Stress-constrained versus fracture-based topology optimization: A comparative study

Stress-constrained and fracture-based topology optimization are both popular methods to enhance fracture resistance in engineering structures and materials. However, their comparative advantages and applicability to various design scenarios remain underexplored. In this study, we revisit both formulations and systematically compare them by analyzing their underlying physics and capabilities. The stress-constrained formulation incorporates material strength surfaces as constraints, while the fracture-based formulation models both crack nucleation and propagation using a strongly coupled phase-field fracture theory. We then assess their optimized structures across several benchmark design scenarios accounting for various fracture behaviors. Our comparisons reveal several key insights. First, both formulations perform equivalently in design domains under uniform stress states, where the strength surface governs fracture nucleation. Second, the fracture-based formulation consistently produces feasible solutions in design domains with boundary defects and large pre-cracks, where the critical energy release rate becomes crucial in fracture nucleation. In this scenario, the stress-constrained formulation operates by eliminating stress concentrations; however, it may underestimate the fracture resistance of a structure due to the lack of information on the critical energy release rate. Third, the fracture-based formulation is preferable when the design priority is structural toughness maximization that involves both fracture nucleation and propagation. Finally, despite some limitations in the design performance, the stress-constrained formulation offers better computational efficiency and simpler implementation. These findings shed light on the similarities and differences between the two formulations and provide guidelines for selecting the suitable approach for practical design problems.