Many natural and man-made
materials exhibit self-similar hierarchical microstructures on several length scales.
The effective macroscopic mechanical properties of such materials or composites are
affected by the number of hierarchical levels and the topology of microstructures.
Although the effective mechanical properties can be determined numerically using
homogenization techniques, the computational costs can become prohibitively high as
the level of hierarchy increases. This paper proposes an analytical approach to
predicting the effective stiffness of a class of materials and structures with self-similar
hierarchical microstructures. For each microstructural configuration, a simple
relationship between the effective stiffness and the hierarchical level is established
and verified against results of finite element analysis or data in the literature.
It is found that the simple relationships we have developed provide quite
accurate stiffness predictions of various hierarchical materials and composites
including the Menger sponge. For composites, the predicted effective stiffness is
accurate even when one of the phases is near its incompressibility limit, with
its Poisson ratio close to 0.5. Inspired by the Menger sponge and informed
by our topology optimization result, we propose a lighter yet stiffer “cross
sponge”.
Centre for Innovative Structures and
Materials
School of Civil, Environmental and Chemical Engineering
RMIT University
GPO Box 2476
Melbourne VIC 3001
Australia
Centre for Innovative Structures and
Materials
School of Civil, Environmental and Chemical Engineering
RMIT University
GPO Box 2476
Melbourne VIC 3001
Australia
Centre for Innovative Structures and
Materials
School of Civil, Environmental and Chemical Engineering
RMIT University
GPO Box 2476
Melbourne VIC 3001
Australia
Centre for Innovative Structures and
Materials
School of Civil, Environmental and Chemical Engineering
RMIT University
GPO Box 2476
Melbourne VIC 3001
Australia