Constitutive laws, nonlinear viscoelasticity

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. 

C. Guillope and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal., Th. Meth.. ppl., 15 (1990) 849-869. 

Drapaca CS, Sivaloganathan S, Tenti G (2007) Nonlinear constitutive laws in viscoelasticity. Math Mech Solids 12 475-501... 

Drapaca, C.S., Sivaloganathan, S. and Tenti, G. (2007) Nonlinear Constitutive Laws in Viscoelasticity. Mathematics and Mechanics of Solids, 12,475-501. 

Transient Response Creep. The creep behavior of the polsrmeric fluid in the nonlinear viscoelastic regime has some different features from what were found with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation... 

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. 

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. 

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 nonlinear constitutive law due to Schapery may be linearized by assuming that the nonlinearizing parameters 8 y d g2 have a value of unity. In addition, the stress-dependent part of the exponent in the definition of the shift function is set to zero. Consequently, the constitutive law reduces to the hereditary integral form commonly used to describe a linear viscoelastic material. 

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." °... 

In the following, we are motivated to develop a general nonlinear theory of viscoelasticity because, in the practical application of tire industry, mbber materials are used under conditions which do not comply with the infinitesimal deformation assumptions of the linear theory. For these materials, the range of deformation beyond which superposition and thereby linearity holds is extremely limited. Anyway, one of the first requirements for a nonlinear constitutive law is that, for a very small deformation, the model reduces to the corresponding linear model [120]. 

The effects of a number of environmental factors on viscoelastic material properties can be represented by a time shift and thus a shift factor. In Chapter 10, a time shift associated with stress nonlinearities, or a time-stress-superposition-principle (TSSP), is discussed in detail both from an analytical and an experimental point of view. A time scale shift associated with moisture (or a time-moisture-superposition-principle) is also discussed briefly in Chapter 10. Further, a time scale shift associated with several environmental variables simultaneously leading to a time scale shift surface is briefly mentioned. Other examples of possible time scale shifts associated with physical and chemical aging are discussed in a later section in this chapter. These cases where the shift factor relationships are known enables the constitutive law to be written similar to Eq. 7.53 with effective times defined as in Eq. 7.54 but with new shift factor functions. This approach is quite powerful and enables long-term predictions of viscoelastic response in changing environments. 

Failure, however defined, should be a part of a complete constitutive description of a material as discussed in the previous sections. In other, words, the key to dealing effectively with the failure of time dependent or viscoelastic polymers lies in treating failure properties as a termination of a nonlinear viscoelastic process. Perhaps, for this reason, a number of investigators have suggested that modulus and strength laws should be related to each other for polymers (eg., Landel, (1964)). 

General Regimes of Response. The nonlinear viscoelastic response of polymers, of course, follows some of the same classifications as does the linear response. Hence, the behavior above the glass temperature and into the terminal zone is fluid behavior, and often follows time-temperature superposition. The phenomenology of polymer melts and solutions is commonly described by constitutive laws that relate the stress and strain histories to each other (59,69). A brief description of the K-BKZ model (70-72) is provided as it seems to capture most of the behaviors of polymer melts and solutions subjected to large deformations or high deformation rates. At the same time the nonlinear form of the reptation... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

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