The theory of elasticity is concerned with the mechanics of deformable media which, after the removal of the forces producing deformation, completely recover their original shape and give up all the work expended in the deformation.The first attempts to develop the theory of elasticity on the basis of the concept of a continuous medium, which enables one to ignore its molecular structure and describe macroscopic phenomena by the methods of mathematical analysis, date back to the first half of the eighteenth century.The fundamental contribution to the classical theory was made by R. Hooke, C. L. M. H. Navier, A. L. Cauchy, G. Lame, G. Green, B. P. E. Clapeyron. In 1678 Hooke established a law linearly con- necting stresses and strains.After Navier established the basic equations in 1821 and Cauchy developed the theory of stress and strain, of great importance in the development of elasticity theory were the investigations of B. de Saint Venant. In his classical work on the theory of torsion and bending Saint Venant gave the solution of the problems of torsion and bending of prismatic bars on the basis of the general equations of the theory of elasticity. In these investigations Saint Venant devised a semi-inverse method for the solution of elasticity problems, formulated the famous Saint Venant's principle, which enables one to obtain the solution of elasticity problems. Since then much effort has been made to develop the theory of elasticity and its applications, a number of general theorems have been proved, the general methods for the integration of differential equations of equilibrium and motion have been proposed, many special problems of fundamental interest have been solved. The development of new fields of engineering demands deeper and more extensive studies of the theory of elasticity. High velocities call for the formulation and solution of complex vibrational problems. Lightweight metallic structures draw particular attention to the question of elastic stability. The concentration of stress entails dangerous consequences, which cannot safely be ignored.