  We study an operation in matroid theory that allows one to transition a given
matroid into another with more bases via relaxing a \emph{stressed subset}.
This framework provides a new combinatorial characterization of the class of
split matroids. Moreover, it permits to describe an explicit matroid
subdivision of a hypersimplex, which in turn can be used to write down concrete
formulas for the evaluations of any valuative invariant on these matroids. This
shows that evaluations on split matroids depend solely on the behavior of the
invariant on tractable subclass of Schubert matroids. We address systematically
the consequences of our approach for several invariants. They include the
volume and Ehrhart polynomial of base polytopes, the Tutte polynomial,
Kazhdan--Lusztig polynomials, the Whitney numbers of the first and second kind,
spectrum polynomials and a generalization of these by Denham, chain polynomials
and Speyer's $g$-polynomials, as well as Chow rings of matroids and their
Hilbert--Poincar\'e series. The flexibility of this setting allows us to give a
unified explanation for several recent results regarding the listed invariants;
furthermore, we emphasize it as a powerful computational tool to produce
explicit data and concrete examples.
