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The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

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Abstract

We investigate a family of isotropic volumetric-isochoric decoupled strain energies

$$\begin{aligned} F\mapsto W_{\mathrm{eH}}(F):=\widehat{W}_{\mathrm{eH}}(U):=\left \{ \begin{array}{l@{\quad}l} \frac{\mu}{k} e^{k\|\operatorname {dev}_n\log{U}\|^2}+\frac{\kappa}{{2 {\widehat {k}}}} e^{\widehat{k}[\operatorname {tr}(\log U)]^2}&\text{if}\ \det F>0,\\ +\infty&\text{if}\ \det F\leq0, \end{array} \right . \end{aligned}$$

based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the infinitesimal shear modulus, \(\kappa=\frac{2\mu+3\lambda}{3}>0\) is the infinitesimal bulk modulus with λ the first Lamé constant, \(k,\widehat{k}\) are additional dimensionless material parameters, F=∇φ is the gradient of deformation, \(U=\sqrt{F^{T} F}\) is the right stretch tensor and is the n-dimensional deviatoric part of the strain tensor logU. For small elastic strains, W eH approximates the classical quadratic Hencky strain energy

$$\begin{aligned} F\mapsto W_{\mathrm{H}}(F):=\widehat{W}_{\mathrm{H}}(U)&:={\mu}\| \operatorname {dev}_n\log U\|^2+\frac{\kappa}{2}\bigl[\operatorname{tr}( \log U)\bigr]^2, \end{aligned}$$

which is not everywhere rank-one convex. In plane elastostatics, i.e., n=2, we prove the everywhere rank-one convexity of the proposed family W eH, for \(k\geq\frac{1}{4}\) and \(\widehat{k}\geq\frac{1}{8}\). Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for n=2,3 and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family W eH is not preserved in dimension n=3 and that the energies

$$\begin{aligned} F\mapsto\frac{\mu}{k}e^{k\|\log U\|^2},\qquad F\mapsto\frac{\mu }{k}e^{\frac{k}{\mu} \bigl(\mu\|\operatorname {dev}_n\log U\|^2+\frac{\kappa}{2}[\operatorname {tr}(\log U)]^2 \bigr)}, \quad F \in\mathrm{GL}^+(n), n\in \mathbb {N}, n\geq2 \end{aligned}$$

are also not rank-one convex.

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Notes

  1. Although every such Riemannian metric is uniquely characterized by three coefficients, the geodesic distance to SO(n) in fact depends on only two of them, corresponding to the two material parameters μ and κ.

  2. Truesdell writes [251]: “It is important to realize that since each of the several material tensors [the strain tensors like , , logU, UU −1] is an isotropic function of any one of the others, an exact description of strain in terms of any one is equivalent to a description in terms of any other; only when an approximation is to be made may the choice of a particular measure become important.”

  3. Such an assumption is especially suitable for only slightly compressible materials or under small elastic strains [98].

  4. In Hencky’s first paper [99], the constitutive law is proposed, which is Cauchy-elastic, tensorially correct, but not hyperelastic. This has been corrected by Hencky in later papers. Incidentally, Becker’s law (1.6) is also Cauchy-elastic, tensorially correct, but hyperelastic only for ν=0 [33, 46] (see also [176, 265]).

  5. Note that (1.7) is the uniaxial specification of (1.6), and (1.6) is closely resembling (1.5)2. A small calculation [176] shows τ Becker=Vτ H, where τ is the corresponding Kirchhoff stress τ=(detF)⋅σ=D logV W(logV) and V is the left stretch tensor. Moreover . Hence, for small elastic strains , Becker’s law coincides with Hencky’s model to first order in the nonlinear strain measure .

  6. In the German metal forming literature the logarithmic strain is also called “Umformgrad”. In [138, page 17] Ludwik uses the “effective specific elongation” \(\alpha=\int_{\ell_{0}}^{\ell} \frac{d \ell}{\ell }=\ln\frac{\ell}{\ell_{0}}\). It can be motivated by considering the summation over the infinitesimal increase in length as referred to the current length, i.e., \(\ln \frac{\ell}{\ell _{0}}=\lim_{N\rightarrow\infty} \sum_{i=0}^{N-1}\frac{\ell _{i+1}-\ell_{i}}{\ell_{i}}\) [94, 261]. The scalar Hencky-type measure \(\| \operatorname {dev}_{3}\log U\|\) is sometimes used as “equivalent strain” in order to represent the degree of plastic deformation [184, 185]. Its use for severe shearing has been questioned in [224]. In our opinion the problematic issue is not the logarithmic measure itself, but its degenerate (sublinear) growth behavior for large strains. The opposing views may be reconciled by using \(e^{\|\operatorname {dev}_{3}\log U\|}\) as “exponentiated equivalent strain” measure.

  7. I.e., an amorphous metal which is very nearly isotropic with superior elastic deformability up to 1–2 % distortional strain, but which shows no ductility, in contrast to polycrystalline metals which typically show elastic strains up only to 0.1–0.2 %. Recently, Murphy [163] (see also [266]) has postulated a linear Cauchy stress-strain relation for some strain measure and gets as well W H as a preferred solution. His corresponding “strain measure” E is then \(E:=\frac{1}{\det V}\cdot \log V\), so that , which is Hencky’s relation in disguise. However, VE(V) is not invertible, thus E does not really qualify as a strain measure.

  8. Tarantola noted [245, page 15] that “Cauchy originally defined the strain as , but many lines of thought suggest that this was just a guess, that, in reality is just the first order approximation to the more proper definition  , i.e., in reality, ”.

  9. In the one dimensional case \({\varphi(x_{1},t)=(\varphi_{1}(x_{1},t), x_{2}, x_{3})^{T} \Rightarrow F=\nabla} \varphi=\operatorname {diag}({\varphi }_{1,x_{1}},1,1)\Rightarrow D={\mathrm{sym}}(\dot{F}{F}^{-1})=\operatorname {diag}(\frac{\dot{\varphi}_{1,x_{1}}}{\varphi_{1,x_{1}}},0,0 )\) and \(\int _{0}^{t} \frac{\dot{\varphi}_{1,x_{1}}}{\varphi_{1,x_{1}}} ds=\log|\varphi_{1,x_{1}}|+C\cong\log U\).

  10. Computing the rates \(\frac{\mathrm{d}}{{\mathrm {dt}}} \log U\) is more complicated because, in addition to the principal strains being a function of time, the principal directions also change in time [69, 93, 112, 120].

  11. Since GL+(3) is an open subset of \(\mathbb {R}^{3\times3}\), in accordance with [15, page 352] we say that W is rank-one convex on GL+(3) if it is convex on all closed line segments in GL+(3) with end points differing by a matrix of rank one, i.e.,

    $$\begin{aligned} W\bigl( F+(1-\theta) \xi\otimes\eta\bigr)\leq\theta W( F)+(1-\theta) W(F+ \xi\otimes\eta), \end{aligned}$$
    (1.9)

    for all F∈GL+(3), θ∈[0,1], and for all ξ, \(\eta\in\mathbb{R}^{3}\), with F+η∈GL+(3) for all t∈[0,1]. In other words, the energy function W is rank-one convex on GL+(3) if and only if the function tW(F+η) is convex \(\forall \xi, \eta\in\mathbb{R}^{3}\), on all closed line segments in the set {t:F+η∈GL+(3)}.

  12. The condition \(D^{2}_{F} W(F)(\xi\otimes\xi,\xi\otimes\xi)>0 \forall\xi\in\mathbb{R}^{3}\setminus\{0\}\), i.e., the convexity of tW(F+ξ) for all \(\xi\in\mathbb{R}^{3}\) with F+ξ∈GL+(3) for all t∈[0,1], is a necessary condition for the existence of at least one longitudinal acceleration wave [4, 213, 270].

  13. The domain where the Hencky energy W H is rank-one convex is included in the domain for which the eigenvalues λ 1,λ 2,λ 3 of U satisfy \(\lambda_{1}^{2}\leq e^{2} \lambda_{2}\lambda_{3},\lambda_{2}^{2}\leq e^{2} \lambda_{3}\lambda_{1}, \lambda_{3}^{2}\leq e^{2} \lambda_{1}\lambda_{2} \) (see Corollary 5.10). Moreover, this domain is included in the domain defined by \(\|\operatorname {dev}_{3}\log U\|^{2}\leq\frac {4}{3}\). Numerical computations reveal that the exponentiated Hencky energy is rank-one convex in a domain for which \(\|\operatorname {dev}_{3}\log U\|^{2}\leq a\) with \(a> \frac{4}{3}\) (see Sect. 6.3).

  14. In this paper we also show that for planar elastostatics \(F\mapsto e^{\|\log U\|^{2}}\) is not rank-one convex, a surprising observation which is difficult to obtain, since ellipticity is lost for extremely large principal stretches only.

  15. The idea of considering the exponential function in modelling of nonlinear elasticity is not entirely new. In fact \(W(F)=\frac{\mu}{2 k} [e^{k (I_{1}-3)}-1 ]\), where \(I_{1}=\operatorname {tr}(F F^{T})\), is a Fung-type model which is often used in the biomechanics literature to describe the nonlinearly elastic response of biological tissues [25, 85]. In the limit \(\lim_{k\rightarrow0}\frac{\mu}{2 k} [e^{k (I_{1}-3)}-1 ]=\frac{\mu}{2}(I_{1}-3)\), we recover the Neo-Hookean energy for elastic incompressible materials. Another Fung-type energy [25, 85] is .

  16. Richter in 1949 [197] already considers the following complete set of isotropic invariants: \(K_{1}=\operatorname {tr}(\log U), K_{2}^{2}= \operatorname {tr}((\operatorname {dev}_{3}\log U)^{2})\) and \(\operatorname {tr}((\operatorname {dev}_{3}\log U)^{3})\), see also [139]. A similar list of invariants was used by Lurie [139, page 189]: K 1, K 2 and \(\widetilde{K}_{3}=\arcsin(K_{3})\).

  17. The energy (1.12) does not satisfy the tension-compression symmetry.

  18. The numerical results given by Hennan and Anand [98] correspond to the large volumetric strain range 0.75≤detF≤1.16 (−0.3≤logdetF≤0.15) but small shear strain range \(\|\operatorname {dev}_{3}\log V\|\leq0.035\).

  19. Since \(\operatorname {tr}(\sigma)=0\) one might rather expect the stronger statement \(B=\Bigl( {\scriptsize\begin{matrix} B_{11} & B_{12} & 0 \cr B_{12} & B_{22} & 0 \cr 0 & 0 & 1 \end{matrix}} \Bigr)\), i.e., B 33=1, as well as detB=1. However, this is not true in general for isotropic energies, e.g., it is not satisfied for Neo-Hooke or Mooney-Rivlin type materials.

  20. In the literature, all these concepts are defined using strict inequalities for λ i λ j , ij. In this paper these common cases will be denoted by TE+, OF+, E+ and PC+, respectively.

  21. These inequalities appear also, but not as strict inequalities, in the following theorem:

    Theorem 2.1 ([16, Theorem 6.5]) Let \({W}:\mathrm{GL}^{+}(n)\rightarrow\mathbb{R}\) be an objective-isotropic function of class C 2 with the representation in terms of the singular values of U via \(W(F)=\widehat{W}(U)=g(\lambda_{1},\lambda_{2},\ldots ,\lambda_{n})\) . Let F∈GL+(n) be given with the n-tuple of singular values λ 1,λ 2,…,λ n . Then D 2 W(F)[H,H]≥0 for every \(H\in \mathbb {R}^{n\times n}\) if and only if the following conditions hold simultaneously:

    1. (i)

      \(\sum_{i,j=1}^{n} \frac{\partial^{2} g}{\partial\lambda _{i} \partial\lambda_{j}}a_{i}a_{j}\geq0\) for every \((a_{1},a_{2},\ldots ,a_{n})\in \mathbb {R}^{n}\) (convexity of g);

    2. (ii)

      for every \(i\neq j, \underbrace{\frac{\frac{\partial g}{\partial\lambda_{i}}-\frac {\partial g}{\partial\lambda_{j}}}{\lambda_{i}-\lambda_{j}}\geq 0}_{\text{``}\mathrm{OF}\mbox{-}\mathrm{inequality}\text{''}}\ \text{if}\ \lambda_{i}\neq \lambda_{j}, \frac{\partial^{2} g}{\partial\lambda_{i}^{2}}-\frac {\partial^{2} g}{\partial \lambda_{i}\partial\lambda_{j}}\geq0 \ \text{if}\ \lambda _{i}=\lambda_{j}\).

    3. (iii)

      \(\frac{\partial g}{\partial\lambda_{i}}+\frac {\partial g}{\partial\lambda_{j}}\geq0\) for every ij.

    Hence, if the function FW(F) is convex in F∈GL+(n), then the OF-inequalities hold true. However, the convexity of FW(F) is physically not acceptable, since it precludes buckling.

  22. Similarly, as shown in [134] the energy \(C\mapsto\frac{\mu}{4} [\|C\|^{2}-2 \log(\det C)-3 ]\) is convex in C and indeed polyconvex. The convexity in C has been used by Fung [85] to invert the second Piola-Kirchhoff stress tensor S 2=2D C [W(C)].

  23. This is suggested by the formula presented in [29, page 736]: \(e^{\alpha\cdot\widehat{A}}=\bigl( {\scriptsize\begin{matrix} \cosh\alpha& \sinh\alpha \cr \sinh\alpha& \cosh\alpha \end{matrix}} \bigr)\ \text{for} \ \widehat{A}=\bigl( {\scriptsize\begin{matrix} 0 & \alpha \cr \alpha& 0 \end{matrix}} \bigr)\).

  24. In terms of the Young’s modulus and the shear modulus ν is given by \(\nu=\frac{E}{2 \mu}-1\), while in terms of the Young’s modulus and the bulk modulus κ it is given by \(\nu =\frac{1}{2}-\frac{E}{6 \kappa}\).

  25. We use that \(\kappa=\frac{2 \mu (1+\nu)}{3 (1-2\nu)}\), \(\nu=\frac{3 \kappa-2 \mu}{2(3 \kappa+\mu)}\).

  26. In [159] it is claimed that the classical elasticity formulation is applicable only for \(\frac{1}{5}<\nu<\frac{1}{2}\).

  27. We use the definition of polyconvexity given by Ball [15] (see also [216, 220]). Polyconvexity implies LH-ellipticity and may lead to an existence theorem based on the direct methods of the calculus of variations, provided that proper growth conditions are satisfied [18, 20, 96, 167, 168].

  28. For this special material the energy is elliptic for \(\rho<\frac{\lambda_{1}}{\lambda_{2}}<\frac{1}{\rho}\), \(\rho=2-\sqrt{3}=0.268\).

  29. This means \(e^{k \|\operatorname {dev}_{3} \log (a U)\|^{2}}=e^{k \|\operatorname {dev}_{3}\log U\|^{2}}\) for all a>0.

  30. The invariance under inversion of an energy W is the tension-compression symmetry W(F)=W(F −1).

  31. Hutchinson and Neale [116] have considered the energy \(\|\operatorname {dev}_{3}\log U\|^{N}\) for 0<N≤1.

  32. In [257] Vallée et al. have given a proof without using a Taylor expansion.

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Acknowledgements

This paper is inconceivable without the stimulus of Albert Tarantola’s book “Elements for Physics” [244]. We would like to thank Prof. Krzysztof Chelminski (TU Warsaw) for helping us in the study of rank-one convexity of the function \(e^{\|\log U\|^{2}}\) in the planar case, Prof. David Steigman (UC Berkeley) who indicated to us reference [116], Prof. Bernard Dacorogna (EPFL-Lausanne) for sending us reference [58] and Prof. Miroslav Šilhavý (Academy of Sciences of the Czech Republic, Prague) for comments on rank-one convexity. The interest in considering nonlinear scalar functions of \(\|\operatorname {dev}_{3}\log U\|^{2}\) arose after an insightful comment by Prof. Reuven Segev (Ben-Gurion University of the Negev, Beer-Sheva) on the presentation of the first author at the 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM July 2013) in Montreal. Discussion with Prof. Chandrashekhar S. Jog (Indian Institute of Science, Bangalore) on the TSTS-M+ condition and with Robert Martin (Duisburg-Essen University) on response of rubber are also gratefully acknowledged.

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Correspondence to Patrizio Neff.

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In memory of Albert Tarantola (⋆ 1949–† 2009), lifelong advocate of logarithmic measures.

Appendix

Appendix

1.1 A.1 Some Useful Identities

  • \(\operatorname {tr}(B^{-1}X B)=\operatorname {tr}(X)\) for any invertible matrix B.

  • \(\operatorname {dev}_{n}(B^{-1}X B)=B^{-1}X B-\frac{1}{n} \operatorname {tr}(B^{-1}X B)=B^{-1}(\operatorname {dev}_{n} X) B\) for any invertible matrix B.

  • .

  • The norm of the deviator in \(\mathbb {R}^{n\times n}\):

    $$\begin{aligned} \left \Vert \operatorname {dev}_n\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \xi_1&0&\cdots&0\\ 0&\xi_2&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\xi_n\\ \end{array} \right ) \right \Vert ^2&=\sum_{i=1}^n \xi_i^2-\frac{1}{n}\Biggl(\sum_{i=1}^n \xi_i\Biggr)^2 =\frac{n-1}{n}\sum _{i=1}^n \xi_i^2- \frac{2}{n}\sum_{i,j=1,i< j}^n \xi_i\xi_j \\ &=\frac{1}{n}\Biggl[(n-1)\sum_{i=1}^n \xi_i^2-2\sum_{i,j=1,i<j}^n \xi_i\xi_j\Biggr] \\ &=\frac{1}{n}\sum_{i,j=1,i<j}^n \bigl( \xi_i^2-2 \xi_i\xi_j+ \xi_j^2\bigr)=\frac{1}{n}\sum _{i,j=1,i<j}^n (\xi_i-\xi_j)^2. \end{aligned}$$
    (A.1)
  • From [165, page 200] we have: \(\frac{\|X\| ^{p}}{z^{\alpha}}\) is convex in (X,z) if \(\frac{\alpha+1}{\alpha}\geq \frac{p}{p-1}\Leftrightarrow p\geq\alpha+1\).

  • \(\log U=\sum_{i=1}^{n} \log\lambda_{i} N_{i}\otimes N_{i}\), where N i are the eigenvectors of U and λ i are the eigenvalues of U.

  • , convergent for .

  • \(\log V=\sum_{i=1}^{n} \log\widehat{\lambda}_{i} \widehat {N}_{i}\otimes\widehat{N}_{i}\), where \(\widehat{N}_{i}\) are the eigenvectors of V and \(\widehat{\lambda}_{i}\) are the eigenvalues of V.

  • , convergent for .

  • $$\begin{aligned} \left ( \begin{array}{c@{\quad}c} F_{11} & F_{12} \\ F_{21} & F_{22} \\ \end{array} \right )^{-1}=\frac{1}{F_{11}F_{22}-F_{12}F_{21}}\left ( \begin{array}{c@{\quad}c} F_{22} & -F_{12} \\ -F_{21} & F_{22} \\ \end{array} \right )\quad\Rightarrow\quad\|F^{-1}\|^2 \overset{n=2}{=} \frac{1}{(\det F)^2}\|F\|^2. \end{aligned}$$
  • $$\begin{aligned} F=\left ( \begin{array}{c@{\quad}c} F_{11} & F_{12} \\ F_{21} & F_{22} \\ \end{array} \right ),\qquad U^2=F^T F=\left ( \begin{array}{c@{\quad}c} F_{11}^2+F_{21}^2 & F_{11} F_{12}+F_{21} F_{22} \\ F_{11} F_{12}+F_{21} F_{22} & F_{12}^2+F_{22}^2 \\ \end{array} \right ). \end{aligned}$$

    The eigenvalues of U 2 are:

    $$\begin{aligned} \mu_1&=\frac{1}{2} \Bigl(F_{11}^2+F_{12}^2+F_{21}^2+F_{22}^2- \sqrt{ \bigl(F_{11}^2+F_{12}^2+F_{21}^2+F_{22}^2 \bigr){}^2-4 (F_{12} F_{21}-F_{11} F_{22} ){}^2} \Bigr) \\ &=\frac{1}{2} \bigl(\|F\|^2-\sqrt{\|F\|^4-4(\det F)^2} \bigr), \\ \mu_2&=\frac{1}{2} \Bigl(F_{11}^2+F_{12}^2+F_{21}^2+F_{22}^2+ \sqrt{ \bigl(F_{11}^2+F_{12}^2+F_{21}^2+F_{22}^2 \bigr){}^2-4 (F_{12} F_{21}-F_{11} F_{22} ){}^2} \Bigr) \\ &=\frac{1}{2} \bigl(\|F\|^2+\sqrt{\|F\|^4-4(\det F)^2} \bigr). \end{aligned}$$

    The principal stretches of F, i.e., the eigenvalues of \(U=\sqrt{F^{T} F}\), which are the same as the eigenvalues of \(V=\sqrt{F F^{T}}\), are \(\lambda_{1}(F)=\sqrt{\mu_{1}}, \lambda_{2}(F)=\sqrt{\mu_{2}}\).

  • Taking the pure stretch under shear stress \(F_{1}= \bigl( {\scriptsize\begin{matrix} \cosh\frac{t}{2} & \sinh\frac{t}{2} & 0 \cr \sinh\frac{t}{2} & \cosh\frac{t}{2} & 0 \cr 0 & 0 & 1 \end{matrix}} \bigr)\) and the simple glide \(F_{2}= \bigl( {\scriptsize\begin{matrix} 1 & t& 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \end{matrix}} \bigr)\), the corresponding rates \(L_{1}(t)=\frac{{\mathrm{d}}}{{\mathrm{dt}}}F_{1}\cdot F_{1}^{-1}\neq \frac{{\mathrm{d}}}{{\mathrm{dt}}}F_{2}\cdot F_{2}^{-1}=L_{2}(t)\) are different, as is \(\log U_{1}(t)\neq \log\sqrt{F_{2}^{T}F_{2}}=\log U_{2}(t)\) and \(\frac{{\mathrm {d}}}{{\mathrm{dt}}}\log U_{1}\neq\frac{{\mathrm{d}}}{{\mathrm {dt}}}\log U_{2}(t)\). However, D 1(t)=sym L 1(t)=sym L 2(t)=D 2(t). This shows that \(\frac {{\mathrm{d}}}{{\mathrm{dt}}}\log U(t)=D(t)\) is true only for coaxial families U(t).

1.2 A.2 Vallée’s Formula

Lemma A.1

(Vallée’s formulaFootnote 32 (see also [130, 210, 256, 257]))

Let us consider S∈Sym(3) and let \(\varPsi:\mathrm {Sym}(3)\rightarrow \mathbb {R}\) be a differentiable isotropic scalar value function. We define W(S)=Ψ(exp(S)). Then, the following chain rules hold:

$$\begin{aligned} \begin{aligned} &D_S\bigl[\varPsi\bigl(\exp(S)\bigr) \bigr]=\exp(S)\cdot D \varPsi\bigl(\exp(S)\bigr), \qquad D_SW(S)=D \varPsi\bigl(\exp(S)\bigr)\cdot\exp(S), \\ &D_C \varPsi(C)=D W(\log C)\cdot C^{-1},\qquad C\cdot D_C\varPsi(C)=D W(\log C), \end{aligned} \end{aligned}$$
(A.2)

while it is generally not true that D C [logC].H=〈C −1,H〉.

Proof

Let us first remark that

(A.3)

Further we consider the Taylor expansion of the function Ψ(exp(S))

$$\begin{aligned} &\varPsi\bigl(\exp(S+H)\bigr) \\ &\quad=\varPsi\bigl(\exp(S)+D\bigl(\exp(S)\bigr). H+\cdots\bigr) \\ &\quad=\varPsi\bigl(\exp(S)\bigr)+\bigl\langle D \varPsi\bigl(\exp (S)\bigr), D\bigl( \exp(S)\bigr). H\bigr\rangle +\cdots \\ &\quad=\varPsi\bigl(\exp(S)\bigr)+\biggl\langle D \varPsi\bigl(\exp (S)\bigr), H+ \frac{1}{2}(S H+H S)\biggr\rangle \\ &\qquad+\biggl\langle D \varPsi\bigl(\exp(S)\bigr),\frac{1}{6}(S S H+S H S+H S S)+\cdots\biggr\rangle +\cdots \\ &\quad=\varPsi\bigl(\exp(S)\bigr)+\bigl\langle D \varPsi\bigl(\exp (S)\bigr), H\bigr\rangle +\frac{1}{2} \bigl\langle D \varPsi\bigl(\exp(S)\bigr),S H+H S\bigr\rangle \\ &\qquad+\frac{1}{6}\bigl\langle D \varPsi\bigl(\exp(S)\bigr),SSH+S H S+H S S\bigr\rangle +\cdots \\ &\quad=\varPsi\bigl(\exp(S)\bigr)+\bigl\langle D \varPsi\bigl(\exp (S)\bigr), H\bigr\rangle +\frac{1}{2} \bigl[\bigl\langle S^T D \varPsi\bigl(\exp(S) \bigr), H\bigr\rangle +\bigl\langle D \varPsi\bigl(\exp(S)\bigr) S^T,H \bigr\rangle \bigr] \\ &\qquad+\frac{1}{6} \bigl[S^T S^T \bigl\langle D \varPsi\bigl(\exp(S)\bigr),H\bigr\rangle +\bigl\langle S^T D \varPsi \bigl( \exp(S)\bigr) S^T,H\bigr\rangle \\ &\qquad+\bigl\langle D \varPsi\bigl(\exp(S)\bigr) S^T S^T,H\bigr\rangle \bigr]+\cdots. \end{aligned}$$
(A.4)

Since S∈Sym(3), it follows

$$\begin{aligned} \varPsi\bigl(\exp(S+H)\bigr)&=\varPsi\bigl(\exp(S)\bigr)+\bigl\langle D \varPsi\bigl( \exp(S)\bigr), H\bigr\rangle \\ &\quad+\frac{1}{2} \bigl[\bigl\langle S D \varPsi\bigl(\exp(S)\bigr), H \bigr\rangle +\bigl\langle D \varPsi\bigl(\exp(S)\bigr) S,H\bigr\rangle \bigr] \\ &\quad+\frac{1}{6} \bigl[S S \bigl\langle D \varPsi\bigl(\exp (S)\bigr),H \bigr\rangle +\bigl\langle S D \varPsi\bigl(\exp(S)\bigr) S,H\bigr\rangle \\ &\qquad+\bigl\langle D \varPsi\bigl(\exp(S)\bigr) S S,H\bigr\rangle \bigr]+\cdots. \end{aligned}$$
(A.5)

On the other hand, since is a isotropic tensor function and obvious exp(S) is also isotropic, we have that (exp(S)) is also a isotropic tensor function and therefore it holds

$$\begin{aligned} D \varPsi\bigl(\exp(S)\bigr)\cdot S=S\cdot D \varPsi\bigl(\exp(S)\bigr). \end{aligned}$$
(A.6)

Therefore,

(A.7)

Using again the isotropy of (exp(S)), we obtain

$$\begin{aligned} \varPsi\bigl(\exp(S+H)\bigr) &=\varPsi\bigl(\exp(S)\bigr)+\bigl\langle \exp(S)\cdot D \varPsi\bigl(\exp(S)\bigr),H\bigr\rangle + \cdots. \end{aligned}$$
(A.8)

We recall that we simultaneously have

$$\begin{aligned} \varPsi\bigl(\exp(S+H)\bigr) &=\varPsi\bigl(\exp(S)\bigr)+\bigl\langle D_S \varPsi\bigl(\exp(S)\bigr),H\bigr\rangle +\cdots, \end{aligned}$$
(A.9)

for all H∈Sym(3). Thus, we deduce

$$\begin{aligned} \begin{aligned} &\bigl\langle D_S\varPsi\bigl(\exp(S)\bigr),H\bigr\rangle =\bigl\langle \exp(S)\cdot D \varPsi\bigl(\exp(S)\bigr),H\bigr\rangle , \\ &\bigl\langle D_SW(S),H\bigr\rangle =\bigl\langle \exp(S)\cdot D \varPsi\bigl(\exp(S)\bigr),H\bigr\rangle . \end{aligned} \end{aligned}$$
(A.10)

Choosing S=logC, the relations (A.2)3 also results and the proof is complete. □

1.3 A.3 LH-Ellipticity for Functions of the Type Fh(detF)

We consider a function \(h:\mathbb {R}\to \mathbb {R}\) and we analyze when the function Fh(detF) is LH-elliptic as a function of F, \(F\in \mathbb {R}^{3\times3}\). We recall that

$$\begin{aligned} D(\det F).H=\det F\cdot \operatorname {tr}\bigl(H F^{-1}\bigr)=\langle \operatorname {Cof}F,H \rangle. \end{aligned}$$
(A.11)

Using the first Frechét-formal derivative, we compute the derivative

$$\begin{aligned} D\bigl(h(\det F)\bigr).(H,H)=h^{\prime}(\det F)\cdot\langle \operatorname {Cof}F, H \rangle, \end{aligned}$$
(A.12)

and the second derivative will be

$$\begin{aligned} &D^2\bigl(h(\det F)\bigr).(H,H) \\ &\quad=h^{\prime\prime}(\det F)\cdot\langle \operatorname {Cof}F, H\rangle ^2+h^{\prime}( \det F)\bigl\langle D(\operatorname {Cof}F).H, H\bigr\rangle \\ &\quad=h^{\prime\prime}(\det F)\cdot\langle \operatorname {Cof}F, H\rangle ^2+h^{\prime}( \det F) \bigl\{ \bigl\langle \langle \operatorname {Cof}F,H\rangle F^{-T},H\bigr\rangle \\ &\qquad+\det F\bigl\langle -F^{-T}H^T F^{-T},H \bigr\rangle \bigr\} , \\ &\quad=h^{\prime\prime}(\det F)\cdot\langle \operatorname {Cof}F, H\rangle ^2+h^{\prime} (\det F) \det F \bigl\{ \bigl\langle F^{-T},H\bigr\rangle ^2 \\ &\qquad-\bigl\langle F^{-T}H^T F^{-T},H\bigr\rangle \bigr\} . \end{aligned}$$
(A.13)

Hence, for \(\xi,\eta\in \mathbb {R}^{3}\) we have

$$\begin{aligned} &D^2\bigl(h(\det F)\bigr).(\xi\otimes\eta,\xi\otimes\eta)) \\ &\quad=h^{\prime\prime}(\det F)\cdot\bigl\langle \operatorname {Cof}F, (\xi \otimes\eta) \bigr\rangle ^2+h^{\prime} (\det F) \det F \bigl\{ \bigl\langle F^{-T},(\xi\otimes\eta)\bigr\rangle ^2 \\ &\qquad-\bigl\langle F^{-T}(\xi\otimes\eta)^T F^{-T},(\xi\otimes\eta)\bigr\rangle \bigr\} . \end{aligned}$$
(A.14)

On the other hand

This leads to the surprising simplification

$$\begin{aligned} D^2\bigl(h(\det F)\bigr).&(\xi\otimes\eta,\xi\otimes \eta)=h^{\prime\prime}(\det F)\cdot\bigl\langle \operatorname {Cof}F, (\xi \otimes\eta)\bigr\rangle ^2. \end{aligned}$$
(A.15)

In conclusion, Fh(detF) is LH-elliptic if and only if th(t) is convex since \(\langle \operatorname {Cof}F, (\xi\otimes\eta )\rangle^{2}\) is positive.

From [55, page 213] we know more:

Proposition A.2

Let \(W:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}\) be quasiaffine but not identically constant and \(h:\mathbb{R}\rightarrow\mathbb{R}\) be such that W(F)=h(detF). Then

$$\begin{aligned} W \textit{polyconvex}\quad\Leftrightarrow\quad W \textit {quasiconvex}\quad \Leftrightarrow\quad W \textit{rank one convex}\quad\Leftrightarrow \quad h\ \textit{convex}. \end{aligned}$$
(A.16)

1.4 A.4 Convexity for Functions of the Type tξ((logt)2)

We consider a generic function \(\xi:\mathbb {R}_{+}\to \mathbb {R}_{+}\) and we find a characterization of the convexity for the function tξ((logt)2). In the following let ζ denote the function \(\zeta:\mathbb {R}_{+}\to \mathbb {R}_{+}\) , ζ(t)=(logt)2. We deduce

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}}{\mathrm{dt}}{\xi}\bigl((\log t)^2\bigr)&= \xi^{\prime}\bigl((\log t)^2\bigr) 2 \frac{1}{t} \log t, \\ \frac{{\mathrm{d}}^2 }{{\mathrm{dt}}^2}\xi\bigl((\log t)^2\bigr)&=2\frac{\mathrm{d}}{\mathrm{dt}} \biggl(\xi^{\prime}\bigl((\log t)^2\bigr) 2 \frac{1}{t} \log t \biggr) \\ &=4 \xi^{\prime\prime}\bigl((\log t)^2\bigr) \frac{1}{t^2} ( \log t)^2-2 \xi^{\prime}\bigl((\log t)^2\bigr) \frac{1}{t^2} \log t+2 \xi^{\prime}\bigl((\log t)^2\bigr) \frac{1}{t^2} \\ &=2\frac{1}{t^2} \bigl[{2} \xi^{\prime\prime}\bigl((\log t)^2 \bigr) (\log t)^2 +\xi^{\prime}\bigl((\log t)^2 \bigr) (1- \log t) \bigr], \end{aligned} \end{aligned}$$
(A.17)

where \(\xi^{\prime}=\frac{d \xi}{d\zeta}\). Hence, the function tξ((logt)2) is

  • convex on [1,∞) as a function of t if and only if \(2\frac{d^{2}\xi(\zeta)}{d\zeta^{2}} \zeta+ {\frac{d\xi(\zeta )}{d\zeta} (}1- \sqrt{\zeta})\geq 0\), for all \(\zeta\in \mathbb {R}_{+}\).

  • convex on (0,1) as a function of t if and only if \(2\frac{d^{2}\xi(\zeta)}{d\zeta^{2}} \zeta+ {\frac{d\xi(\zeta )}{d\zeta}(1}+ \sqrt{\zeta})\geq 0\), for all \(\zeta\in \mathbb {R}_{+}\).

1.5 A.5 Connecting \(\operatorname {dev}_{3}\log U\) with \(\operatorname {dev}_{2}\log U\)

For U ∈GL(2), we define the lifted quantity

$$\begin{aligned} U=\left ( \begin{array}{c@{\quad}c@{\quad}c} &U^\sharp& 0\\ & & 0\\ 0&0&(\det U^\sharp)^{1/2} \end{array} \right )\in\mathrm{GL}(3). \end{aligned}$$
(A.18)

We remark that

$$\begin{aligned} \det \left ( \begin{array}{c@{\quad}c@{\quad}c} &U^\sharp& 0\\ & & 0\\ 0&0&(\det U^\sharp)^{1/2} \end{array} \right )=\det U^\sharp\bigl(\det U^\sharp\bigr)^{1/2}=\bigl(\det U^\sharp \bigr)^{3/2}, \end{aligned}$$
(A.19)

which implies (detU)1/3=[detU (detU )1/2]1/3=[(detU )3/2]1/3=(detU )1/2. Moreover, we obtain

$$\begin{aligned} \operatorname {dev}_3\log U&=\log\frac{U}{\det U^{1/3}}=\log\frac{U}{(\det U^\sharp)^{1/2}}=\log \left ( \begin{array}{c@{\quad}c@{\quad}c} & \frac{U^\sharp}{(\det U^\sharp)^{1/2}}& 0\\ & & 0\\ 0&0&1 \end{array} \right ) \\ &=\left ( \begin{array}{c@{\quad}c@{\quad}c} & \log\frac{U^\sharp}{(\det U^\sharp)^{1/2}}& 0\\ & & 0\\ 0&0&0 \end{array} \right )=\left ( \begin{array}{c@{\quad}c@{\quad}c} &\operatorname {dev}_2 \log U^\sharp& 0\\ & & 0\\ 0&0&0 \end{array} \right ). \end{aligned}$$
(A.20)

In general, for \(A^{\sharp}\in \mathbb {R}^{2\times2}\) and \(\alpha\in \mathbb {R}\) we have

$$\begin{aligned} \left\|\operatorname {dev}_3\left ( \begin{array}{c@{\quad}c@{\quad}c} &A^\sharp& 0\\ & & 0\\ 0&0&\alpha \end{array} \right )\right\|^2&=\left\| \left ( \begin{array}{c@{\quad}c@{\quad}c} &A^\sharp& 0\\ & & 0\\ 0&0&\alpha \end{array} \right )\right\|^2-\frac{1}{3} [\operatorname {tr}\left[ \left ( \begin{array}{c@{\quad}c@{\quad}c} &A^\sharp& 0\\ & & 0\\ 0&0&\alpha \end{array} \right )\right]^2 \\ &=\bigl\| A^\sharp\bigr\| ^2+\alpha^2-\frac{1}{3} \bigl[\operatorname {tr}\bigl(A^\sharp\bigr)+\alpha\bigr]^2 \\ &=\bigl\| A^\sharp\bigr\| ^2-\frac{1}{3} \bigl[\operatorname {tr}\bigl(A^\sharp\bigr)\bigr]^2-\frac{2}{3}\alpha \operatorname {tr}\bigl(A^\sharp\bigr) -\frac{1}{3} \alpha^2+ \alpha^2 \\ &=\bigl\| \operatorname {dev}_2 A^\sharp\bigr\| ^2-\frac{2}{3} \alpha \operatorname {tr}\bigl(A^\sharp\bigr) +\frac{2}{3} \alpha^2. \end{aligned}$$
(A.21)

Thus

$$\begin{aligned} \left\|\operatorname {dev}_3\left ( \begin{array}{c@{\quad}c@{\quad}c} &A^\sharp& 0\\ & & 0\\ 0&0&\alpha \end{array} \right )\right\|^2&=\bigl\| \operatorname {dev}_2 A^\sharp\bigr\| ^2 \end{aligned}$$
(A.22)

if and only if α=0 or \(\alpha=\operatorname {tr}(A^{\sharp})\). Hence, we deduce \({\|\operatorname {dev}_{3}\log U\|^{2}}={\|\operatorname {dev}_{2}\log U^{\sharp}\|^{2}}\), for U of the form (A.18). Since U ∈PSym(2), we can assume that \(U^{\sharp}= \bigl({\scriptsize\begin{matrix}\lambda_{1}&0\cr 0&\lambda_{2}\end{matrix}} \bigr)\), \(\lambda_{1},\lambda_{2}\in \mathbb {R}_{+}\). Then, the lifted quantity U lies in \(\operatorname {PSym}(3)\) and \(U = \bigl({\scriptsize\begin{matrix}\lambda_{1}&0&0\cr 0&\lambda_{2}&0\cr 0&0&(\lambda_{1} \lambda_{2})^{1/2}\end{matrix}} \bigr)\).

The next problem is if for a given deformation \(\varphi^{\sharp}=(\varphi_{1}^{\sharp}, \varphi_{2}^{\sharp}):\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) such that \(U^{\sharp}=\sqrt{(\nabla\varphi^{\sharp})^{T} \nabla\varphi^{\sharp}}\) we can construct an ansatz \(\varphi:\mathbb {R}^{3}\rightarrow \mathbb {R}^{3}\) such that \(U=\sqrt{\nabla\varphi^{T} \nabla\varphi}\), where U is the lifted quantity associated to U . For this it is necessary to have φ=(φ 1(x 1,x 2),φ 2(x 1,x 2),x 3 α(x 1,x 2)) and \(\alpha_{,x_{1}}=0, \alpha_{,x_{2}}=0\). Checking the compatibility equation we see that this is possible if and only if \(\det\nabla \varphi^{\sharp}=K=\mathit{const.}\), which implies \(\varphi_{3,x_{3}}=K\). In the incompressible case det∇φ=1, an appropriate ansatz is therefore

$$\begin{aligned} \varphi(x_1,x_2,x_3)=\bigl( \varphi_1^\sharp(x_1,x_2), \varphi_2^\sharp(x_1,x_2),x_3 \bigr), \end{aligned}$$
(A.23)

since

$$\begin{aligned} U^2&={\nabla\varphi^T \nabla\varphi}= \left ( \begin{array}{c@{\quad}c@{\quad}c}&(\nabla\varphi^\sharp)^T \nabla \varphi^\sharp&0\\ &&0\\ 0&0&1 \end{array} \right )=\left ( \begin{array}{c@{\quad}c@{\quad}c}&(\nabla\varphi^\sharp)^T \nabla \varphi^\sharp&0\\ &&0\\ 0&0&\det[(\nabla\varphi^\sharp)^T \nabla\varphi^\sharp] \end{array} \right ) \\ &=\left ( \begin{array}{c@{\quad}c@{\quad}c}&(U^\sharp)^2&0\\ &&0\\ 0&0&(\det[(U^\sharp)^{1/2}])^2 \end{array} \right )=\left ( \begin{array}{c@{\quad}c@{\quad}c}&U^\sharp&0\\ &&0\\ 0&0&(\det U^\sharp)^{1/2} \end{array} \right )^2, \end{aligned}$$
(A.24)

with detU =1.

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Neff, P., Ghiba, ID. & Lankeit, J. The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity. J Elast 121, 143–234 (2015). https://doi.org/10.1007/s10659-015-9524-7

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