Rank one convexity of the exponentiated Hencky-logarithmic strain energy in finite elastostatics

We investigate a family of isotropic volumetric-isochoric decoupled strain energies F 7→ WeH(F ) := ŴeH(U) :=    μ k e ‖ devn logU‖ 2 + κ 2k̂ e [tr(logU)] 2 if det F > 0, +∞ if detF ≤ 0, based on the Hencky-logarithmic (true, natural) strain tensor logU , 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, k̂ are dimensionless parameters, F = ∇φ is the gradient of deformation, U = √ F TF is the right stretch tensor and devn logU = logU − 1 n tr(logU) · 1 is the deviatoric part (the projection onto the traceless tensors) of the strain tensor logU . For plane elastostatics, i.e. n = 2, we prove the rank-one convexity of this family. Moreover, we show that the corresponding Cauchy (true) stress-true strain relation is invertible and we discuss the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor. We also prove that the rank-one convexity of the energies belonging to this family is not preserved in three-dimension and that the energy F 7→ e ‖ logU‖ 2 , F ∈ GL(n), n ∈ N, n ≥ 2 is not rank-one convex. An immediate application in multiplicative elasto-plasticity is proposed.