Viscoelasticity, linear

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. 

I IEP roughly matches the maximum in the elastic modulus corresponding to the viscoelastic linear region in a 2 vol% dispersion, with or without magnetic field. 

The high-strain experiments do not verify the viscoelastic linearity. 

Before phase separation, viscoelastic linearity is respected because the viscoelastic values are the same in the low-strain and high-strain experiments. The difference begins to appear at the beginning of phase separation. For... 

When we progress from the foregoing qualitative discussion of structure-property relationships to the quantitative specification of mechanical properties, we enter a jungle that has been only partially explored. The most convenient point of departure into this large and complex subject is provided by the topic of "linear viscoelasticity." Linear viscoelasticity represents a relatively simple extension of classical (small-strain) theory of elasticity. In situations where linear viscoelasticity applies, the mechanical properties can be determined from a few experiments and can be specified in any of several equivalent formulations (11). 

K-factor See coefficient of thermal conductivity, kieselguhr See diatomaceous earth, kinematic viscosity See viscosity, kinematic, kinetic A branch of dynamics that is concerned with the relations between the movement of bodies and the forces acting on them. See Avogadro s law reactor technology thermogravimetric analysis viscoelasticity, linear viscoelasticity, nonlinear. 

Describing and predicting viscoelastic properties of polymer materials or adhesively bonded joints on the basis of analytical mathematical equations are justified only in the limits of linear viscoelasticity. Linear viscoelasticity is typically limited to strain levels below 0.5%. Furthermore, linear viscoelastic behavior is associated to the Boltzmann superposition principle, the correspondence principle, and the principle of time-temperature superposition. 

In the case of gel-like samples (G > G" in the viscoelastic linear region), this test is frequently used to determine the yield point (yield stress) and flow point (flow stress). The yield point corresponds to the limiting value of the linear viscoelastic region. The flow point corresponds to the stress where G = G". 

Frequency sweeps are oscillatory tests performed at variables frequencies, keeping the amplitude and temperature at a constant value. For controlled shear strain tests, a sinusoidal strain is fixed with an amplitude in the viscoelastic linear region. These tests are used to investigate the time-dependent shear behavior. 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

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. 

Many investigators beheve that the Bingham model accounts best for observations of electrorheological behavior (116,118), but other models have also been proposed (116,119). There is considerable evidence that ER materials behave as linear viscoelastic fluids while under the influence of electric field (120) thus it appears that these materials maybe thought of as elastic Bingham fluids. 

Figure 16 (145). For an elastic material (Fig. 16a), the resulting strain is instantaneous and constant until the stress is removed, at which time the material recovers and the strain immediately drops back to 2ero. In the case of the viscous fluid (Fig. 16b), the strain increases linearly with time. When the load is removed, the strain does not recover but remains constant. Deformation is permanent. The response of the viscoelastic material (Fig. 16c) draws from both kinds of behavior. An initial instantaneous (elastic) strain is followed by a time-dependent strain. When the stress is removed, the initial strain recovery is elastic, but full recovery is delayed to longer times by the viscous component.

Figure 36 is representative of creep and recovery curves for viscoelastic fluids. Such a curve is obtained when a stress is placed on the specimen and the deformation is monitored as a function of time. During the experiment the stress is removed, and the specimen, if it can, is free to recover. The slope of the linear portion of the creep curve gives the shear rate, and the viscosity is the appHed stress divided by the slope. A steep slope indicates a low viscosity, and a gradual slope a high viscosity. The recovery part of Figure 36 shows that the specimen was viscoelastic because relaxation took place and some of the strain was recovered. A purely viscous material would not have shown any recovery, as shown in Figure 16b. 

One might consider combining the Hertzian linear viscoelastic bulk related phenomenon inside the contact area with the results provided by Greenwood and... 

Johnson [109] for linear viscoelastic effects inside the cohesive zone. For growing cracks... 

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

The premise of the above analysis is the fact that it has treated the interfacial and bulk viscoelasticity equally (linearly viscoelastic experiencing similar time scales of relaxation). Falsafi et al. make an assumption that the adhesion energy G is constant in the course of loading experiments and its value corresponds to the thermodynamic work of adhesion W. By incorporating the time-dependent part of K t) into the left-hand side (LHS) of Eq. 61 and convoluting it with the evolution of the cube of the contact radius in the entire course of the contact, one can generate a set of [LHS(t), P(0J data. By applying the same procedure described for the elastic case, now the set of [LHS(t), / (Ol points can be fitted to the Eq. 61 for the best values of A"(I) and W. 

Golden, J.M. and Graham, G.A.C., Boundary Value Problems in Linear Viscoelasticity. Springer-Verlag, Heidelberg, 1988. 

Other ideas proposed to explain the 3/4 power-law dependence include effects due to viscoelasticity, non-linear elasticity, partial plasticity or yielding, and additional interactions beyond simply surface forces. However, none of these ideas have been sufficiently developed to enable predictions to be made at this time. An understanding of this anomalous power-law dependence is not yet present. 

The EMT analysis indicated that the stress relaxes in proportion to the number of bonds removed. The initial linear decrease of E/Eq with is intuitively appealing and is the basis for many linear constitutive theories of polymers. An example is the Doi-Edwards theory of viscoelasticity of linear polymer melts [49] in which... 

This type of response is referred to as non-linear viscoelastic but as it is not amenable to simple analysis it is often reduced to the form... 

This equation is the basis of linear viscoelasticity and simply indicates that, in a tensile test for example, for a fixed value of elapsed time, the stress will be directly proportional to the strain. The different types of response described are shown schematically in Fig. 2.1. 

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

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