Thermoviscoelastic Boundary Value Problems

As pointed out in Sect. 1.7, the viscoelastic functions of many materials depend strongly on temperature. The simplest realistic way of incorporating this dependence is to assume that the material is thermorheologically simple (TRS) in the sense defined in Sect. 1.7. This implies a non-linear dependence on the temperature field which renders the solution of most problem categories very difficult, in particular those where the temperature field is not given a priori but must be determined as part of the solution. A way out of this is to adopt a fully linear theory, as developed for example by Christensen (1982), Chap. 3. The assumption behind such a theory is that the effects of temperature variation on the viscoelastic functions is sufficiently small that its product with the field variables can be neglected. In many cases, this is very restrictive on the allowed range of temperature variation. A fully linear theory will not be considered here. We remark however that such a theory is susceptible to treatment by the Correspondence Principle-based methods, discussed in Chap. 2. 

The Edelstein method applies to materials that are aging and with spatial inhomogeneity - due to the dependence of its mechanical properties on temperature and other environmental variables. This is clear from F. Williams (1975 a) analysis. However, the final result is an integral equation which, in all but the simplest cases, must be solved numerically. 

Graham and Williams (1972) give generalized Papkovich-Neuber and other representations of solutions for aging materials. 

There is some work on TRS materials which does not take the temperature field as given, but seeks to solve for it. Hunter (1967) manages to obtain solutions for the temperature field in a semi-infinite rod subjected to forced oscillations. It is necessary however to assume that the loss tangent depends linearly on the temperature field. Lockett and Morland (1967) assume that temperature and mechanical fields are decoupled, so that the temperature field may be solved for independently and specified as input to the mechanical equations. 

For a survey that includes Russian contributions to this topic, see Il iushin and Pobedria (1970). 

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This chapter has been used to illustrate the method discussed in Chap. 2, which may be used to solve thermoviscoelastic boundary value problems involving temperature fields simultaneously varying with position and time. The method relies upon the solution of integral equations in terms of Neumann series expansions. 

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