elasticity and viscoelasticity - elsevier · 2013. 6. 26. · viscoelasticity considers in addition...

CHAPTER 2

Elasticity andViscoelasticity

C H A P T E R 2.1

Introduction to Elasticityand ViscoelasticityJEAN LEMAITRE

Universit!e Paris 6, LMT-Cachan, 61 avenue du Pr!esident Wilson, 94235 Cachan Cedex, France

For all solid materials there is a domain in stress space in which strains arereversible due to small relative movements of atoms. For many materials likemetals, ceramics, concrete, wood and polymers, in a small range of strains, thehypotheses of isotropy and linearity are good enough for many engineeringpurposes. Then the classical Hooke’s law of elasticity applies. It can be de-rived from a quadratic form of the state potential, depending on twoparameters characteristics of each material: the Young’s modulus E andthe Poisson’s ratio n.

c* ¼ 1

2rAijklðE;nÞsijskl ð1Þ

eij ¼ r@c*

@sij¼ 1 þ n

Esij �

nEskkdij ð2Þ

E and n are identified from tensile tests either in statics or dynamics. A greatdeal of accuracy is needed in the measurement of the longitudinal andtransverse strains (de � �10�6 in absolute value).

When structural calculations are performed under the approximation ofplane stress (thin sheets) or plane strain (thick sheets), it is convenient towrite these conditions in the constitutive equation.

� Plane stress ðs33 ¼ s13 ¼ s23 ¼ 0Þ:

e11

e22

e12

264

375 ¼

1

E� n

E0

1

E0

Sym1 þ n

E

2666666664

3777777775

s11

s22

s12

264

375 ð3Þ

Handbook of Materials Behavior ModelsCopyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 71

� Plane strain ðe33 ¼ e13 ¼ e23 ¼ 0Þ:s11

s22

s12

264

375 ¼

lþ 2m l 0

lþ 2m 0

Sym 2m

264

375

e11

e22

e12

264

375

with

l ¼ nEð1 þ nÞð1 � 2nÞ

m ¼ E

2ð1 þ nÞ

8>>><>>>:

ð4Þ

For orthotropic materials having three planes of symmetry, nineindependent parameters are needed: three tension moduli E1; E2; E3

in the orthotropic directions, three shear moduli G12; G23; G31, andthree contraction ratios n12; n23; n31. In the frame of orthotropy:

e11

e22

e33

e23

e31

e12

266666666666666666666664

377777777777777777777775

¼

1

E1� n12

E1� n13

E10 0 0

1

E2� n23

E20 0 0

1

E30 0 0

1

2G230 0

Sym1

2G310

1

2G12

266666666666666666664

377777777777777777775

s11

s22

s33

s23

s31

s12

266666666666666666666664

377777777777777777777775

ð5Þ

Nonlinear elasticity in large deformations is described in Section 2.2,with applications for porous materials in Section 2.3 and for elastomersin Section 2.4.

Thermoelasticity takes into account the stresses and strains induced bythermal expansion with dilatation coefficient a. For small variations oftemperature y for which the elasticity parameters may be consideredas constant:

eij ¼1 þ n

Esij �

nEskkdij þ aydij ð6Þ

For large variations of temperature, E; n; and a will vary. In rateformulations, such as are needed in elastoviscoplasticity, for example, the

Lemaitre72

derivative of E; n; and a must be considered.

’eij ¼1 þ n

E’sij �

nE’skkdij þ a’ydij þ

@

@y1 þ n

E

� �sij �

@

@ynE

�skkdij þ

@a@y

ydij

� �’y

ð7ÞViscoelasticity considers in addition a dissipative phenomenon due to

‘‘internal friction,’’ such as between molecules in polymers or between cells inwood. Here again, isotropy, linearity, and small strains allow for simplemodels. Quadratric functions for the state potential and the dissipativepotential lead to either Kelvin-Voigt or Maxwell’s models, depending upon thepartition of stress or strains in a reversible part and in an irreversible part.They are described in detail for the one-dimensional case in Section 2.5 andrecalled here in three dimensions.

� Kelvin-Voigt model:

sij ¼ lðekk þ yl’ekkÞdij þ 2mðeij þ ym’eijÞ ð8ÞHere l and m are Lame’s coefficients at steady state, and yl and ym are twotime parameters responsible for viscosity. These four coefficients may beidentified from creep tests in tension and shear.

� Maxwell model:

’eij ¼1 þ n

E’sij þ

st1

� �� n

E’skk þ

skk

t2

� �dij ð9Þ

Here E and n are Young’s modulus and Poisson’s ratio at steady state, andt1 and t2 are two other time parameters. It is a fluidlike model:equilibrium at constant stress does not exist.

In fact, a more general way to write linear viscoelastic constitutivemodels is through the functional formulation by the convolution product asany linear system. The hereditary integral is described in detail for theone-dimensional case, together with its use by the Laplace transform, inSection 2.5.

eijðtÞ ¼Z t

o

Jijklðt � tÞ dskl

dtdtþ

Xn

p¼1

Jijklðt � tÞDspkl ð10Þ

JðtÞ� �

is the creep functions matrix, and Dspkl are the eventual stress steps.

The dual formulation introduces the relaxation functions matrix RðtÞ� �

sijðtÞ ¼Z t

o

Rijklðt � tÞ dekl

dtdtþ

Xn

p¼1

RijklDepkl ð11Þ

When isotropy is considered the matrix, J½ and R½ each reduce to twofunctions: either JðtÞ, the creep function in tension, is identified from a creep

2.1 Introduction to Elasticity and Viscoelasticity 73

test at constant stress; JðtÞ ¼ eðtÞ=s and K, the second function, from thecreep function in shear. This leads to

eij ¼ ð J þ KÞ � Dsij

Dt� K�

Dskk

Dtdij ð12Þ

where � stands for the convolution product and D for the distributionderivative, taking into account the stress steps.

Or MðtÞ, the relaxation function in shear, and LðtÞ, a function deducedfrom M and from a relaxation test in tension RðtÞ ¼ sðtÞ=e; LðtÞ ¼MðR � 2MÞ=ð3M � RÞ

sij ¼ L � DðekkÞDt

dij þ 2M � Deij

dtð13Þ

All of this is for linear behavior. A nonlinear model is described inSection 2.6, and interaction with damage is described in Section 2.7.

Lemaitre74

C H A P T E R 2.2

Background onNonlinear ElasticityR. W. OGDEN

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

Contents2.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.2.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.2.3 Stress and Equilibrium . . . . . . . . . . . . . . . . . . . . . . 77

2.2.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.2.5 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.2.6 Constrained Materials . . . . . . . . . . . . . . . . . . . . . . . 80

2.2.7 Boundary-Value Problems . . . . . . . . . . . . . . . . . . . . 82

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.2.1 VALIDITY

The theory is applicable to materials, such as rubberlike solids and certain softbiological tissues, which are capable of undergoing large elastic deformations.More details of the theory and its applications can be found in Beatty [1]and Ogden [3].

2.2.2 DEFORMATION

For a continuous body, a reference configuration, denoted by Br, is identifiedand @Br denotes the boundary of Br. Points in Br are labeled by theirposition vectors X relative to some origin. The body is deformed quasi-statically from Br so that it occupies a new configuration, denoted B, with


boundary @B. This is the current or deformed configuration of the body. Thedeformation is represented by the mapping w : Br ! B, so that

x ¼ vðXÞ X 2 Br; ð1Þwhere x is the position vector of the point X in B. The mapping v is called thedeformation from Br to B, and v is required to be one-to-one and to satisfyappropriate regularity conditions. For simplicity, we consider only Cartesiancoordinate systems and let X and x, respectively, have coordinates Xa and xi,where a; i 2 f1; 2; 3g, so that xi ¼ wiðXaÞ. Greek and Roman indices refer,respectively, to Br and B, and the usual summation convention for repeatedindices is used.

The deformation gradient tensor, denoted F, is given by

F ¼ Grad x Fia ¼ @xi=@Xa ð2ÞGrad being the gradient operator in Br. Local invertibility of v and its inverserequires that

05 J � det F51 ð3Þwherein the notation J is defined.

The deformation gradient has the (unique) polar decompositions

F ¼ RU ¼ VR ð4Þwhere R is a proper orthogonal tensor and U, V are positive definite andsymmetric tensors. Respectively, U and V are called the right and left stretchtensors. They may be put in the spectral forms

U ¼X3

i¼1

liuðiÞ � uðiÞ V ¼

X3

i¼1

livðiÞ � vðiÞ ð5Þ

where vðiÞ ¼ RuðiÞ; i 2 f1; 2; 3g, li are the principal stretches, uðiÞ the uniteigenvectors of U (the Lagrangian principal axes), vðiÞ those of V (the Eulerianprincipal axes), and � denotes the tensor product. It follows from Eq. 3 thatJ ¼ l1l2l3:

The right and left Cauchy-Green deformation tensors, denoted C and B,respectively, are defined by

C ¼ FTF ¼ U2 B ¼ FFT ¼ V2 ð6Þ

2.2.3 STRESS AND EQUILIBRIUM

Let rr and r be the mass densities in Br and B, respectively. The massconservation equation has the form

rr ¼ rJ ð7Þ

Ogden76

The Cauchy stress tensor, denoted r, and the nominal stress tensor, denotedS, are related by

S ¼ JF�1r ð8Þ

The equation of equilibrium may be written in the equivalent forms

div r þ rb ¼ 0 Div S þ rrb ¼ 0 ð9Þ

where div and Div denote the divergence operators in B and Br, respectively,and b denotes the body force per unit mass. In components, the secondequation in Eq. 9 is

@Sai

@Xaþ rrbi ¼ 0 ð10Þ

Balance of the moments of the forces acting on the body yields simplyrT ¼ r, equivalently STFT ¼ FS: The Lagrangian formulation based on theuse of S and Eq. 10, with X as the independent variable, is used henceforth.

2.2.4 ELASTICITY

The constitutive equation of an elastic material is given in the equivalentforms

S ¼ HðFÞ ¼ @W

@FðFÞ r ¼ GðFÞ � J�1FHðFÞ ð11Þ

where H is a tensor-valued function, defined on the space of deformationgradients F, W is a scalar function of F and the symmetric tensor-valuedfunction G is defined by the latter equation in Eq. 11. In general, the form ofH depends on the choice of reference configuration and it is referred to as theresponse function of the material relative to Br associated with S. For a givenBr, therefore, the stress in B at a (material) point X depends only on thedeformation gradient at X. A material whose constitutive law has the form ofEq. 11 is generally referred to as a hyperelastic material and W is called astrain-energy function (or stored-energy function). In components, (11)1 hasthe form Sai ¼ @W=@Fia, which provides the convention for ordering of theindices in the partial derivative with respect to F.

If W and the stress vanish in Br, so that

WðIÞ ¼ 0@W

@FðIÞ ¼ O ð12Þ

where I is the identity and O the zero tensor, then Br is called a naturalconfiguration.

2.2 Background on Nonlinear Elasticity 77

Suppose that a rigid-body deformation x* ¼ Qx þ c is superimposed onthe deformation x ¼ vðXÞ, where Q and c are constants, Q being a rotationtensor and c a translation vector. The resulting deformation gradient, F * say,is given by F* ¼ QF: The elastic stored energy is required to be independentof superimposed rigid deformations, and it follows that

WðQFÞ ¼ WðFÞ ð13Þ

for all rotations Q. A strain-energy function satisfying this requirement is saidto be objective.

Use of the polar decomposition (Eq. 4) and the choice Q ¼ RT in Eq. 13shows that WðFÞ ¼ WðUÞ: Thus, W depends on F only through the stretchtensor U and may therefore be defined on the class of positive definitesymmetric tensors. We write

T ¼ @W

@Uð14Þ

for the (symmetric) Biot stress tensor, which is related to S byT ¼ ðSR þ RTSTÞ=2.

2.2.5 MATERIAL SYMMETRY

Let F and F0 be the deformation gradients in B relative to two differentreference configurations, Br and B0

r respectively. In general, the response ofthe material relative to B0

r differs from that relative to Br, and we denote by Wand W 0 the strain-energy functions relative to Br and B0

r. Now let P ¼ Grad X0

be the deformation gradient of B0r relative to Br, where X0 is the position

vector of a point in B0r. Then F ¼ F0P: For specific P we may have W 0 ¼ W,

and then

WðF0PÞ ¼ WðF0Þ ð15Þ

for all deformation gradients F0. The set of tensors P for which Eq. 15 holdsforms a multiplicative group, called the symmetry group of the material relativeto Br. This group characterizes the physical symmetry properties ofthe material.

For isotropic elastic materials, for which the symmetry group is the properorthogonal group, we have

WðFQÞ ¼ WðFÞ ð16Þ

for all rotations Q. Since the Q’s appearing in Eqs. 13 and 16 are independent,the combination of these two equations yields

WðQUQTÞ ¼ WðUÞ ð17Þ

Ogden78

for all rotations Q. Equation 17 states that W is an isotropic function of U. Itfollows from the spectral decomposition (Eq. 5) that W depends on U onlythrough the principal stretches l1; l2, and l3 and is symmetric in thesestretches.

For an isotropic elastic material, r is coaxial with V, and we may write

r ¼ a0I þ a1B þ a2B2 ð18Þ

where a0; a1, and a2 are scalar invariants of B (and hence of V) given by

a0 ¼ 2I1=23

@W

@I3a1 ¼ 2I

�1=23

@W

@I1þ I1

@W

@I2

� �a2 ¼ �2I

�1=23

@W

@I2ð19Þ

and W is now regarded as a function of I1; I2, and I3, the principal invariantsof B defined by

I1 ¼ trðBÞ ¼ l21 þ l2

2 þ l23; ð20Þ

I2 ¼ 12 ½I2

1 � tr ðB2Þ ¼ l22l

23 þ l2

3l21 þ l2

1l22 ð21Þ

I3 ¼ det B ¼ l21l

22l

23 ð22Þ

Another consequence of isotropy is that S and r have the decompositions

S ¼X3

i¼1

tiuðiÞ � vðiÞ r ¼

X3

i¼1

sivðiÞ � vðiÞ ð23Þ

where si; i 2 f1; 2; 3g are the principal Cauchy stresses and ti the principalBiot stresses, connected by

ti ¼@W

@li¼ Jl�1

i si ð24Þ

Let the unit vector M be a preferred direction in the reference configurationof the material, i.e., a direction for which the material response is indifferentto arbitrary rotations about the direction and to replacement of M by �M.Such a material can be characterized by a strain energy which depends on Fand the tensor M � M [2, 4, 5] Thus, we write WðF; M � MÞ. The requiredsymmetry (transverse isotropy) reduces W to dependence on the five invariants

I1; I2; I3; I4 ¼ M � ðCMÞ I5 ¼ M � ðC2MÞ ð25Þ

where I1; I2; and I3 are defined in Eqs. (20)–(22). The resulting nominalstress tensor is given by

S ¼ 2W1FT þ 2W2ðI1I � CÞFT þ 2I3W3F�1 þ 2W4M � FM

þ 2W5ðM � FCM þ CM � FMÞ ð26Þwhere Wi ¼ @W=@Ii; i ¼ 1; . . . ; 5.


When there are two families of fibers corresponding to two preferreddirections in the reference configuration, M and M0 say, then, in addition toEq. 25, the strain energy depends on the invariants

I6 ¼ M0 � ðCM0Þ I7 ¼ M0 � ðC2M0Þ I8 ¼ M � ðCM0Þ ð27Þ

and also on M � M0 (which does not depend on the deformation); see Spencer[4, 5] for details. The nominal stress tensor can be calculated in a similarway to Eq. 26.

2.2.6 CONSTRAINED MATERIALS

An internal constraint, given in the form CðFÞ ¼ 0, must be satisfiedfor all possible deformation gradients F, where C is a scalar function. Twocommonly used constraints are incompressibility and inextensibility, forwhich, respectively,

CðFÞ ¼ detF � 1 CðFÞ ¼ M � ðFTFMÞ � 1 ð28Þ

where the unit vector M is the direction of inextensibility in Br. Since anyconstraint is unaffected by a superimposed rigid deformation, C must be anobjective scalar function, so that CðQFÞ ¼ CðFÞ for all rotations Q.

Any stress normal to the hypersurface CðFÞ ¼ 0 in the (nine-dimensional)space of deformation gradients does no work in any (virtual) incrementaldeformation compatible with the constraint. The stress is thereforedetermined by the constitutive law (11)1 only to within an additivecontribution parallel to the normal. Thus, for a constrained material, thestress-deformation relation (11)1 is replaced by

S ¼ HðFÞ þ q@C

@F¼ @W

@Fþ q

@C

@Fð29Þ

where q is an arbitrary (Lagrange) multiplier. The term in q is referred to asthe constraint stress since it arises from the constraint and is not otherwisederivable from the material properties.

For incompressibility and inextensibility we have

S ¼ @W

@Fþ qF�1 S ¼ @W

@Fþ 2qM � FM ð30Þ

respectively. For an incompressible material the Biot and Cauchy stresses aregiven by

T ¼ @W

@U� pU�1 det U ¼ 1 ð31Þ

Ogden80

and

r ¼ F@W

@F� pI det F ¼ 1 ð32Þ

where q has been replaced by �p, which is called an arbitrary hydrostaticpressure. The term in a0 in Eq. 18 is absorbed into p, and I3 ¼ 1 in theremaining terms in Eq. 18. For an incompressible isotropic material theprincipal components of Eqs. 31 and 32 yield

ti ¼@W

@li� pl�1

i si ¼ li@W

@li� p ð33Þ

respectively, subject to l1l2l3 ¼ 1.For an incompressible transversely isotropic material with preferred

direction M, the dependence on I3 is omitted and the Cauchy stress tensoris given by

r ¼ � pI þ 2W1B þ 2W2ðI1B � B2Þ þ 2W4FM � FM

þ 2W5ðFM � BFM þ BFM � FMÞ ð34ÞFor a material with two preferred directions, M and M0, the Cauchy stresstensor for an incompressible material is

r ¼ � pI þ 2W1B þ 2W2ðI1B � B2Þ þ 2W4FM � FM

þ 2W5ðFM � BFM þ BFM � FMÞ þ 2W6FM0 � FM0

þ 2W7ðFM0 � BFM0 þ BFM0 � FM0Þþ W8ðFM � FM0 þ FM0 � FMÞ ð35Þ

where the notation Wi ¼ @W=@Ii now applies for i ¼ 1; 2; 4; . . . ; 8.

2.2.7 BOUNDARY-VALUE PROBLEMS

The equilibrium equation (second part of Eq. 9), the stress-deformationrelation (Eq. 11), and the deformation gradient (Eq. 2) coupled with Eq. 1 arecombined to give

Div@W

@F

� �þ rrb ¼ 0 F ¼ Grad x x ¼ vðXÞ X 2 Br ð36Þ

Typical boundary conditions in nonlinear elasticity are

x ¼ nðXÞ on @Bxr ð37Þ

STN ¼ sðF;XÞ on @Btr ð38Þ

where n and s are specified functions, N is the unit outward normal to @Br,


and @Bxr and @Bt

r are complementary parts of @B. In general, s may dependon the deformation through F. For a dead-load traction s is independent of F.For a hydrostatic pressure boundary condition, Eq. 38 has the form

s ¼ �JPF�TN on @Btr ð39Þ

Equations 36–38 constitute the basic boundary-value problem in non-linear elasticity.

In components, the equilibrium equation in Eq. 36 is written

Aaibj@2xj

@Xa@Xbþ rrbi ¼ 0 ð40Þ

for i 2 f1; 2; 3g, where the coefficients Aaibj are defined by

Aaibj ¼ Abjai ¼@2W

@Fia@Fjbð41Þ

When coupled with suitable boundary conditions, Eq. 41 forms a system ofquasi-linear partial differential equations for xi ¼ wiðXaÞ. The coefficientsAaibj are, in general, nonlinear functions of the components of thedeformation gradient.

For incompressible materials the corresponding equations are obtained bysubstituting the first part of Eq. 30 into the second part of Eq. 9 to give

Aaibj@2xj

@Xa@Xb� @p

@xiþ rrbi ¼ 0 det @xi@Xað Þ ¼ 1 ð42Þ

where the coefficients are again given by Eq. 41.In order to solve a boundary-value problem, a specific form of W needs to

be given. The form of W chosen will depend on the particular materialconsidered and on mathematical requirements relating to the properties ofthe equations, an example of which is the strong ellipticity condition.Equations 40 are said to be strongly elliptic if the inequality

AaibjmimjNaNb > 0 ð43Þholds for all nonzero vectors m and N. Note that Eq. 43 is independentof any boundary conditions. For an incompressible material, thestrong ellipticity condition associated with Eq. 42 again has the form ofEq. 43, but the incompressibility constraint now imposes the restrictionm � ðF�TNÞ ¼ 0 on m and N.

REFERENCES

1. Beatty, M. F. (1987). Topics in finite elasticity: Hyperelasticity of rubber, elastomers and

biological tissues } with examples. Appl. Mech. Rev. 40; 1699–1734.

Ogden82

2. Holzapfel, G. A. (2000). Nonlinear Solid Mechanics. Chichester: Wiley.

3. Ogden, R. W. (1997). Non-linear Elastic Deformations. New York: Dover Publications.

4. Spencer, A. J. M. (1972). Deformations of Fibre-Reinforced Materials. Oxford: Oxford University

Press.

5. Spencer, A. J. M. (1984). Constitutive theory for strongly anisotropic solids. In Continuum

Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures No. 282,

pp. 1–32, Spencer, A. J. M., ed., Wien: Springer-Verlag.


C H A P T E R 2.3

Elasticity of PorousMaterialsN. D. CRISTESCU

231 Aerospace Building, University of Florida, Gainesville, Florida


2.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.3.3 Identification of the Parameters . . . . . . . . . . . . . . 85

2.3.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.3.1 VALIDITY

The methods used to determine the elasticity of porous materials and/orparticulate materials as geomaterials or powderlike materials are distinct fromthose used with, say, metals. The reason is that such materials possess poresand=or microcracks. For various stress states these may either open or closed,thus influencing the values of the elastic parameters. Also, the stress-straincurves for such materials are strongly loading-rate-dependent, starting fromthe smallest applied stresses, and creep (generally any time-dependentphenomena) is exhibited from the smallest applied stresses (see Fig. 2.3.1 forschist, showing three uniaxial stress-strain curves for three loading rates and acreep curve [1]). Thus information concerning the magnitude of the elasticparameters cannot be obtained:

� from the initial slope of the stress-strain curves, since these are loading-rate-dependent;

� by the often used ‘‘chord’’ procedure, obviously;� from the unloading slopes, since significant hysteresis loops are

generally present.

Handbook of Materials Behavior ModelsCopyright # 2001 by Academic Press. All rights of reproduction in any form reserved.84

2.3.2 FORMULATION

The elasticity of such materials can be expressed as ‘‘instantaneousresponse’’ by

D ¼’T

2Gþ 1

3K� 1

2G

� �1

3ðtr ’TÞ

%1 ð1Þ

where G and K are the elastic parameters that are not constant, D is the strainrate tensor, T is the stress tensor, tr( ) is the trace operator, and 1 is the unittensor. Besides the elastic properties described by Eq. 1, some othermechanical properties can be described by additional terms to be added toEq. 1. For isotropic geomaterials the elastic parameters are expected todepend on stress invariants and, perhaps, on some damage parameters, sinceduring loading some pores and microcracks may close or open, thusinfluencing the elastic parameters.

2.3.3 IDENTIFICATION OF THE PARAMETERS

The elastic parameters can be determined experimentally by two procedures.With the dynamic procedure, one is determining the travel time of the two

FIGURE 2.3.1 Uniaxial stress-strain curves for schist for various loading rates, showing time

influence on the entire stress-strain curves and failure (stars mark the failure points).

2.3 Elasticity of Porous Materials 85

elastic (seismic) extended longitudinal and transverse waves, whichare traveling in the body. If both these waves are recorded, then theinstantaneous response is of the form of Eq. 1. The elastic parameters areobtained from

K ¼ r v2p �

4

3v2

S

� �G ¼ rv2

S ð2Þ

where vS is the velocity of propagation of the shearing waves, vp the velocity ofthe longitudinal waves, and r the density.

The static procedure takes into account that the constitutive equations forgeomaterials are strongly time-dependent. Thus, in triaxial tests performedunder constant confining pressure s, after loading up to a desired stress state t(octahedral shearing stress), one is keeping the stress constant for a certaintime period tc [2, 3]. During this time period the rock is creeping. When thestrain rates recorded during creep become small enough, one is performing anunloading–reloading cycle (see Fig. 2.3.2). From the slopes

1

3Gþ 1

9K

� ��1

� 1

6Gþ 1

9K

� ��1

ð3Þ

of these unloading–reloading curves one can determine the elastic parameters.For each geomaterial, if the time tc is chosen so that the subsequent unloadingis performed in a comparatively much shorter time interval, no significantinterference between creep and unloading phenomena will take place. Anexample for schist is shown in Figure 2.3.3, obtained in a triaxial test with fiveunloading–reloading cycles.

FIGURE 2.3.2 Static procedures to determine the elastic parameters from partial unloading

processes preceded by short-term creep.

Cristescu86

If only a partial unloading is performed (one third or even one quarter ofthe total stress, and sometimes even less), the unloading and reloading followquite closely straight lines that practically coincide. If a hysteresis loop is stillrecorded, it means that the time tc was chosen too short. The reason forperforming only a partial unloading is that the specimen is quite ‘‘thick’’ andas such the stress state in the specimen is not really uniaxial. During completeunloading, additional phenomena due to the ‘‘thickness’’ of the specimenwill be involved, including, e.g., kinematic hardening in the oppositedirection, etc.

Similar results can be obtained if, instead of keeping the stress constant,one is keeping the axial strain constant for some time period during which theaxial stress is relaxing. Afterwards, when the stress rate becomes relativelysmall, an unloading–reloading is applied to determine of the elasticparameters. This procedure is easy to apply mainly for particulate materials(sand, soils, etc.) when standard (Karman) three-axial testing devices are usedand the elastic parameters follow from

K ¼ 1

3

DtDe1 þ 2De2

G ¼ 1

2

DtDe1 � De2

ð4Þ

where D is the variation of stress and elastic strains during the unloading–reloading cycle. The same method is used to determine the bulk modulus K inhydrostatic tests when the formula to be used is

K ¼ DsDev

ð5Þ

with s the mean stress and ev the volumetric strain.Generally, K is increasing with s and reaching an asymptotic constant value

when s is increasing very much and all pores and microcracks are closed

FIGURE 2.3.3 Stress-strain curves obtained in triaxial tests on shale; the unloadings follow a

period of creep of several minutes.


under this high pressure. The variation of the elastic parameters with t ismore involved: when t increases but is still under the compressibility–dilatancy boundary, the elastic parameters are increasing. For higher values,above this boundary, the elastic parameters are decreasing. Thus theirvariation is related to the variation of irreversible volumetric strain, which, inturn, is describing the evolution of the pores and microcracks existing in thegeomaterial. That is why the compressibility–dilatancy boundary plays therole of reference configuration for the values of the elastic parameters so longas the loading path (increasing s and=or t) is in the compressibility domain,the elastic parameters are increasing, whereas if the loading path is in thedilatancy domain (increasing under constant s), the elastic parameters aredecreasing. If stress is kept constant and strain is varying by creep, in thecompressibility domain volumetric creep produces a closing of pores andmicrocracks and thus the elastic parameters increase, and vice versa in thedilatancy domain. Thus, for each value of s the maximum values of the elasticparameters are reached on the compressibility–dilatancy boundary.

2.3.4 EXAMPLES

As an example, for rock salt in uniaxial stress tests, the variation of the elasticmoduli G and K with the axial stress s1 is shown in Figure 2.3.4 [4]. Thevariation of G and K is very similar to that of the irreversible volumetric

FIGURE 2.3.4 Variation of the elastic parameters K and G and of irreversible volumetric strain

in monotonic uniaxial tests.

Cristescu88

strain eIV . If stress is increased in steps, and if after each increase the stress in

kept constant for two days, the elastic parameters are varying duringvolumetric creep, as shown in Figure 2.3.5. Here D is the ratio of the appliedstress and the strength in uniaxial compression sc ¼ 17:88 MPa. Again, asimilarity with the variation of eI

V is quite evident. Figure 2.3.6 shows for adifferent kind of rock salt the variation of the elastic velocities vP and vS intrue triaxial tests under confining pressure pc ¼ 5 MPa (data by Popp,Schultze, and Kern [5]). Again, these velocities increase in the compressibilitydomain, reach their maxima on the compressibility–dilatancy boundary, andthen decrease in the dilatancy domain.

For shale, and the conventional (Karman) triaxial tests shown inFigure 2.3.3, the values of E and G for the five unloading–reloading cyclesshown are: E ¼ 9:9, 24.7, 29.0, 26.3, and 22.3 GPa, respectively, whileG ¼ 4:4, 10.7, 12.1, 10.4, and 8.5 GPa.

For granite, the variation of K with s is given as [2]

KðsÞ :¼K0 � K1 1 � s

s0

� �; if s � s0

K0; if s � s0

8><>: ð6Þ

with K0 ¼ 59 GPa, K1 ¼ 48 GPa, and s0 ¼ 0:344 GPa, the limit pressure whenall pores are expected to be closed.

FIGURE 2.3.5 Variation in time of the elastic parameters and of irreversible volumetric strain in

uniaxial creep tests.


The same formula for alumina powder is

KðsÞ :¼ K1 � pa exp �bspa

� �ð7Þ

with K1 ¼ 1� 107 kPa the constant value toward which the bulk modulustends at high pressures, a ¼ 107, b ¼ �1:2 � 10�4, and pa ¼ 1 kPa. Also foralumina powder we have

E sð Þ :¼ E1 � pab expð�dsÞ ð8Þwith E1 ¼ 7� 105 kPa, b ¼ 6:95� 105, and d ¼ 0:002.

For the shale shown in Figure 2.3.3, the variation of K with s for0 � s � 45 MPa is

K sð Þ :¼ �0:78s2 þ 65:32s� 369 ð9Þ

REFERENCES

1. Cristescu, N. (1986). Damage and failure of viscoplastic rock-like materials. Int. J. Plasticity

2 (2): 189–204.

2. Cristescu, N. (1989). Rock Rheology, Kluver Academic Publishing.

3. Cristescu, N. D., and Hunsche, U. (1998). Time Effects in Rock Mechanics, Wiley.

4. Ani, M., and Cristescu N. D. (2000). The effect of volumetric strain on elastic parameters for

rock salt. Mechanics of Cohesive-Frictional Materials 5 (2): 113–124.

5. Popp, T., Schultze, O., and Kern, H. ( ). Permeation and development of dilatancy and

permeability in rock salt, in The Mechanical Behavior of Salt (5th Conference on Mechanical

Behavior of Salt), Cristescu, N. D., and Hardy, Jr., H. Reginald, eds., Trans Tech Publ.,

Clausthal-Zellerfeld.

FIGURE 2.3.6 The maximum of vs takes place at the compressibility–dilatancy boundary

(figures and hachured strip); changes of vp and vs as a function of strain (’e ¼ 10�5 s�1,

pc ¼ 5 Mpa, T ¼ 308C), showing that the maxima are at the onset of dilatancy (after

Reference [4]).

Cristescu90

C H A P T E R 2.4

Elastomer ModelsR. W. OGDEN

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK


2.4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.4.3 Description of the Model. . . . . . . . . . . . . . . . . . . . . 93

2.4.4 Identification of Parameters . . . . . . . . . . . . . . . . . . 93

2.4.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.4.6 Table of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 94

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.4.1 VALIDITY

Many rubberlike solids can be treated as isotropic and incompressible elasticmaterials to a high degree of approximation. Here describe the mechanicalproperties of such solids through the use of an isotropic elastic strain-energyfunction in the context of finite deformations. For general background onfinite elasticity, we refer to Ogden [5].

2.4.2 BACKGROUND

Locally, the finite deformation of a material can be described in terms of thethree principal stretches, denoted by l1; l2; and l3. For an incompressiblematerial these satisfy the constraint

l1l2l3 ¼ 1 ð1Þ

The material is isotropic relative to an unstressed undeformed (natural)configuration, and its elastic properties are characterized in terms of a


strain-energy function Wðl1; l2; l3Þ per unit volume, where W dependssymmetrically on the stretches subject to Eq. 1.

The principal Cauchy stresses associated with this deformation aregiven by

si ¼ li@W

@li� p; i 2 f1; 2; 3g ð2Þ

where p is an arbitrary hydrostatic pressure (Lagrange multiplier). Byregarding two of the stretches as independent and treating the strain energy asa function of these through the definition #Wðl1; l2Þ ¼ Wðl1; l2; l

�11 l�1

2 Þ,we obtain

s1 � s3 ¼ l1@ #W

@l1s2 � s3 ¼ l2

@ #W

@l2

ð3Þ

For consistency with the classical theory, we must have

#Wð1; 1Þ ¼ 0;@2 #W

@l1@l2ð1; 1Þ ¼ 2m;

@ #W

@lað1; 1Þ ¼ 0;

@2 #W

@l2a

ð1; 1Þ ¼ 4m;

a 2 f1; 2gð4Þ

where m is the shear modulus in the natural configuration. The equations inEq. 3 are unaffected by superposition of an arbitrary hydrostatic stress. Thus,in determining the characteristics of #W, and hence those of W, it suffices to sets3 ¼ 0 in Eq. 3, so that

s1 ¼ l1@ #W

@l1s2 ¼ l2

@ #W

@l2

ð5Þ

Biaxial experiments in which l1; l2 and s1; s2 are measured then providedata for the determination of #W. Biaxial deformation of a thin sheet where thedeformation corresponds effectively to a state of plane stress, or the combinedextension and inflation of a thin-walled (membranelike) tube with closedends provide suitable tests. In the latter case the governing equations arewritten

P* ¼ l�11 l�1

2

@ #W

@l2F* ¼ @ #W

@l1� 1

2l2l

�11

@ #W

@l2

ð6Þ

where P* ¼ PR=H, P is the inflating pressure, H the undeformed membranethickness, and R the corresponding radius of the tube, while F* ¼ F=2pRH,with F the axial force on the membrane (note that the pressure contributes tothe total load on the ends of the tube). Here l1 is the axial stretch and l2 theazimuthal stretch in the membrane.

Ogden92

2.4.3 DESCRIPTION OF THE MODEL

A specific model which fits very well the available data on various rubbers isthat defined by

W ¼XN

n¼1

mnðlan

1 þ lan

2 þ lan

3 � 3Þ=an ð7Þ

where mn and an are material constants and N is a positive integer, which formany practical purposes may be taken as 2 or 3 [3]. For consistency withEq. 4 we must have XN

n¼1

mnan ¼ 2m ð8Þ

and in practice it is usual to take mnan > 0 for each n ¼ 1; . . . ;N.In respect of Eq. 7, the equations in Eq. 3 become

s1 � s3 ¼XN

n¼1

mnðlan

1 � lan

3 Þ s2 � s3 ¼XN

n¼1

mnðlan

2 � lan

3 Þ ð9Þ

2.4.4 IDENTIFICATION OF PARAMETERS

Biaxial experiments with s3 ¼ 0 indicate that the shapes of thecurves of s1 � s2 plotted against l1 are essentially independent of l2 formany rubbers. Thus the shape may be determined by the pure shear test withl2 ¼ 1, so that

s1 � s2 ¼XN

n¼1

mnðlan

1 � 1Þ � s2 ¼XN

n¼1

mnðlan

3 � 1Þ ð10Þ

for l1 � 1; l3 � 1. The shift factor to be added to the first equation in Eq. 10when l2 differs from 1 is XN

n¼1

mnð1 � lan

2 Þ ð11Þ

Information on both the shape and shift obtained from experiments at fixedl2 then suffice to determine the material parameters, as described in detail inReferences [3] or [4].

Data from the extension and inflation of a tube can be studied on this basisby considering the combination of equations in Eq. 6 in the form

s1 � s2 ¼ l1@ #W

@l1� l2

@ #W

@l2¼ l1F* � 1

2l2

2l1P* ð12Þ

2.4 Elastomer Models 93

2.4.5 HOW TO USE IT

The strain-energy function is incorporated in many commercial FiniteElement (FE) software packages, such as ABAQUS and MARC, and can beused in terms of principal stretches and principal stresses in the FE solution ofboundary-value problems.

2.4.6 TABLE OF PARAMETERS

Values of the parameters corresponding to a three-term form of Eq. 7 are nowgiven in respect of two different but representative vulcanized naturalrubbers. The first is the material used by Jones and Treloar [2]:

a1 ¼ 1:3; a2 ¼ 4:0; a3 ¼ �2:0;

m1 ¼ 0:69; m2 ¼ 0:01; m3 ¼ �0:0122 Nmm�2

The second is the material used by James et al. [1], the material constantshaving been obtained by Treloar and Riding [6]:

a1 ¼ 0:707; a2 ¼ 2:9; a3 ¼ �2:62;

m1 ¼ 0:941; m2 ¼ 0:093; m3 ¼ �0:0029 Nmm�2

For detailed descriptions of the rubbers concerned, reference should be madeto these papers.

REFERENCES

1. James, A. G., Green, A., and Simpson, G. M. (1975). Strain energy functions of rubber.

I. Characterization of gum vulcanizates. J. Appl. Polym. Sci. 19: 2033–2058.

2. Jones, D. F., and Treloar, L. R. G. (1975). The properties of rubber in pure homogeneous strain.

J. Phys. D: Appl. Phys. 8: 1285–1304.

3. Ogden, R. W. (1982). Elastic deformations of rubberlike solids, in Mechanics of Solids

(Rodney Hill 60th Anniversary Volume) pp. 499–537, Hopkins, H. G., and Sevell, M. J., eds.,

Pergamon Press.

4. Ogden, R. W. (1986). Recent advances in the phenomenological theory of rubber elasticity.

Rubber Chem. Technol. 59: 361–383.

5. Ogden, R. W. (1997). Non-Linear Elastic Deformations, Dover Publications.

6. Treloar, L. R. G., and Riding, G. (1979). A non-Gaussian theory for rubber in biaxial strain.

I. Mechanical properties. Proc. R. Soc. Lond. A369: 261–280.

Ogden94

C H A P T E R 2.5

Background onViscoelasticityKOZO IKEGAMI

Tokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo 101-8457, Japan

Contents2.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.5.2 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . 95

2.5.3 Static Viscoelastic Deformation. . . . . . . . . . . . . . . 98

2.5.4 Dynamic Viscoelastic Deformation . . . . . . . . . 100

2.5.5 Hereditary Integral . . . . . . . . . . . . . . . . . . . . . . . . 102

2.5.6 Viscoelastic Constitutive Equation by the

Laplace Transformation . . . . . . . . . . . . . . . . . . . . 103

2.5.7 Correspondence Principle . . . . . . . . . . . . . . . . . . 104

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.5.1 VALIDITY

Fundamental deformation of materials is classified into three types: elastic,plastic, and viscous deformations. Polymetric material shows time-dependentproperties even at room temperature. Deformation of metallic materials is alsotime-dependent at high temperature. The theory of viscoelasticity can beapplied to represent elastic and viscous deformations exhibiting time-dependent properties. This paper offers an outline of the linear theoryof viscoelasticity.

2.5.2 MECHANICAL MODELS

Spring and dashpot elements as shown in Figure 2.5.1 are used to representelastic and viscous deformation, respectively, within the framework of the


linear theory of viscoelasticity. The constitutive equations between stress sand stress e of the spring and dashpot are, respectively, as follows:

s ¼ ke s ¼ Zdedt

ð1Þ

where the notations k and Z are elastic and viscous constants, respectively.Stress of spring elements is linearly related with strain. Stress of dashpotelements is related with strain differentiated by time t, and the constitutiverelation is time-dependent.

Linear viscoelastic deformation is represented by the constitutive equationscombining spring and dashpot elements. For example, the constitutiveequations of series model of spring and dashpot shown in Figure 2.5.2 isas follows:

sþ Zk

dsdt

¼ Zdedt

ð2Þ

This is called the Maxwell model. The constitutive equation of the parallelmodel of spring and dashpot elements shown in Figure 2.5.3 is as follows:

s ¼ keþ Zdedt

ð3Þ

This is called the Voigt or Kelvin model.

FIGURE 2.5.1 Mechanical model of viscoelasticity.

Ikegami96

There are many variations of constitutive equations giving linearviscoelastic deformation by using different numbers of spring and dashpotelements. Their constitutive equations are generally represented by thefollowing ordinary differential equation:

p0sþ p1dsdt

þ p2d2sdt2

þ . . .þ pndnsdtn

¼ q0eþ q1dedt

þ q2d2edt2

þ . . .þ qndnedtn

ð4Þ

The coefficients p and q of Eq. 4 give the characteristic properties of linearviscoelastic deformation and take different values according to the number ofspring and dashpot elements of the viscoelastic mechanical model.

FIGURE 2.5.2 Maxwell model.

2.5 Background on Viscoelasticity 97

2.5.3 STATIC VISCOELASTIC DEFORMATION

There are two functions representing static viscoelastic deformation; one iscreep compliance, and another is the relaxation modulus. Creep complianceis defined by strain variations under constant unit stress. This is obtained bysolving Eqs. 2 or 3 for step input of unit stress. For the Maxwell model andthe Voigt model, their creep compliances are represented, respectively, bythe following expressions. For the Maxwell model, the creep compliance is

et

Zþ 1

k¼ 1

k

t

tþ 1

�ð5Þ

where tM ¼ Z=k, and this is denoted as relaxation time. For the Voigt model,the creep compliance is

e ¼ 1

k1 � exp � kt

Z

� �� ¼ 1

k1 � exp � t

tk

� �� ð6Þ

where tK ¼ Z=k, and this is denoted as retardation time.Creep deformations of the Maxwell and Voigt models are illustrated in

Figures 2.5.4 and 2.5.5, respectively. Creep strain of the Maxwell model

FIGURE 2.5.3 Voigt (Kelvin) model.

Ikegami98

FIGURE 2.5.4 Creep compliance of the Maxwell model.

FIGURE 2.5.5 Creep compliance of the Voigt model.


increases linearly with respect to time duration. The Voigt model exhibitssaturated creep strain for a long time.

The relaxation modulus is defined by stress variations under constant unitstrain. This is obtained by solving Eqs. 2 or 3 for step input of unit strain. Forthe Maxwell and Voigt models, their relaxation moduli are represented by thefollowing expressions, respectively. For the Maxwell model,

s ¼ k exp � kt

Z

� �¼ k exp � t

tM

� �ð7Þ

For the Voigt model,

s ¼ k ð8ÞRelaxation behaviors of the Maxwell and Voigt models are illustrated in

Figures 2.5.6 and 2.5.7, respectively. Applied stress is relaxed by Maxwellmodel, but stress relaxation dose not appear in Voigt model.

2.5.4 DYNAMIC VISCOELASTIC DEFORMATION

The characteristic properties of dynamic viscoelastic deformation arerepresented by the dynamic response for cyclically changing stress or strain.

FIGURE 2.5.6 Relaxation modulus of the Maxwell model.

Ikegami100

The viscoelastic effect causes delayed phase phenomena between input andoutput responses. Viscoelastic responses for changing stress or strain aredefined by complex compliance or modulus, respectively. The dynamicviscoelastic responses are represented by a complex function due to the phasedifference between input and output.

Complex compliance J� of the Maxwell model is obtained by calculatingchanging strain for cyclically changing stress with unit amplitude. Substitu-ting changing complex stress s ¼ expðiotÞ, where i is an imaginary unit andois the frequency of changing stress, into Eq. 2, complex compliance J� isobtained as follows:

J� ¼ 1

k� i

1

oZ¼ 1

k� i

1

kotM¼ J0 � iJ00 ð9Þ

where the real part J0 ¼ 1=k is denoted as storage compliance, and theimaginary part J00 ¼ 1=kotM is denoted as loss compliance.

The complex modulus Y� of the Maxwell model is similarly obtained bycalculating the complex changing strain for the complex changing strain

FIGURE 2.5.7 Relaxation modulus of the Voigt model.


e ¼ expðiotÞ as follows:

Y� ¼ kðotMÞ2

1 þ ðotMÞ2þ ik

otM

1 þ ðotMÞ2¼ Y 0 þ iY 00 ð10Þ

where Y 0 ¼ kððotMÞ2=ð1 þ ðotMÞ2ÞÞ and Y 00 ¼ kðotM=ð1 þ ðotMÞ2ÞÞ. Thenotations Y 0 and Y 00 are denoted as dynamic modulus and dynamic loss,respectively. The phase difference d between input strain and output stress isgiven by

tan d ¼ Y 00

Y 0 ¼1

otM

ð11Þ

This is called mechanical loss.Similarly, the complex compliance and the modulus of the Voigt model are

able to be obtained. The complex compliance is

J� ¼ 1

k

1

1 þ ðotKÞ2

" #� i

1

k

otK

1 þ ðotKÞ2

" #¼ J0 � iJ00 ð12Þ

where J0 ¼ 1

k

1

1 þ ðotKÞ2

" #and J00 ¼ 1

k

otK

1 þ ðotKÞ2

" #

The complex modulus is

Y� ¼ k þ iotK ¼ Y 0 þ iY 00 ð13Þ

where Y 0 ¼ k and Y 00 ¼ kotK.

2.5.5 HEREDITARY INTEGRAL

The hereditary integral offers a method of calculating strain or stress variationfor arbitrary input of stress or strain. The method of calculating strainfor stress history is explained by using creep compliance as illustrated inFigure 2.5.8. An arbitrary stress history is divided into incremental constantstress history ds0 Strain variation induced by each incremental stress historyis obtained by creep compliance with the constant stress values. InFigure 2.5.8 the strain induced by stress history for t05t is represented bythe following integral:

eðtÞ ¼ s0 JðtÞ þZ t

0

Jðt � t0 Þ ds0

dt0dt0 ð14Þ

Ikegami102

This equation is transformed by partially integrating as follows:

eðtÞ ¼ sðtÞJð0Þ þZ t

0

sðt0 Þ dJðt � t0Þdðt � t0Þ dt0 ð15Þ

Similarly, stress variation for arbitrary strain history becomes

sðtÞ ¼ e0YðtÞ þZ t

0

Yðt � t0 Þ ds0

dt0dt0 ð16Þ

Partial integration of Eq. & gives the following equation:

sðtÞ ¼ eðtÞYð0Þ þZ t

0

sðt0 Þ dYðt � t0Þdðt � t0Þ dt0 ð17Þ

Integrals in Eqs. 14 to 17 are called hereditary integrals.

2.5.6 VISCOELASTIC CONSTITUTIVE EQUATIONBY THE LAPLACE TRANSFORMATION

The constitutive equation of viscoelastic deformation is the ordinarydifferential equation as given by Eq. 4. That is,

Xn

k¼0

pkdksdtk

¼Xm

k¼0

qkdkedtk ð18Þ

FIGURE 2.5.8 Hereditary integral.


This equation is written by using differential operators P and Q,

Ps ¼ Qe ð19Þ

where P ¼Pn

k¼0

pkdk

dtkand Q ¼

Pmk¼0

qkdk

dtk.

Equation (1?) is represented by the Laplace transformation as follows.Xn

k¼0

pksk %s ¼Xn

k¼0

qksk%e ð20Þ

where %s and %e are transformed stress and strain, and s is the variable ofthe Laplace transformation. Equation 20 is written by using the Laplacetransformed operators of time derivatives %P and %Q as follows:

%s ¼%Q

%P%e ð21Þ

where %P ¼Pn

k¼0

pksk and %Q ¼Pmk¼0

qksk.

Comparing Eq. 21 with Hooke’s law in one dimension, the coefficient %Q= %Pcorresponds to Young’s modulus of linear elastic deformation. This factimplies that linear viscoelastic deformation is transformed into elasticdeformation in the Laplace transformed state.

2.5.7 CORRESPONDENCE PRINCIPLE

In the previous section, viscoelastic deformation in the one-dimensional statewas able to be represented by elastic deformation through the Laplacetransformation. This can apply to three-dimensional viscoelastic deformation.The constitutive relations of linear viscoelastic deformation are divided intothe relations between hydrostatic pressure and dilatation, and betweendeviatoric stress and strain.

The relation between hydrostatic pressure and dilatation is represented byXm

k¼0

p0kdks0ijdtk

¼Xn

k¼0

q00kdkeii

dtkð22Þ

P00sii ¼ Q00eii ð23Þ

where P00 Pmk¼0

p00kdk

dtkand Q00 ¼

Pn

k¼0

q00kdk

dtk. In Eq. 22 hydrostatic pressure is (1/3)

sii and dilatation is eii.

Ikegami104

The relation between deviatoric stress and strain is represented by

Xm

k¼0

p0kdks0ijdtk

¼Xn

k¼0

q0kdke0ijdtk

ð24Þ

P0s0ij ¼ Q0e0ij ð25Þ

where P0 ¼Pmk�0

p0kdk

dtkand Q0 ¼

Pn

k¼0

q0kdk

dtk. In Eq. 24 deviatoric stress and strain

are s0ij and e0ij, respectively.The Laplace transformations of Eqs. 22 and 24 are written, respectively, as

follows:

%P00 %sii ¼ %Q00%eii ð26Þ

where %P00 ¼ %P00ðsÞ and %Q00 ¼ %Q00sðsÞ, and

%P0 %s0ij ¼ %Q0%e0ij ð27Þ

where %P0 ¼ %P0ðsÞ and %Q0 ¼ %Q0ðsÞ.The linear elastic constitutive relations between hydrostatic pressure and

dilatation and between deviatoric stress and strain are represented as follows:

sii ¼ 3Keii ð28Þ

s0ij ¼ 2Ge0ii ð29Þ

Comparing Eq. 17 with Eq. 19, and Eq. 18 with Eq. 20, the transformedviscoelastic operators correspond to elastic constants as follows:

3K ¼%Q00

%P00ð30Þ

2G ¼%Q0

%P0ð31Þ

where K and G are volumetric coefficient and shear modulus, respectively.For isotropic elastic deformation, volumetric coefficient K and shear

modulus G are connected with Young’s modulus E and Poisson’s ratio n asfollows:

G ¼ E

2ð1 þ nÞ ð32Þ

K ¼ E

3ð1 � 2nÞ ð33Þ


Using Eqs. 30–33, Young’s modulus E and Poisson’s ratio are connected withthe Laplace transformed coefficient of linear viscoelastic deformationas follows:

E ¼ 3 %Q0 %Q00

2 %P0 %Q00 þ %P00 %Q0 ð34Þ

n ¼%P0 %Q00 � %P00 %Q0

2 %P0 %Q00 þ %P00 %Q0 ð35Þ

Linear viscoelastic deformation corresponds to linear elastic deformationthrough Eqs. 30–31 and Eqs. 34–35. This is called the correspondenceprinciple between linear viscoelastic deformation and linear elastic deforma-tion. The linear viscoelastic problem is the transformed linear elastic problemin the Laplace transformed state. Therefore, the linear viscoelastic problem isable to be solved as a linear elastic problem in the Laplace transformed state,and then the elastic constants of solved solutions are replaced with theLaplace transformed operator of Eqs. 30–31 and Eqs. 34–35 by usingthe correspondence principle. The solutions replaced the elastic constantsbecome the solution of the linear viscoelastic problem by inversing theLaplace transformation.

REFERENCES

1. Bland, D. R. (1960). Theory of Linear Viscoelasticity, Pergamon Press.

2. Ferry, J. D. (1960). Viscoelastic Properties of Polymers, John Wiley & Sons.

3. Reiner, M. (1960). Deformation, Strain and Flow, H. K. Lewis & Co.

4. Flluege, W. (1967). Viscoelasticity, Blaisdell Publishing Company.

5. Christensen, R. M. (1971). Theory of Viscoelasticity: An Introduction, Academic Press.

6. Drozdov, A. D. (1998). Mechanics of Viscoelastic Solids, John Wiley & Sons.

Ikegami106

C H A P T E R 2.6

A Nonlinear ViscoelasticModel Based onFluctuating ModesRACHID RAHOUADJ AND CHRISTIAN CUNAT

LEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la For#et-de-Haye, 54500 Vandoeuvre-l"es-

Nancy, France

Contents2.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.6.2 Background of the DNLR . . . . . . . . . . . . . . . 108

2.6.2.1 Thermodynamics of Irreversible

Processes and Constitutive Laws . . . 108

2.6.2.2 Kinetics and Complementary

Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.6.2.3 Constitutive Equations of

the DNLR . . . . . . . . . . . . . . . . . . . . . . . . 112

2.6.3 Description of the Model in the Case

of Mechanical Solicitations . . . . . . . . . . . . . . 113

2.6.4 Identification of the Parameters . . . . . . . . . 113

2.6.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.6.6 Table of Parameters. . . . . . . . . . . . . . . . . . . . . 115

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.6.1 VALIDITY

We will formulate a viscoelastic modeling for polymers in the temperaturerange of glass transition. This physical modeling may be applied using integralor differential forms. Its fundamental basis comes from a generalization of theGibbs relation, and leads to a formulation of constitutive laws involvingcontrol and internal thermodynamic variables. The latter must traduce


different microstructural rearrangements. In practice, both modal analysisand fluctuation theory are well adapted to the study of the irreversibletransformations.

Such a general formulation also permits us to consider variousnonlinearities as functions of material specificities and applied perturbations.

To clarify the present modeling, called ‘‘the distribution of nonlinearrelaxations’’ (DNLR), we will consider the viscoelastic behavior in the simplecase of small applied perturbations near the thermodynamic equilibrium. Inaddition, we will focus our attention upon the nonlinearities induced bytemperature and frequency perturbations.

2.6.2 BACKGROUND OF THE DNLR

2.6.2.1 THERMODYNAMICS OF IRREVERSIBLE

PROCESSES AND CONSTITUTIVE LAWS

As mentioned, the present irreversible thermodynamics are based on ageneralization of the fundamental Gibbs equation to systems evolving outsideequilibrium. Note that Coleman and Gurtin [1], have also applied thispostulate in the framework of rational thermodynamics. At first, a set ofinternal variables (generalized vector denoted z) is introduced to describe themicrostructural state. The generalized Gibbs relation combines the two lawsof thermodynamics into a single one, i.e., the internal energy potential:

e ¼ eðs; e; n; . . . ; zÞ ð1Þ

which depends on overall state variables, including the specific entropy, s.Furthermore, with the positivity of the entropy production being alwaysrespected, one obtains for open systems:

TdDis

dt¼ Tss ¼ �Js :rT �

Xn

k¼1

Jk :rmk þ A � ’z � 0 ð2Þ

where the nonequilibrium thermodynamic forces may be separated into twogroups: (i) the gradient ones, such as the gradient of temperature gradientrT, and the gradient of generalized chemical potential rmk; and (ii) Thegeneralized forces A, or affinities as defined by De Donder [2] for chemicalreactions, which characterize the nonequilibrium state of a uniform medium.

The vectors Js, Jk, and ’z correspond to the dual, fluxes, or rate-type variables.

To simplify the formulation of the constitutive laws, we will now considerthe behavior of a uniform representative volume element (RVE without any

Rahouadj and Cunat108

gradient), thus:

Tss ¼ A � ’z � 0 ð3Þ

The equilibrium or relaxed state (denoted by the index r) is currentlydescribed by a suitable thermodynamic potential (cr) obtained via theLegendre transformation of Eq. 1 with respect to the control or state variable(g). In this particular state, the set of internal variables is completelygoverned by (g):

cr ¼ crðg; zrðgÞÞ ¼ crðgÞ ð4Þ

Our first hypothesis [3] states that it is always possible to define athermodynamic potential c only as a function of g and z, even for systemsoutside equilibrium:

c ¼ cðg; zÞ ð5Þ

Then, we assume that the constitutive equations may be obtained as functionsof the first partial derivatives of this potential with respect to the dualvariables, and depend consequently on both control and internal variables;i.e., b ¼ bðg; zÞ and A ¼ Aðg; zÞ. In fact, this description is consistent with theprinciple of equipresence, as postulated in rational thermodynamics. There-fore, the thermodynamic potential becomes in a differential form:

dck ¼Xq

m¼1

bmdgm �Xr

j¼1

Aj dzj ð6Þ

Thus the time evolution of the global response, b, obeys a nonlineardifferential equation involving both the applied perturbation g and theinternal variable z (generalized vector):

’b ¼ au : ’gþ b : ’z ð7aÞ

’A ¼ �tb : ’g� g : ’z ð7bÞ

This differential system resumes in a general and condensed form theannounced constitutive relationships. The symmetrical matrix au ¼ @2c=@g@gis the matrix of Tisza, and the symmetrical matrix g ¼ @2c=@z@z traduces theinteraction between the dissipation processes [3]. The rectangular matrixb ¼ @2c=@z@g expresses the coupling effect between the state variables andthe dissipation variables.

In other respects, the equilibrium state classically imposes the thermo-dynamic forces and their rate to be zero; i.e., A ¼ 0 and ’A ¼ 0. From Eq. 7bwe find, for any equilibrium state, that the internal variables’ evolution resultsdirectly from the variation of the control variables:

’zr ¼ �g�1 : tb : ’g ð8Þ

2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes 109

According to Eqs. 7b and 8, the evolution of the generalized force becomes

’A ¼ �g : ð’z � ’zrÞ ð9Þ

and its time integration for transformation near equilibrium leads to thesimple linear relationship

A ¼ �gðz � zrÞ ð10Þ

where g is assumed to be constant.

2.6.2.2 KINETICS AND COMPLEMENTARY LAWS

To solve the preceding three equations (7a–b, 10), with the unknown varia-bles being b, z, zr, and A, one has to get further information about the kineticrelations between the nonequilibrium driving forces A and their fluxes ’z.

2.6.2.2.1 First-Order Nonlinear Kinetics and Relaxation Times

We know that the kinetic relations are not submitted to the samethermodynamic constraints as the constitutive ones. Thus we shall considerfor simplicity an affine relation between fluxes and forces. Note that this well-known modeling, early established by Onsager, Casimir, Meixner, de Donder,De Groot, and Mazur, is only valid in the vicinity of equilibrium:

’z ¼ L �A ð11Þ

and hence, with Eq. 10:

’z ¼ �L �g � ðz � zrÞ ¼ �t�1 :ðz � zrÞ ð12Þ

According to this nonlinear kinetics, Meixner [4] has judiciously suggested abase change in which the relaxation time operator t is diagonal. Here, weconsider this base, which also represents a normal base for the dissipationmodes. In what follows, the relaxation spectrum will be explicitly defined onthis normal base. To extend this kinetic modeling to nonequilibriumtransformations, which is the object of the nonlinear TIP, we also suggestreferring to Eq. 12 but with variable relaxation times. Indeed, each relaxationtime is inversely proportional to the jump frequency, u, and to the probabilitypj ¼ expð�DFþ;r

j =RTÞ of overcoming a free energy barrier, DFþ;rj . It follows

that the relaxation time of the process j may be written:

trj ¼ 1=u expðDFþ;r

j =RTÞ ð13Þ

where the symbol (þ) denotes the activated state, and the index (r) refers tothe activation barrier of the REV near the equilibrium.


The reference jump frequency, u0 ¼ kBT=h, has been estimated fromGuggenheim’s theory, which considers elementary movements of translationat the atomic level. The parameters h, kB, and r represent the constants ofPlank, Boltzmann, and of the perfect gas, respectively, and T is the absolutetemperature. It seems natural to assume that the frequency of the microscopicrearrangements is mainly governed by the applied perturbation rate, ’g,through a shift function að’gÞ:

u ¼ u0=að’gÞ ð14Þ

Assuming now that the variation of the activation energy for each process isgoverned by the evolution of the overall set of internal variables leads us tothe following approximation of first order:

DFþj ¼ DFþ;r

j þ Kz :ðz � zrÞ ð15Þ

In the particular case of a viscoelastic behavior, this variation of the freeenergy becomes negligible. The temperature dependence obviously intervenesinto the basic definition of the activation energy as

DFþ;rj ¼ DEþ;r � T DSþ;r

j ð16Þ

where the internal energy DEþ;r is supposed to be the same for all processes. Itfollows that we may define another important shift function, noted aðTÞ,which accounts for the effect of temperature. According to the Arrheniusapproximation, DEþ;r being quasi-constant, this shift function verifies thefollowing relation:

ln aðT;TrefÞ ¼ DEþ;rð1=T � 1=TrefÞ ð17Þ

where Tref is a reference temperature. For many polymers near the glasstransition, this last shift function obeys the WLF empiric law developed byWilliam, Landel, and Ferry [5]:

lnðaTÞ ¼ c1ðT � TrefÞ=½c2 þ ðT � TrefÞ ð18Þ

In summary, the relaxation times can be generally expressed as

tjðTÞ ¼ trj ðTrefÞaðT;TrefÞ að’gÞ aðz; zrÞ ð19Þ

and the shift function aðz; zrÞ becomes negligible in viscoelasticity.

2.6.2.2.2 Form of the Relaxation Spectrum near the Equilibrium

We now examine the distribution of the relaxation modes evolving during thesolicitation. In fact, this applied solicitation, g, induces a state of fluctuationswhich may be approximately compared to the corresponding equilibrium one.According to prigogine [6], these fluctuations obey the equipartition of theentropy production. Therefore, we can deduce the expected distribution in


the vicinity of equilibrium as

p0j ¼ B

ffiffiffiffitr

j

qwith

Xn

j¼1

p0j ¼ 1 and B ¼ 1

�Xn

j¼1

ffiffiffiffitr

j

qð20Þ

where trj is the relaxation time of the process j, p0

j its relative weight in theoverall spectrum, and n the number of dissipation processes [3].

As a first approximation, the continuous spectrum defined by Eq. 20 maybe described with only two parameters: the longest relaxation timecorresponding to the fundamental mode, and the spectrum width. Notethat a regular numerical discretization of the relaxation time scale usinga sufficiently high number n of dissipation modes, e.g., 30, gives asufficient accuracy.

2.6.2.3 CONSTITUTIVE EQUATIONS OF THE DNLR

Combining Eqs. 7a and 12 gives, whatever the chosen kinetics,

’b ¼ au : ’g� b � ðz � zrÞ :t�1z ¼ au : ’g� ð#a � #arÞ :t�1

b ð21aÞ

To simplify the notation, tb will be denoted t. In a similar form and afterintroducing each process contribution in the base defined above, one has

’bm ¼Xn

p¼1

aump ’gp �

Xn

j¼1

bjm � p0j b

rm

tjð21bÞ

where the indices u and r denote the instantaneous and the relaxedvalues, respectively.

Now we shall examine the dynamic response due to sinusoidallyvarying perturbations gn ¼ g0expðiotÞ, where o is the applied frequency,and i2 ¼ �1, i.e., ’gn ¼ iogn. The response is obtained by integrating theabove differential relationship. Evidently, the main problem encounteredin the numerical integration consists in using a time step that mustbe consistent with the applied frequency and the shortest time ofrelaxation. Furthermore, a convenient possibility for very small pertur-bations is to assume that the corresponding response is periodic and outof phase:

bn ¼ b0expðiot þ jÞ and ’bn ¼ iobn ð22Þ

where j is the phase angle. In fact, such relations are representative of variousphysical properties as shown by Kramers [7] and Kronig [8].

The coefficients of the matrices of Tisza, au and ar, and the relaxationtimes, tj, may be dependent on temperature and=or frequency. In uniaxial


tests of mechanical damping, these Tisza’s coefficients correspond to thestorage and loss modulus E0 (or G0) and E00 (or G00), respectively.

2.6.3 DESCRIPTION OF THE MODEL IN THECASE OF MECHANICAL SOLICITATIONS

We consider a mechanical solicitation under an imposed strain e. Here, theperturbation g and the response b are respectively denoted e and s. Accordingto Eqs. 19 and 21b, the stress rate response, ’s, may be finally written

’s ¼Xn

j¼1

p0j au : ’e�

Xn

j¼1

sj � p0j ar : e

að’eÞ aðe; erÞ aðT;TrefÞtjðTrefÞð23Þ

As an example, for a pure shear stress this becomes

’s12 ¼Xn

j¼1

p0j Gu’e12 �

Xn

j¼1

sj 12 � p0j Gr e12

að’eÞ aðe; erÞ aðT;Tref ÞtGj ðTrefÞ

ð24Þ

In the case of sinusoidally varying deformation, the complex modulus isgiven by

G � ðoÞ ¼ Gu þ ðGr � GuÞXn

j¼1

p0j

1

1 þ iotGj

ð25Þ

It follows that its real and imaginary components are, respectively,

G0ðoÞ ¼ Gu þ ðGr � GuÞXn

j¼1

p0j

1

1 þ o2ðtGj Þ

2 ð26Þ

G00ðoÞ ¼ ðGr � GuÞXn

j¼1

p0j

otj

1 þ o2ðtGj Þ

2 ð27Þ

2.6.4 IDENTIFICATION OF THE PARAMETERS

The crucial problem in vibration experiments concerns the accuratedetermination of the viscoelastic parameters over a broad range of frequency.Generally, to avoid this difficulty one has recourse to the appropriate principleof equivalence between temperature and frequency, assuming implicitlyidentical microstructural states. A detailed analysis of the literature hasbrought us to a narrow comparison of the empirical model of Havriliak and


Negami (HN) [9] with the DNLR. The HN approach appears to besuccessful for a wide variety of polymers; it combines the advantagesof the previous modeling of Cole and Cole [10] and of Davidsonand Cole [11]. For pure shear stress the response given by this HNapproach is

G� ¼ GuHN þ ðGr

HN � GuHNÞ

1

½1 þ ðiotHNÞab

ð28Þ

where GuHN; Gr

HN; a; and b are empirical parameters. Thus the real andimaginary components are, respectively,

G0 ¼ GuHN þ ðGr

HN � GuHNÞ

cosðbyÞ

½1 þ 2oataHNcosðap=2Þ þ o2at2ab=2 ð29Þ

G00 ¼ ðGrHN � Gu

HNÞsinðbyÞ

½1 þ 2oataHNcosðap=2Þ þ o2at2ab=2 ð30Þ

The function y is defined by

y ¼ tan�1 oataHNsinðap=2Þ1 þ oataHNcosðap=2Þ

� �ð31Þ

Eqs. 28 to 30 are respectively compared to Eqs. 25 to 27 in order to establish acorrespondence between the relaxation times of the two models:

logðtGrj Þ ¼ logðtHNÞ þ jL=n þY ð32Þ

where Y, L, and n are a scale parameter, the number of decades of thespectrum, and the number of processes, respectively. A precise empiricalconnection is obtained by identifying the shift function for the time scalewith the relation

tGj ¼ að’gÞtGr

j ¼ aðoÞtGrj ¼

tanðbyÞotHN

� �tGr

j ð33Þ

This involves a progressive evolution of the difference of modulus as afunction of the applied frequency:

ðGr � GuÞ ¼ ðGrHN � Gu

HNÞfG ð34Þ

The function fG is given by

fG ¼ cosðbyÞð1 þ tan2ðbyÞÞ

½1 þ 2oataHNcosðap=2Þ þ o2at2aHN

b=2ð35Þ


2.6.5 HOW TO USE IT

In practice, knowledge of the only empirical parameters of HN’s modeling(and=or Cole and Cole’s and Davidson and Cole’s) permits us, in theframework of the DNLR, to account for a large variety of loading histories.

2.6.6 TABLE OF PARAMETERS

As a typical example given by Hartmann et al. [12], we consider the case of apolymer whose chemical composition is 1PTMG2000=3MIDI=2DMPD*. Themaster curve is plotted at 298 K in Figure 2.6.1. The spectrum is discretized

FIGURE 2.6.1 Theoretical simulation of the moduli for PTMG ( J).*

FIGURE 2.6.2 Theoretical simulations of the shift function aðoÞ and of fG for PTMG.*

* PTMG: poly (tetramethylene ether) glycol; MIDI: 4,40-diphenylmethane diisocyanate; DMPD:

2,2-dimethyl-1, 3-propanediol with a density of 1.074 g=cm3, and glass transition Tg ¼ �408C.


with L ¼ 6, a scale parameter Y equal to �5.6, and 50 relaxation times. Theparameters Gr

HN ¼ 2:14 MPa, GuHN ¼ Gu ¼ 1859 MPa, tHN ¼

1.649 10�7 s, a ¼ 0:5709 and b ¼ 0:0363 allow us to calculate the shiftfunction aðoÞ and the function fG which is necessary to estimate the differencebetween the relaxed and nonrelaxed modulus, taking into account theexperimental conditions. Figure 2.6.1 illustrates the calculated viscoelasticresponse, which is superposed to HN’s one. The function fG and the shiftfunction aðoÞ illustrate the nonlinearities introduced in the DNLR modeling(Fig. 2.6.2).

REFERENCES

1. Coleman, B. D., and Gurtin, M. (1967). J. Chem. Phys. 47 (2): 597.

2. De Donder, T. (1920). Le,con de thermodynamique et de chimie physique, Paris: Gauthiers-

Villars.

3. Cunat, C. (1996). Rev. Gcn. Therm. 35: 680–685.

4. Meixner, J. Z. (1949). Naturforsch., Vol. 4a, p. 504.

5. William, M. L., Landel, R. F., and Ferry, J. D. (1955). The temperature dependence of

relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Amer.

Chem. Soc. 77: 3701.

6. Prigogine, I. (1968). Introduction "a la thermodynamique des processus irr!eversibles, Paris:

Dunod.

7. Kramers, H. A. (1927). Atti. Congr. dei Fisici, Como, 545.

8. Kronig, R. (1926). J. Opt. Soc. Amer. 12: 547.

9. Havriliak, S., and Negami, S. (1966). J. Polym. Sci., Part C, No. 14, ed. R. F. Boyer, 99.

10. Cole, K. S., and Cole, R. H. (1941). J. Chem. Phys. 9: 341.

11. Davidson, D. W., and Cole, R. H. (1950). J. Chem. Phys. 18: 1417.

12. Hartmann, B., Lee, G. F., and Lee, J. D. (1994). J. Acoust. Soc. Amer. 95 (1).


C H A P T E R 2.7

Linear Viscoelasticitywith DamageR. A. SCHAPERY

Department of Aerospace Engineering and Engineering Mechanics, University of Texas,

Austin, Texas

Contents2.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.7.3 Description of the Model. . . . . . . . . . . . . . . . . . . 119

2.7.4 Identification of the Material Functions

and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2.7.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.7.1 VALIDITY

This paper describes a homogenized constitutive model for viscoelasticmaterials with constant or growing distributed damage. Included are three-dimensional constitutive equations and equations of evolution for damageparameters (internal state variables, ISVs) which are measures of damage.Anisotropy may exist without damage or may develop as a result ofdamage. For time-independent damage, the specific model covered here isthat for a linearly viscoelastic, thermorheologically simple material in whichall hereditary effects are expressed through a convolution integral with onecreep or relaxation function of reduced time; nonlinear effects of transientcrack face contact and friction are excluded. More general cases that accountfor intrinsic nonlinear viscoelastic and viscoplastic effects as well asthermorheologically complex behavior and multiple relaxation functions arepublished elsewhere [10].


2.7.2 BACKGROUND

As background to the model with time-dependent damage, consider first theconstitutive equation with constant damage, in which e and s representthe strain and stress tensors, respectively,

e ¼ fSdsg þ eT ð1Þ

where S is a fully symmetric, fourth order creep compliance tensor and eT isthe strain tensor due to temperature and moisture (and other absorbedsubstances which affect the strains). The braces are abbreviated notation for alinear hereditary integral. Although the most general form could be used,allowing for general aging effects, for notational simplicity we shall use thefamiliar form for thermorheologically simple materials,

f fdgg ¼Z t

o�f ðx� x0Þ @g

@t0dt0 ¼

Z x

o�fðx� x0Þ @g

@x0dx0 ð2Þ

where it is assumed f ¼ g ¼ o for t5o and

x �Z t

o

dt00=aT½Tðt00Þ x0 ¼ xðt0Þ ð3Þ

Also, aTðTÞ is the temperature-dependent shift factor. If the temperature isconstant in time, then x� x0 ¼ ðt � t0Þ=aT: Physical aging [12] may be takeninto account by introducing explicit time dependence in aT; i.e., useaT ¼ aTðT; t00Þ in Eq. 3. The effect of plasticizers, such as moisture, may alsobe included in aT: When Eq. 2 is used with Eq. 1, f and g are components ofthe creep compliance and stress tensors, respectively.

In certain important cases, the creep compliance components areproportional to one function of time,

S ¼ kD ð4Þ

where k is a constant, dimensionless tensor and D ¼ DðxÞ is a creepcompliance (taken here to be that obtained under a uniaxial stress state).Isotropic materials with a constant Poisson’s ratio satisfy Eq. 4. If such amaterial has mechanically rigid reinforcements and=or holes (of any shape), itis easily shown by dimensional analysis that its homogenized constitutiveequation satisfies Eq. 4; in this case the stress and strain tensors in Eq. 1should be interpreted as volume-averaged quantities [2]. The Poisson’s ratiofor polymers at temperatures which are not close to their glass-transitiontemperature, Tg, is nearly constant; except at time or rate extremes, somewhatabove Tg Poisson’s ratio is essentially one half, while below Tg it is commonlyin the range 0.35–0.40 [5].

Schapery118

Equations 1 and 4 give

e ¼ fDdðksÞg þ eT ð5ÞThe inverse is

s ¼ kIfEdeg � kIfEdeTg ð6Þwhere kI ¼ k�1 and E ¼ EðxÞ is the uniaxial relaxation modulus in which,for t > o,

DdEf g ¼ EdDf g ¼ 1 ð7ÞIn relating solutions of elastic and viscoelastic boundary value problems,and for later use with growing damage, it is helpful to introduce thedimensionless quantities

eR � 1

ERfEdeg eR

T � 1

ERfEdeTg uR � 1

ERfEdug ð8Þ

where ER is an arbitrary constant with dimensions of modulus, called thereference modulus; also, eR and eR

T are so-called pseudo-strains and uR isthe pseudo-displacement. Equation 6 becomes

s ¼ CeR � CeRT ð9Þ

where C � ERkI is like an elastic modulus tensor; its elements are calledpseudo-moduli. Equation 9 reduces to that for an elastic material by takingE ¼ ER; it reduces to the constitutive equation for a viscous material if E isproportional to a Dirac delta function of x. The inverse of Eq. 9 gives thepseudo-strain eR in terms of stress,

eR ¼ #Ssþ eRT ð10Þ

where #S ¼ C�1 ¼ k=ER: The physical strain is given in Eq. 5.

2.7.3 DESCRIPTION OF THE MODEL

The correspondence principle (CPII in Schapery [4, 8]) that relates elastic andviscoelastic solutions shows that Eqs. 1–10 remain valid, under assumptionEq. 4, with damage growth when the damage consists of cracks whose facesare either unloaded or have loading that is proportional to the external loads.With growing damage k; C, and #S are time-dependent because they arefunctions of one or more damage-related ISVs; the strain eT may also dependon damage. The fourth-order tensor k must remain inside the convolutionintegral in Eq. 5, just as shown. This position is required by thecorrespondence principle. The elastic-like Eqs. 9 and 10 come from Eq. 5,and thus have the appropriate form with growing damage. However, with

2.7 Linear Viscoelasticity with Damage 119

healing of cracks, pseudo-stresses replace pseudo-strains because k mustappear outside the convolution integral in Eq. 5 [8].

The damage evolution equations are based on viscoelastic crack growthequations or, in a more general context, on nonequilibrium thermodynamicequations. Specifically, let WR and WR

C denote pseudo-strain energy densityand pseudo-complementary strain energy density, respectively,

WR ¼ 1

2CðeR � eR

TÞðeR � eRTÞ � F ð11Þ

WRC ¼ 1

2#Sssþ eR

Tsþ F ð12Þ

so that

WRC ¼ �WR þ seR ð13Þ

and

s ¼ @WR

@eReR ¼ @WR

C

@sð14Þ

The function F is a function of damage and physical variables that causeresidual stresses such as temperature and moisture.

For later use in Section 2.7.4, assume the damage is fully defined by a set ofscalar ISVs, Sp (p ¼1, 2, . . . P) instead of tensor ISVs. Thermodynamic forces,which are like energy release rates, are introduced,

fp � � @WR

@Spð15Þ

or

fp �@WR

C

@Spð16Þ

where the equality of these derivatives follows directly from the totaldifferential of Eq. 13.

Although more general forms could be used, the evolution equations for’Sp � dSp=dx are assumed in the form

’Sp ¼ ’SpðSq; fpÞ ð17Þ

in which ’Sp may depend on one or more Sq (q ¼ 1, . . . P), but on only oneforce fp. The entropy production rate due to damage is non-negative ifX

p

fp ’Sp � O ð18Þ

thus satisfying the Second Law of Thermodynamics. It is assumed that whenj fpj is less than some threshold value, then ’Sp ¼ O.

Schapery120

Observe that even when the stress vanishes, there may be damage growthdue to F. According to Eqs. 12 and 16,

fp ¼@WR

C

@Sp¼ 1

2

@ #S

@Spssþ @eR

T

@Spsþ @F

@Sp

ð19Þ

which does not vanish when r ¼ o, unless @F=@Sp ¼ 0.The use of tensor ISVs is discussed and compared with scalar ISVs by

Schapery [10]. The equations in this section are equally valid for tensor andscalar ISVs.

2.7.4 IDENTIFICATION OF THE MATERIALFUNCTIONS AND PARAMETERS

The model outlined above is based on thermorheologically simple behaviorin that reduced time is used throughout, including damage evolution(Eq. 17). In studies of particle-reinforced rubber [4], this simplicitywas found, implying that even the microcrack growth rate behaviorwas affected by temperature only through viscoelastic behavior of therubber. If the damage growth is affected differently by temperature (andplasticizers), then explicit dependence may be introduced in the rate(Eq. 17). In the discussion that follows, complete thermorheologicalsimplicity is assumed.

The behavior of particle-reinforced rubber and asphalt concrete has beencharacterized using a power law when fp > o,

’Sp ¼ ð fpÞap ð20Þ

where ap is a positive constant. (For the rubber composite two ISVs, witha1 ¼ 4:5 and a2 ¼ 6, were used for uniaxial and multiaxial behavior, whereasfor asphalt one ISV, with a ¼ 2:5, was used for uniaxial behavior.) Acoefficient depending on Sp may be included in Eq. 20; but it does not reallygeneralize the equation because a simple change of the variable Sp may beused to eliminate the coefficient.

Only an outline of the identification process is given here, but details areprovided by Park et al. [3] for uniaxial behavior and by Park and Schapery [4]and Ha and Schapery [1] for multiaxial behavior. Schapery and Sicking[11] and Schapery [9] discuss the model’s use for fiber composites. The effectsof eT and F are neglected here.

(a) The first step is to obtain the linear viscoelastic relaxation modulusEðxÞ and shift factor aT for the undamaged state. This may be done


using any standard method, such as uniaxial constant strain rate testsat a series of rates and temperatures. Alternatively, for example, uniaxialcreep tests may be used to find DðxÞ, after which EðxÞ is derivedfrom Eq. 7.

(b) Constant strain rate (or stress rate) tests to failure at a series ofrates or temperatures may be conveniently used to obtain the additional dataneeded for identification of the model. (However, depending on thecomplexity of the material and intended use of the model, unloading andreloading tests may be needed [7].) Constant strain rate tests often arepreferred over constant stress rate tests because meaningful post-stress peakbehavior (prior to significant strain localization) may be found from theformer tests.For isothermal, constant strain rate, R, tests, the input is Rt ¼ #Rx; where#R ¼ RaT and x ¼ t=aT. Inasmuch as the model does not depend ontemperature when reduced time is used, all stress vs. reduced time responsecurves depend on only one input parameter #R, regardless of temperature.Thus, one may obtain a complete identification of the model from a series oftests over a range of #R using one temperature and different rates or one rateand different temperatures; both types of tests may be needed in practice for #Rto cover a sufficiently broad range. One should, however, conduct at leasta small number of both types of tests to check the thermorheologicallysimple assumption.

(c) Convert all experimental values of displacements and strainsfrom step (b) tests to pseudo-quantities using Eq. 8. This removes intrinsicviscoelastic effects, thus enabling all subsequent identification steps to bethose for a linear elastic material with rate-dependent damage. If controlledstrain (stress) tests are used, then one would employ WRðWR

CÞ in theidentification. However, mixed variables may be input test parameters, suchas constant strain rate tests of specimens in a test chamber at a series ofspecified pressures [4]. In this case it is convenient to use mixed pseudo-energy functions in terms of strain and stress variables. Appropriateenergy functions may be easily constructed using methods based on linearelasticity theory.

(d) The procedure for finding the exponent a and pseudo Young’s modulusin terms of one damage parameter is given by Park et al. [3]. After this, theremaining pseudo-moduli or compliances may be found in terms of one ormore ISVs, as described by Park and Schapery [4] using constant strain ratetests of bar specimens under several confining pressures. The materialemployed by them was initially isotropic, but it became transversely isotropicas a result of damage. Identification of the full set of five pseudo-moduli andthe pseudo-strain energy function, as functions of two ISVs, is detailed by Haand Schapery [1].

Schapery122

2.7.5 HOW TO USE IT

Implementation of user-defined constitutive relations based on this model in afinite element analysis is described by Ha and Schapery [1]. Included arecomparisons between theory and experiment for overall load-displacementbehavior and for local strain distributions. The model employed assumes thematerial is locally transversely isotropic with the axis of isotropy assumedparallel to the local maximum principal stress direction, accounting for priorstress history at each point. A procedure is proposed by Schapery [10] thatenables use of the same model when transverse isotropy is lost due to rotationof the local maximum principal stress direction.

REFERENCES

1. Ha, K., and Schapery, R. A. (1998). A three-dimensional viscoelastic constitutive model for

particulate composites with growing damage and its experimental validation. International

Journal of Solids and Structures 35: 3497–3517.

2. Hashin, Z. (1983). Analysis of composite materials } a survey. Journal of Applied Mechanics

105: 481–505.

3. Park, S. W., Kim, Y. R., and Schapery, R. A. (1996). A viscoelastic continuum damage

model and its application to uniaxial behavior of asphalt concrete. Mechanics of Materials

24: 241–255.

4. Park, S. W., and Schapery, R. A. (1997). A viscoelastic constitutive model for particulate

composites with growing damage. International Journal of Solids and Structures 34: 931–947.

5. Schapery, R. A. (1974). Viscoelastic behavior and analysis of composite materials, in

Mechanics of Composite Materials, pp. 85–168, vol. 2, Sendeckyi, G. P., ed., New York:

Academic.

6. Schapery, R. A. (1981). On viscoelastic deformation and failure behavior of composite

materials with distributed flaws, in 1981 Advances in Aerospace Structures and Materials,

pp. 5–20, Wang, S. S., and Renton, W. J., eds., ASME, AD-01.

7. Schapery, R. A. (1982). Models for damage growth and fracture in nonlinear viscoelastic

particulate composites, in: Proc. Ninth U.S. National Congress of Applied Mechanics, Book No.

H00228, pp. 237–245, Pao, Y. H., ed., New York: ASME.

8. Schapery, R. A. (1984). Correspondence principles and a generalized J integral for large

deformation and fracture analysis of viscoelastic media, in: International Journal of Fracture

25: 195–223.

9. Schapery, R. A. (1997). Constitutive equations for special linear viscoelastic composites with

growing damage, in Advances in Fracture Research, pp. 3019–3027, Karihaloo, B. L., Mai, Y.-

W., Ripley, M. I., and Ritchie, R. O., eds., Pergamon.

10. Schapery, R. A. (1999). Nonlinear viscoelastic and viscoplastic constitutive equations with

growing damage. International Journal of Fracture 97: 33–66.

11. Schapery, R. A., and Sicking, D. L. (1995). On nonlinear constitutive equations for elastic and

viscoelastic composites with growing damage, in Mechanical Behavior of Materials, pp. 45–76,

Bakker, A., ed., Delft: Delft University Press.

12. Struik, L. C. E. (1978). Physical Aging in Amorphous Polymers and Other Materials,

Amsterdam: Elsevier.


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