Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time

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1 Tamkang Journal of Science and Engineering, Vol. 13, No. 2, pp (2010) 117 Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time Praveen Ailawalia 1 * and Rajneesh Kumar 2 1 Department of Mathematics, M. M. Engineering College, Maharishi Markandeshwar University, Mullana, District Ambala, Haryana, India 2 Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India, Abstract The analytic expressions for the displacements, microrotation, stresses and temperature distribution on the free surface of micropolar thermoelastic medium possessing cubic symmetry with one relaxation time as a result of time harmonic inclined load have been obtained. The inclined load is assumed to be a linear combination of a normal load and a tangential load. The Fourier transform has been employed to solve the problem. The variations of the displacements, microrotation, stresses and temperature distribution with the horizontal distance have been shown graphically. Key Words: Micropolar, Thermoelastic, Cubic Symmetry, Time Harmonic Inclined Load, Fourier Transforms 1. Introduction *Corresponding author. praveen_2117@rediffmail.com The classical uncoupled theory of thermoelasticity predicts two phenomena not compatible with physical observations. First, the equation of heat conduction of this theory does not contain any elastic terms, second, the heat equation is of a parabolic type, predicting infinite speeds of propagation for heat waves. Biot [1] introduced the theory of coupled thermoelasticity to overcome the first shortcoming. The governing equations for this theory are coupled, eliminating the first paradox of the classical theory. However, both theories share the second shortcoming since the heat equation for the coupled theory is also parabolic. Two generalizations to the coupled theory were introduced. The first is due to Lord and Shulman [2] who introduced the theory of generalized thermoelasticity with one relaxation time by postulating a new law of heat conduction to replace the classical Fourier law. This new law contains the heat flux vector as well as its time derivative. It contains also a new constant that acts as a relaxation time. The heat equation of this theory is of the wave-type, ensuring finite speeds of propagation for heat and elastic waves. The remaining governing equations for this theory, namely, the equations of motion and constitutive relations remain the same as those for the coupled and uncoupled theories. The second generalization to the coupled theory of thermoelasticity is what is known as the theory of thermoelasticity with two relaxation times or the theory of temperature-rate-dependent thermoelasticity. Muller [3], in a review of the thermodynamics of thermoelastic solids, 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 [5] obtained another version of the constitutive equations. These equations were also obtained independently and more explicitly by Suhubi [6]. This theory contains two constants that act as relaxation times and modify all the equations of the coupled theory, not only the heat equation. The classical Fourier s law of heat conduction is not violated if the me-

2 118 Praveen Ailawalia and Rajneesh Kumar dium under consideration has a center of symmetry. The classical theory of elasticity is inadequate to represent the behavior of some modern engineering structures like polycrystalline materials and materials with fibrous or coarse grain. The study of these materials requires incorporating the theory of oriented media. Micropolar Elasticity termed by Eringen [7] is used to describe the deformation of elastic media with oriented particles. A micropolar continuum is a collection of interconnected particles in the form of small rigid bodies undergoing both translational and rotational motions. The force at a point of a surface element of bodies of these materials is completely characterized by a stress vector and a couple stress vector at that point. The linear theory of micropolar thermoelasticity was developed by extending the theory of micropolar continua to include thermal effects by Eringen [8] and Nowacki [9]. Following various methods, the elastic fields of various loadings, inclusion and inhomogeneity problems, and interaction energy of point defects and dislocation arrangement have been discussed extensively in the past. Generally all materials have elastic anisotropic properties which mean the mechanical behavior of an engineering material is characterized by the direction dependence. However the three dimensional study for an anisotropic material is much more complicated to obtain than the isotropic one, due to the large number of elastic constants involved in the calculation. In particular, transversly isotropic and orthotropic materials, which may not be distinguished from each other in plane strain and plane stress, have been more regularly studied. A review of literature on micropolar orthotropic continua shows that Iesan [10 12] analysed the static problems of plane micropolar strain of a homogeneous and orthotropic elastic solid, torsion problem of homogeneous and orthotropic cylinders in the linear theory of micropolar elasticity and bending of orthotropic micropolar elastic beams by terminal couple. Nakamura et al. [13] applied finite element method for orthotropic micropolar elasticity. Kumar and Choudhary [14 18] have discussed various problems in orthotropic micropolar continua. Because a wide class of crystals such as W, Si, Cu, Ni, Fe, Au, Al etc., which are some frequent by used substances, belong to cubic materials. The cubic materials have nine planes of symmetry whose normals are on the three coordinate axes and on the coordinate planes making an angle /4 with the coordinate axes. With the chosen coordinate system along the crystalline directions, the mechanical behavior of a micropolar cubic crystal can be characterized by three independent elastic constants. Minagawa et al. [19] discussed the propagation of plane harmonic waves in a cubic micropolar medium. Kumar and Rani [20] studied time harmonic sources in a thermally conducting cubic crystal. Kumar and Ailawalia [21 24] discussed some source problems in micropolar medium with cubic symmetry. Kuo [25] and Garg et al. [26] have discussed the problem of inclined load in the theory of elastic solids. Kumar and Ailawalia [27,28] studied the response of moving inclined load in 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 analyze 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 inclined load in micropolar thermoelastic medium possessing cubic symmetry. The present investigation is to determine the components of displacement, microrotation, stresses and temperature distribution in micropolar thermoelastic medium with cubic symmetry due to time harmonic inclined load. The solution is obtained by introducing potential functions after employing integral transformation technique. The integral transform is inverted using a numerical method. 2. Problem Formulation We consider a homogeneous, micropolar generalized thermoelastic solid half-space with cubic symmetry. A rectangular coordinate system (x, y, z) having origin on the surface y = 0 and y-axis pointing vertically into the medium is considered. Suppose that an inclined line load F 0 is acting along the interface on the y-axis and its inclination with z-axis is. If we restrict our analysis to plane strain parallel to

3 Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time 119 (7) t22, t21, m23 are the components of normal force stress, tangential force stress and tangential couple stress respectively. A1, A2, A3, A4, B3 are characteristic constants of the material, n = (A1 + 2A2)aT, at is coefficient of linear expansion, r is the density and j is the microinertia, K* is the coefficient of thermal conductivity, C* is the specific heat at constant strain, t0 is the 2 2 thermal relaxation time and Ñ 2 = x y Introducing the dimensionless variables defined by the expressions Figure 1. Geometry of the problem. r xy-plane with displacement vector u = ( u1, u2,0) and r microrotation vector f = (0,0, f 3 ) then the field equations and constitutive relations for micropolar thermoelastic solid with cubic symmetry in the absence of body forces, body couples and heat sources can be written by following the equations given by Minagawa et al. [19] and Lord-Shulman [2] as, (8) (1) (9) in equations (1)-(4) we obtain (dropping the primes), (2) (10) (3) (11) (4) (5) (12) (6)

4 120 Praveen Ailawalia and Rajneesh Kumar Applying the Fourier transform with respect to x defined by (13) (25) Introducing potential functions defined by (14) ~ ~ on equations (21)-(24) and eliminating T and f 3 from the resulting expressions we get, in equations (10)-(13), q(x, y, t) and Y(x, y, t) are scalar potential functions we obtain, (26) (15) (27) (16) (17) (28) (18) The roots of equations (26) and (27) are given by (29) (19) Assuming time harmonic behavior as The solution of equations (26) and (27) satisfying radiation conditions is given by (20) (30) we obtain from equations (14)-(17), (31) (32) (21) (33) (22) (23) (24) (34)

5 Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time Boundary Conditions (38) We consider a normal line load F1 acting in the positive y direction on the surface y = 0 along the z-axis and a tangential load F2 acting at the origin in the positive x direction, then the boundary conditions on the surface y = 0 are (39) (40) (35) d(x) is Dirac delta function, h is the heat transfer coefficient h for isothermal boundary and h 0 for insulated boundary. Applying Fourier transform defined by (25) on the boundary conditions (35) and using (5)-(8), (14), (20) and (30)-(33), we obtain the expressions for displacement components, microrotation, force stress, couple stress and temperature distribution for micropolar thermoelastic solid with cubic symmetry as, (41) (42) (36) (43) (37) 4. Particular Cases Case (4.1): Neglecting micropolarity effect (i.e B3 = j = 0) we obtain the corresponding expressions for displacements, stresses and temperature distribution as, (44) (45) Figure 2. Components of the inclined load.

6 122 Praveen Ailawalia and Rajneesh Kumar (46) (47) (55) (48) (56) Case (4.3): Thermoelastic micropolar solid (49) Case (4.2): Neglecting thermal effect, the expressions for displacements, microrotation and stresses are obtained as, (4.3a) Taking A1 = l + 2m + K, A2 = l, A3 = m + K, A4 = m, B3 = g (57) in equations (36)-(42), (44)-(48) and (50)-(55) we obtain the corresponding expressions in isotropic micropolar thermoelastic medium, isotropic thermoelastic medium and isotropic micropolar medium. 5. Inclined Line Load (5.1): For an inclined line load F0 we have (see Figure 1) (50) (51) (52) (53) (58) Using (58) in (36)-(42), (44)-(48) and (50)-(55), we obtain the corresponding expressions in micropolar thermoelastic medium possessing cubic symmetry, thermoelastic medium possessing cubic symmetry and micropolar medium possessing cubic symmetry for time harmonic inclined load respectively. (5.2): Using (57) and (58) in (36)-(42), (44)-(48) and (50)-(55), we obtain the corresponding expressions in isotropic micropolar thermoelastic medium, isotropic thermoelastic medium and isotropic micropolar medium for time harmonic inclined load respectively. 6. Inversion of the Transform (54) To obtain the solution of the problem in the physical

7 Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time 123 domain, we must invert the transform in (36) (42), (44) (48) and (50) (55). These expressions are functions of y and the parameter of Fourier transform, hence are of the form ~ f (, y). To get the function f (x, y) in the physical domain we invert the Fourier transform using, (59) (60) f e and f o are respectively even and odd parts of the function ~ f (, y). The method for evaluating this integral is described by Press et al. [29] which involves the use of Rhomberg s integration with adaptive step size. This also uses the results from successive refinements of the extended trapezoidal rule followed by extrapolation of the results to the limit when the step size tends to zero. 7. Numerical Results and Discussions For numerical computations, we take the following values of relevant parameters for micropolar medium with cubic symmetry as, tangential couple stress M 23 =(m 23 / F 0 ) and temperature distribution T * =(T / F 0 ) for a micropolar thermoelastic medium possessing cubic symmetry cubic crystal (MTECC) and micropolar thermoelastic isotropic solid (MTEIS) have been studied and the variations of these components with distance x have been shown by (a) solid line ( ) for MTECC and =0, (b) dashed line ( ) for MTEIS and =0, (c) solid line with centered symbol (* * * *) for MTECC and =30, (d) dashed line with centered symbol(* * *) for MTEIS and =30, (e) solid line with centered symbol ( ) for MTECC and =60, (f) dashed line with centered symbol (o---o---o) for MTEIS and =60, (g) solid line with centered symbol ( ) for MTECC and =90, (h) dashed line with centered symbol ( ) for MTEIS and =90. These variations are shown in Figures 3 8. The computations are carried out for y = 1.0 in the range 0 x 10.0 at t = 0.1 and for one value of non-dimensional frequency = Discussions for Various Cases For the comparison with micropolar isotropic solid, following Eringen [30] and Dhaliwal and Singh [31], we take the following values of relevant parameters for the case of Magnesium crystal like material as, The values of tangential displacement U 1 =(u 1 / F 0 ), normal displacement U 2 =(u 2 / F 0 ), tangential force stress T 21 =(t 21 / F 0 ) normal force stress T 22 =(t 22 / F 0 ), The values of tangential displacement decrease sharply and then oscillate with increase in horizontal distance. The degree of sharpness, however, decreases with increase in the angle of inclination of source with normal direction. Also, very close to the point of application of source, the value of tangential displacement decreases with increase in the orientation of source for both MTECC and MTEIS. These variations of tangential displacement are shown in Figure 3. The values of normal displacement for MTEIS are large as compared to the values for MTECC. Close to the point of application of source, the values of normal displacement for MTECC are quite close to each other for different orientations of source but the difference in values at the same point is significant for MTEIS. These variations of normal displacement for different values of angle of inclination are shown in Figure 4. It is observed from Figures 5 and 6 that the variations

8 124 Praveen Ailawalia and Rajneesh Kumar Figure 3. Variation of tangential displacement U1(= u1/f0) with distance x. (Insulated boundary) Figure 4. Variation of normal displacement U2(= u2/f0) with distance x. (Insulated boundary) of tangential force stress are more oscillatory in nature in comparison to the variations of normal force stress. Also the variations of both the quantities are similar in nature for MTECC and MTEIS for a particular value of angle of inclination. Near the point of application of source, the values of normal force stress increases with increase in the orientation of applied source on the surface of solid. These variations of tangential force stress and normal force stress are shown in Figures 5 and 6 respectively. Figure 5. Variation of tangential force stress T21(= t21/f0) with distance x. (Insulated boundary) Figure 6. Variation of normal force stress T22(= t22/f0) with distance x. (Insulated boundary) The variations of tangential couple stress are less oscillatory in nature as compared to the variations of all other quantities. These oscillations decrease in nature with increase in angle of inclination of source with normal direction for both MTECC and MTEIS. The values of tangential couple stress are large for MTEIS as compared to the values for MTECC. These variations of tangential couple stress are shown in Figure 7. Contrary to the discussions given above for tangen-

9 Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time 125 Figure 7. Variation of tangential couple stress M 23 (= m 23 /F 0 ) with distance x. (Insulated boundary) Figure 8. Variation of temperature distribution T * (= T/F 0 ) with distance x. (Insulated boundary) tial couple stress, the values of temperature distribution are more for MTECC in comparison to the values for MTEIS. The variations of temperature distribution being oscillatory in nature are more for MTECC. Also it is observed that the magnitude of these oscillations decrease with increase in angle of inclination of source, applied on the surface of solid. 9. Conclusion The properties of a body depend largely on the direction of symmetry and the inclination of applied source. Frequency plays an important role in the study of deformation of a body. The values of normal force stress and tangential couple stress increases with increase in angle of inclination of source, near the point of application of source, but the values of tangential displacement and temperature distribution decreases with the orientation of source at the same point. References [1] Biot, M., Thermoelasticity and Irreversible Thermodynamics, J. Appl. Phys., Vol. 27, pp (1956). [2] Lord, H. W. and Shulman, Y., A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids, Vol. 15, pp (1967). [3] Muller, J., M., The Coldness a Universal Function in Thermoelastic Bodies, Arch. Ration. Mech. Anal., Vol. 41, pp (1971). [4] Green, A. E. and Laws, N., On the Entropy Production Inequality, Arch. Ration. Mech. Anal., Vol. 45, pp (1972). [5] Green, A. E. and Lindsay, K. A., Thermoelasticity, J. Elasticity, Vol. 2, pp. 1 5 (1972). [6] Suhubi, E, S., Thermoelastic Solids in A C Eringen (Ed.) Continuum Physics, Vol. 2, Academic Press, Newyork,, Part 2, Chapter 2 (1975). [7] Eringen, A. C., Linear Theory of Micropolar Elasticity, J. Math. Mech., Vol. 15, pp (1966). [8] Eringen, A. C., Foundations of Micropolar Thermoelasticity Course of Lectures No. 23, CISM Udine, Springer (1970). [9] Nowacki, M., Couple-Stresses in the Theory of Thermoelasticity, Proc. IUTAM Symposia, Vienna, Editors H., Parkus and L. I Sedov, Springer-Verlag, pp (1966). [10] Iesan, D., The Plane Micropolar Strain of Orthotropic Elastic Solids, Arch. Mech., Vol. 25, pp (1973). [11] Iesan, D., Torsion of Anisotropic Elastic Cylinders, ZAMM, Vol. 54, pp (1974). [12] Iesan, D., Bending of Orthotropic Micropolar Elastic Beams by Terminal Couples, An. St. Uni. Iasi, Vol.

10 126 Praveen Ailawalia and Rajneesh Kumar XX, pp (1974). [13] Nakamura, S., Benedict, R. and Lakes, R., Finite Element Method for Orthotropic Micropolar Elasticity, Int. J. Engg. Sci., Vol. 22, pp (1984). [14] Kumar, R. and Choudhary, S., Influence and Green s Function for Orthotropic Micropolar Continua, Archives of Mechanics, Vol. 54, pp (2002). [15] Kumar, R. and Choudhary, S., Dynamical Behavior of Orthotropic Micropolar Elastic Medium, Journal of Vibration and Control, Vol. 8, pp (2002). [16] Kumar, R. and Choudhary, S., Mechanical Sources in Orthotropic Micropolar Continua, Proc. Indian. Acad. Sci. (Earth Plant. Sci.), Vol. 111, pp (2002). [17] Kumar, R. and Choudhary, S., Response of Orthotropic Micropolar Elastic Medium Due to Various Sources, Meccanica, Vol. 38, pp (2003). [18] Kumar, R. and Choudhary, S., Response of Orthotropic Micropolar Elastic Medium Due to Time Harmonic Sources, Sadhana, Vol. 29, pp (2004). [19] Minagawa, S., Arakawa, K. and Yamada, M., Dispersion Curves for Waves in a Cubic Micropolar Medium with Reference to Estimations of the Material Constants for Diamond, Bull. JSME., Vol. 24, pp (1981). [20] Kumar, R. and Rani, L., Elastodynamics of Time Harmonic Sources in a Thermally Conducting Cubic Crystal, Int. J. Appl. Mech. Engg., Vol. 8, pp (2003). [21] Kumar, R. and Ailawalia, P., Behaviour of Micropolar Cubic Crystal Due to Various Sources, Journal of Sound and Vibration, Vol. 283, pp (2005). [22] Kumar, R. and Ailawalia, P., Deformation in Micropolar Cubic Crystal Due to Various Sources, Int. J. Solids. Struct., Vol. 42, pp (2005). [23] Kumar, R. and Ailawalia, P., Mechanical/Thermal Sources at Thermoelastic Micropolar Medium without Energy Dissipation Possessing Cubic Symmetry, International Journal of Thermophysics, Vol. 28, pp (2007). [24] Kumar, R. and Ailawalia, P., Deformations in Micropolar Thermoelastic Medium Possessing Cubic Symmetry Due to Inclined Loads, Mechanics of Advanced Materials and Structures, Vol. 15, pp (2008). [25] Kuo, J. T., Static Response of a Multilayered Medium under Inclined Surface Loads, J. Geophysical Research, Vol 74, pp (1969). [26] Garg, N. R., Kumar, R., Goel, A. and Miglani, A., Plane Strain Deformation of an Orthotropic Elastic Medium Using Eigen Value Approach, Earth Planets Space, Vol. 55, pp. 3 9 (2003). [27] Kumar, R. and Ailawalia, P. Moving Inclined Load at Boundary Surface, Applied Mathematics and Mechanics, Vol. 26, pp (2005). [28] Kumar, R. and Ailawalia, P. Interactions Due to Inclined Load at Micropolar Elastic Half-Space with Voids, Int. J. Appl. Mech. Engg., Vol. 10, pp (2005). [29] Press, W. H., Teukolsky, S. A., Vellerling, W. T. and Flannery, B. P., Numerical Recipes, Cambridge: Cambridge University Press (1986). [30] Eringen, A. C., Plane Waves in Non-Local Micropolar Elasticity, Int. J. Engg. Sci., Vol. 22, pp (1984). [31] Dhaliwal, R. S. and Singh, A., Dynamic Coupled Thermoelasticity, p. 726, Hindustan Publication Corporation, New Delhi, India (1980). Manuscript Received: Apr. 10, 2006 Accepted: Sep. 20, 2007

Thermoelastic Interactions without Energy Dissipation Due to Inclined Load

Tamkang Journal of Science and Engineering, Vol. 11, No. 2, pp. 109 118 (2008) 109 Thermoelastic Interactions without Energy Dissipation Due to Inclined Load Rajneesh Kumar 1 * and Leena Rani 2 1 Department