Modulus, dynamic

Viscoelasticity is studied using dynamic mechanical analysis. When we apply a small oscillatory strain and measure the resulting stress. Purely elastic materials have stress and strain in phase, so that the response of one caused by the other is immediate. In purely viscous material the phase delay between stress and strain reach 90 degree phase lag. 

Viscoelastic materials exhibit behavior somewhere in the middle of these two types of material, exhibiting some lag in strain [1,5]. 

Following the classical treatments of the dynamic modulus G, it can be used to represent the relations between the oscillating stress and strain  

Where a(, and so are the amplitudes of stress and strain and 8 is the phase shift between them [1,5]. 

Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system - an open circuit. As a result, only the spring connected in parallel to the dashpot will contribute to the total strain in the system [23-26], 

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A dynamic modulus can also be evaluated by following a procedure similar to that used for the dynamic compUance above. The modulus is most easily approached by considering a Maxwell element in which case the differential equation analog to Eq. (3.77) is... 

The dynamic mechanical properties of VDC—VC copolymers have been studied in detail. The incorporation of VC units in the polymer results in a drop in dynamic modulus because of the reduction in crystallinity. However, the glass-transition temperature is raised therefore, the softening effect observed at room temperature is accompanied by increased brittleness at lower temperatures. These copolymers are normally plasticized in order to avoid this. Small amounts of plasticizer (2—10 wt %) depress T significantly without loss of strength at room temperature. At higher levels of VC, the T of the copolymer is above room temperature and the modulus rises again. A minimum in modulus or maximum in softness is usually observed in copolymers in which T is above room temperature. A thermomechanical analysis of VDC—AN (acrylonitrile) and VDC—MMA (methyl methacrylate) copolymer systems shows a minimum in softening point at 79.4 and 68.1 mol % VDC, respectively (86). 

Similar information can be obtained from analysis by dynamic mechanical thermal analysis (dmta). Dmta measures the deformation of a material in response to vibrational forces. The dynamic modulus, the loss modulus, and a mechanical damping are deterrnined from such measurements. Detailed information on the theory of dmta is given (128). 

Butyl-type polymers exhibit high damping, and the viscous part of the dynamic modulus is uniquely broad as a function of frequency or temperature. Molded mbber parts for damping and shock absorption find wide appHcation in automotive suspension bumpers, auto exhaust hangers, and body mounts. 

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]. 

The strength of the filler skeleton may be characterized by the complex dynamic modulus measured at low frequencies [24]. The authors note that when c > ccr the yield point should be viewed as a sum of two components ... 

Fig. 5. Typical forms of frequency dependences of dynamic modulus. The content of the filler increases upon the transition from durve 1 to 2 and to 3. The discussion of regions I-VI, displayed on the curves, see text...

Fig. 10. Concentration dependence of a modulus in the region of low-frequency plateau (i.e. yield stress , measured by a dynamic modulus). Dispersion medium poly (butadiene) with M = 1.35 x 105 (7), silicone oil (2) polybutadiene with M = 1 x I04 (3). The points are taken from Ref. [6], The straight line through these points is drawn by the author of the present paper. In the original work the points are connected by a curve in another manner...

A typical behavior of amplitude dependence of the components of dynamic modulus is shown in Fig. 14. Obviously, even for very small amplitudes A it is difficult to speak firmly about a limiting (for A -> 0) value of G, the more so that the behavior of the G (A) dependence and, respectively, extrapolation method to A = 0 are unknown. Moreover, in a nonlinear region (i.e. when a dynamic modulus depends on deformation amplitude) the concept itself on a dynamic modulus becomes in general not very clear and definite. 

The existence of the G (A) dependence even in the region of very small amplitudes is explained by a brittle pattern of fracture of a filler s structure, so that measuring virtually frequency (and amplitude) dependences of a dynamic modulus, a researcher always deals with a material in which the structure is partially fractured. 

The discussion of the results of measuring the dynamic properties of filled polymers is based very often on the idea of correlation of the G" and t functions, which is not always expressed directly. However, due to a very sharp dependence of a dynamic modulus on the amplitude, it is not clear how to understand this correlation. 

Moreover, if for pure polymer melts the correlation of the behavior of the functions ri (co) andrify) under the condition of comparing as y takes place, as a general rule, but for filled polymers such correlation vanishes. Therefore the results of measuring frequency dependences of a dynamic modulus or dynamic viscosity should not be compared with the behavior of the material during steady flow. 

As regards a qualitative pattern of influence of the filler on dynamic properties of melts of filled polymers, the situation in many respects is the similar described above for yield stress and viscosity. Indeed, the interpretation of the field of yield stress, estimated by a dynamic modulus, was given in an appropriate section. 

Experimentally DMTA is carried out on a small specimen of polymer held in a temperature-controlled chamber. The specimen is subjected to a sinusoidal mechanical loading (stress), which induces a corresponding extension (strain) in the material. The technique of DMTA essentially uses these measurements to evaluate a property known as the complex dynamic modulus, , which is resolved into two component parts, the storage modulus, E and the loss modulus, E . Mathematically these moduli are out of phase by an angle 5, the ratio of these moduli being defined as tan 5, Le. 

FIGURE 1.7 Construction of a master curve of dynamic modulus /a versus log (frequency) by lateral shifting of experimental results made over a small frequency range but at several different temperatures. 

The plastic strain at fracture decreases markedly with time as the cement ages also the elastic modulus increases (Wilson, Paddon Crisp, 1979 Barton et al., 1975). There is an increase in dynamic modulus with time (Barton et al., 1975). 

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). 

An apparatus for measuring the dynamic modulus and hysteresis of elastomers. The stress-strain oscillogram is shown on a ground-glass screen by means of an optical system. Now superseded by modem computer controlled servo hydraulic and dynamic mechanical thermal analysis machines. Roll Bending... 

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