General time-dependent modulus

Rather than considering the displacement x t) in dependence on the forces in the past, tp t < t), one can ask conversely for the functional dependence of the force tp[t) on the previous displacements x t < t). The solution is obvious We just have to exchange the susceptibility a t — t ) against the generalized time-dependent modulus a(t — t ) and represent tp t) as a sum of relaxation curves ... 

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7 ... 

Several effects may lead to this result of heating rate-dependent decrease of modulus (i) a nonuniform through-thickness temperature gradient, or (ii) a thermal lag between the specimen and the DMA temperature measurement, or (iii) a general time dependence, in addition to the temperature dependence of the polymer properties. 

If the time dependent modulus of the material, E(t), is expressed in a Prony series (generalized Maxwell model) representation (Eq. 6.31 or Eq. 5.22), then the simple algebraic form of the function leads to explicit expressions for the storage and loss moduli from solution to Eq. 6.49. 

Show that the complex moduli, E and E", can be represented as indicated in Eq. 6.51 in the case where the time dependent modulus, E(t) is given by a generalized Maxwell model. 

The power law for the time dependent modulus can be transformed into a law valid for the frequency domain, either by applying the general relations of linear response or by using the ( -dependent form of the fluctuation-dissipation theorem. The result is... 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

Dynamic oscillatory shear measurements of polymeric materials are generally performed by applying a time dependent strain of y(t) = y0sin(cot) and the resultant shear stress is a(t) = y0[G sin(a)t) + G"cos(cot)], with G and G" being the storage and loss modulus, respectively. 

Studies of rheokinetics over the whole range of polyester curing is based (as for other materials) on a dynamic method, i.e., on measurements of the time dependence of the dynamic modulus at a fixed frequency, from which the time dependence of the degree of conversion (3(t). The observed dependence P(t) for polyester resins can be analyzed by an equation of the type used for other materials. Thus the following general equation was proposed for the kinetics of curing polyester and epoxy resins 69 72... 

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. 

Notice that the above equation is simply a time-dependent generalization of Hooke s law [Eq. (7.98)]. For viscoelastic solids, G(t) relaxes to a finite "value, called the equilibrium shear modulus G q (see Fig. 7.22, top curve) —... 

Time dependent shear modulus in the linear response regime denoted as g (f) of the quiescent system generalized one if including dependence on shear rate... 

These assumptions are not always justifiable when applied to plastics unless modification has occurred. The classical equations cannot be used indiscriminately. Each case must be considered on its own merits, with account being taken of such factors as the mode of deformation, the service temperature and environment, the fabrication method, and so on. In particular, it should be noted that the past traditional equations that have been developed for other materials, principally steel, use the relationship that stress equals the modulus times strain, where the modulus is constant. Except for thermoset reinforced plastics and certain engineering plastics, many plastics do not generally have a constant modulus. Different approaches have been used for the nonconstant situation some are quite accurate. The drawback is that most of these methods are quite complex, involving numerical techniques that are not attractive to designers. One method that has been widely accepted is this so-called pseudoelastic design method. In this method appropriate values of such time-dependent properties as the modulus are selected and substituted into the standard equations. 

A suitable property to describe the mechanical behaviour of a pol3mier under time-dependent enforcement is the complex Young s modulus. For a polymer, which, in general, is composed of crystalline and amorphous parts (for a comprehensive description of polymer... 

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