Equilibrium problem for a thermoelastic Kirchhoff–Love plate with a delaminated flat rigid inclusion

Abstract

A new mathematical model describing an equilibrium of a thermoelastic heterogeneous Kirchhoff–Love plate is considered. A corresponding nonlinear variational problem is formulated with respect to a two-dimensional domain with a cut. This cut corresponds to an interfacial crack located on a given part of the boundary of a flat rigid inclusion. The flat inclusion is described by a cylindrical surface. Due to the presence of the flat rigid inclusion in the plate, restrictions of the functions describing displacements to the corresponding curves satisfy special constraints having a linear form. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual non-penetration of opposite crack faces. Solvability of the problem is proved. Under the assumption that the solution of the variational problem is smooth enough, an equivalent differential formulation is found. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Non-smooth variational problems and applications

“…From a mathematical point of view, the developed non-smooth variational methods are based on the Lagrange multiplier approach and dual optimization techniques. The variational problems under consideration are subjected to gradient constraint [ 1 ] and unilateral constraints [ 2 , 3 ], they obey non-differentiable objectives and may lose the property of coercivity [ 4 ]. These features result in non-smooth optimization, quasi-variational inequalities [ 5 ] and hemi-variational inequalities [ 6 ].…”

“…From the point of view of physics, the variational approach is applied to continuum mechanics of solids as well as incompressible fluids given by the Navier–Stokes [ 9 ] and Stokes [ 6 ] models. The different types of models for solids describe elastic junctions [ 2 ], thermo-elastic composites under mechanical vibration [ 12 ], dynamic behaviour of the Euler–Bernoulli beams [ 8 ] and thermo-elastic Kirchhoff–Love plates [ 3 ]. There are considered bodies that exhibit power-law hardening like the Norton–Hoff and Ramberg–Osgood materials [ 5 ], and ideal elasto-plastic behaviour [ 1 ].…”

“…By this, nonlinear boundary conditions are of the first importance for physically consistent modelling. The theme issue studies the unilateral contact appearing for inclusions subject to delamination with cracks [ 2 , 3 ], non-smooth slip [ 9 ], frictional contact [ 4 , 13 ] and degenerating Robin-type transmission conditions for the thin reactive heat-conducting inter-phases [ 11 ]. The rate-independent evolutionary systems driven by non-convex energies have been suggested [ 10 ], which are successful to model properly jump discontinuities in time during quasi-brittle crack propagation.…”