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CHAPTER 2

Elasticity andViscoelasticity


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C H A P T E R 2.1

Introduction to Elasticityand Viscoelasticity JEAN LEMAITREUniversit !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 Hookes 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 Youngs modulus E andthe Poissons ratio n.

c*

1

2r AijklE;ns ijs kl 1

eij r@ c *@ s ij

1 n

Es ij

nE

s kkdij 2

E and n are identied from tensile tests either in statics or dynamics. A greatdeal of accuracy is needed in the measurement of the longitudinal andtransverse strains ( de % 106 in absolute value).

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

Plane stress s 33 s 13 s 23 0:

e11e22

e12

264

375

1E

nE

0

1E

0

Sym1 n

E

2666666664

3777777775

s 11s 22

s 12

264

375

3

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Plane strain e33 e13 e23 0:

s 11s 22s 12264 375

l 2m l 0

l 2m 0Sym 2m264 375

e11e22e12264 375

with

l nE

1 n1 2n

mE

21 n

8>>>>>:4

For orthotropic materials having three planes of symmetry, nine

independent parameters are needed: three tension moduli E 1; E2; E3in 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

1E1

n12E1

n13E1

0 0 0

1E2

n23E2

0 0 0

1E3 0 0 0

12G23

0 0

Sym1

2G310

12G12

266666666666666666664

377777777777777777775

s 11

s 22

s 33

s 23

s 31

s 12

266666666666666666666664

377777777777777777777775

5

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 coefcient a . For small variations of temperature y for which the elasticity parameters may be consideredas constant:

eij 1 n

Es ij

nE

s kkdij ayd ij 6

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

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

eij 1 n

Es ij

nE

s kkdij a yd ij @

@ y1 n

E

s ij

@ @ y

nE

s kkdij

@ a@ y

yd ij

!y

7

Viscoelasticity considers in addition a dissipative phenomenon due tointernal 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 Maxwells 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 and

recalled here in three dimensions.Kelvin-Voigt model:

s ij l ekk yl ekkdij 2meij ymeij 8

Here l and mare Lames coefcients at steady state, and yl and ym are twotime parameters responsible for viscosity. These four coefcients may beidentied from creep tests in tension and shear.Maxwell model:

eij 1 n

E s ij st 1

nE s kk

s kkt 2 d

ij 9

Here E and n are Youngs modulus and Poissons ratio at steady state, andt 1 and t 2 are two other time parameters. It is a uidlike 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.

eijt Z t

o J ijkl t t

ds kldt

dt Xn

p1 J ijkl t t D s

pkl 10

J t is the creep functions matrix, and D s pkl are the eventual stress steps.

The dual formulation introduces the relaxation functions matrix Rt s ijt Z

t

oRijklt t

dekldt

dt

X

n

p1Rijkl D e

pkl 11

When isotropy is considered the matrix, J and R each reduce to twofunctions: either J t, the creep function in tension, is identied from a creep

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test at constant stress; J t et=s and K , the second function, from thecreep function in shear. This leads to

eij J K Ds ij

Dt K

Ds kk

Dtdij 12

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

Or Mt, the relaxation function in shear, and Lt, a function deducedfrom M and from a relaxation test in tension Rt s t=e; Lt MR 2M=3M R

s ij LDekk

Dtdij 2M

Deijdt

13

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.

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C H A P T E R 2.2

Background onNonlinear ElasticityR. W. O GDENDepartment of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

Contents2.2.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 .2 .2 Deformat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.2.3 Stress and Equilibrium . . . . . . . . . . . . . . . . . . . . . . 772.2.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.2.5 Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 782.2.6 Constrained Materials . . . . . . . . . . . . . . . . . . . . . . . 80

2.2.7 Boundary-Value Problems .................... 82References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 conguration, denoted by B r , is identi ed

and @ B

r denotes the boundary of B

r . Points inB

r are labeled by theirposition vectors X relative to some origin. The body is deformed quasi-statically from B r so that it occupies a new con guration, denoted B , with



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

x vX X 2 B r ; 1

where x is the position vector of the point X in B . The mapping v is called thedeformation from B r 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 X a and x i,where a ; i 2 f 1; 2; 3g, so that x i wi X a . Greek and Roman indices refer,respectively, to B r and B , and the usual summation convention for repeatedindices is used.

The deformation gradient tensor , denoted F, is given by

F Grad x Fia @ x i=@ X a 2Grad being the gradient operator in B r . Local invertibility of v and its inverserequires that

05 J det F5 1 3

wherein the notation J is dened.The deformation gradient has the (unique) polar decompositions

F RU VR 4

where R is a proper orthogonal tensor and U, V are positive de nite andsymmetric tensors. Respectively, U and V are called the right and left stretchtensors. They may be put in the spectral forms

U X3

i1l iui ui V X

3

i1l i vi vi 5

where vi Rui; i 2 f 1; 2; 3g, l i are the principal stretches, ui the uniteigenvectors of U (the Lagrangian principal axes ), vi those of V (the Eulerian principal axes), and denotes the tensor product. It follows from Eq. 3 that J l 1 l 2l 3:

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

C FT F U2 B FFT V2 6

2.2.3 STRESS AND EQUILIBRIUM

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

r r r J 7

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

S J F1r 8

The equation of equilibrium may be written in the equivalent forms

div r r b 0 Div S r r b 0 9

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

@ Sai@ X a

r r bi 0 10

Balance of the moments of the forces acting on the body yields simplyr T r , equivalently ST FT 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 HF @ W @ F

F r GF J 1FHF 11

where H is a tensor-valued function, dened on the space of deformationgradients F, W is a scalar function of F and the symmetric tensor-valuedfunction G is dened by the latter equation in Eq. 11. In general, the form of H depends on the choice of reference con guration and it is referred to as theresponse function of the material relative to B r associated with S. For a givenB r , 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 of Eq. 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 Sa i @ 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 B r , so that

W I 0@ W @ F

I O 12

where I is the identity and O the zero tensor, then B r is called a naturalconguration.

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Suppose that a rigid-body deformation x * Qx c is superimposed onthe deformation x vX, 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 * QF: The elastic stored energy is required to be independentof superimposed rigid deformations, and it follows thatW QF W F 13

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 R T in Eq. 13shows that W F W U: Thus, W depends on F only through the stretchtensor U and may therefore be de ned on the class of positive de nite

symmetric tensors. We writeT

@ W @ U

14

for the (symmetric) Biot stress tensor , which is related to S byT SR R T ST =2.

2.2.5 MATERIAL SYMMETRY

Let F and F0 be the deformation gradients in B relative to two differentreference con gurations, B r and B 0r respectively. In general, the response of the material relative to B 0r differs from that relative to B r , and we denote by W and W 0 the strain-energy functions relative to B r and B 0r . Now let P Grad X

0

be the deformation gradient of B 0r relative to B r , where X0 is the position

vector of a point in B 0r . Then F F0P: For speci c P we may have W 0 W ,and then

W F0P W F0 15

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 B r . This group characterizes the physical symmetry properties of the material.

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

W FQ W F 16

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

W QUQ T W U 17

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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 l 1; l 2, and l 3 and is symmetric in these

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

r a0I a1B a2B2 18

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

a0 2I 1=23

@ W @ I 3

a1 2I 1=23

@ W @ I 1

I 1@ W @ I 2 a2 2I 1=23 @ W @ I 2 19

and W is now regarded as a function of I 1; I 2, and I 3 , the principal invariants

of B dened byI 1 trB l 21 l

22 l

23; 20

I 2 12 I 21 tr B

2 l 22l23 l

23l

21 l

21 l

22 21

I 3 det B l 21l22l

23 22

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

S X3

i1 tiui

vi r

X3

i1s

i vi

vi 23

where s i; i 2 f 1; 2; 3g are the principal Cauchy stresses and ti the principalBiot stresses, connected by

ti @ W @ l i

J l 1i s i 24

Let the unit vector M be a preferred direction in the reference con gurationof 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 W F; M M. The requiredsymmetry ( transverse isotropy) reduces W to dependence on the ve invariants

I 1; I 2; I 3; I 4 M CM I 5 M C2M 25

where I 1; I 2; and I 3 are de ned in Eqs. (20) (22). The resulting nominalstress tensor is given by

S 2W 1FT 2W 2I 1I CFT 2I 3W 3F1 2W 4M FM

2W 5M FCM CM FM 26where W i @ W =@ I i; i 1; . . . ; 5.



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

I 6 M0 CM0 I 7 M0 C2M0 I 8 M CM0 27and 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 CF 0, must be satis edfor all possible deformation gradients F, where C is a scalar function. Twocommonly used constraints are incompressibility and inextensibility, forwhich, respectively,

CF detF 1 CF M FT FM 1 28

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

Any stress normal to the hypersurface CF 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 HF q@ C@ F

@ W @ F

q@ C@ F

29

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 @ W @ F

qF1 S @ W @ F

2qM FM 30

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

T @ W @ U

pU1 det U 1 31

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and

r F@ W @ F

pI det F 1 32

where q has been replaced by p, which is called an arbitrary hydrostatic pressure. The term in a0 in Eq. 18 is absorbed into p, and I 3 1 in theremaining terms in Eq. 18. For an incompressible isotropic material theprincipal components of Eqs. 31 and 32 yield

ti @ W @ l i

pl 1i s i l i@ W @ l i

p 33

respectively, subject to l 1l 2l 3 1.For an incompressible transversely isotropic material with preferred

direction M, the dependence on I 3 is omitted and the Cauchy stress tensoris given by

r pI 2W 1B 2W 2I 1B B2 2W 4FM FM 2W 5FM BFM BFM FM 34

For a material with two preferred directions, M and M0, the Cauchy stresstensor for an incompressible material is

r pI 2W 1B 2W 2I 1B B2 2W 4FM FM

2W 5FM BFM BFM FM 2W 6FM0

FM0

2W 7FM0 BFM0 BFM0 FM0 W 8FM FM0 FM0 FM 35

where the notation W i @ W =@ I i now applies for i 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 r r b 0 F Grad x x vX X 2 B r 36

Typical boundary conditions in nonlinear elasticity are

x nX on @ B x r 37

ST N sF; X on @ B tr 38where n and s are speci ed functions, N is the unit outward normal to @ B r ,



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and @ B x r and @ Btr are complementary parts of @ B . In general, s may depend

on 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 JPFT N on @ B tr 39Equations 36 38 constitute the basic boundary-value problem in non-linear elasticity.

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

A aib j@ 2 x j

@ X a @ X b r r bi 0 40

for i 2 f 1; 2; 3g, where the coef cients A aib j are de ned by

A a ib j A b ja i @ 2

W @ Fia @ F jb41

When coupled with suitable boundary conditions, Eq. 41 forms a system of quasi-linear partial differential equations for x i wi X a . The coef cientsA aib j are, in general, nonlinear functions of the components of thedeformation gradient.

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

A a ib j@ 2 x

j@ X a @ X b @ p@ x i r r b

i 0 det @ x i@ X a 1 42

where the coef cients are again given by Eq. 41.In order to solve a boundary-value problem, a speci c 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 of the equations, an example of which is the strong ellipticity condition.Equations 40 are said to be strongly elliptic if the inequality

A a ib jmim jN a N b > 0 43

holds 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 of Eq. 43, but the incompressibility constraint now imposes the restrictionm FT N 0 on m and N.

REFERENCES

1. Beatty, M. F. (1987). Topics in nite elasticity: Hyperelasticity of rubber, elastomers andbiological tissues } with examples. Appl. Mech. Rev.40; 16991734.

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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. 132, Spencer, A. J. M., ed., Wien: Springer-Verlag.



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C H A P T E R 2.3

Elasticity of PorousMaterialsN. D. CRISTESCU231 Aerospace Building, Univ ersity of Florida, Gainesv ille, Florida

Contents2.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.3.3 Identication of the Parameters . . . . . . . . . . . . . . 852 .3 .4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 in uencing the values of the elastic parameters. Also, the stress-strain

curves 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 signi cant hysteresis loops aregenerally present.

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


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2.3.2 FORMULATIONThe elasticity of such materials can be expressed as instantaneousresponse by

D

T2G

1

3K

12G 13 tr T%1 1

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, thusinuencing 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 timeinuence on the entire stress-strain curves and failure (stars mark the failure points).

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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 are

obtained fromK r v 2 p

43

v 2S G r v 2S 2where v S is the velocity of propagation of the shearing waves, v p the velocity of the 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 con ning 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

13G

1

9K 1

1

6G

19K

13

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 signi cantinterference between creep and unloading phenomena will take place. Anexample for schist is shown in Figure 2.3.3, obtained in a triaxial test with veunloading reloading cycles.

FIGURE 2.3.2 Static procedures to determine the elastic parameters from partial unloadingprocesses preceded by short-term creep.

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If only a partial unloading is performed (one third or even one quarter of the 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 specimen

will 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 13

D tD e1 2D e2

G 12

D tD e1 D e2

4

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 D sD ev

5

with s the mean stress and ev the volumetric strain.Generally, K is increasing with s and reaching an asymptotic constant valuewhen 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 aperiod of creep of several minutes.



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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 con guration 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 the

compressibility 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 elastic

moduli G and K with the axial stress s 1 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 strainin monotonic uniaxial tests.

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strain eI V . 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 s c 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 v P and v S intrue triaxial tests under con ning pressure pc 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 ve unloading reloading cyclesshown are: E 9:9, 24.7, 29.0, 26.3, and 22.3 GPa, respectively, whileG 4:4, 10.7, 12.1, 10.4, and 8.5 GPa.

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

K s : K 0 K 1 1

ss 0 ; if s s 0

K 0; if s s 08>:

6

with K 0 59 GPa, K 1 48 GPa, and s 0 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 inuniaxial creep tests.



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The same formula for alumina powder is

K s : K 1 pa exp bs

pa 7with K 1 1 107 kPa the constant value toward which the bulk modulustends at high pressures, a 107, b 1:2 104, and pa 1 kPa. Also foralumina powder we have

E s : E1 pab expds 8

with E1 7 105 kPa, b 6:95 105, and d 0:002.For the shale shown in Figure 2.3.3, the variation of K with s for

0 s 45 MPa is

K s : 0:78s 2 65:32s 369 9

REFERENCES1. 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 Cohesiv e-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 Behav ior of Salt (5th Conference on MechanicalBehavior of Salt), Cristescu, N. D., and Hardy, Jr., H. Reginald, eds., Trans Tech Publ.,

Clausthal-Zellerfeld.

FIGURE 2.3.6 The maximum of v s takes place at the compressibility dilatancy boundary(gures and hachured strip); changes of v

pand v

sas a function of strain ( e 105 s1 ,

pc 5 Mpa, T 308 C), showing that the maxima are at the onset of dilatancy (afterReference [4]).

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C H A P T E R 2.4

Elastomer ModelsR. W. O GDENDepartment of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

Contents2.4.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.4.3 Description of the Model. . . . . . . . . . . . . . . . . . . . . 932.4.4 Identi cation of Parameters . . . . . . . . . . . . . . . . . . 932.4.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.4.6 Table of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 94References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 nite deformations. For general background on

nite elasticity, we refer to Ogden [5].

2.4.2 BACKGROUND

Locally, the nite deformation of a material can be described in terms of thethree principal stretches, denoted by l 1; l 2; and l 3 . For an incompressiblematerial these satisfy the constraint

l 1l 2l 3 1 1

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



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strain-energy function W l 1; l 2; l 3 per unit volume, where W dependssymmetrically on the stretches subject to Eq. 1.

The principal Cauchy stresses associated with this deformation are

given by

s i l i@ W @ l i

p; i 2 f 1; 2; 3g 2

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 de nition #W l 1; l 2 W l 1; l 2; l 11 l

12 ,

we obtain

s 1 s 3 l 1@ #

W @ l 1 s2 s 3 l 2

@ #W @ l 2 3

For consistency with the classical theory, we must have

#W 1; 1 0;@ 2 #W

@ l 1@ l 21; 1 2m;

@ #W @ l a

1; 1 0;@ 2 #W @ l 2a

1; 1 4m;

a 2 f 1; 2g4

where m is the shear modulus in the natural con guration. 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 suf ces to sets 3 0 in Eq. 3, so that

s 1 l 1@ #W @ l 1

s 2 l 2@ #W @ l 2

5

Biaxial experiments in which l 1; l 2 and s 1; s 2 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 in ation of a thin-walled (membranelike) tube with closedends provide suitable tests. In the latter case the governing equations arewritten

P* l 11 l12

@ #W @ l 2

F * @ #W @ l 1

12

l 2l 11@ #W @ l 2

6

where P* PR=H , P is the in ating pressure, H the undeformed membranethickness, and R the corresponding radius of the tube, while F * 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 l 1 is the axial stretch and l 2 theazimuthal stretch in the membrane.

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2.4.3 DESCRIPTION OF THE MODEL

A specic model which ts very well the available data on various rubbers is

that de ned byW X

N

n1mnl

an1 l

an2 l

an3 3=an 7

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

n1mnan 2m 8

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

s 1 s 3 XN

n1mnl

an1 l

an3 s 2 s 3 X

N

n1mnl

an2 l

an3 9

2.4.4 IDENTIFICATION OF PARAMETERS

Biaxial experiments with s3

0 indicate that the shapes of thecurves of s 1 s 2 plotted against l 1 are essentially independent of l 2 formany rubbers. Thus the shape may be determined by the pure shear test withl 2 1, so that

s 1 s 2 XN

n1mnl

an1 1 s 2 X

N

n1mnl

an3 1 10

for l 1 1; l 3 1. The shift factor to be added to the rst equation in Eq. 10when l 2 differs from 1 is

XN

n1mn1 l

an2 11

Information on both the shape and shift obtained from experiments at xedl 2 then suf ce to determine the material parameters, as described in detail inReferences [3] or [4].

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

s 1 s 2 l 1@ #W @ l 1

l 2@ #W @ l 2

l 1F * 12

l 22l 1P* 12

2.4 Elastomer Models 93


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2.4.5 HOW TO USE IT

The strain-energy function is incorporated in many commercial Finite

Element (FE) software packages, such as ABAQUS and MARC, and can beused in terms of principal stretches and principal stresses in the FE solution of boundary-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 rst is the material used by Jones and Treloar [2]:

a1 1:3; a2 4:0; a3 2:0;m1 0:69; m2 0:01; m3 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 0:707 ; a2 2:9; a3 2:62;m1 0:941 ; m2 0:093 ; m3 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 : 20332058.

2. Jones, D. F., and Treloar, L. R. G. (1975). The properties of rubber in pure homogeneous strain. J. Phys. D: Appl. Phys.8: 12851304.

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 : 361383.

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 : 261280.

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C H A P T E R 2.5

Background onViscoelasticityK OZO IKEGAMITokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo 101-8457, Japan

Contents2.5.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.5.2 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . 952.5.3 Static Viscoelastic Deformation. . . . . . . . . . . . . . . 982.5.4 Dynamic Viscoelastic Deformation . . . . . . . . . 1002.5.5 Hereditary Integral . . . . . . . . . . . . . . . . . . . . . . . . 1022.5.6 Viscoelastic Constitutive Equation by the

Laplace Transformation . . . . . . . . . . . . . . . . . . . . 1032.5.7 Correspondence Principle . . . . . . . . . . . . . . . . . . 104R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 6

2.5.1 VALIDITY

Fundamental deformation of materials is classi ed 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



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

s ke s Zdedt

1

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:

s Zk

dsdt

Zdedt

2

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 ke Zdedt 3

This is called the Voigt or Kelvin model.

FIGURE 2.5.1 Mechanical model of viscoelasticity.

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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:

p0s p1dsdt

p2d2sdt2

. . . pndnsdtn

q0e q1dedt

q2d2edt2

. . . qndnedtn

4

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

FIGURE 2.5.2 Maxwell model.

2.5 Background on Viscoelasticity 97


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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 dened 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

etZ

1k

1k

tt

1 5where t M Z=k, and this is denoted as relaxation time. For the Voigt model,the creep compliance is

e 1k

1 exp ktZ ! 1k 1 exp tt k ! 6

where t K Z=k, and this is denoted as retardation time.Creep deformations of the Maxwell and Voigt models are illustrated inFigures 2.5.4 and 2.5.5, respectively. Creep strain of the Maxwell model

FIGURE 2.5.3 Voigt (Kelvin) model.

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FIGURE 2.5.4 Creep compliance of the Maxwell model.

FIGURE 2.5.5 Creep compliance of the Voigt model.



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

The relaxation modulus is de ned by stress variations under constant unit

strain. 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 k exp ktZ k exp tt M 7

For the Voigt model,s k 8

Relaxation behaviors of the Maxwell and Voigt models are illustrated inFigures 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.

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The viscoelastic effect causes delayed phase phenomena between input andoutput responses. Viscoelastic responses for changing stress or strain aredened 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 expio t, where i is an imaginary unit andois the frequency of changing stress, into Eq. 2, complex compliance J isobtained as follows:

J 1k

i1

oZ

1k

i1

kot M J 0 iJ 00 9

where the real part J 0 1=k is denoted as storage compliance, and the

imaginary part J 00

1=kot M 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.



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e expio t as follows:

Y kot M2

1 ot M2 ik

ot M

1 ot M2 Y

0 iY 00 10

where Y 0 kot M2=1 ot M2 and Y 00 kot M=1 ot M2. 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 Y 00

Y 0

1ot M

11

This is called mechanical loss.Similarly, the complex compliance and the modulus of the Voigt model areable to be obtained. The complex compliance is

J 1k

11 ot K 2" # i 1k ot K 1 ot K 2" # J 0 iJ 00 12

where J 0 1k

11 ot K 2" #and J 00 1k ot K 1 ot K 2" #

The complex modulus is

Y k iot K Y 0 iY 00 13

where Y 0 k and Y 00 kot K .

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 d s 0 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 t05 t is represented bythe following integral:

et s 0 J t Z t

0 J t t0

ds 0

dt0dt0 14

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This equation is transformed by partially integrating as follows:

et s t J 0 Z t

0s t0

dJ t t0dt t0

dt0 15

Similarly, stress variation for arbitrary strain history becomes

s t e0Y t Z t

0Y t t0

ds 0

dt0dt0 16

Partial integration of Eq. & gives the following equation:

s t etY 0 Z t

0s t0

dY t t0dt t0

dt0 17

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

k0 pk d

ksdtk

Xm

k0qk d

kedtk 18

FIGURE 2.5.8 Hereditary integral.



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

Ps Qe 19

where P Pn

k0 pk d

k

dtk and Q Pm

k0qk d

k

dtk.

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

Xn

k0 pksk % s X

n

k0qksk%e 20

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

% s %Q%P

%e 21

where %P Pn

k0 pksk and %Q P

m

k0qksk.

Comparing Eq. 21 with Hooke s law in one dimension, the coef cient %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 by

Xm

k0 p0k

dks 0ijdtk

Xn

k0q00k

dkeiidtk

22

P00s ii Q00eii 23

where P 00Pm

k0 p00k d

k

dtk and Q 00 Pn

k0q00k d

k

dtk. In Eq. 22 hydrostatic pressure is (1/3)s ii and dilatation is eii.

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The relation between deviatoric stress and strain is represented by

Xm

k0

p0kdks 0ijdtk

Xn

k0

q0kdke0ijdtk

24

P0s 0ij Q0e0ij 25

where P 0 Pm

k0 p0k

dk

dtkand Q 0 P

n

k0q0k

dk

dtk. In Eq. 24 deviatoric stress and strain

are s 0ij and e0ij, respectively.

The Laplace transformations of Eqs. 22 and 24 are written, respectively, asfollows:

%P00% s ii %Q00%eii 26

where %P00 %P00s and %Q00 %Q00s s, and%P0% s 0ij %Q

0%e0ij 27

where %P0 %P0s and %Q0 %Q0s.The linear elastic constitutive relations between hydrostatic pressure and

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

s ii 3Keii 28

s 0ij 2Ge0ii 29

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

3K %Q00

%P0030

2G %Q

0

%P0 31

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

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

G E

21 n 32

K E

31 2n 33



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

E 3 %Q0%Q002 %P0%Q00 %P00%Q0 34

n %P0%Q00 %P00%Q0

2 %P0%Q00 %P00%Q0 35

Linear viscoelastic deformation corresponds to linear elastic deformationthrough Eqs. 30 31 and Eqs. 34 35. This is called the correspondence

principle 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.

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C H A P T E R 2.6

A Nonlinear ViscoelasticModel Based onFluctuating ModesR ACHID R AHOUADJ AND CHRISTIAN CUNATLEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la For #et-de-Haye, 54500 Vandoeuvre-l "es-

Nancy, France

Contents2.6.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.6.2 Background of the DNLR . . . . . . . . . . . . . . . 108

2.6.2.1 Thermodynamics of IrreversibleProcesses and Constitutive Laws . . . 108

2.6.2.2 Kinetics and ComplementaryLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.6.2.3 Constitutive Equations of the DNLR . . . . . . . . . . . . . . . . . . . . . . . . 112

2.6.3 Description of the Model in the Caseof Mechanical Solicitations. . . . . . . . . . . . . . 113

2.6.4 Identi cation of the Parameters . . . . . . . . . 1132.6.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.6.6 Table of Parameters. . . . . . . . . . . . . . . . . . . . . 115References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 integral

or 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



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

Such a general formulation also permits us to consider variousnonlinearities as functions of material speci cities and applied perturbations.To clarify the present modeling, called the distribution of nonlinear

relaxations (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 T HERMODYNAMICS OF IRREVERSIBLEPROCESSES 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 rst, a set of internal 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 es; e; n; . . . ; z 1

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

T dD isdt

Ts s Js : r T Xn

k1 Jk : r mk A z ! 0 2

where the nonequilibrium thermodynamic forces may be separated into twogroups: (i) the gradient ones, such as the gradient of temperature gradientr T, and the gradient of generalized chemical potential r mk; and (ii) Thegeneralized forces A, or af nities as de ned 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, uxes, 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


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gradient), thus:

T s s A

z ! 0 3

The equilibrium or relaxed state (denoted by the index r ) is currentlydescribed by a suitable thermodynamic potential ( c r ) 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):

c r c r g; zr g c r g 4

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

c c g; z 5

Then, we assume that the constitutive equations may be obtained as functionsof the rst partial derivatives of this potential with respect to the dualvariables, and depend consequently on both control and internal variables;i.e., b bg; zand A Ag; z. 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:

dc k Xq

m1bmdgm X

r

j1 A j dz j 6

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 au : g b : z 7a

A tb : g g : z 7b

This differential system resumes in a general and condensed form theannounced constitutive relationships. The symmetrical matrix au @2c =@g@gis the matrix of Tisza, and the symmetrical matrix g @2c =@z@z traduces theinteraction between the dissipation processes [3]. The rectangular matrixb @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 0 and

A 0. From Eq. 7bwe nd, for any equilibrium state, that the internal variables evolution resultsdirectly from the variation of the control variables:

zr g 1 : tb : g 8

2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes 109


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

A g :z zr 9

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

A gz zr 10

where g is assumed to be constant.

2.6.2.2 K INETICS 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 uxes

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 af ne relation between uxes 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 L A 11

and hence, with Eq. 10:

z L g z zr t 1 :z zr 12

According to this nonlinear kinetics, Meixner [4] has judiciously suggested abase change in which the relaxation time operator t is diagonal. Here, we

consider this base, which also represents a normal base for the dissipationmodes. In what follows, the relaxation spectrum will be explicitly de ned 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 probability p j expD F ;r j =RT of overcoming a free energy barrier, D F

;r j . It follows

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

t r j 1=u expD F ;r j =RT 13

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



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The reference jump frequency, u0 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 of

Plank, 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 ag:

u u0=ag 14

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 rst order:

D F j D F

;r j K z :z z

r 15

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

D F ;r j D E ;r T D S ;r j 16

where the internal energy D E ;r is supposed to be the same for all processes. Itfollows that we may de ne another important shift function, noted aT ,which accounts for the effect of temperature. According to the Arrheniusapproximation, D E ;r being quasi-constant, this shift function veri es thefollowing relation:

ln aT ; T ref D E ;r 1=T 1=T ref 17

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

lnaT c1T T ref =c2 T T ref 18

In summary, the relaxation times can be generally expressed ast jT t r j T ref aT ; T ref agaz; z

r 19

and the shift function az; zr 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 uctuations

which may be approximately compared to the corresponding equilibrium one.According to prigogine [6], these uctuations obey the equipartition of theentropy production. Therefore, we can deduce the expected distribution in



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the vicinity of equilibrium as

p0 j B

ffiffiffiffit r j

q with

Xn

j

1

p0 j 1 and B 1

0Xn

j

1

ffiffiffiffit r j

q 20

where t r j is the relaxation time of the process j, p0 j its relative weight in the

overall spectrum, and n the number of dissipation processes [3].As a rst approximation, the continuous spectrum de ned by Eq. 20 may

be 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 suf ciently high number n of dissipation modes, e.g., 30, gives asuf cient accuracy.

2.6.2.3 C ONSTITUTIVE EQUATIONS OF THE DNLR

Combining Eqs. 7a and 12 gives, whatever the chosen kinetics,b au : g b z zr :t 1z a

u : g #a #ar :t 1b 21a

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

bm Xn

p1aumpg p X

n

j1

b jm p0 j br m

t j21b

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 g0expio t, where o is the applied frequency,and i2 1, i.e., gn iog n. The response is obtained by integrating the

above 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 of relaxation. Furthermore, a convenient possibility for very small pertur-bations is to assume that the corresponding response is periodic and outof phase:

bn b0expio t j and bn iob n 22

where j is the phase angle. In fact, such relations are representative of various

physical properties as shown by Kramers [7] and Kronig [8].The coef cients of the matrices of Tisza, au and ar , and the relaxationtimes, t j, may be dependent on temperature and =or frequency. In uniaxial



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tests of mechanical damping, these Tisza s coef cients 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 nally written

s

Xn

j1

p0 j au : e

Xn

j1

s j p0 j ar : e

aeae; er aT ; T ref t jT ref

23

As an example, for a pure shear stress this becomes

s 12 Xn

j1 p0 j G

ue12 Xn

j1

s j 12 p0 j Gr e12

aeae; er aT ; T ref t G j T ref 24

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

G o Gu Gr GuXn

j1 p0 j

11 iot G j

25

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

G0o Gu Gr GuXn

j1 p0 j

11 o 2t G j

2 26

G00o Gr Gu

Xn

j1

p0 jot j

1 o2t

G j

2 27

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 dif culty one has recourse to the appropriate principle

of 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



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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 Davidson

and Cole [11]. For pure shear stress the response given by this HNapproach is

G GuHN Gr HN G

uHN

11 iot HNa b

28

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

G0 GuHN Gr HN G

uHN

cosby

1 2oat

aHNcosap =2 o 2

at 2

a b=2 29

G00 Gr HN GuHN

sinby 1 2o a t aHNcosap =2 o 2a t 2a

b=2 30

The function y is dened by

y tan 1o a t aHNsinap =2

1 o a t aHNcosap =2 31Eqs. 28 to 30 are respectively compared to Eqs. 25 to 27 in order to establish acorrespondence between the relaxation times of the two models:

logt Gr j logt HN jL =n Y 32

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

t G j agtGr j ao t

Gr j

tan byot HN t Gr j 33

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

Gr Gu Gr HN GuHN f G 34

The function f G is given by

f G cosby 1 tan 2by

1 2o a t aHNcosap =2 o 2a t 2aHNb=2 35



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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 ao and of f G for PTMG.*

* PTMG: poly (tetramethylene ether) glycol; MIDI: 4,4 0-diphenylmethane diisocyanate; DMPD:2,2-dimethyl-1, 3-propanediol with a density of 1.074 g =cm 3 , and glass transition T g 408C.



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with L 6, a scale parameter Y equal to 5.6, and 50 relaxation times. Theparameters Gr HN 2:14 MPa, GuHN Gu 1859 MPa, t HN 1.649 10

7 s, a 0:5709 and b 0:0363 allow us to calculate the shift

function ao and the function f G 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 f G and the shiftfunction ao 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: 680685.4. 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).



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C H A P T E R 2.7

Linear Viscoelasticitywith DamageR. A. SCHAPERYDepartment of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, Texas

Contents2.7.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.7.3 Description of the Model. . . . . . . . . . . . . . . . . . . 1192.7.4 Identi cation of the Material Functions

and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212.7.5 How to Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3

2.7.1 VALIDITY

This paper describes a homogenized constitutive model for viscoelastic

materials 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 of damage. For time-independent damage, the speci c 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 account

for intrinsic nonlinear viscoelastic and viscoplastic effects as well asthermorheologically complex behavior and multiple relaxation functions arepublished elsewhere [10].



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2.7.2 BACKGROUND

As background to the model with time-dependent damage, consider rst the

constitutive equation with constant damage, in which e and s representthe strain and stress tensors, respectively,

e fSds g eT 1

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 the

familiar form for thermorheologically simple materials,

f fdg g Z t

o f x x0

@ g @ t0

dt0 Z x

o f x x0

@ g @ x0

dx0 2

where it is assumed f g o for t5 o and

x Z t

odt00=aT T t00 x0 xt0 3

Also, aT T is the temperature-dependent shift factor. If the temperature is

constant in time, then x x0 t t0=aT : Physical aging [12] may be takeninto account by introducing explicit time dependence in aT ; i.e., useaT aT T ; t00 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 of the creep compliance and stress tensors, respectively.

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

S kD 4

where k is a constant, dimensionless tensor and D Dx 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 satis es 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-transition

temperature, T g , is nearly constant; except at time or rate extremes, somewhatabove T g Poisson s ratio is essentially one half, while below T g it is commonlyin the range 0.35 0.40 [5].

Schapery118


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Equations 1 and 4 give

e fDdks g eT 5

The inverse iss kI f Edeg kI f EdeT g 6

where kI k1 and E Ex is the uniaxial relaxation modulus in which,for t > o,

DdEf g EdDf g 1 7

In relating solutions of elastic and viscoelastic boundary value problems,and for later use with growing damage, it is helpful to introduce thedimensionless quantities

eR1

ERf Edeg eRT

1ER

f EdeT g uR1

ERf Edug 8

where ER is an arbitrary constant with dimensions of modulus, called thereference modulus; also, eR and eRT are so-called pseudo-strains and u

R isthe pseudo-displacement. Equation 6 becomes

s CeR CeRT 9

where C ERkI is like an elastic modulus tensor; its elements are called

pseudo-moduli. Equation 9 reduces to that for an elastic material by takingE 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 #

Ss eRT 10

where#

S C1 k=ER: The physical strain is given in Eq. 5.

2.7.3 DESCRIPTION OF THE MODELThe 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 convolution

integral 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


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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 growth

equations or, in a more general context, on nonequilibrium thermodynamicequations. Speci cally, let W R and W RC denote pseudo-strain energy densityand pseudo-complementary strain energy density, respectively,

W R 12

CeR eRT eR eRT F 11

W RC 12

#

Sss eRT s F 12

so that

W RC W R se R 13and

s @ W R

@ eReR

@ W RC@ s

14

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 de ned by a set of scalar ISVs, S p ( p 1,2, . . . P) instead of tensor ISVs. Thermodynamic forces,which are like energy release rates, are introduced,

f p @ W R

@ S p15

or

f p@ W RC@ S p

16

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

Although more general forms could be used, the evolution equations forS p dS p=dx are assumed in the form

S p S pSq; f p 17

in which S p may depend on one or more Sq (q 1, . . . P), but on only oneforce f p. The entropy production rate due to damage is non-negative if

X p

f p S p O 18

thus satisfying the Second Law of Thermodynamics. It is assumed that whenj f pj is less than some threshold value, then S p O.

Schapery120


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

f p @ W RC@ S p

12

@ #

S@ S p ss

@ eRT @ S p s

@ F@ S p 19

which does not vanish when r o, unless @ F=@ S p 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 f p > o,

S p f pa p 20

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

Only an outline of the identi cation 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 ber composites. The effectsof eT and F are neglected here.

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



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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 nd Dx, after which Ex is derived

from Eq. 7.(b) Constant strain rate (or stress rate) tests to failure at a series of rates or temperatures may be conveniently used to obtain the additional dataneeded for identi cation 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 signi cant strain localization) may be found from theformer tests.

For isothermal, constant strain rate, R, tests, the input is Rt #Rx; where#R RaT and x 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 identi cation of the model from a series of tests 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 suf ciently broad range. One should, however, conduct at leasta small number of both types of tests to check the thermorheologically

simple 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 identi cation steps to bethose for a linear elastic material with rate-dependent damage. If controlledstrain (stress) tests are used, then one would employ W RW RC in theidenti cation. However, mixed variables may be input test parameters, suchas constant strain rate tests of specimens in a test chamber at a series of specied 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 nding 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 con ning pressures. The materialemployed by them was initially isotropic, but it became transversely isotropic

as a result of damage. Identi cation of the full set of ve pseudo-moduli andthe pseudo-strain energy function, as functions of two ISVs, is detailed by Haand Schapery [1].

Schapery122


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2.7.5 HOW TO USE IT

Implementation of user-de ned constitutive relations based on this model in a

nite 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 forparticulate composites with growing damage and its experimental validation. International Journal of Solids and Structures35: 34973517.

2. Hashin, Z. (1983). Analysis of composite materials } a survey. Journal of Applied Mechanics105 : 481505.

3. Park, S. W., Kim, Y. R., and Schapery, R. A. (1996). A viscoelastic continuum damagemodel and its application to uniaxial behavior of asphalt concrete . Mechanics of Materials24: 241255.

4. Park, S. W., and Schapery, R. A. (1997). A viscoelastic constitutive model for particulatecomposites with growing damage . International Journal of Solids and Structures 34: 931947.

5. Schapery, R. A. (1974). Viscoelastic behavior and analysis of composite materials, inMechanics of Composite Materials, pp. 85168, vol. 2, Sendeckyi, G. P., ed., New York:Academic.

6. Schapery, R. A. (1981). On viscoelastic deformation and failure behavior of compositematerials with distributed aws, in 1981 Advances in Aerospace Structures and Materials,pp. 520, Wang, S. S., and Renton, W. J., eds., ASME, AD-01.

7. Schapery, R. A. (1982). Models for damage growth and fracture in nonlinear viscoelasticparticulate 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 largedeformation and fracture analysis of viscoelastic media, in: International Journal of Fracture25: 195223.

9. Schapery, R. A. (1997). Constitutive equations for special linear viscoelastic composites withgrowing 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 withgrowing damage. International Journal of Fracture 97: 3366.

11. Schapery, R. A., and Sicking, D. L. (1995). On nonlinear constitutive equations for elastic andviscoelastic composites with growing damage, in Mechanical Behavior of Materials, pp. 4576,


lemaitre ch2

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