Small Viscoelasticity Approximation

In Sect. 2.5, we discussed briefly the case where the ratio of viscoelastic (time-dependent terms in the functions G(t), J(t)) to purely elastic effects (constant terms in these functions) is small. The possibility of expressing solutions as power series in this ratio was noted. In the present section, we shall consider solutions to first order in this parameter. It turns out that the problem of the moving load greatly simplifies, to the extent that explicit expressions for all quantities of interest may be written down, in contrast to the highly implicit equations which emerged from the exact analysis in the previous section. 

We shall confine our attention to the steady state problem, with limiting friction, the general equations for which are given in Sect. 3.5. It should be emphasized however that this approximation, if valid, brings about great simplification in a wide variety of problems. 

To first order, it follows from (3.7.1) that T x,y), defined by (3.5.24b) is given by 

Let us now consider the special case of a standard linear model, putting [see (3.6.15, 16)] 

It should be noted that (3.7.23) is essentially as general as an arbitrary discrete or continuous spectrum model, in the present context. This is because all quantities must be linear in the viscoelastic functions, so that the results for a more general model are simply sums of terms of the form that will now be derived. Explicit results can be obtained for the problem with friction, in terms of Whittaker functions. However, these will not be introduced in the present work. We refer to Golden (1979a, 1986a) for further details. In the frictionless case (from (3.7.11) we see that d,o = 0)  

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An expression for the hysteretic friction coefficient in the small viscoelasticity approximation is derived in the next section. 

We consider (3.8.6) in the small viscoelasticity approximation. If e is chosen to be unity, then the portion Vo(x) in (3.7.4) cancels the term to first order. 

In the small viscoelasticity approximation, however, the entirely explicit formula (3.8.15), or for standard linear solid, (3.8.16), can be given for lubricated contact. The expression for a more general spectrum model is a sum of terms of the form given by (3.8.16). 

We consider plane contact and crack problems in this chapter, without neglecting inertial effects. Such problems are typically far more difficult than the non-inertial problems discussed in Chaps. 3 and 4, and require different techniques for their solution. This is an area still in the development stage so that it will not be possible to achieve the kind of synthesis or unification which is desirable. We confine our attention to steady-state motion at uniform velocity V in the negative x direction. We begin by deriving boundary relationships between displacement and stress. These are applied to moving contact problems in the small viscoelasticity approximation, and to Mode III crack problems without any approximation. 

In the small deformation approximation, it is assumed that the deformations undergone by the material are small, at least in the recent past. Approximations of different orders can be developed. The approximation of first order for an incompressible fluid is given by Boltzmann s equation of linear viscoelasticity,... 

Equation 94 holds regardless of the overtone order and is a simple result. However, Eq. 94 is not reproduced when applying the small-load approximation (Eq. 51) and using the load impedance of a viscoelastic film as expressed in Eq. 72 ... 

We consider the moving load problem in this section to the extent of deriving an expression for the coefficient of hysteretic friction in the small viscoelasticity and small velocity approximations, respectively. 

Sect. 5.4 contains approximate formulae for the coefficient of hysteretic friction. The approximations apply in the cases of small viscoelasticity (5.4.4,13) and small velocity (5.4.22, 23, 26). 

Golden, J.M. (1978) Hysteretic friction in the small velocity approximation. Wear 50, 259-273 Golden, J.M. (1979a) The problem of a moving rigid punch on an unlubricated viscoelastic halfplane. Q. J. Mech. Appl. Math. 32, 25-52... 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

Later on we shall see that the superposition principle is is, for polymers, only seldomly obeyed linear viscoelasticity is only met at very small stresses and deformations, at loading levels occurring in practice the behaviour may strongly deviate from linearity. However, the superposition principle provides a useful first-order approximation. 

Thixotropy is the tendency of certain substances to flow under external stimuli (e.g., mild vibrations). A more general property is viscoelasticity, a time-dependent transition from elastic to viscous behavior, characterized by a relaxation time. When the transition is confined to small regions within the bulk of a solid, the substance is said to creep. A substance which creeps is one that stretches at a time-dependent rate when subjected to constant stress and temperature. The approximately constant stretching rates at intermediate times are used to characterize the creeping characteristics of the material. 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

The engineering property that is of interest for most of these applications, the modulus of elasticity, is the ratio of unit stress to corresponding unit strain in tension, compression, or shear. For rigid engineering materials, unique values are characteristic over the useful stress and temperature ranges of the material. This is not true of natural and synthetic rubbers. In particular, for sinusoidal deformations at small strains under essentially isothermal conditions, elastomers approximate a linear viscoelastic... 

Figure 20 shows the result for a flow curve, where a small positive separation parameter was necessary to fit the flow curve and the linear viscoelastic moduli simultaneously. The data are compatible with the (ideal) concept of a yield stress, but fall below the fit curves for very small shear rates. This indicates the existence of an additional decay mechanism neglected in the present approach [32, 33]. Again, the A-formula describes the experimental data correctly for approximately four decades. For higher shear rates, an effective Herschel-Bulkley law... 

It has been remarked that time (frequency) - temperature reduced data on carbon black filled rubbers exhibit increased scatter compared to similar data on unfilled polymers. Payne (102) ascribes this to the effects of secondary aggregation. Possibly related to this are the recent observations of Adicoff and Lepie (174) who show that the WLF shift factors of filled rubbers giving the best fit are slightly different for the storage and loss moduli and that they are dependent on strain. Use of different shift factors for the various viscoelastic functions is not justified theoretically and choice of a single, mean ar-funetion is preferred as an approximation. The result, of course, is increased scatter of the experimental points of the master curve. This effect is small for carbon black... 

In practice, large deformations are generally applied. The linear region tends to be very small, up to a strain of 10 4 10 3. With increasing strain, the system becomes more and more viscoelastic, and its structure gradually breaks down then yielding occurs, accompanied by a stress overshoot, and the system finally shows plastic flow. The yield stress, or firmness, is the most important quantity in practice, and it approximately scales with solid fat content squared. Other variables affecting the modulus have a comparable effect on the yield stress. 

In principle, viscoelastic constants can be extracted from the experimental data by fitting Eq. 70 (or any of the more complicated equations below) to the data. For a small film fhickness, certain approximations hold which make the derivation more transparent. If kfdf is much less than unity, the tangent in Eq. 70 can be Taylor-expanded to third order as tan(x) l/3x, resulting... 

When Ay is very small (Ay < 0.1%), the stress response caused by the sinusoidal straining given by Equation 1 is approximately sinusoidal, and the viscoelastic behavior falls in the region of linear viscoelasticity. In this case, the phase angle difference 8 between the stress wave and strain wave is constant throughout the cycle, and the stress response can be expressed by ... 

When we use the value of Eo obtained with the relatively small amplitude of the superimposed strain (Ay)s, o-e ax -min from Equation 9 is larger than the experimental value of (Td ax < min (where aa indicates the viscoelastic stress). This is caused by the fact that the modulus observed with a high frequency, smaller strain amplitude is larger than that observed with a low frequency, larger strain amplitude. Assuming that the actual <7eimax IS approximately equal to (raj axy adjust the values of E 0) by a factor f given by Equation 10 ... 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

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