ON THE GENERAL EQUATIONS, DOUBLE HORMONIC EQUATION AND EIGEN-EQUATION IN THE PROBLEMS OF IDEAL PLASTICITY

摘要/Abstract

摘要： In this paper the outcome ofaxisymmetric problems of ideal plasticity in paper [39], [19] and [37] is directly extended to the three-dimensional problems of ideal plasticity, and get at the general equation in it. The problem of plane strain for material of ideal rigid-plasticity can be solved by putting into double harmonic equation by famous Pauli matrices of quantum electrodynamics different from the method in paper [7]. We lead to the eigen equation in the problems of ideal plasticity, taking partial tenson of stress-increment as eigenfunctions, and we are to transform from nonlinear equations into linear equation in this paper.

关键词: gauge equivalence, r- matrix, integrable system

Abstract: In this paper the outcome ofaxisymmetric problems of ideal plasticity in paper [39], [19] and [37] is directly extended to the three-dimensional problems of ideal plasticity, and get at the general equation in it. The problem of plane strain for material of ideal rigid-plasticity can be solved by putting into double harmonic equation by famous Pauli matrices of quantum electrodynamics different from the method in paper [7]. We lead to the eigen equation in the problems of ideal plasticity, taking partial tenson of stress-increment as eigenfunctions, and we are to transform from nonlinear equations into linear equation in this paper.

Key words: gauge equivalence, r- matrix, integrable system

沈惠川. ON THE GENERAL EQUATIONS, DOUBLE HORMONIC EQUATION AND EIGEN-EQUATION IN THE PROBLEMS OF IDEAL PLASTICITY[J]. Applied Mathematics and Mechanics (English Edition), 1986, 7(1): 65-76.

Shen Hui-chuan. ON THE GENERAL EQUATIONS, DOUBLE HORMONIC EQUATION AND EIGEN-EQUATION IN THE PROBLEMS OF IDEAL PLASTICITY[J]. Applied Mathematics and Mechanics (English Edition), 1986, 7(1): 65-76.

参考文献

[1] Bjorken,J.D.and S.D.Drell,Relativistic Quantum Mechanics, McGraw-Hill(1964);Relativistic Quantum Fields, McGraw-Hill(1964).
[2] Bohm,D.,Quantum Theory, Prentice-Hall Inc., New York (1951).
[3] Boussinesq,J.,Applications des Potentiels á l’étnde de lequilibre et du Mouvement des Solides Elastiques,Paris (1885).
[4] Chien Weizang,Calculus of Veriations and Finite Unit,(I).Science Press,Beijing (1980),578-598. (in Chinese).
[5] Dirac,P.A.M.,The Principles of Quantum Mechanics,Oxford (1958).
[6] Eliushen,A.A.,Plasticity,National (1948).(in Russian).
[7] Fan Jiachan,The exact solutions of von Mises yielding criterion for ideally plastic body under uui uniform pressure in case of plane strain,Appl.Math.Mech.,3 (1982),529-535.
[8] Flügge,S.,Practical Quantum Mechanics,Springer-varlag (1974).
[9] Galerkin,B.G.,To the general solution of solving a elasticity for three-dimension use stress and displacement functions,Report.Acad Scien.USSR.Groap.A (1931), 281-286. (in Russian).
[10] Hill,R.,The Mathematical Theory of Plasticity,Oxford,Clarendon Press (1950).
[11] Hodge,P.G.,Plastic Analysis of Structures,McGraw-Hill (1959).
[12] Hodge,P.G.,Limit Analysis of Rotationally Symmetric Plates and Shells,Prentice-Hall Inc.(1963).
[13] Johnson,W.and P.B.Mellor,Engineering Plasticity,VNR (1973).
[14] Kachanov,L.M.,Foundations of the Theory of Plasticity,North-Holland Publishing Company (1971).
[15] Kachanov,L.M.,Foundations of the Theory of Plasticity,National (1950).(in Russian).
[16] Krutkov,Y.A.,The Stress Function Tensor and General Solutions for Statics of Elasticity,M.,Acad.Scien.USSR,(1949).
[17] Kyuichiro Washizu,The Theory of Plasticity,Iwanami(1957).(in Japanese.).
[18] Landau,L.D.and E.M.Lifshitz,Quantum Mechanics,Nonrelativistic Theory,Addison Wesley,Reading,Mass(1958).
[19] Lin Hongsun,On the problem of axial-symmetric plastic deformation,Acta Physica Sinica,10,2(1954),89-104. (in Chinese).
[20] Martin,J.B., Plasticity: Fundamentals and General Results,MIT Press(1975).
[21] Von Mises,R.,Mechanik der festen korper in plastisch deformablem znstand,G?ttinger Nachr.,Math.Phys.Klasses.H4,(1913),582.
[22] Nadai,A.,Theory of Flow and Fracture of Solids,McGraw-Hill(1950).
[23] Olszak,W.,Z.Mróz and P.Perzyna,Recent Trends in the Development of the Theory of Plasticity,Pergamon Press-PWN.(1963).
[24] Olszak,W.,P.Perzyna and A.Sawczuk,Teoria Plastycznos’ci,PWN(1965).
[25] Prager,W.and P.G.Hodge,Theory of Perfectly Plastic Solids,John wiley(1951).
[26] Save,M.A.and C.E.Massonnet,Plastic Analysis and Design of Plates,Shells and Disks,NHPC(1972).
[27] Sawczuk,A.and T.Jaeger,Grenztragfahigkeits Theorie der Platten,Springer-Verlag (1963).
[28] Schiff,L.I.,Quantum Mechanics,(3rd ed),McGraw-Hill(1968).
[29] Slater,R.A.C.,Engineering Plasticity,Macmillan Press,LTD.(1977).
[30] Shen Huichuan,Dynamical stress function tensor,Appl.Math.Mech,3,6(1982),899-904.
[31] Shen Huichuan,Dynamical stress function tensor and several homogeneous solutions of elastostatics,Journal of China University of Science and Technology,14,Supplement 1(1984),95-102. (in Chinese).
[32] Shen Huichuan,General solution of elastic dynamics,Nature Journal,6,9(1985),853-858;Nature Journal,7,8(1984),633-634;7,10(1984),756. (in Chinese).
[33] Shen Huichuan,The general solution of the peristaltic fluid dynamics,Nature journal,7,10(1984),799;7,12(1984),940. (in Chinese).
[34] Shen Huichuan,The fission of spectrum line of monochromatic elastic wave,Appl.Math.Mech,5,4(1984),1509-1519.
[35] Shen Huichuan,The solution of deflection of elastic thin plate by the joint action of dynamical lateral pressure,force in central surface and external field on the elastic base,Appl.Math.Mech.5,6(1984),1791-1801.
[36] Shen Huichuan,The relation of von Kármán equation for elastic large deflection problem and Schrödinger equation for quantum eigenvalues problem,Appl.Math.Mech.6,8(1985),761-775.
[37] Shen Huishen,On the general equation of axisymmetric problems of ideal plasticity,Appl.Math.Mech.5,4(1984).
[38] Shen Huiahen and Jin Yon.ie, Upper-bound solutions of auaymmetric combined backward-forward extrusion,.lournal of shanghai liaotong University (1984), 87-99. (in Chinese).
[39] Symonds, P. S., On the general equations of problems of axial symmetry in the theory of plasticity, Quar. Appl. Math.,.,4 (1949), 448-452.
[40] Tomonaga, S., Quantum Mechanics, Misuzu (1978).
[41] van tier Waerden, B. L., Group ?7reory and Quatran Mechanics. Springer-verlag (1974).
[42] Yukawa Hideki and R. Toyoda, Quantum Mechanics, Iwanami (1978).