# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 3 Elastoplastic Analysis
### Elastoplastic Constitutive Model  
For its implementation in computational mechanics, it is convenient to present the previous elastoplastic theory in matrix formulation [23]. Thus, generically, the yield criterion of Equation (34) can be defined as  
\[F(\mathbf{\sigma},\kappa)=f(\mathbf{\sigma},\kappa)-\sigma_{Y}(\kappa)=0, \tag{39}\]where \(\kappa\) is a generic hardening parameter (which in this work will be assumed as the effective plastic deformation, Equation (36)). By differentiating Equation (39), it is possible to write  
\[dF=\left(\frac{\partial F}{\partial\sigma}\right)\cdot d\sigma+\frac{\partial F }{\partial\kappa}\cdot d\kappa=0, \tag{40}\]  
which is equivalent to Equation (37) but for a material with hardening. Thus, Equation (40) can be written as function of the plastic multiplier \(d\lambda\),  
\[\mathbf{a}^{T}\cdot d\sigma-A\cdot d\lambda=0, \tag{41}\]  
where the vector \(\mathbf{a}\) is termed the flow vector, \(\mathbf{a}=\partial f/\partial\sigma\). The hardening parameter \(A\) depends on the hardening rule [23], and it can be written as  
\[A=-\frac{1}{d\lambda}\cdot\frac{\partial F}{\partial\kappa}\cdot d\kappa. \tag{42}\]  
From decomposition equation, Equation (35), it possible to write \(\epsilon^{e}=\epsilon-\epsilon^{p}\). Thus, with Equation (38), \(d\epsilon^{p}=d\lambda\cdot\mathbf{a}\), and using the Hooke law,  
\[d\sigma=\mathbf{c}\cdot d\epsilon^{e}=\mathbf{c}\cdot(d\epsilon-d\epsilon^{p })=\mathbf{c}\cdot(d\epsilon-d\lambda\cdot\mathbf{a}), \tag{43}\]  
it is possible to develop Equation (41) as  
\[\mathbf{a}^{T}\cdot(\mathbf{c}\cdot(d\epsilon-d\lambda\cdot\mathbf{a}))-A \cdot d\lambda=0\Longrightarrow d\lambda=\frac{\mathbf{a}^{T}\cdot\mathbf{c} \cdot d\epsilon}{\mathbf{a}^{T}\cdot\mathbf{c}\cdot\mathbf{a}+A}. \tag{44}\]  
Substituting Equation (44) in Equation (43) permits us to obtain  
\[d\sigma=\mathbf{c}\cdot d\epsilon-\frac{\mathbf{a}^{T}\cdot\mathbf{c}\cdot d \epsilon}{\mathbf{a}^{T}\cdot\mathbf{c}\cdot\mathbf{a}+A}\cdot\mathbf{c}\cdot \mathbf{a}=\left(\mathbf{c}-\frac{\mathbf{c}\cdot\mathbf{a}\cdot\mathbf{a}^{T }\mathbf{c}}{\mathbf{a}^{T}\cdot\mathbf{c}\cdot\mathbf{a}+A}\right)\cdot d \epsilon=\mathbf{c}^{p}\cdot d\epsilon, \tag{45}\]  
Figure 2: (**a**) Flow vector representation. (**b**) Stress returning procedure.  
with \(\mathbf{c}^{p}\) being the elastoplastic tangential constitutive matrix. The elastic constitutive matrix can be defined as  
\[\mathbf{c}=\begin{bmatrix}\frac{1}{L}&\frac{-\
u}{L}&\frac{-\
u}{L}&0&0&0\\ \frac{-\
u}{L}&\frac{1}{L}&\frac{-\
u}{L}&0&0&0\\ \frac{-\
u}{L}&\frac{-\
u}{L}&\frac{-\
u}{L}&0&0&0\\ 0&0&0&\frac{2\cdot(\
u+1)}{L}&0&0\\ 0&0&0&0&\frac{2\cdot(\
u+1)}{L}&0\\ 0&0&0&0&0&\frac{2\cdot(\
u+1)}{L}\end{bmatrix}^{-1}, \tag{46}\]  
where \(E\) is the elastic modulus of the material (observed in the linear elastic phase) and \(\
u\) the Poisson's coefficient.  
The hardening parameter \(A\) can be obtained by means of the hardening hypothesis [23] considering the associated flow rule. In this work, the analysed nonlinear materials exhibit a \"linear-elastic/linear plastic\" hardening behaviour. Thus, as shown in the literature [23], the hardening parameter \(A\) can be written as  
\[A=\frac{E_{T}}{1-\frac{E_{T}}{E}}, \tag{47}\]  
where \(E_{T}\) is the tangent modulus (tangent to the stress-strain curve in the plastic regime).