﻿ 低周疲劳下加筋板裂纹扩展行为有限元分析
 舰船科学技术  2023, Vol. 45 Issue (7): 10-17    DOI: 10.3404/j.issn.1672-7649.2023.07.003 PDF

1. 高性能船舶技术教育部重点实验室(武汉理工大学)，湖北 武汉 430063;
2. 武汉理工大学 船舶与能动学院，湖北 武汉 430063

Finite element analysis of crack growth of stiffened plates under low cycle fatigue
JIN Si-yuan2, YANG Ping1,2, HU Kang2, PENG Zi-ya2
1. Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China;
2. Depts. of Naval Architecture, Ocean and Structural Engineering, School of Naval Architecture and Power, Wuhan University of Technology, Wuhan 430063, China
Abstract: In order to study low -cut fatigue load under the hull and tendon increase plate crack expansion, use node release method to study the expansion of low -we weekly fatigue cracking expansion under different working conditions, explore the height of the gluten and spacing to the crack expansion process. The influence of the field, accumulating plastic strain, and crack closed effect. The results showed that the increase in the height of the gluten bar can enhance the residual pressure stress field, resulting in an enhancement of the closed effect, and the cumulative plastic strain at the tip of the cracks also decreased, thereby reducing the fissure expansion rate. The increase in the spacing of the gluten strip has reduced the structure stiffness near the crack, the closed effect weakens, and the fissure expansion rate increases.
Key words: stiffened plate     low cycle fatigue     crack growth
0 概　述

1 理论分析 1.1 应力和位移场

 图 1 加筋板理论简化模型 Fig. 1 Simplified theoretical model of stiffened plate

 ${v_i}^a = \frac{\sigma }{E}2{y_i} + \frac{2}{E}\int_0^a {{K_{2a}}} \frac{{\partial {K_{2F}}}}{{\partial F}}{\rm{d}}a 。$ (1)

 ${K_{2F}} = \frac{F}{{\sqrt {\text{π} a} }}\left( {1 - \alpha {y_i}\frac{\partial }{{\partial {y_i}}}} \right)\left( {\frac{a}{{\sqrt {{a^2} - {\textit{z}}_i^2} }} + \frac{a}{{\sqrt {{a^2} - \bar {\textit{z}}_i^2} }}} \right) 。$ (2)

 $v_i^b = \sum\limits_{j = 1}^{2n} {\left( {\delta _{ij,0}^b + \frac{2}{E}\int_0^a {{K_{2Qj}}\frac{{\partial {K_{2F}}}}{{\partial F}}{\rm{d}}a} } \right)} 。$ (3)

 ${K_{2Qj}} = \frac{{{Q_j}}}{{\sqrt {\text{π} a} }}\left( {1 - \alpha {y_i}\frac{\partial }{{\partial {y_i}}}} \right)\left( {\frac{a}{{\sqrt {{a^2} - {\textit{z}}_i^2} }} + \frac{a}{{\sqrt {{a^2} - \bar {\textit{z}}_i^2} }}} \right)。$ (4)
1.2 累积塑性应变理论

 $\dot D = {\left( {\frac{{{\sigma ^2}{R_V}}}{{2Eb{{\left( {1 - D} \right)}^2}}}} \right)^{{S_0}}} \cdot \dot \varepsilon _{{p}}^{{{acc}}} ，$ (5)

 $\dot \varepsilon _{{p}}^{{{acc}}}{\text{ = }}\frac{{m{\sigma ^{m - 1}}\dot \sigma }}{{{K^m}{{\left( {1 - D} \right)}^{2m}}}} ，$ (6)

 ${k_y} = d{\left( {\frac{{{t_w}}}{t}} \right)^3} + 1 。$ (7)

 $_{ } 角钢d=0.98-0.14\left(L/W\right) ，$ (8)
 $扁钢d=0.\text{12}-0.\text{02}\left(L/W\right) ，$ (9)
 $\text{T}型材\left\{\begin{array}{l}d=0.8\left(1.5\leqslant L/W\leqslant3\right)，\\ d=1-0.133\left(L/W\right)\left(3\leqslant L/W\leqslant 5\right)。\end{array} \right.$ (10)

 $\dot D = {\left( {\frac{{{\sigma ^2}{R_V}}}{{2Eb{{\left( {1 - D} \right)}^2}}}} \right)^{{S_0}}} \cdot \frac{{m{\sigma ^{m - 1}}\dot \sigma }}{{{k_{\text{y}}}{K^m}{{\left( {1 - D} \right)}^{2m}}}} ，$ (11)

 $D{\text{ = }}1 - {\left( {1 - \frac{N}{{{N_{\text{F}}}}}} \right)^{\tfrac{1}{{2{\text{m}} + 2{S_0} + 1}}}}。$ (12)

 $\begin{split} \varepsilon _p^{acc} =& {\varepsilon _F} - \left( {{\varepsilon _F} - {\varepsilon _0}} \right){\left( {1 - \frac{N}{{{N_F}}}} \right)^{\frac{{2{S_0} + 1}}{{2m + 2{S_0} + 1}}}}= {\varepsilon _{\text{F}}} - \\ & \frac{{\left( {\sigma _{{\text{max}}}^m - \sigma _{{\text{min}}}^m} \right)\left( {2m + 2{S_0} + 1} \right){N_F}}}{{k_y^m{K^m}\left( {2{S_0} + 1} \right)}}{\left( {1 - \frac{N}{{{N_F}}}} \right)^{\frac{{2{S_0} + 1}}{{2m + 2{S_0} + 1}}}} \end{split}。$ (13)

1.3 裂纹扩展及闭合效应

Elber于1971年提出了裂纹闭合理论，采用裂纹闭合参数U表示裂纹闭合的水平，在一个载荷循环周期内，裂纹闭合参数U可以表示为：

 $U{\text{ = }}\frac{{{\sigma _{\max }} - {\sigma _{{\text{op}}}}}}{{{\sigma _{\max }} - {\sigma _{\min }}}}{\text{ = }}\frac{{\Delta {\sigma _{{\text{eff}}}}}}{{\Delta \sigma }} 。$ (14)

 $\frac{{{\rm{d}}a}}{{{\rm{d}}N}} = b{\left( {COD} \right)^{\tfrac{1}{p}}}。$ (15)

 $COD = f\left( {U\Delta \sigma ,\;\beta ,\;\frac{{2c}}{a}} \right) 。$ (16)

2 有限元分析 2.1 几何模型

 图 2 加筋板试验件模型 Fig. 2 Stiffened plate test part model
2.2 有限元模型及边界条件

 图 3 有限元模型 Fig. 3 Finite element model
2.3 材料模型

2.4 收敛性验证 2.4.1 循环次数的影响

 图 4 循环次数的影响 Fig. 4 The impact of the number of cycles
2.4.2 裂纹扩展单元尺寸的影响

 图 5 裂纹扩展单元的影响 Fig. 5 The impact of the crack expansion unit

2.5 结果分析 2.5.1 裂纹尖端尾迹区残余压应力场分析

 图 6 不同时刻的裂尖附近应力场分布 Fig. 6 The distribution of stress field near the tip of different moments

 图 7 不同裂纹扩展长度下的残余压应力场分布 Fig. 7 Different crack expansion length of residual pressure stress field distribution
2.5.2 裂纹尖端累积塑性应变分析

 图 8 累积塑性应变分布 Fig. 8 Cumulative plastic strain distribution
2.5.3 裂纹张开位移分析

 图 9 裂纹张开位移分布 Fig. 9 Rapture Tienca Rotary Distribution

2.5.4 加筋条刚度对加筋板裂纹扩展的影响

2.5.4.1 加筋条高度的影响

1）筋条高度对加筋板张开载荷的影响

 图 10 筋条高度的影响 Fig. 10 The impact of height

2）筋条高度对加筋板裂纹闭合参数的影响

3）筋条高度对加筋板累计塑性应变以及残余应力场的影响

2.5.4.2 加筋条间距的影响

1）间距对加筋板张开载荷的影响

 图 11 筋条间距的影响 Fig. 11 Impact of Bar Spacing

2）间距对加筋板闭合效应的影响

3）间距对累计塑性应变的影响

4）间距对加筋板残余应力场的影响

3 结　语

1）筋条在加筋板扩展过程中起到了很好的止裂作用。具体表现在同种载荷工况下，筋条的存在能一定程度增强裂纹尖端的残余压应力场，使得裂纹的闭合效应增强，从而降低裂纹扩展速率。同时由于筋条的影响，使得加筋板裂纹区域的刚度增加，导致其在循环载荷的作用下裂纹尖端的累积塑性应变相对于板的整体减小。

2）对于加筋条的刚度来讲，筋条的高度越大，构件的整体刚度越大，断裂性能越好。筋条的高度越高，可以增强残余压应力场，导致闭合效应增强，裂纹尖端的累积塑性应变也减小，从而降低裂纹扩展速率。

3）筋条的间距影响相对以上因素较小，间距的增大使得裂纹附近的结构刚度降低，闭合效应减弱，裂纹扩展速率增加。

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