摘要
功能梯度材料作为一种特殊的复合材料,其材料的微观组成和性能随位置状态呈现连续变化,并且功能梯度材料可将性能各异的材料按照设计的意愿在结构内部非均匀、连续地合成新型材料,以适用于各种特殊的工程结构极限需要。现在,其用途已由原来的航天工业扩大到核能源、燃气涡轮发动机和土木工程等诸多领域。然而功能梯度材料以其优良的特性被广泛应用于各领域的同时,其材料的连续变化特性也给功能梯度材料的力学行为研究带来了一定的难题。本文从更符合实际的角度,不需要事先对梯度材料参数作特殊的假设,对任意梯度的功能梯度材料的相关动静态力学问题提出了一种积分方程解法,主要包括以下几个方面:
(1)对任意梯度变化的功能梯度轴对称结构,我们提出了一种新的简单有效的解析方法—积分方程方法,分别对功能梯度圆环、圆筒、空心圆球、以及转动圆盘等结构的纯弹性问题进行了研究。首先将所考虑的问题简化为Fredholm积分方程,继而通过数值求解积分方程给出其应力场和位移的分布情况。通过对一些典型的梯度变化情况进行计算,我们发现某些梯度可使环向应力的最大值出现在结构内部,而不是在结构表面,并能有效缓和结构内应力的分布。因此在实际结构优化设计中,可根据本文提出的方法以及不同的使用条件来对具体的梯度变化形式以及参数大小进行设计,从而使得结构在实际服役环境中更为安全。
(2)采用本文提出的积分方程方法对具有任意梯度变化的匀速转动极正交各向异性功能梯度圆环的弹性场进行了分析,重点讨论了正交各向异性参数和材料梯度参数对弹性场、尤其是环向应力的影响。结果表明,在不同边界条件下,极正交各向异性功能梯度圆环中的环向应力并非总大于径向应力,梯度的影响规律也并不一样,而这是和均匀材料结构所不同的地方。因此,在对功能梯度正交各向异性圆环结构优化设计时,需要对不同边界条件下的结构建立各自对应的破坏准则。此方法为功能梯度正交各向异性轴对称问题的优化设计提供了一定的参考依据。
(3)进一步,本文对功能梯度轴对称结构相关的耦合问题如热-力耦合、力-电耦合问题进行了分析。对材料性能按特殊的幂函数形式变化的情况,我们推导得到了相关问题的封闭解。而对任意梯度变化的材料性能,采用积分方程方法求得了结构内电弹性场的分布情况。结果表明,选择合适的材料以及梯度参数一方面可有效缓和应力分布,另一方面可使功能梯度压电器件达到其最好的性能,从而提高压电器件的可靠性和使用寿命。
(4)最后对均匀压电(和磁电弹)材料为基底及其表面为功能梯度压电涂层(和磁电弹涂层)结构在承受冲击载荷情况下裂尖处的瞬态响应进行了分析。考虑了两种不同的加载情况:实际的冲击载荷直接作用的材料表面的情况和载荷作用在裂纹表面的情况。通过积分变换法将所考虑的问题简化为对奇异积分方程的求解,并分别得到了四种常用的断裂参数。研究表明梯度参数对裂纹的动态响应因子有重大影响,尤其当冲击载荷直接作用于功能梯度涂层表面时。同时,本文对不同的动态断裂参数(动态应力、应变强度因子、能量释放率和能量密度因子)分别进行了计算和比较。对智能材料而言,由于与一些实验结果较为吻合,应变强度因子可作为有效的断裂判据。
As a class of new advanced composite materials, functionally graded materials whose material properties continuously vary along one or multi spatial directions can be tailored to satisfy particular engineering application by technologically manufacturing process. In recent years, functionally graded materials have been widely used in nuclear energy reactor, turbine motors, aerospace engineering, and civil engineering etc. However, its continuous material properties bring out some difficulties in analyzing their mechanical behaviors. From a more practical point of view, we do not need to assume the gradient to be special functions and the integral equation method is presented to treat static and dynamic mechanical problems of functionally graded materials with any gradient. This thesis mainly includes the following items:
(1) Firstly, for functionally graded axisymmetrical structures with arbitrarily varying material properties, a simple and efficient approach is put forward to study the elastic problem of functionally graded circular disk, cylinder, sphere and sandwich rotating disk. The associated elastic problem is reduced to a Fredholm integral equation. By solving the resulting equation, the distribution of the stress components and displacement can be determined. As examples, some typically varying material properties are considered. The obtained results indicate that change in the gradient of the functionally graded axisymmetrical structure does not induce a substantial variation of the radial stress, but strongly affects the circumferential stress. And for some special gradient parameters, the distribution curves of circumferential stress become more gently and maximum value of circumferential stress occurs at an interior position of the structure, not at the surfaces like homogeneous materials. Thus, in practical design, the method presented in this thesis may help engineering designers to choose appropriate gradients and materials to acquire an optimal state and to ensure the structure to be safer in the service-environment.
(2) With above-mentioned method we study the elastic problem of rotating functionally graded hollow polar orthotropic circular disks whose relevant material properties arbitrarily varying along the radial direction. Emphasis is placed on the influence of orthotropy and gradient on the elastic field in particular the circumferential stress. Numerical results are presented for two particular cases:free boundaries and clamped-free boundaries. The numerical results show that, for different boundary conditions, the circumferential stress is not always larger than radial stress, and the influences of gradient parameter on circumferential stress are different, too. So, during the design of functionally graded orthotropic rotating disk, for different boundary conditions, individual failure criterion needs to be established. The obtained results are helpful for the optimization on design of functionally graded orthotropic rotating disk for the purpose of optimal design.
(3) Furthermore, some coupling problems of functionally graded axisymmetrical structures are investigated such as thermoelastic and electroelastic problems. For the gradient of a power-law behavior, a closed form solution is derived. For general gradient, an integral equation method is suggested to reduce the problem to a Fredholm integral equation, and the response of the coupling field can be determined. The obtained results show that choosing appropriate materials and gradient parameters can make not only the stresses more gently, but also the functionally graded piezoelectric devices to achieve the best performance, and then enhance the reliability and service life of devices.
(4) The transient response of the dynamic fracture parameters for an interfacial crack of functionally graded piezoelectric or magnetoelectric material coated by a homogeneous piezoelectric or magnetoelectric material substrate on the surface is analyzed. Two different loading situations are mainly considered:one is a realistic situation when impacts are suddenly applied on the material surface; and the other is on the crack surfaces. With the integral transform method, the problem is reduced to solving singular integral equations, and four different dynamic fracture parameters are obtained. It is found that when applied electromechanical impacts are exerted on the material surface, the gradient index causes the transient response to be significantly amplified or reduced depending on negative or positive gradient index. At the same time, by comparing the four dynamic fracture parameters, dynamic stress intensity factor, strain intensity factor, energy release rate and energy density factor, it is found that the stain intensity factor seems to be an efficient fracture parameter for smart materials, due to its consistency with some experimental observations.
引文
[1]新材料与新工艺.军民两用技术与产品,2007,05:19
[2]张志民.复合材料结构力学.北京:北京航空航天大学出版社,1993
[3]Emilio Carlos, N.S., Matthew, C.W., Glaucio, H.P. Modeling bamboo as a functionally graded material:lessons for the analysis of affordable materials. J. Mater. Sci.,2006,41:6991-7004
[4]王宝林,韩杰才,张幸红.非均匀材料力学.北京:科学出版社,2003
[5]Niino, M., Hrai, T., Watanabe, R. Journal of Japan Society of Composite Materials,1987,13:257-264
[6]王保林,韩杰才,张幸红.非均匀材料力学.北京:科学出版社,2003
[7]Rabin, B.H., Shiota, I. Funtionally gradient materials Mater. Res. Soc. Bull., 1995,20:14-18
[8]余茂黎,魏明坤.功能梯度材料的研究动态.功能材料,1992,23:184-191
[9]李智慧,何小凤,李运刚.功能梯度材料的研究现状.河北理工学院学报,2007,29(1):45-50
[10]Koizu, M. FGM activities in Japan. ComPosite:Part B,1997,28B:1-4
[11]鲁先孝,马玉璞,林新志.功能梯度材料在隐身方面的应用.材料开发与利用,2007,2:52-56
[12]Chin Emest, S.C. Foreword army focused research team on functionally graded armor ComPosites. Mater. Sci. Eng. A,1999,259:155-161
[13]PomPe, W., Worch, H., Epple, M. Functionally graded materials for biomedical Application. Mater. Sci. Eng. A,2003,362:40-60
[14]Hedia, H.S. Effect of coating thickness and its material on the stress distribution for dental inPlants. J. Medical Eng. Tech.,2007,31(4):253-262
[15]Hedia, H.S. Design of functionally graded dental implant in the Presence of cancellous bone. J. Biomedical Mater. Res. Part B,2005,75(1):74-80
[16]Huang, M., Wang, R., Thompson, V. et al. Bioinspired design of dental multilayers. J. Mater. Sci.,2007,31:54-62
[17]Takagi, K., Li, J.F., Yokoyama, S. et.al. Design and fabrication of functionally graded PZT/Pt piezoelectric bimorph actuator. Sci. Tech. Advanced Mater.,2002, 3:217-222
[18]Zhu, X., Zhu, J., Zhou, S. et al. Microstructures of themonomorph piezoelectric ceramic actuators with functionally gradient Sensors and Actuators A:Physical, 1999,74:198-202
[19]Zhu, X.H., Meng, Z.Y. Operational principle, fabrication and displacement characteristics of a functionally gradient piezoelectric ceramic actuator. Sensors and Actuators A:Physical,1996,48:169-173
[20]Eshelby, J.D. The determination of the elastic filed of ellipsoidal inclusion and related problems. Proc. Roy. Soc.,1957, A241:376-396
[21]Samadhiya, R., Mukherjee, A., Schmauder, S. Characterization of discretely graded materials using acoustic wave Propagation. Computational Mater. Sci., 2006,37:20-28
[22]He, X.Q., Ng, T.Y., Sivashanker, S. Active control of FGM plates with integrated Piezoelectric sensors and actuators. Int. J. Solids Struct.,2002,38: 1641-1655
[23]Lengauer, W., Dreyer, K. Functionally graded hardmetals. J. Alloys Compounds,2002,338:194-212
[24]Timoshenko, S.P., Goodier, J.N. Theory of Elasticity,3rd edn. New York: McGraw-Hill,1970
[25]Shi, Z.F., Zhang, T.T., Xiang, H.J. Exact solutions of heterogeneous elastic hollow cylinders. Compos. Struct.,2007,79:140-147
[26]Horgan, C.O., Chan, A.M. The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. J. Elasticity,1999,55: 43-59
[27]Tutuncu, N., Ozturk, M. Exact solutions for stress in functionally grade pressure vessels. Compos., Part B Eng.,2001,32:683-686
[28]Zhang, X., Hasebe, N. Elasticity solution for a radially nonhomogeneous hollow circular cylinder. J. Appl. Mech. ASME,1999,66:598-606
[29]Dryden, J., Jayaraman, K. Effect of inhomogeneity on the stress pipes. J. Elasticity,2006,83:179-189
[30]Chen, W.Q. Stress distribution in a rotating elastic functionally graded material hollow sphere with spherical isotropy. J. Strain Anal. Eng. Des.,2000,35(1): 13-20
[31]You, L.H., Zhang, J.J., You, X.Y. Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials. Int. J. Press. Vessels Piping,2005,82(5):347-354
[32]You, L.H., You, X.Y., Zhang, J.J., et al. On rotating circular disks with varying material properties. Z angew Math. Phys.,2007,58:1068-1084
[33]Guven, U., and Baykara, C. On stress distributions in functionally graded isotropic spheres subjected to internal pressure. Mech. Res. Commu.,2001,28(3): 277-281
[34]Batra, R.C. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAA J.,2008,46(8):2050-2057
[35]Horgan, C.O., Chan, A.M. The stress response of functionally graded isotropic linearly elastic rotating disks. J. Elasticity,1999,55:219-230
[36]Zenkour, A.M. Elastic deformation of the rotating functionally graded annular disk with rigid casing. J. Mater. Sci.,2007,42:9717-9724
[37]Bayat, M., Saleem, M., Sahari, B.B., et al. Analysis of functionally graded rotating disks with variable thickness. Mech. Res. Commun.,2008,35:283-309
[38]Eraslan, A.N., Akis, T. On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems. Acta. Mech., 2006,181:43-63
[39]Guven, U., (?)elik, A. Transverse vibrations of functionally graded isotropic linearly elastic rotating solid disk. Mech. Res. Commun.,2001,28:271-276
[40]Zenkour, A.M. Stress distribution in rotating composite structures of functionally graded solid disks. J. Mater. Process. Technol.,2009,209: 3511-3517
[41]Ozturk, M., Erdogan, F. Mode I crack problem in inhomogeneous orthotropic medium. Int. J. Eng. Sci.,1997,35(9):869-883
[42]Sankar, B.V. An elasticity solution for functionally graded beams. Comps. Sci. Tech.,2001,61(5):689-696
[43]Zhu, H., Sankar, B.V. A combined Fouier Galerkin method for the analysis of functioanlly graded beams. J. Appl. Mech.,2004,71(2):421-424
[44]黄德进,丁皓江,陈伟球.任意载荷作用下各向异性功能梯度梁的解析解和半解析解.中国科学(G辑:物理学力学天文学),2009,39(6):830-842
[45]Ding, H.J., Huang, D.J., Chen, W.Q. Elasticity solutions for plane anisotropic functionally graded beams. Int. J. Solids Struct.,2007,44(1):176-196
[46]Agarwal, S., Chakraborty, A., Gopalakrishnan, S. Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions.
Compos. Struct.,2006,72(1):91-104
[47]Nemirovskii, Y.V., Yankovskii, A.P.. Rational profiling of reinforced rotating disks. Mech. Compos. Mater.,2002,38:1-15
[48]Tang, S. Elastic stresses in rotating anisotropic disks. Int. J. Mech. Sci.,1969, 11:509-517
[49]Tununcu, N. Effect of anisotropy on stresses in rotating disks. Int. J. Mech. Sci., 2000,37:873-881
[50]Ari-Gur, J., Stavsky, Y. On rotating polar-orthotropic circular disks. Int. J. Solids Strut.,1981,17:57-67
[51]Jain, R., Ramachandra, K., Simha, K.R.Y. Rotating anisotropic disk of uniform strength. Int. J. Mech. Sci.,1999,41:639-648
[52]Koo, K.N. Vibration analysis and critical speeds of polar orthotropic annular disks in rotation. Compos. Struct.,2006,76:67-72
[53]Liang, D.S., Wang, H.J., Chen, L.W. Vibration and stability of rotating polar orthotropic annular disks subjected to a stationary concentrated transverse load. J. Sound Vib.,2002,250:795-811
[54]Chrisrensen, R.M., Wu, E.M. Optimal design of anisotropic (fiber-reinforced) flywheels. J. Compos. Mater.,1977,11:395-404
[55](?)allioglu, H., Topcu, M., Tarakcilar, A.R. Stress analysis of an orthotropic rotating disk under thermal loading. J. Reinf. Plastic Comp.,2004,23: 1859-1867
[56](?)allioglu, H. Thermal stress analysis of curvilinearly orthotropic rotating disks. J. Thermoplastic Comp. Mater.,2007,20:357-369
[57]Tokovyy, Y.V., Ma, C. C. Thermal stresses in anisotropic and radially inhomogeneous annular domains. J. Thermal Stresses,2008,31(9):892-913
[58]Tokovyy, Y., Ma, C. C. An explicit-form solution to the plane elasticity and thermoelasticity problems for anisotropic and inhomogeneous solids. Int. J. Solids Struct.,2009,46(21):3850-3859
[59]Oral, A., Anlas, G. Effects of radially varying moduli on stress distribution of nonhomogeneous anisotropic cylindrical bodies. Int. J. Solids Struct.,2005, 42(20):5568-5588
[60]Tarn, J.Q. Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. Int. J. Solids Struct.,2001,38(46-47): 8189-8206
[61]Ting, T.C.T. Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. R. Soc. Lond.,1996, A 452: 2397-2421
[62]Ting, T.C.T. New solutions to pressuring, shearing, torsion and extension of a circular tube or bar. Proc. R. Soc. Lond.,1999, A 455:3527-3542
[63]Alshits, V.I., Kirchner, H.O.K. Cylindrically anisotropic, radially inhomogeneous elastic materials. Proc. R. Soc. Lond.,2001, A 457:671-693
[64]Tarn, J.Q., Chang, H.H. Torsion of cylindrical orthotropic elastic circular bars with radial inhomogeneity:some exact solutions and end effects. Int. J. Solids Struct.,2008,45:303-319
[65]Reddy, T.Y., Srinath, H. Elastic stresses in a rotating anisotropic annular disk of variable thickness and variable density. Int. J. Mech. Sci.,1974,16:85-89
[66]Durodola, J.F., Attiam O. Deformation and stresses in functionally graded rotating disks. Compos. Sci. Technol.,2000,60:987-995
[67]Wang, X., Sudak, L.J. Three-dimensional analysis of multi-layered functionally graded anisotropic cylindrical panel under thermomechanical loading. Mech. Mater.,2008,40:235-254
[68]李春雨,邹振祝,段祝平.正交各向异性功能梯度材料Ⅲ型裂纹尖端动态应力场.应用数学和力学,2000,21(6):590-596
[69]Zhang, L.H., Kim, J. H. A complex variable approach for asymptotic Mode-Ⅲ crack-tip fields in an anisotropic functionally graded material. Eng. Fract. Mech., 2009,76(16):2512-2525
[70]Batra, R.C., Jin, J. Natural frequencies of a functionally graded anisotropic rectangular plate. J. Sound Vib.,2005,282(1-2):509-516
[71]Guven, U., Celik, A., Baykara, C. On transverse vibrations of functionally graded polar orthotropic rotating solid disk with variable thickness and constant radial stress. J. Reinf. Plast. Compos.,2004,23(12):1279-1284
[72]边祖光,陈伟球,丁皓江.正交各向异性功能梯度圆柱壳的自由振动.应用力学学报,2004,21(3):75-78
[73]刘俊俏,李星.SH波在正交各向异性功能梯度无限长条中心裂缝处的散射.固体力学学报,2006,27(4):369-373
[74]Ramirez, F., Heyliger, P.R., Pan, E. Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Compos. Part B-Eng., 2006,37(1):10-20
[75]Sladek, J., Sladek, V., Solek, P. Elastic analysis in 3D anisotropic functionally graded solids by the MLPG. Cmes-Computer Modeling Eng.& Sci.,2009,43(3): 223-251
[76]Sladek, J., Sladek, V., Hellmich, C., et al. Heat conduction analysis of 3-D axisymmetric and anisotropic FGM bodies by meshless local Petrov-Galerkin method. Comput. Mech.,2007,39(3):323-333
[77]Hetnarski, R.B., Eslami, M.R. Thermal stresses-advanced theory and applications. Springer,2009
[78]Noda, N. Thermal stresses in functionally graded materials. J. Thermal Stresses, 1999,22(4):477-512
[79]Obata, Y., Noda, N. Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material. J. Thermal Stresses,1994, 17(3):471-487
[80]Kim, K.S., Noda, N. A Green's function approach to the deflection of a FGM plate under transient thermal loading. Arch. Appl. Mech.,2002,72(2-3):127-137
[81]Tanaka, K., Tanaka, Y., Watanabe, H., et al. An improved solution to thermoelastic material design in functionally gradient materials:scheme to reduce thermal stresses. Comput. Methods Appl. Mech. Eng.,1993,109(3-4): 377-389
[82]Tanaka, K., Watanabe, H., Sugano, Y., et al. A multicriterial material tailoring of a hollow cylinder in functionally gradient materials:Scheme to global reduction of thermoelastic stresses. Comput. Methods Appl. Mech. Eng.,1996, 135(3):369-380
[83]Ravichandran, K.S. Thermal residual stresses in functionally graded material system. Mater. Sci. Eng. A,1995,201(1-2):269-276
[84]Finot, M., Suresh, S., Bull, M.C., et al. Curvature changes during thermal cycling of a compositionally graded Ni-A12O3 multi-layered material. Mater. Sci. Eng. A,1996,205(1-2):59-71
[85]Chiu, C.C. Residual stresses in ceramic coatings as determined from the curvature of a coated strip. Mater. Sci. Eng. A,1992,150(1):139-148
[86]Zimmerman, R.W., Lutz, M.P. Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. J. Thermal Stresses,1999,22(2): 177-188
[87]Lutz, M.P., Zimmerman, R.W. Thermal stress and effective thermal expansion coefficient of a functionally gradient sphere. J. Therm. Stress,1996,19:39-54
[88]Jabbari, M., Sohrabpour, S., Elsami, M.R. Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. Press. Vessels Piping,2002,79(7):493-497
[89]Jabbari, M., Sohrabpour, S., Eslami, M.R. General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state loads. J. Appl. Mech.-Transact. ASME,2003,70(1):111-118
[90]Elsami, M.R., Babaei, M.H., Poultangari, R. Thermal and mechanical stresses in a functionally graded thick sphere. Int. J. Press. Vessels Piping,2005,82: 522-527
[91]Tokovyy, Y.V., Ma, C.C. Analysis of 2D non-axisymmetric elasticity and thermoelasticity problems for radially inhomogeneous hollow cylinders. J. Eng. Math.,2008,61:171-184
[92]You, L.H., Ou, H., Li, J. Stress analysis of functionally graded thick-walled cylindrical vessels. AIAA J.,2007,45:2790-2798
[93]El-Naggar, A.M., Abd-Alla, A.M., Fahmy, M.A., et al. Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Heat Mass Transfer., 2002,39:41-46
[94]Kordkheili, S.A.H., Naghdabadi, R. Thermoelastic analysis of a functionally graded rotating disk. Compos. Struct.,2007,79:508-516
[95]Bayat, M., Saleem, M., Sahari, B.B., et al. Mechanical and thermal stresses in a functionally graded rotating disk with variable thickness due to radially symmetry loads. Int. J. Press. Vessels Piping,2009,86:357-372
[96]Sumi, N., Sugano, Y. Thermally Induced Stress Waves in Functionally Graded Materials with Temperature-dependent Material Properties. J. Therm. Stress, 1997,20:281-294
[97]Ohyoshi, T., Sui, G.J. Backscatter of elastic wave from a thermally affected inhomogeneous layer. J. Therm. Stress,1997,20:363-371
[98]Ootao, Y., Tanigawa, Y. Transient thermoelastic analysis for a functionally graded hollow cylinder. J. Therm. Stress,2006,29:1031-1046
[99]Praveen, G.N., Reddy, J.N. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct.,1998,35: 4457-4476
[100]Hosseini, S.M., Akhlaghi, M., Shakeri, M. Transient heat conduction in functionally graded thick hollow cylinders by analytical method. Heat Mass Transfer.,2007,43:669-675
[101]Zhao, J., Ai, X., Li, Y.Z., et al. Thermal shock resistance of functionally gradient solid cylinders. Mater. Sci. Eng. A,2006,418 99-110
[102]Liew, K.M., Kitiporncai, S.X., Zhang, Z., et al. Analysis of thermal stress behavior of functionally graded hollow circular cylinders. Int. J. Solids Struct., 2003,40:2355-2380
[103]Ootao, Y., Akai, T., Tanigawa, Y. Three-dimensional transient thermal stress analysis of a nonhomogeneous hollow circular due to a moving heat source in the axial direction. J. Therm. Stress,1995,18:497-512
[104]Berlincourt, D.A. Piezoelectric crystals and ceramics. Ultrasonic transducer materials. New York:Plenum Press,1971,63-124
[105]Jaffe, B., Cook, W.R. Jr, Jaffe, H. Piezoelectric ceramics. Academic. London, 1971
[106]Tiersten, H.F. Linear piezoelectric plate vibrations. Plenum. New York,1969
[107]Rao, S.S., Sunar, M. Piezoelectricity and its use in disturbance sensing and control of flexible structures:a survey. Appl. Mech. Rev.,1994,47:113-123
[108]Saravanos, D.A., Heyliger, P. R. Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Appl. Mech. Rev.,1999,52: 305-320
[109]Adelman, N.T., Stavsky, Y., Segal, E. Axisymmetric vibrations of radially polarized piezoelectric ceramic cylinders. J. Sound Vibration,1975,38:245-254
[110]Adelman, N.T., Stavsky, Y., Segal, E. Vibrations of radially polarized composite piezoelectric cylinders and disks. J. Sound Vib.,1975,43:37-44
[111]Paul, H.S., Raju, D.P. Asymptotic analysis of the modes of wave propagation in a piezoelectric solid cylinder. J. Acoustical Soc. Am.,1982,71:255-263
[112]Paul, H.S., Venkatesan, M. Vibrations of a hollow circular cylinder of piezoelectric ceramics. J. Acoustical Soc. Am.,1987,82:952-956
[113]Rajapakse, R.K.N.D., Zhou, Y. Stress analysis of piezoceramic cylinders. Smart Mater. Struct.,1997,6:169-177
[114]Chen, W.Q. Problems of radially polarized piezoelastic bodies. Int. J. Solids Struct.,1999,36:4317-4332
[115]Galic, D., Horgan, C.O. Internally pressurized radially polarized piezoelectric cylinders. J. Elasticity,2002,66:257-272
[116]Galic, D., Horgan, C.O. The stress response of radially polarized rotating piezoelectric cylinders. ASME J. Appl. Mech.,2003,70:426-435
[117]Moffett, M.B., Powers, J.M., Clay, W.J.Jr. Ultrasonic microprobe hydrophones J. Acoust. Soc. Am.,1988,84:1186-1193
[118]Meyer, R.J., Weitzing, H., Xu, Q., et al. Lead zirconate titanate hollow-sphere transducers. J. Am. Ceram. Soc.,1994,77:1669-1672
[119]Lockwood, G.R., Turnbull, D.H., Forster, F.S. Fabrication of high frequency spherically shaped PZT transducers. IEEE Trans. Ultrason. Ferroelectr. Freq. Control,1994,41:231-235
[120]Alkoy, S., Dogan, A., Hladky, A.C., et al. Miniature piezoelectric hollow sphere transducers. IEEE Trans. Ultrason. Ferroelectr. Freq. Control,1999,44: 1067-1076
[121]Alkoy, S., Hladky, A.C., Dogan, A., et al. Design, performance and modeling of piezoceramic hollow-sphere microprobe hydrophones. Meas. Sci. Technol., 2009,20:095204
[122]Toupin, R.A. Piezoelectric relations and the radial deformation of a polarized spherical shell. J. Acoust. Soc. Am.,1959,31:315-318
[123]Sinha, D.K. Note on the radial deformation of a piezoelectric, polarized spherical shell with a symmetrical temperature distribution. J. Acoust. Soc. Am., 1962,34:1073-1075
[124]Hamonic, B., Debus, J.C., Decarpigny, J.N., et al. Analysis of a radiating thin-shell sonar transducer using the finite element method. J. Acoust. Soc. Am., 1989,86:1245-1253
[125]Alkoy, S., Meyer, R.J.Jr, Hughes, W.J., et al. Piezoelectric hollow spheres for microprobe hydrophones. Ferroelectrics,1999,226:11-25
[126]Loza, I.A.S., N.A. Axisymmetrical vibrations of a hollow piezoceramic sphere with radial polarization. Sov. Appl. Mech.,1984,20:113-117
[127]Kim, J.O., Lee, J.G., Chun, H.Y. Radial vibration characteristics of spherical piezoelectric transducers. Ultrasonics,2005,43:531-537
[128]Chiroiu, V., Munteanu, L. On the free vibrations of a piezoceramic hollow sphere. Mech. Res. Commun.,2007,34:123-129
[129]Birman, V., Griffin, S., Knowles, G. Axisymmetric dynamics of composite spherical shells with active piezoelectric/composite stiffeners. Acta Mech.,2000, 141:71-83
[130]Chen, W.Q., Ding, H.J., Xu, R.Q. Three-dimensional static analysis of multi-layered piezoelectric hollow spheres via the state space method. Int. J. Solids Struct.,2001,38:4921-4936
[131]Saadatfar, M.R., A. Stress in piezoelectric hollow sphere with thermal gradient. J. Mech. Sci. Technol.,2008,22:1450-1467
[132]Heyliger, P., Wu, Y.C. Electroelastic fields in layered piezoelectric spheres. Int. J. Eng. Sci.,1999,37:143-161
[133]Dai, H.L., Wang, X. Stress wave propagation in laminated piezoelectric spherical shells under thermal shock and electric excitation. Eur. J. Mech. A /Solids,2005,24:263-276
[134]Li, H.Y., Liu, Z.X., Lin, Q.R. Spherical-symmetric steady-state response of piezoelectric spherical shell under external excitation. Appl. Math. Mech.,2000, 21:947-956
[135]Suresh, S. Graded materials for resistance to contact deformation and damage. Science,2001,29:2447-2451
[136]Wu, C.C.M., Kahn, M., Moy, W. Piezoelectric ceramics with functional gradients:a new application in material design. J. Am. Ceram. Soc.,1996,79: 809-812
[137]Alkoy, S. Fabrication and properties of thin-shell monolithic piezoelectric ceramic transducers. J. Mater. Sci.,2007,42:6742-6748
[138]Tarn, J.Q., Chang, H.H. Extension, torsion, bending, pressuring, and shearing of piezoelectric circular cylinders with radial inhomogeneity. J. Intell. Mater. Syst. Struct.,2005,16:631-641
[139]Babaei, M.H., Chen, Z.T. Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft. Arch. Appl. Mech.,2008,78:489-500
[140]Hou, P.F., Wang, H.M., Ding, H.J. Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder. Int. J. Eng. Sci.,2003,41:1849-1868
[141]Ootao, Y., Akai, T., Tanigawa, Y. Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow cylinder. J. Thermal Stresses 2008,31:935-955
[142]Alibeigloo, A. Static analysis of a functionally graded cylindrical shell with piezoelectric layers as sensor and actuator. Smart Mater. Struct.,2009,18: 065004
[143]Du, J.K., Jin, X.Y., Wang, J., et al. Love wave propagation in functionally graded piezoelectric material layer. Ultrasonics,2007,46:13-22
[144]Yu, J.G.W., B., Chen, G.Q. Wave characteristics in functionally graded piezoelectric hollow cylinders. Arch. Appl. Mech.,2009,79:807-824
[145]Ding, H.J., Wang, H.M., Chen, W.Q. Analytical solution for the electroelastic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere. Arch. Appl. Mech.,200373:49-62
[146]Delale, F., Erdogan, F. The crack problem for a nonhomogeneous plane. J. Appl. Mech.,1983,50:609-614
[147]Erdogan, F. The crack problem for bonded nonhomogeneous materials under antiplane shear loading. J. Appl. Mech.,1985,52(823-828)
[148]Erdogan, F., Ozturk, M. Diffusion problems in bonded nonhomogeneous materials with an interface cut. Int. J. Engng Sci.,1992,30(10):1507-1523
[149]Delale, F., Erdogan, F. On the mechanical modeling of the interfacial region in bonded half-planes. J. Appl. Mech.,1988,55:317-324
[150]Delale, F., Erdogan, F. Interface crack in a nonhomogeneous elastic medium. Int. J. Engng Sci.,1988,26(6):559-568
[151]Erdogan, F., Kaya, A.C., Josehp, P.F. The crack problem in bonded nonhomogeneous materials. J. Appl. Mech.,1991,58:410-418
[152]Ozturk, M., Erdogan, F. The axisymmetric crack problem in a nonhomogeneous medium. J. Appl. Mech.,1993,60:406-413
[153]Ozturk, M., Erdogan, F. An axisymmetric crack in bonded materials with a nonhomogeneous interfacial zone under torsion. J. Appl. Mech.,1995,62: 116-125
[154]Erdogan, F. Fracture mechanics of functionally graded materials. Comp. Eng., 1995,5(7):753-770
[155]Erdogan, F., Ozturk, M. Periodic cracking of functionally graded coatings. Int. J. Engng. Sci.,1995,33:2179-2195
[156]Chen, Y.F., Erdogan, F. The interface problem for a nonhomogeneous coating bonded to a homogeneous substrate. J. Mech. Phys. Solids,1996,44(5):71-78
[157]Choi, H.J. A periodic array of cracks in a functionally graded nonhomogeneous medium loaded under in-plane normal and shear. Int. J. Fract., 1997,88:107-128
[158]Chan, Y.S., Paulino, G.H., Fannjiang, A.C. The crack problem for nonhomogeneous materials under antiplane shear loading-displacement based formulation. Int. J. Solids Struct.,2001,38:2989-3005
[159]Paulino, G.H., Fannjiang, A.C., Chan, Y.S. Gradient elasticity theory for a mode III crack in a functionally graded material. Mater. Sci. Forum.,1999, 308-311:971-976
[160]Ueda, S. The surface crack problem for a layered plate with a functionally graded nonhomogeneous interface. Int. J. Fract.,2001,110:189-204
[161]Konda, N., Erdogan, F. The mixed mode crack problem in a nonhomogeneous elastic medium. Eng. Fract. Mech.,1994,47(4):533-541
[162]Choi, H.J., Jin, T.E., Lee, K.Y. Collinear cracks in a layered half-plane with a graded nonhomogeneous interfacial zone-part1:mechanical response. Int. J. Fract.,1998,94:103-112
[163]Choi, H.J. Bonded dissimilar strips with a crack perpendicular to the functionally graded interface. Int. J. Solids Struct.,1996,33:4101-4117
[164]Choi, H.J. An analysis of cracking in a layered medium with a functionally graded nonhomogeneous interface. ASME, J. Appl. Mech.,1996,63:479-486
[165]Wang, B.L., Mai, Y.W., Sun, Y.G. Anti-plane fracture of a functionally graded material strip. Eur. J. Mech. A/Solids,2003,22(3):357-368
[166]陈宜周.弹性功能梯度材料板条中周期裂纹的反平面问题.力学学报,2004,36(4):501-506
[167]程站起,华利民,卫兴.功能梯度材料涂层平面运动裂纹分析.力学季刊,2008,29(1):78-84
[168]程站起,仲政.功能梯度板条Ⅲ型运动裂纹问题研究.应用力学学报,2008,25(1):62-66
[169]Bao, G., Wang, L. Multiple cracking in functionally graded ceramic/metal coatings. Int. J. Solids Struct.,1995,32:2853-2871
[170]Babaei, R., Lukasiewicz, S.A. Frictional cotact problem of crack in FGMs under mechanical and time-dependent thermal stresses. J. Thermal Stresses,1997, 20:439-461
[171]Wang, X.D., Meguid, S.A. On the dynamic crack propagation in an interface with spatially varying elastic properties. Int. J. Fract.,1995,69:87-99
[172]Babaei, R., Lukasiewicz, S.A. Dynamic response of a crack in a functionally graded material between two dissimilar half planes under anti-plane shear impact load. Eng. Fract. Mech.,1998,60:479-487
[173]王保林,杜善义,韩杰才.非均匀复合材料中反平面裂纹的动态断裂力学研究.复合材料学报,1998,15(4):119-127
[174]王保林,韩杰才,杜善义.动态载荷下功能梯度复合材料的圆币形裂纹问题.固体力学学报,1999,20(3):219-225
[175]Li, X.F., Guo, S.H. Effects of nonhomogeneity on dynamic stress intensity factors for an anti-plane interface crack in a functionally graded material bonded to an elastic semi-strip. Comput. Mater. Sci.,2006,38:432-441
[176]Li, X.F., Fan, T.Y. Dynamic analysis of a crack in a functionally graded material sandwiched between two elastic layers under anti-plane loading. Compos. Struct.,2007,79:211-219
[177]Li, C.Y., Weng, G.J., Duan, Z.P. Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading. Int. J. Solids Struct.,2001, 38:7473-7485
[178]Li, C., Duan, Z., Zou, Z. Torsional impact response of a penny-shaped interface crack in bonded materials with a graded materials interlayer. J. Appl. Mech.,2002,69:303-308
[179]Meguid, S.A., Wang, X.D., Jiang, L.T. On the dynamic propagation of a finite crack in functionally graded materials. Eng. Fract. Mech.,2002,69:1753-1768
[180]Jiang, L.Y., Wang, X.D. On the dynamic crack propagation in an interphase with patially varying elastic properties under inplane loading. Int. J. Fract.,2002, 114:225-244
[181]Chen, Z.X., Liu, Z., Zou, Z. Transient internal crack problem for a nonhomogeneous orthotropic strip (Mode I). Int. J. Engrg. Sci.,2002,40: 1761-1774
[182]Li, Y., Ramesh, K.T., Chin, E.S.C. Dynamic characterization of layered and graded structures under impulsive loading. Int. J. Solids Struct.,2001,38: 6045-6061
[183]Lucato, S.L.S., Bahr, H.A., Pham, V.B., et al. Electrically driven cracks in piezoelectric ceramics:experiments and fracture mechanics analysis. J. Mech. Phys. Solids,2002,50:2333-2353
[184]Furuta, A., Uchino, K. Dynamic observation of crack propagation in piezoelectric multilayer actuators. J. Am. Ceram. Soc.,1993,76:1615-1617
[185]Qiu, J.H., Tani, J., Ueno, T., et al. Fabrication and high durability of functionally graded piezoelectric bending actuators. Smart Mater. Struct.,2003, 12:115-121
[186]Takagi, K., Li, J.-F., Yokoyama, S., et al. Fabrication and evaluation of PZT/Pt piezoelectric composites and functionally graded actuators. J. Eur. Ceram. Soc., 2003,23:1577-1583
[187]Li, C., Weng, G.J. Antiplane crack problem in functionally graded piezoelectric materials. ASMEJ. Appl. Mech.,2002,69:481-488
[188]Wang, B.L. A mode Ⅲ crack in a functionally graded piezoelectric material. Mech. Res. Commun.,2003,30:151-159
[189]Li, C.Y., Weng, G.J. Yoffe-type moving crack in a functionally graded piezoelectric material. Proc. R. Soc. Lond. A,2002,458:381-399
[190]Jin, B., Zhong, Z. A moving mode-Ⅲ crack in functionally graded piezoelectric material:permeable problem. Mech. Res.Comm.,2002,29: 217-224
[191]Kwon, S.M., Lee, K.Y. An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip. Arch. Appl. Mech.,2003,73:348-366
[192]胡克强,仲政,李国强.功能梯度压电板条中的电渗透型运动裂纹.同济大学学报(自然科学版),2004,32(12):1631-1636
[193]胡克强,仲政,金波.功能梯度压电板条中电绝缘型运动裂纹的电弹性场.力学季刊,2003,24(3):371-378
[194]Chen, J., Liu, Z., Zou, Z. Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes. Arch. Appl. Mech., 2003,72:686-696
[195]Yong, H.D., Zhou,Y.H. Transient anti-plane crack problem for two bonded functionally graded piezoelectricmaterials. Arch. Appl. Mech.,2006,76:497-509
[196]Nan, C.W. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B,1994,50:6082-6088
[197]Wang, B.L., Mai, Y.W., Sun, Y.G. Crack tip field in piezoelectric/piezomagnetic media Eur. J. Mech. A/Solids,2003,22:591-602
[198]Gao, C.F., Tong, P., Zhang, T.Y. Fracture mechanics for a mode III crack in a magnetoelectroelastic solid. Int. J. Solids Struct.,2004,41:6613-6629
[199]Sih, G.C., Jones, R., Song, Z.F. Piezomagnetic and piezoelectric poling effects on mode Ⅰ and Ⅱ crack initiation behavior of magnetoelectroelastic materials. Theor. Appl. Fract. Mech.,2003,40:161-186
[200]Tian, W.Y., Rajapakse, R.K.N.D. Fracture analysis of magnetoelectroelastic solids using path independent integrals. Int. J. Fract.,2005,131:311-335
[201]Wang, X., Shen, Y.P. The general solution of three-dimensional problems in magnetoelectroelastic media Int. J. Eng. Sci.,2002,40:1069-1080
[202]Spyropoulos, C.P., Sih, G.C., Song, Z.F. Magnetoelectroelastic composite with poling parallel to plane of line crack under out-of-plane deformation. Theor. Appl. Fract. Mech.,2003,39:281-289
[203]Li, X.F. Dynamic analysis of a cracked magnetoelectroelastic medium under antiplane mechanical and inplane electric and magnetic impacts. Int. J. Solids Struct.,2005,42:3185-3205
[204]Yong, H.D., Zhou, Y.H. Transient response of a cracked magnetoelectroelastic strip under anti-plane impact. Int. J. Solids Struct.,2007,44:705-717
[205]Zhong, X.C., Li, X.F. A finite length crack propagating along the interface of two dissimilar magnetoelectroelastic materials. Int. J. Eng. Sci.,2006,44: 1394-1407
[206]Feng, W.J., Su, R.K.L. Dynamic internal crack problem of a functionally graded magneto-electro-elastic strip. Int. J. Solids Struct.,2006,43:5196-5216
[207]Chen, J., Soh, A.K., Liu, J., et al. Transient anti-plane crack problem of a functionally graded piezoelectric strip bonded to elastic layers. Acta. Mech., 2004,169:87-100
[208]刘五祥,仲政.任意梯度分布功能梯度板的柱形弯曲分析.武汉理工大学学报,2008,30(4):106-109
[209]Santare, M.H., Thamburaj, P., Gazonas, G.A. The use of gradient elements in the study of elastic wave propagation in continuously nonhomogeneous materials. Int. J. Solids Struct.,2003,40:5621-5634
[210]Chakraborty, A., Gopalakrishnan, S., Reddy, J.N. A new beam finite element for the analysis of functionally graded materials. Int. J. Mech. Sci.,2003,45: 519-539
[211]Rallirez, F., Heyliger, P.R., Pan, E. Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Compos. Part B,2006, 37:10-20
[212]Baganas, K. Wave propagation and profile reconstruction in inhomogeneous elastic media. Wave Motion,2005,42:261-273
[213]Zuiker, J.R. Functionally graded materials:choice of micromechanics model and limitations in property variation. Comp. Engng.,1995,5(7):807-819
[214]Peng, X.L., Li, X.F., Lee, K.Y. Interface crack problem of functionally graded piezoelectric materials:effects of the position of electromechanical impact and gradient. Acta. Mech.,2009,207:69-82
[215]Peng, X.L., Li, X.F. Transient response of the crack-tip field in a magnetoelectroelastic half-space with a functionally graded coating under impacts. Arch. Appl. Mech.,2009,79:1099-1113
[216]Baker, C.T.H. Expansion methods. In:Delves, L.M., Walsh, J. (eds.) Numerical Solution of Integral Equations. Oxford:Clarendon,1974,80-96
[217]徐芝纶.弹性力学.第三版.北京:高等教育出版社,1987,93-95
[218]Li, X.F., Peng, X.L., Kang, Y.A. Pressurized hollow spherical vessels with arbitrary radial nonhomogeneity. AIAA J.,2009,47:2262-2265
[219]Li, X.F., Peng, X.L. A pressurized functionally graded hollow cylinder with arbitrarily varying material properties. J. Elasticity,2009,96:81-95
[220]Atkinson, K.E., Shampine, L.F. Algorithm 876:Solving fredholm integral equations of the second kind in MATLAB. ACM Trans. Math. Soft.,2008,34: 21
[221]Lekhnitskii, S.G. Theory of Elasticity of an Anisotropic Body. Moscow:Mir, 1981
[222]Baker, C.T.H. The numerical treatment of integral equations. Oxford: Clarendon Press,1977
[223]Atkinson, K. The numerical solution of integral equations of the second kind. Cambridge:Cambridge University Press,1997
[224]Ootao, Y., Tanigawa, Y. Three-dimensional transient thermal stresses of functionally graded rectangular plate due to partial heating. J. Therm. Stress, 1999,22(1):35-55
[225]Ootao, Y., Tanigawa, Y. Transient thermoelastic problem of functionally graded thick strip due to nonuniform heat supply. Compos. Struct.,2004,63(2): 139-146
[226]Tanigawa, Y., Oka, N., Akai, T., et al. One-dimensional transient thermal stress problem for nonhomogeneous hollow circular cylinder and its optimization of material composition for thermal stress relaxation. JSME Int. J.. Series A, Solid Mech. Mater.Eng.,1997,40(2):117-127
[227]Shackelford James, F., Alexander, W. Materials Science and Engineering Handbook. Boca Raton:CRC Press LLC,2001
[228]Bhowmick, S., Misra, D., Nath Saha K. Approximate solution of limit angular speed for externally loaded rotating solid disk. Int. J.Mech. Sci.,2008,50(2): 163-174
[229]Stehfest, H. Numerical inversion of Laplace transforms. Commun. ACM, 1970,13:47-49
[230]Davies B., M.B. Numerical inversion of the Laplace transform:a surey and comparison of methods. J. Comput. Phy.,1979,33:1-32
[231]Miller, M.K., Guy, W.T. Numerical inversion of the Laplace transform of Jacobi ploynomials. SIAM J. Num. Ana.,1966,3:624-635
[232]Tokovyy, Y.V., Ma, C.C. Analytical solutions to the planar non-axisymmetric elasticity and thermoelasticity problems for homogeneous and inhomogeneous annular domains. Int. J. Eng. Sci.,2009,47:413-437
[233]Li, X.F., Lee, K.Y. Elastic response of a piezoelectric ceramic plate with two penny-shaped electrodes at the opposite surfaces under an electric load. J. Intell. Mater. Syst. Struct.,2007,18:743-751
[234]Ghorbanpour, A., Golabi, S., Saadatfar, M. Stress and electric potential fields in piezoelectric smart spheres. J. Mech. Sci. Technol.,2006,20:1920-1933
[235]Crump, K.S. Numerical inversion of Laplace transforms using a Fourier series approximation. J. Assoc. Comp. Mach.,1976,23:89-96
[236]Li, X.-F., Duan, X.-Y. Comparison of dynammic response of a piezoelectric ceramic containing two parallel cracks via two methods of Laplace inversion. Int. J. Fract.,2003,122:L131-L136
[237]Li, X.-F., Lee, K.Y. Dynamic behavior of a piezoelectric ceramic layer with two surface cracks. Int. J. Solids Struct.,2004,41:3193-3209
[238]Pak, Y.E. Crack extension force in a piezoelectric material. J. Appl. Mech., 1990,57:647-653
[239]Park, S., Sun, C.T. Fracture criteria for piezoelectric ceramics. J. Am. Ceram. Soc.,1995,78:1475-1480
[240]Schneider, G.A., Heyer, V. Influence of the electric field on vickers indentation crack growth in BaTiO3. J. Eur. Ceram. Soc.,1999,19:1299-1306
[241]Li, X.F., Lee, K.Y. Crack growth in a piezoelectric material with a Griffith crack perpendicular to the poling axis. Philos. Mag.,2004,84:1789-1820
[242]Kanninen, M.F., Popelar, C.H. Advanced Fracture Mechanics. New York: Oxford University Press,1985