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New integral estimates for deformations in terms of their nonlinear strains. (English) Zbl 0491.73023

MSC:
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
46N99 Miscellaneous applications of functional analysis
30C20 Conformal mappings of special domains
53A05 Surfaces in Euclidean and related spaces
53A15 Affine differential geometry
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[1] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 373-403. · Zbl 0368.73040
[2] Federer, H., Geometric Measure Theory, Springer-Verlag, 1969. · Zbl 0176.00801
[3] Fong, J., & W. Penn, Construction of a strain-energy function for an isotropic elastic material. Trans. Society of Rheology 19 (1975), 99-113. · doi:10.1122/1.549389
[4] Flory, P., & Y. Tatara, The elastic free energy and the elastic equation of state: elongation and swelling of polydimethylsiloxane networks, J. Polymer Science (physics) 13 (1975), 683. · doi:10.1002/pol.1975.180130403
[5] Gilbarg, D., & N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1977. · Zbl 0361.35003
[6] Green, A. E., & W. Zerna, Theoretical elasticity, Oxford Univ. Press, 1968.
[7] Hughes, T. J. R., T. Kato, & J. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63 (1977), 273-294. · Zbl 0361.35046 · doi:10.1007/BF00251584
[8] John, F., Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. · Zbl 0102.17404 · doi:10.1002/cpa.3160140316
[9] John, F., Bounds for deformations in terms of average strains, in Inequalities III, O. Shisha, editor, Academic Press, 1972. · Zbl 0292.53003
[10] John, F., Uniqueness of nonlinear elastic equilibrium for prescribed boundary displacements and sufficiently small strains, Comm. Pure Appl. Math. 25 (1972), 617-634. · Zbl 0287.73009 · doi:10.1002/cpa.3160250505
[11] John, F., & L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. · Zbl 0102.04302 · doi:10.1002/cpa.3160140317
[12] Knowles, J., Finite elastostatic fields with unbounded deformation gradients, in Finite Elasticity ? AMD vol. 27, Amer. Soc. Mech. Eng., 1977. · Zbl 0388.73036
[13] Murnaghan, F., Finite deformations of an elastic solid, John Wiley and Sons, 1951. · Zbl 0045.26504
[14] Necas, J., Theory of locally monotone operators modeled on the finite displacement theory for hyperelasticity, in Beiträge zur Analysis 8, Berlin, 1976. · Zbl 0332.73046
[15] Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. · Zbl 0088.07601
[16] Oden, J. T., Existence theorems for a class of problems in nonlinear elasticity, J. Math. Anal. Appl. 69 (1979), 51-83. · Zbl 0413.73023 · doi:10.1016/0022-247X(79)90178-1
[17] Oden, J. T., & Kikuchi, N. Existence theory for a class of problems in nonlinear elasticity: finite plane strain of a compressible hyperelastic body. Journal de Annals de Toulouse, No. 3, 1979 (to appear).
[18] Oden, J. T., & R. E. Showalter, Existence theory in nonlinear elasticity, Proc. of an NSF workshop, Austin, Texas, 1977.
[19] Ogden, R. W., Volume changes associated with the deformations of rubber-like solids, J. Mech. Phys. Solids 24 (1976), 313-338. · Zbl 0356.73041 · doi:10.1016/0022-5096(76)90007-7
[20] Peng, S., & R. Landel, Stored energy function and compressibility of compressible materials under large strain, J. Applied Physics 46 (1975), 2599-2604. · doi:10.1063/1.321936
[21] Reshetnyak, Y. G., Stability theorems in certain aspects of differential geometry and analysis, Mat. Zametki 23 (1978), 7773-7781.
[22] Sburlan, S. F., The Dirichlet problem of elastic equilibrium, Rev. Roumaine Sci. Tech. Sér. Méc. Appl. 19 (1974), 833-847. · Zbl 0374.73021
[23] Stoppelli, F., Un teorema di esistenza e di unicità relativo allé equazione dell’ elastostatica isoterma per deformazioni finite, Ricerche Mat. 3 (1954), 247-267. · Zbl 0058.39701
[24] Strauss, M. J., Variations of Korn’s and Sobolev’s inequalities, in Berkeley symposium on partial differential equations, AMS symposia, vol. 23, 1971. · Zbl 0259.35008
[25] Treloar, L. R. G., The physics of rubber elasticity, Oxford University Press, 1975. · Zbl 0347.73042
[26] Truesdell, C., The mechanical foundations of elasticity and fluid dynamics, reprinted from J. of Rational Mech. and Anal., (vols. 1-3, 1952-4) by Gordon and Breach, 1966.
[27] van Buren, W., On the existence and uniqueness of solutions to boundary value problems in finite elasticity, Ph. D. Thesis, Dept., of Math. Carnegie-Mellon Univ., 1968.
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