Crack on the boundary of a thin elastic inclusion inside an elastic body

Mathematics and Mechanics of Solids

Leugering

2013

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.

Section: Khludnev and Leugeringmentioning

confidence: 90%

On Timoshenko thin elastic inclusions inside elastic bodies

Mathematics and Mechanics of Solids

Leugering

2013

“…We omit this proof since it can be done similar to that in [12]. The solution of the problem (12)-(13) is unique what can be checked by contradiction arguments.…”

Section: Break Between Rigid and Semirigid Inclusionsmentioning

confidence: 93%

“…Note that it suffices to check (3), (6), (7). The second, third and fourth boundary conditions of (5) can be verified as those of [12]; the details are omitted. We substitute the test functions…”

Section: Theorem 2 Problem Formulationsmentioning

confidence: 99%

“…A coercivity proof of the functional E on the set K 1 can be provided as that in [12]. Since the set K 1 is weakly closed in the space H 1 Γ (Ω γ ) 2 × H 2 (γ s ), we obtain a solution existence of the problem (17)- (18).…”

Section: Fig 2 Zero Angle Between Inclusionsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Junction problem for rigid and semirigid inclusions in elastic bodies

Попова

2016

Arch Appl Mech

The paper is concerned with equilibrium problems for 2D elastic bodies having thin inclusions with different properties. In particular, rigid and semirigid inclusions are considered. It is assumed that inclusions have a joint point, and we analyze a junction problem for these inclusions. Existence of solutions is proved for each equilibrium problem, and different equivalent formulations of these problems are discussed. In particular, junction conditions at the joint point are found. A delamination of the inclusions is also assumed. This means that we have a crack, and inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. We discuss a convergence to zero and infinity of a rigidity parameter of the semirigid inclusion and analyze limit problems.

On delaminated thin Timoshenko inclusions inside elastic bodies

Itou

Math Methods in App Sciences

2014

In the paper, we consider equilibrium problems for 2D elastic bodies with thin inclusions modeled in the frame of Timoshenko beam theory. It is assumed that a delamination of the inclusion takes place thus providing a presence of cracks between the inclusion and the elastic body. Nonlinear boundary conditions at the crack faces are imposed to prevent a mutual penetration between the faces. Different problem formulations are analyzed: variational and differential. Dependence on physical parameters characterizing the mechanical properties of the inclusion is investigated. The paper provides a rigorous asymptotic analysis of the model with respect to 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. Also anisotropic inclusions with parameters are analyzed when parameters tend to zero and infinity. In particular, in the limit, we obtain the so called semi‐rigid inclusions. Copyright © 2014 John Wiley & Sons, Ltd.