In this paper we present a mixed finite element formulation for modeling the mechanical behavior of the cardiac tissue. The modeling may be divided into a passive and an active part. The passive properties are in this work described by a transversely isotropic, hyperelastic material model. Moreover, the myocardium is assumed to be almost incompressible. In advanced models the active contraction is strongly connected to electro-physiological processes in the muscle cells. In this paper we simplify the active stress contribution by modeling it as a linear function in time. Due to the fact that the heart muscle undergoes large deformations during contraction, both material and geometrical non-linearities must be taken into account in the analysis. Numerical results are presented for different activation patterns applied to a box-shaped test specimen, which represents a piece of the myocardial wall. The proposed mixed method offers reasonable results in the modeling of the mechanical behavior of the test specimen.