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

We investigate a family of isotropic volumetric-isochoric decoupled strain energies F bar right arrow W-eH(F) := (W) over cap (eH)(U) := {mu/k(+infinity)(ek parallel to devn log U parallel to 2) + kappa/2 (k) over cape((k) over cap [tr(logU)]2) if det F > 0, if det F <= 0, based on the Hencky-logarithmic (true, natural) strain tensor logU, where mu > 0 is the infinitesimal shear modulus, kappa = 2 mu+3 lambda/3 > 0 is the infinitesimal bulk modulus with lambda the first Lame constant, k, (k) over cap are additional dimensionless material parameters, F = del phi is the gradient of deformation, U = root F-T F is the right stretch tensor and dev(n) logU = logU - 1/n tr(logU) . 1 is the n-dimensional deviatoric part of the strain tensor logU. For small elastic strains, W-eH approximates the classical quadratic Hencky strain energy F bar right arrow W-H(F) := (W) over cap (H)(U) := mu parallel to dev(n) log U parallel to(2) + kappa/2[tr(logU)](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 >= 14 and (k) over cap >= 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 F bar right arrow mu/ke(k parallel to log U parallel to 2), F bar right arrow mu/ke(k/mu(mu parallel to devn log U parallel to 2 +kappa/2[tr(logU)]2) , F is an element of GL(+) (n), n is an element of N, n >= 2 are also not rank-one convex.