A novel topology optimization formulation for enhancing fracture resistance with a single quasi-brittle material

Optimal design of structures for fracture resistance is a challenging subject. This appears to be largely due to the strongly nonlinear governing equations associated with explicitly modeling fracture propagation. We propose a topology optimization formulation, in which low weight structures are obtained with significantly increased resistance to brittle fracture, in which crack propagation is explicitly modeled with the phase field approach. By contrast to our previous work, several important features are included which greatly assist the optimizer in dealing with the strongly discontinuous brittle fracture process, including a new objective function, which provides additional path information to the optimizer. Increased local control of the topology is introduced via a smoothed threshold function in the phase field fracture formulation and a constraint relaxation continuation scheme is proposed to alleviate some difficulty during the initial optimization iterations. The derivation of the analytical, path-dependent sensitivities for the relevant functions is provided and the results from two benchmark numerical examples are presented which demonstrate the effectiveness of the proposed method.