Stress and Strain

When a deforming force is applied to a solid body, the body develops an internal restoring force in response, exactly equal in magnitude and opposite in direction to the applied force once the body settles into (quasi-)equilibrium. Stress is defined as this restoring force per unit cross-sectional area of the body:

where $F$ is the magnitude of the deforming (or, equivalently, the internal restoring) force and $A$ is the area over which it acts. Since force has SI unit newton ($\text{N}$) and area has SI unit $\text{m}^2$, stress has SI unit $\text{N/m}^2$, which is given the special name pascal ($\text{Pa}$) -- the same unit used for pressure, and indeed stress and pressure have exactly the same dimensional formula, $[\text{ML}^{-1}\text{T}^{-2}]$.

Deformation itself is measured by strain, defined generally as the ratio of the change produced in a dimension of the body to its original value of that dimension. Because it is a ratio of two similar quantities, strain is a pure number and has no unit and no dimensions.

Depending on how the deforming force is applied, three distinct kinds of stress -- and three matching kinds of strain -- arise:

1. Tensile (or compressive) stress and longitudinal strain. When a force is applied along the length of a rod or wire, either pulling its two ends apart (tensile) or pushing them together (compressive), the body's length changes while its shape (cross-sectional profile) stays essentially the same. If a rod of original length $L$ and cross-sectional area $A$ is stretched (or compressed) by a force $F$ applied normal to, and centred on, its end faces, the tensile (or compressive) stress is $F/A$, and the longitudinal strain is

where $\Delta L$ is the change in length. If the applied force instead squeezes a solid uniformly along one axis in the middle of a large body rather than stretching it (a less common everyday case), the same idea applies with the sign of $\Delta L$ reversed.

2. Shearing (or tangential) stress and shearing strain. Here the deforming force is applied tangentially -- parallel to, rather than perpendicular to, the surface of the body -- typically to one face of a body while the opposite face is held fixed. This changes the body's shape without changing its volume: imagine a deck of cards, with the bottom card fixed and a sideways force applied to the top card, causing the whole stack to lean over into a slanted, parallelogram-like shape while each individual card's own dimensions stay unchanged. If the tangential force $F$ acts over area $A$ of the top face, and this causes the top face to be displaced sideways by $\Delta x$ relative to the bottom (fixed) face, with the body's height (measured perpendicular to the force) equal to $L$, then the shearing stress is $F/A$, and the shearing strain is

where $\theta$ is the (necessarily small, for the elastic-limit regime this chapter deals with) angle through which the sheared face has turned, in radians. …