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Straightening wrinkles. (English) Zbl 1323.74013
Summary: We consider the elastic deformation of a circular cylindrical sector composed of an incompressible isotropic soft solid when it is straightened into a rectangular block. In this process, the circumferential line elements on the original inner face of the sector are stretched while those on the original outer face are contracted. We investigate the geometrical and physical conditions under which the latter line elements can be contracted to the point where a localized incremental instability develops. We provide a robust algorithm to solve the corresponding two-point boundary value problem, which is stiff numerically. We illustrate the results with full incremental displacement fields in the case of Mooney-Rivlin materials and also perform an asymptotic analysis for thin sectors.

74B10 Linear elasticity with initial stresses
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[1] Aron, M.; Christopher, C.; Wang, Y., On the straightening of compressible, nonlinearly elastic, annular cylindrical sectors, Math. Mech. Solids, 3, 131-145, (1998) · Zbl 1001.74531
[2] Aron, M., Some remarks concerning a boundary-value problem in non-linear elastostatics, J. Elast., 60, 165-172, (2000) · Zbl 1001.74012
[3] Aron, M., Combined axial shearing, extension, and straightening of elastic annular cylindrical sectors, IMA J. Appl. Math., 70, 53-63, (2005) · Zbl 1151.74313
[4] Biot, A. M., Surface instability of rubber in compression, Appl. Sci. Res. A, 12, 168-182, (1963) · Zbl 0121.19004
[5] Coman, C.; Destrade, M., Asymptotic results for bifurcations in pure bending of rubber blocks, Q. J. Mech. Appl. Math., 61, 395-414, (2008) · Zbl 1147.74022
[6] Destrade, M.; Ni Annaidh, A.; Coman, C. D., Bending instabilities of soft biological tissues, Int. J. Solids Struct., 46, 4322-4330, (2009) · Zbl 1176.74068
[7] Destrade, M.; Ogden, R. W., Surface waves in a stretched and sheared incompressible elastic material, Int. J. Non-Linear Mech., 40, 241-253, (2005) · Zbl 1349.74202
[8] Destrade, M.; Ogden, R. W., On the third- and fourth-order constants of incompressible isotropic elasticity, J. Acoust. Soc. Am., 128, 3334-3343, (2010)
[9] Destrade, M., Ogden, R.W., Sgura, I., Vergori, L., 2014. Straightening: existence, uniqueness and stability. Proc. R. Soc. Lond. A, in press. · Zbl 1371.74041
[10] Destrade, M.; Murphy, J. G.; Ogden, R. W., On deforming a sector of a circular cylindrical tube into an intact tubeexistence, uniqueness, and stability, Int. J. Eng. Sci., 48, 1212-1224, (2010) · Zbl 1231.74040
[11] 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
[12] Ericksen, J. L., Deformations possible in every isotropic, incompressible, perfectly elastic body, Z. Angew. Math. Phys., 5, 466-489, (1954) · Zbl 0059.17509
[13] Fu, Y. B., 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
[14] Fu, Y. B.; Lin, Y. P., A WKB analysis of the buckling of an everted neo-Hookean cylindrical tube, Math. Mech. Solids, 7, 483-501, (2002) · Zbl 1072.74027
[15] Goriely, A., Vandiver, R., Destrade, M., 2008. Nonlinear Euler buckling. Proc. R. Soc. Lond. Ser. A 464, 3003-3019. · Zbl 1152.74342
[16] Haughton, D. M., Flexure and compression of incompressible elastic plates, Int. J. Eng. Sci., 37, 1693-1708, (1999) · Zbl 1210.74111
[17] Haughton, D. M., A practical method for the evaluation of eigenfunctions from compound matrix variables in finite elastic bifurcation problems, Int. J. Non-Linear Mech., 46, 795-799, (2011)
[18] Norris, A.N., Shuvalov, A.L., 2012. Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism. Proc. R. Soc. Lond. Ser. A 468, 467-484. · Zbl 1364.74018
[19] Ogden, R. W., Nonlinear elastic deformations, (1997), Dover New York · Zbl 0541.73044
[20] Roccabianca, S.; Bigoni, D.; Gei, M., Long-wavelength bifurcations and multiple neutral axes in elastic multilayers subject to finite bending, J. Mech. Mater. Struct., 6, 511-512, (2011)
[21] Schallamach, A., How does rubber slide?, Wear, 17, 191, (1971)
[22] Shuvalov, A.L., 2003. A sextic formalism for three-dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials. Proc. R. Soc. Lond. Ser. A 459, 1611-1639. · Zbl 1058.74044
[23] Tadmor, E. B.; Miller, R. E.; Elliott, R. S., Continuum mechanics and thermodynamics, (2012), University Press Cambridge · Zbl 1257.82002
[24] Truesdell, C.; Noll, W., The non-linear field theories of mechanics, (2004), Springer Berlin · Zbl 0779.73004
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