Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

Since the reference configuration is unaffected by superimposed ri.gid-body m·o- tions.. we know that the second Piola-Kirchhoff stress tensor and the right CauchyGreen tensor simply transform according to s+ = S and c+ = C. We conclude that the stress relation (5.82) is independent of the observer.

EXAMPLE ·5.·5 Investigate if the elastic material given by

(5.83) implies u+ = J(E+). By recalling the transformations for the Cauchy stress tensor u and the Green-Lagrange strain tensor E, which satisfy eqs. (5.41 )..:and (5.43ht and knowing that J is the same function for the two (different) :motions x and x+ (it is for the same elastic material), we conclude that

(5.84)

Note that this relation is only true for Q = I; thus,. constitutive equation (5.83) does not satisfy the principle of material frame-indifference. II

Isotropic Cauchy-elastic materials.. We assume that the Cauchy stress tensor u depends on the left Cauchy-Green tensor b = FFT. The constitutive equation (5.72) may then be written .in the alternative form

where ~ is a tensor-valued function of the symmetric second-order tensor b associated with the Cauchy stress tensor tr.

ln order to "find the restriction imposed on the response function ~ by the assumption of material...frame-:indifference, let .u·+ = ~(b+), where the response function lJ is the same for the two motions x and x+. We now use (5.44h on the one hand and (5.41), {5.85) on the other hand .in order to obtain

(5.86) Combining eqs. (5.86) 1 and (5..86)2 we find the fundamental :invariance relation

(5.87)

for the function ~ and for every tensor band orthogonal tensor Q. Hence, constitutive

equation (5.85) is independent of the observer if~ satisfies restriction (5.87)~

A specific elastic material which may be described by the constitutive equation in the form (5.85), with property (5.87), is said to be isotropic. A tensor-valued function such as ~ (b) is said to be isotropic if it satisfies relations of type (5 ..87). He-nee, we refer to ~ (b) as a tensor-valued isotropic tensor .function of one variable b.

From the physical point of view the condition o.f :isotropy is expressed by the prop- erty that the material exhibits no preferred directions. In fact, the stress response of an isotropic elastic material is not .affected by the choice of the reference configuration.

For a piece of wood, for example, which .is of cellular structure, in the direction of the grain differ from those in .other directions, so th~.--riiaterial certainly is

not isotropic. _/./

The isotropic tensor function ~(.b), which satisfies (5.~7)", may be represented in the explicit form

for each b, which is known as the Rivlin-Ericksen representation theorem. ·For a proof of this crucial .relation see RIVLIN and ERICKSEN :[1955], TRUESDELL and

NOLL [1992, Section 12], SPENCER .[1980, Appendix] and GURTIN [198la, pp. 233235]. Note that the representation theorem (5.8-8) represents a fundamental requirement for the mathematical form of the stress relation.

Here, aa, a = 0, 1, 2, are three scalar functions called response coefficients or

material functions. He-nee, in general, for an isotropic material only three parameters are needed in order to describe the stress state. The scalar functions aa depend on the three invariants of tensor b and therefore on the current deformation state. The invariants are defined with respect to eqs. (1.170)-( 1.172) as

where /\~ are the three eigenvalues of the -symmetric spatial tensor b, see eq. (2.117).

In eq. (5.9.1), relation (2.79)2 was used.

Representation (5.88) is the most general form of .a constitutive -equation for iso- tropic elastic materials also known as the first representation theorem for isotropic tensor functions. From the constitutive equation (5.88) we deduce that the principal directions of the Cauchy stress tensor u and the left Cauchy-Green tensor b coincide. Hence, for isotropic elastic materials the two symmetric tensors u and b are said to be

coaxial in eve.ry configuration.

202 5 Some Aspects of Objectivity

By analogy with (5.S-8), the constitutive equation

characterizes the behavior of a viscous fluid, in particular, of a so-called ·Reiner-RivUn Huid. Therein, p and d are the spatial mass density and the rate of defonnation tensor, while ctn, a = 0, 1, 2, are scalar functions of the .invariants Ia, a = 1., 2_, ·3, given .in

(5.89)-(5.91) with d replacing b.

ln order to ·find .an alternative explicit re.presentation for (5."88), we recall the Cayley.. Hamilton. equation ( 1..174). Since any tensor satisfies its own characteristic equation, we may write Cl.174) as ba - 11b2 + l 2 b - 1:11 = 0 and find, by multiplying this equation with b- 1, that

(5."93)

Eliminating b2 from (5.88) in favor of b- 1 we obtain an alternative represe~tation of a constitutive equation for isotropic -elastic materials, i.e..

where /3<"' .a = O_, l~ -1, are three scalar func~ions (response coefficients) which, in tenns of the three invariants of b_, are expressed as

Representation (5.94) .is also known as the second representation theorem for

isotropic tensor functions, see, for example, GURTIN [1981.a, p. 235].

Incomp·ressible Cauchy-elastic -materials. .If the Cauchy-elastic material is incom-

pressible, then the stress relation .is detem1ined only up to an arbitrary scalar JJ which can be identified as a pressure-like -quantity. The constitutive equations (5.72), (5.76) and (5.82) are then replaced by

where the tensor-valued tensor functions 11(F) 1 g(U) and j](C) need only be defined for the. kinematic constrain.ts detF = 1, det:U = 1 and detC = 1, respective.Iy. The indeterminate tenns -pl and -pc- 1 are known as reaction stresses, which do no work in any motion compatible with above constraints. For incompressible materials the (indeterminate) scalar p required to maintain incompressibility may only be found by means of the equilibrium conditions and the boundary conditions and is not specified

204 5 Some Aspec.ts of Objectivity

Note that the Newtonian viscous fluid is simply a special case of the Reiner-. Rivlin fluid (5.92) obtained by choosing the response coefficients eta, a= 0, 1, 2,

u = -p(p)I characterizing the material properties of an elastic ·fluid.

2.By recalling the transfonnation rules (5.41), (5.31) and (5.40h, (5.45h (from Sections 5. l and 5.2), show that the constitutive equations

ir = Ju + ulT + ad

are acceptable forms which conform to the principle of material frame-indiffer- ence, where a is a material parameter and a_, I and d denote the spatial acceleration field, the spatial ·velocity gradient and the .rate of deformation tensor, respectively.

······..

3.Consider a material which is isotropic and Cauchy-elastic. Let a uniform extension (or compression) in all three directions according to relation (2·~129), which -corresponds to a triaxial stress state, ·be given.

By detenninin.g the deformation gradient F and the left Cauchy-Green tensor b (with respect to a set of some orthonormal basis vectors ea) find the most general representation for the Cauchy stress tensor CT.

4.By means of a pull-back of (5.88) to the reference configuration .and a scaling with the .inverse volume ratio J- 1 , show that for isotropic elastic materials the second Piola-Kirchhoff stress tensor S is coaxial with the .right Cauchy-stress tensor C if.and only ~"{ u is coaxial with b.

.206 6 Hyperelastic l\tlaterials

ln Chapter 6 we discuss phenomenological constitutive equations which interrelate the stress components and the strain components within a non.Ii.near regime. Since we are studying so-called purely mechanical theories, thermodynam.ic variables such as the ·entropy and the temperature are ignored. However, they are taken into account and elaborated on within Chapter 7. No attempt is made to present a comprehensiv·e list of the various important contributions on constitutive modeling to date. This text

.discusses some selected material models essential in science, industrial engineering practice and in the :field of biomech.anics, where structures exhibit large strain behavior, very often within the coupled thermodynam.ic regime.

Sections 6.1-6.8 consider hyperelastic materials in general, and cover a wide range of important types of material such as isotropic and transversely isotropic materials., in- compressible and compressible hyperelastic materials and composite materials, in par- ticular. Some important specifications for rubber-like (or other) materials are also presented. The remaining Sections 6..9-6.. 11 focus attention on .inelastic materials.. Based on the concept of .internal variables., viscoelastic materials and isotropic hyperelastic materials with damage at finite strains are introduced. Plastic and viscoplastic mate~ rials which have the ability to undergo irreversible or .permanent deformations are not co.nsiderd in this text.

6.1Gen·eral Remarks on Constitutive Equations

It is the aim of constitutive .theories to develop maLhemat.ical models for representing the real behavior of matter. Constitutive theories of materials are very important but they are a difficult subject in modern nonlinear continuum mechanics. We make no attempt to conduct a comprehensive review of the large number of constitutive theories. For more on formulating nonlinear-constitutive theories see, for example, TRUES.DELL and NOLL [1992] and the excellent contributions by Rivlin in the 1940s and 1950s collected by BARENBLATT and JOSEPH [1997],.

In particular, we present a nonlinear constitutive theory suitable to describe a wide variety of physical phenomena in which the strains may be large, i.e. finite. For the case of a (hyper)elastic material the resulting theory is called finite (hyper)efasticity theory or just finite (hyper)elasticity for which nonlinear continuum mechanics is the fundamental basis (see GREEN and ADKINS [1970] for an analytic.a] treatment .and inter alia LE TALLEC [1994] for numerical solution techniques).

Constitutive equations for hyperelastic materials. A so-called hyperelastic material (or in the literature often called a Green-elastic ·material) postulates the existence of a Helmholtz free..energy function \JI, which is defined per unit reference volume rather than per unit mass.

For the case in which \JI = W(F) is solely .a function of ·F or some strain tensor, as introduced in Sect.ion 2.4, the Helmholtz free-energy function is referred to as the strain-energy function or stored-energy funcUon (see Section 4.4, p. :159). Subse- quently, we often use the common terminology ·strain energy or stored energy. The strain-energy function \II = \JI (F) is a typical example of a scalar-valued function of one tensor variable F, which we assume to be .continuous.

A concept of importance in elasticity is polyconvexity of strain-energy functions.

The global existence theory of solutions, for example, is based on the condition of po:lyconvex.ity of strain-energy functions. For an extensive discussion on the underlying issue, see BALL [ 1977], CIARLET [.1988, Chapters 4, 7], "MARSDEN and HUGHES f.1.994, Section 6.4] and SlLHAVY [1997, Sections ]7.5, 18.5].

We now restrict attention to homogeneous materials in which the distributions of the internal constituents are assumed to be uniform on the continuum scale. For this

type of ideal material the strain-energy function ~1' depends only upon the deformation

gradient F. Of course, for so-called heterogeneous materials (a material that is not homogeneous) \JI will depend additionally upon the position of a po.int in the medium.

A hypereh1stic material is defined as a subclass of an elastic material, as given in

eqs. (5.77)a and (5.72), whose response functions 6 and g have physical express.ions of the form

and by use of relation (3.9) for the symmetric Cauchy stress tensor, i.e. u == ,J- 1PFT =

T

·U '

These types of equation we already know as (purely mechanical) constitutive equations

(or equations of They establish .an axiomatic or empirical model as the basis for

approximating the behavior of a real materiaL Such a model we call a material model or a constitutive model. As is clear from the ·Constitutive equations (6.1) and (6.2) the stress response of hypere:lastic materials is derived from a given scalar-valued ene1~gy function, which implies that hyperelasticity has a conservative structure.

The derivative of the s.calar-va:lued function \}I with respect to the tensor variable F determines the gradient of wand -is understood according to the definition introduced

in ( 1.239)..It is a second-order tensor which we know as the first Pio stress tensor P,. The derivation .requires that the component function w(F;.A) is differentiable with respect to .all components Fa...i.

208 6 Hyperelastic ·Materials

A so-ca1led perfectly elastic material is by definition a material which produces locally no entropy (see TRUESDELL and NOLL [.1992, Section 80]). In other words, we use subsequently the term 'perfectly' for a certain class of materials which has the. special merits that for every admissible process the internal dissipation Vint .is zero (naturally, damage, viscous mechanisms and plastic deformations are excluded). We

will consider perfectly elastic materials up to Section 6.9.

We may derive the constitutive equation (6.1) directly from the Clausius-Planck form of the second Jaw of thermodynamics (4.154) whkh degenerates to an equality for the class of perfectly elastic materials. With the expression (4.. 154.) for .Vint the time

differentiation of the strain-energy function, i.e.

at every point of the continuum body and for all times during the process.

As F and hence Fcan be chosen arbitrarily, the expressions in parentheses must be zero. Therefore, as a consequence of the second law of thermodynam.ics, the physical expression (6. I) holds. We often say that P is the thermodynamic "force lvork conju- gate to F. This procedure goes back to COLEMAN and NOLL [1963] and COLEMAN and GURTIN [ 1967], and in the literature is sometimes referred to as the Coleman-Noll

:procedure.

For convenience, throughout this text we require that the strain-energy function vanishes in the reference configuration, .i.e. where F = I. We express this assumption

From the physical observation we know that the strain-energy function \ll inpreases with deformation. In addition to (6.4), we therefore require that

which restricts the ranges of adm.issible functions occurriQg in expressions for the strain energy.

Tbe strain-energy function \{1 attains its global minimum for F = I at .thermodynamic equilibrium (in fact, fro.m (6.4), ·w(I) is zero). We assume that \JI has no other stationary points in the strain space. Relations (6.4) and (6.5) ensure that the stress in the reference configuration, which we call the .residual stress, is zero. We say that the reference configuration is stressMfree.

For the behavior at finite strains we require additionally that the scalar-valued function '1i must satisfy so-called growth conditions. This implies that \JI tends to +oo if