Preface -- 1 Introduction -- 2 Basic concepts of the linear theory of elasticity -- 2.1 Stresses -- 2.2 Linearstrains -- 2.3 Constitutive relations -- 2.4 Boundary value problems -- 2.4.1 Static statements -- 2.4.2 Dynamic problems -- 2.5 Simplified models -- 2.5.1 Elastic rods and strings -- 2.5.2 Beam models -- 2.5.3 Membranes -- 2.5.4 Plane stress and strain states -- 3 Conventional variational principles -- 3.1 Classical variational approaches -- 3.1.1 Energy relations -- 3.1.2 Direct principles -- 3.1.3 Complementary principles -- 3.2 Variational principles in dynamics -- 3.3 Generalized variational principles -- 3.3.1 Relations among variational principles -- 3.3.2 Semi-inverse approach -- 3.4 Finite dimensional discretization -- 3.4.1 Ritz method -- 3.4.2 Galerkin method -- 3.4.3 Finite element method -- 3.4.4 Boundary element method -- 4 The method of integrodifferential relations -- 4.1 Basic ideas -- 4.1.1 Analytical solutions in linear elasticity -- 4.1.2 Integral formulation of Hooke's law -- 4.2 Family of quadratic functionals -- 4.3 Ritz method in the MIDR -- 4.3.1 Algorithm of polynomial approximations -- 4.3.2 2D clamped plate - static case -- 4.4 2D natural vibrations -- 4.4.1 Eigenvalue problem -- 4.4.2 Free vibrations of circular and elliptic membranes -- 5 Variational properties of the integrodifferential statements -- 5.1 Variational principles for quadratic functionals -- 5.2 Relations with the conventional principles -- 5.3 Bilateral energy estimates -- 5.4 Body on an elastic foundation -- 5.4.1 Variational principle for the energy error functional -- 5.4.2 Bilateral estimates -- 6 Advance finite element technique -- 6.1 Piecewise polynomial approximations -- 6.2 Smooth polynomial splains -- 6.2.1 Argyris triangle -- 6.2.2 Stiffness matrix for the Argyris triangle.

6.2.3 C2 approximations for a triangle element -- 6.3 Finite element technique in linear elasticity problems -- 6.4 Mesh adaptation and mesh refinement -- 7 Semi-discretization and variational technique -- 7.1 Reduction of PDE system to ODEs -- 7.1.1 Beam-oriented notation -- 7.1.2 Semi-discretization in the displacements -- 7.1.3 Semi-discretization in the stresses -- 7.2 Analysis of beam stress-strain state -- 7.3 2D elastic beam vibrations -- 8 An asymptotic approach -- 8.1 Classical variational approach -- 8.2 Integrodifferential approach -- 8.2.1 Basic ideas of asymptotic approximations -- 8.2.2 Beam equations - general case of loading -- 8.3 Elastic beam vibrations -- 8.3.1 Statement of an eigenvalue problem -- 8.3.2 Longitudinal vibrations -- 8.3.3 Lateral vibrations -- 8.4 3D static problem -- 9 A projection approach -- 9.1 Projection formulation of linear elasticity problems -- 9.2 Projections vs. variations and asymptotics -- 10 3D static beam modeling -- 10.1 Projection algorithms -- 10.2 Cantilever beam with the triangular cross section -- 10.3 Projection beam model -- 10.4 Characteristics of a beam with the triangular cross section -- 11 3D beam vibrations -- 11.1 Integral projections in eigenvalue problems -- 11.2 Natural vibrations of a beam with the triangular cross section -- 11.3 Forced vibrations of a beam with the triangular cross section -- A Vectors and tensors -- B Sobolev spaces -- Bibliography -- Index.