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Differential approximation for viscoelasticity. (English) Zbl 0820.73031
The paper pertains to the field of evolution equations containing hereditary effect due to the time non-locality which produces dissipation, and the goal is to devise approximate equations which accurately reproduce dissipation. R. C. MacCamy [J. Math. Anal. Appl. 179, No. 1, 120-134 (1993; Zbl 0790.34059)] studied the equations of parabolic nature. Here the authors are dealing with the hyperbolic case by taking as a model the displacement problem $$P(\mu,\lambda)[b,u_ 0,u_ 1]$$ of the linear, isotropic, and homogeneous viscoelasticity. The constitutive equation of the viscoelastic body is $$\sigma(x,t) = {\partial\over \partial t} \int^ t_{-\infty} L(\mu(t-r), \lambda(t - r)) [u(x,r)]dr$$, where $$\sigma$$ is the stress tensor, $$u$$ denotes the displacement, $$\mu$$ and $$\lambda$$ are functions on $$[0,\infty)$$ characterizing the viscoelastic body, and $$L(\mu,\lambda)[u] = 2\mu E[u] + 2\lambda \text{tr }E [u]I$$, $$E[u] = [\nabla u + \nabla u^ T]/2$$. It is assumed that $$u$$ is zero up to time $$t = 0$$, and the density is taken equal to one. Under these conditions the problem $$P(\mu,\lambda) [b,u_ 0,u_ 1]$$ takes the form $$\ddot u(x,t) = {\partial\over \partial t} \int^ t_ 0 \text{div }L(\mu(t - r),\lambda(t - r))[u (r,t)] dr + b(x,t)$$ in $$\Omega$$, $$u(x,t) = 0$$ on $$\partial \Omega$$, $$u(\cdot,0) = u_ 0$$, $$\dot u(\cdot,0) = u_ 1$$, where the dot indicates time derivative, $$b$$ is the body force, $$\Omega$$ is a bounded three- dimensional region representing a fixed configuration of the body, and $$u_ 0$$, and $$u_ 1$$ are given functions on $$\Omega$$. The corresponding approximate problem is studied by using the procedure of the above mentioned work of MacCamy. The paper is of real theoretical and practical interest.

##### MSC:
 74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) 45K05 Integro-partial differential equations
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