## 3 Generation of Cross-Correlated Random Fields
### Framework of Elastic-Ideal Plastic Material Behavior
#### 4.1.2 Solution Procedure  
Elastic Predictor  
In the initial step, the deformation of the body was assumed to be purely elastic, which is described in the intermediate configuration, as shown before. Following the research of [PERSON] et al., who derived the framework of the finite strain elasto-plasticity by combining the multiplicative decomposition of the deformation gradient with hyperelastic strain-energy density functions [65; 66; 67], the elastic deformation is obtained by evaluating the hyperelastic strain-energy density function \(\Psi_{\text{el}}\) with respect to the elastic part of the deformation. In this study, the elastic material behavior of the SFRC is described by the potential [57]  
\[\Psi_{\text{el}}(\bar{\mathbf{C}}_{\text{el}})=\frac{1}{2}\Lambda (J-1)^{2}-\mu\ln(J)+\frac{1}{2}\mu\big{(}\text{tr}\,\bar{\mathbf{C}}_{\text{ el}}-3\big{)}+\\ \big{[}\alpha+\beta\big{(}\text{tr}\,\bar{\mathbf{C}}_{\text{el} }-3\big{)}+\gamma(I_{4}-1)\big{]}(I_{4}-1)-\frac{1}{2}\alpha(I_{5}-1). \tag{39}\]Here, \(\Lambda\), \(\mu\), \(\alpha\), \(\beta\), and \(\gamma\) are the independent material parameters characterizing the transversely isotropic material behavior. In addition to the well-known invariants \(I_{1}\), \(I_{2}\), and \(I_{3}\) for the description of the isotropic material behavior [68; 69], the quantities \(I_{4}\) and \(I_{5}\) give the pseudo-invariants, which are defined for a symmetric second-order tensor \(\mathbf{B}\) by [70; 71]  
\[I_{4}=\mathbf{a}\cdot\mathbf{B}\cdot\mathbf{a}\qquad\text{and}\qquad I_{5}= \mathbf{a}\cdot\mathbf{B}^{2}\cdot\mathbf{a}. \tag{40}\]  
The vector \(\mathbf{a}\) gives the fiber orientation and, therefore, holds the transversely isotropic characteristics of the material. In this case, it was assumed that the fibers are mainly orientated along the tensile test specimen. Hence, \(a_{1}=1\) and \(a_{2}=a_{3}=0\) hold. With respect to the used transversely isotropic potential and the tensor derivatives, given by  
\[\frac{\partial I_{4}}{\partial\mathbf{B}} =\mathbf{a}\otimes\mathbf{a} \frac{\partial I_{5}}{\partial\mathbf{B}} =\mathbf{a}\cdot\mathbf{B}\otimes\mathbf{a}+\mathbf{a}\otimes \mathbf{B}\cdot\mathbf{a}\] \[\frac{\partial\mathbf{B}}{\partial\mathbf{B}} =\mathbb{I}^{s}=\frac{1}{2}(\delta_{ik}\delta_{lj}+\delta_{il} \delta_{jk}) \frac{\partial\mathbf{B}^{-1}}{\partial\mathbf{B}} =-\frac{1}{2}(B_{ik}^{-1}B_{lj}^{-1}+B_{il}^{-1}B_{jk}^{-1}), \tag{41}\]  
the second Piola-Kirchhoff stress tensor in the intermediate configuration reads  
\[\bar{\mathbf{S}}=\mu(\mathbf{I}-\bar{\mathbf{C}}_{\text{el}}^{-1 })+\lambda J(J-1)\bar{\mathbf{C}}_{\text{el}}^{-1}+\] \[2\beta(I_{4}-1)\mathbf{I}+2[\alpha+\beta(\text{tr}\,\bar{ \mathbf{C}}_{\text{el}}-3)+2\gamma(I_{4}-1)]\mathbf{a}\otimes\mathbf{a}- \alpha(\bar{\mathbf{C}}_{\text{el}}\cdot\mathbf{a}\otimes\mathbf{a}+\mathbf{ a}\otimes\bar{\mathbf{C}}_{\text{el}}\cdot\mathbf{a}). \tag{42}\]  
Checking Yield Condition  
To be able to evaluate the yield condition, provided in Equation (34), in the intermediate configuration, the deviatoric part of the Mandel stress must be computed to derive the corresponding von Mises stress. If the resulting von Mises stress is lower than the yield strength of the material, the body shows only elastic deformation. Therefore, no further action is required. The stress measure can be updated with the trial value obtained by the elastic predictor step. If, however, the von Mises stress is larger than the yield strength, the body shows not only elastic, but also plastic deformation. In this case, the determination of the resulting plastic deformation is covered by the plastic corrector step. Plastic Corrector Step (Return Mapping)  
Since the yield condition is given in the intermediate configuration, this also holds for the computation of the updated plastic part of the deformation gradient. As shown before, an associative plastic flow rule introduced by [PERSON] et al. [56] was used to describe the evaluation of the plastic deformation.