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Accurate Evaluation of Interlaminar Stresses in Composite Laminates via Mixed One-Dimensional Formulation

A. G. de Miguel

Polytechnic University of Turin, 10129 Turin, Italy

*Ph.D. Student, Mul Group, Department of Mechanical and Aerospace Engineering; .

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E. Carrera

Polytechnic University of Turin, 10129 Turin, Italy

†Professor of Aerospace Structures and Aeroelasticity, Mul Group, Department of Mechanical and Aerospace Engineering; .

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A. Pagani

Polytechnic University of Turin, 10129 Turin, Italy

‡Assistant Professor, Mul Group, Department of Mechanical and Aerospace Engineering; .

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and

E. Zappino

Polytechnic University of Turin, 10129 Turin, Italy

§Assistant Professor, Mul Group, Department of Mechanical and Aerospace Engineering; .

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Published Online:30 Sep 2018https://doi.org/10.2514/1.J057189

Abstract

This paper presents a novel mixed one-dimensional formulation based upon Reissner’s mixed variational theorem for the accurate stress analysis of general multilayered beam problems. Carrera’s unified formulation is recalled to generate a class of theory of structure that assumes both displacements and stresses over the cross section of the beam. On the other hand, the finite element method is employed to obtain the governing equations in weak form using standard Lagrangian interpolation. Independent assumptions are taken for each layer of the laminated beam through a layerwise distribution of the unknowns, and the interlaminar continuity of the transverse stresses is imposed a priori by ensuring piecewise continuous fields in the thickness direction. A set of locally defined, hierarchical Legendre polynomials is chosen for the expansion of the variables over the cross-section surface, enabling the user to refine the model in particular zones of interest and to control the accuracy of the stress analysis through the polynomial order of the theory of structure. The numerical assessment of the proposed mixed elements for multilayered problems is carried out by comparison against benchmark solutions of the literature, including plate formulations and elasticity solutions. The free-edge problem is also addressed to show the three-dimensional capabilities of the model.

References

[1] Reissner E., “On a Certain Mixed Variational Theorem and a Proposed Application,” International Journal for Numerical Methods in Engineering, Vol. 20, No. 7, 1984, pp. 1366–1368. doi:https://doi.org/10.1002/nme.1620200714 IJNMBH 0029-5981 Crossref Google Scholar
[2] Tsai S. W., Composites Design, 4th ed., Think Composites, Dayton, OH, 1988. Google Scholar
[3] Reddy J. N., “Mechanics of Laminated Composite Plates and Shells,” Theory and Analysis, 2nd ed., CRC Press, Boca Raton, FL, 2004. Crossref Google Scholar
[4] Carrera E., “C0 Reissner-Mindlin Multilayered Plate Elements Including Zig-Zag and Interlaminar Stress Continuity,” International Journal for Numerical Methods in Engineering, Vol. 39, No. 11, 1996, pp. 1797–1820. doi:https://doi.org/10.1002/(SICI)1097-0207(19960615)39:11<1797::AID-NME928>3.0.CO;2-W IJNMBH 0029-5981 Crossref Google Scholar
[5] Carrera E., “ $C_{Z}^{0}$ Requirements—Models for the Two Dimensional Analysis of Multilayered Structures,” Composite Structures, Vol. 37, No. 3, 1997, pp. 373–383. doi:https://doi.org/10.1016/S0263-8223(98)80005-6 COMSE2 0263-8223 Crossref Google Scholar
[6] Jones R. M., Mechanics of Composite Materials, 2nd ed., Taylor and Francis, London, 1998. Google Scholar
[7] Timoshenko S. P., “On the Transverse Vibrations of Bars of Uniform Cross Section,” Philosophical Magazine, Vol. 43, No. 4, 1922, pp. 125–131. doi:https://doi.org/10.1080/14786442208633855 Google Scholar
[8] Whitney J., “The Effect of Transverse Shear Deformation on the Bending of Laminated Plates,” Journal of Composite Materials, Vol. 3, No. 3, 1969, pp. 534–547. doi:https://doi.org/10.1177/002199836900300316 JCOMBI 0021-9983 Crossref Google Scholar
[9] Mindlin R. D., “Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” Journal of Applied Mechanics, Vol. 18, No. 1, 1951, pp. 31–38. JAMCAV 0021-8936 Crossref Google Scholar
[10] Lo K., Christensen R. and Wu E., “A High-Order Theory of Plate Deformation. Part 2: Laminated Plates,” Journal of Applied Mechanics, Vol. 44, No. 4, 1977, p. 669. doi:https://doi.org/10.1115/1.3424155 JAMCAV 0021-8936 Crossref Google Scholar
[11] Vlasov V., Thin-Walled Elastic Beams, National Technical Information Service, Alexandria, VA, 1984. Google Scholar
[12] Reddy J., “A Refined Nonlinear Theory of Plates with Transverse Shear Deformation,” International Journal of Solids and Structures, Vol. 20, No. 9, 1984, pp. 881–896. doi:https://doi.org/10.1016/0020-7683(84)90056-8 IJSOAD 0020-7683 Crossref Google Scholar
[13] Lekhnitskii S. G., “Strength Calculation of Composite Beams,” Vestnik Inzhen I Tekhnikov, Vol. 9, 1935, pp. 137–148. Google Scholar
[14] Ambartsumian S., “On a General Theory of Anisotropic Shells,” Journal of Applied Mathematics and Mechanics, Vol. 22, No. 2, 1958, pp. 305–319. doi:https://doi.org/10.1016/0021-8928(58)90108-4 JAMMAR 0021-8928 Crossref Google Scholar
[15] Murakami H., “Laminated Composite Plate Theory with Improved In-Plane Responses,” Journal of Applied Mechanics, Vol. 53, No. 3, 1986, pp. 661–666. doi:https://doi.org/10.1115/1.3171828 JAMCAV 0021-8936 Crossref Google Scholar
[16] Oñate E., Elijo A. and Oller S., “Simple and Accurate Two-Noded Beam Element for Composite Laminated Beams Using a Refined Zigzag Theory,” Computer Methods in Applied Mechanics and Engineering, Vols. 213–216, No. 3, 2012, pp. 362–382. doi:https://doi.org/10.1016/j.cma.2011.11.023 CMMECC 0045-7825 Crossref Google Scholar
[17] Tessler A., Di Sciuva M. and Gherlone M., “Refinement of Timoshenko Beam Theory for Composite and Sandwich Beams Using Zigzag Kinematics,” NASA TR 215086, 2007. Google Scholar
[18] Icardi U. and Sola F., “Assessment of Recent Zig-Zag Theories for Laminated and Sandwich Structures,” Composites Part B: Engineering, Vol. 97, No. 7, 2016, pp. 26–52. doi:https://doi.org/10.1016/j.compositesb.2016.04.058 Crossref Google Scholar
[19] Reddy J. N., “A Generalization of Two-Dimensional Theories of Laminated Composite Plates,” Communications in Applied Numerical Methods, Vol. 3, No. 3, 1987, pp. 173–180. doi:https://doi.org/10.1002/cnm.1630030303 Crossref Google Scholar
[20] Pagano N., “Exact Solutions for Composite Laminates in Cylindrical Bending,” Journal of Composite Materials, Vol. 3, No. 3, 1969, pp. 398–411. doi:https://doi.org/10.1177/002199836900300304 JCOMBI 0021-9983 Crossref Google Scholar
[21] Shimpi R. P. and Ghugal Y. M., “A New Layerwise Trigonometric Shear Deformation Theory for Two-Layered Cross-Ply Beams,” Composites Science and Technology, Vol. 61, No. 9, 2001, pp. 1271–1283. doi:https://doi.org/10.1016/S0266-3538(01)00024-0 CSTCEH 0266-3538 Crossref Google Scholar
[22] Surana K. and Nguyen S., “Two-Dimensional Curved Beam Element with Higher-Order Hierarchical Transverse Approximation for Laminated Composites,” Computers and Structures, Vol. 36, No. 3, 1990, pp. 499–511. doi:https://doi.org/10.1016/0045-7949(90)90284-9 CMSTCJ 0045-7949 Crossref Google Scholar
[23] Carrera E., “Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells,” Archives of Computational Methods in Engineering, Vol. 9, No. 2, 2002, pp. 87–140. doi:https://doi.org/10.1007/BF02736649 Crossref Google Scholar
[24] Engrand D., “Local Effects Calculations in Composite Plates by a Boundary Layer Method,” Local Effects in the Analysis of Structures, Studies in Applied Mechanics, edited by Ladevèze P., Vol. 12, Elsevier, Atlanta, GA, 1985, pp. 199–214. doi:https://doi.org/10.1016/B978-0-444-42520-1.50013-X Crossref Google Scholar
[25] Noor A. K. and Burton W. S., “Stress and Free Vibration Analyses of Multilayered Composite Plates,” Composite Structures, Vol. 11, No. 3, 1989, pp. 183–204. doi:https://doi.org/10.1016/0263-8223(89)90058-5 COMSE2 0263-8223 Crossref Google Scholar
[26] Malik M. and Bert C. W., “Differential Quadrature Analysis of Free Vibration of Symmetric Cross-Ply Laminates with Shear Deformation and Rotatory Inertia,” Shock and Vibration, Vol. 2, No. 4, 1995, pp. 321–338. doi:https://doi.org/10.1155/1995/703928 SHVIE8 Crossref Google Scholar
[27] Reissner E., “On a Mixed Variational Theorem and on Shear Deformable Plate Theory,” International Journal for Numerical Methods in Engineering, Vol. 23, No. 2, 1986, pp. 193–198. doi:https://doi.org/10.1002/nme.1620230203 IJNMBH 0029-5981 Crossref Google Scholar
[28] Reissner E., On a Certain Mixed Variational Theorem and on Laminated Elastic Shell Theory, Springer, Berlin, 1986, pp. 17–27. doi:https://doi.org/10.1007/978-3-642-83040-2_2 Google Scholar
[29] Toledano A. and Murakami H., “A High-Order Laminated Plate Theory with Improved in-Plane Responses,” International Journal of Solids and Structures, Vol. 23, No. 1, 1987, pp. 111–131. doi:https://doi.org/10.1016/0020-7683(87)90034-5 IJSOAD 0020-7683 Crossref Google Scholar
[30] Carrera E., “Evaluation of Layerwise Mixed Theories for Laminated Plates Analysis,” AIAA Journal, Vol. 36, No. 5, 1998, pp. 830–839. doi:https://doi.org/10.2514/2.444 AIAJAH 0001-1452 Link Google Scholar
[31] Carrera E., “Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 1: Governing Equations,” AIAA Journal, Vol. 37, No. 9, 1999, pp. 1107–1116. doi:https://doi.org/10.2514/2.821 AIAJAH 0001-1452 Link Google Scholar
[32] Carrera E., “An Assessment of Mixed and Classical Theories on Global and Local Response of Multilayered Orthotropic Plates,” Composite Structures, Vol. 50, No. 2, 2000, pp. 183–198. doi:https://doi.org/10.1016/S0263-8223(00)00099-4 COMSE2 0263-8223 Crossref Google Scholar
[33] Jing H.-S. and Liao M.-L., “Partial Hybrid Stress Element for the Analysis of Thick Laminated Composite Plates,” International Journal for Numerical Methods in Engineering, Vol. 28, No. 12, 1989, pp. 2813–2827. doi:https://doi.org/10.1002/nme.1620281207 IJNMBH 0029-5981 Crossref Google Scholar
[34] Rao K. M. and Meyer-Piening H.-R., “Analysis of Thick Laminated Anisotropic Composite Plates by the Finite Element Method,” Composite Structures, Vol. 15, No. 3, 1990, pp. 185–213. doi:https://doi.org/10.1016/0263-8223(90)90031-9 COMSE2 0263-8223 Crossref Google Scholar
[35] Carrera E. and Demasi L., “Two Benchmarks to Assess Two-Dimensional Theories of Sandwich, Composite Plates,” AIAA Journal, Vol. 41, No. 7, 2003, pp. 1356–1362. doi:https://doi.org/10.2514/2.2081 AIAJAH 0001-1452 Link Google Scholar
[36] Wu C.-P. and Li H.-Y., “An RMVT-Based Third-Order Shear Deformation Theory of Multilayered Functionally Graded Material Plates,” Composite Structures, Vol. 92, No. 10, 2010, pp. 2591–2605. doi:https://doi.org/10.1016/j.compstruct.2010.01.022 COMSE2 0263-8223 Crossref Google Scholar
[37] Tessler A., “Refined Zigzag Theory for Homogeneous, Laminated Composite, and Sandwich Beams Derived from Reissner’s Mixed Variational Principle,” Meccanica, Vol. 50, No. 10, 2015, pp. 2621–2648. doi:https://doi.org/10.1007/s11012-015-0222-0 MECCB9 0025-6455 Crossref Google Scholar
[38] Jing H.-S. and Liao M.-L., “Multiterm Extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Plates,” Journal of Applied Mechanics, Vol. 79, No. 6, 2012, Paper 061018. doi:https://doi.org/10.1115/1.4006495 JAMCAV 0021-8936 Google Scholar
[39] Dhanesh N., Kapuria S. and Achary G. G. S., “Accurate Prediction of Three-Dimensional Free Edge Stress Field in Composite Laminates Using Mixed-Field Multiterm Extended Kantorovich Method,” Acta Mechanica, Vol. 228, No. 8, 2017, pp. 2895–2919. doi:https://doi.org/10.1007/s00707-015-1522-0 AMHCAP 0001-5970 Crossref Google Scholar
[40] Carrera E., “Developments, Ideas, and Evaluations Based Upon Reissner’s Mixed Variational Theorem in the Modeling of Multilayered Plates and Shells,” Applied Mechanics Reviews, Vol. 54, No. 4, 2001, pp. 301–329. doi:https://doi.org/10.1115/1.1385512 AMREAD 0003-6900 Crossref Google Scholar
[41] Whitney J. M., “Stress Analysis of Thick Laminated Composite and Sandwich Plates,” Journal of Composite Materials, Vol. 6, No. 3, 1972, pp. 426–440. doi:https://doi.org/10.1177/002199837200600301 JCOMBI 0021-9983 Crossref Google Scholar
[42] Washizu K., Variational Methods in Elasticity and Plasticity, Pergamon, New York, 1988. Google Scholar
[43] Reissner E., “On the Theory of Bending of Elastic Plates,” Journal of Mathematics and Physics, Vol. 23, Nos. 1–4, 1944, pp. 184–191. doi:https://doi.org/10.1002/sapm1944231184 Crossref Google Scholar
[44] Murakami H., Reissner E. and Yamakawa J., “Anisotropic Beam Theories with Shear Deformation,” Journal of Applied Mechanics, Vol. 63, No. 3, 1996, pp. 660–668. doi:https://doi.org/10.1115/1.2823347 JAMCAV 0021-8936 Crossref Google Scholar
[45] Murakami H. and Yamakawa J., “Dynamic Response of Plane Anisotropic Beams with Shear Deformation,” Journal of Engineering Mechanics, Vol. 123, No. 12, 1997, pp. 1268–1275. doi:https://doi.org/10.1061/(ASCE)0733-9399(1997)123:12(1268) JENMDT 0733-9399 Crossref Google Scholar
[46] Carrera E., Giunta G. and Petrolo M., Beam Structures: Classical and Advanced Theories, Wiley, Chichester, U.K., 2011. Crossref Google Scholar
[47] Carrera E., Filippi M. and Zappino E., “Laminated Beam Analysis by Polynomial, Trigonometric, Exponential and Zig-Zag Theories,” European Journal of Mechanics-A/Solids, Vol. 41, Sept.–Oct. 2013, pp. 58–69. doi:https://doi.org/10.1016/j.euromechsol.2013.02.006 Crossref Google Scholar
[48] Szabó B. and Babuška I., Finite Element Analysis, Wiley, New York, 1991. Google Scholar
[49] Carrera E., de Miguel A. and Pagani A., “Hierarchical Theories of Structures Based on Legendre Polynomial Expansions with Finite Element Applications,” International Journal of Mechanical Sciences, Vol. 120, No. 1, 2017, pp. 286–300. doi:https://doi.org/10.1016/j.ijmecsci.2016.10.009 IMSCAW 0020-7403 Crossref Google Scholar
[50] Pagani A., de Miguel A., Petrolo M. and Carrera E., “Analysis of Laminated Beams via Unified Formulation and Legendre Polynomial Expansions,” Composite Structures, Vol. 156, Nov. 2016, pp. 78–92. doi:https://doi.org/10.1016/j.compstruct.2016.01.095 COMSE2 0263-8223 Crossref Google Scholar
[51] Pagano N., “Influence of Shear Coupling in Cylindrical Bending of Anisotropic Laminates,” Journal of Composite Materials, Vol. 4, No. 3, 1970, pp. 330–343. doi:https://doi.org/10.1177/002199837000400305 JCOMBI 0021-9983 Crossref Google Scholar
[52] Cho M. and Parmerter R., “Efficient Higher Order Composite Plate Theory for General Lamination Configurations,” AIAA Journal, Vol. 31, No. 7, 1993, pp. 1299–1306. doi:https://doi.org/10.2514/3.11767 AIAJAH 0001-1452 Link Google Scholar
[53] Carrera E. and Demasi L., “Classical and Advanced Multilayered Plate Elements Based Upon PVD and RMVT. Part 1: Derivation of Finite Element Matrices,” International Journal for Numerical Methods in Engineering, Vol. 55, No. 2, 2002, pp. 191–231. doi:https://doi.org/10.1002/nme.492 IJNMBH 0029-5981 Crossref Google Scholar
[54] Groh R. and Weaver P., “On Displacement-Based and Mixed-Variational Equivalent Single Layer Theories for Modelling Highly Heterogeneous Laminated Beams,” International Journal of Solids and Structures, Vol. 59, Supplement C, 2015, pp. 147–170. doi:https://doi.org/10.1016/j.ijsolstr.2015.01.020 IJSOAD 0020-7683 Crossref Google Scholar
[55] Pipes R. and Pagano N., “Interlaminar Stresses in Composite Laminates Under Uniform Axial Extension,” Journal of Composite Materials, Vol. 4, No. 4, 1970, pp. 538–548. doi:https://doi.org/10.1177/002199837000400409 JCOMBI 0021-9983 Crossref Google Scholar
[56] Pipes R. and Pagano N., “Interlaminar Stresses in Composite Laminates—An Approximate Elasticity Solution,” Journal of Applied Mechanics, Vol. 41, No. 3, 1974, pp. 668–672. doi:https://doi.org/10.1115/1.3423368 JAMCAV 0021-8936 Crossref Google Scholar
[57] Wang A. and Crossman F. W., “Some New Results on Edge Effect in Symmetric Composite Laminates,” Journal of Composite Materials, Vol. 11, No. 1, 1977, pp. 92–106. doi:https://doi.org/10.1177/002199837701100110 JCOMBI 0021-9983 Crossref Google Scholar

Volume 56, Number 11November 2018

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Acknowledgments

This research has been carried out within the project FULLCOMP (Fully Analysis, Design, Manufacturing, and Health Monitoring of Composite Structures) funded by the Marie Sklodowska-Curie actions grant agreement number 642121. The H2020 European Training Networks are gratefully acknowledged.

Received7 February 2018
Accepted8 July 2018
Published online30 September 2018

Accurate Evaluation of Interlaminar Stresses in Composite Laminates via Mixed One-Dimensional Formulation

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