Components made of short glass fiber reinforced (SGFR) thermoplastics are increasingly used in the automotive industry, and more frequently subjected to fatigue loadings during their service life. The determination of a predictive fatigue criterion is therefore a serious issue for the designers, and requires the knowledge of the local mechanical response under a large range of environmental conditions (temperature and relative humidity). As the cyclic behavior of polymeric material is reckoned to be highly nonlinear, even at room temperature, an accurate constitutive model is a preliminary step for confident fatigue design. The injection molding process induces a complex fiber orientation distribution (FOD), which affects both the mechanical response and the fatigue life of SGFR thermoplastics. This paper presents an extension of the constitutive behavior proposed by the authors in a previous work [Launay et al., Int J Plasticity, 2011], in order to take into account the influence of the local FOD on overall anisotropic elastic and viscoplastic properties. The proposed model is written in a general 3D anisotropic framework, and is validated on tensile samples with various FOD and loading histories: monotonic tensions, creep and/or relaxation steps, cyclic loadings. In Part II of this paper [Launay et al., Int J Fatigue, 2012], this constitutive model will be applied to the simulation of different fatigue samples subjected to multiaxial cyclic loadings.