Implicit integration under mixed controls of a breakage model for unsaturated crushable soils

Yida Zhang, Giuseppe Buscarnera*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


This paper discusses a series of stress point algorithms for a breakage model for unsaturated granular soils. Such model is characterized by highly nonlinear coupling terms introduced by breakage-dependent hydro-mechanical energy potentials. To integrate accurately and efficiently its constitutive equations, specific algorithms have been formulated using a backward Euler scheme. In particular, because implementation and verification of unsaturated soil models often require the use of mixed controls, the incorporation of various hydro-mechanical conditions has been tackled. First, it is shown that the degree of saturation can be replaced with suction in the constitutive equations through a partial Legendre transformation of the energy potentials, thus changing the thermomechanical state variables and enabling a straightforward implementation of a different control mode. Then, to accommodate more complex control scenarios without redefining the energy potentials, a hybrid strategy has been used, combining the return mapping scheme with linearized constraints. It is shown that this linearization strategy guarantees similar levels of accuracy compared with a conventional strain-suction-controlled implicit integration. In addition, it is shown that the use of linearized constraints offers the possibility to use the same framework to integrate a variety of control conditions (e.g., net stress and/or water-content control). The convergence profiles indicate that both schemes preserve the advantages of implicit integration, that is, asymptotic quadratic convergence and unconditional stability. Finally, the performance of the two implicit schemes has been compared with that of an explicit algorithm with automatic sub-stepping and error control, showing that for the selected breakage model, implicit integration leads to a significant reduction of the computational cost. Such features support the use of the proposed hybrid scheme also in other modeling contexts, especially when strongly nonlinear models have to be implemented and/or validated by using non-standard hydro-mechanical control conditions.

Original languageEnglish (US)
Pages (from-to)887-918
Number of pages32
JournalInternational Journal for Numerical and Analytical Methods in Geomechanics
Issue number6
StatePublished - Apr 25 2016


  • Breakage
  • Constitutive modeling
  • Hydro-mechanical coupling
  • Stress-point algorithms
  • Unsaturated soils

ASJC Scopus subject areas

  • Computational Mechanics
  • General Materials Science
  • Geotechnical Engineering and Engineering Geology
  • Mechanics of Materials


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