Incompatible Element Method for Couple Stress Plasticity and Numerical Study of Some Scale Effects Phenomena

For Fleck-Hutchinson couple stress plasticity, C 1 continuous element is required because this kind of theory fits within the framework of reduced couple stress theory where the first-order strain gradients are involved. Based on the general couple stress theory, a method for constructing C 0 -continuous incompatible element for couple stress theory is derived presently. A 4-node plane incompatible element is developed according this method. Numerical results indicate that this element performs well and some valuable phenomena are revealed. Introduction Recent experiments [1, 2] have shown materials will display strong scale effects when the scale of non-inform plastic deformation field associated their intrinsic length scale is on the order of microns. The classical continuum mechanics cannot predict such scale-dependent phenomena for their constitutive models possess no any material intrinsic length parameter. In order to explain such scale effect phenomena, Fleck and Hutchinson [1, 3] developed a couple stress theory of strain gradient plasticity, which incorporates the rotation gradient of deformation into constitutive model, and introduces a material characteristic length parameter related to the rotation gradient. Theoretical predictions agree well with the micro-torsion [1] and micro-bending experiments [2]. In finite element implementation of Fleck-Hutchinson couple stress plasticity, the higher order nature of theory requires that both the displacement and its first-order derivatives to be continuous across the adjacent elements’ boundaries. Presently, the mixed-type finite element method [4] is the main numerical tool for analysis of strain gradient theory. According to the mixed-type element formulation, C 0 -continous element can be developed because displacement gradients are treated as nodal degrees of freedom and kinematic constraints between displacements and rotation gradients are enforced via the Lagrange multiplier method. In order to treat materials in incompressible status, an essential assumption of Fleck-Hutchinson stain gradient plasticity, special mixed element formulation where the hydrostatic stress p is treated as an independent variable should be adopted. Therefore, redundant nodal variables are involved and the band character of the global stiffness matrix is destroyed for mixed-type element formulations. Furthermore, this approach will lead to a non-positive-definite global stiffness matrix with zero diagonal blocks that affect the efficiency of solving equations. Noticed that Fleck-Huchinson couple stress plasticity falls within the classification of the reduced couple stress theory [5, 6], the kinematic relationships between material displacement u and the associated material rotation θ should be satisfied, ( ) 1/ 2 curl = θ u . In the general couple stress theory [7-9], micro-rotation ω , an independent kinematic quantity with no direct dependence Key Engineering Materials Online: 2004-10-15 ISSN: 1662-9795, Vols. 274-276, pp 649-654 doi:10.4028/www.scientific. net/KEM.274-276.649 © 2004 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific. net. (Semanticscholar.org-13/03/20,17:14:10) on u, is introduced. This enables the C 0 -continuous element to be developed based on the general couple stress theory. The systemic method to construct the incompatible element for couple stress theory is developed in present. The resultant element can be directly applied for studying the incompressible materials without any special treatment. Review of Couple Stress Theories In this section, general couple stress theory is outlined firstly, and then, we show how it degenerates to the reduced couple stress theory. Cauchy stresses ij t are decomposed into the symmetric part ij σ and the anti-symmetric part ij τ . Couple stresses are denoted by ij μ . Equilibrium equation and moment equilibrium equation are ( ) , , , , 0 , 1/ 2 0 ij j ij j ij j jk ijk pi p t e σ τ τ μ = + = + = . (1) The stress traction T and couple stress traction m on the boundary are respectively ( ) i ij ij j T n σ τ = + and i ji j m n μ = , where n is the unit normal vector on the boundary. Displacement-strain relationships are ( ) , , / 2 ij i j j i u u ε = + , ( ) , , / 2 ij ijk k j i i j e u u γ ω = − − , , ij i j χ ω = . (2) The strain energy density W is assumed to depend on the strain tensor ij ε , the curvature tensor ij χ and the relative rotation tensor ij γ , it reads ( ) 0 0 0 , , ij ji ji ij ij ij ij ij ij ji ij ji W d d d ε χ γ ε χ γ σ ε μ χ τ γ = + + ∫ ∫ ∫ . (3) When the rotation of microstructure is restricted to be equal to that of the macro-medium around it, 0 ij γ = . Hence, ( ) , ij ij W W ε χ ≡ , the general couple stress theory degenerates to the reduced one. A deformation theory of phenomenological strain gradient plasticity is developed by Fleck and Hutchinson [3] based on the reduced couple stress theory. The strain energy density W is assumed to depend not only upon usual von Mises effective strain ( ) ( ) ( ) 2 / 3 1/ 3 e ij ij ii jj ε ε ε ε ε = − but also upon the analogous effective curvature ( ) 2 / 3 e ij ij χ χ χ = . The combined strain measure is given by 2 2 2 e e l = + ε ε χ . Here, the material characteristic length l enters from dimensional considerations. The overall effective stress σ is defined as 2 2 2 e e σ σ l μ − = + , where 3 / 2 e ij ij σ s s = , 3 / 2 e ij ij μ μ μ = and / 3 ij ij ij kk s σ δ σ = − . The effective stress σ and stain ε are related by / σ W ε = ∂ ∂ . From above it follows that 2 1 3 3 ij ij ij ii jj s   = −     σ ε ε ε ε ε , 2 2 3 ij ji l = σ μ χ ε . (4) 650 Advances in Engineering Plasticity and Its Applications