# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 2 Meshless Methods
### Discrete System of Equations  
Being a discrete numerical technique, RPIM uses the Galerkin weak formulation to construct the global system of equations. Thus, considering the solid domain \(\Omega\), with a body force \(\mathbf{b}\), bounded by a surface boundary \(\Gamma\), submitted to external forces \(\overline{t}\), on the natural boundary \(\Gamma_{t}\), and displacement constrains at the essential boundary \(\Gamma_{u}\), the Galerkin weak form can be expressed as  
\[\int_{\Omega}\delta\mathbf{\varepsilon}^{T}\cdot\mathbf{\sigma}\cdot d\Omega-\int_{ \Omega}\delta\mathbf{u}(\mathbf{x}_{I})^{T}\cdot\mathbf{b}\cdot d\Omega-\int _{\Gamma_{t}}\delta\mathbf{u}(\mathbf{x}_{I})^{T}\cdot\mathbf{\bar{t}}\cdot d \Gamma=0, \tag{23}\]  
recalling that because \(u(\mathbf{x}_{I})=\Phi(\mathbf{x}_{I})^{T}\cdot\mathbf{u}\), it is possible to interpolate all the components of the field variable with one operation:  
\[\mathbf{u}(\mathbf{x}_{I})= \begin{bmatrix}u(\mathbf{x}_{I})\\ v(\mathbf{x}_{I})\\ w(\mathbf{x}_{I})\end{bmatrix} =\mathbf{H}(\mathbf{x}_{I})\cdot\mathbf{u}=\]  
\[= \begin{bmatrix}\phi_{1}(\mathbf{x}_{I})&0&0&\ldots&\phi_{n}(\mathbf{x}_{I}) &0&0\\ 0&\phi_{1}(\mathbf{x}_{I})&0&\ldots&0&\phi_{n}(\mathbf{x}_{I})&0\\ 0&0&\phi_{1}(\mathbf{x}_{I})&\ldots&0&0&\phi_{n}(\mathbf{x}_{I})\end{bmatrix}  \cdot \begin{bmatrix}u_{1}\\ v_{1}\\ w_{1}\\ \vdots\\ u_{n}\\ w_{n}\end{bmatrix}  \tag{24}\]  
and express the deformation vector as  
\[\mathbf{\varepsilon}(\mathbf{x}_{I})=\mathbf{L}\cdot\mathbf{u}(\mathbf{x}_{I})= \begin{bmatrix}\frac{\partial}{\partial x}&0&0\\ 0&\frac{\partial}{\partial y}&0\\ 0&0&\frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}&0\\ 0&\frac{\partial}{\partial z}&\frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}&0&\frac{\partial}{\partial x}\end{bmatrix} \cdot \Phi(\mathbf{x}_{I})\cdot\mathbf{u}=\mathbf{B}(\mathbf{x}_{I})\cdot\mathbf{u} =\mathbf{B}(\mathbf{x}_{I})\cdot \begin{bmatrix}u_{1}\\ v_{1}\\ w_{1}\\ \vdots\\ u_{n}\\ v_{n}\\ w_{n}\end{bmatrix}  \tag{25}\]being \(\mathbf{B}(\mathbf{x}_{l})\) the deformation matrix of integration point \(\mathbf{x}_{l}\), which can be defined as  
\[\mathbf{B}(\mathbf{x}_{l})= \begin{bmatrix}\frac{\partial\mathbf{p}_{l}(\mathbf{x }_{l})}{\partial x}&0&0&\cdots&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{ \partial x}&0&0\\ 0&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial y}&0&\cdots&0&\frac{ \partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial y}&0\\ 0&0&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial x}&\cdots&0&0&\frac {\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial x}\\ \frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial y}&\frac{\partial \mathbf{p}_{l}(\mathbf{x}_{l})}{\partial x}&0&\cdots&\frac{\partial\mathbf{p} _{l}(\mathbf{x}_{l})}{\partial y}&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l} )}{\partial x}&0\\ 0&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial x}&\frac{\partial \mathbf{p}_{l}(\mathbf{x}_{l})}{\partial y}&\cdots&0&\frac{\partial\mathbf{p} _{l}(\mathbf{x}_{l})}{\partial x}&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l} )}{\partial y}\\ \frac{\partial\mathbf{p}_{l}(\mathbf{x}_{l})}{\partial z}&0&\frac{\partial \mathbf{p}_{l}(\mathbf{x}_{l})}{\partial x}&\cdots&\frac{\partial\mathbf{p} _{l}(\mathbf{x}_{l})}{\partial z}&0&\frac{\partial\mathbf{p}_{l}(\mathbf{x}_{ l})}{\partial x}\end{bmatrix}  \tag{26}\]  
where \(n\) is the number of nodes inside the influence domain of integration point \(\mathbf{x}_{l}\). Then, with the linear strain-stress relation \(\sigma(\mathbf{x}_{l})=\mathbf{c}(\mathbf{x}_{l})\cdot\mathbf{c}(\mathbf{x}_ {l})\), stress at an integration point \(\mathbf{x}_{l}\) can be defined by  
\[\sigma(\mathbf{x}_{l})=\mathbf{c}(\mathbf{x}_{l})\cdot\mathbf{\varepsilon}( \mathbf{x}_{l})=\mathbf{c}(\mathbf{x}_{l})\cdot\mathbf{B}(\mathbf{x}_{l}) \cdot\mathbf{u} \tag{27}\]  
where \(\mathbf{c}(\mathbf{x}_{l})\) is the material constitutive matrix of integration point \(\mathbf{x}_{l}\).