Strength-based Approach to Fracture

The strength-based approach to fracture holds that fracture occurs when $\sigma(\vec r)> \sigma_{UTS}$. This poses a problem when there are flaws in the material because $\sigma(\vec r)$ is difficult to track/calculate beyond simple loading conditions and/or simple geometries of a sample.

• Results from elasticity at a vacancy: $\frac{\sigma_{max}}{\sigma_{applied}} = 3$
  • Defects knock UTS down by a factor of 3.
• Stress concentration for a crack: $\frac{\sigma_{max}}{\sigma_{applied}}\approx 2\sqrt{\frac{a}{\rho}}$, where $a$ is the semi-major axis of the crack and $\rho$ is the radius of curvature of the crack.

Issues with a strength-based approach:

• $\sigma_{max}$ depends on the shape of the flaw, which is difficult to determine.
• Assumes linear elasticity everywhere, which is wrong.
• Assumes the strength of the material (that we have measured) does not depend on the specific sample we use. But in reality, strengths vary from sample to sample due to varying flaw size, shape, and distribution.

Conclusion: abandon strength-based approach, turn to energy-based approach

Energy-based Approach to Fracture

When a crack advances, the elastic potential energy is released.

• No crack: material under tension stores elastic energy
• Crack: elastic energy is released, which creates a net negative energy, which is the driving force for the propagation of a crack
  • What stops the propagation of a crack?

When a crack advances, more surface area is created, which costs energy.

• A material needs to absorb chemical energy to break bonds so that new surfaces can be made.

The tradeoff between the release of elastic energy and the absorption of chemical energy defines when a crack grows.

• Elastic energy decreases: $\triangle U_{elastic} =-\beta \frac{\sigma^2}{E}wa^2$, where $\beta$ is a geometrically dependent constant, $\sigma$ is the stress on a sample, $E$ is the elastic modulus of a sample, $w$ is the width of a sample, and $a$ is the crack’s length parameter
  • Surface cracks: length = $a$
  • Internal cracks: length = $2a$
• Surface energy increases: $\triangle U_{surface}=2\gamma(2aw)$, where $\gamma$ is the surface energy (the energy required to make 1 square meter of a material)
• Critical conditions for crack growth: $\triangle U_{total}=0$ —> $\sigma_{critical}=\sqrt(\frac{4\gamma E}{\beta a})$, where $\beta = 2\pi$
  • Similar approach: $\sigma_{critical}=\frac{K_c}{f\sqrt(\pi a)}$, where $f$ is a geometric factor, similar to $\beta$

Fracture Terminology