A derivation of degenerate von Kármán equations for strongly anisotropic plates
Author:
Robert G. Root
Journal:
Quart. Appl. Math. 57 (1999), 19-36
MSC:
Primary 74K20; Secondary 35Q72, 74B20, 74G10
DOI:
https://doi.org/10.1090/qam/1672167
MathSciNet review:
MR1672167
Full-text PDF Free Access
References |
Similar Articles |
Additional Information
- Melvyn S. Berger, On vonKármán’s equations and the buckling of a thin elastic plate. I. The clamped plate, Comm. Pure Appl. Math. 20 (1967), 687–719. MR 221808, DOI https://doi.org/10.1002/cpa.3160200405
- Philippe G. Ciarlet, A justification of the von Kármán equations, Arch. Rational Mech. Anal. 73 (1980), no. 4, 349–389. MR 569597, DOI https://doi.org/10.1007/BF00247674
- P. G. Ciarlet, Plates and junctions in elastic multi-structures, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 14, Masson, Paris; Springer-Verlag, Berlin, 1990. An asymptotic analysis. MR 1071376
- Philippe G. Ciarlet and Patrick Rabier, Les équations de von Kármán, Lecture Notes in Mathematics, vol. 826, Springer, Berlin, 1980 (French). MR 595326
- Gaetano Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, Boundary problems in differential equations, Univ. of Wisconsin Press, Madison, 1960, pp. 97–120. MR 0111931
- Morton E. Gurtin, Topics in finite elasticity, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 35, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 599913
- W. M. Greenlee, Degeneration of a compound plate system to a membrane-plate system: a singularly perturbed transmission problem, Ann. Mat. Pura Appl. (4) 128 (1981), 153–167. MR 640780, DOI https://doi.org/10.1007/BF01789471
- S. G. Lekhnitskiĭ, Theory of elasticity of an anisotropic body, Translated from the second Russian edition, “Mir”, Moscow, 1981. MR 610607
J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983
- Robert Griffith Root, Boundary value problems for degenerate elliptic-parabolic fourth order equations, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)–University of Delaware. MR 2636629
- Robert G. Root, Boundary value problems for degenerate elliptic-parabolic equations of the fourth order, Trans. Amer. Math. Soc. 324 (1991), no. 1, 109–134. MR 986699, DOI https://doi.org/10.1090/S0002-9947-1991-0986699-9
- Robert G. Root, Existence theory for higher order elliptic-parabolic equations with an application to elasticity, J. Math. Anal. Appl. 154 (1991), no. 1, 255–272. MR 1087973, DOI https://doi.org/10.1016/0022-247X%2891%2990085-E
- Robert G. Root, Boundary value problems for degenerate von Kármán equations, Quart. Appl. Math. 57 (1999), no. 1, 1–17. MR 1672163, DOI https://doi.org/10.1090/qam/1672163
- J. J. Stoker, Nonlinear elasticity, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0413654
T. von Kárman, Festigkeitsprobleme im Maschinbau, Encyl. der Math. Wissenschaften 4 (4), Leipzig, 348–352 (1907–1914)
- R. J. Weinacht, Asymptotic distribution of eigenvalues for a class of degenerate elliptic operators of the fourth order, Rend. Mat. (7) 6 (1986), no. 1-2, 159–170 (1988). MR 973614
- R. J. Weinacht, Degenerate elliptic equations and spongy elastic plates, Problems of applied analysis (Oberwolfach, 1985) Methoden Verfahren Math. Phys., vol. 33, Peter Lang, Frankfurt am Main, 1987, pp. 59–73. MR 917499
- Eberhard Zeidler, Nonlinear functional analysis and its applications. IV, Springer-Verlag, New York, 1988. Applications to mathematical physics; Translated from the German and with a preface by Juergen Quandt. MR 932255
M. S. Berger, On von Kármán’s equations and the buckling of a thin elastic plate, I, Communications on Pure and Applied Mathematics 20, 687–719 (1967)
P. G. Ciarlet, A justification of the von Kármán equations, Archive for Rational Mechanics and Analysis 73, 349–389 (1980)
P. G. Ciarlet, Plates and junctions in elastic multi-structures, Recherches en Mathématiques Appliqués, Vol. 14, Masson, Paris, 1990
P. G. Ciarlet and P. Rabier, Les équations de von Kármán, Springer, Berlin and New York, 1980
G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of the second order, in Proceedings of a Symposium on Boundary Value Problems in Differential Equations (R. E. Langer, ed.), Univ. Wisconsin Press, Madison, WI, 1960, pp. 97–120
M. E. Gurtin, Topics in Finite Elasticity, CBMS-NSF Series #35, SIAM, Philadelphia, 1981
W. M. Greenlee, Degeneration of a compound plate system to a membrane-plate system: A singularly perturbed transmission problem, Ann. Mat. Appl. (4) 128, 153–167 (1981)
S. G. Lekhnitskii, Theory of elasticity of an anisotropic body, Mir, Moscow, 1981
J. E. Marsden and T. J. R. Hughes, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983
R. G. Root, Boundary value problems for degenerate elliptic-parabolic fourth order equations, University of Delaware, doctoral dissertation, 1988
R. G. Root, Boundary value problems for degenerate elliptic-parabolic equations of the fourth order, Transactions of the American Mathematical Society 324, No. 1 (1991)
R. G. Root, Existence theory of higher order elliptic-parabolic equations with an application to elasticity, Journal of Mathematical Analysis and Its Applications 54, No. 1 (1991)
R. G. Root, Boundary value problems for degenerate von Kármán equations, Quart. Appl. Math. 57, 1–17 (1999)
J. J. Stoker, Nonlinear Elasticity, Gordon and Breach, New York, 1968
T. von Kárman, Festigkeitsprobleme im Maschinbau, Encyl. der Math. Wissenschaften 4 (4), Leipzig, 348–352 (1907–1914)
R. J. Weinacht, Asymptotic distribution of eigenvalues for a class of degenerate elliptic operators of the fourth order, Rend. Mat. (7) 6, No. 1-2, 159–170 (1986)
R. J. Weinacht, Degenerate elliptic equations and spongy elastic plates, in Methoden und Verfahren der mathematischen Physik, Problems of Applied Analysis, Vol. 33, Lang, Frankfurt am Main, 1987, pp. 59–74
E. Zeidler, Nonlinear functional analysis and its applications, vol. 4, Applications to Mathematical Physics, Springer, Berlin and New York, 1988
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74K20,
35Q72,
74B20,
74G10
Retrieve articles in all journals
with MSC:
74K20,
35Q72,
74B20,
74G10
Additional Information
Article copyright:
© Copyright 1999
American Mathematical Society