{"id":"f3cd6706-3654-4cef-92af-96a82ff71779","arxiv_id":"1909.13624","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a tetragonal periodic metamaterial, the micro-scale stiffness tensor is determined as the least upper bound (Löwner supremum) of unit-cell apparent stiffnesses under affine Dirichlet conditions, and the meso-scale tensor follows from the harmonic mean relation with the homogenized macro…","lead":"This paper presents a recipe for computing two scale-independent stiffness tensors in the relaxed micromorphic model directly from the geometry of a periodic microstructure. The method could make this generalized continuum model easier to use for predicting wave band gaps and size effects in mechanical metamaterials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper does not prove that the four tetragonal unit cells in Fig. 16 exhaust all admissible cells; a stiffer KUBC cell would invalidate the reported Cmicro and hence Ce. The numerical values in Eq. (50) are therefore conditional on that enumeration.","rationale":"The paper is a serious attempt to fix the scale-independent short-range parameters of the relaxed micromorphic model from first-order homogenization. The harmonic-mean link (15), the energy bound (30), and the Löwner-supremum construction are internally coherent, and the fact that the paper itself flags the positive-definiteness of Cmicro-Cmacro as an open problem shows appropriate transparency. The numerical values in Table 2 suggest that positive definiteness holds for the reported metamaterial, so that issue is secondary. The load-bearing gap is the exhaustiveness of the admissible unit-cell set: Theorem 1 and Eq. (50) identify Cmicro with a supremum computed from only four cells, and the paper gives no proof that no other fundamental domain of the square lattice, invariant under the tetragonal group, yields a KUBC apparent tensor not dominated by C0_micro. Since the central claim is a rigorous determination, this missing enumeration is a real soft spot. The proposed computational enumeration would settle it. Because the reader's verdict was already CONDITIONAL and this concern supports that classification without overturning the paper's contribution, the verdict should remain unchanged. The reader's weakest_assumption named this exhaustiveness gap and also the positive-definiteness open problem; I share the first as the primary concern but regard the second as less threatening here, hence partial agreement.","tokens_in":24555,"tokens_out":12187,"duration_ms":116536,"concrete_test":"Systematically enumerate all unit cells of the periodic cross lattice that are fundamental domains of area a^2 and 2a^2 and whose domain is invariant under the tetragonal point group, including all translations of the origin modulo the lattice. For each such cell, compute the KUBC apparent stiffness tensor C^V_KUBC with the same FE-HMM discretization used for Table 2, and test whether the reported Cmicro from Eq. (50) dominates every such tensor in the Löwner order. If any admissible cell yields <C^V_KUBC E,E> > <Cmicro E,E> for some strain E, then the reported identification violates (44); if all are dominated, the exhaustiveness gap is closed and the conditional verdict can be upgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identification Cmicro = C0_micro (Eq. 50) rests on Theorem 1's inequality (44), which requires Cmicro to dominate the KUBC apparent stiffness of every admissible tetragonal unit cell. The paper computes the Löwner supremum only over the four cells in Fig. 16, selected from square cells of side a and rotated squares of side a√2 in Fig. 15. It states that other quadrilaterals such as rectangles and parallelograms are also valid unit cells, and it claims without proof that only these four cells capture the tetragonal symmetry under affine Dirichlet boundary conditions. No argument establishes that every fundamental domain of the square lattice whose KUBC tensor is tetragonal belongs to this list. If a further admissible cell V has a strain E with <C^V_KUBC E,E> > <C0_micro E,E>, then the reported Cmicro violates (44) and Ce = Cmicro(Cmicro-Cmacro)^{-1}Cmacro is not justified. The second open point, strict positive definiteness of Cmicro-Cmacro, is explicitly left open in Section 5; for the numerical values in Table 2 the difference appears positive definite, so the enumeration gap is the more serious threat to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24780,"tokens_out":9800,"duration_ms":91187,"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":[{"comment":"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.","section":"Theorem 1 and Section 3.2 (Fig. 16, Eq. (50))"},{"comment":"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.","section":"Theorem 1 / Eq. (43) and Section 5"},{"comment":"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.","section":"Abstract and Conclusion vs. Section 2.3"}],"minor_comments":[{"comment":"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.","section":"Section 2.3 / Theorem 1"},{"comment":"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.","section":"Table 2"},{"comment":"There are numerous typographical and grammatical errors (e.g., in Section 2.1 the sentence beginning 'Since' is garbled). A careful proofread is recommended.","section":"Throughout"},{"comment":"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).","section":"Section 1.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is part of an ongoing program by the authors, and the same Cmicro/Ce values are used in several companion papers. Before the identification can be relied upon, the enumeration gap and the positivity issue should be resolved. The variational core of the paper is sound, so I do not see grounds for rejection, but the overclaim in the abstract should be corrected in any revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is the strongest attempt I know to get scale-independent Cmicro from first-order homogenization, and the Löwner-supremum construction is genuinely new. But the abstract and Theorem 1 promise more than the body delivers. What is actually shown is that Cmicro must dominate every KUBC apparent stiffness tensor of admissible tetragonal cells; the numerical identification then sets Cmicro equal to the computed supremum. That is a defensible heuristic, not a rigorous determination.\n\nWhat the paper does well: the derivation of inequality (30) from the model energy and the Hill-Mandel lemma is clean, and the four reported KUBC tensors are internally consistent. The idea of using an extended Neumann principle to cut down the unit-cell family is sensible, and the concrete numbers feed into the companion band-gap paper. The harmonic mean formula (15) is prior work by the same group, properly cited; the self-citation there is not a problem. The paper also deserves credit for stating its own open problems clearly, especially the positive-definiteness issue in Section 5.\n\nThe main soft spot is the unit-cell enumeration. The paper shows four cells in Fig. 16, notes that rectangles and parallelograms are also valid unit cells, then asserts without proof that only these four capture tetragonal symmetry under affine Dirichlet boundary conditions. If a stiffer admissible tetragonal cell exists, the reported Cmicro violates (44). This is a load-bearing gap, not a cosmetic one. The second gap is that Ce = Cmicro(Cmicro–Cmacro)^{-1}Cmacro requires Cmicro–Cmacro to be strictly positive definite. Section 5 leaves that open; for the numerical values it seems to hold, but the general claim is unproved. The abstract's \"rigorously determine\" thus overstates what is established. These are addressable in revision, and the paper would be stronger if it either proved the enumeration or explicitly presented the method as selecting an upper-bound-optimal Cmicro within a chosen family.\n\nNo code or data were provided, so I could not independently reproduce the numbers; for a methods paper that is a minor inconvenience rather than a fatal flaw. The citation pattern is reasonable and the argument is coherent on its own terms.\n\nThis paper is for people working on micromorphic parameter identification and metamaterial wave propagation. It deserves a serious referee. I would send it to review with a request to tighten the claims: either prove or relabel the unit-cell enumeration and either prove or assume the positive-definiteness condition. With those changes, the core contribution stands.","headline":"A genuinely new identification route for the relaxed micromorphic model, but the advertised 'rigorous determination' is really a bounding inequality plus a convenient choice.","tokens_in":25393,"tokens_out":2564,"would_cite":true,"duration_ms":30005,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74A30","74A35","74A60","74B05","74M25","74Q15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["relaxed micromorphic model","parameter identification","Löwner matrix supremum","tensor harmonic mean","apparent stiffness","periodic homogenization","tetragonal symmetry","metamaterial"],"falsifier":"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.","tokens_in":24342,"feed_emoji":"⚙️","tokens_out":13852,"duration_ms":112303,"temperature":0.7,"pith_summary":"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.","feed_headline":"One unit-cell calculation fixes micromorphic short-range stiffness","feed_subtitle":"The micro-scale tensor is the least upper bound of unit-cell stiffnesses; macro-scale is from periodic homogenization.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $C_\\mathrm{micro}$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the exact tensor-harmonic-mean relation $C_\\mathrm{macro}=C_\\mathrm{micro}(C_\\mathrm{micro}+C_e)^{-1}C_e$ that links the two identified tensors.","marker":"[6]"},{"why":"Establishes the apparent stiffness under affine Dirichlet data and the size hierarchy showing kinematically uniform boundary conditions overestimate the effective stiffness, motivating the maximal-stiffness view.","marker":"[36]"},{"why":"Gives the energy-equivalence statement used to define the apparent stiffness tensor under affine Dirichlet boundary conditions.","marker":"[33]"},{"why":"Provides the standard formula for the average displacement gradient under affine Dirichlet data, used in deriving the bound on $C_\\mathrm{micro}$.","marker":"[82]"},{"why":"Formulates the extended symmetry principle used to restrict admissible unit cells to tetragonal-symmetric ones.","marker":"[67]"},{"why":"Gives an early textbook statement of the same symmetry principle, cited to require $C_\\mathrm{micro}$ to share the material's tetragonal symmetry.","marker":"[80]"},{"why":"Introduces the Löwner matrix ordering and the supremum problem used to select the least upper bound $C^0_\\mathrm{micro}$.","marker":"[14]"},{"why":"Supplies the two-scale finite-element computational scheme used to obtain the converged plane-strain values for $C_\\mathrm{macro}$ and for the apparent stiffnesses.","marker":"[20]"}],"fun_headline_variants":["Micromorphic constants from one unit-cell stiffness bound","Unit-cell supremum fixes micromorphic short-range constants","Homogenization pins both micromorphic scale tensors","Short-range micromorphic stiffness from a single cell","Scale-independent micromorphic parameters via one cell"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Micromorphic constants from one unit-cell stiffness bound","Unit-cell supremum fixes micromorphic short-range constants","Homogenization pins both micromorphic scale tensors","Short-range micromorphic stiffness from a single cell","Scale-independent micromorphic parameters via one cell"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1332,"prompt_tokens":953,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":306}},"tokens_in":569,"tokens_out":379,"duration_ms":4557,"temperature":1.0,"reasoning_tokens":306,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:44:47.667242+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Transparent anisotropyfortherelaxedmicromorphicmodel: macroscopicconsistencyconditionsandlongwavelengthasymptotics","cited_arxiv_id":null,"evidence_quote":"Supplies the exact tensor-harmonic-mean relation $C_\\mathrm{macro}=C_\\mathrm{micro}(C_\\mathrm{micro}+C_e)^{-1}C_e$ that links the two identified tensors."},{"cited_title":"Application of variational concepts to size effects in elastic heterogeneous bodies.Journal of the Mechanics and Physics of Solids, 38(6):813–841, 1990","cited_arxiv_id":null,"evidence_quote":"Establishes the apparent stiffness under affine Dirichlet data and the size hierarchy showing kinematically uniform boundary conditions overestimate the effective stiffness, motivating the maximal-stiffness view."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the standard formula for the average displacement gradient under affine Dirichlet data, used in deriving the bound on $C_\\mathrm{micro}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formulates the extended symmetry principle used to restrict admissible unit cells to tetragonal-symmetric ones."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives an early textbook statement of the same symmetry principle, cited to require $C_\\mathrm{micro}$ to share the material's tetragonal symmetry."},{"cited_title":"Mathematical morphology on tensor data using the Löwner ordering","cited_arxiv_id":null,"evidence_quote":"Introduces the Löwner matrix ordering and the supremum problem used to select the least upper bound $C^0_\\mathrm{micro}$."},{"cited_title":"The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE2 method","cited_arxiv_id":null,"evidence_quote":"Supplies the two-scale finite-element computational scheme used to obtain the converged plane-strain values for $C_\\mathrm{macro}$ and for the apparent stiffnesses."}],"review_version":1}