Mechanics of Materials

Stress is the internal resistance of a material to deformation, defined as force per unit area. Strain is the measure of deformation representing the displacement of particles in the body relative to their reference position:

Where:

• σ = stress (Pa or N/m²)
• F = force (N)
• A = cross-sectional area (m²)
• ε = strain (dimensionless)
• ΔL = change in length (m)
• L₀ = original length (m)

Hooke's Law states that the strain in a solid is proportional to the applied stress within the elastic limit of that material:

Where:

• σ = stress (Pa)
• E = Young's modulus or modulus of elasticity (Pa)
• ε = strain (dimensionless)

Young's modulus is a measure of the stiffness of a solid material. It defines the relationship between stress and strain in a material.

Yield criteria determine when a material begins to yield (deform plastically). Two common criteria are:

1. Von Mises Yield Criterion:

2. Tresca Yield Criterion:

Where:

• σᵥₘ = von Mises stress (Pa)
• σₜᵣₑₛ = Tresca stress (Pa)
• σ₁, σ₂, σ₃ = principal stresses (Pa)

Yielding occurs when the calculated stress exceeds the yield strength of the material.