Identification of Simple Shear in Plane Strain Deformation

[+] Author and Article Information
G. Damamme

CEA-DAM, Centre Ile de France, BP12, 91680 Bruyères-le-Châtel Cedex, France

A. Colin, C. Desrayaud, F. Montheillet

Centre SMS, CNRS UMR 5146, Ecole Nationale Supérieure des Mines de Saint-Etienne, 158 Cours Fauriel, 42023 Saint-Etienne Cedex 2, France

J. Eng. Mater. Technol 131(1), 011106 (Dec 19, 2008) (6 pages) doi:10.1115/1.3030940 History: Received February 10, 2008; Revised September 06, 2008; Published December 19, 2008

A theoretical analysis of plane strain incompressible velocity fields is first carried out. Two types of local fields are distinguished according to the sign of det(L), where L is the velocity gradient tensor, whereas det(L)=0 is associated with simple shear. A geometrical interpretation is used to illustrate the various decompositions of L. Finally, it is suggested that the condition det(L)=0 can be used to predict the occurrence of shear bands during metal forming processes.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

(a) Diagram showing the radial and orthoradial components of the velocity field associated with the elongation (or contraction) rate and the rotation rate of segment OM. (b) Diagram used to bring into evidence the two types of velocity field, according to whether circle (C) intersects or does not intersect the ε̇-axis.

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Figure 2

Vector representation of tensor L and its various decompositions

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Figure 3

Diagrams illustrating the two types of plane strain incompressible velocity fields: (a) L⃗ lies outside cone (K), i.e., det L<0, two directions do not rotate; (b) L⃗ lies inside cone (K), i.e., det L>0, the rotation rate of any direction is nonzero

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Figure 4

Maps of det(L)/‖L‖ for various steps of the edge-on forging of a circular cylinder. The nominal strains are (a) 0.24, (b) 0.80, (c) 1.14, and (d) 1.49. The upper half of the workpiece is shown.

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Figure 5

Maps of the normalized equivalent strain rate for various steps of the edge-on forging of a circular cylinder. The nominal strains are (a) 0.24, (b) 0.80, (c) 1.14, and (d) 1. 49. The upper half of the workpiece is shown.



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