Calculus of Variations and Geometric Measure Theory

R. Scala - N. Van Goethem

Dislocations at the continuum scale: functional setting and variational properties

created by scala on 27 Nov 2013
modified on 11 Dec 2014



Inserted: 27 nov 2013
Last Updated: 11 dec 2014

Year: 2013


Considering the existence of solutions to a minimum problem for dislocations in finite elasticity \cite{SVG2014a}, in the present paper we analyze the first variation of the energy at the minimum points with respect to Lipschitz variations of the lines. The appropriate functional spaces needed to describe dislocations is the first purpose of this work. The equilibrium of forces at optimality and the explicit expression of these forces is the second aim following from the preceding theoretical developments. By this procedure, one recovers the well-known Peach-K\"ohler force, and show that it must be balanced by a defect-induced configurational force. In the modeling application, in order to consider complex structures such as dislocation clusters, countable families of dislocations are represented by means of integer-valued $1$-currents at the continuum scale in the spirit of \cite{SVG2014a}.

Keywords: dislocations, Variational problems, elasticity, Peach-Kohler force, divergence free measures