2.13. Strain Energy

The work done by external forces in causing deformation is stored within the body in the form of strain energy. In an ideal elastic process, no dissipation of energy takes place, and all the stored energy is recoverable upon unloading. The concept of elastic strain energy, introduced in this section, is useful as applied to the solution of problems involving both static and dynamic loads. It is particularly significant for predicting failure in members under combined loading.

Strain Energy Density for Normal and Shear Stresses

We begin our analysis by considering a rectangular prism of dimensions dx, dy, dz subjected to uniaxial tension. The front view of the prism is represented in Fig. 2.21a. If the stress is applied very slowly, as is generally the case in this text, it is reasonable to assume that equilibrium is maintained at all times. In evaluating the work done by stresses σ_x on either side of the element, it is noted that each stress acts through a different displacement. Clearly, the work done by oppositely directed forces (σ_x dy dz) through positive displacement (u) cancel one another. The net work done on the element by force (σ_x dy dz) is therefore