N. Cristescu

A general constitutive equation for transient and stationary creep of rock salt

Abstract A constitutive equation, which besides creep describes volumetric dilatancy and/or compressibility, is subjected to some restrictions if we would like it to describe both stationary and transient creep. These general formulations are discussed here. A procedure to determine a viscoplastic potential for stationary creep from experimental data is presented. An example is given for rock salt, but the procedure is quite general and can be used for any creeping geomaterial, possessing the property of being either dilatant or compressible. Comparison of the model prediction with the data is presented. Long-term failure, or creep-failure, is also considered. Typical mining examples of creep, wall convergence for both short-term failure and long-term failure are given for vertical shafts or boreholes. Other examples will be given elsewhere.

Plastic flow through conical converging dies, using a viscoplastic constitutive equation

Abstract The influence of the working speed in drawing or extrusion processes is shown when a viscoplastic constitutive equation is used and when the yield condition is replaced by a yield inequality. It is shown that the drawing (or extrusion) pressure is increased when the speed of the process or when the viscosity coefficient of the material is increased, or when the diameter of the die is decreased or when the mean yield stress of the billet is reduced. This increase is more significant for higher reduction ratios and semi-cone angles. It is shown that the optimum semi-cone angle is also dependent on the speed of the process as is the critical semi-cone angle, which corresponds to the possible formation of a dead-zone near to the die surface. The pressure on the die surface is variable and depends on the speed of the process also. From the numerical examples given it is evident that in many practical circumstances the influence of speed may be significant even for very low working speeds.

An elastic/viscoplastic model for transient creep of rock salt

Abstract Using the true triaxial test results on Gorleben rock salt, a new elastic/viscoplastic model for transient creep is formulated. Both yield function and viscoplastic potential are determined from experimental data. Singularity and asymptotic properties of the yield surfaces and viscoplastic potential are considered. A new procedure to determine the viscoplastic potential is proposed. The model prediction is matching quite well the experimental data. The present model is an improvement of previous ones (Cristescu and Hunsche, 1992; Cristescu, 1994a). The model is not only in a better agreement with the data but also could be easily incorporated into a finite element program. Incorporated in a finite element program the model has given good agreement with the data. As an example the stress distribution around a vertical cylindrical cavity is analyzed.

A procedure to determine nonassociated constitutive equations for geomaterials

Abstract A procedure for determining a phenomenological elastic/viscoplastic nonassociated constitutive equation for geomaterials is presented. For this purpose, triaxial test data obtained with either a “true” or a classical triaxial device are necessary. The constitutive equation is aimed at describing such geomaterial properties as creep, irreversible compressibility or dilatancy, work-hardening, damage, and failure. Long-term failure can also be described with this model. According to the procedure, first the elastic parameters are determined from unloading tests (which follow short-term creep tests), then the yield function is determined, and finally the viscoplastic potential. No a priori assumption is made concerning the form of the yield function or of the viscoplastic potential; their expressions are obtained from the data by using the procedure suggested here. Examples for sand and rock salt are given. Comparisons of the model predictions with the experimental data are discussed.

Elastic/viscoplastic constitutive equations for rock

Abstract The experimental data needed for the formulation of a constitutive equation to describe the creep of rocks in uniaxial or triaxial tests are reviewed and the influence of loading rate on dilatancy is pointed out. The assumptions which have to be made in obtaining a constitutive equation from these experimental data are specified. A general constitutive equation is then formulated for rocks, which can describe their elastic and viscoplastic properties for any triaxial compressive stress state. Dilatancy and volumetric compressibility of rocks are defined in mathematical terms and so are the concepts of the compressibility/ dilatancy boundary and that of damage. Various time-dependent effects which can be modelled with such a constitutive equation are illustrated. It is shown, for instance, how the loading rate influences the dilatancy and/or the compressibility as well as the various stress-strain curves and the ultimate failure. The peculiarities of the creep phenomenon as described by the model are discussed. Numerical examples are given for several rocks. Various mining engineering problems have already been solved with such constitutive equations.

A new anisotropic failure criterion for transversely isotropic solids

A coordinate-free formulation of a failure criterion for transversely isotropic solids is proposed. In the three-dimensional stress space the criterion is represented by an elliptic paraboloid. The anisotropic form of the proposed criterion is based on generalization of the second invariant of the deviatoric stress and of the mean stress obtained through the introduction of a unique fourth-order tensor. For isotropic conditions, the criterion reduces to the Mises–Schleicher failure condition. It is shown that the criterion satisfactorily predicts the strength anisotropy of transversely isotropic rocks subjected to an axisymmetric stress state. The procedure for the identification of the parameters of the criterion from a few simple laboratory tests is outlined.

Nonassociated elastic/viscoplastic constitutive equations for sand

Abstract Elastic/viscoplastic constitutive equations for sands are proposed. Since an associated constitutive equation is unable to fit well the data, a nonassociated constitutive equation is chosen. It is shown how the equation of the stabilization boundary, the viscoplastic potential, and other constitutive parameters can be determined from the data. Failure,, dilatancy, and/or compressibility are included in the constitutive equation. Prediction of the model compares favorably with the experimental evidence.

Damage and failure of viscoplastic rock-like materials

Abstract Damage and failure by dilatancy of viscoplastic rock-like materials are considered. A damage parameter is introduced which is the total energy release due to microcracking when dilatancy takes place. This is expressed with the irreversible part of the volumetric stress power. The damage parameter can be used to describe the influence of the loading rate on failure, the failure due to creep tests and the influence of the loading history on failure, etc. Failure around underground openings occurring after a certain time interval following excavation can also be predicted. The damage parameter is a part of the constitutive equation and as such can be easily applied to various problems in which deformation of rocks by creep is followed by failure.

A procedure for determining the constitutive equations for materials exhibiting both time-dependent and time-independent plasticity

Abstract A procedure is given in order to determine various functions and constants entering in a constitutive equation exhibiting both time-dependent and time-independent plasticity, starting from a set of experimental data. As typical experiments one has chosen the symmetric longitudinal impact of two identical bars, since for such kinds of problems a great deal of experimental data are available. The unloading process is also considered since it is mainly during unloading that interesting aspects concerning “plasticity” may be put into evidence. A “relaxation boundary” which plays a main role in time-dependent plasticity is introduced. Several computed examples are given in order to show the influence of various functions or constants entering in the constitutive equation.

Viscoplastic creep of rocks around horizontal tunnels

The aim of this note is to present a mathematical model to describe dilatancy and/or compressibility of the rocks surrounding a horizontal tunnel, during their deformation by creep. The constitutive equation proposed for the purpose can describe both dilatancy and compressibility depending on the stress state existing at the particular point of the rock under consideration. Due to excavation, the primary stress state in the rock changes to a secondary one. Generally around the tunnel there are areas where the rock is in a dilatant state, while in others the rock is compressible, and finally in others the rock is in an elastic state. Mathematical criteria are given to outline the boundaries of such areas. The boundaries between the domains may also vary in time, since generally the stress state slowly varies with time. Since dilatancy can be related to the damage and failure of rocks, the constitutive equation can be used to predict damage and failure, localization of the damage, etc. (TRRL)