A strain gradient generalized continuum approach for modelling elastic scale effects

This work follows a generalized continuum framework [C. Sansour, A unified concept of elastic–viscoplastic cosserat and micromorphic continua, J. Phys. IV Proc. 8 (1998) 341–348] to derive a first strain gradient formulation which features a generalized deformation description, new strain and stress measures. As a consequence of these new quantities a corresponding generalized variational principle is formulated and its underlying equilibrium equations are derived. The approach is completed by Dirichlet boundary conditions for the displacement field and its derivatives. The basic idea behind this generalized continuum theory is the consideration of a micro- and a macro-space which span together the generalized space. The approach is appealing from theoretical as well as numerical point of view as it allows for the consideration of classical material laws and circumvents the use of otherwise cumbersome representation theorems. The resulting expressions for first and second order stresses are obtained by numerical integration over the micro-space. In this way material information of the micro-space, which is here only the geometrical specifications of the micro-continuum, can naturally enter the constitutive law. Moreover, non-linear material behaviour can be considered in a straightforward manner. In this work conventional hyperelasticity will be considered.Four applications in the context of linear and non-linear hyperelasticity demonstrate the potential of the proposed method. In particular, the use of a moving least square-based meshfree method facilitates a pure displacement-based approximation scheme, as it can provide C1C1 continuity which is required for this strain gradient formulation.