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Viscoelastic pulse propagation and stable probability distributions. (English) Zbl 0625.73037
The purpose of the present note is to point out an interesting connection between viscoelastic pulse propagation and the theory of infinitely divisible and stable probability distributions. Velocity pulses in viscoelastic materials have the same shapes as infinitely divisible probability densities, and when the viscoelastic material response is of the kind that is typical for stiff polymeric materials, the velocity pulse takes the shape of a stable probability density.
In Section 2 we explain the equations and boundary conditions governing one-dimensional wave propagation in a semi-infinite body of linearly viscoelastic material, and obtain the Laplace transform of the velocity V(y,t) for the case in which the input velocity is a Dirac delta, $$V(0,t)=\delta (t)$$. The viscoelastic material is assumed to have a creep compliance whose slope is completely monotone. We consider in particular the case of a power-law compliance, a model that is a good approximation to the behaviour of some polymeric materials.
In Section 3 we explain the definitions of infinitely divisible and stable probability densities for positive random variables, and show the connection with viscoelastic pulse shapes. At any fixed station y, the velocity V as a function of time is an infinitely divisible probability density. When the compliance has the power-law form, the pulse shape (in time, at a given station) is one of the stable densities.
The remainder of the paper concerns the pulse shape in a power-law material. In the limiting cases of perfectly elastic and perfectly viscous behavior the Laplace transform can be inverted explicitly, and in the general case asymptotic approximations valid for small and large times can be obtained. We show this in Section 4.
In Section 5 we explain an approach to the calculation of the pulse shape, based on an approximation to the inversion integral, and in Section 6 we show that the approximate inversion integral can be reduced to a real integral over a finite interval by integrating along a steepest descent path. This integral can be evaluated numerically. In Section 7 we discuss the numerical results.

##### MSC:
 74J99 Waves in solid mechanics 74H50 Random vibrations in dynamical problems in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010) 74J10 Bulk waves in solid mechanics
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