The paper deals with the analysis of the rank-one, quasi-convex, poly-convex, and convex envelopes of functions \(W(F)\) defined on \(m\times n\) matrices \(F\) of the form \(\varphi(q(F))\), where \(q\) is a quadratic form. Under suitable rank-one compatibility conditions on the quadratic form, these envelopes are computed in some special cases. A detailed computation of the envelopes for the James–Ericksen energy \[ \kappa_1(c_{11}+c_{22}-2)^2+\kappa_2c_{12}^2+\kappa_3 \biggr(\bigr({c_{11}-c_{22}\over 2}\bigr)^2-\varepsilon^2\biggr)^2 \] (where \(n=m=2\), \(C=F^TF\) and \(\varepsilon\) is the strain tensor) is performed.