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3、i t y C o l l e c t i o n o f S c h o l a r l y a n d A c a d e mi c Pa p e r s :HU S C A PPHYSICAL REVIEW E 93,022406(2016)Self-adaptive formation of uneven node spacings in wild bambooHiroyuki Shima*Department of Environmental Sciences,University of Yamanashi,4-4-37 Takeda,Kofu,Yamanashi 400-8510,
4、JapanMotohiro SatoDivision of Engineering and Policy for Sustainable Environment,Faculty of Engineering,Hokkaido University,Kita-13,Nishi-8,Kita-ku,Sapporo,Hokkaido 060-8628,JapanAkio InoueFaculty of Environmental and Symbiotic Sciences,Prefectural University of Kumamoto,3-1-100 Tsukide,Higashi-ku,K
5、umamoto,Kumamoto 862-8502,Japan(Received 24 September 2015;revised manuscript received 2 January 2016;published 10 February 2016)Bamboo has a distinctive structure wherein a long cavity inside a cylindrical woody section is divided intomany chambers by stiff diaphragms.The diaphragms are inserted at
6、 nodes and thought to serve as ring stiffenersfor bamboo culms against the external load;if this is the case,the separation between adjacent nodes shouldbe configured optimally in order to enhance the mechanical stability of the culms.Here,we reveal the hithertounknown blueprint of the optimal node
7、spacings used in the growth of wild bamboo.Measurement data analysistogether with theoretical formulations suggest that wild bamboos effectively control their node spacings as wellas other geometric parameters in accord with the lightweight and high-strength design concept.DOI:10.1103/PhysRevE.93.02
8、2406I.INTRODUCTIONBamboo is a fascinating tropical plant native to East andSoutheast Asia.A salient feature of bamboo is its growth rate:the fastest among all the worlds woody plants 1.Severalspecies of bamboo grow 1 m per day,with no need forfertilizers or watering.The rapid maturity of bamboo grov
9、esaffords them a clear advantage for survival over competingtree species,yielding low-cost and easily available materialsthat can be used in the construction and pulp industries 25.Moreover,a significant capability of bamboo for biomass ac-cumulationhasbeensuggestedinthepast68,especiallyforcarbon st
10、orage,which is beneficial to the global carbon cycle911.Withtheseecofriendlyproperties,bamboohascroppedup as a green alternative to other woody raw materials 5.From a mechanical standpoint,bamboo can be considereda nature-derived smart material in which a high stiffness andlight weight coexist 3,121
11、4.Its stiffness against tensileloading is due to the longitudinal arrays of cellulose fibersembedded within a ligneous matrix 15,16.The volume frac-tion of the fibers is maximum at the outer surface of the culmwalland decreases monotonically inthe inward direction 17.This gradient distribution of fi
12、bers is effective in reinforcingthe whole bamboo in a similar manner as functionally gradedcomposite materials 1823.Another important feature ofbamboo,its light weight,is a direct consequence of its hollowandcylindricalstructure.Hollownessisingeneralpreferredforcarrying loads compared with solid cou
13、nterparts of the samecross-sectionalarea24.Inaddition,hollowcylindersuselessmaterial and are lighter while resisting the same bending ortorsional loads.One possible disadvantage caused by hollowness is easeof collapse when a strong bending force is applied.In order*hshimayamanashi.ac.jpto overcome t
14、his limitation,bamboos form many“nodes”atintervalsalongtheculm(seeFig.1).Anodeisacombinationofanexternalridgeattheoutersurfaceandaninternaldiaphragmembedded in the hollow cavity(Fig.2).Since a diaphragmacts as a ring stiffener for the region near the node,sequentialinsertion of many diaphragms with
15、an appropriate spacingleads to improvement in the mechanical stiffness of the wholebamboo culm 19,2530.However,this argument poses aquestion:What node spacings are optimal for adapting tobending damage caused by external forces?Obviously,the stability of the culm under bending isenhanced by installi
16、ng diaphragms as densely as possible.However,the excessive density of diaphragms makes thewhole culm heavier and inhibits the growth rate of bamboo.Itis expected,therefore,that there is an optimum configurationof node spacings that bamboos have acquired inherently aftermillions of years of evolution
17、.Taking a closer look,in fact,one may find that the spacing between adjacent nodes(i.e.,theinternode length)is not constant but,rather,is longest slightlybelow the middle part of the culm and gradually diminishesfromtheretowardtheendsoftheculm(seeRefs.31and32,for instance).The purpose of this articl
18、e is to elucidate theorigin of inhomogeneity in internode lengths in wild bamboousing the language of structural mechanics.Combining theo-retical results with experimental bamboo geometry data,wedemonstrate that the inhomogeneity of internode lengths is aconsequence of the self-adaptive property of
19、wild bamboo forsecuring mechanical stability against bending.II.MECHANICS OF HOLLOW CYLINDERSA.Ovalization under pure bendingA bamboo culm is an orthotropic hollow cylinder dividedintochambersbystiffdiaphragmspositionedatthenodes.Het-erogeneity and anisotropy in the internal microstructure of thewoo
20、dyportionmaycontributetothemechanicalsuperiorityof2470-0045/2016/93(2)/022406(9)022406-12016 American Physical SocietyHIROYUKI SHIMA,MOTOHIRO SATO,AND AKIO INOUEPHYSICAL REVIEW E 93,022406(2016)FIG.1.Left:Wild bamboo grove.Right:Notation and terminol-ogy specifying each segment of a whole bamboo“cul
21、m.”The stem ofbambooistechnicallyknownasaculmsincebambooisthevernacularorcommontermformembersofaparticulartaxonomicgroupoflargewoody grasses(family Poaceae)8.bamboo culms.To make a concise argument,nevertheless,weuse a simplified theoretical model to formulate the mechanicsof a bamboo culm under ben
22、ding.Our primary hypothesis is that a bamboo culm is a long,straight,thin-walled cylinder made from a homogeneous andelasticmaterial.Thestudyofthemechanicalstabilityofelasticcylinders has a long history 3336 and is well established forthecaseofhollowcylinderswithcircularcrosssections.Whenthe cylinde
23、r is bent uniformly,the longitudinal tension andcompression that resist the applied bending moment tend toflatten the cross section 33 as sketched in Fig.3.The gradualchange in the cross section leads to a reduction in the bendingstiffnessofthecylinderwithincreasingappliedmoment.Afterthe bending mom
24、ent reaches a maximum value,the structurebecome unstable and so the object suddenly forms a kink 37.Theinitialbendingbehaviorofthecylinderischaracterizedby a uniform ovalization of the cross section 38,39.In theunloaded state,the cylinders axis is straight and its crosssection is perfectly circular.
25、However,if a small moment isapplied,the axis attains a constant curvature and ovalizes.The degree of ovalization is quantified by the dimensionlessparameter,called the oblateness.Given an initially circularcross section of radius r,the product r is equal to thereduction in the minor radius of the ov
26、al obtained afterdeformation.For later use,we introduce another dimensionless param-eter,?,by considering the curvature of the cross-sectionalcurve.An illustrative diagram is given in Fig.4.In general,the curvature of the cross-sectional curve is written d/ds.Here,is the angle between the tangent at
27、 a point on theFIG.2.Left:Segment of a bamboo culm.Right:Magnified viewof the section near a diaphragm.FIG.3.Side and cross-sectional views of an elastic hollowcylinder.(a)A straight cylinder before bending.(b)A deformedcylinder after bending.The degree of ovalization in the cross sectionis quantifi
28、ed by the dimensionless parameter,called the oblateness.curve and the vertical line(i.e.,the y axis);s is the arc lengthmeasured from the x axis to the point.Assuming that thecross-sectional deformation is small and inextensional in thecircumferential direction,we havedds=1r+3?rcos2sr.(1)Here,the fi
29、rst term on the right-hand side is the originalcurvature,and the second term represents the change in thecurvescurvatureobtainedafterovalization.Weseelaterthat?servesasabasicvariableforthestrainenergyofthethin-walledcylinder under bending.The relation between?and our oblateness is deducedfrom the ar
30、gument below.Equation(1)implies that =(s/r)+(3?/2)sin(2s/r).We also have dy=cos ds byelementary trigonometry.Combining the two results,it ispossible to prove thaty(s)=?s0cos?sr+3?2sin2 sr?d s.(2)FIG.4.Coordinate system used for analysis of ovalization underbending.A quarter round of the cross sectio
31、n is depicted in the x-yplane.022406-2SELF-ADAPTIVE FORMATION OF UNEVEN NODE.PHYSICAL REVIEW E 93,022406(2016)For relatively small?,the Taylor-series expansion is appliedto the integrand in Eq.(2)to obtainy(s)r=?3?2?3239105?+?3?91059147?+.,(3)where sin(s/r).Specifically,at the apex characterizedby =
32、1,we have y=r(1 ),with being oblateness.Therefore,it follows that and?are interdependent throughthe relation=?+35?2935?3+.(4)Equation(4)means that?is nearly equal to if ovalization issufficiently slight so that?1.B.Strain energy of deformed cylindersWe are in a position to formulate the strain energ
33、y ofbamboo culms under pure bending.We first remember thatbamboo is an anisotropic material owing to its structure,which is a lignin matrix reinforced with fibers aligned inthe longitudinal direction of the culm.These fibers providea higher stiffness and strength along the axial directioncompared to
34、 those along the transverse axis.As a result,theYoungs modulus parallel to the cylinder axis,designated E?,differsfromthatinthecircumferentialdirection,E.Intypicalbamboos,the ratio E?/Ehas been estimated to be a fewtensorless19,21,23,4042.Bearingthisanisotropyinmind,we formulate the total strain ene
35、rgy per unit length,U,of acylinder whose centroidal axis has been deformed into an arcof curvature 38:U=F?(,?()+F(?(),(5)F?(,?()=E?2r3w2?1 32?()12?()2+2116?()3?,(6)F(?()=3E8w3r?()2.(7)Here,w is the wall thickness,and the notation?()indicatesexplicitlythat?isafunctionof.Intheaboveexpressions,F?accoun
36、ts for the strain energy of longitudinal stretching,andFrepresents the strain energy of circumferential bending.Note that Fis simply a parabolic function of?withno higher order terms.This is because Fis determinedby the ovalization-induced change in the curvature of thecross-sectional curve,which is
37、 directly proportional to?seeEq.(1).On the other hand,F?involves higher order termswith respect to?,since it is determined by the second momentof area of the deformed cross section,which is proportional to?r/20y2ds.UsingtheexpansionofEq.(3),thesecondmomentof area is written by a power series in?.In
38、our computation,the terms up to the order of?3have been taken into accountto obtain the expression of Eq.(6).Given a value of,the optimum value of?(and that of)is determined by the equilibrium condition of U/?=0.It follows from Eqs.(5)(7)that at the equilibrium state,?should satisfyE?Er4w22?1?+23632
39、4?=1.(8)Therefore,bysolvingEq.(8)withrespectto?andsubstitutingthe solution into Eqs.(5)(7),we can write U as a function of alone.We then numerically calculate the bending moment,m=dU/d,and its dependence on.The results indicatethat m reaches a maximum at a curvature of=?EE?12wr2 0.485,(9)which corre
40、sponds to?=0.235 and the oblateness of=0.268?415.(10)Interestingly,the characteristic value of given by Eq.(10)is independent of any of the parameters defining the cylinderselasticity and geometry 33.In other words,the maximumof m is reached when the shorter radius of the ovalized crosssection becom
41、es equal to approximately 73%of the radius ofthe original circular cross section,regardless of the elasticconstants,initial circular radius,and wall thickness of thecylinder.If we apply a bending moment that exceeds themaximum,goesbeyondthevalueinEq.(10)andthecylinderwill collapse owing to loss of m
42、echanical resilience againstdeformation.C.Cap-induced stiffeningOur previous discussion has been pertinent to sufficientlylong cylinders under theopen-ended condition;i.e.,thosewithno cap at the two ends.If the cylinder is relatively short andrigid circular caps are fixed at both ends of the cylinde
43、r,thenthe ovalization can be largely suppressed as a whole becauseof the boundary condition that the cross sections at the twoends keep their original circular shapes.This“held-circular”effect at the edges propagates in the axial direction from theedgestothemiddleofthecylinder;therefore,thecloseracr
44、osssection is to either of the two ends,the more the ovalization atthe cross section is suppressed.Itisunderstoodintuitivelythattheefficiencyofcap-inducedsuppression of ovalization depends on the cylinder length?,cylinder radius r,and wall thickness w.In the followingdiscussion,we demonstrate how to
45、 quantify the efficiency ofsuppression in terms of geometric parameters such as?,r,andw.Figure 5(a)shows the curved profile of a portion of anopen-ended,long,hollow cylinder which has been subjectedto a pure bending moment.The applied bending momentcorresponds to a certain value of no more than 4/15
46、.Theamount of ovalization in the cross section is constant over thewholecylinder,includingthetwoends.Thisspatialuniformityin oblateness is broken if we attach caps at the two ends.The held-circular effect driven by the caps is schematicallyillustrated in Fig.5(b);since the ends preserve the original
47、circular shape,there are likely to be transition zones in thevicinity of the two ends,within which increases from 0 atthe ends to the given value over a central region.022406-3HIROYUKI SHIMA,MOTOHIRO SATO,AND AKIO INOUEPHYSICAL REVIEW E 93,022406(2016)FIG.5.(a)Curved profile of a portion of an open-
48、ended,long,hollow cylinder.(b)A closed-ended long cylinder.(c)A closed-endedmedium-length cylinder.(d)A closed-ended short cylinder.(e)The relationship between the curvature radius R and the cylinder length?underthe conditions in(d).It is obvious that,for cylinders which are shorter thanthe one depi
49、cted in Fig.5(b),the central region in which remains effectively constant is also shorter;see Fig.5(c).Indeed,for even shorter cylinders,the held-circular effect maywell dominate behavior over the entire length of the cylinder.InFig.5(d),acylinderhasbeensketchedwhichissoshortthatthegeneratorontheout
50、ersideremainsapproximatelystraight.This suggests that a cylinder having this geometry should fallin the range over which the held-circular effect is dominant.Specialattentionispaidtothecaseinwhichthecentralzoneshrinks to a point on the generator,as depicted in Fig.5(d).In this extreme situation,the