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Determination of the principal axes of elliptic cross sections in strain ellipsoids by a variational method |
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| Authors: | Mostafa E Mostafa |
| Institution: | Nuclear Materials Corporation, Cairo, Egypt |
| Abstract: | A method is presented for the determination of the shape and orientation of the ellipse on a central plane section across a triaxial ellipsoid (such as a strain ellipsoid) of known orientation and magnitude. Conditional extremal values of longitudinal strain (λ) are obtained in the plane section and these are the principal axes of the strain ellipse. In solving this variational problem, the arbitrary cosntants (two undetermined Lagrange multipliers) are found to have physical meaning in themselves as proved in a complementary geometric solution. The two roots of the first constant directly determine the magnitude of the principal axes of the strain ellipse in terms of the normal to the cross section. The two arbitrary constants together define shear strains and orientation of these axes. Strain computation uses simple, single-line equations in either frames of strained or unstrained states. |
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