Softening of the Euler buckling criterion under discretisation of compliance
D.J. Carter, D.J. Dunstan, W. Just, O.F. Bandtlow, A. San Miguel
SSoftening of the Euler buckling criterion under discretisation of compliance
D.J. Carter
School of Physics and Astronomy, Queen Mary University of London, London E1 4NS
D.J. Dunstan ∗ School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, United Kingdom
W. Just and O.F. Bandtlow
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
A. San Miguel
Universit´e de Lyon, F-69000 Lyon, France and Institut Lumi`ere Mati`ere,CNRS, UMR 5306, Universit´e Lyon 1, F-69622 Villeurbanne, France (Dated: December 1, 2020)Euler solved the problem of the collapse of tall thin columns under unexpectedly small loads in1744. The analogous problem of the collapse of circular elastic rings or tubes under external pressurewas mathematically intractable and only fully solved recently. In the context of carbon nanotubes,an additional phenomenon was found experimentally and in atomistic simulations but not explained:the collapse pressure of smaller diameter tubes deviates below the continuum mechanics solution[Torres-Dias et al ., Carbon , 145 (2017)]. Here, this deviation is shown to occur in discretizedstraight columns and it is fully explained in terms of the phonon dispersion curve. This reveals anunexpected link between the static mechanical properties of discrete systems and their dynamicsdescribed through dispersion curves.
The tendency of tall thin columns to collapse under un-expectedly small loads was already known to the ancientGreeks - it is suggested that their use of entasis (Fig.1a)was to strengthen columns [1], and the Romans basedtheir structures on short fat columns, as in the Pont duGard (Fig.1b). The phenomenon was explained by Eu-ler in a classic work [2]; his explanation is now usuallyexpressed in terms of the elastic energy required for alateral deflection of the column compared with the workdone by the advance of the load for the same deflection.
FIG. 1. (a) Entasis in Greek columns (exaggerated). (b)Schematic of the Pont du Gard. (c) A continuous pillar, buck-ling (dashed). (d) A discretized pillar, buckling (dashed). (e)The collapse of a continuous elastic ring. (f) A discretisedelastic ring (a polygon). (g) A continuous tube. (h) An ex-ample of a discretised tube (a nanotube).
The collapse of circular tubes under external pressurefollows the same simple physics, but is mathematicallyintractable. Although it was of great importance in early steam engine boilers, and later in oil wells, engineers hadto rely on empirical testing [3, 4], as only partial theoret-ical solutions were available [5]. The full solution for acontinuous elastic ring was given only in 2011 [6, 7]. Mostrecently, it was observed that small-diameter atomisticrings or tubes collapse at pressures lower than the con-tinuum solution. If the continuum solution is normalisedagainst the bending stiffness of the tube wall, D , and thediameter, d , of the tube, the collapse pressure, P C = Dd ,can be expressed as P N C = P C d D = 3. In a recent study ofthe collapse pressure of single-wall carbon nanotubes [8],a surprising result was reported. Experimental measure-ments and simulations (molecular dynamics modellingand density-functional modelling) were in agreement infinding P N C = 3(1 − β d ), with a value of β about 0.4nm(this is therefore the diameter of the smallest stable nan-otubes). Both the analytic form of this expression andthe value of β were wholly unexplained.Nanotubes are not continuous elastic rings, as theyhave a radius of only a few atoms. To find the collapsepressure of discretised elastic rings, Sun et al. [9] modelledthem as polygons of area A with rigid sides joined byangular springs at the vertices (spring constant k ), underan external pressure, P . The total energy is written as afunction of the angles of the hinges, (cid:80) kθ i + P A , andminimised. Results were similar, but still unexplained.Here we set out to find an explanation by investigatingthe simpler problem of an atomistic or discretised col-umn. The same phenomenon occurs; the collapse force isreduced for small numbers of segments. Neither numeri-cal nor analytical analysis reveal any explanation. a r X i v : . [ c ond - m a t . o t h e r] N ov �� �� � → � →θ � θ � θ � θ � θ ν = � θ ν = � θ ν = � θ ν = � θ � θ � θ � + � θ � + � FIG. 2. A pillar along the x -axis is discretised to have compli-ance only at N angular hinges of stiffness c = a S . The lowerpart of the graph shows a chain of rigid rods with N +1 anglesshown by ν . Buckling occurs causing y -axis displacement y ν . Recasting the problem in terms of the Euler buckling ofan infinitely long column, constrained to buckle at finitewavelengths, does provide an explanation. This approachreveals a link between the static mechanical properties ofdiscrete systems and phonon dispersion curves.In continuum mechanics, a column of length L withunconstrained ends and a bending stiffness, D (definedby E = DR − where R is the radius of curvature and E is the stored elastic energy per unit length) has a bucklingforce of F C = π DL . This force is to be applied alongthe column axis. The normalised collapse force, F CN = F C L D , is thus π ≈ . S = D , at a number, N , of points (atoms orhinges with angular springs) does indeed give a reductionin the normalised collapse force for small N .We used the numerical method of Sun et al. [9] for poly-gons, energy minimised, but for straight columns underan endload (Fig.2). Note that we have divided the col-umn into N equal intervals and then moved all the com-pliance in each interval to the centre of that interval. That gives N + 1 rigid links or rods, joined by the com-pliant hinges, and the two end rods are half the lengthof the others. The calculations confirm that the collapseforce is reduced at small N for the discretised columns,as for the polygons. However, the calculations give nohint of the reason for this behaviour.Indeed, for small N , it is not difficult to find analyticexpressions for the collapse force, e.g. for N = 1, thenormalised collapse force is 4 and for N = 6 it is 9.646.One might think, by considering the case of N = 1, thatthe compliance has been moved from where, in a con-tinuous column, much of it is wasted (regions of smallercurvature), to the point of maximum curvature. Thatexplanation is, however valid only for N = 1. For larger N , it is easy to show that as much compliance has beenmoved to where it is less useful as to where it is moreuseful.An alternative to the energy calculation is to calculatethe lateral oscillation frequency of the column as a func-tion of the longitudinal compression or tension. Com-pression softens the oscillatory mode. The collapse forceis found by setting the frequency (found via eigenvaluesfor a matrix of equations of motion) to zero and solvingfor the force. These results of course agree with the en-ergy calculation, but again do not give any hint as to thephysical reason for the phenomenon.The explanation is found by removing the restrictionof the column length, L . Consider an infinite chain ofpoint-mass hinges of mass ρa , connecting light rods oflength a , with angular springs in the hinges of compli-ance Sa and under a tension T . The equation of motionfor the n th hinge is readily set up, as Eq.(1a) where the y -coordinate is perpendicular to the chain. Eq.(1b) –(1d) show its development using Bloch’s theorem [10] toobtain the phonon dispersion curve ω ( k ).¨ y n = Ta ρ ( y n − − y n + y n +1 ) − a ρS ( y n − − y n − + 6 y n − y n +1 + y n +2 ) (1a)¨ y n e ia kn u n ( x ) = u n ( x ) e ia kn (cid:18) Ta ρ (cid:0) e − ia k + e ia k − (cid:1) − a ρS (cid:0) e − i a k − e − ia k − e ia k + e − i a k − (cid:1)(cid:19) (1b)¨ y n = λ = − ω = − a ST − a k )) sin (cid:0) a k ] (cid:1) a ρS (1c) ω ( k ) = 2 sin 12 a k (cid:115) a ST − a ka ρS (1d)This dispersion relation, Eq.(1d), is interesting becauseit has solutions for negative tension, i.e. compression. Asthe problem is set up, these solutions are unphysical, be-cause an infinite or indefinitely long system will collapse even under an indefinitely small compression. However,we can extract interesting special cases which are phys-ically realisable. First, we are not interested in runningwaves, but in standing waves. We can set the wavelength λ of a standing wave by imposing constraints y = 0 ev-ery half-wavelength along the chain, to constrain the po-sitions of the nodes. This also prevents buckling of thechain under compression at wavelengths longer than λ ,and under compressive forces less than required for buck-ling at this wavelength. We can do this for the specialcases where the half-wavelength λ/ π/k is an inte-ger multiple of the length a , i.e. k = π/ ( N a ), andeven more specifically we set the nodes halfway betweentwo hinges (previously our unconstrained endpoints). Inthis way the section between two adjacent nodes repli-cates the model of Fig.2, and of course it has the fre-quency given by Eq.(1d). The compressive force at whichEq.(1d) gives zero frequency is, precisely, the Euler buck-ling force for the finite, half-wavelength system. Substi-tuting L = λ = πk into Eq.(1d), we have an expressionfor the frequency as a function of L , with a still as a vari-able. We may now express a in terms of N , the numberof atoms and L , the length: a = LN , but there is no needyet to make N an integer. Keeping N as a real number,we solve for the collapse tension T C or compressive force F C = − T C by setting the frequency equal to zero, andwe obtain F C = − T C = 2 N L S (cid:16) − cos πN (cid:17) . (2)Normalising, F NC = 2 N (cid:16) − cos πN (cid:17) . (3)Expanding the cosine term as 1 − π N + . . . gives F NC = π (cid:18) − π N + O ( N − ) (cid:19) . (4)Dropping the higher-order terms, this can be written forcolumns as F NC = π (cid:18) − β N (cid:19) (5)with β = π /
12. For comparison with polygons andnanotubes, noting that P C corresponds to a tangentialforce in the circumscribed circle of F C = P C R , and thatthe length of the circle (2 πR ) corresponds to two bucklingwavelengths rather than the half-wavelength for columns,the equivalent expressions are F NC = 34 π (cid:18) − β N (cid:19) (6)Each problem has its own value of β .Eq.(3) is plotted in Fig.3. For comparison with thenumerical and analytic solutions, we may pick out thevalues of Eq.(3) where N is an integer: And of coursethey agree exactly. However, Eq.(3) provides both the explanation of the functional form of the dependence ofthe collapse force on N and the explanation of the valueof the parameter β . The functional form is very closeto N − (Eq.(5)) until N is small enough that the higherterms in the cosine expansion become important, whichis only at N = 1 (Fig.3). The value of β in Eq.(5) issimply given by β = π = 0 .