Research ArticleA Two-Dimensional Generalized ElectromagnetothermoelasticDiffusion Problem for a Rotating Half-Space
Jingrui Zhang and Yanyan Li
School of Astronautics Beijing Institute of Technology Beijing 100081 China
Correspondence should be addressed to Jingrui Zhang zhangjingruibiteducn
Received 13 November 2013 Revised 26 February 2014 Accepted 27 February 2014 Published 9 April 2014
Academic Editor Filippo de Monte
Copyright copy 2014 J Zhang and Y Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In the context of the theory of generalized thermoelastic diffusion a two-dimensional generalized electromagnetothermoelasticproblem with diffusion for a rotating half-space is investigated The rotating half-space is placed in an external magnetic fieldwith constant intensity and its bounding surface is subjected to a thermal shock and a chemical potential shock The problemis formulated based on finite element method and the derived finite element equations are solved directly in time domain Thenondimensional temperature displacement stress chemical potential concentration and induced magnetic field are obtainedand illustrated graphically The results show that all the considered variables have a nonzero value only in a bounded region andvanish identically outside this region which fully demonstrates the nature of the finite speeds of thermoelastic wave and diffusivewave
1 Introduction
Biot [1] proposed the coupled thermoelasticity to amenda defect in uncoupled thermoelasticity that elastic defor-mation has no effect on temperature However this theoryshares another defect in uncoupled thermoelasticity in thatit predicts infinite speed for heat propagation which isphysically impossible To overcome such defect the gener-alized thermoelastic theories have been developed by Lordand Shulman (L-S) [2] and Green and Lindsay (G-L) [3]Both theories can characterize the so-called second soundeffect that is heat propagates in medium with a finitespeed The L-S theory was later extended by Dhaliwal andSherief [4] to the case of anisotropic media Based on thesegeneralized thermoelastic theories many efforts have beendevoted to dealing with the generalized dynamic problemsSherief and Dhaliwal [5] studied a one-dimensional thermalshock problem by the Laplace transform technique and itsinverse transform Dhaliwal and Rokne [6] solved a thermalshock problem of a half-space with its plane boundary eitherheld rigidly fixed or stress-free and an approximate small-time solution was obtained by using the Laplace transformmethod Sherief and Anwar [7] considered the thermoelasticproblem of a homogeneous isotropic thick plate of infinite
extent with heating on a part of the surface by means of statespace approach together with Laplace and Fourier integraltransforms and their inverse counterparts
Investigation of the propagation of electromagnetother-moelastic waves in a thermoelastic solid has attracted muchattention due to its extensive potential applications in diversefields such as geophysics for understanding the effect of theEarthrsquos magnetic field on seismic waves damping of acousticwaves in a magnetic field and emissions of electromagneticradiations from nuclear devices Sharma and Chand [8]analyzed a one-dimensional transient magnetothermoelasticproblem by introducing a potential function Ezzat et al [9]researched a two-dimensional electromagnetothermoelasticplane wave problem of a medium of perfect conductivityin terms of normal mode analysis Sherief and Helmy [10]dealt with a two-dimensional electromagnetothermoelasticproblem for a finitely conducting half-space by Laplaceand Fourier transforms Tianhu et al [11] studied theelectromagnetic-thermoelastic interactions in a semi-infiniteperfectly conducting solid by hybrid Laplace transform-finiteelement method Ezzat and Youssef [12] solved the problemof generalized magnetothermoelasticity in a perfectly con-ducting medium by means of Laplace and Fourier transformtechniques Sharma and Thakur [13] studied the effect of
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 964218 12 pageshttpdxdoiorg1011552014964218
2 Mathematical Problems in Engineering
rotation on Rayleigh-Lamb waves in magnetothermoelasticmedia Othman and Song [14] studied the effect of rotationon plane waves of generalized electromagnetothermovis-coelasticity with two relaxation times Guan [15] studied atwo-dimensional of a rotating half-space by using Laplacetransform and its numerical inversion He and Jia [16]studied a two-dimensional of a rotating half-space by usingthe normal mode analysis Othman and Song [17] investi-gated reflection of magnetothermoelastic waves in a rotatingmedium Recently Deswal and Kalkal [18] considered a two-dimensional generalized electromagnetothermoviscoelasticproblem for a half-space with diffusion whose surface issubjected to mechanical and thermal loads by introducingpotential functions along with the normal modes based onG-L theory
Diffusion can be defined as the random walk of anensemble of particles from regions of high concentration toregions of lower concentration There is now a great deal ofinterest in the study of this phenomenon due to its diverseapplications in geophysics and industry In geology diffusionprinciple has been applied to measuring the diffusion coef-ficients of various cations in minerals which are present inthe Earthrsquos crust In diffusion bonding diffusion techniqueis used to join metallic or nonmetallic materials together Inheat treatment ofmetals the surface characteristics ofmetalssuch as wear and corrosion resistance and hardness canbe improved by carburizing through diffusion In integratedcircuit fabrication diffusion is used to introduce dopantsin controlled amounts into the semiconductor substrate Inparticular diffusion is used to form the base and emitterin bipolar transistors integrated resistors the sourcedrainregions in MOS transistors and dope polysilicon gates inMOS transistors In the above cases temperature plays a vitalrole in the process of diffusion and it is urgent to explore theinteractions among diffusion field strain field temperaturefield and so forth Normally the diffusion process is modeledby what is known as Fickrsquos law which does not take intoconsideration the mutual interplay between the introducedsubstance and the substrate or the effect of temperature onthe interplay Nowacki [19ndash21] put forward the theory ofthermoelastic diffusion in which the coupled thermoelasticmodel was formulated and infinite speed of propagation ofthermoelastic wave was predicted Recently Sherief et al[22] extended this theory associating with L-S model toa generalized thermoelastic diffusion theory that predictsfinite speeds of propagation for thermoelastic and diffusivewaves Following this theory Sherief and Saleh [23] studieda one-dimensional problem of a half-space by using Laplacetransform and its numerical inversion Singh [24] analyzedthe reflection problem of SV wave from free surface inan elastic solid Aouadi [25] examined the thermoelasticdiffusion problem for an infinite elastic body with a sphericalcavity Xia et al [26] worked on the dynamic response of aninfinite body with a cylindrical cavity by using finite elementmethod
In the present work a two-dimensional generalizedelectromagnetothermoelastic problem with diffusion for arotating half-space is studied in the context of the theoryof the generalized thermoelastic diffusion The problem is
formulated based on finite element method and the derivedfinite element equations are solved directly in time domainThe variations of the considered variables are obtained andillustrated graphically
2 Basic Equations
The linear electrodynamic equations of slowly movingmedium for a homogeneous and perfectly conducting elasticsolid are given by Maxwellrsquos equations as follows
nabla times ℎ = 119869 + 1205760 (1)
nabla times 119864 = minus1205830ℎ (2)
119864 = minus1205830( times 119867) (3)
nabla sdot ℎ = 0 (4)
where 119867 is the applied external magnetic field intensityvector ℎ is the inducedmagnetic field vector119864 is the inducedelectric field vector 119869 is the current density vector 119906 is thedisplacement vector 120583
0and 1205760are the magnetic permeability
and electric permeability respectively and nabla is Hamiltonrsquosoperator
In the absence of body force and inner heat source thegeneralized electromagnetothermoelastic diffusive governingequations based on the generalized thermoelastic diffusiontheory put forth by Sherief et al [22] can be written as
120590119894119895= 2120583120576
119894119895+ 120575119894119895(1205821120576119896119896minus 1205741120579 minus 1205742119875) (5)
In the above equations a superimposed dot denotes thederivative with respect to time a comma followed by a suffixdenotes material derivative and the summation conventionis used 120590
119894119895are the components of the stress tensor 120576
119894119895are
the components of the strain tensor 119906119894are the components
of displacement vector 120581119894119895are the coefficients of thermal
conductivity 119878 is the entropy density 119865119894are the components
of Lorentz force Ω is the angular velocity 120578119894is the flow of
the diffusing mass vector 119902119894are the components of heat flux
vector 1205910is the thermal relaxation time 120591 is the diffusion
relaxation time 119879 is the absolute temperature 1198790is the initial
reference temperature 120588 is themass density119862119864is the specific
heat at constant strain 120572119905is the coefficient of linear thermal
expansion 120572119888is the coefficient of linear diffusion expansion
119863119894119895are the coefficients of diffusion 119862 is the concentration of
diffusive material 120582 120583 are Lamersquos constants 119875 is the chemicalpotential ldquo119886rdquo is a measure of thermodiffusion effect and ldquo119887rdquois a measure of diffusive effect
We consider the problem of a homogeneous isotropicand perfectly conducting thermoelastic rotating half-space(119909 ge 0) A magnetic field with constant intensity 119867 =
(0 01198670) acts parallel to the bounding surface (taken as
the direction of the 119911-axis) At the same time angularvelocity Ω = (0 0 Ω) in half-space goes around the 119911-axis Considering rotating effect the equation of motion isincluded in the centripetal acceleration related to time andCoriolis accelerationΩtimes (Ω times 119906) item 2Ω times The surface ofthe half-space is subjected at time 119905 = 0 to a thermal shockand a chemical potential that are functions of 119910 and 119905 Thusall the variables will be functions of time 119905 and coordinates 119909and 119910 Due to the application ofH this results in an inducedmagnetic field ℎ and an induced electric field 119864 in the half-space when it undergoes deformation
It can be noted from (19)ndash(22c) that the induced electricfield the induced magnetic field and the Lorentz forceare functions of the components of displacement whichimplies that the generalized electromagnetothermoelasticproblems with diffusion can then be treated as a generalizedthermoelastic one with diffusion Once the components ofdisplacement are obtained the induced electric field and theinduced magnetic field can be calculated from (19) and (20)respectively
Generally speaking for generalized multifield problemsthe involved physical fields such as electromagnetic fieldtemperature field strain field and diffusion field wouldcouple with each other whichmakes the governing equationsof such problems usually too complex to get the solutionsby analytical method so that numerical methods would bepowerful tools to solve such problems One feasible waycan be the integral transform techniques By means of thismethod the partial differential governing equations can beconverted into ordinary differential equations and solved intransform domain By applying inverse transform the solu-tions of the problem in time domain can then be obtainedHowever this method encounters loss of precision believedto be caused by discretization error and truncation errorintroduced inevitably in the process of numerical inverseLaplace and Fourier transforms which leads to identifyingheat wave front and prediction of the second sound effect theresult of the [27] is thus not so obvious as that the result of [56] showing clear step in the temperature field for small timeAn alternative choice to such problems is the hybrid Laplacetransform-finite element method presented by Chen andWeng [28 29] The same as depicted above this method alsoencounters loss of precision and the step of the temperaturein the heat wave front is not obvious either Therefore theapplicability of the integral transform techniques as wellas the hybrid Laplace transform-finite element method togeneralized thermoelastic problems is limited To avoid thedefects of the above methods we are inspired to formulateour problem by finite element method and directly solve thederived nonlinear finite element equations in time domainas reported by Tian et al in [30] in which the obtainedresults show that this method can achieve a high calculation
4 Mathematical Problems in Engineering
precision The diffusion problem done by Xia et al [26] wasjust solved by using this method recently
3 Finite Element Formulations
Rewrite (5) (6) and (7) in matrix form as follows
120590 = [1198620] 120576 minus 120574
1 120579 minus 120574
2 119875
119862 = 1205742119879
120576 + 119889120579 + 119899119875
120588119878 = 1205741119879
120576 + 1198881120579 + 119889119875
(23)
The generalized heat conduction law and Fickrsquos law of massdiffusion can be written in matrix form as
119902 + 1205910 119902 = minus [120581] 120579
1015840
+ 120591 = [119863] 11987510158401015840
(24)
where 1205791015840 = 120579119894 11987510158401015840 = 119875
119894119894
According to finite elementmethod the half-space can bedivided into elements and nodal points and any variable con-sidered within an element can be approximated by the valuesof nodal points together with shape functions To this endwe introduce two sets of shape functions to approximate thedisplacement the temperature and the chemical potential onthe element level
119906 = [119873119890
1] 119906119890 120579 = 119873
119890
2119879120579119890 119875 = 119873
119890
2119879119875119890
(25)
where 119906119890 120579119890 and 119875119890 are the vectors of nodal displace-
ment temperature and chemical potential respectively [1198731198901]
and 1198731198902 are shape functions they are
[119873119890
1] = [
1198731
0 1198732
0 sdot sdot sdot 119873119899
0
0 1198731
0 1198732
0 sdot sdot sdot 119873119899
]
119873119890
2119879= 1198731 119873
2sdot sdot sdot 119873
119899
(26)
where 119899 denotes the number of nodes in the gridIn terms of 120576
In the absence of body force and inner heat sourceconsidering the Lorentz force 119865
119894 the virtual displacement
principle of the generalized electromagnetothermoelasticproblems with diffusion can be formulated as
int119881
[120575120576119879120590 minus 120588119879
0( 119878 + 120591
0119878) 120575 120579 + 120575120579
1015840119879
119902 + 1205910119902
+1205751198751015840119879
(120578 + 120591 120578) minus 120575119875119879( + 120591)] 119889119881
= int119881
120575119906119879(119865 minus 120588 [ + Ω times (Ω times 119906)
+ 2Ω times ]) 119889119881
+ int119860120590
120575119906119879 119889119860 + int
119860119902
120575120579 119902119889119860
+ int119860120578
120575119875 120578 119889119860
(30)
where represents the traction vector 119902 the heat fluxvector and 120578 the mass flux vector and the variables witha superimposed bar mean that they are given on surface 119860
120590
represents the area of the stress tensor119860119902represents the area
of the heat flux vector and119860120578represents the area of the mass
flux vectorSubstituting (23)ndash(25) (27) and (29) into (30) we obtain
int119881
120575120576119879120590 119889119881
= int119881
(120575119906119890119879[1198611]119879) [[1198620] [1198611] 119906119890 minus 1205741 119873119890
2119879120579119890
minus 1205742 119873119890
2119879119875119890] 119889119881
= 120575119906119890119879int119881
[1198611]119879([[1198620] [1198611] 119906119890 minus 1205741 119873119890
Once the initial conditions and the boundary conditionsare specified the finite element equation in (32) can besolved directly in time domain In the process of numericalcalculation and finite element solution the space domainand time domain are discrete In the calculation because thesurface of 119874119860 is subjected to a thermal shock and a chemicalpotential shock this part of the unit is divided in a moredetailed way the entire model is divided into 1535 units and3222 nodes similarly the initial time step is set to 119905 = 3times10
minus7the variable threshold is set at 119905 = 1times 10
minus7 which ensures theaccuracy and convergence and also saves a lot of calculatedtime
4 Numerical Results and Discussions
The schematic of the considered half-space as well as theapplied loads on its bounding surface is shown in Figure 1(a)The bounding surface is assumed to be traction-free and thethermal shock and the chemical potential shock applied onthe bounding Surface have respectively the following form
120579 = 1205790119867(119905)119867 (119871 minus
Due to the symmetries of geometrical shape and boundaryconditions the problem can be treated as a plane strain prob-lem and only half of the half-space needs to be consideredThe model for simulation is shown in Figure 1(b) where119874119860119861119862 outlines the region for implementing the simulationand119874119863 represents the regionwithinwhich the thermal shockand the chemical potential shock are applied
Mathematical Problems in Engineering 7
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
00
02
04
06
08
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
t = 005t = 01
t = 015
Figure 2 Nondimensional temperature distribution along 119874119860
The half-space is taken to be copper material and thematerial properties are
120582 = 776 times 1010 kg(ms2) 120583 = 386 times 10
10 kg(ms2)
120588 = 8954 kgm3 120581 = 386W(mK)
119863 = 85 times 10minus9 kg sm3 120572
119888= 198 times 10
minus4m3kg
120572119905= 178 times 10
minus5 Kminus1 119862119864= 3831 J(kgK)
119886 = 12 times 104m2 (s2 K) 119887 = 9 times 10
5m5 (kg s2) (36)
To simplify the simulation we introduce the following nondi-mensional variables
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
rotation on Rayleigh-Lamb waves in magnetothermoelasticmedia Othman and Song [14] studied the effect of rotationon plane waves of generalized electromagnetothermovis-coelasticity with two relaxation times Guan [15] studied atwo-dimensional of a rotating half-space by using Laplacetransform and its numerical inversion He and Jia [16]studied a two-dimensional of a rotating half-space by usingthe normal mode analysis Othman and Song [17] investi-gated reflection of magnetothermoelastic waves in a rotatingmedium Recently Deswal and Kalkal [18] considered a two-dimensional generalized electromagnetothermoviscoelasticproblem for a half-space with diffusion whose surface issubjected to mechanical and thermal loads by introducingpotential functions along with the normal modes based onG-L theory
Diffusion can be defined as the random walk of anensemble of particles from regions of high concentration toregions of lower concentration There is now a great deal ofinterest in the study of this phenomenon due to its diverseapplications in geophysics and industry In geology diffusionprinciple has been applied to measuring the diffusion coef-ficients of various cations in minerals which are present inthe Earthrsquos crust In diffusion bonding diffusion techniqueis used to join metallic or nonmetallic materials together Inheat treatment ofmetals the surface characteristics ofmetalssuch as wear and corrosion resistance and hardness canbe improved by carburizing through diffusion In integratedcircuit fabrication diffusion is used to introduce dopantsin controlled amounts into the semiconductor substrate Inparticular diffusion is used to form the base and emitterin bipolar transistors integrated resistors the sourcedrainregions in MOS transistors and dope polysilicon gates inMOS transistors In the above cases temperature plays a vitalrole in the process of diffusion and it is urgent to explore theinteractions among diffusion field strain field temperaturefield and so forth Normally the diffusion process is modeledby what is known as Fickrsquos law which does not take intoconsideration the mutual interplay between the introducedsubstance and the substrate or the effect of temperature onthe interplay Nowacki [19ndash21] put forward the theory ofthermoelastic diffusion in which the coupled thermoelasticmodel was formulated and infinite speed of propagation ofthermoelastic wave was predicted Recently Sherief et al[22] extended this theory associating with L-S model toa generalized thermoelastic diffusion theory that predictsfinite speeds of propagation for thermoelastic and diffusivewaves Following this theory Sherief and Saleh [23] studieda one-dimensional problem of a half-space by using Laplacetransform and its numerical inversion Singh [24] analyzedthe reflection problem of SV wave from free surface inan elastic solid Aouadi [25] examined the thermoelasticdiffusion problem for an infinite elastic body with a sphericalcavity Xia et al [26] worked on the dynamic response of aninfinite body with a cylindrical cavity by using finite elementmethod
In the present work a two-dimensional generalizedelectromagnetothermoelastic problem with diffusion for arotating half-space is studied in the context of the theoryof the generalized thermoelastic diffusion The problem is
formulated based on finite element method and the derivedfinite element equations are solved directly in time domainThe variations of the considered variables are obtained andillustrated graphically
2 Basic Equations
The linear electrodynamic equations of slowly movingmedium for a homogeneous and perfectly conducting elasticsolid are given by Maxwellrsquos equations as follows
nabla times ℎ = 119869 + 1205760 (1)
nabla times 119864 = minus1205830ℎ (2)
119864 = minus1205830( times 119867) (3)
nabla sdot ℎ = 0 (4)
where 119867 is the applied external magnetic field intensityvector ℎ is the inducedmagnetic field vector119864 is the inducedelectric field vector 119869 is the current density vector 119906 is thedisplacement vector 120583
0and 1205760are the magnetic permeability
and electric permeability respectively and nabla is Hamiltonrsquosoperator
In the absence of body force and inner heat source thegeneralized electromagnetothermoelastic diffusive governingequations based on the generalized thermoelastic diffusiontheory put forth by Sherief et al [22] can be written as
120590119894119895= 2120583120576
119894119895+ 120575119894119895(1205821120576119896119896minus 1205741120579 minus 1205742119875) (5)
In the above equations a superimposed dot denotes thederivative with respect to time a comma followed by a suffixdenotes material derivative and the summation conventionis used 120590
119894119895are the components of the stress tensor 120576
119894119895are
the components of the strain tensor 119906119894are the components
of displacement vector 120581119894119895are the coefficients of thermal
conductivity 119878 is the entropy density 119865119894are the components
of Lorentz force Ω is the angular velocity 120578119894is the flow of
the diffusing mass vector 119902119894are the components of heat flux
vector 1205910is the thermal relaxation time 120591 is the diffusion
relaxation time 119879 is the absolute temperature 1198790is the initial
reference temperature 120588 is themass density119862119864is the specific
heat at constant strain 120572119905is the coefficient of linear thermal
expansion 120572119888is the coefficient of linear diffusion expansion
119863119894119895are the coefficients of diffusion 119862 is the concentration of
diffusive material 120582 120583 are Lamersquos constants 119875 is the chemicalpotential ldquo119886rdquo is a measure of thermodiffusion effect and ldquo119887rdquois a measure of diffusive effect
We consider the problem of a homogeneous isotropicand perfectly conducting thermoelastic rotating half-space(119909 ge 0) A magnetic field with constant intensity 119867 =
(0 01198670) acts parallel to the bounding surface (taken as
the direction of the 119911-axis) At the same time angularvelocity Ω = (0 0 Ω) in half-space goes around the 119911-axis Considering rotating effect the equation of motion isincluded in the centripetal acceleration related to time andCoriolis accelerationΩtimes (Ω times 119906) item 2Ω times The surface ofthe half-space is subjected at time 119905 = 0 to a thermal shockand a chemical potential that are functions of 119910 and 119905 Thusall the variables will be functions of time 119905 and coordinates 119909and 119910 Due to the application ofH this results in an inducedmagnetic field ℎ and an induced electric field 119864 in the half-space when it undergoes deformation
It can be noted from (19)ndash(22c) that the induced electricfield the induced magnetic field and the Lorentz forceare functions of the components of displacement whichimplies that the generalized electromagnetothermoelasticproblems with diffusion can then be treated as a generalizedthermoelastic one with diffusion Once the components ofdisplacement are obtained the induced electric field and theinduced magnetic field can be calculated from (19) and (20)respectively
Generally speaking for generalized multifield problemsthe involved physical fields such as electromagnetic fieldtemperature field strain field and diffusion field wouldcouple with each other whichmakes the governing equationsof such problems usually too complex to get the solutionsby analytical method so that numerical methods would bepowerful tools to solve such problems One feasible waycan be the integral transform techniques By means of thismethod the partial differential governing equations can beconverted into ordinary differential equations and solved intransform domain By applying inverse transform the solu-tions of the problem in time domain can then be obtainedHowever this method encounters loss of precision believedto be caused by discretization error and truncation errorintroduced inevitably in the process of numerical inverseLaplace and Fourier transforms which leads to identifyingheat wave front and prediction of the second sound effect theresult of the [27] is thus not so obvious as that the result of [56] showing clear step in the temperature field for small timeAn alternative choice to such problems is the hybrid Laplacetransform-finite element method presented by Chen andWeng [28 29] The same as depicted above this method alsoencounters loss of precision and the step of the temperaturein the heat wave front is not obvious either Therefore theapplicability of the integral transform techniques as wellas the hybrid Laplace transform-finite element method togeneralized thermoelastic problems is limited To avoid thedefects of the above methods we are inspired to formulateour problem by finite element method and directly solve thederived nonlinear finite element equations in time domainas reported by Tian et al in [30] in which the obtainedresults show that this method can achieve a high calculation
4 Mathematical Problems in Engineering
precision The diffusion problem done by Xia et al [26] wasjust solved by using this method recently
3 Finite Element Formulations
Rewrite (5) (6) and (7) in matrix form as follows
120590 = [1198620] 120576 minus 120574
1 120579 minus 120574
2 119875
119862 = 1205742119879
120576 + 119889120579 + 119899119875
120588119878 = 1205741119879
120576 + 1198881120579 + 119889119875
(23)
The generalized heat conduction law and Fickrsquos law of massdiffusion can be written in matrix form as
119902 + 1205910 119902 = minus [120581] 120579
1015840
+ 120591 = [119863] 11987510158401015840
(24)
where 1205791015840 = 120579119894 11987510158401015840 = 119875
119894119894
According to finite elementmethod the half-space can bedivided into elements and nodal points and any variable con-sidered within an element can be approximated by the valuesof nodal points together with shape functions To this endwe introduce two sets of shape functions to approximate thedisplacement the temperature and the chemical potential onthe element level
119906 = [119873119890
1] 119906119890 120579 = 119873
119890
2119879120579119890 119875 = 119873
119890
2119879119875119890
(25)
where 119906119890 120579119890 and 119875119890 are the vectors of nodal displace-
ment temperature and chemical potential respectively [1198731198901]
and 1198731198902 are shape functions they are
[119873119890
1] = [
1198731
0 1198732
0 sdot sdot sdot 119873119899
0
0 1198731
0 1198732
0 sdot sdot sdot 119873119899
]
119873119890
2119879= 1198731 119873
2sdot sdot sdot 119873
119899
(26)
where 119899 denotes the number of nodes in the gridIn terms of 120576
In the absence of body force and inner heat sourceconsidering the Lorentz force 119865
119894 the virtual displacement
principle of the generalized electromagnetothermoelasticproblems with diffusion can be formulated as
int119881
[120575120576119879120590 minus 120588119879
0( 119878 + 120591
0119878) 120575 120579 + 120575120579
1015840119879
119902 + 1205910119902
+1205751198751015840119879
(120578 + 120591 120578) minus 120575119875119879( + 120591)] 119889119881
= int119881
120575119906119879(119865 minus 120588 [ + Ω times (Ω times 119906)
+ 2Ω times ]) 119889119881
+ int119860120590
120575119906119879 119889119860 + int
119860119902
120575120579 119902119889119860
+ int119860120578
120575119875 120578 119889119860
(30)
where represents the traction vector 119902 the heat fluxvector and 120578 the mass flux vector and the variables witha superimposed bar mean that they are given on surface 119860
120590
represents the area of the stress tensor119860119902represents the area
of the heat flux vector and119860120578represents the area of the mass
flux vectorSubstituting (23)ndash(25) (27) and (29) into (30) we obtain
int119881
120575120576119879120590 119889119881
= int119881
(120575119906119890119879[1198611]119879) [[1198620] [1198611] 119906119890 minus 1205741 119873119890
2119879120579119890
minus 1205742 119873119890
2119879119875119890] 119889119881
= 120575119906119890119879int119881
[1198611]119879([[1198620] [1198611] 119906119890 minus 1205741 119873119890
Once the initial conditions and the boundary conditionsare specified the finite element equation in (32) can besolved directly in time domain In the process of numericalcalculation and finite element solution the space domainand time domain are discrete In the calculation because thesurface of 119874119860 is subjected to a thermal shock and a chemicalpotential shock this part of the unit is divided in a moredetailed way the entire model is divided into 1535 units and3222 nodes similarly the initial time step is set to 119905 = 3times10
minus7the variable threshold is set at 119905 = 1times 10
minus7 which ensures theaccuracy and convergence and also saves a lot of calculatedtime
4 Numerical Results and Discussions
The schematic of the considered half-space as well as theapplied loads on its bounding surface is shown in Figure 1(a)The bounding surface is assumed to be traction-free and thethermal shock and the chemical potential shock applied onthe bounding Surface have respectively the following form
120579 = 1205790119867(119905)119867 (119871 minus
Due to the symmetries of geometrical shape and boundaryconditions the problem can be treated as a plane strain prob-lem and only half of the half-space needs to be consideredThe model for simulation is shown in Figure 1(b) where119874119860119861119862 outlines the region for implementing the simulationand119874119863 represents the regionwithinwhich the thermal shockand the chemical potential shock are applied
Mathematical Problems in Engineering 7
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
00
02
04
06
08
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
t = 005t = 01
t = 015
Figure 2 Nondimensional temperature distribution along 119874119860
The half-space is taken to be copper material and thematerial properties are
120582 = 776 times 1010 kg(ms2) 120583 = 386 times 10
10 kg(ms2)
120588 = 8954 kgm3 120581 = 386W(mK)
119863 = 85 times 10minus9 kg sm3 120572
119888= 198 times 10
minus4m3kg
120572119905= 178 times 10
minus5 Kminus1 119862119864= 3831 J(kgK)
119886 = 12 times 104m2 (s2 K) 119887 = 9 times 10
5m5 (kg s2) (36)
To simplify the simulation we introduce the following nondi-mensional variables
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
In the above equations a superimposed dot denotes thederivative with respect to time a comma followed by a suffixdenotes material derivative and the summation conventionis used 120590
119894119895are the components of the stress tensor 120576
119894119895are
the components of the strain tensor 119906119894are the components
of displacement vector 120581119894119895are the coefficients of thermal
conductivity 119878 is the entropy density 119865119894are the components
of Lorentz force Ω is the angular velocity 120578119894is the flow of
the diffusing mass vector 119902119894are the components of heat flux
vector 1205910is the thermal relaxation time 120591 is the diffusion
relaxation time 119879 is the absolute temperature 1198790is the initial
reference temperature 120588 is themass density119862119864is the specific
heat at constant strain 120572119905is the coefficient of linear thermal
expansion 120572119888is the coefficient of linear diffusion expansion
119863119894119895are the coefficients of diffusion 119862 is the concentration of
diffusive material 120582 120583 are Lamersquos constants 119875 is the chemicalpotential ldquo119886rdquo is a measure of thermodiffusion effect and ldquo119887rdquois a measure of diffusive effect
We consider the problem of a homogeneous isotropicand perfectly conducting thermoelastic rotating half-space(119909 ge 0) A magnetic field with constant intensity 119867 =
(0 01198670) acts parallel to the bounding surface (taken as
the direction of the 119911-axis) At the same time angularvelocity Ω = (0 0 Ω) in half-space goes around the 119911-axis Considering rotating effect the equation of motion isincluded in the centripetal acceleration related to time andCoriolis accelerationΩtimes (Ω times 119906) item 2Ω times The surface ofthe half-space is subjected at time 119905 = 0 to a thermal shockand a chemical potential that are functions of 119910 and 119905 Thusall the variables will be functions of time 119905 and coordinates 119909and 119910 Due to the application ofH this results in an inducedmagnetic field ℎ and an induced electric field 119864 in the half-space when it undergoes deformation
It can be noted from (19)ndash(22c) that the induced electricfield the induced magnetic field and the Lorentz forceare functions of the components of displacement whichimplies that the generalized electromagnetothermoelasticproblems with diffusion can then be treated as a generalizedthermoelastic one with diffusion Once the components ofdisplacement are obtained the induced electric field and theinduced magnetic field can be calculated from (19) and (20)respectively
Generally speaking for generalized multifield problemsthe involved physical fields such as electromagnetic fieldtemperature field strain field and diffusion field wouldcouple with each other whichmakes the governing equationsof such problems usually too complex to get the solutionsby analytical method so that numerical methods would bepowerful tools to solve such problems One feasible waycan be the integral transform techniques By means of thismethod the partial differential governing equations can beconverted into ordinary differential equations and solved intransform domain By applying inverse transform the solu-tions of the problem in time domain can then be obtainedHowever this method encounters loss of precision believedto be caused by discretization error and truncation errorintroduced inevitably in the process of numerical inverseLaplace and Fourier transforms which leads to identifyingheat wave front and prediction of the second sound effect theresult of the [27] is thus not so obvious as that the result of [56] showing clear step in the temperature field for small timeAn alternative choice to such problems is the hybrid Laplacetransform-finite element method presented by Chen andWeng [28 29] The same as depicted above this method alsoencounters loss of precision and the step of the temperaturein the heat wave front is not obvious either Therefore theapplicability of the integral transform techniques as wellas the hybrid Laplace transform-finite element method togeneralized thermoelastic problems is limited To avoid thedefects of the above methods we are inspired to formulateour problem by finite element method and directly solve thederived nonlinear finite element equations in time domainas reported by Tian et al in [30] in which the obtainedresults show that this method can achieve a high calculation
4 Mathematical Problems in Engineering
precision The diffusion problem done by Xia et al [26] wasjust solved by using this method recently
3 Finite Element Formulations
Rewrite (5) (6) and (7) in matrix form as follows
120590 = [1198620] 120576 minus 120574
1 120579 minus 120574
2 119875
119862 = 1205742119879
120576 + 119889120579 + 119899119875
120588119878 = 1205741119879
120576 + 1198881120579 + 119889119875
(23)
The generalized heat conduction law and Fickrsquos law of massdiffusion can be written in matrix form as
119902 + 1205910 119902 = minus [120581] 120579
1015840
+ 120591 = [119863] 11987510158401015840
(24)
where 1205791015840 = 120579119894 11987510158401015840 = 119875
119894119894
According to finite elementmethod the half-space can bedivided into elements and nodal points and any variable con-sidered within an element can be approximated by the valuesof nodal points together with shape functions To this endwe introduce two sets of shape functions to approximate thedisplacement the temperature and the chemical potential onthe element level
119906 = [119873119890
1] 119906119890 120579 = 119873
119890
2119879120579119890 119875 = 119873
119890
2119879119875119890
(25)
where 119906119890 120579119890 and 119875119890 are the vectors of nodal displace-
ment temperature and chemical potential respectively [1198731198901]
and 1198731198902 are shape functions they are
[119873119890
1] = [
1198731
0 1198732
0 sdot sdot sdot 119873119899
0
0 1198731
0 1198732
0 sdot sdot sdot 119873119899
]
119873119890
2119879= 1198731 119873
2sdot sdot sdot 119873
119899
(26)
where 119899 denotes the number of nodes in the gridIn terms of 120576
In the absence of body force and inner heat sourceconsidering the Lorentz force 119865
119894 the virtual displacement
principle of the generalized electromagnetothermoelasticproblems with diffusion can be formulated as
int119881
[120575120576119879120590 minus 120588119879
0( 119878 + 120591
0119878) 120575 120579 + 120575120579
1015840119879
119902 + 1205910119902
+1205751198751015840119879
(120578 + 120591 120578) minus 120575119875119879( + 120591)] 119889119881
= int119881
120575119906119879(119865 minus 120588 [ + Ω times (Ω times 119906)
+ 2Ω times ]) 119889119881
+ int119860120590
120575119906119879 119889119860 + int
119860119902
120575120579 119902119889119860
+ int119860120578
120575119875 120578 119889119860
(30)
where represents the traction vector 119902 the heat fluxvector and 120578 the mass flux vector and the variables witha superimposed bar mean that they are given on surface 119860
120590
represents the area of the stress tensor119860119902represents the area
of the heat flux vector and119860120578represents the area of the mass
flux vectorSubstituting (23)ndash(25) (27) and (29) into (30) we obtain
int119881
120575120576119879120590 119889119881
= int119881
(120575119906119890119879[1198611]119879) [[1198620] [1198611] 119906119890 minus 1205741 119873119890
2119879120579119890
minus 1205742 119873119890
2119879119875119890] 119889119881
= 120575119906119890119879int119881
[1198611]119879([[1198620] [1198611] 119906119890 minus 1205741 119873119890
Once the initial conditions and the boundary conditionsare specified the finite element equation in (32) can besolved directly in time domain In the process of numericalcalculation and finite element solution the space domainand time domain are discrete In the calculation because thesurface of 119874119860 is subjected to a thermal shock and a chemicalpotential shock this part of the unit is divided in a moredetailed way the entire model is divided into 1535 units and3222 nodes similarly the initial time step is set to 119905 = 3times10
minus7the variable threshold is set at 119905 = 1times 10
minus7 which ensures theaccuracy and convergence and also saves a lot of calculatedtime
4 Numerical Results and Discussions
The schematic of the considered half-space as well as theapplied loads on its bounding surface is shown in Figure 1(a)The bounding surface is assumed to be traction-free and thethermal shock and the chemical potential shock applied onthe bounding Surface have respectively the following form
120579 = 1205790119867(119905)119867 (119871 minus
Due to the symmetries of geometrical shape and boundaryconditions the problem can be treated as a plane strain prob-lem and only half of the half-space needs to be consideredThe model for simulation is shown in Figure 1(b) where119874119860119861119862 outlines the region for implementing the simulationand119874119863 represents the regionwithinwhich the thermal shockand the chemical potential shock are applied
Mathematical Problems in Engineering 7
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
00
02
04
06
08
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
t = 005t = 01
t = 015
Figure 2 Nondimensional temperature distribution along 119874119860
The half-space is taken to be copper material and thematerial properties are
120582 = 776 times 1010 kg(ms2) 120583 = 386 times 10
10 kg(ms2)
120588 = 8954 kgm3 120581 = 386W(mK)
119863 = 85 times 10minus9 kg sm3 120572
119888= 198 times 10
minus4m3kg
120572119905= 178 times 10
minus5 Kminus1 119862119864= 3831 J(kgK)
119886 = 12 times 104m2 (s2 K) 119887 = 9 times 10
5m5 (kg s2) (36)
To simplify the simulation we introduce the following nondi-mensional variables
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
precision The diffusion problem done by Xia et al [26] wasjust solved by using this method recently
3 Finite Element Formulations
Rewrite (5) (6) and (7) in matrix form as follows
120590 = [1198620] 120576 minus 120574
1 120579 minus 120574
2 119875
119862 = 1205742119879
120576 + 119889120579 + 119899119875
120588119878 = 1205741119879
120576 + 1198881120579 + 119889119875
(23)
The generalized heat conduction law and Fickrsquos law of massdiffusion can be written in matrix form as
119902 + 1205910 119902 = minus [120581] 120579
1015840
+ 120591 = [119863] 11987510158401015840
(24)
where 1205791015840 = 120579119894 11987510158401015840 = 119875
119894119894
According to finite elementmethod the half-space can bedivided into elements and nodal points and any variable con-sidered within an element can be approximated by the valuesof nodal points together with shape functions To this endwe introduce two sets of shape functions to approximate thedisplacement the temperature and the chemical potential onthe element level
119906 = [119873119890
1] 119906119890 120579 = 119873
119890
2119879120579119890 119875 = 119873
119890
2119879119875119890
(25)
where 119906119890 120579119890 and 119875119890 are the vectors of nodal displace-
ment temperature and chemical potential respectively [1198731198901]
and 1198731198902 are shape functions they are
[119873119890
1] = [
1198731
0 1198732
0 sdot sdot sdot 119873119899
0
0 1198731
0 1198732
0 sdot sdot sdot 119873119899
]
119873119890
2119879= 1198731 119873
2sdot sdot sdot 119873
119899
(26)
where 119899 denotes the number of nodes in the gridIn terms of 120576
In the absence of body force and inner heat sourceconsidering the Lorentz force 119865
119894 the virtual displacement
principle of the generalized electromagnetothermoelasticproblems with diffusion can be formulated as
int119881
[120575120576119879120590 minus 120588119879
0( 119878 + 120591
0119878) 120575 120579 + 120575120579
1015840119879
119902 + 1205910119902
+1205751198751015840119879
(120578 + 120591 120578) minus 120575119875119879( + 120591)] 119889119881
= int119881
120575119906119879(119865 minus 120588 [ + Ω times (Ω times 119906)
+ 2Ω times ]) 119889119881
+ int119860120590
120575119906119879 119889119860 + int
119860119902
120575120579 119902119889119860
+ int119860120578
120575119875 120578 119889119860
(30)
where represents the traction vector 119902 the heat fluxvector and 120578 the mass flux vector and the variables witha superimposed bar mean that they are given on surface 119860
120590
represents the area of the stress tensor119860119902represents the area
of the heat flux vector and119860120578represents the area of the mass
flux vectorSubstituting (23)ndash(25) (27) and (29) into (30) we obtain
int119881
120575120576119879120590 119889119881
= int119881
(120575119906119890119879[1198611]119879) [[1198620] [1198611] 119906119890 minus 1205741 119873119890
2119879120579119890
minus 1205742 119873119890
2119879119875119890] 119889119881
= 120575119906119890119879int119881
[1198611]119879([[1198620] [1198611] 119906119890 minus 1205741 119873119890
Once the initial conditions and the boundary conditionsare specified the finite element equation in (32) can besolved directly in time domain In the process of numericalcalculation and finite element solution the space domainand time domain are discrete In the calculation because thesurface of 119874119860 is subjected to a thermal shock and a chemicalpotential shock this part of the unit is divided in a moredetailed way the entire model is divided into 1535 units and3222 nodes similarly the initial time step is set to 119905 = 3times10
minus7the variable threshold is set at 119905 = 1times 10
minus7 which ensures theaccuracy and convergence and also saves a lot of calculatedtime
4 Numerical Results and Discussions
The schematic of the considered half-space as well as theapplied loads on its bounding surface is shown in Figure 1(a)The bounding surface is assumed to be traction-free and thethermal shock and the chemical potential shock applied onthe bounding Surface have respectively the following form
120579 = 1205790119867(119905)119867 (119871 minus
Due to the symmetries of geometrical shape and boundaryconditions the problem can be treated as a plane strain prob-lem and only half of the half-space needs to be consideredThe model for simulation is shown in Figure 1(b) where119874119860119861119862 outlines the region for implementing the simulationand119874119863 represents the regionwithinwhich the thermal shockand the chemical potential shock are applied
Mathematical Problems in Engineering 7
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
00
02
04
06
08
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
t = 005t = 01
t = 015
Figure 2 Nondimensional temperature distribution along 119874119860
The half-space is taken to be copper material and thematerial properties are
120582 = 776 times 1010 kg(ms2) 120583 = 386 times 10
10 kg(ms2)
120588 = 8954 kgm3 120581 = 386W(mK)
119863 = 85 times 10minus9 kg sm3 120572
119888= 198 times 10
minus4m3kg
120572119905= 178 times 10
minus5 Kminus1 119862119864= 3831 J(kgK)
119886 = 12 times 104m2 (s2 K) 119887 = 9 times 10
5m5 (kg s2) (36)
To simplify the simulation we introduce the following nondi-mensional variables
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
Once the initial conditions and the boundary conditionsare specified the finite element equation in (32) can besolved directly in time domain In the process of numericalcalculation and finite element solution the space domainand time domain are discrete In the calculation because thesurface of 119874119860 is subjected to a thermal shock and a chemicalpotential shock this part of the unit is divided in a moredetailed way the entire model is divided into 1535 units and3222 nodes similarly the initial time step is set to 119905 = 3times10
minus7the variable threshold is set at 119905 = 1times 10
minus7 which ensures theaccuracy and convergence and also saves a lot of calculatedtime
4 Numerical Results and Discussions
The schematic of the considered half-space as well as theapplied loads on its bounding surface is shown in Figure 1(a)The bounding surface is assumed to be traction-free and thethermal shock and the chemical potential shock applied onthe bounding Surface have respectively the following form
120579 = 1205790119867(119905)119867 (119871 minus
Due to the symmetries of geometrical shape and boundaryconditions the problem can be treated as a plane strain prob-lem and only half of the half-space needs to be consideredThe model for simulation is shown in Figure 1(b) where119874119860119861119862 outlines the region for implementing the simulationand119874119863 represents the regionwithinwhich the thermal shockand the chemical potential shock are applied
Mathematical Problems in Engineering 7
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
00
02
04
06
08
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
t = 005t = 01
t = 015
Figure 2 Nondimensional temperature distribution along 119874119860
The half-space is taken to be copper material and thematerial properties are
120582 = 776 times 1010 kg(ms2) 120583 = 386 times 10
10 kg(ms2)
120588 = 8954 kgm3 120581 = 386W(mK)
119863 = 85 times 10minus9 kg sm3 120572
119888= 198 times 10
minus4m3kg
120572119905= 178 times 10
minus5 Kminus1 119862119864= 3831 J(kgK)
119886 = 12 times 104m2 (s2 K) 119887 = 9 times 10
5m5 (kg s2) (36)
To simplify the simulation we introduce the following nondi-mensional variables
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
Once the initial conditions and the boundary conditionsare specified the finite element equation in (32) can besolved directly in time domain In the process of numericalcalculation and finite element solution the space domainand time domain are discrete In the calculation because thesurface of 119874119860 is subjected to a thermal shock and a chemicalpotential shock this part of the unit is divided in a moredetailed way the entire model is divided into 1535 units and3222 nodes similarly the initial time step is set to 119905 = 3times10
minus7the variable threshold is set at 119905 = 1times 10
minus7 which ensures theaccuracy and convergence and also saves a lot of calculatedtime
4 Numerical Results and Discussions
The schematic of the considered half-space as well as theapplied loads on its bounding surface is shown in Figure 1(a)The bounding surface is assumed to be traction-free and thethermal shock and the chemical potential shock applied onthe bounding Surface have respectively the following form
120579 = 1205790119867(119905)119867 (119871 minus
Due to the symmetries of geometrical shape and boundaryconditions the problem can be treated as a plane strain prob-lem and only half of the half-space needs to be consideredThe model for simulation is shown in Figure 1(b) where119874119860119861119862 outlines the region for implementing the simulationand119874119863 represents the regionwithinwhich the thermal shockand the chemical potential shock are applied
Mathematical Problems in Engineering 7
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
00
02
04
06
08
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
t = 005t = 01
t = 015
Figure 2 Nondimensional temperature distribution along 119874119860
The half-space is taken to be copper material and thematerial properties are
120582 = 776 times 1010 kg(ms2) 120583 = 386 times 10
10 kg(ms2)
120588 = 8954 kgm3 120581 = 386W(mK)
119863 = 85 times 10minus9 kg sm3 120572
119888= 198 times 10
minus4m3kg
120572119905= 178 times 10
minus5 Kminus1 119862119864= 3831 J(kgK)
119886 = 12 times 104m2 (s2 K) 119887 = 9 times 10
5m5 (kg s2) (36)
To simplify the simulation we introduce the following nondi-mensional variables
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
Figure 9 Nondimensional concentration distribution along 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10N
ondi
men
siona
l che
mic
al p
oten
tial P
Nondimensional coordinate y
t = 005t = 01
t = 015
Figure 10 Nondimensional chemical potential distribution along119874119862
The nondimensional temperature displacement stresschemical potential concentration and induced magneticfield are illustrated in Figures 2ndash12 respectively droppingthe asterisk at the upper right corner of the nondimensionalvariables for convenience It should be pointed out that wavereflection from any edges is excluded in the simulation
Figures 2 and 3 show the distributions of the nondi-mensional temperature along 119874119860 and 119874119862 respectively InFigure 2 when the time 119905 is given the distance of the heatpropagation in the 119909 direction should be 119909 = V
ℎ119905 where V
ℎis
nondimensional heat wave velocity When 120591 = 002 we canachieve V
ℎ= 007 The heat propagation in the 119909 direction at
the time 119905 = 05 119905 = 01 is 119909 = 035 119909 = 07 respectivelyFrom Figure 2 a distinct temperature step on thermal wavefront distribution on 119874119860 can be readily seen but it becomesindistinct along with the passage of time In Figure 3 within
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
Figure 12 Nondimensional induced magnetic field distributionalong 119874119860
0 le 119910 le 02 the temperature keeps constant all along whichis consistent with the thermal boundary condition along119874119863As shown in Figures 2 and 3 the temperature increases withthe passage of time
Figure 4 shows the distributions of the nondimensionalhorizontal displacement along119874119860 Due to the thermal shockthe parts of the half-space near the bounding surface expandtoward the unconstrained direction thus yielding negativedisplacement It can be found that different parts on 119874119860
undergo different deformations Some undergo expansionand some undergo compression while the rest remain undis-turbed resulting in the displacement shift from negative topositive gradually With the passage of time the expansionparts enlarge and move inside dynamically making thenegative-positive region transform dynamically We should
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
Non
dim
ensio
nal i
nduc
ed m
agne
tic fi
eld h
Nondimensional coordinate x
Ω = 00 t = 01Ω = 80 t = 01
minus060
minus055
minus050
minus045
minus040
minus035
minus030
minus025
minus020
minus015
minus010
minus005
000
005
Figure 13 Nondimensional induced magnetic field distributionalong 119874119860
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate x
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 14 Nondimensional temperature distribution along 119874119860
be aware that the vertical displacement for 119874119860 is always zerobecause of the symmetries
Figures 5 and 6 show the distributions of the nondimen-sional horizontal and vertical displacements on 119874119862 respec-tively As seen from Figure 5 the horizontal displacementon 119874119862 is negative which means that 119874119862 undergoes thermalexpansion deformation andmoves toward the unconstraineddirection As shown in Figure 6 the vertical displacementalong 119874119862 is positive This can be interpreted as follows Dueto the symmetries the vertical displacement of point ldquo119900rdquo isalways zero This implies that point ldquo119900rdquo can not be allowedto move up and down which prevents all the other pointson 119874119862 from moving downwards thus leading to positivedisplacement The vertical displacement on 119874119862 firstly goesup and then goes down It can also be seen from Figures 5
10 Mathematical Problems in Engineering
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000
01
02
03
04
05
06
07
08
09
10
Nondimensional coordinate y
Non
dim
ensio
nal t
empe
ratu
re120579
Ω = 00 t = 01Ω = 80 t = 01
Figure 15 Nondimensional temperature distribution along 119874119862
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
and 6 that the magnitudes of the displacement increase withthe passage of time
Figure 7 shows the distributions of nondimensional stress120590119909119909
along 119874119860 Due to the symmetries the other two compo-nents of stress namely 120590
119910119910and 120590119909119910 are always zero along119874119860
It can be observed that 120590119909119909
is all negative which is known ascompressive stress
Figures 8 9 10 and 11 show the distributions of nondi-mensional chemical potential and concentration along 119874119860
and 119874119862 respectively It can be observed that the speed ofdiffusive wave is larger than that of thermoelastic wave whichcan be deduced by comparing the distance of diffusive wavetraversing across the half-space with that of thermoelasticwave along the same direction for a given time
Figure 12 shows the distributions of the nondimensionalinduced magnetic field Due to the mutual interaction
Figure 18 Nondimensional stress distribution along 119874119860
between the applied external magnetic field and the elasticdeformation this results in an induced magnetic field inthe half-space As shown in Figure 12 the magnitude of ℎincreases with the passage of time
It can be readily seen from Figures 2ndash12 that all theconsidered variables have a nonzero value only in a boundedregion and the value vanishes outside this region whichis totally dominated by the nature of the finite speeds ofthermoelastic wave and diffusive wave
From Figures 13 14 15 16 17 18 19 20 21 and 22 it canbe readily seen that rotation acts to decrease the magnitudeof the real part of displacement stress and inducedmagneticfield and not to affect themagnitude of temperature chemicalpotential and concentration
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
Figure 20 Nondimensional chemical potential distribution along119874119862
5 Concluding Remarks
A two-dimensional generalized electromagnetothermoelas-tic problem with diffusion for a rotating half-space is studiedin the context of the theory of generalized thermoelasticdiffusion by means of finite element methodThe nondimen-sional temperature displacement stress chemical potentialconcentration and induced magnetic field are obtained Theresults show that (1) all the considered variables have anonzero value only in a bounded region and vanish identi-cally outside this region which is governed by the nature ofthe finite speeds of thermoelastic wave and diffusive wave (2)the speed of diffusive wave is larger than that of thermoelasticwave (3) rotation acts to decrease the magnitude of thereal part of displacement stress and induced magnetic field
Figure 22 Nondimensional concentration distribution along 119874119862
and not to affect the magnitude of temperature chemicalpotential and concentration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M A Biot ldquoThermoelasticity and irreversible thermodynam-icsrdquo Journal of Applied Physics vol 27 no 3 pp 240ndash253 1956
[2] HW Lord and Y Shulman ldquoA generalized dynamical theory ofthermoelasticityrdquo Journal of the Mechanics and Physics of Solidsvol 15 no 5 pp 299ndash309 1967
[3] A E Green and K A Lindsay ldquoThermoelasticityrdquo Journal ofElasticity vol 2 no 1 pp 1ndash7 1972
12 Mathematical Problems in Engineering
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
[4] R S Dhaliwal and H H Sherief ldquoGeneralized thermoelasticityfor anisotropic mediardquo Quarterly of Applied Mathematics vol38 no 1 pp 1ndash8 1980
[5] HH Sherief and R S Dhaliwal ldquoGeneralized one-dimensionalthermal shock problem for small timesrdquo Journal of ThermalStresses vol 4 no 3-4 pp 407ndash420 1981
[6] R S Dhaliwal and J G Rokne ldquoOne-dimensional thermalshock problem with two relaxation timesrdquo Journal of ThermalStresses vol 12 no 2 pp 259ndash279 1989
[7] H H Sherief and M N Anwar ldquoState-space approach to two-dimensional generalized thermoelasticity problemsrdquo Journal ofThermal Stresses vol 17 no 4 pp 567ndash586 1994
[8] J N Sharma and D Chand ldquoTransient generalised magne-tothermoelastic waves in a half-spacerdquo International Journal ofEngineering Science vol 26 no 9 pp 951ndash958 1988
[9] M A Ezzat M I Othman and A S El-KaramanyldquoElectromagneto-thermoelastic plane waves with thermalrelaxation in a medium of perfect conductivityrdquo Journal ofThermal Stresses vol 38 pp 107ndash120 2001
[10] H H Sherief and K A Helmy ldquoA two-dimensional problemfor a half-space in magneto-thermoelasticity with thermalrelaxationrdquo International Journal of Engineering Science vol 40no 5 pp 587ndash604 2002
[11] H Tianhu S Yapeng and T Xiaogeng ldquoA two-dimensionalgeneralized thermal shock problem for a half-space inelectromagneto-thermoelasticityrdquo International Journal ofEngineering Science vol 42 no 8-9 pp 809ndash823 2004
[12] M A Ezzat and H M Youssef ldquoGeneralized magneto-thermoelasticity in a perfectly conducting mediumrdquo Interna-tional Journal of Solids and Structures vol 42 no 24-25 pp6319ndash6334 2005
[13] J N Sharma andM DThakur ldquoEffect of rotation on Rayleigh-Lambwaves inmagneto-thermoelasticmediardquo Journal of Soundand Vibration vol 296 no 4-5 pp 871ndash887 2006
[14] M I A Othman and Y Song ldquoEffect of rotation on planewaves of generalized electromagneto-thermoelasticity with tworelaxation timesrdquo Applied Mathermatical Modeling vol 43 pp712ndash727 2007
[15] M Z Guan ldquoA two-dimensional generalized electromagneto-thermoelastic coupled problem for a rotating half-space byusing Laplace transform and its numerical inversionrdquo ChinaScience and Technology Information vol 24 pp 336ndash338 2007
[16] T-H He and W-W Jia ldquoTwo-dimensional generalizedelectromagneto-thermoelastic coupled problem for a rotatinghalf-spacerdquo Engineering Mechanics vol 26 no 2 pp 196ndash2022009
[17] M I A Othman and Y Song ldquoReflection of magneto-thermo-elastic waves from a rotating elastic half-spacerdquo InternationalJournal of Engineering Science vol 46 no 5 pp 459ndash474 2008
[18] S Deswal and K Kalkal ldquoA two-dimensional generalizedelectro-magneto-thermoviscoelastic problem for a half-spacewith diffusionrdquo International Journal of Thermal Sciences vol50 no 5 pp 749ndash759 2011
[19] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids Irdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 55ndash64 1974
[20] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 129ndash135 1974
[21] W Nowacki ldquoDynamical problems of thermoelastic diffusionin solids IIIrdquo Bulletin de lrsquoAcademie Polonaise des Sciences Seriedes Sciences Techniques vol 22 pp 266ndash275 1974
[22] H H Sherief F A Hamza and H A Saleh ldquoThe theory ofgeneralized thermoelastic diffusionrdquo International Journal ofEngineering Science vol 42 no 5-6 pp 591ndash608 2004
[23] H H Sherief and H A Saleh ldquoA half-space problem in thetheory of generalized thermoelastic diffusionrdquo InternationalJournal of Solids and Structures vol 42 no 15 pp 4484ndash44932005
[24] B Singh ldquoReflection of SV waves from the free surface of anelastic solid in generalized thermoelastic diffusionrdquo Journal ofSound and Vibration vol 291 no 3ndash5 pp 764ndash778 2006
[25] M Aouadi ldquoA problem for an infinite elastic body with aspherical cavity in the theory of generalized thermoelasticdiffusionrdquo International Journal of Solids and Structures vol 44no 17 pp 5711ndash5722 2007
[26] R-H Xia X-G Tian and Y-P Shen ldquoThe influence of dif-fusion on generalized thermoelastic problems of infinite bodywith a cylindrical cavityrdquo International Journal of EngineeringScience vol 47 no 5-6 pp 669ndash679 2009
[27] H H Sherief and F A Megahed ldquoA two-dimensional thermoe-lasticity problem for a half space subjected to heat sourcesrdquoInternational Journal of Solids and Structures vol 36 no 9 pp1369ndash1382 1999
[28] T-C Chen and C-I Weng ldquoGeneralized coupled transientthermoelastic plane problems by Laplace transform-finitemethodrdquo Journal of AppliedMechanics Transactions ASME vol55 no 2 pp 377ndash382 1988
[29] C Tei-Chen andW Cheng-I ldquoCoupled transient thermoelasticresponse in an axi-symmetric circular cylinder by Laplacetransform-finite element methodrdquo Computers and Structuresvol 33 no 2 pp 533ndash542 1989
[30] X Tian Y Shen C Chen and T He ldquoA direct finite elementmethod study of generalized thermoelastic problemsrdquo Interna-tional Journal of Solids and Structures vol 43 no 7-8 pp 2050ndash2063 2006
