Neural network constitutive model for crystal structures

Abstract

Neural network constitutive models (NNCMs) for crystal structures are proposed based on computationally generated high-fidelity data. Stress, and tangent modulus data are generated under various strain states using empirical potentials and first-principles 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.

Appendix A: Strain energy density function of anisotropic crystal structures

Hyperelasticity is the constitutive relation in which the material status can be described only with the given strain. A strain energy density is a function of strain, and the stress and the tangent modulus are obtained by differentiating the strain energy density with respect to strain.

For a crystal structure, Smith and Rivlin introduced form-invariant quantities based on symmetry transformation [44], and the form of the strain energy function was determined for each of the crystal classes. The polynomial-based strain energy density function for anisotropic materials [45] can be expressed as.

$$ \varPsi = c_{ij \ldots k} L_{1}^{i} L_{2}^{j} \ldots L_{N}^{k} $$

(1)

where ${c}_{ij\cdots k}$ are material constants, L_i is the element of the polynomial basis associated with the given crystal class, and N is the number of polynomial bases.

Based on this model, a new anisotropic hyperelastic model for crystal structures can be derived by limiting the order of polynomial bases to three.

$$ \varPsi = \mathop \sum \limits_{i = 1}^{N} c_{i} L_{i} + \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} c_{ij} L_{i} L_{j} + \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{k = 1}^{N} c_{ijk} L_{i} L_{j} L_{k} $$

(2)

where $c_{i}$, $c_{ij}$, and $c_{ijk}$ are the material constants, and $L_{i}$ is the element of the polynomial basis, which is expected to be a function of the form invariants.

Based on the abovementioned model, Kim et al. presented a specific hyperelastic model for crystal structures, using FCC and diamond cubic systems [15]. They directly employed the nine strain invariants of the crystal structure as polynomial bases of the model. These nine strain invariants were as follows:

$$ \begin{aligned}&J_{1} = L_{1} = \sum G_{11} = G_{11} + G_{22} + G_{33} \\ &J_{2} = L_{2} = \sum G_{11} G_{22} = G_{22} G_{33} + G_{33} G_{11} + G_{11} G_{22}\\ &J_{3} = L_{3} = G_{11} G_{22} G_{33}\\ &J_{4} = L_{4} = \sum G_{23}^{2} = G_{23}^{2} + G_{31}^{2} + G_{12}^{2}\\ &J_{5} = L_{5} = \sum G_{23}^{2} G_{31}^{2} = G_{23}^{2} G_{31}^{2} + G_{31}^{2} G_{12}^{2} + G_{12}^{2} G_{23}^{2}\\ & J_{6} = L_{6} = G_{23} G_{31} G_{12}\\ &J_{7} = L_{7} = \sum G_{11} \left( {G_{31}^{2} + G_{12}^{2} } \right){ } = G_{11} \left( {G_{31}^{2} + G_{12}^{2} } \right) \\ &\quad + G_{22} \left( {G_{12}^{2} + G_{23}^{2} } \right) + G_{33} \left( {G_{23}^{2} + G_{31}^{2} } \right)\\ &J_{8} = L_{8} = \sum G_{11} G_{31}^{2} G_{12}^{2} = G_{11} G_{31}^{2} G_{12}^{2} \\ &\quad + G_{22} G_{12}^{2} G_{23}^{2} + G_{33} G_{23}^{2} G_{31}^{2}\\ &J_{9} = L_{9} = \sum G_{23}^{2} G_{22} G_{33} = G_{23}^{2} G_{22} G_{33} \\ &\quad + G_{31}^{2} G_{33} G_{11} + G_{12}^{2} G_{11} G_{22} \end{aligned}$$

(3)

where

$$ G_{ij} = \frac{{\partial x_{m} }}{{\partial X_{i} }}\frac{{\partial x_{m} }}{{\partial X_{j} }} - \delta_{ij} $$

(4)

Here, $X_{i}$ is the undeformed coordinate, $x_{i}$ is the deformed coordinate, and $\delta_{ij}$ is Kronecker delta. $G_{ij}$ is twice the Green-Lagrangian strain $E_{ij} \left( {G_{ij} = 2E_{ij} } \right)$. In the abovementioned equations, the first three polynomial bases $\left( {L_{1} ,L_{2} {\text{, and }}L_{3} } \right)$ are related to normal strains; the next three polynomial bases $\left( {L_{4} ,L_{5} {\text{, and }}L_{6} } \right)$ only include shear strain terms and the last three polynomial bases $\left( {L_{7} ,L_{8} {\text{, and }}L_{9} } \right)$ consist of both normal and shear strain terms.