The computational modelling of fiber-reinforced structures is of prime importance for automotive products like tires, airsprings or hoses and for biomechanical applications like arteries. In the first part of the paper, we outline the development of invariant-based constitutive models suitable for the fiber-reinforced structure within the context of the phenomenological theory. The polyconvex strain energy function of fiber composites is adopted from the literature and split into the energy of the fiber, matrix, and the fiber-matrix interaction respectively. Experimental data from the literature are used to validate the model. In the second part of the paper, the residual stress and damage effects are incorporated into the constitutive relations by exploiting an invariant-based representation and using an isotropic/anisotropic damage formulation, respectively. It is assumed that residual stress arises in fiber-matrix composites due to manufacturing processes and affects the overall lifetime. The proposed formulation is implemented into commercial finite element code ABAQUS/Standard and examined by means of several applications, showing a high robustness and reliability.