Non-linear effects in seismic waves in high-energy earthquakes: A two-dimensional analysis for non-homogeneous isotropic media with a view towards the study of the 2009 L’Aquila earthquake

It is usually accepted in geophysics (and in civil engineering) that linear models can be used for describing an earthquake and the consequent seismic waves’ propagation. However, the large deforma- tion experienced by the soil in these situations suggests that such a paradigm should be considered in a more critical way. In fact, we claim that, in the vicinity of some discontinuities (that are common in all the geophysical applications of continuum models), the corresponding strain concentrations make the hypothesis of small deformation to be inadequate. In this paper we verify the inappropriateness of the linear paradigm in a simple but reasonable case, with a view to a future application of this study to the effects of the 2009 L’Aquila earthquake. To this aim, we start with an analysis which is restricted to a two- dimensional body (i) with an inhomogeneity that resembles the Aterno River Valley, central Italy and (ii) with a non-linearity that is the most simple one, choosing the strain energy to be quadratic in the non-linear measures of deformation. More precisely, we consider a 2D piecewise homogeneous domain and a material that is viscoelastic isotropic and geometrically non-linear. We apply, to the bottom of such a domain, a seismic excitation and calculate the differences in the response between the linear and the geometrically non-linear cases. Using a suitably designed numerical model, we prove that, as conjectured, not only such differences are, indeed, originated in the vicinity of the pre-defined geometrical discontinuities but also that they propagate in the rest of the domain. Moreover we find numerically predictions of the frequency ratios and ground acceleration time dependence and amplitude that produce, in the case of non-linear mod- els, predictions which are closer to experimental evidence than those obtained using the corresponding linear model.