# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 3 Elastoplastic Analysis  
The mathematical theory of plasticity provides a theoretical description for the relationship between stress and strain for a material exhibiting an elastoplastic behaviour [23]. Material plasticity is characterized by an irreversible deformation. Before occuring, plastic flow, the material shows a linear elastic behaviour, for which a linear relation following the generalized Hooke's law is valid. After the yield stress \(\sigma_{Y}\) is reached (establishedby the yield criterion), plastic behaviour occurs and the stress field starts following a flow rule. In order to implement material plasticity, it is necessary to establish three fundamental concepts:  
* a yield criterion, indicating the stress level at which plastic flow initiates, and an initial yield surface, defining the elastic limit in a multiaxial stress state;
* a hardening law, describing the evolution of the subsequent yield surfaces; and
* a flow rule, relating the plastic deformation to the current stresses in the material and the stress increments outside the yield surface.  
Concerning the yield criterion, in this work, the [PERSON] yield criterion is considered:  
\[F(\mathbf{\sigma})=\sqrt{\frac{1}{2}\big{[}(\sigma_{xx}-\sigma_{yy})^{2}+(\sigma_{ yy}-\sigma_{zz})^{2}+(\sigma_{zz}-\sigma_{xx})^{2}+6(\sigma_{yz}^{2}+\sigma_{ zx}^{2}+\sigma_{xy}^{2})\big{]}}-\sigma_{Y}=0. \tag{34}\]  
Assuming small strains, the total strain \(\mathbf{\varepsilon}\) can be decomposed into two components [34, 35],  
\[\mathbf{\varepsilon}=\mathbf{\varepsilon}^{p}+\mathbf{\varepsilon}^{e}, \tag{35}\]  
where \(\mathbf{\varepsilon}^{p}\) is the plastic component of the total strain and \(\mathbf{\varepsilon}^{e}\) the elastic component. Concerning the hardening law, this work assumes that the yield surface grows linearly from the initial yield surface, only changing its size and expanding uniformly with the increase of the effective plastic strain, \(\mathbf{\varepsilon}^{p}\):  
\[\mathbf{\varepsilon}^{p}=\int d\mathbf{\varepsilon}^{p}=\int\Big{[}\frac{2}{3}d\mathbf{ \varepsilon}_{ij}^{p}\cdot d\mathbf{\varepsilon}_{ij}^{p}\Big{]}^{\frac{1}{2}} \tag{36}\]  
where \(d\mathbf{\varepsilon}^{p}\) is the plastic component of strain occurring during a strain increment.  
The flow rule serves as a mathematical description of the evolution of the infinitesimal increments of the plastic strain \(d\mathbf{\varepsilon}^{p}\) in the course of the load history of the body [36]. As the incremental load increases, and the yield function \(f\) reaches a value equal to the yield stress of the material, \(\sigma_{Y}\), the frontier of elasticity starts to end and plasticity begins. The variation of the yield function can be written as a function of the variation of the stress field  
\[df=\left(\frac{\partial f}{\partial\sigma}\right)^{T}\cdot d\mathbf{\sigma}, \tag{37}\]  
where \(\partial f/\partial\sigma\) is the gradient of \(f\), and therefore an orthogonal vector to the yield surface for a certain stress increment \(d\mathbf{\sigma}\). Applying the Prandtl-Reuss flow rule, it is possible to write  
\[d\mathbf{\varepsilon}^{p}=d\lambda\cdot\frac{\partial f}{\partial\sigma}, \tag{38}\]  
where \(d\lambda\) is the plastic multiplier (a scalar). The vector \(\partial f/\partial\sigma\) is a vector normal to the yield surface at the current stress state, as shown in Figure 2a [23, 36], also known as a flow vector: \(\mathbf{a}=\partial f/\partial\sigma\).