# The Quick Determination of a Fibrous Composite's Axial Young's Modulus via the FEM
## 4 Discussion  
The values of the elastic constants that define the constitutive law of a composite made of a polymer matrix reinforced with glass or carbon fibers can be determined by several analytical methods developed by researchers, which are, however, very laborious. In most of the proposed methods, it is necessary to determine strain and stress fields for a specimen considered representative of the material. Boundary methods, which use variational calculations, impose restrictive boundary conditions that make the obtained results less accurate. Experimental methods remain the most reliable methods but have the disadvantage of the time in which they can be carried out and the resources involved (and thus costs). This paper shows that the FEM can be a simple and sufficiently accurate method for the quick estimation of some of the elastic constants. In this work, we dealt with the modulus of elasticity, but it is obvious that other constants can be determined if suitable models are used. In this paper, with the help of simple formulas, we managed to calculate [PERSON]'s modulus. We can conclude that the presented method has the advantage  
Figure 4: The eigenfrequencies for the beam clamped at one end.  
Figure 3: The eigenfrequencies for the beam clamped at both ends.  
of speed, simplicity and feasibility for obtaining quick estimates of the values of elastic constants in the design phases of a mechanical system. The accuracy of the results can be improved by considering a FEM model with more elements, which will more accurately describe the existing situation for each practical application [49; 50; 51]. This method provides fast and accurate estimates of [PERSON]'s moduli, which are important data in the design process. Obviously, the assumptions for bars in the theory of straight bars must be respected. The errors that may appear are due to the FEM, within the limits accepted for the approximation method, and the deviation from classical assumptions. The method allows for obtaining quick and useful results. Obviously, the need to know the mechanical properties expressed by the elastic constants of the materials is a main objective when designing a mechanical system. Among these materials, polymer composites reinforced with aligned cylindrical fibers are distinguished, mainly due to their numerous practical applications. Obviously, experimental measurements provide the most reliable results, but this operation is expensive and time-consuming. This is why fast methods for estimating these properties according to the properties of the component phases are necessary. Based on micromechanical models, analytical formulas were determined to obtain engineering constants. But, this approach requires the determination of the stress and strain fields for the studied system. The same requirement is implied if homogenization theory is used. Other theoretical models have been proposed that involve the knowledge of particular loading situations but that provide upper and lower bounds for the elastic constants, which means significant errors in the determination of these constants, which can sometimes be far from the real values [7; 8; 9; 10; 11; 12; 13]. The method used in the present paper considers the homogenized material and uses classical calculation methods to determine the natural frequencies of such a homogenized bar. With the FEM, the values of the natural frequencies are determined for the real system, consisting of the matrix material and the reinforcing fibers. By comparing the two values, [PERSON]'s modulus can be easily determined. In this way, fast and accurate estimates of this quantity are obtained [52; 53]. In practice, complex situations may arise, determined by practical applications, in which effects that cannot be neglected, such as thermal effects, humidity, etc., may occur. In order to solve such problems, classical models must be improved to be able to take into account these effects, which introduce new parameters.