臺灣博碩士論文加值系統

本文旨在探討含壓電材料結構之奇異電彈場，包括兩個主題：一為含壓電材料楔形結構尖端之應力奇異性分析；另一則為含電極壓電片之電彈場分析。在壓電楔形部分，首先利用廣義Lekhnitskii公式配合特徵函數展開法，探討壓電材料-壓電材料、壓電材料-複合材料以及壓電材料-等向性材料等三種楔形結構，經由特定邊界條件的設定，可求得應力奇異性階數。若壓電體之極化軸以及複合材料之纖維指向特定方向，奇異性階數可分離為面內以及面外，此時特徵方程式可進一步化簡。奇異性階數會受到壓電體極化方向、複合材料之纖維方向、楔形角等因素影響，經由本文的數值分析，可探討降低奇異性階數的條件，甚至使應力奇異性消失，以作為安全設計之參考。
當壓電體沿z軸極化時，壓電楔形之靜電場將與面外彈性場耦合。針對此一特殊案例，本文引入梅林轉換求解雙壓電材料接合楔形在十種不同邊界條件下之面外應力奇異性階數，結果可退化為壓電材料-等向性介電質、壓電材料-等向性導體以及單一壓電材料楔形。分析結果發現，當單一壓電材料楔形之邊界條件為自由-接地邊界以及固定-絕緣邊界(即C-D邊界條件組合)，其應力奇異性為複數，且虛部可達到0.5以上。對於兩邊界條件皆為自由-絕緣邊界(即A-A邊界條件組合)，本文再利用複變函數配合特徵函數展開法做更進一步的探討，除求解雙壓電材料楔形尖端之面外應力奇異場、角函數以及強度因子之外，並證明當a≠b、R2^2-4R1<0、a≠0且b≠0時，面外應力奇異性階數必為複數。經由以上分析，發現壓電材料楔形之面外應力奇異性階數與彈性體楔形有很大的不同，因為後者之面外應力奇異性階數恆為實數。
在含半無限長壓電片方面，分析模型有二，案例一為面外彈性場與面內靜電場耦合；案例二則為面內彈性場與面內靜電場耦合。經由傅立葉轉換配合Wiener-Hopf法，發現兩個案例在電極尖端附近皆呈現r^-1/2之奇異性。最後引入能量密度因子理論，探討電極尖端之破壞行為，分析結果可作為疊層壓電致動器之安全設計參考。

Based on the generalized Lekhnitskii complex potential associated with eigenfunction expansion, the stress singularities occurring near the apex piezoelectric-piezoelectric, piezoelectric-composite and piezoelectric-conductor bonded wedges are studied. The stress singularity orders, which depend on the poling and fiber orientations and wedge angles, are obtained numerically through solving the eigenvalue problems. If the piezoelectric is polarized in some specific directions, the singularity orders can be decoupled into inplane and antiplane cases. The numerical results can be used as a guideline in the design of the piezoelectric structures.
For the special case that piezoelectric is polarized along z-axis, the characteristic equations governing the antiplane stress singularities are obtained analytically by using the Mellin transform. The results can be reduced to one-piezoelectric-wedge, piezoelectric-composite bonded wedge and piezoelectric-conductor wedge. In the case of one-piezoelectric-wedge under free-ground and clamped-insulated boundary conditions (i.e., C-D boundary conditions), the antiplane stress singularity order is found to be complex. The imaginary part of the singularity order may be greater than 0.5. In the case of free-insulated boundary at both edges (i.e., A-A boundary conditions), the singular electro-elastic fields near the apex of piezoelectric bonded wedge are investigated further by using the complex variable technique associated with eigenfunction expansion. The angular functions and intensity factors are obtained analytically. The results show that the antiplane stress singularity orders are always complex if a≠b, R2^2-4R1<0, a≠0 and b≠0. This is quite different from the elastic material wedge, in which the antiplane stress singularity orders are always real.
There are two models to be analyzed in the piezoelectric layer with semi-infinite electrode, i.e., the anti-plane deformation with in-plane electrical field and the in-plane electroelastic field. Based on the Fourier transform and the Wiener—Hopf technique, the electroelastic fields near the electrode tip are found to exhibit the square root singularity. The energy density factor criterion is applied to examine the fracture behavior near the electrode tip.

摘要....................................................................I
英文摘要................................................................II
誌謝....................................................................III
目錄....................................................................IV
表目錄..................................................................VIII
圖目錄..................................................................IX
符號說明................................................................XI
第一章 緒論............................................................1
1.1前言............................................................1
1.2文獻回顧........................................................4
1.2.1彈性體楔形結構之相關文獻回顧....................................4
1.2.2含壓電楔形結構之相關文獻回顧....................................6
1.2.3含電極壓電片之相關文獻回顧......................................7
1.3研究動機與本文架構..............................................9
1.3.1含壓電材料楔形之研究動機........................................9
1.3.2含電極壓電片之研究動機..........................................10
1.3.3本文架構........................................................10
第二章 壓電材料之廣義Lekhnitskii複變函數公式...........................12
2.1廣義平面問題假設與基本方程式....................................12
2.2應力函數、電位移函數以及複變函數的引入..........................20
2.3各種特殊退化問題之探討..........................................25
2.3.1退化條件A：.....................................................25
2.3.2退化條件B：.....................................................26
2.3.3退化條件C(無壓電效應)：.........................................28
2.3.4退化條件D：.....................................................29
第三章 含壓電材料楔形之應力奇異性......................................31
3.1特徵函數展開法、邊界與連續條件以及特徵方程式....................34
3.2特殊退化問題的探討..............................................40
3.3材料矩陣之座標轉換..............................................43
3.4與文獻結果之比較................................................47
3.5數值分析之新結果................................................50
3.5.1退化條件A.......................................................50
3.5.2退化條件B.......................................................57
3.5.3廣義平面問題....................................................63
第四章 含壓電材料楔形之面外應力奇異性探討..............................66
4.1梅林轉換公式....................................................66
4.1.1以應力及電位移函數為基礎之基本公式..............................66
4.1.2以位移及電位函數為基礎之基本公式................................69
4.2以梅林轉換求含壓電楔形之面外應力奇異性階數......................71
4.2.1非耦合為面外彈性場以及面內靜電場................................76
4.2.2退化為單一壓電材料楔形..........................................77
4.2.3壓電體-等向性介電質接合楔形.....................................80
4.2.4壓電體-等向性導體接合楔形.......................................82
4.3複變函數公式....................................................84
4.4以複變函數求解含壓電楔形在A-A邊界條件組合下之應力奇異性.........85
4.4.1等楔形角........................................................86
4.4.2不等楔形角......................................................89
4.4.3數值範例分析....................................................96
第五章 含電極壓電片之電彈場探討........................................103
5.1問題描述........................................................103
5.2案例一：面外彈性場與面內電場耦合................................106
5.2.1邊界條件........................................................107
5.2.2傅立葉轉換以及Wiener-Hopf法.....................................110
5.2.3傅立葉逆轉換....................................................115
5.3案例二：面內彈性場與面內靜電場耦合..............................124
5.3.1邊界條件........................................................124
5.3.2傅立葉轉換與狀態空間(State space)...............................125
5.3.3Wiener-Hopf法...................................................129
5.3.4電極尖端附近之電彈場漸近解(Asymptotic solution)及其強度因子.....134
5.4能量密度理論....................................................139
5.4.1案例一之破壞分析................................................139
5.4.2案例二之破壞分析................................................140
第六章 結論............................................................142
6.1利用廣義Lekhnitskii公式求解含壓電楔形之應力奇異性階數...........142
6.2利用梅林轉換求解含壓電楔形之面外應力奇異性......................144
6.3A-A邊界條件下含壓電楔形之面外應力奇異性階數.....................144
6.4含電極壓電片之電彈場分析........................................145
6.5本論文所用數學技巧之綜合評論....................................146
6.6未來研究方向....................................................147
第七章 本文貢獻........................................................148
參考文獻................................................................150
附錄....................................................................160

[1]Williams M.L., “Stress singularities resulting from various boundary conditions in angular corners of plates in tension”, Journal of Applied Mechanics, Vol. 19, pp. 526-528, 1952.
[2]Tranter C.J., “The use of the Mellin transform in finding the stress distribution in an infinite wedge”, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 1, pp. 125-130, 1948.
[3]Godfrey D.E.R., “Generalized plane stress in an elastic wedge under isolated loads”, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 8 (Part 2), pp. 226-236, 1954.
[4]Bogy D.B., “Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading”, Transactions of the ASME, Journal of Applied Mechanics, Vol. 35, pp. 460-466, 1968.
[5]Hein V.L. and Erdogan F., “Stress singularities in a two-material wedge”, International Journal of Fracture Mechanics, Vol. 7, pp. 317-330, 1971.
[6]Dundurs J., “Discussion”, Transaction of the ASME, Journal of Applied Mechanics, Vol. 36, pp. 650-652, 1969.
[7]Bogy D.B., “On the problem of edge-bonded elastic quarter-planes loaded at the boundary”, International Journal of Solids and Structures, Vol. 6, pp. 1287-1313, 1970.
[8]Bogy D.B., “Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions”, Transactions of the ASME, Journal of Applied Mechanics, Vol. 38, pp. 377-386, 1971.
[9]Bogy D.B. and Wang K.C., “Stress singularities at interface corners in bonded dissimilar isotropic elastic materials”, International Journal of solids and Structures, Vol. 7, pp. 993-1005, 1971.
[10]Ma C.C. and Hour B.L., “Analysis of dissimilar anisotropic wedge subjected to antiplane shear deformation”, International Journal of Solids and Structures, Vol. 25, pp. 1295-1309, 1989.
[11]Munz D., Fett T. and Yang Y.Y., “The regular stress term in bonded dissimilar materials after a change in temperature”, Engineering Fracture Mechanics, Vol. 44, pp. 185-194, 1993.
[12]Munz D. and Yang Y.Y., “Stresses near the edge of bonded dissimilar materials described by two stress intensity factors”, International Journal of Fracture, Vol. 60, pp. 169-177, 1993.
[13]Yang Y.Y. and Munz D., “Determination of the regular stress term in a dissimilar materials joint under thermal loading by the Mellin transform”, Journal of Thermal Stresses, Vol. 17, pp. 321-336, 1994.
[14]Muskhelishvili N.I., Some Basic Problems of the Mathematical theory of Elasticity, Noordhoff, Groningen Holland, 1953.
[15]Theocaris P.S., “The order of singularity at a multi-wedge corner of a composite plate”, International Journal of Engineering Science, Vol. 12, pp. 107-120, 1974.
[16]Dempsey J.P. and Sinclair, G.B., “On the stress singularities in the plane elasticity of the composite wedge”, Journal of Elasticity, Vol. 9, pp. 373-391, 1979.
[17]Dempsey J.P. and Sinclair, G.B., “On the singular behavior at the vertex of a bi-material wedge”, Journal of Elasticity, Vol. 11, pp. 317-327, 1981.
[18]Chen D.H. and Nisitani H., “Singular stress field in two bonded wedges”, Transactions of Japan Society of Mechanical Engineers Series A, Vol. 58, pp. 457-464, 1992.
[19]Bogy D.B., “The plane solution for anisotropic elastic wedge under normal and shear loading”, Vol. 39, p. 1103-1109, 1972.
[20]Kuo M.C. and Bogy D.B., “Plane solutions for the displacement and traction-displacement problems for anisotropic elastic wedges”, Transactions of the ASME, Journal of Applied Mechanics, Vol. 41, pp. 197-202, 1974.
[21]Kuo M.C. and Bogy D.B., “Plane solutions for traction problem on orthotropic unsymmetrical wedges and symmetrically twinned wedges”, Transactions of the ASME, Journal of Applied Mechanics, Vol. 41, pp. 203-208, 1974.
[22]Ma C.C., “Plane solution of thermal stresses for anisotropic bimaterial elastic wedges”, Journal of Thermal Stresses, Vol. 18, pp. 219-245, 1995.
[23]Ting T.C.T, “Explicit solution and invariance of the singularities at an interface crack in anisotropic composites”, International Journal of Solids and Structures, Vol. 22, pp. 965-983, 1986.
[24]Stroh A.N., “Steady state problems in anisotropic elasticity”, Journal of Mathematical Physics, Vol. 41, pp. 77-103, 1962.
[25]Chen H.P., “Stress singularities in anisotropic multi-material wedges and junctions”, International Journal of Solids and Structures, Vol. 35, 1057-1073, 1998.
[26]Wang S.S. and Choi I., “Boundary-layer effects in composite laminates: part I-free-edge stress singularities”, Transactions of the ASME, Journal of Applied Mechanics, Vol. 49, pp. 541-548, 1982.
[27]Wang S.S. and Choi I., “Boundary-layer effects in composite laminates: part II-free-edge stress solutions and basic characteristics”, Transactions of the ASME, Journal of Applied Mechanics, Vol. 49, pp. 549-560, 1982.
[28]Delale F., “Stress singularities in bonded anisotropic materials”, International Journal of Solids and Structures, Vol. 20, pp. 31-40, 1984.
[29]Lin K.Y. and Hartmann H.H., “Numerical analysis of stress singularities at a bonded anisotropic wedge”, Engineering Fracture Mechanics, Vol. 32, pp. 211-224, 1989.
[30]Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963.
[31]Ting T.C.T, Anistropic Elasticity: Theory and Applications, Oxford University Press, New York, 1996.
[32]Chue C.H. and Liu C.I., “A general solution on stress singularities in anisotropic wedge”, International Journal of Solids and Structures, Vol. 38, pp. 6889-6906, 2001.
[33]Chue C.H. and Liu C.I., “Disappearance of free-edge stress singularity in composite laminates”, Composite Structures, Vol. 56, pp. 111-129, 2002.
[34]Chue C. H. and Liu C.I., “Stress singularities in a bimaterial anisotropic wedge with arbitrary fiber orientation”, Composite Structures, Vol. 58, pp. 49-56, 2002.
[35]Chue C.H., Tseng C.H. and Liu C.I., “On stress singularities in an anisotropic wedge for various boundary conditions”, Composite Structures, Vol. 54, pp. 87-102, 2001.
[36]Pageau S.S., Joseph P.F. and Biggers S.B. Jr., “Finite element analysis of anisotropic materials with singular inplane stress fields”, International Journal of Solids and Structures, Vol. 32, pp. 571-591, 1995.
[37]Pageau S.S., Joseph P.F. and Biggers S.B. Jr., “A finite element analysis of the singular stress fields in anisotropic materials loaded in antiplane shear”, International Journal for Numerical Methods in Engineering, Vol. 38, pp. 81-97, 1995.
[38]Pageau S.S. and Biggers S.B. Jr., “A finite element approach to three-dimensional singular stress states in anisotropic muti-material wedges and junctions”, International Journal of Solids and Structures, Vol. 33, pp. 33-47, 1996.
[39]Sosa H., “Plane problems in piezoelectric media with defects”, International Journal of Solids and Structures, Vol. 28, pp. 491-505, 1991.
[40]Chen T. and Lai D., “An exact correspondence between plane piezoelectricity and generalized plane strain in elasticity”, Proceeding of Royal Society London A, Vol. 453, pp. 2689-2713, 1997.
[41]Chen T. and Yen W.J., “Piezoelectric analogy of generalized torsion in anisotropic elasticity”, Journal of Elasticity, Vol. 49, 239-256, 1998.
[42]Barnet D.M. and Lothe J., “Dislocations and line charges in anisotropic piezoelectric insulators”, Physica Status Solidi (b), Vol. 67, 105-111, 1975.
[43]Chung M.Y. and Ting T.C.T., “Line force, charge, and dislocation in anisotropic piezoelectric composite wedges and spaces”, Transactions of the ASME Journal of Applied Mechanics, Vol. 62, pp. 423-428, 1995.
[44]Ding H., Wang G. and Chen W., “General solution of plane problem of piezoelectric media expressed by harmonic functions”, Applied Mathematics and Mechanics (English Edition), Vol. 18, pp. 757-764, 1997.
[45]Xu X.L. and Rajapakse R.K.N.D., “On singularities in composite piezoelectric wedges and junctions”, International Journal of Solids and Structures, Vol. 37, 3253-3275, 2000.
[46]Chen B.J., Xiao Z.M. and Liew K.M., “Electro-elastic stress analysis for a wedge-shaped crack interacting with a screw dislocation in piezoelectric solid”, International Journal of Engineering Science, Vol. 40, pp. 621-635, 2002.
[47]Chen B.J., Xiao Z.M. and Liew K.M., “A screw dislocation in a piezoelectric bi-material wedge”, International Journal of Engineering Science, Vol. 40, pp. 1665-1685, 2002.
[48]Furuta A. and Uchino K., “Dynamic observation of crack propagation in piezoelectric multilayer actuators”, Journal of American Ceramic Society, Vol. 76, pp. 1615-1617, 1993.
[49]Aburatani H., Harada S., Uchino K., Furuta A. and Fuda Y., “Destruction Mechanisms in Ceramic Multilayer Actuators”, Japanese Journal of Applied Physics, Vol. 33, pp. 3091-3094, 1994.
[50]He L.H. and Ye R.Q., “Concentration of electric field near electrodes on piezoelectric layer”, Theoretical and Applied Fracture Mechanics, Vol. 33, 101-106, 2000.
[51]Li X.F. and Duan X.Y., “Electrodeelastic analysis of a piezoelectric layer with electrodes”, International Journal Fracture, Vol. 111, pp. L73-L78, 2001.
[52]Shindo Y., Narita F. and Sosa H., “electroelastic analysis of piezoelectric ceramics with surface electrodes”, International Journal of Engineering Science, Vol. 36, pp. 1001-1009, 1998.
[53]Ru C.Q., “Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes”, Journal of the Mechanics and Physics of Solids, Vol. 48, pp. 693-708, 2000.
[54]Ye R.Q. and He L.H., “Electric field and stress concentrations at the edge of parallel electrodes in piezoelectric ceramics”, International Journal of Solids and Structures, Vol. 38, 6941-6951, 2001.
[55]Sneddon I.N., Crack Problems in the Mathematical Theory of Elasticity, North Carolina State College, Departments of Mathematics and Engineering Research, Raleigh, N. C., 1961.
[56]Sneddon I.N., Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966.
[57]Sneddon I.N., Crack Problems in the Classical Theory of Elasticity, Wiley, New York, 1969.
[58]Sneddon I.N., The Use of Integral Transforms, McGraw, New York, 1972.
[59]Mandal B.N. and Mandal N., Advances in Dual Integral Equations, Chapman and hall, Boca Raton, 1999.
[60]Noble B., Method Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon Press, New York, 1958.
[61]Knauss W.G., “Stresses in an infinite strip containing a semi-infinite crack”, Transaction of the ASME, Journal of Applied Mechanics, Vol. 33, pp. 356-362, 1966.
[62]Nilsson F., “Dynamic stress-intensity factors for finite strip problems”, International Journal of Fracture Mechanics, Vol. 8, pp. 403-411, 1972.
[63]Georgiadis H.G., “Complex-variable and integral-transform methods for elastodynamic solutions of cracked orthotropic strips”, Engineering Fracture Mechanics, Vol. 24, pp. 727-735, 1986.
[64]Georgiadis H.G. and Papadopoulos G.A., “Determination of SIF in a cracked plane orthotropic strip by the Wiener-Hopf technique”, International Journal of Fracture, Vol. 34, pp. 57-64, 1987.
[65]Wang C.Y. and Mason J.J., “A solution method for finding dynamic stress intensity factors for arbitrarily oriented cracks in transversely isotropic materials”, International Journal of Fracture, Vol. 106, pp. 217-243, 2000.
[66]Li S. and Mataga P.A., “Dynamic crack propagation in piezoelectric materials-Part I. Electrode solution”, Journal of the Mechanics and Physics of Solids, Vol. 44, pp. 1799-1830, 1996.
[67]Li S. and Mataga P.A., “Dynamic crack propagation in piezoelectric materials-Part II. Vacuum solution”, Journal of the Mechanics and Physics of Solids, Vol. 44, pp. 1831-1866, 1996.
[68]Griffith A.A., “The phenomena of rupture and flow in solids”, Philosophical Transactions of the Royal Society of London A, Vol. 221, pp. 163-198, 1921.
[69]Erdogan F. and Sih G.C., “On the crack extension in plates under plane loading and transverse shear”, Transaction of the ASME, Journal of Basic Engineering, Vol. 85, pp. 519-527, 1963
[70]Sih G.C., “Strain energy density factor applied to mixed mode crack problems”, International Journal of Fracture, Vol. 10, pp. 305-322, 1974.
[71]Sih G.C. (Ed.), Mechanics of Fracture Vol. I-VII, Noordhoff International Publishing, Leyden, 1973-1981.
[72]Sih G.C., “Fracture mechanics of engineering structural components”, in Sih G.C. and Faria L. (Eds.), Fracture Mechanics Methodology Vol. I, Martinus Nijhoff Publishers, The Netherlands, pp. 35-101, 1984.
[73]Pak Y.E., “Crack extension force in a piezoelectric material”, Transaction of the ASME, Journal of Applied Mechanics, Vol. 57, pp. 647-653, 1990
[74]Pak Y.E., “Linear electro-elastic fracture mechanics of piezoelectric materials”, International Journal of Fracture, Vol. 54, pp. 79-100, 1992.
[75]Suo Z., Kuo C.M., Barnett D.M. and Willis J. R., “Fracture mechanics for piezoelectric ceramics”, Journal of the Mechanics and Physics of Solids, Vol. 40, pp. 739-765, 1992.
[76]Park S. and Sun C.T., ”Fracture criteria for piezoelectric ceramics”, Journal of American Ceramic Society, Vol. 78, pp. 1475-1480, 1995.
[77]Xu X.L. and Rajapakse R.K.N.D., “Analytical solution for an arbitrarily oriented void/crack and fracture of piezoceramics”, Acta Materialia, Vol. 47, pp. 1735-1747, 1999.
[78]Zuo J.Z. and Sih G.C., “Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics”, Theoretical and Applied Fracture Mechanics, Vol. 34, pp. 17-33, 2000.
[79]Sih G.C. and Zuo J.Z, “Multi-scale behavior of crack initiation and growth in piezoelectric ceramics”, Theoretical and Applied Fracture Mechanics, Vol. 34, pp. 123-141, 2000.
[80]Sih G.C., “A field model interpretation of crack initiation and growth behavior in ferroelectric ceramics: change of poling direction and boundary condition”, Theoretical and Applied Fracture Mechanics, Vol. 38, 1-14, 2002.
[81]Sih G.C. and Song Z.F., “Damage analysis of tetragonal perovskite structure ceramics implicated by asymptotic field solutions and boundary conditions”, Theoretical and Applied Fracture Mechanics, Vol. 38, 15-3, 2002.
[82]Berlincourt D.A., Curran D.R. and Jaffe, H., Piezoelectric and piezoceramic materials and their function in transducers. In: Masson, W.P. (ed.) Physical Acoustics, I-A., Academic Press, New York, pp. 169—270, 1964.
[83]Li X.F. and Fan T.Y., “Semi-infinite antiplane crack in a piezoelectric material”, International Journal of Fracture, Vol. 102, pp. L55-L60, 2000.
[84]Parton V.Z. and Kudryavtsev B.A., Electromagnetoelasticity, Gordon and Breach Science Publishers, New York, 1988.
[85]Luke Y., The Special Functions and Their Approximations, Academic Press, New York, 1969.
[86]Bahar L.Y., “A state space approach to elasticity”, Journal of the Franklin Institute, Vol. 299, pp. 33-41, 1975.
[87]Sosa H.A. and Hahar L.Y., “Transition from Airy stress function to state space formulation of elasticity”, Journal of the Franklin Institute, Vol. 329, pp. 817-828, 1992.
[88]Sosa H.A., “On the modeling of piezoelectric laminated structures”, Mechanics Research Communications, Vol. 19, pp. 541-546, 1992.
[89]Bak J. and Newman D.J., Complex Analysis, Springer, New York, 1997.
[90]Fulks W., Complex Analysis: An Introduction, Marcel Dekker, New York, 1993.