Top

Published in:

2017 | OriginalPaper | Chapter

4. Theory of the Elastica and a Selection of Its Applications

Author : Oliver M. O’Reilly

Published in: Modeling Nonlinear Problems in the Mechanics of Strings and Rods

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The theory of the elastica is discussed in this chapter. In addition to classical buckling problems, several applications of this theory to rod-like bodies adhering to rigid substrates are discussed.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Link, Writhe, and Twist

next chapter Kirchhoff’s Rod Theory

These developments are discussed at length in Bolza’s marvelous textbook [30].

The interested reader is referred to the works of Bigoni et al. [26], Maddocks and his collaborators [169, 215, 224‐226] and Majidi and his coworkers [219, 220, 268, 269] for discussions of, and references to, these results.

We shall give a prescription for this quantity later on in the related context of more general rod theories (cf. Eqns. (5.36)₃, (5.37), and (7.98)₃).

As usual, for ease of exposition and without loss of generality, we assume that there is at most one such point.

For additional details on this matter, see [263, 264] and Exercises 1.3 and 1.4

A translation of Euler’s original work [106] is readily available and was published by Oldfather et al. [254].

Discussions on how to calculate this constant can be found in [172, 175].

Our emphasis of the key role played by the material momentum balance law in specifying the adhesion boundary condition is heavily influenced by the works [217, 219, 220, 264, 292].

See the papers [116, 117, 217, 219, 220, 290, 291, 364] and references therein.

See Eqn. (4.6).

For background on elliptic integrals and functions, Byrd and Friedman’s classic handbook [41] and Lawden’s concise textbook [200] are recommended. Integrals of the form (4.70) can also be evaluated using symbolic manipulation packages such as MATHEMATICA.

See Example 288.50 for the integral \(\int _{\psi }^{ \frac{\pi }{ 2} } \frac{d\theta } {\sqrt{a+b\sin \left (\theta \right )}}\) where \(b > \left \vert a\right \vert > 0\) in [41].

See Example 288.00 for the integral \(\int _{\psi }^{ \frac{\pi }{ 2} } \frac{d\theta } {\sqrt{a+b\sin \left (\theta \right )}}\) where a > b > 0 in [41].

Eqn. (4.131) can also be inferred from [32, Eqn. (2.6)].

As noted earlier, for background on elliptic integrals and functions, the handbook [41] and the textbook [200] are recommended.

Referring the reader to [8, 163, 265, 375] for further details, it is known that a constant moment is not necessarily conservative.

Our manipulations of Π _f are inspired by related work in a recent paper by Farjoun and Neu [108].

These methods are discussed in Section 9.3 of Chapter 9

The contribution of ρ ₀ f to this expression follows from Eqn. (4.139) with some minor modifications to one of the limits of integration.

Alternatively, the adhesion may be represented as a surface potential by also adding W _ad ℓ to Π. This is accomplished by eliminating W _ad in the second integrand and adding it to the first integrand.

Compatibility conditions of the form (4.152) for adhesion problems can be found in [219] and [318] and for problems where the rod passes through a sleeve, as in the elastica arm scale, in [27, 32]. They express the restrictions that variations in \(\theta \left (\gamma ^{\pm }\right )\) and γ are not always independent.

If the rod were extensible, then \(\left [\!\left [\mathbf{n} \cdot \mathbf{r}^{'}\right ]\!\right ]_{\gamma }\) would be due to the jump in the stretch of the centerline across the discontinuity. For examples where this situation arises, see [181] and [218].

Further background on Legendre’s treatment of the second variation can be found in the superb texts by Bolza [30] and Gelfand and Fomin [113].

Our definition of the conjugate point differs from the traditional definition as the latter applies to the case where the rod is clamped at both of its ends.

See, in particular, [113, Theorem 3 in Section 26]. Choosing w(0) = 0 implies that the boundary term J ₂ defined in Eqn. (4.176) will vanish.

Gelfand and Fomin’s proof in [113] pertains to the fixed-fixed case. It requires some minor modifications to deal with the fixed-free case of interest here and these modifications are outlined in [289].

That is, the problem of a terminally loaded fixed-free strut.

This example is adopted from [219]. It is the simplest illustrative example of a buckling problem featuring adhesion that we could find.

This is equivalent to the classical result for the buckling load of a fixed-free strut.

An analytic expression, featuring Airy functions, for w(ξ) can be established for Eqn. (4.191)₃ when θ ^∗ = −90^∘.

Additional perspectives on buckling can be found in Section 5.17 of Chapter 5. We also refer the reader to the seminal text by Timoshenko and Gere [345, Chapter 2].

The interested reader is referred to the works of Eshelby [102, Page 142] and Kienzler and Herrmann [182, 183] for discussions on material forces in the context of Bernoulli-Euler beam theory.

As can be seen from Section 9.3.2, the variations used to establish the corner condition (9.25) correspond to varying γ. For further details on calculus of variations problems of this type see [30, Section 10].

Alexander, J.C., Antman, S.S.: The ambiguous twist of Love. Quarterly of Applied Mathematics 40 (1), 83–92 (1982/83)

16.

Autumn, K., Sitti, M., Liang, Y.A., Peattie, A.M., Hansen, W.R., Sponberg, S., Kenny, T.W., Fearing, R., Israelachvili, J.N., Full, R.J.: Evidence for van der Waals adhesion in gecko setae. Proceedings of the National Academy of Sciences 99 (19), 12,252–12,256 (2002). URL http://dx.doi.org/10.1073/pnas.192252799

17.

Batista, M.: Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions. International Journal of Solids and Structures 51 (13), 2308–2326 (2014). URL http://dx.doi.org/10.1016/j.ijsolstr.2014.02.036

25.

Bigoni, D., Bosi, F., Corso, F.D., Misseroni, D.: Instability of a penetrating blade. Journal of the Mechanics and Physics of Solids 64, 411 – 425 (2014). URL http://dx.doi.org/10.1016/j.jmps.2013.12.008

26.

Bigoni, D., Bosi, F., Misseroni, D., Dal Corso, F., Noselli, G.: New phenomena in nonlinear elastic structures: from tensile buckling to configurational forces. In: D. Bigoni (ed.) Extremely Deformable Structures, pp. 55–135. Springer-Verlag, Vienna (2015). URL http://dx.doi.org/10.1007/978-3-7091-1877-1_2

27.

Bigoni, D., Corso, F.D., Bosi, F., Misseroni, D.: Eshelby-like forces acting on elastic structures: Theoretical and experimental proof. Mechanics of Materials 80, 368–374 (2015). URL http://dx.doi.org/10.1016/j.mechmat.2013.10.009

30.

Bolza, O.: Lectures on the Calculus of Variations, third edn. Chelsea, New York (1973)

31.

Born, M.: Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen. Dieterichsche Universitäts-Buchdruckerei, Göttingen (1906)MATH

32.

Bosi, F., Misseroni, D., Dal Corso, F., Bigoni, D.: An elastica arm scale. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 470 (2169), 20140,232 (2014). URL http://dx.doi.org/10.1098/rspa.2014.0232

33.

Bosi, F., Misseroni, D., Dal Corso, F., Bigoni, D.: Development of configurational forces during the injection of an elastic rod. Extreme Mechanics Letters 4, 83–88 (2015). URL http://dx.doi.org/10.1016/j.eml.2015.04.007

34.

Bosi, F., Misseroni, D., Dal Corso, F., Bigoni, D.: Self-encapsulation, or the ‘dripping’ of an elastic rod. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 471 (2179), 20150,195 (2015). URL http://dx.doi.org/10.1098/rspa.2015.0195

41.

Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, Die Grundlehren der mathematischen Wissenschaften, vol. 67. Springer-Verlag, New York (1971). Second edition, revised

65.

Coleman, B.D., Dill, E.H.: Flexure waves in elastic rods. Journal of the Acoustical Society of America 91 (5), 2663–2673 (1992). URL http://dx.doi.org/10.1121/1.402974

66.

Coleman, B.D., Dill, E.H., Swigon, D.: On the dynamics of flexure and stretch in the theory of elastic rods. Archive for Rational Mechanics and Analysis 129 (2), 147–174 (1995). URL http://dx.doi.org/10.1007/BF00379919

91.

Domokos, G., Holmes, P., Royce, B.: Constrained Euler buckling. Journal of Nonlinear Science 7 (3), 281–314 (1997). URL http://dx.doi.org/10.1007/BF02678090

92.

Domokos, G., Ruina, A.: A circle construction based on elastostatics and hydrodynamics. Mechanics Research Communications 20 (3), 181–185 (1993). URL http://dx.doi.org/10.1016/0093-6413(93)90054-R

102.

Eshelby, J.D.: The continuum theory of lattice defects. In: F. Seitz, D. Turnbull (eds.) Solid State Physics, vol. 3, pp. 79–144. Academic Press (1956). URL http://dx.doi.org/10.1016/S0081-1947(08)60132-0

106.

Euler, L.: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti: Additamentum 1 De Curvis Elasticis. Leonhardi Euleri Opera Omnia, Series prima (Opera mathematica), Vol. XXIV, Auctoritate et impensis Societatis Scientiarum Naturalium Helveticae. Orell Füssli, Zürich (1952). An English translation of this work can be found in [254].

108.

Farjoun, Y., Neu, J.: The tallest column: A dynamical system approach using a symmetry solution. Studies in Applied Mathematics 115 (3), 319–337 (2005). URL http://dx.doi.org/10.1111/j.1467-9590.2005.00316.x

109.

Faruk Senan, N.A., O’Reilly, O.M., Tresierras, T.N.: Modeling the growth and branching of plants: A simple rod-based model. J. Mech. Phys. Solids 56 (10), 3021–3036 (2008). URL http://dx.doi.org/10.1016/j.jmps.2008.06.005

113.

Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice-Hall, Englewood Cliffs, N. J. (1964)MATH

116.

Glassmaker, N., Jagota, A., Hui, C., Noderer, W., Chaudhury, M.: Biologically inspired crack trapping for enhanced adhesion. Proceedings of the National Academy of Sciences 104 (26), 10,786–10,791 (2007). URL http://dx.doi.org/10.1073/pnas.0703762104

117.

Glassmaker, N.J., Hui, C.Y.: Elastica solution for a nanotube formed by self-adhesion of a folded thin film. Journal of Applied Physics 96 (6), 3429–3434 (2004). URL http://dx.doi.org/10.1063/1.1779974

118.

Goldstein, R.E., Goriely, A.: Dynamic buckling of morphoelastic filaments. Physical Review E 74, 010,901 (2006). URL http://dx.doi.org/10.1103/PhysRevE.74.010901

121.

Goriely, A.: The Mechanics and Mathematics of Biological Growth. Springer-Verlag, New York (2017)MATH

122.

Goriely, A., Neukirch, S.: The mechanics of attachment in twining plants. Physical Review Letters 97, 184,302 (2006). URL http://dx.doi.org/10.1103/PhysRevLett.97.184302

132.

Green, A.E., Naghdi, P.M.: A derivation of jump condition for entropy in thermomechanics. Journal of Elasticity 8 (2), 119–182 (1978). URL http://dx.doi.org/10.1007/BF00052481

145.

Guillon, T., Dumont, Y., Fourcaud, T.: A new mathematical framework for modelling the biomechanics of growing trees with rod theory. Mathematical and Computer Modelling 55 (9–10), 2061–2077 (2012). URL http://dx.doi.org/10.1016/j.mcm.2011.12.024

163.

van der Heijden, G.H.M., Peletier, M.A., Planqué, R.: On end rotation for open rods undergoing large deformations. Quarterly of Applied Mathematics 65 (2), 385–402 (2007). URL http://dx.doi.org/10.1090/S0033-569X-07-01049-X

168.

Hess, W.: Ueber die Beigung und Drillung eines unendlich dünnen elastischen Stabes mit zwei gleichen Widersänden, auf dessen freies Ende eine Kraft und ein um die Hauptaxe ungleichen Widerstandes drehendes Kräftepaar einwirkt. Mathematische Annalen 25 (1), 1–38 (1885). URL http://dx.doi.org/10.1007/BF01446419

169.

Hoffman, K.A., Manning, R.S., Maddocks, J.H.: Link, twist, energy, and the stability of DNA minicircles. Biopolymers 70 (2), 145–157 (2003). URL http://dx.doi.org/10.1002/bip.10430

172.

Israelachvili, J.R.: Intermolecular and Surface Forces: With Applications to Colloidal and Biological Systems (Colloid Science), second edn. Academic Press, San Diego (1992)

175.

Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and the contact of elastic solids. Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences 324 (1558), 301–313 (1971). URL http://dx.doi.org/10.1098/rspa.1971.0141

181.

Kendall, K.: The adhesion and surface energy of elastic solids. Journal of Physics D: Applied Physics 4 (8), 1186–1195 (1971). URL http://dx.doi.org/10.1088/0022-3727/4/8/320

182.

Kienzler, R., Herrmann, G.: Mechanics in Material Space: With Applications to Defect and Fracture Mechanics. Springer-Verlag, Berlin (2000)CrossRefMATH

183.

Kienzler, R.R., Herrmann, G.G.: On material forces in elementary beam theory. ASME Journal of Applied Mechanics 53 (3), 561–564 (1986). URL http://dx.doi.org/10.1115/1.3171811

185.

Kirchhoff, G.: Über des gleichgewicht und die Bewegung eines unendlich dünnen elastichen Stabes. Crelles Journal für die reine und angewandte Mathematik 56, 285–313 (1859). URL http://dx.doi.org/10.1515/crll.1859.56.285

200.

Lawden, D.F.: Elliptic Functions and Applications, Applied Mathematical Sciences, vol. 80. Springer-Verlag, New York (1989)

211.

Lotz, J.C., O’Reilly, O.M., Peters, D.M.: Some comments on the absence of buckling of the ligamentous human spine in the sagittal plane. Mechanics Research Communications 40 (1), 11–15 (2012). URL http://dx.doi.org/10.1016/j.mechrescom.2011.11.010

213.

Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, fourth edn. Cambridge University Press, Cambridge (1927)

215.

Maddocks, J.H.: Stability of nonlinearly elastic rods. Archive for Rational Mechanics and Analysis 85 (4), 311–354 (1984). URL http://dx.doi.org/10.1007/BF00275737

216.

Mahadevan, L., Keller, J.B.: Periodic folding of thin sheets. SIAM Journal on Applied Mathematics 55 (6), 1609–1624 (1995). URL http://dx.doi.org/10.1137/S0036139994263410

217.

Majidi, C.: Remarks on formulating an adhesion problem using Euler’s elastica. Mechanics Research Communications 34 (1), 85–90 (2007). URL http://dx.doi.org/10.1016/j.mechrescom.2006.06.007

218.

Majidi, C., Adams, G.G.: Adhesion and delamination boundary conditions for elastic plates with arbitrary contact shape. Mechanics Research Communications 37 (2), 214–218 (2010). URL http://dx.doi.org/10.1016/j.mechrescom.2010.01.002

219.

Majidi, C., O’Reilly, O.M., Williams, J.A.: On the stability of a rod adhering to a rigid surface: Shear-induced stable adhesion and the instability of peeling. Journal of the Mechanics and Physics of Solids 60 (5), 827–843 (2012). URL http://dx.doi.org/10.1016/j.jmps.2012.01.015

220.

Majidi, C., O’Reilly, O.M., Williams, J.A.: Bifurcations and instability in the adhesion of intrinsically curved rods. Mechanics Research Communications 49, 13–16 (2013). URL http://dx.doi.org/10.1016/j.mechrescom.2013.01.004

224.

Manning, R.S.: Conjugate points revisited and Neumann-Neumann problems. SIAM Review 51 (1), 193–212 (2009). URL http://dx.doi.org/10.1137/060668547

226.

Manning, R.S., Rogers, K.A., Maddocks, J.H.: Isoperimetric conjugate points with application to the stability of DNA minicircles. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454 (1980), 3047–3074 (1998). URL http://dx.doi.org/10.1098/rspa.1998.0291

230.

Marshall, J.S., Naghdi, P.M.: A thermodynamical theory of turbulence. I. Basic developments. Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 327 (1595), 415–448 (1989). URL http://dx.doi.org/10.1098/rsta.1989.0001

254.

Oldfather, W.A., Ellis, C.A., Brown, D.M.: Leonhard Euler’s elastic curves. Isis 20 (1), 72–160 (1933). URL http://www.jstor.org/stable/224885

263.

O’Reilly, O.M.: The energy jump condition for thermomechanical media in the presence of configurational forces. Continuum Mechanics and Thermodynamics 18 (6), 361–365 (2007). URL http://dx.doi.org/10.1007/s00161-006-0036-3

264.

O’Reilly, O.M.: A material momentum balance law for rods. Journal of Elasticity 86 (2), 155–172 (2007). URL http://dx.doi.org/10.1007/s10659-006-9089-6

265.

O’Reilly, O.M.: Intermediate Engineering Dynamics: A Unified Treatment of Newton-Euler and Lagrangian Mechanics. Cambridge University Press, Cambridge (2008)CrossRefMATH

266.

O’Reilly, O.M.: Some perspectives on Eshelby-like forces in the elastica arm scale. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 471 (2174), 20140,785 (2015). URL http://dx.doi.org/10.1098/rspa.2014.0785

268.

O’Reilly, O.M., Peters, D.M.: On the stability analyses of three classical buckling problems for the elastic strut. Journal of Elasticity 105 (1–2), 117–136 (2011). URL http://dx.doi.org/10.1007/s10659-010-9299-9

269.

O’Reilly, O.M., Peters, D.M.: Nonlinear stability criteria for tree-like structures composed of branched elastic rods. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 468 (2137), 206–226 (2012). URL http://dx.doi.org/10.1098/rspa.2011.0291

273.

O’Reilly, O.M., Tresierras, T.N.: On the evolution of intrinsic curvature in rod-based models of growth in long slender plant stems. International Journal of Solids and Structures 48 (9), 1239–1247 (2011). URL http://dx.doi.org/10.1016/j.ijsolstr.2010.12.006

274.

O’Reilly, O.M., Tresierras, T.N.: On the static equilibria of branched elastic rods. International Journal of Engineering Science 49 (2), 212–227 (2011). URL http://dx.doi.org/10.1016/j.ijengsci.2010.11.008

282.

Pamp, A., Adams, G.G.: Deformation of bowed silicon chips due to adhesion and applied pressure. Journal of Adhesion Science and Technology 21 (11), 1021–1043 (2007). URL http://dx.doi.org/10.1163/156856107782105963

286.

de Payrebrune, K.M., O’Reilly, O.M.: On constitutive relations for a rod-based model of a pneu-net bending actuator. Extreme Mechanics Letters 8 (C), 38–46 (2016). URL http://dx.doi.org/10.1016/j.eml.2016.02.007

289.

Peters, D.M.: Nonlinear stability criteria for elastic rod structures. Ph.D. thesis, University of California at Berkeley (2011). URL https://escholarship.org/uc/item/6km3q6b7

290.

Plaut, R.H., Dalrymple, A.J., Dillard, D.A.: Effect of work on contact of an elastica with a flat surface. Journal of Adhesion Science and Technology 15 (5), 565–581 (2001). URL http://dx.doi.org/10.1163/156856101300189934

291.

Plaut, R.H., Williams, N.H., Dillard, D.A.: Elastica analysis of the loop tack adhesion test for pressure sensitive adhesives. Journal of Adhesion 76 (1), 37–53 (2001). URL http://dx.doi.org/10.1080/00218460108029616

292.

Podio-Guidugli, P.: Peeling tapes. In: P. Steinmann, G.A. Maugin (eds.) Mechanics of Material Forces, Advances in Mechanics and Mathematics, vol. 11, pp. 253–260. Springer-Verlag, New York (2005). URL http://dx.doi.org/10.1007/0-387-26261-X_25

298.

Reid, W.T.: Riccati Differential Equations. Academic Press, New York (1972)MATH

312.

Santillan, S.T., Virgin, L.N.: Numerical and experimental analysis of the static behavior of highly deformed risers. Ocean Engineering 38 (13), 1397–1402 (2011). URL http://dx.doi.org/10.1016/j.oceaneng.2011.06.009

313.

Santillan, S.T., Virgin, L.N., Plaut, R.H.: Static and dynamic behavior of highly deformed risers and pipelines. ASME Journal of Offshore Mechanics and Artic Engineering 132 (2), 021,401 (2010). URL http://dx.doi.org/10.1115/1.4000555

318.

Seifert, U.: Adhesion of vesicles in two dimensions. Physical Review A 43 (12), 6803–6814 (1991). URL http://dx.doi.org/10.1103/PhysRevA.43.6803

322.

Silk, W., Erickson, R.O.: Kinematics of hypocotyl curvature. American Journal of Botany 65 (3), 310–319 (1978). URL http://www.jstor.org/stable/2442271

324.

Silk, W., Wang, L.L., Cleland, R.E.: Mechanical properties of the rice panicle. Plant Physiology 70 (2), 460–464 (1982)CrossRef

345.

Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability, second edn. McGraw-Hill, New York (1961)

350.

Truesdell, C.: The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788. Leonhardi Euleri Opera Omnia, Series secunda (Opera mechanica et astronoca), Vol. XI, sectio secunda. Auctoritate et impensis Societatis Scientiarum Naturalium Helveticae. Orell Füssli, Zürich (1960)

364.

Williams, J.A.: Adhesional instabilities and gecko locomotion. Journal of Physics D: Applied Physics 48 (1), 015,401 (2015). URL http://dx.doi.org/10.1088/0022-3727/48/1/015401

368.

Yamamoto, H., Yoshida, M., Okuyama, T.: Growth stress controls negative gravitropism in woody plant stems. Planta 216 (2), 280–292 (2002). URL http://dx.doi.org/10.1007/s00425-002-0846-x

372.

Zhou, X., Majidi, C., O’Reilly, O.M.: Flexing in motion: A locomotion mechanism for soft robots. International Journal of Non-Linear Mechanics 74, 7–17 (2015). URL http://dx.doi.org/10.1016/j.ijnonlinmec.2015.03.001

373.

Zhou, X., Majidi, C., O’Reilly, O.M.: Soft hands: An analysis of some gripping mechanisms in soft robot design. International Journal of Solids and Structures 64–65, 155–165 (2015). URL http://dx.doi.org/10.1016/j.ijsolstr.2015.03.021

374.

Zhou, X., O’Reilly, O.M.: On adhesive and buckling instabilities in the mechanics of carbon nanotubes bundles. ASME Journal of Applied Mechanics 82 (10), 101,007 (2015). URL http://dx.doi.org/10.1115/1.4030976

375.

Ziegler, H.: Principles of Structural Stability, second edn. Birkhaüser, Basel (1977). URL http://dx.doi.org/10.1007/978-3-0348-5912-7

Title: Theory of the Elastica and a Selection of Its Applications
Author: Oliver M. O’Reilly
Publisher: Springer International Publishing
Book: Modeling Nonlinear Problems in the Mechanics of Strings and Rods
Print ISBN: 978-3-319-50596-1

Electronic ISBN: 978-3-319-50598-5

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-50598-5_4

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Premium Partners