Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

• Published in: Journal of optimization theory and applications Vol. 155; no. 1; pp. 54 - 78
• Main Authors: Khludnev, A., Leugering, G., Specovius-Neugebauer, M.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-10-2012; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Boundary value problems; Calculus of Variations and Optimal Control; Optimization; Composite materials; Crack propagation; Cracks; Criteria; Elastic bodies; Energy; Engineering; Equilibrium; Inclusions; Inequalities; Mathematical problems; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Optimization algorithms; Perturbation methods; Studies; Theory of Computation; Rigid inclusion; Nonpenetration condition; Variational inequality; Inclusion shape; Crack; Control problem
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-012-0053-2

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.