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Biaxial mechanical characterization of bat wing skin

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Bioinspir. Biomim. 10 (2015) 036004

doi:10.1088/1748-3190/10/3/036004

PAPER

Biaxial mechanical characterization of bat wing skin RECEIVED

7 August 2014 REVISED

A J Skulborstad1, S M Swartz2 and N C Goulbourne1

19 November 2014

1

ACCEPTED FOR PUBLICATION

16 December 2014 PUBLISHED

20 April 2015

2

Department of Aerospace Engineering, University of Michigan, FXB Building, 1320 Beal Avenue, Ann Arbor, MI 48109, USA Department of Ecology and Evolutionary Biology and School of Engineering, Brown University, Box G-B206, Providence, RI 02912, USA

E-mail: [email protected] Keywords: biaxial mechanical testing, bat wing skin, tissue mechanics, DIC, fiber kinematics, large deformation

Abstract The highly flexible and stretchable wing skin of bats, together with the skeletal structure and musculature, enables large changes in wing shape during flight. Such compliance distinguishes bat wings from those of all other flying animals. Although several studies have investigated the aerodynamics and kinematics of bats, few have examined the complex histology and mechanical response of the wing skin. This work presents the first biaxial characterization of the local deformation, mechanical properties, and fiber kinematics of bat wing skin. Analysis of these data has provided insight into the relationships among the structural morphology, mechanical properties, and functionality of wing skin. Large spatial variations in tissue deformation and non-negligible fiber strains in the cross-fiber direction for both chordwise and spanwise fibers indicate fibers should be modeled as two-dimensional elements. The macroscopic constitutive behavior was anisotropic and nonlinear, with very low spanwise and chordwise stiffness (hundreds of kilopascals) in the toe region of the stress–strain curve. The structural arrangement of the fibers and matrix facilitates a low energy mechanism for wing deployment and extension, and we fabricate examples of skins capturing this mechanism. We propose a comprehensive deformation map for the entire loading regime. The results of this work underscore the importance of biaxial field approaches for soft heterogeneous tissue, and provide a foundation for development of bio-inspired skins to probe the effects of the wing skin properties on aerodynamic performance.

1. Introduction Biological tissues exhibit great diversity in composition and functionality. The wing skin of bats is highly flexible and extensible; strains of over 100% have been reported [1]. This contrasts with the much stiffer constituents of insect and bird wings [2, 3]. In concert with the skeletal structure and musculature, the skin enables large changes in camber and wingspan during forward and hovering flight. Although several studies have investigated the aerodynamics and kinematics of bats [4–9, among others], few have investigated the complex histology and mechanical response of the wing skin. In this work, we present the first study of the mechanical properties of bat wing skin under biaxial extension, and gain insights in connections between the morphological structure, mechanical properties, and functionality of the wing skin. Among the few histological analyses conducted to date, Gupta studied several bat species and found that © 2015 IOP Publishing Ltd

the wing skin is organized into several layers [10]. The external dorsal and ventral surfaces are epidermal. Deep to the surface layers are thicker dermal layers containing small collagen and elastin fibrils as well as smooth muscle fibers. Finally, a central hypodermal layer contains loose connective tissue containing blood vessels, nerve processes, striated muscle fibers, and larger elastin bundles or fibers [10]. Holbrook and Odland characterized the net-like fiber ultrastructure of the central layer of Tadarida brasiliensis, the Brazilian free-tailed bat, under light microscopy, transmission electron microscopy (TEM), and scanning electron microscopy [11]. In that species, the fiber architecture in the plagiopatagium (PLP), between the body and the fifth digit, consists of parallel chordwise fibers extending from the humerus, radius, and ulna to the trailing edge, and spanwise fibers roughly perpendicular to the chordwise fibers [11]. The bundles in the T. brasiliensis wing are composed of numerous elastin

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Figure 1. The plagiopatagium of G. soricina wing skin contains spanwise corrugations. The fiber architecture, including roughly parallel chordwise and spanwise fibers, is revealed under polarized light at higher magnification and shown at the right of the image. Adapted from image by Jorn Cheney and Andrew Bearnot [12].

fibers enclosed by a loose sheath of collagen, as revealed by TEM and amino acid analysis [11]. Here, we study Glossophaga soricina, Pallas’ longtongued bat, a small primarily nectar-feeding bat with exceptional hovering abilities. Weighing 10–15 g, it has a wingspan of approximately 24 cm, and a chord (leading to trailing edge) of roughly 4 cm (figure 1) [12]. The PLP is the largest area of the wing, and therefore generates the majority of lift [4]. The thickness of the PLP is between 160 and 260 μm with a mean of approximately 200 μm. The PLP may be considered as a composite that consists of a heterogeneous matrix composed of ground substance and microscopic collagen and elastin fibrils. It possesses embedded mesoscopic fiber architecture consisting of two families of approximately perpendicular long, straight fibers, with one family oriented predominantly in the chordwise direction, and the other predominantly spanwise [1]. Based on optical properties, the spanwise fibers are primarily composed of elastin, and have diameters of approximately 50–70 μm and spacing of 1000–2000 μm [13, 14]. The chordwise fibers are muscles known as plagiopatagiales, with diameters and spacing of approximately 70–200 μm and 300–900 μm, respectively [13, 14]; the structures observed in the wing comprise striated skeletal muscle fibers and collagenous tendon. Like most soft tissue, the skin is considered incompressible due to its high water content [15, 16]. The wing skin exhibits spanwise corrugations between the chordwise fibers, in which the degree of wrinkling depends in part on the degree to which the wing skeleton extends the skin. The tissue, subtended from the arm and forearm to the connective tissue-reinforced trailing edge, is pretensioned (chordwise) by the frame of the body, humerus, radius, and fifth digit. Figure 1 shows the spanwise corrugations of the PLP, and the fiber architecture of chordwise and spanwise fibers under polarized light [12]. 2

To date, there is a single study of the mechanical response of bat wing skin under tensile loading, in which three regions of the wing skin of nine bat species were tested with unconstrained uniaxial tension [1]. These tests showed greatest stiffness in the direction parallel to adjacent wing bones, or roughly chordwise. The spanwise direction, perpendicular to wing bones, exhibited the greatest extension. Uniaxial tests are useful in providing some qualitative and quantitative mechanical information, however, they do not represent the multi-axial loading conditions experienced in vivo and are not sufficient to capture axial coupling exhibited by many soft tissues in which the material response along one axis depends on the deformation state of the other axis [17]. Although multi-axial testing is required to fully characterize anisotropic tissues from a theoretical point of view (the strain energy depends on at least three independent measures of deformation, for example invariants I1, I4, and I6 of the right Cauchy–Green tensor), biaxial testing is typically employed when testing thin tissues [18]. In this work, finite deformation biaxial testing was performed with in situ digital image correlation (DIC) of the tissue surface and polarized image correlation (PIC), a hybrid technique recently developed by the authors that incorporates polarized filters to visualize and compute strains in the underlying mesoscopic fiber architecture during biaxial deformation [13]. The primary objectives of this work are to (1) present the first biaxial characterization of the local deformation, mechanical properties, and fiber kinematics of bat wing skin, and (2) identify connections between the structural morphology, mechanical properties, and overall functionality of the wing skin. In particular, we propose a map connecting deformation mechanisms and structural constituents to regions of the macroscopic constitutive curves, investigate the physiological operating regime, and identify key properties for bio-inspired skins to systematically explore structural and material effects on aerodynamic performance metrics. A greater understanding of the tissue mechanics may enable the development of bioinspired skins that better help meet objectives of micro-aerial vehicle design aligned with capabilities of bats, including low weight, low noise, and high maneuverability including hovering. The paper is organized into five sections. In section 2, specimen handling and preparation is outlined, the custom biaxial testing device is described, the PIC technique is detailed, and all testing protocols are summarized. Section 3 outlines results of biaxial testing experiments, which includes local strain field response, mechanical properties, and fiber kinematics. In section 4, we propose a deformation map, discuss the wing skin mechanical properties in the context of likely physiological (in vivo) operating regimes, and provide a description of novel bio-inspired skins that have potential use for micro-aerial vehicles. Section 5

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Figure 2. (a) Biaxial test set up, indicating loops of suture tethering sample to force transducers. (b) Close-up image of speckling on skin sample surface for DIC and (c) snapshot of Green strain field for central area of the speckled image, indicated by the green shading in (b). Legend shows a histogram of the spatial distribution of strain.

2. Methods

and for establishing experimental protocols. In total, eleven samples were used to determine protocols, three samples tore during preparation, and six samples were tested with established protocols for characterization.

2.1. Tissue sample preparation Wing skin was taken from ten naturally deceased or euthanized adult G. soricina from an experimental colony housed at Brown University. None were aged or showed signs of any tissue damage or injury to the wings. Experiments were conducted in accordance with approved protocols (Brown University IACUC). Bat wing skin is delicate and perishable, requiring careful handling and rapid preparation; therefore we employ a specialized handling protocol that is outlined in detail in our previous work [14]. Briefly, frozen G. soricina were thawed in a bath of physiological saline solution at room temperature. Due to transportation and handling requirements, freezing the specimens was necessary. The effects of freezing on tissue mechanical properties are not fully understood. Some studies have shown significant differences in fresh and thawed tissue properties, while others have shown no or marginal differences in properties [20–25]. A 24 mm square frame was affixed to the tissue surface in the PLP region of the wing to keep the tissue from recoiling to a highly compact wrinkled configuration upon excision from the wing due to residual strains. These residual stretches were on the order of 1.5–3. The framed sample was excised, and polypropylene surgical sutures were looped through the sample edges to minimize edge effects (figure 2(a)) [26]. Samples were speckled with white paint particles to capture local strains with in situ DIC during mechanical testing, with spatial resolution on the order of the largest fiber diameters (approximately 165 μm) [14]. This is particularly important in heterogeneous materials where the scale of the heterogeneity is apparent at the mesoscale. Typically, one PLP sample was excised from each wing. Smaller samples from the distal dactylopatagium were also excised for use in exploratory stages

2.2. Biaxial mechanical testing apparatus and procedure Biaxial testing is required to capture axial coupling in the mechanical response of wing membrane skin and to preserve realistic fiber kinematics. A specialized apparatus was developed for large deformation biaxial testing of soft wing tissue as described in detail previously [13]. Briefly, each sample was tethered to four moving carriages with loops of suture attached to each side of the sample (figure 2). We employed tethered sutures to reduce edge effects such as stress shielding induced by clamping methods [25]. Stepper motors displace the moving carriages, and each axis can be controlled independently. Custom software enables displacement control, force control, strain control, or invariant control. For example, to implement Green strain control, stretch in the two loading directions is averaged over a small central region of the sample in real time and the stage controller updates the driver velocities to maintain a prescribed Green strain ratio, Ex:Ey. Force transducers attached to the moving carriages record force in both loading directions. A CCD camera with a macro lens, with resolution of 11 μm per pixel, was positioned above the setup to record images of the deforming sample for 2D image correlation deformation computations. This is a crucial feature for capturing accurate local deformations. The image correlation software Aramis™ was used to compute strains. This involves tracking the displacements of small regions called subsets through subsequent deforming images [26–28]. The unique grayscale distribution of each subset is found in the subsequent images by computing the correlation coefficient over a nearby region and finding the location of the maximum correlation [26–28]. Strains were calculated for 15 pixel square subsets by

summarizes key conclusions and suggests areas for future work.

3

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Table 1. Biaxial testing protocols. Samples were tested with either displacement control or Green strain control. Protocols are given as ratios of prescribed displacements (dx:dy) or strains (Ex:Ey). Sample

Test type

Test conditions

Alignment

Displacement protocols (dx:dy)

D1 D2 D3 P1 P2a C1 C2a

DIC DIC DIC PIC PIC DIC/PIC DIC/PIC

Displacement Displacement Green strain Displacement Displacement Displacement Displacement

on-axis on-axis on-axis on-axis off-axis on-axis off-axis

1:0, 3:1, 2:1, 1:1, 1:2, 1:3, 0:1 1:0, 3:1, 2:1, 1:1, 1:2, 1:3, 0:1 Ex:Ey = 1:0, 2:1, 1:1, 1:2, 0:1 dx:dy = 1:1 2:1, 3:2, 1:1, 2:3, 1:2 2:1, 1:1, 1:2 1:1 1:1

a

P2 and C2 are the same sample tested with two test types.

averaging with the surrounding eight subsets. With a spatial strain resolution of approximately 165 μm, we were able to resolve approximate fiber strains, but could not resolve the interfaces between fiber and matrix. The standard deviation error or noise floor for the strain was approximately 0.3% Green strain. To achieve the requisite surface for 2D image correlation, the inherent 3D corrugations of the skin were removed by displacing the spanwise sample edges prior to testing (a stretch of approximately 1.1). This added a small amount of stress in the spanwise fibers in the 2D testing reference configuration. For PIC, polarizing filters were incorporated into the apparatus above and below the sample to reveal the birefringent subsurface fiber architecture during biaxial testing [13]. The grayscale distribution of the polarized images showing the tissue subsurface fiber network provided the requisite image pattern for image correlation [13]. In addition, the matrix material exhibits small-scale inherent variations in grayscale distribution due to microscopic birefringent components so that both the mesoscopic fiber network strains and the matrix strains are resolvable. The same procedure is used to compute strains for the subsurface images as is used for the speckled surface images. The PIC technique is similar to other methods to track the deformation of subsurface or embedded constituents; for example, injected fluorescent nano-particles have been used for subsurface deformation tracking in polymers [29]. The PIC technique enables characterization of the embedded fiber kinematics and examination of the role of the fiber architecture in the overall mechanical response of the tissue. Traditional tests with ambient light use DIC to capture surface deformations of the speckled surface of the tissue. By utilizing both testing techniques, we can gain insight into the deformation mechanisms of the tissue components and their relation to the macroscopic properties and function of the tissue. Three testing conditions were implemented in this study: (1) DIC of the sample surface, (2) PIC of the subsurface fiber architecture, and (3) PIC integrated with DIC for simultaneous surface and subsurface strain fields. Custom LabView™ codes coordinate stage movement with force readings and image capture so that strain fields are synchronized with force 4

Figure 3. Off-axis sample alignment configuration with fibers oriented approximately 30°–35° to the loading axes. Symbols αx and αy denote angles of spanwise and chordwise fibers to the horizontal loading axis, respectively, and β denotes the relative angle between the spanwise and chordwise fibers.

measurements. Time-resolved force measurements were integrated with strain values averaged over a small central region (approximately 5 mm by 5 mm), away from the edges and possible edge effects, to determine macroscopic constitutive behavior of the skin [25]. Cauchy stresses were calculated from the force measurements, stretches, and initial sample dimension, with the incompressibility assumption. Thickness deformation was determined from the initial thickness (approximately 200 μm, in section 1) using the incompressibility assumption. We did observe some stress concentrations around the suture attachments, but they did not propagate into the central region where strains were calculated. Table 1 lists the conditions of all samples tested and protocols employed for each test. Samples were tested with spanwise and chordwise fibers approximately aligned with the loading axes, x and y, respectively, or off-axis with the material axis at an angle of approximately 30°–35° to the loading axes (figure 3). All tests, with the exception of those performed on sample D3, were conducted using displacement d x : d y control protocols with various ratios of displacements prescribed. Sample D3 was tested under Green strain (Ex:Ey) control. Tests were carried out at low displacement rates of 0.01–0.03 mm s−1 or strain rates of approximately 0.0025 s−1, depending on the

(

)

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type of testing. Although soft tissues are often cyclically tested until the stress–strain results converge (preconditioning) [15], preliminary work indicated that the mechanical response of these wing skin samples did not converge, therefore samples could not be preconditioned. Samples were tested several times, and because wing skin is viscoelastic, like most soft tissues, variations in the stress–strain curves result from viscoelastic effects for a given test protocol. The loading/unloading response curves show notable hysteresis. For presentation purposes, only loading curves are shown in the following sections to demonstrate constitutive features and property ranges for the tissue. Wing skin undergoes complex cyclic loading in vivo due to variation in flight kinematics, and here, each test may be considered as a functionally relevant representative constitutive response. The testing order of each sample and which tests were used for property calculations are outlined in the appendix. During testing, samples were periodically moistened with a small amount of saline solution to minimize drying.

3. Results 3.1. Strain field response Local deformation fields (figures 2(c), 4–6) facilitate an investigation of mesoscopic mechanics that are relevant to the skin tissue. Enabled by PIC, experimental strain field responses capture spatial variation of strain and its evolution with applied deformation, which provide insight into the behavior of individual components, their interactions, and their effect on the mechanical response of the composite. Figure 4 shows chordwise and spanwise strain fields in a central third region of sample C1 under equibiaxial displacement for a prescribed Green strain of 6.5%. The left images show strain fields only, and the right images show the same strain fields superimposed on the underlying fiber architecture. Prominent heterogeneities on the mesoscopic scale coincide with locations of the fibers and demonstrate a strong effect of the fiber network on the composite’s deformation [13]. Non-uniformity in multiple aspects of the fiber composition and architecture, including alignment, spacing, and fiber diameter both along fiber lengths and among fibers, induces secondary effects responsible for some spatial variability in the heterogeneity. For example, the slightly greater strain in the spanwise fibers in the lower right corner of figure 4(a) is likely due to slight spatial variation in fiber composition, and the slightly greater strains in some chordwise fibers on the left side of figure 4(b) is likely due to variations in the fiber structure, with thicker and closely paired/branching fibers showing greater strains. In addition, regions associated with the matrix material also show small variations in strain, which highlights the fact that the matrix itself is heterogeneous, with smaller-scale fibrils. Some regions of 5

the matrix have strain magnitudes as larger or larger than those observed in the fibers, which emphasizes that all tissue components are highly compliant. Greater overall chordwise strain in spanwise fibers than in surrounding matrix regions indicates that spanwise fibers are soft components embedded in a slightly stiffer collagen-filled matrix. Additional evidence supports this hypothesis. The primary constituent of spanwise-oriented fibers is elastin, a material of low elastic modulus, and we found that these fibers display strain-induced birefringence [13]; materials that exhibit strain-induced birefringence are typically isotropic and lack birefringence at rest due to lack of ordered substructure, but become ordered and birefringent as they are strained and their substructure becomes ordered [30]. It follows that longitudinal (along the fiber axis) and cross-fiber (perpendicular to the fiber axis) elastin behavior is at least initially softer than that of the matrix. A composite consisting of compliant fibers in a relatively stiffer matrix appears to be a distinctive feature of bat wing skin, especially the PLP [13]. This condition is in direct contrast to that observed in most collagen-reinforced soft tissues, and fiber-reinforced materials in general, in which fibers add stiffness to a relatively soft matrix [31, 32]. This points to the elastin fibers having a role other than reinforcement, and the soft elastin fibers could function to facilitate the extension of the wing with relatively low force. The relative stiffness behavior of the chordwise fibers is more complex. In the spanwise strain field, locations of chordwise fibers show both slightly greater and smaller strains than the surrounding matrix material. This indicates that sub-regions of passive (unstimulated, non-contracting) muscle fibers may be either stiffer or softer than the surrounding matrix in the cross-fiber direction. We expect skeletal muscle fibers in their passive state to have low stiffness in the fiber and cross-fiber directions [33, 34]. Further, the chordwise fibers are heterogeneous, composed of both muscle fibers and collagenous tendon, which may contribute to these variations. Upon activation of muscles, their stiffness increases dramatically, hence muscle recruitment may enable modulation of stiffness of the tissue composite [35, 36]. Muscle bundles appear to be more compliant in their cross-fiber direction than elastin fibers in their cross-fiber direction. This may be explained partly by the subset size used to calculate strain; because the size of the subsets are on the order of the larger chordwise fibers (muscles), strains computed for locations corresponding to smaller spanwise elastin fibers include contributions from the surrounding matrix material [13]. Our finding that the cross-fiber strains are non-negligible illustrates the important role the fibers play in enabling extension of the wing in the presence of the relatively stiff matrix and indicates that a more complete description of fibers as 2D elements rather than 1D elements may be important in modeling the tissue

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Figure 4. (a) Chordwise and (b) spanwise strain fields (8 × 8 mm) in sample C1 under equibiaxial displacement at a prescribed Green strain of 6.5% show variations due to fiber architecture. Left images show strain fields only. Right images show strain fields overlaid on the fiber architecture. (c) Simplified schematic of network of serial and parallel springs, with soft cross-fiber springs indicated in bold.

6

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Figure 5. (a) Spanwise direction strain fields (approx. 4 mm by 3 mm) and (b) chordwise direction strain fields (approx. 3 mm by 4 mm). Left images show surface (DIC) fields of the matrix behavior and right images show sub-surface (PIC) images including fiber structure behavior. Similar surface and subsurface fields indicate a strong interaction of tissue components through the thickness.

Figure 6. Spanwise and chordwise standard deviation in Green strains plotted against average Green strain in a small central area (3 × 3 mm) of sample P1 tested under equibiaxial displacement protocol. Linear regression lines are indicated with R2 fit values. Strain field images are shown at different strain levels. Small loss of correlation areas developed with increasing strain.

behavior. In a highly simplified analogy, the wing can be thought of as a network of soft and stiff springs in series and parallel (figure 4(c)). In each direction, a soft spring represents the low but non-negligible cross-fiber constitutive behavior and is in series with stiffer matrix spring elements. This set of series elements is in parallel with another set of series element representing cross-fiber behavior and longitudinal 7

fiber behavior. In this representation, the low-stiffness elastin behavior in its cross-fiber direction enables greater chordwise deformation than would otherwise be achievable if the fibers were absent from the tissue. In both the chordwise and spanwise direction, similar strains in fibers and matrix in the fiber direction indicate strong interaction of the components along their interface throughout the tissue. The non-

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homogeneity in strain is due to the transverse fiber behavior, as indicated earlier. We did not observe any mechanisms that indicate substantial non-affine fiber behavior, such as significant fiber sliding or buckling. Figure 5 shows representative surface (DIC) and subsurface (PIC) strain fields of a small region in sample C1 under equibiaxial displacement for a prescribed Green strain of 6.5%. The surface images show the surface matrix deformation, while the subsurface images also show the deformation of the embedded fiber structure. The skin samples are thin (mean thickness of 200 μm) and we do not expect much difference in material behavior through the thickness. The strain fields and strain histograms for the tissue surface and subsurface show consistency; strains are similar for surface matrix material on top of fibers and subsurface fibers. This likely indicates highly interactive, although not perfectly bonded, fiber and matrix components through the thickness, and confirms our expectation. Figure 6 shows how the spanwise and chordwise standard deviation in strain varies with increasing average strain for a small central area of sample P1 tested with an equibiaxial displacement protocol. Strain field insets give a spatial visualization of the quantitative variation given by the curves. The standard deviation increases at a constant rate for both chordwise and spanwise directions as indicated by the linear fit R2 values. This demonstrates that variation in strain across the sample is a constant fraction of mean strain at all applied strain magnitudes tested. The chordwise standard deviation is approximately 63% of the mean value, and the standard deviation in the spanwise direction is approximately 38% of the mean strain. These values highlight the large dispersion in the deformation profile of the skin composite that would not be captured at larger scales. The variation in strain is primarily due to spatial compositional variation, as discussed with regard to figure 4. It follows that the greater chordwise strain variation (slope of 0.63) is due to greater spatial compositional changes in that direction. The limiting case for these curves would be serial elements with equivalent constitutive behavior, giving zero standard deviation. A leveling off of the curve would indicate that the relative constitutive behavior of the components converges at large strains, and conversely, an increasing slope would indicate that the relative constitutive behavior of the components diverges further at larger strains. The relatively constant slope shown in figure 6 for the wing skin tissue indicates that the relative constitutive behavior of the components remains fairly stable for the given loading state. The analysis demonstrates the importance of the mesoscopic scale in understanding the mechanics of bat wing skin structure. 3.2. Mechanical properties We define mechanical properties as determined by averaging tissue deformation over a relatively small 8

central region of the samples, away from potential edge effects. Figures 7–9 show chordwise and spanwise stress–strain curves for various biaxial displacement protocols for samples D1, D2, and D3, respectively. The inlays show Green strain responses for chordwise and spanwise directions under prescribed displacement ratios. The stress–strain curves exhibit highly nonlinear behavior, following a J-shaped curve typical of soft biological tissues. The curve consists of three primary regions: a shallow toe region, a transitional heel, and a linear upturn region. The tissue is highly anisotropic, with the transition to the stiff upturn region at a lower strain in the chordwise direction than the highly extensible spanwise direction. Figures 7–9 also demonstrate the presence of axial coupling in skin mechanical behavior; differences among curves for each sample illustrate that the constitutive state of one axis depends on the state of the deformation along the other axis. We observe that, in general, as more spanwise strain is applied, the chordwise stress–strain curves transitions from the toe region at smaller chordwise strains, and vice versa: as more chordwise strain is applied, the spanwise stress–strain curves transition from the toe region at smaller spanwise strains. In addition, we observe a relatively high degree of interspecimen variability in the mechanical response of the tissue. For example, the slope of the curves in the toe and upturn regimes (quantified below) varies greatly, and indicates a large range of functionally relevant mechanical properties. Table 2 lists average maximum stresses and strains of four tissue samples (D1, D2, D3, P1) tested with equibiaxial displacement to failure. The skin of the PLP exhibits much greater extensibility in the spanwise direction, with an average maximum strain of 28% compared to 14.5% for the chordwise direction. These values fall at the low end of maximum strains measured for wing membrane skin uniaxially loaded to failure from nine bat species (13–232%) [1]. The relatively low maximum strains may be influenced by several factors including interspecies variability, different testing protocols, and the fact that samples were strained to remove inherent wrinkling before testing, and this pre-strain was applied primarily in the spanwise direction. The substantial difference in chordwise and spanwise strain ranges is consistent with the relative extensibility of single elastin fibers, which have been found to reach 130–150% strain before breaking [30, 37], and skeletal muscle fibers which can strain 35–55% before failure [38]. The overall trend of greater spanwise extension is consistent with the literature [1, 39], and supports the hypothesis that a primary function of the spanwise elastin fiber network is to permit the large degree of extension and refolding of the wing [1, 11]. The ultimate stresses developed in the tissue are on the order of 1 MPa, with ultimate stress in the spanwise direction slightly greater than in the chordwise direction. This is an order of magnitude smaller than ultimate stresses reported for skin of

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Figure 7. Stress–strain data for sample D1 in (a) chordwise and (b) spanwise directions. In (a), the number following each color indicates the test number followed by the prescribed displacement ratio dspan:dchord. The inset in (b) shows the Green strain response of the sample chordwise and spanwise directions under prescribed displacement ratios, indicated next to each curve.

Figure 8. Stress–strain data for sample D2 in (a) chordwise and (b) spanwise directions. In (a), the number following each color indicates the test number followed by the prescribed displacement ratio dspan:dchord. The inset in (b) shows the Green strain response of the sample chordwise and spanwise directions under prescribed displacement ratios, indicated next to each curve.

Figure 9. Stress–strain data for sample D3 in (a) chordwise and (b) spanwise directions. In (a), the number following each color indicates the test number followed by the prescribed strain ratio Espan:Echord. The inset in (b) shows the Green strain response of the sample chordwise and spanwise directions under prescribed displacement ratios, indicated next to each curve.

Table 2. Maximum Green strain and ultimate Cauchy stress for wing tissue loaded equibiaxially to failure. (Four samples: D1, D2, D3, P1.)

Mean Std. dev.

Max strainchord (%)

Max strainspan (%)

Ultimate stresschord (MPa)

Ultimate stressspan (MPa)

14.5 3.70

28.0 6.98

1.79 1.11

2.86 1.58

9

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Table 3. Tangent moduli of bat wing skin toe (t) and upturn (u) portions of stress–strain curve. The tests used to compute means are specified in the appendix.

Mean Std. dev.

Etchord (MPa)

Etspan (MPa)

Euchord (MPa)

Euspan (MPa)

0.233 0.0246

0.115 0.0403

49.1 40.9

142 94.5

some other vertebrate species uniaxially tested to failure, and falls within the low end of the range reported previously for bats (1.14–26.38 MPa) [1, 40–44]. The nonlinear constitutive behavior of the skin can be characterized by the elastic tangent modulus (linear slope) for the shallow toe region and linear upturn region of the stress–strain curves. Table 3 lists the mean and standard deviation values for five samples tested with various loading protocols. The tangent moduli in the toe region (Et) are on the order of hundreds of kilopascals. The initial toe tangent moduli in the chordwise direction for sample P1 are an order of magnitude larger than those of the other samples, ranging between 2400 and 7900 Pa, and are excluded from table 3. Visual inspection of the fiber structure under polarized light indicates that the diameters of the fibers in this sample are on the larger end of the established range for the species, highlighting interspecimen variability. It may also be that the degree of specimen preservation differed between this and other samples. In the linear upturn region, the maximum tangent moduli (Eu) are three to four orders of magnitude larger than those of the toe region, and fall within the range of reported moduli for bat skin tested uniaxially [1]. Anisotropy of the wing skin samples subjected to equibiaxial strain is characterized by an anisotropic index, AI, defined as the chordwise tangent modulus divided by the spanwise tangent modulus. Table 4 lists equibiaxial response properties of bat wing skin and other soft tissues ordered by peak elongation [45–50]. The wing skin sample has an initial toe tangent modulus of the same order of magnitude as that of several other soft tissues from the cardiac and vascular system under similar equibiaxial strain, (table 4) [47–49]. In their respective upturn regions, bat wing skin has tangent moduli 1–3 orders of magnitude lower than pericardium, annulus fibrosa, sclera, and abdominal aorta [45–48]. The anisotropic index for bat wing skin has a value of 1.93 in the shallow toe region, indicating that the chordwise direction is initially nearly twice as stiff as the spanwise direction. At larger strains, specifically at a strain of 11.5% Green strain, the anisotropic index is 14.4. This indicates that the anisotropy of the material increases significantly with increasing stretch. While nearly all tissues studied to date show increasing anisotropy with stretch, none show as dramatic a change as bat wing skin (table 4). The dramatic change in anisotropy may provide a passive mechanism to guide evolving wing shape during flight (further discussion in section 4.2). 10

3.3. Fiber orientation and kinematics Enabled by the PIC technique, visualization of the embedded fiber architecture facilitates quantitative determination of fiber spatial distributions, which have been shown to significantly affect the constitutive behavior of tissues [32, 51]. In this section, we present fiber orientation measurements and kinematics for two samples (P1 and P2) under various loading protocols, and identify the main deformation mechanisms of the fibers. Three angles were measured in the samples’ central region, away from edges, at various stages of the loading trajectory: chordwise fiber angle (αy), spanwise fiber angle (αx), and relative angle between spanwise and chordwise fibers (β) as shown in figure 3. Measurements were typically taken from 5 to 6 spanwise fibers and 10 to 15 chordwise fibers, yielding approximately 70 intersections per sample image. Distributions of the fiber alignment measurements are characterized by mean, median, and standard deviation. Location and variance equivalence tests (Kruskal–Wallis and Brown–Forsythe tests, respectively) were used to determine whether statistically significant changes in angle measurements occur with the deformation state relative to the initial sample configuration (p-value < 0.05) assuming independent samples [52–54]. Figure 10 shows histograms of the spatial distributions for spanwise and chordwise fibers relative to the x-axis for an equibiaxial test performed on sample P1 (on-axis). Initial distributions are dark gray and distributions at an applied equibiaxial stretch of 1.2 are white, with overlaps in lighter gray. Figure 11 shows the corresponding plots for an equibiaxial test performed on sample P2 (off-axis). These figures highlight a substantial spread in fiber orientation for both families of fibers with standard deviations of 3°–8°. Figure 12 shows mean angle values for the orientation of the chordwise and spanwise fibers relative to the x-axis and the relative angle between the chordwise and spanwise fibers for equibiaxial tests performed on sample P1 (on-axis) and sample P2 (off-axis). Measurements that differ statistically from initial values (p-values < 0.05) are indicated by black stars. Note that we confirmed that the relative angle between the chordwise and spanwise fibers is approximately 90°. Small, statistically significant rotations of 0°–2° occur only at large applied stretches of 1.2, and we conclude that the primary deformation mechanism for the fibers is stretch, with small rotations providing a secondary mechanism of deformation at large stretches. Similar results of small but statistically significant angle changes at large applied stretches occurred for some non-equibiaxial test cases, but other tests showed no significant changes in angle. It is likely that wing membrane skin operates physiologically in the toe region, well outside of the extreme upturn region where small fiber rotations may occur (see section 4.2).

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Table 4. Equibiaxial response properties of bat wing membrane skin in comparison to other soft tissues.

Tangent moduli (MPa)

Anisotropic index (—)

Material

Peak elongation (Green strain)

Peak stress (MPa)

toe

upturn

toe

upturn

Sclera [44] Annulus fibrosus [45] Abdominal aorta [46] Bat wing skin chord span Pericardium [47] Mitral valve leaflet [48] Myocardium [49]

0.035 0.06, 0.09 0.11, 0.12 0.12

0.35 1, 1.5 120 N m−1 0.04 0.015 0.323, 0.107 80–200 N m−1 0.0098, 0.0069

— 13.5, 14.1 0.133, 0.118 0.131 0.068 0.327, 0.104 0.0893, 0.0938 —

2.8, 3.3 43.3, 133 3.9, 3.3 3.72 0.258 15.0, 2.80 3.43, 9.96 0.049, 0.0775

— 1.04 1.13 1.93

1.06 3.07 1.18 14.4

3.14 0.95 —

5.34 2.61 1.58

0.16 0.2, 0.4 0.313

Figure 10. (a) Spanwise and (b) chordwise fiber angle spatial distribution histograms at prescribed stretches of 1 (dark gray) and 1.2 (white), with overlapping regions in lighter gray for sample P1 (on-axis) tested equibiaxially. The plots show a large spread in fiber orientation and small changes with applied stretch.

Figure 11. (a) Spanwise and (b) chordwise fiber angle spatial distribution histograms at prescribed stretches of 1 (dark gray) and 1.2 (white), with overlapping regions in lighter gray for sample P2 (off-axis) tested equibiaxially. The plots show a large spread in fiber orientation and small changes with applied stretch.

4. Discussion Biaxial mechanical characterization demonstrates that the constitutive behavior of bat wing skin is heterogeneous, nonlinear, and highly anisotropic. These data provide the requisite foundation to identify connections between structural morphology, mechanical properties, and overall functionality of the wing skin as a deployable morphing surface. In this section, we first propose a map connecting deformation mechanisms to regions of the macroscopic constitutive curves (section 4.1); and subsequently discuss the 11

implications of the constituents and mechanical properties on the functionality of the wing skin in the context of the physiological (in vivo) mechanical operating regimes (section 4.2) and bio-inspired skins for micro-aerial vehicles (section 4.3). 4.1. Deformation map The characteristic J-shaped stress–strain curve has three major regimes as outlined in section 3.2: from an initially flattened reference configuration, we observed an initial shallow toe region, a second transitional, heel region, and a third steep, linear upturn region. In this

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Figure 12. Mean chordwise, spanwise, and relative angle versus applied stretch for (a) sample P1 (on-axis) and (b) sample P2 (offaxis) tested equibiaxially. Statistically significant changes from initial values (p-value < 0.05) are indicated by black stars. Standard deviations are indicated by error bars. Small statistical rotations indicate stretch as the primary deformation mechanism of the fibers.

Figure 13. Proposed macroscopic deformation mechanisms map for spanwise constitutive behavior under spanwise stretch. Lm is the initial wrinkled length of the matrix material. Lef is the flat length of the spanwise elastin fiber. The matrix remains wrinkled in the first regime (I) and becomes flat and contributes to load carrying at the beginning of the second regime (II).

section, we consider the full response of the tissue from the initially wrinkled reference configuration and distinguish the three major regimes as I, II, and III. We suggest that the constituent structural elements of wing skin provide the basis for this behavior (figure 13). By integrating tissue structure information with mechanical characterization results, we outline major deformation mechanisms at work in bat wing skin. Consider an in vitro square section of the PLP, where the wing is somewhat open but retains characteristic spanwise wrinkles (figure 1). In this reference state, spanwise elastin fibers and chordwise muscle fibers must be in tension, given that upon excision the wing sample recoils in both directions to a compact, heavily corrugated state. This configuration of mismatched states is akin to wrinkling of thin films on pre-stretched substrates (see for example [55]). It can be visualized as follows: first, consider an undeformed fiber network of straight spanwise and chordwise fibers; second, stretch the network in the 12

spanwise direction; next, attach the fiber network to a flat undeformed matrix; finally, release the composite. In this last step, when the skin is released from a flat state, the skin wrinkles due to the mismatch in the prestretch between the spanwise fibers and the matrix. This wrinkled reference state differs from the flattened reference state required for 2D DIC experimental characterization. From the wrinkled reference state, skin samples can be extended along the span until the sample is just flat, i.e. matrix corrugations are removed while holding length constant in the chordwise direction. During this action, the spanwise elastin fibers dominate spanwise load carrying while the matrix flattens. As the matrix flattens, a macroscopic organizational change, the matrix sub-structure does not deform. This step represents the first region (I) of the spanwise stress– strain curve for the initially wrinkled tissue. Recall that the initial wrinkled state of the matrix arises from a mismatch in the natural configurations of the fiber and the matrix. This kinematic instability may provide a low energy mechanism for skin extension. Upon further spanwise stretch, the flat matrix, which was found to be slightly stiffer than the spanwise elastin fibers (section 3.1), engages and begins to contribute to load carrying. This step corresponds to the initial portion of the transitional region (II) in the spanwise stress–strain curve. The microscopic structure of the matrix in G. soricina is not known in detail, however, one dominant element is collagen in the form of microfibrils, which could be an important stiffening component [10, 56]. Recall that the 2D DIC experimental characterization has a flattened reference state such that the initial shallow toe region of those curves correspond to the end of the shallow region (I)/beginning transition region (II) of the constitutive curve for the initially wrinkled tissue sample considered in this section. A 3D experimental study would include the wrinkled (I) regime. As the matrix and spanwise elastin fibers continue to stretch, the stress–strain curve reaches the steep

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linear region (III). We suggest that elastin fibers facilitate extension of the wing membrane skin with relatively little force, which in turn, increases the aerodynamically active surface area of the wing during downstroke. Elastin fibers may also contribute to the dynamics of passive modulation of wing shape. In addition, the fibers may provide stiffness to the skin when it undergoes considerable stretch, as suggested by Holbrook and Odland, as in region III of the curve; this function requires appreciable stiffening of the fibers as they approach their limiting extensibility before or in concert with reaching limiting extensibility of the matrix [11]. The wing skin is not wrinkled in the chordwise direction in the same manner as it is in the spanwise direction, and is essentially flat in this direction the unloaded configuration. Under chordwise stretch, the matrix and chordwise fibers deform together with no relative sliding (section 3.1), such that all components carry load through all three regimes (I, II, and III) of the chordwise stress–strain curve. In the first region (I), the compliant transverse behavior of the spanwise elastin fibers is the main factor that enables composite stretch with low force (section 3.1). Small regions of great compliance in the heterogeneous matrix material also contribute. The stiffening behavior of the composite in the transition region (II) and steep linear region (III) of the stress–strain curve is a result of a combination of stiffening contributions from all components (matrix, spanwise fibers, and chordwise fibers), which make up a network of series and parallel springs (section 3.1). Component sub-structure composition and mechanisms including microscopic collagen fibrils are likely important stiffening factors in these regimes. 4.2. Physiological regime The in vivo mechanical operating regime for bat wing skin is not known, although some inferences can be made from observations of wing skin during flight. From a strain energy perspective, operating in the initial shallow region (I) is favorable. However, it may be aerodynamically necessary to operate in stiffer regimes during some maneuvers. Albertani et al used visual image correlation (VIC) to measure in-flight Green strain ranges for Artibeus jamaicensis, a larger fruit-eating species in the same family as G. sorcina, and reported a physiological strain range of around 10% in the spanwise direction and 3% in the chordwise direction, although strains were computed from an unknown but tensile initial reference state [57]. Furthermore, the dots on the wing surface tracked for VIC were placed fairly far apart, such that substantial inherent curvature induced between dots during flight is not accounted for in strain estimation, introducing an unknown amount of error. In addition, it was possible to assess only a small portion of the complex wingbeat cycle in that study. The true strain range is 13

almost certainly greater than the reported values in Albertani et al but much less than the median uniaxial failure strain of 75% measured by Swartz et al for A. jamaicensis [1, 57]. G. soricina is capable of similar in vitro deformations of 3 and 10% strain in the chordwise and spanwise directions, respectively. These levels of strain correspond to the shallow toe region of the constitutive curves (figures 7–9). Thus, G. soricina could likely operate in the shallow toe region with very small stiffnesses on the order of hundreds of kPa (table 2). Stiffness anisotropy (section 3.2) may also be energetically beneficial to bats, providing a passive mechanism for evolving shape control. Low stiffness in the spanwise direction complements the kinematics of the wing skeletal system, enabling large spanwise extension of the wing skin while maintaining the chord, to facilitate increases in lifting surface with relatively low force. Higher chordwise stiffness and increasing AI provides a higher and increasing energy barrier to chordwise length changes, which may favor passive maintenance of appropriate pressure-induced camber for flight [58]. The presence of muscle fibers in chordwise orientation in the PLP suggests that muscle activation is likely important in the function of the wing skin. Muscle contraction generally generates force when the muscle ends are fixed (isometric contraction), or generates displacement when the muscle is free to shorten. Although the leading edge of the PLP is rigidly secured by the humerus and radius, the trailing edge is reinforced with connective tissue, which allows for potential chordwise displacement. Thus activation of the plagiopatagiales muscles could modulate chordwise stiffness and/or length to some extent, and either of these factors could affect camber and hence wing shape during aerodynamic loading. The ability to actively tailor these properties could be advantageous for a flying animal [4, 11, 59]. For example, during downstroke the animal extends its arms and legs maintaining the chord; activation of the plagiopatagiales stiffens the chordwise direction of the skin membrane, which gives a greater energy barrier to chordwise deflection. The effect of muscle activation would be greatest in the initial shallow regime of the passive tissue rather than the higher stiffness steep linear region. 4.3. Bio-inspired skins for micro-aerial vehicles The results of the mechanical characterization provide a baseline for the development of bio-inspired skins for micro-aerial vehicles with objectives aligned with the capabilities of bats; these include high maneuverability, multiple flight modes, light weight, and quiet flight. Potential skins should be compliant to provide low energy shape changes, but also stiff enough to support a range of aerodynamic loads. We have shown that stiffness values on the order of hundreds of kPa

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Figure 14. Bio-inspired skin based on the plagiopatagium of bats for possible application in micro-aerial vehicles demonstrating (a) components attached at discreet points to form a layered structure, and (b) stress–strain behavior similar to biological counterpart.

correspond to likely operating regimes. In addition, compliant membranes have been shown to possess delayed stall at higher angles of attack than their rigid counterparts [58]. We have identified the mismatch in natural states of the spanwise elastin fibers and matrix material as the source of the wrinkled skin structure. This can be replicated in synthetic skins to provide a passive low energy mechanism for wing deployment and extension. Our fabrication procedure has four steps. First, we form a very compliant and thin fiber structure of silicone (Dragon Skin-10, Smooth-on Inc., Easton, Pa). This may be accomplished using a mold or by cutting away material from a thin silicone sheet. Second, we stretch the fiber structure to a prescribed deformation in the fiber direction, and affix it to a glass plate. Next, we attach the other components (the matrix and other fiber family). This can be done by pouring uncured matrix material (Dragon Skin-20, Smooth-on, Inc. Easton, Pa) over the fiber families to create a single continuous skin, or by attaching cured sheet(s) of matrix at discreet points along the fibers to create a layered structure (figure 14(a)). Finally, the fully cured composite is released from the flat plate, and wrinkles arise naturally due to the mismatch in natural states of the components. Synthetic skins should also incorporate nonlinearity and anisotropy, which we have proposed offers some passive shape control; because of the Jshaped nonlinear stress–strain relationship, as the skin deforms, it is increasingly difficult for the wing skin to change shape. The anisotropy preferentially guides the evolving wing skin deformation to increasing span because the lower spanwise stiffness gives a lower energy barrier to spanwise length changes (section 4.2). To date, flexible isotropic films [60, 61], 14

and stretchable isotropic membranes [58, 62–65] have been used for biomimetic flight studies. This work has outlined and created new constructs that replicate wing skin structure in a highly compliant and anisotropic form (figure 14(b)), and can advance biomimetic micro-aerial vehicle design. Just as the effects of bio-inspired wing kinematics on performance of engineered flappers have been studied in wind tunnel tests [60, 62, 65], one may investigate the influence of highly compliant, nonlinear, and anisotropic membranes on aerodynamic performance by quantifying lift, drag, thrust, and power, etc. It is also feasible to implement surface features such as wrinkling and active components representing plagiopatagiales. Simple rectangular wings fixed at the leading and trailing edges [e.g. 58] or more complex flapping structures that more closely mimic the skeletal kinematics of G. soricina are possible [e.g. 62]. We expect that different sets of properties improve functional performance at distinct portions of the wingbeat (upstroke versus downstroke), various flight speeds, and for distinct flight modes. Active elements representing muscle fibers could provide the mechanism to transition between these states, as suggested by Cheney et al [59]. Shape memory polymers and dielectric elastomers [66] are examples of stiffness modulating elements that could be implemented as active elements.

5. Summary This work represents the first biaxial characterization of the local deformation, mechanical properties, and fiber kinematics of bat wing skin. Through visualization of the underlying fiber architecture during mechanical testing, the PIC technique has enabled the investigation of local component deformations and kinematics. Analysis of these results has provided insight into the relationships between the structural morphology, mechanical properties, and functionality of the wing skin. Examination of the strain field response showed large spatial variation in the deformation of the tissue and non-negligible fiber strains in the cross-fiber direction for both chordwise and spanwise fibers, which indicates that cross-fiber behavior influences the overall constitutive response of the composite. Thus, future modeling efforts should consider the fibers as 2D, not 1D, elements. Fiber kinematics confirmed stretch as the dominant deformation mechanism for the fibers, with minimal fiber reorientation at even large stretches. The macroscopic constitutive behavior was nonlinear, with very low stiffness, on the order of hundreds of kilopascals, in the toe region. As observed previously, the wing membrane skin is highly anisotropic with greater stiffness in the chordwise direction, and it shows a dramatic increase in anisotropy with applied equibiaxial strain. This anisotropy may provide a mechanism

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to passively influence the evolving wing shape under aerodynamic loads. The results of this work underscore the importance of biaxial field approaches for soft heterogeneous tissue, and provide a foundation for development of synthetic skins to probe the effects of the wing skin properties on aerodynamic performance metrics. Future work will investigate single fiber behavior to gain additional insight into relative component behavior, interfacial properties, smallerscale deformation mechanisms, and how these three factors affect macroscopic composite properties.

Acknowledgments We acknowledge support of the Air Force Office of Scientific Research (Grant #F023809), the National Science Foundation (NSF IOS 1145549), and we thank Jorn Cheney and Andrew Bearnot for their many contributions to this project.

Appendix (See table A1.) Table A1. Outline of displacement control testing protocols (dx:dy) for samples D1, D2, P1, P2, C1, and C2. Sample D3 was tested under Green strain control protocols (Ex:Ey) where x and y are the loading axes. Tests used for determining average mechanical properties and fiber kinematics are denoted with * and ∧, respectively. Test

D1

D2

D3(Ex: Ey)

P1

P2

C1

C2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1:1 1:1 1:1* 1:1 1:1 1:1 2:1 2:1* 1:2 1:2* 1:2 1:2 0:1* 0:1 1:0* 1:0 1:3 1:3* 3:1* 3:1 1:1 1:1* 1:2 1:2 2:1 1:1 1:1

1:1 1:1 1:1* 1:1* 1:1* 1:1* 2:1* 1:2* 0:1* 1:0* 1:3 1:3* 3:1* 1:1* 1:1* 1:2* 2:1* 1:1*

1:1 1:1* 1:1 1:2* 2:1* 0:1* 1:0* 1:1 1:1*(dx:dy)

1:1*∧ 1:2*∧ 2:1*∧ 2:3∧ 3:2∧ 1:1*∧

1:1∧ 1:1∧ 1:1 2:1∧ 2:1∧ 1:2

1:1*

1:1

15

References [1] Swartz S M, Groves M S, Kim H D and Walsh W R 1996 Mechanical properties of bat wing membrane skin J. Zoology 239 357–78 [2] Smith C, Herbert R, Wootton R and Evans K 2000 The hind wing of the desert locust (Schistocerca gregaria Forskal): II. Mechanical properties and functioning of the membrane J. Exp. Biol. 203 2933–43 (PMID:10976030) [3] Cameron G J, Wess T J and Bonser R H C 2003 Young’s modulus varies with differential orientation of keratin in feathers J. Struct. Biol. 143 118–23 [4] Vaughan T A 1966 Morphology and flight characteristics of molossid bats J. Mammal. 47 249–60 [5] Norberg U M 1972 Bat wing structures important for aerodynamics and rigidity (Mammalia, Chiroptera) Zoomorphology 73 45–61 [6] Thollesson M and Norberg U M 1991 Moments of inertia of bat wings and body J. Exp. Biol. 158 19–35 (PMID:9319563) [7] Bullen R D and McKenzie N L 2007 Bat wing airfoil and planform structures relating to aerodynamic cleanliness Aust. J. Zoology 55 237–47 [8] Kostandov M, Willis D J, Pivkin I V, Swartz S M and Laidlaw D H 2008 Qualitative visual comparison of three simulations of fluid flow around a flying bat Master’s Thesis, Brown University [9] Riskin D K et al 2008 Quantifying the complexity of bat wing kinematics J. Theor. Biol. 254 604–15 [10] Gupta B B 1967 The histology and musculature of plagiopatagium in bats Mammalia 31 313–21 [11] Holbrook K A and Odland G F 1978 A collagen and elastic network in the wing of the bat J. Anat. 126 21–36 (PMID:1235709) [12] Skulborstad A J and Goulbourne N C 2014 A structurally motivated constitutive model for bat wing skin Proc. 2014 ASME IMECE (Quebec, Canada) IMECE2014-39525 [13] Skulborstad A J, Wang Y, Davidson J D, Swartz S M and Goulbourne N C 2013 Polarized image correlation for large deformation fiber kinematics Exp. Mech. 53 1405–13 [14] Cheney J Department of Ecology and Evolutionary Biology, Brown University, Providence RI, unpublished data [15] Fung Y C 1981 Biomechanics: Mechanical Properties of Living Tissues (New York: Springer) [16] Lanir Y and Fung Y C 1974 Two-dimensional mechanical properties of rabbit skin: II. Experimental results J. Biomech. 7 171–82 [17] Sacks M S and Sun W 2003 Multiaxial mechanical behavior of biological materials Annu. Rev. Biomed. Eng. 5 251 [18] Holzapfel G A and Ogden R W 2009 On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework Math. Mech. Solids 14 474–89 [19] Brunon A, Bruyere-Garnier K and Coret M 2010 Mechanical characterization of liver capsule through uniaxial quasi-static tensile tests until failure J. Biomech. 43 2221–7 [20] Woo S, Orlando C, Camp J and Akeson W 1986 Effects of postmortem storage by freezing on ligament tensile behavior J. Biomech. 19 399–404 [21] Tamura A, Omori K, Miki K, Lee J, Yang K and King A 2002 Mechanical characterization of porcine abdominal organs Stapp Car Crash J. 46 55–69 (PMID:16686009) [22] Santago A, Kemper A, McNally C, Sparks J and Duma S 2009 Freezing affects the mechanical properties of bovine liver Biomed. Sci. Instrum. 45 24–9 (PMID:19369734) [23] Van Ee C, Chasse A and Myers B 1998 The effect of postmortem time and freezer storage on the mechanical properties of skeletal muscle SAE Technical Paper 983155 [24] Venkatasubramanian R T, Grassl E D, Barocas V H, Lafontaine D and Bischof J C 2005 Effects of freezing and cryopreservation on the mechanical properties of arteries Ann. Biomed. Eng. 34 823–32 [25] Sun W, Sacks M S and Scott M J 2005 Effects of boundary conditions on the estimation of the planar biaxial mechanical properties of soft tissues J. Biomech. Eng. 127 709–15

Bioinspir. Biomim. 10 (2015) 036004

A J Skulborstad et al

[26] Peters W H and Ranson W F 1982 Digital imaging techniques in experimental stress analysis Opt. Eng. 21 213427 [27] Sutton M A, Wolters W J, Peters W H, Ranson W F and McNeill S R 1983 Determination of displacements using an improved digital correlation method Image Vis. Comput. 1 133–9 [28] Chu T C, Ranson W F and Sutton M A 1985 Applications of digital-image-correlation techniques to experimental mechanics Exp. Mech. 25 232–44 [29] Berfield T A, Patel J K, Shimmin R G, Braun P V, Lambros J and Sottos N R 2006 Fluorescent image correlation for nanoscale deformation measurements Small 2 631–5 [30] Aaron B B and Gosline J M 1981 Elastin as a random‐network elastomer: a mechanical and optical analysis of single elastin fibers Biopolymers 20 1247–60 [31] Meyers M A, Chen P-Y, Lin A Y-M and Seki Y 2008 Biological materials: structure and mechanical properties Prog. Mater. Sci. 53 1–206 [32] Gasser T C, Ogden R W and Holzapfel G A 2006 Hyperelastic modelling of arterial layers with distributed collagen fibre orientations J. R. Soc. Interface 3 15–35 [33] Calvo B, Ramirez A, Alonso A, Grasa J, Soteras F, Osta R and Munoz M J 2010 Passive nonlinear elastic behavior of skeletal muscle: experimental results and model formulation J. Biomech. 43 318–25 [34] Winters T M, Sepulveda G S, Cotter P S, Kaufman K R, Lieber R L and Ward S R 2009 Correlation between isometric force and intramuscular pressure in rabbit tibialis anterior muscle with an intact anterior compartment Muscle Nerve 40 79–85 [35] Hawkins D and Bey M 1997 Muscle and tendon force-length properties and their interactions in vivo J. Biomech. 30 63–70 [36] Grover J P, Corr D T, Toumi H, Manthei D M, Oza A L, Vanderby R Jr and Best T M 2007 The effect of stretch rate and activation state on skeletal muscle force in the anatomical range Clin. Biomech. 22 360–8 [37] Carton R W, Dainauskas J and Clark J W 1962 Elastic properties of single elastic fibers J. Appl. Physiol. 17 547–51 (PMID:13876985) [38] Kovanen V, Suominen H and Heikkinen E 1984 Mechanical properties of fast and slow skeletal muscle with special reference to collagen and endurance training J. Biomech. 17 725–35 [39] Swartz S M, Iriarte-Díaz J, Riskin D K, Song A, Tian X, Willis D and Breuer K S 2007 Wing structure and the aerodynamic basis of flight in bats Aerospace Sciences Meeting and Exhibition (Reno, Nevada) AIAA Paper 2007-42 [40] Dunn M G and Silver F H 1983 Viscoelastic behavior of human connective tissues: relative contribution of viscous and elastic components Connective Tissue Res. 12 59–70 [41] Shadwick R E, Russell A P and Lauff R F 1992 The structure and mechanical design of rhinoceros dermal armour Phil. Trans. R. Soc. B 337 419–28 [42] Harkness M R and Harkness R D 1964 Some mechanical properties of collagenous frameworks and their functional significance Proc 4th Int. Congress Biorheol Part 4 ed A L Copley (NY: Interscience) pp 477–89 [43] Jayne B C 1988 Mechanical behaviour of snake skin J. Zoology 214 125–40 [44] Greven H, Zanger K and Schwinger G 1995 Mechanical properties of the skin of Xenopus laevis (Anura, Amphibia) J. Morphol. 224 15–22 [45] Eilaghi A, Flanagan J G, Tertinegg I, Simmons C A, Wayne Brodland G and Ross Ethier C 2010 Biaxial mechanical testing of human sclera J. Biomech. 43 1696–701

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[46] O’Connell G D, Sen S and Elliott D M 2012 Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration Biomech. Model. Mechanobiol. 11 493–503 [47] Vande Geest J P, Sacks M S and Vorp D A 2006 The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta J. Biomech. 39 1324–34 [48] Sacks M S and Chuong C J 1998 Orthotropic mechanical properties of chemically treated bovine pericardium Ann. Biomed. Eng. 26 892–902 [49] May-Newman K and Yin F C 1995 Biaxial mechanical behavior of excised porcine mitral valve leaflets Am. J. Physiol.-Heart C 269 H1319–27 (PMID:7485564) [50] Sacks M S and Chuong C J 1993 Biaxial mechanical properties of passive right ventricular free wall myocardium J. Biomech. Eng. 115 202 [51] Wang Y, Son S, Swartz S M and Goulbourne N C 2012 A mixed Von Mises distribution for modeling soft biological tissues with two distributed fiber properties Int. J. Solids Struct. 49 2914–23 [52] Statistical data analysis guide Mathematica v. 8 Wolfram [53] Devore J and Farnum N 2005 Applied Statistics for Engineers and Scientists (USA: Thompson Brooks/Cole) [54] Brown M B and Forsythe A B 1974 Robust tests for the equality of variances J. Am. Stat. Assoc. 69 364–7 [55] Cao Y and Hutchinson J W 2012 Wrinkling phenomena in neo-Hookean film/substrate bilayers J. Appl. Mech. 79 031019 [56] Quay W B 1970 Integument and derivatives Biology of Bats vol 2 ed W A Wimsatt (New York: Academic) pp 1–56 [57] Albertani R, Hubel T, Swartz S M, Breuer K S and Evers J 2011 In-flight wing-membrane strain measurements on bats Exper. Appl. Mech. 6 437–45 [58] Song A, Tian X, Israeli E, Galvao R, Bishop K, Swartz S and Breuer K 2008 Aeromechanics of membrane wings with implications for animal flight AIAA J. 46 2096–106 [59] Cheney J A, Konow N, Middleton K M, Breuer K S, Roberts T J, Giblin E L and Swartz S M 2014 Membrane muscle function in the compliant wings of bats Bioinspiration Biomimetics 9 025007 [60] Gerdes J W, Cellon K C, Bruck H A and Gupta S K 2013 Characterization of the mechanics of compliant wing designs for flapping-wing miniature air vehicles Exp. Mech. 53 1561–71 [61] Zdunich P, Bilyk D, MacMaster M, Loewen D, DeLaurier J, Kornbluh R, Low T, Stanford S and Holeman D 2007 Development and testing of the mentor flapping-wing micro air vehicle J. Aircr. 44 1701–11 [62] Chen P, Joshi S, Swartz S M, Breuer K S, Bahlman J W and Reich G W 2014 Bat inspired flapping flight. SciTech (National Harbor, Maryland) AIAA Paper 2014–1120 [63] Ifju P G, Jenkins D A, Ettinger S, Lian Y, Shyy W and Waszak M R 2002 Flexible-wing-based micro air vehicles AIAA Pap. 705 1–11 [64] Albertani R, Stanford B, Hubner J P and Ifju P G 2007 Aerodynamic coefficients and deformation measurements on flexible micro air vehicle wings Exp. Mech. 47 625–35 [65] Bahlman J W, Swartz S M and Breuer K S 2013 Design and characterization of a multi-articulated robotic bat wing Bioinspiration Biomimetics 8 016009 [66] Hays M R, Morton J, Dickinson B, Chakravarty U K and Oates W S 2012 Aerodynamic control of micro air vehicle wings using electroactive membranes J. Intell. Mater. Syst. Struct. 24 862–78