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**Transient impact response analysis of an elastic-plastic beam.**
*(English)*
Zbl 1480.74035

Summary: A hybrid, numerical-analytical model is presented to investigate the transient response of a simply supported elastic-plastic beam subjected to impact of a sphere. The model couples a finite difference model based on the Rayleigh’s beam theory with a theoretical contact model, named refined Stronge’s model. The elastic-plastic transient impact responses of the beam and the wave propagation are solved by the finite difference model. The local elastic-plastic contact behavior is analyzed by the refined Stronge’s model. The presented model is verified by the experiments. The comparisons of the numerical results with the experimental results show that the presented model is valid and can predict accurately the transient impact response and impact-induced wave propagation. The comparison of the calculating efficiency with the 3D finite element model shows that the presented model is highly efficient and suitable for parametric study. The effects of the various impact parameters, such as impactor mass, velocity, plasticity and impact location on the impact response, the energy loss and the coefficient of restitution are investigated. It has been found that the impact-induced wave propagation influences significantly the impact force response, the contact time, the energy loss and the coefficient of restitution. A jumping phenomenon of the energy loss and the Newton coefficient of restitution has been found, and it is resulted from the impact-induced wave propagation. The presented model appears to be suitable and convenient for the impact response analysis of elastic-plastic beam, especially for the study of impact-induced wave effects.

### MSC:

74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

### Keywords:

Rayleigh beam; elastic-plastic impact; transient response; finite difference method; contact model### Software:

LS-DYNA
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\textit{L. Zhang} et al., Appl. Math. Modelling 55, 616--636 (2018; Zbl 1480.74035)

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### References:

[1] | Goldsmith, W., Impact: The Theory and Physical Behavior of Colliding Solids (1960), Edward Arnold: Edward Arnold London · Zbl 0122.42501 |

[2] | Yigit, A. S.; Christoforou, A. P.; Majeed, M. A., A nonlinear visco-elastoplastic impact model and the coefficient of restitution, Nonlinear Dyn., 66, 509-521 (2011) |

[3] | Stronge, W. J., Impact Mechanics (2000), Cambridge University Press: Cambridge University Press London · Zbl 0961.74002 |

[4] | Johnson, K. L., Contact Mechanics (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0599.73108 |

[5] | Thorton, C., Coefficient of restitution for collinear collisions of elastic- perfectly plastic spheres, J. Appl. Mech., 64, 383-386 (1997) · Zbl 0883.73072 |

[6] | Vuquoc, L.; Zhang, X.; Lesburg, L., A normal force-displacement model for contacting spheres accounting for plastic deformation: force-driven formulation, J. Appl. Mech., 67, 363-371 (2000) · Zbl 1110.74732 |

[7] | Jackson, R. L.; Green, I., A finite element study of elasto-plastic hemispherical contact against a rigid flat, ASME J. Tribol., 127, 343-354 (2005) |

[8] | Brake, M. R., An analytical elastic-perfectly plastic contact model, Int. J. Solids Struct., 49, 3129-3141 (2012) |

[9] | Burgoyne, H. A.; Daraio, C., Strain-rate-dependent model for the dynamic compression of elastoplastic spheres, Phys. Rev. E., 89, 256-266 (2014) |

[10] | Marsh, D. M., Plastic flow in glass, Proc. R. Soc. A, 279, 420-435 (1964) |

[11] | Machado, M.; Moreira, P.; Flores, P.; Lankarani, H. M., Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory, Mech. Mach. Theory, 53, 99-121 (2012) |

[12] | Alves, J.; Peixinho, N.; Silva, M. T.D.; Flores, P.; Lankarani, H. M., A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids, Mech. Mach. Theory, 85, 172-188 (2015) |

[13] | Abrate, S., Modeling of impacts on composite structures, Compos. Struct., 51, 129-138 (2001) |

[14] | Pashah, S.; Massenzio, M.; Jacquelin, E., Prediction of structural response for low velocity impact, Int. J. Impact Eng., 35, 119-132 (2008) · Zbl 1198.70007 |

[15] | Pashah, S.; Massenzio, M.; Jacquelin, E., Structural response of impacted structure described through anti-oscillators, Int. J. Impact Eng., 35, 471-486 (2008) |

[16] | Ivañez, I.; Barbero, E.; Sanchez-Saez, S., Analytical study of the low-velocity impact response of composite sandwich beams, Compos. Struct., 111, 459-467 (2014) |

[17] | Lifshitz, J. M.; Gov, F.; Gandelsman, M., Instrumented low-velocity impact of CFRP beams, Int. J. Impact Eng., 16, 201-215 (1995) |

[18] | Seifried, R.; Schiehlen, W.; Eberhard, P., Numerical and experimental evaluation of the coefficient of restitution for repeated impacts, Int. J. Impact Eng., 32, 508-524 (2005) |

[19] | Malekzadeh, K.; Khalili, S. M.R.; Veysi Gorgabad, A., Dynamic response of composite sandwich beams with arbitrary functionally graded cores subjected to low-velocity impact, Mech. Adv. Mater. Struct., 22, 605-618 (2015) |

[20] | Boroujerdy, M. S.; Kiani, Y., Low velocity impact analysis of composite laminated beams subjected to multiple impacts in thermal field, Z. Angew. Math. Mech., 96, 843-856 (2015) |

[21] | Lee, E.; Symonds, P. S., Large plastic deformations of beams under transverse impact, ASME J. Appl. Mech., 19, 308-314 (1952) |

[22] | Stronge, W. J.; Yu, T., Dynamic Models for Structural Plasticity (1993), Springer: Springer New York |

[23] | Lellep, J.; Torn, K., Shear and bending response of a rigid-plastic beam subjected to impulsive loading, Int. J. Impact Eng., 31, 1081-1105 (2005) |

[24] | Ghaderi, S. H.; Hokamoto, K.; Fujita, M., Analysis of stationary deformation behavior of a semi-infinite rigid-perfect plastic beam subjected to moving distributed loads of finite width, Int. J. Impact Eng., 36, 115-121 (2009) |

[25] | Jones, N., Structural Impact (2012), Cambridge University Press: Cambridge University Press Cambridge |

[26] | Yu, T. X., Elastic effect in the dynamic plastic response of structures, (Jones, N.; Wierzbicki, T., Structural Crashworthiness and Failure (1993), Elsevier: Elsevier Liverpool), 341-384 |

[27] | Doyle, J. F., Wave Propagation in Structures: An FFT-Based Spectral Analysis Methodology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0715.73022 |

[28] | Schwieger, H., Central deflection of a transversely struck beam, Exp. Mech., 10, 166-169 (1970) |

[29] | Yigit, A. S.; Christoforou, A. P., Limits of asymptotic solutions in low-velocity impact of composite plates, Compos. Struct., 81, 568-574 (2007) |

[30] | Christoforou, A. P.; Yigit, A. S.; Majeed, M., Low-velocity impact response of structures with local plastic deformation: characterization and scaling, J. Comput. Nonlinear Dyn., 8, Article 011012-1-011012-10 (2013) |

[31] | Kenny, S.; Pegg, N.; Taheri, F., Finite element investigations on the dynamic plastic buckling of a slender beam subject to axial impact, Int. J. Impact Eng., 27, 1-17 (2002) |

[32] | Sadighi, M.; Pouriayevali, H., Quasi-static and low-velocity impact response of fully backed or simply supported sandwich beams, J. Sandw. Struct. Mater., 10, 499-524 (2008) |

[33] | Kabir, M. Z.; Shafei, E., Analytical and numerical study of FRP Retrofitted RC beams under low velocity impact, Trans. A Civil Eng., 16, 415-428 (2009) |

[34] | Ivañez, I.; Sanchez-Saez, S., Numerical modelling of the low-velocity impact response of composite sandwich beams with honeycomb core, Compos. Struct., 106, 716-723 (2013) |

[35] | Meybodi, M. H.; Saber-Samandari, S.; Sadighi, M.; Bagheri, M. R., Low-velocity impact response of a nanocomposite beam using an analytical model, Lat. Am. J. Solids Struct., 12, 333-354 (2015) |

[36] | Yu, T. X.; Yang, J. L.; Reid, S. R.; Austin, C. D., Dynamic behaviour of elastic-plastic free-free beams subjected to impulsive loading, Int. J. Solids Struct., 33, 2659-2680 (1996) · Zbl 0900.73118 |

[37] | Yu, T. X.; Yang, J. L.; Reid, S. R., Interaction between reflected elastic flexural waves and a plastic ‘hinge’ in the dynamic response of pulse loaded beams, Int. J. Impact Eng., 19, 457-475 (1997) |

[38] | Yang, J. L.; Xi, F., Experimental and theoretical study of free-free beam subjected to impact at any cross-section along its span, Int. J. Impact Eng., 28, 761-781 (2003) |

[39] | Wang, H.; Yin, X.; Qi, X.; Deng, Q.; Yu, B.; Hao, Q., Experimental and theoretical analysis of the elastic-plastic normal repeated impacts of a sphere on a beam, Int. J. Solids Struct., 109, 131-142 (2017) |

[40] | Graff, K. F., Wave Motion in Elastic Solids (1991), Dover Publications: Dover Publications New York |

[41] | Khan, A. S.; Huang, S., Continuum Theory of Plasticity (1995), John Wiley & Sons: John Wiley & Sons New York · Zbl 0856.73002 |

[42] | Nr, C. M.; Lee, L. H.N., Dynamic behavior of inelastic cylindrical shells at finite deformation, Int. J. Non Linear Mech., 9, 193-207 (1974) · Zbl 0284.73064 |

[43] | Qi, X.; Yin, X., Experimental studying multi-impact phenomena exhibited during the collision of a sphere onto a steel beam, Adv. Mech. Eng., 8, 1-16 (2016) |

[44] | Rao, S. S.; Yap, F. F., Mechanical Vibrations (1995), Addison-Wesley: Addison-Wesley Reading, MA |

[45] | Hallquist, J., LS-DYNA Keyword User’s Manual, Version: 970 (2003), Livermore Software Technology Corporation: Livermore Software Technology Corporation California |

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