Historic introduction. Definition of variational problems, demonstration of the equivalence of the integration of a differential equation and seeking the minimum of a suitable functional.
Functionals and operators in the Hilbert space. Positive operators and their physical meaning. Energy space of positive definitive operators. Essential and natural boundary conditions of differential equations. Generalized or weak solution to the problem of the minimum of the energy functional.
Variational principles of the linear elasticity. Fundamental relations, extremes of functionals, classical variational principles. (Lagrange, Castiligliano, Reisssner, Hu-Washizu).
Application of the variational principles to the derivation of governing equations of selected loaded simple bodies.
Methods of weighted residuals and direct variational methods. Interior and boundary trial function methods. Collocation method, min-max method, least squares method, orthogonality methods. Trefftz boundary method.
Method of boundary integral equations in the linear elasticity. Betti reciprocal theorems. Fundamental solution for Laplace operator.
Green tensor. Somigliani formulas. Fundamental solution of the elastostatics. Derivation of the boundary integral equations of the mixed boundary-value problem of the elastostatics.
Numerical methods for solutions of boundary integral equations.
Solution of the problems of the fracture mechanics.
Physical and mathematical aspects of the stability problems. Stability of elastic systems, energy criterion of stability, bifurcation and limit points. Eigenvalue problem and its relation to the free vibration analysis problems and stability problems.
Nonlinear systems and stability criterions. Thermodynamic approach to stability.
Time reserve.