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Discovering plasticity models without stress data

Moritz Flaschel, Siddhant Kumar, Laura De Lorenzis

arXiv:2202.04916v1cs.CE

TL;DR

Plasticity discovery is challenging because existing data-driven methods can require stress labels, while stress is difficult to measure and path-dependent behavior requires histories of loading data. EUCLID uses sparse regression and physics constraints to discover plasticity models from displacement and force data without stress measurements. The paper demonstrates interpretable model discovery, with performance depending on whether the experiment sufficiently excites relevant deformation modes.

  • Problem

    Data-driven material modeling methods often require stress-strain labels that are difficult to obtain experimentally, and plasticity adds dependence on loading history.

    Source: Introduction · pp. 1–2

  • Method

    EUCLID uses sparse regression over an interpretable candidate-model library and physics constraints to discover plasticity models from displacement and reaction-force data without stress data.

    Source: Introduction · p. 2, Optimization problem · p. 5

  • Results

    EUCLID discovers interpretable plasticity models from displacement and net reaction-force data without stress data, as demonstrated for plasticity alongside prior hyperelasticity work.

    Source: Discussion · p. 11

  • Takeaways & Limitations

    EUCLID offers a physics-constrained, data-efficient alternative to supervised methods and enables parsimonious model selection rather than an a priori choice of plasticity model.

    Source: Discussion · p. 11

    For the tested Tresca model, comparatively low shear deformation in the loading setup impaired shear-region fitting; adding shear deformation improved results.

    Source: §1.4 · p. 18

Abstract

from arXiv · show

We propose a new approach for data-driven automated discovery of material laws, which we call EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), and we apply it here to the discovery of plasticity models, including arbitrarily shaped yield surfaces and isotropic and/or kinematic hardening laws. The approach is unsupervised, i.e., it requires no stress data but only full-field displacement and global force data; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a potentially large catalogue of candidate functions; it is one-shot, i.e., discovery only needs one experiment. The material model library is constructed by expanding the yield function with a Fourier series, whereas isotropic and kinematic hardening are introduced by assuming a yield function dependency on internal history variables that evolve with the plastic deformation. For selecting the most relevant Fourier modes and identifying the hardening behavior, EUCLID employs physics knowledge, i.e., the optimization problem that governs the discovery enforces the equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity promoting regularization is deployed to generate a set of solutions out of which a solution with low cost and high parsimony is automatically selected. Through virtual experiments, we demonstrate the ability of EUCLID to accurately discover several plastic yield surfaces and hardening mechanisms of different complexity.

Introduction

Existing data-driven material models often require stress-labeled data and can be black boxes, while plasticity adds dependence on loading history. EUCLID addresses this by discovering plasticity models from displacement and reaction-force data using sparse candidate libraries and physics constraints.

Introduction

  • Existing supervised or curve-fitting methods require strain-stress pairs, although experimental stress tensors are difficult to measure and force data provide only incomplete stress projections.The passage also notes that generating tensorial training pairs through multiscale simulation remains computationally expensive.Source: Introduction · p. 1
  • Plasticity makes learning more challenging because stress depends on material-point history, and supervised approaches need stress-strain paths across many loading histories.The passage notes that such methods often fail to extrapolate beyond training-path duration.Source: Introduction · pp. 1–2
  • EUCLID discovers interpretable plasticity models from displacement and reaction-force data from a single experiment, without stress data or an a priori material-model choice.The framework selects relevant features from a library of candidate material models using physics constraints.Source: Introduction · p. 2
  • The candidate library expands the yield function with Fourier terms and represents isotropic and kinematic hardening through history variables evolving with plastic deformation.The history variables are the accumulated plastic multiplier and back stress.Source: Introduction · p. 2

Results

EUCLID represents plasticity with a Fourier-based yield-function library and hardening variables, then uses equilibrium constraints and sparse optimization to identify models. In virtual benchmarks, it exactly identifies several models without noise and produces accurate surrogates when exact recovery fails, with important exceptions.

Material model library

  • A Fourier-series yield-function library represents smooth yield surfaces of arbitrary shape, while Voce isotropic and Armstrong-Frederick kinematic laws model hardening.The yield function depends on stress, accumulated plastic multiplier, and back stress; isotropy requirements exclude selected sine and cosine terms.Source: Material model library · pp. 2, 4

Optimization problem

  • EUCLID fits candidate material parameters by enforcing weak-form equilibrium against displacement and reaction-force data, including bulk and loaded-boundary force balance.The two-dimensional inverse problem uses data to discover plasticity models applicable to three-dimensional stress states.Source: Optimization problem · pp. 4–5
  • An ℓp penalty promotes sparse yield-function parameters; multiple initial guesses and threshold-based selection favor low-cost, parsimonious solutions despite non-convexity.The method uses p = 1/4; smaller p and larger λp promote sparsity more aggressively but increase non-convexity.Source: Optimization problem · p. 5

Benchmarks

  • The virtual benchmark uses eight elasto-plastic models and noisy synthetic displacements from one tension-compression experiment on a perforated plate.The plate has two elliptic holes; the protocol records 2,250 load steps, and tested displacement-noise levels range from 0 to 0.5 µm.Source: Benchmarks · pp. 8–9
  • Without displacement noise, EUCLID exactly identifies both yield-function forms and parameters for VM, F2, F1, and NC.When exact recovery fails, discovered models generally remain accurate surrogates, except at higher noise for TR* and NC; Tresca also fits shear regions non-optimally without noise.Source: Benchmarks · p. 10
  • When exact yield-function recovery fails, false-positive features have small coefficients and false negatives generally have little behavioral impact, supporting accurate surrogate models.Tresca is the exception in shear-stress regions even without noise; adding shear deformation to the data drastically improves its results.Source: Benchmarks · p. 10

Discussion

EUCLID discovers interpretable plasticity models from displacement and reaction-force data without stress data. The authors identify extensions to more general plasticity and other material behaviors, and application to experimental data, as future directions.

Discussion

  • EUCLID discovers interpretable plasticity models using displacement and net reaction-force data, without stress data.The authors describe it as a physics-constrained alternative to supervised methods and use sparse regression for parsimonious model selection instead of an a priori plasticity-model choice.Source: Discussion · p. 11
  • The authors identify pressure sensitivity and anisotropy as targets for extending plasticity discovery.They also name visco-elasticity, visco-plasticity, damage, and combinations of these behaviors as possible extensions.Source: Discussion · p. 11
  • Applying EUCLID to experimental data in two- and three-dimensional settings is a future goal.The authors specify digital image and volume correlation data, respectively.Source: Discussion · p. 11

Methods

The supplementary information provides detailed descriptions of the method. It covers the material model library, data generation, and the objective and optimization procedures.

Methods

  • SI Section 1 describes the numerical implementation of the material model library.Source: Methods · p. 11
  • SI Section 2 details how the data are generated.Source: Methods · p. 11
  • SI Sections 3 and 4 discuss the objective function formulation and optimization procedure, respectively.Source: Methods · p. 11

1 Material model library and numerical implementation aspects

The library uses a Fourier expansion to represent yield surfaces and hardening, while numerical implementation uses return mapping and associated derivatives. Validation compares a smooth Fourier approximation of Tresca with the original model and identifies a shear-data limitation.

1.1 Candidate material models and physical requirements

  • A Fourier parameterization describes arbitrarily shaped smooth yield surfaces; isotropy permits only a_i modes i = 0, 3, 6, ... and requires b_i = 0, while tension–compression symmetry permits only even-numbered modes.Hiso(γ) describes isotropic hardening. With only θ0 and θ1 active, convexity requires θ0 ≥ 10|θ1|.Source: §1.1 · pp. 14–15

1.2 Return mapping algorithm and consistent tangent modulus

  • The return-mapping update accepts an elastic trial stress when f ≤ 0; otherwise it iteratively solves for the corrected stress, plastic multiplier, and back stress.The evolution laws use implicit Euler discretization, and the tangent modulus is not needed for the inverse problem.Source: §1.2 · pp. 15–16

1.3 Derivatives of the yield function

  • Derivatives of the yield function with respect to stress, accumulated plastic multiplier, and back stress support return mapping and tangent-modulus calculations.Source: §1.3 · p. 16

1.3.1 Derivatives with respect to the stress tensor

  • For repeated relative principal stresses, the implementation approximates the corresponding derivative summands as zero; exact ties are considered unlikely in simulations.Source: §1.3.1 · p. 17

1.3.2 Derivatives with respect to the accumulated plastic multiplier

  • The derivative with respect to accumulated plastic multiplier includes the derivative of Hiso(γ), which is described as trivial.Source: §1.3.2 · p. 18

1.3.3 Derivatives with respect to the back stress

  • The derivative with respect to back stress is obtained using the chain rule and the back-stress definition.Source: §1.3.3 · p. 18

1.4 Validation of the Fourier approximation of the Tresca yield surface

  • Fourier approximations of non-smooth models use eleven active features and avoid complex stress updates; Tresca and its approximation give matching forward predictions and are recovered in good agreement.Tresca fitting remains non-optimal in shear regions under the considered loading, while adding horizontal displacement to the dataset improves it.Source: §1.4 · p. 18

2 Data generation

The data-generation procedure starts with simulated full-field displacements and adds noise to emulate measurement noise. The resulting displacement series are temporally denoised before use.

2 Data generation

  • Finite-element simulations provide the displacement measurements, with independent zero-mean Gaussian noise added to emulate measurement noise.The noise has standard deviation σ > 0.Source: §2 · p. 18
  • The added noise is applied to each displacement component at each node and time step.Source: §2 · p. 18
  • Noisy displacements are denoised using a temporal Savitzky–Golay filter with quadratic polynomial order and a 50-step moving window.Source: §2 · p. 18

3 Weak formulation of linear momentum balance and choice of test functions

EUCLID enforces linear momentum balance in weak form using displacement and reaction-force data. The resulting free-node and boundary residuals provide the costs minimized to identify material parameters.

3 Weak formulation of linear momentum balance and choice of test functions

  • Displacements are interpolated on a four-node bilinear quadrilateral mesh, and the small-strain tensor is computed from their symmetric gradient.Each element uses four quadrature points.Source: §3 · p. 19
  • Weak-form linear momentum balance constrains the material parameters in the bulk and at the boundary, while avoiding the noise sensitivity of the strong form’s double spatial derivatives.Source: §3 · p. 19
  • With test functions approximated using the displacement basis and held constant over time, balance at free degrees of freedom equates nodal internal forces to zero.The test functions use the Bubnov–Galerkin method.Source: §3 · p. 19
  • At displacement-controlled degrees of freedom, measured net reaction forces for each boundary subset must balance the corresponding summed internal nodal forces.Source: §3 · pp. 19–20
  • Summed squared residuals for free-node and reaction-force balance form two costs, minimized together with a reaction-balance weight of λr = 100.The weight was chosen heuristically to make the bulk and boundary terms contribute similarly.Source: §3 · p. 20

4 Optimization of the ℓp-regularized problem

EUCLID searches over regularization strengths and selects a sparse material model from solutions meeting a cost threshold. Its optimization uses staged initialization and can encounter stress-update convergence problems near physical constraints.

4 Optimization of the ℓp-regularized problem

  • Optimization first estimates yield-surface size using a von Mises model with linear isotropic and kinematic hardening, then uses 100 random starts for the unregularized search.The best unregularized solution initializes the regularized search; a trust-region reflective Newton solver approximates gradients by finite differences.Source: §4 · p. 20
  • The regularized problem is solved in parallel across λp values from 2^-5 to 2^15, yielding one candidate solution per value.Source: §4 · p. 20
  • After discarding solutions above a cost threshold, EUCLID selects the sparsest remaining solution and thresholds small coefficients to zero.The authors set Cth = 1.01 Cmin and θth = 0.005 θ0.Source: §4 · p. 21
  • Stress-update failures near the physical constraint occurred for some unregularized random starts, but not in the regularized problem; smaller load steps were unavailable for the fixed inverse data.Interpolating additional time points was suggested as a possible remedy but not tested.Source: §4 · p. 21, Figure S5 · p. 25

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