Evolution Equations for Defects in Finite Elasto-Plasticity

Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)

Abstract

The paper deals with continuous models of elasto-plastic materials with microstructural defects such as dislocations and disclinations. The basic assumptions concern the existence of plastic distortion and so-called plastic connection with metric property and the existence of the free energy function. This is dependent on the Cauchy-Green strain tensor, and its gradient with respect to the plastically deformed anholonomic configuration, and on the dislocation and disclination densities. The defect densities are defined in terms of the incompatibility of the plastic distortion and non-integrability of the plastic connection. The evolution of plastic distortion and disclination tensor has been postulated under the appropriate viscoplastic and dissipative type equations, which are compatible with the principle of the free energy imbalance. The associated small distortion model is provided. The present model and the previous ones have been also compared.

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Notes

Acknowledgements

This work was supported by a grant of Ministery of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0083 within PNCDI III.

References

  1. Arsenlis A, Parks DM (1999) Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Materialia 47(5):1597–1611Google Scholar
  2. Bilby BA (1960) Continuous distribution of dislocations. In: Sneddon IN, Hill R (eds) Progress in Solid Mechanics, North-Holland, Amsterdam, pp 329–398Google Scholar
  3. Clayton JD, McDowell DL, Bammann DJ (2006) Modeling dislocations and disclinations with finite micropolar elastoplasticity. International Journal of Plasticity 22(2):210–256Google Scholar
  4. Cleja-Ţigoiu S, Ţigoiu V (2011) Strain gradient effect in finite elasto-plastic damaged materials. International Journal of Damage Mechanics 20(4):484–514Google Scholar
  5. Cleja-Ţigoiu S, Paşcan R, Ţigoiu V (2016) Interplay between continuous dislocations and disclinations in elasto-plasticity. International Journal of Plasticity 79:68–110Google Scholar
  6. Cleja-Ţigoiu S (2007) Material forces in finite elasto-plasticity with continuously distributed dislocations. International Journal of Fracture 147(1):67–81Google Scholar
  7. Cleja-Ţigoiu S (2010) Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature. International Journal of Fracture 166(1):61–75Google Scholar
  8. Cleja-Ţigoiu S (2014) Dislocations and disclinations: continuously distributed defects in elastoplastic crystalline materials. Archive of Applied Mechanics 84(9):1293–1306Google Scholar
  9. Cleja-Ţigoiu S (2017) Finite elasto-plastic models for lattice defects in crystalline materials. In: dell’Isola F, Sofonea M, Steigmann D (eds) Mathematical Modelling in Solid Mechanics, Springer, Singapore, Advanced Structured Materials, vol 69, pp 43–57Google Scholar
  10. Cleja-Tigoiu S, Maugin GA (2000) Eshelby’s stress tensors in finite elastoplasticity. Acta Mechanica 139(1):231–249Google Scholar
  11. de Wit R (1970) Linear theory of static disclinations. In: Simmons JA, de Wit R, Bullough R (eds) Fundamental Aspects of Dislocation Theory, Nat. Bur. Stand. (U.S.) Spec. Publ., vol 317, I, pp 651–673Google Scholar
  12. de Wit R (1973a) Theory of disclinations: II. Continuous and discrete disclinations in anisotropic elasticity. J Res Nat Bur Stand - A Phys Chem 77A(1):49–100Google Scholar
  13. de Wit R (1973b) Theory of disclinations: III. Continuous and discrete disclinations in isotropic elasticity. J Res Nat Bur Stand - A Phys Chem 77A(3):359–368Google Scholar
  14. de Wit R (1973c) Theory of disclinations: IV. Straight disclinations. J Res Nat Bur Stand - A Phys Chem 77A(5):607–658Google Scholar
  15. de Wit R (1981) A view of the relation between the continuum theory of lattice defects and noneuclidean geometry in the linear approximation. International Journal of Engineering Science 19(12):1475–1506Google Scholar
  16. Eringen AC (2002) Nonlocal Continuum Field Theories. Springer, New YorkGoogle Scholar
  17. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: Theory and experiment. Acta Metallurgica et Materialia 42(2):475–487Google Scholar
  18. Fressengeas C, Taupin V, Capolungo L (2011) An elasto-plastic theory of dislocation and disclination fields. International Journal of Solids and Structures 48(25):3499–3509Google Scholar
  19. Gurtin ME (2002) A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 50(1):5–32Google Scholar
  20. Gurtin ME, Fried E, Anand L (2010) The Mechanics and Thermodynamics of Continua. University Press, CambridgeGoogle Scholar
  21. Kossecka E, de Wit R (1977) Disclination kinematics. Archives of Mechanics 29:633–650Google Scholar
  22. Kröner E (1990) The differential geometry of elementary point and line defects in Bravais crystals. International Journal of Theoretical Physics 29(11):1219–1237Google Scholar
  23. Kröner E (1992) The internal mechanical state of solids with defects. International Journal of Solids and Structures 29(14):1849–1857Google Scholar
  24. Lazar M, Maugin GA (2004a) Defects in gradient micropolar elasticity. I: screw dislocation. Journal of the Mechanics and Physics of Solids 52(10):2263–2284Google Scholar
  25. Lazar M, Maugin GA (2004b) Defects in gradient micropolar elasticity. II: edge dislocation and wedge disclination. Journal of the Mechanics and Physics of Solids 52(10):2285–2307Google Scholar
  26. Lazar M, Maugin GA, Aifantis EC (2006) On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. International Journal of Solids and Structures 43(6):1404–1421Google Scholar
  27. Maugin GA (1994) Eshelby stress in elastoplasticity and ductile fracture. International Journal of Plasticity 10(4):393–408Google Scholar
  28. Maugin GA (2006) Internal variables and dissipative structures. J Non-Equilib Thermodyn 15:173–192Google Scholar
  29. Minagawa S (1979) A non-Riemannian geometrica theory of imperfections in a Cosserat continuum. Arch Mech 31(6):783–792Google Scholar
  30. Schouten J (1954) Ricci Calculus. Springer, BerlinkGoogle Scholar
  31. Teodosiu C (1982) Elastic Models of Crystal Defects. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  32. Yavari A, Goriely A (2012) Riemann–Cartan geometry of nonlinear dislocation mechanics. Archive for Rational Mechanics and Analysis 205(1):59–118Google Scholar
  33. Yavari A, Goriely A (2013) Riemann-Cartan geometry of nonlinear disclination mechanics. Mathematics and Mechanics of Solids 18(1):91–102Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

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