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Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization

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Pith's one-line read The short-range elastic parameters of the relaxed micromorphic model can be fixed by first-order homogenization alone, through a Löwner-supremum choice of the micro-scale stiffness tensor.

desk verdict A genuinely new identification route for the relaxed micromorphic model, but the advertised 'rigorous determination' is really a bounding inequality plus a convenient choice. read the letter →

arxiv 1909.13624 v1 pith:ZCLUGOAP submitted 2019-08-18 physics.app-ph

Patrizio Neff (1) , Bernhard Eidel (2) , Marco Valerio d'Agostino (3) , Angela Madeo (3) ((1) Universität Duisburg-Essen , (2) Universität Siegen , (3) INSA-Lyon) This is my paper · ORCID
classification physics.app-ph MSC 74A3074A3574A6074B0574M2574Q15
keywords relaxedmicromorphicmodelparameteridentificationLöwnermatrixsupremumtensorharmonicmeanapparentstiffnessperiodichomogenizationtetragonalsymmetrymetamaterial
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 tries to show that the scale-independent short-range elastic parameters of the relaxed micromorphic model can be determined for a periodic microstructure using only classical first-order homogenization computations, with no fitting to size-dependent experiments. It argues that the micro-scale stiffness tensor $C_\mathrm{micro}$ should be the smallest tensor that dominates, in energy, the apparent stiffness of every admissible tetragonal unit cell under affine Dirichlet boundary conditions, namely the Löwner matrix supremum. With that tensor in hand, the meso-scale tensor $C_e$ follows from the exact relation $C_\mathrm{macro} = C_\mathrm{micro}(C_\mathrm{micro}+C_e)^{-1}C_e$, where $C_\mathrm{macro}$ is the usual periodic homogenization result. If correct, the method turns the parameter identification of generalized continua into a routine unit-cell calculation and explains why the model keeps bounded stiffness at small sample sizes.

What carries the argument

The carrying mechanism is the relaxed micromorphic energy $$W(\nabla u,P,\mathrm{Curl}\,P)=\tfrac12\langle C_e\,\mathrm{sym}(\nabla u-P),\mathrm{sym}(\nabla u-P)\rangle+\tfrac12\langle C_\mathrm{micro}\,\mathrm{sym}P,\mathrm{sym}P\rangle+\tfrac12\langle C_c\,\mathrm{skew}(\nabla u-P),\mathrm{skew}(\nabla u-P)\rangle+\tfrac{\mu $L_c^{2}$}{2}\|\mathrm{Curl}\,P\|^2,$$ whose use of $\mathrm{Curl}\,P$ rather than $\nabla P$ as the curvature measure makes the infinite-sample limit reduce to linear elasticity with stiffness $C_\mathrm{macro}=C_\mathrm{micro}(C_\mathrm{micro}+C_e)^{-1}C_e$ and the single-cell limit reduce to linear elasticity with stiffness $C_\mathrm{micro}$. This split lets the paper identify $C_\mathrm{micro}$ as the Löwner matrix supremum of apparent unit-cell stiffnesses $C^V_{\mathrm{KUBC}}$ under affine Dirichlet data, restricted to unit cells whose symmetry obeys the extended symmetry principle; the standard energy-equivalence lemma supplies the apparent stiffness, and $C_\mathrm{macro}$ comes from periodic boundary conditions. The four-cell numerical study then picks the componentwise maxima of the Lamé-type coefficients, giving $\lambda_\mathrm{micro}=5.270$ GPa, $\mu_\mathrm{micro}=8.927$ GPa, and $\mu^*_\mathrm{micro}=8.332$ GPa.

What would settle it

Compute the apparent stiffness of a tetragonal-symmetric unit cell not among the four, such as a larger square or a differently offset rotated square, under affine Dirichlet boundary conditions; if for any strain $E$ its energy exceeds $\langle C_\mathrm{micro}E,E\rangle$, the reported $C_\mathrm{micro}$ is not the true supremum. Alternatively, search for a strain $E$ with $\langle(C_\mathrm{micro}-C_\mathrm{macro})E,E\rangle=0$; such a zero eigenvalue would make the formula for $C_e$ singular.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for a given periodic microstructure with tetragonal symmetry, the micro-scale tensor $C_\mathrm{micro}$ in the relaxed micromorphic model satisfies $\langle C_\mathrm{micro} E,E\rangle \ge \langle C^V_{\mathrm{KUBC}} E,E\rangle$ for every symmetric strain $E$ and every admissible tetragonal unit cell $V$, where $C^V_{\mathrm{KUBC}}$ is the apparent stiffness computed under affine Dirichlet (kinematically uniform) boundary conditions; the least such tensor, the Löwner matrix supremum $C^0_\mathrm{micro}$, is the natural identification of $C_\mathrm{micro}$. Together with the macro-scale tensor $C_\mathrm{macro}$ obtained from periodic homogenization, the meso-scale tensor is fixed by the tensor-harmonic-mean relation $C_\mathrm{macro}=C_\mathrm{micro}(C_\mathrm{micro}+C_e)^{-1}C_e$, equivalently $C_e=C_\mathrm{micro}(C_\mathrm{micro}-C_\mathrm{macro})^{-1}C_\mathrm{macro}$, so the two scale-independent short-range tensors of the model are determined by first-order homogenization alone. The paper evaluates these tensors for an aluminum/air tetragonal metamaterial in plane strain, computing $C_\mathrm{micro}$ from four unit cells and $C_\mathrm{macro}$ from periodic boundary conditions.

Load-bearing premise

The calculation assumes that the four unit cells tested are the only admissible tetragonal cells, so that the stiffness tensor built from them is the true least upper bound, and that the micro-scale tensor remains strictly stiffer than the macro-scale tensor so that the inversion producing the meso-scale tensor is well defined.

Editorial extensions

If this is right

  • For a homogeneous unit cell, $C_\mathrm{micro}=C_\mathrm{macro}$, so $C_e\to\infty$ and the relaxed micromorphic model reduces to classical linear elasticity with stiffness $C_\mathrm{macro}$, satisfying the requirement that a homogeneous microstructure leaves the response invariant.
  • For infinitely rigid inclusions, $C_\mathrm{micro}\to\infty$ and $C_\mathrm{macro}\to C_e$, so the model reduces to a Cosserat-type response, giving a sensible rigid-microstructure limit.
  • Both $C_\mathrm{micro}$ and $C_\mathrm{macro}$ are independent of the characteristic length $L_c$, so the scale-independent short-range parameters are available from static first-order unit-cell computations before any dynamical or size-dependent fitting.
  • The same two tensors, through $C_e=C_\mathrm{micro}(C_\mathrm{micro}-C_\mathrm{macro})^{-1}C_\mathrm{macro}$, determine the meso-scale stiffness that enters long-wavelength response and band-gap predictions in the companion wave-propagation paper.
  • The energy bound $C_\mathrm{micro}\ge C^V_{\mathrm{KUBC}}$ keeps the stored elastic energy of the relaxed micromorphic model bounded for arbitrarily small sample sizes, avoiding the unbounded stiffness of second-gradient and Eringen-Mindlin-type formulations.

Reading between the lines

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

  • Editorial inference: the same scheme should transfer to three-dimensional and lower-symmetry periodic microstructures, provided the family of admissible unit cells satisfying the symmetry principle can be enumerated; the bottleneck is the exhaustive list of cells, not the homogenization steps.
  • Editorial inference: the computed numerical values depend on the four-cell candidate list; checking convergence of the Löwner supremum as more tetragonal cells are added would turn the reported stiffnesses into a certified bound.
  • Editorial inference: if strict positive definiteness of $C_\mathrm{micro}-C_\mathrm{macro}$ is established for sufficiently contrasted microstructures, the relation for $C_e$ implies a quantitative separation: the softer the inclusions, the closer $C_\mathrm{micro}$ can approach $C_\mathrm{macro}$, and the larger $C_e$ becomes.
  • Editorial inference: the identification of $C_\mathrm{micro}$ as a supremum suggests an experimental test: measure the apparent stiffness of progressively smaller samples under affine loading and check that the extrapolated maximum equals the computed $C_\mathrm{micro}$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper proposes a first-order homogenization procedure for identifying the scale-independent stiffness tensors of the relaxed micromorphic continuum model for a given periodic microstructure. The authors consider affine Dirichlet (KUBC) loadings on candidate unit cells and, invoking a Hill-Mandel energy equivalence, derive inequality (30), which states that the micro-scale stiffness tensor Cmicro must dominate the apparent stiffness tensor C^V_KUBC in the energy norm for every admissible cell. They combine this bound with an extended Neumann's principle restricting admissible cells to those whose KUBC response is tetragonal, and introduce C0_micro as the Löwner matrix supremum of the family. Theorem 1 then asserts the bound <Cmicro E,E> >= <C0_micro E,E> and the linking relation Cmacro = Cmicro(Cmicro+Ce)^-1 Ce. Numerically, for an aluminum/air cross-shaped metamaterial, the authors compute Cmacro by periodic homogenization (Table 1) and C^V_KUBC for four square cells (Table 2), then set Cmicro to the componentwise maxima (Eq. (50)) and solve for Ce via Eq. (15). The paper concludes that the short-range parameters are determined by first-order homogenization alone and reports applications in companion papers.

Significance. If correct, the method would be practically valuable: it replaces the ambiguous extended homogenization postulates for micromorphic continua (steps b-e listed in the introduction) with standard first-order KUBC computations, and it provides a transparent inequality (44) that relates the micromorphic micro-stiffness to classical apparent stiffnesses. The variational derivation of (30) and the Hill-Mandel step are clean and are not circular: the bound is derived from the model energy and the energy equivalence, not from the numerical values. The numerical results are internally consistent: the four reported KUBC tensors in Table 2 satisfy the expected hierarchy with respect to PBC values, and the componentwise-max selection in Eq. (50) indeed yields a Löwner upper bound for those four cells. However, the strength of the claim depends on two points that are not settled in the manuscript: the exhaustiveness of the four-cell enumeration and the strict positive definiteness of Cmicro - Cmacro. Both are flagged in the paper itself (Section 3.2 and Section 5), and both are load-bearing for Eq. (50) and Eq. (15).

major comments (3)
  1. [Theorem 1 and Section 3.2 (Fig. 16, Eq. (50))] The set of 'admissible unit-cells' in Theorem 1 is never defined mathematically, and the paper does not prove that the four cells in Fig. 16 exhaust the family of tetragonal-symmetric cells under KUBC. The text in Section 3.2 states that rectangles and parallelograms are also valid tessellations and then asserts, without proof, that only four cells capture the tetragonal symmetry under affine Dirichlet boundary conditions. Since the numerical C0_micro in Eq. (50) is the componentwise maximum of the four computed tensors, the existence of any further admissible cell with a larger mu, mu+lambda, or mu* would violate inequality (44) and change the reported Cmicro and Ce. The paper should either prove the exhaustiveness claim (or precisely define the family and prove that its supremum equals the four-cell value) or explicitly present the numerical identification as conditional on the enumeration.
  2. [Theorem 1 / Eq. (43) and Section 5] The central formula Ce = Cmicro(Cmicro-Cmacro)^-1 Cmacro requires that Cmicro - Cmacro be positive definite. Section 5 shows only that Q(E,E) >= 0 and states that strict positivity is an open problem; no assumption or proof is supplied in Theorem 1. As stated, the theorem therefore does not rigorously establish the existence of Ce for a general periodic microstructure. For the specific numerical example the difference appears positive definite from Table 2, but the theorem's generality and the abstract's claim of rigorous determination are not supported. Please add explicit sufficient conditions (e.g., on contrast or geometry) under which strict positivity holds, or restrict the theorem to the verified numerical setting.
  3. [Abstract and Conclusion vs. Section 2.3] The manuscript claims to 'rigorously determine' Cmicro, but the derivation establishes only the necessary bound (44). The actual identification Cmicro = C0_micro is a choice among infinitely many tensors satisfying the bound; Remark 1 explicitly permits any other positive definite tensor dominating all C^V_KUBC. The conclusion's statement that Cmicro 'can be identified with' the Löwner supremum therefore overstates the mathematical content of the paper. Please rephrase the claims so that the upper-bound/selection nature of the procedure is accurately described, and clarify what would be needed to prove uniqueness.
minor comments (4)
  1. [Section 2.3 / Theorem 1] Please provide a formal definition of 'admissible unit-cell' used in Theorem 1, for example in terms of fundamental domains of the lattice whose KUBC response is tetragonal.
  2. [Table 2] Table 2 would benefit from a column indicating which row corresponds to which cell (a)-(d) of Fig. 16; currently only the PBC row is clearly labeled.
  3. [Throughout] There are numerous typographical and grammatical errors (e.g., in Section 2.1 the sentence beginning 'Since' is garbled). A careful proofread is recommended.
  4. [Section 1.3] The statement 'Lc -> infinity corresponds to using P = grad u' is heuristic; consider making this limit precise, for instance via the variational formulation in Eq. (17).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Löwner-supremum identification is a transparent construction derived from the energy inequality and Hill–Mandel equivalence, not a fitted value renamed as a prediction.

full rationale

The paper's derivation chain is self-contained. Section 2.1–2.3 derives the central bound (44) as a necessary consequence of the model energy inequality (23) together with the Hill–Mandel energy equivalence (29), giving <Cmicro E,E> ≥ <C^V_KUBC E,E> for admissible unit cells. The numerical identification in Eqs. (47)–(50) is then an explicit construction of the Löwner matrix supremum from the computed KUBC tensors in Table 2; this is a parameter-identification rule, openly introduced as an optimal choice rather than as an independent prediction of a quantity already contained in the inputs. The relation Cmacro = Cmicro(Cmicro+Ce)^{-1}Ce used to obtain Ce is not merely a self-citation: it follows from the Lc→0 equilibrium equations displayed in Eqs. (13)–(15), and the cited prior work [6] is parameter-free and externally checkable. Two genuine gaps are acknowledged in the paper and affect correctness rather than circularity: the exhaustiveness of the four tetragonal unit cells in Fig. 16 is asserted without proof, so a stiffer admissible KUBC cell would violate (44); and Section 5 explicitly leaves open the strict positive definiteness of Cmicro−Cmacro needed for Ce. Neither gap makes the derivation equivalent to its own inputs by construction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central identification rests on the relaxed micromorphic framework, the harmonic mean link to Cmacro, a symmetry restriction, and a finite candidate set. Cmicro is not derived from independent physics; it is chosen as the least upper bound of computed apparent stiffnesses. The strict positivity needed for Ce is admitted open in the paper.

free parameters (1)
  • Cmicro Lamé parameters (λ, μ, μ*) in Eq. (50) = λ = 5.270 GPa, μ = 8.927 GPa, μ* = 8.332 GPa
    Chosen as componentwise maxima of the KUBC apparent stiffness entries in Table 2, per Eq. (47). This is the Löwner supremum over the four considered unit cells, not an independently predicted constant; the paper itself notes in Remark 1 that larger Cmicro satisfying (44) are allowed.
assumptions (6)
  • domain assumption The suitably homogenized model for the transition scale is of relaxed micromorphic type.
    Section 1: 'we therefore postulate for this work that the suitably homogenized model will be of the micromorphic type'. If this is false, the identified Ce and Cmicro do not describe the actual transition-scale response.
  • domain assumption The exact micro-macro harmonic mean relation Cmacro = Cmicro(Cmicro+Ce)^{-1}Ce holds.
    Equation (15) from prior work by the same group; the paper imports it as exact, but it is a theorem of the relaxed micromorphic model, not of the real microstructure.
  • domain assumption The unit-cell material is linear elastic with uniformly positive definite stiffness, including the air phase.
    Section 3.1 says C(ξ) is the elasticity tensor of aluminum or air; air stiffness is never specified and the homogenization and variational arguments require positive definiteness.
  • ad hoc to paper The four unit cells in Fig. 16 exhaust the tetragonal-symmetric admissible unit cells for the Löwner supremum.
    Section 3.2 and Fig. 15 scan only square cells of side a and rotated squares of side a√2; no proof covers other tetragonal tiling cells.
  • domain assumption Cmicro - Cmacro is positive definite so that Ce = Cmicro(Cmicro-Cmacro)^{-1}Cmacro is well defined.
    Section 5 states strict positivity of Q(E,E) is still open; the paper nevertheless uses the formula to compute Ce.
  • domain assumption Extended Neumann's principle restricts all relaxed-model tensors to tetragonal symmetry.
    Section 2.3 invokes this principle to reduce candidate unit-cells; symmetry constraints are standard but are an input assumption.

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Cite this review

Pith. "Pith review of Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization." pith.science (2026). https://pith.science/paper/ZCLUGOAP

@misc{pith2026190913624,
  author       = {Pith},
  title        = {Pith review of: Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZCLUGOAP}},
  note         = {Machine review of arXiv:1909.13624}
}
read the original abstract

We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann's principle on the overall representation of anisotropy. We explain our idea of "maximal" stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper [16] to describe the wave propagation including band-gaps in the same tetragonal metamaterial.

Figures

Figures reproduced from arXiv: 1909.13624 by the authors.

Figure 1
Figure 1. Infinite periodic structure. The effective properties are given by the effective stiffness tensor [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Finite sized sample of a periodic structure. The scale-separation hypothesis does not anymore apply. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Long-range fluctuation field P, short-range elastic interaction scale e and averaged out micro-fluctuation. The static equilibrium equations are the Euler-Lagrange equations to (1) and (2), which read in strong form6 Div [Ce sym (∇u − P) + Cc skew (∇u − P)] = f, (4) Ce sym (∇u − P) + Cc skew (∇u − P) − Cmicro sym P − µ L2 c Curl Curl P = 0. While the relaxed micromorphic model allows for balance equations in the cla… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Shear test on a block. Vertical sides free. Admissible configuration [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Qualitative relation between the Cosserat model and Toupin’s version of gradient elasticity. Cosserat [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: "Smaller is stiffer": N × N array of unit cells. The apparent stiffness CKUBC (stiffness under applied kinematically uniform boundary conditions) decreases with increasing size to reach in the limit of infinite size Cmacro. This stiffness-delta between the smallest dis…
Figure 7
Figure 7. Figure 7: (Stiffness in affine loading w.r.t. displacement [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Scale-independent response governed by two springs in series. If [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: (Stiffness in non-affine loading, like torsion) Qualitative difference of the relaxed micromorphic model [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: The process of homogenization. Following the classical Hill-Mandel lemma, we demand energy equiva [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Affine Dirichlet loading (KUBC) of the unit-cell [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Qualitative result for the apparent stiffness [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Difference between affine Dirichlet boundary conditions (KUBC) and periodic Dirichlet boundary con [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Geometry of the unit-cell and elastic parameters for Aluminum. [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Identification of candidate unit-cell variants that fulfill the extended Neumann’s principle for (a) [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: The four regular square unit-cells respecting tetragonal symmetry. [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

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Pith tools

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