Crack Theory

#### Introduction

• Cracks are small imperfections within a material that are created when it splits apart.
• These tend to propagate and cause further cracking by creating new surfaces, eventually splitting and causing the material to fail.
• Cracks can be caused by a variety of reasons, mostly to do with an applied load in the wrong position or direction.
• These can include:
• Machining, Forging and Welding defects.
• Thermal expansion and contraction
• Corrosion pitting
• Incorrect procedure during forming processes
• When a crack begins to grow, the material releases its strain energy.
• Strain energy is found through $SE = \frac{\sigma^2}{2E}$ or $SE = \frac{1}{2} \sigma \epsilon$
• Maximum strain energy is often defined as Resilience and can be found with a stress strain diagram.
• This is covered in detail later.

#### Crack propagation

• Crack propagation occurs the crack approaches the Critical Crack Length (Covered later).
• At that length, an imperfection concentrates the load on the object at the bond, which would fail under the concentrated load.
• This then concentrates double the stress at the next bond, and so on until the entire material fails.
• In relation to strain energy, the released strain energy from a crack is released in the area adjacent to the crack.
• The released energy is now concentrated at the tip of the crack, which builds up.
• Thus, most of the time cracks get larger and grow faster.
• However if the energy drops below a threshold, the crack will halt its growth.

#### Critical Crack Length

• This is the length of a crack that will cause the growth of the crack to proceed through the entire material.
• Brittle materials have a very short critical crack length.
• It can be found by:
(1)
\begin{align} L_g = \frac{1}{\pi} \times \frac{\frac{"Work of Fracture"}{"Unit Area of Crack Surface"}}{\frac{"Strain Energy Stored"}{"Unit Volume of Material"}} \end{align}
(2)
\begin{align} L_g = \frac{2WE}{\pi \sigma^2} \end{align}
• Where:
• $L_g$ is the critical crack length (m)
• W is the work of fracture for each surface (J/$m^2$)
• E is Young's Modulus (Pa)
• $\sigma$ is the average tensile stress (Pa)

#### Failure due to cracking

• Note that the longer $L_g$ is the less likely a material will fail.
• And by its corollary, Brittle materials have a very short critical crack length.
• Once the $L_g$ has been exceeded, failure is inevitable if stress levels are maintained.
• A common cause is fatigue failure from cyclic loading, which could cause a small imperfection.

#### Preventing and Repairing Cracks

• Cracks can be prevented by:
• Rounding off corners to allow more even distribution of load
• Perforating(punching a hole in) areas susceptible to tearing open to stop crack growth
• Lowering the stress at susceptible areas.
• Reinforcing the material (e.g. reinforced concrete to reduce brittleness)
• Adding expansion joints to prevent expansion cracking
• Placing things perpendicular to the crack shear plane to block growth.
• Cracks can be repaired by:
• Crushing small cracks together in different directions in hope of stopping it themselves.
• Changing the volume of the material through injection etc. to squeeze the cracks shut.
• Welding or using Adhesives to patch up the cracks.
• Welding for metals and adhesives for polymers.
page revision: 1, last edited: 06 Jul 2011 12:40