Аннотації
25.12.2017
Отримано точний розв’язок рівнянь теорії пружності для круглих плит з осесиметричним навантаженням. Розглянута задача згину круглих плит, які перебувають під дією нормально доданих сил до будь-якого закону навантаження і з будь-якими типами їх опирання. Показано, що згин круглої плити під дією осесиметричного навантаження веде до зміни температурного поля.
Получено точное решение уравнений теории упругости для круглых плит нагруженных осесимметрично. Рассмотрена задача изгиба круглых плит, находящихся под действием нормально приложенных сил к любому закону нагрузки и к любым типам их сопротивления. Показано, что изгиб круглой плиты под действием осесимметричной нагрузки ведет к появлению температурного поля.
Here are considered the problems of the theory of elasticity for bodies of rotation loaded axially symmetrically. The exact solutions of the displacement of the equations of the theory elasticity for round plates loaded axially symmetrically. It is shown that the solution of the problem of bending of circular plates under the action of an arbitrary axisymmetric load of intensity P requires the specification of a normal axial stress. To determine it, we use a method based on translational approximations of the solution of this problem, proposed by one of the authors. The solutions are found in a closed form to determine the stresses at any point of the body under consideration, which is loaded by an asymmetrically uniformly distributed load-pressure. The problem of the bending of circular plates under the action of normally applied forces is considered. The law of loading of round plates, as well as the type of their resistance, can be arbitrary, which means, any. When solving the bending problem of circular plates under the action of an arbitrary axisymmetric load, a function f is used, analogous to the thermoelastic displacement potential. For an exact solution of this problem, a polytropic, namely, non-isothermal and non-adiabatic thermodynamic process is considered. The function f must satisfy the equation for the volumetric expansion, by means of which the temperature change that arises under the action of the load and which depends on the boundary conditions of the resistance is determined. It is shown that during the deformation, the temperature of the body point changes and as a result, there is an absorption or release of heat by an elastic non-insulated body when it interacts with the surrounding medium. In particular, the compressed zone of the plate generates heat, and the stretched zone absorbs it. It is shown that when stress is removed, the stresses and temperature stresses disappear and the plate returns to the unstrained and undeformed state. The deformation process is very slow, that is, it will be thermodynamically reversible. When solving the problem of the theory of elasticity for bending of circular plates loaded asymmetrically, the formulas for stresses and displacements were obtained for the first time, and the possibility of determining the stress-strain state of any point of the rotating body in question is of great practical importance.
1. Голденблат И.И. Нелинейные проблемы теории упругости. – М.: Наука, 1969. – 336 с.2. Жермен П. Курс механики сплошных сред. – М.: Высшая школа, 1983. – 399 с.3. Ландау Л.Д. и Лившиц Е. М. Теория упругости. – М.: Наука. 1987. – Т. VII. –246 c.4. Мелан Э., Паркус Г. Термоупругие напряжения, вызываемые стационарными температурными полями. – М.: Физматгиз, 1958. – 167 с.5. Тимошенко С.П., Гудьер Дж. Теория упругости. – М.: Наука, 1979. – 560 с.6. Фен Дж. Машины, энергия, энтропия. – М.: Мир, 1986. – 336 с.
1. Holdenblat I. Nelineynye problemy teorii upruhosty (Nonlinear problems of the theory of elasticity). ‑ M .: Nauka, 1969. ‑ 336 p.2. Germain P. Kurs mechaniki sploshnyh sred (Course of mechanics of continuous media). ‑ M .: Vycsha Shkola, 1983. ‑ 399 p.3. Landau L., Livshits E. Theoriya upruhosty (Theory of elasticity). ‑ M .: Nauka. 1987. ‑ T. VII. ‑ 246 C.4. Melan E., Parkus G. Termoupruhye napryazhenia, vyzyvaemye statsyonarnymy temperaturnыmy polem (Thermoelastic stresses caused by stationary temperature fields). ‑ M.: Fyzmathyz, 1958. ‑ 167 p.5. Timoshenko S., Huder J. Teoria upruhosty (Theory of elasticity). ‑ M .: Nauka, 1979. ‑ 560 p.6. Feng J. Mashiny, energya, entropya (Machines, energy, entropy). ‑ M .: Mir, 1986. ‑ 336 p.