Three-dimensional free vibration analysis of functionally graded rectangular plates using the B-spline Ritz method

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1 The 01 World Congress on Advances in Civil, Environmental, and Materials Research (ACEM 1) Seoul, Korea, August 6-30, 01 Three-dimensional free viration analsis of functionall graded rectangular plates using the B-spline Rit method *Harunou Nagino 1), Motohiro Sato ), Tomisaku Miusawa 3) and Takashi Mikami 4) 1) Department of Civil and Environmental Engineering, Oita National College of Technolog, 1666 Maki, Oita , Japan ) Division of Engineering and Polic for Sustainale Environment, Facult of Engineering, Hokkaido Universit, Kita 13 Nishi 8, Kita-ku, Sapporo , Japan 3) Civil Engineering and Environmental Design Course, Daido Universit, Hakusui-cho 40, Minami-ku, Nagoa , Japan 4) Hokkaido Universit, Kita 13 Nishi 8, Kita-ku, Sapporo , Japan 1) nagino@oita-ct.ac.jp ABSTRACT This paper presents the three-dimensional free viration analsis of functionall graded rectangular plates with aritrar oundar conditions using the B-spline Rit method ased on the theor of elasticit. The Young s modulus of the plate is assumed to var through the thickness direction, and the Poisson s ratio and densit are constant. The convergence and accurac of the present method were investigated. Rapid, stale convergence and high accurac were otained the present method. Furthermore, the effects of thickness-to-length ratio, Young s modulus ratio and oundar condition on the fundamental frequenc parameters, mode shapes, strain energies and kinetic energies of functionall graded plate were also carried out. 1. INTRODUCTION Functionall graded materials (FGMs) were first proposed a group of Japanese material scientists (Koiumi 1997). The material propert of FGMs varies smoothl and continuousl through the thickness from the surface to other surface. FGMs have received considerale attention in modern technolog as a structural element or a material ingredient due to increasing performance demands, and are epected as a future high performance composite material in man engineering fields as aerospace, nuclear, marine, and civil engineering, etc. Therefore, an understanding of the dnamic ehavior of the structure memers made of FGMs is ver important in structural design. 1), ) Associate Professor, 3) Professor, 4) Vice President

2 The free viration analsis of structural memers ased on the three-dimensional (3-D) theor of elasticit does not rel to hpotheses involving kinematics of deformation. Therefore, 3-D free viration analsis provides realistic results ased on the theor, and it can not e predicted shear deformation theories (Mindlin 1951, Redd 1985). The studies on the 3-D free virations of rectangular plates made of FGMs are limited to four edges simpl-supported condition (Vel 004). Recentl, Malekadeh (009) proposed the semi-analtical differential quadrature method, and analed 3-D free virations of functionall graded rectangular plates with one pair of parallel simpl-supported edges. This paper presents the 3-D free viration analsis of functionall graded rectangular plates with aritrar oundar conditions using the B-spline Rit method. The Young s modulus of the plate is assumed to var through the thickness direction, and the Poisson s ratio and densit are constant. The analsis is ased on the linear, small-deformation 3-D theor of elasticit. The B-spline Rit method has een proposed Nagino (008). This method is formulated the Rit procedure with triplicate series of normalied B-spline functions (Boor 197) as displacement amplitude components. The geometric oundar conditions are numericall satisfied the method of artificial spring (Kao 1974). To demonstrate the convergence and accurac of the present method, several eamples are solved, and the results are compared with the 3-D finite element solutions. Furthermore, a detailed investigation of the effects of thickness-to-length ratio, Young s modulus ratio and oundar condition on the fundamental frequenc parameters, mode shapes, strain energies and kinetic energies of functionall graded plate are also carried out.. FORMULATION.1. Material properties of FGMs The Young s modulus E( ) in the functionall graded plate is assumed to var eponentiall in the thickness direction as follows: where E t and E ( ) = E ep p h, ln E t p =, (1) E E are the Young s moduli at top and ottom surface, respectivel. Poisson s ratio ν and densit ρ are constant. The distriution of Young s modulus E( ) in the thickness direction is shown in Fig Analtical model Consider a functionall graded rectangular plate with length a, width and uniform thickness h in Fig.. The stress free surface are assumed at = 0 and =h. The plate is defined with respect to a right-handed orthogonal coordinate sstem (,, ). The periodic displacement components at an point are defined the in-plane components u, v and the transverse component w in the, and direction, respectivel.

3 1.00 / h E t / E = 10 E t / E = 5 E t / E = 1 E t / E = 1 / 5 E t / E = 1 / E () Fig. 1. The distriution of Young s modulus in the thickness direction, v, w h O a E t E Fig.. Coordinate sstem and geometr of functionall graded plate, u.3. Formulation of eigenvalue equation ased on the B-spline Rit method The strain energ U of the functionall graded rectangular plate can e epressed in integral form as U 1 a h = ( )d d d σε σε + σε + τγ + τ γ + τγ, () in which ε, ε, ε, γ, γ, γ and σ, σ, σ, τ, τ, τ are strain and stress components, respectivel. In the 3-D theor of elasticit, strain and stress components of the functionall graded rectangular plate are defined as u ε =, ε v w =, ε =, γ u v = +, γ v w = +, γ w u = +, (3)

4 σ = { µ ( ) + G ( )} ε + µ ( ) ε + µ ( ) ε, σ = µ () ε + {() µ + G()} ε + µ () ε, σ = µ ( ) ε + µ ( ) ε + { µ ( ) + G( )} ε, τ = G ( ) γ, τ = G ( ) γ, τ = G ( ) γ, (4) ν E ( ) µ ( ) =, (1 ν )(1 + ν ) E( ) G ( ) =, (5) (1 + ν ) where E( ) is Young s modulus, ν is Poisson s ratio, and G ( ) is shear modulus. The kinetic energ T of the plate can e written as T ρ a h u v w = ddd t t t, (6) in which ρ is the mass densit per unit volume. Here, for simplicit and convenience in mathematical formulation, the following nondimensional coordinate sstem (ξ, η, ζ ) are introduced as ξ =, a η =, ζ =. (7) h The periodic displacement components can e epressed non-dimensional displacement amplitude function U, V and W in ξ, η and ζ coordinates and the temporal coordinate t as i u( ξηζ,,, t) au( ξηζ,, ) e ωt =, i v( ξηζ,,, t) av( ξηζ,, ) e ωt =, i w( ξηζ,,, t) aw( ξηζ,, ) e ωt =, (8) where ω denotes the circular frequenc and i = 1 is an imaginar constant. The assumed spatial displacement field is ased on a separale assumption for displacement amplitude functions. The functions are epressed as the summation of a triplicate series of B-spline functions as follows: iξ iη iζ U( ξ, η, ζ ) A N ( ξ) N ( η) N ( ζ ) =, m= 1 n= 1 r= 1 iξ iη iζ m= 1 n= 1 r= 1 mnr m, kξ n, kη r, kζ V( ξ, η, ζ ) B N ( ξ) N ( η) N ( ζ ) =, iξ iη iζ m= 1 n= 1 r= 1 mnr m, kξ n, kη r, kζ W( ξ, η, ζ ) = C N ( ξ) N ( η) N ( ζ ), (9) mnr m, kξ n, kη r, kζ

5 in which N m, k ξ ( ξ ), N n, k η ( η) and N r, k ζ ( ζ ) are one-dimensional (1-D) normalied B- spline functions with the degree of spline function ( k ξ 1), ( k η 1) and ( k ζ 1). A mnr, B mnr and C mnr are unknown spline coefficients. The appearing in Eq. (9) are defined as: i k = mξ + kξ, i k = mη + kη and i k = mζ + kζ, where m ξ, m η, m ζ and k ξ, k η, ξ η ζ kζ are the numer of knots and the order of spline function in the ξ, η and ζ directions, respectivel. Sustituting Eqs. (7), (8) and (9) into Eqs. () and (6), the maimum strain energ U ma and maimum kinetic energ T ma of the plate can e written in a non-dimensional coordinate sstems as ahe U ( )d d d ma = σ ε + σ ε + σε + τγ + τ γ + τ γ ζ η ξ ahe T = { } [K P ]{ }, (10) 3 ah Tma = ρω ( U + V + W )d ζ d η d ξ ρω ah T = { } [M]{ }, (11) where [K P ] and [M] are the stiffness and mass matrices of the plate, respectivel, and { } is unknown coefficient vector in the following: { δ δ δ } T { } = { } { } { }, (1) A B C in which the column vectors { δ A }, { δ B } and { δ C } as { } { } T δ A = A111 A11 A11 i A11 A1 i A1 i i A ζ ζ η ζ iξ iηi, ζ { } { } T δ B = B111 B11 B11 i B11 B1 i B1 i i B ζ ζ η ζ iξ iηi, ζ { } { } T δ C = C111 C11 C11 i C11 C1 i C1 i i C ζ ζ η ζ iξ iηi. (13) ζ The oundar conditions at the four edges ( = 0, a and = 0, ) of a functionall graded rectangular plate would e satisfied as follows: Simpl supported: v = w = 0, σ = 0 at ( = 0, a), u = w = 0, σ = 0 at ( = 0, ). (14)

6 Clamped edge: u = v = w = 0 at ( = 0, a), u = v = w = 0 at ( = 0, ). (15) Free edge: σ = τ = τ = 0 at ( = 0, a), σ = τ = τ = 0 at ( = 0, ). (16) The oundar conditions for top and ottom stress free surfaces of the plate can e epressed σ = τ = τ = 0 at ( = 0, h). (17) To deal with the geometric oundar conditions at the four edges ( = 0, a and = 0, ), the method of artificial spring (Kao 1974) is used. In this method, three tpes of spring coefficient α, β and γ corresponding to the geometric oundar conditions u, v, w are introduced at each oundar edges of the plate. The energ contriution L due to the spring is given 1 1 L= u + v + w + u + v + w h a h ( α β γ )d d ( α β γ )d d = 0, a = 0,. (18) Sustituting Eqs. (7), (8) and (9) into Eq. (18), the maimum artificial spring energ L of the plate can e given in a non-dimensional coordinate sstems as ma 3 ah ah Lma = ( αu βv γw )dζ d η ( αu βv γw )dζ dξ ξ= 0,1 η= 0,1 ahe T = { } [K L]{ }, (19) where [K L] is the stiffness matri for the artificial springs. For the geometric oundar condition at the four edges (ξ = 0, 1 and η = 0, 1), the non-dimensional spring parameters are assumed to e ero, the oundar edges will ecome the stress free condition. If the spring parameter is assumed to e infinite, the oundar edges will lead to procedure the clamped condition. The total potential energ Π of the functionall graded plate can e epressed as Π= ( U + L ) T. (0) ma ma ma In Eq. (0), minimiing the total potential energ Π with respect to the unknown spline coefficient vector { } i.e.:

7 Π = 0, (1) { } which lead to the following the eigenvalue equation in matri form: [K UU ] [K UV] [K UW ] [M UU ] [0] [0] { δ A} {0} [K VU] [K VV ] [K VW ] [0] [M VV] [0] Ω { δ B} = {0}, () [K WU ] [K WV ] [K WW ] [0] [0] [M WW ] { δc} {0} Ω = ω, (3) a E ρ where Ω is the non-dimensional frequenc parameter, [K IJ ] and [M IJ ] (I, J = U, V, W) are the su-stiffness matrices and su-mass matrices of the plate, respectivel. The sie of matri in Eq. () is 3( m k )( m + k )( m + k ). 3. NUMERICAL RESULTS ξ + ξ η η ζ ζ The fundamental natural frequencies of the functionall graded rectangular plates with aritrar oundar conditions were analed. Firstl, the convergence and accurac of the present method were investigated. Net, the effects of thickness-tolength ratio h / a, Young s modulus ratio E t / E and oundar condition on the fundamental frequenc parameters, mode shapes, strain energies and kinetic energies of functionall graded plate were also carried out. Poisson s ration ν = 0.3 and aspect ratio / a = 1 were used in the numerical calculations. For the definition of the oundar conditions of the plate, for eample, the smols SF-CS, identifies a plate with edges (ξ = 0, 1) and (η = 0, 1) having simpl supported edge (S), free edge (F), clamped edge (C) and simpl supported edge (S), respectivel. The non-dimensional 8 artificial spring parameters are used to e 10 and the placement of the knots was set to the Cheshev-Gauss-Loatto points, namel, the shifted Cheshev points (Nagino 008) in the following analsis. All computations are performed in doule precision on a personal computer, and all of the fundamental frequenc parameter Ω 1st = ω 1st a ρ / E and viration modes are organied to four significant digits Convergence and comparison studies The Rit method provides theoreticall accurate solutions, and, generall, the natural frequencies otained the Rit procedure are the upper ounds of the eact frequencies. The effects of the degree of spline function (k ξ 1) (k η 1) (k ζ 1) and the numer of knot m ξ m η m ζ on the convergence of the present method are investigated. Moreover, the accurac of the present method is also investigated.

8 Tale 1. Convergence of fundamental frequenc parameter Ω 1st for SS-SS functionall graded plates: E / E = 100 t h / a ( kξ 1) ( kη 1) ( kζ 1) mξ mη mζ D.O.F. Ω 1st (Error %) D FEM (C3D8) ( 0.63 %) ( 0.7 %) D FEM (C3D8) ( 0.09 %) ( 0.03 %) Tale. Convergence of fundamental frequenc parameter functionall graded plates: E / E = 100 t Ω 1st for CC-CC h / a ( kξ 1) ( kη 1) ( kζ 1) mξ mη mζ D.O.F. Ω 1st (Error %) D FEM (C3D8) ( 0.40 %) ( 0.15 %) D FEM (C3D8) ( %) ( %)

9 Tale 3. Convergence of fundamental frequenc parameter Ω 1st for CF-FF functionall graded plates: E / E = 100 t h / a ( kξ 1) ( kη 1) ( kζ 1) mξ mη mζ D.O.F. Ω 1st (Error %) D FEM (C3D8) ( 0.1 %) ( 0.06 %) D FEM (C3D8) ( %) ( %) Tales 1, and 3 show the effects of the degree of spline function (k ξ 1) (k η 1) (k ζ 1) and the numer of knot m ξ m η m ζ on the convergence of the fundamental frequenc parameter Ω 1st for functionall graded square plate having SS-SS, CC-CC and CF-FF respectivel. The thickness-to-length ratio h / a are set as 0. (moderatel thick plate) and 0.5 (thick plate), and the Young s modulus ratio E t / E = 100 is used. The results are compared with the 3-D finite element solutions Aaqus 6.1. Here, C3D8 means first-order solid element. The error (%) is defined as follows: Ω Ω FEM Present Error (%) = 100 ΩPresent, (4) where Ω Present is convergence value otained from present method in Tales 1, and 3. Tales 1, and 3 show that stale convergence can e otained. It is found that the fundamental frequenc parameter Ω 1st rapidl converge using the degree of spline function The results in Tales 1, and 3 show ecellent agreements in all cases; thus, high accurac is otained. The B-spline Rit method ields highl accurate results with few degrees of freedom (D.O.F.) for the fundamental frequencies and displacement mode shapes of the functionall graded rectangular plate. In addition, stale and rapidl converging and ecellent upper ound solutions are otained the present method regardless of the thickness-to-length ratio and oundar conditions.

10 Ω 1st E t / E = 100 E t / E = 10 E t / E = 1 E t / E = 1/10 E t / E = 1/100 Ω 1st (FGM) / Ω 1st (Isotropic) E t / E = 100 E t / E = 10 E t / E = 1 E t / E = 1/10 E t / E = 1/ h / a (a) h / a Ω 1st () Ω1st (FGM) / Ω 1st (Isotropic) Fig. 3. The effects of thickness-to-length ratio and Young s modulus ratio on the fundamental frequenc parameter of CC-CC functionall graded square plates. (a) E t / E = 1 / 10, h / a = 0.1 () E t / E = 1, h / a = 0.1 (c) E t / E = 10, h / a = 0.1 (d) E t / E = 1 / 10, h / a = 0.3 (e) E t / E = 1, h / a = 0.3 (f) E t / E = 10, h / a = 0.3 Fig. 4. The fundamental modes of CC-CC functionall graded square plates. 3.. Results and discussions The B-spline Rit method is applied to investigate the fundamental free viration of the functionall graded square plate with aritrar oundar conditions. Fig. 3 shows the effects of thickness-to-length ratio h / a and Young s modulus ratio E t / E on the fundamental frequenc parameter Ω 1st of CC-CC functionall graded

11 E t / E = 100 E t / E = 10 E t / E = 1 E t / E = 1/10 E t / E = 1/100 U 0.4 E t / E = 100 E t / E = 10 E 0. t / E = 1 E t / E = 1/10 E t / E = 1/ h / a (a) Bending component U s h / a () Transverse shear component Fig. 5. The effects of thickness-to-length ratio and Young s modulus ratio on the strain energies of fundamental mode for CC-CC functionall graded square plates E t / E = 100 E t / E = 10 E t / E = 1 E t / E = 1/10 E t / E = 1/ T In h / a (a) In-plane component T Trans 0.4 E t / E = 100 E t / E = 10 E 0. t / E = 1 E t / E = 1/10 E t / E = 1/ h / a () Transverse-plane component Fig. 6. The effects of thickness-to-length ratio and Young s modulus ratio on the kinetic energies of fundamental mode for CC-CC functionall graded square plates. square plates. In addition, the fundamental modes of CC-CC functionall graded square plates are depicted in Fig.4. The thickness-to-length ratio h / a varies from 0.01 to 0.5. The Young s modulus ratio E t / E is set as 100, 10, 1, 1/10, and 1/100. Note that E t / E = 1 means an isotropic plate. It is found that the fundamental frequencies parameter Ω 1st increases with increment of thickness-to-length ratio h / a and Young s modulus ratio E t / E. Fig. 5 shows the effects of thickness-to-length ratio h / a and Young s modulus ratio E t / E on the strain energies of fundamental viration mode for CC-CC functionall graded square plates. The numerical calculation condition is the same as the Fig. 3. Here, the ending component U and the transverse shear component U s are defined

12 U U 1 a h = ( σε + σε + τ γ )ddd, (5) ( )ddd. (6) a h s = τ γ + τγ It is seen that the ending component U decreases with increment of thickness-tolength ratio h / a, however, the ending component U of functionall graded plates are greater than the ones of isotropic plates. On the other hand, with increasing thicknessto-length ratio h / a, the transverse shear component U s also increases. But, the transverse shear component U s of functionall graded plates are smaller than the ones of isotropic plates. Thus, it seems that functionall graded materials are strong to transverse shear deformation. Fig. 6 depicts the effects of thickness-to-length ratio h / a and Young s modulus ratio E t / E on the kinetic energies of fundamental mode for CC-CC functionall graded square plates. The numerical calculation condition is the same as the Fig. 3. Here, the In-plane component T In and the transverse-plane component T Trans are defined ρω a h ( T )d d d In = u + v, (7) ρω a h TTrans = d d d w (8) It is seen that the In-plane component T In increases with increment of thickness-tolength ratio h / a and Young s modulus ratio E t / E. On the other hand, the transverseplane component T Trans decreases with increment of thickness-to-length ratio h / a and Young s modulus ratio E t / E Ω 1st SS-SS 1 CC-CC CC-FF h / a Fig. 7. The effects of thickness-to-length ratio and oundar condition the fundamental frequenc parameter of functionall graded square plates.

13 (a) h / a = 0.1, SS-SS () h / a = 0.1, CC-CC (c) h / a = 0.1, CC-FF (d) h / a = 0.3, SS-SS (e) h / a = 0.3, CC-CC (f) h / a = 0.3, CC-FF Fig. 8. The fundamental mode shapes of functionall graded thick plates with SS-SS, CC-CC and CC-FF. Fig. 7 shows the effects of thickness-to-length ratio h / a and oundar condition the fundamental frequenc parameter Ω 1st of functionall graded square plates. The thickness-to-length ratio h / a varies from 0.01 to 0.5. The Young s modulus ratio E t / E is set as 10. In addition, Fig.8 depicts the fundamental mode shapes of functionall graded square plates having SS-SS, CC-CC, CC-FF. From Fig. 7, with increasing the thickness-to-length ratio h / a, the fundamental frequencies parameter Ω 1st also increases regardless of the oundar conditions. 4. COCLUSIONS The 3-D free viration analsis of functionall graded rectangular plates with aritrar oundar conditions has een presented. The governing eigenvalue equation is formulated the B-spline Rit method ased on the linear, small-deformation 3-D theor of elasticit. Rapid, stale convergence and high accurac were otained the present method. Furthermore, the effects of thickness-to-length ratio, Young s modulus ratio and oundar condition on the fundamental frequenc parameters, mode shapes, strain energies and kinetic energies of functionall graded plate were also investigated. The fundamental frequencies parameter increases with increment of thickness-tolength ratio and Young s modulus ratio. Therefore, when the top surface sets higher Young s modulus, the functionall graded plates can increase the natural frequenc than the isotropic plates. Moreover, functionall graded materials are strong to

14 transverse shear deformation. Finall, with increasing the thickness-to-length ratio, the fundamental frequencies parameter also increases regardless of the oundar conditions. ACKNOWLEDGMENT This stud was supported the Ministr of Education, Science, Sports and Culture (MEXT), Grant-in-Aid for Young Scientists (B) (No ) and Japan Societ for the Promotion of Science (JSPS), Grant-in-Aid for Young Scientists (A) (No ). REFERENCES Boor, C.D. (197). On calculating with B-spline, Journal of Approimation Theor, Vol. 6, Kao, R. (1974). On the treatment of oundar conditions the method of artificial parameters, International Journal for Numerical Methods in Engineering, Vol. 8, Koiumi, M. (1997). FGM activities in Japan, Composites Part B, Vol. 8, 1-4. Malekadeh, P. (009). Three-dimensional free viration analsis of thick functionall graded plates on elastic foundations, Composite Structures, Vol.89, Mindlin, R.D. (1951). Influence of rotator inertia and shear on fleural motions of isotropic, elastic plates, ASEM Journal of Applied Mechanics, Vol.73, Nagino, H., Mikami, T., Miusawa, T. (008). Three-dimensional free viration analsis of isotropic rectangular plates using the B-spline Rit method,journal of Sound and Viration, Vol.317, Redd, J.N., Phan, N.D. (1985). Stailit and Viration of isotropic, orthotropic and laminated plates according to a high-order shear deformation theor,journal of Sound and Viration, Vol.98, Vel, S.S., Batra, R.C. (004). Three-dimensional eact solution for the viration of functionall graded rectangular plates, Journal of Sound and Viration, Vol.7,