Plastic Deformation of Boise Sandstone

INTRODUCTION The inelastic stress-strain behavior of rocks has urged engineers to seek for the possible application of plasticity to engineering pro¬jects such as tunnels, oil wells, dams, etc. for a long time. Early efforts in this direction have tentatively used the classical theories to solve some axisymmetric problems, in which the rocks were treated as perfectly plastic and nonfrictional materials (Talobre, J. A., 1957). Meanwhile, one of the fundamental assumptions is that the volume strain of elastoplastic material is purely elastic. Drucker and Prager (1952) developed a perfectly plastic and frictional model with a generalized form of Mohr-Coulomb law, which serves as both failure criterion and yield criterion. Although the associated flow rule gives unacceptably large dilatancy, this model is a very important step to deal with the frictional behavior of geological materials. As a further step, Roscoe et al (1963) and Schofield and Wroth (1968) proposed a Cam-clay model, which accounts for the yielding before the material reaches a failure envelope and reflects the plastic volume strain as a strain hardening factor, and the open end of the Drucker-Prager envelope is capped by a family of yield surfaces, the envelope is made conforming with crit¬ical void ratio line. The proposed shapes of the cap have been based on experimental data and on convenience of mathematical descrip¬tions. Drucker, Gibson and Henkel (1957) introduced a spherical cap for soils, while in Cam-clay model the cap is semi-logarithmic yield function derived from the result of triaxial tests. DiMaggio and Sandler (1971) proposed an elliptical model for sand, in which the hardening function depends exponentially on plastic strain. Sandler et al (1976) used a plane cap model to express the behavior of a wide range of geological materials of which the nonlinear hysteretic nature is significant. Mizuno and Chen (1982) illustrated the physical mean¬ing of cap models and their adaption to finite element calculation. In recent years, Carroll et al (1972, 1980) have studied the com¬paction of porous materials under hydrostatic pressure and discussed the mechanism of the nonlinear response thoroughly. Analytical and numerical analysis of the hollow sphere model by Curran and Carroll (1979) predicted an initial yielding surface, dependent on the initial porosity, and showed a strong coupling between hydrostatic and deviatoric effects. All these main features from the abovementioned theoretical research are visualized by the experimental data on Boise sandstone (1984). The combination of such previous work would be logically the construction of a plasticity model for porous rocks which, following the tradition of assigning the name of birthplace to a special model, is called Cal-Rock model (Carrol, M. M., 1984). The results of triaxial tests along different loading paths reveal the compaction characteristics for deviatoric and hydrostatic stress separately. the fitting of the nonlinear stress-strain curves leads to a construction of yield surfaces. Moreover, an uniaxial strain test gives the strength envelope which is the critical state line. The whole model is composed in the frame of elasticity theory. The concept and representation of plasticity used here are in the spirit of Naghdi¬Casey theory (Casey and Naghdi, 1984 and 1984, Naghdi and Trapp, 1975). In order to find the way of using such a plasticity model in engineering problems, the stress field in a thick-walled cylinder of porous material is calculated. This is an axisymmetrical problem with plane strain, which is an idealization of oil well in an infinite medium. Because the constitutive relation is highly nonlinear, a numerical method has been utilized to obtain the solution. CONSTRUCTION OF CAL-ROCK MODEL (1). Elastic-plastic constitutive equations The basic assumption is that the yield surface in stress space depends on one parameter, namely the plastic volume strain, as the data on Boise sandstone suggested. Starting from this point, the theory formulated in stress space and in strain space by Naghdi and his co-workers (1975, 1981) can be simplified. The simplified yield function is