Twodimensional generalized thermoelastic problem for anisotropic halfspace
Debkumar Ghosh^{1} , Abhijit Lahiri^{2} , Ibrahim A. Abbas^{3}
^{1, 2}Department of Mathematics, Jadavpur University, Kolkata, 700032, India
^{3}Department of Mathematics, Sohag University, Sohag, Egypt
^{1}Corresponding author
Mathematical Models in Engineering, Vol. 3, Issue 1, 2017, p. 2740.
https://doi.org/10.21595/mme.2017.18236
Received 8 February 2017; accepted 12 March 2017; published 30 June 2017
JVE Conferences
This paper concerns with the study of wave propagation in fibre reinforced anisotropic half space under the influence of temperature and hydrostatic initial stress. LordShulman theory is applied to the heat conduction equation. The resulting equations are written in the form of vector matrix differential equation by using Normal Mode technique, finally which is solved by Eigen value approach.
Keywords: eigenvalue, generalized thermoelasticity, normal mode, vectormatrix differential equation.
1. Introduction
Fibrereinforced composite(FRC) materials are usually low weight and high strength used in construction engineering. The physical property of FRC material is governed by the theory of elasticity for different materials with the direction along the direction of fibre. Green [1] studied wave propagation in anisotropic elastic plates. Abbas and Othman [2] discussed the distribution of wave propagation under hydrostatic initial stress of fibrereinforced materials in anisotropic halfspace. Baylies and Green [3] analyse the flexural waves in fibrereinforced laminated plates. Rogerson [4] discussed effect of penetration in a sixply composite laminates.
Most of the thermoelasticity and generalized thermoelasticity (coupled or uncoupled) problems have been solved by potential function approach. This method is not always suitable as discussed by Dhaliwal and Sherief [5] and Sherief and Anwar [6]. These may be summarized by the initial conditions and the boundary conditions for physical problems which are directly concern with the material quantities under consideration and not with the potential function. Also, the potential function representations are not convergent always while the physical problems in natural variables constitute convergent solution. So, the alternative method of potential function approach is eigenvalue approach. In this method, we obtain a vectormatrix differential equation from the basic equations which reduces finally to an algebraic eigenvalue problem and the solutions for the field variables are obtained by determining the eigenvalues and eigenvectors from the corresponding coefficient matrix. In this theory, body forces and/or heat sources are also accommodated as in Das and Lahiri [7], Bachher et al. [8]. Now, two different models of generalized thermoelastisities are extensively used. One is Lord and Shulman (LS) [9] theory and the other is Green and Lindsay (GL) [10] theory. Introducing one relaxtion time parameter in LS theory the heat conduction equation becomes hyperbolic type without violating conventional Fourier’s law. Whereas the GL theory modified the heat conduction equation as well as the equation of motion in coupled thermoelasticitywo relaxation time parameters. There are other three models (Model I, II and III by Green and Nagdhi [1113]) for generalized thermoelasticity concerned to the theory of with or without energy dissipation.
2. Development of governing equations
The stressstrain relation and the governing equations of motion without body forces and heat sources are written as follow:
$+2\left({\mu}_{L}{\mu}_{T}\right)\left({a}_{i}{a}_{k}{e}_{kj}+{a}_{j}{a}_{k}{e}_{ki}\right)+\beta {a}_{k}{a}_{m}{e}_{km}{a}_{i}{a}_{j}{\beta}_{ij}\left(T{T}_{0}\right){\delta}_{ij},i,j,k,m=\mathrm{1,2},3,$
We consider the problem of a elastic halfspace $(x\ge 0)$ in fibrereinforced anisotropic material with $a\equiv ({a}_{1},{a}_{2},{a}_{3})$ where ${a}_{1}^{2}+{a}_{2}^{2}+{a}_{3}^{2}=1$ as in I. A. Abbas [14], where the displacements are given:
We consider the direction of fibre as $a\equiv (\mathrm{1,0},0)$ with $x$axis as prefered direction, and Eqs. (15), reduces as given:
with:
where ${\alpha}_{11}$, ${\alpha}_{22}$ are linear thermal expansion coeeficients.
To transform the above governing equations in nondimensional forms, we introduce the nondimensional variables as follows:
$\left({{\sigma}^{\text{'}}}_{11},{{\sigma}^{\text{'}}}_{12},{{\sigma}^{\text{'}}}_{22}\right)=\frac{1}{\rho {c}_{1}^{2}}\left({\sigma}_{11},{\sigma}_{12},{\sigma}_{22}\right),{c}_{1}^{2}=\frac{{A}_{11}}{\rho}.$
Using nondimensional Eq. (13), the governing equations reduces to (eleminating primes for convenience):
where:
$\left({\epsilon}_{2},{\epsilon}_{3}\right)=\frac{{T}_{0}{\beta}_{11}}{{A}_{11}\rho {c}_{e}}\left({\beta}_{11},{\beta}_{22}\right),{\epsilon}_{1}=\frac{{K}_{11}}{{K}_{22}}.$
3. Solution procedure
3.1. Normal mode analysis: formulation of vectormatrix differential equation
For the solution of the Eqs. (1419), physical variables can be decomposed using normal modes Eq. (20) in the following form:
where $i=\sqrt{1}$, $\omega $ is the angular frequency and a is the wave number along $x$axis.
Using Eq. (20), Eqs. (1419) reduces to omitting ‘*’ for convenience:
where:
$M53=\frac{\frac{{R}_{p}}{2}{B}_{1}{B}_{4}}{{B}_{4}+\frac{{R}_{p}}{2}},M54=\frac{ia{B}_{3}}{{B}_{4}+\frac{{R}_{p}}{2}},M62=ia{\epsilon}_{3}(\omega +{t}_{0}{\omega}^{2}),$
$M63=\omega +{t}_{0}{\omega}^{2}+{\epsilon}_{2}{a}^{2},M64={\epsilon}_{2}^{2}\left(\omega +{t}_{0}{\omega}^{2}\right).$
Eqs. (2426) can be written in the form of vectormatrix differential equation as [2, 8]:
where $\overrightarrow{W}$ =${\left[uvT{u}^{\text{'}}{v}^{\text{'}}{T}^{\text{'}}\right]}^{T}$ and $A=\left[\begin{array}{ll}{L}_{11}& {L}_{12}\\ {L}_{21}& {L}_{22}\end{array}\right]$. Where ${L}_{11}$ is null matrix and ${L}_{12}$ identity matrix of order 3×3 respectively and ${L}_{21}$ and ${L}_{22}$ are given by:
${L}_{21}=\left(\begin{array}{ccc}M41& 0& 0\\ 0& M52& M53\\ 0& M62& M63\\ & & \end{array}\right),{L}_{22}=\left(\begin{array}{ccc}0& M45& 1\\ M54& 0& 0\\ M64& 0& 0\\ & & \end{array}\right).$
3.2. Solution of the vectormatrix differential equation
To solve the vectormatrix differential Eq. (27), we apply the method of eigenvalue approach,
The characteristic equation of the matrix $\overrightarrow{A}$ is given by:
The roots of the characteristic Eq. (28) are $\lambda ={\lambda}_{i}$, $i=$ 1, 2, 3 which are of the form $\lambda =\pm {\lambda}_{1}$, $\lambda =\pm {\lambda}_{2}$ and $\lambda =\pm {\lambda}_{3}$ and they are also eigenvalues of the matrix.
The eigenvector, $\overrightarrow{W}$ corresponding to the eigenvalue $\lambda $ can obtained as:
where ${\delta}_{1}={a}_{2}{b}_{3}{a}_{3}{b}_{2}$, ${\delta}_{2}={a}_{3}{b}_{1}{a}_{1}{b}_{3}$, ${\delta}_{3}={a}_{1}{b}_{2}{a}_{2}{b}_{1}$.
As in Lahiri et al. [7], the general solution of Eq. (27) which is regular as can be written as:
Hence the field variables can be written as the following:
$v={A}_{1}{x}_{12}{e}^{{\lambda}_{1}x}+{A}_{2}{x}_{22}{e}^{{\lambda}_{2}x}+{A}_{3}{x}_{32}{e}^{{\lambda}_{3}x},$
$T={A}_{1}{x}_{13}{e}^{{\lambda}_{1}x}+{A}_{2}{x}_{23}{e}^{{\lambda}_{2}x}+{A}_{3}{x}_{33}{e}^{{\lambda}_{3}x}.$
The simplified form of Eqs. (2123) can be written as:
where:
${R}_{12}\left(x\right)=\left[{\lambda}_{2}{x}_{21}+{B}_{1}ia{x}_{22}{x}_{23}\right]{e}^{{\lambda}_{2}x},$
${R}_{13}\left(x\right)=\left[{\lambda}_{3}{x}_{31}+{B}_{1}ia{x}_{32}{x}_{33}\right]{e}^{{\lambda}_{3}x},$
${R}_{21}\left(x\right)=\left[{\lambda}_{1}{B}_{1}{x}_{11}+{B}_{2}ia{x}_{12}{B}_{3}{x}_{13}\right]{e}^{{\lambda}_{1}x},$
${R}_{21}\left(x\right)=\left[{\lambda}_{2}{B}_{1}{x}_{21}+{B}_{2}ia{x}_{22}{B}_{3}{x}_{23}\right]{e}^{{\lambda}_{2}x},$
${R}_{21}\left(x\right)=\left[{\lambda}_{3}{B}_{1}{x}_{31}+{B}_{2}ia{x}_{32}{B}_{3}{x}_{33}\right]{e}^{{\lambda}_{3}x},$
${R}_{31}\left(x\right)=\left[{B}_{4}ia{x}_{11}{\lambda}_{1}{x}_{12}\right]{e}^{{\lambda}_{1}x},$
${R}_{32}\left(x\right)=\left[{B}_{4}ia{x}_{21}{\lambda}_{2}{x}_{22}\right]{e}^{{\lambda}_{2}x},$
${R}_{33}\left(x\right)=\left[{B}_{4}ia{x}_{31}{\lambda}_{3}{x}_{32}\right]{e}^{{\lambda}_{3}x}.$
4. Boundary conditions
Considering the problem of a halfspace $\varphi $, defined as follows:
In order to determine the arbitrary constants ${A}_{i}^{\text{'}}s$, $i=$ 1, 2, 3, we consider the boundary conditions as follows.
4.1. Case 1
a) Mechanical Boundary condition:
For stressfree surface $x=$0, ${\sigma}_{11}=0$, ${\sigma}_{12}=0.$
b) Thermal Boundary condition:
where $\nu $ is Biot’s number.
4.2. Case 2
a) Mechanical Boundary condition:
For stressfree surface $x=$0, ${\sigma}_{11}={P}_{1}+{P}_{2}{e}^{\omega t+iay}$, ${\sigma}_{12}=0$.
b) Thermal Boundary condition:
5. Numerical analysis
5.1. Case 1
5.1.1. Distribution of different stress components
Fig. 1 represents distribution of normal stress ${\sigma}_{11}$ for $y=$0.3.
For fixed time $t$, ${\sigma}_{11}$ gradually increases as $x$ increases. For fixed $x$ numerical values of ${\sigma}_{11}$ gradually decreases as t increases.
Fig. 2 represents distribution of normal stress ${\sigma}_{12}$ for $y=$0.2.
For fixed time $t$, ${\sigma}_{12}$ gradually decreases as $x$ increases. For fixed x numerical values of ${\sigma}_{12}$ gradually increases as $t$ increases.
Fig. 3 represents distribution of normal stress ${\sigma}_{22}$ for $y=$0.5.
For fixed time $t$, ${\sigma}_{22}$ gradually decreases as $x$ increases. For fixed $x$ numerical values of ${\sigma}_{12}$ gradually increases as $t$ increases.
Fig. 1. Stress component ${\sigma}_{11}$ at $y=$0.3 for different values of $t$ verses $x$
Fig. 2. Stress component ${\sigma}_{12}$ at $y=$0.2 for different values of $t$ verses $x$
Fig. 4 represents distribution of normal stress ${\sigma}_{11}$ for different values of $x$ and $y$ for fixed $t=$0.1 and $\omega =$0.5.
$x$ numerical value of ${\sigma}_{11}$ gradually decreases as $y$ increases. For fixed $y$ the numerical value of ${\sigma}_{11}$ gradually increases as $x$ increases. ${\sigma}_{11}$ is maximum when $x=$1 and $y=$0.
Fig. 5 represent distribution of normal stress ${\sigma}_{12}$ for different values of $x$ and $y$ for fixed $t=$0.4 and $\omega =$5.
For fixed $x$ numerical value of ${\sigma}_{12}$ gradually decreases as $y$ increases. For fixed $y$ the numerical value of ${\sigma}_{12}$ gradually decreases as $x$ increases. Significant changes occur in the region 0.2$\le x\le $0.6 and 0.6$\le y\le $1.0.
Fig. 6 represent distribution of stress component ${\sigma}_{22}$ at for different values of $x$ and $y$ for fixed $t=$0.1 and 0.1.
Fig. 4. Stress component ${\sigma}_{11}$ at $t=$0.1 and $\omega =$0.5 verses $x$ and $y$
Fig. 5. The variation of stress component ${\sigma}_{12}$ at $t=$0.4 and $\omega =$3 verses $x$ and $y$
Fig. 6. Stress component ${\sigma}_{22}$ at $t=$0.1 and $\omega =$0.1 verses $x$ and $y$
For fixed $x$ numerical value of ${\sigma}_{22}$ gradually increases as $y$ increases. For fixed $y$ numerical value of ${\sigma}_{22}$ gradually increases as $x$ increases.
Fig. 7 represent distribution of normal stress ${\sigma}_{12}$ for different values of $x$ and $t$ for fixed $y=$0.2 and 1.
For fixed $x$, nominal decreasing of numerical values of ${\sigma}_{12}$ has been seen as $t$ increases, while For fixed $t$, numerical values of ${\sigma}_{12}$ decreases gradually as $x$ increases. numerical values of ${\sigma}_{12}$ minimum at $x=$1 and 0.02$\le t\le $0.1
Fig. 8 represent distribution of normal stress ${\sigma}_{22}$ for different values of $x$ and $t$ for fixed $y=$0.5 and 2.
For fixed $x$, nominal decreasing of ${\sigma}_{22}$ has been seen as t increases. For fixed $t$, numerical values of ${\sigma}_{12}$ decreases as $x$ increases. Also, significant changes occur in the region 0.6$\le x\le $1.0 and 0$\le t\le $1.0.
Fig. 7. Stress component ${\sigma}_{12}$ at $y=$0.2 and $\omega =$1 verses $x$ and $t$
Fig. 8. The distribution of stress component ${\sigma}_{22}$ at $y=$0.5 and $\omega =$2 verses $x$ and $t$
5.1.2. Distribution of temperature
Fig. 9 represent distribution of temperature, $T$ for different values of $x$ and $y$ for fixed $t=$0.3 and 2.
For fixed $x$ numerical value of $T$ gradually decreases as $y$ increases. For fixed $y$ numerical value of $T$ gradually increases as $x$ increases. $T$ in minimum at $x=$0 and significant changes occurs in the region 0.6$\le x\le $ 1.0 and 0 $\le y\le $1.0
Fig. 10 represent distribution of temperature, $T$ for different values of $x$ and $t$ for fixed $y=$0.1 and 1.5.
Fig. 9. The variation of $T$ at $t=$0.3 and $\omega =$2 verses $x$ and $y$
For fixed $t$ numerical value of $T$ nominally increases as $x$ increases. For fixed $x$ numerical value of $T$ gradually increases as $t$ increases.
Fig. 11 represent distribution of temperature, $T$ for different values of $y$ and $t$ for fixed $x=$0.5 and 3.
For fixed $t$ numerical value of $T$ nominally increases as $y$ increases. For fixed $x$ numerical values of $T$ decreases as $t$ increases.
Fig. 10. Variation of $T$ at $y=$0.1 and $\omega =$1.5 verses $x$ and $t$
Fig. 11. Variation of $T$ at $x=$0.5 and $\omega =$3 verses $y$ and $t$
5.2. Case 2
5.2.1. Distribution of different stress components
Fig. 12 represents distribution of normal stress ${\sigma}_{11}$ for $y=$0.3, $t=$0.01 and 5 for different numerical values of ${R}_{p}$.
For fixed ${R}_{p}$, ${\sigma}_{11}$ gradually decreases as $x$ increases. For fixed x numerical values of ${\sigma}_{11}$ gradually decreases as ${R}_{p}$ increases.
Fig. 13 represents distribution of normal stress ${\sigma}_{11}$ for $y=$0.3, $t=$0.01 and 5 for different fractional values of ${R}_{p}$.
For fixed time ${R}_{p}$, ${\sigma}_{11}$ gradually increases as $x$ increases. For fixed $x$ numerical values of ${\sigma}_{11}$ gradually decreases as ${R}_{p}$ increases. Significant changes occurred for 0$\le x\le $ 0.4.
Fig. 14 represents the distribution of stress component ${\sigma}_{12}$ at $y=$0.2, $t=$0.03 and 0.3 for different fractional values ${R}_{p}$ of verses $x$ for $P1=$1.
For fixed time ${R}_{p}$, ${\sigma}_{11}$ gradually decreases as $x$ increases. For fixed x numerical values of ${\sigma}_{11}$ increases as ${R}_{p}$ increases. Significant changes occurred for 0$\le x\le $ 0.4.
Fig. 15 represents the distribution of stress component ${\sigma}_{12}$ at $y=$0.3, $t=$0.1 and 4 for different integral values ${R}_{p}$ for $P1=$0.
For fixed $Rp$, ${\sigma}_{12}$ gradually decreases as $x$ increases. For fixed $x$ numerical values of ${\sigma}_{12}$ increases as ${R}_{p}$ increases.
Fig. 16 represents the distribution of stress component ${\sigma}_{22}$ at $y=$0.2, $t=$0.6 and 0.4 for different values ${R}_{p}$ for $P1=$0.
For fixed ${R}_{p}=$0.9, ${\sigma}_{22}$ gradually increases as $x$ increases but for other fixed values of ${R}_{p}$, ${\sigma}_{22}$ gradually decreases as $x$ increases. For fixed $x$, ${\sigma}_{22}$ gradually increases as $x$ increases for ${R}_{p}=$0.9 but for other fixed values of ${R}_{p}$, ${\sigma}_{22}$ gradually decreases as $x$ increases.
Fig. 17 represents the distribution of stress component ${\sigma}_{11}$ for different values of $t$ for fixed $y=$0.2 and 1.5.
Fig. 12. Stress component ${\sigma}_{11}$ at $y=$0.3, $t=$0.01 and $\omega =$ 5 for different values ${R}_{p}$ of verses $x$
Fig. 13. Stress component ${\sigma}_{11}$ at $y=$0.2 and $\omega =$ 0.2 for different values ${R}_{p}$ of verses $x$
Fig. 14. Stress component ${\sigma}_{12}$ at $y=$0.2, $t=$0.03 and $\omega =$ 0.3 for different values ${R}_{p}$ of verses $x$ for $P1=$1
Fig. 15. Stress component ${\sigma}_{12}$ at $y=$0.3, $t=$0.1 and $\omega =$ 4 for different values ${R}_{p}$ of verses $x$ for $P1=$0
For fixed $t$, the numerical value of ${\sigma}_{11}$ gradually increases as $x$ increases. For fixed $x$, the numerical value of ${\sigma}_{11}$ gradually increases as $t$ increases.
Fig. 18 The distribution of stress component ${\sigma}_{12}$ for fixed $y=$0.3 and 0.5 for different values of $t$.
For fixed $t$, the numerical value of ${\sigma}_{12}$ gradually decreases as $x$ increases. For fixed $x$, the numerical value of ${\sigma}_{12}$ gradually increases as $t$ increases.
Fig. 16. Stress component ${\sigma}_{22}$ at $y=$0.2, $t=$0.6 and $\omega =$ 0.4 for different values ${R}_{p}$ of verses $x$ for $P1=$0
Fig. 17. Stress component ${\sigma}_{11}$ at $y=$0.2 and $\omega =$ 1.5 for different values of $t$ verses $x$
Fig. 18. Stress component ${\sigma}_{12}$ at $y=$0.3 and $\omega =$ 0.5 for different values of $t$ verses $x$
Fig. 19. Stress component ${\sigma}_{22}$ at $y=$0.5 and $\omega =$ 0.2 for different values of $t$ verses $x$
Fig. 19 represents the distribution of stress component ${\sigma}_{22}$ at fixed $y=$0.5 and 0.2 for different values of $t$.
For fixed $t$, the numerical value of ${\sigma}_{12}$ gradually increases as $x$ increases, but for fixed $x$, the numerical value of ${\sigma}_{12}$ gradually decreases as t increases.
Fig. 20 represents distribution of normal stress ${\sigma}_{11}$ for different values of $x$ and for fixed $t=$0.3 and $y=$0.3.
For fixed $x$ numerical value of ${\sigma}_{11}$ remain constant as increases, but for fixed $\omega $ the numerical value of ${\sigma}_{11}$ gradually decreases as $x$ increases.
Fig. 21 represents distribution of normal stress ${\sigma}_{12}$ for different values of $x$ and for fixed $y=$0.3 and $t=$0.5.
For fixed $x$, nominal increasing of numerical values of ${\sigma}_{12}$ has been seen as increases, while for fixed $\omega $, numerical values of ${\sigma}_{12}$ increases gradually as $x$ increases after $x=$0.6 (approx.). Numerical values of ${\sigma}_{12}$ minimum at $x=$0.5 (approx.) and 1$\le t\le $3.
Fig. 22 represents distribution of stress component ${\sigma}_{22}$ at for different values of $x$ and for fixed $t=$0.1 and $y=$0.4.
For fixed $\omega $ numerical value of ${\sigma}_{22}$ gradually decreases as $x$ increases. Numerical values of ${\sigma}_{22}$ minimum at $x=$1 and 0$\le \omega \le $1.
Fig. 20. Stress component ${\sigma}_{11}$ at $y=$0.3 and $t=$0.3 verses $x$ and $\omega $
Fig. 21. Stress component ${\sigma}_{12}$ at $y=$0.3 and $t=$0.5 verses $x$ and $\omega $
Fig. 22. Stress component ${\sigma}_{22}$ at $y=$0.4 and $t=$0.1 verses $x$ and $\omega $
5.2.2. Distribution of temperature
Fig. 23 represents distribution of temperature, $T$ for different values of $x$ and $\omega $ for fixed $t=$ 0.1 and $y=$ 0.4.
For fixed $x$ numerical value of $T$ gradually increases in the region 0$\le \omega \le $0.3 (approx.) For fixed $\omega $ numerical value of $T$ nominally increases as $x$ increases.
Fig. 23. The variation of $T$ at $y=$0.4 and $t=$0.1 verses $x$ and $\omega $
6. Conclusion
We consider the physical parameters in SI units given in Dhaliwal and Singh following below to obtain the numerical result to observe the effect of wave propagation:
${\mu}_{T}=2.46\times 1{0}^{10}N/{m}^{2},{\mu}_{L}=5.66\times 1{0}^{10}N/{m}^{2},$
$\alpha =1.28\times 1{0}^{10}N/{m}^{2},\beta =220.90\times 1{0}^{10}N/{m}^{2},$
${\alpha}_{11}=0.017\times 1{0}^{4}de{g}^{1},{\alpha}_{22}=0.015\times 1{0}^{4}de{g}^{1},$
$l=0.5,{T}_{0}=293K,{c}_{e}=0.787\times 1{0}^{3}JK{g}^{1}de{g}^{1}$
${K}_{11}=0.0921\times 1{0}^{10}J{m}^{1}{s}^{1}de{g}^{1}$
${K}_{22}=0.0963\times 1{0}^{10}J{m}^{1}{s}^{1}de{g}^{1}$
${P}_{1}=0or1,{P}_{2}=0.1,{P}_{3}=0.2.$
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