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Version: publishedVersion × Subject: viscoelasticity ×

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    A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Farshbaf Shaker, Mohammad Hassan; Heinemann, Christian
    In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous 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 as well as boundedness of the phase field variable which results in a doubly 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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    Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farshbaf-Shaker, M. Hassan; Heinemann, Christian
    Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variable. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [4], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, wellposedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.
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    A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Scala, Riccardo; Schimperna, Giulio
    We consider a three-dimensional viscoelastic body subjected to external forces. Inertial effects are considered; hence the equation for the displacement field is of hyperbolic type. The equation is complemented with Dirichlet and Neuman conditions on a part the boundary, while on another part the body is in adhesive contact with a solid support. The boundary forces acting on the latter part due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a nonlinear ODE which describes the evolution of the delamination order parameter z. Following the lines of a new approach introduced by the authors in a preceding paper and based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solutions to the resulting PDE system. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.
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    A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Scala, Riccardo
    We consider a system of two viscoelastic bodies attached on one edge by an adhesive where a delamination process occurs. We study the dynamic of the system subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coefficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs.