Qing Hua Qin Department of Mechanical Engineering, University of Sydney Sydney, NSW 2006, Australia Tel." ; Fax.

Transcription

1 International Journal of Fracture 82: R41-R46, R Kluwer Academic Publishers. Printed in the Netherlands. USING GSC THEORY FOR EFFECTIVE THERMAL EXPANSION AND PYROELECTRIC COEFFICIENTS OF CRACKED PIEZOELECTRIC SOLIDS Qing Hua Qin Department of Mechanical Engineering, University of Sydney Sydney, NSW 2006, Australia Tel." ; Fax." Many brittle materials, such as concrete, piezoelectricity, have a large number of preexisting microcracks, The determination of their effective material properties has been the focus of considerable research. Current research into the development of effective material properties has mostly concentrated on the Taylor's method[l], the self-consistent method[2], the differential theory[3], the Mori-Tanaka method[4] and generalized selfconsistent(gsc) method[5]. For the themmelectroelastic problems, the first theoretical study appears to be that of Nemflmm and co-workers which is applicable to two phase thermopiezoelectric laminates that are connected in series or parallel[6]. After that, Dunn [7] evaluated the effective properties of two phase composites using dilute, self-consistent, Mori-Tanaka and differential micromechanical models. Benveniste[8] and Dunn[9] showed that the effective thermal-stress constants and p)qoelectric coefficients are related to the corresponding isothermal electroelastic moduli in two-phase media. Chen[10] further obtained some simple algebraic formulae tbr the prediction of overall thermoelectroelastic moduli of multiphase fibrous composites with the self-consistent and Mori-Tanaka methods. He indicated that when the phases have equal transverse shear rigidities, the overall moduli estimated by both methods are identical with exact solutions given by Chen[1 IJ for composites with arbitrary transverse geometry. More recently, Yu and his colleagues[12-14] obtained some new results of cracked thermopiezoelectric materials, additional references to work in this area can be found therein. This paper constitutes a continuation of analysis completed by the authors of those paper[12-14] on cracked thermopiezoelectric materials. In contrast to our previous studies, the present work focus on developing a GSC them T for efli:ctive thermal expansion and p3aoelectric coefficients of piezoelectric medium containing microcracks with given length and same orientation, The derivation is based on the extended Stroh formalism and a recently developed explicit solution of thermal-, electric- and elastic fields for a cracked in an infinite piezoelectric solid. In the analysis, a representative area element is adopted, which contains a microcrack surrounded by an elliptic matrix in a solid x~dth effective properties. Numerical results are given tbr a piezoelectric ceramic BatiO3. BASIC FORMULA TIONS Let us consider a two-dimensional thermopiezoelectric solid, where the material is transversely isotropic and coupling between h>plane stresses and inplane electric fields takes place. Choosing the xs-axis as the poling direction, the plane strain constitute equations are expressed by:

2 R42 with Z~ = 2A =,{[l+{t(i-3)]au'nj +AUjn'}dl (11) Mlere superscript,,m,, denotes the moduli associated with the uncracked material, N is the crack number within the area A, AU~ is the jump of elastic displacement(or electric potential) across the crack surfaces, lk the length of k-th crack, n = { 0, 1 } r is the normal local to the crack surface, and H(x) is the Heaviside step function. For simplicity, we assume that all cracks have the same length and orientation later. Hence, the effective thermal expansion and psaoelectric coefficients may be given by[13]: or*0 = c~(~)0 +Z ~ 7"0 = ~' (~)0 _ E(I~Z ~ (12) From (10) to (1.2) it is immediately seen that tile determination of the effective thermal expansion and pyroelectric constants requires a knowledge of the jump AU across the cracks in a RAE when (8) or (9) is prescribed. For a microcrack-weakened medium an exact solution of AU is not feasible and approximate methods are usually devised to determine AU and thus the effective properties. Tile approximation of Z through use of the generalized self-consistent theory is the main subject of this paper. GSC THEORY The calculation on AU is based on an effective crack model shov, al in fig. 1. This model has been described in[12], and for the reader's convenience, we repeat it as E,, EA ' ' ~ c l ' a C ~ / Figure 1. Schematic of representative area element with a crack follows. Tlle model contains a single crack of length 2a embedded in an elliptical matrix with modulus E At while the surrounding material outside the ellipse has an effective modulus E' as-yet-unknown(fig. 1). The major axes of the elliptical inclusion is chosen as 11=a+5, 12 =~5 (13) where ~5 is related to tile crack density ~ by[5]: a 2 = x(a +5)~5 - Naz/A (14) Int Journ of Fracture 82 (199@)

3 R43 t/ i 1 t - 0 t~13 = 0 C4d el5 0 ; O0 D 1 0 C15 --Kll O t {2) or inversely Ell t g ~ Sll SI 3 2cl~ = 0 -E l 0 -E3 gsl and simply, in matrix form H : EZ- y0, Z = FI1 + 0{0 SI3 0 0 g31 $ g.~3 0 s44 gl5 0 0 gl~ Ol, 0 g,3 0 0 ~ G~I] k C{II 0, where I1={ol, o33 ol,~ D1 DJr={01 03 o5 DI DJ r, Z={Z~ Z= 2Z12 Z3~ Z32}r={c~ c3 2c5 -E~ -Es} r, and y=ecc The effective materia! properties of the cracked body are defined as I-]: ~;'Z-~,T or Z= v'ii+c~'~ (6) where overbar denotes the area average ofa quantily over a representative area element(rae) A, i.e., 1 (-)al (7) (i) = ~-, and where the superscript "*" stands for the effective value. To obtain the relations between the thermal and electroelastic moduli of a cracked medium, an auxiliary uniform temperature problem is considered in which following boundary conditions are prescribed: or O(S) = 0, H(S)=O (8) 0 (x): 0% u(~=0; (9) where "S" stands for the extenml surface, the superscript "0" denotes constant, U--:{uj u: +}r. In the case of cracked body, the average strain and stress are defined on the basis of integral average are[12] as 0 O) (4) (5) Z = -A fz~'da + Z ~ 2-MdA : A dl H," (io) In~ Jouv~ of 7~o~upe 82 {799C)

4 R44 Since it is difficult to find the elastic displacements and electric potential analytically for the model, an approximate method[5] will be adopted to estimate tile jump AU. Ill this case, the corresponding energy functional can be defined as: where ~^" and f2* stand for the domain with E M and E*, respectively, I-~ 'q --- { 0 y ~ 93M } r 0. Among all possible solutions, tile exact U leads to Jmin. For convenience, let U ^ and U" be the elastic displacements and electric potential for a crack embedded in a matrix material and in an effective medium, respectively. These solutions can be obtained by way of the method given in[ 12]: Irte[Xv(~)~ -' (co o +Bo)- X0 {(~,~ -a 2 )"2- z. }10 x~ > 0 U=[Re[AV(~)B_,(C~O+Bo)_Ao{(~,~_~:),,~_~,}] o ~2 <0 AU =(a 2 -x, = )"2(eta -b)0 x,] <a (17) where the symbol "~" stands for complex conjugate, o 'q/0, and A, B, A0, B0, b and F are the well-defined matrices or vectors given in[12]. The approximate U is then assumed to be the linear superposition of the above solutions, i.e., U = xmu M +~*U* (18) where X M and X* are two 3 order diagonal constant matrices to be determined by the minimum potential principle. Substituting (18) into (l 5) and rniafimizhlg the result yields {1},2 {I}=A~X*J 2 (d'j (19) where ((1)l i )jk =f*'~^*.ls ~,K,,~.,.,'-" ll^~rrm ~.,,'--, A~ +.Is" f E" rl M ~Z M,4~ (J,K not summed) UKm-- J,iV" K,m~ ( 00,2 ).rx = 2Is,,E~,,U[,,UKV.,,, ds + 2Is. E*vx,, U'j., UMu.,, ds (.I,I< not summed) (20) E M * " * ', * ( ~= )j~: = Is ~,,,x,.uj,,uk,,.ds + Is.EuK,.Uj,U~:,.ds (J, Knot summed) d M = CMO -b ^~, d" = C*o -b*, }T X* = {~11 ~(22 33 in which C as defined in[12] can be rewritten as C = -2 Im( AB-' ) (22) where "Im" stands for the imagina~2 part. Solving (l 9) for X v and X" and substituting the results into (18) the jump AU in ( t I ) can be determined. The effective values of or* in ( 12)t can thus be found as (16) u,, = ~,, P, a lo#' +,=~. v=mx., (c,~r. ' "+c',o; ~ b~ )/ 2 (23) J Int Journ of Fracture 82 (1996)

5 R45 It can be seen fiom (23) that the determination of cf requires a knowledge of the corresponding effective conductivity and isothemlal electroelastic moduli, wlfich has been discussed in [12]. As illustration we consider a cracked piezoelectric ceramic, BatiO3[12-14]. The normalized a3~/a(~33 and k~/~.<~) are presented in Figures 2 and 3 for various values of crack density e. The dilute and self-consistent solutions are also given in these figures fox comparison. It is found again that the three methods can provide ahnost the same results for ~.<0.2. As far as we know, however, no reference results are available presently. Which model is more reasonable is still unknown and needs experimental support. REFERENCES [1] D. Fanella and D. Krajcinovic, Engineering Fracture Mechanics 29 (1988) [2] B. Budiansky and R.J. O'Connell, International Journal of Solids and Structures 12 (1976) [3] Z. Hashin, Journal of the Mechanics and Physics of Solids 36 (1988) [4] Y. Benveniste, Mechanics Research Communications 13 (1986) [5] Y. Huang, K.K. Hu and A. Chandra, Journal of the Mechanics and Physics of Solids 42 (1994) [6] R.E. Newnham, D.P. Skinner and L.E. Cross, Materials Research Bulletin 13 (1978) [7] M.L. Dunn, Journal of Applied Physics 73 (1993) [8] Y. Benveniste, Journal of Applied Mechanics 60 (1993) [9] M.L. Dunn, Proceedings, Royal Society of London 441A (1993) [10] T. Chen, International Journal of Solids and Structures 31 (1994) [ 11 ] T. Chen, Journal of Applied Physics and Solids 41 (1993) [12] S.W. Yu and Q.H. Qin, Theoretical Applied Fracture Mechanics (1966) in press. [ 13] Y.W. Mai, Q.H. Qin and S.W. Yu, to be submitted. [14] Q.H. Qin and S.W. Yu, in Proceedings, 9th International Conference on Fracture, Sydney 1997, accepted. 16 January1997 E, ~E ~ Fizure i. Schematic of representative area element with a crack.!rid ~To~{v~ o # E~og~e 82 (7<QOOJ

6 R ,-- Dilute r.,,- = --A-- G SC-method ~.r -r'r" -, if" Self-consistent.r-r" ~ll ~l~r'_ o 4 U ~r'g" A A ua 2 0, I, I J, I J I,, I ,8 1.0 Crack density Figure 2. Effective thermal expansion constant vs. c. ~' 6 oo "~ e-- Dilute / --A--- G SC-method --m-- Sell-consistent jmtilajjliiiill[aaaa, I ~ I, l, I ~ I Crack density If" Figure 3. Effective pyroelectric constant vs. c. Int Journ of Fracture 82 (1996)