Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements

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1 J. Appl. Comput. Mec., 3(4) (017) 74-8 DOI: /JACM ISSN: jacm.scu.ac.ir Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements Noa Fouda 1, Tawfik El-midany, A.M. Sadoun 3,4 1 Production Engineering and mecanical Design Dept, Mansoura University Al Mansura, Egypt, foudanoa@yaoo.com Production Engineering and mecanical Design Dept, Mansoura University Al Mansura, Egypt, foudanoa@yaoo.com 3 Mecanical Engineering Dept, King Abdulaziz University Jedda, Saudi Arabia, assadoun@kau.edu.sa 4 Mecanical Production and Design Dept, Zagazig University, Al Zagazig, Egypt, a.sadoun76@yaoo.com Received April ; Revised June 8 017; Accepted for publication June Corresponding autor: A.M. Sadoun, a.sadoun76@yaoo.com Copyrigt 017 Said Camran University of Avaz. All rigts reserved. Abstract. Tis study presents te effect of porosity on mecanical beaviors of a power distribution functionally graded beam. Te Euler-Bernoulli beam is assumed to describe te kinematic relations and constitutive equations. Because of tecnical problems, particle size sapes and micro-voids are created during te fabrication wic sould be taken into consideration. Two porosity models are proposed. Te first one describes properties in te explicit form as linear functions of te porosity parameter. Te second is a modified model wic presents porosity and Young s modulus in an implicit form were te density is assumed as a function of te porosity parameter and Young s modulus as a ratio of mass wit porosity to te mass witout porosity. Te modified proposed model is more applicable tan te first model. Te finite element model is developed to solve te problem by using te MATLAB software. Numerical results are presented to sow te effects of porosity on mecanical beaviors of functionally graded beams. Keywords: Mecanical Beaviors; Porous material; Functionally graded material; Beam Analysis; Finite Element Metod. 1. Introduction Functionally graded materials (FGMs) represent a new generation of materials wit revolutionary properties, composed of a mixture of two different materials (metal and ceramic) and ave great practical applications in engineering and industrial fields [1]. Te constituent mixture varies smootly toug a specified spatial direction to avoid te stress concentration induced by discontinuity of material properties. During fabrication processes of FGM, te large difference in solidification temperatures between material constituents may cause micro voids or porosities witin te materials during sintering []. Aqida et al. [3] presented te causes of te porosity formation wic are air bubbles entering te melt matrix material, te water vapor on te particles surfaces, te gas entrapment during te mixing process, te evolution of ydrogen, and te srinkage during te solidification. Kim et al. [4] fabricated FG te nano/micro porous titanium surfaces using anodizing and found tat te amorpous Publised Online: June

2 Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements 75 titanium dioxide nanotubular layer significantly improves te ydropilicity. Wattanasakulpong et al. [5] discussed te effect of porosities appening inside FGM samples fabricated by a multistep sequential infiltration tecnique. Nowadays, lots of researcers are interested in studying te mecanical beavior of FG porous beams. Tere are two different models of te porosity in te literature. Te first model is derived from te rule of mixtures wic is discussed in te next sections. Ji et al. [6] used a simple mixture rule to provide a unified description of te pysical properties of polypase composites in terms of component properties, volume fractions, and microstructures, and ten exploited te proposed model for porous materials as a special class of two-pase composites. Wattanasakulpong and Ungbakorn [7] studied linear and nonlinear vibrations of elastically restrained FG beams aving porosities by using te differential quadrature metod (DQM). Wattanasakulpong and Caikittiratana [8] illustrated te vibration beavior of porous FG Timosenko beams by using Cebysev collocation metod and assumed a beam wit even and uneven distributions of porosities over te cross-section. Atmane et al. [9] presented te effects of te tickness stretcing and te porosity on mecanical beaviors of FG beams resting on elastic foundations and derived closed form solutions by using te Navier tecnique. Ebraimi and Moktari [10] obtained vibrational beavior of a rotating porous FG beam by using DQM. Ebraimi and Jafari [11] and Ebraimi et al. [1] analyzed termo-mecanical vibrations of FG porous beams under various termal loadings by employing a semi-analytical differential transform metod. Safiei et al. [13] studied nonlinear vibrations of porous imperfect FG tapered microbeams based on te modified couple stress and Euler Bernoulli teories. Ebraimi and Barati [14] presented a iger order refined beam model to investigate te vibration of viscoelastic nanocrystalline silicon nanobeams wit porosities. Te second model for te porosity assumes tat te porous material properties vary troug te tickness of te structure according to te specific continuous function. Te following groups of researc exploited tis model to investigate te mecanical beavior of te porous structures. Magnucki and Stasiewicz [15] investigated te elastic buckling of a porous isotropic beam by means of te Finite Element Metod (COSMOS). Jabbari et al. [16] studied te termal buckling of a sandwic piezoelectric circular plate made of te porous material and concluded tat increasing te porosity of a porous plate in saturated and unsaturated conditions as different results in relation to te stability of te plate; te stability of saturated and unsaturated porous plates decreases and increases, respectively, wen te porosity is increased. Xue et al. [17] presented a constitutive model for FG porous sape memory alloys. Cen et al. [18] investigated te elastic bending and buckling beavior of Timosenko FG porous beams and found tat an increase in te porosity coefficient leads to lower critical buckling loads of functionally graded porous beams. Kitiporncai et al. [19] studied te buckling and te vibration of FG porous nanocomposite beams were te internal pores and grapene platelets (GPLs) were layer-wise distributed in te matrix according to different patterns. According to te widespread literature reviews, we found tat no researcers ave attempted to address te softening in modulus of elasticity wen expressed as a ratio of te mass of a porous beam to te mass of a non-porous one. Terefore, te effect of te porosity on functionally graded structures needs more investigation. Te present study is intended to fill tis gap in te literature by considering a modified porosity model. Te material graduation is assumed to be a continuous power function distributed troug te beam tickness. Te static, buckling and vibration beavior of te porous FG beam are ten studied. Te manuscript is organized as follows: Section describes te matematical modelling of functionally graded porous materials. Te governing equations of a porous FG beam wit Euler-Bernoulli kinematic assumptions are presented in Section 3. A numerical finite element model is developed to solve te governing equations. Section 4 demonstrates te model validation and sows numerical results. Finally, Section 5 summarizes te concluding remarks.. Functional graded Formulation.1 Material graduation Functions Functionally graded materials (FGM) are a special class of composites manufactured from a mixture of two or more constituent materials (especially ceramics and metals). Te mixture as a continuous variation of properties relative to a specific spatial direction (commonly troug te tickness direction). Te simplest and te most accepted omogenization metod to estimate te effective properties at micromecanics level is known as Voigt rule [0] wic states tat te volume fraction of materials are graded across te beam tickness z by te following functions [1]: V c 1 z ( ) 0 k (1a) V c V m 1 (1b) were V is te volume fraction, k is a nonnegative power exponent, is a total beam tickness, and subscripts c and m represent te ceramic and metal, respectively. Terefore, te Young s modulus and te mass density of FGM can be described by te following relations: 1 z k E ( z ) [ Ec E m ]( ) E m (a) 1 z k ( z ) [ c m ]( ) m (b) Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8

3 76 Fouda et. al., Vol. 3, No. 4, 017 Te constituent metal of te FG beam in te present manuscript is te steel [E m= 10 GPa, and te ceramic is te alumina [ E c = 390 GPa, c = 3.96 g/cm 3, and te beam tickness is presented in Fig.1 wic sows tat at 0 m = 7.8 g/cm3 and m is 0.3] is 0.3]. Te material properties distribution troug k te beam is completely te pure ceramic, at k 10 te beam is approximately te pure metal, wile at k 1 properties cange linearly from te metal at te bottom surface to te ceramic at te top surface. It is wort noting tat, te exponent parameter as an inverse effect on te Young s modulus rater tan te density. Tat means, as te exponent increases from 0 to 10 (te ceramic ric pase to te metal ric pase), te Young s modulus decreases gradually from 390 GPa to 10 GPa, wile te density increases from 3.96 g/cm 3 to 7.8 g/cm 3. c Fig. 1. Te variation of te Young modulus and te density at different material distributions (k).. Porosity models In te present analysis, two models of porosity are presented. Te first one is considered previously by many autors, and te oter one is adopted according to experimental observations. A geometrical description of a functionally graded simplysupported beam of te lengt (L), te widt (b), and te tickness () wit a random porosity distribution is sown in Fig.. Fig.. A porous functionally graded simply-supported beam. a) Classical Model By considering an FGM beam wit te porosity fraction ( 1) distributed consistently between te ceramic and te metal, te modified rule of mixtures is proposed as [7, 9, 10]: E ( z ) Ec ( V c ) E m ( V m ) (3a) ( z ) c ( v c ) m ( v m ) (3b) By substituting Eq. (1) into Eq. (3), te equivalent modulus and density functions, including te porosity parameters, can be written as: 1 z k Ec E m E ( z ) [ Ec E m ]( ) E m [ ] (4a) 1 z k c m ( z ) [ c m ]( ) m [ ] (4b) Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8

4 Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements 77 b) Adopted model According to te experimental observation by Sarkar et al. [], te actual strengts of a composite are lower tan tose obtained troug te teoretical estimation by applying te simple rule of mixtures. Terefore, te modification of te rule of mixture is needed. Bert [3] predicted te elastic moduli of solids wit te oriented porosity wit a semi-empirical approac and concluded tat te linear variation of te porosity is insufficient to consider a reduction in te rigidity of te structure. Hardin et al. [4] found tat te elastic modulus decreases nonlinearly wit te porosity and tat te steel exibits a critical porosity level above wic it loses all te stiffness. Watcman et al. [5] stated tat Gibson and Asby proved tat te strengt of a strut in te scaffold is te same as te strengt of te bulk material and tis is related to te strengt of te cellular solid (scaffold) by its relative density. Sabree et al. [6] exploited Gibson and Asby model to study te mecanical properties of porous ceramic scaffolds. Zok and Levi [7] proved tat te effect of te matrix porosity on te modulus can be described by an empirical relationsip related te moduli wit te density. For a porous body, tere are many teoretical models linking te elastic properties wit te porosity and consequently wit te apparent density [8]. In te present analysis, it is assumed tat te porosity and te density are inter-related by an explicit function given in Eq. (4.b). On te oter and, te elastic modulus is related explicitly to te density ratio and implicitly to te porosity as follows: 1 z k Ec E m mo m E ( z ) [ Ec E m ]( ) E m ( )[ ] (5) m were te equivalent mass along te tickness can be calculated by: m o ( z ) dz at 0 (6a) m ( z ) dz at (6b) Te effect of te porosity on te equivalent mass, (m), and different material exponents, (k), is presented in Fig. 3. As sown, te equivalent mass is reduced as te porosity increased. However, te equivalent mass is increased by increasing te material exponent due to te increase in metal pase in te constituents. o Fig. 3. Variation of te equivalent mass of functionally graded material versus te material distribution and te porosity. Te variations of rigidities, D z ( z ) dz, of te porous function graded beams for classical and modified models wit different material distributions are illustrated in Fig. 4. It is concluded tat te rigidity of te FG beam is increased by increasing te material graduation or decreasing te porosity percentage. Te effects of te porosity and te material graduation for te two models sow similar beavior, owever, te porosity in te adopted model is more significant tan te classical model. At material exponent k=4, te rigidity of te classical model is decreased by.7% as te porosity increases to 0.. At te same conditions, te rigidity of te modified model is decreased by 36.36% wic indicates te reliability of tis model. 3. Problem Formulation 3.1 Matematical Problem Regarding a tin beam (L/ >0), te geometrical fit conditions of te Euler Bernoulli teory describe te in-plane (u) and te transverse (w) displacements by: dw o u ( x, z, t ) uo ( x, t ) z (7a) dx w ( x, z, t ) w o ( x, t ) (7b) were u o and w o are te mid-plane axial and transverse displacements, respectively, and t denotes te time. Te governing equations of motion can be described by [9]: Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8

5 78 Fouda et. al., Vol. 3, No. 4, 017 (a) Classical Model (b) Adopted Model Fig. 4. Variation of te equivalent rigidity of functionally graded material versus te material distribution and porosity factor for a classical and adopted models. dn dx f u dz (8a) t M w 0 w q ( N ) dz x x x t (8b) were q(x,t) denotes te magnitude of te distributed vertical load on te beam, N is te applied axial compressive force, is te material density, and M is te bending moment. Te governing equations of motion can be presented in terms of displacements as: 3 uo w o u 3 E ( z ) dz ze ( z ) dz f dz x x t 3 4 uo w o W o w ze ( z ) dz ( ) 3 z E z dz q N 4 dz x x x x t (9a) (9b) 3. Numerical Problem Te variational form of te equilibrium equations in terms of te displacements for te porous FG beam can be represented by: T L 0 0 o o o o o o o o o o ) E( z ) u u w w u w u w w w m( z z dz t t t t x x x x x x w L o w o w o N f uo qw odxdt [ N u o V w o M 0 x x x 0 T 0 To discretize te porous FG beam to elements, te in-plane and transverse displacement components at te mid-plane of a beam-element can be described by [0, 30, 31]: ( e ) 0 M { d } K b { d } K { d } { F } Q (1) Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8 ] (10) i i 1 1 (11a) i 1 u ( x, t ) N U ( t ) N U ( t ) N U ( t ) 4 ( e ) o (, ) k k (11b) K 1 W x t N W N W N N W N were U, W and are te plane displacement, te transverse displacement and te slope at te nodal points, respectively, and N i is a group of Lagrangian interpolation sape functions for in-plane displacements, and N k is a set of Hermetian interpolation sape functions for transverse displacements. By substituting and integrating Eq. (11) into Eq. (10), te following equation of motion can be represented in te matrix form as:

6 Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements 79 were M is te mass stiffness matrix, K b is te geometrical buckling matrix, K is te stiffness matrix, d is te displacement of te nodal value, d is te acceleration, F is te concentrated force vector, and Q is te distributive force vector. For te static problem, te buckling stiffness and mass matrices are neglected, and terefore, te following equilibrium equation is solved. K { d } { F} Q (13) However, for te buckling and te free vibration, te eigenvalue problems are solved using te following relations: K { d } K b { d } (14a) K { d } [ M ]{ d } (14b) were { d } is te eigenvectors, is te eigenvalues (critical buckling loads), and system (natural frequencies). is te eigenvalues of dynamic 4. Numerical Results In tis section, numerical results of te static buckling and te free vibration of a porous FG beam are presented for te two proposed models. Te material constituents and teir distributions are presented in section. Te geometrical dimensions of te beam are b L= m. 4.1 Static Analysis In tis analysis, te system of linear equations (Eq. 13) are solved and te non-dimensional center deflection (te maximum 4 for te simply supported beam) is calculated by W max 100 Ec I / ql (max( w 0 )). Figure (5) illustrates te porosity and material graduation effects on te bending deflection of te beam under te uniform distributed load. It is wort noting tat te deflection in bot models is increased by increasing te material graduation and te porosity percentage due to te decrease in te equivalent modulus of te elasticity. As te porosity increases from 0 to 0. at k=, te deflection increases from to.3155 (by 7%) for te classical model and from to.984 (65%) for te adopted model. Tis means tat te adopted model is more sensitive to te porosity tan te classical model. Note tat te cange in te classical model is only due to te linear effect of te porosity. However, te cange in te adopted model is due to te coupling effect of te porosity and te density. In addition, from te engineering viewpoint, increasing te porosity to 0. makes te structure very weak wic is consistent wit te adopted model. Fig. 5. Effect porosity and material graduation on te static bending of te FG beam. 4. Buckling Analysis Wen a slender structure is loaded in compression, it may lose its ability to carry te load wile it reaces to a critical load value, known as te critical buckling load. Terefore, it is motivating to illustrate te effect of te porosity on te critical buckling load. To calculate te critical load, te eigenvalue problem (Eq. 14a) is solved to find te smallest eigenvalue of te structure. Te effects of te porosity and material distribution on te non-dimensional buckling load N NL / E c I min ( ) are presented in Fig. (6). As depicted in te above-mentioned figure, te buckling load is decreased by increasing te porosity and te material graduation for te two proposed models. Moreover, as te porosity increases from 0 to 0. at k=, te buckling load decreases by 1% for te classical model and by 39% for te adopted model wic sows tat te adopted model is more pronounced. Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8

7 80 Fouda et. al., Vol. 3, No. 4, 017 Fig. 6. Effect porosity and material graduation on te critical buckling load of te FG beam. 4.3 Dynamic Analysis Te free vibration of a porous FG beam is investigated in tis subsection. Te eigenvalue problem (Eq. 14b) is solved to find te first tree natural frequencies of te porous beam structure. Te non-dimensional natural frequency is calculated according to te formula, i i L c c w A / E I, i = 1,,3. It is known tat te natural frequency of a structure is a function of bot density and elasticity, but tey ave inverse effects (te natural frequency decreases as te density increases or as te stiffness decreases). Fig. 7. Effect porosity and material graduation on te fundamental natural frequency of te FG beam. Fig. 8. Effect porosity and material graduation on te nd natural frequency of te FG beam. As can be seen in Fig. 1, for a functionally graded material, as te material exponent increases te equivalent young s modulus decreases and te density increases. As a result, te natural frequencies decrease as sown in Figs. (7-9). Figure 7 presents te variation in te first natural frequency wit respect to te material distribution and te porosity percentage for te Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8

8 Bending, Buckling and Vibration of a Functionally Graded Porous Beam Using Finite Elements 81 classical and te adopted model, respectively. As illustrated in te above-mentioned figure, te first natural frequency is decreased by increasing te material exponent k for bot models. Altoug te porosity as trivial effects on te frequency in te classical model, it as a noteworty effect on te frequency for te adopted model. For tis case, by fixing te material exponent at k=1, and increasing te porosity from 0 to 0., te first natural frequency decreases by % for te classical model and by 14% for te adopted model. Te effects of te porosity and te material distribution parameter on te second and te tird natural frequencies sow similar beavior as presented in Figs. 8 and Conclusions Fig. 9. Effect porosity and material graduation on te 3 rd natural frequency of te FG beam. A modified porosity model is presented to study te static bending, te buckling and free vibrations of a porous functionally graded beam. Te porosity effects are studied using two models. Te material graduation is assumed to be distributed troug te beam tickness according to a nonlinear power function. Kinematic fit conditions of Euler-Bernoulli beams wit elastic Hookean constitutive equations are implemented. Te finite element metod is used to solve te problem. Te most significant findings of te obtained results can be summarized as follows: a) A model is proposed to consider te variation in te elasticity as an implicit function of te porosity and an explicit function of te density wic is more applicable from te experimental viewpoint compared to te classical model. b) Te static deflection is increased by increasing bot te porosity and te material distribution parameter for te two presented models. c) Te critical buckling load is decreased by increasing bot te porosity and te material distribution parameter for te two models. d) Te frequency is decreased by increasing te material exponent for bot models. However, te frequency is more pronounced by te porosity in te proposed model, wile te porosity effect on te frequency in te classical model is insignificant. References [1] Eltaer, M.A., Kairy, A., Sadoun, A.M., Omar, F.A. Static and buckling analysis of functionally graded Timosenko nanobeams, Applied Matematics and Computation, 9, 014, pp [] Zu, J., Lai, Z., Yin, Z., Jeon, J., Lee, S. Fabrication of ZrO NiCr functionally graded material by powder metallurgy, Materials Cemistry and Pysics, 68(1), 001, pp [3] Aqida, S.N., Gazali, M.I., Hasim, J. Effects of porosity on mecanical properties of metal matrix composite: an overview, Jurnal Teknologi, 40, 004, pp [4] Kim, H.S., Yang, Y., Ko, J.T., Lee, K.K., Lee, D.J., Lee, K.M., Park, S.W. Fabrication and caracterization of functionally graded nano micro porous titanium surface by anodizing, Journal of Biomedical Materials Researc Part B: Applied Biomaterials, 88B, 009, pp [5] Wattanasakulpong, N., Prusty, B.G., Kelly, D.W., Hoffman, M. Free vibration analysis of layered functionally graded beams wit experimental validation, Materials & Design, 36, 01, pp [6] Ji, S., Gu, Q., Xia, B. Porosity dependence of mecanical properties of solid materials, Journal of Materials Science, 41, 006, pp [7] Wattanasakulpong, N., Ungbakorn, V. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams wit porosities, Aerospace Science and Tecnology, 3, 014, pp [8] Wattanasakulpong, N., Caikittiratana, A. Flexural vibration of imperfect functionally graded beams based on Timosenko beam teory, Cebysev collocation metod, Meccanica, 50(5), 015, pp [9] Atmane, H.A., Tounsi, A., Bernard, F. Effect of tickness stretcing and porosity on mecanical response of a functionally graded beams resting on elastic foundations, International Journal of Mecanics and Materials in Design, 13, 015, pp Journal of Applied and Computational Mecanics, Vol. 3, No. 4, (017), 74-8

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