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

cs.CEOct 23, 2026

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

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

Organizations: Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, MD, USA · Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD, USA · Department of Civil and Systems Engineering, Johns Hopkins University, Baltimore, MD, USA

Abstract

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.

Figures & tables

Figure 1: Schematic of the latent-state transition and stress emission in the GP-SSM constitutive update: the latent state

\bm{z}_{t-1}

is advanced through the mechanics-informed transition of Eq. ( 16 ) to obtain

\bm{z}_{t}

, which together with the current strain

\bm{\varepsilon}_{t}

is mapped by the Gaussian-process stress map

g

to the stress

\bm{\sigma}_{t}

Figure 2: Effect of temporal downsampling on representative axial stress–strain training sequences, shown as

\sigma_{11}

against

\epsilon_{11}

at several confining pressures. Black lines denote the full-resolution sequence after plateau truncation; red markers denote the

N_{\mathrm{tpts}}=100

points retained by evenly spaced index selection. The elastic branch, peak, and post-peak plateau are all preserved in the reduced sequence.

Figure 3: Training dynamics of the GP-SSM (representative fold): (a) Negative-ELBO training loss versus epoch; (b) Train and validation RMSE evaluated every 100 epochs, where train RMSE is computed non-autoregressively on $N_{\mathrm{tpts}}=100$ downsampled points per sequence and validation RMSE uses closed-loop autoregressive rollout on $N_{\mathrm{vpts}}=500$ points per sequence, with predicted stress fed back as the invariants at each step. The marked point (epoch 3100) indicates the checkpoint selected by early stopping on the EMA-smoothed validation RMSE.

Figure 4: Offline GP-SSM predictions of the axial stress–strain response under triaxial compression with reported levels of confinement, shown as $|\sigma_{11}|$ against $|\epsilon_{11}|$ for each confining pressure (labeled in MPa). Black dashed lines denote the reference RVE-scale stress histories. Solid blue lines denote the GP-SSM posterior mean, and the shaded bands show the 68% and 95% credible intervals. Training sequences (white background) are evaluated non-autoregressively using the reference previous-step stress invariants. Validation and test sequences (yellow background) are evaluated in closed-loop autoregressive rollout over their complete preprocessed time histories, without temporal downsampling, with predicted stress fed back to compute the invariants at the next step. RMSE and mean predictive standard deviation are reported for each sequence.

Figure 5: End-to-end pipeline from experimental stress–strain data and GP-SSM training to parameter export, VUMAT implementation, and finite element deployment in Abaqus/Explicit.

State variable index	Meaning
1–2	latent state components $z_{1},z_{2}$
3–8	accumulated six-component strain vector
9	previous first stress invariant $I_{1}$
10	previous deviatoric invariant $J_{2}$

Table 1: State variables used by the Abaqus/Explicit VUMAT implementation.

Figure 6: Changing-confinement loading path and response: (a) the two-step loading protocol, with confining pressure $P$ ramped from $0$ to $26~\mathrm{MPa}$ in Step 1 and continuing to rise smoothly from $26$ to $31.2~\mathrm{MPa}$ during the axial displacement/velocity loading of Step 2. (b) axial stress–strain response $|S_{11}|$ versus $|E_{11}|$ , colored by the instantaneous confining pressure, comparing the GP-SSM model against the KCC reference model, with confining-pressure values annotated at representative points along the curve. (c) lateral stress evolution versus time, showing close agreement between the GP-SSM and the reference KCC model as the confining pressure rises through both loading steps. (d) evolution of the latent state components $z_{1}$ and $z_{2}$ with equivalent strain $\bar{\varepsilon}$ , showing monotonic saturation consistent with accumulating material history.

Figure 7: Single-element response under fixed confinement with two unloading–reloading excursions: (a) The two-step protocol, with confining pressure $P$ ramped from $0$ to $33~\mathrm{MPa}$ in Step 1 and held constant in Step 2 while an axial displacement $u$ is imposed and twice reversed. (b) Axial response, $|S_{11}|$ against $|E_{11}|$ showing two elastic unloading events. The solid line is the GP-SSM prediction colored by current confining pressure. The dashed line is the KCC reference under the same load path. (c) Lateral stress versus time, showing the ramp to $33~\mathrm{MPa}$ and subsequent maintenance of confinement, with brief fluctuations during unloading–reloading. (d) Latent components $z_{1}$ and $z_{2}$ versus equivalent strain, remaining fixed during unloading–reloading below the historical maximum and resuming evolution once that maximum is exceeded.

Figure 8: Structural-level finite element response of the multi-element cylindrical model under $24~\mathrm{MPa}$ confinement, comparing the GP-SSM-driven simulation against the KCC reference solution. (a) axial reaction force versus imposed end displacement, with an inset showing the deformed mesh. (b) element-level axial stress–strain trajectories for both models, illustrating the heterogeneous loading and unloading histories that develop across the cylinder. (c) histogram of the final axial strain $|E_{11}|$ at $25~\mathrm{mm}$ end displacement over all elements for the GP-SSM (blue) and KCC (orange) simulations. (d) layer-averaged final axial strain $|E_{11}|$ as a function of axial distance from the localizing end, showing that both models concentrate the deformation within roughly the first $400~\mathrm{mm}$ of the cylinder and recover the same elastically unloaded plateau beyond it, with the GP-SSM predicting a slightly more intense and more sharply terminated localization band.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Value
Number of stress–strain histories ( $N_{\text{hist}}$ )	20
Train/validation/test split ( $N_{\text{train}}/N_{\text{val}}/N_{\text{test}}$ )	12/5/3 sequences
Cross-validation folds	2
Latent dimension ( $d_{z}$ )	2
Transition MLP architecture ( $f_{\theta}$ )	$5\rightarrow 64\rightarrow 64\rightarrow 64\rightarrow 2$
Transition activation	tanh hidden layers, softplus output

Appendix

Table 2: Implementation settings used in the present study.

Figure 9: Non-uniqueness of the KCC maximum-surface calibration. (a) Fits of $\Delta\sigma_{m}(p)$ (Eq. ( 2 )) to twenty M7 peak triaxial strengths (circles), each fixing $a_{0}$ at the value shown and refitting the rest; parentheses give the RMS residual relative to the optimum ( $0.69$ MPa). Values of $a_{0}$ spanning nearly an order of magnitude fit the data almost equally well, separating only below about $10$ MPa. (b) RMS residual over the $(a_{0},a_{2})$ plane, with the optimum (star) and the contour at twice its residual (red), which encloses the fits of (a): a long correlated valley rather than an isolated minimum, since $a_{0}$ and $1/a_{2}$ are constrained mainly through the asymptote $a_{0}+1/a_{2}$ .

State Space Models

State space models represent sequences through evolving latent states, enabling efficient temporal modeling, system identification, and forecasting.

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Neural Surrogate Modeling State Space Models

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Data-driven constitutive models for path-dependent plasticity are commonly formulated using nonlinear recurrent neural networks, whose sequential state evolution limits parallel training and whose predictions may depend on the discretization of the applied strain path. We introduce a Constitutive State Space (CSS) model that reformulates structured state-space dynamics as an incremental constitutive operator. The strain increment is decomposed into magnitude and direction: the loading direction drives the latent state-space system, while the increment magnitude enters the zero-order-hold discretization of its continuous-time linear recurrence. This mechanics-tailored construction guarantees stationarity under zero increments, strongly reduces sensitivity to strain-path resolution, and retains the parallel-scan structure of S5 for efficient training on long constitutive histories. The CSS and Minimal State Cell (MSC) architectures are compared for four multiaxial path-dependent material models including isotropic J2 plasticity, pressure-sensitive foam plasticity, and combined isotropic-kinematic hardening. CSS matches or exceeds the prediction accuracy of the MSC, including one order of magnitude lower validation losses for the plastically incompressible materials. Importantly, CSS maintains low errors across large changes in strain-path discretization, whereas the MSC error increases substantially when evaluated at coarser resolutions than used for training. CSS trains substantially faster and requires fewer strain-stress pairs to attain comparable or better accuracy. Analysis of the learned state further reveals latent structure consistent with the dimensionality of the underlying physical constitutive models. These results establish mechanics-tailored structured state-space dynamics as a computational framework for efficient and discretization-robust data-driven constitutive modeling.

Abstract

State Space Models

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