臺灣博碩士論文加值系統

本文發展異向性材料與功能材料之狀態空間數學模式，以解析壓電彈性力學與廣義熱彈性力學等相關問題。此數學架構以位移向量和應力向量為基本變量，將基本方程式表示為簡單之矩陣形式，保留了原方程式的特性。依此推得之狀態方程式與輸出方程式，數學結構簡潔優美，得以運用許多不適用於傳統模式的數學方法解析問題，各物理問題數學模式之間有明顯的對應性和類比性。文中亦利用變分原理推導出狀態方程式與輸出方程式，藉此，可進一步瞭解此狀態空間架構之數學性質。本文並應用狀態空間模式探討若干直角座標系與圓柱座標系下之物理問題，包括複合材料層板應力分析，異向性材料熱彈性力學反應，壓電材料圓柱體之廣義平面問題，壓電材料試體之有效長度，彈性波在功能材料中之傳播特性等，盡可能地求得各問題的確解。研究顯示狀態空間數學模式能有效處理異向性材料與功能材料之壓電熱彈性力學問題。

A state space formalism for piezothermoelasticity and generalized thermoelasticity of anisotropic materials and functionally graded materials is developed. By taking the displacement vector and the stress vectors as the fundamental field variables, the basic equations for the problems can be expressed in a concise matrix form which retains the characteristics of the original field equations without recourse to elimination of the unknown variables. Accordingly, the state equation and the output equation are expressed in remarkably neat yet explicit representations so that many mathematical methods which are usually not amenable to the conventional approaches are applicable to determining the analytic solution for a problem. Furthermore, correspondence and analogy among various theories appear explicitly in the state space setting. To examine further the mathematical properties of the system matrices inherent in the state space framework, the state equation and the output equation are also derived through the variational formulation. A number of problems in Cartesian coordinate and cylindrical coordinate systems, such as the stress analysis of composite laminates, the responses of anisotropic bodies in generalized thermoelasticity, the generalized plane problems of piezoelectric circular cylinders, the effective lengths of piezoelectric specimens, and the propagation behaviors of elastic waves in functionally graded materials, are studied following the state space approach. Whenever possible, we seek for the exact solutions for the problems under study. The present study shows that the state space formalism is an effective and systematic approach for problems of piezothermoelasticity of anisotropic bodies and functionally graded materials.

Abstract (in Chinese) iv
Abstract v
Acknowledgments vi
Table of Contents viii
List of Tables xiii
List of Figures xiv
Nomenclature xvi
Chapter 1 Introduction 1
1.1 Background and Motivation 1
1.2 Objectives and Scope 3
Part I State Space Formalism in Cartesian Coordinates and Applications 5
Chapter 2 State Space Formulation in Cartesian Coordinate System 6
2.1 Basic Equations in Matrix Form 6
2.1.1 State equation and output equation 7
2.1.2 Structures and identities of relevant matrices 9
2.2 Variational Formulation 11
2.2.1 Lagrangian formulation 11
2.2.2 Hamiltonian formulation 13
2.3 Characteristics of Hamiltonian System 16
2.3.1 Solvability conditions 17
2.3.2 Symplectic orthogonality 20
2.4 Correspondences in the State Space Setting 22
2.4.1 Correspondences among various theories 22
2.4.2 Correspondence with Stroh’s formalism 24
2.5 Summary and Concluding Remarks of Chapter 2 26
Chapter 3 Exact Stress Analysis of Composite Laminates 27
3.1 State Space Formulation 27
3.1.1 Problem statement 28
3.1.2 Laminates under extension, torsion, and bending 30
3.2 Torsion of Cross-ply Laminates and Orthotropic FGM 32
3.2.1 Exact solution for homogeneous bars 32
3.2.2 Exact solution for FGM bars 35
3.2.3 Exact solution for cross-ply laminates 37
3.3 Summary and Concluding Remarks of Chapter 3 46
3.A Proof of Zero Resultant Shear Forces over Cross-section 46
3.B Eigen Relation 47
Chapter 4 Generalized Thermoelasticity with Thermal Relaxation 49
4.1 State Space Formulation 50
4.2 Nondimensionalization 52
4.3 Perturbation with Multiple Scales 54
4.4 Plane Harmonic Waves 55
4.4.1 Strong coupling 55
4.4.2 Weak coupling 57
4.4.3 The case c0n ≈ cΘ0(á) 62
4.5 Results and Discussions 65
4.6 Summary and Concluding Remarks of Chapter 4 70
4.A Solvability Condition 72
4.B Higher-order Solutions to Eqs. (4.20) and (4.21) 73
Part II State Space Formalism in Cylindrical Coordinates and Applications 75
Chapter 5 State Space Formulation in Cylindrical Coordinate System 76
5.1 Basic Equations in Matrix Form 76
5.1.1 State equation and output equation 76
5.1.2 Structures and identities of relevant matrices 78
5.2 Variational Formulation 80
5.2.1 Lagrangian formulation 80
5.2.2 Hamiltonian formulation 82
5.3 Characteristics of Hamiltonian System 84
5.3.1 Solvability conditions 86
5.3.2 Symplectic orthogonality 89
5.4 Correspondences in the State Space Setting 91
5.5 Summary and Concluding Remarks of Chapter 5 92
Chapter 6 Generalized Plane Problems of Piezoelectric Circular Cylinders 93
6.1 State Space Formulation 94
6.2 Exact Solution for Power-Law Radial Inhomogeneity 96
6.2.1 Homogeneous solution 97
6.2.2 Particular solutions 98
6.3 Summary and Concluding Remarks of Chapter 6 100
Chapter 7 Effective Lengths of Tensile and Torsional Specimens of Piezoelectric Materials 101
7.1 State Space Formulation 102
7.2 Solution by Matrix Algebra 103
7.2.1 Characteristics of the eigensolution 104
7.2.2 Characteristic decay length 106
7.3 Results and Discussions 107
7.4 Summary and Concluding Remarks of Chapter 7 112
7.A Matrices for Orthorhombic Material of Class mm2 in Axisymmetric Problem 112
Chapter 8 Characteristics of Elastic Waves in Circular Tubes or Bars of Functionally Graded Materials 114
8.1 Two-dimensional Problems of ElasticWaves 114
8.1.1 Power series approximation 116
8.1.2 Piecewise-constant approximation 117
8.2 Results and Discussions 119
8.3 Summary and Concluding Remarks of Chapter 8 120
8.A Eigen Relation 123
Chapter 9 Conclusions and Further Research 125
9.1 Conclusions 125
9.2 Further Research 126
Bibliography 128
Appendix A Alternative Formulation in Matrix Form 134
Appendix B Displacement Field for Generalized Plane Problems 136
Appendix C Relations between the Two State Space Formulations 139
Appendix D Approximate Methods 141
D.1 Power Series Approximation 141
D.2 Piecewise Constant Approximation 145
Vita 148

Achenbach, J.D., Wave Propagation in Elastic Solids. North-Holland, Amsterdam, 1973.
Anwar, N., Sherief, H., State space approach to generalized thermoelasticity. Journal of Thermal Stresses 11, 353--365, 1988.
Arfken, G.B., Weber, H.J., Mathematical Methods for Physicists, 5th ed. Academic Press, San Diego, 2001.
Arnold, V.I., Mathematical Methods of Classical Mechanics. Springer-Verlag, Berlin, 1978.
Bahar, L.B. Hetnarski, R.B., State space approach to thermoelasticity. Journal of Thermal Stresses 1, 135--145, 1978.
Boley, B.A., Weiner, J.H., Theory of Thermal Stresses. Wiley, New York, 1960.
Brebbia, C.A., Telles, J.C.F., Wrobel, L.C., Boundary Element Techniques, Springer-Verlag, Berlin, 1984.
Chadwick, P., Thermoelasticity, the dynamic theory. Progress in Solid Mechanics, Vol. 1, edited by Sneddon, I.N. and Hill, R., North-Holland, Amsterdam, 1960.
Chandrasekharaiah, D.S., Thermoelasticity with second sound: a review. ASME Applied Mechanics Review 39, 355--376, 1986.
Chandrasekharaiah, D.S., Hyperbolic thermoelasticity: a review of recent literature. ASME Applied Mechanics Review 51, 705--729, 1998.
Chang, H.H., Tarn, J.Q., A state space formalism and perturbation method for generalized thermoelasticity of anisotropic bodies. International Journal of Solids and Structures 44, 956--975, 2007a.
Chang, H.H., Tarn, J.Q., A state space approach for exact analysis of composite laminates and functionally graded materials. International Journal of Solids and Structures, 2007b. available online
Chen, T., Chung, C.T., Lin, W.L., A revisit of a cylindrically anisotropic tube subjected to pressuring, shearing, torsion, extension and a uniform temperature change. International Journal of Solids and Structures 37, 5143--5159, 2000.
Chester, M., Second sound in solids. Physical Review 131, 2013--2015, 1963.
Chou, F.H., Achenbach, J.D., Three-dimensional vibrations of orthotropic cylinders. ASCE Journal of the Engineering Mechanics Division 98, 813--822, 1972.
Christensen, R.M., Mechanics of Composite Materials. John Wiley, New York, 1979.
Courant, R., Hilbert, D., Methods of Mathematical Physics. Volume I, Interscience, New York, 1955.
Dhaliwal, R.S., Sherief, H.H., Generalized thermoelasticity for anisotropic media. Quarterly of Applied Mathematics 38, 1--8, 1980.
Ezzat, M.A., El-Karamany, A.S., Samaan, A.A., State-space formulation to generalized thermoviscoelasticity with thermal relaxation. Journal of Thermal Stresses 24(9), 823--846, 2001.
Hawwa, M.A., Nayfeh, A.H., The general problem of thermoelastic waves in anisotropic periodically laminated composites. Composite Engineering 5, 1499--1517, 1995.
Hildebrand, F.B., Methods of Applied Mathematics, 2nd ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
Hildebrand, F.B., Advanced Calculus for Applications, 2nd ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
Horgan, C.O., Recent developments concerning Saint-Venant's principle: an update. ASME Applied Mechanics Review 42, 295--303, 1989.
Horgan, C.O., Recent developments concerning Saint-Venant's principle: a second update. ASME Applied Mechanics Review 49, S101--S111, 1996.
Horgan, C.O., Chan, A.M., Torsion of functionally graded isotropic linearly elastic bars. Journal of Elasticity 52, 181--199, 1998.
Jones, R.M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975.
Korn, G.A., Korn, T.M., Mathematical Handbook for Scientists and Engineers. MacGraw-Hill, New York, 1968.
Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco, 1963.
Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Body. Translated from the 1977 Russian edition, Mir, Moscow, 1981.
Lord, H.W., Shulman, Y., A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids 15, 299--309, 1967.
Ma, L.F., Chen, Y.H., Zhang, S.Y., On the explicit formulations of circular tube, bar and shell of cylindrically piezoelectric material under pressuring loading. International Journal of Engineering Science 39, 369--385, 2001.
Martin, P.A., Berger, J.R., Waves in wood: free vibrations of a wooden pole. Journal of the Mechanics and Physics of Solids 49, 1155--1178, 2001.
Mian, M.A., Spencer, A.J.M., Exact solutions for functionally graded and laminated elastic materials. Journal of the Mechanics and Physics of Solids 46, 2283--2295, 1998.
Mirsky, I., Axisymmetric vibrations of orthotropic cylinders. Journal of the Acoustical Society of America 36, 2106--2112, 1964.
Nayfeh, A., Introduction to Perturbation Techniques. Wiley, New York, 1981.
Nowacki, W., Dynamic Problems of Thermoelasticity. Noordhoff, Leyden, The Netherlands, 1975.
Nowacki, W., Thermoelasticity. 2nd ed. Pergamon Press, Oxford, 1986.
Nye, J.F., Physical Properties of Crystals. Oxford University Press, Oxford, 1957.
Pease, M.C., Methods of Matrix Algebra. Academic Press, New York, 1965.
Rogers, T.G., Watson, P., Spencer, A.J.M., An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness. Proceedings of the Royal Society of London A 437, 199--213, 1992.
Rooney, F., Ferrari, M., Tension, bending, and flexure of functionally graded cylinders. International Journal of Solids and Structures 38, 413--421, 2001.
Sneddon, I.S., The Linear Theory of Thermoelasticity. CISM 119, Springer-Verlag, Wien-New York, 1974.
Stroh, A.N., Dislocations and cracks in anisotropic elasticity. Philosophical Magazine 3, 625--646, 1958.
Stroh, A.N., Steady state problems in anisotropic elasticity. Journal of Mathematical Physics 41, 77--103, 1962.
Tanigawa, Y., Some basic thermoelastic problems for nonhomogeneous structural materials. ASME Applied Mechanics Review 48, 287--300, 1995.
Tarn, J.Q., An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated cylindrical shells. Journal of the Mechanics and Physics of Solids 42, 1633--1650, 1994.
Tarn, J.Q., Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. International Journal of Solids and Structures 38, 8189--8206, 2001a.
Tarn, J.Q., Exact solutions for electroelastic analysis of generalized plane strain and torsion of piezoelectric cylinders. The Chinese Journal of Mechanics 17, 149--156, 2001b.
Tarn, J.Q., A state space formalism for anisptropic elasticity Part I: Rectilinear anisotropy. International Journal of Solids and Structures 39, 5143--5155, 2002a.
Tarn, J.Q., A state space formalism for anisotropic elasticity Part II: Cylindrically anisotropy. International Journal of Solids and Structures 39, 5157--5172, 2002b.
Tarn, J.Q., A state space formalism for piezothermoelasticity. International Journal of Solids and Structures 39, 5173--5184, 2002c.
Tarn, J.Q., Exact solutions of a piezoelectric circular tube or bar under extension, torsion, pressuring, shearing, uniform electric loading and temperature change. Proceedings of the Royal Society of London A 458, 2349--2367, 2002d.
Tarn, J. Q., Stress singularity in an elastic cylinder of cylindrically anisotropic materials. Journal of Elasticity 69, 1--13, 2002e.
Tarn, J.Q., Chang, H.H., Extension, torsion, bending, pressuring and shearing of piezoelectric circular cylinders with radial inhomogeneity. A special issue on Mechanics of Piezoelectric, Magnetoelectric and Ferroelectric Materials. Journal of Intelligent Material Systems and Structures 16, 631--641, 2005.
Tarn, J.Q., Chang, H.H., Effective lengths of tensile and torsional specimens of piezoelectric materials. Journal of Mechanics 22(1), 27--34, 2006.
Tarn, J.Q., Huang, L.J., Saint-Venant end effects in multilayered piezoelectric laminates. International Journal of Solids and Structures 39, 4979--4998, 2002.
Tarn, J.Q., Wang, Y.M., An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated plates. International Journal of Solids and Structures 31, 231--246, 1994.
Tarn, J.Q., Wang, Y.M., Laminated composite tubes under extension, torsion, bending, shearing and pressuring: a state space approach. International Journal of Solids and Structures 38, 9053--9075, 2001.
Tarn, J.Q., Wang, Y.M., Heat conduction in a cylindrically anisotropic tube of a functionally graded material. The Chinese Journal of Mechanics 19, 365--372, 2003.
Tarn, J.Q., Wang, Y.M., End effects of heat conduction in circular cylinders of functionally graded materials and laminated composites. International Journal of Heat and Mass Transfer 47, 5741--5747, 2004.
Tiersten, H.F., Linear Piezoelectric Plate Vibrations. Plenum Press, New York, 1969.
Ting, T.C.T., Anisotropic Elasticity, Theory and Applications. Oxford University Press, Oxford, 1996a.
Ting, T.C.T., Pressuring, shearing, torsion and extension of a circular tube or bar of a cylindrically anisotropic material. Proceedings of the Royal Society of London A 452, 2397--2421, 1996b.
Ting, T.C.T., New solutions to pressuring, shearing, torsion, and extension of a cylindrically anisotropic elastic circular tube or bar. Proceedings of the Royal Society of London A 455, 3527--3542, 1999.
Ting, T.C.T., Recent developments in anisotropic elasticity. International Journal of Solids and Structures 37, 401--409, 2000.
Tsai, C.L., Daniel, I.M., Yaniv G., Torsional response of rectangular composite laminates. ASME Journal of Applied Mechanics 57, 383--387, 1990.
Verma, K.L., Hasebe, N., Wave propagation in plates of general anisotropic media in generalized thermoelasticity. International Journal of Engineering Science 39, 1739--1763, 2001.
Verma, K.L., On the propagation of waves in layered anisotropic media in generalized thermoelasticity. International Journal of Engineering Science 40, 2077--2096, 2002.
Wang, S.S., Choi, I., Boundary-layer effects in composite laminates I. free-edge stress singularities. ASME Journal of Applied Mechanics 49, 541--548, 1982a.
Wang, S.S., Choi, I., Boundary-layer effects in composite laminates II. free-edge stress solutions and basic characteristics. ASME Journal of Applied Mechanics 49, 549--560, 1982b.
Wang, Y.M., Tarn, J.Q., Hsu, C.K., State space approach for stress decay in laminates. International Journal of Solids and Structures 37, 3535--3553, 2000.
Wu, K.C., Generalization of the Stroh formalism to 3-dimensional anisotropic elasticity. Journal of Elasticity 51, 213--225, 1998.
Yang, Y.Y., Time-dependent stress analysis in functionally graded materials. International Journal of Solids and Structures 37, 7593--7608, 2000.
Yin, W.L., Thermal stresses and free-edge effects in laminated beams: a variational approach using stress functions. ASME Journal of Electronic Packaging 113, 68--75, 1991.
Yin, W.L., Refined variational solutions of the interfacial thermal stresses in a laminated beam. ASME Journal of Electronic Packaging 114, 193--198, 1992.
Yin, W.L., Free-edge effects in anisotropic laminates under extension, bending and twisting, part I: a stress-function-based variational approach. ASME Journal of Applied Mechanics 61, 410--415, 1994a.
Yin, W.L., Free-edge effects in anisotropic laminates under extension, bending and twisting, part II: eigenfunction analysis and the results for symmetric laminates. ASME Journal of Applied Mechanics 61, 416--421, 1994b.
Yuan, F.G., Hsieh, C.C., Three-dimensional wave propagation in composite cylindrical shells. Composite Structures 42, 153--167, 1998.
Zhong, W., A New Systematic Methodology for Theory of Elasticity. Dalian University of Technology, Dalian, China, 1995. (in Chinese)
Zhong, W., Williams, F.W., Physical interpretation of the symplectic orthogonality of the eigensolutions of a Hamiltonian or symplectic matrix. Computers and Structures 49, 749--750, 1993.