Abstract
Professor J.F. Bell's empirical result regarding the rotation factor in the polar decomposition of the deformation gradient for the finite twist–extension of a thin-walled polycrystalline cylindrical metal tube is examined. The correct expression for the rotation is derived and used to show how Bell's result should be interpreted. Some implications for his incremental plasticity equations are also discussed. In particular, they are shown to satisfy appropriate invariance requirements when cast in terms of the variables actually measured by Bell in his experiments. Further consequences of his equations consistent with his data are also derived. Finally, it is shown that his theory furnishes a consistent constitutive statement about the response of isotropic solids provided that the Cauchy stress is constrained to be symmetric.
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Beatty, M.F., 'Hyperelastic Bell materials: retrospection, experiment, theory', in: Fu, Y.B. and Ogden, R.W. (eds), Nonlinear Elasticity, Theory and Applications, London Mathematical Society Lecture Note Series 283, Cambridge University Press, 2001, pp. 58-96.
Beatty, M.F. and Hayes, M.A., 'Deformations of an elastic, internally constrained material. Part I: Homogeneous deformations', J. Elasticity 29 (1992) 1-84.
Bell, J.F., 'Material objectivity in an experimentally based incremental theory of large finite plastic strain', Int. J. Plasticity 6 (1990) 293-314.
Bell, J.F., 'Laboratory experiments on thin-walled tubes at large finite strain — symmetry, coaxiality, rigid body rotation, and the role of invariants, for the applied stress σ = RT TR , the Cauchy stress σ* = [III V]−1 FT TR , and the left Cauchy-Green stretch tensor V = FR T', Int. J. Plasticity 11 (1995) 119-144.
Bell, J.F. and Baesu, E., 'On the symmetry and coaxiality of pertinent stress and stretch tensors during non-proportional loading at finite plastic strain', Acta Mech. 115 (1996) 1-14.
Chadwick, P., Continuum Mechanics: Concise Theory and Problems, Dover, New York, 1999.
Lu, J. and Papadopoulos, P., 'On the direct determination of the rotation tensor from the deformation gradient', Math. Mech. Solids 2 (1997) 17-26.
McMeeking, R.M., 'The finite strain torsion test of a thin-walled tube of elastic-plastic material', Int. J. Solids Struct. 18 (1982) 199-204.
Noll, W., 'A mathematical theory of the mechanical behavior of continuous media', Arch. Ration. Mech. Anal. 2 (1958) 197-226.
Noll, W., 'A new mathematical theory of simple materials', Arch. Ration. Mech. Anal. 48 (1972) 1-50.
Ogden, R.W., 'Inequalities associated with the inversion of elastic stress-deformation relations and their implications', Math. Proc. Cambridge Phil. Soc. 81 (1977) 313-324.
Ogden, R.W., Nonlinear Elastic Deformations, Ellis Horwood, Chichester, 1984.
Rajagopal, K.R., and Srinivasa, A.R., 'Mechanics of the inelastic behavior of materials. Part II: Inelastic response', Int. J. Plasticity 14 (1998) 969-995.
Sellers, H.S. and Douglas, A.S., 'A physical theory of finite plasticity from a theoretical perspective', Int. J. Plasticity 6 (1990) 329-351.
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Steigmann, D.J. An Analysis of Professor J.F. Bell's Research on the Finite Twist and Extension of Thin-walled Polycrystalline Cylindrical Tubes. Meccanica 38, 395–404 (2003). https://doi.org/10.1023/A:1024606431746
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DOI: https://doi.org/10.1023/A:1024606431746