We introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase-field method. In the basic formulation we have a PDE-constrained optimization problem, where the finite element and design analysis are solved simultaneously. The starting point of the relaxation is a reformulation of the material problem involving linear and 0-1 constraints only. The 0-1 constraints are then relaxed and approximated by a Cahn-Hilliard-type penalty in the objective functional, which yields convergence of minimizers to 0-1 designs as the penalty parameter decreases to zero. A major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter. The relaxation scheme yields a large-scale optimization problem with a high number of linear inequality constraints. We discretize the problem by finite elements and solve the arising finite-dimensional programming problems by a primal-dual interior point method. Numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach.