Axially symmetric plastic deformations in soils

1961 ◽  
Vol 254 (1036) ◽  
pp. 1-45 ◽  
Keyword(s):  
Plastic Flow ◽  
Plane Surface ◽  
Velocity Fields ◽  
Natural Soil ◽  
Flow Rule ◽  
Foundation Engineering ◽  
Axially Symmetric ◽  
Plastic Deformations ◽  
Von Karman ◽  
Von Kármán

A theoretical investigation is given of quasi-static axially symmetric plastic deformations in soils.The mechanical behaviour of a natural soil is approximated by that of an ideal soil which obeys Coulomb’s yield criterion and associated flow rule, with restriction to rigid, perfectly plastic deformations. There are considerable variations in the structure of the associated stress and velocity field equations for the various plastic regimes, but it is noteworthy that real families of characteristics occur in all non-trivial cases. Attention is focused on those plastic régimes agreeing with the heuristic hypothesis of Haar & von Kármán as being seemingly of application to certain classes of problems, in particular to those of indentation. The stress and velocity fields are then hyperbolic with identical families of characteristics, and the stress field is statically determinate under appropriate boundary conditions. In applications of the theoretical analysis, attention is confined to situations involving only the Haar & von Kármán plastic regimes. First, possible velocity fields are obtained for the incipient plastic flow of a right circular cylindrical sample of soil subjected to uni-axial compressive stress parallel to its axis. Secondly, a complete solution is obtained for the incipient plastic flow in a semi-infinite region of soil, bounded by a plane surface, due to load applied through a flat-ended,smooth, rigid, circular cylinder; numerical results obtained for this problem include the variation of yield-point load with angle of internal friction of the ideal soil. These applications relate to problems of the mechanical testing of soil samples and of load-bearing capacity in foundation engineering.

On the plastic flow of metals under conditions of axial symmetry

1955 ◽  
Vol 233 (1193) ◽  
pp. 267-287 ◽  
Keyword(s):  
Stress Field ◽  
Plastic Flow ◽  
Plane Surface ◽  
Yield Criterion ◽  
Circumferential Stress ◽  
Flow Rule ◽  
Axially Symmetric ◽  
Principal Stresses ◽  
Infinite Body ◽  
Plastic Stress

This paper is concerned with the axially symmetric plastic flow of a rigid-plastic nonhardening material which obeys the Tresca yield criterion of constant maximum shearing stress and the associated flow rule. A general discussion of the basic equations is given. The discussion shows that the hypothesis of Haar and von Kármán is likely to be of great importance in the solution of axially symmetric problems. This conclusion is substantiated by the remainder of the work which considers problems in which the hypothesis is satisfied, i.e. problems in which the circumferential stress is equal to one of the principal stresses in the meridional planes. Possible plastic velocity fields in a circular cylindrical bar stressed to yielding in compression or tension are obtained in §3. Section 4 examines plastic stress fields in the neighbourhood of stress-free conical surfaces. In the final sections of the paper, the plastic stress field and a permissible deformation mode for the problem of the indentation of the plane surface of a semi-infinite body by a circular flat-ended rigid punch are obtained. It is shown that the plastic stress field near the punch can be extended into the rigid region without violating the yield criterion.

Large Symmetric Deflections of Annular Plates

1958 ◽  
Vol 25 (4) ◽  
pp. 449-452
Author(s):  
G. A. Wempner ◽  
R. Schmidt
Keyword(s):  
Series Solution ◽  
Numerical Procedure ◽  
Axially Symmetric ◽  
Annular Plates ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán

Abstract This paper presents a series solution of the von Karman equations for axially symmetric deflections of annular plates. A numerical procedure is described for evaluating the coefficients in the series. Results obtained by this procedure are given for a plate subjected to axial edge loads. A method is presented for demonstrating convergence and appraising the errors.

scholarly journals Elastoplastic deformation of a rotating hollow cylinder with a rigid casing

2019 ◽  
pp. 120-135
Author(s):  
A N Prokudin ◽  
S V Firsov
Keyword(s):  
Plastic Flow ◽  
Hollow Cylinder ◽  
Outer Surface ◽  
Rotation Speed ◽  
Exact Analytical Solution ◽  
Plastic Region ◽  
Flow Rule ◽  
Elastoplastic Material ◽  
Plastic Deformations ◽  
Rotating Hollow Cylinder

A rotating hollow cylinder with fixed ends is considered, the inner surface of which is free of stresses, and the outer one is fixed from radial movements. It is assumed that the cylinder is made of an ideal isotropic elastoplastic material, and the deformations in it are small and represent the sum of elastic and plastic deformations. Stresses are associated with elastic deformations by Hooke's law. Plastic deformations are determined using the Tresca - Saint-Venant condition and the plastic flow rule associated with it. The cylinder rotation speed first monotonically increases to a maximum value, and then decreases to zero. By using the elastic solution, the dependence is found for the critical rotation speed at which the plastic flow begins. It is established that, depending on the thickness of the cylinder and the Poisson's ratio, plastic flow can begin, either on the inner or on the outer surface of the cylinder. In addition, 3 plastic regions appear in the cylinder at the loading stage, and 4 plastic regions appear at the unloading stage. These regions correspond to two faces and two edges of the Treska prism. For each plastic region, an exact analytical solution of the determining system of equations is found. The system of conditions at the boundaries between the regions providing continuity of the obtained solutions throughout the cylinder is given. Two cases are considered, i.e. the case with a plastic flow which starts first on the inner, and then on the outer surface of the cylinder. Analytical expressions are obtained for rotational speeds at which new regions appear. The relationship between the nucleation rates of the secondary and primary plastic flow is established. The value of the maximum rotation speed sufficient for a complete transition of the cylinder to the state of the secondary plastic flow was also found. It has been revealed that the adding of a rigid casing can significantly increase the resource of an exploited part.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

Sign in / Sign up

Export Citation Format

Share Document