1 Introduction

Since the beginning of the century, the use of composite materials in the form of plate and beam has expanded considerably to this day whether in the automotive, construction, and more recently in aeronautic. The composite materials have significant advantages over traditional materials. In conventional multilayer structures, layered composite materials are used to improve the performance (mechanical, thermal, acoustic...) of the structure (plate). The disadvantage of this type is to create stress concentrations at the interfaces due to the change of mechanical and thermal properties. In the late 80s, a team of Japanese researchers has proposed to overcome these difficulties by developing new materials known as functionally graded materials (FGM). Functionally graded materials are composites in which the material properties vary continuously as a known function of the spatial position; these materials are usually associated with particulate composite where the volume fraction of particles varies in one or several directions. The initial development of FGMs is designed to serve as a thermal barrier¹. Today, there have been more and more numerous modern engineering applications of FGM, like the spacecraft, rocking engine casings and packaging materials in the microelectronics industry, biomaterials (dental implants) and others^2,3.

The metallurgical field has also been the subject of recent research work dealing with the behavior of functionally graded steels^4,5. The effect of a band in FGM around a hole in a homogeneous plate was analyzed for the reduction of the SCF under biaxial loading⁶. Many researchers were interested in calculating the stress concentration factor (SCF) in FGM plates for two types of holes (circular, elliptical)^7-12. In the present work, we used the finite element method to calculate the stress concentration factor at the edge of a notch in a homogeneous ceramic plate and TiB FGM plate for different combinations ceramic-metal. The calculation of FGM's stress concentration SC involves three directions (x, y) and the direction of the radius of the notch. To reduce the SCF, we used a FGM layer around a notch in a homogeneous ceramic under uniaxial load. The graded finite elements are implemented in the FE software Abaqus¹³ to verify the UMAT used subroutine (Appendix).

2 Problem Formulation

A Functionally graded material plate with a semicircular lateral notch is subjected to uniaxial tensile load equal to σ₀ =100 MPa (Figure 1). Geometric characteristics of the plate are:

Figure 1 Geometric model of the plate with a lateral notch circular, (a) the entire plate, (b) half of the plate. 

Width W = 20mm; height H = 2W and thickness denoted t=2 mm. The lateral notch has a radius r = a.

The origin of the coordinates coincides with the center of the notch lateral. For reasons of symmetry of loading and geometry, half of the plate has been studied (Figure 1). The boundary conditions of the model were imposed by constraining the y-displacements (U2 = 0) at y = 0 and rotations around the axes ox and oz (UR3 = 0 and UR2= 0) at y = 0 in the Oxz plane of symmetry of the full model. The properties of the different constituents of the FGM (metal, ceramic) used are indicated in Table 1.

The Figure 2 shows the law graph for different combinations of metal-ceramic. The FGM material is expressed by the following exponential law:

Figure 2 Exponential variation of Young's modulus for different combinations of the constituents. 

With:

Where E₂ = E(0) is the Young's modulus of the metal and E₁ the Young's modulus of ceramic.

Different functionally graded material configurations can be obtained by varying the gradation direction of the constituent materials as:

ξ= x and L=W for x-FGM (Figure 3a).

Figure 3 Direction of the FGM Young's modulus FGM (a) x-FGM variation, (b) y-FGM variation, (c) r-FGM variation. 

ξ= y and L=W for y-FGM (Figure 3b).

ξ= x2+y2 and L=W2 for r-FGM (Figure 3c).

3 Finite Element Modeling

For modeling the Young's modulus in a desired direction, the subroutine UMAT Abaqus was used¹⁴. The routine is written in FORTRAN language and runs in parallel with the Abaqus solver. It allows us to establish an algorithm to calculate the variables used by the solver Abaqus. The routine was coded so that the material stiffness matrices are established with appropriate material properties, namely, the Young's modulus. In this study, the Poisson's ratio is assumed constant, since it has been shown that variations in the Poisson ratio are much less important than the Young's modulus¹⁵. Calculating the stiffness matrix requires the use of the Gaussian quadrature¹⁶.

A half symmetric model of a 20 × 20 mm² plate and a thickness t with 2 mm radius center notch was used for verification as shown in Figure 4. The mesh was refined around the center notch and was graded in the direction moving away from the notch toward the outer boundaries. The initial mesh consisted of 13810 C3D20R quadratic brick, full integration stress elements with 276200 nodes. The base Young’s modulus is TiB with E₁ =375 GPa and E₂ is the metal Young’s modulus. The Poisson’s ratio was held constant equal to 0.3.

Figure 4 Model verification mesh with 13810 C3D20R elements. 

4 Results and Analyses

The finite element method has allowed us to calculate the normalized stress concentration of a notched plate noted Kt for different directions of FGM and a constant normalized thickness (t/W=0.1). The stress concentration allows increasing the stress at the notch root. The factor Kt is defined by the ratio of the maximum stress to the applied stress σ₀:

In the case of a homogeneous and isotropic plate, Kt is called stress concentration factor. The Figure 5 shows the normalized distribution of σ₂₂ stress versus x from the edge of the notch for the tree direction of FGM and the normalized distribution of σ₂₂ stress of ceramic.

Figure 5 Variation of normalized normal stresses vs normalized x for different gradation direction of Ti-TiB FGM. 

In our case, the stress σmax represents the normal stress along the y axis at the edge of the notch at y = 0 and noted σ_22max. The value of σ_22max is taken in the middle of the thickness where it is maximum (Figure 6). The maximum values are naturally obtained at notch edge (Figure 6). We deduce that the direction of FGM along r gives the minimum value. We note that FGM distribution management for the rest of the study. The gain of normalized stress concentration (Kt) of an r-FGM plate at the edge of the notch is 44% compared to a homogeneous ceramic plate.

Figure 6 Strain max contours for different gradation direction distribution of FGM (a) x-FGM variation, (b) y-FGM variation, (c) r-FGM variation. 

The field of deformation shape of the plate is similar to the graduated FGM modulus for the three directions (Figure 7). The maximum deformation is lower in the case of r-FGM.

Figure 7 Stresses contour for different gradation direction distribution of FGM (a) x-FGM variation, (b) y-FGM variation, (c) r-FGM variation. 

4.1 Parametric study

The profile of the stress distribution σ₂₂ normalized along the x-axis is the same for the various combinations of metal-ceramic. The decrease of the stress concentration factor at the edge of the notch is proportional to the decrease in the E₁/E₂ report. For a normalized depth graduation of FGM between 0.4 and 0.7, the normalized stress tend towards the value 1, for a standardized depth graduation of FGM superior to 0.7, the stresses decrease slightly (Figure 8). The greatest reduction of the stress concentration factor at the edge of the notch is obtained for the combination AL-TiB and is about 60% compared to the TiB ceramics (Figure 9).

Figure 8 Variation of normalized normal stresses vs normalized x for different combinations of the FGM. 

Figure 9 Variation of the stress concentration factor Kt based on the normalized x for different ratios E1 / E2. 

Figure 10 shows the evolution of K_t as a function of the normalized notch radius for different combinations of the FGM. K_t increases with increasing radius of the notch for all combinations. E₂ is the Young modulus of ceramic.

Figure 10 Variation of stress concentration factor as a function of normalized radius of notch for different ratios E1 / E2. 

4.2 Effect of the FGM layer

To analyze the effect of the FGM layer, a ceramic plate in TiB contains a r-FGM layer in Ti-TiB with a width h (h=b-a) around a circular lateral notch of radius a (Figure 11).

Figure 11 A geometric model FGM layer in a ceramic plate. 

We observe that an increase of the width h corresponds to a decrease of the Kt value at the rim of the notch and, at the same time, we have an increase of the interface Kt (Figure 12). The Kt at the rim of the notch is noted Ka and the other Kb. The use of a standard FGM layer Ti-TiB (h/W=0.4) leads to a significant decrease of Ka with a gain of 41% and a gain of 64% for the Kb with respect to a ceramic plate. We observe that the value of Kt = 2.02 of a complete r-FGM is close to that of Ka (Ka = 2.2) of a plate with the same size of the layer.

Figure 12 Variation of normalized normal stresses vs normalized x for different normalized witdh of FGM layer. 

Figure 13 shows the evolution of Ka and Kb depending on the width of the layer. The values of Ka and Kb tend respectively to 2.2 and 1.32. The gains of Ka and Kb are summarized in Table 2 which are respectively 41% and 64%.

Figure 13 Variation of Ka and Kb depending on the normalized width of FGM layer. 

Table 2 The gain of SCF Ka and Kb for different normalized width of layer. 

Normalized h/W	Gain (%)
	ΔKa/Kt	ΔKb/Kt
0.1	25.54	46.50
0.2	32.26	58.60
0.3	36.83	60.75
0.4	41	64

5 Conclusion

The use of FG-layer is another way to increase the performance of the notched structures.

The use of a r-FGM or r-FGM layer in a ceramic notched plate allowed to draw the following conclusions: