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Spectral perturbations in linear viscoelasticity of the Boltzmann type. (English) Zbl 0598.73033
The authors study vibration frequencies of linear viscoelastic materials of Boltzmann type, i.e. parameters \(\zeta\) for which second order systems of the form \[ u_{tt}(\cdot,t)=Au(\cdot,t)-\epsilon \cdot \int^{t}_{-\infty}B(t-s)u(\cdot,s)ds \] possess solutions of the form \(u(\cdot,t)=u_ 0\cdot e^{-\zeta t}\). Here u(\(\cdot,t)\) maps a bounded subset of \({\mathbb{R}}^ 3\) into \({\mathbb{R}}^ 3\), and A and B(s) are suitable elliptic operators. The main results of the paper are the existence of analytic branches of the \(\zeta\) (\(\epsilon)\) for \(\epsilon\) near zero, counting multiplicities properly, and formulae for the linear terms in the corresponding expansions. The proofs use a reformulation of the problem as an abstract evolution equation, reduction to a finite dimensional problem, and the Weierstrass preparation theorem.
Reviewer: H.Engler
MSC:
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
47A10 Spectrum, resolvent
74H45 Vibrations in dynamical problems in solid mechanics
45K05 Integro-partial differential equations
47Gxx Integral, integro-differential, and pseudodifferential operators
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References:
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