Available online Journal of Scientific and Engineering Research, 2016, 3(6): Research Article

Transcription

1 vailable online 6 (6):88-99 Research rticle IN: 9-6 CODEN(U): JERBR -D Problem of Generalized Thermoelastic Medium with Voids under the Effect of Gravity: Comparison of Different Theories Mohamed I Othman Ezaira RM Edeeb Department of Mathematics Faculty of cience Zagazig University P.O. Box 59 Zagazig Egypt bstract The purpose of this paper is to study the -D problem of generalized thermoelastic medium with voids under the effect of gravity within the framework of the Green-Lindsay Lord hulman and classical coupled theories. The normal mode analysis is used in our problem. Numerical results with comparisons between theories are illustrated graphically. Comparisons are made between the three theories in the presence and absence of gravity and also with and without voids. Keywords gravity; Lord hulman; Green-Lindsay; thermoelasticity; normal mode analysis voids. UDC Introduction The effect of mechanical and thermal distribution of an elastic body is studied within the framework of the theory of thermoelasticity. The generalized theory of thermoelasticity is one of the modified versions of classical uncoupled and coupled theory of thermoelasticity that has been developed in order to remove the paradox of physical impossible phenomena of infinite velocity of thermal signals in the classical coupled thermoelasticity theory stated by Biot []. The coupled CD theory of thermo-elasticity was extended by including the thermal relaxation times into the constitutive equation by Lord and hulman [] and Green and Lindsay []. These theories eliminate the paradox of an infinite velocity of the heat propagation and were termed the generalization theories of thermoelasticity there are the following differences between the two theories: i. The Lord-hulman L- theory involves one relaxation time of the thermo-elastic while the Green and Lindsay G-L theory takes into account two relaxation times. ii. In the L- theory the energy equation involves the first and second derivatives of the strain with respect to time whereas the corresponding equation in the G-L theory needs only the first derivative of this strain with respect to time. iii. In the linear case according to the approach of the G-L theory the heat cannot propagate with a finite speed unless the stresses depend on the temperature velocity whereas according to the L- theory the heat can propagate with a finite speed even though the stresses are independent of the temperature velocity. iv. The two theories are structurally different from one another and one cannot be obtained as a particular case of the other. Theory of linear elastic materials with voids is an important development of the classical theory of elasticity; this theory deals with materials which have a distribution of small voids where the volume of void is included among the kinematics variables and investigate various types of geological and biological materials since the classical theory of elasticity is not sufficient. The theory reduces to the classical theory in the limiting case of 88

2 Othman MI & Edeeb ERM 6 (6):88-99 the volume of void tending to zero. Cowin and Nuiato [] developed a theory of linear elastic materials with voids to study mathematically the mechanical behavior of porous solids. Puri and Cowin [5] studied the behavior of plane waves in a linear elastic material with voids. Iesan [6] developed the linear theory of thermoelastic materials with voids. Dhaliwal and Wang [7] developed a heat flux dependent theory of thermoelasticity with voids. Ciarletta and carpetta [8] discussed some results on thermoelasticity of dielectric materials with voids. Othman et al. [9] studied a -D problem of a rotating thermoelastic solid with voids under thermal loading due to laser pulse and initial stress type III. Othman and Edeeb [ ] investigated the -D problem of a rotating thermoelastic solid with voids thermal loading due to laser pulse and two-temperature under three theories. The effect of gravity on the wave propagation in an elastic solid medium was first considered by Bromwich [] treating the force of gravity as a type of body force. Othman et al. [] studied the effect of gravity on the generalized thermoelastic medium with temperature dependent properties for different theories. ilawalia and Narah [] studied the effect of rotation in generalized thermoelastic solid under the influence of gravity with an overlying infinite thermoelastic fluid. In the present work we have formulated the generalized thermoelastic medium with voids for three theories under the influence of gravity and solve for the components of displacement stresses temperature distribution. The normal mode method was used to obtain the exact expression for the considered variables. Comparisons are carried out between the considered variables as calculated from the generalized thermoelastic pours medium based on the L- G-L and CD theories in the absence and presence of gravity. In addition a comparison made between the three theories with and without voids.. Formulation of the Problem and Basic Equations We consider a homogeneous isotropic elastic body with voids in a half-space z under the effect of a constant gravitational field of acceleration g. We are interested in plane strain in the plane with displacement components u u such that u u ( x z t ) u u ( x z t ). Case : The basic governing equations of a linear thermoelastic medium with voids under the effect of gravity based on the L- G-L and CD theories are The stress-strain relation written as: σ ij [ λekk + b (+ ) T ] δij μ e ij t = () eij ( u i j +u j i ) () The dynamical equations of an elastic medium are given by e T u u μ u + ( λ + μ) b (+ ) + ρ g = ρ () x x t x x t + ( + ) e (+ ) T u u = () μ u λ μ b ρ g ρ z z t z x t The equation of voids is b e b m( ) T = ρ (5) t t t The heat conduction equation K T ρ C E (+ ) T + β T(+ n ) e mt(+ n ) t t + t. (6) Where σ ij are the components of stress tensor eij are the components of strain are the Lame' constants ( ) t such that t is the coefficient of thermal expansion ij is the Kronecker delta 89

3 Othman MI & Edeeb ERM 6 (6):88-99 b m are the material constants due to the presence of voids is the density C E is the specific heat at constant strain n is a parameter are the thermal relaxation times K is the thermal conductivity T is the reference temperature is chosen so that ( T T) T is the change in the volume fraction field. For a two dimensional problem in -plane Eq. () can be written as: u u u u ij [ ( ) b ( ) ] ij ( ) i j. x z t z x For the purpose of numerical evaluation we introduce dimensions variables ( x z' ) = ( xz ) c ( ) σ u u = ( u u c ) ij σ ij = σ ij T ' = T ' c T t' = '= c CE c. K Using the above dimensions quantities Eqs. ()-(6) become e T u u u 5 + (+ ) = (8) x x t x x t e T u u u 5 + (+ ) = (9) z z t z x t (7) g ' t g c 6e 7 8 9(+ ) T = () t t t T e T (+ n ) (+ ) + (+ n ). t t t t t t where bc c mt 8 9 T c gc mc C E c 5 b 6 K cc E () c 7. We define displacement potentials R and Q which relate to displacement components u and u as u R x Q z u R z Q x u u e R ( ) Q. z x Using Eq. () in Eqs. (8)-() we obtain: 5 ( )R Q (+ ) T = () t x t 5 R ( )Q = x t R ( ) ( ) T t t t R n T T (+ ) (+ ). t t t t t The components of stress tensor are C E () () (5) (6) (7) 9

4 Othman MI & Edeeb ERM 6 (6):88-99 u u u σ xx = [ ] (8) x z x σ σ zz Where u u u = [ ] T (9) x z z = u u. z x () bc T ( ).. Normal Mode nalysis The solution of the considered physical variable decomposed in terms of normal modes as the following form ij { RQ Tσ }( x zt ) = { R Q T σ }(z) exp[i( ωt + cx )]. ij Where R Q T σ ij are the amplitudes of the functions RQ T σ ij is the complex time constant i and c is the wave number in the x -direction. Using () in Eqs. ()-(7) we obtain 5T (D ) R Q + = () 6 7 R +(D ) Q = () (D c ) R +(D ) + = () (D c ) R +(D ) =. (5) Where c 5 ic ( i) 8 c 7 i8 9 9 ( i ( i n ) ) Eliminating () 5 6 ic 7 c 5 c i. Q and between Eqs. ()-(5) we get the following ordinary differential equation of the eighth order which satisfied with R : 8 6 [D BD + BD BD + B] R (z) =. Where B (6) B c c c c 5 8 c c B c c B c c c. c In a similar manner we get 8 6 B B B B R Q T Equation (7) factored as [D D + D D + ]{ (z) (z) (z) (z)} =. (7) k k k k R Q T [(D )(D )(D )(D )]{ (z) (z) (z) (z)} =. (8) Where k n ( n ) are the roots of the Eq. (7) The solution of Eq. (7) bound as z is given by: d D. dz 9

5 Othman MI & Edeeb ERM 6 (6):88-99 n k (9) R = M e n n k Q = H M e () n n k () = H M e = n H nm ne k z. () Where M n (n ) are constants. To obtain the components of the displacement vector from (9) and () in () u n = H nm ne k z () u n = H 5nM ne k z () From Eqs. ()-() in (8)-() to obtain the components of the stresses k xx = H 6nM ne zz (5) k 7n n (6) = H M e k 8n n. (7) = H M e 6 Where H n = H = ( k ) n k n c 9 k n n n [ 5( k n 8) + 9] [ ( ) + ( H ] n n + n k H H H n = H n = ic k nhn H 5n = ( k n ichn ) H 6n = (i ch n k nh 5n ) + i ch n + H n H n H 7n = (i ch n k nh 5n ) k nh 5n + H n H n H 8n = ( k nh n ich 5n).. Boundary Conditions In this section we need to consider the boundary conditions at z in order to determine the constants Mn ( n ). () The mechanical boundary condition i( t + cx) σ zz = Pe σ =. (8) z () The thermal boundary condition that the surface of the half-space is subjected to i(ωt + cx) T = Pe. (9) Where the magnitude of the applied force in the half-space is P and P is the applied constant temperature to the boundary. Using the expressions of the variables into the above boundary conditions (8) (9) we obtain H 7nM n = P () H8nM n = () 5 9

6 Othman MI & Edeeb ERM 6 (6):88-99 k nh nm n = () H nm n = P. () Invoking boundary conditions ()-() at the surface z of the plate we obtain a system of four equations. fter applying the inverse of matrix method we get the values of the four constants M n (n ). Case : The solution of wave propagation of a generalized thermoelastic medium without voids under the effect of gravity is: by putting b m equal to zero then the basic governing Eqs. () and ()-(6) of a linear thermoelastic medium without voids under the effect of gravity can be written as: e T u u μ u + ( λ + μ) (+ ) + ρ g = ρ () x t x x t e T u u μ u + ( λ + μ) (+ ) ρ g = ρ (5) z t z x t T e K T ρc E (+ ) + T(+ n ) t t t t u u u u ij [ ( ) ( ) ] ij ( ) i j. x z t z x The dimensions of Eqs. ()-(6) have the form e T u u u E E E E + (+ ) + = (8) x t x x t e T u u u E E E E + (+ ) = (9) z t z x t T e T (+ ) + (+ n ). t t t t Where T gc c K c ce E E E E. ce Using Eq. () in Eqs. (8)-(5) we obtain: ( F E ) R E Q E (+ ) T t x t E R ( E )Q = x t R Τ Τ (+ ). t t t (6) (7) (5) = (5) The components of stress tensor are u u u σ xx = E5[ ] E6( i) Τ (5) x z x u u u σ zz = E5[ ] E6( i) Τ (55) x z z σ u z u = [ ]. (56) x (5) (5) 9

7 Othman MI & Edeeb ERM 6 (6):88-99 T Where E E ( i) F E 5 6 Using () in Eqs. (5)-(5) we obtain F (D F ) R Q F Τ = (57) 5R F6 Q = (58) F + (D ) 7 8 E (D c ) R + (D E ) Τ = (59) where F Fc E ice E ( i) 5 6 F F F F F F F ice F c E i n c i 7 F8 Eliminating Q and order which satisfied with ( ). T between Eqs. (57)-(59) we get the following ordinary differential equation of sixth R 6 [D I D + I D I ] R (z) =. (6) I F F F F F Where F F6 F8 F F8 F F6 F7 c. In a similar manner we get 6 R Q T Equation (6) factored as I F F F F F F F F F F F F F c [D I D + I D I ]{ (z) (z) (z)} =. (6) k k k R Q Τ [(D )(D )(D )]{ (z) (z) (z)} =. (6) n Where k (n ) are the roots of the Eq. (6). The solution of Eq. (6) bound as z is given by: k R = M n e (6) n n (6) n= Q = L M e k k T = LnM ne (65) where M n (n ) are some constants. To obtain the components of the displacement vector from (6) and (6) in () n n n (66) u = L M e k z u n = LnM ne k z. (67) From Eqs. (6)-(65) in (5)-(56) to obtain the components of the stress vector 9

8 Othman MI & Edeeb ERM 6 (6):88-99 k xx = L5nM ne (68) k zz = H 6nM ne (69) k 7n n. (7) = H M e F5 Where L n = ( k F ) n 6 F7 k n c n ( kn 8) L = ( ) Ln = i c + k nln Ln = ( k n i cln ) L5n = E5(i cln k nln ) + i cln E6Ln H6n = E5 cl ( k i L n k L ne ) L n n L = ( k L icl ). 7n n n n 5. Boundary Conditions In this section we need to consider the boundary conditions at z in order to determine the constants M (n ). n () The mechanical boundary conditions i ( t + cx ) σ = Pe σ (7) zz () The thermal boundary condition that the surface of the half-space subjected to i ( t+cx ). T = P e Where P is the magnitude of the applied force in of the half-space and P is the applied constant temperature to the boundary. Using the expressions of the variables into the above boundary conditions (7) (7) we obtain L6nM n = P (7) L7nM n = (7) LnM n = P. (75) Invoking boundary conditions (7)-(75) at the surface z of the plate we obtain a system of three equations. fter applying the inverse of matrix method we can get the values of the three constants M n (n ). 6. Numerical results and discussion In order to illustrate the obtained theoretical results in the preceding section following Dhaliwal and ingh [5] the magnesium material chosen for purposes of numerical evaluations. The constants of the problem taken as 5. N /. m.78 N / m K.7 W / m deg t.78 N / m T 98 K.7 Kg / m CE. J / Kg deg.68 N / m deg.58 / s. The voids parameters are 5.75 m 6.75 N / m m N / m deg.787 N / m s. The comparisons carried out for 6 b.89 N / m N (7) 95

9 Othman MI & Edeeb ERM 6 (6):88-99 x.5 t. c. i.6 p. p.5 s.5 s z 7. The computations were carried out at t.. The numerical technique outlined above was used for the distribution of the real part of the displacement u the stresses zz and the change in the volume fraction field with the distance z for the problem under consideration. ll the considered variables depend not only on the variables t z and x but also on the thermal relaxation times and. The results are shown in Figs. -6. The graphs show the six curves predicted by three different theories of thermoelasticity (CD L- and G- L). In these figures the solid lines represent the solution in the CD theory the dashed lines represent the solution with the G-L theory and the dashed-dotted lines represent the solution with the L- theory. Here all the variables are taken in non-dimensional forms and we consider four cases: () Equations of the CD theory when n. () Lord and hulman L- theory when n () Green and Lindsay G-L theory when n. () The three theories in the absence of a gravity field from the above mentioned by taking g. Figs. - show comparisons among the considered variables in the absence and presence of the gravity effect ( g g 9.8). Fig. shows that the distribution of the vertical displacement u always begins from a positive value with gravity in the context of the three theories CD L- and G-L it decreases in the range z.5 then increases in the range z 7 while increases in the range.5 z. However without gravity it begins from zero and increases in the range z.5 then decreases in the range.5 z.5. Fig. depicts the distribution of the change in the volume fraction field with and without gravity under the three theories. The values of with gravity are decreasing in the range z.8 then increasing in the range.8 z.5. The values of without gravity is greater than that with gravity in the range z.8 while the vice versa in the range.8 z 6. Fig. shows that the distribution of the stress component begins from zero in the context of the three theories and satisfies the boundary conditions at z. The values of with gravity are greater than that without gravity. Figs. -6 show the comparisons among the considered variables in the absence and presence of voids in the case of material with gravity. Fig. exhibits that the distribution of the vertical displacement u begins from positive values in the presence and absence of voids. In the context of the three theories we notice that the values of u with voids are greater than that without voids in the range z.5 while the vice versa in the range.5 z 5.5 then converges to zero. Fig. 5 determines the distribution of stress component in the context the three theories in the presence and absence of voids. It explained that the distribution of increases with the increase of the values of voids in the range z then decreases with the increase of the void in the range z 6. Fig. 6 demonstrates that the distribution of the stress component in the context of the three theories begins from zero and satisfies the boundary conditions at z with and without voids. It is noticed that the value of with voids is greater than that without voids in the range z.7 and the vice versa in the range.7 z Conclusion By comparing the figures that were obtained for the three thermoelastic theories important phenomena are observed:. The values of all physical quantities converge to zero with increasing distance z and all functions are continuous. zz zz 96

10 Othman MI & Edeeb ERM 6 (6): The normal mode analysis technique has been used is applied to a wide range of problems in thermodynamics and thermoelasticity.. ll the physical quantities satisfy the boundary conditions.. The gravity has a significant effect on the variation of the considered physical quantities since they make great changes in the behavior of the functions also the same observation of the absence and presence of the voids in the thermoelastic solid. x u g= CD L- G-L g= z Figure : The displacement component u distribution against z with and without gravity x g= g= CD -6 L- G-L z Figure : The displacement of volume fraction field against z with and without gravity x - CD L- G-L 5 g=9.8 g= z Figure : The displacement of the stress tensor σ against z with and without gravity 97

11 Othman MI & Edeeb ERM 6 (6): x - CD L- G-L 8 with voids u 6 without voids z Figure : The displacement component u distribution against z with and without voids. -. withouth voids zz with voids -. CD L- G-L z Figure 5: The displacement of the stress tensor σ zz against z with and without voids 5 x - CD L- G-L with voids without voids z Figure 6: The displacement of the stress tensor σ against z with and without void 98

12 Othman MI & Edeeb ERM 6 (6):88-99 References []. Biot M. (956). Thermoelasticity and irreversible thermodynamics. Journal of pplied Physics []. Lord H.W. & hulman Y. (967). generalized dynamical theory of thermo- elasticity. Mechanical Journal of Physical olids []. Green.E. & Lindsay K.. (97). Thermoelasticity. J. Elasticity -7. []. Cowin.C. & Nuiato J.W. (98). Linear elastic materials with voids. Journal of Elasticity 5-7. [5]. Puri P. & Cowin.C. (985) Plane waves in linear elastic materials with voids. Journal of Elasticity [6]. Iesan D. (986). theory of thermoelastic materials with voids. cta Mechanica [7]. Dhaliwal R.. & Wang J. (995) heat-flux dependent theory of thermo-elasticity with voids. cta Mechanica. -9. [8]. Ciarletta M. & carpetta E. (995) ome results on thermoelasticity for dielectric materials with voids. Journal of pplied Mathematics and Mechanics [9]. Othman M.I.. Zidan M.E.M. & Hilal M.I.M. (5). -D problem of a rotating thermoelastic solid with voids under thermal loading due to laser pulse and initial stress type III. Journal of Thermal tresses []. Othman M.I.. & Edeeb E.R.M. (6). -D problem of a rotating thermo-elastic solid with voids and thermal loading due to laser pulse under three theories. Journal of Computational and Theoretical Nanoscience []. Othman M.I.. & Edeeb E.R.M. (6). Two-temperature generalized rotation-thermoelastic medium with voids and initial stress: comparison of different theories Journal of cientific and Engineering Research -5. []. Bromwich T.J. (898). On the influence of gravity on elastic waves and in particular on the vibrations of an elastic globe. The Proceedings of the London Mathematical ociety 98-. []. Othman M.I.. Elmaklizi J.D. & aied.m. (). Generalized thermo-elastic medium with temperature dependent properties for different theories under the effect of gravity field. International Journal of Thermophysics []. ilawalia P. & Narah N.. (9). Effect of rotation in generalized thermo-elastic solid under the influence of gravity with an overlying infinite thermo-elastic fluid. pplied of Mathematics and Mechanics [5]. Dhaliwal R.. & ingh. (98). Dynamic coupled thermoelasticity. Hindustan Publication Corp New Delhi India. 99