# The Quick Determination of a Fibrous Composite's Axial Young's Modulus via the FEM
## 1 Introduction
It is clear that, to determine the mechanical properties of composites, the best method, which has been intensively used by researchers, is to experimentally measure the eigenfrequencies of a material specimen and use these measurements to calculate the values of the engineering constants [33; 34; 35; 36]. But, this method involves time and resources to perform the measurements, which is why it is desirable to use a quick method to determine properties in the first phase of design. Concrete is one of the materials that deserves special attention due to its extraordinary application in civil engineering. In this field, the current practice is to reinforce the concrete with cylindrical iron fibers. The FEM proved to also, in this case, be a suitable method for determining the mechanical properties of the homogenized material, and the elastic constants of the homogenized material were determined and verified experimentally [37]. A very good agreement was found. Another area where intense development is taking place is the use of natural fibers to reinforce a polymer matrix. However, studying these fibers is more complicated because they have a viscoelastic behavior [38], and the developed models are generally sophisticated. The analytical methods used are more elaborate and require more computational effort. Another interesting method used in the study of composite materials reinforced with glass or carbon fibers, which exhibit a viscoelastic behavior, is presented in [39]. In developing the method, the time factor must also be introduced, which will describe the flow rate of the material if the temperature at which it is used increases. Other researchers have developed methods for determining elastic coefficients for different types of materials used in engineering applications [40; 41]. A study on identifying material properties using an inverse method is presented in [42]. In this paper, the authors try to present the advantages of using the classical formulas from Vibration Mechanics to determine [PERSON]'s modulus. A FEM model is used to calculate the natural frequencies of a beam. Two sets of boundary conditions are considered: the beam is clamped at both ends, and the beam is clamped at one of the ends. The material from which the beam is made is a polymer composite reinforced with carbon fibers. With the standard assumptions of the classical theory of straight bars, knowing the values of natural frequencies determined based on the formulas provided in the literature and knowing the values of the natural frequencies determined with the FEM model, [PERSON]'s modulus can be determined. In this paper, the method is used to determine [PERSON]'s modulus for a unidirectional carbon fiber-reinforced polymer composite material. The main advantage of this method is that it provides quick estimates of these values. Obviously, the costs involved are also reduced. If compared to experimental methods, the method is much cheaper and more convenient. Also, the time required to obtain these values is reduced. The analysis focuses on the study of longitudinal vibrations through conditions imposed on the displacements in the FEM, allowing only the axial displacement of the bar, considering a set of special boundary conditions. In this way, computing resources are used optimally. The method can be applied to a wide class of materials in order to determine the properties of the homogenized material.