Effect of Magnetic Field and a Mode-I Crack 3D-Problem in Micropolar Thermoelastic Cubic Medium Possessing Under Three Theories

Transcription

1 Journal of Solid Mechanics Vol. 5, No. () pp Effect of Magnetic Field and a Mode-I Crack D-Problem in Micropolar Thermoelastic Cubic Medium Possessing Under Three Theories Kh. Lotf,,, N. Yahia Department of Mathematics, Facult of Science, Zagazig Universit, Zagazig Egpt, P.O. Bo 59 Department of Mathematics, Facult of Science and Arts, Al-mithnab, Qassim Universit, P.O. Bo 9, Buridah 59, Al-mithnab, Kingdom of Saudi Arabia Received Ma ; accepted Jul ABSTRACT A model of the equations of two dimensional problems in a half space, whose surface in free of micropolar thermoelastic medium possesses cubic smmetr as a result of a Mode-I Crack is studied. There acts an initial magnetic field parallel to the plane boundar of the half- space. The crack is subjected to prescribed temperature and stress distribution. The formulation in the contet of the Lord-Şhulman theor LS includes one relaation time and Green-Lindsa theor GL with two relaation times, as well as the classical dnamical coupled theor CD. The normal mode analsis is used to obtain the eact epressions for the displacement, microrotation, stresses and temperature distribution. The variations of the considered variables with the horizontal distance are illustrated graphicall. Comparisons are made with the results in the presence of magnetic field. A comparison is also made between the three theories for different depths.. All rights reserved. Kewords: GLtheor; Magneto-thermoelasticit; Mode-I Crack; Microrotation; Micropolar thermoelastic medium INTRODUCTION T HE linear theor of elasticit is of paramount importance in the stress analsis of steel, which is the commonest engineering structural material. To a lesser etent, linear elasticit describes the mechanical behavior of the other common solid materials, e.g. concrete, wood and coal. However, the theor does not appl to the behavior of man of the new snthetic materials of the clastomer and polmer tpe, e.g. polmethl- methacrlate (Perspe), polethlene and polvinl chloride. The linear theor of micropolar elasticit is adequate to represent the behavior of such materials. For ultrasonic waves i.e. for the case of elastic vibrations characterized b high frequencies and small wavelengths, the influence of the bod microstructure becomes significant. This influence of microstructure results in the development of new tpe of waves, not in the classical theor of elasticit. Metals, polmers, composites, soils, rocks, concrete are tpical media with microstructures. More generall, most of the natural and manmade materials including engineering, geological and biological media possess a microstructure. Eringen and Şuhubi [] and Eringen [] developed the linear theor of micropolar elasticit. Thermoelasticit theories, which admit a finite speed for thermal signals, have been receiving a lot of attention for the past four decades. In contrast to the conventional coupled thermoelasticit theor based on a parabolic heat equation (Biot, []), which predicts an infinite speed for the propagation of heat, these theories involve a hperbolic Corresponding author. address: khlotf_@ahoo.com, khlotf_@ahoo.com (Kh. Lotf).. All rights reserved.

2 Kh. Lotf and N. Yahia 5 heat equation and are referred to as generalized thermoelasticit theories. Two generalizations to the coupled theor were introduced. The first is due to Lord and Şhulman [] who introduced the theor of generalized thermoelasticit with one relaation time b postulating a new law of heat conduction to replace the classical Fourier's law. Othman [5] constructs the model of generalized thermoelasticit in an isotropic elastic medium under the dependence of the modulus of elasticit on the reference temperature with one relaation time. The second generalization to the coupled theor of thermoelasticit is what is known as the theor of thermoelasticit with two relaation times or the theor of temperature rate dependent thermoelasticit, and was proposed b Green and Lindsa [6]. It is based on a form of the entrop inequalit proposed b Green and Laws [7]. Green and Lindsa [6] obtained another version of the constitutive equations. These equations were also obtained independentl and more eplicitl b Şuhubi [8]. This theor contains two constants that act as relaation times and modifies all the equations of the coupled theor, not onl the heat equation. The classical Fourier's law of heat conduction is not violated if the medium under consideration has a center of smmetr. Othman [9] studied the relaation effects on thermal shock problems in elastic half space of generalized magneto- thermoelastic waves under three theories. Following various methods, the elastic fields of various loadings, inclusion and inhomogeneit problems, and interaction energ of point defects and dislocation arrangement have been discussed etensivel in the past. Generall, all materials have elastic anisotropic properties which mean the mechanical behavior of an engineering material characterized b the direction dependence. The three dimensional stud for an anisotropic material is much more complicated to obtain than the isotropic one,however, due to the large number of elastic constants involved in the calculation. In particular, transversel isotropic and orthotropic materials, which ma not be distinguished from each other in plane strain and plane stress, have been more regularl studied. A review of literature on micropolar orthotropic continua shows that Iesan [], [][, [] analzed the static problems of plane micropolar strain of a homogeneous and orthotropic elastic solid, torsion problem of homogeneous and orthotropic clinders in the linear theor of micropolar elasticit and bending of orthotropic micropolar elastic beams b terminal couple. Nakamura et al. [] applied the finite element method for orthotropic micropolar elasticit. Recentl, Kumar and Choudhar [], [5], [6], [7], [8] have discussed various problems in orthotropic micropolar continua. Singh and Kumar [9] and Singh [] have also studied the plane waves in micropolar generalized thermoelastic solid. Othman and Lotf [] studied two-dimensional problem of generalized magneto-thermoelasticit under the effect of temperature dependent properties. Othman and Lotf [] studied transient disturbance in a half-space under generalized magneto-thermoelasticit with moving internal heat source. Othman and Lotf [] studied the plane waves in generalized thermo-microstretch elastic half-space b using a general model of the equations of generalized thermo-microstretch for a homogeneous isotropic elastic half space. Othman and Lotf [] studied the generalized thermo-microstretch elastic medium with temperature dependent properties for different theories. Othman and Lotf [5-6] studied the effect of magnetic field and inclined load in micropolar thermoelastic medium possessing cubic smmetr under three theories. The normal mode analsis was used to obtain the eact epression for the temperature distribution, thermal stresses, and the displacement components. In the recent ears, considerable efforts have been devoted the stud of failure and cracks in solids. This is due to the application of the latter generall in industr and particularl in the fabrication of electronic components. Most of the studies of dnamical crack problem are done using the equations of coupled or even uncoupled theories of thermoelasticit [7-]. This is suitable for most situations where long time effects are sought. However, when short time are important, as in man practical situations, the full sstem of generalized thermoelastic equations must be used []. The purpose of the present paper is to determine the normal displacement, normal force stress, and tangential couple stress in a micropolar elastic solid with cubic smmetr. The normal mode analsis used for the problem of generalized thermo-microstretch for an infinite space weakened b a finite linear opening Mode-I crack is solving for the considered variables. The distributions of the considered variables are represented graphicall. A comparison is carried out between the temperature, stresses, the couple stress, the microotation and displacement components as calculated from the generalized thermoelasticit LS, GL and CD theories for the propagation of waves in semiinfinite elastic solids with cubic smmetr. FORMULATION OF THE PROBLEM We consider a homogeneous, micropolar generalized thermoelastic solid half-space with cubic smmetr. We consider rectangular coordinate sstem (,, z) having origin on the surface = and -ais pointing verticall into the medium. A magnetic field with constant intensit H=(,,H ) acts parallel to the bounding plane (Table as the

3 55 Effect of Magnetic Field and a Mode-I Crack D-Problem direction of the z-ais). Due to the application of initial magnetic field H, there are results of an induced magnetic field h and an induced electric field E. The simplified linear equations of electrodnamics of slowl moving medium for a homogeneous, Thermall and electricall conducting elastic solid are, z H Geometr of the problem. curl h J E (). curl E h () div h () E. u H where u is the partied velocit of the medium, and the small effect of temperature gradient on J is ignored. The dnamic displacement vector is actuall measured from a stead state deformed position and the deformation is supposed to be small. The components of the magnetic intensit vector in the medium are H, H, H H h,, z (5) z The electric intensit vector is normal to both the magnetic intensit and the displacement vectors. Thus, it has the components ().. E H, E H u, E z (6) The current densit vector J be parallel to E, thus:.... h h J μ H ε v J μ H ε u Jz (7) h H,, e (8) where e is the dilatation. If we restrict our analsis to plane strain parallel to -plane with displacement vector u ( u, v,) and microotation vector,, then the field equations and constitutive relations for micropolar thermoelastic solid with cubic smmetr in the absence of bod forces. Bod couples and heat sources can be written b following the equations given b Minagawa et al., Green Lindsa [6] and Othman and Baljeet as: u u v e u T u A A A A A A H H T t t t t, (9)

4 Kh. Lotf and N. Yahia 56 v u u e v T v A A A A A A H H T t t t t, v u B A A A A j, t K T C ( n t ) T T ( n nt )e t t u v T A A T t, t u v T A A T t, t u v A A, v u A A, m z B, mz B, () () () () () (5) (6) (7) (8) where ij and m, are the components of force stress and coupled stress, A A ij linear epansion, is the densit, j is the microinertia, specific heat at constant strain; t and t are the thermal relaation times, and,, is coefficient of K is the coefficient of thermal conductivit, T. T C is the For thermoelastic micropolar isotropic medium A, A, A, A, B are characteristic constants of the material defied as: A k, A, A k, A, B (9) For simplification, we shall use the following non- dimensional variables: Co T ij i i, u i ui, t t, t t, t t, T, ij, Co To To To C C F, F h mij m,, g g, F, F, h, C T To T H. ij () C C A where, C. K Eqs. (9)- () take the following form (dropping the dashed for convenience) A A A A u u A u e T RH T t A A A t t A A A A A u e T RH T t A A A t t () ()

5 57 Effect of Magnetic Field and a Mode-I Crack D-Problem u A A C A A C jc T u T n to ( n nt ) t t t T B B B t () () Introducing potential functions defined b q q u, v, (5) in Eqs. () (), where q(,, t) and (,, t) are scalar potential functions, we obtain. q t T t t a t a a a a t n t T ( n nt ) t t t t q (6) (7) (8) (9) we can obtain from Eqs. (8) and (5) h q () where a A a A A a ( A A ) C a jc To R H C R o o,,,,,, H o o o H A A B B K () The solution of the considered phsical variables can be decomposed in terms of normal modes as the following form: [,,e,,q,m,t](,,t) [ (), (),e (), (),q (),m (),T ()]ep( t ia) () ij ij ij ij where is a comple constant and a is the wave number in the - direction. Using Eq. (), Eqs. (6) () (D a b )q nt () (D a a ) a () a (D a ) (D a a a5 ) (5) (D a n ) T (D a ) q (6) where

6 Kh. Lotf and N. Yahia 58 a t b, a, a, n, n n t, w( n n wt ) a a Eliminating,T between Eqs. (6)-() we obtain and (D B D B )(q, T ) (7) D B D B, ψ (8) where B a n b n (9) B a ( n b n ) a n b () B a ( a a ) a a a () 5 B a ( a a a a a ) a ( a a ) () The solution of Eqs. (7) and (8), which are bounded for >, are given b q T k j M j(a, ) e j () k j M j (a, ) e j () kn M n (a, ) e (5) n k n M n(a, ) e n (6) where M j(a, ), M j (a, ), M n(a, ) and M n (a, ) are some parameters depending on a and. k j, ( j,) are the roots of the characteristic equation of Eq. (7) and k n, ( n,) are the roots of the characteristic equation of Eq. (8). Using Eqs. ()- (6) into Eqs. (7) and (8) we get the following relations T k j R jm j(a, ) e j (7) k n R n M n(a, ) e n (8) where R K a b n (9),, R, K, a a a (5)

7 59 Effect of Magnetic Field and a Mode-I Crack D-Problem The roots K, and K, of Eqs. (7) and (8), respectivel are given b K, B B B (5) K, B B B (5) APPLICATION The plane boundar subjects to an instantaneous normal point force and the boundar surface is isothermal, the boundar conditions at the vertical plan z and in the beginning of the crack at are p( ), a T f ( ), a and T, a, m, (5) Using Eq.(), (5), (6)-(9) on the non-dimensional boundar conditions and using Eq.(), (5), (7)-(8), we obtain the epressions of displacements, force stress, coupled stress and temperature distribution for micropolar generalized thermoelastic medium with magnetic field as follows: Displacement of an eternal mode-i crack. k k k k u () iame iame k Me kme (5) k k k k v () k Me kme ia(me kme ) (55) () s M e s M e s M e s M e (56) k k k k () r M e r M e r M e r M e (57) k k k k k k m z () B (krme krme ) w / Co (58) k k T () R M e R M e (59) where s ( a A / C A k / C R ( wt ), s ( a A / C A k / C R ( wt )) o o o o s iak ( A A ) / Co, s iak ( A A ) / Co, r - iak ( A A ) / C, r - iak ( A A ) / C, o o r (a A R (A-A ) Ak ) / Co, r (a A R (A-A ) Ak ) / C o. (6)

8 Kh. Lotf and N. Yahia 6 Invoking the boundar conditions (5) at the surface of the plate, we obtain a sstem of four equations. After appling the inverse of matri method, we have the values of the four constants M j, j,, and M n, n,. Hence, we obtain the epressions of displacements, force stress, coupled stress and temperature distribution for micropolar generalized thermoelastic medium. NUMERICAL RESULTS AND DISCUSSIONS In order to illustrate the theoretical results obtained in preceding section and to compare these in the contet of various theories of thermoelasticit, we now present some numerical results. In the calculation process, we take the case of magnesium crstal (Eringen) as material subjected to mechanical and thermal disturbances for numerical calculations considering the material medium as that of copper. Since, is comple, then we take i. The other constants of the problem are taken as, ζ, The phsical constants used are: p, f 5.7 gm / cm, j. cm, 9. dne / cm, T C,. dne / cm, K. dne / cm,.779 dne, K cal cm C C cal gm C.6 / sec,. /. a, t. The results are shown in Figs. -6. The graph shows the three curves predicted b different theories of thermoelasticit. In these Figures, the solid lines represent the solution in the Coupled theor, the dotted lines represent the solution in the generalized Lord and Shulman theor and dashed lines represent the solution derived using the Green and Lindsa theor. These Figures represent the solution obtained using the coupled theor (CD theor: n, n, n, t, t ), the generalized theor with one relaation time (Lord-Şhulman theor LS theor: n, n, n, t., t ), and generalized theor with two relaation times (Green-Lindsa theor (GL theor: n, n, n, t., t. ). We notice that the results for the temperature, the displacement and stresses distribution when the relaation time is including in the heat equation are distinctl different from those when the relaation time is not mentioned in heat equation, because the thermal waves in the Fourier's theor of heat equation travel with an infinite speed of propagation as opposed to finite speed in the non-fourier case. This demonstrates clearl the difference between the coupled and the generalized theories of thermoelasticit. Figs. -6. The graph shows the two cases when the magnetic field acts on the material and when absence, from this Figures., we obtain the effect of magnetic field caused rearranged the atoms in the medium and compact the curves. For the value of, namel., were substituted in performing the computation. It should be noted (Fig.) that in this problem, the crack's size, is taken to be the length in this problem so that (when H eist) and )when H. ), represents the plane of the crack that is smmetric with respect to the plan. It is clear from the graph that T has maimum value at the beginning of the crack ( ), it begins to fall just near the crack edge ( ), where it eperiences sharp decreases (with maimum negative gradient at the crack's end). The value of temperature quantit converges to zero with increasing the distance. Fig., the horizontal displacement, u, begins with increase then smooth decreases again to reach its maimum magnitude just at the crack end. Beond it u falls again to tr to retain zero at infinit. Fig., the vertical displacement v, we see that the displacement component v alwas starts from the negative value and terminates at the zero value. Also, at the crack end to reach minimum value, beond reaching zero at the double of the crack size (state of particles equilibrium). The displacements u and v show different behaviours, because of the elasticit of the solid tends to resist vertical displacements in the problem under investigation. Both of the components show different behaviours, the former tends to increase to maimum just before the end of the crack. Then it falls to a minimum with a highl negative gradient. Afterwards it rises again to a maimum beond about the crack end.

9 Temperature(T) 6 Effect of Magnetic Field and a Mode-I Crack D-Problem... H esist CD LS GL H = Distance() Fig. Variation of temperature distribution T with different theories. The stresse component, reach coincidence with negative value (Fig.) and satisf the boundar condition at,reach the maimum value near the end of crack ( ) and converges to zero with increasing the distance. Fig. 5, shows that the stress component satisf the boundar condition at X and had a different behaviour. It decreases in the start and start decreases (maimum) in the contet of the three theories until reaching the crack end ( ). These trends obe elastic and thermoelastic properties of the solid under investigation. Fig.6, the tangential coupled stress m satisfies the boundar condition at X.It decreases in the start and start increases (maimum) in the contet of the three theories then smooth decreasing until reaching the crack end. The effect of the magnetic field down load the magnitude of the front of wave propagation...5. H =. CD LS GL.5 U..5 H eist Fig Distance() Variation of displacement distribution u with different theories. - - H eist o - V -6-8 H = Distance() CD LS GL Fig. Variation of displacement distribution v with different theories.

10 Kh. Lotf and N. Yahia 6.5. H = H eist CD LS GL Distance() Fig. Variation of stress distribution with different theories. - - H = H esist CD LS GL Distance() Fig. 5 Variation of stress distribution at with different theories..5 H =. CD LS GL m Hesist Distance() Fig. 6 Variation of tangential couple stress theories. m with different Fig. 7-5 show the comparison between the temperature T, displacement components u, V, the force stresses components,, and the tangential coupled stress m, the case of different three values of, (namel =., =. and =.) under GL theor. It should be noted (Fig.7) that in this problem. It is clear from the graph that T has maimum value at the beginning of the crack ( ), it begins to fall just near the crack edge ( ), where it eperiences sharp decreases (with maimum negative gradient at the crack's end). Graph lines for both values of show different slopes at crack ends according to -values. In other words, the temperature line for =. has the highest gradient when compared with that of z =. and z=. at the first of the range. In addition, all lines begin to

11 T 6 Effect of Magnetic Field and a Mode-I Crack D-Problem coincide when the horizontal distance is beond the double of the crack size to reach the reference temperature of the solid. These results obe phsical realit for the behaviour of copper as a polcrstalline solid, it shown also in Fig: 8. Fig. 9, the horizontal displacement u, despite the peaks (for different vertical distances =., =. and =.) occur at equal value of, the magnitude of the maimum displacement peak strongl depends on the vertical distance. It is also clear that the rate of change of u decreases with increasing as we go farther apart from the crack Fig.. On the other hand, Fig. shows atonable increase of the vertical displacement, v, near the crack end to reach maimum value beond reaching zero at the double of the crack size (state of particles equilibrium). Fig., the vertical stresses Graph lines for both values of show different slopes at crack ends according to - values. In other words, the component line for =. has the lowest gradient when compared with that of =. and =. at the edge of the crack. In addition, all lines begin to coincide when the horizontal distance is beond the double of the crack size to reach zero after their relaations at infinit. Variation of has a serious effect on both magnitudes of mechanical stresses. These trends obe elastic and thermoelastic properties of the solid under investigation. Fig., shows that the stress component satisf the boundar condition, the line for z =. has the highest. These trends obe elastic gradient when compared with that of z =. and z=. and converge to zero when and thermoelastic properties of the solid, it is shown also in Fig.. Fig. 5, the tangential coupled stress m it increases in the start and start decreases in the contet of the three values of until reaching the crack end, for =. has the highest gradient when compared with that of z =. and z=. at the edge of the crack. All lines begin to coincide when the horizontal distance is beond the edge of the crack..5.. =. =. = Distance() Fig. 7 Temperature distribution T with variation of distances under GL theor. =. =. = T Distance() Fig. 8 Temperature distribution T with variation of distances under GL theor in D.

12 V U Kh. Lotf and N. Yahia 6.5. =. =. = Distance() Fig. 9 Displacement distribution u with variation of distance under GL theor. =. =. = U..5 Distance() Fig. Displacement distribution u with variation of distance under GL theor in (D) Distance() =. =. =. Fig. Displacement distribution v with variation of distance under GL theor =. =. = Distance() Fig. Stress distribution with variation of distance under GL theor.

13 65 Effect of Magnetic Field and a Mode-I Crack D-Problem =. =. = Distance() Fig. Stress distribution with variation of distance under GL theor =. =. =..5 Distance().5 Fig. Stress distribution with variation of distance under GL theor =. =. =..6 m Distance() Fig. 5 Tangential couple stress under GL theor. m with variation of distance Finall The Figs. 6- shows the D plots for the phsical components under Lord-Şhulman theor with the effect of magnetic field. Fig. 6 Shows the variation of the temperature and the depth with increasing under the effects of magnetic field, which it decreases with increasing of wave speed until approaching to zero, as well it decreases with decreasing of the value of depth. Fig. 7 Shows the variation of two horizontal displacement u and the depth with increasing under the effects of magnetic field. The value of the displacement u components has an oscillator behavior with magnetic field in the whole range of the magnetic field. Fig. 8 Shows the variation vertical displacement v and the depth with increasing (when the waves translate from a side to another through the crack (as two laers)) waves under the effects of magnetic field. The effect of magnetic on vertical displacement v wave which it decreases and increases periodicall, as well it decreases with increasing of the value of wave speed. Fig. 9 Show the variation of normal stress in D under the effect of magnetic field. The effect of magnetic field on normal stress which it oscillator behavior with increasing of magnetic field, as well it decreases with

14 Kh. Lotf and N. Yahia 66 increasing of the value of wave speed. Fig. Show the variation of stress component. The value of the stress component has an oscillator behavior with magnetic field in the whole range of the phase velocit. Fig., the tangential coupled stress m in D, it increases in the start and start decreases in the contet of the three values of until reaching the crack end, The thermo-dnamical heat distrbution in D Fig. 6 Temperature distribution T with variation of distances under GL theor and magnetic field. The displacment distrbution u in D u Fig. 7 D displacement component u with variation of distances under GL theor and magnetic field. The displacment distrbution v in D.5 v Fig. 8 D displacement component v with variation of distances under GL theor and magnetic field.

15 67 Effect of Magnetic Field and a Mode-I Crack D-Problem The stress distrbution in D Fig. 9 D normal stress component with variation of distances under GL theor and magnetic field The stress distrbution in D Fig. D shear stress component with variation of distances under GL theor and magnetic field. The stress distrbution m in D m Fig. D tangential couple stress m with variation of distances under GL theor and magnetic field. CONCLUSIONS Due to the complicated nature of the governing equations of the elasticit-electromagnetic theor, the work done in this field is unfortunatel limited in number. The method used in this stud provides a quite successful in dealing with such problems. This method gives eact solutions in the elastic medium without an assumed restrictions on the actual phsical quantities that appear in the governing equations of the problem considered. Important phenomena are observed in all these computations:. The curves in the contet of the (CD), (L-S) and (G-L) theories decrease or incressing eponentiall with increasing, this indicate that the thermoelastic waves are unattenuated and nondispersive, where purel thermoelastic waves undergo both attenuation and dispersion.

16 Kh. Lotf and N. Yahia 68. The presence of magnetic field plas a significant role in all the phsical quantities.. The curves of the phsical quantities with (G-L) theor in most of figures are greater in comparison with those under (CD and (L-S) theories.. Analtical solutions based upon normal mode analsis for themoelastic problem in solids have been developed and utilized. 5. A linear opening mode-i crack has been investigated and studied for copper solid. 6. Temperature, radial and aial distributions were estimated at different distances from the crack edge. 7. The stresses distributions, the tangential coupled stress and the values of microstress were evaluated as functions of the distance from the crack edge. 8. Crack dimensions are significant to elucidate the mechanical structure of the solid. 9. Cracks are stationar and eternal stress is demanded to propagate such cracks.. It can be concluded that a change of volume is attended b a change of the temperature while the effect of the deformation upon the temperature distribution is the subject of the theor of thermoelasticit.. The value of all the phsical quantities converges to zero with an increase in distance and all functions are continuous.. Deformation of a bod depends on the nature of forced applied as well as the tpe of boundar conditions.. It is clear from all the figures that all the distributions considered have a non-zero value onl in a bounded region of the half-space. Outside of this region, the values vanish identicall and this means that the region has not felt thermal disturbance et.. The results presented in this paper should prove useful for researchers in material science, designers of new materials. 5. Stud of the phenomenon of relaation time and magnetic is also used to improve the conditions of oil etractions. REFERENCES [] Eringen A. C., Suhubi E. S., 96, Nonlinear theor of simple micropolar solids, International Journal of Engineering Science :-8. [] Eringen A. C., 966, Linear theor of micropolar elasticit, Journal of Applied Mathematics and Mechanics 5: [] Biot M. A., 956, Thermoclasticit and irreversible thermodnamics, Journal of Applied Phsics 7:-5. [] Lord H. W., Shulman Y., 967, A generalized dnamical theor of thermoelasticit, Journal of the Mechanics and Phsics of Solids 5:99-6. [5] Othman M. I. A.,, Lord-shulman theor under the dependence of the modulus of elasticit on the reference temperature in two-dimensional generalized thermo- elasticit, Journal of Thermal Stresses 5:7-5. [6] Green A. E., Lindsa K.A., 97, Thermoelasticit, Journal of Elasticit :-7. [7] Green A. E., Laws N., 97, On the entrop production inequalit, Archive for Rational Mechanics and Analsis 5:7-5. [8] Suhubi E. S., 975, Thermoelastic Solids in Continuum Phsics, Part, Chapter, Academic Press, New York. [9] Othman M. I. A.,, Relaation effects on thermal shock problems in an elastic half-space of generalized magnetothermoelastic waves, Mechanics and Mechanical Engineering 7: [] Iesan D., 97, The plane micropolar strain of orthotropic elastic solids, Archives of Mechanics 5: [] Iesan D., 97, Torsion of anisotropic elastic clinders, Journal of Applied Mathematics and Mechanics 5: [] Iesan D., 97, Bending of Orthotropic Micropolar Elastic Beams b Terminal Couples, An State Universit Lasi : - 8. [] Nakamura S., Benedict R., Lakes R., 98, Finite element method for orthotropic micropolar elasticit, International Journal of Engineering Science :9-. [] Kumar R., Choudhar S.,, Influence and green's function for orthotropic micropolar containua, Archives of Mechanics 5: [5] Kumar R., Choudhar S.,, Dnamical behavior of orthotropic micropolar elastic medium, Journal of vibration and control 5:5-69. [6] Kumar R., Choudhar S.,, Mechanical sources in orthotropic micropolar continua, Proceedings of the Indian Academ of Sciences ():-. [7] Kumar R., Choudhar S.,, Response of orthotropic micropolar elastic medium due to various sources, Meccanica 8:9-68. [8] Kumar R., Choudhar S.,, Response of orthotropic micropolar elastic medium due to time harmonic sources, Sadhana 9:8-9.

17 69 Effect of Magnetic Field and a Mode-I Crack D-Problem [9] Singh B., Kumar R., 998, Reflection of plane wave from a flat boundar of micropolar generalized thermoelastic halfspace, International Journal of Engineering Science 6: [] Singh B.,, Reflection of plane sound wave from a micropolar generalized thermoelastic solid half-space, Journal of sound and vibration 5: [] Othman M.I. A., Lotf KH., 9, Two-dimensional Problem of generalized magneto-thermoelasticit under the Effect of temperature dependent properties for different theories, Multidiscipline Modeling in Materials and Structures 5:5-. [] Othman M.I. A., Lotf KH., Farouk R.M., 9, Transient disturbance in a half-space under generalized magnetothermoelasticit due to moving internal heat source, Acta Phsica Polonica A 6:86-9. [] Othman M.I. A, Lotf KH.,, On the plane waves in generalized thermo-microstretch elastic half-space, International Communication in Heat and Mass Transfer 7:9-. [] Othman M.I. A, Lotf KH., 9, Effect of magnetic field and inclined load in micropolar thermoelastic medium possessing cubic smmetr, International Journal of Industrial Mathematics (): 87-. [5] Othman M.I. A, Lotf KH.,, Generalized thermo-microstretch elastic medium with temperature dependent properties for different theories, Engineering Analsis with Boundar Elements :9-7. [6] Lotf Kh.,, Two temperature generalized magneto-thermoelastic interactions in an elastic medium under three theories, Applied Mathematics and Computation 7: [7] Dhaliwal R., 98, Eternal Crack due to Thermal Effects in an Infinite Elastic Solid with a Clindrical Inclusion, Thermal Stresses in Server Environments, Doi:.7/ _. [8] Hasanan D., Librescu L., Qin Z., Young R., 5, Thermoelastic cracked plates carring nonstationar electrical current, Journal of Thermal Stresses 8: [9] Ueda S.,, Thermall induced fracture of a piezoelectric Laminate with a crack normal to interfaces, Journal of Thermal Stresses 6:-. [] Elfalak A., Abdel-Halim A. A., 6, A mode-i crack problem for an infinite space in thermoelasticit, Journal of Applied Sciences 6: