By Allen C. Pipkin
This e-book comprises notes for a one-semester path on viscoelasticity given within the department of utilized arithmetic at Brown collage. The direction serves as an advent to viscoelasticity and as a exercise session within the use of assorted ordinary mathematical equipment. The reader will quickly locate that he must do a little paintings at the facet to fill in info which are passed over from the textual content. those are notes, now not a totally specific rationalization. moreover, a lot of the content material of the direction is within the difficulties assigned for answer by means of the coed. The reader who doesn't at the least try and clear up a superb a few of the difficulties is probably going to overlook many of the aspect. a lot that's recognized approximately viscoelasticity isn't mentioned in those notes, and references to unique assets should not supply, so it will likely be tricky or most unlikely to exploit this booklet as a reference for having a look issues up. Readers short of whatever extra like a treatise should still see Ferry's Viscoelastic houses of Polymers, Lodge's Elastic drinks, the volumes edited through Eirich on Rheology, or any factor of the Transactions of the Society of Rheology. those works emphasize actual elements of the topic. at the mathematical facet, Gurtin and Sternberg's lengthy paper at the Linear concept of Viscoelasticity (ARMA II, 291 (I962» continues to be the simplest reference for proofs of theorems.
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Extra info for Lectures on Viscoelasticity Theory
Is obtained by setting Since the Laplace transform of as well. Since 1D in the expression for also, as expected, and then J* l/G* = l/G e = Je. Ge G2 O. and the loss modulus is The dynamic viscosity G2 /m so G(t) sJ(s) = dynamic viscosity G2 /m Since Ggexp(-t/T). Notice that the viscosity infinity. J(t) Gl l/(s+c) is the transform of ~o is is unity, J* = and radians beThe is zero. (T+t)/~ o ,where sG(s) = ~oS/(ST+l), or, with G s/(S+T- 1 ). ~/2 t The loss tangent is infinite. The storage modulus ~o.
In that limit, the frequency is m(p~/4). In one cycle of oscillation, the logarithm of the amplitude decreases by 2 p~ /2 ~ = 5p, independent of e. m = (FC)2 and the damping constant is m. Real solids with small losses. For solid polymers we can model the behavior of the modulus in the vicinity of a time t by a power law, G(t) = G(t ) (t/t ) -pet ) 0 0 0 0 present problem, the most appropriate time to We can guess that in the to use in this approximation must be something of the order of the period of oscillation, but we don't really know exactly, so it is not entirely obvious what values of C and p we should use in the preceding results in order to get the right answer for a given real material.
Show that the former 2 + FCs p sP, and thus the integrand, has a branch cut along the negative real In the cut plane, the integrand has poles at FC s = -r sin a ~ ir cos a, where and The inversion contour can be x deformed into a loop integral along the two Rides of the branch cut, plus circles around the two poles. The poles con- tribute a damped oscillation at the frequency W = r cos a, decaying as exp[-(r sin a)tJ. The integrals along the two sides of the branch cut can be combined into a single real integral, FC sin(prr) rr 00 fo x 4 xl+Pexp(_xt)dx 2+p 2 2p + 2FCx cos(prr) + (FC) x In spite of the complicated appearance of this integral, it is evident that as a function of t, it is a linear combination of pure decays 51 exp(-xt) with various values of x.