## 2 Natural Neighbour Radial Point Interpolation Method
### Material Homogenisation Technique  
First presented by [PERSON] [11], the concept of Representative Volume Element (RVE) aims to characterise the microstructure of a material through a representative sub-region. An RVE represents a statistically significant sub-region of the material that captures the essential microstructural features. It is crucial for the RVE to be statistically representative, including a sufficient sampling of all microstructural heterogeneities within the multi-phase material. The appropriate size of the RVE is a critical parameter in multi-scale modelling. It needs to be large enough to capture the relevant heterogeneities but remain small compared to the overall macroscopic domain size. This section uses the RVE concept to present a micromechanical approach for determining the effective elastic properties of an one-phase material with distinct volume fractions. An RVE, with volume domain \(\hat{\Omega}\), is associated with a macroscale material point \(\mathbf{x}_{I}\) (whose Cartesian coordinates \(\mathbf{x}_{I}\) are defined at the macroscale). Then, at the microscale, there are infinitesimal points \(\bar{\mathbf{x}}\in\hat{\Omega}\), whose set represent the macroscale point \(\mathbf{x}\). A local deformation at point \(\mathbf{x}\) leads to a perturbation of the RVE equilibrium. In the deformed configuration, the Cartesian coordinates of material point \(\mathbf{x}\) are represented by \(\mathbf{X}\), which can be written as \(\mathbf{X}=\phi(\mathbf{x})\), with the motion of \(\mathbf{x}\) represented by function \(\phi\). Thus, it is possible to define the displacement of material point \(\mathbf{x}\) with:  
\[\mathbf{u}(\mathbf{x})=\phi(\mathbf{x})-\mathbf{x}=\mathbf{X}-\mathbf{x} \tag{35}\]  
which leads to,  
\[\mathbf{X}=\mathbf{x}+\mathbf{u}(\mathbf{x}) \tag{36}\]  
This expression allows us to define the deformation gradient, \(\mathbf{F}(\mathbf{x})\), as a function of the displacement:  
\[\mathbf{F}(\mathbf{x})=\frac{\partial\mathbf{X}}{\partial\mathbf{x}}=\frac{ \partial(\mathbf{x}+\mathbf{u}(\mathbf{x}))}{\partial\mathbf{x}}=\mathbf{I}+ \frac{\partial\mathbf{u}(\mathbf{x})}{\partial\mathbf{x}} \tag{37}\]  
where \(\mathbf{I}\) is the second-order identity tensor. Assuming the microscale coordinates of a deformed point belonging to the RVE defined as \(\bar{\mathbf{X}}\), and if a macroscale deformation gradient at material point \(\mathbf{x}\) (i.e., to an RVE) is applied, a microscopic displacement field \(\mathbf{\tilde{u}}(\mathbf{\tilde{X}})\) is produced [48],  
\[\mathbf{\tilde{u}}(\mathbf{\tilde{X}})=(\mathbf{F}(\mathbf{X})-\mathbf{I}) \cdot\mathbf{\tilde{x}}+\mathbf{\tilde{u}}(\mathbf{\tilde{X}}) \tag{38}\]  
in which the first term is the linear displacement field and the second term is the displacement fluctuation field. The RVE displacement field \(\mathbf{\tilde{u}}(\mathbf{\tilde{x}})\) is obtained with the equilibrium equations assuming a specific macroscale deformation gradient to the RVE. Then, with \(\mathbf{\tilde{u}}(\mathbf{\tilde{x}})\), the corresponding RVE's strain and stress fields are obtained for each RVE's integration point, \(\mathbf{\tilde{x}}_{I}\in\hat{\Omega}\).