Ideal Flow in Plasticity

[+] Author and Article Information
Kwansoo Chung

Department of Materials Science and Engineering, Seoul National University, 56-1 Shinlim-dong, Kwanak-ku, Seoul 151-742, Korea

Sergei Alexandrov

 Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Prospect Vernadskogo, 119526 Moscow, Russia

The constitutive law of the deformation theory in Eq. 44 is similar to that of hyperelasticity, while the relationship in Eq. 45 may be considered the Castigliano Theorem I, which is originally for the linear elasticity but extended here for the deformation theory of plasticity (or hyperelasticity).

Appl. Mech. Rev 60(6), 316-335 (Nov 01, 2007) (20 pages) doi:10.1115/1.2804331 History:

Ideal plastic flows constitute a class of solutions in the classical theory of plasticity based on, especially for bulk forming cases, Tresca’s yield criterion without hardening and its associated flow rule. They are defined by the condition that all material elements follow the minimum plastic work path, a condition which is believed to be advantageous for forming processes. Thus, the ideal flow theory has been proposed as the basis of procedures for the direct preliminary design of forming processes, which mainly involve plastic deformation. The aim of the present review is to provide a summary of both the theory of ideal flows and its applications. The theory includes steady and nonsteady flows, which are divided into three sections, respectively: plane-strain flows, axisymmetric flows, and three-dimensional flows. The role of the method of characteristics, including the computational aspect, is emphasized. The theory of ideal membrane flows is also included but separately because of its advanced applications based on finite element numerical codes. For membrane flows, restrictions on the constitutive behavior of materials are significantly relaxed so that more sophisticated anisotropic constitutive laws with hardening are accounted for. In applications, the ideal plastic flow theory provides not only process design guidelines for current forming processes under realistic tool constraints, but also proposes new ultimate optimum process information for futuristic processes.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

Schematic of the process design scheme sequentially utilizing the direct design theory based on the ideal flow theory, analytical methods, and experimental trials

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Figure 2

Corner trajectories for stretching an initially square flat plate

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Figure 3

The Tresca hexagon

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Figure 4

A meridian plane of the axisymmetric ideal flow: Eulerian cylindrical (rz) and Lagrangian principal line (ζν) coordinate systems

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Figure 5

Ideal die and the field of characteristics for the plane-strain extrusion (drawing)

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Figure 6

Ideal die with the minimum length and the field of characteristics

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Figure 7

Notations for (a) initial, (b) intermediate and (c) final shapes in the axisymmetric ideal flow bending

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Figure 8

Coordinate systems for an intermediate shape in the axisymmetric ideal flow bending

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Figure 9

Illustration of the condition for the existence of the solution

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Figure 10

Notations for (a) initial and (b) final shapes in the plane-strain ideal flow bending

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Figure 11

Plane-strain ideal compression of Specimen B by a frictionless die A: initial state

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Figure 12

Plane-strain ideal compression of Specimen B with a frictionless die A: intermediate status

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Figure 13

Final and intermediate shapes in the plane-strain ideal flow compression with a frictionless die of a constant width l=1.0

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Figure 14

Final and intermediate shapes in the plane-strain ideal flow compression with a frictionless die of varying width l=h

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Figure 15

The final part configuration analytically and numerically obtained along with the characteristic lines

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Figure 16

Evolution of the (half) part shape from the optimum rectangular shape to the prescribed final shape

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Figure 17

Distribution of the (a) hζ and (b) φ values along the boundary line

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Figure 18

Schematic view of a as a projection of f on the x1-x2 plane in the final configuration

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Figure 19

Evolution of cross-sectional shapes in the axisymmetric ideal sheet forming process

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Figure 20

Evolution of intermediate configuration obtained for the membrane ideal forming: (a) initial cutout stage, (b)–(e) intermediate shapes at α=0.2, 0.4, 0.6, and 0.8, respectively, and (f) final rectangular cup with round corners (prescribed)

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Figure 21

Histories of nodal forces in the axisymmetric ideal sheet forming process when α is (a) 0.4, (b) 0.8, and (c) 1.0, respectively

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Figure 22

Ideal sheet forming application for tube hydro-forming: (a) the final tube geometry prescribed and the resulting thickness strain contour and (b) the predicted ideal initial tube cross section and length




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