Failure theories are crucial for predicting when materials will break or deform under stress. They help engineers choose the right materials and design safe structures. Different theories work best for different materials, from ductile metals to brittle ceramics.

Understanding these theories is key to preventing catastrophic failures in real-world applications. By applying the right failure criteria, engineers can ensure that products and structures can withstand the loads they'll face during their lifetime.

Failure Theories

Maximum Stress and Strain Theories

• Maximum normal stress theory predicts failure when the maximum principal stress exceeds the ultimate tensile strength of the material
• Maximum shear stress theory (Tresca criterion) predicts failure when the maximum shear stress exceeds half the yield strength of the material
  • Assumes that yielding begins when the maximum shear stress reaches a critical value
  • Commonly used for ductile materials (mild steel, aluminum)
• Distortion energy theory (von Mises criterion) predicts failure when the distortion energy exceeds the distortion energy at yield in a simple tension test
  • Considers the effect of all principal stresses on the yielding of the material
  • Widely used for ductile materials (structural steel, copper)

Coulomb-Mohr Theories

• Coulomb-Mohr theory predicts failure based on a combination of normal and shear stresses acting on a particular plane
  • Considers the effect of both normal and shear stresses on failure
  • Suitable for materials with different strengths in tension and compression (cast iron, concrete)
• Modified Mohr theory is an extension of the Coulomb-Mohr theory that accounts for the effect of the intermediate principal stress
  • Provides more accurate predictions for materials with different strengths in tension and compression
  • Applicable to a wider range of materials and loading conditions (composite materials, rocks)

Fracture Mechanics

Fracture Mechanics Concepts

• Fracture mechanics is the study of the propagation of cracks in materials
  • Deals with the analysis of stress and strain fields around crack tips
  • Helps predict the critical crack size and the remaining life of a component
• Stress intensity factor ($K$) is a measure of the stress state near the tip of a crack
  • Depends on the applied stress, crack size, and geometry of the component
  • Used to determine the critical crack size and the fracture toughness of the material ($K_{Ic}$)

Brittle-Ductile Transition

• Brittle-ductile transition is the temperature range over which a material's behavior changes from brittle to ductile
  • Brittle materials fail suddenly with little plastic deformation (glass, ceramics)
  • Ductile materials undergo significant plastic deformation before failure (metals)
  • The transition temperature depends on the material, strain rate, and the presence of notches or cracks

Stress States

Plane Stress and Plane Strain

• Plane stress is a state of stress in which one of the principal stresses is zero
  • Occurs in thin plates loaded in their plane (pressure vessels, aircraft skins)
  • Allows for simplification of stress analysis by assuming a two-dimensional stress state
• Plane strain is a state of strain in which one of the principal strains is zero
  • Occurs in thick components under loading (thick-walled cylinders, foundation piles)
  • Requires consideration of all three principal stresses for accurate analysis
  • Results in higher constraint and reduced ductility compared to plane stress conditions