Abstract
The paper deals with the deformation of an ideal plastic solid that is initially rigid, i.e., elastic deformations are neglected. In addition, the solid flows by means of the mechanism of extended slip, for which the rotation-rate vector field remains continuous and the strain-rate tensor is solenoidal. The Tresca yield criterion applies to such a solid and with an associated flow-rule is represented in a manner that includes both bi-axial and tri-axial states of strain. Two new theorems are proved, and a second-order partial differential equation is derived for the first invariant of the stress tensor (hydrostatic pressure); the analogue of a similar published equation for the bi-axial strain case. To illustrate the methodology, the above theory is applied to the tri-axial problem of a thick metal plate clamped round a circle and deflected by means of pressure. It is shown, from the exact solution, that the errors due to the use of the approximate membrane formula for a clamped thin plate are small, even for a 6mm thick plate clamped on a circle 100mm in diameter. Surprisingly, there is an initial thickening of the plate and it is shown that, regardless of the plate thickness, the pressure passes through a maximum at a deflection equal to \({1/\sqrt{3}}\) of the radius of the clamping circle.
Similar content being viewed by others
References
Bish R.L.: Plastic shear deformation of a thin strain-hardening disc: variational principles. Phil. Mag. A 91, 1–15 (2011)
Bish R.L.: The flow of a solid by viscous slip between elastic domains. Acta Mech. 81, 39–45 (1990)
Bish R.L.: Rotation-rate continuity in bi-axial plastic deformation. Z. Angew. Math. Mech. 84, 266–279 (2004)
Bish R.L.: Rotationally continuous plastic flow. Q. J. Mech. Appl. Math. 52, 645–662 (1999)
Hill R.: On the problem of uniqueness in the theory of a rigid-plastic solid. J. Mech. Phys. Solids 4, 47–225 (1956)
Hill R.: On the problem of uniqueness in the theory of a rigid-plastic solid. J. Mech. Phys. Solids 5, 1–8 (1956)
Hill R.: On the problem of uniqueness in the theory of a rigid-plastic solid. J. Mech. Phys. Solids 5, 153–161 (1956)
Hill R.: On the problem of uniqueness in the theory of a rigid-plastic solid. J. Mech. Phys. Solids 5, 302–307 (1957)
Dewhurst P.: On the non-uniqueness of the machining process. Proc. Roy. Soc. A 360, 587–610 (1978)
Johnson, W., Sowerby, R., Venter, R.D.: Plane-strain slip-line fields for metal deformation processes, pp. 24–25. Pergamon Press, Oxford (1982)
Kachanov, L.M. (1971) Foundations of the Theory of Plasticity. North-Holland, Amsterdam, pp. 70–77
Prager W.: Recent developments in the mathematical theory of plasticity. J. Appl. Phys. 20, 235–241 (1949)
Phillips A., Wu H.C.: Int. J. Solids Struct. 9, 15–30 (1973)
Richmond O., Morrison H.L.: Streamlined wire drawing dies of minimum length. J. Mech. Phys. Solids 15, 195–197 (1967)
Parsons D.H.: Plastic flow with axial symmetry using the Mises flow criterion. Proc. London Math. Soc. Ser. III 6, 610–625 (1956)
Besdo D.: Principal- and slip-line methods of numerical analysis in plane and axially-symmetric deformations of rigid/plastic media. J. Mech. Phys. Solids 19, 313–328 (1971)
Lippman H.: Statics and dynamics of axially-symmetric plastic flow. J. Mech. Phys. Solids 13, 29–39 (1965)
Taylor G.I., Quinney H.: The plastic distortion of metals. Proc. Roy. Soc. 230, 323–362 (1931)
Johnson W., Duncan J.L.: Use of the biaxial test extensometer. Sheet Metal Ind. 42, 271–275 (1965)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bish, R.L. Tri-axial deformation of a plastic-rigid solid. Acta Mech 223, 655–668 (2012). https://doi.org/10.1007/s00707-011-0580-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-011-0580-1