Elasticity

Some basic equations of the theory of elasticity are presented in this chapter, together with terminology and definitions needed to understand the theory. An especially important result is Somigliana’s displacement formula, which gives the displacements at a point \(\boldsymbol{\xi }\) inside a region V, but not on its boundary S, in terms of the counterparts in elasticity for the single and double layer potentials in potential theory with density functions equal to (a) the tractions on S and (b) the displacements on the same boundary. Expressions are also given for the stresses and tractions at points \(\boldsymbol{\xi }\) inside V, as well as limit values for the displacement formula for two dimensions for the case that \(\boldsymbol{\xi }\rightarrow S\) . When the boundary S represents a crack, comparable formulas are developed for the displacements, stresses, and tractions at a point \(\boldsymbol{\xi }\) inside V in terms of the displacement discontinuities along the crack. These formulas are attributed to Volterra.