Engineering Transactions, 51, 1, pp. 63–85, 2003

Numerical Simulation of Finite Deformations of a Dynamically Loaded Elasto-Viscoplastic Circular Membrane

W. Dornowski
Institute of Fundamental Technological Research, Polish Academy of Sciences

The present paper deals with theoretical modelling and numerical simulation of a thin circular plate subjected to impulsive loading. To this end, the convective description is applied. The kinematical hypothesis used for theoretical description of the transient response includes membrane deformations only. This assumption is valid in the range of large deformations. The dynamical response of the material is described by Perzyna's elasto-viscoplastic constitutive relations. The theory is completed by an algorithm of the explicit finite difference method. With respect to the conditional stability of that method, the stability criterion is given. Basing on experimental data, an identification of material parameters is carried out. Some comparisons with the corresponding theoretical and experimental results are presented. Satisfactory agreement of the results has been found. Finally, an example of the plastic strain localization in a membrane is presented.
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J.H. ARGYRIS, J.ST. DOLTSINIS, M. KLEIBER, Incremental formulation in nonlinear mechanics and large strain elasto-plasticity, Comp. Meth. Appl. Mech. Eng., 14, 259–294, 1978.

G. BĄK, W. DORNOWSKI, Analytical-numerical method for description of moderately large deflections of impulsively loaded elastic-viscoplastic plates [in Polish], Archives Civ. Engng., 3–4, 303–315, 1991.

W.E. BAKER, Approximate techniques for plastic deformation of structures under impulsive loading, Shock Vib. Dig., 7, 7, 107–117, 1975.

C.T. CHON, P.S. SYMONDS, Large dynamic plastic deflections of plates by mode method, J. Engng. Mech. Div., Proceedings, ASCE, 103, (EM1), 169–187, 1977.

W. DORNOWSKI, G. BĄK, Analytical-numerical solutions for elastic-viscoplastic circular and rectangular plates under impulsive loading in range of moderately large deflections [in Polish], Archives Civ. Engng., 3–4, 373–392, 1991.

W. DORNOWSKI, Finite deformations of elasto-plastic membranes loaded dynamically [in Polish], Buli. MUT, 2, 498, 85–97, 1994.

W. DORNOWSKI, Interaction of membrane and bending forces in pieces at nonlinear vibrations, Engng. Trans., 42, 1–2, 33–59, 1994.

W. DORNOWSKI, Numerical simulation of plastic flow processes under dynamic cyclic loadings [in Polish], MUT report, 2598/99, Warszawa 1999.

W. DORNOWSKI, P. PERZYNA, Localization phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings, Computer Assisted Mechanics and Engineering Sciences, 7, 117–160, 2000.

T.A. DUFFEY, The large deflection dynamic response of clamped circular pieces subject to explosive loading, Sandia Laboratories Research Report No. SC-RR-67-532, 1967.

A.E. GREEN, W. ŻERNA, Theoretical elasticity, Second Edition, Clarendon Press, Oxford 1960.

C. GUEDES SOARES, A mode solution for the finite deflections of a circular piąte loaded impulsively, Engng. Trans., 29, 99–114, 1981.

R. HILL, Aspects of invariance in solid mechanics, Advances in Applied Mechanics, 18, 1–75, 1978.

M. KLEIBER, C. WOŹNIAK, Nonlinear mechanics of structures, PWN Warszawa, Kluwer Academic Publishers Dordrecht/Boston/London, 1991.

P. KŁOSOWSKI, K. WOŹNICA, D. WEICHERT, Dynamics of elasto-viscoplastic plates and shells, Archive Appl. Mech., 65, 326–345, 1995.

J.W. LEECH, E.A. WITMER, T.H.H. PIAN, Numerical calculation technique for large elastic-plastic transient deformations of thin shells, AIAA J., 12, 6, 2352–2359, 1968.

H. LIPPMANN, Kinetics of the axisymmetric rigid-plastic membrane supplied to initial impact, Int. J. Mech. Sci., 16, 297–303, 945–947, 1974.

J.E. MARSDEN, T.J.R. HUGHES, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983.

J.B. MARTIN, P.S. SYMONDS, Mode approximations for impulsively loaded rigid-plastic structures, J. Engng. Mech. Div., Proceedings, ASCE, 92, (EM5), 43–66, 1966.

A. NEEDLEMAN, V. TYERGAARD, Finite element analysis of localization plasticity, [in:] Finite Elements, Vol. V: Special problems in solid mechanics, J.T. ODEŃ and G.F. CAREY [Eds.], Prentice-Hall, Englewood Cliffs, New Jersey 1984.

G.N. NURICK, H.T. PEARCE, J.B. MARTIN, The deformation of thin plates subjected to impulsive loading, [in:] Inelastic Behaviour of Plates and Shells, L. BEVILACQUA [Ed.], Springer Verlag, New York 1986.

N. PERRONE, P. BHADRA, Simplified large deflection mode solutions for impulsively loaded viscoplastic circular membranes, J. Appl. Mech., 51, 505–509, 1984.

P. PERZYNA, Internal state variable description of dynamic fracture of ductile solids, Int. J. Solids Struct., 22, 7, 797–818, 1986.

R.D. RICHTMYER, K.W. MORTON, Difference methods for initial-value problems, John Wiley, New York, 1967.

M. STOFFEL, R. SCHMIDT, D. WEICHERT, Shock wave-loaded plates, Int. J. Solids Struct., 2001 (in press).

P.S. SYMONDS, T. WIERZBICKI, Membrane mode solution for impulsively loaded circular plates, J. Appl. Mech., 46, 58–64, 1979.

R.G. TEELING-SMITH, G.N. NURICK, The deformations and tearing of thin circular plates subjected to impulsive loads, Int. J. Impact Engng, 11, l, 77–91, 1991.

E.A. WITMER, H.A. BALMER, J.W. LEECH, T.H.H. PIAN, Large dynamic deformations of beams, circular rings, circular plates and shells, AIAA J., l, 8, 1848–1857, 1963.

C. WOŹNIAK, Nonlinear theory of shells [in Polish], PWN, Warszawa 1966.

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