User-Defined-Models for the Itasca Codes FLAC, FLAC3D, UDEC, 3DEC, PFC2D and PFC3D

It is well known that salt rock responses to loading by elastic and visco-plastic deformation. The stress-deformation behaviour which is characterized by time-dependent ductile deformation without any visible macroscopic fracture is denoted as creep. Creep tests under constant stress conditions reveal that in general creep may be subdivided into the following three phases:

1. Primary creep – also denoted as transient or non-stationary creep,
2. Secondary or stationary creep, and
3. Tertiary creep or creep failure.

These three phases of creep are in a close relation one with the others, and as a result of intracrystalline deformation processes they pass from one to the other. Primary creep is characterized by high deformation rates. Decisive causes for primary creep are the dislocations which are present within the lattice structure and which start to move when stress increases. With growing deformation, the motion capacity of the present dislocations diminishes. If deformation continues, new dislocations will be produced within the lattice. Thus, the density of dislocations rises, and this rising density will cause an increasing resistance against deformation itself so that for maintaining a constant deformation rate an increasingly higher force is necessary or the deformation rate will decrease even when load is kept constant. This material hardening which increases with increasing deformation is counteracted by the recovery of dislocations. Out of this process, stationary creep develops by the fact that formation rate and recovery rate tend to approach equal values. In this phase of creep the density of dislocations, the deformation resistance and consequently also the creep rates devolve to constants (Blum 1978). When damaging processes and the softening processes which are linked to them and which start in the stress space above the dilatancy threshold (Hunsche 1998) achieve a critical value, creep will pass into its tertiary phase so that we can observe creep failure.

Here, a constitutive model is presented which is based on these physical processes and which describes all three creep phases in the scope of a creep model. In the present report, the theoretical considerations are illustrated together with the derivation of the corresponding parameters of the constitutive law for two Stassfurt rock salt varieties taken from the Sondershausen mine and the Asse site and with numerical recalculations of the respective laboratory tests. These numerical calculations were carried out with the explicit finite difference program FLAC (Itasca 2000) into which the constitutive model has been implemented as DLL file.

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Dissertation Ralf-Michael Günther (German)
Download UDM and Examples Flac3D 3.10
Download UDM and Examples Flac3D 4.00

UBCSAND is a 2-dimensional effective stress plasticity model for use in advanced stress-deformation analyses of geotechnical structures. The model was developed primarily for sand-like soils having the potential for liquefaction under seismic loading (e.g., sands and silty sands with a relative density less than about 80%). The model predicts the shear stress-strain behavior of the soil using an assumed hyperbolic relationship and estimates the associated volumetric response of the soil skeleton using a flow rule that is a function of the current stress ratio, η. The model can be used in a fully-coupled fashion where the mechanical and groundwater flow calculations are performed in parallel.

The yield surface defined in the model follows a Mohr-Coulomb formulation with a cohesion intercept equal to zero. The active yield surface is defined by the mobilized friction angle Φ_mob. The yield surface hardens during loading through a hyperbolic relationship between plastic shear strain and stress ratio η, where η is defined as the maximum shear stress divided by the mean effective stress in the loading plane. Φ_mob reaches a maximum at the specified maximum friction angle, Φ_f. Internal functions are used to relax Φ_mob in response to unloading or loading reversals.

To simply the use of UBCSAND in preliminary evaluations, a set of input parameters have been developed to represent the response of a hypothetical generic sand. These parameters provide reasonable estimates of stiffness and capture the liquefaction response in terms of the cyclic resistance ratio (CRR) as presented by the 1996/98 MCEER Workshops (Youd and Idriss et al., 2001), the effect of initial overburden stress on liquefaction as captured by the K_σ factor (Youd and Idriss et al., 2001), and a generally conservative interpretation of the effects of static bias as represented by Idriss and Boulanger (2003).

The UBCSAND constitutive model has been developed at the University of British Columbia by Prof. Peter M. Byrne. The many contributors to this effort include Drs. Michael Beaty, Ernie Naesgaard, and Humberto Puebla. It is considered a research tool that continues to evolve, although several of the versions have been used successfully on practical projects. All of the various versions have followed the same basic formulation but may have significant differences in the particular details of its implementation. The UDM C++ code uses the 904aR revision developed in 2008. This version was derived from the 904a code that was in widespread use since 2002.

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Download UDM and Examples

The NTUA-SAND Model (Andrianopoulos et al 2010a, b, 2011) is a bounding surface plasticity model with a vanished elastic region, developed to accurately simulate the rate-independent dynamic response of non-cohesive soils under small, medium and large cyclic strain amplitudes. This is achieved using a single set of values for the model constants, irrespective of initial stress and density conditions, as well as loading direction. The model is equally efficient in simulating the monotonic response.

The model builds on the constitutive efforts of Manzari & Dafalias (1997) and Papadimitriou & Bouckovalas (2002) and adopts three open cone-type non-circular surfaces, with their apex at the origin of stress space. These surfaces, named critical state surface, bounding surface and dilatancy surface, correspond to the deviatoric stress ratios at critical state, peak strength and phase transformation, respectively. The aperture of these surfaces is explicitly related to the state parameter ψ (Been and Jefferies, 1985), thus allowing the incorporation of the Critical State Theory of Soil Mechanics. The non-linear soil response under small to medium cyclic strain amplitudes is governed by a Ramberg-Osgood type hysteretic formulation, aiming at accurately simulating the shear modulus degradation and the hysteretic damping increase with cyclic shear strain. Furthermore, an empirical index of the directional effect of fabric evolution scales the plastic modulus, aiming at accurately simulating the rates of excess pore pressure build-up and permanent strain accumulation leading to liquefaction or cyclic mobility. To ensure numerical stability, the UDM employs the modified-Euler integration scheme with automatic error control and sub-stepping (Sloan et al. 2001).

A comprehensive yet practical constitutive model for sand (Wang 1990; Wang et al., 1990) was developed within the general framework of bounding surface plasticity.  The original model was written in general stress and strain tensors for three-dimensional conditions.  To implement the model into FLAC, formulations were derived that conform to plane-strain conditions (Wang and Makdisi, 1999).  While incorporating fewer parameters than the original model, this version retains the capability of simulating the generation of pore water pressure and soil liquefaction.

The bounding surface hypo-plasticity model includes two features that are not present in the more routinely used equivalent-linear procedures:

1. a failure surface defined by a friction angle to represent the ultimate state for sandy soils;
2. the soil modulus for each stress increment is defined as a function of stress-loading level in terms of the "distance" to the failure surface and to the maximum pre-stressed surface in the stress tensor space.  Thus the stress-strain relationship is fully nonlinear under both monotonic and cyclic loading conditions. 

The model also is capable of simulating shear-stress-induced volumetric change.  Under undrained conditions, the tendency for volumetric change produces excess pore water pressure.  This feature enables the prediction of liquefaction due to shear-stress loading and the potential for settlement after dissipation of the induced excess pore pressure. 

NorSand-M is an isotropically hardening - isotropically softening generalised critical state model that captures a wide range of particulate soil behaviour, the ‘M’ corresponding to the monotonic version of the constitutive model. From a users’ viewpoint, NorSand offers three main attributes: (i) it uses few material properties which are easily measured in conventional triaxial laboratory tests; (ii) the initial state parameter is readily measured; and (iii) it has been validated for a range of stress paths, including plane strain, and offers second-order detail in many aspects of soil behaviour. It therefore finds use as a general and validated soil model suitable for practical engineering.

The model is provided as a FISH-version for FLAC users. The DLL version will follow shortly.

The Hein-Model is an elasto-viscoplastic creep law to simulate porous materials, e.g. crusched salt. The elastic behavior is determined by porosity depending elastic properties, beginning with low bulk and shear modulus at the beginning, reaching the values of the compacted material at total compaction. The viscoplastic part is a combined hydrostatic - deviatoric law.