# A Multifeatured Data-Driven Homogenization for Heterogeneous Elastic Solids
## 2 Overview of Methods
### Homogenization Technique
#### 2.1.2 Micro-Level Equilibrium Equation and Homogenization  
At the micro-level, the equilibrium equation can be simply shown as,  
\[\
abla\cdot\mathbf{\sigma}_{m}=0\ \ \text{in}\ \ \Omega_{m} \tag{9}\]  
where \(\mathbf{\sigma}_{m}\) is the stress at the micro-level, and \(\Omega_{m}\) is the volume of the RVE of microstructures. As described in the preceding section, the Dirichlet boundary is applied in the micro-level domain. In this study, the stress-strain relationship at the micro-level is described either by linear or nonlinear elasticity. In the case of linear elasticity, the relationship between stress and strain can be obtained as,  
\[\mathbf{\sigma}_{m}=\ \mathbb{E}\colon\mathbf{\epsilon}_{m} \tag{10}\]  
where \(\mathbb{E}\) denotes the rank-four tensor of elastic modulus, which can be given as,  
\[\mathbb{E}_{ijkl}=K\delta_{ij}\delta_{kl}+\mu\bigg{(}\delta_{ik}\delta_{jl}+ \delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl}\bigg{)} \tag{11}\]where \(K=\lambda+\frac{2}{3}\mu\), and \(\mu\) and \(\lambda\) are first and second [PERSON]'s coefficients.  
For the nonlinear elastic model, the following hypoelastic constitutive law [31] is considered:  
\[\sigma_{ij}=S_{ij}+\sigma_{kk}\frac{\delta_{ij}}{3}\] \[S_{ij}=\frac{2}{3}\sigma_{e}\frac{e_{ij}}{\varepsilon_{e}}\] \[\sigma_{kk}=\frac{E}{1-2\
u}\frac{1}{3}\varepsilon_{kk}\] \[e_{ij}=\varepsilon_{ij}-\frac{1}{3}\varepsilon_{kk}\delta_{ij} \tag{12}\] \[\varepsilon_{e}=\sqrt{\frac{2}{3}\varepsilon_{ij}\varepsilon_{ij}}\] \[\frac{\sigma_{e}}{\sigma_{0}}=\begin{cases}\sqrt{\frac{1+\mu^{2} }{(n-1)^{2}}-\left(\frac{n}{n-1}-\frac{\varepsilon_{e}}{\varepsilon_{0}} \right)^{2}}-\frac{1}{n-1}&\varepsilon_{e}\leq\varepsilon_{0}\\ \left(\frac{\varepsilon_{e}}{\varepsilon_{0}}\right)^{\frac{1}{n}}&\varepsilon _{e}\geq\varepsilon_{0}\end{cases}\]  
where \(\sigma_{m}\) is the micro stress, and \(\varepsilon_{m}\) is the strain at the micro-level. It is worth mentioning that \(E\) and \(\
u\) denote [PERSON]'s modulus and [PERSON]'s ratio, respectively. The nonlinear FEM procedure is employed, which leads to the discretized weak form as,  
\[\mathbf{K}_{m}du\ =\ -\mathbf{f}_{m}^{int} \tag{13}\]  
where \(\mathbf{K}_{m}=\int_{\Omega_{m}}\mathbf{B}_{m}^{T}\,\mathbf{D}_{m}\)\(\mathbf{B}_{m}\) and \(\mathbf{f}_{m}^{int}=\int_{\Omega_{m}}\mathbf{B}_{m}^{T}\,\mathbf{\sigma}_{m}\) are the micro stiffness matrix, and micro internal force, respectively. \(\mathbf{B}_{m}\) is the derivative of micro shape function and \(\mathbf{D}_{m}\) is the micro consistent tangent stiffness matrix.  
For homogenization of micro-level responses, The nodes should be partitioned into two groups [29]: interior nodes of the RVE, \(x_{a}\in\mathcal{V}\), and exterior nodes, \(x_{b}\in\partial\mathcal{V}\) (Note: \(\mathcal{V}\) is the RVE associated with microstructures). Therefore, the internal force vector \(f(\mathbf{u})\) and associated tangent \(K(\mathbf{u})\) of the discretized microstructure can be partitioned as,  
\[\mathbf{f}=\begin{bmatrix}\mathbf{f}_{a}\\ \mathbf{f}_{b}\end{bmatrix},\ \ \ \ \ K=\begin{bmatrix}\mathbf{K}_{aa}&K_{ab}\\ K_{ba}&K_{bb}\end{bmatrix} \tag{14}\]  
The homogenized tangent stiffness matrix can be obtained as,  
\[\widetilde{\mathbf{K}}_{bb}=\mathbf{K}_{bb}-\,\mathbf{K}_{ba}\mathbf{K}_{aa}^{-1}\mathbf{K}_{ab} \tag{15}\]  
\[\mathbb{C}=\frac{1}{|\mathcal{V}|}\,\mathbb{D}\widetilde{\mathbf{K}}_{bb}\mathbb{D }^{T} \tag{16}\]  
where \(\mathbb{C}\) is the homogenized tangent stiffness. Finally, for the homogenization of stress, we have,  
\[\bar{\mathbf{\sigma}}=\,\frac{1}{|\mathcal{V}|}\mathbb{D}^{T}\mathbf{f}_{a} \tag{17}\]  
where \(\mathcal{V}\) is the volume of the RVE of microstructures.