When considering multi-physics applications described by hyperbolic models, flow regimes are not characterized by one Mach number only. An example is given by the simulation of elastic materials where in addition to the standard acoustic Mach number, a shear Mach number depending on the shear modulus, describing the elastic shear stiffness of the material, can be defined. Consequently, not only one but several scales are present in the model which correspond to different magnitudes in the Mach number and introduce numerical challenges with respect to stability, resolution and efficiency of the numerical solver. The stability of standard explicit finite volume schemes requires a CFL condition that depends on the largest characteristic speed which leads to vanishing time steps in weakly compressible flows and very rigid materials. Therefore, to avoid vanishing time steps for such applications, implicit or implicit-explicit time integrators are necessary. Moreover, the monitoring of fast sound and shear waves is usually less in the focus of a numerical simulation than the usually slower material contact waves. Focusing only on material waves yields a less restrictive, Mach number independent CFL condition, which allows for large time steps and is thus advantageous when slow dynamics are observed over a long time. In this talk we address the construction of a semi-implicit scheme for systems of hyperbolic conservation laws at the example of non-linear elasticity in an Eulerian framework. The basis of the scheme is given by a relaxation approach introduced by Jin & Xin allowing for an efficient and easily implementable finite volume scheme that captures accurately material waves throughout all Mach number regimes.