Viscoelastic Stress Analysis in Two and Three Dimensions

The various approaches to the solution of viscoelastic boundary value problems discussed in the last chapter for bars and beams carry over to the solution of problems in two and three dimensions. In particular, if the solution to a similar problem for an elastic material already exists, the correspondence principle may be invoked and with the use of Laplace or Fourier transforms a solution can be found. Such solutions can be used with confidence but one must be cognizant of the general equations of elasticity and the methods of solutions for elasticity problems in two and three dimensions as well as any assumptions that might often be applied. To provide all of the necessary information and background for multidimensional elasticity theory is beyond the scope of this text but the procedures needed will be outlined in the following sections. This chapter will focus on developing the equations, assumptions and procedures one must use to solve two and three dimensional viscoelastic boundary value problems. The problem of an elastic thick walled cylinder will be used as a vehicle to demonstrate how to obtain the solution of a more difficult reinforced viscoelastic thick walled cylinder. In the process, we first demonstrate how the elasticity solution is developed and then apply the correspondence principle to transform the solution to the viscoelastic domain. Several extensions to this problem will be discussed and additional practice is provided in the homework problems at the end of the chapter.