Nonlinear constitutive law

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. 

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 purpose of the nonlinear constitutive law is to provide the evolution of the remanent strain history given the stress or total strain history. Consistent with the facts that domain switching gives rise to deviatoric strains and ferroelastic ceramics exhibit kinematic hardening effects, it is assumed that the material responds elastic-ally within a switching (yield) surface described by... 

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]. 

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. 

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. 

Obviously, in electrochemical experiments, the first condition is almost always fulfilled. However, the requirement of appropriate feedback mechanisms (i.e., appropriate nonlinear evolution laws) seems to constitute a severe restriction on the possible reaction mechanisms that give rise to pattern formation. From this point of view, it is astonishing that nearly all electrochemical systems exhibit dynamic instabilities. 

Finite element analysis with polymer-specific material or constitutive laws Accommodates complex geometries. Can handle nonlinearity in material behavior and large strains. Rapid analysis possible. Can predict very complicated polymer behavior, including filled polymers and complex temperature-loading histories. Requires the most computing power. Requires the most material testing. 

Remark To a large extent these criteria can be applied in the case of nonlinear models. In that case it has to be further checked that the constitutive laws touched by inversion in the bond graph representation (elements passed through by bicausality or of which causality changes with respect to the causal representation) are invertible in the domain of definition of the involved variables. 

Behavior of Entangled Polymer Melts and Solutions Steady-State Behavior. Before discussing the constitutive law given by the K-BKZ theory, it is important to discnss the nonlinear behavior of polymers that is observed. One snch behavior is that in steady state, ie, when the material response has quit changing after the application of a stress or deformation rate. For such a situation and recalling that a Newtonian fluid follows a viscosity law in which the stress is proportional to the strain rate... 

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... 

Recent years have seen great advances in nonlinear analysis of frame structures. These advances were led by the development and implementation of force-based elements (Spacone et al. 1996), which are superior to classical displacement-based elements in tracing material nonlinearities such as those encountered in reinforced concrete beams and columns. In the classical displacement-based frame element, the cubic and linear Hermitian polynomials used to interpolate the transverse and axial displacement fields, respectively, are only approximations of the actual displacement fields in the presence of non-uniform beam cross-section and/or nonhnear material behaviour. On the other hand, force-based frame element formulations stem from equilibrium between section and nodal forces, which can be enforced exactly in the case of a frame element. The exact flexibiUty matrix can be computed for an arbitrary (geometric) variation of the cross-section and for any section/material constitutive law. Thus, force-based elements enable, at no significant additional computational costs, a drastic reduction in the number of elements required for a given level of accuracy in the simulated response of a EE model of a frame structure. 

ECS does not have any specific guidelines on how to model material nonlinear-ities, except for general considerations on the concrete and steel constitutive laws (Eurocode 8, 2003). Material properties should be based on mean values of the mechanic characteristics for existing structures. 

Spread-plasticity models are classical finite elements where material nonlinearities are modeled at each integration point. Besides the classical two-node displacement-based beam elements, force-based two-node force-based elements have seen a widespread use both in research and commercial software (McKenna, 1997 MIDAS, 2006 Zimmermann, 1985-2007). The assumed force fields in a two-mode force-based element are exact within classical beam theories, such as the Euler-Bernoulli and Timoshenko theories (Marini Spacone, 2006 Spacone et al, 1996). This implies that only one element per structural member is used. The element implementation is not trivial and it implies element iterations, but these steps are transparent to the user. The section constitutive law is the source of material nonlinearities. 

Among the large variety of smart materials discussed today, piezoelectric and magnetostrictive materials can be described by linearised constitutive laws that are given below. Other widely used material types, such as electrostric-tive or shape memory materials, exhibit strongly nonlinear behaviour, the modelling of which may become quite demanding. 

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 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." °... 

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. 

In this work a numerical simulation of silo discharge processes is presented. A new advanced nonlinear hypoplastic constitutive law for die bulk solids with a new approach for the viscosity is adapted, and dynamic effects are especially studied. The dynamical behaviour of the dischaige process is described by a system of nonlinear differential equations in the Eulerian reference frame. Via the Finite Element Method (FEM), the velocity field of the flowing material, its density and pressure distribution can be calculated without the need of re-meshing the FE grid. The numerical simulation examples are chosen to be similar to an experimental test-silo for comparing the results with measured values. 

In case the applied load is increased beyond a certain level, the stress-strain relation is no longer proportional. The material behavior then becomes nonlinear, as illustrated in Fig. 2, which provides an out-of-scale description of a tension test for a steel specimen by means of the stress-strain curve, frequently referred also as the material s constitutive law. The green lines denote unloading and may coincide with the corresponding loading curve, in which case the... 

Remark 2 In the PDEM there is no need for the stochastic process of concern to be Markovian. Actually, in most cases the process is not Markov. Eor instance, in engineering practice a complex stmcture may be modeled by the finite element method where nonlinear constitutive relationship of the material, say, the stochastic damage constitutive law for concrete, is embedded. In this case, usually quite a few of internal variables, say, the damage variables, are involved, and thus the response processes are not Markovian (Li et al. 2014). 

The constitutive law describing the behavior of the steel material is the uniaxial Menegotto-Pinto model (1973). This computationally efficient nonlinear law is capable to model both kinematic and isotropic hardening as well as the Bauschinger effect, allowing for accurate simulation and reproduction of experimental results. The response of the steel material is defined by the following nonlinear equation ... 

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... 

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