SUMMARYIn this paper, the non-coaxial relation between the principal plastic strain increments and the principal stresses, which results from the internal friction in geomaterials, is analyzed, and the phenomenon of the unbalanced development of plastic flow in two conjugate directions is discussed. A non-coaxial, unbalanced plastic flow model for Coulomb frictional materials is developed and used to determine the orientation of shear band in geomaterials. It is shown that the unbalanced index r of plastic flow has important effect on the orientation of the shear band, and the orientation determined by the conventional plastic flow theory is only a special case of the proposed model when r = 0. This result soundly explains the reason that the geomaterials with the same internal friction angle and dilatancy angle can have very different shear band orientations. In addition, the difference between the intrinsic and apparent dilatancy angles is analyzed, and it is emphasized that the dilatancy angle commonly used in practice is indeed the apparent dilatancy angle.

SUMMARYIn this paper, the interaction of double-periodical cracks is accurately solved based on the isolating analysis procedure, superposition principle, pseudo-traction method, Chebyshev polynomial expansion and cracksurface collocation technique. The jump displacement crossing crack faces, the average additional strain and therefore the effective compliance of the double-periodically cracked plate are directly determined. The numerical results for axial-symmetrically distributed double-periodical cracks, general double-periodical cracks with one collinear direction as well as two sets of double-periodical cracks with same size and square distribution are given in this paper. And the partial typical numerical results are compared with the previous works. The analysis shows that the anisotropy induced by the general double-periodical cracks is generally not orthogonal anisotropy. Only when the double-periodical cracks are axial-symmetrically distributed, is the anisotropy orthogonal. In this special cases, the effective engineering constants (consist of effective elastic modulus, the effective Poisson's ratio, the effective shear modulus) of cracked plate versus crack spacing, in the plane stress and plane strain conditions, respectively, are analysed.

Emergence and evolution of en echelon crack arrays play an essential role in the failure of geomaterials, regardless of the scale of deformation. This paper derives analytically an approximation for the critical crack size/spacing ratio at which a uniformly growing en echelon crack array yields to a non-uniform growing pattern of crack growth. Numerical results show that the critical crack size/spacing ratio ("a/d where a is the half crack length and d is the distance between the centres of two adjacent cracks) depends on the orientation of the crack array , the number of cracks N in the array, as well as the loading conditions. In general, decreases with N, but depends non-linearly on . When en echelon crack arrays are under far-"eld shearing along the direction of the crack array, our prediction for is about 0.5715, comparing to the experimental observation of about 0.5. For rock joints containing en echelon cracks, the interpreted critical crack size at the onset of the alternating evolution of crack growth are 0.66 and 0.89 for "26 and 163, comparing to our predictions of 0.5695 and 0.6112, respectively.

In this paper, the internal structure of shear band is investigated, and a model of the shear band with an echelon crack structure is developed. The model assumes the shear band to be composed of two conjugate sets of echelon cracks, such that the smaller echelon cracks are embedded in the space of the larger ones. The additional strain induced by the echelon cracks and the anisotropic development of the compliance tensor in the shear band zone are analyzed. The critical crack density at the onset of shear band is obtained by applying the bifurcation condition. Deviating from previous approaches, the new procedure evaluates the thickness of shear band based on the geometrical characteristics of echelon crack arrays and the failure probability of grain boundaries in the longitudinal direction at the onset of shear band. Parametric analysis shows that grain size, internal friction angle, dilation angle, and failure probability of grain boundaries are the dominant factors that account for the shear band thickness. The calculated results are consistent with the experimental data available in the literature. The model soundly explains that the measurements of the shear band thickness are generally scattered, ranging from 4 to 30 (or even more) times the grain size.