## 1 Introduction  
Functionally graded materials (FGMs) represent a special category of composite materials, usually made of two materials, with very different properties (Figure 1). The material properties vary continuously (Figure 2) between the extreme surfaces of the material, where the properties are those of the respective materials in their pure state [1, 2, 3]. Today, the most used materials used in the construction of FGMs are ceramic materials and metals. Their volume fractions, in the thickness direction, varies continuously, according to a material law \(F(z)\), as Figure 2 shows. Figure 1: The adopted coordinate system. The material law function \(F(z)\) defines the variation of the elastic, physical and mechanical properties of the material (Young's Modulus, Poisson's ratio, density, thermal coefficient, etc.). Several material laws are known [3,4], having the names: Power Law, Reuss Law, Voigt Law, Local Representative Volume Element (LRVE) Law, Tamura Law, Mori-Tanaka Law, Exponential Law, Sigmoid Law and others. The best-known and most used material law, in practice and in the literature, is the Power Law ([PERSON] et al., 2009), which for Young's Modulus, \(F(z)\equiv E(z)\), has the expression [2],  
\[E(z)=E_{b}+(E_{t}-E_{b})\Big{(}0.5+\frac{z}{\hbar}\Big{)}^{k} \tag{1}\]  
where the indices \"\(t\)\" and \"\(b\)\" mean the top face or side of the material with higher properties, respectively the bottom face or side of the material with low properties and \(k\) is the power coefficient (values from '0' to '\(\infty\)'). In Figure 3, the variation of the material law function for \(E(z)\) is presented for different values of the power coefficient. Of course, there are many other material pairs such as: polymer-elastomer, organic-inorganic glasses, etc. which can be used [4,5]. The name of such materials appeared in the mid-1980s, being introduced by scientific researchers from Japan, who were working on the creation of ultra-resistant materials at high temperatures for aerospace constructions. The appearance of functionally graded materials attracted the attention of many scientific researchers [6, 7, 8, 9, 10, 11], whose efforts were channeled in two directions: the realization (manufacturing) of these materials and the calculation of the structures from these materials. The two directions interfere and stimulate each other, especially in the field of materials  
Figure 3: Variation of material Power Law for different values of power coefficient \(k\). Figure 2: The variation of the properties on the thickness of the material. design. In both directions the results are remarkable, but not finished; both manufacturing and calculation being in continuous development [12; 13; 14; 15]. The field of use of FGMs [16; 17] is very broad, such as biomedical equipment and instruments, medical prostheses, solar panels, aerospace constructions, sensor construction, optics, electronics and many others. The attention given to FGM is fully justified [18]. The present paper comes from the concern (purpose) to find models and calculation methodologies as simple as possible, as efficient as possible, based on the current knowledge of analytical and numerical analysis of structures [19], using widely accessible commercial analysis programs with the finite element method. In the field of calculation of plates from FGMs [20; 21; 22], the concept of multilayer plate is often used. This concept is grounded in the calculus of functionally graded plates (FGPs), providing substantiated answers to a series of questions regarding the number of layers (their thickness), the influence of Poisson's ratio, the influence of the power coefficient from the material law, etc. The calculation model and methodology, based on the concept of a multilayer plate, are validated by the calculus compared to the analytical solution of the plate bending rigidity, the maximum transverse displacement, the calculus of the stresses and the calculus of free vibration frequencies.