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# Shape sensitivity of enery

## Crack problems, J-integral, and GJ-integral

In crack problems, the elastic system is described on the domain $\Omega$ with a crack $\Sigma$, where $\Omega =D\setminus \Sigma$ by a domain $D$ with the Lipschitz property and a surface (curve in 2-d case) $\Sigma$ with the edge $\partial \Sigma$. The boundary of $\Omega$ has not the Lipschitz property. The definition of the GJ-integral (2.5) changes slightly. $\Omega$ is devided into domains $\Omega ^+, \Omega ^-$ with the Lipschitz property and the surface (line) $\Pi$ such that $\Sigma \subset \Pi$ and $D=\Omega ^+\cup \Omega ^-\cup (\Pi \cap D)$. In crack problem, if $\omega \subset D$, (2.11) is written as \begin{eqnarray*} \int _{\omega \cap \Omega }\nabla _x\widehat{W}(u)\cdot \mu \, dx &=&\int _{\partial (\omega \cap \Omega ^+)}\widehat{W}(u)^+(\mu \cdot n^+)\,ds\\ &&+\int _{\partial (\omega \cap \Omega ^-)}\widehat{W}(u)^-(\mu \cdot n^-)\,ds,\\ &&-\int _{\omega \cap \Omega }\widehat{W}(u)\textrm{div}\mu \, dx,\\ &=&\int _{\partial \omega }\widehat{W}(u)(\mu \cdot n)\,ds+ \int _{\omega \cap \Sigma }[\! [\widehat{W}(u)]\! ](\mu \cdot n^+)\,ds,\\ &&-\int _{\omega \cap \Omega }\widehat{W}(u)\textrm{div}\mu \, dx. \end{eqnarray*} since $W(u)^+=W(u)^-$ on $\Pi \setminus \Sigma$, where $\widehat{W}(u)^±$ is the trace of $\widehat{W}(u)|_{\Omega ^±}$ on $\partial (\omega \cap \Omega ^{±})$, $n^±$ the outward unit normal to $\Omega ^±$ respectively, and $[\! [\widehat{W}(u)]\! ]$ stands for $\widehat{W}(u)^+-\widehat{W}(u)^-$ on $\Sigma$. Also the first term in the right-hand side of (2.12) is written as $$\int _{\partial (\omega \cap \Omega )}\widehat{T}(u)(\nabla u\cdot \mu )ds =\int _{\partial \omega }\widehat{T}(u)(\nabla u\cdot \mu )ds +\int _{\omega \cap \Sigma }[\! [\widehat{T}(u)(\nabla u\cdot \mu )]\! ]ds$$ where $[\! [\widehat{T}(u)(\nabla u\cdot \mu )]\! ]=n_j^+ \left ([\partial \widehat{W}(u)/\partial \zeta _{ij}]^+(\nabla u_i\cdot \mu )^+-[\partial \widehat{W}(u)/\partial \zeta _{ij}]^-(\nabla u_i\cdot \mu )^-\right )$. In the crack problem, for $\omega \subset D$, (2.5b) is defined by \begin{eqnarray} P_{\omega }(u,\mu )& :=&\int _{\partial \omega }\left \lbrace \widehat{W}(u)(\mu \cdot n)-\widehat{T}(u)(\nabla u\cdot \mu )\right \} \,ds, \notag \\ &&+\int _{\omega \cap \Sigma }\left \{[\! [\widehat{W}(u)]\! ](\mu \cdot n)- [\! [\widehat{T}(u)(\nabla u\cdot \mu )]\! ]\right \} ds. \tag{2.15} \end{eqnarray} If $\mu$ is the vector $\mu _C$ obtained from the crack extension, then $\mu _C\cdot n=0$ on $\omega \cap \Sigma$ and $\widehat{T}(u)=0$ on $\omega \cap \Sigma$ by stress free on the crack surface so the last term in ((2.15) ) vanishes.

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