A dual-stage constitutive modeling framework based on finite strain data-driven identification and physics-augmented neural networks

TL;DR

This work addresses the challenge of deriving hyperelastic constitutive laws from experimentally accessible data by proposing a dual-stage framework that first builds a stress–strain dataset via Data-Driven Identification (DDI) from full-field displacement data and boundary forces, and then calibrates a Physics-Augmented Neural Network (PANN) that respects thermodynamic and material-symmetry constraints. It introduces three finite-strain DDI formulations (updated Lagrangian, original total Lagrangian, and adapted total Lagrangian) and a polyconvex, invariants-based PANN for isotropic elasticity, validated on synthetic data and extended to noisy scenarios with 3D FE applications. The results show that the framework can accurately recover material behavior from plane-stress data, remains robust under measurement noise, and yields competent 3D predictions, highlighting its potential for automated, data-driven constitutive modeling. This approach provides a path toward automated, physics-consistent constitutive modeling from experiments, with future directions including anisotropy and viscoelastic/plastic extensions.

Abstract

In this contribution, we present a novel consistent dual-stage approach for the automated generation of hyperelastic constitutive models which only requires experimentally measurable data. To generate input data for our approach, an experiment with full-field measurement has to be conducted to gather testing force and corresponding displacement field of the sample. Then, in the first step of the dual-stage framework, a new finite strain Data-Driven Identification (DDI) formulation is applied. This method enables to identify tuples consisting of stresses and strains by only prescribing the applied boundary conditions and the measured displacement field. In the second step, the data set is used to calibrate a Physics-Augmented Neural Network (PANN), which fulfills all common conditions of hyperelasticity by construction and is very flexible at the same time. We demonstrate the applicability of our approach by several descriptive examples. Two-dimensional synthetic data are exemplarily generated in virtual experiments by using a reference constitutive model. The calibrated PANN is then applied in 3D Finite Element simulations. In addition, a real experiment including noisy data is mimicked.

Figures (16)

• Figure 1: Dual-stage framework: Using DDI in order to generate material database $\mathcal{D}^*$ of stress-strain states based only on measurable full-field data $\boldsymbol{u}(\boldsymbol{X},t)$ and global testing force $\boldsymbol{F}(t)$ from a (virtual) experiment as well as boundary conditions. PANN serves as constitutive model trained on determined material database and can be used for prediction of unseen data.
• Figure 2: Starting point of DDI: (a) continuous specimen $\mathcal{B}$ under loading conditions with known stress vector $\boldsymbol{t_{\!\boldsymbol{f}}}(t)$ and unknown stress vector $\boldsymbol{t_{\boldsymbol{\zeta}}}(t)$, known displacement field $\boldsymbol{u}$ without rigid body translation and unknown material data base consisting of material states $(e_{kl}^{*z},\sigma_{kl}^{*z})$, which are associated to the material points $\boldsymbol{X}$ at time $t$ by the continuous mapping $\hat{s}(\boldsymbol{X}, t)$ as well as (b) specimen discretized into subdomains ${}^\tau\!\Omega^e$ under loading conditions with known nodal forces ${}^\tau\!\!\boldsymbol{f}^\alpha$ and unknown nodal forces ${}^\tau\!\boldsymbol{\zeta}^\beta$ at snapshot $\tau$ and unknown material data base consisting of material states $(e_{kl}^{*z},\sigma_{kl}^{*z})$, which are associated to the quadrature point $g$ of linear element $e$ at snapshot $\tau$ by the discrete mapping $s(g, \tau)$.
• Figure 3: Uniaxial tensile test of compressible neo-Hooke material with $E:=1MPa, \nu := 0.3$: (a) homogenous specimen under loading conditions and $11$-component of (b) Euler-Almansi and Green-Lagrange as well as (c) 2nd Piola-Kirchhoff and Cauchy stress tensor.
• Figure 4: Illustration of the PANN for modeling the constitutive behavior of isotropic elastic solids. Note that the hidden layer of the ICNN may be multilayered.
• Figure 5: Benchmark test: (a) boundary conditions with $u_2^\text{max} = 50mm$ for uniaxial tensile test of specimen with reference thickness $h_0 = 5mm$ to be investigated as well as (b) domain of interest, given by $100mm \times 100mm \times 5mm$, for DDI, which contains the interior $\Omega_0$ with the two ellipsoidal holes as well as the boundaries of prescribed and unknown nodal forces.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5