# The Quick Determination of a Fibrous Composite's Axial Young's Modulus via the FEM
## 1 Introduction  
Determining Young's modulus for a material is a step during the design of a mechanical system. Its determination can be achieved experimentally or using different calculation formulas, methods for which there is a rich body of literature. For fibrous composite materials in particular, numerous methods have been developed to determine this value, with greater or lesser precision. The main methods developed are based on micromechanical models or homogenization theory. Their use involves determining the fields of strains and stresses for the respective system. Another method used by researchers, based on variational principles, generally provides upper and lower bounds for the modulus of elasticity. Obviously, in this case, the result is given by the difference between the upper and lower values of the modulus, and this method can lead, for certain concentrations of the composite phases, to errors, which are sometimes significant [1]. These limits were established in the cited paper for an orthotropic and a transversally isotropic composite. These methods are based on the consideration of particular cases of loading and boundaryconditions and can present significant errors [2; 3]. Micromechanical models are theoretically more accurate but require knowledge of the fields of stresses and strains for the material [4; 5]. The results obtained in these studies have been developed and verified experimentally, and the results agree very well [6; 7]. Fiber-reinforced composites represent an important class of materials with many applications in practice, and, as a result, numerous studies have been carried out on their mechanical properties [8; 9; 10; 11; 12; 13; 14; 15; 16]. These studies have generated numerous problems related to the methods of solving the equations obtained and the numerical results obtained [17; 18; 19; 20]. Experimental methods are obviously the most reliable methods to determine the values of the elastic constants for a material. However, their use requires time and resources that can be significant in certain cases. This is why, in the first phase of a project, it is more advantageous to work with some estimates obtained quickly that have a satisfactory degree of confidence. Following this, in the final part of the project, more in-depth investigations should be carried out regarding the properties of the materials that have been decided to be used. The method of a representative volume element for a composite material with short fibers was analyzed in [21]. The method can be applied in many applications, for example, in reinforced composites with aligned cylindrical fibers. The finite element method (FEM) has become an established method in the analysis of deformable solids and has been successfully used by researchers to determine the mechanical behavior of composite materials. Generally, this method is applied for static analysis, mainly to determine the stress and strain fields inside a composite. In this paper, the FEM was used to study the vibrations of a composite and determine the natural frequencies of the considered specimen. The advantage of this method is that it can provide fast results for a very wide class of materials with different topologies, geometries, and mechanical properties. The ability of the FEM to obtain results very close to reality was demonstrated by experimental verification [22]. The FEM proved, once again, to be a useful method for studying the mechanical properties of composite materials made up of a polymer matrix reinforced with aligned cylindrical fibers [23]. It should be mentioned that most calculation methods involve the determination of the fields of stresses and strains for a certain load case (boundary conditions). Obviously, for almost all cases encountered in practice, this calculation involves a numerical method. For the determination of strains and stresses for a deformable solid, the method used with the greatest success at the moment is the FEM.