Viscoelastic constitutive relation

Lefebvre et developed a generalized Fickean diffusion model using the free-volume concept. A finite-element model that accounts for Schapery s nonlinear viscoelastic constitutive relation(25) and the nonlinear diffusion model of Lefebvre et was discussed by Roy and Reddy. ( 4,45)... 

Equation (46) provides a general viscoelastic constitutive relation that can be applied to either plane stress, plane strain, or axisymmetric problems. For plane strain, the out-of-plane strain component 33 is identically zero. The corresponding stress component, a33, may be obtained from Eq. (46) by setting 33 = 0. Since for the plane stress case a33 is identically zero, the corresponding strain component 33 can be evaluated from Eq. (41) in the form... 

Polymeric materials, such as rubber, exhibit a mechanical response which cannot be properly described neither by means of elastic nor viscous effects only. In particular, elastic effects account for materials which are able to store mechanical energy with no dissipation. On the other hand, a viscous fluid in a hydrostatic stress state dissipates energy, but is unable to store it. As the experimental results reported in Part 1 have shown, filled rubber present both the characteristics of a viscous fluid and of an elastic solid. Viscoelastic constitutive relations have been introduced with the intent of describing the behavior of such materials able to both store and dissipate mechanical energy. 

With this aim we introduce the ABAQUS FEA finite viscoelasticity constitutive relation 1 and we investigate the resulting material behavior by means of two prototype experiments. Section 4.8.2 of the ABAQUS Theory Manual [173] gives the constitutive relation for modeling nonlinear viscoelastic effects in the form ... 

In linear viscoelasticity, the creep function and the relaxation function are interrelated and each would permit the derivation of the viscoelastic constitutive relation under the condition that the principle of time invariance can be applied, meaning that the material is influenced only by a t) and s(f) and no other influencing variables are being effective. In summary, in the idealized case of the isothermal linear viscoelasticity under the conditions of static stress or strain, the linear viscoelastic behavior of a material may adequately be described by the creep function F t), the relaxation function R t), the retardation spectrum or the relaxation spectrum. 

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

In this paper we present a constitutive relation for predicting the rheology of short glass fibers suspended in a polymeric matrix. The performance of the model is assessed through its ability to predict the steady-state and transient shear rheology as well as qualitatively predict the fiber orientation distribution of a short glass fiber (0.5 mm, L/D < 30) filled polypropylene. In this approach the total extra stress is equal to the sum of the contributions from the fibers (a special form of the Doi theory), the polymer and the rod-polymer interaction (multi-mode viscoelastic constitutive relation). 

The concept behind the model is that the contribution from the rods to the stress primarily occurs while the rods are changing their orientation. After the rods have reached a steady-state in their orientation then-contribution to the stress is at a minimum. However the enhanced steady-state rheology and the viscoelastic properties can be predict by superimposing the rod contribution onto a viscoelastic constitutive relation fit to the bulk steady-state rheology. 

In this approach the contribution to the extra stress of the matrix and the rod-matrix interaction is captured using a multi-mode viscoelastic constitutive relation. For the model predictions in the paper, we chose to use the Phan-Thien Tanner equation (PTT) [13]. The ability to indirectly capture the rod-matrix interaction is based on the presence of the fiber retarding the long relaxation... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

All the constitutive relation that we have discussed in this chapter include some relaxation equations for the internal tensor variables which ought to be considered to be independent variables in the system of equations for the dynamics of a viscoelastic liquid. 

However, in the earlier times, the constitutive relation for a viscoelastic liquid were formulated when the equations for relaxation processes could not be written down in an explicit form. In these cases the constitutive relation was formulated as relation between the stress tensor and the kinetic characteristics of the deformation of the medium (Astarita and Marrucci 1974). 

Instead of velocity gradients, displacement gradients can be used in relation (8.38). In this form, relations of the kind (8.38) are established on the basis of the phenomenological theory of so-called simple materials (Coleman and Nolle 1961). To put the theory into practice, function (8.38) should be, for example, represented by an expansion into a series of repeated integrals, so that, in the simplest case, one has the first-order constitutive relation (8.37). Let us note that the first person who used functional relations of form (8.38) for the description of the behaviour of viscoelastic materials was Boltzmann (see Ferry 1980). 

One can see that there are several forms for the representation of the constitutive relation of a viscoelastic liquid. Of course, we ought to say that all the types of constitutive relation we discussed in this section are equivalent. We can use any of them to describe the flow of viscoelastic liquids. However, the description of the flow of a liquid in terms of the internal variables allows one to use additional information, if it is available, about microstructure of the material, and, in fact, appears to be the simplest one for derivation and calculation. We believe that the form, which includes the internal variables, reflects a deeper penetration into the mechanisms of the viscoelastic behaviour of materials. From this point of view, all the representations of deformed material can be unified and classified. 

To calculate characteristics of linear viscoelasticity, one can consider linear approximation of constitutive relations derived in the previous section. The expression (9.19) for stress tensor has linear form in internal variables x"k and u"k, so that one has to separate linear terms in relaxation equations for the internal variables. This has to be considered separately for weakly and strongly entangled system. 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

Although the microscopic theory remains to be the real foundation of the theory of relaxation phenomena in polymer systems, the mesoscopic approach has and will not lose its value. It will help to understand the laws of diffusion and relaxation of polymers of various architecture. The information about the microstructure and microdynamics of the material can be incorporated in the form of constitutive relation, thus, allowing to relate different linear and non-linear effects of viscoelasticity to the composition and chemical structure of polymer liquid. 

Even in the apparently linear range, the response to stress should be considered as viscoelastic rather than elastic. Most polymers that behave in a linear, viscoelastic manner at small strains (< 1 %) behave in a nonlinear fashion at strains of the order of 1 % or more. However, in a fibrous composite, the resin may behave quite differently than it would in bulk. Stress and strain concentrations may exceed the limiting values for linearity in localized regions. Thus the composite may exhibit nonlinearity (Ashton, 1969 Trachte and DiBenedetto, 1968), as is the case with particulate-filled polymers (Section 12.1.2). Although nonlinearity at low strains is characteristic, Halpin and Pagano (1969) have predicted constitutive relations for isotropic linear viscoelastic systems, and verified their prediction using specimens of fiber-reinforced rubbers. 

The loading rate and time affect the deformations in polymeric solids. As already pointed out, viscoelastic materials show simultaneously the face of an elastic solid and that of a flowing viscous liquid. This implies that the simplest constitutive relation for a polymeric solid should, in general, contain time and frequency as variables in addition to stress and strain. 

Since by definition the viscoelastic material in shear exhibits both the behavior governed by equation 6 and than represented by equation 8, the constitutive relation for the linear viscoelastic solid can be written as... 

Hooke s law describes the behaviour of a linear elastic solid and Newton s law that of a linear viscous liquid. A simple constitutive relation for the behaviour of a linear viscoelastic solid is obtained by combining these two laws ... 

Schaffer and Adams< 2) carried out a nonlinear viscoelastic analysis of a unidirectional composite laminate using the finite-element method. The nonlinear viscoelastic constitutive law proposed by Schapery<26) was used in conjunction with elastoplastic constitutive relations to model the composite response beyond the elastic limit. 

For simple viscoelastic materials, the main characteristic of the constitutive relations is the presence of a time derivative of the stress tensor, associated to a relaxation process. In the case of an incompressible Maxwell material, the constitutive relation is given through the evolution equation... 

A quite general integral series representation of the internal energy y/ was proposed by Pipkin and Rogers [161]. Dai [185] used the first term of such an integral series to describe the nonhomogeneous deformation of a nonlinearly viscoelastic slab. The constitutive relation they obtained is... 

---