Mechanical behavior of idealized, stingray-skeleton-inspired tiled composites as a function of geometry and material properties

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Abstract

Tilings are constructs of repeated shapes covering a surface, common in both manmade and natural structures, but in particular are a defining characteristic of shark and ray skeletons. In these fishes, cartilaginous skeletal elements are wrapped in a surface tessellation, comprised of polygonal mineralized tiles linked by flexible joints, an arrangement believed to provide both stiffness and flexibility. The aim of this research is to use two-dimensional analytical models to evaluate the mechanical performance of stingray skeleton-inspired tessellations, as a function of their material and structural parameters. To calculate the effective modulus of modeled composites, we subdivided tiles and their surrounding joint material into simple shapes, for which mechanical properties (i.e. effective modulus) could be estimated using a modification of traditional Rule of Mixtures equations, that either assume uniform strain (Voigt) or uniform stress (Reuss) across a loaded composite material. The properties of joints (thickness, Young’s modulus) and tiles (shape, area and Young’s modulus) were then altered, and the effects of these tessellation parameters on the effective modulus of whole tessellations were observed. We show that for all examined tile shapes (triangle, square and hexagon) composite stiffness increased as the width of the joints was decreased and/or the stiffness of the tiles was increased; this supports hypotheses that the narrow joints and high tile to joint stiffness ratio in shark and ray cartilage optimize composite tissue stiffness. Our models also indicate that, for simple, uniaxial loading, square tessellations are least sensitive and hexagon tessellations most sensitive to changes in model parameters, indicating that hexagon tessellations are the most “tunable” to specific mechanical properties. Our models provide useful estimates for the tensile and compressive properties of 2d tiled composites under uniaxial loading. These results lay groundwork for future studies into more complex (e.g. biological) loading scenarios and three dimensional structural parameters of biological tilings, while also providing insight into the mechanical roles of tessellations in general and improving the design of bioinspired materials.

Introduction

The tiling of surfaces with repeated geometric elements is a common structural motif in biological tissues and one that transcends phylogeny. Structural tilings have evolved independently in multiple systems and at a variety of size scales: from the micron-scale plates in the layers of nacre in mollusc shells (Barthelat and Zhu, 2011), to the sub-millimeter mineralized tiles (tesserae) sheathing the cartilages of sharks and rays (Seidel et al., 2016), to the macroscopic plates in the body armors of boxfish (Yang et al., 2015) and turtle shells (Chen et al., 2015, Krauss et al., 2009) as shown in Fig. 1a. The mechanical characteristics of tiled natural composites are typically impressive amalgamations of those of their mineralized and organic component parts, resulting in natural armors that can be both lightweight and puncture resistant, but also flexible and tough (Chen et al., 2015, Krauss et al., 2009, Liu et al., 2010, Liu et al., 2014, Martini and Barthelat, 2016, Rudykh et al., 2015, Yang et al., 2013, Yang et al., 2012). The shapes and materials of the tiling subunits, their spatial arrangement, and their physical interactions control composite functional properties, guiding deformation and hindering damage propagation (Krauss et al., 2009, Liu et al., 2010, Vernerey and Barthelat, 2010, Yang et al., 2015). Analytical and experimental models of suture behavior, for instance, show that simple adjustments to the geometry and/or attachment areas of sutural teeth can be used to tune the mechanical properties (e.g. stiffness, strength, toughness), deformation or failure behaviors of a structured composite (Krauss et al., 2009, Li et al., 2013, Lin et al., 2014).

The surface tiling of the skeleton of sharks and rays (elasmobranch fishes) has been recognized for over a century as a diagnostic character of all living members of this group, but the functional significance of this feature remains unclear. The tiled layer of elasmobranch cartilage, like most natural tilings, is comprised of hard inclusions/tiles (tesserae; Fig. 1b) joined by unmineralized collagen fibers (Fig. 2c; see also Seidel et al., 2016). However, elasmobranch tesserae lack the interdigitations found in many other biological tilings, such as those seen in turtle osteoderms or boxfish scutes (Fig. 1a) (Chen et al., 2015, Krauss et al., 2009, Yang et al., 2015). Furthermore, unlike the dermal scales of fishes, armadillo and some mammals, arrays of tesserae lack appreciable gaps or overlaps, and so can be considered “true tessellations” (Bruet et al., 2008, Chen et al., 2015, Wang et al., 2016, Yang et al., 2012). Elasmobranch tesserae also represent an intermediate size class of biological tiles, being typically hundreds of microns in size, an order of magnitude larger than mollusc nacre platelets and at least an order of magnitude smaller than most scales and osteoderms (Chen et al., 2015, Olson et al., 2012). The tessellation of the elasmobranch skeleton is believed to manage stress distribution in a way that can minimize damage to the cartilage and also provide both flexibility and stiffness (Fratzl et al., 2016, Lin et al., 2014, Liu et al., 2010), the latter being somewhat counterintuitive considering the lack of obvious interlocking features between tesserae. The correlation between the structural and material aspects of tesserae and the mechanical properties of the skeleton at a larger scale remain undemonstrated. In particular, although elasmobranch tessellation is apparently largely comprised of hexagonal tiles (Dean and Schaefer, 2005, Dean et al., 2016, Seidel et al., 2016), other shapes are possible (Fig. 1b); however, the role of tile shape in the mechanics of the tessellated composite (i.e. at the level of the skeletal tissue) has never been investigated.

In the current paper, our objectives are to analytically model biologically-inspired tessellated composites constructed with different tile types (triangle, square and hexagon) to observe the effects of (1) tile shape, (2) joint/tile size and (3) joint/tile material properties on the mechanical behavior (specifically, the effective stiffness) of the composite material (variables shown in Fig. 2c). Our results establish a baseline for future analyses of tessellations with more complicated (e.g. biologically relevant) morphologies (e.g. 3D tessellations) and loading conditions (e.g. bending, shear and multi-axial loading). The results presented in this study improve our understandings of the functional significance of the tesseral morphologies observed in elasmobranch skeletons, while also framing form-function laws for engineered tiled composites.

Section snippets

Modified Rule of Mixtures model

To estimate the mechanical characteristics of our tessellated composites, we modify traditional Rule of Mixtures methods, which allow calculation of the contributions of constituent phases to the net stiffness of a composite. These methods permit the modeling of different materials arranged either in parallel (Voigt iso-strain model) or in series (Reuss iso-stress model), taking into account their volume fractions (VF) and stiffnesses (E1 and E2) (Bayuk et al., 2008). Geometrical

Results and discussion

FEA and analytical calculations showed general agreement in their estimates of composite model stiffness as a function of E1/E2 and a given t/√A value (Appendix Fig. A.7). This supports our conjecture that our analytical models of a single tile and its surrounding joint material can be used to approximate the behavior of a larger tiled array, in a manner similar to FE models employing periodic boundary conditions (see Appendix A). Furthermore, our results were largely consistent, regardless of

Conclusions

All examined models show stiffening of the composite when joint widths are minimized and/or tile stiffness is maximized. On average, however, the effective modulus of the square array is least sensitive and that of the hexagon array most sensitive to changes in model parameters. This suggests that square arrays would be less sensitive to structural/material variation (e.g. a wide range of E1/E2 values results in the same effective modulus, particularly when joints are thick), whereas hexagon

Acknowledgments

We would like to thank the organizers of the ‘Articulated Structures and Dermal Armor’ symposium at the 2015 International Conference on Mechanics of Biomaterials and Tissues for the opportunity to publish in this volume. We also thank Callie Crawford, Andrew Gillis, Matt Kolmann, James Michaelson in collaboration with the Virtual Museum of Natural History, Michael Porter, Tristan Stayton and Adam Summers for providing the scan data for the images in Fig. 1. We also thank Bas Overvelde for his

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