Abstract
An accurate prediction of the three-dimensional (3D) and often singular stress field at the free edges of composite laminates has been the subject of intense research since the pioneering numerical work of Pipes and Pagano (J. Compos. Mater. 4:538–548, 1970). In this work, we present an accurate 3D elasticity-based solution for determining the free edge stress field in laminates under uniaxial extension, bending, twisting and thermal loading, using the mixed-field multiterm extended Kantorovich method (MMEKM), which is a powerful iterative analytical method, developed recently by the author group. Unlike the existing 3D elasticity-based solutions, the present solution satisfies all the boundary conditions and the interfacial continuity conditions exactly at all points. The results are compared with various existing solutions reported for symmetric cross-ply and angle-ply laminates. New results are presented for the free edge stress field in antisymmetric angle-ply laminates for the four different load cases. The results show a rapid convergence and excellent accuracy of the MMEKM solution in all cases. The solution successfully captures the stress singularity at the free edge by predicting a steadily increasing peak value with increasing number of terms. The presented results will serve as useful benchmarks for assessing the accuracy of other approximate 3D and 2D theory-based solutions.
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References
Andakhshideh A., Tahani M.: Free-edge stress analysis of general rectangular composite laminates under bending, torsion and thermal loads. Eur. J. Mech. A/Solids 42, 229–240 (2013)
Andakhshideh A., Tahani M.: Interlaminar stresses in general thick rectangular laminated plates under in-plane loads. Compos. Part B 47, 58–69 (2013)
Becker W.: Closed-form solution for the free-edge effect in cross-ply laminates. Compos. Struct. 26, 39–45 (1993)
Becker W.: Free-edge stress concentration in angle-ply laminates. Arch. Appl. Mech. 65, 38–43 (1995)
Cho M., Kim H.S.: Iterative free-edge stress analysis of composite laminates under extension, bending, twisting and thermal loadings. Int. J. Solids Struct. 37, 435–459 (2000)
Cho M., Yoon J.Y.: Free-edge interlaminar stress analysis of composite laminates by extended Kantorovich method. AIAA J. 37, 656–660 (1999)
Davi G., Milazzo A.: Bending stress fields in composite laminate beams by a boundary integral formulation. Comput. Struct. 71, 267–276 (1999)
Esquej R., Castejon L., Lizaranzu M., Carrera M., Miravete A., Miralbes R.: A new finite element approach applied to the free edge effect on composite materials. Compos. Struct. 98, 121–129 (2013)
Flanagan G.: An efficient stress function approximation for free-edge stresses in laminates. Int. J. Solids Struct. 31, 941–952 (1994)
Hayashi T.: Analytical study of interlaminar shear stresses in a laminated composite plate. Trans. Jpn. Soc. Aeronaut. Space Sci. 10, 43–48 (1967)
Herakovich C.T.: On thermal edge effects in composite laminates. Int. J. Mech. Sci. 18, 129–134 (1976)
Icardi U., Bertetto A.M.: An evaluation of the influence of geometry and of material properties at free edges and at the corners of composite laminates. Comput. Struct. 57, 555–571 (1995)
Jones R.M.: Mechanics of Composite Materials. Taylor & Francis, New York (1999)
Kant T., Swaminathan K.: Estimation of transverse/interlaminar stresses in laminated composites - a selective review and survey of current developments. Compos. Struct. 49, 65–75 (2000)
Kapuria, S., Dhanesh, N.: Three-Dimensional extended Kantorovich solution for accurate prediction of interlaminar stresses in composite laminated panels with interfacial imperfections. J. Eng. Mech. ASCE 141, 04014140 (pp. 1–11) (2015)
Kapuria S., Dumir P.C., Sengupta S.: Three dimensional solution for shape control of a simply supported rectangular hybrid plate. J. Thermal Stresses 22, 159–176 (1999)
Kapuria, S., Kumari, P.: Multiterm extended Kantorovich method for three-dimensional elasticity solution of laminated plates. J. Appl. Mech. 79, 061018 (pp. 1–9) (2012)
Kapuria, S., Kumari, P.: Extended Kantorovich method for coupled piezoelasticity solution of piezolaminated plates showing edge effects. Proc. R. Soc. A 469, 20120565 (pp. 1–19) (2013)
Kassapoglou C., Lagace P.A.: Closed form solution for the interlaminar stress field in angle-ply and cross-ply laminates. J. Compos. Mater. 21, 292–308 (1987)
Kerr A.D.: An extension of the Kantorovich method. Quart. Appl. Math. 26, 219–229 (1968)
Kim H.S., Cho M., Kim G.I.: Free-edge strength analysis in composite laminates by the extended Kantorovich method. Compos. Struct. 49, 229–235 (2000)
Kim H.S., Cho M., Lee J., Deheeger A., Grediac M., Mathias J.D.: Three dimensional stress analysis of a composite patch using stress functions. Int. J. Mech. Sci. 52, 1646–1659 (2010)
Kim H.S., Lee J., Cho M.: Free-edge interlaminar stress analysis of composite laminates using interface modeling. J. Eng. Mech. ASCE 138, 973–983 (2012)
Kim T., Atluri S.N.: Interlaminar stresses in composite laminates under out-of-plane shear/bending. AIAA J. 32, 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. Comput. Mech. 16, 83–97 (1995)
Krishna Murty A.V., Hari Kumar H.K.: Modelling of symmetric laminates under extension. Compos. Struct. 11, 15–32 (1989)
Lekhnitskii S.G.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963)
Lessard L.B., Schmidt A.S., Shokrieh M.M.: Three-dimensional stress analysis of free-edge effects in a simple composite cross-ply laminate. Int. J. Solids Struct. 33, 2243–2259 (1996)
Liang W.Y., Tseng W.D., Tarn J.Q.: Exact analysis of stress fields in composite laminates under extension. J. Mech. 30, 477–489 (2014)
Lindemann J., Becker W.: Analysis of the free-edge effect in composite laminates by the boundary finite element method. Mech. Compos. Mater. 36, 207–214 (2000)
Lovinger Z., Frostig Y.: High order behavior of sandwich plates with free edges-edge effects. Int. J. Solids Struct. 41, 979–1004 (2004)
Mittelstedt C., Becker W.: Interlaminar stress concentrations in layered structures: Part I-A selective literature survey on the free-edge effect since 1967. J. Compos. Mater. 38, 1037–1062 (2004)
Mittelstedt C., Becker W.: Free-edge effects in composite laminates. Appl. Mechan. Rev. 60, 217–244 (2007)
Nosier A., Bahrami A.: Free edge stresses in antisymmetric angle-ply laminates in extension and torsion. Int. J. Solids Struct. 43, 6800–6816 (2006)
Nosier A., Bahrami A.: Interlaminar stresses in antisymmetric angle-ply laminates. Compos. Struct. 78, 18–33 (2007)
Ottavio M.D., Vidal P., Valot E., Polit O.: Assessment of plate theories for free-edge effects. Compos. Part B 48, 111–121 (2013)
Pagano N.J.: Free edge stress fields in composite laminates. Int. J. Solids Struct. 14, 401–406 (1978)
Pagano N.J.: Stress fields in composite laminates. Int. J. Solids Struct. 14, 385–400 (1978)
Phillips E.A., Herakovich C.T., Graham L.L.: Damage development in composites with large stress gradients. Compos. Sci. Technol. 61, 2169–2182 (2001)
Pipes R.B., Pagano N.J.: Interlaminar stresses in composite laminates under uniform axial extension. J. Compos. Mater. 4, 538–548 (1970)
Puppo A.H., Evensen H.A.: Interlaminar shear in laminated composites under generalized plane stress. J. Compos. Mater. 4, 204–220 (1970)
Raju I.S., Crews J.H.: Interlaminar stress singularities at a straight free edge in composite laminates. Comput. Struct. 14, 21–28 (1981)
Salamon N.J.: Interlaminar stresses in a layered composite laminate in bending. Fiber Sci. Technol. 11, 305–317 (1978)
Sandhu, R.S., Wolfe, W.E., Sierakowski, R.L., Chang, C.C., Chu, H.R.: Finite element analysis of free-edge delamination in laminated composite specimens. U.S. Air Force Wright Laboratory report, WL–TR–91–3022 (1991)
Sarvestani H.Y., Naghashpour A.: Analysis of free edge stresses in composite laminates using higher order theories. Indian J. Mat. Sci. 253018, 1–15 (2014)
Shames I.H., Dym C.L.: Energy and Finite Element Methods in Structural Mechanics. Hemisphere Publishing, New York (1985)
Stolarski H.K., Chiang M.Y.M.: On the significance of the logarithmic term in the free edge stress singularity of composite laminates. Int. J. Solids Struct. 25, 75–93 (1989)
Tahani M., Nosier A.: Edge effects of uniformly loaded cross-ply composite laminates. Mater. Des. 24, 647–658 (2003)
Tahani M., Nosier A.: Free edge stress analysis of general cross-ply composite laminates under extension and thermal loading. Compos. Struct. 60, 91–103 (2003)
Tian Z., Zhao F., Yang Q.: Straight free-edge effects in laminated composites. Finite Elem Anal Des 41, 1–14 (2004)
Tong P., Pian T.H.H.: On the convergence of the finite element method for problems with singularity. Int. J. Solids Struct. 9, 313–321 (1973)
Vidal P., Gallimard L., Polit O.: Assessment of variable separation for finite element modeling of free edge effect for composite plates. Compos. Struct. 123, 19–29 (2015)
Wang A.S.D., Crossman F.W.: Edge effects on thermally induced stresses in composite laminates. J. Compos. Mater. 11, 300–312 (1977)
Wang A.S.D., Crossman F.W.: Some new results on edge effect in symmetric composite laminates. J. Compos. Mater. 11, 92–106 (1977)
Wang S.S., Yuan F.G.: A singular hybrid finite element analysis of boundary-layer stresses in composite laminates. Int. J. Solids Struct. 19, 825–837 (1983)
Yin W.L.: Free-edge effects in anisotropic laminates under extension, bending and twisting, Part I :A stress-function-based variational approach. Trans. ASME 61, 410–415 (1994)
Yin W.L.: Free-edge effects in anisotropic laminates under extension, bending and twisting, Part II: Eigenfunction analysis and the results for symmetric laminates. Trans. ASME 61, 416–421 (1994)
Yin W.L.: The effect of temperature gradient on the free-edge interlaminar stresses in multi-layered structures. J. Compos. Mater. 31, 2460–2477 (1997)
Zhang D., Ye J.Q., Sheng H.Y.: Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading. Compos. Struct. 76, 314–325 (2006)
Zhang D., Ye J.Q., Sheng H.Y.: Free-edge and ply cracking effect in angle-ply laminated composites subjected to in plane loads. J. Eng. Mechan. ASCE 133, 1268–1277 (2007)
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Dhanesh, N., 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 228, 2895–2919 (2017). https://doi.org/10.1007/s00707-015-1522-0
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DOI: https://doi.org/10.1007/s00707-015-1522-0