Stress and Strain

The elastic response of a body is described quantitatively using two paired concepts: stress and strain. When a deforming force acts on a body and changes its shape or size, internal restoring forces are set up inside the material to oppose that change. In the body's new, deformed equilibrium state, these internal restoring forces exactly balance the externally applied deforming force (equal in magnitude, opposite in direction).

Stress is defined as the internal restoring force acting per unit area of the body:

where $F$ is the magnitude of the internal restoring force (numerically equal to the externally applied deforming force at equilibrium) and $A$ is the cross-sectional area over which it acts. The SI unit of stress is $\text{N m}^{-2}$, also called the pascal (Pa), and its dimensional formula is $[L^{-1}M^1T^{-2}]$ — exactly the same dimensions as pressure, since both are force per unit area.

Strain measures how much the body has deformed, relative to its own original size — it is defined as the ratio of the change produced in some dimension of the body to that dimension's original value:

Because it is a ratio of two quantities of the same kind (both lengths, or both volumes), strain is a pure number with no units and no dimensions.

Depending on exactly how the deforming force acts, three distinct pairs of stress-and-strain arise, matching the three ways a body's dimensions can change (length, volume, shape):

1. Tensile / compressive stress and strain. When a deforming force $F$ acts along the length of a rod or wire, perpendicular to its cross-sectional area $A$, it either stretches the body (tensile, or longitudinal, stress, Fig. 6.1(a)) or, if the two ends are pushed together instead, compresses it (compressive stress, Fig. 6.1(b)):

The corresponding strain — the tensile or longitudinal strain — is the fractional change in length: if $L$ is the original length and $\Delta l$ the change in length,

2. Volume (hydraulic) stress and strain. When the deforming force $F$ acts perpendicular to, and uniformly over, the entire surface of a body — as when a body is immersed in a fluid under pressure, Fig. 6.2 — it changes the body's size without changing its shape. This is called volume stress or hydraulic/hydrostatic stress:

If $V$ is the original volume and $\Delta V$ the change in volume produced,

$\text{Volume strain} = \dfrac{\Delta V}{V}$ …