﻿ 混凝土随机损伤本构关系工程参数标定与应用
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 同济大学学报(自然科学版)  2017, Vol. 45 Issue (8): 1099-1107.  DOI: 10.11908/j.issn.0253-374x.2017.08.001 0

### 引用本文

LI Jie, FENG Decheng, REN Xiaodan, WAN Zengyong. Calibration and Application of Concrete Stochastic Damage Model[J]. Journal of Tongji University (Natural Science), 2017, 45(8): 1099-1107. DOI: 10.11908/j.issn.0253-374x.2017.08.001.

### 文章历史

1. 同济大学 土木工程学院，上海 200092;
2. 东南大学 土木工程学院，江苏 南京 210096

Calibration and Application of Concrete Stochastic Damage Model
LI Jie1, FENG Decheng1,2, REN Xiaodan1, WAN Zengyong1
1. College of Civil Engineering, Tongji University, Shanghai 200092, China;
2. College of Civil Engineering, Southeast University, Nanjing 210096, China
Abstract: The stochastic damage model for concrete is introduced in the paper. The fundamental parameters of the model are calibrated through the concrete constitutive model in the Chinese design code. In order to make the model convenient for engineering applications, some modifications are made to account for the complicated interaction effects between reinforcement steels and concrete in realistic structures. Then the model is implemented into the structural analysis software ABAQUS and OpenSees through user subroutines, and used to simulate different kinds of structural member experiments. The results indicate that the proposed model can reflect the mechanical behaviors of concrete and reinforced concrete members and structures, and it offers an effective way for nonlinear analysis of structures in engineering.
Key words: concrete    damage    randomness    parameter calibration    steel-concrete interaction

1 混凝土随机损伤本构关系 1.1 混凝土细观随机断裂模型

 图 1 并联弹簧系统 Fig.1 Parallel element system

 ${D^ \pm }{\rm{ = }}\frac{{A_D^ \pm }}{{{A^ \pm }}}$ (1)

 ${D^ \pm }\left( {{\varepsilon ^{{\rm{e}} \pm }}} \right) = \frac{1}{N}\sum\limits_{i = 1}^N {H\left( {{\varepsilon ^{{\rm{e}} \pm }} - \Delta _i^ \pm } \right)}$ (2)

 $H\left( x \right) = \left\{ \begin{array}{l} 0,x \le 0\\ 1,x > 0 \end{array} \right.$ (3)

 ${D^ \pm }\left( {{\varepsilon ^{{\rm{e}} \pm }}} \right) = \int_0^1 {H\left[ {{\varepsilon ^{{\rm {e}} \pm }} - {\Delta ^ \pm }\left( x \right)} \right]} {\rm{d}}x$ (4)

 ${d^ \pm }\left( {{\varepsilon ^{{\rm{e}} \pm }}} \right) - E\left\{ {\int_0^1 {H\left[ {{\varepsilon ^{{\rm{e}} \pm }} - {\mathit{\Delta }^ \pm }\left( x \right)} \right]{\rm{d}}x} } \right\}$ (5)

 ${d^ \pm } = \int_0^{{\varepsilon ^{{\rm{e}} \pm }}} {\frac{1}{{\sqrt {2\pi } {\zeta ^ \pm }}}} \exp \left( { - \frac{{{z^ \pm }}}{2}} \right){\rm{d}}{\varepsilon ^{{\rm{e}} \pm }}$ (6)

 ${z^ \pm } = \frac{{\ln {\varepsilon ^{{\rm{e}} \pm }} - {\lambda ^ \pm }}}{{{\zeta ^ \pm }}}$ (7)

 ${\sigma ^ \pm } = \left( {1 - {d^ \pm }} \right){E_c}{\varepsilon ^{{\rm{e}} \pm }}$ (8)

1.2 塑性变形的考量

 $\left\{ \begin{array}{l} {\varepsilon ^{{\rm{p}} + }} = 0\\ {\varepsilon ^{{\rm{p}} - }} = {\left( {\frac{{\xi _{\rm{p}}^ - {d^ - }}}{{1 - {d^ - }}}} \right)^ {n_{\rm{p}}^ - }}{\varepsilon ^{{\rm{e}} - }} \end{array} \right.$ (9)

 图 2 经验塑性变形 Fig.2 Emparical plastic strain
1.3 一维本构关系到多维本构关系的拓展

 $\mathit{\boldsymbol{\sigma }} = \left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{D}}} \right): \mathit{\boldsymbol{\bar \sigma }}\\ = (\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{D}}) : \mathit{\boldsymbol{C}}_0 : (\mathit {\boldsymbol{\varepsilon}} - \mathit{\boldsymbol{\varepsilon}}^{\rm{P}})$ (10)

 $\mathit{\boldsymbol{D}} = {d^ + }{\mathit{\boldsymbol{P}}^ + } + {d^ - }{\mathit{\boldsymbol{P}}^ - }$ (11)

 $\left\{ \begin{array}{l} {Y^ + } = \frac{1}{2}{{\mathit{\boldsymbol{\bar \sigma }}}^ + }:{\mathit{\boldsymbol{C}}_0}:\mathit {\boldsymbol{\bar \sigma }}\\ {Y^ - } = \frac{1}{{2{b_0}}}{\left( {\alpha \mathit{\boldsymbol{\bar I}} + \sqrt {3{{\mathit {\boldsymbol{\bar J}}}_2}} } \right)^2} \end{array} \right.$ (12)

 $\left\{ \begin{array}{l} \varepsilon _{{\rm{eq}}}^{{\rm{e + }}}{\rm{ = }}\sqrt {\frac{{2{Y^ + }}}{{{E_0}}}} \\ \varepsilon _{{\rm{eq}}}^{{\rm{e - }}}{\rm{ = }}\frac{1}{{\alpha - 1}}\sqrt {\frac{{2{Y^ - }}} {{{b_0}}}} \end{array} \right.$ (13)

 ${D^ \pm }\left( {\varepsilon _{{\rm{eq}}}^{{\rm{e}} \pm }} \right){\rm{ = }}\int_0^1 {H\left[ {\varepsilon _{{\rm{eq}}}^{{\rm{e}} \pm } - {\mathit{\Delta }^ \pm }\left( x \right)} \right]{\rm {d}}x}$ (14)

 ${d^ \pm }\left( {\varepsilon _{{\rm{eq}}}^{{\rm{e}} \pm }} \right){\rm{ = }}\mathit{E}\left\{ {\int_0^1 {H\left[ {\varepsilon _{{\rm{eq}}}^{{\rm{e}} \pm } - {\mathit{\Delta }^ \pm }\left( x \right)} \right]{\rm{d}}x} } \right\}$ (15)

2 细观随机断裂模型参数的工程标定

 ${\sigma ^ \pm }{\rm{ = }}\left( {1 - {d^ \pm }} \right){E_c}\varepsilon _{{\rm{eq}}}^{{\rm{e}} \pm }$ (16)

 $\frac{{\partial {\sigma ^ \pm }}}{{\partial {\varepsilon ^ \pm }}} = \frac{{\partial {\sigma ^ \pm }}}{{\partial {\varepsilon ^{{\rm{e}} \pm }}}}\frac{{\partial {\varepsilon ^{{\rm{e}} \pm }}}} {{\partial {\varepsilon ^ \pm }}}{\rm{ = }}{E_c}\left[ {1 - {d^ \pm } - {\varepsilon ^{{\rm{e}} \pm }}f\left( {{\varepsilon ^{{\rm{e}} \pm }}} \right)} \right]\frac{{\partial {\varepsilon ^{{\rm{e}} \pm }}}}{{\partial {\varepsilon ^ \pm }}}$ (17)

 ${f_{{\rm{t/c}}}}{\rm{ = }}\left( {1 - {d^ \pm }} \right){E_{\rm{c}}}\varepsilon _{{\rm{t/c}}}^{\rm {e}}$ (18)

 $1 - {d^ \pm } - \varepsilon _{{\rm{t/c}}}^{\rm{e}}f\left( {\varepsilon _{{\rm{t/c}}}^{\rm{e}}} \right) = 0$ (19)

 $\left\{ \begin{array}{l} {\zeta ^ \pm } = \frac{{{E_{\rm{c}}}\varepsilon _{{\rm{t/c}}}^{\rm{e}}}}{{\sqrt {2\pi } {f_{{\rm {t}}/{\rm{c}}}}}}\exp \left\{ { - \frac{1}{2}{{\left[ {{\mathit{\Phi }^{ - 1}}\left( {1 - \frac{{{f_ {{\rm{t/c}}}}}}{{{E_{\rm{c}}}\varepsilon _{{\rm{t/c}}}^{\rm{e}}}}} \right)} \right]}^2}} \right\}\\ {\lambda ^ \pm }{\rm{ = }}\ln \varepsilon _{{\rm{t/c}}}^{\rm{e}} - {\zeta ^ \pm }{\mathit{\Phi }^{ - 1}}\left( {1 - \frac{{{f_{{\rm{t/c}}}}}}{{{E_{\rm{c}}}\varepsilon _{{\rm{t/c}}}^{\rm{e}}}}} \right) \end{array} \right.$ (20)

 图 3 均值损伤演化 Fig.3 Mean value of damage evolution

 图 4 细观随机断裂模型与规范本构模型对比 Fig.4 Comparison between stochastic fracture model and code model

3 钢筋-混凝土复合效应的考虑

3.1 受拉刚化效应

 $\frac{{{\sigma ^ + }}}{{{f_{\rm{t}}}}}{\rm{ = }}\left( {1 - \theta } \right)\exp \left[ {\frac {{270}}{{\sqrt \theta }}\left( {{\varepsilon _{\rm{t}}} - {\varepsilon ^ + }} \right)} \right] + \theta$ (21)

 $d_{{\rm{ts}}}^ + {\rm{ = }}1 - \frac{{{\varepsilon _{\rm{t}}}}}{{{\varepsilon ^ + }}}\left[ {\left( {1 - \theta } \right)\left( {1 - {d^ + }} \right) + \theta } \right]$ (22)

3.2 箍筋约束效应

 $d_{{\rm{con}}}^ - {\rm{ = }}\int_0^1 {H\left[ {\gamma {\varepsilon ^{e - }} - {\mathit{\Delta }^ - }\left( x \right)} \right]} {\rm{d}}x$ (23)

 $\gamma {\rm{ = }}1 - {\left( {\frac{{0.87\varepsilon _{\rm{p}}^ - }}{{\varepsilon _{\rm{p}}^ - + 1\;000\beta }}} \right)^\alpha }$ (24)

 \left\{ \begin{align} & \alpha =-0.253f_{1}^{'}+0.862 \\ & \beta =0.713f_{1}^{'}+4.14 \\ \end{align} \right. (25)

 图 5 箍筋约束混凝土应力-应变曲线 Fig.5 Stress-strain curves of confined concrete
3.3 正交软化效应

 $\mathit{\boldsymbol{\sigma = }}\left( {1 - {d^ + }} \right){\mathit{\boldsymbol{\bar \sigma }}^ + } + \beta \left( {1 - {d^ - }} \right){\mathit{\boldsymbol{\bar \sigma }}^ - }$ (26)

 $\beta = \sqrt {\frac{1}{{1 + 400\varepsilon _{{\rm{eq}}}^ + }}}$ (27)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\sigma }}{\rm{ = }}\left( {\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{D}}^s}} \right):{\mathit{\boldsymbol{C}}_0}:{\varepsilon ^{\rm{e}}}\\ {\mathit{\boldsymbol{D}}^s}{\rm{ = }}{d^ + }{\mathit{\boldsymbol{P}}^ + } + \left[ {1 - \beta \left( {1 - {d^ - }} \right)} \right]{\mathit{\boldsymbol{P}}^ - } \end{array} \right.$ (28)
4 随机损伤本构模型的工程应用

4.1 钢筋混凝土柱反复加载试验

 图 6 文献[27]试件BG-8 Fig.6 Specimen BG-8 of literature [27]
 图 7 柱底剪力V-柱顶位移Δ曲线 Fig.7 Load-top displacement curve
4.2 钢筋混凝土梁倒塌试验

 图 8 倒塌试验示意图(单位:mm) Fig.8 Collapse test setup(unit: mm)
 图 9 中柱荷载位移曲线 Fig.9 Loaddisplacement curve of middle column
4.3 钢筋混凝土双向板加载试验

 图 10 钢筋混凝土双向板 Fig.10 Reinforced concrete two-way slab

 图 11 受拉损伤分布 Fig.11 Distribution of tensile damage
 图 12 荷载位移曲线 Fig.12 Loaddisplacement curve
4.4 钢筋混凝土剪力墙反复加载试验

 图 13 剪力墙几何与配筋信息(单位：mm) Fig.13 Geometric and reinforce information of the shear wall(unit: mm)

 图 14 剪力墙荷载位移曲线 Fig.14 Loaddisplacement curve of the shear wall
5 结论

(1) 混凝土随机损伤本构关系具有客观的物理基础和完整的理论基础，可以科学地反映混凝土的受力力学行为；利用我国混凝土设计规范规定的本构关系标定了混凝土随机损伤本构模型的基本参数，所得结果可应用于实际工程；

(2) 通过修正素混凝土损伤本构模型，可以考虑实际结构中复杂的钢筋混凝土相互作用效应；

(3) 通过与不同类型试验结果的对比，表明混凝土随机损伤本构模型可以理想地模拟混凝土结构和钢筋混凝土结构的受力力学行为，具有实用性和可靠性.

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