Thermoelastic Interactions without Energy Dissipation Due to Inclined Load

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 11, No. 2, pp (2008) 109 Thermoelastic Interactions without Energy Dissipation Due to Inclined Load Rajneesh Kumar 1 * and Leena Rani 2 1 Department of Mathematics, Kurukshetra University, Kurukshetra Haryana, India 2 M. M. Engineering College, Kurukshetra University, Mullana (Ambala) , Haryana, India Abstract The linear theory of thermoelasticity without energy dissipation is employed to investigate the disturbance due to inclined line load which is assumed to be a linear combination of a normal load and a tangential load. Laplace-Fourier transforms are applied to the basic equations to form a vector matrix differential equation, which is then solved by using eigenvalue approach. The displacements, stresses and temperature distribution so obtained in the physical domain are computed numerically and illustrated graphically for magnesium-like material for an insulated boundary. Key Words: Without Energy Dissipation, Laplace and Fourier Transforms, Eigen Value Approach, Mechanical Sources 1. Introduction Thermoelasticity theories that admit a finite speed for thermal signals (second sound) have aroused much interest in the last three decades. In contrast to the conventional coupled thermoelasicity theory based on parabolic heat equation [1], which predicts an infinite speed for the propagation of heat, these theories involve a hyperbolic heat equation and are referred to as generalized thermoelasticity theories. Among these generalized theories, the theory proposed by Lord and Shulman [2] and the one developed by Green and Lindsay [3] have been subjected to a large number of investigations. In view of some experimental evidence available in favor of finiteness of heat propagation speed, generalized thermoelasticity theories are considered to be more realistic than the conventional theory in dealing with practical problems involving very large heat fluxes at short intervals, like those occurring in laser units and energy channels. For review of the relevant literature, see [4 6]. In the decade of 1990 s Green and Naghdi proposed *Corresponding author. rajneesh_kuk@rediffmail.com three new thermoelastic theories based on entropy equality rather than the usual entropy inequality [7 10]. The constitutive assumptions for the heat flux vector are different in each theory. Thus, they obtained three theories that they called thermoelasticity of type I, thermoelasticity of type II and thermoelasticity of type III, respectively. When the theory of type I is linearized we obtain the classical system of thermoelasticity. The theory of type II does not allow the dissipation of the energy and it is usually known as thermoelasticity without energy dissipation and thermoelasticity of type III permits the propagation of thermal signals at both infinite and finite speeds. Chandrasekharaiah [11,12] established uniqueness results in the theory of thermoelasticity without energy dissipation. Li and Dhaliwal [13] studied thermal shock problem in thermoelasticity without energy dissipation. Scott [14] discussed energy dissipation of inhomogeneous plane waves in thermoelasticity. Chandrasekharaiah, Chandrasekharaiah and Srinath [15,16] studied thermoelastic waves in unbounded body with cylindrical and spherical cavity respectively. Iesan [17] studied theory of thermoelasticity without energy dissipation.

2 110 Rajneesh Kumar and Leena Rani Quintanilla [18] considered spatial behaviour in thermoelasticity without energy dissipation. Quintanilla [19, 20] discussed instability, non-existence in the nonlinear theory of thermoelasticity and existence in thermoelasticity without energy dissipation. Wang and Slattery [21] considered thermoelasticity without energy dissipation for initially stressed bodies. Thermoelasticity without energy dissipation of nonsimple materials was discussed by Quintanilla [22]. When the source surface is very long in one direction in comparison with the others, the use of two-dimensional approximation is justified and consequently calculations are simplified to a great extent and one gets analytical solution in a closed form. A very long stripsource and a very long line source are examples of such two-dimensional sources. Love [23] obtained expressions for the displacements due to a line source in an isotropic elastic medium. Maruyama [24] obtained the displacement and stress fields corresponding to long strike-slip faults in a homogeneous isotropic half-space. Okada [25, 26] provided a compact analytic expressions for the surface deformation and internal deformation due to inclined shear and tensile faults in a homogeneous isotropic half-space. Several authors [27 30] discussed the problems of inclined load in the theory of elastic solids. Kumar and Rani [31,32] investigated the response to inclined load in thermoelastic body with voids. We study the general plane strain problem of thermoelastic half-space due to different sources. The integral transform technique has been used to solve it. The results of the problem may be applied to a wide class of geophysical problems involving temperature change. The physical applications are encounter in the context of problems like ground explosions and oil industries etc. This problem is also useful in the field of geomechanics, where the interest is about the various phenomenon occurring in the earthquakes and measuring of displacement, stress and temperature distribution due to the presence of certain sources. The present investigation is to determine the components of displacements, stresses and temperature distribution in a thermoelastic halfspace due to an inclined line load in the form of Laplace and Fourier transforms, which are converted to the original solution numerically. The deformation due to other sources such as strip loads, continuous line loads etc. can also be similarly obtained. The deformation at a 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. 2. Basic Equations Following Green and Naghdi [7] the field equations and constitutive relations in thermoelastic solid without energy dissipation in the absence of body forces and heat sources can be written as: and (1) (2) (3) where, -Lame s constants, T-temperature change, =(3 +2 ) t, t -linear thermal expansion, u-displacement vector, t ij -stress tensor,,c e -density and specific c heat at constant strain, respectively, K * e ( 2 ) is 4 a material characteristic constant of the theory, ij - Kronecker delta, T 0 uniform temperature; a superposed dot represents differentiation with respect to time t, gradient operator and 2 is Laplacian operator. 3. Formulation and Solution of the Problem We consider a homogeneous, isotropic, thermoelastic half-space in the undeformed state at uniform temperature T 0. The rectangular Cartesian co-ordinate system (x,y,z) having origin on the plane surface z = 0 with z-axis pointing vertically into the medium is introduced. Suppose that an inclined line load F 0 per unit length is acting on y-axis and its inclination with z-direction is (Figure 1). For two-dimensional problem, we assume u = (u, 0, w) (4)

3 Thermoelastic Interactions without Energy Dissipation Due to Inclined Load 111 on equations (1) (2) and with the help of (4) and (7) (after suppressing the asterisks) we obtain (10) (11) (12) where primes in equations (10) (12) represent the first and second order differentiation with respect to z, respectively and Figure 1. Inclined load over a thermoelastic half-space. in equations (1) (3). The initial and regularity conditions are given by The equations (10) (12) can be written as (5) (13) (6) where To transform the equations (1) (3) into nondimensional form, we define the following variables (7) (8) where L is standard length. Applying the Laplace and Fourier transforms To solve the equation (13), we take (14) so that and (15) (9) which leads to an eigenvalue problem. The charac-

4 112 Rajneesh Kumar and Leena Rani terstic equation corresponding to the matrix A is given by det[a-qi] = 0, (16) which on expansion leads to The solution of equation (13) is given by (17) The eigenvalues of the matrix A are characterstic roots of equation (17) given as q ( =1,2,3). The eigenvector X (, p) corresponding to the eigenvalues q can be determined by solving the homogeneous equation [A-qI] X(,p) = 0, (18) The set of eigenvectors X (, p), ( = 1,2,3,4,5,6) may be obtained as (19) where B ( = 1,2,3,4,5,6) are arbitrary constants. Thus equation (19) represents the solution of the general problem in the plane strain case of homogeneous thermoelasticity by employing the eigenvalue approach and therefore can be applied to a broad class of problems in the Laplace and Fourier transforms. The transformed displacements and temperature distribution that satisfy the Regularity condition (6) are given by (20) (21) (22) where 4. Application Consider a normal line load F 1 per unit length, acting in the positive z-direction on the plane boundary z = 0 along the y-axis and a tangential line load F 2, per unit length, acting at the origin in the positive x-direction then the boundary conditions are (23) Figure 2. Normal and tangential loadings.

5 Thermoelastic Interactions without Energy Dissipation Due to Inclined Load 113 where () is the Dirac delta function, (x) and (x) specify the vertical and horizontal load distribution functions, respectively, along x-axis, h is heat transfer coefficient, F 1 and F 2 are magnitude of forces. F1 Using equations (7) and (8) along with F1 T, F 2 = F 2 in the boundary conditions (23) (suppressing the T0 asterisks for convenience) and applying the Laplace and Fourier transforms defined by (9), we obtain the transformed boundary conditions as 0 in equation (23). The Laplace and Fourier transforms with respect to the pair (x, ) forthecaseofauniform strip load of unit amplitude and width 2a applied at originofthecoordinatesystem(x=z=0)indimensionless form after suppressing the asterisks becomes (27) Case (1c). Linearly distributed force The solution due to linearly distributed force applied on the half-space surface is obtained by setting (24) where ~ ( ) and ~ ( ) are the Fourier transforms of (x) and (x) respectively. Making use of equations (3) (8) (suppressing the asterisks for convenience) and applying the Laplace and Fourier transforms defined by (9) in the transformed boundary conditions (24) and substituting the values of ~ u, w ~ ~ and T from equations (20) (22), we obtain the expressions for displacement components, stresses and temperature distribution which are presented in Appendix A. Inclined line load For an inclined line load F 0, per unit length, we have (see Figure 1) F 1 =F 0 cos, F 2 =F 0 sin. (25) Case (1a). Concentrated force In this case with (x) = (x), (x) = (x) ~ ( ) 1, ~ ( ) 1 (26) Case (1b). Uniformly distributed force The solution due to uniformly distributed force applied on the half-space surface is obtained by setting in equation (23) where 2a is the width of strip load. Using equations (7) and (8) (suppressing the asterisks) and applying the transforms defined by equation (9), we get (28) Using equation (25) in equation (A.1) and with the aid of equations (26) (28), we obtain the expressions for displacements, stresses and temperature distribution for different sources applied on the surface of thermoelastic half-space. Particular case: Neglecting the thermal effect we obtain the corresponding expressions in elastic medium. Sub-case 1: If h 0 in equation (A.1), we recover the corresponding expressions of displacements, stresses, and temperature distribution for the insulated boundary. Sub-case 2: If h in equation (A.1), we obtain the corresponding expressions of displacements, stresses, and temperature distribution for the isothermal boundary. 5. Inversion of the Transforms To obtain the solution of the problem in the physical domain, we must invert the transforms in equation (A.1). These expressions are functions of z, the parameters of

6 114 Rajneesh Kumar and Leena Rani Laplace and Fourier transforms p and, respectively, and hence are of the form ~ f (, z, p). To get the function f(x, z, t) in the physical domain, first we invert the Fourier transform using (29) where f e and f 0 are, respectively, even and odd parts of the function ~ f (,z,p). Thus, expression (29) gives us the Laplace transform f(x, z, p) of the function f(x,z,t). Following Honig and Hirdes [33], the Laplace transform function f(x, z, p) can be inverted to f(x, z, t). The last step is to calculate the integral in equation (29). The method for evaluating this integral is described by Press et al. [34], which involves the use of Romberg 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. 6. Numerical Results and Discussion Following Dhaliwal and Singh [35] we take the case of magnesium crystal-like material for numerical calculations. The physical constants used are Figure 3 depicts the variation of normal displacement w with distance x. For thermoelastic mediun, near the source application, the values of normal displacement w increase as the angle of inclination increases from =0 to =90. The values of w for =30,45, 60 lies between the corresponding displacements for normal line load and tangential line load in the whole range. The values of w for =90 show small variation about zero and become zero ultimately. For elastic medium, the values of normal displacement for =0 to = 90 show opposite oscillatory behaviour in the whole range. Figure 4 depicts the variation of temperature distribution T with distance x. The values of T for =0 are larger than rest of the angles of inclination in the whole range. The values of T decrease slowly with increase in distance x in the range 0 x 10 for all the angles of inclination. Figure 5 shows the variation of normal stress t zz with distance x. For thermoelastic medium, near the source application, the values of t zz increase as the angle of inclination decreases from =90 to =0. The values of normal stress initially decrease and then follow oscillatory pattern about zero in the whole range for =30, 45, 60 respectively, whereas for =0, the values of normal displacement decrease in the range 0 x 10, de- = Nm -2, = Nm -2, = kgm -3,c e = Jkg -1 K -1, = Nm -2 K -1,F 0 =1, T 0 = 298 K. The comparison of the values of normal displacement w, boundary temperature field T and normal stress t zz with distance x for elastic and thermoelastic medium for concentrated force (CF), uniformly distributed force (UDF) and linearly distributed force (LDF) are shown graphically in Figures 3 11 for different values of =0, 30, 45, 60, 90. The computations are carried out for non-dimensional time t = 0.1 at z = 1.0 in the range 0 x 10. For all the three forces solid and dashed lines without center symbols corresponds to thermoelastic medium and with center symbols corresponds to elastic medium. The results for distributed force are presented for dimensionless width a = 1. Figure 3. Variation of normal displacement w with distance x (concentrated force).

7 Thermoelastic Interactions without Energy Dissipation Due to Inclined Load 115 lies between the corresponding values of =0 to = 90. For elastic medium the values of normal displacement start with a small decrease and depict oscillatory pattern about zero in the whole range. Figure 7 depicts the variation of temperature distribution T with distance x. Intially as the angle of inclination increases the values of temperature decreases. The values of T for =0 remain constant in the whole range whereas for =30 to =90 the values of T depicts os- Figure 4. Variation of temperature T with distance x (concentrated force). Figure 6. Variation of normal displacement w with distance x (uniformly distributed force). Figure 5. Variation of normal stress t zz with distance x (concentrated force). picts opposite oscillatory pattern in the range 1.1 x 2.5 and same pattern in the range 2.6 x 10 in comparison to rest of the angles of inclination. For elastic medium, the values of normal displacement increase in the ranges 0 x 2, 3.6 x 5, 7 x 9 and decrease in rest of the ranges for all angles of inclinations. Figure 6 depicts the variation of normal displacement w with distance x. For thermoelastic medium, the values of normal displacement w for =30, 45, 60 Figure 7. Variation of temperature T with distance x (uniformly distributed force).

8 116 Rajneesh Kumar and Leena Rani cillatory pattern about zero in the range 0 x 10. Figure 8 shows the variation of normal stress t zz with distance x. For all the angles of inclination the values of normal stress for elastic medium are greater than thermoelastic medium in the ranges 0 x 2.5, 4.5 x 6, 7.5 x 9.3 and less in rest of the ranges for all angles of inclinations. The variation of normal displacement and temperature distribution for linearly distributed force as well as for uniformly distributed force are same with difference in their magnitude. These variations are shown in Figures 9 and 10, respectively. Figure 11 shows the variation of normal stress t zz with distance x. Initially, The values of normal stress for thermoelastic medium start with a small increase whereas for elastic medium start from zero and then become oscillatory about zero in the whole range for all the angles of inclination. Figure 8. Variation of normal stress t zz with distance x (uniformly distributed force). Figure 10. Variation of temperature T with distance x (linearly distributed force). Figure 9. Variation of normal displacement w with distance x (linearly distributed force). Figure 11. Variation of normal stress t zz with distance x (linearly distributed force).

9 Thermoelastic Interactions without Energy Dissipation Due to Inclined Load Conclusion 1. At the point of source application the values of normal displacement for concentrated force show opposite pattern for elastic and thermoelastc medium whereas show same oscillatory pattern for uniformly distributed force and linearly distributed force for all the angles of inclination. 2. It is evident from the figures that the values of normal displacement, stresses and temperature distribution for =45 lies between the corresponding values of =0 and =90. where h * * Appendix A (A.1) References [1] Chadwick, P., Thermoelasticity, the Dynamic Theory, Progress in Solid Mechanics, Vol. 1 (Sneddon, I. N., Hill. R., eds.), p. 265, Amsterdam: North-Holland (1960). [2] Lord, H. W. and Shulman, Y., A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids, Vol. 15, pp (1967). [3]Green,A.E.andLindsay,K.A., Thermoelasticity, J. Elasticity, Vol. 2, pp. 1 7 (1972). [4] Chandrasekharaiah, D. S., Thermoelasticity with Second Sound: A Review, Appl. Mech. Rev. Vol. 39, pp (1986). [5] Ignaczak, J., Generalized Thermoelasticity and its Applications. In: Thermal stresses, Vol. III (Hetnarski, R. B. ed.). Chapter 4. Oxford: Elsevier (1989). [6] Joseph, D. D. and Preziosi, L., Heat Waves, Revs. Mod. Phys. Vol. 61, pp (1989) and addendum Vol. 62, pp (1990). [7] Green, A. E. and Naghdi, P. M., A Re-Examination of the Basic Postulates of Thermomechanics, Proc. R. Soc. London, Ser. A, Vol. 432, pp (1991). [8] Green, A. E. and Naghdi, P. M., On Undamed Heat Waves in Elastic Solid, J. Thermal Stresses, Vol. 15, pp (1992). [9] Green, A. E. and Naghdi, P. M., Thermoelasticity without Energy Dissipation, J. Elasticity, Vol. 31, pp (1993). [10] Green, A. E. and Naghdi, P. M., A Unified Procedure for Construction of Theories of Deformable Media. I. Classical Continuum Physics, II. Generalized Continua, III. Mixtures of Interacting Continua, Proc. R. Soc. London A, Vol. 448, pp , , (1995).

10 118 Rajneesh Kumar and Leena Rani [11] Chandrasekharaiah, D. S., A Note on the Uniqueness of Solution in the Linear Theory of Thermoelasticity without Energy Dissipation, J. Elasticity, Vol. 43, pp (1996). [12] Chandrasekharaiah, D. S., A Uniqueness Theorem in the Theory of Thermoelasticity without Energy Dissipation, J. Thermal Stresses, Vol. 19, pp (1996). [13] Li, H. and Dhaliwal, R. S., Thermal Shock Problem in Thermoelasticity without Energy Dissipation, Indian J. pure appl. Math., Vol. 27, pp (1996). [14] Scott, N. H., Energy and Dissipation of Inhomogeneous Plane Waves in Thermoelasticity, Wave motion, Vol. 23, pp (1996). [15] Chandrasekharaiah, D. S. and Srinath, K. S., Axisymmetric Thermoelastic Interaction without Energy Dissipation in an Unbounded Body with Cylindrical Cavity, J. Elasticity, Vol. 46, pp (1997). [16] Chandrasekharaiah, D. S. and Srinath, K. S., Thermoelastic Waves without Energy Dissipation in an Unbounded Body with Spherical Cavity, Internat. J. Math and Math. Sci., Vol. 23, pp (2000). [17] Iesan, D., On the Theory of Thermoelasticity without Energy Dissipation, J. Thermal Stresses, Vol. 21, pp (1998). [18] Quintanilla, R., On the Spatial Behaviour in Thermoelasticity without Energy Dissipation, J. Thermal Stresses, Vol. 22, pp (1999). [19] Quintanilla, R., Instability, Non-existence in the Nonlinear Theory of Thermoelasticity without Energy Dissipation, Continuum Mech. Thermodyn., Vol. 13, pp (2001). [20] Quintanilla, R., On Existence in Thermoelasticity without Energy Dissipation, J. Thermal Stresses, Vol. 25, pp (2002). [21] Wang, J. and Slattery, S. P., Thermoelasticity without Energy Dissipation for Initially Stressed Bodies, International Journal of Mathematics and Mathematical Sciences, Vol. 31, pp (2002). [22] Quintanilla, R., Thermoelasticity without Energy Dissipation of Nonsimple Materials. Jour. Appl. Mathematics Mechanics (ZAMM), Vol. 83, pp (2003). [23] Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, U.S.A. (1944). [24] Maruyama, T., On Two-Dimensional Elastic Dislocations in an Infinite and Semi-Infinite Medium, Bull. Earthq. Res. Inst. Vol. 44, pp (1966). [25] Okada, Y., Surface Deformation due to Inclined Shear and Tensile Faults in a Homogeneous Isotropic HalfSpace, Bull. Seismol. Soc. Am. Vol. 75, pp (1985). [26] Okada, Y., Internal Deformation due to Shear and Tensile Faults in a Half-Space, Bull. Seismol. Soc. Am. Vol. 82, pp (1992). [27] Kuo, J. T., Static Response of a Multilayered Medium under Inclined Surface Loads, Journal of Geophysical research, Vol. 74, pp (1969). [28] Gerstle, F. P. and Pearsall, G. W., The Stress Response of an Elastic Surface to a High Velocity, Unlubricated Punch, ASME Jounal of Applied Mechanics, Vol. 41, pp (1974). [29] Kumar, R., Miglani, A. and Garg, N. R., Plane Strain Problem of Poroelasticity using Eigenvalue Approach, Proceeding Indian Acad. Sci. (Earth Planet Sci.), Vol. 109, pp (2000). [30] Kumar, R., Miglani, A. and Garg, N. R., Response of an Anisotropic Liquid-Saturated Porous Medium due to Two-Dimensional Sources, Proceeding Indian Acad. Sci. (Earth Planet Sci.), Vol. 111, pp (2002). [31] Kumar, R. and Rani, L., Response of Thermoelastic Half-Space with Voids due to Inclined Load, (Accepted for publication in International Journal of Applied Mechanics and Engineering, IJAME, POLAND). [32] Kumar, R. and Rani, L., Deformation due to Inclined Load in Thermoelastic Half-Space with Voids, Archives of Mechanics, POLAND, Vol. 57, pp (2005) [33] Honig, G. and Hirdes, U., A Method for the Numerical Inversion of Laplace Transform, Journal of Computational and Applied Mathematics, Vol. 10, pp (1984). [34] Press,W. H., Teukolshy, S. A., Vellerling, W. T. and Flannery, B. P., Numerical Recipes in FORTRAN (2nd edn.), Cambridge University Press, Cambridge (1986). [35] Dhaliwal, R. S. and Singh, A., Dynamic Coupled Thermoelasticity, p. 726, Hindustan Publ. Corp., New Delhi, India (1980). Manuscript Received: Mar. 4, 2005 Accepted: Apr. 24, 2006