Point Singularities in Incompatible Elasticity

Abstract

The equations of stress equilibrium and strain compatibility/incompatibility are discussed for fields with point singularities in a planar domain. The sufficiency (or insufficiency) of the smooth maps, obtained by restricting the singular fields to the domain away from the singularities, in completely characterizing the equations of equilibrium and compatibility/incompatibility over the entire domain, is established and illustrated with examples. The uniqueness of the solution to the stress problem of incompatible linear elasticity, allowing for singular fields, is proved. The uniqueness fails when the problem is considered solely in terms of the restricted maps. As further applications of our framework, a general stress solution, in response to point supported body force and defect fields, is derived and a generalized notion of the force acting on a defect is developed.

Appendix A: Proof of the Existence of Extension in Lemma 2.3

We first establish the existence result when $\operatorname{deg}(T_{0})<0$. Consider $\vartheta \in \mathcal{D}(\Omega )$ such that $\operatorname{supp}(\vartheta ) \subset {B}_{r}$ and $\vartheta (\boldsymbol{x}) =1$ for all $\boldsymbol{x} \in {B}_{r/2}$. Given $\lambda >1$, define $\vartheta _{\lambda } \in \mathcal{D}(\Omega )$ as $\vartheta _{\lambda } (\boldsymbol{x})=\vartheta (\lambda \boldsymbol{x})$. Hence, $\operatorname{supp}(\vartheta _{\lambda }) \subset B_{r/\lambda }$ and $\vartheta _{\lambda }(\boldsymbol{x}) =1$ for all $\boldsymbol{x} \in {B}_{{r}/{2\lambda }}$. For any $\phi \in \mathcal{D}(\Omega )$, $(1- \vartheta _{2^{j}})\phi \in \mathcal{D}(\Omega -O)$. We consider the sequence of distributions ${T}^{j}\in \mathcal{D}'(\Omega )$ as ${T}^{j}= (1- \vartheta _{2^{j}}){T_{0}}$. Hence, $T^{j} (\phi )={T_{0}}((1- \vartheta _{2^{j}})\phi )$ for all $\phi \in \mathcal{D}(\Omega )$. For any $\phi \in \mathcal{D}(\Omega )$, we have $(T^{j+1}-T^{j}) (\phi )=(\phi T_{0})(-\vartheta _{2^{j+1}}+ \vartheta _{2^{j}}) = 2^{-nj} \left ((\phi T_{0})|_{{B}_{r}} \right )_{2^{-j}} (\vartheta -\vartheta _{2})$. Given $\operatorname{deg}(T_{0})<0$, for $k \in \mathbb{R}$ such that $\operatorname{sd}(T_{0})< k <n$, we can conclude that

$$ \lim _{j\to \infty } 2^{-nj} \left ((\phi T_{0})|_{\mathcal{B}_{r}} \right )_{2^{-j}} (\vartheta -\vartheta _{2})=0. $$

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Also, there exists $j_{0}\in \mathbb{N}$ and $c_{0}\in \mathbb{R}$ such that, for any $j>j_{0}$, $(T^{j+1}-T^{j}) (\phi ) < c_{0} 2^{j(k-n)}$. Hence the sequence $T^{j} (\phi )$ is a Cauchy sequence and we can define a distribution $T\in \mathcal{D}'(\Omega )$ as $T(\phi )=\lim _{j\to \infty } T^{j} (\phi )$. Clearly, $T\in \mathcal{D}'(\Omega )$ such that $T|_{\Omega -O}=T_{0}$. It can be shown that $\operatorname{deg}(T)=\operatorname{deg}(T_{0})$ [1]. Further, if $T_{1} \in \mathcal{D}'(\Omega )$ is another extension of $T_{0}$ then $(T_{1}-T) \in \mathcal{E}(\Omega )$. Consequently $\operatorname{deg}(T_{1})\geq 0$ if $(T_{1}-T)\neq 0$ (Lemma 2.4). This is a contradiction. Hence $T$ is the unique extension of $T_{0}$ such that $\operatorname{deg}(T)=\operatorname{deg}(T_{0})$. Next, we consider the case when $\operatorname{deg}(T_{0}) \geq 0$. Let $\rho$ be the greatest integer smaller (or equal) than $\operatorname{deg}(T_{0})$ and let $x^{\alpha }={x_{1}}^{\alpha _{1}}{x_{2}}^{\alpha _{2}}\dots {x_{n}}^{ \alpha _{n}}$ for any $\alpha =(\alpha _{1},\alpha _{2}\dots \alpha _{n})$. The function $\phi \in \mathcal{D}(\Omega )$ can be uniquely decomposed as

$$ \phi =\sum _{|\alpha | \leq \rho } w^{\alpha } \partial ^{\alpha } \phi (O)+ \sum _{|\alpha |=\rho +1} x^{\alpha } \psi _{\alpha } $$

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where $\psi _{\alpha } \in \mathcal{D}(\Omega )$ and $w^{\alpha }\in \mathcal{D}(\Omega )$ is as introduced in Lemma 2.4. We have $\operatorname{deg}(x^{\alpha }T_{0}) <0$ for $|\alpha |=\rho +1$ [1]. Let $x^{\alpha }T \in \mathcal{D}'(\Omega )$ be the unique extension of $x^{\alpha }T_{0} \in \mathcal{D}'(\Omega -O)$ such that $\operatorname{deg}(x^{\alpha }T)=\operatorname{deg}(x^{\alpha }T_{0})$. The distribution $T \in \mathcal{D}'(\Omega )$, defined as $T(\phi )=\sum _{|\alpha |=\rho +1} x^{\alpha }T(\psi _{\alpha })$ satisfies $T|_{\Omega -O}=T_{0}$. It can be shown that $\operatorname{deg}(T)=\operatorname{deg}(T_{0})$ [1]. Let $T_{1} \in \mathcal{D}'(\Omega )$ be an extension of $T_{0}$ such that $\operatorname{deg}(T_{1})=\operatorname{deg}(T_{0})$. Then $(T_{1}-T) \in \mathcal{E}(\Omega )$ with $\operatorname{deg}(T_{1}-T)\leq \operatorname{deg}(T_{0})$, and we can write $T_{1}-T=\Sigma _{|\alpha | \leq \operatorname{deg}({T_{0}})} q^{ \alpha } \partial ^{\alpha }\delta _{O}$, where $q^{\alpha }\in \mathbb{R}$ (Lemma 2.4).