On weak solutions of boundary value problems within the surface elasticity of Nth order

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 Nth 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 to H1(V)boolean AND HN(Ss) where Ss subset of S equivalent to partial differential V and V subset of R3.

Automated summary

This study investigates the existence and uniqueness of weak solutions to boundary value problems describing an elastic body with weakly nonlocal surface elasticity. The chosen model includes surface strain energy as a quadratic function and the virtual work principle is extended for higher-order strain gradient media. Energy functional spaces of Sobolev type are introduced to characterize the smoothness of solutions. Compared with classical linear elasticity solutions, weak solutions for solids with surface stresses are shown to be smoother on the boundary.