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The quasistatic bending response is presented for a simply supported functionally graded rectangular plate subjected to a through-the-thickness temperature field under the effect of various theories of generalized thermoelasticity, namely, classical dynamical coupled theory, Lord and Shulman's theory with one relaxation time, and Green and Lindsay's theory with two relaxation times. The generalized shear deformation theory obtained by the first author is used. Material properties of the plate are assumed to be graded in the thickness direction according to a simple exponential law distribution in terms of the volume fractions of the constituents. The numerical illustrations concern quasistatic bending response of functionally graded square plates with two constituent materials are studied using the different theories of generalized thermoelasticity

In the recent years, functionally graded materials (FGMs) have gained considerable attention in many engineering applications. FGMs are considered as a potential structural material for future high-speed spacecraft and power generation industries. FGMs are new materials, microscopically inhomogeneous, in which the mechanical properties vary smoothly and continuously from one surface to the other [

The effect of thermal loading on the displacement and stress fields for FGM plates and shells has been studied by a number of authors. For example, Wetherhold et al. [

The theory of thermoelasticity that includes the effect of temperature change has been well established. According to this theory, the temperature field is coupled with the elastic strain field. This theory covers a wide range of extensions of classical dynamical coupled thermoelasticity. Lord and Shulman [

Lord and Shulman [

In this paper, a generalized nonclassical dynamic coupled thermoelasticity analysis is carried out on a functionally graded material plate. Through-the-thickness temperature distribution varying according to exponential law is considered and then temperature and stress behavior are presented for the mentioned plate. The governing equations of FGM plate for two-dimensional generalized thermoelastic problems are derived within the framework of the classical coupled theory, Lord and Shulman's theory, and Green and Lindsay's theory. The material properties of the functionally graded plate are assumed to vary continuously through the thickness according to an exponential law distribution of the volume fraction of the constituents. A generalized shear deformation theory is presented to obtain the governing equations. An exact solution for the coupled governing equations under simply supported boundary conditions is obtained. Numerical results are provided to show the influence of the material properties, and a temperature field on the displacement and stresses.

We consider a solid rectangular plate of length

The displacements of a material point located at

The strain components will be

In addition, the stress-strain-temperature relations for the linear thermoelastic materials are given, according to the generalized theories of thermoelasticity, by

In details, we can rewrite the stress components in the deferent theories of generalized thermoelasticity as follows:

It is clear that, by setting

Note that the stress-temperature modulus

To solve the problem, we obtain the stress and moment resultants for the FGM plate by integrating the stress components given in (

By using Hamilton's principle, the governing equations can be obtained in the form

The exact closed form solution of (

We present exact results for a simply supported FG square plate subjected to a transient thermal load. Since it is common in high-temperature applications to employ a ceramic top layer as a thermal barrier to a metallic structure, we choose the constituent materials of the FG plate to be Aluminum (Al) and Silicon (SiC) having the following material properties:

Dimensionless results for FGM square plates according to various theories of thermoelasticity with different time parameters.

Theory | ||||||
---|---|---|---|---|---|---|

0.05 | C-T | 0.333122102 | 0.383966985 | 0.569639832 | 0.393610131 | |

L-S | 0.333122105 | 0.383966988 | 0.569639832 | 0.393610133 | ||

G-L | 0.333122105 | 0.383966675 | 0.569639365 | 0.393609811 | ||

0.10 | C-T | 0.319915220 | 0.368744318 | 0.546983655 | 0.377986612 | |

L-S | 0.319915222 | 0.368744321 | 0.546983655 | 0.377986614 | ||

G-L | 0.319915222 | 0.368743701 | 0.546982732 | 0.377985978 | ||

0.20 | C-T | 0.268738109 | 0.309755974 | 0.459333491 | 0.317481666 | |

L-S | 0.268738111 | 0.309755977 | 0.459333491 | 0.317481667 | ||

G-L | 0.268738111 | 0.309754800 | 0.459331743 | 0.317480460 | ||

0.50 | C-T | |||||

L-S | ||||||

G-L | ||||||

0.80 | C-T | |||||

L-S | ||||||

G-L | ||||||

1.00 | C-T | |||||

L-S | ||||||

G-L |

It is well known that G-L theory is accurate to predict temperature, displacement, and stresses, so some results have been plotted in Figures

Through-the-thickness variation of dimensionless temperature

Through-the-thickness variation of dimensionless longitudinal stress

Through-the-thickness variation of dimensionless transverse shear stress

Through-the-thickness variation of dimensionless in-plane shear stress

Dimensionless displacement

Dimensionless longitudinal stress

Dimensionless transverse shear stress

Dimensionless in-plane shear stress

Now, we discuss the effect of relaxation times in G-L model. Let the first relaxation time

In this paper, the numerical illustrations concern quasistatic bending response of FG square plates are studied in the context of the generalized thermoelasticity theories. A refined shear deformation theory is used for this purpose. Material properties of the plate are assumed to be graded in the thickness direction according to a simple exponential law distribution in terms of the volume fractions of the constituents. An exact solution for the present problem is obtained. Numerical results are provided to show the influence of the material properties, and a temperature field on the displacement and stresses.

From these results, we can conclude that

the results of G-L model give an accurate prediction comparing with those obtained by the other two models;

the results of L-S model agree well with those obtained by C-T model;

the temperature may be independent of the parameters used in the different theories;

for higher values of the relaxation times, L-S and G-L models may be failed to get accurate solution comparing with the C-T model.

The investigators would like to express their appreciation to the Deanship of Scientific Research at King AbdulAziz University for their financial support of this study, Grant no. 180/428.