Plane strain drawing and extrusion of a rigid-perfectly plastic material through concave dies

1968 ◽  
Vol 10 (4) ◽  
pp. 231-238 ◽  
Author(s):  
R. Sowerby ◽  
W. Johnson ◽  
S.K. Samanta
Keyword(s):  
Plane Strain ◽  
Plastic Material ◽  
Perfectly Plastic

The Upper Bound Approach to Plane Strain Problems Using Linear and Rotational Velocity Fields—Part II: Applications

1986 ◽  
Vol 108 (4) ◽  
pp. 307-316 ◽  
Author(s):  
Betzalel Avitzur ◽  
Waclaw Pachla
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
Metal Cutting ◽  
Failure Modes ◽  
Plastic Material ◽  
Rotational Velocity ◽  
Optimization Procedure ◽  
Velocity Fields ◽  
Perfectly Plastic ◽  
Strip Drawing

Following Part I which investigated an upper bound approach to plane strain deformation of a rigid, perfectly plastic material, this Part II considers the same approach as applied to actual forming operations. The processes of drawing and extrusion, of metal cutting and of rolling are analyzed, and explicit equations are developed to calculate the surfaces of velocity discontinuity (shear boundaries), velocity discontinuities, and the upper bound on power for these processes. Both the simple, unielement velocity fields as well as the more complex multielement fields are explored. The upper bound solution is shown to be a function of the independent (input) and pseudoindependent (assumed) process parameters as minimized by an optimization procedure. Rules concerning the assumption of pseudoindependent parameters are presented and the optimization procedure is discussed. Final conclusions lead the way for the application of upper bound analyses to such industrial processes as sheet and strip drawing, extrusion, forging, rolling, leveling, ironing and machining, and to the investigation of such flow failure modes as central bursting, piping and end splitting (alligatoring).

On the stability of supported excavations

1984 ◽  
Vol 21 (2) ◽  
pp. 338-348 ◽  
Author(s):  
A. M. Britto ◽  
O. Kusakabe
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
Plastic Material ◽  
Cohesive Soil ◽  
Perfectly Plastic ◽  
Centrifuge Tests ◽  
Bentonite Slurry ◽  
Undrained Conditions ◽  
Elastic Perfectly Plastic ◽  
The Stability

Unsupported plane strain trenches and axisymmetric shafts cannot be excavated to great depths in a purely cohesive soil. Therefore, it is standard practice to provide some form of support. Timber supports with struts are conventional and quite common. Bentonite slurry support has become more popular in recent years especially in the construction of diaphragm walls. In this paper the effect of rigid lateral support and slurry support on the stability (mode of failure) for both plane strain and axisymmetric excavations are investigated under undrained conditions. When immediate failure is of interest in saturated clays the changes in the water content can be neglected and the soil can be treated as a [Formula: see text] material. For the purposes of the analyses presented here the lateral support is assumed to be rigid and the soil is idealized as an elastic perfectly plastic material with cohesion Cu. The results from upper bound calculations, finite element collapse analyses, and centrifuge tests are presented. The analogy between deep footing failure and base failure of excavation allows the solutions for the footing problem to be interpreted for trench excavations. It is found that slurry support is more effective than rigid lateral support for axisymmetric excavations. The slurry support reduces the amount of surface settlement and also stabilises the trench against base failure. For excavations with rigid lateral support the possibility of base failure is greatly increased. The results are presented in the form of stability charts. Keywords: limit analysis, slurry support, stability number, supported excavation, upper bound solution.

Plane-Strain Compression of a Three Layer Strip Containing Viscoplastic Material with Saturation Stress

2009 ◽  
Vol 623 ◽  
pp. 89-103 ◽  
Author(s):  
Wiktoria Miszuris
Keyword(s):  
Plane Strain ◽  
Plastic Material ◽  
Plane Strain Compression ◽  
Material Structure ◽  
Saturation Stress ◽  
Viscoplastic Material ◽  
Closed Form Solutions ◽  
Layer Material ◽  
Perfectly Plastic ◽  
Friction Surfaces

The plane strain compression of a long symmetric strip consisted of a three layer material between rigid, parallel, rough plates is under consideration. Two possible geometrical configurations of the layers are examined (a) a viscoplastic material is situated between two layers consisting of a rigid/perfectly plastic material, (b) a rigid/perfectly plastic material lies between two viscoplastic layers. It is assumed throughout the paper that the viscoplastic law is bounded in that sense that it reaches its critical value (saturation stress) as the strain rate tends to infinity. Exploiting closed form solutions obtained, qualitative differences between them and the known from literature solutions for three layer material structure with classic viscoplastic material are discussed. Asymptotic behaviour of solutions in the vicinity of maximum friction surfaces is analysed for any configuration.

The Upper Bound Approach to Plane Strain Problems Using Linear and Rotational Velocity Fields—Part I: Basic Concepts

1986 ◽  
Vol 108 (4) ◽  
pp. 295-306 ◽  
Author(s):  
Betzalel Avitzur ◽  
Waclaw Pachla
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
Rotational Motion ◽  
Plastic Material ◽  
Rotational Velocity ◽  
Linear Motion ◽  
Velocity Discontinuity ◽  
Perfectly Plastic ◽  
Plane Strain Deformation ◽  
Strip Drawing

This paper investigates an upper bound approach to plane strain deformation of a rigid, perfectly plastic material. In this approach the deformation region is divided into a finite number of rigid triangular bodies that slide with respect to one another. Neighboring rigid body zones are analyzed in specific cases where the zones are (1) both in rotational motion, (2) one in linear, the other in rotational motion and (3) both in linear motion. Specific equations are presented that describe surfaces of velocity discontinuity (shear boundaries) between the moving bodies, and the velocity discontinuities and shear power losses for each of the three cases. The shape of the surface of velocity discontinuity is uniquely determined by the velocity ratios of neighboring bodies, their relative directions of motion and, where applicable, the positions of their centers of rotation. Where one or both neighboring bodies exhibit rotational motion, the surface of velocity discontinuity is found to be a cylindrical surface. In the case of two neighboring bodies, each with linear motion, the surface of velocity discontinuity is found to be planar. The velocity discontinuity is found to be constant along the entire surface of velocity discontinuity. The characteristics of the surfaces of velocity discontinuity in plane strain deformation are investigated. The upper-bound approach to plane strain problems can be successfully adapted to real metal forming processes, including sheet and strip drawing, extrusion, forging, rolling, leveling, ironing, and machining.

scholarly journals Plastic forming processes of transverse non-homogeneous composite metallic sheets

2021 ◽  
Vol 11 (1) ◽  
pp. 294-302
Author(s):  
Gal Davidi
Keyword(s):  
Inner Core ◽  
Plastic Material ◽  
Radial Solution ◽  
Algebraic Differential Equation ◽  
Forming Process ◽  
Perfectly Plastic ◽  
Plastic Forming ◽  
Validity Limit ◽  
Significant Achievement

Abstract In this work an analysis of the radial stress and velocity fields is performed according to the J 2 flow theory for a rigid/perfectly plastic material. The flow field is used to simulate the forming processes of sheets. The significant achievement of this paper is the generalization of the work by Nadai & Hill for homogenous material in the sense of its yield stress, to a material with general transverse non-homogeneity. In Addition, a special un-coupled form of the system of equations is obtained where the task of solving it reduces to the solution of a single non-linear algebraic differential equation for the shear stress. A semi-analytical solution is attained solving numerically this equation and the rest of the stresses term together with the velocity field is calculated analytically. As a case study a tri-layered symmetrical sheet is chosen for two configurations: soft inner core and hard coating, hard inner core and soft coating. The main practical outcome of this work is the derivation of the validity limit for radial solution by mapping the “state space” that encompasses all possible configurations of the forming process. This configuration mapping defines the “safe” range of configurations parameters in which flawless processes can be achieved. Several aspects are researched: the ratio of material's properties of two adjacent layers, the location of layers interface and friction coefficient with the walls of the dies.

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