Contribution of Grain Boundary Pores to the High-Temperature Background of Internal Friction in Metals with Ultrafine Gr
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ribution of Grain Boundary Pores to the High-Temperature Background of Internal Friction in Metals with Ultrafine Grain V. G. Kul’kov* Moscow Power Engineering Institute (MPEI), Volzhsky Branch, Volzhsky, 404101 Russia *e-mail: [email protected] Received April 10, 2020; revised April 29, 2020; accepted May 27, 2020
Abstract—The energy of activation of background internal friction is found, based on the solution to the inhomogeneous diffusion equation of vacancies in an annular region of grain conjugation. The dependence of the internal friction logarithm on the inverse temperature is presented as a graph with two or three linear sections with different angles of inclination to the coordinate axes, testifying to the existence of different energies of activation of the background at different temperatures. DOI: 10.3103/S1062873820090221
INTRODUCTION The growing interest of researchers in porous materials is due to their having a number of interesting physical properties (e.g., low density, increased diffusion permeability, plasticity, adsorption and catalytic activity, and sound absorption [1–3]). It is known that porous materials have elevated amounts of internal friction with respect to bulk [4] and film structures [5]. In most cases, the pores have different geometric shapes ranging from long cylindrical [6] to spherical and lenticular [7], and they can be found in grain boundary regions [8, 9]. The aim of this work was to study the mechanism of internal friction in a polycrystalline material with pores on its grain boundaries.
excess vacancies. On the inner and outer circles, these vacancies extend into pores, so the excess concentration of vacancies on them vanishes. The inhomogeneous diffusion equation for vacancies in the annular region can be written as
∂C ( r , t ) ∂C ( r , t ) = D 1 ∂ r + A exp (i ωt ) . r ∂r ∂t ∂r
R
ANALYTICAL APPROACH We shall consider the cross sections of pores by a grain boundary to be circles, and the pores themselves to be of the same size, uniformly distributed across the boundary region. Let us consider one such pore. We assume the calculated region to be annular, as is shown in Fig. 1. Here, the inner circle is the cross section of a pore with radius R − ΔR, and the ring is a pore-free segment of the boundary with outer radius R, on which the neighboring grains conjugate. If we assume the pore radius to be R0 = R − ΔR, we find R from the condition of equality of the ratios of the pore-free part area to the total area for the whole boundary and for 2 the chosen region: ν = (1 − Δ ) , where Δ = ΔR R . Then R = R0 1 − Δ. A normal time-varied stress acts on the boundary. The annular region of grain conjugation is a distributed, periodically acting source of 1043
ΔR
Fig. 1. Boundary plane section of the pores. The pores are highlighted in white.
(1)
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KUL’KOV
Here, C(r, t) is an excess concentration of vacancies within the boundary, relative to the equilibrium value; D is the coefficient of the grain boundary diffusion of vacancies; r is the polar radius; and
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