Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T13:14:40.911Z Has data issue: false hasContentIssue false
  • English
  • Français

On a symbolic algebra for Hencky-Prandtl nets

Published online by Cambridge University Press:  24 October 2008

R. Hill
R. Hill
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

In the classical theory of plane deformations in isotropic plastic media, the field equations are hyperbolic and the orthogonal families of characteristics are known as Hencky-Prandtl nets. Their distinctive geometry has been given symbolic expression by Collins (1968), in an algebra of infinite matrices associated with canonical series representations of the general solution. This has become the standard technique when investigating boundary-value problems, both analytically and numerically. The basic framework of the algebra is here reorganized and developed. A systematic approach then leads to new identities which are shown to be fundamental in the algebraic hierarchy.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 2 , September 1983 , pp. 341 - 350
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Collins, I. F.The algebraic geometry of slipline fields with applications to boundary-value problems. Proc. Roy. Soc. London A 303 (1968), 317338.Google Scholar
(2)Collins, I. F.Geometric properties of some slipline fields for compression and extension. J. Mech. Phys. Solids 16 (1968), 137152.CrossRefGoogle Scholar
(3)Collins, I. F.Integral equation formulation of slipline field problems. Applications of Numerical Methods to Forming Processes; Am. Soc. Mech. Engng. 28 (1978), 129141.Google Scholar
(4)Collins, I. F. Boundary-value problems in plane-strain plasticity. Mechanics of Solids: The Rodney Hill 60th Anniversary Volume (ed. Hopkins, H. G. and Sewell, M. J.), pp. 135184 (Pergamon Press, Oxford, 1982).CrossRefGoogle Scholar
(5)Dewhurst, P. and Collins, I. F.A matrix technique for constructing slipline field solutions to a class of plane-strain plasticity problems. Int. J. Numerical Methods in Engineering 7 (1973), 357378.CrossRefGoogle Scholar
(6)Ewing, D. J. F.A series method for constructing slipline fields. J. Mech. Phys. Solids 15 (1967), 105114.CrossRefGoogle Scholar
(7)Geiringer, H.Fondements mathmatiques de la thorie des corps plastiques isotropes. Memorial des Sciences Mathmatiques 86 (1937), 189.Google Scholar
(8)Hill, R.On the vectorial superposition of Hencky-Prandtl nets. J. Mech. Phys. Solids 15 (1967), 255262.CrossRefGoogle Scholar
(9)Johnson, W., Sowerby, R. and Venter, R. D.Plane-strain slipline fields for metal deformation processes, Chap. 6 (Pergamon Press, Oxford, 1982).Google Scholar