On equality of relaxations for linear elastic strains

Abstract

We use the quadratic rank-one convex envelope $qr_e(f)$ for $f:M_s^{n} \to \mathbb R$ defined on the space of linear elastic strains with $n\geq 2$ to study conditions for equality of semiconvex envelopes. We also use the corresponding quadratic rank-one convex hull $qr_e(K)$ for compact sets $K\subset M_s^{n}$ to give a condition for equality of semiconvex hulls. We show that $L^e_c(K)=C(K)$ if and only if $qr_e(K)=C(K)$, where $L^e_c(K)$ is the closed lamination convex hull on linear strains. We also establish that for functions satisfying $f(A)\geq c|A|^2-C_1$ for $A\in M_s^{n}$, $R_e(f)=C(f)$ if and only if $qr_e(f)=C(f)$.