Behavior of the bulk energy density in the limit

Let $(\kappa ,g,G)\in Std((0,1))$ and $m\mapsto (\kappa _m,f_m)\in Sid((0,1))$ be given such that

$$\lim_{m\rightarrow \infty }\sup\limits_{x\in \,(0,1)\,\backsl... ...ght\vert +\left\vert \nabla f_m(x)-G(x)\right\vert \right) =0.$$ (3)

In particular, we follow Choksi & Fonseca [2] in not requiring here that
$\liminf_{m\rightarrow \infty }\kappa_m$ $= \kappa$. To distinguish this weaker convergence (2.3), we write here $(f_m,\nabla f_m)\rightarrow (g,G)$ in place of the notation $(\kappa _m,f_m)\rightarrow (\kappa ,g,G)$ used in Part One for the convergence that includes the requirement $\liminf_{m\rightarrow \infty }\kappa _m=\kappa$. From the definition of Std, we may choose positive constants $\bar{m},\bar{M}$ such that $0<\bar{m}<G(x)<\bar{M}$ for all $x\in (0,1)\backslash \kappa$. Because $(f_m,\nabla f_m)\rightarrow (g,G)$, we may choose a positive integer N such that $\frac{\bar{m}}2\leq \nabla f_m(x)\leq 2\bar{M}$ for all $x\in (0,1)\,\backslash \,(\kappa \cup \kappa _m)$ and for all m>N. Since W is continuous, its restriction $W\mid _{[\frac{\bar{m}}2,2\bar{M}% ]}\,$is uniformly continuous, and we conclude that

$$\lim\limits_{m\rightarrow \infty }\int\limits_0^1W(\nabla f_m(x))\,dx=\int\limits_0^1W(G(x))\,dx.$$ (4)

Thus, the bulk energies for the approximating simple deformations form a convergent sequence whose limit is independent of the choice of sequence $(\kappa _m,f_m)$ $\rightarrow (g,G)$; in particular, the limiting bulk energy depends only upon G, the deformation without disarrangements.