Linear viscoelasticity constitutive equation

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

The Nagdi and Murch theory of viscoelastic-plasticity contains many of the same caveats as in plasticity theory and, in fact, reduces to the two limiting cases of plasticity for non-viscoelastic materials and linear viscoelasticity for non-yielded materials. The only portions needed here are the linear viscoelastic constitutive equations given in Chapter 6 and a generic failure law given by,... 

Equation (5.12) would be identical to the linear viscoelastic constitutive equation if the kernels Lq and L] were independent of f(5). Because f(C) is present in the equation, linear transforms cannot be utilized on the constitutive equation to demonstrate that the assumed linear elastic solution for the stresses was valid. If it can be shown that equation (5.12) will satisfy the equations of equilibrium and compatibility whenever the linear elastic solution is valid, a type of correspondence principle will have been developed similar to what has already been done in the theory of linear viscoelasticity. The validity of the elastic solution can be demonstrated by substituting the constitutive equation directly into the equations of equilibrium and compatibility. It should be immediately obvious that no complications can arise in such a procedure since the only spatially dependent quantities in the constitutive equation are the stresses and the strains ij. For the two dimensional problem only one equation of compatibility is present,... 

Kwon Y, Leonov AI (1995) Stability constraints in the formulation of viscoelastic constitutive equations. J Non-Newton Fluid Mech 58 25—46 Laius LA, Kuvshinskii EV (1963) Structure and mechanical properties of oriented amorphous linear polymers. Fizika Tverdogo Tela 5(11) 3113—3119 (in Russian)... 

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... 

Duct flows of nonnewtonian fluids are described by the governing equations (Eq. 10.24-10.26), by the constitutive equation (Eq. 10.27) with the viscosity defined by one of the models in Table 10.1, or by a linear or nonlinear viscoelastic constitutive equation. To compare the available analytical and experimental results, it is necessary to nondimensionalize the governing equations and the constitutive equations. In the case of newtonian flows, a uniquely defined nondimensional parameter, the Reynolds number, is found. However, a comparable nondimensional parameter for nonnewtonian flow is not uniquely defined because of the different choice of the characteristic viscosity. 

We introduce the change of variables into Equations 11.6a-c and neglect nonlinear terms in the perturbation variables 0, 4, and n. (The nonlinearities here are quadratic, but they will not be quadratic for the energy equation or for a viscoelastic constitutive equation like the PTT model.) We thus obtain the following linear equations ... 

There have been attempts to explain the onset of melt fracture by using hnear stability theory, along the hnes discussed in the preceding chapter for the spinline. The procedure is the same in principle, but the computational problem to solve the relevant linear eigenvalue problem for channel flow with any viscoelastic constitutive equation is a very difficult one. Based on a number of successful solutions it appears that plane laminar flow of viscoelastic liquids at very low Reynolds numbers is stable to infinitesimal disturbances. 

Figure 10 outlines the dynamic moduli as function of the frequency o) for different values of static prestrain e q [71]. At lower frequencies (ca O) the storage modulus tends to a finite nonzero value with a nonzero derivative. This behavior cannot be described by (linear or nonUnear) standard viscoelastic constitutive equations. The data collated by [71] suggest a non-monotonic dependence of the storage modulus upon the static prestrain e q from 0 = 0-65 to G 0 = 0.75, the storage modulus S considerably decreases, but it increases again at e 0 = 0.95. A similar, but less accentuated, trend is shown by the loss modulus. Experiments collated in [72, 73] and more recently in [74] are in agreement with Lee and Kim s results. 

Hanyga A, Seredynska M (2007) Multiple-integral viscoelastic constitutive equations. Int J Non Linear Mech 42 722-732... 

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. 

We expect that the classical framework of linear viscoelasticity also applies at the gel point. The relaxation spectrum for the critical gel is known and can be inserted into Eq. 3-3. The resulting constitutive equation will be explored in a separate section (Sect. 4). Here we are mostly concerned about the material parameters which govern the wide variety of critical gels. 

Whilst obtaining this is the ultimate goal for many rheologists, in practice it is not possible to develop such an expression. However, our mechanical analogues do allow us to develop linear constitutive equations which allow us to relate the phenomena of linear viscoelastic measurements. For a spring the relationship is straightforward. When any form of shear strain is placed on the sample the shear stress responds instantly and is proportional to the strain. The constant of proportionality is the shear modulus... 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

The set of relations (8.27) determines the fluxes as quasi-linear functions of forces. The coefficients in (8.27) are unknown functions of the thermodynamic variables and internal variables. We should pay special attention to the fourth relation in (8.27) which is a relaxation equation for variable The viscoelastic behaviour of the system is determined essentially by the relaxation processes. If the relaxation processes are absent (all the =0), equations (8.27) turn into constitutive equations for a viscous fluid. 

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. 

Pokrovskii VN, Pyshnograi GV (1990) Non-linear effects in the dynamics of concentrated polymer solutions and melts. Fluid Dyn 25 568-576 Pokrovskii VN, Pyshnograi GV (1991) The simple forms of constitutive equation of polymer concentrated solution and melts as consequence of molecular theory of viscoelasticity. Fluid Dyn 26 58-64... 

Pyshnograi GV (1996) An initial approximation in the theory of viscoelasticity of linear polymers and non-linear effects. J Appl Mech Techn Phys 37(1) 123—128 Pyshnograi GV (1997) The structure approach in the theory of flow of solutions and melts of linear polymers. J Appl Mech Techn Phys 38(3) 122—130 Pyshnograi GV, Pokrovskii VN (1988) Stress dependence of stationary shear viscosity of linear polymers in the molecular field theory. Polym Sci USSR 30 2624—2629 Pyshnograi GV, Pokrovskii VN, Yanovsky YuG, Karnet YuN, Obraztsov IF (1994) Constitutive equation on non-linear viscoelastic (polymer) media in zeroth approximation by parameter of molecular theory and conclusions for shear and extension. Phys — Doklady 39(12) 889-892... 

Linear Viscoelasticity Theory. FTMA is based on linear viscoelasticity theory. A one dimensional form of constitutive equation for linear viscoelastic materials which are isotropic, homogeneous, and hereditary (non-aging) is given by (21) ... 

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

Constitutive equations of the Maxwell-Wiechert tjq)e have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of pol3maer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple spring-dashpot mechanical analogy leading to the linear equation ... 

Polymeric fluids are the most studied of all complex fluids. Their rich rheological behavior is deservedly the topic of numerous books and is much too vast a subject to be covered in detail here. We must therefore limit ourselves to an overview. The interested reader can obtain more thorough presentations in the following references a book by Ferry (1980), which concentrates on the linear viscoelasticity of polymeric fluids, a pair of books by Bird et al. (1987a,b), which cover polymer constitutive equations, molecular models, and elementary fluid mechanics, books by Tanner (1985), by Dealy and Wissbrun (1990), and by Baird and Dimitris (1995), which emphasize kinematics and polymer processing flows, a book by Macosko (1994) focusing on measurement methods and a book by Larson (1988) on polymer constitutive equations. Parts of this present chapter are condensed versions of material from Larson (1988). The static properties of flexible polymer molecules are discussed in Section 2.2.3 their chemistry is described in Flory (1953). 

Doi and Edwards (1978a, 1979, 1986) developed a constitutive equation for entangled polymeric fluids that combines the linear viscoelastic response predicted by de Gennes... 

Many constitutive equations have been proposed in addition to those indicated above, which are special cases of fluids with memory. Most of these expressions arise from the generalization of linear viscoelasticity equations to nonlinear processes whenever they obey the material indifference principle. However, these generalizations are not unique, because there are many equations that reduce to the same linear equation. It should be noted that a determined choice among the possible generalizations may be suitable for certain types of fluids or special kinds of deformations. In any case, the use of relatively simple expressions is justified by the fact that they can predict, at least qualitatively, the behavior of complex fluids. 

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