Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Numerical investigation of insect wing fracture behaviour H. Rajabi a,b,n, A. Darvizehc, A. Shafieib, D. Taylord, J.-H. Dirkse a
Institute of Zoology, Functional Morphology and Biomechanics, Christian-Albrechts-University, Kiel, Germany Department of Mechanical Engineering, The University of Guilan, Rasht, Iran Department of Mechanical Engineering, Faculty of Engineering, Anzali Branch, Islamic Azad University, Bandar Anzali, Iran d Trinity Centre for Bioengineering, Trinity College Dublin, Dublin, Ireland e Department of New Materials and Biosystems, Max Planck Institute for Intelligent Systems, Stuttgart, Germany b c
art ic l e i nf o
a b s t r a c t
Article history: Accepted 31 October 2014
The wings of insects are extremely light-weight biological composites with exceptional biomechanical properties. In the recent years, numerical simulations have become a very powerful tool to answer experimentally inaccessible questions on the biomechanics of insect flight. However, many of the presented models require a sophisticated balance of biomechanical material parameters, many of which are not yet available. In this article we show the first numerical simulations of crack propagation in insect wings. We have used a combination of the maximum-principal stress theory, the traction separation law and basic biomechanical properties of cuticle to develop simple yet accurate finite element (FE) models of locust wings. The numerical results of simulated tensile tests on wing samples are in very good qualitative, and interestingly, also in excellent quantitative agreement with previously obtained experimental data. Our study further supports the idea that the cross-veins in insect wings act as barriers against crack propagation and consequently play a dominant role in toughening the whole wing structure. The use of numerical simulations also allowed us to combine experimental data with previously inaccessible data, such as the distribution of the first principal stress through the wing membrane and the veins. A closer look at the stress-distribution within the wings might help to better understand fracture-toughening mechanisms and also to design more durable biomimetic micro-air vehicles. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Locust wing Crack propagation Stress distribution Critical crack length Finite element
1. Introduction Insect wings appear as delicate and fragile structures, yet they are able to efficiently withstand external mechanical stress during the lifetime of an insect. During normal flight, the flapping or gliding wings of insects can experience different types of dynamic loading, they can actively or passively change their shape, and are subjected to aerodynamic forces significantly larger than the insects’ weight for thousands of loading cycles (Dirks et al., 2013; Du and Sun, 2010; Ellington, 1984; Ennos, 1989; Wootton, 1992; Wootton et al., 2003). Still many fundamental questions regarding the biomechanics of insect flight remain unanswered (Herbert et al., 2000; Sane, 2003). A common problem within the insect flight biomechanics community is that due to the small size of the specimens, and their rapid movements, often direct experimental studies of the
n Corresponding author at: Institute of Zoology, Functional Morphology and Biomechanics, Christian-Albrechts-University, Kiel, Germany. Tel: þ 49 431 8804505; Fax: þ 49 431 8801389. E-mail addresses: [email protected], [email protected] (H.-n. Rajabi).
wing dynamics and biomechanical properties are still very challenging and difficult. However, with the decreasing costs of computing power, numerical simulations have become more and more popular in studies on insect flight. In the recent years various finite element models have been used to study the biomechanical behaviour of wings from dragonflies, locusts and other insects; this work has been reviewed and discussed in detail by Wootton et al. (Dirks et al., 2013; Du and Sun, 2010; Ellington, 1984; Ennos, 1989; Wootton, 1992; Wootton et al., 2003). So far, these numerical models have been mostly used to simulate dynamic properties of the insect wing; for example the umbrella-like unfolding in the locust hind-wing (Herbert et al., 2000; Sane, 2003), or the vibrational and deflection behaviour of wings (Darvizeh et al., 2011, 2009; Kesel et al., 1998; Rajabi et al., 2011; Vanella et al., 2008). Interestingly, for some of these cases it has been shown that a relatively sophisticated and fine-tuned model is required to replicate the behaviour of the wing; and even small changes of the parameters can easily result in notable deviations of the simulation from the model (Herbert et al., 2000). Other studies however, focusing more on static material properties of the wings, have shown that even simple numerical models can suffice to simulate natural biomechanical “behaviour”
http://dx.doi.org/10.1016/j.jbiomech.2014.10.037 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Rajabi, H., et al., Numerical investigation of insect wing fracture behaviour. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.037i
H. Rajabi et al. / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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of the wing and simulations can be even used to supplement some otherwise difficult or even impossible mechanical experiments (Combes and Daniel, 2003a; Ganguli et al., 2009). Here we show that a relatively simple finite element model can be devised which is capable of closely replicating the fracture behaviour of a locust hind wing. In a previous study, we examined the fracture behaviour of locust wings (Dirks and Taylor, 2012a). Our results showed that the cross veins present in the wings act as barriers to crack propagation, effectively increasing the material's toughness by a factor of 50%. The objective of the present article was to devise a way to simulate these results numerically and to use the numerical model to investigate other factors which are difficult to study experimentally.
2. Materials and methods
tensile force
initial notch
Fig. 1. Sketch of the experimental and simulated set-up for tensile tests on a locust hind wing.
2.1. Finite element modelling using image processing technique To create a 2D meshed model of the test samples, images were captured from video recordings of locust hind wings under tensile loading (N ¼ 11, adult female Schistocerca gregaria locusts, see Dirks and Taylor, 2012a). We chose representative recordings showing typical vein-inhibited crack growth (suppl. video 1) or recordings where the sample failed due to the development of secondary cracks (suppl. Video 2). The images were recoded into binary format, where veins and membranes are displayed by black and white colours, respectively. A custom made Matlab code (R2012a, Mathworks, Natick, MA, USA) was then used to generate a mesh from the veins and membranes. The mesh was then imported into the explicit FE solver ABAQUS/Standard Version 6.1. The wings were modelled using four-node quadrilateral shell elements of 0.121 mm in size. The models contain 13964, 12250, 11563 nodes and 13724, 12024, 11329 elements, respectively. Supplementary material related to this article can be found online at http://dx. doi.org/10.1016/j.jbiomech.2014.10.037.
The first stage of the failure of a material is crack initiation. Crack initiation refers to the beginning of the process of degradation of an element. At the first step of crack initiation, the software checks the value of the stress at the element on the crack tip as well as the stress in all elements remote from the initial notch. When the highest principal stress in an element of the structure equals or exceeds the uniaxial ultimate tensile (Sut) or compressive strength (Suc) the element fails (Budynas and Nisbett, 2008). The general state of the stress at any point of the body can be defined by three principal stresses, σ 1 4 σ 2 4 σ 3 with failure occurring when
σ 1 Z Sut :
ð1Þ
It is important to note that the crack initiation criterion predicts only the initial start of a crack. How the elements split into two parts and the magnitude of separation are governed by a traction-separation law, as described in the next section.
2.2. Material properties and boundary conditions Only very limited datasets are available on the mechanical properties of insect cuticle, (Dirks and Taylor, 2012b; Vincent and Wegst, 2004). In particular there are no experimental data on the wing vein strength or fracture toughness. However, previous experimental tensile tests on insect wings indicate that the deformation of cuticle usually occurs in an elastic manner (Dirks and Taylor, 2012a; Rajabi and Darvizeh, 2013; Wootton et al., 2000). Therefore, we used a linear elastic material model to describe the mechanical behaviour of the wing veins and membranes. Our earlier experiments with hind wings from S. gregaria locusts showed that the wing membrane has a mean tensile strength 52.21 MPa, a stiffness of 1.86 GPa and a fracture toughness of 1.04 MPa√m, which corresponds well with values published elsewhere (Dirks and Taylor, 2012a; Smith et al., 2000). As there are currently no experimental datasets available for the Young's modulus of longitudinal and cross veins in locust hind wings, we have chosen a stiffness of 3 GPa and a strength of 52.21 MPa, which corresponds the stiffness of leg cuticle and tensile strength of the membrane respectively (Dirks and Taylor, 2012b). The fracture toughness of the veins was considered to be 1.57 MPa√m (Dirks and Taylor, 2012a).The material density and the Poisson's ratio of both veins and membrane cuticle were taken as 1200 k m 3 and 0.49, respectively (Combes and Daniel, 2003b; Smith et al., 2000; Vincent and Wegst, 2004). The thickness of the membranes was assumed to be constant 1.7 mm over the entire wing structure (Smith et al., 2000). The thickness of the simulated veins (with rectangular cross sections) was adapted to equal the cross sections of the circular hollow real cross-veins. After assigning the appropriate thickness to the models, the model was set up to simulate the fracture toughness test used in our experimental work (Dirks and Taylor, 2012a). Specifically, a rectangular sample of wing had a sharp notch induced into it from one edge; the sample was then loaded in axial tension with a crosshead speed of 0.2 mm/min with uniform displacement boundary conditions across the upper and lower edges (see Fig. 1).
2.4. Traction-separation law When a crack-like defect appears in a stressed material, a damage region may be developed due to stress concentration ahead of the crack tip. The damaged region, which is a consequence of plasticity or micro-cracking, is known as the cohesive zone (Anderson, 1991). FE modelling of crack propagation can be achieved by introducing the cohesive zone elements between all continuum elements. These cohesive elements are used to represent the cohesive forces that act against element separation (Achintha and Burgoyne, 2013). When failure takes place, these interface elements open up and the continuum elements will be separated (Brocks et al., 2003). The fracture of a material can be characterized by traction-separation laws, i.e. they are used to describe the constitutive behaviour of a material in the cohesive zone. For our simulations, and to reduce computational complexity, we used a linear traction-separation law (Camacho and Ortiz, 1996), which is determined by the characteristic toughness and characteristic strength of the material and can be written in the following form δ t ¼ t0 1 : ð2Þ
δσcr
In the above, t is the cohesive traction, t 0 is the traction at fracture, δ is the crack opening displacement and δσ cr is the critical opening displacement. Based on this linear softening model, the cohesive force is linearly and irreversibly decreased to zero as the crack opening displacement is increased. When the critical displacement is reached, the cohesive elements lose their stiffness and the crack growth occurs in a direction perpendicular to the maximum principal stress. In our simulations, the experimentally measured values of tensile strength and fracture toughness of the wing materials reported above are taken as the characteristic strength and characteristic toughness, respectively.
2.3. Crack initiation criterion
3. Results
Our previous experiments showed that locust wing behaves like a brittle material as regards its crack propagation characteristics (Dirks and Taylor, 2012a). To simulate the failure of brittle biological materials, such as bone, eggshell and mollusc shells, previous studies have successfully applied the maximum principal stress theory (Darvizeh et al., 2014, 2013; Doblaré et al., 2004; Faghih Shojaei et al., 2012; Willinger et al., 2000). We have therefore also chosen the maximum principal stress theory to simulate the crack initiation in the insect wings.
An example of our results for the simulated crack propagation through a locust hind wing is shown in video 1 (see suppl. video 1 for the experimental test). To illustrate the results we chose representative screenshots of the simulation before the crack hits the first cross vein, at each cross vein and before complete failure of the wing (see Fig. 2). The stress-strain values from the same
Please cite this article as: Rajabi, H., et al., Numerical investigation of insect wing fracture behaviour. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.037i
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Fig. 2. Selected frames from numerical simulation of crack-propagation and tensile stress in a S. gregaria hind-wing membrane and wing veins under tensile load. A) Initially, the stress is evenly distributed across the sample (the arrow indicates the initial position of the crack tip). B) When hitting the first cross-vein the initial crack is delayed. C to F) Further tensile loading drives the crack through the wing structure. Each cross-vein further delays the crack propagation. Although the stress-concentration is generally largest right in front of the crack, also the vein pattern distributes stress within the structure.
experiment and the corresponding numerical simulation are shown in Fig. 3. These figures illustrate that the fracture patterns and the mechanical stresses obtained by the numerical simulation are in very good qualitative and quantitative agreement with the experimental results. As discussed in our previous study (Dirks and Taylor, 2012a), each peak in the stress-strain curves corresponds to the point where the crack is temporarily stopped by a reinforcing cross-vein. Finding these peaks in the simulated stressstrain curves was rather to be expected, given an accurate enough numerical model and a sufficient difference in stiffness between the veins and the membrane. However, more interestingly, also the absolute peak stress values of the simulation correspond
extremely well with an average error of 6.53% to the experimental results, even though only very basic material-property assumptions were made. Fig. 4 shows that a pre-existing crack with 484 mm length, hence smaller than the “critical crack length” (CCL) does not notably propagate. Only at tensile stresses approaching the UTS the wing shows catastrophic failure close to the grips. Notable differences between the experimental data and the results of the simulations are a direct and more linear increase of stress immediately after start of the simulated tensile stress. This is likely to be a result of uniformly applied stress in the simulation, preventing experimental artefacts such as sample “alignment” after the start of the test.
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To test the robustness of our model we performed several simulations using different fracture energies and relative wing sizes. The results show the expected qualitative and quantitative changes and thus illustrate the robustness of our simulation in respect to morphological, geometrical and biomechanical variations within the wing samples (see suppl. Fig. 1).
25
4
3
simulation
14
experiment 5 2
12 6
stress (MPa)
10 15
stress
8 1
10
crack length (mm)
20
4. Discussion 4.1. Stress distribution in the wing One of the major advantages of numerical biomechanical simulations is their ability to reveal information which cannot be obtained experimentally, and thus provide insight into the underlying physical and biological processes involved. In this case, we were able to study the quantitative and qualitative stressdistribution within the wing membrane and the wing veins. Besides the crack propagation, the colour code of Fig. 2 also shows the numerically simulated distribution of the first principal stress in the veins and membranes at six time points during tensile loading. The pattern of the veins and artefacts from the fixation result in only few and small local stress concentrations (upper left corners). Once the crack starts to propagate and reaches the first
6
5
crack length
4
0
2 0
5
10
15
20
strain (%)
Fig. 3. Comparison of stress-strain curve and corresponding crack length from numerical simulation and experiment. With increasing strain the stress on the wing membrane increases until the crack starts to propagate. Each cross-vein temporarily stops the crack propagation and increases the stress on the structure. When the crack breaks through a cross vein, the stress decreases and the crack continues to propagate. The results of the numerical simulation show a remarkable similarity to the experimental results.
1 mm Fig. 5. A crack propagating towards a longitudinal vein is deflected and keeps running towards the cross-sectional veins.
Fig. 4. A wing under tensile load with a crack smaller than the critical crack length fails due to secondary cracks developing near stress concentrations. Our numerical model is also able to reproduce these non-trivial failure modes.
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vein (see Fig. 5). This behaviour can frequently be observed when performing experimental tensile tests on locust wings.
5. Conclusion
Fig. 6. Magnified section of the numerical results of stress-distribution in the wing. The stress is distributed into nearby cells even through intact longitudinal and cross veins. Scale bar 1 mm, for colour-coding see Fig. 1.
vein (see Fig. 2B and video 2), a high stress is carried by the crossvein and transmitted into the neighbouring wing cells (see Fig. 6). As the results show, the cross veins effectively transfer the tensile stress to the longitudinal veins which are thicker than the cross veins. It is further interesting to note that in some parts of the wing, the cross-veins are “branching” from the longitudinal veins like a symmetric crossroad intersection rather than alternating T-junctions (or an overlapping brick-pattern). This morphological adaptation might help to avoid local stress concentrations, when the cross-veins transmit the stress into the neighbouring cells. Part of the tensile stress is however still transmitted into the nearby wing cells. Hence, even longitudinal veins obviously cannot completely “shield” the neighbouring cells above and below the propagating crack and only help to distribute stress into another cell. The results presented in this article are based on morphological features and biomechanical properties of adult S. gregaria locust hind wings. Results of our previous study however have illustrated possible morphological similarities between the vein-patterns found hind-wings of S. gregaria and the wings of dragonflies (Dirks and Taylor, 2012a). So far, there is very limited data available on the fracture properties of other insect wings, however we hope to address and expand our model to other species in future studies. 4.2. Critical crack length In our previous study we were able to use our experimental results to calculate a CCL for the locust hind wing membrane. At a given stress σ, defects smaller than the CCL will not self-propagate and can be calculated using the fracture toughness KC of a material pffiffiffiffiffiffi K C ¼ F σ π a: ð3Þ At a tensile stress near the ultimate tensile strength (UTS), our earlier work showed that the critical crack length of the wing membrane is 566 mm for an edge crack. Our experiments also showed that if the length of the initial crack is less than this CCL, the crack does not propagate and tensile failure mainly occurs near the grips with high stress concentrations (see Fig. 4). The simple numerical model presented in this study is also capable of reproducing this failure mode. This finding is important because it defines an optimal separation for the veins to achieve maximum strength for the wing structure. In addition, our numerical model is also capable of showing that cracks briefly propagating towards a longitudinal vein are deflected and continue their growth parallel to the longitudinal
The results of this study show for the first time that even a simple finite element model can be used to comprehensively simulate the relatively complex tensile failure behaviour of an insect wing made of cuticle. Using established approaches to model crack initiation and propagation, and basic assumptions of the material properties, our model was found to be capable of qualitatively and quantitatively reproducing the crack propagation through a cross-vein reinforced wing. Our model allows us to obtain experimentally inaccessible information, such as the stressdistribution within the wing membrane. As the lifetime of biologically inspired micro-air vehicles (MAVs) is still partly limited by the durability of the artificial wing membrane (Bontemps et al., 2012; Ellington, 1999; Ma et al., 2013), we believe that our numerical model thus could be used to easily design and optimize wing-vein patterns to improve the durability of flapping MAVs. Given the availability of better experimental data on the wing veins' stiffness, strength and fracture toughness, future work is planned to address several experimentally not easily accessible questions regarding the biomechanical properties of insect wings. In particular we will study the crack propagation during tensile fatigue and the effect of crack orientation within the wings.
Conflicts of interest statement The authors declare there are no conflicts of interest to disclose.
Acknowledgements This study was financially supported by German Academic Exchange Service (DAAD), the University of Guilan and the Max Planck Society. In addition, the authors would like to thank Mr. Shahab Eshghi for his help in developing the finite element models.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2014.10.037. References Achintha, M., Burgoyne, C., 2013. Fracture energy of the concrete–FRP interface in strengthened beams. Eng. Fract. Mech 110, 38–51. Anderson, T.L., 1991. Fracture Mechanics Fundamentals and Applications. CRC Press, Florida. Bontemps, A., Vanneste, T., Paquet, J.-B., Dietsch, T., Grondel, S., Cattan, E., 2012. Design and performance of an insect-inspired nano air vehicle. Smart Mater. Struct. 22, 014008. Brocks, W., Cornec, A., Scheider, I., 2003. Computational aspects of nonlinear fracture mechanics (No. GKSS/WMS/02/05), GKSS Forschunszentrum Geesthacht GmbH. GKSS-Forschungszentrum Geesthacht GMBH. Budynas, R.G., Nisbett, J.K., 2008. Shigley's Mechanical Engineering Design, eighth ed. McGraw-Hill Book Company, New Jersey. Camacho, G.T., Ortiz, M., 1996. Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33, 2899–2938. Combes, S., Daniel, T., 2003a. Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending. J. Exp. Biol. 206, 2989–2997. Combes, S., Daniel, T., 2003b. Flexural stiffness in insect wings I. Scaling and the influence of wing venation. J. Exp. Biol. 206, 2979–2987. Darvizeh, A., Ansari, R., Momen, A, Rajabi, H., 2014. Experimental and numerical studies of the mechanical behavior of mollusk shells. In: Presented at the BiAnnual International Conference on Experimental Solid Mechanics (X-Mech2014), Tehran.
Please cite this article as: Rajabi, H., et al., Numerical investigation of insect wing fracture behaviour. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.037i
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Darvizeh, A., Rajabi, H., Fatahtooei Nejad, S., Khaheshi, A., Haghdoust, P., 2013. Biomechanical properties of hen's eggshell: experimental study and numerical modeling. World Acad. Sci. Eng. Technol. 78, 468–471. Darvizeh, A., Rajabi, H., Khaheshi, A., Sobhani, M.K., Etedadi, J., 2011. Morphological and numerical investigations of butterfly wing composite structure. In: Presented at the International Bionic Engineering Conference, Boston. Darvizeh, M., Darvizeh, A., Rajabi, H., Rezaei, A., 2009. Free vibration analysis of dragonfly wings using finite element method. Int. J. Multiphys. 3, 101–110. Dirks, J.-H., Parle, E., Taylor, D., 2013. Fatigue of insect cuticle. J. Exp. Biol. 216, 1924–1927. Dirks, J.-H., Taylor, D., 2012a. Veins improve fracture toughness of locust wings. PLoS ONE 7, e43411. Dirks, J.-H., Taylor, D., 2012b. Fracture toughness of locust cuticle. J. Exp. Biol. 215, 1502–1508. ́ J.M., Gómez, M.J., 2004. Modelling bone tissue fracture and Doblaré, M., Garcıa, healing: a review. Eng. Fract. Mech. 71, 1809–1840. Du, G., Sun, M., 2010. Effects of wing deformation on aerodynamic forces in hovering hoverflies. J. Exp. Biol. 213, 2273–2283. Ellington, C.P., 1984. The aerodynamics of hovering insect flight. VI. Lift and power requirements. Philos. Trans. R. Soc. Lond. B Biol. Sci 305, 145–181. Ellington, C.P., 1999. The novel aerodynamics of insect flight: applications to microair vehicles. J. Exp. Biol. 202, 3439–3448. Ennos, A.R., 1989. Comparative functional morphology of the wings of Diptera. Zool. J. Linnean Soc. 96, 27–47. Faghih Shojaei, M., Mohammadi, V., Rajabi, H., Darvizeh, A., 2012. Experimental analysis and numerical modeling of mollusk shells as a three dimensional integrated volume. J. Mech. Behav. Biomed. Mater. 16, 38–54. Ganguli, R., Gorb, S.N., Lehmann, F.-O., Mukherjee, S., Mukherjee, S., 2009. An experimental and numerical study of calliphora wing structure. Exp. Mech. 50, 1183–1197. Herbert, R.C., Young, P.G., Smith, C.W., Wootton, R.J., Evans, K.E., 2000. The hind wing of the desert locust (Schistocerca gregaria Forskål). III. A finite element analysis of a deployable structure. J. Exp. Biol. 203, 2945–2955.
Kesel, A., Philippi, U., Nachtigall, W., 1998. Biomechanical aspects of the insect wing: an analysis using the finite element method. Comput. Biol. Med. 28, 423–437. Ma, K.Y., Chirarattananon, P., Fuller, S.B., Wood, R.J., 2013. Controlled flight of a biologically inspired, insect-scale robot. Science 340, 603–607. Rajabi, H., Darvizeh, A., 2013. Experimental investigations of the functional morphology of dragonfly wings. Chinese Phys. B 22, 088702. Rajabi, H., Moghadami, M., Darvizeh, A., 2011. Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing. J. Bionic Eng. 8, 165–173. Sane, S.P., 2003. The aerodynamics of insect flight. J. Exp. Biol. 206, 4191–4208. Smith, C.W., Herbert, R.C., Wootton, R.J., Evans, K.E., 2000. The hind wing of the desert locust (Schistocerca gregaria Forskål). II. Mechanical properties and functioning of the membrane. J. Exp. Biol. 203, 2933–2943. Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E., Balachandran, B., 2008. Influence of flexibility on the aerodynamic performance of a hovering wing. J. Exp. Biol. 212, 95–105. Vincent, J.F.V., Wegst, U.G.K., 2004. Design and mechanical properties of insect cuticle. Arth. Struct. Dev 33, 187–199. Willinger, R., Diaw, B.M., Kang, H.-S., 2000. Finite element modelling of skull fractures caused by direct impact. Int. J. Crashworthiness 5, 249–258. Wootton, R.J., 1992. Functional morphology of insect wings. Annu. Rev. Entomol. 37, 113–140. Wootton, R.J., Evans, K.E., Herbert, R.C., Smith, C.W., 2000. The hind wing of the desert locust (Schistocerca gregaria Forskål). I. Functional morphology and mode of operation. J. Exp. Biol. 203, 2921–2931. Wootton, R.J., Herbert, R.C., Young, P.G., Evans, K.E., 2003. Approaches to the structural modelling of insect wings. Philos. Trans. R. Soc. Lond. B Biol. Sci 358, 1577–1587.
Please cite this article as: Rajabi, H., et al., Numerical investigation of insect wing fracture behaviour. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.037i
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Numerical investigation of insect wing fracture behaviour H. Rajabi a,b,n, A. Darvizehc, A. Shafieib, D. Taylord, J.-H. Dirkse a
Institute of Zoology, Functional Morphology and Biomechanics, Christian-Albrechts-University, Kiel, Germany Department of Mechanical Engineering, The University of Guilan, Rasht, Iran Department of Mechanical Engineering, Faculty of Engineering, Anzali Branch, Islamic Azad University, Bandar Anzali, Iran d Trinity Centre for Bioengineering, Trinity College Dublin, Dublin, Ireland e Department of New Materials and Biosystems, Max Planck Institute for Intelligent Systems, Stuttgart, Germany b c
art ic l e i nf o
a b s t r a c t
Article history: Accepted 31 October 2014
The wings of insects are extremely light-weight biological composites with exceptional biomechanical properties. In the recent years, numerical simulations have become a very powerful tool to answer experimentally inaccessible questions on the biomechanics of insect flight. However, many of the presented models require a sophisticated balance of biomechanical material parameters, many of which are not yet available. In this article we show the first numerical simulations of crack propagation in insect wings. We have used a combination of the maximum-principal stress theory, the traction separation law and basic biomechanical properties of cuticle to develop simple yet accurate finite element (FE) models of locust wings. The numerical results of simulated tensile tests on wing samples are in very good qualitative, and interestingly, also in excellent quantitative agreement with previously obtained experimental data. Our study further supports the idea that the cross-veins in insect wings act as barriers against crack propagation and consequently play a dominant role in toughening the whole wing structure. The use of numerical simulations also allowed us to combine experimental data with previously inaccessible data, such as the distribution of the first principal stress through the wing membrane and the veins. A closer look at the stress-distribution within the wings might help to better understand fracture-toughening mechanisms and also to design more durable biomimetic micro-air vehicles. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Locust wing Crack propagation Stress distribution Critical crack length Finite element
1. Introduction Insect wings appear as delicate and fragile structures, yet they are able to efficiently withstand external mechanical stress during the lifetime of an insect. During normal flight, the flapping or gliding wings of insects can experience different types of dynamic loading, they can actively or passively change their shape, and are subjected to aerodynamic forces significantly larger than the insects’ weight for thousands of loading cycles (Dirks et al., 2013; Du and Sun, 2010; Ellington, 1984; Ennos, 1989; Wootton, 1992; Wootton et al., 2003). Still many fundamental questions regarding the biomechanics of insect flight remain unanswered (Herbert et al., 2000; Sane, 2003). A common problem within the insect flight biomechanics community is that due to the small size of the specimens, and their rapid movements, often direct experimental studies of the
n Corresponding author at: Institute of Zoology, Functional Morphology and Biomechanics, Christian-Albrechts-University, Kiel, Germany. Tel: þ 49 431 8804505; Fax: þ 49 431 8801389. E-mail addresses: [email protected], [email protected] (H.-n. Rajabi).
wing dynamics and biomechanical properties are still very challenging and difficult. However, with the decreasing costs of computing power, numerical simulations have become more and more popular in studies on insect flight. In the recent years various finite element models have been used to study the biomechanical behaviour of wings from dragonflies, locusts and other insects; this work has been reviewed and discussed in detail by Wootton et al. (Dirks et al., 2013; Du and Sun, 2010; Ellington, 1984; Ennos, 1989; Wootton, 1992; Wootton et al., 2003). So far, these numerical models have been mostly used to simulate dynamic properties of the insect wing; for example the umbrella-like unfolding in the locust hind-wing (Herbert et al., 2000; Sane, 2003), or the vibrational and deflection behaviour of wings (Darvizeh et al., 2011, 2009; Kesel et al., 1998; Rajabi et al., 2011; Vanella et al., 2008). Interestingly, for some of these cases it has been shown that a relatively sophisticated and fine-tuned model is required to replicate the behaviour of the wing; and even small changes of the parameters can easily result in notable deviations of the simulation from the model (Herbert et al., 2000). Other studies however, focusing more on static material properties of the wings, have shown that even simple numerical models can suffice to simulate natural biomechanical “behaviour”
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Please cite this article as: Rajabi, H., et al., Numerical investigation of insect wing fracture behaviour. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.037i
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of the wing and simulations can be even used to supplement some otherwise difficult or even impossible mechanical experiments (Combes and Daniel, 2003a; Ganguli et al., 2009). Here we show that a relatively simple finite element model can be devised which is capable of closely replicating the fracture behaviour of a locust hind wing. In a previous study, we examined the fracture behaviour of locust wings (Dirks and Taylor, 2012a). Our results showed that the cross veins present in the wings act as barriers to crack propagation, effectively increasing the material's toughness by a factor of 50%. The objective of the present article was to devise a way to simulate these results numerically and to use the numerical model to investigate other factors which are difficult to study experimentally.
2. Materials and methods
tensile force
initial notch
Fig. 1. Sketch of the experimental and simulated set-up for tensile tests on a locust hind wing.
2.1. Finite element modelling using image processing technique To create a 2D meshed model of the test samples, images were captured from video recordings of locust hind wings under tensile loading (N ¼ 11, adult female Schistocerca gregaria locusts, see Dirks and Taylor, 2012a). We chose representative recordings showing typical vein-inhibited crack growth (suppl. video 1) or recordings where the sample failed due to the development of secondary cracks (suppl. Video 2). The images were recoded into binary format, where veins and membranes are displayed by black and white colours, respectively. A custom made Matlab code (R2012a, Mathworks, Natick, MA, USA) was then used to generate a mesh from the veins and membranes. The mesh was then imported into the explicit FE solver ABAQUS/Standard Version 6.1. The wings were modelled using four-node quadrilateral shell elements of 0.121 mm in size. The models contain 13964, 12250, 11563 nodes and 13724, 12024, 11329 elements, respectively. Supplementary material related to this article can be found online at http://dx. doi.org/10.1016/j.jbiomech.2014.10.037.
The first stage of the failure of a material is crack initiation. Crack initiation refers to the beginning of the process of degradation of an element. At the first step of crack initiation, the software checks the value of the stress at the element on the crack tip as well as the stress in all elements remote from the initial notch. When the highest principal stress in an element of the structure equals or exceeds the uniaxial ultimate tensile (Sut) or compressive strength (Suc) the element fails (Budynas and Nisbett, 2008). The general state of the stress at any point of the body can be defined by three principal stresses, σ 1 4 σ 2 4 σ 3 with failure occurring when
σ 1 Z Sut :
ð1Þ
It is important to note that the crack initiation criterion predicts only the initial start of a crack. How the elements split into two parts and the magnitude of separation are governed by a traction-separation law, as described in the next section.
2.2. Material properties and boundary conditions Only very limited datasets are available on the mechanical properties of insect cuticle, (Dirks and Taylor, 2012b; Vincent and Wegst, 2004). In particular there are no experimental data on the wing vein strength or fracture toughness. However, previous experimental tensile tests on insect wings indicate that the deformation of cuticle usually occurs in an elastic manner (Dirks and Taylor, 2012a; Rajabi and Darvizeh, 2013; Wootton et al., 2000). Therefore, we used a linear elastic material model to describe the mechanical behaviour of the wing veins and membranes. Our earlier experiments with hind wings from S. gregaria locusts showed that the wing membrane has a mean tensile strength 52.21 MPa, a stiffness of 1.86 GPa and a fracture toughness of 1.04 MPa√m, which corresponds well with values published elsewhere (Dirks and Taylor, 2012a; Smith et al., 2000). As there are currently no experimental datasets available for the Young's modulus of longitudinal and cross veins in locust hind wings, we have chosen a stiffness of 3 GPa and a strength of 52.21 MPa, which corresponds the stiffness of leg cuticle and tensile strength of the membrane respectively (Dirks and Taylor, 2012b). The fracture toughness of the veins was considered to be 1.57 MPa√m (Dirks and Taylor, 2012a).The material density and the Poisson's ratio of both veins and membrane cuticle were taken as 1200 k m 3 and 0.49, respectively (Combes and Daniel, 2003b; Smith et al., 2000; Vincent and Wegst, 2004). The thickness of the membranes was assumed to be constant 1.7 mm over the entire wing structure (Smith et al., 2000). The thickness of the simulated veins (with rectangular cross sections) was adapted to equal the cross sections of the circular hollow real cross-veins. After assigning the appropriate thickness to the models, the model was set up to simulate the fracture toughness test used in our experimental work (Dirks and Taylor, 2012a). Specifically, a rectangular sample of wing had a sharp notch induced into it from one edge; the sample was then loaded in axial tension with a crosshead speed of 0.2 mm/min with uniform displacement boundary conditions across the upper and lower edges (see Fig. 1).
2.4. Traction-separation law When a crack-like defect appears in a stressed material, a damage region may be developed due to stress concentration ahead of the crack tip. The damaged region, which is a consequence of plasticity or micro-cracking, is known as the cohesive zone (Anderson, 1991). FE modelling of crack propagation can be achieved by introducing the cohesive zone elements between all continuum elements. These cohesive elements are used to represent the cohesive forces that act against element separation (Achintha and Burgoyne, 2013). When failure takes place, these interface elements open up and the continuum elements will be separated (Brocks et al., 2003). The fracture of a material can be characterized by traction-separation laws, i.e. they are used to describe the constitutive behaviour of a material in the cohesive zone. For our simulations, and to reduce computational complexity, we used a linear traction-separation law (Camacho and Ortiz, 1996), which is determined by the characteristic toughness and characteristic strength of the material and can be written in the following form δ t ¼ t0 1 : ð2Þ
δσcr
In the above, t is the cohesive traction, t 0 is the traction at fracture, δ is the crack opening displacement and δσ cr is the critical opening displacement. Based on this linear softening model, the cohesive force is linearly and irreversibly decreased to zero as the crack opening displacement is increased. When the critical displacement is reached, the cohesive elements lose their stiffness and the crack growth occurs in a direction perpendicular to the maximum principal stress. In our simulations, the experimentally measured values of tensile strength and fracture toughness of the wing materials reported above are taken as the characteristic strength and characteristic toughness, respectively.
2.3. Crack initiation criterion
3. Results
Our previous experiments showed that locust wing behaves like a brittle material as regards its crack propagation characteristics (Dirks and Taylor, 2012a). To simulate the failure of brittle biological materials, such as bone, eggshell and mollusc shells, previous studies have successfully applied the maximum principal stress theory (Darvizeh et al., 2014, 2013; Doblaré et al., 2004; Faghih Shojaei et al., 2012; Willinger et al., 2000). We have therefore also chosen the maximum principal stress theory to simulate the crack initiation in the insect wings.
An example of our results for the simulated crack propagation through a locust hind wing is shown in video 1 (see suppl. video 1 for the experimental test). To illustrate the results we chose representative screenshots of the simulation before the crack hits the first cross vein, at each cross vein and before complete failure of the wing (see Fig. 2). The stress-strain values from the same
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Fig. 2. Selected frames from numerical simulation of crack-propagation and tensile stress in a S. gregaria hind-wing membrane and wing veins under tensile load. A) Initially, the stress is evenly distributed across the sample (the arrow indicates the initial position of the crack tip). B) When hitting the first cross-vein the initial crack is delayed. C to F) Further tensile loading drives the crack through the wing structure. Each cross-vein further delays the crack propagation. Although the stress-concentration is generally largest right in front of the crack, also the vein pattern distributes stress within the structure.
experiment and the corresponding numerical simulation are shown in Fig. 3. These figures illustrate that the fracture patterns and the mechanical stresses obtained by the numerical simulation are in very good qualitative and quantitative agreement with the experimental results. As discussed in our previous study (Dirks and Taylor, 2012a), each peak in the stress-strain curves corresponds to the point where the crack is temporarily stopped by a reinforcing cross-vein. Finding these peaks in the simulated stressstrain curves was rather to be expected, given an accurate enough numerical model and a sufficient difference in stiffness between the veins and the membrane. However, more interestingly, also the absolute peak stress values of the simulation correspond
extremely well with an average error of 6.53% to the experimental results, even though only very basic material-property assumptions were made. Fig. 4 shows that a pre-existing crack with 484 mm length, hence smaller than the “critical crack length” (CCL) does not notably propagate. Only at tensile stresses approaching the UTS the wing shows catastrophic failure close to the grips. Notable differences between the experimental data and the results of the simulations are a direct and more linear increase of stress immediately after start of the simulated tensile stress. This is likely to be a result of uniformly applied stress in the simulation, preventing experimental artefacts such as sample “alignment” after the start of the test.
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To test the robustness of our model we performed several simulations using different fracture energies and relative wing sizes. The results show the expected qualitative and quantitative changes and thus illustrate the robustness of our simulation in respect to morphological, geometrical and biomechanical variations within the wing samples (see suppl. Fig. 1).
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3
simulation
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experiment 5 2
12 6
stress (MPa)
10 15
stress
8 1
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4. Discussion 4.1. Stress distribution in the wing One of the major advantages of numerical biomechanical simulations is their ability to reveal information which cannot be obtained experimentally, and thus provide insight into the underlying physical and biological processes involved. In this case, we were able to study the quantitative and qualitative stressdistribution within the wing membrane and the wing veins. Besides the crack propagation, the colour code of Fig. 2 also shows the numerically simulated distribution of the first principal stress in the veins and membranes at six time points during tensile loading. The pattern of the veins and artefacts from the fixation result in only few and small local stress concentrations (upper left corners). Once the crack starts to propagate and reaches the first
6
5
crack length
4
0
2 0
5
10
15
20
strain (%)
Fig. 3. Comparison of stress-strain curve and corresponding crack length from numerical simulation and experiment. With increasing strain the stress on the wing membrane increases until the crack starts to propagate. Each cross-vein temporarily stops the crack propagation and increases the stress on the structure. When the crack breaks through a cross vein, the stress decreases and the crack continues to propagate. The results of the numerical simulation show a remarkable similarity to the experimental results.
1 mm Fig. 5. A crack propagating towards a longitudinal vein is deflected and keeps running towards the cross-sectional veins.
Fig. 4. A wing under tensile load with a crack smaller than the critical crack length fails due to secondary cracks developing near stress concentrations. Our numerical model is also able to reproduce these non-trivial failure modes.
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vein (see Fig. 5). This behaviour can frequently be observed when performing experimental tensile tests on locust wings.
5. Conclusion
Fig. 6. Magnified section of the numerical results of stress-distribution in the wing. The stress is distributed into nearby cells even through intact longitudinal and cross veins. Scale bar 1 mm, for colour-coding see Fig. 1.
vein (see Fig. 2B and video 2), a high stress is carried by the crossvein and transmitted into the neighbouring wing cells (see Fig. 6). As the results show, the cross veins effectively transfer the tensile stress to the longitudinal veins which are thicker than the cross veins. It is further interesting to note that in some parts of the wing, the cross-veins are “branching” from the longitudinal veins like a symmetric crossroad intersection rather than alternating T-junctions (or an overlapping brick-pattern). This morphological adaptation might help to avoid local stress concentrations, when the cross-veins transmit the stress into the neighbouring cells. Part of the tensile stress is however still transmitted into the nearby wing cells. Hence, even longitudinal veins obviously cannot completely “shield” the neighbouring cells above and below the propagating crack and only help to distribute stress into another cell. The results presented in this article are based on morphological features and biomechanical properties of adult S. gregaria locust hind wings. Results of our previous study however have illustrated possible morphological similarities between the vein-patterns found hind-wings of S. gregaria and the wings of dragonflies (Dirks and Taylor, 2012a). So far, there is very limited data available on the fracture properties of other insect wings, however we hope to address and expand our model to other species in future studies. 4.2. Critical crack length In our previous study we were able to use our experimental results to calculate a CCL for the locust hind wing membrane. At a given stress σ, defects smaller than the CCL will not self-propagate and can be calculated using the fracture toughness KC of a material pffiffiffiffiffiffi K C ¼ F σ π a: ð3Þ At a tensile stress near the ultimate tensile strength (UTS), our earlier work showed that the critical crack length of the wing membrane is 566 mm for an edge crack. Our experiments also showed that if the length of the initial crack is less than this CCL, the crack does not propagate and tensile failure mainly occurs near the grips with high stress concentrations (see Fig. 4). The simple numerical model presented in this study is also capable of reproducing this failure mode. This finding is important because it defines an optimal separation for the veins to achieve maximum strength for the wing structure. In addition, our numerical model is also capable of showing that cracks briefly propagating towards a longitudinal vein are deflected and continue their growth parallel to the longitudinal
The results of this study show for the first time that even a simple finite element model can be used to comprehensively simulate the relatively complex tensile failure behaviour of an insect wing made of cuticle. Using established approaches to model crack initiation and propagation, and basic assumptions of the material properties, our model was found to be capable of qualitatively and quantitatively reproducing the crack propagation through a cross-vein reinforced wing. Our model allows us to obtain experimentally inaccessible information, such as the stressdistribution within the wing membrane. As the lifetime of biologically inspired micro-air vehicles (MAVs) is still partly limited by the durability of the artificial wing membrane (Bontemps et al., 2012; Ellington, 1999; Ma et al., 2013), we believe that our numerical model thus could be used to easily design and optimize wing-vein patterns to improve the durability of flapping MAVs. Given the availability of better experimental data on the wing veins' stiffness, strength and fracture toughness, future work is planned to address several experimentally not easily accessible questions regarding the biomechanical properties of insect wings. In particular we will study the crack propagation during tensile fatigue and the effect of crack orientation within the wings.
Conflicts of interest statement The authors declare there are no conflicts of interest to disclose.
Acknowledgements This study was financially supported by German Academic Exchange Service (DAAD), the University of Guilan and the Max Planck Society. In addition, the authors would like to thank Mr. Shahab Eshghi for his help in developing the finite element models.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2014.10.037. References Achintha, M., Burgoyne, C., 2013. Fracture energy of the concrete–FRP interface in strengthened beams. Eng. Fract. Mech 110, 38–51. Anderson, T.L., 1991. Fracture Mechanics Fundamentals and Applications. CRC Press, Florida. Bontemps, A., Vanneste, T., Paquet, J.-B., Dietsch, T., Grondel, S., Cattan, E., 2012. Design and performance of an insect-inspired nano air vehicle. Smart Mater. Struct. 22, 014008. Brocks, W., Cornec, A., Scheider, I., 2003. Computational aspects of nonlinear fracture mechanics (No. GKSS/WMS/02/05), GKSS Forschunszentrum Geesthacht GmbH. GKSS-Forschungszentrum Geesthacht GMBH. Budynas, R.G., Nisbett, J.K., 2008. Shigley's Mechanical Engineering Design, eighth ed. McGraw-Hill Book Company, New Jersey. Camacho, G.T., Ortiz, M., 1996. Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33, 2899–2938. Combes, S., Daniel, T., 2003a. Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending. J. Exp. Biol. 206, 2989–2997. Combes, S., Daniel, T., 2003b. Flexural stiffness in insect wings I. Scaling and the influence of wing venation. J. Exp. Biol. 206, 2979–2987. Darvizeh, A., Ansari, R., Momen, A, Rajabi, H., 2014. Experimental and numerical studies of the mechanical behavior of mollusk shells. In: Presented at the BiAnnual International Conference on Experimental Solid Mechanics (X-Mech2014), Tehran.
Please cite this article as: Rajabi, H., et al., Numerical investigation of insect wing fracture behaviour. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.037i
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Darvizeh, A., Rajabi, H., Fatahtooei Nejad, S., Khaheshi, A., Haghdoust, P., 2013. Biomechanical properties of hen's eggshell: experimental study and numerical modeling. World Acad. Sci. Eng. Technol. 78, 468–471. Darvizeh, A., Rajabi, H., Khaheshi, A., Sobhani, M.K., Etedadi, J., 2011. Morphological and numerical investigations of butterfly wing composite structure. In: Presented at the International Bionic Engineering Conference, Boston. Darvizeh, M., Darvizeh, A., Rajabi, H., Rezaei, A., 2009. Free vibration analysis of dragonfly wings using finite element method. Int. J. Multiphys. 3, 101–110. Dirks, J.-H., Parle, E., Taylor, D., 2013. Fatigue of insect cuticle. J. Exp. Biol. 216, 1924–1927. Dirks, J.-H., Taylor, D., 2012a. Veins improve fracture toughness of locust wings. PLoS ONE 7, e43411. Dirks, J.-H., Taylor, D., 2012b. Fracture toughness of locust cuticle. J. Exp. Biol. 215, 1502–1508. ́ J.M., Gómez, M.J., 2004. Modelling bone tissue fracture and Doblaré, M., Garcıa, healing: a review. Eng. Fract. Mech. 71, 1809–1840. Du, G., Sun, M., 2010. Effects of wing deformation on aerodynamic forces in hovering hoverflies. J. Exp. Biol. 213, 2273–2283. Ellington, C.P., 1984. The aerodynamics of hovering insect flight. VI. Lift and power requirements. Philos. Trans. R. Soc. Lond. B Biol. Sci 305, 145–181. Ellington, C.P., 1999. The novel aerodynamics of insect flight: applications to microair vehicles. J. Exp. Biol. 202, 3439–3448. Ennos, A.R., 1989. Comparative functional morphology of the wings of Diptera. Zool. J. Linnean Soc. 96, 27–47. Faghih Shojaei, M., Mohammadi, V., Rajabi, H., Darvizeh, A., 2012. Experimental analysis and numerical modeling of mollusk shells as a three dimensional integrated volume. J. Mech. Behav. Biomed. Mater. 16, 38–54. Ganguli, R., Gorb, S.N., Lehmann, F.-O., Mukherjee, S., Mukherjee, S., 2009. An experimental and numerical study of calliphora wing structure. Exp. Mech. 50, 1183–1197. Herbert, R.C., Young, P.G., Smith, C.W., Wootton, R.J., Evans, K.E., 2000. The hind wing of the desert locust (Schistocerca gregaria Forskål). III. A finite element analysis of a deployable structure. J. Exp. Biol. 203, 2945–2955.
Kesel, A., Philippi, U., Nachtigall, W., 1998. Biomechanical aspects of the insect wing: an analysis using the finite element method. Comput. Biol. Med. 28, 423–437. Ma, K.Y., Chirarattananon, P., Fuller, S.B., Wood, R.J., 2013. Controlled flight of a biologically inspired, insect-scale robot. Science 340, 603–607. Rajabi, H., Darvizeh, A., 2013. Experimental investigations of the functional morphology of dragonfly wings. Chinese Phys. B 22, 088702. Rajabi, H., Moghadami, M., Darvizeh, A., 2011. Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing. J. Bionic Eng. 8, 165–173. Sane, S.P., 2003. The aerodynamics of insect flight. J. Exp. Biol. 206, 4191–4208. Smith, C.W., Herbert, R.C., Wootton, R.J., Evans, K.E., 2000. The hind wing of the desert locust (Schistocerca gregaria Forskål). II. Mechanical properties and functioning of the membrane. J. Exp. Biol. 203, 2933–2943. Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E., Balachandran, B., 2008. Influence of flexibility on the aerodynamic performance of a hovering wing. J. Exp. Biol. 212, 95–105. Vincent, J.F.V., Wegst, U.G.K., 2004. Design and mechanical properties of insect cuticle. Arth. Struct. Dev 33, 187–199. Willinger, R., Diaw, B.M., Kang, H.-S., 2000. Finite element modelling of skull fractures caused by direct impact. Int. J. Crashworthiness 5, 249–258. Wootton, R.J., 1992. Functional morphology of insect wings. Annu. Rev. Entomol. 37, 113–140. Wootton, R.J., Evans, K.E., Herbert, R.C., Smith, C.W., 2000. The hind wing of the desert locust (Schistocerca gregaria Forskål). I. Functional morphology and mode of operation. J. Exp. Biol. 203, 2921–2931. Wootton, R.J., Herbert, R.C., Young, P.G., Evans, K.E., 2003. Approaches to the structural modelling of insect wings. Philos. Trans. R. Soc. Lond. B Biol. Sci 358, 1577–1587.
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