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

Abstract

We combine the rate formulation for the objective, corotational Zaremba–Jaumann rate $\frac{D^{ZJ}}{Dt}\left[\sigma \right]=H^{ZJ}\left(\sigma \right).D,D=sym{D}_{\xi }v,$ operating on the Cauchy stress $\sigma$, the Eulerian rate of deformation $D$ and the spatial velocity $v$ with the novel “corotational stability postulate” (CSP)$〈\frac{D^{ZJ}}{Dt}\left[\sigma \right],D〉>0\forall D\in Sym\left(3\right)\setminus \left\{0\right\}$to show that for a given isotropic Cauchy-elastic constitutive law $B\mapsto \sigma \left(B\right)$ in terms of the left Cauchy–Green tensor $B=F{F}^T$, the induced fourth-order tangent stiffness tensor $H^{ZJ}\left(\sigma \right)$ is positive definite if and only if for $\hat{\sigma }(logB):=\sigma \left(B\right)$, the strong True-Stress–True-Strain monotonicity condition (TSTS-M$++$) in the logarithmic strain is satisfied: (TSTS-M++)$sym{D}_{logB}\hat{\sigma }(logB)\in {Sym}_4^{++}\left(6\right)$$⟹〈\hat{\sigma }(log{B}_1)-\hat{\sigma }(log{B}_2),log{B}_1-log{B}_2〉>0\forall B_1,B_2\in {Sym}^{++}\left(3\right),B_1\ne B_2.$ 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 $H^{ZJ}\left(\sigma \right)$, since $H^{ZJ}\left(\sigma \right)$ is accordingly positive definite. Notably, (TSTS-M$++$) is one way to characterize the fundamental notion of “stress increases with strain”. 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.