成功大學博碩士論文系統

簡易檢索 / 詳目顯示

回結果列表

研究生：	梁文宇 Liang, Wen-Yu
論文名稱：	複合層板拉伸與彎曲之精確解析 Exact Analysis for Composite Laminates under Extension and Bending
指導教授：	譚建國 Tarn, Jiann-Quo 朱聖浩 Ju, Shen-Haw
學位類別：	博士 Doctor
系所名稱：	工學院 - 土木工程學系 Department of Civil Engineering
論文出版年：	2016
畢業學年度：	105
語文別：	英文
論文頁數：	100
中文關鍵詞：	複合層板、自由邊界效應、廣義平面應變、板、狀態空間架構、辛正交、拉伸、彎曲、扭矩、單斜晶材料、正向性材料、哈密頓、特徵函數展開法。
外文關鍵詞：	Composite laminates, Free-edge effect, Generalized plane strain, Plate, State space formalism, Symplectic orthogonality, Extension, Bending, Torsion, Monoclinic material, Hamiltonian, Eigenfunction expansions.
相關次數：	點閱：167 下載：9
分享至:	分享至facebook 分享至twitter

查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

本文以Hamiltonian狀態空間法解析複合材料層板受拉伸與彎曲後之位移與應力場量。首先經由Lagrangian系統中的Legendre變換，無須特別分類組合各場變量，將卡氏座標之彈性力學基本方程式建構成以位移向量與其共軛之應力向量為主要變數之狀態方程式以及輸出方程式。此法有效且有系統地解決複合材料層板相關問題。解析流程牽涉特徵函數展開法、傳遞矩陣法，以及Hamiltonian矩陣中各特徵向量間之辛正交關係式。無論複合材料層疊數目多寡，所求得之場量皆能滿足異向性彈性力學基本方程式、矩形截面四周之邊界條件、層與層交界面之連續條件，以及外力作用端面處之邊界條件。本文所得之解析解與有限元素法之數值解相比後，可發現兩者有合理之一致性，對於研究複合材料層板之自由邊界效應是重要的，藉此，為數值模擬與材料特性研究提供適用之基準。

This work used the Hamiltonian state space approach for the exact analysis of displacement and stress fields in the multilayered laminated composite plates of elastic materials under extension and bending. Without grouping the field variables in an ad hoc process, the basic equations of elasticity in the Cartesian coordinates are formulated into a state equation and an output equation which are composed of the primary variables in terms of the generalized displacements and the conjugate generalized tractions by means of Legendre’s transformation from the Lagrangian system. The present approach is effective and systematic. The solution process involves an eigenfunction expansion technique, transfer matrix method, and symplectic orthogonality between the eigenvectors of a Hamiltonian matrix, and all the basic equations of anisotropic elasticity, the traction-free conditions on the bounding planes of the rectangular section, the boundary conditions at the end sections where the external loadings are applied, and the interfacial continuity conditions in the multilayered laminated systems, are satisfied exactly, regardless of the number of layers. Comparisons of the stress fields between the proposed analytical and finite element solutions show good agreement. The present approach is important with regard to studying the free-edge effects, in addition to obtaining solutions which serve as useful benchmarks for numerical modeling and material characterization of composite laminates.

Abstract (in chinese)	I
Abstract	II
Acknowledgments	III
Table of Contents	IV
List of Tables	VI
List of Figures	VII
Nomenclature	IX
Chapter 1	Introduction	1
1.1	Background	1
1.2	Literature review	4
1.3	Objectives and thesis outline	12
Chapter 2	Hamiltonian State Space Formulation	15
2.1	Basic equation in matrix form	15
2.2	State Equation and Output Equation	18
2.3	Formulation for problem of piezoelectric laminates	22
2.4	Symplectic orthogonality	29
Chapter 3	Solution Approach of Hamiltonian State Space Formulation	36
3.1	Particular solution	36
3.2	Eigensolution	44
3.3	Eigensolution for zero eigenvalue	47
3.4	Complete solution for the state space equations	48
3.5	Satisfaction of the edge conditions	49
3.6	Antiplane problem	53
3.7	Solution for monoclinic materials	54
3.8	Relations between the parameters and the applied loads at the end sections	57
Chapter 4	Exact Analysis for Cross-ply Laminates under Extension and Bending	59
Chapter 5	Theoretical study of singularities at the interface of composite laminates	75
5.1	Background	75
5.2	Problem statement	77
5.3	Displacement and stress fields at interface of laminates	78
5.4	Notch SIFs	80
5.5	Least-squares collocation methods to find g	81
5.6	Numerical results	82
Chapter 6	Conclusions and Further Research	89
6.1	Conclusions	89
6.2	Further Research	90
Bibliography	92

List of Tables
Table (4.1). Elastic constants of selected materials	67
Table (4.2). The dimensionless eigenvalues   of the eigensystem	68
Table (5.2). Eigenvalues   for the Singular order	83

List of Figures
Figure (2.1). Free-edge problem for a multilayered laminated plate of rectangular	22
Figure (4.1). The free-edge problem for a composite laminate of rectangular section subjected to an axial force P, a bending moment M at the end sections	63
Figure (4.2)-(a), (b): Finite element mesh with 117120 20-node isoparametric elements (1444467 degrees of freedom)	63
Figure (4.3). The contour lines at zero value of symmetric mode	64
Figure (4.4). Through-thickness variation of the non-dimensional inplane stress     and shear stress   at the free-edge   under extension.	69
Figure (4.5). Through-thickness variation of the non-dimensional inplane stress   and shear stress   at the free-edge   under vertical bending moment.	70
Figure (4.6). Distribution of the non-dimensional interlaminar stresses   and  at the  -interface and inplane stress   at the middle surface of   under extension.	71
Figure (4.7). Distribution of the non-dimensional interlaminar stresses   and  at the  -interface and inplane stress   at the middle surface of   under bending.	71
Figure (4.8). Comparison of the interlaminar normal and shear stresses between the present study and other published approaches. (Extension)	73
Figure (4.9). Comparison between the present study and finite element analyses. (Extension)	74
Figure (4.10). Comparison between the present study and finite element analyses. (Bending)	74
Figure (5.1). The free-edge problem for an angle-ply laminate of rectangular section with stacking sequences of [ ] subjected to an axial force  , a bending moment  , and a torque   at the end sections.	77
Figure (5.2). Geometry of a sharp V-notch with a number of materials (The origin of interest is located at the notch tip and the center of the notch surface is in the negative x direction, where angles in the anti-clockwise direction is positive.)	78
Figure (5.3). The contour result of the displacement u	84
Figure (5.4). The contour result of the displacement v	84
Figure (5.5). The contour result of the displacement w	85
Figure (5.6). The contour result of the stress  	85
Figure (5.7). The contour result of the stress  	86
Figure (5.8). The contour result of the stress  	86
Figure (5.9). The contour result of the stress  	87
Figure (5.10). The contour result of the stress  	87
Figure (5.11). The contour result of the stress  	88
Figure (5.12). Through-thickness variation of the stresses at the free-edge (extension)	88
                                    

Andakhshideh, A., Tahani, M., Free-edge stress analysis of general rectangular composite laminates under bending, torsion and thermal loads. European Journal of Mechanics A/Solids 42, 229-240, 2013.
Arnold, V.I., Mathematical Methods of Classical Mechanics. Springer, Berlin, 1978.
Becker, W., Closed-form solution for the free-edge effect in crossply laminates. Composite Structures 26(1-2), 39-45, 1993.
Batra, R.C., Liang, X.Q., The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Composite Structures 63, 203-216, 1997.
Bahar, L.Y., A state space approach to elasticity. Journal of the Franklin Institute 299(1), 33-41, 1975.
Chang, H.H., Tarn, J.Q., Three-dimensional elasticity solutions for rectangular orthotropic plates. Journal of Elasticity 108, 49-66. 2012.
Chaudhuri, R.A., Xie, M., Free-edge stress singularity in a bimaterial laminate. Composite Structures 40(2), 129-136, 1998.
Cho, M., Kim, S.H., Iterative free-edge stress analysis of composite laminates under extension, bending, twisting and thermal loadings. International Journal of Solids and Structures 37, 435-459, 2000.
Demasi, L., Three-dimensional closed form solutions and exact thin plate theories for isotropic plates. Composite Structures 80, 183-195, 2007.
Demasi, L., Three-dimensional closed form solutions and ∞³ theories for orthotropic plates. Mechanics of Advanced Materials and Structures 17, 20-39, 2010.
Delale, F., Stress singularities in bounded anisotropic materials. International Journal of Solids and Structures 20(1), 31-40, 1984.
Dhanesh, D., Kapuria, S., Achary, G.G.S., Accurate prediction of three-dimensional free edge stress field in composite laminates using mixed-field multiterm extended Kantorovich method. Acta Mech, 1-25, 2016.

D’Ottavio, M., Vidal, P., Valot, E., Polit, O., Assessment of plate theories for free-edge effects. Composites: Part B 48, 111-121, 2013.
Gopinathan, S V., Varadan, V.V., Varadan, V.K., A review and critique of theories for piezoelectric laminates. Smart Materials and Structures 9, 24-48, 2000.
Goldstein, H., Classical Mechanics. Addison-Wesley, Reading , 1965.
Goodsell, J., Pipes, R.B., Free-Edge interlaminar stresses in angle-ply laminates: A family of analytic solutions. Journal of Applied Mechanics 83(5), 1-5, 2016.
Hayashi, T., Analytical Study of Interlaminar Shear Stresses in a Laminated Composite Plate. Transaction of Japan Society for Aeronautical Engineering and Space Science 10, 43-48, 1967.
Hein, V.L., Erdogan, F., Stress singularities in a two-material wedge. International Journal of Fracture Mechanics 7(3), 317-330, 1971.
Heyliger, P., Exact solutions for simply supported laminated piezoelectric plates. ASME Journal of Applied Mechanics 64, 299-306, 1997.
Huang, B., Kim, H.S., Wang, J., Du, J., Free edge stress prediction for magneto-electro-elastic laminates using a stress function based equivalent single layer theory. Composites Science and Technology 123, 205-211, 2016.
Ju, S.H., Finite element calculation of stress intensity factors for interface notches. Computer Methods in Applied Mechanics and Engineering 199(33-36), 2273-2280, 2010.
Han, C., Wu, Z., Niu, Z., Accurate prediction of free-edge and electromechanical coupling effects in cross-ply piezoelectric laminates. Composite Structures 113, 308-315, 2014.
Mittelstedt, C., Becker, W., Interlaminar stress concentrations in layered structures: Part I – A Selective literature survey on the free-edge effect since 1967. Journal of Composite Materials 38, No12, 1037-1026, 2004.
Mittelstedt, C., Becker, W., Free-edge effects in composite laminates. Applied Mechanics Reviews 60, 217-245, 2007.

Kant, T., Swaminathan, K., Estimation of transverse/interlaminar stresses in laminated composites－a selective review and survey of current developments. Composite Structures 49. 65-75, 2000.
Kapuria, S., Dhanesh, N., Free-edge interlaminar stress analysis of composite laminates using interface modeling. International Journal of Mechanics Sciences 113, 148-161, 2016.
Kapuria, S., Kumari, P., Multiterm extended Kantorovich method for three-dimensional elasticity solution of laminated plates. Journal of Applied Mechanics 79, 1-9, 2012.
Kumari, P., Kapuria, S., Rajapakse, R.K.N.D., Three-dimensional extended Kantorovich solution for Levy-type rectangular laminated plates with edge effects. Composite Structures 107, 167-176, 2014.
Kumari, P., Behera, S., Kapuria, S., Coupled three-dimensional piezoelasticity solution for edge effects in Levy-type rectangular piezolaminated plates using mixed field extended Kantorovich method. Composite Structures 140, 491-505, 2016.
Kassapoglou. C. and Lagace, P.A., An efficient method for the calculation of interlaminar stresses in composite materials, Journal of Applied Mechanics 53, 744-750, 1986.
Kassapoglou, C. and Lagace, P.A., Closed Form Solutions for the Interlaminar Stress Field in Angle-Ply and Cross-Ply Laminates, Journal of Composite Materials 21, 292-308, 1987.
Kassapoglou, C., Determination of Interlaminar Stresses in Composite Laminates under Combined Loads, Journal of Reinforced Plastics and Composites 9, 33-58, 1990.
Kim, S.H., Lee, J., Cho, M., Free-edge interlaminar stress analysis of composite laminates using interface modeling. Journal of engineering mechanics 138(8), 974-983, 2012.
Kim, T., Atluri, S.N., Interlaminar stresses in composite laminates under out-of-plane shear/bending. AIAA Journal 32(8), 1700-1708, 1994.
Kim, T., Atluri, S.N., Analysis of edge stresses in composite laminates under combined thermo-mechanical loading, using a complementary energy approach. Computational Mechanics 16, 83-97, 1995.
Kim, T.W., Im, S., Boundary layer in wedges of laminated composite strips under generalized plane deformation-Part I: asymptotic solutions. International Journal of Solids and Structures 32(5), 609-628. 1995.
Kim, T.W., Im, S., Boundary layer in wedges of laminated composite strips under generalized plane deformation-Part II: complete numerical solutions. International Journal of Solids and Structures 32(5), 629-645. 1995.
Ko, C.C., Lin, C.C., Method for calculating interlaminar stresses in symmetric laminates containing a circular hole. AIAA Journal 30(1), 197-204, 1992.
Ko, C.C., Lin, C.C., Interlaminar stresses around a hole in symmetric cross-ply laminates under bending/torsion. AIAA Journal 31(6), 1118-1124, 1993.
Kerr, A.D., An extended Kantorovich method for the solution of eigenvalue problems. International Journal of Solids and Structures 5, 559-572. 1969.
Pan, E., Exact solution for simply supported and multilayered magneto-electro-elastic plates. Journal of Applied Mechanic 68, 608-618, 2001.
Pagano, N.J., On the Calculation of Interlaminar Normal Stress in Composite Laminate. Journal of Composite Materials 8. 65-81, 1974.
Pagano, N.J., Stress Fields in Composite Laminates. International Journal of Solids and Structures 14, 385-400, 1978a.
Pagano, N.J., Free Edge Stress Fields in Composite Laminates. International Journal of Solids and Structures 14, 401-406, 1978b.
Pagano, N.J., Exact solutions for composite laminates in cylindrical bending. Journal of Composite Materials 3, 398-411, 1969.
Pagano, N.J., Exact solutions for rectangular bidirectional composites and sandwich plates. Journal of Composite Materials 4, 20-34, 1970.
Pipes, R.B. and Pagano, N.J., Interlaminar stresses in composite laminates-an approximate elasticity solution. Journal of Applied Mechanics 41. 668-672, 1974.
Puppo, A.H. and Evensen, H.A., Interlaminar shear in laminated composites under Generalized Plane Stress. Journal of Composite Materials 4, 204-220, 1970.

Ren, J.G., Exact solutions for laminated cylindrical shells in cylindrical bending. Composites Science and Technology 29, 169-187, 1987.
Ting, T.C.T., Anisotropic Elasticity, Oxford University Press, New York, 1996.
Ting, T.C.T, Chou, S.C., Edge singularities in anisotropic composites. International Journal of Solids and Structures 17(11), 1057-1068, 1981.
Liang, W.Y., Ju, S.H., Tarn, J.Q., Analytic determination of stress fields in cross-ply symmetric composite laminates. International Journal of Solids and Structures 94-95, 87-99, 2016.
Liang, W.Y., Tseng, W.D., Tarn, J.Q., Exact analysis of stress fields in composite laminates under extension. Journal of Mechanics 30, 477-489, 2014.
López, C., Firmo, J.P., Correia, J.R., Tiago, C., Fire protection systems for reinforced concrete slabs strengthened with CFRP laminates. Construction and Building Materials 47, 324-333, 2013.
Meyer-Piening, H.R., Application of the elasticity solutions to linear sandwich beam, plate and shell analyses. Journal of Sandwich Structures and Materials 6, 295-312, 2004.
Morton, S.K. and Webber, J.P.H., Interlaminar failure due to mechanical and thermal stresses at the free edges of laminated plates. Composites Science and Technology 47, 1-13, 1993.
Noor, A.K., Burton, W.S., Three-dimensional solutions for antisymmetrically laminated anisotropic plates. Journal of Applied Mechanics 57, 182-188, 1990.
Rose, C.A. and Herakovich, C.T., An approximate solution for interlaminar stresses in composite laminates, Composites Engineering 3, 271–285, 1993.
Savoia, M., Reddy, J.N., A variational approach to three-dimensional elasticity solutions of laminated composite plates. Journal of Applied Mechanics 59, S166-S175, 1992.
Stroh, A.N., Steady state problems in anisotropic elasticity, Journal of Mathematical Physics 41, 77-103, 1962.

Stolarski, H.K., Chiang, M.Y.M., On the significance of the logarithmic term in the free edge stress singularity of composite laminates. International Journal of Solids and Structures 25(1), 75-93, 1989.
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vettenling, W.T., Numerical Recipes, the Art of Scientific Computing, Cambridge University Press, New York, 1986.
Tahani, M., Nosier, A., Free edge stress analysis of general cross-ply composite laminates under extension and thermal loading, Composite Structures 60, 91-103, 2003.
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., Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. International Journal of Solids and Structures 38, 8189-8206, 2001.
Tarn, J.Q., A state space formalism for anisotropic 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: Cylindrical 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 loadings and temperature change. Proceedings of the Royal Society of London A 458, 2349-2367, 2002d.
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., Tseng, W.D, Chang, H.H., A circular elastic cylinder under its own weight. International Journal of Solids and Structures 46, 2886-2896, 2009a.
Tarn, J.Q., Chang, H.H., Tseng, W.D., Axisymmetric deformation of a transversely isotropic cylindrical body: a Hamiltonian state space approach. Journal of Elasticity 97, 131-154, 2009b.
Tarn, J.Q., Chang, H.H., Tseng, W.D., A Hamiltonian state space approach for 3D analysis of circular cantilevers. Journal of Elasticity 101, 207-237, 2010.
Tarn, J.Q., Tseng, W.D., Exact analysis of curved beams and arches with arbitrary end conditions: a Hamiltonian state space approach. Journal of Elasticity 107, 39-63, 2012.
Tarn, J.Q., Chang, H.H., A Hamiltonian state space approach to anisotropic elasticity and piezoelasticity. Acta Mechanica 224, 1271-1284, 2013.
Tseng, W.D., Tarn, J.Q., A circular elastic cylinder under extension. Journal of Mechanics 27, 399-407, 2011.
Tseng, W.D., Tarn, J.Q., Chang, J. H., Laminated tubes under extension, internal and external pressure. Journal of Mechanics 30, 455-466, 2014a.
Tseng, W.D., Tarn, J.Q., Chang, C.C., A cantilever subjected to axial force, bending moment, and shear force. Journal of Mechanics 30, 549-559, 2014b.
Tseng, W.D., Tarn, J.Q., Three-dimensional solution for the stress field around a circular hole in a plate. Journal of Mechanics 30, 611-624, 2014.
Tseng, W.D., Tarn, J.Q., A state space solution approach for problems of cylindrical tubes and circular plates. Journal of Mechanics 31, 557-572, 2015a.
Tseng, W.D., Tarn, J.Q., Exact elasticity solution for axisymmetric deformation of circular plates. Journal of Mechanics 31, 617-629, 2015b.
Tseng, W.D., Tarn, J.Q., State space approach to some problems of elasticity in cylindrical coordinates. Golden Light Academic Publishing, Saarbrücken, 2015c.
Tseng, W.D., Tarn, J.Q., Tung, L.Y., Liang, W.Y., Free vibration of circular plates with various edge boundary conditions by 3-D elasticity theory. Journal of Robotics and Mechanical Engineering Research 1, 28-35, 2015.
Wang, Y., Xu, R.Q., State space formulation of magneto-electro-elasticity in Hamiltonian system and applications. Composite Structures 133, 607-620, 2015.
Wang, S.S., Choi, I., Boundary-layer effects in composite laminates: Part 1: Free-edge stress singularities. Journal of Applied Mechanics 49, 541-548, 1982a.

Wang, S.S., Choi, I., Boundary-layer effects in composite laminates: Part 2: Free-edge stress solutions and basic characteristics. Journal of Applied Mechanics 49, 549-560, 1982b.
Werner, H., A Three-dimensional solution for rectangular plate bending free of transversal normal stresses. Communications in Numerical Methods in Engineering 15, 295-302, 1999.
Wu, C.P., Kuo, H.C., Interlaminar stress analysis for laminated composite plates based on a local higher-order lamination theory. Composite Structures 20(4), 237-247, 1992.
Wu, Z., Han, C., Niu, Z., A 3D exact analysis of the boundary layer effect of asymmetric piezoelectric laminates with electromechanical coupling. International Journal of Solids and Structures 72, 118-129, 2015.
Yin, W.L., Free-edge effects in anisotropic laminates under extension, bending and twisting, Part I: A stress-function-based variational approach, Journal of Applied Mechanics 61, 410-415, 1994.
Ye, J.Q., Laminated Composite Plates and Shells: 3D Modelling. Springer, Berlin, 2003.
Ye, J.Q., Sheng, H.Y., Qin, Q.H., A state space finite element for laminated composites with free edges and subjected to transverse and in-plane loads. Computers and Structures 82, 1131-1141, 2004.
Zenkour, A.M., Three-dimensional Elasticity solution for uniformly loaded cross-ply laminates and sandwich plates. Journal of Sandwich Structures and Materials 9, 213-238, 2007.
Zhang, C., Yeh, H.Y., A variational approach for straight edge effect stresses in laminated composites, Composite Structures 43, 243-254, 1998.
Zhang, D., Ye, J., Sheng, H.Y., Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading. Composite Structures 76, 314-325, 2006.

Zhang, D., Ye, J., Lam, D., Free-edge and ply cracking effect in angle-ply laminated composites subjected to in-plane loads. Journal of Engineering Mechanics 133(12), 1268-1277, 2007.
Zhong W.X., A New Systematic Methodology for Theory of Elasticity. Dalian University of Technology Press, Dalian, 1995.
Zwiers, R.I., Ting, T.C.T., Spilker, R.L., On the logarithmic singularity of free-edge stress in laminated composites under uniform extension, Journal of Applied Mechanics 49(3). 561-569, 1982.

校內：2018-12-04公開
校外：2021-12-04公開

簡易檢索 / 詳目顯示

相關論文