Infinitesimal Strain

An elastic material responds to a stress by a change of volume and shape, or strain, which stays constant as long as the stress is maintained. Materials for which strains are completely reversible and proportional to the stresses that cause them are called ideally elastic and are said to follow Hooke’s law. Many actual materials are nearly ideally elastic as long as the stress-induced strains are small. Strain in such materials is the usual means of observing stress, which itself is an abstraction and not directly observable. Strain, as treated in continuum mechanics, is also an abstraction, but one that more closely approaches observable reality. The element of abstraction comes from treating the deformed body as a continuum, with the implication that neighboring material points in an undeformed body remain arbitrarily close neighbors after deformation. Let one point on a stretchable material line (imagine a rubber band) be held in place at the origin of a one-dimensional coordinate system and stretched, throughout but not necessarily uniformly (the rubber band may vary in thickness), by pulling on its free end with the position Δ x. Let the end, as a consequence of the stretching, be moved by the displacement Δ u. Then any original length element Δ x of the line will be changed to a new length Δx+Δu, say, the particular segment starting at the points P before and P′ after the deformation.