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A generalized version of a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate using the concept of noninteger derivatives. (English) Zbl 06209393
Summary: A low velocity impact between a rigid sphere and transversely isotropic strain-hardening plate supported by a rigid substrate is generalized to the concept of noninteger derivatives order. A brief history of fractional derivatives order is presented. The fractional derivatives order adopted is in Caputo sense. The new equation is solved via the analytical technique, the Homotopy decomposition method (HDM). The technique is described and the numerical simulations are presented. Since it is very important to accurately predict the contact force and its time history, the three stages of the indentation process, including (1) the elastic indentation, (2) the plastic indentation, and (3) the elastic unloading stages, are investigated.

MSC:
74Mechanics of deformable solids
65Numerical analysis
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References:
[1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[2] V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508-518, 2005. · Zbl 1061.34003 · doi:10.1016/j.jmaa.2004.07.039
[3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[5] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, part II,” Geophysical Journal International, vol. 13, no. 5, pp. 529-539, 1967. · doi:10.1111/j.1365-246X.1967.tb02303.x
[6] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[7] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[8] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008. · Zbl 1152.37001
[9] A. Yildirim, “An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 4, pp. 445-450, 2009.
[10] S. G. Samko, A. A. Kilbas, and O. I. Maritchev, “Integrals and derivatives of the fractional order and some of their applications,” Nauka i TekhnIka, Minsk, 1987 (Russian). · Zbl 0617.26004
[11] I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367-386, 2002. · Zbl 1042.26003
[12] A. Atangana, “New class of boundary value problems,” Information Sciences Letters, vol. 1, no. 2, pp. 67-76, 2012.
[13] A. Atangana, “Numerical solution of space-time fractional derivative of groundwater flow equation,” in Proceedings of the International Conference of Algebra and Applied Analysis, vol. 2, no. 1, p. 20, Istanbul, Turkey, June 2012.
[14] G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,” Applied Mathematics Letters, vol. 18, no. 7, pp. 817-826, 2005. · Zbl 1075.60068 · doi:10.1016/j.aml.2004.09.012
[15] G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367-1376, 2006. · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[16] M. Davison and C. Essex, “Fractional differential equations and initial value problems,” The Mathematical Scientist, vol. 23, no. 2, pp. 108-116, 1998. · Zbl 0919.34005
[17] C. F. M. Coimbra, “Mechanics with variable-order differential operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692-703, 2003. · Zbl 1103.26301 · doi:10.1002/andp.200310032
[18] I. Andrianov and J. Awrejcewicz, “Construction of periodic solutions to partial differential equations with non-linear boundary conditions,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 4, pp. 327-332, 2000. · Zbl 0977.35031 · doi:10.1515/IJNSNS.2000.1.4.327
[19] C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons Jr., “A new perturbative approach to nonlinear problems,” Journal of Mathematical Physics, vol. 30, no. 7, pp. 1447-1455, 1989. · Zbl 0684.34008 · doi:10.1063/1.528326
[20] B. Delamotte, “Nonperturbative (but approximate) method for solving differential equations and finding limit cycles,” Physical Review Letters, vol. 70, no. 22, pp. 3361-3364, 1993. · Zbl 1051.65505 · doi:10.1103/PhysRevLett.70.3361
[21] M. Shariyat, Automotive Body: Analysis and Design, K. N. Toosi University Press, Tehran, Iran, 2006.
[22] R. Ollson, “Impact response of orthotropic composite plates predicted form a one-parameter differential equation,” American Institute of Aeronautics and Astronautics Journal, vol. 30, no. 6, pp. 1587-1596, 1992. · Zbl 0825.73112 · doi:10.2514/3.11105
[23] A. S. Yigit and A. P. Christoforou, “On the impact of a spherical indenter and an elastic-plastic transversely isotropic half-space,” Composites, vol. 4, no. 11, pp. 1143-1152, 1994. · doi:10.1016/0961-9526(95)91288-R
[24] A. S. Yigit and A. P. Christoforou, “On the impact between a rigid sphere and a thin composite laminate supported by a rigid substrate,” Composite Structures, vol. 30, no. 2, pp. 169-177, 1995. · doi:10.1016/0263-8223(94)00037-9
[25] A. P. Christoforou and A. S. Yigit, “Characterization of impact in composite plates,” Composite Structures, vol. 43, pp. 5-24, 1998.
[26] A. P. Christoforou and A. S. Yigit, “Effect of flexibility on low velocity impact response,” Journal of Sound and Vibration, vol. 217, no. 3, pp. 563-578, 1998. · doi:10.1006/jsvi.1998.1807
[27] A. S. Yigit and A. P. Christoforou, “Limits of asymptotic solutions in low-velocity impact of composite plates,” Composite Structures, vol. 81, pp. 568-574, 2007. · doi:10.1016/j.compstruct.2006.10.006
[28] A. Atangana and A. Secer, “A note on fractional order derivatives and Table of fractional derivative of some specials functions,” Abstract Applied Analysis. In press. · Zbl 1276.26010
[29] T. H. Solomon, E. R. Weeks, and H. L. Swinney, “Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow,” Physical Review Letters, vol. 71, pp. 3975-3978, 1993. · doi:10.1103/PhysRevLett.71.3975
[30] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Connecticut, UK, 2006.
[31] R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance, vol. 190, pp. 255-270, 2008. · doi:10.1016/j.jmr.2007.11.007
[32] A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, “Fractional diffusion in inhomogeneous media,” Journal of Physics, vol. 38, no. 42, pp. L679-L684, 2005. · Zbl 1082.76097 · doi:10.1088/0305-4470/38/42/L03
[33] F. Santamaria, S. Wils, E. de Schutter, and G. J. Augustine, “Anomalous diffusion in purkinje cell dendrites caused by spines,” Neuron, vol. 52, no. 4, pp. 635-648, 2006. · doi:10.1016/j.neuron.2006.10.025
[34] H. G. Sun, W. Chen, and Y. Q. Chen, “Variable order fractional differential operators in anomalous diffusion modelling,” Journal of Physics A, vol. 388, pp. 4586-4592, 2009.
[35] A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-vries equations,” Abstract Applied Analysis, vol. 2013, Article ID 947986, 8 pages, 2013. · Zbl 1291.35273 · doi:10.1155/2013/947986
[36] A. Atangana and J. F. Botha, “Analytical solution of groundwater flow equation via Homotopy Decomposition Method,” Journal of Earth Science & Climatic Change, vol. 3, p. 115, 2012.
[37] M. Shariyat, R. Ghajar, and M. M. Alipour, “An analytical solution for a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate,” Journal of Engineering Mathematics, vol. 75, pp. 107-125, 2012. · Zbl 1254.74091 · doi:10.1007/s10665-011-9505-1
[38] J. Awrejcewicz, V. A. Krysko, O. A. Saltykova, and Yu. B. Chebotyrevskiy, “Nonlinear vibrations of the Euler-Bernoulli beam subjected to transversal load and impact actions,” Nonlinear Studies, vol. 18, no. 3, pp. 329-364, 2011. · Zbl 1229.37103
[39] C. C. Poe Jr. and W. Illg, “Strength of a thick graphite/epoxy rocket motor case after impact by a blunt object.,” in Test Methods for Design Allowable for Fibrous Composites, C. C. Chamis, Ed., vol. 2, pp. 150-179, ASTM, Philadelphia, Pa, USA, 1989, ASTM STP 1003.
[40] C. C. Poe Jr., “Simulated impact damage in a thick graphite/epoxy laminate using spherical indenters,” NASA TM 100539, 1988.