Notes on Failure Theories in Materials

Introduction to Failure in Materials

• Loads on objects can lead to material failure.
• Predicting static failure involves understanding stress levels required for failure.

Definition of Failure

• Ductile Materials: Failure = onset of plastic deformation.
• Brittle Materials: Failure = fracture.
• Easy to define for uniaxial stress (tensile test):
  • Ductile: at yield strength.
  • Brittle: at ultimate strength.

Complex Stress States

• Tri-axial stress state complicates failure predictions.
• No universal method; we must choose from various failure theories based on experimental validation.
• Ductile vs. brittle materials require different failure theories.

Purpose of Failure Theories

• Allow prediction of material failure by comparing stress state with material properties (yield/ultimate strengths).
• Stress state described using three principal stresses.

Maximum Principal Stress Theory

• Simplest theory: failure occurs when max/min principal stresses reach yield/ultimate strengths.
• Not effective for ductile materials.

Key Observations for Ductile Materials

• Hydrostatic stresses do not cause yielding.
• Yielding in ductile materials caused by shear deformation (deviatoric stresses), not hydrostatic stresses.

Hydrostatic and Deviatoric Stresses

• Hydrostatic stresses: equal principal stresses, no shear stresses.
• Deviatoric stresses: calculated by subtracting hydrostatic component from principal stresses.

Moore's Circle

• Useful for visualizing stress states:
  • Hydrostatic stress configuration leads to a single point.
  • Deviatoric component impacts yielding.

Better Failure Theories for Ductile Materials

Tresca Criterion (Maximum Shear Stress Theory)

• Yielding occurs when max shear stress equals shear stress at yielding in a tensile test.
• Depicts hydrostatic stress independence.

Von Mises Criterion (Maximum Distortion Energy Theory)

• Yielding occurs when distortion energy equals that at yielding in uniaxial tensile test.
• Considers effect of deviatoric stresses, independent of hydrostatic stress.

Yield Surfaces Comparison

• Yield surfaces represent failure theories in principal stress space.
• Principal Stress Ordering: sigma A, sigma B, sigma C (with sigma C = 0 for plane stress).
• Tresca Yield Surface: determined by max/min principal stress differences.
• Von Mises Yield Surface: elliptical shape, calculated based on principal stress differences.

Comparison of Theories

• Maximum principal stress theory is less reliable.
• Tresca and von Mises theories align closely with experimental results, but von Mises is often preferred for better accuracy.
• Both theories remain consistent when extended to three dimensions.

Failure Theories for Brittle Materials

• Brittle failure defined by fracture, not yielding.
• Requires separate tension/compression ultimate strengths.

Coulomb-Mohr Theory

• Sensitive to hydrostatic stresses and distinguishes between tensile and compressive strengths.
• Failure envelope formed by tangent lines to Mohr's circles from tensile and compressive tests.

Modified Mohr Theory

• Variation that improves fit with experimental data.
• Applicable for brittle materials, especially when considering hydrostatic effects.

Conclusion

• Overview of common failure theories, with availability of more specific theories for unique scenarios.
• Further discussions available on support channels like Patreon.