Energetic and dynamic properties of grain boundaries play vital roles in the mechanical and plastic behaviors of polycrystalline materials. In this thesis, continuum models are developed incorporating the underlying discrete dislocation structures of grain boundaries, and simulations are performed to study the energetic and dynamic properties of both low angle and high angle grain boundaries.

In the first part, we develop a continuum model to compute the energy of low angle grain boundaries for any given degrees of freedom based on a continuum dislocation structure. Numerical method is developed to minimize the grain boundary energy associated with the dislocation structure subject to the constraint of Frank's formula for dislocations with all possible Burgers vectors. Comparisons with atomistic simulations show excellent agreements between the results of the two models. We use our continuum model to systematically study the energy of low angle grain boundaries in fcc Al.

In the second part, we derive a continuum model for the dynamics of low angle grain boundaries with any shape in two dimensions. The dynamics of grain boundaries is driven by both the long-range elastic energy field and local surface energy effect, and incorporates both the motion of curved grain boundary and the dislocation structure evolution on the grain boundary.
These evolutions of the grain boundary and its dislocation structure lead to shrinking of the grain boundary and grain rotation by both the coupling and sliding motions. The change of the shape of the grain boundary is naturally accounted for in our continuum model. Simulations are performed for circular and non-circular grain boundaries with different dislocation structures.

In the third part, we present a continuum equation of motion for grain boundaries derived from the underlying discrete disconnection mechanism. The model applies generally to grain boundaries with any misorientation angle. Simulations based on the continuum equation are performed for the relaxation of perturbed grain boundaries and grain boundary motion under applied shear stress with pinned junction points. We also present an equation of motion for the junctions where multiple grain boundaries meet in polycrystals. The resulting equation of motion naturally exhibits junction drag -- a widely observed phenomena in junction dynamics in solids and liquids.