Viscoelasticity constitutive tensor

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. 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

Among the equations that govern a viscoelastic problem, only the constitutive equations differ formally from those corresponding to elastic relationships. In the context of an infinitesimal theory, we are interested in the formulation of adequate stress-strain relationships from some conveniently formalized experimental facts. These relationships are assumed to be linear, and field equations must be equally linear. The most convenient way to formulate the viscoelastic constitutive equations is to follow the lines of Coleman and Noll (1), who introduced the term memory by stating that the current value of the stress tensor depends upon the past history... 

Equations (1.1) and (1.2) are not closed because of the presence of the extra stress tensor x. Therefore, one more equation is required, which is provided by a viscoelastic constitutive model [35, 49, 79). According to nonequilibritun thermodynamics, the most thermodynamically consistent way to describe the constitutive model is in terms of internal (structural) variables for which separate evolution equations are to be described [49]. The simplest case is when a single, second-order... 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

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). 

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. 

Polymer solutions may have the memory effects observed in viscoelastic phenomena. This requires additional relaxation terms in the constitutive equations for the viscous pressure tensor, which may be affected by the changes in the velocity gradient. Besides this, the orientation and stretching of the macromolecules may have an influence on the flow. 

As remarked earlier, the nonlinear viscoelastic behavior of entangled wormy micellar solutions is similar to that of entangled flexible polymer molecules. Cates and coworkers (Cates 1990 Spenley et al. 1993, 1996) derived a full constitutive equation for entangled wormy micellar solutions, based on suitably modified reptation ideas. The stress tensor obtained from this theory is (Spenley et al. 1993)... 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

While the stress tensor component tfor purely viscous fluids can be determined from the instantaneous values of the rate of deformation tensor 4, the past history of deformation together with the current value of 4, may become an important factor in determining t, for viscoelastic fluids. Constitutive equations to describe stress relaxation and normal stress phenomena are also needed. Unusual effects exhibited by viscoelastic fluids include rod climbing (Weis-senberg effect), die swell, recoil, tubeless siphon, drag, and heat transfer reduction in turbulent flow. 

Although bone is a viscoelastic material, at the quasi-static strain rates in mechanical testing and even at the ultrasonic frequencies used experimentally, it is a reasonable first approximation to model cortical bone as an anisotropic, linear elastic solid with Hooke s law as the appropriate constitutive equation. Tensor notation for the equation is written as ... 

Where S J = the partial tensor of stress, = the partial tensor of strain, spherical tensor of stress, = spherical tensor of strain, G = shear modulus, K = volume modulus, ov = stress tensor, = Dirac delta function and = strain tensor. The 3D constitutive equation of viscoelastic mass undergoing static-dynamic coupling loading is derived as follows (Li et al. 2006) ... 

The viscoelastic properties of a material are entirely characterized by the constitutive equation whidi relates the stress tensor Oap(t) to the velodty... 

Example 3 illustrates the use of thermodynamics principles in formulating constitutive equations for a poro-viscoelastic medium. The ultimate purpose here is also to develop solutions for a long horizontally aligned tunnel with a circular cross-section embedded in a poro-viscoelastic massif. The setting of the problem is similar to Example 2 discussed above except that the spherical cavity is replaced by a long lined tunnel (Dufour et al. 2009). We start by restricting to small strain problems where the strain tensor of a viscoelastic material can be decomposed into an elastic part (denoted by superscript e ) and a viscoelastic part (superscript V ) ... 

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... 

Here p is the density, t the time, Xi the three Cartesian coordinates, and o,- the components of velocity in the respective directions of these coordinates. In equation 2, the index j may assume successively the values 1, 2, 3 gj is the component of gravitational acceleration in the j direction, and atj the appropriate component of the stress tensor (see below). (A third equation, describing the law of conservation of energy, can be omitted for a process at constant temperature the discussion in this chapter is limited to isothermal conditions.) Now, many experiments are purposely designed so that both sides of equation 1 are zero, and so that in equation 2 the inertial and gravitational forces represented by the first and last terms are negligible. In this case, the internal states of stress and strain can be calculated from observable quantities by the constitutive equation alone. For infinitesimal deformations, the appropriate relations for viscoelastic materials involve the same geometrical form factors as in the classical theory of equilibrium elasticity they are described in connection with experimental methods in Chapters 5-8 and are summarized in Appendix C. 

In the framework of nonlinear viscoelasticity, Fosdick and Yu [165] proposed their own constitutive equation. They assumed that the second Piola-Kirchhoff stress tensor is given by... 

Before leaving the integral constitutive equations, we remark diat a class of equations has been proposed in wMch the functions 01 and 02 of eq 4.4.1S depend on the invariants of the strain rate tenscv D, rather than the strain tensor B (Bird et al., 1968). Many of the sinqiler examples of diese equations do not reduce to the equation of linear viscoelasticity at sniall strains (Gross and Maxwell, 1972 Astarita and Marrucci, 1974) and this class of equations has not been much favored lately. 

The constitutive viscoelastic law is developed for the rheological model presented in Figure 2, where masses oscillate together with the springs. Separating the stress tensor into pressure and deviatoric parts. 

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. 

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