Piskozub Y. Structural-modular method of jump functions to study the deformation of bimaterials with cracks and physically nonlinear thin inclusions.

This dissertation work develops general mathematical models and methods for investigating mechanical fields in bodies with thin physically linear or nonlinear inhomogeneities considering possible non-ideality of contact interaction and surface effects as well as arbitrary type and regime of quasi-static loading-unloading. Based on the general relations of theory of elasticity, a system of thin physically nonlinear inclusion models is constructed. Using the principle of conjugation of continua of different dimensions, the relations of the problem of conjugation of limit values of analytical functions, and the concept of the jump function method, a structural-modular method of jump functions (SMMJF) is proposed which allows one to solve the problems by considering the nonlinear constitutional equations and contact conditions. Combined plane-antiplane problems for a piecewise homogeneous bulk with frictional slip at the contact boundary under multistep loading-unloading conditions are investigated. An incremental approach is applied to determine the a priori unknown slip zones to account for residual stresses during multistep loading-unloading by force and dislocation factors. Energy dissipation, critical load in presence of limitations on the size of slip zones are calculated. The effect of dislocation loading on a bulk with thin orthotropic microinclusion in the presence of additional surface stresses at the contact boundary is investigated within the concept of mechanics of a deformable solid. For a multilayer model of a thin interphase inhomogeneity, the "safety zones" of points of application of concentrated force factors have been studied. A method for solving the problems of deformation of thin interfacial nonlinear deformable inclusions in an arbitrary quasistatic multistep loading-unloading process was developed. This allowed us by the developed SMMJF to obtain the resulting systems of singular integral equations (SSIE) with variable coefficients-functions that arise when physical nonlinearity of the inclusion material. An incremental iteration scheme is developed and tested in detail for solving such SSIE . We made specific calculations of the main parameters of the stress strain state for different Ramberg-Osgood, elastic-plastic and other types of strain diagrams. The results of the work can be used for designing new materials with desired performance properties, predicting the deflected mode of action and optimizing the loading of bodies with thin ribbonlike inhomogeneities both in composite mechanics and in micro- and nanomechanics.