﻿ 基于多尺度分析的混凝土随机损伤本构关系
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 同济大学学报(自然科学版)  2017, Vol. 45 Issue (9): 1249-1257.  DOI: 10.11908/j.issn.0253-374x.2017.09.001 0

### 引用本文

LIANG Shixue, LI Jie, YU Feng. A Multi-scale Analysis-based Stochastic Damage Model of Concrete[J]. Journal of Tongji University (Natural Science), 2017, 45(9): 1249-1257. DOI: 10.11908/j.issn.0253-374x.2017.09.001.

### 文章历史

1. 同济大学 土木工程学院, 上海 200092;
2. 浙江理工大学 建筑工程学院, 浙江 杭州 310018;
3. 同济大学 土木工程防灾国家重点实验室, 上海 200092

A Multi-scale Analysis-based Stochastic Damage Model of Concrete
LIANG Shixue1,2, LI Jie1,3, YU Feng
1. College of Civil Engineering, Tongji University, Shanghai 200092, China;
2. School of Civil Engineering and Architecture, Zhejiang Sci-Tech University, Hangzhou 310018, China;
3. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: The relationship between microscopic and macroscopic material properties was established by multi-scale method, that's to say, the macroscopic damage evolution could be obtained from the micro-cell simulation results. Then, two kinds of typical micro-cells generated for the macroscopic tensile and shear damage evolution. In order to apply the damage evolution from the micro-cell analysis to the engineering simulations, a pragmatic damage evolution law was put forward for the tensile and shear damage and the damage evolution parameters were obtained. A comparative study of the simulation and the experimental results testified the the proposed model whose results agree well with the experimental results.
Key words: concrete    multi-scale    stochastic damage evolution    damage law    damage model

1 多尺度随机损伤演化 1.1 多尺度损伤表示

 图 1 宏观与细观结构 Fig.1 Macro-structure and micro structure of concrete
 $\mathit{\boldsymbol{y}} = \frac{\mathit{\boldsymbol{x}}}{\lambda }$ (1)

 $\nabla {\mathit{\boldsymbol{\sigma }}^\lambda } = 0\;在\;\mathit{\Omega }\;中$ (2)
 ${\mathit{\boldsymbol{\sigma }}^\lambda }\mathit{\boldsymbol{n}} = \mathit{\boldsymbol{t}}\;在\;{\mathit{\Gamma }_{\rm{t}}}\;上$ (3)
 $\mathit{\boldsymbol{u}} = \mathit{\boldsymbol{\bar u}}\;在\;{\mathit{\Gamma }_{\rm{u}}}\;上$ (4)
 ${\mathit{\boldsymbol{\sigma }}^\lambda }\mathit{\boldsymbol{n}} = \mathit{\boldsymbol{h}}\;在\;{\mathit{\Gamma }_{\rm{c}}}\;上$ (5)

 图 2 含缺陷材料的均匀化 Fig.2 Homogenization of micro-cracked material
 $\mathit{\boldsymbol{\bar \sigma }} = \frac{1}{{{V_y}}}\oint_{\partial {\mathit{\Omega }_y}} {\left( {{\mathit{\boldsymbol{t}}^\lambda } \otimes \mathit{\boldsymbol{x}}} \right){\rm{d}}\mathit{\Omega }}$ (6)
 $\mathit{\boldsymbol{\bar \varepsilon = }}\frac{1}{{2{V_y}}}\oint_{\partial {\mathit{\Omega }_y}} {\left( {{\mathit{\boldsymbol{u}}^\lambda } \otimes \mathit{\boldsymbol{n}} + \mathit{\boldsymbol{n}} \otimes {\mathit{\boldsymbol{u}}^\lambda }} \right){\rm{d}}\mathit{\Gamma }}$ (7)

 $\left\langle {{\mathit{\boldsymbol{\sigma }}^\lambda }} \right\rangle = \frac{1}{{{V_y}}}\int_{{\mathit{\Omega }_y}} {{\mathit{\boldsymbol{\sigma }}^\lambda }{\rm{d}}\mathit{\Omega }}$ (8)
 $\left\langle {{\mathit{\boldsymbol{\varepsilon }}^\lambda }} \right\rangle = \frac{1}{{{V_y}}}\int_{{\mathit{\Omega }_y}} {{\mathit{\boldsymbol{\varepsilon }}^\lambda }{\rm{d}}\mathit{\Omega }}$ (9)

 $\mathit{\boldsymbol{\bar \sigma = }}\left\langle {{\mathit{\boldsymbol{\sigma }}^\lambda }} \right\rangle$ (10)

 $\left\langle {{\varepsilon^\lambda }} \right\rangle = \bar \varepsilon -\frac{1}{{2{V_y}}}\oint_{{\mathit{\Gamma }_{\rm{c}}}} {\left( {{\mathit{\boldsymbol{n}} } \otimes \mathit{\boldsymbol{u}}^\lambda + {\mathit{\boldsymbol{u}}^\lambda } \otimes \mathit{\boldsymbol{n}}} \right){\rm{d}}\mathit{\Gamma }}$ (11)

 ${{\bar \psi }^e} = \frac{1}{{{V_y}}}\left( {\int_{{\mathit{\Omega }_y}} {{\psi ^\lambda }{\rm{d}}\mathit{\Omega }} + \frac{1}{2}\oint_{{\mathit{\Gamma }_{\rm{c}}}} {{\mathit{\boldsymbol{u}}^\lambda }\mathit{\boldsymbol{h}}{\rm{d}}\mathit{\Gamma }} } \right)$ (12)

 ${d^ \pm } = 1 - \frac{{{{\bar \psi }^{e \pm }}}}{{\bar \psi _0^{e \pm }}}$ (13)

 $\bar \psi _0^{e \pm } = \frac{1}{2}{\varepsilon ^{e \pm }}{E_0}{\varepsilon ^{e \pm }}$ (14)

1.2 受拉损伤演化

 ${d^ + } = 1 - \frac{{{{\bar \psi }^{e + }}}}{{\bar \psi _0^{e + }}}$ (15)

1.3 受剪损伤演化

 ${d_{\rm{s}}} = 1 - \frac{{{{\bar \psi }^{{e_{\rm{s}}}}}}}{{\psi _0^{{e_{\rm{s}}}}}}$ (16)

 图 3 单向受压损伤表示 Fig.3 Shear damage under compression

 $\left\{ \begin{array}{l} {{\sigma '}_x} = \sigma {\cos ^2}\alpha \\ {{\sigma '}_y} = \sigma {\sin ^2}\alpha \\ \tau ' = \left( {\sigma \sin 2\alpha } \right)/2 \end{array} \right.$ (17)

 $\left\{ \begin{array}{l} {{\varepsilon '}_x} = \frac{1}{E}\left[ {\sigma {{\cos }^2}\alpha - \nu \sigma {{\sin }^2}\alpha } \right]\\ {{\varepsilon '}_y} = \frac{1}{E}\left[ {\sigma {{\sin }^2}\alpha - \nu \sigma {{\cos }^2}\alpha } \right] \end{array} \right.$ (18)

 $\tau = \left( {1 - {d_{\rm{s}}}} \right)G\gamma$ (19)

 $\begin{array}{l} \varepsilon = \left( {1 - \frac{{1 + v}}{2}{{\sin }^2}2\alpha } \right)\frac{\sigma }{E} + \left[ {\frac{{1 + v}}{{2\left( {1 - {d^ - }} \right)}}{{\sin }^2}2\alpha } \right]\frac{\sigma }{E} = \\ \;\;\;\;\;\left[ {1 + \left( {\frac{{{d^ - }}}{{1 - {d^ - }}}} \right)\frac{{1 + v}}{2}{{\sin }^2}2\alpha } \right]\frac{\sigma }{E} \end{array}$ (20)

 $\sigma = \left( {1 - {d^ - }} \right)E\varepsilon$ (21)

 ${d^ - } = \frac{{\left( {1 - \beta } \right){d_{\rm{s}}}}}{{1 - \beta {d_{\rm{s}}}}}$ (22)
 $\beta = 1 - \frac{{1 + v}}{2}{\sin ^2}2\alpha$ (23)

d定义为与受压应力状态对应的受剪损伤变量.由式(22) 可知：ds=0时，d=0且ds=1时，d=1，说明d满足作为损伤变量的基本条件.

2 细观单元分析

2.1 混凝土细观建模

 图 4 混凝土材料的三种描述方式 Fig.4 Three reconstruction models of concrete

 $R\left( {{\tau _1},{\tau _2}} \right) = \exp \left( { - {{\left( {\frac{{{\xi _1}}}{{{b_1}}}} \right)}^2} - {{\left( {\frac{{{\xi _2}}}{{{b_2}}}} \right)}^2}} \right)$ (24)

BAZANT和PIJAUDIER-CABOT[23]的研究指出，混凝土材料性质的相关长度可以视为其骨料对材料非局部化的影响区域，并将其定义为混凝土3倍最大骨料粒径dm.根据REN等[24]的混凝土试验级配，最大骨料粒径为dm=8 mm.据此，本文采用的相关长度[25]b1=b2=24 mm.

 图 5 目标功率谱密度 Fig.5 Target power spectral density function
 图 6 随机谐和函数建模功率谱密度 Fig.6 Reconstructed power spectral density function
 图 7 混凝土断裂能样本 Fig.7 Samples of fracture energy of concrete
2.2 单轴受拉模拟

 图 8 单轴受拉基本单元数值模型 Fig.8 Numerical model of uniaxial tension

 图 9 试件破坏模拟与试验结果 Fig.9 Tensile failure simulation and test failure mode of concrete

 图 10 单轴受拉应力应变曲线及样本均值 Fig.10 Uniaxial tensile stress-strain relationship
 图 11 样本受拉应力损伤曲线 Fig.11 Tensile damage curves
2.3 受剪模拟

 图 12 受剪破坏模拟与试验结果 Fig.12 Shear failure simulation

 图 13 受剪应力应变曲线及样本均值 Fig.13 Uniaxial tensile stress-strain relationship
 图 14 样本受剪应力损伤曲线 Fig.14 Tensile damage curves
3 实用随机损伤演化公式

 ${d^ \pm } = 1 - \frac{{{\rho _{{\rm{t}}/{\rm{c}}}}}}{{{\alpha _{{\rm{t}}/{\rm{c}}}}x_{{\rm{t}}/{\rm{c}}}^{1.1{n_{\rm{t}}}/{n_{\rm{c}}}} + {\rho _{{\rm{t}}/{\rm{c}}}}}}$ (25)
 ${\rho _{{\rm{t}}/{\rm{c}}}} = \frac{{{f_{{\rm{t}}/{\rm{c}}}}}}{{E{\varepsilon _{{\rm{t}}/{\rm{c}}}}}}$ (26)
 ${n_{{\rm{t}}/{\rm{c}}}} = \frac{{E{\varepsilon _{{\rm{t}}/{\rm{c}}}}}}{{E{\varepsilon _{{\rm{t}}/{\rm{c}}}} - {f_{{\rm{t}}/{\rm{c}}}}}}$ (27)
 ${x_{{\rm{t}}/{\rm{c}}}} = \frac{\varepsilon }{{{\varepsilon _{{\rm{t}}/{\rm{c}}}}}}$ (28)

 $\left\{ \begin{array}{l} d\left( 0 \right) = 0\;\;\;初始状态下，混凝土无损伤\\ d\left( \infty \right) = 1\;\;\;完全损伤材料的损伤为1\\ \dot d > 0\;\;\;\;\;\;\;\;损伤不可恢复 \end{array} \right.$ (29)

 ${d^ \pm }\left( 0 \right) = 1 - \frac{{{\rho _{{\rm{t}}/{\rm{c}}}}}}{{{\rho _{{\rm{t}}/{\rm{c}}}}}} = 0$ (30)
 ${d^ \pm }\left( \infty \right) = 1 - \frac{{{\rho _{{\rm{t}}/{\rm{c}}}}}}{\infty } = 1$ (31)
 $\begin{array}{*{20}{c}} {\dot d^ \pm } = \frac{{{\alpha _{{\rm{t}}/{\rm{c}}}}\left( {1.1{n_{{\rm{t/c}}}} - 1} \right){\rho _{{\rm{t}}/{\rm{c}}}}x_{{\rm{t}}/{\rm{c}}}^{\left( {1.1{n_{{\rm{t/c}}}} - 1} \right)}}}{{{{\left( {{\alpha _{{\rm{t}}/{\rm{c}}}}x_{{\rm{t}}/{\rm{c}}}^{1.1{n_{\rm{t/c}}}} + {\rho _{{\rm{t}}/{\rm{c}}}}} \right)}^2}}} > 0\\ {若\;{\alpha _{{\rm{t}}/{\rm{c}}}} \ge 0} \end{array}$ (32)

 图 15 实用受拉损伤演化曲线 Fig.15 Pragmatic tensile damage law of concrete
 图 16 实用受剪损伤演化曲线 Fig.16 Pragmatic shear damage law of concrete

 $f\left( x \right) = \left\{ \begin{array}{l} \frac{1}{{\sqrt {2{\rm{\pi }}} {\sigma _{{\alpha _{{\rm{t}}/{\rm{c}}}}}}x}}{{\rm{e}}^{ - \frac{{\left( {{\rm{In}}x - {\mu _{{\alpha _{{\rm{t}}/{\rm{c}}}}}}} \right)}}{{2\sigma _{{\alpha _{{\rm{t}}/{\rm{c}}}}}^2}}}},\;\;\;\;x > 0\\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;其他 \end{array} \right.$ (33)

 图 17 受拉损伤发展参数概率密度分布 Fig.17 Probability distribution of αt
 图 18 受剪损伤发展参数概率密度分布 Fig.18 Probability distribution of αc

 $\sigma = \left( {1 - {d^ \pm }} \right)E\left( {\varepsilon - {\varepsilon ^p}} \right)$ (34)

 $\left\{ \begin{array}{l} {\varepsilon ^{p + }} = 0\\ {\varepsilon ^{p - }} = {\xi _p}{\varepsilon ^{e - }} \end{array} \right.$ (35)
4 数值算例

 图 19 单轴拉压数值模型与边界条件 Fig.19 Numerical models and boundary conditions

 图 20 单轴受拉模拟结果 Fig.20 Simulation results of uniaxial tensile test
 图 21 单轴受压模拟结果 Fig.21 Simulation results of uniaxial compressive test
5 结论

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