Contents

# Shape sensitivity of energy

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

### Griffith's energy balance theory

In the 2-dimensional case, cracks are curves and a smooth crack extension $\{\Sigma (t)\}$ along a curve $\Pi$ is defined more simply (does not need the geodesic coordinate), and $\partial \Sigma (t)$ is the set of points (crack tip). Let $\Sigma _0$ be the initial crack. Griffith's criterion [Gr21, Gr24]: Crack propagation occurs when there is a crack extension $\Sigma (t)\supset \Sigma _0$ such that $$\mathcal{G}(f,g,\Sigma (\cdot ))\ge C_F$$ where the constant $C_F$ is called the fracture toughness value. Here, it is not necessary to know the actual crack extension $\Sigma ^*(t)$, because the actual crack extension must satisfy $\mathcal{G}(f,g,\Sigma ^*(\cdot ))\ge \mathcal{G}(f,g,\Sigma (\cdot ))$. Therefore, for a line crack, the risk of cracking can be evaluated by calculating the J-integral (2.16a) (see [Oh09] for more).

Notes
The energy criterion proposed by Griffith can be said that “If the excessive energy is sufficient enough to break the material bonding so as to create the new crack surfaces, fracture takes place.”[4.1, Sumi].
Griffith's criterion
The discussion here is based on Bui[2.1 Griffith's theory of fracture, Bui04]. During a crack growth of the length amount $\ell (t)$, the energy required for fracture is $C_F\ell (t)$. Griffith suppose that the total energy does not change. $$\mathcal{E}(u(t);f,g,\Omega (t))+\mathcal{E}_k(u(t);\Omega (t))+C_F\ell (t)=0$$ where $\mathcal{E}_k(u(t);\Omega (t))$ is kinetic energy. Suppose that an equilibrium state is reached for the stationary crack $\Sigma$, with the extemal load $(f,g)$ so that any infinitesimal change of the equilibrium state with $\ell (t) \gt 0$ under $\mathcal{E}_k(u(t);\Omega (t)) = 0$ implies $$\mathcal{E}(u(t);f,g,\Omega (t))+C_F\ell (t)=0$$ This is the criterion of crack initiation at the equilibrium state of the solid at rest just at some critical load, which means \begin{eqnarray} \left .\frac{d}{dt}\mathcal{E}(u(t);f,g,\Omega (t))\right |_{t=0}+C_F\ell '(0)&=&0\notag \\ -\left .\frac{d}{dt}\mathcal{E}(u(t);f,g,\Omega (t))\right |_{t=0}&=&C_F\ell '(0)\notag \\ \mathcal{G}(f,g,\Sigma (\cdot ))&=&C_F\tag{2.A3} \end{eqnarray} Defining the quasi-static energy release rate $\mathcal{G}=C_F$, one gets the crack initiation criterion as $\mathcal{G}=C_F$. Crack propagation under $\mathcal{E}_k(u(t);\Omega (t)) = 0$ is called a controlled or stable propagation. By using (2.A2), the crack initiation criterion (2.A3) is　 $$R_{\Omega }(u,e_1 \beta )=C_F\tag{2.A4}$$ Personal additions. If $R_{\Omega }(u,e_1 beta) \lt C_F$, then we have from (2.A2) for small $t$ $$\mathcal{E}(u;f,g,\Omega )-\mathcal{E}(u(t);f,g,\Omega (t)) \lt C_F\ell (t)$$ which means that the energy required to realise virtual crack growth cannot be supplied. In other words, if $R_{\Omega }(u,e_1 \beta ) \lt C_F$, the crack cannot grow. If $R_{\Omega }(u,e_1 \beta ) \gt C_F$, then we have for small $t$ \begin{eqnarray*} \mathcal{E}_k(u(t);\Omega (t))-\mathcal{E}_k(u;\Omega ) &=&\mathcal{E}(u;f,g,\Omega )-\mathcal{E}(u(t);f,g,\Omega (t))-C_F\ell (t)\\ & \gt &0 \end{eqnarray*} which can be transformed in either kinetic energy or fracture energy to provide an additional propagation or both energies.

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