Load-deformation relations in second-order elasticity. (English) Zbl 0830.73011
Summary: A method for determining overall load-deformation relations for small- strain elastic deformations has been adapted so that second-order elastic effects can be included in certain overall quantities, such as bending stiffness, without explicit determination of the second-order solution. The approach uses Betti’s reciprocal theorem. The strains are considered small enough that third-order effects can be neglected, but there is no restriction on the magnitude of the displacements. The method is applied to obtain the resultant force and moment to second order for cylindrical bars with arbitrary cross section under torsion, extension and torsion, bending by an end load, and twisting and bending to helical form.
MSC:
| 74B20 | Nonlinear elasticity |
References:
| [1] | A. Signorini, Sulle deformazioni termoelastiche finite, Proc. 3rd Int. Conf. Appl. Mech.,2, 80-89 (1930). · JFM 56.0687.02 |
| [2] | F. D. Murnaghan,Finite deformation of an elastic solid, Amer. J. Math.,59, 235-260 (1937) · JFM 63.0748.01 · doi:10.2307/2371405 |
| [3] | F. D. Murnaghan,Finite Deformation of an Elastic Solid. John Wiley & Sons, Inc., New York 1951. · Zbl 0045.26504 |
| [4] | A. E. Green and R. T. Shield,Finite extension and torsion of cylinders, Phil. Trans. Roy. Soc., London,A244, 47-86 (1951). · Zbl 0043.39202 · doi:10.1098/rsta.1951.0015 |
| [5] | M. Misicu,Echilibrul mediilor continue cu deformari mari, Study. Cercet Mec. Metal,4, 31-53 (1953). |
| [6] | R. S. Rivlin,The solution of problems in second order elasticity theory, J. Rat. Mech. Anal.,2, 53-81 (1953). · Zbl 0050.18601 |
| [7] | J. E. Adkins, A. E. Green and R. T. Shield,Finite plane strain, Phil. Trans. Roy. Soc., London,A246, 181-213 (1953). · Zbl 0058.39602 · doi:10.1098/rsta.1953.0013 |
| [8] | R. S. Rivlin and C. Topakoglu,A theorem in the theory of finite elastic deformations, J. Rat. Mech. Anal.,3, 581-589 (1954). · Zbl 0056.42603 |
| [9] | A. E. Green and E. B. Spratt,Second-order effects in the deformation of elastic bodies, Proc. Roy. Soc.,A224, 347-361 (1954). · Zbl 0055.18105 · doi:10.1098/rspa.1954.0163 |
| [10] | P. L. Sheng,Secondary Elasticity, Chinese Assoc. Adv. Sci., Monograph Series 1, I, No. 1, Taipei 1955. |
| [11] | A. E. Green and J. E. Adkins,Large Elastic Deformations, Oxford Univ. Press, Oxford 1960 (2nd ed. 1970). |
| [12] | C. Truesdell and W. Noll,The Non-linear Field Theories of Mechanics, Handbuch der Physik III/3, Springer, Berlin 1965. · Zbl 0779.73004 |
| [13] | Z. Bai,Overall Results in Second-Order Elasticity, Ph.D. dissertation, Univ. Ill. Urbana-Champaign, Urbana, Illinois 1991. |
| [14] | R. T. Shield,Load-displacement relations for elastic bodies, J. Appl. Math. Phys. (ZAMP),18, 682-693 (1967). · Zbl 0149.42901 · doi:10.1007/BF01602041 |
| [15] | W. S. Blackburn and A. E. Green,Second-order torsion and bending of isotropic elastic cylinders, Proc. Roy. Soc., London,A240, 408-422 (1957). · Zbl 0077.38103 · doi:10.1098/rspa.1957.0095 |
| [16] | W. S. Blackburn,Second-order effects in the torsion and bending of transversely isotropic incompressible elastic beams, Quart. J. Appl. Math.,11, 142-158 (1958). · Zbl 0084.40201 · doi:10.1093/qjmam/11.2.142 |
| [17] | R. T. Shield and S. Im,Small strain deformations of elastic beams and rods including large deflections, J. Appl. Math. Phys. (ZAMP),37, 491-513 (1986). · Zbl 0589.73039 · doi:10.1007/BF00945427 |
| [18] | R. T. Shield,A consistent theory for elastic deformations with small strains, J. Appl. Mech.,51, 717-723 (1984). · Zbl 0549.73026 · doi:10.1115/1.3167714 |
| [19] | Z. Bai and R. T. Shield,Identities for Torsion of Cylinders, J. Appl. Mech.,61, 499-500 (1994). · Zbl 0862.73027 · doi:10.1115/1.2901483 |
| [20] | R. T. Shield,Finite extension and torsion of thin elastic strips, Proc. IUTAM Symp. Finite Elasticity, 331-346 (1980). |
| [21] | R. T. Shield,An energy method for certain second-order effects with application to torsion of elastic bars under tension, J. Appl. Mech.,47, 75-81 (1980). · Zbl 0436.73007 · doi:10.1115/1.3153641 |
| [22] | A. E. H. Love,A Treatise on the Mathematical Theory of Elasticity, Cambridge Univ. Press, 4th ed., 1927. Reprinted by Dover, New York 1944. · JFM 53.0752.01 |
| [23] | D. E. Parker,The role of Saint Venant’s solution in rod and beam theories, J. Appl. Mech.,46, 861-866 (1979). · Zbl 0418.73044 · doi:10.1115/1.3424668 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
