Prescribing Surface Strains to change Gauss curvature

Prescribing  Surface Strains to change Gauss curvature

To change Gauss curvature K of a surface we need to strain each differential shell element by virtue of Egregium theorem ( K is invariant if strain is zero in isometry mappings).

Can someone help with a geometrical problem where imposed strains are to be defined (at differential shell element level of a right circular cylinder) to obtain a surface of revolution with Gauss curvature +1 or -1 (sphere or pseudosphere) ? Or their isometric equivalents? Cylindrical coordinates may be used.

Essentially I wish to be able to use a more accurate version of non-linear geometry extension of  Von Kármán’s equations. This is because they are not in a form to be solved readily as a pde or an ode.

Background to this is a question about such non-linear deformations I posed below in Mathoverflow/Stackexchange:

http://mathoverflow.net/questions/248409/a-flat-strip-cannot-be-twisted-isometrically

Regards

Narasimham