﻿ 三维裂纹前缘应力强度因子数值计算方法
 舰船科学技术  2019, Vol. 41 Issue (2): 9-13 PDF

1. 中国船舶科学研究中心 深海载人装备国家重点实验室，江苏 无锡 214082;
2. 沈阳新松机器人自动化股份有限公司 移动机器人事业部，辽宁 沈阳 110169

Numerical calculation method for stress intensity factor of 3-D crack front
HUANG Ru-xu1, YANG Xiao-long2, LU Bo1, WAN Zheng-quan1
1. State Key Laboratory of Deep-sea Manned Vehicles, China Ship Scientific Research Center, Wuxi 214082, China;
2. Business Group of Mobile Robot, SIASUN Robot and Automation Co., Ltd., Shenyang 110169, China
Abstract: This paper proposed an effective finite element method for calculating stress intensity factor of 3D crack front using 20-node singularity element and quarter point displacement method. The surface cracked plate SIF calculation has been carried out by numerical method and Newman-Raju formula. The effect of crack front mesh parameters on the SIF results of the whole surface crack front were discussed. The results indicate that the calculation result of the proposed method is stable and the relative errors are within 2%. The SIF results of the SE(B) specimen single edge crack have been calculated by the numerical method with a maximum 4.7% error compared with the analytical solution results. The method proposed herein can be used in the calculation of KI, KII, KIII for cracked structures.
Key words: stress intensity factor     3D crack     quarter point displacement method     finite element method
0 引　言

1 三维裂纹应力强度因子有限元法

 图 1 奇异单元类型及其围绕裂纹前缘形式 Fig. 1 Singular element types and form of singular elements around crack tip

Shaw和Henshell[7]以及Barsoum等[8]证明了通过将蜕化的20节点等参单元的中间节点移动至1/4位置处便可以获得 $1/\sqrt r$ 奇异性，应力强度因子由此可通过裂纹尖端点a和1/4位置处节点（记为b点）的位移来估算，称为1/4节点位移法。

 \begin{split} &\left[ {{u_i}} \right] = \left[ {\begin{array}{*{20}{c}} {{u}}\\ {{v}}\\ {{w}} \end{array}} \right] =\\ &\quad \left[ \begin{aligned} & \displaystyle\frac{{{K_I}}}{{4\mu }}\sqrt {\displaystyle\frac{r}{{2{\text{π}} }}} \left[ {\left( {2\kappa \!-\! 1} \right)\cos \displaystyle\frac{\theta }{2} - \cos \displaystyle\frac{{3\theta }}{2}} \right] \!+\!\\ &\quad \displaystyle\frac{{{K_{II}}}}{{4\mu }}\sqrt {\displaystyle\frac{r}{{2{\text{π}} }}} \left[ {\left( {2\kappa \!+\! 3} \right)\sin \displaystyle\frac{\theta }{2} \!+\! \sin \frac{{3\theta }}{2}} \right]\\ & \displaystyle\frac{{{K_I}}}{{4\mu }}\sqrt {\displaystyle\frac{r}{{2{\text{π}}}}} \left[ {\left( {2\kappa \!+\! 1} \right)\sin \displaystyle\frac{\theta }{2} \!-\! \sin \frac{{3\theta }}{2}} \right] \!-\!\\ &\quad \displaystyle\frac{{{K_{II}}}}{{4\mu }}\sqrt {\frac{r}{{2{\text{π}} }}} \left[ {\left( {2\kappa \!-\! 3} \right)\cos \frac{\theta }{2} + \cos \frac{{3\theta }}{2}} \right]\\ & \!-\! \displaystyle\frac{{\upsilon ''z}}{E}\left( {{\sigma _{xx}} \!+\! {\sigma _{yy}}} \right) \!+\! \displaystyle\frac{{2{K_{III}}}}{\mu }\sqrt {\left( {\frac{r}{{2{\text{π}} }}} \right)} \sin \frac{\theta }{2} \end{aligned} \right]{\text{，}} \end{split} (1)

3种类型应力强度因子表达式分别为：

 ${K_I} = \frac{{2\mu }}{{\kappa + 1}}\sqrt {\frac{{2{\text{π}}}}{{{r_{a - b}}}}} \left( {{{\rm{v}}_b} - {{\rm{v}}_a}} \right){\text{，}}$ (2)
 ${K_{II}} = \frac{{2\mu }}{{\kappa + 1}}\sqrt {\frac{{2{\text{π}} }}{{{r_{a - b}}}}} \left( {{{\rm{u}}_b} - {{\rm{u}}_a}} \right){\text{，}}$ (3)
 ${K_{III}} = \mu \sqrt {\frac{{\text{π}} }{{2{r_{a - b}}}}} \left( {{{\rm{w}}_b} - {w_a}} \right){\text{。}}$ (4)

2 三维裂纹体有限元模型

 图 2 含半椭圆表面裂纹平板结构有限元模型 Fig. 2 Surface cracked plate finite element model

 图 3 裂纹前缘有限元网格划分示意图 Fig. 3 Mesh around crack front

 图 4 裂纹前缘各节点局部坐标系 Fig. 4 Local coordinate of the crack front
3 网格参数对SIF计算结果的影响

 ${K_I} = Y\left( {\frac{a}{t},\frac{a}{c},\frac{c}{{2W}},\varphi } \right)\left( {{\sigma _t} + H{\sigma _b}} \right)\frac{{\sqrt {{\text{π}} a} }}{{E\left( k \right)}}{\text{，}}$ (5)

3.1 单元层数ntip对SIF计算结果的影响

 图 5 不同单元层数裂纹前缘应力强度因子分布 Fig. 5 SIFs along crack front for different ntip

3.2 单元夹角θtip对SIF计算结果的影响

 图 6 不同单元夹角裂纹前缘应力强度因子分布 Fig. 6 SIFs along crack front for different θtip

3.3 单元层渐比pr对SIF计算结果的影响

 ${p_r} = \frac{{{l_{i - 1}}}}{{{l_i}}}\left( {2 \text{≤} i \text{≤} {n_{tip}}} \right){\text{。}}$ (6)

 图 7 不同单元层渐比裂纹前缘应力强度因子分布 Fig. 7 SIFs along crack front for different pr

3.4 裂纹前缘半径rtip对SIF计算结果的影响

 图 8 不同前缘半径裂纹前缘应力强度因子分布 Fig. 8 SIFs along crack front for different rtip

4 三点弯曲试样SIF计算结果验证

 图 9 SE（B）试样图 Fig. 9 SE(B) specimen

 图 10 SEB试样模型及应力分布 Fig. 10 FE model, stress contour of the specimen

 $\begin{split} {K_I} =& \frac{F}{{t\sqrt W }}\left[ {\frac{{6{{\left( {\displaystyle\frac{a}{W}} \right)}^{1/2}}}}{{\left( {1 + 2\displaystyle\frac{a}{W}} \right){{\left( {1 - \displaystyle\frac{a}{W}} \right)}^{3/2}}}}} \right]\times\\ & \left[ {1.99 \!-\! \frac{a}{W}\left( {1 \!-\! \frac{a}{W}} \right)\left( {2.15 \!-\! 3.93\frac{a}{W} \!+\! 2.7{{\left( {\frac{a}{W}} \right)}^2}} \right)} \right] {\text{。}} \end{split}$ (7)

 图 11 裂纹前缘应力强度因子分布曲线（a=16 mm） Fig. 11 SIF of crack front for a=16 mm

 图 12 裂纹前缘中点应力强度因子计算结果 Fig. 12 SIF results of middle point of crack front
5 结　语

1）提出的20节点奇异单元裂纹网格划分方法计算裂纹前缘应力强度因子结果精度高，研究结果表明网格划分参数对应力强度因子的计算结果影响不大；

2）表面裂纹应力强度因子数值计算结果与Newman-Raju解析结果的最大误差不超过2%，具有很高的计算精度；

3）三点弯曲试样单边穿透裂纹应力强度因子数值计算结果与解析结果的最大误差为4.7%，且随着裂纹扩展，误差越来越小；

4）裂纹前缘单元层数ntip可取3～10之间的任意整数值，裂纹前缘单元夹角θtip取45°，即环绕裂纹前缘的每层单元数目为8个，裂纹前缘单元层渐比pr取1.0，裂纹前缘半径rtip可在0.05～0.50 a之间取值。

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