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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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