Uniformity and homogeneity of elastic rods, shells and Cosserat three-dimensional bodies

Marcelo Epstein; Manuel de León

Archivum Mathematicum (1996)

  • Volume: 032, Issue: 4, page 267-280
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Epstein, Marcelo, and de León, Manuel. "Uniformity and homogeneity of elastic rods, shells and Cosserat three-dimensional bodies." Archivum Mathematicum 032.4 (1996): 267-280. <http://eudml.org/doc/18469>.

@article{Epstein1996,
abstract = {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.},
author = {Epstein, Marcelo, de León, Manuel},
journal = {Archivum Mathematicum},
keywords = {Cosserat media; rods; shells; uniformity; homogeneity; non-holonomic frame bundles; non-holonomic G-structures; connections; Cosserat media; rods; shells; uniformity; homogeneity; non-holonomic frame bundles; non-holonomic G-structures; connections},
language = {eng},
number = {4},
pages = {267-280},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Uniformity and homogeneity of elastic rods, shells and Cosserat three-dimensional bodies},
url = {http://eudml.org/doc/18469},
volume = {032},
year = {1996},
}

TY - JOUR
AU - Epstein, Marcelo
AU - de León, Manuel
TI - Uniformity and homogeneity of elastic rods, shells and Cosserat three-dimensional bodies
JO - Archivum Mathematicum
PY - 1996
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 032
IS - 4
SP - 267
EP - 280
AB - 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.
LA - eng
KW - Cosserat media; rods; shells; uniformity; homogeneity; non-holonomic frame bundles; non-holonomic G-structures; connections; Cosserat media; rods; shells; uniformity; homogeneity; non-holonomic frame bundles; non-holonomic G-structures; connections
UR - http://eudml.org/doc/18469
ER -

References

top
  1. S.S. Antman, Nonlinear Problems of Elasticity, Applied Math. Sci. 107, Springer-Verlag, New York, 1995. (1995) Zbl0820.73002MR1323857
  2. G. Capriz, Continua with Microstructure, Springer Tracts in Natural Philosophy, 35, 1989. (1989) Zbl0676.73001MR0985585
  3. H. Cohen C.N. DeSilva, Nonlinear Theory of Elastic Directed Surfaces, J. Math. Phys. 7 (6) (1967), 960-966. (1967) 
  4. L.A. Cordero C.T.J. Dodson M. de León, Differential Geometry of Frame Bundles, Mathematics and Its Applications, Kluwer, Dordrecht, 1989. (1989) 
  5. E. Cosserat F. Cosserat, Théorie des corps Déformables, Hermann, Paris, 1909. (1909) 
  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. (1993) MR1277160
  7. M. de León M. Epstein, Material bodies of higher grade, C. R. Acad. Sc. Paris, 319, Sér. I, (1994), 615-620. (1994) MR1298293
  8. M. de León M. Epstein, The geometry of uniformity in second-grade elasticity, Acta Mechanica, 114 1 (1996), 217-224. (1996) MR1375732
  9. M. de León P.R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Math. Ser. 152, Amsterdam, 1989. (1989) MR1021489
  10. M. Elzanowski M. Epstein, On the symmetry group of second-grade materials, Int. J. Non-Linear Mechanics, 27, 4 (1992), 635-638. (1992) MR1175083
  11. M. Elzanowski S. Prishepionok, Locally Homogeneous Configurations of Uniform Elastic Bodies, Reports on Mathematical Physics, 31 (1992), 229-240. (1992) MR1232644
  12. M. Elzanowski S. Prishepionok, Connections on higher order frame bundles, Proceedings Colloquium on Differential Geometry, July 25-30, 1994, Debrecen, Hungary. (1994) 
  13. M. Elzanowski, Mathematical Theory of Uniform Elastic Structures, Monografie, Politechnika Swietokrzyska, Kielce, 1995. (1995) 
  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. (1994) MR1377280
  15. M. Epstein M. de León, Homogeneity conditions for generalized Cosserat media, Journal of Elasticity, 43 (1996), 180-201. (1996) MR1415542
  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. (1995) 
  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. (1995) MR1424749
  18. M. Epstein M. de León, Geometrical Theory of Uniform Cosserat Media, Preprint IMAFF-CSIC, 1996. (1996) 
  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. (1997) 
  20. M. Epstein M. de León, Uniformity and homogeneity of deformable surfaces, to appear in C. R. Acad. Sci. Paris, (1996). (1996) MR1375733
  21. M. Epstein M. de León, On uniformity of shells, Preprint IMAFF-CSIC, 1996. (1996) MR1375733
  22. J.L. Ericksen, Uniformity in Shells, Arch. Rational Mech. Anal. 37 (1970) 73-84. (1970) Zbl0194.57204MR1553543
  23. J.L. Ericksen, Symmetry transformations for thin elastic shells, Arch. Rational Mech. Anal. 47 (1972) 1-14. (1972) Zbl0242.73045MR0359504
  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. (1976) MR0468445
  25. A. Fujimoto, Theory of G -structures, Publications of the Study Group of Geometry, Vol. I. Tokyo, 1972. (1972) Zbl0235.53035MR0348664
  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. (1965) MR0183156
  27. A.E. Green P.M. Naghdi, Some remarks on the linear theory of shells, Q. J. Mech. Appl. Math. 18 (1965) 257-276. (1965) MR0189334
  28. S. Kobayashi K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience Publishers, New York, 1963. (1963) MR0152974
  29. I. Kolář, Generalized G -structures and G -structures of higher order, Boll. Un. Mat. Ital. (4) 12, no. 3, suppl (1975), 245-256. (1975) MR0445424
  30. I. Kolář P.W. Michor J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. (1993) MR1202431
  31. I. Kolář, Canonical forms on the prolongations of principal fibre bundles, Rev. Roumaine Math. Pures Appl. 16 (1971), 1091-1106. (1971) MR0301668
  32. P. Libermann, Sur la géométrie des prolongements des spaces fibrés vectoriels, Ann. Inst. Fourier, 14 (1) (1964), 145-172. (1964) MR0167925
  33. P. Libermann, Surconnexions et connexions affines spéciales, C. R. Acad. Sc. Paris, 261 (1965), 2801-2804. (1965) Zbl0129.36603MR0190869
  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. (1967) 
  35. P. Libermann, Sur les groupoides différentiables et le “presque parallelisme”, Instituto Nazionale di Alta Matematica, Symposia Mathematica, X (1972), 59-93. (1972) Zbl0265.53030MR0370668
  36. P. Libermann, Parallélismes, J. Differential Geometry, 8 (1973), 511-539. (1973) Zbl0284.53023MR0353189
  37. J.E. Marsden, Lectures on Geometric Methods in Mathematical Physics, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1981. (1981) Zbl0485.70001MR0619693
  38. J.E. Marsden T.J.R. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, New Jersey, 1983. (1983) 
  39. G.A. Maugin, The Method of Virtual Power in Continuum Mechanics-Applications to Coupled Fields, Acta Mechanica, 30 (1980), 1-70. (1980) MR0563729
  40. G.A. Maugin, Material Inhomogeneities in Elasticity, Chapman & Hall, London, 1993. (1993) Zbl0797.73001MR1250832
  41. G.A. Maugin, On the Structure of the Theory of Polar Elasticity, Preprint 1996. (1996) MR1639983
  42. P. Molino, Théorie des G -Structures: Le Probléme d’Equivalence, Lecture Notes in Math., 588, Berlin-New York, Springer, 1977. (1977) Zbl0357.53022MR0517117
  43. W. Noll, Materially Uniform Simple Bodies with Inhomogeneities, Arch. Rational Mech. Anal., 27, (1967), 1-32. (1967) MR0225530
  44. V. Oproiu, Connections in the semiholonomic frame bundle of second order, Rev. Roum. Math. Pures et Appl., XIV, 5 (1969), 661-672. (1969) Zbl0182.24104MR0248671
  45. J.F. Pommaret, Lie Pseudogroups and Mechanics, Mathematics and Its Applications, Gordon and Breach, New York, 1988. (1988) Zbl0677.58003MR0954613
  46. R.A. Toupin, Elastic materials with couple-stresses, Arch. Rational Mech. Anal. 11, (1962), 385-414. (1962) Zbl0112.16805MR0144512
  47. R.A. Toupin, Theories of Elasticity with Couple-stress, Arch. Rational Mech. Anal. 17, (1964), 85-112. (1964) Zbl0131.22001MR0169425
  48. C. Truesdell W. Noll, The Non-Linear Field Theories of Mechanics, Handbuch der Physik, Vol. III/3, Berlin-New York, Springer, 1965. (1965) MR0193816
  49. C.C. Wang C. Truesdell, Introduction to rational elasticity, Noordhoff International Publishing, Leyden, 1973. (1973) MR0468442
  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. (1967) Zbl0187.48802MR0229420
  51. C.C. Wang, Material Uniformity and Homogeneity in Shells, Arch. Rational Mech. Anal. 47 (1972) 343-368. (1972) Zbl0266.73054MR0359505
  52. P.Ch. Yuen, Higher order frames and linear connections, Cahiers de Topologie et Géometrie Différentielle, XII, 3 (1971), 333-371. (1971) Zbl0222.53033MR0307102

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.