Analytical Solution of the Boundary Value Problem of Delaminated Doubly-Curved Composite Shells
Delamination is one of the major failure modes in laminated composite structures. Delamination tips are mostly captured by spatial numerical models in order to predict crack growth. This paper presents some mechanical models of delaminated composite shells based on shallow shell theories. The mechanical fields are based on a third-order displacement field in terms of the through-thickness coordinate of the laminated shell. The undelaminated and delaminated parts are captured by separate models and the continuity and boundary conditions are also formulated in a general way providing a large size boundary value problem. The system of differential equations is solved by the state space method for an elliptic delaminated shell having simply supported edges. The comparison of the proposed and a numerical model indicates that the primary indicator of the model is the deflection, the secondary is the widthwise distribution of the energy release rate. The model is promising and suitable to determine accurately the J-integral distribution along the delamination front. Based on the proposed model it is also possible to develop finite elements which are able to replace the computationally expensive spatial models of delaminated structures.
[1] L. N. Phillips, Ed., Design with Advanced Composite Materials. Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer-Verlag, The Design Council, 1989.
[2] J. N. Reddy, Mechanics of laminated composite plates and shells - Theory and analysis. Boca Raton, London, New York, Washington D.C.: CRC Press, 2004.
[3] D. F. Adams, L. A. Carlsson, and R. B. Pipes, Experimental characterization of advanced composite materials, Third ed. Boca Raton, London, New York, Washington, D.C.: CRC Press, 2000.
[4] T. L. Anderson, Fracture Mechanics - Fundamentals and Applications, Third ed. Boca Raton, London, New York, Singapore: CRC Press, Taylor & Francis Group, 2005.
[5] J. Jumel, M. K. Budzik, and M. E. R. Shanahan, “Beam on elastic foundation with anticlastic curvature: Application to analysis of mode I fracture tests,” Engineering Fracture Mechanics, vol. 78, no. 18, pp. 3253–3269, 2011.
[6] J. Jumel, M. K. Budzik, and M. E. R. Shanahan, “Process zone in the single cantilever beam under transverse loading. part I: Theoretical analysis,” Theoretical and Applied Fracture Mechanics, vol. 56, no. 1, pp. 7–12, 2011.
[7] A. Szekrényes, “Improved analysis of unidirectional composite delamination specimens,” Mechanics of Materials, vol. 39, pp. 953–974, 2007.
[8] M. G. Andrews and R. Massabò, “The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers,” Engineering Fracture Mechanics, vol. 74, pp. 2700-2720, 2007.
[9] M. Pelassa and R. Massabò, “Explicit solutions for multi-layered wide plates and beams with perfect and imperfect bonding and delaminations under thermo-mechanical loading,” Meccanica, vol. 50, no. 10, pp. 2497–2524, 2015.
[10] B. D. Davidson, L. Yu, and H. Hu, “Determination of energy release rate and mode mix in three-dimensional layered structures using plate theory,” International Journal of Fracture, vol. 105, pp. 81–104, 2000.
[11] B. V. Sankar and V. Sonik, “Pointwise energy release rate in delaminated plates,” AIAA Journal, vol. 33, no. 7, pp. 1312–1318, 1995.
[12] C. K. Hirwani, S. K. Panda, and T. R. Mahapatra, “Nonlinear finite element analysis of transient behavior of delaminated composite plate,” Journal of Vibration and Acoustics, vol. 140, no. 2, p. 021001, 2018.
[13] A. Szekrényes, “Bending solution of third-order orthotropic Reddy plates with asymmetric interfacial crack,” International Journal of Solids and Structures, vol. 51, pp. 2598–2619, 2014.
[14] N. Nanda and S. Sahu, “Free vibration analysis of delaminated composite shells using different shell theories,” International Journal of Pressure Vessels and Piping, vol. 98, pp. 111 – 118, 2012.
[15] S. K. Panda and B. N. Singh, “Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel using nonlinear FEM,” Finite Elements in Analysis and Design, vol. 47, pp. 378–386, 2011.
[16] V. K. Singh and S. K. Panda, “Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels,” Thin-Walled Structures, vol. 85, pp. 431–349, 2014.
[17] L. P. Kollár and G. S. Springer, Mechanics of Composite Structures. Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paolo: Cambridge University Press, 2003.
[18] A. Szekrényes, “Nonsingular crack modelling in orthotropic plates by four equivalent single layers,” European Journal of Mechanics A/Solids, vol. 55, pp. 73–99, 2016.
[19] A. Szekrényes, “Semi-layerwise analysis of laminated plates with nonsingular delamination - the theorem of autocontinuity,” Applied Mathematical Modelling, vol. 40, pp. 1344–1371, 2016.
[20] N. Nanda, “Static analysis of delaminated composite shell panels using layerwise theory,” Acta Mechanica, vol. 225, no. 10, pp. 2893–2901, 2014.
[21] A. Szekrényes, “The role of transverse stretching in the delamination fracture of softcore sandwich plates,” Applied Mathematical Modelling, vol. 63, pp. 611–632, 2018.
[22] K. N. Shivakumar and I. S. Raju, “An equivalent domain integral method for three-dimensional mixed-mode fracture problems,” Engineering Fracture Mechanics, vol. 42, no. 6, pp. 935–959, 1992.
[23] R. H. Rigby and M. H. Aliabadi, “Decomposition of the mixed-mode J-integral - revisited,” International Journal of Solids and Structures, vol. 35, no. 17, pp. 2073–2099, 1998.
[24] J. Petrolito, “Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements,” Applied Mathematical Modelling, vol. 38, no. 24, pp. 5858–5869, 2014.
[25] M. Izadi and M. Tahani, “Analysis of interlaminar stresses in general cross-ply laminates with distributed piezoelectric actuators,” Composite Structures, no. 92, pp. 757–768, 2010.
[26] A. Szekrényes, “The system of exact kinematic conditions and application to delaminated first-order shear deformable composite plates,” International Journal of Mechanical Sciences, vol. 77, pp. 17–29, 2013.
[27] A. Szekrényes, “Antiplane-inplane shear mode delamination between two second-order shear deformable composite plates,” Mathematics and Mechanics of Solids, pp. 1–24, 2015.
[28] A. Szekrényes, Nonsingular delamination modeling in orthotropic com- posite plates by semi-layerwise analysis. Hungarian Academy of Sciences, D.Sc. dissertation, December 2017.
[29] Y. Jianqiao, Laminated Composite Plates and Shells - 3D modelling. London, Berlin, Heidelberg, New York, Hong Kong, Milan, Paris, Tokyo: Springer, 2003.
[1] L. N. Phillips, Ed., Design with Advanced Composite Materials. Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer-Verlag, The Design Council, 1989.
[2] J. N. Reddy, Mechanics of laminated composite plates and shells - Theory and analysis. Boca Raton, London, New York, Washington D.C.: CRC Press, 2004.
[3] D. F. Adams, L. A. Carlsson, and R. B. Pipes, Experimental characterization of advanced composite materials, Third ed. Boca Raton, London, New York, Washington, D.C.: CRC Press, 2000.
[4] T. L. Anderson, Fracture Mechanics - Fundamentals and Applications, Third ed. Boca Raton, London, New York, Singapore: CRC Press, Taylor & Francis Group, 2005.
[5] J. Jumel, M. K. Budzik, and M. E. R. Shanahan, “Beam on elastic foundation with anticlastic curvature: Application to analysis of mode I fracture tests,” Engineering Fracture Mechanics, vol. 78, no. 18, pp. 3253–3269, 2011.
[6] J. Jumel, M. K. Budzik, and M. E. R. Shanahan, “Process zone in the single cantilever beam under transverse loading. part I: Theoretical analysis,” Theoretical and Applied Fracture Mechanics, vol. 56, no. 1, pp. 7–12, 2011.
[7] A. Szekrényes, “Improved analysis of unidirectional composite delamination specimens,” Mechanics of Materials, vol. 39, pp. 953–974, 2007.
[8] M. G. Andrews and R. Massabò, “The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers,” Engineering Fracture Mechanics, vol. 74, pp. 2700-2720, 2007.
[9] M. Pelassa and R. Massabò, “Explicit solutions for multi-layered wide plates and beams with perfect and imperfect bonding and delaminations under thermo-mechanical loading,” Meccanica, vol. 50, no. 10, pp. 2497–2524, 2015.
[10] B. D. Davidson, L. Yu, and H. Hu, “Determination of energy release rate and mode mix in three-dimensional layered structures using plate theory,” International Journal of Fracture, vol. 105, pp. 81–104, 2000.
[11] B. V. Sankar and V. Sonik, “Pointwise energy release rate in delaminated plates,” AIAA Journal, vol. 33, no. 7, pp. 1312–1318, 1995.
[12] C. K. Hirwani, S. K. Panda, and T. R. Mahapatra, “Nonlinear finite element analysis of transient behavior of delaminated composite plate,” Journal of Vibration and Acoustics, vol. 140, no. 2, p. 021001, 2018.
[13] A. Szekrényes, “Bending solution of third-order orthotropic Reddy plates with asymmetric interfacial crack,” International Journal of Solids and Structures, vol. 51, pp. 2598–2619, 2014.
[14] N. Nanda and S. Sahu, “Free vibration analysis of delaminated composite shells using different shell theories,” International Journal of Pressure Vessels and Piping, vol. 98, pp. 111 – 118, 2012.
[15] S. K. Panda and B. N. Singh, “Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel using nonlinear FEM,” Finite Elements in Analysis and Design, vol. 47, pp. 378–386, 2011.
[16] V. K. Singh and S. K. Panda, “Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels,” Thin-Walled Structures, vol. 85, pp. 431–349, 2014.
[17] L. P. Kollár and G. S. Springer, Mechanics of Composite Structures. Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paolo: Cambridge University Press, 2003.
[18] A. Szekrényes, “Nonsingular crack modelling in orthotropic plates by four equivalent single layers,” European Journal of Mechanics A/Solids, vol. 55, pp. 73–99, 2016.
[19] A. Szekrényes, “Semi-layerwise analysis of laminated plates with nonsingular delamination - the theorem of autocontinuity,” Applied Mathematical Modelling, vol. 40, pp. 1344–1371, 2016.
[20] N. Nanda, “Static analysis of delaminated composite shell panels using layerwise theory,” Acta Mechanica, vol. 225, no. 10, pp. 2893–2901, 2014.
[21] A. Szekrényes, “The role of transverse stretching in the delamination fracture of softcore sandwich plates,” Applied Mathematical Modelling, vol. 63, pp. 611–632, 2018.
[22] K. N. Shivakumar and I. S. Raju, “An equivalent domain integral method for three-dimensional mixed-mode fracture problems,” Engineering Fracture Mechanics, vol. 42, no. 6, pp. 935–959, 1992.
[23] R. H. Rigby and M. H. Aliabadi, “Decomposition of the mixed-mode J-integral - revisited,” International Journal of Solids and Structures, vol. 35, no. 17, pp. 2073–2099, 1998.
[24] J. Petrolito, “Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements,” Applied Mathematical Modelling, vol. 38, no. 24, pp. 5858–5869, 2014.
[25] M. Izadi and M. Tahani, “Analysis of interlaminar stresses in general cross-ply laminates with distributed piezoelectric actuators,” Composite Structures, no. 92, pp. 757–768, 2010.
[26] A. Szekrényes, “The system of exact kinematic conditions and application to delaminated first-order shear deformable composite plates,” International Journal of Mechanical Sciences, vol. 77, pp. 17–29, 2013.
[27] A. Szekrényes, “Antiplane-inplane shear mode delamination between two second-order shear deformable composite plates,” Mathematics and Mechanics of Solids, pp. 1–24, 2015.
[28] A. Szekrényes, Nonsingular delamination modeling in orthotropic com- posite plates by semi-layerwise analysis. Hungarian Academy of Sciences, D.Sc. dissertation, December 2017.
[29] Y. Jianqiao, Laminated Composite Plates and Shells - 3D modelling. London, Berlin, Heidelberg, New York, Hong Kong, Milan, Paris, Tokyo: Springer, 2003.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:79134", author = "András Szekrényes", title = "Analytical Solution of the Boundary Value Problem of Delaminated Doubly-Curved Composite Shells", abstract = "Delamination is one of the major failure modes in laminated composite structures. Delamination tips are mostly captured by spatial numerical models in order to predict crack growth. This paper presents some mechanical models of delaminated composite shells based on shallow shell theories. The mechanical fields are based on a third-order displacement field in terms of the through-thickness coordinate of the laminated shell. The undelaminated and delaminated parts are captured by separate models and the continuity and boundary conditions are also formulated in a general way providing a large size boundary value problem. The system of differential equations is solved by the state space method for an elliptic delaminated shell having simply supported edges. The comparison of the proposed and a numerical model indicates that the primary indicator of the model is the deflection, the secondary is the widthwise distribution of the energy release rate. The model is promising and suitable to determine accurately the J-integral distribution along the delamination front. Based on the proposed model it is also possible to develop finite elements which are able to replace the computationally expensive spatial models of delaminated structures.
", keywords = "J-integral, Lévy method, third-order shell theory, state space solution.", volume = "13", number = "9", pages = "582-10", }
