December  2002, 1(4): 565-573. doi: 10.3934/cpaa.2002.1.565

On equality of relaxations for linear elastic strains

1. 

School of Mathematical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom

Received  October 2001 Revised  July 2002 Published  September 2002

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 $qr_e(f)=C(f)$.
Citation: Kewei Zhang. On equality of relaxations for linear elastic strains. Communications on Pure & Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565
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