Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

Khludnev, A. M.; Leugering, Günter; Specovius–Neugebauer, Maria

doi:10.1007/s10957-012-0053-2

Cited by 15 publications

(14 citation statements)

References 12 publications

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“…The claim follows by using the convergences (19), (22) and the following "ε/2"-argument: For a given ε > 0 we may choose a small λ > 0 such that b − b λ < ε/2 due to (19). For every such λ we choose a small τ > 0 such that b λ − b τ,λ < ε/2 due to (22).…”

Section: Existence Results For the Time-continuous Systemmentioning

confidence: 99%

A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

Farshbaf-Shaker

Heinemann

2015

Math. Models Methods Appl. Sci.

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In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

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Section: Existence Results For the Time-continuous Systemmentioning

confidence: 99%

A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

Farshbaf-Shaker

Heinemann

2015

Math. Models Methods Appl. Sci.

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“…From (17), it follows that the curve Γ c satisfy the relations −hσ ν − m ν 0, −hσ ν + m ν 0, or −hσ ν |m ν |. Integrating the first equation (9) by parts and using (12) and (16), we obtain Consequently, on the curve Γ c , the following equalities hold:…”

Section: Formulation Of the Boundary-value Problemmentioning

confidence: 96%

“…Note that this approach allows for the possibility of mutual penetration of the crack faces [4]. Variational methods have been used to study the large class of mathematical problems of cracks with nonlinear nonpenetration conditions (in the form of systems of equalities and inequalities) defined on the curve (surface) which describes cracks in elastic bodies [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Equilibrium problem for a Timoshenko plate with an oblique crack

Lazarev

2013

J Appl Mech Tech Phy

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The condition of mutual nonpenetration of the crack faces is proposed for a Timoshenko plate with an oblique crack, whose initial state is defined by a surface the normal to which makes a small angle with the middle plane. Unique solvability of the variational problem of plate equilibrium with the nonpenetration conditions for the crack faces specified on the curve describing the crack is proved. A differential formulation of the problem equivalent to the original formulation for sufficiently smooth solutions is proposed. For the one-dimensional case (beam with a cut), an analytical solution is obtained, and the cases of longitudinal tension and compression are examined.

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“…We believe that such a11 inttueuc:e is pussil,le and eau l,,e used for the control of crack propagation in elastic media. Indeed, the dependence of the Griffith funr.tional with respect to shape changes of an elastic or rigid inclusion has been considered in 112,11). This research has been triggered by numcrical studies on optimization an control of crack growth also for the case of cohesivc crack theories in 121 , 22, 18).…”

Section: Lntroductionmentioning

confidence: 99%