# Heat Conduction in a Functionally Graded Plate Subjected to Finite Cooling/Heating Rates: An Asymptotic Solution

## Abstract

**:**

## 1. Introduction

## 2. Basic Equations

_{0}which can be assumed to be zero without loss of generality. The temperature then gradually changes to −T

_{a}and −T

_{b}at the surfaces x = 0 and x = b of the plate, respectively. We use a linear ramp function to describe the variations of the boundary temperatures. The initial and boundary conditions for the heat conduction problem are thus

_{a}and t

_{b}are two temporal parameters describing the rates of temperature variation (cooling/heating rates) at the plate surfaces. The one-dimensional heat conduction in the plate is governed by the following basic equation

## 3. A Multi-Layered Material Model and the Discrete Temperature Solution

_{n}(n = 1, 2, …, N) and the two boundaries of the plate are x

_{0}= 0 and x

_{N+1}= b. When N becomes large, each layer may approximately be regarded as a homogeneous layer with constant properties k

_{n}(thermal conductivity), ρ

_{n}(density), c

_{n}(specific heat), and κ

_{n}= k

_{n}/(ρ

_{n}c

_{n}) (thermal diffusivity). Let T

_{n}denote the temperatures at x = x

_{n}(n = 0, 1, 2, …, N+1). The temperature within the nth layer under the initial condition, Equation (1), has the form [6,7]

_{nl}is a constant given by

_{n}(τ) (n = 1, 2, ..., N) are determined from the heat flux continuity conditions across the interfaces.

_{n}(τ). The corresponding Laplace transformed equations have the form:

_{n}(τ). In Equation (8), the nonzero coefficients a

_{mn}(s) have the same expressions as those for sudden cooling conditions (t

_{a}→ 0, t

_{b}→ 0) in Reference [5], and b

_{m}(s) have the forms:

_{a}and τ

_{b}are nondimensional temporal parameters defined by

_{n}(s) are given by

_{n}(τ) for short times as follows (τ << 1):

_{a}and τ

_{b}should also be small, i.e., the cooling/heating rates are finite but still relatively high.

## 4. A Closed Form Short Time Solution

_{n+1}– x

_{n}→ 0 (n = 0, 1, 2, …, N) and N → ∞. The limits of ${L}_{n}^{(0)}$ and ${P}_{n}^{(0)}$ in Equations (14) and (15) can be found as follows [5]:

_{1}(x) and Ω

_{2}(x) are defined by

_{a}and τ

_{b}approach zero,${T}^{(1)}(x,\tau )$ and ${T}^{(2)}(x,\tau )$ reduce to

_{b}= 0 in the boundary condition Equation (2b) by taking k(x) = k

_{0}, c(x) = c

_{0}, ρ(x) = ρ

_{0}and κ(x) = κ

_{0}in Equations (17–20). The complete solution of temperature in this case is

_{a}) versus nondimensional coordinate (x/b) at various nondimensional time τ. The temporal parameter τ

_{a}is 0.001. The asymptotic solution almost coincides with the complete solution in the entire plate for nondimensional times up to τ = 0.05. Those solutions are also in good agreement in the region of x/b < 0.8 for times up to τ = 0.10. For times up to τ = 0.15, the solutions agree well with each other in the region of x/b < 0.6. The asymptotic solution approximately satisfies the boundary condition at x = b for τ < 0.05. Figure 2b and Figure 2c show similar results when the parameter τ

_{a}is increased to 0.05 and 0.1, respectively. It appears that τ

_{a}or the rate of boundary temperature variation does not significantly influence the applicability region of the short time solution. It is expected that the short time solution for an FGM plate will also be approximately valid for nondimensional times up to τ = 0.10 if the material property gradation is not extremely steep.

**Figure 2.**Temperature field: asymptotic and complete solutions for a homogeneous plate. (

**a**) τ

_{a}= 0.001; (

**b**) τ

_{a}= 0.05; (

**c**) τ

_{a}= 0.1.

## 5. Effects of Cooling Rate on the Thermal Stress Intensity Factor for an Edge Crack in an FGM Plate

_{0}the temperature variation.

_{I}denotes the TSIF, K

^{*}the nondimensional TSIF, and α

_{0}= α(0). In Equation (28) F(1) is a function of time τ.

Materials | Young’s modulus (GPa) | Poisson’s ratio | CTE
(10^{−6}/K) | Thermal conductivity (W/m-K) | Mass
density (g/cm^{3}) | Specific heat (J/g-K) |
---|---|---|---|---|---|---|

TiC | 400 | 0.2 | 7.0 | 20 | 4.9 | 0.7 |

SiC | 400 | 0.2 | 4.0 | 60 | 3.2 | 1.0 |

_{a}. The crack length is a/b = 0.1 and the material gradation profile index is p = 0.2. The TSIF under the sudden cooling condition (τ

_{a}= 0, and hence infinite cooling rate) is also included. For a given cooling rate (T

_{a}/τ

_{a}), the TSIF initially increases with time, rapidly reaches the peak value and then decreases with time. The peak TSIF decreases significantly with a decrease in the cooling rate (increasing τ

_{a}). Moreover, the time at which the TSIF reaches its peak increases with a decrease in the cooling rate.

## 5. Concluding Remarks

## References

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**MDPI and ACS Style**

Jin, Z.
Heat Conduction in a Functionally Graded Plate Subjected to Finite Cooling/Heating Rates: An Asymptotic Solution. *Materials* **2011**, *4*, 2108-2118.
https://doi.org/10.3390/ma4122108

**AMA Style**

Jin Z.
Heat Conduction in a Functionally Graded Plate Subjected to Finite Cooling/Heating Rates: An Asymptotic Solution. *Materials*. 2011; 4(12):2108-2118.
https://doi.org/10.3390/ma4122108

**Chicago/Turabian Style**

Jin, Zhihe.
2011. "Heat Conduction in a Functionally Graded Plate Subjected to Finite Cooling/Heating Rates: An Asymptotic Solution" *Materials* 4, no. 12: 2108-2118.
https://doi.org/10.3390/ma4122108