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.