﻿ 夹持边界条件下表面裂纹应力强度因子求解
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Solution of stress intensity factor of surface cracked geometry with clamped ends
Cao Shusen, He Xiaofan , Yang Boxiao, Liu Wenting
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Determination of the stress intensity factor (SIF) of surface cracked geometry with clamped ends is necessary for crack growth analysis in laboratory. Based on the feature of the clamped ends, the clamped condition was simplified as a combined uniform tension stress and bending moment act with the rotation angles of the specimen ends retained to be zero. The elastic potential energy of the equivalent model was derived based on Newman-Raju expression of SIF for surface crack under free uniform tension and bending moment, and then the function relationship between the uniform tension stress and the bending moment was deduced using Castigliano's first theorem. By superposition, the SIF expression of the equivalent model was given. To verify the validity of the expression, SIF solutions of some representative surface cracked geometry with clamped ends were obtained by Abaqus. Comparisons indicate that the expression is accurate enough. In addition, the relationship of crack shape and length-to-thickness ratio to the modifying factor for clamped ends was analyzed.
Key words: surface crack     stress intensity factor     clamped ends     superposition principle     finite element method

1 等效模型的建立 1.1 试验机夹持边界条件分析

 图 1 疲劳试验夹持条件Fig. 1 Clamped ends condition in fatigue tests
1.2 等效模型

 图 2 等效模型建立Fig. 2 Establishment of the equivalent model
2 基于等效模型的应力强度因子求解 2.1 应力强度因子叠加

2.2 均匀拉伸和弯矩作用下应力强度因子解

Newman-Raju公式[2]给出了在均匀拉伸应力和弯矩共同作用下表面裂纹在全范围内应力强度因子解,见下式:

 图 3 含裂纹截面示意图Fig. 3 Schematic diagram of the intersecting surface including the crack
2.3 等效弯矩求解

 图 4 裂纹等比扩展Fig. 4 Crack growth in proportion

2.4 基于等效模型的应力强度因子

3 等效模型的有限元验证 3.1 有限元验证的目的

3.2 有限元建模

 图 5 自由拉伸条件下试件模型Fig. 5 Specimen model unde free uniform tension
 图 6 夹持条件的建模Fig. 6 Model of the clamped ends conditions

 图 7 整体网格划分图Fig. 7 Mesh method of the whole model
 图 8 含裂纹处网格划分Fig. 8 Mesh mehtod of the part including crack
3.3 夹持边界条件对表面方向和深度方向应力强度因子的影响

 a/mm K/(MPa) θ=0° θ=90° 夹持 均匀拉伸 差别/% 夹持 均匀拉伸 差别/% 4 372 393 -5.4 295 297 -0.8 5 422 463 -8.9 388 391 -0.7 6 483 562 -14 518 520 -0.4 7 582 747 -22.1 744 746 -0.3 注：h=20 mm;w=7.5 mm;t=5 mm;b=4 mm.
 图 9 自由拉伸和夹持条件下K-θ曲线(h=20 mm,w=7.5 mm,t=5 mm,a=6 mm,b=4 mm)Fig. 9 K-θ curves under free uniform tension and clamped ends condition (h=20 mm,w=7.5 mm,t=5 mm,a=6 mm,b=4 mm)
3.4 夹持边界条件下应力强度因子解的检验

 图 10 公式(19)与有限元计算所得修正因子结果比较(w=10 mm,t=5 mm,θ=0°)Fig. 10 Comparison of the correction factor from equation solutions and finite element solutions (w=10 mm,t=5 mm,θ=0°)

4 影响因素分析

 图 11 不同参数对修正因子的影响Fig. 11 Curves of correction factor under deferent variables

5 结 论

1) 建立了夹持边界条件下表面裂纹应力强度因子求解的等效模型,即将夹持条件等效为均匀拉伸与弯矩的共同作用,并且试件端部转角为0°,给出了等效弯矩和均匀拉伸应力的关系,采用叠加原理给出了夹持边界条件下应力强度因子解,其形式为自由均匀拉伸载荷作用应力强度因子解乘以一个与试件和裂纹尺寸有关的修正因子;

2) 与Abaqus有限元数值解的计算对比表明,基于该等效模型的夹持边界条件下的应力强度因子解是合理的;

3) 分析了试件几何和裂纹尺寸对修正因子的影响,当裂纹尺寸较大或者试件长厚比(h/t)较小时,夹持条件会对沿宽度方向的应力强度因子产生比较明显的影响.

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#### 文章信息

Cao Shusen, He Xiaofan, Yang Boxiao, Liu Wenting

Solution of stress intensity factor of surface cracked geometry with clamped ends

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(11): 1637-1642.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0717