An unconstrained implicit stress integration algorithm for the HS model

doi:10.1016/j.compgeo.2026.108400

An unconstrained implicit stress integration algorithm for the HS model

2026-07-04

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DOI：10.1016/j.compgeo.2026.108400

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The Hardening Soil (HS) model is widely used in numerical analyses of geotechnical problems, including deep excavations, tunneling, and foundations. However, its implicit implementation under general three-dimensional stress states still faces two non-smoothness problems. At the constitutive function level, the Mohr–Coulomb criterion exhibits a non-unique outward normal at deviatoric plane corners, and the dependence of stiffness moduli on the minimum principal stress may cause derivative jumps during principal stress permutation. At the complementarity-constraint level, the double-hardening double-yield surface structure introduces two sets of Karush–Kuhn–Tucker (KKT) conditions, requiring repeated active set enumeration and activation mode switching in conventional return mapping algorithms. To address these issues, an implicit scheme combining constitutive smooth reconstruction with unconstrained stress integration is proposed. A smooth deviatoric shape function, a generalized confining pressure, and a shear hardening correction coefficient are introduced to reformulate the HS model into a three-dimensional smooth form that preserves the original triaxial compression response and calibration framework. The KKT conditions are then replaced by the Chen–Harker–Kanzow–Smale (CHKS) smoothing complementarity function, by which the constrained stress integration is recast as an unconstrained nonlinear system and solved uniformly using a line search Newton method. This formulation inherently bypasses explicit loading/unloading discrimination. The proposed scheme is implemented through the user defined material subroutine (UMAT) interface of ABAQUS/Standard and is evaluated through laboratory element test simulations, a cantilever excavation benchmark, and a strip footing robustness example. At the material point level, the UMAT results agree well with Plaxis solutions and available laboratory data. At the boundary value problem level, the cantilever excavation reproduces the Plaxis HS reference solution, whereas the strip footing example demonstrates robust convergence under localized plastic deformation. Residual histories of representative load steps indicate rapid terminal convergence of the global Newton iterations, approaching the quadratic reference trend.

An unconstrained implicit stress integration algorithm for the HS model

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