考虑应变梯度效应的三点弯梁模型解析研究——第一部分:局部化带的传播

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

为分析应变软化和由此带来的应变局部化问题,将梯度塑性理论引入裂纹带模型。以拉应变局部化区域代替裂纹带,在三点弯梁裂纹带(具有一定尺寸的带宽由特征长度确定)内部存在着不均匀分布的拉应变,这与实验结果相符。对拉应变进行积分,得到了拉应变局部化区域的张拉位移的理论表达式,结果表明:该位移与拉应力成线性规律,拉应变局部化区域的宽度越大,弹性模量越小或降模量越小,则该位移越大。此外,采用应力平衡条件、应变软化的本构关系及平截面假定,还得到了拉应变局部化区域的扩展规律,结果表明:下降模量越大、三点弯梁高度越小及弹性模量越小,则在相同的拉应力的情况下,拉应变局部化区扩展的长度越小;抗拉强度对拉应变局部化区扩展长度的最大值没有影响。此外,还研究了梁中部横截面内中性轴到具有最大承载能力的点的距离的变化规律。

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.