STATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION

Transcription

1 STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros H-355 Hungary e-mal: Abstract The object of the present paper s to analyse the statc behavour of the pezoelectrc two-layered beams wth nterlayer slp The Euler-Bernoull hypothess s assumed to hold for each layer and a lnear consttutve equaton between the horzontal slp and the nterlamnar shear force s consdered For each beam component the appled electrc feld s constant and the prescrbed mechancal loads are not present INTRODUCTION Pezoelectrc beams are commonly used n many engneerng applcatons such as control elements and transducers Pezoelectrc beams n unmorph and bmorph confguratons n partcular are wdely used for systems where actuaton and/or sensng are needed [] The present paper deals wth the statc analyss of twolayered pezoelectrc beams wth nterlayer slp The connecton of the beam components n aal drecton s weak but n normal drecton s perfect There s no separaton n normal drecton between the beam components The nterlayer slp n aal drecton s defned on the common boundary of beam components as the dfference of the aal components of the dsplacement feld The consdered twolayered pezoelectrc beam confguraton s shown n Fg The cross-secton of beam component B s a rectangle A of whch sdes are b h ( ) The centre of the cross-secton A s C ( ) and the Y -weghted centre of the whole crosssecton s C as shown n Fg The Young modulus of beam component B s denoted by Y ( ) From Fg t follows that c CC c c CC c c c c A Y A hb A h b () () Accordng to the Euler-Bernoull beam theory the dsplacements are as follows [3] dw u u ( ) z v w w( ) ( y z) B ( ) (3)

2 constant Fgure Two-layered pezoelectrc beam wth weak shear connecton where u v and w are the dsplacements n y and z drectons The net consttutve equaton wll be used Y e3 Ez (4) Here s the normal stress s the normal stran E s the z component of the z electrc feld vector and e s the pezoelectrc constant of the thckness polarzed 3 beam component E n terms of potental z s obtaned as (Fg ) E E (5) z z h h n beam components B and B The top and bottom surfaces of pezoelectrc layers are metalzed Net we use e3 e ( ) By the use of Eqs (4) (5) the epresson of the normal stress can be gven as du d w Y z e ( ) ( ) y z B h (6) We defne the followng stress resultants By smple computatons we obtan (7) N d A M z d A ( ) A A du d w N ( ) c eb ( ) (8) du d w M ( ) c IY ( ) e cb ( ) (9)

3 Here ( ) There s no appled mechancal load n aal drecton I z da A ths means that N N N () The nterlayer slp s s obtaned as the dfference of aal component of dsplacements defned on the common boundary of beam components B and B that s s( ) u ( ) u ( ) () From Eqs () and () we have du A Y ds b e e () du ds b e e Substtutng of Eqs () (3) nto Eq (8) gves (3) Here N c c e c e c ds d w b N c c e c e c ds d w b A Y (4) (5) (6) From Eqs (9) () (3) we can derve the epresson of the total bendng moment actng on the whole cross-secton A A A as ds d w M M M c IY ec ec b IY I Y IY (7) The whole cross-secton s loaded by the shear force V whch s d d d V c IY (8) 3 M s w 3 The nterlayer shear force S actng n aal drecton s a lnear functon of the slp ts epresson s S ks (9)

4 where k s the slp modulus [3] The force equlbrum equaton for beam component B can be formulated as 3 dn d s d w ks c ks 3 () Combnaton of Eq (8) wth Eq () yelds where d s V IY s c () IY k IY IY c () IY In problem shown n Fg V and s() thus we have for s s( ) s( ) K snh (3) The constant K s obtaned from the net boundary condton (Fg ) DETERMINATION OF SLIP AND DEFLECTION On the whole beam M ( ) From ths equaton we get Integraton of Eq (5) provdes N ( L) (4) d w ds IY c c e ce b (5) that s dw dw IY IY c s( ) s() b ce ce (6) dw IY c s( ) b ce ce (7) snce dw (Fg ) A repeated ntegraton gves the result

5 cosh ( ) IYw ck bc e c e (8) snce w() Lengthy but elementary computatons yeld IY ds b N( L) ce ce IY L c (9) Combnaton of Eq (3) wth Eq (9) leads to the formula of K b c e K c L c e cosh (3) The fnal epresson of the slp functon can be wrtten n the net form s b c e c e snh c cosh L ( ) (3) Substtutng K nto equaton (8) provdes the deflecton of two-layered pezoelectrc beam n terms of appled voltage b( ce ce ) cosh w( ) IY cosh L (3) c For bmorph beam h h c c e e e we have eb snh s( ) cosh L ecb cosh w( ) IY coshl (33) (34) 3 NUMERICAL EXAMPLE The followng data have been used n numercal computatons: h h b 5 m L 5 m Y Y 5 GPa e e 76 N/mV 4 The slp functons and deflecton functons for k Pa k 6 8 k k m V Pa 3 Pa 4 Pa k5 Pa are determned The graphs of slp and deflecton functons for k k k 3 k and 4 k are shown n Fg and Fg 3 5 respectvely

6 Fgure Plots of slp functons Fgure 3 Plots of deflecton functons 4 CONCLUSIONS Ths paper presents an analytcal soluton for two-layered pezoelectrc cantlever beam wth mperfect shear connecton Smple formulas are derved to obtan the deflecton and slp functons An eample llustrates the applcaton of derved formulas and the effect of slp modulus to the deformaton and slp ACKNOWLEDGEMENTS Ths research was supported by the Natonal Research Development and Innovaton Offce NKFIH K57 REFERENCES [] YANG J: The Mechancs of Pezoelectrc Structures World Scentfc Publshng New Jersey 6 [] YANG J: An Introducton to the Theory of Pezoelecrcty Sprnger Berln 6 [3] ECSEDI I BAKSA A: Statc analyss of composte beams wth weak shear connecton Appled Mathematcal Modellng 35(4)