Stress and Strain

Stress and Strain

When you apply a force to a solid object, the object does not remain completely unchanged. It deforms — its shape or size changes, at least a little. The study of how solids respond to such forces begins with two fundamental ideas: stress, which measures the intensity of the internal forces within the material, and strain, which measures the resulting deformation relative to the original dimensions.

The Concept of Stress

Imagine a solid rod being pulled from both ends. Inside the rod, the atoms are being pulled apart. To describe this internal condition, we consider an imaginary cross-section through the rod. The total force acting across that section is the same as the external force applied. But what matters for the material's response is not the total force — it is the force per unit area.

Stress is a vector quantity, but in most elementary treatments we work with its magnitude. Its SI unit is the pascal (Pa), where $1 \text{ Pa} = 1 \text{ N m}^{-2}$. Because solids can withstand very large forces, you will often see stress expressed in megapascals ($1 \text{ MPa} = 10^6 \text{ Pa}$) or gigapascals ($1 \text{ GPa} = 10^9 \text{ Pa}$).

The direction of the force relative to the surface matters enormously. This gives us three distinct types of stress.

Normal stress occurs when the force is perpendicular to the surface. If the force pulls outward, stretching the object, it is called tensile stress. If the force pushes inward, compressing the object, it is called compressive stress. In both cases, the formula is the same: $\sigma = F/A$, where $\sigma$ (sigma) is the usual symbol for normal stress.

Shear stress occurs when the force is parallel (tangential) to the surface. Imagine a book lying flat on a table; if you push the top cover sideways while holding the bottom cover still, the pages slide past each other. That sliding action is shear. The shear stress is $\tau = F/A$, where $\tau$ (tau) denotes shear stress, and the area $A$ is the area of the surface being sheared.

The Concept of Strain

When a solid is under stress, it deforms. Strain is a measure of this deformation — it tells us how much the object has changed relative to its original size or shape. Because strain is a ratio of two lengths (or two angles), it is a dimensionless quantity.

Longitudinal strain describes the change in length of an object under normal stress. If a rod of original length $L$ stretches or compresses by an amount $\Delta L$, then

This strain is positive for elongation (tensile) and negative for contraction (compressive).

Shear strain describes the angular deformation caused by shear stress. Consider a rectangular block whose top face is displaced sideways by a distance $x$ relative to the bottom face, while the height of the block is $h$. The shear strain $\gamma$ (gamma) is defined as

$\text{Shear strain} = \gamma = \frac{x}{h} = \tan \theta$ …