Constitutive equation nonlinear viscoelastic

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

The nonlinear viscoelastic models (VE), which utilize continuum mechanics arguments to cast constitutive equations in coordinate frame-invariant form, thus enabling them to describe all flows steady and dynamic shear as well as extensional. The objective of the polymer scientists researching these nonlinear VE empirical models is to develop constitutive equations that predict all the observed rheological phenomena. 

Together with Eq. 3.3-17, Eq. 3.3-16 is the White-Metzner constitutive equation, which has been used frequently as a nonlinear viscoelastic model. Of course, for small deformations, X(i) = dx/dt, and the single Maxwell fluid equation (Eq. 3.3-9) is obtained. 

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... 

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. 

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. 

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)... 

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. 

Experimental data for polymer solutions have been reported by Osaki, Tamura, Kurata, and Kotaka (60), by Booij (12), and by Macdonald (50). Osaki et al. used polystyrene in toluene, polymethylmethacrylate in diethylphthalate, and poly-n-butylmethacrylate in diethylphthalate. Booij s data were for aluminum dilaurate in decalin and a rubbery ethylene-propylene copolymer in decalin. Macdonald s experiments were performed on several polystyrenes in several Aroclors and on polyisobutylene in Primol. Shortly after the original publication of the Japanese group, Macdonald and Bird (51) showed that a nonlinear viscoelastic constitutive equation was capable of describing quantitatively their data on both the non-Newtonian viscosity and the superposed-flow material functions. Other measurements and continuum model calculations have been described by Booij (12 a). 

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. 

In the first experiments over an extended frequency range, the biaxial viscoelastic as well as uniaxial viscoelastic properties of wet cortical human and bovine femoral bone were measured using both dynamic and stress relaxation techniques over eight decades of frequency (time) [Lakes et al, 1979]. The results of these experiments showed that bone was both nonlinear and thermorheologically complex, that is, time-temperature superposition could not be used to extend the range of viscoelastic measurements. A nonlinear constitutive equation was developed based on these measurements [Lakes and Katz, 1979a]. 

Nonlinear viscoelastic behaviour. To see the characteristic features of the constitutive equation (7.195), we approximate V (0 by... 

Though the theoretical constitutive equation (7.195) explains many features of nonlinear viscoelasticity, there are some discrepancies which are worth discussing. 

R. M. Shay and J. M. Caruthers, A New Nonlinear Viscoelastic Constitutive Equation for Predicting Yield in Amorphous Solid Poljmiers , J. Rheol. 30, 871-827 (1986). 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

K. S. Cho and S. Y. Kim, A Thermodynamic theory on the Nonlinear Viscoelasticity of Glassy Polymers, 1. Constitutive Equation Macromol. Theory Simul. 9, 328-335 (2000). 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

The uniaxial nonlinear viscoelastic constitutive equation of Schapery( > can be written for an isotropic material as... 

Schapery, R. A. (1997). "Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics." Mechanics of Time-Dependent Materials 1 209-240. 

For other method to analyze the nonlinear viscoelastic properties, we assume a constitutive equation of Fourier series type as [8]... 

There are numerous other constitutive equations of both differential and integral type for polymer melts, and some do a better job of matching data from a variety of experiments than does the PTT equation. The overall structure of the differential equations is usually of the form employed here The total stress is a sum of individual stress modes, each associated with one term in the linear viscoelastic spectrum, and there is an invariant derivative similar in structure to the one in the PTT equation, but with different quadratic nonlinearities in t and Vv. The Giesekus model, for example, which is also widely used, has the following form ... 

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 ... 

The linear viscoelastic phenomena described in the preceding chapter are all interrelated. From a single quite simple constitutive equation, equation 7 of Chapter 1, it is possible to derive exact relations for calculating any one of the viscoelastic functions in shear from any other provided the latter is known over a sufficiently wide range of time or frequency. The relations for other types of linear deformation (bulk, simple extension, etc.) are analogous. Procedures for such calculations are summarized in this chapter, together with a few remarks about relations among nonlinear phenomena. 

The restriction of linear viscoelastic behavior—small deformations—is more serious for equations 56 to 63 than for most of the preceding treatments, because with constantly increasing strain or stress the nonlinear regime may soon be reached. Then formulation in terms of a more complicated constitutive equation is necessary. 

The interrelationships for linear viscoelasticity in Sections B to F are accepted with almost the confidence given deductions from the laws of thermodynamics. Relations from nonlinear viscoelasticity theory are less well established. Many nonlinear constitutive equations have been proposed. Some predict certain relations which are in close accord with experiment and can be accepted with confidence but fail in other respects. A very thorough analysis with emphasis on viscoelastic liquids is provided by the treatise of Bird, Armstrong, and Hassager." °... 

Behavior in simple extension is, of course, a relatively elementary aspect of nonlinear viscoelasticity. Other types of deformation, and combinations of deformations, can provide additional information which must be described by appropriate constitutive equations and eventually interpreted on a molecular basis. Some investigations of time-dependent properties in pure-shear - and biaxial extension of thin, flat specimens - of soft rubbery polymers have been reported. 

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