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1 i~ -- r! ---~~~-- ~-~ ~ ~~ -! International Journal of Fracture 101: L3-L8, 2000., Kluwer Academic Publishers. Printed in the Netherlands. L3 MICROCRACKING IN PIEZOELECRICS WEAKENS HE ELECRO- MECHANICAL COUPLING AND CHANGES IS DIRECIONALIY Igor Sevostianov.t, Department of Mechanical Engineering, ufts University Medford, MA, USA " isevos01@tufts.edu Mark Kachanov Department of Mechanical Engineering, ufts University Medford, MA, USA! mkachano@tufts.edu Abstract. Microcracking in piezoelectrics is found to produce two major effects on the effective piezoelectric properties: it weakens the electromechanical coupling and it changes its "directionality", i.e. the direction of mechanical (electrical) response to the applied electrical (mechanical) loads. he latter effect implies that microcracking in piezoelectric sensors and actuators reduces the accuracy of the devices. We quantify the mentioned loss of accuracy. his can also be viewed as a guantification of the "piezoelectric fatigue". I 1. Introduction. In a purely elastic anisotropic material, microcracking weakens the anisotropy, provided the orientational distribution of microcracks is more or less random (Mauge and Kachanov, 1994). he underlying reason is that a crack normal to the "stiffer" matrix direction produces a larger contribution to the overall compliance than a crack of the same size that is normal to the "softer" direction; hence, a set of cracks of diverse orientations reduces the anisotropy. In the problem of effective conductivity, the effect of random micro cracking is similar (it reduces the matrix anisotropy) and for the same reason: microcracks normal to the "more conductive" direction reduce the conductivity to a greater extent than the ones normal to the "less conductive" direction. he piezoeffect at the macro-level is necessarily anisotropic, since it is characterized by a 3-rd rank tensor and such a tensor is zero in the isotropic case. Since random microcracking generally, reduces anisotropies, it can be expected to weaken the piezo-effect as well. his hypothesis is examined in the present work. Another effect identified in the present work is that micro cracking alters!. "directionality" of the electromechanical coupling. his implies that microcracking reduces the accuracy of various piezoelectric devices (sensors and actuators).

2 ~ - 4'"' "c" L4 2. Basic equations and numerical procedure. We consider a 2-D transversely-isotropic piezoelectric material. he numerical calculations were done for the values of material constants that correspond to piezoceramics PZ-4 (see able 1). «~ S13 S44 d31 d33 d1s E E : I '". ABLE 1. Values of physical constants for piezoceramics PZ-4 (Sij in 10-12m2/N; dij in la-login; Eij in 10-8Plm) he constitutiv equations have the form E xx = SIlO" xx + Sl30" zz + d31ez Ezz = S310" xx + S330" zz + d33ez Exz =S440"xz +d1sex Dx = dlso" xz + EIIEx Dz = d310" xx + d330" zz + E33Ez where E ij, 0" ij are elastic strains and stresses, D x and D z are electric displacements, Ex and E z are components of the electric field, S ij are elastic compliances in two index notation (measured at constant electric field, so that they are not affected by the piezoeffect), E~ are dielectric permeabilities (measured at constant stress field) and dij are piezoelectric constants. We examine the effect of microcracking on the effective piezoelectric response by finite elements calculations performed for a number of microcrack arrays. he results (averages over realizations of the micro crack statistics) are plotted in terms of the crack density parameter p = (1/ A)Il;, where 21i is i-th.- crack length and A is the reference area of averaging... he focus of our analysis is the impact of microcracking on the dimensionless electromechanical parameters coupling. that he have strength the structure of this coupling is characterized by,6 ~

3 ~ - L5 d~!/ 8kl&mm Of them, the following four coefficients are of key importance (see Grinchenko et ai, 1989):. 2 d d 2 d2 i kt 2 = dl5 ' ks 2 = ' k2 31 = d31 ' k & Higher (lower) values of these constants ~orresp.ond to stronger (weaker) coupling. Higher values of some of them combined With lower values of the other ones correspond to a mixed effect, that depends on the load/response directionality ~'..,' d d..",., 1 r"\1 2 -' dls f2\ k2 = 31 33, '.. \1J k - \:) S, -.. ' I S ~ 0 50 " S 12 co ~ :.~.: -+4 Ell 33. ~ -. ~~ - -- ') ':::0-/,:: 3 4 G) k2 djl 0 k 2 dj3 \ ') ;' -I S S II E33 33 E "', 1.00 " ',':,~ ~~~ ' ~~ ~ I I I G) -=-. n I I (b) - ~4 ~ , Fig.l. Changes in dimensionless constants characterizing the strength of the electromechanical coupling, as microcrack density p incrcases. he constants are normalized to their,'alues in the absence of microcracks.

4 - - L6 he following three orientational distributions of microcracks were examined: (1) randomly oriented cracking (simulated by a set of cracks of discrete 'orientations with step n/12); (2) cracks parallel to the "elastically stiffer" x- direction; and (3) cracks parallel to the "elastically softer" y-direction. 3. Results. Figs 1,2 show the effect of the three orientational distributions of microcracks on the effective piezoelectric properties. his effect is illustrated by. changes in dimensionless constants kt,ks,ki3 andk33 as the microcrack density '-.."i c"', increases. hese constants characterize the strength of the electromechanical ;;. coupling, whereas ratios dls/d33, dis/d31, and d33/d31 characterize the "directionality" of the coupling. he constants shown in curves of Figs 1,2 are normalized to their values for the matrix without cracks. 1.4 'Q dls/d (i) G) d1s/d 'CV 3 0 d33/d3! , 1.2 r:'\ \...?;)...'" Q (c) """."'"", ,,' electromechanical Fig. 2. Changes coupling, ratios as d1s/d33, microcrack dis/d31,andd33/d31, density p increases. he characterizing parameters are the normalized "directionality" to their of values the " in the absence of micro cracks.

5 I - 4. Physical implications. In the problems of the purely elastic effective properties and the dielectric effective permeabilities, the impact of microcracking is qualitatively obvious (increase in compliance and permeability) and has been addressed in literature. ~. As far as the piezoelectric effective properties are concerned, it appears that.. two distinct physical mechanisms can be identified: (A)Microcracking weakens the strength of the piezoeffect (the strength of the electromechanical coupling);,..j 1,;t; (B) Microcracking changes the "directionality" of the electromechanical [f~~i~)r;:c.." coupling, as measured by changes in ratios f". - [~"'::. c dis/d33, dis/d31,andd33/d31. c Our analysis quantifies the "piezoelectric fatigue" due to microcracking, by quantitatively relating the effects (A) and (B) to the microcrack density. he mechanism (B) appears to be particularly important in this respect: it reduces the accuracy of various piezoelectric devices. Indeed, for piezoelectrics used in various actuators, micro cracking changes the mechanical response to the applied electric field - as illustrated, for example, by significant changes (that are dependent on the microcrack orientations) in the ratio E zz / E xx of strains induced by Ez component of the electric field, see the curves for d33/d31 in Fig. 2. Similarly, micro cracking reduces the accuracy of piezoelectric sensors, particularly if they are subjected to non-uniaxial stress states. Our analysis allows one to quantify these losses of accuracy. In more detail, the results presented in Figs l,2 can be interpreted as follows. Fig. la shows that randomly oriented micro cracking reduces the values of all fo~ dimensionless parameters that characterize the strength of the electromechanical coupling. Fig. 1 b illustrates the case when microcracks are normal to the "elastically stiffer" direction of the matrix. In this case, only one parameter (k31) undergoes significant changes, whereas parameter k33 remains unaffected and parameters ks and kt change only slightly. Fig. lc illustrates the case of microcracks normal to the "elastically softer" direction of the matrix. he results are quite interesting: decrease of parameters = ks, k31 and k33 is followed by their unexpected increase. A possible explanation is that, in this case of orientational distribution, microcracking enhances the purely elastic and purely conductive anisotropy, and this may lead to the increase ~ of the piezoeffect (since the latter, being characterized by 3-rd rank tensor, is ~~cc ~~~j' inherently ariisotropic). Figs 2 (a-c) show the changes in "directionality" of the electromechanical coupling as microcrack density p increases. An interesting observation, related to L7

6 - " L8 the case when microcracks are normal to the "elastically stiffer" direction of the matrix (Fig. 2b), is that the point of maximum of ratio dls/d33 corresponds, approximately, to the point where the purely elastic and purely conductive properties become isotropic. Acknowledgment: his work was supported by the National Science Foundation. through grant to ufts University. REFERENCES.: Grinchenko, v.., Ulitko, A. F. and Shulga, A. N., 1989, Electroelasticity (Naukova Dumka, Kiev, in Russian). Mauge, C. and Kachanov, M. (1994). Effective elastic properties of anisotropic materials with arbitrarily oriented cracks, JMech.Phys.Solids 42, "'- ~