J. A. Fernández Merodo

Discretization Techniques for Transient, Dynamic and Cyclic Problems in Geotechnical Engineering: Second Order Equation

The second order equations are the form more widely used both in solids and in soil dynamics. This Chapter is devoted to present the Discretization techniques which are used. The solution strategy consists on (i) perform a discretization in space from which a system of ordinary differential equations (ODE) is obtained, and (ii) discretize this system in time. An important aspect is that of imposing suitable boundary conditions. Here we will present a simple technique which can be used for elastic or viscoplastic materials assuming that the wave is planar in the neighbourhood of the boundary. Finally, we will present some applications to cyclic and dynamic problems.

Mathematical Models for Transient, Dynamic and Cyclic Problems in Geotechnical Engineering

Depending on their dependence on time, loading can be classified as static or transient. Two special cases of transient loading are (i) cyclic loading and (ii) dynamic loading. While accelerations are basic in the latter, they can be neglected in some problems in the former. It is also important to notice that there exist dynamic problems which are not cyclic (propagation of a fast landslide), and cyclic problems which are not dynamic (response of a marine foundation to forces induced by tides). In the case of geomaterials, and more specifically soils, there are cases where the problem is linear, such as the small amplitude vibrations of foundations, while in others, such as liquefaction of an earth dam during an earthquake, non linearity is a key factor.