Manuel K. Rausch

Vahidullah Tac, Manuel K. Rausch, Francisco Sahli-Costabal et al.

We develop a fully data-driven model of anisotropic finite viscoelasticity using neural ordinary differential equations as building blocks. We replace the Helmholtz free energy function and the dissipation potential with data-driven functions that a priori satisfy physics-based constraints such as objectivity and the second law of thermodynamics. Our approach enables modeling viscoelastic behavior of materials under arbitrary loads in three-dimensions even with large deformations and large deviations from the thermodynamic equilibrium. The data-driven nature of the governing potentials endows the model with much needed flexibility in modeling the viscoelastic behavior of a wide class of materials. We train the model using stress-strain data from biological and synthetic materials including humain brain tissue, blood clots, natural rubber and human myocardium and show that the data-driven method outperforms traditional, closed-form models of viscoelasticity.

L. River Spencer, Manuel K. Rausch, Chad M. Landis et al.

Magneto-active elastomers exhibit large, nonlinear deformations under combined mechanical loading and magnetic fields, and their effective behavior is strongly governed by microstructural heterogeneity. Predictive modeling of these materials is challenging because their response involves strong magneto-mechanical coupling, large deformations, and the nearly incompressible behavior of elastomeric matrices. Existing multiscale approaches often rely on staggered strategies or formulations that do not robustly treat near-incompressibility in strongly coupled settings. This work presents a fully coupled computational homogenization framework for nearly incompressible magnetoelastic composites in which the mechanical deformation and magnetostatic fields are solved monolithically on a representative volume element (RVE). The microscale problem uses a mixed finite-element discretization with Lagrangian displacement degrees of freedom and a N'ed'elec-based magnetic vector potential, enabling a curl-conforming representation of magnetic induction together with periodic boundary constraints for both mechanical and magnetic fields. Near-incompressibility is treated using J-bar stabilization, in which the volumetric response is controlled by the cell-averaged dilatation while the isochoric response is evaluated using a scaled deformation gradient. The constitutive behavior is derived from an additive free-energy decomposition with hyperelastic, vacuum magnetic, and saturation-type magnetization contributions. The resulting formulation enables robust three-dimensional RVE simulations of heterogeneous magneto-elastic composites with complex particle distributions under large deformations and strong coupling. Numerical examples show how particle interactions, microstructural arrangement, and inclusion compressibility influence deformation patterns and the effective magneto-mechanical response.

Vahidullah Tac, Manuel K Rausch, Ilias Bilionis et al.

Many natural materials exhibit highly complex, nonlinear, anisotropic, and heterogeneous mechanical properties. Recently, it has been demonstrated that data-driven strain energy functions possess the flexibility to capture the behavior of these complex materials with high accuracy while satisfying physics-based constraints. However, most of these approaches disregard the uncertainty in the estimates and the spatial heterogeneity of these materials. In this work, we leverage recent advances in generative models to address these issues. We use as building block neural ordinary equations (NODE) that -- by construction -- create polyconvex strain energy functions, a key property of realistic hyperelastic material models. We combine this approach with probabilistic diffusion models to generate new samples of strain energy functions. This technique allows us to sample a vector of Gaussian white noise and translate it to NODE parameters thereby representing plausible strain energy functions. We extend our approach to spatially correlated diffusion resulting in heterogeneous material properties for arbitrary geometries. We extensively test our method with synthetic and experimental data on biological tissues and run finite element simulations with various degrees of spatial heterogeneity. We believe this approach is a major step forward including uncertainty in predictive, data-driven models of hyperelasticity

Asghar A. Jadoon, Aryan Tyagi, L. River Spencer et al.

Multiscale topology optimization (TO) of hyperelastic materials remains computationally prohibitive due to the repeated solution of microscale boundary value problems. In this work, we present a concurrent multiscale topology optimization framework that overcomes this limitation by leveraging physics-augmented neural networks (PANNs) as surrogate constitutive models. The proposed approach enables the simultaneous optimization of macroscale material distribution and microscale descriptors, within a unified nonlinear finite strain setting. The surrogate models are constructed using input-specific neural networks (ISNNs) that enforce key physical principles directly within the architecture, including convexity and material symmetry through invariant-based representations and structural tensors. This ensures thermodynamic consistency and numerical stability while accurately representing homogenized anisotropic hyperelastic responses. The trained PANNs replace the microscale boundary value problem and provide efficient evaluations of stresses and consistent tangent moduli using analytical first and second derivatives of the neural network, enabling tractable large-scale multiscale optimization. The framework is demonstrated on representative microstructures exhibiting transversely isotropic, cubic anisotropic, and nearly incompressible isotropic behavior. The results show that the proposed method captures complex multiscale interactions and enables physically meaningful spatial tailoring of material properties, while significantly reducing computational cost compared to classical FE$^2$ approaches. These findings establish PANNs as a powerful tool for high-fidelity multiscale topology optimization of nonlinear anisotropic materials.

Adrian Buganza Tepole, Asghar Jadoon, Manuel Rausch et al.

Accurate constitutive models of soft materials are crucial for understanding their mechanical behavior and ensuring reliable predictions in the design process. To this end, scientific machine learning research has produced flexible and general material model architectures that can capture the behavior of a wide range of materials, reducing the need for expert-constructed closed-form models. The focus has gradually shifted towards embedding physical constraints in the network architecture to regularize these over-parameterized models. Two popular approaches are input convex neural networks (ICNN) and neural ordinary differential equations (NODE). A related alternative has been the generalization of closed-form models, such as sparse regression from a large library. Remarkably, all prior work using ICNN or NODE uses the invariants of the Cauchy-Green tensor and none uses the principal stretches. In this work, we construct general polyconvex functions of the principal stretches in a physics-aware deep-learning framework and offer insights and comparisons to invariant-based formulations. The framework is based on recent developments to characterize polyconvex functions in terms of convex functions of the right stretch tensor $\mathbf{U}$, its cofactor $\text{cof}\mathbf{U}$, and its determinant $J$. Any convex function of a symmetric second-order tensor can be described with a convex and symmetric function of its eigenvalues. Thus, we first describe convex functions of $\mathbf{U}$ and $\text{cof}\mathbf{U}$ in terms of their respective eigenvalues using deep Holder sets composed with ICNN functions. A third ICNN takes as input $J$ and the two convex functions of $\mathbf{U}$ and $\text{cof}\mathbf{U}$, and returns the strain energy as output. The ability of the model to capture arbitrary materials is demonstrated using synthetic and experimental data.

Vahidullah Taç, Amirhossein Amiri-Hezaveh, Manuel K. Rausch et al.

We propose a new framework for identifying mechanical properties of heterogeneous materials without a closed-form constitutive equation. Given a full-field measurement of the displacement field, for instance as obtained from digital image correlation (DIC), a continuous approximation of the strain field is obtained by training a neural network that incorporates Fourier features to effectively capture sharp gradients in the data. A physics-based data-driven method built upon ordinary neural differential equations (NODEs) is employed to discover constitutive equations. The NODE framework can represent arbitrary materials while satisfying constraints in the theory of constitutive equations by default. To account for heterogeneity, a hyper-network is defined, where the input is the material coordinate system, and the output is the NODE-based constitutive equation. The parameters of the hyper-network are optimized by minimizing a multi-objective loss function that includes penalty terms for violations of the strong form of the equilibrium equations of elasticity and the associated Neumann boundary conditions. We showcase the framework with several numerical examples, including heterogeneity arising from variations in material parameters, spatial transitions from isotropy to anisotropy, material identification in the presence of noise, and, ultimately, application to experimental data. As the numerical results suggest, the proposed approach is robust and general in identifying the mechanical properties of heterogeneous materials with very few assumptions, making it a suitable alternative to classical inverse methods.