A method for calculating the axisymmetric response of a two-layered half-space under dynamic concentrated loading

The transient response of a linearly elastic structure made of a layer overlaying a half-space, subjected to a normal point force on its free surface, is investigated. Both welded contact and frictionless sliding are considered at the interface. This paper presents a method to calculate the response, for a wide range of loading durations (T), on the axis of symmetry of the configuration (in the layer and the half-space). Equations of the boundary–value problem are manipulated in an integral transform domain and the Cagniard–de Hoop method is used. The final form of the exact analytical solution is a sum of contributions corresponding to the rays of the generalized ray theory; little computational effort need be developed for evaluating each contribution. While this theory has only been used to obtain early-time responses, long-time responses—up to 30 times the transit time of P-waves in the layer—have been calculated for this study. This work was conducted to help characterize the stress transmitted in the human lung (half-space) when the thoracic wall (layer) is subjected to a non-penetrating impact. Depending on T, multiple reflections of waves in the layer or typical low-frequency response are observed. The influence of the contact condition with respect to T is elucidated.