# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 3 Elastoplastic Analysis
### Stress-Returning Algorithm  
In this work, an incremental procedure is adopted, in which an incremental stress leads to an incremental strain. Thus, trial stresses are a sum of a previous stress state with an incremental stress state. Consequently, it is possible to obtain trial stress states beyond the yield surface. For such case, the trial stress state must be forced to return to the yield surface. In this work, a \"backward-Euler\" procedure is adopted [23], Figure 2b.  
Considering a trial stress state \(\boldsymbol{e}^{trial}(\mathbf{x}_{I})\) at a given integration point \(\mathbf{x}_{I}\), the trial stress state is constituted by the previous stress state on the integration point, \(\boldsymbol{\sigma}(\mathbf{x}_{I})\), and the incremental stress state of the present load increment (or iteration), \(d\boldsymbol{\sigma}(\mathbf{x}_{I})\),  
\[\boldsymbol{\sigma}^{trial}(\mathbf{x}_{I})=\boldsymbol{\sigma}(\mathbf{x}_{ I})+d\boldsymbol{\sigma}(\mathbf{x}_{I}), \tag{48}\]  
if the trial stress state does not respect the yield criterion, Equation (39), being observed:  
\[f(\boldsymbol{\sigma}^{trial}(\mathbf{x}_{I}),\kappa)>\sigma_{Y}(\kappa), \tag{49}\]  
then, it is necessary to return the stress state to the yield surface. As Figure 2b shows, consider a two-dimensional Westgaard stress space in which a stress state represented by point B has passed beyond the yield surface. In order to avoid extra, unnecessary computations, instead calculating the intersection stress state represented with point A, it is assumed a \"forward\" yield function \(f_{B}\), containing the stress state B,  
\[f_{B}=\overline{\sigma_{B}}-\sigma_{Y}^{*}, \tag{50}\]  
in which \(\sigma_{Y}^{*}\) is the updated yield stress, accounting for the hardening parameter  
\[\sigma_{Y}^{*}=\sigma+[E_{T}\cdot\overline{\epsilon_{i-1}^{p}}], \tag{51}\]  
where \(\overline{\epsilon_{i-1}^{p}}\) is the accumulated effective plastic strain from the previous increment, \((i-1)\).  
Thus, the flow vector is calculated from stress state B, \(\mathbf{a}_{B}\), and the plastic multiplier \(d\lambda\) is given by  
\[d\lambda=\frac{f_{\text{B}}}{\mathbf{a}_{B}^{T}\cdot\mathbf{c}\cdot\mathbf{a} _{B}+H^{\prime}}. \tag{52}\]Now it is possible to estimate the stress in point \(C\):  
\[\mathbf{\sigma}_{C}=\mathbf{\sigma}_{B}-d\lambda\cdot\mathbf{\mathrm{c}}\cdot\mathbf{\mathrm{a}}_ {B}. \tag{53}\]  
As can be observed, stress state \(\mathbf{\sigma}_{C}\) is an estimated stress state. In order to achieve accurately stress state \(\mathbf{\sigma}_{C}\), it would be required to use the flow vector calculated directly in point \(C\),  
\[\mathbf{\sigma}_{C}=\mathbf{\sigma}_{B}-d\lambda\cdot\mathbf{\mathrm{c}}\cdot\mathbf{\mathrm{a }}_{C}. \tag{54}\]  
However, stress state \(\mathbf{\sigma}_{C}\) is unknown, naturally hindering the calculation of \(\mathbf{\mathrm{a}}_{C}\). Therefore, an iterative procedure must be implemented to reach point \(C\) from point \(B\). For each \"stress return iteration\", the effective plastic strain is updated in order to apply Equation (51) and pass to the next \"stress return iteration\", until point \(C\) is satisfactorily approximated, concluding the iterative process.