Numerical treatment of cosserat based rate independent strain gradient plasticity theories

Main Article Content

Juan David Gómez C.

Keywords

non–classical continuum theories, Cosserat continuum medium theory, couple stress theory, small scale inelastic response, finite element analysis, constitutive modeling, integration algorithms

Abstract

The current trend towards miniaturization in the microelectronics industryhas pushed for the development of theories intended to explain the behaviorof materials at small scales. In the particular case of metals, a class ofavailable non–classical continuum mechanics theories has been recently employedin order to explain the wide range of observed behavior at the micronscale. The practical use of the proposed theories remains limited due to issuesin its numerical implementation. First, in displacement–based finite elementformulations the need appears for higher orders of continuity in the interpolationshape functions in order to maintain the convergence rate upon meshrefinement. This limitation places strong restrictions in the geometries of theavailable elements. Second, the available inelastic constitutive models for smallscale applications have been cast into deformation theory formulations limitingthe set of problems to those exhibiting proportional loading only. In thisarticle two contributions are made for the particular case of a Cosserat couplestress continuum. First it describes a numerical scheme based on a penaltyfunction/reduced integration approach that allows for the proper treatment ofthe higher order terms present in Cosserat like theories. This scheme results in a new finite element that can be directly implemented into commercial finiteelement codes. Second, a flow theory of plasticity incorporating size effects isproposed for the case of rate independent materials overcoming the limitationsin the deformation theory formulations. The constitutive model and its correspondingtime–integration algorithm are coupled to the new proposed finiteelement and implemented in the form of a user element subroutine into thecommercial code ABAQUS. The validity of the approach is shown via numericalsimulations of the microbending experiment on thin Nickel foils reportedin the literature.

PACS: 81.40.Lm, 81.40.Jj, 46

MSC: 82B21, 65N30

Downloads

Download data is not yet available.
Abstract 485 | PDF Downloads 295

References

[1] N. A. Fleck and J. W. Hutchinson. A Phenomenological Theory for Strain Gradient Effects in Plasticity. Journal of the Mechanics and Physics of Solids, ISSN 0022–5096, 41(12), 1825–1857 (1993).

[2] N. A. Stelmashenko, M. G. Walls, L. M. Brown and Yu V. Milman. Microindentations on W and Mo oriented single crystals : an STM study. Acta Metallurgica et Materialia, ISSN 1359–6462, 41(10), 2855–2865 (1993).

[3] Qing Ma and David R. Clarke. Size dependent hardness of silver single crystals.Journal of Materials Research, ISSN 0884-2914, 10(4), 853–863 (1995).

[4] W. J. Poole, M. F. Ashby and N. A. Fleck. Micro–hardness of annealed and workhardened copper polycrystals. Scripta Mettallurgica et Materialia, ISSN 1359–6462, 34(4), 559–564 (1996).

[5] Ranjana Saha, XueZhenyu, Huang Young and William Nix. Indentation of a soft metal film on a hard substrate: strain gradient hardening effects. Journal of the mechanics and physics of solids, ISSN 0022–5096, 49(9), 1997–2014 (2001).

[6] A. A. Elmustafa and D. S. Stone. Indentation size effect in polycrystalline F. C. C. metals. Acta Materialia, ISSN 1359–6454, 50(14), 3641–3650 (2002).

[7] J. S. St¨olken and A. G. Evans. A microbend test method for measuring the plasticity length scale. Acta Materialia, ISSN 1359–6462, 46(14), 5109–5115 (1998).

[8] P. Shrotriya, S. M. Allameh, J. Lou, T. Buchheit and W. O. Soboyejo. On the measurement of the plasticity length scale parameter in LIGA nickel foils. Mechanics of materials, ISSN 0167–6636, 35(3–6), 233–243 (2003).

[9] N. Fleck and John W. Hutchinson. Strain Gradient Plasticity in Advances in Applied mechanics, John W. Hutchinson and Theodore Y. Wu (Series Editors), ISBN 0120020335, 33, 295–361 (1997). Referenciado en 101, 102, 116, 119
[10] H. Gao, Y. Huang and W. D. Nix. Modeling Plasticity at the Micrometer Scale. Naturwissenschaften, ISSN 0028–1042, 86(11), 507–515 (1999).

[11] H. Gao, Y. Huang, W. D. Nix and J. W. Hutchinson. Mechanism–based strain gradient plasticity: I Theory. Journal of the mechanics and physics of solids, ISSN 0022–5096, 47(6), 1239 1263 (1999).

[12] Zdenek P. Bazant. Size Effect in Blunt Fracture: Concrete, Rock, Metal . Journal of EngineeringMechanics, ISSN 0733–9399, 110(4), 518–535 (1984).

[13] E. C. Aifantis. On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science, ISSN 0020–7225, 30(10), 1279–1299 (1992).

[14] R. De Borst and H. M¨uhlhaus. Gradient–dependent plasticity: formulation and algorithmic aspects. International journal for numerical methods in engineering, ISSN 0029–5981, 35(3), 521–539 (1992).

[15] N. A. Fleck, G. M. Muller, M. F. Ashby and J. W. Hutchinson. Strain gradient plasticity: theory and experiment . Acta metallurgica et materialia, ISSN 0956- 7151, 42(2), 475–487 (1994).

[16] Y. Guo, Y. Huang, H. Gao, Z. Zhuang and K. C. Hwang. Taylor–based nonlocal theory of plasticity: numerical studies of the micro-indentation experiments and crack tip fields. International journal of solids and structures, ISSN 0020–7683, 38(42), 7447–7460 (2001).

[17] J. L. Bassani. Incompatibility and a simple gradient theory of plasticity. Journal of the mechanics and physics of solids, ISSN 0022–5096, 49(9), 1983–1996 (2001).

[18] Zdenek P. Bazant. Scaling of dislocation–based strain–gradient plasticity. Journal of the Mechanics and Physics of Solids, ISSN 0022–5096, 50(3), 435–448 (2002).

[19] Abu Al–Rub and George Z. Voyiadjis. Analytical and Experimental Determination of the Material Intrinsic Length Scale of Strain Gradient Plasticity Theory From Micro–And Nano–Indentation experiments. International Journal of Plasticity, ISSN 0749–6419, 20(6), 1139–1182 (2004).

[20] E. Cosserat and F. Cosserat. Th´eorie des corps d´eformables. Paris: A. Hermann & Fils, 1909.

[21] E. Aero and E. Kuvshinsky. Fundamental equations of the Theory of elastic media with rotationally interacting particles. Soviet Physics–Solid State, ISSN 0038–5654, 2, 1272–1281 (1961).

[22] R. D. Mindlin. Micro–structure in Linear Elasticity. Archive for Rational Mechanics and Analysis, ISSN 0003–9527, 16(1), 51–78 (1964).

[23] R. De Borst. A generalization of J2–flow Theory for polar continua. Computer Methods in Applied Mechanics and Engineering, ISSN 0045–7825, 103, 347–362 (1993).

[24] R. D. Mindlin and H. F. Tiersten. Effects of Couple-Stresses in Linear Elasticity. Communicated by C. Truesdell, Archive for Rational Mechanics and Analysis, ISSN 0003–9527, 11(1), 415–448 (1962).

[25] R. Toupin. Elastic Materials with Couple–Stresses. Archive for Rational Mechanics and Analysis, ISSN 0003–9527, 11(1), 385–414 (1962).

[26] R. D. Mindlin. Second gradient of strain and surface–tension in linear elasticity. International Journal of Solids and Structures, ISSN 0020–7683, 1(), 417–438 (1965).

[27] L. R. Herrmann. Mixed Finite Elements for Couple–Stress Analysis in Mixed and Hybrid Finite Element Methods. Eds. S. N. Alturi, R. H. Gallagher and O. C. Zienkiewicz. John Wiley and Sons, 1983.

[28] J. Y. Shu, W. E. King and N. Fleck. Strain Gradient Plasticity: Size Dependent Deformation of Bicrystals. Journal of the Mechanics and Physics of Solids, ISSN 0022–5096, 47, 297–324 (1999).

[29] J. Y. Shu, W. E. King and N. A. Fleck. Finite elements for materials with strain gradient effects. International journal for numerical methods in engineering, ISSN 0029–5981, 44(3), 373–391 (1999).

[30] Z. Xia and J. Hutchinson. Crack Tip Fields in Strain Gradient Plasticity. Journal of the Mechanics and Physics of Solids, ISSN 0022–5096, 44(10), 1621–1648 (1996).

[31] Y. Wei and J. W. Hutchinson. Steady–state crack growth and work of fracture for solids characterized by strain gradient plasticity. Journal of the Mechanics and Physics of Solids, ISSN 0022–5096, 45(8), 1253–1273 (1997).

[32] M. R. Begley and J. W. Hutchinson. The mechanics of size–dependent indentation. Journal of the Mechanics and Physics of Solids, ISSN 0022-5096, 46(10), 2049–2068 (1998).

[33] George Z. Voyiadjis and Abu Al–Rub. Gradient plasticity theory with variable length scale parameter . International journal of solids and structures, ISSN 0020– 7683, 42(14), 3998–4029 (2005).

[34] Y. C. Fung and Pin Tong. Classical and Computational Solid Mechanics (Advanced Series in Engineering Science), ISBN 978–9810241247. World Scientific Publishing Company, 2001.