We initiate a study of homeomorphisms f with constant principal
strains (cps) between smoothly bounded planar domains D, D′.
An initial result shows that in order
for there to be such a mapping of a given Jordan domain D onto
D′, a certain condition of an isoperimetric nature must be
satisfied by the latter. Thereafter, we establish the fundamental
fact that principal strain lines (characteristics) of such
mappings necessarily have well-defined tangents where they meet
∂D. Using this, we obtain information about the boundary
values of the Jacobian transformation of f, and finally we
determine the class of all cps-homeomorphisms of a half-plane
onto itself.