The fracture properties of ferroelectrics directly determine their processability and reliability of devices made of them. However, both experimentally and theoretically reported fracture toughness of piezoelectric ceramic materials remains nearly the same as that reported 30 years ago, limiting the application of piezoelectric devices in situation where high reliability is required. Here, we try to reveal the parameters that could be used to optimize the fracture performance of ferroelectrics. Specifically, stress-strain curves, intrinsic fracture toughness and long-crack fracture toughness of three typical PZT ceramics were measured by uniaxial compression method, crack-tip opening displacement (COD) technique and single-side V-notch beam (SEVNB) technique, respectively. It is shown that the intrinsic fracture toughness is positively correlated with the Young’s modulus of the material, which suggests that improving the Young’s modulus of ferroelectrics is an effective way to improve their intrinsic fracture toughness. The long-crack fracture toughness is related to the intrinsic toughness and extrinsic ferroelastic domain switching/phase transformation toughening, which also suggests that optimizing the ferroelastic switching behavior of piezoelectric ceramics can improve their extrinsic effect. Compared to the hard doped PZT, the soft doped PZT has low coercive stress, high remanent strain and high shielding toughness. The fracture patterns observed in different PZT materials are related to the different ferroelastic switching behavior of the materials. Soft PZT ceramics exhibit intergranular fracture, while hard PZT with weak ferroelastic switching behavior exhibits transgranular fracture. In conclusion, fracture toughness of ferroelectrics is enhanced by optimizing Young’s modulus and toughening of ferroelastic switching.

Piezoelectric ceramics have been widely used in modern industries working as actuators, transducers, etc., due to their unique electromechanical coupling performance, ultra-fast response and small size^{[1-2]}. After several decades of intensive investigations, the functional property of piezoelectric ceramic materials have been significantly improved, and some new piezoelectric ceramic systems were also developed^{[3-4]}. In comparison, the mechanical property of piezoelectric ceramic materials, e.g., fracture toughness and strength^{[5]}, are also very important, as the mechanical performance directly determines their processability, and the reliability of piezoelectric devices^{[6-7]}. However, although tremendous investigations related to the fracture behavior of piezoelectric ceramic materials have been conducted by researchers in the last 30 years, both experimentally and theoretically^{[8-9]}, the reported fracture toughness ${{K}_{\text{Ic}}}$ of piezoelectric ceramic materials remains near the same as that reported 30 years ago, in range of 0.5– 1.1 MPa·m^{1/2} for unpoled PZT ceramics^{[10⇓⇓⇓-14]}. To a certain extent, this situation has limited the application of piezoelectric devices in situation where high reliability is required.

At the moment, most of the fracture behaviors of piezoelectric ceramic materials observed in experiments or problems encountered in the industries could be well explained, e.g., the crack length-dependent fracture toughness (R-curve)^{[15-16]} and the fracture toughness anisotropy of the materials after texturing^{[10-11,17-18]}. As reviewed by Schneider^{[17]}, Kuna^{[19]}, Webber^{[20]}, et al., only a small number of problems are not well explained yet, for instance, the strange fracture surface mode-some piezoelectric ceramic materials undergo intergranular fracture, while some are dominated by transgranular fracture^{[21]}, and the different fracture mode in terms of hard and soft PZT^{[22]}. Therefore, the next step of investigation on the fracture behavior of piezoelectric ceramic materials should focuse on proposing ideas or schemes that could be used to optimize the fracture behavior of ferroelectric ceramic materials. However, different from metals and other ceramics such as alumina ceramics, revealing the parameters that have influence on the fracture behavior of ferroelectric ceramic materials is a difficult task. This is because, on the one hand, the measured fracture toughness of piezoelectric ceramic materials is technique dependeddifferent charactering techniques normally lead to different fracture toughness values^{[18,23-24]}. On the other hand, piezoelectric ceramic materials with different ferroelastic domain switching behavior have different toughening performance^{[10,13,20]}.

Hence, to reveal the parameters that could be referred for designing piezoelectric ceramic materials with better reliability, the constitutive behavior and the fracture performance should be both characterized. Moreover, materials with different performance but similar compositions should be characterized in comparison, which we tried to accomplish here. In this work, three types of commercially available PZT ceramics (PZT-4, -5, and -8) with different ferroelastic domain switching behaviors were chosen as model materials. The stress- strain curves of ceramics were firstly characterized during a uniaxial compression process to determine the parameters that have influence on the fracture behavior of ceramics. Then the intrinsic fracture toughness (also known as crack- tip fracture toughness) of ceramics, ${{K}_{\text{I}0}}$, was determined by the crack opening displacement (COD) technique. The long-crack fracture toughness of ceramics was also measured by the single-edge V-notch beam (SEVNB) technique. The scanning electron microscopy (SEM) was used to characterize the fracture mode of the crack surface. Based on the measured data, the different fracture performances of the three types of ceramics were discussed in comparison, by combining the measured fracture toughness, the constitutive behavior, and the intrinsic fracture toughness of these materials.

1 Experimental

1.1 Material description

The material compositions of the used PZT-4, -5, and -8 are Pb_{0.95}Sr_{0.05}(Sb_{0.33}Nb_{0.77})_{0.1}Zr_{0.45}Ti_{0.45}O_{3}-1%Fe, Pb_{0.98}Sr_{0.02}(Zn_{0.33}Nb_{0.77})_{0.1}Zr_{0.45}Ti_{0.45}O_{3}, and PbZr_{0.53}Ti_{0.47}O_{3}-0.6%Fe (in mol). They were fabricated by a standard solid-state synthesis process (S1). Their grain size and density are tabulated in Table S1.

1.2 Stress-strain curve measurement

The testing setup for characterizing the stress-strain behavior of the materials during uniaxial compression can be found in our previously published work^{[11]} and S2.

1.3 Fracture toughness testing

Fig. 1 reveals an optical microscope image of a V-notch. The long-crack fracture toughness ${{K}_{\text{Ivnb}}}$ of the specimen was calculated by^{[23]}:

where, $F$ is the fracture stress, ${{S}_{1}}$ is the outer span, ${{S}_{2}}$ is the inner span, $B$ is the specimen thickness, $W$ is the specimen depth and $\alpha =a/W$, $a$ is the average V-notch depth, ${{Y}^{*}}$ is a dimensionless correction factor based on the ratio of $a/W$. The details of the test setup and the geometric dimensions of samples can be found in S3.

The crack-tip opening displacement (COD) technique was used to characterize ${{K}_{\text{I}0}}$. If no additional transverse or conical cracks appear and the residual stress field relaxes due to residual stress after the Vickers indentation is performed, the crack remains open. The details of measurement and computation are shown in S4.

Due to the toughening effect of the ferroelectric domain, the measured values gradually become larger than the estimated values, and the successive data points vary from the fitted straight line (Fig. 2(a)).

Figure 2.Measured COD plotted versus the calculated COD(a) PZT-4; (b) PZT-5; (c) PZT-8

In addition to the preceding computation method mentioned above, the crack-tip fracture toughness can also be obtained by Irwin fitting^{[25]}. That is, only ${{A}_{0}}$ (the coefficient based on the ratio of a/b) is used. This approach has the advantage of not requiring half-diagonal length data for crack lengths and indentations. However, this evaluation is limited to the near-tip crack opening displacement field^{[25]}. Fig. S1 shows the measured crack opening displacement COD as a function of the distance $x$ from the crack tip and the least-squares fitting curves with x_{max}=10 μm.

2 Results and discussions

2.1 Stress-strain curves of the tested materials

Fig. 3 depicts the measured stress-strain curves during mechanical compression. At first, all of the materials reveal a linear stress-strain behavior, and then follow a significant nonlinear regime due to ferroelastic switching (in addition to non-180° ferroelastic domain switching, ferroelastic phase transformation can also induce nonlinear deformation for ferroelectrics, and both mechanisms can be called ferroelastic switching^{[26]}), during which the strain accumulates quickly with the increase of the applied stress. With further increase of compressive stress, the stress-strain curves gradually change from nonlinearity to linearity, due to the decrease of the volume of domains that could be switched by the applied stresses. Additionally, based on the obtained stress-strain curves, the coercive stress $~{{\sigma }_{c}}$, the remanent strain ${{\varepsilon }_{r}}$, the Young’s modulus $E$ and the Poisson’s ratio $\nu $ of the tested materials can be determined and tabulated in Table S5.

Figure 3.Stress-strain curves of PZT obtained during mechanical compression(a) PZT-4; (b) PZT-5; (c) PZT-8

2.2 Fracture toughness and fracture mode of the tested materials

Fig. 4 depicts the measured ${{K}_{Ivnb}}$ and ${{K}_{I0}}$ for unpoled PZT ceramics with different compositions. The ${{K}_{Ivnb}}$ of PZT-5 is approximately 1.10 MPa∙m^{1/2}, with a value similar to the results reported by others on PZT 151^{[10,27⇓-29]}, e.g., by Schneider with a value of 1.0 MPa·m^{1/2}^{[29]}. In comparison, the ${{K}_{Ivnb}}$ for PZT-4 is 0.91 MPa·m^{1/2}; for PZT-8, its ${{K}_{Ivnb}}$ is 0.78 MPa·m^{1/2}. As for ${{K}_{I0}}$, the highest value is for PZT-4; the lowest value is for PZT-5; the intermediate value is for PZT-8. Additionally, for PZT-4, PZT-5 and PZT-8, the ${{K}_{I0}}$ values determined by Eq. (S4-1)-Eq (S4-2) are 0.63, 0.52, and 0.56 MPa·m^{1/2}, respectively; which are slightly lower than the values of 0.75, 0.59, and 0.6 MPa·m^{1/2} obtained via Irwin parabola. This is because ${{A}_{1}}$ (the coefficient based on the ratio of a/b) is positive for all the cracks evaluated in this work. Therefore, according to Eq. (S4-1), the ${{K}_{I0}}$ for the same $u(x)$ is always higher if only ${{A}_{0}}$ is considered (Irwin parabola)^{[25]}. Even the ${{K}_{I0}}$ values determined by Irwin parabola could be higher than the observed ${{K}_{IC}}$ value^{[30]}. Thus, the three-term- approximation should be the first choice.

Figure 4.Fracture toughness ${{K}_{Ivnb}}$ and intrinsic fracture toughness ${{K}_{I0}}$ for PZT-4, -5, and -8

Fig. S2 displays the photography of the fracture surface characterized by SEM. For PZT-5, the fracture morphology is intergranular fracture dominated (~90%);while for PZT-4 and PZT-8, most parts of the fracture morphology are transgranular (~70% for PZT-4 and ~95% for PZT-8). It is in accordance with the previous reports of Guillon, et al^{[22]}. They found that the fracture mode of PZT piezoceramics is related to the dopants， PZT with soft additives is mainly intergranular while PZT with hard additives is rather trangranular.

2.3 Discussions

As the intrinsic toughness is controlled by the atomic interactions of the materials, and is generally believed to be proportional to their theoretical tensile strength. Additionally, the theoretical tensile strength of a material is thought to be one tenth of its Young’s modulus. For PZT-4, -5, and -8, 1 : 0.7 : 0.8 for both ${{E}_{initial}}$ and ${{K}_{I0}}$. This demonstrates that to improve the intrinsic fracture toughness of piezoelectric ceramics, we have to find strategy to improve their Young’s modulus.

Additionally, from Fig. 4, it can be seen that the measured values of ${{K}_{Ivnb}}$ and ${{K}_{I0}}$ are controversial, e.g., the largest ${{K}_{Ivnb}}$ was observed in PZT-5, while it reveals the smallest ${{K}_{I0}}$. According to the previous investigations^{[23]}, such controversy can be rationalized by the toughening mechanism of ferroelastic switching. Under external load, stress concentrates near the crack tip with distributions dominated by the ${{K}_{appl}}$-stress field. Due to ferroelastic switching, this ${{K}_{appl}}$-stress field will trigger a process zone with dimensions related to the stress field distribution, the coercive stress ${{\sigma }_{c}}$, and the intrinsic toughness ${{K}_{I0}}$ of the materials. For unpoled PZT ceramics, the domains with random orientations switch into the direction along the applied tensile stresses. This process zone, obviously, will shield the ${{K}_{appl}}$-stress field. With the growth of the crack, the process zone triggered by the ${{K}_{appl}}$-stress field becomes the crack wake. For PZT-4 and -5, the switched domain remained in the wake of crack, leading to the increase of the closure shielding stresses at the crack tip (Fig. 5(a)). To promote further increment of the crack, a larger applied stress is required. Therefore, under controlled load, the fracture toughness of piezoelectric ceramic materials normally reveals an R-curve behavior, and when the crack propagates a certain distance, a plateau value $K_{IR}^{max}$ can be achieved. Obviously, the value $K_{IR}^{max}$ is nearly equal to the sum of ${{K}_{I0}}$ and $K_{\mu }^{max}$ ($K_{IR}^{max}$ means the largest fracture toughness of ferroelectrics under controlled load; $K_{\mu }^{max}$ presents the maximum shielding toughness). That is $K_{IR}^{max}={{K}_{I0}}+K_{\mu }^{max}$.

Figure 5.Illustration of the toughening mechanism(a) PZT-4 and PZT-5; (b) PZT-8

The calculated shielding toughness $K_{\mu }^{max}$ (S7) for PZT-4, -5, and -8 was 0.17, 0.24, and 0 MPa∙m^{1/2}, respectively. Then the $K_{IR}^{max}$ for PZT-4, -5, and -8 should be 0.80, 0.76, and 0.56 MPa∙m^{1/2}, respectively, which are smaller than the test value of 0.91, 1.10, and 0.78 MPa∙m^{1/2}. This phenomenon was possibly induced by other toughening mechanism that was not taken into consideration in Eq.(S7-1), for instance, the shielding effect of process zone near the crack tip, crack deflection toughening, the kink of the crack during propagation, and the crack-interface grain bridging. Taken PZT-8 as an example, the calculated $K_{\mu }^{max}$ is zero based on Eq. (S7-1) as the switched domains switch back in the crack wake (Fig. 5(b)). However, the measured ${{K}_{Ivnb}}$ is larger than ${{K}_{I0}}$, with a value of 0.22 MPa∙m^{1/2}.

The difference between the observed intergranular and transgranular crack surfaces in different PZT is supposed to relate to the ferroelastic switching as well. In order to express our point of view more conveniently, it is firstly assumed that the tensile strength of the grain boundaries is weaker than the grains. Then for polycrystalline materials, if the grain boundaries are aligned with the crack growth path, the crack would propagate along with the grain boundary (Fig. S3(a)). However, in most cases, the grain interfaces are not parallel to the main crack, and hence, if the crack propagates along the grain interface, the crack tip is expected to experience mixed loading, and more energy is consumed (Fig. S3(b)). With the increase of the grain size, obviously, the influence of the crack kink on the stress redistribution increases. When the stress field near the crack tip increases to critical strength of the grains, the crack passes through the grains, leading to transgranular cracks. However, if the size of the grains is smaller than a critical value, the influence of the crack kink on the stress redistribution is limited. As a result, for materials with small grain size, the crack mode is normally intergranular dominated^{[21]}.

However, for PZT ceramics, apart from the crack kink, the ferroelastic switching can also influence the stress field within the few grains near the crack tip. As the ferroelastic switching in ferroelectrics is accomplished by domain nucleation and domain wall motion, and the existence of grain boundaries affects the continuity of the ferroelastic switching process, the stress on the grain interface could be enhanced or decreased, depending on the crystallographic direction of the grains near the crack tip^{[21]}. For PZT-5, as its ${{\sigma }_{c}}$ is significantly smaller than those of PZT-4 and PZT-8, these effects may dominate the fracture mode, leading to intergranular fracture. However, for PZT-4 and PZT-8, their ${{\sigma }_{c}}$ is larger. Hence, the effect of ferroelastic domain switching on the stress redistribution at the grain interface is limited, and therefore, transgranular fracture is dominated in them. Guillon^{[22]} suggested that soft additives may cause grain boundary segregation in soft PZT, which changed the grain boundary energy and weakened grain boundaries. To further understand the effect of domain inversion on crack propagation modes in ferroelectric materials, in situ experiments and phase field modeling are required.

3 Conclusions

In summary, by using three typical PZT ceramics as model materials, the correlation between their constitutive and fracture behavior was investigated. Their deformation behavior in the direction parallel or perpendicular to the applied stress were characterized under uniaxial compression at room temperature. Their fracture toughness was measured by SEVNB technique and ${{K}_{I0}}$ was measured by COD technique, and the fracture mode was characterized by SEM. The different fracture performance of the three types of materials were discussed in comparison, by combining the measured fracture toughness, the constitutive behavior, and the intrinsic toughness of these types of materials. The main conclusions are as follows:

1) For PZT-4, -5, and -8, 1 : 0.7 : 0.8 for both ${{E}_{initial}}$ and ${{K}_{I0}}$. This demonstrates that improving the Young’s modulus of ferroelectrics is an effective way to improve their intrinsic fracture toughness.

2) The ${{K}_{Ivnb}}$ of PZT-4, -5, and -8 are 0.91, 1.10, and 0.78 MPa·m^{1/2}, respectively. The controversy between ${{K}_{Ivnb}}$ and ${{K}_{I0}}$ revealed can be rationalized by the toughening mechanism of ferroelastic switching. These results demonstrate that by optimizing the ferroelastic switching behavior of piezoelectric ceramics, their extrinsic effect of $K_{\mu }^{max}$ could be improved.

3) The difference between the observed intergranular and transgranular crack surfaces in different PZT is also related to the ferroelastic switching. Both the fracture mode and ferroelastic switching related to the grain size, so the fracture toughness of PZT ceramics with defined composition could be optimized by changing the grain size.

Supporting Materials

Supporting materials related to this article can be found at
https://doi.org/10.15541/jim20220638.

Correlation between Constitutive Behavior and Fracture Performance of PZT Ceramics

WANG Xueyao^{1}, WANG Wugang^{2}, LI Yingwei^{1}, PENG Qi^{1}, LIANG Ruihong^{2}

(1. School of Civil Engineering, Wuhan University, Wuhan 430072 Wuhan, China; 2. Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, China)

S1 Preparation of ceramics

The materials were fabricated by a standard solid-state synthesis process using high-purity powders as raw ingredients. The weighted powder was first processed for 4 h in a planetary ball mill, after which the slurry was taken out and dried, pressed into large pieces, and synthesized for 2 h at 850 ℃. The powder was then crushed and re-milled in a planetary ball mill with the appropriate amount of water for 6 h. After extraction and drying, PVA is added to the powder for granulation, which is then pressed into rectangular strips using a uniaxial press. The pressed sample was sintered at 1200-1300 ℃ for 2-3 h after being heated to 650 ℃ for 1 h to volatilize PVA.

Before testing, all samples were annealed at 450 ℃ for 1 h to release the residual stress induced during cutting. According to the SEVNB technique reported by Kübler^{[1]}, the used samples were machined into bar shape with dimensions of 3 mm×4 mm×27 mm. For each data point of the fracture toughness, at least five measurements were taken, then the average value and the standard deviation were calculated to ensure representative results. The dimension of the samples used to characterize the stress-strain curves is 4 mm×4 mm×10 mm.

Table 1.

Microstructural properties of the investigated materials

Microstructural properties of the investigated materials

Material

Density /(g·cm^{-3})

Grain size/μm

PZT-4

7.7

(2.5±0.5)

PZT-5

7.8

(5.0±0.5)

PZT-8

7.6

(2.5±0.5)

S2 Details of stress-strain test

The deformation in the direction along and perpendicular to the applied stress were both measured uring uniaxial compression. The compressive stress is applied to the specimens by a screw-driven testing machine, together with a spherical hinge to avoid any bias stress. Strains gauges are glued on the four 4 mm×10 mm surfaces of the specimens to measure the longitudinal and transversal deformation. The signals of the stress and strain are input into an A/D data acquisition card monitored by a computer. During testing, a preload of about 20 N was applied on the specimens firstly, and then the strain signals were checked, based on which the position of the specimens was adjusted to assure uniaxial compression without bending. The loading rate is set to be 1 MPa/s, and unloading starts with rate of 1 MPa/s when the applied load reaches 400 MPa.

S3 Details of fracture toughness test

The V-shaped notch was introduced at the center of the 3 mm×27 mm surfaces of the bar samples by a cutting machine provided by Wuhan Dislocation Technology Company. A servo motor guides the straight reciprocating motion of a razor blade. Diamond particles were pasted on the razor blade to aid the cutting speed, which intended to introduce a V-notch on the surface of the sample. The moving speed of the razor blade and the cutting depth of the V-shaped notch are both controlled automatically. In addition, the cutting machine can control and monitor the contact force between the blade and the sample. Under the contact force of about 1 N, the slow cutting speed can greatly avoid the damage to the notch. During operation, a V-shaped notch with a depth of ~1 mm was cut on the sample first, and then replaced the old blade by a new one to finish the cutting process. After cutting, no obvious damage was observed near the notch by optical microscope.

The geometric dimensions of the V-notch crack were characterized under optical microscope. The determined depth d, the radius of the V-notch root r, and the angle θ between the two surfaces vary from 0.89 mm to 1.18 mm, 3 μm to 8 μm, and 24° to 30°, respectively, satisfying the requirements of the SEVNB technique^{[S1]}. The specimens were then fractured by four-point bending with an inner span of 10 mm and an outer span of 20 mm. During testing, the applied load increased monotonically at a rate of 0.5 mm/min until the sample fractured. After fracture, crack surfaces were characterized by scanning electron microscopy (SEM).

For Eq. (1), ${{Y}^{*}}$ is a dimensionless correction factor based on the ratio of $a/W$, which has the form:

First, the surface of the sample containing the indentation is meticulously polished in a series of steps using 10–0.1 mm diamond paste and ultrasonically cleaned after polishing. The cracks were induced by Vickers indentation with applied load of 49 N for 10 s. The gold was then sputtered onto the same surfaces as the conductive layer. A magnification of 70000 times was taken with an SEM at the crack‐tip region and 35000 times at sufficiently large crack opening displacement. The step size between the images were increased with increasing distance to the crack‐tip and chosen magnification. Finally, the semi-automatic procedure method developed by Vögler^{[S2]} was used to calculate the crack opening displacement $u$ as a function of the crack tip distance $x$. For every image, the crack was selected via a gray threshold and divided into 70 equally large segments. For each segment, the average crack opening displacement was calculated^{[S2]}.

The crack opening displacement field can be characterized using the equation derived by Fett, et al^{[S3]} for semi-elliptical cracks caused by Vickers indentation:

and $x$ represents the distance from the crack tip, a represents the length of the crack measured from the center of the indentation to the crack tip, $b$ represents the half-diagonal length of the indentation, ${E}'$ represents the effective Young's modulus of the material. It is sufficient to use the coefficients ${{A}_{0}}$–${{A}_{2}}$ for cracks between 1.4<$a/b$<3.5 in the calculation. Because the crack lies in the plane of the strain, the effective Young's modulus is used^{[S3-S4]}:

${E}'=E/(1-{{v}^{2}})$

where $v$ is the Poisson's ratio.

Assuming ${{K}_{I0}}$= 1 MPa·m^{1/2}, and combining the measured $a$, $b$, ${Y}'$, $x$, we may get the computed crack displacement ${{u}_{comp}}$ by utilizing formulas (S4-1, 2), then plot together with the measured crack displacement ${{u}_{meas}}$ results. And the actual crack-tip fracture toughness of the material can be derived from the slope of the fitted line through the origin dot, as illustrated by the red line in the Fig. 2.

Fig. S1 shows that the measured COD gradually deviates from the Irwin parabola fitting curve, which is mainly related to two reasons: (i) The nonlinearity of the materials will affect the crack-tip opening displacement; (ii) The crack-tip opening displacement is affected by the indentation residual stress field. However, it should be pointed out that since the ferroelastic switching has been completed at the near-tip (the region within 10 μm), the residual stress field caused by the indentation will not affect the opening displacement in this region (the opening displacement at the near-tip is controlled by the near-tip stress field). Therefore, only the region within 10 μm of the crack tip is selected for fitting.

Figure S1.COD profiles for (a) PZT-4, (b) PZT-5, and (c) PZT-8The Irwin parabola is represented by the red line

S5 Mechanical properties of the investigated materials

By dividing the strain with stress, the loading part of the stress-strain curve can be replotted as $d\varepsilon /d\sigma $ as a function of the applied stress, and based on which, we can figure out an inflection point that is normally defined as coercive stress of ${{\sigma }_{c}}$^{[S5]}. At ${{\sigma }_{c}}$, obviously, the materials present the maximum ferroelastic domain switching speed. For PZT-4, -5, and -8, the determined ${{\sigma }_{c}}$ are ~120, ~55, ~170 MPa, respectively.

During unloading, the strain displays linear behavior first and then follows a nonlinear performance. For PZT-4 and -5, significantly remanent strain was observed (Fig. 3(a, b)). The smallest critical stress was observed for PZT-5, with values of ~25 MPa (Fig. 3(b)); the largest critical stress was observed for PZT-8, with values of ~125 MPa (Fig. 3(c)); intermediate critical stress of ~40 MPa was determined for PZT-4 (Fig. 3(a)). When the applied stresses reach 400 MPa, the measured longitudinal strain ${{\varepsilon }_{33}}$ for PZT-4, -5, and -8 are 0.55%, 0.73%, and 0.64%, respectively. The measured strains at zero stress from the data of ${{\varepsilon }_{33}}$ are normally defined as the remanent strain ${{\varepsilon }_{r}}$, which are 0.18% and 0.32% for PZT-4 and -5, respectively. However, for PZT-8, the plastic deformation observed during loading was completed recovered during unloading (Fig. 3(c)). That is, the measured ${{\varepsilon }_{r}}$ is zero.

Additionally, based on the obtained stress-strain curves, the Young’s modulus $E$ and the Poisson’s ratio $\nu $ of the tested materials can be determined. The Poisson's ratio $\nu $ is calculated from the ratio of transverse strain to longitudinal strain. The $E$ recorded during initial loading (${{E}_{\text{initial}}}$) represents the Young’s modulus of the material in the unpoled state; the $E$ determined during initial unloading from the maximum stress level (${{E}_{\text{unloading}}}$) represents the Young’s modulus along the loading direction for the mechanically textured sample. Similarly, two different values of the Poisson’s ratio can be determined. For the material in the unpoled state, the initial Poisson’s ratio ${{\nu }_{\text{initial}}}$ for PZT-4, -5, and -8 are 0.4, 0.33, and 0.12, respectively; the Poisson’s ratio determined during initial unloading from the maximum stress level (${{\nu }_{\text{unloading}}}$) are 0.35, 0.33 and 0.27 for the mechanically textured PZT-4, -5, and -8, respectively.

The determined values of coercive stress $~{{\sigma }_{c}}$, the remanent strain ${{\varepsilon }_{r}}$, the Young’s modulus $E$ and the Poisson’s ratio $\nu $ of the tested materials tabulated in Table S2.

S6 SEM micrographs of the fracture surfaces

It can be observed that the fracture surface morphology of the materials was consistent and without evolving during crack propagation from the micrograph characterized by SEM. Therefore, a representative fracture morphology micrograph was selected for each material.

S7 Proportionality factor

According to the previous investigations, the shielding fracture toughness $K_{\mu }^{max}$ can be estimated by the following equation^{[S6]}:

Here λ is the proportionality factor induced by the domain switching behavior difference under tensile and compressive stresses. Near room temperature, the tested material is composed of the rhombohedral (R) phase and the tetragonal (T) phase, and the fraction of the R phase and T phase are nearly equal, then λ is nearly equal to 1.41^{[S7]}. The other parameters have been determined based on the stress-strain curves displayed above.

Table 2.

Determined values of ${{\sigma }_{c}}$, ${{\varepsilon }_{r}}$, ${{E}_{\mathbf{initial}}}$, ${{E}_{\mathbf{unloading}}}$, ${{\nu }_{\mathbf{initial}}}$, and ${{\nu }_{\mathbf{unloading}}}$

[S1] KUBLER J. Fracture toughness of ceramics using the SEVNB method a joint VAMSA/ESIS round robin. Fract. Mech. Ceram., 2002, 13: 437.

[S2] VOGLER M, FETT T, RODEL J. Crack-tip toughness of lead-free (1−x)(Na_{1/2}Bi_{1/2})TiO_{3}-xBaTiO_{3} piezoceramics. J. Am. Ceram. Soc., 2018, 101(12): 5304.

[S3] SCHNEIDER G A, FETT T. Computation of the stress intensity factor and COD for submicron sized indentation cracks. J. Ceram. Soc. Japan, 2006, 114(1335): 1044.

[S4] FETT T, KOUNGA NJIWA A B, RODEL J. Crack opening displacements of Vickers indentation cracks. Eng. Fract. Mech., 2005, 72(5): 647.

[S5] Zhou D, KAMLAH M, MUNZ D. Effects of bias electric fields on the non-linear ferroelastic behavior of soft lead zirconate titanate piezoceramics. J. Am. Ceram. Soc., 2005, 88: 867.

[S6] DENKHAUS S M, VOGLER M, NOVK N, et al. Short crack fracture toughness in (1−x)(Na^{1/2}Bi^{1/2})TiO_{3}-xBaTiO_{3} relaxor ferroelectrics. J. Am. Ceram. Soc., 2017, 100(10): 4760.

[S7] LI F X, FANG D N, SOH A K. Theoretical saturated domain- orientation states in ferroelectric ceramics. Scr. Mater., 2006, 54(7): 1241.