yieldcriteria

Yield criteria are mathematical formulations that define when a material yields, i.e., begins plastic deformation, under complex loading. They specify a yield surface in stress space; once the stress state reaches or crosses this surface, irreversible deformation occurs. In many models the surface evolves with accumulated plastic strain (hardening), while the plastic flow direction is determined by a flow rule. Yield criteria are central to constitutive models used in structural analysis, metal forming, and geomechanics, and are commonly implemented in finite element codes.

Prominent isotropic metal criteria include von Mises (based on the deviatoric part of stress, J2) and Tresca (maximum shear). Drucker-Prager extends von Mises to account for hydrostatic pressure. Mohr-Coulomb is widely used for soils and rocks, combining cohesion and internal friction angle to yield under pressure. Anisotropic materials may use Hill or other advanced surfaces; some criteria (Rankine) emphasize principal stresses for tension or compression limits.

Yield criteria are typically expressed as f(σ, α) ≤ 0, with α representing internal variables for hardening. A plastic flow rule relates plastic strain increments to the gradient ∂f/∂σ; associated flow uses the same function f, while non-associated flows use a different potential to capture dilatancy or other effects.

Limitations include that no single criterion captures all materials and loading conditions. Rate dependence, temperature, damage, and large strains may require more complex models. Nonetheless, yield criteria remain a foundational element of plasticity theory and are essential for predicting strength and failure under multiaxial loading.