RESEARCH ARTICLE

How does a slender tibia resist buckling? Effect of material,structural and geometric characteristics on buckling behaviour ofthe hindleg tibia in stick insect postembryonic developmentMaximilian Schmitt1,2, Thies H. Buscher1, Stanislav N. Gorb1 and Hamed Rajabi1,*

ABSTRACTDuring the lifespan of the stick insect Carausius morosus, their longand narrow tibiae experience substantial compressive loads. Themechanical load on the tibiae increases as the weight of the insectrises. The increase in body weight is accompanied by a notableincrease in the insect’s body size and, accordingly, by an increase inthe length of the tibiae. Both of these changes can raise the risk ofbuckling of the tibiae. In this study, we tracked changes in the materialand geometric properties of the hindleg tibia of C. morosus duringgrowth. The results show that although buckling (either by Eulerbuckling or local buckling) is the dominant failure mode undercompression, the tibia is very capable of maintaining its bucklingresistance in each postembryonic developmental stage. This isessentially the result of a compromise between the increasingslenderness of the tibia and its increasing material stiffness. Theuse of an optimal radius to thickness ratio, a soft resilin-dominatedcore, and chitin fibres oriented in both longitudinal and circumferentialdirections are presumably additional strategies preventing buckling ofthe tibia. This study, providing the first quantitative data on changes inthe biomechanical properties of cuticle during the entire life of aninsect, is expected to shed more light on the structure–property–function relationship in this complex biological composite.

KEYWORDS: Carausius morosus, Biomechanics, Material gradient,Cuticle, Stiffness, Slenderness

INTRODUCTIONThe term ‘compression member’ is employed in engineering todescribe structures that are designed to resist compressive forces(Budynas and Nisbett, 2015). Such structures are widely used incivil and mechanical engineering for load-carrying and supportingpurposes. Although structurally very efficient, compressionmembers have a major disadvantage: when slender, they aresusceptible to failure by buckling.Buckling is a complex mechanical phenomenon arising from the

instability of a structure subjected to compressive stresses. Inslender compression members, buckling takes place spontaneouslyand at stresses far below the strength of the material of which thestructure is made (Beer et al., 2006). Hence, it is known to be one of

the most catastrophic failure modes (Levy and Salvadori, 1992).This is somewhat reflected in a relatively large factor of safety ofengineering structures subjected to axial compression, considered tobe about four times higher than that of a similar structure undertension (Grote and Feldhusen, 2011). This higher factor of safety isoften achieved by three key strategies: (i) the use of materials withhigher Young’s moduli, (ii) adding extra material to the cross-section and (iii) adjusting the geometric parameters of the member(Clausen, 1851; Keller, 1960; Galambos, 1998).

Buckling, however, is not only limited to metals and traditionalengineering materials. Many biological systems have evolvedstrategies to overcome the problem of failure by buckling (Sakeset al., 2016). An example is the hollow stem of the bamboo plant,known as bamboo culm (Harries et al., 2017). The culm issubdivided into sections of regular size by so-called nodes,recognizable by the presence of external circular ridges. While thehollow structure increases the structural stiffness of the culm for agiven weight, the nodes improve its resistance to local buckling(Kappel et al., 2004). The latter is achieved by providing extrareinforcement and reducing the effective length of the culm.

A foam-like infill is another strategy utilized by some biologicalsystems. The foam core, which provides internal support againstkinking, can be found in plant stems, echidna spines and porcupinequills, all of which have closed-cell foam-filled cores (Karam andGibson, 1994). The foam may be additionally stiffened by thin,solid, radially extended ribs (Vincent and Owers, 1986). In somecases, such as in hedgehog spines, the foam may form a honeycombstructure, resulting in a remarkable increase in buckling resistancecompared with an equivalent hollow cylinder (Karam and Gibson,1994).

In comparison to the above-mentioned structural examples, somemembers of parasitic hymenopteran families (e.g. Aulacidae) employan adaptive behaviour as a buckling-prevention mechanism in theirlong ovipositor (Sakes et al., 2016). Although the ovipositor in thisgroup shows several morphological adaptations (Vilhelmsen et al.,2001), it is additionally supported by the insects’ hindlegs (Sakeset al., 2016). During penetration into a substrate, the ovipositor is heldbetween the legs, reducing its unsupported length.

Similar to many other body parts, insect legs, and in particulartheir tibiae, experience excessive bending and compressive stressesduring their lifespan (Parle et al., 2016a). Taking into account theirthin-walled hollow structure, such stresses increase the risk ofbuckling failure in tibiae, and thus potentially reduce insectsurvival. Therefore, it is reasonable to expect that insect tibiae areadapted to better resist buckling. Our knowledge about the bucklingbehaviour of insect tibiae, however, is limited to only a few recentreports (Parle et al., 2016a,b; Parle and Taylor, 2017). Althoughthese studies suggest that tibiae in some insects have undergoneadaptation to reduce the likelihood of buckling, we are not fullyReceived 26 October 2017; Accepted 14 December 2017

1Functional Morphology and Biomechanics, Institute of Zoology,Kiel University, Am Botanischen Garten 1-9, D-24098 Kiel, Germany.2Westphalian Institute for Biomimetics, Department of Mechanical Engineering,Westphalian University of Applied Sciences, Munsterstrasse 265, 46397 Bocholt,Germany.

*Author for correspondence (hrajabi@zoologie.uni-kiel.de; harajabi@hotmail.com)

H.R., 0000-0002-1792-3325

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aware of the mechanisms underlying this ability (Parle et al.,2016b). Here, therefore, we utilized a combination of mechanicaltesting and modern imaging techniques to address these questionsexperimentally. The well-established model organism in insectlocomotion (Graham, 1972; Schütz and Dürr, 2011), sensoryphysiology (Tichy, 1979) and control (Dean, 1991), the stick insectCarausius morosus (Sinéty 1901), was used in this study. Stickinsects (Phasmatodea) in general possess considerably elongatedextremities, usually with very long and slender tibiae. It is currentlyunknown whether, during daily activities, the tibiae are in danger ofbuckling. But, especially in case of a possible drop down fromelevated vegetation in which most species dwell (Büscher and Gorb,2017), they seem to be notably prone to buckling.As these stick insects are hemimetabolous, their tibiae scale up

over time from the first nymphal stage to the adult. We trackedchanges in the geometric characteristics, material properties andmechanical behaviour of the hindleg tibia, which is assumed toexperience highest forces during the insect locomotion, comparedwith the other legs (Pfeiffer et al., 1991). We aimed to answer thefollowing questions. (i) How do geometric parameters of the tibia,such as length, radius, thickness, slenderness ratio and cross-sectional shape, change during development? (ii) Does the Young’smodulus of the tibia cuticle change during the insect’s lifetime? (iii)How do the geometry of the tibia and the Young’s modulus of itsmaterial influence the buckling resistance of the tibia at eachpostembryonic developmental stage? (iv) Which biomechanicalstrategies influence the buckling behaviour of the tibia?

MATERIALS AND METHODSSpeciesThe stick insect C. morosus has six nymphal stages followed by theadult stage. Stick insects were taken from our laboratory culture andraised separately, in order to precisely track the nymphal stage andobserve moulting events. They were fed with fresh blackberryleaves and water ad libitum, and kept under a natural day/nightcycle.We always used hindleg tibiae of those specimens which werein the second week after moulting. For this purpose, the insects were

first anaesthetized with CO2, and then either the right or the lefthindleg tibia was cut using a sharp razor blade. Afterwards, theinsects were quarantined until the cut legs had been autotomizedthoroughly by the insects, and then returned to the laboratory cultureafter recovery. All procedures comply with ethical guidelines at KielUniversity.

Mechanical testingCompression tests were performed using a ZwickiLine uniaxialcompression testing machine (Zwick Roell, Ulm, Germany)equipped with a 5 N load cell (Xforce P load cell, Zwick Roell)(Fig. 1B). The distal end of the freshly cut hindleg tibiae was placedon a 3D-printed sample holder which had a 1 mm-deep hole in themiddle. The specimens were fixed in the hole using a small drop ofhot wax. The proximal end of the specimens was placed in contactwith a piece of rough sandpaper adhered to the load cell. Thisenabled us to more precisely control the straightness of specimensprior to the experiment. Compression tests were always performed atan increasing displacement of 5 mmmin−1. Force and displacementwere continuously recorded by the software testXpert (v3.5, ZwickRoell) throughout the experiment. We finished each experimentwithin a maximum of 15 min after removal of the tibiae from theinsects’ body. The whole testing process was filmed using a NikonD5300 digital camera (Nikon Corp., Tokyo, Japan) equipped with amacro lens (Canon Macro Lens EF 100 mm, Canon Inc., Tokyo,Japan). Mechanical tests were conducted on 10–17 hindleg tibiae ateach postembryonic developmental stage (Table 1).

We estimated the effective Young’s modulus (E) of the tibiacuticle using the following equation:

E ¼ FL

Ad; ð1Þ

where F is the measured force at any point of the linear portion of theforce–displacement curve and δ is the corresponding displacement.A and L are the average cross-sectional area and length of testedtibiae, respectively.

0 0.2 0.4 0.6 0.8 1.0

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Load cell

Tibia specimen

Displacement (mm)

Forc

e (m

N)

sample holderElastic deformation

Onset of Eulerbuckling

Formation of a plastic hinge

Loadingdirection

Movingstroke

Onset of localbuckling

A B Fig. 1. Uniaxial compression tests on thehindleg tibia of the stick insect Carausiusmorosus. (A) Representative force–displacement curves of specimens at the firstfive nymphal stages (L1–L5, red dashed line)and those at the last nymphal (L6) and adultstage (black dashed line). (B) Schematicdiagram of the compression testing machine.Tibia specimens were fixed on a 3D-printedsample holder and subjected to adisplacement-controlled compressive load.

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The stress at buckling was calculated by dividing the force atbuckling (Fb) by the cross-sectional area of specimens at thebuckling site:

sb ¼ Fb

A: ð2Þ

The cross-sectional area at the buckling site was taken as the averagecross-sectional area of at least three specimens at that region (see‘Micro-computed tomography scanning’, below). The bucklingsites were identified from the recorded videos.When loaded in axial compression, a column-like structure could

fail at either a material or a structural level. Failure at the materiallevel is essentially due to fracture or plasticity. In such cases, failureoccurs when the stress at any point of the structure exceeds theultimate compressive strength (σuc) or the yield strength (σy) of thematerial of which it is made. Failure at a structural level, however,could occur by either Euler or local buckling. The stress required tocause failure by Euler buckling can be theoretically predicted usingthe following formula (Timoshenko, 1953):

sEuler ¼ p2EI

AL2e; ð3Þ

where I and Le are the minimum second moment of area andeffective length of the column, respectively. The stress needed toinduce local buckling can be estimated by (Timoshenko and Gere,1961; Rees, 1997):

slocal ¼ E

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1� n2Þp t

r; ð4Þ

where t and r are the thickness and outer radius of the column,respectively. ν is Poisson’s ratio of the column material, and here itis assumed to be 0.3.

Micro-computed tomography scanningMicro-computed tomography (µCT) scanning was used to measurethe geometric parameters of the hindleg tibia of the stick insect at allseven postembryonic developmental stages. The tibiae were dried inan ascending ethanol series prior to scanning. All tibiae werescanned using a Skyscan 1172 µCT scanner (Bruker, Kontich,Belgium) with a resolution of 1–3 µm at a peak voltage of 40 kV anda current of 250 µA. The scans were performed on at least threetibiae at each developmental stage.The obtained data were used to calculate the second moment of

area (I ), cross-sectional area (A), thickness (t) and radius (r) of thespecimens. Measurements were performed using BoneJ (Doubeet al., 2010) and ImageJ software (v1.5i, National Institutes ofHealth, Bethesda, MD, USA).

The slenderness ratio (SR) of specimens was calculated bydividing the length (L) to minimum radius of gyration (rg):

SR ¼ L

rg; ð5Þ

where rg is obtained by:

rg ¼ffiffiffiI

A

r: ð6Þ

We used the data from µCT scans to measure the area and secondmoment of area for more than 2000 cross-sections of each scannedspecimen. Eqn 6 was then used to calculate the radius of gyration ofeach cross-section. The minimum obtained value was taken as theminimum radius of gyration of that specimen. The minimum radiusof gyration of specimens at each developmental stage was assumedto be equal to the average of those of at least three tibiae at that stage.

Scanning electron microscopyFor scanning electron microscopy (SEM), three tibiae at eachdevelopmental stagewere first air dried and then broken into smallerpieces. The fractures were mounted on SEM stubs using carbonLeit-tabs (Plano GmbH, Wetzlar, Germany). They were sputtercoated with a ∼10 nm-thick gold–palladium layer. The brokensections of the tibiae were imaged using a Hitachi S-4800 scanningelectron microscope (Hitachi High-Technologies, Tokyo, Japan) at3 kV accelerating voltage.

Confocal laser scanning microscopyConfocal laser scanning microscopy (CLSM) was utilized to obtainan understanding of the material composition of the tibia cuticle.Weused the method described by Michels and Gorb (2012) andpreviously employed by us to demonstrate the presence of the softand hard cuticle in insect exoskeletons (Rajabi et al., 2015, 2017a,2018). For this purpose, three freshly cut tibiae at eachdevelopmental stage were first washed in distilled water. Theywere then carefully cut into pieces of ∼0.5 mm thickness using asharp razor blade. The obtained sections were mounted in glycerinebetween a glass slide and a coverslip. We used a Zeiss LSM 700microscope (Carl Zeiss Microscopy, Jena, Germany) to visualizethe cross-section of the specimens. The microscope was equippedwith four stable solid-state lasers (405, 488, 555 and 639 nm) andfour emission filters (BP420–480, LP490, LP560 and LP640 nm).

StatisticsStatistical analyses were performed according to the protocolsuggested by Nayak and Hazra (2011). Because the obtained datawere unpaired and non-parametric, the Kruskal–Wallis H test wasutilized to compare datasets. The analysis was followed by Dunn’s

Table 1. Results of mechanical and geometric measurements at each developmental stage

Developmental stageNo. of tibiaspecimens

Criticalforce (mN)

Stress atbuckling (MPa)

Effective Young’smodulus (GPa)

Slendernessratio

Radius tothickness ratio

Area ratio of hardto soft cuticle

L1 10 4.4±2.1 66.7±14.2 1.18±0.28 67.7±11.4 6.84±0.67 0.39±0.03L2 14 13.8±8.4 39.5±14.0 0.80±0.26 52.7±13.3 8.10±0.65 0.34±0.02L3 11 32.2±12.1 49.3±18.7 1.46±0.71 52.4±14.6 9.51±0.79 0.44±0.03L4 11 38.6±21.6 45.5±22.5 1.79±1.03 66.5±11.5 9.16±1.35 0.46±0.05L5 17 75.7±25.6 67.3±23.6 2.84±0.59 66.0±11.1 13.13±1.21 0.54±0.03L6 10 93.2±28.8 65.6±24.2 4.46±0.58 101.8±28.3 4.32±0.41 0.65±0.01Adult 16 126.8±19.9 37.4±15.9 3.65±0.79 106.7±15.9 4.23±0.24 0.57±0.03

Data are means±s.d.

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post hoc test when appropriate. The values in the text are means±s.d., unless stated otherwise.

RESULTSRepresentative force–displacement curves obtained from uniaxialcompression tests on the hindleg tibiae of the stick insectC. morosus are shown in Fig. 1A. Prior to buckling, the tibiaeexhibited a linear elastic behaviour. Once buckling began, the slopeof the force–displacement curve decreased (onset of buckling). Thebuckling of the tibiae at different postembryonic developmentalstages appeared to occur in two different modes (shown by blackand red dashed lines in Fig. 1A). The first buckling mode wasassociated with the sideways bending of the whole sample (Fig. 1A,onset of Euler buckling). This buckling mode, known as Eulerbuckling, resulted in instability of the tibia. In this state, any furtherloading caused the formation of a plastic hinge and, finally, thecollapse of the specimen. This buckling mode was characteristic ofspecimens at the last nymphal (L6) and adult stages. The hindlegtibiae of individuals in the first five nymphal stages (L1–L5),however, buckled in a different way. In these specimens, theinstability was initiated by formation of a localized kink (Fig. 1A,onset of local buckling). The initial local buckling was subsequentlyfollowed by out-of-straightness of the tibia. In contrast to Eulerbuckling, localized buckling resulted in a sudden reduction of theresistance of the tibia to the applied displacement. Movies 1 and 2show the onset and evolution of buckling in two tibia specimens atthe fifth nymphal stage (L5) and adult stage, respectively.

Fig. 2A shows the critical buckling force of the hindleg tibia ateach postembryonic developmental stage. The specimens at the firstnymphal stage (L1) had the lowest critical force, 4±2 mN, and this

increased to a value of 39±21 mN at the fourth nymphal stage (L4).From this stage, critical force increased again, but at a slightlyhigher rate until it reached a maximum value of 127±20 mN at theadult stage.

The stress at buckling at each postembryonic developmental stageis given in Fig. 2B. The stress varied from a minimum value of11.26 MPa at the fourth nymphal stage (L4) to a maximum value of112.67 MPa at the fifth nymphal stage (L5). The mean stress, incontrast, was found to vary in a rather narrow range from 37.42 MPaat the adult stage to 67.33 MPa at the fifth nymphal stage (L5).However, statistical analysis indicated no significant differencebetween the stresses at different stages (H=15.62, N=89, P>0.05).

We estimated the stress in the tibia in a simple standing posture bydividing the insect body weight by the sum of the cross-sectionalarea of six tibiae. The stress induced by the body weight at eachstage is given in Fig. 2C. Similar to the stress at buckling, there wasno significant difference between the stresses at the different stages(H=16.58, N=89, P>0.05).

The effective Young’s modulus was used as a measure of thestiffness of the tibia cuticle (Fig. 2D). The effective Young’smodulus slightly decreased from 1.2±0.3 GPa at the first nymphalstage (L1) to 0.8±0.3 GPa at the second nymphal stage (L2).However, statistical analysis showed no significant differencebetween the moduli at these two stages (N=24, P>0.05). Therewas also no significant difference in the effective Young’s modulusof specimens at the sixth nymphal stage (L6, 4.5±0.6 GPa) and theadult stage (3.7±0.8 GPa) (N=26, P>0.05). Except for the slightdifferences between the aforementioned groups, the general trend inthe effective Young’s modulus was an upward increase duringgrowth of the insect.

Crit

ical

forc

e (m

N)

0

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L1 L2 L3 L4 L5 L4 L5L6 Adult

A

L1 L2 L3 L6 Adult

Stre

ss a

t buc

klin

g (M

Pa)

0

80

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L1 L2 L3 L4 L5 L6 Adult

Effe

ctiv

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s m

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Pa)

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C D

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eigh

t (kP

a)

60

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L1 L2 L3 L4 L5 L6 AdultStage

180

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Fig. 2. Data obtained from uniaxialcompression tests on the hindleg tibia at allpostembryonic developmental stages.(A) Critical buckling force increases in thecourse of insect development. (B) Stress atbuckling of the tibia. According to statisticalanalysis, there is no significant differencebetween the stresses at differentdevelopmental stages. (C) Stress acting on thetibia as a result of insect body weight was alsofound to remain almost constant during insectgrowth. (D) Effective Young’s modulus of thetibia cuticle. The general trend is an increase inthe effective Young’s modulus with thedevelopment of the insect. The borders of theboxes indicate the 25th and 75th percentiles,the line within them marks the median, and thewhiskers (error bars) define the 10th and 90thpercentiles.

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Fig. 3 represents the geometric parameters of the hindleg tibiaat each postembryonic developmental stage. A gradual changeoccurred in the length and cross-sectional shape of the tibia duringinsect growth (Fig. 3E). The data obtained from µCT analysis(available on request) were used to calculate the slenderness ratio ofthe tibia, presented in Fig. 3A. Statistical analysis revealed nosignificant difference between slenderness ratios of the tibia at thefirst five nymphal stages (L1–L5) (N=67, P>0.05). We also found

no significant difference between the slenderness ratios at the lastnymphal stage (L6) and the adult stage (N=26, P>0.05). However,there was a significant difference between the slenderness ratios ofthe tibiae in the first five nymphal stages (L1–L5) and the last twostages (L6 and adult) (H=58.23, N=93, P<0.05).

The radius to thickness (r/t) ratio of the tibia, shown in Fig. 3B,was found to increase from 6.84±0.67 at the first nymphal stage (L1)to 13.13±1.21 at the fifth nymphal stage (L5). From L5 to the sixth

E

B

Rad

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ickn

ess

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a fra

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har

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sof

t cut

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0.3

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Fig. 3. Data from geometric analysis of the hindleg tibia at all postembryonic developmental stages. (A) Slenderness ratio of the tibia. Although theslenderness ratio remains almost constant at the first five nymphal stages (L1–L5), a significant increase is observed from the fifth (L5) to the sixth (L6) nymphal andthe adult stage. (B) Ratio of the radius to the thickness of the hindleg tibia at each developmental stage. (C) Area of the hard, sclerotized cuticle and soft, resilin-dominated cuticle as a fraction of the total cross-sectional area. (D) Area ratio of the hard to soft cuticle, which appears to follow the same trend as Young’s modulus.The borders of the boxes indicate the 25th and 75th percentiles, the line within them marks the median, and the whiskers (error bars) define the 10th and 90thpercentiles. (E) Three-dimensionally (3D) reconstructed models of the hindleg tibia at all developmental stages. The cross-section of the tibia at several points isshown beside each model. A gradual change can be seen in the length and cross-sectional shape of the tibia during growth. Scale bar: 1 mm.

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nymphal stage (L6), the r/t ratio suddenly dropped to 4.32±0.37, butit remained almost constant from the sixth nymphal stage (L6) to theadult stage (N=16, P>0.05).The results of CLSM revealed the presence of two distinguishable

layers with different autofluorescence characteristics in the hindlegtibia cuticle (Fig. 4D,E). According toMichels and Gorb (2012), theblue autofluorescence of the inner layer indicates a high proportionof resilin, the pliant protein. The outer layer, in contrast, shows astrong green autofluorescence, which suggests that this layer mainlyconsists of sclerotized chitinous material. Fig. 3C shows the area ofthe hard sclerotized and soft resilin-dominated cuticle as a fractionof total area obtained from measurements on tibia cross-sections.Using these data, we calculated the area ratio of the hard to softcuticle at all stages (Fig. 3D). Interestingly, the obtained data seemto follow the same trend as that observed for the effective Young’s

modulus. The area ratio of the hard to soft cuticle slightly decreasedfrom 0.39±0.03 at the first nymphal stage (L1) to 0.34±0.02 at thesecond nymphal stage (L2). After this, the area ratio continuouslyincreased, reaching a maximum value of 0.65±0.01 at the sixthnymphal stage (L6). The area ratio of the hard to soft cuticle at theadult stage (0.57±0.03) was slightly lower than that measured in L6.

Fig. 4B,C,F shows representative SEM images of the hindlegtibia cuticle. The obtained images suggest that the tibia cuticleconsists of up to five main layers: (i) epicuticle, (ii) outer exocuticle,(iii) mesocuticle, (iv) inner exocuticle and (v) endocuticle. Theepicuticle is the outermost layer, which is usually very thin, with athickness of up to 0.5 µm in adult individuals. It covers a very denseexocuticular layer with a lamellar organization. The thickness of thislayer in adult specimens was 1.0–2.8 µm. Below the exocuticle,there is a less dense layer, which is probably the mesocuticle, with a

A B

C D

F

E

B

C

D

F

Exo

Exo

Endo

Meso

Epi

Exo

Exo

Meso

Fig. 4. Microstructure and material composition of the hindleg tibia cuticle. (A) Scheme of a tibia cross-section (see E) showing the approximate position ofeach given image (B,C,D,F). (B) Scanning electron microscopy (SEM) image showing the outer exocuticle (Exo), mesocuticle (Meso), inner exocuticle andendocuticle (Endo) in a cross-section of a specimen at the fifth nymphal stage (L5). (C) SEM image illustrating the lamellar microstructure of the outer and innerexocuticle and the unorganized mesocuticle in a cross-section of the tibia at the fifth nymphal stage (L5). The thin epicuticle layer covering the tibia is alsovisible in this image. (D) Confocal laser scanning microscopy (CLSM) image of a cross-section of an adult specimen showing the presence of a material gradientalong the thickness of the tibia. (E) CLSM image of the whole cross-section of an adult specimen showing the hard sclerotized and the soft resilin-dominatedcuticle in green and blue, respectively. (F) SEM image showing the perpendicularly arranged chitin fibres within the endocuticle of a specimen at the adult stage.Scale bars: A, 100 µm; B, 10 µm; C, 2 µm; D, 10 µm; E, 100 µm; and F, 5 µm.

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thickness ranging from 2.0 to 3.6 µm in adult individuals. Becauseof its less-organized structure, it was not possible to unambiguouslydetermine the arrangement of chitin microfibres in this layer. Wefound another exocuticle layer just beneath the mesocuticle.Although less dense than the outer exocuticle, the two seem tohave a similar layered structure. The inner exocuticle reached athickness of up to 2 µm in the adult stick insects. The innermostlayer in the cuticle microstructure is a less-dense endocuticle,consisting of alternating thin helicoidal and thicker unidirectionallayers. In some regions, the unidirectional layers within theendocuticle had the same orientation and, together with helicoidallayers, formed so-called locust-like layering (Neville and Luke,1969) (Fig. 4B). In some other regions, however, they changedorientation by 90 deg, resulting in the formation of a pseudo-orthogonal layering (Neville and Luke, 1969) (Fig. 4F). Theendocuticle, with a thickness of 8–16 µm in adult individuals, is thethickest layer in the tibia cuticle. Although we were able to confirmthe presence of the aforementioned layers in tibiae at allpostembryonic developmental stages, their thickness was found tobe very variable. The thickness of each particular layer was alsohighly dependent on the measurement point on the tibia, even in asingle individual.The measured mechanical and geometric characteristics of the

tibia during development of the stick insect C. morosus aresummarized in Table 1.

DISCUSSIONA variety of biological factors may influence the mechanicalbehaviour of tibiae of the stick insect C. morosus. An example isthe increase in the body size (Fig. S1B), which has an inevitableimpact on the length of the tibia (Fig. 3E; Fig. S1C). The increase inbody size further results in an increase in body mass (Fig. S1A) and,consequently, raises the force acting on each tibia. Although one mayexpect that the added body mass at each postembryonicdevelopmental stage increases the mechanical stress in the tibiae, asimple estimation suggests that this is not the case (Fig. 2C). Indeed,we found no significant difference between the stresses applied to thetibiae by the weight of the insect at different developmental stages.This may be a major reason why the buckling resistance of thehindleg tibia also remains constant during insect growth (Fig. 2B).A remarkable difference, however, was observed between the

stress at buckling and the stress imposed on the hindleg tibia bythe body weight (Fig. 2B,D). Comparison of the two stresses at theadult stage, for instance, suggests that the tibia possesses a largefactor of safety of 127.2±23.2 in standing posture. Using kinematicdata from the literature, we additionally determined the factor ofsafety of the tibia when subject to compressive stresses during dailyactivities. Data from ground reaction force measurements on freelywalking adult stick insects showed that compressive forces in theinsect hindlegs can reach a peak value of 9.6 mN (Dallmann et al.,2016). Assuming the entire force to be transmitted to the tibia, thisresults in a factor of safety of 13.7±2.2 for the tibia when walking.This is about twice the maximum factor of safety measured for thetibia of other arthropods, which ranged from 1.7 to 7 (Parle et al.,2016a). Therefore, this finding raises the question of why the tibianeeds such a high load-carrying capacity.A recent study reported frequent intentional drops or accidental

falls of the wingless stick insect Extatosoma tiaratum, whileclimbing vegetation (Zeng et al., 2015). This behaviour, which mayserve as a defence reaction, is also likely to be valid for the winglessspecies C. morosus. Immediately after falling from a height, thestick insects are able to reorientate and stabilize their body during

the plunge (Zeng et al., 2017), and land on their tarsi (M.S., T.H.B.,S.N.G. and H.R., unpublished observation). However, landing onthe tarsi may induce much higher stresses in the tibiae than thoseapplied in normal standing posture (Heitler, 1977). Although furtherinvestigations are needed, this may explain the large factor of safetyconsidered in the design of the hindleg tibia.

It is well known that the resistance of a structure to buckling isessentially influenced by its geometric characteristics, in particularslenderness, and the stiffness of the material of which the structure ismade (Cedolin, 2010). The increase of the slenderness ratio, leadingto a reduced buckling resistance, seems to be an inevitable part ofinsect growth. In the case of the stick insect C. morosus, this isreflected in the significant increase of the slenderness ratio of thetibia from the first five nymphal stages (L1–L5) to the last twostages (L6 and adult) (Fig. 3A). However, except for the slightdecrease in the effective Young’s modulus, the development of thestick insect is accompanied by a significant increase in the stiffnessof its tibia cuticle (Fig. 2D). The increase in the effective Young’smodulus appears to compensate for the negative effect of theincreasing slenderness, enabling the tibia to retain its bucklingresistance during insect growth (Fig. 2B).

The change of the effective Young’s modulus during insectpostembryonic development was found to follow the same trend asthat of the area proportion of the hard sclerotized to soft resilin-dominated cuticle (Figs 2D and 3D). Taking into account thepredominant role of the sclerotized cuticle in stiffening insect cuticle(Hepburn, 1985; Rajabi et al., 2017b), the similar trend betweenthese two parameters appears to be very reasonable. The slightdecrease in the effective Young’s modulus from the sixth nymphalstage (L6) to the adult stage (Fig. 2D) can also be explained by thepresence of a lower area proportion of the hard to soft cuticle at theadult compared with the L6 stage (Fig. 3C,D). Taking into accountthe remarkably longer life period of C. morosus at the adult stagethan at all nymphal stages together (Roth, 1917), it is reasonable tohypothesize that the deposition of the resilin-dominated endocuticleat the adult stage is also more prolonged.

We were able to distinguish two distinctly different force–displacement curves in the results from the uniaxial compressiontests on tibia specimens. While the specimens at the first fivenymphal stages (L1–L5) suddenly lost their load-carrying capacityafter buckling, those at the last nymphal (L6) and adult stagesshowed a gradual reduction in their ability to support the externalload (red and black dashed lines in Fig. 1A, respectively). Thisdifference appears to be essentially the result of the occurrence oftwo different buckling modes in these two groups of specimens (i.e.local buckling and Euler buckling, respectively). This finding isalso consistent with the observation of the considerably lower r/tratio of the tibiae at the sixth nymphal (L6) and adult stage whencompared with that of the other living stages (L1–L5) (Fig. 3B).Although the higher r/t ratio of the tibiae in the L1–L5 group causesthem to become more resistant to Euler buckling, this subsequentlymakes them more prone to local buckling.

Here, we adapted the method suggested by Taylor and Dirks(2012) to find the optimum value of the r/t ratio of the hingleg tibiaat each postembryonic developmental stage. This optimum ratioshould provide the highest load-carrying capacity for a givenweight. As stated, the failure of the tibia under compression isessentially the result of either Euler or local buckling. Fig. 5A showsgraphs of these two buckling modes (blue and grey dashed lines),plotted as a function of r/t for adult stick insects. The bucklingstress, given by the Euler curve, increases with the increase of the r/tratio. In contrast, the stress at buckling predicted by the local

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buckling curve decreases as the r/t ratio increases. The optimum r/tis reached where the two curves intersect. For the tibia of adultspecimens, this optimum value occurs at a point where the r/t ratio isequal to 5.72. As seen in Fig. 5A, the experimentally measuredvalues of r/t ratio, 4–4.46 (black ellipse), are close to thetheoretically predicted optimum value. Fig. 5B compares theexperimental r/t values with those calculated theoretically at eachpostembryonic developmental stage. A good agreement was foundbetween the two sets of data at all stages, except at the fifth nymphalstage (L5). Our analysis, in general, suggests the presence of abiomechanical adaptation in the hindleg tibia towards an optimumr/t ratio to reduce the probability of failure by buckling. Theoptimization of the r/t ratio has already been employed in the design

of steel circular hollow tubes used in wind turbine towers to controland improve their buckling behaviour (Guo et al., 2013).

As shown in Fig. 5B, the experimental r/t ratios of tibiae at thefirst five nymphal stages (L1–L5) are higher than the theoreticallypredicted optimum values. In contrast, the r/t ratios obtained frommeasurements on the tibiae at the sixth nymphal (L6) and adultstages are lower than the optimum ones. This finding highlights thedependency of the failure mode on the r/t ratio at each stage andfurther supports our observation that the buckling mode changesduring insect growth.

In the CLSM images, we were able to observe the presence of amaterial gradient along the thickness of the tibia cuticle (Fig. 4D,E).In addition to toughening the whole structure of the tibia, the softresilin-dominated core can improve its resistance to buckling(Karam and Gibson, 1995; Suresh, 2001). The hard sclerotizedcoverage, in contrast, enhances the stability of the tibia and protectsit against mechanical damage (Rajabi et al., 2016). Interestingly, inmany regions, it was possible to identify a clear border between thehard and soft cuticle layers (Fig. 4E). This observation confirms ourprevious finding that a realistic model of insect cuticle, in addition toconsideration of a material gradient, should take the discontinuity atthe interface of layers into account (Rajabi et al., 2017b). Such amaterial discontinuity in the insect cuticle may be a biomechanicalstrategy to avoid the propagation of an already existing crack fromone layer to another.

Natural fibre-reinforced composites, such as the tibia cuticle, arewell known for their light weight, high structural integrity,reasonable stiffness, strength and toughness (Mayer and Sarikaya,2002; Dirks and Taylor, 2012; Keogh et al., 2015). However, thebehaviour of such composites under loading is also influenced bytheir fibre orientation (Laranjeira et al., 2010; Martinie and Roussel,2011). It has previously been reported that hollow cylinders withfibres oriented in the axial direction have an optimum bendingstrength (compared with cylinders with fibres at a particular angle).In contrast, hollow cylinders reinforced with fibres in thecircumferential direction show optimum compression resistance(Li et al., 2006). In the tibia cuticle, we demonstrated the presence ofboth fibre orientations (Fig. 4F). During the insect lifetime, the tibiaexperiences a combination of bending and compression. Therefore,it is reasonable to expect that the cuticle in this body part is adaptedto resist both loading conditions by aligning fibres in these twoorientations.

Comparison of our results with those of previous measurementson tibiae of the adult stick insects loaded in bending (Parle et al.,2016b) indicates that the bending stiffness of the tibia cuticle islarger than its stiffness in compression (5.5±4.13 and 3.65±0.79 GPa, respectively). The difference between the measuredproperties is even more striking when comparing the strength of thetibia in bending and compression: 116.0±66.1 and 37.4±15.9 MPa,respectively. This suggests that, although compression is a typicalloading condition for the tibia, especially when an insect is standingupright, bending stresses in the leg during daily activities mayexceed those caused by compression. Future studies are planned toassess macro- and micro-morphological differences betweenforelegs, midlegs and hindlegs of the stick insect C. morosus andto evaluate their potential influence on the buckling resistance ofeach leg under combined bending and compression.

ConclusionsIn this study, we tracked changes in the material, structural andgeometric properties of the hindleg tibia of the stick insect C.morosus at all postembryonic developmental stages. We aimed to

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Fig. 5. Theoretical prediction of the optimal radius to thickness (r/t) ratioof the hindleg tibia when loaded in compression. Comparisons are shownbetween the theoretical and experimental data at all postembryonicdevelopmental stages. (A) Stress at buckling versus r/t ratio for the tibia at theadult stage. The load-carrying capacity of the tibia predicted by Euler buckling(blue dashed line) and local buckling (grey dashed line) is plotted as a functionof r/t ratio. The optimum value of r/t occurs at the point where the two curvesintersect (red dashed line). Assuming a yield strength of about 85 MPa (Parleet al., 2015), the horizontal black dashed line gives the stress at which the tibiacuticle may fail as a result of plasticity. The experimental data obtained frommeasurements on tibiae of adult stick insects fall within the black ellipse.(B) Comparison of the theoretically predicted optimal (red) and experimentallymeasured (black) r/t values at each developmental stage. Except at the fifthnymphal stage (L5), the experimental r/t values are close to the optimum value.

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understand how the observed changes can influence the load-carrying capacity of the tibia under compression. Based on theobtained data, the following conclusions can be drawn. (i) Althoughthe geometric characteristics of the hindleg tibia change remarkablyduring growth, the tibia is able to maintain its resistance to buckling.This is the result of a compromise between the increasingslenderness of the tibia and its increasing stiffness. (ii) The failureof the tibia under compression mainly occurs in two differentbuckling modes: local buckling, in the first five nymphal stages, andEuler buckling, in the sixth nymphal and the adult stage. (iii) Theoccurrence of either of these buckling modes at each developmentalstage is likely to be influenced by the ratio of radius to thickness (r/t)of the tibia at that stage. This ratio at most developmental stages isclose to an optimum value, representing a good capability to resistboth Euler and local buckling. (iv) The presence of a soft, resilin-dominated endocuticle with fibres orientated in both longitudinaland circumferential directions is likely to be a biomechanicaladaptation that provides the tibia with an enhanced bucklingresistance.

AcknowledgementsThe authors would like to thank Dr Clemens Schaber (Kiel University) and DrAlexander Kovalev (Kiel University) for their technical help. We thank Dr JanMichels(Kiel University) for his helpful comments on the CLSM images. We acknowledgethe helpful discussions with Prof. David Taylor (Trinity College Dublin), HalvorT. Tramsen (Kiel University), Mohsen Jafarpoor (Guilan University), Prof. TobiasSeidl (Westphalian University of Applied Sciences) and Ahmad R. Mojdehi (VirginiaPolytechnic Institute & State University). Thanks also to Prof. Volker Durr (BielefeldUniversity) and Marek Noheijl (Recany nad Labem, Czech Republic) for providingspecimens for this study.

Competing interestsThe authors declare no competing or financial interests.

Author contributionsConceptualization: S.N.G., H.R.; Methodology: T.H.B., S.N.G., H.R.; Software:M.S.; Validation: M.S., T.H.B.; Formal analysis: M.S.; Investigation: M.S.;Resources: T.H.B., S.N.G.; Data curation: M.S., T.H.B., H.R.; Writing - original draft:M.S., H.R.; Writing - review & editing: T.H.B., S.N.G., H.R.; Visualization: M.S.,T.H.B., H.R.; Supervision: T.H.B., S.N.G., H.R.; Project administration: S.N.G., H.R.;Funding acquisition: S.N.G., H.R.

FundingThis study was financially supported by Federal State Funding at Kiel University(Christian-Albrechts-Universitat zu Kiel) to H.R.

Data availability3D reconstructed models of the hindleg are available on request from thecorresponding author.

Supplementary informationSupplementary information available online athttp://jeb.biologists.org/lookup/doi/10.1242/jeb.173047.supplemental

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