Solution of certain variational problems of thermal resilience for thin shells considering the selection of optimum conditions for localized heat treatment

One of the possible ways of stating and solving the selection problem for optimum temperature fields for localized axisymmetric heating of shells is investigated. The minimum of shell elastic energy is taken as the optimization criterion. An infinite cylindrical shell was considered in a similar formulation in [1], The corresponding variational problem is formulated for the functional of elastic energy with additional limitations imposed on the function of twist angle at specified shell sections. The variational problem is reduced to an isoperimetric by the use of singular functionals of the -function kind. The related Euler equation is obtained, and this together with the problem resolvent equation constitute a complete set of equations for determining the extremum temperature field with related stress-strain state of the shell. Cylindrical, conical, and spherical shells are considered separately. A numerical analysis is made for the simplest conditions of localized heating of cylindrical and conical shells.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 47–54, July–August, 1968.