Mathematical modeling of stationary thermoelastic state in a half plane containing an inclusion and a crack due to local heating by a heat flux
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Date
2020-01-01
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Видавництво Львівської політехніки
Lviv Politechnic Publishing House
Lviv Politechnic Publishing House
Abstract
Розглянуто двовимiрнi стацiонарнi задачi теплопровiдностi та термопружностi для
напiвнескiнченного пружного тiла, що мiстить включення та трiщину. Для цього побудовано математичнi моделi цих двовимiрних задач у виглядi системи сингулярних
iнтегральних рiвнянь (СIР) першого та другого роду. Числовий розв’язок системи iнтегральних рiвнянь одержано методом механiчних квадратур у разi пружної пiвплощини, що локально нагрiвається тепловим потоком i мiстить кругове виключення та
теплоiзольовану прямолiнiйну трiщину. Отримано графiчнi залежностi коефiцiєнтiв
iнтенсивностi напружень (КIН), якi характеризують розподiл iнтенсивностi напруженнь у вершинах трiщини, залежно вiд пружних та термопружних характеристик
включення та матрицi, вiд вiдносного положення трiщини та включення. Отриманi
результати використанi для визначення критичних значень теплового потоку, за якого трiщина починає рости. Ця модель є розвитком вiдомих моделей двовимiрних стацiонарних задач теплопровiдностi та термопружностi для кусково-однорiдних тiл з
трiщинами.
The two-dimensional stationary problems of heat conduction and thermoelasticity for a semi-infinite elastic body containing an inclusion and a crack are considered. For this purpose, mathematical models of these two-dimensional problems in the form of a system of singular integral equations (SIEs) of the first and the second kinds are constructed. The numerical solution of the system of integral equations in the case of a half plane containing an inclusion and thermally insulated crack due to local heating by a heat flux is obtained using the method of mechanical quadratures. We present graphical dependencies of stress intensity factors (SIFs), which characterize the distribution of intensity of stresses on the tops of a crack, on the elastic and thermoelastic characteristics of an inclusion and a matrix, as well as on a relative position of a crack and an inclusion. The obtained results are subsequently used to determine the critical values of a heat flux at which a crack starts to grow. This model is the development of known models of two-dimensional stationary problems of heat conduction and thermoelasticity for piecewise-homogeneous bodies with cracks.
The two-dimensional stationary problems of heat conduction and thermoelasticity for a semi-infinite elastic body containing an inclusion and a crack are considered. For this purpose, mathematical models of these two-dimensional problems in the form of a system of singular integral equations (SIEs) of the first and the second kinds are constructed. The numerical solution of the system of integral equations in the case of a half plane containing an inclusion and thermally insulated crack due to local heating by a heat flux is obtained using the method of mechanical quadratures. We present graphical dependencies of stress intensity factors (SIFs), which characterize the distribution of intensity of stresses on the tops of a crack, on the elastic and thermoelastic characteristics of an inclusion and a matrix, as well as on a relative position of a crack and an inclusion. The obtained results are subsequently used to determine the critical values of a heat flux at which a crack starts to grow. This model is the development of known models of two-dimensional stationary problems of heat conduction and thermoelasticity for piecewise-homogeneous bodies with cracks.
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Keywords
термопружнiсть, коефiцiєнт iнтенсивностi напружень, сингулярне iнтегральне рiвняння, включення, теплопровiднiсть, трiщина, тепловий потiк, stress intensity factor, singular integral equation, inclusion, heat conduction, thermoelasticity, crack, heat flux
Citation
Zelenyak V. M. Mathematical modeling of stationary thermoelastic state in a half plane containing an inclusion and a crack due to local heating by a heat flux / Zelenyak V. M. // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2020. — Vol 7. — No 1. — P. 88–95.