ISSN 3041-1815. Physicochemical Mechanics of Materials. 2026.
Volume 62, Issue 4
Construction of partial solutions of equations of the theory of elasticity taking into account the action of volume forces
Revenko V. P. 
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine, Lviv, Ukraine. (https://ror.org/05dry3213)
Keywords
three-dimensional body, volume forces, partial solutions of Navier’s equations, volume solution, cylinder, disk.
Cite as
Revenko V. P. Construction of partial solutions of equations of the theory of elasticity taking into account the action of volume forces. Physicochemical Mechanics of Materials. 2026. 62(4), 132-138.
Abstract
A linear model of the theory of elasticity is considered, taking into account volume forces. The volume force potential satisfies the three-dimensional Laplace equation. A general solution of the system of Navier’s differential equations is given as the sum of the homogeneous and partial solutions. New physically justified partial solutions are constructed in the Cartesian coordinate system, when the potential is a three-dimensional harmonic function. Volumetric deformations and stresses are recorded. It is shown that there is a linear relationship between the first invariants of a purely volume stress state and the volume force potential. A purely volume partial solution is found in a cylindrical coordinate system in the axisymmetric case. Analytical formulas for expressing purely volume deformations and stresses are given. A practical problem of rotation of an isotropic cylinder around its axis is considered when there is no volume force acting in the direction of the axis of rotation. Its partial solution is constructed, the boundary conditions are written as well as the way to satisfy them is proposed. For a thin disk, the boundary condition is solved and the distribution of displacements and stresses is determined. The permissible angular velocity of disk rotation is calculated.