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Continuum damage mechanics has been proven to be a valuable tool for the determination of material deterioration of composites. Within this framework, a continuum damage model for determination of effective response of fiber reinforced composites is presented. It is assumed that damage evolves on planes with normal orthogonal to the fiber direction and thus is capable of describing the matrix dominated failure of composites. The approach is based on a tensorial representation of damage and on a coordinate-free determination of thermodynamic and dissipative potentials. To simplify the expression of the thermodynamic potential, certain damage tensor components are used to describe the initial material anisotropy. Furthermore, dissipative potentials are formulated which are assumed to be isotropic functions of the thermodynamic forces associated with damage tensors. Parameters for the CDM model are determined using a microscale homogenization method, in which a representative cell of the periodic microstructure consists of fibers, matrix, and two crack fields which are parallel to the fiber direction and evolve independently. The assumption of two independent crack fields in an initially transversely isotropic media makes the material behaviour monoclinic. A representative cell is discretized by the finite element method and the CDM model is fitted by the least squares method to the results obtained. It is concluded from the obtained results that the proposed damage model is capable of describing elastic properties of the damage material. Finally, the CDM model is implemented into the general purpose finite element program ABAQUS in order to capture the effective response of composites. The proposed approach is illustrated by simulating damage growth at the meso-level by calculating the response of a unidirectional laminate loaded in tension. The obtained results show that the essential aspects of the mechanical behaviour of a continuous fiber composite loaded in tension can be captured by the present damage model.