## 3 Boundary Conditions and Material Models
### Multiscale Failure Analysis  
#### 3.3.1 Failure and Softening Formulation for Fiber  
In this paper, the fiber is treated as a transversely isotropic material in micromechanical model, thus some failure criteria for laminated composites can be applied to fiber failure analysis. A widely used polynomial failure criterion for composite materials proposed by [PERSON] and [PERSON] [20] is used. The criterion can be expressed as:  
\[F_{i}\sigma_{i}+F_{ij}\sigma_{i}\sigma_{j}+F_{ijk}\sigma_{i}\sigma_{j}\sigma_{k }\geq 1\quad i,j,k=1,\ldots,6 \tag{12}\]  
where \(\sigma_{i}\), \(\sigma_{j}\), and \(\sigma_{k}\) are stress components. \(F_{i}\), \(F_{ij}\), and \(F_{ijk}\) are components of the lamina strength tensors in the principal material axes. The third-order tensor \(F_{ijk}\) is usually ignored from a practical standpoint due to the large number of material constants required. Then, the general polynomial criterion can be reduced to a general quadratic criterion given by  
\[\begin{array}{ll}F_{1}\sigma_{1}+F_{2}\sigma_{2}+F_{3}\sigma_{3}+2F_{12} \sigma_{1}\sigma_{2}+2F_{13}\sigma_{1}\sigma_{3}+2F_{23}\sigma_{2}\sigma_{3}+ F_{11}{\sigma_{1}}^{2}\\ +F_{22}{\sigma_{2}}^{2}+F_{33}{\sigma_{3}}^{2}+F_{44}{\sigma_{4}}^{2}+F_{55}{ \sigma_{5}}^{2}+F_{66}{\sigma_{6}}^{2}\geq 1\end{array} \tag{13}\]  
For the mechanical properties of the dry fibers, the fiber manufacturer only provides tension and compressive strength parameters; therefore, Equations (12) and (13) are modified as  
\[\begin{array}{ll}F_{1}\sigma_{1}+F_{2}\sigma_{2}+F_{3}\sigma_{3}+2F_{12} \sigma_{1}\sigma_{2}+2F_{31}\sigma_{3}\sigma_{1}+2F_{23}\sigma_{2}\sigma_{3}+ F_{11}{\sigma_{1}}^{2}\\ +F_{22}{\sigma_{2}}^{2}+F_{33}{\sigma_{3}}^{2}\geq 1\end{array} \tag{14}\]  
where the tensors in Equations (13) and (14) can be determined as follows.  
\[\begin{array}{ll}F_{1}=\frac{1}{X_{\rm C}}-\frac{1}{X_{\rm T}},\ F_{2}=\frac {1}{Y_{\rm C}}-\frac{1}{Y_{\rm T}},\ F_{3}=\frac{1}{Z_{\rm C}}-\frac{1}{Z_{\rm T }}\\ F_{11}=\frac{1}{X_{\rm T}X_{\rm C}},\ F_{22}=\frac{1}{Y_{\rm T}Y_{\rm C}},\ F_{33}=\frac{1}{Z_{\rm T}Z_{\rm C}}\\ F_{12}=\frac{-1}{2\sqrt{X_{\rm T}X_{\rm C}Y_{\rm T}Y_{\rm C}}},\ F_{23}=\frac{-1}{2 \sqrt{Y_{\rm T}Y_{\rm C}Z_{\rm T}}C},\ F_{31}=\frac{-1}{2\sqrt{Z_{\rm T}Z_{\rm C }X_{\rm T}X_{\rm C}}}\end{array} \tag{15}\]  
where \(X_{\rm T}\), \(Y_{\rm T}\), and \(Z_{\rm T}\) are the tensile strength in 1-direction, 2-direction, and 3-direction, respectively. \(X_{\rm C}\), \(Y_{\rm C}\), and \(Z_{\rm C}\) are the compression strength in 1-direction, 2-direction, and 3-direction, respectively.  
The modified Tsai-Wu Criterion is an expression that only considers the tensile and compressive strength, and we use the criterion to determine the initial failure of the fiber.  
Post-initial failure in fiber direction is modeled by a gradual unloading model, where one or more of the elastic material properties of a lamina are set to zero or a small fraction of the original value once failure is detected. The degradation equations are given in Reference [21]. Furthermore, in the finite element models, the material softening laws with Tsai-Wu failure criteria have been implemented using user defined subroutines USDFLD (User subroutine to redefine field variables at a material point) of Abaqus (Version 6.11, Dassault systems simulia Corp, Providence, RI, USA, 2011).  
For a micromechanical analysis, the properties of fibers are essential, the material data of 12K Toray T700 50C are used in the analysis of this paper as shown in Table 1. The elastic data are cited from Reference [13], tensile strength is obtained from data sheet of the fiber manufacturer, compressive strength are calculated with an empirical correction \(k=0.8\), and the empirical correction is determined according to the ratio of compressive strength and tensile strength of T300 fiber, the data are summarized in the World-Wide Failure Exercise [22].