Viscoelasticity nonlinear theory

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." °... 

Rivlin and Ericksen (1955) proposed a nonlinear viscoelasticity theory,in which the components of stress at time t in an element of material depend on the gradient of displacement, velocity, acceleration, second acceleration,..., and (n- l)th acceleration in that element at time t. The fluid described by the constitutive equation based on this theory is referred to as a memoryless fluid, the reason being that the components of stress at time t are independent of those experienced up to the time t. A general representation of this constitutive equation is given by (Rivlin and Ericksen 1955)... 

Further discussions on nonlinear viscoelastic theory are given by Green and Adkins (1960) and Hilton (1975). 

In the model, in order to describe the frozen stress and its activation in SMP, a linear viscoelastic theory was used as the first trial. Since the linear theory is limited to small deformations, subsequently a nonlinear viscoelastic theory was adopted for large deformations. The linear viscoelastic model was found to predict the characteristics of SMPs, especially the strain fixity and recovery properties for small deformations with some discrepancy between the experimental and calculated values. The main source for the error was found to be the reduced rigidity of SMPs due to the thermal treatment. This should be avoided for better shape memory performance of SMPs. 

In another research, the thermo-mechanical behavior of SMPs was described by both linear and nonlinear viscoelastic theories [4]. In this woik four element mechanical units consisting of spring, dashpot and frictional device were used to derive a constitutive differential equation. In order to determine the material properties by a constitutive differential equation the modulus, viscosity and other parameters were assumed to decay exponentially with temperature. Liu et al. [5] developed a constitutive equation for SMPs based on thermodynamic concepts of entropy and internal energy. They adopted the concept of frozen strain and demonstrated the utility of the model by simulating the stress and strain... 

Recently, McLeish and Larson [95] developed a nonlinear viscoelastic theory for an idealized branched polymer with multiple branches but only two branch points. This molecular structure, called the pom-pom (described in Section 10.9.2), is a generalization of the H polymer in that each of the two branch points of the pom-pom is permitted to have an arbitrary number of branches, q see Fig. 9.4. The pom-pom model contains three basic time constants the backbone reptation time T, the backbone stretch time T, and the arm relaxation time x,. These time constants are given in terms of the molecular parameters of the pom-pom molecule as ... 

P. E. Rouse. The theory of nonlinear viscoelastic properties of dilute solutions of scaling polymers. J Chem Phys 27 1273-1280, 1953. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

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... 

Although attempts to measure and interpret nonlinear behavior are potentially useful, there are few reports in the literature on the measurement of the nonlinear viscoelastic properties of foods. This has been due to a lack of both suitable instrumentation and suitably developed theory nonlinear behavior, the predominant form of which is the exhibition of normal stresses, and a dependence of viscosity on shear rate, is much more complex than linear behavior (Gunasekaran and Ak, 2002). 

Linear viscoelastic behavior is actually observed with polymers only in very restricted circumstances involving homogeneous, isotropic, amorphous specimens subjected to small strains at temperatures near or above Tg and under test conditions that are far removed from those in which the sample may be broken. Linear viscoelasticity theory is of limited use in predicting service behavior of polymeric articles, because such applications often involve large strains, anisotropic objects, fracture phenomena, and other effects which result in nonlinear behavior. The theory is nevertheless valuable as a reference frame for a wide range of applications, just as the thermodynamic equations for ideal solutions help organize the observed behavior of real solutions. 

At small stresses and strains, glassy PC exhibits linear viscoelastic behavior. The limit of applicability of the theory of linear viscoelasticity has been investigated by Yannas et al. over the temperature range 23 °C-130 °C. The critical strain at which, within the precision of their measurement, deviations from the linear theory occur has been found to diminish from about 1.2% at 23 °C to about 0.7 % at 130 °C. According to Jansson and Yannas the transition from linear to nonlinear viscoelastic behavior is marked by the onset of significant rotation around backbone bonds. 

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)... 

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

In the last chapter we discussed the relation between stress and strain (or instead rate-of-strain) in one dimension by treating the viscoelastic quantities as scalars. When the applied strain or rate-of-strain is large, the nonlinear response of the polymeric liquid involves more than one dimension. In addition, a rheological process always involves a three-dimensional deformation. In this chapter, we discuss how to express stress and strain in three-dimensional space. This is not only important in the study of polymer rheological properties in terms of continuum mechanics " but is also essential in the polymer viscoelastic theories and simulations studied in the later chapters, into which the chain dynamic models are incorporated. 

The molecular approach which we will see eventually proved to be most successful in treating negative is based on the work of Doi [23]. Doi noted that the well established phenomenological theories for thermotropes (which he termed TLP for Ericksen, Leslie and Parodi [68]) which is successful in describing many dynamic phenomena in MLC nematics, is limited for polymeric liquid crystals in that it does not predict nonlinear viscoelasticity. Doi s approach determines the phenomenological coefficients from molecular parameters, so that the effects of, for example, molecular weight and concentration can be treated. He considers a single molecule (the test rod ) and notes that as concentration increases, constraints on its motion are imposed by collisions with other rods. This constraint can be modeled as a tube... 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

More evidence comes from the study of viscoelasticity, which has been done extensively in the past and established the characteristic aspects common to all flexible polymers. The reptation model has succeeded in explaining many of these features and also predicting some of the behaviour in nonlinear viscoelasticity. In this chapter we shall describe the reptation theory for viscoelasticity in detail, and discuss the validity of the reptation model in solutions and melts. 

Below the glass temperatin-e, the nonlinear viscoelastic response of polymeric materials has been much less widely studied than has the behavior of melts and solutions. One reason for this is the lack of an adequate theory of behavior. Therefore the discussion about amorphous materials below the glass tem-peratiu e focuses on recent measin-ements of the nonlinear response as well as... 

Issues of Material Compressibility. There is a full theory of compressible and nonlinear viscoelastic materials that would be equivalent to the compressible finite deformation elasticity theory described above (eq. 39), but more complicated because of the need to develop an expansion of the time-dependent strain potential function as a series of multiple integrals (108,109). One such formahsm is discussed briefiy under Lustig, Shay and Caruthers Model. Here a simphfied model that is based upon the K-BKZ framework with a VL-like kernel function (98) is examined. 

Fig. 60. Comparison of experimental (circles) creep and recovery behavior of a glass-reinforced phenolic resin with the predictions from the Schapery (147-149) nonlinear viscoelastic model. o Experimental Data A Predicted Recovery Data (Nonlinear Theory). After Schapery (147), with permission.

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). 

A thermodynamically consistent theory for nonlinear viscoelasticity was first proposed by Schapery.(26) Jhe law can be derived from fundamental principles using the concepts of irreversible thermodynamics. A comprehensive review of the thermodynamics basis of Schap-ery s theory has been presented by Hiel et... 

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