## 5 Homogenized Viscoelastic Coefficients
### The Homogenized Method  
One benefit of the homogenization theory is that it makes it possible to investigate differential equations with coefficients that vary quickly or periodically. After averaging procedures, engineering constants (which are helpful in engineering practice) are produced. As a result, a material with a periodic structure can be considered homogeneous. Thus, an equation with constant coefficients is used in the modeling instead of a differential equation with periodic coefficients that vary greatly. This is how microstructured materials (composite materials are also included in this class) are included in the continuum notion. In this section, the creep response of a unidirectional composite reinforced with fibers is examined using a computational approach. The classic presentation of the homogenization method developed by [PERSON] [183] (Figure 7).  
Considering the valuable hypothesis from the theory of elasticity, the stress tensor \(\sigma^{\delta}\) for a repeating unit cell (RVE) that forms the composite of size \(\delta\) must satisfy the classic equations:  
\[\begin{array}{c}\frac{\partial\sigma_{1i}^{s}}{\partial x_{1}}+\frac{\partial \tau_{1i}^{s}}{\partial x_{2}}+\frac{\partial\tau_{1i}^{s}}{\partial x_{3}}=f_{ 1}(x);\\ \frac{\partial\tau_{1i}^{s}}{\partial x_{1}}+\frac{\partial\tau_{2i}^{s}}{ \partial x_{2}}+\frac{\partial\tau_{2i}^{s}}{\partial x_{3}}=f_{2}(x);\\ \frac{\partial\tau_{1i}^{s}}{\partial x_{1}}+\frac{\partial\tau_{2i}^{s}}{ \partial x_{2}}+\frac{\partial\tau_{3i}^{s}}{\partial x_{3}}=f_{3}(x),\end{array} \tag{106}\]  
The tensor is symmetric, so \(\sigma_{ij}^{\delta}=\sigma_{ji}^{\delta}\), for \(i,j=1,2,3\).  
The contour conditions that the displacements must satisfy are as follows:  
\[u^{\delta}\Big{|}_{\partial_{1}\Omega}=\widetilde{u}\enspace. \tag{107}\]  
The boundary conditions on contour \(\partial_{2}\Omega\), \((\partial_{1}\Omega\cup\partial_{2}\Omega=\partial\Omega)\) of the RVE are as follows:  
\[\begin{array}{c}\sigma_{1i}^{s}n_{1}+\tau_{2i}^{s}n_{2}+\tau_{3i}^{s}n_{3}=T _{1}(x);\\ \tau_{2i}^{s}n_{1}+\sigma_{2i}^{s}n_{2}+\tau_{2i}^{s}n_{3}=T_{2}(x);\\ \tau_{3i}^{s}n_{1}+\tau_{3i}^{s}n_{2}+\sigma_{3i}^{s}n_{3}=T_{3}(x).\end{array} \tag{108}\]  
The Hook's Law has the following form:  
\[\begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \tau_{23}\\ \tau_{31}\\ \tau_{12}\end{array}=\begin{bmatrix}C_{1111}&C_{1122}&C_{1133}\\ C_{2211}&C_{2222}&C_{2233}&0\\ C_{3311}&C_{3322}&C_{3333}\\ &C_{2323}\\ 0&&C_{3131}\\ &&C_{2122}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{23}\\ \gamma_{31}\\ \gamma_{13}\end{bmatrix}^{\delta}. \tag{109}\]  
or,  
\[\sigma^{\delta}=C\varepsilon^{\delta}\enspace. \tag{110}\]  
Using the averaging methods, these coefficients are obtained. The method to obtain is presented in [139; 189]. It results in two ways to obtain these coefficients:  
1. Using the local equations, the strain and stress field, and the averages are determined, obtaining the homogenized coefficients after averaging;
2. Using the variational formulation and determining a special function \(w^{kh}\) that can also help us determine the homogenized coefficients. In the first way, the field of stress and strain should be obtained. Rarely can the exact stress and strain field be determined, and therefore, a numerical method must be applied  
Figure 7: The aspect of the structure that must be homogenized.  
to obtain these data. The most commonly used method in application is the finite element method (FEM).