Propagation of elastic waves in a flexomagnetic solid

Swarnava Ghosh

REVIEW 4 major objections 5 minor 7 references

Propagation of elastic waves in a flexomagnetic solid

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Flexomagnetism plus microstructure makes elastic waves dispersive and can let shear waves outrun compression waves; true wave freezing still does not occur.

desk verdict Closed-form dispersion relations for flexomagnetic waves, but the transverse channel has a real error and the small-exchange elimination is unquantified; correct the equations and this becomes publishable. read the letter →

arxiv 2602.20485 v1 pith:UTXOJ64P submitted 2026-02-24 cond-mat.mtrl-sci physics.app-ph

Swarnava Ghosh This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.app-ph MSC 74J0574A6074F15

keywords flexomagnetismelasticwavepropagationstrain-gradientelasticitymicrostructurelengthscaledispersionrelationattenuationgroupvelocityzero

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

The paper develops a continuum theory of elastic waves in a linear, isotropic flexomagnetic solid that also carries microstructure and nonlocal strain-gradient elasticity. It derives closed-form expressions for the frequency, phase velocity, and group velocity of longitudinal and transverse waves. Those formulas make the flexomagnetic coefficient and the two material length scales directly visible in wave behavior. If the theory is right, ordinary solids that couple strain gradients to magnetization should show dispersive waves, wavenumber-dependent attenuation, and a regime in which transverse waves travel faster than longitudinal waves, none of which classical linear elasticity permits. The same analysis concludes that true wave freezing, where a wave stops without spreading, cannot happen in this model.

What carries the argument

The central objects are the dispersion relations in Equations (32) and (47), obtained from a variational principle for an energy density that includes strain-gradient elasticity, micro-inertia, exchange coupling, and a Lifshitz-invariant flexomagnetic term. The decisive algebraic step is eliminating magnetization under the small-exchange, quasi-static assumption, giving M = −(k² f_m/(a_m+1))u for longitudinal waves and the analogous transverse relation; this turns flexomagnetism into an effective k²-dependent correction inside the elastic wave frequency. The phase-velocity ratio formula with its critical wavenumber then converts the magnetic coupling into a concrete prediction about which wa

What would settle it

Measure the acoustic dispersion of longitudinal and transverse waves in a thin flexomagnetic film, such as chromium oxide, with independently characterized microstructural length λ and nonlocal length l; if the longitudinal-to-transverse phase-velocity ratio never falls below unity for l below the predicted critical length, or if no evanescent attenuation branch appears where Equation (62) predicts, the central claim fails—and observing a true frozen wave packet would directly contradict the paper's wave-freezing conclusion.

Watch

Extended reading notes

Core claim

For an isotropic linear elastic flexomagnetic solid with micro-inertia length λ, nonlocal elastic length l, and flexomagnetic coefficient fm, the paper derives the longitudinal frequency as ω_l = k[(c(a_m+1)(k²l²+1) − μ0 k² f_m²)/(ρ(a_m+1)(k²λ²/3 + 1))]^{1/2}, with an analogous transverse expression. From these dispersion relations it concludes that flexomagnetism lowers wave speeds relative to the nonmagnetic case, introduces wavenumber dependence that classical elasticity lacks, allows transverse phase velocities to exceed longitudinal ones for certain length scales and wavenumbers, produces an evanescent attenuation branch, and can yield zero- or negative-group-velocity modes only when th

Load-bearing premise

The load-bearing assumption is that magnetization follows the elastic motion quasi-statically and that exchange stiffness is negligible, so M is eliminated algebraically before Equation (30) and Equation (46); if magnetization inertia or exchange coupling matters in real nanoscale flexomagnets, the dispersion relations in Equations (32) and (47) and all derived phenomena would change, and the paper itself notes in Section 4 that dynamic flexomagnetism is unexplored.

Editorial extensions

If this is right

Longitudinal and transverse waves in flexomagnetic solids become dispersive, with phase and group velocities depending on wavenumber, the flexomagnetic coefficient, the microstructural length, and the nonlocal elastic length.
For nonlocal lengths below a critical value, there is an intermediate wavenumber window in which transverse phase velocity exceeds longitudinal phase velocity, reversing the classical ordering.
One of the two wavenumber roots for a given real frequency is always imaginary, so the theory predicts evanescent waves and an attenuation band whose extent depends on the flexomagnetic and length-scale parameters.
Zero- and negative-group-velocity modes can appear when flexomagnetism is present and the nonlocal length is sufficiently small, but not for typical larger nonlocal lengths such as 10 angstroms.
Wave freezing, defined as a zero group velocity with a stationary inflection point in the dispersion curve, is not realized in this linear flexomagnetic model.

Reading between the lines

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

Because the flexomagnetic correction enters as a k² subtraction inside the dispersion relation, measuring the frequency cutoff or the attenuation band in a nanoscale flexomagnetic film could yield a direct estimate of the flexomagnetic coefficient without separate magnetic measurements.
The paper deliberately neglects magnetization dynamics and the author flags dynamic flexomagnetism as unexplored; at high wavenumbers, dynamic magnetization inertia could shift the predicted attenuation and group-velocity behavior, so those high-k predictions should be viewed as provisional.
The same derivation pattern should extend to anisotropic and coupled flexoelectric–flexomagnetic solids, where additional couplings may widen, narrow, or eliminate the transverse-faster-than-longitudinal window.
The zero- and negative-group-velocity conditions depend on nanometer-scale length parameters, suggesting that nanoscale patterning or curvature of flexomagnetic films could be used to engineer slow or backward elastic waves.

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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

Quick take: this is a direct analog of the flexoelectric/gradient-elasticity wave papers, with the flexomagnetic coefficient swapped in. That doesn't make it worthless—the explicit dispersion relations, attenuation ranges, and ZGV/NGV criteria for flexomagnetic solids are stated in closed form here for the first time, and the reduction to Papargyri-Beskou when f_m=0 is a good benchmark. But two technical issues need to be fixed before the results can be trusted.

First, the exchange term. The energy (1) is quadratic in ∇M, so the variational derivative should give a term linear in M_{,jj}. Instead, Eqs (23), (26), and (29) contain A(∂²M/∂x²)² (and Eq (29) has A k⁴ M²). That is dimensionally inconsistent with the stated energy, and if corrected to the linear form the Fourier-space relation becomes (a_m+1 + A k²)M + f k² u = 0, so the dispersion relations acquire A-dependent terms. The paper drops the exchange term by appealing to small exchange stiffness, but gives no material parameters or wavenumber bounds to justify that. At the wavenumbers shown in the figures (up to ~1 Å⁻¹) and with typical exchange lengths of a few nm, A k² can be of order (a_m+1)≈126, so the central results have an unstated domain of validity.

Second, the transverse equation (46) has a spurious +1. For transverse waves, ∇·M = 0 and the magnetic scalar potential drops out of the field equations, so the magnetic constitutive equation is a_m M - f^m_44 u_{,xx} = 0, not (a_m+1)M - f^m_44 u_{,xx}=0. This error propagates into Eq (47) and into the derived phase-velocity ratios, attenuation conditions, and ZGV/NGV criteria for transverse waves. That's a load-bearing mistake, since the claim that transverse phase velocities can exceed longitudinal ones is one of the headline results.

These are fixable. The longitudinal dispersion (32) is plausible under the stated assumptions, and the k→0 limits and the f_m=0 reduction check out. The paper is also honest about the unexplored dynamic flexomagnetism in Section 4. But as it stands, the transverse results are unreliable and the small-exchange argument is hand-wavy.

Who should read it: someone working in flexomagnetic continuum theories or gradient-elastic wave propagation, who wants a template to fix and extend. I would not cite it until the equations are corrected.

Recommendation: send to peer review. It's a legitimately relevant topic, the core idea is sound, and the issues are correctable. A good referee would catch the transverse +1 and ask for a quantitative exchange estimate.

Referee Report

4 major / 5 minor

Summary. The manuscript develops a continuum theory of elastic waves in an isotropic linear elastic flexomagnetic solid with microstructure and strain-gradient elasticity. It derives closed-form longitudinal and transverse dispersion relations, Eqs. (32) and (47), after eliminating the magnetization algebraically under a small-exchange-stiffness assumption, and then discusses dispersion, attenuation of evanescent waves, zero- and negative-group-velocity modes, and wave freezing. The formulas are shown to reduce to the classical elastic limit and to the Papargyri-Beskou gradient-elastic limit in special cases.

Significance. The topic is timely and the paper provides transparent analytical formulas with material constants taken from the literature rather than fitted, which is a strength. The limiting reductions to classical and Papargyri-Beskou results are useful checks. However, the derivation contains internal inconsistencies and algebraic errors that affect load-bearing parts of the central claims: the exchange term in the field equations is incompatible with the stated energy, the transverse magnetic equation uses the wrong coefficient, and the attenuation and negative-group-velocity ranges are derived incorrectly. These issues must be fixed before the paper can be considered reliable.

major comments (4)

[§2, Eqs. (23), (26), (29)] The exchange term is inconsistent with the energy (1). With g_{ijkl}=A δ_{ik}δ_{jl}, the variational derivative of (1/2)μ0 g_{ijkl} M_{i,j}M_{k,l} is linear, −A M_{i,jj}, whose Fourier-space form is A k² M. Equations (26) and (29) instead contain A(∂²M/∂x²)² and A k⁴M², which are dimensionally and structurally incompatible with the stated energy. The later 'small exchange stiffness' drop therefore rests on an ill-defined term. Moreover, no quantitative criterion is given: at k≈1 Å⁻¹ and an exchange length √A≈1 nm, A k²≈100, comparable to a_m+1≈126, so the algebraic elimination in Eq. (30) and the resulting Eqs. (32)/(47) are not obviously valid in the high-k regime where attenuation and negative-group-velocity effects are claimed.
[§3.2, Eq. (46)] The transverse magnetic field equation is written as (a_m+1)M + f^m_44 k²u = 0, but for transverse waves ∇·M=0, so the scalar-potential term ψ,i in Eq. (42) vanishes and the coefficient should be a_m, not a_m+1. The '+1' in the longitudinal case comes from eliminating ψ. Unless a non-constant ψ is explicitly retained, Eq. (46) and hence Eq. (47) are not derived from Eq. (42). The numerical effect is small for a_m=125, but the derivation should be corrected.
[§3.4, Eqs. (59)–(63)] The attenuation analysis is not algebraically correct. The discriminant in Eqs. (59)–(60) contains [̄c l² − μ0 ̄f_m²/(1+a_m)]², which has incompatible dimensions; the quadratic in k² has coefficient a′ = ̄c l² − μ0 ̄f_m²/(1+a_m), so the discriminant term is linear in a′, not quadratic. In addition, with k² = −q, the positivity condition gives 1/(l² − μ0 ̄f_m²/(̄c A)) < q < 3/λ² for a′>0, so the inequality in Eq. (63) is reversed. As written, the attenuation range and the ω→0 and ω→∞ limits in Section 3.4 are incorrect.
[§3.5, Eqs. (69)–(71)] The negative-group-velocity range ignores the reality condition for ω. For λ=0, propagating waves require cA − βk² > 0, i.e. k² < γ/(3β) when β>0. Thus V_g≤0 is obtained on the finite interval [√(γ/(6β)), √(γ/(3β))), not on (√(γ/(6β)), ∞) as stated in Eq. (71). The same omission affects Eq. (75) for λ>0. Negative group velocity may still exist, but the stated wavenumber range is wrong and must be corrected.

minor comments (5)

[Eqs. (37), (39), (52), (54)] In the k→∞ limits, the numerator should contain l² rather than l; e.g. Eq. (37) should read c(a_m+1)l² − μ0 f_m². The printed 'c(a_m+1)l' is dimensionally inconsistent with a velocity squared.
[§3.1] The reference 'Figure 3.1' should be 'Fig. 1'.
[Table 1] The units of f_m, f^m_12, and f^m_44 should be specified and checked against the energy (1); the values listed in A are not obviously compatible with the SI form of the flexomagnetic energy term.
[Eqs. (27)–(30)] The ansatz introduces M(t) and mentions the LLG equation, but the subsequent derivation simply eliminates M algebraically. The paper should state explicitly that a quasi-static approximation is being made and discuss its range of validity.
[after Eq. (63)] The sentence beginning 'In the absence of flexomagnetism...' ends with 'and .'; the missing expression should be supplied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dispersion relations are derived from a stated energy functional with literature-sourced constants and reduce to the known Papargyri-Beskou limit when flexomagnetism is absent.

full rationale

The derivation chain is self-contained as a model-based analysis. Starting from the internal energy (Eq. 1) with flexomagnetic, strain-gradient, and microstructural terms, the paper derives the linearized field equations (Eqs. 22-24), substitutes plane-wave ansatze, eliminates the magnetization in the stated small-exchange-stiffness limit (Eq. 30), and solves for the longitudinal and transverse frequencies (Eqs. 32, 47). The material parameters in Table 1 are taken from prior literature (Eliseev et al., Sidhardh and Ray, Wang et al., Makushko et al.), not fitted to the paper's own dispersion predictions, so the predictions are not statistically forced. The f_m=0 limit recovers exactly the Papargyri-Beskou et al. dispersion, an external benchmark, and the k-to-0 limit recovers the classical velocities sqrt(c/rho) and sqrt(c44/rho). The stipulated assumptions (small exchange stiffness; no magnetization dynamics; no piezomagnetism) are explicit modeling restrictions and are acknowledged limitations, including the closing remark that dynamic flexomagnetism has not been systematically investigated. They may affect the quantitative validity of the dispersion at high wavenumbers, but they do not make the derivation circular. The only self-citation (Ghosh 2026) appears in background literature on flexoelectricity and is not load-bearing for the wave-propagation result. Possible internal inconsistencies in the exchange term's Fourier form are correctness concerns, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; flexomagnetic coefficients are taken from prior literature. The central derivation depends on a small-exchange/quasi-static magnetization assumption and on two hand-swept length scales λ and l.

free parameters (2)

microstructural length λ = 0–40 Å (swept)
Chosen by hand in Table 1 and the figures; central dispersion, attenuation, and zero/negative group velocity conditions depend on it.
nonlocal elastic interaction length l = 0–100 Å (swept)
Chosen by hand in Table 1 and the figures; controls dispersion, critical wavenumber, and phase-velocity ratio.

assumptions (5)

domain assumption The internal energy Eq (1), with linear elasticity, strain-gradient elasticity, exchange, and Lifshitz flexomagnetic coupling, is the correct free energy for a flexomagnetic solid.
Basis of all field equations; taken from prior continuum theories rather than derived in this paper.
domain assumption The material is isotropic and all material tensors take the special forms in Eq (21).
Underlies the scalar dispersion relations; many candidate flexomagnetic materials are anisotropic, as the paper acknowledges in Section 4.
ad hoc to paper Magnetization is quasi-static and exchange stiffness is small, so the A-terms are dropped and M is eliminated algebraically.
Stated before Eq (30) and Eq (45) without quantitative justification; dynamic flexomagnetism is left to future work in Section 4.
domain assumption The magnetic field is curl-free and represented by a scalar potential ψ; applied magnetic field and piezomagnetism are zero.
Used in Eqs (12)-(14) and Eq (20); restricts the analysis to magnetostatic regimes.
standard math Plane-wave ansatz Eq (27)/(44) with subsequent linearization.
Standard Fourier-mode analysis for linear PDEs.

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Cite this review

Pith. "Pith review of Propagation of elastic waves in a flexomagnetic solid." pith.science (2026). https://pith.science/paper/UTXOJ64P

@misc{pith2026260220485,
  author       = {Pith},
  title        = {Pith review of: Propagation of elastic waves in a flexomagnetic solid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTXOJ64P}},
  note         = {Machine review of arXiv:2602.20485}
}

read the original abstract

Flexomagnetism is the coupling between magnetism and strain gradients and is a technologically relevant phenomenon. We present a theory of elastic wave propagation in a linear elastic flexomagnetic material with microstructure and strain gradient elastic interactions. The expressions of frequency, phase velocity, and group velocity of longitudinal and transverse waves are derived and are shown to depend on the flexomagnetic coefficient and microstructure. We also show that the effect of flexomagnetism and microstructure can lead to some interesting phenomena in wave propagation, which are not observed in classical linear elasticity theory of waves. Specifically, in contrast to classical linear elastic materials, where wave propagation is non-dispersive, flexomagnetic materials with microstructure can exhibit both normal and abnormal dispersion. It is also noteworthy that, in flexomagnetic materials with gradient elasticity, the phase velocities of transverse waves can exceed those of longitudinal waves, which is atypical in classical elasticity. Furthermore, waves can also attenuate for a certain range of wavenumbers that depend on the flexomagnetic coefficient and microstructural parameters. Finally, we explore the possibility of waves exhibiting zero group velocity modes, where waves are non-propagating but have strong local energy confinement, negative group velocity modes, where the wave packet moves in the opposite direction to that of wave propagation, and the phenomenon of wave freezing, where a propagating wave stops in space without diffusing or spreading.

Figures

Figures reproduced from arXiv: 2602.20485 by the authors.

**Figure 1.** A semi-infinite elastic solid with microstructure and non-local elastic interaction. The lengthscale of the microstructure is λ, and the non-local elastic interactions are restricted to a spherical region of radius l. The solid exhibits flexomagnetism. The wave is assumed to propagate in the x-direction. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** The effect of the length parameters λ and l on the wave frequency is shown for different values of wave number k. (a) shows the longitudinal case, and (b) shows the transverse case. The wave frequencies are normalized by the classical phase velocities of the longitudinal and transverse cases, respectively. velocity of transverse waves depends on f m 44 and fm > fm 44 the influence of flexomagnetism on the phase velo… view at source ↗

**Figure 3.** The effect of length parameters λ and l on the phase velocities of (a)-(b) longitudinal and (c)-(d) transverse waves is shown for different values of wave number k. The phase velocities are normalized by the classical phase velocities. (b) and (d) shows an inset of (a) and (c), respectively, from wave numbers 0 to 1 Angstrom−1 , denoted by the black box in (a) and (c). between abnormal or normal dispersion to non-di… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The effect of magnetism on the phase velocities of (a) longitudinal and (b) transverse waves is shown for different values of wave number k. The phase velocities are normalized by the classical phase velocities. wavenumber k, and is given by V l p V t p = c(am + 1)(k…

**Figure 5.** Figure 5: The effect of the length parameters λ and l on the group velocities of (a)-(b) longitudinal and (c)-(d) transverse waves is shown for different values of wave number k. The group velocities are normalized by the classical phase velocities. (b) and (d) shows an inset of…

**Figure 6.** Figure 6: The effect of magnetism on the group velocities of (a) longitudinal and (b) transverse waves is shown for different values of wave number k. The phase velocities are normalized by the classical phase velocities. and k 2 2 (ω) = 2ω 2ρ c¯− λ2 3 ω2ρ − r c¯− λ2 3 ω2ρ 2 + …

**Figure 7.** Figure 7: The effect of the length parameters λ and l on the ratio of the group velocity and phase velocity of (a)-(b) longitudinal and (c)-(d) transverse waves are shown for different values of wave number k. (b) and (d) shows an inset of (a) and (c), respectively, from wave nu…

**Figure 8.** Figure 8: The effect of magnetism on the ratio of group velocity to phase velocity of (a) longitudinal and (b) transverse waves is shown for different values of wave number k. The phase velocities are normalized by the classical phase velocities. (a) (b) [PITH_FULL_IMAGE:figure…

**Figure 9.** Figure 9: The effect of the nonlocal elastic interaction length l on the ratio of the phase velocities of longitudinal and transverse waves is shown in (a). In (b), the phase diagram shows the region in k-l space. The region depicted by the color gray is where phase velocities o…

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Works this paper leans on

7 extracted references · 2 canonical work pages

[2004]

Generalization of stoney’s equation for flexoelectric thin films on elastic substrates

Swarnava Ghosh. Generalization of stoney’s equation for flexoelectric thin films on elastic substrates. arXiv preprint arXiv:2601.09160 ,

arXiv
[2006]

Anomalous tensorial properties of anisotropic 2d materials

Elizabeth J Dresselhaus, Sanjay Govindjee, and Kranthi K Mandadapu. Anomalous tensorial properties of anisotropic 2d materials. arXiv preprint arXiv:2508.20055 ,

arXiv
[2011]

URL https://link.aps.org/doi/10.1103/PhysRevB.84.174112

doi: 10.1103/PhysRevB.84.174112. URL https://link.aps.org/doi/10.1103/PhysRevB.84.174112. Eugene A Eliseev, Anna N Morozovska, Maya D Glinchuk, and R Blinc. Spontaneous flexoelectric/flexomagnetic effect in nanoferroics. Physical Review B—Condensed Matter and Materials Physics , 79(16):165433,

work page doi:10.1103/physrevb.84.174112
[2017]

URL https://link.aps.org/doi/10.1103/PhysRevB.96.064402

doi: 10.1103/PhysRevB.96.064402. URL https://link.aps.org/doi/10.1103/PhysRevB.96.064402. Yechao Ling, Xiao Yu, Shijun Yuan, Anpeng He, Zhida Han, Jun Du, Qi Fan, Shicheng Yan, and Qingyu Xu. Flexomagnetic effect enhanced ferromagnetism and magnetoelectrochemistry in freestanding high-entropy alloy films. ACS nano , 17(17):17299–17307,

work page doi:10.1103/physrevb.96.064402
[2018]

Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations

Adam Sky, David Codony, Stephan Rudykh, Andreas Zilian, Stéphane Bordas, and Patrizio Neff. Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations. arXiv preprint arXiv:2511.22756,

arXiv
[2019]

Wave-freezing and other phenomena in temporal metasurfaces driven by nonlocal inter- actions

Kshiteej J Deshmukh. Wave-freezing and other phenomena in temporal metasurfaces driven by nonlocal inter- actions. arXiv preprint arXiv:2404.09060 ,

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Acoustic band structure of periodic elastic composites

Manvir S Kushwaha, Peter Halevi, Leonard Dobrzynski, and Bahram Djafari-Rouhani. Acoustic band structure of periodic elastic composites. Physical review letters , 71(13):2022,

2022

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [2004]

Generalization of stoney’s equation for flexoelectric thin films on elastic substrates

Swarnava Ghosh. Generalization of stoney’s equation for flexoelectric thin films on elastic substrates. arXiv preprint arXiv:2601.09160 ,

arXiv

[2] [2006]

Anomalous tensorial properties of anisotropic 2d materials

Elizabeth J Dresselhaus, Sanjay Govindjee, and Kranthi K Mandadapu. Anomalous tensorial properties of anisotropic 2d materials. arXiv preprint arXiv:2508.20055 ,

arXiv

[3] [2011]

URL https://link.aps.org/doi/10.1103/PhysRevB.84.174112

doi: 10.1103/PhysRevB.84.174112. URL https://link.aps.org/doi/10.1103/PhysRevB.84.174112. Eugene A Eliseev, Anna N Morozovska, Maya D Glinchuk, and R Blinc. Spontaneous flexoelectric/flexomagnetic effect in nanoferroics. Physical Review B—Condensed Matter and Materials Physics , 79(16):165433,

work page doi:10.1103/physrevb.84.174112

[4] [2017]

URL https://link.aps.org/doi/10.1103/PhysRevB.96.064402

doi: 10.1103/PhysRevB.96.064402. URL https://link.aps.org/doi/10.1103/PhysRevB.96.064402. Yechao Ling, Xiao Yu, Shijun Yuan, Anpeng He, Zhida Han, Jun Du, Qi Fan, Shicheng Yan, and Qingyu Xu. Flexomagnetic effect enhanced ferromagnetism and magnetoelectrochemistry in freestanding high-entropy alloy films. ACS nano , 17(17):17299–17307,

work page doi:10.1103/physrevb.96.064402

[5] [2018]

Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations

Adam Sky, David Codony, Stephan Rudykh, Andreas Zilian, Stéphane Bordas, and Patrizio Neff. Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations. arXiv preprint arXiv:2511.22756,

arXiv

[6] [2019]

Wave-freezing and other phenomena in temporal metasurfaces driven by nonlocal inter- actions

Kshiteej J Deshmukh. Wave-freezing and other phenomena in temporal metasurfaces driven by nonlocal inter- actions. arXiv preprint arXiv:2404.09060 ,

arXiv

[7] [2025]

Acoustic band structure of periodic elastic composites

Manvir S Kushwaha, Peter Halevi, Leonard Dobrzynski, and Bahram Djafari-Rouhani. Acoustic band structure of periodic elastic composites. Physical review letters , 71(13):2022,

2022