Linear viscoelasticity material constants

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... 

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. 

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. 

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... 

Transient Response Constant Rate of Deformation. The constant rate of deformation response of the linear viscoelastic material was discussed above. From that discussion, the stress-deformation response looks nonlinear even when the material is linear viscoelastic. For the nonlinear material the response will not be simply described by the linear viscoelastic laws. However, the curves will look similar at low strain rates. At higher strain rates, a stress overshoot is observed. 

Again, the viscoelastic solution for stress is exactly the same as the elastic solution stress. As stated earlier, in general, if the linear elastic solution for stresses for a given boundary value problem does not contain elastic constants, the solution for stresses in a viscoelastic body with equivalent geometry and equivalent loads is identical to that for the elastic body. This means that the stress analysis of most problems considered in elementary solid mechanics such as beams in bending, bars in torsion or axial load, pressure vessels, etc. will have the same solution for stress in a linear viscoelastic material as in a linear elastic material. Further, stress analysis of combined axial, bending, torsion and pressure loads can be handled easily using superposition. 

Equation (3.26) gives a relaxation modulus that is independent of the applied strain magnitude, and will obey the homogeneity requirement of linearity for all scalars, with any arbitrary strain input. Equation (3.26) is not linear however as norms are not superposable except in the most trivial examples. The hypothetical material represented by Equation (3.26) is therefore non-linear, but for many types of tests used for material characterization it could not be distinguished from a linear viscoelastic material. In fact, the parameters n and P appearing in this constitutive equation can be adjusted so that the time derivative of the stress for a constant strain rate input is proportional to the relaxation modulus a commonly cited property of a linear viscoelastic material [12,37]. Careful examination of the stress output to various strain inputs confirms the non-linear nature of this equation and indicate it is within the range of this simple equation to describe the onedimensional response of solid propellants at small strains. To demonstrate this ability, the stress output for a variety of strain inputs have been determined for different values of n and p. 

These data are illustrated in Figure 3.4 through 3.9. In these calculations the ratio n/P has been kept constant therefore all of the equations would exhibit the same relaxation modulus. However as clearly indicated by these figures, the behavior to other inputs is different for the different values of n and P. This feature of giving the same output for one test, yet a different output for other tests, is characteristic of non-linear systems. If the material were linear, this feature would be impossible since one test dictates the results of all other tests for linear systems. Characterization of non-linear materials is therefore a difficult task as many tests must be used and one never knows if the chosen representation is complete. Individuals familiar with the behavior of linear viscoelastic materials and the non-linear behavior of... 

Transient Response Constant Rate of Deformation. The constant rate of deformation response of the linear viscoelastic material was discussed above. Prom that discussion, the stress-deformation response looks nonlinear even when the material is linear viscoelastic. For the nonlinear material the response will not be simply described by the linear viscoelastic laws. However, the curves will look similar at low strain rates. At higher strain rates, a stress overshoot is observed, which cannot occur for the linear viscoelastic material. Figure 26 shows the effect of increasing the strain rate on the transient stress-time response (which is related to the strain) for a polymer solution (81). As seen, the stress overshoot becomes weaker with decreasing strain rates (when the material response may be linearly viscoelastic) however, as strain rate increases, there is an onset of... 

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.

The viscoelastic creep modulus may be determined at a given temperature by dividing the constant applied stress by the total strain prevailing at a particular time. Since the creep strain increases with time, the viscoelastic creep modulus must decrease with time (Fig. 2-23). Below its critical stress for linear viscoelasticity, the viscoelastic creep modulus versus time curve for a material is independent of the applied stress. In other words, the family of strain versus time curves for a material at a given temperature and several levels of applied stress may be collapsed to a single viscoelastic creep-modulus-time-curve if the highest applied stress is less than the critical value. 

It is clear that this data treatment is strictly valid providing the tested material exhibits linear viscoelastic behavior, i.e., that the measured torque remains always proportional to the applied strain. In other words, when the applied strain is sinusoidal, so must remain the measured torque. The RPA built-in data treatment does not check this y(o )/S (o)) proportionality but a strain sweep test is the usual manner to verify the strain amplitude range for constant complex torque reading at fixed frequency (and constant temperature). 

An important and sometimes overlooked feature of all linear viscoelastic liquids that follow a Maxwell response is that they exhibit anti-thixo-tropic behaviour. That is if a constant shear rate is applied to a material that behaves as a Maxwell model the viscosity increases with time up to a constant value. We have seen in the previous examples that as the shear rate is applied the stress progressively increases to a maximum value. The approach we should adopt is to use the Boltzmann Superposition Principle. Initially we apply a continuous shear rate until a steady state... 

The reduced expressions of Table 4 form a set of universal viscoelastic functions. Given the polymer molecular weight, material constants [Jg, Je, etc.), and one extremal relaxation/retardation time, one should be able to predict, roughly, the nature of the system response (within the framework of the linear models) from Eqs. (T 1)—(T 6) and Fig. 2—5. 

Hooke s Law, which states that a proportional relationship exists between stress and strain, usually holds for a viscoelastic material at a small strain. This phenomenon is called linear viscoelasticity (LVE). Within the LVE region, the viscoelastic parameters G and G" remain constant when the amplitude of the applied deformation is changed. Consequently, parameters measured within the LVE region are considered material characteristics at the observation time (frequency). 

Real (viscoelastic) materials give an intermediate response that is an exponential curve. The exponential time constants associated with the curve are used to approximate the relaxation times of the material itself. Thus, the shape of the output curve is analyzed to give viscoelastic information, although this model fitting is only strictly legitimate in the linear viscoelastic region. Workers have shown that the mechanical parts of the models (springs and dashpots) can be associated with specific parts of a food s makeup. 

According to the change of strain rate versus stress the response of the material can be categorized as linear, non-linear, or plastic. When linear response take place the material is categorized as a Newtonian. When the material is considered as Newtonian, the stress is linearly proportional to the strain rate. Then the material exhibits a non-linear response to the strain rate, it is categorized as Non Newtonian material. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. This kind of materials are known as thixotropic deformation is observed when the stress is independent of the strain rate [2,3], In some cases viscoelastic materials behave as rubbers. In fact, in the case of many polymers specially those with crosslinking, rubber elasticity is observed. In these systems hysteresis, stress relaxation and creep take place. 

The quantities r and r] in equation (8.34) depend on the invariants of the tensor rik in accordance with equation (8.32). We ought to note that the behaviour of a non-linear viscoelastic liquid in a non-steady state would be different, if a dependence of the material parameters r and r] on the tensor velocity gradients or on the stress tensor is assumed. This is a point which is sometimes ignored. In any case, if r and r) are constant, equation (8.34) belongs to the class of equations introduced and investigated by Oldroyd (1950). 

A tapping mode (also called intermittent contact mode) it is a non linear resonance mode. In this case, the oscillation amplitude is larger and the mean position of the tip is closer to the surface. The tip almost touches the surface at each oscillation. In this mode, friction can be avoided as well as the sample deformation and wear. Adhesion is also avoided thanks to the extremely short time of contact . The height of the sample is generally controlled so that the oscillation amplitude remains constant. The phase shift of the oscillation is then characteristic of the system dissipation, which is very useful for characterizing viscoelastic materials. 

Differences between solid-like and liquid-bke complex fluids show up in all three of the shearing measurements discussed thus far the shear start-up viscosity t), the steady-state viscosity rj(y), and the linear viscoelastic moduli G co) and G (o). The start-up stresses a = y/ +()>, t) of prototypical liquid-like and solid-like complex fluids are depicted in Fig. 1-6. For the liquid-like fluid the viscosity instantaneously reaches a steady-state value after inception of shear, while for the solid-like fluid the stress grows linearly with strain up to a critical shear strain, above which the material yields, or flows, at constant shear stress. 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

Ageing plays a key role in the non-linear viscoelastic behaviour of polymers. When left at rest and at constant temperature, there is continuous stiffening. However, when the aged material is slightly heated or mechanically deformed, it is deaged and softened (Struik, 1978, 1983). 

Whereas the general linear transport equation (11) usually provides an adequate description of transport processes, there are cases of practical interest where a generalization is needed Since the transport coefficient is a constant, equation (11) implies that the response— the flux J— momentarily follows the driving force T, which is not always the case. Consider, for example, the response of a viscoelastic material (such as a gel) to an applied stress or the response of a dielectric material to an applied electric field, both of which generally lag behind the driving force. Such delayed responses may be interpreted as... 

The relative magnitudes of the two moduli provide significant information regarding strength of internal association or structure in fluids and dispersions. These moduli are measured as a function of strain, frequency, or time. For some dispersions, the magnitudes of G and G" may remain constant as a function of either frequency or strain. Such materials are referred to as linearly viscoelastic. 

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