Extended finite element method for simulating the mechanical behavior of multiple random holes and inclusions in functionally graded material

Analysis of mechanical behavior of a structure containing defects such as holes and inclusions is essential in many engineering applications. In many structures, the discontinuities may have a significant influence on the reduction of the structural stiffness. In this work, we consider the effect of multiple random holes and inclusions in functionally graded material (FGM) plate and apply the extended finite element method with enrichment functions to simulate the mechanical behavior of those discontinuous interfaces. The inclusions also have FGM properties. Numerical examples are considered and their obtained results are compared with the COMSOL, the finite element method software.

Extended finite element method for simulating the mechanical behavior of multiple random holes and inclusions in functionally graded material Kim Bang Tran, The Huy Tran, Quoc Tinh Bui and Tich Thien Truong Abstract-Analysis of mechanical behavior of a structure containing defects such as holes and inclusions is essential in many engineering applications.
In many structures, the discontinuities may have a significant influence on the reduction of the structural stiffness.In this work, we consider the effect of multiple random holes and inclusions in functionally graded material (FGM) plate and apply the extended finite element method with enrichment functions to simulate the mechanical behavior of those discontinuous interfaces.The inclusions also have FGM properties.Numerical examples are considered and their obtained results are compared with the COMSOL, the finite element method software.Index Terms-XFEM, hole, inclusion, multiple.

INTRODUCTION
n many engineering applications, a structure may contain defects such as holes and inclusions.Therefore, analysis of mechanical behavior of a structure containing defects plays a vital role in ensuring reliability when operating and avoiding the resonance phenomenon that can cause large displacements, may lead to structural failure.Due to the complexity of physical and mechanical characteristics of those structure, analytical and experimental methods are difficult in obtaining the solution.The analytical methods are limited to models with simple boundary conditions and configuration, while the experimental method is often time-consuming, costly and only be done for a certain number of models.Moreover, the accuracy of result depends on samples, machine tools, experience, and so on.Therefore, some efficient numerical methods have been developed to simulate complex structures.
When modeling the interface problems by means of the finite element methods (FEM), the defects face must be coincided with the edge of the elements and the FEM has encountered many difficulties.To overcome these difficulties, the extended finite element method (XFEM) was developed to solve those problems.Some advantages of the XFEM over FEM may be pointed out as follows: conformal mesh when modeling discontinuity such as hole and inclusion is not required, discontinuities are represented by enrichment functions, any special element is no longer required.
Functionally graded materials (FGM) is a composite material whose mechanical properties change with a mathematical function.This function can contain a variable that is the coordinate of a point on an object.Because the material properties change throughout the body, FGM is of great interest in various technical fields.The main advantage of FGM is that there is no boundary between two different materials and therefore will not lead to stress in the body, despite the fact that the material properties change drastically.

I
Most of the problem of discontinuous interfaces such as holes and inclusion is investigated with homogeneous material [1][2][3][4][5], or the FGM structure contains only one type of discontinuity [7][8].In this paper, we present the extended finite element method for simulating the mechanical behavior of multiple random holes and inclusions in a finite FGM plate with varying elastic properties in the transverse direction.The inclusions also have FGM properties.Poisson's ratio is held constant and Young's modulus is considered to vary across the radius and x-axis.

Level set method for holes and inclusions detection
In XFEM, the level set method is used to detect discontinuous boundaries.According to [2], a boundary of a hole or an inclusion can be considered a discontinuous boundary.The level set value on the boundary of a hole or an inclusion will be zero.The physical description of the elements containing the boundary of a hole or an inclusion will be described by the enrichment function.
To compute the normal level set function ϕ, consider Γ is the geometry of the hole or inclusion.At any point x, we define the scattering point xΓ on the boundary so that the distance || x -xΓ || is the smallest.The level set function ϕ can be expressed as follows min The appearance of a hole or an inclusion with a particular boundary Γ can be detected by the level set value .In the whole body, < 0 at any point located inside the domain bounded by and > 0 at any point located outside the domain.

Enrichment functions for discontinuities
To describe the physical properties of a material discontinuity element such as an inclusion, we will use the absolute enrichment function.According to [2], this function can be defined as the absolute value of the signed distance function as follow ( ) ϕ(x) can be calculated by interpolating the nodal signed distances within an element ( ) Where Ni is shape function at node i To describe the physical properties of a geometric discontinuity element such as hole, we use Heaviside function.According to [2], a node that lies outside the hole will have H(x) = 1 and a node that lies inside the hole will have

XFEM for discontinuity problems
According to [2], the displacement field of a two-dimensional element with discontinuity will be of the following form N is the total number of nodes and n is the number of nodes under the element; u is the transposing element at the nodes of the element as in the finite element method, a is the degree of freedom added at the enriched nodes and g(x) is the enrichment function.
With the displacement field approximated to be enriched, the linear system of equations has the following form = K u f (6) With the stiffness matrix K of the enriched elements will be calculated according to the formula , , If an element contains boundary of a hole, g(x) = H(x) and g(x) = χ(x) if boundary of an inclusion passes through an element.
As the FGM has the properties changing throughout the body, it is important to implement the following critical steps in the calculation process for XFEM.Divide the elements into multiple triangular child domains to locate the points that take the Gaussian integral.Repeat on each Gauss point: -Calculate the deformation matrix B at the Gauss point under consideration -Calculate the material matrix D at the point Gauss is considering -Calculate the element's stiffness matrix -Assemble the element stiffness matrix into the global stiffness matrix -Finish the iteration on Gaussian points.

Problem 1
Considers a rectangular isotropic FGM plate with material variation in the Cartesian x-direction, the dimensions and are depicted in Fig. 3.The external load is q = 1 N/m 2 .The lower edge of the plate is clamped and plane strain state is assumed.The plate contains an four circular holes and two circular inclusions.All holes and inclusions have equal radius R = 0.085 m and different positions as depicted in table 1.The Poisson's ratio of the matrix is assumed to be constant = 0.In order to assess the accuracy and ef ciency of the XFEM for modeling arbitrary discontinuities, we compare the finite element method (FEM) solution to that obtained by XFEM.The mesh size for the nite element solution was chosen so that further re ning the mesh does not produce signi cant change in the solution, thus the FEM solution could be taken as the reference solution, with which we compare the accuracy of the XFEM solution The computed results obtained by the XFEM and the FEM are listed in Table 2 including the percentage errors.As expected, the minimum and the maximum displacement obtained by the XFEM matches well with those derived from the FEM.The stress and displacement field of the plate are sketched in subsequent Figs.4-6.

Problem 2
In the next problem, the dimensions, boundary condition, loading are taken from the previous case.The plate now contains an three circular holes and four circular inclusions.All holes and inclusions have different radius and different positions as depicted in table 3. The Poisson's ratio of the matrix is assumed to be constant = 0.3 and the elastic modulus E of the matrix varies exponentially from the left to the right edge as follow  The computed results obtained by the XFEM and the FEM are listed in Table 4 including the percentage errors.The minimum and the maximum displacement obtained by the XFEM matches well with those derived from the FEM.The stress and displacement field of the plate are sketched in subsequent Figs.8-10.

CONCLUSION
A extended finite element method (XFEM) has been proposed for discontinuity problem analysis under the effect of multiple random holes and inclusions in functionally graded material.This method is convenient in treating those defects without meshing the internal boundaries because of the enrichment function.Several numerical examples are considered when many holes and inclusions with different sizes appear in the body.
The obtained solutions show a good agreement of between the extended finite element method and the traditional finite element method.The presented approach has shown several advantages and it is promising to be extended to more complicated problems such as modeling the body containing different discontinuity boundaries.

Figure 1
Figure 1 Signed distance function

Figure 3 .
Figure 3. FGM plate with material variation in the x-direction with four circular holes and two circular inclusions

Figure 4 .
Comparison of displacement results ux between XFEM and FEM XFEM (Matlab) FEM (Comsol) Figure 5.Comparison of displacement results uy between XFEM and FEM XFEM (Matlab) FEM (Comsol) Figure 6.Comparison of stress results yy between XFEM and FEMTABLE 2. COMPARISON OF RESULTS BETWEEN XFEM

Figure 7 .
Figure 7. FGM plate with material variation in the x-direction with three circular holes and four circular inclusionsWe compare the finite element method (FEM) solution to that obtained by XFEM.

TABLE 4 .
COMPARISON OF RESULTS BETWEEN XFEM AND FEM