Deformation Model
Spring Systems
F = -kx
The original formulation of Hooke's Law for helical springs and elastic elements. Force is proportional to displacement, with the negative sign indicating restoring force direction.
F
Force
N (Newtons)
k
Spring Constant
N/m
x
Displacement
m (meters)
Spring Constants
F = -kx| Material | Value | Unit |
|---|---|---|
| Steel Spring (Light) | 100-500 | N/m |
| Steel Spring (Medium) | 500-2,000 | N/m |
| Steel Spring (Heavy) | 2,000-10,000 | N/m |
| Rubber Band | 10-50 | N/m |
| Suspension Coil | 15,000-50,000 | N/m |
Applications
- • Mechanical springs in suspensions, valves, and actuators
- • Vibration isolation and damping systems
- • Force measurement (spring scales, load cells)
Deformation Model
Tensile Stress-Strain
σ = Eε
Uniaxial stress-strain relationship for rods, bars, and structural members. Young's modulus (E) represents material stiffness in the elastic region before yielding.
σ
Stress
Pa (Pascals)
E
Young's Modulus
Pa (GPa typical)
ε
Strain
dimensionless (m/m)
Young's Modulus Values
σ = Eε| Material | Value | Unit |
|---|---|---|
| Steel (Structural) | 200 | GPa |
| Aluminum (6061) | 68.9 | GPa |
| Copper (Pure) | 110-130 | GPa |
| Titanium (Grade 5) | 113.8 | GPa |
| Concrete | 25-30 | GPa |
Applications
- • Structural beam and column design
- • Axial loading of rods, cables, and fasteners
- • Thermal expansion stress calculations
Deformation Model
Shear Stress-Strain
τ = Gγ
Shear deformation relationship for torsional loading, bolted joints, and shear panels. Shear modulus (G) is typically 40% of Young's modulus for isotropic materials.
τ
Shear Stress
Pa (Pascals)
G
Shear Modulus
Pa (GPa typical)
γ
Shear Strain
dimensionless (radians)
Shear Modulus Values
τ = Gγ| Material | Value | Unit |
|---|---|---|
| Steel (Structural) | 79.3 | GPa |
| Aluminum (6061) | 26 | GPa |
| Copper (Pure) | 44-48 | GPa |
| Brass | 35-40 | GPa |
| Rubber | 0.0003-0.003 | GPa |
Applications
- • Torsional shaft and spring design
- • Bolted and riveted joint analysis
- • Shear wall and diaphragm calculations
Deformation Model
Elastic Limits
σ ≤ σ_y (elastic)
Boundaries of linear elastic behavior. Proportional limit marks end of perfect linearity; yield strength defines onset of permanent (plastic) deformation.
σ_p
Proportional Limit
MPa
σ_y
Yield Strength
MPa
σ_u
Ultimate Strength
MPa
Yield/Strength Values
σ_y, σ_p| Material | Value | Unit |
|---|---|---|
| Steel A36 | 250 | MPa (σ_y) |
| Aluminum 6061-T6 | 276 | MPa (σ_y) |
| Copper (Annealed) | 70 | MPa (σ_y) |
| Titanium Grade 5 | 830 | MPa (σ_y) |
| Concrete (C30) | 30 | MPa (σ_c) |
Applications
- • Material selection for elastic service conditions
- • Safety factor determination (σ_working = σ_y / SF)
- • Failure prevention in mechanical components