Sinusoidal deformations

It turns out that stress relaxation following a simple shear deformation is seldom employed experimentally. A more common technique is to measure the steady state response to small sinusoidal deformations as a function of angular frequency to. The dynamic storage modulus G (to) and loss modulus G"(to) in small sinusoidal deformations are related to G(t) ... 

Figure H3.2.2 Responses of an ideal elastic, viscous, and viscoelastic material to a sinusoidal deformation. 8, phase angle y, shear strain co, angular frequency o, shear stress.

Prior to the experiments, the polymer sample is fixed between two clamps and the sample dimensions are measured carefully. One of the two clamps is fixed, while the other can be moved up and down. To this clamp, a sinusoidal deformation with an amplitude e and a frequency to is applied and the corresponding force is measured, which can be transferred into the stress a a with the sample dimensions. From these two quantities the complex modulus of elasticity E is measured in dependence of sample temperature T and oscillation frequency to. 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

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... 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

It is evident from the above description that G (oj) and are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. The loss parameters G" oj) and on the other hand, reflect the nonrecoverable use of applied mechanical energy to cause viscous flow in the material. At a specified frequency and temperature, the dynamic response of a polymer in shear deformation can be summarized by any one of the following pairs of parameters G ( w) and G"( w), J ioj) and or absolute modulus G and tan 6. 

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 ... 

Because the surface light scattering method is less well known than the bulk scattering one, we will give some details about its principle. In the surface scattering method, a laser beam impinges on the surface. Because of thermal motion, the surface is not perfectly flat and scatters light in all directions around the specularly reflected beam. The surface displacement C can be described as a superposition of sinusoidal deformations C = where r is the position in the equilibrium plane of the surface and q is... 

The rheological behavior of a viscoelastic material can be investigated by applying a small-amplitude sinusoidal deformation. The behavior can be described by a mechanical model, called the Maxwell model [33], consisting of an elastic spring with the Hookean constant, G , and a dashpot with the viscosity, r/<,. The variation of storage modulus (G ) and loss modulus (G") with shear frequency, O), are given by the equations... 

Fig. 8.3 rop of a rubber curtain constrained at one edge with an imposed sinusoidal deformation z(0,y) =A(0) sin(i7(0)j) and power laws describing the evolution of the wavelength with the... 

The amplitude of the sinusoidal deformation is made small (1.0 x 10 cm, approximately 0.03 to 0.05 percent strain) to ensure linear viscoelastic response. The frequency is maintained constant for each data point. 

The viscoelastic behaviour of concentrated suspensions can be studied using several different methods (4, 7). The most widely used method consists of subjecting the material to a continuously oscillating strain over a range of frequencies and then measuring the peak value of the stress, ao, and the phase difference between the stress and strain, 8. A sinusoidal deformation is usually employed. 

A very good way to characterize and differentiate between elastomers and rigid plastics is by the measurement of dynamic mechanical properties. A most convenient method to study dynamic mechanical properties is to impose a small, sinusoidal shear or tensile strain and measure the resulting stress. Dynamic mechanical properties are most simply determined for a small sinusoidally varying strain, for which the response is a sinusoidally varying stress. An increase in frequency of the sinusoidal deformation is equivalent to an increase in strain rate. 

The moduli K and K are associated with mean curvature C -I- C2 and Gaussian curvature C1C2, respectively. Their roles are very different. Consider an initially plane surface subject to a sinusoidal deformation defined by some wave vector k. Cl measured parallel to k will vary sinusoidally, whilst C2 measured perpendicularly will be identically zero. The elastic energy involved in such a deformation is proportional to K and is independent of K (since C1C2 = 0). The role of K is thus essentially to limit the amplitude of thermal curvature fluctuations. 

When a Hookean solid is subjected to sinusoidally varying shear displacements of small amplitudes, at frequencies low enough so that inertia is not important, the shear stress will remain proportional to the shear strain. Thus stress and strain signals will be in phase. When, however, a Newtonian liquid is subjected to the same kind of sinusoidal deformation, the stress will be proportional to the strain rate. Since the strain rate is the time derivative of the strain, it will be 90 degrees out of phase with the strain. For the Newtonian fluid, therefore, stress and... 

Many technologies and natural phenomena involve processes of fast expansion or compression of fluid interfaces covered with surfactant adsorption layers. The dynamic system properties depend on the mechanisms and rate of equilibrium restoration after a deformation. At small magnitudes of deformation the mechanical relaxation of an interface can be described by the complex dilational viscoelastic modulus [1,2]. For sinusoidal deformations it is deflned as the ratio of complex amplitudes of interfacial tension variation and the relative surface area variation f (I ty) = dy /din A being a function of frequency. This modulus may include... 

The influence of the vinyl content on the viscoelastic behaviour of polybutadienes is shown in Figure 4. Measurements of tan S, the phase angle between stress and strain under sinusoidal deformation, have been performed by Dynamic Mechanical Spectrometry (Rheometrics). Looking at the shift along the temperature axis due to the different vinyl content, a maximum vinyl content of 72% has been chosen, since beyond this limit the polymer can hardly be regarded as a rubber. 

The complex modulus evolves from a complex variables treatment of the sinusoidal deformation, which is well documented in various references on DMA [e.g., McCrum et al. (1967) Murayama (1978) Ferry (1980) Nielsen and Landel (1994)]. 

In nonresonance forced vibration methods, either a sinusoidal deformation or load is applied to the specimen, and the response of either the force or deformation is detected, respectively. The storage modulus ( ), loss modulus ( "), and the phase angle (S) between the deformation and force are calculated. The tensile, flexural, and shear modes are generally used, and most of the commercially available apparatuses use nonresonance forced vibration... 

Figure 9 shows an example of a nonresonance forced vibration apparatus. A sinusoidal deformation is applied to the specimen, and the amplitude of the deformation is measured using a strain transducer, and the magnitude of the response force is measured using a stress transducer. If the deformation and force are recorded on a recorder, the output data are as shown in Fig. lOA. 

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]. 

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