In this work the control of underactuated mechanical systems with dry friction on actuated and unactuated joints is investigated. A new interconnection-and-damping-assignment passivity-based-control (IDA-PBC) design is presented, which includes the adaptive estimation of the friction forces and the introduction of a nonlinear dissipative term in the closed-loop system dynamics. As a result, the traditional IDA-PBC is complemented with an additional matching condition and the control law is augmented with a new term that accounts for the Coulomb friction forces on all joints. Two adaptive control paradigms are considered for comparison purposes and stability conditions are discussed. The control design is detailed for two demonstrative examples: the disk-on-disk system; the Acrobot system. The effectiveness of the proposed design is demonstrated with numerical simulations.