# Stresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings

^{1}

^{2}

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## Abstract

**:**

_{3}N

_{4}-based ceramics and is assumed to be of semi-infinite extent. The load is a distributed loading over a rectangular area that is parallel to the external surface of the FGM and either on its external surface or within its interior space. The point-load analytical solutions or so-called Yue’s solutions are used for the numerical integration over the distributed loaded area. The loaded area is discretized into 200 small equal-sized rectangular elements. The numerical integration is carried out with the regular Gaussian quadrature. Weak and strong singular integrations encountered when the field points are located on the loaded plane, are resolved with the classical methods in boundary element analysis. The numerical integration results have high accuracy.

## 1. Introduction

_{3}N

_{4}-based ceramics of semi-infinite extent is used as the FGM for the stress and displacement analysis. The rectangular loading area is parallel to the boundary of the semi-infinite FGM space and the uniform normal loads are chosen. The displacements and stresses induced in the graded materials are presented. The comparison of elastic fields is made for two different positions of the rectangular loads.

## 2. Numerical Method for Analysis of Mechanical States in FGMs

#### 2.1. The Point-Load Solution Suitable for the FGM

^{–15}MPa). Consequently, the first elastic solid becomes a void space of upper semi-infinite extent. The Yue’s solution for a point load in a multilayered elastic solid of infinite extent is automatically degenerated into the generalized Mindlin solution for a point load in a layered half-space.

**Figure 1.**Functionally Graded Material (FGM) half-space subjected to loads on a rectangular area (a = 2 mm, b = 1 mm).

#### 2.2. The Numerical Method for Analysis of the FGM due to Distributed Loadings

_{ijk}*(Q, P) and u

_{ik}*(Q, P) are the point-load solutions of the layered medium; σ

_{ijk}*(Q, P) are stresses for the field point Q due to the unit force along the k direction at the source point P; u

_{ik}*(Q, P) displacements for the field point Q along the i direction due to the unit force along the k direction at the source point P; t

_{k}(P) is the traction at the source point P.

_{k}(P), the integrals shown in Expressions (1) and (2) cannot be integrated into analytical forms. The 2D integrals in (1) and (2) have to be calculated numerically. A discretization technique, similar to that used in boundary element methods [16], is adopted. The loading area S is discretized into quadrilateral elements. In each element, the interpolation functions between the global and local coordinates are introduced. Thus, the integral on each element is executed in local coordinates and is calculated by using the regular Gaussian quadrature.

## 3. Displacements and Stresses in FGMs under Rectangular Loadings

#### 3.1. General

_{0}exp(αy) where y = 0 is either the boundary of the half plane or the plane of the crack. In Kassir [5], it is assumed that μ = μ

_{0}׀y׀

^{m}, (0 < m < 1).Generally, it is easy to obtain the analytical solutions of FGMs for the above-mentioned assumption of the material properties. Actually, the properties of FGMs are distributed in complex forms. Thus, the proposed numerical method is used much effectively to analyze the mechanical response of the actual FGMs because it can also take into account the depth variations in both Young’s modulus and Poisson’s ratio.

_{3}N

_{4}-based materials given in Pender et al. [18] are further used for the stress analysis. The Si

_{3}N

_{4}-based graded materials were fabricated with controlled, unidirectional gradients in elastic modulus from the surface to the interior. The elastic parameters shown in Figure 2 were estimated by using several photomicrographs and image analysis software. The FGM had a constant Poisson’s ratio 0.22 and its elastic modulus is described by a piecewise linear interpolation as follows

_{0}= 225.1 GPa and ν = 0.22). This modulus value is equal to the average value of the FGM modulus.

_{3}N

_{4}. It can be found that the tensile strength of Si

_{3}N

_{4}is 810 MPa and the compressive strength of Si

_{3}N

_{4}is more than 1.5 GPa. Herein, it is assumed that p(x,y) = 100 MPa. For this case, the FGMs may be in the elastic state and the proposed method can be used for analysis of the displacement and stress fields.

#### 3.2. The Loading Area at h = 0 mm

_{x}, u

_{y}and u

_{z}along y = 0.5 mm at the depths z = 0, 0.125 and 0.225 mm. From these figures, it can be observed that the absolute values of displacements decrease as the depth increased for the two cases. At a given depth, the absolute values of the displacements of Case 1 are smaller than the ones of Case 2.

_{xx}, σ

_{yy}and σ

_{zz}along y = 0.5 mm at depths z = 0, 0.125 and 0.225 mm. It can be observed that the normal stresses are discontinuous at the points x = ± 1.0 mm and z = 0 mm across the loading area. At a given depth and among −1.0 mm < x < 1.0 mm, the absolute values of σ

_{xx}and σ

_{yy}for Case 1 are smaller than the those of the corresponding normal stresses for Case 2. At the depth z = 0.225 mm, this influence is not obvious. Figure 6 shows the variations of the three normalized shear stresses σ

_{xy}, σ

_{xz}and σ

_{yz}along y = 0.5 mm at the depths z = 0, 0.125 or 0.225 mm. It can be observed that there are no obvious influences of material heterogeneities on the distributions of stresses except σ

_{yz}. At the depth z = 0.225 mm and among −1.0 mm < x < 1.0 mm, the σ

_{yz}values have obvious differences between Cases 1 and 2.

**Figure 4.**Normalized displacements (

**a**) u

_{x}; (

**b**) u

_{y}; (

**c**) u

_{z}along y = 0.5 mm at z = 0, 0.125, 0.225 mm for the loading area h = 0 mm.

**Figure 5.**Normalized stresses (

**a**) σ

_{xx}; (

**b**) σ

_{yy}; (

**c**) σ

_{zz}along y = 0.5 mm at z = 0, 0.125, 0.225 mm for the loading area h = 0 mm.

**Figure 6.**Normalized stresses (

**a**) σ

_{xx}; (

**b**) σ

_{yy}; (

**c**) σ

_{zz}along y = 0.5 mm at z = 0, 0.125, 0.225 mm for the loading area h = 0 mm.

#### 3.3. The Loading Area at h = 0.13 mm

_{x}and among −1.0 mm ≤ x ≤ 1.0 mm for u

_{y}and u

_{z}.

**Figure 7.**Normalized displacements (

**a**) u

_{x}; (

**b**) u

_{y}; (

**c**) u

_{z}along y = 0.5 mm at z = 0, 0.13, 0.225 mm for the loading area h = 0.13 mm.

_{xx}and σ

_{yy}for Cases 1 and 2 have obvious differences for −1.0 mm ≤ x ≤ 1.0 mm at the depths z = 0 mm and 0.13 mm while the stresses has negligible differences at z = 0.225 mm. However, the values of σ

_{zz}at any depths for Cases 1 and 2 have no obvious differences. Figure 9 shows the variations of the three normalized shear stresses along y = 0.5 mm for different depths z = 0, 0.13 and 0.225 mm. It can be found that there are small differences between Cases 1 and 2 except the shear stress σ

_{xy}. The σ

_{xy}values have large differences between Cases 1 and 2.

**Figure 8.**Normalized stresses (

**a**) σ

_{xx}; (

**b**) σ

_{yy}; (

**c**) σ

_{zz}along y = 0.5 mm at z = 0, 0.13, 0.225 mm for the loading area h = 0.13 mm.

**Figure 9.**Normalized stresses (

**a**) σ

_{xy}; (

**b**) σ

_{xz}; (

**c**) σ

_{yz}along y = 0.5 mm at z = 0, 0.13, 0.225 mm for the loading area h = 0.13 mm.

#### 3.4. Comparison of Elastic Fields for the Two Loading Positions

_{z}values for h = 0.13 mm are smaller than the ones for h = 0 mm. This is because the materials above the loading plane h = 0.13 mm constrain the deformation of the FGMs. However, u

_{x}and u

_{y}have small differences for two loading positions.

_{xx}, σ

_{yy}and σ

_{zz}are the maximums at loading plane positions h = 0 mm and h = 0.13 mm, respectively. Away from the loading plane positions, the values of these stress components become small. For the two loading positions, the absolute values of σ

_{xy}are the maximums at z = 0 mm, i.e., the boundary of semi-infinite extent. At x = ±1.0 mm, σ

_{xz}= 0 and σ

_{yz}= 0 for the loading position h = 0 mm while σ

_{xz}and σ

_{yz}has a jump for the loading position h = 0.13 mm.

## 4. Conclusions

_{3}N

_{4}-based ceramics due to the rectangular loading. The displacements and stresses are presented and compared to those of a homogeneous elastic solid of semi-infinite extent. It was found that the heterogeneity of FGM has an evident influence on the elastic fields of the semi-infinite elastic solids. This capability to exactly calculate the complete elastic field induced in FGM is important to the understanding of the FGM mechanical behavior and in the design of FGM properties with depth.

## Acknowledgements

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**MDPI and ACS Style**

Xiao, H.-T.; Yue, Z.-Q.
Stresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings. *Materials* **2012**, *5*, 210-226.
https://doi.org/10.3390/ma5020210

**AMA Style**

Xiao H-T, Yue Z-Q.
Stresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings. *Materials*. 2012; 5(2):210-226.
https://doi.org/10.3390/ma5020210

**Chicago/Turabian Style**

Xiao, Hong-Tian, and Zhong-Qi Yue.
2012. "Stresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings" *Materials* 5, no. 2: 210-226.
https://doi.org/10.3390/ma5020210