工程科学与技术   2020, Vol. 52 Issue (4): 117-123

1. 江西省水利科学研究院，江西 南昌 330029;
2. 河海大学 水利水电学院，江苏 南京 210098

LIU Zhi1, ZHAO Lanhao2, WU Xiaobin1, HU Guoping1, ZHOU Qingyong1
1. Jiangxi Provincial Inst. of Water Sci., Nanchang 330029, China;
2. College of Water Conservancy and Hydropower Eng., Hohai Univ., Nanjing 210098, China

1 混凝土四参数损伤模型

 $F\left( {I_1',J_2',{{\rm{\varepsilon}} _0}} \right) = A\frac{{J_2'}}{{{{\rm{\varepsilon}} _0}}} + B\sqrt {J_2'} + C{{\rm{\varepsilon}} _1} + DI_1' = 0$ (1)

 ${{\rm{\varepsilon}} ^ * } = A\frac{{J_2'}}{{{{\rm{\varepsilon}} ^ * }}} + B\sqrt {J_2'} + C{{\rm{\varepsilon}} _1} + DI_1'$ (2)

ABCD4个参数与破坏准则使用参数相同，求解式（2），且考虑到 ${{{\varepsilon}} ^ * } \ge 0$ ，可得多轴应力状态下等效应变 ${{\rm{\varepsilon}} ^ * }$

 ${{\rm{\varepsilon}} ^ * } = \frac{{(B\sqrt {J_2'} + C{{\rm{\varepsilon}} _1} + DI_1') + \sqrt {{{(B\sqrt {J_2'} + C{{\rm{\varepsilon}} _1} + DI_1')}^{\rm{2}}}{\rm{ + 4}}AJ_{\rm{2}}'} }}{2}$ (3)

 ${\rm{\sigma}} = \left( {1 - {d_{\rm{t}}}} \right){E_{\rm{c}}}{\rm{\varepsilon}}$ (4)
 ${d_{\rm{t}}} = \left\{\!\!\!\! {\begin{array}{*{20}{l}} {1 - \dfrac{{{f_{{\rm{t}},{\rm{r}}}}}}{{{E_{\rm{c}}}{{\rm{\varepsilon }}_{t,{\rm{r}}}}}}(1.2 - 0.2{x^5}),\;x \le 1;}\\ {1 - \dfrac{{{f_{{\rm{t}},{\rm{r}}}}}}{{{E_{\rm{c}}}{{\rm{\varepsilon }}_{t,{\rm{r}}}}\left[ {{a_{\rm{t}}}{{(x - 1)}^{1.7}} + x} \right]}},\;x > 1} \end{array}} \right.$ (5)
 $x = \frac{{\rm{\varepsilon}} }{{{{\rm{\varepsilon}} _{{\rm{t}},{\rm{r}}}}}}$ (6)

 ${\rm{\sigma}} = \left( {1 - {d_{\rm{c}}}} \right){E_{\rm{c}}}{{\varepsilon}}$ (7)
 ${\;\;\;\;\;\;\;\;\;d_{\rm{t}}} = \left\{\!\!\!\! {\begin{array}{*{20}{l}} {1 - \dfrac{{n{f_{{\rm{c}},{\rm{r}}}}}}{{{E_{\rm{c}}}{{\rm{\varepsilon }}_{{\rm{c}},{\rm{r}}}}\left( {n - 1 + {x^n}} \right)}},\;x \le 1;}\\ {1 - \dfrac{{{f_{{\rm{c}},{\rm{r}}}}}}{{{E_{\rm{c}}}{{\rm{\varepsilon }}_{{\rm{c}},{\rm{r}}}}\left[ {{a_{\rm{c}}}{{(x - 1)}^2} + x} \right]}},\;x > 1} \end{array}} \right.$ (8)
 $n = \frac{{{E_{\rm{c}}}{{\rm{\varepsilon}} _{{\rm{c}},{\rm{r}}}}}}{{{E_{\rm{c}}}{{{\varepsilon}} _{{\rm{c}},{\rm{r}}}} - {f_{{\rm{c}},{\rm{r}}}}}}$ (9)
 $x = \frac{{\rm{\varepsilon}} }{{{{\rm{\varepsilon}} _{{\rm{c}},{\rm{r}}}}}}$ (10)

2 应力卸载残余应变量值

${k_{\rm{p}}}$ ${k_{{\rm{un}}}}$ 关系如图1所示。当 ${k_{{\rm{u}}{\rm{n}}}} \approx 5.23$ 时， ${k_{\rm{p}}} = {k_{{\rm{u}}{\rm{n}}}}$ ，即卸载刚度 $E = 0$ ，因此取临界值 ${k_{{\rm{u}}{\rm{n}}}} = 4.5$ ；当 ${k_{{\rm{u}}{\rm{n}}}} \ge$ $4.5$ 时， ${{\rm{\varepsilon}} _{\rm{p}}}{\rm{ = 0}}{\rm{.85}}{{\rm{\varepsilon}} _{{\rm{u}}{\rm{n}}}}$

 图1 残余应变临界值 Fig. 1 Critical value of residual strain

3 滞回规则特征点及路径

3.1 完全加卸载循环

 $\frac{{{{{\varepsilon}} _{{\rm{r}}{\rm{e}}}}}}{{{{\rm{\varepsilon}} _{\rm{r}}}}} = \frac{{{{{\varepsilon}} _{{\rm{u}}{\rm{n}}}}}}{{{{\rm{\varepsilon}} _{\rm{r}}}}} + {k_{\rm{r}}}$ (11)

 ${\;\;\;\;\;\;\;\; \rm{\sigma}} = {\xi _1}{{\rm{e}}^{{\xi _2}\left( {1 - \frac{{{\rm{\varepsilon}} - {{\rm{\varepsilon}} _{\rm{p}}}}}{{{{\rm{\varepsilon}} _{{\rm{u}}{\rm{n}}}} - {{\rm{\varepsilon}} _{\rm{p}}}}}} \right)}}{E}\left( {{\rm{\varepsilon}} - {{\rm{\varepsilon}} _{\rm{p}}}} \right)$ (12)
 ${\!\!\!\!\!\! \rm{\sigma}} = \frac{{{{\varepsilon}} - {{{\varepsilon}} _{\rm{p}}}}}{{{{{\varepsilon}} _{{\rm{r}}{\rm{e}}}} - {{\rm{\varepsilon}} _{\rm{p}}}}}{{{\sigma}} _{{\rm{r}}{\rm{e}}}}$ (13)

 $d = {d_{{\rm{u}}{\rm{n}}}} + \frac{{{d_{{\rm{r}}{\rm{e}}}} - {d_{{\rm{u}}{\rm{n}}}}}}{{{{{\varepsilon}} _{\rm{p}}} - {{{\varepsilon}} _{{\rm{u}}{\rm{n}}}}}}\left( {{{\varepsilon}} - {{{\varepsilon}} _{{\rm{u}}{\rm{n}}}}} \right)$ (14)

3.2 局部重新加载循环

 ${\;\;\;\;\;\;\;\;{\rm{\varepsilon}} _{{\rm{r}}{\rm{x}}}} = {{\rm{\varepsilon}} _{{\rm{u}}{\rm{n}}}} + \left( {{{\rm{\varepsilon}} _{{\rm{r}}{\rm{e}}}} - {{\rm{\varepsilon}} _{{\rm{u}}{\rm{n}}}}} \right){\left( {\frac{{{{\rm{\sigma}} _{{\rm{u}}{\rm{n}}}} - {{\rm{\sigma}} _{\rm{u}}}}}{{{{\rm{\sigma}} _{{\rm{u}}{\rm{n}}}}}}} \right)^{{n_{{\rm{p}}{\rm{u}}}}}}$ (15)

 ${\!\!\!\!\!\!\!\!\!\!\!\! \rm{\sigma}} = \frac{{{\rm{\varepsilon}} - {{\rm{\varepsilon}} _{\rm{u}}}}}{{{{\rm{\varepsilon}} _{{\rm{r}}{\rm{x}}}} - {{\rm{\varepsilon}} _{\rm{u}}}}}{{\rm{\sigma}} _{{\rm{r}}{\rm{x}}}}$ (16)

3.3 局部卸载循环

 ${{\rm{\varepsilon}} _{{\rm{ux}}}} = {{\rm{\varepsilon}} _{{\rm{un}}}} + \left( {{{\rm{\varepsilon}} _{{\rm{re}}}} - {{\rm{\varepsilon}} _{{\rm{un}}}}} \right){\left( {\frac{{{{\rm{\sigma}} _{\rm{x}}} - {{\rm{\sigma}} _{\rm{u}}}}}{{{{\rm{\sigma}} _{{\rm{re}}}} - {{\rm{\sigma}} _{\rm{u}}}}}} \right)^{{n_{{\rm{pr}}}}}}$ (17)

 ${{\sigma}} = {{{\eta}} _1}{{\rm{e}}^{{{{\eta}} _2}\left( {1 - \frac{{{{\varepsilon}} - {{{\varepsilon}} _{\rm{p}}}}}{{{{{\varepsilon}} _{\rm{x}}} - {{{\varepsilon}} _{\rm{p}}}}}} \right)}}E\left( {{{\varepsilon}} - {{{\varepsilon}} _{\rm{p}}}} \right)$ (18)

4 损伤模型在程序中的实现

 图5 损伤模型数值实现流程 Fig. 5 Flow chart of numerical realization of damage model

1）当 ${\rm{\varepsilon}} _n^ * \ge {\rm{\varepsilon}} _{n - 1}^ * = {\rm{\varepsilon}} _{\max }^ *$ 时，则 ${S\!_n} = 1$ ，记录 ${\rm{\varepsilon}} _{{\rm{re}}}^ *$

2）当 ${\rm{\varepsilon}} _{\rm{p}}^ * \le {\rm{\varepsilon}} _n^ * \le {\rm{\varepsilon}} _{n - 1}^ * \le {\rm{\varepsilon}} _{\max }^ *$ 时，且 ${S\!_{n - 1}} = 1$ ${S\!_{n - 1}} = 2$ ，则记录 ${S\!_n} = 2$ ${\rm{\varepsilon}} _{\rm{u}}^ * = {\rm{\varepsilon}} _n^ *$

3）当 ${\rm{\varepsilon}} _{\rm{p}}^ * \le {\rm{\varepsilon}} _{n - 1}^ * \le {\rm{\varepsilon}} _n^ * \le {\rm{\varepsilon}} _{{\rm{re}}}^ *$ 时，且 ${S\!_{n - 1}} \ne 1$ ，则记录 ${S\!_n} = 3$ ${\rm{\varepsilon}} _x^ * = {\rm{\varepsilon}} _n^ *$

4）当 ${\rm{\varepsilon }}_{\rm{p}}^ * \le {\rm{\varepsilon }}_n^ * \le {\rm{\varepsilon }}_{n - 1}^ * \le {\rm{\varepsilon }}_x^ * <{\rm{\varepsilon }}_{\max }^ *$ 时，且 ${S\!_{n - 1}} = 3$ ${S\!_{n - 1}} =$ $4$ ，则记录 ${S\!_n} = 4$ ${\rm{\varepsilon}} _{\rm{u}}^ * = {\rm{\varepsilon}} _n^ *$

5）当 ${\rm{\varepsilon}} _n^ * \le {\rm{\varepsilon}} _{n - 1}^ *$ ，且 ${\rm{\varepsilon}} _n^ * \le {\rm{\varepsilon}} _{\rm{p}}^ *$ ${\rm{\varepsilon}} _n^ * \ge {\rm{\varepsilon}} _{n - 1}^ *$ ，且 ${S\!_{n - 1}} = 5$ 时，记录 ${S\!_n} = 5$

5 算例验证 5.1 单轴循环拉伸荷载作用数值验证

 图6 等参单元2维平面示意图 Fig. 6 2–dimensional plane diagram of isoparametric elements

 图7 循环拉伸荷载下应力应变曲线对比 Fig. 7 Comparison of stress–strain curves under cyclic tensile load

 图8 循环拉伸荷载下损伤历程对比 Fig. 8 Comparison of damage history under cyclic tensile load

5.2 Koyna重力坝震况验证

 图9 Koyna重力坝示意图 Fig. 9 Diagram of Koyna gravity dam

 图10 Koyna地震波加速度时程曲线 Fig. 10 Acceleration time histories of Koyna seismic wave

 图11 不同时刻本文模型仿真Koyna坝体损伤分布 Fig. 11 Damage distribution of Koyna dam at different times in the simulation results of the model in this paper

 图12 不同时刻文献试验中Koyna坝体损伤分布[16] Fig. 12 Damage distribution of Koyna dam at different times in literature test[16]

 图13 不同时刻Koyna坝体损伤分布[17] Fig. 13 Damage distribution diagram of Koyna dam at different times[17]

6 结　语