本篇論文分為兩部分，第一部份探討RZT (Refined zigzag theory)應用於脫層之三明治樑(cracked sandwich beam, CSB)，第二部分為RZT理論應用於功能梯度複合樑，皆以RZT理論為基礎。在論文第一部分中，除了描述RZT之理論架構外，也給出裂縫尖端之連續條件以及CSB的邊界條件，可解出脫層三明治樑之位移、應力、柔度以及能量釋放率。在RZT理論中，考慮因層間剪力模數差異所造成的軸向位移之轉折現象，而FSDT (First-order shear deformation theory)假設各層具有相同之剪應變，此簡化造成FSDT所計算之柔度小於實際值。藉由比對有限元素分析結果，也證實本論文利用RZT所得到的理論解析解比FSDT更正確。在第一部分的最後，本研究針對楊氏係數、剪力模數、各層厚度等各種參數對脫層三明治樑力學參數的影響，並就力學角度提出解釋。一般三明治樑因表層與中間層剛性差異容易在接合面上產生脫層現象，解決方式之一是引入功能梯度材料，消除介面之材料性質差異以降低應力。因此本篇論文在第二部分將功能梯度材料應用於上下對稱之三明治樑，並將RZT理論擴展至適用於功能梯度複合樑，探討各種受力及邊界條件以及材料性質變化，並與有限元素分析比較，結果顯示本論文提出以RZT為基礎之理論解之準確度相當高。由本論文提出之方法，可在少量的計算資源下，獲得準確的位移及應力。

In this thesis, the refined zigzag theory is used to solve the deflections and stresses of composite sandwich beam. According to the configurations of the sandwich beam, two parts are included in this study. In the first part, the cracked sandwich beam (CSB) is investigated by using the Refined Zigzag Theory (RZT). To solve the displacements, stresses, compliance and energy release reates, the continuity conditions at the crack tip and the boundary conditions are derived. By comparing the results between the theroretical predictions and finite element computations, the solutions by RZT are more accurate than those by FSDT (First-order Shear Deformation Theory). In the present study, parameters such as elastic modulus, shear modulus, layer thickness are considered to investigate their effects on the energy release rates of the CSB.
In the second part, the RZT is extended to investigated the sandwich beam with functionally graded material. Various boundary conditions, loading conditions and spacial distribution models are considered to investigated their effects on the FGM sandwich beam. It reveals that the solutions by RZT agree very much with those by FEM and has low computational resources. Material properties equation in thickness direction and the effect of isotropic or orthotropic materials. In general, FGM is an isotropic material. Although FGM with orthotropic material has not developed on the market, it is expected that the numerical simulation results for FGM with orthotropic material presented of this paper will be applied to this new material in the future.

[1] S. P. Timoshenko, "LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 41, no. 245, pp. 744-746, 1921.
[2] K. H. Lo, R. M. Christensen, and E. M. Wu, "A High-Order Theory of Plate Deformation—Part 1: Homogeneous Plates," Journal of Applied Mechanics, vol. 44, no. 4, pp. 663-668, 1977.
[3] K. H. Lo, R. M. Christensen, and E. M. Wu, "A High-Order Theory of Plate Deformation—Part 2: Laminated Plates," Journal of Applied Mechanics, vol. 44, no. 4, pp. 669-676, 1977.
[4] J. N. Reddy, "A Simple Higher-Order Theory for Laminated Composite Plates," Journal of Applied Mechanics, vol. 51, no. 4, pp. 745-752, 1984.
[5] C.-T. Sun and J. M. Whitney, "Theories for the Dynamic Response of Laminated Plates," AIAA Journal, vol. 11, no. 2, pp. 178-183, 1973.
[6] Y. Frostig, M. Baruch, O. Vilnay, and I. Sheinman, "High‐Order Theory for Sandwich‐Beam Behavior with Transversely Flexible Core," Journal of Engineering Mechanics, vol. 118, no. 5, pp. 1026-1043, 1992.
[7] C. N. Phan, Y. Frostig, and G. A. Kardomateas, "Analysis of Sandwich Beams With a Compliant Core and With In-Plane Rigidity—Extended High-Order Sandwich Panel Theory Versus Elasticity," Journal of Applied Mechanics, vol. 79, no. 4, pp. 041001-041001-11, 2012.
[8] M. di Sciuva, "Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: An evaluation of a new displacement model," Journal of Sound and Vibration, vol. 105, no. 3, pp. 425-442, 1986.
[9] R. C. Averill, "Static and dynamic response of moderately thick laminated beams with damage," Composites Engineering, vol. 4, no. 4, pp. 381-395, 1994.
[10] A. Tessler, M. Di Sciuva, and M. Gherlone, "A Refined Zigzag Beam Theory for Composite and Sandwich Beams," Journal of Composite Materials, vol. 43, no. 9, pp. 1051-1081, 2009.
[11] M. Di Sciuva, M. Gherlone, L. Iurlaro, and A. Tessler, "A class of higher-order C0 composite and sandwich beam elements based on the Refined Zigzag Theory," Composite Structures, vol. 132, pp. 784-803, 2015.
[12] R. M. J. Groh and A. Tessler, "Computationally efficient beam elements for accurate stresses in sandwich laminates and laminated composites with delaminations," Computer Methods in Applied Mechanics and Engineering, vol. 320, pp. 369-395, 2017.
[13] L. A. Carlsson and J. W. Gillespie, "Mode-II Interlaminar Fracture of Composites," in Composite Materials Series, vol. 6, K. Friedrich Ed.: Elsevier, 1989, pp. 113-157.
[14] L. A. Carlsson, L. S. Sendlein, and S. L. Merry, "Characterization of Face Sheet/Core Shear Fracture of Composite Sandwich Beams," Journal of Composite Materials, vol. 25, no. 1, pp. 101-116, 1991.
[15] L. A. Carlsson, "On the Design of the Cracked Sandwich Beam (CSB) Specimen," Journal of Reinforced Plastics and Composites, vol. 10, no. 4, pp. 434-444, 1991.
[16] A. Quispitupa, C. Berggreen, and L. A. Carlsson, "On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization," Engineering Fracture Mechanics, vol. 76, no. 4, pp. 594-613, 2009.
[17] M. Naebe and K. Shirvanimoghaddam, "Functionally graded materials: A review of fabrication and properties," Applied Materials Today, vol. 5, pp. 223-245, 2016.
[18] S. Xiang, Y.-x. Jin, Z.-y. Bi, S.-x. Jiang, and M.-s. Yang, "A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates," Composite Structures, vol. 93, no. 11, pp. 2826-2832, 2011.
[19] Y. Y. Yang, "Stress analysis in a joint with a functionally graded material under a thermal loading by using the Mellin transform method," International Journal of Solids and Structures, vol. 35, no. 12, pp. 1261-1287, 1998.
[20] F. Delale and F. Erdogan, "The Crack Problem for a Nonhomogeneous Plane," Journal of Applied Mechanics, vol. 50, no. 3, pp. 609-614, 1983.
[21] B. V. Sankar, "An elasticity solution for functionally graded beams," Composites Science and Technology, vol. 61, no. 5, pp. 689-696, 2001.
[22] A. Mahi, E. A. Adda Bedia, A. Tounsi, and I. Mechab, "An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions," Composite Structures, vol. 92, no. 8, pp. 1877-1887, 2010.
[23] A. Bessaim, M. S. A. Houari, A. Tounsi, S. R. Mahmoud, and E. A. A. Bedia, "A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets," Journal of Sandwich Structures & Materials, vol. 15, no. 6, pp. 671-703, 2013.
[24] Tahar Hassaine Daouadji, Abdelaziz Hadj Henni, Abdelouahed Tounsi, and Adda Bedia El Abbes, “A New Hyperbolic Shear Deformation Theory for Bending Analysis of Functionally Graded Plates,” Modelling and Simulation in Engineering, vol. 2012, Article ID 159806, 10 pages, 2012.
[25] L. Hadji, Z. Khelifa, and A. B. El Abbes, "A new higher order shear deformation model for functionally graded beams," KSCE Journal of Civil Engineering, vol. 20, no. 5, pp. 1835-1841, 2016.
[26] S. Kapuria, M. Bhattacharyya, and A. N. Kumar, "Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation," Composite Structures, vol. 82, no. 3, pp. 390-402, 2008.
[27] H.-T. Thai and T. P. Vo, "A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates," Applied Mathematical Modelling, vol. 37, no. 5, pp. 3269-3281, 2013.
[28] A. M. Zenkour, "A simple four-unknown refined theory for bending analysis of functionally graded plates," Applied Mathematical Modelling, vol. 37, no. 20, pp. 9041-9051, 2013.