Layer structures around an edge dislocation in a smectic phase under shear are studied with both phase field and order parameter models. It is shown that, contrast to a crystal solid, the conventional picture of the Peach-Koehler force experienced by dislocations when the sample is under a shear stress cannot be readily applied to the smectic phases. Under a uniform shear flow, we obtain the phase field and order parameter solutions around an edge dislocation. The solutions elucidate properties such as the layer distortion range around the dislocation and scaling of inter-dislocation interaction on dislocation separation. Calculations on energy dissipation indicate the extreme shear-thinning behavior that an edge dislocation induces a shear stress independent of the shear rate. Finally in a bulk sample with dislocation forming loops and networks, we argue that the uniform flow component around the dislocation is important to the energy dissipation and we show that its scaling exponent with the shear rate is very close to results from many previous rheology measurements.
