Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity

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1 COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY ( 65-7 (6 Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity H. M. Youssef A. A. El-Bary Mathematical Department Faculty of Engineering Umm Al-Qurach University PO Makkah Saudi Arabia yousefanne@yahoo.com Basic Applied Science Dept. Arab Academy for Science Technology Alexria Egypt aaelbary@aast.edu (Rec. September 5 Abstract: One-dimensional generalized thermoelastic mathematical model with variable thermal conductivity for heat conduction problem is constructed for a layered thin plate. The basic equations are transformed by Laplace transform solved by a direct method. The solution was applied to a plate of swich structure which is thermally shocked traction free in the outer sides. The inverses of Laplace transforms are obtained numerically. The temperature the stress the displacement distributions are represented graphically. ey words: thermoelasticity generalized thermoelasticity thermal shock composite materials variable thermal conductivity Notation: λ μ Lame s constants u i components of displacement vector ρ density F i body force vector C E specific heat at constant strain thermal conductivity t time κ diffusivity T temperature Q heat source T reference temperature q heat flux θ temperature increment θ = T T τ relaxation time σ ij components of stress tensor i = 3 the cartesian coordinates e components of strain tensor ij. INTRODUCTION Lord Shulman [] obtained the governing equations of generalized thermoelasticity involving one relaxation time for isotropic homogeneous media which is called the first generalization to the coupled theory of elasticity. These equations predict finite speeds of propagation of heat displacement distribution. The corresponding equations for an isotropic case were obtained by Dhaliwal Sherief []. Due to complexity of the governing equations mathematical difficulties associated with their solution several simplifications have been used. For example some authors [3 4] use the framework of coupled thermoelasticity where the relaxation time is taken as zero resulting in a parabolic system of partial differential equations. The solution of this system exhibits infinite speed of propagation of heat signals contradictory to physical observation. Some other authors use still further going simplifications by ignoring the inertia effects in a coupled theory [5] or by neglecting the coupling effect. The second generalization to the coupled theory of elasticity is what is known as the theory of thermoelasticity with two relaxation times or the theory of temperature-rate- -dependent thermoelasticity. Müller [6] 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

2 66 H. M. Youssef A. A. El-Bary Laws [7]. Green Lindsay obtained an explicit version of the constitutive equations in [8]. These equations were also obtained independently by Suhubi [9]. This theory contains two constants that act as relaxation times 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 medium under consideration has a center of symmetry. Eraby Suhubi [] studied wave propagation in a cylinder. Ignaczak [] [] studied a strong discontinuity wave obtained a decomposition theorem. It is usual to assume in thermal stress calculations that material properties are independent of temperature. However significant variations occur over the working temperature range of the engineering ceramics particularly in the coefficient of thermal conductivity. Godfrey has reported decreases of up to 45 per cent in the thermal conductivity of various samples of silicon nitride (between 4 C. The question arises: what are the effects of these variations on the stress displacement distributions in metal components [3]. Modern structural elements are often subjected to temperature changes of such magnitude that their material properties may no longer be regarded as having constant values even in an approximate sense. The thermal mechanical properties of materials vary with temperature so that the temperature dependent material properties must be taken into consideration in the thermal stress analysis of these elements. This work deals with a plate consisting of layers of various materials each of which is homogeneous isotropic. When this plate which is initially at rest having a uniform temperature is suddenly heated at the free surfaces a heat flow occurs in the plate change in thermal the mechanical field is brought about.. THE GOVERNING EQUATIONS We will consider the thermal conductivity variable as where ( ( θ ( θ = = + ( θ = T T such that T T T << is called the thermal conductivity is a small value is the thermal conductivity when does not depend on temperature ( =. Now we will consider the mapping = ( θ dθ ( θ where is a new function expressing the heat conduction. By substitution from ( in ( we have Then we have = + θ θ θ ( d. = θ + θ (3 ( = θ + θ (4 i i = θ (5 ii i i where [ ] = [] [] i = []. t The heat equation reads ( i = θ (6 x i θ = + τ θ γt e i 3 i + = t κ (7 where κ is the diffusivity. Using Eqs. (5 (6 the heat equations will take the form γ T ii τ e t t κ = + + i = 3. (8 The equations of motion are or we have ( ρ u = λ+ μ u + μ u γθ + ρ F (9 i j ji i j j i i ρ u λ μ u μ u γ ρ F. ( i i = + j ji + i j j + i ( + θ By neglecting the small values the body forces we arrive at ( ρ u = λ+ μ u + μ u γ. ( The constitutive relation i j ji i j j i ( e σ = μe + λ γ δ. ( ij ij kk ij After evaluation of the temperature increment θ can be obtained by solving equation (3 to give + + θ =.

3 Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material FORMULATION OF THE PROBLEM The coordinate system is chosen that the x-axis is taken perpendicularly to the layer the y- z-axes in parallel. We are dealing with one-dimensional generalized thermoelasticity with one relaxation time. We consider that the displacement components for one dimension medium have the form ( The strain components are The heat equation u = u x t u = u =. x y z u e= exx = ( x eyy = ezz = exy =. γ T e = τ + +. (3 x τ τ κ The constitutive relation takes the form u σ = ( λ+ μ γ. (4 x The equation of motion u = ( + ρu λ μ γ. (5 x x For simplicity we use the following non-dimensional variables [4]: / / λ+ μ x' λ+ μ u' x = u = ρ κ ρ κ / / λ+ μ t ' λ+ μ t = β = ρ κ μ ( 3λ+ ( λ+ μ τ' σ ' μ αt τ = σ = = '. ρ κ λ+ μ λ+ μ For convenience we drop the primes. Hence we obtain D = + τ [ + ε e] (6 t t u σ = = e x (7 u e u = =. (8 x x x x By eliminating between Eqs. (7 (8 we get where ε = D σ = e (9 ( 3 + ( λ+ μ T T λ μ α κ. Taking Laplace transform as define st f ( s = f( t e dt. Then Eqs. (6 (7 (9 will take the forms where D =. x By eliminating e we get ( D s s h e + τ θ = ε ( σ = ( e θ ( s e D σ = ( D s σ = s θ ( ( = ε( s+ τs σ D ε + s+ τ s θ = Using the above two equations we obtain where 4 ( (3 (4 D LD + M θ = (5a 4 ( D LD + M σ = (5b ( τ ( ε L = s + s+ s + + ( τ + ε s+ s s ( τ ( ε M = s s+ s +.

4 68 H. M. Youssef A. A. El-Bary 4. APPLICATION Considering a layered of swich structure such as shown in Fig. where layers I I are made from the same metal the layer is a different metal. Layer is put in the middle of the plate its thickness is a half of that of the plate. x = x = x = x = ( τ ( ε M = s s+ s +. 3 In region I where x The solution of the Eqs. (4 (5 takes the form c( k s cosh ( kx ( cosh ( I = + + c k s k x (3 I. cs ( kx c s ( k x σ = cosh + cosh. (3 I I Fig.. In region I where x The solution of the Eqs. (4 (5 takes the form ( ( ( ( I = A k s cosh k x + A k s cosh k x (6 ( ( I σ = A s cosh k x + A s cosh k x (7 where the parameters k k satisfy the equation where: 4 I I k Lk + M = ( τ ( ε ε( τ I I I I L = s + s+ s + + s+ s s ( τ ( ε I I I M = s s+ s +. In region where x The solution of the Eqs. (4 (5 takes the form ( cosh ( = B p s p x + ( ( + B p s cosh p x ( ( (8 σ = Bs cosh px + B s cosh p x (9 where the parameters p p satisfy the equation 4 p L p + M = ( τ ( ε ε( τ L = s + s+ s + + s+ s s 5. THE BOUNDARY CONDITIONS ( The thermal boundary conditions which takes the form where = θ + θ. θ = θ Ht ( for x= ± = =± s for x (3 ( The mechanical boundary conditions σ = for x =± which takes the form σ = for x =±. (33 (3 The continuity conditions (i (ii I = at x = I σ = σ at x = σ I = at x = I = σ at x =. (34 Introducing the pervious conditions into Eqs. (4-9 we obtain ( I I = = s k ( k ( k s k s cosh ( kx cosh ( kx cosh cosh ( k k ( kx I I s cosh cosh σ = σ = cosh cosh ( k x (35 (36

5 Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material 69 ( p s ( ( ( = s k k p p cosh p k p k p cosh ( px + sinh ( p s ( ( ( s k k p p cosh p k p k p sinh cosh ( px s σ = cosh ( k k ( p p ( p k p k p cosh ( px + sinh s cosh ( k k ( p p ( p k p k p sinh cosh ( px. We can get the displacement by using Eq. (4 such that so we obtain u I I ( k u = s Dσ ( ( (37 (38 ksinh kx ksinh kx = u = (39 s k cosh cosh u p = s k k p p cosh( p ( ( k p k p sinh ( px sinh + p s k k p p cosh( p ( ( k p k p sinh sinh ( px. (4 6. THE INVERSION OF THE LAPLACE TRANSFORM In order to invert the Laplace transform in Eqs. (35-4 we adopt a numerical inversion method based on a Fourier series expansion [5]. By this method the inverse f ( t of the Laplace transform f ( s is approximated by ct N e ikπ ikπt f( t = f ( c + Re f c+ exp t k = t t < t < t where N is a sufficiently large integer representing the number of terms in the truncated Fourier series chosen in the form inπ inπt exp( ct Re f c + exp ε t t where ε is a prescribed small positive number that corresponds to the degree of accuracy required. The parameter c is a positive free parameter that must be greater than the real part of all the singularities of f (. s The optimal choice of c was obtained according to the criteria described in [5]. 7. NUMERICAL RESULTS The copper material the type 36 stainless steel are chosen for the purposes of numerical evaluations [6 7]. Table. Materials parameters Parameter Copper I I Stainless steel α T ρ 8954 kg m kg m 3 C E 383. m s 56 m s 386 kg m s kg m s 3 T μ/λ τ. s. s ε The computations were carried out for the value of time t =.3 for the length = (unit length. The numerical values of the temperature displacement stress are represented graphically where the solid line in the temperature distribution shows the case when the

6 7 H. M. Youssef A. A. El-Bary thermal conductivity is constant while the doted line illustrates the case of thermal conductivity variable. 8. CONCLUSIONS. Figure shows the temperature field distribution with respect to the dimension x we can see that for the three layers the temperature increases when the thermal conductivity is constant decreases when it is variable. The differences between the two plates appears to be more significant in the central layers than in the surface layers. Fig. 4. The stress distribution The three curves demonstrate that the effect of the thermal conductivity on all the fields in different materials is clear we have to take it into account in any analysis of heat conduction. Fig.. The temperature distribution. Figure 3 shows the displacement field distribution with respect to the dimension x we can see that for the three layers the absolute value of the displacement increases when the thermal conductivity is constant decreases when it is variable. Fig. 3. The displacement distribution 3. Figure 4 shows the stress field distribution with respect to the dimension x we can see that for the three layers the absolute value of the displacement increases when the thermal conductivity is constant decreases when it is variable. References [] H. Lourd Y. Shulman A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids (967. [] R. Dhaliwal H. Sherief Generalized thermoelasticity for anisotropic media Quart. Appl. Math (98. [3] A. Roberts On the steady motion of a line load over a coupled thermoelastic half-space subsonic case Quart. J. Mech. Appl. Math (97. [4] F. Ziegler Ebene Wellenausbreitung im Halbraum bei Zufallserregung Spannungs-und Temperaturfeld Acta Mechanica (966. [5] Y. Yang C. Chen Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials Int. J. Eng. Sci (986. [6] I. Müller The coldness a universal function in thermoelastic solids Arch. Rat. Mech. Anal (97. [7] A. Green N. Laws On the entropy production inequality Arch. Rat. Anal (97. [8] A Green. Lindsay Thermoelasticity J. Elast. (97. [9] E. Şuhubi Thermoelastic solids in: A. C. Eringen (ed Cont. Phys Academic Press New York Chapter (987. [] S. Erbay E. Şuhubi Longitudinal wave propagation in a generalized thermo-elastic cylinder J. Therm. Stresses 9 79 (986. [] J. Ignaczak A strong discontinuity wave in thermoelastic with relaxation times J. Thermal Stresses 8 5 (985. [] J. Ignaczak Decomposition theorem for thermoelasticity with finite wave speeds J. Thermal Stresses 4 (978. [3] D. P. H. Hasselman R. A. Heller Thermal Stresses in Sever Environments Plenum Press New York (98. [4] N. M. El-Maghraby H. M. Youssef State Space Approach to Generalized Thermoelastic Problem with Thermo-Mechanical Shock J. Applied Mathematics Computation 56 Issue P (4.

7 Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material 7 [5] G. Hanig U. Hirdes A method for the numerical inversion of Laplace transform J. Comp. Appl. Math. 3-3 (984. [6] A. A. El-Bary H. M. Youssef Thermal Shock Problem for one dimension generalized thermoelastic layered composite material accepted for publication in Math. Comp. App. (5 [7] S. Minagawa Nonlinear-eigenvalue problem for heat conduction in a one-dimensional coupled thermoelasticlayered composite Int. J. Eng. Sci (987. PROF. HAMDY M. YOUSSEF. April 97 PhD in Mathematics Alexria University Egypt 99; Nationality: Egyptian; yousefanne@yahoo.com; Membership: The Egyptian Mathematical Society; Current affiliation: Umm Al-Qurah University Saudi Arabia; Professional: Theory of Thermoelasticity; Publications: papers in international Journals 5 papers in international conferences; The most important paper: Theory of Two-Temperature Generalized Thermoelasticity IMA Journal of Applied Mathematics (5-8; Honors: Who s Who in Science Engineering 6-7; Outsting Intellectuals of the th Century 6 Cambridge University; Membership of IBC Research Council COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY ( 65-7 (6