Summary
We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.
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1929 [1] Friedrichs, K. O., Ein Verfahren der Variationsrechnung das Minimum eines Integrals das Maximum eines anderen Ausdruckes darzustellen. Ges. Wiss. Gött., Nachr. 13–20.
1948 [1] Diaz, J. B., & H. J. Greenberg, Upper and lower bounds for the solution of the first boundary value problem of elasticity. Quart. Appl. Math. 6, 326–31.
1957 [1] Synge, J. L., The Hypercircle in Mathematical Physics. Cambridge.
1960 [1] Ericksen, J. L., Tensor Fields. Handbuch der Physik, Vol. III/1. Berlin-GöttingenHeidelberg: Springer.
1964 [1] Mikhlin, S., Variational Methods in Mathematical Physics. New York: Pergamon.
1965 [1] Truesdell, C, & W. Noll, The non-linear Field Theories of Mechanics. Handbuch der Physik, Vol. III/3. Berlin-Heidelberg-New York: Springer.
1966 [1] Truesdell, C., The Elements of Continuum Mechanics. Berlin-Heidelberg-New York: Springer.
1968 [1] Brezis, H., & M. Sibony, Méthodes d'approximation pour les opérateurs monotones. Arch. Rational Mech. Anal. 28, 1, 59–82.
1969 [1] Mikhlin, S., Numerische Realisierung von Variationsmethoden. Berlin: Akademie-Verlag.
1970 [1] Nečas, J., & I. Hlaváček, On inequalities of Korn's Type. Arch. Rational Mech. Anal. Part I, 36, 305–311; Part II, 36, 312–334.
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Communicated by C. Truesdell
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Villaggio, P. Energetic bounds in finite elasticity. Arch. Rational Mech. Anal. 45, 282–293 (1972). https://doi.org/10.1007/BF00251377
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DOI: https://doi.org/10.1007/BF00251377