Integration of the Strains to Obtain Displacements

There are two aspects to the following discussion of strains and displacements. The first aspect is an outline of the process that is the general integration of the six strains to obtain the three displacements. The second aspect is the redirection of the series of equations developed during the process of obtaining the displacements towards the second goal, which is the partial differential equations that relate the strains to each other. The equations that relate the strains are called the compatibility equations. In this textbook, the compatibility equations are of more immediate concern than the process of integrating the strains to obtain the displacements. As is proved in Endnote (1) of Chapter 3 there are six second order compatibility equations that occur in two sets or three equations of similar form. The second of the two sets of three compatibility equations is rederived here because the form of those compatibility equations is less obvious than that of the first set.

Throughout the process begun below for obtaining the displacements from the strains, it is of course presumed that the strains are known functions of the cartesian coordinates, and if necessary, time as well. The process begins with the first order partial differential equations that are the linear form of the strain–displacement equations.