The isotropic relaxed micromorphic model in polar coordinates and its application to an elastostatic axisymmetric extension problem

Esmaeal Ghavanloo; Patrizio Neff

REVIEW 3 major objections 4 minor 42 references

The isotropic relaxed micromorphic model in polar coordinates and its application to an elastostatic axisymmetric extension problem

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives the relaxed micromorphic model in polar coordinates and solves an axisymmetric extension problem exactly, obtaining closed-form displacement and microdistortion fields.

desk verdict A useful polar-coordinate formulation and closed-form solution for the planar relaxed micromorphic model, but the advertised 3D long-cylinder benchmark is not supported because the plane-strain reduction ignores out-of-plane microdistortion. read the letter →

arxiv 2411.18213 v2 pith:7536W4OO submitted 2024-11-27 math.AP

Esmaeal Ghavanloo , Patrizio Neff This is my paper · ORCID

classification math.AP MSC 74A3574G0533C10

keywords relaxedmicromorphicmodelpolarcoordinatesaxisymmetricextensionclosed-formsolutionmodifiedBesselfunctionsconsistentcouplingboundaryconditionsize-dependentelasticitygeneralizedcontinua

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 fills a gap: until now the relaxed micromorphic model had no general formulation in orthogonal curvilinear coordinates. The authors write the isotropic model's governing equations in polar coordinates and solve an elastostatic axisymmetric extension problem, a long circular cylinder under uniform radial boundary displacement. The solution is closed form, built from modified Bessel functions, and uses the consistent coupling boundary condition $P\cdot\tau = Du\cdot\tau$ on the boundary. They show that the classical linear elasticity solution reappears as the characteristic length $L_c$ tends to zero or infinity, and use the formulas to study how material parameters and $L_c$ shift the displacement profile. If correct, the result supplies an analytical benchmark for checking and calibrating finite element implementations of the relaxed micromorphic model.

What carries the argument

The machinery is the polar-coordinate representation of the relaxed micromorphic model's differential operators, together with the consistent coupling boundary condition. The paper writes the gradient, divergence, and curl of the relevant tensor fields in $(r,\theta)$ coordinates, producing the governing equations (12)--(20). For the axisymmetric problem it introduces the combinations $X = du_r/dr - P_{rr}$, $Y = u_r/r - P_{\theta\theta}$, and $Z = P_{\theta\theta}+P_{rr}$, reducing the system to a single inhomogeneous Bessel-type equation $d^2Z/dr^2 + (1/r)\,dZ/dr - aZ + b = 0$; the solution enters the displacement through $du_r/dr + u_r/r = C_1 A + D_1 B I_0(\sqrt{a}r)$. The boundary condition $P\cdot\tau = Du\cdot\tau$ fixes the remaining constants and eliminates the off-diagonal microdistortion.

What would settle it

Measure the interior radial displacement of a long cylinder under uniform radial boundary displacement, with the model's parameters first calibrated independently from bending or wave-propagation tests; formula (68) predicts a specific profile controlled by the modified Bessel functions and by $R/L_c$, so a systematic mismatch across radii would falsify the model or the boundary condition. A cheaper check is to solve the full two-field variational problem numerically without the $u_\theta=0$ ansatz and compare whether the computed minimizer indeed has vanishing angular displacement and the closed-form field.

Watch

Extended reading notes

Core claim

The central claim is that the isotropic relaxed micromorphic model can be formulated explicitly in polar coordinates and that, for the axisymmetric extension of a long cylinder, its equilibrium equations reduce to a linear system of ODEs whose solution is closed form. Under the axisymmetric ansatz ($u_\theta=0$, all fields depending only on $r$), the displacement $u_r(r)$ and the diagonal microdistortion components $P_{rr}$, $P_{\theta\theta}$ are expressed through $I_0$ and $I_1$ modified Bessel functions, while the off-diagonal components $P_{r\theta}$ and $P_{\theta r}$ vanish under the consistent coupling boundary condition. In the limits $L_c\to 0$ and $L_c\to\infty$, the solution degenerates to the classical linear elastic field $u_r = U_0 r/R$, with explicit expressions for $P_{\theta\theta}$ and $P_{rr}$. The paper also shows that the Cosserat couple modulus $\mu_c$ drops out of this axisymmetric problem, so the result isolates the size-dependent effects carried by the characteristic length and the micro-moduli.

Load-bearing premise

The derivation assumes the consistent coupling boundary condition $P\cdot\tau = Du\cdot\tau$ on the boundary and the axisymmetric plane-strain ansatz $u_\theta=0$ with all fields depending only on $r$; if the physical boundary condition on $P$ differs, or the true minimizer develops angular or off-axis structure, the closed-form solution will not describe that setting.

Editorial extensions

If this is right

If the solution is correct, it gives a benchmark for finite element codes: with the listed parameter sets, an implementation should reproduce the closed-form displacement and microdistortion profiles.
The limits $L_c\to 0$ and $L_c\to\infty$ both recover classical linear elasticity, so code validation can target those endpoints as separate checks.
The Cosserat couple modulus does not affect the axisymmetric extension solution, which simplifies parameter identification for this geometry.
The explicit dependence on $\lambda_m$ and $\mu_m$ shows how micro-moduli bend the displacement profile relative to the classical field, with the model predicting smaller or larger displacements depending on the parameter ratio $\beta_1$.
The closed-form expression makes the role of the characteristic length $L_c$ explicit, so it can be used to study size effects without solving the full boundary-value problem numerically.

Reading between the lines

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

A natural extension is to repeat the derivation in cylindrical or spherical coordinates, where the same operator identities are available, giving benchmarks for geometries closer to actual metamaterial specimens.
The solution implicitly predicts a size-dependent effective stiffness for the cylinder: for a fixed boundary displacement, the interior displacement profile changes with $R/L_c$, so measuring interior strains on cylinders of different radii could test the model without requiring full-field measurements.
Replacing the consistent coupling boundary condition by $P\cdot\tau = 0$ would produce a different solution, so the gap between the two closed forms is a quantitative measure of how much the boundary condition on $P$ matters.
The closed form assumes $u_\theta=0$ and radial-only dependence; solving the full two-field variational problem numerically without that ansatz would certify whether the true minimizer is indeed axisymmetric.

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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

This paper does two things: it derives the 2D relaxed micromorphic equations in polar coordinates and then solves an axisymmetric extension problem in closed form using modified Bessel functions. The derivation is long but plausible, and the final formulas are new as far as I can tell. The classical-elasticity limit cases come out right, which is a good internal check. If you work on the planar model, this gives you a concrete benchmark for finite element verification. That part is solid.

The main problem is the jump from the planar model to the long-cylinder problem. Section 3 is explicitly about the two-dimensional relaxed micromorphic model; P is a 2×2 tensor and u has only radial and angular components. Then Section 4 says that for a long cylinder, plane-strain conditions are warranted and drops all out-of-plane components of P, including Pzz. That does not follow. In the full 3D energy, Pzz enters through tr(Du−P) and trP, and a constant Pzz costs no CurlP energy. For fixed in-plane data, stationarity with respect to a constant Pzz gives a nonzero value for generic parameters, including the ones in Table 1. So the true 3D plane-strain minimizer has out-of-plane microdistortion, and Eqs. (68)–(70) are not the solution of the 3D long-cylinder problem. They may well be the solution of the explicitly 2D planar problem, but then the benchmark claim needs to be restated. This is a load-bearing gap, not a cosmetic one.

The novelty claim in Section 1 is also a bit stronger than the evidence. The paper says the general curvilinear-coordinate formulation is absent from the literature, but Refs. [30] and [31] already solve cylindrical torsion and bending problems in cylindrical coordinates. Those are special problems, not the full general formulation, so the gap is narrower than claimed. I would soften the wording.

Minor issues: there is no independent verification of the closed form against finite elements or any other numerical method; the “numerical results” are just evaluations of the derived expressions. Some algebraic steps are skipped, especially in the reduction from (29)–(34) to the Bessel equation. That is not fatal for a paper like this, but a supplementary page of key manipulations would help referees trust the algebra.

Who is this for? Researchers using the relaxed micromorphic model who want an analytical benchmark for planar polar problems. It deserves a serious referee, not a desk rejection. The derivation is substantial and the flaw is fixable: either add a rigorous reduction argument showing when Pzz vanishes, or explicitly frame the solution as a planar 2D benchmark. I would send it to review with the expectation of major revision.

Referee Report

3 major / 4 minor

Summary. The paper derives the governing equations of the isotropic relaxed micromorphic model in two-dimensional polar coordinates and applies them to an axisymmetric extension problem for a long circular cylinder under uniform radial displacement. The authors assume plane strain, set uθ=0 and all fields independent of θ and z, and restrict the microdistortion to its in-plane components. Using the consistent coupling boundary conditions, they reduce the problem to linear ODEs and obtain closed-form expressions involving modified Bessel functions for the displacement and in-plane microdistortion components. They then study the limits Lc→0 and Lc→∞, which recover classical elasticity, and present numerical results for three parameter sets from the literature.

Significance. If the polar-coordinate equations and the closed-form solution are correct, the paper provides a useful analytical benchmark for validating finite element implementations of the planar relaxed micromorphic model. The derivation is self-contained, does not fit parameters to data, and the limit-case recovery of classical elasticity is a valuable internal check. However, the claimed application to a three-dimensional long cylinder is not justified by the presented reduction: the independent microdistortion field has out-of-plane components that are silently discarded. The significance of the paper as a 3D benchmark therefore depends on an additional reduction argument or on an explicit reframing as a planar 2D model.

major comments (3)

[§4.1, Eqs. (29)-(34) and Eq. (9)] The reduction from the three-dimensional model (1)-(4) to the plane-strain problem drops all out-of-plane microdistortion components without justification. For the axisymmetric ansatz u=(u_r(r),0,0) with ∂_z=0, the energy still depends on c=P_zz through the terms µe|sym(Du-P)|², λe tr²(Du-P), µm|sym P|² and λm tr²(P). The stationarity condition with respect to a constant mode c is λe(s-p-c) = [2(µe+µm)+λm]c + λm p, where s=u_r'+u_r/r and p=P_rr+P_θθ, so c = [λe(s-p)-λm p]/[λe+2(µe+µm)+λm]. For the parameters in Table 1 this is generically nonzero, and a constant P_zz produces no Curl P penalty. Hence the true 3D long-cylinder minimizer is not the solution of Eqs. (29)-(34) unless the constraint P_zz=0 is imposed and justified. The statement that plane-strain conditions are warranted concerns only the displacement field, not the independent microdistortion field, and the use of the 2D bulk moduli in Eq. (37) effectively adopts the planar model rather than proving a 3D reduction.
[§3.2, Eqs. (23)-(28)] For a cylindrical lateral surface the tangent space is spanned by e_θ and e_z, but the consistent coupling boundary conditions are written only for τ=e_θ. Applying the general condition (7) with τ=e_z to the axisymmetric displacement gives P·e_z = Du·e_z = 0 at r=R, i.e., P_rz(R)=P_θz(R)=P_zz(R)=0. These additional conditions do not appear in the 2D formulation and cannot be recovered from the e_θ condition used to obtain Eq. (36). If the intended setting is the full 3D model, this is a missing boundary condition; if the intended setting is the planar 2D model, the paper should state this explicitly and should not present the solution as the long-cylinder limit of the 3D model.
[§1 and §3] The introduction claims that the general formulation of the relaxed micromorphic model in orthogonal curvilinear coordinates is absent from the literature, but Section 3 actually derives only the two-dimensional polar-coordinate formulation: Eqs. (8)-(9) restrict P to the r-θ plane and the Curl operator in Eq. (14) is the planar 2D curl. The title and abstract should be adjusted to state that the contribution is a planar polar-coordinate formulation, or the full 3D cylindrical-coordinate formulation should be derived. This is not merely a wording issue because the axisymmetric extension problem is framed for a 3D long cylinder rather than for a planar domain.

minor comments (4)

[§1, last paragraph] There is a typo: 'relaxed meromorphic model' should read 'relaxed micromorphic model'.
[§4.1, Eq. (63)] The text says 'subtracting Eqs. (34) from (33)', but Eqs. (33) and (34) are the Pθr and Pθθ equations; the algebra appears to subtract different equations. Please check the equation numbering and clarify the manipulation.
[§4.1, Eqs. (29)-(68)] Several key algebraic reductions (e.g., from Eqs. (31)-(34) to Eqs. (40)-(47), and from Eqs. (54)-(60) to (61)-(62)) are described only as 'after simplification'. Given that the closed-form solution is the main result, a detailed derivation or a symbolic-verification appendix would substantially improve reproducibility.
[Figure 1 caption] The phrase 'Due to the length of the cylinder, plane-strain conditions are warranted' presents an assumption as a consequence; this should be flagged as an additional modeling assumption, especially because of the microdistortion field's independence.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the polar-coordinate equations and the closed-form Bessel solution are derived within the paper from the stated model, and the imported prior-work items are modeling inputs, not outputs of the derivation.

full rationale

The paper's claimed derivation chain is self-contained. Equations (12)-(20) are obtained by substituting textbook differential operators and the constitutive laws (3)-(4) into the equilibrium equations. The axisymmetric reduction (29)-(34) is an explicit ansatz (u_theta=0, r-only dependence), and the closed forms (53), (58), (68)-(70) follow by algebraic manipulation of the resulting ODE system, with constants fixed by the Dirichlet condition (35) and the consistent-coupling conditions (36). No parameter is fitted to the solution, and no output quantity is used to define an input. The consistent-coupling condition P*tau = Du*tau (Eq. 7) is imported from Ref. [7] (same research group), and the plane-strain claim in Sec. 4.1 is an unproven reduction of the 3D model; these are external modeling assumptions or correctness risks, not circular reductions. The numerical parameter sets in Table 1 come from earlier identification papers [35,36], but they are not fitted here, and the formulas are evaluated predictively. Self-citations are numerous, but the load-bearing algebra and the Bessel solution do not reduce to those citations. The plane-strain truncation of P components is a physical assumption that could be challenged, but that is a modeling-validity concern rather than a circularity concern.

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

No new fitted parameters or entities are introduced. The paper's central claim rests on the validity of the underlying relaxed micromorphic model, the homogenization relations, and the consistent coupling boundary condition, all inherited from prior work, plus the axisymmetric ansatz that reduces the problem to one dimension.

assumptions (5)

domain assumption The relaxed micromorphic energy functional and constitutive relations (Eqs. 1-4) are taken from prior work as the starting point.
The entire derivation assumes this model; the paper does not re-derive it.
domain assumption The homogenization relations between macro and micro moduli (Eq. 38) are used to simplify the solution.
These relations come from previous papers by the authors (Refs. [4], [23], [24]) and are applied without proof here.
domain assumption The consistent coupling boundary condition P·τ = Du·τ (Eq. 7) is the correct higher-order boundary condition.
This is a modeling proposal from Ref. [7] by the same group, not a universally accepted law.
standard math The solution of the ODEs can be expressed with modified Bessel functions I0, I1, K0, K1.
Standard theory of linear ODEs with Bessel equations.
domain assumption The axisymmetric ansatz (uθ=0, all fields depend only on r) is valid.
Used without proof in §4.1 to reduce the PDEs to ODEs.

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Cite this review

Pith. "Pith review of The isotropic relaxed micromorphic model in polar coordinates and its application to an elastostatic axisymmetric extension problem." pith.science (2026). https://pith.science/paper/7536W4OO

@misc{pith2026241118213,
  author       = {Pith},
  title        = {Pith review of: The isotropic relaxed micromorphic model in polar coordinates and its application to an elastostatic axisymmetric extension problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7536W4OO}},
  note         = {Machine review of arXiv:2411.18213}
}

read the original abstract

In this paper, we consider the isotropic relaxed micromorphic model in polar coordinates and use this representation to solve explicitly an elastostatic axisymmetric extension problem involving a linear system of ordinary differential equations. To obtain an analytical solution, modified Bessel functions are utilized and closed-form solutions for the displacement and microdistortion are obtained. We show how certain limit cases (classical linear elasticity), which are naturally included in the relaxed micromorphic model, can be efficiently achieved. Furthermore, numerical results are calculated and the effects of various parameters are examined. The results can be used to calibrate and check corresponding finite element codes.

Figures

Figures reproduced from arXiv: 2411.18213 by the authors.

**Figure 2.** Profile of non-dimensional radial displacement for classical elasticity and the relaxed micromorphic model [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Comparison between the relaxed micromorphic and the classical models for [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Comparison between the relaxed micromorphic and the classical models for [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Variations of parameter δ as a function of radial position for β1 = {1, 2, 3, 4, 5} [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Variations of parameter δ as a function of radial position for β2 = {1.5, 2.5, 3.5, 4.5, 5.5}. Furthermore, we showed that the classical linear elasticity model can be obtained as limit-cases of the relaxed micromorphic model, highlighting its versatility and robustnes…

Discussion (0). Continue with ORCID to comment.

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A micromechanics-based micromorphic model for granular materials and prediction on dispersion behaviors

C. Xiu, X. Chu, J. Wang, W. Wu, and Q. Duan. “A micromechanics-based micromorphic model for granular materials and prediction on dispersion behaviors”. Granular Matter 22 (2020), 1–22

2020
[42]

A non-classical Bernoulli-Euler beam model based on a simplified micro- morphic elasticity theory

G. Zhang, X.-L. Gao, C. Zheng, and C. Mi. “A non-classical Bernoulli-Euler beam model based on a simplified micro- morphic elasticity theory”. Mechanics of Materials 161 (2021), 103967. 13

2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

G. Rizzi, G. H¨ utter, H. Khan, I.-D. Ghiba, A. Madeo, and P. Neff. “Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)”. Mathematics and Mechanics of Solids 27.3 (2022), 507–553

work page 2022

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Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua

G. Rizzi, G. H¨ utter, A. Madeo, and P. Neff. “Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua”.Continuum Mechanics and Thermodynamics 33 (2021), 1505– 1539

work page 2021

[3] [1]

Hierarchy of generalized continua issued from micromorphic medium constructed by homogenization

S. Alavi, J. Ganghoffer, H. Reda, and M. Sadighi. “Hierarchy of generalized continua issued from micromorphic medium constructed by homogenization”. Continuum Mechanics and Thermodynamics 35.6 (2023), 2163–2192

work page 2023

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Exploring wave propagation in heterogeneous metastructures using the relaxed micromorphic model

R. Alberdi, J. Robbins, T. Walsh, and R. Dingreville. “Exploring wave propagation in heterogeneous metastructures using the relaxed micromorphic model”. Journal of the Mechanics and Physics of Solids 155 (2021), 104540

work page 2021

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Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results

H. Askes and E. C. Aifantis. “Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results”.International Journal of Solids and Structures 48.13 (2011), 1962–1990

work page 2011

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Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics

G. Barbagallo, A. Madeo, M. V. d’Agostino, R. Abreu, I.-D. Ghiba, and P. Neff. “Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics”. International Journal of Solids and Structures 120 (2017), 7–30. 11

work page 2017

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Effects of micro-structure on the stress concentration at a spherical cavity

J. L. Bleustein. “Effects of micro-structure on the stress concentration at a spherical cavity”. International Journal of Solids and Structures 2.1 (1966), 83–104

work page 1966

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Additive manufacturing and size-dependent mechanical properties of three-dimensional microarchitected, high-temperature ceramic metamaterials

H. Cui, R. Hensleigh, H. Chen, and X. Zheng. “Additive manufacturing and size-dependent mechanical properties of three-dimensional microarchitected, high-temperature ceramic metamaterials”. Journal of Materials Research 33.3 (2018), 360–371

work page 2018