Stress Calculation Based On Cauchy Born Rule And Neural Network

In material physics,the first Piola-Kirchhoff?PK?stress is a relatively important research property.The atomic-continuous medium model based on the Cauchy Born rule for calculating the first PK stress at finite temperature is an already developed method which is superior to traditional molecular dynamics simulation?MD?and Monte Carlo simulation?MC?.At the same time,the neural network in deep learning and its derivative algorithms have achieved good results in both industry and academia.We hope to apply the neural network method to the calculation of the first PK stress,in which the data set is simulated by molecular dynamics.Based on the above ideas,this article has done the following work:?1?Completion of the formula for calculating the first PK stress based on the Cauchy Born rule for the three-component alloy Co₂FeAl of the L2₁structure:intro-ducing the virial term,using the quasi-harmonic approximation method for the second-order Taylor expansion of the virial term.The stress is approximated and then converted into reciprocal space by Fourier transform for simplification.The results calculated by the Cauchy Born rule are compared with those of the molecular dynamics simulation.The error is analyzed to show the validity of the Cauchy Born rule for calculating the first PK stress:the relative error is within 8%.In this process,it is verified that the accuracy of the approximate value in the reciprocal space is improved,and the influence of the atomic mass on the approximation result is explored.?2?Review the reference system invariance and material symmetry in the physical model,and generate the invariant subgroup and the base vector to perform the first and last layers of the neural network according to the constraints that the BCC structure of the material Fe needs to satisfy.At the same time,the deformation of material Fe at zero temperature and the corresponding stress data were obtained by molecular simulation method.The training set,verification set and test set were divided,and the model with the lowest loss function on the verification set was selected to evaluate the test set:relative error is within 3%.