The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity [article]

Patrizio Neff and Ionel-Dumitrel Ghiba and Johannes Lankeit
2014 arXiv   pre-print
We investigate a family of isotropic volumetric-isochoric decoupled strain energies F W__ eH(F):=W__ eH(U):={< a r r a y >. based on the Hencky-logarithmic (true, natural) strain tensor U, where μ>0 is the infinitesimal shear modulus, κ=2μ+3λ/3>0 is the infinitesimal bulk modulus with λ the first Lamé constant, k,k are dimensionless parameters, F=∇φ is the gradient of deformation, U=√(F^T F) is the right stretch tensor and dev_nU =U-1/n tr(U)· 11 is the deviatoric part of the strain tensor U.
more » ... r small elastic strains, W__ eH approximates the classical quadratic Hencky strain energy F W__ H(F):=W__ H(U):=μ dev_n U^2+κ/2 [ tr( 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__ eH, for k≥1/4 and k≥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.
arXiv:1403.3843v2 fatcat:xrm6p6azbvgcded4vevhilzeca