脆塑性迭代逼近算法的改进

金俊超,景来红,杨风威,宋志宇,尚朋阳

Improvement of multi-step brittle-plastic approach

Jun-chao JIN,Lai-hong JING,Feng-wei YANG,Zhi-yu SONG,Peng-yang SHANG

表 5 基于泊松效应的改进塑性位势跌落方法的合理性

Tab.5 Rationality of improved calculation method based on plastic potential theory considering Poisson’s effect

破坏类型 峰值应力 应力跌落过程的主应力增量 合理性 单轴拉伸破坏 $\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\\end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} }\\ 0 \\ 0 \\ \end{array} \right]$ $ \left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right]\left[\begin{array}{c} - \Delta \varepsilon _1^{\rm{p}} \\ v\Delta \varepsilon _1^{\rm{p}} \\ v\Delta \varepsilon _1^{\rm{p}} \\ \end{array} \right] = \left[ \begin{array}{c} - E\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ 0 \\ \end{array} \right] $ 正确。仅主拉伸方向应力发生改变与事实相符 单轴压缩破坏 $\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\\end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ \sigma _3^{\rm{p} } \\\end{array} \right]$ $\left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c} } {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array} } \right] \left[\begin{array}{c} v\Delta \varepsilon _3^{\rm{p} } \\ v\Delta \varepsilon _3^{\rm{p} } \\ - \Delta \varepsilon _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c}- \lambda \Delta \varepsilon _3^{\rm{p} } \\- \lambda \Delta \varepsilon _3^{\rm{p} }\\- \left( {\lambda +2G} \right)\Delta \varepsilon _3^{\rm{p} } \end{array} \right]$ 正确。仅主压缩方向应力发生改变，与事实相符 二向纯剪破坏 $\left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ \sigma _2^{\rm{p} } \\ \sigma _3^{\rm{p} } \\ \end{array} \right] = \left[ \begin{array}{c} \sigma _1^{\rm{p} } \\ 0 \\ - \sigma _1^{\rm{p} } \\\end{array} \right]$ $ \left[ \begin{array}{c} \Delta {\sigma _1} \\ \Delta {\sigma _2} \\ \Delta {\sigma _3} \\ \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {\lambda +2G}&\lambda &\lambda \\ \lambda &{\lambda +2G}&\lambda \\ \lambda &\lambda &{\lambda +2G} \end{array}} \right] \left[ \begin{array}{c} \left( {1 - v} \right)\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ - \left( {1 - v} \right)\Delta \varepsilon _1^{\rm{p}} \\\end{array} \right] = \left[\begin{array}{c} 2\left( {1 - v} \right)G\Delta \varepsilon _1^{\rm{p}} \\ 0 \\ - 2\left( {1 - v} \right)G\Delta \varepsilon _1^{\rm{p}} \\ \end{array} \right] $ 正确。只有1和3方向应力发生变化，与事实相符