  We calculate the rates of inelastic collisions for ultracold alkali-metal
atoms in radiofrequency-dressed traps, using coupled-channel scattering
calculations on accurate potential energy surfaces. We identify an rf-induced
loss mechanism that does not exist in the absence of rf radiation. This
mechanism is not suppressed by a centrifugal barrier in the outgoing channel,
and can be much faster than spin relaxation, which is centrifugally suppressed.
We explore the dependence of the rf-induced loss rate on singlet and triplet
scattering lengths, hyperfine splittings and the strength of the rf field. We
interpret the results in terms of an adiabatic model of the collision dynamics,
and calculate the corresponding nonadiabatic couplings. The loss rate can vary
by 10 orders of magnitude as a function of singlet and triplet scattering
lengths. $^{87}$Rb is a special case, where several factors combine to reduce
rf-induced losses; as a result, they are slow compared to spin-relaxation
losses. For most other alkali-metal pairs, rf-induced losses are expected to be
much faster and may dominate.
