  A continuum model of the two dimensional low angle grain boundary motion and
the dislocation structure evolution on the grain boundaries has been developed
in Ref. [48]. The model is based on the motion and reaction of the constituent
dislocations of the grain boundaries. The long-range elastic interaction
between dislocations is included in the continuum model, and it maintains a
stable dislocation structure described by the Frank's formula for grain
boundaries. In this paper, we develop a new continuum model for the coupling
and sliding motions of grain boundaries that avoids the time-consuming
calculation of the long-range elastic interaction. In this model, the
long-range elastic interaction is replaced by a constraint of the Frank's
formula. The constrained evolution problem in our new continuum model is
further solved by using the projection method. Effects of the coupling and
sliding motions in our new continuum model and relationship with the classical
motion by curvature model are discussed. The continuum model is validated by
comparisons with discrete dislocation dynamics model and the early continuum
model [48] in which the long-range dislocation interaction is explicitly
calculated.
