We study the Γ-limit of Ambrosio-Tortorelli-type functionals Dε(u, v), whose dependence on the symmetrised gradient e(u) is different in Au and in e(u) − Au, for a C-elliptic symmetric operator A, in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano in [43]. In particular we prove that G(S)BD functions with bounded A-variation are (S)BD.