Upper-Bound Solutions of Axisymmetric Forming Problems—II

1964 ◽  
Vol 86 (4) ◽  
pp. 326-332 ◽  
Author(s):  
Shiro Kobayashi
Keyword(s):  
Velocity Field ◽  
Upper Bound ◽  
Wire Drawing ◽  
Velocity Fields ◽  
Upper Bound Method ◽  
Extended Treatment ◽  
Flow Through

The two developments in the upper-bound method are given. These are an extended treatment of the velocity field for outward flow of a cylindrical deforming region, and the introduction of a new velocity field for the flow through conical dies. The upper-bound calculation based on these velocity fields is presented in the examples of compression of a cylinder, wire-drawing, and bar-extrusion.

The Study of Distorted Grid Patterns for Flow-Through Conical Converging Dies by the Multi-triangular Velocity Field

1984 ◽  
Vol 106 (2) ◽  
pp. 150-160 ◽  
Author(s):  
J. Pan ◽  
W. Pachla ◽  
S. Rosenberry ◽  
B. Avitzur
Keyword(s):  
Velocity Field ◽  
Upper Bound ◽  
Cone Angle ◽  
Wire Drawing ◽  
Perfectly Plastic ◽  
Distorted Grid ◽  
Upper Bound Technique ◽  
Independent Process ◽  
Flow Through ◽  
Reduction In Area

A variety of velocity fields may be used to analyze the intermediate and final distorted grids for the so-called “flow-through” metal forming processes such as wire drawing, rolling, extrusion, etc. In this paper the triangular velocity field describes the flow of homogeneous, perfectly plastic Mises’ material through a conical converging die. The traditional triangular velocity field was treated and the solution extended. The shape of the distorted grids was uniquely determined by the minimization of the power (drawing or extrusion stresses) required to cause its distortion for a given set of independent process parameters, i.e., process geometry-reduction in area and semi-cone angle, and friction. Actual power (forming stress requirements) was estimated by the upper-bound technique. For the unitriangular velocity field, the power was minimized with respect to the shape of the workpiece (the shape of the triangle). For the multitriangular velocity field, the power was minimized with respect to the shape and the number of triangles. Further, the number of triangles was treated as a real number. Thus, the accurate lower upper-bound was found and the reasonable solution in predicting real distortion grid patterns was then obtained. The analysis determines the severity of the distortion as a function of process geometry and friction.

Analysis of Strip Rolling by the Upper Bound Approach

1987 ◽  
Vol 109 (4) ◽  
pp. 338-346 ◽  
Author(s):  
B. Avitzur ◽  
W. Gordon ◽  
S. Talbert
Keyword(s):  
Velocity Field ◽  
Rigid Body ◽  
Upper Bound ◽  
Velocity Fields ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Total Power ◽  
Strip Rolling ◽  
Upper Bound Technique ◽  
Independent Process

The process of strip rolling is analyzed using the upper bound technique. Two triangular velocity fields, one with triangles in linear rigid body motion and the other with triangles in rotational rigid body motion, are developed. The total power is determined as a function of the four independent process parameters (relative thickness, reduction, friction and net front-back tension). The results of these two velocity fields are compared with the established solution from Avitzur’s velocity field of continuous deformation. Upon establishing the validity of the triangular velocity field as an approach to the strip rolling problem, recommendations are suggested on how this approach can be used to study the split end or alligatoring defect.

Direct Upper Bound Solution to Rectangular-to-Polygonal Extrusion Through Square Die

1999 ◽  
Vol 121 (2) ◽  
pp. 195-201 ◽  
Author(s):  
S. K. Sahoo ◽  
P. K. Kar ◽  
K. C. Singh
Keyword(s):  
Steady State ◽  
Velocity Field ◽  
Upper Bound ◽  
Velocity Fields ◽  
Extrusion Pressure ◽  
Admissible Velocity ◽  
Bound Solution ◽  
Upper Bound Solution ◽  
Aspect Ratios

This paper is concerned with an attempt to find an upper bound solution for the problems of steady-state extrusion of asymmetric polygonal section bars through rough square dies. A class of kinematically admissible velocity fields is examined, reformulating the SERR technique, to get the velocity field that gives the lowest upper bound. This velocity field is utilized to compute the non-dimensional average extrusion pressure at various area reductions for different billet aspect ratios.

Limit Analysis of Flow Through Inclined Converging Planes

1980 ◽  
Vol 102 (2) ◽  
pp. 109-117 ◽  
Author(s):  
M. Kiuchi ◽  
B. Avitzur
Keyword(s):  
Lower Bound ◽  
Plane Strain ◽  
Upper Bound ◽  
Limit Analysis ◽  
Velocity Fields ◽  
The Body ◽  
Design Flow ◽  
Upper And Lower Bounds ◽  
Lower Upper ◽  
Flow Through

A variety of mathematical models may be used to analyze plastic deformation during a metal-forming process. One of these methods—limit analysis—places the estimate of required power between an upper bound and a lower bound. The upper- and lower-bound analysis are designed so that the actual power or forming stress requirement is less than that predicted by the upper bound and greater than that predicted by the lower bound. Finding a lower upper-bound and a higher lower-bound reduces the uncertainty of the actual power requirement. Upper and lower bounds will permit the determination of such quantities as required forces, limitations on the process, optimal die design, flow patterns, and prediction and prevention of defects. Fundamental to the development of both upper-bound and lower-bound solutions is the division of the body into zones. For each of the zones there is written either a velocity field (upper bound) or a stress field (lower bound). A better choice of zones and fields brings the calculated values closer to actual values. In the present work, both upper- and lower-bound solutions are presented for plane-strain flow through inclined converging dies. For the upper bound, trapezoidal velocity fields, uni-triangular velocity fields, and multi-triangular velocity fields have been dealt with and the solutions compared to previously published work on cylindrical velocity fields. It was found that in different domains of the various combinations of the process parameters, different patterns of flow (cylindrical, triangular, etc.) provide lower upper-bound solutions. The lower-bound solution for plane-strain flow through inclined converging planes is newly developed.

An Upper Bound Solution for a Two-Layer Cylinder Subject to Compression and Twist

2007 ◽  
Vol 345-346 ◽  
pp. 37-40 ◽  
Author(s):  
Gow Yi Tzou ◽  
Sergei Alexandrov
Keyword(s):  
Velocity Field ◽  
Upper Bound ◽  
Additional Condition ◽  
Velocity Fields ◽  
Upper Bound Theorem ◽  
Predictive Capacity ◽  
Admissible Velocity ◽  
Bound Solution ◽  
Bound Theorem ◽  
Formal Requirements

The choice of a kinematically admissible velocity field has a great effect on the predictive capacity of upper bound solutions. It is always advantageous, in addition to the formal requirements of the upper bound theorem, to select a class of velocity fields satisfying some additional conditions that follow from the exact formulation of the problem. In the case of maximum friction law, such an additional condition is that the real velocity field is singular in the vicinity of the friction surface. In the present paper this additional condition is incorporated in the class of kinematically admissible velocity fields chosen for a theoretical analysis of two - layer cylinders subject to compression and twist. An effect of the angular velocity of the die on process parameters is emphasized and discussed.

A Three-Dimensional Upper Bound Elemental Technique for Forging Analysis

1996 ◽  
Vol 210 (4) ◽  
pp. 353-359 ◽  
Author(s):  
R S Lee ◽  
C T Kwan
Keyword(s):  
Velocity Field ◽  
Upper Bound ◽  
Metal Flow ◽  
Three Dimensional ◽  
Velocity Fields ◽  
Connecting Rod ◽  
Total Energy Consumption ◽  
Pure Lead ◽  
Forming Load ◽  
Theoretical Predictions

In this paper, two kinematically admissible velocity fields are derived for the proposed three-dimensional arbitrarily triangular and trapezoidal prismatic upper bound elemental technique (UBET) elements. These elements are applied to the portions between the circular shaped part and the straight rod part with three-dimensional metal flow in connecting rod forging, and then the capability of the proposed elements are shown. From the derived velocity fields, the upper bound loads on the upper die and the velocity field are determined by minimizing the total energy consumption with respect to some chosen parameters. Experiments with connecting rod forging were carried out with commercial pure lead billets at ambient temperature. The theoretical predictions of the forming load is in good agreement with the experimental results. It is shown that the proposed UBET elements in this work can effectively be used for the prediction of the forming load and velocity field in connecting rod forging.

Upper-Bound Solutions of Axisymmetric Forming Problems—I

1964 ◽  
Vol 86 (2) ◽  
pp. 122-126 ◽  
Author(s):  
Shiro Kobayashi
Keyword(s):  
Upper Bound ◽  
Velocity Fields ◽  
Curved Surfaces ◽  
Admissible Velocity ◽  
Extended Treatment ◽  
Discontinuity Surfaces ◽  
Cylindrical Parts ◽  
Conical Surfaces

The Kudo method for obtaining average pressures in some axisymmetric forming problems by the use of velocity fields having conical surfaces as discontinuity surfaces is reviewed in the paper. Following this it is shown that the extended treatment of admissible velocity fields is possible if the conical surfaces suggested by Kudo are replaced by curved surfaces. The application of these velocity fields to forming problems is then illustrated by simple examples of compression of cylindrical parts, and the improved upper bound to forming pressures is obtained.

A Generalized Velocity Field for Plane Strain Extrusion Through Arbitrarily Curved Dies

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
H. Haghighat ◽  
P. Amjadian
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Angular Velocity ◽  
Plane Strain ◽  
Upper Bound ◽  
Velocity Fields ◽  
Finite Element Code ◽  
Admissible Velocity ◽  
Paper Plane ◽  
Good Agreement

In this paper, plane strain extrusion through arbitrarily curved dies is investigated analytically, numerically, and experimentally. Two kinematically admissible velocity fields based on assuming proportional angles, angular velocity field, and proportional distances from the midline in the deformation zone, sine velocity field, are developed for use in upper bound models. The relative average extrusion pressures for the two velocity fields are compared to each other and also with the velocity field of a reference for extrusion through a curved die. The results demonstrate that the angular velocity field is the best. Then, by using the developed analytical model, optimum die lengths which minimize the extrusion loads are determined for a streamlined die and also for a wedge shaped die. The corresponding results for those two die shapes are also determined by using the finite element code and by doing some experiments and are compared with upper bound results. These comparisons show a good agreement.

Plane Strain Forging—A Lower Upper Bound Approach

1975 ◽  
Vol 97 (1) ◽  
pp. 119-124 ◽  
Author(s):  
V. Nagpal ◽  
W. R. Clough
Keyword(s):  
Velocity Field ◽  
Process Parameters ◽  
Upper Bound ◽  
Dead Zone ◽  
Incompressible Material ◽  
Velocity Fields ◽  
Admissible Velocity ◽  
Special Cases ◽  
Rectangular Strip ◽  
General Velocity

A general kinematically admissible velocity field applicable to forging of a rectangular strip of a incompressible material is presented. Generalized shape of any dead zone, if assumed, can be obtained in terms of process parameters from this velocity field. Two different upper bound solutions for average forging pressure are obtained from simple velocity fields which are special cases of proposed general velocity field. Numerical results of the solutions show improvement over previous upper bound solutions published in literature over a certain range of process parameters.

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