Volume 42 Issue 12
Dec.  2021
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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

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

doi: 10.21656/1000-0887.420045
  • Received Date: 2021-02-20
  • Accepted Date: 2021-02-20
  • Rev Recd Date: 2021-05-27
  • Available Online: 2021-11-23
  • Publish Date: 2021-12-01
  • An efficient method was developed to determine the first- and high-order terms of asymptotic solutions of plastic stress and displacement near V-notch tips under anti-plane shear in power-law hardening materials. Through introduction of the asymptotic series expansions of stress and displacement fields around the V-notch tip into the fundamental equations of the elastoplastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions were established. Then the interpolating matrix method was employed to solve the resulting nonlinear and linear ODEs. Consequently, the high-order stress exponents and the associated eigen-solutions were obtained. The presented method, being capable of dealing with the V-notches with arbitrary opening angles and strain hardening indexes under anti-plane shear, has the advantages of great versatility and high accuracy. Typical examples were given to demonstrate the accuracy and effectiveness of this method.

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  • [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
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