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Fracture features offerroelectric ceramicsIvan Parinov a & Lyubov Parinova ba Mechanics and Applied Mathematics ResearchInstitute, Rostov-on-Don University , Rostov-on-Don,344090, Russiab Rostov-on-Don State University , Rostov-on-Don,344090, RussiaPublished online: 09 Mar 2011.

To cite this article: Ivan Parinov & Lyubov Parinova (1998) Fracturefeatures of ferroelectric ceramics, Ferroelectrics, 211:1, 41-49, DOI:10.1080/00150199808232332

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Ferroeledrics, 1998, Vol. 211, pp. 41 -49 Reprints available directly from the publisher Photocopying permitted by license only

0 1998 OPA (Overseas Publishers Association) Amsterdam B.V. Published under license

under the Gordon and Breach Science Publishers imprint.

Printed in India.

FRACTURE FEATURES OF FERROELECTRIC CERAMICS

IVAN PARINOV a** and LYUBOV PARINOVAb

a Mechanics and Applied Mathematics Research Institute, Rostov-on-Don

Rostov-on-Don 344090, Russia University, Rostov-on-Don 344090, Russia; Rostov-on-Don State University,

(Received 10 October 1997)

Some mechanisms of the strength and fracture toughness alteration for the ferroelectric ceramics (FC) using the early computer simulations are considered. The microcracking processes and effects of the crack bridging are studied. Finally, the growth features of the steady-state crack into ferroelectrics domain structure are investigated.

Keywords: Ferroelectric ceramics; domain structure; microcracking

INTRODUCTION

Design and creation of new FC compositions which to work in extreme conditions demand following studies their fracture resistance. The change of the latter is caused by the fracture characteristics and toughening mechanisms. The new problems are added by the calculation of the essential structural heterogeneities due to the existence of the grains, voids, microcracks, additions, domain structure, etc. with complex influences which occur by manufacture and work of the ceramic components. Hence, the prediction of the ceramics behaviour and optimization their strength properties are impossible without wide microstructural studies, selection and investigation of the fracture peculiarities and consideration of the joint effects.

*Corresponding author.

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42 I. PARINOV AND L. PARINOVA

The aim of this article consists in the study of the formation and degradation mechanisms of the strength and fracture toughness for FC compositions by computer simulations based on joint consideration of the material manufacture and fracture. Specific attention directs to the effects of microcracking, crack bridging and features of the steady-state crack propagation into FC domain structure.

MICROCRACKING TOUGHENING

It is known, [2*31 that the microcracking process zone near macrocrack relaxes the surplus stresses and causes the crack draging. However, the microcraking processes in the polycrystalline materials have not simple character. So, the ceramic microcracking by cooling and poling and microcracks into process zone ahead of crack tip directly make worse the fracture resistance. The computer models for ceramic compositions have estimated these opposed effects of microcracking. [42 51 They are suitable for grain sizes with the prevalent influence of the spontaneous microcracking ( ie . , at I > I : , I : is the critical length of the cracked boundary). In this size region the microcracks do not contribute to material toughness which is decreased with grain growth. However, more interesting size region ( I < 1:) remained without attention where a toughness increase was occured.

For analysis of these microcracking effects it is necessary to take into account the internal and applied stresses and to introduce the idea of the threshold stress for microcrack formation. Then, in the conditions of the saturated microcracking into process zone (Le., when there is failure of half intergranular boundaries, approximately) and steady-state crack, the energy balance relation has form:[31

K,"/K: = (S/8 + (4/S - 11S/80)1/2)/(1 - S/4) (1)

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FRACTURE FEATURES OF CERAMICS 43

K r , KE is the fracture toughness of the material with and without microcracking, respectively, v is Poissons's ratio, p is the number of facets per grain, g is the number of grains in unit volume, x- 1 is a dilatational factor, m = 0.25-2 is the coefficient depending on the grain geometry and the ratio, k = 02/o ,(oI, o2 are the principal tensions).

The numerical results are represented in the Table I. As PZT structure with initial presspowder porosity, C ; , it is considered the lattice model with square cell size, 6. [4-61 Obviously, the grain size is decreased but the parameter of F , / I is increased vs initial porosity, C:. The number of facets per grain, p . and characteristic faset size, 1, have behaviour which is similar to the grain size one. Then, the number of grains in unit volume, g , rises with C j due to the shrinkage processes which to cause the volumetric decrease thank to the closed porosity formation. The simulated facet densities, pg13 = 4.86- 7.32 for different variants coincide well with the value (pg13 = 6) for a typical equiaxed structure with 12 facets per grain. ['I The numerical results for K r / K : (at m = 2, k = 13) repeat qualitatively the known trend of fracture toughness growth due to the microcracking into process zone for grain sizes from interval: 0.41; < I < 1;.

The considerable values of K ? / K : witness on the importance of the stress state calculation in the toughening processes. Besides, the special selection of the parameters of m and k can help to model perfection in accordance with experimental data. Finally, correspondent variations of the structural characteristics (Ci, p , g and I ) optimize FC structure for achievement of the maximum fracture resistance.

CRACK BRIDGING EFFECTS

Behaviour of the toughness Tin the dependence on the crack length, c (T- curve) is the very important characteristic for estimation of the toughening effects caused by ceramic microstructure. [91 The crack bridging has small

TABLE I Computer simulation results

Properties C j = 0% C j = 10% C j = 20% C ; = 30% C j = 40% C j = 50%

P 15.44 14.24 13.15 11.50 11.30 10.36 86' 0.21 0.22 0.22 0.27 0.29 0.34

1.32 1.29 1.27 1.20 1.18 1.11 1.14 1.16 1.18 1.25 1.27 1.36

P d 7.32 6.63 6.03 5.40 5.39 4.86 K,mIKf 2.94 2.85 2.82 2.76 2.76 2.76

W p,lI

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44 I. PARINOV A N D L. PARINOVA

influence on the FC fracture resistance to compare with materials possessing the prevalant effects of the thermal expansion anisotropy (e.g., noncubic ceramic, A1203). [*I This has been shown by the computer simulations for fine grained PZT compositions (D, = 10 pm) with various porosity: [lo] T,/ TO- 1.1, where T , is the toughness saturation value, To is the intrinsic toughness. Nevertheless, for BaTi03 compositions possessing the coarse grained structure with grain size D, up to 500 pm the toughening due to the crack bridging may be actual.

Then, we compute the inert strength am ( ie . , the strength in the absence of any kinetic effects) from the conditions for Vickers halfpenny indentation cracks produced at load P to proceed to unlimited failure under action of an applied tensile stress a,. At equilibrium crack we have: [91

where $J is a geometry-dependent coefficient appropriate to penny-like cracks, xo is a coefficient denoting the intensity of the residual elastic-plastic contact deformations. Generally, a,(c) has two peaks which are divided by the value c = d (where d is the characteristic bridge spacing). The first maximum (at c < d ) is associated with the residual contact field, that dominates. The second maximum (at c > d ) is associated with the bridging, that dominates. The appropriate heights of these barriers are caused by the load P. After first barrier (at c < d ) the crack becomes unstable. However, it propagates spontaneously upto catastrophical fracture if the second barrier (at c > d ) turns out below the first one. Otherwise, the macrofracture is possible only when load to be sufficient for overcoming by crack the second barrier. Hence, a bigger maximum defines the ceramic strength om.

Secondary recrystallization and following cooling during ceramic production by hot-pressing method to causes the grain growth, inter- granular boundaries microcracking and significant alterations of the ceramic strength and fracture toughness. The studies of the bridging effects show an absence of the simple dependences of the toughness T and applied stress a, (k., strength a,) vs grain size (see Fig. 1). These results are explained by additive microcracks influences localized at the intergranular boundaries. Next, note that in the region of c < d the effect of increasing d will be to extent correspondent portion of the T(c) curve further downward. If we scale up d such that T(c) intersects the c-axis before the condition c = d is satisfied, then pre-existing flaws would become amenable to unstable crack extension without any external load applied. In this case, the spontaneous

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FRACTURE FEATURES OF CERAMICS

(4 (b)

45

CRACK LENGTH, c

I

CRACK LENGTN, c

FIGURE 1 The qualitative trends in the a) toughness and b) applied stress vs crack length. Solid and dashed curves show dependences before and after grain growth, relatively. Moreover, there are regions c < d (I) and c > d (11). The local minima are correspondent to c = d.

microcracking will prevail over the effects of crack bridging and defines the ceramic strength.

CRACK GROWTH INTO FERROELECTRICS DOMAIN STRUCTURE

The effects of the ferrohardness on the FC fracture toughness due to the twinning have been studied.["' It has been shown, that in the case of the ferrosoft ceramic this mechanisms causes the prevalent toughening to compare with microcracking near crack, branching and bridging of the crack. Then, three mechanisms of the crack draging at 90" domain boundaries, namely (i) by crack plane orientation change, (ii) by crack interaction with noncoherent boundary, and (iii) by boundary with absorbed impurity, have been considered, too. [ I 2 . 13' Here we study the possibility of the steady-state crack growth parallel to an internal interface ( i e . , domain or phase boundary) of crystallite with lamellar domain structure or coexisting phases.

Consider a composite system which consists of the homogeneous layer and substrate (Fig. 2). Let, their elastic properties (E ,v) to be equal and homogeneous tensition o0 = EE, where E is the microstrain caused by sintering temperature and FC phase parameters. Then, typically the crack initiation occurs into substrate and it extends parallel to the interface (Fig. 2). When the crack is localized into substrate at the depth, Ah, below interface, then the layer that to be above crack is bended and some elastic strain energy retains after crack propagation. For its estimation we take into account that the contributions of the strain energies due to the compression force P and moment Mare additive. [I4] Next, we have the asymptotic strain

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46 I. PARINOV AND L. PARINOVA

FIGURE 2 Used cbmposite structure consists of the layer (1) and substrate (2); (a) The tension a0 is caused by microstrain E in the homogeneous layer; (b) equivalent force system of P and M. The other numbers designate: a neutral axis (3), an interface (4) and a crack (5).

energy release rate for steady-state crack with length of a using a composite beam approach: [I5]

G = - ( P da a aI(2EA) + M2a/(2EZh3)) (4)

where the dimensionless moment of inertia, Z, of the beam per unit width and the effective cross section, A , are

z = (A + 1)3/12, A = h( i +A), x = z / h ( 5 )

The load P and bending moment M (per unit thickness) at the uniform stress a0 have forms

Then, the dimensional analysis gives relations for Modes I and 11 stress intensity factors (SIF): [I5]

K~ = C ~ P ~ - ' / ~ ~ ( X ) + c ~ M ~ - ~ / ~ ~ ( x ) KII = C 3 P h - q ( x ) + c4Mh-3/2g(x) (7)

where Ci are unknown constants and J g are unknown functions. The comparison of the Eqs. (4) and (7) with GE = K: + KtI gives

With the premise that the factors Ci in relations (8) contribute only to the constant terms, while g andfcontribute to the variables, Eqs. (8) become

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FRACTURE FEATURES OF CERAMICS 41

Cf + C: = 0.5, Cz + Ci = 0.5, g = Z-1/2, f = (h/A>”’ (9)

For final definition of the constants, Ci, it is necessary the assumption about strict solubility this problem for case either P = 0 or M = 0. The validity of the last condition in the present formulation of the problem, construction and numerical solution of the correspondent integral equation for SIF by half-infinite crack propagation near free surface have been represented. [I4] The unknown constants have values CI = 0.434 and C3 = 0.558. Besides, there is equation ClC2 + C,C, = 0. Then, we have

C2 = C3 = 0.558, C 4 = -C1 = -0.434 (10)

Next, inserting the necessary parameters from relations (9, (6), (9) and (10) into Eq. (7) gives the normalized K, and K I ~

K1/(c7ohi’*) = (0.434 + 0.966A/(A + 1))/(1 + K ~ ~ / ( c q h ’ / ~ ) = (0.558 - 0.752A/(A + 1))/(1 + A)”2 (11)

The numerous experimental data for different bimaterial systems and loadings have shown “49151 that the crack trajectory correspondent to the simple Mode I stress intensity demonstrates a surprising stability and the fracture resistance values have a fine reproducibility to compare with results for mixed mode. Furthermore, as condition of steady-state crack into ferroelectrics domain structure we select the following one: KrI = 0. Further, from second Eq. (1 1) it is found the depth A,, at which the crack appears to a steady-state trajectory, namely A, = z,/h = 2.876. The existence of the asymptotical limit permits to find the critical layer thickness, h,, below which the complete fracture is inhibited. This value is obtained by equating the asymptotic magnitude of KI equal to the analogous fracture toughness KIc. In condition of the crack growth into substrate along steady-state trajectory KrI = 0, Eq. (1 1) provides the predicition

h, = 0.755( 1 + A s ) ( K ~ c / ~ o ) 2 (12)

As it has been shown by experiments, [I6] the intracrystalline strains have values from 5 x lop5 to 2 x into morphotropic transition regions in dependence on the sintering temperature of the PZT ceramic in paraelectric phase. These strains into grains of tetragonal phase within the error are temperature independent, they exceed the microstrains in paraelectric phase and appear to be (6 f 3) x Therefore, in computer simulations we consider microstrains interval E = 5 x 1OP5-9 x lop4 for definition of the

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48 I. PARINOV AND L. PARINOVA

uniform tension a. which causes the crack growth. Then, we select the values of the E = 60-100 GPa and KI , = 0.4-1.4 M P a d 2 [17]. Finally, we define the layer thickness in the form h = nhd, where hd-0.2 pm is the domain size for grain one g = lOpm [''I, n is the domain number in crystallite under uniform tension, ao. Obviously, the coordinate z, for steady-state crack growth locates into grain if h,,, = 3.4pm, then nmax = 17. Further, the formulae (1 1) and (12) give

KI = 0, 5 8 5 ~ o h ~ / ~ , h, = 2.926(K1,/ao)~ (13)

where KI is the SIF for steady-state crack growth at the depth of A, h and h, is the layer thickness limit defining inhibited fracture. The selection of the parameters correspondent to the maximum of KI and minimum of h, gives E = 9 x lop4, KI, = 0.4 MPam'/2, h = 3.4pm, E = 100 GPa. The numerical result is KI = 0.097 M P u ~ ' / ~ , and h, = 57.8 pm. Obviously, KI << KI, and h, >> g for considered PZT parameters. Hence, the steady-state crack growth is impossible parallel to or along interfaces into crystallite. The initiated crack at this boundary propagates catastrophically until it meets a fixed 90" domain boundary. [I2, 13]

CONCLUSIONS

The possibilities of the early computer simulations [4-61 are expanded on the FC toughening due to the saturated microcracking near crack into size region of 0.41: < 1 < 1:. The simulated results confirm the known trend of the fracture toughness rise with grain size in the limits of this region. [2931

The qualitative trends of the FC toughness and strength vs crack length in the conditions of the crack bridging are found. These parameters have not simple dependencies on the grain size thank to the effects of microcraks at the grain boundaries. By the definite selection of the characteristic bridge spacing, d, it is possible that the microcracking will find ceramic strength covering the effects of the crack bridging.

Finally, it is shown the impossibility of the stable crack growth parallel to 180" domain or phase boundary into FC crystallite.

Acknowledgements

This research was supported by Russian Foundation of Fundamental Research under Grants No.95-01-00072 and 96-01-01790, and by Russian Goskomvuz under Grant in Ural Metallurgy Program.

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[I71 Pisarekno, G. G. (1987). Prochnost Piezokeramiki, Naukova Dumka, Kiev, (in Russian). [I81 Pertzev, N. A. and Ark, A. G. (1991). Solid State Physics, 33, 3077 (in Russian).

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