## 3 Generation of Cross-Correlated Random Fields
### Framework of Elastic-Ideal Plastic Material Behavior  
#### 4.1.1 Constitutive Equations  
The constitutive model of the elastic-ideal plastic material behavior comprises the following equations:  
* Elastic constitutive equation;
* Flow rule;
* Yield condition. which need to satisfy the Kuhn-Tucker complementary conditions and the consistency condition [54]. The plastic constitutive material model introduced here was based on the research provided by [PERSON] et al. [55] and [PERSON] et al. [56]. It was adapted to the transversely isotropic material by using the strain-energy density function by [PERSON] et al. [57]. Furthermore, hardening was not considered. Hence, the material was assumed to show the ideal plastic behavior [24]. Since the modeling approach covers finite deformation, the constitutive model needs to be formulated with respect to the nonlinear continuum mechanical framework. Therefore, the deformation is described by the multiplicative decomposition of the deformation gradient, also known as Kroner-Lee decomposition. The decomposition is given by [58; 59; 60; 61]  
\[\mathbf{F}=\mathbf{F}_{\mathrm{el}}\cdot\mathbf{F}_{\mathrm{pl}}. \tag{20}\]  
Here, \(\mathbf{F}_{\mathrm{el}}\) holds the elastic and \(\mathbf{F}_{\mathrm{pl}}\) the plastic deformation, respectively. Since the deformation gradient was used to map continuum mechanical quantities from the reference to the current configuration and vice versa, this decomposition revealed the definition of an additional configuration, the so-called intermediate configuration [54; 58; 60]. As shown in Figure 6, with \(\mathbf{F}_{\mathrm{pl}}\), the quantities were mapped from the reference to the intermediate configuration and with \(\mathbf{F}_{\mathrm{el}}\) from the intermediate to the current configuration. Accordingly,  
\[J=J_{\mathrm{el}}J_{\mathrm{pl}}\quad\mathrm{with}\quad J=\det\mathbf{F}= \sqrt{\det\mathbf{C}} \tag{21}\]  
holds, with \(\mathbf{C}=\mathbf{F}^{\mathrm{T}}\cdot\mathbf{F}\) being the right Cauchy-Green tensor, defined in the reference configuration. It also splits into an elastic and a plastic part written by the decomposition [62]:  
\[\mathbf{C}=\mathbf{F}_{\mathrm{pl}}^{\mathrm{T}}\cdot\mathbf{C}_{\mathrm{el}} \cdot\mathbf{F}_{\mathrm{pl}}, \tag{22}\]  
where  
\[\mathbf{C}_{\mathrm{el}}=\mathbf{F}_{\mathrm{el}}^{\mathrm{T}}\cdot\mathbf{F}_ {\mathrm{el}},\qquad\mathbf{C}_{\mathrm{pl}}=\mathbf{F}_{\mathrm{pl}}^{ \mathrm{T}}\cdot\mathbf{F}_{\mathrm{pl}} \tag{23}\]  
Assuming \(J_{2}\)-plasticity and, hence, plastic incompressibility leads to  
\[\det\mathbf{C}_{\mathrm{pl}}=(J_{\mathrm{pl}})^{2}=1 \tag{24}\]  
and  
\[\det\mathbf{C}=J^{2}=(J_{\mathrm{el}})^{2}=\det\mathbf{\bar{C}}_{\mathrm{el}}, \tag{25}\]  
respectively. With respect to the introduced configurations, \(\mathbf{\bar{C}}_{el}\), which is located in the intermediate configuration, contains the elastic deformation, whereas \(\mathbf{C}_{pl}\) was assigned to the reference configuration and includes the plastic deformation. A crucial part of the plasticity framework is the plastic flow rule providing information about the irreversible deformation of a body. The fundamental equation is the second law of thermodynamics provided by the Clausian-Duhem inequality for isothermal processes, which is written as [63]  
\[-\Psi+\mathbf{S}:\frac{1}{2}\dot{\mathbf{C}}\geq 0, \tag{26}\]  
where the dot indicates the derivative with respect to time, \(\mathbf{S}\) is the second Piola-Kirchhoff stress tensor, and \(\Psi\) the Helmholtz free energy. Introducing the Helmholtz free energy as a function of \(\mathbf{C}\), the associated rate can be expressed by  
\[\Psi=\Psi_{\mathrm{el}}=\frac{\partial\Psi_{\mathrm{el}}}{\partial\dot{ \mathbf{C}}_{\mathrm{el}}}:\dot{\mathbf{C}}_{\mathrm{el}}. \tag{27}\]  
Since, in this case, both kinematic and isotropic hardening were omitted, only the part \(\Psi_{\mathrm{el}}\) of the Helmholtz free energy \(\Psi\), which was assigned to the elastic deformation, contributes to Equation (26).