ProjektWorkAccess

Faculty of Civil Engineering Institute for Structural Analysis

Project Work

FORMULATION ANDIMPLEMENTATION OFISOTROPIC AND ANISOTROPICELASTIC CONSTITUTIVEMODELS IN TERMS OF FEM

Mahmud Al Harun

2

Project Work

Formulation and Implementation of 3D Isotropicand Anisotropic Elastic Constitutive Models for Fi-nite Deformations in Terms of the Finite ElementMethod (FEM)

Submitted by:

Mahmud Al Harun

Supervised by:Prof. Dr.-Ing. Michael Kaliske

and:

Dipl.-Ing. Robert Fleischhauer

Submitted on:07 April, 2015

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4

ABSTRACT

The main purpose of this work is the formulation of an approach to the modellingof isotropic and anisotropic material’s hyperelastic material behaviour at large defor-mation. In order to model the nonlinear mechanical behaviour of anisotropic solidsundergoing large elastic deformation, appropriate constitutive models are required.According to the material composition, different materials show different mechani-cal properties when they are loaded. Some materials, such as anisotropic materi-als, have different composition and strength in three corresponding orthonormal di-rections. The basic focus of this project work is to show numerical simulations ofisotropic and anisotropic material responses on the basis of continuum mechanicsand Finite Element Method (FEM). To describe the nonlinear continuum mechanicshyperelastic material model, named finite hyperelasticity theory [1], is considered.For the isotropic material the material behaviour depends mainly on first three invari-ants of right Cauchy-Green deformation tensor but for transversely isotropic mate-rial the energy density function has two more additional terms. Moreover for purelyanisotropic material, Helmholtz energy function depends on the fourth invariants andmixed fourth invariants, with respect to right or left Cauchy-Green deformation tensor.In particular, the formulation pursued here is based on a model for elastic deforma-tion as a transformation of local reference configuration for each material element.For instance wood is one of the anisotropic material. The stress-strain curve of woodvaries largely according to loading direction, such as tangential, radial and axial di-rections. For axial loading stress-strain curve has elastic and plastic portion whichfollows nonlinear continuum mechanics.

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6

CONTENTS

1 Introduction 9

2 Continuum Mechanics Basis 11

2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Kirchhoff Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Second Piola-Kirchhoff Stress . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Cauchy Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Nominal Stress / First Piola-Kirchhoff Stress . . . . . . . . . . . . . . . 17

2.2.5 Biot Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.6 True Stress and Engineering Stress . . . . . . . . . . . . . . . . . . . . 17

2.3 Balance Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Balance of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Discretization and Shape Function . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Isotropic Hyperelasticity 29

4 Anisotropic Hyperelasticity 35

5 Examples 41

6 Conclusion and Outlook 45

7

8 Contents

1 INTRODUCTION

Since wooden-type materials are denoted by a heterogeneous microstructure andtherefore show different structural behavior when loaded in different directions, asuitable constitutive model is needed for an adequate numerical analysis in terms ofthe FEM. The task of this project work is the implementation of an anisotropic elastic-plastic material model into the existing FEM code. The task is fulfilled by processinga hyperelastic material implementation for the isotropic material. Then the hypere-lastic model is processed to describe anisotropic and orthotropic effects of woodentype materials to be implemented into the FEM code. Stresses and tangent mod-uli of the anisotropic elastic material description are consistently formulated, neces-sary to achieve an efficient FEM implementation and convergence of the solution isachieved stepwise by Newton-Raphson iteration method. Finally, the achieved consti-tutive model implementation is verified with the help of convergence studies.

During quasi-static transverse compression, wood shows three regions [12] [13] [14].At low strain, the deformation is linear and elastic. This region ends in a collapse re-gion of relatively constant stress. The collapse is initiated by elastic or plastic bucklingof cell walls or by fracture of cell walls. At very high strains, the collapsed cell wallscontact other cell walls and the stress increases rapidly during wood densification.Although all wood exhibits these general features, key details of compression prop-erties are dependent on various anatomical features of the wood specimen, such asdensity, moisture content, percentage of late wood material, ray volume, annual ringgrowth etc. and on loading direction with respect to radial and tangential directions.

For the case of tensile loading, the hyperelastic material shows nonlinear behaviour.Hyperelastic material can be used for general tensile loadings and for this purpose,proper numerical simulations are required. Different material models are used fordifferent purposes of hyperelastic material, such as large deformation, large displace-ment.

A hyperelastic or Green elastic material is a type of constitutive model for ideally elas-tic material for which the stress-strain relationship derives from strain energy densityfunction. The hyperelastic material is a special case of a Cauchy elastic material.For the analysis of material behaviour, the work adopted Helmholtz free energy func-tion and from this function, other required material property, such as stress, strain,displacement, residual energy etc. are derived. For many materials, linear elasticmodels do not accurately describe the observed material behavior. The simplest hy-perelastic material model is Saint Venant-Kirchhoff model which is just an extensionof the linear elastic material model to the nonlinear hyperelastic materials. For non-linear problem, Newton type iterations are used. The speed of the solution dependson convergence behaviour.

First of all Saint Venant-Kirchhoff model is considered and according to it, isotropicmaterial equations are considered. And for isotropic material, strain energy densityfunction is considered with respect to invariants and the result is checked for resid-ual norm, energy norm, convergence of iteration and other standard condition forunit deformation gradient. The solution type is based on implicit solution. The load isprescribed by displacement driven method and during each iteration process all prop-erties, i.e. elastic moduli matrix, stiffness matrix, force etc. are updated. By using

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FORTRAN program, equations are written and by using FEAP, the specimen boundaryconditions and applied loads are specified. By using visual studio, the solution is builtand the output results are generated. Similarly, for anisotropic material model elasticlimit behaviour is verified. The stresses, residual energy and other required data aregenerated with respect to standard conditions, such as material equilibrium condition.

10 Chapter 1 Introduction

2 CONTINUUM MECHANICSBASIS

Continuum mechanics provides the basis for modelling the kinematics and mechani-cal behaviour of continuums. Solid materials with a hyperelastic material behaviourare considered. This chapter outlines the principle equations of nonlinear continuummechanics that describe the fundamental geometric mappings, basic stress mea-sures, kinematics, variational principles, balance equations of a solid body undergo-ing finite deformations, shape functions and true stress and engineering stress. Apartfrom the reiteration of the basic relations in continuum mechanics, it is also aimed tointroduce the notations used in the next chapters.

2.1 KINEMATICS

Consider a material body, see Fig. 2.1, β consisting of a continuous set of materialspoints P ∈ β in a three-dimensional Euclidean space R3. The surface of β is denoted by∂β.

The reference configuration β0 ∈ R3 - also called material or Lagrangian configuration- is defined by the position X of the material points P at time t = t0, i.e.

X =XJEJ ; J = 1, 2, 3.

The actual configuration βt ∈ R3 - also called spatial or Eulerian configuration- is de-fined by the position x of the material points P ∈ βt at time t, i.e.

x = xjej ; j = 1, 2, 3 (2.1)

with the cartesian basis ej , j = 1, 2, 3 of the actual configuration. The motion of thebody β is described by the one-to-one mapping of the material points P of β0 into βt.

ϕ : β0 7→ βt. (2.2)

The transformation ϕ maps, at fixed time t ∈ R+, points X ∈ β0 of the reference con-figuration onto points x ∈ βt of the actual configuration, i.e.

ϕ(X, t) : X 7→ x = ϕ(X, t). (2.3)

with xj=xj(X1, X2, X3, t).

11

dA

da

dV

dv

β0

E3 , e3

E2 , e2

E1 , e1

dX

X dx

x

x = (X; t)

F

cof[F]

det[F]

βt

Figure 2.1: Representation of reference and actual configuration and correspondinggeometric mappings (transport theorems).

which is assumed to be reversible

ϕ−1(x, t) : x 7→ X = ϕ−1(x, t) (2.4)

with Xj=Xj(x1, x2, x3, t). As one of the most important kinematic quantities in contin-uummechanics, the deformation gradient is introduced, which is the partial derivativeof the deformation mapping x=ϕ(X,t) with respect to the material coordinates X, i.e.

F = F(X, t) := Gradx = ∇xx =∂x

∂X. (2.5)

The deformation gradient is a two-point transformation tensor; one base vector isdefined with respect to Eulerian configuration and the other is defined with respect tothe Lagrangian configuration

F(X, t) = F ijei ⊗ Ej . (2.6)

The arrangement of the deformation gradient components is denoted by

F ij =

∂xi

∂Xj= xi,j

12 Chapter 2 Continuum Mechanics Basis

⇒

F ij

=

∂x1

∂X1

∂x1

∂X2

∂x1

∂X3

∂x2

∂X1

∂x2

∂X2

∂x2

∂X3

∂x3

∂X1

∂x3

∂X2

∂x3

∂X3

. (2.7)

Since the mapping, in Eq. (2.5), must be one-to-one, the deformation gradient is notallowed to be singular, thus, its inverse

F−1(X, t) = gradX = ∇xX =∂X

∂x= F

−1ji

Ej ⊗ ei (2.8)

must exist. Furthermore, the associated components matrix representation appearsas

⇒

F−1j

i=∂Xj

∂xi= Xj ,i

⇒

F−1j

i=

∂X1

∂x1∂X1

∂x2

∂X1

∂x3

∂X2

∂x1

∂X2

∂x2

∂X2

∂x3

∂X3

∂x1

∂X3

∂x2

∂X3

∂x3

. (2.9)

A sufficient condition for the existence of the inverse of F(X,t) is that the determinantof F(X,t), the Jacobian J , is not equal to zero, i.e. J = detF(X, t) 6=0. Together with thecontinuity of the mapping ϕ the condition is enforced

J = detF(X, t) > 0 (2.10)

for all notations of the body β. Otherwise the body could interpenetrate itself, i.e. itcould undergo deformations that are unphysical. On the basis of deformation gradientthree fundamental geometric mappings - also called transport theorem - are importantin continuum mechanics. A graphical representation of the transformation theoremsis given in Fig. 2.1. Introducing the displacement vector u(X,t) as difference betweenthe position vector of the actual and reference configuration

u(X, t) = ϕ(X, t)−X (2.11)

leads to the following alternative representation of the deformation gradient as

F = Grad[X+ u(X, t)] = 1+Gradu (2.12)

where 1 denotes the second order identity tensor.

Further strain measures can be defined by using the polar decomposition of the localdeformation gradient

F = RU = VR (2.13)

2.1 Kinematics 13

where the rotation tensor R is a proper orthogonal tensor (with R−1 = RT ), whichcauses a rotation of the material elements. The symmetric tensors U and V arethe so-called right stretch and left stretch tensor of deformation, respectively. Then,further strain measures can be obtained by the multiplicative combination of the form

C := FTF = (RU)T (RU) = U2 (2.14)

b := FFT = (VR)(VR)T = V2 (2.15)

with CAB = F aAgabF

bBand bab= F a

AG−1ABF b

B

where C is the right Cauchy-Green tensor and b is the left Cauchy-Green tensor (Fin-ger deformation tensor). Furthermore, suitable measures can be found by using halfof the difference of the squares of the infinitesimal material and spatial line elements,i.e. 1

2 (dx·dx - dX·dX). The definition of the Green-Lagrange strain tensor E and Euler-Almansi strain tensor e, can be expressed,

E :=1

2(C− 1) (2.16)

e :=1

2(1− b−1) (2.17)

with EAB = 12 (CAB − δAB) and eab = 1

2 (δab − b−1ab).

2.2 STRESS MEASURES

Numerous definitions and names of stress tensors have been proposed in the litera-ture. Each definition has advantages and disadvantages. In the following, differentstress tensors, used for practical nonlinear analysis, are discussed. Such stress mea-sures that are widely used in continuum mechanics are:

⋄ The Kirchhoff stress (τ)

⋄ The Nominal stress (N)

⋄ The First Piola-Kirchhoff stress (P = NT )

⋄ The Second Piola-Kirchhoff stress (S)

⋄ The Cauchy stress (σ)

⋄ The Biot stress (T) etc.

Considering the situation shown in Fig. 2.2, in the reference configuration Ω0, theoutward normal to a surface element dΓ0 is N≡n0 and the traction acting on thatsurface is t0 leading to a force vector df0. In the deformed configuration Ω, the surfaceelement changes to dΓ with outward normal n and traction vector t leading to a forcedf . Note that this surface can either be a hypothetical cut inside the body or an actualsurface. The quantity F is the deformation gradient tensor.


n

F-1.df

df

Ω0

d

N =n0

d

df0

Ø

Ω

Figure 2.2: Different stress measures initial and current configuration.

2.2.1 Kirchhoff Stress

Often it is convenient to work with the so-called Kirchhoff stress tensor , τ , which differsfrom the Cauchy stress tensor by the volume ratio J . This stress is used widely innumerical algorithms in metal plasticity (where there is no change in volume duringplastic deformation). It is a contravariant spatial tensor field parametrized by spatialcoordinates and is defined by

τ = Jσ

⇒ τij = Jσij . (2.18)

2.2.2 Second Piola-Kirchhoff Stress

The second Piola-Kirchhoff stress tensor (S) is introduced which does not admit a phys-ical interpretation in terms of surface traction. The contravariant material tensor fieldis symmetric and parametrized by material coordinates. Therefore, it often representsa very useful stress measure in computational mechanics and in the formulation ofconstitutive equations, in particular, for solids.

The second Piola-Kirchhoff stress tensor is obtained by the pull-back operation on thecontravariant spatial tensor field τs by the motion χ, which is expressed as,

S = χ−1∗

(τs) = F−1τF−T (2.19)

2.2 Stress Measures 15

⇒ Sij = F−1imF−1jnτmn. (2.20)

Hence, the Kirchhoff stress tensor is the push-forward of S, i.e.

τ = χ∗(Ss) = FSFT (2.21)

⇒ τij = FimFjnSmn. (2.22)

Using the Eq. (2.18), (2.20) and (2.22), the Piola transformation relating the two stressfields is obtained S and σ, i.e.

S = JF−1σF−T = F−1P = ST

⇒ Sij = JF−1imF−1jnσmn = F−1

imPmj = Sji. (2.23)

2.2.3 Cauchy Stress

The Cauchy stress or true stress (σ) is a measure of the force acting on an element ofarea in the deformed configuration. This tensor is symmetric and defined via theinverse of Eq. (2.23)

σ = J−1FSFT

⇒ σij = J−1FimFjnSmn. (2.24)

Again, from the Fig. 2.2, it can be found,

df = t · dΓ = σT · ndΓ (2.25)

or

t = σT ·n. (2.26)

where t is the traction force and n is the normal to the surface on which the trac-tion acts. The relation in Eq. (2.26) is used for traction boundary conditions. Theprescribed boundary traction must be self-equilibrating.


2.2.4 Nominal Stress / First Piola-Kirchhoff Stress

The nominal stress N=PT is the transpose of the first Piola-Kirchhoff stress P and isdefined via

df = t0 · dΓ0 = NT · n0dΓ0 = P · n0dΓ0 (2.27)

ort0 = NT · n0 = P · n0. (2.28)

This stress is unsymmetric and is a two point tensor like the deformation gradient.This is because it relates the force in the deformed configuration to an oriented areavector in the reference configuration

P = FS

⇒ Pij = FimSmj . (2.29)

2.2.5 Biot Stress

In addition to the above stress tensors, another important quantity prevails. It is amaterial stress tensor formally defined as

TB = RTP

⇒ TBij= RmiPmj . (2.30)

The non-symmetric tensor TB, which is not in general positive definite, is known asthe Biot stress tensor. With Eq. (2.29) and the polar decomposition F = RU and fromEq. (2.30) that TB = RT (FS) = US. Herein, R and U denote the rotation tensor (withdetR = 1) and the (positive definite) symmetric right stretch tensor, respectively. Theyare according to the polar decomposition of the deformation gradient F.

2.2.6 True Stress and Engineering Stress

When a material is loaded, the material shape also changes. Hence, true stress andengineering stress are different by definition. Engineering stress is the ratio of forceand initial area of the specimen. True stress is the ratio of force and true area orcurrent area. True area means the reduced area of the material during uniaxial tensileloading. In each tensile loading step, the area changes its shape. The both stressescan be shown as

2.2 Stress Measures 17

S =F

A0

and

τ =F

A(2.31)

whereas, S is the engineering stress; τ denotes true stress; F is the applied force; A0

represents the initial area of the specimen before loading, while A is the present areaincluding area reduction due to loading.

Figure 2.3: Schematic stress vs. strain diagram describing true stress and engineeringstress for a hyperelastic material.

Fig. 2.3 is obtained by plotting the stress vs. strain, which reflects more accuratelythe true behaviour of a material. The decrease in engineering stress beyond thetensile point occurs because of the definition of engineering stress. The original areaA0 is used in the calculations, but this is not precise because the area continuallychanges. True stress and true strain are often not required. When the yield strengthis exceeded, the material deforms. The component has failed because it has no longerthe original intended shape. A hyperelastic material is different from other materialsdue to they are non-linearly elastic, isotropic and strain-rate independent.

2.3 BALANCE PRINCIPLES

Conservation laws and balance principles for physical quantities constitute the phys-ical basis of continuum mechanics. They are material-independent, i.e. they arevalid for every continuum. In detail, four balance equations and one inequality are


provided: conservation of mass, balance of linear momentum, balance of angular mo-mentum, balance of energy and the entropy inequality, also referred to as Clausius-Duhem-Inequality.

2.3.1 Balance of Momentum

The balance of momentum for a solid body consists of two parts: balance of linearmomentum and balance of angular momentum. The balance of linear momentummay be expressed by integrating the surface and body loads over the body. The bal-ance of the linear momentum states that the rate of change of the linear momentumL is equal to the external resultant force fext, i.e. L = fext, with

L :=

∫

βt

ρvdv (2.32)

and

fext :=

∫

βt

ρfdv +

∫

∂βt

tda (2.33)

where f is the volume acceleration and t the traction vector acting on the surface ∂βt.Here, ρ is the mass density per unit volume and ∂βt is the surface area of the body,both for the current configuration. The mass density in the current configuration isrelated to the reference configuration mass density, ρ0, through

ρ0 = Jρ. (2.34)

Considering Cauchy’s theorem, Gauss divergence theorem yielding

∫

∂βt

tda =

∫

∂βt

σnda =

∫

βt

divσdv. (2.35)

The balance of linear momentum describes the traditional equilibrium of a body (orany part of a body) and is obtained by equating the resultant, fext, to the rate ofchange of the body momentum, L. Accordingly, the Eq. (2.35) and from the conver-sion law of mass, the global form of the spatial balance of linear momentum equationcan be found as,

∫

βt

ρvdv =

∫

βt

(ρf + divσ)dv

⇒∫

βt

[divσ + ρ(f − v)]dv = 0 (2.36)

and the local form of the spatial balance of linear momentum

2.3 Balance Principles 19

ρv = divσ + ρf

⇒ divσ + ρ(f−v) = 0 (2.37)

where for all x ∈ βt. This fundamental statement is also called Cauchy’s first equationof motion.

Similar relations may be constructed for the balance of angular momentum and leadto the requirement

σ = σT (2.38)

that is, the Cauchy stress tensor is symmetric and, thus, has only six independentcomponents.

The balance of momentum may also be written for the reference configuration usingresults deduced above. Accordingly, the integrals may be written with respect to thereference body as

∫

β0

ρ0fdV +

∫

∂β0

t0dS =

∫

β0

ρ0vdV. (2.39)

Considering Gauss divergence theorem, the following equation can be found

∫

βt

divσdv =

∫

∂βt

σnda =

∫

∂βt

σda =

∫

∂β0

PNdA =

∫

β0

DivPdV (2.40)

and the local form of the material balance appears as

DivP+ ρ0(f − v) = 0. (2.41)

for all X ∈ β0.

In these relations Div is the divergence with respect to the reference configurationcoordinates

DivP =∂PaA∂XA

ea. (2.42)

The symmetry of the Cauchy stress tensor, σ, leads to the corresponding requirementon P

FPT = PFT (2.43)


and subsequently to the symmetry of the second Piola-Kirchhoff stress tensor

S = ST . (2.44)

2.3.2 Variational Principles

In general it is mostly impossible to find an analytical solution of the boundary valueproblem. Instead discretization methods based on the direct methods of the calculusof variation can be applied. The discrete methods of the calculus of variation lead hereto the nonlinear, so-called weak form of equilibrium. In view of an iterative solutionof the weak form by using the Newton-Raphson iteration scheme, the linearization ofthe weak form is required. The weak form is the basis for the application of the finiteelement method. Here, the standard displacement formulation is focused on.

Multiplying the strong form of equilibrium by a suitable vector-valued test function δϕ,with δϕ = 0 on ∂β0ϕ, and integrating over the volume of the body β0 yields

∫

β0

(DivP+ ρ0f) · δϕdV = 0 (2.45)

with the boundary condition of the element and exploiting the identity

DivP · δϕ = Div(δϕ.P)−P : Gradδϕ (2.46)

and according to the Gauss divergence theorem, the weak form of equilibrium isobtained as

G(ϕ, δϕ) := Gint −Gext = 0

⇒ G =

∫

β0

P : δFdV −¨∫

β0

ρ0f · δϕdV +

∫

∂β0t

t · δϕdA«

= 0 (2.47)

with δF = Grad[δϕ].

The weak form of equilibrium is formally equivalent to the principle of virtual work.The test function is then the vector of virtual deformations, Gint the work of internalforces and Gext the work of external forces acting on the body β0. Solving the bound-ary value problem is formally equivalent to finding a stationary point ϕ of a function,called the total energy or potential energy Π with,

Π(ϕ) =

∫

β0

ψ(F)dV −∫

β0

〈ρ0f , ϕ〉 dV −∫

∂β0t

〈t, ϕ〉 dA→ stat. (2.48)

2.3 Balance Principles 21

The associated Euler-Lagrange equations (the stationary conditions) are obtained asfollows: The classical methods of the calculus of variations are based on a family ofvariations ϕs, is given in the simple additive form

ϕs = ϕ+ sδϕ (2.49)

and associated deformation gradient

Fs =Gradϕ+sGradδϕ

⇒ δFs = Gradδϕ = δF (2.50)

where δϕ is an arbitrary continuous test function that vanishes at the boundary ∂β0ϕand s ∈ R is a scalar. The potential Π(ϕs) becomes stationary for s = 0, i.e.

Π(ϕ) = Π(ϕs) |s=0→ stat. (2.51)

if

δΠ(ϕ, δϕ) =d

ds|s=0 Π(ϕs) = 0 (2.52)

holds. Inserting the family of variations the Eq. (2.49) into Eq. (2.51) yields

Π(ϕs) =

∫

β0

ψ(Fs)dV −∫

β0

〈ρ0f , ϕs〉 dV −∫

∂β0t

〈t, ϕ〉 dA. (2.53)

The derivatives of the last equation yield

d

dsΠ(ϕs) =

∫

β0

〈∂Fψ(Fs), δFs〉 dV −∫

β0

〈ρ0f , δϕ〉 dV −∫

∂β0t

〈t, ϕ〉 dA (2.54)

so that Eq. (2.52) appears as

δΠ(ϕ, δϕ) =d

ds|s=0 Π(ϕs)

⇒∫

β0

〈P, δF〉 dV −∫

β0

〈ρ0f , δϕ〉 dV −∫

∂β0t

〈t, ϕ〉 dA = 0. (2.55)

With the identity P:δF = Div(Pδϕ) - DivP·δϕ and the Gauss divergence theorem canbe obtained as

δΠ(ϕ, δϕ) = −∫

β0

〈(DivP+ ρ0f) , δϕ〉 dV +

∫

∂β0t

〈(PN−t) , δϕ〉 dA = 0 (2.56)


which must hold for arbitrary test function δϕ with δϕ = 0 on ∂β0ϕ. Using the funda-mental lemma of the calculus of variations Eq. (2.56) is satisfied, if and only if

DivP+ρ0f = 0 (2.57)

for all X ∈ β0 and

PN = t (2.58)

for ∂β0t. Since δϕ can be varied independently in the interior of β0 and on the boundary∂β0t. Alternatively, in terms of the symmetric stress tensor S the weak form is

δΠ(ϕ, δϕ) =

∫

β0

S :1

2δCdV −∫

β0

ρ0f ·δϕdV −∫

∂β0t

t · δϕdA = 0 =: G(ϕ, δϕ) (2.59)

where δE = 12δC and δC = (δF)TF+ FT(δF) have been taken into account.

2.4 DISCRETIZATION AND SHAPE FUNCTION

According to finite element method the material is discretized into finite elements toachieve the solution accurately. It is known as discretization. From the divergencetheorem, the relation between Cauchy stress and virtual displacement (δu) appears

divσ · δu = div(δu · σ)− σ : grad(δu). (2.60)

For the solution of the discretization, an arbitrary continuous test function (δϕ) is intro-duced. During Newton-Raphson iteration step the test function tends to the minimumvalue and when the test function reach to a minimum value the solution is achieved.The relation between Cauchy stress and the test function is expressed by

div(σ · δϕ) = div(σ) · δϕ+ σ : grad(δϕ). (2.61)

The test function vanishes (δϕ = 0) at the boundary condition ∂β0ϕ. Then the aboveequation can be expressed in weak form, such as

∫

β0

div (σ · δϕ) dv =

∫

β0

div (δϕ · σ) dv −∫

∂β0

σ : grad(δϕ)dv = 0. (2.62)

A variational equation or theorem may be solved using the direct method of the cal-culus of variations. In direct method of the calculus of variations the dependent vari-ables are expressed as a set of trial functions multiplying parameters. This reducesa steady state problem to an algebraic process and a transient problem to a set of

2.4 Discretization and Shape Function 23

B (Total body)

B e (Discrete body)

U (Small element)

U e

U e

U e (Boundary node)

U e

Figure 2.4: Discrete element of a plane body.

ordinary differential equations. In the finite element method, the region is dividedinto finite elements and perform the approximations on each element.

The finite element method is a numerical method for finding approximate solutions ofpartial differential equations. The physical domain β0 is approximated by βh0 , whichdenotes geometrical approximation of the considered body divided into n-elementdomains

β0 ≈ βh0 =n⋃

e=1

βe0 (2.63)

and integrals are defined as

∫

β0

(·)dβ ≈∫

βh0

(·)dβ =n∑

e=1

∫

βe0

(·)dβ. (2.64)

The method in here is exemplarily applied to the linearized weak form of equilibriumin the three dimensional reference. The similar construction is performed for theboundaries. With this construction the parts of the variational equation or theoremare evaluated element by element.

Furthermore, the isoparametric concept is taken into account, i.e. the geometry aswell as the displacement field ϕ are approximated by the same ansatz functions.Choosing eight-noded cubical finite elements, see Fig. 2.5, the approximation of thegeometry in the reference configuration is given by

X = X (ξ, η, ζ) =n∑

I=1

NI(ξ, η, ζ)XI (2.65)

with n = 8. The values XI represent the nodal coordinates in the reference configu-ration. The ansatz functions in terms of the natural coordinates of the isoparametricspace Ωe, i.e. ξ ∈ [0, 1], η ∈ [0, 1] and ζ ∈ [0, 1]. NI are the shape function.


The displacement of a discrete element will be defined as the multiplication of nodenumber and displacement on that specific node.

uk = NIuIk. (2.66)

The test function for that specific node becomes

δϕk = NIδϕIk

(2.67)

whereas∂NI

∂Xt= βIt (2.68)

andβIt δϕ

Ik= gradδϕkt. (2.69)

ξ

η

ζ

Ωe

X1

X2

X3

x = (X; t)

J1

2

34

5 6

78

12

34

5 6

78

Be0

Figure 2.5: Illustration of a eight-noded cubic finite element in the parametrized spaceΩe and reference configuration B0

e.

At the boundary condition the test function δϕ = 0 and comparing the Eq. (2.62) andEq. (2.69), it can be obtained

∫

β

div(σ · δϕ)dv =

∫

∂β

δϕ : (σn)da (2.70)

where

t = σn

⇒∫

β

σ : grad (δϕ) dv =

∫

∂β

δϕ: tda

2.4 Discretization and Shape Function 25

⇒ fint − fext = RIk

⇒ δϕIk

∫

βE

σkt : βIt dv

E −∫

∂β

NI .tkda

= 0 = RIk· δϕI

k. (2.71)

This is the resulting equilibrium of a body. For a body in equilibrium the resulting sumof all external applied force equals to the internal resisting force of the body.

2.5 NOTATION

In the next part, following symbols are used.

W (E) = Strain Energy Density Function with respect to E

W (C) = Strain Energy Density Function with respect to C

F = Deformation gradient

E = Green-Lagrange strain

– E = 12 (C− I)

I = Second order identity tensor

C = Right Cauchy-Green deformation tensor

– C = FTgF = 2E+ I

– g is the metric tensor in the current configuration which is equivalent toKronecker delta δij

b= Left Cauchy-Green deformation tensor

– b = FFT = FGFT

– G is the metric tensor in the reference configuration

J = Jacobian scalar value

– J = detF =√detC

P = First Piola-Kirchhoff stress tensor

S = Second Piola-Kirchhoff stress tensor

– S= ∂W (E)∂E

= ∂W (E)∂C

: ∂C∂E

= 2∂W (C)∂C

σ = Cauchy stress tensor

τ = Kirchhoff stress tensor

C = Elasticity moduli fourth order tensor

– C = ∂S∂E

= ∂S∂C

: ∂C∂E

= 4 ∂S∂C


I1 = First invariant

– I1 = trace(C) = trC

I2 = Second invariant

– I2 =12 [(trC)2- tr(C2)]

I3 = Third invariant

– I3 = detC

I4 = Fourth invariant

– I4 = n0 · Cn0; where as n0 refers initial orientation vector on fiber direction

I5 = Fifth invariant

– I5 = n0 · C2n0; whereas n0 refers initial orientation vector on fiber direction

2.5 Notation 27


3 ISOTROPICHYPERELASTICITY

In this chapter, the strain energy density function is restricted by a particular propertythat the material may possess, namely isotropy. This property is based on the physi-cal idea that the response of the material, when studied in a stress-strain experiment,is the same in all directions. A material is isotropic if its mechanical and thermal prop-erties are the same in all directions. Isotropic materials can have a homogeneous ornon-homogeneous microscopic structures. For example, steel is an isotropic materialbut its microstructure is non-homogeneous.

One of the consequences of elastic deformation is the accumulation of potential en-ergy in the deformed body, which is referred to as strain energy in the context ofdeformable solids. The potential energy associated with a deformed configurationonly depends on the final deformed shape, and not on the deformation path overtime that brought the body into its current configuration. The independence of thestrain energy on the prior deformation history is a characteristic property of so-calledhyperelastic materials. This property is closely related with the fact that elastic materialsare conservative: the total work done by the internal elastic forces in a deformationpath depends solely on the initial and final configuration, not the path itself. Differentparts of a deforming body undergo shape changes of different severity. As a conse-quence, the relation between deformation and strain energy is better defined on alocal scale.

Hyperelastic constitutive laws are used to model material that responds elasticallywhen subjected to very large strains. They account large shape changes. A hypere-lastic material has a nonlinear behaviour, which means that its response to the load isnot directly proportional to the deformation. An elastic material is hyperelastic if thereis a scalar function denoted by W = W (E) : Rn×n→ R, called strain energy function orstored energy function. For hyperelastic material, Saint Venant-Kirchhoff model is thesimplest model.

The St.Venant-Kirchhoff is an improvement of linear elasticity model by using rotation-ally invariant Green strain E. The St.Venant-Kirchhoff model is a rotationally invariantmodel; deformations that differ by a rigid body transformation are guaranteed tohave the same strain energy. As a consequence a St.Venant-Kirchhoff material ex-hibits plausible material response in many large deformation scenarios where linearelasticity would not be applicable. Although the St.Venant-Kirchhoff model offers sig-nificant benefits over a linear elastic model, its scope is limited to a certain degreedue its poor resistance to forceful compression: as a St.Venant-Kirchhoff elastic bodyis compressed, starting from its undeformed configuration, it reacts with a restorativeforce which initially grows with the degree of compression. However, once a criticalcompression threshold is reached (58% of undeformed dimensions, when compressionoccurs along a single axis) the strength of the restorative force reaches a maximum.Further compression will be met with decreasing resistance, in fact the restorativeforce will vanish as the object is compressed all the way down to zero volume (anindication of this is that when F = 0 we also have P = 0). Continued compression pastthe point of zero volume (forcing the material to invert) will then create a restorative

29

force that pushes the body towards complete inversion (reflection) along one or moreaxes. The St.Venant-Kirchhoff model possesses well-known limitations, particularlysome instabilities when subjected to pure compression.

According to the Saint Venant Kirchhoff model for isotropic material, the work energydensity function becomes

W (E) =λ

2(trE)2 + µtr(E2) (3.1)

where λ and µ are Lame coefficients of the material. Those coefficients are simplyrelated to the physically meaningful Young modulus E and Poisson coefficient ν asfollows: λ = Eν

(1+ν)(1−2ν)and µ = E

2(1+ν). For different values of ν, Lame coefficients

have the following ranges,

if

0 ≤ ν ≤ 1

2

then first Lame-coefficient becomes,

0 ≤ λ ≤ ∞

and the second Lame-coefficient is in the range of

E

2≥ µ ≥ E

3.

Another parameter which is related to λ and µ is the bulk modulus, K, which is definedby

K = λ+2

3µ =

E

3(1− 2ν). (3.2)

The bulk modulus K tends to infinity as ν approaches to 12 .

The Saint Venant-Kirchhoff model is a classical nonlinear model for compressible hy-perelastic materials often used for metals. Note that this material model is suitable forlarge displacement but it is not recommended to use it for large compressive strains.For large deformations a modified Saint Venant Kirchhoff model is obtained and the formwill be

W (E) =κ

2(lnJ)2 + µtr(E2) (3.3)

where bulk modulus κ > 0 and J = det F. The proposed material model circumventsthe serious drawbacks of the classical St.Venant-Kirchhoff model when used for large

30 Chapter 3 Isotropic Hyperelasticity

compressive strains. Neo-Hookean model is nonlinear behaviour of material goinglarge deformations. This model is based on statistical thermodynamics and usablefor rubber and plastics. For situations where large elastic deformations are involvedthe Neo-Hookean, modified Neo-Hookean or Ogden models should be used. Thosemodels are nonlinear material model. Hyperelastic materials are important to modelusing nonlinear methods and even at small strains error can be noticeable betweenlinear elastic model and nonlinear material models. Nonlinear materials can exhibitnon-symmetric stress responses when loaded in the opposite direction. Hyperelasticmaterial response can be hard to predict without modelling especially under complexloading conditions.

If a scalar-valued tensor function is an invariant under a rotation, it may be expressedin terms of the principal invariants of its arguments (for example, b or C), which isa fundamental result for isotropic scalar functions, known as the representation theorem

for invariants [15] or [16].

The residual stress, in the reference configuration, is zero and the reference config-uration is stress free and it is represented by W = W (I) = W (F) = 0, I is the identitytensor. Having this in mind, the strain energy may be expressed as a set of indepen-dent invariants of the symmetric Cauchy-Green tensors C or b.

W =W [I1(C), I2(C), I3(C)] =W [I1(b), I2(b), I3(b)] (3.4)

where as relation between Green-Lagrange strain and right Cauchy-Green deforma-tion tensor is

E =1

2(C− I) (3.5)

tr(E) =1

2[tr(C)− 3] (3.6)

(tr(E))2 =1

4[(tr(C))2 − 6tr(C) + 9] =

1

4(I21 − 6I1 + 9) (3.7)

tr(E2) =1

4[tr(C2)− 2tr(C) + 3] =

1

4(I21 − 2I1 − 2I2 + 3). (3.8)

Then the strain energy density function for isotropic material will be

W (E)iso =λ

2(trE)2 + µtr(E2)

⇒W (C)iso =λ

8(I21 − 6I1 + 9) +

µ

4(I21 − 2I1 − 2I2 + 3). (3.9)

The derivatives of the first two invariants with respect to the metric tensor (g) in thecurrent configuration. By derivating the following chain rule can be obtained

I1 = trC ⇒ ∂I1

∂g= b (3.10)

31

⇒ ∂I1

∂gij= bij

I2 =1

2[I21 − tr(C2)]

⇒ ∂I2

∂g= I1b− bgb (3.11)

⇒ ∂I2

∂gij= I1bij − bimgmnbnj . (3.12)

Stress measures contain the amount of force per unit of area. In finite deformationproblems, care must be taken to describe the configuration to which stress is mea-sured. The second Piola-Kirchhoff stress, S, is a stress measure with respect to thereference configuration and has components

S = SijEiETj . (3.13)

Consider Eq. (3.1) and general form of second Piola-Kirchhoff stress tensor, S, interms of Green-Lagrangean strain, which characterizes isotropic hyperelastic materialat finite strains, i.e.

S =∂W (E)

∂E=λ

2· 2(trE) · I + 2µET = λ(trE)I+ 2µE (3.14)

⇒ Sij = λ(Emnδmn)δij + 2µEji

.

Expression can be obtained for the Piola-Kirchhoff stress tensors P (which is non-symmetric) and S (which is symmetric). From the chain rule first Piola-Kirchhoff stresstensor P will be

P = FS. (3.15)

The Cauchy stress, σ, and the Kirchhoff stress, τ , are defined with respect to the cur-rent configuration. They are related through the determinant of the deformation gra-dient. The Kirchhoff stress, τ , which is related to the first and second Piola-Kirchhoffstresses, will be

τ = FSFT = PFT. (3.16)

And the Cauchy stress tensor σ becomes


σ =1

JFSFT =

1

JPFT =

τ

J. (3.17)

The quantity C characterizes the gradient of functions S and relates the work conju-gate pairs of stress and strain tensors. It measures the change in stress which resultsfrom a change in strain and is referred to as the elasticity tensor in the material de-scription or the referential tensor of elasticities. If the existence of a scalar valuedenergy function W (hyperelasticity) is assumed, then S may be derived from the firstpartial derivative of energy density function. Hence elastic moduli C (fourth ordersymmetric tensor) will be

Cijkl =∂Sij

∂Ekl= λδmnδij [

1

2δmkδnl +

1

2δmlδnk] + 2µ[

1

2δjkδil +

1

2δjlδik]

= 2µIijkl + λδij [1

2δnkδnl +

1

2δnlδnk]

= 2µIijkl + λ[δijδkl]

= 2µI+ λ[1⊗ 1] (3.18)

where as elasticity tensor can be expressed in Voigt’s notation. In Eq. (3.18) theidentity tensor can be expressed according to Voigt’s notation as,

I =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0.5 0 00 0 0 0 0.5 00 0 0 0 0 0.5

and 1⊗ 1 =

1 1 1 0 0 01 1 1 0 0 01 1 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

For practical application of the theory of elasticity it is convenient to write the tensorequations in matrix notation. The coefficients of first and second tensors can be writ-ten as column and square matrices, respectively. A matrix representation of higherorder tensors is, in general, not possible. But for elasticity tensor one may find somekind of matrix representation if one considers the sub-symmetries

Cijkl = Cjikl = Cijlk = Cjilk

by these symmetric relation, tensors may reduce the number of indices using the fol-lowing scheme:

33

11 → 1, 22 → 2, 33 → 3, 23 → 4, 31 →5, 12 → 6

32 → 4, 13 → 5, 21 → 6

Furthermore, from the main symmetry Cijkl = Cklij , it follows

Cmn = Cnm

So, the elasticity tensor can be expressed by Voigt’s notation as,

[C]ijkl =

C11 C12 C13 C14 C15 C16

C22 C23 C24 C25 C26

C33 C34 C35 C36

C44 C45 C46

sym. C55 C56

C66

The elasticity tensor in the spatial description or the spatial tensor of elasticities,denoted by c, is defined as the push-forward operation of C times a factor J−1 (see[17]), in other texts the definition of c frequently excludes the factor J−1. It is the Piolatransformation of C on each large index so that by applying push-forward operation,the following result is obtained

c = J−1χ(C)

⇒ cijkl=J−1FiIFjJFkKFlLCIJKL. (3.19)


4 ANISOTROPICHYPERELASTICITY

Anisotropy is property of being directionally dependent, as opposed to isotropy, whichimplies identical properties in all directions. It can be defined as a difference, whenmeasured along different axes, in a material’s physical and mechanical properties.Wood is easier to split along its grain than against it. The material strength of woodvaries according to different fiber directions, such as longitudinal, tangential, radialdirection.

For transversely isotropic material the work energy density function can be shown as

W (C) =W (I1, I2, I3, I4, I5). (4.1)

For anisotropic material the strain energy density function depends on fourth invari-ant. The three orthogonal fiber direction is perpendicular to each other, such asright-hand rules. During loading, shear stress develops in the plane and the shearresistance force depends on the surface contact. For anisotropic material, shear forcein all three direction varies according to the fiber orientation. The fibers stay closelyand the deformation of fibers depends on the combining effect of all neighbouringfibers. It means the deformation of longitudinal fiber is resisted by longitudinal fiberand the neighbouring radial and tangential fiber also. It can be mentioned as couplingeffect of the fiber. For anisotropic elastic materials, there is coupling between shearstresses of one direction with shear stresses in neighbouring directions. Hence, forthe work energy density function of the anisotropic material, a coupling term mustbe introduced and the mixed fourth invariant is required to be introduced. And thegeneral form of the energy density function depends on the pure fourth invariant andmixed fourth invariant. It can be represented as,

W (C) =W (I4LL, I4RR, I4TT , I4RL, I4RT , I4LT ). (4.2)

Then the strain energy density function for anisotropic material will be

W (E)aniso = [λ

2(trE)2 + µtr(E2)]aniso. (4.3)

This relation with invariants becomes

W (C)aniso = λLL(I4LL − 1)2 + λRR(I4RR − 1)2 + λTT (I4TT − 1)2

+µRL(I24RL) + µLT (I

24LT ) + µRT (I

24RT )

35

+λRL(I4LLI4RR − 1)2 + λLT (I4LLI4TT − 1)2 + λRT (I4RRI4TT − 1)2. (4.4)

The terms λ and µ are the first and second Lame constants, respectively. The Lameconstants are material properties related to the elastic modulus and Poisson ratio.The second Lame constant is identical to the modulus of rigidity (G). The relationbetween stress, strain and Lame constants can be shown as,

[σ]i = [f(λ, µ)](i,j) · [nk]j . (4.5)

Here, σ represents the normal stresses and the shear stresses. And nk represents theorientation vector in three orthonormal direction.

Normal stresses in the three principal directions are all influenced equally by a com-mon term: the change in volume (λεi) and a unique term influenced strains in thesame direction as the stress and the rigidity of the body (2µεi). Any change in volumenecessarily induces stresses in all three directions. The shear and normal stresses areboth influenced by an identical term, that is related to the rigidity modulus and thecorresponding strain. Essentially, normal and shear stresses have the same relation-ship to strain, except that normal stresses include the effect of changes in volume.This validates the discussion in class about normal stresses leading to changes involume and shear stresses leading only to changes in shape.

Here, right Cauchy-Green deformation tensor C depends on deformation gradient andon metric tensor on the current configuration, g. The right Cauchy-Green tensor canbe expressed as

C = FTgF =⇒ Cij = FmigmnFnj (4.6)

and left Cauchy-Green deformation tensor depends on deformation gradient, F andon metric tensor on the reference configuration, G. So, the left Cauchy-Green tensorappears as

b = FG−1FT = FFT

⇒ bij = FimFjm. (4.7)

Again, I4II = fourth invariant in the three directions of fibers;

In longitudinal direction, I4LL= nL0 · CnL0

⇒ I4LL = nL0i · CijnL0j

whereas nL0 = initial orientation vector in longitudinal direction

– nL= current orientation vector in longitudinal direction = F · nL0

– radial direction, I4RR = nR0 · CnR0

– tangential direction, I4TT = nT0 · CnT0.

36 Chapter 4 Anisotropic Hyperelasticity

Again, due to coupling nL0 , nR0 , nT0 are perpendicular to each other. Hence themixed fourth invariants are required to introduce.

Derivative of the invariants with respect to g becomes

I4LL = (nL0 · CnL0)ij = nL0i · CijnL0j (4.8)

⇒ I4LL = nL0i · ((FT gF )nL0)j = nL0i · FmigmnFnjnL0j (4.9)

⇒ ∂I4LL∂gkl

=∂

∂gkl[nL0i · FmigmnFnjnL0j ]

⇒ ∂I4LL∂gkl

= nL ⊗ nL. (4.10)

Similarly, for radial and tangential direction the derivatives are shown as

∂I4RR

∂g= nR ⊗ nR (4.11)

and∂I4TT

∂g= nT ⊗ nT . (4.12)

For the mixed fourth invariants, the invariants can be represented as

I4RL = (nR0 · CnL0)ij = nR0i · CijnL0j (4.13)

⇒ I4RL = nR0i.((FT gF )nL0)j = nR0i.FmigmnFnjnL0j (4.14)

⇒ ∂I4RL

∂gkl=

∂

∂gkl[nR0i.FmigmnFnjnL0j ]

⇒ ∂I4RL

∂gkl=

1

2(nR ⊗ nL + nL ⊗ nR). (4.15)

Similarly,

∂I4LT∂g

=1

2(nL ⊗ nT + nT ⊗ nL) (4.16)

∂I4RT

∂g=

1

2(nR ⊗ nT + nT ⊗ nR) (4.17)

holds. So, the Kirchhoff stress tensor will be the first derivative of Eq. (4.4)

37

τ = 2∂W (C)

∂g= 4λLL(I4LL − 1)nL ⊗ nL + 4λRR(I4RR − 1)nR ⊗ nR

+4λTT (I4TT − 1)nT ⊗ nT + 2µRLI4RL(nR ⊗ nL + nL ⊗ nR)

+2µLT I4LT (nL ⊗ nT + nT ⊗ nL) + 2µRT I4RT (nR ⊗ nT + nT ⊗ nR)

+4λRL(I4RRI4LL − 1) · (I4RRnL ⊗ nL + I4LLnR ⊗ nR)

+4λLT (I4LLI4TT − 1) · (I4LLnT ⊗ nT + I4TTnL ⊗ nL)

+4λRT (I4RRI4TT − 1) · (I4RRnT ⊗ nT + I4TTnR ⊗ nR). (4.18)

With the indices the Eq. (4.18) becomes

⇒ τij = 4λLL(I4LL − 1)nLinLj + 4λRR(I4RR − 1)nRinRj

+4λTT (I4TT − 1)nT inTj + 2µRLI4RL(nRinLj + nLinRj)

+2µLT I4LT (nLinTj + nT inLj) + 2µRT I4RT (nRinTj + nT inRj)

+4λRL(I4RRI4LL − 1) · (I4RRnLinLj + I4LLnRinRj)

+4λLT (I4LLI4TT − 1) · (I4LLnT inTj + I4TTnLinLj)

+4λRT (I4RRI4TT − 1) · (I4RRnT inTj + I4TTnRinRj). (4.19)

The elastic moduli C will be the second derivative of energy density function or thefirst derivative of the Kirchhoff stress tensor

Cijkl = 2∂τij

∂gkl

⇒ C = 8λLLnL ⊗ nL ⊗ nL ⊗ nL + 8λRRnR ⊗ nR ⊗ nR ⊗ nR

+8λTTnT ⊗ nT ⊗ nT ⊗ nT


+2µRL(nR ⊗ nL + nL ⊗ nR) · (nR ⊗ nL + nL ⊗ nR)

+2µLT (nL ⊗ nT + nT ⊗ nL) · (nL ⊗ nT + nT ⊗ nL)

+2µRT (nR ⊗ nT + nT ⊗ nR) · (nR ⊗ nT + nT ⊗ nR)

+8λRL(I4RRI4LL − 1) · (nL ⊗ nL ⊗ nR ⊗ nR + nR ⊗ nR ⊗ nL ⊗ nL)

+8λRL(I4RRnL ⊗ nL + I4LLnR ⊗ nR) · (I4RRnL ⊗ nL + I4LLnR ⊗ nR)

+8λLT (I4LLI4TT − 1) · (nL ⊗ nL ⊗ nT ⊗ nT + nT ⊗ nT ⊗ nL ⊗ nL)

+8λLT (I4LLnT ⊗ nT + I4TTnL ⊗ nL) · (I4LLnT ⊗ nT + I4TTnL ⊗ nL)

+8λRT (I4RRI4TT − 1) · (nR ⊗ nR ⊗ nT ⊗ nT + nT ⊗ nT ⊗ nR ⊗ nR)

+8λRT (I4RRnT ⊗ nT + I4TTnR ⊗ nR) · (I4RRnT ⊗ nT + I4TTnR ⊗ nR). (4.20)

With indices the above relation can be expressed as

⇒ Cijkl = 8λLLnLinLjnLknLl + 8λRRnRinRjnRknRl + 8λTTnT inTjnTknT l

+2µRL(nR ⊗ nL + nL ⊗ nR)ij · (nR ⊗ nL + nL ⊗ nR)kl

+2µLT (nL ⊗ nT + nT ⊗ nL)ij · (nL ⊗ nT + nT ⊗ nL)kl

+2µRT (nR ⊗ nT + nT ⊗ nR)ij(nR ⊗ nT + nT ⊗ nR)kl

+8λRL(I4RRI4LL − 1)(nLinLjnRknRl + nRinRjnLknLl)

+8λRL(I4RRnL ⊗ nL + I4LLnR ⊗ nR)ij · (I4RRnL ⊗ nL + I4LLnR ⊗ nR)kl

+8λLT (I4LLI4TT − 1)(nLinLjnTknT l + nT inTjnLknLl)

39

+8λLT (I4LLnT ⊗ nT + I4TTnL ⊗ nL)ij · (I4LLnT ⊗ nT + I4TTnL ⊗ nL)kl

+8λRT (I4RRI4TT − 1)(nRinRjnTknT l + nT inTjnRknRl)

+8λRT (I4RRnT ⊗ nT + I4TTnR ⊗ nR)ij · (I4RRnT ⊗ nT + I4TTnR ⊗ nR)kl. (4.21)


5 EXAMPLES

TENSION TEST

A cube sample is considered with dimension 10mm × 10mm × 10mm and tensile loadis applied on the cube according to different fiber directions in different boundaryconditions. The material has the same dimension but due to anisotropic property, thestrength is different in all three direction.

Time = 1.00E+00

12

3

1.49E+03

1.51E+03

1.53E+03

1.55E+03

1.56E+03

1.58E+03

1.60E+03

1.61E+03

1.63E+03

1.65E+03

1.67E+03

1.68E+03

1.48E+03

_________________ S T R E S S 3

Time = 1.00E+00Time = 1.00E+00

Figure 5.1: Tensile loading in a cube in longitudinal direction (at first load step).

The cube is defined as eight noded cubic solid block. The whole cube possesses samematerial property (one material set). Finite displacement is defined and both SaintVenant isotropic and anisotropic analysis are performed. The cube is meshed fiveequivalent parts in each direction. The sum of force and displacement are stored asoutput value. For each Newton iteration step, tangent matrix and residual matrix isformed. Time increment is set as one. Firstly, the third direction is considered as Lon-gitudinal direction and the tensile load is applied. For Longitudinal direction, boundarycondition in third dimension is fixed and the element displacement is allowed to somearbitrary limit by FEAP with the EBoundary and EDisplacement command respectively.For the Radial and Tangential direction similar condition is applied also.

The anisotropic hyperelastic material was loaded in all three directions, such as Lon-gitudinal, Radial and Tangential fiber directions. The material strength is different inall different loading conditions. The strength of the material also depends on density,humidity, porosity, weakness, defects of the block, direction of failure, types of load-

41

Time = 1.00E+00

12

3

1.49E+03

1.51E+03

1.53E+03

1.55E+03

1.56E+03

1.58E+03

1.60E+03

1.61E+03

1.63E+03

1.65E+03

1.67E+03

1.68E+03

1.48E+03

_________________ S T R E S S 3

Time = 1.00E+00Time = 1.00E+00Time = 1.00E+00Time = 2.00E+00

12

3

3.28E+03

3.32E+03

3.36E+03

3.40E+03

3.44E+03

3.48E+03

3.52E+03

3.56E+03

3.60E+03

3.64E+03

3.68E+03

3.72E+03

3.24E+03

_________________ S T R E S S 3


12

3

5.32E+03

5.39E+03

5.46E+03

5.54E+03

5.61E+03

5.69E+03

5.76E+03

5.84E+03

5.91E+03

5.99E+03

6.06E+03

6.14E+03

5.24E+03

_________________ S T R E S S 3


12

3

7.62E+03

7.74E+03

7.86E+03

7.99E+03

8.11E+03

8.23E+03

8.35E+03

8.48E+03

8.60E+03

8.72E+03

8.85E+03

8.97E+03

7.49E+03

_________________ S T R E S S 3


12

3

1.02E+04

1.05E+04

1.07E+04

1.09E+04

1.11E+04

1.14E+04

1.16E+04

1.18E+04

1.20E+04

1.23E+04

1.25E+04

1.27E+04

1.00E+04

_________________ S T R E S S 3


12

3

1.32E+04

1.36E+04

1.40E+04

1.43E+04

1.47E+04

1.51E+04

1.55E+04

1.59E+04

1.62E+04

1.66E+04

1.70E+04

1.74E+04

1.28E+04

_________________ S T R E S S 3


12

3

1.66E+04

1.72E+04

1.78E+04

1.84E+04

1.90E+04

1.96E+04

2.02E+04

2.08E+04

2.14E+04

2.20E+04

2.26E+04

2.32E+04

1.60E+04

_________________ S T R E S S 3


12

3

2.04E+04

2.13E+04

2.22E+04

2.31E+04

2.40E+04

2.50E+04

2.59E+04

2.68E+04

2.77E+04

2.86E+04

2.96E+04

3.05E+04

1.94E+04

_________________ S T R E S S 3


12

3

2.46E+04

2.60E+04

2.73E+04

2.87E+04

3.00E+04

3.14E+04

3.27E+04

3.41E+04

3.54E+04

3.68E+04

3.81E+04

3.95E+04

2.33E+04

_________________ S T R E S S 3


12

3

2.94E+04

3.14E+04

3.33E+04

3.52E+04

3.72E+04

3.91E+04

4.10E+04

4.30E+04

4.49E+04

4.69E+04

4.88E+04

5.07E+04

2.75E+04

_________________ S T R E S S 3

Time = 1.00E+01Time = 1.00E+01

Figure 5.2: Deformed shape of the cube during longitudinal loading in tenth load step.

ing etc. Different standard values are taken for the determination of material strengthand the fiber strength varies according to the fiber directions. The Lame constantsare shown in Table. 5.1.

For example, wood is an anisotropic material. Wood exhibits its highest strengthin tension parallel to the grain. Tensile strength parallel to the grain of small clearspecimens is approximately 2 to 3 times greater than compressive strength parallel tograin, about 1.5 times greater than static bending strength and 10 to 12 times greaterthan shear strength. The mechanical properties of wood are known to be greatlyinfluenced by its anatomical structure. To apply timber as structural components,such as roof trusses, the tension properties of the timber are particularly important.During tensile loading in wood, failure mainly occurs due to shear failure at joint orfasteners.

A graph is plotted Force vs. Displacement (see Fig. 5.3) and the material shows max-imum stiffness in longitudinal fiber loading condition. The main fiber is located inlongitudinal direction, so during applied load it shows maximum resistance to deform.For the tension loading, failure will occur by the fiber tension failure. During radial andtangential loading fiber also resist the distortion of the material. The radial stiffnessof anisotropic material is higher than tangential loading. During loading, some shearstress also generates. Shear stress in a hyperelastic material is resisted by the com-bined effect of all three orthonormal fibres. For the shear resistance, coupling effectprevails. Also the curve is convex type which is the characteristic behaviour of SaintVenant-Kirchhoff material model type.

To solve the anisotropic hyperelastic material, implicit algorithm is used and in eachiteration step, stiffness matrix is updated. The loading type is static and the durationof the load is for some period of time. Hence implicit solution is convenient thanexplicit solution method. For the implicit solution method, all the material propertiesis modified in each iteration step and during next iteration modified property matrixis used. For each load step, when solution is achieved, then iteration finishes and

42 Chapter 5 Examples

0

500000

1e+006

1.5e+006

2e+006

2.5e+006

0 1 2 3 4 5 6 7 8

Forc

e (N

)

Displacement (mm)

Force vs. Displacement graph for anisotropic material

LongitudinalRadial

Tangential

Figure 5.3: Force vs. Displacement graph for anisotropic material in all three fiberdirection.

next load step starts. The time and speed of convergence depends on the solutiontype and the material model, is used. For isotropic hyperelastic material, solutiontime is faster than anisotropic hyperelastic material. By FEAP input file, iteration stepis defined and in each load step, solution is achieved. The solution to a nonlinearproblem is commonly computed using a sequence of linear approximation by usingpopular scheme of Newton’s method.

For the longitudinal fiber direction, the number of iteration (initial and final load step)and energy norm are shown in Fig. 5.4. For the isotropic material, the number ofiteration was four and for anisotropic material iteration step was four also, but thesolution time for anisotropic material is higher. For the isotropic material convergenceis achieved quickly. The duration and iteration step also depend on material model.

From the logarithmic graph, it is observed that the solution is achieved similar to superlinear, which is the characteristic behaviour of Newton’s solution. For the materialresidual energy norm, solution is achieved after certain iteration step. The initialenergy norm in initial loading step is lower than the final energy norm. During theload implementation, residual energy norm is increased in each load step gradually.During each load step, additional load is added as well as residual energy is alsoincreased. During final load step, solution is achieved swiftly and the final energynorm is very low.

43

1.00E-15

1.00E-14

1.00E-13

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

1.00E+07

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

) N ( elac

S .g

oL

mro

N y

gren

E

Number of Iteration to Achieve Solution

Enegy Norm vs. Number of Iteration Logarithmic Graph

Initial Load Step

Final Load Step

Figure 5.4: Energy Norm vs. Number of iteration Logarithmic Graph.

Material Type Lame Constants (MPa) Initial orientation vector

Anisotropic Material

Longitudinal direction (L) λLL = 1200

nL0

=

100

Radial direction (R) λRR = 1000

nR0

=

010

Tangential direction (T) λTT = 1300

nT0

=

001

L and R λRL = 600, µRL = 600 —L and T λLT = 700, µLT = 700 —R and T λRT = 500, µRT = 500 —

Table 5.1: Parameter values that are used for the computation.

44 Chapter 5 Examples

6 CONCLUSION ANDOUTLOOK

A model formulation and numerical implementation is developed for the anisotropicelastic material models at finite displacement and small deformation based on incre-mental minimization principles. Hyperelastic materials are important to model usingnonlinear methods and the response of the hyperelastic material is complex under dif-ferent loading conditions. For the analysis of hyperelastic materials, proper modellingand numerical equations are required to derive. The material, including heteroge-neous micro structure, show different structural behaviour when loaded in differentdirections, a suitable constitutive model is needed for an adequate numerical analysisin terms of the FEM. Anisotropic material has different material property according tothe loading response. A hyperelastic material implementation was the starting pointfor extending stepwise this model upto elasticity. Initially an isotropic hyperelasticmaterial model was considered and for the analysis of the material, Saint Venant-Kirchhoff model was taken into consideration. According to Kirchhoff model, the mate-rial property equations, such as work energy density function, Kirchhoff stress, elasticmoduli were determined. Similarly, the hyperelastic model was processed to describeanisotropic and orthotropic effects of wooden-type materials. For the anisotropic hy-perelastic material, work energy density function was taken related to the fourth in-variants. For the corresponding equations, for example Kirchhoff stress and elasticmoduli equations were developed. Stresses and tangent moduli of the anisotropicelastic material description were formulated. The achieved constitutive model imple-mentation was verified with the help of convergence studies.

According to finite element method, the anisotropic material was divided into finitecubic blocks and then displacement prescribed load was imposed in three differentorthonormal directions. The applied force was resisted according to the elastic mod-uli equations. For the loading behaviour, it was obtained that the longitudinal fiberstrength was highest and the tangential strength was the minimum resisting forceduring deformation. The solution process was controlled by Newton-Raphson itera-tion step and the convergence was achieved in each load step. The tangent moduliwas also checked in each load step and the iteration step was kept as low as possible.For the longitudinal loading of anisotropic material the iteration step was maximumfour steps. For solving the isotropic material required time was less compared toanisotropic material. For different strength properties, the solution time for longitu-dinal, tangential and radial was different. The speed of solution depends on solutiontype, convergence type, material properties and other related properties. The solutioncan be found when the local equilibrium was achieved, which means the differencebetween external force and internal force is closed to zero. During loading, stresses,energy norm, residual norm etc. were checked.

A hyperelastic material implementation was the starting point for extending stepwiseupto elasto-plasticity. During the task, only the elastic part loading condition was con-sidered. So the obtained result curve is only valid for elastic response of hyperelasticmaterial behaviour. To achieve the total material behaviour, which is nonlinear, plas-tic response of the material should be considered too. For implementation of plasticmaterial property, some other additional properties, such as work hardening law and

45

plastic flow rule etc. are needed to consider. When a hyperelastic material is in plasticdeformation, initially the material responses as elastic, but after some time plastic de-formation starts, depending on the time increment and load increment rate. For theapplication of hyperelastic material as tension member, more research is requiredand the full behaviour of the material is required to derive. After achieving the totalsolution hyperelastic material will be implemented in large scale construction.

46 Chapter 6 Conclusion and Outlook

BIBLIOGRAPHY

[1] Holzapfel,G.A. [2000], Nonlinear Solid Mechanics, Austria, (Chapter 4,6).

[2] Vera, E. [2010], Design of Polyconvex Energy Functions for All Anisotropy classes,University of Duisburg-Essen, Germany, (Chapter 2).

[3] GÃuktepe, S. [2007], Micro-Macro Approaches to Rubbery and Glassy Polymers:Predictive Micromechanically-Based Models and Simulations, (Chapter 2).

[4] Sifakis, E.D. [2012], The classical FEM method and discretization methodology,Part 1.

[5] Jakel, R. [2010], Aalysis of Hyperelastic Material with Mechanica, presentation,TU Chemnitz, Germany.

[6] Zienkiewicz, O.C. and Taylor, R.L. [2005], The-Finite-Element-Method-Fundamentals, 6th Edition, (Chapter 1:2).

[7] Nairn, J.A. [2006], Numerical Simulations of Transverse Compression and Densi-fication in Wood, University of Utah, USA.

[8] Svendsen, B. [2000], On the modelling of anisotropic elastic and inelastic be-haviour at large deformations, University of Dortmund, Germany.

[9] Michael Raulli, Dilation and Lame Constants, http://3-11-mechanics-of-materials-fall-1999.

[10] Taylor, R.L. [2002], FEAP-A Finite Element Analysis Program, Version 7.4, (Chap-ter 6,13).

[11] Govindjee, S., Nonlinear Continuum Mechanics, Useful Definitions or Concepts,Presentation WS 06-07, http://www.ce.berkeley.edu/sanjay/contmech/2006/ds1.

[12] Bodig, J. [1963], The Peculiarity of Compression of Conifers in Radial Direction.Forest Products J., 13:438.

[13] Bodig, J. [1965], The effect of Anatomy on the Initial Stress-Strain Relationship inTransverse Compression. Forest Products J., 15:197âAS202.

[14] Bodig, J. [1966], Stress-Strain Relationship for Wood in Transverse Compression.J. Materials, 1:645âAS666.

[15] Truesdell, C., and Noll, W. [1992], The non-linear field theories of mechanics, 2ndedn., Springer-Verlag, Berlin, (Section 10).

[16] Gurtin, M.E. [1981a], An Introduction to Continuum Mechanics, Academic Press,Boston, (p. 231).

[17] Marsden, J.E., and Huges, T.J.R.[1994], Mathematical Foundations of Elasticity,Dover, New York, (Section 3.4).

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