On Existence of Optimal Solutions to Boundary Control Problem for an Elastic Body with Quasistatic Evolution of Damage

  • Peter I. KogutEmail author
  • Günter Leugering
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 211)


We study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We use the damage field \(\zeta =\zeta (t,x)\) as an internal variable which measures the fractional decrease in the stress-strain response. When \(\zeta =1\) the material is damage-free, when \(\zeta =0\) the material is completely damaged, and for \(0<\zeta <1\) it is partially damaged. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation, whereas the model for the stress in elastic body is given as \(\varvec{\sigma }=\zeta (t,x) A\mathbf {e}({\mathbf {u}})\). The optimal control problem we consider in this paper is to minimize the appearance of micro-cracks and micro-cavities as a result of the tensile or compressive stresses in the elastic body.


Weak Solution Optimal Control Problem Displacement Field Elastic Body Radon Measure 
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  1. 1.
    Bouchitte, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. 3, 139–168 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buttazzo, G., Varchon, N.: On the optimal reinforcement of an elastic membrane. Riv. Mat. Univ. Parma. 4(7), 115–125 (2005)MathSciNetGoogle Scholar
  3. 3.
    Buttazzo, G., Kogut, P.I.: Weak optimal controls in coefficients for linear elliptic problems. Revista Matematica Complutense 24, 83–94 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society, Providence, RI (2002)zbMATHGoogle Scholar
  5. 5.
    Kogut, P.I., Leugering, G.: Optimal \(L^1\)-control in coefficients for Dirichlet elliptic problems: \(H\)-optimal solutions. ZAA 31(1), 31–53 (2011)MathSciNetGoogle Scholar
  6. 6.
    Kogut, P.I., Leugering, G.: Optimal \(L^1\)-control in coefficients for Dirichlet elliptic problems: \(W\)-optimal solutions. J. Optim. Theory Appl. 150(2), 205–232 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kuttler, K.L.: Quasistatic evolution of damage in an elastic-viscoplastic material. Electron. J. Differ. Eqns. 147, 1–25 (2005)Google Scholar
  8. 8.
    Lions, J.-L.: Quelques Méthodes de Résolution des Problèms aux Limites Non Linéares. Dunon, Paris (1969)Google Scholar
  9. 9.
    Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)Google Scholar
  10. 10.
    Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura. Appl. 146, 65–96 (1987)Google Scholar
  11. 11.
    Zhikov, V.V., Pastukhova, S.E.: Homogenization of degenerate elliptic equations. Siberian Math. J. 49(1), 80–101 (2006)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Differential EquationsDnipropetrovsk National UniversityDnipropetrovskUkraine
  2. 2.Institüt für Angewandte Mathematik Lehrstuhl II UniversitätErlangenGermany

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