Lossless multi-scale constitutive elastic relations with artificial intelligence

Abstract

Abstract A seamless and lossless transition of the constitutive description of the elastic response of materials between atomic and continuum scales has been so far elusive. Here we show how this problem can be overcome by using artificial intelligence (AI). A convolutional neural network (CNN) model is trained, by taking the structure image of a nanoporous material as input and the corresponding elasticity tensor, calculated from molecular statics (MS), as output. Trained with the atomistic data, the CNN model captures the size- and pore-dependency of the material’s elastic properties which, on the physics side, derive from its intrinsic stiffness as well as from surface relaxation and non-local effects. To demonstrate the accuracy and the efficiency of the trained CNN model, a finite element method (FEM)-based result of an elastically deformed nanoporous beam equipped with the CNN as constitutive law is compared with that obtained by a full atomistic simulation. The trained CNN model predicts the elasticity tensor in the test dataset with a root-mean-square error of 2.4 GPa (3.0% of the bulk modulus) when compared to atomistic calculations. On the other hand, the CNN model is about 230 times faster than the MS calculation and does not require changing simulation methods between different scales. The efficiency of the CNN evaluation together with the preservation of important atomistic effects makes the trained model an effective atomistically informed constitutive model for macroscopic simulations of nanoporous materials, optimization of nanostructures, and the solution of inverse problems.