Hereditary integral equation

As discussed previously, the relation between stress and strain for linear viscoelastic materials involves time and higher derivatives of both stress and strain. While the differential equation method can be quite general, a hereditary integral method has proved to be appealing in many situations. This hereditary integral equation approach is attributed to Boltzman and was only one of his many accomplishments. In the late nineteenth century, when the method was first introduced, considerable controversy arose over the procedure. Now, it is the method of choice for the mathematical expression of viscoelastic constitutive (stress-strain) equations. For an excellent discussion of these efforts of Boltzman, see Markovitz (1977). 

Notice that the factor is a constant in the first integral, which allowed us to write it as part of the integrand. The second integral is obtained from the first by making the substitution s = t — tf si often referred to as the time lapse). As seen from equation (15), the response of the system is completely characterized by the function i/r(t) = (C/t)g /, which is therefore called the response function (or sometimes the aftereffect function to highlight the fact that the response is delayed). Moreover, it is seen that the (delayed) response may be expressed as a convolution or hereditary integral. 

Many processes in pharmaceutics are related to transport, and the appHcations of the outlined theory are therefore numerous. Notwithstanding their practical importance, the special instances of the general transport equation (11) listed in Table 2 are assumed to be relatively familiar, and will therefore not be discussed further in this chapter. Instead, we focus our attention on applications of hereditary integrals and linear response theory, in particular on dynamic mechanical analysis (DMA) and impedance spectroscopy. 

Equation (10) cannot be applied until A, the equivalent relaxation time for the fluid, is known. However, A is defined by the linear Maxwell model, and actual polymer solutions exhibit marked nonlinear viscoelastic properties [5,6,7]. For both fresh and shear degraded solutions of Separan AP 30 polyacrylamide, which exhibit pronounced drag reduction in turbulent flow, Chang and Darby [8] have measured the nonlinear viscosity and first normal stress functions, and Tsai and Darby [6] have reported transient elastic properties of similar solutions, A nonlinear hereditary integral function containing six parameters has been proposed to represent the measured properties [8], The apparent viscosity function predicted by this model is ... 

In applied viscoelasticity not all the constitutive equations are formulated by an a-priori defined internal energy y/, but the constitutive model is expressed directly by the functional relation between the stress and the strain through an hereditary integral. In rheology this class of constitutive models is called Rivlin-Sawyers models Fxmg s [164], Fosdick and Yu s [165] and many other models currently used belong to this constitutive class. 

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. 

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. 

Equation (2.10.4) contains hereditary integrals. More will be introduced below. We need to assume that C(0 is not only receding but either contracting or stationary so that if z 6 C(0 it will have been in the contact region at all previous times. 

Equation (4.2) allows for no fading memory viscoelasticity, only permanent strain-time memory. If the kernel functionals of equation (4.1) were allowed to take on terms like (j lkl then the equation could contain two types of memory phenomenon the fading memory viscoelasticity contained in the hereditary integral representation and the permanent memory behavior registered in the norms. 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

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