Electrostrictive Metamaterial Study: Shaping Fields to Exceed Intrinsic Material Limits

Electrostrictive Metamaterial Study: Shaping Fields to Exceed Intrinsic Material Limits A. Aman Panda1, B. Khushbu Dash2,∗, C. Nachiketa Mishra3 1Dept. of Mechanical Engineering, Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, Chennai, India 2Dept. of Chemistry, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai, India 3Dept. of Science & Humanities, Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, Chennai, India ∗Corresponding author. Email: d [email protected] Abstract Electrostrictive actuators are widely sought for their reliability in micro-positioning and adaptive optics. They promise precise motion but typically face a three-way trade-off. Pursuing larger stroke increases internal stress and off-axis deformation. Material tuning in relaxor ferroelectrics, such as PMN–PT–BT, can yield large strain. However, device-level gains are often limited by fatigue and parasitics. A central challenge is to improve gain, reliability, and directional control simultaneously. Here we show that two electrostrictive metamaterials, Design-1 and Design-2, resolve this trade-off via shape-based field routing. These designs utilize architected PMN–PT–BT unit cells with compliant-hinge geometries. We analyzed their performance using a fully coupled ferroelectroelastic finite-element model. Both designs exhibited simultaneous gains compared to a solid slab. Motion per unit input energy rose by ∼ 16 × for Design-1 and ∼ 11 × for Design-2. Stroke per unit internal stress roughly doubled. Additionally, directionality strengthened by ∼ 3 × . Specifically, the strongly guided surface fraction (ratio > 5) grew from ∼ 22 . 7% to ∼ 61–64 . 5%. Furthermore, this geometric field-routing shortened high-stress tails. This enabled safe-bias operation at ∼ 2 . 4–3 . 2 × lower average polarization. It also boosted the actuation-axis Qeff by ∼ 1 . 7 × for Design-2. These results establish geometric control as a strategy to surpass material limits, offering a recipe for fatigue-tolerant and robust precision devices. Keywords: electrostriction, metamaterials, continuum mechanics, relaxor ferroelectrics, PMN–PT–BT, compliant-hinge geometry, stress-quiet actuation, safe-bias operation, precision actuation 1 Introduction The ability to convert electrical energy into mechanical motion is a cornerstone of modern technology, driving devices from precision actuators to acoustic transducers [ 1 , 2 ]. This capability, known as electromechanical coupling, is primarily governed by two distinct phenomena in solidstate materials: piezoelectricity and electrostriction [ 3 , 4 ]. Piezoelectricity is a linear effect, where mechanical strain ( S ) is directly proportional to the applied electric field ( E ), described as Sij = dkijEk . However, this powerful effect is restricted by crystal symmetry. It can only exist in the 20 out of 32 crystallographic point groups that lack a center of inversion [ 2 , 5 ]. In contrast, electrostriction is a universal, quadratic phenomenon present in all dielectric materials, regardless of their crystal symmetry. It is fundamentally described as a coupling between strain and electric polarization ( P ), Sij = QijklPkPl . In linear dielectrics, this is often phenomenologically defined by the electric field as Sij =MijklEkEl[6, 7]. 1 In acentric materials, where both piezoelectric and electrostrictive strains may be observed, the linear piezoelectric effect dominates the electromechanical response [ 8 , 9 ]. This is due to the relative magnitudes of the material coefficients. The piezoelectric coefficient d(on the order of 300–600 pC N−1 for common lead zirconate titanate (PZT) ceramics [ 10 ]) typically yields a much larger strain than the electrostrictive coefficient M (often 10 −16 m2V−2 or less) at practical field levels [11, 12]. However, this dynamic shifts completely in relaxor ferroelectrics, such as lead magnesium niobate–lead titanate (PMN-PT). In these materials, the electrostrictive response can be dominant, particularly when operated in their high-temperature, centrosymmetric paraelectric phase. In this phase, the linear piezoelectric effect is absent by symmetry [ 8 , 13 – 15 ]. More importantly, the mechanism is distinct. Conventional piezoelectricity relies on the movement of domain walls, a process that results in significant hysteresis, creep, and material fatigue [ 2 , 16 ]. Conversely, electrostriction is proportional to the square of the field. It produces strain independent of the field’s polarity [ 8 , 15 ] (Fig. 1) and exhibits very low hysteresis (often ¡5 % ) [ 17 , 18 ]. This makes it far more suitable for high-precision applications like micro-positioning devices [ 19 – 23 ] and adaptive optics [ 24 , 25 ]. These applications cannot tolerate the non-linearity (hysteresis) and instability (creep and fatigue) of traditional piezoelectrics [11]. The reason this electrostrictive response becomes exceptionally large in relaxors is due to their unique nanoscale structure. Relaxors contain polar nanoregions (PNRs) that are easily reoriented or grown by an external field [ 6 , 7 , 26 ]. This high degree of polarizability leads to a colossal macroscopic dielectric permittivity ( ϵr ), often exceeding 10 3 [ 27 ]. Because the field-related electrostriction coefficient M scales with the square of permittivity ( M≈Q·ϵ2 0 ( ϵr− 1) 2 ), the resulting strain in relaxors becomes massive. This offers a path to large, stable, and precise actuation that conventional piezoelectrics cannot match [13, 26, 28, 29]. Historically, efforts to further enhance this response have focused on material-level optimization[ 9 , 12 , 20 , 21 , 23 , 30 , 31 ]. This includes difficult and time-consuming chemical processes, such as precisely engineering compositions near a morphotropic phase boundary (MPB) [ 9 , 26 ] or the complex, costly growth of large-scale single crystals[ 30 , 32 ]. A powerful alternative to these chemistry-bound limitations is the use of metamaterials. These are artificial structures Figure 1: Macroscopic electrostriction showing deformation independent of electric field polarity. (a) No field ( E = 0). (b) Field in + x . (c) Field in −x . A positive Qxxxx causes longitudinal expansion and transverse contraction, while a negative Qxxxx reverses these effects. 2 whose geometry is engineered to produce effective properties not found in the constituent bulk material. Initial studies in metamaterials led to breakthroughs like materials with a negative Poisson’s ratio[ 33 , 34 ] and negative refractive indices [ 35 ]. These advances enabled the design of invisibility cloaks [ 36 ] and perfect lenses [ 37 ]. Over the past decade, these ideas crossed into the electromechanical media. Piezoelectric metamaterials now manipulate elastic waves [ 38 ], tune programmable vibrational bandgaps [ 39 – 41 ], and harvest energy from ambient vibrations [ 42 , 43 ]. Most notably, architectures have been used to engineer effective piezoelectric coefficients that are dramatically enhanced[44]. Despite this broad progress, two significant gaps remain. First, the field remains overwhelmingly piezo-centric. Explicitly electrostriction-focused architectures are comparatively rare. Only a few theoretical works suggest that composite designs could magnify the apparent electrostrictive response beyond the limits of the constituent materials [ 45 , 46 ]. Second, a primary application for high-response electrostrictive materials is actuation, but a larger stroke almost always comes at the cost of high internal mechanical stress and unwanted off-axis motion. This stress often concentrates at sharp geometric features or electrode edges. This leads to micro-cracking, mechanical fatigue, and catastrophic device failure long before the material’s intrinsic strain limit is reached [4, 19, 47]. In this study, we introduce and investigate two metamaterial designs, Design-1 and Design-2, derived through exploration, that function as “stress-quiet” actuators. These architectures use geometry to strategically manage internal field distributions. This approach achieves an enhanced electrostrictive response with very low off-axis motion, while simultaneously reducing peak internal stresses. The behavior of these geometries was analyzed through device-level, surface-averaged metrics and compared to a monolithic slab design using a fully-coupled 2dimensional finite element model (FEM) in COMSOL Multiphysics. In this work, we: (i) outline the approach and modeling choices; (ii) describe the geometry design strategy for routing fields and releasing stress; (iii) present the results and discussion, including quantitative gains in actuation per stress, energy-aware motion, and effective electrostrictive coefficients; (iv) discuss the conclusions, the implications of stress-quiet actuation, the limitations of this study, and future work; and (v) detail the methods used, including finite element model parameters and material properties. 2 Overview of the Approach To isolate the effect of geometry on the electro-mechanical response, we study three distinct unit cell designs: 1. A solid, monolithic slab (as a baseline). 2. Design 1 (D1), an architected cell. 3. Design 2 (D2), a second architected cell. As illustrated in Fig. 2, all three geometries are composed of the same electrostrictive material—a ternary solid solution of lead magnesium niobate (PMN), lead titanate (PT), and barium titanate (BT) (PMN–PT–BT)—and are subjected to identical electrical boundary conditions. The specific material parameters for PMN–PT–BT are held constant across all three designs and are listed in Table 5 (Sec. 6). For this comparison, the bottom portions are held at ground ( 0 V ), while the top portions are driven by a sinusoidal voltage, V0(t) = Vmax sin(2πt), with Vmax = 200 V. This comparative setup allows us to attribute any differences in electro-mechanical performance (such as actuation, internal stress, and motion directionality) purely to the unit cell’s shape and architecture. 3 Figure 2: Study setup: a slab, Design 1 (D1) and Design 2 (D2), all driven by the same voltage V0=Vmax sin(2πt), where Vmax = 200 V. 3 Design and field distribution strategy Our core strategy is to use geometry—specifically compliant-hinge pathways—to steer the internal electric field and polarization P. This allows us to tailor the site of electrostrictive strain generation and control how it translates into macroscopic displacement (Fig. 3). Compliant joints are a standard way to guide motion and strain flow. They aggregate many small local deformations into a single useful output, a design idea formalized in the compliant-mechanisms literature [48]. In our baseline slab, the field is mostly uniform. Consequently, the polarization is spatially delocalized. Under the quadratic electrostriction law (S ∝ P 2 ), this activates strain broadly rather than concentrating it where it is most useful. By contrast, our ring–link layout with narrow, compliant hinges structures the field. It channels Pinto selected paths, such as the flexure hinges and the auxetic-strut motif (Fig. 3). In these regions, local increases in Ptranslate via the quadratic law into larger local S. The hinges then aggregate these local strains into a coherent global motion. Interface-engineered oxides demonstrate this principle at the materials scale. In these materials, localized strain and symmetry breaking at the interface drive exceptionally large electrostriction. This underscores why routing fields is critical [31]. This choice of architecture, rather than material change, follows a common pattern in architected systems. Geometry can guide and concentrate energy along intended routes while relaxing it where it is less useful. In piezo systems, circular auxetic substrates illustrate how negative-Poisson-ratio layouts direct deformation to productive regions without altering the underlying material [ 43 ]. Likewise, GRIN (gradient-index) metamaterials—structures whose effective properties vary smoothly in space—are used to guide and focus waves [ 40 ]. We apply this same idea at quasi-static bias. In our designs, the compliant ring-and-hinge network routes polarization and strain toward the actuation axis. Simultaneously, it softens the response near free edges and tight curvatures. This supports cleaner, more directed motion. Quantitative evidence for this behavior is reported 4 Figure 3: Geometric control of electric field distribution for enhanced electrostrictive response. in the next section. 4 Results and Discussion 4.0.1 Electrostriction gain and safe-bias efficiency We first verify that the device-level electrostriction follows the expected quadratic law. To quantify this, we fit no-intercept lines to the surface-averaged strain and polarization-squared terms (Fig. 4), where the slopes yield the effective electrostrictive coefficients, Qeff (Table 1). The relations exhibit high linearity ( R2≈ 1 . 000) for all geometries, confirming the macroscopic S∝P2 relationship is preserved. The architected geometry of Design 2 (D2) enhances the actuation-axis coefficient, Qeff 11 , to a value ∼ 1 . 72 × that of the non-architected slab. This increase indicates a higher strain generation per unit of polarization squared, a result of the geometry-driven field routing. Next, we quantify the effect of geometry on electromechanical efficiency. The architected unit 012 3 4 ⟨ P 2 y ⟩ S ⟩(C 2 ⟩m ⟨4 ) ×10 ⟨4 ⟨2 ⟨1 0 ⟨ S xx ⟩ S ⟩( – ) ×10 ⟨6 Slab Design -1 Design-2 (a) ⟨Sxx⟩Svs ⟨P2 y⟩S 01234 ⟨ P 2 y ⟩ S ⟩(C 2 ⟩m ⟨4 ) ×10 ⟨4 0 2 4 ⟨ S yy ⟩ S ⟩( – ) ×10 ⟨6 Slab Design -1 Design-2 (b) ⟨Syy ⟩Svs ⟨P2 y⟩S 012 ⟨ P x P y ⟩ S ⟩(C 2 ⟩m ⟨4 ) ×10 ⟨9 0 2 4 ⟨ S xy ⟩ S ⟩( – ) ×10 ⟨11 ⟨5 05 ×10 ⟨20 ⟨1 0 1 ×10 ⟨21 Slab Design -1 Design-2 (c) ⟨Sxy ⟩Svs ⟨PxPy⟩S Figure 4: No-intercept linear fits on surface-averaged quantities; slopes and R2 are reported in Table 1. 5 Table 1: No-intercept fits on surface-averaged channels (Fig. 4). Entries correspond to Q11 , Q12 , and Q44. Reported are Qeff (m4/C2), R2, and 95% confidence intervals. Q11 Geometry Qeff R295% CI Slab 1.330 000 ×10−21.000 000 ±1.362 ×10−11 Design 1 1.042 843 ×10−21.000 000 ±3.868 ×10−8 Design 2 2.288 095 ×10−21.000 000 ±3.879 ×10−8 Q12 Geometry Qeff R295% CI Slab −6.059 998 ×10−31.000 000 ±1.981 ×10−11 Design 1 3.749 780 ×10−31.000 000 ±1.990 ×10−8 Design 2 3.948 585 ×10−41.000 000 ±4.290 ×10−8 Q44 Geometry Qeff R295% CI Slab 1.936 361 ×10−20.999 996 ±7.360 ×10−6 Design 1 2.380 184 ×10−20.999 991 ±1.990 ×10−5 Design 2 −1.385 478 ×10−10.999 955 ±2.696 ×10−4 cells demonstrate improved conversion of input energy to displacement. The energy-aware figure of merit, ⟨|u|⟩S/Uin , increases by ∼ 16 × for Design 1 (D1) and ∼ 11 × for D2 compared to the slab (Table 2). This improved efficiency permits operation at lower internal polarization levels. As shown in Fig. 5b, at peak drive ( t∗ = 0 . 25), comparable stroke is achieved with an average actuation-axis polarization ⟨|Py|⟩S that is 2 . 4 × (D1) and 3 . 2 × (D2) lower than the slab. These results show that geometric architecting can simultaneously increase the effective electrostrictive gain and improve the motion per unit of input energy, thereby widening the safe operational window by reducing the required average polarization. 4.0.2 Stress-quiet actuation We now quantify the surface stress distribution at peak drive ( t∗ = 0 . 25) and evaluate the stroke obtained per unit of average stress. For this quantification, we convert the finite-element fields at t∗ = 0 . 25 into an area-weighted stress distribution. For every surface triangle e with vertices at positions xi , xj , and xℓ and nodal von Mises values σVM,i , σVM,j , and σVM,ℓ , we formed an element-level stress by nodal averaging σ(e) VM =σVM,i +σVM,j +σVM,ℓ 3,(1) Table 2: Energy-aware motion. Surface-averaged displacement magnitude ⟨|u|⟩S , electric energy Ue , elastic energy Uelastic , total input energy per cycle Uin ≈Ue + Uelastic , and the efficiency proxy ⟨|u|⟩S/Uin. Geometry Ue(J) Uelastic (J) Uin (J) ⟨|u|⟩S(µm) ⟨|u|⟩S Uin (µm/J) (⟨|u|⟩S/Uin) Slab Slab 2.994 ×10−61.799 ×10−92.996 ×10−66.279 ×10−32.096 ×1031.000 Design-1 1.167 ×10−72.283 ×10−11 1.168 ×10−73.953 ×10−33.386 ×10416.154 Design-2 1.591 ×10−74.124 ×10−11 1.591 ×10−73.747 ×10−32.355 ×10411.235 6 and its physical area A(e)=1 2xj−xi×xℓ−xi.(2) Using a pooled 0–99.5% stress range common to all three geometries, we then computed, for each bin Bk, pk=1 Atot X e:σ(e) VM∈Bk A(e), Atot =X e A(e),(3) so that the plotted quantity is percent of total area per bin. On top of each curve we overlaid two area-weighted scalars, ⟨σVM⟩=1 Atot X e A(e)σ(e) VM,(4) σVM,95 :X e:σ(e) VM≤σVM,95 A(e)= 0.95 Atot,(5) which are shown in the plot (Fig. 5a) as the dashed (area-weighted mean) and dotted (areaweighted 95th-percentile) vertical lines. The resulting distributions (Fig. 5a) differentiate the slab from the architected cells. The slab histogram is broader and possesses a longer high-stress tail, indicating that a larger fraction of its surface operates near the local material limit. In contrast, Design 1 shifts the majority of its surface area into lower-stress bins, and its area-weighted 95th-percentile stress is lower than that of the slab. This result suggests the compliant load paths are effective at redistributing stress. Design 2 exhibits the same trend to a lesser degree. This suppression of the high-stress tail implies that, for an equivalent drive phase, the architected geometries achieve actuation while exposing a smaller surface area to large von Mises stresses. Normalizing the motion by the average stress confirms this mechanical benefit: the figure of merit ⟨|u|⟩S/⟨σVM⟩S improves for both D1 and D2 (Table 3). This indicates more displacement per unit of internal loading, which is consistent with safer biasing and potentially improved fatigue life. 0 100 200 300 400 500 600 700 σ VM (×10 3 N/m 2 ) 0 20 40 60 % of area per bi Slab Desig -1 Desig -2 Mea 95%h Perce %ile (a) Slab Design-1 Design-2 0 5 10 15 20 25 Values (×10 −3 ) 2.4x lower |⟨ P ⟩| s slab 3.2x lower |⟨ P ⟩| s slab |⟨ P ⟩| ⟨C/m 2 ) |⟨ u ⟩| ⟨ μ m) (b) Figure 5: (a) Elementwise von Mises stress distribution across geometries at peak drive ( t∗ = 0 . 25). Surface triangles are binned over a shared 0–99.5% stress range, and bars report percent of total area per bin. Dashed lines mark the area-weighted mean stress; dotted lines mark the area-weighted 95th-percentile stress. Both architected designs shift area into lower-stress bins and shorten the high-stress tail compared with the slab, evidencing stress reduction by geometry. (b) Safe-bias actuation: motion maintained with lower average polarisation. 7 Table 3: Actuation per stress. Surface-averaged displacement ⟨|u|⟩S , surface-averaged von Mises stress ⟨σVM⟩S, and the figure of merit ⟨|u|⟩S/⟨σVM⟩S. Geometry ⟨|u|⟩S(µm) ⟨σVM⟩S(N m−2)⟨|u|⟩S ⟨σVM⟩S(µm m2N−1) ⟨|u|⟩S |⟨σVM⟩S| Slab Design-1 3.953 ×10−38.982 ×1044.401 ×10−81.942 Design-2 3.747 ×10−38.290 ×1044.520 ×10−81.995 Slab 6.279 ×10−32.771 ×1052.266 ×10−81.000 4.0.3 Directionality To ensure a consistent comparison across meshes of different densities, directionality was quantified using the same surface triangles as in the stress analysis (Sec. 4.0.2). Thus, the element areas A(e) and total surface area Atot follow Eqs. (2) and (3) , with quantities weighted by each triangle’s physical area. For triangle e with nodal displacements ( ux,i, uy,i ), ( ux,j, uy,j ), (ux,ℓ, uy,ℓ) at t∗= 0.25, we form element-averaged components, ¯u(e) x=ux,i +ux,j +ux,ℓ 3,¯u(e) y=uy,i +uy,j +uy,ℓ 3,(6) and the corresponding elementwise ratio is r(e)≡|¯u(e) y| |¯u(e) x|,(7) so that every surface triangle contributes one directionality value and is counted in proportion to its physical area. For any threshold τ (e.g. τ = 1 , 2 , 5 , 100), the area fraction reported in Table 4 is obtained by reusing the area-fraction form (3), but with the selection made on r(e): area-fraction(r > τ) = 1 Atot X e:r(e)>τ A(e).(8) This definition ensures the reported percentages represent the ”percent of device surface with predominantly vertical motion” and not ”percent of mesh nodes,” which would otherwise bias the result toward the more finely meshed designs. The arrow plots in Fig. 6 show the underlying displacement directions used to build these statistics. (a) (b) (c) Figure 6: Displacement direction fields in (a) slab, (b) D1, and (c) D2. The length of the arrows are proportional to the |u|. 8 The resulting distributions (Fig. 7) and metrics (Table 4) quantify the difference in motion directionality. The slab exhibits a vertical preference ( r > 1 on ≈ 86% of its area), but only ≈ 23% of its area achieves r > 5, and ≈ 0 . 5% exceeds r > 100. In contrast, both architected geometries show motion that is almost entirely vertical ( r > 1 on 100% of the area; r > 2 on ≈ 99%). These designs also possess large high-directionality regions, with r > 5 on 61–65% of the area and r > 100 on ≈ 9%. This confirms that the geometric features responsible for stress redistribution (Sec. 4.0.2) also channel motion into the primary actuation axis, providing a mechanism-level explanation for the low-parasitic response. 4.0.4 Convergence and key design drivers Finally, we analyze the optimization convergence and identify the most influential design variables (shown in Fig. 10). The Nelder–Mead routine converged for both architectures; the running-best objective in Fig. 9 rises monotonically and then plateaus, confirming successful maximization of the objective (Eq 11) defined in Sec. 6. To rank the influence of each parameter on the objective, we used a correlation-based sensitivity: for each design variable xj and objective J (Eq 11), we compute the Pearson coefficient rj= corr(xj,J) = cov(xj,J) σxjσJ ,(9) and convert it to a unitless importance score by taking magnitude and normalizing to the largest value, Ij=|rj| maxk|rk|∈[0,1].(10) We use |rj| because only the strength of influence is needed for ranking (the sign simply indicates whether Jincreases or decreases with xj). Fig. 8 reports these normalized importances. In Design 1, angular variables dominate ( α2 highest, then α1 ), while t0 and r2 are weaker—implying performance is set mainly by angular control of compliant/field pathways. In Design 2, t0 is the strongest driver (with θ2 and θ1 secondary), and r2, θ4, θ5 contribute little. Practically, this analysis suggests the search can be narrowed around the dominant variables and that geometry-driven alignment—angles in Slab Design-1 Design-2 10 −1 10 0 10 1 10 2 10 3 r = | u y |/| u x | samples median IQR r=1 Figure 7: Area-weighted directionality r = |uy|/|ux| at t∗ = 0 . 25. Violins show surfaceelement distributions; squares/whiskers mark median/IQR; dashed line r = 1. Both tails trimmed by 0.5% (by area) for display. Table 4: Directionality metrics at peak phase ( t∗ = 0 . 25), computed on surface elements and weighted by element area. High-end outliers were trimmed symmetrically by 0 . 5% of total area before plotting; the r > 100 column is reported on the full (untrimmed) data. Geometry % area % area % area % area (r > 1) (r > 2) (r > 5) (r > 100) Slab 85.859 58.586 22.727 0.50 Design-1 100.000 99.240 61.434 9.04 Design-2 100.000 98.819 64.495 8.98 9 [21] Qinghu Guo, Dongxu Li, Zhonghua Yao, Huajun Sun, Hua Hao, Hanxing Liu, and Shujun Zhang. A supercritical relaxor phase boundary for ultrahigh electrostrictive properties. Inorganic Chemistry Frontiers, 10:6918–6927, 2023. doi:10.1039/D3QI01636F. Published 13 Oct 2023. [22] Vincenzo Esposito, Henrik Bruus, Nini Pryds, and Igor Lubomirsky. Micropump with electrostrictive material actuation, 2025. International Bureau of the World Intellectual Property Organization. [23] Victor B. Tinti, Jae Kyeong Han, Victor Frederiksen, Huaiyu Chen, Jesper Wallentin, Ilya Kantor, Anders Lyksborg-Andersen, Thomas W. Hansen, Gihoon Bae, Weiguo Song, Eugen Stamate, Daniele Zanetti de Florio, Henrik Bruus, and Vincenzo Esposito. Oxygendefective electrostrictors for soft electromechanics. Science Advances, 10(35):eadq3444, 2024. doi:10.1126/sciadv.adq3444. [24] K. Uchino. Deformable mirror using the pmn electrostrictor. Applied Optics, 20(17): 3077–3081, 1981. doi:10.1364/AO.20.003077. [25] Pierre-Yves Madec. Overview of deformable mirror technologies for adaptive optics and astronomy. In J.-P. V´eran B. L. Ellerbroek, E. Marchetti, editor, Proc. SPIE 8447, Adaptive Optics Systems III, volume 8447 of SPIE Proceedings, page 844705, Amsterdam, Netherlands, 2012. European Southern Observatory. doi:10.1117/12.926909. [26] L. E. Cross, S. J. Jang, R. E. Newnham, S. Nomura, and K. Uchino. Large electrostrictive effects in relaxor ferroelectrics. Ferroelectrics, 23(1):187–191, 1980. doi:10.1080/00150198008018801. [27] Craig L. Hom and Natarajan Shankar. A fully coupled constitutive model for electrostrictive ceramic materials. Journal of Intelligent Material Systems and Structures, 5(6):795–801, 1994. doi:10.1177/1045389X9400500610. [28] V. S. Vikhnin, R. Blinc, and R. Pirc. Mechanisms of electrostriction and giant piezoelectric effect in relaxor ferroelectrics. Journal of Applied Physics, 93(12):9947–9952, June 2003. doi:10.1063/1.1575915. [29] Chang Won Ahn, Gangho Choi, Ill Won Kim, Jae-Shin Lee, Ke Wang, Younghun Hwang, and Wook Jo. Forced electrostriction by constraining polarization switching enhances the electromechanical strain properties of incipient piezoceramics. NPG Asia Materials, 9:e346, 2017. doi:10.1038/am.2016.210. [30] Andrew J. Bell. Factors influencing the piezoelectric behaviour of pzt and other “morphotropic phase boundary” ferroelectrics. Journal of Materials Science, 41(1):13–25, 2006. doi:10.1007/s10853-005-5913-9. [31] H. Zhang, N. Pryds, D.-S. Park, N. Gauquelin, S. Santucci, D. V. Christensen, D. Jannis, D. Chezganov, D. A. Rata, A. R. Insinga, I. E. Castelli, J. Verbeeck, I. Lubomirsky, P. Muralt, D. Damjanovic, and V. Esposito. Atomically engineered interfaces yield extraordinary electrostriction. Nature, 609(7928):695–700, 2022. doi:10.1038/s41586-022-05073-6. [32] Hao Liu, Yun Chen, Zhi Du, Qiong Zhang, Xiang Qiu, Jing Guo, Ke Lin, Yanbo Wang, Xianlin Dong, Duanwu Cao, Yanting Wang, Seung-Hyub Baek, Inwhan Jeon, Jihong Wen, Xiangdong Ding, Jun Sun, Juli´an A. Rodriguez, Wangyang Li, Xuegeng Dong, Ke-Zun Guo, Ramamoorthy Ramesh, Xiaoqing Pan, and Xiangdong Weng. Giant piezoelectric voltage coefficient in grain-oriented modified batio3 single crystals. Nature Communications, 7: 13089, 2016. doi:10.1038/ncomms13089. 16 [33] Roderic S. Lakes. Foam structures with a negative poisson’s ratio. Science, 235(4792): 1038–1040, 1987. doi:10.1126/science.235.4792.1038. [34] Leon Mizzi, Elrashid Mahdi, Aleksandr Titov, Ruben Gatt, David Attard, Kenneth E. Evans, Joseph N. Grima, and Eng-Poh S. Tan. Mechanical metamaterials with star-shaped pores exhibiting negative and zero poisson’s ratio. Materials Design, 146:196–207, 2018. doi:10.1016/j.matdes.2018.03.011. [35] Jason Valentine, Shuang Zhang, Thomas Zentgraf, Eric Ulin-Avila, Dmitri A. Genov, Guy Bartal, and Xiang Zhang. Three-dimensional optical metamaterial with a negative refractive index. Nature, 455(7211):376–379, 2008. doi:10.1038/nature07247. [36] David Schurig, John J. Mock, Bradley J. Justice, Steven A. Cummer, John B. Pendry, Anthony F. Starr, and David R. Smith. Metamaterial electromagnetic cloak at microwave frequencies. Science, 314(5801):977–980, 2006. doi:10.1126/science.1133628. [37] J. B. Pendry. Negative refraction makes a perfect lens. Physical Review Letters, 85(18): 3966–3969, 2000. doi:10.1103/PhysRevLett.85.3966. [38] Y. Y. Chen, G. K. Hu, and G. L. Huang. Adaptive elastic metamaterials with tunable negative mass density by piezoelectric shunting. Journal of Intelligent Material Systems and Structures, 27(17):2469–2475, 2016. doi:10.1177/1045389X16671331. [39] Yanhui Xiao, Jihong Wen, and Xuefeng Wen. Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators. Journal of Physics D: Applied Physics, 45(19):195401, 2012. doi:10.1088/0022-3727/45/19/195401. [40] Geon Lee, Dongwoo Lee, Jeonghoon Park, Yeongtae Jang, Miso Kim, and Junsuk Rho. Piezoelectric energy harvesting using mechanical metamaterials and phononic crystals. Communications Physics, 5:94, 2022. doi:10.1038/s42005-022-00869-4. [41] Christopher Sugino, Massimo Ruzzene, and Alper Erturk. Nonreciprocal piezoelectric metamaterial framework and circuit strategies. Phys. Rev. B, 102:014304, Jul 2020. doi:10.1103/PhysRevB.102.014304. [42] Haixin Huang, Yanhua Zhou, Yan Tan, and Jian Jin. A graded metamaterial for broadband and high-capability piezoelectric energy harvesting. Energy, 254:124381, 2022. doi:10.1016/j.energy.2022.124381. [43] Pourya Eghbali, Jos´e da Silva Sinke, E. de Jong, and Derk M. Brouwer. Study in circular auxetic structures for efficiency enhancement in piezoelectric vibration energy harvesting. Scientific Reports, 10(1):16338, 2020. doi:10.1038/s41598-020-73425-1. [44] Jikun Yang, Zhanmiao Li, Xudong Xin, Xiangyu Gao, Wei Chen, Shuang Ren, Yao Chen, Fei Wang, Zhaoqing Liu, Shiyuan Liu, Ping Tan, Qiaofeng Dai, and Jianfeng Song. Designing electromechanical metamaterial with full nonzero piezoelectric coefficients. Science Advances, 5(11):eaax1782, 2019. doi:10.1126/sciadv.aax1782. [45] Katia Bertoldi, Vincenzo Vitelli, Johan Christensen, and Martin van Hecke. Flexible mechanical metamaterials. Nature Reviews Materials, 2:17066, 2017. doi:10.1038/natrevmats.2017.66. [46] M. J. A. Smith, B. T. Kuhlmey, C. Martijn de Sterke, C. Wolff, M. Lapine, and C. G. Poulton. Electrostriction enhancement in metamaterials. Phys. Rev. B, 91:214102, Jun 2015. doi:10.1103/PhysRevB.91.214102. 17 [47] T. R. Shrout and Sei-Joo Jang. Relaxor ferroelectrics for electrostrictive transducers. Technical Report ADA248671, U.S. Department of Defense (DTIC), 1990. URL https: //apps.dtic.mil/sti/tr/pdf/ADA248671.pdf. [48] Larry L. Howell. Compliant Mechanisms. Wiley-Interscience, New York, 2001. ISBN 978-0471384786. [49] Craig L. Hom and Natarajan Shankar. A finite element method for electrostrictive ceramic devices. International Journal of Solids and Structures, 33(12):1757–1779, 1996. doi:10.1016/0020-7683(95)00123-9. [50] COMSOL Multiphysics. Electrostrictive disc: Modeling nonlinear strain in electrostrictive materials using the mems module. Application library example, COMSOL, 2023. [51] Craig L. Hom, Steven M. Pilgrim, Natarajan Shankar, Keith Bridger, Mona Massuda, and Stephen R. Winzer. Calculation of quasi-static electromechanical coupling coefficients for electrostrictive ceramic materials. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 41(4):542–551, 1994. doi:10.1109/58.294116. 18