This paper deals with the pure bending of thin and thick metallic sheets under plane strain condition. Its main objective is to construct an efficient, robust, incremental numerical tool that can be used to simulate the bending process and predict the sheets’ bendability. In this investigation, the material constitutive law describing the mechanical behavior of the sheet is defined by an extended Gurson model. This model generalizes Gurson's original one to account for both plastic anisotropy and mixed (isotropic and kinematic) hardening of the matrix. In this model, only the growth phase of voids was considered (without taking into account nucleation). Here, the extended model was coupled with the kinematics of the pure bending process and with the equilibrium equations, and then implemented in an incremental numerical tool in order to predict the bendability of the sheets studied. This bendability was then verified using two different criteria. Part I of this paper focuses on the constitutive law, fulfillment of the force equilibrium, the thickness evolution and the shift of the neutral fiber, together with a comparison and discussion of the choices made by other authors regarding these parameters. The validation of the model developed in this paper will be carried out in Part II through simulations of the pure bending process.