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Constrained neural softening functions for the Mullins effect in parameter-dependent materials

Dominik K. Klein, Jonas Fey, Maximilian Kannapinn, Oliver Weeger

arXiv:2610.34565v1cs.CE

TL;DR

Neural constitutive models for the Mullins effect need to balance flexible softening representations with physical consistency and robust calibration. This paper develops parameter-dependent neural softening functions and a sufficient ellipticity condition within a pseudo-elastic framework. Constrained formulations exclude physically unreasonable predictions by construction and show narrow error distributions across repeated calibrations.

  • Problem

    Neural Mullins models are scarce, and the constraints on neural softening functions and their effects on model behavior have not been systematically investigated.

    Source: §1 · p. 2

  • Method

    The study formulates parameter-dependent neural softening functions in a pseudo-elastic framework, using a multiplicative decomposition and structural constraints, including a sufficient ellipticity condition.

    Source: §1 · p. 2, §5 · p. 12

  • Results

    Constrained formulations exclude physically unreasonable predictions by construction and show narrow error distributions across repeated calibrations.

    Source: §5 · p. 12

  • Takeaways & Limitations

    Neural softening functions offer a potential unified extension of the classical pseudo-elastic framework that reduces the need to select a classical softening function for each material.

    Source: §5 · p. 12

    The models consider isotropic materials that remain isotropic under softening and neglect recovery and other additional effects.

    Source: §2 · p. 3, §4.1 · p. 7

Abstract

from arXiv · show

Accurately capturing the Mullins effect across varying material compositions remains challenging, as constitutive models must unify flexible representations of stress softening with physical consistency and robust parameter calibration. In this work, we investigate parameter-dependent physics-augmented neural network constitutive models within the pseudo-elastic modeling framework, focusing on structural constraints that ensure physically admissible neural softening functions. This paper has three key contributions: (1) We formulate different neural softening functions within the pseudo-elastic framework, including a multiplicative decomposition of the softening function that enables parameter-dependent ansatz functions satisfying common mechanical constitutive conditions by construction. (2) We derive a sufficient condition for ellipticity of pseudo-elastic constitutive models, which can be enforced in the constitutive models by construction. (3) We systematically investigate how flexibility and structural constraints influence the performance of neural softening functions in terms of prediction accuracy and calibration robustness. For this, we consider multiaxial experimental data of carbon black-filled EPDM rubber synthesized with varying amounts of curatives. The findings of our work suggest that neural softening functions are a promising extension of the classical pseudo-elastic framework and, when subject to suitable constraints, can improve predictive accuracy for highly nonlinear, parameter-dependent materials.

1 Introduction

Neural softening functions for parameter-dependent Mullins models remain underexplored, and their constraints and calibration behavior have not been systematically assessed. This work formulates such functions in a pseudo-elastic framework and derives a sufficient ellipticity condition that can be enforced by construction.

1 Introduction

  • Accounting for the Mullins effect matters for reliable predictions because mechanical preconditioning can significantly affect constitutive-model predictive capability.Heterogeneous deformation in engineering components can also produce spatially heterogeneous softening in initially homogeneous materials.Source: §1 · p. 2
  • The study focuses on isotropic pseudo-elastic models with parameter-dependent material behavior and excludes recovery and other additional effects.The authors describe this reduced model complexity as a limitation intended to clarify suitable neural softening-function formulations.Source: §1 · p. 2
  • Neural softening functions in the pseudo-elastic framework had not been proposed, and their constraints and effects on model behavior had not been systematically investigated.Existing neural Mullins models were scarce, and thermodynamic consistency was not always ensured by construction.Source: §1 · p. 2
  • The study formulates and evaluates neural softening functions, including a multiplicative decomposition that supports parameter-dependent ansatz functions satisfying common constitutive conditions by construction.The evaluation uses multiaxial data for carbon black-filled EPDM rubber synthesized with varying curative amounts.Source: §1 · p. 2
  • A sufficient ellipticity condition for pseudo-elastic models is translated into a monotonicity constraint on the neural softening function that can be imposed by construction.The condition accounts for different tangent operators during primary loading and unloading or reloading.Source: §1 · pp. 2–3

2 Fundamentals of parameterized pseudo-elasticity

The framework assumes an incompressible, isotropic body whose material behavior depends on a parameter. It combines a polyconvex energy potential with a history-dependent softening function and considers structural constraints on both components.

2 Fundamentals of parameterized pseudo-elasticity

  • The model assumes an incompressible body that remains isotropic under softening, with material behavior characterized by parameter s.For the examples, s is scalar, though the theoretical framework and model could be extended to vector-valued parameters.Source: §2 · p. 3

2.1 Energy potential

  • The energy potential depends on invariants K̄1 and K̄2 and parameter s; convexity and component-wise monotonicity in the invariants are used as sufficient conditions for polyconvexity.It is also normalized to vanish in the undeformed configuration, which ensures nonnegative energy and zero stress there.Source: §2.1 · p. 3

2.2 Softening function

  • The softening function is decomposed multiplicatively into η1 and η2, with η1 depending on the energy potential and parameter and η2 on the energy-history gap.The factors are nonnegative and bounded in [0, 1], so the combined softening function is also bounded in that range.Source: §2.2 · p. 4
  • A monotonicity constraint on η1 provides a sufficient condition for ellipticity when the energy potential is polyconvex, but ellipticity is not imposed as mandatory.Thermodynamic consistency is associated with a separate monotonicity constraint and normalization that preserves the unsoftened response at the history maximum.Source: §2.2 · p. 4

2.3 Parametric dependency

  • For η1, the study compares unrestricted, jointly increasing, and jointly decreasing variants in energy and parameter, while leaving mixed monotonicity constraints for future work.The energy-potential parameter dependence is additionally restricted by a heuristic monotonicity condition; the investigation is limited to one experimental dataset.Source: §2.3 · pp. 4–5

3 Formulation of neural softening functions

The section builds neural softening functions from feed-forward neural networks and incorporates mathematical restrictions into their architectures. It develops formulations with different levels of flexibility and a priori constitutive-condition satisfaction.

3.1 Neural network architectures

  • The architectures are scalar-valued feed-forward networks, with single- and two-layer forms and tanh or sigmoid activations.The trainable parameters are weights and biases.Source: §3.1 · p. 5
  • The network architectures incorporate restrictions including zero normalization, boundedness, and monotonicity to construct softening functions.The paper introduces the decomposed form η = 1 − η1(ψ0, s)η2(∆ψ), with sufficient conditions for thermodynamic consistency and ellipticity.Source: §3.1 · p. 5, §3.2 · p. 5

3.2 Neural softening functions

  • The Ogden–Roxburgh model uses constant η1 and a constrained single-layer tanh representation for η2.Its parameters ensure η2 ∈ [0, 1], ∂∆ψη2 ≥ 0, and η2(0) = 0.Source: §3.2 · p. 6
  • The formulations extend the Ogden–Roxburgh reference with more flexible neural ansatz functions for the softening response.The paper interprets Ogden–Roxburgh itself as a single-layer tanh network.Source: §3.2 · p. 6, Figure 3 · p. 6
  • PANN–const retains constant η1 but replaces the single-layer η2 representation with a more flexible two-layer tanh network.The two-layer η2 representation is constrained to ensure boundedness, monotonicity, and zero normalization.Source: §3.2 · p. 6, Figure 3 · p. 6
  • PANN–dec and PANN–inc use two-layer tanh networks for η1, bounded between zero and one when ψ0 and s are non-negative.Their η1 functions are decreasing and increasing, respectively.Source: §3.2 · p. 6
  • PANN–unr omits η1 weight and bias constraints, while PANN–sm uses sigmoid activation and a decreasing η1.PANN–unr lacks constructed boundedness and monotonicity; PANN–sm is bounded between zero and one.Source: §3.2 · p. 6
  • Ogden–Roxburgh and PANN–const/dec/sm satisfy all considered conditions a priori; PANN–inc lacks ellipticity and PANN–unr satisfies only zero normalization.Figure 4 presents condition fulfillment for the investigated models using a polyconvex energy potential.Source: §3.2 · p. 6, Figure 4 · p. 6

4 Application to experimental data

The application tests neural softening representations on curative-varied EPDM data and finds that constrained PANN–dec predicts accurately across the tested load cases and parameter values.

4.1 Considered data

  • The data cover five carbon-black-filled EPDM compositions with varying curative content; only the first cycle at each new maximum strain is considered.Pure shear cases are interpolation in deformation space, and the data are split into calibration, validation, and test sets.Source: §4.1 · p. 7, Table 1 · p. 7
  • The pseudo-elastic framework omits rate- and temperature-dependence and additional softening during repeated cycles at a fixed maximum deformation.The authors identify these omitted effects as limitations of the modeling approach.Source: §4.1 · p. 7

4.2 Considered models and calibration strategy

  • Calibration fits the virgin energy potential first, then holds it fixed while fitting each softening model, with 100 random initializations per model.The energy potential is identical across softening models, so their differences arise from the softening function.Source: §4.2 · p. 8, §4.3 · p. 8

4.3 Model performance on experimental data

  • PANN–const, PANN–inc, and PANN–dec have narrow MSE distributions and outperform Ogden–Roxburgh on calibration, validation, and test datasets.Exploratory PANN–unr and PANN–sm show wider MSE spreads, especially on validation and test data.Source: Figure 5 · p. 8, §4.3 · p. 9
  • Replacing Ogden–Roxburgh’s single-layered η2 with PANN–const’s two-layer network reduces MSE and improves stress predictions.PANN–const can represent a steep initial decrease followed by a slower decrease in softening versus Δψ; both models lack direct parameter dependence.Source: §4.3 · p. 9, Figure 6 · p. 9
  • PANN–dec mainly exploits parameter dependence, while PANN–inc is nearly independent of the parameter and reduces softening at small energy values.PANN–dec has η1 ≈0.97 for EPDM–0.4 and ≈0.82 for EPDM–1.9 and EPDM–2.4; PANN–inc rises from ≈0.85 at ψ0 = 0 MPa to ≈1 for ψ0 ≥5 MPa.Source: §4.3 · p. 10, Figure 7 · p. 10
  • Models satisfying the same constitutive conditions can still differ outside calibration, and unrestricted or SM architectures show unreliable extrapolation.Only about a third of PANN–unr instances yield physically reasonable predictions; some predict negative η1 beyond calibration, while PANN–sm instances can show rapid softening-function collapse.Source: §4.3 · p. 10
  • PANN–dec accurately predicts primary, unloading, and reloading paths across load cases and parameter values, with the largest deviations for EPDM–0.4 equibiaxial unloading and reloading.Its pure-shear predictions also agree well with experimental data; equibiaxial predictions are excellent for larger curative contents.Source: §4.3 · pp. 10–11, Figure 8 · p. 11

5 Conclusion

The paper formulates parameter-dependent neural softening functions for pseudo-elastic Mullins models and derives conditions that can ensure ellipticity. Its findings highlight the value of structural constraints for physically reasonable predictions and calibration robustness, while identifying effects and datasets for future investigation.

5 Conclusion

  • The study introduces a multiplicative softening-function decomposition for parameter dependence and derives sufficient conditions for pseudo-elastic ellipticity.The ellipticity conditions include polyconvexity of the energy potential and monotonicity of the neural softening function.Source: §5 · p. 12
  • Neural softening functions extend the classical pseudo-elastic framework while retaining common constitutive conditions and may reduce the need to select a classical function for each material.The evaluation uses multiaxial data for carbon black-filled EPDM rubber with varying curative content.Source: §5 · p. 12
  • Constrained softening formulations exclude physically unreasonable predictions by construction and show narrow error distributions across repeated calibrations.Unconstrained models can achieve low calibration errors but frequently produce physically unreasonable predictions.Source: §5 · p. 12
  • Models satisfying the same constitutive conditions can perform vastly differently, as illustrated by hyperbolic-tangent and sigmoid softening functions.The paper reports this contrast despite the functions satisfying the same conditions.Source: §5 · p. 12
  • Future work should address temperature dependence, recovery, softening-induced anisotropy, viscoelastic coupling, and additional experimental datasets.The additional datasets are intended to clarify formulations for different material behaviors.Source: §5 · p. 12

A Ellipticity in pseudo-elastic Mullins modeling

This appendix derives an ellipticity condition for the considered incompressible pseudo-elastic Mullins framework. It gives sufficient, but not necessary, conditions for ellipticity and shows how they apply to multiplicatively decomposed softening functions.

A Ellipticity in pseudo-elastic Mullins modeling

  • Unloading or reloading denotes elastic paths, while primary paths are those on which softening occurs; the distinction is needed to calculate the consistent tangent modulus.The normalization gives η ≤ 1 during unloading or reloading and η = 1 on primary paths.Source: A Ellipticity in pseudo-elastic… · p. 13
  • The ellipticity condition is derived for both unloading or reloading paths and primary loading paths, using the corresponding consistent tangent modulus.The derivation inserts a traveling-wave ansatz into the linearized balance of momentum; the condition corresponds physically to real-valued wave speeds.Source: A Ellipticity in pseudo-elastic… · p. 13
  • Sufficient, but not necessary, conditions for ellipticity include a polyconvex unsoftened energy potential and bounded, appropriately monotone softening components.For η = 1 − η₁(ψ₀)η₂(Δψ), the stated conditions require η₁, η₂ ∈ [0, 1], ∂ψ₀η₁ ≤ 0, and ∂Δψη₂ ≥ 0.Source: A Ellipticity in pseudo-elastic… · pp. 13–14
  • The sufficient conditions apply to multiplicatively decomposed softening functions, and PANN–const, PANN–dec, and PANN–sm are elliptic by construction under the stated assumptions.The classical Ogden–Roxburgh model is also elliptic when it uses a polyconvex energy potential.Source: A Ellipticity in pseudo-elastic… · p. 14
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