FLAC3D Theory and Background • Constitutive Models

Plastic Model Group

All plastic models potentially involve some degree of permanent, path-dependent deformations (failure)—a consequence of the non-linearity of the stress-strain relations. The different models are characterized by their yield function, hardening/softening functions, and flow rule. The yield functions for each model define the stress combination for which plastic flow takes place. These functions or criteria are represented by one or more limiting surfaces in a generalized stress space, with points below or on the surface being characterized by an incremental elastic or plastic behavior, respectively. The plastic flow formulation rests on basic assumptions from plasticity theory that the total strain increment may be decomposed into elastic and plastic parts, with only the elastic part contributing to the stress increment by means of an elastic law (see the discussion in Incremental Equations of the Theory of Plastic Flow). In addition, both plastic and elastic strain increments are taken to be coaxial with the current principal axes of the stresses (only valid if elastic strains are small compared to plastic strains during plastic flow). The flow rule specifies the direction of the plastic strain increment vector as that normal to the potential surface; it is called associated if the potential and yield functions coincide, and non-associated otherwise. See Vermeer and de Borst (1984) for a more detailed discussion on the theory of plasticity.

For the Von-Mises, Drucker-Prager, Mohr-Coulomb, ubiquitous-joint, strain-hardening/softening, SUBI (bilinear-hardening/softening-ubiquitous-joint), MohrT (Mohr-Coulomb tension crack), ubiquitous-anisotropic, COMBA (columnar-basalt) and concrete models, a shear yield function and a non-associated shear flow rule are used. For the double-yield and CYSoil models, shear and volumetric yield functions, non-associated shear flow and associated volumetric flow rules are included. The CHSoil model is a simplified version of the CYSoil model that provides built-in friction hardening and dilation hardening/softening laws, and does not include a volumetric cap. In addition, the failure envelope for each of the above models is characterized by a tensile yield function with associated flow rule.

The modified Cam-Clay, PH (plastic-hardening), Soft-Soil and NorSand model formulation rests on a combined shear and volumetric yield function and associated flow rule.

The two types of Hoek-Brown model in FLAC3D and 3DEC provide different formulations to represent yielding. The basic Hoek-Brown model using a nonlinear shear yield function and a plasticity flow rule that varies as a function of the stress level is referred to as the Hoek-Brown-PAC model. For the traditional Hoek-Brown model, plastic flow is handled in a manner similar to that in the Mohr-Coulomb model in which a dilation angle is specified. Also, a tensile yield function similar to that used with the Mohr-Coulomb model is included with the Hoek-Brown model.

Note also that all plasticity models are formulated in terms of effective stresses, not total stresses.

The plasticity models with strain softening or with non-associated flow rule can produce localization (i.e., the development of families of discontinuities such as shear bands in a material that starts as a continuum). Note that localization is grid-dependent since there is no intrinsic length scale incorporated in the formulations. This is an important consideration when creating a grid for a plasticity analysis, and is discussed more fully in Localization, Physical Instability and Path-Dependence.

As discussed in Implementation, in the numerical implementation of the models, an elastic trial (or “elastic guess”) for the stress increment is first computed from the total strain increment using the incremental form of Hooke’s law. The corresponding stresses are then evaluated. If they violate the yield criteria (i.e., the stress-point representation lies above the yield function in the generalized stress space), plastic deformations take place. In this case, only the elastic part of the strain increment can contribute to the stress increment; the latter is corrected by using the plastic flow rule to ensure that the stresses lie on the composite yield function. This section describes the yield and potential functions, flow rules, and stress corrections for the different plasticity models.