带裂纹非均匀各向异性材料的双周期弹性焊接第一基本问题
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摘要
双周期弹性问题作为构建各向异性损伤理论的基础问题,是弹性和断裂力学理论的重要研究课题.利用复变函数理论提出并讨论两种各向异性材料组成的无限板的平面弹性第一基本问题,板内含有的双周期分布裂纹群以及焊接界面都假设是任意光滑的曲线.运用Lekhnitskii各向异性板的复变函数理论,将求解该平面弹性问题划归为寻求满足对应边值问题的解析函数;然后构造Sherman变换得到解析函数的广义表达式;进一步利用广义Plemelj公式将问题转化为一组正则型奇异积分方程的解,并在数学上严格证明积分方程的唯一可解性.
Doubly periodic plane elasticity problem is very useful in elasticity and fracture of isotropic and anisotropic elastic materials.In this paper,the first fundamental plane elasticity welding problem is investigated.Employing the complex variable function method,solving this problem is transferred into seeking analytic functions which fit certain boundary value problems.Further,under some general restrictions,Sherman's transform for doubly-periodic elasticity theory of homogeneous anisotropic materials to that of inhomogeneous materials is developed.Then,using the general representation for the solution,the boundary value problem is reduced to a normal type singular integral equation with a Weierstrass zeta kernel along the boundary of cracks and welding interface,and the unique solvability of which is established.
Doubly periodic plane elasticity problem is very useful in elasticity and fracture of isotropic and anisotropic elastic materials.In this paper,the first fundamental plane elasticity welding problem is investigated.Employing the complex variable function method,solving this problem is transferred into seeking analytic functions which fit certain boundary value problems.Further,under some general restrictions,Sherman's transform for doubly-periodic elasticity theory of homogeneous anisotropic materials to that of inhomogeneous materials is developed.Then,using the general representation for the solution,the boundary value problem is reduced to a normal type singular integral equation with a Weierstrass zeta kernel along the boundary of cracks and welding interface,and the unique solvability of which is established.
引文
[1]路见可.双周期平面弹性理论中的复Airy函数[J].数学杂志,1986,6(3):319-330.
[2]郑可.双周期平面弹性基本问题[J].数学物理学报,1988,8(1):95-104.
[3]郑可.带裂缝的双周期平面弹性基本问题[J].数学物理学报,1988,8(3):321-326.
[4]郑可.几类带双周期裂缝的无限平面弹性问题[J].数学杂志,1990,10(3):325-328.
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[6]郑可.双周期平面弹性混合问题[J].应用数学与计算数学学报,1997,20(1):77-85.
[7]李星.三维各向同性不同材料带双周期孔洞的弹性焊接混合边值问题[J].数学物理学报,1992,12(4):397-406.
[8]李星.双周期裂纹场的不同材料弹性平面焊接问题[J].数学物理学报,1993,13(2):124-132.
[9]李星.双周期裂纹场平面弹性焊接的数学问题[J].应用数学和力学,1993,14(12):1085-1092.
[10]LI Xing.The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain problem[J].International Journal of Fracture,2001,109(4):403-411.
[11]LI Xing.Applications of doubly quasi-periodic boundary value problems in elasticity theory[M].Aachen:Shaker Verlag,2001.
[12]LI Xing.Complete plane strain problem of a nonhomogeneous elastic body with a doubly-periodic set of cracks[J].ZAMM-Journal of Applied Mathematics and Mechanics,2001,81(6):377-391.
[13]LI Xing.General solution for complete plane strain problem of a nonhomogeneous body with a doubly-periodic set of inlays[J].ZAMM-Journal of Applied Mathematics and Mechanics,2006,86(9):682-690.
[14]LI Xing.Modified doubly-periodic second fundamental complete plane strain problem with relative displacements[J].Journal of Applied Functional Analysis,2007,2:99-114.
[15]时朋朋,李星.含双周期裂纹的一维六方准晶电弹性全平面应变基本问题[J].应用力学学报,2014,31(2):251-256.
[16]SHI Pengpeng.Interaction between the doubly periodic interfacial cracks in a layered periodic composite:Simulation by the method of singular integral equation[J].Theoretical and Applied Fracture Mechanics,2015,78(1):25-39.
[17]SHI Pengpeng.On the plastic zone size of solids containing doubly periodic rectangular-shaped arrays of cracks under longitudinal shear[J].Mechanics Research Communications,2015,67:39-46.
[18]LEKHNITSKII S G.Theory of elasticity of an anisotropic body[M].Moscow:Mir Publishers,1981.
[2]郑可.双周期平面弹性基本问题[J].数学物理学报,1988,8(1):95-104.
[3]郑可.带裂缝的双周期平面弹性基本问题[J].数学物理学报,1988,8(3):321-326.
[4]郑可.几类带双周期裂缝的无限平面弹性问题[J].数学杂志,1990,10(3):325-328.
[5]郑可.带裂缝的双周期各向异性平面弹性基本问题[J].应用数学与计算数学学报,1993,7(1):14-20.
[6]郑可.双周期平面弹性混合问题[J].应用数学与计算数学学报,1997,20(1):77-85.
[7]李星.三维各向同性不同材料带双周期孔洞的弹性焊接混合边值问题[J].数学物理学报,1992,12(4):397-406.
[8]李星.双周期裂纹场的不同材料弹性平面焊接问题[J].数学物理学报,1993,13(2):124-132.
[9]李星.双周期裂纹场平面弹性焊接的数学问题[J].应用数学和力学,1993,14(12):1085-1092.
[10]LI Xing.The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain problem[J].International Journal of Fracture,2001,109(4):403-411.
[11]LI Xing.Applications of doubly quasi-periodic boundary value problems in elasticity theory[M].Aachen:Shaker Verlag,2001.
[12]LI Xing.Complete plane strain problem of a nonhomogeneous elastic body with a doubly-periodic set of cracks[J].ZAMM-Journal of Applied Mathematics and Mechanics,2001,81(6):377-391.
[13]LI Xing.General solution for complete plane strain problem of a nonhomogeneous body with a doubly-periodic set of inlays[J].ZAMM-Journal of Applied Mathematics and Mechanics,2006,86(9):682-690.
[14]LI Xing.Modified doubly-periodic second fundamental complete plane strain problem with relative displacements[J].Journal of Applied Functional Analysis,2007,2:99-114.
[15]时朋朋,李星.含双周期裂纹的一维六方准晶电弹性全平面应变基本问题[J].应用力学学报,2014,31(2):251-256.
[16]SHI Pengpeng.Interaction between the doubly periodic interfacial cracks in a layered periodic composite:Simulation by the method of singular integral equation[J].Theoretical and Applied Fracture Mechanics,2015,78(1):25-39.
[17]SHI Pengpeng.On the plastic zone size of solids containing doubly periodic rectangular-shaped arrays of cracks under longitudinal shear[J].Mechanics Research Communications,2015,67:39-46.
[18]LEKHNITSKII S G.Theory of elasticity of an anisotropic body[M].Moscow:Mir Publishers,1981.