Nonlinear Structural Materials Module

Transcription

1 Nonlinear Structural Materials Module User s Guide VERSION 4.4

2 Nonlinear Structural Materials Module User s Guide COMSOL Protected by U.S. Patents 7,59,58; 7,596,474; 7,623,99; and 8,457,932. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement ( and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see Version: November 203 COMSOL 4.4 Contact Information Visit the Contact COMSOL page at to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at Other useful links include: Support Center: Product Download: Product Updates: COMSOL Community: Events: COMSOL Video Center: Support Knowledge Base: Part number: CM02290

3 Contents Chapter : Introduction About the Nonlinear Structural Materials Module 6 What Can the Nonlinear Structural Materials Module Do? Where Do I Access the Documentation and Model Libraries? Overview of the User s Guide 0 Chapter 2: Nonlinear Structural Materials Theory Hyperelastic Material Theory 2 Working with Hyperelastic Materials Lagrangian Formulation Modeling Large Deformations Modeling Elastic Deformations Thermal Expansion Large Strain Plasticity Theory Isochoric Elastic Deformation Nearly Incompressible Hyperelastic Materials Theory for the Predefined Hyperelastic Material Models Elastoplastic Material Theory 32 Introduction to Small and Large Plastic Strains Plastic Flow for Small Strains Isotropic Plasticity Yield Function Hill Orthotropic Plasticity CONTENTS 3

4 Hardening Models Plastic Flow for Large Strains Numerical Solution of the Elastoplastic Conditions References for Elastoplastic Materials Creep and Viscoplasticity Theory 47 About Creep Fundamental Creep Material Models Solver Settings for Creep and Viscoplasticity Creep Material Models for Metals and Crystalline Solids Viscoplasticity References for Nonlinear Structural Materials 59 Chapter 3: Nonlinear Structural Materials Working with Nonlinear Structural Materials 62 Adding a Material to a Solid Mechanics Interface Hyperelastic Material Linear Elastic Material Creep Viscoplasticity Plasticity C ONTENTS

5 Introduction This guide describes the Nonlinear Structural Materials Module, an optional add-on package for COMSOL Multiphysics designed to assist you to model structural behavior that includes nonlinear materials. The module is an add-on to the Structural Mechanics Module or the MEMS Module and extends it with support for modeling nonlinear materials, including hyperelasticity, creep, plasticity, and viscoplasticity. This chapter introduces you to the capabilities of this module. A summary of the physics interfaces and where you can find documentation and model examples is also included. The last section is a brief overview with links to each chapter in this guide. In this chapter: About the Nonlinear Structural Materials Module Overview of the User s Guide 5

6 About the Nonlinear Structural Materials Module These topics are included in this section: What Can the Nonlinear Structural Materials Module Do? Where Do I Access the Documentation and Model Libraries? The Physics Interfaces and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual What Can the Nonlinear Structural Materials Module Do? The Nonlinear Structural Materials Module is an optional package that extends the Structural Mechanics Module to studies that include structural mechanics with nonlinear material behavior. It is designed for researchers, engineers, developers, teachers, and students that want to simulate nonlinear structural materials, including a full range of possible multiphysics couplings. The module provides an extensive set of nonlinear structural material models, including the following materials: Predefined and user-defined hyperelastic materials: neo-hookean, Mooney-Rivlin, Saint-Venant Kirchhoff, Arruda-Boyce, Ogden, and others. Small-strain and large-strain plasticity models using different hardening models. User-defined plasticity, flow rule and hardening models. Predefined and user-defined creep and viscoplastic material models: Norton, Garofalo, Anand, potential, volumetric, deviatoric, and others. NOTE ABOUT MATERIALS The material property groups (including all associated properties) can be added to models from the Material page. See Materials in the COMSOL Multiphysics Reference Manual. 6 CHAPTER : INTRODUCTION

7 Where Do I Access the Documentation and Model Libraries? A number of Internet resources provide more information about COMSOL, including licensing and technical information. The electronic documentation, topic-based (or context-based) help, and the Model Libraries are all accessed through the COMSOL Desktop. If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different guide. However, if you are using the Help system in COMSOL Multiphysics, these links work to other modules (as long as you have a license), model examples, and documentation sets. THE DOCUMENTATION AND ONLINE HELP The COMSOL Multiphysics Reference Manual describes all core physics interfaces and functionality included with the COMSOL Multiphysics license. This book also has instructions about how to use COMSOL and how to access the electronic Documentation and Help content. Opening Topic-Based Help The Help window is useful as it is connected to many of the features on the GUI. To learn more about a node in the Model Builder, or a window on the Desktop, click to highlight a node or window, then press F to open the Help window, which then displays information about that feature (or click a node in the Model Builder followed by the Help button ( ). This is called topic-based (or context) help. To open the Help window: In the Model Builder, click a node or window and then press F. On any toolbar (for example, Home or Geometry), hover the mouse over a button (for example, Browse Materials or Build All) and then press F. From the File menu, click Help ( ). In the upper-right part of the COMSOL Desktop, click the ( ) button. ABOUT THE NONLINEAR STRUCTURAL MATERIALS MODULE 7

8 To open the Help window: In the Model Builder, click a node or window and then press F. On the main toolbar, click the Help ( ) button. From the main menu, select Help>Help. Opening the Documentation Window To open the Documentation window: Press Ctrl+F. From the File menu select Help>Documentation ( ). To open the Documentation window: Press Ctrl+F. On the main toolbar, click the Documentation ( ) button. From the main menu, select Help>Documentation. THE MODEL LIBRARIES WINDOW Each model includes documentation that has the theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. You can use the step-by-step instructions and the actual models as a template for your own modeling and applications. In most models, SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available. Once the Model Libraries window is opened, you can search by model name or browse under a module folder name. Click to highlight any model of interest and a summary of the model and its properties is displayed, including options to open the model or a PDF document. The Model Libraries Window in the COMSOL Multiphysics Reference Manual. 8 CHAPTER : INTRODUCTION

9 Opening the Model Libraries Window To open the Model Libraries window ( ): From the Home ribbon, click ( ) Model Libraries. From the File menu select Model Libraries. To include the latest versions of model examples, from the File>Help menu, select ( ) Update COMSOL Model Library. On the main toolbar, click the Model Libraries button. From the main menu, select Windows>Model Libraries. To include the latest versions of model examples, from the Help menu select ( ) Update COMSOL Model Library. CONTACTING COMSOL BY For general product information, contact COMSOL at info@comsol.com. To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to support@comsol.com. An automatic notification and case number is sent to you by . COMSOL WEBSITES COMSOL website Contact COMSOL Support Center Product Download Product Updates COMSOL Community Events COMSOL Video Gallery Support Knowledge Base ABOUT THE NONLINEAR STRUCTURAL MATERIALS MODULE 9

10 Overview of the User s Guide The Nonlinear Structural Materials Module User s Guide gets you started with modeling of nonlinear structural materials using COMSOL Multiphysics. The information in this guide is specific to this module. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual. As detailed in the section Where Do I Access the Documentation and Model Libraries? this information can also be searched from the COMSOL Multiphysics software Help menu. TABLE OF CONTENTS AND INDEX To help you navigate through this guide, see the Contents and Index. NONLINEAR MATERIALS Nonlinear Structural Materials chapter describes the features available with this add-on module Hyperelastic Material, Linear Elastic Material, Creep, Viscoplasticity, and Plasticity. These features are described in this guide. For all other features, see the Solid Mechanics interface either in the Structural Mechanics Module User s Guide or in the MEMS Module User s Guide for details. Or search online. NONLINEAR MATERIALS THEORY Nonlinear Structural Materials Theory chapter describes the Hyperelastic Material Theory, Elastoplastic Material Theory, and Creep and Viscoplasticity Theory features available with this module. 0 CHAPTER : INTRODUCTION

11 2 Nonlinear Structural Materials Theory The Nonlinear Structural Materials Module contains materials which are used in combination with the Solid Mechanics interface and related multiphysics interfaces such as Thermal Stress in combination with either the Structural Mechanics Module or the MEMS Module. In this chapter: Hyperelastic Material Theory Elastoplastic Material Theory Creep and Viscoplasticity Theory References for Nonlinear Structural Materials

12 Hyperelastic Material Theory In this section: Working with Hyperelastic Materials Lagrangian Formulation Modeling Large Deformations Modeling Elastic Deformations Thermal Expansion Large Strain Plasticity Theory Isochoric Elastic Deformation Nearly Incompressible Hyperelastic Materials Theory for the Predefined Hyperelastic Material Models Hyperelastic Material If you have the Structural Mechanics Module, see Theory for the Solid Mechanics Interface in the Structural Mechanics Module User s Guide. If you have the MEMS Module, see Theory for the Solid Mechanics Interface in the MEMS Module User s Guide. The links to the nodes described in the Structural Mechanics Module User s Guide and MEMS Module User s Guide do not work in the PDF, only from the on line help. Working with Hyperelastic Materials A hyperelastic material is defined by its elastic strain energy density W s, which is a function of the elastic strain state. It is often referred to as the energy density. The hyperelastic formulation normally gives a nonlinear relation between stress and strain, as opposed to Hooke s law in linear elasticity. Most of the time, the right Cauchy-Green deformation tensor C is used to describe the current state of strain (although one could use the left Cauchy-Green tensor B, the 2 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

13 deformation gradient tensor F, and so forth), so the strain energy density is written as W s (C). For isotropic hyperelastic materials, any state of strain can be described in terms of three independent variables common choices are the invariants of the right Cauchy-Green tensor C, the invariants of the Green-Lagrange strain tensor, or the principal stretches. If you have the Structural Mechanics Module, see Geometric Nonlinearity Theory for the Solid Mechanics Interface in the Structural Mechanics Module User s Guide. If you have the MEMS Module, see Geometric Nonlinearity Theory for the Solid Mechanics Interface in the MEMS Module User s Guide. Lagrangian Formulation The formulation used for structural analysis in COMSOL Multiphysics is totally Lagrangian, for both small and large deformations. This means that the computed stress and deformation state is always referred to the material configuration (material frame), rather than to current position in space (spatial frame). Likewise, material properties are always given for material particles and with tensor components referring to a local coordinate system based on the material frame. This has the advantage that spatially varying material properties can be evaluated just once for the initial material configuration and do not change as the solid deforms and rotates. The gradient of the displacement vector u, which is mentioned frequently in the theory, is always computed with respect to material coordinates. In 3D Cartesian coordinates, this means u u X v X u Y v Y u Z v Z w w w X Y Z HYPERELASTIC MATERIAL THEORY 3

14 The displacement is considered as a function of the material coordinates (X, Y, Z), but it is not explicitly a function of the spatial coordinates (x, y, z). It is thus only possible to compute derivatives with respect to the material coordinates. Modeling Large Deformations Consider a certain physical particle, initially located at the coordinate X. During deformation, this particle follows a path x xx t Here, x is the spatial coordinate and X is the material coordinate. For simplicity, assume that undeformed and deformed positions are measured in the same coordinate system. Using the displacement u it is then possible to write x X+ ux t The deformation gradient tensor F shows how an infinitesimal line element, dx, is mapped to the corresponding deformed line element dx by dx x dx X FdX The deformation gradient F contains the complete information about the local straining and rotation of the material. It is a two-point tensor (or a double vector), which transforms as a vector with respect to each of its indices. It involves both the reference and present configurations. In terms of the displacement gradient, F can be written as F x u+ I X The deformation of the material (stretching) causes changes in the material density. The ratio between current and initial volume (or mass density) is given by dv dv detf J Here, 0 is the initial density and is the current density after deformation. The determinant of the deformation gradient tensor F is related to volumetric changes with 4 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

15 respect to the initial state. A pure rigid body rotation implies J, also, an incompressible material is represented by J. These are called isochoric processes. The determinant of the deformation gradient tensor is always positive (a negative mass density is unphysical). The relation 0 /J implies that for J there is compression, and for J there is elongation. Note that J0, so F is invertible. The variable solid.rho represents a reference or initial density 0, and not the current density. Since the deformation tensor F is a two-point tensor, it combines both spatial and material frames. It is not symmetric. Applying a singular value decomposition on the deformation gradient tensor gives an insight into how much stretch and rotation a unit volume of material has suffered. The right polar decomposition is defined as F RU where R is a proper orthogonal tensor ( detr, and R R T ), and U is the right stretch tensor given in the material frame. The internal variables for the deformation gradient tensor are named solid.fdxx, solid.fdxy, and so on; for the rotation tensor they are named solid.rotxx, solid.rotxy, and so on; and for the right stretch tensor solid.ustchxx, solid.ustchxy, and so on. An upper case index refers to the material frame, and a lower case index refers to the spatial frame. The right Cauchy-Green deformation tensor C defined by C F T F U 2 is a symmetric and positive definite tensor, which accounts for the strain but not for the rotation. Also, the Green-Lagrange strain tensor is a symmetric tensor -- C I 2 HYPERELASTIC MATERIAL THEORY 5

16 Some authors prefer to use the left Cauchy-Green deformation tensor BFF T, which is also symmetric and positive definite but it is defined in the spatial frame. The internal variables for the right Cauchy-Green deformation tensor in local coordinate system are named solid.cl, solid.cl2, and so on; and for the Green-Lagrange strain tensor in local coordinates solid.el, solid.el2, and so on. Modeling Elastic Deformations The elastic deformation tensor is the basis for all strain energy formulations in hyperelastic materials and it is derived by removing the inelastic deformation from the total deformation tensor. Normally, the total deformation is multiplicatively decoupled into elastic and inelastic deformations F F el F in The elastic deformation tensor is computed as F el FF in (2-) so the inelastic deformations are removed from the deformation gradient tensor. The elastic right Cauchy-Green deformation tensor is then computed from the elastic deformation gradient C el T F elfel and the elastic Green-Lagrange strain tensor is computed as: el -- C 2 el I 6 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

17 The inelastic deformation tensor F in is derived from inelastic processes, such as thermal expansion or plasticity. The internal variables for the elastic right Cauchy-Green deformation tensor in the local coordinate system are named solid.cel, solid.cel2, and so on; and for the elastic Green-Lagrange tensor in local coordinates solid.eel, solid.eel2, and so on. Furthermore, the elastic, inelastic, and total volume ratio are related as detf detf el detf in or J J el J in The internal variables for the elastic, inelastic, and total volume ratio are named solid.jel, solid.ji, and solid.j. Thermal Expansion If thermal expansion is present, a stress-free volume change occurs. This is a pure volumetric change, so the multiplicative decomposition of the deformation tensor in Equation 2- implies J el detf detf th J J th Here, the thermal volume ratio, J th, depends on the thermal stretch th, which for linear thermal expansion in isotropic materials can be written in terms of the isotropic coefficient of thermal expansion, iso, and the absolute change in temperature J th 3 th and th + iso T T ref Here, the term iso TT ref is the thermal strain. The isotropic thermal gradient is therefore a diagonal tensor defined as F th th I. The internal variables for the thermal stretch and the thermal volume ratio are named solid.stchth, and solid.jth. HYPERELASTIC MATERIAL THEORY 7

18 Large Strain Plasticity Theory When large strain plasticity occurs, the plastic deformation gradient in not necessarily a diagonal tensor, as when thermal expansion occurs. The multiplicative decomposition of the deformation gradient tensor in Equation 2- implies F el FF p Here, the plastic deformation tensor F p depends on the plastic flow rule, yield function, and plastic potential. Plastic Flow for Large Strains Isochoric Elastic Deformation It turns out to be convenient for some classes of hyperelastic materials to split the strain energy density into volumetric (also called dilatational) and isochoric (also called distortional or volume-preserving) contributions. The elastic deformation tensor is then multiplicatively decomposed into the volumetric and isochoric components F el F evol F el with F evol as the volumetric elastic deformation (a diagonal tensor) and F el the isochoric elastic deformation gradient. Isochoric deformation means that the volume ratio is kept constant during deformation, so the isochoric elastic deformation is computed by scaling it by the elastic volume ratio. The elastic volume ratio is defined by J el detf el detf evol and the volumetric deformation as F evol J 3 el I By using J el it is possible to define the isochoric-elastic deformation gradient /3 F el J el the isochoric-elastic right Cauchy-Green tensor F el 8 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

19 T 2/3 C el F el Fel J el and the isochoric-elastic Green-Lagrange strain tensor el -- C 2 el I This scaling changes the eigenvalues of the tensor, but not its principal directions, so the original and isochoric tensors remain coaxial to each other. C el Some authors call and the modified tensors. Note that F el C el detf el and detc el The internal variables for the isochoric-elastic Cauchy-Green deformation tensor in local coordinate system are named solid.ciel, solid.ciel2, and so on. The other two invariants normally used together with J el are the first and second invariant of the isochoric-elastic right Cauchy-Green deformation tensor In these equations: I 2 C el I C el and I 2 C el 2/3 I C el tracec el J el I C el 2 2 4/3 -- I 2 Cel tracec el J el I 2 C el Since I 3 C el detc el, the third invariant is never explicitly used. The internal variables for the invariants J el, I C el, and I 2 C el are named solid.jel, solid.iciel, and solid.i2ciel. The invariants of the isochoric (modified) elastic Green-Lagrange strain tensor are related to the invariants of the isochoric-elastic right Cauchy-Green deformation tensor HYPERELASTIC MATERIAL THEORY 9

20 I 2 el I el trace el I 2 el trace el -- I 2 C el 3 -- I 4 2 C el 2I C el + 3 I 3 el det el -- I 8 C el I 2 C el The internal variables for the invariants of the isochoric elastic Green-Lagrange strain tensor are named solid.ieiel, solid.i2eiel, and solid.i3eiel. Nearly Incompressible Hyperelastic Materials As mentioned in Working with Hyperelastic Materials, isotropic hyperelastic materials are described by their elastic strain energy density, W s, written in terms of at most three independent variables. Common choices are the invariants of the elastic right Cauchy-Green tensor, the elastic Green-Lagrange strain tensor, the isochoric variants of these tensors, or the principal elastic stretches. Once the strain energy density is defined, the second Piola-Kirchhoff stress in the local coordinate system is computed as S W s 2 C In the general case, the expression for the energy W s is symbolically evaluated down to the components of C using the invariants definitions prior to the calculations of the components of the second Piola-Kirchhoff stress tensor. The differentiation is performed in components on the local coordinate system. If the Nearly incompressible material check box is selected for the Hyperelastic Material node, the total elastic energy function is presented as: W s W iso + W vol where W iso is the isochoric strain energy density and W vol is the volumetric strain energy density. 20 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

21 The isochoric strain energy density can be entered as an expression involving the following: Components of the isochoric elastic right Cauchy-Green tensor C el in the local coordinate system. Principal invariants of the isochoric elastic right Cauchy-Green tensor C el. Principal invariants of the isochoric elastic Green-Lagrange strain. The volumetric strain energy density, W vol, can only be defined as an expression of the elastic volumetric deformation. The most commonly used form is: el W vol J el -- J 2 el 2 (2-2) where is the initial bulk modulus. From here, the volumetric stress (pressure) is calculated as When the formula in Equation 2-2 is used, it becomes The second Piola-Kirchhoff stress is then given by p p W vol J p p J S p p JC W iso + 2 C (2-3) and the Cauchy stress tensor by J FSFT p p I 2J W isof + F T C THE LOCKING PROBLEM A numerical scheme is said to exhibit locking if the accuracy of the approximation deteriorates as a parameter tends to a limiting value (Ref. ). Finite elements in solid mechanics are said to lock when exhibiting an unphysical response to deformation (Ref. 2). Locking can occur for many different reasons. For linear elastic materials, this typically happens as Poisson s ratio tends to 0.5, or the bulk modulus is much larger than the shear modulus. Numerical errors arise because the shape functions are unable to properly describe the volume preserving deformation. HYPERELASTIC MATERIAL THEORY 2

22 To avoid the locking problem in computations, the mixed formulation replaces p p in Equation 2-3 with a corresponding interpolated pressure help variable p w, which adds extra degrees of freedom to the ones defined by the displacement vector u. The general procedure is the same as when the Nearly incompressible material check box is selected for the Linear Elastic Materials node. Theory for the Predefined Hyperelastic Material Models Different hyperelastic material models are constructed by specifying different elastic strain energy expressions. This module has several predefined material models and also has the option to enter user-defined expressions for the strain energy density. NEO-HOOKEAN The strain energy density for the compressible version of the Neo-Hookean material is written in terms of the elastic volume ratio J el and the first invariant of the elastic right Cauchy-Green deformation tensor I C el (Ref. 0). W s -- I 2 3 logj el + -- logj 2 el 2 Here, and are the Lamé coefficients. Note that the natural logarithm is used. The nearly incompressible version uses the isochoric invariant bulk modulus W s -- I J 2 el 2 I C el and the initial ST VENANT-KIRCHHOFF One of the simplest hyperelastic material models is the St Venant-Kirchhoff material, which is an extension of a linear elastic material into the hyperelastic regime. The elastic strain energy density is written with two parameters (the two Lamé coefficients) and two invariants of the elastic Green-Lagrange strain tensor, I el and I 2 el Here, and are the Lamé parameters. W s I2 2 2I 2 22 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

23 The nearly incompressible version uses the isochoric invariants I el and I 2 el, and the initial bulk modulus is calculated from the Lamé parameters W s I 2 2I J 2 el 2 MOONEY-RIVLIN, TWO PARAMETERS Only a nearly incompressible version is available, and the elastic strain energy density is written in terms of the two isochoric invariants of the elastic right Cauchy-Green deformation tensors I C el and I 2 C el, and the elastic volume ratio J el W s C 0 I 3+ C 0 I J 2 el 2 The material parameters C 0 and C 0 are related to the Lamé parameter C 0 C 0 MOONEY-RIVLIN, FIVE PARAMETERS Rivlin and Saunders (Ref. 2) proposed a phenomenological model for small deformations in rubber-based materials on a polynomial expansion of the first two invariants of the elastic right Cauchy-Green deformation, so the strain energy density is written as an infinite series W s C mn I 3 m I 2 3 m 0 n 0 with C 00. This material model is sometimes also called polynomial hyperelastic material. In the first-order approximation, the material model recovers the Mooney-Rivlin strain energy density while the second-order approximation incorporates second-order terms W s C 0 I 3+ C 0 I 2 3 W s C 0 I 3+ C 0 I 2 3+ C 20 I C 02 I C I 3I 2 3 The nearly incompressible version uses the isochoric invariants of the elastic right Cauchy-Green deformation tensors n HYPERELASTIC MATERIAL THEORY 23

24 I C el and I 2 C el and it adds a contribution due to the elastic volume ratio. The strain energy density is then computed from W s C mn I 3 m I 2 3 n + -- J 2 el 2 m 0 n 0 Here, is the initial bulk modulus. 2 2 MOONEY-RIVLIN, NINE PARAMETERS The Mooney-Rivlin, nine parameters material model is an extension of the polynomial expression to third order terms and the strain energy density is written as 3 3 W s C mn I 3 m I 2 3 n + -- J 2 el 2 m 0 n 0 YEOH Yeoh proposed (Ref. ) a phenomenological model in order to fit experimental data of filled rubbers, where Mooney-Rivlin and Neo-Hookean models were to simple to describe the stiffening effect in the large strain regime. The strain energy was fitted to experimental data by means of three parameters, and the first invariant of the elastic right Cauchy-Green deformation tensors I C el W s c I 3 + c 2 I c 3 I 3 3 Since the shear modulus depends on the deformation, and it is calculated as W s W s 2 + I I 2 2c + 4c 2 I 3+ 6c 3 I 3 2 this imposes a restriction on the coefficients c c 2 c 3, since 0. The nearly incompressible version uses the isochoric invariant of the elastic right Cauchy-Green deformation tensor I C el, and it adds a contribution from the elastic volume ratio W s c I 3 c 2 I 3 2 c 3 I J 2 el 2 24 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

25 OGDEN The Neo-Hookean material model usually fits well to experimental data at moderate strains, but fails to model hyperelastic deformations at high strains. In order to model rubber-like materials at high strains, Ogden adapted (Ref. ) the energy of a Neo-Hookean material to W s N p p p p p el + el2 + p el3 3 Here p and p are material parameters, and el, el2, and el3 are the principal elastic stretches such as J el el el2 el3. The Ogden model is empirical, in the sense that it does not relate the material parameters p and p to physical phenomena. The parameters p and p are obtained by curve-fitting measured data, which can be difficult for N. The most common implementation of Ogden material is with N, so four parameters are needed. The nearly incompressible version uses the isochoric elastic stretches and the initial bulk modulus 3 eli eli J el W s N p p p p p el + el2 + p el J 2 el 2 The isochoric elastic stretches define a volume preserving deformation, since el el2 el3 el el2 el3 J el STORAKERS The Storakers material (Ref. 2 and Ref. 5) is commonly used to model highly compressible foams. The strain energy density is written in a similar fashion as in Ogden material: W s N k k k 2 el k k + k el2 + el3 3 + k k k J el The initial shear and bulk moduli can be computed from the parameters k and k as HYPERELASTIC MATERIAL THEORY 25

26 N k and 2 k k N k k for constantparameters k, the initial bulk modulus becomes, so a stable material requires and. In this case, the Poisson's ratio is given by, which means that for a Poisson s ratio bigger than is needed. VARGA The Varga material model (Ref. ) describes the strain energy in terms of the elastic stretches as W s c el + el2 + el3 3+ c 2 el el2 + el2 el3 + el el3 3 The nearly incompressible version uses the isochoric elastic stretches defined as and the initial bulk modulus 3 eli eli J el W s c el + el2 + el3 3 + c 2 el el2 + el2 el3 + el el J 2 el 2 The simplest Varga model is obtained by setting c and c 2 W s el + el2 + el J 2 el 2 ARRUDA-BOYCE The other hyperelastic materials described are phenomenological models in the sense that they do not relate the different material parameters (normally obtained by curve-fitting experimental data) to physical phenomena. Arruda and Boyce (Ref. 3) derived a material model based on Langevin statistics of polymer chains. The strain energy density is defined by p W s 0 c p I 3 p p 26 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

27 Here, 0 is the initial macroscopic shear modulus, I C el is the first invariant of the elastic right Cauchy-Green deformation tensor, and the coefficients c p are obtained by series expansion of the inverse Langevin function. Arruda and Boyce truncated the series and used only the first five terms as listed in Table 2-: TABLE 2-: FIRST FIVE COEFFICIENTS OF ARRUDA-BOYCE MATERIAL MODEL C C 2 C 3 C 4 C 5 /2 /20 N /050 N 2 9/7000 N 3 59/ N 4 The strain energy density is then defined as p W s 0 c p I 3 p p Other authors (Ref. ) use only the first three coefficients of the series. The number of segments in the polymeric chain is specified by the parameter N so the material model is described by only two parameters, 0 and N. This material model is sometimes also called the eight-chain model since it was derived for N. The nearly incompressible version uses the isochoric invariant bulk modulus 5 5 p W s 0 c p I 3 p + -- J 2 el 2 p I C el and the initial GENT Many hyperelastic material models are difficult to fit to experimental data. Gent material (Ref. 3 and Ref. 4) is a simple phenomenological constitutive model based on only two parameters, and j m, which defines the strain energy density as: W s -- j 2 m I 3 log Here, is the shear modulus and j m is a limiting value for I 3, which takes care of the limiting polymeric chain extensibility of the material. j m HYPERELASTIC MATERIAL THEORY 27

28 Since the strain energy density does not depend on the second invariant I 2, the Gent model is often classified as a generalized Neo-Hookean material. The strain energy density tends to be the one of incompressible Neo-Hookean material as. The nearly incompressible formulation uses the isochoric invariants initial bulk modulus j m I C el and the W s -- j 2 m I 3 log J 2 el 2 j m Gent material is the simplest model of the limiting chain extensibility family. BLATZ-KO The Blatz-Ko material model was developed for foamed elastomers and polyurethane rubbers, and it is valid for compressible isotropic hyperelastic materials (Ref. ). The elastic strain energy density is written with three parameters and the three invariants of the elastic right Cauchy-Green deformation tensor, I C el, I 2 C el, and I 3 C el W s -- 2 I I I Here, is an interpolation parameter bounded to, µ is the shear modulus, and is an expression of Poisson s ratio. When the parameter, the strain energy simplifies to a similar form of the Mooney-Rivlin material model In the special case of, the strain energy reduces to a similar form of the Neo-Hookean model I 2 I 3 W s -- 2 I 3 -- I W s I 3 -- I J 2 2 GAO Gao proposed (Ref. 6) a simple hyperelastic material where the strain energy density is defined by two parameters, a and n, and two invariants of the elastic right Cauchy-Green deformation tensors C el : 28 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

29 W s n ai + n I Here, the invariant I - C el is calculated as: tracec el I I 2 C el I 3 C el Gao proposed that the material is unconditionally stable when the parameters are bounded to n and a, and related these parameters under small strain to the Young s module and Poisson s ratio by: E 3 n n 2 8a and 2n n 2n + Since n and it is bounded to n, this material model is stable for materials with an initial Poisson s ratio in the range of MURNAGHAN The Murnaghan potential is used in nonlinear acoustoelasticity. Most conveniently it is expressed in terms of the three invariants of the elastic Green-Lagrange strain tensor, I el, I 2 el, and I 3 el W s I 2 2 2I l + 2mI3 3 2mI I 2 + ni 3 Here, l, m, and n are the Murnaghan third-order elastic moduli, which can be found experimentally for many commonly encountered materials such as steel and aluminum, and and are the Lamé parameters. USER DEFINED When a material model is user-defined, an expression for the elastic strain energy W s is entered, which can include any expressions involving the following: Components of C el, the elastic right Cauchy-Green deformation tensor in the local material coordinate system. Principal invariants of C el I 2 C el I C el tracec el I 2 Cel tracec el HYPERELASTIC MATERIAL THEORY 29

30 I 3 C el detc el The internal variables for these invariants are named solid.icel, solid.i2cel, and solid.i3cel. Components of the elastic Green-Lagrange strain tensor el in the local coordinate system. Principal elastic stretches el, el2, and el3, which are the square-root of the eigenvalues of the elastic right Cauchy-Green deformation tensor C el. The internal variables for the principal elastic stretches are named solid.stchelp, solid.stchelp2, and solid.stchelp3. Invariants of the elastic Green-Lagrange strain tensor. Since el -- C 2 el I the invariants of el are written in terms of the invariants of C el : I 2 el I el trace el I 2 el trace el -- I 2 C el 3 -- I 4 2 C el 2I C el+ 3 I 3 el det el -- I 8 3 C el I 2 C el + I C el The internal variables for these invariants are named solid.ieel, solid.i2eel, and solid.i3eel. When the Nearly incompressible material check box is selected for the Hyperelastic Material node, the elastic strain energy is decoupled into the volumetric and isochoric components. Enter the volumetric strain energy W vol, which can be an expression involving the elastic volume ratio 30 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

31 J el detf el Also enter the isochoric strain energy, W iso, as an expression involving the invariants of the isochoric elastic right Cauchy-Green tensor I C el and I 2 C el or the invariants of the isochoric elastic Green-Lagrange strain I el, I 2 el, and I 3 el The internal variables for J el, I C el, and I 2 C el are named solid.jel, solid.iciel, and solid.i2ciel. The internal variables for I el, I 2 el, and I 3 el are named solid.ieiel, solid.i2eiel, and solid.i3eiel. The strain energy density must not contain any other expressions involving displacement or their derivatives. This excludes components of the displacement gradient u and deformation gradient FuI tensors, their transpose, inversions, as well as the global material system components of C and. If they occur, such variables are treated as constants during symbolic differentiations. HYPERELASTIC MATERIAL THEORY 3

32 Elastoplastic Material Theory In this section: Introduction to Small and Large Plastic Strains Plastic Flow for Small Strains Isotropic Plasticity Yield Function Hill Orthotropic Plasticity Hardening Models Plastic Flow for Large Strains Numerical Solution of the Elastoplastic Conditions References for Elastoplastic Materials Working with Nonlinear Structural Materials Introduction to Small and Large Plastic Strains There are two implementations of plasticity available in COMSOL Multiphysics. One is based on the additive decomposition of strains, which is the most suitable approach in the case of small strains, and the other one is based on the multiplicative decomposition of the deformation gradient, which is more suitable when large plastic strains occurs. The additive decomposition of strains is used when small plastic strain is selected as the plasticity model. The stress-strain relationship is then written as Here: 0 C: 0 th p c is the Cauchy stress tensor is the total strain tensor 0 and 0 are the initial stress and strain tensors 32 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

33 th is the thermal strain tensor p is the plastic strain tensor c is the creep strain tensor, and C is the fourth-order elasticity tensor. The elastic strain tensor is computed by removing inelastic and initial strains from the total strain tensor el 0 th p c In the special case of zero initial strains and no creep or thermal expansion, the elastic strain tensor simplifies to el p (2-4) If or p are large, the additive decomposition might produce incorrect results; however, the additive decomposition of strains is widely use for metal and soil plasticity. When Small plastic strain is selected as the plasticity model for the Plasticity node, and the Include geometric nonlinearity check box is selected on the study settings window, the second Piola-Kirchhoff stress is used instead of the Cauchy stress, and the additive decomposition of strains is understood as the summation of Green-Lagrange strains. When large plastic strain is selected as the plasticity model, the total deformation gradient tensor is multiplicatively decomposed into elastic deformation gradient and plastic deformation gradient F F el F p The plastic deformation is removed from the total deformation gradient, F el FF p so the elastic Green-Lagrange strain tensor is computed from the elastic deformation gradient tensor F el el -- F 2 el T Fel I ELASTOPLASTIC MATERIAL THEORY 33

34 and the plastic Green-Lagrange strain tensor is computed from the plastic deformation gradient tensor T p -- F 2 p Fp I As opposed to the small strain formulation described in Equation 2-4, the total, plastic and elastic Green-Lagrange strain tensors are related as el T F p F p p Under multiplicative decomposition, the elastic right Cauchy-Green tensor and the plastic right Cauchy-Green tensor are defined as C el T F elfel T and C p F p Fp When Large plastic strain is selected as the plasticity model for the Plasticity node, the Include geometric nonlinearity check box in the study settings is automatically selected and becomes unavailable. Plastic Flow for Small Strains The flow rule defines the relationship between the increment of the plastic strain tensor and the current state of stress, for a yielded material subject to further p loading. When Small plastic strain is selected as the plasticity model for the Plasticity node, the direction of the plastic strain increment is defined by p Q p Here, is a positive multiplier (also called the consistency parameter or plastic multiplier) which depends on the current state of stress and the load history, and Q p is the plastic potential. The dot (for ) means the rate at which the plastic strain tensor p changes with respect to Q p /. It does not represent a true time derivative. Some authors call this formulation rate independent plasticity. 34 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

35 The direction of the plastic strain increment is perpendicular to the surface (in the p space of principal stresses) defined by the plastic potential Q p. The plastic multiplier is determined by the complementarity or Kuhn-Tucker conditions 0, F y 0 and F y 0 (2-5) where F y is the yield function. The yield surface encloses the elastic region defined by F y <0. Plastic flow occurs when F y 0. If the plastic potential and the yield surface coincide with each other Q p F y, the flow rule is called associated, and the rate in Equation 2-6 is solved together with the conditions in Equation 2-5. p F y (2-6) For a non-associated flow rule, the yield function does not coincide with the plastic potential, and together with the conditions in Equation 2-5, the rate in Equation 2-7 is solved for the plastic potential Q p (often, a smoothed version of F y ). p Q p (2-7) The evolution of the plastic strain tensor (with either Equation 2-6 or p Equation 2-7, plus the conditions in Equation 2-5) is implemented at Gauss points in the plastic element elplastic. Isotropic Plasticity For isotropic plasticity, the plastic potential Q p is written in terms of at most three invariants of Cauchy s stress tensor Q p Q p I J 2 J 3 where the invariants of the stress tensor are I J 2 J 3 trace --dev:dev 2 detdev ELASTOPLASTIC MATERIAL THEORY 35

36 so that the increment of the plastic strain tensor p can be decomposed into p Q p Q p I Q J p 2 Q J p I J 2 J 3 The increment in the plastic strain tensor includes in a general case both deviatoric p and volumetric parts. The tensor is symmetric given the following properties p I I J dev J devdev 2 --J 3 2 I (2-8) A common measure of inelastic deformation is the effective plastic strain rate, which is defined as pe 2 -- p: p 3 (2-9) The trace of the incremental plastic strain tensor, which is called the volumetric plastic strain rate, depends only on the reliance of the plastic potential on the first pvol invariant I (), sincej 2 / and J 3 / are deviatoric tensors pvol trace p trace Q p Qp I For metal plasticity under the von Mises and Tresca criteria, the volumetric plastic strain rate is always zero because the plastic potential is independent of the invariant I. This is known as J2 plasticity. Plasticity The effective plastic strain and the volumetric plastic strain are available in the variables solid.epe and solid.epvol. 36 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

37 Yield Function When an associated flow rule is applied, the yield function must be smooth, that is, continuously differentiable with respect to the stress. In COMSOL Multiphysics, the following form is used: F y ys where ys is the yield stress. The predefined form of the effective stress is the von Mises stress, which is commonly used in metal plasticity: mises 3J dev:dev 2 Other expressions can be defined, such as Tresca stress, Hill orthotropic plasticity, or another user-defined expression. The Tresca effective stress is calculated from the difference between the largest and the smallest principal stress tresca 3 A user-defined yield function can by expressed in terms of invariants of the stress tensor such as the pressure (volumetric stress) p -- I 3 the effective (von Mises) stress mises, or other invariants, principal stresses, or stress tensor components. Plasticity Hill Orthotropic Plasticity Hill (Ref. 4, Ref. 5) proposed a quadratic yield function (and associated plastic potential) in a local coordinate system given by the principal axes of orthotropy a i ELASTOPLASTIC MATERIAL THEORY 37

38 Q p F G H L M N 2 (2-0) The six parameters F, G, H, L, M, and N are related to the state of anisotropy. As with isotropic plasticity, the elastic region Q p 0 is bounded by the yield surface Q p 0. Hill demonstrated that this type of anisotropic plasticity is volume preserving, this is, given the associated flow rule p Q p the trace of the plastic strain rate tensor is zero, which follows from the expressions for the diagonal elements of p p p22 p33 Q p G 33 + H 22 Q p F H Q p F G so the plastic volumetric strain rate is zero pvol trace p p + p22 + p33 0 Hill plasticity is an extension to J2 (von Mises) plasticity, in the sense that it is volume preserving. Due to this assumption, six parameters are needed to define orthotropic plasticity, as opposed to orthotropic elasticity, where nine elastic coefficients are needed. EXPRESSIONS FOR THE COEFFICIENTS F, G, H, L, M, N Hill noticed that the parameters L, M, and N are related to the yield stress in shear with respect to the axes of orthotropy a i, thus they are positive parameters L , M , N ys ys3 2 2 ys2 38 CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY

39 Here, ysij represents the yield stress in shear on the plane ij. The material parameters ys, ys2, and ys3 represent the tensile yield stress in the direction, a, a 2, and a 3, and they are related to Hill s parameters F, G, and H as G+ H 2 ys H + F 2 ys F + G 2 ys3 or equally 2F ys2 2 ys3 2 ys 2G ys3 2 ys 2 ys2 2H ys 2 ys2 2 ys3 Note that at most, only one of the three coefficients F, G, and H can be negative. In case of hardening, these coefficients (either Hill s coefficients or the shear and tensile yield stresses) are renamed with the initial prefix. In order to define a yield function and plastic potential suitable for isotropic or kinematic hardening, the average initial yield stress ys0 is calculated from the Hill s parameters F, G, and H (this is equivalent to the initial yield stress ys0 in von Mises plasticity) ys F + G + H ys ys2 ys3 (2-) Defining Hill s effective stress as (Ref. 5) 2 hill 2 ys0 F G H L 23 + M N 2 makes it possible to write the plastic potential in a similar way to von Mises plasticity. Hardening is then applied on the average yield stress variable ys0, by using the plastic potential Here, the average yield stress Q p hill ys ELASTOPLASTIC MATERIAL THEORY 39

40 ys ys0 + h pe now depends on the initial yield stress ys0, the hardening function h, and the effective plastic strain pe. Plasticity Hardening Models The plasticity model implements three different kinds of hardening models for elastoplastic materials: Perfect plasticity (no hardening) Isotropic hardening Kinematic hardening PERFECT (OR IDEAL) PLASTICITY In this case the plasticity algorithm solves either the associated or non-associated flow rule for the plastic potential Q p p Q p with the yield function F y ys0 In the settings for plasticity you specify the effective stress for the yield function from a von Mises stress, a Tresca stress, Hill effective stress, or a user-defined expression. When Large plastic strain is selected as the plasticity model for the Plasticity node, either the associate or non-associated flow rule is applied as written in Equation CHAPTER 2: NONLINEAR STRUCTURAL MATERIALS THEORY