Mathematical representation of linear viscoelasticity

The models discussed here, which are phenomenological and have no direct relation with chemical composition or molecular structure, in principle enable the response to a complicated loading pattern to be deduced from a single creep (or stress-relaxation) plot extending over a long time interval. Interpretation depends on the assumption in linear viscoelasticity that the total deformation can be considered as the sum of independent elastic (Hookean) and viscous (Newtonian) components. In essence, the simple behaviour is modelled by a set of either integral or differential equations, which are then applicable in other situations. 

The models discussed here, which are phenomenological and have no direct relation with chemical composition or molecular structure, in principle enable the response to a complicated loading pattern to be deduced from a single creep (or stress-relaxation) plot extending 

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The mathematical representation of the elastic behavior of oriented heterogeneous solids can be somewhat improved through a more appropriate choice of the boundary conditions such as proposed by Hashin and Shtrikman [66] and Stern-stein and Lederle [86]. In the case of lamellar polymers the formalisms developed for reinforced materials are quite useful [87—88]. An extensive review on the experimental characterization of the anisotropic and non-linear viscoelastic behavior of solid polymers and of their model interpretation had been given by Hadley and Ward [89]. New descriptions of polymer structure and deformation derive from the concept of paracrystalline domains particularly proposed by Hosemann [9,90] and Bonart [90], from a thermodynamic treatment of defect concentrations in bundles of chains according to the kink and meander model of Pechhold [10—11], and from the continuum mechanical analysis developed by Anthony and Kroner [14g, 99]. 

A simple way to illustrate the viscoelastic properties of materials subjected to small deformations is to evaluate the stress that results from combining a linear spring that obeys Hooke s law and a simple fluid that obeys Newton s law of viscosity. An example of such combination is the mathematical representation of the Maxwell element. Even though this model is inadequate for quantitative correlation of polymer properties, it illustrates the quahtative nature of real behavior. Furthermore, it can be generahzed by the concept of a distribution of relaxation times so that it becomes adequate for quantitative evaluation. Maxwell s element is a simple one combining one viscous parameter and one elastic parameter. Mechanically, it can be visualized as a Hookean spring and a Newtonian dashpot in series ... 

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