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Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. (English) Zbl 0805.49028
This paper deals with phenomena in media which are governed by linear elliptic systems of equations with complex coefficients, such as conducting, transparent or viscoelastic media which may be homogeneous or space-periodic mixture of composite materials. Anyhow, they are characterized by complex valued tensors. The authors formulate four equivalent systems of equations describing the electromagnetic field or viscoelastic field and construct functionals of which the Euler equations coincide with the classical ones. Thus they obtain two pairs of mutually conjugate variational principles which are equivalent to each other, two minimax and two minimal ones, all similar to Dirichlet’s and Thompson’s variational principles for electrostatic problems. The functionals constructed are proportional to the energy dissipation averaged over the period of oscillation. The treatment of viscoelasticity problems leads to functionals and variational principles of just the same form. When either of them is used in the homogenization theory, the same bounds on the effective properties of composite materials are obtained.

MSC:
49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
78A25 Electromagnetic theory, general
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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