In the context of finite deformations, a purely kinematical treatment of Volterra dislocations is presented. These dislocations correspond to deformations that have a jump discontinuity across a singular surface in a fixed reference configuration of the continuum, but whose right Cauchy-Green deformation tensor field is continuous and has second partial derivatives that are continuous. Analytical and geometrical proofs of Weingarten's theorem for finite deformations are given. Tune-dependent Volterra dislocations are also considered.
[1] Weingarten, G. Sulle superficie di discontinuità nella teoria della elasticita dei corpi solidi. Atti della Reale Accademia dei Lincei Rendiconti, Ser 5, Vol. X, 57-60 (1901).
2.
[2] Volterra, V. Sur l'équilibre des corps élastiques multiplement connexes. Annales École Normale, Ser. 3, Vol. XXIXV, 401-518 (1907) = Opere Matematiche, Vol. 3 153-242, Accademia Nazionale dei Lincei, Rome, 1957.
3.
[3] Volterra, V. and Volterra, E. Sur les Distorsions des Corps Élastiques (Théorie et Applications), Mémorial des Sciences Mathématiques, Fascicule CXLVI, Gauthier-Villars, Paris, 1960.
4.
[4] Love, A. E. H.A Treatise on the Mathematical Theory of Elasticity, 4th edn., Cambridge University Press, Cambridge , 1927.
5.
[5] Nabarro, F R. N. (ed.) Dislocations in Solids, Vols. 1-6, North-Holland, Amsterdam , 1979-83.
6.
[6] Cottrell, A. H.Dislocations and Plastic Flow in Crystals, Oxford University Press, Oxford , 1953.
7.
[7] Nabarro, F. R. N.Theory of Crystal Dislocations, Oxford University Press, Oxford , 1967 (also Dover, 1987).
8.
[8] Gurtin, M. E.The linear theory of elasticity. Handbuch der Physik, Vol.2, ed. S. Flügge , pp. 1-295, Springer, Berlin , 1972.
9.
[9] Teodosiu, C.Elastic Models of Crystal Defects, Springer, Berlin , 1982.
10.
[10] Goodier, J. N.On the theory of dislocations and an application to the plane stress problem of a system of forces acting on a hole. Proceedings of the 5th International Congress of Applied Mechanics, ed. J. P. den Hartog & H. Peters , pp. 129-133, Wiley, New York (and Chapman and Hall, London), 1939.
11.
[11] Zubov, L. M.Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, Springer, Berlin , 1997.
12.
[12] Shield, R. T.The rotation associated with large strains . SIAM Journal of Applied Mathematics, 25, 483-491 (1973).
13.
[13] Naghdi, P M. and Vongsampigoon, L.Small strain accompanied by moderate rotation . Archive for Rational Mechanics and Analysis, 80, 263-294 (1982).
14.
[14] Truesdell, C. and Toupin, R. A.The classical field theories. Handbuch der Physik, III/1, ed. S. Flügge , pp. 226-793, Springer, Berlin , 1960.
15.
[15] Fosdick, R. L.Remarks on compatibility. Modern Developments in the Mechanics of Continua, ed. S. Eskinazi , 109-127, Academic Press, New York , 1966.
16.
[16] Blume, J. A.Compatibility conditions for a left Cauchy-Green strain field . Journal of Elasticity21, 271-308 (1989).
17.
[17] Cosserat, E. and F. Sur la théorie de l'éasticité. Annales de la Faculte des Sciences de l'Universite de Toulouse, 10, 1-116 (1896).
18.
[18] Kosinski, W.Field Singularities and Wave Analysis in Continuum Mechanics, Ellis Horwood, New York (also Wiley, and PWN-Polish Scientific Publishers), 1986.
19.
[19] Forsyth, A. R.Theory of Functions of a Complex variable, Vol. 1, 3rd edn., Cambridge University Press, Cambridge , 1918 (also Dover, New York, 1965).
20.
[20] Lamb, H. : Hydrodynamics, 6th edn., Cambridge University Press, Cambridge , 1932 (also Dover, New York, 1945).
