Decomposition of Hereditary Integrals

It is on these, rather than (2.3.8, 12), that the approach developed in the following sections will rest on these equations and (2.3.9) or (2.3.13). We remark that relations (2.3.9) and (2.3.13) could have been written down directly, at least if the proportionality assumption is made. Essentially, the point is that made in the context of (1.8.23) which we restate here within the present simpler, more concrete framework. Consider (2.3.9) for example. The proportionality assumption means that there is only one hereditary integral in the theory, and the equations of the theory are identical to the elastic equations if displacements are replaced by quantities of the form of v(r, t) where l t) is proportional to //(/), for example. It follows that elastic solutions are applicable if displacements are replaced by i (r, t) and corresponding quantities for the other components. This is precisely the content of equation (2.3.9). A similar argument applies to (2.3.13). It is not necessary even to assume proportionality for certain special problems, though these problems are difficult to characterize in fundamental terms. They are mainly problems where all the dependence on material properties can be grouped into one function. 

The fact that (2.3.9) and (2.3.13) are manifestly correct is the essential justification of (2.3.5) and (2.3.6). The arguments leading up to (2.3.9) and (2.3.13) could strictly have been omitted. However, they provide a link between reasoning based on formal elastic-viscoelastic correspondence as incorporated in (2.1.10), (2.1.11) and (2.3.1) and the methodology which is developed below. 

We now focus on the option characterized by (2.3.7-10) and (2.3.16), though the developments are equally applicable to the second option. An integral equation will be derived for y(r, 0, once we have shown how this 

We consider a problem in this section which arises under several guises through the book, in particular, in the context of deriving an integral equation for the quantity Let us first phrase the problem in fairly abstract 

General Theorems and Methods of Solution of Boundary Value Problems 

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The decomposition of hereditary integrals described in Sect. 2.4 for non-aging materials and the general techniques based on that decomposition may be extended to the case of aging viscoelastic bodies. A form of this decomposition has been used by Graham (1980) to develop solutions for frictional contact problems that involve varying contact area. 

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