TABLE OF CONTENTS The difficulty in obtaining the analytical solutions of elastodynamic problems is also encountered when attempting to solve static problems, as for instance that of determining the stresses in a three-dimensional body induced by forces applied on small portions of it. As an example, let us consider a cylindrical rod inserted in a tensile test machine, see Figure 1.23. If a numerical simulation of the experiment is performed, by using the finite element method (regarding the finite element method see Chapter 3, section 3.4), quite enlightening results are obtained concerning the displacement field and the stress distribution, especially sufficiently near the grips. Major information can be suitably summarized as follows:
Near the loaded parts of the rod, in a portion extending no more than a fraction of the rod radius in each direction, the displacement field, and even more conspicuously the stress field, are found to vary in all directions, in connection with the detailed space distribution of the contact forces exerted by the grips. Such fields characterize the so called local response of the system rod plus grips.
In the current portion of the rod, sufficiently far from the grips, the stress field is found to reduce practically to only one non-vanishing component, namely the uniform axial stress ? zz (r, ?, z) = T/S, where cylindrical coordinates r, ?, z are used.
