In this work we propose an anisotropic stored energy function which satisfies a priori the Legendre–Hadamard condition, which is strongly related to the material stability of the constitutive equations. In the linearized case this condition implies positive wave speeds. The Legendre–Hadamard condition plays also an important role for the (local) existence of solutions in the neighborhood of stationary points. We apply the proposed hyperelastic energies to soft tissues and compare the formulation with existing models which have been used for the calculation of medial collateral ligament and arterial walls. In our numerical and analytical investigations we discuss the distribution of wave speeds for a sequence of deformation states containing some essential stress–strain characteristics of the compared models.