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Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs

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Pith's one-line read The full Lattice Spring Model with softening, hardening, and perfectly plastic springs is equivalent to a state-dependent sweeping process.

desk verdict A genuinely useful sweeping-process formulation for lattice elastoplasticity with softening, but the paper's main numerical claim rests on an unproven existence step that the author explicitly defers. read the letter →

arxiv 2508.16333 v2 pith:MKT4YZQ7 submitted 2025-08-22 math.OC cond-mat.softmath-phmath.MP

Ivan Gudoshnikov This is my paper · ORCID
classification math.OCcond-mat.softmath-phmath.MP MSC 74C0574N3047J2047J26
keywords state-dependentsweepingprocesselastoplasticitywithsofteningnon-uniquenessofsolutionsshearbandstrainlocalizationlatticespringmodelimplicitcatch-upalgorithmplasticitybifurcation
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

This paper tries to show that the quasi-static evolution of a lattice of elasto-plastic springs—including the mathematically troublesome softening case—can be written exactly as a state-dependent sweeping process, a differential inclusion with a moving convex set. If the equivalence is right, the same numerical and analytical machinery applies to hardening, perfect plasticity, and softening, with the difference between them reduced to a parameter. The paper demonstrates the payoff by analytically exhibiting coexisting solutions in a two-spring softening system and by simulating shear-band formation with strain localization in regular lattices. This matters because softening plasticity is normally ill-posed: stress-strain relations lose monotonicity, models become mesh-dependent, and solutions can branch. A single finite-dimensional formulation that still produces well-defined evolutions up to spring failure would give a rigorous route into ductile-fracture initiation.

What carries the argument

The load-bearing object is the state-dependent moving set C(y, q): for each spring, a convex polytope of admissible stresses whose upper and lower thresholds shift with the damage variable q and with the applied displacement load. The sweeping-process inclusion dot-y in -N_C(y,q)(y), together with the linear compatibility and equilibrium constraints, encodes the whole lattice. The numerical scheme is the implicit catch-up map, whose fixed points at each time step are the discrete solutions; stability of these fixed points under iteration is what the paper uses to explain bifurcation and branching in softening.

What would settle it

Take the two-spring softening system of Section 5, set both springs to softening parameters with no complete failure, and choose the initial condition on the distributed saddle solution satisfying (106)-(107). Run the implicit catch-up iterations with a small time step; if for any step before failure the projection map has no fixed point or the iteration diverges or cycles, then the claimed sweeping-process formulation does not produce the evolution asserted for softening.

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Extended reading notes

Core claim

At the center is a finite-dimensional equivalence (Theorem 4.1): the full displacement-stress-damage system (LSM1)-(LSM6) for a lattice built of springs of any mixture—hardening, perfectly plastic, softening—holds if and only if the pair (y, q) of reduced stress and damage variables solves the state-dependent sweeping process (58)-(59), with the moving set C(y, q) the polytope of admissible stresses. The equivalence is two-sided, which is stronger than earlier sweeping-process formulations for perfectly plastic lattices. For the toy lattice of two springs in series, the paper solves the sweeping process explicitly and finds coexisting solutions when both springs soften: a distributed solutio

Load-bearing premise

The implied existence of a well-defined solution for softening lattices: at every time step the implicit catch-up map must have a fixed point that the iteration can actually reach, even though the standard existence condition behind the cited theorem fails exactly for softening springs.

Editorial extensions

If this is right

  • Changes in the plastic regime become parameter changes: beta crossing 1 turns a hardening spring into a softening one, so one solver and one theory cover all three cases.
  • Softening lattices admit multiple coexisting evolutions; the catch-up iteration's fixed points classify them as stable or saddle, giving a route to predict which branch a load path selects.
  • Shear bands with strain localization appear as emergent, symmetry-breaking events in symmetric lattices, and their orientation is controlled by lattice geometry and defects, so the model can reproduce ductile-fracture precursors.
  • Evolution can be computed reliably only up to the first spring's complete failure; past that point the sweeping process has no admissible continuation, matching the idea of material separation.
  • The two-spring toy model provides a minimal analytically solvable case where the nonsmooth bifurcation underlying softening can be studied rigorously.

Reading between the lines

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

  • If the promised local existence theorem for softening is established, the discrete model could serve as a regularized testbed in which shear-band width is controlled by lattice spacing, allowing mesh-dependence questions to be studied by refinement.
  • The observed relation between fixed-point stability and energy minima suggests a conjecture the paper does not prove: each iteration of the catch-up map decreases an underlying energy toward a local minimum; a cycling example would refute it, and a proof would unify variational and sweeping-process approaches.
  • The degeneracy in the plasticity-modulus formula means many parameter pairs produce the same observable modulus, so calibration from macroscopic stress-strain curves alone cannot identify them; extra physical or atomistic information would be needed to predict damage evolution and failure location.
  • A measure-valued state-dependent sweeping process for continuous media, which the paper sketches as conceivable, would complement the existing continuum variational theory for softening and could yield two-sided error estimates and optimal-control formulations for ductile-fracture initiation.
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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

4 major / 5 minor

Summary. The paper develops a state-dependent sweeping process formulation for quasi-static elasto-plastic lattice spring models whose springs may harden, soften, or be perfectly plastic. After analyzing a single spring (§3), it formulates the full lattice equations (LSM1)–(LSM6) and proves in Theorem 4.1 an equivalence between these equations and the state-dependent sweeping process (58)–(59), including reconstruction formulas (60)–(65). It then introduces implicit catch-up algorithms (§4.3–4.4), analyzes a two-spring toy model (§5) where non-uniqueness is asserted in Proposition 5.1, and presents numerical simulations of rectangular and triangular lattices (§6) showing shear-band localization in softening springs. The author is explicit that for softening and perfect plasticity the standard existence theorem (Theorem 1.2) does not apply because condition (93) fails, and that a local existence theorem is future work (§7).

Significance. If the central equivalence and the existence gap were closed, the paper would provide a unified finite-dimensional framework for rate-independent plasticity with softening, connecting the mechanical problem to a state-dependent sweeping process and enabling a rigorous study of non-uniqueness and shear-band bifurcations. The two-sided equivalence proof of Theorem 4.1 is detailed and self-contained, the toy-model analysis addresses a genuine non-uniqueness phenomenon, and the numerical experiments are accompanied by available code and videos. The derivation does not appear to presuppose its conclusions; the main weakness is not circularity but a missing existence theorem in the softening regime. As it stands, the advertised claim that the sweeping process 'solves' the evolution with softening outruns the proved results.

major comments (4)
  1. [§4.2, Remark 4.5, §7] The paper's central claim is that the state-dependent sweeping process solves the evolution for arbitrary mixtures including softening, but no existence theorem is proved in this regime. Theorem 1.2 requires the Lipschitz condition (93), and Remark 4.5 explicitly notes that (93) cannot hold when β_i ≤ 1 (softening/perfectly plastic springs). Section 7 defers a local existence result to future work. Since Theorem 4.1 only establishes equivalence for a given solution of (58)–(59), and Algorithm 1/2 is defined by fixed points of (91)/(99), the Section 6 simulations are currently not known to be solutions of the time-continuous problem. Please either prove existence of a fixed point for each time step under a suitable non-degeneracy assumption, or explicitly restrict the claims to the discrete fixed-point problem and state the absence of existence as a limitation in the abstract and conclusi
  2. [§5.3, Proposition 5.1] The analytical non-uniqueness result is only sketched. The proof says 'To save space we will omit complete derivations' and reports that equations were solved manually. Formulas (108)–(109) and the case analysis are central to the claim of coexisting solutions and to the bifurcation picture in Figure 8. Please provide a complete derivation (or put it in a supplementary file) with the inequality checks (110) and all case distinctions, rather than a sketch.
  3. [§6, Algorithm 2] The numerical experiments do not report the time-step size, the tolerance used in Algorithm 2, the maximum number of iterations, or the initial-guess policy; Remark 4.6 states that iteration may diverge when there are no stable fixed points. For the shear-band runs in Figs. 10–16, please report for every run that the convergence criterion was met at each time step, and provide data underlying the iteration-count plots. This is necessary to distinguish genuine solutions from algorithm failure and to support the claim that modifying the initial guess yields a second solution (Fig. 3 vs Fig. 10d).
  4. [§4.2, Theorem 4.1 proof] In the sweep-to-LSM direction the proof says 'we do not consider the state of complete failure' and uses implication (41). The theorem statement, however, does not state this exclusion. If the equivalence is intended to cover complete-failure states, the proof is incomplete; if not, the statement and the abstract's 'arbitrary placement' should be qualified. This is not merely cosmetic, since complete failure is reached in the softening simulations (Remark 3.1).
minor comments (5)
  1. [§1, §4.1.2, §6] Several typos and language issues: 'Given finctions' should be 'Given functions'; 'not not alter' in §4; 'modyfing' and 'incuded' in §6. A careful proofreading pass is needed.
  2. [§4.2, equations (67)–(74)] The notation for the moving set is very dense. A short table collecting the symbols in (55), (60), (70), (72), (73)–(74) would improve readability.
  3. [§5, Figures 8–9] The captions do not define the coordinate axes or the meaning of the wedges and red/black points. Please add explicit axis labels and a legend.
  4. [§6] The numerical setup is incomplete: no time-step size, tolerance, or iteration limits are given. These should be stated for reproducibility, especially because Algorithm 2's convergence is not guaranteed in the softening case.
  5. [§4.1] The paper relies heavily on [Gud+23] for definitions and algebraic properties. Please state explicitly in Section 4 which results are taken from that reference and which are new, so that the equivalence theorem can be checked without reading the full prior paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the sweeping-process formulation is a proven equivalence, and the shear-band/non-uniqueness outputs are not fitted inputs; the acknowledged missing existence theorem for softening is a completeness gap, not circularity.

full rationale

Theorem 4.1 gives a two-sided derivation: equations (LSM1)-(LSM6) are transformed into (58)-(59) by direct substitution of Hooke's law, compatibility, equilibrium and the normal-cone form of the flow rule, and the reverse direction reconstructs (LSM4)-(LSM5) and (LSM3) from the sweeping-process inclusion. Neither direction postulates the target behavior; the equivalence is algebraic and is proved in the text, with the lattice-construction material from [Gud+23] used as algorithmic/geometric background rather than as the source of the softening results. The plastic parameters (beta_i, gamma_i, etc.) are assigned before the simulations, and the non-symmetric shear bands (Fig. 10), the bifurcation of fixed points (Fig. 8, Observation 5.1), and the non-uniqueness in Proposition 5.1 are outputs of the model, not parameters fitted to those outputs. The self-citations [Gud+23, GM21] provide the lattice-rigidity and pseudoinverse identities and an external uniqueness theorem for perfect plasticity; they are not invoked to forbid alternatives or to define the softening sweeping process, so they do not constitute circularity. The paper explicitly flags that the standard existence condition (93) fails when beta_i <= 1 (Remark 4.5), that Algorithm 1 can diverge when no stable fixed point exists (Remark 4.6), and that a local existence theorem for softening is future work (Section 7); Proposition 5.1 is also only sketched. These are genuine correctness/completeness limitations of the softening claim, but they are not self-referential reductions of the derivation to its own inputs. The absence of circularity is therefore consistent with a nonzero-risk assessment on other grounds.

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

No new particles, forces, or dimensions are introduced; the 'damage variable' z is a standard internal variable in plasticity. The load-bearing postulates are the linear polytope constitutive ansatz, the small-strain rigidity assumptions, and, for softening, the unproven existence of sweeping process solutions.

free parameters (2)
  • spring constitutive parameters (E, a, b, yield stress) in numerical examples = e.g., E=2, a=0.4, b=-0.3, sigma_y0=0.4 for softening (113)
    Chosen by hand in Section 6 to place each spring in the softening, hardening or perfectly plastic regime. They are not fitted to data, but they select the regime that generates the observed shear bands.
  • lattice discretization and loading steps = e.g., 12x18 nodes, time step not specified
    The numerical demonstrations use specific lattice sizes and displacement load steps; no convergence study with respect to these is reported, leaving sensitivity of the shear band patterns unquantified.
assumptions (5)
  • standard math Convex analysis: normal cone calculus, projections, Moore-Penrose pseudoinverse properties (Lemma 2.1, Prop 4.1)
    Used throughout the derivation of the sweeping process, e.g., equations (14)-(17), (52)-(54).
  • domain assumption Constitutive ansatz: material state is (stress, internal variable) with the specific linear state-dependent polytope (22), (37)
    The paper restricts to linear isotropic hardening/softening with the particular set (22); the central claim holds only within this constitutive class.
  • domain assumption Quasi-static equilibrium and small-strain linearized geometric compatibility (LSM4)-(LSM6)
    The model is rate-independent and uses fixed reference geometry (43)-(44); large deformations are excluded.
  • ad hoc to paper Assumptions 1-4: kinematic determinacy (45)-(47), nonzero self-stress space (53), and regularity of loads (Assumption 3)
    These guarantee the algebraic reductions in Theorem 4.1; they are stated as assumptions rather than derived.
  • ad hoc to paper Existence of solutions to the state-dependent sweeping process in the softening case
    Theorem 1.2's condition (93) fails for softening (Remark 4.5). The paper assumes the implicit catch-up map has fixed points for the numerical simulations; a local existence theorem is deferred to future work (Section 7).

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Pith. "Pith review of Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs." pith.science (2026). https://pith.science/paper/MKT4YZQ7

@misc{pith2026250816333,
  author       = {Pith},
  title        = {Pith review of: Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MKT4YZQ7}},
  note         = {Machine review of arXiv:2508.16333}
}
read the original abstract

Plasticity with softening and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of the set of admissible stresses. We present a state-dependent sweeping process which solves the evolution of elasto-plastic Lattice Spring Models with arbitrary placement of softening, hardening and perfectly plastic springs. Using numerical simulations of regular grid lattices with softening we demonstrate the emergence of non-symmetric shear bands with strain localization. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions. These solutions correspond to fixed points in the implicit catch-up algorithm and we observe a discontinuous bifurcation with the exchange of stability of those fixed points.

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