A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain

Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba-Jaumann objective derivative of the Cauchy stress σ, i.e. {equation} {{ D}ZJ}{{ D} t}[σ] = {{ d}}{{ d}{t}}[σ] - W \, σ + σ \, W, W = { skew}( F \, F⁻¹) {equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor F = D φ, the left Cauchy-Green tensor B = F \, FT, and the strain-rate tensor D = sym( F \, F⁻¹), we show that {equation} {eqCPSdef} {alignedat}{2} ∀ \,D∈{ Sym}(3) \! \! \{0\}: ~ {{{ D}ZJ}{{ D} t}[σ]},{D} > 0 & log B σ(log B) \;is strongly Hilbert-monotone & { sym} { D}log B σ(log B) ∈{ Sym}⁺⁺_4(6) (TSTS-M⁺⁺), {alignedat} {1} {equation} where Sym⁺⁺₄(6) denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the True-Stress True-Strain strict Hilbert-Monotonicity (TSTS-M⁺) for B ↦ σ(B) = σ(log B), i.e. {equation} σ(log B_1)-σ(log B_2),{log B_1-log B_2} > 0 ∀ \, B_1≠ B_2∈{ Sym}⁺⁺(3) \, . {equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case.