The determination of the elastic properties of composite materials (multiphase aggregates, polycrystals, and porous or cracked solids) from the elastic properties of the components may be approached in several ways. The problem may be treated statistically, via scattering theory, through variational principles, or by the assumption of specific geometries for the material under consideration. Each of these methods is reviewed in turn. The widely used Voigt-Reuss-Hill average can be a poor approximation for both two-phase composites and polycrystals, and its replacement by the two Hashin-Shtrikman bounds is recommended. For pore-containing or crack-containing media, specific geometry models must be considered if useful results are to be obtained. If aggregate theory is used to estimate the moduli of individual components of a composite whose bulk properties are known, the shear moduli of the component phases must be matched (within a factor of 2 or 3) for the method to be useful. Results for nonlinear composites (which allow calculation of the pressure variation of aggregate moduli) have been obtained for only a few special cases.

The Elastic Properties of Composite Materials

Constitutive law for the shear strength of rock joints based on three-dimensional surface parameters

A new 2D discontinuity roughness parameter and its correlation with JRC

Thirteen natural rock profiles (Barton and Choubey, 1977) are analyzed for their fractal properties. Most of the profiles were found to approximate fractal curves but some also showed features of specific wavelengths and amplitudes superimposed on fractal characteristics. The profiles showed fractal dimensions from 1.1 to 1.5 covering a range of selfsimilar and self-affine curves. The analysis results suggest a negative correlation between fractal dimension,D, and amplitude,A. Joint roughness coefficients (JRC) show a positive correlation with amplitude,A, and a negative correlation with fractal dimension,D. A numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles. Smooth profiles (low JRC, highD, lowA) display many small contact regions whereas rough fractures (high JRC, lowD, highA) display few large contact areas. The agreement with published experimental data supports the suggested correlations between JRC and the fractal parameters,A andD. It is suggested that observed scale effects in JRC and joint dilation can be explained by small differential strain discontinuities across fractures, which originate at the time of fracture formation.

Natural fracture profiles, fractal dimension and joint roughness coefficients

One of the primary objectives of this work is to better understand the frictional behavior of joints under shear loads, including the creation of damage zones. Discontinuities have an important influence on the deformational behavior of rock systems. The choice of a general criterion to determine the shear strength of rough rock joints is a general problem that has been investigated for many years. Numerous shear models have been proposed in the last decades to relate shear-strength to measurable joint parameters, but their limitations have to be recognized. The problem is how to measure and then to express the roughness with a number (e.g. JRC) or a mathematical expression in order to introduce the morphology of the joint into a shear strength criterion. In the frame of this work it has been pointed out that the geometry of roughness influences the size and distribution of contact areas during shearing. In order to locate and estimate the contact area during the shearing, it was argued that only the zones of the surface faced to the shear direction, and steeper than a threshold inclination are involved in the shearing. An empirical relation between the potential contact area and the minimal apparent dip inclination of the surface is proposed. The close agreement between this empirical description of the potential contact area, and experimental points permits to predict the real contact area involved in the phenomena. A new constitutive law, relating stress and displacements, is proposed to model the shear resistance of joints under constant normal load conditions. It is based on the empirical surface description, and on the results from more than fifty constant-normal-load direct-shear tests performed on both replicas of tensile joints, and induced tensile fractures for seven rock types. It is shown that this constitutive model is able to describe experimental shear tests realized in laboratory. Moreover, the parameters required in the model can be easily obtained through standard laboratory tests. The proposed model was also used to estimate the JRC value. The expression obtained to evaluate the joint roughness coefficient is capable of predicting the JRC. It was successfully compared with JRC values obtained by back analysis of shear tests. In the current research no attention was paid to investigate the influence of the scale on the shearing. The results have been validated only in the range of the samples tested in laboratory. Further studies are needed to explore the applicability of the proposed model in field conditions.

C.J. Haas

Papers

The fractal dimension as a measure of the roughness of rock discontinuity profiles

Installation induced stresses for grouted roof bolts

The effective modulus of rock as a composite material