On the solution of a non-linear parabolic equation with a floating boundary arising in a problem of plastic impact of a beam
Author:
Thomas C. T. Ting
Journal:
Quart. Appl. Math. 21 (1963), 133-150
DOI:
https://doi.org/10.1090/qam/153176
MathSciNet review:
153176
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: The deformation of a cantilever beam with strain rate sensitivity subjected to impact loading at its base has been studied in [11] by an approximate method in which the inertia forces in the plastic region are neglected. If these forces are taken into account, the equation of motion in the plastic region is a fourth order non-linear parabolic differential equation with a floating boundary, i.e. one whose position varies with time and must be found as part of the solution. A numerical solution of this equation is presented here. The results show that the bending moment in the plastic region varies nearly linearly. This result implies that the shear force is nearly constant in the plastic region, and hence that the inertia forces in the plastic region are small in comparison with the shear force in the same region.
L. W. Ehrlich, A numerical method of solving a heat flow problem with moving boundry, J. Assoc. Comput. Mach. 5 (1958) 161
- Jim Douglas Jr. and T. M. Gallie Jr., On the numerical integration of a parabolic differential equation subject to a moving boundary condition, Duke Math. J. 22 (1955), 557–571. MR 78755
D. Young and L. Ehrlich, On the numerical solution of linear and non-linear parabolic equations on the Ordvac, Interim Technical Report No. 18, Office of Ordnance Research Contract DA-36-034-ORD-1486, University of Maryland, February 1956.
- S. D. Conte and W. C. Royster, A study of finite difference approximations to a fourth order parabolic differential equation, Ballistic Research Laboratories, Aberdeen Proving Ground, Md., Rep. no., 1955. Rep. no. 959. MR 0090136
- George G. O’Brien, Morton A. Hyman, and Sidney Kaplan, A study of the numerical solution of partial differential equations, J. Math. Physics 29 (1951), 223–251. MR 0040805
- F. B. Hildebrand, Methods of applied mathematics, Precntice-Hall, Inc., New York, N. Y., 1952. MR 0057300
- Jim Douglas Jr., On the relation between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations, J. Soc. Indust. Appl. Math. 4 (1956), 20–37. MR 80368
- J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc. 43 (1947), 50–67. MR 19410
- M. L. Juncosa and David Young, On the Crank-Nicolson procedure for solving parabolic partial differential equations, Proc. Cambridge Philos. Soc. 53 (1957), 448–461. MR 88804, DOI https://doi.org/10.1017/s0305004100032436
- E. C. Du Fort and S. P. Frankel, Stability conditions in the numerical treatment of parabolic differential equations, Math. Tables Aids Comput. 7 (1953), 135–152. MR 59077, DOI https://doi.org/10.1090/S0025-5718-1953-0059077-7
Thomas C. T. Ting and P. S. Symomds, Impact of a cantilever beam with strain rate sensitivity, Proceedings of the Fourth National Congress of Applied Mechanics (June 1962), ASME, New York 1962, Vol. 2, p. 1153
L. W. Ehrlich, A numerical method of solving a heat flow problem with moving boundry, J. Assoc. Comput. Mach. 5 (1958) 161
J. Douglas, Jr. and J. M. Gallie, Jr., On the numerical intergation of a parabolic differential equation subject to a moving boundry condition, Duke Math J., 22 (1955) 557
D. Young and L. Ehrlich, On the numerical solution of linear and non-linear parabolic equations on the Ordvac, Interim Technical Report No. 18, Office of Ordnance Research Contract DA-36-034-ORD-1486, University of Maryland, February 1956.
S. D. Conte and W. C. Royster, A Study of finite difference approximationa to a fourth order parabolic differential equation, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, Report No. 959, Oct. 1955
G. G. O’Brien, M. A. Hyman and S. Kaplan, A study of the numerical solution of partial differential equations, J. Math. Phys. 29 (1951) 223
F. B. Hildebrand, Methods of Applied Mathematics, Prentice-Hall, Inc. 1954
J. Douglas, Jr., On the realtion between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations, J. Soc. Indust. Appl. Math. 4 (1956) 20
J. Crank and P. Nicolson, A Practical method for the numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb. Phil. Soc., 43 (1947) 50
M. L. Juncosa and D. Young, On the Crank-Nicolson procedure for solving parabolic partial differential equations, Proc. Camb. Phil. Soc. 53 (1957) 448
E. C. DuFort and S. P. Frankel, Stability conditions in the numerical treatment of parabolic differential equations, Math. Tables and Other Aids to Comp. 7 (1953) 135
Thomas C. T. Ting and P. S. Symomds, Impact of a cantilever beam with strain rate sensitivity, Proceedings of the Fourth National Congress of Applied Mechanics (June 1962), ASME, New York 1962, Vol. 2, p. 1153
Additional Information
Article copyright:
© Copyright 1963
American Mathematical Society