On the Use of Singular Yield Conditions and Associated Flow Rules

1953 ◽  
Vol 20 (3) ◽  
pp. 317-320
Author(s):  
William Prager
Keyword(s):  
Plastic Deformation ◽  
Closed Form ◽  
Circular Hole ◽  
Work Hardening ◽  
Yield Condition ◽  
Flow Rule ◽  
System Of Equations ◽  
Hardening Material ◽  
Perfectly Plastic ◽  
Yield Conditions

Abstract It is well known that the use of Tresca’s yield condition frequently leads to a simpler system of equations for the stresses in a plastic solid than the use of the yield condition of Mises. However, in most cases where Tresca’s yield condition has been used, the flow rule associated with the Mises condition has been retained. Following Koiter, it is shown that further simplification results from the use of the flow rule associated with the Tresca condition. The reason for this is discussed in connection with two examples concerning the finite enlargement of a circular hole in an infinite sheet of perfectly plastic or work-hardening material. The second example is probably the first nontrivial case in which a problem of finite plastic deformation of a work-hardening material has been treated in closed form by the use of incremental stress-strain relations.

A New Method of Analyzing Stresses and Strains in Work-Hardening Plastic Solids

1956 ◽  
Vol 23 (4) ◽  
pp. 493-496
Author(s):  
William Prager
Keyword(s):  
Work Hardening ◽  
Yield Condition ◽  
Transverse Load ◽  
Flow Rule ◽  
Total Stress ◽  
Stress Strain ◽  
Hardening Material ◽  
Characteristic Features ◽  
Associated Flow Rule ◽  
Yield Conditions

Abstract For work-hardening plastic solids, segmentwise linear yield conditions and the associated flow rules constitute a reasonable compromise between the mathematically convenient but physically unsound total stress-strain laws and the physically sound but mathematically inconvenient incremental laws. They allow total stress-strain laws to be used in the small, but retain the characteristic features of incremental laws in the large. The use of a segmentwise linear yield condition and the associated flow rule is illustrated by the analysis of the bending moments and deflections of a simply supported circular plate that is made of a work-hardening material and subjected to a uniformly distributed transverse load.

Finite Elastic-Plastic Expansion of a Cylindrical Tube

1973 ◽  
Vol 2 (4) ◽  
pp. 216-222
Author(s):  
B. Slevinsky ◽  
J. B. Haddow
Keyword(s):  
Plastic Deformation ◽  
Yield Condition ◽  
Cylindrical Tube ◽  
Flow Rule ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
End Conditions ◽  
Cylindrical Tubes ◽  
Limiting Case ◽  
Plastic Flow Rule

A numerical method for the analysis of the isothermal elastic-plastic expansion, by internal pressure, of cylindrical tubes with various end conditions is presented. The Tresca yield condition and associated plastic flow rule are assumed and both non-hardening and work-hardening tubes are considered with account being taken of finite plastic deformation. Tubes which undergo further plastic deformation on unloading are also considered. Expansion of a cylindrical cavity from zero radius in an infinite medium is considered as a limiting case.

Rigid-Plastic Response of Floating Plates

1988 ◽  
Vol 32 (03) ◽  
pp. 168-176
Author(s):  
John Anastasiadis ◽  
Paul C. Xirouchakis
Keyword(s):  
Rigid Body ◽  
Yield Condition ◽  
Body Motion ◽  
Phase Iii ◽  
Flow Rule ◽  
Rigid Plastic ◽  
Flexural Response ◽  
Tresca Yield Condition ◽  
Perfectly Plastic ◽  
R Ratio

This paper presents the exact formulation and solution for the static flexural response of a rigid perfectly plastic freely floating plate subjected to lateral axisymmetric loading. The Tresca yield condition is adopted with the associated flow rule. The plate response is divided into three phases: Initially the plate moves downward into the foundation as a rigid body (Phase I). Subsequently the plate deforms in a conical mode in addition to the rigid body motion (Phase II). At a certain value of the load a hinge-circle forms which may move as the pressure increases further (Phase III). The nature of the solution during the third phase depends upon the parameter α = a/R (ratio of radius of loaded area to the plate radius). When α = αs≅ 0.46 the hinge-circle remains stationary under increasing load. For α < αs the hinge-circle shrinks, whereas for α > αs the hinge-circle expands with increasing pressure. The application of the present results to the problem of laterally loaded floating ice plates is discussed.

Transient Thermal Stresses in Elasto-Plastic Discs

1969 ◽  
Vol 11 (4) ◽  
pp. 384-391 ◽  
Author(s):  
H. Odenö
Keyword(s):  
Temperature Distribution ◽  
Thermal Stresses ◽  
Plastic Material ◽  
Yield Condition ◽  
Transient Temperature ◽  
Flow Rule ◽  
Axially Symmetric ◽  
Exterior Surface ◽  
Tresca Yield Condition ◽  
Perfectly Plastic

A thin circular disc of elastic-perfectly plastic material, subjected to an axially symmetric transient temperature distribution, is treated analytically. All material parameters are assumed to be independent of the temperature. Poisson's ratio is taken to be one-half. The Tresca yield condition with associated flow rule is employed. The temperature distribution is that which appears when the outer rim surface of the disc receives a rapid temperature increase and it is solved approximately by the collocation method. The analysis shows that under certain circumstances, plastic deformation will occur in a moving annular region. This region starts to develop at the exterior surface and moves inward, while changing its width. After a certain finite time its width shrinks to zero. Except for a residual constant state of strain, the strain field is then again elastic. An application to the method of separating the ring and the shaft in a shrink-fit is carried out numerically. The residual stresses in the ring are calculated.

Plastic Deformation when Cutting into an Inclined Plane

1967 ◽  
Vol 9 (1) ◽  
pp. 1-10 ◽  
Author(s):  
W. B. Palmer
Keyword(s):  
Plastic Deformation ◽  
Plastic Flow ◽  
Work Hardening ◽  
Slip Line ◽  
Inclined Plane ◽  
Line Field ◽  
Hardening Material ◽  
Slip Line Field ◽  
Theory Of Plasticity

Plastic flow and tool forces were observed as an orthogonal tool cut slowly into an inclined plane of En 9 steel. A slip-line field is constructed which represents the observed flow, and on the basis of the theory of plasticity for work-hardening material estimates of stress are consistent with observed tool forces.

Incremental Stress-Strain Law Applied to Work-Hardening Plastic Materials

1959 ◽  
Vol 26 (4) ◽  
pp. 594-598
Author(s):  
Chintsun Hwang
Keyword(s):  
Work Hardening ◽  
Yield Condition ◽  
Transverse Load ◽  
Rational Approach ◽  
Total Stress ◽  
Stress Strain ◽  
Hardening Material ◽  
Plastic Materials ◽  
Von Mises ◽  
Supported Work

Abstract For problems involving work-hardening plastic materials, the incremental stress-strain law is considered to be a more rational approach than the conventional total stress-strain law. Up to the present the incremental stress-strain law was not subject to widespread use because it is mathematically inconvenient to handle. In this paper a method is developed in which the incremental law is applied to a work-hardening material in plane stress corresponding to the yield condition of von Mises. The method is illustrated by an analysis of the plastic bending of a simply supported work-hardening circular plate under uniformly distributed transverse load. The resulting difference-differential equations are solved by the NCR 304 digital computer.

The Elastic, Plastic Bending of a Simply Supported Plate

1961 ◽  
Vol 28 (3) ◽  
pp. 395-401 ◽  
Author(s):  
G. Eason
Keyword(s):  
Yield Condition ◽  
Constant Load ◽  
Flow Rule ◽  
Plastic Bending ◽  
Simply Supported ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
Perfectly Plastic ◽  
Elastic Perfectly Plastic ◽  
Von Mises

In this paper the problem of the elastic, plastic bending of a circular plate which is simply supported at its edge and carries a constant load over a central circular area is considered. The von Mises yield condition and the associated flow rule are assumed and the material of the plate is assumed to be nonhardening, elastic, perfectly plastic, and compressible. Stress fields are obtained in all cases and a velocity field is presented for the case of point loading. Some numerical results are given comparing the results obtained here with those obtained when the Tresca yield condition is assumed.

The Yield Condition and Flow Rule for a Metal Subjected to Finite Elastic Volume Change

1971 ◽  
Vol 93 (4) ◽  
pp. 708-712 ◽  
Author(s):  
J. B. Haddow ◽  
T. M. Hrudey
Keyword(s):  
Plastic Deformation ◽  
Hydrostatic Pressure ◽  
Plastic Flow ◽  
Volume Change ◽  
High Hydrostatic Pressure ◽  
Elastic Strain ◽  
Yield Condition ◽  
Flow Rule ◽  
Elastic Plastic ◽  
Plastic Flow Rule

A theory for elastic-plastic deformation with finite elastic strain is outlined. The results of this theory are specialized to consider a metal subjected to high hydrostatic pressure which produces finite elastic volume change. Drucker’s postulate is used to obtain the form of the yield condition and the associated plastic flow rule.

Plastic Wave Speeds in Isotropically Work-Hardening Materials

1977 ◽  
Vol 44 (1) ◽  
pp. 68-72 ◽  
Author(s):  
T. C. T. Ting
Keyword(s):  
Work Hardening ◽  
Wave Speed ◽  
Three Dimensional ◽  
Yield Condition ◽  
Elastic Response ◽  
Plastic Wave ◽  
Wave Speeds ◽  
Shear Wave Speed ◽  
Perfectly Plastic ◽  
Von Mises

Plastic wave speeds in materials whose elastic response is linear and isotropic while the plastic flow is incompressible and isotropically work-hardening are obtained. One of the three plastic wave speeds is identical to the elastic shear wave speed regardless of the form of the yield condition. The other two plastic wave speeds, cf and cs, are determined for materials obeying the von Mises yield condition. The dependence of cf and cs on the stress state and the direction of propagation is investigated in detail. The largest and smallest cf and cs, and the directions along which they occur are also presented. For materials obeying the Tresca’s yield condition, it is shown that one can obtain the corresponding results by simply specializing the results for the von Mises materials. Unlike in one-dimensional analyses where the plastic wave speed becomes zero for perfectly plastic solids, the three-dimensional analyses show that the ratio of cf to c1, where c1 is the elastic dilatation wave speed, is always larger than 3/7 for the von Mises materials and 1/2 for the Tresca’s materials. For most materials under moderate loadings, this ratio is much higher.

A Comparison of Flow and Deformation Theories in a Radially Stressed Annular Plate

1973 ◽  
Vol 40 (1) ◽  
pp. 283-287 ◽  
Author(s):  
P. C. T. Chen
Keyword(s):  
Internal Pressure ◽  
Closed Form ◽  
Deformation Theory ◽  
Annular Plate ◽  
Yield Condition ◽  
Flow Theory ◽  
Quantitative Comparison ◽  
The Other ◽  
Perfectly Plastic ◽  
Elastic Perfectly Plastic

Two mathematically consistent solutions to the strains and displacement in a partly plastic, annular plate stressed by internal pressure are obtained according to the deformation theory of Hencky and to the flow theory of Prandtl-Reuss. In both cases, the material is assumed to be elastic, perfectly plastic and obeying the Mises yield condition. It is shown that one solution is expressed in closed form and the other, in terms of simple integrals. A quantitative comparison of two theories is given and the effect of compressibility is discussed.

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