Strict convexity of yield surfaces of some weakly-textured materials

Let Sym${}_0$ be the space of traceless symmetric second-order tensors. We say that a polycrystalline elastic–plastic material is weakly-textured if its yield function $f:{\text{Sym}}_0\to R$ is the sum of a texture-independent isotropic part $f_{\text{iso}}$ and an anisotropic part which is linear in the relevant texture coefficients. Let $c>0$ and $S≔f^{-1}\left(c\right)\subset {\text{Sym}}_0$ be the yield surface of the weakly-textured material in question. We present a sufficient condition (*), namely that ${\nabla }^{2}f\left(S\right)$ be positive definite for each $S\in S$, for a smooth yield surface $S$ to be strictly convex in Sym${}_0$. We apply this sufficient condition to weakly-textured materials with yield functions that satisfy the following conditions: (i) the yield functions $f$ and $f_{\text{iso}}$ are smooth; (ii) ${\nabla }^2{f}_{\text{iso}}\left(S\right)$ is positive definite for each $S$ in $S_{\text{iso}}≔f_{\text{iso}}^{-1}\left(c\right)\subset {\text{Sym}}_0$. We prove that the yield surface $S\subset {\text{Sym}}_0$ of such weakly-textured material is strictly convex if the texture coefficients in $f$ are sufficiently small. As illustration for practical applications, by appealing to condition (*) we study the strict convexity of the yield surface pertaining to a weakly-textured orthorhombic aggregate of cubic crystallites which has a quadratic yield function of the type proposed by Hill in 1948. Moreover, we show that all 35 samples of cold-rolled and annealed low-carbon steel sheets studied by Stickels and Mould have their quadratic yield functions and corresponding yield surfaces strictly convex.