Graph model of three-dimensional elastic solids in cartesian coordinates

The theory of graphs represents an unsophisticated section of mathematics with a wide range of applications. It is based on the simple ideas and elements such as points and lines. The theory of graphs builds a rich diversity of forms from that, providing efficient tools for construction of models and means of solution to a wide range of problems. The method of a numerical analysis of the mechanical fields in the deformable body, based on the graph model of an elastic medium in the form of the directed graph, is considered. According to the method applied the elastic medium along coordinate planes divides into separate elements. In line with this notion we establish an elementary cell configuration, a subgraph of an element, by installing hypothetical meters on an element of a solid. Derivation of cell equations, which is based on conversion of an element to a cell, relies on an invariant. We use the deformation energy as the invariant. A procedure to determine parameters of the elementary cell is described. The graph of a whole body is built following the same rule as in an elementary cell. With the use of a unit cell having 24 degrees of freedom, the strain field is approximated by linear polynomials (with corresponds to approximated of the displacement fields by quadratic polynomials). The standard finite-element method requires 60 degrees of freedom (elements with 20 nodes) for the same purpose. The proposed graphical approach thus reduces the number of equations that describe the model. Kirchhoff’s laws (apex and contour) realized in the analyzer are shown to correspond to equations of equilibrium and strain compatibility in the elastic body The equations are of no use when determining the stress-strained state of the body in the explicit form with its model.