Thermal buckling analysis of ceramic-metal functionally graded plates

Transcription

1 Vol. No () Natural Science hermal bucling analysis of ceramic-metal functionally graded plates Ashraf M. Zenour * Daoud S. Mashat Department of Mathematics Faculty of Science King AbdulAziz University Jeddah Saudi Arabia; *Corresponding Author: zenour@au.edu.sa Department of Mathematics Faculty of Science Kafrelsheih University Kafr El-Sheih Egypt Received 7 March ; revised April ; accepted 6 April. ABSRAC hermal bucling response of functionally graded plates is presented in this paper using sinusoidal shear deformation plate theory (). he material properties of the plate are assumed to vary according to a power law form in the thicness direction. Equilibrium and stability equations are derived based on the. he non-linear governing equations are solved for plates subjected to simply supported boundary conditions. he bucling analysis of a functionally graded plate under various types of thermal loads is carried out. he influences of many plate parameters on bucling temperature difference will be investigated. Numerical results are presented for the demonstrating its importance and accuracy in comparison to other theories. Keywords: hermal Bucling; Non-Linear Strains; Functionally Graded Material; Sinusoidal Plate heory; hermal Load. INRODUCION he rapid development of composite materials and structures in recent years has drawn ineased attention from many engineers and researchers. hese materials are broadly used in aerospace mechanical nuclear marine and structural engineering. In conventional laminated composite structures homogeneous elastic laminas are bonded together to obtain enhanced mechanical and thermal properties. However the abrupt change in material properties aoss the interface between different materials can result in large inter-laminar stresses leading to delimitation acing and other damage mechanisms which result from the abrupt change of the mechanical properties at the interface between the layers. o remedy such defects functionally graded materials (FGMs) within which material properties vary continuously have been proposed. he concept of FGM was proposed in 984 by a group of materials scientists in Sendai Japan for thermal barriers or heat shielding properties. Initially FGM was designed as a thermal barrier material for aerospace application and fusion reactors. Later on FGM was developed for the military automotive biomedical and semiconductor industries and as a general structural element in high thermal environments. FGM is one of the advanced high temperature materials capable of withstanding extreme temperature environments. FGMs are composite and mioscopically heterogeneous in which the mechanical properties vary smoothly and continuously from one surface to the other. his is achieved by gradually varying the volume fraction of the constituent materials. ypically these materials are made from a mixture of ceramics and metal or a combination of different materials. he ceramic constituent of the material provides the high-temperature resistance due to its low thermal conductivity and protects the metal from oxidation. he ductile metal constituent on the other hand prevents fracture caused by stresses due to high-temperature gradient in a very short period of time. Further a mixture of a ceramic and a metal with a continuously varying volume fraction can be easily manufactured [-4]. A comprehensive wor on the FGMs was presented in the literature. he response of FG ceramic-metal plates has been investigated by Praveen and Reddy [5] using a plate finite element. hey investigated the static and dynamic thermoelastic responses of the FGMs by varying the volume fraction using a simple power law distribution. Reddy [6] developed the Navier s solutions for FG plates using the third-order shear deformation plate theory (SD) and an associated finite element model. Amini et al. [7] desibed a method for three-dimensional free vibration analysis of rectangular FGM plates Copyright SciRes.

2 A. M. Zenour et al. / Natural Science () resting on an elastic foundation using Chebyshev polynomials and Ritz s method. his analysis has been based on a linear small-strain three-dimensional elasticity theory. Analysis of FG plates under static and dynamic loads has been presented by Slade et al. [8] using the meshless local Petrov-Galerin method and Reissner- Mindlin theory to desibe the plate bending problem. Kim et al. [9] investigated finite element computation of fracture parameters in FGM assemblages of arbitrary geometry with stationary acs. In Altenbach and Eremeyev [] a viscoelastic FG polymer foam has been studied using a new plate theory based on the direct approach. he large deflection response of simply supported rectangular FG plates under normal pressure loading has been analyzed by Ovesy and Ghannadpour [] using a finite strip method. In Han [] a numerical method was proposed for analyzing transient waves in plates of FGM excited by impact loads. he bending problem of transverse load acting on FGM rectangular plate using both two-dimensional trigonometric and three-dimensional elasticity solutions was investigated by Zenour [3]. Zenour [45] studied the bending response bucling and free vibration of simply supported FG sandwich plate using the. Zenour [6] presented the derivation of equations for free vibration of FG plates expressing the displacement components by trigonometric series representation through the plate thicness. Other researches into FGMs have included the nonlinear analysis of FG plates [7] large deformation analysis of FG shells [8] static and vibration analysis of FG beams [9]. In view of the advantages of FGMs a number of investigations dealing with thermal bucling had been published in the scientific literature. In recent years the mechanical and thermal bucling analysis of FG ceramic-metal plates has been presented by Zhao et al. [] using the first-order shear deformation plate theory in conjunction with the Ritz method. A two-dimensional global higher-order deformation theory has been employed by Matsunaga [] for thermal bucling of plates made of FGMs. Morimoto et al. [3] presented the thermal bucling analysis of FG rectangular plates subjected to partial heating in a plane and uniform temperature rise through its thicness. In Ref. [4] Shariat and Eslami presented the thermal bucling analysis of rectangular FG plates with geometrical imperfections using the classical plate theory to derive the equilibrium stability and compatibility equations of an imperfect FGM. hermal bucling of rectangular and circular plates compose of FGM was also studied based on the first- and higher-order shear deformation plate theory [5-7]. Various plate theories depending upon the throughthicness displacement pattern considered have been used to determine thermal bucling loads of composite plates. he classical plate theory [4] which is based on Kirchhoff s hypothesis overestimates the thermal bucling load when applied to even moderately thic plates. his is particularly true for composite plates in which transverse shear moduli are small in comparison to the in-plane Young s moduli [8]. In such cases it becomes necessary to tae into account shear deformation effects. hus various improved plate theories such as first-order shear deformation [56] higher order shear deformation [56] and sinusoidal shear deformation [ ] plate theories have been developed to predict the behavior of plates with thicness shear deformation. In this article thermal bucling analysis of rectangular FG ceramic-metal plates is investigated. he material properties of the FG plates are assumed to vary continuously through the thicness according to a simple power law distribution of the volume fraction of the constituents. he is used to obtain the bucling of the plate under different types of thermal loads. he thermal loads are assumed to be uniform linear and non-linear distribution through the thicness. Additional numerical results are presented for FGM plates that show the effects of various parameters on thermal bucling response.. MAHEMAICAL MODEL Consider a rectangular plate of length a width b and thicness h made of FGM. he plate is subjected to a thermal load ( x y z ). he material properties of the FGM plate such as Young s modulus E and thermal expansion coefficients are assumed to be functions of the volume fraction of the constituent materials. he FGM plate is supported at four edges defined in the ( x yz ) coordinate system with x- and y-axes located in the middle plane ( z ) and its origin placed at the corner of the plate. he modulus of elasticity E the coefficient of thermal expansion and Poisson s ratio are assumed as [5] where Ez ( ) E E V ( z) V v( z) v m cm m cm Ecm Ec Em cm c m z V h and E m and m denote the elastic moduli and the coefficient of thermal expansion of metal; E c and c denote the elastic moduli and the coefficient of thermal () () Copyright SciRes.

3 97 A. M. Zenour et al. / Natural Science () expansion of ceramic and is the volume fraction exponent. he value of equal to zero represents a fully ceramic plate. he above power law assumption reflects a simple rule of mixtures used to obtain the effective properties of the ceramic-metal plate. he rule of mixtures applies only to the thicness direction. he density of the plate varies according to the power law and the power law exponent may be varied to obtain different distributions of the component materials through the thicness of plate. Note that the volume fraction of the metal is high near the bottom surface of the plate and that of ceramic high near the top surface. In addition Eq. indicates that the bottom surface of the plate ( z h/) is metal-rich whereas the top surface ( z h/) of the plate is ceramic-rich. For simplicity is assumed constant aoss the plate thicness. he displacements of a material point located at (x y z) in the FGM plate might better be illustrated as [9 3]: w u( x y z) uz ( z) x w u( x y z) vz ( z) (3) y u ( x y z) w 3 where u v and w are the displacements of the middle surface along the axes x y and z respectively and and are the rotations about the y and x-axes and account for the effect of transverse shear. he coefficient of or which is given by ( z) should be odd function of z. All of the generalized displacements ( uvw ) are functions of the (x y). he displacement of the classical thin plate theory () can easily be obtained by setting ( z). he displacements of the first-order shear deformation plate theory () are obtained by setting ( z) z. In addition the higher-order shear deformation plate theory () [6] is obtained by setting 4 z ( z) z. (4) 3 h Also the is obtained by setting [45]: h z ( z) sin h. (5) Note that the present as well as is simplified by enforcing traction-free boundary conditions at the plate faces. he accounts according to a cosine-law distribution of the transverse shear deformation through the thicness of the FGM plate. he and contain the same number of dependent unnowns. No transversal shear correction factors are needed for both and because a correct representation of the transversal shearing strain is given. he non-linear strain components ij compatible with the displacement field in Eq.3 are z ( z) (6) ( z) 3 (7) 3 3 where u w v w 3 3 v u w w w w w. (8) he stress-strain relations for the FGM plate are given by Ez ( ) (9) Ez ( ) { 3 3 } { 3 3 } ( ) where ( x y z ) is the temperature rise through the thicness. he stress and moment resultants of the FGM plate can be obtained by integrating Eq.9 over the thicness and are written as and Ni A B C i A Mi B D F i B () Si C F G i C N A B C M B D F ( ) S C F G () Copyright SciRes.

4 A. M. Zenour et al. / Natural Science () where i and Q3 3 H ( ) Q 3 3. () (3) In Eqs.- N N and N and M M and M are the basic components of stress resultants and stress couples; S S and S are additional stress couples associated with the transversal shear effects; and Q 3 and Q 3 are transversal shear stress resultants. he coefficients A B C... etc. are defined by h/ { A B D } E( z){ z z }d z h/ h/ h/ h/ h/ h/ ( )( ( ) / 3) d. h { C F G } ( z) E( z){ z ( z)}d z { A B C } ( z) E( z) ( x y z){ z ( z)}d z H E z z z 3. EQUILIBRIUM AND SABILIY EQUAIONS (4) he total potential energy of a plate subjected to thermal loads is defined as [7] V Um Ub Uc U (5) where Um Ub U c and U are membrane strain energy bending strain energy coupled strain energy and thermal strain energy. he strain energy for FGM plate based on the is defined as given below in Eq.6. he equilibrium and stability equations of FGM plates may be derived by the variational approach. he expansion of V about the equilibrium state by the aylor series is 3 V V V V (7)! 3!.... he governing equations of equilibrium can be derived by using the first variation V. he non-linear equilibrium equations associated with the present are N N N N M M M N w N w N w S S Q3 S S Q3. (8) o establish the stability equations the condition V used to derive the stability equations of many practical plate bucling problems is also used here. he external load acting on the original configuration is considered to be the itical bucling temperature if the above equation ( V ) is satisfied. Assuming that the state of stable equilibrium of a general plate under thermal load may be designated by u v w. he displacements of the neighboring state are u u u v v v w w w (9) where u v w and are arbitrarily small inement of displacements. he stability equations are represented by using the above total displacement given in Eq.9 in the equation V and collecting the second-order terms. hey read N N N N M M M N w N w N w S S Q 3 S S Q 3 () where the supersipt refers to the state of stability and the supersipt refers to the state of equilibrium conditions. he terms N N and N are the pre-bucling force resultants obtained as A A N N N. () 4. EXAC SOLUIONS FOR HERMAL BUCKLING OF FGM PLAES Rectangular plates are generally classified in accordance V v dv. (6) Copyright SciRes.

5 97 A. M. Zenour et al. / Natural Science () with the type support used in the absent of the body forces and lateral loads except the external temperature load. he following boundary conditions are imposed at the side edges v w N M S atx a u w N M S aty b () Following Navier solution procedure we assume the following solution form for ( u v w ) that satisfies the simply-supported boundary conditions u Umn cos( x)sin( y) v Vmn sin( x)cos( y) w X mn cos( x)sin( y) Ymn sin( x)cos( y) Wmn sin( x)sin( y) (3) mn where m a n b; m and n are mode numbers; Umn Vmn Wmn X mn and Y mn are arbitrary parameters to be determined subjected to the condition that the solution in Eq.3 satisfies the conditions in Eq.. Substituting Eq.3 into Eq. one obtains where { } denotes the column [ L]{ } (4) t { } { Umn Vmn Wmn Xmn Ymn} (5) and elements Lrs Lsr of the coefficient matrix [ L ] are given by: L A [ ( ) ] L A ( ) L B 3 ( ) L C 4 [ ( ) ] L5 C ( ) L A [( ) ] L B 3 ( ) L L 4 5 L C 5 [( ) ] L D A 33 ( )[ ( ) ( )] L F 34 ( ) L F 35 ( ) L G H 44 [ ( ) ] ( ) L45 G ( ) L G H (6) 55 [( ) ] ( ). For non-trivial solutions of Eq.4 the determinant L should be zero. his equation ( L ) is stated for the determination of the lowest itical load. In the following the solutions of the equation L for different types of thermal loading conditions are presented. he plate is assumed simply supported in bending and rigidly fixed in extension. he temperature change is varied only in the thicness direction. 4.. hermal Bucling for FGM Plates under Uniform emperature Rise he initial uniform temperature of the plate is assumed to be i. he temperature is uniformly raised to a final value f in which the plate bucles. he temperature change is f i. Substituting Eq.6 into the equation L the bucling temperature change using the shear deformation theories is obtained as ( ns m)[ Pa( ) P ( ns m)] ( )[ ( ) ( )] where a A Pa P n s m P A H P A G C ( ) P PD B H ( ) P PD AF B BG FC h/ A ( zez ) ( )d z sa/ b. h/ (7) (8) he itical bucling temperature change is the smallest value of which is obtained when m = and n =. herefore ( s )[ Pa ( ) P ( s )]. a A ( )[ Pa ( ) P ( s )] (9) For the classical plate theory the itical bucling temperature difference is given as s AD B aaa ( ) ( )( ). (3) 4.. hermal Bucling for FGM Plates Subjected to a Graded emperature Change aoss the hicness For an FG plate the temperature change is not uniform. he temperature varies according to the power law variation. Usually the temperature rises much higher at Copyright SciRes.

6 A. M. Zenour et al. / Natural Science () the ceramic side than that in the metal side of the plate. In this case the temperature through the thicness is given by ( z) V (3) where m is the temperature of the bottom surface which is metal-rich and is the power law exponent ( ). Similar to the previous loading case solving the equation L the bucling temperature difference ( h/) ( h/) using the shear deformation plate theories can be determined and then we can obtain the itical bucling as where s Pa P s ( )[ ( ) ( )] ma a A ( )[ Pa ( ) P ( s )] A h/ h/ m (3) A ( zv ) E( z)d z. (33) Also the itical bucling temperature difference for the classical plate theory is deduced as ( s )( AD B) A m a AA ( ) A. (34) Note that the value of equal to unity represents a linear temperature change aoss the thicness. While the value of excluding unity represents a non-linear temperature change through the thicness. 5. RESULS AND DISCUSSION he general approach outlined in the previous sections for the thermal bucling analysis of the homogeneous and FGM plates under uniform linear and non-linear temperature rises through the thicness is illustrated in this section using the. he correlation between the present theory and different higher- and first-order shear deformation theories and classical plate theory is established. o illustrate the proposed method a ceramic-metal FG plate is considered. he combination of materials consists of aluminum and alumina. he Young s modulus and the coefficient of thermal expansion for alumina 6 are Ec 38 GPa c 7.4 / C and for aluminum are Em 7 GPa m 6 3 / C respectively. Note that Poisson's ratio is selected constant for both aluminum and alumina and it equal to.3. he shear correction factor for is set equal to 5/6. For the linear and non-linear temperature rises through the thicness the temperature rises 5 C in the metal-rich surface of the plate (i.e. m 5C ). We will assume in all analyzed cases (unless otherwise stated) that a/ b a/ h and 3. Numerical results of the present investigation are given in ables -4 and Figures -4. In ables and the side to thicness ratio of the plate is set as a/ h. In these tables the itical bucling temperature difference of the plate under uniform and linear temperature rises is shown for different values of the power law index using various plate theories. he results obtained as per the present and are compared with the corresponding ones presented by Javaheri and Eslami [3]. Excellent agreement is achieved between the two solutions. It is seen that for all theories the itical temperature difference ineases monotonically as the aspect ratio a/ b ineases. Moreover the itical bucling deeases until it reaches minimum values and then ineases as the values of the volume fraction exponent ineases. ables 3 and 4 exhibit the 3 itical temperature difference t for different values of the aspect ratio a/ b the temperature exponent and the power law index under non-linear temperature loads at a/ h and 5 respectively. he nonlinearity temperature exponent is taen here as 5 and. he effect of a/ b on the itical bucling t is similar to that in the case of uniform and linear temperature difference aoss the thicness. As the power law index ineases the itical bucling t deeases to reach lowest values and then ineases excluding t of the rectangular plates for. Also it is noticed that t ineases as the nonlinearity index ineases. In general the values of the itical temperature difference calculated by using the shear deformation theories are lower than those calculated by using the classical plate theory indicating the shear deformation effect. he without using any shear correction factor gives results very close to and closer than those obtained using. he itical bucling temperature difference t of the ceramic-metal FG rectangular plate ( 5) versus the side-to-thicness ratio a/ h calculated by all theories under a uniform linear and non-linear temperature load are shown in Figure. For plates with small a/ h ratio very large differences between the results of both and and those of both and are observed. Moreover the differences between the higherorder shear deformation theories ( and ) and are lower than those between any of them and. However for a large value of the side-to-thicness ratio the difference between the values predicted by the shear deformation theories and is low significant because the plate is essentially thin. Because of permitting shear deformation in and the plate becomes more flexible and thus the itical bucling temperatures calculated by these theories are smaller than those cal- Copyright SciRes.

7 974 A. M. Zenour et al. / Natural Science () able. Critical bucling temperature change index and aspect ratio a/ b. of FGM plate under uniform temperature rise for different values of power law heory a/ b a/ b a/ b 3 a/ b 4 a/ b (7.88) (7.99) (4.688) (4.747) (85.5) (85.495) (44.648) (45.34) (.667) (.88) (7.939) (7.943) (9.835) (9.859) (39.64) (39.78) (67.5) (67.5) (.634) (3.69) (7.6) (7.65) (8.3) (8.64) (36.3) (36.38) (6.395) (6.758) (93.65) (94.454) (7.46) (7.469) (8.636) (8.673) (37.) (37.346) (63.68) (63.488) (96.) (97.) he results in parenthesis are obtained in [3]. able. Critical bucling temperature change index and aspect ratio a/ b. of FGM plate under linear temperature rise for different values of power law heory a/ b a/ b a/ b 3 a/ b 4 a/ b (4.77) (4.98) (75.376) (75.495) (6.55) (6.99) (79.97) (8.684) (43.334) ( ) (5.53) (5.5) (7.83) (7.868) (64.936) (65.4) (6.748) (7.58) (83.) (84.3) (3.89) (3.899) (.64) (.659) (53.7) (53.95) (97.73) (97.698) (5.56) (53.977) (4.364) (4.375) (4.65) (4.9) (57.6) (57.39) (.9) (3.646) (6.47) (63.8) he results in parenthesis are obtained in [3]. culated by. he itical bucling temperature difference t as a function of a/ b for various values of the power law index under a uniform linear and non-linear temperature loads is depicted in Figure. It is observed that with ineasing the plate aspect ratio / a b the itical bucling temperature difference also ineases gradually whatever the material gradient index is. Since the ceramic plate is weaer than the metallic one thus the itical bucling temperature of the first plate is higher Copyright SciRes.

8 A. M. Zenour et al. / Natural Science () able 3. Critical bucling temperature change t of FGM plate under non-linear temperature rise for different values of index aspect ratio a/ b and temperature exponent a/ h. heory a/ b a/ b a/ b able 4. Critical bucling temperature change t of FGM plate under non-linear temperature rise for different values of index aspect ratio a/ b and temperature exponent a/ h 5. heory a/ b a/ b a/ b than that of the second. For the FGM plate t deeases as the metallic constituent in the plate ineases. Figure 3 investigates the itical bucling temperature difference t of homogeneous and FG plates versus the side-to-thicness ratio a/ h under various types of temperature loads. Figure 4 gives similar for FG plates versus the aspect ratio a/ b. he bucling temperature of the homogeneous plate is considerably higher than Copyright SciRes.

9 976 A. M. Zenour et al. / Natural Science () Figure. Critical bucling temperature difference t due to uniform linear and non-linear temperature rise aoss the thicness versus the side-tothicness ratio a/ h. Figure. Critical bucling temperature difference t due to uniform linear and non-linear temperature rise aoss the thicness versus the aspect ratio a/ b. Copyright SciRes.

10 A. M. Zenour et al. / Natural Science () Figure 3. Critical bucling temperature difference t due to uniform linear and non-linear temperature rise aoss the thicness versus the side-tothicness ratio a/ h. under uniform linear and nonlinear thermal loading through the thicness is investigated in this paper. he constituent materials are graded from the ceramic surface to the metallic surface according to the power law variation. he is used to deduce the equilibrium and stability equations for a simply supported functionally graded rectangular plate under thermal loading. he results obtained using are compared with those obtained using and ; and compared with published ones. he numerical results of itical bucling temperature difference using are very close to those of and the two theories have similar trends for all cases of loading. he itical bucling temperature difference is proportional to the plate aspect ratio. he thicer plates need a temperature to bucle higher than that the thinner plates need it. he itical bucling temperature differences of functionally graded plates are generally lower than the corresponding ones for homogeneous ceramic plates. 7. ACKNOWLEDGEMENS he investigators would lie to express their appreciation to the Deanship of Scientific Research at King AbdulAziz University for its financial support of this study Grant No. 3-38/49. Figure 4. Critical bucling temperature difference t due to uniform linear and non-linear temperature rise aoss the thicness versus the aspect ratio a/ b. that for the FGM one especially for the comparatively thicer plates. Again because of the thicer plates are stronger than the thinner ones thus the itical bucling temperature of the first type is higher than that of the second one. Note that t of the plate under uniform temperature rise is smaller than that of the plate under linear temperature rise and the latter is smaller than that of the plate under non-linear temperature rise. 6. CONCLUSIONS he thermal bucling analysis for ceramic-metal FG plates REFERENCES [] Koizumi M. (997) FGM activities in Japan. Composites Part B: Engineering 8(-) -4. [] Bhangale R.K. and Ganesan N. (5) A linear thermoelastic bucling behavior of functionally graded hemispherical shell with a cut-out at apex in thermal environment. International Journal of Structural Stability and Dynamics 5() [3] Javaheri R. and Eslami M.R. () Bucling of functionally graded plates under in-plane compressive loading. ZAMM Journal of Applied Mathematics and Mechanics 8(4) [4] Chung Y.-L. and Chang H.-X. (8) Mechanical behavior of rectangular plates with functionally graded coefficient of thermal expansion subjected to thermal loading. Journal of hermal Stresses 3(4) [5] Praveen G.N. and Reddy J.N. (998) Nonlinear transient thermoelastic analysis of functionally graded ceramicmetal plates. International Journal of Solids and Structures 35(33) [6] Reddy J.N. () Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering 47(-3) [7] Amini M.H. Soleimani M. and Rastgoo A. (9) hree-dimensional free vibration analysis of functionally graded material plates resting on an elastic foundation. Smart Materials and Structures 8(8) -9. [8] Slade J. Slade V. Hellmich Ch. and Eberhardsteiner J. (7) Analysis of thic functionally graded plates by local integral equation method. Communications in Numerical Methods in Engineering 3(8) Copyright SciRes.

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