Springbackward Phenomenon of a Transversely Isotropic Functionally Graded Composite Cylindrical Shell

Document Type : Research Paper


1 Department of Mathematics, RTM Nagpur University, Nagpur, India.

2 Deptt. of Mathematics, Shri Lemdeo Patil Mahavidyalya, Nagpur, India.

3 Deptt. of Mathematics, RTM Nagpur University, Nagpur, India.


This study provides an approach to predict the springback phenomenon during post-solidification cooling in a functionally graded hybrid composite cylindrical shell with a transverse isotropic structure. Here, the material properties are given with a general parabolic power-law function. During the theoretical analysis, an appropriate transformation is introduced in the equilibrium equation, which is resulting in a hypergeometrical differential equation. Thermoelastic solutions are obtained and analyzed for a homogeneous, nonhomogeneous, and elastic-plastic state. The solution is validated by applying it to the multilayered functionally graded cylindrical shell using the transfer or propagator matrix method.


Main Subjects

[1] O’Neill, J. M., Rogers, T. G., and Spencer, A. J. M., “Thermally induced distortions in the moulding of laminated channel sections”, Math. Engg. Ind., Vol. 2, pp. 65-72, 1988.             
[2] Wawner, T. O., and Gundel, D. B., “Investigation of the Reaction Kinetics between SiC Fibers and selectively Alloyed Titanium Matrices”, School of Engineering and Applied Science Technical Repot (Grant No. NAG-1-745, Department of Materials Science, University of Virginia, Charlottesville, VA, 1991.
[3] Birman, V., “Stability of functionally graded hybrid composite plates”, Composites Engineering, Vol. 5, pp. 913-921, 1995.                                                                                                          
[4] Ootao, Y., and Tanigawa, Y., “Three-dimensional transient thermal stresses of functionally graded rectangular plate due to partial heating”, Journal of Thermal Stresses, Vol. 22, pp. 35-55, 1999.                                                
[5] Reddy, J.N., “Analysis of functionally graded plates”, Int. J. Numer. Meth. Engg. , Vol. 47, pp. 663–684, 2000. [6] Ye, G. R., Chen, W. Q., and Cai, J. B., “A uniformly heated functionally graded cylindrical shell with transverse isotropy”,  Mechanics Research Communications, Vol. 28, pp. 535-542, 2001.                                               
[7] Kieback, B., Neubrand, A., and Riedel, H., “Processing techniques for functionally graded materials”, Mater Sci. Engg., A362, pp. 81-105, 2003.                                                         
[8] Sugano, Y., Chiba, R., Hirose, K., and Takahashi, K.,, “Material design for reduction of thermal stress in a functionally graded material rotating disk”, JSME International Journal Series A Solid Mechanics and Material Engineering, Vol. 47, pp. 189-197, 2004.                                   
[9] Eraslan, A. N., and Akis, T., “Elastoplastic response of a long functionally graded tube subjected to internal pressure”, Turkish J. Eng. Env. Sci., Vol. 29, pp. 361-368, 2005.              
[10] Ohmichi, M., and Noda, N., “The effect of oblique functional gradation to thermal stresses in the functionally graded infinite strip”, Acta Mechanica, Vol. 196, pp. 219-237, 2007.                                                                                  
[11] Huang, Y. H.,  and Han, X., “Transient Analysis of Functionally Graded Materials Plate using Reduced-Basis Methods”, Computational Mechanics, Proceedings of International Symposium on Computational Mechanics, China, 2007.                                                                      
[12] Bobaru, F., “Designing optimal volume fractions for functionally graded materials with temperature dependent material properties”, J. Appl. Mech, Vol. 74, pp. 861-875, 2007.   
[13] You, L. H., Wang, J. X., and Tang, B. P., “Deformations and stresses in annular disks made of functionally graded materials subjected to internal and/or external pressure”, Meccanica, Vol. 44, pp. 283-292, 2008.
[14] Paulino, G.H., “Multiscale and functionally graded materials”, In: Proceedings of the international conference FGM IX, Hawaii, 2008.                                                                                  
[15] Chien-Ching Ma, and Yi-Tzu Chen, “Theoretical analysis of heat conduction problems of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes”, Acta Mechanica, Vol. 221, Number 3-4, Page 223, 2011.                                                         
[16] Birman, V., Keil, T., and Hosder, S., “Functionally graded materials in Engineering, In: Structural interfaces and attachments in Biology”, Springer, New York, 2012.                             
[17] Chiba, R., and Sugano, Y., “Optimisation of material composition of functionally graded materials based on multiscale thermoelastic analysis”, Acta Mechanica, Vol. 223, pp. 891-909, 2012.
[18] Lamba, N. K., Khobragade, N. W., “Uncoupled thermoelastic analysis for a thick cylinder with radiation”, Theoretical and Applied Mechanics Letters, Vol. 2, pp. 21-35, 2012.                
[19] Gaikwad, K., “Two-dimensional steady-state temperature distribution of a thin circular plate due to uniform internal energy generation”, Cogent Mathematics, Vol. 3, 1135720, 2016.                                         
[20] Matveenko, V. P., Fedorov, A. Yu., and Shardakov, I. N., “Analysis of stress singularities at singular points of elastic solids made of functionally graded materials”, Doklady Physics, Vol. 61, pp. 24- 28, 2016.                            [21] Williams, T. O., Arnold, S. M., and Pindera, M. J., “An analytical/Numerical correlation study of the multiple concentric cylinder model for the thermoplastic response of metal matrix composites”, NASA Contractor Report 191142, Lewis Research Center, Cleveland, Ohio, 1993.                                                                                          
[22] Spencer, A. J. M., Watson, P., and Rogers, T. G., “Thermoelastic Distortions in laminated anisotropic tubes and channel section”, Journal of Thermal Stresses, Vol. 15, pp. 129-141, 1992.                                             
[23] Varghese V., and Khobragade, N.W., “Mathematical analysis of functionally graded hybrid composite channel section in the interfacial zone during post-solidification cooling”, Adv. And Appl in fluid Mechanics, Vol. 3, pp. 41-55, 2008.                                                   
[24] Arnold, S. M., Arya, V. K., and Melis, M. E., “Elastic/Plastic analysis of advanced composites investigating the use of the complaint layer concept in reducing residual stresses resulting from processing”, NASA Technical Memorandum 103204, Lewis Research Center, 1990.
[25] Abramowitz, M., and Stegun, I. A., (Editors), “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables”, National Bureau of Standards Applied Mathematics, Washington, 1964.