Determination of the midsurface of a deformed shell from prescribed fields of surface strains and bendings

We show how to determine the midsurface of a deformed thin shell from the following set of data: known geometry of the undeformed midsurface, the surface strains and the surface bendings. It is assumed that the two latter fields had been obtained beforehand by solving a problem posed for the so-called intrinsic field equations of the non-linear theory of thin shells. Two different methods of determining the deformed midsurface in space are worked out: (a) directly from its first and second fundamental form using some results from mathematical analysis; (b) integrating the system of first-order PDEs for the surface deformation gradient. In both cases the corresponding integrability conditions are discussed; it is shown that they are equivalent to the compatibility conditions of the non-linear theory of thin shells.