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### Effective Characteristics of Composites with Non-linearly Elastic Components

#### Abstract

Majority of fibers used in composite production are linearly elastic in the main range of possible acting stresses. Some nonlinearity can occur at the stresses close to strength. Some polymeric fibers have wider range of non-linearity. Metallic fibers usually have significant range of non-linearity and/or plasticity. In active loading non-linearity is not distinguished from plasticity. Many polymeric binders have some non-linearity of stress-strain diagrams at tension, at compression, and especially at shear. It contributes to the effective nonlinearity of stress-strain diagrams, especially for angle-plies. Metallic matrices have some non-linearity and plasticity. Modeling of fracture of composites based on linearly-elastic theory might be not completely adequate. Development of non-linear models was done mostly for the metal-matrix composites. It was based mainly on applications of methods of theory of plasticity. The model presented in the report was initially developed by the first author at the beginning of seventies. Report is devoted to more advanced version of the model. There are several difficulties to overcome. Using condition that the strains in the constituents of composite and effective strains of composite are much smaller than unity, it is possible to describe diagrams in form of power series of strains. However, the typical shape of diagrams makes it inconvenient because it is necessary to keep many terms to prevent diagrams from decreasing in practical range of strains. Keeping many terms means too many constants have to be found experimentally. Power series in stresses is more convenient, because for practical purposes it is enough to keep linear and cubic terms. The role of quadratic term is to describe differences of diagrams of tension and compression. Inverse dependence can contain many terms, which can be recalculated from few initial characteristics. Another difficulty is related to complex stress state and necessity to use tensor form describing nonlinear dependence of each component of strain on each component of stress. The mixed invariant of stress tensor and tensor of the fourth rank characterized the medium was used as an additional argument converting linear law to nonlinear one. Approximate solutions of boundary problems for non-linear equations are used.