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### ✅ **Stress & Strain**

- **Normal (tensile) stress**  
  σ = Ft / A0

- **Shear stress**  
  τ = Fs / A0

- **Normal (tensile) strain**  
  ε = ΔL / L0

- **Shear strain**  
  γ = Δx / y

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### ✅ **Elastic Behavior**

- **Hooke’s Law (tension)**  
  σ = E * ε

- **Shear modulus**  
  G = E / (2 * (1 + ν))

- **Bulk modulus**  
  K = E / (3 * (1 - 2 * ν))

- **Bulk pressure**  
  P = -K * (ΔV / V0)

- **Poisson’s ratio**  
  ν = -ε_lateral / ε_longitudinal

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### ✅ **Plastic Deformation**

- **Offset strain at yield point**  
  ε_p = 0.002

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### ✅ **True Stress & Strain**

- **True stress**  
  σ_T = σ * (1 + ε)

- **True strain**  
  ε_T = ln(li / l0)

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### ✅ **Ductility**

- **Percent elongation**  
  %EL = 100 * (Lf - L0) / L0

- **Percent area reduction**  
  %RA = 100 * (A0 - Af) / A0

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### ✅ **Resilience & Toughness**

- **Resilience (elastic energy)**  
  Ur = 0.5 * σ_y * ε_y

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### ✅ **Useful Deformation Equations**

- **Axial deflection**  
  δ = (F * L0) / (E * A0)

- **Torsion angle**  
  α = (2 * M * L0) / (π * r04 * G)

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### ✅ **Hardness Estimation**

- **Tensile strength from Brinell hardness**  
  TS ≈ 3.45 * HB

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### ✅ **Material Statistics**

- **Mean**  
  x̄ = sum(xi) / n

- **Standard deviation**  
  s = sqrt(sum((xi - x̄)2) / (n - 1))

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### ✅ **Safety Factor**

- **Working stress**  
  σ_working = σ_y / N