Elastic deformation: Deformation that is fully reversible when the load is removed.
Plastic deformation: Permanent deformation that remains after the load is removed.
Elastic limit (or proportional limit): Maximum stress at which the material still obeys Hooke’s law.
Yield point: Stress at which plastic deformation begins.
Ultimate tensile strength (UTS): Maximum stress a material can sustain before necking.
Fracture point: Stress at which the material finally breaks.
2. Stress–Strain Curve
The stress–strain curve summarises the mechanical response of a material under tension or compression.
Suggested diagram: Typical stress–strain curve showing elastic region, yield point, ultimate tensile strength and fracture.
3. Elastic Region
In the elastic region the relationship between stress and strain is linear and described by Hooke’s law:
\$\sigma = E\,\varepsilon\$
where \$E\$ is the Young’s modulus (a measure of stiffness).
4. Plastic Region
Beyond the yield point the material deforms plastically. The stress–strain relationship becomes non‑linear and the material will not return to its original shape when the load is removed.
5. Important Material Properties
Property
Symbol
Typical Units
Physical Meaning
Young’s Modulus
\$E\$
Pa (N·m⁻²)
Stiffness in the elastic region
Yield Strength
\$\sigma_y\$
Pa
Stress at which plastic deformation starts
Ultimate Tensile Strength
\$\sigma_{UTS}\$
Pa
Maximum stress material can sustain
Fracture Stress
\$\sigma_f\$
Pa
Stress at which the material breaks
Modulus of Resilience
\$U_r\$
J·m⁻³
Energy per unit volume absorbed elastically
Modulus of Toughness
\$U_t\$
J·m⁻³
Energy per unit volume absorbed up to fracture
6. Energy Considerations
The area under the stress–strain curve up to a given strain represents the energy per unit volume stored or dissipated.
Elastic energy (modulus of resilience): \$Ur = \int{0}^{\varepsilony} \sigma\, d\varepsilon = \frac{1}{2}\sigmay \varepsilony = \frac{\sigmay^2}{2E}\$
Toughness (total energy to fracture): \$Ut = \int{0}^{\varepsilon_f} \sigma\, d\varepsilon\$
7. Factors Influencing Elastic and Plastic Behaviour
Material type: Metals typically show a distinct yield point; polymers may exhibit a gradual transition.
Temperature: Raising temperature generally reduces \$E\$ and \$\sigma_y\$, making materials more ductile.
Strain rate: Faster loading can increase apparent strength (strain‑rate hardening).
Microstructure: Grain size, dislocation density, and phase composition affect both elastic and plastic limits.
8. Practical Applications
Design of springs – requires materials with high \$E\$ and a large elastic limit.
Safety components (e.g., car crumple zones) – rely on controlled plastic deformation to absorb impact energy.
Structural beams – must stay within the elastic region under service loads to avoid permanent deflection.
Metal forming processes (rolling, forging) – exploit plastic behaviour to shape components.
9. Summary Checklist
Identify the elastic limit on a stress–strain curve.
Apply Hooke’s law to calculate stress or strain in the elastic region.
Distinguish between yield strength, ultimate tensile strength and fracture stress.
Calculate modulus of resilience and understand its significance for energy absorption.
Explain how temperature and strain rate affect elastic and plastic behaviour.