Like in the well-known free-edge effect situation, stress fields in the vicinity of free corners of layered plates exhibit a distinct three-dimensional and singular behavior and thus represent an important technical situation. Nevertheless, there are only few thorough investigations available concerning stress concentrations near free laminate corners. Since numerical analyses of stress concentration phenomena in composite laminates are computationally expensive, the present contribution is devoted to a simple closed-form higher-order theory approach for the calculation of displacements, strains, and stresses in the vicinity of a rectangular corner of a symmetric cross-ply laminate under thermal load. Appropriate representations for the displacement field in the manner of a single-layer theory with unknown inplane components and assumed trigonometric functions through the thickness yield closed-form expressions for the strains and stresses throughout the whole laminate. The inplane displacement functions are determined by equilibrium considerations in an integrated form with the solution of some resultant characteristic equations. Boundary conditions are fulfilled in an integral sense. The present approach can be applied easily, requires little computational effort and is in good conformity with comparative finite-element calculations and other closed-form analyses.
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References
1.
1. Jones, R.M. (1975). Mechanics of Composite Materials, Scripta Book Co., Washington .
2.
2. Tsai, S.W. and Hahn, H.T. (1980). Introduction to Composite Materials, Technomic Publishing Co., Inc., Lancaster et al .
3.
3. Pipes, R.B. and Pagano N.J. (1970). Interlaminar Stresses in Composite Laminates under Uniform Axial Extension , Journal of Composite Materials, 4: 538–548 .
4.
4. Bar-Yoseph, P. and Pian, T.H.H. (1981). Calculation of Interlaminar Stress Concentration in Composite Laminates , Journal of Composite Materials, 15: 225–239 .
5.
5. Becker, W. (1993). Closed-Form Solution for the Free-Edge Effect in Cross-Ply Laminates , Composite Structures, 26: 39–45 .
6.
6. Becker, W. (1994). Closed-Form Analysis of the Free Edge Effect in Angle-Ply Laminates , Journal of Applied Mechanics, 61: 209–211 .
7.
7. Becker, W. and Kress, G. (1994). Stiffness Reduction in Laminate Coupons due to the Free-Edge Effect , Composites Science and Technology, 52: 109–115 .
8.
8. Hayashi, T. (1967). Analytical Study of Interlaminar Shear Stresses in a Laminated Composite Plate , Trans. Japan Soc. Aero. Space Sci., 10(17): 43–48 .
9.
9. Hsu, P.W. and Herakovich, C.T. (1977). A Perturbation Solution for Interlaminar Stresses in Bidirectional Laminates , In: Composite Materials: Testing and Design (Fourth Conference), ASTM STP 617, American Society for Testing and Materials, pp. 296–316 .
10.
10. Hsu, P.W. and Herakovich, C.T. (1977). Edge Effects in Angle-Ply Composite Laminates , Journal of Composite Materials, 11: 422–428 .
11.
11. Kassapoglou, C. and Lagace, P.A. (1986). An Efficient Method for the Calculation of Interlaminar Stresses in Composite Materials , Journal of Applied Mechanics, 53: 744–750 .
12.
12. Kassapoglou, C. and Lagace, P.A. (1987). Closed Form Solutions for the Interlaminar Stress Field in Angle-Ply and Cross-Ply Laminates , Journal of Composite Materials, 21: 292–308 .
13.
13. Kassapoglou, C. (1989). The Effect of Combined Loading and Stacking Sequence on the Interlaminar Stress Field at Free Edges of Composite Laminates , 30th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics and Materials Conference, Mobile, Alabama, April 3rd–5th 1989 – A Collection of Technical Papers, Part 4, pp. 2089–2097 .
14.
14. Kim, D.M. and Hong, C.S. (1992). A Simple Sublaminate Approach to the Design of Thick Composite Laminates for Suppression of Free-Edge Delamination , Composite Science and Technology, 43: 147–158 .
15.
15. Kress, G. and Becker, W. (1995). Effective Tensile Stiffness of Finite-Width Quasi-Isotropic Laminates , Journal of Reinforced Plastics and Composites, 14: 1043–1053 .
16.
16. Krishna Murty, A.V. and Hari Kumar, H.K. (1989). Modelling of Symmetric Laminates under Extension , Composite Structures, 11: 15–32 .
17.
17. Morton, S.K. and Webber, J.P.H. (1993). Interlaminar Failure due to Mechanical and Thermal Stresses at the Free Edges of Laminated Plates , Composites Science and Technology, 47: 1–13 .
18.
18. Pagano, N.J. and Pipes, R.B. (1971). The Influence of Stacking Sequence on Laminate Strength , Journal of Composite Materials, 5: 50–57 .
19.
19. Pagano, N.J. and Pipes, R.B. (1973). Some Observations on the Interlaminar Strength of Composite Laminates , International Journal of Mechanical Science, 15: 679–688 .
20.
20. Pagano, N.J. (1974). On the Calculation of Interlaminar Normal Stress in Composite Laminate , Journal of Composite Materials, 8: 65–81 .
21.
21. Pagano, N.J. (1978). Stress Fields in Composite Laminates , International Journal of Solids and Structures, 14: 385–400 .
22.
22. Pagano, N.J. (1978). Free Edge Stress Fields in Composite Laminates , International Journal of Solids and Structures, 14: 401–406 .
23.
23. Pagano, N.J. and Soni, S.R. (1983). Global-Local Laminate Variational Model , International Journal of Solids and Structures, 3: 207–228 .
24.
24. Pipes, R.B. and Pagano, N.J. (1974). Interlaminar Stresses in Composite Laminates – An Approximate Elasticity Solution , Journal of Applied Mechanics, 41: 668–672 .
25.
25. Puppo, A.H. and Evensen, H.A. (1970). Interlaminar Shear in Laminated Composites under Generalized Plane Stress , Journal of Composite Materials, 4: 204–220 .
26.
26. Rose, C.A. and Herakovich, C.T. (1993). An Approximate Solution for Interlaminar Stresses in Composite Laminates , Composites Engineering, 3(3): 271–285 .
27.
27. Soni, S.R. (1982). Stress and Strength Analysis of Composite Laminates at Delamination, Progress in Science and Engineering of Composites , In: Proceedings of the International Conference on Composite Materials, Tokyo, pp. 251–260 .
28.
28. Soni, S.R. and Pagano, N.J. (1983). Elastic Response of Composite Laminates, In: Hashin, Z. and Herakovich, C.T. (eds), Mechanics of Composite Materials – Recent Advances, pp. 227–242, Pergamon Press, New York .
29.
29. Tang, S. (1975). A Boundary Layer Theory – Part I: Laminated Composites in Plane Stress , Journal of Composite Materials, 9: 33–41 .
30.
30. Tang, S. and Levy, A. (1975). A Boundary Layer Theory – Part II: Extension of Laminated Finite Strip , Journal of Composite Materials, 9: 42–52 .
31.
31. Wang, J.T.S. and Dickson, J.N. (1978). Interlaminar Stresses in Symmetric Composite Laminates , Journal of Composite Materials, 12: 390–402 .
32.
32. Webber, J.P.H. and Morton, S.K. (1993). An Analytical Solution for the Thermal Stresses at the Free Edges of Laminated Plates , Composites Science and Technology, 46: 175–185 .
33.
33. Yin, W.-L. (1994). Simple Solutions of the Free-Edge Stresses in Composite Laminates under Thermal and Mechanical Loads , Journal of Composite Materials, 28(6): 573–586 .
34.
34. Zhang, C. and Yeh, H.-Y. (1998). A Variational Approach for Straight Edge Effect Stresses in Laminated Composites , Composite Structures, 43: 243–254 .
35.
35. Altus, E., Rotem, A. and Shmueli, M. (1980). Free Edge Effect in Angle Ply Laminates – A New Three Dimensional Finite Difference Solution , Journal of Composite Materials, 14: 21–30 .
36.
36. Bauld, N.R., Goree, J.G. and Tzeng, L.-S. (1985). A Comparison of Finite-Difference and Finite-Element Methods for Calculating Free Edge Stresses in Composites , Computers & Structures, 20(5): 897–914 .
37.
37. Hong, C.S. and Kim, K.S. (1982). An Analysis of Free-Edge Delamination in Laminated Composite under Uniform Axial Strain , In: Progress in Science and Engineering of Composites, Proceedings of the International Conference on Composite Materials, Tokyo, pp. 261–268 .
38.
38. Pipes, R.B. (1980). Boundary Layer Effects in Composite Laminates , Fibre Science and Technology, 13: 49–71 .
39.
39. Whitcomb, J.D., Raju, I.S. and Goree, J.G. (1982). Reliability of the Finite Element Method for Calculating Free Edge Stresses in Composite Laminates , Computers & Structures, 15(1): 23–37 .
40.
40. Frick, A. and Klamser, H. (1990). Untersuchung der Verschiebungs- und Spannungsverteilungen an unbelasteten Randern multidirektionaler Laminate (in german), published in: Fortschritt-Berichte VDI, Issue 1, No. 183, VDI Publishing House Dusseldorf, Germany .
41.
41. Isakson, G. and Levy, A. (1971). Finite-Element Analysis of Interlaminar Shear in Fibrous Composites , Journal of Composite Materials, 5: 273–276 .
42.
42. Joo, J.W. and Sun, C.T. (1992). A Failure Criterion for Laminates Governed by Free Edge Interlaminar Shear Stress , Journal of Composite Materials, 26(10): 1510–1522 .
43.
43. Raju, I.S. and Crews, J.H. (1981). Interlaminar Stress Singularities at a Straight Free Edge in Composite Laminates , Computers & Structures, 14(1–2): 21–28 .
44.
44. Rohwer, K. (1982). On the Determination of Edge Stresses in Layered Composites , Nuclear Engineering and Design, 70: 57–65 .
45.
45. Rohwer, K. (1982). Stresses and Deformations in Laminated Test Specimens of Carbon Fiber Reinforced Composites, Scientific Report No. DFVLR-FB 82-15, DFVLR, Institut fur Strukturmechanik, Braunschweig, Germany.
46.
46. Rybicki, E.F. (1971). Approximate Three-Dimensional Solutions for Symmetric Laminates under Inplane Loading , Journal of Composite Materials, 5: 354–360 .
47.
47. Spilker, R.L. (1980). A Traction-Free-Edge Hybrid-Stress Element for the Analysis of Edge Effects in Cross-Ply Laminates , Computers & Structures, 12: 167–179 .
48.
48. Spilker, R.L. and Chou, S.C. (1980). Edge Effects in Symmetric Composite Laminates: Importance of Satisfying the Traction Free Edge Condition , Journal of Composite Materials, 14: 2–20 .
49.
49. Wang, A.S.D. and Crossman, F.W. (1977). Some New Results on Edge Effect in Symmetric Composite Laminates , Journal of Composite Materials, 11: 92–106 .
50.
50. Wang, A.S.D. and Crossman, F.W. (1977). Edge Effects on Thermally Induced Stresses in Composite Laminates , Journal of Composite Materials, 11: 300–312 .
51.
51. Wang, S.S. and Yuan, F.G. (1983). A Singular Hybrid Finite Element Analysis of Boundary-Layer Stresses in Composite Laminates , International Journal of Solids and Structures, 19(9): 825–837 .
52.
52. Wang, S.S. and Yuan, F.G. (1983). A Hybrid Finite Element Approach to Composite Laminate Elasticity Problems with Singularities , Journal of Applied Mechanics, 50: 835–844 .
53.
53. Wu, C.M.L. (1990). Nonlinear Thermal and Mechanical Analysis of Edge Effects in Angle-Ply Laminates , Computers & Structures, 35(6): 705–717 .
54.
54. Yang, H.T.Y. and He, C.C. (1994). Three-Dimensional Finite Element Analysis of Free Edge Stresses and Delamination of Composite Laminates , Journal of Composite Materials, 28(15): 1394–1412 .
55.
55. Yeh, J.R. and Tadjbakhsh, I.G. (1986). Stress Singularity in Composite Laminates by Finite Element Method , Journal of Composite Materials, 20: 347–364 .
56.
56. Ory, H., Reimerdes H.-G. and Dieker S. (1984). Berechnung von interlaminaren Spannungen in mehrschichtigen Faserverbundwerkstoffen mit Hilfe von Ubertragungsmatrizen (in german) , Zeitschrift fur Flugwissenschaften und Weltraumforschung8(6): 392–404 .
57.
57. Lindemann, J. and Becker, W. (2000). Analysis of the Free-Edge Effect in Composite Laminates by the Boundary Finite Element Method , Mechanics of Composite Materials, 36(3): 355–366 .
58.
58. Anquez, L. (1986). A Numerical Analysis of Singular Stress Fields at the Free Edge of Layered Composites, In: Brandt, A.M. and Marshall, I.H. (eds), Brittle Matrix Composite, Vol. 1, pp. 287–303, Elsevier Applied Science, London, New York .
59.
59. Bogy, D.B. (1972). The Plane Solution for Anisotropic Elastic Wedges under Normal and Shear Loading , Journal of Applied Mechanics, pp. 1103–1109 .
60.
60. Delale, F. (1984). Stress Singularities in Bonded Anisotropic Materials , International Journal of Solids and Structures, 20(1): 31–40 .
61.
61. Dempsey, J.P. and Sinclair, G.B. (1981). On the Singular Behavior at the Vertex of a Bi-Material Wedge , Journal of Elasticity, 11(3): 317–327 .
62.
62. Hein, V.L. and Erdogan, F. (1971). Stress Singularities in a Two-Material Wedge , International Journal of Fracture Mechanics, 7: 317–330 .
63.
63. Kuo, M.C. and Bogy, D.B. (1974). Plane Solutions for the Displacement and Traction-Displacement Problems for Anisotropic Elastic Wedges , Journal of Applied Mechanics, pp. 197–202 .
64.
64. Wang, S.S. and Choi, I. (1982). Boundary-Layer Effects in Composite Laminates, Part 1: Free-Edge Stress Singularities , Journal of Applied Mechanics, 49: 541–548 .
65.
65. Wang, S.S. and Choi, I. (1982). Boundary-Layer Effects in Composite Laminates, Part 2: Free-Edge Stress Solutions and Basic Characteristics , Journal of Applied Mechanics, 49: 549–560 .
66.
66. Wang, S.S. (1983). Elasticity Solutions for a Class of Composite Laminate Problems with Stress Singularities, In: Hashin, Z. and Herakovich, C.T. (eds), Mechanics of Composite Materials – Recent Advances, pp. 259–281, Pergamon Press, New York .
67.
67. Zwiers, R.I., Ting, T.C.T. and Spilker, R.L. (1982). On the Logarithmic Singularity of Free-Edge Stress in Laminated Composites under Uniform Extension , Journal of Applied Mechanics, 49: 561–569 .
68.
68. Herakovich, C.T., Post, D., Buczek, M.B. and Czarnek, R. (1985). Free Edge Strain Concentrations in Real Composite Laminates: Experimental-Theoretical Correlation , Journal of Applied Mechanics, 52: 787–793 .
69.
69. Pipes, R.B. and Daniel, I.M. (1971). Moire Analysis of the Interlaminar Shear Edge Effect in Laminated Composites , Journal of Composite Materials, 5: 255–259 .
70.
70. Pipes, R.B., Kaminski, B.E. and Pagano, N.J. (1973). Influence of the Free Edge upon the Strength of Angle-Ply Laminates , Analysis of the Test Methods for High Modulus Fibers and Composites, ASTM STP, 521: 218–228 .
71.
71. Whitney, J.M. and Browning, C.E. (1972). Free-Edge Delamination of Tensile Coupons , Journal of Composite Materials, 6: 300–303 .
72.
72. Whitney, J.M. (1973). Free-Edge Effects in the Characterization of Composite Materials, ASTM STP 521, American Society for Testing and Materials, pp. 167–180.
73.
73. Wortler, M. (1989). Interlaminare Spannungskonzentrationen in Faserverbundwerkstoffen (in german), Dissertation Thesis, published in: Fortschritt-Berichte VDI, Issue 1, No. 169, VDI Publishing House, Dusseldorf, Germany.
74.
74. Herakovich, C.T. (1989). Free Edge Effects in Laminated Composites, In: Herakovich, C.T. and Tarnopol’skii, Y.M. (eds), Handbook of Composites, – Structure and Design, Vol. 2, pp. 187–230, Elsevier Science Publishers, B.V. Amsterdam .
75.
75. Kant, T. and Swaminathan, K. (2000). Estimation of Transverse/Interlaminar Stresses in Laminated Composites – a Selective Review and Survey of Current Developments , Composite Structures, 49: 65–75 .
76.
76. Rohwer, K. (1996). Modelle und Methoden zur Berechnung von Laminaten aus unidirektionalen Faserverbunden (in german), Habilitation Thesis, published in: Fortschritt-Berichte VDI, Issue 1, No. 264, VDI Publishing House, Dusseldorf, Germany.
77.
77. Stiftinger, M.A. (1996). Semi-Analytical Finite Element Formulations for Layered Composite Shells with Consideration of Edge Effects, Dissertation Thesis, University of Vienna, Austria, published in: Fortschritt-Berichte VDI, Issue 18, No. 203, VDI Publishing House, Dusseldorf, Germany.
78.
78. Becker, W., Jin, P.P. and Neuser, P. (1999). Interlaminar Stresses at the Free Corners of a Laminate , Composite Structures, 45: 155–162 .
79.
79. Mittelstedt, C. and Becker, W. (2003). Three-Dimensional Closed-Form Analysis of the Stress Field at Rectangular Corners of Layered Plates, accepted for publication in Archive of Applied Mechanics, in press.
80.
80. Mittelstedt, C. and Becker, W. (2003). Closed-Form Analysis of the Free-Corner Effect in Composite Laminates , In: Adali, S., Morozov, E.V. and Verijenko, V.E. (eds), Proceedings of the Fourth International Conference on Composite Science and Technology (ICCST/4), 21–23 January 2003, pp. 199–204 , Durban, South Africa.
