# Constitutive Models

Introduction

Numerical solution schemes face several difficulties when implementing constitutive models to represent geomechanical material behavior. There are three characteristics of geo-materials that cause particular problems.

One is physical instability. Physical instability can occur in a material if there is the potential for softening behavior when the material fails. When physical instability occurs, part of the material accelerates and stored energy is released as kinetic energy. Numerical solution schemes often have difficulties at this stage because the solution may fail to converge when a physical instability arises.

A second characteristic is the path dependency of nonlinear materials. In most geomechanical systems, there are an infinite number of solutions that satisfy the equilibrium, compatibility, and constitutive relations that describe the system. A path must be specified for a “correct” solution to be found. For example, if an excavation is made suddenly (e.g., by explosion), then the solution may be influenced by inertial effects that introduce additional failure of the material. This may not be seen if the excavation is made gradually. The numerical solution scheme should be able to accommodate different loading paths in order to properly apply the constitutive model.

A third characteristic is the nonlinearity of the stress-strain response. This includes the nonlinear dependence of both the elastic stiffness and the strength envelope on the confining stress. This can also include behavior after ultimate failure that changes character according to the stress level (e.g., different post-failure response in the tensile, unconfined, and confined regimes). The numerical scheme needs to be able to accommodate these various forms of nonlinearity.

The difficulties faced in numerical simulations in geomechanics—physical instability,
path dependence, and implementation of extremely nonlinear constitutive models—can all be
addressed by using the *explicit, dynamic solution* scheme provided in FLAC3D. This
scheme allows the numerical analysis to follow the evolution of a geologic system in a
realistic manner, without concerns about numerical instability problems. In the explicit,
dynamic solution scheme, the full dynamic equations of motion are included in the
formulation. By using this approach, the numerical solution is stable even when the
physical system being modeled is unstable. With nonlinear materials, there is always the
possibility of physical instability (e.g., the sudden collapse of a slope). In real life,
some of the strain energy in the system is converted into kinetic energy, which then
radiates away from the source and dissipates. The explicit, dynamic solution approach
models this process directly, because inertial terms are included—kinetic energy is
generated and dissipated.

In contrast, schemes that do not include inertial terms must use some numerical procedure to treat physical instabilities. Even if the procedure is successful at preventing numerical instability, the path taken may not be a realistic one. The numerical scheme should not be viewed as a black box that will give “the solution.” The way the system evolves physically can affect the solution. The explicit, dynamic solution scheme can follow the physical path. By including the full law of motion, this scheme can evaluate the effect of the loading path on the constitutive response.

The explicit, dynamic solution scheme also allows the implementation of strongly nonlinear constitutive models because the general calculation sequence allows the field quantities (forces/stresses and velocities/displacements) at each element in the model to be physically isolated from one another during one calculation step. The general calculation sequence of FLAC3D is described in the section of Theoretical Background. The implementation of elastic/plastic constitutive models within the framework of this scheme is discussed in the Incremental Formulation section.

The mechanical constitutive models available in FLAC3D range from linearly elastic models to highly nonlinear plastic models. The basic constitutive models are listed below. A short discussion of the theoretical background and simple example tests for each model follow this listing.

Section Outline

- Constitutive Models in FLAC3D and 3DEC
- Incremental Formulation
- Null Model Group
- Elastic Model Group
- Plastic Model Group
- Drucker-Prager Model
- Mohr-Coulomb Model
- Ubiquitous-Joint Model
- Anisotropic-Elasticity Ubiquitous-Joint Model
- Strain-Softening/Hardening Mohr-Coulomb (SSoft) Model
- Bilinear Strain-Softening/Hardening Ubiquitous-Joint (SUBI) Model
- Double-Yield Model
- Modified Cam-Clay Model
- Hoek-Brown Model
- Hoek-Brown-PAC Model
- Cap-Yield (CYSoil) Model*
- Simplified Cap-Yield (CHSoil) Model*
- Plastic-Hardening (PH) Model*
- Swell Model*
- Mohr-Coulomb Tension Crack (MohrT) Model*
- Soft-Soil Model*
- NorSand Model*

- Creep Constitutive Models in 3DEC and FLAC3D
- Description of Creep Constitutive Models
- Solving Creep Problems
- Input Instructions for Creep Modeling
- Verification and Example Problems
- Maxwell/Kelvin/Burgers Model: Parallel-Plate Viscometer
- WIPP-Type Models: Parallel-Plate Viscometer
- Power Model: Cylindrical Cavity
- Power-Mohr Model: Cylindrical Cavity
- WIPP Model: Cylindrical Cavity
- Power Model: Spherical Cavity
- Maxwell Model: Oedometer Test
- Kelvin Model: Oedometer Test
- Maxwell/Burgers Model: Compression Test
- Burgers-Mohr/Power-Mohr Model: Loading/Unloading Compression Test
- WIPP-Drucker Model: Comparison with Drucker-Prager Model
- WIPP-Drucker Model: Compression Test Showing Localization
- WIPP-Salt Model: Hydrostatic Compression Test
- WIPP-Salt Model: Shear Compression Test
- Soft-Soil-Creep Model: Triaxial Consolidated Undrained Loading

- References

- Model Tests and Examples
- Oedometer Test with Mohr-Coulomb Model
- Uniaxial Compressive Strength of a Jointed Material Sample
- Isotropic Consolidation Test with Double-Yield Model
- Isotropic Consolidation Test with Modified Cam-Clay Model
- Triaxial Compression Test with Hoek-Brown Model
- Triaxial Compression Test with Hoek-Brown-PAC Model
- Isotropic Compression Test with CYSoil Model
- Oedometer Test with CYSoil Model
- Drained Triaxial Test with CYSoil Model — Constant Dilation
- Drained Triaxial Test with CYSoil Model — Dilation Hardening
- Undrained Triaxial Test with CYSoil Model
- Drained Triaxial Compression Test with Simplified Cap-Yield (CHSoil) Model
- Comparison between Mohr-Coulomb Model and Plastic-Hardening model
- Comparison of Plastic-Hardening Model without and with Small-Strain Stiffness
- Isotropic Compression Test with Plastic-Hardening Model
- Drained Triaxial Compression Test with Plastic-Hardening Model
- Undrained Triaxial Compression Test with Plastic-Hardening Model
- Oedometer Test with Plastic-Hardening Model
- Single Zone Swell Test
- Single Zone Loading-Unloading Test with MohrT Model
- Undrained Triaxial Test with Soft-Soil Model
- Undrained Triaxial Test with NorSand Model

- References

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