Complex viscoelastic functions shear stress

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

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. 

Since the linear viscoelasticity of a material is described with a material function G(t), any experiment which gives full information on G(t) is sufficient it is not necessary to give the stresses corresponding to various strain histories. We will restrict the discussion to incompressible isotropic materials. In this case, different types of deformation such as elongation and shear give equivalent information in the range of linear viscoelasticity. Several types of experiments measure relaxation modulus, creep compliance, complex modulus etc which are equivalent to the relaxation modulus (1). 

As mentioned earlier, the DMTA technique measures molecular motion in adhesives, and not heat changes as with DSC. Many adhesives exhibit time-dependent, reversible viscoelastic properties in deformation. Hence a viscoelactic material can be characterized by measuring its elastic modulus as a function of temperature. The modulus depends both on the method and the time of measurement. Dynamic mechanical tests are characterized by application of a small stress in a time-varying periodic or sinusoidal fashion. For viscoelastic materials when a sinusoidal deformation is applied, the stress is not in phase with displacement. A complex tensile modulus E ) or shear modulus (G ) can be obtained ... 

Based on the measurement of the stress, a, resulting on the application of periodic strain, e, with equipment as shown in Fig. 4.155, one can develop a simple formalism of viscoelasticity that permits the extraction of the in-phase modulus, G, the storage modulus, and the out-of-phase modulus, G", the loss modulus. This description is analogous to the treatment of the heat capacity measured by temperature-modulated calorimetry as discussed with Fig. 4.161 of Sect. 4.5. The ratio G7G is the loss tangent, tan 6. The equations for the stress o are easily derived using addition theorems for trigonometric functions. A complex form of the shear modulus, G, can be used, as indicated in Fig. 4.160. 

Some of the manifestations of viscoelasticity are delayed relaxation of stress after cessation of flow phase shift between stress and strain rate in oscillatory shear flow shear thinning (decrease of viscosity) at shear rates exceeding the reciprocal of the longest relaxation time and normal stress differences in shear flow, whose magnitudes are related to the relaxation time spectrum. A very convenient observation for experimentalists is that there is a close similarity between the shear viscosity and first normal stress difference as functions of shear rate and the corresponding parameters, complex viscosity and storage modulus, as functions of frequency in a small amplitude oscillatory shear. 

The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillatory driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which is attached a stress transducer. On account of the energy dissipated by the viscoelastic polymer system, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions, p, G, G" and others [33-40]. 

We have reported [66] a limited study of spread polymethyl methacrylates and polyethylene oxide. Figure 12.19 shows the variation in surface tension, shear viscosity and dilational modulus obtained from SQELS data as a function of surface concentration. The viscoelastic moduli both show maximum values at finite values of the surface concentration. As the capillary waves generate oscillatory stress and strain, these are related via the complex dynamic modulus of the surface... 

In the above unsteady tests, if one keeps the level of imposed stress and strain low enough, the measured material functions show an independence from these applied stimuli levels, exhibiting only a dependence on time (or frequency). This type of response indicates linear viscoelastic behavior. The primary modes of deformation employed in these tests are either shear or extension. If there is no volume change accompanying the deformation, a single modulus or compliance, whether real or complex, but a function of time (or frequency) and temperature only, suffices to characterize the material behavior. We will define moduli and compliances further below. Let us now start examining these and other key topics in linear viscoelasticity. 

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