东北大学学报:自然科学版  2016, Vol. 37 Issue (8): 1127-1132  
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引用本文 [复制中英文]

周立明, 孟广伟, 李锋, 郭桂凯. 含裂纹复合材料的Cell-based光滑扩展有限元法[J]. 东北大学学报:自然科学版, 2016, 37(8): 1127-1132.
[复制中文]
ZHOU Li-ming , MENG Guang-wei , LI Feng , GUO Gui-kai . Cell-based Smoothed Extended Finite Element Method for Composite Materials with Cracks[J]. Journal Of Northeastern University Nature Science, 2016, 37(8): 1127-1132. DOI: 10.3969/j.issn.1005-3026.2016.08.014.
[复制英文]

基金项目

国家重大科学仪器设备开发专项(2012YQ030075);国家自然科学基金资助项目(51305157);吉林省科技厅基金资助项目(20160520064JH)

作者简介

周立明(1982-), 男, 吉林白山人,吉林大学副教授, 博士;
孟广伟(1959-), 男, 吉林长春人, 吉林大学教授, 博士生导师。

文章历史

收稿日期: 2015-03-09
含裂纹复合材料的Cell-based光滑扩展有限元法
周立明, 孟广伟, 李锋, 郭桂凯    
吉林大学 机械科学与工程学院,吉林 长春 130025
摘要: 为克服有限元法(FEM)某些固有的缺陷,提高计算精度,将Cell-Based光滑有限元法(CSFEM)与扩展有限元法(XFEM)相结合,提出光滑扩展有限元法(CS-XFEM).用该方法对含中心裂纹和斜裂纹的正交各向材料板进行模拟,并与FEM,XFEM和BXFEM (bimaterial extended finite element method)计算结果进行对比.数值算例结果表明,CS-XFEM兼具CSFEM和XFEM两者优点:单元网格与裂纹面相互独立,裂尖不必是单元节点,裂尖处网格也不需要加密,域内积分可转化为边界积分,形函数不需求导,对网格质量要求低;因此是分析断裂问题的简洁高效的数值计算方法.
关键词光滑扩展有限元法    正交各向异性    扩展有限元法    应力强度因子    
Cell-based Smoothed Extended Finite Element Method for Composite Materials with Cracks
ZHOU Li-ming, MENG Guang-wei, LI Feng, GUO Gui-kai    
School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China
Abstract: To overcome some inherent flaws and improve accuracy of the finite element method (FEM), a novel numerical method called cell-based smoothed extended finite element method (CS-XFEM) was presented. It combined the cell-based smoothed finite element method (CSFEM) and the extended finite element method (XFEM). The CS-XFEM was used to simulate an orthotropic plate containing center crack or inclined crack, and then was compared with FEM, XFEM and bimaterial extended finite element method (BXFEM). The result shows that the CS-XFEM has the advantages of both the CSFEM and XFEM: the meshes are independent to the crack surface; the end of crack needn't to be a node and the meshes around the end needn't to be fined; the CS-XFEM can transform domain integration into boundary integration, therefore, the derivatives of the shape functions are not needed and the mesh size needn't to be regular. The CS-XFEM is a simple and efficient numerical method to analyze fracture problems.
Key Words: numerical calculation    CS-XFEM    orthotropic    XFEM    stress intensity factor    

有限元是目前解决工程实际问题最有效的数值方法,但其存在某些固有的缺陷[1-2]:①求解裂纹类强间断问题需细分网格;②模拟大变形问题时网格需不断地重构;③处理夹杂问题时需沿夹杂和基体的界面处划分网格;④刚度矩阵过刚,位移解偏小等.为克服前三点困难提出了扩展有限元,为改进解的精度提出了光滑有限元.

XFEM由Belytschko等[3]提出,是目前求解含断裂问题最有效的数值方法.XFEM基于单位分解法,在位移场中引入扩展项,其计算网格独立于结构的任何内部细节点,具有计算精度高、网格划分简单等特点.Moës,Sukumar等[4-5]将该方法推广到了三维,Asadpoure等[6]利用该方法研究了正交材料中的静态裂纹问题.Esnaashari等[7]提出了求解裂纹问题的BXFEM.Motamedi等[8]在动态裂纹扩展方面进行了研究.方修君等[9]将XFEM嵌套于ABAQUS软件中,对含裂纹混凝土结构进行了研究;余天堂[10]将XFEM与线性互补法相结合,求解了裂纹面非线性接触问题.

SFEM(smoothed finite element method)是Liu等[11]将光滑应变措施引入有限元法,改进有限元法刚度结构的一种方法,具有形函数简单、对网格要求低、计算效率高等优点,现已广泛应用于各个领域[12-15].

本文基于CSFEM,结合XFEM,提出了CS-XFEM(cell-based smoothed element method),对含中心裂纹、斜裂纹的正交各向材料板进行了模拟,并与FEM,XFEM和BXFEM计算结果进行了对比.

1 复合材料断裂力学

正交各向异性材料的正轴应变与应力关系为

(1)

式中:分别为应力和应变列阵;S为材料的柔度矩阵,二维空间中为

(2)

式中E, νG分别为弹性模量、泊松比和剪切弹性模量.

图 1 含裂纹正交各向异性体 Fig.1 An arbitrary cracked orthotropic body

图 1所示,考虑一个等厚度、均匀的正交各向异性体含一条穿透裂纹的情况,满足力边界和位移边界 ,(x, y)为全局坐标,(x′, y′)为局部坐标,(r, θ)为极坐标,假定弹性主方向与参考坐标轴一致时,平面应力状态下应力函数F应满足的变形协调方程为

(3)

特征方程的根为λ1, 1, λ2, 2. 1, 2分别为λ1λ2的共轭复数,则裂纹尖端应力场和位移场的渐近解[6]如下:

I型:

(4)
(5)
(6)
(7)
(8)

II型:

(9)
(10)
(11)
(12)
(13)

式中:Re表示取实部;KIKII分别为I型和II型裂纹的应力强度因子;

(14)
(15)
2 Cell-based光滑扩展有限元法

Cell-based光滑扩展有限元法的位移模式与扩展有限元表达形式一致,即

(16)

式中:I为节点(图 2中‘•’),J为被裂纹完全贯穿单元的节点(图 2中‘□’),K为裂尖单元的节点(图 2中‘○’);NIu(x), NJa(x)和NKb(x)分别为相应节点的形函数,uI, aJbK分别为相应节点的位移;NCS-FEM, NCS-c, NCS-f分别为节点I, J, K的集合.H(x)为Heaviside函数:

(17)

式中:x*为裂纹面节点坐标;n为外法向向量.

Fl(x)为裂尖处扩展函数:

图 2 光滑域的划分 Fig.2 Division of smooth domain

图 2所示,将求解域Ω离散为Ne个四边形单元,节点个数为Nd, Ω=∪i=1NeΩie, ΩieΩje=, ij, 为空集,再将Ωei划分为nc个光滑区域,共Ns个光滑子域.

应变满足:

(19)

式中 Iu(xk), Ja(xk)和 Kb(xk)分别为相应I, J, K节点的光滑应变矩阵,可统一表示为

(20)

式中:

(21)
(22)
(23)

式中:h=x, yl=1, 2, 3, 4;Nseg为边界Γsk的个数;Ngau为每段边界高斯点的个数;wm, n为高斯权函数;nxny为积分段外法向向量的分量;xm, n为第m段边界处的第n个高斯点;Ask为第k光滑区域的面积:

(24)

离散方程为

(25)

式中:

(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)

式中:

(35)

式中:C为弹性矩阵;为体力;为面力;N为有限元形函数.

3 交互积分

考虑两种独立的平衡状态:状态1(σij(1), εij(1), ui(1))为真实物理场状态,状态2(σij(2), εij(2), ui(2))为辅助物理场状态.叠加状态1和状态2可得到另一状态的J积分[16]

(36)

式中:δ1j为克罗内克函数;A为求解域;q为任一可微函数.

整理式(36),得

(37)

式中:M(1+2)为交互积分,

(38)
(39)

式(38)可化为

(40)

式中:KI(1)KII(1)为真实场下的I型和II型应力强度; KI(2)KII(2)为辅助场下的I型和II型应力强度;

(41)
(42)
(43)

式中Im表示取虚部.

KI(2)=1, KII(2)=0,式(38)为

(44)

KI(2)=0, KII(2)=1,式(38)为

(45)
4 数值算例 4.1 算例1

含中心裂纹的正交各向异性材料板受均布载荷作用,裂纹长度为2a,单位板厚、几何构型、加载方式,以及网格划分为4 900时的情况如图 3所示.材料参数:E11=114.8 GPa,E22=11.7GPa,G12=9.66 GPa,ν12=0.21.

图 3 含中心裂纹的复合材料板几何模型和网格划分 Fig.3 Geometry and mesh generation for a composite plate with central crack

表 1给出了FEM,XFEM,CS-XFEM和BXFEM求解含中心裂纹复合材料板应力强度因子KI的结果,其中ncell为单元数.从表中可以看出CS-XFEM具有较高的计算精度,与BXFEM和XFEM所得结果十分接近,远高于FEM求解精度;也可看出,积分区域c的选取对计算结果影响不大.CS-XFEM不仅具有XFEM的优点:单元与裂纹面相互独立, 裂尖不必为单元节点,裂尖处也不需要网格加密,还具有CSFEM形函数简单、对网格要求低的特点.

表 1 含中心裂纹的复合材料板的应力强度因子KI Table 1 Stress intensity factor KI of a composite plate with central crack

图 4给出了CS-XFEM得到的应力云图,很明显地表现出了应力场的不连续性和正交特性效应,从而也说明了CS-XFEM的正确性.

4.2 算例2

含斜裂纹的正交各向异性材料板受均布载荷作用,裂纹长度为a, φ=45°,单位板厚、几何构型、加载方式和单元划分如图 5所示.材料参数:E11=0.81GPa,E22=11.84GPa,G12=0.63GPa,ν12=0.38.

表 2给出了CS-XFEM和XFEM求解应力强度因子的结果,可见两者精度基本一致,证明了CS-XFEM的正确性与有效性.图 6给出了c/a=0.3,0.4,0.5,0.6,0.7,0.8时,采用CS-XFEM计算所得的应力强度因子KIKII,可见CS-XFEM对c/a不敏感,具有较高的求解精度.

图 4 应力云图 Fig.4 Stress nephogram
图 5 含斜裂纹复合材料板几何模型和网格划分 Fig.5 Geometry and meshes geration for a composite plate with inclined crack
表 2 CS-XFEM和XFEM结果比较 Table 2 CS-XFEM and XFEM results compared
图 6 应力强度因子与c/a的关系 Fig.6 Relationship between stress intensity factors and c/a
5 结论

1) CS-XFEM的计算精度同XFEM和BXFEM精度基本相同,远高于FEM求解精度.

2) CS-XFEM兼具CSFEM和XFEM的优点.

3) CS-XFEM对c/a的取值不敏感.

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