BME Advanced Mechanics 5 — Questions and Answers
Question 1: Which theory of failure predicts that fracture occurs when the maximum principal stress reaches the uniaxial tensile strength, making it most suitable for:
- Ductile metals under combined loading
- Brittle materials like ceramics and cast iron (Correct answer)
- Polymers at elevated temperatures
- Composites under shear loading
Correct answer: Brittle materials like ceramics and cast iron
The maximum normal stress (Rankine) criterion applies to brittle materials because they fracture with minimal plastic deformation when the principal stress reaches the tensile strength.
Question 2: For a thick-walled cylinder under internal pressure p_i with inner radius a and outer radius b, the hoop stress is maximum at:
- The outer surface (r = b)
- The mid-thickness (r = (a+b)/2)
- The inner surface (r = a) (Correct answer)
- Uniformly throughout the wall
Correct answer: The inner surface (r = a)
By the Lamé equations, hoop stress decreases radially outward and is highest at the inner wall r = a.
Question 3: In torsion of non-circular cross-sections, Prandtl's stress function φ satisfies which governing equation?
- ∇²φ = 0 (Laplace equation)
- ∇²φ = −2Gθ (Poisson's equation) (Correct answer)
- ∇⁴φ = 0 (biharmonic equation)
- ∂²φ/∂x² = Gθ
Correct answer: ∇²φ = −2Gθ (Poisson's equation)
Prandtl's stress function satisfies the Poisson equation ∇²φ = −2Gθ within the cross-section, with φ = 0 on the boundary.
Question 4: The Airy stress function approach in 2D elasticity ensures equilibrium automatically. The compatibility condition requires the Airy function φ to satisfy:
- ∇²φ = 0
- ∇⁴φ = 0 (Correct answer)
- ∇²φ = p/E
- ∂⁴φ/∂x⁴ = 0
Correct answer: ∇⁴φ = 0
The Airy stress function must satisfy the biharmonic equation ∇⁴φ = 0 in the absence of body forces to ensure strain compatibility.
Question 5: A shaft is subjected to combined bending moment M and torque T. The von Mises equivalent stress for design is:
- √(σ² + τ²)
- √(σ² + 3τ²) (Correct answer)
- σ + τ
- √(σ² − τ²)
Correct answer: √(σ² + 3τ²)
The von Mises criterion gives σ_eq = √(σ² + 3τ²) for combined normal stress σ and shear stress τ in a shaft.
Question 6: Hardness testing (e.g., Vickers, Brinell) is related empirically to which mechanical property?
- Young's modulus
- Fracture toughness
- Ultimate tensile strength (Correct answer)
- Endurance limit
Correct answer: Ultimate tensile strength
Empirically, UTS (MPa) ≈ 3.45 × HB (Brinell hardness) for steels, making hardness a convenient non-destructive proxy for tensile strength.
Question 7: The concept of 'stress triaxiality' is important in fracture mechanics because higher triaxiality:
- Reduces the yield stress required for plastic deformation
- Suppresses plastic flow and promotes brittle cleavage fracture (Correct answer)
- Increases ductility by spreading deformation uniformly
- Has no effect on fracture mode
Correct answer: Suppresses plastic flow and promotes brittle cleavage fracture
High stress triaxiality (hydrostatic tension) constrains shear deformation, suppressing plasticity and making the material behave more brittlely.
BME Advanced Mechanics 5
Which theory of failure predicts that fracture occurs when the maximum principal stress reaches the uniaxial tensile strength, making it most suitable for: