A Fourier integral approach to an aeolotropic medium

• Main Author: Quinlan, Patrick Michael
• Format: Others
• Published: 1949
• Online Access: https://thesis.library.caltech.edu/1388/1/Quinlan_pm_1949.pdf; Quinlan, Patrick Michael (1949) A Fourier integral approach to an aeolotropic medium. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CTQV-V186. https://resolver.caltech.edu/CaltechETD:etd-04142009-143250 <https://resolver.caltech.edu/CaltechETD:etd-04142009-143250>

Chapter I: The equations of equilibrium in terms of the displacement components for an axially symmetric aeolotropic medium are developed from the strain-energy function of the medium. Then follows a discussion of the literature of the subject, and an outline of the scope of the present thesis. Charter II: The solution is carried through using Fourier Integral technique for the two dimensional plane strain case. Stresses and displacements are obtained for a concentrated line load. Chapter III: The results of Chapter II are applied to determine the surface settlements, vertical pressures, and shears for a symmetrically loaded strip called "the unit strip" of width two units. The following special load distributions are investigated: concentrated, uniform, parabolic, inverted parabolic, hollow wall, and rigid wall. Extension is then made to a strip of any arbitrary width 2a, and settlements are obtained by means of influence factors, (Graph I). An examination is made of the influence of the type of load distribution, demonstrating St. Venant's principle of equipollent loads. Chapter IV: The equations of Chapter I are solved for an axially symmetric loading by transforming to polar co-ordinates and using Fourier-Bessel Integral technique. The solution is carried through for the concentrated load case, and the results check those given by Mitchell (6). Chapter V: An investigation similar to that made in Chapter III is made for a loaded circular area of unit radius. The results are then extended to a circle of any arbitrary width a. Surface settlements are obtained quickly by means of influence factors (Graph II). In the latter part of the Chapter series expansions are obtained for the stresses and displacements at any point in the mass, and application is made to some of the more practical load distributions. Chapter VI: Corresponding results for an elastic isotropic medium, to those given in above chapters, are obtained by the application of a limiting technique to above results. The ease with which the results are obtained is striking. A discussion is given of the infinite surface displacements that are usually obtained in two-dimensional problems. Chapter VII: In this chapter a review is made of the literature of the three constant medium. The physical significance of the assumptions and the measure of fulfillment of these assumptions by some types of wood, and by some crystals, is examined. Some errors are noted, and corrected. Finally all are shown to be just particular cases of the medium of Chapter II, without having the redeeming feature of simplicity over the more general theory. Chapter VIII: Results for Orthotropic plates are deduced from those given in Chapter II by a change of constants. Chapter IX: Typical problems in soil mechanics connected with a loaded column, and with a loaded wall, are worked out in detail. Graph III shows for a particular case the effect aeolotropy may have on the vertical stress distributions in a loaded soil. A brief outline is made of some other problems in an aeolotropic medium capable of solution by the methods of this thesis. Appendix F: Practical methods are given for the determination of the required constants. The value of skew samples is shown. The results obtained in this thesis for an aeolotropic medium, apart from the concentrated case given by Mitchell(6), are new. A good test of the accuracy of the work is provided by the known isotropic elastic results obtained by a limiting procedure in Chapter VI. As far as the author is aware, some of the results of Chapter VI are new also. The direct application of Fourier Integral technique to the displacement equations of equilibrium is very rare in elastic problems. This thesis illustrates the power and simplicity of such an approach. Finally, as shown in Chapter IX the results are very readily adapted to practical use.