Phase-field fracture models represent cracks as smooth fields rather than sharp discontinuities. The crack is a region where the material has lost its ability to bear stress, described by a continuous damage variable that transitions from intact to broken. The approach is elegant and handles crack nucleation, propagation, and branching within a single framework. But it has a gap: it cannot distinguish between crack initiation (which depends on material strength) and crack propagation (which depends on fracture energy). Standard models couple the two, making it impossible to set strength and toughness independently.
An eigenstrain formulation decouples them (arXiv:2603.21811). The crack generates a local eigenstrain — a stress-free deformation that the material would undergo if unconstrained. The eigenstrain is computed at each integration point through a return-mapping algorithm, the same type used in plasticity. No additional global degrees of freedom are needed. The strength surface (when does the crack initiate?) and the fracture energy (how much energy does propagation cost?) become independent parameters.
Two strength criteria are demonstrated: one with independent tensile and shear strengths, and a Drucker-Prager criterion where strength increases with confining pressure. The model captures dynamic crack branching — a crack accelerating under load that spontaneously splits into two — which requires the interplay between strength and toughness to produce correctly.
The structural insight: a crack is not a single phenomenon. It is two phenomena sewn together at the crack tip: an initiation event controlled by stress (the material exceeds its strength) and a propagation event controlled by energy (the crack has enough energy to extend). The standard phase-field model forces both to be aspects of one parameter. The eigenstrain formulation recognizes that they are independent and lets the physics decide when each matters. The crack tip is where stress-governed and energy-governed failure meet.