Experiment-informed finite-strain inverse design of spinodal metamaterials

TL;DR

The study tackles the challenge of designing spinodal metamaterials for large deformations where nonlinear mechanisms complicate predictions and data are scarce. It introduces a physics‑enhanced forward model built from two convex PICNNs that yield a nonconvex energy density $W(\varepsilon,\boldsymbol{\Theta})$ whose derivative gives the stress, and it uses gradient-based optimization to perform inverse design with sparse experimental data. A dataset of $N=107$ morphologies fabricated into $321$ samples across three loading directions supports training and validation, while nonlinear FE analyses connect deformation pathways to surface curvature via energy absorption metrics and the normal participation factor $\langle \eta \rangle$. The framework achieves accurate predictions and demonstrates the ability to reach unseen target responses, offering a scalable design route for high-energy-absorption metamaterials and enabling extension to other architected materials and loading scenarios.

Abstract

Spinodal metamaterials, with architectures inspired by natural phase-separation processes, have presented a significant alternative to periodic and symmetric morphologies when designing mechanical metamaterials with extreme performance. While their elastic mechanical properties have been systematically determined, their large-deformation, nonlinear responses have been challenging to predict and design, in part due to limited data sets and the need for complex nonlinear simulations. This work presents a novel physics-enhanced machine learning (ML) and optimization framework tailored to address the challenges of designing intricate spinodal metamaterials with customized mechanical properties in large-deformation scenarios where computational modeling is restrictive and experimental data is sparse. By utilizing large-deformation experimental data directly, this approach facilitates the inverse design of spinodal structures with precise finite-strain mechanical responses. The framework sheds light on instability-induced pattern formation in spinodal metamaterials -- observed experimentally and in selected nonlinear simulations -- leveraging physics-based inductive biases in the form of nonconvex energetic potentials. Altogether, this combined ML, experimental, and computational effort provides a route for efficient and accurate design of complex spinodal metamaterials for large-deformation scenarios where energy absorption and prediction of nonlinear failure mechanisms is essential.

Figures (6)

• Figure 1: Spinodal morphology design space defined by a three-parameter $\boldsymbol{\Theta}$ representation (center). Three design subspaces, defined by the non-zero dimensionality of the $\boldsymbol{\Theta} = [\theta_1,\theta_2,\theta_3]$ vector are represented by color clouds encompassing the designs used for training. The one-dimensional (1D) subspace corresponding to lamellar morphologies was represented by a non-zero $\theta_3$, while the 2D and 3D subspaces subsequently added non-zero $\theta_2$ and $\theta_3$ parameters and corresponded to columnar and cubic morphologies, respectively. Three representative designs of increasing norm $|\boldsymbol{\Theta}|$ are presented within each subspace, along with pole figures denoting the directional probability of normal vectors $\boldsymbol{n}$.
• Figure 2: Results and analysis of nanomechanical experiments. (a) In situ snapshots $i$--$iv$ showing the progression of deformation for the $\boldsymbol{\Theta}=[0^\circ,0^\circ,33^\circ]$ morphology along the $\hat{\boldsymbol{e}}_1$ direction, up to 40% strain. Scale bars, 20 µm. (b) Qualitative structure-to-response relations enabled by spherical pole figures denoting the directional surface-normal distributions for representative lamellar (left), columnar (center), and cubic (right) morphologies---accompanied by corresponding stress-strain responses along the $\hat{\boldsymbol{e}}_i$ directions. The pole figures serve as a proxy for structural anisotropy, with higher surface-normal distributions along a given direction correlating to a more compliant response. The lamellar and columnar morphologies exhibited negative-stiffness regions corresponding to nonlinear buckling (as marked for the lamellar sample shown in (a)), along with stiffening at large deformations due to self-contact of shells. (c) Range of finite stress-strain behaviors across the training data as observed from ex situ compressions along the three principal directions $\hat{\boldsymbol{e}}_1$ (left), $\hat{\boldsymbol{e}}_2$ (center), $\hat{\boldsymbol{e}}_3$ (right) highlighted on a generic spinodal morphology. The black lines denote the stress bounds across the training dataset, while color-coded responses correspond to the three representative morphologies shown in (b).
• Figure 3: Physics-enhanced deep learning framework. (a) Left: The uniaxial compression stress response (as a function of applied strain $\varepsilon$) of spinodal metamaterials is modeled as the derivative of a deep neural network-based nonconvex energy density potential $W(\varepsilon,\boldsymbol{\Theta})$. The model consists of two potentials convex in $\varepsilon$ and given by separate neural networks: $W_1(\varepsilon,\boldsymbol{\Theta})$ with the energy and stress vanishing at $\varepsilon=0$ by construction; $W_2(\varepsilon,\boldsymbol{\Theta})$ with the energy $(v+W_1(b,\boldsymbol{\Theta}))$ and vanishing stress at $\varepsilon=b$. Both $b(\boldsymbol{\Theta})$ and $v(\boldsymbol{\Theta})$ are also given by neural networks. The nonconvex potential $W$ is obtained by a combination of $W_1$ and $W_2$. Right: The stress is obtained by differentiating $W$ with respect to $\varepsilon$. Also shown are the derivatives of $W_1$ and $W_2$ for reference. (b) Schematic of the partial input convex neural network (PICNN) architecture for $W_1$ and $W_2$ and their combination thereof. The PICNN architecture predicts an energy which is convex with respect to the strain $\varepsilon$ (via convex path) and parameterized by the design parameters $\boldsymbol{\Theta}$ (via nonconvex path). See SI Appendix Machine Learning Framework section for details.
• Figure 4: Forward model results. (a) Stress-strain plots showing the ML-predicted (teal) versus experimental ground truth (red) curves for three representative spinodal design in the test dataset (i.e., sample from outside the training dataset). For reference, we show the derivatives $\partial W_1/\partial\varepsilon$ (dark blue) and $\partial W_2/\partial\varepsilon$ (light blue). (b) Distribution of predicted vs. ground truth values on the test dataset for normalized stress, normalized incremental stiffness (i.e., slope of curve), and normalized cumulative energy absorbed (i.e., area under curve) at all strain increments. The dashed line represents the ideal line with zero intercept and unit slop; $R^2$ denotes the corresponding goodness-of-fit.
• Figure 5: Finite-strain simulations and normal participation factor. (a) In situ snapshots for $\boldsymbol{\Theta}=[0^\circ,23^\circ,37^\circ]$ loaded in the $\hat{\boldsymbol{e}}_1$ direction for $1\%$, $10\%$, and $20\%$ strain points. Scale bars, 10 µm. (b) Comparison between in situ experiment (green line) and simulation (blue line) for the normalized stress-strain curves up to 20% strain. (c) Distribution of energy mechanisms as a function of strain, including plastic dissipation, elastic strain energy, and frictional dissipation. Up to 20 % strain, friction effects are negligible and plastic dissipation is the dominant mode of dissipation. (d) Geometric representations of the normal participation factor $\eta$ (NPF) for three representative cases and its correlation to the equivalent plastic strain $\bar{\varepsilon}_p$. The dark purple regions correspond to $\eta \approx 1$ which correlates to regions that undergo high plastic deformation (gold regions) for the cases shown in (i) and (ii), while diminished correlation occurs in the buckling-prone case in (iii). (e) Linear correlation between total energy dissipation and $\eta$, where the data points corresponding to the three cases in (d) are indicated, showing a loss of correlation for geometries that undergo buckling events.