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On weak solutions of boundary value problems within the surface elasticity of N th order
Author(s) -
Eremeyev Victor A.,
Lebedev Leonid P.,
Cloud Michael J.
Publication year - 2021
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.202000378
Subject(s) - uniqueness , sobolev space , elasticity (physics) , mathematical analysis , mathematics , weak solution , boundary value problem , surface (topology) , elastic energy , linear elasticity , weak formulation , surface energy , strain energy , physics , geometry , thermodynamics , finite element method
A study of existence and uniqueness of weak solutions to boundary value problems describing an elastic body with weakly nonlocal surface elasticity is presented. The chosen model incorporates the surface strain energy as a quadratic function of the surface strain tensor and the surface deformation gradients up to N th order. The virtual work principle, extended for higher‐order strain gradient media, serves as a basis for defining the weak solution. In order to characterize the smoothness of such solutions, certain energy functional spaces of Sobolev type are introduced. Compared with the solutions obtained in classical linear elasticity, weak solutions for solids with surface stresses are smoother on the boundary; more precisely, a weak solution belongs toH 1 ( V ) ∩ H N ( S s )whereS s ⊂ S ≡ ∂ V and V ⊂ R 3 .