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

Abstract

We investigate a family of isotropic volumetric-isochoric decoupled strain energiesbased on the Hencky-logarithmic (true, natural) strain tensor log 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 devn log U = log U − 1 n tr(log U ) • 1 1 is the deviatoric part of the strain tensor log U . For small elastic strains, W eH approximates the classical quadratic Hencky strain energywhich is not everywhere rank-one convex. In plane elastostatics, i.e. n = 2, we prove the everywhere rankone 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 and that the energiesare not rank-one convex.

Homogeneous Cauchy stress induced by non‐homogeneous deformations

“…where V = √ F F T is the left stretch tensor. Then σ eH is invertible, while W eH is not rank-one convex [3][4][5].…”

Rank‐one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two‐dimensional case

“…Such energy functions play an important role in nonlinear elasticity theory [6,7], where an additive volumetric-isochoric split…”

“…bi-SO(2)-invariant) as well as isochoric is rank-one convex if and only if it is polyconvex. A function W : GL + (2) → R is called isochoric ifSuch energy functions play an important role in nonlinear elasticity theory [6,7], where an additive volumetric-isochoric splitof the elastic energy potential W into an isochoric part W iso and a volumetric part W vol is oftentimes assumed. This constitutive requirement is equivalent [8] to the existence of a function p :…”

An extended exponentiated Hencky energy for transverse isotropy

“…Recently it was discovered in [2] that the incorporated isotropic, logarithmic invariants can be uniquely motivated by purely geometric considerations based on the geodesic distance of the deformation gradient to the special orthogonal group SO(n). Building on those achievements the exponentiated Hencky energy was developed in [3] and subsequent contributions, showing remarkable properties in view of mathematical aspects and performance [4]. The formal extension to transversely isotropic problems appears attractive in the context of directional dependent strain stiffening materials.…”