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A general method of stress analysis for a generalized linear yield criterion under plane strain and plane stress

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Abstract

The present paper deals with the system of equations comprising a yield criterion together with the stress equilibrium equations under plane strain and plane stress conditions. This system of equations can be studied independently of any flow rule, and its solution supplies the distribution of stress. It is shown that this problem of plasticity theory is reduced to a purely geometric problem. The stress equilibrium equations are written relative to a coordinate system in which the coordinate curves coincide with the trajectories of the principal stress directions. The general solution of the system is constructed giving a relation connecting the two scale factors for the coordinate curves. This relation is used for developing a method for finding the mapping between the principal lines and Cartesian coordinates with the use of a solution of a hyperbolic system of equations. In particular, the mapping between the principal lines and Cartesian coordinates is given in parametric form with the characteristic coordinates as parameters. Finally, a boundary value problem for the region adjacent to an external boundary which coincides with a principal stress trajectory is formulated and its general solution is given.

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References

  1. Besdo, D.: Principal- and slip-line methods of numerical analysis in plane and axially-symmetric deformations of rigid/plastic media. J. Mech. Phys. Solids 19, 313–328 (1971)

    Article  ADS  MATH  Google Scholar 

  2. Minfeng, H.: Coordinates of principal stresses for elastic plane problem. Appl. Math. Mech. 18(2), 157–162 (1997)

    Article  MATH  Google Scholar 

  3. Haderka, P., Galybin, A.N.: The stress trajectories method for plane plastic problems. Int. J. Solids Struct. 48, 450–462 (2011)

    Article  MATH  Google Scholar 

  4. Richmond, O., Alexandrov, S.: Nonsteady planar ideal plastic flow: general and special analytical solutions. J. Mech. Phys. Solids 48(8), 1735–1759 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Tam, K.-M.M., Mueller, C.T.: Additive manufacturing along principal stress lines. 3D Print Addit. Manuf. 4(2), 63–81 (2017)

    Article  Google Scholar 

  6. Gao, G., Li, Y.B., Pan, H., Chen, L.M., Liu, Z.Y.: An effective members-adding method for truss topology optimization based on principal stress trajectories. Eng. Comput. 34(6), 2088–2104 (2017)

    Article  Google Scholar 

  7. Lippmann, H.: Principal line theory of axially-symmetric plastic deformation. J. Mech. Phys. Solids 10, 111–122 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Sadowsky, M.A.: Equiareal pattern of stress trajectories in plane plastic strain. ASME J. Appl. Mech. 63, A74–A76 (1941)

    MathSciNet  MATH  Google Scholar 

  9. Alexandrov, S., Harris, D.: Geometry of principal stress trajectories for a Mohr–Coulomb material under plane strain. Z. Angew. Math. Mech. 97, 473–476 (2017)

    Article  MathSciNet  Google Scholar 

  10. Alexandrov, S.E., Goldstein, R.V.: Trajectories of principal stresses in the plane—stressed state of material obeying the Tresca and Coulomb–Mohr yield conditions. Dokl. Phys. 59, 460–462 (2014)

    Article  ADS  Google Scholar 

  11. Coburn, N.: The linear yield condition in the plane plasticity problem. Duke Math. J. 10, 455–462 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  12. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs (1969)

    Google Scholar 

  13. Druyanov, B.: Technological Mechanics of Porous Bodies. Clarendon Press, New York (1993)

    MATH  Google Scholar 

  14. Torkamani, M.A.M.: A linear yield surface in plastic cyclic analysis. Comput. Struct. 22, 499–516 (1986)

    Article  Google Scholar 

  15. Billington, E.W.: Generalized isotropic yield criterion for incompressible materials. Acta Mech. 72, 1–20 (1988)

    Article  MATH  Google Scholar 

  16. Ma, G., Hao, H., Miyamoto, Y.: Limit angular velocity of rotating disc with unified yield criterion. Int. J. Mech. Sci. 43, 1137–1153 (2001)

    Article  MATH  Google Scholar 

  17. Zhang, Y.-Q., Hao, H., Yu, M.-H.: A unified characteristic theory for plastic plane stress and strain problems. ASME J. Appl. Mech. 70, 649–654 (2003)

    Article  ADS  MATH  Google Scholar 

  18. Zhao, D., Fang, Q., Li, C., Liu, X., Wang, G.: Derivation of plastic specific work rate for equal area yield criterion and its application to rolling. J. Iron Steel Res. Int. 17, 34–38 (2010)

    Article  Google Scholar 

  19. Altenbach, H., Kolupaev, V., Yu, M.: Yield criteria of hexagonal symmetry in the \(\pi \)-plane. Acta Mech. 224, 1527–1540 (2013)

    Article  MATH  Google Scholar 

  20. Dats, E.P., Murashkin, E.V., Gupta, N.K.: On yield criterion choice in thermoelastoplastic problems. Proc. IUTAM 23, 187–200 (2017)

    Article  Google Scholar 

  21. Clausen, J., Damkilde, L., Andersen, L.: An efficient return algorithm for non-associated plasticity with linear yield criteria in principal stress space. Comput. Struct. 85, 1795–1807 (2007)

    Article  Google Scholar 

  22. Huang, J., Griffiths, D.V.: Observations on return mapping algorithms for piecewise linear yield criteria. Int. J. Geomech. 8, 253–265 (2008)

    Article  Google Scholar 

  23. Manola, M.M.S., Koumousis, V.K.: Ultimate state of plane frame structures with piecewise linear yield conditions and multi-linear behavior: a reduced complementarity approach. Comput. Struct. 130, 22–33 (2014)

    Article  Google Scholar 

  24. Hill, R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)

    MATH  Google Scholar 

  25. Spencer, A.J.M.: A theory of the kinematics of ideal soils under plane strain conditions. J. Mech. Phys. Solids 12, 337–351 (1964)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Chakrabarty, J.: Theory of Plasticity. McGraw-Hill, Singapore (1987)

    Google Scholar 

  27. Druyanov, B., Nepershin, R.: Problems of Technological Plasticity. Elsevier, Amsterdam (1994)

    Google Scholar 

  28. Kim, Y.-J., Schwalbe, K.-H.: Compendium of yield load solutions for strength mis-matched DE(T), SE(B) and C(T) specimens. Eng. Fract. Mech. 68, 1137–1151 (2001)

    Article  Google Scholar 

  29. Alexandrov, S.: Upper Bound Limit Load Solutions for Welded Joints with Cracks. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

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Acknowledgements

This research was made possible by Grants 16-49-02026 from RSCF (Russia) and INT/RUS/RSF/P-17 from DST (India).

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Prospect Vernadskogo, Moscow, Russia, 119526

    Sergei Alexandrov

  2. Metal Forming Laboratory, Mechanical Engineering Department, IIT Bombay, Mumbai, 400076, India

    Prashant Date

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  1. Sergei Alexandrov
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  2. Prashant Date
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Correspondence to Sergei Alexandrov.

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Communicated by Andreas Öchsner.

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Alexandrov, S., Date, P. A general method of stress analysis for a generalized linear yield criterion under plane strain and plane stress. Continuum Mech. Thermodyn. 31, 841–849 (2019). https://doi.org/10.1007/s00161-018-0743-6

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