Stochastic Modeling of Anisotropic Strength Surfaces from Atomistic Simulations

Abstract

This work develops a unified framework for inferring, representing, and statistically characterizing an anisotropic strength surface directly from molecular dynamics data. Large-scale tensile loading simulations are used to generate failure data across all principal stress ratios and loading orientations, facilitated by a data-driven mapping between imposed strain-rate tensors and resulting stresses. The orientation-dependent strength surface is then represented using a constrained parametric formulation in which the surface parameters vary smoothly with loading angle through a low-dimensional functional encoding. To deploy the framework, we specifically consider the case of monocrystalline graphene, which is a prototypical two-dimensional material that has been extensively characterized, both experimentally and computationally, in the literature. For defective graphene, multiple random realizations of vacancy defect distributions are used to construct a stochastic ensemble of angular strength surfaces. Because each anisotropic strength surface requires substantial atomistic sampling to construct, the resulting ensemble is inherently limited in size, motivating the use of compact encoding, dimensionality reduction, and probabilistic modeling to characterize strength variability. Dimensionality reduction via Principal Component Analysis reveals a condensed latent representation of the fitted, encoded surfaces, where a Gaussian mixture model is employed to capture defect-induced variability, including rare outlier behaviors arising from clustered vacancy defects. Sampling from this probabilistic model enables the generation of new, physically admissible strength surfaces and the construction of confidence intervals in both parameter space and stress space. (Abstract shortened to meet arXiv limits.)

Figures (19)

• Figure 1: Schematic of a graphene sheet, with armchair and zigzag directions indicated, as well as examples of single vacancy (SV) and double vacancy (DV) defects. Visualized with the OVITO package ovito.
• Figure 2: Schematic overview of the strength surface construction workflow. The top row represents a conventional plane stress strength surface obtained from MD simulations in which the principal stress axes are fixed relative to the lattice. In this setting, failure data lie on a two-dimensional envelope in $(\sigma_1, \sigma_2)$ space and are represented by a strength surface parameterized by constant coefficients $(\alpha, k)$. The bottom row represents generalized data generation in which the principal axes rotate relative to the lattice, introducing an explicit angular dependence $\theta$. The resulting MD failure data populate a three-dimensional space $(\sigma_1, \sigma_2, \theta)$, and fitting yields orientation-dependent strength parameters $\alpha(\theta)$ and $k(\theta)$, represented as continuous functions of loading angle. The fixed-axis strength surface is recovered as a slice of this angularly dependent surface at constant $\theta$, illustrating the natural increase in model complexity required to capture anisotropic fracture behavior.
• Figure 3: Schematic of the stochastic modeling and sampling framework for physically admissible orientation-dependent strength surfaces. Starting from fitted continuous strength parameters $\alpha(\theta)$ and $k(\theta)$, an encoder maps the functions to a discrete coefficient vector $\mathbf z$ (here realized via a truncated Fourier representation). Principal Component Analysis (PCA) is then applied to $\mathbf z$ to obtain a reduced-order latent representation $\boldsymbol \eta$ with dimension $d < 2n$. The distribution of latent variables is modeled using a Gaussian mixture model (GMM), enabling probabilistic sampling in the reduced space. Sampled latent vectors $\hat{\boldsymbol \eta}$ are lifted back to the full coefficient space via inverse PCA and decoded to recover continuous parameter functions $(\hat{\alpha}(\theta), \hat{k}(\theta))$. The encoding, dimensionality reduction, and decoding steps collectively ensure that all sampled strength surfaces remain smooth and physically admissible.
• Figure 4: Visualization of the pristine graphene sheet at failure for (a) 0° uniaxial tension (armchair); (b) 90° uniaxial tension (zigzag); (c) pure biaxial tension.
• Figure 5: Uniaxial tensile strength along varying loading angles, where 0° and 60° correspond to the armchair orientations, and 30° and 90° are the zigzag directions.