## 4 Analytical Methods
### Results and Applications (Mori-Tanaka Theory)  
Numerous works have addressed the problem of determining elastic constants using analytical boundary methods. Particular loading situations of RVE were sought in which stress and strain could be determined, and stress energy or strain energy was calculated. By comparing with the exact values of these quantities, upper or lower limits of the elastic constants were determined. Additionally, numerous studies have used these results for the study of certain new materials. An example of using the method to determine the physical properties of a composite is presented in [152]. The fibers are considered to be uniform. A comparison with other methods used shows that the inner and upper edges are the best estimate provided, even if not spectacular, so the method proves to be useful. A phenomenological-mechanistic model is used to study the strength of fiber-reinforced composite beams [153]. A composite panel is composed of thin transversely isotropic layers (laminas). Each of the laminae is composed of unidirectional layers. The predictions obtained follow the experimental load-strain curves in four-point bending until the failure process begins to dominate. A study of the influence of microstructural parameters in the case of composites is described in [154]. The stiffness moduli of composites are determined as a function of fiber orientation and under arbitrary loading conditions. Compared to micromechanical theories, the method proves to be more practical for predicting the stiffness modulus of fiber composites in terms of accuracy and simplicity. Some studies [155; 156; 157; 158; 159] apply analytical methods to determine upper and lower bounds of elastic constants for various engineering applications. Two different approaches to the study of unidirectional composites are presented in [160]. The stress concentration factors around broken fibers in a three-dimensional model with a hexagonal matrix are determined by the two methods. Parametric studies are first performed, and the results are compared with experiments and other existing theories. A 3D finite element model for predicting damage initiation, propagation, and fracture toughness of TC33/epoxy unidirectional carbon fiber-reinforced polymer (CFRP) laminates under biaxial loading is developed in [161]. The carbon fibers are assumed to be linearly elastic, and the matrix is considered to have an elastoplastic behavior defined by the extended Drucker-Prager plastic yielding model. An analytical determination of the out-of-plane modulus of thin QI-CFRP specimens is made using a modified form of three-dimensional laminate theory [162]. The modulus calculated in this way is the lower bound of the apparent bending moduli, and that calculated with three-dimensional laminate theory is the upper bound. The Mori-Tanaka method has proven to be a very efficient method for estimating elastic constants. A confirmation is the rich literature that uses this method and the engineering applications that apply it in design calculations. Some significant contributions that used the method are presented below. For glass-reinforced polyvinyl chloride (PVC) plastics, the Mori-Tanaka method is used, and the results are compared with 3D numerical calculations for creep deformation and creep failure [163]. The comparison validates the Mori-Tanaka model as a practical tool to predict the effect of fiber length and fiber volume fraction on creep deformation and creep failure. A composite reinforced with short fibers, differently oriented in the matrix, is studied to determine the creep behavior [164]. The fibers are assumed to have the same length. According to the Mori-Tanaka mean field theory, and considering that the effect of an imperfect interface can be expressed by a modified Eshelby tensor, the effect of the weakened interface on the creep behavior for a nearly unidirectional composite is determined. The Mori-Tanaka method has been analyzed and used in numerous applications [165; 166; 167; 168; 169].