DruckerPragerin

The Drucker–Prager model is a mathematical framework used in continuum mechanics and material science to describe the yield behavior of isotropic materials under combined hydrostatic pressure and deviatoric stress. It extends the von Mises yield criterion by incorporating a hydrostatic pressure term, making it particularly useful for modeling geomaterials, soils, and certain metals that exhibit pressure-dependent yielding.

The model was independently developed by Peter Drucker and William Prager in the 1920s and 1930s, though its formalization as a yield criterion became widely recognized later. The Drucker–Prager yield function is expressed as:

σ₀ = αI₁ + √(J₂)

where σ₀ is the yield stress, I₁ is the first invariant of the stress tensor (representing hydrostatic pressure), J₂ is the second invariant of the deviatoric stress tensor, and α is a material-specific parameter that accounts for pressure sensitivity. The term αI₁ introduces a linear dependence on pressure, allowing the model to capture phenomena like increased strength under confinement, which is critical in geotechnical engineering.

The Drucker–Prager model is often implemented in finite element analyses to simulate plastic deformation in materials like concrete, rock, and certain alloys. Its simplicity compared to more complex models, such as the Mohr–Coulomb criterion, makes it computationally efficient while still capturing essential pressure-dependent behaviors. However, it assumes isotropic material behavior and does not account for strain hardening or softening, limiting its applicability in advanced simulations.

Variations of the model, such as the modified Drucker–Prager criterion, have been proposed to improve accuracy by incorporating additional terms or parameters. Despite its limitations, the Drucker–Prager model remains a foundational tool in computational mechanics for analyzing yield and plastic flow in pressure-sensitive materials.