Constitutive State-Space Modeling of Path-Dependent Plasticity: A Resolution-Consistent and Parallelizable Computational Framework

• Learned constitutive operators map strain–stress histories to local material responses, making stable state evolution, limited discretization sensitivity, and long-history efficiency relevant numerical requirements.Source: §1 · p. 2
• Fully data-driven recurrent models use learned activation or gating mechanisms for history-dependent plasticity, while later MSC variants pursued approximate resolution invariance (Bonatti and Mohr, 2021; Bonatti and Mohr, 2022).Source: §1 · p. 2
• The MSC processes increments sequentially, has only approximate resolution consistency for sufficiently small increments, and can gain diminishing returns from increasing state dimension after its encoder saturates.These limitations motivate treating increment magnitude directly in state evolution.Source: §1 · p. 3
• CSS decomposes each strain increment into magnitude and direction, using direction to drive the latent system and magnitude in its zero-order-hold discretization.The adapted recurrence preserves parallel prefix-scan evaluation during training.Source: §1 · p. 3
• The study assesses CSS against MSC for accuracy, computational scaling, data efficiency, model capacity, and learned-state structure at the constitutive-update level.Source: §1 · p. 3

2 Computational formulation of data-driven constitutive models

• The constitutive task is to predict current stress from the strain history, represented incrementally by the previous state and current strain increment.The state is postulated to retain salient history information sufficient to compute current stress.Source: §2 · pp. 3–4

2.1 Minimal State Cell (MSC) model

• The MSC feeds the previous state and strain direction through internal layers to produce a candidate state and a magnitude-dependent gate for updating the state.Source: §2.1 · p. 4
• The MSC then computes stress from the updated state using a linear output layer.Source: §2.1 · p. 5

2.2 Constitutive State Space (CSS) model

• CSS decomposes each increment into magnitude and direction, then routes the direction through an input gate and affine encoder into a stack of S5 blocks.Source: §2.2 · pp. 5–6
• Each S5 block has a linear recurrent core whose complex-valued state evolves by an affine ODE discretized with zero-order hold.Source: §2.2 · p. 6
• The zero-order-hold recurrence coefficients depend on strain-increment magnitude, tying latent evolution to the prescribed increment size.Source: §2.2 · p. 6
• CSS decodes the latent output nonlinearly between blocks and maps the final block output to updated stress; its memory comprises the states of all blocks.Source: §2.2 · p. 6
• For a fixed loading direction, zero-order hold gives the exact discrete evolution of the linear core over the increment magnitude, though full-network resolution robustness is assessed numerically because decoding is nonlinear.Source: §2.2 · p. 7

3.1 Materials

• The four benchmark materials are copper and dual-phase steel with J2 isotropic hardening, metallic foam with pressure-sensitive plasticity, and low-carbon steel with combined isotropic-kinematic hardening.Source: §3.1 · p. 7

3.2 Strain path sampling

• Each six-dimensional strain path has 126 points and is generated as a bounded random walk with a roughened, subdivided main path.Noise is added to intermediate points, and each of the 25 rough-path segments is further divided into five subsegments.Source: §3.2 · pp. 8–9

3.3 Dataset construction

• Single-element Abaqus/Explicit simulations generate stress histories paired with strain paths, and each material dataset contains 10,000 sequence pairs.The 10,000 pairs are split into 8,000 training, 1,000 validation, and 1,000 testing examples.Source: §3.3 · pp. 9–10

• Material-point evaluations under prescribed strain histories isolate constitutive-update behavior from mesh dependence, equilibrium iterations, contact, and other solver effects.Source: §4 · p. 10
• Controlled changes in strain-path discretization keep the underlying loading path and reference response unchanged, enabling direct assessment of resolution sensitivity.Source: §4 · p. 11
• Prediction quality is evaluated on held-out testing data using MSE and component-wise NRMSE, with dataset-level NRMSE averaged across stress paths.NRMSE is a dimensionless relative error based on component RMSE and mean absolute value.Source: §4 · p. 11

4.1.1 Hyperparameter study for the MSC model

• For foam, DP steel, and copper, MSC performance degrades sharply below seven state variables, while LC steel improves through 12, matching the physics-based state counts.Beyond seven variables, additional gains are limited for foam, DP steel, and copper.Source: §4.1.1 · p. 11, Figure 4 · p. 12
• MSC median MSE improves from 1.58 × 10−4 at w = 15 to 2.27 × 10−5 at w = 80; depth effects are non-monotonic, with d = 6 best.Models with 50,000–100,000 parameters reach a median MSE of 2.3 × 10−5, while compact models around 30,000 parameters trade size for accuracy.Source: §4.1.1 · p. 12, Figure 4 · p. 12

4.1.2 Hyperparameter study for the CSS model

• For CSS with one block, validation MSE improves up to s = 8 for copper, DP steel, and foam, and tends to stagnate beyond s = 16 for LC steel.These plateaus are near the respective data-generating models’ state counts of seven and 12.Source: Figure 5 · p. 14, §4.1.2 · p. 14
• At fixed state-variable count, more CSS blocks benefit all four materials; foam accuracy improves by more than one order of magnitude with four blocks instead of one.Within the 15,000–34,000 parameter constraint, latent input dimension h has only a mild effect; above a critical minimum p, block count has the strongest influence.Source: §4.1.2 · pp. 14–15
• CSS test loss decreases approximately linearly with parameter count in logarithmic space, indicating scaling behavior across the tested models.The large-model study reaches up to 105k parameters.Source: §4.1.2 · pp. 16–17, Figure 7 · p. 16
• In the larger CSS search, p = 30 is the computational optimum except for LC steel, where p = 40 gives the lowest MSE; h = 60 performs best, with limited gains over h = 30.Six blocks are favored for copper and DP steel, versus eight for metallic foam and mixed-hardening LC steel.Source: §4.1.2 · p. 16, Figure 8 · p. 17

4.2 Comparison of the MSC and CSS model prediction accuracy

• CSS validation losses are approximately one order of magnitude lower than MSC’s for three plastically incompressible materials, while both reach about 10^-5 for metallic foam.The comparison uses the same batch size and trains for up to 3,000 epochs.Source: §4.2 · p. 18, Figure 9 · p. 18
• CSS component-wise NRMSE medians fall between 10^-3 and 10^-2 across materials, while MSC metallic-foam relative errors are approximately 1%.For the other MSC von Mises materials, normal-stress relative errors are lower than shear-stress errors; their NRMSE profiles are qualitatively similar.Source: §4.2 · pp. 18–19, Figure 10 · p. 19, Figure 11 · p. 20
• On a 95%-quantile low-carbon-steel test path, both models agree satisfactorily with the reference, though MSC shows deviations of a few MPa near the sequence end.The path is selected from the testing set as a high-error case; the reference comes from a single-element FE analysis.Source: §4.2 · p. 20, Figure 12 · p. 21, Figure 13 · p. 22

4.3.1 Stationarity

• Both models keep stress constant when 50 zero-increment steps are appended to random test paths.The passage reports that stationarity is satisfied by construction and verified empirically.Source: §4.3.1 · p. 22, Figure 14 · p. 23

4.3.2 Resolution invariance

• Resolution testing resamples the original 125-segment paths by interpolation to datasets with 250 through 10,000 increments.Models are trained and evaluated across combinations of these resolutions without rerunning finite-element simulations.Source: §4.3.2 · p. 23
• MSC test error can rise by several orders of magnitude when testing paths are coarser than training paths.A model trained on 500-increment paths has MSE 6 × 10^-6 on matching tests but 2 × 10^-2 on the corresponding 125-increment representation.Source: §4.3.2 · p. 23, Figure 15 · p. 24
• CSS maintains test MSE of order 10^-6 across training and testing resolutions, attributed partly to its zero-order-hold discretization.The evaluation varies the representation of the same paths; CSS’s learned locally linear response also reproduces that response at coarser increments.Source: §4.3.2 · pp. 24–25, Figure 15 · p. 24

4.4 Parallel training efficiency

• CSS’s linear recurrence supports parallel evaluation through an all-prefix scan, whereas MSC’s state-dependent nonlinear recurrence must be evaluated sequentially.The scan has runtime scaling O(log(np)) in the number of processors; parallel sequence evaluation is useful when the full input sequence is known during training.Source: §4.4 · p. 25
• CSS training time scales approximately as n^0.62 with sequence length, versus n^0.98 for MSC, and its speed advantage grows from roughly 2.8× at 125 increments to 8× at 2,000.At 2,000 increments, MSC requires nearly 12 days while CSS takes under 1.5 days on the same dataset and hardware.Source: §4.4 · p. 26, Figure 16 · p. 26
• Neither architecture provides parallel-in-time acceleration during step-by-step finite-element analysis, where future increments are unavailable.The computational benefit described is reduced offline training cost for learning from long histories.Source: §4.4 · p. 27

4.5 Training-data efficiency

• MSC’s test error decreases with more training paths but is lowest at intermediate resolutions of 500–1,000 increments, not the longest sequences.With 8,000 paths, its MSE is 4.39 × 10^-6 at 1,000 increments and 6.50 × 10^-6 at 2,000; with 2,000 paths, it is 8.69 × 10^-5 at 125 increments versus 2.12 × 10^-5 at 1,000.Source: §4.5 · p. 27, Figure 17 · p. 28
• With 8,000 paths at 125 increments, CSS achieves 1.74 × 10^-6 MSE from 1 million pairs, versus MSC’s best 4.39 × 10^-6 from 8 million pairs.This is an 8× reduction in total training data at improved accuracy.Source: §4.5 · pp. 28–29, Figure 17 · p. 28
• CSS test error decreases monotonically with more training paths across the tested resolutions, reaching 1.30×10^-6 MSE at 1,000 increments with 8,000 paths.Across other tested resolutions at 8,000 paths, its MSE remains between 1.47×10^-6 and 1.74×10^-6.Source: §4.5 · p. 28, Figure 17 · p. 28

4.6 Physical interpretability

• Im χ2 correlates with equivalent stress (r = −0.86) and equivalent plastic strain (r = −0.91), while Re χ2 correlates with them at r = 0.74 and r = 0.80, respectively.Across the loading history, −Im χ2 tracks equivalent plastic strain with good fidelity.Source: §4.6 · p. 29
• PCA shows that the copper CSS trajectories occupy a seven-dimensional subspace, matching the number of state variables in the physics-based model; larger state spaces yield the same conclusion.Only the first seven components carry appreciable variance; this result persists for p ∈ [5, 6, 7, 8, 10, 12].Source: §4.6 · p. 29, Figure 18 · p. 30
• Among shear stresses, Re χ0 correlates with σ23 (r = −0.82), Im χ3 with σ12 (r = −0.72), and Im χ0 and Im χ3 with σ13 (r = −0.59 and 0.58); no state variable strongly correlates with normal stresses.Figure 18a reports Pearson correlations across 1,000 test paths for a model with eight real-valued state variables; panel 18b shows the Im χ2–equivalent-strain relationship for one path.Source: §4.6 · p. 29, Figure 18 · p. 30

5 Conclusions

• CSS decomposes strain increments into magnitude and direction, using the magnitude in zero-order-hold discretization to make updates stationary at zero increments and amenable to parallel prefix scans.This adapts state-space evolution to incremental constitutive mechanics rather than directly applying S5 to stress–strain sequences.Source: §5 · p. 30
• Across four benchmark materials, CSS matched or exceeded MSC accuracy, maintained low test errors across investigated training and testing resolutions, and avoided the MSC’s several-orders-of-magnitude error increase at coarser resolutions.For plastically incompressible materials, validation loss fell by up to approximately one order of magnitude.Source: §5 · p. 31
• CSS trained up to eight times faster than MSC for histories of up to 2,000 increments and required fewer strain–stress pairs for comparable or better accuracy.Source: §5 · p. 31
• Correlation and PCA analyses found state components associated with equivalent plastic strain and an effective learned-state dimensionality consistent with the underlying physics-based model.CSS improves accuracy, resolution robustness, and computational efficiency without prescribing phenomenological internal variables or their evolution equations.Source: §5 · p. 31
• Future work includes strain-rate and temperature dependence, finite-element boundary-value deployment, transfer and multi-task learning, and representative-volume-element surrogates.These extensions would broaden applicability or test scalability, data needs, and resolution robustness in settings not assessed here.Source: §5 · p. 31

CRediT authorship contribution statement

• The listed author contributions span conceptualization, methodology, software, validation, analysis, and writing.Source: CRediT authorship contribution statement · p. 31

Declaration of generative AI in the manuscript preparation process

• The authors used ChatGPT 5.5 for selected rephrasing and state that they reviewed and edited the output.Source: Declaration of generative AI in the… · p. 32