5: Stress-strain relationships

Section 1: Fundamentals - 5: Stress-Strain Relationships

Understanding the relationship between stress and strain is fundamental to predicting how structural materials deform and fail under load. Stress ($\sigma$) quantifies the internal force intensity within a material, defined as force per unit area ($\sigma = F/A$). Strain ($\epsilon$) measures the material's deformation response, defined as the change in length per original length ($\epsilon = \Delta L / L_0$). Both have normal (acting perpendicular to a plane, causing tension/compression) and shear (acting parallel to a plane, causing sliding) components.

The graphical representation of this relationship is the stress-strain curve, typically obtained from a tensile test. For ductile materials like mild steel, the curve exhibits distinct regions:

1. Linear Elastic Region: Stress is proportional to strain (Hooke's Law: $\sigma = E\epsilon$). The slope is the Modulus of Elasticity (Young's Modulus, $E$), a fundamental material stiffness property (e.g., ~200 GPa for steel). Deformation here is fully recoverable upon unloading. The highest stress in this linear zone is the Proportional Limit.
2. Yield Point: A distinct point (yield strength, $F_y$) where significant plastic (permanent) deformation begins with little or no stress increase. Some materials show a plateau.
3. Strain Hardening Region: Stress increases again with further strain as the material becomes harder but less ductile.
4. Ultimate Tensile Strength (UTS): The maximum stress the material can withstand.
5. Necking and Fracture: Localized reduction in cross-section (necking) occurs, leading to failure at the fracture stress.

Brittle materials (e.g., concrete, cast iron) lack a distinct yield point and significant plastic region. Their stress-strain curve is nearly linear up to sudden fracture at the ultimate strength.

Poisson's Ratio ($\nu$) describes the lateral strain ($\epsilon_{lat}$) that occurs perpendicular to the applied axial load: $\nu = -\epsilon_{lat} / \epsilon_{axial}$. For most metals, $\nu \approx 0.3$.

Why is this critical?

• Design Limits: Structures are typically designed so stresses remain within the elastic range (below yield) to avoid permanent deformation under service loads.
• Material Selection: The curve defines key properties: stiffness ($E$), yield strength ($F_y$), ultimate strength ($F_u$), and ductility (measured by % elongation or reduction in area at fracture).
• Predicting Behavior: The relationship underpins all structural analysis methods for calculating deflections and internal forces. Understanding plasticity is vital for assessing collapse mechanisms and ductile failure modes.