Computational Implementation of Cosserat Based Strain Gradient Plasticity Theories

The current trend in microelectronics towards miniaturization has pushed for an interest in theories intended to explain the behavior of materials at small scales. In particular, an increase in yield strength with decreasing size has been experimentally observed in several materials and under different loading conditions. A class of non-classical continuum mechanics theories has been recently employed in order to explain the wide range of observed size dependent phenomena. The theories are non-classical in the sense that they bring about additional kinematic variables. In the numerical treatment of such theories two issues are clearly identified. First, in a displacement based finite element approach the need appears for higher orders of continuity in the interpolation functions or else alternative formulations must be used. Second, if nonlinear-inelastic material response is expected the theories should be recast in rate form and the corresponding integration algorithms should complement the implementation. In this article we address both problems for the particular case of a Cosserat couple stress theory. We describe alternatives for the numerical treatment and then we extend the framework to the case of a rate independent inelastic-non-linear material behavior. The equations are presented in its flow theory form together with integration algorithms. The development of microelectronics and other small scale related problems have pushed for a recent interest in continuum mechanics theories incorporating length scales. This class of theories is mainly intended to explain size dependent response observed in a variety of materials and testing conditions. Classical continuum mechanics theories have no internal length scale and they implicitly assume that the wavelength of the imposed deformation field is many times larger than the representative volume element (RVE) of the material. This means that there is no deformation localization or that gradients of strain are rapidly smoothed out. Under plastic conditions and non-uniform deformation fields it appears that localization phenomena occur at specimen sizes which are already many times larger than this RVE. In other words, the response becomes size dependent at volumes where continuum mechanics theories are still applicable. Since classical continuum mechanics theories have no internal length scale, they are unable to predict the wide range of observed size dependent phenomena. In the strain gradient theories available in the literature the length scale appear as an additional material parameter that enhances the resistance with the gradients of strain. Theories incorporating length scale (l) theories, when some characteristic dimension representative of the plastic …