Linear viscoelastic materials

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. 

When a linear viscoelastic material is subjected to a constant stress, a, at time, /i, then the creep strain, e(t), at any subsequent time, t, may be expressed as... 

For a linear viscoelastic material in which the strain recovery may be regarded as the reversal of creep then the material behaviour may be represented by Fig. 2.49. Thus the time-dependent residual strain, Sr(t), may be expressed as... 

J.L. Leblanc, Fourier Transform rheometry A new tool to investigate intrinsically non-linear viscoelastic materials, Ann. Trans. Nordic Rheol. Soc., 13, 3-21, 2005. 

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

For positive exponent values, the symbol m with m > 0 is used. The spectrum has the same format as in Eq. 8-1, H X) = H0(X/X0)m, however, the positive exponent results in a completely different behavior. One important difference is that the upper limit of the spectrum, 2U, has to be finite in order to avoid divergence of the linear viscoelastic material functions. This prevents the use of approximate solutions of the above type, Eqs. 8-2 to 8-4. 

In this section we deal with perhaps the most conceptually difficult of all the responses observed in linear viscoelastic materials. This is the response of a material to an oscillating stress or strain. This is an area that illustrates why rheological techniques can be considered as mechanical spectroscopy. When a sample is constrained in, say, a cone and plate assembly, an oscillating strain at a given frequency can be applied to the sample. After an initial start-up period, a stress develops in direct response to the applied strain due to transient sample and instrumental responses. If the strain has an oscillating value with time the stress must also be oscillating with time. We can represent these two wave-forms as in Figure 4.6. 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

Note 4 Generally, a linear viscoelastic material has a spectrum of retardation times, which are reciprocals of = 0, 1in the polynomial Q D). 

A general description of the fundamental relationships governing the dynamic response of linear viscoelastic materials may be found in several sources (28, 37, 93). In general, sinusoidally applied strains (stresses) result in sinusoidal stresses (strains) that are out of phase. Measurements may be made under uniaxial, shear, or dilational loading conditions, and the resultant complex moduli or compliance and loss-phase angle are computed. Rotating radius vectors are usually taken to represent the... 

An approach suggested by Williams et al. (117) uses an energy balance equation for initiation of flaw growth in a linearly viscoelastic material. A spherical flaw geometry was selected for simplicity since the expressions for the critical values of applied stress to cause fracture were similar for several flaw geometries. The critical conditions are based on a power (energy rate) balance... 

Thus, we may give a good description of a linear viscoelastic material in terms of relaxed, and unrelaxed elastic constants and a distribution of relaxation times (- this is not necessarily the same distribution for each elastic constant ). These all have to be found from experiments. In general it is possible to find some of the relaxed and unrelaxed elastic constants and to estimate the distribution of relaxation times. 

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. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

Stresses in viscoelastic materials "remember" deformation prehistory and so are not an unambiguous function of instantaneous deformations however, they may be expressed by a functional. For a linear viscoelastic material, the relationship between stresses and deformations... 

The calculation of residual stresses in the polymerization process during the formation of an amorphous material was formulated earlier.12 The theory was based on a model of a linear viscoelastic material with properties dependent on temperature T and the degree of conversion p. In this model the effect of the degree of conversion was treated by a new "polymerization-time" superposition method, which is analogous to the temperature-time superposition discussed earlier. 

Apply the Boltzmann superposition principle for the case of a continuous stress application on a linear viscoelastic material to obtain the resulting strain y(t) in terms of J(t — t ) and ih/dt, the stress history. Consider the applied stress in terms of small applied At,-, as shown on the accompanying figure. 

When a sinusoidal strain is imposed on a linear viscoelastic material, e.g., unfilled rubbers, a sinusoidal stress response will result and the dynamic mechanical properties depend only upon temperature and frequency, independent of the type of deformation (constant strain, constant stress, or constant energy). However, the situation changes in the case of filled rubbers. In the following, we mainly discuss carbon black filled rubbers because carbon black is the most widespread filler in rubber products, as for example, automotive tires and vibration mounts. The presence of carbon black filler introduces, in addition, a dependence of the dynamic mechanical properties upon dynamic strain amplitude. This is the reason why carbon black filled rubbers are considered as nonlinear viscoelastic materials. The term non-linear viscoelasticity will be discussed later in more detail. 

Linear Viscoelasticity Theory. FTMA is based on linear viscoelasticity theory. A one dimensional form of constitutive equation for linear viscoelastic materials which are isotropic, homogeneous, and hereditary (non-aging) is given by (21) ... 

It consists in developing numerical and analytical methods to invert a linear viscoelastic material fimction to determine the molecular weight distribution. There are several reasons to pursue such an objective many commercial polymers are slightly or not at aU soluble in usual solvents, thus techniques like gel permeation chromatography or light scattering are inapplicable. Rheological characterization can be performed on-line and in real time and it is also a less cots-effective technique. 

Y]sf Findlay, JS Lai, K Onaran. Creep and Relaxation of Non-Linear Viscoelastic Materials. Amsterdam, North-Holland, 1976, 71. 

As a general rule, the larger the losses, the larger are the discrepancies between these two solutions to the free vibration problem. However, it is possible to have greater discrepancies even for a low loss material due to changes in the damped sinusoidal term. For a more complex linear viscoelastic material consisting of a finite number of elementary viscoelastic elements, the solution would include a sum of decreasing exponential terms and damped sinusoidal waves. 

Ferguson, R., paper presented at lUPAC, Montreal, 1991. Findley, W. N., Lai, S. S., and Onaran, K., Creep Relaxation of Non Linear Viscoelastic Materials , North Holland, Amsterdam (1976). 

This phase angle difference represents the net effect of the nonelastic contribution, and its physical significance corresponds to that of constant 8 measured for the linear viscoelastic material. 

As discussed earlier for a Hookean solid, stress is a linear function of strain, while for a Newtonian fluid, stress is a linear function of strain rate. The constants of proportionality in these cases are modulus and viscosity, respectively. However, for a viscoelastic material the modulus is not constant it varies with time and strain history at a given temperature. But for a linear viscoelastic material, modulus is a function of time only. This concept is embodied in the Boltzmann principle, which states that the effects of mechanical history of a sample are additive. In other words, the response of a linear viscoelastic material to a given load is independent of the response of the material to ary load previously on the material. Thus the Boltzmann principle has essentially two implications — stress is a linear function of strain, and the effects of different stresses are additive. 

The most commonly used model is the Boltzmann superposition principle, which proposes that for a linear viscoelastic material the entire loading history contributes to the strain response, and the latter is simply given by the algebraic sum of the strains due to each step in the load. The principle may be expressed as follows. If an equation for the strain is obtained as a function of time under a constant stress, then the modulus as a function of time may be expressed as... 

First, we need a rule to predict the effect of time-varying loads on a viscoelastic model. When a combination of loads is applied to an elastic material, the stress (and strain) components caused by each load in turn can be added. This addition concept is extended to linear viscoelastic materials. The Boltzmann superposition principle states that if a creep stress ai is... 

It is wasteful of material to design a product to be in the linear viscoelastic region. The pseudo-elastic design method, for non-linear viscoelastic materials, gives a more reasonable design. The process requires an initial design,... 

For a linear viscoelastic material, the stress also varies sinusoidally, but leads in phase by an angle 8... 

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. 

Figure 7.2 shows the effect of applying a stress a, e.g. a tensile load, to a linear viscoelastic material at time t = 0. The resulting strain e t) can be divided into three parts ... 

Just as real materials may have behaviour close to ideal yielding behaviour but with some features similar to those of viscoelastic materials, materials that exhibit behaviour close to the ideal linear viscoelastic may exhibit features similar to those of yield, particularly for high stresses and long times of application. If the term e i) in equation (7.4) is not zero, a linear viscoelastic material will not reach a limiting strain on application of a fixed stress at long times the strain will simply increase linearly with time. The material may also depart from linearity at high or even moderate stresses, so that higher stresses produce disproportionately more strain. These are features characteristic of yield. 

The principal limitation of Wright s static displacement model is that it does not consider the accumulation of deformations due to the passage of a number of waves. This problem has been approached by Schapery and Dunlap (1978), modeling the soil as a linearly viscoelastic material. Their analysis also included the effect of energy adsorption of the seafloor on the wave characteristics. 

Kum Kumar, M. V. R., Narasimhan, R. Analysis of spherical indentation of linear viscoelastic materials. Curr. Sci. 87 (2004) 1088-1095. 

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