Zastosowanie metody perturbacji do analizy plastycznego płynięcia ośrodka typu Coulomba w stanie osiowej symetrii
Application of the perturbation method to the analysis of the plastic flow of a Coulomb type body in an axially symmetric state
This paper is concerned with the plastic f10w of a perfect rigid-plastic medium of the Coulomb type in the case of axial symmetry. The influence of the gravity force and the adjoint law of f10w being assumed as a physical law. The solution is obtained by assuming that the state of stress corresponds to the corners or the Tresca (Coulomb) hexagon for which the Haar-Karman principle is satisfied. Then, the static and kinematic equations are of the hyperbolic type and the characteristics of the two set s of equations coincide.
The problems are solved by the perturbation method with reference to the initial characteristics as proposed in Ref. 9 for a Tresca body and adopted for a body of the Coulomb type.
The fields of stress and velocity are obtained for the problems under consideration. It is found that in some strain regions the field of velocity does not satisfy the relations resulting from the yield condition and the associate law of f10w for the Haar-Karman corners of the Tresca hexagon, and requiring that the principal strain rates have different signs in an axial plane. Then, the static field gives a lower estimate of the limit lo ad and the kinematic field an upper one. It is also found by numerical computation that the difference between the two estimates of the limit load may be considerable.
References
A.D. Cox, G. Eason, H.G. Hopkins, Axially symmetric plastic deformations in soils, Phil. Trans. Royal Soc. London, Ser. A, 1036, 254 (1961), 1–45.
R.T. Shield, On the plastic flow of metals under conditions of axial symmetry, Proc. Roy. Soc., Ser. A, 233 (1955), 267 -287.
В. Г. Верезапцев, Осесимметричная задача теории предельного равновесия сыпучей среды. Москва 1952.
Д. Д. Ивлев, Приближенное решение методом малого параметра плоских упруго-пластических задач теории идеальной пластичности, Вест. Моск. уиив., Серия мех. астр. физ. хим., 5 (1957), 17-26.
A. L. Kuznetsov, The problem of torsion and plane strain of nonhomogeneous plastic solids, Arcn. Mech. Stos., 10 (1958), 447-462.
A.J.M. Spencer, Perturbation methods in plasticity, I. Plane strain of non-homogeneous plastic solids, J. Mech. Phys. Sol., 9 (1961), 279-288.
A.J.M. Spencer, Perturbation methods in plasticity. II. Plane strain of slightly-irregular bodies, S. Mech. Phys. Sol., 10 (1962), 17-27.
A.J.M. Spencer, Perturbation methods in plasticity, III Plane strain of ideal soils and plastic solids with body forces, S. Mech. Phys. Sol., 10 (1962), 165-178.
A.J. M. Spencer, The approximate solution of certain problems of axially-symmetric plastic flow, J. Mech. Phys. Sol., 12 (1964), 231-244.
O. Richmond, H.L. Morrison, Application of a perturbation technique based on the method of charakteristics to axisymmetric plasticity, J. Appi. Mech., 38 (1968).
Z. Mróz, Graphical solution of axially symmetric problems of plastic flow, J. Appi. Mat. Phys. (ZAMP), 18 (1967), 219-236.
Z. Mróz, K. Kwaszczynńska, Axially symmetric plastic flow of soils treated by the graphical method, Arch. Inzyn. Wow., 14 (1968), 27-37.
K. Kwaszczyńska, Z. Mróz, A. Drescher, Analysis of compression of short cylindres of Coulomb material, Int. J. Mech. Sci. Perg. Press, 11 (1969), 145-158.
R.T. Shield, Mixed boundary value problems in soil mechanics, Quart. Appi. Mat., 11 (1953), 61-75.
Praca zbiorowa pod redakcja Z. ORSZAKA, Teoria plastycznoici, PWN, Warszawa 1965.