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Research Papers: Applications

A Fast Approximate Method for Heat Conduction in an Inhomogeneous Half-Space Subjected to Frictional Heating

[+] Author and Article Information
Xiujiang Shi

School of Mechatronics Engineering,
Harbin Institute of Technology,
Harbin 150001, Heilongjiang, China;
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

Liqin Wang

School of Mechatronics Engineering,
Harbin Institute of Technology,
Harbin 150001, Heilongjiang, China

Qinghua Zhou

School of Aeronautics and Astronautics,
Sichuan University,
Chengdu 610065, Sichuan, China

Qian Wang

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received August 25, 2017; final manuscript received November 18, 2017; published online February 9, 2018. Assoc. Editor: Sinan Muftu.

J. Tribol 140(4), 041101 (Feb 09, 2018) (13 pages) Paper No: TRIB-17-1332; doi: 10.1115/1.4038953 History: Received August 25, 2017; Revised November 18, 2017

This paper reports a new three-dimensional model for heat conduction in a half-space containing inhomogeneities, applicable to frictional heat transfer, together with a novel combined algorithm of the equivalent inclusion method (EIM) and the imaging inclusion approach for building this model. The influence coefficients (ICs) for temperature and heat flux are obtained via converting the frequency response function (FRF) and integrating Green's function. The model solution is based on the discrete convolution and fast Fourier transform (DC-FFT) algorithm using the ICs, convenient for solving problems involving multiple elliptical inhomogeneities with arbitrary orientations. A group of parametric studies are conducted for understanding the thermal fields in the inhomogeneous half-space due to surface frictional heating, influenced by the properties of the inhomogeneity, its depth, and orientation.

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Figures

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Fig. 1

Conduction of surface heating to an inhomogeneous half-space: (a) Half space containing an elliptical inhomogeneity is subjected to surface frictional heating and (b) equivalent inclusion and eigentemperature gradient ∇Ti*

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Fig. 2

Decomposition of the equivalent inclusion model for half-space heat conduction: (a) Equivalent inclusion model subjected to a distributed surface heat flux, (b) a homogeneous half-space subject to the surface heat flux, and (c) disturbance due to the inhomogeneity, which is converted to an equivalent inclusion and eigentemperature gradient ∇Ti*

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Fig. 3

Image inclusion approach for solving the inhomogeneous half-space heat conduction problem: (a) Half space with an inclusion, (b) a full space with the inclusion, and (c) mirror of (b). (d) Homogeneous half-space with the surface heat flux due to solutions (b) and (c).

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Fig. 4

Flowchart showing the basic scheme for solving frictional heat transfer in an inhomogeneous half-space

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Fig. 5

Three-dimensional heat conduction in a half-space containing a spherical inhomogeneity, caused by a distributed Hertzian-type heat flux. This problem was used to validate the developed method.

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Fig. 6

2D axisymmetric abaqus model for the same problem shown in Fig. 5: (a) The problem and the bottom boundary condition and (b) loading and mesh discretization

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Fig. 7

Temperature contours in the xoz plane by the current method (a) and FEM (b), for km = 0.025 W/(mm °C), qf = 1.0, D* = 100μm, H* = 100 μm

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Fig. 8

Temperature profiles analyzed by the new method and the FEM, for qf = 1.0, D* = 100 μm: (a) H*/D* = 0.7 and (b) H*/D* = 1.75

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Fig. 9

Average relative errors of the temperatures solved for cases with different inhomogeneity depth/diameter ratios by the new method and the FEM, for qf = 1.0, D* = 100 μm: (a) k*/km = 1/4 and (b) k*/km = 4

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Fig. 10

Temperature profiles for cases with a spherical inhomogeneity of different diameters, H* = 100 μm: (a) D* = 60 μm, (b) D* = 80 μm, and (c) D* = 100 μm

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Fig. 11

Surface temperature profiles along the x axis for the cases of different inhomogeneity diameters for qf = 1.0, H* = 90 μm: (a) k*/km = 1/4 and (b) k*/km = 4

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Fig. 12

Surface temperature profiles along the x-axis for the inhomogeneity at different depths for qf = 1.0, D* = 100 μm: (a) k*/km = 1/4 and (b) k*/km = 4

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Fig. 13

Maximum surface temperature change ratio as a function of H*/D* for qf = 1.0, D* = 100 μm: (a) k*/km = ¼ and (b) k*/km = 4

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Fig. 14

Temperature profiles along the x-axis affected by the inhomogeneity conductivity ratio for qf = 1.0, H* = 90 μm, and D* = 60 μm

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Fig. 15

Surface temperature changing ratio as a function of inhomogeneity/matrix conductivity ratio for qf = 1.0, H* = 90 μm, and D* = 60 μm

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Fig. 16

Euler angles for an ellipsoidal inhomogeneity, for α=0 deg, β=90 deg: (a) Euler angles and coordinate transformation, (b) γ=0 deg, (c) γ=45 deg, and (d) γ=90 deg

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Fig. 17

Effect of inhomogeneity orientation on the temperature distribution: (a) temperature profiles along the z axis, for k*/km=4, H*=50 μm, a=25 μm, b=12.5μm, c=12.5μm, α=0 deg, β=90 deg; temperature contours, (b) γ=0 deg, (c) γ=45 deg, and (d) γ=90 deg

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Fig. 18

Temperature profiles disturbed by multiple inhomogeneities, for H*=40μm, D*=40μm: (a) three spherical inhomogeneities along the z-axis, perpendicular to the surface, dz=60μm, (b) three spherical inhomogeneities parallel to the x-axis, dx=60μm, (c) temperature profiles along the z-axis for case a, and (d) temperature profiles along the z axis for case b

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Fig. 19

Discretization of the computation region: (a) the xoy plane and (b) the xoz plane

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