zbMATH — the first resource for mathematics

Bending instabilities of soft biological tissues. (English) Zbl 1176.74068
Summary: Rubber components and soft biological tissues are often subjected to large bending deformations while “in service”. The circumferential line elements on the inner face of a bent block can contract up to a certain critical stretch ratio \(\lambda _{\text{cr}}\) (say) before bifurcation occurs and axial creases appear. For several models used to describe rubber, it is found that \(\lambda _{\text{cr}}=0.56\), allowing for a 44% contraction. For models used to describe arteries it is found, somewhat surprisingly, that the strain-stiffening effect promotes instability. For example, the models used for the artery of a seventy-year old human predict that \(\lambda _{\text{cr}}=0.73\), allowing only for a 27% contraction. Tensile experiments conducted on pig skin indicate that bending instabilities should occur even earlier there.

74G60 Bifurcation and buckling
74L15 Biomechanical solid mechanics
92C10 Biomechanics
Full Text: DOI
[1] Biot, M. A.: Surface instability of rubber in compression, Appl. sci. Res. 12, 168-182 (1963) · Zbl 0121.19004
[2] Biryukov, S. V.: Impedance method in the theory of elastic surface waves, Sov. phys. Acoust. 31, 350-354 (1985)
[3] Coman, C.; Destrade, M.: Asymptotic results for bifurcations in pure bending of rubber blocks, Quart. J. Mech. appl. Math. 61, 395-414 (2008) · Zbl 1147.74022 · doi:10.1093/qjmam/hbn009
[4] Davis, J.; Kaufman, K. R.; Lieber, R. L.: Correlation between active and passive isometric force and intramuscular pressure in the isolated rabbit tibialis anterior muscle, J. biomech. 36, 505-512 (2003)
[5] Demiray, H.; Levinson, M.: On a class of finite deformations of elastic soft tissues, Bull. math. Biol. 44, 175-192 (1982) · Zbl 0481.73040
[6] Destrade, M.; Gilchrist, M. D.; Prikazchikov, A.; Saccomandi, G.: Surface instability of sheared soft tissues, J. biomech. Eng. 130, 061007 (2008)
[7] Destrade, M.; Ogden, R. W.: Surface waves in a stretched and sheared incompressible elastic material, Int. J. Non-lin. Mech. 40, 241-253 (2005) · Zbl 1349.74202
[8] Destrade, M.; Scott, N. H.: Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint, Wave motion 40, 347-357 (2004) · Zbl 1163.74339 · doi:10.1016/j.wavemoti.2003.09.003
[9] Dorfmann, A.; Trimmer, B. A.; Jr, W. A. Woods: A constitutive model for muscle properties in a soft-bodied arthropod, J. roy. Soc. interf. 4, 257-269 (2007)
[10] Dowaikh, M. A.; Ogden, R. W.: On surface waves and deformations in a pre-stressed incompressible elastic solid, IMA J. Appl. math. 44, 261-284 (1990) · Zbl 0706.73018 · doi:10.1093/imamat/44.3.261
[11] Franceschini, G.; Bigoni, D.; Regitnig, P.; Holzapfel, G. A.: Brain tissue deforms similarly to filled elastomers and follows consolidation theory, J. mech. Phys. sol. 54, 2592-2620 (2006) · Zbl 1162.74303 · doi:10.1016/j.jmps.2006.05.004
[12] Fu, Y. B.: An explicit expression for the surface-impedance matrix of a generally anisotropic incompressible elastic material in a state of plane strain, Int. J. Non-lin. Mech. 40, 229-239 (2005) · Zbl 1349.74204
[13] Gent, A. N.: A new constitutive relation for rubber, Rub. chem. Tech. 69, 59-61 (1996)
[14] Gent, A. N.; Cho, I. S.: Surface instabilities in compressed or bent rubber blocks, Rub. chem. Tech. 72, 253-262 (1999)
[15] Goriely, A.; Destrade, M.; Ben Amar, M.: Instabilities in elastomers and soft tissues, Quart. J. Mech. appl. Math. 59, 615-630 (2006) · Zbl 1107.74018 · doi:10.1093/qjmam/hbl017
[16] Green, A.E., Zerna, W., 1954. Theoretical Elasticity. University Press, Oxford. Reprinted by Dover, New York, 1992. · Zbl 0056.18205
[17] Hamilton, M. F.; Ilinskii, Y. A.; Zabolotskaya, E. A.: Separation of compressibility and shear deformation in the elastic energy density, J. acoust. Soc. am. 116, 41-44 (2004)
[18] Haughton, D. M.: Flexure and compression of incompressible elastic plates, Int. J. Eng. sc. 37, 1693-1708 (1999) · Zbl 1210.74111 · doi:10.1016/S0020-7225(98)00141-4
[19] Haughton, D. M.: Evaluation of eigenfunctions from compound matrix variables in non-linear elasticity I. Fourth order systems, J. comput. Phys. 227, 4478-4485 (2008) · Zbl 1136.74009 · doi:10.1016/j.jcp.2008.01.003
[20] Haughton, D. M.; Orr, A.: On the eversion of compressible elastic cylinders, Int. J. Sol. struct. 34, 1893-1914 (1997) · Zbl 0944.74517 · doi:10.1016/S0020-7683(96)00122-9
[21] Ho, J.; Kleiven, S.: Dynamic response of the brain with vasculature: a three-dimensional computational study, J. biomech. 40, 3006-3012 (2007)
[22] Horgan, C. O.; Saccomandi, G.: A description of arterial wall mechanics using limiting chain extensibility constitutive models, Biomech. model. Mechanobiol. 1, 251-266 (2003)
[23] Horgan, C. O.; Saccomandi, G.: Phenomenological hyperelastic strain-stiffening constitutive models for rubber, Rubber chem. Tech. 79, 152-169 (2006)
[24] Humphrey, J. D.: Continuum biomechanics of soft biological tissues, Proc. roy. Soc. lond. 459, 3-46 (2003) · Zbl 1116.74385 · doi:10.1098/rspa.2002.1060
[25] Kanner, L. M.; Horgan, C. O.: Plane strain bending of strain-stiffening rubber-like rectangular beams, Int. J. Sol. struct. 45, 1713-1729 (2008) · Zbl 1159.74372 · doi:10.1016/j.ijsolstr.2007.10.022
[26] Maikos, J. T.; Elias, R. A. I.; Shreiber, D. I.: Mechanical properties of dura mater from the rat brain and spinal cord, J. neurotrauma 25, 38-51 (2008)
[27] Ogden, R. W.: On isotropic tensors and elastic moduli, Proc. cambr. Phil. soc. 75, 427-436 (1974) · Zbl 0327.73007
[28] Ogden, R.W., 1984. Non-Linear Elastic Deformations. Ellis Horwood, Chichester. Reprinted by Dover, New York, 1997. · Zbl 0541.73044
[29] Rivlin, R. S.: Large elastic deformations of isotropic materials V: The problem of flexure, Proc. roy. Soc. lond. 195, 463-473 (1949) · Zbl 0036.24901 · doi:10.1098/rspa.1949.0004
[30] Roccabianca, S., Gei, M., Bigoni, D., 2009. Plane strain bifurcations of elastic layered structures subject to finite bending: theory vs. experiments, submitted for publication. · Zbl 1425.74087
[31] Shuvalov, A. L.: A sextic formalism for three-dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials, Proc. roy. Soc. lond. 459, 1611-1639 (2003) · Zbl 1058.74044 · doi:10.1098/rspa.2002.1075
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.