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

基于Eshelby-Stroh公式各向异性弹性体接触问题研究

颜灯灯 李成刚 申景金 王艳 王春明 宋伟山

颜灯灯, 李成刚, 申景金, 等 . 基于Eshelby-Stroh公式各向异性弹性体接触问题研究[J]. 北京航空航天大学学报, 2018, 44(6): 1273-1282. doi: 10.13700/j.bh.1001-5965.2017.0400
引用本文: 颜灯灯, 李成刚, 申景金, 等 . 基于Eshelby-Stroh公式各向异性弹性体接触问题研究[J]. 北京航空航天大学学报, 2018, 44(6): 1273-1282. doi: 10.13700/j.bh.1001-5965.2017.0400
YAN Dengdeng, LI Chenggang, SHEN Jingjin, et al. Study on contact problem of anisotropic elastic body based on Eshelby-Stroh formalism[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(6): 1273-1282. doi: 10.13700/j.bh.1001-5965.2017.0400(in Chinese)
Citation: YAN Dengdeng, LI Chenggang, SHEN Jingjin, et al. Study on contact problem of anisotropic elastic body based on Eshelby-Stroh formalism[J]. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(6): 1273-1282. doi: 10.13700/j.bh.1001-5965.2017.0400(in Chinese)

基于Eshelby-Stroh公式各向异性弹性体接触问题研究

doi: 10.13700/j.bh.1001-5965.2017.0400
基金项目: 

江苏省自然科学基金 BK20141414

详细信息
    作者简介:

    颜灯灯  男, 硕士研究生。主要研究方向:触觉传感器

    李成刚  男, 博士, 副教授, 硕士生导师。主要研究方向:工业机器人技术、并联机器人与传感器技术

    申景金  男, 博士, 讲师, 硕士生导师。主要研究方向:生物材料本构关系、接触力学、机器人分析与控制

    通讯作者:

    李成刚, E-mail:lichenggang@nuaa.edu.cn

  • 中图分类号: TB121

Study on contact problem of anisotropic elastic body based on Eshelby-Stroh formalism

Funds: 

Natural Science Foundation of Jiangsu Province BK20141414

More Information
  • 摘要:

    针对线性各向异性弹性体小变形接触问题,将弹性体按是否与刚体压头发生接触进行划分,基于Eshelby-Stroh公式求解各个部分的位移函数和应力函数,进一步通过应力函数积分得到载荷值。考虑到求解结果存在交接处应力突变和非接触区域应力不近似于零的问题,采用整体位移约束法和线性叠加原理,通过迭代方式使位移函数和应力函数逼近理想解,解决了圆柱压头和倒圆角楔形压头与弹性体的接触问题。基于圆柱压头求得的载荷值接近弹性半空间法的求解结果,当级数总项数为400时,计算结果的相对误差仅为0.52%。基于圆柱压头和倒圆角楔形压头求得的载荷值与ABAQUS仿真结果较为吻合:圆柱压头载荷值的相对误差为0.67%;倒圆角楔形压头,对6个不同的圆角值进行计算,载荷的相对误差都小于2%。

     

  • 图 1  压头与弹性体的接触模型

    Figure 1.  Contact model of indenter and elastic body

    图 2  当前解与弹性半空间解的比较

    Figure 2.  Comparison between present solution and solution of elastic half space

    图 3  K=500时接触区域应力的相对误差

    Figure 3.  Relative error of stress in contact zone at K=500

    图 4  ABAQUS仿真接触应力

    Figure 4.  Contact stress of ABAQUS simulation

    图 5  圆柱压头与有界弹性体接触应力分布

    Figure 5.  Distribution of contact stress between cylindrical indenter and bounded elastic body

    图 6  θ=10°, R=0, 50, 100, 150, 200, 250 mm时的计算与仿真应力分布及两者的绝对误差

    Figure 6.  Computed and simulated stress distribution and absolute error between them at θ=10° and R=0, 50, 100, 150, 200, 250 mm

    表  1  K=400时的迭代结果

    Table  1.   Iterative results at K=400

    迭代次数 中间点位移/mm 中间点应力/MPa 载荷/N
    0 -0.56 -33.19 1 413.16
    1 -1.49 -37.27 1 969.28
    2 -4.36 -52.47 3 405.87
    3 -6.21 -60.30 4 400.09
    4 -7.11 -64.14 4 899.41
    5 -7.48 -65.59 5 116.40
    6 -7.65 -66.36 5 224.26
    7 -7.72 -66.63 5 268.43
    8 -7.76 -66.79 5 290.35
    9 -7.77 -66.84 5 298.72
    10 -7.78 -66.87 5 303.15
    11 -7.78 -66.88 5 304.71
    12 -7.78 -66.89 5 305.04
    13 -7.78 -66.89 5 305.36
    14 -7.78 -66.89 5 305.54
    15 -7.78 -66.89 5 305.60
    下载: 导出CSV

    表  2  不同R值下的压痕深度与仿真接触宽度

    Table  2.   Indentation depth and simulation contact width at different R values

    圆角半径/mm 压痕深度/mm 仿真接触宽度/mm 接触宽度相对误差/%
    0 20.24 99.67 0.33
    50 19.47 99.56 0.44
    100 18.56 99.51 0.49
    150 17.53 99.48 0.52
    200 16.37 99.50 0.50
    250 15.00 99.55 0.45
    下载: 导出CSV

    表  3  不同R值下的计算载荷与仿真载荷

    Table  3.   Calculation load and simulation load at different R values

    圆角半径/mm 计算载荷/N 仿真载荷/N 载荷相对误差/%
    0 11 947 11 785 1.38
    50 11 867 11 762 0.89
    100 11 679 11 571 0.94
    150 11 354 11 227 1.13
    200 10 865 10 723 1.32
    250 10 151 9 991 1.60
    下载: 导出CSV
  • [1] TÖYRÄS J, LYYRA-LAITINEN T, NⅡNIM K M, et al.Estimation of the Young's modulus of articular cartilage using an arthroscopic indentation instrument and ultrasonic measurement of tissue thickness[J].Journal of Biomechanics, 2001, 34(2):251-256. doi: 10.1016/S0021-9290(00)00189-5
    [2] KORHONEN R K, SAARAKKALA S, TOYRAS J, et al.Experimental and numerical validation for the novel configuration of an arthroscopic indentation instrument[J].Physics in Medicine & Biology, 2003, 48(11):1565-1576. http://med.wanfangdata.com.cn/Paper/Detail?id=PeriodicalPaper_JJ027207656
    [3] DIMITRIADIS E K, HORKAY F, MARESCA J, et al.Determination of elastic moduli of thin layers of soft material using the atomic force microscope[J].Biophysical Journal, 2002, 82(5):2798-2810. doi: 10.1016/S0006-3495(02)75620-8
    [4] WANNINAYAKE I B, DASGUPTA P, SENEVIRATNE L D, et al.Air-float palpation probe for tissue abnormality identification during minimally invasive surgery[J].IEEE Transactions on Biomedical Engineering, 2013, 60(10):2735-2744. doi: 10.1109/TBME.2013.2264287
    [5] 郭渊, 关志东, 刘德博, 等.复合材料静压痕与落锤冲击初始损伤对比试验[J].北京航空航天大学学报, 2009, 35(8):1018-1021. http://bhxb.buaa.edu.cn/CN/abstract/abstract8770.shtml

    GUO Y, GUANG Z D, LIU D B, et al.Comparison between quasi-static indentation testing and drop-weight impact testing on delamination on set damage[J].Journal of Beijing University of Aeronautics and Astronautics, 2009, 35(8):1018-1021(in Chinese). http://bhxb.buaa.edu.cn/CN/abstract/abstract8770.shtml
    [6] STOLZ M, GOTTARDI R, RAITERI R, et al.Early detection of aging cartilage and osteoarthritis in mice and patient samples using atomic force microscopy[J].Nature Nanotechnology, 2009, 4(3):186-192. doi: 10.1038/nnano.2008.410
    [7] LIAO Q, HUANG J, ZHU T, et al.A hybrid model to determine mechanical properties of soft polymers by nanoindentation[J].Mechanics of Materials, 2010, 42(12):1043-1047. doi: 10.1016/j.mechmat.2010.09.005
    [8] MUSKHELISHVILI N I.Some basic problems of the mathematical theory of elasticity[J].Mathematical Gazette, 1953, 48(365):351. http://www.springerlink.com/content/978-94-017-3034-1
    [9] ENGLAND A H, SIH G C.Complex variable methods in elasticity[M].London:Wiley-Interscience, 1971:318.
    [10] GLADWELL G M L, ENGLAND A H.Orthogonal polynomial solutions to some mixed boundary-value problems in elasticity theory[J].Quarterly Journal of Mechanics & Applied Mathematics, 1977, 30(2):175-185. http://qjmam.oxfordjournals.org/content/30/2/175.short
    [11] OKUMURA M, HASEBE N, NAKAMURA T.Crack due to wedge-shaped punch with friction[J].Journal of Engineering Mechanics, 1990, 116(10):2173-2185. doi: 10.1061/(ASCE)0733-9399(1990)116:10(2173)
    [12] STROH A N.Dislocations and cracks in anisotropic elasticity[J].Philosophical Magazine, 1958, 3(30):625-646. doi: 10.1080/14786435808565804
    [13] STROH A N.Steady state problems in anisotropic elasticity[J].Studies in Applied Mathematics, 1962, 41(1):77-103. doi: 10.1002/sapm196241177/pdf
    [14] FAN C W, HWU C.Punch problems for an anisotropic elastic half-plane[J].Journal of Applied Mechanics, 1996, 63(1):69-76. doi: 10.1115/1.2787211
    [15] HWU C, FAN C W.Sliding punches with or without friction along the surface of an anisotropic elastic half-plane[J].Quarterly Journal of Mechanics & Applied Mathematics, 1998, 51(1):159-177. http://qjmam.oxfordjournals.org/content/51/1/159.short
    [16] BATRA R C, JIANG W.Analytical solution of the contact problem of a rigid indenter and an anisotropic linear elastic layer[J].International Journal of Solids and Structures, 2008, 45(22-23):5814-5830. doi: 10.1016/j.ijsolstr.2008.06.016
    [17] JIANG W, BATRA R C.Indentation of a laminated composite plate with an interlayer rectangular void[J].Composites Science and Technology, 2010, 70(6):1023-1030. doi: 10.1016/j.compscitech.2010.02.030
    [18] VEL S S, BATRA R C.The generalized plane strain deformations of thick anisotropic composite laminated plates[J].International Journal of Solids and Structures, 2000, 37(5):715-733. doi: 10.1016/S0020-7683(99)00040-2
    [19] TING T C T.Anisotropic elasticity theory and applications[M].Oxford:Oxford University Press, 1996:134-142.
  • 加载中
图(6) / 表(3)
计量
  • 文章访问数:  534
  • HTML全文浏览量:  32
  • PDF下载量:  358
  • 被引次数: 0
出版历程
  • 收稿日期:  2017-06-12
  • 录用日期:  2017-08-31
  • 网络出版日期:  2018-06-20

目录

    /

    返回文章
    返回
    常见问答