Discovering constitutive equations of crystal structures by sparse identification

Abstract

In this study, the symbolic constitutive equations of single crystal structures were discovered using the sparse identification (SPID) method with an energy criterion. Six materials belonging to three crystal systems were targeted: cubic structures (Si, Ge), tetragonal crystal structures (rutile TiO2, body-centered-tetragonal with four-atom rings (BCT-4) ZnO), and hexagonal crystal structures (wurtzite GaN, wurtzite ZnO). Using ab initio calculation, the stress and tangent modulus of the six crystal structures were generated as reference data under various deformed states. First, the candidates for constitutive equations were constructed from the form-invariants of crystal, which are based on the generated tangent modulus datasets considering the symmetric conditions of crystal structures. Then, using the SPID method, the insignificant candidate terms were eliminated based on the energy criterion. The coefficients of the remaining terms were then viewed as the discovered material constants of each single crystal structure. Subsequently, the prediction accuracy of the SPID constitutive model based on the discovered material constants was investigated under the test-deformed state, which was not included in the modeling process. Furthermore, the discovered SPID models were embedded in nonlinear finite element analyses and verified.