XXIV ICTAM, 21-26 August 2016, Montreal, Canada
ENHANCEMENT OF BUCKLING STRENGTH IN THE TAPERED SKELETAL ELEMENTS
OF MARINE SPONGES
Michael A. Monn1 and Haneesh Kesari ∗1
1
Brown University School of Engineering, Brown University, Providence, Rhode Island, USA
Summary Biological structures, such as stems and bones, possess some very distinct mechanical designs. By comparing variants of these
designs using synthetic materials, scientists have shown that bio-inspired designs can lead to an enhancement of the materials’ properties.
However, a major criticism of pursuing bio-inspired engineering has been that there are scarcely any examples of structural biomaterials that
have been rigorously shown to contain a close-to-optimal design. To address this criticism, we discuss the skeletal elements of the marine
sponge Tethya aurantia called spicules, which are monolithic silica rods that are 1–2 mm long, 30–50 µm thick. The spicules’ mechanical
effectiveness is primarily limited by their buckling strength. Spicules possess a distinct taper along their length. Using recent mathematical
results, mechanical testing and computational modeling we show that the spicules’ tapered shape is very close to the shape that has the
greatest buckling strength.
INTRODUCTION
Structural biomaterials (SBs), such as bones and shells, possess some remarkable mechanical properties, such as high specific toughness and stiffness [1, 2]. Underlying the SBs’ remarkable properties are the structures’ unique mechanical designs,
which are a consequence of millions of years of evolution. These distinct designs have serendipitously inspired engineers to
come up synthetic materials that possess enhanced mechanical properties [3]. Currently, engineers are interested in making
the process of discovering new ideas and designs from nature more systematic. However, these efforts are being excessively
undermined by the idea that evolutionary adaptations do not create perfect designs, but rather only provide enhancements to
make biological materials perform well enough. Furthermore, there are scarcely any examples of SBs that have been rigorously shown to contain a close to optimal design. For this reason, it is natural to wonder whether evolution is even capable of
producing a “perfect” structure.
To provide a balanced perspective on the benefits of looking to nature for inspiration we discuss the form and function of
the skeletal elements of the marine sponge Tethya aurantia. These elements, called spicules, are axisymmetric, monolithic
silica rods that are 1–2 mm long, and 30–50 µm thick. The spicules are distributed in the sponges’ compliant collagenous body
to enhance its stiffness [4]. These spicules possess a distinct taper along their length, which we found be a highly uniform
feature across individual specimens. The spicules are internally homogeneous and we found that their deformation is linearly
elastic and that they break cleanly into two pieces in an abrupt, brittle fashion. From these two observations and the slender
nature of the spicules, we infer that they primarily fail through a buckling instability, which is triggered by the compressive
tractions that are distributed along their lengths.
The spicules most certainly experience various, complex compressive traction distributions along their length. However,
we show that of all possible compressive load distributions the case where the forces are concentrated at the ends is the most
critical. Furthermore, we found using numerical modeling that the compressive loads on the sponge body mostly appear as
concentrated forces on the spicule ends. Thus, the spicule’s performance is limited by the maximum compressive force that
it can support on its ends prior to encountering the buckling instability. It was proven only recently that columns that possess
what we term the Clausen profile can withstand the largest compressive end forces for a given volume of material before
buckling [5]. We found that the tapered profiles of the spicules are very close to the Clausen profile. In addition to maximizing
the buckling strength, we also show that small imperfections in the Clausen profile result in only minor decreases in the critical
buckling load. This design is therefore not only optimal, but also robust against potential fabrication constraints. Thus, our
results demonstrate that in at least some cases evolution can be powerful enough to lead us to the best solutions.
HYPOTHESIS AND SPICULE SHAPE MEASUREMENTS
Buckling is a phenomenon that occurs when an, often slender, element under compressive forces transitions from a purely
compressed state to a state characterized by large lateral deflections. This sudden transition, or “bifurcation”, in the deformation behavior results in a loss of the structure’s capacity to transmit loads and is characterized by increased compliance and
the presence of tensile stresses due to bending.
For an axisymmetric column with radius r that can vary with the position z along its length L. Of all possible r(z) over
[0, L], for which the total volume V is constant, the shape r̂(z) that results in the maximum load transmitted before the onset
of buckling is given by
∗ Corresponding
author. Email: Haneesh [email protected]
r̂(θ) = Lα sin(θ)
1
L
θ − sin(2θ)
z(θ) =
π
2
(1a)
(1b)
where the parameter θ varies from 0 to π, and α is the column’s aspect ratio [5]. This shape provides a 33% enhancement to
the critical buckling load over that of an equal volume cylindrical column.
The Clausen profile is a very gradually tapered shape; a feature that serves as the motivation for our study of the spicules.
Specifically, we hypothesize that the taper of the spicules maximizes structural support provided to the sponge by maximizing
their critical buckling load.
To test our hypothesis, we quantify the taper of 32 spicules by extracting the boundary of the spicule profile from stitched
SEM images. We consider four candidate models – constant cross-section (CCS), ellipsoid, double-cone, and the Clausen
profile – to describe the shape of the spicules. For each spicule measured, we fit the four models to the boundary data points
using the aspect ratio as the fitting parameter and minimizing the sum of squared errors (SSE) between the data and the
model. In order to select the best fit model we compute Akaike’s Information Criterion (AIC) for the four candidates for each
spicule [6]. We find that of the four candidate models considered, there is very strong support that Clausen profile is the best
descriptor of the spicule shape. This similarity between the Clausen profile and the measured taper of the spicules reinforces
our hypothesis that the tapered shape of the spicules is an adaptation for making the sponge body stiffer.
Table 1: Summary of model comparison metrics across the 32 spicules
SSE × 1000
mean
Clausen (1)
ellipsoid
CCS
double cone
0.16 ±
0.28 ±
0.77 ±
2.13 ±
SD
0.08
0.17
0.40
0.84
AIC
mean
-3608.3 ±
-3475.7 ±
-3205.6 ±
-2932.9 ±
SD
160.8
187.8
145.9
102.9
ROBUSTNESS OF THE CLAUSEN PROFILE
Finally, we investigate the sensitivity of the enhancement in buckling load to deviations from the perfect Clausen column
shape. Specifically we perturb the Clausen profile while enforcing the constraint that the volume of the perturbed and unperturbed columns must be equal. We numerically compute the buckling load for the perturbed shape and compare it to the
critical buckling load of the unperturbed Clausen profile. By generating 104 different randomly perturbed shapes, we estimate
the worst case reduction of the critical buckling load as a function of the magnitude of the perturbation.
We repeat this process for an equal volume CCS column and show that the critical buckling load of the Clausen profile is
affected less by imperfections in its cross-sectional area.
References
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