Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain

We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate {align} {{ D}ZJ}{{ D} t} [σ] = HZJ(σ).D, D = { sym} { D} v\,, {align} operating on the Cauchy stress σ, the Eulerian strain rate D and the spatial velocity v with the novel {corotational stability postulate} (CSP){equation} {{ D}ZJ}{{ D} t}[σ], D > 0 ∀ \, D∈{ Sym}(3)\{0\} {equation} to show that for a given isotropic Cauchy-elastic constitutive law B ↦ σ(B) in terms of the left Cauchy-Green tensor B = F FT, the induced fourth-order tangent stiffness tensor HZJ(σ) is positive definite if and only if for σ(log B):=σ(B), the strong monotonicity condition (TSTS-M⁺⁺) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M⁺⁺) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor HZJ(σ). The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M⁺⁺) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions.