# The Quick Determination of a Fibrous Composite's Axial Young's Modulus via the FEM
## 2 Materials and Methods  
The FEM is used to determine the natural frequencies of a composite material using all of the matrix material and fibrous material information. At the same time, the classical methods for determining the natural frequencies of a straight bar made of the same material, which represents a homogenized material whose properties are determined by the properties of the phases used, are considered. By comparing these values with those obtained using the FEM, the elastic constant values for the homogenized material can be obtained. To facilitate the presentation, some known results from the literature on straight-beam vibrations are presented very briefly. The classical assumptions from the theory of straight bars are considered fulfilled. We mention that we present only the important ones in the context of the paper [43]. Two cases are considered in our study: a beam clamped at both ends and a beam clamped at one end. Based on the relationships for determining the eigenfrequencies for a beam with the mentioned boundary conditions, it is possible to determine Young's modulus for the homogenized material. These values represent a quick estimate of [PERSON]'s modulus. The FEM is used to determine these natural frequencies by considering the matrix and reinforcing fibers included in it. This can be accomplished relatively easily and accurately using this method. In this way, through a relatively simple calculation, considering the classical results obtained in the study of vibrations of straight bars to be valid, [PERSON]'s modulus can be easily determined. Figure 1 shows the two types of analyzed beams. The software used is Hypermesh 2020. The type of finite element used is hexahedral. Consider the longitudinal vibration of a straight beam. These vibrations are described by the following differential equations:  
\[\frac{\partial^{2}u}{\partial x^{2}}=\frac{\rho}{E}\frac{\partial^{2}u}{ \partial t^{2}} \tag{1}\]  
Figure 1: Beam (a) clamped at both ends; (b) clamped at one end. If we choose a solution having the form  
\[u(x,t)=\Phi(x)\sin(pt+\phi) \tag{2}\]  
and introduce it into Equation (1), we obtain a differential equation that gives the amplitude of the natural vibrations (eigenfunctions):  
\[\frac{d^{2}\Phi}{dx^{2}}+p^{2}\frac{\rho}{E}\Phi=0 \tag{3}\]  
The initial conditions, considered at the moment \(t=0\), are  
\[u(x,0)=f(x);\;\;\frac{\partial u}{\partial t}\Big{|}_{t=0}=g(x) \tag{4}\]  
Equation (3) gives the solution:  
\[\Phi(x)=C_{1}\sin\alpha x+C_{2}\cos\alpha x \tag{5}\]  
where  
\[\alpha^{2}=p^{2}\frac{\rho}{E} \tag{6}\]  
For a beam clamped at both ends, the boundary conditions are  
\[\Phi(0)=0;\;\;\Phi(l)=0 \tag{7}\]  
and it results easily in  
\[C_{2}=0;\;C_{1}\sin\alpha l=0 \tag{8}\]  
We easily obtain  
\[\alpha=\frac{n\pi}{l};\;n=1,2,3,\ldots\ldots. \tag{9}\]  
Using notation (6), the eigenpulsations are given by the following relation:  
Figure 1:0  
From Equation (10), the Young's modulus of the homogenized material clamped at both ends can be determined:  
Figure 1:1  
Using Equation (11) results in  
Figure 1:2  
In a similar manner, if we consider the beam clamped only on one end, the literature [43] offers the following for the eigenpulsation:  
Figure 1:3  
or  
Figure 1:4So, the Young's modulus of the homogenized material clamped at one end can be determined with the following relation:  
Figure 1:5  
Using Equation (14) results in  
Figure 1:6  
Using the FEM, it is possible to obtain a number of eigenpulsations that give us more values for \(E\). The average can be calculated in order to obtain a better estimation of \(E\).