Energy Considerations in the One-dimensional Case

Note that K t, t ) may be taken to be symmetric in its arguments. Differentiating V t), we obtain 

From (1.1.12), we have that the rate of increase of mechanical energy density at a point, due to the work done by boundary forces, is a t)e t)y neglecting the kinetic energy term. This must be equal to the rate of increase of stored elastic energy plus the rate of energy dissipation, per unit volume. Comparing with (1.3.4), we deduce that 

This relationship does not in general determine K(t, / ), so that knowledge of constitutive behaviour does not determine the stored energy. This is unsatisfactory since, in principle at least, knowledge of the constitutive relationship combined with the dynamical equations determines the behaviour of the material under any system of forces. At least, this is true if a uniqueness theorem can be proved. This topic is referred to briefly in Sect. 1.8. If it is the case that K(t, t ) cannot be deduced from G(0, then this implies that the distribution of stored energy in the material is not determined even if the complete history of stresses and strains are known. This question is discussed in some detail by Christensen (1982), Chap. 3, with references to the literature. 

Observe that (1.3.10) shows clearly that the presence of mechanical energy dissipation is a direct consequence of the dependence of G t) upon time. If such dependence is not present, the theory reduces to linear elastic theory. 

Note that V t) was chosen to be purely quadratic, without linear terms. It may be checked that the presence of such contributions would lead to terms in the resulting constitutive relationship which are independent of the field quantities. This would correspond to non-zero stress in an undeformed medium, for example. Such phenomena will not be considered. 

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