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考虑端部摩擦的中心直裂纹巴西圆盘断裂参数解析解

刘建 乔兰 李庆文 赵国彦

刘建, 乔兰, 李庆文, 赵国彦. 考虑端部摩擦的中心直裂纹巴西圆盘断裂参数解析解[J]. 工程科学学报. doi: 10.13374/j.issn2095-9389.2021.06.07.006
引用本文: 刘建, 乔兰, 李庆文, 赵国彦. 考虑端部摩擦的中心直裂纹巴西圆盘断裂参数解析解[J]. 工程科学学报. doi: 10.13374/j.issn2095-9389.2021.06.07.006
LIU Jian, QIAO Lan, LI Qing-wen, ZHAO Guo-yan. Analytical solutions of fracture parameters for a centrally cracked Brazilian disk considering the loading friction[J]. Chinese Journal of Engineering. doi: 10.13374/j.issn2095-9389.2021.06.07.006
Citation: LIU Jian, QIAO Lan, LI Qing-wen, ZHAO Guo-yan. Analytical solutions of fracture parameters for a centrally cracked Brazilian disk considering the loading friction[J]. Chinese Journal of Engineering. doi: 10.13374/j.issn2095-9389.2021.06.07.006

考虑端部摩擦的中心直裂纹巴西圆盘断裂参数解析解

doi: 10.13374/j.issn2095-9389.2021.06.07.006
基金项目: 国家自然科学基金与山东联合基金重点资助项目(U1806209);中央高校基本科研业务费专项资金资助项目(FRF-TP-19-021A3);北京科技大学青年教师学科交叉研究资助项目(FRF-IDRY-19-002)
详细信息
    通讯作者:

    E-mail: lanqiao@ustb.edu.cn

  • 中图分类号: O346.1

Analytical solutions of fracture parameters for a centrally cracked Brazilian disk considering the loading friction

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  • 摘要: 运用权函数法推导出考虑加载端摩擦的四种形式分布载荷加载下,中心直裂纹巴西圆盘试样在任意I/II复合型断裂模式下I、II型应力强度因子及T应力的解析解,并探究了端部摩擦及载荷分布角度对断裂参数的影响。研究结果表明:(1)当中心裂纹相对长度β较小时,纯I型、纯II型断裂的YIYIIT*(分别是量纲为一的I型、II型应力强度因子及T应力)均随摩擦系数及载荷分布角度增大而减小;但是,当β较大时,摩擦系数增大可使纯I型YI增大,而载荷分布角度增大可使纯II型T*增大。(2)接触载荷分布形式为常数函数时,载荷分布角度对断裂参数的影响最显著,而四次函数下其对断裂参数的影响相对最小。(3)当β较小时,纯II型加载角度随载荷分布角度增大而减小;当β较大时,其随载荷分布角度增大而增大;摩擦系数增大可使纯II型加载角度增大。

     

  • 图  1  CCBD试样与BD试样示意图。(a) CCBD试样;(b) BD试样

    Figure  1.  Schematic of (a) centrally cracked Brazilian disk and (b) Brazilian disk specimens

    图  2  圆盘边界上(r=R)式(13)计算结果与q(θ)的对比

    Figure  2.  Comparison between the calculated results based on Eq. (13) and the applied normal pressures q(θ)

    图  3  均布载荷加载下本文结果与Dong等[4]及李一凡等[21]结果的对比。(a)YI;(b) YII;(c) T*

    Figure  3.  Comparison between the results of this study and the results of Dong et al.[4] and Li et al.[21] under uniformly distributed pressure: (a) YI; (b) YII; and (c) T*

    图  4  纯I型及纯II型断裂的几何参数随摩擦系数的变化特征。(a) β=0.2:纯I型YI;(b)β=0.2:纯I型T*;(c)β=0.2:纯II型YII;(d) β=0.2:纯II型T*;(e)β=0.8:纯I型YI;(f)β=0.8:纯I型T*;(g) β=0.8:纯II型YII;(h)β=0.8:纯II型T*

    Figure  4.  Variations in the YI, YII and T* of pure mode I and II fractures versus friction coefficient μ: (a) β=0.2: pure mode-I YI; (b) β=0.2: pure mode-I T*; (c) β=0.2: pure mode-II YII; (d) β=0.2: pure mode-II T*; (e) β=0.8: pure mode-I YI; (f) β=0.8: pure mode-I T*; (g) β=0.8: pure mode-II YII; (h) β=0.8: pure mode-II T*

    图  5  纯I型及纯II型断裂的几何参数随载荷分布角度的变化特征。(a) β=0.2:纯I型YI;(b)β=0.2:纯I型T*;(c)β=0.2:纯II型YII;(d) β=0.2:纯II型T*;(e)β=0.8: 纯I型YI;(f)β=0.8:纯I型T*;(g) β=0.8:纯II型YII;(h)β=0.8:纯II型T*

    Figure  5.  Variations in the YI, YII, and T* of pure mode I and II fractures versus the load distribution angle α: (a) β = 0.2: pure mode-I YI; (b) β = 0.2: pure mode-I T*; (c) β = 0.2: pure mode-II YII; (d) β = 0.2: pure mode-II T*; (e) β = 0.8: pure mode-I YI; (f) β = 0.8: pure mode-I T*; (g) β = 0.8: pure mode-II YII; (h) β = 0.8: pure mode-II T*

    图  6  载荷分布角度对纯II型加载角度的影响。(a)β=0.2;(b) β=0.8

    Figure  6.  Effect of the load distribution angle on the critical loading angle for pure mode II fractures: (a) β = 0.2; (b) β = 0.8

    表  1  分布载荷加载下巴西圆盘应力解析解的系数

    Table  1.   Series coefficients Cn of the stress analytical solutions for the Brazilian disk subjected to distributed pressures

    Distribution formf(θ)/qmaxqmaxC0Cn (n=±1, ±2, ···)  
    Uniform1$ {[2(\sin \alpha + \mu (1 - \cos \alpha ))]^{ - 1}} $$ \dfrac{{2\alpha }}{{\text{π}} }{q_{\max }} $$ {q_{\max }}\dfrac{{\sin 2n\alpha - 2\mu {{(\sin n\alpha )}^2}}}{{n{\text{π}} }} $
    Elliptical$ {\left(1 - {\left(\dfrac{\theta }{\alpha }\right)^2}\right)^{1/2}} $$ {[{\text{π}} ({J_1}(\alpha ) + \mu {H_1}(\alpha ))]^{ - 1}} $$ \dfrac{\alpha }{2}{q_{\max }} $$ {q_{\max }}\dfrac{{{J_1}(2n\alpha ) - \mu {H_1}(2n\alpha )}}{{2n}} $
    Parabolic$ 1 - {\left(\dfrac{\theta }{\alpha }\right)^2} $$ \begin{gathered} {\alpha ^2}[4\mu (1 + {\alpha ^2}/2 - \alpha \sin \alpha - \hfill \\ \cos \alpha ) + 4\sin \alpha - 4\alpha \cos \alpha {]^{ - 1}} \hfill \\ \end{gathered} $$ \dfrac{{4\alpha }}{{3{\text{π}} }}{q_{\max }} $$ \begin{gathered} {q_{\max }}[\mu (2\alpha n\sin 2n\alpha + \cos 2n\alpha - 2{\alpha ^2}{n^2} - 1) \hfill \\ + \sin 2n\alpha - 2\alpha n\cos 2n\alpha ]/(2{\text{π}} {\alpha ^2}{n^3}) \hfill \\ \end{gathered} $
    Quartic polynomial$ {\left(1 - {\left(\dfrac{\theta }{\alpha }\right)^2}\right)^2} $$ \begin{gathered} {\alpha ^4}[16(3 - {\alpha ^2})\sin \alpha - 48\alpha \cos \alpha \hfill \\ + 2\mu ({\alpha ^4} + 4{\alpha ^2} + 24 + \hfill \\ 8({\alpha ^2} - 3)\cos \alpha - 24\alpha \sin \alpha ){]^{ - 1}} \hfill \\ \end{gathered} $$ \dfrac{{16\alpha }}{{15{\text{π}} }}{q_{\max }} $$ \begin{gathered} {q_{\max }}[(3\mu - 4{\alpha ^2}{n^2}\mu - 6\alpha n)\cos 2n\alpha + \hfill \\ (6\alpha n\mu + 3 - 4{\alpha ^2}{n^2})\sin 2n\alpha - \hfill \\ 2\mu ({\alpha ^4}{n^4} + {\alpha ^2}{n^2} + 3/2)]/(2{\text{π}} {n^5}{\alpha ^4}) \hfill \\ \end{gathered} $
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  • [1] Atkinson C, Smelser R E, Sanchez J. Combined mode fracture via the cracked Brazilian disk test. Int J Fract, 1982, 18(4): 279 doi: 10.1007/BF00015688
    [2] Fett T. T-stresses in rectangular plates and circular disks. Eng Fract Mech, 1998, 60(5-6): 631 doi: 10.1016/S0013-7944(98)00038-1
    [3] Fett T. Stress intensity factors and T-stress for single and double-edge-cracked circular disks under mixed boundary conditions. Eng Fract Mech, 2002, 69(1): 69 doi: 10.1016/S0013-7944(01)00078-9
    [4] Dong S M, Wang Y, Xia Y M. Stress intensity factors for central cracked circular disk subjected to compression. Eng Fract Mech, 2004, 71(7-8): 1135 doi: 10.1016/S0013-7944(03)00120-6
    [5] Akbardoost J, Rastin A. Comprehensive data for calculating the higher order terms of crack tip stress field in disk-type specimens under mixed mode loading. Theor Appl Fract Mech, 2015, 76: 75 doi: 10.1016/j.tafmec.2015.01.004
    [6] Smith D J, Ayatollahi M R, Pavier M J. The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Eng Mater Struct, 2001, 24(2): 137 doi: 10.1046/j.1460-2695.2001.00377.x
    [7] Dong Z, Tang S B, Lang Y X. Hydraulic fracture prediction theory based on the minimum strain energy density criterion. Chin J Eng, 2019, 41(4): 436

    董卓, 唐世斌, 郎颖娴. 基于最小应变能密度因子断裂准则的岩石裂纹水力压裂研究. 工程科学学报, 2019, 41(4):436
    [8] Liu H Y. Initiation mechanism of cracks of rock in compression and shear considering T-stress. Chin J Geotech Eng, 2019, 41(7): 1296

    刘红岩. 考虑T应力的岩石压剪裂纹起裂机理. 岩土工程学报, 2019, 41(7):1296
    [9] Ayatollahi M R, Aliha M R M. Mixed mode fracture in soda lime glass analyzed by using the generalized MTS criterion. Int J Solids Struct, 2009, 46(2): 311 doi: 10.1016/j.ijsolstr.2008.08.035
    [10] Aliha M R M, Ayatollahi M R. Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int J Solids Struct, 2012, 49(13): 1877 doi: 10.1016/j.ijsolstr.2012.03.029
    [11] Ayatollahi M R, Aliha M R M. On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of rock materials. Eng Fract Mech, 2008, 75(16): 4631 doi: 10.1016/j.engfracmech.2008.06.018
    [12] Hua W, Dong S M, Peng F, et al. Experimental investigation on the effect of wetting-drying cycles on mixed mode fracture toughness of sandstone. Int J Rock Mech Min Sci, 2017, 93: 242 doi: 10.1016/j.ijrmms.2017.01.017
    [13] Yin T B, Wu Y, Wang C, et al. Mixed-mode I + II tensile fracture analysis of thermally treated granite using straight-through notch Brazilian disc specimens. Eng Fract Mech, 2020, 234: 107111 doi: 10.1016/j.engfracmech.2020.107111
    [14] Awaji H, Sato S. Combined mode fracture toughness measurement by the disk test. J Eng Mater Technol, 1978, 100(2): 175 doi: 10.1115/1.3443468
    [15] Dorogoy A, Banks-Sills L. Effect of crack face contact and friction on Brazilian disk specimens—A finite difference solution. Eng Fract Mech, 2005, 72(18): 2758 doi: 10.1016/j.engfracmech.2005.05.005
    [16] Ayatollahi M R, Aliha M R M. Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading. Comput Mater Sci, 2007, 38(4): 660 doi: 10.1016/j.commatsci.2006.04.008
    [17] Xu J G, Dong S M, Hua W. Effect of confining pressure on stress intensity factors determined by cracked Brazilian disk. Rock Soil Mech, 2015, 36(7): 1959

    徐积刚, 董世明, 华文. 围压对巴西裂纹圆盘应力强度因子影响分析. 岩土力学, 2015, 36(7):1959
    [18] Hua W, Xu J G, Dong S M, et al. Effect of confining pressure on stress intensity factors for cracked Brazilian disk. Int J Appl Mechanics, 2015, 7(3): 1550051 doi: 10.1142/S1758825115500519
    [19] Hua W, Li Y F, Dong S M, et al. T-stress for a centrally cracked Brazilian disk under confining pressure. Eng Fract Mech, 2015, 149: 37 doi: 10.1016/j.engfracmech.2015.09.048
    [20] Hou C, Wang Z Y, Liang W G, et al. Determination of fracture parameters in center cracked circular discs of concrete under diametral loading: A numerical analysis and experimental results. Theor Appl Fract Mech, 2016, 85: 355 doi: 10.1016/j.tafmec.2016.04.006
    [21] Li Y F, Dong S M, Hua W. T-stress for central cracked Brazilian disk subjected to compression. Rock Soil Mech, 2016, 37(11): 3191

    李一凡, 董世明, 华文. 中心裂纹巴西圆盘压缩载荷下T应力研究. 岩土力学, 2016, 37(11):3191
    [22] Hou C, Wang Z Y, Liang W G, et al. Investigation of the effects of confining pressure on SIFs and T-stress for CCBD specimens using the XFEM and the interaction integral method. Eng Fract Mech, 2017, 178: 279 doi: 10.1016/j.engfracmech.2017.03.049
    [23] Tang S B. Stress intensity factors for a Brazilian disc with a central crack subjected to compression. Int J Rock Mech Min Sci, 2017, 93: 38 doi: 10.1016/j.ijrmms.2017.01.003
    [24] Dong Z, Tang S B. Effect of confining pressure and diametrical force on stress intensity fracture in central cracked disk. Chin J Comput Mech, 2018, 35(2): 168 doi: 10.7511/jslx20170221001

    董卓, 唐世斌. 围压与径向荷载共同作用下巴西盘裂纹应力强度因子的解析解. 计算力学学报, 2018, 35(2):168 doi: 10.7511/jslx20170221001
    [25] Markides C F, Pazis D N, Kourkoulis S K. Stress intensity factors for the Brazilian disc with a short central crack: Opening versus closing cracks. Appl Math Model, 2011, 35(12): 5636 doi: 10.1016/j.apm.2011.05.013
    [26] Hondros G. The evaluation of Poisson's ratio and the modulus of materials of low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete. Australian J Appl Sci, 1959, 10(3): 243
    [27] Fairhurst C. On the validity of the ‘Brazilian’ test for brittle materials. Int J Rock Mech Min Sci Geomech Abstr, 1964, 1(4): 535 doi: 10.1016/0148-9062(64)90060-9
    [28] Hung K M, Ma C C. Theoretical analysis and digital photoelastic measurement of circular disks subjected to partially distributed compressions. Exp Mech, 2003, 43(2): 216
    [29] Wei X X, Chau K T. Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test. Int J Solids Struct, 2013, 50(14-15): 2395 doi: 10.1016/j.ijsolstr.2013.03.026
    [30] Japaridze L. Stress-deformed state of cylindrical specimens during indirect tensile strength testing. J Rock Mech Geotech Eng, 2015, 7(5): 509 doi: 10.1016/j.jrmge.2015.06.006
    [31] Yu J H, Shang X C, Wu P F. Influence of pressure distribution and friction on determining mechanical properties in the Brazilian test: Theory and experiment. Int J Solids Struct, 2019, 161: 11 doi: 10.1016/j.ijsolstr.2018.11.002
    [32] Markides C F, Kourkoulis S K. The stress field in a standardized Brazilian disc: The influence of the loading type acting on the actual contact length. Rock Mech Rock Eng, 2012, 45(2): 145 doi: 10.1007/s00603-011-0201-2
    [33] Kourkoulis S K, Markides C F, Chatzistergos P E. The Brazilian disc under parabolically varying load: Theoretical and experimental study of the displacement field. Int J Solids Struct, 2012, 49(7-8): 959 doi: 10.1016/j.ijsolstr.2011.12.013
    [34] Yu J H, Shang X C. Analysis of the influence of boundary pressure and friction on determining fracture toughness of shale using cracked Brazilian disc test. Eng Fract Mech, 2019, 212: 57 doi: 10.1016/j.engfracmech.2019.03.009
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  • 收稿日期:  2021-06-07
  • 网络出版日期:  2021-08-20

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