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arxiv: 2606.08095 · v1 · pith:JTQJBDXBnew · submitted 2026-06-06 · 🧮 math.NA · cs.NA

Strain localization in softening plasticity without modifying standard constitutive models: a deformable Cosserat approach

Andrea Panteghini , M.B. Rubin This is my paper

Pith reviewed 2026-06-27 19:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords strain localizationsoftening plasticityCosserat continuumfinite element methodelastoplastic modelsinternal length scaleshear bandssoil mechanics
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The pith

A deformable Cosserat model lets standard elastoplastic constitutive laws simulate strain localization without any changes to their stress-update algorithms or tangent operators.

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

The paper establishes a formulation that confines all dissipation to the macro-continuum while restricting the micro-continuum to linear elastic director-field contributions. This separation preserves the original constitutive structure so that existing elastoplastic models function as unmodified black-box components. The internal length scale emerges directly from the micro-continuum and controls the development and interaction of localization bands instead of acting as a diffusive regularizer. Implementation requires only linear additions to the residual and tangent operators inside standard finite-element codes. Benchmark calculations on shallow foundations with Tresca and Matsuoka-Nakai models demonstrate mesh-convergent load-displacement curves, dissipated energy, and shear-band patterns even under highly unstable post-peak softening.

Core claim

The central claim is that a deformable Cosserat continuum with strict separation of dissipative macro-mechanisms and energetic micro-mechanisms allows any standard elastoplastic model formulated for a classical Cauchy continuum to be used without modification of its stress-update algorithm or consistent tangent, while the naturally arising internal length scale from the micro-continuum governs localization pattern development, interaction and selection.

What carries the argument

Deformable Cosserat continuum that confines dissipation to the macro-continuum and restricts the micro-continuum to linear elastic director-field terms.

If this is right

  • Existing elastoplastic models can be inserted unchanged into the Cosserat framework and still produce regularized localization.
  • The internal length scale controls band spacing and interaction without requiring artificial diffusion terms.
  • Load-displacement responses, total dissipated energy, and shear-band geometries converge with mesh refinement even for nonlinearly interacting, unstable localization processes.
  • Only linear additions to the residual and tangent operators are needed, preserving the structure of conventional finite-element implementations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of dissipation and elasticity could be tested by comparing the approach against a fully coupled Cosserat plasticity model on the same mesh to quantify any algorithmic overhead introduced by the black-box assumption.
  • The method may extend naturally to three-dimensional problems where multiple intersecting bands interact, since the length scale is set by the micro-continuum rather than mesh size.
  • Because the length scale is tied to the director-field stiffness, varying that stiffness independently offers a route to explore how material microstructure influences band selection without altering the macro constitutive law.

Load-bearing premise

The added Cosserat contributions remain purely energetic and never enter the dissipative constitutive response, so they leave standard stress-update algorithms untouched.

What would settle it

A single-element or single-band test in which the Cosserat formulation with an unmodified black-box constitutive model produces a different post-peak load-displacement curve or dissipated-energy value than the same model run with its internal variables explicitly coupled to the director strain.

Figures

Figures reproduced from arXiv: 2606.08095 by Andrea Panteghini, M.B. Rubin.

Figure 1
Figure 1. Figure 1: Schematic representation of the kinematics of the deformable Cosserat model in a 2D setting. The vectors [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the adopted yield criteria in the octahedral plane. (a) The solid line is the Matsuoka-Nakai criterion. For comparison the dashed [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mesh 1 (coarse mesh). The arrow indicates the edge of the shallow footing. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Strip footing on a Tresca soil under perfect plasticity: normalized load–displacement curves. The Cauchy response (obtained by enforcing [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Strip footing on a Tresca soil with exponential isotropic softening: (a) normalized load–displacement curves and (b) total dissipated [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial distribution of κ for the strip footing on Tresca soil with exponential softening, r = 2 · 10−3 . (a)–(b) Mesh 2 and Mesh 3 at peak load; (c)–(d) Mesh 2 and Mesh 3 at the end of the analysis (¯u/B = 0.1). The localization pattern, consisting of two dominant shear bands with a slightly diffuse intermediate zone, is essentially coincident across meshes, indicating full mesh objectivity. persist in th… view at source ↗
Figure 7
Figure 7. Figure 7: Spatial distribution of κ for the strip footing on Tresca soil with exponential softening, r = 10−3 . (a)–(b) Mesh 3 and Mesh 4 at peak load; (c)–(d) at ¯u/B = 0.02; (e)–(f) at the end of the analysis (¯u/B = 0.1). In addition to the main shear bands, a network of secondary bands develops. Both primary and secondary structures are nearly coincident across meshes, demonstrating mesh-independent prediction o… view at source ↗
Figure 8
Figure 8. Figure 8: Spatial distribution of κ for the strip footing on Tresca soil with exponential softening, r = 6 · 10−4 . (a)–(b) Mesh 3 and Mesh 4 at peak load; (c)–(d) at ¯u/B = 0.02; (e)–(f) at the end of the analysis (¯u/B = 0.1). The main shear bands are well captured and essentially coincident, whereas minor differences remain in the finer secondary structures, indicating that a finer discretization is required to f… view at source ↗
Figure 9
Figure 9. Figure 9: Influence of the micro-elastic parameters [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Strip footing on Matsuoka–Nakai soil under perfect plasticity ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between |u˙| obtained with Mesh 2 for the Cauchy Continuum (a-c-e) and for r = 10−4 (b-d-f) at different loading levels for the Nγ problem. (a) Cauchy, ¯u/B = 0.05, (b) r = 10−4 , ¯u/B = 0.05. (c) Cauchy, ¯u/B = 0.07, (d) r = 10−4 , ¯u/B = 0.07. (e) Cauchy, ¯u/B = 0.1 (end of the analysis), (f) r = 10−4 , ¯u/B = 0.1 (end of the analysis) mesh-independent manner even in the presence of strong so… view at source ↗
Figure 12
Figure 12. Figure 12: Convergence of the solution for the strip footing on Matsuoka–Nakai soil with softening, obtained using the Riks procedure for [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Convergence of the ϕ profiles for r = 4.5 · 10−4 . (a) Mesh 3, stress peak (point P of Fig. 12a), (b) Mesh 4, stress peak (point P of Fig. 12a). (c) Mesh 3, point A of Fig. 12a, (d) Mesh 4, point A of Fig. 12a. (d) Mesh 3, point B of Fig. 12a, (e) Mesh 4, point B of Fig. 12a. (c) Mesh 3, point L of Fig. 12a, (d) Mesh 4, point L of Fig. 12a. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Influence of the internal length r = ℓ/B on the load–displacement response for the Matsuoka–Nakai model with softening (Mesh 3). Smaller values of r lead to a more pronounced softening and to an unstable post-peak response, whereas larger values produce a smoother and more stable behavior. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Spatial distribution of the angle of shearing resistance [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Spatial distribution of the angle of shearing resistance [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Spatial distribution of the angle of shearing resistance [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Spatial distribution of the angle of shearing resistance [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Spatial distribution of the modulus of the micro-stress [PITH_FULL_IMAGE:figures/full_fig_p028_19.png] view at source ↗
read the original abstract

This paper presents a formulation for strain localization in softening plasticity based on a deformable Cosserat model. The approach enables the direct use of standard elastoplastic constitutive models formulated for a classical Cauchy continuum, without modifying the stress update algorithm or consistent tangent operator. A key feature of the framework is the strict separation of dissipative and energetic mechanisms: all dissipation is confined to the macro-continuum, while the micro-continuum contributes only through linear elastic terms associated with the director field. As a result, the constitutive structure of the elastoplastic model is preserved, and existing models can be employed as black-box components. The internal length scale arises naturally from the micro-continuum and governs the development, interaction and selection of localization patterns, rather than acting as a diffusive parameter. The formulation is easy to implement within standard finite element frameworks, requiring only additional linear contributions to the residual and tangent operators. The performance of the approach is assessed through benchmark problems involving shallow foundations on soil, a demanding test due to complex and unstable localization mechanisms. Both Tresca and Matsuoka-Nakai plasticity models are considered, including cases with highly unstable post-peak responses. Numerical results show convergence of load-displacement responses, dissipated energy and shear-band patterns upon mesh refinement, even in the presence of nonlinear interacting localization processes. These findings demonstrate a robust and physically consistent approach for the analysis of strain localization in softening plasticity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a deformable Cosserat continuum formulation for strain localization in softening plasticity that permits direct use of unmodified standard elastoplastic constitutive models (e.g., Tresca, Matsuoka-Nakai) formulated for Cauchy continua. Dissipation is confined to the macro-scale symmetric strain while the micro-director field contributes only linear-elastic energetic terms; the internal length emerges from the micro-continuum rather than as a diffusive regularization parameter. The approach requires only additional linear terms in the global residual and tangent. Benchmark results on shallow-foundation problems demonstrate mesh-convergent load-displacement curves, dissipated energy, and shear-band patterns even under unstable post-peak response.

Significance. If the claimed strict energetic/dissipative separation holds, the method offers a practical route to regularization that preserves existing, often complex and validated stress-update algorithms as black-box components. This is valuable in computational geomechanics where constitutive-model reuse is essential. The benchmarks are demanding and the reported convergence of interacting localization patterns is a positive indicator of robustness.

major comments (2)
  1. [Formulation and implementation sections] The central claim of black-box compatibility rests on the assertion that the local constitutive integration receives only the symmetric macro strain and that the yield function, flow rule, and consistent tangent remain independent of micro-rotation and curvature. The manuscript must explicitly show (in the section describing the stress-update procedure and the weak-form residuals) that no hidden coupling enters the return-mapping algorithm through the director-field terms or the global tangent assembly.
  2. [Numerical examples and discussion] The internal-length interpretation as governing pattern selection rather than diffusion is load-bearing for the physical claim. The paper should demonstrate, via a parameter study or analytic argument, that the micro-continuum stiffness parameters control band width and interaction independently of any numerical diffusion introduced by the finite-element discretization.
minor comments (2)
  1. Notation for the micro-director field and its curvature measure should be introduced with a clear table or diagram to avoid confusion with standard Cosserat rotation tensors.
  2. [Numerical results] The abstract states convergence upon mesh refinement; the main text should report quantitative measures (e.g., L2 norms of strain or band-width evolution) rather than qualitative statements alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Formulation and implementation sections] The central claim of black-box compatibility rests on the assertion that the local constitutive integration receives only the symmetric macro strain and that the yield function, flow rule, and consistent tangent remain independent of micro-rotation and curvature. The manuscript must explicitly show (in the section describing the stress-update procedure and the weak-form residuals) that no hidden coupling enters the return-mapping algorithm through the director-field terms or the global tangent assembly.

    Authors: We agree that the separation must be shown with greater explicitness. The formulation applies the standard elastoplastic return mapping exclusively to the symmetric macro-strain tensor; the micro-rotation and curvature enter only the linear-elastic director terms of the weak form (Eqs. 8-10) and are assembled as additional linear contributions to the global residual and tangent. No coupling reaches the local constitutive routine. To satisfy the request we will add a short dedicated paragraph (and a clarifying flowchart) in the stress-update and implementation sections that isolates the macro-strain input to the black-box model and confirms that the consistent tangent operator remains unmodified. revision: yes

  2. Referee: [Numerical examples and discussion] The internal-length interpretation as governing pattern selection rather than diffusion is load-bearing for the physical claim. The paper should demonstrate, via a parameter study or analytic argument, that the micro-continuum stiffness parameters control band width and interaction independently of any numerical diffusion introduced by the finite-element discretization.

    Authors: We accept that an explicit demonstration strengthens the physical claim. While the reported mesh-convergent results already indicate that regularization is not controlled by discretization, we will add a parameter study in the numerical-examples section that varies the micro-stiffness coefficients (bending and curvature moduli) at fixed mesh size and documents the resulting changes in band width and pattern interaction. A short analytic argument based on the characteristic length extracted from the linearized micro-continuum equations will also be included to confirm independence from numerical diffusion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central claim is that a deformable Cosserat formulation permits black-box use of standard Cauchy elastoplastic models by confining all dissipation to the macro-continuum and restricting micro-director contributions to linear elastic terms. No equations, parameters, or steps are shown to reduce by construction to fitted inputs, self-citations, or renamed known results. The separation is asserted as a modeling choice whose validity is demonstrated through numerical benchmarks rather than derived from prior self-referential results. The framework is presented as independent of specific model parameters and implementable via standard FE additions to residuals and tangents.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters, axioms, or invented entities beyond the stated separation of mechanisms; the internal length scale is described as arising naturally but is likely introduced as a model parameter in the micro-continuum.

free parameters (1)
  • internal length scale
    Described as arising naturally from the micro-continuum but functions as a regularization parameter in practice.
axioms (1)
  • domain assumption The micro-continuum contributes only through linear elastic terms associated with the director field
    Explicitly stated as enabling the black-box use of standard models.

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