×

zbMATH — the first resource for mathematics

Circumferential buckling instability of a growing cylindrical tube. (English) Zbl 1270.74088
Summary: A cylindrical elastic tube under uniform radial external pressure will buckle circumferentially to a non-circular cross-section at a critical pressure. The buckling represents an instability of the inner or outer edge of the tube. This is a common phenomenon in biological tissues, where it is referred to as mucosal folding. Here, we investigate this buckling instability in a growing elastic tube. A change in thickness due to growth can have a dramatic impact on circumferential buckling, both in the critical pressure and the buckling pattern. We consider both single- and bi-layer tubes and multiple boundary conditions. We highlight the competition between geometric effects, i.e. the change in tube dimensions, and mechanical effects, i.e. the effect of residual stress, due to differential growth. This competition can lead to non-intuitive results, such as a tube growing to be thinner and yet buckle at a higher pressure.

MSC:
74G60 Bifurcation and buckling
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Amar, M.B.; Goriely, A., Growth and instability in elastic tissues, J. mech. phys. solids, 53, 2284-2319, (2005) · Zbl 1120.74336
[2] Demirkoparan, H.; Pence, T.J., Swelling of an internally pressurized nonlinearly elastic tube with fiber reinforcing, Int. J. solids struct., 44, 4009-4029, (2007) · Zbl 1178.74041
[3] Dion, B.; Naili, S.; Renaudeaux, J.P.; Ribreau, C., Buckling of elastic tubes: study of highly compliant device, Med. biol. eng. comput., 33, 2, 196-201, (1995)
[4] Euler, L., 1759. Sur la force des colonnes. Mem. Acad., Berlin.
[5] Fu, Y., Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness, Int. J. non-linear mech., 33, 1111-1122, (1998) · Zbl 1342.74063
[6] Fung, Y.C., What are the residual stresses doing in our blood vessels?, Ann. biomed. eng., 19, 237-249, (1991)
[7] Goriely, A.; Ben Amar, M., Differential growth and instability in elastic shells, Phys. rev. lett., 94, 19, 198103, (2005)
[8] Goriely, A.; Moulton, D.E., New trends in the physics and mechanics of biological systems, () · Zbl 1270.74088
[9] Goriely, A.; Vandiver, R., On the mechanical stability of growing arteries, IMA J. appl. math., 75, 4, 549-570, (2010) · Zbl 1425.74324
[10] Goriely, A.; Vandiver, R.; Destrade, M., Nonlinear Euler buckling, Proc. R. soc. A, 464, 2099, 3003-3019, (2008) · Zbl 1152.74342
[11] Goriely, A.; Moulton, D.E.; Vandiver, R., Elastic cavitation, tube hollowing, and differential growth in plants and biological tissues, Euro. phys. lett., 91, 18001, (2010)
[12] Hrousis, C.A.; Wiggs, B.R.; Drazen, J.M.; Parks, D.M.; Kamm, R.D., Mucosal folding in biologic vessels, J. biomech. eng., 124, 4, 334-341, (2002)
[13] Huber, H.L.; Koessler, K.K., The pathology of bronchial asthma, Arch. intern. med., 30, 689-760, (1922)
[14] Humphrey, J.D., Continuum biomechanics of soft biological tissues, Proc. R. soc. London A, 459, 3-46, (2003) · Zbl 1116.74385
[15] James, A.L.; Paré, P.D.; Hogg, J.C., The mechanics of airway narrowing in asthma, Am. rev. respir. dis., 139, 1, 242-246, (1989)
[16] Lambert, R.K., Role of bronchial basement membrane in airway collapse, J. appl. physiol., 71, 666-673, (1991)
[17] Lambert, R.K.; Codd, S.L.; Alley, M.R.; Pack, R.J., Physical determinants of bronchial mucosal folding, J. appl. physiol., 77, 1206-1216, (1994)
[18] Lee, M.M.L.; Chien, S., Morphologic effects of pressure changes on canine carotid artery endothelium as observed by scanning electron microscopy, Anat. rec., 194, 1-14, (1978)
[19] Liao, D.; Zhao, J.; Yang, J.; Gregersen, H., The oesophageal zero-stress state and mucosal folding from a giome perspective, World J. gastroenterol., 13, 9, 1347-1351, (2007)
[20] Lu, X.; Zhao, J.; Gregersen, H., Small intestinal morphometric and biomechanical changes during physiological growth in rats, J. biomech., 38, 417-426, (2005)
[21] Moreno, R.H.; Hogg, J.C.; Paré, P.D., Mechanics of airway narrowing, Am. rev. respir. dis., 133, 1171-1180, (1986)
[22] Moulton, D.E.; Goriely, A., Anticavitation and differential growth in elastic shells, J. elasticity, 102, 117-132, (2011) · Zbl 1273.74181
[23] Moulton, D.E., Goriely, A. Possible role of differential growth in airway wall remodeling in asthma. J. Appl. Physiol., in press.
[24] Ogden, R.W., Non-linear elastic deformation, (1984), Dover New York · Zbl 0551.73043
[25] Prikazchikov, D.A.; Rogerson, G.A., On surface wave propagation in incompressible, transversely isotropic, pre-stressed elastic half-spaces, Int. J. eng. sci., 42, 10, 967-986, (2004) · Zbl 1211.74130
[26] Redington, A.E.; Howarth, P.H., Airway wall remodelling in asthma, Thorax, 52, 310-312, (1997)
[27] Renaudeaux, J.P.; Dion, B., Dispositif d’injection de médicaments en phase liquide, Brevet, 91, 01790, (1991)
[28] Rodriguez, E.K.; Hoger, A.; McCulloch, A., Stress-dependent finite growth in soft elastic tissues, J. biomech., 27, 455-467, (1994)
[29] Tadjbakhsh, I.; Odeh, F., Equilibrium states of elastic rings, J. math. anal. appl., 18, 59-74, (1967) · Zbl 0148.19505
[30] Vandiver, R.M., 2009. Morphoelasticity: the mechanics and mathematics of elastic growth. Ph.D. Thesis, University of Arizona.
[31] Vandiver, R.; Goriely, A., Tissue tension and axial growth of cylindrical structures in plants and elastic tissues, Europhys. lett., 84, 58004, (2008)
[32] Vandiver, R.; Goriely, A., Morpho-elasto-dynamics: the long-time dynamics of elastic growth, J. biol. dyn., 3, 2, 180-195, (2009) · Zbl 1342.92035
[33] Wiggs, B.R.; Hrousis, C.A.; Drazen, J.M.; Kamm, R.D., On the mechanism of mucosal folding in normal and asthmatic airways, J. appl. physiol., 83, 1814-1821, (1997)
[34] Yang, W.; Fung, T.C.; Chian, K.S.; Chong, C.K., Instability of the two-layered thick-walled esophageal model under the external pressure and circular outer boundary condition, J. biomech., 40, 481-490, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.