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- W2949506844 abstract "We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$ Fmapsto W_{_{rm eH}}(F):=widehat{W}_{_{rm eH}}(U):=left{begin{array}{lll} frac{mu}{k},e^{k,|{rm dev}_nlog {U}|^2}+frac{kappa}{2, {widehat{k}}},e^{widehat{k},[{ rm tr}(log U)]^2}&text{if}& { rm det} F>0, +infty &text{if} &{ rm det} Fleq 0, end{array}right.quad $$ based on the Hencky-logarithmic (true, natural) strain tensor $log U$, where $mu>0$ is the infinitesimal shear modulus, $kappa=frac{2mu+3lambda}{3}>0$ is the infinitesimal bulk modulus with $lambda$ the first Lame constant, $k,widehat{k}$ are dimensionless parameters, $F=nabla varphi$ is the gradient of deformation, $U=sqrt{F^T F}$ is the right stretch tensor and ${rm dev}_nlog {U} =log {U}-frac{1}{n} {rm tr}(log {U})cdot 1!!1$ is the deviatoric part of the strain tensor $log U$. For small elastic strains, $W_{_{rm eH}}$ approximates the classical quadratic Hencky strain energy $$ Fmapsto W_{_{rm H}}(F):=widehat{W}_{_{rm H}}(U):={mu},|{rm dev}_nlog U|^2+frac{kappa}{2},[{rm tr}(log U)]^2, $$ 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 $W_{_{rm eH}}$, for $kgeq frac{1}{4}$ 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 $W_{_{rm eH}}$ is not preserved in dimension $n=3$." @default.
- W2949506844 created "2019-06-27" @default.
- W2949506844 creator A5046930021 @default.
- W2949506844 creator A5071892237 @default.
- W2949506844 creator A5077215828 @default.
- W2949506844 date "2014-03-15" @default.
- W2949506844 modified "2023-10-18" @default.
- W2949506844 title "The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity" @default.
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