The invariant integral of physical mesomechanics as the foundation of mathematical physics: Some applications to cosmology, electrodynamics, mechanics and geophysics

Phe general invariant integral based on the energy conservation law is introduced into physical mesomechanics, with taking into account the cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter. Phe physical mesomechanics thus becomes a mega-mechanics embracing most of the scales of nature. Some basic laws following from the general invariant integral are indicated, including Coulomb’s law of electricity generalized for moving electric charges, Newton’s law of gravitation generalized for coupled gravitational/cosmic field, the Archimedes’ law of buoyancy generalized for bodies partially submerged in water, and others. Using the invariant integral the temperature track behind moving cracks and dislocations is found out, and the coupling of elastic and thermal energies is set up in fracturing and plastic flow, namely for opening mode cracks and edge dislocations. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding invariant integrals. As applied to the horizontal drilling and hydrofracturing of boreholes in the Earth’ crust, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a diskshape fracture issuing the borehole, are derived in the fluid extraction regime. A theory of fracking in shale gas/oil reservoirs is suggested for three basic regimes of the drill mud permeation into the multiply fractured rock region, with calculating the shape and volume of this region in terms of the geometry parameters and pressures of rock, drill mud and shale gas. Phe method of functional equations in the theory of a complex variable and the boundary layer method are also used to solve these problems.