留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

反平面塑性V形切口尖端应力和位移渐近解

李聪 胡斌 牛忠荣

李聪,胡斌,牛忠荣. 反平面塑性V形切口尖端应力和位移渐近解 [J]. 应用数学和力学,2021,42(12):1258-1275 doi: 10.21656/1000-0887.420045
引用本文: 李聪,胡斌,牛忠荣. 反平面塑性V形切口尖端应力和位移渐近解 [J]. 应用数学和力学,2021,42(12):1258-1275 doi: 10.21656/1000-0887.420045
LI Cong, HU Bin, NIU Zhongrong. Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1258-1275. doi: 10.21656/1000-0887.420045
Citation: LI Cong, HU Bin, NIU Zhongrong. Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1258-1275. doi: 10.21656/1000-0887.420045

反平面塑性V形切口尖端应力和位移渐近解

doi: 10.21656/1000-0887.420045
基金项目: 国家自然科学基金 (11272111)
详细信息
    作者简介:

    李聪(1989—), 女, 讲师, 博士(通讯作者. E-mail:Licong@ahjzu.edu.cn)

  • 中图分类号: O344.3

Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear

  • 摘要:

    提出了一种确定幂硬化材料反平面V形切口尖端应力和位移渐近解的主导项和高阶项的有效方法。首先通过在弹塑性理论基本方程中引入V形切口尖端应力场和位移场的渐近级数展开,建立以应力和位移为特征函数的非线性和线性常微分方程组。然后采用插值矩阵法求解常微分方程组,可得到多阶应力特征指数和其相对应的特征函数。该方法具有通用性强、精度高等优点,可处理任意开口角度和应变硬化指数的V形切口。典型算例验证了该方法的准确性和有效性。

  • 图  1  反V形切口尖端区域

    Figure  1.  An anti-plane V-notch tip

    图  2  插值矩阵法迭代计算Ⅲ型裂纹第1阶应力指数$ {s_1} $收敛情况

    Figure  2.  First-order stress exponent s1 in the iterative calculation of the mode Ⅲ crack with the interpolating matrix method

    图  3  幂硬化反平面Ⅲ型裂纹第1阶应力和位移特征角函数

    Figure  3.  The 1st-order stress and displacement eigen-functions for the mode Ⅲcrack

    图  4  幂硬化反平面Ⅲ型裂纹高阶应力和位移特征角函数,m=7

    Figure  4.  The high-order stress and displacement eigen-functions for the mode Ⅲ crack with m=7

    图  5  反平面切口第1阶应力特征角函数(m=7)

    Figure  5.  The 1st-order stress eigen-functions for the mode Ⅲ V-notch with m=7

    图  6  反平面切口第2阶应力特征角函数(m=7)

    Figure  6.  The 2nd-order stress eigen-functions for the mode Ⅲ V-notch with m=7

    图  7  反平面裂纹第1阶位移和应力特征角函数

    Figure  7.  The 1st-order stress and displacement eigen-functions for the mode Ⅲ V-notch

    图  8  反平面V形切口第1阶应力特征角函数(m=5)

    Figure  8.  The 1st-order stress and displacement eigen-functions for the mode Ⅲ V-notch (m=5)

    表  1  反平面Ⅲ型裂纹前4阶应力特征指数随硬化指数m的变化

    Table  1.   Stress exponent $ {s_k} $ of the mode Ⅲ crack with various m values

    $ {s_k} $methodm=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $present (Q=20)−0.249 999−0.166 666−0.124 999−0.090 907−0.071 426
    present (Q =40)−0.249 999−0.166 667−0.125 000−0.090 909−0.071 429
    present (Q =80)−0.249 999−0.166 667−0.125 000−0.090 909−0.071 429
    ref. [14]−0.250 000−0.166 667−0.125 000−0.090 909−0.071 429
    $ {s_{\text{2}}} $present (Q =20)0.249999#0.499998#0.4374900.3657590.314434
    present (Q =40)0.249 999#0.500 001#0.435 7610.363 7580.312 255
    present (Q =80)0.249 999#0.500 001#0.435 6660.363 6460.312 133
    ref. [14]0.250 000#0.500 000#0.435 6600.363 6360.312 094
    $ {s_{\text{3}}} $present (Q =20)0.574 1500.501 5590.624 995#0.727 256#0.700 294×
    present (Q =40)0.572 9480.500 0860.625 000#0.727 272#0.695 939×
    present (Q =80)0.572 8790.500 0040.625 000#0.727 272#0.695 695×
    ref. [14]0.572 8760.500 0000.625 000#0.727 273#0.695 617×
    $ {s_{\text{4}}} $present (Q =20)0.749 997#1.166 662#0.999 979×0.822 425×0.785 686#
    present (Q =40)0.749 997#1.166 669#0.996 522×0.818 425×0.785 719#
    present (Q =80)0.749 997#1.166 669#0.996 332×0.818 201×0.785 719#
    ref. [14]0.750 000#1.166 667#0.996 320×0.818 182×0.785 714#
    下载: 导出CSV

    表  2  反平面V形切口($ {\text{2}}\alpha {\text{ = 3}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化

    Table  2.   Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 3}}{{\text{0}}^ \circ } $ for various m values

    $ {s_k} $methodm=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $ present(Q=20) −0.232204 −0.157527 −0.119464 −0.087783 −0.069413
    present(Q=40) −0.232206 −0.157527 −0.119465 −0.087784 −0.069415
    present(Q=80) −0.232206 −0.157527 −0.119465 −0.087784 −0.069415
    ref. [17] −0.232206 −0.157564 −0.119502 −0.087824 −0.069457
    $ {s_{\text{2}}} $ present(Q=20) 0.232204# 0.472581# 0.554001 0.472991 0.413208
    present(Q=40) 0.232206# 0.472581# 0.551878 0.470411 0.410316
    present(Q=80) 0.232206# 0.472581# 0.551876 0.470409 0.410314
    ref. [17] 0.551851 0.470288 0.409963
    $ {s_{\text{3}}} $ present(Q=20) 0.696612× 0.623794 0.597320# 0.702264# 0.763543#
    present(Q=40) 0.696618× 0.621968 0.597325# 0.702272# 0.763565#
    present(Q=80) 0.696618× 0.621972 0.597325# 0.702272# 0.763565#
    ref. [17] 0.622051
    $ {s_{\text{4}}} $ present(Q=20) 0.700023 1.102689# 1.227466× 1.033765× 0.895829×
    present(Q=40) 0.698686 1.102689# 1.223221× 1.028606× 0.890047×
    present(Q=80) 0.698682 1.102689# 1.223217× 1.028602× 0.890043×
    ref. [17] 0.698604
    下载: 导出CSV

    表  3  反平面V形切口($ {\text{2}}\alpha {\text{ = 15}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化

    Table  3.   Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 15}}{{\text{0}}^ \circ } $ for various m values

    $ {s_k} $methodm=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $ present(Q=20) −0.080378 −0.060350 −0.049587 −0.039967 −0.033919
    present(Q=40) −0.080383 −0.060356 −0.049564 −0.039974 −0.033925
    present(Q=80) −0.080383 −0.060356 −0.049593 −0.039974 −0.033925
    ref. [17] −0.080384 −0.060369 −0.049614 −0.040000 −0.033956
    $ {s_{\text{2}}} $ present(Q=20) 0.080378# 0.181051# 0.247935# 0.319736# 0.373109#`
    present(Q=40) 0.080383# 0.181068# 0.247970# 0.319792# 0.373175#`
    present(Q=80) 0.080383# 0.181068# 0.247965# 0.319792# 0.373175#`
    ref. [17]
    $ {s_{\text{3}}} $ present(Q=20) 0.241134# 0.422452# 0.545457# 0.679439# 0.780137#
    present(Q=40) 0.241149# 0.422492# 0.545534# 0.679558# 0.780275#
    present(Q=80) 0.241149# 0.422492# 0.545523# 0.679558# 0.780275#
    ref. [17]
    $ {s_{\text{4}}} $ present(Q=20) 1.589836 1.556131 1.520007 1.470114 1.426214
    present(Q=40) 1.587821 1.553694 1.516597 1.464221 1.417315
    present(Q=80) 1.587820 1.553693 1.516597 1.464220 1.417314
    ref. [17] 1.587794 1.553562 1.516332 1.463944 1.416870
    下载: 导出CSV

    表  4  反平面V形切口($ {\text{2}}\alpha {\text{ = 6}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化(Q=40)

    Table  4.   Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 6}}{{\text{0}}^ \circ } $ for various m values (Q=40)

    $ {s_k} $methodm=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $ present −0.209034 −0.144787 −0.111379 −0.083025 −0.066275
    ref. [17] −0.209035 −0.144820 −0.111421 −0.083068 −0.066323
    $ {s_{\text{2}}} $ present 0.209034# 0.434361# 0.556895# 0.613636 0.545403
    ref. [17] 0.613578 0.545247
    $ {s_{\text{3}}} $ present 0.627102# 0.775407 0.702317 0.664200# 0.729025#
    ref. [17] 0.775369 0.702299
    $ {s_{\text{4}}} $ present 0.851750 1.013509# 1.225169# 1.310297× 1.157081×
    ref. [17] 0.851659
    下载: 导出CSV

    表  5  反平面V形切口($ {\text{2}}\alpha {\text{ = 9}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化(Q=40)

    Table  5.   Stress exponent $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 9}}{{\text{0}}^ \circ } $ for various m values (Q=40)

    $ {s_k} $methodm=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $ present −0.178394 −0.126571 −0.099142 −0.075360 −0.060999
    ref. [17] −0.178395 −0.126599 −0.099179 −0.075403 −0.061047
    $ {s_{\text{2}}} $ present 0.178394# 0.379713# 0.495710# 0.602880# 0.670989#
    ref. [17]
    $ {s_{\text{3}}} $ present 0.535182# 0.885997# 0.899889 0.809746 0.736893
    ref. [17] 0.899826 0.809670 0.736783
    $ {s_{\text{4}}} $ present 1.040992 0.970537 1.090562# 1.281120# 1.402977#
    ref. [17] 1.040889 0.970465
    下载: 导出CSV

    表  6  反平面V形切口($ {\text{2}}\alpha {\text{ = 12}}{{\text{0}}^ \circ } $)前4阶应力特征指数随硬化指数m的变化(Q=40)

    Table  6.   Stress exponents $ {s_k} $ of the mode Ⅲ V-notch with $ {\text{2}}\alpha {\text{ = 12}}{{\text{0}}^ \circ } $ for various m values (Q=40)

    $ {s_k} $methodm=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $ present −0.137145 −0.099974 −0.080061 −0.060462 −0.051585
    ref. [17] −0.137146 −0.100000 −0.080094 −0.062500 −0.051628
    $ {s_{\text{2}}} $ present 0.137145# 0.299922# 0.400305# 0.483696# 0.567435#
    ref. [17]
    $ {s_{\text{3}}} $ present 0.411435# 0.699818# 0.880671# 1.027854# 1.013736
    ref. [17] 1.013494
    $ {s_{\text{4}}} $ present 1.279549 1.222471 1.162660 1.082304 1.186455#
    ref. [17] 1.279434 1.222375 1.162546 1.082129
    下载: 导出CSV

    表  7  反平面V形切口前4阶应力特征指数随硬化指数m和张角2α的变化 ( Q =40)

    Table  7.   Stress exponents $ {s_k} $ of the mode Ⅲ V-notch for various m values and $ {\text{2}}\alpha $ (Q=40)

    $ {s_k} $$ {\text{2}}\alpha $/(°)m=3m=5m=7m=10m=13
    $ {s_{\text{1}}} $ 0 −0.316987 −0.194766 −0.140313 −0.098797 −0.076224
    30 −0.313242 −0.193473 −0.139669 −0.098487 −0.076043
    90 −0.300944 −0.188989 −0.137387 −0.097374 −0.075386
    150 −0.274774 −0.178251 −0.131633 −0.094459 −0.073631
    $ {s_{\text{2}}} $ 0 −0.049037 −0.026170 −0.017810 −0.012034 −0.009085
    30 −0.000702 0.008722 0.009252 0.008153 0.006996
    90 0.138739 0.116011 0.095654 0.074849 0.061274
    150 0.274774# 0.321572 0.272393 0.220346 0.184745
    $ {s_{\text{3}}} $ 0 0.218913× 0.142426× 0.104693× 0.074729× 0.058054×
    30 0.311838× 0.210917× 0.158173× 0.114793× 0.090035×
    90 0.300944× 0.421011× 0.328695× 0.247072× 0.197934×
    150 0.380293 0.534753# 0.658165# 0.535151× 0.443121×
    $ {s_{\text{4}}} $ 0 0.285422 0.213317 0.169289 0.129282 0.104644
    30 0.313242# 0.285697 0.228736 0.176303 0.143628
    90 0.638404 0.50008 0.410023 0.323996 0.268649
    150 0.824322× 0.821395× 0.676419× 0.623227 0.531023
    下载: 导出CSV
  • [1] 李聪, 牛忠荣, 胡宗军, 等. 求解双材料裂纹结构全域应力场的扩展边界元法[J]. 应用数学和力学, 2019, 40(8): 926-937. (LI Cong, NIU Zhongrong, HU Zongjun, et al. Computation of total stress fields for cracked bi-material structures with the extended boundary element method[J]. Applied Mathematics and Mechanics, 2019, 40(8): 926-937.(in Chinese)
    [2] HUTCHINSON J W. Singular behavior at the end of a tensile crack in a hardening material[J]. Journal of the Mechanics and Physics of Solids, 1968, 16(1): 13-31. doi: 10.1016/0022-5096(68)90014-8
    [3] RICE J R, ROSENGREN G F. Plane strain deformation near a crack tip in a power-law hardening material[J]. Journal of the Mechanics and Physics of Solids, 1968, 16(1): 1-12. doi: 10.1016/0022-5096(68)90013-6
    [4] KUANG Z, XU X. Stress and strain fields at the tip of a sharp V-notch in a power-hardening material[J]. International Journal of Fracture, 1987, 35: 39-53. doi: 10.1007/BF00034533
    [5] XIA L, WANG T C. Singular behaviour near the tip of a sharp V-notch in a power law hardening material[J]. International Journal of Fracture, 1993, 59(1): 83-93. doi: 10.1007/BF00032219
    [6] LI Y C, WANG T C. High-order asymptotic field of tensile plane-strain nonlinear crack problems[J]. Scientia Sinica(Series A), 1986, 29(9): 941-955.
    [7] SHARMA S M, ARAVAS N. On the development of variable-separable asymptotic elastoplastic solutions for interfacial cracks[J]. International Journal of Solids & Structures, 1993, 30(5): 695-723.
    [8] XIA L, WANG T C, SHIH C F. Higher-order analysis of crack tip fields in elastic power-law hardening materials[J]. Journal of the Mechanics and Physics of Solids, 1993, 41(4): 665-687. doi: 10.1016/0022-5096(93)90022-8
    [9] YUAN F G, YANG S. Crack-tip fields in elastic-plastic material under plane stress mode Ⅰ loading[J]. International Journal of Fracture, 1997, 85(2): 131-155. doi: 10.1023/A:1007361116709
    [10] YANG S, CHAO Y J, SUTTON M A. Higher order asymptotic crack tip fields in a power-law hardening material[J]. Engineering Fracture Mechanics, 1993, 45(1): 1-20. doi: 10.1016/0013-7944(93)90002-A
    [11] CHAO Y J, YANG S. Higher order crack tip fields and its implication for fracture of solids under mode Ⅱ conditions[J]. Engineering Fracture Mechanics, 1996, 55(5): 777-794. doi: 10.1016/0013-7944(96)00054-9
    [12] RICE J R. Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear[J]. Journal of Applied Mechanics, 1967, 34(2): 287. doi: 10.1115/1.3607681
    [13] AMAZIGO J C. Fully plastic crack in an infinite body under anti-plane shear[J]. International Journal of Solids and Structures, 1974, 10(9): 1003-1015. doi: 10.1016/0020-7683(74)90008-0
    [14] YANG S, YUAN F G, CAI X. Higher orderasymptotic elastic-plastic crack-tip fields under antiplane shear[J]. Engineering Fracture Mechanics, 1996, 54(3): 405-422. doi: 10.1016/0013-7944(95)00191-3
    [15] YANG S, YUAN F G, CHIANG M Y M. Analytical forms of higher-order asymptotic elastic-plastic crack-tip fields in a linear hardening material under antiplane shear[J]. International Journal of Fracture, 1996, 80(1): 59-71. doi: 10.1007/BF00036480
    [16] YUAN F G, YANG S. Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear[J]. International Journal of Fracture, 1994, 69(1): 1-26.
    [17] WANG T J, KUANG Z B. Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading[J]. International Journal of Fracture, 1999, 96(4): 303-329. doi: 10.1023/A:1018657316810
    [18] ZAPPALORTO M, LAZZARIN P. Analytical study of the elastic-plastic stress fields ahead of parabolic notches under antiplane shear loading[J]. International Journal of Fracture, 2007, 148(2): 139-154. doi: 10.1007/s10704-008-9185-7
    [19] LAZZARIN P, ZAPPALORTO M. Plastic notch stress intensity factors for pointed V-notches under antiplane shear loading[J]. International Journal of Fracture, 2008, 152(1): 1-25. doi: 10.1007/s10704-008-9260-0
    [20] ZAPPALORTO M, LAZZARIN P. A unified approach to the analysis of nonlinear stress and strain fields ahead of mode Ⅲ-loaded notches and cracks[J]. International Journal of Solids and Structures, 2010, 47(6): 851-864. doi: 10.1016/j.ijsolstr.2009.11.021
    [21] ARAVAS N, BLAZO D H. Higher order terms in asymptotic elastoplastic mode Ⅲ crack tip solutions[J]. Acta Mechanica, 1991, 90(1/4): 139-153.
    [22] LOGHIN A, ZHANG N, JOSEPH P F. A nonlinear finite element eigenanalysis of antiplane shear including higher order terms[J]. Engineering Fracture Mechanics, 2000, 66(5): 441-454. doi: 10.1016/S0013-7944(00)00031-X
    [23] ZHANG N, JOSEPH P F. A nonlinear finite element eigenanalysis of singular stress fields in bimaterial wedges for plane strain[J]. International Journal of Fracture, 1998, 94(3): 299-319.
    [24] LOGHIN A, JOSEPH P F. Asymptotic solutions for mixed mode loading of cracks and wedges in power law hardening materials[J]. Engineering Fracture Mechanics, 2001, 68(14): 1511-1534. doi: 10.1016/S0013-7944(01)00050-9
    [25] PATWARDHAN P A, NALAVDE R A, KUJAWSKI D. An estimation of Ramberg-Osgood constants for materials with and without Loder’s strain using yield and ultimate strengths[J]. Procedia Structure Integrity, 2019, 17: 750-757. doi: 10.1016/j.prostr.2019.08.100
    [26] NIU Z R, GE D L, CHENG C Z, et al. Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials[J]. Applied Mathematical Modelling, 2009, 33(3): 1776-1792. doi: 10.1016/j.apm.2008.03.007
  • 加载中
图(14) / 表(7)
计量
  • 文章访问数:  318
  • HTML全文浏览量:  130
  • PDF下载量:  28
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-02-20
  • 录用日期:  2021-02-20
  • 修回日期:  2021-05-27
  • 网络出版日期:  2021-11-23
  • 刊出日期:  2021-12-01

目录

    /

    返回文章
    返回