Herein, we propose new 8-node hexahedral, 6-node prismatic, and 5-node pyramidal 3D solid finite elements enriched by cover functions. To resolve the linear dependence problem, piecewise linear shape functions are adopted as the partition of unity functions. Various mesh patterns are considered for investigation of the linear dependence problem, and no rank deficiency is observed. We demonstrate convergence behavior and effectiveness of the new enriched elements, and adaptive use of cover functions through six problems: an ad hoc problem, a tool jig problem, a slender beam problem, a gear problem, and a connecting rod problem. (C) 2018 Elsevier Ltd. All rights reserved.