On Timoshenko thin elastic inclusions inside elastic bodies

Abstract

The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Finally, exemplary networks of Timoshenko beams are considered as inclusions as well.

Junction problem for rigid and semirigid inclusions in elastic bodies

“…The results are concerned with a solution existence and qualitative properties of solutions. We can mention many other papers related to equilibrium problems with thin elastic and rigid inclusions and cracks; see [6,7,[10][11][12][13][14]30]. Optimal control problems for such models can be found in [15,[33][34][35].…”

Optimal size of a rigid thin stiffener reinforcing an elastic plate on the outer edge

“…Such linear models allow the opposite crack faces to penetrate to each other, such state leads to inconsistency with practical situations . Since the beginning of 1990, the crack theory with non‐penetration conditions of inequality type is under active study . Using the variational methods, various problems for bodies with rigid inclusions have been successfully formulated and investigated; see for example .…”

On the mechanical interplay between Timoshenko and semirigid inclusions embedded in elastic bodies

“…In particular, we can describe inclusions in the frame of Euler‐Bernoulli and Timoshenko beam models as well as many limiting models obtained by passing to limits as parameters of inclusions are changing. An analysis of limit models was fulfilled in . At this point, we can mention other approaches to describe inclusions in elastic bodies .…”