Research Collection

Master Thesis

Numerical Implementation of an Anisotropic Finite-Deformation Plasticity Model

Author(s): Mettler, Linus Ken

Publication Date: 2012

Permanent Link: https://doi.org/10.3929/ethz-a-009774144

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Master Thesis

Numerical Implementation of an Anisotropic Finite-Deformation

Plasticity Model

Linus Ken Mettler

Prof. Dr. Sanjay Govindjee Advisor

Structural Engineering, Mechanics and Materials Department of Civil and Environmental Engineering

UC Berkeley, CA 94720, USA

Prof. Dr. Jürg Dual Supervisor

Institute of Mechanical Systems Department of Mechanical and Process Engineering

ETH Zürich, CH-8092, Switzerland

December 2012

Preface

This report covers the contents of a master thesis conducted in partial satisfaction of the requirements for the degree of Master of Science in Mechanical Engineering, ETH Zürich, Switzerland. This research project has been carried out under the supervision of Prof. Dr. Sanjay Govindjee at the Department of Civil and Environ- mental Engineering, University of California at Berkeley, USA.

Acknowledgements

I wish to thank Professor Sanjay Govindjee (UC Berkeley), my research advisor, for his extraordinary support and for devoting many instructive hours to discuss the topics at hand. I also owe my gratitude to Professor Jürg Dual (ETH Zürich) for arranging this project and making my stay at Berkeley possible, and to Professor Edoardo Mazza (ETH Zürich) who has sparked my sustained interest in the field of continuum mechanics through an array of excellent and inspiring courses at the Center of Mechanics, ETH Zürich. During my stay, I enjoyed the friendly support of Gerd Brandstetter (UC Berkeley) in various matters. Also, I am thankful to Professor Panayiotis Papadopoulos (UC Berkeley) and Professor Jia Lu (U Iowa) for the constructive discussion of their 2001 publication - the main subject of this research.

Zürich, December 2012

Linus Mettler

i

Abstract

This report outlines the numerical implementation of a finite-deformation elastic- plastic constitutive model proposed by Papadopoulos and Lu (2001) for orthotropic materials. The model is based on the rate-type Green-Naghdi plasticity theory, and a logarithmic strain measure is used. A clever choice of tensor basis for constitutive modelling leads to a significantly simplified algorithm: the generally coupled problem of 6 dimensions is reduced to 2 dimensions.

A thorough derivation of the algorithmic procedure is provided. The tensor no- tation is replaced with a 2×2 matrix notation in order to exploit the particular structure of the orthotropic constitutive equations. The Papdopoulos-Lu model is implemented in a finite-element program and its capability and accuracy is sub- sequently demonstrated on an array of numerical examples. Apart from reducing the cost of solving the local incremental elastic-plastic problem, the implementation exhibits favorable convergence properties allowing for much larger time integration steps than other models (e.g. the hypo-elastic Hill model).

Finally, an intrinsically plane-stress orthotropic plasticity algorithm is proposed. It is constitutively based on the model by Papadopoulos and Lu, however, does not make use of their tensor basis. It also leads to 2 coupled equations and therefore comes at similar computational cost as the 3-dimensional problem, while an expen- sive plane stress iteration procedure can be avoided.

Keywords: Anisotropy, Plasticity, Finite deformation, Green-Naghdi theory

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Contents

Preface i

Abstract iii

1 Introduction 1

2 Constitutive Theory of Green and Naghdi 3

2.1 History Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Stored Energy and Yield Function . . . . . . . . . . . . . . . . . . . . 4

2.3 Loading/Unloading Conditions . . . . . . . . . . . . . . . . . . . . . 4

2.4 Plastic Evolution and Consistency Condition . . . . . . . . . . . . . . 5

3 Derivation of Papadopoulos-Lu Model 7

3.1 Volumetric and Deviatoric Parts of Potential . . . . . . . . . . . . . . 7

3.2 Tensor Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Hardening Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 Comparison with Conventional Material Parameters . . . . . . . . . . 13

4 Algorithmic Implementation 19

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Radial Return of Trial Stress . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Algorithmic Evaluation of Consistency . . . . . . . . . . . . . . . . . 24

4.4 Update of History Variables and Stress . . . . . . . . . . . . . . . . . 26

4.5 Consistent Algorithmic Tangent Moduli . . . . . . . . . . . . . . . . . 28

4.6 Cockroft-Lathem Damage Parameter . . . . . . . . . . . . . . . . . . 30

4.7 Summary of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Evaluation of the Numerical Implementation 33

5.1 Elastic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Plastic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Stability and Accuracy Analysis 45

6.1 Nonlinear B-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Finite Step Accuracy: Iso-Error Maps . . . . . . . . . . . . . . . . . . 51

7 Summary and Conclusions 55

References 57

A Excerpt of Code: plas6o.f 59

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vi CONTENTS

B Plane Stress Orthotropic Plasticity Algorithm 65 B.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 B.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 B.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

1 Introduction

Background

Anisotropic materials account for a large number of applications in material sci- ence and structural engineering. The anisotropy arises from orientation-dependent composition, such as the microscopic texture of sheet metals imposed by manufac- turing, or the structured nature of biological tissue and fiber-reinforced composites. Even the assumption that initially isotropic materials, owing their properties to the random orientation of a large number of individual crystal grains, remain isotropic throughout the deformation is only approximate. Due to the elongation and rota- tion of the grains towards a particular orientation depending on the strain path this assumption becomes increasingly inaccurate for large deformations.

In the past decades, numerous anisotropic criteria for initial yield (e.g. Hill) or failure (e.g. Tsai-Wu) have been developed and extensively tested, in particular for composite materials. However, limited attention was given to the characterization of strongly anisotropic elastic-plastic behavior under large plastic deformation which is, for example, of importance in sheet metal forming.

Overview of the Contents

At the center of this thesis is an article by Papadopoulos and Lu [23] which outlines an anisotropic finite deformation plasticity model based on the Green-Naghdi fi- nite plasticity theory. The theory is rate-type, rate-independent: the elastic-plastic stress-strain relationship is established in its time derivative but the constitutive parameters are independent of the strain rate (therefore neglecting viscid effects).

Anisotropy, provided that it does not evolve in the course of plastic deforma- tion1, is incorporated using structural tensors. The constitutive model accounts for anisotropic behavior in the yield condition, flow rule, hardening law as well as the elastic stress-strain relation. It includes isotropic and kinematic hardening. Each material symmetry group imposes a distinct structure on the constitutive equations. This is exploited by choosing a particular tensor basis to construct the constitutive equations, leading to an extraordinarily simple and efficient algorithmic implemen- tation. Although attention in this work is restricted to orthorhombic systems the theory may be similarly applied to other (selected) crystal groups.

For a numerical evaluation the algorithm is subsequently incorporated into the current version of FEAPr [30], a fortran-based finite-element program licensed and distributed by the University of California. To include the new algorithm in FEAP does not only constitute an advancement because of its numerical efficiency; the model is built on a sound thermodynamic basis leading to a very robust algorithm that is in accordance with important principles of physics. A slight drawback to the model is, however, that the material parameters used for the constitutive model

1 This assumption may not always be adequate, see discussion in chapter 7.

1