Viscoelasticity integral formulation

One may also use a memory function m(t) within the integral formulation of linear and nonlinear viscoelasticity this memory fimction m(t), which is the derivative of the relaxation function GKt), is not a measurable function. 

Three-dimensional formulations for T including nonlinear viscoelastic integral forms are contained in the papers of White [W9], White and Tanaka [W23], and White and Lobe [W21], These all have the general form of Eq. (70) with [compare Eq. (47)]... 

Depending on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. 

This idea can be used to formulate an integral representation of linear viscoelasticity. The strategy is to perform a thought experiment in which a step function in strain is applied, e t) = Cq H t), where H t) is the Heaviside step function, and the stress response a t) is measured. Then a stress relaxation modulus can be defined by E t) = <7(t)/ o Note that does not have to be infinitesimal due to the assumed superposition principle. To develop a model capable of predicting the stress response from an arbitrary strain history, start by decomposing the strain history into a sum of infinitesimal strain increments ... 

The RPA can also do a stress relaxation test, Figure 37 shows RPA stress relaxation decay curves from three truck tread stocks of the same formulation. Each curve indicates different qualities of mix caused by variations in mixing work history (variations in total energy at Banbury dump). As the energy at dump for these tread mixes increases, the peak torque decreases, the slope of the log-log plot of the stress relaxation curve becomes steeper, the regression line intercept decreases, the integrated area under the curve decreases, and the time to a given % drop decreases [133]. Therefore a simple stress relaxation test is a fast and effective w ay to measure variations in viscoelastic properties for both raw rubber and mixed stocks. 

This idea can be used to formulate an integral representation of linear viscoelasticity. The strategy is to perform a thought experiment in which a step fimction in strain is applied, e(f)= b- H(f), where H(f) is the Heaviside step... 

There are increasingly cases where a CHILE approach may not be adequate. A good example is the off-tool post-cure of a structure. This wUl be done in cases where the tool or perhaps the autoclave cannot handle the required final hold temperatures, or where there is a desire to maximize use of expensive tooling and equipment. In this situation, the CHILE approach will be unable to properly predict the relaxation of stresses after the first hold, and a full viscoelastic approach is required. Whereas most full viscoelastic approaches are integral in formulation, and present significant issues in terms of implementation and use for industrial use, recent work has provided a differential viscoelastic formulation that is backwardly compatible with the CHILE approach, thus allowing the user to reasonably seamlessly move back and forth between the current industrially validated CHILE approach and the required viscoelastic capability for the more complex problems of tomorrow. 

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). 

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. 

Single hereditary formulation has proven to reproduce all the crucial aspects of rubber behavior (hysteresis, relaxation and creep). In the simplest situatirm, the current value of the stress is the sum of two different contributions a purely elastic term depending oti the current value of the strain and a hereditary integral depending oti the whole strain history. In the linear model of viscoelasticity, introduced by Bernstein et al. [166], the stress dependence on the strain histoiy is assumed to be linear, i.e.. 

The integral in (1.2.5) is really a special case (where e t) is differentiable) of a Stieltjes integral. Gurtin and Sternberg (1962) base their rigorous formulation of Linear Viscoelasticity on constitutive equations which have this Stieltjes form. We adopt a convenient notation of theirs, and write (1.2.5) as... 

If a material exhibits linear viscoelastic behaviour as defined by equations (28) to (31) its behaviour can be described by sets of linear differential equations. The developments in linear viscoelasticity prior to 1943, from the standpoint of the integral equations of Boltzmann" and the various differential formulations, have been comprehensively reviewed by Leaderman. Further insight into the development of the subject is provided by the treatise of Alfrey. The generalized mathematical structure of linear viscoelasticity has been developed by Gross. ... 

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