A review on data-driven constitutive laws for solids

Jan Niklas Fuhg; Govinda Anantha Padmanabha; Nikolaos Bouklas; Bahador Bahmani; WaiChing Sun; Nikolaos N. Vlassis; Moritz Flaschel; Pietro Carrara; Laura De Lorenzis

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A review on data-driven constitutive laws for solids

Jan Niklas Fuhg, Govinda Anantha Padmanabha, Nikolaos Bouklas, Bahador Bahmani, WaiChing Sun, Nikolaos N. Vlassis, Moritz Flaschel, Pietro Carrara, Laura De Lorenzis

arXiv:2405.03658v1May 6, 2024cs.CEcs.LGphysics.app-ph

TL;DR

Data-driven constitutive modeling must exploit increasingly large and diverse observations while addressing interpretability, generalization, and trustworthiness. This review organizes machine-learning and model-free approaches across path-independent and path-dependent solid behavior, learning processes, data requirements, sampling, verification, and validation. It concludes that physics constraints can improve robustness, interpretability, generalization, and trustworthiness, while certification metrics and benchmarks remain open needs.

Problem
Constitutive modeling needs methods that can use large experimental and computational datasets while addressing generalization, overfitting, interpretability, and trustworthiness.
Method
The review classifies data-driven constitutive approaches by machine-learning versus model-free methods, interpretability, learning process, data requirements, path dependence, sampling, and validation.
Results
Physics constraints are associated with gains in robustness, interpretability, generalization, and overall trustworthiness across different classes of data-driven constitutive models.
Takeaways & Limitations
Data-driven constitutive modeling can connect richer observations to predictive simulations, multiscale calculations, multifidelity information, and material-variability analysis.
Takeaways & Limitations
General metrics for deciding when an ML model is trustworthy and benchmarks for safe use in high-risk applications remain lacking.

Abstract

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This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.

1 Introduction

The review maps data-driven constitutive modeling methods for path-independent and path-dependent solids, emphasizing how large datasets, interpretability, physics constraints, and data requirements shape their use. It distinguishes machine-learning-based from model-free approaches and surveys their benefits, drawbacks, generalization, and validation challenges.

1 Introduction: Modern full-field experiments and multiscale computations have shifted constitutive modeling from limited observations toward large-data regimes.This creates opportunities for richer constitutive laws while increasing the challenge of analyzing available data.
1 Introduction: The review classifies data-driven approaches into machine-learning methods and model-free methods, with further distinctions based on interpretability and learning process.Its taxonomy also considers supervised versus unsupervised learning and physics-informed constraints.
1 Introduction: Model-free methods incorporate material observations directly into the solution of solid-mechanics governing equations, whereas ML-based methods produce a constitutive mapping that can later operate independently of the original data.This distinction concerns whether the dataset remains part of the solution after learning.
1 Introduction: Interpretable, parsimonious constitutive representations can reduce overfitting and support generalization beyond the training range when the underlying physics remains consistent.The review contrasts these with uninterpretable or partially interpretable approaches that encode constitutive relations without physically explainable input-output relationships.
1 Introduction: The review maps state-of-the-art techniques rather than ranking them, examining their mechanics knowledge, generalization, data hunger, advantages, drawbacks, and links to experimental design.It also organizes discussion by path-independent and path-dependent behavior and points toward integrated experiment–modeling frameworks.

2 DD methods outline and earlier reviews

The review organizes data-driven constitutive approaches by interpretability, learning strategy, and model construction, while contrasting their strengths, limitations, and application requirements.

Interpretable approaches: Interpretable methods use symbolic or sparse regression to discover analytical constitutive expressions whose functional dependencies and parameters can be physically explained.Symbolic regression combines mathematical building blocks through genetic algorithms, whereas sparse regression selects expressions from a predefined candidate library.
Uninterpretable approaches: Neural networks support complex regression and history-dependent modeling through architectures including feed-forward, recurrent, convolutional, attention, and graph networks.Their popularity is supported by widespread automatic-differentiation implementations.
Uninterpretable approaches: Reinforcement learning extends data-driven modeling toward control and experimental design by learning actions from states, rewards, and repeated environment interactions.This addresses settings where predefined input-output mappings are insufficient for dynamic, uncertain decision-making.
Other machine-learning methods: Support vector regression is robust to outliers and easy to implement, with dimensionality-independent computational complexity and good generalization when well fitted.Convex loss yields a unique training solution, but performance degrades for big, noisy datasets and lacks a probabilistic fitting interpretation.
Closing remarks: No single method is universally best: method selection depends on dataset size and source, interpretability needs, dimensionality, data noise, and response complexity.The review also notes that neural networks are flexible and accessible, whereas methods with steeper learning curves may offer application-specific advantages.
Model-free approaches: Model-free methods directly use discrete material observations in forward computations, making them data-hungry and limiting their treatment of dissipative constitutive laws.They must represent material behavior without introducing postulated internal or history variables unless sufficiently large datasets are available.

3 Data sampling and design of experiments

The review frames sampling and experimental design as central to data-driven constitutive modeling, comparing one-shot and adaptive strategies while emphasizing unresolved challenges for sequential responses.

Overview: Sampling strategies seek fewer but more relevant observations that capture the major complexities of constitutive mappings, because performance depends on available data.The review distinguishes one-shot from sequential sampling approaches.
Experimental design: Experimental design parallels sampling but additionally must account for laboratory feasibility, including equipment, time, and material constraints.Deep reinforcement learning is reviewed as a potential tool for optimizing and informing experimental setups.
One-shot sampling: One-shot sampling fixes sample size and locations in a single stage, requiring the input domain to be known for temporal and non-temporal variables.Grid sampling is simple and supports sensitivity studies and numerical integration, but its collapsing property can duplicate effective evaluations.
One-shot sampling: Latin hypercube sampling is preferred over grid or uniform random sampling when designs must be both space-filling and non-collapsing.Grid designs can duplicate projected coordinates, while random designs may cluster and leave regions unsampled.
Temporal sampling: Temporal sampling remains difficult because sequential material responses require space-filling sequences, and reliable goal-oriented sampling is still an open issue.This challenge is especially relevant for responses depending on three-dimensional loading paths.
Sequential/adaptive sampling: Adaptive sampling addresses unknown sample sizes and placements by adding data where surrogate uncertainty exceeds a tolerance and retraining the model.A cited multiscale framework uses kriging variance as an error estimate during finite-element material-point calculations.

4 DD modeling for path-independent CLs

Path-independent constitutive laws are reviewed across machine-learning, multiscale, reduced-order, model-free, and sparse-regression approaches for elastic and heterogeneous materials. The section emphasizes energy-based formulations, data-efficient training, and careful finite-strain parametrization.

4 DD modeling for path-independent CLs: Energy-based mappings produce scalar strain-energy outputs, whose strain derivatives recover stress and simplify physical-constraint enforcement.The scalar output can simplify training, while differentiation yields the stress–strain relation and supports thermodynamic consistency.
4 DD modeling for path-independent CLs: Multiscale neural networks encode homogenization by chaining analytically characterized composite building blocks from microscopic to effective material properties.The Deep Material Network uses a binary-tree structure whose parameters are trained on two- or three-dimensional data.
4 DD modeling for path-independent CLs: Unsupervised neural constitutive models can learn from full-field displacements and global reaction forces, avoiding experimentally scarce or simulation-costly stress data.This energy-based characterization strategy addresses the high data demands of supervised stress–strain training.
4 DD modeling for path-independent CLs: Reduced-order microstructural simulations and clustering decrease the cost of two-scale simulations while retaining high computational accuracy.The offline stage partitions representative-volume-element geometry into clusters with similar strain concentration.
4 DD modeling for path-independent CLs: Finite-strain models require deliberate choices among deformation and stress measures because parametrization changes the learning problem’s size and complexity.The reviewed measures include F, E, C and their conjugate stresses P or S; finite-strain modeling also introduces geometric nonlinearity and frame considerations.
4 DD modeling for path-independent CLs: Sparse-regression methods such as EUCLID discover interpretable hyperelastic energy functions from displacement and global reaction-force data without stress labels.Physics-motivated losses based on linear momentum compensate for the absence of labeled stress–strain pairs.

5 DD modeling for path-dependent CLs

Path-dependent constitutive laws require data-driven models that account for the entire deformation history rather than only the current strain. The review focuses on hypo- and elastoplasticity, viscoelasticity, damage, fracture, fatigue, and coupled multiphysics behavior.

5 DD modeling for path-dependent CLs: Path-dependent stress depends on current strain and the entire deformation history, creating a one-to-many strain–stress mapping.A single deformation state can therefore correspond to potentially infinite stress states, unlike path-independent behavior.
5 DD modeling for path-dependent CLs: The review covers data-driven approaches for hypo- and elastoplasticity, viscoelasticity, damage, fracture, fatigue, and multiphysics constitutive coupling.These topics are treated as distinct path-dependent material behaviors and coupled physical processes.

5.1 Plasticity

Data-driven plasticity models progress from history-based neural networks toward formulations with internal variables, modular submodels, and embedded physical consistency. The review emphasizes trade-offs among extrapolation reliability, interpretability, data requirements, and thermodynamic consistency.

DD plasticity modeled after hypoplasticity: History-based neural networks predict path-dependent stress evolution from observable stress and strain histories but may fail on complex or out-of-domain loading paths.Early approaches used quasi-sequential inputs and outputs to distinguish loading and unloading behavior.
DD plasticity modeled after hypoplasticity: Introducing internal variables gives neural networks additional state information for modeling viscoplastic strain rates and internal-variable evolution.The reviewed formulation uses current strain, internal variables, and current stress as inputs, with current viscoplastic strain and internal-variable rates as outputs.
DD plasticity modeled after hypoplasticity: Physics-informed formulations improve structural consistency by enforcing isotropy, using tangent-stiffness factors, or encoding thermodynamic potentials directly.TANNs derive stress from a learned Helmholtz free-energy density, so identifying thermodynamic laws is not left entirely to the machine-learning model.
DD plasticity modeled after hypoplasticity: Thermodynamics-based neural networks obtain the stress update by differentiating the predicted Helmholtz free-energy density at the next time step.This scalar-energy formulation supports implicit thermodynamic consistency and simplifies the learning target.
DD plasticity modeled after elastoplasticity: Modular elastoplastic models improve interpretability, reduce reliance on big data, and simplify thermodynamic enforcement by assigning data-driven models to selected constitutive components.They can also shift learning toward one-to-one mappings such as yield-function representations, provided component-specific data are available.
DD plasticity modeled after elastoplasticity: The review’s taxonomy links weaker constitutive constraints with greater data requirements, while method selection depends on dataset size, data source, interpretability, dimensionality, complexity, and noise.No single method is identified as best for all constitutive-modeling settings.

5.2 Viscoelasticity

Viscoelastic constitutive modeling spans uninterpretable and interpretable data-driven approaches, including neural networks, recurrent architectures, symbolic regression, and sparse regression. The literature addresses history dependence, limited data, and links between learned architectures and material evolution equations.

Uninterpretable ML approaches: Recurrent neural networks naturally represent history dependence because their hidden states carry information from previous time steps.This structural analogy motivates their use for sequential viscoelastic updates of stresses and internal variables.
Uninterpretable ML approaches: Neural networks learn viscoelastic responses ranging from simplified scalar laws based on 1D creep data to complete rate-dependent constitutive frameworks.Inputs may include temperature, initial stress, or elastic properties; some models are unsuitable as full constitutive laws for finite-element simulation.
Interpretable ML approaches: A neural approximation of associative and hereditary memory preserves interpretability by connecting the network architecture to integrodifferential material equations.The model is trained on 1D experimental data and tested on non-monotonic loading paths.
Interpretable ML approaches: Interpretable viscoelastic approaches include symbolic regression and sparse regression, with EUCLID identifying linear viscoelastic laws from phenomenological libraries and unlabeled data.The generalized-standard-material formulation extends this idea to broader classes that include viscoelastic behavior.
Model-free approaches: Model-free viscoelasticity has been demonstrated in a one-dimensional state space using differential representations of material history and truss-structure analyses.The cited approach adapts a model-free plasticity framework to viscoelastic response.

5.3 Damage and fracture

Data-driven damage and fracture models use neural, physics-informed, graph-based, and model-free methods to represent degradation, cracking, and history dependence. A recurring challenge is enforcing physically meaningful damage evolution, especially irreversibility and unloading/reloading behavior.

Diffuse damage: Damage irreversibility is central to history-dependent constitutive modeling because it enables physically sound unloading and reloading branches.The damage rate is constrained to be nonnegative, making the evolution law consistent with the assumed irreversibility condition.
Diffuse damage: Most reviewed damage approaches use neural networks with damage implicitly embedded in the constitutive mapping, while no interpretable approaches are yet available.The review covers parameter identification, neural constitutive laws, and model-free methods.
Diffuse damage: Recurrent neural networks support history-dependent multiscale modeling of damaging poroelastic materials across micro-, meso-, and macro-scales.The framework uses discrete-element and finite-element data and addresses unloading/reloading discrimination and material objectivity.
Diffuse damage: Physics-informed neural networks can impose damage irreversibility, zero stress at zero strain, critical-damage failure, and automatic loading-state detection.The cited coupled damage/plasticity approach nevertheless requires very large training datasets for reasonable accuracy.
Fracture: Fracture studies in the reviewed literature focus on linear-elastic behavior, including neural solutions for crack problems with or without crack-face contact.The constrained crack formulation converges faster than the unconstrained counterpart, while its cost is lower than singular-element FEM only in the constrained case.

5.4 Multiphysics

Multiphysics constitutive modeling must capture coupled mechanical and nonmechanical variables while respecting thermodynamics and often unknown causal relationships. Data-driven approaches range from neural surrogates and model-free methods to symbolic, causal, and probabilistic descriptor-discovery frameworks, each with distinct scope and computational trade-offs.

Overview: Multiphysics constitutive laws require additional thermal, hydraulic, or chemical descriptors beyond stresses and strains, making identification more challenging than in solid mechanics.The review focuses on geomechanics and highlights the difficulty of selecting descriptors, learning dependencies, and satisfying thermodynamic laws.
Known descriptors, unknown functional form: Multiscale first-principles approaches can avoid constitutive functional forms and physical violations but incur high computational costs, especially when multiple physics have different resolutions.Finding an explicit functional form can improve computational efficiency and interpretability in this setting.
Known descriptors, unknown functional form: When descriptors are known but functional forms are unknown, neural surrogates and differentiable approximators can replace costly fine-scale computations.Examples include poroelastic and unsaturated-soil models learned from multiscale simulations.
Known descriptors, unknown functional form: Symbolic regression becomes less computationally efficient as the number of material descriptors grows or the target law is expressed by ODEs or PDEs rather than algebraic equations.This limits straightforward interpretable-model discovery for complex multiphysics constitutive systems.
Unknown descriptors and causal structure: Causal and probabilistic methods address cases where descriptor relationships or even the descriptor set are unknown.A probabilistic framework can recover classical granular-material concepts such as the fabric tensor, but assumes a static causal graph structure.

6 Evaluation, verification, validation and their challenges

The review proposes objective metrics for evaluating data-driven constitutive models and emphasizes separate verification and validation, especially because opaque, over-parameterized models may generalize poorly and remain difficult to certify.

6.1 Performance metrics: A lack of standardized tests and agreed evaluation metrics complicates objective comparison of data-driven constitutive approaches.The review proposes a framework of metrics rather than attempting to establish field-wide consent.
6.1 Performance metrics: Performance should be assessed across accuracy, precision, physical consistency, interpretability, generalization, cost, robustness, stability, confidence, data hunger, numerical behavior, reliability, and trustworthiness.These metrics cover prediction quality, physics, computational demands, uncertainty, data requirements, reproducibility, and overall task-specific performance.
6.1 Performance metrics: Metrics should be defined for each application to compare models or datasets and to support objective verification, validation, and trust assessment.The same framework can compare different approaches, training data sources, and whether a candidate constitutive law reproduces the investigated behavior.
6.2 Verification and validation: Verification checks whether the mathematical and numerical model is correctly implemented, whereas validation assesses predictive reproduction across applications and outside the training domain.Extrapolation and limited data make validation particularly important for solid-mechanics constitutive modeling.
6.2 Verification and validation: Data-driven models require independent verification and validation because opaque decision logic, many parameters, and overfitting can undermine reliability and generalization.Regularization and other validation techniques address overfitting, but certification criteria and safety benchmarks remain unresolved.

7 Conclusions

The review maps how data-driven constitutive methods can expand material modeling while retaining useful physical structure. It identifies physics constraints, integration, path dependence, low-data learning, and community benchmarks as central priorities for trustworthy progress.

7 Conclusions: Data-driven constitutive modeling can exploit larger datasets, broaden modeling flexibility, combine multifidelity information, streamline predictive simulations, enable multiscale calculations, and support uncertainty analysis.These opportunities are presented as potential benefits rather than an exhaustive ranking of existing methods.
7 Conclusions: Physics constraints can preserve phenomenological modeling benefits while improving robustness, interpretability, generalization, and trustworthiness of expressive automated approaches.The review connects these gains to mechanistic knowledge formalized as constraints across different classes of constitutive models.
7 Conclusions: Integrating data-driven models with advanced experiments and established computational engineering remains an open challenge, including the performance costs of automatic differentiation in nonlinear finite-element solvers.The workflow must connect full-field experimental data to constitutive laws and practical engineering simulation tools.
7 Conclusions: Discovering irreversible mechanisms and internal variables directly from data remains underexplored, while path-dependent learning also makes experimental sampling more complex and restricted.The difficulty arises because the learning task concerns quantities dependent on internal variables such as hardening variables or yield functions.
7 Conclusions: The field should pursue robust low-data methods and establish curated datasets with benchmark problems for evaluating constitutive approaches across interpolation, extrapolation, and validation settings.The review proposes hyperelasticity benchmarks using RVE data and complementary full-field displacement and reaction-force datasets from distinct boundary-value problems.