Uncoupled thermoelastic problem of a functionally graded thermosensitive rectangular plate with convective heating

Abstract

The main objective of the present paper is to study the temperature and thermal stress analysis of a functionally graded rectangular plate with temperature-dependent thermophysical characteristics of materials under convective heating. The nonlinear heat conduction equation is reduced to linear form using Kirchhoff’s variable transformation. Analytic solution of the heat conduction equation is obtained in the transform domain by developing an integral transform technique for convective-type boundary conditions. Goodier’s displacement function and Boussinesq harmonic functions are used to obtain the displacement profile and its associated thermal stresses. A mathematical model is prepared for functionally graded ceramic–metal-based material. The results are illustrated numerically and depicted graphically for both thermosensitive and nonthermosensitive functionally graded plate. During this study, one observed that notable variations are seen in the temperature and stress profile, due to the variation in the material parameters.

Appendices

Appendix A

Following [24], we define the integral transform and its inversion formula of the temperature function $\theta (x, y,t)$ with respect to the space variable x, in $0\le x\le a$ as

$$\begin{aligned} \left( {\begin{array}{l} \hbox {Integral} \\ \hbox {Transform} \\ \end{array}} \right) \bar{{\theta }}(\beta _n ,y, t)= & {} \int \limits _{{x}'=0}^a S(\beta _n , {x}') \bar{{\theta }}({x}', y, t) \mathrm{d}{x}' \end{aligned}$$

(A1)

$$\begin{aligned} \left( {\begin{array}{l} \hbox {Inversion} \\ \hbox {Formula} \\ \end{array}} \right) \theta (x, y,t)= & {} \sum _{n=1}^\infty S(\beta _n , x) \bar{{\theta }}(\beta _n , y,t) \end{aligned}$$

(A2)

Here, $S(\beta _n , x)$ is the kernel of the transform given by

$$\begin{aligned} S(\beta _n , x)=A_1 (\beta _n \cos \beta _n x+\varepsilon _1 \sin \beta _n x) \end{aligned}$$

(A3)

where

$$\begin{aligned} A_1 =\left[ {\sqrt{2}/\sqrt{\left( {\beta _n^2 +\varepsilon _1^2 } \right) \left( {a+\frac{\varepsilon _2 }{\beta _n^2 +\varepsilon _2^2 }} \right) +\varepsilon _1 }} \right] \end{aligned}$$

(A4)

Here, ${\beta _n{^{\prime }}}\hbox {s}$ are the positive roots of the transcendental equation

$$\begin{aligned} \tan \beta _n a=\frac{\beta _n (\varepsilon _1 +\varepsilon _2 )}{\beta _n^2 -\varepsilon _1 \varepsilon _2 } \end{aligned}$$

(A5)

Similarly, the integral transform and its inversion formula of the temperature function $\bar{{\theta }}(\beta _n ,y, t)$ with respect to the space variable y, in $0\le y\le b$ are defined as [24]

$$\begin{aligned} \left( {\begin{array}{l} \hbox {Integral} \\ \hbox {Transform} \\ \end{array}} \right) \bar{{\bar{{\theta }}}}(\beta _n ,\alpha _m , t)= & {} \int \limits _{{y}'=0}^b S(\alpha _m , {y}') \bar{{\theta }}(\beta _n , {y}', t) \mathrm{d} {y}' \end{aligned}$$

(A6)

$$\begin{aligned} \left( {\begin{array}{l} \hbox {Inversion} \\ \hbox {Formula} \\ \end{array}} \right) \bar{{\theta }}(\beta _n ,y, t)= & {} \sum \limits _{m=1}^\infty S(\alpha _m , y) \bar{{\bar{{\theta }}}}(\beta _n ,\alpha _m , t) \end{aligned}$$

(A7)

Here, $S(\alpha _m , y)$ is the kernel of the transform given by

$$\begin{aligned} S(\alpha _m , y)=A_2 (\alpha _m \cos \alpha _m y+\varepsilon _1 \sin \alpha _m y) \end{aligned}$$

(A8)

where

$$\begin{aligned} A_2 =\left[ {\sqrt{2}/\sqrt{\left( {\alpha _m^2 +\varepsilon _1^2 } \right) \left( {a+\frac{\varepsilon _2 }{\alpha _m^2 +\varepsilon _2^2 }} \right) +\varepsilon _1 }} \right] \end{aligned}$$

(A9)

Here, ${\beta _n{^{\prime }}}\hbox {s}$ are the positive roots of the transcendental equation

$$\begin{aligned} \tan \alpha _m b=\frac{\alpha _m (\varepsilon _1 +\varepsilon _2 )}{\alpha _m^2 -\varepsilon _1 \varepsilon _2 }. \end{aligned}$$

(A10)

Appendix B

The volume fraction distribution of metal obeying simple power law with exponent $\gamma$ is given as [5]

$$\begin{aligned} f_m (x)=1-x^{\gamma }\quad \hbox {for } \gamma \ge 0 \end{aligned}$$

(B1)

where $f_m (x)$ is the local volume fraction of metal in a functionally graded plate and $\gamma$ is a parameter that describes the volume fraction of metal.

The thermal conductivity of the functionally graded material dependent on x is expressed using the thermal conductivities of metals $k_m$ and of ceramics $k_c$ with the volume fractions of metals $f_m (x)$ and ceramics $1-f_m (x)$ as follows:

$$\begin{aligned} k(x,T)=k_m (T)f_m (x)+k_c (T)(1-f_m (x)) \end{aligned}$$

(B2)

Inverse transformation We substitute Eq. (B2) in Eq. (15) to obtain the inverse transformation of Eq. (15) as

$$\begin{aligned} \theta (T)= \int _{T_0 }^{T } {(k_m (T)f_m (x)+k_c (T)(1-f_m (x))} \mathrm{d} T \end{aligned}$$

(B3)

To determine T from Eq. (B3), we analyze the discrete values of $\theta$ in a thin layer where the volume fraction and the material properties are assumed to be constants for each layer. Hence, we obtain the following approximation:

$$\begin{aligned} \theta (T)= f_m (x) \int _{_{T_0 } }^{T } {k_m (T) \mathrm{d} T+(1-f_m (x) \int \limits _{T_0 }^T k_c (T))} \mathrm{d} T \end{aligned}$$

(B4)

Following Noda [1], we assume the temperature-dependent thermal conductivity as

$$\begin{aligned} k (T) = k_0 \exp (\varpi _1 T) , \varpi _1 <0 \end{aligned}$$

Hence, Eq. (A4) becomes

$$\begin{aligned} \theta = (1/\varpi _1 ) [[\exp (\varpi _1 T)-\exp (\varpi _1 T_0 )] u(x)] \end{aligned}$$

(B5)

where $u(x)=[f_m (x)(k_{m_0 } -k_{c_0 } ) + k_{c_0 } ]$.

Using Eq. (B5) in Eq. (22), we obtain

$$\begin{aligned} T(x,y, t)= & {} \frac{1}{\varpi _1 } \log _e [\exp (\varpi _1 T_0 )+h (x,y, t)] =\frac{1}{\varpi _1 }\log _e \left[ {\exp (\varpi _1 T_0 ) \left( {1+\frac{h (x,y, t)}{\exp (\varpi _1 T_0 )}} \right) } \right] \end{aligned}$$

(B6)

$$\begin{aligned} h (x,y, t)= & {} \sum _{m=1}^\infty \sum _{n=1}^\infty \{(\varpi _1 / u(x)) S(\beta _n , x) S(\alpha _m , y)\times [E_1 \exp (-\omega t)+E_2 \exp (\omega t)+E_3 \exp (-A_2 t)]\}\nonumber \\ \end{aligned}$$

(B7)

We use the following logarithmic expansion:

$$\begin{aligned}&\log _e [(h (x, y, t)/\exp (\varpi _1 T_0 ))+1] = [h (x, y, t)/\exp (\varpi _1 T_0 )] \nonumber \\&\quad +\, (1/2) [(h (x, y, t))/\exp (\varpi _1 T_0 )]^{2} + (1/3) [(h (x, y, t))/\exp (\varpi _1 T_0 )]^{3} + \cdots \end{aligned}$$

(B8)

We observe that $[h (x, y, t)]^{L}$ given in Eq. (B7) converges to zero as L tends to infinity.

Also the truncation error in Eq. (B8) is observed as $4.113\times 10^{-5}.$

Hence, for the sake of brevity, neglecting the terms with order more than one, we obtain

$$\begin{aligned} \log _e [(h (x, y, t)/\exp (\varpi _1 T_0 ))+1] \cong [h (x, y, t)/\exp (\varpi _1 T_0 )] \end{aligned}$$

Hence, Eq. (B6) becomes

$$\begin{aligned} T(x, y, t)\cong & {} T_0 +\bigg [\{1/u (x)\exp (\varpi _1 T_0 )\}\left\{ \sum _{m=1}^\infty \sum _{n=1}^\infty \{S(\beta _n , x) S(\alpha _m , y) \right. \\&\left. \times \, [E_1 \exp (-\omega t)+E_2 \exp (\omega t)+E_3 \exp (-A_2 t)]\}\phantom {T(x, y, t)\cong T_0 +[\{1/u (x)\exp (\varpi _1 T_0 )\}\left\{ \sum _{m=1}^\infty \sum _{n=1}^\infty \{S(\beta _n , x) S(\alpha _m , y) \right. }\right\} \bigg ] \end{aligned}$$