Abstract
The layerwise theory of Reddy is used to study the low velocity impact response of laminated plates. Forced-vibration analysis is developed by the modal superposition technique. Six different models are introduced for representation of the impact pressure distribution. The first five models, in which the contact area is assumed to be known, result in a nonlinear integral equation similar to the one obtained by Timoshenko in 1913. The resulting nonlinear integral equation is discretised using a time-element scheme. Two different interpolation functions, namely: (i) Lagrangian and (ii) Hermite, are used to express the impact force. The Hermitian polynomial-based representation, obviously more sophisticated, is introduced to verify the Lagrangian-based representation. Due to its modular nature the present numerical technique is preferable to the existing numerical methods in the literature. The final loading model, in which the time dependence of the contact area is taken into account according to the Hertzian contact law, resulted in a relatively more complicated but more realistic, nonlinear integral equation. The analytical developments concerning this model are all new and are reported for the first time in this paper. Also a simple, but accurate, numerical technique is developed for solving our new nonlinear integral equation which results in the time-history of the impact force. Our numerical results are first tested with a series of existing example problems. Then a detailed study concerning all the response quantities, including the in-plane and interlaminar stresses, is carried out for symmetric and antisymmetric cross-ply laminates and important conclusions are reached concerning the usefulness and accuracy of the various plate theories.
Similar content being viewed by others
References
Abrate S 1991 Impact on laminated composite materials.Appl. Mech. Rev. 44: 155–190
Bogdanovich A E, Yarve E V 1989 Numerical analysis of laminated plates subjected to impact loading.Proc. of the American Society for Composites, 4th Technical Conference (Blacksburg, VA: Virginia Polytechnic Inst.) pp. 399–409
Bogdanovich A E, Yarve E V 1990 Calculation of damage zones in laminated composite plates subjected to low velocity impact.Proc. of the American Society for Composites, 5th Technical Conference, East Lansing, Michigan, pp 1–11
Byun C, Kapania R K 1991 Prediction of interlaminar stresses in laminated plates using global orthogonal polynomials.Enhanced analysis techniques for composite materials (eds) L Schwer, J N Reddy, A Mal, ASME Winter Annual Meeting, Atlanta, pp. 113–124
Byun C, Kapania R K 1992 Nonlinear impact response of thin imperfect laminated plates using a reduction method.Compos. Eng., Int. J. 2: 391–410
Cairns D S, Lagace P A 1989 Transient response of graphite epoxy and kevlar epoxy laminates subjected to impact.AIAA J. 27: 879–884
Chaudhuri R A, Seide P 1987 An approximation semi-analytical method for prediction of interlaminar shear stresses in arbitrary laminated thick plates.Comput. Struct. 25: 627–636
Chen J K, Sun C T 1985 Dynamic large deflection response of composite laminates subjected to impact.Compos. Struct. 4: 59–73
Conwag H D 1956 The pressure distribution between two elastic bodies in contact.Z. Angew. Math. Phys. 7: 460–465
Goldsmith W 1960Impact: The theory and physical behaviour of colliding solids (London: Edward Arnold)
Greszczuk L B 1975 Response of isotropic and composite materials in particle impact.Am. Soc. Test. Mater., Spec. Tech. Publ. 568: 183–211
Greszczuk L B 1982 Damage in composite materials due to low velocity impact. InImpact dynamics (eds) J A Zukas, T Nicholas, H F Swift, L B Greszczuk, D R Curran (New York: John Wiley)
Kapania R K, Raciti S 1989 Recent advances in analysis of laminated beams and plates. Part II: Vibrations and wave propagation.AIAA J. 27: 935–949
Kapania R K, Yang T Y 1986 Formulation of an imperfect quadrilateral doubly-curved shell element for post-buckling analysis.AIAA J. 24: 310–311
Kant T, Mallikarjuna 1991 Nonlinear dynamics of laminated plates with a higher-order theory andC 0 finite elements.Int. J. Nonlinear Mech. 26: 335–343
Nosier A, Kapania R K, Reddy J N 1992 Low velocity impact response of laminated plates. Center for Composite Materials and Structures, Report 92-20, VIP & SU, Blacksburg, VA
Obst A W, Kapania R K 1992 Nonlinear static and transient analysis of laminated beams.Compos. Eng., Int. J. 2: 375–390
Pagano N J 1969 Exact solutions for composite laminates in cylindrical bending.J. Compos. Mater. 3: 398–411
Petersen B R 1985Finite element analysis of composite plate impacted by a projectile. PhD dissertation, University of Florida
Pintado P, Vogler T J, Morton J 1991 Impact damage development in thick composite laminates.Compos. Eng. 1: 195–221
Qian Y, Swanson S R 1990 Experimental measurements of impact response in carbon/epoxy plates.AIAA J. 28: 1069–1074
Ramkumar R L, Chen P C 1983 Low-velocity impact response of laminated plates.AIAA J. 21: 1448–1452
Reddy J N 1980 A penalty plate bending element for the analysis of laminated anisotropic composite plates.Int. J. Numer. Methods Eng. 15: 1187–206
Reddy J N 1983 Geometrically nonlinear transient analysis of laminated composite plates.AIAA J. 21: 621–629
Reddy J N 1987 A generalization of two-dimensional theories of laminated composite plates.Commun. Appl. Numer. Methods 3: 173–180
Reddy J N 1990 A general third-order nonlinear theory of plates with moderate thickness.J. Nonlinear Mech. 25: 677–686
Sankar B V, Sun C T 1985a Efficient numerical algorithm for transverse impact problems.Comput. Struct. 20: 1009–1012
Sankar B V, Sun C T 1985b Low velocity impact response of laminated beams subjected to initial stresses.AIAA J. 23: 1962–1969
Sun C T 1977 An analytical method for evaluation of impact damage energy of laminated composites.Am. Soc. Test. Mater., Spec. Tech. Publ. 617: 427–440
Sun C T, Chattopadhyay S 1975 Dynamic response of anisotropic laminated plates under initial stress for impact of a mass.J. Appl. Mech., ASME 97: 693–698
Sun C T, Chen J K 1985 On the impact of initially stressed composite laminates.J. Compos. Mater. 19: 490–504
Sun C T, Liou W J 1989 Investigation of laminated composite plates under impact dynamic loading using a three-dimensional hybrid stress finite element method.Comput. Struct. 33: 879–884
Swanson S R, Smith N L, Qian Y 1991 Analytical and experimental strain response in impact of composite cylinders.Compos. Struct. 18: 75–108
Tan T M, Sun C T 1982 Wave propagation in graphite/epoxy laminates due to impact. NASA CR-168057
Timoshenko S P 1913 Zur Frage Nach der Wirkung Eines Stosses auf Einen Balken.Z. Math. Phys. 62: 198–209
Timoshenko S, Goodier J N 1970Theory of elasticity (New York: McGraw-Hill)
Thangjitham S, Librescu L, Cederbaum G 1987 Low-velocity impact response of orthotropic plates using a higher-order theory.Proc. of 28th AIAA/ASME/ASCE/-AHS/ASC Structures, Structural Dynamics and Materials Conference, pp. 448–457
Yang S H, Sun C T 1981 Indentation law for composite laminates. NASA CR-65460
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nosier, A., Kapania, R.K. & Reddy, J.N. Forced vibration and low-velocity impact of laminated composite plates. Sadhana 19, 509–541 (1994). https://doi.org/10.1007/BF02812166
Issue Date:
DOI: https://doi.org/10.1007/BF02812166