# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 3 Elastoplastic Model  
Plasticity models depend on the nature of the materials whose behavior is being simulated. In Figure 4, some elastoplastic models are observed.  
In Figure 4a, we can see that the elastic part is linear, characterized by the Young Modulus \(E\) and by \(\alpha_{1}=\tan^{-1}E\). When the yield stress, \(\sigma_{Y}\), is reached, the stress level is maintained while the deformation, of plastic nature only, increases. In Figure 4b, we have two slopes. The first slope, characterized by \(\alpha_{1}\), is the elastic part, while the second slope, when the yield stress is reached, characterized by \(\alpha_{2}=\tan^{-1}E_{T}\), is the plastic part, with \(E_{T}\) being the tangent modulus. In the case of Figure 4c, the stress relation after yield is no longer linear. This work uses a linear elastic-linear plastic model similar to Figure 4b.  
In this work, a small strain formulation is used to model the elastoplastic behavior of structures. In small strains, the total strain \(\epsilon\) is separated into two components [27; 28]. This is called the decomposition law:  
\[\epsilon=\epsilon^{p}+\epsilon^{\epsilon} \tag{8}\]  
where \(\epsilon^{p}\) is the plastic component of the total strain and \(\epsilon^{\epsilon}\) is the elastic component.  
The yield criterion, which assesses if the material is under an elastic or plastic regime, can be generally defined as a function of the stress, \(\sigma\), and the load history:  
\[F(\sigma,\kappa)=f(\sigma,\kappa)-\sigma_{Y}(\kappa)=0 \tag{9}\]where \(f(\sigma,\kappa)\) is a yield function that depends on the stress state, \(\sigma\); a hardening parameter, \(\kappa\); and the yield stress of the material, \(\sigma_{Y}(\kappa)\), which can also be hardening-dependent. In this study, to evaluate if the stress level surpasses the yield surface, the von [PERSON] yield criterion is used:  
\[\overline{\sigma}^{2}=\sigma_{Y}^{2}=3J_{2} \tag{10}\]  
where \(\overline{\sigma}\) is the effective stress, and \(J_{2}\) is the second invariant of the deviatoric stress tensor.  
An isotropic hardening rule is considered. This means that there is a uniform expansion of the initial yield surface with the increase in the effective plastic strain, \(\overline{\epsilon}^{p}\):  
\[\overline{\epsilon}^{p}=\int d\overline{\epsilon}^{p}=\int\left[\frac{2}{3}d \overline{\epsilon}_{ij}^{p}d\overline{\epsilon}_{ij}^{p}\right]^{\frac{1}{2}} \tag{11}\]  
in which \(d\epsilon^{p}\) is the plastic component of the strain occurring during a strain increment.  
For the completeness of an elastoplastic model, a flow rule is necessary. It serves as a mathematical description of the evolution of the infinitesimal increments in plastic strain \(d\epsilon^{p}\) in the course of the load history of the body [29]. The Prandtl-Reuss flow rule is applied:  
\[d\epsilon^{p}=d\lambda\frac{\partial f}{\partial\sigma} \tag{12}\]  
where \(d\lambda\) is the plastic multiplier (i.e., proportional constant), and \(\frac{\partial f}{\partial\sigma}\) is the \(f\) gradient, and is therefore an orthogonal vector to the yield surface [29; 30]. Depending on the value of \(d\lambda\), the material may be subjected to distinct situations, as shown in Table 1.