## 3 Generation of Cross-Correlated Random Fields
### Framework of Elastic-Ideal Plastic Material Behavior
#### 4.1.3 Variational Formulation and Consistent Tangent Modulus Tensor  
For a body \(\mathcal{B}\), the local form of the balance of momentum with respect to the reference configuration reads  
\[\mathrm{Div}\,\mathbf{P}+\rho_{0}\mathbf{b}=\rho_{0}\bar{u}. \tag{47}\]  
Here, \(\mathbf{P}\) is the first Piola-Kirchhoff stress tensor, \(\rho_{0}\) is the density, \(\mathbf{b}\) are the volume forces, and \(\mathbf{u}\) is the displacement field. To solve this differential equation, Neumann and Dirichlet boundary conditions were used. They are given by  
\[\mathbf{u} =\bar{\mathbf{u}}\ \mathrm{on}\ \partial\mathcal{B}_{\cap} \tag{48}\] \[\mathbf{P}\cdot\mathbf{N} =\bar{\mathbf{t}}\ \mathrm{on}\ \partial\mathcal{B}_{\sigma}, \tag{49}\]  
where \(\partial\mathcal{B}\) is the boundary surface of \(\mathcal{B}\) and \(\mathbf{N}\) gives the normal to the boundary surface \(\partial\mathcal{B}_{\sigma}\). To ensure that the boundary conditions cover the complete surface of the body, \(\partial\mathcal{B}=\partial\mathcal{B}_{\cap}\cup\partial\mathcal{B}_{\sigma}\) and \(\partial\mathcal{B}_{\cap}\cap\partial\mathcal{B}_{\sigma}=\varnothing\) hold. Applying standard variational calculus on Equation (47) and \(\mathbf{P}=\mathbf{F}\cdot\mathbf{S}\) leads to  
\[F(\mathbf{u},\delta\mathbf{u})=\int_{\mathcal{B}}\frac{1}{2}\mathbf{S}:\delta \mathbf{C}\mathrm{d}V+\int_{\mathcal{B}}\rho_{0}(\bar{\mathbf{u}}-\mathbf{b}) \cdot\delta\mathbf{u}\mathrm{d}V-\int_{\partial\mathcal{B}_{\sigma}}\bar{ \mathbf{t}}\cdot\delta\mathbf{u}\mathrm{d}S=0 \tag{50}\]  
for the equilibrium state of the body \(\mathcal{B}\). Based on the definition:  
\[\mathbf{C}=\mathbf{F}^{\mathrm{T}}\cdot\mathbf{F} \tag{51}\]  
the virtual right Cauchy-Green deformation tensor, expressed with respect to the virtual deformation gradient \(\delta\mathbf{F}=\mathrm{Grad}\,\delta\mathbf{u}\), is given by  
\[\delta\mathbf{C}=(\delta\mathbf{F}^{\mathrm{T}}\cdot\mathbf{F}+\mathbf{F}^{ \mathrm{T}}\cdot\delta\mathbf{F}). \tag{52}\]  
To solve this nonlinear equation within the FEM framework, a standard Newton-Raphson iteration scheme was used. To ensure a quadratic convergence rate, a consistent linearization of the principal of virtual displacement in Equation (50) is crucial. Therefore, in this paper, the perturbation technique developed by [PERSON] (1972) was used. This approach is widely used and has been extensively investigated with respect to convergence properties and the application within the finite strain framework [PERSON] (1972); [PERSON] (1972); [PERSON] (1973). Due to the symmetry of \(\mathbf{S}\), the linear increment of \(F\) can be formulated as \[\Delta F(\mathbf{u},\delta\mathbf{u},\Delta\mathbf{u})=\frac{1}{2}\int_{\mathcal{B}} (\delta\mathbf{C}:\Delta\mathbf{S}+\Delta\delta\mathbf{C}:\mathbf{S})\mathrm{d}V, \tag{53}\]  
where \(\Delta\delta\mathbf{C}=(\Delta\mathbf{F}^{\mathrm{T}}\cdot\delta\mathbf{F}+ \delta\mathbf{F}^{\mathrm{T}}\cdot\Delta\mathbf{F})\) is the linearized virtual right Cauchy-Green deformation tensor given in terms of the incremental deformation gradient \(\Delta\mathbf{F}=\operatorname{Grad}\Delta\mathbf{u}\). To obtain the incremental second Piola-Kirchhoff stress tensor \(\Delta\mathbf{S}\), the consistent tangent modulus tensor is required. In the reference configuration, it is expressed by  
\[\mathbb{C}=2\frac{\partial\mathbf{S}}{\partial\mathbf{C}}, \tag{54}\]  
which can be written as  
\[\mathbb{C}=2\frac{\Delta\mathbf{S}}{\Delta\mathbf{C}}=\sum_{i,j,l=1}^{3}2 \frac{\Delta S_{ij}}{\Delta C_{kl}}\mathbf{E}_{i}\otimes\mathbf{E}_{j}\otimes \mathbf{E}_{k}\otimes\mathbf{E}_{l}. \tag{55}\]  
The incremental change of the right Cauchy-Green tensor was obtained by applying a perturbation technique to the deformation gradient.