Summary
It is shown that the Orowan-type kinematic relations as well as the Schmid yield criterion can be derived basing oneself on a formula defining the mean curvature of glide surfaces for a principal congruence of edge dislocation lines accompanied with a particular distribution of secondary point defects. Moreover, it appears that this mean curvature has the physical meaning of a mesoscopic material parameter defining a relation between the evolution of the dislocation state and plastic deformation. It is pointed out that the existence of Orowan-type relations puts kinematic constraints, dependent on the isometry group of glide surfaces, on the dislocation density tensor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Frank, F. C., Steeds, J. W.: Crystal dislocations. In: The physics of metals 2. Defects (Hirsch, P. B. ed.), pp. 43–97. London: Cambridge University Press 1975.
Hull, D., Bacon, D. J.: Introduction to dislocations. Oxford: Pergamon Press 1984.
Fridman, J. B.: Mechanical properties of metals, vol. I Moscow: “Maszinostroenie” 1974 (in Russian).
Eisenhart, L. P.: Riemannian geometry. Princeton: Princeton University Press 1964.
Orlov, A. N.: Introduction to the theory of defects in crystals. Moscow: “Vyscaja skola” 1983 (in Russian).
Bilby, B. A., Bullough, R., Garder, L. R., Smith, E.: Continuous distributions of dislocations. IV Single glide and plane strain. Proc. Roy. Soc.244, 538–557 (1958).
Christian, J. W.: Transformation in metals and alloys, Part I. Oxford: Pergamon Press 1975.
Yang, W., Lee, W. B.: Mesoplasticity and its applications. Berlin: Springer 1993.
Kondo, K.: Non-Riemannian geometry of imperfected crystals from a macroscopic viewpoint. RAAG Memoirs1, 6–17 (1955).
Kröner, E.: The continuized crystal — a bridge between micro- and macromechanics? ZAMM66, T284-T292 (1986).
Kröner, E.: The differential geometry of elementary point and line defects in Bravais Crystals. Int. J. Theor. Phys.29, 1219–1237 (1990).
Trzęsowski, A.: Geometrical and physical gauging in the theory of dislocations. Rep. Math. Phys.32, 71–98 (1993).
Trzęsowski, A.: Dislocations and internal length measurement in continuized crystals, Part I and II. Int. J. Theor. Phys.33, 931–966 (1994).
Kadić, A., Edelen, D. G. B.: A gauge theory of dislocations and disclinations. Berlin: Springer 1983.
Turski, Ł.: Variational principle for equilibrium and incompatibility equations in dislocation theory. Bull. Acad. Polon. Sci., Serie Sci. Techn.XXVII, 289–294 (1966).
Trzęsowski, A.: On the geometry of diffusion processes in continuized dislocated crystals. Fortschr. Phys.43, 565–584 (1995).
Oding, I. A.: The theory of dislocations in metals. Warsaw: PWT 1961, (in Polish).
Trzęsowski, A.: Kinematics of edge dislocations, Part I and II. Int. J. Theor. Phys.36, 2877–2911 (1997).
Friedman, A.: Isometric embedding of Riemannian manifolds into Euclidean space. Rev. Mod. Phys.37, 201–203 (1965).
Trzęsowski, A.: On the isotropy of continuized dislocated crystals, Part I and II. Int. J. Theor. Phys.36, 177–208 (1997).
Schouten, J. A.: Ricci-calculus. Berlin: Springer 1954.
Sikorski, R.: Introduction to differential geometry. Warsaw: PWN 1972 (in Polish).
Von Westenholz, C.: Differential forms in mathematical physics. Amsterdam: North-Holland 1972.
Hicks, N. J.: Notes on differential geometry. Toronto: Van Nostrand 1965.
Trzęsowski, A.: Geometry of crystal structure with defects, Part. I. Int. J. Theor. Phys.26 311–333 (1987).
Perzyna, P.: Thermodynamics of inelastic materials. Warsaw: PWN 1978, (in Polish).
Gairola, B. K. D.: Gauge theory of dislocations. In: Continuum models of discrete systems, Proceedings of CMDS-7, (Anthony, K.-H., Wagner, H.-J., eds.), pp. 579–590. Switzerland: Trans Tech Publications 1993.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Trzęsowski, A. On the geometric origin of Orowan-type kinematic relations and the Schmid yield criterion. Acta Mechanica 141, 173–192 (2000). https://doi.org/10.1007/BF01268676
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01268676