Boltzman superposition principle

In previous chapters, relaxation and creep testing was introduced and the relaxation modulus and creep compliance were defined as the stress output for a constant strain input (relaxation) and the strain output for a constant stress input (creep). A question naturally arises as how the output could be found if a variable input of either strain or stress were to occur. One could, of course, attempt to solve a general differential equation if the variation is specified but such an approach could, in some cases, be quite tedious. 

The Boltzman superposition principle (or integral) is applicable to stress analysis problems in two and three-dimensions where the stress or strain input varies with time, but first the approach will be introduced in this section only for one-dimensional or a uniaxial representation of the stress-strain (constitutive) relation. The superposition integral is also sometimes referred to as Duhamel s integral (see W.T. Thompson, Laplace Transforms, Prentice Hall, 1960). 

Consider a variable stress input as shown in Fig. 6.1 with the thought of seeking a method to find the strain output. First assume that the variable 

Obviously, if sufficiently small steps are selected over corresponding small time intervals, the curve can be fitted to any degree of accuracy desired. 

Recall from Chapter 3 that the creep response can be represented by a creep compliance due to a step input at time zero as, 

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This differential form can be integrated to give the integral form of the model which can also be derived from the Boltzman superposition principle using the concept of fading memory of viscoelastic liquids ... 

The stress relaxation modulus and the creep compliance are both manifestations of the same dynamic process at the molecular level and are closely related. This relationship, however, is not a simple reciprocal relations that would be expressed as G t) = but rather in an integral equation that is derived from the Boltzman superposition principle. It relates recoverable compliance, to r/o, zero shear viscosity [22]. 

Boltzman Superposition Principle (3,10). This principle states that the stress at a given point in the polymer is a function of the entire strain history at that point. Therefore, to each strain term in equation 9 is added an integral that represents contributions to the stress at a given time from strain increments at earlier times. 

The Boltzman Superposition Principle is one starting point for inclusion of structural relaxation losses. An equally valid starting point is to include in equation 9 time derivatives (first-order and higher) of stress and strain. It can he shown that this approach is equivalent to the above integral representation (10). Finally, modified stress-strain relations, to describe viscoelastic response, have also been formulated using fractional derivatives (11). 

Although it is a powerful means of investigating molecular structure and of basic characterization, and provides a general indication of the influence of M on flow behavior, the restrictions imposed by linear viscoelasticity make it inapplicable to a wide variety of practical problems. Nonlinearity is often associated with the phenomenon of shear thinning , that is, a reduction of the viscosity with shear rate in steady flow, characteristic of many polymer melts at intermediate shear rates [15]. This contrasts with the Newtonian behavior implied by the Boltzman superposition principle for steady flow [Eq. (54), in a liquid, G(s) must vanish as s coj. 

Several examples are in order to demonstrate the utility of the Boltzman superposition principle. 

The creep compliance of a Kelvin element is D(t) = l-e j. Using the Boltzman superposition principle, find an equation for the strain vs. time in a constant stress rate test. Sketch your results, i.e., e vs. t. 

In Chapter 6, it was shown that the Boltzman superposition principle could be used to derive an integral constitutive law for a linear viscoelastic material as... 

The discussion that follows, of sound propagation in a lossy polymer, is limited to the case where the stress-strain relation in the polymer is linear. The effect of loss mechanisms on the mechanical response of polymers is included by modifying the stress-strain relations (eq. 9). At small strains, at which the behavior of the polymer is linear, the stress-strain relations are modified according to the Boltzman Superposition Principle (3,10). This principle states that the stress at a given point in the polymer is a function of the entire strain history at that point. Therefore, to each strain term in equation 9 is added an integral that represents contributions to the stress at a given time from strain increments at earlier times. 

Discuss the difference between superposition linearity and proportional linearity and the relation of each to the Boltzman superpositon principle. 

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