Contents

# Shape sensitivity of enery

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

### J-integral

Let $D=(-1,1)\times (-1,1)$, $\Sigma =\{(x_1,x_2):-1\le x_1\le 0, x_2=0\}$, $\Omega =D\setminus \Sigma$ and $\Gamma _D=\{(x_1,x_2):\, x_1=1, -1 \lt x_2 \lt 1\}$. Consider the straight crack extension $$\Omega (t)=D\setminus \Sigma (t), \Sigma (t)=\{(x_1,x_2):-1\le x_1\le t, x_2=0\} .$$ Here $t \gt 0$ stands for the crack increment. Let us denote the energy at the crack increment $t$ by $$\mathcal{E}(u(t);f,\Omega (t)) :=\int _{\Omega (t)}\left \{\widehat{W}(u(t))-f\cdot u(t)\right \} dx$$ When the elastic plate is homogineous and $f=0$ near the crack tip $\partial \Sigma =\gamma _{\Sigma }$, G.P. Cherepanov[Ch67] and J. Rice[Ri68] showed that \begin{eqnarray} \mathcal{G}(f,\Sigma (\cdot ))&=&J, J:=\int _{C}\left (\widehat{W}(u)dx_2-\widehat{T}(u) \frac{\partial u}{\partial x_1}ds\right ) \tag{2.16a} \end{eqnarray} \begin{eqnarray} \mathcal{G}(f,\Sigma (\cdot ))&:=& \lim _{t→0+}t^{-1}[\mathcal{E}(u;f,\Omega )-\mathcal{E}(u(t);f,\Omega (t))] \tag{2.16b} \end{eqnarray} where $C$ is a closed curve surrounding the crack tip $\gamma _{\Sigma }$. Here $\mathcal{G}(f,\Sigma (\cdot ))$ given in (2.16b) is called the strain energy release rate and the right-hand side of (2.16a) is the original J-integral ((2.16) is proved mathematically in [D-D81, Oh81]). If $C$ is parametrized by arc length $s$, that is, $C=\{(x_1(s),x_2(s));\, a\le x\le b\}$, then the outward unit norma $n=(n_1,n_2)$ at $(x_1(s_0),x_2(s_0))$ is equivalent to $$n=\left (\frac{dx_2}{ds}(s_0),-\frac{dx_1}{ds}(s_0)\right )$$ which means that $dx_2=n_1ds=(n\cdot e_1)ds$ with the unit vector $e_1$ in the $x_1$-direction. Let $\omega$ be the domain such that $\partial \omega =C$. Since $f=0$ in $\omega$, $\nabla e_1=0$ and $\nabla C_{ijkl}=0$ by homogeneity, we have $R_{\omega }(u,e_1)=0$, so that the J-integral (2.16a) is rewritten by GJ-integral (2.5) as \begin{eqnarray} J&=&P_{\omega }(u,e_1)    \textrm{See \eqref{eqn:GP-intCrack}} \tag{2.17} \end{eqnarray} If the elastic plate is isotropic, then near the crack tip, the following exapansion hold using the polar coordinate $(r,\theta )$ \begin{eqnarray} \vec u(x)&=& \sum _{i=1}^3 \vec{S}_i^C(r,\theta )+\vec{u}_R(x),  \vec{u}_R\in H^2(\textrm{near }\gamma _{\Sigma };\mathbb{R}^2)\tag{2.18a} \end{eqnarray} \begin{eqnarray} \vec{S}_i^C(r,\theta )&:=& \frac{K_i(\gamma _{\Sigma })}{2G}\sqrt{\frac{r}{2\pi }} \vec{\Phi }_i(\theta ) \textrm{ for }i=1,2 \tag{2.18b} \end{eqnarray} where the constant $K_i(\gamma _{\Sigma })$ are called the stress intensity factors (SIF) at $\gamma _{\Sigma }$ (for mathematical proof see e.g.[Chapter 7, Na94]) and \begin{eqnarray} \vec{\Phi }_1(\theta ) &;=& \left (\array{\cos (\theta /2)\left ({\kappa -1+2\sin ^2(\theta /2)}\right )\\ ~~~~ \sin (\theta /2)\left ({\kappa +1- 2\cos ^2(\theta /2)}\right )}\right ) \tag{2.19a} \end{eqnarray} \begin{eqnarray} \vec{\Phi }_2(\theta ) &:=& \left (\array{\sin (\theta /2)\left ({\kappa +1+2\cos ^2(\theta /2)}\right )\notag \\ -\cos (\theta /2)\left ({\kappa -1-2\sin ^2(\theta /2)}\right ) } \right ) \tag{2.19b} \end{eqnarray} where $\kappa =3-4\nu$ (plane strain); $\kappa =(3-\nu )/(1+\nu )$ (plane stress) with the Poisson ratio $\nu$. Hence substituting (2.18a) in J-integral (2.16a) , we obtain $$\mathcal{G}(f,\Sigma (\cdot ))=\frac{\kappa +1}{8G}(K_1(\gamma _{\Sigma })^2+K_2(\gamma _{\Sigma })^2) \tag{2.20}$$ Here we notice that $u\notin H^{3/2}(\textrm{near }\gamma _{\Sigma };\mathbb{R}^2)$, so that we cannot apply Theorem 2.6 to $u$ near $\gamma _{\Sigma }$. From Theorem 2.2 we note that if $f\neq 0$ in the region enclosed in $C$ or the plate is not homogeneous, the J-integral (2.16a) depends on $C$''. $\mathcal{G}$ is the important parameter in Griffith's energy balance theory which is the basis of fracture mechanics (see [Bu04, Oh09, Sumi] for details).

(a) Polar coordinate system near the crack tip (front) $\gamma _{\Sigma }$
(b) In-plane deformed configurations near the crack tip $\gamma _{\Sigma }$: $K_1(\gamma _{\Sigma })\neq 0$ (mode I) indicates that in-plane deformed configurations near the crack tip is opening, and $K_2(\gamma _{\Sigma })\neq 0$ (mode II) corresponds to shearing of the crack face.

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