In Chap.  2 , we have established the displacement representation theorem within the framework of elastodynamics, which universally applies to how elastic bodies, as continuous media, respond to external loads. Building on this foundation, in this chapter, we will further exploring the displacement representation theorem for elastic bodies with dislocations occurring on faults. This is crucial for representing the displacement fields generated by simplified seismic source models. Special consideration is necessary because elastic bodies containing faults no longer meet the conditions of displacement being single-valued and continuous, typical of a continuous medium.