The maximum allowable stress (material strength) and the energy required to propagate a crack (material fracture toughness) are the two fundamental properties that a minimal model of material and structural failure must capture. Classical approaches address these features separately: perfect plasticity models strength, while Griffith’s theory models toughness. Cohesive fracture models and gradient damage models can represent both mechanisms within a unified framework, but each presents important limitations. Sharp-interface cohesive models face mathematical and numerical difficulties when cracks evolve freely [2]. Conversely, formulating damage models that admit arbitrary strength surfaces under multiaxial loading while remaining consistent with Griffith’s crack propagation criterion remains challenging [3]. We recently proposed [1] a variational phase-field model of fracture capable of accommodating arbitrary closed convex strength domains. In contrast to Ambrosio–Tortorelli–type formulations, the phase-field variable does not degrade the elastic stiffness. Instead, the elastic energy displays linear growth outside the strength domain, which progressively shrinks to zero as the phase-field variable approaches unity. Analysis of a three-dimensional model problem reveals a sharp-interface limit in the form of a cohesive law, explicitly derived from the phase-field formulation. In this presentation, we summarize the main features of this strength-based phase-field framework, see also [5], and present first numerical results obtained using convex optimization solvers to handle arbitrary—including non-smooth—convex strength criteria. We also discuss a fundamental classification of classical strength criteria based on jump compatibility conditions. REFERENCES [1] Bourdin B., Marigo J.-J., Maurini C., Zolesi C., A variational approach to fracture incorporating any convex strength criterion, Preprint, 2025. [2] Rodella A., Marigo J.-J., Maurini C., Vidoli S., Sharp-interface cohesive fracture models with consistent bulk energies: Numerical investigations, Journal of the Mechanics and Physics of Solids, 2026, 106543. [3] Vicentini F., Zolesi C., Carrara P., Maurini C., De Lorenzis L., On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states, International Journal of Fracture, 2024. [4] Bleyer J., Automating the formulation and resolution of convex variational problems: applications from image processing to