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Buckling analysis of piezoelectric laminates based on higher-order shear deformation theory

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https://www.eduzhai.net International Journal of Composite M aterials 2013, 3(4): 92-99 DOI: 10.5923/j.cmaterials.20130304.02 Buckling Analysis of Piezolaminated Plates Using Higher Order Shear Deformation Theory Rajan L. Wankhade*, Kamal M. Bajoria Department of Civil Engineering, Indian Institute of Technology, Bombay, M umbai, India Abstract Buckling analysis of piezolaminated p late subjected to comb ined action of electro -mechanical loading is considered for the present work. Higher order shear deformat ion theory based finite element method is used to perform the analysis. An isoparametric eight noded element is emp loyed in the fin ite element fo rmulat ion. Studies are conducted for different ply o rientation and plate aspect ratios of piezolaminated plate with different electric condition of the piezoelectric layer. Buckling loads are found out for simp ly supported piezoelectric laminated plates. Piezo laminated plates consist of cross ply symmetric and antisymmetric o rientation of laminates with piezolayer attached at the top and bottom of the plate. Results presented for buckling analysis of plates are verified with other nu merical solution availab le in the literature and further results for future references are provided. Keywords Buckling Analysis, Piezo laminated Plate, Higher Order Shear Deformat ion Theory 1. Introduction Over the last two decades plenty of work has been carried out on analysis of piezoelectric laminated structures including smart beams, colu mns and plates. The increased use of piezo laminated smart structures in structural application is stimulat ing development of different methods and solution techniques for the accurate analysis of piezo laminated structures. Some of these methods are Galerkin method, Fin ite element method, and Reddy’s higher order shear deformation theory. A lso the experimental investigation of smart piezo laminated structures has attracted some attention. The need of such smart structures with high stiffness and low weight in engineering applications has led to the gradual replacement of many isotropic materials/structures with light weight piezo-co mposites. The piezoelectric material possesses a coupled electro mechanical property having a d irect and converse piezoelectric effect. The availability of these piezoelectric ceramics in the form of thin sheet makes them well suited for use as distributed sensors and actuators to control structural response in shape, vibration and buckling. Berlincourt et al.[1] and Mason[2] dis cussed the fu n damen t als o f p iezo elect ricity in b rief. Takag i[3] presented definition of smart materials and structures with notable arguments and detailed co mments. The limitations of classical p late theo ry and first o rder shear defo rmat ion * Corresponding author: rajanw04@gm ail.com (Rajan L. Wankhade) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved theories forced the development of higher order theories to avoid the use of shear correction factors, to include correct cross sectional warping and to get the realistic variation of the transverse shear strains and stresses through the thickness of plate. Tzou and Zhou[4] studied linear dynamics and distributed control of piezoelectric laminated continua. Donthireddy and Chandrashekhara[5] developed a mathe matica l mode l based on a layerwise theory for laminated co mposite beams with piezoelectric actuators to study its shape control. Soares et al.[6] analysed piezolaminated composite plates using refined finite element models based on higher order displacement fields and studied structural optimizat ion. Artel and Becker[7] investigated numerically the effect of electro mechanical coupling on the interlaminar stresses and the electric field strengths at free edges of piezo laminated plates. Oh and Lee[8] studied geometric nonlinear analysis of cylindrical piezolaminated shells based on the multifield layerwise theory using finite element method for the thermal and piezoelastic loading. Tzou and Gad re[9, 10] carried out theoretical analysis for vibrat ion control of a mu ltilayered thin shell coupled with piezoelectric actuators. Batra and Liang[11] analysed the steady state vibrations of a simp ly supported rectangular elastic laminated plate with embedded PZT layers. They used three dimensional linear theory of elasticity. Oh et al.[12] worked on postbuckling and vibration characteristics of piezolaminated composite plate subject to thermo-p iezoelectric loads. Shen[13, 14] presented post buckling of shear deformable laminated plates with piezoelectric actuators under complex loading condition including thermo-electro-mechanical loading. Wang[15] International Journal of Composite M aterials 2013, 3(4): 92-99 93 investigated elastic buckling of colu mn structures with a pair of piezoelectric layers surface bonded on both sides of the columns. The governing equation coupling the piezoelectric effect was derived based on the assumed distribution of the electric potential in the fle xura l direction of the pie zoelectric layer and an eigenvalue problem was solved using the direct difference method. Liew et al[16] carried out stability of piezoelectric FGM rectangular p lates subjected to non-uniformly distributed load, heat and voltage. For carrying out analysis element free Galerkin method was emp loyed. In the present work stability analysis of piezolaminated plates is carried out considering higher order shear deformation theory. Piezo laminated plate studied is subjected to combined action of electro mechanical loading. Linear through-the-thickness electric potential distribution is assumed for each piezoelectric sub-layer. Temperature field is assumed to be uniform for the orthotropic layers of the laminate and for p iezolayer. Finite element formu lation considers an isoperimetric eight noded rectangular element. Parametric studies are conducted to demonstrate the influence of boundary condition, ply orientation and plate aspect ratio on stability of p iezolaminated plate with different electric condition of the piezo layer. Nu merical results are presented considering simply supported and clamped piezoelectric laminated plate. 2. Finite Element Formulation Fin ite element formu lation is based on higher order shear deformation theory wh ich do not need shear correction factor as that of first order shear deformation theory. And hence for considering the effect of shear deformation displacement field of HOST is assumed as follows u = u0 + zθ x + z 2u * 0 + z 3θ * x v = v0 + zθ y + z 2v0* + z 3θ * y w = w0 (1) Where, u, v and w are the displacement of any point in the plate domain in x, y and z d irection respectively. u0, v0 and w0 are the d isplacement of midpoint of normal. θx , θ y are the rotations of normal at the middle plane in x and y direction about y and x axis res p ectiv ely . u0*, v0* w0* , θ * x and θ * y are higher order terms which accounts cubic variation of normal. Figure 1. shows a piezo laminated composite plate provided with p iezoelectric patches at top and bottom ssurface of the plate. Fiber can be o riented with reference to the horizontal axes and is modelled in finite element formulat ion. Figure 1. Piezolaminated composite plate provided with piezoelectric patches at top and bottom surface 94 Rajan L. Wankhade et al.: BucklingAnalysis of Piezolaminated Plates Using Higher Order Shear Deformation Theory Strain-Displacement Relations The strains associated with the displacement model as per higher order shear deformation theory are given by ε x = ε x0 + zχ x + z 2ε * x0 + z 3 χ * x ε y = ε y0 + zχ y + z 2ε * y 0 + z 3 χ * y εz =ε z0 + zχ z + z 2ε * z0 + z 3 χ * y γ xy = ε xy0 + zχ xy + z 2ε * xy 0 + z 3 χ * xy γ xz =ϕx + zχ xz + z 2ϕ * x + z 3 χ * xz γ yz = ϕ y + zχ yz + z 2ϕ * y + z 3 χ * yz (2) for i = 1,2,3,4 For middle edge node N5 (ξ ,η) = 1 (1− ξ 2 ) (1−η) 2 N7 (ξ ,η) = 1 (1− ξ 2 ) (1+η) 2 N6 (ξ ,η) = 1 (1+ ξ ) (1−η 2 ) 2 N8 (ξ ,η) = 1 (1− ξ ) (1−η 2 ) 2 (4) Displacement fiel d The displacement field associated with the eight nodded element can be written as Where, [ ] ε x0 ε y0 ε z0 ε xy0 T =    ∂u 0 ∂x ∂v0 ∂y θx ∂u 0 ∂y + ∂v0 ∂x T   = χx χ y χz χxy T    ∂θ x ∂x ∂θ y ∂y 2w0* ∂θ x ∂y + ∂θ y ∂x T   8 ∑ u = Ni ui i =1 8 ∑ v = Ni vi i =1 [ ] ε * x0 ε * y0 ε * xy0 T =   ∂u * 0  ∂x ∂v0* ∂y ∂uo* + ∂v0*   T ∂y ∂x  8 ∑ w = Ni wi (5) i =1 [χ * x χ * y ]χ * xy T =    ∂θ * x ∂x ∂θ * y ∂y ∂θ * x + ∂θ * y T  ∂y ∂x  [ ] φx* φ * y T =  3θ  * x ∂w0* ∂x 3θ * y ∂w0* ∂y T   [ ] ψ x ψ y T =  2u0  + ∂θ z ∂x 2v0 + ∂θ z ∂y T   (3) Shell Element and Shape functions 8-node rectangular element is emp loyed as shown in figure Strain wi thin the Element Strains associated with the displacement field can be written as follow, a. Middle p lane membrane strains ε=p ε L p + ε N p (6)  3×1 in which ε L p and ε N p are linear and non-linear components of middle plane membrane strains and combin ing linear and non-linear terms membrane strains are given as,  ∂u    1  ∂w  2     ∂x   2  ∂x       ∂v ε p =   ∂y     +   1 2  ∂w ∂y  2     (7)  3×1     ∂u + ∂v    ∂y ∂x         ∂w   ∂x . ∂w ∂y        Curvature strains/Bending strains Figure 2. Eight-nodded rectangular element Shape functions are given as, For corner nodes: Ni (ξ ,η) = 1 4 (1 + ξξi ) (1 + ηη i )(−1+ ξξi + ηη i ) , Curvature strains are linearly related to bending displacement as, ε L b = ε  xxb   ε yyb  ε xyb   International Journal of Composite M aterials 2013, 3(4): 92-99 95   ∂θ x    ∂x   =z ∂θ y   = z.K (8)  ∂y    ∂θ x + ∂θ y    ∂y ∂x  Shear strains, γ  2 ×1 = γ γ xz yz    = θ x  θ  y + + ∂w  ∂x ∂w    ∂y  (9) Thus  ε  6×1 = ε p  ε3×b1     3×1 = ε  ε L p L b    + ε  N p  0    (10) Hence, ε =εL+εN  6×1 6×1 6×1 (11) and shear strains can separately be written as, {} γ 2×1 = γ γ xz yz    (12) In which ε p ,{ε b }and {γ } are membrane, bending and shear components of strains respectively. ε  is co mbined 6×1 γ strain vector of memb rane and bending strains. 2×1 is a vector containing shear strains. Subscript ‘p’ stands for in-plane, ‘b’ for bending, ‘L’ for linear and subscript ‘N’ stands for non-linear. Displacement-strain relation Strains are related with displacements using strain displacement matrix as follows, ε  = B δe  (13) Electro-Mechanical Coupling For piezo laminated plates two constitutive relationships exist including the effect of mechanical and electrical loading as given by eq. 14. Variat ion of temperature effect is neglected in formulat ion. {D} = [e]{ε} + [g]{E p } { } {σ} = [C]{ε} − [e]t E p (14) And hence,   σ x'   σ y'  σ z' σ x'y'        =   Q1'1' Q2'1' Q3'1' Q1'4' σ x'z'     0 Q1'2' Q2'2' Q3'2' Q2'4' 0 Q1'3' Q2'3' Q3'3' Q3'4' 0 Q1'4 ' Q2'4' Q3'4' Q4'4' 0 0 0 0 0 Q5'5' 0 0 0           ε x' ε y' ε z'      0   ε x'y'   Q5'6'   ε x'z'   (15) σ y'z'     0 0 0 0 Q6'5' Q6'6'   ε y'z'    0 0 ez1    0 0 −   0 0 0 0 ez 2 ez3 0        Exp'    E p y'      0  ey5 0     Ezp'   ex6 0 0  Where, {D} is electric displacement vector,[e] is dielectric permittiv ity matrix, ε is the strain vector, {g} is the dielectric matrix. {E} is the electric field vector, [σ ] is the stress vector and[C] is the elastic matrix for constant electric fie ld. Electrical Potenti al Function φ ' a and φ ' s are the electric displacement at any point in the actuator and the sensor layers, respectively, the electrical potential functions in terms of the nodal potential vector a re given by [ ] { } φa' = N pa φ e a [ ] { } φs' = N ps φ e s (16) [ ] [ ] Where, N pa and N ps are the shape function matrices for the actuator and sensor layers, respectively. { } { } φae and φ e s are the nodal electric potential vector for the actuator and sensor layers respectively and can be given as follow. { }φ e a ={φa1 φa2 φa3.................φan } T { }φ e s ={φs1 φs2 φs3.................φsn } T (17) Stiffness Matri x Equations Element stiffness matrix can be written as, [ ]{ } [ ] { } [ ] [ ] K e δ e + Kσe δ e = F1e + Faec (18) In which [ ] [ ] [ ] [ ][ ] [ ][ ] [ ] Ke = K e d + K e aa −1 K e aa K e ad + K e ds K e ss −1 K e sd (19) { } [ ][ ] { } Faec = K e da K e aa −1 Qae (20) Where, [ ] ∫ K e d = [B]T [C][B]dV , V [ ] [ ] ∫ K e da = K e ad T = [B]T [e][Ba ]dV , Va 96 Rajan L. Wankhade et al.: BucklingAnalysis of Piezolaminated Plates Using Higher Order Shear Deformation Theory [ ] ∫ K e aa = [Ba ]T [g][Ba ]dV , traction and applied electric charge on actuator the equation for external work done can be written as [ ] [ ] ∫ { }∫ ∫ Va K e ds = K e sd T = [B]T [e][Bs ]dV= {R} Vs A u − T σ− ( x, y) dA +   ϕa' A −e qa ( x, y) dA (25) [ ] ∫ and K e ss = [Bs ]T [g][Bs ]dV Vs (21) Stability equati on The geometric stiffness matrix is associated with the x and Thus the element equation in the global stiffness matrix y coordinates. The characteristic equation of stability is can be written as [ ] [ ] [ ] { } [K]{δ} + Kσ δ = F1 + Fac Equili brium and incremental Equations written as, (22) ([K ] + λ [Kσ ]) {δ} = F (26) ([K ] + λ [Kσ ]) ({δ}+ {dδ}) = {F} (27) Virtual displacement princip le is employed to obtain Where, the lowest magnitude of eigen value g ives crit ical equilibriu m equation. Equ ilibriu m between internal and buckling load and the vector {dδ }represents the buckled external fo rces has to be satisfied. If Ψ represents the mode shape. And the eigen value problem is solved to get vector of the sum of the internal and external forces. buckling loads. {Ψ} = {R} − {P} (23) ([K ] + λ [Kσ ]) {dδ} = 0 (28) Where, {R} represents the external forces due to imposed load and {P} is a vector of internal resisting fo rces. The equilibriu m state is achieved when {Ψ} = 0 . 3. Results and Discussions {Ψ=} {R} + 1 2 ∫ {ε }T V {σ } dV ∫ − 1 2 Eap T{Da} dV Va (24) { } ∫ ∫ − 1 2 Vs Esp T {Ds} dV + V εN T {σ0}dV Where V, Va and Vs are the area of the entire structure, sensor layer and actuator layer respectively. Considering the work done by external forces due to the applied surface Buckling analysis of piezolaminated plates have been carried out for different p ly orientation considering symmetric and antisymmetric cross-ply lamination schemes. Buckling of smart piezo laminated plate is examined subjected to uniform axial co mpressive force at the edges. Furthermore fo r piezo laminated plate, crit ical mechanical buckling loads are obtained for different thickness to span ratios of piezolaminated plates and for closed and open loop electric condition. 3.1. Buckling of Si mply Supported S quare Multilayered Plate (00/900…)n _ Table 1. Non-dimensional critical buckling load ( N = σ crb2 / E2 h2 ) for simply supported multilayered cross ply laminated plate a/h theory Layer 5 present Owen and Li [17] 10 present Owen and Li [17] 20 present 3 Owen and Li [17] 50 present Owen and Li [17] 100 present Owen and Li [17] 5 present Owen and Li [17] 10 present Owen and Li [17] 20 present 5 Owen and Li [17] 50 present Owen and Li [17] 100 present Owen and Li [17] 3 4.5479 4.5597 5.4031 5.4026 5.6578 5.6679 5.7546 5.7521 5.7641 5.7680 4.6011 4.6078 5.4221 5.4208 5.6734 5.6730 5.7527 5.7531 5.7675 5.7683 Degree of Orthotropy ( E1 / E2 ) 10 7.1726 7.1758 9.9543 9.9590 11.0735 11.0727 11.4457 11.4414 11.5081 11.5048 7.5455 7.5445 10.1611 10.1609 11.1364 11.1370 11.4585 11.4531 11.5018 11.5086 20 9.4143 9.4178 15.3227 15.3201 18.3872 18.3879 19.5111 19.5135 19.7069 19.7000 10.2731 10.2727 15.9969 15.9976 18.6426 18.6413 19.5615 19.5605 19.7136 19.7140 30 10.8521 10.8573 19.6863 19.6872 25.2481 25.2476 27.4985 27.4983 27.8731 27.8727 12.1089 12.0834 20.9499 20.9518 25.7910 25.7923 27.6044 27.6037 27.9037 27.9028 40 11.8872 11.8867 23.3341 23.3330 31.6928 31.6949 35.3947 35.3957 36.0231 36.0220 12.8271 12.8344 25.1145 25.2150 32.5842 32.6137 35.4832 35.5819 36.0621 36.0740 International Journal of Composite M aterials 2013, 3(4): 92-99 97 Effect of orthotropy is studied for simply supported square mult ilayered plate subjected to a uniform uniaxial co mpressive force at the edges. Laminated plate is having cross ply orientation of layers. Critical buckling loads are tabulated in Tab le 1 for effect of orthotropy of individual layers, the ratio of span to thickness (a/h) and the number of layers of simply supported square multilayered co mposite plates. Material property adopted is as follow. E1 = 3 to 40, E3 = 1, G12 = G13 = 0.6, E2 E2 E2 E2 G23 E2 = 0.5 and µ12 = µ13 = µ 23 = 0.25 Results are found in good agreement with that of refined fin ite element solution given by Owen and Li [17]. 3.2. Buckling of Si mply Supported S quare Piezoelectric Laminated Plate (p/00/900/900/00/p) The piezoelectric laminate with piezolayer attached at the top and bottom of the plate is subjected to a uniaxial uniform edge compressive force Nx. Hence a square piezoelectric laminated plate of thickness of 0.01 m and side length ‘a’ is having simp ly supported boundary on all four edges. The laminate consists of a (p/00/900/900/00/p) Graphite-Epo xy sublaminate provided with two PZT-5A attached on outer surface of the plate. Each piezoelectric layer has thickness of 0.1h, whereas each elastic layer has a thickness of 0.2h. Elastic and pie zoelectric materia l properties are as shown in table 2. Table 2. Elastic and piezoelectric properties for Graphite-Epoxy and PZT-5 Elastic Properties Properties Grphite-Epoxy E11 (Gpa) 181 E22 (Gpa) 10.3 E33 (Gpa) 10.3 G12 (GPa) 7.17 G23 (GPa) 2.87 G32 (GPa) 7.17 υ12 0.28 υ 23 0.28 υ 32 0.33 PZT -5A 61.0 61.0 53.2 22.6 21.1 21.1 0.35 0.38 0.38 Piezoelectric Properties Prop ert ies d31 (10-12 m/V) d32 (10-12 m/V) d33 (10-12 m/V) d15 (10-12 m/V) d 24 (10-12 m/V) g11 (10-8 F/m) g22 (10-8 F/m) g33 (10-8 F/m) Grphit-Epoxy 0 0 0 0 0 0.0031 0.0027 0.0027 PZT -5A -171 -171 374 584 584 1.53 1.53 1.5 The effect of span to thickness ratio on critical uniaxial buckling load (Nxa2/E2.h3) of simply supported square laminated plate with different electric condition is shown in figure 3. 25 20 15 Ncr 10 Open Loop Closed Loop 5 Without piezo effect 0 0 20 40 60 80 100 a/h Figure 3. Effect of span to thickness ratio on normalized critical buckling load of simply supported symmetric cross ply piezolaminated plate (p/00/900/900/00/p) Fro m figure 3 it can be observed that critical buckling load increases for piezoelectric effect. Also piezoelectricity has litt le effect on the buckling load of the laminates with closed-circuits than that of open loop. There is an increase of 7.9 % in critical buckling load fo r piezo laminated plate than that of without piezoeffect. Hence critica l buckling load can be considerably increases if plate is subjected to piezoeffect. For lesser vaues of a/h ratio % variation in critical buckling load is found to be 3.3 % fo r open loop electric condition than that of closed loop. 3.3. Buckling of Si mply Supported S quare Piezoelectric Laminated Plate (p/00/900/00/900/p) Simp ly supported square piezolaminates with piezolayer attached at the top and bottom of the plate is subjected to a uniaxial uniform edge co mpressive force Nx. Piezoelectric laminated plate is having a th ickness of 0.01 m and side length ‘a’. The laminate consists of antisymmetric p ly orientation of Graphite-Epo xy sublaminate provided with two PZT-5A attached on outer surfaces of the plate. Material properties of Graphite-Epo xy and PZT-5A are same as 98 Rajan L. Wankhade et al.: BucklingAnalysis of Piezolaminated Plates Using Higher Order Shear Deformation Theory previous example. Each piezoelectric layer has thickness of 0.1h, whereas each elastic layer has a thickness of 0.2h. Figure 4 shows effect of span to thickness ratio on critical mechanical buckling load (Nxa2/E2.h3) of piezo laminated plate with different electric conditions. 25 20 15 10 Nc r 5 Open Loop Closed Loop 0 0 50 100 150 a/h Figure 4. Effect of span to thickness ratio on normalized critical buckling load of simply supported antisymmetric cross ply piezolaminated plate (p/00/900/00/900/p) Significant increase in critical buckling load is observed for open loop electric condition than that of closed loop. Thus piezoelectricity has little effect on the buckling load of the laminates with closed-circuits in this case. The increase in crit ical buckling load is found to be 8.09 %, 11.05 % and 10.19 % for open loop circu it than that of closed loop for a/h = 10, 20 and 100 respectively. 4. Conclusions A fin ite isoparametric element for the mechanical displacement field is co mbined with an electric potential field to consider piezoelectric effect. Parametric study is conducted to demonstrate the influence of boundary condition, ply orientation and plate aspect ratio on stability of piezo laminated plate with d ifferent electric condition of the piezo layer. For sy mmetric cross ply piezo laminated plate % increase in crit ical buckling load is found to be 7.9 % than that of without piezoeffect. Thus considering piezoeffect bucklibg load can be considerably increased. Appendix: Stiffness Coefficients of the Laminated Plate According to The Higher Order Shear Deformation Theory The stiffness coefficients according to the HOST are as follows Q11 = C11 cos 4 θ + 2 (C12 + 2C33 )sin 2 θ cos 2 θ + C22 sin 4 θ ( ) Q12 = C12 cos4 θ + sin 4 θ + (C11 + C22 − 4C33 )sin 2 θ cos2 θ ( ) ( ) Q13 = C11 − C12 − 2C33 sinθ cos3 θ + C12 − C22 + 2C33 cosθ sin3 θ ( ) Q2=2 C11 sin4 θ + 2 C12 + 2C33 sin2 θ cos2 θ + C22 cos4 θ Q23 = (C11 − C12 − 2C33 )sin 3 θ cosθ + (C12 − C22 + 2C33 )cos3 θ sin θ Q33 = (C11 − 2C12 + C22 − 2C33 )sin 2 θ cos 2 θ ( ) + C33 cos4 θ + sin 4 θ Q44 = C44 cos2 θ + C55 sin 2 θ Q45 = (C44 − C55 )sin θ cosθ Q55 = C44 sin 2 θ + C55 cos2 θ Qij = Q ji for i, j = 1,2,…5 ( ) Where, C x'x' = Ex' 1 −υ y'z'υz'y' υ* ; ( ) C y'y' = E y' 1−υ x'z'υ z'x' υ* ; ( ) υυ υ C = E 1 − z'z' z' ; x' y' y'x' * ( ) Cx'y' = Ex' υ y'x' +υ z'x'υ y'z' ; υ* ( ) Cx'z' = Ex' υz'x' +υ y'x'υz'y' ; υ* ( ) C y'z' = E y' υ z'y' +υ x'y'υ z'x' υ* G x' y' = C x'' y' k ; Gz'x' = C z'x' k ; Gy'z' = C y'z' k ; And ( ) υ* = 1− υx'y'υ y'x' +υ y'z'υz'y' +υz'x'υx'z' + 2υ y'x'υz'y'υx'z' REFERENCES [1] D. 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