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The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity. (English) Zbl 1130.74008
Summary: In an attempt to describe cork-pulling, we model a cork as an incompressible rubber-like material and consider that it is subject to a helical shear deformation superimposed onto a shrink fit and a simple torsion. It turns out that this deformation field provides an insight into the possible appearance of secondary deformation fields for special classes of materials. We also find that these latent deformation fields are woken up by normal stress differences. We present some explicit examples based on neo-Hookean, generalized neo-Hookean and Mooney-Rivlin forms of the strain-energy density. Using a simple exact solution found in the neo-Hookean case, we conjecture that it is advantageous to accompany the usual vertical axial force by a twisting moment, in order to extrude a cork from the neck of a bottle efficiently. Then we analyse departures from the neo-Hookean behaviour by exact and asymptotic analyses. In that process, we are able to give an analytic example of secondary (or latent) deformations in the framework of nonlinear elasticity.
##### MSC:
 74B20 Nonlinear elasticity 74G75 Inverse problems in equilibrium solid mechanics 74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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##### References:
 [1] Atkin, R.J. & Fox, N. 2005 An introduction to the theory of elasticity Mineola, NY: Dover Publications. · Zbl 1231.74001 [2] Currie, P.K. & Hayes, M. 1981 On non-universal finite elastic deformations. Proc. IUTAM Symp. on Finite Elasticity (eds. Carlson, D.E. & Shield, R.T.), pp. 143–150, The Hague, The Netherlands: Martinus Nijhoff [3] Ericksen, J.L. 1954 Deformations possible in every isotropic incompressible perfectly elastic body. Z. Angew. Math. Phys. 5, 466–486, (doi:10.1007/BF01601214). [4] Ericksen, J.L. 1955 Deformations possible in every compressible isotropic perfectly elastic body. J. Math. Phys. 34, 126–128. [5] Fosdick, R.L. & Kao, B.G. 1978 Transverse deformations associated with rectilinear shear in elastic solids. J. Elast. 8, 117–142, (doi:10.1007/BF00052477). · Zbl 0393.73041 [6] Fosdick, R.L. & Serrin, J. 1973 Rectilinear steady flow of simple fluids. Proc. R. Soc. A 332, 311–333, (doi:10.1098/rspa.1973.0028). · Zbl 0261.76001 [7] Fu, Y.B. & Ogden, R.W. 2001 Nonlinear elasticity: theory and applications. Cambridge, UK: Cambridge University Press. · Zbl 0962.00003 [8] Gibson, L.J., Easterling, K.E. & Ashby, M.F. 1981 The structure and mechanics of cork. Proc. R. Soc. A 377, 99–117, (doi:10.1098/rspa.1981.0117). [9] Hill, J.M. 1975 The effect of precompression on the load–deflection relations of long rubber bush mountings. J. Appl. Polym. Sci. 19, 747–755, (doi:10.1002/app.1975.070190312). [10] Horgan, C.O. 1995 Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81, (doi:10.1137/1037003). [11] Horgan, C.O. & Saccomandi, G. 2003 Helical shear for hardening generalized neo-Hookean elastic materials. Math. Mech. Solids 8, 539–559, (doi:10.1177/10812865030085007). [12] Horgan, C.O. & Saccomandi, G. 2003 Coupling of anti-plane shear deformations with plane deformations in generalized neo-Hookean isotropic, incompressible, hyperelastic materials. J. Elast. 73, 221–235, (doi:10.1023/B:ELAS.0000029990.92029.a7). · Zbl 1061.74007 [13] Knowles, J.K. 1976 On finite anti-plane shear for incompressible elastic materials. J. Aust. Math. Soc. B 19, 400–415. · Zbl 0363.73045 [14] Mollica, F. & Rajagopal, K.R. 1997 Secondary deformations due to axial shear of the annular region between two eccentrically placed cylinders. J. Elast. 48, 103–123, (doi:10.1023/A:1007484731059). · Zbl 0917.73017 [15] Ogden, R.W. 1997 Non-linear elastic deformations. New York, NY: Dover. [16] Polignone, D.A. & Horgan, C.O. 1991 Pure torsion of compressible nonlinearly elastic circular cylinders. Quart. Appl. Math. 49, 591–607. · Zbl 0751.73014 [17] Rivlin, R.S. 1948 Large elastic deformations of isotropic materials IV. Phil. Trans. R. Soc. A 241, 379–397, (doi:10.1098/rsta.1948.0024). · Zbl 0031.42602 [18] Saccomandi, G. 2004 Phenomenology of rubber-like materials. Mechanics and thermomechanics of rubberlike solids (eds. Saccomandi, G. & Ogden, R.W.), pp. 91–134, Vienna, Austria: Springer [19] Wineman, A. 2005 Some results for generalized neo-Hookean elastic materials. Int. J. Nonlin. Mech. 40, 271–279, (doi:10.1016/j.ijnonlinmec.2004.05.007). · Zbl 1349.74064
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