On the Variational Limits of Lattice Energies on Prestrained Elastic Bodies

Abstract

We study the asymptotic behavior of the discrete elastic energies in the presence of the prestrain metric G, assigned on the continuum reference configuration $\Omega$. When the mesh size of the discrete lattice in $\Omega$ goes to zero, we obtain the variational bounds on the limiting (in the sense of $\Gamma$-limit) energy. In the case of the nearest-neighbour and next-to-nearest-neighbour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to G.

Appendix

$\Gamma$-convergence

We now recall the definition and some basic properties of $\Gamma$-convergence , that will be needed in the sequel.

Definition 10.A.1

Let $\{I_\epsilon \}, I: X \rightarrow \overline{\mathbb {R}} = \mathbb {R} \cup \{-\infty , \infty \}$ be functionals on a metric space X. We say that $I_\epsilon$ $\Gamma$-converge to I (as $\epsilon \rightarrow 0$), if and only if

(i) For every $\{u_\epsilon \}, u \in X$ with $u_\epsilon \rightarrow u$, we have: $I(u) \le \liminf _{\epsilon \rightarrow 0} I_\epsilon (u_\epsilon ).$

(ii) For every $u \in X$, there exists a sequence $u_\epsilon \rightarrow u$ such that $I(u) = \lim _{\epsilon \rightarrow 0}I_\epsilon (u_\epsilon )$.

Theorem 10.A.2

([BD98], Chap. 7) Let $I_\epsilon , I$ be as in Definition 10.A.1 and assume that there exists a compact set $K \subset X$ satisfying:

$$ \inf _X I_\epsilon = \inf _KI_\epsilon , \qquad \forall \epsilon .$$

Then $\lim _{\epsilon \rightarrow 0} (\inf _X I_\epsilon ) = \min _X I$, and moreover if $\{u_\epsilon \}$ is a converging sequence such that

$$ \lim _{\epsilon \rightarrow 0} I_\epsilon (u_\epsilon ) = \lim _{\epsilon \rightarrow 0} (\inf _X I_\epsilon ), $$

then $u=\lim u_\epsilon$ is a minimum of I, i.e., $I(u) = \min _X I$.

Theorem 10.A.3

([BD98], Chap. 7) Let $\Omega$ be an open subset of $\mathbb {R}^n$. Any sequence of functionals $I_\epsilon :L^2(\Omega ,\mathbb {R}^n)\rightarrow \overline{\mathbb {R}}$ has a subsequence which $\Gamma$-converges to some lower semicontinuous functional $I:L^2(\Omega ,\mathbb {R}^n)\rightarrow \overline{\mathbb {R}}$. Moreover, if every subsequence of $\{I_\epsilon \}$ has a further subsequence that $\Gamma$-converges to (the same limit) I, then the whole sequence $I_\epsilon$ $\Gamma$-converges to I.

Convexity and Quasiconvexity

In this section $f:\mathbb {R}^{m\times n}\rightarrow {\mathbb {R}}$ is a function assumed to be Borel measurable, locally bounded and bounded from below. Recall that the convex and quasiconvex envelopes of f, i.e., $Cf, Qf:\mathbb {R}^{m\times n}\rightarrow {\mathbb {R}}$ are defined by

$$\begin{aligned} \begin{aligned} Cf(M)&= \text {sup} \left\{ g(M); ~~g:\mathbb {R}^{m \times n} \rightarrow \mathbb {R},~~ g \text { convex, } g \le f\right\} , \\ Qf(M)&= \text {sup} \left\{ g(M); ~~g:\mathbb {R}^{m \times n} \rightarrow \mathbb {R},~~ g \text { quasiconvex, } g \le f\right\} . \end{aligned} \end{aligned}$$

We say that f is quasiconvex, if

$$\begin{aligned} f(M) \le {\int \!\!\!\!\!\!-}_{D} f(M + \nabla \phi (x))~\mathrm {d}x, \qquad \forall M \in \mathbb {R}^{m \times n},\quad \forall \phi \in W^{1, \infty }_0 (D, \mathbb {R}^{m}), \end{aligned}$$

on every open bounded set $D\subset \mathbb {R}^{n}$.

Theorem 10.A.4

([Dac08], Chap. 6)

• (i) When $m=1$ or $n=1$ then f is quasiconvex if and only if f is convex.

• (ii) For any open bounded $D\subset \mathbb {R}^n$ there holds

  $$Qf(M) = \inf \left\{ {\int \!\!\!\!\!\!-}_D f(M+\nabla \phi (x))~\mathrm {d}x; ~ \phi \in W_0^{1,\infty }(D,\mathbb {R}^m)\right\} .$$

• (iii) Assume that, for some $n_1+n_2 = n$ we have

  $$f(M) = f_1(M_{n_1}) + f_2(M_{n_2}), \qquad \forall M\in \mathbb {R}^{m\times n},$$

  where $M_{n_1}$ stands for the principal minor of M consisting of its first $n_1$ columns, while $M_{n_2}$ is the minor of M consisting of its $n_2$ last columns. Assume that $f_1, f_2$ are Borel measurable and bounded from below. Then

  $$Cf= Cf_1 + Cf_2, \qquad Qf= Qf_1 + Qf_2.$$

The following classical results explain the role of convexity and quasiconvexity in the integrands of the typical integral functionals.

Theorem 10.A.5

([Dac08]) Let $\Omega$ be a bounded open set in $\mathbb {R}^{n}$ and let $f: \mathbb {R}^{m\times 1} \rightarrow \mathbb {R}$ be lower semicontinuous (lsc) . Then the functional

$$ I(u) = \int _{\Omega }f(u(x))~\mathrm {d}x, \qquad \forall u \in L^{2}(\Omega , \mathbb {R}^{m}),$$

is sequentially lower semi-continuous with respect to the weak convergence in $L^{2}(\Omega , \mathbb {R}^{m})$ if and only if f is convex.

Theorem 10.A.6

([Dac08], Chap. 9) Let $\Omega$ be a bounded open set in $\mathbb {R}^{n}$ and let $f: \Omega \times \mathbb {R}^{m\times n} \rightarrow \mathbb {R}$ be Caratheodory, and satisfy the uniform growth condition

$$\begin{aligned}&\exists C_1, C_2>0, \quad \forall x\in \Omega , \quad \forall M\in \mathbb {R}^{m\times n}, \\ \nonumber&C_1|M|^2 - C_2\le f(x, M) \le C_2(1 + |M|^{2}). \end{aligned}$$

(10.A.1)

Assume that the quasiconvexification Qf of f with respect to the variable M, is also a Caratheodory function. Then for every $u\in W^{1,2}(\Omega , \mathbb {R}^m)$ there exists a sequence $\{u_\epsilon \}\in u+ W_0^{1,2}(\Omega , \mathbb {R}^m)$ such that, as $\epsilon \rightarrow 0$

$$\begin{aligned} u_\epsilon \rightharpoonup u \quad \text{ weakly } \text{ in } W^{1,2} \quad \text{ and } \quad \int _{\Omega }f(x, \nabla u_\epsilon (x))~\mathrm {d}x \rightarrow \int _{\Omega }Qf(x, \nabla u(x))~\mathrm {d}x. \end{aligned}$$