Physics-informed deep neural network enabled discovery of size-dependent deformation mechanisms in nanostructures

Abstract

The surface effects based on the Gurtin-Murdoch interface model are incorporated into a physics-informed deep neural network (DNN) to enable the discovery of the size-dependent mechanical response in nanoscale structures for the first time. The DNN directly deals with the minimization of a cost function with contributions from total potential energies and penalties representing the displacement boundary conditions. Following this spirit, the Gurtin-Murdoch interface model is directly implemented into the DNN cost function through additional surface energies associated with the surfaces of the nanostructures without resorting to the implementation of Young-Laplace equations explicitly employed in the analytical approaches. The DNN technique is first validated on both displacement and stress fields vis-à-vis the analytical solution for the modified Kirsch problem with the Gurtin-Murdoch interface under far-field shear loading. Then, the DNN technique is critically and vigorously assessed against the Q4, Q8, and Q9 finite-element results of nanocylinders under diametrical loading. Notably, the accuracy of the DNN is demonstrated to be comparable to the analytical, Q8 and Q9 finite element-based results without the apparent stress discontinuities noticeable in the Q4 element. The numerical example indicates that the selection of collocation points in the DNN approach can be made completely random using the Monte-Carlo simulation without sacrificing the accuracy of the predicted stress field. An inverse load identification problem is also considered to demonstrate the special strength of the DNN approach, namely, both the forward and inverse problems can be handled using the same framework. The present contribution provides a compelling alternative and independent means of identifying surface elasticity effect and hence is a good tool in assessing the finite-element method as well as other approaches’ predictive capability for this class of materials.