Abstract
We study the possibility for an isotropic elastic body to support forms of instability induced by shear stress states which are reminiscent of the planar Couette and the Taylor–Couette patterns observed in the flow of viscous fluids. Here, we investigate the emergence of bifurcating periodic deformations for an infinitely long compressible elastic block confined between and attached to parallel plates which are subject to a relative shear displacement. We specialize our analysis by considering a generalized form of the Blatz–Ko strain energy function and show through numerical representative examples that planar Couette modes are always preferred with respect to the twisting Taylor–Couette modes. Finally, we introduce a suitably restricted form of the strong ellipticity condition for the incremental elasticity tensor and discuss its significance in this bifurcation problem.
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Barkley D., Tuckerman L.S.: Mean flow of turbulent–laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137 (2007)
Beatty M.F.: Topics in finite elasticity: hyperelasticity of rubber, elastomers and biological tissues—with examples. Appl. Mech. Rev. 40(12), 1699–1734 (1987)
Chen Y.-C., Haughton D.M.: Stability and bifurcation of inflation of elastic cylinders. Proc. R. Soc. Lond. A 459, 137–156 (2003)
Del Piero G.: Lower bounds for the critical loads of elastic bodies. J. Elast. 10, 135–143 (1980)
Deshpande V.S., Fleck N.A.: Multi-axial yield behaviour of polymer foams. Acta mater. 49, 1859–1866 (2001)
Lockett F.J., Rivlin R.S.: Stability in Couette flow of a viscoelastic fluid. Part I. J. Mecanique 7, 475–498 (1968)
Ma T., Wang S.: Stability and bifurcation of the Taylor problem. Arch. Rational Mech. Anal. 181, 149–176 (2006)
Fosdick, R., Foti, P., Fraddosio, A., Marzano, S.: The Taylor instability for an incompressibile isotropic elastic tube (in preparation)
Fosdick, R., Foti, P., Fraddosio, A., Piccioni, M.D.: A procedure for the lower bound estimate of the critical load for compressible elastic solids (2009, Forthcoming)
Haughton D.M.: On non-linear stability in unconstrained non-linear elasticity. Int. J. Non-Linear Mech. 39, 1181–1192 (2004)
Healey T.J., Montes-Pizarro E.L.: Global bifurcation in nonlinear elasticity with an application to barrelling states of cylindrical columns. J. Elast. 71, 33–58 (2003)
Henrion D., Lasserre J.B.: GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi. ACM Trans. Math. Soft. 29, 165–194 (2003)
Ryzhak E.I.: Korn’s constant for a parallelepiped with a free face or pair of faces. Math. Mech. Solids 4, 35–55 (1999)
Simpson H.C., Spector S.J.: On bifurcation in finite elasticity: buckling of a rectangular rod. J. Elast. 92, 277–326 (2008)
Smith M.M., Rivlin R.S.: Stability in Couette flow of a viscoelastic fluid. Part II. J. Mecanique 11, 69–94 (1972)
Taylor G.I.: Stability of a viscous fluid contained between two rotating cylinders. Phil. Trans. A 223, 289–343 (1923)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. In: Encyclopaedia of Physics, vol. III/3. Springer, Berlin (1965)
Van Hove L.: Sur l’extension de la condition de Legendre du calcul des variations aux integrals multiples a plusieurs fonctions inconnues. Proc. Kön. Ned. Akad. Wet. 50, 18–23 (1947)
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Fosdick, R., Foti, P., Fraddosio, A. et al. Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids. Z. Angew. Math. Phys. 61, 537–554 (2010). https://doi.org/10.1007/s00033-009-0020-4
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DOI: https://doi.org/10.1007/s00033-009-0020-4