Finite Element Analysis of the Generalized Thermoelastic ...

Ibrahim A. Abbas JOURNAL OF PHYSICS VOL. 1 NO. 2 July (2012) PP. 4 - 10Finite Element Analysis of the GeneralizedThermoelastic Interactions in an Elastic HalfSpace Subjected to a Ramp-Type HeatingIbrahim A. AbbasAbstract — A finite element method is used to study thethermoelastic interactions in an elastic half spacesubjected to a ramp-type heating. The governingequations are taken in a unified system from which thefield equations for coupled thermoelasticity as well asfor generalized thermoelasticity can be easily obtainedas particular cases. Due attention has been paid to thefinite time of rise of temperature, displacement andstress. The problem has been solved numerically using afinite element method (FEM). The non-dimensionalgoverning equations are solved by (FEM). Numericalresults for the temperature distribution, displacementand thermal stress are represented graphically. Acomparison is made with the results predicted by thethree theoriesKeywords: Finite element method, Generalized, thermoelasticityTI. INTRODUCTIONhe generalized theories of thermoelasticity, which admitthe finite speed of thermal signal, have been the centerof interest of active research during last three decades.These theories remove the paradox of infinite speed of heatpropagation inherent in the conventional coupled dynamicaltheory of thermoelasticity introduced by Biot [1]. Hedeveloped the coupled theory of thermoelasticity to dealManuscript received May 12, 2012 and accepted June 1,2012.Department of Mathematics, Faculty of Science and Arts -Khulais, King Abdulaziz University, Jeddah, Saudi Arabia.Department of mathematics, Faculty of Science, Sohag University,Sohag, Egypt.E-mail: ibrabbas7@yahoo.com.with a defect of the uncoupled theory that mechanicalcauses have no effect on the temperature. However, thistheory shares a defect of the uncoupled theory in that itpredicts infinite speeds of propagation for heat waves. Thisimplies that the thermal wave propagates with an infinitespeed, which is physically unrealistic. To overcome such anabsurdity, generalized thermoelasticity theories have beenpropounded by Lord Shulman [2] as well as Green andLindsay [3] advocating the existence of finite thermal wavespeed in solids. These theories have been developed byintroducing one or two relaxation times in the thermoelasticprocess , either by modifying Fourier's heat conductionequation or by correcting the energy equation andNeumann-Duhamel relation, with an aim to eliminate theparadox of infinite speed. According to these generalizedtheories, heat propagation can be visualized as a wavephenomenon rather than a diffusion one; in the literature , itis usually referred to as the second sound effect. Potentialapplications for these new theories have been reported inthe survey article by Ignaczak [4] and references therein.During the last three decades a number of investigations [5-8] have been carried out using the aforesaid theories ofgeneralized thermoelasticity. Chandrasekharaiah [9] studiedone-dimensional waves in a homogeneous isotropic halfspacedue to sudden inputs of temperature and stress/strainon the boundary by employing the Laplace transformmethod in the context of thermoelasticity without energydissipation. In a paper by Abd-alla and Abbas [10], aproblem of generalized magnetothermoelasticity for aninfinitely long, perfectly conducting cylinder has beenstudied. Note that in most of the earlier studies, mechanicalor thermal loading on the bounding surface was consideredto be in the form of a shock. However, the sudden jump ofthe load is merely an idealized situation because it isimpossible to realize a pulse described mathematically by astep function; even a very rapid rise time (of order 10 -9 )may be slow in terms of the continuum. This is particularlytrue in the case of second sound effects when the thermalrelaxation times for typical metals are less than 10 -9 s. It isthus felt a finite rise time of external load (mechanical orthermal) applied on the surface should be considered whileCopyright ©1996-2012 Researchpub.org. All rights Reserved. 4

Ibrahim A. Abbas JOURNAL OF PHYSICS VOL. 1 NO. 2 July (2012) PP. 4 - 10( )( x )∂u x,0 ∂θ,0u ( x,0)= = 0, θ ( x,0)= = 0,∂t∂t⎧ 0 t ≤ 0⎪ tσ ( 0, t) = 0, θ ( 0, t) = ⎨θ1 0 < t ≤ t0⎪ t0⎪⎩ θ1 t > t0III. FINITE ELEMENT METHOD(11)(12)In order to investigate the thermoelastic interactions inan elastic half space subjected to a ramp-type heating, the(FEM) [19-20] is adopted due to its flexibility in modelinglayered structures and its capability in obtaining full fieldnumerical solution. The governing equations (8) and (9) arecoupled with initial and boundary conditions (10) and (11).The numerical values of the dependent variables likedisplacement u and the temperature θ are obtained at theinteresting points which are called degrees of freedom. Theweak formulations of the non-dimensional governingequations are derived. The set of independent test functionsto consist of the displacement δ u and the temperatureδθ is prescribed. The governing equations are multipliedby independent weighting functions and then are integratedover the spatial domain with the boundary. Applyingintegration by parts and making use of the divergencetheorem reduce the order of the spatial derivatives andallows for the application of the boundary conditions. Thesame shape functions are defined piecewise on the elements.Using the Galerkin procedure, the unknown fields u and θand the corresponding weighting functions areapproximated by the same shape functions. The last steptowards the finite element discretization is to choose theelement type and the associated shape functions. On theother hand, the time derivatives of the unknown variableshave to be determined by Newmark time integration method[15].In particular, the equation of motion become2∂δu ⎡∂u ⎛ ∂θ⎞⎤∂ u∫ − β1 ⎜θ+ t1 ⎟ dx + δu dx =2∂x ⎢∂x ∂t ⎥ ∫⎣ ⎝ ⎠⎦∂t(13)Γ⎡∂u⎛δu⎢− β ⎜θ+⎣∂x⎝The energy equation has the form⎡Γ∂θ⎞⎤⎟∂t⎥⎠⎦1t1 .−⎛+ nt⎞⎤dx +⎛+ t⎞dx =⎝ ⎠⎦⎝ ⎠2 2∂δθ ∂θ ∂u∂ u∂θ ∂ θ∫ ⎢ β2 ⎜ 2δθ2 ⎟⎥∫ ⎜ 2 2 ⎟Γ ∂x ∂x ∂t ∂t Γ ∂t ∂t⎣⎡∂θ⎛ ∂u∂ u ⎞⎤δθ ⎢ − β ⎜ + ⎟⎥⎣ ∂x ⎝ ∂t ∂t⎠⎦Γ22nt2 .2Γ(14)IV. NUMERICAL RESULTS AND DISCUSSIONIn order to illustrate the problem, the copper materialwas chosen for purposes of numerical evaluations. Thephysical data which given as [15]10 1 2 10 1 2λ = 7.76× 10 ( kg)( m) − ( s) − , µ = 3.86×10 ( kg)( m) − ( s) − ,ρ = = ×3 −3 −6 −18.954x10 ( kg)( m) , αt17.8 10 ( K) .2 −1 −3( )T0 = 300 K , K = 3.86×10 ( kg)( m)( K) ( s) ,c = × m K se2 2 −1 −23.831 10 ( ) ( ) ( ) ,Before going to the analysis the grid independence test hasbeen conducted and the results are presented in table 1. Thegrid size has been refined and consequently the values ofdifferent parameters as observed from table 1 get stabilized.Further refinement of mesh size over 5000 elements doesnot change the values considerably, which is thereforeaccepted as the grid size for computing purposeTable 1. Grid independent test ( ( t = 0.3, t = 0.4, x = 1.0)MeshsizeθCD LS GLu × 10−2−5× 10θ−2−5× 10u × 100θ−2× 10−5u × 1050 5.554107 1.370979 6.395838 8.985454 6.445057 6.672534100 5.559483 1.373254 6.566698 9.419929 6.615257 6.8633691000 5.561257 1.374003 6.622675 9.561751 6.671027 6.9258264000 5.561274 1.374011 6.623204 9.563090 6.671555 6.9264165000 5.561274 1.374011 6.623216 9.563123 6.671567 6.926430The field quantities, temperature, displacement and stressdepend not only on the time t and space x, but also depend onrise-time parameter t0and on thermal relaxation timeparameters t 1, t 2and n. It has been observed that in all threetheories of CD, LS and GL, the finite rise-time parameter hasa significant effect on the temperature quantities. Here all thevariables/parameters are taken in nondimensional forms. Theresults for temperature, displacement and stress has beencarried out by taking θ₁=1. Figures 1, 2 and 3 exhibit thevariation of the temperature, displacement and stress withspace x for the classical dynamical coupled theory CD andfour different values of time t. Figures 4, 5 and 6 show thevariations of the temperature, displacement and stress underLord-Shulman's theory LS, i.e., when there is one thermalrelaxation time ( t2= 0.1,0.2,0.3,0.4 ) and for t0= 0.4 att=0.3. Similar results are shown on figures 7, 8 and 9 for onet = and lettingvalue of the relaxation time ( 20.1)( )t0= 0.2,0.4,0.6 for t=0.3. Figures 10, 11 and 12 displaythe variation of three theories effects on the temperature,displacement and stress. In this case, one may takeCopyright ©1996-2012 Researchpub.org. All rights Reserved. 6

Ibrahim A. Abbas JOURNAL OF PHYSICS VOL. 1 NO. 2 July (2012) PP. 4 - 10( t1 = t2 = n = 0)for CD while ( t1 0, t20.1, n 1)and ( t 0.2, t 0.1, n 0)1 2= = = for LS= = = for GL. It is obvious fromfigures 1,4,7 and 10 that the temperature decreases with theincrease of the space but they increase when decreasing therelaxation times or decreasing the rise-time parameter. Nosignificant difference in the value of temperature is noticedfor the LS and GL theories. It is obvious from figures 2,5,8and 11 that the displacement is negative at x=0 where itsmagnitude is maximum. The displacement increases from thenegative value to a positive value. In the positive values, thedisplacement have a peak value that depends on the values ofthe time, rise-time and relaxation times. It is obvious fromfigures 3,6,9 and 12 which gives the stress variation atdifferent instants of time with the space. Its magnitudeincreases from zero to a maximum value ( which is positivevalue for t < t and negative value for t t00> ); after thatdecreases rapidly as x increases.Fig. 2: The displacement distribution for CD at t 0= 0.4and different value of t.Fig. 3: Stress distribution for CD at t 0= 0.4and different value of t .Fig. 1. The Temperature distribution for CD at t 0= 0.4and different value of t.Fig. 4: Temperature distribution for LS at t 0= 0.4and t = 0.3 with different value of t 2.Copyright ©1996-2012 Researchpub.org. All rights Reserved. 7

Ibrahim A. Abbas JOURNAL OF PHYSICS VOL. 1 NO. 2 July (2012) PP. 4 - 10Fig. 5: The desplacement distribution for LS at t 0= 0.4and t = 0.3 with different value of t 2.Fig. 8: The desplacement distribution for LS att2= 0.1 and t = 0.3Fig. 6: Stress distribution for LS at at t 0= 0.4and t = 0.3 with different value of t 2.Fig. 9: Stress distribution for LS att2= 0.1 and t = 0.3Fig. 7: Temperature distribution for LS att2= 0.1 and t = 0.3Figure 10: The variation of temperature for threetheories at t 0= 0.2 and t = 0.3Copyright ©1996-2012 Researchpub.org. All rights Reserved. 8

Ibrahim A. Abbas JOURNAL OF PHYSICS VOL. 1 NO. 2 July (2012) PP. 4 - 10with a cavity. Forsch Ingenieurwes 71 (2007) 215-222.[18] I.A. Abbas, A. N. Abd-alla, Effects of thermalrelaxations on thermoelastic interactions in aninfinite orthotropic elastic medium with a cylindricalcavity, Arch. Appl. Mech. 78 (2008) 283-293.[19] J.N. Reddy, An Introduction to the finite elementmethod, second ed., McGraw-Hill, New York, 1993.[20] R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts andapplications of finite element analysis, third ed.,John Wiley, New York, 1989Copyright ©1996-2012 Researchpub.org. All rights Reserved. 10