Analytical solutions for three-dimensional steady and transient heat conduction problems of a double-layer plate with a local heat source

International Journal of Heat and Mass Transfer 89 (2015) 652–666

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Analytical solutions for three-dimensional steady and transient heat conduction problems of a double-layer plate with a local heat source Hao-Jie Jiang, Hong-Liang Dai ⇑ State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China Department of Engineering Mechanics, College of Mechanical & Vehicle Engineering, Hunan University, Changsha 410082, China

a r t i c l e

i n f o

Article history: Received 30 January 2014 Received in revised form 4 May 2015 Accepted 24 May 2015

Keywords: Analytical solution Three-dimensional temperature field Double-layer plate Local heat source Steady heat conduction Transient heat conduction

a b s t r a c t In this paper, analytical solutions for three-dimensional steady and transient heat conduction problems of a double-layer plate with a local heat source are presented, the double-layer structure includes a coating layer and FGM layer. The Poisson method and layer wise (LW) approach are applied to solve the three-dimensional steady heat conduction problem. Meanwhile, to solve the three-dimensional transient heat conduction problem of the double-layer plate, the method of separation of variables (SOV) and LW approach are together applied. The aim of this research is to understand the influences of selected coating material, local heat source and structural parameters on temperature distribution of the double-layer plate, and to guide engineers designing the double-layer structures in thermal environment. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Double-layer structures are widely used in aerospace, nuclear reactors, internal combustion engines and other fields, and these structures often lie in thermal environment. Therefore, it is very important to investigate the three-dimensional heat conduction problem for the double-layer (Coating/FGM) plate. For the temperature-related problems of FGM structures, Ding et al. [1] gave a solution of dynamic thermoelastic problem for cylindrical shells. Sadowski et al. [2] carried on theoretical prediction and experimental verification of temperature distributions for FGM cylindrical plates which subjected to thermal shock. Skoczen´ [3] obtained FGM structural members via low temperature strain. Without considering energy dissipation, Mallik and Kanoria [4] dealt with the problem of thermoelastic interactions in a FGM structure due to the presence of periodically varying heat sources. By using Green’s function technique [5,6], the heat source and heat flux for FGM structures were determined. Utilizing the principle of virtual displacements, Brischetto et al. [7] analyzed a simply supported FGM rectangular plate subjected to thermo-mechanical loadings. Using the homogenization method, Shabana and Noda [8] carried on numerical evaluation of the thermomechanical ⇑ Corresponding author at: Department of Engineering Mechanics, College of Mechanical & Vehicle Engineering, Hunan University, Changsha 410082, China. Tel.: +86 731 88664011; fax: +86 731 88711911. E-mail address: [email protected] (H.-L. Dai). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.094 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

effective properties for FGM. Shen et al. [9–14] studied a series of mechanical behaviors for FGM structures in thermal environment. Feng and Jin [15] investigated the fracture behavior of FGM plate containing parallel surface cracks which subjected to a thermal shock. Based on the three-dimensional linear theory of elasticity, Li et al. [16] presented free vibration analysis of FGM rectangular plates with simply supported and clamped edges in thermal environment. Using the higher-order shear deformation plate theory, Sun and Luo [17] studied the wave propagation of an infinite FGM plate in thermal environment. By using the infinitesimal theory of magnetothermoelasticity, Dai et al. [18–20] studied thermomechanical behaviors of FGM cylindrical and spherical structures. Applying the finite difference method, Hein et al. [21] analyzed the heat conduction problem with spatially varying parameters. Assuming material properties were temperature dependent and graded in the radial direction, Malekzadeh et al. [22] presented a three-dimensional free vibration analysis of the FGM truncated conical shells placed in thermal environment. Due to nonuniform heat supply, Ootao and Ishihara [23] gave a three-dimensional solution for transient thermoelastic problem of a functionally graded rectangular plate with piecewise exponential law. For effects of coating on resistance of high temperature, Limarga et al. [24] studied high-temperature vibration damping behavior of thermal barrier coating materials. Based on theory and experiment, the tribological properties and wear behaviors [25–33] of variety of coating layers from room temperature to high temperature had been investigated. Based on infrared pyrometry combing with

H.-J. Jiang, H.-L. Dai / International Journal of Heat and Mass Transfer 89 (2015) 652–666

653

Nomenclature x, y, z a, b h, H y0 yd t i, j, k k1 , k2 , k3

Cartesian coordinates [m] plate dimension in x and y direction [m] thickness of coating layer and FGM layer [m] location of laser point in y direction [m] width of local heat source strip [m] time [s] number of half wave in x, y and z direction thermal conductivity coefficient in x, y and z direction [W/m K] F k3 heat conduction coefficients of the FGM layer at z direction [W/m K] 0 0 kim , kic , ði ¼ 1; 2; 3Þ heat conduction coefficients of metal and ceramic materials at three different directions [W/m K] T, T 0 , DT temperature, room temperature and temperature increment [K]

specific robot spray trajectories, Xia et al. [34] detected and recorded temperature evolution continuously during preheating, spraying and cooling stages. Hodhod et al. [35] determined the effect of different coating types with different thicknesses on the residual load capacities of reinforced concrete loaded column models. Using thermal spray coatings, Matthews et al. [36,37] gave the roles of microstructure in the high temperature oxidation mechanism of Cr3C2ANiCr composite coatings. Nguyen et al. [38] studied the influence of cerium oxide argon-annealed coating on the alloy oxidation behavior at 1100 °C. Hernández Rossette et al. [39] investigated the unsteady aerodynamic and aero-thermal performance of a first stage gas turbine bucket with thermal barrier coating and internal cooling configuration. By means of several material characterizations before and after annealing processes, Antonaia et al. [40] discussed the stability of WAAl2O3 coating at high temperature. For evaluating the thermal shielding efficiency of intumescent coating, Han et al. [41] deposited a cone calorimeter as heater source which coupled with a thermocouple as detector of the temperature for steel plates. Ohtsu et al. [42] investigated the effect of the heating temperature on the characteristics of the surface layer in a simple treatment process using calcium-hydroxide slurry. Zhang et al. [43] developed a computational model to predict the temperature profile over an organic coating on a metal surface. Applying with detonation-gun spray technology, Kaur et al. [44] studied Cr3C2ANiCr coating on T22 boiler steel subjected to high-temperature oxidation and oxidation-erosion environment. Yu et al. [45] showed that the gas temperature could improve the coating deposition efficiency. By a potentially simple, scalable, non-vacuum technique, Srivastava et al. [46] analyzed high temperature oxidation and corrosion behavior of Ni/NiACoAAl composite coatings. Abyzov et al. [47] developed a composite material from particles of diamond in a copper matrix of good thermal and mechanical properties. Arizmendi-Morquecho et al. [48] analyzed the high temperature behavior of recycled fly ash cenospheres which deposited thermal barrier coatings. Barshilia et al. [49] found that the multi-functional ZnO coating was reliable for high temperature photothermal conversion applications. Selvakumar and Barshilia [50] presented the state-of-the-art of the physical vapor deposited solar selective coatings. By means of scanning electron microscope [51,52], the surface topography of composite coatings were observed and analyzed. Freni et al. [53] presented novel experimental methods for verification of both hydrothermal and mechanical stabilities of adsorbent coatings. Aydin [54] gave combined effects of thermal barrier coating and blended with diesel fuel on usability of vegetable oils in diesel engines. In short, coatings of structure play great roles in improving the heat resistance.

DT c , DT F temperature increment for coating layer and FGM layer [K] DT b temperature increment at adhesive position [K] c, q specific heat [J/kg K] and density of the double-layer plate [kg/m3] cm , cc specific heat of metal and ceramic materials for FGM layer [J/kg K] qm , qc density of metal and ceramic materials for FGM layer [kg/m3] F0 heat flow absorbed by the coating layer [J] 0 00 000 f , f , f heat flux density [J/m2] f heat generation [J/m2] l, n, N arbitrary discrete layer, functionally graded index and the total number of discrete layer

On the other hand, a few researchers have reported the problems of temperature field. Barik et al. [55] studied stationary plane contact of a heat conducting FGM punch and a rigid insulated half-space. Using Legendre polynomials and Euler differential equations system, Karampour [56] obtained steady temperature distribution for the Poro-FGM spherical vessel. Based on a graded element model, Cao et al. [57] obtained fundamental solution for steady-state heat transfer in FGM. By the methods of Laplace transform and inverse Laplace theorem, Zhou et al. [58] gave exact solutions of the temperature distribution for both the FGM strip and the well stirred fluid. Based on the elasticity theory, Malekzadeh et al. [59] presented the transient analysis of rotating multi-layered FGM cylindrical shells in thermal environment. Applying with Hermitian transfinite element method, Shariyat et al. [60–62] obtained temperature-dependent behaviors for FGM structures under thermal loads. Based on the Lord-Shulman theory, Zhou et al. [63] studied the transient thermoelastic response of FGM rectangular plates. By the methods of Laplace and finite cosine transformations, Ootao and Tanigawa [64–67] studied the three-dimensional transient thermal problems exactly. Using the modified Durbin’s numerical inversion method, Keles and Conker [68] solved transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres analytically. By experimental and mixed finite element methods, Aksoylar et al. [69] gave a nonlinear transient analysis for FGM cylinder under blast loads. Jiang and Dai [70] carried out a three-dimensional thermodynamic analysis of simply supported high strength and low alloy rectangular steel plates under laser shock processing. However, as far as we know, analytical solutions of three-dimensional steady and transient heat conduction problem for a double-layer (coating/FGM) plate with local heat source has not been found in literatures. The aim of this study is to give analytical solutions of three-dimensional steady and transient heat conduction problems for a double-layer (coating/FGM) plate with local heat source. In this study, the Poisson method, LW approach and SOV method are used to solve the three-dimensional temperature field, which could guide engineers designing the double-layer structure adapt to high-temperature environment.

2. Basic formulations of the problem Consider a double-layer plate with local heat source under a three-dimensional temperature field (as shown in Fig. 1). The Cartesian coordinate system oxyz is set on the top surface ðz ¼ 0Þ of the double-layer plate, here a, b, h and H in Fig. 1 represent the double-layer structure’s length, width, thicknesses of material

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Fig. 1. Geometrical configuration of a double-layer (coating/FGM) plate with local heat source.

I and thicknesses of the total structure, respectively. In addition, y0 is the location of local heat source, yd the width of local heat source. Material I and material II represent, respectively, the coating material and FGM for the double-layer structure. 2.1. Steady heat conduction analysis The three-dimensional steady heat conduction equation of the double-layer plate in this model can be written as

k1 ðzÞ

@ 2 DTðx; y; zÞ @ 2 DTðx; y; zÞ @ 2 DTðx; y; zÞ þ k2 ðzÞ þ k3 ðzÞ 2 2 @x @y @z2

¼ f ðx; y; zÞ

0

f ðx; y; zÞ ¼ f ðx; yÞdðzÞ

ð2Þ

where dðzÞ is the Dirac delta function, and the uniform heat flux 0 distribution f ðx; yÞ is selected as the form of rectangular constant energy density

f ðx; yÞ ¼



F 0 =4ayd

at z ¼ 0;

0

others

y0 6 y 6 y0 þ yd

ð3Þ

where F 0 is the local heat flow absorbed by the coating layer. The temperature increment function of the total structure DTðx; y; zÞ can be expressed as



DTðx; y; zÞ ¼

  dDT c ðx; y; zÞ F dDT F ðx; y; zÞ ¼ k 3   dz dz z¼h z¼h

DT c ðx; y; zÞ ð0 6 z 6 hÞ DT c ðx; y; zÞ ðh < z 6 HÞ

ð4Þ

ð6bÞ

where DT b ðx; yÞ is the temperature increment at adhesive position, F

k3 and k3 are the heat conduction coefficients of the coating layer and FGM layer at z direction, respectively. The solution scheme diagram for inhomogeneous equation (1) accompany with above-mentioned temperature boundary conditions can be seen in Fig. 2. For the coating layer, assuming the temperature increments are 0 at all six faces firstly (see Fig. 2(a)). Then the first part of the temperature increment function DT 1c ðx; y; zÞ yields

ð1Þ

where k1 ðzÞ, k2 ðzÞ and k3 ðzÞ denote, respectively, the heat conduction coefficients along x, y and z directions. For the coating layer, the corresponding k1 ðzÞ, k2 ðzÞ and k3 ðzÞ are constants. For the FGM layer, three heat conduction coefficients k1 ðzÞ, k2 ðzÞ and k3 ðzÞ vary along the z direction. DTðx; y; zÞ is the corresponding temperature increment function relative to room temperature T 0 , i.e. DT ¼ T  T 0 , and the heat source term f ðx; y; zÞ in Eq. (1) is defined as [70]

0

k3

DT 1c ðx; y; zÞ ¼

1 X 1 X 1 X ipx jp y kpz Eijk sin sin sin a b h i¼1 j¼1 k¼1

ð7Þ

Substituting Eq. (7)into Eq. (1), utilizing the orthogonal behavior of the trigonometric function, it can be obtained that

Eijk ¼

2 a2 byd hkijk

Z

a

0

Z

b

0

Z

h

F 0 sin 0

ipx jp y kpz dx dy dz sin sin a b h

ð8Þ

where

kijk

"    2  2 # 2 i j k ¼ k1 þ k2 þ k3 p2 a b h

ð9Þ

and

Z

a

sin

ipx a dx ¼ ½ð1Þiþ1 þ 1 a ip

sin

jp y dy ¼ b

0

Z

b

0

Z

h

sin

sin

y0

¼ Z

y0 þyd

ð10aÞ

jpy dy b

  b jpðy0 þ yd Þ jpy0 cos  cos jp b b

kpz h dz ¼ ½ð1Þkþ1 þ 1 h kp

ð10bÞ

ð10cÞ

where the subscript c and F here represent the abbreviation of coating layer and FGM layer, respectively. Similar with Ref. [70], the values of temperature increment on the boundaries of the double-layer plate can be written as follows

Through above integral, then the first part of the temperature increment function for the coating layer DT 1c ðx; y; zÞ is obtained

DTð0; y; zÞ ¼ DTða; y; zÞ ¼ 0

DT 1c ðx; y; zÞ ¼

ð5aÞ

1 X 1 X 1 X i¼1 j¼1 k¼1

DTðx; 0; zÞ ¼ DTðx; b; zÞ ¼ 0

ð5bÞ

DTðx; y; HÞ ¼ 0

ð5cÞ

All faces are insulated except the local heat source zone, i.e. at surface z ¼ 0. The following two temperature-dependent equations for adhesive position that between the coating layer and FGM layer should be considered

DT c ðx; y; hÞ ¼ DT F ðx; y; hÞ ¼ DT b ðx; yÞ

0

ð6aÞ

8F 0 ijkabyd p3 kijk

  jpy0 jpðy0 þ yd Þ ipx jpy kpz sin   cos þ cos sin sin b a b h b ð11Þ From Fig. 2(b), the heat conduction equation of the second part for the coating layer can be written as

k1

@ 2 DT 2c ðx; y; zÞ @ 2 DT 2c ðx; y; zÞ @ 2 DT 2c ðx; y; zÞ þ k þ k ¼0 2 3 @x2 @y2 @z2

ð12Þ

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Fig. 2. General decomposition of the Poisson problem on rectangular domain (a) Zero boundary value Poisson problem; (b) General Dirichlet problem.

where DT 2c ðx; y; zÞ is the temperature increment function of the second part for the coating layer. The corresponding boundary temperature except the local heat source zone are expressed as

According to the above decomposition of the Poisson problem, the whole temperature increment function for the coating layer DT c ðx; y; zÞ can be obtained as

DT 2c ð0; y; zÞ ¼ DT 2c ða; y; zÞ ¼ 0

ð13aÞ

DT 2c ðx; 0; zÞ ¼ DT 2c ðx; b; zÞ ¼ 0

ð13bÞ

DT c ðx; y; zÞ ¼ DT 1c ðx; y; zÞ þ DT 2c ðx; y; zÞ   1 X 1 X 1 X 8F 0 jpy0 jpðy0 þ yd Þ ¼  cos þ cos b abyd ijkp3 kijk b i¼1 j¼1 k¼1

DT 2c ðx; y; hÞ ¼ DT b ðx; yÞ

ð13cÞ

DT 2c ðx; y; 0Þ ¼ 0

1 X 1 X ipx jp y g 1 ðzÞ sin sin a b i¼1 j¼1

g 1 ðzÞ ¼ T s esz

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi u 2 uk ip þ k jp 2 t 1 a 2 b ¼ k3

ð21Þ

ð15Þ

k1 ðzÞ

ð16Þ

Thus, the unknown function g 1 ðzÞ can also be expressed as

g 1 ðzÞ ¼ C 1 coshðs1 zÞ þ C 2 sinhðs1 zÞ

1 X 1 X

ipx jpy ½C 1 coshðs1 zÞ þ C 2 sinhðs1 zÞ sin sin a b j¼1 ð18Þ

Considering Eq. (13d), it can be easily obtained that C 1 ¼ 0, yields

DT 2c ðx; y; zÞ ¼

1 X 1 X ipx jpy C 2 sinhðs1 zÞ sin sin a b i¼1 j¼1

C2 ¼

4

Ra Rb 0

0

DT b sin ipax sin jpby dx dy ab sinhðs1 hÞ

l

heat conduction coefficients of arbitrary layer ki ði ¼ 1; 2; 3Þ can be



l

ki ðzÞ ¼

n h 0 0 0 þ z=ðH  hÞ ðkim  kic Þ þ kic ; hH

i ¼ ð1; 2; 3Þ

ð23Þ

where

z¼

Nh  H H  h þ l; N1 N1

ðl ¼ 1; 2; . . . NÞ

ð24Þ 0

0

where n is the functionally graded index, kim and kic represent the heat conduction coefficients of bottom layer and top layer (metal and ceramic are selected in this paper) along three different directions, respectively. N is the total number of discrete layer, l the arbitrary discrete layer. Then the heat conduction equation of arbitrary layer can be written as

ð19Þ l

k1 ðzÞ

And based on Eq. (13c), unknown constant C 2 can be determined

@ 2 DT F ðx;y; zÞ @ 2 DT F ðx; y; zÞ @ 2 DT F ðx; y;zÞ 000 þ k2 ðzÞ þ k3 ðzÞ ¼ f ðx;y; zÞ @x2 @y2 @z2 ð22Þ

The LW method is applied as the heat conduction coefficients of the FGM layer k1 ðzÞ, k2 ðzÞ and k3 ðzÞ vary along the z coordinate, which cannot be expanded in the Fourier series form. The discrete

ð17Þ

The temperature increment function of the second part DT 2c ðx; y; zÞ can be written as

i¼1

ipx jp y sin sinhðs1 zÞ a b

For the FGM layer, the three-dimensional steady heat conduction equation is

where T s is a constant, and substituting Eqs. (14) and (15) into Eq. (12), yields

DT 2c ðx; y; zÞ ¼

 sin

ð14Þ

where

s1;2

1 X 1 ipx jpy k pz X 16DT b sin sin þ a b h ijp2 sinhðs1 hÞ i¼1 j¼1

ð13dÞ

It can be easily obtained that the temperature increment function DT 2c ðx; y; zÞ which satisfies the Eqs. (13a–d) is of the form [7]

DT 2c ðx; y; zÞ ¼

 sin

@ 2 DT lF ðx;y; zÞ @ 2 DT lF ðx;y;zÞ @ 2 DT lF ðx;y; zÞ l l 000 þ k2 ðzÞ þ k3 ðzÞ ¼ f ðx; y;zÞ @x2 @y2 @z2 ð25Þ

The arbitrary layer’s boundary values of the temperature incre-

ð20Þ

ment function DT lF ðx; y; zÞ for the FGM layer are

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DT lF ð0; y; zÞ ¼ DT lF ða; y; zÞ ¼ 0

ð26aÞ

DT lF ðx; 0; zÞ ¼ DT lF ðx; b; zÞ ¼ 0

ð26bÞ

DT 0F ðx; y; hÞ

ð26cÞ

Thus, the second part of the arbitrary layer’s temperature increment function for the FGM layer DT l2F ðx; y; zÞ can be obtained as

DT l2F ðx; y; zÞ ¼

¼ DT b ðx; yÞ

1 X 1 X

½C l1 coshðsl zÞ þ C l2 sinhðsl zÞ sin

i¼1 j¼1

ipx jpy sin a b ð34Þ

DT NF ðx; y; HÞ ¼ 0

ð26dÞ

where DT 0F ðx; y; hÞ is the top temperature increment function of the FGM layer that bonded to the coating layer, and DT NF ðx; y; HÞ the bottom temperature increment of FGM (i.e. Nth discrete FGM layer). As the normal heat density of the coating layer equals to FGM layer on the bonding position, the continuous condition for the normal heat density can be written as

  0  dT c ðx; y; zÞ 0 dT F ðx; y; zÞ k3 ¼ k3    dz dz z¼h

ð27Þ

where

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u

 2 u l ip 2 l uk1 a þ k2 jbp sl ¼ t l k3

ð35Þ

The second part of the arbitrary discrete layer’s temperature boundary condition for the FGM layer can be written as

DT l2F ðx; y; zÞ ¼ DT b ðx; yÞ sin

z¼h

And with two continuity boundary conditions for arbitrary discrete FGM layer [23]

DT Fb ðx; y; zl Þ ¼ DT Ft ðx; y; zlþ1 Þ l

k3b

ð28aÞ

@ DT Fb ðx; y; zl Þ lþ1 @ DT Ft ðx; y; zlþ1 Þ ¼ k3t @z @z

ð28bÞ

where DT Fb ðx; y; zl Þ is the temperature increment for bottom of lth FGM layer, DT Ft ðx; y; zl Þ the temperature increment for top surl

000

00

f ðx; y; zÞ ¼ f ðx; yÞsin

ðz  HÞp 2ðh  HÞ

where 00

f ðx; yÞ ¼

1 X 1 X 16DT b i¼1 j¼1

 sin

abijp

"

2

k1

ð29Þ

#  2  2 ip jp  k2 þ k 3 s2 a b

ipx jp y sin a b

The first part of the temperature increment function for arbitrary discrete layer DT l1F ðx; y; zÞ which satisfies the temperature boundary conditions can be assumed as

DT l1F ðx; y; zÞ

1 X 1 X 1 X ipx jp y ¼ Elijk sin sin a b i¼1 j¼1 k¼1

 sin

ðN  1Þðz  hÞkp Hh

ð31Þ

Utilizing the orthogonal behavior of the trigonometric function, the coefficient

Elijk

Elijk

2.2. Transient heat conduction analysis The three-dimensional transient heat conduction equation for the double-layer structure can be written as

cðzÞqðzÞ

@ DT @ 2 DT @ 2 DT @ 2 DT  k1 ðzÞ  k2 ðzÞ  k3 ðzÞ 2 2 @t @x @y @z2

¼ f ðx; y; z; tÞ

#  2  2 32DT b ip jp 2 ¼  k2 þ k3 s k1 abijp2 ðH  hÞ a b

DTðx; 0; z; tÞ ¼ DTðx; b; z; tÞ ¼ 0

ð39bÞ

DTðx; y; H; tÞ ¼ 0

ð39cÞ

The temperature-dependent boundary conditions of adhesive position between the coating layer and FGM layer are

DT c ðx; y; h; tÞ ¼ DT F ðx; y; h; tÞ ¼ DT b ðx; y; tÞ  dDT c ðx; y; z; tÞ  dz

l l 2 2 @ 2 DT lF ðx; y; zÞ l @ DT F ðx; y; zÞ l @ DT F ðx; y; zÞ þ k2 þ k3 ¼0 2 2 @x @y @z2

ð33Þ

ð40aÞ



F dDT F ðx; y; z; tÞ

¼ k3

dz

z¼h

The second part of heat conduction equation (see Fig. 2(b)) for the FGM layer can be written as

ð38Þ

ð39aÞ



ð40bÞ z¼h

For the coating layer, the temperature increment function DT c ðx; y; z; tÞ can be expanded as

DT c ðx; y; z; tÞ ¼

1 X 1 X 1 X

Hijk ðtÞX i ðxÞY j ðyÞZ k ðzÞ

ð41Þ

i¼1 j¼1 k¼1

ð32Þ

l

DT c ðx; y; z; tÞ ð0 6 z 6 hÞ DT F ðx; y; z; tÞ ðh < z 6 HÞ

DTð0; y; z; tÞ ¼ DTða; y; z; tÞ ¼ 0

k3

4kðN  1ÞðH  hÞ ðN1Þk   2 h i2  ð1Þ 2 l ip 2 l jp l ðN1Þkp 2 ½4k ðN  1Þ  1p k1 a  k2 b  k3 Hh

k1



The temperature boundary conditions of the double-layer plate can be written as follows

can be obtained as

"

ð37Þ

where cðzÞ and qðzÞ denote, respectively, the specific heat and density of the whole double-layer structure. cðzÞ and qðzÞ are constants for the coating layer. For the FGM layer, cðzÞ and qðzÞ vary along the z coordinate, the heat source term expression f ðx; y; z; tÞ is similar as Eq. (3), other symbols represent the same meaning as the steady heat conduction analysis. The temperature increment function of the double-layer plate DTðx; y; z; tÞ can be expressed as

DTðx; y; z; tÞ ¼ ð30Þ

ð36Þ

Substituting Eq. (34) into Eq. (36), the coefficients C l1 and C l2 can be obtained (see in Appendix A).

l

face of ðl þ 1Þth FGM layer, k3b and k3t the heat conduction coefficients on z direction for bottom of lth FGM layer and top surface of ðl þ 1Þth FGM layer, respectively. Applying with the same decomposition principle like the coating layer, similarly, considering the first part of the solution (see Fig. 2(a)), the expression 000 f ðx; y; zÞ on Eq. (25) can be expressed as

ðzl  HÞp 2ðh  HÞ

where X i ðxÞ, Y j ðyÞ and Z k ðzÞ are trigonometric eigenfunctions which are expressed as

X i ðxÞ ¼ sin

ipx a

ð42aÞ

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Y j ðyÞ ¼ sin Z k ðzÞ ¼ sin

jp y b

ð42bÞ

ð4k þ 1Þpz 2h

ð42cÞ

1 X 1 X

DT F ðx; y; z; tÞ ¼

WðtÞX i ðxÞY j ðyÞZ 1 ðzÞ

i¼1 j¼1

þ

1 X 1 X

HðtÞX i ðxÞY j ðyÞZ 2 ðzÞ

ð53Þ

i¼1 j¼1

And the inhomogeneous term f ðx; y; z; tÞ can be expanded in the form of eigenfunction [70,71] 1 X 1 X 1 X

where the meaning of X i ðxÞ and Y j ðyÞ are the same as Eqs. (42a) and (42b), the expression of Z 1 ðzÞ and Z 2 ðzÞ are

ð43Þ

Z 1 ðzÞ ¼ sin

ðz  HÞp hH

ð54aÞ

Utilizing the orthogonal behavior of the trigonometric eigenfunction, /ijk ðtÞ can be easily obtained as

Z 2 ðzÞ ¼ sin

ðz  HÞp 2ðh  HÞ

ð54bÞ

f ðx; y; z; tÞ ¼

/ijk ðtÞX i ðxÞY j ðyÞZ k ðzÞ

i¼1 j¼1 k¼1

8 abh

/ijk ðtÞ ¼

Z

a

0

Z

b

Z

0

h

f ðx; y; z; tÞX i ðxÞY j ðyÞZ k ðzÞdx dy dz

ð44Þ

0

Substituting Eq. (44) into Eq. (43), and considering Eqs. (37) and (41), yields

dHijk ðtÞ þ xijk Hijk ðtÞ ¼ Xijk ðtÞ dt

  dWðtÞ dHðtÞ þ x0ij ðzÞWðtÞ þ ½Z 2 ðzÞ=Z 1 ðzÞ þ x00ij ðzÞHðtÞ dt dt ¼ A1 B½Z 2 ðzÞ=Z 1 ðzÞ

ð45Þ

where

64F 0 ; ayd hijp2 cq

00 ij ðzÞ

x

xijk ¼ vijk =cc qc ; Xijk ðtÞ ¼ /ijk ðtÞ=cc qc

ð55Þ

where

A1 ¼

 2  2  2 ! ip jp ð4k þ 1Þp k1 þ k2 þ k3 a b 2h

vijk ¼

Substituting Eqs. (53) and (54a,b) into Eq. (50), yields

B ¼ cos

jpðy0 þ yd Þ jpy0  cos b b

ð56aÞ

 2  2   !, ip jp p 2 cðzÞqðzÞ k1 ðzÞ þ k2 ðzÞ þ k3 ðzÞ a b ðh  HÞ

¼

ð46Þ

X i ðxÞ, Y j ðyÞ and Z k ðzÞ are not zero for arbitrary values of x, y and z, and with the initial boundary condition DT c ðx; y; z; 0Þ ¼ 0, yields

Hijk ð0Þ ¼ 0

ð47Þ

ð56bÞ  2  2  ip jp p þ k2 ðzÞ þ k3 ðzÞ a b 2ðh  HÞ

x00ij ðzÞ ¼ k1 ðzÞ

2 !, cðzÞqðzÞ ð56cÞ

According to Eqs. (45) and (47), yields

Hijk ðtÞ ¼

Z

t

Xijk ðsÞexijk ðtsÞ ds

ð48Þ

0

Substituting Eq. (48) into Eq. (41), the coating layer’s temperature increment function DT c ðx; y; z; tÞ can be expressed as n

ð4kþ1Þp2 . o 2 jp 2  k1 ðiapÞ þk2 ð b Þ þk3 cc qc ðtsÞ 64F 0 2h e ds ayd hijp2 cq i¼1 j¼1 k¼1 0   jpðy0 þ yd Þ jpy0 ipx jpy ð4k þ 1Þpz  cos  cos sin sin ð49Þ sin b b a b 2h

DT c ðx; y;z; tÞ ¼

1 X 1 X 1 Z X

2

2

2

@ DT F @ DT F @ DT F @ DT F  k1 ðzÞ  k2 ðzÞ  k3 ðzÞ @t @x2 @y2 @z2

000

¼ f ðx; y; z; tÞ

x

ð57aÞ

ð50Þ

 2  2  2 !, ip jp p cðzl Þqðzl Þ þ k2 ðzl Þ þ k3 ðzl Þ a b 2ðh  HÞ

x00ij ðzl Þ ¼ k1 ðzl Þ

ð57bÞ where the subscript l denotes the arbitrary discrete layer, the expressions of zl and ki ðzl Þ show the same as Eqs. (23) and (24), the expressions cðzl Þ and qðzl Þ are

cðzl Þ ¼



where 000

00

f ðx; y; z; tÞ ¼ f ðx; y; tÞ sin

ðz  HÞp 2ðh  HÞ

ð58aÞ



n h þ zl =ðH  hÞ ðqm  qc Þ þ qc hH

qðzl Þ ¼

ð52Þ

where cm and cc are, respectively, specific heat of metal and ceramic materials in present model, qm and qc are density of metal and ceramic materials, respectively. Substituting Eqs. (56)–(58) into Eq. (55), with the initial condition Wl ð0Þ ¼ 0, yields

  1 X 1 X 64F 0 jpðy0 þ yd Þ jpy0 f ðx; y; tÞ ¼ cos  cos b ayd hijp2 b i¼1 j¼1 ipx jpy sin a b

n h þ zl =ðH  hÞ ðcm  cc Þ þ cc hH

ð51Þ

00

 sin

 2  2   !, ip jp p 2 cðzl Þqðzl Þ k1 ðzl Þ þk2 ðzl Þ þk3 ðzl Þ a b ðhHÞ

0 ij ðzl Þ ¼

t

For the FGM layer, the expression of the heat conduction equation is

cðzÞqðzÞ

As x0ij ðzÞ and x00ij ðzÞ are related to the z coordinate, they can be divided into

And the continuity boundary condition is the similar as Eqs. (28a,b) for discrete FGM layer, only adding the time variable for temperature increment. In order to satisfy Eqs. (39c) and (40a), the temperature increment function for the FGM layer DT F ðx; y; z; tÞ can be expanded as

Wl ðtÞ ¼

ð58bÞ

A1 B h x0 ðz Þt tðxijk þx0ij ðzl Þ t x0 ðz Þ 1  e ij l  e ð1 þ e ij l Þ x0ij ðzl Þ e

tðxijk þx0ij ðzl ÞÞ

x0ij ðzl Þt

 ð1 þ exijk t Þð1 þ e

Þ

x00ij ðzl Þ Z 2 ðzl Þ xijk Z 1 ðzl Þ ð59Þ

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See the right side of Eq. (59),

Z 2 ðzl Þ Z 1 ðzl Þ

show as

0 0

type when zl ¼ H.

Based on L’Hospital’s rule, the expression Wl ðtÞ can be rewritten as

Wl ðtÞ ¼

 A1 B x0 ðz Þt tðxijk þx0ij ðzl Þ t x0 ðz Þ ð1 þ e ij l Þ 1  e ij l  e 0 2xij ðzl Þ  00 tðxijk þx0ij ðzl ÞÞ x0 ðz Þt xij ðzl Þ e ð1 þ exijk t Þð1 þ e ij l Þ

xijk

ð60Þ

method and Newmark method be considered to solve this temperature problem. Based on the symmetry of the structure and loading, a quarter of coating/FGM plate is considered in the computation, i.e. the calculated region is 0 6 n 6 1=2, 0 6 g 6 1=2, 0 6 n 6 1=2 (n ¼ x=a, g ¼ y=b, f ¼ z=H). The region is divided into l  l  l square portions, implying that the length of square L is equal to 1=2l. For the first order partial derivative function:

Substituting Eqs. (59) and (60) into Eq. (53), yields

DT F ðx; y; zl ; tÞ ¼



DT 1F ðx; y; zl ; tÞ 1 6 l 6 N  1 DT 2F ðx; y; zl ; tÞ l ¼ N

3. Numerical examples and discussion In the following examples, some of the basic parameters are shown in Tables 1 and 2, the heat conduction coefficients and density of the FGM are selected (the top ceramic material is Si3 N4 , the bottom metal material is SUS304) are shown as [61]: k3c ¼ 13:723 W=m K, qc ¼ 2370 kg=m3 , k3m ¼ 15:379 W=m K, 3 qm ¼ 8166 kg=m . Example 1. This example mainly discusses the steady temperature increment distribution for the double-layer (coating/FGM) plate along z coordinate. When considering the temperature dependence of thermal conductivity, the effective thermal conductivity of the coating layer and FGM layer can be expressed respectively as [60–62]

kei ¼ ki0 þ ki1 DT þ ki2 DT 2 þ ki3 DT 3 l

ð62Þ



n h h 0 0 0 0 þ z=ðH  hÞ ðkim0 þ kim1 DT þ kim2 DT 2 þ kim3 DT 3 Þ hH  i 0 0 0 0  kic0 þ kic1 DT þ kic2 DT 2 þ kic3 DT 3 Þ 0

0

0

0

þ ðkic0 þ kic1 DT þ kic2 DT 2 þ kic3 DT 3 Þ; 0

i ¼ ð1; 2; 3Þ

ð63Þ

0

where the coefficient kij , kimj and kicj ðj ¼ 1; 2; 3Þ are the same as reference [71]. Considering thermal resistance between the coating layer and FGM layer as well as discrete FGM layers, yields

DT c  Rb ¼ 0 f

DT 0F

; RbF ¼

DT lF

1 ½DTði þ 1; j; k; tÞ  DTði  1; j; k; tÞ 2L

ð65aÞ

DT ;g ði; j; k; tÞ ¼

1 ½DTði; j þ 1; k; tÞ  DTði; j  1; k; tÞ 2L

ð65bÞ

DT ;f ði; j; k; tÞ ¼

1 ½DTði; j; k þ 1; tÞ  DTði; j; k  1; tÞ 2L

ð65cÞ

ð61Þ

where expression of DT 1F ðx; y; z; tÞ and DT 2F ðx; y; z; tÞ are shown in Appendix B.

kei ðzÞ ¼

DT ;n ði; j; k; tÞ ¼

 DT lþ1 F 000

ð64Þ

f

With two above-mentioned factors(temperature dependence of thermal conductivity and thermal resistance), heat conduction equation cannot be solved analytically. Thus the finite difference Table 1 Principal geometry parameters of double-layer (coating/FGM) plate. Length a (m)

Width b (m)

Thickness of coating layer h (m)

Total thickness H (m)

1

0.5

0.005

0.02

For the second order partial derivative function:

DT ;nn ði; j; k; tÞ ¼

1 L2

½DTði þ 1; j; k; tÞ  2DTði; j; k; tÞ þ DTði  1; j; k; tÞ ð66aÞ

DT ;gg ði; j; k; tÞ ¼

1 L2

½DTði; j þ 1; k; tÞ  2DTði; j; k; tÞ þ DTði; j  1; k; tÞ ð66bÞ

DT ;ff ði; j; k; tÞ ¼

1 L2

½DTði; j; k þ 1; tÞ  2DTði; j; k; tÞ þ DTði; j; k  1; tÞ ð66cÞ

The high order nonlinear equilibrium relative to temperature DT should be linearized while considering material parameters changing accompany with temperature. In arbitrary step of iteration J, the high order nonlinear terms relative to DT should be linearized, and the form can be adopted as follows:

ðDT N ÞJ ¼ ðDT  DT N1 ÞJ ¼ ðDTÞJ  ðDTÞN1 JP

ð67Þ

where N is a constant, ðDTÞJp is the mean value of the first two iteration. For different step of iteration, ðDTÞJp can be adopted as

J¼1:

ðDTÞJp ¼ DT 1

J¼2:

ðDTÞJp ¼ ½DT 1 þ ðDTÞJ1 =2

JP3:

ðDTÞJp ¼ ½ðDTÞJ1 þ ðDTÞJ2 =2

ð68Þ

Fig. 3 denotes influence of number of the Fourier series item on the steady temperature increment distribution for the double-layer plate along z coordinate at point ðx ¼ a=2; y ¼ b=2Þ. From Fig. 3, it is seen easily that the convergence of the present serial solution is achieved by using only 81 terms, and the full convergence is demonstrated with 100 terms. And in the subsequent examples of steady heat conduction analysis, the number of Fourier series item is always selected as 100. Fig. 4 gives effect of temperature dependence and thermal resistance on steady temperature increment distribution for the double-layer plate when F 0 ¼ 102 J and F 0 ¼ 104 J . From this figure, it can be found that this two factors basically do not affect the on temperature when F 0 ¼ 102 J, and have little effect on tem-

Table 2 Basic parameters of coating and local heat source. Heat conduction coefficient (W/m K)

Discrete layer number

Location of local heat source (m)

Width of heat source (m)

k1 = 0.04 k2 = 0.04 k3 = 0.06

150

b/4

b/20

perature when F 0 ¼ 104 J. In other words, this investigated heat source is too small to greatly change heat conductivity of plate, namely, the temperature dependence and thermal resistance can be ignored in present model. Fig. 5 shows influence of heat flow F 0 on the steady temperature increment distribution for the double-layer plate’s along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. From Fig. 5, the steady

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0.08

0.025 9 Terms, coating layer 9 Terms, FGM layer

25 Terms, coating layer 25 Terms, FGM layer

0.020

0.06

F0=100J, coating layer

F0=300J, coating layer

F0=100J, FGM layer

F0=300J, FGM layer

F0=500J, coating layer

F0=700J, coating layer

F0=500J, FGM layer

F0=700J, FGM layer

0.015 Δ T [K]

ΔT [K]

0.04

0.02

0.010 0.005

0.00

-0.02 0.0

49 Terms, coating layer 49 Terms, FGM layer

81 Terms, coating layer 81 Terms, FGM layer

0.000

100 Terms, coating layer 100 Terms, FGM layer

-0.005

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6 * Z =Z/H

Z*=Z/H Fig. 3. Effect of number of Fourier series item on the double-layer plate’s steady temperature increment distribution along z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

2

F0=10 J 0.012

0.024

0.008

Δ T [K]

0.000 1.6

temperture independence coating layer temperture independence FGM layer temperture dependencecoating layer temperture dependence FGM layer

coating layer N=50, FGM layer

coating layer N=150, FGM layer

coating layer N=250, FGM layer

coating layer N=400, FGM layer

0.018 4

F0=10 J

1.2 0.8

ΔT [K]

0.004

1.0

Fig. 5. Effect of heat flow F 0 on the double-layer plate’s steady temperature increment distribution along z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

0.030

0.016

0.8

0.012

0.006

0.4 0.0

0.0

0.000

0.2

0.4

0.6

* Z =Z/H

0.8

1.0 0.0

0.2

0.4

0.6

0.8

1.0

∗

Z =Z/H Fig. 4. Effect of temperature dependence and thermal resistance on steady temperature increment distribution for the double-layer plate when F 0 ¼ 102 J and F 0 ¼ 104 J.

Fig. 6. Effect of number of discrete layers N on the double-layer plate’s steady temperature increment distribution along z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

temperature increment increases with the increasing of heat flow F 0 . The steady temperature increment increases along the z coordinate, and arrives at the maximum value at z ¼ h (adhesive position), then the steady temperature increment decreases gradually along the z coordinate. Fig. 6 depicts influence of number of discrete layers N on the steady temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. From Fig. 6, one knows that the steady temperature increment increases with the increasing of discrete layer N, it can be concluded that the single complete structure could sustain a more high temperature. And the maximum value of the steady temperature increment occurs at the interface of the structure (z ¼ h). Fig. 7 reveals influence of local heat source zone on the steady temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. Seen from Fig. 7, when y0 < b=2, the steady temperature increment decreases with the increasing of y0 , and the maximum temperature increment still happens at z ¼ h. When y0 ¼ b=2, the temperature increment has a sudden change at the coating layer of the double-layer plate, the magnitude of temperature is about 15 times comparing with the case when y0 < b=2, and the maximum steady temperature increment occurs at z ¼ 0:1H. This bizarre phenomenon can

be explained that the local heat source is just at the point ðx ¼ a=2; y ¼ b=2Þ. Fig. 8 shows influence of the local heat source strip’s width (i.e. yd ) on the steady temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. Seen from Fig. 8, the change trend of the steady temperature increments of the coating layer for the double-layer plate is merely the same when yd ¼ b=20 and yd ¼ b=5. The steady temperature increment of the coating layer for the double-layer plate has a sudden increase which is similar with a parabolic type when yd ¼ b=2 and yd ¼ 7b=10, the amplitude of the former steady temperature increment for the coating layer is larger than the latter, but it is just opposite on the FGM layer. Fig. 9 depicts influence of thickness ratio H=h (the ratio of the total plate thickness to coating layer thickness) on the steady temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. From Fig. 9, the steady temperature increment decreases with the increasing of thickness ratio H=h. Thus, the thicker the coating layer is, the greater the steady temperature changes. Fig. 10 demonstrates influence of the length–width ratio a=b on the steady temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. It can be

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0.32

0.016

H/h=4, coating layer H/h=4, FGM layer

y 0=b/2

H/h=10, coating layer H/h=10, FGM layer

0.24

0.012 0.16

FGM layer

0.008

0.00

coating layer

0.015

coating layer

ΔT [K]

ΔT [K]

0.08

y 0=b/4 y 0=b/3

0.010

0.000 H/h=20, coating layer H/h=20, FGM layer

0.005

FGM layer

0.0

0.2

0.4

0.6

H/h=40, coating layer H/h=40, FGM layer

-0.004

0.000 0.0

0.004

y 0=b/5

0.8

1.0

∗

Z =Z/H

0.2

0.4

0.6

Z∗=Z/H

0.8

1.0

Fig. 9. Effect of thickness ratio H=h on the double-layer plate’s steady temperature increment distribution along the z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

Fig. 7. Effect of location of heat source y0 on the double-layer plate’s steady temperature increment distribution along the z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

0.016

a/b=2, coating layer a/b=2, FGM layer

a/b=6, coating layer a/b=6, FGM layer

0.012

0.08 yd=b/2

0.06

yd=7b/10

ΔT [K]

0.008

0.04 0.02

0.004

ΔT [K]

coating layer

0.00 0.016

FGM layer

0.000

FGM layer

0.012

yd=b/20

coating layer

-0.004 0.0

yd=b/5

0.008

a/b=20, coating layer a/b=20, FGM layer

a/b=10, coating layer a/b=10, FGM layer

0.2

0.4

0.6

0.8

1.0

∗

Z =Z/H

0.004 Fig. 10. Effect of length–width ratio a=b on the double-layer plate’s steady temperature increment distribution along the z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

0.000 0.0

0.2

0.4

0.6

0.8

1.0

∗

Z =Z/H Fig. 8. Effect of heat source strip’s width yd on the double-layer plate’s steady temperature increment distribution along the z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

seen from Fig. 10 that the temperature increment decreases with the increasing of length–width ratio a=b. Accordingly, the length– width ratio a=b is a key factor which could guide designing such a steady temperature problem of the double-layer plate. Fig. 11 reveals influence of the selected coating material  (mainly the fact of heat conduction coefficient k3 ) on the steady temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ. From Fig. 10, it can be easily obtained that the temperature increment decreases  with the increasing of coefficient k3 . Example 2. This example mainly investigates the steady temperature increment distribution for the double-layer plate along x coordinate at the point ðy ¼ b=2; z ¼ hÞ and the steady temperature increment distribution for the double-layer plate with the movement of the heat source location y0 along the width direction at the point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ.

Fig. 12 depicts influence of the heat flow F 0 on the steady temperature increment distribution for the double-layer plate along x coordinate at the point ðy ¼ b=2; z ¼ hÞ. It is seen from Fig. 12 that the steady temperature increment increases with the increasing of heat flow F 0 . For arbitrary curve, the temperature increment has a sudden increase near x ¼ 0, remains unchanged on the whole, and finally decreases rapidly near x ¼ a. Fig. 13 shows the steady temperature increment distribution with the movement of heat source location y0 along the y coordinate at the point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ. From Fig. 13, when location of heat source y0 moves on both ends of width boundary, the temperature increment at the point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ arrives at a certain peak value, and maintains flat when the location of heat source y0 moves on other location. When y0 > 19b=20, the temperature increment has an abrupt decreasing at point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ as the location of heat source exceeds the boundary of the double-layer plate. Example 3. In order to validate the present method on the transient heat conduction analysis, degenerate the double-layer plate into a single layer whose parameters are the same as Araya and Gutierrez [72,73], dimensions of the single layer are

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0.016

∗

∗

k 3 =k3, coating layer

k3 =5k3 , coating layer

FGM layer

FGM layer

1400

1000 800

0.008

ΔT [K]

ΔT [K]

0.012

0.004

0.000

600 400

0.2

200

∗

∗

k3 =20k3 , coating layer

k 3 =10k3, coating layer

FGM layer

FGM layer

0.0

Araya and Gutierrez [72] Gutierrez and Araya [73] Present

1200

0.4

0.6

0.8

0

1.0

0.0

0.1

0.2

0.3

∗

0.4

0.5

0.6

0.7

∗

Z =Z/H

Y =y/b  k3

Fig. 11. Effect of heat conduction coefficient for selected coating material on the double-layer plate’s steady temperature increment distribution along the z coordinate at point ðx ¼ a=2; y ¼ b=2Þ.

Fig. 14. Comparison of the present method with Araya and Gutierrez [72,73] in the y coordinate on the top surface ðz ¼ 0Þ.

0.025 25

y=b/2, z=h, F0=700J

0.020

5 terms of Fourier series 15 terms of Fourier series 25 terms of Fourier series 45 terms of Fourier series 65 terms of Fourier series

y=b/2, z=h, F0=500J

0.010

y=b/2, z=h, F0=300J

0.005

y=b/2, z=h, F0=100J

15

ΔT [K]

ΔT [K]

20

0.015

10 5

0.000 0

0.0

0.2

0.4

0.6

0.8

1.0

∗

0

X =X/a

10

20

30

40

50

Time (s) Fig. 12. Effect of heat flow F 0 on the double-layer plate’s steady temperature increment distribution along the x coordinate at point ðy ¼ b=2; z ¼ hÞ.

Fig. 15. Effect of number of Fourier series item on the double-layer plate’s transient temperature increment distribution with the increasing of time at point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ.

0.3 2

0.2

1.8 F0=10 J 1.2 0.6 temperature independence coating layer temperature independence FGM layer temperature dependence coating layer temperature dependence FGM layer

0.0

0.0

-0.1

Δ T [K]

ΔT [K]

0.1

F0=100J

120

F0=300J -0.2

F0=500J

60

F0=700J

0

-0.3 0.0

-0.6 180 F0=104J

0.2

0.4

0.6

0.8

1.0

∗

Y =y0/b Fig. 13. The double-layer plate’s steady temperature increment distribution with the move of heat source location y0 along the width direction at point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ.

-60 0.0

0.2

0.4

0.6 * Z =Z/H

0.8

1.0

Fig. 16. Effect of temperature dependence and thermal resistance on transient temperature increment distribution for the double-layer plate when F 0 ¼ 102 J and F 0 ¼ 104 J.

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15 F0=100J, coating layer

y0=b/2

0

F0=100J, FGM layer F0=300J, coating layer

10

-800

F0=300J, FGM layer

FGM layer

F0=500J, coating layer

5

-1600

F0=700J, coating layer F0=700J, FGM layer

coating layer

ΔT [K]

Δ T [K]

F0=500J, FGM layer

0

-2400

10 coating layer

0 -5 0.0

0.2

0.4

0.6

0.8

1.0

-10

∗

Z =Z/H

FGM layer

-20

Fig. 17. Effect of heat flow F 0 on the double-layer plate’s transient temperature increment distribution along z coordinate at t ¼ 15 s.

y0=b/5

-30 0.0 0:02 m  0:04 m  0:00625 m along X, Y and Z directions, respectively. Y coordinate in reference[73] is corresponding to x coordinate in present model, laser position is at 0:75Ly , where Ly the tool insert dimension in the Y direction, laser speed 0.1 m/s and laser net power 50 W, the comparison of temperature distribution along y coordinate on top surface (i.e. along X coordinate in reference [73]) is shown in Fig. 14, it can be found that the temperature has a sudden decrease across the y coordinate on top surface. And seen from the curves of the figure, the results are nearly the same, which indicates that the present method is feasible.

0.6

0.8

Fig. 19. Effect of location of heat source y0 on the double-layer plate’s transient temperature increment distribution along the z coordinate at t ¼ 15 s.

Fig. 15 shows influence of number of Fourier series item on the transient temperature increment distribution for the double-layer plate at the point ðx ¼ a=2; y ¼ b=2; z ¼ hÞ. From Fig. 15, one knows, the convergence of the present series solution is achieved by using only 45 terms, and the full convergence is demonstrated with 65 terms in the example. When t > 15 s, the transient temperature remains unchanged by and large, which denotes that the temperature distribution achieves steady state basically. Accordingly, the time of transient temperature is selected as t ¼ 15 s in the following transient heat conduction analysis.

10

10

8

8

6

6 4 coating layer

2

coating layer

2

0

FGM layer

-2 0.0 10

0.2

0.4

0.8

N=150

-2 1.0 0.0 10

8

8

6

6

0.2

0.4

0.6

0.8

1.0

4 coating layer

coating layer

2

2 FGM layer

0 -2 0.0

FGM layer

0

N=50 0.6

0.2

0.4

0

N=250 0.6

0.8

1.0

Z =Z/H

4

ΔT [K]

0.4 ∗

Examples 4. This example mainly studies the transient temperature increment distribution for the double-layer plate along z coordinate, and all parameters are the same as the analysis of the steady heat conduction problem.

4

0.2

y0=b/3

y0=b/4

1.0

-2 0.0

FGM layer

0.2

0.4

N=400 0.6

0.8

1.0

∗

Z =Z/H Fig. 18. Effect of number of discrete layers N on the double-layer plate’s transient temperature increment distribution along z coordinate at t ¼ 15 s.

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along z coordinate at t ¼ 15 s. It is seen from Fig. 17 that the transient temperature increment increases with the increasing of the heat flow F 0 . The transient temperature of the coating layer presents similarly with the sinusoid type, and the maximum transient temperature increment happens at z ¼ 0:75H. Fig. 18 demonstrates effect of number of discrete layers N on the transient temperature increment distribution for the double-layer plate along z coordinate at t ¼ 15 s . Watching from Fig. 18, the transient temperature distribution of the double-layer plate shows the same trend with different discrete layers N. Fig. 19 denotes influence of the location of heat source y0 on the transient temperature increment distribution for the double-layer plate along z coordinate at t ¼ 15 s. From Fig. 19, the transient temperature is merely the same when y0 ¼ b=5 and y0 ¼ b=4. The transient temperature achieved is a negative value when y0 ¼ b=3 and y0 ¼ b=2, the former magnitude is less than the latter. This phenomenon can be attributed to concentration of a local heat source near y0 ¼ b=2. Fig. 20 reveals influence of the heat source strip’s width yd on the transient temperature increment distribution for the double-layer plate along z coordinate at t ¼ 15 s. It can be seen from Fig. 20 that the change trend of the transient temperature distribution is similar when yd ¼ b=20 and yd ¼ b=5 at the coating layer, and the former temperature is a little greater than the later on the FGM layer. The magnitude of transient temperature decreases a lot when yd ¼ b=2 and yd ¼ 7b=10, and the closer it gets to yd ¼ b=2, the greater the transient temperature decreases. Fig. 21 demonstrates influence of thickness ratio H=h (the ratio of the total plate thickness to coating layer thickness) on the transient temperature increment distribution for the double-layer plate along z coordinate at t ¼ 15 s. Seen from Fig. 21 that the transient temperature increment on the coating layer decreases with the increasing of thickness ratio H=h. On the other hand, the transient temperature increment on the FGM layer increases with the increasing of thickness ratio H=h (except at H=h ¼ 10, it can be speculated that this thickness ratio for double-layer plate could sustain less temperature).

yd=b/2

0

yd=7b/10 -120 -240

ΔT [K]

-360 coating layer -480

FGM layer

12 FGM layer 9 6

coating layer

3

yd=b/20 yd=b/5

0 -3 0.0

0.2

0.4

∗

0.6

0.8

1.0

Z =Z/H Fig. 20. Effect of heat source strip’s width yd on the double-layer plate’s transient temperature increment distribution along the z coordinate at t ¼ 15 s.

Fig. 16 gives effect of temperature dependence and thermal resistance on transient temperature increment distribution for the double-layer plate when F 0 ¼ 102 J and F 0 ¼ 104 J. From the Fig. 16, this two factors basically do not affect the on temperature when F 0 ¼ 102 J, but have great influence on temperature increment when F 0 ¼ 104 J. Namely, the temperature dependence and thermal resistance must be taken into account when applied with high energy heat source, but they can be ignored when the order of magnitude for heat source is 102 J for present model. Fig. 17 depicts influence of the heat flow F 0 on the transient temperature increment distribution for the double-layer plate

10 8

6 H/h=4, coating layer H/h=4, FGM layer

5

H/h=10, coating layer H/h=10, FGM layer

4

6

3

4

2

2

1 0

ΔT [K]

0 -2 0.0 12

0.2

0.4

0.6

0.8

1.0

H/h=20, coating layer H/h=20, FGM layer

0.0 25 20

10 8

0.2

0.4

0.6

0.8

1.0

H/h=40, coating layer H/h=40, FGM layer

15 0.04

6

10

0.02

4

0.00

5

-0.02

2

0.0

-0.04 0.0000.0050.0100.0150.0200.025

0

0 0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

∗

Z =Z/H Fig. 21. Effect of thickness ratio H=h on the double-layer plate’s transient temperature increment distribution along the z coordinate at t ¼ 15 s.

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4

0.6 0.4

3

FGM layer

0.2 coating layer

coating layer n=0.5 n=1 n=5 n=20

2

ΔT [K]

-0.2

a=10b

a=6b

Δ Τ [Κ]

0.0 a=20b

1

10.5 0

7.0 -1 0.0

3.5 coating layer

0.2

0.4

0.6

0.8

1.0

Z*=Z/H FGM layer

0.0

a=2b

-3.5 0.0

0.2

0.4

∗

0.6

0.8

1.0

Fig. 24. Effect of functionally graded index n on the double-layer plate’s transient temperature increment distribution along the z coordinate at ðx ¼ a=2; y ¼ b=2Þ and t ¼ 15 s.

Z =Z/H Fig. 22. Effect of length–width ratio a=b on the double-layer plate’s transient temperature increment distribution along the z coordinate at t ¼ 15 s.

Fig. 22 shows effect of length–width ratio a=b on the transient temperature increment distribution for the double-layer plate along z coordinate at t ¼ 15 s. From Fig. 22, both the transient temperature increments of the double-layer plate decrease with the increasing of length–width ratio a=b. Fig. 23 depicts effect of the selected coating material (mainly  the fact of heat conduction coefficient k3 ) on the transient temperature increment distribution for the double-layer plate along z coordinate at t ¼ 15 s. From Fig. 23, it can be easily found that the temperature increment of the coating layer decreases with  the increasing of heat conduction coefficient k3 , but it is just the opposite changing trend on the FGM layer. Fig. 24 demonstrates effect of the functionally graded index n on the transient temperature increment distribution for the double-layer plate along z coordinate at the point ðx ¼ a=2; y ¼ b=2Þ and t ¼ 15 s. It can be seen from Fig. 24 that the transient temperature of FGM layer for double-layer plate

12

∗

k3 =k3, coating layer FGM layer

9

∗

k3 =5k3, coating layer FGM layer ∗

Δ T [K]

6

4. Conclusions Analytical solutions of the steady and transient temperature increment distribution for a double-layer (coating/FGM) plate are presented. By means of numerical examples, some conclusions can be drawn as follows (1) Both the amplitude of steady and transient temperature increments for the double-layer (coating/FGM) plate vary a lot on which the local heat source applied, it can be attributed to the temperature concentration. (2) Both the steady and transient temperature increments of the double-layer plate decrease with the increasing of length– width ratio a=b, which could provide optimal design for the double-layer plate under the hostile temperature environment. (3) The temperature increment of the FGM layer for the double-layer plate decreases with the increasing of the  thickness ratio H=h and heat conduction coefficient k3 on the steady heat conduction analysis, but it is just opposite on the transient heat conduction process. (4) The temperature dependence of thermal conductivity as well as the effect of the thermal boundary resistance can be ignored in this present model because of low heat source.

k3 =10k3, coating layer FGM layer

Conflict of interest

∗

k3 =20k3, coating layer FGM layer

None declared.

3

0

-3 0.0

decreases with the increasing of functionally graded index n, which could help us in designing high temperature-resistance structure.

Acknowledgments

0.2

0.4

0.6

0.8

1.0

Z*=Z/H 

Fig. 23. Effect of heat conduction coefficient for selected coating material k3 on the double-layer plate’s transient temperature increment distribution along the z coordinate at t ¼ 15 s.

The authors wish to thank reviewers for their valuable comments and the research is supported by the National Natural Science Foundation of China (11372105), New Century Excellent Talents Program in University (NCET-13-0184), State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (71475004) and Hunan Provincial Innovation Foundation for Postgraduate (CX2013B148).

H.-J. Jiang, H.-L. Dai / International Journal of Heat and Mass Transfer 89 (2015) 652–666

Appendix A

  ðzlþ1  HÞp ðzl  HÞp C l1 ¼ 16DT b sin sinhðsl zl Þ  sin  sinhðslþ1 zlþ1 Þ 2ðh  HÞ 2ðh  HÞ  2  lþ1 l = ijp ½coshðs zlþ1 Þ sinhðs zl Þ  coshðsl zl Þ sinhðslþ1 zlþ1 Þ   ðzl  HÞp ðzlþ1  HÞp C l2 ¼ 16DT b sin coshðslþ1 zlþ1 Þ  sin  coshðsl zl Þ 2ðh  HÞ 2ðh  HÞ   = ijp2 ½coshðslþ1 zlþ1 Þ sinhðsl zl Þ  coshðsl zl Þ sinhðslþ1 zlþ1 Þ Appendix B

DT 1F ðx; y; z; tÞ ¼

1 X 1 X A1 B x0 ðz Þt f1  e ij l  et ½xijk þ x0ij ðzl Þ½1 0 x ðz Þ ij l i¼1 j¼1

þ et x0ij ðzl Þ  et x0ij ðzl Þt

 ð1 þ e

x00ij ðzl Þ ½xijk þ x0ij ðzl Þð1 þ exijk t Þ xijk

Þg  X i ðxÞY j ðyÞZ 2 ðzl Þ þ

1 X 1 X

A1 B

i¼1 j¼1



DT 2F ðx; y; z; tÞ ¼

1  exijk t

xijk

1 X 1 X i¼1 j¼1

X i ðxÞY j ðyÞZ 2 ðzl Þ

A1 B x0 ðz Þt f1  e ij l  et ½xijk þ x0ij ðzl Þ½1 2x0ij ðzl Þ

þ et x0ij ðzl Þ  et x0ij ðzl Þt

 ½1 þ e

x00ij ðzl Þ ½xijk þ x0ij ðzl Þð1 þ exijk t Þ xijk

gX i ðxÞY j ðyÞZ 1 ðzl Þ þ

1 X 1 X A1 B i¼1 j¼1



1  exijk t

xijk

X i ðxÞY j ðyÞZ 2 ðzl Þ

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