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Data-Driven Discovery of Composition-Dependent Constitutive Models for Hyperelasticity and Viscoelasticity of Digital Materials
Josué García-Ávila, Beijun Shen, Manuel K. Rausch, Mary C. Boyce, Adrián Buganza-Tepole
TL;DR
Digital materials vary in nonlinear, rate-dependent response across compositions, while classical models can be difficult to identify flexibly across material families. This paper combines a structured finite-viscoelastic formulation with data-driven constitutive functions conditioned on material features. The framework captures composition- and rate-dependent response, with the NODE energy and positive-MLP flow pairing learning a smooth manifold across compositions and rates without further regularization.
Problem
Across material families, specifying strain-energy functions and internal-variable kinetics remains challenging, while unconstrained neural stress models may lack physical guarantees and interpretability.
Source: §1 · p. 3Method
The framework retains multiplicative finite-viscoelastic kinematics and learns composition-conditioned equilibrium energies and nonequilibrium flow laws using analytical parameters or NODEs and positive neural networks.
Source: §1 · p. 3Results
The NODE energy paired with a positive-MLP flow rule learns a smooth manifold across compositions and rates without further regularization.
Source: §8 · p. 24Takeaways & Limitations
The framework captures nonlinear, strain-rate-dependent digital-material response while retaining a thermodynamic structure.
Source: §1 · p. 3Different semi-closed evolution-law parameter combinations can yield nearly indistinguishable responses, leaving parameters potentially non-identifiable from limited mechanical tests.
Source: §7.2.2 · p. 23
Abstract
from arXiv · showhide
Digital materials fabricated by multi-material 3D printing are designed as controlled mixtures of stiff and compliant constituents, yielding effective responses that span more than an order of magnitude in apparent stiffness and exhibit strongly nonlinear, composition-dependent, and rate-dependent dissipative behavior. Classical finite-strain viscoelastic models represent such behavior with closed-form strain energy functions for equilibrium and non-equilibrium stresses as well as evolution of internal variables, which may limit flexibility when a single constitutive model is expected to generalize across materials and loading rates. Here, we present a data-driven multi-material constitutive modeling framework that generalizes a formulation by Bergstr\"om and Boyce. The proposed framework retains the structure of the classical model, namely multiplicative kinematics, invariant-based strain-energy functions, and a scalar dissipative evolution law directed along the normalized nonequilibrium deviatoric stress. For the equilibrium branch, the data-driven discovery framework either directly predicts closed-form model parameters as functions of composition or automatically constructs a polyconvex strain-energy function using neural ordinary differential equations (NODEs). The nonequilibrium branch kinetics are learned similarly, either by directly identifying closed-form parameters across compositions or by using appropriately constrained artificial neural networks. Using multi-rate uniaxial compression data across multiple material compositions, we show that the proposed formulation captures rate-dependent stiffness and hysteresis across compositions while preserving thermodynamic consistency.
1. Introduction
Composition-dependent, rate-dependent digital materials challenge constitutive models to balance flexible fitting with physical structure. The paper proposes a unified, data-driven finite-viscoelastic framework that retains thermodynamic structure while learning constitutive behavior across materials.
1. Introduction
- Voxel-based composites can exhibit effective responses distinct from their constituents, motivating models that describe behavior continuously across compositions rather than fitting each mixture independently.Composition and spatial distribution can tune the range of material responses.Source: §1 · p. 2
- Determining both the strain-energy form and internal-variable kinetics remains difficult across material families, despite thermodynamically rigorous model structures.Multiple nonequilibrium branches and phenomena such as softening or hardening increase nonlinearity and parameter-identification challenges.Source: §1 · p. 3
- Unconstrained neural stress models may violate physical and thermodynamic requirements, while direct stress prediction can be difficult to interpret and unreliable outside training data.The cited concerns include objectivity, material symmetries, increasing energy monotonicity, and the second law.Source: §1 · p. 3
- Prior physics-augmented viscoelastic approaches face challenges in enforcing non-negative dissipation and controlling optimization cost, while existing digital-material and latent-state models retain stated structural limitations.The cited examples use prescribed QLV structure or infer latent states, and are not developed for continuous composition families.Source: §1 · p. 3
- The proposed framework preserves multiplicative kinematics and stress-directed dissipative flow while learning composition-dependent analytical parameters or constitutive functions with NODEs and positive neural networks.It is designed to capture composition-dependent rate response across more than an order of magnitude in apparent stiffness and nearly three orders of magnitude in strain rate.Source: §1 · p. 3, Figure 1 · p. 4
2. Constitutive framework for finite-strain viscoelasticity
The framework represents finite-strain viscoelasticity with branch-specific internal kinematics, a composition-conditioned free energy, and a thermodynamically constrained dissipative flow law.
2.1 Branch-wise multiplicative kinematics
- Each nonequilibrium branch carries an internal viscous deformation in a multiplicative kinematic structure, with branch A at equilibrium and branches B and C nonequilibrium.The multiplicative structure represents internal history through branch-specific memory variables.Source: §2.1 · p. 4
2.2. Free-energy decomposition and multi-material dependence
- The free energy combines an equilibrium contribution with summed nonequilibrium contributions, while material descriptors Z condition the multi-material model.For this material set, Z is reduced to composition P because hardness H and P are correlated.Source: §2.2 · pp. 4–5
- The first Piola–Kirchhoff stress follows from the free-energy derivative, with equilibrium, nonequilibrium, and incompressibility-pressure components.The multiplier p* enforces J = 1.Source: §2.2 · p. 5
2.3. Thermodynamic restriction and dissipative structure
- Thermodynamic admissibility is enforced by expressing branch flow as a scalar rate along the normalized deviatoric-stress direction and constraining dissipation to be non-negative.The scalar rate can depend on effective driving stress, internal-state measures, and material descriptors.Source: §2.3 · p. 6
3. Multi-material constitutive modeling
The framework conditions equilibrium and nonequilibrium constitutive functions on material descriptors while retaining a finite-strain viscoelastic structure. It combines closed-form and learned representations for elastic response and dissipation.
3.1. Equilibrium branch
- Material descriptors condition a shared equilibrium model through either Arruda–Boyce parameters or a more flexible NODE representation.The conditioned Arruda–Boyce model represents composition-dependent chain-network parameters.Source: §3.1 · p. 7, §3.1.1 · p. 7
3.1.1 Closed-form multi-material conditioning of the hyperelastic response
- Closed-form equilibrium alternatives include neo-Hookean, Fung, and Arruda–Boyce models, whose forms encode different strain-stiffening behavior.Neo-Hookean has a constant energy derivative and no strain stiffening; Fung gives exponential stiffening, while Arruda–Boyce represents finite chain extensibility.Source: §3.1.1 · p. 7
3.1.2 Data-driven hyperelastic response based on NODEs
- The equilibrium NODE evolves a learned state from I1 − 3 and uses its terminal value to define the energy gradient, with constraints yielding a monotonic, convex response.The construction also separates the small-strain shear modulus from the deformation-dependent strain-stiffening contribution.Source: §3.1.2 · pp. 8–9
3.2. Nonequilibrium branch
- Nonequilibrium branches are treated as effective constitutive modes because available mechanical data cannot uniquely identify their underlying molecular mechanisms.The framework uses as few branches as needed to describe the data robustly.Source: §3.2 · p. 9
3.2.1. Data-driven and closed-form modeling of the stored elastic energy in non-equilibrium branches
- Each nonequilibrium branch stores energy through its own elastic state and scalar function α(r), represented by an Arruda–Boyce-type form or a NODE over the branch elastic invariant.The branch free energy is restricted to the elastic first isochoric invariant for the adopted isotropic incompressible case.Source: §3.2.1 · pp. 9–10
3.2.2. Closed-form and data-driven branch dissipation
- Branch dissipation follows a non-negative scalar evolution law along the driving-stress direction, with closed-form or learned kinetics conditioned on material features.The generalized power law separates material-dependent and memory-dependent contributions to mitigate gradient cancellation and parameter confounding during training.Source: §3.2.1 · p. 10, §3.2.2 · pp. 10–11
4. Uniaxial incompressible case
The homogeneous uniaxial calibration case specializes the three-dimensional formulation to incompressible deformation. The branch evolution is expressed through scalar internal viscous stretches.
4. Uniaxial incompressible case
- The calibration case assumes incompressible uniaxial deformation and an isochoric viscous distortion for each nonequilibrium branch.The total deformation has det F = 1, and each viscous distortion has det F_v(r) = 1.Source: §4 · p. 12
- The formulation derives branch elastic invariants and effective driving stresses for this uniaxial specialization.The deviatoric branch stress and normalized flow direction are also specialized for the scalar evolution equation.Source: §4 · p. 12
- Each branch’s internal viscous stretch evolves in scalar form using the 11-components of its viscous distortion rate and rate-of-deformation tensor.The resulting evolution equation combines the branch-wise dissipative structure with the uniaxial specialization.Source: §4 · p. 12
5. Materials and experimental data
The experiments use five PolyJet digital materials spanning elastomer-rich to Vero-rich compositions. Multi-rate compression curves provide the data for constitutive-model training.
5.1 Base materials
- Five materials range from neat Agilus 30 Black (A30) through mixtures A50, A70, A85, and A95 with increasing Vero content.A30 is compliant and recovers most applied deformation, whereas A95 is stiffer and dissipates more energy.Source: §5.1 · p. 13, Table 2 · p. 13
- The composition descriptor P is the Agilus Black percentage, defined as 100 − φv, where φv is estimated Vero volume fraction.The specimens combine Agilus 30 Black and Vero Ultra White voxel-scale mixtures.Source: §5.1 · p. 13, Table 2 · p. 13
5.2 Mechanical testing
- Room-temperature uniaxial compression used lubricated steel platens and optical displacement tracking to keep deformation close to homogeneous and exclude machine compliance from the recorded response.The stiffer A85 and A95 specimens were smaller to keep reaction forces within the load-cell capacity.Source: §5.2 · p. 13
- Each test used one load–unload cycle to about 67% engineering strain, at nominal rates of 0.001, 0.01, 0.1, or 0.5 s−1.Fresh specimens were used, most rates were repeated, and stress–true strain curves formed the training data.Source: §5.2 · p. 13
6. Model training
Training calibrates the equilibrium branch first, then fits nonequilibrium branches to time histories across compositions and rates. The loss and integration procedures account for the different stress scales and sequential internal-state updates.
6.1 Equilibrium branch training
- The equilibrium branch is calibrated first from the lowest-rate loading–unloading response, treated as an approximation rather than a direct measurement of full relaxation.This approximation is most reliable for compliant compositions and less exact for A70–A95, where nonequilibrium contributions may remain at 0.001 s−1.Source: §6.1 · p. 13
- Equilibrium parameters are optimized with shuffled mini-batches and Adam, then frozen as the contribution used while training branches B and C.The equilibrium fit minimizes mean-squared error between predicted and experimental nominal stresses.Source: §6.1 · p. 14
6.2 Nonequilibrium branch training
- Nonequilibrium training steps sequentially through each measured stretch–time history, updating internal viscous stretches for each branch.The material descriptor is embedded once per curve and reused at all time steps.Source: §6.2 · p. 14
- The nonequilibrium loss averages curve-wise normalized errors so each history contributes equally despite differences in stress range or sample count.The normalization uses each curve’s experimental stress range.Source: §6.2 · p. 14
7. Results and discussion
The results examine how the framework and its branch structures, energy representations, and constraints capture composition-dependent equilibrium and viscoelastic responses across loading rates while preserving convexity and nonnegative dissipation. They also show how semi-closed evolution laws shape parameter interpretation and identifiability.
7.1. Equilibrium response across multiple materials
- The first invariant I1 captures the main equilibrium nonlinearities, and the conditioned NODE enforces a nonnegative, monotone ∂ψ/∂I1 and convex energy while learning smooth responses at unseen compositions.Composition affects both the initial-stiffness offset and the NODE vector field; A30 has ∂ψ/∂I1 on the order of 0.1 MPa, while A95 exceeds 4 MPa at large deformations.Source: §7.1 · p. 15, Figure 3 · p. 16
- NODE gives the closest overall equilibrium fit and best mean held-out performance, while regularized Arruda–Boyce is comparable and improves on other closed-form models; neo-Hookean best predicts A50 Black individually.NODE extrapolates closest to data, especially toward soft A30, whereas neo-Hookean's limited flexibility can stabilize individual extrapolations but misses strain stiffening.Source: §7.1 · pp. 15–17, Figure 5 · p. 18
7.2. Multi-material nonequilibrium response
- Across 20 datasets, the learned positive flow rule captures responses across four rates and five materials; AB- and NODE-based energies predict similarly overall, though unregularized AB can show high-rate chain-locking artifacts.The artifact appears for A30, A50, and A70 Black; a physically meaningful penalty on normalized chain stretch χchain removes it, while NODE does not show the same issue.Source: §7.2.1 · p. 19, §7.2 · p. 19, Figure 6 · p. 20
7.2.1 Comparison of closed-form and NODE energies with a learned evolution law
- Higher strain rates increase stored nonequilibrium energy and dissipation, with stronger effects in lower-Agilus-content materials; dissipation can exceed total stored elastic energy.Energy quantities are evaluated at the shared maximum deformation; lower Agilus content also yields larger equilibrium and nonequilibrium energy contributions.Source: §7.2.1 · p. 19, Figure 6 · p. 20
- Two nonequilibrium branches capture distinct mechanisms: B stores most rate-sensitive energy and can overshoot into tension, while C stores little elastic energy and dissipates rapidly.Branch C's larger stress despite small elastic deformation reflects its greater elastic stiffness; both branches can develop tension during unloading while the material remains in compression.Source: §7.2.1 · pp. 20–22, Figure 8 · p. 22
- On held-out A60 Black, regularized AB and NODE perform comparably across rates, but AB degrades at 0.5 s^-1 with underestimated loading and an unloading artifact, whereas NODE unloads more smoothly.Both models use the same positive neural-network flow rule; AB does slightly better at the three lower rates.Source: §7.2.1 · p. 20, Figure 7 · p. 21
7.2.2 Semi-closed evolution laws: interpretability and identifiability
- Among semi-closed evolution laws, Option 5 trains more stably, while Option 2 reaches the lowest error after 1000 epochs; distinct parameter combinations can yield nearly indistinguishable responses.Compensating parameter trade-offs and redistribution of viscous features across branches indicate non-identifiability; Option 2 makes branch B flow linear in nonequilibrium stress, with inverse viscosity K(B) proportional to P.Source: §7.2.2 · p. 23
- For Option 2, branch C is nearly linear for A30 Black but its exponent rises as Agilus content decreases, reaching m(C) ∼6 for A95 Grey.For semi-form dissipation, the energy representation also substantially changes the result: Option 2 reverses viscous hardening between branches B and C depending on whether NODE or AB energy models are used.Source: §7.2.2 · p. 23
- Option 5 identifies near-linear stress dependence for A30 and exponents m(B), m(C) ∼3 for A95, alongside non-monotonic viscous flow-hardening in branch C.The initial increase in f(C) suppresses viscous flow and indicates hardening at small viscous stretch; its subsequent decrease facilitates flow and signals a transition toward softening.Source: §7.2.2 · p. 23
- Because Option 5 lacks a state-dependent function in branch B, f(C) alone absorbs deformation dependence.The functional form can capture observed stress-stretch behavior while compromising identifiability; proposed safeguards include monotonicity, bounded exponents, or restrictions on branch activity, but overly strong assumptions risk model bias.Source: §7.2.2 · p. 23
- If monotonicity of the state function is enforced, Option 5 is excluded, while Option 2 requires two strain-hardening branches and decreasing viscosity with higher Agilus Black content.Under this constraint, Option 2 also implies power-law behavior in branch C, with a greater exponent at lower Agilus Black content.Source: §7.2.2 · p. 23
8. Conclusion and future work
The framework combines composition-conditioned closed-form constitutive laws or physically constrained neural models, balancing interpretability against flexibility. Its learned nonequilibrium branches separate into distinct dissipative roles, while multiaxial loading and damage or fracture remain future extensions.
8. Conclusion and future work
- The framework’s equilibrium and nonequilibrium models balance interpretability against flexibility, with NODEs and positive neural networks learning constitutive laws conditioned on material features.Arruda-Boyce and NODE energies capture strain stiffening; Arruda-Boyce requires careful regularization, while NODEs support robust interpolation and extrapolation across material responses.Source: §8 · p. 24
- NODE energy paired with a positive-MLP flow rule learns a smooth manifold across compositions and rates without further regularization.The framework retains multiplicative kinematics and a flow rule directed along normalized nonequilibrium deviatoric stress.Source: §8 · pp. 24–25
- The learned nonequilibrium response separates into a fast, mostly viscous branch and a slower branch that stores most nonequilibrium elastic energy and retains loading history.This separation was not imposed in advance and appears with both closed-form and NODE branch elasticity; its micromechanical origins are not investigated.Source: §8 · p. 25
- Future work includes extending the framework to multiaxial loading data and to damage and fracture of digital materials.Source: §8 · p. 25
Authorship contribution
The listed contributors span conceptualization, methodology, investigation, analysis, data curation, software, validation, visualization, writing, supervision, and resources.
Authorship contribution
- JG-A contributed to conceptualization, methodology, investigation, formal analysis, data curation, software, validation, visualization, and writing.JG-A contributed to both the original draft and review and editing.Source: Authorship contribution · p. 25
- BS contributed to formal analysis, data curation, and writing through review and editing.Source: Authorship contribution · p. 25
- MKR contributed to methodology, funding acquisition, resources, and writing through review and editing.Source: Authorship contribution · p. 25
- MCB contributed to methodology, supervision, resources, and writing through review and editing.Source: Authorship contribution · p. 25
- ABT contributed to conceptualization, methodology, supervision, funding acquisition, resources, and writing.ABT contributed to both the original draft and review and editing.Source: Authorship contribution · p. 25