nach oben

Journal of Elasticity

Erschienen in:

01.12.2015

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

verfasst von: Patrizio Neff, Ionel-Dumitrel Ghiba, Johannes Lankeit

Erschienen in: Journal of Elasticity | Ausgabe 2/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq4_HTML.gif

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.

Nächster Artikel Mathematical Modeling of Weak and Strong Piezoelectric Interfaces

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

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.

Truesdell writes [251]: “It is important to realize that since each of the several material tensors [the strain tensors like

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq22_HTML.gif

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq23_HTML.gif

, logU, U−U ⁻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.”

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

In Hencky’s first paper [99], the constitutive law

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq34_HTML.gif

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]).

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=V⋅τ _H, where τ is the corresponding Kirchhoff stress τ=(detF)⋅σ=D _logV W(logV) and V is the left stretch tensor. Moreover

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq35_HTML.gif

. Hence, for small elastic strains

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq36_HTML.gif

, Becker’s law coincides with Hencky’s model to first order in the nonlinear strain measure

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq37_HTML.gif

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.

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

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq43_HTML.gif

, which is Hencky’s relation in disguise. However, V↦E(V) is not invertible, thus E does not really qualify as a strain measure.

Tarantola noted [245, page 15] that “Cauchy originally defined the strain as

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq44_HTML.gif

, 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

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq45_HTML.gif

, i.e., in reality,

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq46_HTML.gif

”.

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 $\int _{0}^{t} \frac{\dot{\varphi}_{1,x_{1}}}{\varphi_{1,x_{1}}} ds=\log|\varphi_{1,x_{1}}|+C\cong\log U$.

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].

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+tξ⊗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 t↦W(F+tξ⊗η) is convex $\forall \xi, \eta\in\mathbb{R}^{3}$, on all closed line segments in the set {t:F+tξ⊗η∈GL⁺(3)}.

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 t↦W(F+tξ⊗ξ) for all $\xi\in\mathbb{R}^{3}$ with F+tξ⊗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].

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).

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.

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

https://static-content.springer.com/image/art%3A10.1007%2Fs10659-015-9524-7/MediaObjects/10659_2015_9524_IEq88_HTML.gif

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 $\widetilde{K}_{3}=\arcsin(K_{3})$.

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

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 $\|\operatorname {dev}_{3}\log V\|\leq0.035$.

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.

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

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:

(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);

(ii)

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

(iii)

$\frac{\partial g}{\partial\lambda_{i}}+\frac {\partial g}{\partial\lambda_{j}}\geq0$ for every i≠j.

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

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)].

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

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 $\nu =\frac{1}{2}-\frac{E}{6 \kappa}$.

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

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

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].

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

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

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

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

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

Adamov, A.A.: Comparative analysis of the two-constant generalizations of Hooke’s law for isotropic elastic materials at finite strains. J. Appl. Mech. Tech. Phys. 42(5), 890–897 (2001) ADSCrossRefMATH

Al-Mohy, A., Higham, N., Relton, S.: Computing the Fréchet derivative of the matrix logarithm and estimating the condition number. SIAM J. Sci. Comput. 35(4), C394–C410 (2013) MATHMathSciNetCrossRef

Alexander, H.: The tensile instability of an inflated cylindrical membrane as affected by an axial load. Int. J. Mech. Sci. 13(2), 87–95 (1971) CrossRef

Altenbach, H., Eremeyev, V., Lebedev, L., Rendón, L.A.: Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech. 80(3), 217–227 (2010) MATHADSCrossRef

Anand, L.: On H. Hencky’s approximate strain energy function for moderate deformations. J. Appl. Mech. 46, 78–82 (1979) MATHADSCrossRef

Anand, L.: Moderate deformations in extension-torsion of incompressible isotropic elastic materials. J. Mech. Phys. Solids 34, 293–304 (1986) ADSCrossRef

Annaidh, A.N., Destrade, M., Gilchrist, M.D., Murphy, J.G.: Deficiencies in numerical models of anisotropic nonlinearly elastic materials. Biomech. Model. Mechanobiol. 12(4), 781–791 (2012) CrossRef

Arghavani, J., Auricchio, F., Naghdabadi, R.: A finite strain kinematic hardening constitutive model based on Hencky strain: general framework, solution algorithm and application to shape memory alloys. Int. J. Plast. 27(6), 940–961 (2011) MATHCrossRef

Aron, M.: On certain deformation classes of compressible Hencky materials. Math. Mech. Solids 11, 467–478 (2006) MATHMathSciNetCrossRef

10.

Aubert, G.: Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2. J. Elast. 39(1), 31–46 (1995) MATHMathSciNetCrossRef

11.

Aubert, G., Tahraoui, R.: Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3. Rend. Circ. Mat. Palermo 34(3), 460–488 (1985) MATHMathSciNetCrossRef

12.

Aubert, G., Tahraoui, R.: Sur la faible fermeture de certains ensembles de contraintes en élasticité non-linéaire plane. Arch. Ration. Mech. Anal. 97(1), 33–58 (1987) MATHMathSciNetCrossRef

13.

Baker, M., Ericksen, J.L.: Inequalities restricting the form of the stress-deformation relation for isotropic elastic solids and Reiner-Rivlin fluids. J. Wash. Acad. Sci. 44, 33–35 (1954) MathSciNet

14.

Ball, J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Knops, R.J. (ed.) Herriot Watt Symposion: Nonlinear Analysis and Mechanics, vol. 1, pp. 187–238. Pitman, London (1977)

15.

Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) MATHCrossRefMathSciNet

16.

Ball, J.M.: Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51(3), 699–728 (1984) MATHMathSciNetCrossRef

17.

Ball, J.M.: Some open problems in elasticity. In: Newton, P., et al. (eds.) Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002) CrossRef

18.

Balzani, D., Neff, P., Schröder, J., Holzapfel, G.A.: A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43(20), 6052–6070 (2006) MATHCrossRefMathSciNet

19.

Balzani, D., Schröder, J., Gross, D., Neff, P.: Modeling of Anisotropic Damage in Arterial Walls Based on Polyconvex Stored Energy Functions. In: Owen, D.R.J., Onate, E., Suarez, B. (eds.) Computational Plasticity VIII, Fundamentals and Applications, Part 2, pp. 802–805. CIMNE, Barcelona (2005)

20.

Balzani, D., Schröder, J., Neff, P., Holzapfel, G.A.: Materially stable constitutive equations for arterial walls based on polyconvex energies—application to damage modeling and residual stresses. J. Biomech. 39, 409 (2006) CrossRef

21.

Batra, R.C.: On the coincidence of the principal axes of stress and strain in isotropic elastic bodies. Lett. Appl. Eng. Sci. 3, 435–439 (1975) MATH

22.

Batra, R.C.: Deformation produced by a simple tensile load in an isotropic elastic body. J. Elast. 6(1), 109–111 (1976) MATHCrossRef

23.

Batra, R.C., dell’Isola, F., Vidoli, S.: A second-order solution of Saint-Venant’s problem for a piezoelectric circular bar using Signorini’s perturbation method. J. Elast. 52(1), 75–90 (1998) MATHMathSciNetCrossRef

24.

Bažant, Z.P.: Easy-to-compute tensors with symmetric inverse approximating Hencky finite strain and its rate. J. Eng. Mater. Trans. ASME 120, 131–136 (1998) CrossRef

25.

Beatty, M.: Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples. Appl. Mech. Rev. 40(12), 1699–1734 (1987) ADSCrossRef

26.

Beatty, M.F.: Introduction to nonlinear elasticity. In: Carroll, M.M., Hayes, M.A. (eds.) Nonlinear Effects in Fluids and Solids, pp. 13–112. Springer, Berlin (1996) CrossRef

27.

Becker, G.F.: The finite elastic stress-strain function. Am. J. Sci. XLVI, 337–356 (1893). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/becker_latex_new1893.pdf CrossRef

28.

Bell, J.F.: Mechanics of Solids: Volume 1: The Experimental Foundations of Solid Mechanics. Handbuch der Physik. Springer, Berlin (1973)

29.

Bernstein, D.S.: Matrix Mathematics. Princeton University Press, New Jersey (2009) MATHCrossRef

30.

Bertram, A., Böhlke, T., Šilhavý, M.: On the rank 1 convexity of stored energy functions of physically linear stress-strain relations. J. Elast. 86, 235–243 (2007) MATHCrossRefMathSciNet

31.

Bhatia, R., Holbrook, J.: Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413(2), 594–618 (2006) MATHMathSciNetCrossRef

32.

Bîrsan, M., Neff, P., Lankeit, J.: Sum of squared logarithms: an inequality relating positive definite matrices and their matrix logarithm. J. Inequal. Appl. 2013(1), 168 (2013) CrossRefMathSciNetMATH

33.

Blume, J.: On the form of the inverted stress-strain law for isotropic hyperelastic solids. Int. J. Non-Linear Mech. 27(3), 413–421 (1992) MATHMathSciNetADSCrossRef

34.

Böhlke, T., Bertram, A.: On ellipticity of finite isotropic linear elastic laws. Preprint der Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg (1), 1–18 (2002)

35.

Borwein, J.M., Vanderwerff, J.D.: Convex Functions. Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010) MATHCrossRef

36.

Bruhns, O.T.: Die Berücksichtigung einer isotropen Werkstoffverfestigung bei der elastisch-plastischen Blechbiegung mit endlichen Formänderungen. Ing.-Arch. 39(1), 63–72 (1970) CrossRef

37.

Bruhns, O.T.: Elastoplastische Scheibenbiegung bei endlichen Formänderungen. Z. Angew. Math. Mech. 51, T101–T103 (1971)

38.

Bruhns, O.T., Thermann, K.: Elastisch-plastische Biegung eines Plattenstreifens bei endlichen Formänderungen. Ing.-Arch. 38(3), 141–152 (1969) CrossRef

39.

Bruhns, O.T., Xiao, H., Mayers, A.: Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 457, 2207–2226 (2001) MATHADSCrossRef

40.

Bruhns, O.T., Xiao, H., Mayers, A.: Finite bending of a rectangular block of an elastic Hencky material. J. Elast. 66(3), 237–256 (2002) MATHCrossRefMathSciNet

41.

Bruhns, O.T., Xiao, H., Meyers, A.: Hencky’s elasticity model with the logarithmic strain measure: a study on Poynting effect and stress response in torsion of tubes and rods. Arch. Mech. 52(4), 489–509 (2000) MATHMathSciNet

42.

Buliga, M.: Four applications of majorization to convexity in the calculus of variations. Linear Algebra Appl. 429(7), 1528–1545 (2008) MATHMathSciNetCrossRef

43.

Cai, Z., Starke, G.: First-order system least squares for the stress-displacement formulation: linear elasticity. SIAM J. Numer. Anal. 41(2), 715–730 (2003) MATHMathSciNetCrossRef

44.

Cai, Z., Starke, G.: Least-squares methods for linear elasticity. SIAM J. Numer. Anal. 42(2), 826–842 (2004) MATHMathSciNetCrossRef

45.

Carroll, M.M.: Controllable states of stress for compressible elastic solids. J. Elast. 3(1), 57–61 (1973) CrossRefMathSciNet

46.

Carroll, M.M.: Must elastic materials be hyperelastic? Math. Mech. Solids 14(4), 369–376 (2009) MATHCrossRef

47.

Chen, Y.C.: Stability and bifurcation of homogeneous deformations of a compressible elastic body under pressure load. Math. Mech. Solids 1(1), 57–72 (1996) MATHMathSciNet

48.

Ciarlet, P.G.: Three-Dimensional Elasticity, 1st edn. Studies in Mathematics and Its Applications, vol. 1. Elsevier, Amsterdam (1988) MATH

49.

Cleja-Ţigoiu, S.: Yield criteria in anisotropic finite elasto-plasticity. Arch. Mech. 57(2–3), 81–102 (2005) MATHMathSciNet

50.

Cleja-Ţigoiu, S.: Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity. Int. J. Plast. 19(11), 1917–1964 (2003) MATHCrossRef

51.

Criscione, J.C.: Direct tensor expression for natural strain and fast, accurate approximation. Compos. Struct. 80(25), 1895–1905 (2002) MathSciNetCrossRef

52.

Criscione, J.C., Humphrey, J.D., Douglas, A.S., Hunter, W.C.: An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. J. Mech. Phys. Solids 48(12), 2445–2465 (2000) MATHADSCrossRef

53.

Curnier, A., Rakotomanana, L.: Generalized strain and stress measures, critical survey and new results. Eng. Trans. (Warsaw) 39, 461–538 (1991) MathSciNet

54.

Dacorogna, B.: Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete Contin. Dyn. Syst. 1(2), 257–263 (2001) MATHMathSciNetCrossRef

55.

Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Applied Mathematical Sciences, vol. 78. Springer, Berlin (2008) MATH

56.

Dacorogna, B., Koshigoe, H.: On the different notions of convexity for rotationally invariant functions. Ann. Fac. Sci. Toulouse 2, 163–184 (1993) MATHMathSciNetCrossRef

57.

Dacorogna, B., Marcellini, P.: Implicit Partial Differential Equations. Birkhäuser, Boston (1999) MATHCrossRef

58.

Dacorogna, B., Maréchal, P.: A note on spectrally defined polyconvex functions. In: Carozza, M., et al. (eds.) Proceedings of the Workshop “New Developments in the Calculus of Variations”, pp. 27–54. Edizioni Scientifiche Italiane, Napoli (2006)

59.

Davis, C.: All convex invariant functions of Hermitian matrices. Arch. Math. 8, 276–278 (1957) MATHMathSciNetCrossRef

60.

de Boer, R.: Die elastisch-plastische Biegung eines Plattenstreifens aus inkompressiblem Werkstoff bei endlichen Formänderungen. Ing.-Arch. 36(3), 145–154 (1967) CrossRef

61.

de Boer, R., Bruhns, O.T.: Zur Berechnung der Eigenspannungen bei einem durch endliche Biegung verformten inkompressiblen Plattenstreifen. Acta Mech. 8(3–4), 146–159 (1969) CrossRef

62.

dell’Isola, F., Ruta, G.C., Batra, R.C.: A second-order solution of Saint-Venant’s problem for an elastic pretwisted bar using Signorini’s perturbation method. J. Elast. 49(2), 113–127 (1997) MathSciNetCrossRefMATH

63.

dell’Isola, F., Ruta, G.C., Batra, R.C.: Generalized Poynting effects in predeformed prismatic bars. J. Elast. 50(2), 181–196 (1998) MATHMathSciNetCrossRef

64.

Destrade, M., Murphy, J., Saccomandi, G.: Simple shear is not so simple. Int. J. Non-Linear Mech. 47(2), 210–214 (2012) ADSCrossRef

65.

Diani, J., Gilormini, P.: Combining the logarithmic strain and the full-network model for a better understanding of the hyperelastic behavior of rubber-like materials. J. Mech. Phys. Solids 53(11), 2579–2596 (2005) MATHADSCrossRef

66.

Dłużewski, P.: Anisotropic hyperelasticity based upon general strain measures. J. Elast. 60(2), 119–129 (2000) MATHCrossRef

67.

Dłuzewski, P., Jurczak, G., Antúnez, H.: Logarithmic measure of strains in finite element modelling of anisotropic deformations of elastic solids. Comput. Assist. Mech. Eng. Sci. 10, 69–79 (2003) MATH

68.

Dłużewski, P., Traczykowski, P.: Numerical simulation of atomic positions in quantum dot by means of molecular statics. Arch. Mech. 55(5–6), 393–406 (2003) MATH

69.

Dui, G.S.: Some basis-free formulae for the time rate and conjugate stress of logarithmic strain tensor. J. Elast. 83(2), 113–151 (2006) MATHMathSciNetCrossRef

70.

Ebbing, V., Schröder, J., Neff, P.: Approximation of anisotropic elasticity tensors at the reference state with polyconvex energies. Arch. Appl. Mech. 79, 651–657 (2009) MATHADSCrossRef

71.

Edelstein, W., Fosdick, R.: A note on non-uniqueness in linear elasticity theory. Z. Angew. Math. Phys. 19(6), 906–912 (1968) MATHMathSciNetCrossRef

72.

Ernst, E.: Ellipticity loss in isotropic elasticity. J. Elast. 51(3), 203–211 (1998) MATHMathSciNetCrossRef

73.

Fiala, Z.: Time derivative obtained by applying the Riemannian manifold of Riemannian metrics to kinematics of continua. C. R., Méc. 332(2), 97–102 (2004) MATHCrossRef

74.

Fiala, Z.: Geometry of finite deformations, linearization, and incremental deformations under initial stress/strain. In: Proceedings of International Conference Engineering Mechanics 2008 (2008). 20 pp.

75.

Fiala, Z.: Logarithmic strain in 1 versus 3(2) dimensions. In: Proceedings of International Conference Engineering Mechanics 2010 (2010) 11 pp.

76.

Fiala, Z.: Geometrical setting of solid mechanics. Ann. Phys. 326(8), 1983–1997 (2011) MATHMathSciNetADSCrossRef

77.

Fitzgerald, J.E.: A tensorial Hencky measure of strain and strain rate for finite deformations. J. Appl. Phys. 51, 5111–5115 (1980) ADSCrossRef

78.

Flory, P.J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961) MathSciNetCrossRef

79.

Fosdick, R., Piccioni, M.D., Puglisi, G.: A note on uniqueness in linear elastostatics. J. Elast. 88(1), 79–86 (2007) MATHMathSciNetCrossRef

80.

Fosdick, R., Šilhavý, M.: Generalized Baker-Ericksen inequalities. J. Elast. 85(1), 39–44 (2006) MATHCrossRefMathSciNet

81.

Fosdick, R., Volkmann, E.: Normality and convexity of the yield surface in nonlinear plasticity. Q. Appl. Math. 51, 117–127 (1993) MATHMathSciNet

82.

Freed, A.D.: Natural strain. J. Eng. Mater. Technol. 117(4), 379–385 (1995) CrossRef

83.

Freed, A.D.: Hencky strain and logarithmic rates in Lagrangian analysis. Int. J. Eng. Sci. 81, 135–145 (2014) MathSciNetCrossRef

84.

Fu, Y.B., Ogden, R.W.: Nonlinear Elasticity: Theory and Applications, vol. 281. Cambridge University Press, Cambridge (2001) CrossRefMATH

85.

Fung, Y.C.: Inversion of a class of nonlinear stress-strain relationships of biological soft tissues. J. Biomech. Eng. 101(1), 23–27 (1979) CrossRef

86.

Gao, X.L.: Finite deformation elasto-plastic solution for the pure bending problem of a wide plate of elastic linear-hardening material. Int. J. Solids Struct. 31(10), 1357–1376 (1994) MATHCrossRef

87.

Gao, X.L.: Finite deformation continuum model for single-walled carbon nanotubes. Int. J. Solids Struct. 40(26), 7329–7337 (2003) MATHCrossRef

88.

Gao, X.L., Atluri, S.N.: An exact finite deformation elasto-plastic solution for the outside-in free eversion problem of a tube of elastic linear-hardening material. IMA J. Appl. Math. 58(3), 259–275 (1997) MATHMathSciNetADSCrossRef

89.

Gearing, B.P., Anand, L.: Notch-sensitive fracture of polycarbonate. Int. J. Solids Struct. 41(3), 827–845 (2004) MATHCrossRef

90.

Germain, S., Scherer, M., Steinmann, P.: On inverse form finding for anisotropic hyperelasticity in logarithmic strain space. Int. J. Struct. Chang. Solids - Mech. Appl. 2(2), 1–16 (2010)

91.

Ghiba, I.D., Müller, B., Neff, P., Starke, G.: On the stress-strain invertibility in nonlinear elasticity (2015, in preparation)

92.

Glüge, R., Kalisch, J.: Graphical representations of the regions of rank-one-convexity of some strain energies. Tech. Mech. 32, 227–237 (2012)

93.

Gurtin, M.E., Spear, K.: On the relationship between the logarithmic strain rate and the stretching tensor. Int. J. Solids Struct. 19(5), 437–444 (1983) MATHMathSciNetCrossRef

94.

Hanin, M., Reiner, M.: On isotropic tensor-functions and the measure of deformation. Z. Angew. Math. Phys. 7(5), 377–393 (1956) MATHMathSciNetCrossRef

95.

Hartig, E.: Der Elastizitätsmodul des geraden Stabes als Funktion der spezifischen Beanspruchung. Z. Civilingenieur XXXIX, 113–138 (1893). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hartig_elastizitaetsmodul.pdf

96.

Hartmann, S., Neff, P.: Polyconvexity of generalized polynomial type hyperelastic strain energy functions for near incompressibility. Int. J. Solids Struct. 40(11), 2767–2791 (2003) MATHMathSciNetCrossRef

97.

Henann, D.L., Anand, L.: Fracture of metallic glasses at notches: effects of notch-root radius and the ratio of the elastic shear modulus to the bulk modulus on toughness. Acta Mater. 57(20), 6057–6074 (2009) CrossRef

98.

Henann, D.L., Anand, L.: A large strain isotropic elasticity model based on molecular dynamics simulations of a metallic glass. J. Elast. 104(1–2), 281–302 (2011) MATHMathSciNetCrossRef

99.

Hencky, H.: Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Z. Angew. Math. Mech. 4(4), 323–334 (1924) CrossRefMATH

100.

Hencky, H.: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Tech. Phys. 9, 215–220 (1928). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1928.pdf (see also the technical translation NASA TT-21602) MATH

101.

Hencky, H.: Das Superpositionsgesetz eines endlich deformierten relaxationsfähigen elastischen Kontinuums und seine Bedeutung für eine exakte Ableitung der Gleichungen für die zähe Flüssigkeit in der Eulerschen Form. Ann. Phys. 2, 617–630 (1929). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky_superposition1929.pdf MATHCrossRef

102.

Hencky, H.: Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern? Z. Phys. 55, 145–155 (1929). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky1929.pdf MATHADSCrossRef

103.

Hencky, H.: The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. J. Rheol. 2, 169–176 (1931). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/henckyjrheology31.pdf ADSCrossRef

104.

Hencky, H.: The elastic behavior of vulcanized rubber. Rubber Chem. Technol. 6(2), 217–224 (1933). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hencky_vulcanized_rubber.pdf CrossRef

105.

Hill, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc., Math. Phys. Eng. Sci. 193(1033), 281–297 (1948) MATHADSCrossRefMathSciNet

106.

Hill, R.: The Mathematical Theory of Plasticity. Clarendon, Oxford (1950) MATH

107.

Hill, R.: On constitutive inequalities for simple materials—I. J. Mech. Phys. Solids 16(4), 229–242 (1968) MATHADSCrossRef

108.

Hill, R.: Constitutive inequalities for isotropic elastic solids under finite strain. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 314(1519), 457–472 (1970) MATHADSCrossRef

109.

Hill, R.: Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1–75 (1978) MATHCrossRefMathSciNet

110.

Hill, R.: On the theory of plane strain in finitely deformed compressible materials. Math. Proc. Camb. Philos. Soc. 86, 161–178 (1979) MATHCrossRefMathSciNet

111.

Hocine, B., Chevalier, L., Idjeri, M.: A constitutive model for isotropic rubber-like materials based on the logarithmic strain invariants approach (2014, submitted)

112.

Hoger, A.: The material time derivative of logarithmic strain. Int. J. Solids Struct. 22(9), 1019–1032 (1986) MATHMathSciNetCrossRef

113.

Hoger, A.: The stress conjugate to logarithmic strain. Int. J. Solids Struct. 23(12), 1645–1656 (1987) MATHMathSciNetCrossRef

114.

Horgan, C., Murphy, J.: Constitutive modeling for moderate deformations of slightly compressible rubber. J. Rheol. 53, 153 (2009) ADSCrossRef

115.

Horgan, C., Murphy, J.: On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers. Int. J. Solids Struct. 46(16), 3078–3085 (2009) MATHCrossRef

116.

Hutchinson, J.W., Neale, K.W.: Finite strain J ₂-deformation theory. In: Carlson, D.E., Shield, R.T. (eds.) Proceedings of the IUTAM Symposium on Finite Elasticity, pp. 237–247. Nijhoff, Dordrecht (1982). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hutchinson_ellipticity80.pdf

117.

Imbert, A.: Recherches théoriques et expérimentales sur l’élasticité du Caoutchouc. University-de Lyon, Ph.D.-thesis, 1880, https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/imbert_rubber.pdf

118.

Jog, C.S.: Foundations and Applications of Mechanics: Continuum Mechanics, vol. 1. CRC Press, Boca Raton (2002)

119.

Jog, C.S.: On the explicit determination of the polar decomposition in n-dimensional vector spaces. J. Elast. 66(2), 159–169 (2002) MATHMathSciNetCrossRef

120.

Jog, C.S.: The explicit determination of the logarithm of a tensor and its derivatives. J. Elast. 93(2), 141–148 (2008) MATHMathSciNetCrossRef

121.

Jog, C.S., Patil, K.D.: Conditions for the onset of elastic and material instabilities in hyperelastic materials. Arch. Appl. Mech. 83, 1–24 (2013) CrossRefMATH

122.

Johnson, B., Hoger, A.: The dependence of the elasticity tensor on residual stress. J. Elast. 33(2), 145–165 (1993) MATHMathSciNetCrossRef

123.

Jones, D.F., Treloar, L.R.G.: The properties of rubber in pure homogeneous strain. J. Phys. D, Appl. Phys. 8(11), 1285 (1975) ADSCrossRef

124.

Kakavas, P.A.: Prediction of the nonlinear Poisson function using large volumetric strains estimated from a finite hyperelastic material law. Polym. Eng. Sci. 40(6), 1330–1333 (2000) CrossRef

125.

Kearsley, E.A.: Asymmetric stretching of a symmetrically loaded elastic sheet. Int. J. Solids Struct. 22(2), 111–119 (1986) CrossRef

126.

Kijowski, J., Magli, G.: Relativistic elastomechanics as a Lagrangian field theory. J. Geom. Phys. 9(3), 207–223 (1992) MATHMathSciNetADSCrossRef

127.

Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast. 5(3–4), 341–361 (1975) MATHMathSciNetCrossRef

128.

Knowles, J.K., Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal. 63(4), 321–336 (1976) MathSciNetCrossRefMATH

129.

Knowles, J.K., Sternberg, E.: On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elast. 8(4), 329–379 (1978) MATHMathSciNetCrossRef

130.

Kochkin, A.P.: Stress-strain dependence in the nonlinear theory of elasticity. Indian J. Pure Appl. Math. 17(4), 564–579 (1986) MATHMathSciNet

131.

Krawietz, A.: A comprehensive constitutive inequality in finite elastic strain. Arch. Ration. Mech. Anal. 58, 127–149 (1975) MATHMathSciNetCrossRef

132.

Lankeit, J., Neff, P., Nakatsukasa, Y.: The minimization of matrix logarithms: on a fundamental property of the unitary polar factor. Linear Algebra Appl. 449(0), 28–42 (2014) MATHMathSciNetCrossRef

133.

Lehmann, Th., Liang, H.: The stress conjugate to logarithmic strain logV. Z. Angew. Math. Mech. 73(12), 357–363 (1993) MATHMathSciNetCrossRef

134.

Lehmich, S., Neff, P., Lankeit, J.: On the convexity of the function C→f(detC) on positive definite matrices. Math. Mech. Solids 19, 369–375 (2014) MATHMathSciNetCrossRef

135.

Lewis, A.S.: Convex analysis on the Hermitian matrices. SIAM J. Optim. 6(1), 164–177 (1996) MATHMathSciNetCrossRef

136.

Lewis, A.S.: The mathematics of eigenvalue optimization. Math. Program., Ser. B 97(1–2), 155–176 (2003) MATHMathSciNet

137.

Lewis, A.S., Overton, M.L.: Eigenvalue optimization. Acta Numer. 5, 149–190 (1996) MathSciNetADSCrossRefMATH

138.

Ludwik, P.: Elemente der technologischen Mechanik. Springer, Berlin (1909). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/ludwik.pdf MATHCrossRef

139.

Lurie, A.I.: Non-linear Theory of Elasticity. Elsevier, Amsterdam (1990) MATH

140.

Marsden, J.E., Hughes, J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983) MATH

141.

Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics. Springer, Berlin (2011) CrossRefMATH

142.

Martin, R., Neff, P.: Some remarks on monotonicity of primary matrix functions on the set of symmetric matrices. Arch. Appl. Mech. (2015, accepted)

143.

Marzano, S.: An interpretation of Baker-Ericksen inequalities in uniaxial deformation and stress. Meccanica 18(4), 233–235 (1983) MATHMathSciNetCrossRef

144.

Merodio, J., Neff, P.: A note on tensile instabilities and loss of ellipticity for a fiber-reinforced nonlinearly elastic solid. Arch. Mech. 58(3), 293–303 (2006) MATHMathSciNet

145.

Merrill, G.: Biographical memoir George Ferdinand Becker. Mem. Nat. Acad. Sci., XXI (1927) (see http://en.wikipedia.org/wiki/George_Ferdinand_Becker)

146.

Meyers, A.: On the consistency of some Eulerian strain rates. Z. Angew. Math. Mech. 79(3), 171–177 (1999) MATHMathSciNetCrossRef

147.

Meyers, A., Xiao, H., Bruhns, O.T.: Choice of objective rate in single parameter hypoelastic deformation cycles. Compos. Struct. 84(17), 1134–1140 (2006) CrossRef

148.

Miehe, C., Apel, N., Lambrecht, M.: Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials. Comput. Methods Appl. Mech. Eng. 191(47), 5383–5425 (2002) MATHMathSciNetADSCrossRef

149.

Miehe, C., Méndez Diez, J., Göktepe, S., Schänzel, L.M.: Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory. Int. J. Solids Struct. 48(13), 1799–1817 (2011) CrossRef

150.

Miehe, C., Göktepe, S., Méndez Diez, J.: Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space. Int. J. Solids Struct. 46, 181–202 (2009) MATHCrossRef

151.

Mielke, A.: Finite elastoplasticity, Lie groups and geodesics on SL(d). In: Newton, P. (ed.) Geometry, Mechanics and Dynamics. Volume in Honour of the 60th Birthday of J.E. Marsden, pp. 61–90. Springer, Berlin (2002)

152.

Mielke, A.: Necessary and sufficient conditions for polyconvexity of isotropic functions. J. Convex Anal. 12(2), 291–314 (2005) MATHMathSciNet

153.

Mihai, A., Goriely, A.: Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials. Proc. R. Soc. Lond. 467(2136), 3633–3646 (2011) MATHMathSciNetADSCrossRef

154.

Mihai, A., Goriely, A.: Numerical simulation of shear and the Poynting effects by the finite element method: an application of the generalised empirical inequalities in non-linear elasticity. Int. J. Non-Linear Mech. 49, 1–14 (2013) CrossRef

155.

Moakher, M.: Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24(1), 1–16 (2002) MATHMathSciNetCrossRef

156.

Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26(3), 735–747 (2005) MATHMathSciNetCrossRef

157.

Moon, H., Truesdell, C.: Interpretation of adscititious inequalities through the effects pure shear stress produces upon an isotropic elastic solid. Arch. Ration. Mech. Anal. 55(1), 1–17 (1974) MATHMathSciNetCrossRef

158.

Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11(9), 582–592 (1940) MATHADSCrossRef

159.

Mott, P.H., Roland, C.M.: Limits to Poisson’s ratio in isotropic materials. Phys. Rev. B 80(13), 132104 (2009), 4 pp. ADSCrossRef

160.

Müller, Ch., Bruhns, O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22(9), 1658–1682 (2006) MATHCrossRef

161.

Murnaghan, F.D.: The compressibility of solids under extreme pressures. In: Theodore von Karman Anniversary Volume, Calif. Inst. Techn., Pasadena, pp. 121–136. Univ. of Berkeley Press, Berkeley (1941). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/murnaghan.pdf

162.

Murnaghan, F.D.: The compressibility of media under extreme pressures. Proc. Natl. Acad. Sci. USA 30(9), 244 (1944) MATHMathSciNetADSCrossRef

163.

Murphy, J.: Linear isotropic relations in finite hyperelasticity: some general results. J. Elast. 86(2), 139–154 (2007) MATHCrossRefMathSciNet

164.

Nadai, A.: Plastic behavior of metals in the strain-hardening range. Part I. J. Appl. Phys. 8(3), 205–213 (1937) MATHADSCrossRef

165.

Neff, P.: Mathematische Analyse multiplikativer Viskoplastizität. Ph.D. thesis, Technische Universität Darmstadt. Shaker Verlag, Aachen, 2000. ISBN:3-8265-7560-1, https://www.uni-due.de/~hm0014/Download_files/cism_convexity08.pdf

166.

Neff, P.: Some results concerning the mathematical treatment of finite plasticity. In: Deformation and Failure in Metallic Materials, pp. 251–274. Springer, Berlin (2003) CrossRef

167.

Neff, P.: Critique of “Two-dimensional examples of rank-one convex functions that are not quasiconvex” by M.K. Benaouda and J.J. Telega. Ann. Pol. Math. 86(2), 193–195 (2005) MATHMathSciNetCrossRef

168.

Neff, P.: A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci. 44, 574–594 (2006) MATHMathSciNetCrossRef

169.

Neff, P., Eidel, B., Martin, R.: The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky (a new translation of Hencky’s original German articles) (2014). arXiv:1402.4027

170.

Neff, P., Eidel, B., Martin, R.: Geometry of logarithmic strain measures in solid mechanics. The Hencky energy is the squared geodesic distance of the deformation gradient to SO(n) in any left-invariant, right-O(n)-invariant Riemannian metric on GL(n) (2015, in preparation)

171.

Neff, P., Eidel, B., Osterbrink, F., Martin, R.: The Hencky strain energy ∥logU∥² measures the geodesic distance of the deformation gradient to SO(3) in the canonical left-invariant Riemannian metric on GL(3). PAMM 13(1), 369–370 (2013) CrossRef

172.

Neff, P., Eidel, B., Osterbrink, F., Martin, R.: A Riemannian approach to strain measures in nonlinear elasticity. C. R. Acad. Sci. 342, 254–257 (2014)

173.

Neff, P., Ghiba, I.D.: The exponentiated Hencky-logarithmic strain energy. Part III: Coupling with idealized isotropic finite strain plasticity. Contin. Mech. Thermodyn., the special issue in honour of D.J. Steigmann (2015, to appear). arXiv:1409.7555

174.

Neff, P., Jeong, J., Fischle, A.: Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mech. 211(3–4), 237–249 (2010) MATHCrossRef

175.

Neff, P., Lankeit, J., Madeo, A.: On Grioli’s minimum property and its relation to Cauchy’s polar decomposition. Int. J. Eng. Sci. 80, 207–217 (2014) MathSciNetCrossRef

176.

Neff, P., Münch, I., Martin, R.: Rediscovering G.F. Becker’s early axiomatic deduction of a multiaxial nonlinear stress-strain relation based on logarithmic strain. Math. Mech. Solids (2014, to appear). arXiv:1403.4675v2, doi:10.1177/1081286514542296

177.

Neff, P., Nakatsukasa, Y., Fischle, A.: A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm. SIAM J. Matrix Anal. Appl. 35, 1132–1154 (2014) MATHMathSciNetCrossRef

178.

Norris, A.: Eulerian conjugate stress and strain. J. Mech. Mater. Struct. 3(2), 243–260 (2008) MathSciNetCrossRef

179.

Ogden, R.W.: Compressible isotropic elastic solids under finite strain-constitutive inequalities. Q. J. Mech. Appl. Math. 23(4), 457–468 (1970) MATHCrossRef

180.

Ogden, R.W.: On constitutive relations for elastic and plastic materials. Ph.D. Thesis, Cambridge University (1970)

181.

Ogden, R.W.: Inequalities associated with the inversion of elastic stress-deformation relations and their implications. Math. Proc. Camb. Philos. Soc. 81, 313–324 (1977) MATHMathSciNetCrossRef

182.

Ogden, R.W.: Non-linear Elastic Deformations, 1st edn. Mathematics and Its Applications. Ellis Horwood, Chichester (1983)

183.

Ogden, R.W., Saccomandi, G., Sgura, I.: Fitting hyperelastic models to experimental data. Comput. Mech. 34(6), 484–502 (2004) MATHCrossRef

184.

Onaka, S.: Equivalent strain in simple shear deformation described by using the Hencky strain. Philos. Mag. Lett. 90(9), 633–639 (2010) ADSCrossRef

185.

Onaka, S.: Appropriateness of the Hencky equivalent strain as the quantity to represent the degree of severe plastic deformation. Mater. Trans. 53(8), 1547–1548 (2012) CrossRef

186.

Ortiz, M., Radovitzky, R.A., Repetto, E.A.: The computation of the exponential and logarithmic mappings and their first and second linearizations. Int. J. Numer. Methods Eng. 52(12), 1431–1441 (2001) MATHMathSciNetCrossRef

187.

Panov, A.D., Shumaev, V.V.: Using the logarithmic strain measure for solving torsion problems. Mech. Solids 47(1), 71–78 (2012) CrossRef

188.

Pennec, X.: Left-invariant Riemannian elasticity: a distance on shape diffeomorphisms? In: 1st MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical, Statistical and Registration Methods for Modeling Biological Shape Variability, pp. 1–13 (2006)

189.

Pennec, X., Stefanescu, R., Arsigny, V., Fillard, P., Ayache, N.: Riemannian elasticity: a statistical regularization framework for non-linear registration. In: Medical Image Computing and Computer-Assisted Intervention----MICCAI 2005, pp. 943–950. Springer, Berlin (2005) CrossRef

190.

Perić, D., Owen, D.R.J., Honnor, M.E.: A model for finite strain elasto-plasticity based on logarithmic strains: computational issues. Comput. Methods Appl. Mech. Eng. 94(1), 35–61 (1992) MATHADSCrossRef

191.

Plesěk, J., Kruisová, A.: Formulation, validation and numerical procedures for Hencky’s elasticity model. Comput. Struct. 84, 1141–1150 (2006) CrossRef

192.

Poirier, J.-P., Tarantola, A.: A logarithmic equation of state. Phys. Earth Planet. Inter. 109, 1–8 (1998) ADSCrossRef

193.

Poisson, S.D.: Traité de Mécanique, vol. 2 (1811)

194.

Raoult, A.: Non-polyconvexity of the stored energy function of a St.Venant-Kirchhoff material. Apl. Mat. 6, 417–419 (1986) MathSciNetMATH

195.

Reinhardt, W.D., Dubey, R.N.: Application of objective rates in mechanical modeling of solids. J. Appl. Mech. 63(3), 692–698 (1996) MATHMathSciNetADSCrossRef

196.

Richter, H.: Das isotrope Elastizitätsgesetz. Z. Angew. Math. Mech. 28(7–8), 205–209 (1948). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/richter_isotrop_log.pdf MATHMathSciNetCrossRef

197.

Richter, H.: Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Formänderungen. Z. Angew. Math. Mech. 29(3), 65–75 (1949). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/richter_deviator_log.pdf MATHMathSciNetCrossRef

198.

Richter, H.: Zum Logarithmus einer Matrix. Arch. Math. 2, 360–363 (1950). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/richter_log.pdf MATHCrossRefMathSciNet

199.

Richter, H.: Zur Elastizitätstheorie endlicher Verformungen. Math. Nachr. 8, 65–73 (1952). https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/richter_endliche_verzerrungen52.pdf MATHMathSciNetCrossRef

200.

Rivlin, R.: Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 241(835), 379–397 (1948) MATHMathSciNetADSCrossRef

201.

Rivlin, R.: Restrictions on the strain-energy function for an elastic material. Math. Mech. Solids 9(2), 131–139 (2004) MATHMathSciNetCrossRef

202.

Rivlin, R.S.: Some restrictions on constitutive equations. In: Domingos, J.J., Nina, M.N.R., Whitelaw, J.H. (eds.) Proc. Int. Symp. on the Foundations of Continuum Thermodynamics, pp. 229–258. Macmillan, London (1974)

203.

Rolph, W.D., Bathe, K.J.: On a large strain finite element formulation for elasto-plastic analysis. In: Constitutive Equations: Macro and Computational Aspects, Winter Annual Meeting, pp. 131–147. ASME, New York (1984)

204.

Rosakis, P.: Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109(1), 1–37 (1990) MATHMathSciNetCrossRef

205.

Rosakis, P.: Characterization of convex isotropic functions. J. Elast. 49, 257–267 (1998) MATHMathSciNetCrossRef

206.

Rougée, P.: Mécanique des Grandes Transformations. Mathématiques et Applications, vol. 25. Springer, Berlin (1997) MATH

207.

Rougée, P.: An intrinsic Lagrangian statement of constitutive laws in large strain. Compos. Struct. 84(17), 1125–1133 (2006) CrossRefMathSciNet

208.

Sansour, C.: On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues. Int. J. Solids Struct. 38(50), 9221–9232 (2001) MATHMathSciNetCrossRef

209.

Sansour, C.: On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. Eur. J. Mech. A, Solids 27(1), 28–39 (2008) MATHMathSciNetADSCrossRef

210.

Sansour, C., Kollmann, F.G.: On theory and numerics of large viscoplastic deformation. Comput. Methods Appl. Mech. Eng. 146(3), 351–369 (1997) MATHADSCrossRef

211.

Sawyers, K.N., Rivlin, R.: Instability of an elastic material. Int. J. Solids Struct. 9(5), 607–613 (1973) MATHMathSciNetCrossRef

212.

Sawyers, K.N., Rivlin, R.: A note on the Hadamard criterion for an incompressible isotropic elastic material. Mech. Res. Commun. 5, 211–224 (1978) MATHMathSciNetCrossRef

213.

Sawyers, K.N., Rivlin, R.: On the speed of propagation of waves in a deformed compressible elastic material. Z. Angew. Math. Phys. 29, 245–251 (1978) MATHMathSciNetCrossRef

214.

Scheidler, M.: Time rates of generalized strain tensors Part I: Component formulas. Mech. Mater. 11(3), 199–210 (1991) CrossRef

215.

Scheidler, M.: Time rates of generalized strain tensors, Part II: Approximate basis-free formulas. Mech. Mater. 11(3), 211–219 (1991) CrossRef

216.

Schröder, J., Neff, P.: Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct. 40(2), 401–445 (2003) MATHCrossRefMathSciNet

217.

Schröder, J., Neff, P.: Poly, quasi and rank-one convexity in mechanics. In: CISM-Course Udine. Springer, Berlin (2009)

218.

Schröder, J., Neff, P.: On the construction of polyconvex transversely isotropic free energy functions. In: IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, 21 August 2001 (2001). Organizer: C. Miehe, University Stuttgart

219.

Schröder, J., Neff, P., Balzani, D.: A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids Struct. 42(15), 4352–4371 (2005) MATHCrossRefMathSciNet

220.

Schröder, J., Neff, P., Ebbing, V.: Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J. Mech. Phys. Solids 56(12), 3486–3506 (2008) MATHMathSciNetADSCrossRef

221.

Schwarz, A., Schröder, J., Starke, G.: A modified least-squares mixed finite element with improved momentum balance. Int. J. Numer. Methods Eng. 81(3), 286–306 (2010) MATHMathSciNet

222.

Sendova, T., Walton, J.R.: On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain. Int. J. Non-Linear Mech. 40(2–3), 195–212 (2005) MATHMathSciNetADSCrossRef

223.

Seth, B.R.: Generalized strain measure with applications to physical problems. In: Rainer, M.A. (ed.) Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, pp. 162–172. Pergamon, Oxford (1961)

224.

Shrivastava, S., Ghosh, C., Jonas, J.: A comparison of the von Mises and Hencky equivalent strains for use in simple shear experiments. Philos. Mag. Lett. 92(7), 779–786 (2012) ADSCrossRef

225.

Sidoroff, R.: Sur les restrictions à imposer à l’énergie de déformation d’un matériau hyperélastique. C. R. Acad. Sci. Paris 279, 379–382 (1974) MATHMathSciNet

226.

Šilhavý, M.: The Mechanics and Thermomechanics of Continuous Media. Springer, Berlin (1997) CrossRefMATH

227.

Šilhavý, M.: Convexity conditions for rotationally invariant functions in two dimensions. In: Sequeira, et al. (eds.) Applied Nonlinear Analysis. Kluwer Academic, New York (1999)

228.

Šilhavý, M.: On isotropic rank one convex functions. Proc. R. Soc. Edinb. 129, 1081–1105 (1999) MATHCrossRef

229.

Šilhavý, M.: Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Math. Bohem. 126(2), 521–529 (2001) MATHMathSciNet

230.

Šilhavý, M.: Convexity Conditions for Rotationally Invariant Functions in Two Dimensions. Springer, Berlin (2002) CrossRefMATH

231.

Šilhavý, M.: Monotonicity of rotationally invariant convex and rank 1 convex functions. Proc. R. Soc. Edinb. 132, 419–435 (2002) MATHCrossRefMathSciNet

232.

Šilhavý, M.: An O(n) invariant rank 1 convex function that is not polyconvex. Theor. Appl. Mech. 28, 325–336 (2002) CrossRefMathSciNetMATH

233.

Šilhavý, M.: On SO(n)-invariant rank 1 convex functions. J. Elast. 71, 235–246 (2003) MATHCrossRefMathSciNet

234.

Silling, S.A.: Numerical studies of loss of ellipticity near singularities in an elastic material. J. Elast. 19(3), 213–239 (1988) MATHMathSciNetCrossRef

235.

Simo, J.C.: Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput. Methods Appl. Mech. Eng. 99(1), 61–112 (1992) MATHMathSciNetADSCrossRef

236.

Simpson, H., Spector, S.: On copositive matrices and strong ellipticity for isotropic elastic materials. Arch. Ration. Mech. Anal. 84(1), 55–68 (1983) MATHMathSciNetCrossRef

237.

Simpson, H., Spector, S.: On bifurcation in finite elasticity: buckling of a rectangular rod. J. Elast. 92(3), 277–326 (2008) MATHMathSciNetCrossRef

238.

Skrzypek, J., Wróblewski, A.: Application of logarithmic strains to changing principal directions via progressing transformations. J. Struct. Mech. 13(3–4), 283–299 (1985) CrossRef

239.

Smith, C.W., Wootton, R.J., Evans, K.E.: Interpretation of experimental data for Poisson’s ratio of highly nonlinear materials. Exp. Mech. 39(4), 356–362 (1999) CrossRef

240.

Starke, G.: An adaptive least-squares mixed finite element method for elasto-plasticity. SIAM J. Numer. Anal. 45(1), 371–388 (2007) MATHMathSciNetCrossRef

241.

Starke, G., Schwarz, A., Schröder, J.: Analysis of a modified first-order system least squares method for linear elasticity with improved momentum balance. SIAM J. Numer. Anal. 49(3), 1006–1022 (2011) MATHMathSciNetCrossRef

242.

Tabor, D.: The bulk modulus of rubber. Polymer 35(13), 2759–2763 (1994) CrossRef

243.

Tanner, R., Tanner, E.: Heinrich Hencky: a rheological pioneer. Rheol. Acta 42(1–2), 93–101 (2003) CrossRef

244.

Tarantola, A.: Elements for Physics: Quantities, Qualities, and Intrinsic Theories. Springer, Heidelberg (2006) CrossRef

245.

Tarantola, A.: Stress and strain in symmetric and asymmetric elasticity (2009). Preprint arXiv:0907.1833

246.

https://www.uni-due.de/mathematik/ag_neff/neff_hencky/treloarjonesdata.zip

247.

Treloar, L.R.G.: Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70 (1944) CrossRef

248.

Treloar, L.R.G.: The elasticity and related properties of rubbers. Rep. Prog. Phys. 36(7), 755 (1973) ADSCrossRef

249.

Treloar, L.R.G.: The Physics of Rubber Elasticity. Oxford University Press, London (1975) MATH

250.

Truesdell, C.: Das ungelöste Hauptproblem der endlichen Elastizitätstheorie. Z. Angew. Math. Mech. 36(3–4), 97–103 (1956) MATHMathSciNetCrossRef

251.

Truesdell, C.: The Mechanical Foundations of Elasticity and Fluid Dynamics, vol. 8. Gordon & Breach, New York (1966) MATH

252.

Truesdell, C., Moon, H.: Inequalities sufficient to ensure semi-invertibility of isotropic functions. J. Elast. 5(34), 183–189 (1975) MATHMathSciNet

253.

Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3. Springer, Heidelberg (1965)

254.

Truesdell, C., Toupin, R.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/1. Springer, Heidelberg (1960)

255.

Truesdell, C., Toupin, R.: Static grounds for inequalities in finite strain of elastic materials. Arch. Ration. Mech. Anal. 12(1), 1–33 (1963) MATHMathSciNetCrossRef

256.

Vallée, C.: Lois de comportement élastique isotropes en grandes déformations. Int. J. Eng. Sci. 16(7), 451–457 (1978) MATHCrossRef

257.

Vallée, C., Fortuné, D., Lerintiu, C.: On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity. C. R., Méc. 336(11), 851–855 (2008) MATHCrossRef

258.

Wang, Y., Aron, M.: A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J. Elast. 44(1), 89–96 (1996) MATHMathSciNetCrossRef

259.

Weber, G., Anand, L.: Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comput. Methods Appl. Mech. Eng. 79(2), 173–202 (1990) MATHADSCrossRef

260.

Wilber, J.P., Criscione, J.C.: The Baker-Ericksen inequalities for hyperelastic models using a novel set of invariants of Hencky strain. Int. J. Solids Struct. 42(5–6), 1547–1559 (2005) MATHMathSciNetCrossRef

261.

Xiao, H.: Hencky strain and Hencky model: extending history and ongoing tradition. Multidiscipline Mod. Mat. Struct. 1(1), 1–52 (2005) CrossRef

262.

Xiao, H., Bruhns, O.T., Meyers, A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech. 124(1–4), 89–105 (1997) MATHMathSciNetCrossRef

263.

Xiao, H., Bruhns, O.T., Meyers, A.: Existence and uniqueness of the integrable-exactly hypoelastic equation ${\tau}^{\circ}=\lambda(\operatorname {tr}D){\mbox{I}}+ 2\mu{D}$ and its significance to finite inelasticity. Acta Mech. 138(1–2), 31–50 (1999) MATHCrossRef

264.

Xiao, H., Bruhns, O.T., Meyers, A.: Elastoplasticity beyond small deformations. Acta Mech. 182(1–2), 31–111 (2006) MATHCrossRef

265.

Xiao, H., Chen, L.S.: Hencky’s elasticity model and linear stress-strain relations in isotropic finite hyperelasticity. Acta Mech. 157(1–4), 51–60 (2002) MATHCrossRef

266.

Zhang, Y., Li, H., Xiao, H.: Further study of rubber-like elasticity: elastic potentials matching biaxial data. Appl. Math. Mech. 35, 13–24 (2014) MathSciNetCrossRef

267.

Zhilin, P.A., Altenbach, H., Ivanova, E.A., Krivtsov, A.: Material strain tensor. In: Altenbach, H., Forest, S., Krivtsov, A. (eds.) Generalized Continua as Models for Materials, pp. 321–331. Springer, Berlin (2013) CrossRef

268.

Zimmermann, J., Stommel, M.: The mechanical behavior of rubber under hydrostatic compression and the effect on the results of finite element analyses. Arch. Appl. Mech. 83(2), 293–302 (2013) MATHADSCrossRef

269.

Zubov, L.M., Rudev, A.N.: Necessary and sufficient criteria for ellipticity of the equilibrium equations of a non-linearly elastic medium. J. Appl. Math. Mech. 59(2), 197–208 (1995) MATHMathSciNetCrossRef

270.

Zubov, L.M., Rudev, A.N.: A criterion for the strong ellipticity of the equilibrium equations of an isotropic nonlinearly elastic material. J. Appl. Math. Mech. 75, 432–446 (2011) MATHMathSciNetCrossRef

Titel: The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity
verfasst von: Patrizio Neff
Ionel-Dumitrel Ghiba
Johannes Lankeit
Publikationsdatum: 01.12.2015
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 2/2015
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-015-9524-7

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 2/2015

Mathematical Modeling of Weak and Strong Piezoelectric Interfaces

An Improved Proof of Instability of Some Euler Elasticas

On the Asymptotic Derivation of Winkler-Type Energies from 3D Elasticity

Time Dependent Void Growth in Elastomers Undergoing Chemo-mechanical Evolution

Premium Partner

Marktübersichten