Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion

Lazarev, N. P.

doi:10.1007/s00033-014-0488-4

Cited by 44 publications

(29 citation statements)

References 11 publications

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“…We can choose (ū,v) ∈ K 1 and multiply (19), (21) bȳ u − u,v − v, respectively. Integrating over Ω γ and γ s , respectively, we obtain…”

Section: Theorem 4 Problem Formulationsmentioning

confidence: 99%

“…Assuming that in (29),ṽ =ṽ ,1 = 0 as x 1 = 0, x 1 = −1 (this provides ρ = (c 1 , 0), c 1 = const, due to conditions ρ 1 (0) = c 1 , ρ 2 (0) =ṽ(0), ρ 2,1 (0) = v ,1 (0)) we obtain (21) and the first relation of (30); see below. Going back to (29) the boundary conditions (24) follow, and moreover, the identity (25) is derived.…”

Section: Theorem 4 Problem Formulationsmentioning

confidence: 99%

“…The conditions mentioned do not allow the opposite crack faces to penetrate each other. The book [8] contains results for crack models with the non-penetration conditions for a wide class of constitutive laws; see also [16,17,[19][20][21]29]. In the frame of crack models with inequality-type boundary conditions at the crack faces, a lot of results for elasticity models are obtained by now [9].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Junction problem for rigid and semirigid inclusions in elastic bodies

Khludnev

Попова

2016

Arch Appl Mech

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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.

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“…We can choose (ū,v) ∈ K 1 and multiply (19), (21) bȳ u − u,v − v, respectively. Integrating over Ω γ and γ s , respectively, we obtain…”

Section: Theorem 4 Problem Formulationsmentioning

confidence: 99%

Section: Theorem 4 Problem Formulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Junction problem for rigid and semirigid inclusions in elastic bodies

Khludnev

Попова

2016

Arch Appl Mech

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“…Using the shape‐topological sensitivity analysis, the existence of a solution of an optimal control problem concerning the best choice of the location and shape of elastic inclusions was proved in []. We can mention results concerning explicit formulae of derivatives of the energy functionals with respect to the rigid inclusion shapes …”

Section: Introductionmentioning

confidence: 99%

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous two‐dimensional bodies with a crack

Lazarev

Everstov

2018

Z Angew Math Mech

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A two‐dimensional model describing equilibrium of a cracked inhomogeneous body with a rigid inclusion is studied. We assume that the Signorini condition, ensuring non‐penetration of the crack faces, is satisfied. For a family of corresponding variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional on a suitable Sobolev space, with the location parameter of the inclusion is chosen as a control parameter.

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“…Problems for different models of elastic bodies containing rigid inclusions and cracks with both linear and nonlinear boundary conditions has been under active study; see [4,7,8,11,14,31,35]. It is known that various problems for bodies with rigid inclusions may be successfully formulated and investigated using variational methods, see for example [11,23,27,32,34]. In particular, a framework for two-dimensional elasticity problems with nonlinear Signorini-type conditions on a part of boundary of a thin delaminated rigid inclusion is proposed in [11].…”

Section: Introductionmentioning

confidence: 99%

Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

Lazarev

2015

Z. Angew. Math. Mech.

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The equilibrium problems for two-dimensional elastic body with a rigid delaminated inclusion are considered. In this case, there is a crack between the rigid inclusion and the elastic body. Non-penetration conditions on the crack faces are given in the form of inequalities. We analyze the dependence of solutions and derivatives of the energy functionals on the thickness of rigid inclusion. The existence of the solution to the optimal control problem is proved. For that problem the cost functional is defined by derivatives of the energy functional with respect to a crack perturbation parameter while the thickness parameter of rigid inclusion is chosen as the control function.

show abstract

Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion

Cited by 44 publications

References 11 publications

Junction problem for rigid and semirigid inclusions in elastic bodies

Junction problem for rigid and semirigid inclusions in elastic bodies

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous two‐dimensional bodies with a crack

Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

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