Hencky’s logarithmic strain and dual stress–strain and strain–stress relations in isotropic finite hyperelasticity

Dedicated to Professor Wei-zang Chien on the occasion of his 90th birthday
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Abstract

It has been known that the Kirchhoff stress tensor τ and Hencky’s logarithmic strain tensor h may be useful in formulations of isotropic finite elasticity and elastoplasticity. In this work, a straightforward proof is presented to demonstrate that, for an isotropic hyperelastic solid, the just-mentioned stress–strain pair τ and h are derivable from two dual scalar potentials with respect to each other. These results establish a simple, explicit dual formulation of isotropic finite hyperelasticity. As a result, they supply a complete solution to the problem of finding out the inverted stress–strain relation for isotropic hyperelastic solids, raised by J.A. Blume [Int. J. Non-linear Mech. 27 (1992) 413]. Moreover, an explicit form of such an inverted hyperelastic stress–strain relation is derived in terms of the powers I, τ and τ2.

Introduction

Usually, an isotropic elastic solid undergoing finite deformations is defined by a stress–strain relation that prescribes the dependence of a stress measure T on a strain measure E, i.e.,T=T(E).For a hyperelastic solid, certain restrictions concerning the specific stress power should be imposed. As a result, the stress–strain relation for isotropic finite hyperelastic solids can be derived from an isotropic scalar potential known as the strain-energy function.

On the other hand, an isotropic elastic solid may be equally well defined by an inverted stress–strain relation, i.e., a strain–stress relation, which gives the dependence of a strain measure E on a stress measure T, i.e.,E=E(T).Because of the foregoing restrictions, the strain–stress relation for isotropic hyperelastic solids should also be derived from a scalar potential.

In continuum mechanics, there are a great many stress and strain measures for consideration. In formulating an elastic relation, the stress measure T and the strain measure E may be freely chosen among them in principle. The Cauchy stress tensor σ and the Cauchy–Green tensor B=FFT and the stretch tensor V=FFT are commonly used in literature. With the pair (σ,B) or (σ,V), the explicit stress–strain relation σ=σ̂(B) for isotropic hyperelastic solids in terms of the strain-energy function is well-known (see, e.g., Truesdell and Noll, 1965; Gurtin, 1981; Ogden, 1984). It may be clear that the form of a hyperelastic stress response function T(E) or a hyperelastic strain response function E(T) relies on the choice of the stress–strain pair (T,E). Generally, for a stress–strain pair (T,E) that need not be work-conjugate, the explicit form of the hyperelastic stress–strain relation T=T(E) may not be so clear or so simple. In particular, for a chosen stress measure T, a given form of stress response function T(E) may not be well-defined in the sense of hyperelasticity for every strain measure E. Most recently, Chiskis and Parnes (2000) have studied an interesting particular example in this respect. Let the stress measure T be the Cauchy stress σ, i.e., T=σ, and let the strain measure E to be determined. They consider a Hookean type elastic relation linear in E, i.e.,σ=Λ(trE)I+2GE,where Λ and G are the Lamé elastic constants evaluated at small deformations. They demonstrate that the elastic relation (1) is hyperelastic if and only if E is of the formE=Λ2GV(2G/Λ)I.The latter requires that E be dependent on the Lamé elastic constants Λ and G. It does not appear that such an E qualifies as a strain measure in a pure kinematic sense, since it specifies different straining states for the same deformation of material bodies with different Lamé constants.

In a paper, Blume (1992) raised and investigated the problem of finding out an explicit representation for the strain–stress relation1 B=B(σ) for isotropic hyperelastic solids. She derived conditions on the form of such an inverted hyperelastic constitutive relation, and, in the incompressible case, achieved an explicit general representation for a hyperelastic strain–stress relation in terms of a generating scalar potential. However, it appears that no explicit results have been derived for the general compressible case. Moreover, Blume (1992) noted that, even for a simple incompressible case, it appears to be difficult to derive an explicit form of the hyperelastic strain–stress relation in terms of the powers σr with r=0, 1, 2.

The known stress measure closest to the Cauchy stress (true stress) σ is the Kirchhoff stress τ=Jσ, also known as the weighted Cauchy stress. We shall show that, if we replace the Cauchy stress σ with the Kirchhoff stress τ=Jσ in the aforementioned issues raised by Chiskis and Parnes (2000) and Blume (1992), respectively, then simple, complete solutions for them would be possible. With the replacement of the Cauchy stress σ by the Kirchhoff stress τ=Jσ, Eq. (1) becomesτ=Jσ=Λ(trE)I+2GE.Then arises the question as to what strain measure makes the above Hookean type elastic relation hyperelastic. Moreover, replacing the Cauchy stress σ by the Kirchhoff stress τ=Jσ, we may reformulate the foregoing Blume’s problem as follows. Let B=B(τ) be an isotropic hyperelastic strain–stress relation. Find a general explicit expression for the strain response function B(τ) in terms of a free scalar potential Σ=Σ(τ). Note that the Kirchhoff stress τ is just the Cauchy stress scaled by the Jacobian (volume ratio) J=detF. They differ only by a scalar factor J. Since the stress power is just the inner product of the Kirchhoff stress τ and the stretching D and since the notion of hyperelasticity is concerned directly with the stress power, it may be expected that the Kirchhoff stress τ should be more pertinent than the Cauchy stress σ in finite hyperelastic formulation.

In this work, we shall show that, with the Kirchhoff stress tensor τ and Hencky’s logarithmic strain measure h (see Eqs. , below) we may arrive at a simple, explicit dual formulation of stress–strain and strain–stress relations for isotropic finite hyperelasticity. We demonstrate in a straightforward manner that, for an isotropic hyperelastic solid, the foregoing stress–strain pair τ and h are derivable from two dual scalar potentials with respect to each other. These results supply a complete solution to the foregoing problem raised by Blume (1992). In particular, with reference to the foregoing issue treated by Chiskis and Parnes (2000), we show that the linear stress–strain relation between the Kirchhoff stress and Hencky strain is hyperelastic for any given Lamé constants. Moreover, using the eigenprojection method based on Sylvester’s formula, we derive an explicit form of the hyperelastic strain–stress relation in terms of the three powers τr with r=0, 1, 2.

Usefulness of the inverted stress–strain relation for hyperelastic solids has been pointed out by Blume (1992); refer to the relevant references therein. In addition, in recent years, a hyperelastic strain–stress relation in terms of the Hencky strain and Kirchhoff stress has been found essential to formulations of finite inelasticity theories (see, e.g., Bruhns et al., 1999, Bruhns et al., 2001b; Xiao et al., 1997a, Xiao et al., 1997b, Xiao et al., 1999, Xiao et al., 2000). Generally, Hill, 1968, Hill, 1970, Hill, 1978 found that Hencky’s logarithmic strain measure has inherent advantages over other strain measures in his study of a priori constitutive inequalities2 and treated the Hencky strain, its rate and its work-conjugate stress as basic measures for strain, strain rates and stresses, etc. Recently, certain significant properties of the Hencky strain or natural strain have been indicated by Freed (1995), Bažant (1998), and Xiao et al. (1997b). Now, the Hencky strain has found applications in finite elasticity and inelasticity; refer to, e.g., the relevant references mentioned above, as well as de Boer (1967), de Boer and Bruhns (1969), Bruhns and Thermann (1969), Bruhns, 1970, Bruhns, 1971, Stören and Rice (1975), Raniecki and Nguyen (1984), Eterovic and Bathe (1990), Weber and Anand (1990), Miehe et al. (1994), Stumpf and Schieck (1994), Schieck and Stumpf (1995), Bonet and Wood (1997), Kollmann and Sansour (1997), Miehe (1998), and many others.

Section snippets

Dual stress–strain and strain–stress relations

Let τ be the Kirchhoff stress tensor and h Hencky’s logarithmic strain tensor.

Explicit strain–stress relation in terms of the powers τr

Since Σ(τ) is isotropic, it is a symmetric function of the three eigenvalues τi of τ, i.e.,Σ(τ)=ψ(τ123).Hence we have7Στ=∑i=13ψτinini.Thus, Eq. (23) has the spectral formB=∑i=13e2(ψ/τi)nini.The latter yieldsλi=e(ψ/τi).

Sometimes, we need to put Eq. (23) in the usual power formB=b0I+b1τ+b2τ2,where the coefficients bs with s=0, 1, 2 are three symmetric functions of the three eigenvalues τi of τ. Usually, it does not appear to be easy to work

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