## 4 Analytical Methods  
### Estimation of Bound  
Variational approaches that provide upper and lower bounds on the modulus of elasticity or other elastic constants (such as bulk modulus, shear modulus, Poisson's ratio, etc.) are among the most often used techniques. The difference between the two limits indicates how accurate the estimates are in this method. Additionally, it can be detrimental for specific concentrations of the composite material's phases. They are typically fairly imprecise and are derived for certain types of boundary conditions. The elastic constants for a polymeric material reinforced with parallel, cylindrical fibers arranged in a rectangular pattern are found in this presentation. The process is straightforward, and the desired values can be achieved for applications with high precision. Numerous methods for predicting the constraints on the elastic/viscoelastic parameter of a multi-phase (or two-phase) composite are provided in the literature. The majority of the studies that have been conducted to evaluate the limits of the fiber-reinforced composite's effective elastic/viscoelastic properties make the assumption that both phases exhibit isotropic material behavior. Scarce effort has been made in the literature to address situations where the reinforcing phase, for example, has anisotropic or transversally isotropic characteristics. An illustration of this would be a graphite/epoxy composite, in which the fiber exhibits anisotropic behavior, while the matrix material is thought to be isotropic. Let us take a quick look at some of the approaches found in the literature to gain a better understanding of these issues. The problem of parallel fibers that are sufficiently long to allow for the disregard of end effects is taken into consideration while calculating the bounds on elastic/viscoelastic moduli. A cylindrical specimen with a much larger cross-section than the fiber cross-section might be used to illustrate the material. Since the fiber direction and the specimen's longitudinal axis coincide, it may be assumed that the fiber runs continuously along the specimen's length because end effects are disregarded. Additionally, it is expected that the specimen is transversally isotropic and statistically homogeneous. Predicting the bound on an elastic/viscoelastic property of such a specimen based on its geometrical characteristics and the elastic/viscoelastic moduli of its constituents is the problem to be examined. Let us first assume that the two parts are linear and isotropic in order to illustrate the challenges of obtaining the constraints on the characteristic values of a composite using an analytical approach. Assume that the lower bounds of the bulk and shear moduli are represented by \(K_{23}^{+}\) and \(G_{23}^{+}\), respectively, and the upper bounds by \(K_{23}^{-}\) and \(G_{23}^{+}\). It should be noted that the upper and lower bounds on the characteristic parameters are denoted by \"\(+\)\" and \"\(-\)\" upper indices in this section and the ones that follow, as in \(E_{11}^{+},E_{11}^{-},v_{12}^{+},v_{12}^{-}\). The aforementioned bounds can be predicted using a variety of methods found in the literature. The constraints on the bulk modules \(K_{23}\), the longitudinal [PERSON]'s modulus \(E_{11}\), and [PERSON]'s ratio \(\
u_{1}=\
u_{12}=\
u_{13}\) can be obtained by applying [PERSON]'s theory [130; 131; 132]. The volume ratios of fiber \(\
u_{f}\) and matrix \(\
u_{m}\) are used to calculate these bounds:  
\[K_{23}^{+}=\frac{v_{f}K_{f}(K_{m}+G_{m})+v_{m}K_{m}\left(K_{f}+G_{f}\right)}{ v_{f}\left(K_{m}+G_{f}\right)+v_{m}\left(K_{f}+G_{f}\right)}\;; \tag{58}\]\[K_{23}^{-}=\frac{v_{f}K_{f}(K_{m}+G_{m})+v_{m}K_{m}\Big{(}K_{f}+G_{m}\Big{)}}{v_{f} (K_{m}+G_{m})+v_{m}\Big{(}K_{f}+G_{m}\Big{)}}. \tag{59}\]  
In this relationship, the fiber and matrix phases are represented by the plane-strain bulk modulus \(K_{\lambda}\) and the shear modulus \(G_{\lambda}\), respectively, where \(\lambda=f,m\) (fiber-matrix).