An Eulerian rate formulation for finite strain analysis based on the Jaumann rate of stress is proposed and is shown to be in accordance with the definition of a hypoelastic material model. Integrability conditions of the proposed hypoelastic model in the sense of Cauchy and Green elasticity are derived. A hyperelastic potential may be used to derive the model. Derived integrability conditions show that the proposed model is exactly integrable if the Jaumann rate of the Kirchhoff stress is used. Two deformation cases, rectilinear shear and closed path four-step loading, are solved. Contrary to classical hypoelasticity, results obtained do not show any sign of shear oscillation and/or elastic dissipation under simple shear motion and closed path elastic loading. The proposed model is further extended to finite strain viscoelasticity through specifying a general nonlinear viscous flow rule on the Eulerian background. Multiplicative decomposition of the deformation gradient into its elastic and inelastic parts is employed for a generalized Maxwell model and the elastic part of the left Green-Cauchy deformation tensor is consistently introduced on the Eulerian configuration. Exact deviatoric/volumetric decoupled forms for kinematic and kinetic variables are further obtained with no assumption for viscous incompressibility. The proposed finite strain viscoelastic model is then used to solve the problem of cyclic simple shear for various large shear amplitudes. Results obtained are in good agreement with reported experimental data. The proposed model is consistent with the thermodynamics of irreversible phenomena, i.e. it yields non-dissipative (reversible) stress response for purely elastic deformations and the dissipation

(irreversibility) of the model is solely determined by its viscous

(inelastic) stress response. Furthermore, the proposed model is formulated and integrated in the fixed background and no eigenvalue extraction is needed. This makes the model a good candidate for finite element implementation.