Steady-State Heat Distribution in Bimaterial with an Interface Crack: Part 1
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-State Heat Distribution in Bimaterial with an Interface Crack: Part 1 A. V. Glushkoa,*, A. S. Ryabenkoa,**, and A. S. Chernikovaa,*** a
Voronezh State University, Voronezh, 394018 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
Received January 17, 2018; revised November 19, 2018; accepted February 8, 2019
Abstract—The transmission problem describing a steady-state temperature distribution in a plane consisting of two half-planes occupied by different materials with exponential internal thermal conductivities with a single finite crack along the interface is considered. The compatibility conditions for the boundary functions are formulated under which the problem has a unique classical solution. Closedform representations of the classical solution are found. The weak solution to the problem is studied without making additional assumptions, and asymptotic expansions are constructed for the weak solution and its first derivatives near the ends of the crack. Keywords: transmission problem, classical solution, boundary conditions, steady-state heat equation, crack, asymptotics DOI: 10.1134/S096554251906006X
1. INTRODUCTION Mathematical models describing the characteristics of materials with cracks [1–5] have been an issue of much interest over the last several decades. A direction in the study of such problems is associated with the investigation of thermal processes in materials with cracks [6–8]. Such models are numerous and rely heavily on the properties of the materials involved, the geometry of the domains filled with the materials, the number of cracks, and their positions. As early studies in this area, we mention [9], where a crack at the interface of a nonhomogeneous coating bonded to a homogeneous substrate was analyzed, and [10, 11], which investigate the influence of mechanical and thermal shock loading on collinear cracks in a layered half-plane with a step boundary. Among the mechanical works dealing with this subject, we note [12, 13]. The main method for analyzing a large number of models in works of these authors is based on numerical computations. Mathematical aspects of the modeling of materials with cracks were addressed in [14–19]. More recently, the temperature field and the heat flux distribution near a crack have been investigated using asymptotic methods [20–30]. Specifically, a steady-state heat distribution in functionally graded materials filling the entire plane with a single finite crack was studied in [20–24], while an unsteady heat distribution in similar geometry was addressed in [25]. The methods described in [20–24] were developed in [26–30], where a steady-state heat distribution in a functionally graded bimaterial with a finite crack was studied. This paper is the first in a series of works devoted to a problem modeling heat conduction in a plane consisting of two half-planes filled with different materials, which leads to a system of equations with classical transmission conditions. The boundary conditions are specified so that a
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