The equations of motion for hyperelastic materials are hyperbolic, if the specific energy is a rank-one convex function of the deformation gradient. This condition is not easy (almost impossible) to check even in the case of isotropic elastic materials, where the specific energy depends only on the invariants of the right or left Cauchy-Green deformation tensor. We will consider the Eulerian formulation of the hyperelasticity for isotropic solids. These equations are invariant under rotation. The consequence of that are immediate : for hyperbolicity it is sufficient to consider only 1D case. Indeed, the normal characteristic direction can always be transformed by rotation to the one of Cartesian basis vectors (we have to use three composed rotations defined by the Euler angles between the Cartesian basis and a natural local basis on characteristic surface). So, the problem to assure the hyperbolicity of the one-dimensional system for arbitrary strains and shears becomes the basic one. This 1D problem stays complex because the number of unknowns involved in such a formulation is large (14 scalar partial differential equations). For a specific energy in separable form (it is the sum of the hydrodynamic energy depending only on the entropy and the density, and the shear energy which does not depend on the density), and under classical hypotheses about the pressure behavior (the hydrodynamic sound speed should be positive), we reduce the problem of hyperbolicity to a simpler one : show that a symmetric 3x3 matrix is positive definite on a one-parameter family of unit-determinant deformation gradient compact surfaces. Some explicit forms of the stored energy are formulated which guarantee the hyperbolicity of equations for the motion of hyperelastic materials. This talk is partially based on the paper [1]. Numerical applications to the high velocity impact problems are also discussed. References 1. S. Ndanou, N. Favrie and S. L. Gavrilyuk, 2014 Criterion of Hyperbolicity in Hyperelasticity in the Case of the Stored Energy in Separable Form, J. Elasticity, v . 115, p1-25. (joint work with N. Favrie and S. Ndanou)