Proportionality assumption

Under the assumption that (1.8.5) holds, there may be as many as 21 independent components of Gijkiit). The presence of this many independent functions significantly complicates the mathematical analysis of many problems. There are also, naturally, major experimental problems involved in the determination of so many functions. Therefore, if approximate or exact interrelationships between the various functions can be established, considerable simplifications may result. A rather strong assumption, from the physical point of view, which greatly simplifies the mathematics is to put all the Gijki t) proportional to one function  

A consequence of the proportionality assumption, which will be of some importance, is that if the quantities 

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A linear system of reservoirs is one where the fluxes between the reservoirs are linearly related to the reservoir contents. A special case, that is commonly assumed to apply, is one where the fluxes between reservoirs are proportional to the content of the reservoirs where they originate. Under this proportionality assumption the flux f,y from reservoir i to reservoir j is given by... 

This would seem to indicate that the proportionality assumption has no physical basis. However, in many cases the bulk modulus is much larger than the shear modulus (except perhaps near / = 0) so that, for practical purposes, we can take V = Y- In other words, K(t) is so much larger than G(t) that it does not matter what shape we assume for K(t). This is the case for amorphous polymers at temperatures well above their glass transition temperatures. Also for many rigid plastics, for example, amorphous polymers in the glassy state, v is constant but less than j-, typically having values in the range 0.35-0.41 [Schapery (1974)]. In such cases, the proportionality assumption would seem to have approximate validity. In summary, therefore, this assumption, while motivated primarily by the need for mathematical simplicity, is a reasonable approximation for many materials. 

Note that the Correspondence Principle does not apply if the material is aging. This includes the case where temperature variation destroys the convolution form of the hereditary integrals. However, an extension of the proportionality assumption, described in Sects. 1.8, 1.9, provides for aging materials with only one independent relaxation function. For such materials there exist useful analogies between viscoelastic and appropriate elastic solutions. These are referred to in Sect. 2.12. 

This will only be possible under the proportionality assumption discussed in Sect. 1.8. Applying this assumption, it is easy to show that f(t), g t) are connected by a simple formula. One can show furthermore that W/(r), Oij(r) are given by their static elasticity form while the time functions are easily determined from the boundary conditions. 

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. 

We have therefore managed to eliminate the embarrassing integral on the right of (3.1.3 c) without resorting to the proportionality assumption. This exceptional case is an example of that referred to in Sect. 2.3, where all the time-depen-dent viscoelastic quantities can be grouped into a single factor. 

The problem will be solved for the case where the viscoelastic half-plane is characterized by a discrete spectrum model (Sect. 1.6). The more general continuous spectrum model is discussed by Golden (1977). The proportionality assumption (Sect. 1.9) will be adopted for the material so that a unique Poisson s ratio exists. Therefore, from (1.6.25, 28, 29), (3.5.20) and (3.5.22), we have... 

For a material with definite Poisson s ratio (the proportionality assumption), (3.1.15) and (2.12) give that... 

I. Method of Solution. The method of solution is based on the viscoelastic Kolosov-Muskhelishvili equations, adapted to a half-space. Explicit solutions to the first and second boundary value problems are presented in detail. In these cases no restrictions on material behaviour are necessary. In the case of mixed boundary value problems where surface friction is present, it is necessary to make the proportionality assumption. Limiting frictional contact problems are... 

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