Failure Theories

Why it matters

Failure theories are crucial in machine design as they help predict the conditions under which materials or components might fail. Understanding these theories ensures that engineers can design machines that are both safe and efficient, preventing costly and dangerous failures in real-world applications.

Key ideas

• Failure Theories: These are mathematical models that predict the failure of materials under different states of stress. They are essential for designing components that can withstand operational loads without failure.
• Types of Failure: The two primary types of failure are ductile and brittle. Ductile failure involves significant plastic deformation, while brittle failure occurs with little to no plastic deformation.
• Common Theories:
  • Maximum Normal Stress Theory: Suitable for brittle materials, it compares tensile and compressive principal stresses with their respective strengths; brittle materials can have very different tensile and compressive strengths. More appropriate interaction criteria may be needed under combined stresses.
  • Maximum Shear Stress Theory (Tresca): Used for ductile materials, it suggests failure occurs when the maximum shear stress reaches the shear yield strength.
  • Distortion Energy Theory (von Mises): Also for ductile materials, it predicts failure when the distortion energy per unit volume reaches a critical value.

Tresca predicts first yield when the largest principal-stress difference equals the uniaxial yield strength: max(|σ1-σ2|, |σ2-σ3|, |σ3-σ1|) = σ_y. Thus τ_max = σ_y/2 at first yield. Both Tresca and von Mises describe ductile yielding, not fatigue, buckling or all possible fracture mechanisms. Include all three principal stresses, including the zero out-of-plane stress in plane stress.

Formulas

• Maximum Normal Stress Theory: σ_max = σ_ult
  • σ_max: Maximum normal stress (Pa)
  • σ_ult: Ultimate tensile strength (Pa)
• Maximum Shear Stress Theory: τ_max = τ_y
  • τ_max: Maximum shear stress (Pa)
  • τ_y: Shear yield strength (Pa)
• Distortion Energy Theory: σ_v = σ_y
  • σ_v: von Mises stress (Pa)
  • σ_y: Yield strength (Pa)

Worked example

Given: A steel rod with a yield strength of 250 MPa is subjected to principal stresses of 150 MPa, 100 MPa, and -50 MPa.

1. Calculate von Mises stress using the formula: σ_v = sqrt(0.5 * ((σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2))
   • σ1 = 150 MPa, σ2 = 100 MPa, σ3 = -50 MPa
2. Substitute the values: σ_v = sqrt(0.5 * ((150 - 100)^2 + (100 + 50)^2 + (-50 - 150)^2)) σ_v = sqrt(0.5 * (2500 + 22500 + 40000)) σ_v = sqrt(0.5 * 65000) σ_v = sqrt(32500) σ_v = 180.28 MPa
3. Compare with yield strength: Since σ_v = 180.28 MPa < 250 MPa, von Mises predicts no first yield at this load. The achieved von Mises factor is 250/180.28 ≈ 1.387. Tresca gives equivalent stress 150 - (-50) = 200 MPa and factor 1.25. Whether the design is acceptable depends on the required factor and other failure modes.

Final Answer: 180.28 MPa

Common mistakes

• Confusing the application of failure theories for ductile and brittle materials.
• 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 comparing them with material strengths. Practice problems on identifying suitable failure theories for given materials and conditions.

Quick check

1. What is the primary difference between ductile and brittle failure?
2. Which failure theory is most suitable for ductile materials?
3. How is von Mises stress calculated?

Answers: 1. Amount of plastic deformation before failure. 2. Distortion Energy Theory (von Mises). 3. Using the formula σ_v = sqrt(0.5 * ((σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2)).