Mechanical characterization of piezoelectric materials: A perspective on deformation behavior across different microstructural length scales

Piezoelectric materials (hereafter abbreviated as PEMs) have extensive applications in various solid-state devices due to the growing interdependence between various multidisciplinary fields. The dielectric, piezoelectric, and electromechanical properties (Fig. 1) of PEM can be tailored by the microstructural length scales (from atomistic and grain level) and hence are promising candidates for sensors, actuators, semiconductors, ultrasonic biomedical transducers, many electronics, and energy harvesting devices.^1,2 With the advent of micro- and nanotechnology, there is a continuous thrust to employ these PEM as thin films (approximately few nm in size) in various micro-electromechanical (MEMS) and nano-electromechanical systems (NEMS). Consequently, the piezoelectric community has successfully incorporated the multifunctionality of PEM (i.e., polarization characteristics, dielectric, elastic, and piezo/ferroelectric properties) at reduced length scales (i.e., in thin films, MEMS, and NEMS devices). For example, modern multilayer piezoelectric actuators (MPEAs) consist of stacked layers of thin films of conventional PEM such as barium titanate (BT) and lead zirconate titanate (PZT) embedded with metal electrodes (Fig. 2). These MPEAs, compared to their contemporary bimorphs, can offer small/fine movement (of order approximately sub-μm to nm range) in many optical devices, biomedical positioners, and robotic arms where precise positioning is paramount.^3,4

The applications of PEM extend across various interdisciplinary fields, including defense, aerospace, automobile, textile, optoelectronics, and medical science. The PEM used in the above applications are mostly a ceramic compound (single or polycrystals) having a perovskite $(ABO3)$ crystal structure (Fig. 1). A cubic symmetry of these perovskites can be realized above the Curie temperature (⁠$Tc$⁠) where the piezoelectric effect disappears, while tetragonal, rhombohedral, and monoclinic non-centro symmetry is possible below $Tc$⁠, thus the piezoelectric effect is evident. It can be noted that the existence of tetragonal, rhombohedral, and monoclinic phases also depends on the stoichiometric elemental composition of PEM relative to the morphotropic phase boundary (MPB). Often with the design and fabrication of PEM and their subsequent applications in MEMS and NEMS devices, the incorporation of thin layers of ceramics between metal electrodes is a challenging task. Moreover, obtaining high (desired) piezoelectric characteristics (i.e., high electric field and large elongations) in a very short response time (approximately, a few milliseconds) and at relatively low voltages is also of utmost importance from the functionality point of view. However, the above functionality is closely associated with the microstructural features in PEM,⁵ which can be controlled either during fabrication or by employing post-fabrication treatments such as aliovalent cation doping (both donor and acceptor), changing stoichiometric composition (relative to MPB), introducing mechanical stresses during poling process, patterning/grating nano metal oxide film and thermal annealing, etc. These microstructural aspects in PEM span across several length scales (Fig. 1), starting from individual perovskite unit cells (Å size) to grains and grain boundaries (GB) (of μm or sub-μm size). The intermediate length scale between unit cell and grains is nano-sized defects, such as dislocations, ferroelectric domains (also known as polarization vectors), and polar nano regions (PNR) (of sub-nm size), thought to have a pronounced influence on the mechanical properties of PEM, as indicated by past and recent studies.^6–32 While ferroelectric domains alone contribute to the dielectric and piezoelectric properties, dislocation–ferroelectric domain interactions control the mechanical properties of PEM,^{7,8,13,15–18,23,26,29–32} and recent studies have shown that the mechanical properties can also be tailored by taming a systematic network of dislocations^23,26 and ferroelectric domains.^{5,22,24,25,27,28,32} However, the defect-based multifunctionality in PEM is still in the nascent stage^29–31 and is also a timely topic.

The applications of piezoelectric ceramics as a bulk (i.e., at mesoscale, say, pallets of few mm) in military impact fuzes, fuel injection systems, and naval hydrophones are widely recognized. The main function of these PEM devices is to initiate the detonation upon impact, timely closer and opening of valve, and extraction of underwater acoustic signals (response), respectively. Conventional PEM such as BT and PZT, as well as advanced $Ba(ZrxTi1−x)O3−(BayCa1−y)TiO3$ (BZT–BCT) and potassium sodium niobate (KNN)-based Pb-free ceramics have been successfully tested for the above applications and previously several studies have reported their mechanical properties via bulk deformation testing that includes uniaxial compression (both room and controlled temperature environments),^{6,7,9,10,12,18,19,20} R-curve behavior,^12,17,19,20 and macro-scale indentation experiments.^6,33–36 The R-curve (sometimes also referred to as the tearing resistance curve), a curve representing a change in the fracture resistance with respect to crack extension is of paramount importance in fracture mechanics, particularly in ceramics as a high magnitude of stress field is induced at the growing crack tip. This stress field may arise due to thermoelastic mismatch between target ceramic and the material that is in contact with the ceramic, surface damage due to point contact, and phase transformation by the application of high loads. Note that one should always consider the magnitude of R-curves till the stable crack generation (or stable tearing) and the further extension after the onset of brittle fracture is not accountable. The arguments in all the above studies are mainly based on the hypothesis with no microstructural evidence, which questions the validity of these studies. This is due to the limitations associated with the conventional macro-scale deformation experiments in unraveling the early-stage plasticity, observed at the μm or sub-μm scale. The advances in mechanical testing techniques such as micro-pillar compression^23,37,38 and micro- and nano-indentation^24,39–41 have led the piezoelectric community to probe the mechanical response of individual grains and domains of PEM. In the recent past, with further advancements in micro- and nanotechnology, PEMs have extended the freedom of applications in MEMS and NEMS devices. However, most of these PEM, during their applications in MEMS and NEMS, undergo high-frequency contact loading environments. Such high-stress conditions are detrimental enough to cause the deterioration of functional properties. As indicated earlier, multifunctionality in PEM is directly related to the microstructural features present at sub-μm length scales; therefore, it is also imperative to understand the collective mechanical response of clusters of dislocations and ferroelectric domains. On this front, the instrumented indentation techniques (IITs) such as nano- and pico-indentation are helpful in probing the nanoscale plasticity in piezoelectric ceramics as well as capturing deformation-induced phase transitions and the defect-initiated plastic deformation in PEM. Several studies in the past reported nanoindentation response of several Pb-based and Pb-free piezoelectric ceramics and evidenced dislocation-mediated plasticity^{11,15,16,18,23,26,42–44} and ferroelectric activities.^{5,8,11,13,14,16,18,22,23,25,27–29,41} Particularly, the works of Schneider et al.¹¹ and Kathavate et al.⁴¹ have provided direct observations of dislocation and ferroelectric distortion within and around the indentation zone, respectively, which was a significant open problem in the literature.

With the advent of small-scale technology in recent decades, nanoindentation technique is emerging to be a versatile tool to characterize the mechanical properties of materials at lower length scales. The first existence of IITs as a load (vis-à-vis depth) sensing instrument was found during the 1970s in Russia^45,46 while the incorporation of displacement (piezo) sensors (both open-loop and closed-loop feedback) for precise depth sensing and loading made it possible to monitor in situ mechanical response (i.e., load, P vs displacement, h curves) of the test material. The precise loading of the indenter tip in the conventional nanoindenter was typically based on a closed-loop system, while the incorporation of open-loop displacement sensors in the advanced nanoindenter is advantageous to probe the material's creep displacement at room temperature.⁴⁷ Nowadays, a nanoindenter can generate the P–h curves with displacement and load resolution as small as ∼5 nm and a few nN, respectively, thus making them suitable to probe the deformation response of underlying microstructural features (say, an individual grain or clusters of dislocations/ferroelectric domains). Some of the unique advantages of nanoindentation techniques over bulk deformation testing techniques and conventional indentation are as follows: (a) mechanical properties such as hardness, H, and elastic modulus, E, can be directly measured from the analysis of P–h curves [Fig. 3(a)], thus eliminating the tedious process of measuring the dimensions of indentation impression, (b) a fully plastic regime in the material can be obtained without fracture in the load range of several nN to a few mN [Figs. 3(b) and 3(c)] since the indentation impressions are confined to a very small area (say, up to a few μm), it is recognized as a non-destructive technique. Three types of indenters Berkovich, conical, and cube-corners are generally used in nanoindentation experiments. The difference in these indenters arises mainly from the geometry, the shape of the indenter [Fig. 3(c)],⁴⁸ and their suitability on the test materials into which these indenters are being punched. Among all the indenters, Berkovich indenters are quite popular due to their geometrical self-similarity and the ability to manufacture sharper tips (∼50 nm tip radius). The typical centerline to face angle, ψ, for the Berkovich tip is ∼65.3°, while it is ∼35.3° for cub-corner indenters, as shown in Fig. 3(c). Jang and Pharr⁴⁹ demonstrated the marked influence of ψ on the fracture toughness and crack propagation in $[100]$ oriented Si and Ge single crystals. Using cube-corner and pyramidal indenter (ψ, 35.3° to 85°), they observed a gradual decrease in the crack length and well-developed radial cracks with an increase in ψ. Despite the same projected area at a given load, the sharper indenter assists the more displacement of the material and consequently, the greater magnitude of stresses, thereby guiding crack propagation. Therefore, one may note that in addition to the intrinsic properties, a nanoindentation response of a material is also dependent on the type/geometry and size of the indenter.

The International Standards Organization (ISO-14577 parts 1–4)⁵⁰ and American Society of Testing of Materials (ASTM E2546)⁵¹ provided important information related to the sample preparation techniques, allowable surface roughness, and standard methods for performing nanoindentation tests on the different materials, while the Oliver and Pharr (O&P)^52,53 method directs the guidelines and a procedure for the analysis of nanoindentation data. As indicated earlier, H and E values of the test material can be directly estimated from the P–h curves, the first step in the estimation of H and E values begins with confirming the validity of nanoindentation experiments by fitting the obtained P–h curves according to the following O&P equation:⁵² 

Here, P represents the indentation load, while h and $hf$ specifies the instantaneous and final penetration depth. The power law index, m is indicative of the validity of nanoindentation experiments and values of which should essentially fall in the range of 1–2. For instance, the typical m values reported in the literature for the flat punch are equal to 1, while they are ∼1.2–2 for all the other indenters.^52,53 The next step in determining the H and E values according to below equations (2) and (3)^52,53 is to evaluate the following parameters [as shown in Figs. 3(a) and 3(c)]: (i) maximum indentation load, $Pmax$⁠; (ii) maximum displacement, $hmax$⁠, and contact depth, $hc$⁠; and (iii) unloading stiffness, S, slope of the initial part (elastic) of unloading curve $(dPdh)$⁠,

The factor 24.5 in Eq. (2) considers the ideal Berkovich tip, which further reflected from the area function calibration polynomial series.^52,53 Where $Ac$ represents the projected contact area. During indentation, pile-up and sink-in [Fig. 3(b)] around the imprint edges are observed in high strain-hardening and low strain-hardening materials, respectively.⁴⁶ One of the limitations of the O&P method is that it does not account for pile-up corrections. However, H and E values can be determined after eliminating the sink-in effects based on corrected $hc$ as per the below Eq. (4):^52,53

Here, the typical value for indenter geometric constant β for the conical indenter is 0.72, 0.75 for the spherical tip and Berkovich indenters, and 1 for flat punch.^52,53 Finally, the elastic modulus of specimen, $Es$⁠, can be calculated from $Er$ according to the following Eq. (5):⁵² 

where $ϑ$ represents Poisson's ratio and subscripts i and s specify the indenter and specimen, respectively. The reported values of E and $ϑ$ for the diamond tip in the literature are ∼1170–1200 GPa and ∼0.07, respectively.^46,54

In summary, the next step after the materials design process is to establish the structure–property correlationship. In view of this, nanoindentation has gained considerable attention of materials scientists to characterize the mechanical properties of materials, particularly, at reduced length scales, and, therefore, a useful tool for probing the mechanical response of an individual grain and/or clusters of ferroelectric domains and domain walls (DWs) in piezoceramics.

How can we visualize the atoms on the surface of the materials? Until the discovery of the scanning tunneling microscope (STM) in 1981, probing of an individual atom was subtle. It was the first time in the history of materials science, the visualization of a single atom was made available to materials scientists. STM works on the principle of the passage of tunneling current between the biased tip and target sample. Soon it was realized that the flow of tunneling current when tip and samples are a few distance apart creates the collateral forces. This limitation of STM led to the discovery of the atomic force microscope (AFM), thinking that these unwanted forces can be used to probe atomic interactions, and hence the name atomic force microscope.⁵⁵ The main advantage of AFM over STM is that AFM can probe the atomic interactions on non-conductive surfaces without creating contact between the tip and surface (i.e., non-contact mode). Both STM and AFM were invented by Binnig et al.^55–57 in 1981 and 1986, respectively. The schematic in Fig. 4 demonstrated the basic principle of STM and AFM to characterize the atoms or atomic interactions on the surface. Recently, nanoindentation experiments using AFM tip become an increasingly interesting topic to probe the nanomechanical properties of thin brittle materials [for example, one atom thick graphene, MoS₂, and hexagonal-boron nitride (h-BN) thin films].⁵⁸ 

A piezoresponse force microscope (PFM) is a channel equipped in AFM in addition to its working principle, wherein the application of the direct current (DC) field probes the movement of nano-assembled ferroelectric domains in piezoceramics and multiferroic materials using bias-induced surface deformation. In 1992, Güthner and Dransfeld⁵⁹ invented PFM to probe the electromechanical characteristics in piezoelectric materials and semiconductors. In recent times, applications of PFM are also extended to the biological field to probe the mechanobiology in teeth, bone, and lungs.⁶⁰ PFM technique allows the characterization of ferroelectric activities such as ferroelectric domain switching, identification, and the movement of ferroelectric DW over an area as high as 100 × 100 μm² with the help of a conductive cantilever tip. Piezoresponse force spectroscopy (SS-PFM) is an additional advantage, which gives quantitative information on piezoelectric and dielectric properties. The conductive cantilever with curvature ∼10–20 nm, generally made of dense, hard, and lustrous coating (for example, Ti/Ir) performs two main functions: (a) collects local actuation strain on the surface upon electrical excitation (i.e., actuation or converse piezoelectric effect) and (b) probes the electromechanical properties via controlling mechanical motion (i.e., sensor or direct piezoelectric effect).⁶¹ 

PFM, within its predictive capabilities, can provide an ample amount of qualitative and quantitative information about piezoceramics. For instance, the PFM phase image provides information about the distribution of various ferroelectric domains in a crystal, as shown in Fig. 5(a).²⁸ The density of the DW can be quantitatively estimated while analyzing the PFM phase images by performing a line scan [Fig. 5(b)].²⁸ On the other hand, SS-PFM quantitatively demonstrates the switching ability of ferroelectric domains and local hysteresis in the material using amplitude vs electric field/bias (butterfly loop) and phase vs electric field/bias (hysteresis loop) curves, respectively, as presented in Fig. 5(c).²⁸ Furthermore, the PFM amplitude image shown in Fig. 5(d)^,62 demonstrates the piezo-activeness of PEM. Generally, higher amplitude values upon DC excitation indicate higher piezo-activeness/coefficient of the material. The information about the analysis and interpretation of PFM images is well documented in Refs. 28 and 62–64 Some of the key points to understand on this front are (i) the different contrasts and contours present in PFM phase images indicate the state/orientation of various ferroelectric domains (i.e., “$c−$”, “$c+$⁠,” and “$a$” domains), (ii) “$a$” domains are normal to “$c$” domains (i.e., separated by 90° or ferroelastic DW), while two adjacent “$c+$” and “$c−$” domains are separated by 180° DW (also known as ferroelectric DW), schematically represented in Fig. 5(e), and (iii) local piezoelectric charge coefficient, $d33∗$⁠, values (pm/V) can be obtained after averaging the slope of positive and negative sides of the butterfly curve in Fig. 5(c).²⁸ 

In summary, PFM is an effective tool that characterizes the ferroelectric domain interactions within the piezo-crystals. With the advent of technology, applications of PFM are extended in the field of mechanobiology, crystal orientation determination via vector-PFM (V-PFM), resonance-PFM (R-PFM) for probing coupling characteristics in piezoelectric materials with relatively weaker piezo-charge coefficients, and in situ conductive/AFM-nanoindentation in the controlled environment (i.e., electromechanical).

Although the mechanical properties of various PEM and related characterization techniques across different length scales are well documented in the previous studies presented in Sec. I, there is an emerging need to review all such aspects comprehensively, thus providing the prospects for further development in the field. Therefore, we overviewed the previous literature (until December 2021) and assimilated different mechanical testing techniques and mechanical properties of PEM considering microstructural activities.

Keeping this in view, the present Tutorial is organized in the following manner: Sec. II presents the mechanical behavior of Pb-based piezoelectric materials, subsequently followed by Sec. III outlining the mechanical characteristics of Pb-free piezoceramics. A general discussion based on the authors' observations, current challenges, and key prospects is highlighted in Sec. IV. Section V provides a summary of the Tutorial with some important conclusions from the review of past literature. Complex mathematical formulations and simulations are avoided, rather emphasis is given only to the experimental characterization, thereby serving the purpose of the Tutorial.

In this section, we overview some mechanical testing techniques and changes in mechanical properties of various Pb-based piezoceramics such as PZT (including undoped, hard-doped, and soft-doped), lead magnesium niobate-lead titanate (PMN-PT), and lead zinc niobate-lead titanate (PZN-PT).

PZT represents a solid solution of lead zirconate $(PbZrO3)$ and lead titanate $(PbTiO3)$ and is a potential material for automobile (fuel injection systems), piezo-generator and piezo-transducer applications.^27,28 The stoichiometric composition $(PbZr0.52Ti0.48O3)$ is widely preferred due to the existence of tetragonal non-centrosymmetry (i.e., composition near to MPB);^1–2 however, the emerging need to enhance the multifunctional properties of PZT necessitates to alter the microstructural aspects of undoped PZT by adding suitable acceptor and donor dopants. PZT obtained by doping acceptor dopants such as $K+$⁠, $Na+$ (for replacing “A” site cation) and $Al3+$⁠, $Mn3+$⁠, $Fe3+$ (for replacing “B” site cation) are referred to as hard-doped PZT (PZT-H) in the sense that their ferroelectric domain configuration is difficult to reorient upon the application of the electric field (i.e., difficult to polarize).^7,27,28,65 On the other hand, soft-doped PZT (PZT-S) are obtained by doping suitable donor dopants such as $La3+$⁠, $W6+$ (for replacing “A” site cation) and $Sb5+$⁠, $Nb5+$ (for replacing “B” site cation) that are easy to polarize and exhibit large polarization values compared to the counterpart PZT-H.^7,27,28,65

The mechanical characterization of PZT has always been a subject of interest to the piezoelectric research community. It was Grekov and Kramarov⁶⁶ in 1978 and Okazaki⁶⁷ in 1983, who reported the mechanical characterizations of bulk PZT via compression tests. They mainly argued the orientation dependence (anisotropy) of mechanical strength in PZT due to the combined influence of electric and mechanical stress fields (i.e., coupling characteristics). However, from the theoretical perspective, the initial work on the fracture mechanisms in PZT is thought to be reported by Parton⁶⁸ in 1976, followed by Bruce et al.⁶⁹ in 1978 and a notable review article by Cook et al.⁷⁰ Subsequently, Pisarenko et al.⁷¹ investigated the anisotropy in fracture toughness, $KC$ in $Ba$-doped (i.e., PZT-H), $Nb$ and $W$-doped (i.e., PZT-S) PZT via double torsion (DT) (Fig. 6) and Vickers indentation. All the samples exhibited similar $KC$ values (∼1.50 ± 0.20 MPa m^1/2) obtained via DT and Vickers indentation within the standard deviation. Furthermore, $KC$ values obtained for the loading direction parallel to the ferroelectric domain orientations (⁠$kc∥$⁠) were slightly higher than those obtained when loaded perpendicular to ferroelectric domain orientations, (⁠$kc⊥$⁠), indicating polarization-dependent nature of $KC$ in PEM. The anisotropy in $KC$⁠, determined as $kc∥kc⊥$ was found to be ∼0.88 for all the samples, suggesting the difference in crack propagation characteristics due to ferroelectric domain reorientation near the crack tip upon unloading due to relaxation of stresses.

In 1990, the pioneering work of Mehta and Virkar⁶ reported the fracture behavior of single-phase tetragonal PZT (⁠$Zr/Ti=0.54/0.46$⁠) with different ferroelectric domain configurations via the single edge notched beam method (SENB) and Vickers indentation. PZT specimens were annealed at different temperatures below and above the $Tc$ (⁠$Tc=350°C$ for PZT) and the ferroelectric domain configurations are systematically varied [Fig. 7(a)]. The reported $KC$ values for PZT samples decrease with an increase in temperature [Fig. 7(b)]. For instance, the $KC$ values for as poled (AP) PZT at room temperature was ∼1.85 MPa m^1/2, while it was ∼1 MPa m^1/2 for the samples annealed above the $Tc$⁠, indicating the dependence of ferroelectric domain configurations on the mechanical properties [Fig. 7(b)]. Their observations also revealed the ferroelectric domain switching process (or domain-twinning) as a viable toughening mechanism in PZT, which is applicable for all piezoceramics. For instance, cracks originating from the corners of indentation imprint in the perpendicular direction are relatively shorter than those emanating in the parallel direction of polarization (Fig. 8). This could be due to significant reorientation of ferroelectric domains ahead of the crack tip that propagates perpendicular to the poling direction. On the other hand, ferroelectric domains exhibit a 90° switching (probed via x-ray diffraction, XRD, technique in the indentation/fractured vicinity) during the propagation of cracks that are parallel to the polarization. The work of Mehta and Virkar⁶ has set nice guidelines for the PEM community regarding the ferroelectric distortion during mechanical testing of PEM.

Following this, static compressive and fatigue strength of PZT under different poling (poled vs unpoled) and electrical conditions (i.e., electrically conducting vs insulating) has been reported by Okazaki and Tanimoto⁷² via uniaxial static and cyclic compression tests (Fig. 9). Interestingly, both static and fatigue strengths showed a marked dependence on the electrical loading conditions during the tests, as poled PZT samples in insulating environments exhibited higher strength and fatigue life than in the electrical conducting environment. The microstructural examination revealed the intergranular cracks during the failure of poled/insulated PZT samples, while transgranular fracture was realized in poled/conducting combination. This could be due to the lattice strain incompatibilities associated with ferroelectric domain switching during the electromechanical loading, which was later substantiated by the studies of Schneider et al.^8,11 and Kathavate et al.^5,28,41 Among all the samples, unpoled PZT exhibited relatively better fatigue than poled PZT (Fig. 10). Subsequently, several studies^73–79 have also indicated the significant influence of the nature of the specimen (i.e., poled vs unpoled) and electrical conditions on the fracture behavior and mechanical properties (mainly compressive strength) of PZT.

Although the above studies reported the mechanical properties and fracture behavior of PZT under electromechanical conditions, the same was not very well understood in thermo-mechanical environments until 2006, though PZT often can undergo such kind of loading conditions. Wang et al.⁸⁰ reported the mechanical properties of PZT-S (La-doped) in a thermo-mechanical environment via the dynamic mechanical analysis (DMA) technique. They demonstrated the memory effect in PZT and hypothesized the domain switching dependent deformation behavior of PZT. Following this, the pioneer works of Webber, Rödel, and co-workers^17,19,81 cover the profound discussion on the compressive strength and R-curve behavior during temperature-controlled uniaxial compression testing of PZT-S (mainly La-doped). The novel test setup developed by Webber, Rödel, and co-workers^17,19,81 is shown in Fig. 11(a). They mainly argued that the R-curve behavior in PZT could be primarily attributed to non-180° domain switching at the crack tip. As evident in Fig. 12(a), elastic modulus, E (or materials stiffness), showed a marked dependency on the temperature initially up to a compressive stress of $100MPa$⁠, where complete ferroelectric domain switching can be realized. However, beyond the critical stresses, $σcrit$ (say, $100MPa$ in this case), ferroelectric domain switching saturates, therefore, the plateau region at maximum E values. While upon unloading, E decreases again (by 45%–50%) depending on the test temperature, possibly because of domain back-switching upon the relaxation of compressive stresses. Meanwhile, the compression experiments [Fig. 11(b)] performed by Schäufele and Härdtl⁸² on PZT-H and PZT-S also confirm the above fact that $σcrit$ are necessary for the domain switching process. The plateau region at the maxima at all the temperatures indicates almost similar E values (∼120 ± 5 GPa) within the standard deviation [Fig. 12(a)], while the discernable difference (∼10%) in remanent strain, $εr$ values are evident [see the difference between the loading and unloading point on the curve in Fig. 12(a)]. The R-curve behavior shown in Fig. 12(b) and different $Kc$ values [i.e., initial $Kc$ (⁠$Kcini$⁠), maximum $Kc$ (⁠$Kcmax$⁠), and shielding $Kc$ (⁠$Kcshield$⁠)] in Fig. 12(c) also showed a marked dependence of the test temperature. The $Kcini$ values are more or less similar at all the temperatures ∼0.52 ± 0.8 MPa m^1/2, while there is a significant difference (∼20%) between the plateau region of $Kcmax$ values across all the test temperatures, which further also reflected in the difference in $Kcshield$ values (i.e., $Kcmax−Kcini$⁠). These differences in $Kc$ are mainly attributed to the ferroelastic domain switching, which eventually controls the crack growth (i.e., R-curve behavior) during temperature-controlled compressive loading. An increase in temperature increases the degree of randomly oriented ferroelectric domains,^{5,6,17,19,20,22,25,27,28,41,81} which further leads to reducing the energy barrier to overcome ferroelastic domain switching and subsequently decreases the $εr$ values. Only a tiny fraction of ferroelectric domains is available for the switching and back-switching at elevated temperatures during the crack propagation, thus evidencing the combined coupling effect (i.e., ferro electro-elastic) on the mechanical behavior of PZT.

Although the above studies correlate the mechanical properties to ferroelectric activities during the mechanical loading, the lack of microstructural evidence poses a way forward to look for nanoscale characterization of these piezoceramics. Toward this front, nanoindentation experiments are likely to probe the collective response of clusters of ferroelectric domains and dislocations. The in situ monitoring of indentation load, P vs displacement, h characteristic (i.e., P vs h curves) makes nanoindentation a versatile technique to probe the small-scale plasticity (or elastic–plastic nature) (i.e., in the nm regime) in ceramics. Keeping this in view, Hurtado-Macias et al.,⁸³ in the case of PZT-S (with Zr/Ti ratio 43/57, 45/55, and 53/47), hypothesized the dislocation–ferroelectric domain interaction during nanoindentation. Nanoindentation experiments performed with three different indenters (i.e., conical, cube-corner, and Berkovich) and a load regime from 500 μN to 800 mN revealed that H in PZT-S is dependent on the indentation size, $hc$ (vis-à-vis indentation load, P). The nature of P vs h curves also indicated the existence of significant pop-in events (i.e., displacement bursts) during the indentations made with sharp conical and Berkovich indenters [Fig. 13(a)]. The existence of displacement bursts during indentation is indicative of simultaneous processes such as plasticity initiation, strain transfer across grain boundaries (in the case of polycrystals), and fracture events at imprint corners and beneath the indentation (or indentation vicinity). Dislocation emission and migration appear to be important mechanisms for the occurrence of such events. However, the statistical (Weibull) distribution of the above competing mechanisms is necessary to describe the pop-in events. Similar observations are also reported by Kathavate et al.²² during the nanoindentation experiments on PZT piezoceramics (undoped), wherein they attributed it to the ferroelectric domain switching process during the indentation. As evident from Fig. 13(b), H values in all the PZT-S showed a decreasing trend with an increase in $hc$⁠, indicating a strong indentation size effect (ISE) in H. The ISE are strongly attributed to the geometrically necessary dislocations (GND), whose density increases with an increase in strain gradient during the indentation. Besides this, they also argued the possibility of ferroelectric domain switching during indentation. However, the hypothesis proposed by Hurtado-Macias et al.⁸³ is mainly based on the mathematical formulations and theory proposed by Nix and Gao⁸⁴ and Swadener et al.,⁸⁵ and no microstructural evidence was reported.

Motivated by the above, recently, Kathavate and co-workers^{22,27,28,32,41} have provided a profound discussion on the nanoindentation response of different domain configured PZT (undoped), PZT-H, and PZT-S along with the direct observations of ferroelectric distortion in the indentation vicinity^32,41 using PFM. The state of partially and fully disturbed (i.e., PD and FD, respectively) ferroelectric domains in the as poled (AP) PZT was achieved by systematically annealing the specimen below and above the $Tc$ (i.e., 0.8$Tc$ and 1.2$Tc$⁠, respectively) [Fig. 7(a)], while the nanoindentation experiments were performed with the Berkovich indenter in the load range of 1 to 5 mN. The elastic–plastic nature of PZT-H and PZT-S can be realized from the P vs h curves shown in Figs. 14(a) and 14(b), respectively, while Fig. 14(c) represents the depth dependence of H values in PZT-S obtained using micro- and nanoindentation. They have also demonstrated the quantitative illustration of ISE in nanoindentation H in PZT-H and PZT-S in light of differences in ferroelectric domain configurations using empirical and semi-empirical models.²⁷ The H values in all the PZT show the trend; $HFD>HPD>HAP$⁠, indicating that ferroelectric domain configurations markedly effect the nanomechanical properties of PZT. The maximum penetration depth, $hmax$ at $Pmax=5mN$ is ∼180–200 nm, indicating the collective response of ferroelectric domains within an individual grain during indentation. The $Kc$ values obtained for PZT-H and PZT-S shown in Fig. 7(b) exhibit the trend; $KCAP>KCPD>KCFD$⁠, suggesting ferroelectric domain induced toughening in PZT. The characterization of ferroelectric domains using PFM revealed the existence of non-90° and non-180° domain walls (DW) in PD and FD PZT-H and PZT-S,²⁸ while the trend in H values was realized as an outcome of ferroelectric domain switching during loading and reorientation (or rearrangement) upon the relaxation of indentation stresses^32,41 and activation of defect dipoles and oxygen vacancies,²⁸ which further corroborates with the previous studies.^{6,17,19,81,82} It is evident from the Fig. 15(a) that the degree of ferroelectric distortion is maximum in the indentation crack wake zone (during crack growth), possibly due to the domain switching activities, while some fraction of the ferroelectric domains shows rearrangement at the crack tip after the removal of indentation load, as described schematically in Fig. 15(b). Table I below summarizes the $Kc$ values for various PZT and another Pb-based piezoceramics in different ferroelectric domain states reported in the literature. However, the shortcoming of these studies poses some important questions at this juncture; (i) Does ferroelectric distortion in the sub-surface vicinity during indentation matter?, and (ii) Among dislocation-mediated plasticity and ferroelectric domains assisted toughening, which is more dominant in the plastic deformation in PZT (in all the piezoceramic)? We believe the PEM research community is extensively working on the above problems and could foresee the potential to develop in situ small-scale testing techniques to probe the real-time electromechanical behavior of PZT. Although this section covers the essential aspects of the mechanical behavior of PZT across various length scales, we advise readers to go through the cited references^86–92 for further understanding of the subject.

The solid solutions of $(x)(PbCyNb1−y)O3−(1−x)(PbTiO3)$ (where C = Mg in the case of PMN-PT and C = Zn for PZN-PT) have attracted significant attention of the piezoelectric community due to their giant piezoelectric response (i.e., piezoelectric charge coefficient, d₃₃ > 3000 pC/N, electromechanical coupling coefficient, k₃₃ = 95% and piezoelectric strain, $εp=1.5%$⁠). Such ultrahigh piezoelectric properties of PEM make them promising candidates for next-generation high-performance sensors, actuators, and transducers.^93–95 Most importantly, unlike other PEM, PMN-PT, and PZN-PT can endure complex loading conditions, which further suits them for various demanding applications. For instance, in underwater acoustic transducers, these PEM are constantly preloaded with uniaxial compressive stresses to overcome the tension, wherein the simultaneous effect of internal stresses, external dynamic environment, and hydrostatic pressure can be realized.^96–98 Furthermore, the coupled thermomechanical environment in impact fuzes can induce significant ferroelectric domain switching and subsequent movement of MPB in PMN-PT and PZN-PT.^99,100 Besides this, there are multiple occasions where mechanical loading alone could be detrimental to structural distortion (i.e., movement of MPB) and subsequent deterioration in piezoelectric and mechanical properties of PMN-PT and PZN-PT (associated inherent brittleness could also be one of the reasons in many instances). Therefore, it is imperative to understand the mechanical behavior of PMN-PT and PZN-PT ceramics to suit them for the applications mentioned above.

The stress-induced electromechanical response of PMN-PT and PZN-PT (particularly, in single crystals) via uniaxial compression tests and microindentation studies has been investigated in the past,^101–108 indicating two dominant deformation mechanisms in piezoceramics: (a) ferroelectric domain switching (or polarization switching) and (b) subsequent phase transformations. For instance, Wan et al.^107,108 investigated the effect of crystallographic orientation on the stress-induced phase transformation and ferroelectric activities in PMN-0.32 PT single crystals. Among the three crystallographic orientations (i.e., [001], [011] and [111] [001] orientation undergoes phase transformation, while [011] and [111] orientations undergo significant ferroelectric domain switching during loading and reorientation of ferroelectric domains upon unloading. The $[001]$ oriented crystals exhibited systematic rhombohedral (R) to orthorhombic (O) phase transformations initially (up to compressive stress of 15–20 MPa) and then to tetragonal (T) phase with further increase in stresses which affect the electromechanical response. Interestingly, they witnessed an increase in piezoelectric (i.e., $d33$ values) and mechanical response (i.e., strain values, ɛ) [Figs. 16(a) and 16(b), respectively] with an increase in compressive stresses up to a critical limit (say, up to ∼20 MPa in this case). They attributed this to the reversible polarization rotations (i.e., phase transformation) from R to O upon loading and unloading, which eventually favors the enhancement in electromechanical response, while the electromechanical response is expected to disappear gradually at higher loads (say, beyond 20 MPa in Fig. 16), indicating the irreversible phase transformation from O to T. In our opinion, it can be construed that the stress-induced mechanical response of PMN-PT and PZN-PT ceramics and subsequent phase transformation depends on the composition near MPB, which further corroborates with the studies reported by Ahart et al.¹⁰⁹ 

In another study, Webber and co-workers¹¹⁰ have systematically demonstrated the effect of grain structure and polarization orientations on the R-curve behavior of PMN-0.29 PT single and polycrystals via uniaxial compression tests. The compact tension (CT) specimens of single-crystal PMN-0.29 PT were produced by the solid-state crystal growth technique and poled along $[100]$ and $[01¯1]$ crystallographic orientations and perpendicular cracks were realized at the surface [Figs. 17(a) and 17(b)]. The R-curve behavior in single crystals indicated the orientation dependence of crack growth (i.e., anisotropic behavior) during mechanical loading, as initial $Kc$ values (i.e., $Kc$ values at $Δa=0$⁠) of $[01¯1]$ oriented crystals were ∼25% higher than the $[100]$ oriented crystals, while this difference further grows up to four times at the maximum plateau region of $Kc$ values [Fig. 17(c)]. This further suggests that the $[100]$ orientation is highly favorable for the crack growth indicating the pronounced ferroelastic activities across $[100]$ orientation during mechanical loading. On the other hand, polycrystalline samples exhibited comparable (even slightly higher) $Kc$ values to $[01¯1]$ oriented crystals [Fig. 17(d)]. This behavior in polycrystals is plausible because of multiple simultaneous events such as crack bridging, crack deflection, and micro-cracks occurring during the crack growth, which is unlikely in single crystals. Similar behavior is also reported by Shang and Tan¹⁰⁶ and Ozgul et al.¹¹¹ during (micro)indentation-induced domain switching in PMN-0.35 PT and fatigue behavior of PZN-PT single crystals, respectively. In particular, Ozgul et al.¹¹¹ showed the extraordinary synergistic effects between DW pinning and induced thermal energy during ferroelectric domain switching when piezoceramic is subjected to fatigue cycling. It can be anticipated from the above studies that the effect of ferroelastic activities (i.e., domain switching and reorientation) is much more pronounced than the grain size during plastic deformation or toughening of these piezoelectric ceramics.

While the above studies have mainly emphasized the toughening behavior of PMN-PT and PZN-PT, recently, some focused studies have unraveled the elastic–plastic behavior and nanomechanical properties of these ceramics via the nanoindentation technique.^{5,16,18,25,112,113} For instance, Wong and Zeng^16,18 shed light on elastic–plastic deformation of PZN-(6–7)%PT single crystals during nanoindentation, particularly at high P values (∼50 mN). They argued that PEM deformation is mainly dependent on the relaxation parameter, which eventually comprises elastic, elastic–plastic, and plastic contributions. The material pile-up and local damage are evident at low P values (∼20 mN), which is manifested as the pop-ins in the P vs h curves [Fig. 18(a)]. Furthermore, the relaxation parameter calculated from the unloading part of P vs h curves increases significantly with an increase in P, possibly due to the stress-induced depolarization (i.e., depoling due to non-90° and non-180° DW) in poled samples. Interestingly, all the materials poled at different electric fields do not show discernible differences in P vs h curves (i.e., $hmax$ at $Pmax$ is ∼670–680 nm) [Fig. 18(b)], indicating that the poling (may be up to a certain limit, say, ∼15 kV/cm in this case) has a negligible influence on the deformation behavior of PZN-PT. On the other hand, the orientation of ferroelectric domains (or crystal orientation in this case) significantly affects the trends in P vs h curves, thereby indicating the ability of small-scale deformation testing techniques to probe the collective response of clusters of ferroelectric domains.

The nanomechanical properties of $[100]$ and $[110]$ oriented PMN-PT single crystals were probed by Jiang et al.¹¹² via nanoindentation with spherical and Berkovich tips using continuous stiffness measurement (CSM). The calculated indentation stress, $σi$ and strain, $εi$ values for $[100]$ oriented crystals were found to be 3.7 GPa and ∼4.8%, respectively, while it was 4.4 GPa and ∼6% for $[110]$ oriented crystals, suggesting an anisotropy ratio ∼0.8. The marked influence of ferroelastic activities could be seen, as the noticeable deformation bursts are evident in P vs h curves during nanoindentation. To our understanding, it can be realized that the lattice strain incompatibilities arising due to polarization switching (eventually due to lattice distortion) during indentation [Fig. 19(a)] may lead to the phase transformation and consequently affect the mechanical response in ferroelectric crystals. The nanoindentation experiments performed by Kathavate and co-workers^5,25 on different ferroelectric domains configured polycrystalline PMN-PT support the above fact (i.e., AP, PD, and FD). Importantly, unlike PZT, the trend in H values in PMN-PT is realized as $HPD>HFD>HAP$ [Fig. 19(b)] and could be an outcome of the activation of PNR in PD samples during DE of PMN-PT samples, which impedes the polarization switching process during indentation. Apart from the piezoelectric response, polycrystals of PMN-PT also exhibited higher H and E values (∼12 and 160 GPa, respectively) compared to conventional PZT (∼10 and 135 GPa, respectively), possibly due to the differences in ferroelectric domain structure (which in PMN-PT is closely spaced and denser than the PZT), and thus offering a greater elastic recovery upon the relaxation of indentation loads. Readers are encouraged to go through the cited literature^113–118 for a detailed understanding of differences in crystal structure and ferroelectric domain patterns in various piezoceramics. Notably, one of the important observations from the studies of Kathavate and co-workers^5,25 is the existence of normal and reverse ISE (RISE) in PMN-PT. Normal ISE in crystalline metals is a commonly observed phenomenon due to the presence of high GND density beneath the indenter, because of a high plastic strain gradient (particularly at a lower indentation size). While in the case of ISE in PEM, a few studies report that this could be an outcome of collective processes of GNDs,^23,83 ferroelastic activities,^{5,22,23,25,27,28,32,41,83} and flexoelectric component.¹¹⁹ 

On the other hand, RISE in crystalline ceramics could be attributed to indentation-induced cracking, as proposed by Li and Bradt.¹²⁰ However, the close examination of nanoindentation imprints by Kathavate and co-workers^5,25 reveals no cracking at the imprint edges or corners. This further suggests that the occurrence of RISE in polycrystalline PMN-PT may be due to the surface irregularities induced during the sample preparation process.⁵ The above argument was later substantiated by a detailed quantitative estimation of ISE and RISE using empirical models.²⁵ 

Besides the mechanical properties, pressure-induced phase transformation during indentation in piezoceramics is an active topic of interest as it considerably affects the electromechanical properties of piezoceramics. Most recently, Man et al.¹²¹ observed pressure-induced phase transformation for $[100]$ and $[110]$ oriented PMN-0.33 PT single crystals during nanoindentation experiments carried out at different strain rates, $ε˙$ (0.01, 0.05, 0.1, 0.2, and 0.3 s⁻¹). Initially, the volume fraction of T phase in both the orientations decreases with increasing $ε˙$ (up to 0.1 s⁻¹) and then saturates (Fig. 20), highlighting the dependence of phase transformation on the critical value of $ε˙$⁠. This further indicates the ability of PMN-PT to transform from T to R at $ε˙$⁠, which is well versed with the observations of Juliano and co-workers.¹²² As indicated earlier, one of the deformation mechanisms in piezoceramics during indentation is ferroelectric domain switching and reorientation. At lower $ε˙$⁠, there is sufficient time for the material underneath indentation for the domain switching process and thus facilitating phase transformation. On the other hand, beyond a critical $ε˙$⁠, either the lack of sufficient time for domain switching or saturation of the ferroelectric domain switching process might incur less lattice distortion, thereby leading to saturation in the phase transformation process. Similar observations are also reported by Ahart and co-workers and Ganesh and Cohen^109,123 during the phase transformation in PMN-PT and PbTiO₃ (PT) in high-pressure environments (5–25 GPa).

Although the above studies extensively reported the mechanical behavior, ferroelectric distortion, and pressure-induced phase transformation during the mechanical loading of PZT, PMN-PT, and PZN-PT piezoceramics, it is believed that small-scale deformation techniques are still not able to characterize the elastic instabilities during pressure-induced phase transformation in these ceramics. The extent of the reported literature opens up new possibilities for developing such techniques that can probe the fluctuations during indentation, which is of paramount importance to study the relaxation behavior of piezoceramics during unloading.

This section highlights the mechanical behavior of various Pb-free piezoceramics such as barium titanate (BaTiO₃, abbreviated as BT), strontium titanate (SrTiO₃) (abbreviated as ST), BZT–BCT, sodium potassium niobate–niobate tantalate (NKN–NT), and KNN-based ceramics.

BT is the first material used as a piezoceramic initially with T_c = 120 °C, d₃₃ = 190 pC/N, and k₃₃ = 50%.¹²⁴ Until the year 2000, mechanical characterization of BT was a less understood subject due to the complex interplay between clusters of different ferroelectric domains (i.e., “$a$” and “$c$” domains), and limited studies have reported the deformation behavior of BT crystals under mechanical loading conditions. For instance, Kim et al.¹²⁵ reported changes in ferroelectric domain patterns during the uniaxial compression loading of BT. External compressive stresses were induced in the test material during its transformation from cubic to a tetragonal structure. As a result, a change in the ferroelectric domain morphology was realized [Figs. 21(a) and 21(b)]. Notably, before applying uniaxial compressive stresses, the banded ferroelectric domain network [Fig. 21(a)] has been transformed into a lamellar one [Fig. 21(b)] after inducing the external stresses, indicating the formation of 90° DW. The results of this study reveal stress-induced ferroelectric domain formation in piezoceramics. For example, if the internal stresses (or simply residual stresses along the poled surfaces) arise due to mutual interaction between two adjacent grains or GB may tend to form a systematic network of ferroelectric domains with 180° DW. On the other hand, if these internal stresses are lower than the externally applied load, then strip-like ferroelectric domain morphology is expected with the disappearance of 180° DW. Note that grain size has a negligible influence on the formation of the ferroelectric domain network during mechanical loading.

Following this, Muñoz-Saldaña et al.⁸ observed stress (uniaxial compressive) induced movement of DW motion and the formation of new ferroelectric domains in $[001]$ orientation of BT single crystals. It is evident from Figs. 22(a)–22(d) that the compressive stresses parallel to the poled orientation (i.e., $[001]$⁠) leads to the formation of new strip-like ferroelectric domains followed by their lateral growth in $[101]$ orientation. The domain size increases with increasing stresses [Fig. 22(e)] up to coercive stresses, $σc$⁠, while no further growth in ferroelectric domain is possible beyond $σc$⁠. Although the above studies demonstrate the effect of purely mechanical stresses on the electromechanical behavior of BT, its deformation behavior under coupled loading conditions is not known. Following this, Ramamurty and co-workers³³ investigated the deformation behavior of BT ceramics using a spherical indentation under coupled loading conditions [Fig. 23(a)] which was achieved by different indenter and material combinations [Fig. 23(a)] (i.e., poled BT with conducting indenter, poled BT with insulating indenter, and unpoled BT). The P–h response for all the conditions presented in Fig. 23(b), suggests that the deformation behavior of BT is highly sensitive to the combination of loading conditions employed. The S values (i.e., dP/dh measured at the upper portion of the unloading curve) for poled–insulating condition is higher than the poled–conducting condition by ∼0.4%, while among all the cases, poled BT has exhibited higher S than unpoled BT by ∼4.2%, which is precisely opposite to the trends observed in PZT.³³ These experimental observations corroborated well with the analytical models proposed by Giannakopoulos and Suresh.¹²⁶ Importantly, due to the larger indent size (i.e., in the μm regime), the indenter could probe the collective response of 10–12 grains during indentation and, hence, the effect of grain orientations ferroelectric domains is minimal on the deformation of BT. However, no microstructural evidence has been provided to support the above argument. As the overall deformation behavior of bulk material depends on the deformation of an individual grain, work of Ramamurty et al.³³ opened up new pathways to probe the micro- or nanoscale mechanical response of an individual grain or clusters of ferroelectric domains in BT. In view of this, nanoindentation experiments performed by Schneider and Sharma and co-workers^11,14,15,119 on BT for probing the ferroelectric domain activities in the indentation vicinity are notable. The P vs h response for single-crystal BT is presented in Fig. 24(a). The initiation of the pop-in event at P_max = 1 mN is visible with a frequent appearance at higher P (i.e., beyond 4 mN), indicating the elastic–plastic and plastic nature of the test specimen. We would like to shed more light on the above interesting aspect. Owing to its tetragonal symmetry, the stress-induced movement of ferroelectric DW in BT is favored by four crystallographic directions (each direction being mutually perpendicular to the adjacent one), thereby losing the elastic strain energy associated with ferroelectric domains during the process. For example, for $[001]$ oriented single-crystal BT, switching of ferroelectric domains along $[110]$ orientation is highly possible (including positive and negative axes), and so is the crack growth (i.e., fracture/cleavage), thereby forming a zone of switched ferroelectric domains [Fig. 24(b)]. During the switching process, the elastic strain energy associated with ferroelectric domains guides the crack growth beneath and around the indentation zone. Consequently, noticeable displacement bursts are evident during the indentation. Notably, this ferroelectric domain switching process depends on the intensity of applied stresses, wherein complete domain switching is predominant in the crack wake and tip zone. In contrast, ferroelectric domains reorient (or switch back) away from the crack tip, thus exhibiting a systematic rearrangement. Furthermore, in an attempt to measure the $Kc$ values from submicron cracks originating at the imprint corners in BT, Scholz et al.¹²⁷ performed nanoindentation experiments in the load regime of 0.6–10 mN and indicated that $Kc$ values in ceramics depend on crack length and indent size ratio, particularly, in the lower indentation size regime. However, their correlation with ferroelastic activities in the indentation zone was not reported in great details. Schneider and Sharma and co-workers^{11,14,15,119,127} has put forth a detailed understanding of the different deformation mechanisms in PEM (particularly in BT) during mechanical loading (a) dislocation-mediated slip, (b) ferroelastic activities, and (c) flexoelectric component, thereby providing microstructural evidence via small-scale deformation techniques (i.e., nanoindentation and AFM/PFM).

Although Schneider and co-workers^11,14,15,119 were successful in probing the collective response of the bulk (or clusters) of ferroelectric domains, a detailed understanding of the mechanical properties of the individual ferroelectric domain (i.e., “$a$” or “$c$” alone) was far from complete. In view of this, nanoindentation experiments performed by Park et al.¹³ on BT provide significant insights into the mechanical behavior and stress-induced switching of an individual “$a$” and “$c$” domains. The P vs h response of both the domains is presented in Fig. 25, indicating no discernible differences in residual indentation depth, $hf$ initially, however, scrutiny of nanoindentation data indicates that the “$a$” domains (H = 13.94 ± 0.26 GPa and E = 147.58 ± 1.92 GPa) are harder and stiffer than the “$c$” domains (H = 13.02 ± 0.5 GPa and E = 135.28 ± 2.55 GPa) by ∼7% and ∼10%, respectively. Although the experimental and simulation results agreed with each other (within an error of ± 10%), unlike simulated P vs h curves, experimentally observed P vs h curves could not identify the directions of poling (i.e., crystallographic orientations of domains), an important shortcoming of the above study.

The conventional PEM such as BT and PZT are also subjected to the cyclic (or dynamic) environment during actual operation in MEMS and NEMS devices. Therefore, at smaller length scales, the role of fatigue behavior becomes increasingly evident due to smaller number of grains (eventually ferroelectric domains) participating in the deformation process. Recently, Li and co-workers²⁹ have demonstrated the fatigue behavior of BT during cyclic loading. Cyclic nanoindentation experiments on BT micropillars (with diameters of 0.58, 2, and 5 μm) were performed with a triangular load function (i.e., with no dwell time at peak load) with a loading and unloading rate of 10 MPa/s at a frequency of 2 Hz. Figures 26(a) and 26(b) illustrate the $σi$ vs $εi$ response over 1 × 10⁶ cycles and residual strain accumulation in BT, respectively. All the $σi$ vs $εi$ curves show a systematic overlapping with each other from second cycle onward with no visible degradation. The recoverable strain was ∼1% (which is huge for BT) at a stress of ∼203 MPa with residual strain as small as ∼0.08%. Furthermore, residual strain accumulation in BT is almost stable and lower than NiTi (shape memory alloy) [Fig. 26(b)]. The above observations agree well with the scanning electron microscope (SEM) micrographs of the BT micropillar before and after fatigue test [Figs. 26(c) and 26(d)], which further indicate no discernible differences, including no damage after fatigue tests. The enhanced fatigue resistance of the BT micropillar could be attributed to the ferroelectric domain switching during loading, particularly, 90° switching of DW and reorientation upon the relaxation of loads. This work of Li et al.²⁹ has opened new guidelines for the defect-based multifunctionality in PEM. Notably, Mathews and co-workers²³ also emphasized similar aspects of defect-based plasticity, indicating the combined effect of dislocations and ferroelastic activities on the strength and size effects in BT ceramics.

This section summarizes the mechanical behavior of BT ceramics considering ferroelastic activities, dislocation-mediated plasticity, and the flexoelectric effect. One of the long-standing challenges in the mechanical characterization of these BT ceramics (and any other PEM) is probing the superelastcity at lower length scales. We believe the piezoelectric research community is extensively working to extend the capabilities of small-scale deformation testing techniques to encounter the above challenge.

ST, another important Pb-free piezoceramics after BT, has found a unique niche in widespread applications such as high-temperature superconductors (i.e., semiconductive ceramics), microwave capacitors, and scanning microscope.⁹ As most ST ceramics are used in high-temperature environments, the temperature-dependent deformation behavior studies of ST have attracted significant attention from researchers. For instance, Fischer et al.¹²⁸ demonstrated the effect of crystallographic structure on the elastic properties of hot-pressed polycrystalline ST, when subjected to 3 GPa pressure. A cylindrical specimen (diameter 8 mm and height 4.10 mm) was placed in a high precision ultrasonic interferometer (ANU-TECH). The predetermined pressure (for loading the specimen) was generated via a liquid-cell piston-cylinder arrangement equipped with the interferometer. Their results indicated that the E values are markedly dependent on (i) applied pressure where linear dependency is observed, (ii) surrounding acoustic environment, and (iii) structural phase transition during unloading. However, the effect of phase transformation on the elastic properties was less pronounced than in BT ceramics. Koizumi and co-workers¹²⁹ demonstrated the temperature-dependant plasticity in ST above 1200 °C, a temperature so critical for superconductor materials.

The temperature-dependant plasticity studies on ST^130,131 show that ST can exhibit good ductile to brittle transition (DBTT)^132–134 and inverse DBTT⁹ characteristics under uniaxial compression [Fig. 27(a)]. These studies demonstrated the marked influence of crystallographic orientations and dislocation-mediated plasticity on the DBTT and inverse DBTT behavior of ST. Here, we attempt to discuss this in detail. Scrutiny of all the regions in Fig. 27(a) suggests that flow stress, $σflow$⁠, values for all the orientations almost overlap with each other in region A, indicating that at a lower temperature $σflow$ is independent of crystallographic orientations, while in the high temperature regime (regions B and C), it strongly depends on the crystallographic orientations. This deformation behavior in region C and A has two different mechanisms; homogeneous plastic deformation (in region C) followed by inhomogeneous and localized plastic deformation through shear bands [Fig. 27(b)]. The primary reason to support the above argument could be due to the easy activation of $a⟨100⟩{100}$ dislocation slip system, as supported by the transmission electron microscopy (TEM) observations. Interestingly, some of the edge dislocations later turned into screw dislocations at lower temperatures during the plastic deformation. However, the series of works performed by Refs. 9 and 131–134 did not report any evidence of temperature-assisted variation in dislocations (i.e., from edge to screw) and the existence of ferroelectric activities during the deformation of ST.

Most of the thin films in a superconductor, field-insensitive thermometers, and microwave capacitors made of ST ceramics undergo a contact loading conditions in their usage. Therefore, the estimation of mechanical properties such as H (which is also known as an indirect measure of strength), $Kc$⁠, and E becomes increasingly evident from the design perspective. A series of experimental studies are conducted by Durst, Rödel and Webber, and co-workers^{21,26,30,31,38,42–44} using bulk compression and nanoindentation also provide significant insights into the deformation behavior of ST (in both room temperature and controlled environment). Their studies mainly emphasize defect-based multifunctionality, such as dislocation engineering (taming a network of dislocations via mechanical imprint), subsequent plastic deformation and ferroelectric/elastic activities (bulk polarization) as predominant deformation mechanisms. For instance, during the temperature-dependent deformation of $[001]$ single-crystal ST via uniaxial bulk compression tests subjected to 1% plastic strain, $εp$⁠, it was found that dislocation density, $ρd$ increases (by an order one) with an increase in operational temperature (from 25–300 °C),²¹ while a concurrent decrease in yield strength and E was realized. This was primarily attributed to the activation of $⟨110⟩{110}$ slip system oriented 45° to the loading directions (Fig. 28), which can be observed under polarized light microscopy viewing from $(100)$ and $(010)$ faces. These observations agreed with those reported previously by Sigle and co-workers.^9,131–134 In another notable work, Fang et al.⁴² demonstrated the plastic deformation in single-crystal ST with different concentrations of Sr vacancies (i.e., one with Sr/Ti ∼ 1 and the other with Sr/Ti ∼ 1.04) at micro- and nanoscale using nanoindentation technique. A competing mechanism between dislocation nucleation at the nanoscale and dislocation propagation dominating the plasticity at the microscale is highlighted. Furthermore, nanoindentation response of single-crystal $(001)$ oriented ST at room temperature (25 °C) and in a controlled environment (350 °C) was investigated by Javaid et al.⁴⁴ They attributed the onsets of plastic deformation and ISE to the network of dislocation around the Berkovich impression revealed through the sequential etch-pit technique. ISE are more pronounced in ST at 25 °C compared to 350 °C [Fig. 29(a)], possibly due to the difference in dislocation pile-up densities around the indentation imprint along $⟨100⟩$ and $⟨110⟩$ directions. The extension of dislocation pile-up around the indentation at 25 °C was comparatively shorter than that of 350 °C [Figs. 29(b) and 29(c), respectively], thereby affecting the size effects in mechanical properties in ST. Most of the studies in the past reported the direct observations of dislocation assisted plasticity in ST at various length scales. However, the effect of ferroelectric/elastic activities and vacancy concentration on the mechanical properties are not understood in great details.

In recent years, Pb-free piezoceramics have occupied almost 70% of the piezoelectric market commercially due to the restrictions imposed by many environmental agencies on Pb-based ceramics. This has opened abundant compositional space in the periodic table to develop and fabricate high-performance ceramics. Considering the recent reports on the deformation mechanisms of various Pb-free ceramics, Vögler et al.²⁰ demonstrated the temperature-dependent fracture behavior of BZT–BCT piezoceramics, indicating the marked influence of ferroelastic activities during uniaxial compressive loading. Decreasing trend in $Kc$ values with an increase in test temperature was evident due to phase transformation during loading. The ferroelastic measurements also revealed that the degree of back-switching (i.e., reorientation) in ferroelectric domains increases with an increase in test temperature, indicating the lower remanent strain, $εr$ and consequently, the lower $Kc$ (i.e., size of hysteresis on $σ$ vs $ε$ curves in compression). However, no direct observations of ferroelastic activities were presented. The microindentation response and fracture behavior of KNN and sodium bismuth titanate (NBT) based Pb-free piezoceramics has been reported by Wang et al.²⁴ very recently, indicating the contribution of ferroelectric domains in toughening of piezoceramics. Besides this, they also illustrated the stress-induced phase transformation as one of the viable toughening mechanisms in Pb-free piezoceramics. However, their arguments were mainly based on the hypothesis, and no microstructural evidence was produced. Therefore, the comprehensive understanding of pressure-induced phase transformation in toughening of piezoceramics remains far from complete. In this regard, the implementation of PFM around and ahead of the indentation crack is expected to provide more insights into the state of ferroelectric domains and related elasto-electric activities.

A close look at the previous literature enables us to mention that the first work on the fracture behavior of KNN-based ceramics was reported by Zhang et al.,¹³⁵ where they have investigated the differences in indentation crack growth in humid and hydrogen-charging environments. They indicated the potential range of relative humidity (RH) as around 70%–90%, where crack growth is pronounced upon relaxing indentation stresses under humid conditions. On the other hand, crack growth and hydrogen-induced delayed cracking decrease with an increase in hydrogen concentration, suggesting the pronounced influence hydrogen concentration on fracture behavior. Although they argued that “$a$” domains could switch to “$c$” domains during the crack growth at sustained load in hydrogen-charging environment, a clear explanation of why ferroelectric domain switching is evident in the hydrogen-charging environment and reorientation of ferroelectric domains upon unloading is missing. Following this, $Kc$ values of Li-doped KNN (LKNN) and Ni-doped LKNN (LKNN/Ni) composites using Vickers indentation were reported by Zhang and co-workers,¹³⁶ indicating the enhancement in $Kc$ of LKNN/Ni composite (∼2.92 MPa m^1/2) compared to monolithic LKNN (∼1.5 MPa m^1/2). The work of Zhang and co-workers¹³⁶ has opened up new guidelines to tailor the electromechanical properties of piezoceramics via the metal–particle dispersion technique. Notably, the effect of ferroelectric domain switching on the fracture behavior of Pb-free piezoceramics is much more pronounced and has been demonstrated by Martin and Kakimoto¹³⁷ using a three-point bending test and Vickers indentation. Interestingly, it was found in the literature that Pb-free ceramics, mainly KNN based and doped with Li, Ta, and Sb, exhibit higher $Kc$ values than conventional PZT (a Pb-based piezoceramic) probed via single edge V-notched beam (SVNB),^138,139 Vickers indentation,³⁵ and near-tip crack opening displacement.¹⁴⁰ Table II below summarizes the $Kc$ values for various Pb-free piezoceramics along with the respective employed testing techniques. Although the above studies have indicated the pronounced influence of ferroelectric domains on the crack growth a fracture behavior of ST, BZT–BCT, and KNN-based Pb-free piezoceramics, such behavior in NBT was rarely observed. The $Kc$ values probed via Vickers indentation for NBT ceramics were in the range of 1.40–2 MPa m^1/2 (still higher than PZT ∼0.8 MPa m^1/2) and showed a distinct dependence on the grain size and grain morphology.^141–144 Notably, poled NBT samples exhibited slightly higher $Kc$ values (∼1.8–2 MPa m^1/2) than the unpoled ones (∼1.6 MPa m^1/2).¹⁴⁴ We encourage readers to go through the cited references (Refs. 145–149) for additional understanding related to the mechanical behavior of various piezoceramics.

We have described the different mechanical characterization methods for Pb-based and Pb-free piezoceramics reported in the literature and presented the variations among reported results/techniques based on our understanding. A comprehensive look at the previous literature helps us to recognize the present challenges in the field and opens up new strategies for future development of characterization techniques. Keeping this in view, we brief out the main issues in the current scenario and possible future development in Sec. IV.

The motivation for including this section is to report the scientific challenges that the PEM community did not consider collectively. Given this, we highlight some critical aspects related to the current issues in the mechanical testing of piezoceramics, which may be helpful to facilitate the modifications in existing ones or develop new techniques.

First, characterizing the elastic or elastic–plastic behavior is a long-standing challenge due to the inherent brittleness associated with piezoceramics. Although small-scale deformation techniques such as nano- and picoindentation can probe the elastic–plastic nature, issues related to elastic instabilities are still a significant problem. For instance, the occurrence of strain bursts during nanoindentation still requires a detailed understanding from the ferroelastic activities point of view. In line with this, defining the unified deformation mechanism among dislocation assisted or ferroelectric domain induced plasticity is essential. Because most previous studies were limited to single crystals, they reported that dislocation-mediated plasticity is the predominant deformation mechanism. On the other hand, some recent studies on polycrystal demonstrated ferroelectric/elastic activities as the primary deformation mechanism in piezoceramics. Moreover, suppose ferroelastic activities to be the primary deformation mechanism in polycrystalline Pb-based and Pb-free piezoceramics. In that case, difficulties in probing the ferroelectric domain interaction at GB are still a significant problem. Therefore, we could foresee the potential developments in the in situ testing, mainly small-scale deformation testing techniques that can probe the complex interplay between GB and ferroelectric activities during mechanical loading.

Another critical issue in the mechanical deformation of piezoceramics is the GB migration and its interaction with dislocations and ferroelectric domains. Having said that GB movement in crystalline materials can be realized at high temperatures, it has been recognized that GB can also migrate at room temperature.^150–152 Such behavior at the nanoscale could be attributed to nucleation and motion of dislocations, dislocation pile-up at GB, and re-emission of dislocations. However, such behavior in polycrystalline piezoceramics, particularly ferroelectric activities, is not understood. Therefore, high stress-strain driven GB indentation experiments, particularly nanoindentation, can provide significant insights. In line with this, although the network of mechanically imprinted dislocations is favorable for enhancing the mechanical performance of PEM,^{21,26,30,31,38,42–44} a comprehensive understanding of the effect of these dislocation networks on the piezoelectric and dielectric performance is far from complete. Thus far, only a notable work by Höfling and co-workers³⁰ highlights some unique aspects of controlling the ferroelectric domain structure via dislocation engineering technique, thereby tailoring the piezoelectric and mechanical performance of Pb-free PEM. In the same vein, we expect some novel techniques such as in situ synchrotron XRD to probe and image the dislocation–ferroelectric domain interactions during mechanical loading. A few studies indicated the potential use of the above technique to characterize local ferroelectric domain switching in PZT-S,¹⁵³ PZT thin films,¹⁵⁴ bismuth ferrous oxide–lead titanate BiFeO₃–PbTiO₃¹⁵⁵ and NKN (Pb-free) piezoceramics.¹⁵⁶ The main rationale behind adopting the synchrotron diffraction was to understand the effect of intrinsic strain (i.e., lattice strain) contribution and ferroelectric domain switching process on the phase formation in piezoceramics.

The utmost important factor in piezoceramics, considering the ferroelectric activities, is $TC$⁠, where ferroelectric domain configuration changes. Previous studies have indicated the marked influence of $TC$ on the $εr$ and deformation behavior during temperature-controlled uniaxial compression testing.^{6,17,20,21,37,81} On a futuristic note, it would be interesting to probe the actual response of clusters of ferroelectric domains and dislocation interactions in a controlled environment, which was lagging in the literature. The temperature-controlled nanoindentation experiments will be helpful in this regard. Moreover, nanoindentation experiments coupled with AFM using a cantilever tip of ∼20–30 nm (as small as an individual ferroelectric domain) will clearly understand the switching behavior of ferroelectric domains. To date, there is only one report that investigates the nanomechanical properties of individual “$a$” and “$c$” domains.¹³ 

Most of these piezoceramics undergo extreme conditions in military impact fuzes and high actuation strain applications. However, their behavior under high strain-rate conditions (or extreme conditions) is not investigated in detail.^157,158 Therefore, future high strain experiments on polycrystalline piezoceramics, particularly, in Pb-free piezoceramics, can be helpful to design them for high-frequency transducer applications. In line with this, extracting electricity under extreme loading conditions can also be viewed as one of the possible electromechanical mechanisms.¹⁵⁷ We have extensively reviewed the literature, particularly, regarding nanoindentation studies on piezoceramics, and explored all the possibilities to report the ferroelectric/elastic activities in the indentation vicinity, which is still an important aspect. The advent of small-scale testing techniques such as focused ion beam-SEM (FIB-SEM) and in situ nanoindentation experiments coupled with SEM or TEM may help probe the ferroelectric/elastic activities, thus understanding the plastic deformation in piezoceramics. It is also recommended to use bonded interface indentation technique (BIIT) with micro/nano indentations to probe the ferroelectric activities underneath the indentation (i.e., in the sub-surface deformation zone).

Recently, a few studies have shown the in situ conductive nanoindentation response of various thin films such as $Ge1Sb4Te7$ for phase change memory devices,¹⁵⁹ BT nanopillars embedded in the polymer matrix,¹⁶⁰ $ScxAl1−xN$ thin films,¹⁶¹ Sr-doped PZT thin films,¹⁶² and relaxor ferroelectric single crystals.¹⁶³ A notable review article by George et al.¹⁶⁴ highlights the significance and applications of the in situ conductive nanoindentation technique on various materials. However, such behavior in polycrystalline Pb-free piezoceramics is largely unexplored. Therefore, the authors believe that still there is ample room for debate on the above aspects to understand the real-time behavior and new techniques to probe the mechanical and piezoelectric response of various ceramics and thin films. Along similar lines, poling in piezoceramics also significantly influences fracture characteristics. This ultimately entails a dilemma of how much coercive electric field, $Ec$ is necessary to make these piezoceramics suitable to withstand the mechanical loading, mainly when the orientation of ferroelectric domains will induce a high amount of electrical and mechanical stresses in the material. Therefore, the role of poling on the mechanical properties cannot be ruled out and optimization in $Ec$ is indeed a requirement.

Dislocation-toughened piezoceramic is a long-standing debate among the PEM community. Such cases are sporadic in the case of piezo³¹ and other ceramics,^165,166 particularly when it has been stated that it is possible to move the dislocations in piezoceramics, particularly, in single crystals, despite their low $Kc$⁠. It would be interesting to see how the manipulation of $ρd$ at the crack tip will help to enhance the toughness in piezoceramic? Therefore, besides nanoindentation, developments in techniques that can nucleate the dislocations in piezoceramic would be the key priority. The key competing mechanisms will be the emission and propagation of dislocations at the tip of short vs long indentation cracks. Lastly, with caveat, we mention that the synthesis and fabrication route also influence the mechanical properties of piezoceramic, which is beyond the scope of the present Tutorial. We could see potential developments in synthesis techniques for enhancing mechanical performance. Therefore, we advise readers to go through the respective cited research paper for detailed information related to fabrication techniques.

In summary, this Tutorial mainly emphasizes the various contributing microstructural length scales, defect-based functionality, different deformation mechanisms, methodology, and recent progress in the deformation techniques that can probe the mechanical behavior of piezoceramic. After a detailed review of several research articles, it can be construed that there is a continuing surge in the development of smart materials with multifunctional characteristics for miniaturized devices. Undoubtedly, PEMs are promising candidates; thus, this eventually opens up the notion of probing the functional and engineering characteristics of PEM, which can be viewed as a motivation for the present Tutorial. Therefore, an attempt has been made to report all such currently available mechanical characterization techniques with formal analysis, observed trends, variation among reported mechanical properties, and critics from the microstructural viewpoint, thereby highlighting the shortcomings of each method.

The inherent brittleness associated with the piezoceramic urges the PEM community to employ a uniaxial compression test for probing the compressive strength and fracture behavior at a bulk scale. This practice is almost five decades old and has been implemented on several Pb-based (mainly on PZT) and Pb-free (on BT) piezoceramics (polycrystals), where the reported observations indicated the anisotropy in mechanical properties and ferroelectric domain switching and reorientation of the main deformation mechanism. Moreover, the fracture behavior in piezoceramics has also been studied by the DT technique.⁷¹ This further paved a motivation for the mechanical characterization of single crystals to eliminate the contributing effect of grain orientations, adjacent grain, and GB in mechanical properties. A few studies^72,80 also reported the fatigue behavior in piezoceramics using cyclic compression tests to suit them for high-frequency stress applications. Notably, piezoceramics can withstand up to 1 × 10⁶ fatigue cycles despite their brittle nature under compressive loading conditions. In many instances, piezoceramic undergoes the coupled loading environment (i.e., electromechanical and thermomechanical). Therefore, several studies^{17,20,21,33,34,37,44,71,72,77,78,81} reported the mechanical behavior via uniaxial compression tests under electrical (i.e., poled/conducting, poled/insulating, and unpoled combinations) and temperature-controlled environments. Arguably, unpoled piezoceramics exhibit a better set of mechanical properties than the poled ones while decreasing trends in $Kc$ values can be realized at higher temperatures. Although identifying the main deformation mechanisms such as dislocation-mediated plasticity and ferroelectric/elastic activities was a success, commensurate reports with good experimental evidence of such microstructural activities were lagging in the literature. Furthermore, it was not clear from the bulk deformation experiments that, like metals, does piezoceramic also exhibit significant size and scale effects? If yes, why? The deformation techniques that can probe micro-plasticity in these piezoceramics, such as uniaxial compression experiments on micropillars and indentation techniques, provided valuable insights into the above arguable aspects. Strikingly, the extent of micro-compression and micro-indentation testing has provided important observations related to the deformation behavior of individual grains (of μm or sub-μm size) in piezoceramics. This has helped the PEM community design and fabricate the piezoceramics for miniaturized devices such as microwave capacitors, tribo generators, and superconductors, mostly thick/thin coatings. Such aspects related to Pb-based and Pb-free piezoceramics are outlined in the present Tutorial.

Probing micro/nanoscale aspects of plasticity in piezoceramics is a long-standing challenge due to its low fracture toughness. IITs have made this task simpler, revealing the elastic–plastic nature of these ceramics. It has been observed that researchers use the Oliver–Pharr method to extract the mechanical properties of nanoindentation data for PEM and FE materials. However, it should be noted that the O–P method is not applicable when there is an electric field during nanoindentation because, the contact stiffness equation (which is the core assumption of the O–P method) differs for the PEM and FEM materials, as the indentation response is influenced by the 21 elastic, 18 piezoelectric, and six dielectric properties that are independent of materials properties. Kalinin and colleagues have done pioneering works in establishing models and developed relations for contact stiffness for quantifying the indentation response of the PEM and FEM under various indenter geometries.^167–171 

Interestingly, in situ fracture characteristics in these ceramics can be probed via IITs by continuously monitoring the P–h curves. In line with this, in situ conductive nanoindentation experiments will also be an added advantage for probing the real-time electromechanical characteristics of these piezoceramics. Moreover, the characterization of nanoscale microstructural activities (say, dislocation nucleation and/or ferroelectric distortion) has been increasingly more straightforward with the help of IITs and AFM. It is also imperative to understand the fatigue behavior under cyclic nanoindentation loading considering the applications of piezo-thin films in MEMS/NEMS devices, particularly, when the role of ferroelectric activities becomes increasingly apparent.²⁹ This significantly impacts the design and development of piezo-thin films for tribo-nanogenerators and the devices where the shape memory effect is prevalent. Furthermore, characterization of the collective mechanical response of nanoscale defects has been possible with IITs. These IITs have also been proven beneficial for characterizing the stress-induced phase transformation in piezoceramics, which is necessary from the reliability viewpoint of these ceramics in high-stress operational environments.

In conclusion, the mechanical characterization of piezoceramics is a vast and long-standing topic. We have extensively reviewed the literature and summarized our understanding comprehensively. Based on the analysis of observed trends and shortcomings highlighted, this Tutorial outlines the current challenges in the field. On an optimistic note, we conclude that probing micro/nanoscale plasticity in piezoceramics is possible despite their inherent brittleness. Moreover, the lesson learned from the deformation behavior of piezoceramic across various length scales is only a beginning, and discussion on the futuristic outlooks are expected to pave new pathways for the further developments of small-scale deformation testing techniques to provide significant insights into nanoscale plasticity in piezoceramics.

V.S.K. gratefully acknowledges the financial support received during his doctorate from the Council of Scientific and Industrial Research (CSIR), HRDG, EMR-I, New Delhi, Government of India, against the CSIR-SRF Scheme [Grant/File No. 09/1022(0110)/2020-EMR-I]. M.S.R.N.K. acknowledges the Science and Engineering Research Board (SERB), Government of India, for a Core Research Grant (No. CRG/2020/1902). V.S.K. and Y.Z. also acknowledge the financial support of the National Science Foundation (Award No. 1929646).

The authors have no conflicts to disclose.

V. S. Kathavate: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). K. Eswar Prasad: Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Mangalampalli S. R. N. Kiran: Supervision (equal); Writing – review & editing (equal). Yong Zhu: Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.