# Different types of fracture problems

Fracture mechanics began during the First World War when Griffith showed by energy balance and experimentation that the extremely low strength of glass in comparison with theoretical values was due to cracks in the glass (see Wikipedia
). Griffith find a crack size that weakens the strength of glass by considering the fracture in solids as a physical process which requires energy to create additional crack surface. *Griffith theory* can be said to deal with *fracture initiation problem*, as it evaluates danger crack size.

Brittle fractures have also occurred in metals at low temperatures such as the fracture of the Liberty
in the 1950s due to welds with low fracture toughness.

So we need *crack arrest design*[11.1.2 Fracture Control Design, Sumi], which stops brittle cracks within the local structure and prevents catastrophic failure of the entire structure. Crack arrest can be designed by placing materials with high fracture toughness in the direction of crack growth, or by locating structures, such as rivets, that eliminate stess intensity factor. For crack arrest design, it is necessary to predict the crack growth path, which is called the *crack path problem*, and is a mathematically difficult problem due to the discontinuity of the stress intensity factor and energy release rate at the kinked cracks (see e.g.[7.Crack Paths in Brittle Solids, Sumi]).

*Fatigue fracture* is phenomena in which microcracks with repeated loading in a material lead to crack growth and finally to failure of the material. The simplistic thinking is that crack grow by large loading and stop with rapid decrease, resulting in the crack sizes to increase and then to break. Unfortunately, fatigue fracture appears to be more complex phenomena (see e.g.[p.25, Bu04], [5.Fatigue Crack Propagation, Sumi]).

## Various fracture problems

Note for further study.

- Surface crack:
- Let $\Omega $ be a domain without crack, and $\Sigma $ the crack surface assuming the part of the surface $\Pi $. In surface crack problem, $\partial \Sigma \cap \partial \Omega \not =\emptyset $. In internal crack problems $\partial \Sigma \cap \partial \Omega =\emptyset $, stress-free on $\Sigma $ generally, but in surface crack problem, surface force on $\Sigma $ can occur, as in pressure vessels (see e.g., [Kn15, Oh86])
- Paris law in fatigue fracture:
- See [p.26, Bu04], [(5.5), Sumi].
- Fatigue Crack Paths:
- In fatigue fracture, this is necessary because the cracks grow little by little (see e.g., [Fatigue Crack Paths, Sumi]).
- LBB:
- In 3D fracture, there is the leak-before-break (LBB) c concept that a crack would grow through the wall, resulting in a leak, before the crack grow to unstable size. This is 3D fatigue crack path (shape) problem.
- Non penetration condition:
- Conditions to avoid overlapping crack surfaces appearing in linear fracture mechanics, which are researched by A. Khludnev et al. (e,g., [K-K99], [Ko06]).

# Definitions for mathematisation

- $\Omega $:
- Domain without crack. The boundary $\partial \Omega $ has Lipschitz property, that is expressed the graph of Lipschitz continuous function.
- $\Sigma $:
- The crack. that is discontinuous surface of the displacement. It considers the main crack in a fracture phenomenon.
- $\Omega _{\Sigma }$:
- $\Omega _{\Sigma }=\Omega \setminus \Sigma $.
- $\Pi $:
- Virtual surface on which cracks will grow. $\Omega =\Omega ^+\cup (\Pi \cap \Omega )\cup \Omega ^-$, and $\Sigma \subset \Pi $. Here $\Omega ^{±}$ has the boundary $\partial \Omega ^{±}$ has Lipschitz property.
- $(\sigma , \epsilon , u )$ :
- $\sigma =(\sigma _{ij}), \varepsilon =(\varepsilon _{ij}), u=(u_1,…, u_d)$ are stress tensor, strain tensor, displacement vector, respectively.
- Infinitesimal deformation:
- $\epsilon _{ij}=(\partial _j u_i+\partial _i u_j)/2$. In linear elastic fracture mechanics (LEFM), infinitesimal deformation is assumed. If the size of the plastic zone near the crack tip is sufficiently small (Small Scale Yielding), LEFM is applicable with some modification. Cannot be assumed for ductile fracture or finite deforming materials. In
*brittle fracture*, LEFM is valid. - Linear elasticity:
- $\sigma _{ij}=C_{ijkl}\varepsilon _{kl}$, where $C_{ijkl}$ is called Hooke's tensor. The material is isotropic, $C_{ijkl}=\lambda \delta _{ij}\delta _{kl}+ G(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})$, Lame's constants $\lambda ,G$ .
- The unit normal vector $\nu $:
- is the outward unit normal vector $n^-$ of $\partial \Omega ^-$, that is the normal vector crossing $\Sigma $ from the negative side to the positive side.
- $(\sigma ^{±}, \varepsilon ^{±}, u^±)$:
- The traces of $(\sigma , \varepsilon , u)$ on $\Sigma $ from the positive side or the negative side. For example, $u|_{\Sigma }^±(x)=\lim _{\epsilon →+0}u(x±\epsilon \nu (x)), x\in \Sigma $.

# Mathematical fracture problems

- Initial crack:
- Cracks found during production or inspection.
- Fracture initiation:
- Problem of considering the conditions when the crack will grow. Griffith criterion gives the fracture initiation condition. The engineering environment is that of the moment when the crack grows.
- Griffith criterion:
- Let $\mathcal{E}_{elast}(\Sigma ( \ell ) ), \mathcal{E}_{ext}(\Sigma ( \ell )), \mathcal{E}_{kin}(\Sigma ( \ell ) )$ be the elastic strain energy , the potential energy of external loads, the kinetic energy, and $\Sigma (\ell )$ the virtual crack growth with crack length $\ell \ge \ell _0$ as the parameter. $\ell _0$ the length of the initial crack. Griffith postulated that $$ \mathcal{F}(\Sigma (\ell )):= \mathcal{E}_{elast}(\Sigma (\ell ))+\mathcal{E}_{ext}(\Sigma (\ell ))+\mathcal{E}_{kin}(\Sigma (\ell ))+\mathcal{E}_{\Sigma }(\Sigma (\ell ) )=0$$ where he assumed that $\mathcal{E}_{\Sigma }(\Sigma (\ell ) )=C_F\ell $. If $\mathcal{E}_{kin}(\Sigma (\ell ))=0$, the fracture is called
*quasistatic fracture*. $\mathcal{G}(\Sigma (\ell _0+\cdot )):=-\frac{\partial }{\partial \ell }(\mathcal{E}_{elast}(\Sigma (\ell ))+\mathcal{E}_{ext}(\Sigma (\ell )))|_{\ell =\ell _0}$ is called the quasistatic*energy release rate (ERR)*. The criterion of fracture initiation is that $\mathcal{G}(\Sigma (\ell _0+\cdot ))=C_F$. Here we notice that $\mathcal{E}_{ext}(\Sigma (\ell )))$ does not change with respect to $\Sigma (\ell )$, that is called the critical load.