This thesis presents a consistent modeling approach to the phenomenon of deformation localization and material failure, based on high-order finite elements. Strong discontinuities are incorporated into the finite element formulation in an embedded manner. As opposed to previous concepts, the presented approach is consistently deduced regarding its possible application in the context of high-order finite elements. Put forth by a novel reassessment of the strong discontinuity kinematics, an extended p-adaptive formulation is established. Three-dimensional numerical investigations show that the proposed approach facilitates a minimization of potential locking effects. Compared to a corresponding low-order approach, a significantly enhanced numerical resolution of the failure process can be observed.