Linear Hereditary Constitutive Laws

The primary requirement which will be imposed here on the constitutive equation is that it be linear. This implies that the stress, for example, must be given as a linear functional of the strain. In general, this will involve an integral or sum over the strain at various space and time points. We impose the condition that the law be local, which is to say that the stress at a certain position depends only on the strain at that position. This amounts to excluding action at a distance forces such as might arise for example if the material were susceptible to electromagnetic fields which in turn were dependent on the mechanical state of the material. Furthermore, the Principle of Causality implies that the stress depends only on present and past values of the strain. We write 

Since the functions f t x), g(t, f x) characterize the response of the material, their dependence upon x implies that the material is spatially inhomogeneous. For most of the present volume, a homogeneous material will be assumed. However, there are important problem classes for which inhomogeneity cannot be neglected, notably where it arises from space-dependent environmental effects, in particular temperature. This type of inhomogeneity is discussed in Sect. 1.7. 

The space dependence of f(t x), g(t,t x) will not effect the considerations of this section and so explicit reference to it will be dropped. An alternative way of writing (1.2.1) is as follows  

If g(t, t ),f(t) are given, it is always possible to choose G(t, t ) such that (1.2.3) is satisfied. This is clear if one observes that the first condition determines G(t,t ) only up to an arbitrary function of t, which can be chosen so that the second condition is satisfied. 

Integrals over the history of strain (or stress) as occur in (1.2.1,2) are sometimes referred to as hereditary integrals. Materials whose constitutive equations contain such hereditary integrals are described as having memory. 

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The nonlinear constitutive law due to Schapery may be linearized by assuming that the nonlinearizing parameters 8 y d g2 have a value of unity. In addition, the stress-dependent part of the exponent in the definition of the shift function is set to zero. Consequently, the constitutive law reduces to the hereditary integral form commonly used to describe a linear viscoelastic material. 

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