The traction problem for incompressible materials. (English) Zbl 0574.73013

The nonlinear behavior of incompressible materials is examined using both a variational approach and a power series approach. Second order sufficient conditions for the existence of a power series solution are determined. A systematic procedure for finding the power series solution is also presented.
The paper is theoretical. It will therefore probably be of greatest interest to theoreticians working in continuum mechanics and elasticity theory.
Reviewer: R.L.Huston


74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
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[1] J. M. Ball and D. G. Schaeffer, Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 315 – 339. · Zbl 0568.73057
[2] D. R. J. Chillingworth, J. E. Marsden, and Y. H. Wan, Symmetry and bifurcation in three-dimensional elasticity. I, Arch. Rational Mech. Anal. 80 (1982), no. 4, 295 – 331. · Zbl 0509.73018
[3] D. R. J. Chillingworth, J. E. Marsden, and Y. H. Wan, Symmetry and bifurcation in three-dimensional elasticity. II, Arch. Rational Mech. Anal. 83 (1983), no. 4, 363 – 395. , https://doi.org/10.1007/BF00963840 Y. H. Wan and J. E. Marsden, Symmetry and bifurcation in three-dimensional elasticity. III. Stressed reference configurations, Arch. Rational Mech. Anal. 84 (1983), no. 3, 203 – 233. · Zbl 0536.73011
[4] David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2) 92 (1970), 102 – 163. · Zbl 0211.57401
[5] G. Fichera, Existence theorems in elasticity, Handbuch der Physik, Bd. VIa/2, Springer-Verlag, Berlin and New York, 1972, pp. 347-389.
[6] Arthur E. Fischer and Jerrold E. Marsden, Linearization stability of nonlinear partial differential equations, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 219 – 263. · Zbl 0274.58003
[7] A. E. Green and E. B. Spratt, Second-order effects in the deformation of elastic bodies, Proc. Roy. Soc. London. Ser. A. 224 (1954), 347 – 361. · Zbl 0055.18105
[8] G. Grioli, Mathematical theory of elastic equilibrium, Ergeb. Math. Grenzgeb., Band 7, Springer-Verlag, Berlin and New York, 1962. · Zbl 0102.17004
[9] M. Gurtin, The linear theory of elasticity, Handbuch der Physik, vol. VIa/2, Springer-Verlag, Berlin, 1972, pp. 1-295.
[10] J. Marsden and T. Hughes, The mathematical foundations of elasticity, Prentice-Hall, Englewood Clifs, N.J., 1982. · Zbl 0545.73031
[11] J. E. Marsden and Y. H. Wan, Linearization stability and Signorini series for the traction problem in elastostatics, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), no. 1-2, 171 – 180. · Zbl 0533.73022
[12] A. Signorini, Sulle deformazioni termoelastiche finite, Proc. 3rd Internat. Congr. Appl. Mech., vol. 2, 1930, pp. 80-89. · JFM 57.1080.05
[13] F. Stoppelli, Sulle esistenza di soluzioni delli equazioni del\? elastostatica isoterma rel case di sollectizioni detate di assi di equilibria, Richerche Mat. 6 (1957), 241-287; 7 (1958), 71-101, 138-152.
[14] Carlo Tolotti, Orientamenti principali di un corpo elastico rispetto alla sua sollecitazione totale, Atti Accad. Italia. Mem. Cl. Sci. Fis. Mat. Nat. (7) 13 (1942), 1139 – 1162=Ist. Naz. Appl. Calcolo (2) no. 145 (1942) (Italian). · Zbl 0060.43004
[15] C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. · Zbl 0779.73004
[16] C. C. Wang and C. Truesdell, Introduction to rational elasticity, Noordhoff International Publishing, Leyden, 1973. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics of Continua. · Zbl 0308.73001
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