Elasticity and Viscoelasticity CHAPTER2

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1 Elasticity and Viscoelasticity CHAPTER2

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3 CHAPTER2.1 Introduction to Elasticity and Viscoelasticity JEAN LEMAITRE Universit!e Paris 6, LMT-Cachan, 61 avenue du Pr!esident Wilson, Cachan Cedex, France For all solid materials there is a domain in stress space in which strains are reversible due to small relative movements of atoms. For many materials like metals, ceramics, concrete, wood and polymers, in a small range of strains, the hypotheses of isotropy and linearity are good enough for many engineering purposes. Then the classical Hooke s law of elasticity applies. It can be derived from a quadratic form of the state potential, depending on two parameters characteristics of each material: the Young s modulus E and the Poisson s ratio n. c * ¼ 1 2r A ijklðe;nþs ij s kl ð1þ e ij ¼ * ¼ 1 þ ij E s ij n E s kkd ij ð2þ Eandn are identified from tensile tests either in statics or dynamics. A great deal of accuracy is needed in the measurement of the longitudinal and transverse strains (de 10 6 in absolute value). When structural calculations are performed under the approximation of plane stress (thin sheets) or plane strain (thick sheets), it is convenient to write these conditions in the constitutive equation. Plane stress ðs 33 ¼ s 13 ¼ s 23 ¼ 0Þ: e 11 e 22 e E 7 5 ¼ 6 4 Sym n E 1 E 1 þ n E s 11 s 22 s ð3þ Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 71

4 72 Lemaitre Plane strain ðe 33 ¼ e 13 ¼ e 23 ¼ 0Þ: s 11 l þ 2m l s 22 5 ¼ 4 l þ 2m 0 54 Sym 2m s 12 with e 11 e 22 e ne >< l ¼ ð1 þ nþð1 2nÞ >: m ¼ E 2ð1 þ nþ For orthotropic materials having three planes of symmetry, nine independent parameters are needed: three tension moduli E 1 ; E 2 ; E 3 in the orthotropic directions, three shear moduli G 12 ; G 23 ; G 31, and three contraction ratios n 12 ; n 23 ; n 31. In the frame of orthotropy: e n 12 n 3 s E 1 E 1 E 1 e 22 1 n 23 s E 2 E 2 e s 33 E ¼ 3 1 ð5þ e G 23 s 23 1 Sym 0 e 31 2G s e 12 2G 12 Nonlinear elasticity in large deformations is described in Section 2.2, with applications for porous materials in Section 2.3 and for elastomers in Section 2.4. Thermoelasticity takes into account the stresses and strains induced by thermal expansion with dilatation coefficient a. For small variations of temperature y for which the elasticity parameters may be considered as constant: e ij ¼ 1 þ n E s ij n E s kkd ij þ ayd ij ð6þ For large variations of temperature, E; n; and a will vary. In rate formulations, such as are needed in elastoviscoplasticity, for example, the s 12 ð4þ

5 2.1 Introduction to Elasticity and Viscoelasticity 73 derivative of E; n; and a must be considered. e ij ¼ 1 þ n E s ij n E s kkd ij þ a yd ij 1 þ n s n s kk d yd ij y ð7þ Viscoelasticity considers in addition a dissipative phenomenon due to internal friction, such as between molecules in polymers or between cells in wood. Here again, isotropy, linearity, and small strains allow for simple models. Quadratric functions for the state potential and the dissipative potential lead to either Kelvin-Voigt or Maxwell s models, depending upon the partition 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 ¼ lðe kk þ y l e kk Þd ij þ 2mðe ij þ y m e ij Þ ð8þ Here l and m are Lame s coefficients at steady state, and y l and y m are two time parameters responsible for viscosity. These four coefficients may be identified from creep tests in tension and shear. Maxwell model: e ij ¼ 1 þ n s ij þ s n E t 1 E s kk þ s kk d ij ð9þ t 2 Here E and n are Young s modulus and Poisson s ratio at steady state, and t 1 and t 2 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 constitutive models is through the functional formulation by the convolution product as any linear system. The hereditary integral is described in detail for the one-dimensional case, together with its use by the Laplace transform, in Section 2.5. Z t e ijðtþ ¼ J ijkl ðt tþ ds Xn kl dt þ J ijkl ðt tþds p kl ð10þ o dt p¼1 p J ðtþ is the creep functions matrix, and Ds kl are the eventual stress steps. The dual formulation introduces the relaxation functions matrix R ðtþ Z t s ijðtþ ¼ R ijkl ðt tþ de Xn kl dt þ R ijkl De p kl ð11þ o dt p¼1 When isotropy is considered the matrix, ½ JŠ and ½RŠ each reduce to two functions: either J ðtþ, the creep function in tension, is identified from a creep

6 74 Lemaitre test at constant stress; J ðtþ ¼ e ðtþ =s and K, the second function, from the creep function in shear. This leads to e ij ¼ðJþKÞ Ds ij Dt K Ds kk Dt d ij ð12þ where stands for the convolution product and D for the distribution derivative, taking into account the stress steps. Or M ðtþ, the relaxation function in shear, and L ðtþ, a function deduced from M and from a relaxation test in tension R ðtþ ¼ s ðtþ =e; L ðtþ ¼ MðR 2MÞ=ð3M RÞ s ij ¼ L Dðe kkþ d ij þ 2M De ij ð13þ Dt dt All of this is for linear behavior. A nonlinear model is described in Section 2.6, and interaction with damage is described in Section 2.7.

7 CHAPTER2.2 Background on Nonlinear Elasticity R. W. OGDEN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK Contents Validity Deformation Stress and Equilibrium Elasticity Material Symmetry Constrained Materials Boundary-Value Problems References VALIDITY The theory is applicable to materials, such as rubberlike solids and certain soft biological 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] DEFORMATION For a continuous body, a reference configuration, denoted by B r, is identified r denotes the boundary of B r. Points in B r are labeled by their position vectors X relative to some origin. The body is deformed quasistatically from B r so that it occupies a new configuration, denoted B, with Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 75

8 76 Ogden This is the current or deformed configuration of the body. The deformation is represented by the mapping w : B r! B, so that x ¼ vðxþ X 2 B r ; ð1þ where x is the position vector of the point X in B. The mapping v is called the deformation from B r to B, and v is required to be one-to-one and to satisfy appropriate regularity conditions. For simplicity, we consider only Cartesian coordinate systems and let X and x, respectively, have coordinates X a and x i, where a; i 2f1; 2; 3g, so that x i ¼ w i ðx a Þ. Greek and Roman indices refer, respectively, to B r and B, and the usual summation convention for repeated indices is used. The deformation gradient tensor, denoted F, is given by F ¼ Grad x F ia i =@X a ð2þ Grad being the gradient operator in B r. Local invertibility of v and its inverse requires that 05 J det F51 ð3þ wherein the notation J is defined. The deformation gradient has the (unique) polar decompositions F ¼ RU ¼ VR ð4þ where R is a proper orthogonal tensor and U, V are positive definite and symmetric tensors. Respectively, U and V are called the right and left stretch tensors. They may be put in the spectral forms U ¼ X3 l i u ðiþ u ðiþ V ¼ X3 l i v ðiþ v ðiþ ð5þ i¼1 where v ðiþ ¼ Ru ðiþ ; i 2f1; 2; 3g, l i are the principal stretches, u ðiþ the unit eigenvectors of U (the Lagrangian principal axes), v ðiþ those of V (the Eulerian principal axes), and denotes the tensor product. It follows from Eq. 3 that J ¼ l 1 l 2 l 3 : The right and left Cauchy-Green deformation tensors, denoted C and B, respectively, are defined by C ¼ F T F ¼ U 2 B ¼ FF T ¼ V 2 ð6þ i¼ STRESS AND EQUILIBRIUM Let r r and r be the mass densities in B r and B, respectively. The mass conservation equation has the form r r ¼ rj ð7þ

9 2.2 Background on Nonlinear Elasticity 77 The Cauchy stress tensor, denoted r, and the nominal stress tensor, denoted S, are related by S ¼ JF 1 r ð8þ The equation of equilibrium may be written in the equivalent forms div r þ rb ¼ 0 Div S þ r r b ¼ 0 ð9þ 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 second equation in Eq. 9 ai þ r b i ¼ 0 ð10þ a Balance of the moments of the forces acting on the body yields simply r T ¼ r, equivalently S T F T ¼ FS: The Lagrangian formulation based on the use of S and Eq. 10, with X as the independent variable, is used henceforth ELASTICITY The constitutive equation of an elastic material is given in the equivalent forms S ¼ ðfþ r ¼ GðFÞ J 1 FHðFÞ ð11þ where H is a tensor-valued function, defined on the space of deformation gradients F, W is a scalar function of F and the symmetric tensor-valued function G is defined by the latter equation in Eq. 11. In general, the form of H depends on the choice of reference configuration and it is referred to as the response function of the material relative to B r associated with S. For a given B r, therefore, the stress in B at a (material) point X depends only on the deformation 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 a strain-energy function (or stored-energy function). In components, (11) 1 has the form S ai ia, which provides the convention for ordering of the indices in the partial derivative with respect to F. If W and the stress vanish in B WðIÞ ¼0 ðiþ ð12þ where I is the identity and O the zero tensor, then B r is called a natural configuration.

10 78 Ogden Suppose that a rigid-body deformation x * ¼ Qx þ c is superimposed on the deformation x ¼ vðxþ, where Q and c are constants, Q being a rotation tensor 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 independent of superimposed rigid deformations, and it follows that WðQFÞ ¼WðFÞ ð13þ for all rotations Q. A strain-energy function satisfying this requirement is said to be objective. Use of the polar decomposition (Eq. 4) and the choice Q ¼ R T in Eq. 13 shows that WðFÞ ¼WðUÞ: Thus, W depends on F only through the stretch tensor U and may therefore be defined on the class of positive definite symmetric tensors. We write T for the (symmetric) Biot stress tensor, which is related to S by T ¼ðSR þ R T S T Þ= MATERIAL SYMMETRY Let F and F 0 be the deformation gradients in B relative to two different reference configurations, B r and B 0 r respectively. In general, the response of the material relative to B 0 r 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 0 r. Now let P ¼ Grad X0 be the deformation gradient of B 0 r relative to B r, where X 0 is the position vector of a point in B 0 r. Then F ¼ F0 P: For specific P we may have W 0 ¼ W, and then WðF 0 PÞ¼WðF 0 Þ ð15þ for all deformation gradients F 0. The set of tensors P for which Eq. 15 holds forms a multiplicative group, called the symmetry group of the material relative to 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 WðFQÞ ¼WðFÞ ð16þ for all rotations Q. Since the Q s appearing in Eqs. 13 and 16 are independent, the combination of these two equations yields WðQUQ T Þ¼WðUÞ ð17þ

11 2.2 Background on Nonlinear Elasticity 79 for all rotations Q. Equation 17 states that W is an isotropic function of U. It follows from the spectral decomposition (Eq. 5) that W depends on U only through 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 ¼ a 0 I þ a 1 B þ a 2 B 2 ð18þ where a 0 ; a 1, and a 2 are scalar invariants of B (and hence of V) given by a 0 ¼ 2I 3 a 1 ¼ 3 þ I 1 a 2 ¼ 2I and W is now regarded as a function of I 1 ; I 2, and I 3, the principal invariants of B defined by I 1 ¼ trðbþ ¼l 2 1 þ l2 2 þ l2 3 ; ð20þ I 2 ¼ 1 2 ½I2 1 tr ðb2 ÞŠ ¼ l 2 2 l2 3 þ l2 3 l2 1 þ l2 1 l2 2 ð21þ I 3 ¼ det B ¼ l 2 1 l2 2 l2 3 ð22þ Another consequence of isotropy is that S and r have the decompositions S ¼ X3 t i u ðiþ v ðiþ r ¼ X3 s i v ðiþ v ðiþ ð23þ i¼1 where s i ; i 2f1; 2; 3g are the principal Cauchy stresses and t i the principal Biot stresses, connected by t i ¼ Jl 1 i s i i Let the unit vector M be a preferred direction in the reference configuration of the material, i.e., a direction for which the material response is indifferent to 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 F and the tensor M M [2, 4, 5] Thus, we write WðF; M MÞ. The required symmetry (transverse isotropy) reduces W to dependence on the five invariants I 1 ; I 2 ; I 3 ; I 4 ¼ M ðcmþ I 5 ¼ M ðc 2 MÞ ð25þ where I 1 ; I 2 ; and I 3 are defined in Eqs. (20) (22). The resulting nominal stress tensor is given by S ¼ 2W 1 F T þ 2W 2 ði 1 I CÞF T þ 2I 3 W 3 F 1 þ 2W 4 M FM þ 2W 5 ðm FCM þ CM FMÞ ð26þ where W i i ; i ¼ 1;...; 5. i¼1

12 80 Ogden When there are two families of fibers corresponding to two preferred directions in the reference configuration, M and M 0 say, then, in addition to Eq. 25, the strain energy depends on the invariants I 6 ¼ M 0 ðcm 0 Þ I 7 ¼ M 0 ðc 2 M 0 Þ I 8 ¼ M ðcm 0 Þ ð27þ and also on M M 0 (which does not depend on the deformation); see Spencer [4, 5] for details. The nominal stress tensor can be calculated in a similar way to Eq CONSTRAINED MATERIALS An internal constraint, given in the form CðFÞ ¼0, must be satisfied for all possible deformation gradients F, where C is a scalar function. Two commonly used constraints are incompressibility and inextensibility, for which, respectively, CðFÞ ¼detF 1 CðFÞ ¼M ðf T FMÞ 1 ð28þ where the unit vector M is the direction of inextensibility in B r. Since any constraint is unaffected by a superimposed rigid deformation, C must be an objective scalar function, so that CðQFÞ¼CðFÞ for all rotations Q. Any stress normal to the hypersurface CðFÞ ¼0 in the (nine-dimensional) space of deformation gradients does no work in any (virtual) incremental deformation compatible with the constraint. The stress is therefore determined by the constitutive law (11) 1 only to within an additive contribution parallel to the normal. Thus, for a constrained material, the stress-deformation relation (11) 1 is replaced by S @F ð29þ where q is an arbitrary (Lagrange) multiplier. The term in q is referred to as the constraint stress since it arises from the constraint and is not otherwise derivable from the material properties. For incompressibility and inextensibility we have þ qf 1 S þ 2qM ð30þ respectively. For an incompressible material the Biot and Cauchy stresses are given by pu 1 det U ¼ 1 ð31þ

13 2.2 Background on Nonlinear Elasticity 81 and r ¼ pi det F ¼ 1 where q has been replaced by p, which is called an arbitrary hydrostatic pressure. The term in a 0 in Eq. 18 is absorbed into p, and I 3 ¼ 1inthe remaining terms in Eq. 18. For an incompressible isotropic material the principal components of Eqs. 31 and 32 yield t i pl 1 i s i ¼ i p ð33þ respectively, subject to l 1 l 2 l 3 ¼ 1. For an incompressible transversely isotropic material with preferred direction M, the dependence on I 3 is omitted and the Cauchy stress tensor is given by r ¼ piþ2w 1 B þ 2W 2 ði 1 B B 2 Þþ2W 4FM FM þ 2W 5 ðfm BFM þ BFM FMÞ ð34þ For a material with two preferred directions, M and M 0, the Cauchy stress tensor for an incompressible material is r ¼ piþ2w 1 B þ 2W 2 ði 1 B B 2 Þþ2W 4 FM FM þ 2W 5 ðfm BFM þ BFM FMÞþ2W 6 FM 0 FM 0 þ 2W 7 ðfm 0 BFM 0 þ BFM 0 FM 0 Þ þ W 8 ðfm FM 0 þ FM 0 FMÞ where the notation W i i now applies for i ¼ 1; 2; 4;...; 8. ð35þ BOUNDARY-VALUE PROBLEMS The equilibrium equation (second part of Eq. 9), the stress-deformation relation (Eq. 11), and the deformation gradient (Eq. 2) coupled with Eq. 1 are combined to þ r r b ¼ 0 F ¼ Grad x x ¼ vðxþ X 2 B r ð36þ Typical boundary conditions in nonlinear elasticity are x ¼ nðxþ x r ð37þ S T N ¼ sðf; XÞ t r ð38þ where n and s are specified functions, N is the unit outward normal r,

14 82 Ogden x r r are complementary parts 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 ¼ JPF T N t r ð39þ Equations constitute the basic boundary-value problem in nonlinear elasticity. In components, the equilibrium equation in Eq. 36 is 2 x j A aibj þ r b i ¼ 0 b for i 2f1; 2; 3g, where the coefficients A aibj are defined W ð40þ A aibj ¼ A bjai ¼ jb When coupled with suitable boundary conditions, Eq. 41 forms a system of quasi-linear partial differential equations for x i ¼ w i ðx a Þ. The coefficients A aibj are, in general, nonlinear functions of the components of the deformation gradient. For incompressible materials the corresponding equations are obtained by substituting the first part of Eq. 30 into the second part of Eq. 9 to give A 2 x @x i þ r r b i ¼ 0 det ð@x a Þ ¼ 1 ð42þ where the coefficients are again given by Eq. 41. In order to solve a boundary-value problem, a specific formofw needs to be given. The form of W chosen will depend on the particular material considered 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 aibj m i m j N a N b > 0 ð43þ holds for all nonzero vectors m and N. Note that Eq. 43 is independent of any boundary conditions. For an incompressible material, the strong ellipticity condition associated with Eq. 42 again has the form of Eq. 43, but the incompressibility constraint now imposes the restriction m ðf T NÞ¼0onmand N. REFERENCES 1. Beatty, M. F. (1987). Topics in finite elasticity: Hyperelasticity of rubber, elastomers and biological tissues } with examples. Appl. Mech. Rev. 40;

15 2.2 Background on Nonlinear Elasticity 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.

16 CHAPTER2.3 Elasticity of Porous Materials N. D. CRISTESCU 231 Aerospace Building, University of Florida, Gainesville, Florida Contents Validity Formulation Identification of the Parameters Examples References VALIDITY The methods used to determine the elasticity of porous materials and/or particulate materials as geomaterials or powderlike materials are distinct from those used with, say, metals. The reason is that such materials possess pores and=or microcracks. For various stress states these may either open or closed, thus influencing the values of the elastic parameters. Also, the stress-strain curves for such materials are strongly loading-rate-dependent, starting from the smallest applied stresses, and creep (generally any time-dependent phenomena) is exhibited from the smallest applied stresses (see Fig for schist, showing three uniaxial stress-strain curves for three loading rates and a creep curve [1]). Thus information concerning the magnitude of the elastic parameters cannot be obtained: from the initial slope of the stress-strain curves, since these are loadingrate-dependent; by the often used chord procedure, obviously; from the unloading slopes, since significant hysteresis loops are generally present. 84 Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved.

17 2.3 Elasticity of Porous Materials 85 FIGURE 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) FORMULATION The elasticity of such materials can be expressed as instantaneous response by T D ¼ 2G þ 1 3K 1 1 2G 3 ðtr TÞ 1 ð1þ % where G and K are the elastic parameters that are not constant, D is the strain rate tensor, T is the stress tensor, tr( ) is the trace operator, and 1 is the unit tensor. Besides the elastic properties described by Eq. 1, some other mechanical properties can be described by additional terms to be added to Eq. 1. For isotropic geomaterials the elastic parameters are expected to depend on stress invariants and, perhaps, on some damage parameters, since during loading some pores and microcracks may close or open, thus influencing the elastic parameters 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

18 86 Cristescu elastic (seismic) extended longitudinal and transverse waves, which are traveling in the body. If both these waves are recorded, then the instantaneous response is of the form of Eq. 1. The elastic parameters are obtained from K ¼ r v 2 p 4 3 v2 S G ¼ rv 2 S ð2þ where 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 for geomaterials are strongly time-dependent. Thus, in triaxial tests performed under constant confining pressure s, after loading up to a desired stress state t (octahedral shearing stress), one is keeping the stress constant for a certain time period t c [2, 3]. During this time period the rock is creeping. When the strain rates recorded during creep become small enough, one is performing an unloading reloading cycle (see Fig ). From the slopes 1 3G þ K 6G þ 1 1 ð3þ 9K of these unloading reloading curves one can determine the elastic parameters. For each geomaterial, if the time t c is chosen so that the subsequent unloading is performed in a comparatively much shorter time interval, no significant interference between creep and unloading phenomena will take place. An example for schist is shown in Figure 2.3.3, obtained in a triaxial test with five unloading reloading cycles. FIGURE Static procedures to determine the elastic parameters from partial unloading processes preceded by short-term creep.

19 2.3 Elasticity of Porous Materials 87 FIGURE Stress-strain curves obtained in triaxial tests on shale; the unloadings follow a period of creep of several minutes. 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 follow quite closely straight lines that practically coincide. If a hysteresis loop is still recorded, it means that the time t c was chosen too short. The reason for performing only a partial unloading is that the specimen is quite thick and as such the stress state in the specimen is not really uniaxial. During complete unloading, additional phenomena due to the thickness of the specimen will be involved, including, e.g., kinematic hardening in the opposite direction, 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 the axial stress is relaxing. Afterwards, when the stress rate becomes relatively small, an unloading reloading is applied to determine of the elastic parameters. This procedure is easy to apply mainly for particulate materials (sand, soils, etc.) when standard (Karman) three-axial testing devices are used and the elastic parameters follow from K ¼ 1 Dt G ¼ 1 Dt ð4þ 3 De 1 þ 2De 2 2 De 1 De 2 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 in hydrostatic tests when the formula to be used is K ¼ Ds ð5þ De v with s the mean stress and e v 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

20 88 Cristescu under this high pressure. The variation of the elastic parameters with t is more 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 their variation is related to the variation of irreversible volumetric strain, which, in turn, is describing the evolution of the pores and microcracks existing in the geomaterial. That is why the compressibility dilatancy boundary plays the role of reference configuration for the values of the elastic parameters so long as 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 the dilatancy domain (increasing under constant s), the elastic parameters are decreasing. If stress is kept constant and strain is varying by creep, in the compressibility domain volumetric creep produces a closing of pores and microcracks and thus the elastic parameters increase, and vice versa in the dilatancy domain. Thus, for each value of s the maximum values of the elastic parameters are reached on the compressibility dilatancy boundary 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 [4]. The variation of G and K is very similar to that of the irreversible volumetric FIGURE Variation of the elastic parameters K and G and of irreversible volumetric strain in monotonic uniaxial tests.

21 2.3 Elasticity of Porous Materials 89 FIGURE Variation in time of the elastic parameters and of irreversible volumetric strain in uniaxial creep tests. strain e I 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 during volumetric creep, as shown in Figure Here D is the ratio of the applied stress and the strength in uniaxial compression s c ¼ 17:88 MPa. Again, a similarity with the variation of e I V is quite evident. Figure shows for a different kind of rock salt the variation of the elastic velocities v P and v S in true triaxial tests under confining pressure p c ¼ 5 MPa (data by Popp, Schultze, and Kern [5]). Again, these velocities increase in the compressibility domain, reach their maxima on the compressibility dilatancy boundary, and then decrease in the dilatancy domain. For shale, and the conventional (Karman) triaxial tests shown in Figure 2.3.3, the values of E and G for the five unloading reloading cycles shown are: E ¼ 9:9, 24.7, 29.0, 26.3, and 22.3 GPa, respectively, while G 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] 8 >< KðsÞ :¼ K 0 K 1 1 s s 0 >: K 0 ; if s s 0 ; if s s 0 with K 0 ¼ 59 GPa, K 1 ¼ 48 GPa, and s 0 ¼ 0:344 GPa, the limit pressure when all pores are expected to be closed. ð6þ

22 90 Cristescu FIGURE The maximum of v s takes place at the compressibility dilatancy boundary (figures and hachured strip); changes of v p and v s as a function of strain ( e ¼ 10 5 s 1, p c ¼ 5 Mpa, T ¼ 308 C), showing that the maxima are at the onset of dilatancy (after Reference [4]). The same formula for alumina powder is KðsÞ :¼ K 1 p a exp b s p a ð7þ with K 1 ¼ kpa the constant value toward which the bulk modulus tends at high pressures, a ¼ 10 7, b ¼ 1:210 4, and p a ¼ 1 kpa. Also for alumina powder we have EðsÞ :¼ E 1 p a b expð dsþ ð8þ with E 1 ¼ kpa, b ¼ 6: , and d ¼ 0:002. For the shale shown in Figure 2.3.3, the variation of K with s for 0 s 45 MPa is KðsÞ :¼ 0:78s 2 þ 65:32s 369 ð9þ REFERENCES 1. Cristescu, N. (1986). Damage and failure of viscoplastic rock-like materials. Int. J. Plasticity 2 (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): 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.

23 CHAPTER2.4 Elastomer Models R. W. OGDEN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK Contents Validity Background Description of the Model Identification of Parameters How to Use It Table of Parameters References VALIDITY Many rubberlike solids can be treated as isotropic and incompressible elastic materials to a high degree of approximation. Here describe the mechanical properties of such solids through the use of an isotropic elastic strain-energy function in the context of finite deformations. For general background on finite elasticity, we refer to Ogden [5] BACKGROUND Locally, the finite deformation of a material can be described in terms of the three principal stretches, denoted by l 1 ; l 2 ; and l 3. For an incompressible material these satisfy the constraint l 1 l 2 l 3 ¼ 1 ð1þ The material is isotropic relative to an unstressed undeformed (natural) configuration, and its elastic properties are characterized in terms of a Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 91

24 92 Ogden strain-energy function Wðl 1 ; l 2 ; l 3 Þ per unit volume, where W depends symmetrically on the stretches subject to Eq. 1. The principal Cauchy stresses associated with this deformation are given s i ¼ l i p; i 2f1; 2; 3g i where p is an arbitrary hydrostatic pressure (Lagrange multiplier). By regarding two of the stretches as independent and treating the strain energy as a function of these through the definition #Wðl 1 ; l 2 Þ¼Wðl 1 ; l 2 ; l 1 1 l 1 2 Þ, we #W s 1 s 3 ¼ l 1 s 2 s 3 ¼ l 2 2 For consistency with the classical theory, we must have #Wð1; 1Þ 2 2 ð1; 1Þ a ð1; 1Þ ¼0; a 2f1; 2 #W ð1; 1Þ ¼4m; where m is the shear modulus in the natural configuration. The equations in Eq. 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 set s 3 ¼ 0 in Eq. 3, so 2 a ð4þ s 1 ¼ l 1 s 2 ¼ l 2 ð5þ Biaxial experiments in which l 1 ; l 2 and s 1 ; s 2 are measured then provide data for the determination of #W. Biaxial deformation of a thin sheet where the deformation corresponds effectively to a state of plane stress, or the combined extension and inflation of a thin-walled (membranelike) tube with closed ends provide suitable tests. In the latter case the governing equations are written P * ¼ l 1 1 l 1 #W F * l 2l 1 2 where P * ¼ PR=H, P is the inflating pressure, H the undeformed membrane thickness, 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 to the total load on the ends of the tube). Here l 1 is the axial stretch and l 2 the azimuthal stretch in the membrane. ð6þ

25 2.4 Elastomer Models DESCRIPTION OF THE MODEL A specific model which fits very well the available data on various rubbers is that defined by W ¼ XN m n ðl a n 1 þ la n 2 þ la n 3 3Þ=a n ð7þ n¼1 where m n and a n are material constants and N is a positive integer, which for many practical purposes may be taken as 2 or 3 [3]. For consistency with Eq. 4 we must have X N n¼1 m n a n ¼ 2m and in practice it is usual to take m n a n > 0 for each n ¼ 1;...; N. In respect of Eq. 7, the equations in Eq. 3 become s 1 s 3 ¼ XN n¼1 m n ðl a n 1 la n 3 Þ s 2 s 3 ¼ XN IDENTIFICATION OF PARAMETERS n¼1 ð8þ m n ðl a n 2 la n 3 Þ ð9þ Biaxial experiments with s 3 ¼ 0 indicate that the shapes of the curves of s 1 s 2 plotted against l 1 are essentially independent of l 2 for many rubbers. Thus the shape may be determined by the pure shear test with l 2 ¼ 1, so that s 1 s 2 ¼ XN n¼1 m n ðl a n 1 1Þ s 2 ¼ XN n¼1 m n ðl a n 3 1Þ ð10þ for l 1 1; l 3 1. The shift factor to be added to the first equation in Eq. 10 when l 2 differs from 1 is X N n¼1 m n ð1 l a n 2 Þ ð11þ Information on both the shape and shift obtained from experiments at fixed l 2 then suffice to determine the material parameters, as described in detail in References [3] or [4]. Data from the extension and inflation of a tube can be studied on this basis by considering the combination of equations in Eq. 6 in the form s 1 s 2 ¼ l 1 l 2 ¼ l 1 F * 1 2 l2 2 l 1P * ð12þ

26 94 Ogden 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 be used in terms of principal stretches and principal stresses in the FE solution of boundary-value problems TABLE OF PARAMETERS Values of the parameters corresponding to a three-term form of Eq. 7 are now given in respect of two different but representative vulcanized natural rubbers. The first is the material used by Jones and Treloar [2]: a 1 ¼ 1:3; a 2 ¼ 4:0; a 3 ¼ 2:0; m 1 ¼ 0:69; m 2 ¼ 0:01; m 3 ¼ 0:0122 Nmm 2 The second is the material used by James et al. [1], the material constants having been obtained by Treloar and Riding [6]: a 1 ¼ 0:707; a 2 ¼ 2:9; a 3 ¼ 2:62; m 1 ¼ 0:941; m 2 ¼ 0:093; m 3 ¼ 0:0029 Nmm 2 For detailed descriptions of the rubbers concerned, reference should be made to 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: Jones, D. F., and Treloar, L. R. G. (1975). The properties of rubber in pure homogeneous strain. J. Phys. D: Appl. Phys. 8: Ogden, R. W. (1982). Elastic deformations of rubberlike solids, in Mechanics of Solids (Rodney Hill 60th Anniversary Volume) pp , 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: 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:

27 CHAPTER2.5 Background on Viscoelasticity KOZO IKEGAMI Tokyo Denki University, Kanda-Nishikicho 2-2, Chiyodaku, Tokyo , Japan Contents Validity Mechanical Models Static Viscoelastic Deformation Dynamic Viscoelastic Deformation Hereditary Integral Viscoelastic Constitutive Equation by the Laplace Transformation Correspondence Principle References VALIDITY Fundamental deformation of materials is classified into three types: elastic, plastic, and viscous deformations. Polymetric material shows time-dependent properties even at room temperature. Deformation of metallic materials is also time-dependent at high temperature. The theory of viscoelasticity can be applied to represent elastic and viscous deformations exhibiting timedependent properties. This paper offers an outline of the linear theory of viscoelasticity MECHANICAL MODELS Spring and dashpot elements as shown in Figure are used to represent elastic and viscous deformation, respectively, within the framework of the Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 95

28 96 Ikegami FIGURE Mechanical model of viscoelasticity. linear theory of viscoelasticity. The constitutive equations between stress s and stress e of the spring and dashpot are, respectively, as follows: s ¼ ke s ¼ Z de ð1þ dt where the notations k and Z are elastic and viscous constants, respectively. Stress of spring elements is linearly related with strain. Stress of dashpot elements is related with strain differentiated by time t, and the constitutive relation is time-dependent. Linear viscoelastic deformation is represented by the constitutive equations combining spring and dashpot elements. For example, the constitutive equations of series model of spring and dashpot shown in Figure is as follows: s þ Z ds k dt ¼ Z de ð2þ dt This is called the Maxwell model. The constitutive equation of the parallel model of spring and dashpot elements shown in Figure is as follows: s ¼ ke þ Z de ð3þ dt This is called the Voigt or Kelvin model.

29 2.5 Background on Viscoelasticity 97 FIGURE Maxwell model. There are many variations of constitutive equations giving linear viscoelastic deformation by using different numbers of spring and dashpot elements. Their constitutive equations are generally represented by the following ordinary differential equation: ds p 0 s þ p 1 dt þ p d 2 s 2 dt 2 þ...þ p d n s n dt n de ¼ q 0 e þ q 1 dt þ q d 2 e 2 dt 2 þ...þ q d n e n dt n ð4þ The coefficients p and q of Eq. 4 give the characteristic properties of linear viscoelastic deformation and take different values according to the number of spring and dashpot elements of the viscoelastic mechanical model.

30 98 Ikegami FIGURE Voigt (Kelvin) model STATIC VISCOELASTIC DEFORMATION There are two functions representing static viscoelastic deformation; one is creep compliance, and another is the relaxation modulus. Creep compliance is defined by strain variations under constant unit stress. This is obtained by solving Eqs. 2 or 3 for step input of unit stress. For the Maxwell model and the Voigt model, their creep compliances are represented, respectively, by the following expressions. For the Maxwell model, the creep compliance is e t Z þ 1 k ¼ 1 t k t þ 1 ð5þ where t M ¼ Z=k, and this is denoted as relaxation time. For the Voigt model, the creep compliance is e ¼ 1 k kt 1 exp Z ¼ 1 k t 1 exp t k where t K ¼ Z=k, and this is denoted as retardation time. Creep deformations of the Maxwell and Voigt models are illustrated in Figures and 2.5.5, respectively. Creep strain of the Maxwell model ð6þ

31 2.5 Background on Viscoelasticity 99 FIGURE Creep compliance of the Maxwell model. FIGURE Creep compliance of the Voigt model.

32 100 Ikegami increases linearly with respect to time duration. The Voigt model exhibits saturated creep strain for a long time. The relaxation modulus is defined by stress variations under constant unit strain. This is obtained by solving Eqs. 2 or 3 for step input of unit strain. For the Maxwell and Voigt models, their relaxation moduli are represented by the following expressions, respectively. For the Maxwell model, s ¼ k exp kt Z ¼ k exp t t M For the Voigt model, s ¼ k ð8þ Relaxation behaviors of the Maxwell and Voigt models are illustrated in Figures and 2.5.7, respectively. Applied stress is relaxed by Maxwell model, but stress relaxation dose not appear in Voigt model. ð7þ DYNAMIC VISCOELASTIC DEFORMATION The characteristic properties of dynamic viscoelastic deformation are represented by the dynamic response for cyclically changing stress or strain. FIGURE Relaxation modulus of the Maxwell model.

33 2.5 Background on Viscoelasticity 101 FIGURE Relaxation modulus of the Voigt model. The viscoelastic effect causes delayed phase phenomena between input and output responses. Viscoelastic responses for changing stress or strain are defined by complex compliance or modulus, respectively. The dynamic viscoelastic responses are represented by a complex function due to the phase difference between input and output. Complex compliance J of the Maxwell model is obtained by calculating changing strain for cyclically changing stress with unit amplitude. Substituting changing complex stress s ¼ expðiotþ, where i is an imaginary unit and o is the frequency of changing stress, into Eq. 2, complex compliance J is obtained as follows: J ¼ 1 k i 1 oz ¼ 1 k i 1 kot M ¼ J 0 ij 00 ð9þ where the real part J 0 ¼ 1=k is denoted as storage compliance, and the imaginary part J 00 ¼ 1=kot M is denoted as loss compliance. The complex modulus Y of the Maxwell model is similarly obtained by calculating the complex changing strain for the complex changing strain

34 102 Ikegami e ¼ expðiotþ as follows: Y ¼k ðot M Þ 2 1 þðot M Þ 2 þ ik ot M 1 þðot M Þ 2 ¼ Y 0 þ iy 00 ð10þ where Y 0 ¼ kððot M Þ 2 =ð1 þðot M Þ 2 ÞÞ and Y 00 ¼ kðot M =ð1 þðot M Þ 2 ÞÞ. The notations Y 0 and Y 00 are denoted as dynamic modulus and dynamic loss, respectively. The phase difference d between input strain and output stress is given by tan d ¼ Y00 Y 0 ¼ 1 ð11þ ot M 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 þðot K Þ 2 i 1 k " # ot K 1 þðot K Þ 2 " # " # where J 0 ¼ 1 1 k 1 þðot K Þ 2 and J 00 ¼ 1 ot K k 1 þðot K Þ 2 The complex modulus is ¼ J 0 ij 00 ð12þ Y ¼k þ iot K ¼ Y 0 þ iy 00 ð13þ where Y 0 ¼ k and Y 00 ¼ kot K HEREDITARY INTEGRAL The hereditary integral offers a method of calculating strain or stress variation for arbitrary input of stress or strain. The method of calculating strain for stress history is explained by using creep compliance as illustrated in Figure An arbitrary stress history is divided into incremental constant stress history ds 0 Strain variation induced by each incremental stress history is obtained by creep compliance with the constant stress values. In Figure the strain induced by stress history for t 0 5t is represented by the following integral: eðtþ ¼s 0 JðtÞþ Z t 0 Jðt t 0 Þ ds0 dt 0 dt 0 ð14þ

35 2.5 Background on Viscoelasticity 103 FIGURE Hereditary integral. This equation is transformed by partially integrating as follows: Z t eðtþ ¼sðtÞJð0Þþ sðt 0 Þ djðt t0 Þ 0 dðt t 0 Þ dt0 Similarly, stress variation for arbitrary strain history becomes sðtþ ¼e 0 YðtÞþ Yðt t 0 Þ ds0 0 dt 0 dt 0 Partial integration of Eq. & gives the following equation: Z t sðtþ ¼eðtÞYð0Þþ sðt 0 Þ dyðt t0 Þ 0 dðt t 0 Þ dt0 Integrals in Eqs. 14 to 17 are called hereditary integrals. Z t ð15þ ð16þ ð17þ VISCOELASTIC CONSTITUTIVE EQUATION BY THE LAPLACE TRANSFORMATION The constitutive equation of viscoelastic deformation is the ordinary differential equation as given by Eq. 4. That is, X n k¼0 p k d k s dt k ¼ Xm k¼0 q k d k e dt k ð18þ

36 104 Ikegami This equation is written by using differential operators P and Q, Ps ¼ Qe where P ¼ Pn d k Pm d k p k and Q ¼ q k¼0 dtk k k¼0 dt k. Equation (1?) is represented by the Laplace transformation as follows. X n k¼0 p k s k %s ¼ Xn k¼0 q k s k %e ð19þ ð20þ 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 Laplace transformed operators of time derivatives %P and %Q as follows: Q %s ¼ % %e ð21þ %P where %P ¼ Pn p k s k and %Q ¼ Pm q k s k. k¼0 k¼0 Comparing Eq. 21 with Hooke s law in one dimension, the coefficient %Q=%P corresponds to Young s modulus of linear elastic deformation. This fact implies that linear viscoelastic deformation is transformed into elastic deformation in the Laplace transformed state CORRESPONDENCE PRINCIPLE In the previous section, viscoelastic deformation in the one-dimensional state was able to be represented by elastic deformation through the Laplace transformation. This can apply to three-dimensional viscoelastic deformation. The constitutive relations of linear viscoelastic deformation are divided into the relations between hydrostatic pressure and dilatation, and between deviatoric stress and strain. The relation between hydrostatic pressure and dilatation is represented by X m p 0 k k¼0 d k s 0 ij dt k ¼ Xn k¼0 q 00 d k e ii k dt k ð22þ P 00 s ii ¼ Q 00 e ii ð23þ Pm 00 where P p 00 d k k k¼0 dt k and Q00 ¼ Pn s ii and dilatation is e ii. k¼0 q 00 k d k dtk. In Eq. 22 hydrostatic pressure is (1/3)

37 2.5 Background on Viscoelasticity 105 The relation between deviatoric stress and strain is represented by X m k¼0 d k s 0 p 0 ij k dt k ¼ Xn d k e 0 q 0 ij k dt k k¼0 P 0 s 0 ij ¼ Q0 e 0 ij ð24þ ð25þ where P 0 ¼ Pm p 0 d k k k 0 dt k and Q0 ¼ Pn q 0 d k k k¼0 dtk. In Eq. 24 deviatoric stress and strain are s 0 ij and e0 ij, respectively. The Laplace transformations of Eqs. 22 and 24 are written, respectively, as follows: %P 00 %s ii ¼ %Q 00 %e ii ð26þ where %P 00 ¼ %P 00 ðsþ and %Q 00 ¼ %Q 00 sðsþ, and %P 0 %s 0 ij ¼ %Q 0 %e 0 ij ð27þ where %P 0 ¼ %P 0 ðsþ and %Q 0 ¼ %Q 0 ðsþ. The linear elastic constitutive relations between hydrostatic pressure and dilatation and between deviatoric stress and strain are represented as follows: s ii ¼ 3Ke ii ð28þ s 0 ij ¼ 2Ge0 ii ð29þ Comparing Eq. 17 with Eq. 19, and Eq. 18 with Eq. 20, the transformed viscoelastic operators correspond to elastic constants as follows: Q 3K ¼ % 00 ð30þ %P 00 Q 2G ¼ % 0 ð31þ %P 0 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 as follows: G ¼ E 2ð1 þ nþ ð32þ K ¼ E 3ð1 2nÞ ð33þ

38 106 Ikegami Using Eqs , Young s modulus E and Poisson s ratio are connected with the Laplace transformed coefficient of linear viscoelastic deformation as follows: E ¼ 3 %Q 0 %Q 00 2%P 0 % Q 00 þ %P 00 % Q 0 ð34þ n ¼ % P 0 %Q 00 %P 00 %Q 0 2%P 0 % Q 00 þ %P 00 % Q 0 ð35þ Linear viscoelastic deformation corresponds to linear elastic deformation through Eqs and Eqs This is called the correspondence principle between linear viscoelastic deformation and linear elastic deformation. The linear viscoelastic problem is the transformed linear elastic problem in the Laplace transformed state. Therefore, the linear viscoelastic problem is able to be solved as a linear elastic problem in the Laplace transformed state, and then the elastic constants of solved solutions are replaced with the Laplace transformed operator of Eqs and Eqs by using the correspondence principle. The solutions replaced the elastic constants become the solution of the linear viscoelastic problem by inversing the Laplace 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.

39 C H A P T E R2.6 A Nonlinear Viscoelastic Model Based on Fluctuating Modes RACHID RAHOUADJ AND CHRISTIAN CUNAT LEMTA, UMR CNRS 7563, ENSEM INPL 2, avenue de la For#et-de-Haye, Vandoeuvre-l"es- Nancy, France Contents Validity Background of the DNLR Thermodynamics of Irreversible Processes and Constitutive Laws Kinetics and Complementary Laws Constitutive Equations of the DNLR Description of the Model in the Case of Mechanical Solicitations Identification of the Parameters How to Use It Table of Parameters References VALIDITY We will formulate a viscoelastic modeling for polymers in the temperature range of glass transition. This physical modeling may be applied using integral or differential forms. Its fundamental basis comes from a generalization of the Gibbs relation, and leads to a formulation of constitutive laws involving control and internal thermodynamic variables. The latter must traduce Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. All rights of reproduction in any form reserved. 107

40 108 Rahouadj and Cunat different microstructural rearrangements. In practice, both modal analysis and fluctuation theory are well adapted to the study of the irreversible transformations. Such a general formulation also permits us to consider various nonlinearities as functions of material specificities and applied perturbations. To clarify the present modeling, called the distribution of nonlinear relaxations (DNLR), we will consider the viscoelastic behavior in the simple case of small applied perturbations near the thermodynamic equilibrium. In addition, we will focus our attention upon the nonlinearities induced by temperature and frequency perturbations BACKGROUND OF THE DNLR THERMODYNAMICS OF IRREVERSIBLE PROCESSES AND CONSTITUTIVE LAWS As mentioned, the present irreversible thermodynamics are based on a generalization of the fundamental Gibbs equation to systems evolving outside equilibrium. Note that Coleman and Gurtin [1], have also applied this postulate in the framework of rational thermodynamics. At first, a set of internal variables (generalized vector denoted z) is introduced to describe the microstructural state. The generalized Gibbs relation combines the two laws of 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 always respected, one obtains for open systems: T dd is dt ¼ Ts s ¼ J s : rt Xn k¼1 J k : rm k þ A z 0 ð2þ where the nonequilibrium thermodynamic forces may be separated into two groups: (i) the gradient ones, such as the gradient of temperature gradient rt, and the gradient of generalized chemical potential rm k ; and (ii) The generalized forces A, or affinities as defined by De Donder [2] for chemical reactions, which characterize the nonequilibrium state of a uniform medium. The vectors J s, J k, and z correspond to the dual, fluxes, or ratetype variables. To simplify the formulation of the constitutive laws, we will now consider the behavior of a uniform representative volume element (RVE without any