Skip to main content
SpringerLink
Log in

On the general equation and the general solution in problems for plastodynamics with rigid-plastic material

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This work is the continuation of the discussion of refs. [1–2]. We discuss the dynamics problems of ideal rigid — plastic material in the flow theory of plasticity in this paper. From introduction of the theory of functions of complex variable under Dirac-Pauli representation we can obtain a group of the so-called “general equations” (i.e. have two scalar equations) expressed by the stream function and the theoretical ratio. In this paper we also testify that the equation of evolution for time in plastodynamics problems is neither dissipative nor disperive, and the eigen-equation in plastodynamics problems is a stationary Schrödinger equation, in which we take partial tensor of stress-increment as eigenfunctions and take theoretical ratio as eigenvalues. Thus, we turn nonlinear plastodynamics problems into the solution of linear stationary Schrödinger equation, and from this we can obtain the general solution of plastodynamics problems with rigid-plastic material.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Shen Hui-chuan, On the general equations, double harmonic equation and eigen-equation in the problems of ideal plasticity,Appl. Math. Mech.,7, 1 (1986).

    Google Scholar 

  2. Shen Hui-chuan, General solution of ideal plasticity problems,Nature Journal,8, 11 (1985), 847–848. (in Chinese)

    Google Scholar 

  3. Yang Gui-tong and Xiong Zhu-hua,Plastodynamics, Qinghua Univ. Press (1984), 276–277. (in Chinese)

  4. Chien Wei-zang,Calculus of Variations and Finite Unit (I). Science Press (1980), 578–598. (in Chinese)

  5. Chien Wei-zang, Generalized Variation Principles, Intellectual Press (1985), 276–304. (in Chinese)

  6. Washizu, K.,Plasticity, Iwanami (1957). (in Japanese)

  7. Kachanov, L.M.,Foundations of the Theory of Plasticity, North-Holland (1971).

  8. Eliushen, A.A.,Plasticity, National Tech. Press (1948). (in Russian)

  9. Hill, R.,The Mathematical Theory of Plasticity, Oxford (1956).

  10. Hodge, P.G.,Plastic Analysis of Structures, McGraw-Hill (1959).

  11. Johnson, W. and P.B. Mellor,Engineering Plasticity, VNR (1973).

  12. Martin, J.B.,Plasticity: Fundamentals and General Results, MIT Press (1975).

  13. Nadai, A., Theory of Flow and Fracture of Solids, McGraw-Hill (1950).

  14. Olszak, W., Z. Mróz and P. Perzyna,Recent Trends in the Development of the Theory of Plasticity, pergamon (1963).

  15. Prager, W. and P.G. Hodge,Theory of Perfectly Plastic Solids, John Wiley (1951).

  16. Save, M.A. and C.E. Massonnet, Plastic analysis and design of plates, shells and disks, NHPC (1972).

  17. Sawczuk, A. and T. Jaeger,Grenztragfähigkeits Theorie der Platten, Springer-Verlag (1963).

  18. Slater, R.A.C.,Engineering Plasticity, McMillan Press, LTD (1977).

  19. Von Mises R., Mechanik der festen Korper in plastich-deformablen Zuständ, Den Gesellsch. der Wjssensch. Zu Gottingen,Math-phys. Klasses. H.4 (1913), 582.

  20. Djrac, P.A.M.,The Principles of Quantum Mechanics, Oxford (1958).

  21. Flügge, S.,Practical Quantum Mechanics, Springer-Verlag (1974).

  22. Symonds, P.S., On the general equations of problems of axial symmetry in the theory of plasticity,Quar. Appl. Math.,6, 4 (1949), 448–452.

    Google Scholar 

  23. Shen Hui-shen, On the general equation of axisymmetric problems of ideal plasticity,Appl. Math. Mech.,5, 4 (1984).

    Google Scholar 

  24. Lin Hong-sun, On the problem of axial-symmetric plastic deformation,Acta Physica Sinica,10, 2 (1954), 89–104. (in Chinese)

    Google Scholar 

  25. Shen Hui-chuan, The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics, (I),Appl. Math. Meth.,7, 4 (1986).

    Google Scholar 

  26. Shen Hui-chuan, Exact solution of Navier-Stokes equations—The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics, (II),Appl. Math. Mech.,7, 6 (1986).

    Google Scholar 

  27. Shen Hui-chuan, Chaplygin equation in three-dimensional non-constant isentropic flow — The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics, (III),Appl. Math. Mech.,7, 8 (1986).

    Google Scholar 

  28. Shen Hui-chuan, Solutions of magneto hydrodynamics equations—The theory of functions of a complex variable under Dirac-Pauli representation and its application in fluid dynamics, (IV),Appl. Math. Mech.,7, 9 (1986).

    Google Scholar 

  29. Tomonaga, S.,Quantum Mechanics, Misuzu (1978). (in Japanese)

  30. Yukawa, H.,Quantum Mechanics, Iwanami (1978). (in Japanese)

  31. Landau, L.D. and E.M. Lifshitz,Quantum Mechanics, Addison-Wesley, Reading, Mass (1958).

    Google Scholar 

  32. Böhm, D.,Quantum Theory, Prentice-Hall Inc., New York (1951).

    Google Scholar 

  33. Schiff, L.I.,Quantum Mechanics, (3rd ed.) McGraw-Hill (1968).

  34. Shen Hui-chuan, Dynamical stress function tensor,Appl. Math. Mech.,3, 6 (1982), 899–904.

    Google Scholar 

  35. Shen Hui-chuan, Dynamical stress function tensor and several homogeneous solution of elastostatics,Journal of China University of Science and Technology,14, Supplement 1, JCUST 84016 (1984), 95–102. (in Chinese)

    Google Scholar 

  36. Shen Hui-chuan, General solution of elastodynamics,Appl. Math. Mech.,6. 9 (1985), 853–858;Nature J. 7, 8 (1984), 633–634.7, 10 (1984), 756.

    Google Scholar 

  37. Shen Hui-chuan, The fission of spectrum line of monochromatic elastic wave,Appl. Math. Mech.,5, 4 (1984), 1509–1519.

    Google Scholar 

  38. Shen Hui-chuan, 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.

    Google Scholar 

  39. Shen Hui-chuan, The relation of von Kármán equation for elastic large deflection problem and Schrödinger equation for quantum eigenvalues problemAppl. Math. Mech.,6, 8 (1985), 761–775.

    Google Scholar 

  40. Shen Hui-chuan, Again on the relation of von Kármán equation for elastic large deflection problem and Schrödinger equation for quantum eigen values problem,Appl. Math. Mech. (to be published)

  41. Shen Hui-chuan, The schrödinger equation of thin shell theories,Appl. Math. Mech.,6, 10 (1985).

    Google Scholar 

  42. Shen Hui-chuan, The schrödinger equation in theory of plates and shells with orthorhombic anisotropy,Appl. Math. Mech., (to be published)

Download references

Author information

Authors and Affiliations

  1. Department of Earth and Space Science, University of Science and Technology of China, Hefei

    Shen Hui-chuan

Authors
  1. Shen Hui-chuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Chien Wei-zang

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hui-chuan, S. On the general equation and the general solution in problems for plastodynamics with rigid-plastic material. Appl Math Mech 8, 45–56 (1987). https://doi.org/10.1007/BF02014498

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02014498

Keywords

Search

Navigation