Correction to: “On thermoelastic problem of a thermosensitive functionally graded rectangular plate with instantaneous point heat source” by V. R. Manthena, G. D. Kedar, Journal of Thermal Stresses, vol. 42, issue 7, pp. 849-862, 2019

This article refers to:

On thermoelastic problem of a thermosensitive functionally graded rectangular plate with instantaneous point heat source

Response to the comment on the paper “On thermoelastic problem of a thermosensitive functionally graded rectangular plate with instantaneous point heat source, V. R. Manthena, G. D. Kedar, Journal of Thermal Stresses, 2019, vol. 42, no. 7, pp. 849-862”, by Asterios Pantokratoras.

The authors would like to thank A. Pantokratoras for his critical analysis of the above mentioned paper. Our response to his queries is as follows:

(1) The following dimensionless parameters are used. (B1) $$\begin{array}{c}T*=\frac{T}{T_{m0}},T_0*=\frac{T_0}{T_{m0}},(x*,y*,z*)=\frac{(x,y,z)}{b},(a*,b*,c*)=\frac{(a,b,c)}{b},t*=\frac{\kappa t}{b^2},\rho *=\frac{\rho }{{\rho }_{m0}},\\ {\epsilon }_i*=\frac{{\epsilon }_{i}b}{k_{m0}},{\varpi }_i*=\frac{{\varpi }_i{b}^2}{\kappa },i=1,2,\omega *=\frac{\omega b^2}{\kappa },E_{j0}*=\frac{E_{j0}}{E_{m0}},{\alpha }_{j0}*=\frac{{\alpha }_{j0}}{{\alpha }_{m0}},j=m,c*.\end{array}$$(B1) where $T_{m0},{\rho }_{m0},E_{m0},{\alpha }_{m0}$ being the reference values of temperature, density, Young’s modulus, coefficient of linear thermal expansion of the metal, $\kappa$ being thermal diffusivity, $\omega ,{\varpi }_i$ being the frequency.

The material properties $k(x,T),C(x,T),$ internal heat generation $g(x,y,z,t),$ and heat flow $f(y,z,t)$ are taken as (B2) $$\begin{array}{c}k(x,T)=k_{m0}k*(x*,T*)\\ C(x,T)=C_{m0}C*(x*,T*)\\ g(x,y,z,t)=q_{0}g*(x*,y*,z*,t*)\\ f(y,z,t)=q_{1}f*(y*,z*,t*)\end{array}$$(B2)

In Eqs. (B1)(B1) $$\begin{array}{c}T*=\frac{T}{T_{m0}},T_0*=\frac{T_0}{T_{m0}},(x*,y*,z*)=\frac{(x,y,z)}{b},(a*,b*,c*)=\frac{(a,b,c)}{b},t*=\frac{\kappa t}{b^2},\rho *=\frac{\rho }{{\rho }_{m0}},\\ {\epsilon }_i*=\frac{{\epsilon }_{i}b}{k_{m0}},{\varpi }_i*=\frac{{\varpi }_i{b}^2}{\kappa },i=1,2,\omega *=\frac{\omega b^2}{\kappa },E_{j0}*=\frac{E_{j0}}{E_{m0}},{\alpha }_{j0}*=\frac{{\alpha }_{j0}}{{\alpha }_{m0}},j=m,c*.\end{array}$$(B1) and (B2)(B2) $$\begin{array}{c}k(x,T)=k_{m0}k*(x*,T*)\\ C(x,T)=C_{m0}C*(x*,T*)\\ g(x,y,z,t)=q_{0}g*(x*,y*,z*,t*)\\ f(y,z,t)=q_{1}f*(y*,z*,t*)\end{array}$$(B2) , the quantities with asterisks are dimensionless, $q_0,q_1$ are functions having relevant dimensions, $k_{m0},C_{m0}$ are the thermal conductivity, specific heat capacity of the metal having relevant dimensions.

Using Eqs. (B1)(B1) $$\begin{array}{c}T*=\frac{T}{T_{m0}},T_0*=\frac{T_0}{T_{m0}},(x*,y*,z*)=\frac{(x,y,z)}{b},(a*,b*,c*)=\frac{(a,b,c)}{b},t*=\frac{\kappa t}{b^2},\rho *=\frac{\rho }{{\rho }_{m0}},\\ {\epsilon }_i*=\frac{{\epsilon }_{i}b}{k_{m0}},{\varpi }_i*=\frac{{\varpi }_i{b}^2}{\kappa },i=1,2,\omega *=\frac{\omega b^2}{\kappa },E_{j0}*=\frac{E_{j0}}{E_{m0}},{\alpha }_{j0}*=\frac{{\alpha }_{j0}}{{\alpha }_{m0}},j=m,c*.\end{array}$$(B1) and (B2)(B2) $$\begin{array}{c}k(x,T)=k_{m0}k*(x*,T*)\\ C(x,T)=C_{m0}C*(x*,T*)\\ g(x,y,z,t)=q_{0}g*(x*,y*,z*,t*)\\ f(y,z,t)=q_{1}f*(y*,z*,t*)\end{array}$$(B2) , the heat conduction Eq. (1), the initial and boundary conditions (2), Kirchhoff’s variable defined in Eq. (10) in the above paper becomes dimensionless (ignoring asterisks for convenience).

The internal heat generation $g(x,y,z,t)$ and heat flow $f(y,z,t)$ defined after Eq. (12) also become dimensionless ($x_0,y_0,z_0$ being dimensionless constants), in which $Q_0=(q_0{b}^2/k_{m0}{T}_{m0})$ and $Q_1=(q_{1}b/k_{m0}{T}_{m0})$ are respectively the dimensionless Pomerantsev reference number and Kirpichev reference number.

Using Eq. (10), the dimensionless heat conduction Eq. (1) becomes (B3) $$\frac{{\partial }^2\Theta }{\partial x^2}+\frac{{\partial }^2\Theta }{\partial y^2}+\frac{{\partial }^2\Theta }{\partial z^2}+g(x,y,z,t)=\rho \frac{\partial \Theta }{\partial t}$$(B3)

The value $(1/\kappa )$ which appears from Eq. (11) onwards should be $\rho ,$ and the sentence defining thermal diffusivity $\kappa$ is to be ignored.

(2) In numerical results and discussion section, the paragraph,

“For numerical computations, we introduce the following nondimensional parameters. $$\begin{array}{c}\overline{T}=\frac{T}{T_0},\overline{X}=\frac{x}{b},\overline{Y}=\frac{y}{b},\overline{Z}=\frac{z}{b},\tau =\frac{\kappa t}{b^{{}_2}},\overline{\omega }=\frac{\omega }{12(1+\nu ){\alpha }_0{T}_{0}b},\\ ({\overline{M}}_x,{\overline{M}}_y,{\overline{M}}_{xy})=\frac{(M_x,M_y,M_{xy})}{E_0{b}^3},({\overline{\sigma }}_{xx},{\overline{\sigma }}_{yy},{\overline{\sigma }}_{xy})=\frac{({\sigma }_{xx},{\sigma }_{yy},{\sigma }_{xy})}{E_0{\alpha }_0{T}_0}\end{array}$$ with parameters $a=4cm,b=2cm,$ surrounding temperature $T_0=320K.$” should be read as

“For numerical computations, we define, (B4) $$\begin{array}{c}\overline{T}=T,\overline{X}=x,\overline{Y}=y,\overline{Z}=z,\tau =t,\overline{w}=w,\\ ({\overline{M}}_x,{\overline{M}}_y,{\overline{M}}_{xy})=(M_x,M_y,M_{xy}),({\overline{\sigma }}_{xx},{\overline{\sigma }}_{yy},{\overline{\sigma }}_{xy})=({\sigma }_{xx},{\sigma }_{yy},{\sigma }_{xy})\end{array}$$(B4) where the variables/functions on the right hand side of eqn. (B4) are dimensionless, with parameters $a=4,b=2,T_0=320,T_{m0}=330.$”

The deflection in Eqs. (3), (5), and (7) should be w.