Small Strain Viscoelasticity Theory

The one-dimensional viscoelastic response is schematically illustrated in Fig. 2.12. Note that the stress relaxation is a phenomenon that appears under a constant strain condition, while creep is one that appears under a constant stress condition. The response shown is represented by a model based on an excitation-response theory together with a data management procedure. Note that we assume an isotropic material response. 

---

Small Strain Viscoelasticity Theory Let US consider a step input function x(t) = ... 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

Linear viscoelastic behavior is actually observed with polymers only in very restricted circumstances involving homogeneous, isotropic, amorphous specimens subjected to small strains at temperatures near or above Tg and under test conditions that are far removed from those in which the sample may be broken. Linear viscoelasticity theory is of limited use in predicting service behavior of polymeric articles, because such applications often involve large strains, anisotropic objects, fracture phenomena, and other effects which result in nonlinear behavior. The theory is nevertheless valuable as a reference frame for a wide range of applications, just as the thermodynamic equations for ideal solutions help organize the observed behavior of real solutions. 

When we progress from the foregoing qualitative discussion of structure-property relationships to the quantitative specification of mechanical properties, we enter a jungle that has been only partially explored. The most convenient point of departure into this large and complex subject is provided by the topic of "linear viscoelasticity." Linear viscoelasticity represents a relatively simple extension of classical (small-strain) theory of elasticity. In situations where linear viscoelasticity applies, the mechanical properties can be determined from a few experiments and can be specified in any of several equivalent formulations (11). 

The accurate applicability of linear viscoelasticity is limited to certain restricted situations amorphous polymers, temperatures near or above the glass temperature, homogeneous, isotropic materials, small strains, and absence of mechanical failure phenomena. Thus, the theory of linear viscoelasticity is of limited direct applicability to the problems encounted in the fabrication and end use of polymeric materials (since most of these problems involve either large strains, crystalline polymers, amorphous polymers in a glass state failure phenomena, or some combination of these disqualifying features). Even so, linear viscoelasticity is a most important subject in polymer materials science—directly applicable in a minority of practical problems, but indirectly useful (as a point of reference) in a much wider range of problems. 

The theory behind linear viscoelasticity is simple and appealing. It is important to realize, however, that the applicability of the model for fluoropolymers is restricted to strains below the yield strain. One example comparing predictions based on linear viscoelasticity and experimental data for PTFE with 15 vol% glass fiber in the very small strain regime is shown in Fig. 11.4. 

Our discussion of the viscoelastic properties of polymers is restricted to the linear viscoelastic behavior of solid polymers. The term linear refers to the mechanical response in whieh the ratio of the overall stress to strain is a function of time only and is independent of the magnitudes of the stress or strain (i.e., independent of stress or strain history). At the onset we concede that linear viscoelastie behavior is observed with polymers only under limited conditions involving homogeneous, isotropie, amorphous samples under small strains and at temperatures close to or above the Tg. In addition, test conditions must preclude those that ean result in specimen rupture. Nevertheless, the theory of linear viseoelastieity, in spite of its limited use in predicting service performance of polymeric articles, provides a useful reference point for many applications. 

It has long been recognised that the mechanical properties of polymers are time-dependent. The behaviour at very small strains (less than 0 5%) can be described by the theory of linear viscoelasticity. Conventionally the stress a at time t is related to the strain e at all previous instants by the equation... 

There are other models based on springs and dashpots such as the simple Kelvin-Voigt model for viscoelastic solid and the Burgers model. Reader is referred to Refs. [1-5] for details. Other elementary models are the dumbbell, bead-spring representations, network, and kinetic theories. However, the most notable limitation of all these models is their restriction to small strain and strain rates [2, 3]. 

By the late 1940s, experimental researchers led by Leaderman and Tobolsky found that the small-strain properties of elastomers and thermoplastic melts were described by the theory of linear viscoelasticity developed by Boltzmann [B26] in the 1870s. This view is made clear in the monographs of Leaderman [L5] and Mark and Tobolsky [M7], which were published in this period. In the next two decades, this subject was developed in monographs by Tobolsky [T5] and, later. Ferry [F3]. 

A theory of this kind neglects viscoelasticity, viscosity or birefringence. Solutions of these equations can be found [12] which connect states of small strain (e < to states of large strain (e > C2) across moving interfaces. These solutions have abrupt changes of strain across the interface, and the interface can come to rest, given proper boundary conditions at the ends of the fiber. 

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

Creep obeys viscoelastic theory at small strains and it is possible to apply predictive methods using data from dynamic mechanical analysis to obtain creep data. Here, remarkable amounts of data can be obtained from... 

Schapery [16, 17] has used the theory of the thermodynamics of irreversible processes to produce a model that may be viewed as a further extension of Leaderman s. Schapery continued Leaderman s technique of replacing the stress by a function of stress /(a) in the superposition integral, but also replaced time by a function of time, the reduced time ip. The material is assumed to be linear viscoelastic at small strains, with a creep compliance function of the form [17]... 

Within the asymptotic limit of infinitesimally small strain (and/or strain rate), modulus and viscosity functions of polymers are nowadays understood with respect to the theory of linear viscoelasticity [1] and, in principle, conversion between the various functions is possible, whatever is the mode of deformation [2]. The current practice of linear viscoelastic concepts reveals however that such conversions are... 

In defining the constitutive relations for an elastic solid, we have assumed that the strains are small and that there are linear relationships between stress and strain. We now ask how the principle of linearity can be extended to materials where the deformations are time dependent. The basis of the discussion is the Boltzmann superposition principle. This states that in linear viscoelasticity effects are simply additive, as in classical elasticity, the difference being that in linear viscoelasticity it matters at which instant an effect is created. Although the application of stress may now cause a time-dependent deformation, it can still be assumed that each increment of stress makes an independent contribution. From the present discussion, it can be seen that the linear viscoelastic theory must also contain the additional assumption that the strains are small. In Chapter 11, we will deal with attempts to extend linear viscoelastic theory either to take into account non-linear effects at small strains or to deal with the situation at large strains. 

UHMWPE specimens subjected to very small strains, while exploration of hyperelasticity theory led to the conclusion that it is often safer to use a more sophisticated constitutive model when modeling UHMWPE. The use of linear viscoelasticity theory led to a reasonable prediction for the response of the material during a uniaxial compression test however, even small changes to the strain rate rendered the previously identified material parameters unsatisfactory. Isotropic J2-plasticity theory provided excellent predictions under monotonic, uniaxial, constant-strain rate, constant-temperature conditions, but it was unable to predict reasonable results for a cyclic test. The augmented Hybrid Model was capable of predicting the behavior of UHMWPE during a uniaxial tension test, a cyclic uniaxial fully... 

In the model, in order to describe the frozen stress and its activation in SMP, a linear viscoelastic theory was used as the first trial. Since the linear theory is limited to small deformations, subsequently a nonlinear viscoelastic theory was adopted for large deformations. The linear viscoelastic model was found to predict the characteristics of SMPs, especially the strain fixity and recovery properties for small deformations with some discrepancy between the experimental and calculated values. The main source for the error was found to be the reduced rigidity of SMPs due to the thermal treatment. This should be avoided for better shape memory performance of SMPs. 

Ward has discussed attempts to extend linear viscoelastic theory to take into account non-linear effects at small strains and to deal with the situation of large strains. 

A simpler and effective approach was given by Schapery [149]. The theory has proven to describe reasonably well the non-linear viscoelastic behaviour for several polymers [150]. The constitutive equation in terms of strain is restricted to small strains by the rmderlying thermodynamic theory. For the unidirectional case is given as... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

---