THREE-POINT BENDING MODEL CONSIDERING STRAIN GRADIENT EFFECTS——PART Ⅰ: PROPAGATION OF TENSILE LOCALIZED BAND

The gradient-dependent plasticity is introduced into the crack band model to analyze the strain-softening and consequent strain localization behavior for quasi-brittle materials, such as rock, concrete and ceramics. The present tensile strain localization zone, whose thickness depends on characteristic length of quasi-brittle materials, is substituted for the previous crack band in three-point bending beam. The old crack band model is improved as the non-uniform tensile plastic strain is concentrated in the zone with certain size beyond the peak stress, which is in agreement with experimental observations. An analytical solution for tensile displacement is proposed by integrating the tensile strain. It is found that the displacement is proportional to tensile stress. In addition, the larger the width of the zone, the larger the displacement is. However, the lower elastic modulus or the softening modulus leads to larger displacement. Besides, the regularity of tensile strain localization zone propagation is presented considering the stress equilibrium condition, strain softening constitutive relation and the assumption that plane cross sections of a beam remain plane under pure bending. The results show that under the same tensile stress, the extending length increases with softening modulus; lower depth of the rock beam or elastic modulus leads to larger length. The tensile strength has no influence on the maximum of extending length. Moreover, the evolution regularity of the depth of elastic tensile region is studied.