Theories of Failure

Why it matters

Understanding the theories of failure is crucial for engineers to predict when a material or structure will fail under various loading conditions. This knowledge helps in designing safer and more efficient structures, ensuring reliability and longevity in applications ranging from bridges to aircraft.

Key ideas

• Failure Theories: These are mathematical models that predict the failure of materials under complex stress states. They are essential for ensuring that structures can withstand the loads they encounter in real-world applications.
• Types of Failure: Materials can fail due to yielding (plastic deformation) or fracture (breaking). Theories of failure help predict both types.
• Common Theories:
  • Maximum Normal Stress Theory: Predicts failure when the maximum normal stress in a material reaches the ultimate tensile strength.
  • Maximum Shear Stress Theory (Tresca): Predicts initial yielding of an isotropic ductile material when maximum shear stress reaches σ_y/2, calibrated to uniaxial yield.
  • Distortion Energy Theory (von Mises): Proposes that failure occurs when the distortion energy per unit volume reaches a critical value.
  • Mohr's Theory: Considers both normal and shear stresses, predicting failure based on the Mohr's circle.

Choosing a criterion

Tresca and von Mises predict initial yielding under the stated idealizations, not every failure mode. Maximum-normal-stress criteria need both tensile and compressive strengths when applicable to brittle materials. Fatigue, buckling, creep, and fracture need separate assessment.

In three dimensions, σ_v = √{[(σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²]/2}. The two-stress expression below assumes plane stress, σ₃ = 0.

Formulas

• Maximum Normal Stress Theory: σ_max = σ_ult
  • σ_max: Maximum normal stress (Pa)
  • σ_ult: Ultimate tensile strength (Pa)
• Tresca yield criterion: max(|σ₁−σ₂|, |σ₂−σ₃|, |σ₃−σ₁|) = σ_y
  • τ_max: Maximum shear stress (Pa)
  • σ_y: Uniaxial yield strength (Pa); equivalently τ_max = σ_y/2
• Distortion Energy Theory: σ_v = √(σ_1² - σ_1σ_2 + σ_2²) = σ_y
  • σ_v: von Mises stress (Pa)
  • σ_1, σ_2: Principal stresses (Pa)
  • σ_y: Yield strength (Pa)

Worked example

Given: A steel bar with yield strength σ_y = 250 MPa is subjected to principal stresses σ_1 = 150 MPa and σ_2 = 100 MPa, with the third principal stress zero (plane stress).

1. Calculate von Mises stress using the Distortion Energy Theory:
   • Formula: σ_v = √(σ_1² - σ_1σ_2 + σ_2²)
   • Substitute values: σ_v = √((150 MPa)² - (150 MPa)(100 MPa) + (100 MPa)²)
   • Calculation: σ_v = √(22500 - 15000 + 10000) MPa
   • σ_v = √17500 MPa
   • σ_v ≈ 132.29 MPa
2. Compare with yield strength:
   • Since σ_v < σ_y, this idealized von Mises model does not predict initial yielding. It does not establish safety against all failure modes.

Final Answer: 132.29 MPa

Common mistakes

• Confusing the different failure theories and applying the wrong one for a given situation.
• Incorrectly calculating principal stresses or von Mises stress.
• Forgetting to compare calculated stresses with material properties like yield strength.

For GATE ME

Questions often involve calculating stresses using different failure theories and determining whether a material will fail. Practice problems that require understanding the conditions under which each theory applies and how to compute stresses accurately.

Quick check

1. What is the main difference between Tresca and von Mises theories?
2. How do you determine if a material will fail using the Maximum Normal Stress Theory?
3. What is the significance of principal stresses in failure theories?

Answers: 1. Tresca uses maximum shear stress, von Mises uses distortion energy. 2. Compare maximum normal stress to ultimate tensile strength. 3. They are used to calculate von Mises stress and assess failure.