Choosing the right constitutive model in DeepEX
From simplified non-linear soil springs to advanced FEM formulations
The constitutive model is the link between soil data and predicted behaviour. It governs how stiffness changes with stress and strain, how plastic deformation develops, and how the ground responds during unloading and reloading. In excavation analysis, those choices can materially affect calculated wall movements, bending moments, support reactions, and surface settlements.
A more advanced model, however, is not automatically a better model. Its value depends on whether the expected ground behaviour is relevant to the problem and whether the required parameters can be established with reasonable confidence. A well-calibrated, simpler formulation may be more dependable than a sophisticated model built on uncertain inputs.
Two modelling levels in DeepEX
The Non-Linear analysis uses independent elastoplastic soil springs to represent the relationship between wall movement and soil reaction. FEM uses continuum constitutive laws to represent stress, strain, yielding, hardening, and deformation throughout the soil mass. These are complementary levels of modelling, not interchangeable versions of the same method.
A practical way to choose
Before selecting a model, answer three questions:
What is the engineering objective: overall equilibrium and structural demand, serviceability movements, rock-mass behaviour, or cyclic response?
Which mechanisms need to be represented: stress-dependent stiffness, unloading and reloading, small-strain stiffness, volumetric hardening, or load reversal?
Can the additional parameters be supported by laboratory tests, field measurements, correlations, or back-analysis?
The appropriate model is generally the simplest formulation that captures the mechanisms governing the decision and can be calibrated with the available data.
Soil-response models for Non-Linear analysis
For each soil, the spring model is selected in the C. Elasto-plastic tab of the Edit Soil Types dialog. These options define the relationship between wall displacement and soil reaction; they are not full continuum constitutive laws (Deep Excavation LLC, 2026b).
Elastic-Plastic - Linear Load-Reload
Soil reaction increases linearly with wall movement until the limiting active or passive resistance is reached. A constant stiffness governs loading, unloading, and reloading. The model is transparent and robust, but it does not reproduce progressive stiffness reduction or a distinct unloading/reloading stiffness.
Best suited to: preliminary models, routine checks with limited stiffness data, sensitivity studies, and cases where structural forces and equilibrium are more important than refined movement predictions.
Use with care when: wall movement or settlement is a key design criterion; the selected constant stiffness can dominate the result.
Exponential
The Exponential model introduces stress-dependent stiffness and a separate unloading/reloading modulus. This better reflects the tendency for soil stiffness to increase with confinement and for unloading/reloading to be stiffer than primary loading. In the Non-Linear analysis, the pre-yield curve is represented by multiple linear branches rather than by a full continuum hardening law. (Deep Excavation LLC, 2026b; Duncan & Chang, 1970)
Best suited to: staged braced or anchored excavations, displacement-sensitive designs, profiles with stiffness increasing significantly with depth, and sequences involving excavation and support installation.
Subgrade Modulus
This option follows the Winkler approach, with the ground represented by independent springs defined through a coefficient of subgrade reaction. That coefficient is not an intrinsic soil property: it depends on the loaded area, wall geometry, soil profile, deformation mode, and discretisation.
Best suited to: projects with locally calibrated subgrade-reaction values, monitored-excavation back-analysis, established local design procedures, and controlled parametric studies.
Use with care when: values calibrated for one wall geometry or site are transferred directly to a substantially different problem.
HS-Small - Approximated Procedure
This spring-based approximation represents the high stiffness observed at very small strains and its progressive reduction as strain develops. The feature is particularly relevant to serviceability because movements around excavations often begin while much of the soil remains within the small-strain range. (Benz, 2007; Deep Excavation LLC, 2026b)
Best suited to: excavations near movement-sensitive buildings, utilities, railways, or existing foundations, especially where small-strain stiffness is supported by geophysical testing, laboratory data, or calibrated correlations.
Use with care when: the spring approximation is treated as equivalent to the full Small-Strain Soil Hardening continuum model used in FEM.
Hoek-Brown
The Hoek-Brown option represents the nonlinear increase in rock-mass strength with confinement and may be more suitable than a linear Mohr-Coulomb envelope when the rock mass can be treated as an equivalent continuum. (Hoek et al., 2002)
Best suited to: walls socketed into weathered or jointed rock, rock cuts, and rock masses characterised using intact strength, Geological Strength Index (GSI), and disturbance conditions.
Use with care when: persistent discontinuities, faults, bedding, or wedges govern the response; these mechanisms may require explicit structural treatment.
Constitutive models for Finite Element Analysis
The FEM model is assigned from the A. General tab through the FEM Properties dialog. DeepEX offers two routes:
Automatic correlation with simplified models. DeepEX converts the selected simplified soil model and its parameters into an equivalent FEM formulation. This is an efficient starting point and supports consistency between Non-Linear and FEM design sections, but the resulting model and correlated parameters should still be reviewed whenever deformation predictions are important.
Direct FEM model selection. By clearing the automatic-correlation option, the engineer can choose a constitutive model directly and define its dedicated strength, stiffness, hardening, state, and drainage parameters. This route offers greater control but requires stronger calibration.
Mohr-Coulomb
A linear-elastic, perfectly plastic baseline model. A single Young's modulus and Poisson's ratio describe the pre-yield response; cohesion and friction angle define failure, and dilatancy controls plastic volumetric strain. It does not capture stress-dependent stiffness, small-strain behaviour, or distinct unloading/reloading stiffness.
Best suited to: preliminary FEM analyses, transparent baseline comparisons, strength-focused checks, and projects with limited stiffness information.
Use with care when: surface settlement, wall displacement, or unloading behaviour governs the design.
Modified Mohr-Coulomb
This formulation retains the familiar Mohr-Coulomb strength criterion while introducing stress-dependent stiffness and a distinction between loading and unloading/reloading. It is a practical intermediate step between basic Mohr-Coulomb and full hardening models.
Best suited to: staged excavations where stiffness variation with depth matters and improved deformation estimates are needed without the complete parameter set of a hardening model.
Use with care when: the analysis requires the full shear- and volumetric-hardening mechanisms available in Soil Hardening models.
Soil Hardening - Bower et al. - MC Failure
This model combines stress-dependent stiffness with shear and volumetric hardening while retaining the Mohr-Coulomb failure envelope. It distinguishes triaxial loading, oedometer loading, and unloading/reloading stiffnesses, and includes irreversible shear deformation and cap-type plastic compression. The reformulation was developed to improve numerical robustness in demanding analyses. (Bower et al., 2020)
Best suited to: routine 2D and 3D staged excavations, braced and anchored systems, and analyses where both wall movement and surface settlement matter.
Use with care when: the reference stiffnesses E50,ref, Eoed,ref, and Eur,ref, stress exponent m, reference pressure pref, failure ratio Rf, and strength parameters cannot be established credibly.
Small-Strain Soil Hardening
This model extends the hardening formulation by introducing the very high stiffness exhibited at extremely small strains and its reduction as shear strain develops. It also improves unloading/reloading loops and hysteretic response, which can materially affect predicted movements outside heavily strained zones. (Benz, 2007)
Best suited to: excavations close to sensitive structures, tunnels, utilities, railways, and foundations, as well as low-amplitude cyclic problems where small-strain response is relevant.
Use with care when: the very-small-strain shear modulus G0,ref and stiffness-degradation parameter gamma0.7 rely only on broad, unverified correlations.
Hoek-Brown
The Generalised Hoek-Brown model represents the nonlinear strength envelope of an isotropic jointed rock mass. Typical inputs include intact uniaxial compressive strength sigma_ci, intact-rock constant mi, GSI, disturbance factor D, and rock-mass deformation properties. (Hoek et al., 2002)
Best suited to: excavations in fractured or weathered rock, rock-supported walls, tunnels, shafts, and stress ranges that cannot be represented adequately by a single equivalent cohesion and friction angle.
Use with care when: persistent discontinuities control sliding, toppling, or wedge failure.
Drucker-Prager
A pressure-dependent elastoplastic model with a smooth conical yield surface. It can be viewed as a smooth approximation to the angular Mohr-Coulomb surface and may improve numerical performance in complex 3D stress paths. The response depends strongly on how it is matched to Mohr-Coulomb strength. (Drucker & Prager, 1952)
Best suited to: 3D analyses that benefit from a smooth yield surface, pressure-sensitive frictional materials, and controlled comparisons of failure criteria.
Use with care when: parameters are converted from Mohr-Coulomb without checking whether the adopted matching convention represents the relevant compression, extension, or intermediate stress state.
Soil Hardening - Bower et al.
The full Bower formulation represents stress-dependent stiffness, shear hardening, volumetric hardening, and dilatancy with a smooth three-dimensional failure description. Compared with the MC-failure version, it avoids some corners and numerical singularities associated with the traditional Mohr-Coulomb surface. (Bower et al., 2020)
Best suited to: advanced 2D and 3D excavation analyses, complex staging and stress paths, and projects requiring credible deformation predictions with stable numerical performance.
Use with care when: the additional sophistication is not supported by adequate stiffness and state calibration.
Soil Hardening - Schanz et al. (1999)
The original, widely recognised Hardening Soil model uses a hyperbolic stress-strain relationship with separate stiffnesses for primary triaxial loading, oedometer loading, and unloading/reloading. It represents stress dependency, shear hardening, volumetric hardening, and Mohr-Coulomb strength at failure. (Schanz et al., 1999; Bower et al., 2020)
Best suited to: projects calibrated within the established Hardening Soil framework, comparison with published case histories or other FEM software, and benchmarking against the reformulated Bower implementations.
Use with care when: numerical convergence is difficult under demanding stress paths; the Bower reformulation was developed partly to address such limitations.
Dafalias-Manzari
A SANISAND-family critical-state model developed for granular soils under monotonic and cyclic loading. It accounts for density and confinement, contractive and dilative response, kinematic hardening, fabric change during reversal, cyclic mobility, and pore-pressure accumulation under undrained loading. (Dafalias & Manzari, 2004)
Best suited to: saturated sandy deposits under seismic or repeated loading, liquefaction and cyclic-mobility studies, and advanced performance-based analyses supported by suitable laboratory testing.
Use with care when: the project is a routine static excavation, cohesive soil, or rock mass, or the parameter-intensive calibration is not supported by monotonic and cyclic test data.
Practical model-selection guide
The following table is a starting point, not a universal rule. Final selection should reflect the engineering objective, expected material behaviour, construction sequence, drainage conditions, and confidence in the available parameters.
Engineering objective | Suggested starting model | Calibration demand |
Preliminary or baseline FEM analysis | Mohr-Coulomb | Low |
Routine non-linear excavation analysis | Exponential | Moderate |
Locally calibrated spring analysis | Subgrade Modulus | Moderate |
Stress-dependent response with moderate inputs | Modified Mohr-Coulomb | Moderate |
General staged-excavation deformation analysis | Soil Hardening - Bower et al. - MC Failure | Moderate to high |
Compatibility with established HS calibrations | Soil Hardening - Schanz et al. | High |
Advanced FEM with smooth 3D failure surface | Soil Hardening - Bower et al. | High |
Small movements near sensitive structures | Small-Strain Soil Hardening | High |
Equivalent-continuum rock mass | Hoek-Brown | Moderate to high |
Smooth pressure-dependent yielding in 3D | Drucker-Prager | Moderate to high |
Cyclic response or liquefaction of saturated sand | Dafalias-Manzari | Very high |
Good modelling practice
Model selection is only one part of a defensible analysis. DeepEX guidance also emphasises soil properties, wall friction, initial conditions, geometry, and construction sequence (Deep Excavation LLC, 2026b). In practice:
Select drained or undrained behaviour to reflect permeability, construction duration, and loading conditions.
Establish the initial stress state and overconsolidation conditions carefully.
Reproduce excavation, dewatering, support installation, prestressing, and other construction stages in the correct sequence.
Calibrate stiffness against laboratory tests, in-situ tests, geophysical measurements, or monitored field behaviour whenever possible.
Review interface and wall-friction assumptions, mesh adequacy, and boundary locations.
Use sensitivity analyses for uncertain parameters and compare more than one constitutive model when movements govern the design.
Check whether deformation patterns and load transfer are physically reasonable; numerical convergence alone does not validate the model.
A useful hierarchy
Begin with a transparent baseline, then add complexity only when it represents a mechanism that matters to the decision. Compare the advanced result with the baseline and explain why the difference is physically credible.
Conclusion
DeepEX allows engineers to progress from efficient spring-based representations to advanced continuum formulations within the same modelling environment. Non-Linear models are well suited to staged wall analysis and design iteration, while FEM models can represent stress-dependent stiffness, hardening, small-strain response, nonlinear rock-mass strength, and cyclic sand behaviour.
For many routine excavations, a calibrated hardening model provides a practical balance between realistic deformation predictions and manageable parameter requirements. Mohr-Coulomb remains valuable as a clear baseline. Small-Strain Soil Hardening, Hoek-Brown, Drucker-Prager, and Dafalias-Manzari are most effective when the specific behaviour they represent is relevant and supported by suitable data.
The central principle is simple: choose the model whose assumptions match the ground behaviour and the engineering question - not merely the most advanced option in the list.
References
Benz, T. (2007). Small-strain stiffness of soils and its numerical consequences [Doctoral dissertation, University of Stuttgart]. German National Library. https://d-nb.info/984776931/04
Bower, T. A., Jefferson, A. D., & Cleall, P. J. (2020). A reformulated hardening soil model. Proceedings of the Institution of Civil Engineers - Engineering and Computational Mechanics, 173(1), 11-29. https://doi.org/10.1680/jencm.18.00054
Dafalias, Y. F., & Manzari, M. T. (2004). Simple plasticity sand model accounting for fabric change effects. Journal of Engineering Mechanics, 130(6), 622-634. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:6(622)
Deep Excavation LLC. (2026a, April 27). Exploring constitutive laws for soils in DeepEX: A guide to advanced soil models and analysis. https://www.deepexcavation.com/post/exploring-constitutive-laws-for-soils-in-deepex
Deep Excavation LLC. (2026b, April 27). Section 11: DeepFEM - Finite element analysis. https://www.deepexcavation.com/post/section-11-deepfem-finite-element-analysis
Drucker, D. C., & Prager, W. (1952). Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics, 10(2), 157-165. https://doi.org/10.1090/qam/48291
Duncan, J. M., & Chang, C. Y. (1970). Nonlinear analysis of stress and strain in soils. Journal of the Soil Mechanics and Foundations Division, 96(SM5), 1629-1653. https://doi.org/10.1061/JSFEAQ.0001458
Hoek, E., Carranza-Torres, C., & Corkum, B. (2002). Hoek-Brown failure criterion - 2002 edition. In Proceedings of the Fifth North American Rock Mechanics Symposium and the 17th Tunnelling Association of Canada Conference (pp. 267-273).
Schanz, T., Vermeer, P. A., & Bonnier, P. G. (1999). The hardening soil model: Formulation and verification. In R. B. J. Brinkgreve (Ed.), Beyond 2000 in computational geotechnics: 10 years of PLAXIS (pp. 281-296). A. A. Balkema. https://doi.org/10.1201/9781315138206-27
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