The effect of gravity and inclined load in micropolar thermoelastic medium possessing cubic symmetry under G-N theory

Mohmed I.A. Othmn S.M. Ao-Dh c Hneen A. Alosimi

a Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

b Department of Mathematics, Faculty of Science, Taif University, Taif 888, Saudi Arabia

c Department o f Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

Abstract The problem of the micropolar thermoelastic medium under (G-N) theory of both types II, III under the effect of the gravity was investigated. The normal mode analysis is used to obtain the solution of the physical quantities. Comparisons are made between the results predicted by the (G-N) theory of type II and type III in the presence and absence of gravity and for different values of the angle of inclination.

Keywords: Micropolar thermoelasticity; Green-Naghdi theory; Normal mode method; Gravity; Inclined load.

1.Introduction

Eringen's micropolar theory of elasticity [1] is now well known and does not need much introduction and in this theory, a load across a surface element is transmitted by a force vector along with a couple stress vector. The motion is characterized by six degrees of freedom three of translation and three of the microrotation. Boschi and Iesan [2] , Scarpetta [3] ,Passarella [4] , Eringen and ¸S uhubi [5] , Nowacki [6] and Eringen [7-9] developed the linear theory of micropolar elasticity.Tauchert et al. [10] also derived the basic equations of the linear theory of micropolar thermoelasticity. Dost and Tabarrok [11] presented the micropolar generalised thermoelasticity by using Green-Lindsay theory. One can refer to Dhaliwal and Singh [12] for a review on the micropolar thermoelasticity. Chandrasekhariah [13] formulated a theory of micropolar thermoelasticity which includes heat-flux among the constitutive variables. Othman and Singh [14] studied the effect of rotation on generalized micropolar thermoelasticity in a halfspace under five theories.

There are some engineering materials such as metals which are not suitable for use in experiments concerning second sound propagation because they possess a the relatively high rate of thermal damping, but given the state of recent advances in materials science, it may be possible in the foreseeable future to identify an idealized material for the purpose of studying the propagation of thermal waves at finite speeds.The relevant theoretical developments on the subject are due to Green and Naghdi [15-17] whose developed different theories labeled type I, type II, and type III. The (G-N) theory of type I in the linearized theory is equivalent to the classical coupled thermoelasticity theory, the second (G-N II) does not admit energy dissipation and the third (G-N III) admits dissipation of energy and the heat flux is a combination of type I and type II. Both types II and III imply a finite speed of propagation of heat waves. Othman and Abbas [18] studied the effect of rotation on thermoelastic waves with (G-N)theory in a homogeneous isotropic hollow cylinder. Abo-Dahab et al. [19] obtained an exact magneto-thermoelastic solution for a hollow sphere subjected to initial stress,rotation and magnetic field. Abo-Dahab et al. [20] discussed the effect of rotation on the wave propagation in hollow poroelastic circular cylinders. Hussein et al. [21] investigated the phenomena of initial stress, rotation and magnetic stress effects on isotropic elastic hollow cylinder. Ailawalia et al.[22] studied the effect of hydrostatic initial stress and rotation with Green-Naghdi of type III thermoelastic half-space. Elsirafy, et al. [23] deals the phenomena of voids and rotation parameters influence on P-wave in a thermoelastic half-space under Green-Naghdi theory. Many authors [24-27] studied the effect of magnetic field on thermoelastic medium.

Because a wide class of crystals such as W, Si, Cu, Ni, Fe,Au, Al etc., which are some frequent by using substances, belong to cubic materials. The cubic materials have nine planes of symmetry whose normal are on the three coordinate axes and on the coordinate planes making an angle π/4 with the coordinate axes. With the chosen coordinate system along the crystalline directions, the mechanical behavior of a cubic crystal can be characterized by five independent elastic constants.Minagawa [28] discussed the propagation of plane harmonic waves in a cubic micropolar medium. Kuo [29] has discussed the problem of inclined load in the theory of elastic solids.Recently, Kumar and Ailawalia [30] , studied the response of moving inclined load in the micropolar theory of elasticity.The deformation due to other sources such as strip loads,continuous line loads, etc. can also be similarly obtained. The deformation at any point of the medium is useful to analyse the deformation field around mining tremors and drilling into the crust of the earth. It can also contribute to the theoretical consideration of the seismic and volcanic sources since it can account for the deformation fields in the entire volume surrounding the source region. No attempt has been made so far to study the response of the inclined load in micropolar thermo-elastic medium possessing cubic symmetry.

In the classical theory of elasticity, the gravity effect is generally neglected. The effect of gravity and diffusion on micropolar thermoelasticity with temperature-dependent elastic modium under G-N theory was studied by Othman et al.[31] . The effect of gravity in the problem of propagation of waves in solids, in particular on an elastic globe, was first studied by Bromwich [32] . The influence of the gravitational field and rotation on a generalized thermoelastic medium using a dual-phase-lag model has studied by Othman et al.[33] . Subsequently, an investigation of the effect of gravity on the surface waves, on the propagation of waves in an elastic layer has been studied by De and Sengupta [34] . Othman et al. [34] have studied the effect of hall current and gravity on magneto-micropolar thermoelastic medium with microtemperatures. Ailawalia [36] studied the gravitational effect along with the rotational effect and with two temperatures on generalized thermoelastic medium, respectively. These problems are based on the more realistic elastic model since earth;the moon and all other planets have the strong gravitational effect. Othman et al. [37] investigated the 2D problem of micropolar thermoelastic rotating medium possessing cubic symmetry under the effect of inclined load with GN-III.

The present paper is to investigate the effect of the gravitational field on the plane waves in a micropolar thermoelastic medium under the (G-N) theory of type II and type III. The normal mode analysis is used to obtain the exact expressions for the considered variables. The distributions of the considered variables are presented graphically. A comparison is carried out between the stresses, couple stress,micro-rotation and displacement components as calculated from the micropolar thermo-elastic medium under (G-N)theory of type II and type III in the presence and the absence of gravity and different values of the angle of inclination.

2.Formulation of the problem

We consider a homogeneous micropolar thermoelastic halfspace under the influence of gravity. All the considered quantities are functions of the time variable t and of the coordinatesxandz.

Following Eringen [1] , the field equations for a homogeneous, isotropic micropolar thermoelastic solid without body forces, can be considered in the form,

The constitutive equations can be written as

We consider the normal source acting on the plane surface of micropolar thermo-elastic half-space under the influence of gravity as shown in Fig. 1 .

The system of governing equations of a micropolar thermoelasticity with gravity and without body forces consists of based on (G-N) theory, Refs. [35-37] :

Fig. 1. Geometry of the problem.

where,i,j,r= 1 , 2, 3 ,gis the gravity,Tis the temperature above the reference temperatureT0chosen so that| (T-T0) /T0| ＜＜ 1 , λ, µ are the Laméconstants, the components of the displacement vector u are u = (u1, 0,u3) ,tis the time, σijare the components of the stress tensor,eijare the components of strain tensor,Jthe microinertia moment,mijis the couple stress tensor, δijis the Kronecker delta, φ is the microrotation vector will be φ = (0, φ2, 0) , εijris the alternate tensor, the mass density is ρ, the specific heat at constant strain isCE, the thermal conductivity isK∗andKis the material characteristic of the (G-N) theory of type II and III,γ,γ1,k,α,β are constitutive coefficients ande=is the dilatation.

For simplifications we shall use the following nondimensional variables:

3.Normal mode analysis

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

4.Application

We consider an inclined loadpacting in the direction who make an angle θwith the direction of thex-axis

WhereF1,F2andf(x,t) are given function.

Using (16) , (21) , (26) on the non-dimensional boundary conditions and using (7) , (9) , (12) , (34) we obtain the expressions of displacements, force stress, coupled stress and temperature distribution for micropolar thermoelastic medium:

Invoking the boundary conditions (36) at the surfacez=0 of the plate, we obtain a system of four equations. After applying the inverse of matrix method, we have the values of the four constantsMn,n= 1 , 2, 3 , 4. Hence, we obtain the expressions of displacements, force stress, coupled stress and temperature distribution for micropolar thermoelastic medium.

5.Numerical results

To study the effect of the gravity field and inclined load,we now present some numerical results. For this purpose,copper is taken as the thermoelastic material for which we take the following values of the different physical constants as in Othman et al. [37] .

Fig. 2. Variation of the displacement distribution u 1 with variation of the rotation under (G-N) theory.

Fig. 3. Variation of the stress distribution σzz with variation of the rotation under G-N theory.

The above numerical technique, was used for the distribution of the real parts of the displacement componentu1, the stress component σzz, the couple stress componentmzyand the microrotation component φ2with the distancezin 2D for (G-N) theory of types II and III with and without gravity effect. All the physical quantities are shown graphically in Figs. 2-5 in the case of two different values of gravity(g= 0, 0. 1 , 0. 3 ) at θ= 30°.

Similarly the distribution of the real parts of the physical quantities with distancezin 2D for (G-N) theory of type III with inclined effect are shown graphically in Figs. 6-9 in the case of different values of angle ( θ= 15°, 30°, 45°, 60°) atg= 0. 1 .

Fig. 4. Variation of the tangential couple stress m zy with variation of the rotation under G-N theory.

Fig. 5. Variation of the microrotation component φ2 with variation of the rotation under G-N theory.

Fig. 6. Variation of the displacement distribution u 1 with variation of the angle under GN-III theory.

Fig. 2 shows the distributions of the displacement componentu1in the case of (g= 0, 0. 1 , 0. 3 ) and in the context of(G-N) of both types II and III, it noticed that the distribution ofu1increases with the increase of the gravity. Fig. 3 explains the distributions of the stress component σzzin the case of(g= 0, 0. 1 , 0. 3 ) and in the context of (G-N) theory of both types II and III. The distribution of σzzincreases as the increasing of the gravity with respect to type III and the vice versa for type II. Figs. 4 and 5 show the distributions of the couple stress componentmzyand the microrotation component φ2in the case of (g= 0, 0.1, 0.3) and in the context of (G-N)of both types II and III, it is noticed that the distribution ofmzyincreases with the decrease of the gravity, while the distribution of φ2decreases with the decrease of the gravity. It is explained that all the curves converge to zero, and the rotation has a significant role for the distributions of all physical quantities. Fig. 6 explains the distribution of the displacement componentu1in the case of ( θ= 15°, 30°, 45°, 60°) and in the context of (G-N III). The distribution ofu1is increasing with the increase of θ. Fig. 7 shows the distributions of stress components σzzin the case of ( θ= 15°, 30°, 45°, 60°) and in the context of (G-N III). It noticed that the distribution of stress component σzzincreases with the increase of θ.Figs. 8 and 9 depict the distribution of the couple stress componentmzyand the micro-rotation component φ2in the case of ( θ= 15°, 30°, 45°, 60°) and in the context of (GN III). It is noticed that the distribution of the couple stress componentmzyincreases with the increase of θwhile the distribution of the micro-rotation component φ2decreases with the increase of θ.

Fig. 7. Variation of the stress distribution σzz with variation of the angle under GN-III theory.

Fig. 9. Variation of the microrotation component φ2 with variation of the angle under GN-III theory.

Fig. 10. 3D variation of the displacement u 1 with the variation of x , z .

Fig. 11. 3D variation of the stress distribution σzz with the variation of x , z .

3D curves are representing the complete relations between physical quantities, and both components of distance as shown in Figs. 10 and 11 in the presence of the gravityg= 0. 1 and θ= 15°in the context of (G-N III). These figures are very important to study the dependence of these physical quantities on the two components of the displacement. The curves obtained are highly depending on the distance from origin,all the physical quantities are moving in wave propagation.

6.Conclusion

According to the above results, we can conclude that:

1. The curves of the physical quantities with (G-N III) are most differing from the curves with (G-N II).

2. The used method in the present article is applicable to a wide range of problems in thermoelasticity.

3. The values of all the physical quantities converge to zero by increasing the distancezand all the functions are continuous.

4. The gravity field and the inclined load play a significant role in the distribution of all the physical quantities.

The results presented in this paper should prove useful for researchers in material science, designers of new materials,physicists as well as for those working on the development of thermoelasticity and in practical situations as in geophysics,optics, acoustics, geomagnetic and oil prospecting etc.