Asymptotic behavior elasticity

For the splitting of the quasi-static stress - obtained after the substraction of (Tr from the measured stress a - into its two components, the asymptotic behavior at large strains can be used. It is dominated by the network forces and thus determined by the associated network shear modulus. The properties of the crystallite branch, i.e., the associated elasticity and finite plasticity, follow from step-cycle tests. 

For those composites initially having some of the fibers intact, there will always be some which must be stretched elastically. This will require a stress which will tend towards the value given by Eqn. (60) with / replaced by /, the volume fraction of fibers initially intact. If a relaxation test were carried out, the stress would become asymptotic to the level predicted by Eqn. (60). The remaining broken fibers will interact with the matrix in a complex way, but at a given strain and strain rate, a characteristic stress contribution can be identified in principle. Details have not been worked out. However, the total stress would be the sum of the contribution from the broken and unbroken fibers. If the transient behavior is ignored (i.e., assumed to die away relatively fast compared to the strain rate), a basic model can be constructed. 

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

In case the viscoelastic liquid is only slightly elastic, the general constitutive behavior depicted by Eq. 1 can be substantially simplified, by obtaining an asymptotic series of approximation in W, which can be taken as small under these conditions. As a consequence, one may arrive at the following second-order constitutive model, for example, [2] ... 

Sreenivasan and White (89) recently invoked the theory based on the elastic behavior of stretched polymers of de Gennes (84), who states that coil-stretch transition does not occur in turbulent flows in randomly fluctuating strain rates and that, if moderately stretched, the polymers produce no measurable change in viscosity. Sreenivasan and White (89) were able to explain the onset of drag reduction and maximum drag reduction asymptote on the basis of the elastic theory, and their results qualitatively explain the existing experiments. 

The essential difference in character between the elastic low-temperature behavior and the viscous high-temperature response is shown by plotting the extension as a function of time (see Fig. 5.9). On a log-log scale an elastic response should have an asymptotic slope of 1 at low strains for a system with a well defined Young s modulus, E,... 

One would be in an ad-hoc fashion to assume that, because of the tendency toward interpenetration, near the crack tips the crack surfaces would come in smooth contact and form a cusp, and the resulting contact region would consist of a single uninterrupted zone rather than the sum of a series of discrete zones as implied by the oscillatory nature of the elastic solution (see Comninou [ll], Atkinson [l2]). Another way is to assume that near the crack tip the linear theory is not valid and to use a large deformation nonlinear theory. An asymptotic analysis using such a theory was provided by Knowles and Sternberg [l3] for the plane stress interface crack problem in two bonded dissimilar incompressible Neo-Hookean materials which shows no oscillatory behavior for stresses or... 

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