Annotation of the results obtained in 2022
As a result of studying the phenomenon of the relief formation on the surface of a nanomaterial, a generalization of the Asaro-Tiller-Grinfeld model has been proposed that allows considering the variation of the surface energy due to material deformation. The new method resulted in the analysis of the effect of the elastic constants, which characterize the elastic behavior of the surface layer, on the process of reorganization of surface atoms leading to the formation of nanoscale surface structures. The main driving force of such transformations is a variation of the chemical potential. Due to the reduced stability of the surface atomic layers, it leads to the diffusion of atoms along the surface from regions with a high value of the chemical potential to the regions with a low value. In our study, the change in the chemical potential is associated with the change in the stress field along the perturbed surface. To analyze the stresses, we developed the method based on the constitutive equations of the Gurtin-Murdoch model of surface elasticity in which the surface layer of a solid is considered as a separate substructure with mechanical properties different from the bulk material. In the frame of the research concerning the void formation and growth in multiply twinned particles the theoretical models of a void evolution in pentagonal wires and icosahedral particles are suggested to predict the pore kinetics in the dependence on the following parameters such as the elastic moduli of the particle material, the initial pore and particle radii, vacancy relaxation volume and specific surface energy. These models are based on the new analytical solutions of the boundary-value problems of vacancy diffusion induced by both the Gibbs-Thompson surface tension and the stress state of disclination defects (wedge disclination and Marks-Ioffe stereo-disclination) in hollow cylindrical and spherical bodies. The model of an elastic solid under the plane strain, containing the defect like a nearly circular cylindrical channel incorporating the surface stress and surface tension is constructed. Based on the boundary perturbation technique, the solution of the problem has been reduced to the singular integro-differential equation in the complex displacement for any order approximation. The stress fields obtained in the zero-order approximation by the original Gurtin-Murdoch surface elasticity model and its known simplified versions have been compared. The inaccuracy of each simplification is demonstrated. The same comparative analysis has been performed in the solution of the problem on an interaction of the periodic system of edge dislocations with the free plane boundary under plane strain incorporating surface stresses and surface tension. The stress fields at the boundary have been depicted depending on the distance between dislocations and the boundary using surface elasticity models under consideration. The stress field in the vicinity of a circular nanohole in a plate of a nanometer thickness obtained by the solution of the modified Kirsch problem with the original Gurtin-Murdoch model has been analyzed. The numerical results have been obtained in the form of graphic dependences of stresses on polar coordinates under the uniaxial tension of the plate. It is shown that allowing for the surface tension at the free surface of the hole and the faces of the plate essentially influence on the stress field in the vicinity of the nanohole and its surface. The Ritz method is developed for finding the critical buckling load of a stretched plate of a nanometer thickness with a circular hole, in which the obtained stress field is realized, taking into account the surface tension at the surface of the hole and on the faces of the plate in the transverse direction. The program was written in the Matlab environment. Using this program, the analysis of the results obtained was carried out depending on the parameters of the problem.

 

Publications

---

Annotation of the results obtained in 2023
In this study, a mathematical model of a heteroepitaxial material has been developed to allow for the influence of geometric and physical-mechanical properties of an interfacial layer on the distribution of elastic stresses along the nanostructured interface between two bulk materials under plane strain conditions in accordance with the Gurtin–Murdoch model of surface/interfacial elasticity. Based on the boundary perturbation technique, we derived a solution to the boundary value problem, resulting in the obtaining of asymptotic representations for the stress and strain tensor components at any desired level of approximation. The stress state near the nanostructured interphase of an isotropic heteroepitaxial system has been studied using the first-order approximation. The effects of the perturbation wavelength, interfacial stiffness, relative stiffness of bulk phases, residual interfacial and misfit stresses have been analyzed. A comparative assessment has been conducted, juxtaposing the obtained solution with the outcomes derived from the simplified Gurtin–Murdoch model in earlier studies. A further development of the approach to studying the formation of surface irregularities in the stressed solids under the influence of diffusion processes was proposed for an arbitrary surface relief. The analysis involves examining the stress distribution along the surface of a complex-shape which forms due to the loss of the morphological stability predicted by the extended model. To consider the impact of the film thickness and elastic properties of the substrate on the formation of the surface relief in a heteroepitaxial film coating under the influence of the surface diffusion, the approach based on the utilization of the superposition method was proposed. This method enables to determine the stressed state of the film near the perturbed free surface through a linear combination of previously obtained solutions for a homogeneous solid and a heterogeneous layered material. Applying the established methodology, we investigated how the geometric and physical-mechanical parameters of the film-substrate system impact the formation of the nanosized relief on the film surface. The theoretical research concerning the void evolution in pentagonal nanowires and icosahedral particles due to relaxation of the residual stress has been proceeded in the second stage of the project. The residual stress induced by the five-fold symmetry axes (restricted by the classical crystallography) can be described within the disclination concept. During particle growth, the elastic energy stored by the residual stress is increased until it relaxes through the void formation. The proposed energetic model of void evolution in pentagonal particles (multiply twinned particles) allows anticipating void evolution scenarios in dependence of initial void radius and material parameters (elastic constants, surface tensile and disclination strength). Besides, the critical and optimal parameters of voids in pentagonal particles are determined. Continuous models of the nonlinear buckling of nanorod and nanoplate by the Steigmann-Ogden theory of surface elasticity have been constructed. Based on the Euler–Bernoulli kinematic hypothesis and Timoshenko hypothesis, the constitutive relations for the internal forces and moments under the nonlinear buckling of the nanorod have been derived. New effective models and rigidities were introduced for the description of these relations. The dynamic equations for the nonlinear buckling of the nanorod have been derived as well. Two static problems on buckling of a nanofiber under the dead weight and free (from both sides) and rigid (from one side - cantilever) supports have been solved. The size effect has been revealed, i.e., the surface stresses increase essentially the nanofiber rigid with decreasing the nanofiber thickness to the nanometer size. Comparative analysis of applying two beam theories considered and Gurtin-Murdoch and Steigmann-Ogden surface elasticity models has been carried out. It has been discovered that the Timoshenko beam theory and Steigmann-Ogden surface elasticity soften the size effect. The solution of the problem on the stress-strain state of an elastic body containing a nearly circular hole of a nanometer size under the plane strain conditions has been first obtained. The solution is based on the complete Gurtin-Murdoch surface elasticity model and the boundary perturbation method. The algorithm of the solution is adopted for any approximation and any continually differential function describing the boundary of the hole. Obtained results allow estimating influence of an error of the shape deviation from the circular one on the stress state and improving accuracy as compared with simplified surface elasticity models. The stress field in the Kirsch problem for a plate of a nanometer thickness with a circular hole incorporating surface stresses at the boundary of the hole and on the faces of the plate has been investigated based on the complete Gurtin-Murdoch surface elasticity model. As follows from the analytical solution, the elastic stress field depends on the thickness. The difference between modified and classic (not allowing for the surface effects) problems decreases with the distance from the hole boundary. Besides, numerical studies show that the stress field approaches the Kirsch solution when the hole radius increases. The solution of the problem on an interaction of an edge dislocations row with the flat surface of an elastic body incorporating the stretching and bending rigidity of the surface material according to the Steigmann-Ogden surface elasticity model has been constructed. The comparison of the stress fields at the surface, obtained by the Steigmann-Ogden, Gurtin-Murdoch and all simplified models, using actively in the literature, has been carried out. It was found that maximum values of longitudinal stresses decrease, and normal ones increase due to allowing for the bending rigidity of the surface. An influence of the surface bending rigidity on the image force acting on each dislocation has been studied as well. The results obtained by the Gurtin-Murdoch and Steigmann-Ogden models show that allowing for the surface bending rigidity leads to the decrease of the image force regardless of the Burgers vector direction and the relative distance between neighboring dislocations. It was established that the solution can be used in the case of an isolated dislocation if the distance between neighboring dislocations is two orders of magnitude greater than the distance of the dislocations to the surface.

 

Publications

---