RZT (refined zigzag theory)理論已被證實可計算含脫層三明治複合樑(cracked sandwich beam, CSB)之柔度及能量釋放率，因其考慮轉折位移(zigzag displacement)，RZT所計算之柔度與能量釋放率比一階剪力變形理論(first order shear deformation theory, FSDT)更準確。然而RZT並未考慮裂縫尖端應力奇異性之影響，導致其柔度及能量釋放率之計算仍有所低估，因此本研究發展等效彈簧修正模型，可有效解決柔度之低估現象。此修正理論利用Wiener-Hopf method求得三明治複合樑核心層簡化模型之位移解析解，再透過應變能等效計算等效彈簧之彈簧係數，並藉此計算開口位移量以修正原始RZT理論中的位移連續條件，最後計算含脫層三明治複合樑之柔度與能量釋放率。
為了證實理論解，本研究建立實驗流程，製作含脫層三明治試片，並測試其柔度與能量釋放率。試片上下層為碳纖維板，中間層則為PVC發泡板，其楊氏係數可透過萬能試驗機量測而得。製作完成之三明治試片，置於萬能試驗機執行三點彎曲試驗，透過不同裂縫長度試片之柔度，可反推能量釋放率。
本研究發展之修正RZT理論，與有限元素軟體分析結果比較，在三種不同碳纖維板厚度、各種裂縫長度以及不同幾何參數與材料性質之下，可證實使用修正RZT理論得到的柔度及能量釋放率比起原始RZT更準確。而在裂縫長度較長的情況下，實驗柔度低於理論解，其原因可能來自於試片裂縫面之間的摩擦而導致柔度下降。

The refined zigzag theory (RZT) has been proven to be able to calculate the compliance and energy release rate of a cracked sandwich beam (CSB). The accuracy of RZT is better than that of first order shear deformation theory (FSDT) because the introduction of zigzag displacements. However, the stress singularity at the crack tip is not taken into consideration in RZT, resulting in an underestimation of compliance and energy release rate in RZT. In this study, a modified model composed of the virtual crack and equivalent spring is developed to solve the underestimation of compliance. This model utilize the Wiener-Hopf method to obtain an analytical solution of the core layer of the sandwich beam. The spring constant can be determined through the equivalence of strain energy. The opening displacement of the virtual crack then can be determined to modify the displacement continuity conditions in RZT. Finally, the modified compliance and energy release rate can be calculated.
In order to validate the analytical solution, an experimental procedure is established in this study to measure the compliance and energy release rate of a cracked sandwich beam. The specimen is composed of two carbon fiber plates and one PVC foam core as the face sheets and core, respectively. The Young’s modulus of the materials can be measured by universal testing machine. The specimen is placed on the testing machine to conduct the three-point bending testing. The compliance and energy release can be measured for various specimens with different crack length.
By comparing the RZT, the modified model developed in this study has been proven to be able to calculate the compliance and energy release rate at higher accuracy under various composite thicknesses and crack lengths. The measured compliance in three-point bending testing is lower than that calculated by theory. The possible reason is from the friction between two crack faces.

[1] Quispitupa A, Berggreen C. and Carlsson LA, ''Face/core interface fracture characterization of mixed mode bending sandwich specimens,'' Fatigue & Fracture of Engineering Materials & Structures, Vol. 34, pp. 839-853, 2011.
[2] Timoshenko S. P, ''On the correction for shear of the differential equation for transverse vibration of prismatic bars,'' Philosophical Magazine Series, Vol. 6, No. 41, pp. 744-746, 1921.
[3] Lo K. H, Christensen R. M. and Wu E. M, ''A higher order theory of plate deformation, Part 1: Homogeneous plates,'' Journal of Applied Mechanics, Vol. 44, pp. 663–668, 1977.
[4] Reddy J. N, ''A simple higher order theory for laminated composite plates,'' Journal of Applied Mechanics, Vol. 51, pp. 745–52, 1984.
[5] Lo K. H, Christensen R. M. and Wu E. ''M,Stress solution determination for high order plate theory,'' International Journal of Solids and Structures, Vol. 14, pp. 655-662, 1978.
[6] Mahi A, Bedia E. A. A and Tounsi A, ''A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates,'' Applied Mathematical Modelling, Vol 39, pp. 2489–2508, 2015.
[7] Habib Hebali, Abdelouahed Tounsi, Mohammed Sid Ahmed Houari and Aicha Bessaim, ''New Quasi-3D Hyperbolic Shear Deformation Theory for the Static and Free Vibration Analysis of Functionally Graded Plates,'' Journal of Engineering Mechanics, Vol. 140, No. 2, pp. 374–383, 2014.
[8] Hadji L, Khelifa Z, Daouadji T H and Bedia, ''Static bending and free vibration of FGM beam using an exponential shear deformation theory,'' Coupled Systems Mechanics, Vol 4, pp. 99–114, 2015.
[9] Sayyad A. S, Ghugal Y. M and Naik N. S, ''Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory,'' Curved and Layered Structures, Vol 2, pp. 279–289, 2015.
[10]Trung-Kien Nguyen, Ngoc-Duong Nguyen, Thuc P. Vo and Huu-Tai Thai, ''Trigonometric-series solution for analysis of laminated composite beams,'' Composite Structures, Vol 160, pp. 142-151, 2017.
[11]E. Carrera, M. Filippi and E. Zappino, ''Laminated beam analysis by polynomial, trigonometric, exponential and zig-zag theories,'' European Journal of Mechanics - A/Solids, Vol 41, pp. 58-69, 2013.
[12]Carrera E, Filippi M and Zappino E, ''Free vibration analysis of laminated beam by polynomial, trigonometric, exponential and zig-zag theories,'' Journal of Composite Materials, Vol 48, No 19, pp. 2299-2316, 2014.
[13]Sun C. T. and Whitney J. M, ''Theories for the dynamic response of laminated plates,'' AIAA Journal, Vol 11, pp. 178–183, 1973.
[14]Murakami H, ''Laminated composite plate theory with improved in-plane responses,'' ASME Journal of Applied Mechanics, Vol 53, pp. 661–666, 1986.
[15]Di Sciuva M, ''Multilayered anisotropic plate models with continuous interlaminar stresses,'' Composite Structures, Vol 22, pp. 149-167, 1992.
[16]Tessler A, Di Sciuva M. and Gherlone M, ''A Refined Zigzag Beam Theory for Composite and Sandwich Beams,'' Journal of Composite Materials, Vol 43, pp. 1051–1081, 2009.
[17]Di Sciuva M, Gherlone M, Iurlaro L. and Tessler A. ''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.
[18]Groh R. M. J. and Tessler A. ''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.
[19]Levinson M, ''A new rectangular beam theory,'' Journal of Sound and Vibration, Vol 74, pp. 81-87, 1981.
[20]Anı Ural, Alan T Zehnder and Anthony R Ingraffea, ''Fracture mechanics approach to facesheet delamination in honeycomb: measurement of energy release rate of the adhesive bond,'' Engineering Fracture Mechanics,Vol 70, pp. 93-103, 2003.
[21]Pan S. D, Wu L. Z, Sun Y. G. and Zou Z. G. ''Fracture test for double cantilever beam of honeycomb sandwich panels,'' Material Letters, Vol 62, pp. 523-526, 2008.
[22]Farshidi A, Berggreen C. and Schäuble R. ''Numerical fracture analysis and model validation for disbonded honeycomb core sandwich composites,'' Composite Structures, Vol 210, pp. 231-238, 2019.
[23]L. Cadieu, J.B Kopp, J. Jumel, J. Bega and C. Froustey, ''A fracture behaviour evaluation of Glass/Elium150 thermoplastic laminate withthe DCB test: Influence of loading rate and temperature,'' Composite Structures, Vol 255, 2021.
[24]V. Alfred Franklin and T. Christopher, ''Fracture Energy Estimation of DCB Specimens Made of Glass/Epoxy: An Experimental Study,'' Advances in Materials Science and Engineering, Vol 2013, 2013.
[25]Carlsson L. A, Senlein L.S. and Merry S. L, ''Characterization of face sheet/core shear fracture of composite sandwich beams,'' Journal of Composite Materials, Vol 25, pp. 101–116, 1991.
[26]Ayse Cagla Balaban and Kong Fah Tee, ''Strain energy release rate of sandwich composite beams for different densities and geometry parameters,'' Theoretical and Applied Fracture Mechanics, Vol 101, pp. 191-199, 2019.
[27]Daniel O. Adams, Joseph Nelson, Zack Bluth and Chris Hansen, ''Development and Evaluation of Fracture Mechanics Test Methods for Sandwich Composites,'' University of Utah, Salt Lake City, 2013.
[28]Marcello Manca, Amilcar Quispitupa, Christian Berggreen and Leif A. Carlsson, ''Face/core debond fatigue crack growth characterization using the sandwich mixed mode bending specimen,'' Composites Part A: Applied Science and Manufacturing, Vol 43, pp. 2120-2127, 2012.
[29]Chen, Chung-De and Wei Dai, ''The analysis of mode II strain energy release rate in a cracked sandwich beam based on the refined zigzag theory,'' Theoretical and Applied Fracture Mechanics,2020.
[30]Quispitupa, Amilcar, Berggreen, Christian and CARLSSON, L, ''Face/core interface fracture characterization of mixed mode bending sandwich specimens,'' Fatigue & Fracture of Engineering Materials &Structures, Vol 34, pp. 839-853, 2019.
[31]Victoria Mollón, Jorge Bonhomme, J. Viña and Antonio Argüelles, ''Theoretical and experimental analysis of carbon epoxy asymmetric dcb specimens to characterize mixed mode fracture toughness,'' Polymer Testing, Vol 29, pp. 766-770, 2010.
[32]Matthews T, Ali Moais and Paris A.J, ''Finite element analysis for large displacement J-Integral method for Mode I interlaminar fracture in composite materials,'' Finite Elements in Analysis and Design, Vol 83, pp. 43–48, 2014.
[33]Hu Jong Wan, ''J-Integral Evaluation for Calculating Structural Intensity and Stress Intensity Factor Using Commercial Finite Element (FE) Solutions,'' Advanced Materials Research, Vol 650, pp.379-384, 2013.
[34]Krueger R, "Virtual crack closure technique: History, approach, and applications," ASME Journal of Applied Mechanics, Vol 57, pp. 109–143, 2004.
[35]H. Millwater, D. Wagner, A. Baines and A. Montoya, "A virtual crack extension method to compute energy release rates using a complex variable finite element method," Engineering Fracture Mechanics, Vol 162, pp.95-111, 2016.
[36]Qiao P. and Wang J, ''Mechanics and fracture of crack tip deformable bi-material interface,'' International Journal of Solids and Structures, Vol 41, pp. 7423–7444, 2004.
[37]Wang J. and Qiao P, ''Analysis of beam-type fracture specimens with crack-tip deformation,'' International Journal of Fracture, Vol 132, pp.223-248, 2005.
[38]Qiao P. and Wang J, ''Novel joint deformation models and their application to delamination fracture analysis,'' Composites Science and Technology, Vol 65, pp. 1826-1839, 2005.
[39]Kannien M. F, ''An augmented double cantilever beam model for studying crack propagation and arrest,'' International Journal of Fracture, Vol 9, pp. 83-92, 1973.
[40]Penado F. E, ''A closed from solution for the energy release rate of the double cantilever beam specimen with an adhesive layer,'' Journal of Composite Materials, Vol 27, pp. 383-407, 1993.
[41]Fangliang Chen and Pizhong Qiao, ''Debonding analysis of FRP–concrete interface between two balanced adjacent flexural cracks in plated beams,'' International Journal of Solids and Structures, Vol 46, pp. 2618-2628, 2009.
[42]Knauss W. G, "Stresses in an Infinite Strip Containing a Semi-Infinite Crack," ASME Journal of Applied Mechanics, Vol 33, pp. 356–362, 1966.
[43]Georgiadis H. G., "Determination of SIF in a cracked plane orthotropic strip by the Wiener-Hopf technique," International Journal of Fracture, Vol. 34, pp, 57-64, 1987.
[44]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.
[45]Wang, C. Y. and James J. Mason, "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.
[46]戴維廉, ''RZT 理論 應用於脫層三明治複合樑與功能梯度複合樑之解析解,'' 碩士論文, 國立成功大學, 2019.
[47]陳重德, ''含壓電材料楔型結構以及含電極壓電片之奇異電彈場分析,'' 博士論文, 國立成功大學, 2003.
[48]Srinivas Prasad and Leif A. Carlsson, ''Debonding and crack kinking in foam core sandwich beams—II. Experimental investigation,'' Engineering Fracture Mechanics, Vol 47, pp. 825-841, 1994.
[49]P. F. Liu, B.B. Liao, L.Y. Jia and X.Q. Peng, "Finite element analysis of dynamic progressive failure of carbon fiber composite laminates under low velocity impact," Composite Structures, Vol 149, pp 408-422, 2016.