On the geometry and dynamics of crystalline continua. (English) Zbl 0916.73013

Christodoulou, Demetrios

Ann. Inst. Henri Poincaré, Phys. Théor. 69, No. 3, 335-358 (1998).

Summary: We introduce the concept of a material manifold in continuum mechanics. To describe, in the continuum limit, a crystalline solid containing an arbitrary distribution of dislocations, we endow the material manifold with a structure similar to a Lie group. Starting with a state function defined on the space of local thermodynamic equilibrium states, we formulate the dynamics of crystalline continua in accordance with the least action principle, within the framework of general relativity, in terms of a mapping of the space-time manifold into the material manifold. Besides, we include in our formulation electromagnetic effects.

MSC:

74A99	Generalities, axiomatics, foundations of continuum mechanics of solids
82D25	Statistical mechanics of crystals

Keywords:

Lie group-like structure; material manifold; dislocations; state function; space of local thermodynamic equilibrium states; least action principle; general relativity; space-time manifold; electromagnetic effects

References:

[1]	R. Balian , M. Kléman , J.P. Poirier , ” Physics of Defects ”, North Holland , Amsterdam , 1981 .
[2]	E. Kröner , ” Kontinuums Theorie der Versezungen und Eigenspannungen ” Springer , Berlin , 1958 . · Zbl 0084.40003
[3]	F.R.N. Nabarro , ” Theory of Crystal Dislocations ”, Clarendon Press , Oxford , 1967 .
[4]	F.R.N. Nabarro , ” Dislocations in Solids ”, Vol. 1 , North Holland , Amsterdam , 1979 .