非对称格林函数的指数梯度横向各向同性基体涂层系统（英文）
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摘要
通过研究一组完整的两个位移势,推导了指数梯度横向各向同性基体涂层系统弹性静力学格林函数的解析式。以线积分的形式表示三维点荷载和片荷载格林函数的应力和位移。该公式基于傅里叶级数和汉克尔变换,包括一组完整的转换应力-位移和位移-位势关系。为了便于说明,格林函数被简化为特殊情况,如指数级半空间和均匀双层半空间格林函数。由于被积函数的复杂性,对积分进行了数值计算,并且针对积分的数值计算,提出了一个强有力有效的方法,该方法考虑了积分奇点存在的情况。对现有的均匀双层各向同性和各向同性的半空间数值解进行比较,以确定本方案的准确性。通过典型的数值分析的例子,展示了指数梯度双层半空间格林函数的一般特征,据此可以识别材料性能变化程度。
By virtue of a complete set of two displacement potentials, an analytical derivation of the elastostatic Green's functions of an exponentially graded transversely isotropic substrate–coating system is presented. Three-dimensional point–load and patch–load Green's functions for stresses and displacements are given in line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, with utilizing Fourier series and Hankel transforms. As illustrations, the present Green's functions are degenerated to the special cases such as an exponentially graded half-space and a homogeneous two-layered half-space Green's functions. Because of complicated integrand functions, the integrals are evaluated numerically and for numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities of integration. Comparisons of the existing numerical solutions for homogeneous two-layered isotropic and transversely isotropic half-spaces are made to confirm the accuracy of the present solutions. Some typical numerical examples are also given to show the general features of the exponentially graded two-layered half-space Green's functions that the effect of degree of variation of material properties will be recognized.
By virtue of a complete set of two displacement potentials, an analytical derivation of the elastostatic Green's functions of an exponentially graded transversely isotropic substrate–coating system is presented. Three-dimensional point–load and patch–load Green's functions for stresses and displacements are given in line-integral representations. The formulation includes a complete set of transformed stress–potential and displacement–potential relations, with utilizing Fourier series and Hankel transforms. As illustrations, the present Green's functions are degenerated to the special cases such as an exponentially graded half-space and a homogeneous two-layered half-space Green's functions. Because of complicated integrand functions, the integrals are evaluated numerically and for numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities of integration. Comparisons of the existing numerical solutions for homogeneous two-layered isotropic and transversely isotropic half-spaces are made to confirm the accuracy of the present solutions. Some typical numerical examples are also given to show the general features of the exponentially graded two-layered half-space Green's functions that the effect of degree of variation of material properties will be recognized.
引文
[1]SBURLATI R.Elastic solutions in a functionally graded coating subjected to a concentrated force[J].Journal of Mechanics of Materials and Structures,2012,7:401-412.
[2]ESKANDARI M,SHODJA H M.Green’s functions of an exponentially graded transversely isotropic half-space[J].International Journal of Solids and Structures,2010,47:1537-1545.
[3]KIEBACK B,NEUBRAND A,RIEDEL H.Processing techniques for functionally graded materials[J].Materials Science and Engineering A,2003,362:81-105.
[4]SOBCZAK J,DRENCHEV L.Metallic functionally graded materials:A specific class of advanced composites[J].Journal of Materials Science and Technology,2013,29(4):297-316.
[5]CHO J,ODEN J.Functionally graded material:a parametric study on thermal stress characteristics using the Crank-Nicolson-Galerkin scheme[J].Computer Methods in Applied Mechanics and Engineering,2000,188:17-38.
[6]HORI M,NEMAT-NASSER S.On two micromechanics theories for determining micromacro relations in heterogeneous solids[J].Mechanics of Materials,1999,31:667-682.
[7]RAHMAN S,CHAKRABORTY A.A stochastic micromechanical model for elastic properties of functionally graded materials[J].Mechanics of Materials,2007,39:548-563.
[8]BIRMAN V,BYRD L W.Modeling and analysis of functionally graded materials and structures[J].Applied Mechanics Reviews,ASME,2007,60(5):195-216.
[9]MARTIN P A,RICHARDSON J D,GRAY L J,BERGER JR.On Green’s function for a three-dimensional exponentially graded elastic solid[J].Proceeding of the Royal Society of London,Series A,2002,458:1931-1947.
[10]WANG C D,PAN E,TZENG C S,HAN F,LIAO J J.Displacements and stresses due to a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space[J].International Journal of Geomechanic,2006,6(1):110.
[11]CHAN Y S,GREY L J,KAPLAN T,PAULINO G H.Green’s function for a two-dimensional exponentially graded elastic medium[J].Proceefings of the Royal Society of London,Series A,2004,460:1689-1706.
[12]KASHTALYAN M,RUSHCHITSKY J J.Revisiting displacement functions in threedimensional elasticity of inhomogeneous media[J].International Journal of Solids and Structures,2009,46:3463-3470.
[13]SALLAH O M,GRAY L J,AMER M A,MATBULY M S.Green’s function expantion for exponentially graded elasticity[J].International Journal for Numerical Method in Engineering,2010,82:756-772.
[14]ESKANDARI-GHADI M,AMIRI-HEZAVEH A.Wave propagations in exponentially graded transversely isotropic half-space with potential function method[J].Mechanics of Materials,2014,68:275-292.
[15]KATEBI A,SELVADURAI A P S.Undrained behaviour of a non-homogeneous elastic medium:the influence of variations in the elastic shear modulus with depth[J].Geotechnique,2013,63(13):1159-1169.
[16]NING X,LOVELL M,SLAUGHTER W S.Asymptotic solutions for axisymmetric contact of thin,transversely isotropic elastic layer[J].Wear,2006,260:693-698.
[17]RAHMAN M,NEWAZ G.Boussinesq type solution for a transversely isotropic half-space coated with a thin film[J].International Journal of Engineering Science,2000,38:807-822.
[18]SHODJA H M,ESKANDARI M.Axisymmetric time-harmonic response of a transversely isotropic substrate-coating system[J].International Journal of Engineering Science,2007,45:272-287.
[19]KHOJASTEH A,RAHIMIAN M,PAK R Y S,ESKADARIM.Asymmetric dynamic Green’s functions in a two-layered transversely isotropic half-space[J].International Journal of Engineering science,ASCE,2008,134(9):777-787.
[20]KHOJASTEH A,RAHIMIAN M,PAK R Y S.Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials[J].International Journal of Solids and Structures,2008,45:4952-4972.
[21]KHOJASTEH A,RAHIMIAN M,ESKANDARI M.Threedimensional dynamic Green’s functions in transversely isotropic tri-materials[J].Applied Mathematical Modelling,2013,37:3164-3180.
[22]PAN E,YANG B.Three-dimensional interfacial Green’s functions in anisotropic bimaterials[J].Applied Mathematical Modeling,2003,27:307-326.
[23]ESKANDARI-GHADI M,PAK R Y S,ARDESHIR-BEHRESTAGHI A.Transversely isotropic elastodynamic solution of a finite layer on an infinite subgrade under surface loads[J].Soil Dynamics and Earthquake Engineering,2008,28:986-1003.
[24]KHOJASTEH A,RAHIMIAN M,ESKANDARI M,PAK RY S.Three-dimensional dynamic Green’s functions for a multilayered transversely isotropic half-space[J].International Journal of Solids and Structures,2011,48:1349-1361.
[25]ESKANDARI M,AHMADI S F.Green’s function of a surface-stiffened transversely isotropic half-space[J].International Journal of Solids and Structures,2012,49:3282-3290.
[26]RAHIMIAN M,ESKANDARI-GHADI M,PAK R Y S,KHOJASTEH A.Elastodynamic potential method for transversely isotropic solid[J].Journal of Engineering Mechanics,ASCE,2007,133(10):1134-1145.
[27]LEKHNITSKII S G.Theory of elasticity of an anisotropic elastic body[M].San Francisco:Holden Day,1963.
[28]SNEDDON I N.Fourier transforms[M].New York:Mcgraw-Hill,1951.
[29]SNEDDON I N.The use of integral transforms[M].New York:Mcgraw-Hill,1972.
[30]APSEL R J,LUCO J E.On the Green’s functions for a layered half space[J].Part II.Bulletin of the Seismological Society of America,1983,73(4):931-951.
[31]PAK R Y S,GUZINA B B.Three-dimensional Green’s functions for a multi-layered half-space displacement potentials[J].Journal of Engineering Mechanics,ASCE,2002,128(4):449-461.
[32]RAJAPAKSE R K N D,WANG Y.Green’s functions for transversely isotropic elastic half-space[J].Journal of Engineering Mechanics,ASCE,1993,119(9):1724-1746.
[33]SELVADURAI A P S,KATEBI A.Mindlin’s problem for an incompressible elastic half-space with an exponential variation in the linear elastic shear modulus[J].International Journal of Engineering Science,2013,65:9-21.
[34]POULOS H G,DAVIS E H.Elastic solutions for soil and rock mechanics[M].New York:Wiley,1974.
[35]PAN E.Static response of a transversely isotropic and layered half-space to general surface loads[J].Physics of the Earth and Planetary Interiors,1989,54:353-363.
[2]ESKANDARI M,SHODJA H M.Green’s functions of an exponentially graded transversely isotropic half-space[J].International Journal of Solids and Structures,2010,47:1537-1545.
[3]KIEBACK B,NEUBRAND A,RIEDEL H.Processing techniques for functionally graded materials[J].Materials Science and Engineering A,2003,362:81-105.
[4]SOBCZAK J,DRENCHEV L.Metallic functionally graded materials:A specific class of advanced composites[J].Journal of Materials Science and Technology,2013,29(4):297-316.
[5]CHO J,ODEN J.Functionally graded material:a parametric study on thermal stress characteristics using the Crank-Nicolson-Galerkin scheme[J].Computer Methods in Applied Mechanics and Engineering,2000,188:17-38.
[6]HORI M,NEMAT-NASSER S.On two micromechanics theories for determining micromacro relations in heterogeneous solids[J].Mechanics of Materials,1999,31:667-682.
[7]RAHMAN S,CHAKRABORTY A.A stochastic micromechanical model for elastic properties of functionally graded materials[J].Mechanics of Materials,2007,39:548-563.
[8]BIRMAN V,BYRD L W.Modeling and analysis of functionally graded materials and structures[J].Applied Mechanics Reviews,ASME,2007,60(5):195-216.
[9]MARTIN P A,RICHARDSON J D,GRAY L J,BERGER JR.On Green’s function for a three-dimensional exponentially graded elastic solid[J].Proceeding of the Royal Society of London,Series A,2002,458:1931-1947.
[10]WANG C D,PAN E,TZENG C S,HAN F,LIAO J J.Displacements and stresses due to a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space[J].International Journal of Geomechanic,2006,6(1):110.
[11]CHAN Y S,GREY L J,KAPLAN T,PAULINO G H.Green’s function for a two-dimensional exponentially graded elastic medium[J].Proceefings of the Royal Society of London,Series A,2004,460:1689-1706.
[12]KASHTALYAN M,RUSHCHITSKY J J.Revisiting displacement functions in threedimensional elasticity of inhomogeneous media[J].International Journal of Solids and Structures,2009,46:3463-3470.
[13]SALLAH O M,GRAY L J,AMER M A,MATBULY M S.Green’s function expantion for exponentially graded elasticity[J].International Journal for Numerical Method in Engineering,2010,82:756-772.
[14]ESKANDARI-GHADI M,AMIRI-HEZAVEH A.Wave propagations in exponentially graded transversely isotropic half-space with potential function method[J].Mechanics of Materials,2014,68:275-292.
[15]KATEBI A,SELVADURAI A P S.Undrained behaviour of a non-homogeneous elastic medium:the influence of variations in the elastic shear modulus with depth[J].Geotechnique,2013,63(13):1159-1169.
[16]NING X,LOVELL M,SLAUGHTER W S.Asymptotic solutions for axisymmetric contact of thin,transversely isotropic elastic layer[J].Wear,2006,260:693-698.
[17]RAHMAN M,NEWAZ G.Boussinesq type solution for a transversely isotropic half-space coated with a thin film[J].International Journal of Engineering Science,2000,38:807-822.
[18]SHODJA H M,ESKANDARI M.Axisymmetric time-harmonic response of a transversely isotropic substrate-coating system[J].International Journal of Engineering Science,2007,45:272-287.
[19]KHOJASTEH A,RAHIMIAN M,PAK R Y S,ESKADARIM.Asymmetric dynamic Green’s functions in a two-layered transversely isotropic half-space[J].International Journal of Engineering science,ASCE,2008,134(9):777-787.
[20]KHOJASTEH A,RAHIMIAN M,PAK R Y S.Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials[J].International Journal of Solids and Structures,2008,45:4952-4972.
[21]KHOJASTEH A,RAHIMIAN M,ESKANDARI M.Threedimensional dynamic Green’s functions in transversely isotropic tri-materials[J].Applied Mathematical Modelling,2013,37:3164-3180.
[22]PAN E,YANG B.Three-dimensional interfacial Green’s functions in anisotropic bimaterials[J].Applied Mathematical Modeling,2003,27:307-326.
[23]ESKANDARI-GHADI M,PAK R Y S,ARDESHIR-BEHRESTAGHI A.Transversely isotropic elastodynamic solution of a finite layer on an infinite subgrade under surface loads[J].Soil Dynamics and Earthquake Engineering,2008,28:986-1003.
[24]KHOJASTEH A,RAHIMIAN M,ESKANDARI M,PAK RY S.Three-dimensional dynamic Green’s functions for a multilayered transversely isotropic half-space[J].International Journal of Solids and Structures,2011,48:1349-1361.
[25]ESKANDARI M,AHMADI S F.Green’s function of a surface-stiffened transversely isotropic half-space[J].International Journal of Solids and Structures,2012,49:3282-3290.
[26]RAHIMIAN M,ESKANDARI-GHADI M,PAK R Y S,KHOJASTEH A.Elastodynamic potential method for transversely isotropic solid[J].Journal of Engineering Mechanics,ASCE,2007,133(10):1134-1145.
[27]LEKHNITSKII S G.Theory of elasticity of an anisotropic elastic body[M].San Francisco:Holden Day,1963.
[28]SNEDDON I N.Fourier transforms[M].New York:Mcgraw-Hill,1951.
[29]SNEDDON I N.The use of integral transforms[M].New York:Mcgraw-Hill,1972.
[30]APSEL R J,LUCO J E.On the Green’s functions for a layered half space[J].Part II.Bulletin of the Seismological Society of America,1983,73(4):931-951.
[31]PAK R Y S,GUZINA B B.Three-dimensional Green’s functions for a multi-layered half-space displacement potentials[J].Journal of Engineering Mechanics,ASCE,2002,128(4):449-461.
[32]RAJAPAKSE R K N D,WANG Y.Green’s functions for transversely isotropic elastic half-space[J].Journal of Engineering Mechanics,ASCE,1993,119(9):1724-1746.
[33]SELVADURAI A P S,KATEBI A.Mindlin’s problem for an incompressible elastic half-space with an exponential variation in the linear elastic shear modulus[J].International Journal of Engineering Science,2013,65:9-21.
[34]POULOS H G,DAVIS E H.Elastic solutions for soil and rock mechanics[M].New York:Wiley,1974.
[35]PAN E.Static response of a transversely isotropic and layered half-space to general surface loads[J].Physics of the Earth and Planetary Interiors,1989,54:353-363.