Homogenization of fully nonlinear rod lattice structures: on the size of the RVE and micro structural instabilities

Abstract

This work investigates the possibility of applying two-scale computational homogenization to rod lattice structures emerging, for instance, from additive manufacturing. The influence of the number of unit cells within the representative volume element (RVE), thus, the RVE’s size on the homogenized mechanical response is studied for occurring microscopic structural instabilities. Therein, the macro-scale, described in terms of three-dimensional continuum mechanics, is coupled to the micro-scale described by geometrically exact rods, enabling arbitrary large deformations and rotations. A special feature of the presented framework is that the rods building the lattice structures are not restricted to deform purely elastically but may deform inelastically. The mechanical response of lattice structures is investigated by applying the developed homogenization method to an exemplary lattice. Under special loads the structure reaches an instable state and may buckle. The appearance of instabilities depends on the geometric properties of the lattice’s underlying rods and the RVE’s size.

Two-scale off-and online approaches to geometrically exact elastoplastic rods

“…In the latter, a yield-Table 1 Strain measures and their energetic conjugates are denoted differently in the context of geometrically exact rods and three-dimensional continuum mechanics. This terminology is consistent with [17,23] Geometrically exact rods Three-dimensional continuum mechanics strain measures strain prescriptors strain energetic conjugates stress resultants stress surface in terms of stress resultants is used and a convenient resultant-type hardening potential is introduced. Finally in sect.…”

“…A translation of the rod's strains into continuous strain measures is no more necessary. The thermodynamically consistent extension from the elastic to the inelastic case is given in [16] and further elaborated in [17,18]. The challenge in modelling geometrically exact elastoplastic rods is the lack of knowledge of an appropriate yield-function in terms of the rod's stress resultants and an appropriate resultant-type hardening formulation.…”

“…Resultant-type hardening, which allows for the evolution of the yield-surface [24] has not yet been considered for the case of geometrically exact rods. Hardening is commonly captured with a diagonal hardening tensor [17,18].…”

“…In particular, the focus is on capturing hardening in geometrically exact elastoplastic rods by two different approaches. The frameworks presented here may be further used to investigate on limit loads of slender structures or periodically repeating lattice structures [17]. Both presented approaches make use of two-scale homogenization techniques and distinguish in the following way.…”

“…Before setting the stage by recalling the theory of geometrically exact rods, the terminology used throughout this contribution is introduced. As in Herrnböck et al [17,23] strain measures and their energetic conjugates are denoted differently in the context of three-dimensional continuum mechanics and geometrically exact rods. The different terminology is summarized in table 1.…”