The theory of general and ideal plastic deformations of Treasca solids

Richmond, O.; Alexandrov, Sergei

doi:10.1007/bf01463167

Cited by 27 publications

(10 citation statements)

References 14 publications

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“…The characteristic coordinates are denoted as (α, β). The first equation in (6) corresponds to the α-lines and the second to the β-lines. The ideal flow condition requires that the velocity vector V is tangent to the ξ-coordinate lines (Figure 2).…”

Section: System Of Equations In Characteristic Coordinatesmentioning

confidence: 99%

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General Planar Ideal Flow Solutions with No Symmetry Axis

Alexandrov,

Mokryakov

2023

Materials

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Bulk ideal flows constitute a wide class of solutions in plasticity theory. Ideal flow solutions concern inverse problems. In particular, the solution determines part of the boundary of a region where it is valid. Bulk planar ideal flows exist in the case of (i) isotropic rigid/plastic material obeying an arbitrary pressure-independent yield criterion and its associated flow rule and (ii) the double sliding and rotation model based on the Mohr–Coulomb yield criterion. In the latter case, the intrinsic spin must vanish. Both models are perfectly plastic, and the complete equation systems are hyperbolic. All available specific solutions for both models describe flows with a symmetry axis. The present paper aims at general solutions for flows with no symmetry axis. The general structure of the solutions consists of two rigid regions connected by a plastic region. The characteristic lines between the plastic and rigid regions must be straight, which partly dictates the general structure of the characteristic nets. The solutions employ Riemann’s method in regions where the characteristics of both families are curvilinear. Special solutions that do not have such regions are considered separately. In any case, the solutions are practically analytical. A numerical technique is only necessary to evaluate ordinary integrals. The solutions found determine the tool shapes that produce ideal flows. In addition, the distribution of pressure over the tool’s surface is calculated, which is important for predicting the wear of tools.

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Section: System Of Equations In Characteristic Coordinatesmentioning

confidence: 99%

“…This solution is for Tresca's yield criterion. Papers [5,6] have demonstrated that steady and nonsteady three-dimensional ideal flows exist for the material model comprising this yield criterion and its associated flow rule. Only this material model has been associated with the concept of bulk ideal flows for a long time.…”

Section: Introductionmentioning

confidence: 99%

General Planar Ideal Flow Solutions with No Symmetry Axis

Alexandrov,

Mokryakov

2023

Materials

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“…Hill (1967), who proposed the name ideal flow, generalized the effort to three-dimensional steady flow. Further extension to non-steady three-dimensional flow was made by Wienecke and Richmond (1997) and Richmond and Alexandrov (2002), but without specific application to forming.…”

Section: Introductionmentioning

confidence: 96%

Non-steady plane-strain ideal plastic flow

Chung

Lee

Richmond³

et al. 2005

International Journal of Plasticity

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“…Non-steady ideal membrane flows were then used as the basis of an inverse method for designing sheet forming processes [13]. The general equations of three-dimensional non-steady ideal deformations in a special Lagrangian coordinate system were derived and then specialized to the case of axisymmetric non-steady deformations [14]. Hill has extended the steady rigid-plastic ideal flow to steady elastic incompressible plastic ideal flow, but no specific applications of this theory have been presented [15].…”

Section: Introductionmentioning

confidence: 99%

New approach to the generation of orthogonal body-fitted meshes and its application to non-steady plane-strain ideal plastic flow

Cai

Wang

2020

J Braz. Soc. Mech. Sci. Eng.

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For rigid-perfect plastic solids, ideal plastic flow happens when all material elements follow minimum work paths and the two principal stretch lines are materially embedded. To describe its kinematics effectively, Lagrange convective orthogonal curvilinear system has to be constructed. Therefore, a numerical procedure for generating orthogonal body-fitted meshes has been developed and applied to non-steady plane-strain ideal plastic flow. The mapping between the Cartesian system and the orthogonal system is based on Laplace equations and Cauchy-Riemann condition. The numerical procedure takes into consideration of all the prescribed boundary curves instead of one as the boundary conditions, which has been adopted in earlier studies when using conventional characteristic line method. After each iteration step, the obtained boundary points are adjusted towards the orthogonality conditions until error tolerance requirement is met. For demonstration purpose, the numerical procedure is applied to the optimization of a bulk part under forging. The obtained orthogonal body-fitted meshes are in good agreement with previous methods. Then the optimal initial scale factor is also calculated based on this method and compared with earlier studies. Finally, the evolution of the intermediate shapes and frictional external boundary conditions is also calculated when there are no elastic dead zones.

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The theory of general and ideal plastic deformations of Treasca solids

Cited by 27 publications

References 14 publications

General Planar Ideal Flow Solutions with No Symmetry Axis

General Planar Ideal Flow Solutions with No Symmetry Axis

Non-steady plane-strain ideal plastic flow

New approach to the generation of orthogonal body-fitted meshes and its application to non-steady plane-strain ideal plastic flow

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