In this work, we consider a non linear viscoelastic constitutive model: it involves both non linear elasticity as well as non linear viscous behaviour. The main objective of this model is to reproduce the(γGand)and (γD) curves synthesizing the dependence of both the shear modulus and damping on the excitation level. To do so, the non linear elastic part of the model is described by a hyperbolic law. The non linear viscous part combines a NCQ for linear damping (generalized Maxwell body, Moczo et Kristek, 2005) and a non linear viscous contribution described by a hyperbolic variation with the strain level. Furthermore, this model complies with the thermodynamic principles of continuum mechanics (e.g. derivation from potentials and dissipation function). Starting from this mechanical formulation, the analysis of 1D strong motion amplification is then performed thanks to a discretization by the finite element method. Various numerical issues have been carefully considered to describe the whole algorithmic procedure (for both elastic and viscous non linear components of the constitutive law). Detailed validations of the model have shown its ability to recover low amplitude ground motion response. For larger excitation levels, since the model includes the main features of soil non linear behaviour, the analysis of seismic wave propagation in soil layers leads to interesting results : at the free-surface the amplification peaks are shifted to lower frequency values (when compared to the input motion); higher frequency components are not overdamped as for the equivalent linear model; the amplification level is generally lower.