A non-linear Cosserat continuum-based formulation and moving least square approximations in computations of size-scale effects in elasticity

The Cosserat continuum falls into the group of so-called generalized continua which have the capacity to consider internal lengths and so describe a certain type of size effects. The paper addresses some aspects of the non-linear formulation of the Cosserat continuum and its meshfree approximation. The use of moving least square approximations [P. Lancaster, K. Salkauskas, Mathematics of Computations 37(155) (1981) 141–158] in a non-linear Cosserat continuum-based formulation gives rise to certain implications which are related to the essential boundary condition enforcement as well as to the updating of the rotation field. Here, the enforcement of the displacement boundary conditions is accomplished by modifying the initial variational principle or weak form of equations such that the essential boundary conditions appear as Euler–Lagrange equations. The updating of the rotational degrees of freedom in the meshfree code adopts a multiplicative scheme which is based on the spinor theory and has been already successfully applied to finite elements by the first author. The suitability of the proposed methodology is exemplified in modelling size-scale effects in elasticity. The impact of the Cosserat continuum-based formulation on small-scale structures is demonstrated and the comparison with a classical Green strain tensor-based approach reveals significant differences.