Some new applications of the invariant integrals of mechanics

Some new examples of the use of invariant integrals, a basic example of which in fracture mechanics was proposed in a paper published by the author in JAMM in 1967 for the first time, are given. The key role of L. A. Galin in establishing this scientific technique is briefly covered. The resistance force acting on the edge of a smooth rigid punch moving over the surface of an elastic half-space is found, and this force turns out to be proportional to the indentation force to a power of 3/2. A contact problem is considered for elastic bodies covered with an inextensible flexible film, taking account of contact adhesion. Fundamental problems on the delamination of flexible elastic films and plates from an elastoplastic body, another film or a plate are investigated. A number of problems on the development of cavities under the action of a gas or liquid injected into the cavity is considered. The driving force of erosion gryphons and gryphon cracks is found. The law of interaction between small bodies floating on the surface of a heavy liquid with surface tension is obtained. In one limiting case, it is found to be similar to Ampere's law of attraction between two parallel conductors which carry a current in one direction. The relativistic interaction force for moving electric charges as well as the “gravitational-cosmological” interaction force between masses, that enabled the accelerated expansion of the Universe, the infinite density at the material at the centre of galaxies and the constancy of the orbital velocity of stars in galaxies, are indicated.