Revista EIA, ISSN 1194-1237 Número 2 p.75-88. Agosto 2004Escuela de Ingeniería de Antioquia, Medellín (Colombia)

FRICTIONLE§§ CONTACT IN A LAYEREDPIEZOELECTRIC MEDIUM COMPOSED OF

MATERIAL§ WITH HEXAGONAL SYMMETRYGunlenuo RAum¡2.

AB§TRACTA matrix formulation is presented for the solution of frictionless contact problems on arbitrarily multilayered

piezoelectric half-planes. Different arrangements of elastic and tranwersely orthotropic piezoelectric materials within themultilayered medium are considered. A generalized plane deformation is used to obtain the goveming equilibrium equa-tions for each indiüdual layer. These equations are solved using the infinite Fourier transform technique. The problem isthen reformulated using the locaVglobal stiftress method, in r¡üich a local stiftress matrix relating the stresses and electricdisplacement to the mechanical displacements and electric potential in the transformed domain is formulated for each layer.

Then it is assembled into a global stiftress matrix for the entire half-plane by enforcing interfacial continuity of tractions anddisplacements. This locaVglobal stiftress approach not only eliminates the neces§ty of explicitly finding üe unknownFourier coefficients, but also allows the use of efficient numerical algorithms, many of which have been developed for finiteelement analysis. Unlike finite element methods, the present approach requires minimal input. Application of the mixedboundary conditions reduces the problem to an integral equation. This integral equation is numerically solved for theunknown contact pressure using a technique based on the Chebyshev polynomials.

KEY WORDS : piezoelectricity; contact pressure; multilayered half-plane.

RE§UMENSe presenta una formulación matricial para la solución de problemas de contacto sin fricción en semiplanos

piezoeléctricos elásticos de múltiples capas. Se consideran diferentes disposiciones de materiales piezoeléctricos elásticosy tranwersalmente ortotrópicos dentro del medio de múltiples capas. Se usa una deformación de plano generalizadapara obtener las ecuaciones gobemantes de equilibrio para cada capa individual, que se resuelven con la técnica detransformada de Fourier infinita. Entonces el problema se reformula con el método de rigidez locaVglobal, en el cual se

formula para cada capa una matriz de rigidez local que relaciona los esfrrerzos y el desplazamiento eléctrico con losdesplazamientos mecánicos y el potencial eléctrico en el dominio transformado. En seguida se ensambla en una matriz dengidez global para todo el semiplano imponiendo la continuidad interfacial de tracciones y desplazamientos. Este

enfoque por rigidez locaVglobal no sólo elimina la necesidad de hallar explícitamente los coeficientes de Fourierdesconocidos, sino que también permite el uso de algoritmos numéricos eficientes, muchos de los cuales se desarrollaronpara análisis por elementos finitos. A diferencia de los métodos de elementos finitos, este enfoque requiere una entradaminíma. El uso de condiciones de borde mezcladas reduce el problema a una ecuación integral, que se resuelve para lapresión de contacto desconocida con una técnica basada en los polinomios de Chebyshev.

PALABMS CL,AVE: piezoelectricidad; presión de contacto; semiplano de varias capas.

* lngeniero CiüI. Ph. D. Department of Ciül and Enüronmental Engineering. Tennessee Technological University.Cookeülle, Tennessee 38505

Anículo recibido l9-1V-2O04. Aprobado con reüsión 3-Vlll-2004.Discusión abierta hasta enero 2005.

FNICTIONTgSS CONTACT IN A LAYERED PIEzoELECTRIC MEDIUM CoMPoSED oF MATERIALS wiTH HExAGoNAL SYMMETRY

§. xM§&üm§_Ít§'§*N

The study and analysis of layered smart media(also referred to as intelligent or adaptive) have re-ceived considerable attention in recent years. This isin part because of their potential applicability as sen-

sors and actuators for sensing and controlling the re-sponse of stmctures to a changing enüronment. These

materials have the ability of converting energy fromone form (among magnetic, electric, and mechanicaleneryies) to other. As with other deformable solids, thephenomenon of contact can proüde critical informa-tion related to design and performance of structuralcomponents constructed in part using these materials.The use of piezoelectric lavers or elements as a pres-

sure sensor is one of many types of applications for thistype of problem.ln this context, the sensor applicationcould thenbe combinedwith the actuator applicationto design an active loop to control the deformationand shape of the host structure subjected to contactpressure.

Contact problems in elasticityhave a large andvaried history as seen, for example, in Gladwell. Theextensions to the effects of piezoelectric materials,however, have not yet been made to any great ex-tent. In the present work, we apply the locaVglobaltechnique originally developed for elastic contactproblems by Pindera and Lane in 1993, in which theresponse of layered half-planes was analyzed withinthe framework of the generalized plane deformationformulation. The locaVglobal procedure isbased onthe transfer and flexibility matrix formulation present-

ed by Bufler in 1971 for isotropic layered media withconstant elastic properties, later reformulated interms of the local stiftress matrix by Rowe and Book-er in 1982 and applied to non-homogeneous isotro-pic layered soils. The locaVglobal stiffness approachfacilitates decomposition of the integral equation forthe contact stress distribution on the top surface ofthe layered half-plane into singular and regular parts,

that in turn, can be numerically solved by using thecollocation technique outlined by Erdogan.

In this study, we applya similar type of formula-tion to consider the influence of contact pressure oncontact length for typical piezoelectric solids, anddetermine typical surface and sub-surface distribu-tions for the electric fields under the conditions offrictionless contact.

2. FOR&,¿§UT-&TIüN

?.§ Seor*etry

A multilayered medium composed of an arbi-trary arrangement of elastic and/or piezoelectric lami-nae bonded to a homogeneous elastic or piezoelec-

tric half-plane is considered in the present study. Thelayered medium is infinite in the xy-plane and the load-ing on the top surface is aszumed to be independent ofthe y-coordinate, allowing the problem to be formulat-ed as a planar problem in the xy-coordinate system

that it is also infinite . The variable s u, v , and w representthe mechanical displacements in the x-, y-, and z- di-rections, respectively, and / represents the eiectrostat-ic potential. The assemblage isindentedbya rigid, fric-tionless punch of a parabolic profile. The correspond-ing conñguration is shown in Figure I.

-o

Figure l. Geometry and layer configuration

*"1/$l §.evista EIA

2.X #overmimg eqwmtioms

The theory of linear piezoelectricity has been used throughout this study. Accordingly, the constitutiveequations that describe the electro-mechanical coupling phenomenon are given by

or=CuwS*,-€rjEo

D, = e,o,So, + €ikEk (I)

where o- are the components of the stress tensor, ewarethe components of the piezoelectric constants of thesolid, ¿- are the components of the dielectric tensor, Cro, are the components of the elastic stifhess tensor, Eo

represent the electric ñeld, and So are the components of the inf,nitesimal strain tensor.

The equations of equilibrium and the Gauss equation of electricityin the absence of electric charges andcurrent densities are given by

and

o,i,i I f, =0

Dl,:0,

(2)

(3)

above equations are reduced for an orthotropic

respectively. Here f are the components of the bodyforce and D represent the electric displacement compo-neñts. Auxiliary equations are needed to relate the strain and the electric fields to the displacement and electro-static potential ñelds, respectively. Those relations are given by the mechanical strain-displacement equations

,, = )(r,,, + u,.,) = +l#. *)

and the electric field-electrostatic potential equations

E, =-ú.,=-yOX,

@)

(s)

where il represent the displacement components (¿l in the x-, v in they-, and w inthe z- directions) and Qrepresents the electric potential.

The equations of equilibrium and conservation of electrostatic charge (Gauss equation of electriciry) foran orthotropic material oriented off-axis maybe expressed in terms of the mechanical displacements and theelectrostatic potential as

Ctru,rr * Crsu,u + Cruv,o r C+rr,o + (C* + Crr)w,o i- (et, * err)Q,*, = 0

C ruu,,* * C qru,* t C uur,o * C oot,* + (C * + C *)w,,, * (e ru -t ero)Q,,, = 0

(C,, + C rr)r,o + {C ru + C *)v,* + C rrw,o * C rrw,,, I €tsó,,, + errf,o = 0

(eu + qr)u,- + (qo * %u)v,* * 4sw,o * €,w,o - €rtó,* - €rróu = 0 (6)

here ( )., represents S and ( ),",, represenrs

material oriented on-axis as foilows

Escuelo de lngenierío de Antioquio

p. rr,"

t,

FNICUONI-TSS CONTACT IN A LAYERED PIEZOELECTRIC MEDIUM CoMPoSED oF MATERIALS wITH HEXAGoNAL SYMMETRY

Crru,u+Crru,,, +(C,, tCrr)w,,=+(e,, + err)ó,,, =0Cuuv,o+Coov,,, =0

(C,, + Crr)u,*, r Crrw,o * Crrw,.. t erró,o + errQ,-= :0(e' i err)u,r- I etsw,** * €zzw,r, - srró,o - t rró,r, = 0

The solution of Equations (6) and (Z) for each layer must satist/ the external surface boundary condi_tions as well as the continuity of stresses oL-, oy,, o,,,electric displacement D,, electic potential Q, andmechanical displacementsu, v, and u;. The external surface mixed boundary conditions are expressed as

wt.(x,+hl 2) = f (x) for I x l< c

o,,=0forlxl>c

o*" =0 for -co <.f < +oo

o*=0for-oo<f<+@

D-=AfOr-oo<.x<+oo

and the continuity conditions are given by

uo |x,-hol2¡ = uk*'(x,ltu*rl2) i = x, y, z

o!,¡x,-holz) = o!,*'(x,ho*rl2) k =1,... ,n

»! 1x,-ho/Z) = Do*'(x,ho*rl2) (9)

Hete uk = (uo,v¡,ú¡,ót) represent the displacements and potential at the top or bottom interface of agiven layer /c, and o!, = (ol,o!,,o1) and D! arethe stresses and electric displacement at the upper or lowerinterfaces of the layer in consideration.

A Fourier transform technique is used to solve the system of Equations (6) and (7). Consequently, bytransforming these equations, the equilibrium and electrostatic charge expressions for an orthotropic off-axislayer maybe written as

-€'crrÚ +cssú,,.- 6'Cruv * cqsV,,,-i6eu +Crr)w,=-i6krr+ e,r)qr,- = 0

- 6' C *Ü * C +sü,,= - 6' C uuV * C aaV,,, - i6 (C * + C *)W,, - i€ (eru+ e,o )4r,- = 0

-i6 (C r, + C )ú,, - i € (C * + C *)V,, - 6' C rrfr * C yW,,, - É' r,16+ er, ó,,, = 0

-i6@r, *err)ú,.-i6@ro+ er)V,,- €'rrrfr t e3W,=,+ (2err@-er,@.__, = 0

where ( is the transform variable and an overbar indicates a transformed quantity. For brevity, the transformedequations for on-axis orthotropic materials are omitted.

The solution to Equations (10) in the transformed domain is sought in the form

o)

(8)

(10)

?81 Revista EIA

W(€,r)=W "(é)r€^" ÜG,4= ú "(€)rÉ^'t, (6,4=V,(É)rq^' aG,4=a.(€)r€^' ir r)

where W,G) , ú "(€) , y

"G) , and ó,(6) are the unknown Fourier coefñcients. By substituting these assumed

solutions into the transformed equilibrium and electrostatic charge expressions, Equations (10), the followingmatrix system is obtained for the orthotropic off-axis lamina

,¡ffi&4ffii

(1,2)

(ql-)"2c5) (crc-)"2c4) il.(crr+crr) i)"(e.,+e,r)

(Crc- )"'zCi) (C66- ),2C4) il"(Cru+Co) ü"(eru+e,o)

il.(Crr+Crr) iA(Cru+Co) (Css- 12q) (e,r-).2err)

i)"(er,+ err) i)-(%6+et4) (e,, - 22err) (22 tr, - err)

ú "(é)

V,G)

W,G)

@"G)

For this system of homogeneous equations to have a nontriüal solution, it is necessary to set thedeterminant of the matrix equal to zero. The solution of the resulting equation gives the eigenvalues orcharacteristic roots 2. In the case of a piezoelectric monoclinic material, there are eight eigenvalues ,l andeight unknown Fourier coefficients, lr,(é) , ú,(€) , v ,G) and o,(6) , for each layer. This shows the couplingamong all three mechanical displacements and the electrostatic potential. Materials such as PVDF exhibitorthotropic material symmetry and normally generate only real or imaginary roots (Heyliger and Brooks6

and Ramirez and Heyliger). The form for the displacements in this case can be written in terms of hyperbolicsine and cosine functions and are shown elsewhere (Ramirez and HeyligerT). Materials that have hexagonalsymmetry can generate two pairs of conjugate complex roots and a pair of real roots. The complex roots maybe expresse d as a+ib and -a+ib. The displacement V uncouples from the other displacements and electricpotential for materials that exhibit this hexagonal syrnmetry.

Once the eigenvalues are found, the unknown Fourier coefficients l, , W " , and <D, are written in terms

of ú, from Equation (12). After some mathematical manipulations, the contributions to the displacements andelectrostatic potential maybe written in terms of the trigonometric and hyperbolic functions as follows

ü = sinh(6 az)14 cos(§bz) + G, sin((bz)l + cosh(f az)[F, cos((bz) + G, sit((bz))+

{ cosh(1" É r) + Gr sinhQ"§ z)

V = Fo cosh(§)"oz) + Go sinh(§Loz)

= sinh(éqz)[(C o, F, + C n, Gr) cos((bz) + (-C n, 4 + C u, Gr) sin((bz))+

cosh({az)l(C n, 4 * C o, Gr) cos({bz) + (-C o, F, + C n, Gr) sin((bz))+

R, G, cosh (116 r) + &F, sinhQ.fu)l

W

i

Escuelo de lngenierío de Aniioquio i,,

FRtcrIoNrrss coNTACT IN A LAyERED pIEzoELECTRIC MEDIUM coMposED oF MATERIALS wITH HExAGoNAL syMMETRy

9 = sinh(6 az)[(C u, F, + C u, Gr) cos(( bz) + (-C * F, + C u, G r) sin(§ b z)] +i

cosh((az)l(Cr,I * C*Gr)cos({bz) + (-C* F,+ Cu,Gr)sin((bz)l+

^§rG, cosh(L€r)+[4 sinhQ"(z)l (r¡)

where Cn,,CurCu,,andCu,ateconstantsthatdependonthegeometrymaterialproperties,andthecomplexeigenvalues,t. Similarly, expressions for the stresses and the electric displacement can be obtained using appro-priate differentiation and constitutive laws, and are given by

tÍ__tL-zl-=.. = ffi s inh( § az) cos({ b r) - f, cosh({ az) sin(( b z)l F, +

tE

ffi cosh(( az) sin({ bz) + f, cosh(( az) cos(( bz)l Gr+ p, sinh(1, ( z) G, +

ffi cosh(f az) cos(§ b z) - I sinh( { az) sin(( b z)l F, +

I f, sinh(§ az) sin(( b z) + f , cosh(( az) cos{{ b z)l G, + p, cosher( z) F,

o ,.11, z IT = [ g, co sh( { a z) co s(( b z) - g, sinh(§ az) sín(( b z)] F, +

g

I g, sinb(( az) sin(( b z) + g, cosh(§ az) cos(( b z)f G, + T, sinhe" ( z) F, +

I g, sinh(( az) cos(( b z) - g, cosh( ( az) sin(( b z)l F, +

I g, co sh( f az) sin(( b z) + g, sinh(( az) co s({ b z)] G, + T, coshe=( z) G,

;.-(t.z\T = Q olG o cosh(fl"n z) + Fo sinh(g,o z)l

9

ñ-(t.z\T = [Iq sinh (( a z) c o s({ b z) - h, c o sh(( a z) sin(( b z)] F, +

tqpt, cosh({ az) sin((bz) + 14 sinh(( az) cos(gbz)lG, + X , sir.th(\g z) G, +

fh, c o sh({ az) co s(S b z) - h, sinh({ az) sin({ b z)} F, +

ll4 sinh((az)sin({bz)+ lt, cosh((az)cos((bz)l G, + X, cosh(1, €r) F, G4)

e*l R*vista EIA

where

f = - Cr, + Crr(a C o., - b C n) + err(a C u, - b C ur)

f, = Cu(a C nz + b C A) + err(a C u, * b C ur)

&=Cx(a+Cn)+errCu,gr= Crr(b+Cnr)+errCu,

lq= -err+err(a C n -b C nr)-err(a C Bt-b C 82)

h, = err(a C o2 + b C A) - err(a C u, + b C B)Qo = loC*

2.3 LocaVglabal matrix

In order to reduce the contact problem to a singularintegral equationforthe unlorownpressure distributionalong the contact area, the problem is restated in terms of the interfacial displacements along the common inter-faces separating the indiüdual layersinstead of the unloown Fouriercoefficients{ and Gr. Through thisprocedure,it is posible to construct a local stiftres matrix for each layer relating the lamina stresses and electric displacementat upper and lower surfaces of eachlayer to the correspondingmechanical displacements and electrical potential.First the streses are written in terms of the Fourier coefficients as

lsrc, lcos, Prs* f"0", lt,§, Prc*

-frcns, +frsoc, -frs"*" +frcrc,

,ffi*e{8»

(1s)

-O zz

6i

-o;",

_D,

4i

g§nc" g§hs, Trc^ g§oc" g§,s, Trs*

-gzsns, +g2chcs -gzcns, +g2shcs

l\s'", 4",§, X§* 4"0"" 4t§, Xrc*

-hrcosn +l4src, -4t,§, +l6crc,

Ito", f{§, Prt* -1"0", -lt,*, -Prc*

-frc¡,s, +frsoc, +frsrs, -fzcnc,

-g{¡c, -gsosn -Trc^ Esoc, g¡cosn T.t*+gzshsn -gzcnc, -gzcns, +gzshcs

4t0", 4",,t, Xf * -4"0r, -l\t,*, -Xrco,

-hrcrs, +l4soc, +l4sos, -4"0r,

-t6,,

4i

-tOxz

5

D:éi

4

G2

G3

F2

G1

F3

l*,Escuelo de lngenierío de Anlioquio

FnlcuoNrrss coNTACT IN A LAyERED pTEzoELECTRIC MEDIUM coMposED oF MATERTALS wITH HExAGoNAL syMMETRy

Secondly, and in similar way as that for the stresses, the displacements and electrostatic potential arealso evaluate d at + + to yield

w* +w

-Cnrsrsn+Curcrc" Cn sosr+Curcrc, R.r^

shc, cnsn s¡,

-Cursos,+Curcrc" Corsosr+Curcrc, Src^

and

o

2

ó*+

(16)

Fl

G2

G3

Cn soc,-Crrcos,

vhLs

Corsoc,-Curcos,

Cnrsocr+Curcos, 4 r^

sn s, c¡t

Crrsrcr+Curcosn §r r,

F2

G1

F3

(17)

where the following symbols have been used

,,. ="orn( t ¡".L\ru V r)l\ -,

", =*rn(e ,f;)

", =*r(e uf;)

,, =rirn(e n,f;)

,, =ri"n(e ,l),,=ri"(to!)

where the + superscripts refer to the top and bottom surfaces of each layer. Using matrix operations, it ispossible to solve for the Fourier coefficients { and G in terms of the displacements and the electricalpotentialin Equations (t6) and (17) and then substitute them into Equation (15). This mathematical proce-dure yields the local coefftcient matrix that relates the mechanical displacements and electrical potential tothe transverse stresses and electrical displacement at the top and bottom of each layer. The result can besymbolically written as

8'I §.svista El.A.

(18)ItnÍ tKrÍ,-l J«;ti] _ ftr-t;fI rrli, txti, ) ltul- J Ltrl; J

,&e"4P

1re)

where {Ütt=1wf ti,Úf ,ttf ,óo'li}oand {r}f =¡a)tt6,aili$,oj,ti6,D) ti6¡r

Imposition of the interfacial conditions on mechanical displacements, potential, stresses, and normal

electric displacement along the common interfaces,

{ü};., = {Ú\¡={Ú\oufor k =1,...,(n-l)

{f\1.,+ {f¡; = s

together with the extemal boundary conditions yields a system of equations where the interfacial displacement

components and the electrostatic potential are the onlyunknowns. This global system can be expressed as

tKti, tKli, 0

lK)\, lKl\,+tKli, lK)?,

0 lK)1, lKli,+lKl?,,0 0 lKl:,,00 + tKl,í

{u},{Ú},

:

{Ü}.

{7}i{0}

:

{0}

By enforcing the continuity of displacements and potential along the interfaces of the layered medium,

the redundant continuity equations are eliminated. This results in a reduction in the total number of un-

knownsinthe system.

2.4 Reduction of the contact problem t«r a singular integral equation

Imposition of the top surface mixed boundary condition on the slope of the normal displacement,

w'," = f (x) in the interval -c 1x ( c reduces the contact problem to the following singular integral equa'

tion where the fundamental unknown is the contact pressure p(f)

w'* = $ f" a,,,,,r(á) o,= d6! ¿7T -*

,:.=# f)ruu,'(€)é f)r{tt,''' d,)e-'É. dÉ

Here the t"rm l{,,,¡ (6) is ttre first element of the inverse of the global matrix in Equation (19). The

above integral is divergent because it does not vanish as the transform variable ( approaches t infinity. Using

(20)

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FnlcrloNrrss CoNTACT IN A LAyERED prEzoELECTRrc MEDTuM coMposED oF MATERIALs wrrg HEXAGoNAL syMMETRv

the asymptotic behavior of the local and global stiffness matrices along with the Euler identityexi* = cos x * I sin x , the relation between the Fourier and Hilbert transforms, and the odd-even proper-ties of the trigonometric functions, the singular integral equation is separated into a Cauchy-type integraland an integral with a regular kernel. The final form of these integral equations is written as

*:. _Hi,u

with

f' Kr1*,t¡ p(r)dr + * l:* r,

Ko(x,t) = ) ¡- ü4-sin[(r- x)6]dé0,1)

Q2)

Q1)

where ái,,,, is tne first element of the inverse of the limiting global stiftress matrix [K] as f approaches infinityand H[,,¡(€) = H q,,,¡ G) - Hi,r¡. After the decomposition, the integrals are numerically solved by using theapproach developed by Erdogans that is based on the orthogonal properties of the Chebyshev polynomials.

3. NUMERICAL R§§UIjr§

The methodology described above is applied to several representative problems using two typical piezo-electric materials: PVDF andPZT . The properties of the materials considered in the examples are obtainedfrom Berlincourts and co-workers.

3.I §IomogwNreous piezoelectric hatf-planes

The contact load, B as a function of the con-tact half length, c, for homogeneous PVDF half-planes

at different orientations is generated. The curves ob-tained are for angles of0o, 15o, 30o,45o, 600, and 90o

degrees, and are presented in Figure 2. For the caseof homogeneous half-planes the solution for the con-tact load as a function of the contact length is givenby

P(c)=#;

z,J

cF

and the solution for the contact stress profile is de-scribedby

p(x) Q3)2P(, ,")"'-;t.'-7)

Here P represents the total load, p(x) is thecontact stress, c is the contact half length, and 11i,,¡is the first element of the inverse of the global matrix

02{6Contart Half Length (mm)

Figure 2. Total load - half length response.Homogeneous PVDF

- P\,DF,0Ú

--* P\aD415'---- PVDF,30"

--- PVDr, {5"

--- PVDB 60'

- PVDF 90o /i,t.

/i,/it"''

/t'/ .,' ,

/t/lit1",

s4l Kevista EIA

when f goes to infinity. The above equations indicatethat the contact load versus contact length response

is parabolic, starting atzero and increasing with thesize of the contact length; and the contact stress pro-

file is elliptical and independent of the material prop-erties. The largest magnitude of the contact pressure

occurs at the center of the punch and is equal to 2/n(or 0.6366) times the value of the normalized contactpressure p(x)c/P. A value of 0.6367 is obtained forthe maximum normalized contact pressure using thepresent approach, proüding an additional check onthe solution technique.

The contact load versus contact half lengthresponse curve obtained for the PZT{homogeneoushalf-plane, alongwith the response curve of the PVDF

homogeneous half-plane oriented at 0o degrees, is

shown in Figure 3. It is clear that for a given contactlength, the contact load force is larger for PZ:f A.Thisisprimarilybecause the modulus of elasticityErin the

direction parallel to the applied load is bigger than

the corresponding E, of the PVDF. This illustrates the

- PYDF

--- P7:r4

O' Homgeneous hllf-plues

24Contact llalf l¡ngth (mm)

same result reported by Pindera and Lanez where thethrough-the-thickness modulus of elasticity E, Olaysan important role in controlling the contact load ver-sus contact length response.

3.4 Piczoelectric §ayer embeddedwithin en elestic medium

In general, multilayered composite laminatesare typically constructed by bonding together sev-

eral laminae with different material properties and fi-

ber orientation. The resulting laminated medium canexhibit behaüors that differ from those of its constitu-

ents. We consider a composite half-plane consistingof one laminate bonded to a piezoelectric materialthat in turn is bonded to the half- plane. This simu-lates the case of a sub-surface sensing layer. The toplayer and the half-plane are composed of graphite/epo)ry G300-934) and the in-between layer is eitherPVDF or PZT4.The thicknesses of the top layers are0.00127 m for the graphite/epoxy and either 0.000254

m or 0.00i27 mfor the PVDF and IZT4. The radius ofthe punch is 0.0254 m. In the case of a PVDF piezo-

electric layer, different off-axis angles are considered.The other plies have an orientation equal to 0o de-grees. Contact load versus contact half length re-

sponse curves and contact stress prof,les are obtainedwhen all these parameters are varied. The variationof the electrostatic potential that is sensed by the pi-ezoelectric layer is also investigated.

The amount ofload requiredto generate a given

contact half length depends on the thickness of thepiezoelectric layer that, in this case, plays the role of a

compliant or foreign layer in the layered medium. As

shown in Figure 4, the load required to generate a

given contact length decreases as the thickness of thelayer increases for PVDF. This results shows that the

compliant layer, in this case the PVDF ply, has the effect

of decreasing the overall stiftress of the multilayered

medium. For the PZT4,the opposite phenomenon is

observed and the load required to hold a given con-tact length increases as the thickness of the layer in-

creases. In this case, the compliant layer has the effect

1500

zxG

I? r00o

F

(&&{8»

Figure 3. Total loadHomogeneous

- half length response.PVDF and PZT4

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FnrcttoNI-Bss CoNTACT IN A LAyERED pIEzoELECTRIC MEDTuM coMposrD oF MATERIALS wITH HExAGoNAL syMMETRy

of increasing the stiftress of the composite laminate.The mismatch between the through-thethicknessmoduli of PZiÍ4 and graphite/epoxy is approximately6.26whichisquite differenrfrom the case of PVDF andgraphite/epoxy which is equal to 1.02. Therefore, theamount and type of variation in the response curveswith respect to the change in the thicloress of the com-pliant layer dependson the relative difference betweenthe properties of the compliant layer and the host half-plane. The dependence on the relative difference inthe through-the-thickness properties also influencesthe contact stress profiles as it can be seen in Figure 5.

For the PVDR there are no considerable changes sincethere is no practical mismatch in the through-the-thick-ness direction and this case is therefore not shown. Inthe case of WT4, shown in Figure 5, the thicker thecompliant layer yields stronger deüation of the con-tact stress curye from the elliptical distribution.

- T30o/pvDF/T300, 10?0'/0"1, h = 0.2s{ lm.

- T300/pvDF/T30o, 1010'/0"1, h = r.270 ¡lm

--- T30o/pvDF/T300, 10)0'/0'1, h= 0.2s4¡m.

- - T300/pvDF/'R00, f0'/0'/0'1, h = r.270 mn

iili ,//

,,1

,i",1

24Contact Half I¿ngth (mm)

In Table I, the variation of the electrostaticpotential sensed by either PVDF or PZT wirhin theconfiguration considered in this example is pre-sented. The layer composed by PVDF senses morevoltage that does ¡hePZT4layer. This can be ex-plained by examining the magnitudes of the piezo-electric and dielectric constants , e . dtlrd ár, that aresmaller in the case of PVDF. As the thickness of thepiezoelectric ply increases, the electrostatic poten-tial sensed increases as well. By looking at the consti-tutive equations, it can be concluded that the elec-trostatic potential is proportional to the thickness ofthe piezoelectric lamina.

Table l. Voltage sensed by piezoelectric layer.

Materia r(mm Potential (kV)

PVDF 0.254 0 300 450 60" 90" PZT4

488.8 487.O 485.6 484.4 483.1 163.5

1.27 1547.8 1522.8 1504.8 1487.4 1471.9 713.2

- -13ü/PZIl/T300,

[0 /0 /0"], h = 0.2s,t rm_. T300/IZTJIT300, t0 /0 /0"1, h = 1.270 ¡m.-- EllipticDishibution

c = 2.5,1 mm.

)tc

Figure 5. Contact stress profile. PZT4 compliantlayer

U^ 0.6I

L

tr 0.{

L)

t 300

¡F 200

Figure 4. Total load - half length response.compliant layer

sel

PVDF

ReYista EIA

{&. {8ü)The change in the electrostatic potential is

negligible as the off-axis angle is changed when the

thickness of the piezoelectric layer is small. As this

thickness is increased, the changes in the potential

become more noticeable. The values for the poten-

tial are most likely out of the linear range for this type

of material, and illustrate another difficulty in using

the materiais as pressure sensors. Figures 6 and 7 il-

lustrate how the potential and electric displacement

varies along the x-axis for the layer in consideration.

As expected, it is maximum at the center and van-

ishes asx increases. In Figure 7, the change in sign ofthe electric displacement so that it can satisfythe equi-

librium in the direction of indentation where the sur-

face charge is equal to zero.

¡0 60

Distance X (mm)

Figure 6. Electric potential distribution

¿$. Apptricatioxri {e§}d ssrrclusiorm

The above examples show the efficiency of the

outlined technique for the study of many aspects ofthe contact problem of arbitrarily layered media in-

dented by rigid punches. Numerical results can be

generated to illustrate the effects of both material and

geometry on the electro-mechanical response of the

layered half-plane. The explicit representation of each

layer by the locaVglobal formulation has proved ben-

eficial in the analysis of several classes of problems for

layered media2. By extending this methodologyto in-

clude piezoelectric effects and the coupled response

between elastic and electric fields, newtypes of prob-

lems can be studied. One example includes the shape

control of seat support surfaces. This is very impor-

tant for persons with physical disabilities for whomeffective seating and positioning are important to

avoid tissue injuries due to the pressure.

20 .10 6{)

Distance X (mm)

E 0.01

Q

L

otrr¡

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Figure 7. Electric displacement distribution

lu,

FntctloNrpss coNTACT IN A LAyERED piEZoELECTRTC MEDTuM coMposED oF MATERIALS wITH HExAGoNAL syMMETRy

Rmr§m"§NC§s

[1] Gladwell, G. M. L., Contact Problems in rhe ClassicalTheory of Elasriciry, Sitjhoffand Noordhoff, Alphen aanden Rijn, The Netherlands, 1980.

[2] Pindera, M.-J. and M. S. Lane, "Frictionless Contact ofLayered Half-Planes, Part I: Analysis", Journal of Ap-plied Mechanics, 60, 633-63 8, 1993.

[3] Bufler, H., "Theory of Elasticity of a Multilayered Me-dium ", Joumol of Elasncity, 1,125-t 43, r97 t.

[4] Rowe, R. K. and J. R. Booker, "Finite LayerAnalysis ofNonhomogeneous Soils", Journal of the EngineeringMeclnnics, togJ I 5 - t3z, I 982.

[5] Erdogan, F., 'Approximare Solutions of Systems of Sin-gular Integral Equations", SIAM Journal of AppliedMathemancs, 17, 1 o4t - I O 59, 1969.

[6] Heyliger, P. R. and S. P. Brook, "Exacr Solutions forLaminated Piezoelectric Plates in Cylindrical Bending",ASME J oumal of Applied Mechanics, 63, 9o3-g 10, tgg6.

[7] Ramirez, G. and P. Heyliger, "Frictionless Contacr in aLayered Piezoelectric Half-space", Journal of SmartMaterials and Sfructures, 12, 6tZ-625, Zoo3.

[8] Berlincourt, D. A., D. R. Curran, and H. Jaffe, "piezo-electric and Piezomagnetic Materials and their Func-tion in Transformers", Physical Acoustics, f , pp. 169-27O,1964.

ACKNOWTEDGEMENT§

The discussions and assistance of Professor M.-J.Pindera of the University of Virginia are gratefully ac-knowtedged.

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