含裂纹简支梁在均布荷载作用下的内聚区模型解析函数
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摘要
对于含切口简支梁受均布荷载作用的问题,基于Williams应力函数,通过边界配置法并借用无裂纹体应力边界条件,求得了含高阶项的全场解析解及相应的应力强度因子K_Ⅰ。基于"Duan and Nakagawa’s"模型,通过对首项(奇异项)进行加权积分,消除了裂缝尖端应力呈无穷大的奇异性,得到了内聚区模型的全场解析解。通过对不同解法下典型截面正应力分布的比较,表明内聚区模型解消除了裂缝尖端应力的奇异性,比函数叠加法的结果精度更高,这样的数学力学模型可以从宏观上反映混凝土类材料的断裂特性。
The considering problem is a simply supported beam with an edge-crack under uniform distributed load. Based on the Williams' stress function, the series analytic solution with higher order items and the corresponding stress intensity factor is yielded, in which,the boundary conditions are satisfied by means of boundary collocation method compared to a simply supported beam with no-crack. Based on the ‘Duan and Nakagawa's model,the cohesive zone model analytic solution is obtained by weighted integrating the first item(singular item).By comparing the normal stress distribution in some typical sections between different analytic methods,it is shown that the present solution eliminates the stress singularity at the crack tip, and is more accurate than the result by superposition of analytical functions. Such mathematical and mechanical models can macroscopically represent the fracture characteristics of concrete and other similar materials.
The considering problem is a simply supported beam with an edge-crack under uniform distributed load. Based on the Williams' stress function, the series analytic solution with higher order items and the corresponding stress intensity factor is yielded, in which,the boundary conditions are satisfied by means of boundary collocation method compared to a simply supported beam with no-crack. Based on the ‘Duan and Nakagawa's model,the cohesive zone model analytic solution is obtained by weighted integrating the first item(singular item).By comparing the normal stress distribution in some typical sections between different analytic methods,it is shown that the present solution eliminates the stress singularity at the crack tip, and is more accurate than the result by superposition of analytical functions. Such mathematical and mechanical models can macroscopically represent the fracture characteristics of concrete and other similar materials.
引文
[1]WILLIAMS M L.Bending stress distribution at the base of stationary crack[J].Journal of applied mechanics,1961,28(1):78-82.
[2]胡若邻,黄培炎,郑顺潮.混凝土断裂过程区尺寸的理论推导[J].工程力学,2010,27(6):127-132.(HU Ruolin,HUANG Peiyan,ZHENG Shunchao.Theoretical derivation of the size of fracture process zone of concrete[J].Engineering mechanics,2010,27(6):127-132(in Chinese)).
[3]王元汉,伍佑伦,余飞.板弯曲断裂计算的边界配置解法[J].应用数学和力学,2003,24(6):605-610.(WANG Yuanhan,WUYoulun,YU Fei.Fracture calculation of bending plates by boundary collocation method[J].Applied mathematics and mechanics,2003,24(6):605-610(in Chinese)).
[4]高配琴.关于Wlliiams应力函数边界配置法解的收敛性及有效性探讨[J].西北工业大学学报,1985,3(2):251-260.(GAO Peiqin.An exploration of the convergence and validity of the results of boundary collocation procedures using Williams stress function[J].Journal of northwestern polytechnical university,1985,3(2):251-260(in Chinese)).
[5]熊先仁,袁去惑,杨菊梅,等.应力强度因子的边界配置法程序及工程应用[J].江西科学,1996,14(2):106-113.(XIONG Xianren,YUAN Quhuo,YANG Jumei,et al.The procedure of the mode of boundary collocation of stress intensity factor and its application[J].Jiangxi science,1996,14(2):106-113(in Chinese)).
[6]BARENBLATT G I.The formation of equilibrium cracks during brittle fracture,general ideas and hypotheses,axially-symmetric cracks[J].Journal of applied mathematics&mechanics,1959,23(3):622-636.
[7]HILLERBORG A.Analysis of fracture by means of the fictitious crack model,particularly for fiber reinforced concrete[J].International journal of cement composites,1980,2(4):177-184.
[8]PETERSSON P E.Fracture energy of concrete:method and determination[J].Cement and concrete research,1980,10(1):78-89.
[9]王青原,陈红鸟,张华刚,等.三点弯曲下混凝土梁挠度与裂缝口张开位移关系[J].应用力学学报,2017,34(5):937-944.(WANGQingyuan,CHEN Hongniao,ZHANG Huagang,et al.Relationship of deflection and crack mouth opening displacement of concrete beams under the three-point bending[J].Chinese journal of applied mechanics,2017,34(5):937-944(in Chinese)).
[10]张鹏,胡小飞,姚伟岸.内聚力模型裂纹问题分析的解析奇异单元[J].固体力学学报,2017,38(2):157-164.(ZHANG Peng,HU Xiaofei,YAO Weian.An analytical singular element to study the cohesive zone model for cracks[J].Chinese journal of solid mechanics,2017,38(2):157-164(in Chinese)).
[11]段树金,张彦龙,安蕊梅.基于裂纹尖端二阶弹性解的断裂过程区尺度[J].应用数学和力学,2013,34(6):598-605.(DUAN Shujin,ZHANG Yanlong,AN Ruimei.Fracture process zone size based on secondary elastic crack tip stress solution[J].Applied mathematics and mechanics,2013,34(6):598-605(in Chinese)).
[12]王承强,郑长良.裂纹扩展过程中线性内聚力模型计算的半解析有限元法[J].计算力学学报,2006,23(2):146-151.(WANGChengqiang,ZHENG Changliang.Semi-analytical finite element method for linear cohesive force model in crack propagation[J].Chinese journal of computational mechanics,2006,23(2):146-151(in Chinese)).
[13]卿龙邦,李庆斌,管俊峰.混凝土断裂过程区长度计算方法研究[J].工程力学,2012,29(4):197-201.(QING Longbang,LI Qingbin,GUAN Junfeng.Calculation method of the length of fracture process zone of concrete[J].Engineering mechanics,2012,29(4):197-201(in Chinese)).
[14]卿龙邦,姜军,周明杰,等.基于细观数值模拟的混凝土断裂过程区特性研究[J].混凝土,2016(4):9-12.(QING Longbang,JIANGJun,ZHOU Mingjie,et al.Calculation the properties of fracture process zone on meso-scale in concrete[J].Concrete,2016(4):9-12(in Chinese)).
[15]王元汉.边裂纹物体反平面剪切时的边界配置法[J].华中理工大学学报,1992,20:117-120.(WANG Yuanhan.An edge cracked body in anti-planar shear calculated by the boundary collocation method[J].Journal of Huazhong university of science and technology,1992,20:117-120(in Chinese)).
[16]DUAN S J,NAKAGAWA K.Stress functions with finite stress concentration at the crack tips for central cracked panel[J].Engineering fracture mechanics,1988,29(5):517-526.
[17]DUAN S J,NAKAGAWA K.A mathematical approach of fracture macromechanics for strain-softening material[J].Engineering fracture mechanics,1989,34(5/6):1175-1182.
[18]郭全民,段树金.混凝土简支梁在均布荷载作用下的断裂过程区及自振特性[J].石家庄铁道大学学报(自然学科版),2017,30(2):6-10.(GUO Quanmin,DUAN Shujin.Fracture process zone and natural vibration characteristics of concrete beam under uniform distributed forces[J].Journal of Shijiazhuang Tiedao university(natural science edition),2017,30(2):6-10(in Chinese)).
[19]解沅衡,段树金,侯永康,等.断裂过程区内聚力布对简支梁刚度和自振特性的影响研究[J].石家庄铁道大学学报(自然学科版),2018,31(2):1-6.(XIE Yuanheng,DUAN Shujin,HOU Yongkang,et al.A study on influence of the cohesion in fracture process zone to stiffness and natural vibration frequency of a simply supported beam[J].Journal of Shijiazhuang Tiedao university(natural science edition),2018,31(2):1-6(in Chinese)).
[2]胡若邻,黄培炎,郑顺潮.混凝土断裂过程区尺寸的理论推导[J].工程力学,2010,27(6):127-132.(HU Ruolin,HUANG Peiyan,ZHENG Shunchao.Theoretical derivation of the size of fracture process zone of concrete[J].Engineering mechanics,2010,27(6):127-132(in Chinese)).
[3]王元汉,伍佑伦,余飞.板弯曲断裂计算的边界配置解法[J].应用数学和力学,2003,24(6):605-610.(WANG Yuanhan,WUYoulun,YU Fei.Fracture calculation of bending plates by boundary collocation method[J].Applied mathematics and mechanics,2003,24(6):605-610(in Chinese)).
[4]高配琴.关于Wlliiams应力函数边界配置法解的收敛性及有效性探讨[J].西北工业大学学报,1985,3(2):251-260.(GAO Peiqin.An exploration of the convergence and validity of the results of boundary collocation procedures using Williams stress function[J].Journal of northwestern polytechnical university,1985,3(2):251-260(in Chinese)).
[5]熊先仁,袁去惑,杨菊梅,等.应力强度因子的边界配置法程序及工程应用[J].江西科学,1996,14(2):106-113.(XIONG Xianren,YUAN Quhuo,YANG Jumei,et al.The procedure of the mode of boundary collocation of stress intensity factor and its application[J].Jiangxi science,1996,14(2):106-113(in Chinese)).
[6]BARENBLATT G I.The formation of equilibrium cracks during brittle fracture,general ideas and hypotheses,axially-symmetric cracks[J].Journal of applied mathematics&mechanics,1959,23(3):622-636.
[7]HILLERBORG A.Analysis of fracture by means of the fictitious crack model,particularly for fiber reinforced concrete[J].International journal of cement composites,1980,2(4):177-184.
[8]PETERSSON P E.Fracture energy of concrete:method and determination[J].Cement and concrete research,1980,10(1):78-89.
[9]王青原,陈红鸟,张华刚,等.三点弯曲下混凝土梁挠度与裂缝口张开位移关系[J].应用力学学报,2017,34(5):937-944.(WANGQingyuan,CHEN Hongniao,ZHANG Huagang,et al.Relationship of deflection and crack mouth opening displacement of concrete beams under the three-point bending[J].Chinese journal of applied mechanics,2017,34(5):937-944(in Chinese)).
[10]张鹏,胡小飞,姚伟岸.内聚力模型裂纹问题分析的解析奇异单元[J].固体力学学报,2017,38(2):157-164.(ZHANG Peng,HU Xiaofei,YAO Weian.An analytical singular element to study the cohesive zone model for cracks[J].Chinese journal of solid mechanics,2017,38(2):157-164(in Chinese)).
[11]段树金,张彦龙,安蕊梅.基于裂纹尖端二阶弹性解的断裂过程区尺度[J].应用数学和力学,2013,34(6):598-605.(DUAN Shujin,ZHANG Yanlong,AN Ruimei.Fracture process zone size based on secondary elastic crack tip stress solution[J].Applied mathematics and mechanics,2013,34(6):598-605(in Chinese)).
[12]王承强,郑长良.裂纹扩展过程中线性内聚力模型计算的半解析有限元法[J].计算力学学报,2006,23(2):146-151.(WANGChengqiang,ZHENG Changliang.Semi-analytical finite element method for linear cohesive force model in crack propagation[J].Chinese journal of computational mechanics,2006,23(2):146-151(in Chinese)).
[13]卿龙邦,李庆斌,管俊峰.混凝土断裂过程区长度计算方法研究[J].工程力学,2012,29(4):197-201.(QING Longbang,LI Qingbin,GUAN Junfeng.Calculation method of the length of fracture process zone of concrete[J].Engineering mechanics,2012,29(4):197-201(in Chinese)).
[14]卿龙邦,姜军,周明杰,等.基于细观数值模拟的混凝土断裂过程区特性研究[J].混凝土,2016(4):9-12.(QING Longbang,JIANGJun,ZHOU Mingjie,et al.Calculation the properties of fracture process zone on meso-scale in concrete[J].Concrete,2016(4):9-12(in Chinese)).
[15]王元汉.边裂纹物体反平面剪切时的边界配置法[J].华中理工大学学报,1992,20:117-120.(WANG Yuanhan.An edge cracked body in anti-planar shear calculated by the boundary collocation method[J].Journal of Huazhong university of science and technology,1992,20:117-120(in Chinese)).
[16]DUAN S J,NAKAGAWA K.Stress functions with finite stress concentration at the crack tips for central cracked panel[J].Engineering fracture mechanics,1988,29(5):517-526.
[17]DUAN S J,NAKAGAWA K.A mathematical approach of fracture macromechanics for strain-softening material[J].Engineering fracture mechanics,1989,34(5/6):1175-1182.
[18]郭全民,段树金.混凝土简支梁在均布荷载作用下的断裂过程区及自振特性[J].石家庄铁道大学学报(自然学科版),2017,30(2):6-10.(GUO Quanmin,DUAN Shujin.Fracture process zone and natural vibration characteristics of concrete beam under uniform distributed forces[J].Journal of Shijiazhuang Tiedao university(natural science edition),2017,30(2):6-10(in Chinese)).
[19]解沅衡,段树金,侯永康,等.断裂过程区内聚力布对简支梁刚度和自振特性的影响研究[J].石家庄铁道大学学报(自然学科版),2018,31(2):1-6.(XIE Yuanheng,DUAN Shujin,HOU Yongkang,et al.A study on influence of the cohesion in fracture process zone to stiffness and natural vibration frequency of a simply supported beam[J].Journal of Shijiazhuang Tiedao university(natural science edition),2018,31(2):1-6(in Chinese)).