Size and disorder effects in elasticity of cellular structures: From discrete models to continuum representations

Abstract

Cellular solids usually possess random microstructures that may contain a characteristic length scale, such as the cell size. This gives rise to size dependent mechanical properties where large systems behave differently from small systems. Furthermore, these structures are often irregular, which not only affects the size dependent behavior but also leads to significant property variations among different microstructure realizations. The computational model for cellular microstructures is based on networks of Timoshenko beams. It is a computationally efficient approach allowing to obtain statistically representative averages from computing large numbers of realizations. For detailed analysis of the underlying deformation mechanisms an energetically consistent continuization method was developed which links the forces and displacements of discrete beam networks to equivalent spatially continuous stress and strain fields. This method is not only useful for evaluation and visualization purposes but also allows to perform ensemble averages of, e.g., continuous stress patterns -an analysis approach which is highly beneficial for comparisons and statistical analysis of microstructures with respect to different degrees of structural disorder. (Stefan Liebenstein ) 1 In the following, we denote by 'microstructure' internal sub-structures of a material which are characterized by length scales well below the size of a specimen or device, the geometry of which defines the 'macrostructure'.

Kinematics and constitutive relations in the stress-gradient theory: interpretation by homogenization

“…The subsequent question is whether the present homogenization approach allows quantitative predictions of this size effect. Figure 9 compares the experimental results of Andrews et al [2] and the direct numerical simulations (DNS) of Tekoğlu et al [18] and Liebenstein et al [12] with the predictions of the present homogenization theory. Andrews et al [2] investigated two materials ("Alporas", "Duocell").…”

“…It is known that foam materials exhibit size effects when the specimen size becomes comparable to the cell size of the foam. In particular, negative size effects under uni-axial loading have been observed in experiments with foams [2] and direct numerical simulations with discretely resolved strut structure [12,18]. The observed negative size effect was attributed to a surface layer of incomplete cells which do not carry any load [2,16,20].…”

“…Alternatively, relations (12) and (13) can be written in terms of a "generalized displacement tensor" [17]…”

A free, square, point-loaded ice sheet: A finite element-discrete element approach

“…Size-dependent behaviour of foam-like materials has been investigated in abundance, both experimentally and theoretically [Onck et al, 2001;Diebels and Steeb, 2002;Tekȏglu and Onck, 2005;Tekȏglu et al, 2011;Liebenstein et al, 2018]. A classic general treatise is that of Gibson and Ashby [1999].…”

“…The samples examined had a topology of perturbed hexagons, the cell walls were discretised with Euler-Bernoulli beam finite elements, and their relative densitiesρ 1. In [Liebenstein et al, 2018], a Voronoi-tessellated, rectangular, cellular solid with both a regular and an irregular topology was examined. The cell walls were discretised with Timoshenko beam finite elements and the relative densityρ 1.…”