Limiting behavior of the interfacial energy not associated with $% \Gamma (g)$

We begin by listing the points of $\Gamma (g)$ in increasing order

$$0=x_0<x_1<x_2<\cdots <x_k<\cdots <x_K<x_{K+1}=1,$$

and we consider for each positive integer m as above and $k\in \left\{ 0,\ldots ,K\right\}$ the pair

$$(\kappa _m^{(k)},f_m^{(k)}):=(\kappa _m\cap (x_k,x_{k+1})\,,\,f_m\mid _{(x_k,x_{k+1})})$$

and the triple

$$(\emptyset ,g^{(k)},G^{(k)}):=(\emptyset ,g\mid _{(x_k,x_{k+1})},G\mid _{(x_k,x_{k+1})}).$$

It is easy to see that $(\kappa _m^{(k)},f_m^{(k)})\rightarrow (\emptyset ,g^{(k)},G^{(k)})$, that

$$(\kappa _m^{(k)},f_m^{(k)})\in Sid((x_k,x_{k+1})),\quad (\emp... ...in Std((x_k,x_{k+1})),\quad \Gamma (g^{(k)})=\emptyset ,\quad$$

and

$$\sum\limits_{z\in \,\Gamma (f_m)\,\backslash \,\Gamma (g)}h([... ...=0}^K\sum\limits_{z\in \,\Gamma (f_m^{(k)})}h([f_m^{(k)}](z)).$$ (5)

Therefore, to find the behavior of the left hand side of (2.5) as $m\rightarrow \infty$, it suffices to determine the behavior of $% \sum\limits_{z\in \,\Gamma (f_m^{(k)})}h([f_m^{(k)}](z))$ as $m\rightarrow \infty$ for each $k\in \left\{ 0,\ldots ,K\right\}$. Thus, without loss of generality, we may now replace $(\kappa ,g,G)\in Std((0,1))$ by $% (\emptyset ,g,G)\in Std((0,1)).$

Proposition 2.1   Let $(\emptyset ,g,G)\in Std((0,1))$ be given and let $m\mapsto (\kappa _m,f_m)$ be a sequence of simple deformations such that $(f_m,\nabla f_m)\rightarrow (g,G)$ and, for all positive integers m and $z\in \Gamma (f_m)$, there holds [f_m](z)>0. It follows that

$$\liminf_{m\rightarrow \infty }\sum\limits_{z\in \Gamma (f_m)}h([f_m](z))\geq L\int\limits_0^1(\nabla g(x)-G(x))dx$$ (6)

with L the non-negative number defined in (2.2).

Proof. Because $\kappa =\emptyset$, the function g is continuous, so that [g](z)=0 for all $z\in (0,1)$. By the Trace Theorem and the continuity of g, we may conclude that $\lim_{m\rightarrow \infty }\max \left\{ [f_m](z)\,\mid \,z\in \Gamma (f_m)\right\} =0$. The definition of ``lim inf'' and the assumed positivity of [f_m(z)] tell us that, for each $\varepsilon >0$, we may choose a positive integer $M_\varepsilon$such that

$$\frac{h([f_m](z))}{[f_m](z)}\geq (1-\varepsilon )L$$

for all $m>M_\varepsilon$ and $z\in \Gamma (f_m)$. Multiplying both members of this inequality by [f_m](z) and summing over $z\in \Gamma (f_m)$ we find

$$\begin{eqnarray*}\sum\limits_{z\in \Gamma (f_m)}h([f_m](z)) &\geq &(1-\varepsilo... ... &=&(1-\varepsilon )L\,\int\limits_0^1\,(\nabla g(x)-G(x))\,dx, \end{eqnarray*}$$

and this yields the desired conclusion.

The next result gives us full information about the limiting behavior of the interfacial energy not associated with the jumps of g.

Proposition 2.2   For every $(\emptyset ,g,G)\in Std((0,1))$,

$$\inf \left( \liminf_{m\rightarrow \infty }\sum\limits_{z\in \... ...(f_m)}h([f_m](z))\right) =L\int\limits_0^1(\nabla g(x)-G(x))dx$$ (7)

with L the non-negative number defined in (2.2) and with the infimum taken over the sequences of simple deformations satisfying (i) $% (f_m,\nabla f_m)\rightarrow (g,G)$ and (ii) [f_m](z)>0 for all positive integers m and for all $z\in \Gamma (f_m).$

Before we prove this proposition, we discuss the important and immediate conclusion that we may draw from it and the results in Subsections 2.2 and 2.3:

 

$$\inf \left( \liminf_{m\rightarrow \infty }\Psi _{Sid}((\kappa _m,f_m))\right) \int\limits_0^1(W(G(x))+L(\nabla g(x)-G(x)))\,dx$$ = (8)

$$+\sum\limits_{z\in \Gamma (g)}h([g](z)),$$

with again L the non-negative number defined in (2.2) and with the infimum taken over the sequences of simple deformations satisfying (i) $% (f_m,\nabla f_m)\rightarrow (g,G)$ and (ii) [f_m](z)>0 for all positive integers m and for all $z\in \Gamma (f_m).$ This result captures the flavor of the conclusions obtained by Choksi & Fonseca [2]: (1) it involves the double limit operation `` $\inf \lim \inf$'' to yield a quantity depending only upon (g,G) and not on a particular choice of sequence satisfying (i) and (ii); (2) the bulk term $\int\limits_0^1\left( W(G(x))+L(\nabla g(x)-G(x))\right) \,dx$ on the right-hand side of (2.8) contains two terms in the integrand: one comes from the bulk energy density W and the other comes from the interfacial energy density <tex2html_comment_mark> h for simple deformations. Thus, the diffusion of disarrangements occurring as m tends to infinity causes the interfacial energy for simple deformations to contribute to the limiting bulk energy, while the persistent disarrangements associated with $\Gamma (g)$ cause the initial interfacial energy for simple deformations to contribute to the limiting interfacial energy. If we followed Choksi & Fonseca one step further, we would define

$$\Psi _{Std}(\mathcal{(}g,G)):=\int\limits_0^1(W(G(x))+L(\nabla g(x)-G(x)))\,dx+\sum\limits_{z\in \Gamma (g)}h([g](z)).$$

Instead, we give at the end of this lecture an alternative method for calculating an energy for structured deformation that leads to terms in the integral that can depend non-linearly on $\nabla g(x)-G(x)$, the deformation due to microdisarrangements.