Numerical treatment of cosserat based rate independent strain gradient plasticity theories (2008)
The current trend towards miniaturization in the microelectronic industry has promulgated the development of theories aimed at explaining the behavior of materials used on a small scale. In the particular case of metals, a class of non-classical theories of continuous media mechanics has recently been used in order to explain a wide range of micrometric scale observations. However, the practical use of the proposed theories remains limited due to difficulties in their numerical implementation. First, when these are to be implemented in formulations by finite elements based on displacements, the need for high orders of continuity in interpolation functions is generated in order to maintain the convergence properties in the algorithm. These limitations generate strong restrictions on the geometries of the available elements. On the other hand, the inelastic models available for small-scale applications have been formulated as deformation theories (total) limiting their applicability to problems under proportional load conditions. In this article two contributions are made in the case of a Cosserat continuum with torque voltages. First, a numerical scheme based on a strategy of penalty functions combined with reduced integration is described to appropriately address the problem of higher order terms present in the Cosserat theory. This scheme results in a new finite element that can be directly coupled to commercial distribution programs that accept user subroutines. Secondly, a theory of ﬂ ow of plasticity is proposed incorporating size effects overcoming some of the obstacles of deformation theories. The resulting constitutive model and its corresponding time integration scheme are coupled to the new element formulated and implemented in ABAQUS user subroutines. The validity of the strategy is demonstrated by simulations of the micro ﬂ exion test on nickel sheets reported in the literature.