## 2 Natural Neighbour Radial Point Interpolation Method
### Material Homogenisation Technique
Using the homogenisation principle, the macroscale stress, \(\tilde{\sigma}\), and strain, \(\tilde{\epsilon}\), at material point \(\mathbf{x}_{I}\) can be calculated with the volume average of the stresses, \(\tilde{\sigma}\), and strains, \(\tilde{\epsilon}\), on the RVE's volume \(\hat{\Omega}\),  
\[\tilde{\sigma}(\mathbf{x}_{I})=\frac{1}{\hat{\Omega}}\int_{ \Omega}\tilde{\sigma}(\mathbf{\tilde{x}}_{I})\cdot d\tilde{\Omega}=\frac{1}{ \sum_{I=1}^{n_{Q}}\tilde{\omega}_{I}}\sum_{I=1}^{n_{Q}}\tilde{\sigma}( \mathbf{\tilde{x}}_{I})\cdot\tilde{\omega}_{I} \tag{39}\] \[\tilde{\epsilon}(\mathbf{x}_{I})=\frac{1}{\hat{\Omega}}\int_{ \Omega}\tilde{\epsilon}(\mathbf{\tilde{x}}_{I})\cdot d\hat{\Omega}=\frac{1}{ \sum_{I=1}^{n_{Q}}\tilde{\omega}_{I}}\sum_{I=1}^{n_{Q}}\tilde{\epsilon}( \mathbf{\tilde{x}}_{I})\cdot\tilde{\omega}_{I} \tag{40}\]  
with \(n_{Q}\) being the number of microscale integration points discretising the RVE and \(\tilde{\omega}_{I}\) their corresponding microscale integration weight. A fundamental principle in sub-scale modelling, the Hill-Mandel principle, ensures consistency between the macroscopic and microscopic energy states [11]. It stipulates that for a model to be energetically valid, the deformation energy at the macroscopic level must be equivalent to the average work performed by the stresses at the microscale. Mathematically, this relationship can be expressed as  
\[\tilde{\sigma}(\mathbf{x}_{I})\cdot\tilde{\epsilon}(\mathbf{x}_{I})=\frac{1 }{\hat{\Omega}}\int_{\Omega}\tilde{\sigma}(\mathbf{\tilde{x}}_{I})\cdot \tilde{\epsilon}(\mathbf{\tilde{x}}_{I})\cdot d\hat{\Omega} \tag{41}\]  
which means that the virtual work expressed in Equation (29) is valid for the RVE. Thus, considering a microscopic virtual displacement \(\delta\mathbf{\tilde{u}}\) and neglecting body forces, applying the virtual work principle to the RVE leads to  
\[\delta\mathbf{\tilde{u}}^{T}\cdot\int_{\hat{\Omega}}\mathbf{B}(\mathbf{ \tilde{x}}_{I})^{T}\cdot\mathbf{c}(\mathbf{\tilde{x}}_{I})\cdot\mathbf{B}( \mathbf{\tilde{x}}_{I})\cdot\mathbf{\tilde{u}}\cdot d\hat{\Omega}=\delta \mathbf{\tilde{u}}^{T}\cdot\int_{\tilde{\Gamma}}\mathbf{H}(\mathbf{\tilde{x}} _{I})^{T}\cdot\mathbf{\tilde{t}}\cdot d\tilde{\Gamma} \tag{42}\]  
with \(\mathbf{\tilde{t}}\) being the traction force applied in the RVE's boundary surface \(\tilde{\Gamma}\). Knowing that \(\tilde{\epsilon}(\mathbf{\tilde{x}}_{I})=\mathbf{B}(\mathbf{\tilde{x}}_{I}) \cdot\mathbf{\tilde{u}}\), it is possible to write  
\[\int_{\hat{\Omega}}\mathbf{B}(\mathbf{\tilde{x}}_{I})^{T}\cdot\mathbf{c}( \mathbf{\tilde{x}}_{I})\cdot\tilde{\epsilon}(\mathbf{\tilde{x}}_{I})\cdot d \hat{\Omega}=\int_{\tilde{\Gamma}}\mathbf{H}(\mathbf{\tilde{x}}_{I})^{T} \cdot\mathbf{\tilde{t}}\cdot d\tilde{\Gamma}=\mathbf{\tilde{t}} \tag{43}\]  
This relation allows us to impose macroscale deformation fields into the RVE and obtain corresponding equivalent forces. In order to obtain the homogenised elastic properties of the 2D RVE, the plane strain problem will be assumed. This simplification is valid as long as one of the dimensions is much larger than the other two. In this work, at the macroscale, it will be assumed that the solid \(Oz\) direction is much larger than the \(Ox\) or \(Oy\) directions. Consequently, the 2D RVE also possesses a theoretical infinite \(Oz\) length. For example, if a circular void exists in the RVE, the void will be extended along \(Oz\) direction, creating an infinite cylinder. Thus, the plain strain condition considers that \(\epsilon_{zz}=0\), \(\gamma_{xz}=0\) and \(\gamma_{yz}=0\). However, \(\sigma_{zz}\
eq 0\). Using Voigt notation, it is possible to write the generalised [PERSON]'s law for the plane strain condition as [35]\[ \begin{cases}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \tau_{xy}\end{cases} =\mathbf{c}_{[4\times 4]}\cdot \begin{cases}\epsilon_{xx}\\ \epsilon_{yy}\\ 0\\ \gamma_{xy}\end{cases} = \begin{bmatrix}C_{11}&C_{12}&C_{13}&0\\ C_{21}&C_{22}&C_{23}&0\\ C_{31}&C_{32}&C_{33}&0\\ 0&0&0&C_{44}\end{bmatrix} \cdot