×

zbMATH — the first resource for mathematics

On deforming a sector of a circular cylindrical tube into an intact tube: existence, uniqueness, and stability. (English) Zbl 1231.74040
Summary: Within the context of finite deformation elasticity theory the problem of deforming an open sector of a thick-walled circular cylindrical tube into a complete circular cylindrical tube is analyzed. The analysis provides a means of estimating the radial and circumferential residual stress present in an intact tube, which is a problem of particular concern in dealing with the mechanical response of arteries. The initial sector is assumed to be unstressed and the stress distribution resulting from the closure of the sector is then calculated in the absence of loads on the cylindrical surfaces. Conditions on the form of the elastic strain-energy function required for existence and uniqueness of the deformed configuration are then examined. Finally, stability of the resulting finite deformation is analyzed using the theory of incremental deformations superimposed on the finite deformation, implemented in terms of the Stroh formulation. The main results are that convexity of the strain energy as a function of a certain deformation variable ensures existence and uniqueness of the residually-stressed intact tube, and that bifurcation can occur in the closing of thick, widely opened sectors, depending on the values of geometrical and physical parameters. The results are illustrated for particular choices of these parameters, based on data available in the biomechanics literature.

MSC:
74B20 Nonlinear elasticity
35Q74 PDEs in connection with mechanics of deformable solids
35B32 Bifurcations in context of PDEs
74L15 Biomechanical solid mechanics
92C30 Physiology (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chuong, C.J.; Fung, Y.C., On residual stress in arteries, J. biomech. eng., 108, 189-192, (1986)
[2] Delfino, A.; Stergiopulos, N.; Moore, J.E.; Meister, J.-J., Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation, J. biomech., 30, 777-786, (1997)
[3] Destrade, M.; Gilchrist, M.D.; Motherway, J.A.; Murphy, J.G., Bimodular rubber buckles early in bending, Mech. mater., 42, 469-476, (2010)
[4] Destrade, M.; Nì Annaidh, A.; Coman, C.D., Bending instabilities of soft biological tissues, Int. J. solids struct., 46, 4322-4330, (2009) · Zbl 1176.74068
[5] Fung, Y.C., Biomechanics: mechanical properties of living tissue, (1993), Springer New York
[6] Goriely, A.; Vandiver, R.; Destrade, M., Nonlinear Euler buckling, Proc. roy. soc. A, 464, 3003-3019, (2008) · Zbl 1152.74342
[7] Greenwald, S.E.; Moore, J.E.; Rachev, A.; Kane, T.P.C.; Meister, J.-J., Experimental determination of the distribution of residual strains in the artery wall, J. biomech. eng., 119, 438-444, (1997)
[8] Humphrey, J.D.; Halperin, H.R.; Yin, F.C.P., Small indentation superimposed on a finite equibiaxial stretch: implications for cardiac mechanics, J. appl. mech., 58, 1108-1111, (1991)
[9] Haughton, D.M., Flexure and compression of incompressible elastic plates, Int. J. eng. sci., 37, 1693-1708, (1999) · Zbl 1210.74111
[10] Haughton, D.M.; Ogden, R.W., Bifurcation of inflated circular cylinders of elastic material under axial loading. II. exact theory for thick-walled tubes, J. mech. phys. solids, 27, 489-512, (1979) · Zbl 0442.73067
[11] Haughton, D.M.; Orr, A., On the eversion of compressible elastic cylinders, Int. J. solids struct., 34, 1893-1914, (1997) · Zbl 0944.74517
[12] Hoger, A., On the residual stress possible in an elastic body with material symmetry, Arch. ration. mech. anal., 88, 271-290, (1985) · Zbl 0571.73011
[13] Holzapfel, G.A.; Gasser, T.C.; Ogden, R.W., A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. elasticity, 61, 1-48, (2000) · Zbl 1023.74033
[14] Holzapfel, G.A.; Ogden, R.W., Modelling the layer-specific 3D residual stresses in arteries, with an application to the human aorta, J. roy. soc. interface, 7, 787-799, (2010)
[15] Humphrey, J.D., Cardiovascular solid mechanics, Cells, tissues, and organs, (2002), Springer-Verlag New York
[16] Matsumoto, T.; Sato, M., Analysis of stress and strain distribution in the artery wall consisted of layers with different elastic modulus and opening angle, JSME int. J., C45, 906-912, (2002)
[17] Ogden, R.W., Non-linear elastic deformations, (1984), Ellis Horwood Chichester, Reprinted by Dover, New York, 1997 · Zbl 0541.73044
[18] Ogden, R.W., Nonlinear elasticity, anisotropy and residual stresses in soft tissue, CISM courses and lectures series, vol. 441, (2003), Springer Wien, (Lecture Notes, CISM Course on the Biomechanics of Soft Tissue in Cardiovasular Systems), pp. 65-108 · Zbl 1151.74386
[19] Ogden, R.W.; Schulze-Bauer, C.A.J., Phenomenological and structural aspects of the mechanical response of arteries, (), 125-140
[20] Olsson, T.; Stålhand, J.; Klarbring, A., Modeling initial strain distribution in soft tissues with application to arteries, Biomech. model. mechanobiol., 5, 27-38, (2006)
[21] Rachev, A.; Greenwald, S.E., Residual strains in conduit arteries, J. biomech., 36, 661-670, (2003)
[22] Rachev, A.; Hayashi, K., Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries, Ann. biomed. eng., 27, 459-468, (1999)
[23] Raghavan, M.L.; Trivedi, S.; Nagaraj, A.; McPherson, D.D.; Chandran, K.B., Three-dimensional finite element analysis of residual stress in arteries, Ann. biomed. eng., 32, 257-263, (2004)
[24] Shuvalov, A.L., A sextic formalism for three-dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials, Proc. roy. soc. lond., A459, 1611-1639, (2003) · Zbl 1058.74044
[25] Vaishnav, R.N.; Vossoughi, J., Estimation of residual strains in aortic segments, (), 330-333
[26] Vossoughi, J.; Hedjazi, Z.; Boriss, F.S., Intimal residual stress and strain in large arteries, (), 434-437
[27] Zidi, M.; Cheref, M.; Oddou, C., Finite elasticity modelling of vascular prostheses mechanics, Eur. phys. J. appl. phys., 7, 271-275, (1999)
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.