黏弹性接触界面端附近的奇异应力场
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摘要
研究蠕变加载条件下线黏弹性材料接触界面端附近的奇异应力场问题.考虑接触界面的摩擦,假设界面端的滑移方向不改变,相对滑移量微小,且其与位移同量级,由此线性化局部边界条件,根据对应原理得到Laplace变换域中的界面端应力场,导出时域中奇异应力场的卷积积分表达式.对卷积积分核函数进行数值反演,考虑接触材料的两类组合,一是持久模量具有量级上的差异,另一是持久模量接近相同.算例结果证实核函数可以用准弹性法求得的解析式较准确地近似.在此基础上,利用积分中值定理,并引入各应力分量的修正系数,得到黏弹性奇异应力场的简化式.结合核函数的数值反演结果分析修正系数表达式的取值范围,得到如下结论,若两相接触材料的持久模量相差很大,可以采用准弹性解的解析式较准确地描述界面端的奇异应力场;一般情况下,应力场不存在统一的奇异值和应力强度系数,当采用类似于准弹性解的表达式近似给出黏弹性应力场时,可以估计此近似描述的误差限.文中最后采用有限元分析黏弹性板端部嵌入部位的应力场,算例包括了黏弹性板与弹性金属支承、黏弹性板与黏弹性垫层所形成的滑移接触界面端,利用黏弹性有限元的数值结果验证理论分析所得结论的有效性.
The paper concerns the problem of singular stress filed in viscoelastic contact interface ends under creep loading. The local boundary conditions taking into account the contact friction are linearized by the assumptions of tiny relative slip and invariant slip direction between interfaces. The solution of stress field at the interface end in the Laplace transform domain is obtained based on the correspondence principle, and the convolution integral expressions of the singular stress field in the time domain is developed. The numerical inversion of convolution integral kernel is made by considering two types of combinations of contact materials. One is that the durable modulus has a difference in magnitude, and the other is that the durable modulus is nearly the same. The results of inversion show that kernel functions can be approximated by analytical expressions obtained by the quasi-elastic method with a good accuracy.On this basis, simplified formulas of the viscoelastic singular stress field are developed by using the integral mean value theorem and introducing the correction coefficient of each stress component. The value range of expressions for correction coefficient is investigated in combination with the examination the numerical inversion results of the kernel functions,following conclusions are drawn as follows. If the durable modulus of the two-phase contact material differs greatly,the quasi-elastic solution can be used to describe the singular stress fields near the interface end; in general, there is no uniform singular value and no uniform stress intensity factor for stress fields; when the solution of viscoelastic stress is approximated by formulas similar to the quasi-elastic solution, the error limit can be estimated. In the last part of the paper, the viscoelastic stress analysis of viscoelastic plate at support ends is performed by means of finite element simulation as plane strain problem. The example includes two types of contact interface ends, one is constructed by the viscoelastic plate and an elastic metal support, the other is formed by a viscoelastic plate and viscoelastic cushion layers.The theoretical conclusions obtained in the front part of the paper are validated by the simulation results.
The paper concerns the problem of singular stress filed in viscoelastic contact interface ends under creep loading. The local boundary conditions taking into account the contact friction are linearized by the assumptions of tiny relative slip and invariant slip direction between interfaces. The solution of stress field at the interface end in the Laplace transform domain is obtained based on the correspondence principle, and the convolution integral expressions of the singular stress field in the time domain is developed. The numerical inversion of convolution integral kernel is made by considering two types of combinations of contact materials. One is that the durable modulus has a difference in magnitude, and the other is that the durable modulus is nearly the same. The results of inversion show that kernel functions can be approximated by analytical expressions obtained by the quasi-elastic method with a good accuracy.On this basis, simplified formulas of the viscoelastic singular stress field are developed by using the integral mean value theorem and introducing the correction coefficient of each stress component. The value range of expressions for correction coefficient is investigated in combination with the examination the numerical inversion results of the kernel functions,following conclusions are drawn as follows. If the durable modulus of the two-phase contact material differs greatly,the quasi-elastic solution can be used to describe the singular stress fields near the interface end; in general, there is no uniform singular value and no uniform stress intensity factor for stress fields; when the solution of viscoelastic stress is approximated by formulas similar to the quasi-elastic solution, the error limit can be estimated. In the last part of the paper, the viscoelastic stress analysis of viscoelastic plate at support ends is performed by means of finite element simulation as plane strain problem. The example includes two types of contact interface ends, one is constructed by the viscoelastic plate and an elastic metal support, the other is formed by a viscoelastic plate and viscoelastic cushion layers.The theoretical conclusions obtained in the front part of the paper are validated by the simulation results.
引文
1孙见君,嵇正波,马晨波.粗糙表面接触力学问题的重新分析.力学学报,2018,50(1):68-77(Sun Jianjun,Ji Zhengbo,Ma Chenbo.Reanalysis of the contact mechanics for rough surfaces.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):68-77(in Chinese))
2王雯,吴洁蓓,高志强等.机械结合面切向接触阻尼计算模型.力学学报,2018,50(3):633-642(Wang Wen,Wu Jiebei,Gao Zhiqiang et al.A calculation model for tangential contact damping of machine joint interfaces.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):633-642(in Chinese))
3薛冰寒,林皋,胡志强等.摩擦接触问题的比例边界等几何B可微方程组方法.力学学报,2016,48(3):615-623(Xue Binghan,Lin Gao,Hu Zhiqiang,et al.Analysis of frictional contact problems by SBIGA-BDE method.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):615-623(in Chinese))
4柯燎亮,汪越胜.功能梯度材料接触力学若干基本问题的研究进展.科学通报,2015,60(17):1565-1573(Ke Liaoliang,Wang Yuesheng.Progress in some basic problems on contact mechanics of functionally graded materials.Chin Sci Bull,2015,60(17):1565-1573(in Chinese))
5许金泉.界面力学.2006,北京:科学出版社(Xu Jinquan.The Mechanics of Interface.Beijing:Science Press,2006(in Chinese))
6 Xu JQ,Yoshiharu M.Stress field near the contact edge in fretting fatigue tests.JSME International Journal,Series A,2002,45(2):510-516
7 Whitney TJ,Iarve EV,Brockman RA.Singular stress fields near contact boundaries in a composite bolted joint.International Journal of Solids&Structures,2004,41(7):1893-1909
8 Chen H,Guo Z,Zhou X.Stress singularities of contact problems with a frictional interface in anisotr-opic bimaterials.Fatigue&Fracture of Engineering Materials&Structures,2012,35(8):718-731
9平学成,吴卫星,陈梦成等.铆接界面端三维奇异性应力场的研究.工程力学,2017,34(6):226-235(Ping Xuecheng,Wu Weixing,Chen Mengcheng,et al.Research on 3-D singular stress fields near contact boundarys of riveted assemblies.Engineering Mechanics,2017,34(6):226-235(in Chinese))
10 Hattori T,Nakamura M,Watanabe T.Simulation of fretting-fatigue life by using stress-singularity parameters and fracture mechanics.Tribology International,2003,36(1):87-97
11 Jacob MSD,Arora PR,Saleem M,et al.Fretting fatigue crack initiation:An experimental and theoretical study.International Journal of Fatigue,2007,29(2):1328-1338
12平学成,赵辽翔.不完全接触端微动疲劳强度界面力学评估.机械设计与制造,2014,6:269-272(Ping Xuecheng,Zhao Liaoxiang.Incomplete contact end interfacial mechanics assess of fretting fatigue strength.Machinery Design and Manufacture,2014,6:269-272(in Chinese))
13张志林.飞机座舱透明件设计理论及应用.[博士论文].南京:南京航天航空大学,2005(Zhang Zhilin.Design theory and application for transparency of aircraft canopy and windshield.[PhD Thesis].Nanjing:Nanjing University of Aeronautics and Astronautics,2005(in Chinese))
14刘道启,胡勇,王芳等.载人深潜器观察窗的力学性能.船舶力学,2010,14(7):782-788(Liu Daoqi,HuYong,Wang Fang,et al.Mechanics analysis on deep-sea human occupied vehicle’s viewport windows.Journal of Ship Mechanics,2010,14(7):782-788(in Chinese)
15 Lee SS.Boundary element analysis of the stress singularity at the interface corner of viscoelastic adhesive layers.International Journal of Solids and Structures,1998,35(13):1385-1394
16 Qian ZQ,Akisanya AR,Imbabi MS.Edge effects in the failure of elastic/viscoelastic joints subjected to surface tractions.International Journal of Solids and Structures,2000,37(41):5973-5994
17 Kay N,Barut A,Madenci E.Singular stresses in a finite region of two dissimilar viscoelastic materials with traction-free edges.Computer Methods in Applied Mechanics and Engineering,2002,191(11-12):1221-1244
18郑慧淼,许金泉.黏弹性结合材料界面端奇异场.力学季刊,2008,29(2):187-193(Zheng Huimiao,Xu Jinquan.Viscoelastic singular behavior near an interface edge.Chinese Quarterly of Mechanics,2008,29(2):187-193(in Chinese))
19 Chowdhuri MAK,Xia Z.An analytical model to determine the stress singularity and critical bonding angle for an elastic-viscoelastic bonded joint.Mechanics of Time-Dependent Materials,2012,16(3):343-359
20顾志旭,郑坚,彭威等.固体发动机中轴对称界面端应力奇异性研究.固体火箭技术,2014,37(4):510-515(Gu Zhixu,Zheng Jian,Peng Wei,et al.Study of singular stress fields around axisymmetric joints in solid motor.Journal of Solid Rocket Technology,2014,37(4):510-515(in Chinese))
21 Kuo TL,Hwu CB,Interface corners in linear anisotropic viscoelastic materials.International Journal of Solids and Structures,2013,50(5):710-724
22 Kitamura T,Ngampungpis K,Hirakata H.Stress field near interface edge of elastic-creep bi-material.Engineering Fracture Mechanics,2007,74(10):1637-1648
23 Takahashi Y,Inoue K,Takuma M,et al.Fracture mechanics criterion of time-dependent crack initiation from interface free-edge in adhesively bonded butt joints.Engineering Fracture Mechanics,2017,186(2):368-377
24 Bellman R,Kalaba RE,Lockett J.Numerical Inversion of the Laplace Transform.Americal Elsevier Publish Corporation,1966
25 Swanson SR.Approximate Laplace transform inversion in dynamic viscoelasticity.ASME Journal of Applied Mechanics,1980,47(6):769-774
26彭凡,马庆镇,戴宏亮.黏弹性功能梯度材料裂纹问题的有限元方法.力学学报,2013,45(3):359-366(Peng Fan,Ma Qingzhen,Dai Hongliang.Finite element method for crack problems in viscoelastic functionally graded materials.Chinese Journal of Theoretical and Applied Mechanics,2013,45(3):359-366(in Chinese))
2王雯,吴洁蓓,高志强等.机械结合面切向接触阻尼计算模型.力学学报,2018,50(3):633-642(Wang Wen,Wu Jiebei,Gao Zhiqiang et al.A calculation model for tangential contact damping of machine joint interfaces.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):633-642(in Chinese))
3薛冰寒,林皋,胡志强等.摩擦接触问题的比例边界等几何B可微方程组方法.力学学报,2016,48(3):615-623(Xue Binghan,Lin Gao,Hu Zhiqiang,et al.Analysis of frictional contact problems by SBIGA-BDE method.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):615-623(in Chinese))
4柯燎亮,汪越胜.功能梯度材料接触力学若干基本问题的研究进展.科学通报,2015,60(17):1565-1573(Ke Liaoliang,Wang Yuesheng.Progress in some basic problems on contact mechanics of functionally graded materials.Chin Sci Bull,2015,60(17):1565-1573(in Chinese))
5许金泉.界面力学.2006,北京:科学出版社(Xu Jinquan.The Mechanics of Interface.Beijing:Science Press,2006(in Chinese))
6 Xu JQ,Yoshiharu M.Stress field near the contact edge in fretting fatigue tests.JSME International Journal,Series A,2002,45(2):510-516
7 Whitney TJ,Iarve EV,Brockman RA.Singular stress fields near contact boundaries in a composite bolted joint.International Journal of Solids&Structures,2004,41(7):1893-1909
8 Chen H,Guo Z,Zhou X.Stress singularities of contact problems with a frictional interface in anisotr-opic bimaterials.Fatigue&Fracture of Engineering Materials&Structures,2012,35(8):718-731
9平学成,吴卫星,陈梦成等.铆接界面端三维奇异性应力场的研究.工程力学,2017,34(6):226-235(Ping Xuecheng,Wu Weixing,Chen Mengcheng,et al.Research on 3-D singular stress fields near contact boundarys of riveted assemblies.Engineering Mechanics,2017,34(6):226-235(in Chinese))
10 Hattori T,Nakamura M,Watanabe T.Simulation of fretting-fatigue life by using stress-singularity parameters and fracture mechanics.Tribology International,2003,36(1):87-97
11 Jacob MSD,Arora PR,Saleem M,et al.Fretting fatigue crack initiation:An experimental and theoretical study.International Journal of Fatigue,2007,29(2):1328-1338
12平学成,赵辽翔.不完全接触端微动疲劳强度界面力学评估.机械设计与制造,2014,6:269-272(Ping Xuecheng,Zhao Liaoxiang.Incomplete contact end interfacial mechanics assess of fretting fatigue strength.Machinery Design and Manufacture,2014,6:269-272(in Chinese))
13张志林.飞机座舱透明件设计理论及应用.[博士论文].南京:南京航天航空大学,2005(Zhang Zhilin.Design theory and application for transparency of aircraft canopy and windshield.[PhD Thesis].Nanjing:Nanjing University of Aeronautics and Astronautics,2005(in Chinese))
14刘道启,胡勇,王芳等.载人深潜器观察窗的力学性能.船舶力学,2010,14(7):782-788(Liu Daoqi,HuYong,Wang Fang,et al.Mechanics analysis on deep-sea human occupied vehicle’s viewport windows.Journal of Ship Mechanics,2010,14(7):782-788(in Chinese)
15 Lee SS.Boundary element analysis of the stress singularity at the interface corner of viscoelastic adhesive layers.International Journal of Solids and Structures,1998,35(13):1385-1394
16 Qian ZQ,Akisanya AR,Imbabi MS.Edge effects in the failure of elastic/viscoelastic joints subjected to surface tractions.International Journal of Solids and Structures,2000,37(41):5973-5994
17 Kay N,Barut A,Madenci E.Singular stresses in a finite region of two dissimilar viscoelastic materials with traction-free edges.Computer Methods in Applied Mechanics and Engineering,2002,191(11-12):1221-1244
18郑慧淼,许金泉.黏弹性结合材料界面端奇异场.力学季刊,2008,29(2):187-193(Zheng Huimiao,Xu Jinquan.Viscoelastic singular behavior near an interface edge.Chinese Quarterly of Mechanics,2008,29(2):187-193(in Chinese))
19 Chowdhuri MAK,Xia Z.An analytical model to determine the stress singularity and critical bonding angle for an elastic-viscoelastic bonded joint.Mechanics of Time-Dependent Materials,2012,16(3):343-359
20顾志旭,郑坚,彭威等.固体发动机中轴对称界面端应力奇异性研究.固体火箭技术,2014,37(4):510-515(Gu Zhixu,Zheng Jian,Peng Wei,et al.Study of singular stress fields around axisymmetric joints in solid motor.Journal of Solid Rocket Technology,2014,37(4):510-515(in Chinese))
21 Kuo TL,Hwu CB,Interface corners in linear anisotropic viscoelastic materials.International Journal of Solids and Structures,2013,50(5):710-724
22 Kitamura T,Ngampungpis K,Hirakata H.Stress field near interface edge of elastic-creep bi-material.Engineering Fracture Mechanics,2007,74(10):1637-1648
23 Takahashi Y,Inoue K,Takuma M,et al.Fracture mechanics criterion of time-dependent crack initiation from interface free-edge in adhesively bonded butt joints.Engineering Fracture Mechanics,2017,186(2):368-377
24 Bellman R,Kalaba RE,Lockett J.Numerical Inversion of the Laplace Transform.Americal Elsevier Publish Corporation,1966
25 Swanson SR.Approximate Laplace transform inversion in dynamic viscoelasticity.ASME Journal of Applied Mechanics,1980,47(6):769-774
26彭凡,马庆镇,戴宏亮.黏弹性功能梯度材料裂纹问题的有限元方法.力学学报,2013,45(3):359-366(Peng Fan,Ma Qingzhen,Dai Hongliang.Finite element method for crack problems in viscoelastic functionally graded materials.Chinese Journal of Theoretical and Applied Mechanics,2013,45(3):359-366(in Chinese))