Bending analysis of a functionally graded piezoelectric cantilever beam

Transcription

1 Science in Cina Series G: Pysics Mecanics & Astronomy 7 Science in Cina Press Springer-Verlag Bending analysis of a functionally graded pieoelectric cantilever beam YU Tao & ZHONG Zeng Scool of Aerospace Engineering and Applied Mecanics Tongji University Sangai 9 Cina A new analysis based on Airy stress function metod is presented for a functionally graded pieoelectric material cantilever beam. Assuming tat te mecanical and electric properties of te material ave te same variations along te tickness direction a two-dimensional plane elasticity solution is obtained for te coupling electroelastic fields of te beam under different loadings. Tis solution will be useful in analying FGPM beam wit arbitrary variations of material properties. Te influences of te functionally graded material properties on te structural response of te beam subjected to different loads are also studied troug numerical eamples. functionally graded pieoelectric material cantilever beam bending electroelastic fields Pieoelectric materials ave been widely used as actuators and sensors in smart structures and systems for teir effective capability of conversion between mecanical and electric fields []. Pieoelectric actuators are often made by bonding pieoelectric seets to get better performance but te bonding will cause severe interfacial stress concentration and trigger te initiation and propagation of micro-cracks near te interface wic may lead to failure of te devices. Fortunately functionally graded materials (FGMs) in wic te volume fractions of two or more materials are varied continuously wit te position along certain directions ave been developed []. In recent years anoter attempt of introducing FGMs concepts to pieoelectric materials to make functionally graded pieoelectric materials (FGPMs) as arvested a lot of fruits [4 6]. Te advantage of FGPMs is tat no discernible internal boundaries eist and failures from interfacial stress concentrations developed in conventional bimorps can be avoided. Te gradual cange of material properties can be tailored to different applications and working environments wic makes FGPMs preferable in many applications. As te use of FGPMs increases new metodologies ave to be developed to caracterie tem and to design and analye structural components made wit tese materials. To te best of our knowledge only Si and is coworkers [7 9] studied te response of FGPM beams. But in teir Received December 5; accepted June 6 doi:.7/s Corresponding autor ( ongk@mail.tongji.edu.cn) Supported by te National Natural Science Foundation of Cina (Grant Nos. 4 and 59) Sci Cina-Pys Mec Astron Feb 7 vol. 5 no. 97-8

2 analysis only one or two material parameters were assumed to vary in a power fasion along te tickness direction wile oter parameters kept constant. Te literatures on te response of FGM beam to mecanical and oter loadings are also limited in numbers. In a series of papers Sankar and is co-workers [ ] developed analytical metods for te termomecanical and contact analysis of FGM beams and also for sandwic beams wit FG cores. In tese studies te termomecanical properties of te FGM were all assumed to vary troug te tickness in an eponential fasion. Zu and Sankar [] solved te two-dimensional elasticity equations for a FGM beam subjected to transverse loads using te metod of Fourier analysis combined wit te Galerkin metod in wic te variation of te Young s modulus troug te tickness was given by a polynomial in te tickness coordinate and te Poisson s ratio was assumed to be constant. A new beam element based on te first order sear deformation teory was developed to study te termoelastic beavior of FGM beam structures by Cakraborty and Gopalakrisnan [45]. In tose papers eiter eponential or power-law variations of material properties were used. For FGM beam te present autors [67] ave obtained some analytical solutions adaptive for any kind of material variations. But as far as we know no analytical solutions ave been obtained for FGPM beam wit any kind of variations in material properties. Te objective of tis work is to present a general solution of a clamped functionally graded pieoelectric cantilever beam wit arbitrary graded variations of material properties based on electroelasticity teory. Te influences of different graded function on te structure response of te beam are analyed troug numerical eamples. Summary of te problem and basic equations Figure sows a FGPM cantilever beam of uniform tickness. Te beam is subjected to a uniform pressure q on its top surface a concentrated sear force P and a concentrated moment M at its left end. A Cartesian coordinate system is introduced wit te bottom and top surfaces of te beam lying in te plane = / and = /. Te lengts of te edges of te beam in and y directions are denoted by l and b respectively. In contrast to a omogeneous pieoelectric material all te mecanical and electric material constants are now functions of coordinates. In most of real cases te material properties are varied continuously only in one direction. In te present paper we consider te distribution of material properties as follows: sij = sf ij ( ) dij = dij F( ) λ ij = λ ij F ( ) () were sij dij λ ij are te elastic pieoelectric and dielectric constants respectively; s ij dij λ ij are te corresponding values at te plane were F( ) = ; and F( ) is te graded function. Figure A scematic of clamped functionally graded pieoelectric cantilever beam. 98 YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

3 Generally wen te sie in y-direction of te tree dimensional pieoelectric medium is muc smaller tan te sie in - and -directions as well as all loads are independent of y-coordinate te problem can be simplified as a plane stress problem in te - plane. Similarly te problem can be simplified as a plane strain problem in te - plane if te sie in y-direction of te tree dimensional pieoelectric medium is muc larger tan te sie in - and -directions and all loads are independent of y-coordinate. Here we only consider te plane strain problem since te plane stress solution is easy to obtain using a similar approac. In te absence of body forces and free electric carges te electric and mecanical equilibrium equations are D D + = () and σ τ + = () τ σ + =. Te constitutive relations of FGPM are ε s s σ d E ε s s σ d = + (4) E γ s44 τ d5 σ D d5 λ E = σ D d d + λ E τ. (5) Te relationsips between te strains and te elastic displacements are as follows: u w u w ε = ε = γ = + (6) and te electric fields relate to te electric potential troug te following relations: Φ Φ E = E =. (7) Te strain compatibility equation is ε ε For a cantilever beam te edge conditions are given as follows: Te mecanical boundary conditions: γ + =. (8) P τ d = M σd = σd = b b σ = τ = at at = ; (9) σ = q τ = at = ; () =. () YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no

4 Te electric boundary conditions: D d = at = and = l; () D = at = ±. () Te displacement boundary conditions at te clamped end: w u = w= = at = = l. (4) In eqs. () (4) uw are displacement components; σ σ τ are stress components; y ε εy γ are strain components; E E are electric field components; D D are electric displacement components; and Φ is electric potential. Stress function metod Now Airy stress function U( ) is introduced wit U U σ = σ = τ =. (5) Substituting eqs. (5) (7) and (5) into electric equilibrium equation () we obtain U Φ U U Φ d5 λ + d d + λ =. (6) Substituting eqs. (4) (7) and (5) into strain compatibility equation (8) we ave U 4 4 U U U U U s + s + s44 s + + s 4 Φ Φ Φ = d + d d 5. (7) Suppose U = f( ) + f( ) + f( ) (8) Φ = f ( ) + f ( ) + f ( ). (9) 4 5 For simplification we introduce te following symbols: i d Hi( ) = H ( ) ( )d i = H ( ) i F H ( ) = H ( )d H ( ) = H ( ) d ( i = 4) i i i i T ( ) = H ( ) F( )d T ( ) = H ( ) F( )d YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

5 G ( ) = F( )d G ( ) = F( ) d G ( ) G ( )d = G( ) G( ) G ( ) G( ) = d G ( ) = d G ( ) d F( ) = F( ) F( ) T ( ) = T ( )d T ( ) = T ( )d T ( ) T ( ) N( ) = d N ( ) = d F( ) F( ) T ( ) T ( ) N( ) = d N ( ) = d F( ) F( ) Y ( ) = H ( ) F( )d Y ( ) = H ( ) F( )d. Substituting eqs. (8) and (9) into eqs. (6) and (7) we obtain te following relations: k = d f C s d f( ) d ( ) = λ + d d k d f( ) d f( ) C4 C5 = d + + d d k k d f( ) d f4( ) C d = λ + d d k s d d f( ) d f4( ) C C = + + d d k k d f( ) d d d f( ) d d f5( ) d5 λ f( ) + [ d f( )] + d = d d d d d λ d d d f( ) d d f( ) d f( ) s + s f( ) + s44 + s d d d d d were sλ ( d) d d f ( ) d f ( ) d = d d d f d d + d d 5 [ 5. Solving eqs. () () we can obtain f ( ) f ( ) f ( ) f ( ) f 4 ( ) f 5 ( ). Ten te ( )] stresses strains electric fields and electric displacements can be derived as { [ λ ( ) ( λ ) ( ) ]} () () () σ = C4λ + ( C5λ Cd) C C Cd) F( ) F( ) + λ + ( λ F( ) F( ) + k a C H + C C d H + C + C YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

6 a[ C4λT( ) + ( C5λ Cd) T( ) + C7G( )] a[ C4dT ( ) + ( C5d Cs) T ( ) + C6G ( )] k + F( ) sλ[ C4λ + ( C5λ Cd) ] () + dλ[ C4d + ( C5d Cs) ] + λ( C+ C4 ) C d σ = C λ H ( ) + ( C λ C d ) H ( ) + C + C (4) ( 5 7 τ = [ C λ H ( ) + C λ C d ) H ( ) + C ] λ ( λ [ C H ( ) + C C d ) H ( ) + C ] ( 4 5) ( ) 44[ 4 ( ) ( 5 ) ( ) 7 k ε = C+ C + C+ C s Cλ T + Cλ Cd T + CG( )] k d [ C d T ( ) + ( C d C s ) T ( ) + C G ( )] s C4λ + ( C5λ Cd) + d C4d + ( Cd 5 Cs ) + C+ C4 ε = [ a( C4+ C5) Ca4] + [ a( C+ C) Ca4] + ( s s a k + d a k ) F( )[ C H ( ) + ( C C d ) H ( ) + C + C ] 4 4λ 5λ 7 8 ( sa + da5 )[ C4λT ( ) + ( C5λ Cd) T ( ) + C7G ( )] + ( sa + da6 )[ C4dT( ) + ( C5d Cs) T( ) + C6G( )] + k + [ ( s) λ + sdd] C4λ + ( C5λ Cd) + [ sdλ ( d) d] C4d + ( C5d Cs) + a ( C+ C4 ) C a 4 (7) γ s44[ C4λH( ) + ( C5λ Cd) H ( ) + C7 ] = F( ) + d5[ C4dH( ) + ( C5d Cs) H( ) + C6 ] s44[ CλH( ) + ( Cλ Cd) H ( ) + C ] F( ) + d5[ CdH( ) + ( Cd Cs ) H( ) + C9 ] 4 ( 5 6 E = [ Cd H() + Cd Cs ) H () + C] 9 (5) (6) (8) [ Cd H( ) + ( Cd Cs ) H ( ) + C] (9) E = C4d + ( C5d Cs) Cd Cd Cs) F( ) F( ) + ( F( ) F( ) k a C H + C C d H + C + C { [ λ ( ) ( λ ) ( ) ]} YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

7 were a5[ C4λT ( ) + ( C5λ Cd) T ( ) + C7G ( )] a6[ C4dT( ) + ( C5d Cs) T( ) + C6G( )] k sd C4λ + ( C5λ Cd) F( ) + d d C d + C d C s + d C + C 4 ( 5 ) ( 4 5 4λ ( 5λ 7 D = d F( )[ C H ( ) + C C d ) H ( ) + C ] 4 ( λ F( )[ C d H ( ) + C d C s ) H ( ) C ] 5 λ ( λ + d F( )[ C H ( ) + C C d ) H ( ) C ] F( )[ CdH( ) ( Cd λ + 9 Cs ) H ( ) + C] Cd 4 T Cd 5 Cs T CG 6 C ) C s C C D = + + d5[ C4λT ( ) + ( C5λ Cd) T ( ) + C7G ( )] k k () () + λ [ ( ) + ( ) ( ) + ( )] + () Integrating eq. (6) to we obtain = λ = 44λ + 5 = 5λ + λ a s d d a s d d a d d 4 = 5 = = 5 + λ a s d d s a s d d s a d d s u = ( C + C ) + ( C + C ) + g ( ) k 4 5 k s44[ C4λT ( ) + ( C5λ Cd) T ( ) + C7G ( )] + d5[ Cd 4 T ( ) + ( Cd 5 Cs ) T ( ) + CG 6 ( )] + C+ C4 s C4λ + ( C5λ Cd) + d C4d + ( Cd 5 Cs ). () Integrating eq. (7) to we obtain w= [ a( C4 + C5) Ca4] + [ a( C + C) Ca4] + g( ) 4 4λ 5λ ( s s a k + d a k )[ C Y ( ) + ( C C d ) Y ( ) + C G ( ) + C G ( )] ( sa + da5 )[ C4λT ( ) + ( C5λ Cd) T ( ) + C7G ( )] + ( sa + da6 )[ Cd 4 T ( ) + ( Cd 5 Cs ) T ( ) + CG 6 ( )] + k 4 + [ ( s ) λ + sdd] C4λ + ( C5λ Cd) a C C (4) 4 + [ sdλ ( d) d] C4d + ( C5d Cs ) Ca4 Substituting eqs. (8) () and (4) into te tird equation of eq. (6) we can determine. YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

8 4 g ( ) = a C C C + d (5) 6k k g( ) = a a C + C Ca4 6 s44[ CλT ( ) + ( Cλ Cd) T ( ) + CG ( )] + e. + d5[ CdT ( ) + ( Cd Cs) T ( ) + C9G ( )] Te coefficient in () (6) can be obtained from te boundary conditions (9) (4) sown in Appendi. Numerical eamples and discussion In tis section numerical study of FGPM cantilever beam (l = m =. m) will be made based on te above solution. Te material properties at = are tose of PZT-4 [8] as sown in Table. We assume te graded function F( ) = ep( α / ) were α is te graded inde. In te present study we take te graded inde for tree values:. Table Material properties of PZT-4 Elastic constant ( m /N) 4 Pieoelectric constant ( C/N) Dielectric constant ( 8 F/m) s s s s 44 d d d 5 λ λ (6) Figure sows te variation of pysical quantities wit coordinate wen te cantilever beam bears te uniform pressure at te upper surface ( P = M = q = N / m ). It can be observed tat te displacement w stays nearly uniform along te tickness direction. All te pysical variables of functionally graded materials ( α ) are offset from tose of omogenous materials ( α = ). At te same time curves of functionally graded materials deviate farter from tat of te omogenous materials wit te increase of graded inde. Te stress σ in te FGPM beam is nonlinearly distributed along te tickness direction and te displacement u stays nearly linear along te tickness direction but does not vanis at = wen te graded inde α. Te normal stress σ and te electric displacement D are muc smaller tan oter components of stress and electric displacement respectively. For simplification one can neglect tem in te practical application. Tese observed caracteristics can be used in establising a simplified FGPM beam teory. 4 Concluding remarks An analytical solution is developed for te two dimensional bending problem of a functionally graded pieoelectric material cantilever beam by means of te semi-inverse metod. Assuming tat mecanical and electric properties of te material ave te same dependence on te tickness coordinate we obtained te coupling electroelastic fields of te FGPM beam under pressure on te upper surface as well as te applied concentrated force and couple at te free end. Te obtained solution is valid for arbitrary graded variations of material properties and ence it could serve as a 4 YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

9 Figure Variation of pysical quantities at a FGPM cantilever beam wit coordinate under uniform pressure at te upper surface. (a) Displacement at free end u( ); (b) displacement at free end w( ); (c) stress at clamped end σ ( ); (d) stress at clamped end σ ( ); (e) stress at clamped end τ ( ); (f) electric displacement at clamped end D ( ); (g) electric displacement at clamped end D ( ). YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no

10 basis for establising simplified FGPM teories or as a bencmark result to assess oter approimate metodologies. Te metod developed in te present study can be applied to oter FGPM beams under different boundary conditions. Appendi Te coefficient in () (6) can be obtained from te boundary conditions (9) (4) as follows. From boundary conditions () and () and te first equation of eq. (9) we can derive P q A= H H bk B = H H k P q C = H H bλk C4 = H H λk q C7 = H H H H k q C8 = H H H [ H H H k + H H H H were P C = H H H H bk k = H H H H + H H H H + H H H H. Using boundary conditions () and () we obtain A B C = C = C = C 5 = λ λ d5λ + dλ C6 = C4 T T + C5 T T λ G G 5 d C λ 5 7 d5λ + dλ C9 = C T T + C T T λ G G d C λ (A-) (A-) 6 YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8

11 C = d5 C4λT + C5λT + C7G λ Cd 4 T + Cd 5 T + CG 6. (A-) From boundary condition (9) we ave C C M A H H A H H H H bk = λk M A H H + A H H + H H bk 4 = λk (A-4) were A = a C4λH H C5 H H C8 + λ + + a C4λN N + C5λ N N + C7 G G + a C4dN N C5d N N C6 G G sλ C4λ H H + C5λ H H dλ C4d H H C5dH H + + Cd H H A = a C4λH H + C5λ H H + C7 + a C4λN N + C5λ N N + C7 G G + a C4dN N C5d N N C6 G G sλ C4λ H4 H4 + C5λ H H dλ C4d H4 H4 C5d H H + + Cd H H YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no

12 k = H H H H H( ) H. Using boundary conditions (4) we obtain l l a = C l + C + C 4 k k 4 4λ 5λ 7 8 d = ( s s a k + d a k )[ C Y () + C Y () + C G () + C G ()] ( sa + da5)[ C4λT() + C5λT() + C7G()] k + ( sa + da6 )[ C4dT () + C5dT () + C6G ()] C l C l 4 C 4 k 4k l (A-5) l l s44[ C4λT () + C5λT () + C7G ()] e= C5 C + l k k + d5[ C4dT () + C5dT () + C6G ()] C4 s44[ CλT () + CλT () + CG ()] +. (A-6) + d5[ CdT () + CdT () + C9G ()] Rao S S Sunar M. Pieoelectricity and its use in disturbance sensing and control of fleible structures: A survey. Appl Mec Rev : Tani J Takagi T Qiu J. Intelligent material systems: Application of functional materials. Appl Mec Rev 998 5(8): 55 5 Koiumi M. FGM activities in Japan. Compos Pt B-Eng 997 8: 4[DOI] 4 Zu X H Meng Z Y. Operational principle fabrication and displacement caracteristic of a functionally gradient pieoelectric ceramic actuator. Sens Actuator : 69 76[DOI] 5 Wu C C M Kan M Moy W. Pieoelectric ceramics wit functional gradients: A new application in material design. J Am Ceram Soc : Selley II et al. Functionally graded pieoelectric ceramics. Mater Sci Forum 999: Si Z F. General solution of a density functionally gradient pieoelectric cantilever and its applications. Smart Mater Struct (): 9[DOI] 8 Zang L N Si Z F. Analytical solution of a simply supported pieoelectric beam subjected to a uniformly distributed loading. Appl Mat Mec 4(): 5 [DOI] 9 Si Z F Cen Y. Functionally graded pieoelectric cantilever beam under load. Arc Appl Mec 4 74: 7 47[DOI] Sankar B V. An elasticity solution for functionally graded beams. Compos Sci Tecnol 6: [DOI] Sankar B V Tang J T. Termal stresses in functionally graded beams. AIAA J 4(6): 8 Venkataraman S Sankar B V. Elasticity solution for stresses in a sandwic beam wit functionally graded core. AIAA J 4(): 5 55 Zu H Sankar B V. A combined Fourier series Galerkin metod for te analysis of functionally graded beams. J Appl Mec-T ASME 4 7(): Cakraborty A Gopalakrisnan S. A spectrally formulated finite element for wave propagation analysis in functionally graded beams. Int J Solids Struct 4: 4 448[DOI] 5 Cakraborty A Gopalakrisnan S Reddy J N. A new beam finite element for te analysis of functionally graded materials. Int J Mec Sci 45: 59 59[DOI] 6 Yu T Zong Z. General solution of a clamped functionally graded cantilever-beam under uniform pressure. Acta Mecanica Solida Sinica (in Cinese) 6 7(): 5 7 Zong Z Yu T. Analytical bending solution of functionally graded cantilever-beam. J Tongji Univ (in Cinese) 6 4(4): Ruan X P Danfort S C Safari A et al. Saint-Venant end effects in pieoceramic materials. Int J Solids Struct 7: 65 67[DOI] 8 YU Tao et al. Sci Cina Ser G-Pys Mec Astron February 7 vol. 5 no. 97-8