## 3 Boundary Conditions and Material Models
### The Stiffness Calculation Method Based on Average Stress and Strain  
To evaluate the stiffness properties of a heterogeneous material, it is necessary to calculate the average stress and strain over the unit cells. The constitutive relation of the average stress and strain of the homogeneous composite material [19] is shown in Equation (4). \[\overline{\sigma}_{\alpha}=C_{\alpha\beta}\overline{\varepsilon}_{\beta} \tag{4}\]  
where \(\alpha,\beta=1,\ldots,6\). \(C_{\alpha\beta}\) is the stiffness tensor. Figure 6: Two load cases for unit cells: (**a**) normal load: tension in \(x\) direction; and (**b**) shear load: shear in \(x\)-\(y\) plane. For an orthotropic material, the tensor \(C_{\alpha\beta}\) can be written in the form:  
\[C_{\alpha\beta}=\left[ \begin{array}{cccccc}c_{11}&c_{12}&c_{13}&0&0&0\\ c_{21}&c_{22}&c_{23}&0&0&0\\ c_{31}&c_{32}&c_{33}&0&0&0\\ 0&0&0&c_{44}&0&0\\ 0&0&0&0&c_{55}&0\\ 0&0&0&0&0&c_{66}\end{array} \right] \tag{5}\]  
Then, the relations between the effective engineering constant of composites and the stiffness tensor \(C_{\alpha\beta}\) can be written as  
\[ \begin{array}{l}E_{1}=C_{11}-2C_{12}C_{21}/\left(C_{22}+C_{23}\right)\\ E_{2}=\left[C_{11}\left(C_{22}+C_{23}\right)-2C_{12}C_{21}\right]\left(C_{22}- C_{23}\right)/\left(C_{11}C_{22}-C_{12}C_{21}\right)\\ G_{12}=C_{66},G_{23}=C_{44}\\ v_{12}=C_{12}/\left(C_{22}+C_{23}\right)\\ v_{23}=\left(C_{11}C_{23}-C_{12}C_{21}\right)/\left(C_{11}C_{22}-C_{12}C_{21} \right)\end{array}  \tag{6}\]  
where \(E_{1}\) and \(E_{2}\) are longitudinal and transverse [PERSON]'s modulus. \(v_{12}\) and \(v_{23}\) are [PERSON]'s ratios. \(G_{12}\) is in-plane shear modulus. To obtain the components of the stiffness tensor \(C_{\alpha\beta}\), the six components of strain \(\varepsilon_{ij}^{0}\), \(i,j=1,\dots\), \(3\) are applied by enforcing the following boundary conditions on the displacement components as shown in Figure 6.  
\[ \begin{array}{l}u_{i}(a_{1},y,z)-u_{i}(0,y,z)=a_{1}\varepsilon_{11}^{0}\\ u_{i}(x,a_{2},z)-u_{i}(x,0,z)=a_{2}\varepsilon_{i1}^{0}\\ u_{i}(x,y,a_{3})-u_{i}(x,y,0)=a_{3}\varepsilon_{3}^{0}\end{array}  \tag{7}\]  
The boundary conditions (Equation (7)) means that \(a_{j}\varepsilon_{ij}^{0}\) is the displacement required to enforce a strain of \(\varepsilon_{ij}^{0}\) over a distance \(a_{j}\). Using Equation (7), a surface strain \(\varepsilon_{\beta}^{0}\) can be applied on the unit cell. The relationship between \(\varepsilon_{\beta}^{0}\) and \(\varepsilon_{ij}^{0}\)[19] can be written as  
\[\varepsilon_{\beta}=\varepsilon_{ij}=\varepsilon_{ji} \tag{8}\]  
where \(\beta=i\), if \(i=j\) else \(\beta=9-i-j\). The volume average strain \(\overline{\varepsilon}_{\beta}\) in the unit cells equals to the applied surface strain \(\varepsilon_{\beta}^{0}\),  
\[\overline{\varepsilon}_{\beta}=\frac{1}{V}\int\limits_{V}\varepsilon_{\beta }^{0}\mathrm{d}V=\frac{1}{V}\left[\int\limits_{V_{t}}\varepsilon_{\beta}^{0} \mathrm{d}V+\int\limits_{V_{m}}\varepsilon_{\beta}^{0}\mathrm{d}V\right]= \varepsilon_{\beta}^{0} \tag{9}\]  
where \(V\) is the volume of unit cells, \(V_{t}\) is the volume of fiber, and \(V_{m}\) is the volume of matrix. The corresponding volume average stress \(\overline{\sigma}_{\alpha}\) is described as:  
\[\overline{\sigma}_{\alpha}=\frac{1}{V}\int\limits_{V}\sigma_{\alpha}\mathrm{ d}V=\frac{1}{V}\left[\int\limits_{V_{t}}\sigma_{\alpha}\mathrm{d}V+\int\limits_{V_{m}} \sigma_{\alpha}\mathrm{d}V\right] \tag{10}\]  
where \(\sigma_{\alpha}\) is stress field in the unit cells. For discrete finite elements, Equation (10) can be written by  
\[ \begin{array}{ll}\overline{\sigma}_{\alpha}&=(\sum\limits_{k1=1}^{n}\sigma_{k1 }+\sum\limits_{k2=1}^{m}\sigma_{k2})/(\sum\limits_{k1=1}^{n}V_{k1}+\sum \limits_{k2=1}^{m}V_{k2})\\ &=\sum\limits_{k=1}^{n+m}\sigma_{k}/\sum\limits_{k=1}^{i+j}V_{k}\end{array}