THREE-POINT BENDING MODEL CONSIDERING STRAIN GRADIENT EFFECTS ——PART II: DISPLACEMENT AND ROTATION OF LOCALIZED BAND

The tensile displacement and angle of rotation of tensile strain localized band are analyzed in strain softening stage and in real crack propagation stage based on gradient-dependent plasticity. In elastic stage, the effect of strain gradient can be neglected due to uniform elastic deformation. The relation between the displacement and the rotation is determined by classical elastic theory. After the real crack is initiated in the tensile side of the three-point bending beam, using the stress equilibrium, assumption of plane cross sections and strain softening constitutive relation dependent on strain gradient, a relation between real crack length and tensile strain localized band length is presented. When the real crack just occurs, the tensile stress in the tensile side of the beam is decreased to zero and the tensile localized band length reaches its maximum. In each stage, the distance between neutral axis and tensile strain localized zone decreases monotonously. Both the tensile displacement and angle of rotation of tensile strain localized band are influenced by depth of beam, thickness of localized band, elastic modulus and softening modulus, etc. The opening displacement and angle of rotation increase with softening modulus, elastic modulus and decreasing thickness of localized band. In addition, the tensile displacement increases linearly in elastic stage and strain softening stage. However, the angle of rotation increased nonlinearly in strain softening stage and in real crack propagation stage.