Modeling and Analysis of Size-Dependent Structural Problems by Using Low-Order Finite Elements with Strain Gradient Plasticity

변형률 구배 소성 저차 유한요소에 의한 크기 의존 구조 문제의 모델링 및 해석

  • 박문식 (한남대학교 기계공학과) ;
  • 서영성 (한남대학교 기계공학과) ;
  • 송승 (한남대학교 기계공학과)
  • Received : 2011.05.03
  • Accepted : 2011.07.18
  • Published : 2011.09.01


An elasto-plastic finite element method using the theory of strain gradient plasticity is proposed to evaluate the size dependency of structural plasticity that occurs when the configuration size decreases to micron scale. For this method, we suggest a low-order plane and three-dimensional displacement-based elements, eliminating the need for a high order, many degrees of freedom, a mixed element, or super elements, which have been considered necessary in previous researches. The proposed method can be performed in the framework of nonlinear incremental analysis in which plastic strains are calculated and averaged at nodes. These strains are then interpolated and differentiated for gradient calculation. We adopted a strain-gradient-hardening constitutive equation from the Taylor dislocation model, which requires the plastic strain gradient. The developed finite elements are tested numerically on the basis of typical size-effect problems such as micro-bending, micro-torsion, and micro-voids. With respect to the strain gradient plasticity, i.e., the size effects, the results obtained by using the proposed method, which are simple in their calculation, are in good agreement with the experimental results cited in previously published papers.


Strain Gradient Plasticity;Taylor Dislocation Model;Strain Gradient Invariant;Length Parameter;Strain Gradient Hardening;Non-local Constitutive Theory;Averaged at Nodal Plastic Strain


Supported by : 한국연구재단


  1. Cosserat, E., Cosserat, F., Brocato, M. and Chatzis, K., 1909, Theorie des corps deformables, A. Hermann Paris.
  2. Toupin, R., 1962, "Elastic Materials with Couple- Stresses," Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 385-414.
  3. Mindlin, R., 1964, "Micro-Structure in Linear Elasticity," Archive for Rational Mechanics and Analysis, Vol. 16, No. 1, pp. 51-78.
  4. Koiter, W., 1964, Couple Stresses in the Theory of Elasticity, I and II., Proc. K. Ned. Akad. Wet.(B), Vol. 67, No. 1, pp. 17-44.
  5. Gao, X. L., Park, S. K. and Ma, H. M., 2009, "Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory," Mathematics and Mechanics of Solids, Vol. 14, No. 8, pp. 747-758.
  6. Wang, B., Zhao, J. and Zhou, S., 2010, "A Micro Scale Timoshenko Beam Model Based on Strain Gradient Elasticity Theory," European Journal of Mechanics, A/Solids, Vol. 29, No. 4, pp. 591-599.
  7. Fleck, N. A., Muller, G. M., Ashby, M. F. and Hutchinson, J. W., 1994, "Strain Gradient Plasticity: Theory and Experiment," Acta Metallurgica Et Materialia, Vol. 42, No. 2, pp. 475-487.
  8. McElhaney, K. W., Vlassak, J. J. and Nix, W. D., 1998, "Determination of Indenter Tip Geometry and Indentation Contact Area for Depth-Sensing Indentation Experiments," Journal of Materials Research, Vol. 13, No. 5, pp. 1300-1306.
  9. Stolken, J. S. and Evans, A. G., 1998, "A Microbend Test Method for Measuring the Plasticity Length Scale," Acta Materialia, Vol. 46, No. 14, pp. 5109-5115.
  10. Lloyd, D. J., 1994, "Particle Reinforced Aluminum and Magnesium Matrix Composites," International Materials Reviews, Vol. 39, No. 1, pp. 1-23.
  11. Aifantis, E. C., 1999, "Strain Gradient Interpretation of Size Effects," International Journal of Fracture, Vol. 95, pp. 299-314.
  12. Fleck, N. and Hutchinson, J. , 1997, "Strain Gradient Plasticity," Advances in Applied Mechanics, Vol. 33, pp. 295-361.
  13. Gao, H., Huang, Y., Nix, W. D. and Hutchinson, J. W., 1999, "Mechanism-Based Strain Gradient Plasticity - I. Theory," Journal of the Mechanics and Physics of Solids, Vol. 47, No.6, pp. 1239-1263.
  14. Evans, A. G. and Hutchinson, J. W., 2009, "A Critical Assessment of Theories of Strain Gradient Plasticity," Acta Materialia, Vol. 57, No. 5, pp. 1675-1688.
  15. . Y., King, W. E. and Fleck, N. A., 1999, "Finite Elements for Materials with Strain Gradient Effects," International Journal for Numerical Methods in Engineering, Vol. 44, No.3, pp. 373-391.<373::AID-NME508>3.0.CO;2-7
  16. Soh, A. and Wanji, C., 2004, "Finite Element Formulations of Strain Gradient Theory for Microstructures and the C0-1 Patch Test," International Journal for Numerical Methods in Engineering, Vol. 61, No. 3, pp. 433-454.
  17. Gao, H. and Huang, Y., 2001, "Taylor-Based Nonlocal Theory of Plasticity," International Journal of Solids and Structures, Vol. 38, No. 15, pp. 2615-2637.
  18. Abu Al-Rub, R. K. and Voyiadjis, G. Z., 2005, "A Direct Finite Element Implementation of the Gradient- Dependent Theory," International Journal for Numerical Methods in Engineering, Vol. 63, No. 4, pp. 603-629.
  19. Byon, S. M. and Lee, Y., 2006, "Deformation Analysis of Micro-Sized Material Using Strain Gradient Plasticity," Journal of Mechanical Science and Technology, Vol. 20, No. 5, pp. 621-633.
  20. Abu Al-Rub, R. K. and Voyiadjis, G. Z., 2006, "A Physically Based Gradient Plasticity Theory," International Journal of Plasticity, Vol. 22, No. 4, pp. 654-684.
  21. Fleck, N. A. and Hutchinson, J. W., 2001, "A Reformulation of Strain Gradient Plasticity," Journal of the Mechanics and Physics of Solids, Vol. 49, No. 10, pp. 2245-2271.
  22. Shrotriya, P., Allameh, S. M., Lou, J., Buchheit, T. and Soboyejo, W. O., 2003, "On the Measurement of the Plasticity Length Scale Parameter in LIGA Nickel Foils," Mechanics of Materials, Vol. 35, No. 3-6, pp. 233-243.
  23. Arsenlis, A. and Parks, D. M., 1999, "Crystallographic Aspects of Geometrically-Necessary and Statistically- Stored Dislocation Density," Acta Materialia, Vol. 47, No.5, pp. 1597-1611.
  24. Dassault Systemes Simulia, Inc., 2010, Abaqus v. 6.9, Providence, U.S.A.
  25. Shu, J. Y., 1998, "Scale-Dependent Deformation of Porous Single Crystals," International Journal of Plasticity , Vol. 14, No. 10-11, pp.1085-1107.
  26. Fleck, N. A. and Willis, J. R. 2009, "A Mathematical Basis for Strain-Gradient Plasticity Theory. Part II: Tensorial Plastic Multiplier," Journal of the Mechanics and Physics of Solids,Vol. 57, No. 7, pp. 1045-1057
  27. Idiart, M. I., Deshpande, V. S., Fleck, N. A. and Willis, J. R., 2009, "Size Effects in the Bending of Thin Foils," International Journal of Engineering Science, Vol. 47, No. 11-12, pp. 1251-1264.
  28. Guo, Y., Huang, Y., Gao, H., Zhuang, Z. and Hwang, K. C., 2001, "Taylor-Based Nonlocal Theory of Plasticity: Numerical Studies of the Micro-Indentation Experiments and Crack Tip Fields," International Journal of Solids and Structures, Vol. 38, No. 42-43, pp. 7447-7460.
  29. Byon, S. M., Moon, C. H. and Lee, Y., 2010, "Strain Gradient Plasticity Based Finite Element Analysis of Ultra-Fine Wire Drawing Process," Journal of Mechanical Science and Technology, Vol. 23, No. 12, pp. 3374-3384.
  30. Han, C.-S., Ma, A., Roters, F. and Raabe, D., 2007, "A Finite Element Approach with Patch Projection for Strain Gradient Plasticity Formulations," International Journal of Plasticity, Vol. 23, No.4, pp. 690-710.

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