טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentKatzir Zachi
SubjectA Simple Formula for Dynamic Spherical Cavity Expansion
in Compressible Elastic-Perfectly Plastic
Material with Large Deformations
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Emeritus Miles Rubin
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

In this thesis a simple analytical solution is developed for the problem of constant velocity expansion of a spherical cavity in an infinite compressible elastic- perfectly plastic media that experiences large deformations.  The main goal of the thesis is to formulate a solution for constant spherical cavity expansion which can describe both the transient response of shock loading as well as the asymptotic response of large deformation cavity expansion.

The solution is obtained by using the standard formulation for an incompressible material with a modified value of the shear modulus and with a physical boundary condition on the radial stress at the location of the precursor wave associated with compressible response. Expressions are obtained for the kinematics and the stress field as functions of the radial coordinate and time. In particular, an expression is developed for the contact pressure applied to a spherical cavity which is expanding at constant velocity.

The long time asymptotic solution of the contact pressure is investigated. Examples are used to compare the predictions of this model with full transient numerical simulations, as well as with other analytical results of similarity solutions and with numerical results for asymptotic response. These comparisons indicate that the proposed modified incompressible solution predicts reasonably accurate results for the asymptotic value of the contact pressure for a large range of velocities compared with the predictions of the standard incompressible model. However, it still over-predicts the asymptotic values of the contact pressure for high expansion velocities.

Finally, the transient response of the modified incompressible solution is investigated. It is shown that when the media is assumed to be elastic, the modified incompressible solution predicts the correct value of the shock contact pressure at the cavity surface.  However, the increase in contact pressure with time is not as fast as that of the exact solution.  For an elastic-plastic material, it is shown that the modified incompressible solution predicts reasonably accurate response for both the transient shock decay as well as the transition to the asymptotic response associated with large expansion of the cavity.