Multiaxial Analysis of Linear Viscoelastic Stress

3 Differential Form for the Constitutive Stress-Strain Relationship 701 

The design of pieces that form part of either a structure or a machine often requires an analysis of the distribution of stresses and strains in these pieces. Without taking into account, for the moment, the thermal and calorific effects, the tensions and deformations at each point of the sample should simultaneously satisfy the balance and constitutive equations. 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Owing to the multiaxial character of the problems addressed in this chapter, the field equations depend not only on time but also on the position defined by their coordinates. Finally, it is necessary to stress that the solution of viscoelastic problems requires, as in the elastic case, specification of adequate boundary conditions. In this chapter, in addition to considering both integral and differential multiaxial stress-strain relationships, some viscoelastic problems of interest in technical applications are solved. 

---