaches the experimental UTS when the interface sliding stress becomes large. They considered the initially debonded interface due to high residual stresses, and the interface slips against each other obeying the Coulomb friction law. However, the loading sharing model is required to highlight the effect of high SCF on the UTS when the interface stress τ becomes large. In contrast, the present FEA is considered an initially perfect interface and subsequent interface crack initiation and propagation. Thus, the interface stress transfer and stress concentration effect are more accurately described. Finally, the stress-strain tensile curve of carbon fiber/epoxy composites by FEA is further applied to the progressive failure analysis of composite pressure vessels. As the Young’s modulus and tensile strength in the fiber principal direction dominate the burst strength of the composite vessel, the tensile curve is taken as the longitudinal direction. Fig. 16.11 shows the finite element model of a composite vessel with 74 L capacity, which includes one 30° inner aluminum layer and 10 30° outer composite layers. The mesh model includes 38,958 elements. Table 16.3 lists the winding geometry parameters and Table 16.4 shows the material parameters. The inner radius for the cylinder is 183 mm. Fig. 16.12 shows the fiber principal stresses at different internal pressure. The burst test for the composite vessel is shown in Ref. [25] and the burst pressure by FEA is about 90 MPa, close to the experimental result 94 MPa.

16.4 Concluding remarks Based on the microscopic structures of carbon fiber/epoxy composites, this paper develops a large-scale finite element model with the hexagonal cell to study the progressive failure properties of composites. Different from most of the existing works including Xia et al. [13, 20] and Blassiau et al. [27], all failure modes including fiber breakage, matrix damage evolution, and interface debonding are considered in this work, where Monte Carlo simulation is used for fiber breakage, damage mechanics method is used for matrix damage, and cohesive model is used for interface debonding. From FEA, the LLS model overestimates stress concentration on the first neighboring fibers because it still neglects the matrix damage and

372

FIG. 16.11

16. Micromechanical damage modeling

The finite element model for 74 L-capacity composite vessel.

TABLE 16.3 Geometry parameters for the 74L-capacity composite vessel. Thickness (mm)

Wound angle (°)

Aluminum liner layer

5.0

–

First composite layer

2.1

90

Second composite layer

0.87

12.3

Third composite layer

0.87

15.4

Fourth composite layer

0.87

18.6

Fifth composite layer

2.1

90

Sixth composite layer

0.87

22

Seventh composite layer

0.87

27

Eighth composite layer

0.87

32

Ninth composite layer

0.6

38

Tenth composite layer

0.54

90

373

16.4 Concluding remarks

TABLE 16.4 composites.

6061Al

Mehanical parameters of 6061 Al liner and carbon fiber/epoxy

E1 (GPa) E2 (GPa) E3 (GPa) ν12

ν23

ν13

G12 (GPa) G23 (GPa) G13 (GPa)

70

0.3

0.3

27

Composite 120

70

70

0.3

11.41

11.41

0.33 0.49 0.33 7.92

27

27

3.792

7.92

E is the Young’s modulus, v is the Poisson’s ratio, and G is the shear modulus.

FIG. 16.12

The maximum principal stresses of the composite vessel in the fiber principal direction at internal pressure (A) 40 MPa, (B) 60 MPa, and (C) 80 MPa, respectively.

interface debonding. In addition, the hexagonal cell model leads to a slightly smaller stress concentration for the same fibers around the broken fiber than the square cell model. Finally, the composite tensile strength predicted using the shear-lag model is consistent with the experimental result only when the frictional shear stress is large.

374

16. Micromechanical damage modeling

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Index Note: Page numbers followed by f indicate figures and t indicate tables.

A

ABAQUS command language, 55 ABAQUS-PYTHON script language, 220–221, 306, 308–309, 308f, 345, 345f ABAQUS software, 204, 308–310 ABAQUS-UEXTERNALDB utility subroutine, 53–55, 78–79 ABAQUS-VUMAT, 220–221, 230–231 Acoustic emission (AE), 364–365 Acoustic emission test, 69–71 Adhesive failure analysis, 27, 40–41, 46 Adhesive joints, 176 rate-dependent failure analysis of, 186–188, 186–188f Adhesive layer, damage analysis of blade with thickness of, 302–303, 302f Aluminum layer equivalent plastic strain for, 285 Johnson-Cook model for, 278 plastic deformation of, 288, 289f, 294 strain energy for, 288, 290–291f Anisotropic nonlocal consistent tangent stiffness, 100 Arbitrary Euler-Lagrange (ALE), 298 Arrhenius form, 196 Artificial strain energy (ALLAE), 260, 261f Artificial viscous force, 78 Artificial viscous method, 19 Associated beam theory, 151, 171 Associative intralaminar damage, 69–71 Asymptotic homogenization method, 363 Augmented finite element model, 49–50 Axial compressive load-strain curve, 115, 116f, 120–121, 121f Axial stress distributions, 365–367, 368f on fiber and matrix elements, 365–367, 367f

B

Backward Euler algorithm, 254, 256–257, 274–275

Barenblatt model, 331 Beam theory, 151, 170 Bending strengths, 163–164, 167t Bilinear cohesive/friction coupled model, 145–146 Bilinear cohesive models, 75–76, 230–231, 278–279 Bilinear cohesive zone model, 27–30, 29f, 35–41, 44–46 B-K law, 17–19 Boundary nonlinearities problems, 219 Brittle adhesive failure, 40–41 Bulk elements, 80–81, 100–103

C

Carbon fiber, 179–180 failure criterion and strength distributions for, 360 Carbon fiber/epoxy composite laminates bilinear cohesive models, 75–76 damage models, 51–52 intralaminar damage model, 72–75 nonlocal integral model, 52–53, 76–77 unified finite element model, 77–78 Carbon fiber-reinforced polymer matrix composites, 49 Chang-chang failure criteria, 220, 224–228, 249 Classic nonassociative plasticity theory, 149–150 Clausius-Duhem dissipation inequality, 5 Coarse mesh, 310–312 Coarse mesh model, 37f, 38–40, 39f, 44–45 Coarse mesh size, 135–136 Cohesive elements, 78–79, 103–104 Cohesive/frictional contact algorithm, 169–170, 215–216 Cohesive/friction coupled model, 198–199 for mode-II delamination of composites, 146–150, 146f, 149f numerical algorithm for, 150–151, 152f

377

378 Cohesive interface strengths, 318–320, 318–319f Cohesive length, 9 Cohesive models, 23–24, 75–76, 176, 191–192 advantages for, 1–2 for mixed-mode delamination of composite laminates, 4–9 numerical application of, 2 using finite element analysis (FEA), 8–9 Cohesive segment technique, 339–340, 340f Cohesive softening behavior, 27–29, 34–35 Cohesive strength, 127–128, 131–135, 140–142, 154–163 Cohesive theory, 1–2 Cohesive traction-displacement jump relationship, 28f Cohesive zone length, 34 Cohesive zone model (CZM), 27 bilinear model, 27–30, 29f, 35–41, 44–46 exponential model, 27–32, 31f, 35–41, 44–46 implicit finite element formulation, 32–34 Compliance theory, 151, 170 Composite adhesive joint, 29, 46 Composite flat laminates, 112, 129 Composite laminates, 221–222, 221f viscoelastic cohesive/friction coupled model and numerical algorithm for, 197–200, 197–198f, 201f viscoelastic cohesive model and numerical algorithm for, 193–197 Composite structures, 219 Compressive load-central deflection curve, 115, 120–121 Compressive load-strain curve, 131–135 Contact mechanics, 197–198 Continuum elastoplastic stress-strain model, 333 Convergence issue, solution for, 9 Coulomb friction law, 370–371 Coulomb law, 149–150, 150f Crack band model, 2–3 Crack closure integral-VCCT, 111 Crack growth process, 35, 40f, 45f, 46 Crack propagation, 1–2 length, 111–112, 163–164 rate, 165f Crack tip, 163–164 CZM. See Cohesive zone model (CZM)

Index

D

Damaged stress-strain relationships, of composites, 223, 254–255 Damage evolution behavior, 69–74, 80–81, 85–94, 99–100 Damage evolution laws, 73–74 for matrix tension, 225–226 of composites, 224–228 Damage evolution process, 364–365, 365f Damage mechanics method, 85–94 Damage model, 69–75, 78–79, 94–97, 99–100 Damage-type exponential cohesive model, 2–3 Delamination, 267, 294 buckling model, 110 cases, 2–3 for composite layers, 285, 286f crack length, 116 failure, 288, 289f growth process, 116–124 growth processes for Case-A laminate, 163–164, 164f interface, 235–241, 236–240f unstability behavior, 110, 120–121 Differential-type viscoelastic cohesive model, 191–192 Dirac-delta function, 331–332 Divergence theorem, 222 Dominant failure mechanism, 318–320 Dominating mode-I failure model, 119–120, 124 Double cantilever beam (DCB), 2–3, 11, 13f delamination case, 11–14, 11f load-displacement curves for, 14f material and size parameters for, 12t three mesh models for, 12f Drucker-Prager model, 331–332 Ductile interface fracture, 46 Dynamic penalty method, 230–231, 278–279

E

Elastic damage model, 260–262 Elastic parameters, for PC resin, 309t Elastic-plastic fracture mechanics, 331 Elastic wave dispersion method, 81–82 Elastoplastic damage model, 253 End-loaded-split (ELS), 145–146 End-notched flexure (ENF) composite, 145–146, 153f, 207–211, 207–211f finite element models for, 154f load-displacement curves of, 171–172, 212f

Index

End notch flexure (ENF), 2–3, 11, 127–128 delamination case, 15–17, 15f load-displacement curves for, 16f material and size parameters for, 15t Energy dissipation diagram, 232–235, 235f, 247f Energy dissipation mechanism, 277, 293f failure modes, 288 of GLARE laminates, 277–278, 285–288, 287f total, 292–294, 292f Energy release rate (ERR), 110 Epoxy composite laminates, 141–142 Epoxy matrix, failure criterion and damage modeling for, 360–361 Equivalent plastic strain (PEEQ), 312–313, 314f for aluminum layers, 285, 286f Equivalent plastic strain rate, 255 ERR. See Energy release rate (ERR) Euler algorithm, 350–354 Explicit central difference algorithm, 258–259 Explicit finite element analysis (FEA), 200, 211–213, 216, 230, 277–278, 294 for rate-dependent cohesive failure, 178–179 numerical algorithm using, 256–259 on microscopic RVEs of thermoplastic composites, 308–326 stress-strain curves, 311f Explicit time-stepping algorithm, 216, 274–275 Exponential cohesive model, 2–3, 145–146, 176, 193–194 Exponential cohesive model, for interface debonding, 361–362, 361f Exponential cohesive zone model, 27–32, 31f, 35–41, 44–46 Extended finite element method (XFEM), 338–339 and cohesive model, numerical issues for, 339–341, 340f with embedded cohesive model, 337–339, 337f, 339f Extended finite element model (XFEM), 49–50

F

Failure criteria, of composites, 224–228 Failure surface, 320–326, 322f Fast Fourier Transforms (FFTs), 71–72

379

Fiber breakage mode, 277–278 Fiber compressive failure, 73–74, 100–103 Fiber principal stress, 373f Fiber-reinforced polymer composites, 220 Fiber tensile failure, 73–74, 100–103 Fiber tension, and compression, 224 Fine mesh, 310–312 Fine mesh sizes, 279–282 Finer mesh model, 346–350 Finite critical separation, 149–150 Finite deformation theory, 130 Finite difference algorithm, 277–278, 298 Finite dissipation energy, 331–332 Finite element analysis (FEA), 2–3, 27–29, 46, 50, 53, 64–65, 69–71, 168, 219, 330–331, 358–359, 370f cohesive models using, 8–9 material parameters and, 363–364 numerical convergence using, 2–3 on RVE, micromechanical damage modeling and, 359–364 two steps, 363–364 Finite element model of composites, 230–231, 231f for GLARE laminates, 278–282 for random composite microstructures under transverse tension, 343–350, 345–349f of representative volume element (RVE), 359, 360f real model, 279–282, 280–282f simplified model, 279–282, 280–282f single-edge notched bending (SENB), 341–342, 341f, 343–344f Finite strip method, 116–119 Finite-thickness middle-plane interface element technique, 176–177 Finite volume algorithm, 298 First-order buckling modal, 113–115 First-order shear deformation theory, 151, 171 Fixed ratio mixed mode (FRMM), 2–3, 11 damage evolution, 23f delamination case, 20–22, 21f load-displacement curves for, 22f material and size parameters, 21t Fluid-solid coupling analysis, 299–300 Fluid/solid model, numerical algorithm and, 298–299 FM94/S2 composite layers, material parameters for, 279t

380

Index

Force-time/displacement curves, impact of, 279–285, 283–284f Forward Euler algorithm, 253, 256, 260–262 Four-point bending ENF (4ENF), 145–146 Fracture angle, 226–228 Fracture energy, 339 Fracture mechanics theory, 330–331 Fracture process zone, 331 Frictional contact, 149–150, 149f, 192–193, 197f, 206 Friction angle, 149–150 Friction coefficients, 154–163, 241, 242f

G

Gaussian integration, 15–17, 53–55 Geometrically nonlinear deformation, 128, 141–142 Geometrically nonlinear finite element equation, 115 Glass fiber composite/aluminum metal (GLARE) laminates, 277 energy dissipation mechanisms of, 277–278, 285–288, 287f finite element model for, 278–282 Glass/polyester composites geometry parameters, 153t material parameters, 154t Global buckling load, 116–121, 119t, 124, 131–142 Global load-sharing (GLS) model, 358, 367–370 Graphite/epoxy composite specimens Case-A specimen, 259–267 delamination damage, 267, 268–269f fiber damage evolution, 262, 267f impact force-time curve, 260–262, 261f, 263–265f material parameters, 259t matrix compression, 267, 268–269f matrix tension, 267, 268–269f maximum central displacement, 262, 266f permanent central displacement, 262, 266f plastic parameters, 259t three mesh models, 260, 260f Case-B specimen, 267–274 energy dissipation, 273, 274f impact force-displacement curves, 269–271, 272–273f

impact force-time curves, 269–271, 270–271f material parameters, 267–269, 269t Green’s function-based method, 357–358 Gurson’s type model, for thermoplastic resin, 306–308 Gurson-Tvergaard-Needleman (GTN) model, 330–332 strong discontinuity analysis on strain localization for, 332–337, 335f

H

Hashin failure criteria, 220, 224–228, 249, 322–323, 324–325f, 326 Helmholtz free energy, 5 Hexagonal cell model, 359, 371–373 High-velocity impact problems, of composites, 219 Hourglass method, 241, 243f HS300/ET223 composite laminates, 241, 244t Hybrid composite laminates, 277 Hydrostatic effect, 361

I

Impact force-displacement curves, 232, 234f, 246–247f Impact force-time curves, 232, 233f, 244–245f, 292–294, 292f Implicit finite element analysis, 27–29, 32–34, 50, 112–124 cohesive/friction contact algorithm using, 169–170 Implicit gradient theory, 71–72 Implicit Newton-Raphson algorithm, 115 Infra-red crack opening interferometry measurement, 20–21 In-plane damage mechanism, 94–97 Interface debonding process, 364–365, 366f Interface node bonding technique, 367–370 Interlaminar fracture toughness, 119–120 Intralaminar constitutive model, 223–228 Intralaminar damage, 292–294 of composite layers, 288, 289f Intralaminar damage model, 55–65, 72–75, 248–249 for matrix compression, 228f Intralaminar fiber tension, and compression, 225f Intralaminar matrix tension, 225f Inverse algorithm, 181–182

Index

Inverse method, 176–177, 181, 183, 203–204 Inverse parameter identification method, 204 Isotropic consistent tangent stiffness, 100

J

J2 plasticity model, 331–332 Johnson-Cook model for aluminum layer, 278

K

Kuhn-Tucker loading-unloading condition, 74–75, 257 k-ε turbulence model, 298

L

Large-deformation theory, 199 Laser treatment technique, 20–21 Linear elastic fracture mechanics, 331 Load-displacement curves, 15–17, 154–163 Case-A laminate, 157–159f Case-B laminate, 159–161f Case-C laminate, 161–163f normal stiffness, 154, 156f superimposed tangential traction, 164, 168f viscous constant, 151–154, 155–156f Load-extensometer displacement curve, 42f Load stress-strain curve, 81–82 Localization band, 329–330 Localized damage mechanism, 69–71, 99–100 Local load-sharing (LLS) model, 358, 367–370 Long-term elastic viscoelastic interface, 195 Low-velocity impact problems, of composites, 219

M

Macroscopic progressive failure analysis, 49–50 Mass scaling method, 200 Master-slave node technique, 10, 113–115 Material parameters and finite element analysis, 363–364 of 6061 Al liner and carbon fiber/epoxy composites, 373t MATLAB software, 204 Matrix compression, 226 Matrix compression damage, 235–240, 236–239f, 285, 286f Matrix cracking nodes, 277–278

381

Matrix tension, 225 Matrix tension damage, 235–240, 236–239f, 248, 285, 286f, 294 Mechanical tests, 323, 324t Mesh sensitivity, 50–51, 310–312 Mesh size effect, solution for, 8–9 Microcantilever test method, 81–82 Micromechanical damage modeling, 69–71 Micromechanical failure mechanisms, 357 Microscopic damage mechanics, 305–306 Middle-plane numerical interpolation technique, 8, 33–34, 33f, 103–104, 361–362 Mixed-mode bending fracture (MMB), 2–3, 11 damage evolution, 20f delamination case, 17–19, 17f load-displacement curves for, 18f material and size parameters, 17t Mixed-mode cohesive law, 175–176 Mixed-mode delamination crack propagation, 120–124 Mixed-mode delamination mechanism, 20–21 Mixed-mode delamination model, based on continuum damage mechanics, 6–8 Mixed-mode traction-displacement jump relationship, 75–76 Mode-II shear delamination fracture toughness, of composites, 170–171 Modified beam theory, 151, 171 Mohr-Coulomb failure theory, 220 Mohr-Coulomb frictional contact law, 149–150, 222 Mohr-Coulomb plasticity model, 149–150 Monte Carlo simulation, 359, 371–373 Mori-Tanaka homogenization method, 362–363

N

Newmark time integration algorithm, 230 Newton’s method, 351–352 Newton-Cotes integration, 11 Newton-Raphson algorithm, 11, 15–17 Nonlinear global stiffness equation, 130 Nonlinear stiffness equation, 27–29 Nonlocal continuum damage model, 71–72 Nonlocal finite element model, 71–72, 99–100

382 Nonlocal fracture model, 71–72 Nonlocal gradient model, 330–331 Nonlocal implicit FEA, 99–100 Nonlocal implicit stiffness equation, 72, 77–78 Nonlocal integral model, 52–53, 76–77, 330–331 Nonlocal integral theory, 50–51, 71–72, 76–77, 99–100, 330–331 Nonlocal intralaminar damage model, 77–79, 99–100 Nonlocal progressive failure model, 72–78 Nonpositive definite stiffness matrix, 27–29 Normal cohesive strength, 35 Notched composite laminates, 69–71 Numerical technique buckling and delamination analysis, 113–115 composite laminates under low-velocity impact, 260–262, 269–271, 282–285

O

One-step method, for energy release rate, 10 Open-hole composite laminates, 97–99 Over-notched-flexure (ONF), 145–146

P

Parameter sensitivity analysis, 46 Peel tests, 191 Penalty contact algorithm, 259 Penalty method, 148 Periodic boundary conditions, 309, 345, 363–364 Phantom node method, 339–340 Piola-Kirchoff stress tensor, 130 Plastic composite laminates, 253 Plastic damage model, 260–262 Plastic flow rule, 255–256 Plastic parameters, for PC resin, 309t Plastic yielding function, 307 Polycarbonate (PC), 306 Polymethylmethacrylate (PMMA), 306 Polynomial cohesive models, 2–3 Power law, 17–19 Power-law function, 255 Progressive failure analysis (PFA), 69–71 Prony series, 193, 196 in relaxation moduli, inverse parameter identification of, 179–185, 180t, 180–186f

Index

Puck failure criteria, 220, 224–228, 227f, 249, 254, 278–279, 323–326, 324–325f Python script language, 204

Q

Quasi-brittle carbon fiber/epoxy composites, 64–65 Quasistatic tensile experiment, on glass fiber/PC composite specimen, 311f

R

Rate-dependent failure analysis, of adhesive joints, 186–188, 186–188f Recursive algorithm, 178, 197, 213–215 Representative volume element (RVE), 305–306, 358–359 damage dissipation energy, 310–312, 312f failure criteria and damage models of, 360–362 finite element model of, 359, 360f of composites, 326 tensile stress-strain curves for, 313f three mesh models for, 310f, 310t total strain energy-time curves, 310–312, 312f Restart analysis, 363–364 Return mapping algorithm, 169–170 Riemann-Hilbert problem, 191–192

S

Scalar stiffness degradation (SDEG), 313, 317f Schapery’s linear viscoelastic theory, 176–177, 193–194 Separation, 1–2 Shear bands, 329–331 Shear failure mode, 277–278 Shear-lag theory, 357–358 Shear strain, 309–310 Shear stress, 166–167f, 320–322f Single adhesive lap joint, 35–40 Single-edge notched bending (SENB), 341–342, 341f, 343–344f Single lap joint (SLJ), 36f, 38, 180f Single-leg bending (SLB) composite specimens, 200–206, 202f, 203t, 205–206f experiments, 202f Single-mode delamination model, based on continuum damage mechanics, 5–6 Single-mode traction-displacement jump relationship, 29–30, 75–76

Index

Skin/stiffener adhesive composite panel, 40–46 SLJ. See Single lap joint (SLJ) Smeared crack band model, 71–72, 77 Softening interface, 191–192 Status variables, 235–240, 267, 285 Stiffness degradation behavior, 69–71 Strain-based Hashin failure criterion, 73–74 Strain concentration factors, 363 Strain energy, 285–288, 290–291f Strain method, 331–332 Stress boundary value problem, wtih interface discontinuity, 3–4, 3f Stress concentration factor (SCF), 367–370, 368f, 369t Stress recovery length, 365–367 Stress relaxation concept, 193 Stress update algorithm, 351 Strong discontinuity analysis, 331–332 on strain localization for GTN model, 332–337, 335f Strong discontinuity concept, 331–332 Surface energy, 339, 342 Surface potential function, 361–362 Symmetric composite laminates, 110, 116, 120–124 Symmetric mesh model, 80–81

T

2024-T3 aluminum, material parameters for, 279t T700GC/M21 composite, 231–232, 231t Tensile strains, 309–310 Tensile stress, 320–322f Theoretical model, for composites, 254–256 Thermal residual stress, 364–365, 364f Thermodynamic theory, 5, 330–331 Thermoplastic composites, explicit FEA on microscopic RVEs of, 308–326 Thermoplastic resin, Gurson’s type model for, 306–308 3D finite element model, 113–115 Three mesh models, 151, 155t Through-the-width multiple delamination, 110, 124 Time-dependent elastic damage interface, 191–192 Time integration algorithm, 228–230 Time-stepping algorithm, 230 Traction-displacement jump curve, 127–128 Traction-displacement jump law, 1–2 exponential, 4, 4f

383

Trapezoidal cohesive law, 127–128 Triangular cohesive model, 2, 8 T-shape piecewise blade geometry model of, 298–299, 299f mesh model of, 299f Two-step method, for energy release rate, 10

U

UEXTERNALDB utility subroutine, 72, 99–100 Ultimate tensile strength (UTS), 370–371, 371t Ultrasonic c-scan technique, 235–240, 267 Ultrasonic tests, 285 Unidirectional carbon fiber/epoxy composites, 370f Unified finite element model, 77–78 Unstable snap-back delamination, 116 Unsymmetric case-D laminate, 136–140 Unsymmetric composite laminates, 110, 112, 116, 120–124 UTM5000 electronic test machine, 310–312

V

Variable friction coefficient, 148–149 Virtual crack closure technique (VCCT), 1–2, 11–13, 110–112, 111f delamination analysis of composite laminates, 9–10 with crack propagation, 10f Virtual displacement variation, 337–338 Viscoelastic bilinear cohesive model, 177–178 Viscoelastic cohesive/friction coupled model and numerical algorithm, of composite laminates, 197–200, 197–198f, 201f cohesive/frictional contact algorithm, 215–216 Viscoelastic cohesive model and numerical algorithm, of composite laminates, 193–197 recursive algorithm, 213–215 Viscoelastic interface, 195 Viscoplastic Gurson model, 306 Viscous damping technique, 9 Void model, 307 Void parameters, for PC resin, 309t Void volume fraction (VVF), 312–313, 315–316f, 342, 344f

384 W

Weak discontinuity, 52–53, 331–332 Weibull distribution, 359 Weighted nonlocal consistent tangent stiffness, 103 Williams-Landel-Ferry (WLF) form, 196 Wind power, 297–298

Index

Wind turbine blade, 297–298 Wind velocity, 299–301, 300–301f

Z

Zero-thickness cohesive element, 136, 141–142