2004_hiperelasticity_techreport

Technical Report

Hyperelastic Material Modeling

Manuel J. Garcia R., Oscar E. Ruiz S., Carlos Lopez

Research AssitantsLeidy Yarime Suarez Gonzalez

Mario Gomez BoteroMario Betancur

Laboratorio CAD/CAM/CAEDepartamento de Ingeniera Mecanica

Universidad EAFITMedelln

January 20, 2005

Contents

Summary ix

1 Introduction 1

2 Hyperelastic Cloth 52.1 Linear elastic relation . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Linear elastic relation in matrix form . . . . . . . . . . 72.2 Nonlinear elasticity . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Hyperelastic models . . . . . . . . . . . . . . . . . . . 11

2.3 Characterisation of hyperelastic fabrics . . . . . . . . . . . . . 142.3.1 Uniaxial tension test . . . . . . . . . . . . . . . . . . . 152.3.2 Planar shear test . . . . . . . . . . . . . . . . . . . . . 162.3.3 Biaxial tension test . . . . . . . . . . . . . . . . . . . . 19

2.4 Problem description . . . . . . . . . . . . . . . . . . . . . . . . 202.4.1 Uniaxial test tension of the hyperelastic fabric . . . . . 212.4.2 Methodology to determine the constants of the hyper-

elastic material . . . . . . . . . . . . . . . . . . . . . . 222.4.3 Normalized error value . . . . . . . . . . . . . . . . . . 25

2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Springparticle Model for Hyperelastic Cloth 313.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Multi-Particle vs. FEA methods . . . . . . . . . . . . . 313.1.2 Iteration speed and stability . . . . . . . . . . . . . . . 343.1.3 Cloth geodesics and stretch measure . . . . . . . . . . 353.1.4 Constraints and collision avoidance . . . . . . . . . . . 35

iii

iv Contents

3.1.5 Conclusions for the literature review of cloth simulation 383.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . 383.3 The non-linear pure-tension spring elastic Model . . . . . . . . 39

3.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 Node movement . . . . . . . . . . . . . . . . . . . . . . 423.3.3 Convergence criterion e . . . . . . . . . . . . . . . . . . 46

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.1 Stresses over the woven fabric . . . . . . . . . . . . . . 473.4.2 Deformation energy . . . . . . . . . . . . . . . . . . . . 48

4 Thermoforming Process 514.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4 Fundamentals of the thermoforming process . . . . . . . . . . 534.5 Thermal model . . . . . . . . . . . . . . . . . . . . . . . . . . 544.6 Finite element analysis . . . . . . . . . . . . . . . . . . . . . . 55

4.6.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . 554.6.2 Meshing the model . . . . . . . . . . . . . . . . . . . . 564.6.3 Loading and boundary conditions . . . . . . . . . . . . 564.6.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 584.6.5 Postprocessing . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Original mold . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.8 Proposed mold and results . . . . . . . . . . . . . . . . . . . . 58

5 Thermoforming Contact Analysis 615.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Procedures in a CAD modelling program . . . . . . . . . . . . 625.3 Geometric procedure . . . . . . . . . . . . . . . . . . . . . . . 635.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Conclusions 696.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography 75

Notation

: open set in Rn, : boundary of .N : matrix of dimension nm of nodesF : vector which stores the spring forcesT : total number of springs within the grid() : stress-strain non-linear function which represents the material

properties~Sk : force that every node of the grid producesAk : associated area to spring kNjn : node where a set of springs are joined, see Figure 3.3Nop : any of the nodes opposed to Njn, see Figure 3.3t : number of iterations of the algorithm~R : net force with which every node experimentsp : step sized : deformation of a springlend : final length of a springlin : initial length of a springc : constant defined by the user to compute the step size valuen : normal vector to the surface of the rigid object at the contact

point~R : component of the vector R over the tangent plane to the

surface in the contact pointe : stop criterionV : vector which stores the resultant forcesk : stress of spring ki,j : stress of node i, jEk : energy of spring kE : energy of the whole system

v

Glossary

A : AmpereASTM : American Society for Testing and MaterialsCAD : Computer Aided DesignCNC : Computer Numerical ControlCp : Specific heatFEA : Finite Element AnalysisFEM : Finite Element MethodHyperelastic : Large strain elastic behaviour modeled using a strain energy

potentialISO : International Standards OrganizationK : ConductivityShear : = F/A Intensity of internal forces acting parallel or tangent

to a plane of cutStrain : Unit change due to force in size of body relative to its orig-

inal sizeStress : = F/A Force exerted per unit area at a point within a planeTension : Mode of application of a tensile force normal to the plane of a

joint between rigid adherents and uniformly distributed overthe whole area of the bond line

vii

Summary

This technical report corresponds to the research developed during 2004 bythe CAD/CAM/CAE Laboratory that was sponsored by the University ofEAFIT and Leonisa S.A. This work concentrated on the modeling of hyper-elastic fabrics and its manufacturing process.

The work presents a review of some problems associated with the simu-lation of hyperelastic fabrics using Finite Element Analysis. The precisionand accuracy of experimentally measured material properties are critical forcorrect fabric modeling. In general, a combination of uniaxial tension (com-pression), biaxial tension, and simple shear is required for the characterisa-tion of an hyperelastic material. However, the use of these deformation teststo obtain the mechanical properties of a fabric may be complicated and alsoexpensive. A methodology for characterising the fabric employing one singleexperimental test is presented and the error induced by this approximationis determined. The effects of experimental limitations on the characterisa-tion of material and the pros and cons of choosing a constitutive model arediscussed. The experimental data was obtained by a uniaxial tension testand the material constants were determined by the least squares method. Acomparison of hyperelastic material models is illustrated through an exampleof a rigid body in contact with a hyperelastic fabric in 3D.

Due to the lack of some capabilities of the existing software and plasticdeformation, an hyperelastic fabric model is proposed in this project. Themodel is based on a system of particles joined by hyperelastic springs. Two-and three-dimensional implementations were developed and the model wascompared with finite element analysis.

To complete the cycle, a study and redesign of the thermoforming processof hyperelastic fabrics is presented. The thermoforming consists of male andfemale molds that are heated. The material is pressed down until it adoptsthe shape of the mold. The transient finite element analysis method is used to

ix

x Contents

simulate the heating process. A complete redesign of the molds is proposedbased on improvements of heating times and uniform distribution of thetemperature.

Finally, for the same thermoforming process, a tool to predict the contactsurfaces when pressing the material was developed. Thermal expansion andthickness of the foam was considered in this simulation.

Chapter 1

Introduction

The principal problem in the Elasticity Theory is to find the relation be-tween the stress and the strain in a body under certain forces. HookesLaw is applied when the strains are small. However at large deformations,new expressions to characterise the behaviour of materials like rubber are re-quired. Several constitutive models have been published. Mooney [19] (1940)presented a theory of large elastic deformation, Rivlin (1951) studied largeelastic deformations of rubber, in 1962 BlatzKo [5] presented a new strainenergy function to the deformation of rubbery materials, Yeoh [24] proposeda strain-energy function for the characterisation of carbon-black-filled rub-ber vulcanized in 1990 and in 1971 Ogden constructed an energy function forcharacterisation of rubber-like solids [20]) for nonlinear large elastic deforma-tions based on strain energy density functions. Other constitutive relationsare based on macromolecular network structures. Arruda and Boyce pro-posed a new constitutive model for the deformation of rubber materials, in1992 [2]). These models are characterised by a particular form of the strainenergy function W . In each of these methods, a set of coefficients must bedetermined.

Hyperelasticity is the capability of a material to experience large elasticstrain due to small forces, without losing its original properties [10]. Anhyperelastic material has a nonlinear behaviour, which means that its answerto the load is not directly proportional to the deformation. Several wovenfabrics like Lycra have hyperelastic behaviour and are widely employed bytextile companies.

The modeling and design of hyperelastic materials consists of the selec-tion of an appropriate strain energy function W , and accurate determination

1

2 Chapter 1. Introduction

of material constants for such functions. There is no literature about theapplication of hyperelastic models to the deformation analysis of fabric ma-terials. This paper deals with the study of the deformation of fabrics usingdifferent hyperelastic models.

Chapter 2 describes some of the most used hyperelastic materials models.The strain energy function for different materials is presented. The study inthis chapter focuses on the application of these models to fabrics. Comparedto full three-dimensional hyperelastic materials, fabrics can be considered as2-D-manifolds embedded in a three- dimensional space. They can supporttension and bending forces but they can not stand compression as they fold.Therefore, the standard techniques to identify the hyperelastic propertieshave to be adapted to the fabric case. Section 2.3 introduces a procedure tocharacterise hyperelastic fabric, its advantages and disadvantages are evalu-ated, and presented.

Thermoforming is a particular process that permanently deforms a ma-terial via the application of thermal and structural loads. This transformsthe flat fabric into a curved shell that fits or enhances the contour of thebody. Some facts like wrinkles and breakage of the fabric due to overloadingare usually present in this process. The application of external structuralloads due to the contact of the material with the thermoforming mold, plusthe high temperature causes permanent deformation. Although there is anextensive treatise on plastic deformation of hyperelastic materials, it has notfound its way into any commercial or free software with this capabilities.In particular, no literature was found on permanent deformation of textiles.The first step to achieve this goal was to develop a hyperelastic model offabrics in which plastic deformation can be developed at a further stage.

The forces involved in the thermoforming process are mainly normal tothe surface of the fabric and therefore the fabric acts as a membrane. Themain forces in the fabric are tension with low components in bending, tor-sion, shearing, twisting, and gravitational forces. Chapter 3 proposes analternative model for hyperelastic fabrics with the objective to simulate itsinteraction with a rigid object (the mold). The model is based on a grid ofhyperelastic springs under pure tension loads. Rectangular arrays of springsenable the model to behave orthotropically. Furthermore, it is possible tosimulate the weft and warp of the fabric.

Chapter 4 examines the manufacturing process. The thermoforming de-vice consists of male and female molds that are heated to high temperatures(about 200oC) and compresses the textile. Among the problems found in

3this process are the slow heating rate, nonuniform temperature on the moldsurface, and nonuniform contact with the fabric. These three problems werestudied and simulated using finite element analysis. A complete redesign ofthe molds is presented.

Chapter 2

Hyperelastic Cloth

An elastic material is one that returns to its original (unloaded) shape uponthe removal of applied forces. On the other hand, a plastically deformedsolid does not return to its original shape when unloaded. There is somepermanent deformation. Engineering materials such as crystalline metals areclassified as linear elastic solids whereas rubber-like materials are considerednonlinear elastic solids. The difference is the type of stress-strain behaviouras shown in Figure 2.1. These curves represent the relationship betweenstress and strain to uniaxial loading. The linear elastic range is representedby blue curve in Figure 2.1, a class to which most metals belong. A behaviortypical of rubber-like solids is similar to magenta curve in Figure 2.1, inwhich stresses cannot be described as a linear function of strains. In bothcases however, the curves during loading and unloading follow the same path.The stress is a unique function of the strain or deformation.

2.1 Linear elastic relation

The relationship between stress and deformation is represented by a consti-tutive equation. The most general way to represent a linear relation betweenthe stress tensor ij and the strain tensor kl is given by Hookes law (linearproportional relationship named after Robert Hooke, 1676).

ij = Cijklkl (2.1)

where ij are components of the Cauchy stress tensor, kl are componentsof the strain tensor and Cijkl is called the elastic constants tensor of fourth

5

6 Chapter 2. Hyperelastic Cloth

Elasticregion

(nonlinear)Hyperelastic

Elastic (linear)

unloa

dingloa

ding

loading

unloading

Figure 2.1: Types of stress-strain responses

order.In summary, the behavior of an elastic material is characterized by three

properties which are all interrelated:

1. The material coefficients Cijkl have the symmetries

Cijkl = Cjikl = Cijlk = Cjilk.

These symmetries reduce the number of independent components to36.

2. There is an additional symmetry condition for the tensor C which isCijkl = Cklij. The number of independent components is then reducedto 21.

3. Stress and strain tensors are symmetry, that is, ij = ji and ij = ji.

4. There are 6 independent terms in ij y en ij.

The stress-strain relation for an isotropic linear elastic material can bewritten as

ij = kkij + 2ij, (2.2)

2.1. Linear elastic relation 7

where and are the Lame coefficients and ij the Kronecker delta.

2.1.1 Linear elastic relation in matrix form

Stress can be written as a column matrix [] with

z

yx

yz

z

zxzy

yxz xy yxx

x xy xzy yzsym z

xyzxyxzyz

.

Strain [] is similarly written in matrix form with the exception that afactor of 2 is introduced on the shear terms:

x xy xzy yzsym z

xyz2xy2xz2yz

=

xyzxyxzyz

.

The constitutive relation may then be written in matrix form as

= C or i = Cijj (2.3)xyzxyxzyz

=C11 C12 C13 C14 C15 C16

C22 C23 C24 C25 C26C33 C34 C35 C36

sym C44 C45 C46C55 C56

C66

xyzxyxzyz

.

The last relation holds for arbitrary anisotropic linearly elastic materi-als. The number of independent material constants is further reduced byconsiderations of material symmetry.


2.2 Nonlinear elasticity

In this section we will give a brief introduction to nonlinear elastic materialsunder large deformationsmaterials more generally known as hyperelastic.Among the elastic fabrics, there is a whole range of polymers (rubbers) thatcan be modeled with hyperelastic constitutive equations. among the elasticfabrics. Nonlinear elasticity theory is intended to account for these phenom-ena. The principal features of a nonlinear elastic material are:

1. Large deformation analysis are nonlinear,

2. It does not have permanent deformations,

3. There is no proportionality between stress and strain,

4. The elastic rigid tensor C, is not a matrix of constant coefficients. In-stead, total stress-strain relation is derived from a strain energy func-tion W .

2.2.1 Basic concepts

Some general features of hyperelastic materials are considered and then ex-amples of hyperelastic constitutive models, which are widely used in practice,are given.

Stretch ratio

Figure 2.2(a) shows a beam of initial length Lo and final length L, with anapplied force F . Then, engineering strain E can be expressed as

E =L LoLo

=u

Lo,

where u is the displacement.

Stretch ratio is defined as the ratio of the deformed length L dividedby the initial length Lo:

=L

Lo=

Lo + u

Lo= 1 + E.

2.2. Nonlinear elasticity 9

This is an example of stretch ratio as defined for uniaxial tension of arubber specimen. There are three principal stretch ratios 1, 2, and 3which will provide a measure of the deformation.

Figure 2.2(b) shows a thin square rubber sheet in biaxial tension. Theprincipal stretch ratios 1 and 2 are characterised in a plane deformation.

Lo u

F

L

(a) Stretching of rubber rod

3

12

LLo = = 2

tto

= =

1

3

2

(b) Square rubber sheet in biax-ial tension

Figure 2.2: Principal stretch ratios

Principal invariants

Strain invariants are measures of strain which are independent of the coor-dinate system used to measure the strains. The strain energy function for ahyperelastic material can be written as a function of the principal invariantsI1, I2, I3 of the right CauchyGreen deformation tensor Cijkl.

I1 = 21 +

22 +

23

I2 = 21

22 +

22

23 +

23

21

I3 = 21

22

23

where 1, 2, and 3 are the principal stretch ratios.

Volume ratioJ = 123 = V/Vo.


The volume ratio J is defined as the deformation ratio of the deformedto undeformed volume of material.

Deformation gradient tensor

z yx

DeformedUndeformed

X xDisplacementu

Figure 2.3: Definition of the deformation gradient tensor

If the undeformed configuration is the condition of the body before loadshave been applied to it and the deformed configuration is the location andshape of the body after loads have been applied to it. Then, F is called thedeformation gradient of the actual position x with respect to the position Xof the undeformed configuration. The initial and deformed configuration isshown in Figure 2.3:

Fij =xiXj

.

Strain energy function W (E)

Cauchy stress tensor is not used for analyzing materials undergoing largedeformation because the area in the deformed configuration is generally donot know. Thus, it is necessary to define a stress measure that we can usein the reference configuration. We have to derive the constitutive equationsof hyperelastic materials, in terms of the 2nd Piola Kirchoff stress S and theGreenLagrange strain tensor E.

An elastic material may be represented by a strain energy function W .For hyperelastic materials this takes the following form:


Sij =W

Eij, Eij =

Sij.

W (E)

(S)

E

S

Figure 2.4: Stress-strain curve

The stress-strain curve of hyperelastic materials is shown in Figure 2.4.Notice that the area below the curve is the strain energy function W (E).

2.2.2 Hyperelastic models

This section introduces the concept of hyperelasticity and mentions some as-pects important about hyperelastic models. Hyperelasticity is the capabilityof a material to experience large elastic strain due to small forces, withoutlosing its original properties. An hyperelastic material has a nonlinear be-haviour, which means that its deformation is not directly proportional to theload applied. An elastic material is hyperelastic if there is a scalar function,denoted by W = W (E) : Rnn R called strain energy function (or storedenergy function), such that

Si,j =W (E)

Eij= 2

W (E)

Cij, (2.4)

where Sij are the components of the second Piola-Kirchhoff stress tensor,W is the strain energy function per unit volume undeformed, Eij are thecomponents of the GreenLagrange strain tensor and Cij are the componentsof the right CauchyGreen strain tensor. By algebraic manipulation of (2.4)the components of Cauchy (true) stress tensor [] can be determinate.

i,j = pij + 2W (E)I1

Cij 2W (E)I2

1

Cij, (2.5)


where I1 and I2 are the principal invariants of the [C] tensor. The threestrain invariants of the strain tensor can be expressed as:

I1 = 21 +

22 +

23, I2 =

21

22 +

22

23 +

23

21, I3 =

21

22

23.

The strain energy functions of hyperelastic constitutive models such asMooneyRivlin, neoHookean and ArrudaBoyce are given in the next sub-sections. They are expressed as a function of strain invariants I1, I2, I3 or interms of the principal stretches 1, 2, 3 of strain tensor.

In most of the models, in order to deduce the strain energy functions, ithas been assumed, that the material is isotropic and with constant volume(isometric deformation 123 = 1). Unless indicated otherwise, hyperelas-tic materials are assumed to be nearly or purely incompressible. The mostcommon functions of deformation energy are:

1. MooneyRivlin Model

Mooney and Rivlin proposed a strain energy function W as an infiniteseries in powers of (I1 3) and (I2 3) of the form

W (I1, I2) =nij=0

cij(I1 3)i(I2 3)j, (2.6)

where cij are constants. For example, the Mooney-Rivlin form withtwo parameters is:

W = c10(I1 3) + c01(I2 3).

2. NeoHookean Model

Equation (2.6) shows a series of powers which are usually truncatedin the first terms. Then, taking only the first term of (2.6), the neo-Hookean model is obtained:

W (I1) = c10 (I1 3) .

3. Ogden Model


Ogden [20] deduced a hyperelastic constitutive model for large defor-mations of incompressible rubber-like solids. The strain energy is ex-pressed as a function of principal stretches as

W (1, 2, 3) =nr=0

rr(r1 +

r2 +

r3 3),

with r and r as material constants that can be determined by exper-imental tests.

4. Yeoh Model

The Yeoh model [24] depends only on the first strain invariant I1 . TheStrain energy function W is obtained by

W (I1) =i

n=1

ci0 (I1 3)i .

It applies to the characterisation of elastic properties of carbon-black-filled vulcanized rubber.

5. ArrudaBoyce Model

The constitutive model for the large stretch behavior of elastic rubbermaterials is presented by Arruda and Boyce in [2]. The strain-stressfunction is based on an eight-chain representation of the macromolec-ular structure of the rubber:

W (I1) = G[12(I1 3) + 1

20N(I21 9) +

11

1050N2(I31 27)

]+G

[ 197000N3

(I41 81) +519

673750N4(I51 243)

]+ . . .

where the module G = nk , where n is the chain density , k isBoltzmanns constant, N is the number of rigid links composing a singlechain, and is the temperature . This model is also known as theeighth-chain model since it developed from a representative volumeelement where eight spring leave out from the center of cube to itscorners.


6. Gent Model

The strain energy density in the Gent model is a simple logarithmicfunction of the first invariant I1, involves two material constants, andthe shear modulus and Im which measures a limiting value for I13.Gent proposed the strain energy density

W (I1) =2Im ln

[1 I1 3

Im

].

7. BlatzKo Model

An application of finite elastic theory to the deformation of rubberymaterials is given in [5]. Incompressibility is not assumed. The strainenergy function is cast in terms of the constants , , and f , which canbe determined experimentally,

W (J1, J2, J3) =f

2

[J1 3 + 1 2

{J2/(1 2) 13

}]+(1 f)

2

[J2 3 + 1 2

{J2/(1 2)3 1

}]where is similar to the Poissons ratio, is the shear modulus, andf is a material constant. Also, a new set of invariants is defined byJ1 =

2i , J2 =

2i and J3 = i.

Accurate modeling of hyperelastic materials requires material propertiesdata measured to large strains under different states of stress. Model coeffi-cients are calculated from mechanical test data using least squares fit routinesin the FE software. Suitable tension test methods are discussed in section2.3.

2.3 Characterisation of hyperelastic fabrics

In order for the design predictions to be relevant, it is essential that thematerials properties are determined under test conditions appropriate for theservice conditions. Where combinations of test data are supplied to derivemodel coefficients, these data must be determined at the same temperaturesand strain rates. The material properties required to calculate deviatoriccoefficients can be determined from a combination of test data:

2.3. Characterisation of hyperelastic fabrics 15

1. Uniaxial tension test,

2. Shear test (planar tension),

3. Equibiaxial tension test,

4. Volumetric test.

2.3.1 Uniaxial tension test

The uniaxial tension test determines the properties of the material underplane stress. The most significant requirement is that in order to achieve astate of pure tensile strain, the specimen must be much longer in the directionof stretching, than in the width and thickness dimensions. There also is noan absolute specimen size requirement. But, with finite element analysis onthe specimen geometry, it can be determined that the specimen needs to beat least ten times longer than in width or thickness.

W 10L.

L

W

F

F

Figure 2.5: Uniaxial test

Tensile Strain Strain is normally determined from the relative movementof two gauge marks separated by a known distance


1 =L

L02 = 3 =

12

1 .

Stress state Engineering stress is calculated from the load values measuredby a load cell and the undeformed cross-section using the followingequation

1 = =F

A0, 2 = 3 = 0

where is the stress, F is the applied load, and A0 is the initial areaof the specimen, respectively.

Uniaxial tension test specimen

Various standard tension test methods are specified for plastics [15] and rub-bers [14]. The main differences between the plastic methods and the rubbermethods arise from the test specimen geometry and loading speeds. Tensiontests are performed on bone-shaped test specimens, see Tables 2.1 and 2.2.This arrangement provides for a uniform stress and strain distribution in thecentral gauge section, where these values are at a maximum.

LI3

b1b2

I2I1

L0

L0 gauge lengthL distance between gripsI1 length of narrow parallel-

sided portionI2 distance between broad

parallel-sided portionsI3 overall lengthb1 width of narrow portionb2 width at endsh thickness

2.3.2 Planar shear test

The planar test has the same stress state as the pure shear test. The ex-periment appears at first glance to be nothing more than a very wide tensiletest. The most significant aspect of the specimen is that it is much shorterin the direction of stretching than the width. That is


Standard Type I3 I2 I1 b2 b1 h L0 L

Thermoplastic and thermosetting plasticsISO527-2 1A 150 104

11380 2 200.2 10

0.24 0.2 500.5 115 1

ISO527-2 1B 150 106 120

60 2 200.2 10 0.2

4 0.2 500.5 I2 + 5

ISO527-2 1BA 75 58 2 300.5 100.5 5 0.5 2 250.5 58 + 2ISO527-2 1BA 75 23 2 120.5 4 0.2 2 0.2 2 100.2 23 + 2Rubbers and ElastomersISO37 1 115 - 33 2 25 1 6 0.4 2 0.2 250.5 115ISO37 2 75 - 25 1 12.51 4 0.1 2 0.2 200.5 75ISO37 3 50 - 16 1 8.5 1 4 0.1 2 0.2 100.5 50ISO37 4 35 - 120.5 6 0.5 2 0.1 1 0.1 100.5 35ASTM412

C 115 - 33 2 25 1 6 0.05

1, 3 . . . 3, 3 25 0.25

115

ASTM412

A 140 - 59 2 25 1 12 0.05

1, 3 . . . 3, 3 500.5 140

Thin sheetings and filmsISO527-3 2 150 - - - 10 1 500.5 100

0.5

Table 2.1: Standards specification for tensile testing

W 10L,

where L corresponds to length and w is the width as it is shown in Figure2.6. A minimum ratio of width to gauge length (grip separation) of four isrecommended. Experimental studies [8] on 200mm wide x 60mm long testspecimens gripped at different lengths have shown that stress-strain curvesare unaffected by aspect ratios between four and ten. Thus, the specimen istested in a condition of plane strain rather than the plane stress state thatcharacterises the uniaxial tension test.

F

F

W

L

Figure 2.6: Planar shear test


Probeta Geometra L0

1 [18]

2 [1]

3 [13]

4 [15]10 mm

150 mm

Table 2.2: Sample preparation

Planar strain

1 = =L

L02 =

1 3 = 1

where is the stretch ratio in loading direction.

Stress state The planar stress is calculated from the measured force Fand the specimen width w and thickness t

1 = 2 = 0 3 6= 0.


2.3.3 Biaxial tension test

Equibiaxial tension tests require a stress state with equal tensile stresses alongtwo orthogonal directions. The equal biaxial strain state may be achieved byradial stretching a circular disc as is shown in Figure 2.7.

Figure 2.7: Biaxial specimen

Strain state

1 = 2 = =L

L03 =

2 (2.7)

where is the stretch in perpendicular loading directions.

Stress state The nominal equibiaxial stress contained inside the specimeninner diameter is calculated as:

=P

A0

where is the stress, P is the sum of radial forces, A0 = piDt0 is theinitial area the specimen, D is the original diameter between punchedholes, and t0 is the original thickness.

1 = 2 = 3 = 0.

Successful modeling and design of hyperelastic materials depend on theselection of an appropriate strain energy function, and accurate determina-tion of coefficients in the function. The next section describes the procedurefollowed to find the minimum number of standard tests necessary to obtaingood characterization of the hyperelastic material.


2.4 Problem description

Mechanical properties of the fabrics are essential for the development of newtechnical textile applications and for realistic simulations. This section dealswith the aspect relations with the characterisation of hyperelastic fabrics.

The types of experimental tests to determine the constants of the hypere-lastic model are: uniaxial tension, uniaxial compression, planar shear, biaxialtension, and volumetric test. (See Figure 2.8.) Commonly, all the requiredexperimental tests are not available to characterize an hyperelastic material.The main difficulty to characterise the fabrics is to find out the most suit-able devices for the specific conditions of textile standard tests. Only oneof the tests, the uniaxial tension, could be described as routine or readilyavailable. The high cost of the equipment limits the execution of tests suchas shear and biaxial tension. For example, the biaxial test requires eitheran expensive test machine or a special fixture. The next section discussesstrain mechanical measurements in tension. Also, a methodology has beendeveloped to determine the minimum number of tests that are required tocharacterise an hyperelastic fabric.

StressStrain Curve

Shear Strain

Biaxial tension

Uniaxial tension

0

0.5

1

1.5

2

2.5

3

Stre

ss (M

Pa)

0.2 0.4 0.6 0.8 1Strain (mm/mm)

Figure 2.8: Stress-strain experimental curves for a elastomer

2.4. Problem description 21

2.4.1 Uniaxial test tension of the hyperelastic fabric

The simplest deformation mode from the experimental point-of-view, is theuniaxial tension. For this purpose, it is not recommended that the ASTMD4964 standard test method for tension and elongation of elastic fabrics beapplied. Instead, the norms ASTM 412 [14] for elastomer and rubbers andthe norm ISO 527 [15] for plastics were used. Figures 2.9(a), 2.9(b), and2.9(c) show the stress-strain curve of the fabric hyperelastic at 20C, 80C,and 105C.

Uniaxial Tension Test (T=23 C)

Fiber direction 2

Fiber direction 1

0

1

2

3

4

Stre

ss (M

Pa)

0.2 0.4 0.6 0.8 1Strain (mm/mm)

(a)

Uniaxial Tension Test (T=80C)

0.51

1.52

2.53

3.54

Stre

s (M

Pa)

0 0.2 0.4 0.6 0.8 1Strain (mm/mm)

Fiber direction 2Fiber direction 1

(b)

Uniaxial Tension Test (T=105C)

0.5

1

1.5

2

2.5

3

Stre

ss (P

a)

0 0.2 0.4 0.6 0.8 1Strain (mm/mm)

Fiber Direction 2

Fiber Direction 1

(c)

Figure 2.9: Stress-strain curves for hyperelastic fabric


2.4.2 Methodology to determine the constants of thehyperelastic material

In the characterization of hyperelastic materials, it is desirable to know theerror occurring when all types of experimental tests are used and when onlyone type of test is used. For this purpose, it is proposed that a problemconsisting of a rigid sphere in contact with an hyperelastic material is used.(See figure 2.10(a)). The fabric is represented by a horizontal surface. Thenodes of the periphery were restricted in displacement.

(a) Model 3D

Initial position

Final position Sphere

Hyperelastic Material

(b) Nonlinear FEA

Figure 2.10: Model of the fabric in contact with rigid body

Then, Case A employs uniaxial tension data and Case B employs shear,biaxial, and tension test data, as is shown in Figures 2.11(a) and 2.11(b).The procedure followed to analyze Cases A and B, is shown in Figure 2.12.First, with the strain-stress data and a selected hyperelastic material model(see the section 2.2.2), coefficients of the model were obtained.

Second, a least squares fit procedure was used to calculate the coefficientsin the selected model. This procedure minimizes the sum of squared errorbetween experimental and Cauchy-predicted stress values, EXPi and

THi

(2.5), respectively. For n experimental data lectures, the square error E isdefined by

E2 =ni=1

(1 THi

EXPi)2. (2.8)


, ,...c c0 1

UniaxialTension

Data

(a) Case A

, ,...c c0 1

Biaxial Data

ShearTension

(b) Case B

Figure 2.11: Determination of hyperelastic material constants c0, c1, . . .

The next step consisted of comparing the residual of least squared errorwith a value. If is lower than least squared error, then the constantsare taken for modeling the hyperelastic material. Finally, the finite elementthe model (Figure 2.11(a)) is solved. The solution consists of moving thesphere towards the fabric to determine its deformation. The problem wassolved applying nonlinear analysis with FEM (see figure 2.10(b)). However,although the selected hyperelastic model fits to the strain-stress data, is notnecessarily the solution obtained. Finally, the stability of the hyperelasticmodel was checked. That is, the stretch ratio must be in a correct rangesolution. If the material is stable, the strain and stress results are exported.

1. Case A: Determination of the properties of a hyperelastic material onlyusing experimental data of uniaxial tension.

The hyperelastic models analyzed in this case are shown in Table 2.3.Models with a least squared error (2.8) greater than 30 % is not ac-ceptable for this study. The results of higher order MooneyRivlin andYeoh models give the best fit with the uniaxial tension. The result fromthe ArrudaBoyce model is also acceptable. However, the solution ofthe problem converges only for the Yeoh and ArrudaBoyce models.Table 2.4 shows the results of fitting the hyperelastic constants usingonly the uniaxial tension test.


Material ModelConstants

Material Model

material model:Selection Hyperelastic

BlaztkoArrudaBoyce,OgdenMooneyRivlin,

strainstressdata

End

converge?

Boundaryconditions

test dataExperimental

error <

Hyperelastic

experimental dataFitting

Yes

Finite Element modelfabricrigid body

Analysis

Yes

No

Export results

least squares method

No

Nolinear Finite Element

Figure 2.12: Methodology of the analysis of hyperelastic materials

Model Fitted Data Results

MooneyRivlin (9parame-ters)

Exp DataFitted Data

MooneyRivlin Model (9 parameters)Uniaxial Tension with

Strain (mm/mm)01 2

Stra

in (M

Pa)

0

2

4

No convergence

Table 2.4: Comparison of material models: Case A


Model Fitted Data Results

Yeoh(Order3)

Exp DataFitted Data

Uniaxial Tension with Yeoh (Order 3)

Strain (mm/mm)0 1 2

Stre

ss (M

Pa)

0

2

4

ArrudaBoyce Exp Data

Fitted Data

Uniaxial Tension withArrudaBoyce Model

Strain (mm/mm)0 1 2

Stre

ss (M

Pa)

0

1

2

3

4

5

Table 2.4: Comparison of material models: Case A

2. Case B: Determination of the properties of a hyperelastic material us-ing experimental data of uniaxial tension, biaxial, and shear tension.Hyperelastic models are compared in Table 2.5 . The MooneyRivlinmodel (with 9 parameters) best fits the experimental data. The strain-stress results are shown in Figures 2.13(a), 2.13(b), and 2.13(c). Hy-perelastic material modeling using all three types of test data, achievesresults that adjust exactly to the real problem.

2.4.3 Normalized error value

Let the value of stress for Case A be defined by Ai , i = 1, . . . , n and n is thetotal number of nodes in the domain. Let Bi correspondingly be the stressvalue for Case B. The Normalized Error Value e can be calculated using theexpression


Hyperelastic Model Fit test data LeastSquaredError %

MooneyRivlin

2 Parame-ters

- 60

3 Parame-ters

acceptable 15

5 Parame-ters

good 1

9 Parame-ters

best 0.01

OgdenOrder 1 - 50Order 2 - 54Order 3 - 54

Neo-Hookean - 65ArrudaBoyce acceptable 30

Gent - 880

Yeoh

Order 1 - 60Order 2 - 40Order 3 good 5

BlatzKo - 200

Table 2.3: Results of fitting using uniaxial tension data

e = 100max1in |(Bi Ai )|

max1in |Ai |.

The error found when employing only uniaxial tension test data is ap-proximately 15 %.

2.5 Results

For example, the behaviour of a hyperelastic fabric in a thermoforming pro-cess could be simulated by an exact characterisation, as is shown in Figure2.14(b).

2.5. Results 27

Exp DataFitted Data

Strain (mm/mm)0 1 2

Stre

ss (M

Pa)

0

1

2

3

4

5

T = 20C

(a) Uniaxial tension

Exp DataFitted Data

Strain (mm/mm)0 0.2 0.4 0.6 0.8 1

Stre

ss (M

Pa)

0

0.5

1T=20C

(b) Shear

Exp DataFitted Data

Strain (mm/mm)00.5 1 1.5

Stre

ss (M

Pa)

0

1

2

3

T=20C

(c) Biaxial tension

Figure 2.13: Stress-strain fitting with MooneyRivlin (9 parameters)(T=20C)

Hyperelastic Model Fit test data LeastSquaredError %

MooneyRivlin

2 Parameters - 963 Parameters - 955 Parameters acceptable 309 Parameters good 18

OgdenOrder 1 - 180Order 2 - -Order 3 -

Neo-Hookean - 180ArrudaBoyce - 130

Gent - 880

Yeoh

Order 1 - 180Order 2 - 140Order 3 - 100

Blatz-Ko - 270

Table 2.5: Results of fitting using three tests data

The constants of the hyperelastic fabric are calculated using only uniaxialtension test data. The stress-strain data are fitted using the MooneyRivlin(9 parameters), Yeoh (Order 2 and 3), ArrudaBoyce, and Ogden models.


(a) Thermoformed Mold -hyperelastic fabric

(b) Strain in the direction z

Figure 2.14: Modeling of a thermoformed mold of hyperelastic fabrics

(Table 2.6.) The high-order MooneyRivlin and Yeoh models best fit thetest data. The ArrudaBoyce and Order 1 Ogden models also achieve anacceptable fit. In contrast, Ogden (Orders 2 and 3), neoHookean, Gent,BlatzKo and the lower-order MooneyRivlin and Yeoh models, show largedifferences between the numerical and test data.

Model Fitted Data Constants

MooneyRivlin (9parame-ters)

Exp Data Fitted Data

Uniaxial Tension with Polynomial (Order 3)

Strain (mm/mm)0 0.2 0.4 0.6 0.8

Stre

ss (M

Pa)

0

0.5

1

C10 = 1.1122E + 07C01 = 1.1505E + 07C20 = 4.3547E + 10C11 = 8.7292E + 10C02 = 4.3771E + 10C30 = 1.4695E + 08C21 = 9.5988E + 08C12 = 8.549E + 09C03 = 3.0711E + 09

Table 2.6: Comparison of hyperelastic material models

2.6. Conclusions 29

Model Fitted Data Constants

Yeoh (Or-der 3)

Exp DataFitted Data


Strain (mm/mm)0 0.2 0.4 0.6 0.8

Stre

ss (M

Pa)

0

0.5

1

C10 = 60951C20 = 21463C30 = 17258

ArrudaBoyce Exp Data 1

Fitted Data 1 1

Uniaxial Tension with ArrudaBoyce

Strain (mm/mm)0 0.20.4 0.6 0.8

Stre

ss (M

Pa)

0

0.5

1

= 3891.3L = 0.65907

Yeoh (Or-der 2) Exp Data

Fitted Data


Strain (mm/mm)00.2 0.4 0.6 0.8

Stre

ss (M

Pa)

0

0.5

1

C10 = 52866C20 = 54144

Ogden

Exp Data Fitted Data

Uniaxial Tension with Ogden (Order 3)

Strain (mm/mm)0 0.2 0.4 0.6 0.8

Stre

ss (M

Pa)

0

0.5

1

1.5

1 = 227.241 = 12.6192 = 227.242 = 12.5823 = 227.243 = 12.564

Table 2.6: Comparison of hyperelastic material models

2.6 Conclusions

The derivation of models, such as MooneyRivlin and Gent, is difficult be-cause of the amount of experimental data required to obtain the model coeffi-cients. To improve the accuracy of predictions, it is best to use experimentaldata from a range of experimental tests (uniaxial, biaxial and planar ten-sion). The ArrudaBoyce, neoHookean and Yeoh models offer a physicalinterpretation and provide a better description of general deformation modes


when the parameters are based only on one test. In all cases, it is best toobtain experimental data from the range of strain of interest (this is espe-cially true of the Ogden and polynomial models), and to select the modelcoefficients carefully to ensure stability.

Two different combinations of input data were evaluated: uniaxial tensiononly and combined uniaxial, biaxial, and planar tension, as described inSection 2.4.2. The calculated normalized error shows that there is littledifference between the accuracy of predictions made using uniaxial data onlyand those made using the combined uniaxial, biaxial, and planar tensiondata. Therefore, it appears that the biaxial and planar tests can be omittedwith little effect on accuracy, thus simplifying testing requirements. The useof only uniaxial test data will induce an error of 15 %. In general, it isbetter to obtain data from several experiments involving different kinds ofdeformation. The range of deformation should be restricted to the interestof application in order to determine the model coefficients. For example, thehigher-order models of Ogden and MooneyRivlin present some instabilitieswhen only limited test data are available.

ArrudaBoyce and Yeoh (Order 3) models provide the best fit when theconstants are calculated when only using the uniaxial tension test.

Chapter 3

Springparticle Model forHyperelastic Cloth

This chapter presents a multi-particle method to simulate stretching of elasticcloth and its interaction with a rigid body. The material behaves hyper-elastically and has orthotropic properties.

3.1 Literature Survey

In the modelling of cloth, several issues have been addressed in the existingliterature: (i) multiparticle vs. Finite Element Analysis, (ii) Solvers, (iii)literal vs. penalty simulated constraints, (iv) parametric representation offabric, and some others. In the work presented here we introduce an addi-tional one: (v) linear system vs. Brep driven computation.

3.1.1 Multi-Particle vs. FEA methods

FEA methods

The FEA methods for this problem are a continuum model of the cloth,which must take care of the following aspects: (a) kinematic description ofthe shell, (b) material stress-strain response, (c) force and moment balanceequations, (d) discretization, linearization and (e) solution.

1. Kinematic Description of the Shell: A coordinate frame [1, 2, t,] isplaced at the point , for each quadrangular patch, in the parametric

31

32 Chapter 3. Springparticle Model for Hyperelastic Cloth

space formed by 1 and 2. t is a vector in the thickness direction.This parametric patch is located at the midway between the front andreverse surfaces of the cloth. The normal direction is . The elementdescription allows the modelling of membrane strain, transverse shearstrain and bending strain.

2. Material Stress-strain Response: The fabric presents not only a non-linear relationship between bending moments and bending strain, butalso depends on the history of the load. For typical bending moment-vs.-strain curves see [16], [6], and [17]. In [9] a model of the formM(K) = 1K + 2K

2 + 3K3 + 4K

4 + 5K5+ with K being the

curvature (mts1) and M the flexing moment. The reference is notprecise as to whether the hystheresis characteristic of the stressstrainrelation is modelled by the branches of the quintic equation or not.

3. Force and Moment Balance Equations: The equations for force and mo-mentum are obviously non-linear. Upon linearization, such equationsbecome:

K(, t)

{t

}= Fexternal P (, t) (3.1)

where K(, t) is the stiffness matrix, t is the vector in the thicknessdirection, is the position of the element, and P (, t) are the internalforces to solve. As usual, the solution of the linearized system producesthe updates and t around the linearization point.

4. Solutions and Iteration Speed: The arc length is controlled in [9] inorder to achieve a speedy convergence. It has been found that a smalliteration step is needed and desirable to converge to a first load step.Subsequent steps then may be performed with large arc length intervals.

5. Contact Algorithms: The contact algorithms are in charge of prevent-ing the nodes of the elements crossing the solid obstacles (the humanbody, other cloth, etc.). FEA methods, such as the one proposed in [9],avoid the explicit test of the node positions. One obvious reason forthis strategy is that all the intermediate activities of the algorithm aregeared towards preparing the next iteration of solutions to the A.x = bsystem. Boolean variables are not suited to be included in the A.x = bsystem. For such reason, [9] and many other authors do not explic-itly test for node collision against the solid. Instead, they implement

3.1. Literature Survey 33

penalty functions that make the nodes stay away from the solid. Thisis a desirable action for the purpose of casting the problem in the formA.x = b, but produces appreciable artifacts in the shape of the cloth.In the work developed in this research, one will therefore set up the con-straints in the form of boolean values, therefore avoiding the numericalmanipulation of penalty or potential functions.

Multi-particle Methods

Basic multi-particle methods realise a force balance at each node of a meshrepresenting the cloth. The force balance itself may include stretch, bend,shear, or friction forces. However, these forces are modelled in multi-particlesystems as acting solely upon the nodes of the mesh. One difference betweencontinuum models is that the later ones take into consideration such forces,but act on the whole patch or element. In multi-particle models, in-planeextension, shear strains, and out-of-plane bending are modelled. Out-of-plane twisting and shearing are neglected. As with continuum models, theissues to be addressed are: (a) kinematics of the mesh, (b) material response,(c) equilibrium of nodal forces, (d) iteration step, and (e) constraints.

b is the un-stretched side length of a quadrangular patch whose cornersare the nodes at which the force balance is made. E is the Young modulusof the fabric, b/b is the unitary elongation, A is the cross sectional area ofthe cloth element, h is the cloth thickness, I is the second moment of inertia, is the Poisson module.

1. Kinematics of the mesh: The stretch force is modelled at the nodes as:Fstretch = D

Ehb, with D being a factor to finetune attractionvs. repulsion forces.The shear force is Fshear = B

E h b d/(2(1 + )), with Bbeing a factor to finetune shear strain.The bending force is Fbend = E I /(b2 (1 2)), with beingthe bending deformation.

2. Material properties: See discussion for continuum models.

3. Iteration step: An artificial force at each node is added in such a waythat the computed displacement in the direction of an equilibrium posi-tion is accelerated. In this form, fewer iterations are needed to convergeto the equilibrium state.


4. Constraints and collision detection: Although the collision detection(node vs. solid, cloth vs. cloth) is performed explicitly, and once thecollision is detected, counter-balance forces are added to the affectednode in pretty much the same way the forces to accelerate the stepare calculated. In this form, the solver builds and solves the equationA.x = b, without having to change the A1 in the A.x = b system fromiteration to iteration. Node annihilation caused by boolean conditionsevaluated on such a node immediately triggers row/column elimination.Therefore many authors ( [4]) avoid the explicit elimination of nodesand replace it by penalty functions that cause the nodes to move onlya small amount (if at all) in the constrained directions.

5. Equilibrium of nodal forces: The equilibrium condition for the nodalforces is: W +Fstretch+Fshear+Fbend+Fexternal+Fseams = 0, where Wis the weight of the cloth present at the node, and Fseams is a reactionforce at the nodes, imposed by the sewn cloth pieces and thereforepresent in the node force balance. As per the equation, this is a staticmodel. For dynamic models the equation is: W + Fstretch + Fshear +Fbend + Fexternal + Fseams = Fresultant, with Fresultant determining themovement of the node (by a variable amount) in the direction of theresultant force.

3.1.2 Iteration speed and stability

In the literature surveyed, the issue addressed is: how to cast a multi-particleor continuum cloth model into a A.x = b form so a finite element or equationsolver may be used to determine for the state of the cloth at a determinatetime interval after the simulation start.

In general, the solution of equation 3.2 is the position of the node setx, reacting to the external forces, and responding to the change in potentialenergy of the particle system,

x =M1(Ex

+ F

). (3.2)

This equation, however, has been found difficult to solve in a mass-distributed system (a multi-particle one, for example). Instead, a state-spaceformulation omitting the explicit energy balance has proven to be easier to


approach (Equation 3.3),

d

dt

(xx

)=

d

dt

(xv

)=

(v

M1 f(x, v))

. (3.3)

To solve this equation, in [4] a backward implicit model is used to speed upand stabilise the dynamic simulation. The backward implicit formulation isequivalent to Equation 3.4, where h is the integration time interval.(

I hM1fv

h2M1fx

)v = hM1

(f0 + h

f

xv0

)(3.4)

In Equation 3.4, one solves for v and then computes x = h(v0 +v).When a forward explicit method is used, the h time interval for a stable

integration of Equation 3.5 must be very small,(xv

)= h

(v0

M1f0

)(3.5)

According to [4], it is therefore preferable to use the implicit formulation inEquation 3.4.

3.1.3 Cloth geodesics and stretch measure

An unstretched (flat) quadrangular patch may be modelled as a unit speedparametric surface C(u, v) . In this form, a parametric surface w(u, v) in R3

representing the cloth wrinkled but not warped would keep the unit speedparametrisation. This fact has to do with the fact that the curves w(u, v)and w(u, v) (functions of v and u respectively) are geodesics of the cloth([21]). If one defines wu =

wu

and wv =wv, the material is unstretched if

wu = wv = 1. Otherwise, wu 1 and wv 1 represent the stretchin the u and v directions respectively.

3.1.4 Constraints and collision avoidance

Constraints and collision avoidance are required because there is a spatialrelation between fabrics and the solid object that sustains them, among partsof the fabric with themselves, among parts of garments in contact with eachother or beacuse of the effects of seams in the garment. Such constraints


may be enforced in basically two ways: (i) by introducing potential fieldswhich help to keep nodes of the cloth away from each other, thus avoidinginterpenetration of the fabrics, (ii) by formulating boolean interpenetrationqueries on nodes and applying geometric constraints when the answer to theinterpenetration queries is affirmative.

Potential fields

Under this classification, we have the introduction of ghost springs, exter-nal forces, and penalty functions. An advantage of these methods is thatfor solution engines of the form A.x = b, the added conditions are natu-rally included as additional equations, and therefore the numerical methodsresponsible for the solution do not need to be changed to include the con-straints or collision avoidance. This strategy is followed in [9] and [23].

In [22] geometric constraints are introduced in a Geometric NonlinearFinite Element Method GNFEM, using this matrix equation:

(KT + Kc)q = P + Rc (3.6)

where Kc is the geometric stiffness matrix; Rc is the residual force vector dueto the geometric constraint; is a penalty factor; and KT is the modifiedtangent stiffness matrix, including bending, membrane and initial stress stiff-ness. The cloth is regarded as a thin plate, and represented by shell elementsunder deformation. In this model, the length of the element sides remainsvirtually unchanged, while the bending (buckling) is the predominant phe-nomenon. Therefore, the model works for unstretched cloth (a tablecloth,for example), but it is inaccurate for stretched garments such as underwear,socks, blouses, dresses, etc.

Boolean interpenetration queries

When a node is detected to invade the space of a solid obstacle, the authors in[4] and [3] restrict the movement of the affected node by applying a matrixS to its 3dimensional position x. S has reduced rank (3, 2, 1, 0), andtherefore suppresses 0, 1, 2, or 3 degrees of freedom of the node, respectively.The S matrix represents a parallel, orthonormal projection on a 3, 2, 1, or


0-dimensional space, and it is defined by:

S(x) =

I if ndof(x) = 3 (no restriction) ,

I n nT if ndof(x) = 2 (restriction to a planarspace normal to n) ,

I u uT v vT if ndof(x) = 1 (restriction to thedirection u v ),

0 (total restriction)

(3.7)This is an elegant way to restrict the movement of the node x once therestriction is proven to be needed (collision is detected).

In [11] the formulation is centered on the nodes of the mesh, and thereforethe authors do not seek to integrate the set of all nodes x into a generalmatrix. Each node is followed according to the spring forces applied to it.The solid obstacles are enforced in a way equivalent to [4], but the papermakes no explicit reference to the rank of the space available for each node.

Boolean - potential mixed methods

In [12] the reduced rank space formulation of [4] is used (Equation 3.7). How-ever, a back-bouncing movement of the particle across the obstacle surfaceis provoked by setting up a repulsion force, proportional to the invasion dis-tance along n. In addition, a d damping parameter is used, in order to havethe bouncing die away.

The authors in [4] use a mass value which has 3dimensional components,in order to selectively control the inertia of the particle in one direction (x,y, or z), and therefore the acceleration of the particle in such a direction.Although the method works for avoiding a particle moving in a certain di-rection (by giving it an infinite mass in such a direction), the manipulationof the mass in this form triplicates the storage for the particle masses. Italso may present (in our opinion) undesirable effects in dynamics becausethe mass becomes a 3dimensional entity instead of a scalar.

In [7] the topic of collision detection and geometric constraints is not dealtwith.


3.1.5 Conclusions for the literature review of cloth sim-ulation

According to the literature surveyed, a large effort has been invested bymany authors in casting the problem of cloth simulation into a linearised,continuum model. This model is then written in the form of a A.x = bsystem, and then solved by numerical computation methods.

A disadvantage of this method is that boolean variables expressing col-lision or constraints are not easy to convert into continuum models. Theseboolean variables are then replaced by real-valued penalty functions that areto achieve similar results as the boolean conditions (for example, whetheror not the cloth nodes trespass a solid surface). The simulation in this caseis good enough for the filmmaking or advertising industries, but certainlyunusable for engineering purposes. In the film and advertising industries anappearance of reality is seeked after. In the engineering case a node outsideof the cloth must be outside, and not slightly inside. On the other hand,the addition of resources such as slack or repulsion forces or masses that aredifferent in the different directions seem to introduce artifact effects in thesimulations.

3.2 Problem Description

In multi-particle methods, balance of forces at each particle leds to the fol-lowing equation

W + Fstretch + Fshear + Fbend + Fexternal + Fseams = Fr (3.8)

Where Fr is the resulting force at the particle.

Existing cloth simulation models are in most cases, good enough for vi-sualisation purposes and usually aim at modelling drapes of fabric materialswhere rotational (bending) deformation is large and membrane deformationvery low.

This study use multi-particle methods and is intended to simulate fabricsunder stretching conditions. These situation are typical of garments such asunderwear and blouses which are usually made of materials such as spandexor Lycra. In these garments, the stretch forces are very large comparedto the weight of the cloth or the bending forces. This materials, in general,

3.3. The non-linear pure-tension spring elastic Model 39

presents nonlinear deformations with orthotropic properties. These materialare usually classified as an hyper-elastic material.

In the proposed model, the boolean queries as to whether the node is incontact with a surface or obstacle are to be solved by standard computationalgeometry methods. The reduction of the dimension of the space reachableby such a node because of such constraints is explicitly kept in every node.No penalty or potential functions will be used.

3.3 The non-linear pure-tension spring elas-

tic Model

Consider a rectangular piece of woven fabric to be partitioned into an arrayof points named nodes, numbered from left to right and from bottom to topas shown in Figure 3.1 and stored in matrixN of dimension nm. Beside thepoint location, every node Ni,j has information about the springs connectedto the nodes, its movement restriction, and information about how the nodeis in contact with the surface of a rigid object. The data structure used fora node can be summarise as:

#ifndef _Node_h_

#define _Node_h_

/*-----------------------------------------------------+

| Prototypes of extern functions defined in this file |

+-----------------------------------------------------*/

#include "point.h"

#include "spring.h"

/*--------------------------------------------+

| DEFINITION OF TYPES |

+--------------------------------------------*/

enum Marc{

outside,

contact

};

/*--------------------------------------------+

| DEFINITION OF CLASSES |

+--------------------------------------------*/

class Node {


private:

Point initial_location; /*node location at its initial state */

Point iteration_location;/*node location after being

perturbated */

vector R_around; /*vector which stores the indexes of

the springs around. */

vector Constrains; /*vector which stores the movement

restrictions along axis x,y and z. */

Marc marker; /*marc which indicates if the node is

in contact with the surface or not.*/

};

#endif

N00 N0n

mnN

Nij

S1 S2 S3

S4

S10

Figure 3.1: Numbering nodes and springs on the grid

The links between nodes are represented by spring forces, which are 3D vectors ~Sk R3. Those vectors are numbered consecutively from left toright and from bottom to top as shown in Figure 3.1 and stored in a vectorF = [~S1, ~S2, , ~Sk, , ~S], where is the total number of springs withinthe grid. The overall system with nodes and springs is referred to as a gridin this study.


3.3.1 Methodology

Initially the grid lies on a plane pi and the springs are at equilibrium dueto the absence of external forces. Then the grid is deformed around a rigidobject without get in contact with it. See Figure 3.2. When the grid isreleased, each spring of the grid produces a force ~Sk which is a function ofthe stress (k), the associated area Ak, and is in the direction of the nodesthat the spring is attached to:

~Sk = S ((k)Ak)~Njn ~Nop

~Njn ~Nop. (3.9)

The pair Njn, Nop represent the coordinates of the nodes which connects thespring k in the deformed grid.

To represent the stretch and warp properties of the fabric, the grid hasdifferent material properties along the x and y axes. These are represented bya stressstrain nonlinear functions (), : R R . This nonlinear stress-stain function () is obtained from a laboratory test and is approximatedby a piecewise linear function as is shown in Figure 3.5.

Every node Ni,j of the grid experience a set of forces caused by the sour-rounding springs. The possible configurations for a node are shown in Figure3.3. Any configuration of springs from Figure 3.3 is joined at a node Ni,j,which has been referred to as a join node (Njn). The rest of them have beennamed as opposite nodes (Nop).

Rigid objectnondeformed grid

Initial deformation

Figure 3.2: Deformed grid and the rigid object


Nop

N jn

Nop

Figure 3.3: Possible configurations of the interactions between springs. Two,three and four springs configurations.

The proposed spring elastic model has been programmed as an iterativealgorithm, where the attribute iteration_location of the element Ni,j isupdated at iteration t+ 1, as well as the attribute marker. the term i,jquantifies the quantity of movement of element Ni,j.

1: for each row of matrix N do2: for each column of matrix N do3: N t+1i,j = N

ti,j +

ti,j

4: end for5: end for

Algorithm 1: Definition of the iterative model programmed

At each iteration all the nodes of the grid are moved until a local equi-librium is achieved. The algorithm stops when a global equilibrium state isachieved at iteration t (see Section 3.3.3). The way of moving every nodeNi,j of the grid at iteration t depends on certain constraints which restrictits free movement. This will be discussed in the next section.

3.3.2 Node movement

Once the initial grid is deformed, every node Ni,j experiences a resultant

force ~R which push the node towards a equilibrium position. This force is a


Area Asociated to aspring located at the boundary

Area Asociated to aninterior spring

Figure 3.4: Areas associated to the springs

3D vector and is given by

~Ri,j =k

~Fk . (3.10)

The technique used to get a node Ni,j to a local equilibrium location is

to move it along the direction of ~Ri,j until the resultant force becomes zero

(~Ri,j = 0). The amount of movement in the Ri,j direction is defined by thestep size pi,j, which is a scalar value. Then getting an appropriate value forp becomes a main task of the model. If p is too small or too big, then thealgorithm may not converge.

The criteria chosen to compute the p-value is an average of the defor-mations of each spring which surrounds node Ni,j. The deformation d of aspring k is defined as the difference between the stretched length lst of springk and its unstretched length lun

dk = lst lun .

The step size pi,j for node Ni,j at iteration t is defined as

pi,j =

k around dk

c, (3.11)

where c is a constant defined by the user.


0

2e+06

4e+06

6e+06

8e+06

1e+07

1.2e+07

1.4e+07

0 0.2 0.4 0.6 0.8 1 1.2 1.4

stre

ss (M

pa)

strain

Non-linear functionpiecewise linear function

Figure 3.5: Approximation of the original non-linear function into a piecewiselinear function

The new location of the node Ni,j generated by ~Ri,j force is given by:

N(t+1)i,j = N

(t)i,j +

~R(t)i,j

~R(t)i,jp(t)i,j . (3.12)

The movement of Ni,j can be: (i) free, (ii) restricted by the surface of therigid object, and (iii) constrained by the initial boundary conditions.

Free movement

Nodes involved in this type of movement do not have any restriction. Theyare not in contact with a rigid object, neither are they constrained by theboundary conditions. They can move freely in the 3D space in the directionof vector ~R (see Figure 3.6).

At iteration t a specific node Ni,j can move freely, but at iteration t + 1it could be restricted by the surface of the rigid object.

Movement restricted by the surface

In particular, the spring elastic model being studied does not take into ac-count the forces caused by friction between the woven fabric and the surface.


R

Ni,j

i,j

Figure 3.6: free movement

If at iteration t a node Ni,j is in contact with the surface of the rigidobject, it can experience two kinds of movement depending on the directionof the resulting force ~Ri,j.

When the resulting force in the node over the surface Ni,j points towardthe outside of the object, that is the inner product between the vector normalto the surface n and the resulting force ~Ri,j is positive, then the node moves

in the direction of ~Ri,j and is free of contact with the surface(see figure 3.7a).In the contrary case, when the resulting force points toward the inside of thecontact object, that is the inner product between the vector normal to thesurface n and the resulting force ~Ri,j is negative, then the node Ni,j movesover the surface (see figure 3.7b).

The node Ni,j whose movement is over the surface, gets its local equilib-

rium when the component ~Ri,j of the vector ~Ri,j over the tangent plane tothe surface in the contact point is null. The node is moved in the direction ofcomponent ~Ri,j until this condition is achieved. When it occurs, the positionof the node Ni,j in the tangent plane is projected over the surface to get thefinal location of the node (see figure 3.8).

Constrained movement

The movement in the x, y, and z axes of some specific nodes can be con-strained. Any node can be restricted in more than one axis. Once a nodeNi,j is constrained, the movement in the restricted axis is omitted. Then 3.12only takes into account movement along the nonrestricted axis.


n^

Ni,j

n^

Ni,j

R i,j

R i,j

Figure 3.7: Movement restricted by the surface of the rigid object. On theleft case the resulting force is pointing away of the object and in the rightside case the force is pointing towards the inside of the object

3.3.3 Convergence criterion e

Let V Rs be a vector of dimension s = nm, the total number of nodesin the grid, if k = i n+ j then V is defined as

Vk =

{~R(i,j) if Ni,j is a non-contact node~R(i,j) if Ni,j is a contact node

.

That is Vk contains the magnitude of the resulting force ~Ri,j in the nodes atiteration t. If node Ni,j is in contact with the rigid body then Vk contains

the magnitude of the projection over the surface instead ~Ri,j.The grid will be in theoretical equilibrium when V = 0. Then, The

convergence criteria can be defined as

V e . (3.13)

The infinitynorm of vector V is less than some e R

3.4. Examples 47

R ij R

ij

Rij*Nij

n ^ n ^

Intersection with

Tangent plane

the surface

Figure 3.8: Movement over the surface. The force ~Ri,j over a node Ni,j in

contact with the surface is projected over a tangent plane to obtain ~Ri,j

3.4 Examples

This section presents two application examples of the springparticle modelof hiperelastic cloths. The first one consist of a simulation of contact betweena fabric and a thermoforming mold. The second one simulates the contactbetween a fabric and a woman body.

Figure 3.9 shows the initial and the final state of the interaction with thethermoforming mold. The stop criterion was satisfied after 400 iterations.

The interaction of the fabric with a woman torso is shown in Figure 3.10.Initially, the fabric was deformed to surround the torso. As this is a nonequilibrium state for the fabric it will deform until the resulting forces at thenodes are minimised. The simulation stop after 300 iterations.

3.4.1 Stresses over the woven fabric

When the fabric is taken to its initial deformed state (iteration t = 0), it isequivalent to apply a set of external forces over the nodes. When the mesh(cloth) is released then it moves in the opposite direction of the virtual loads.The final position will minimise the internal stress of the cloth. This modelconsiders only tension and contact forces (no friction). As the properties of afabric usually depend on its orientation, the stresses should be calculated on


(a) initial state (b) final state

Figure 3.9: Interaction between a thermoforming mold and a woven fabric

the weft and warp directions independently. The stress value at a node Ni,jis obtained as an average of the stresses from all the springs which surroundit. The value for the stress of spring k at iteration t is calculated as

k =SkAk

.

Then the stress at a node Ni,j is

i,j =

k around k

q

where q is the number of springs around node Ni,j.Figure 3.11a shows a mapping of the stresses over weft direction of the of

the fabric for the model which simulates the interaction with the torso andFigure 3.11b shows the stresses on the warp direction.

3.4.2 Deformation energy

The deformation energy E of the whole system is calculated as the summationof the energy of each spring. The energy of a single spring Ek is equal to thearea under the stressstrain curve and calculated using its piecewiselinearapproximation,

Ek =

k0

() d ,

3.4. Examples 49

(a) initial state (b) final state

Figure 3.10: Interaction between womans torso and woven fabric

then the total energy E is equal to

E =k

Ek .

The deformation energy was computed for the moldfabric example. Figure3.12 plots E versus the number of iterations. The higher levels of energy areat the initial deformation stage. The level of energy decreases very fast atthe initial iterations and then slow down. It is observed that the methodeffectively goes in the direction of minimum energy.


(a) Stresses over the weft direction ofthe fabric

(b) Stresses over the warp directionof the fabric

Figure 3.11: Stresses over the weft and warp directions of the fabric

Figure 3.12: Quantity of Energy vs Iterations

Chapter 4

Thermoforming Process

4.1 Problem description

Part of the current production process of Leonisa S.A. is the thermoformationof polyurethane for the manufacture of brassieres.

The thermoform process is done with an aluminum mold that containsthe deformation geometry and transfers heat to the material to generate apermanent deformation.

The molds are heated to a temperature in the range of 190C to maximumof 210C, according to the hyperelastic material to be thermoformed.

The following problems have been identified in the current process:

The molds take approximately 45 to 60 minutes to reach the worktemperature of approximately of 200C. During this time, the machineis not producing and therefore represents a high cost for the company.

The working surface of the mold that is in contact with the materialmust have a uniform temperature to guarantee control over the thick-ness and quality of the final product.

Figure 4.1 describes how the present mold system with two heatingplates, the thermoform mold and the insulating table.

The problem can be divided in several parts:

1. The molds take approximately 45 to 60 minutes to reach the workingtemperature of 200C;

51

52 Chapter 4. Thermoforming Process

Figure 4.1: Description of present assembly of the mold

2. The temperature of the mold must be uniform as possible to guaranteequality of the product;

3. The present molds have a high consumption of energy;

4. There is high thermal inertia in the present system.

4.2 Objectives

This part of the study aims to simulate the geometric model of the presentmold and the heat sources, to develop a thermal model of the present mold,to simulate the transient state of the present mold, to establish an alternativeconfiguration of the mold to obtain an improvement in the homogeneity ofthe distribution of temperature, to reduce warm-up time of the mold and toreduce power consumption.

4.3 Methodology

The development of the solution is based on the analysis of heat transfer byfinite elements (FEA). This methodology is divided in three principal phases:

1. Geometric modeling of the mold in the computer using CAD tools,

2. Finite elements modeling, which consists of:

4.4. Fundamentals of the thermoforming process 53

Thermal Int/ Out. Loads and boundary conditions like heat flux,adiabatic surfaces, or convection conditions,

Material properties of the mold and the hyperelastic material, Thermal Simulation. Transient and steady analysis using com-mercial software of finite element analysis, and

Postprocessing. Visualization and analysis of results.

3. Geometric optimization of the mold using CAD and CAE tools.

In the first part, the methodology is applied to the original mold to es-tablish a departure point of this research.The second part develops the samemethodology in the new mold.

The transient thermal behavior of the molds was obtained by finite el-ement analysis using a commercial software of FEA. In this way the realconditions of the molds were used to simulate and take results very close toreality.

4.4 Fundamentals of the thermoforming pro-

cess

Thermoforming is a process which involves the following phases:

1. Heating a sheet of hyperelastic material,

2. Cooling the formed item,

3. Trimming.

The most important part in this study is the heating of the thermoform-ing textile. This phase depends on the type and the intensity of the heatsource, the sheet temperature required for forming, and the properties of thematerial.

In this instance of the thermoforming process, the conduction heatingis used. This process is usually referred to as contact heating because thehyperelastic sheet is held in contact with a polished metal surface. The sheetis held between two heated metal plates until the forming temperature isreached and then is transferred to the mold.


4.5 Thermal model

The analysis of heat transfer is developed in different phases:

1. Heating of the mold from room temperature to reach the temperatureof the work surface;

2. Maintain work temperaturethe mold in this stage develops the pro-ductive process.

The thermal analysis of the mold is developed assuming only heat transferby conduction and convection; and it is concentrated mainly in the analysisof distribution of temperature and the balance of energy according to thefirst law of thermodynamics.

The temperature distribution T in all points of the mold is a functionof space and time, therefore T = f(x, y, z, t). For many problems, the tem-perature distribution in a domain in the subspace R3 is given by the heatbalance equation for a body, which is obtained by the conservation of energyprinciple ( see Equation 4.1)

T

t= k2T + Q

Cpon , (4.1)

where Cp is the specific heat, T is the temperature , K is the conductivity, is the density, and Q is the heat flux.

With the following boundary conditions:

T |1 = T (x, y, z),T

n

2

= q(x, y, z), T + T

n

3

= g(x, y, z)

and an initial condition

T (x, y, z; 0) =T0(x, y, z)

The factors of the analysis of the thermal model are:

1. consumption of energy of heating,

2. manufacture and joint of the components,

4.6. Finite element analysis 55

3. speed of mold heating,

4. energy lost in operation,

5. efficiency of injection of energy of the proposed design, and

6. homogeneity temperature in the mold.

4.6 Finite element analysis

The finite element analysis applies thermal analysis, which calculates thetemperature distribution and related thermal quantities in the thermoform-ing. The thermal quantities of interest are:

temperature distributions, amount of heat lost or gained, thermal gradients, and thermal fluxes.

Thermal simulations play an important role in the design of thermoform-ing molds.

The FEA supports two types of thermal analysis:

1. Steady-state thermal analysis determines the temperature distributionand other thermal quantities under steady-state loading conditions. Asteady-state loading condition is a situation where heat storage effectswhich vary over a period of time, can be ignored.

2. Transient thermal analysis determines the temperature distributionand other thermal quantities under conditions that vary over a periodof time.

4.6.1 Material

The properties of the materials used in the solution of problems are listed inTable 4.1.


Material Density Kgm3

Thermal conductivity WmK Specific heat

JKgK

Aluminum 2702 237 49Hyperelastic textil 30 0.023 1045

Table 4.1: Material properties

4.6.2 Meshing the model

In order to have the geometry from the CAD, the finite element analysisneeds to describe the element to mesh with the model.

The selected element is a tridimensional 10Node Tetrahedral ThermalSolid, this is well-suited to model irregular meshes (such as those producedby various CAD/CAM systems).

The element has one degree of freedom, at each node temperature:

Figure 4.2: 10node tetrahedral thermal solid element

The element is applicable to a tridimensional space, steady-state, or tran-sient thermal analysis (see the figure 4.2).

4.6.3 Loading and boundary conditions

The simulation by Finite Element Analysis needs to define the loads and theboundary conditions in the process of heating of the mold.

Three types of boundary conditions are applied to the mold. In theequations below are explained the boundary conditions in detail:

4.6. Finite element analysis 57

T |1 = T (x, y, z) Dirichlet (4.2)T

n

2

=< T, n > = q(x, y, z) Von-Newman (4.3)

T + T

n

3

= g(x, y, z) Fourier. (4.4)

The heat flux (Q) is generated by electrical resistance that provides theenergy necessary for the process, this is the first boundary condition ex-plained by Von-Newman Equation 4.3.

The second boundary is the energy lost from the lateral walls of themold by the convection between the mold surface and the air, this heatflow depends on the temperature of the surface and the temperature of theenvironment. Convective boundary conditions are applied by the Fourierequation, see 4.4.

The film coefficient (h) was determined in the last study developed in2003 by a experimental process, see figure 4.3

Figure 4.3: Film coeficiente vs. Temperature h(T )


4.6.4 Solution

This process consists of given the boundary conditions like heat flux andheat transfer by convection. The finite element analysis produces a temper-ature distribution in the mold. The solution process uses only heat transferby convection and conduction and gives the temperature in each node at aspecific time.

4.6.5 Postprocessing

After building the model and obtaining the solution, the study still requiresanswers to some critical questions of the redesign process. This part of thework is important because it is very useful for seeing results and mistakes inthe simulation.

Postprocessing means reviewing the results of the thermal analysis. Thisis probably the most important step in the analysis. It is very useful forunderstanding how the applied loads affect the design of the mold, how goodthe finite element mesh is, and so on.

4.7 Original mold

The original mold assume a heat flow of 3000 W on each plate as well as aboundary condition of the convection of the faces exposed to the air.

Transient state analysis from the mold at room temperature to worktemperature, approximately 200 Celsius grades, was developed.

The results obtained by the simulation of the original mold are in Figure4.4, where the mold needs heating time of 50 minutes prior to production.

4.8 Proposed mold and results

The new mold looks to the improvement of energy consumption, even thedistribution of temperature, and emphasizes reduction in warm-up time.

The main elements and the changes in the new mold are:

The energy source is located near of the work surface in the compart-ments for the electrical resistance.

4.8. Proposed mold and results 59

Figure 4.4: Temperature distribution in the original mold

The heat source is isolated by air chambers, that isolate and direct theheat flow towards the material.

Figure 4.5 illustrates how the mold reduces the heating time by about 20minutes with less power to arrive at the work temperature.

The new mold assumes a heat flow of 750 W of each heating chamber.Therefore the new mold requires one 3000 W.

The modeling by finite elements showed that while the power contributedto the system, distribution of temperature is regulated 200 C +/- 1.5 C,which is in accord with the variation of temperature allowed for the process.

Figure 4.6 shows as the mold delays approximately 20 minutes to arriveat the temperature of work and uses less power available to arrive at thetemperature of work.

Other important thing is the geometry of the material inside of the mold,the original mold do not have a constant thickness, therefore the thermoform-ing material could have problems and the quality could not be very good tothe process, because de temperature could be very high where the thicknessis smaller.

A geometry generate by a offset of the mold surface has a constant thick-ness and this could help to improve the quality of the product.


Figure 4.5: Proposed mold

Figure 4.6: Proposed mold: Temperature distribution on work surface

Chapter 5

Thermoforming ContactAnalysis

This chapter presents a procedure to calculate the contact between the maleand female molds during the process of thermoforming of fabrics. Thethermoforming process transforms a flat uniform piece of fabric into a 2Dmanifold of variable thickness. The fabric consists of a sandwich of foambetween two layers of Lycra. The deformed fabric is soft but firm and it isused in brassiere to enhance the shape of the body.

In thermoforming of fabrics the shape is obtained through two basic prin-ciples: heat and pressure. The fabric is pressed against the male and femalemolds previously pre-heated at temperatures of approximately 200oC. Theyare left under pressure for a period of time and then released. The plane fab-ric deforms permanently into a curved shell adopting its final shape after itcools down. The process is very sensitive to pressure, and temperature. Thedistribution of these variables over the surface of the molds will determinethe final thickness at each point of the fabric. The proper control of thesecontinuous variables can be used to produce anatomical shapes or shapesthat help to highlight body contours. However,

2004_hiperelasticity_techreport

Documents

hyperelastic cloth

vector r

uniaxial tension test

uniaxial test tension

problem description

node experimentsp

stress of node i

energy of spring ke