| Title:
|
Uniformity and homogeneity of elastic rods, shells and Cosserat three-dimensional bodies (English) |
| Author:
|
Epstein, Marcelo |
| Author:
|
de León, Manuel |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
32 |
| Issue:
|
4 |
| Year:
|
1996 |
| Pages:
|
267-280 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present a general geometrical theory of uniform bodies which includes three-dimensional Cosserat bodies, rods and shells as particular cases. Criteria of local homogeneity are given in terms on connections. (English) |
| Keyword:
|
Cosserat media |
| Keyword:
|
rods |
| Keyword:
|
shells |
| Keyword:
|
uniformity |
| Keyword:
|
homogeneity |
| Keyword:
|
non-holonomic frame bundles |
| Keyword:
|
non-holonomic G-structures |
| Keyword:
|
connections |
| MSC:
|
53C10 |
| MSC:
|
53C80 |
| MSC:
|
73B99 |
| MSC:
|
73K99 |
| MSC:
|
74A20 |
| MSC:
|
74A35 |
| MSC:
|
74E15 |
| idZBL:
|
Zbl 0881.73003 |
| idMR:
|
MR1441398 |
| . |
| Date available:
|
2008-06-06T21:31:27Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107580 |
| . |
| Reference:
|
[1] S.S. Antman: Nonlinear Problems of Elasticity.Applied Math. Sci. 107, Springer-Verlag, New York, 1995. Zbl 0820.73002, MR 1323857 |
| Reference:
|
[2] G. Capriz: Continua with Microstructure.Springer Tracts in Natural Philosophy, 35, 1989. Zbl 0676.73001, MR 0985585 |
| Reference:
|
[3] H. Cohen C.N. DeSilva: Nonlinear Theory of Elastic Directed Surfaces.J. Math. Phys. 7 (6) (1967), 960-966. |
| Reference:
|
[4] L.A. Cordero C.T.J. Dodson M. de León: Differential Geometry of Frame Bundles.Mathematics and Its Applications, Kluwer, Dordrecht, 1989. |
| Reference:
|
[5] E. Cosserat F. Cosserat: Théorie des corps Déformables.Hermann, Paris, 1909. |
| Reference:
|
[6] M. de León M. Epstein: On the integrability of second order $G$-structures with applications to continuous theories of dislocations.Reports on Mathematical Physics, 33 (3) (1993), 419-436. MR 1277160 |
| Reference:
|
[7] M. de León M. Epstein: Material bodies of higher grade.C. R. Acad. Sc. Paris, 319, Sér. I, (1994), 615-620. MR 1298293 |
| Reference:
|
[8] M. de León M. Epstein: The geometry of uniformity in second-grade elasticity.Acta Mechanica, 114 1 (1996), 217-224. MR 1375732 |
| Reference:
|
[9] M. de León P.R. Rodrigues: Methods of Differential Geometry in Analytical Mechanics.North-Holland Math. Ser. 152, Amsterdam, 1989. MR 1021489 |
| Reference:
|
[10] M. Elzanowski M. Epstein: On the symmetry group of second-grade materials.Int. J. Non-Linear Mechanics, 27, 4 (1992), 635-638. MR 1175083 |
| Reference:
|
[11] M. Elzanowski S. Prishepionok: Locally Homogeneous Configurations of Uniform Elastic Bodies.Reports on Mathematical Physics, 31 (1992), 229-240. MR 1232644 |
| Reference:
|
[12] M. Elzanowski S. Prishepionok: Connections on higher order frame bundles.Proceedings Colloquium on Differential Geometry, July 25-30, 1994, Debrecen, Hungary. |
| Reference:
|
[13] M. Elzanowski: Mathematical Theory of Uniform Elastic Structures.Monografie, Politechnika Swietokrzyska, Kielce, 1995. |
| Reference:
|
[14] M. Epstein M. de León: The Differential Geometry of Cosserat Media.New Developments in Differential Geometry (Debrecen, 1994), Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, 1996, pp. 143-164. MR 1377280 |
| Reference:
|
[15] M. Epstein M. de León: Homogeneity conditions for generalized Cosserat media.Journal of Elasticity, 43 (1996), 180-201. MR 1415542 |
| Reference:
|
[16] M. Epstein M. de León: Differential geometry of continua with structure.Proceedings 8th International Symposium “Continuum Models and Discrete Systems”, Varna, Bulgaria, June 11-16, 1995. World Scientific, Singapore, 1996, pp. 156-163. |
| Reference:
|
[17] M. Epstein M. de León: Geometric characterization of the homogeneity of continua with microstructure.Proceedings Third Meeting on Current Ideas in Mechanics and Related Fields, Segovia, June 19-23, 1995. Extracta Mathematicae 11 (1) (1996), pp. 116-126. MR 1424749 |
| Reference:
|
[18] M. Epstein M. de León: Geometrical Theory of Uniform Cosserat Media.Preprint IMAFF-CSIC, 1996. |
| Reference:
|
[19] M. Epstein M. de León: Shell theory and material uniformity.V Pan American Congress of Applied Mechanics, Mayagüez, Puerto Rico, January 2-6, 1997. |
| Reference:
|
[20] M. Epstein M. de León: Uniformity and homogeneity of deformable surfaces.to appear in C. R. Acad. Sci. Paris, (1996). MR 1375733 |
| Reference:
|
[21] M. Epstein M. de León: On uniformity of shells.Preprint IMAFF-CSIC, 1996. MR 1375733 |
| Reference:
|
[22] J.L. Ericksen: Uniformity in Shells.Arch. Rational Mech. Anal. 37 (1970) 73-84. Zbl 0194.57204, MR 1553543 |
| Reference:
|
[23] J.L. Ericksen: Symmetry transformations for thin elastic shells.Arch. Rational Mech. Anal. 47 (1972) 1-14. Zbl 0242.73045, MR 0359504 |
| Reference:
|
[24] A.C. Eringen, Ch.B. Kafadar: Polar Field Theories.Continuum Physics, Ed. A. Cemal Eringen, vol. IV, Part I, pp. 1-73, Academic Press, New York, 1976. MR 0468445 |
| Reference:
|
[25] A. Fujimoto: Theory of $G$-structures.Publications of the Study Group of Geometry, Vol. I. Tokyo, 1972. Zbl 0235.53035, MR 0348664 |
| Reference:
|
[26] A.E. Green P.M. Naghdi W.L. Wainwright: A General Theory of a Cosserat Surface.Arch. Rational Mech. Anal. 20 (1965) 287-308. MR 0183156 |
| Reference:
|
[27] A.E. Green P.M. Naghdi: Some remarks on the linear theory of shells.Q. J. Mech. Appl. Math. 18 (1965) 257-276. MR 0189334 |
| Reference:
|
[28] S. Kobayashi K. Nomizu: Foundations of Differential Geometry.vol. I, Interscience Publishers, New York, 1963. MR 0152974 |
| Reference:
|
[29] I. Kolář: Generalized $G$-structures and $G$-structures of higher order.Boll. Un. Mat. Ital. (4) 12, no. 3, suppl (1975), 245-256. MR 0445424 |
| Reference:
|
[30] I. Kolář P.W. Michor J. Slovák: Natural Operations in Differential Geometry.Springer-Verlag, Berlin, 1993. MR 1202431 |
| Reference:
|
[31] I. Kolář: Canonical forms on the prolongations of principal fibre bundles.Rev. Roumaine Math. Pures Appl. 16 (1971), 1091-1106. MR 0301668 |
| Reference:
|
[32] P. Libermann: Sur la géométrie des prolongements des spaces fibrés vectoriels.Ann. Inst. Fourier, 14 (1) (1964), 145-172. MR 0167925 |
| Reference:
|
[33] P. Libermann: Surconnexions et connexions affines spéciales.C. R. Acad. Sc. Paris, 261 (1965), 2801-2804. Zbl 0129.36603, MR 0190869 |
| Reference:
|
[34] P. Libermann: Connexions d’ordre supérieur et tenseur de structure.Atti del Convegno Int. di Geometria Differenziale, Bologna, 28-30, September 1967, Ed. Zanichelli, Bologna, 1967, 1-18. |
| Reference:
|
[35] P. Libermann: Sur les groupoides différentiables et le “presque parallelisme”.Instituto Nazionale di Alta Matematica, Symposia Mathematica, X (1972), 59-93. Zbl 0265.53030, MR 0370668 |
| Reference:
|
[36] P. Libermann: Parallélismes.J. Differential Geometry, 8 (1973), 511-539. Zbl 0284.53023, MR 0353189 |
| Reference:
|
[37] J.E. Marsden: Lectures on Geometric Methods in Mathematical Physics.CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1981. Zbl 0485.70001, MR 0619693 |
| Reference:
|
[38] J.E. Marsden T.J.R. Hughes: Mathematical Foundations of Elasticity.Prentice Hall, New Jersey, 1983. |
| Reference:
|
[39] G.A. Maugin: The Method of Virtual Power in Continuum Mechanics-Applications to Coupled Fields.Acta Mechanica, 30 (1980), 1-70. MR 0563729 |
| Reference:
|
[40] G.A. Maugin: Material Inhomogeneities in Elasticity.Chapman & Hall, London, 1993. Zbl 0797.73001, MR 1250832 |
| Reference:
|
[41] G.A. Maugin: On the Structure of the Theory of Polar Elasticity.Preprint 1996. MR 1639983 |
| Reference:
|
[42] P. Molino: Théorie des $G$-Structures: Le Probléme d’Equivalence.Lecture Notes in Math., 588, Berlin-New York, Springer, 1977. Zbl 0357.53022, MR 0517117 |
| Reference:
|
[43] W. Noll: Materially Uniform Simple Bodies with Inhomogeneities.Arch. Rational Mech. Anal., 27, (1967), 1-32. MR 0225530 |
| Reference:
|
[44] V. Oproiu: Connections in the semiholonomic frame bundle of second order.Rev. Roum. Math. Pures et Appl., XIV, 5 (1969), 661-672. Zbl 0182.24104, MR 0248671 |
| Reference:
|
[45] J.F. Pommaret: Lie Pseudogroups and Mechanics.Mathematics and Its Applications, Gordon and Breach, New York, 1988. Zbl 0677.58003, MR 0954613 |
| Reference:
|
[46] R.A. Toupin: Elastic materials with couple-stresses.Arch. Rational Mech. Anal. 11, (1962), 385-414. Zbl 0112.16805, MR 0144512 |
| Reference:
|
[47] R.A. Toupin: Theories of Elasticity with Couple-stress.Arch. Rational Mech. Anal. 17, (1964), 85-112. Zbl 0131.22001, MR 0169425 |
| Reference:
|
[48] C. Truesdell W. Noll: The Non-Linear Field Theories of Mechanics.Handbuch der Physik, Vol. III/3, Berlin-New York, Springer, 1965. MR 0193816 |
| Reference:
|
[49] C.C. Wang C. Truesdell: Introduction to rational elasticity.Noordhoff International Publishing, Leyden, 1973. MR 0468442 |
| Reference:
|
[50] C.C. Wang: On the Geometric Structures of Simple Bodies, a Mathematical Foundation for the Theory of Continuous Distributions of Dislocations.Arch. Rational Mech. Anal., 27, (1967), 33-94. Zbl 0187.48802, MR 0229420 |
| Reference:
|
[51] C.C. Wang: Material Uniformity and Homogeneity in Shells.Arch. Rational Mech. Anal. 47 (1972) 343-368. Zbl 0266.73054, MR 0359505 |
| Reference:
|
[52] P.Ch. Yuen: Higher order frames and linear connections.Cahiers de Topologie et Géometrie Différentielle, XII, 3 (1971), 333-371. Zbl 0222.53033, MR 0307102 |
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