A Perforated-Root Piezoelectric Cantilever Harvester with a Bow-Tie Cellular Substrate: Distributed-Parameter Modelling, Finite-Element Analysis, and Fatigue-Constrained Design

Abstract

Cellular substrates placed beneath the piezoceramic are increasingly used to raise the output of vibration energy harvesters, yet the enhancement is commonly attributed to auxeticity, and the durability penalty of perforation is seldom quantified. This paper presents a perforated-root piezoelectric cantilever in which a doubly periodic array of bow-tie (double-arrowhead) through-holes occupies the high-curvature root beneath the electrode while the distal span remains solid. Using a cell of positive effective Poisson’s ratio, we show that power gain is a substrate-compliance and strain-relocation effect set by the position of the perforation relative to the electrode rather than by a negative Poisson’s ratio. A segmented distributed-parameter model in which the perforated root is homogenised and enters the beam through its longitudinal effective modulus $E_1^{*}$, with the piezoelectric coupling reduced to its plane-stress value, is derived and tested against three independent finite-element measurements on the explicit hole geometry in ANSYS Parametric Design Language (APDL). Two agree closely: a direct axial-tension test returns $E_1^{*}/E_s=0.270$ against 0.268 predicted (0.7%), and the substrate-only fundamental is 21.7 Hz against 21.5 Hz predicted (1.0%). The third does not: with the piezoceramic present, the reduction predicts 47.7 Hz against 36.4 Hz, so the absolute frequency of the complete device is not yet established, and only the trends are relied upon here. At equal overall dimensions and piezoceramic, the baseline cell raises the peak power by about 14% over the solid beam, from 38.6 to 44.0 μW, and lowers the resonance from 57.6 to 50.7 Hz; both effects grow with hole size, reaching 23% and 46.5 Hz at the largest cell examined. A fatigue-constrained formulation, in which the net-section ligament stress bounds the usable fill factor, caps the fill at $f\approx 0.59$ under a 1 g excitation and yields a fill factor–load design map for durable operation.

1. Introduction

Harvesting ambient mechanical vibration to power small autonomous electronic devices has become an active area of energy research, motivated by the goal of reducing or eliminating the batteries that are costly to service and to replace [1]. Among the candidate transduction principles, piezoelectric conversion is the most widely adopted because of its high-power density, favourable scaling and mechanical simplicity [2]. The archetypal device is a piezoceramic-laminated cantilever operated near resonance, whose coupled electromechanical behaviour is accurately captured by the closed-form distributed-parameter model of Erturk and Inman [3,4] and its later extension to nonlinear and parametrically excited regimes [5]. That framework remains the standard basis for predicting harvester response and is the point of departure for the model developed here. Despite this maturity, two obstacles persist: the strain reaching the transducer is exploited inefficiently, and the fundamental resonance of a practical beam usually lies above the band in which ambient vibration is most intense. In parallel, recent advances in high-performance piezoelectric materials have continued to raise the achievable power density and broaden the design space for such harvesters [6,7].

A first strategy for raising the delivered strain reshapes the beam or the piezoceramic. Friswell and Adhikari showed that tailoring the shape of the piezoelectric patch alone can substantially increase the harvested energy [8]. Trapezoidal and hollow substrate geometries have since been optimised to concentrate strain and raise the power density [9], while thickness-graded unimorphs constrained to a fixed first frequency amplify the axial strain in the active layer [10]. Partial coating of the beam has been used to improve the power density at reduced material cost [11] and extended-support layouts even out the strain field along the span [12].

Closer to the present concept, locally removing material through slots, notches or holes both concentrates strain near the transducer and lowers the resonant frequency. Tamrakar et al. reported that a single well-placed slot raises the power density by roughly 70% [13], and a notched variant lowered the frequency while increasing the output voltage [14]. Our earlier work established how slit arrays reshape the modal response of the beam and its energy-harvesting behaviour [15]. Most directly, Su et al. showed that perforating a piezoelectric cantilever produces stress concentration and improves its output [16]; that study, however, perforates the transducer region itself, does not treat the perforation as a designed cellular medium beneath the electrode, and places no durability bound on the hole size.

The complementary objective of lowering the operating frequency has been pursued mainly by adding mass or complexity. Lallart et al. introduced a self-tuning stiffness scheme to track the excitation frequency [17], and Kouritem et al. used sliding proof masses to broaden the bandwidth of a harvester array [18]. Bistable and impacting architectures widen the usable band through nonlinear dynamics [19], as do coupled crossed-beam layouts [20], while a meandered beam with a reduced-width, stress-concentrating section tunes the device downward [21]. Such solutions are effective but generally introduce added mass, moving parts or control overhead. Related nonlinear structural concepts include bistable composite laminated cantilever plates, whose snap-through has been analysed for aeroelastic flutter control [22], and quasi-zero-stiffness devices that combine vibration isolation with energy harvesting using piezoelectric buckled beams [23]; the present compliant-cantilever concept is a linear, root-perforation counterpart to these designs.

A more recent and now dominant substrate-side strategy inserts a cellular, frequently auxetic (negative Poisson ratio) sub-structure beneath the piezoceramic to funnel strain into it. Li et al. demonstrated that an auxetic core in a bimorph raises the transverse stress and the power output [24], and Eghbali et al. popularised bonded auxetic boosters that magnify the power of a conventional cantilever at low frequency [25]. The idea has been realised with hexachiral cells [26], perforated reentrant-chiral honeycombs [27] and three-dimensional metastructure substrates [28], and the subfield has been surveyed comprehensively [29]. Very recent work continues to report gains from negative Poisson ratio honeycomb substrates optimised jointly with the proof mass [30].

A closer reading of this literature, however, exposes an unsettled question of mechanism. The enhancement is routinely credited to the negative Poisson’s ratio of the cell, yet several studies suggest that auxeticity is incidental rather than causal. Chen et al. found that gradient auxetic cells raise the power while producing a more uniform strain field-that is, without increasing the peak stress [31]. In a controlled comparison, Kurt et al. showed that non-auxetic geometries can match the gains of auxetic ones and argued that benchmarking against plain beams is misleading [32]. Fatahi et al. reported enhancement while the equivalent stress in the active layer remained essentially constant [33], and Ranjbarzadeh et al., using homogenization with periodic boundary conditions, concluded that the benefit arises from stiffener-like effective stiffness behaviour rather than from the negative Poisson’s ratio itself [34]. Together, these results point to substrate compliance and the relocation of strain toward the transducer-not auxeticity as the operative effect; what has been missing is a configuration that cleanly isolates the two, together with a transparent model that explains the gain.

A second, largely separate gap concerns durability. Because the bending strain of a cantilever peaks at the clamped root, cyclic fatigue and cracking there govern the harvester’s lifetime, a failure mode reviewed by Salazar et al. [35]; Avvari et al. emphasised that this root fatigue, although decisive, is seldom coupled quantitatively to the power design [36]. Long-term testing confirms progressive cracking and resonance drift under cyclic loading [37], with comparable degradation observed for macro-fibre-composite harvesters over millions of cycles [38]. Interestingly, a cellular substrate can reduce the stress carried by the brittle piezoceramic and extend its life [39]; introducing a perforation, however, deliberately relocates the critical stress into the slender ligaments between the holes, so the usable hole size becomes bounded by the ligament endurance strength. This fatigue limit-rather than the power itself-sets the practical design ceiling, and it has not previously been coupled to the power optimisation of a perforated-substrate harvester.

In this work, we address both gaps with a perforated-root piezoelectric cantilever in which a doubly periodic array of bow-tie (double-arrowhead) through-slits occupies the high-curvature root region directly beneath the piezoceramic, while the distal span remains solid (Figure 1). The novelty of the study is threefold. First, and mechanistically, it uses a bow-tie cell whose effective Poisson’s ratio is positive (Section 2) to show that power enhancement is a substrate-compliance and strain-relocation effect governed by the position of the perforation relative to the electrode and therefore does not require auxeticity. This corroborates and sharpens, with a controlled, root-placed, positive-$\nu$ configuration, the effective stiffness arguments of Kurt et al. [32] and Ranjbarzadeh et al. [30], rather than resolving the debate outright; a single positive-$\nu$ cell adds one consistent data point but does not by itself isolate compliance from auxeticity. Second, methodologically, it develops a segmented distributed-parameter model in which the perforated root is homogenised and enters the beam through its longitudinal effective modulus $E_1^{*}$ and tests it against three independent finite-element measurements on the explicit hole geometry. A direct axial-tension test of the unit cell returns $E_1^{*}/E_s=0.270$ against 0.268 predicted (0.7%), and the substrate-only fundamental is 21.7 Hz against 21.5 Hz predicted (1.0%); with the piezoceramic present, however, the reduction predicts 47.7 Hz against an explicit-hole 36.4 Hz. Cell homogenization and the segmented beam are therefore validated, but the absolute frequency of the complete laminated device is not validated and is identified as an open item rather than claimed. Third, and from a design standpoint, it introduces a fatigue-constrained framework in which the net-section ligament stress bounds the usable fill factor, yielding an excitation-dependent optimum and a fill factor–load design map. At equal overall dimensions and piezoceramic, the proposed device delivers about 14% more power than the solid beam at the baseline cell while simultaneously lowering the resonance, with both effects increasing monotonically with hole size.

The remainder of the paper is organised as follows. Section 2 develops the analytical model the effective properties of the perforated substrate, the reduced piezoelectric coefficient, the segmented composite beam, the coupled electromechanical formulation and the ligament-stress estimate. Section 3 describes the three-dimensional coupled-field finite-element model. Section 4 validates the distributed-parameter model against the finite-element results at the fundamental mode. Section 5 presents the results and discussion the solid-versus-perforated comparison, the effect of hole size, the fatigue-constrained optimisation, load matching and the design map and Section 6 concludes.

2. Analytical Model

2.1. Configuration

The device (Figure 1) is a unimorph cantilever of length $L$ and width $b$, clamped at $x=0$ and free at $x=L$, comprising a metallic substrate of thickness $h_s$ and a piezoceramic patch of thickness $h_p$ bonded over $0\le x\le L_p$. A periodic array of through-thickness bow-tie (double-arrowhead) slits is introduced in the root, i.e., beneath the piezoceramic; the distal span remains solid. Two spanwise segments are therefore distinguished: a composite root over $\left[0,L_p\right]$ (perforated substrate plus piezoceramic) and a solid distal span over $\left[L_p,L\right]$. The unknowns are the transverse displacement $w\left(x,t\right)$ relative to the moving base and the voltage $V\left(t\right)$ across a resistive load $R$; the base undergoes a harmonic acceleration ${\ddot{w}}_b=A\hspace{0.17em}{e}^{j\omega t}$.

2.2. Effective Properties of the Perforated Substrate

The bow-tie perforation is a doubly periodic array of square unit cells of pitch $p_x\times p_z=8\times 8 \mathrm{mm}$, with four along the length and two across the width, giving eight holes in the root. This pitch value is used in both analytical homogenization and the finite-element calculations, and it fixes the relative density and effective properties reported below. Each cell contains a single connected double-arrowhead (hourglass) through-hole: two opposed trapezoidal halves of half-base $c_w$ that taper to a central neck of half-width $g$ and merge there, so the hole is continuous from $z=z_c-c_h$ to $z=z_c+c_h$. The hole size is set by the fill factor

$$f={\displaystyle \frac{2c_h}{p_z}},$$

(1)

the fraction of the cell width spanned by the hole, with $c_w$ and $g$ held fixed (

Table 1. Connected bow-tie cell geometry. Pitch $p_x\times p_z=8\times 8$ mm, half-base $c_w=3.12 \mathrm{mm}$ and neck $g=0.50 \mathrm{mm}$ is fixed; the fill factor $f=2c_h/p_z$ is swept. Four cells span the length and two the width, giving eight holes.

Case	1	2	3	4 (Device)	5	6
Fill $f$	0.45	0.55	0.65	0.74	0.85	0.92
$c_h$ (mm)	1.80	2.20	2.60	2.96	3.40	3.68
$z$-ligament $p_z\left(1-f\right)$ (mm)	4.40	3.60	2.80	2.08	1.20	0.64

). This definition is the one the mechanics requires. The beam is bent about the $z$ axis and the load travels along $x$; to pass a hole it must squeeze through the ligaments left in the $z$ direction, of width $p_z\left(1-f\right)$. Enlarging the hole along $x$ lengthens the detour but never narrows that load path, and leaves $E_1^{*}$ almost unchanged ($0.314\to 0.260$, a factor of 1.2); enlarging it along $z$ pinches the path and $E_1^{*}$ spans $0.555\to 0.087$, a factor of six . The same choice makes the ligament stress of Equation (10) self-consistent, since $1/\left(1-f\right)$ there is the net-section magnification of a section whose load-carrying width has been reduced by the fraction $f$. The relative density $\overline{\rho }$ follows from the cell geometry and is reported alongside $f$; the two are in one-to-one correspondence but are not equal ($f=0.74$ gives $\overline{\rho }=0.634$).

Because the bending half-wavelength of the harvesting mode spans several cells, the perforated band is replaced by an equivalent homogeneous continuum. Its effective plane-stress properties are obtained by energy-based numerical homogenization of the unit cell with periodic boundary conditions: the cell is discretized on a structured plane-stress mesh, three independent unit macro-strains $\left({\epsilon }_1,{\epsilon }_2,{\gamma }_{12}\right)$ are imposed with periodic fluctuations, and the effective stiffness ${\mathbf{C}}^{*}$ is recovered from the cell-averaged strain energy. The procedure recovers the base modulus exactly for a solid cell ($E_1^{*}/E_s=1.000$, ${\nu }_{12}^{*}=0.300$), and its prediction for the baseline cell is confirmed directly by finite element: an axial-tension test on the explicit eight-hole geometry (46,625 elements) returns $E_1^{*}/E_s=0.270$ and $\overline{\rho }=0.634$ against 0.268 and 0.634, with predicted agreement to 0.7% and 0.06%, respectively. The extracted properties (Table 5, Figure 2) show the longitudinal modulus falling steeply with $f$ while the in-plane shear modulus collapses.

Crucially, the effective Poisson ratio is positive for every hole size, ${\nu }_{12}^{*}=+0.17$ down to $+0.01$ (Figure 2b). Although the double-arrowhead is a canonical auxetic motif, its negative Poisson response requires loading along the re-entrant ($z$) axis; the beam is bent along $x$, transverse to that axis, so the governing ${\nu }_{12}^{*}$ is positive and the cell is non-auxetic in the direction that matters for bending. The power gain reported below therefore does not require auxeticity; the positive ${\nu }_{12}^{*}$ shows that a negative Poisson’s ratio is not necessary for the enhancement, though it does not by itself prove that compliance is the sole cause. In the beam model, the perforated band enters through $E_1^{*}$ and $\overline{\rho }$; because Euler–Bernoulli bending is uniaxial only $E_1^{*}$ is carried into the segmented beam, and the remaining constants are reported to characterise the cell and to support the mechanism rather than being used in the one-dimensional response.

2.3. Reduced Piezoelectric Coefficient

A thin piezoceramic laminated to a much stiffer substrate is in a state of plane stress, using parameters of

Table 2. Geometric and material parameters.

$\mathit{L}$	100 mm	${\mathit{E}}_{\mathit{s}},{\mathit{\rho }}_{\mathit{s}}$	100 GPa, 7165 kg m−3
$b$	20 mm	$E_p,{\rho }_p$	61 GPa, 7750 kg m−3
$h_s,h_p$	0.5, 0.4 mm	${\epsilon }_{33}^S/{\epsilon }_0$	826
$L_p$	40 mm	$R$	1 MΩ
$p_x\times p_z$	$8\times 8$ mm	$\zeta$	0.02
$A$	1 m s−2	${\overline{e}}_{31}$	−16.05 C m−2
$K_t$	3.0	${\sigma }_f$	200 MPa (at ×107 cycles)

and the PZT-5A2 of

Table 3. PZT-5A2 constitutive properties (poled through the thickness), following the IEEE Standard on Piezoelectricity [40].

Group	Constants	Values
Elastic stiffness $c_{ij}^E$ (GPa)	$c_{11}=c_{22}$; $c_{12}$; $c_{13}=c_{23}$; $c_{33}$; $c_{44}=c_{55}$; $c_{66}$	120.35; 75.09; 75.18; 110.87; 21.05; 22.58
Piezoelectric $e_{ij}$ (C m−2)	$e_{31}=e_{32}$; $e_{33}$; $e_{15}=e_{24}$	$-5.35$; 15.78; 12.29
Rel. permittivity ${\epsilon }_{ii}^S/{\epsilon }_0$	${\epsilon }_{11}={\epsilon }_{22}$; ${\epsilon }_{33}$ (poling axis)	919; 826
Density ${\rho }_p$ (kg m−3)	-	7750

so the relevant transverse coefficient is the reduced value

$${\overline{e}}_{31}=e_{31}-{\displaystyle \frac{c_{13}}{c_{33}}}\hspace{0.17em}{e}_{33}=-16.05 \mathrm{C}/{\mathrm{m}}^2,$$

(2)

rather than the bare $e_{31}$; the three-dimensional finite-element model captures this automatically through its full constitutive matrix.

2.4. Segmented Composite Beam

The substrate underside is taken as the reference for measuring y and evaluating the section properties of the composite root. The piezoceramic enters the composite beam through its plane-stress Young’s modulus in the length direction, E_p = 1/s₁₁^E ≈ 61 GPa, the reciprocal of the in-plane elastic compliance at constant electric field. This is the narrow-beam (free-edge) reduction, consistent with the coupling coefficient ē₃₁ = E_p d₃₁ used in Section 2.3. Thus, the neutral axis height and flexural rigidity of the composite root are

$$\overline{y}={\displaystyle \frac{E_s^{*}{h}_s^2+E_p{h}_p\left(2h_s+h_p\right)}{2\left(E_s^{*}{h}_s+E_p{h}_p\right)}},$$

(3)

$${\left(EI\right)}_1={\displaystyle \frac{b}{3}}\left[E_s^{*}\left({\left(h_s-\overline{y}\right)}^3+{\overline{y}}^3\right)+E_p\left({\left(h_s+h_p-\overline{y}\right)}^3-{\left(h_s-\overline{y}\right)}^3\right)\right],$$

(4)

with mass per unit length $m_1=b\left({\rho }_s^{*}{h}_s+{\rho }_p{h}_p\right)$. The distal segment is the bare substrate, ${\left(EI\right)}_2=E_{s}b{h}_s^3/12$ and $m_2={\rho }_{s}b{h}_s$.

2.5. Coupled Electromechanical Model and Power

Euler–Bernoulli bending of the piecewise-uniform beam under base excitation,

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}\left[EI\left(x\right){\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right]+m\left(x\right){\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}=-m\left(x\right){\ddot{w}}_b,$$

(5)

is projected onto the mass-normalised modes ${\varphi }_r$ of the segmented beam, ${\left[{EI{\varphi }_r}^{″}\right]}^{″}={\omega }_r^{2}m{\varphi }_r$, obtained numerically. The modal forcing, electromechanical coupling and clamped capacitance are

$${\sigma }_r={\int }_0^L m{\varphi }_r\hspace{0.17em}dx, {\theta }_r={\overline{e}}_{31}\hspace{0.17em}b\left(h_s+h_p/2-\overline{y}\right){\varphi }_r^{\prime \left(L_p\right)}, C_p={\displaystyle \frac{{\epsilon }_{33}^{S}b{L}_p}{h_p}},$$

(6)

so that, with modal damping ${\zeta }_r$,

$${\ddot{\eta }}_r+2{\zeta }_r{\omega }_r{\dot{\eta }}_r+{\omega }_r^2{\eta }_r-{\theta }_{r}V=-{\sigma }_r{\ddot{w}}_b,\hspace{1em}{C}_p\dot{V}+V/R+\sum _r{\theta }_r{\dot{\eta }}_r=0$$

(7)

Although Equation (7) retains all modes, the harvesting band of interest (10–100 Hz) contains only the fundamental, whose modal amplitude at resonance dominates the sum; the results below therefore use the single-mode reduction $r=1$, which reproduces the full-series response near the harvesting resonance to within the plotted line width. For a harmonic base acceleration, the steady-state voltage and average power are

$${\displaystyle \frac{\widehat{V}}{A}}={\displaystyle \frac{-\hspace{0.17em}j\omega {\sum }_r{\theta }_r{\sigma }_r/{\lambda }_r}{\left(1/R+j\omega C_p\right)+j\omega {\sum }_r{\theta }_r^2/{\lambda }_r}}, {\lambda }_r={\omega }_r^2-{\omega }^2+2j{\zeta }_r{\omega }_r\omega , P={\displaystyle \frac{{\left|\widehat{V}\right|}^2}{2R}}$$

(8)

2.6. Ligament Stress

The ligaments are solid substrate material, and the extreme fibre of the substrate is its underside, a distance $\overline{y}$ below the composite neutral axis. The resonant surface bending strain of the root is therefore

$${\epsilon }_s=\kappa \hspace{0.17em}\overline{y},$$

(9)

with curvature $\kappa$ recovered from the modal solution at the start of the perforated band. (Using $\kappa \hspace{0.17em}\left(h_s-\overline{y}\right)$-the strain at the top of the substrate-understates the ligament strain by a factor of about 19 here, because $\overline{y}=0.475 \mathrm{mm}$ sits close to $h_s=0.5 \mathrm{mm}$.) The ligaments carry the section load through the reduced net width, giving the nominal net-section ligament stress

$${\sigma }_{\mathrm{lig}}=K_t\hspace{0.17em}{\displaystyle \frac{E_s\hspace{0.17em}{\epsilon }_s}{1-f}},$$

(10)

where $E_s$ is the (solid) substrate modulus and $K_t$ an apex stress-concentration factor; the $1/\left(1-f\right)$ factor is the net-section magnification, consistent with $f$ being the fraction of load-carrying width removed (Section 2.2). The homogenised modulus $E_1^{*}$ governs the beam-scale bending and is deliberately not reused here, so the local ligament stress is not double-counted. The fatigue constraint ${\sigma }_{\mathrm{lig}}\le {\sigma }_f$ bounds the usable hole size. A harmonic finite-element solve on the explicit geometry at 1 g returns a peak ligament stress of 379 MPa for the baseline cell, confirming both the location and the order of magnitude of Equation (10) and placing the device well above the assumed endurance limit.

The polarisation axis is the thickness direction (axis 3); the finite-element model applies the identical material with polarisation along the thickness ($y$) axis. The distinct constants ($c_{33}<c_{11}$, ${\epsilon }_{33}<{\epsilon }_{11}$, $e_{33}$) therefore lie on the poling axis, consistent with Table 2 (${\epsilon }_{33}^S/{\epsilon }_0=826$) and with Equation (2).

2.7. Modelling Assumptions

The model rests on the following assumptions: (i) linear-elastic substrate and linear, small-signal piezoceramic; (ii) Euler–Bernoulli kinematics, justified by the slenderness $L/h_s\approx 200$; and (iii) perfect substrate–piezoceramic bonding and an ideal clamp; (iv) uniform, transverse, single-axis base excitation; and (v) proportional modal damping.

Two features of the real root are carried explicitly rather than smeared. First, the holes occupy only $32\times 16 \mathrm{mm}$ of the $40\times 20 \mathrm{mm}$ root: there are solid margins of 4 mm at each end and 2 mm along each edge, and those margins are stiff and carry load. The beam therefore has four segments solid root, perforated band, solid root, distal span and the perforated band carries the width-weighted modulus $E_{\mathrm{eff}}=\left[16|/|20|E_1^{*}|+|4|/|20|E_s\right]$, with the perforated strip and the solid edges acting as parallel springs in bending. Smearing $E_1^{*}$ over the whole root instead under-predicts the substrate-only fundamental by 24% (16.4 against a measured 21.7 Hz); with the margins honoured the prediction is 21.5 Hz. Second, the neutral axis differs between the solid margins and the perforated band, so the electromechanical coupling of Equation (6) is integrated piecewise over the electrode, and the curvature entering Equation (9) is evaluated where the holes begin rather than at the clamp, which lies in solid material.

3. Finite-Element Model

A three-dimensional coupled-field model (Figure 3) was built in ANSYS Mechanical APDL as the high-fidelity reference. The substrate was meshed with structural solid elements and the piezoceramic with coupled-field elements carrying the anisotropic elastic, piezoelectric and permittivity data. The resistive load was represented by a circuit element across equipotential electrode sets. A modal analysis classified the modes by participation factor and a harmonic analysis under 1 m s⁻² base acceleration, with constant modal damping and a frequency sweep clustered about each resonance, recovered the electrode voltage, delivered power, tip displacement and layer stresses. The clamped face was fully fixed and the lower electrode grounded. The model is summurised on

Table 4. Summary of the three-dimensional finite-element model.

Item	Setting
Substrate element	3-D structural solid (8-node)
Piezoceramic element	3-D coupled-field solid (UX, UY, UZ, VOLT)
Load element	circuit resistor across the electrodes
Substrate material	isotropic: $E_s=100 \mathrm{GPa},$ $\nu =0.3,$ ${\rho }_s=7165 \mathrm{kg}/{\mathrm{m}}^3$
Piezoceramic material	anisotropic $c^E,$ $e,$ ${\epsilon }^S$ of Table 3
Electrodes	equipotential (voltage-coupled) top and bottom patch faces
Mesh	swept hexahedral, $\sim 1 \mathrm{mm}$ in-plane
Modal analysis	Block–Lanczos, with participation-factor classification
Harmonic analysis	full, constant modal damping $\zeta =0.02,$ clustered sweep
Boundary/load	clamped at $x=0,$ lower electrode grounded, base accel. 1 m s−2, $R=1 \mathrm{M}\mathsf{\Omega }$

.

The first four undamped modes of the perforated-root harvester are shown in Figure 4. The fundamental (harvesting) mode at 36.4 Hz is a first bending mode whose curvature is concentrated in the perforated root beneath the electrode, which is the mechanism exploited in Section 5; the higher modes at 287.8, 425.6 and 804.0 Hz lie well above the harvesting band.

The corresponding von Mises stress field (Figure 5) confirms that the mechanical loading localises in the slender ligaments between the bow-tie slits, where the net section is smallest; this is the region that governs the fatigue-constrained design of Section 5.

4. Model Validation

The distributed-parameter model was tested against three independent finite-element measurements on the explicit hole geometry, each isolating one link in the chain. Throughout, the baseline cell is case 4 ($f=0.74$, $\overline{\rho }=0.634$), the eight-hole layout of the device.

The three checks are summarised in Figure 6.

Check 1: the cell. A static axial-tension test on a $4\times 2$-cell strip with free lateral faces (uniaxial stress, 46,625 elements) measured $\overline{\rho }=0.6336$ and $E_1^{*}/E_s=0.2698$, against 0.634 and 0.268 from the homogenisation: 0.06% and 0.7%. The unit-cell analysis of Section 2.2 is therefore confirmed directly.

Check 2: the segmented beam. A modal analysis of the substrate alone (no piezoceramic, hence no bonding, no electrical degrees of freedom) returned a fundamental of 21.7 Hz. The four-segment beam of Section 2 fed the measured $E_1^{*}$ and, honouring the solid margins, predicts 21.5 Hz: 1.0%. Smearing the perforation over the whole root instead gives 16.4 Hz, in error by 24%; the margins, not the beam theory, account for that difference.

Check 3: the complete device. With the piezoceramic present, the same reduction predicts a short-circuit fundamental of 47.7 Hz, whereas the explicit-hole coupled-field model returns 36.4 Hz. The piezoceramic raises the measured fundamental by a factor of 1.68 ($21.7\to 36.4 \mathrm{Hz}$) but the model by 2.22 ($21.5\to 47.7 \mathrm{Hz}$) the beam gains more stiffness from the patch than the three-dimensional model does. Two mechanisms are candidates and the present results do not separate them. The finite-element model bonds the patch by merging coincident nodes, and the two meshes are not conformal at the interface–the substrate face is perforated and refined, the patch face solid and coarse–so the patch may be only partially engaged and the measured frequency correspondingly soft. Equally, the beam treats the root as a perfect two-layer laminate, whereas the holes are several millimetres across on a 0.5 mm substrate: above a hole there is no substrate at all, and composite action is lost locally in a way the laminate assumption cannot represent. Therefore, we treat 47.7 and 36.4 Hz as two unresolved model predictions rather than as bounds on a physical value, since neither model is independently known to over- or under-predict. A useful additional observation is that the explicit-hole 36.4 Hz is a short-circuit modal result, in which electromechanical stiffening is negligible; it is therefore effectively the elastic frequency of the substrate-piezoceramic laminate. The analytical 47.7 Hz is likewise a purely elastic composite prediction, its eigenvalue containing no electromechanical term. The discrepancy is therefore mechanical, not electromechanical, and the piezoelectric coupling can be ruled out as its cause.

We therefore report on the position plainly. The cell homogenization and the segmented beam are validated to about 1%. The absolute harvesting frequency of the complete laminated device is not: it lies somewhere between 36 and 48 Hz, and settling it requires a coupled-field model with conformal substrate–piezoceramic bonding. The trends reported in Section 5: the direction and relative magnitude of the power gain and frequency shift with hole size rest on the two validated links and are not affected by this open item, but absolute frequencies should be read with it in mind. We emphasise that the power comparisons rest on the two validated links: the +14% gain is a ratio evaluated at each device’s own resonance under the same base acceleration, and both the effective modulus (0.7%) and the substrate-only fundamental (1.0%) are confirmed by finite element. The unresolved 31% concerns the absolute resonance of the complete laminate, which shifts where the peak sits but not the mechanism or the relative trends; a coupled-field model with conformal bonding is the specific step that will resolve it and is ongoing.

5. Results and Discussion

Unless stated otherwise, the excitation is 1 m s⁻² ($\approx 0.1 \mathrm{g}$), the load is $R=1 \mathrm{M}\mathsf{\Omega }$ and $\zeta =0.02$. Every comparison holds the beam length, width, thickness, piezoceramic patch and materials fixed, so that only the substrate topology differs.

5.1. Solid Versus Perforated-Root Harvester

Figure 7 compares the plain solid cantilever with the bow-tie harvester in which the perforation lies in the root, beneath the electrode. The perforated-root device delivers a higher peak power, 44.0 μW against 38.6 μW for the solid beam (a gain of 14% at equal overall dimensions and piezoceramic), and its resonance falls from 57.6 to 50.7 Hz. The response therefore shifts both upward in power and toward lower frequency. The perforated device is lighter than the solid beam by about 37% of the substrate mass over the perforated band; this mass removal is part of the mechanism and is discussed below. Each device is compared at its own resonance under a flat 1 m s⁻² base excitation; because ambient spectra are not flat, the delivered advantage under a specific vibration spectrum will differ from the resonant comparison quoted here.

Observation. The gain originates from a substrate-compliance strain-amplification mechanism. Perforating a span removes both mass and stiffness: the stiffness reduction increases the local bending curvature and hence the strain, while the mass reduction lowers the inertial forcing. In a cantilever, the curvature is greatest near the clamp, exactly where the piezoceramic is bonded; placing the compliant cells there amplifies the strain delivered to the transducer, whereas the mass removed near the clamp contributes little to the modal forcing because the mode-shape amplitude is small there. The strain gain therefore outweighs the mass penalty and the net power rises, while the reduced root rigidity lowers the resonance. The effect is governed by the position of the perforation relative to the electrode, not by the perforation itself.

5.2. Effect of Hole Size

The hole size, parametrized by the fill factor $f=2c_h/p_z$, sets the effective substrate modulus $E_1^{*}$ and the relative density $\overline{\rho }$. Figure 8 and

Table 5. Homogenised effective properties of the connected bow-tie cell (periodic unit-cell analysis of Section 2.2) and peak response. $G_s$ is the solid shear modulus. The homogenisation is confirmed at $f=0.74$ by direct finite-element measurement ($E_1^{*}/E_s=0.270$, $\overline{\rho }=0.634$).

Case	$\mathit{f}$	$\overline{\mathit{\rho }}$	${\mathit{E}}_1^{\mathbf{*}}/{\mathit{E}}_{\mathit{s}}$	${\mathit{\nu }}_{12}^{\mathbf{*}}$	${\mathit{f}}_{\mathbf{res}}$ (Hz)	$\mathit{P}$ (μW)
Solid	-	1.000	1.000	$+0.300$	57.6	38.6
1	0.45	0.775	0.555	$+0.170$	54.5	41.0
2	0.55	0.728	0.459	$+0.137$	53.5	41.8
3	0.65	0.676	0.355	$+0.103$	52.1	42.8
4 (device)	0.74	0.634	0.268	$+0.073$	50.7	44.0
5	0.85	0.582	0.162	$+0.038$	48.5	45.7
6	0.92	0.545	0.087	$+0.013$	46.5	47.4

show the power response for six cell sizes. As the holes grow, the peak power rises monotonically from 41.0 to 47.4 μW (6 to 23% over the solid beam) while the resonance falls from 54.5 to 46.5 Hz, a shift of 11 Hz without any added mass.

It is worth being explicit about why $f$ must be the width fraction. Had the hole instead been enlarged along the beam axis at fixed $c_h$, the effective modulus would have moved only from 0.314 to 0.260: the resonance would have shifted by under 1 Hz and the peak powers would have differed by 0.4%, i.e., the sweep would have shown nothing. The load path is pinched by the $z$ ligaments, not by the extent of the hole along $x$ and only a fill factor defined on the width exposes the design freedom.

Observation. Both directions are favourable for vibration harvesting. The rising power follows directly from the strain-amplification mechanism; the falling resonance is an independent benefit, because ambient mechanical vibration is concentrated at low frequency (10–100 Hz). A same-size solid beam is often too stiff to resonate in this band without an added tip mass, whereas increasing the cell size tunes the device downward without any mass addition. The design thus offers a single geometric parameter that simultaneously increases the power and lowers the operating frequency.

5.3. Fatigue-Constrained Optimisation

The power gain cannot be increased without bound, because softening the root also raises the stress in the slender ligaments. Figure 9a plots the peak power against fill factor together with the ligament stress at three excitation levels; the stress grows steeply with hole size, so the practical limit is set by the substrate endurance strength rather than by power. The bound here is a net-section scaling argument: Equation (10) magnifies the beam-surface strain by $1/\left(1-f\right)$ and the concentration factor $K_t$. The three-dimensional stress field of Figure 5 is a harmonic solution at 1 g on the explicit geometry; it confirms both the location of the critical stress (the inter-slit ligaments, as Equation (10) assumes) and its order of magnitude, with a peak of 379 MPa. A ligament-refined mesh with a peak-stress convergence check would sharpen the fillet-dependent value further.

The endurance limit (${\sigma }_f\approx 200 \mathrm{MPa}$ at ×10⁷ cycles) and stress-concentration factor ($K_t\approx 3.0$) are representative values for a high-strength metallic substrate; the substrate stiffness and density ($E_s=100 \mathrm{GPa}$, ${\rho }_s=7165 \mathrm{kg}/{\mathrm{m}}^3$) follow the authors’ prior work [13] and are not tied to a specific named alloy. Because these inputs are representative rather than measured, the fatigue map is presented as an illustrative design trade-off (a scaling argument), not as an FE-confirmed lifetime prediction. The concentration factor ($K_t\approx 3.0$) is a conservative value for a filleted re-entrant notch (Peterson’s charts), and the endurance limit (${\sigma }_f\approx 200 \mathrm{MPa}$ at ×10⁷ cycles) is representative of a high-strength metallic substrate; both are stated as representative rather than material-specific. We therefore present the fatigue map explicitly as illustrative of the design trade-off only: it fixes the shape of the power-durability boundary, and the absolute fill limit will move once a specific alloy and its measured S-N data are adopted.

Observation. The optimum is excitation dependent. At 0.1 g, the ligament stress stays below the assumed endurance limit even for the largest cell, so the device is power-limited, and the full gain is accessible. Because the stress scales linearly with acceleration, at 1 g the same cells reach the limit, and the design becomes fatigue-limited: the largest usable fill falls to $f\approx 0.59$. The optimal cell size is therefore application-specific, and a harvester intended for high-acceleration environments must trade some of the available power for durability. This constrained optimum, rather than a monotonic sweep, is the design-relevant result.

5.4. Load Matching and Design Map

Figure 9b shows that each cell exhibits a power maximum near $R_{\mathrm{opt}}\approx 1/\left({\omega }_r{C}_p\right)\approx 210 \mathrm{k}\mathsf{\Omega }$, below the 1 MΩ used elsewhere; matching the load raises the harvested power for every size, and $R_{\mathrm{opt}}$ is only weakly dependent on the perforation because $C_p$ is set by the (unchanged) piezoceramic. Combining the two design variables, Figure 10 presents the peak power over the fill factor–load plane with the fatigue boundary overlaid, identifying the operating region that maximises power within the durability constraint: large cells operated near $R_{\mathrm{opt}}$ but inside the fatigue limit.

The load curves deserve a comment. This harvester is strongly coupled: the dimensionless coupling ${\theta }_1^2/\left(C_p{\omega }_1^2\right)\approx 0.15$ exceeds $2\zeta =0.04$ by nearly a factor of four. Each cell therefore shows two power maxima rather than one near 72 kΩ and 641 kΩ, on the short- and open-circuit branches, respectively and the two are equal to four decimal places. The familiar weak-coupling estimate $R_{\mathrm{opt}}\approx 1/\left(\omega C_p\right)$ does not apply, and reporting a single matched load would be arbitrary: either branch delivers the same power, and the choice between them is a circuit design question, not a harvester one. The weak-coupling estimate $R_{\mathrm{opt}}\approx {\displaystyle \frac{1}{\left(\omega C_p\right)}}\approx 210 \mathrm{k}\mathsf{\Omega }$ is therefore not an optimum for this device; it lies in the dip between two equal branches, and we do not equate it to a matched load.

5.5. Discussion

The results establish a transparent design principle for cellular substrate piezoelectric harvesters: the perforation should occupy the high-curvature region beneath the electrode, where reducing the substrate stiffness amplifies the strain delivered to the transducer, so that the device simultaneously gains power and shifts to a lower, more useful resonance. Three further points follow.

First, the enhancement is a compliance and strain-relocation effect and is distinct from a negative Poisson ratio (auxetic) mechanism; the present cell exhibits a positive effective Poisson ratio (Section 2), so the benefit here arises from compliance rather than auxeticity. This corroborates and sharpens the effective stiffness arguments of Kurt et al. [28] and Ranjbarzadeh et al. [34]: the root-placed, positive-$\nu$ configuration adds one controlled data point but does not by itself isolate compliance from auxeticity, which would require a matched-compliance comparison of a positive-$\nu$ and a negative-$\nu$ cell at equal effective bending stiffness. That comparison is a clear next step; the present contribution is the placement principle and the fatigue-constrained design that follows from it. Two design questions follow. Because the mechanism is compliance-driven and shape-agnostic, round holes or slots at matched modulus would give the same gain; the bow-tie was chosen as the canonical auxetic motif to make the mechanistic point, and a round-hole comparison is a natural simplification. A bimorph would centre the neutral axis by symmetry, remove the perforation-induced lever-arm reduction and roughly double the active material, in a natural higher-output extension.

Second, the effective bending behaviour of the perforated root and hence the absolute resonance is governed by the cell’s effective stiffness rather than by its area fraction alone: at $f=0.74$ the cell retains 63% of its area but only 27% of its longitudinal modulus. Two bookkeeping points proved decisive and are worth stating for anyone repeating this class of analysis. The fill factor must be defined on the dimension that pinches the load path; defined on the beam axis instead, the entire sweep collapses and the design freedom disappears. And the perforation must be applied to the area it occupies: the solid margins around the hole array carry load, and smearing the homogenised modulus over the whole root under-predicts the fundamental by 24%. With both handled, the segmented beam matches an explicit-hole substrate model to 1%. The laminated device remains the open case (Section 4).

Third, the harvested power is sensitive to load matching, so performance should be reported at $R_{\mathrm{opt}}$ rather than at an arbitrary load, and the achievable hole size is bounded by ligament fatigue and therefore by the target excitation amplitude.

The study has limitations that also define the next steps. The distributed model is one-dimensional and assumes full strain transfer to the piezoceramic, so the reported power is an upper estimate; the ligament stress is a nominal net-section value whose fillet-dependent peak requires a resolved three-dimensional notch analysis; and the endurance limit and stress-concentration factor are representative rather than material-specific. An experimental demonstration on a laser-cut substrate would provide the strongest confirmation of the predicted power gain and frequency shift, and a spatially graded perforation denser near the clamp where the curvature peaks is a natural extension of the present framework.

6. Conclusions

This study introduced a perforated-root piezoelectric cantilever harvester in which a doubly periodic bow-tie cellular array occupies the high-curvature region of the substrate directly beneath the piezoceramic and developed the distributed-parameter and finite-element models required to design it. Three conclusions follow.

First, power enhancement is a compliance effect rather than an auxetic one. Because the chosen bow-tie cell has a positive effective Poisson’s ratio, the gain observed here cannot be ascribed to a negative Poisson’s ratio; it arises because softening the substrate in the high-curvature root amplifies the strain delivered to the transducer, while the mass removed there contributes little to the modal forcing because the mode-shape amplitude is small near the clamp. The enhancement is therefore governed by the position of the perforation relative to the electrode. Because the chosen cell has a positive effective Poisson’s ratio, this result corroborates and sharpens the effective stiffness arguments of the cellular substrate literature [28,30] rather than resolving the auxeticity debate outright, which would require a matched-compliance positive-versus-negative-$\nu$ comparison.

Second, the segmented distributed-parameter model with the perforated root homogenised and entering the beam through its longitudinal effective modulus $E_1^{*}$ and the piezoelectric coupling reduced to its plane-stress value was tested against three independent finite-element measurements on the explicit hole geometry. Two confirm it: an axial-tension test of the unit cell gives $E_1^{*}/E_s=0.270$ against 0.268 predicted (0.7%), and the substrate-only fundamental is 21.7 Hz against 21.5 Hz predicted (1.0%). The third does not: with the piezoceramic present, the reduction predicts 47.7 Hz against 36.4 Hz. The cell homogenisation and the segmented beam are therefore established; the absolute frequency of the complete laminated device is not and is left as an explicit open item. Either the finite-element bonding of the patch is incomplete, or the perfect-laminate assumption fails where the holes remove the substrate beneath it; a coupled-field model with conformal bonding is required to decide, and until then only the trends should be relied upon.

Third, at equal overall dimensions and piezoceramic, the perforated-root device delivers about 14% more power than the solid beam at the baseline cell, 44.0 against 38.6 μW, while shifting its resonance from 57.6 to 50.7 Hz, toward the band where ambient vibration is most intense. Both effects increase monotonically with hole size, reaching 23% and 46.5 Hz at the largest cell examined provided the fill factor is defined on the cell width, the dimension that pinches the load path. Because perforation relocates the peak stress into the slender ligaments, the usable hole size is bounded by ligament fatigue: under a 1 g constraint, the largest usable fill is $f\approx 0.59$, and a harmonic finite-element solve on the explicit geometry returns 379 MPa at the baseline cell-confirming the location and order of magnitude of the net-section estimate, and placing the device as built outside its own durability bound. Because the harvester is strongly coupled (${\theta }_1^2/C_p{\omega }_1^2\approx 0.15\gg 2\zeta$), the load curve has two equal optima rather than one, near 72 kΩ and 641 kΩ, so performance should be reported at a stated branch within the fatigue limit rather than at an arbitrary resistance.

Several extensions follow naturally. The distributed model is one-dimensional and assumes full strain transfer to the piezoceramic, so the reported power is an upper estimate; a resolved three-dimensional notch analysis would sharpen the fillet-dependent ligament stress, and an experimental demonstration on a laser-cut substrate would provide the strongest confirmation of the predicted power gain and frequency shift. A spatially graded perforation denser near the clamp where the curvature peaks is a promising route to further gains within the same fatigue budget.