  Violation of linearity of the King plot is investigated for a chain of
partially stripped argon isotopes. The nonlinearity originates within the
Standard Model from subtle contributions to the isotope shifts from
next-to-leading order effects, which have never been systematically studied so
far. In light atoms these nonlinear effects are dominated by the quadratic
nuclear recoil ($\propto 1/M^2$ where $M$ is the nuclear mass). Large-scale
relativistic calculations of the linear and quadratic mass shift and the field
shift are performed for the $2P$ fine-structure transitions in Be-like, B-like,
and C-like argon ions. Nonlinearities of the King plots from 5 to 30 kHz are
found, which is four orders of magnitude larger than previous estimates in
comparable systems. Accurate calculations of these effects are vital for
identification of possible nonlinearities originating from physics beyond the
Standard Model.
