** Engineering ** Vol. 3 No. 6 (2011) , Article ID: 5343 , 10 pages DOI:10.4236/eng.2011.36074

Fibre-Reinforced Generalized Thermoelastic Medium under Hydrostatic Initial Stress

^{1}Department of Applied Sciences, Surya School of Engineering and Technology,

Surya World, Punjab, India

^{2}Department of Mathematics, M.M. Engineering College, Maharishi Markandeswar University, Haryana, India

E-mail: praveen_2117@rediffmail.com, shilpy.budhiraja@gmail.com

Received February 22, 2011; revised June 1, 2011; accepted June 10, 2011

**Keywords:** Generalized thermoelastic, Hydrostatic initial stress, Fibre-reinforced, Temperature distribution, Normal mode, Anisotropy

ABSTRACT

The present problem is concerned with the deformation of an infinite fibre-reinforced generalized thermoelastic medium with hydrostatic initial stress under the influence of mechanical force. The normal mode analysis is used to obtain the analytical expressions of the displacement components, force stress and temperature distribution. The numerical results are given and presented graphically for Green-Lindsay [1] theory of thermoela-sticity. Comparisons are made in the presence and absence of hydrostatic initial stress and anisotropy.

1. Introduction

The classical theories of thermo-elasticity involving infinite speed of propagation of thermal signals, contradict physical facts. During the last three decades, nonclassical theories involving finite speed of heat transportation in elastic solids have been developed to remove this paradox. In contrast to the conventional coupled thermo-elasticity theory which involves a parabolic-type heat transport equation, these generalized theories involving a hyperbolic-type heat transport equation are supported by experiments exhibiting the actual occurrence of wave-type heat transport in solids, called sound effect. The extended thermo-elasticity theory proposed by Lord and Shulman [2] incorporates a flux-rate term into Fourier’s law of heat conduction, and formulates a generalized form that involves a hyperbolic-type heat equation admitting finite speed of thermal signals. Muller [3] in a review of the thermodynamics of a thermoelastic solid, proposed an entropy production inequality, with the help of which he considered restrictions on a class of constitutive equations. A generalization of this inequality was proposed by Green and Laws [4]. Green and Lindsay [4] developed temperaturetype-dependent thermo-elasticity (TRDTE) theory by introducing relaxation time factors that does not violate the classical Fourier’s law of heat conduction and this theory also predicts a finite speed for heat propagation.

Barber [5] studied thermoelastic displacements and stresses due to a heat source moving over the surface of a half plane. Sherief [6] obtained components of stress and temperature distributions in a thermoelastic medium due to a continuous source. Dhaliwal et al. [7] investigated thermoelastic interactions caused by a continuous line heat source in a homogeneous isotropic unbounded solid. Chandrasekharaiah and Srinath [8] studied thermoelastic interactions due to a continous point heat source in a homogeneous and isotropic unbounded body. Sharma et al. [9] investigated the disturbance due to a time-harmonic normal point load in a homogeneous isotropic thermoelastic half-space. Sharma and Chauhan [10] discussed mechanical and thermal sources in a generalized thermoelastic half-space. Sharma et al. [11] investigated the steady-state response of an applied load moving with constant speed for infinite long time over the top surface of a homogeneous thermoelastic layer lying over an infinite half-space. Deswal and Choudhary [12] studied a two-dimensional problem due to moving load in generalized thermoelastic solid with diffusion.

Fibre-reinforced composites are used in a variety of structures due to their low weight and high strength. A continuum model is used to explain the mechanical properties of such materials. In the case of an elastic solid reinforced by a series of parallel fibres it is usual to assume transverse isotropy. In the linear case, the associated constitutive relations, relating infinitesimal stress and strain components, have five materials constants. The analysis of stress and deformation of fibrereinforced composite materials has been an important subject of solid mechanics for last three decades. Pipkin [13] and Rogers [14,15] did pioneer works on the subject. Craig and Hart [16] studied the stress boundary-value problem for finite plane deformation of a fibre-reinforced material. Sengupta and Nath [17] discussed the problem of surface waves in a fibre-reinforced anisotropic elastic media.. Singh and Singh [18] discussed the reflection of plane waves at the free surface of a fibre-reinforced elastic half-space. Singh [19] discussed the wave propagation in an incompressible transversely isotropic fibrereinforced elastic media. Singh [20] studied the effects of anisotropy on reflection coefficients of plane waves in fibre-reinforced thermoe-lastic solid. Kumar and Gupta [21] investigated a source problem in fibre-reinforced anisotropic generalized thermo-elastic solid under acoustic fluid layer.

The development of initial stresses in the medium is due to many reasons, for example, resulting from differences of temperature, process of quenching, shot pinning and cold working, slow process of creep, differential external forces, gravity variations, etc. The earth is assumed to be under high initial stresses. It is, therefore, of much interest to study the influence of these stresses on the propagation of stress waves. Biot [22] showed the acoustic propagation under initial stress, which is fundamentally different from that under a stress-free state. He has obtained the velocities of longitudinal and transverse waves along the co-ordinates axis only.

The wave propagation in solids under initial stresses has been studied by many authors for various models. The study of reflection and refraction phenomena of plane waves in an unbounded medium under initial stresses is due to Chattopadhyay et al. [23], Sidhu and Singh [24] and Dey et al. [25]. Montanaro [26] investigated the isotropic linear thermoelasticity with hydrostatic initial stress. Singh et al. [27], Singh [28] and Othman and Song [29] studied the reflection of thermoelastic waves from a free surface under a hydrostatic initial stress in the context of different theories of generalized thermoelasticity. Ailawalia et al. [30] investigated deformation in a generalized thermoelastic medium with hydrostatic initial stress. Ailawalia [31] obtained the components of displacement, stresses, temperature distribution of thermoelastic solid half-space under hydrostatic initial stress subjected to ramp-type heating and loading for G-N theory (type III).

The present paper is concerned with the investigations related to effect of hydrostatic initial stress in fibrereinforced generalized thermoelastic medium. Effects of hydrostatic initial stress and anisotropy are shown graphically on normal displacement, normal force stress and temperature distribution for Green-Lindsay [1] theory of thermoelasticity.

2. Basic Equations and Their Solutions

We consider a homogeneous thermally conducting transversely fibre-reinforced medium with hydros-tatic initial stress of infinite extent with cartesian coordinates system. To analyze the displacement components, stresses and temperature distribution at the interior of the medium, the continuum is divided into two half spaces defined by

1) half space I, ,

2) half space II, ,

if we restrict our analysis to the plane strain parallel to -plane with displacement vector, then the field equations and constitutive relations for such a medium in the absence of body forces and heat sources are written as,

(1)

(2)

(3)

(4)

(5)

(6)

(7)

where

and are material constants, are coefficients of thermal conductivity, are coefficients of linear expansion, are thermal relaxation times, are the components of displacement vector, is the the mass density, is the temperature change of a material particle, is the reference uniform temperature of the body and is the specific heat at constant strain.

For simplification, we shall use the following nondimensional variables

(8)

where

Substituting non-dimensional variables into Equations (1)-(3), we obtain (after dropping the primes)

(9)

(10)

(11)

3. Normal Mode Analysis

The solution of the considered physical variable can be decomposed in terms of normal modes as the following form

(12)

where is the complex time constant and is the wave number in -direction.

Using (12), Equations (9)-(11) take the form

(13)

(14)

(15)

where

(16)

Eliminating and from Equations (13)-(15), we obtain

(17)

where

(18)

(19)

(20)

(21)

The solution of Equation (17) has the form

(22)

(23)

(24)

where, are some parameters depending on and. are the roots of the characteristic equation (17).

Substituting from Equations (22)-(24) into (13)-(15), we obtain the following relations

(25)

(26)

(27)

(28)

where

(29)

Thus the solution of Equations (1)-(3) are

(30)

(31)

(32)

Normal mode analysis of the stress yields the following,

(33)

(34)

(35)

(36)

4. Boundary Conditions

In order to determine the parameters and, we consider the following boundary conditions at

(37)

where is the magnitude of mechanical force.

Using Equations (30)-(34) in boundary condition (37), we get six equations with six unknown parameters and as

(38)

(39)

(40)

(41)

(42)

(43)

Solving equations (38)-(43), the parameters and are derived as follows:

where are defined in the appendix.

5. Particular Cases

5.1. Isotropic Generalized Thermoelastic Medium with Hydrostatic Initial Stress

Substituting, , and, in Equations (30)-(36), we obtain the corresponding expressions of displacement, stress, and temperature distribution in isotropic generalized thermoelastic medium with hydrostatic initial stress.

5.2. Fibre-Reinforced Generalized Thermoelastic Medium

Letting, the expressions (30)-(36) reduce to the case of fibre-reinforced generalized thermoelastic medium.

5.3. Isotropic Generalized Thermoelastic Medium

Substituting, , and, and letting, the expressions (30)-(36) reduce to an isotropic generalized thermoelastic medium.

For all the cases discussed above the components of displacement, stresses and temperature distribution for the region, are obtained by inserting in Equations (30)-(36).

Similarly in the region, the components are obtained by inserting in Equations (30)-(36).

6. Special Cases of Thermoelastic Theory

6.1. Equation of Coupled Thermoelasticity

The equations of the coupled thermoelasticity (C-T theory) are obtained when

(44)

6.2. Lord-Shulman Theory

For the Lord-Shulman (L-S theory)

(45)

6.3. Green-Lindsay Theory

For Green-Lindsay(G-L theory),

(46)

where are the two relaxation times.

6.4. Equations of Generalized Thermoelasticity

The equations of the generalized thermoelasticity without energy dissipation (the linearized G-N theory of type II ) are obtained when

(47)

Equations (1) and (2) are the same and equation (3) takes the form

(48)

where is constant with the dimension of and are characteristic constants of this theory.

7. Numerical Results

With a view to illustrating the analytical procedure presented earlier, we now consider a numerical example for which computational results are given. The results depict the variations of normal displacement, normal force stress and temperature distribution in the context of G-L theory. For this purpose, we take the following values of physical constants as Singh [20]

The computations are carried out on the surface at. The graphical results for normal displacement, normal force stress and temperature distribution are shown in Figures 1-3 with, , , , for a

1) Fibre-reinforced generalized thermoelastic medium with hydrostatic initial stress (FRGTEHIS) by solid line.

2) Fibre-reinforced generalized thermoelastic medium without hydrostatic initial stress (FRGTEW-HIS) by solid line with centered symbol (*).

3) Isotropic generalized thermoelastic medium with hydrostatic initial stress (IGTEHIS) by dashed line.

4) Isotropic generalized thermoelastic medium without hydrostatic initial stress (IGTEWHIS) by dashed line with centered symbol (*).

These graphical results represent the solutions obtained by using the generalized theory with two relaxation times (G-L theory) by taking,.

8. Discussions

The values of normal displacement for the case of FRGTEHIS increase sharply in the range and then

Figure 1. Variations of normal displacement u_{2} with horizontal distance x for mechanical force: (^{________}) FRGTEHIS ^{__} Fibre-reinforced generalized thermoelastic medium with hydrostatic initial stress, (---------) FRGTEWHISFibre-reinforced generalized thermoelastic medium without a hydrostatic initial stress, (_{*}^{___}_{*}^{_____}_{*}) IGTEHIS ^{__} isotropic generalized thermoelastic medium with a hydrostatic initial stress, (_{*}----_{*}----_{*}) IGTEWHIS ^{__ }isotropic generalizedv thermoelastic medium without hydrostatic initial stress.

Figure 2. Variations of normal force stress t_{22} with horizontal distance x for mechanical force: (^{________})FRGTEHIS, (---------)FRGTEWHIS, (_{*}^{___}_{*}^{_____}_{*})IGTEHIS, (_{*}----_{*}------ _{*}) IGTEWHIS. [see Figure 1 for explanation of symbols].

oscillate with distance. In case of IGTEHIS, the variations of normal displacement are very less in magnitude and for FRGTEWHIS, IGTEWHIS, these variations are quite uniform and at a particular point these variations are opposite in nature i.e. when one is on the zenith and the other one is on the lowest point. These variations are shown in Figure 1.

It is observed from Figure 2 that the variations of normal force stress for FRGTEHIS, FRGTEWHIS, IGTEHIS are similar in oscillating manner. The variation of normal force stress for IGTEWHIS is highly oscillating in nature in comparison to the variations obtained for FRGTEHIS, FRGTEWHIS, IGTEHIS.

The deformation of the body effects the change in temperature to a large extent as compared to normal displacement and normal force stress which is evident from Figure 3. Among all the mediums, the variation of temperature distribution is least oscillating for IGTEHIS.

9. Conclusions

1) The affects of anisotropy and hydrostatic initial stress are observed on all the quantities.

2) The variations of the temperature distribution are more oscillatory in nature than those of of normal force stress and normal displacement .

3) The variations for L-S and G-L theory of thermoelasticity are close, although the authors have de-

Figure 3. Variations of temperature distribution T with horizontal distance x for mechanical force: (^{________})FRGTEHIS, (---------)FRGTEWHIS, (_{*}^{___}_{*}^{_____}_{*}) IGTEHIS, (_{*}----_{*}-----_{*})IGTEWHIS. [see Figure 1 for explanation of symbols].

picted the graphical results only for G-L theory.

10. REFERENCES

- A. E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689
- H. W. Lord and Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” Journal of Mechanics and Physics of Solids, Vol. 15, No. 5, September 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5
- I. M. Muller, “The Coldness, Universal Function in Thermoelastic Bodies,” Rational Mechanics Analysis, Vol. 41, No. 5, 1971, pp. 319-332.
- A. E. Green and N. Laws, “On the Entropy Production Inequality,” Archives of Rational Mechanics and Analysis, Vol. 45, No. 1, 1972, pp. 45-47. doi:10.1007/BF00253395
- J. R. Barber, “Thermoelastic Displacements and Stresses Due to a Heat Source Moving over the Surface of a Half Plane,” ASME, Transactions, Journal of Applied Mechanics, Vol. 51, No. 3, 1984, pp. 636-640. doi:10.1115/1.3167685
- H. H. Sherief, “Fundamental Solution of the Generalized Thermoelastic Problem for Short Times,” Journal of Thermal Stresses, Vol. 9, No. 2, 1986, pp. 151-164. doi:10.1080/01495738608961894
- R. S. Dhaliwal, S. R. Majumdar and J. Wang, “Thermoelastic Waves in an Infinite Solid Caused by a Line Heat Source,” International Journal of Mathematics and Mathematical Sciences, Vol. 20, No. 2, 1997, pp. 323-334. doi:10.1155/S0161171297000434
- D. S. Chandrasekharaiah and K. S. Srinath, “Thermoelastic Interactions without Energy Dissipation Due to a Point Heat Source,” Journal of Elasticity, Vol. 50, No. 2, 1998, pp. 97-108. doi:10.1023/A:1007412106659
- J. N. Sharma, R. S. Chauhan and R. Kumar, “TimeHarmonic Sources in a Generalized Thermoelastic Continuum,” Journal of Thermal Stresses, Vol. 23, No. 7, 2000, pp. 657-674. doi:10.1080/01495730050130048
- J. N. Sharma and R. S. Chauhan, “Mechanical and Thermal Sources in a Generalized Thermoelastic Half-Space,” Journal of Thermal Stresses, Vol. 24, No. 7, 2001, pp. 651-675. doi:10.1080/014957301300194823
- J. N. Sharma, P. K. Sharma and S. K. Gupta, “Steady State Response to Moving Loads in Thermoelastic Solid Media,” Journal of Thermal Stresses, Vol. 27, No. 10, 2004, pp. 931-951. doi:10.1080/01495730490440181
- S. Deswal and S. Choudhary, “Two-Dimensional Interactions Due to Moving Load in Generalized Thermoelastic Solid with Diffusion,” Applied Mathematics and Mechanics, Vol. 29, No. 2, 2008, pp. 207-221. doi:10.1007/s10483-008-0208-5
- A. C. Pipkin, “Finite Deformations of Ideal Fiber-Reinforced Composites,” In: G. P. Sendeckyi, Ed, Composites Materials, Academic, New York, 1973, pp. 251-308.
- T. G. Rogers, “Finite Deformations of Strongly Anisotropic Materials,” In: J. F. Hutton, J. R. A. Pearson and K. Walters, Eds., Theoretical Rheology, Applied Science Publication, London, 1975, pp. 141-168.
- T. G. Rogers, “Anisotropic Elastic and Plastic Materials,” In: P. Thoft-Christensen, Ed., Continuum Mechanics Aspects of Geodynamics and Rock Fracture, Mechanics, Reidel, 1975, pp. 177-200.
- M. S. Craig and V. G. Hart, “The Stress Boundary-Value Problem for Finite Plane Deformations of a Fibre-Reinforced Material,” Quarterly Journal of Mechanics and Applied Mathematics, Vol. 32, No. 4, 1979, pp. 473-498. doi:10.1093/qjmam/32.4.473
- P. R. Sengupta and S. Nath, “Surface Waves in Fobre-Reinforced Anisotropic Elastic Media,” Sadhana, Vol. 26, No. 4, 2001, pp. 363-370. doi:10.1007/BF02703405
- B. Singh and S. J. Singh, “Reflection of Plane Waves at the Free Surface of a Fibre-Reinforced Elastic HalfSpace,” Sadhana, Vol. 29, No. 3, 2004, pp. 249-257. doi:10.1007/BF02703774
- B. Singh, “Wave Propagation in an Incompressible Transversely Isotropic Fibre-Reinforced Elastic Media,” Archives of Applied Mechanics, Vol. 77, No. 4, 2007, pp. 253-258. doi:10.1007/s00419-006-0094-9
- B. Singh, “Effect of Anisotropy on Reflection Coefficients of Plane Waves in Fibre-Reinforced Thermoelastic Solid,” International Journal of Mechanics and Solids, Vol. 2, 2007, pp. 39-49.
- R. Kumar and R. Gupta, “Dynamic Deformation in Fibre-Reinforced Anisotropic Generalized Thermoelastic Solid under Acoustic Fluid Layer,” Multidispline Modeling in Materials and Structure, Vol. 5, No. 3, 2009, pp. 283-288. doi:10.1163/157361109789017014
- M. A. Biot, “Mechanics of Incremental Deformations,” John Wiley, New York, 1965.
- A. Chattopadhyay, S. Bose and M. Chakraborty, “Reflection of Elastic Waves under Initial Stress at a Free Surface,” Journal of Acoustical Society of America, Vol. 72, No. 1, 1982, pp. 255-263. doi:10.1121/1.387987
- R. S. Sidhu and S. J. Singh, “Comments on ‘Reflection of Elastic Waves under Initial Stress at a Free Surface’,” Journal of Acoustical Society of America, Vol. 74, No. 5, 1983, pp. 1640-1642. doi:10.1121/1.390130
- S. Dey, N. Roy and A. Dutta, “Reflection and Refraction of P-Waves under Initial Stresses at an Interface,” Indian Journal of Pure and Applied Mathematics, Vol. 16, No. 11, 1985, pp. 1051-1071.
- A. Montanaro, “On Singular Surface in Isotropic Linear Thermoelasticity with Initial Stress,” Journal of Acoustical Society of America, Vol. 106, No. 3, 1999, pp. 1586- 1588. doi:10.1121/1.427154
- B. Singh, A. Kumar and J. Singh, “Reflection of Generalized Thermoelastic Waves from a Solid Half-Space Under Hydrostatic Initial Stress,” Journal of Applied Mathematics and Computation, Vol. 177, No. 1, June 2006, pp. 170-177. doi:10.1016/j.amc.2005.10.045
- B. Singh, “Effect of Hydrostatic Initial Stresses on Waves in a Thermoelastic Solid Half-Space,” Journal of Applied Mathematics and Computation, Vol. 198, No. 2, May 2008, pp. 494-505. doi:10.1016/j.amc.2007.08.072
- M. I. A. Othman and Y. Song, “Reflection of Plane Waves from an Elastic Solid Half-Space under Hydrostatic Initial Stress without Energy Dissipation,” International Journal of Solids and Structures, Vol. 44, No. 17, August 2007, pp. 5651-5664. doi:10.1016/j.ijsolstr.2007.01.022
- P. Ailawalia, S. Kumar and G. Khurana, “Deformation in a Generalized Thermoelastic Medium with Hydrostatic Initial Stress Subjected to Different Sources,” Mechanics and Mechanical Engineering, Vol. 13, No. 1, 2009, pp. 5-24.
- P. Ailawalia, “Effect of Hydrostatic Initial Stress in Green-Naghdi (Type III) thermoelastic Half-Space Subjected to Ramp-Heating and Loading,” Journal of Technical Physics, Vol. 50, No. 4, 2009, 2010, pp. 403-415.

Appendix