Elasticity with Hierarchical Disarrangements: A Field Theory That Admits Slips and Separations at Multiple Submacroscopic Levels

Abstract

The complexity and variety of geometrical changes in physical systems at submacroscopic levels has led to various approaches to the broadening of the classical theory of finite elasticity. One approach, the field theory “elasticity with disarrangements”, employed the multiscale geometry of structured deformations in order to incorporate the effects of disarrangements such as slips and separations at a single submacroscopic level on the macroscopic response of a continuous body. This article extends that field theory by enriching the underlying geometry so as to include the effects of disarrangements at more than one submacroscopic level. The resulting field theory broadens the scope of this approach, sharpens the description of the physical nature of dissipative mechanisms that can arise, and increases the variety of systems of contact forces that can serve as boundary loadings for a body that evolves via multiscale geometrical processes.

Appendix: Proof of the Approximation Theorem for Three-Level Structured Deformations

In this appendix we use the terms “simple deformation” and “piecewise-fit region” in the sense of [12]. Roughly speaking, a simple deformation is a piecewise smooth, injective mapping, and a piecewise-fit region is a finite union of regions without unopened cracks and with finite surface area.

Theorem 1

For each three-level structured deformation$(g,G_{1},G_{2})$from a piecewise-fit region ℬ there exists a double sequence$(n_{1},n_{2})\longmapsto f_{n_{1},n_{2}}$of simple deformations from ℬ for which

$$\begin{aligned} \lim_{n_{1}\rightarrow \infty }\lim_{n_{2}\rightarrow \infty }f_{n _{1},n_{2}} =&g, \\ \lim_{n_{1}\rightarrow \infty }\nabla \lim_{n_{2}\rightarrow \infty }f _{n_{1},n_{2}} =&G_{1}, \\ \lim_{n_{1}\rightarrow \infty }\lim_{n_{2}\rightarrow \infty }\nabla f_{n_{1},n_{2}} =&G_{2}, \end{aligned}$$

where each of the iterated limits$\lim_{n_{1}\rightarrow \infty }$and$\lim_{n_{2}\rightarrow \infty }$is taken in the sense of$L^{\infty }$-convergence.

Proof

Let a three-level structured deformation $(g,G_{1},G_{2})$ and a positive integer $n_{1}$ be given. The three-level accommodation inequality (3) implies that the pair $(g,G_{1})$ is a (two-level) structured deformation. By the Approximation Theorem for two-level structured deformations and properties of the determinant mapping, we may choose a constant $\bar{C}>0$ (that depends only upon the field $G_{1}$) and a simple deformation $f_{n_{1}}$ such that

$$ \Vert g-f_{n_{1}} \Vert _{\infty }< \frac{1}{n_{1}}, \qquad \Vert G_{1}-\nabla f_{n_{1}} \Vert _{\infty }< \frac{ \bar{C}}{n_{1}} ,\quad \text{and}\quad \Vert \det G_{1}-\det \nabla f_{n _{1}} \Vert _{\infty }< \frac{1}{n_{1}}, $$

(103)

where $\Vert \cdot \Vert _{\infty }$ denotes the $L^{\infty }$ norm. By the three-level accommodation inequality (3) and (103) we conclude that

$$ \det \nabla f_{n_{1}}>\det G_{1}-\frac{1}{n_{1}}\geq \det G_{2}-\frac{1}{n _{1}}>c-\frac{1}{n_{1}}, $$

(104)

and we take from this point on $n_{1}>c^{-1}$. (The inequality (104) can be written in terms of fields because the accommodation inequality holds at almost every point in ℬ for one and the same positive number $c$.) We define

$$ \varphi (n_{1}):=2 \biggl(1- \biggl(1-\frac{1}{n_{1}c} \biggr)^{\frac{1}{3}} \biggr)>0 $$

(105)

and note that

$$ \lim_{n_{1}\rightarrow \infty }\varphi (n_{1})=0, $$

(106)

as well as $\varphi (n_{1})>1-(1-\frac{1}{n_{1}c})^{\frac{1}{3}}$, so that

$$ 1-\frac{1}{n_{1}c}> \bigl(1-\varphi (n_{1}) \bigr)^{3}. $$

(107)

We infer from (104), (105), and (107) that

$$\begin{aligned} \det \nabla f_{n_{1}} >& \biggl(1-\frac{1}{n_{1}\det G_{2}} \biggr)\det G_{2} \\ >& \biggl(1-\frac{1}{n_{1}c} \biggr)\det G_{2}> \bigl(1-\varphi (n_{1}) \bigr)^{3}\det G_{2} \\ =&\det \bigl( \bigl(1-\varphi (n_{1}) \bigr)G_{2} \bigr). \end{aligned}$$

(108)

By (106) and the inequality $\det G_{2}>c$, we may choose $N_{1}>c^{-1}$ such that

$$ \bigl(1-\varphi (n_{1}) \bigr)^{3}>\frac{1}{2}\quad \text{for all}\ n_{1}>N_{1}. $$

The estimate (108) then implies

$$ \det \nabla f_{n_{1}}>\det \bigl( \bigl(1-\varphi (n_{1}) \bigr)G_{2} \bigr)>\frac{c}{2}\quad \text{for all}\ n_{1}>N_{1}, $$

and we conclude that the pair $(f_{n_{1},}(1-\varphi (n_{1}))G_{2})$ satisfies the accommodation inequality for (two-level) structured deformation for all $n_{1}>N_{1}$. Thus, for each $n_{1}>N_{1}$ and positive integer $n_{2}$, the Approximation Theorem for (two-level) structured deformations permits us to choose a simple deformation $f_{n_{1},n_{2}}$ such that

$$ \Vert f_{n_{1},n_{2}}-f_{n_{1}} \Vert _{\infty }< \frac{1}{n _{2}}\quad \text{and}\quad \bigl\Vert \nabla f_{n_{1},n_{2}}- \bigl(1- \varphi (n_{1}) \bigr)G_{2} \bigr\Vert _{\infty }< \frac{1}{n_{2}}. $$

Therefore, for each $n_{1}>N_{1}$ we may let $n_{2}$ tend to $\infty$ in the last two inequalities to obtain

$$ \lim_{n_{2}\rightarrow \infty }f_{n_{1},n_{2}}=f_{n_{1}}\quad \text{and}\quad \lim_{n_{2}\rightarrow \infty }\nabla f_{n_{1},n _{2}}= \bigl(1-\varphi (n_{1}) \bigr)G_{2}, $$

and, in view of (106), we may then let $n_{1}$ tend to $\infty$ in each of the last two relations to conclude from (103)

$$\begin{aligned} \lim_{n_{1}\rightarrow \infty }\lim_{n_{2}\rightarrow \infty }f_{n _{1},n_{2}} =&\lim _{n_{1}\rightarrow \infty }f_{n_{1}}=g, \\ \lim_{n_{1}\rightarrow \infty }\nabla \lim_{n_{2}\rightarrow \infty }f _{n_{1},n_{2}} =&\lim_{n_{1}\rightarrow \infty }\nabla f_{n_{1}}=G _{1}, \\ \lim_{n_{1}\rightarrow \infty }\lim_{n_{2}\rightarrow \infty }\nabla f_{n_{1},n_{2}} =&\lim_{n_{1}\rightarrow \infty } \bigl( \bigl(1-\varphi (n_{1}) \bigr)G _{2} \bigr)=G_{2}. \end{aligned}$$

In all of these relations, the limits are taken in the sense of $L^{\infty }$. □