Material forces in finite elasto-plasticity with continuously distributed dislocations

Conference paper

Abstract

In this paper we propose a thermodynamically consistent model for elasto-plastic material with structural inhomogeneities such as dislocations, subjected to large deformations, in isothermal processes. The plastic measure of deformation is represented by a pair of plastic distortion, and plastic connection with non-zero torsion (in order to have the non-zero Burgers vector). The developments are focused on the balance equations (for material forces and for physical force system), derived from an appropriate principle of the virtual power formulated within the constitutive framework of finite elasto-plasticity and on constitutive restrictions imposed by the free energy imbalance. The presence of the material forces (microforce and microstress momentum) is a key point in the exposure, and viscoplastic (generally rate dependent) constitutive representation are derived.

Keywords

Finite deformation Plastic distortion Configuration with torsion Plastic connection Stress momentum Material forces Free energy imbalance Principle of virtual power Dislocations 

AMS Subject Classifications (2000)

74C99 74A20 

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References

  1. 1.
    Acharya A (2004). Constitutive analysis of finite deformation field dislocation mechanics. J Mech Phys Solid 52: 301–316 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bilby BA (1960). Continuous distribution of dislocations. In: Sneddon, IN and Hill, R (eds) Progress in solid mechanics, pp 329–398. North-Holland, Amsterdam Google Scholar
  3. 3.
    Beju I, Soós E, Teodorescu PP (1983) Spinor and non-Euclidean tensor calculus with applications. Ed. Tehnica, Bucureşti Romania, Abacus Press, Tunbridge Wells, Kent, England (romanian version 1979)Google Scholar
  4. 4.
    Brüning M and Ricci S (2005). Nonlocal continuum theory of an isotropically damaged metals. Int J Plast 21: 1346–1382 CrossRefGoogle Scholar
  5. 5.
    Cleja-Ţigoiu S (1990) Large elasto-plastic deformations of materials with relaxed configurations – I. Constitutive assumptions, II. Role of the complementary plastic factor. Int J Eng Sci 28:171–180, 273–284Google Scholar
  6. 6.
    Cleja-Ţigoiu S (2001). A model of crystalline materials with dislocations. In: Cleja-Ţigoiu, S and Ţigoiu, V (eds) Proceedings of 5th international Seminar geometry continua and microstructures, pp 25–36. Sinaia, Romania Google Scholar
  7. 7.
    Cleja-Ţigoiu S (2002a). Couple stresses and non-Riemannian plastic connection in finite elasto-plasticity. ZAMP 53: 996–1013 CrossRefMATHGoogle Scholar
  8. 8.
    Cleja-Ţigoiu S (2002b). Small elastic strains in finite elasto-plastic materials with continuously distributed dislocations. Theor Appl Mech 28(29): 93–112 CrossRefGoogle Scholar
  9. 9.
    Cleja-Ţigoiu S and Maugin GA (2000). Eshelby’s stress tensors in finite elastoplasticity. Acta Mechanica 139: 231–249 CrossRefMATHGoogle Scholar
  10. 10.
    Cleja-Ţigoiu S and Soós E (1990). Elastoplastic models with relaxed configurations and internal state variables. Appl Mech Rev 43: 131–151 Google Scholar
  11. 11.
    Cleja-Ţigoiu S, Fortunée D, Vallée C (2007) Torsion equation in anisotropic elasto-plastic materials with continuously distributed dislocations. Math Mech Solids doi: 10.1177/1081286507079157 Google Scholar
  12. 12.
    Epstein M and Maugin GA (2000). Thermomechanics of volumetric growth in uniform bodies. Int J Plast 16: 951–978 MATHCrossRefGoogle Scholar
  13. 13.
    Fleck NA, Muller GM, Ashby MF and Hutchinson JW (1994). Strain gradient plasticity: theory and experiment. Acta Metall Mater 42: 475–487 CrossRefGoogle Scholar
  14. 14.
    Forest S, Cailletand G and Sievert R (1997). A Cosserat theory for elastoviscoplastic single crystals at finite deformation. Arch Mech 49(4): 705–736 MATHMathSciNetGoogle Scholar
  15. 15.
    Gupta A, Steigmann D, Stölken JS (2006) On the evolution of plasticity and incompatibility. Math Mech Solids. online: doi: 10.1177/1081286506064721
  16. 16.
    Gurtin ME (2000). On the plasticity of single crystal: free energy, microforces, plastic-strain gradients. J Mech Phys Solids 48: 989–1036 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gurtin ME (2002). A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J Mech Phys Solids 50: 5–32 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gurtin ME (2003). On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients. Int J Plast 19: 47–90 MATHCrossRefGoogle Scholar
  19. 19.
    Gurtin ME (2004). A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers and for dissipation due to plastic spin. J Mech Phys Solids 52: 2545–2568 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gurtin ME and Needleman A (2005). Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers. J Mech Phys Solids 53: 1–31 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hirth J and Lothe JP (1982). Theory of dislocations. Krieger Publishing, Malabar, Florida Google Scholar
  22. 22.
    Kondo K, Yuki M (1958) On the current viewpoints of non-Riemannian plasticity theory. In: RAAG memoirs of the unifying study of basic problems in engng and physical sciences by means of geometry II (D), Tokyo, pp 202–226Google Scholar
  23. 23.
    Kröner E (1963). On the physical reality of torque stresses in continuum mechanics. Gauge theory with disclinations. Int J Eng Sci 1: 261–278 CrossRefGoogle Scholar
  24. 24.
    Kröner E (1992). The internal mechanical state of solids with defects. Int J Solids Struct 29: 1849–1857 CrossRefMATHGoogle Scholar
  25. 25.
    Kröner E and Lagoudas DC (1992). Gauge theory with disclinations. Int J Eng Sci 30: 1849–1857 CrossRefGoogle Scholar
  26. 26.
    Le KC and Stumpf H (1996a). A model of elastoplastic bodies with continuously distributed dislocations. Int J Plast 12(5): 611–627 MATHCrossRefGoogle Scholar
  27. 27.
    Le KC and Stumpf H (1996b). Nonlinear continuum with dislocations. Int J Eng Sci 34: 339–358 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Le KC and Stumpf H (1996c). On the determination of the crystal reference in nonlinear continuum theory of dislocation. Proc Roy Soc London A 452: 359–371 MATHCrossRefGoogle Scholar
  29. 29.
    Maugin GA (1999) The thermomechanics of nonlinear irreversible behaviors. World ScientificGoogle Scholar
  30. 30.
    Naghdi PM and Srinivasa AR (1994). Characterization of dislocations and their influence on plastic deformation in single crystal. Int J Eng Sci 32(7): 1157–1182 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Noll W (1967). Materially uniform simple bodies with inhomogeneities. Arch Rat Mech Anal 27: 1–32 CrossRefMathSciNetGoogle Scholar
  32. 32.
    Schouten JA (1954). Ricci-Calculus. Springer-Verlag, Berlin MATHGoogle Scholar
  33. 33.
    Steinmann P (1994). A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int J Solids Struct 31: 1063–1084 MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Steinmann P (1997) Continuum theory of dislocations: impact to single cristal plasticity. In: Owen DRJ, Onãte E, Hinton E Computational plasticity, fundamental and applications. CIME, Barcelona.Google Scholar
  35. 35.
    Steinmann P (2002). On spatial and material settings of hyperelastostatic crystal defects. J Mech Phys Solids 50: 1743–1766 MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Stumpf H and Hackle K (2003). Micromechanical concept for analysis of damage evolution in thermo-viscoplastic and quasi-brittle materials. Int J Solids Struct 40: 1567–1584 MATHCrossRefGoogle Scholar
  37. 37.
    Teodosiu C (1970) A dynamic theory of dislocations and its applications to the theory of the elastic-plastic continuum. In: Simmons JA, de Witt R, Bullough R (eds) Fundamental aspects of dislocation theory, Nat Bur Stand (U.S.), Spec. Publ. 317, II, pp 837–876Google Scholar
  38. 38.
    Wang CC (1967). On the geometric structure of simple bodies, a Mathematical foundation for the theory of continuous distributions of dislocations. Arch Rat Mech Anal 27: 33–94 MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

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

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