ABSTRACT

INTRODUCTION

In 1968 joint elements were introduced in finite element analyses (1) to represent the contribution of discontinuities in the mechanical behavior of rock systems. After some initial studies it became obvious that a series of constraints would have to be introduced on the deformations and stresses in the joint elements to prevent one block from interpenetrating another, to avoid the accumulation of tensile stresses between blocks, and to limit the shear stresses along joints to tolerable values. These additions permitted sophisticated computations of stresses and strains in jointed rocks, with some numerical difficulties, and in well-posed problems valuable results have been obtained. Subsequently, workers in other fields applied similar algorithms to computation of interfaces between dissimilar materials in both static and dynamic problems. Interface constraints have been used by workers employing finite difference, discrete element, and finite element methods. Recently the authors introduced a method called "discontinuous deformation analysis" (2) for backcalculation of deformational modes in blocky rock systems when one is given the displacements of a sufficient set of points. The examples of that paper employed a more thorough approach for meeting the constraints required for nonpenetration of blocks; and the success achieved using this method in complexly jointed systems led us to generalize the equations and procedures so that they might be introduced in finite element, finite difference, and discrete element computations as well. This paper is written to present these general formulations. The problem we address is illustrated in an example in Figure 1. Heretofore, each joint was examined independtly to see if its closing, shearing and decompression were within bounds. If not, corrective forces were applied to improve predicted displacements and stresses for the next iteration. In reality, joints are coupled to other joints through the deformations, displacements and rotations of the blocks they share. Failure to recognize these couplings can lead to selection of inappropriate corrections, causing numerical oscillations. In Figure 1, over- closing of the joint on one side of a block suggests application of correcting forces (C) for the next iteration; but these may cause over- closing of the joint on the opposite side of the block, and so on. Computer programs in existence with which we are acquainted do not adequately deal with this subject. With the development of block theory (3) the primary role of blocks, as opposed to joints, has been explained. As a natural extension of this theory, we have been able to analyze the couplings of corner/corner and corner/face interpenetrations that occur in systems of blocks. Thus it has been possible to greatly reduce the number of steps needed to constrain the joints and to greatly reduce the numerical difficulties.

CORNER INTERPENETRATIONS AND THE CORNER REFERENCE DIAGRAM (CRD)

It has been found necessary and sufficient to discuss the interaction of neighboring corners. This insures that corner to face and face to face interpenetrations are covered. Examination of the corner/corner interpenetration problem produces a system of non-linear inequalities with unknown displacements. These inequalities are not mutually independent; however they can be transformed into linear inequalities yielding, in the final step, a set of linearly independent equations.

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