טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentSosnin Galia
SubjectSelected Problems of Axisymmetrical Elasticity in Spherical
Coordinates
DepartmentDepartment of Civil and Environmental Engineering
Supervisor Professor Oren Vilnay
Full Thesis textFull thesis text - English Version


Abstract

In the present thesis, the problems of axisymmetrically- loaded solid spheres, hollow spheres, and spherical inclusion  are studied. Accurate solutions for the displacements and stresses are presented in terms of the Legendre polynomial.


 It is shown that in the case of a coated inclusion, the approximate Hashin solution is accurate only for the case of a thin coating, and only for limited ranges of solid-coating-inclusion shear modulus relationships. Various possible cases of failure are examined at the critical points inside the inclusion, at a distance from the interface, and along the interface. Failure initiation is analyzed according to the Failure Strength Theories, together with the Davidenkov-Fridman Mechanical Strength Diagram. The Diagram takes into account not only the strength characteristics of the material, but also the compound state of stresses in the three-dimensional space.


A direct solution for the displacements and stresses of a solid sphere compressed by concentrated forces is presented. This solution converges faster than the solution proposed by Lur'e for the displacements only. The case of a continuous loading is also studied.


In addition, the solution for a solid sphere compressed by concentrated forces is obtained by following Leutert's method. The solution is given in spherical coordinates and obtained without Sternberg-Rosenthal's additional  treatment for the singularity. For the first time, Sternberg-Rosenthal's claim regarding Saint-Venant's principle is checked and found to be invalid.


The solution of a hollow sphere subjected to concentrated and a continuous loading on the external boundary is obtained in the present work by applying the solution of a solid sphere under the same load conditions. Stress and displacement analyses are carried out for the cases of thin and thick shells. It is shown that in the case of thin shell, the solution supports the Love-Kirchhoff assumptions.