Just as the concept of stress gives us a measure of force distributions in a deformable body, the concept of strain describes the distribution of deformations locally at every point within the body. In this chapter we will define strains and describe how strains change with directions and with the choice of coordinates, as was done with stresses. Strains will also be related to the displacements of the deformable body. It will be shown that strains must satisfy a set of compatibility equations at every point in a body to ensure that they represent a well-behaved deformation. Since the strains often found in practice are quite small, this book will only consider problems for small strains.

The components of the infinitesimal strain tensor are defined, which represent measures of the relative length changes (longitudinal strains or dilatations) and the angle changes (shear strains) at a considered material point with respect to the chosen coordinate axes. The principal strains (maximum and minimum dilatations) and the maximum shear strains are determined, as well as the areal and volumetric strains. The expressions for the strain components are derived in terms of the spatial gradients of the displacement components. The Saint-Venant compatibility equations are introduced which assure the existence of single-valued displacements associated with a given strain field. The matrix of local material rotations, which accompany the strain components in producing the displacement gradient matrix, is defined. The determination of the displacement components by integration of the strain components is discussed.