Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1886110.31857/S0869-56524896585-591Research ArticleOn the refined stress analysis in the applied elasticity problems accounting of gradient effectsLomakinE. V.<p>Corresponding Member of the Russian Academy of Sciences</p>lomakin@mech.math.msu.suLurieS. A.lomakin@mech.math.msu.suRabinskiyL. N.lomakin@mech.math.msu.suSolyaevY. O.lomakin@mech.math.msu.suLomonosov Moscow State UniversityMoscow Aviation InstituteInstitute of Applied Mechanics of Russian Academy of Sciences2312201948965855912212201922122019Copyright © 2019, Russian academy of sciences2019<p class="a" style="margin-left: 0cm;"><span lang="EN-US">The paper proposes an extension of the approaches of gradient elasticity of deformable media, which consists in using the fundamental property of solutions of the gradient theory - the smoothing of singular solutions of the classical theory of elasticity, converting them into a regular class not only for the problems of micromechanics, where the length scale parameter is of the order of the materials characteristic size, but for macromechanical problems. In these problems, the length scale parameter, as a rule, can be found from the macro-experiments or numerical experiments and does note have an extremely small values. It is shown, by attracting numerical three-dimensional modeling, that even one-dimensional gradient solutions make it possible to clarify the stress distribution in the constrained zones of the body and in the area of the loads application. It is shown that additional length scale parameters of the gradient theory are related with specific boundary effects and can be associated with structural geometric parameters and loading conditions that determine the features of the classical three-dimensional solution.</span></p>second gradient elasticityBernoulli-Euler beamstress statesupport conditionslocal effectsрадиентная теория упругостибалка Бернулли-Эйлеранапряжённое состояниеусловия закреплениялокальные эффекты[Mindlin R.D. Micro-Structure in Linear Elasticity //Archive for Rational Mechanics and Analysis. 1964. V. 16. № 1. P. 51-78. Maranganti R., Sharma P. A Novel Atomistic Approach to Determine Strain-Gradient Elasticity Constants: Tabulation and Comparison for Various Metals, Semiconductors, Silica, Polymers and the (ir) Relevance for Nanotechnologies // J. of the Mechanics and Phy-sics of Solids. 2007. V. 55. № 9. P. 1823-1852.][Mousavi S.M., Aifantis E.C. Dislocation-Based Gradient Elastic Fracture Mechanics for In-Plane Analysis of Cracks // International J. Fracture. 2016. V. 202. № 1. P. 93-110.][Васильев В.В., Лурье С.А. Новое решение осесимметричной контактной задачи теории упругости // Изв. РАН. МТТ. 2017. № 5. С. 12-21.][Reiher J.C., Giorgio I., Bertram A. Finite-Element Analysis of Polyhedra Under Point and Line Forces in Second-Strain Gradient Elasticity // J. Engineering Mechanics. 2016. V. 143. № 2. P. 04016112.][Andreaus U., et al. Numerical Simulations of Classical Problems in Two-Dimensional (non) Linear Second Gradient Elasticity // Intern. J. Engineering Science. 2016. V. 108. P. 34-50.][Васильев В.В., Лурье С.А., Салов В.А. Исследование прочности пластин с трещинами на основе критерия максимальных напряжений в масштабно-зависимой обобщённой теории упругости // Физическая мезомеханика. 2018. Т. 21. № 4.][Lurie S.A., et al. Do Nanosized Rods Have Abnormal Mechanical Properties? On Some Fallacious Ideas and Direct Errors Related to the Use of the Gradient Theories for Simulation of Scale-Dependent Rods // Nanoscience and Technology: An International J. 2016. V. 7. № 4.][Ломакин Е.В. и др. Полуобратное решение задачи чистого изгиба балки в градиентной теории упругости: отсутствие масштабных эффектов // ДАН. 2018. Т. 479. № 4. С. 390-394.][Lurie S., Solyaev Y. Revisiting Bending Theories of Elastic Gradient Beams // Intern. J. Engineering Science. 2018. V. 126. P. 1-21.][Dell’Isola F., Sciarra G., Vidoli S. Generalized Hooke’s Law for Isotropic Second Gradient Materials // Proc. Royal Society A: Mathematical, Physical and Engineering Sciences. 2009. V. 465. № 2107. P. 2177-2196.]