State equations of the draph model of three-dimensional elastic solids in cartesian coordinates

The paper considers the numerical analysis method of mechanical fields in deformable bodies based on the graph model of an elastic medium in the form of the directed graph. According to the applied method the elastic medium along the coordinate planes is divided into separate elements. In line with this notion we have established an elementary cell configuration, a subgraph of the element by installing hypothetical meters on the solid’s element. The derivation of the cell equations which is based on the element conversion into the cell relies on an invariant. The deformation energy is the invariant. The whole body’s graph is built following the same rule as in the elementary cell. Apart from the opportunity to present the system configuration and mutual connections of its variables, the graphs allow to conduct complex mathematical transformations deliberately in the frameworks of the definite algorithm. The matrices which present several structural elements of the graph and the equations which describe the elementary cells both contribute to deriving the constitutive equations of the intact body. There are matrices that set such structural elements of the graph as cycles, paths and chords. The graph of a whole body is built by following the same rule as in the elementary cell. The method is based on transforming the generalized coordinates of a solid body separated into pieces to a system of generalized coordinates of the initial solid body. The nonsingular and mutually inverse matrices do this transform. The specific nature of the graph model lies in the possibility to construct these matrices with no need for their numerical inversion. The derivation of the defining system of equations is based on the use of vertex and contour of Kirchhoff’s laws for graphs, and the properties of the constructed square transformation matrices. The potentials of the graph method are illustrated by solving a test example.