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Description:
In brittle composite materials mesoscopic failure mechanisms like debonding of the matrix-fiber interface or fiber breakage can result in crack deflection and hence in the improvement of the macroscopic damage tolerance. More generally it is known that high values of fracture energy dissipation lead to toughening of the material. This motivates our goal to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus especially on the effect of variation of fiber shapes on the crack paths and thus on the fracture energy. For a systematic approach we set up a shape optimization problem formulation. We derive first order information in the form of a shape gradient or a sub-gradient for different formulations of the shape optimization problem respectively. While the cost function of the optimization problem is represented by the fracture energy the state problem consists in the determination of the potentially discontinuous displacement field in the two dimensional cracked domain. The displacement field includes the information about crack openings which influence the energy dissipation significantly. This effect is due to consideration of cohesiveness. The cohesive tractions and the dissipated energy depend nonlinearly on the crack openings normal as well as tangential to the crack surface. This is an extension to existing approaches. Some of these approaches solely take cohesiveness due to normal crack openings into account. Others rely on Griffith theory and thus model energy dissipation independent of crack openings. Our approach results in an extended set of constitutive equations for description of the behavior of a cracked structure. The cohesive effects and a non-penetration condition, which is imposed to avoid interpenetration of opposite crack sides, lead to additional inequalities. We show that a displacement field which fulfills the constitutive equations can be obtained as solution of a variational inequality or equivalently as minimum of the total energy ...