Extension of the Upper Bound Method to Include Estimation of Stresses

1991 ◽  
Vol 58 (2) ◽  
pp. 493-498 ◽  
Author(s):  
A. Azarkhin ◽  
O. Richmond
Keyword(s):  
Plastic Deformation ◽  
Stress Field ◽  
Upper Bound ◽  
Plastic Material ◽  
Approximate Solutions ◽  
Incompressible Material ◽  
Bulk Deformation ◽  
Upper Bound Method ◽  
Convenient Tool ◽  
Perfectly Plastic

The upper bound method is a convenient tool for evaluating the rate of work in processes involving predominantly plastic deformation of rigid/perfectly plastic material. Since the rate of work for an incompressible material depends only on the deviator portion of the stress, the hydrostatic portion does not enter the formulation and the stress field is not determined. Here we show that this limitation can be overcome by adding a relatively simple postprocessing procedure. We then apply this technique to examples of rigid asperities penetrating a plastic material undergoing subsurface bulk deformation and compare our results with previous approximate solutions.

A Generalization of the Upper Bound Method With Particular Reference to the Problem of a Ploughing Indenter

1989 ◽  
Vol 56 (1) ◽  
pp. 10-14 ◽  
Author(s):  
A. Azarkhin ◽  
O. Richmond
Keyword(s):  
Finite Element ◽  
Upper Bound ◽  
Plastic Material ◽  
Approximate Solutions ◽  
Upper Bound Method ◽  
Numerical Examples ◽  
Perfectly Plastic

An algorithm based on a combination of the upper bound method and finite element repesentation has been developed. The algorithm is applied to the problem of a rigid indenter ploughing through a rigid/perfectly-plastic material. Numerical examples are given and the results are compared with previous approximate solutions. Limitations of the upper bound method are discussed.

On Friction of Ploughing by Rigid Asperities in the Presence of Straining—Upper Bound Method

1990 ◽  
Vol 112 (2) ◽  
pp. 324-329 ◽  
Author(s):  
A. Azarkhin ◽  
O. Richmond
Keyword(s):  
Finite Element ◽  
Friction Factor ◽  
Upper Bound ◽  
Plastic Material ◽  
High Accuracy ◽  
The Other ◽  
Periodic Array ◽  
Combined Loading ◽  
Upper Bound Method ◽  
Perfectly Plastic

Upper bound applications traditionally assume that a rigid/perfectly-plastic material moves by rigid blocks, creating discontinuities of velocity at the interfaces between the blocks. In the present version, the elements (blocks) are plastically deformable and there are no velocity discontinuities between adjacent sides. Since this modification incorporates major features of finite element representation employing arbitrary cells, it allows the use of many parameters for minimization, thus achieving high accuracy. On the other hand, it retains the advantage of upper bound techniques in that the incremental procedure for loading is not necessary, and the results for steady processes are obtained directly. Some energy statements for combined loading are derived and a technique for calculating the ploughing force is presented. Examples for a single fully embedded rigid pyramid and a periodic array of asperities ploughing through the rigid/perfectly plastic material in the presence of subsurface straining are given. The friction factor decreased as the rate of subsurface straining increased, as the pyramid angle of the asperities increased, and as the distance between asperities increased.

A Complete Perfectly Plastic Solution for the Mode I Crack Problem Under Plane Stress Loading Conditions

2006 ◽  
Vol 74 (3) ◽  
pp. 586-589 ◽  
Author(s):  
David J. Unger
Keyword(s):  
Stress Field ◽  
Plane Stress ◽  
Plastic Material ◽  
Crack Problem ◽  
Yield Condition ◽  
Internal Crack ◽  
Mode I ◽  
Mode I Crack ◽  
Loading Conditions ◽  
Perfectly Plastic

A continuous stress field for the mode I crack problem for a perfectly plastic material under plane stress loading conditions has been obtained recently. Here, a kinematically admissible velocity field is introduced, which is compatible with the continuous stress field obtained earlier. By associating these two fields together, it is shown that they constitute a complete solution for the uncontained plastic flow problem around a finite length internal crack, having a positive rate of plastic work. The yield condition employed is an alternative criterion first proposed by Richard von Mises in order to approximate the plane stress Huber-Mises yield condition, which is elliptical in shape, to one that is composed of two intersecting parabolas in the principal stress plane.

Limit Design of a Full Reinforcement for a Circular Cutout in a Uniform Slab

1952 ◽  
Vol 19 (3) ◽  
pp. 397-401
Author(s):  
H. J. Weiss ◽  
W. Prager ◽  
P. G. Hodge
Keyword(s):  
Upper Bound ◽  
Plastic Material ◽  
Yield Criterion ◽  
Curved Beam ◽  
Shearing Stress ◽  
Concentric Ring ◽  
Perfectly Plastic ◽  
Circular Cutout

Abstract A thin square slab with a central circular cutout reinforced by a concentric ring is subjected to uniform tensions Tx and Ty on the exterior edges. It is desired to determine the dimensions of the reinforcement if the slab is not to collapse under any load which could be supported by a similar slab without any cutout or reinforcement. It is assumed that the slab and reinforcement are made of a perfectly plastic material which satisfies the Tresca yield criterion of maximum shearing stress, and that the dimensions of the reinforcement are such that it may reasonably be approximated by a curved beam. Under these assumptions, an upper bound on the necessary thickness of the reinforcement for any given radius is obtained. Certain practical limitations of the theory are discussed.

Causes and Consequences of Nonuniqueness in an Elastic/Perfectly-Plastic Truss

1986 ◽  
Vol 53 (2) ◽  
pp. 235-241 ◽  
Author(s):  
P. G. Hodge ◽  
K.-J. Bathe ◽  
E. N. Dvorkin
Keyword(s):  
Plastic Deformation ◽  
Plastic Material ◽  
Complete Solution ◽  
Physical Modelling ◽  
Equilibrium Equations ◽  
First Order ◽  
Perfectly Plastic ◽  
Points Of View ◽  
Elastic Perfectly Plastic

A complete solution to collapse is given for a three-bar symmetric truss made of an elastic/perfectly-plastic material, using linear statics and kinematics, and the solution is found to be partially nonunique in the range of contained plastic deformation. The introduction of a first-order deviation from symmetry and/or the inclusion of first-order nonlinear terms in the equilibrium equations is found to restore uniqueness. The significance of these effects is analyzed and discussed from mathematical, physical, modelling, computational, and engineering points of view.

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.

On the Prediction of the Strain Rate Intensity Factor in Metal Forming Processes

2008 ◽  
Author(s):  
Elena Lyamina
Keyword(s):  
Intensity Factor ◽  
Strain Rate ◽  
Closed Form ◽  
Upper Bound ◽  
Approximate Solutions ◽  
Closed Form Solutions ◽  
Perfectly Plastic ◽  
Strain Rate Intensity Factor ◽  
Rate Intensity ◽  
Strain Rate Intensity

The strain rate intensity factor is the coefficient of the principal singular term in a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. Such singular behaviour occurs in the case of several rigid plastic models (rigid perfectly plastic solids, the double-shearing model, the double slip and rotation model, some of viscoplastic models). Since it is only possible to introduce the strain rate intensity factor for singular velocity fields, it is obvious that standard finite element codes cannot be used to calculate it. The currently available distributions of the strain rate intensity factor have been found from closed form solutions or with the use of simple approximate solutions (for instance upper bound solutions). Closed form solutions are available for boundary value problems with simple geometry (flow through infinite rough channels, compression of infinite layers between rough plates and so on) and, therefore, are mostly of academic interest. Simple approximate solutions can predict general tendencies in the distribution of the strain rate intensity factor but cannot predict its distribution with a sufficient accuracy for industrial applications. For, the strain rate intensity factor reflects a very local effect inherent in the velocity field whereas simple approximate methods, such as the upper bound method, estimate global parameters, such as the limit load. The purpose of the present research is to propose a special numerical technique for calculating the strain rate intensity factor in the case of plane strain deformation of rigid perfectly plastic materials and to verify it by means of comparison with an analytical solution. The technique is based on the method of Riemann.

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