0
RESEARCH PAPERS

# Robust Design of Cellular Materials With Topological and Dimensional Imperfections

[+] Author and Article Information

Mechanical Engineering Department, The University of Texas at Austin, 1 University Station, C2200 Austin, TX 78712ccseepersad@mail.utexas.edu

Janet K. Allen, Farrokh Mistree

G.W. Woodruff School of Mechanical Engineering, Systems Realization Laboratory, Georgia Institute of Technology, Atlanta, GA 30332-0405

David L. McDowell

G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

Mesoscopic length scales (on the order of tens to hundreds of micrometers in this research) are intermediate between microscopic length scales, which apply to characteristics like gradients of chemical composition and microstructure (e.g., grain boundaries, dislocations, crystal structure), and macroscopic length scales, much greater than the characteristic lengths of heterogeneities, at which homogeneous continuum models are valid.

The plots in Fig. 2 assume relative densities of 20% and doubly periodic structures. $E11∕Es,E22∕Es$ represent effective elastic stiffness for uniaxial loading in the in-plane principal directions. $G12∕Es$ is the effective elastic shear stiffness in the in-plane transverse direction.

Frame finite elements are a superposition of 1D beam and bar finite elements. Frame elements are used to account for transverse loads and bending in cell walls, in addition to axial deformation. Both mechanisms are observed in prismatic cellular materials subject to in-plane loading and elastic deformation.

The area of the unit cell is chosen to correspond to typical size ranges for unit cells fabricated with the thermochemical extrusion process described in Sec. 1.

All elements are assigned a unit depth in the out-of-plane direction. Therefore, the in-plane thickness, $Xi$, of element $i$ is equivalent to its cross-sectional area.

To preserve and enhance the effectiveness of gradient-based optimization algorithms for solving a robust topology design problem, the variation model in Eq. 6 is extrapolated beyond the manufacturable range of cell wall thicknesses, $[XMinMfg,XU]$, to the bounds established for the topology design process, $[XL,XU]$, with $XL$ assumed to be arbitrarily small (i.e., $0⩽XL⩽XU)$.

Some additional approximation and round-off error is associated with the extremely small magnitude of $C33$ for square cells.

The Taylor series approach is also known as worst-case analysis, a term introduced by Parkinson and co-authors (28), because fluctuations are assumed to occur simultaneously in a worst-case combination. It is most accurate for small tolerances and weak or negligible interactions among the factors that fluctuate, and it is based on the assumption that tolerance ranges, rather than statistical distributions, are assigned to relevant factors.

Orthotropic symmetry implies that modifications in the designed quadrant are mirrored immediately to the other three quadrants during the topology design process. Therefore, removing a node in one quadrant is equivalent, in its effect on material properties, of removing any of its three symmetric nodes. Similarly, periodicity implies that each unit cell in the SVE is identical in the designed mesostructure. If only one node is removed from the SVE at a time, removing a node from one unit cell in the SVE is equivalent, in its effect on material properties, of removing the same node from any other unit cell.

The vector of design variables changes for each experiment because a different node is removed in each experiment along with the elements that are connected to it.

Note the relationship between $C33$ and $G12$ recorded in Eq. 5.

Recall that the unit cells have orthotropic symmetry; therefore, only one quadrant of cell wall dimensions is labeled for each unit cell.

1

Corresponding author.

J. Mech. Des 128(6), 1285-1297 (Jan 09, 2006) (13 pages) doi:10.1115/1.2338575 History: Received July 20, 2005; Revised January 09, 2006

## Abstract

A paradigm shift is underway in which the classical materials selection approach in engineering design is being replaced by the design of material structure and processing paths on a hierarchy of length scales for multifunctional performance requirements. In this paper, the focus is on designing mesoscopic material topology—the spatial arrangement of solid phases and voids on length scales larger than microstructures but smaller than the characteristic dimensions of an overall product. A robust topology design method is presented for designing materials on mesoscopic scales by topologically and parametrically tailoring them to achieve properties that are superior to those of standard or heuristic designs, customized for large-scale applications, and less sensitive to imperfections in the material. Imperfections are observed regularly in cellular material mesostructure and other classes of materials because of the stochastic influence of feasible processing paths. The robust topology design method allows us to consider these imperfections explicitly in a materials design process. As part of the method, guidelines are established for modeling dimensional and topological imperfections, such as tolerances and cracked cell walls, as deviations from intended material structure. Also, as part of the method, robust topology design problems are formulated as compromise Decision Support Problems, and local Taylor-series approximations and strategic experimentation techniques are established for evaluating the impact of dimensional and topological imperfections, respectively, on material properties. Key aspects of the approach are demonstrated by designing ordered, prismatic cellular materials with customized elastic properties that are robust to dimensional tolerances and topological imperfections.

<>

## Figures

Figure 1

Examples of ordered, prismatic cellular materials

Figure 2

Effective elastic properties of standard periodic cellular topologies

Figure 3

Outline of the robust topology design method (see Ref. 35)

Figure 4

An initial ground structure for a cellular mesostructure (a) and for a representative unit cell (b). A designed unit cell after topology design (c) and after post-processing (d). A designed cellular mesostructure (e) comprised of a doubly periodic pattern of designed unit cells (d).

Figure 5

Course (a) and fine (b) initial ground structures for cellular mesostructure design

Figure 6

An example of a topological imperfection in a cellular mesostructure

Figure 7

Decision Support Problem for robust topology design of 2D periodic cellular mesostructure with topological and dimensional variation

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections