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ORIGINAL PAPER

Neural network constitutive model for crystal structures Sunyoung Im1 · Hyungjun Kim1 · Wonbae Kim1 · Maenghyo Cho1 Received: 4 June 2020 / Accepted: 8 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract Neural network constitutive models (NNCMs) for crystal structures are proposed based on computationally generated highfidelity data. Stress, and tangent modulus data are generated under various strain states using empirical potentials and firstprinciples calculations. Strain–stress artificial neural network and strain-tangent modulus ANN are constructed. The symmetry conditions are considered for cubic, tetragonal, and hexagonal structures. The NNCMs of six face-centered cubic materials (Cu, Ni, Pd, Pt, Ag, and Au), two diamond cubic materials (Si, Ge), two tetragonal crystal materials (TiO2 , ZnO), and two hexagonal crystal materials (ZnO, GaN) are constructed and tested under the untrained strain state. In particular, the performance of NNCM for cubic structure is better compared with that of the classical model. The suggested NNCM can be embedded into a nonlinear finite element method, and numerical examples are performed to verify the proposed NNCM. Keywords Neural network constitutive model (NNCM) · Crystal structure · Material nonlinearity · Anisotropic hyperelastic model · Machine learning

1 Introduction Continuum mechanics is based on the compatibility of displacement and strain, equilibrium of force and stress, and constitutive law of strain and stress. The first two are uncertainty-free and error-free, and they are derived from axioms. Compatibility is based on geometry, and equilibrium is derived from Newton’s laws of motion. However, the constitutive law is modeled based on experiments and observations. The experiments require physical equipment and human labor, and can only provide a limited type of data. For example, only information about basic behaviors such as uniaxial biaxial, shear, and hydrostatic properties can be obtained from the experiments. With this limited information, the modeling of the behavior of the material requires many assumptions such as linearity, incompressibility, or the existence of strain energy density function. Thus, the constitutive modeling of materials is difficult and will inevitably involve some uncertainty. In recent years, as the interest in nanoscale devices has increased worldwide, single crystals have become extremely

B 1

Maenghyo Cho [email protected] Multiscale Mechanics Design Major, Department of Mechanical Engineering, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul 151-744, Republic of Korea

important. While the mechanical analysis of crystals is very important in designing nano-electromechanical system, few studies have focused on their behavior. Single crystals can be found in nature or are produced artificially. However, they are difficult to grow as their environment must be carefully controlled [1]. As it is not possible to perform experiments such as tension test on macroscale specimens, it