Let $h$ be a $C^1$-function defined on the edge $\partial \Sigma $ of the crack $\Sigma $ given in Section 2.2.2 . Using the geodesic coordinate, we define $\mathcal{E}(v;h)=\tilde{\mathcal{E}}(v;f,g,\Phi _h)$, $\Phi _h(x)=x+h(\gamma (\mathcal{P}(x)))e_{\partial \Sigma ,1}(\gamma (\mathcal{P}(x)))\beta (x)$ where $\beta $ is the cut-off function such that $\beta (x)=1$ at a point $x$ in a neighborhood of $\partial \Sigma $ included in $U(\partial \Sigma )$ given in Section 2.2.2 . Applying Theorem 2.10 to fracture problem with $\mathcal{V}=\{v\in H^1(\Omega ;\mathbb{R}^d):\,v=0~~ \textrm{on }\Gamma _D\}$, $\mathcal{O}=C^1(\partial \Sigma )$, $\varphi =th$ and $\varphi _0=0$, we have \begin{eqnarray*} \frac{d}{dt}\mathcal{E}(u(t);f,g,\Omega (t))|_{t=0}&=& -R_{\Omega }(u,\mu _h) \Omega (t)=D\setminus \Phi _{th}(\Sigma )\\ \mu _h&=&h(\gamma (\mathcal{P}(x)))e_{\partial \Sigma ,1}(\gamma (\mathcal{P}(x))\beta (x) \end{eqnarray*}

At the point $x$ near $\partial \Sigma $, $\mu _h(x)=h(\gamma (\mathcal{P}(x)))e_{\partial \Sigma ,1}(\gamma (\mathcal{P}(x)))$ is the Parallel extension of the vector filed $h(\gamma )e_{\partial \Sigma ,1}(\gamma ), \gamma \in \partial \Sigma $ with the speed $h$. If $\| h\|_{C^1(\partial \Sigma )}$ is large, let $\epsilon h$ be $h$ with sufficiently small $\epsilon \gt 0$.

Let $\omega _0$ be a neiborhood of $\Sigma $, then $J_{\Omega \setminus \overline{\omega _0}}(u,\mu _h)=0$ since $\Phi _{th}(x)=x$ on $\partial D$ and $u$ has the interior regularity so that \begin{eqnarray*} R_{\Omega }(u,\mu _h)&=&R_{\Omega \setminus \overline{\omega _0}}(u,\mu _h) +R_{\omega _0}(u,\mu _h)\\ &=&-P_{\Omega \setminus \overline{\omega _0}}(u,\mu _h) +R_{\omega _0}(u,\mu _h)=J_{\omega _0}(u,\mu _h |\partial \omega _0). \end{eqnarray*} Let $\omega $ be a neighborhood of $\partial \Sigma $.

Neighbourhoods $\omega _0, \omega $ of the crack front $\partial \Sigma $ and the vector field $\mu _h$

Since $u |_{\omega _0\setminus \overline{\omega }}$ is regular except for the direction crossing $\Sigma $, $\widehat{T}^±(u)=0$ and $\mu _h\cdot n=0$ on $\Sigma $, we arrive at \begin{equation} \mathcal{G}(f,g,\Sigma _h(\cdot ))=J_{\omega }(u,\mu _h |\partial \omega )\left (\int _{\partial \Sigma }h(\gamma )d\gamma \right )^{-1}, \Sigma _h(\cdot ):=\{\Phi _{th}(\Sigma )\} _{t\in [0,\epsilon ]} \tag{2.44} \end{equation} which is another proof of Theorem 2.7 .

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Notes

Surface not flat

Similar results are obtained if we notice that on a plane, the shortest line connecting two points is a straight line, but on a general surface, the shortest line is a geodesic. By using geodesic coordinate, a point $x$ near the crack front $\partial \Sigma $ is expressed as $x=\mathcal{P}(x)+\lambda _3(x)e_{\partial \Sigma ,3}(\mathcal{P}(x))$, $\mathcal{P}(x)=g_{\Sigma }(e_{\partial \Sigma ,1}(\gamma (x)),\lambda _1(x))$. Here $[\lambda \mapsto g_{\Sigma }(e_{\partial \Sigma ,1}(\gamma (x)),\lambda )], 0\le \lambda \le \lambda _1(x)$ is the geodesic line passing through $\mathcal{P}(x)$ and crossing $\partial \Sigma $ in the direction $e_{\partial \Sigma ,1}(\gamma (x))$ at $\gamma (x)$ as shown below. Now, we define \begin{equation*} \Phi _h(x):=\mathcal{P}_{h\beta }(x)+\lambda _3(x)e_{\partial \Sigma ,3}(\mathcal{P}_{h\beta }(x)) \end{equation*} where $\mathcal{P}_{h\beta }(x):=g_{\Sigma }(e_{\partial \Sigma ,1}(\gamma (x)),\lambda _1(x)+h(\gamma (x))\beta (x))$ with a cut-off function $\beta $ near $\partial \Sigma $. If $x\notin \textrm{supp}_{\Pi }\beta $, then $$ \mathcal{P}_{h\beta }(x)=g_{\Sigma }(e_{\partial \Sigma ,1}(\gamma (x)),\lambda _1(x))=\mathcal{P}(x)$$ that is, $\Phi _h(x)=\mathcal{P}(x)+\lambda _3(x)e_{\partial \Sigma ,3}(\mathcal{P}(x))=x$, which means that $\Phi _h$ is defined at $x\in \mathbb{R}^d$.

$\mu _h(x)$ is the parallel transport (e.g., [Def.7.4.8, Pr10]) from $\gamma (x)$ to $\mathcal{P}(x)$ along the geodesic, and parallel to its normal when $x$ is near $\partial \Sigma $.

In [Oh81], the product neighborhood $(U,p)$ [Lemma 4.2, Oh81] was used, but for clarity of the geometric image, the geodesic coordinate system was used here. In [Oh81], it is also shown that the the velocity vector of the energy release rate is independent of $(U,p)$ [Lemma 5.3, Oh81].

product neighborhood $(U,p)$ subordinate to the crack front $\partial \Sigma $

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J-integral revisited

Consider the elasticity $\sigma _{ij}(\xi ,u)=C_{ijkl}(\xi )\varepsilon _{kl}(u)$ with $C_{ijkl}(\xi )$ depends on $\xi \in \Omega \subset \mathbb{R}^2$. $\Sigma :=\{(x_1,0): -a\le x_1\le 0\}$, $\Omega :=D\setminus \Sigma $.$\widehat{W}(\xi ,\varepsilon (u)):=\sigma (\xi ,u):\varepsilon (u)/2$
For a given $f\in L^2(\Omega ;\mathbb{R}^2), g\in L^2(\Gamma _N;\mathbb{R}^2)$, find $u\in V(\Omega )$ such that \begin{eqnarray*} \mathcal{E}(u;f,g,\Omega )&=&\min _{v\in V(\Omega )} \mathcal{E}(v;f,g,\Omega )\notag \\ \mathcal{E}(v;f,g,\Omega )&:=& \int _{\Omega }\{\widehat{W}(x,\varepsilon (v))-f\cdot v\} dx -\int _{\Gamma _N}g\cdot v~ds\\ V(\Omega _{\Sigma })&:=& \{v\in H^1(\Omega ;\mathbb{R}^2):v=0~\textrm{on }\Gamma _D\} \end{eqnarray*}

Crack growth $\Sigma (t)=\{(x_1,0):0\le x_1\le \ell (t)\}$ at time $t$.

Since the bilinear forms $a_{\Sigma (t)}(v,w):=\int _{\Omega (t)}\sigma (x,v):\varepsilon (w)dx, \Omega (t)=D\setminus \Sigma (t)$ are coercive on $V(\Omega _{\Sigma (t)})$ and $(f,g)\mapsto u(t)$ are bounded operator from $L^2(\Omega ;\mathbb{R}^2)\times L^2(\Gamma _N;\mathbb{R}^2)$ to $V(\Omega (t))$, we obtain from the similar argument in [Oh81] with $\varphi _t(x)=x+e_1\ell (t)\beta $ \begin{equation} -\left .\frac{d}{dt}\mathcal{E}(u(t);f,g,\Omega (t))\right |_{t=0} =R_{\Omega }(u,\ell '(0) e_1\beta )\tag{2.A1} \end{equation} Therefore, the energy release rate $$ \mathcal{G}(f,g,\Sigma (\cdot )):=\lim _{t→+0}\ell (t)^{-1}[\mathcal{E}(u;f,g,\Omega )-\mathcal{E}(u(t);f,g,\Omega (t))]$$ become the following from (2.A1) \begin{eqnarray} \mathcal{G}(f,g,\Sigma (\cdot ))&=&-\left .\frac{d}{dt}\mathcal{E}(u(t);f,g,\Omega (t))\right |_{t=0}(\ell '(0))^{-1}\notag \\ &=&R_{\Omega }(u,\ell '(0) e_1\beta )(\ell '(0))^{-1}=R_{\Omega }(u,e_1\beta )\tag{2.A2} \end{eqnarray} In Bui[Bu04], (2.A2) is called G-theta integral, and it is written that (2.A2) has been independently given by Ohtsuka (1981)[Oh81], deLorenzi (1982)[Lo82], Destuynder and Djaoua (1981)[D-D81].Also,in [Kn06, Kh-So99], (2.A2) is called Griffith formulaand [Kn-Mi08] for finite elasticity.

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Curve crack growth

(a) Crack tips $\gamma $ and $\phi _t(\gamma )$ (b) Parametrization $\rho $ of $\Pi $ by arc length

Consider the curve crack growth based on the idea of 3D problem[Oh81].

(1): Let $[\lambda _1\mapsto \rho (\lambda _1)]$ be the parametrization of $\Pi $ by arc length such as $\gamma =\rho (0)$, $\rho (\lambda _1)\in \Sigma $ if $\lambda _1\le 0$ and $\rho (\lambda _1)\notin \Sigma $ if $\lambda _1 \gt 0$.
(2): Put $\phi _t(\gamma )=\rho (\ell (t))$. Take a neighbourhood $U(\gamma )$ of $\gamma $ such that the foot of the perpendicular to $\Pi $ from $x\in U(\gamma )$ is uniquely $\rho (\lambda _1(x))\in \Pi $, that is, $x=\rho (\lambda _1(x))+\lambda _3(x)\nu (\rho (\lambda _1(x))$.
(3): $\varphi _t(x)=\rho (\lambda _1(x)+\ell (t)\beta (x))+\lambda _3(x)\nu (\rho (\lambda _1(x)+\ell (t)\beta (x))$ with a cut-off function $\beta $ near $\gamma $, $\textrm{supp}\beta \subset U(\gamma )$.

(a) Neighborhood $U(\gamma )$ and the map $[x\mapsto \varphi _t(x)]$ (b) Vector field $\mu _C$ obtained by parallel extension from crack growth

\begin{eqnarray} \mathcal{G}(f,g,\Sigma (\cdot ))&=&-\left .\frac{d}{dt}\mathcal{E}(u(t);f,g,\Omega (t))\right |_{t=0}(\ell '(0))^{-1}\notag \\ &=&R_{\Omega }(u,\mu _C\beta )(\ell '(0))^{-1}\tag{2.A5} \end{eqnarray} Unfortunately, (2.A5) does not hold if $\Sigma $ grows at an angular bend (see Crack Path ).

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