## 5 Homogenized Viscoelastic Coefficients
### Results and Applications  
The homogenization method has been used extensively since the beginning to determine the creep behavior of unidirectional composites reinforced with continuous fibers. We will show that there are two main methods of using the method, one involving the determination of the real stress and strain field for an RVE and then averaging the property values, and the second using a variational method. The first method is applied in [189], where FEM is used to determine the stresses and strains. It is considered that the constituent fiber is an isotropic linear elastic material and that the matrix is isotropic linear elastic and nonlinear viscoelastic. The results obtained with these calculations for determining the creep behavior are validated with experimental results. Other results using the classical homogenization theory for the analysis of a composite reinforced with carbon fibers, together with experimental tests to validate the results, are presented in [190]. Creep analysis of polymer matrix composites using homogenization theory for time-dependent nonlinear composites is presented in [191]. Studies on carbon fiber-reinforced plastic (CFRP) laminates and glass fiber-reinforced plastic (GFRP) laminates have also been conducted. Experimental validation is obtained on uniaxially loaded specimens. The use of a homogenization method using Eshelby tensors is presented in [192] and enables the calculation of the creep of fiber composites. The micromechanical model is validated through comparison with experimentally obtained data on the flow of the studied material. The conclusion obtained in the study is that the creep of fiber-reinforced composites can be adapted by an appropriate design. Fiber-reinforced concrete is modeled as an aged linear viscoelastic composite material reinforced with ellipsoidal inclusions embedded in a viscoelastic cement matrix in [193]. The time-dependent deformations of the concrete are validated with experimental measurements. In [194], a multi-stage homogenization is performed for a hybrid composite with a polymer matrix reinforced with unidirectional carbon fibers together with silica nanoparticles. After determining the viscoelastic behavior of the silica nanoparticles, a classical micromechanical model is used to analyze the composite, using the homogenization method. The results are in agreement with the experimental determinations made in the paper. Homogenization methods are designed to improve the viscoelastic properties of a randomly oriented chopped glass fiber-reinforced plastic in [195]. Notably, 3D numerical calculations for creep deformations are used. The conclusion obtained is that for a given creep load, the material life increases with increasing fiber volume fraction. The thermomechanical tensile and creep behavior of PEEK 7002/carbon fiber composites in tensile and creep tests, presented in [196], uses a multiscale approach. The study shows a similarity of behavior between the polymer and the composites. An algorithm is proposed to perform such an analysis. A multiscale optimization method for a composite material is presented in [197]. The paper aims to establish the composition of short fiber-reinforced polymer (SFRP) materials that will exhibit a desired behavior in service (mechanical properties to meet the requirements of the manufacturer). The paper illustrates the procedures followed in such an enterprise, which is also described in the paper. Different applications have been addressed for different types of resin and reinforcement materials, with the homogenization theory proving its power in obtaining correct and useful results.