## 2 Fundamentals of Limit Analysis
\Leftrightarrow\dot{p}_{hk}=\dot{\lambda}\frac{\partial f}{\partial\sigma_{hk}} \tag{3}\]  
where \(f(\sigma_{hk})\leq 0\) is the compatibility condition on stress, equality holds for yielding, \(\frac{\partial f}{\partial\sigma_{hk}}\) is the gradient of the yield surface with respect to \(\sigma_{ij}\) and \(\dot{\lambda}\) is the plastic multiplier that accounts for the magnitude of the plastic strain rate, \(\dot{p}_{hk}\). During the loading process, the stress results by the superimposition of the one calculated on the structure supposed to be indefinitely elastic, \(\sigma_{ij}^{E}\), and self-equilibrated stress, \(\zeta_{ij}\), obtained by Equation (1), here called \"eigenstress\" [19], can also be recognized to be the actual residual stress one finds in the structures, whether it is unloaded during the load path so that it can also be called \"actual residual stress\":  
\[\sigma_{ij}=\sigma_{ij}^{E}+\zeta_{ij} \tag{4}\]  
The actual strain field results from the following additive decomposition that can be interpreted through the decomposition (4) yielding to the equation:  
\[\varepsilon_{hk}=C_{ijhk}\sigma_{ij}^{E}+C_{ijhk}\zeta_{ij}+p_{hk}=\varepsilon _{hk}^{E}+\varepsilon_{hk}^{0E}+p_{hk} \tag{5}\]  
Figure 1: Structural domain with the representation of loads, tractions, and constraints. where \(C_{ijlk}\) is the elastic compliance tensor, the inverse of \(S_{ijlk}\). Equation (5) has the following mechanical interpretation: \(\varepsilon_{lk}\), is the sum of the strain in the structure supposed to be indefinitely elastic, \(\varepsilon_{lk}^{E}\), the strain corresponding to the eigenstress, \(\varepsilon_{lk}^{0E}\) and the permanent strain, \(p_{lk}\). The permanent strain \(p_{lk}\) is the discontinuous part of the strain. It is possible to recognize the kinematical compatibility of the strain, namely: \(\varepsilon_{lk}^{E}\) is compatible since it is the elastic solution under the applied loads; the sum \(\left(\varepsilon_{lk}^{0E}+p_{lk}\right)\) is compatible as the strain \(\varepsilon_{lk}^{0E}\) is the elastic response to the discontinuity \(p_{lk}\), acting as an applied dislocation, to ensure its compatibility [2]. The stress \(\zeta_{ij}\) is self-equilibrated stress by definition. To verify the statement, the equilibrium equation of the interior points of the solid body is recalled:  
\[\sigma_{ij,j}+b_{i}=0 \tag{6}\]  
By using Equations (4) and (5), the equilibrium of internal points become  
\[\left(S_{ijlk}\varepsilon_{lk}^{E}+S_{ijlk}\varepsilon_{lk}^{0E}\right)_{j}+b _{i}=0 \tag{7}\]  
The elastic stress is the stress in equilibrium in \(\Omega\), when the constitutive equation is assumed to be indefinitely elastic, subjected to the same loads than the actual structure. Consequently, the elastic stress, \(\sigma_{kk^{\prime}}^{E}\) is in equilibrium with the applied forces so that the following equation holds:  
\[\left(S_{ijlk}\varepsilon_{lk}^{E}\right)_{j}+b_{i}=0 \tag{8}\]  
Comparing Equation (7) with Equation (8), the following equation is obtained, confirming that the elastic-plastic stress is self-equilibrated in the interior of the structure. \[S_{ijlk}\varepsilon_{lk^{\prime},j}^{0E}=0 \tag{9}\]  
Moreover, the self-equilibrated stress satisfies the homogeneous boundary condition on the free boundary, \(\partial\Omega_{f}\), of the structure. Finally, the equilibrium equation involving the stress \(\zeta_{ij,j}\) reduces to Equation (1). The above reasoning can be summarized as follows: the actual elastic-plastic stress solution, resulting from an applied body and boundary loads, differs from the pure elastic stress by self-equilibrated stress. This self-equilibrated stress is due to elastic strain \(\varepsilon_{lk}^{0E}\) that results from the compatibility equation concerning the permanent discontinuous strain \(p_{lk}\) acting as Volterra's dislocation. Dislocations are the kinematical effect equivalent to discontinuities of strain within the structure. A dislocation causes stress in the absence of loads.