Periodic truss structures
Introduction
Cellular structures and materials are ubiquitous in biological systems (Wainwright et al., 1982), structural engineering (Evans et al., 2001) and materials science (Gibson and Ashby, 1997). Broadly, they consist of periodic arrays of plate- or strut-like elements. They can be designed to most efficiently exploit the properties of the constituent elements and/or the intervening spaces in achieving functionality, e.g., bearing loads, enabling fluid flow, facilitating heat transfer, altering optical transmission. They are generally superior to structures in which the elements are distributed in a non-periodic manner, e.g. stochastic foams (Evans et al., 2001). In some cases (e.g. photonic materials), periodicity is essential to achieving functionality. For load bearing applications, strut-like elements in particular are preferred: the load for initiating buckling of a slender strut being much higher than that of a comparable plate with the same mass.
Strut-based cellular structures and materials – hereafter collectively referred to as trusses – are under development for use in an incredibly broad range of technologies, including structural biomedical implants (Murr et al., 2010), aerospace and naval structures (Evans et al., 2001), cushioning and force protection systems (Wadley, 2000), thermal management (Wadley, 2000), actuated structures (Lucato et al., 2004, Hutchinson et al., 2003) and photonic materials (Bückmann et al., 2012, Bauer et al., 2014). Five main classes of fabrication routes have been employed.
- (i)
Investment casting has been used to make laboratory-scale Al-alloy truss structures (Deshpande et al., 2001, Chiras et al., 2002, Wallach and Gibson, 2001). Investment casting is generally the most expensive and least amenable to large-scale production relative to other fabrication routes.
- (ii)
Fabrication schemes based on conventional machining, bending, assembly and brazing of sheet materials have been devised to make metallic trusses (Wadley, 2000, Rathbun et al., 2004). In one version, diamond-shaped holes are punched or laser-machined into thin steel sheet, leaving an X-pattern of narrow struts. The sheet is then bent along lines of nodes to produce one layer of the targeted truss (Wadley et al., 2003).
- (iii)
Metallic trusses can also be made by weaving wires into the desired structure and subsequently brazing the wires together (Wadley, 2000, Kang, 2015). One of the drawbacks is that the weaving operations yield wavy or kinked strut segments between nodes. Moreover, since the nodes are formed by brazing of contacting wires, the integrity of these nodes is likely to be strength-limiting.
- (iv)
Additive manufacturing offers the widest range of topology options. Some of the most notable developments in recent years have been in Ti-alloy trusses, produced by selective electron beam melting (EBM) of fine alloy powders, for biomedical implants (Murr et al., 2010, Murr et al., 2011, Cheng et al., 2012, Li et al., 2014). In another arena, direct laser writing by optical lithography has been used to fabricate polymer truss structures with extremely fine-scale (sub-micrometer) features, for potential use in photonic applications (Bückmann et al., 2012, Bauer et al., 2014).
- (v)
Self-propagating photocuring (SPPC) of photosensitive polymers has found utility in rapid fabrication of polymer trusses for use in impact mitigation and cushioning systems (Jacobsen et al., 2007, Jacobsen et al., 2007, Jacobsen et al., 2008). The main advantage of this process is the short time needed for polymerization (typically less than a minute). One significant limitation is the narrow range of materials that exhibit the requisite physical and chemical properties for SPPC as well as the mechanical properties to produce useful truss structures. It is also restricted to topologies in which all struts intersect one of the external faces. That is, it is inherently a “line-of-sight” curing method.
Despite the broad recognition of the potential of periodic trusses for use in many diverse fields of technology, there are no widely-accepted descriptors of their structure. In the numerous articles on this topic that have appeared in the past two decades, the terminology has been based loosely on descriptions of various polyhedra, but often without explicit connections between truss structure and specific characteristics of the reference polyhedron.
For example, trusses designated as pyramidal are conceptually constructed by placing struts along the four edges of a regular square pyramid at which the triangular faces intersect, but not along the edges of the square base (Evans et al., 2001, Wadley, 2000). Similarly, tetrahedral trusses are formed by placing struts along three non-coplanar edges of a tetrahedron, but not on the other three edges (Wadley, 2000, Rathbun et al., 2004). In other cases, truss structures are constructed by placing struts normal to and at the center of each face of the reference polyhedron (not along the edges), e.g. the truncated octahedral truss (Gurtner and Durand, 2014, Weaire, 1996). Elsewhere, truss structures have been described as being “tetrahedral with three-fold symmetry” or “tetrahedral with six-fold symmetry”, without explicit designations of strut locations (Jacobsen et al., 2008).
In some instances, new words have been devised to describe truss structure. The octet truss, for example, derives from a combination of octahedral and tetrahedral. Here struts are placed along all edges of a series of regular octahedra and tetrahedral arranged to fill three-dimensional space (Deshpande et al., 2001). Other truss structures have been described loosely as “fully triangulated”, “bulk cross” (Kang, 2015), “cross I symmetric”, “G6”, “G7”, “dode-thin”, and “hatched” (Murr et al., 2010, Murr et al., 2011, Cheng et al., 2012). These and the preceding designations are re-visited in a later section of this article.
In addition to the vagaries introduced by using polyhedra as the basis of truss designations, the terminology fails to recognize the fundamentally different nature of polyhedra and of trusses. A polyhedron is a three-dimensional solid whose outer boundaries are defined by plane polygons such that the edge of each polygon belongs to one other polygon. A truss, on the other hand, consists of a set of points (or nodal locations) and a set of lines (or struts) joining certain points. Solid geometry alone lacks the structure needed to completely and unambiguously describe truss structure.
Descriptions of trusses have also frequently invoked terms derived from the field of crystallography. Examples include “body centered cubic” and “diamond”. Indeed, the association between nodal positions of trusses and space lattices in crystallography has led to the characterization of trusses as lattice materials, lattice structures or simply lattices. In addition to the unfortunate conflict with the definitions of lattices in the context of crystallography, the terminology (again) fails to recognize the fundamental differences between space lattices and truss structure: A space lattice defines only an array of regularly-spaced points and provides no information about the connectivity of those points (i.e. topology). Therefore, crystallography alone (like solid geometry) lacks the structure needed to describe truss structure.
Section snippets
Elements of a new language
The principal objective of the article is to present a framework within which trusses can be systematically described and classified. This requires a set of conventions and terminology that, when applied in a consistent manner, yields concise yet unambiguous descriptions of structure types and of specific truss designs. In turn, this goal requires a language of truss structure. As with any language – whether expressed by words in a spoken language or by symbols in mathematics or music – the
Development of the language of truss structure
In the present context, trusses are defined as arrays of straight, interconnected struts with periodic character. They can comprise few (large) repeating units, as found in structural engineering, or aggregates of many (small) repeating cells that, collectively, behave essentially as a material. Their structure is defined completely by: (i) the positions of all nodes in space, and (ii) the connectivity of the nodes by struts. Details of node geometry, strut cross-section, strut waviness and
Generalizations of designations of trusses and structure types
Building upon the pattern established in the preceding examples, a generic terminology is readily developed. As demonstrated in a subsequent section, the terminology can be applied unambiguously to descriptions and classifications of periodic trusses. A summary of the taxonomy is presented in Table 3.
An elementary truss consisting of a linear, square or cubic array of cells is expressed generically as where A is the truss type (SC, BCC or FCC), n is the number of unit cells in each row
Applications
The many disparate truss designations introduced in prior literature can be concisely and unambiguously described using the present system of classification and taxonomy. For example, a pyramidal truss, which contains struts aligned along the four edges at which the triangular faces of a regular square pyramid intersect, is, in general, of the structure type , where the base of the unit cell coincides with the pyramid base. From geometry, the stretch ratio (measured
Merits of compound trusses
To illustrate the merits of compound trusses, we compare the elastic properties of two truss types: and . The latter consists of two elementary trusses that, on their own, act as mechanisms, but together produce a stiff structure.
The elastic properties were computed by finite element (FE) analysis. The FE mesh was created using linear Euler–Bernoulli beam elements, suitable for small-strain analyses with small rotations. The strut slenderness ratio, , was selected to
Concluding discussion
We have presented a system for classification and taxonomy of periodic truss structures. The system employs concepts from crystallography and geometry to describe nodal locations and connectivity of struts. The conventions and terminology yield concise yet unambiguous descriptions of structure types and of specific truss structures. The system captures a broad range of trusses that have been studied in various science and engineering fields and could be expanded to include structures with even
Acknowledgments
This work was supported by the Institute for Collaborative Biotechnologies through Grant W911NF-09-0001 from the US Army Research Office. The content of the information does not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred.
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