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A Gaussian Process State Space Formulation for Materials Constitutive Modeling: Application to Quasi-brittle Materials

Houlin Xu, Chenyang Li, K. T. Ramesh, Michael D. Shields

arXiv:2610.33175v1cs.CEcs.LGcs.NAmath.NAmath.PR

TL;DR

Prescribed, model-specific internal variables make history-dependent constitutive behavior difficult to calibrate from observations. The paper develops a GP state-space approach that learns compact history states from stress–strain data and tests it for quasi-brittle materials. Closed-loop rollouts reproduce held-out histories, and structural simulations transfer material-point-trained behavior to a boundary-value problem.

  • Problem

    Prescribed, model-specific history variables are not directly measurable and can make calibration ill-posed, with different parameterizations matching observed curves yet predicting differently on other paths.

    Source: §1 · p. 2

  • Method

    The model replaces prescribed history variables with a compact learned latent state whose mechanics-informed evolution uses equivalent strain and stress invariants.

    Source: §1 · p. 3, §3.4 · p. 13

  • Results

    Closed-loop rollouts reproduce held-out confinement-dependent loading, peak, softening, and residual responses, while structural simulations transfer material-point-trained updates to a boundary-value problem.

    Source: §8 · pp. 38–39

  • Takeaways & Limitations

    The framework connects learned history states with classical internal-variable mechanics and produces a local update suitable for finite-element calculations.

    Source: §8 · p. 39

    Training histories emphasize moderate-to-high confinement, so predictive accuracy in the low-pressure regime is expected to be weaker and warrants further investigation.

    Source: §7.3 · p. 37

Abstract

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Constitutive models for history-dependent, path-dependent materials typically encode this dependence in prescribed functional forms through internal variables such as damage or plastic strain that are phenomenological, model-dependent, and not directly observable. These rigid forms lack flexibility to match observations, make calibration potentially non-unique, and complicate uncertainty quantification. This paper develops a Gaussian-process state-space model (GP-SSM) that replaces the fixed form of the constitutive model with a flexible GP regression, and the prescribed internal variables with latent state variables learned directly from observable stress-strain histories. As a GP, the model is naturally endowed with a measure of uncertainty and can be constrained to satisfy physical principles. We demonstrate this framework for quasi-brittle materials under triaxial compression, where the latent variable evolution is driven by the current-state stress invariants and the increment of equivalent strain to reflect irreversible, pressure-sensitive history. A sparse multi-output GP then maps the latent state and current strain tensor to the six-component stress response, trained by variational sparse inference. The trained model is exported into Abaqus/Explicit through a user material subroutine. Four verification cases are reported: (1) offline rollout on validation and test histories; (2) a single element under changing confinement; (3) a frozen-memory unloading-reloading extension; and (4) a multi-element structural model compared against the Karagozian and Case Concrete (KCC) model. These results show that the learned state-space update reproduces held-out RVE-scale histories in closed-loop rollout, tracks evolving confinement, supports unloading without violating the irreversibility of the learned history state, and reproduces structural-level behavior consistent with KCC.

1. Introduction

Classical constitutive models represent path dependence with prescribed, often unobservable internal variables, making calibration difficult and potentially non-unique. The paper proposes a compact latent-state GP framework for quasi-brittle materials that learns from stress–strain histories and is designed for finite-element deployment.

1. Introduction

  • Prescribed, model-specific history variables are difficult to measure and can make calibration ill-posed, with different parameterizations reproducing the same stress–strain curves but diverging on other paths.Quasi-brittle response further involves evolving microcracking, frictional sliding, compaction, dilatancy, and confinement-dependent failure.Source: §1 · p. 2
  • The proposed GP-SSM learns a compact latent history state from stress–strain data and combines it with mechanics-informed evolution instead of prescribing phenomenological internal variables.Its low-dimensional memory is intended to separate persistent state size from the complexity of the update.Source: §1 · p. 3
  • Latent-state evolution uses equivalent strain and previous-step stress invariants, while a sparse multi-output GP maps the state and current strain to six stress components.The GP is trained by stochastic variational inference over inducing points and supplies probabilistic uncertainty information for the stress map.Source: §1 · pp. 3–4
  • The model is evaluated as a local, incremental finite-element material update, with verification focused on concrete-like triaxial data and potential applicability to other pressure-sensitive quasi-brittle materials.The implementation is presented as an experiment-to-deployment pipeline and compared with KCC in the paper’s verification cases.Source: §1 · p. 4

2. Background: constitutive modeling paradigms

The section contrasts constitutive approaches by the structure they assume and the source of their stress–strain relations. It then uses classical quasi-brittle models to motivate learning the history-sensitive stress update without prescribing its boundary functions.

2.1. Mechanistic, phenomenological, and data-driven modeling

  • More assumed structure can reduce data needs and support extrapolation, but makes predictions depend more strongly on that structure being correct.Mechanistic models derive relations from an explicit subscale mechanism; phenomenological models postulate and calibrate them, while data-driven models learn them from data.Source: §2.1 · pp. 4–5

2.2. Constitutive models for quasi-brittle materials

  • Most established quasi-brittle models use an elastic predictor followed by a boundary test and, when needed, a correction that updates history variables.Their elastic operator is comparatively general; model-specific physics and calibration are concentrated in the boundary function and its evolution.Source: §2.2 · pp. 6–7

2.3. A representative example: the KCC model

  • KCC defines pressure-dependent yield, maximum, and residual surfaces, then interpolates between them using a fitted damage-like history variable.The interpolation governs hardening up to peak stress and softening thereafter; rate sensitivity adds an empirical dynamic-increase factor.Source: §2.3 · pp. 7–8

2.4. The phenomenological cost of designing the boundary function

  • KCC’s multiple fitted surfaces, damage accumulation, interpolation, and rate dependence require extensive calibration and tie extrapolation to assumed functional forms.Microplane and lattice-facet models use different admissible-stress boundaries but retain the elastic-trial-then-correction logic.Source: §2.4 · p. 8, §2.3 · p. 8
  • The proposed data-driven model learns stress from current strain and internal state, retaining a history-dependent structural skeleton while omitting prescribed boundary functions and return mapping.Its state descriptors are frame-invariant and its accumulated history is irreversible.Source: §2.1 · p. 6, §2.4 · p. 9

3. Latent-state constitutive approach

The approach learns a low-dimensional history state and an incremental stress update from observed material-point stress–strain sequences. Its Markov state supports local solver updates, while invariant descriptors specialize the transition to pressure-sensitive quasi-brittle response.

3.1. Problem statement

  • The learning problem is to infer an incremental constitutive update from finite material-point stress–strain sequences, without measured internal history or prescribed strength, flow, or hardening laws.The model must operate independently at each finite-element integration point using locally available information.Source: §3.1 · pp. 9–10

3.2. The use of a latent state

  • A learned latent state compresses loading history because identical current strains can correspond to different histories and future responses.The state is not uniquely identified: invertible reparameterizations can preserve predictions, so individual coordinates lack a one-to-one physical interpretation without further evidence.Source: §3.1 · p. 10, §3.2 · p. 11

3.3. Markov structure and probabilistic factorization

  • The latent state is Markovian by construction, allowing it and the previous-step stress to serve as a fixed-size solver state rather than retaining the full strain history.The transition is deterministic at deployment; training-time increment-scaled Gaussian perturbations are marginalized during rollout but are not propagated in the deployed update.Source: §3.3 · pp. 11–12

3.4. Specialization to quasi-brittle materials: invariant descriptors

  • For quasi-brittle triaxial loading, the transition uses equivalent strain and previous-step stress invariants to encode loading progression, pressure, and deviatoric intensity.These invariant inputs avoid prescribing classical pressure-dependent functional forms; the GP stress map itself uses rotational augmentation to encourage approximate frame invariance rather than enforcing it exactly.Source: §3.4 · p. 13

4. Gaussian-process state-space constitutive formulation

The GP-SSM pairs an incrementally updated latent representation of material history with a probabilistic stress map, combining an inexpensive state transition with uncertainty-aware stress prediction.

4. Gaussian-process state-space constitutive formulation

  • A deterministic neural transition updates the latent state, while a Gaussian process maps that state and current strain to stress, retaining posterior variance in the stress map.The transition is evaluated at every integration point and increment; this division balances speed with uncertainty quantification.Source: §4 · p. 13

4.1. Latent state evolution

  • The latent increment is driven by deformation progress and a frame-invariant descriptor vector, rather than wall-clock or explicit-solver time.The formulation assumes rate independence and frame-invariant descriptors; its rate-times-increment form also supports different sampling intervals without retraining the transition.Source: §4.1 · pp. 14–15

4.2. Specialization to quasi-brittle materials

  • For the quasi-brittle specialization, equivalent-strain increments drive a nonnegative latent-state rate informed by stress invariants, so the learned history state accumulates during loading.The nonnegative rate represents accumulating damage, compaction, and microcrack growth in this application.Source: §4.2 · p. 15

4.3. Sparse Gaussian-process stress map

  • A sparse variational GP maps latent state and strain to six stress components, using shared inducing locations but separate output kernels and variational distributions.The six outputs have no explicit cross-output covariance, so residual covariance beyond their shared inputs is not represented.Source: §4.3 · p. 16
  • The variational objective balances probabilistic stress fit against posterior complexity, while rotationally augmented copies are not counted as independent physical observations in the KL term.The sparse formulation reduces the leading full-pass cost to O(NM^2 + M^3), compared with O(N^3) for exact GP training.Source: §4.3 · pp. 17–18

4.4. Role of uncertainty

  • Predictive variance separates GP epistemic uncertainty, variation from stochastic latent-state evolution, and learned observation noise.The epistemic term reflects support in joint state–strain space and is the component reducible by collecting additional loading histories.Source: §4.4 · p. 19
  • Finite-element deployment evaluates the GP posterior mean without posterior covariance propagation, so reported finite-element results are not uncertainty-propagating analyses.The framework enables uncertainty propagation, but that analysis is left for future work.Source: §4.3 · p. 18, §4.4 · p. 20

4.5. Framework overview

  • In closed-loop prediction, the updated latent state carries loading history forward and predicted stress supplies the invariants used at the next step.The transition network and stress map are trained jointly, while deployment uses the deterministic latent update and GP posterior mean.Source: §4.5 · p. 20, Figure 1 · p. 20

5. Data Processing and Training

The training pipeline preprocesses and normalizes triaxial histories, then uses strain-progress-based latent evolution, temporal downsampling, and rotational augmentation. Training selects checkpoints by autoregressive validation RMSE before evaluating complete histories and verifying deployment in Abaqus/Explicit.

5.1. Stress–strain histories and normalization

  • Histories are transient-trimmed, initialized with a zero reference state, truncated by equivalent strain, and normalized to establish a common initial condition and limit extrapolation-dominated training.Shared scaling across tensor components preserves their relative structure under rotations, unlike independent component-wise normalization.Source: §5.1 · p. 21

5.2. Temporal downsampling

  • Latent evolution is learned as a rate with respect to equivalent-strain progression, so differently sampled recordings of the same path query and integrate the same rate law.This makes the response time-resolution-independent and permits heterogeneous, unevenly sampled observations without resampling to a common time base.Source: §5.2 · p. 21
  • Training and nonautoregressive evaluation use 100 evenly indexed points per sequence, while optimization-time autoregressive validation uses 500 points.Downsampling follows plateau truncation; the retained points preserve the elastic branch, peak, and post-peak plateau, while final offline evaluation uses complete histories.Source: §5.2 · p. 22, Figure 2 · p. 22

5.3. Rotational augmentation

  • For isotropic RVE-scale material, 20 fixed rotations augment training histories to encourage approximate rotational consistency in the tensorial stress map.Rotational augmentation is limited to training; validation and test histories remain unrotated, and a fixed anisotropic microstructure falls outside the assumption.Source: §5.3 · pp. 23–24

5.4. Training protocol

  • Optimization minimizes the sparse multi-output GP’s negative ELBO jointly with neural transition parameters, using cross-validation folds, early stopping, and stochastic latent-state rollouts.The best checkpoint is chosen by autoregressive validation RMSE in normalized stress units.Source: §5.4 · p. 24
  • Checkpoint selection uses autoregressive validation RMSE; the selected model is then evaluated on complete validation and test histories before Abaqus/Explicit deployment verification.The paper frames the ensuing work as verification of the learned material-point update and its Abaqus implementation, from a single integration point to a multi-element simulation.Source: §5.4 · p. 24

6. Results

The four verification cases evaluate complete-history prediction, finite-element deployment under changing confinement, an unloading–reloading extension, and a multi-element structural response against KCC. The tests use synthetic KCC triaxial histories for training and assess both held-out rollouts and finite-element behavior.

6. Results

  • Training uses 20 synthetic KCC triaxial histories spanning 2–40 MPa confinement, with 12 histories for training, five for validation, and three for testing.KCC internal variables are not training targets; training sequences are reduced to 100 points, while final validation and test evaluations use complete preprocessed histories.Source: §6 · p. 25
  • The selected epoch-3100 checkpoint uses EMA-smoothed validation RMSE, which is evaluated in closed-loop rollout, whereas training RMSE uses reference previous-step stress invariants.The normalized validation RMSE at selection is approximately 0.0347.Source: §6 · p. 25, Figure 3 · p. 26

6.1. Case 1: Offline full-sequence rollout evaluation

  • Full-sequence held-out rollouts had validation RMSE of 1.86–4.04 MPa and test RMSE of 1.73–3.20 MPa, with larger low-confinement errors and smoother sharp post-peak drops.The results support interpolation across unseen confinement levels within the available triaxial loading family, not generalization to other loading paths.Source: §6.1 · p. 26, Figure 4 · p. 27

6.2.1. Deployment protocol

  • The Abaqus/Explicit deployment exports trained parameters to a VUMAT, which updates strain, invariants, latent state, GP stress, and stored material-point state locally.The exported representation contains 10,994 scalar parameters; the local state avoids global history storage.Source: §6.2.1 · p. 28, Figure 5 · p. 28

6.2.2. Case 2: Single-element compression under changing confinement

  • Under confinement rising from 26 to 31.2 MPa during axial loading, the VUMAT response closely matched KCC through peak and softening, while latent components increased monotonically and saturated.The confining-pressure path was not included in training; z2 saturated earlier than z1.Source: §6.2.2 · pp. 29–30, Figure 6 · p. 30

6.3. Case 3: Unloading–reloading extension

  • During unloading and reloading below the historical maximum equivalent strain, the extension freezes latent state and stress invariants, uses linear elastic response, and resumes the GP-SSM update beyond that maximum.The elastic branch uses the initial stiffness, without stiffness degradation or hysteretic dissipation; Case 3 tests latent-state irreversibility rather than calibrated cyclic response.Source: §6.3 · pp. 31–32

6.4. Case 4: Structural application

  • 1.4% higher peak force and about 4.3% lower reaction force at 25 mm distinguish the GP-SSM structural response from KCC, while both models localize deformation within roughly the first 400 mm.The GP-SSM predicts a slightly more intense localization band; the force–displacement curves otherwise closely track KCC.Source: §6.4 · pp. 33–34, Figure 8 · p. 34

7. Discussion

The discussion frames the latent state as a learned constitutive history variable: it preserves incremental memory without prescribing a physical damage or plasticity variable. It also distinguishes this mechanics-informed, compact memory from recurrent surrogates and identifies scope limits and extensions.

7.1. Latent state as learned constitutive history

  • The latent state is a learned history variable sufficient for material-response prediction, not a uniquely identifiable physical damage measure.Its monotonic evolution with equivalent strain and dependence on stress invariants make it a low-dimensional summary of irreversible loading history; frozen-memory elastic unloading further distinguishes it from a memoryless strain-to-stress regressor.Source: §7.1 · pp. 35–36

7.2. Relation to recurrent neural constitutive models

  • Unlike conventional recurrent surrogates, the formulation decouples transition-network capacity from persistent memory and retains only a two-dimensional latent state.The transition is driven by invariants and scaled by equivalent-strain increment, following internal-variable evolution structure more closely than a generic recurrent update.Source: §7.2 · p. 36
  • The latent coordinates are not unique physical variables, but equivalent-strain progression, monotonic accumulation, and I1 and J2 conditioning give them mechanically structured evolution.The current strain tensor remains a separate input to the stress map, preserving tensorial information for stress prediction.Source: §7.2 · p. 36
  • The latent dimension should be the smallest one that makes the update adequately Markovian; too few states can conflate histories, while excess states risk redundancy and overfitting.The number of variables in a conventional model can guide an initial choice but does not imply a one-to-one correspondence.Source: §7.2 · p. 36
  • A sparse multi-output GP separates history evolution from stress prediction and retains epistemic uncertainty for assessing support, extrapolation, and experiment selection.The explicit VUMAT evaluates the posterior mean for inexpensive local updates; the epistemic component ranks proposed state–strain queries, while likelihood noise is a fixed pedestal.Source: §7.2 · pp. 36–37
  • The learned GP reports where data provide weak support, unlike the calibrated KCC alternative, whose calibration ambiguity is not represented in its predictions.The discussion states that adding experiments improves the learned formulation in unsupported regions, whereas KCC calibration ambiguity reflects its assumed functional form.Source: §7.2 · p. 37

7.3. Limitations and future work

  • The low-confinement lateral response is less stable and less specifically trained, while the unloading rule is a minimal, non-learned elastic closure.It freezes latent memory and reuses initial elastic stiffness, so it does not capture nonlinear unloading such as hysteresis or crack-related stiffness softening.Source: §7.3 · p. 37
  • Future extensions could connect latent variables to multiscale mechanisms or adapt their evolution to rate-dependent materials using time, strain rate, temperature, and broader training histories.The proposed rate-dependent extensions target creep, stress relaxation, and rate-sensitive yielding.Source: §7.3 · pp. 37–38

8. Conclusions

The conclusions present a mechanics-informed GP-SSM that learns constitutive history from observable stress–strain data and supports local finite-element updates. The reported verifications span held-out histories, changing confinement, unloading–reloading, and structural response, while identifying distinct local and global validation targets.

8. Conclusions

  • The framework learns latent history from observable stress–strain sequences and builds an incremental constitutive model without prescribed or measured damage, plastic-strain, or hardening variables.Its two-dimensional state evolves through a nonnegative rate scaled by equivalent-strain progression and conditioned on I1 and J2, while the current strain tensor remains in the stress map.Source: §8 · p. 38
  • Closed-loop rollouts reproduce held-out confinement-dependent loading, peak, softening, and residual responses, with wider predictive intervals in more difficult low-confinement cases.The GP posterior supplies an internal indicator of data support and extrapolation risk alongside point predictions.Source: §8 · p. 38
  • The exported Abaqus/Explicit VUMAT tracks pressure increase and reproduces the reference response through peak and post-peak softening under a changing-confinement path not included in training.The deployed evaluator requires no machine-learning library and advances local state variables at integration points.Source: §8 · pp. 38–39
  • Freezing latent memory below the historical maximum equivalent strain preserves irreversible history while allowing stable unloading–reloading.The structural simulation also transfers a material-point-trained update to a boundary-value problem without training on structural force–displacement data.Source: §8 · p. 39
  • The framework's transition law, probabilistic stress map, and finite-element deployment can be extended independently and adapted to other history-dependent material classes.The conclusions describe retraining on corresponding stress–strain histories and adapting invariant conditioning to each class's physics.Source: §8 · p. 39
  • Global force–displacement agreement does not require matching the element-level trajectory ensemble, and localization is treated separately through regularization or nonlocal formulations.The conclusions identify global structural response and local trajectory distributions as complementary validation targets.Source: §8 · p. 39

Appendix A. Implementation details

The appendix records implementation settings and the exported-model file and diagnostic workflow for reproducibility.

Appendix A. Implementation details

  • The implementation uses a two-dimensional latent state, a transition-network hidden width of 64, 150 sparse-GP inducing points, two cross-validation folds, and a 12/5/3 train/validation/test sequence split.The finite-element state stores two latent variables, accumulated strain components, and previous-step stress invariants.Source: Appendix A · p. 40, Table A.2 · p. 40
  • The VUMAT reads exported parameters from a plain-text file and can write diagnostics for comparing Abaqus field output with internal GP-SSM material-point histories.Compared fields include logarithmic strain, stress, and state variables.Source: Appendix A · p. 41

Appendix B. Normalization statistics

The appendix gives normalization statistics for the exported model, including shared strain and stress scales and scalar descriptor statistics.

Appendix B. Normalization statistics

  • The exported-model normalization uses shared component scales sε = 0.023302 and sσ = 34.0975 MPa, with µεeq = 0.032918, sεeq = 0.052752, µI1 = −84.3490 MPa, and sI1 = 75.3799 MPa.Source: Appendix B · p. 41

Appendix C. Non-uniqueness of strength-surface calibration

The KCC maximum-surface fit reproduces twenty M7 peak strengths with a 0.69 MPa RMS residual, yet its parameters are poorly determined. Nearly order-of-magnitude variation in a0 fits similarly, with predictions diverging outside the calibrated range; the GP-SSM instead infers parameters from stress–strain sequences and reports posterior variance where data are lacking.

Appendix C. Non-uniqueness of strength-surface calibration

  • The appendix demonstrates that even a favorable three-parameter calibration to a directly observable target without time integration is ill-posed, not that KCC cannot represent the material.The example fits the maximum surface to twenty triaxial peak strengths generated with the M7 microplane model.Source: Appendix C · p. 42
  • A 0.69 MPa RMS residual fits twenty M7 peak strengths, but nearly order-of-magnitude variation in a0 yields similarly good fits and widely divergent predictions outside the tested pressure range.The objective has a long correlated valley rather than an isolated minimum; more measurements at the same pressures do not resolve it because the residual reflects model-form discrepancy.Source: Figure C.9 · p. 41, Appendix C · p. 42
  • Unlike a manually calibrated fixed-form surface, the GP-SSM infers its parameters by variational inference from observed stress–strain sequences and reports posterior variance where data do not support the model.The GP-SSM has substantially more parameters; its distinction is the inference approach and uncertainty reporting, not a smaller parameter count.Source: Appendix C · p. 42

Appendix D. Training Data Generation

Training histories were generated as controlled single-element Abaqus/Explicit simulations using KCC, with varied confinement and a common two-stage loading protocol. The dataset contains 20 triaxial compression histories spanning 2–40 MPa, partitioned by complete histories into training, validation, and test sets.

Appendix D. Training Data Generation

  • The reference responses come from single-element Abaqus/Explicit simulations using KCC Release III through a user-defined VUMAT, providing controlled stress–strain histories across confinement levels.The one-element setup uses a single integration point to provide a material-point history for each prescribed loading condition.Source: Appendix D · pp. 42–43
  • The reference material parameters are held fixed across simulations, and strain-rate enhancement is disabled, so the responses exclude rate-dependent strength amplification.The specified density is 2321 kg/m3, unconfined compressive strength is 45.2 MPa, Poisson’s ratio is 0.19, and bulk and shear moduli are 16.37 GPa and 12.79 GPa.Source: Appendix D · p. 43
  • Each history applies smoothly increasing equal compressive pressures to reach target confinement, then maintains confinement while imposing axial compression at a nominal strain rate of 0.25 s−1.The second-stage axial velocity rises to 25.4 mm/s by 0.30 s and remains constant; total simulation duration is 0.60 s.Source: Appendix D · p. 43
  • Training uses only the six-component Green–Lagrange strain and total Cauchy stress histories, not KCC internal variables as targets.The recorded components are ordered 11, 22, 33, 12, 23, 13, with tensorial shear strains.Source: Appendix D · p. 43
  • Twenty KCC triaxial-compression histories spanning confining pressures from 2 to 40 MPa cover initial loading, peak strength, and subsequent softening toward residual response.Complete histories were split into 12 training, five validation, and three test histories; validation and test confinement levels are absent from training.Source: Appendix D · pp. 42–43

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