Modelling the flexoelectric effect in solids: a micromorphic approach

Andrew McBride; Denis Davydov; Paul Steinmann

REVIEW 4 major objections 4 minor 55 references

Modelling the flexoelectric effect in solids: a micromorphic approach

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A penalty-coupled micromorphic field lets ordinary C0 finite elements capture flexoelectricity and size effects.

desk verdict A genuinely useful C0 finite-element framework for finite-strain flexoelectricity, but the numerical evidence that it converges to the targeted gradient theory is thinner than the headline claim suggests. read the letter →

arxiv 1909.08695 v2 pith:LXIQQ6GN submitted 2019-08-20 physics.class-ph physics.comp-ph

Andrew McBride , Denis Davydov , Paul Steinmann This is my paper · ORCID

classification physics.class-phphysics.comp-ph

keywords flexoelectricitymicromorphiccontinuafiniteelementmethodstraingradientelectromechanicalcouplingsize-dependentresponsenonlinearelectro-elasticity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Flexoelectricity couples the gradient of deformation to electric polarization, but the governing equations contain second gradients of the motion, which normally force finite element approximations to be globally $C^1$-smooth. This paper claims that a micromorphic reformulation removes that obstacle: an independent micro-deformation field $\bar F$, penalised to stay close to the deformation gradient $F$, supplies the needed gradient information while all fields remain $C^0$. The result is a fully nonlinear, three-dimensional finite element formulation of flexoelectricity that also reproduces size-dependent stiffening. The payoff is that complex flexoelectric geometries become accessible with standard finite element technology rather than specialised $C^1$ elements.

What carries the argument

The load-bearing device is the micromorphic micro-deformation $\bar F$ and its gradient $\bar G = \mathrm{Grad}\,\bar F$, promoted to independent degrees of freedom. The scale-bridging energy $\tfrac12 p[\bar F - F]:[\bar F - F]$ with penalty parameter $p$ lets $\bar G$ approximate the true second gradient $G = \mathrm{Grad}\,\mathrm{Grad}\,\boldsymbol{\varphi}$; as $p\to\infty$ the formulation reproduces gradient elasticity, while finite $p$ keeps the problem $C^0$. The proposed flexoelectric energy $\psi_{\mathrm{flexo}} = \upsilon [f^T E]\cdot \bar G : I$ is linear in $\bar G$, which makes the flexoelectric contribution to the dielectric displacement scale linearly with the flexoelectric coefficient. A parameter-classification diagram shows how the same code reduces to ordinary elasticity, electro-elasticity, micromorphic elasticity and gradient elasticity by sending selected parameters to zero or infinity.

What would settle it

Measure the electric potential across a bent cantilever of a known flexoelectric material (for example a relaxor ceramic) at several beam thicknesses and compare with the model's predicted linear scaling of polarization with the flexoelectric coefficient and concentration of the micro-gradient at the clamped end. If the measured potential does not grow linearly with $\upsilon$ or does not show the predicted size-dependent distribution, the assumed form of Eq. (13) is ruled out. A numerical check is also possible: compute the same cantilever with increasing penalty $p$ and check that the potential converges; the paper reports that near the clamped boundary $\bar F$ cannot be tied to $F$ no matter how large $p$ is, so this is where the approximation should be tested.

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Extended reading notes

Core claim

The central claim is that flexoelectricity can be formulated as a micromorphic continuum problem and solved with conventional $C^0$-continuous finite elements. The authors introduce the micro-deformation $\bar F$ as an independent field whose gradient $\bar G = \mathrm{Grad}\,\bar F$ plays the role of the second gradient of the macroscopic motion; a penalty-like scale-bridging energy $\tfrac12 p[\bar F - F]:[\bar F - F]$ ties $\bar F$ to $F$, so in the limit $p\to\infty$ the micromorphic gradient approaches $G = \mathrm{Grad}\,\mathrm{Grad}\,\boldsymbol{\varphi}$. The flexoelectric energy is taken to be linear in $\bar G$, coupling it to the pulled-back electric field, and the Dirichlet principle supplies the coupled balance equations for the macroscopic motion, the micro-deformation and the electric potential. Numerical examples in three dimensions demonstrate the size-dependent stiffening of a strip with a hole and the flexoelectric potential generated in a bent cantilever.

Load-bearing premise

The load-bearing premise is the constitutive choice in Eq. (13) that flexoelectric energy is linear in the micromorphic gradient $\bar G$, together with the finite penalty coupling $p$ standing in for the true second gradient; the paper itself notes near concentrations the tie $\bar F \approx F$ cannot be sharpened regardless of $p$, and that experimental validation of the full FM-Elasticity model is future work.

Editorial extensions

If this is right

A single $C^0$ finite element code now covers a spectrum of theories—nonlinear elasticity, electro-elasticity, micromorphic elasticity, gradient elasticity and flexoelectric micromorphic elasticity—by taking limits of the same constitutive parameters.
For the bent cantilever, the model predicts that the flexoelectric potential scales linearly with the flexoelectric coefficient and concentrates where the deformation gradient is concentrated, giving a computable signature of where polarization should appear in experiments.
Because micromorphic elasticity is built in, the formulation simultaneously captures size-dependent stiffening, so predictions for small structures include the mechanical size effect that accompanies flexoelectricity.
The converse flexoelectric effect can be added by introducing a micromorphic electric field and a similar scale-bridging energy, extending the same machinery to actuation problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the penalty parameter $p$ is finite in practice, the computed $\bar G$ is a regularised proxy for $G = \mathrm{Grad}\,\mathrm{Grad}\,\boldsymbol{\varphi}$; a systematic study of how the predicted polarization depends on $p$, especially near clamped boundaries, would quantify how much of the response is model artifact.
The linearity of Eq. (13) in $\bar G$ is a strong, testable prediction: if experiments on a single material show a nonlinear relation between strain gradient and polarization, the proposed energy must be replaced by a higher-order term, though the micromorphic machinery itself would survive.
The same penalty-micromorphic idea could be applied directly to other fourth-order coupled problems such as strain-gradient piezoelectricity or flexomagnetic coupling, since the mathematical obstruction—missing $C^1$ continuity—is identical.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Zack, here's my take. The core idea is to introduce a micromorphic deformation gradient field F and penalize its deviation from the macroscopic F, so the flexoelectric term can be written in terms of Grad F rather than second gradients of the displacement. That lets them use standard C0 elements in 3D, which is genuinely new relative to Yvonnet & Liu's C1 Argyris triangles, Abdollahi's meshless 2D, and Mao's linearized mixed formulation. The variational derivation is standard but clean, and they provide all the tangent blocks. The E-elasticity benchmark reproduces Vu et al., which is a good sanity check. The numerical examples show the expected size-dependent stiffening and flexoelectric potentials that scale linearly with the coefficient, as the energy in Eq. (13) implies.

The soft spot is the one they flag themselves in Sec. 4.3.2 and then don't fully address. Figure 10(c) shows that |F−F| does not shrink at the clamped end, and the text admits this persists regardless of p. Since |G| is also largest there, the flexoelectric potential is generated precisely where the micromorphic/macroscopic tie is least reliable. With finite p the scheme solves a regularized problem with an extra parameter, and the p→∞ limit is not demonstrated. There is no p- or h-convergence study. So the claim that the micromorphic approach accommodates the second-gradient flexoelectric theory is not yet supported by numerical evidence that the regularization is actually converged. The authors say the optimal functional setting is open, which is honest, but it means the central claim is conditional.

The other weakness is calibration: the coupled FM-elasticity model is not compared to experiment or to an independent reference solution, and the flexoelectric energy is a proposal. The paper itself calls experimental validation critical future work. For a methods paper that's acceptable, but it limits what one can infer from the computed potentials.

Still, the paper is worth engaging with. The formulation is coherent, the derivations are complete, and the approach will likely be reused. Before building on it, I would want a convergence test in p on a smooth problem and a clearer discussion of whether the clamped-end mismatch is a modeling artifact or a discretization limitation. It deserves a serious referee; with revisions addressing the penalty limit and a benchmark, it would be solid.

Referee Report

4 major / 4 minor

Summary. The paper proposes a micromorphic formulation of nonlinear flexoelectricity. An independent micro-deformation field F and its material gradient G are introduced as additional degrees of freedom, and a scale-bridging penalty term couples F to the macroscopic deformation gradient F. In this way the higher-order gradient content of flexoelectricity is said to be accommodated within standard C0-continuous finite elements. The governing equations are derived from a stationary energy (Dirichlet) principle, the monolithic finite element formulation and tangent blocks are given, and three-dimensional numerical examples (a strip with a hole and a cantilever beam) illustrate size-dependent stiffening and flexoelectric-induced electric potentials. The manuscript also presents a classification of reduced models (M-, E-, EM-, FG-, FM-Elasticity) and discusses the penalty limit p→∞ as a route to gradient elasticity.

Significance. If the numerical consistency and validation gaps were closed, this would be a useful contribution: it offers a practical C0 finite element route to fully nonlinear flexoelectricity in three dimensions, with explicit constitutive forms, a complete tangent structure, and an open-source implementation based on deal.II and automatic differentiation. The variational derivation and the model classification are valuable and clearly presented. The central claim, however, is only partially supported by the evidence in the manuscript: no convergence study is reported, the finite-penalty mismatch between F and F is admitted at the boundary point where |G| and the flexoelectric response peak, and the full FM-Elasticity model is not validated against an independent reference solution or experiment. The paper is therefore a promising starting point rather than a fully established numerical framework for flexoelectricity.

major comments (4)

[Sec. 4.3.2, Fig. 10(c)] The manuscript's own results show a persistent mismatch between the micro-deformation gradient field F and the macroscopic deformation gradient F at X=0, where |G| and the flexoelectric response are largest, and the text states that this mismatch persists 'irrespective as to the choice of the penalty term p'. Because the flexoelectric energy in Eq. (13) is linear in G, the computed electric potential in this region is generated by a field whose tie to the deformation gradient is explicitly unreliable. Since no p- or h-convergence study is presented, the numerical solutions in Section 4 are not established as approximations to the limiting gradient theory. I recommend a quantitative investigation: h-refinement at fixed p, p-refinement, and a comparison with an independent C1 reference solution on a simple geometry, together with an assessment of how the mismatch affects the quantities of interest such as the potential difference between points C and D.
[Sec. 5 (Discussion) and Sec. 4.2] The FM-Elasticity formulation is validated only against benchmark problems for E-Elasticity and M-Elasticity; no validation of the fully coupled flexoelectric response is provided against experiment, an analytical solution, or an independent numerical method. Section 5 explicitly defers experimental validation to future work. Since the central claim is that the model captures flexoelectric polarization and size-dependent effects, at least one quantitative benchmark is needed, for example a comparison with the linearized analytical solution for a bent beam or with the C1 finite element results of Yvonnet and Liu for a simplified geometry. Without such a test, the constitutive choices in Eq. (13) and the resulting potentials remain unverified model predictions.
[Sec. 2.4.3, Eq. (13) and Remark 6] The claim that the micromorphic scheme recovers the flexoelectric gradient theory as p→∞ is not established. The scale-bridging term (Eq. (11)) penalizes the L2 difference between the piecewise polynomial micro-deformation F_h and the continuous deformation gradient F_h = Grad phi_h; because F_h is generally discontinuous across element faces while F_h is continuous, the penalty limit cannot deliver pointwise equality. The persistent gap shown in Fig. 10(c) is consistent with this observation. The authors should either prove a Gamma-convergence or projection-type result for the discrete limit, or reformulate the claim to present FM-Elasticity with finite p as a regularized model in its own right, with p as a constitutive parameter whose influence is systematically studied.
[Secs. 3 and 4] No mesh-refinement or polynomial-order study is reported for the coupled three-field formulation, despite the non-standard choice of tri-quadratic approximations for phi and F with a tri-linear approximation for the electric potential. For a mixed-type nonlinear problem of this structure, stability and convergence of the monolithic Newton iteration cannot be taken for granted. A demonstration of h-convergence (for example, tracking the tip deflection and the potential difference on the cantilever problem) and a brief study of the sensitivity to the penalty parameter p are essential to support the numerical claims of the paper.

minor comments (4)

[Notation, Eq. (13)] The contraction notation in Eq. (13), involving a vector f^T E, a third-order tensor G, and a second-order tensor I, is not covered by the scalar-product definitions in the Notation section, which define only contractions of same-order tensors. The precise index contraction should be written out explicitly.
[References] Reference [46] (Vu, Steinmann and Possart, International Journal for Numerical Methods in Engineering, 70(6):685–704) appears to be a duplicate of reference [40]; the bibliography should be deduplicated and the citation in Sec. 4.2 should point to the intended entry.
[Fig. 10] In panels (c) and (d) of Fig. 10, the axes labelled 'X' and '|G|' are ambiguous because the same x-axis scale is used for different lines; please label the axes explicitly and clarify which quantity is plotted along the horizontal line A–B versus the vertical line C–D.
[Sec. 4.3.2] The sentence 'Several choices for p were investigated and all produced similar behaviour' is too vague to be useful; if these investigations are not reported in detail, the statement should be removed or replaced by a figure or table.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the micromorphic flexoelectric formulation is a self-contained model proposal whose load-bearing constitutive choices are explicitly presented as assumptions, not as derived predictions.

full rationale

The paper's derivation chain is variational and self-contained: it postulates a total potential energy in Eqs. (1)-(3), takes its stationary point in Eq. (4), obtains the Euler equations and boundary conditions in Section 2.3, and then discretizes those equations with C0 finite elements in Section 3. The load-bearing constitutive forms in Section 2.4 are introduced as proposals rather than as consequences of the target result. For example, Eq. (13) says 'We propose here that the flexoelectric contribution takes the form', and the text immediately notes that the additive decomposition and quadratic micromorphic/scale-bridging energies are assumptions, not requirements. The numerical examples are demonstrations of the proposed model, not fitted predictions disguised as independent checks. The statement that the potential scales linearly with the flexoelectric coefficient is a direct algebraic consequence of Eq. (13) and Eq. (A.1), and the paper presents it as such rather than as validation. The p→∞ relation in Remark 6, 'as the penalty-like parameter p→∞ in Eq. (11), G→G', is a formal model-limit statement within the proposed framework, and the classification diagram in Figure 2 is a parameter-restriction hierarchy, not an empirical prediction. The benchmarks used for the E-Elasticity and M-Elasticity submodels come in part from prior work by members of the same group, but those prior results are published constitutive/theoretical frameworks whose assumptions do not include the flexoelectric target, so this is normal self-citation rather than load-bearing circularity. The paper explicitly defers experimental validation of the FM-Elasticity model to future work, stating that 'the validation of the FM-Elasticity model against experiment is critical and will be considered in future work', which reinforces that no experimental prediction is being dressed up as a derivation. The reported boundary-layer mismatch in |F−F| in Section 4.3.2 is a numerical consistency and regularization concern about the finite-penalty approximation, not a circularity: the paper openly reports that the mismatch persists 'irrespective as to the choice of the penalty term p' and calls the optimal functional setting an open challenge. Overall, the central claim is a new constitutive and computational scheme with all assumptions stated, and no step reduces by construction to its own input.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central formulation rests on standard variational calculus plus several constitutive choices. The free parameters are material and regularization constants selected for the examples, not fitted to flexoelectric data. No new physical entities such as particles, forces, or dimensions are introduced.

free parameters (4)

p (penalty-like scale-bridging parameter) = 5000 µ (default); 50 µ in strip example
Controls how closely the micro-deformation gradient G follows the macroscopic deformation gradient. Chosen by hand for numerical examples, not calibrated.
l (intrinsic length scale) = varied; for example L/12, L/6, L/3, L and 0.25 to 2 in the beam
Sets the size-dependent stiffening and flexoelectric coupling. Selected manually to illustrate trends, not measured.
upsilon (flexoelectric coefficient) = varied 0, 0.25, 0.5, 0.75, 1 in the beam example
Scales the proposed flexoelectric energy. No experimental fit is attempted.
alpha, beta, gamma (electro-elastic coupling parameters) = alpha=0.2, beta=2, gamma=-2
Taken from earlier electroelasticity papers as inputs, not derived or fitted in this work.

assumptions (5)

standard math The total potential energy is stationary at equilibrium, the Dirichlet principle.
Invoked in Section 2.2 to derive the governing equations and boundary conditions from Eqs. (1)-(4).
domain assumption The energy density decomposes additively into elastic, flexoelectric, and electric parts, with separate macroscopic, micromorphic, and scale-bridging contributions.
Introduced in Eq. (2); the authors explicitly note the decomposition is an assumption motivated by classical electroelasticity.
ad hoc to paper The flexoelectric energy couples only the micro-deformation gradient G with the electric field in the linear form of Eq. (13).
This is a proposed constitutive choice in Section 2.4.3, not derived from microscale physics. Alternatives are listed in Remark 5.
domain assumption The free space surrounding the body is ignored.
Stated in Section 5; this neglects a contribution that can be significant for electro-active polymers.
domain assumption Isothermal conditions, homogeneous Neumann micromorphic tractions, no body forces, and no free charge are assumed in the numerical examples.
Assumed in Section 4 before the numerical examples are presented.

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Cite this review

Pith. "Pith review of Modelling the flexoelectric effect in solids: a micromorphic approach." pith.science (2026). https://pith.science/paper/LXIQQ6GN

@misc{pith2026190908695,
  author       = {Pith},
  title        = {Pith review of: Modelling the flexoelectric effect in solids: a micromorphic approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXIQQ6GN}},
  note         = {Machine review of arXiv:1909.08695}
}

read the original abstract

Flexoelectricity is characterised by the coupling of the gradient of the deformation and the electrical polarization in a dielectric material. A novel micromorphic approach is presented to accommodate the resulting higher-order gradient contributions arising in this highly-nonlinear and coupled problem within a classical finite element setting. The formulation accounts for all material and geometric nonlinearities, as well as the coupling between the mechanical, electrical and micromorphic fields. The highly-nonlinear system of governing equations are derived using the Dirichlet principle and solved using the finite element method. A series of numerical examples serve to elucidate the theory and to provide insight into this fascinating effect.

Figures

Figures reproduced from arXiv: 1909.08695 by the authors.

**Figure 2.** The relation between the various models of coupled electrical and mechanical elasticity within the micromor [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** The problem of a strip with a hole. Geometry and boundary conditions for (a) M-Elasticity and (b) E- and [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: The finite element mesh of the undeformed strip with a hole geometry, and the final deformed configuration [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: The relationship between the applied displacement [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: As in Sec. 4.1, the symmetry of the problem is exploited. The length scale is fixed as [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 8.** Figure 8: 16 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 8.** Figure 8: The geometry and boundary conditions for the micro-cantilever beam problem. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: The vertical deflection uy along the line A–B for the cantilever beam for various choices of the length scale `. The problem is M-Elasticity. A plot of the deformed shape of the beam for the choices ` ≡ 0 (Elasticity) and ` ≡ 2 is also shown. The deformed shape is colo…

**Figure 10.** Figure 10: The distribution of the potential ϕ over (a) the horizontal line A–B and (b) the vertical line C–D, for various choices of υ. The distribution of the norm of the micro-gradient |G| and the scale transition measure |F − F| along the line A–B is shown in (c). The distri…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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