The corotational stability postulate: Positive incremental Cauchy stress moduli for diagonal, homogeneous deformations in isotropic nonlinear elasticity

Abstract

In isotropic nonlinear elasticity the corotational stability postulate (CSP) is the requirement that (0.1)$〈\frac{D^{\circ }}{Dt}\left[\sigma \right],D〉>0\forall D\in \text{Sym}\left(3\right)\setminus \left\{0\right\},$where $\frac{D^{\circ }}{Dt}$ is any corotational stress rate, $\sigma$ is the Cauchy stress and $D=symL$ is the Eulerian rate of deformation tensor where $L=\dot{F}{F}^{-1}=D_{\xi }v$ is the spatial velocity gradient. For $\hat{\sigma }(logV)≔\sigma \left(V\right)$ it is equivalent almost everywhere to the monotonicity (TSTS-M${}^{+}$) (0.2)$〈\hat{\sigma }(log{V}_1)-\hat{\sigma }(log{V}_2),log{V}_1-log{V}_2〉>0\forall V_1,V_2\in {\text{Sym}}^{++}\left(3\right),V_1\ne V_2.$For hyperelasticity, (CSP) is in general independent of convexity of the mapping $F\mapsto W\left(F\right)$ or $U\mapsto \hat{W}\left(U\right)$. Considering a family of diagonal, homogeneous deformations $t\mapsto F\left(t\right)$ one can, nevertheless, show that (CSP) implies positive incremental Cauchy stress moduli for this deformation family, including the incremental Young’s modulus, the incremental equibiaxial modulus, the incremental planar tension modulus and the incremental bulk modulus. Aside, (CSP) is sufficient for the Baker-Ericksen and tension-extension inequality. Moreover, it implies local invertibility of the Cauchy stress–stretch relation. Together, this shows that (CSP) is a reasonable constitutive stability postulate in nonlinear elasticity, complementing local material stability viz. LH-ellipticity.