This work presents an efficient strategy for dealing with topology optimization associated with the problem of mass minimization under material failure constraints. Although this problem characterizes one of the oldest mechanical requirements in structural design, only a few works dealing with this subject are found in the literature. Several reasons explain this situation, among them the numerical difficulties introduced by the usually large number of stress constraints. The original formulation of the topological problem (existence/non-existence of material) is partially relaxed by following the SIMP (Solid Isotropic Microstructure with Penalization) approach and using a continuous density field ρ as the design variable. The finite element approximation is used to solve the equilibrium problem, as well as to control ρ through nodal parameters. The formulation accepts any failure criterion written in terms of stress and/or strain invariants. The whole minimization problem is solved by combining an augmented Lagrangian technique for the stress constraints and a trust-region box-type algorithm for dealing with side constraints (0<ρ_min≤ρ≤1) . Numerical results show the efficiency of the proposed approach in terms of computational costs as well as satisfaction of material failure constraints. It is also possible to see that the final designs define quite different shapes from the ones obtained in classical compliance problems.