Nonlinear mathematical model of relation of second-rank tensors for composite materials

Analysis of the deformation processes of both long-known and new polymer, composite and synthetic materials used in building structures, parts of apparatuses, machines, as well as power plants revealed their specific properties. It is established that many similar materials have orthotropy of the structure with simultaneous manifestation of deformation anisotropy or heterogeneity. Induced deformation anisotropy or mechanical inhomogeneity is caused by the dependence of stiffness and strength characteristics on the type of stress state. In previous works of the authors, it has been shown that traditional models of deformation of such materials and their mathematical representations lead to gross errors that are clearly manifested in the calculation of various structures. At the same time, the theories of deformation of composite materials with "complicated properties  specially developed for them by other authors in the last 40 years, are very contradictory and have insurmountable disadvantages. The authors of the presented work have previously developed nonlinear energy relations of strain and stress tensors, for determining the constants of which a wide range of experiments is recommended. However, among the experimental tests, it is necessary to involve experiments on complex stress states, many of which are currently practically unrealizable. Therefore, in 2021, a quasi-linear deformation potential was postulated, represented in the main axes of orthotropy of materials. For this option, it turned out to be sufficient to calculate constants according to the simplest experiments. Despite the undoubted advantages of this potential, nevertheless, real nonlinear diagrams were approximated by direct rays using the least squares method, and this, with qualitative adequacy, led to quantitative errors. In this regard, the presented article attempts to avoid the general rules for the formulation of a complete nonlinear potential relationship of strain and stress tensors. In this direction, a nonlinear mathematical model of the connection of two second-rank tensors is postulated, combining the form of the generalized Hooke's law for orthotropic material, the theory of small elastic-plastic deformations and the tensor space technique of normalized stresses. This approach allowed us to determine nonlinear material functions, limiting ourselves to a set of traditional simplest experiments. A remark is made about the uniqueness of solutions to boundary value problems, which boils down to checking the stability of the equations of state in the small Drucker. Within the framework of the proposed mathematical model, widely known experimental diagrams for a carbon-graphite composite are processed, for which nonlinear material functions are obtained.