### Multiscale Analysis of Transverse Cracking in Cross-Ply Laminated Beams Using the Layerwise Theory

Document Type : Research Paper

Authors

Department of Mechanical Engineering, Texas A&M University, College Station

Abstract

A finite element model based on the layerwise theory is developed for the analysis of transverse cracking in cross-ply laminated beams. The numerical model is developed using the layerwise theory of Reddy, and the von Kármán type nonlinear strain field is adopted to accommodate the moderately large rotations of the beam. The finite element beam model is verified by comparing the present numerical solutions with the elasticity solutions available in the literature; an excellent agreement is found. The layerwise beam model is then used to investigate the influence of transverse cracks on material properties and the response in cross-ply laminates using a multiscale approach. The multiscale analysis consists of numerical simulations at two different length scales. In the first scale, a mesoscale, a systematic procedure to quantify the stiffness reduction in the cracked ply is proposed exploiting the laminate theory. In the second scale, a macroscale, continuum damage mechanics approach is used to compute homogenized material properties for a unit cell, and the effective material properties of the cracked ply are extracted by the laminate theory. In the macroscale analysis, a beam structure under a bending load is simulated using the homogenized material properties in the layerwise finite element beam model. The stress redistribution in the beam according to the multiplication of transverse cracks is taken into account and a prediction of sequential matrix cracking is presented.

Keywords

[1] Krajcinovic D., 1979, Distributed damage theory of beams in pure bending, Journal of Applied Mechanics 46(3): 592-596.
[2] Echaani J., Trochu F., Pham X.T., Ouellet M., 1996, Theoretical and experimental investigation of failure and damage progression of graphite-epoxy composites in flexural bending test, Journal of Reinforced Plastics and Composites 15(7): 740-755.
[3] Murri G.B., Guynn, E.G., 1988, Analysis of delamination growth from matrix cracks in laminates subjected to bending loads, Composite Materials: Testing and Design, ASTM Special Technical Publication 972: 322-339.
[4] Ogi K., Smith P.A., 2002, Characterisation of Transverse Cracking in a Quasi-Isotropic GFRP Laminate under Flexural Loading, Applied Composite Materials 9(2): 63-79.
[5] Boniface L., Ogin S.L., Smith P.A., 1991, Strain energy release rates and the fatigue growth of matrix cracks in model arrays in composite laminates, in: Proceedings Mathematical and Physical Sciences 432(1886): 427-444.
[6] Kuriakose S., Talreja R., 2004, Variational solutions to stresses in cracked cross-ply laminates under bending, International Journal of Solids and Structures 41: 2331-2347.
[7] Reddy J.N., 1987, A generalization of two-dimensional theories of laminated composite plates, Communications in Applied Numerical Methods 3(3): 173-180.
[8] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells - Theory and Analysis, CRC Press, Boca Raton, FL.
[9] Robbins D.H., Reddy J.N., 1991, Analysis of piezoelectrically actuated beams using a layer-wise displacement theory, Computers and Structures 41(2): 265-279.
[10] Rosca V.E., Poterasu V.F., Taranu N., Rosca B.G., 2002, Finite-element model for laminated beam-plates composite using layerwise displacement theory, Engineering Transactions 50(3): 165-176.
[11] Reddy J.N., 2002, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, Inc., Hoboken, New Jersey.
[12] Reddy J.N., 2004, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, New York.
[13] Pagano N.J., 1969, Exact solutions for composite laminates in cylindrical bending, Journal of Composite Materials 3: 398-411.
[14] Krajcinovic D.K., 1984, Continuum damage mechanics, Applied Mathematics Reviews 37(1): 1-6.
[15] Lemaitre J., 1984, How to use damage mechanics, Nuclear Engineering and Design 80(2): 233-245.
[16] Kachanov L.M., 1958, Rupture time under creep conditions, Izvestia Academii Nauk SSSR, Otdelenie tekhnicheskich, nauk, 8: 26-31.
[17] Talreja R., 1985, A continuum mechanics characterization of damage in composite materials, in: Proceedings of Royal Society of London, Series A 399(1817): 195-216.
[18] Talreja R., 1985, Transverse cracking and stiffness reduction in composite laminates, Journal of Composite Materials 19(4): 353-375.
[19] Thionnet A., Renard J., 1993, Meso-Macro approach to transverse cracking in laminated composites using Talreja’s model, Composites Engineering 3(9): 851-871.
[20] Li S., Reid R., Soden P.D., 1998, A continuum damage model for transverse matrix cracking in laminated fibre-reinforced composites, in: Philosophical Transactions of the Royal Society London, Series A 356(1746): 2379-2412.
[21] Talreja R., 1990, Internal variable damage mechanics of composite materials, Yielding Damage and Failure of Anisotropic Solids, Mechanical Engineering Publications, London, 509-533.
[22] Reifsnider K.L., Masters J.E., 1978, Investigation of characteristic damage states in composite laminates, ASME Paper, 78WA/Aero-4: 1-10.