A formulation of a Cosserat-like continuum with multiple scale effects

In this work a generalized continuum formulation is introduced which is based on a theoretical framework of a generalized deformation description proposed by Sansour (1998) [25]. That is the deformation field is composed by macro- and micro-components according to the consideration that the generalized continuum consists of a macro- and micro-continuum. It is demonstrated that by specific definition of the topology of the micro-space this generalized deformation formulation allows for the derivation of a generalized variational principle together with corresponding strain measures and underlying equilibrium equations. The approach makes use of a macroscopic rotation field which is considered to be element of the Lie group SO(3) and independent of the macroscopic displacement field. In that way the formulation incorporates three additional rotational degrees of freedom and is closely related to the Cosserat continuum. In contrast to the conventional Cosserat continuum the proposed generalized formulation allows to describe multiple scale effects associated with multiple micro-structural directions, possibly with a different magnitude for each one. The approach considers a geometrically exact description of finite deformation within the macro-continuum, but as a first step linearises the deformation within the micro-continuum. The constitutive law is defined at the microscopic level and the geometrical specification of the micro-continuum is the only material input which goes beyond that needed in a classical description. s meshfree computations demonstrate that this model is able to address fundamental physical phenomena which are related to the underlying micro-structure of the material, in particular scale-effects and oriented material behaviour. Clear differences are revealed between a classical and the non-classical formulation.