CDM-XFEM METHOD FOR CRACK CALCULATION CONSIDERED PLASTIC DISSIPATION OF CONCRETE
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摘要: 混凝土结构在服役期间受外界载荷的影响容易产生裂缝, 导致结构刚度降低、构件承载性能衰退, 而采用准确的计算方法预测混凝土裂缝的发展是治理裂缝的基本前提, 也是保障结构安全的重要手段. 连续损伤力学方法(continuou damage method, CDM)能够描述微裂缝的扩展过程, 但不能表示离散的开裂面, 且存在网格诱导偏差及虚假应力传递的弊端, 扩展有限单元法(mechanics-extended finite element method, XFEM)能够描述宏观裂纹的扩展过程, 但不能反映微裂缝的动态扩展, 两者计算出的裂纹分布与实际差异均较大. 现有的CDM-XFEM方法已经能够模拟混凝土微裂缝及宏观裂缝发展的整个过程, 但忽略了宏观裂缝出现时混凝土产生的塑性应变, CDM与XFEM的能量转化过程欠缺平衡性. 因此, 本文重点考虑能量转化时的塑性耗散, 选取指数型函数为粘结裂缝的牵引-分离模式, 基于能量及应力等效的条件重新构建了CDM与XFEM之间的能量转化方程. 采用广义逆最小二乘法求解能量转化系数, 确定能量转化时的临界位移, 并给出了裂缝面水平集的更新算法及整体计算方法的程序流程. 以双切口混凝土受剪拉开裂试验为例, 采用多种裂缝计算方法与试验进行了对比. 结果表明, 采用考虑混凝土塑性耗散的CDM-XFEM方法算出的裂缝分布及拉力-张开位移曲线与试验结果差异最小, 说明采用考虑混凝土塑性耗散的CDM-XFEM计算方法能够更好地计算混凝土裂缝.Abstract: The concrete structure is easy to appear cracks under external loading during service, which leads to the reduction of structural stiffness and the decline of bearing capacity. Using accurate calculation method to predict the cracks development of concrete is the basic premise for crack control, and also the important measure to ensure the safety of structure. Continuous damage method (CDM) can describe the propagation process of micro cracks, but cannot represent discrete crack surface and contained disadvantages of grid induced deviation and false stress transfer, mechanics-extended finite element method (XFEM) can describe the propagation process of macro cracks, but cannot reflect the dynamic propagation of micro cracks, the cracks distribution calculated by the two methods are all quite different from the actual situation. The existing CDM-XFEM method can effectively simulate the whole process of concrete micro and macro cracks development, but the concrete plastic strain is ignored when the macro cracks appeared, so the energy conversion between CDM and XFEM is lack of balance. In this paper, the plastic dissipation of energy conversion is considered, the exponential function is selected as the traction separation mode of cohesive crack, based on the principle of energy and stress equivalence, the energy conversion equation between CDM and XFEM is reconstructed, the generalized inverse least square method is used to solve the energy conversion coefficient and determine the critical displacement during energy conversion, the updating algorithm of crack level set and overall calculation procedure are given out. Taking the shear tensile cracking experiment of concrete with double incisions as an example, various calculation methods of concrete crack are compared with the experiment. The results show that the crack distribution and tension-displacement curve calculated by CDM-XFEM method considered concrete plastic dissipation is most close to the experiment, which indicates that CDM-XFEM calculation method considered concrete plastic dissipation can better calculate concrete cracks.
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Keywords:
- metro /
- concrete structure /
- crack /
- CDM-XFEM /
- plastic dissipation
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引 言
混凝土结构是建筑、交通、水利等领域的重要基础材料, 其可塑性高、耐久性好是受广泛应用的主要原因. 但混凝土抗拉强度远低于抗压强度[1], 结构在服役期间受外界载荷的影响容易产生裂缝[2], 裂缝的出现不仅会降低结构刚度, 还为外部侵蚀介质的侵入提供了快捷通道, 从而加速内部钢筋锈蚀[3-4]、降低结构承载力. 因此, 采用准确的计算方法预测混凝土的裂缝发展是治理裂缝的基本前提, 也是保障结构安全的重要手段.
基于非线性断裂力学, 混凝土开裂大致分为以下三个过程[5-7]: (1)当材料拉伸应力接近抗拉强度时, 在断裂过程区会形成密集分布的微裂纹, 载荷−变形的关系不再具有线性斜率, 但材料的宏观响应仍然保持稳定; (2)由于载荷增大引起微裂纹的合并和交叉, 以及材料基体中骨料的脱黏, 材料的宏观响应变得不稳定, 载荷−变形的关系位于软化段中, 局部范围出现较大变形, 在应变场中形成了弱不连续性; (3)随着载荷继续增大, 断裂过程区黏聚力逐渐减小至零, 最终形成一个无黏聚应力裂纹的强不连续区域.
为模拟混凝土开裂的上述过程, 出现了较多的数值计算方法, 目前主流方法有三大类: 连续损伤力学方法(continuou damage method, CDM)、扩展有限单元法(mechanics-extended finite element method, XFEM)及两者相结合的方法(CDM-XFEM). 其中CDM方法是假定当等效应力达到抗力准则时, 开始发生初始断裂, 即在损伤单元中, 应力−应变关系会被有效应力−应变关系所取代, 因此, CDM方法只更新裂纹扩展过程中材料的本构关系, 其单元网格保持不变[8-9]. 如Vilppo等[10]从热力学的角度提出了各向异性损伤的本构模型, 分析了混凝土单轴受拉及受压情况下的破坏规律. Poliotti等[11]提出了考虑剪胀参数变化的塑性损伤模型, 分析了不同强度的混凝土受剪切作用下的受力状态. Pereira等[12]提出了一种与有效率相关的非局部损伤模型, 模拟了单切口混凝土受拉出现分叉裂缝的情况. 而XFEM方法是扩充带有不连续性质的形函数来代表计算区域内的间断, 在计算过程中, 不连续场的描述完全独立于网格边界[13-14]. 如Agathos等[15]在XFEM的基础上, 提出采用向量水平集的算法更新裂缝面的演化, 研究了三点受弯作用下含单切口混凝土的裂缝形态. Schatzer等[16]将显−隐式结合的水平集函数融入到XFEM方法中, 分析了含边缘裂口混凝土的裂缝张开度. 为提高裂缝的计算效率及稳定性, Haghani等[17] 开发了基于XFEM和α时间积分结合的混凝土裂缝扩展程序, 研究了振动作用下混凝土大坝的裂缝扩展规律.
相比而言, CDM和XFEM这两种方法各有利弊. CDM方法能够很好地描述裂缝扩展的第一个阶段, 但该方法不能描述离散的开裂面, 同时存在网格诱导偏差及虚假应力传递的弊端[18-20]. XFEM方法虽然能够很好的描述宏观裂纹的扩展, 但不能较好地描述第一阶段中密集分布的微裂纹, 计算出的裂纹分布与实际差异较大 [21]. 因此, 部分学者提出将CDM及XFEM相结合的方法, 该方法基于能量等效原理, 建立CDM与XFEM黏聚裂缝的能量转化关系, 当损伤达到临界值时实现转换 [22]. 如Comi等[23]采用CDM-XFEM方法研究了含双切口的试件受竖向拉伸作用下的开裂情况, Jin等[24]采用该方法研究了含单切口混凝土梁受弯作用下的裂缝形态, Pandey等[25]采用该方法研究了高频疲劳载荷作用下混凝土裂缝的扩展规律.
采用CDM-XFEM准确计算裂缝的关键在于实现CDM与XFEM之间能量转化的平衡性. 首先要采用合适的牵引−分离函数模型描述裂缝张开的变化模式, 如直线型[26]、双折线型[27]、指数型函数[28]等, 其次是选取合适的能量转化点, 将宏观裂缝出现时损伤模型中的剩余能量全部转移到黏结裂缝模型中. 为此, Roth等[29]假定能量转化时, 两者能量等效的分割线为垂线, 且转化点的应力相等, 以此建立了能量平衡方程并求解了牵引−分离函数. Bobinski等[30]假定开裂时仅出现不可恢复变形, 不考虑材料刚度发生退化, 并自定义能量转化时的临界位移, 从而构建了两者能量的转化方程. Wang等[31]基于弹性损伤理论, 假定材料发生开裂时弹性能全部回弹, 从而建立了对应的能量平衡方程.
然而, 相关试验研究证明[32-33], 在实际情况下混凝土出现宏观裂纹时既存在刚度退化现象, 也存在不可逆变形的特征. 因此, 本文考虑CDM与XFEM之间断裂能转化时的塑性耗散, 重新构建能量转化方程, 采用广义逆的最小二乘法求解临界位移及牵引−分离函数的系数, 最终形成考虑混凝土塑性耗散的CDM-XFEM计算方法. 通过与双切口混凝土受剪拉开裂的试验进行对比验证, 结果表明采用考虑塑性耗散CDM-XFEM方法能够有效地预测混凝土裂缝的发展规律.
1. 考虑塑性耗散的CDM-XFEM计算方法
1.1 CDM-XFEM方法的计算过程
CDM-XFEM裂缝计算方法的过程为: 当损伤因子d小于临界值时, 基于连续损伤力学描述微裂缝的发展, 当损伤因子d大于临界值dcrit时, 将损伤模型所需耗散的剩余能量转移到黏结裂缝模型中, 采用扩展有限元单元法描述宏观裂缝的张开规律, 如图1所示, 其计算过程主要包含4个部分.
1.2 混凝土受拉损伤及塑性应变的计算式
本文采用等效应变
$ \bar \varepsilon $ 作为判定混凝土出现损伤的指标, 其计算公式为[34]$$ \left. \begin{gathered} \bar \varepsilon > {r_0},\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{{\rm{then}}} \end{array}}&{D > 0} \end{array} \hfill \\ \bar \varepsilon = \sqrt {\sum\limits_{i = 1}^3 {\left( {{{\left\langle {\varepsilon _i^{{\rm{av}}}} \right\rangle }^2}{\text{ + }}{{\left( {{f_t}/{f_c}} \right)}^2} \cdot {{\left\langle { - \varepsilon _i^{{\rm{av}}}} \right\rangle }^2}} \right)} } \hfill \\ \end{gathered} \right\} $$ (1) 式中,
$ \varepsilon _i^{{\rm{av}}} $ 表示单元平均应变,$ {f_t} $ 和$ {f_c} $ 表示混凝土抗拉强度和抗压强度.受拉弹性损伤因子D的演化准则的计算式[35]为
$$ D = 1 - \sqrt {\frac{{{r_0}}}{{\bar \varepsilon }}\exp \Big[ { - R\left( {\bar \varepsilon - {r_0}} \right)} \Big]} $$ (2) 式中, R为损伤演化参数, 其计算为 [29]
$$ R = \frac{{2E{f_t}{l_{{\rm{rve}}}}}}{{2E{G_F} - {f_t}{l_{{\rm{rve}}}}}} $$ (3) 其中lrve为单元特征长度, 对于C3D8R单元, 一般取值0.02 m[36]. r0为混凝土的初始刚度, 计算式为
$$ {r_0} = \frac{{{f_t}}}{E}\qquad\qquad $$ (4) 对于单次加载−卸载后出现塑性损伤的材料, Hatzigeorgioiu等[37]根据试验结果将弹性损伤因子D修正为Dk, 则变化后的弹性模量Ek表示为
$$ {E_k} = \left( {1 - {D^k}} \right)E $$ (5) 式中, E为初始弹性模量, 则总应变ε可由弹性应变εe和塑性应变εp表示为
$$ \varepsilon = {\varepsilon ^p} + {\varepsilon ^e} = {\varepsilon ^p} + \frac{{\left( {1 - D} \right)E\varepsilon }}{{\left( {1 - {D^k}} \right)E}} $$ (6) 对于混凝土材料, Hatzigeorgioiu等[37]通过数值计算给出了最优值k = 2, 本文采用该数值, 代入式(6)可得出塑性应变的简化计算公式
$$ {\varepsilon ^p} = \frac{D}{{1{\text{ + }}D}}\varepsilon $$ (7) 1.3 黏结裂缝的牵引−分离函数形式
目前CDM-XFEM计算模型中均不考虑黏结裂缝单元的切向牵引力ts, 仅考虑黏结裂缝的法向牵引力tn, 在本文研究中同样只考虑裂缝间的法向牵引力, 其牵引−分离函数体现的是裂缝张开量与裂缝间牵引力的相互关系, 常用的牵引−分离函数主要包括: 线型、双折线型, 多项式型及指数型, 其计算为
$$ \left. \begin{gathered} {t_{n1}} = au \hfill \\ {t_{n2}} = au\;(u < {u_0})\begin{array}{*{20}{c}} {}&{{\rm{or}}} \end{array}\begin{array}{*{20}{c}} {}&{{t_{n2}} = bu\;(u > {u_0})} \end{array} \hfill \\ {t_{n3}} = a{u^3} + b{u^2} + cu + d \hfill \\ {t_{n4}} = a{{\rm{e}}^{bu}} \hfill \\ \end{gathered} \right\} $$ (8) 由式(8)可知, 第1种函数求取的参数最少, 但相关研究表明, 采用线性函数表达牵引−分离法则计算得出的裂缝与实际裂缝相差较大[38]; 第2种和第3种函数需求取的未知变量数分别为3和4, 需引入较多的边界条件才能求取相应系数[39]; 第4种采用了指数型函数来表示, 既体现了裂缝张开与牵引力之间的非线性关系, 求解的未知变量也较少, 因此, 在本文的研究中也选择指数型函数来表达黏结裂缝的牵引−分离关系.
1.4 能量转化的基本假定条件
本文假定能量转化时 (D = Dcr) CDM模型中单元的最大拉应力等于XFEM中裂缝张开至ucr时的牵引力, 即 σ(εcr) = t(ucr), 此时产生的塑性应变为εds, 损伤消耗的能量 (当
$ \bar \varepsilon $ > r0时开始出现损伤) 等于黏结裂缝张开至ucr的能量, 即图2中蓝色的面积等于图3中蓝色的面积; 同时, 损伤模型的剩余能量均全部转移到XFEM粘结裂缝模型中, 即图2中绿色和黄色的累计面积等于图3中黄色的面积, 其中‖u‖代表裂缝的总张开量.1.5 能量转化方程
根据图3所示, 基于能量等效原理, 可得出关系式(9)和式(10)
$$ {\varPsi _1} = {\varPsi _4}\qquad $$ (9) $$ {\varPsi _2}{\text{ + }}{\varPsi _3} = {\varPsi _5} $$ (10) 式(9)中, 能量Ψ1可由r0到εcr的积分面积减去能量Ψ2的面积, 即
$$ {\varPsi _1}{\text{ = }}\int_{{r_0}}^{{\varepsilon _{cr}}} {\sigma \left( {\bar \varepsilon } \right){\rm{d}}} \bar \varepsilon - {\varPsi _2} $$ (11) 根据应变等效原则[29], 即无损伤试件产生的应变ε等价于含损伤试件产生的应变
$ \bar \varepsilon $ , 则等效应力可表示为$$ \sigma \left( {\bar \varepsilon } \right) = E{\left( {1 - D} \right)^2}\bar \varepsilon $$ (12) 将式(2)、式(4)和式(12)代入到式(11)中, 则r0到εcr的积分面积可表示为
$$ \begin{split} & \int_{{r_0}}^{{\varepsilon _{{\rm{cr}}}}} {\sigma \left( {\bar \varepsilon } \right){\rm{d}}} \bar \varepsilon {\text{ = }}\int_{{r_0}}^{{\varepsilon _{{\rm{cr}}}}} {\left\{ {{l_{{\rm{rve}}}}{f_t}\left\{ {\frac{{{r_0}}}{{\bar \varepsilon }}\exp \left[ { - R\left( {\bar \varepsilon - {r_0}} \right)} \right]} \right\}} \right\}{\rm{d}}} \bar \varepsilon = \\&\qquad \exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\left( {\Gamma \left[ {R{r_0}} \right] - \Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right]} \right) \\[-12pt] \end{split} $$ (13) 式(10)中, 能量Ψ2可由为直角三角形面积表示, 即
$$ {\varPsi _2}{\text{ = }}\frac{1}{2}{l_{{\rm{rve}}}}\sigma \left( {{\varepsilon _{{\rm{cr}}}}} \right) \left( {{\varepsilon _{{\rm{cr}}}} - {\varepsilon _{{\rm{ds}}}}} \right) $$ (14) 其中,
${\varepsilon _{{\rm{cr}}}} - {\varepsilon _{{\rm{ds}}}}$ 可表示为$$ {\varepsilon _{{\rm{cr}}}} - {\varepsilon _{{\rm{ds}}}}{\text{ = }}\frac{{\sigma \left( {{\varepsilon _{{\rm{cr}}}}} \right)}}{{{E_k}}} = \frac{{E{{\left( {1 - {D_{{\rm{cr}}}}} \right)}^2}\left( {1 + {D_{{\rm{cr}}}}} \right)}}{{{D_{{\rm{cr}}}}}} $$ (15) 将式(15)代入式(14), 则能量Ψ2的计算式表示为
$$ {\varPsi _2}{\text{ = }}\frac{{f_t^2{l_{{\rm{rve}}}}}}{{2r_0^2}}\frac{{{{\left( {1 - {D_{{\rm{cr}}}}} \right)}^4}\left( {1 + {D_{{\rm{cr}}}}} \right)}}{{{D_{{\rm{cr}}}}}}{\varepsilon _{{\rm{cr}}}} $$ (16) 为简化表示, 将损伤因子Dcr表示为关于εcr的函数
$ A\left( {{\varepsilon _{{\rm{cr}}}}} \right) $ , 则式(16)为$$ \left. \begin{gathered} A\left( {{\varepsilon _{{\rm{cr}}}}} \right){\text{ = }}\sqrt {\frac{{{r_0}}}{{{\varepsilon _{{\rm{cr}}}}}}\exp \Big[ { - R\left( {{\varepsilon _{{\rm{cr}}}} - {r_0}} \right)} \Big]} \hfill \\ {\varPsi _2}{\text{ = }}\frac{{f_t^2{l_{{\rm{rve}}}}{\varepsilon _{{\rm{cr}}}}}}{{2r_0^2}} \cdot \frac{{{A^4}\left( {{\varepsilon _{{\rm{cr}}}}} \right)\Big[ {{\text{2}} - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)} \Big]}}{{1 - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)}} \hfill \\ \end{gathered} \right\} $$ (17) 将式(13)和式(17)代入到式(11), 可得到能量Ψ1的计算公式
$$ \begin{split} & {\varPsi _1}{\text{ = }}\exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\left( {\Gamma \left[ {R{r_0}} \right] - \Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right]} \right) - \\&\qquad \frac{{f_t^2{l_{{\rm{rve}}}}{\varepsilon _{{\rm{cr}}}}}}{{2r_0^2}} \cdot \frac{{{A^4}\left( {{\varepsilon _{{\rm{cr}}}}} \right)\Big[ {{\text{2}} - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)} \Big]}}{{1 - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)}} \end{split}$$ (18) 式中, Γ(x)为伽马函数, 裂缝的牵引分离函数形式采用指数型函数, 即式(8)的tn4, 则能量Ψ4可表示为
$$ {\varPsi _4}{\text{ = }}\int_0^{{u_{{\rm{cr}}}}} {t\left( u \right){\rm{d}}} u = \int_0^{{u_{{\rm{cr}}}}} {\alpha {{\rm{e}}^{ - \beta \left( u \right)}}{\rm{d}}} u = \frac{{\alpha - \alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}}}}{\beta } $$ (19) 将式(18)和式(19)代入到式(9), 则第1个能量等效方程可表示为
$$ \begin{split} & \exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\left( {\Gamma \left[ {R{r_0}} \right] - \Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right]} \right) - \hfill \\&\qquad \frac{{f_t^2{l_{{\rm{rve}}}}{\varepsilon _{{\rm{cr}}}}}}{{2r_0^2}} \cdot \frac{{{A^4}\left( {{\varepsilon _{{\rm{cr}}}}} \right)\Big[ {{\text{2}} - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)} \Big]}}{{1 - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)}}{\text{ = }}\frac{{\alpha - \alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}}}}{\beta } \end{split} $$ (20) 同理, 能量Ψ3和能量Ψ5的方程可表示为
$$ \begin{split} {\varPsi _3} = &\int_{{\varepsilon _{{\rm{cr}}}}}^\infty {\sigma \left( {\bar \varepsilon } \right){\rm{d}}} \bar \varepsilon \hfill = { \int_{{\varepsilon _{{\rm{cr}}}}}^\infty {\left\{ {{l_{\rm{rve}}}{f_t}\left\{ {\frac{{{r_0}}}{{\bar \varepsilon }}\exp \Big[ { - R\left( {\bar \varepsilon - {r_0}} \right)} \Big]} \right\}} \right\}{\rm{d}}} \bar \varepsilon } =\\& {\exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right]} \\[-12pt] \end{split} $$ (21) $$ {\varPsi _5} = \int_{{u_{{\rm{cr}}}}}^\infty {\sigma \left( u \right){\rm{d}}} u = \int_{{u_{{\rm{cr}}}}}^\infty {\alpha {{\rm{e}}^{ - \beta \left( u \right)}}{\rm{d}}} u = \frac{{\alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}}}}{\beta } $$ (22) 将式(17)、式(21)和式(22)代入到式(10), 则第2个能量等效方程可表示为
$$ \begin{split} & \frac{{f_t^2{l_{{\rm{rve}}}}{\varepsilon _{{\rm{cr}}}}}}{{2r_0^2}} \cdot \frac{{{A^4}\left( {{\varepsilon _{{\rm{cr}}}}} \right)\Big[ {{\text{2}} - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)} \Big]}}{{1 - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)}}{\text{ + }} \\&\qquad \exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right] = \frac{{\alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}}}}{\beta } \end{split} $$ (23) 根据1.4节提出的应力等效假定条件, 除了满足能量等效, 转换时的应力条件也应该相等, 即
$$ \left. \begin{array}{l} t\left( {{u_{{\rm{cr}}}}} \right) = \sigma \left( {{\varepsilon _{{\rm{cr}}}}} \right) \\ \alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}} = {f_t}{A^{\text{2}}}\left( {{\varepsilon _{{\rm{cr}}}}} \right) \end{array} \right\} $$ (24) 联立方程式(20)、式(23)及式(24), 最终求出参数α, β和ucr.
2. CDM-XFEM的数值求解算法
2.1 数值求解流程
本文采用ABAQUS软件的用户自定义扩展模块, 编写相应的用户单元子程序 UEL(user defined element). 计算流程总体分3个部分, 即
(1) 计算八节点六面体单元的应力及损伤因子, 以0.7作为临界损伤标准[29], 判定模型是否达到能量转化条件.
(2) 求解能量转化的临界位移, 计算牵引−分离函数的系数, 确定裂缝张开量.
(3) 基于主应力确定裂缝的发展方向, 即裂缝的发展方向垂直于最大主应力方向, 更新裂缝尖端及裂缝面的水平集函数, 具体如图4所示.
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图 4 考虑混凝土塑性耗散的CDM-XFEM方法计算流程Figure 4. Calculation flow of CDM-XFEM method considered concrete plastic dissipation2.2 能量转化系数求解方法
对于牵引−分离函数参数α, β和临界位移ucr, 采用广义逆的最小二乘法进行求解[40], 步骤为: 先将能量等效方程及应力等效方程转为3个待求的非线性方程, 即式(25), 然后对临界应变εcr设置初始值, 代入到式(26)和式(27)进行迭代求算, 最终求出最小范数解, 这里设置的最小范数的约束条件[29]为式(28)
$$ \left. \begin{gathered} {F_1} = \exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\left( {\Gamma \left[ {R{r_0}} \right] - \Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right]} \right) - \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}\frac{{f_t^2{l_{{\rm{rve}}}}{\varepsilon _{{\rm{cr}}}}}}{{2r_0^2}} \cdot \frac{{{A^4}\left( {{\varepsilon _{{\rm{cr}}}}} \right)\Big[ {{\text{2}} - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)} \Big]}}{{1 - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)}} - \frac{{\alpha - \alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}}}}{\beta } \hfill \\ {F_2} = \frac{{f_t^2{l_{{\rm{rve}}}}{\varepsilon _{{\rm{cr}}}}}}{{2r_0^2}} \cdot \frac{{{A^4}\left( {{\varepsilon _{{\rm{cr}}}}} \right)\Big[ {{\text{2}} - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)} \Big]}}{{1 - A\left( {{\varepsilon _{{\rm{cr}}}}} \right)}}{\text{ + }} \hfill \\ \begin{array}{*{20}{c}} {}&{} \end{array}\exp \left( {R{r_0}} \right){f_t}{l_{{\rm{rve}}}}{r_0}\Gamma \left[ {R{\varepsilon _{{\rm{cr}}}}} \right] - \frac{{\alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}}}}{\beta } \hfill \\ {F_3} = \alpha {{\rm{e}}^{ - \beta {u_{{\rm{cr}}}}}} - {f_t}{A^{\text{2}}}\left( {{\varepsilon _{{\rm{cr}}}}} \right) \hfill \\ \end{gathered} \right\} $$ (25) $$ - {\left[ {\begin{array}{*{20}{c}} {\dfrac{{\partial {F_1}}}{{\partial \alpha }}}&{\dfrac{{\partial {F_1}}}{{\partial \beta }}}&{\dfrac{{\partial {F_1}}}{{\partial {u_{cr}}}}} \\ {\dfrac{{\partial {F_2}}}{{\partial \alpha }}}&{\dfrac{{\partial {F_2}}}{{\partial \beta }}}&{\dfrac{{\partial {F_2}}}{{\partial {u_{cr}}}}} \\ {\dfrac{{\partial {F_3}}}{{\partial \alpha }}}&{\dfrac{{\partial {F_3}}}{{\partial \beta }}}&{\dfrac{{\partial {F_3}}}{{\partial {u_{cr}}}}} \end{array}} \right]^{ - 1}}\left\{ {\begin{array}{*{20}{c}} {{F_1}} \\ {{F_2}} \\ {{F_3}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} {\Delta \alpha } \\ {\Delta \beta } \\ {\Delta {u_{cr}}} \end{array}} \right\} $$ (26) $$ \left. \begin{array}{l} \alpha {\text{ = }}\alpha {\text{ + }}\Delta \alpha \\ \alpha {\text{ = }}\beta {\text{ + }}\Delta \beta \\ {u_{{\rm{cr}}}}{\text{ = }}{u_{{\rm{cr}}}}{\text{ + }}\Delta {u_{{\rm{cr}}}} \end{array} \right\} $$ (27) $$ \left\| M \right\| = \sqrt {\sum\limits_{i = 1}^n {F_i^2} } \leqslant {10^{ - {\text{8}}}} $$ (28) 2.3 裂缝尖端及裂缝面水平集更新
裂缝水平集更新的关键是将速度场从裂纹前缘扩展到整个水平集子域[41], 裂缝尖端和裂缝面水平集的速度分量如图5所示, 各个变量的关系式为
$$ V{\text{ = }}{V_\phi }{n_\phi } \times {V_\psi }{n_\psi } $$ (29) 裂缝水平集更新步骤主要如下:
(1) 对裂尖水平集进行初值化, 使其满足以下方程
$$ \frac{{\partial \psi }}{{\partial t}}{\text{ + }}{V_\psi }\left| {\nabla \psi } \right|{\text{ = }}0 \qquad\qquad$$ (30) $$ \frac{{\partial \psi }}{{\partial \tau }}{\text{ + sign}}\left( \psi \right)\left( {\left| {\nabla \psi } \right| - 1} \right){\text{ = }}0 $$ (31) (2) 定义裂缝的初始扩展速度, 即式(32), 本文θ取值45°, R取值0.005 m[42], 再将1维的速度场扩展到3维的水平集子域, 进行正交延展, 即式(33)所示
$$ \left. \begin{gathered} {V_\phi } = \frac{1}{{\Delta t}}2R \sin \Delta \theta \sin \Delta \theta \;\; \\ {V_\psi } = \frac{1}{{\Delta t}}2R \sin \Delta \theta \cos \Delta \theta \;\; \end{gathered} \right\} $$ (32) $$ \left. \begin{gathered} \nabla {V_\psi } \cdot \nabla \psi {\text{ = }}0 \;\; \\ \nabla {V_\psi } \cdot \nabla \phi {\text{ = }}0 \;\; \\ \nabla {V_\phi } \cdot \nabla \phi {\text{ = }}0 \;\; \\ \nabla {V_\phi } \cdot \nabla \psi {\text{ = }}0 \;\; \end{gathered} \right\}\qquad\qquad $$ (33) 根据Peng[43]的方法, 采用Hamilton–Jacobi方程对上述方程进行稳态求解, 使其满足方程式(34), 计算表达式为
$$ \left. \begin{gathered} \frac{{\partial {V_\psi }}}{{\partial \tau }}{\text{ + sign}}\left( \psi \right)\frac{{\nabla \psi }}{{\left| {\nabla \psi } \right|}} \cdot \nabla {V_\psi }{\text{ = }}0 \;\; \\ \frac{{\partial {V_\psi }}}{{\partial \tau }}{\text{ + sign}}\left( \phi \right)\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}} \cdot \nabla {V_\psi }{\text{ = }}0 \;\; \\ \frac{{\partial {V_\phi }}}{{\partial \tau }}{\text{ + sign}}\left( \phi \right)\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}} \cdot \nabla {V_\phi }{\text{ = }}0 \;\; \\ \frac{{\partial {V_\phi }}}{{\partial \tau }}{\text{ + sign}}\left( \psi \right)\frac{{\nabla \psi }}{{\left| {\nabla \psi } \right|}} \cdot \nabla {V_\phi }{\text{ = }}0 \;\; \end{gathered} \right\} $$ (34) (3) 修正速度分量中裂纹面水平集的速度, 修正目的是保证裂缝面内(
$ \psi < 0 $ )的$ {\overline V _\phi } $ 为0, 使已生成裂缝不受水平集更新影响, 保证裂纹前缘平滑[44], 其中$ H\left( \psi \right) $ 为阶跃函数, 修正方程为$$ {\overline V _\phi } = H\left( \psi \right)\frac{{{V_\phi }\psi }}{{{V_\psi }\Delta t}}\qquad\;\; $$ (35) $$ H\left( \psi \right){\text{ = }}\left\{ \begin{gathered} 1,\begin{array}{*{20}{c}} {}&{\psi > 0} \end{array} \hfill \\ 0,\begin{array}{*{20}{c}} {}&{{\rm{else}}} \end{array} \hfill \\ \end{gathered} \right. $$ (36) (4) 采用修正后的速度, 更新裂纹面水平集函数, 再对裂纹面水平集函数进行重初值化, 即
$$ \frac{{\partial \phi }}{{\partial t}}{\text{ + }}{\bar V_\phi }\left| {\nabla \phi } \right|{\text{ = }}0 $$ (37) $$ \frac{{\partial \phi }}{{\partial \tau }}{\text{ + sign}}\left( \phi \right)\left( {\left| {\nabla \phi } \right| - 1} \right){\text{ = }}0 $$ (38) (5) 更新裂尖水平集函数, 使其满足以下条件
$$ \frac{{\partial \psi }}{{\partial t}}{\text{ + }}{V_\psi }\left| {\nabla \psi } \right|{\text{ = }}0 $$ (39) (6) 对水平集函数进行正交化, 使其满足
$ \nabla \phi \cdot \nabla \psi {\text{ = }}0 $ , 并进行再初值化处理, 然而循环至第(2)步, 即$$ \frac{{\partial \psi }}{{\partial \tau }}{\text{ + sign}}\left( \phi \right)\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}} \cdot \nabla \psi {\text{ = }}0 $$ (40) $$ \frac{{\partial \psi }}{{\partial \tau }}{\text{ + sign}}\left( \psi \right)\left( {\left| {\nabla \psi } \right| - 1} \right){\text{ = }}0 $$ (41) 3. 应用算例
3.1 双切口混凝土受剪拉作用开裂试验
基于Nooru-Mohamed[45]的试验, 对比分析不同方法得出的混凝土受剪拉作用下的开裂情况. 如图6(a)所示, 混凝土试件的长度和高度均为200 mm, 厚度为50 mm, 切口位于混凝土两侧中心, 切口深度均为25 mm, 宽度为5 mm, 混凝土的强度为C30, 混凝土底部和右下侧部固定约束, 左上侧和顶部与钢框架连接, 加载钢框架的长度为220 mm, 宽度为120 mm, 如图6(b)所示.
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试验时先在混凝土侧部施加水平力Ph分别到5 kN, 10 kN, 27.5 kN, 然后在顶端施加拉伸力Ps, 试验时在切口位置安装了位移计测试了位移变化情况, 即CTOD, 型号为WPS拉绳式位移计, 测试精度为0.01 mm.
3.2 裂缝分布对比
图7为试验及不同计算方法得出的裂缝分布对比图, 其中各种计算方法采用的最小网格尺寸为4 mm, 可看出采用CDM方法计算得出的裂缝形态在Ph = 5 kN时出现了分叉裂缝, 与试验差别较大, 且裂缝后期的走向还受到网格依耐的影响; 采用XFEM方法计算得出的裂缝仅为一条, 且均位于左端切口附近; 采用原有的CDM-XFEM方法计算得出的裂缝条数与试验相同, 但不同载荷阶段的弯曲程度与试验差距较大, 如Ph = 27.5 kN时试验从右切口延伸出的裂缝的拐点位于50 mm的位置, 但模拟得出裂缝的拐点却位于150 mm的位置. 采用考虑塑性耗散的CDM-XFEM方法计算得出的裂缝条数及走向与试验均较为接近, 虽然裂缝的弯曲程度与试验有部分差异, 但整体发展趋势一致. 以切口部位为中轴线, 裂缝距中轴线的最大垂直距离为H, 当Ph加载到27.5 kN时, 试验、CDM方法、XFEM方法、既有的CDM-XFEM方法、考虑塑性耗散的CDM-XFEM方法算出的H值分别为46.5 mm, 76.5 mm, 69.3 mm, 58.2 mm, 48.2 mm, 由此可知, 采用考虑塑性耗散CDM-XFEM计算出的混凝土裂缝形态与试验差异最小.
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图 7 试验与各种方法算出的裂缝分布对比Figure 7. Comparison of crack distribution calculated by various methods and experiment图8为考虑混凝土塑性耗散的CDM-XFEM方法算出的不同载荷下的三维裂缝面云图, 对比图7(a)可知, 数值计算出的裂缝形态在混凝土前部和后部的裂缝走向相同, 而试验的前部和后部的裂缝走向略有差异, 这是由于混凝土的开裂形态还受到内部骨料分布、骨料形态及ITZ界面力学性能等因素的影响[47-49].
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图 8 考虑塑性耗散的CDM-XFEM计算出的三维裂缝分布云图Figure 8. 3D crack distribution calculated by CDM-XFEM considered plastic dissipation3.3 拉力与切口位移关系曲线对比
图9为侧向载荷Ph为5 kN和10 kN时, 试验与数值计算得出的混凝土切口处张开位移CTOD与竖向拉力Ps的关系曲线, 由图可知, 试验曲线出现拐点的位移分别为15 kN和12 kN, 采用CDM方法计算出的初始刚度大于试验数据, 而采用XFEM方法计算出的初始刚度远大于试验数据, 在后续的软化阶段, CDM或XFEM方法计算出的曲线均与试验差异较大, 原有的CDM-XFEM方法计算出的曲线虽然与试验走向基本一致, 但在软化阶段与试验的吻合程度不如考虑塑性耗散的CDM-XFEM方法计算的曲线. 在整个加载过程中, 试验与CDM方法、XFEM方法、原有的CDM-XFEM方法、考虑塑性耗散的CDM-XFEM方法计算得出的最大位移差值占比分别为26%, 20%, 14%, 8%, 说明采用考虑塑性耗散的CDM-XFEM方法分析混凝土的开裂规律是合理可行的.
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图 9 试验与各种方法算出的拉力-张开位移曲线对比Figure 9. Comparison of the tension-open displacement curves obtained by experiment and various calculation methods4. 结 论
本文通过考虑能量转化时的塑性耗散, 选取裂缝的牵引−分离函数模式, 重新构建了能量转化方程, 最终提出了考虑混凝土塑性耗散的CDM-XFEM裂缝计算方法. 通过与双切口混凝土受剪拉开裂试验进行对比分析, 结果表明:
(1) 采用考虑塑性耗散的CDM-XFEM计算方法能有效模拟混凝土的开裂过程, 与其他计算方法相比, 在裂缝形态方面与试验差异最小;
(2) CDM和XFEM方法计算出的拉力-张开位移曲线均与试验差异较大, 原有的CDM-XFEM方法计算出的曲线虽然与试验走向基本一致, 但在软化阶段与试验的吻合程度不如考虑塑性耗散的CDM-XFEM方法. 说明采用考虑塑性耗散的CDM-XFEM方法分析混凝土的受力开裂是合理可行的.
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图 4 考虑混凝土塑性耗散的CDM-XFEM方法计算流程
Figure 4. Calculation flow of CDM-XFEM method considered concrete plastic dissipation
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图 7 试验与各种方法算出的裂缝分布对比
Figure 7. Comparison of crack distribution calculated by various methods and experiment
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图 8 考虑塑性耗散的CDM-XFEM计算出的三维裂缝分布云图
Figure 8. 3D crack distribution calculated by CDM-XFEM considered plastic dissipation
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