Viscoelasticity dynamic moduli

However, it yields dynamic modulus. Some other techniques were also used to characterize hydrogels, for example, viscoelastic measurements [28, 30, 31] and swelling equilibrium [20]. 

Frictional forces are not proportional to load-friction increases with increasing speed, and the static coefficient of friction is lower than its dynamic one. When two viscoelastic low-modulus materials are run against each other, additional inconsistencies result. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

The dynamic storage modulus of closed-cell PE foams was investigated by dynamic viscoelastic measurement in the compression mode. It was found that dynamic modulus correlated with compression hardness and that the resistance against pressure inside the cells had no effect upon static modulus or dynamic storage modulus. 8 refs. 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

More recently, Lin and Masuda [47] measured the viscoelastic properties of polypropylene melts filled with small (0.15 pm) and larger (4.0 pm) calcium carbonate particles. The dynamic modulus and viscosity were found to rise with filler loading especially at low frequencies. With highly filled compositions (at... 

The results of measurements of the dependencies G (w,t) for three circular frequencies w0 = 27tf0, wi= 4rcwo, and w2 = 16jtwo are shown in Fig. 3.1. The lack of coincidence in the shapes of die time dependencies of the dynamic modulus components for different frequencies is obvious. This phenomenon is especially true for G", because the position of the maximum differs substantially along the time axis. In the most general sense, this reflects the contributions of the main relaxation mechanisms of the material to its measured viscoelastic properties. 

The relaxation spectrum H(0) completely characterizes the viscoelastic properties of a material. H(0) can be found from the measured frequency dependence of the dynamic modulus of elasticity G (co) by means of the following integral equation ... 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Dynamic modulus is a convenient characteristic of viscoelasticity. To analyse the results, it is convenient also to consider the asymptotic behaviour of the dynamic modulus at high and low frequencies. In the latter case... 

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

When calculating the Laplace transform, one finds an enhancement of the dynamic modulus due to the macromolecular coils in the viscoelastic liquid... 

The real and the imaginary components of dynamical modulus of a dilute suspension of macromolecules in a viscoelastic liquid are calculated at values of B shown at the curves and at % = 1. Adapted from the paper of Pokrovskii and Volkov (1978a). 

To extend the theory for higher frequencies, we have to consider the general case, when the micro-viscoelasticity is given by (6.47). Using equations (4.28) and (4.29), after some rearrangement, one can find the dynamic modulus... 

One can notice that the dissipative terms in the dynamic equation (3.11) (taken for the case of zero velocity gradients, z/jj = 0) have the form of the resistance force (D.3) for a particle moving in a viscoelastic liquid, while the memory functions are (with approximation to the numerical factor) fading memory functions of the viscoelastic liquid. The macromolecule can be considered as moving in a viscoelastic continuum. In the case of choice of memory functions (3.15), the medium has a single relaxation time and is characterised by the dynamic modulus... 

On the other hand, the properties of the system as a whole can be calculated and the macroscopic dynamic modulus can be determined. Here the question of the relation between the postulated micro-viscoelasticity and the resulting macro-viscoelasticity appears. The answer requires a properly formulated self-consistency condition. Simple speculations show that equality of the micro- and macro-viscoelasticity cannot be obtained. Nevertheless, it is natural to require the equality of relaxation times of micro- and macroviscoelasticities. It will be shown in this section that this condition can be satisfied. 

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. 

Pokrovskii VN, Volkov VS (1978b) The calculation of relaxation time and dynamical modulus of linear polymers in one-molecular approximation with self-consistency. (A new approach to the theory of viscoelasticity of linear polymers). Polym Sci USSR 20 3029-3037... 

Dynamic Modulus. Measurement of dynamic viscoelasticity was made by the use of a direct-reading dynamic viscoelastometer from the Toyo Measuring Instrument Co., at a frequency of 110 Hz. 

The design of effective sound and vibration damping materials assumes an understanding of the mechanisms controlling the dissipation process and knowledge of candidate material properties. The use of viscoelastic materials as sound and vibration absorbers is wide-spread and well-known. Accurate measurement of the complex dynamic moduli of these materials is therefore vital to the control of acoustic and vibrational energy. This chapter discusses and compares three apparatus used to measure the dynamic modulus of viscoelastic materials. 

To obtain a closed solution it is necessary to assume a viscoelastic model. For example, if a standard solid model is adopted, the following value for the dynamic modulus is found ... 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

When the strain amplitude Is relatively large as In the case of tire cord In a running tire, the viscoelastic behavior Is no longer linear. The stress-strain loop Is not elliptic but distorted (Figure 1). The material properties In the nonlinear regime can not be represented with the real and Imaginary moduli. In the present study, we characterize the viscoelastic properties In nonlinear regime by the effective dynamic modulus and mechanical loss.(J )... 

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