Modeling of One Inclusion in the Infinite Peristatic Matrix Subjected to Homogeneous Remote Loading

Abstract

A basic problem of micromechanics is analysis of one inclusion in the infinite matrix subjected to a homogeneous remote loading. A heterogeneous medium with the bond-based peridynamic properties (see Silling, J Mech Phys Solids 48:175-209, 2000) of constituents is considered. At first, volumetric boundary conditions are set up at the external boundary of a final domain obtained from the original infinite domain by truncation. One also considers a periodization approach (for a dilute concentration of inclusion). At last, a group of the iteration methods is considered where the displacement field is decomposed as linear displacement corresponding to the homogeneous loading of the infinite homogeneous medium and a perturbation field introduced by one inclusion. This perturbation field is found by the iteration method for entirely infinite sample with an initial approximation given by a driving term which has a compact support. The form of the mentioned solutions is adapted for subsequent incorporation into one or another micromechanical approach for peristatic composite materials. The methods are demonstrated by numerical examples for 1D case. A convergence of numerical results for the peristatic composite bar to the corresponding exact evaluation for the local elastic theory is shown.

English teaching in distant education policy development: Ukrainian aspect

“…The classes in Second Life are similar to face-to-face training and include a bulletin board, text reading, grammar, vocabulary, pronunciation, and then the platform can be used as an environment for the implementation of virtual excursions and field trips to meet with students of other nationalities, which makes the virtual environment a full-fledged platform for language learning (Buryachenko, 2019). Teachers, as a rule, have the opportunity to integrate a number of different resources into the virtual environment: electronic textbooks, video, audio, YouTube channels, electronic teaching materials, podcasts, multimedia software for language learning (Buryachenko, 2019). Thus, the Second Life virtual environment is a place to practice the language contains 125 lessons to help learn vocabulary without prior knowledge.…”

“…Voloshinov et al (2020), consider the fundamentals and prospects of distance learning in higher educational institutions of Ukraine. Buryachenko (2019), considers the use of a synchronous 3D virtual environment for the possibility of distance learning a foreign language as one of the conditions for the high-quality training of future specialists for professional activity; Kachmarchyk et al (2019) describe the structure of an e-learning course in a foreign language based on the MOODLE 3 platform (Ginaya et al, 2018).…”

Variational principles and generalized Hill’s bounds in micromechanics of linear peridynamic random structure composites

“…which is a non-local counterpart of (2 2 ) mapping a vector space σ ∈ R n into η ∈ R n . This construction will be performed by a slight modification of the iteration method proposed by Buryachenko [39].…”

“…This means that the non-zero elements of the vectorb p is limited by the indices |p| ≤ d(N v + N v ) where N v and N v are the numbers of nodes in the inclusion v and the external part of interaction interface v l \ v, respectively. The last established feature of the right-hand side of (50) allows us to exploit the iterative methods (see [39] for details), which are especially effective for analyses of infinite banded matrices. The popular iterative methods pass from one iteration to the next by modifying one or a few components of an approximate vector solution at a time.…”

“…In each inner iteration we apply the SOR to solve the linear system (59) under a convergence criterion specified in [39]. When the following convergence criterion is satisfied, we also stop the outer iterative sequence, and then the solution of u 1 is obtained.…”