﻿ 复杂载荷下裂纹SIF计算的子模型技术
 舰船科学技术  2021, Vol. 43 Issue (3): 6-13    DOI: 10.3404/j.issn.1672-7649.2021.03.002 PDF

Research on the sub-modeling technique of SIF calculation of cracks under complex loads
JIANG Wei, HUANG Xiao-ping
State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: Accurate calculation of the stress intensity factors (SIF) of cracks under complex loads must be done firstly using sub-modeling technique in order to evaluate the fatigue behaviour of navel architecture and ocean engineering structures based on fracture mechanics. In view of the complicated and low efficiency problems in adopting sub-modeling technique, the Circumferentially-layering method, Transition matrix method and Crack mapping method are respectively proposed in order to improve the process of creating sub-models in Ansys based on the overall FEM model in MSC.Patran. Plug-ins MPC_arranger_V1.0, FEM_coor_transfer_V1.0 and Crack_mapper_V1.0 are respectively writen in VBA or APDL to accelerate the MPC creation between shell and solid elements, nodes location conversion between different coordinate systems and crack meshing improvement. Based on the related DNV and ABS rules, taking the fatigue hotspot of a B-type LNG fuel tank as an example, the rationality verification of the above methods was performed. The results showed that the plug-ins can greatly and reasonably improve the efficiency of sub-modeling technique. The methods proposed in this article could provide a reference for rapid realization of sub-modeling technique in accurate calculation of the SIF of cracks under complex loads.
Key words: complex loads     crack     SIF     sub-modeling technique     B-type LNG fuel tank
0 引　言

1）减少甚至取消了有限元模型中的复杂传递区域；

2）方便对局部区域进行细节分析并得到精确解；

3）可用于验证原模型的网格划分是否满足要求。

1）子模型和原整体模型相应位置处的单元属性及位置需一致；

2）切割边界只能选在壳单元、体单元内或壳体单元连接处；

3）切割边界应远离应力集中区域；

4）原整体模型相应位置处网格有必要进行足够程度的细化。

1）疲劳寿命评估中，对不同尺寸表面裂纹SIF求解需要使用三维单元[17-18]，也就意味着Ansys子模型及原Patran壳模型的对应位置都要使用体单元建模，即删除原Patran整体模型对应位置处的壳单元，用体单元重建，并用MPC连接重建的体单元和其周围壳单元，方可保证壳体间变形正确传递；

2）对于结合Patran和Ansys的子模型技术的疲劳评估，若疲劳热点较多且位置各异，仅在Ansys APDL中往复调整子模型位置与原模型一致是不明智的，此时建立子模型应优先考虑方便裂纹建立的做法，然后在Ansys外部实现坐标转换。由此可见，这种子模型技术的前处理的实现难度较高，各环节的实现低效易出错。目前较少学者采用子模型技术进行裂纹扩展分析[7]，也很少有人对有限元前处理过程进行有效优化[19]，这在很大程度上限制了子模型技术在船舶疲劳强度校核中的推广。

1 子模型技术简介

 图 1 应力强度因子计算流程 Fig. 1 The procedure of SIF calculation

2）在Ansys中建立与Patran体单元部分对应的子模型，并对裂纹进行网格划分。然后输出Patran中实体单元边界节点的位置信息，经坐标转换及插值施加为Ansys子模型的边界条件，即可实现SIF的求解。

2 壳体边界MPC的创建

 图 2 边界MPC的创建 Fig. 2 Creation of boundary MPC

2.1 平面逐周分层法

 图 3 平面逐周分层法 Fig. 3 Circumferentially-layering in plane

 $\left\{\begin{array}{l}\rho \rm{=}\sqrt{{\left({x}_{P}-{x}_{O}\right)}^{2}+{\left({y}_{P}-{y}_{O}\right)}^{2}};\\ \theta =\left\{\begin{array}{l}\left|\mathrm{arcsin}\left(\dfrac{{y}_{P}-{y}_{O}}{\rho }\right)\right|,\quad\quad\;\;\text{当}{x}_{P}>{x}_{O},{y}_{P}\geqslant {y}_{O};\\ \left|\mathrm{arcsin}\left(\dfrac{{y}_{P}-{y}_{O}}{\rho }\right)\right|+\dfrac{\text{π} }{2},\quad\text{当}{x}_{P}\leqslant {x}_{O},{y}_{P}>{y}_{O};\\ \left|\mathrm{arcsin}\left(\dfrac{{y}_{P}-{y}_{O}}{\rho }\right)\right|+\text{π} ,\;\;\;\;\;\text{当}{x}_{P}<{x}_{O},{y}_{P}\leqslant {y}_{O};\\ \left|\mathrm{arcsin}\left(\dfrac{{y}_{P}-{y}_{O}}{\rho }\right)\right|+\dfrac{3\text{π} }{2},\;\;\;\text{当}{x}_{P}\geqslant {x}_{O},{y}_{P}<{y}_{O};\end{array}\right.\\ z={z}_{P}-{z}_{O}\text{。}\end{array}\right.$ (1)

2.2 曲面逐周分层法

 图 4 圆柱面逐周分层法 Fig. 4 Circumferentially-layering on cylinder

 图 5 椭柱面逐周分层法 Fig. 5 Circumferentially-layering on elliptic cylinder

 $\left\{ \begin{gathered} {\rho _{{A_2}}}{\rm{ = }}\sqrt {{{\left( {{x_P} - {x_{{A_2}}}} \right)}^2} + {{\left( {{y_P} - {y_{{A_2}}}} \right)}^2}} \text{，} \\ {\rho _{{B_2}}}{\rm{ = }}\sqrt {{{\left( {{x_P} - {x_{{B_2}}}} \right)}^2} + {{\left( {{y_P} - {y_{{B_2}}}} \right)}^2}} \text{。} \\ \end{gathered} \right.$ (2)

 图 6 圆球面逐周分层法 Fig. 6 Circumferentially-layering on sphere

 $\left\{ \begin{gathered} R{\rm{ = }}\sqrt {{\rho ^2} + {{\left( {{z_P} - {z_O}} \right)}^2}} \text{，} \\ \phi = \arcsin \left(\frac{{{z_P} - {z_O}}}{R}\right) \text{。} \\ \end{gathered} \right.$ (3)

 图 7 椭球面逐周分层法 Fig. 7 Circumferentially-layering on ellipsoid

 $\left\{ \begin{gathered} {R_{{A_2}}}{\rm{ = }}\sqrt {{\rho _{{A_2}}}^2 + {{\left( {{z_P} - {z_{{A_2}}}} \right)}^2}} \text{，} \\ {R_{{B_2}}}{\rm{ = }}\sqrt {{\rho _{{B_2}}}^2 + {{\left( {{z_P} - {z_{{B_2}}}} \right)}^2}} \text{。} \\ \end{gathered} \right.$ (4)
2.3 注意事项

1）逐周分层是一种方便MPC创建的思想。简言之，即先按边界点周向位置进行排序，再按层向位置归类，反之亦可，本文暂时采用先逐周后分层的手段。此外，在进行周向排序和层向归类时，判据并不唯一，本文仅取其中易于实现的一类判据进行实现。

2）使用逐周分层法具有一定的前提条件。各船级社疲劳评估规范均要求将局部区域的网格细化的较为规整，一是方便热点应力的插值，二是网格过渡和单元拖拽都比较简单，三是方便Ansys中子模型的建立。所以本文所提逐周分层法暂时考虑较为规整的边界。简言之，若整体模型对应位置处网格细化较为合理，这对于大多数复杂结构并不难实现，则推荐使用本文方法，若前期网格细化过于随意，会使切割边界及子模型的建立变得复杂，也会影响本方法的使用。

3）在实际操作中，各种误差均需考虑，如曲面网格光顺程度的影响，可直接在插件中进行设置。本文仅在理论层面进行说明，暂不涉及误差分析。

3 基于矩阵法的坐标变换

1）两坐标系各平面间的相对转角其实并不直观，即便先通过计算方向向量来求得各向转角，也费诸多周折。

2）体单元中或存在成千上万的节点，若每次求解SIF均通过Ansys APDL处理坐标变换，在批量求解各尺寸裂纹的SIF时，也会增加前处理的负担。

 $\begin{split}&\left[\begin{array}{ccc}{x}_{p}& {y}_{p}& {z}_{p}\end{array}\right]\underset{transfer matrix }{\underbrace{{\left[\begin{array}{ccc}{p}_{1}& {q}_{1}& {r}_{1}\\ {p}_{2}& {q}_{2}& {r}_{2}\\ {p}_{3}& {q}_{3}& {r}_{3}\end{array}\right]}^{-1}\left[\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}\right]}}-\\ &\left[\begin{array}{ccc}{x}_{O}& {y}_{O}& {z}_{O}\end{array}\right]=\left[\begin{array}{ccc}{x}_{a}& {y}_{a}& {z}_{a}\end{array}\right]\text{。}\end{split}$ (5)

4 基于映射网格的裂纹建模

5 子模型技术的验证

 图 8 疲劳热点及子模型 Fig. 8 Fatigue hotspot and sub-model

 图 9 横浪工况 Fig. 9 Beam sea load case

 图 10 裂纹位置及构件尺寸 Fig. 10 Crack location and component size

 图 11 I型裂纹SIF Fig. 11 Mode-I SIF of crack
6 结　语

1）针对壳体单元间MPC的创建低效问题，提出更加高效的逐周分层法并编写插件MPC_arranger_V1.0进行实现。给出对平板及各典型曲面结构进行逐周分层的相关公式及逻辑流程，且成功应用于舱体疲劳热点处MPC的快速建立。相对GUI操作，一般视边界点数量可将MPC创建效率提高数百倍，具有一定推广意义。

2）针对不同坐标系间节点位置转换困难问题，提出更加直观的矩阵法并编写插件FEM_coor_transfer_V1.0进行实现，且成功应用于舱体疲劳热点位置处的节点坐标转换。

3）针对非常见构件求解裂纹SIF时自由划分方式的局限性，提出更具优势的映射划分方式并编写插件Crack_mapper_V1.0进行实现，且成功应用于舱体疲劳热点处各尺寸表面裂纹的SIF求解。

 [1] 闫小顺, 黄小平, 梁园华, 等. 波浪载荷作用下海洋结构焊趾表面裂纹的SIF计算[J]. 上海交通大学学报, 2014, 50(2): 288-293. YAN Xiao-shun, HUANG Xiao-ping, LIANG Yuan-hua, et al. Stress intensity factor calculation for surface crack at weld toe of ocean engineering structures under wave loads[J]. Journal of Shanghai Jiaotong University, 2014, 50(2): 288-293. [2] NEWMAN JR J C, RAJU I S. An empirical stress-intensity factor equation for the surface crack[J]. Engineering Fracture Mechanics, 1981, 15(1-2): 185-192. DOI:10.1016/0013-7944(81)90116-8 [3] RHEE H C, HAN S, GIPSON G S. Reliability of solution method and empirical formulas of stress intensity factors for weld toe cracks of tubular joints[C]//Offshore Mechanics and Arctic Engineering. 1991, 3: 441−452. [4] D. BOWNESS, M. M. K. Lee. Prediction of weld toe magnification factors for semi-elliptical cracks in T–butt joints[J]. International Journal of Fatigue, 2000, 22(5). [5] BS7910. Guide to methods for assessing the acceptability of flaws in metallic structures[M]. British Standards Institution, 2005. [6] VERITAS D N. Strength analysis of liquefied gas carriers with independent type B prismatic tanks, Classification notes No. 31.12[J]. DNV: Høvik, 2013. [7] SRACIC M W, ELKE W J. Effect of boundary conditions on finite element submodeling[M]//Nonlinear Dynamics, Volume 1. Springer, Cham, 2019: 163−170. [8] CURRELI C, DI PUCCIO F, MATTEI L. Application of the finite element submodeling technique in a single point contact and wear problem[J]. International Journal for Numerical Methods in Engineering, 2018, 116(10-11): 708-722. DOI:10.1002/nme.5940 [9] STRANG G, FIX G J. An analysis of the finite element method[M]. Englewood Cliffs, NJ: Prentice-hall, 1973. [10] SUN C T, MAO K M. A global-local finite element method suitable for parallel computations[J]. Computers & Structures, 1988, 29(2): 309-315. [11] CORMIER N G, SMALLWOOD B S, SINCLAIR G B, et al. Aggressive submodelling of stress concentrations[J]. International Journal for Numerical Methods in Engineering, 1999, 46(6): 889-909. DOI:10.1002/(SICI)1097-0207(19991030)46:6<889::AID-NME699>3.0.CO;2-F [12] SUN Y, ZHAI J, ZHANG Q, et al. Research of large scale mechanical structure crack growth method based on finite element parametric submodel[J]. Engineering Failure Analysis, 2019, 102: 226-236. DOI:10.1016/j.engfailanal.2019.04.012 [13] SAINT-VENANT B. Mémoire sur la torsion des prismes[J]. Mémoires des Savants étrangers, 1855, 14: 233-560. [14] MISES R V. On saint Venant's principle[J]. Bulletin of The American Mathematical Society, 1945, 51(8): 555-562. DOI:10.1090/S0002-9904-1945-08394-3 [15] 博嘉科技. 有限元分析软件—Ansys融会与贯通[M]. 北京: 中国水利水电出版社, 2002. [16] 夏伟, 胡成, 瞿尔仁. Ansys子模型分析技术在处理应力集中时的应用[J]. 工程与建设, 2006(02): 92-94. DOI:10.3969/j.issn.1673-5781.2006.02.002 [17] SHAHANI A R, HABIBI S E. Stress intensity factors in a hollow cylinder containing a circumferential semi-elliptical crack subjected to combined loading[J]. International Journal of Fatigue, 2007, 29(1): 128-140. DOI:10.1016/j.ijfatigue.2006.01.017 [18] MARENIĆ E, SKOZRIT I, TONKOVIĆ Z. On the calculation of stress intensity factors and J-integrals using the submodeling technique[J]. Journal of Pressure Vessel Technology, 2010, 132(4): 041203. DOI:10.1115/1.4001267 [19] PERIĆ M, TONKOVIĆ Z, MAKSIMOVIĆ K S, et al. Numerical analysis of residual stresses in a T-joint fillet weld using a submodeling technique[J]. FME Transactions, 2019, 47(1): 183-189. DOI:10.5937/fmet1901183P [20] Guide for building and classing liquefied gas carriers with independent tanks[Z]. ABS, 2017. [21] 季林帅. 深潜耐压圆柱壳极限承载力研究[D]. 镇江: 江苏科技大学, 2015. [22] 梁园华, 杨清峡, 闫小顺, 等. 老龄半潜式钻井平台节点疲劳裂纹扩展寿命预报[J]. 海洋工程, 2015, 33(6): 20-25. LIANG Yuan-hua, YANG Qing-xia, YAN Xiao-shun, et al. Fatigue crack growth life prediction for a welded detail on an ageing semi-submersible platform[J]. The Ocean Engineering, 2015, 33(6): 20-25. [23] 方敏. 环肋椭圆柱壳振动特性研究[D]. 武汉: 华中科技大学, 2018. [24] 薛鸿祥, 唐文勇, 曲雪, 等. 4500米级载人深潜器耐压球壳疲劳可靠性分析[J]. 船舶工程, 2013, 35(6): 112-115. XUE Hong-xiang, TANG Wen-yong, QU Xue, et al. Fatigue and reliability analysis of spherical shell for 4500m manned submersible[J]. Ship Engineering, 2013, 35(6): 112-115. [25] 祝慧钞. 潜艇耐压球柱组合壳体极限强度可靠性分析[D]. 哈尔滨: 哈尔滨工程大学, 2016. ZHU Hui-chao. Reliability analysis of ultimate strength of submarine sphere-cylinder combined pressure shell[D]. Harbin: Harbin Engineering University, 2016. [26] 姜筠. 晃荡载荷下MOSS型LNG船支撑系统强度分析[D]. 大连: 大连理工大学, 2014. [27] 孔小兵, 黄小平, 罗盼. 集装箱船纵骨端部焊趾处表面裂纹应力强度因子计算[J]. 中国造船, 2016, 57(2): 67-77. KONG Xiao-bing, HUANG Xiao-ping, LUO Pan. Calculation of stress intensity factors of surface cracks at weld toe of longitudinal stiffener joints in a container ship[J]. Shipbuilding of China, 2016, 57(2): 67-77. DOI:10.3969/j.issn.1000-4882.2016.02.008 [28] 陈景杰, 黄一, 刘刚. 基于奇异元计算分析裂纹尖端应力强度因子[J]. 中国造船, 2010, 51(3): 56-64. DOI:10.3969/j.issn.1000-4882.2010.03.007 [29] 刘帆, 黄小平. 集装箱船典型疲劳评估节点应力强度因子计算[J]. 中国造船, 2015, 56(1): 27-40. LIU Fan, HUANG Xiao-ping. Calculation of stress intensity factor in typical fatigue assessment for container ships[J]. Shipbuilding of China, 2015, 56(1): 27-40. DOI:10.3969/j.issn.1000-4882.2015.01.004