<p>Minkowski sum has long been studied in robot configuration space, contact detection and motion planning. A parametric expression of its boundary is derived in closed-form recently, when two bodies are convex and enclosed by smooth surfaces. It is applicable only when globally linear transformations are applied. However, when the two bodies are subject to locally nonlinear deformations, such expressions will not be valid. This paper proposes GPDMink, a new characterization of Minkowski sum of two convex bodies enclosed by smooth surfaces that are nonlinearly deformable. A closed-form parametric expression of the Minkowski sum boundary is derived when one of the bodies is under nonlinear deformations. An optimized characterization of Minkowski sum boundary is then proposed when both bodies are nonlinearly deformed. The case studies are presented using superquadrics that are subject to concatenations of tapering deformation and linear transformations, which demonstrate the correctness, optimality and efficiency of the proposed method. As an important application for multibody systems, like robot manipulators, a contact detection method that efficiently searches for optimal gradient parameters is studied based on the proposed closed-form Minkowski sum expression. The contact result is then used as a submodule in sampling-based motion planners, allowing robot manipulators to avoid obstacles that are densely placed in the environment.</p>