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Energy minimization and the formation of microstructure in dynamic anti- plane shear. (English) Zbl 0786.73066
The behaviour of a continuum model within the framework of nonlinear viscoelasticity, which provides an insight into the dynamical development of microstructures observed during displacement phase transformations in certain materials, is investigated. The problem of dynamical two- dimensional anti-plane shear with linear viscoelastic damping is considered. Using a transformation due to P. Rybka [Proc. R. Soc. Edinb., Sect. A 121, No. 1/2, 101-138 (1992; Zbl 0758.73004)], the equivalence of this problem to a semilinear degenerate parabolic system can be shown, and the latter system can be treated by the method of semigroup theory to establish existence, uniqueness and regularity of solutions in $$L^ p$$ spaces.
The issues of energy minimization and propagation of strain discontinuities are also discussed, but unfortunately, the authors are unable to obtain analogues of the results on non-minimization of the energy and non-propagation of strain discontinuities in one-dimensional models.
Finally, a finite difference approach to the numerical solution of the dynamical anti-plane shear problem is developed, and some numerical simulations of the dynamical evolution of patterns and microstructure in the above problem are presented.

MSC:
 74A60 Micromechanical theories 74M25 Micromechanics of solids 74A15 Thermodynamics in solid mechanics 74E10 Anisotropy in solid mechanics 74S20 Finite difference methods applied to problems in solid mechanics
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