Complete stress-strain curve of rock specimen in uniaxial compression was derived analytically Using displacement method and energy conservation method, respectively. Phenomenon of shear strain localization initiated just at peak stress was considered in the form of a single shear band intersecting two lateral edges of specimen. Consistent with experimental observations at least qualitatively, the size of shear band and the distributed plastic shear strain in the band were described by gradient-dependent plasticity where an internal length parameter reflects the heterogeneity of rock. For displacement method, the total axial deformation of Specimen was decomposed into elastic and plastic parts. The latter was concerned with shear band thickness, Softening modulus and shear band inclination. For energy conservation method, the adopted assumption was that energy consumed by shear band was equal to the work done by axial stress. It is found that the resulting analytical expressions for complete stress-strain curve according to two kinds of methods are identical; however, the former is even more concise. Comparisons of previous experimental results regarding length effects and the present analytical solutions were presented to check the validation of the present model. Parametric studies show that geometrical size of specimen and constitutive parameters as well as shear band inclination influence the post-peak response.
Experimental results show that failure of rock must localize and thus takes place in a small zone of a specimen. The narrow zone is usually called localized band in which intense and distributed strains are concentrated. Moreover, the thickness of band is found to be dependent on the average grain diameter (Muehlhaus & Vardoulakis 1987) or maximum aggregate diameter (Baźant & Pijaudier-Cabot 1989). According to the different failure mechanisms, localized band can be divided into shear and tensile localized bands.
More attention has been focused on the complete stress-strain curves of quasi-brittle materials in uniaxial tension and compression, for they are especially Important for understanding of constitutive relations of quasi-brittle materials, load-carrying capacity, failure mechanisms, size effects and instability. Experimental tests show that the complete stress-strain curve of a specimen subjected to compressive stress in the axial direction is composed of two parts: ascending and descending branches. According to the different signs of the slope of the post-peak stress-strain diagrarn (Wawersik & Fairhurst 1970), the curves can be classified into Class I corresponding to negative slope and Class II corresponding to positive slope (so-called snap-back phenomenon). Due to the fact that deformation of specimen beyond the peak stress is localized into some narrow zones, it is recognized that the measured slope of softening branch of complete stress-strain curve cannot be seen as a constitutive parameter. In fact, it belongs to a kind of responses of structure composed of localized band and elastic body outside the band.
Recently, a great progress has been achieved on numerical aspect of obtaining the complete stress-strain curves. To avoid the pathological numerical results based on traditional elastoplastic theory, generalizations or modifications from the classical elastoplastic theory must be carried out.