In summary, this thesis focuses on developing an a priori theory for geometric finite element discretizations of a Cosserat rod model, which is derived from incompatible elasticity. This theory will be supported by corresponding numerical experiments to validate the convergence behavior of the proposed method. The main result describes the qualitative behavior of intrinsic H1-errors and L2-errors in terms of the mesh diameter 0 < h ≪ 1 of the approximation scheme. Geometric Finite Element functions uh with its subclasses Geodesic Finite Elements and Projection- based Finite Elements as conforming path-independent and objective discretizations of Cosserat rod configurations were used. Existence, regularity, variational bounds and vector field transport estimates of the Cosserat rod model were derived to ob- tain an intrinsic a-priori theory. In the second part, this thesis concerns the derivation of the Cosserat rod from 3D elasticity featuring prestress together with numerical experiments for microheterogeneous prestressed materials.