Stress and strain

STRESS. When a deforming force is applied to a body, the shape or size (or both) can change because the relative positions of its constituent atoms or molecules shift. Even when this deformation is too small to be visible to the naked eye, an internal RESTORING FORCE develops inside the body in response to the deformation. Stress is defined as this internal restoring force per unit area, $\sigma=\dfrac{F}{A}$; its SI unit is N m$^{-2}$, also called the pascal (Pa), and its dimensional formula is $[ML^{-1}T^{-2}]$. Stress is, in general, a tensor quantity (it can act in different directions on different faces of a small volume element), but the chapter works with two simpler, scalar-like components of it. If DELTA-A is a small cross-sectional area inside the body and F and -F are the equal-and-opposite internal forces the two sides of that area exert on each other, F can be resolved into a component $F_n$ normal (perpendicular) to DELTA-A and a component $F_t$ tangential to it. The NORMAL or LONGITUDINAL STRESS is $\sigma_n=F_n/\Delta A$, and the TANGENTIAL or SHEARING STRESS is $\sigma_t=F_t/\Delta A$. Longitudinal stress is further classified as TENSILE STRESS, when the internal forces on the two sides of DELTA-A pull apart from each other (stretching, as in a wire under a hanging load), or COMPRESSIVE STRESS, when the internal forces push toward each other (squeezing, as in a pillar under a roof load). A third kind of stress is VOLUME STRESS: when a body is acted on everywhere over its surface by a force that at every point is normal to the surface and proportional to the local area -- exactly what happens when a solid is fully immersed in a fluid at pressure P -- the resulting force per unit area, $\sigma_v=F/A$, is numerically the same as the pressure P itself. STRAIN. Strain measures how much a body is deformed by a force, expressed as the fractional change in size: for a rod of natural length L stretched to length $L+\Delta L$, the strain is $\varepsilon=\Delta L/L$, a dimensionless ratio with no unit or dimension. Strain is classified into three types matching the three kinds of stress. (1) LONGITUDINAL STRAIN, $\varepsilon_l=\Delta L/L$, is further split into TENSILE STRAIN (length increased from natural length) and COMPRESSIVE STRAIN (length decreased from natural length). (2) SHEARING STRAIN, $\varepsilon_s$: for a cuboid of height h with its bottom face fixed, if a tangential force along the top edge displaces the top face sideways through a small distance x, the shearing strain is $\varepsilon_s=x/h=\tan\theta\approx\theta$ for small angles, i.e. the shearing strain equals the angle of shear theta itself. (3) VOLUME STRAIN, $\varepsilon_v=\Delta V/V$, the fractional change in volume of a body subjected to a volume stress. ELASTIC LIMIT AND THE STRESS-STRAIN PROFILE. The maximum stress within which a body still regains its original size and shape after the deforming force is removed is called the elastic limit; beyond it, the body acquires a permanent deformation (a stretched rubber band, pulled too far, loses its elasticity and does not return to its original size). Plotting stress (vertical axis) against strain (horizontal axis) for a wire being loaded tr …