On the mechanics of crystalline solids with a continuous distribution of dislocations. (English) Zbl 1288.82055

Christodoulou, Demetrios; Kaelin, Ivo

Adv. Theor. Math. Phys. 17, No. 2, 399-477 (2013).

Summary: We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate to Lie groups. We then consider the static case, which describes crystalline bodies in equilibrium in free space. The mathematical problem in this case is the free minimization of an energy integral, and the associated Euler-Lagrange equations constitute a nonlinear elliptic system of partial differential equations. We solve the problem in the simplest cases of interest.

MSC:

82D20	Statistical mechanics of solids
82D25	Statistical mechanics of crystals
74E15	Crystalline structure
74B20	Nonlinear elasticity
35R03	PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.

Keywords:

crystalline solid; Lie group; minimization of an energy integral; Euler-Lagrange equation; affine group; Heisenberg group; Killing fields