The Displacement Analysis Investigation for Variable Gradient Parameters Functionally Graded Materials Based on Strip Element Method

This paper adopted the strip element method to research the displacement response of functionally graded material structure with variable gradient parameters under mechanical loading. Based on the dynamic equilibrium equation of the system, strip element method theory was established through the principle of virtual work to get the control equations of functionally graded materials plane problem, using the modal superposition method to complete the solution of the equation and obtain the displacement distribution of the FGM plates with variable gradient parameters. The accuracy of the calculation method was verified by comparison with the result of finite element method. The results show that the displacement response of the strip element method is consistent with the finite element method. With the increase of the gradient coefficient, the ceramic content is increased and the material stiffness is also increased; then the displacement response caused by external loading is decreased. With the increase of the gradient coefficient, the influence of the gradient coefficient on the displacement response is decreased gradually.


Introduction
Functionally graded materials (FGM) [1] are a newly developed material in recent years.Their distinguishing feature is that the composition of the material presents a continuous gradient change in the direction of thickness.The feature and function of the material also appear as a gradient change.
A large number of domestic and abroad scholars had done a lot of research on the mechanical problems of functionally graded materials [2,3].Tornabene [4] applied the Generalized Differential Quadrature (GDQ) Method to study four parameters functionally graded and laminated composite shells and panels of revolution.The free vibration of functional gradient materials under mechanical loading and temperature loading was analyzed using Peano-Baker series method by Liu [5].The dynamic characteristic analytical solution of the quadrilateral and rhombic function gradient plate was obtained by using the beam function group [6].Tornabene [7] analyzed the dynamic of doubly curved shell structures, and the natural frequencies and mode shapes of several structures are presented and compared to numerical solutions taken from the literature.Viola [8] studied the dynamic behavior of moderately thick functionally graded parabolic panels of revolution and was concerned with the influence of the three parameters of the power-law distribution on their mechanical behavior.Reddy [9] adopted the first-order shear deformation theory to study the structural response of the functional gradient plate structure under the thermomachine coupling.Based on the first-order shear deformation theory (FSDT), Tornabene [10] focused on the dynamic behavior of moderately thick functionally graded parabolic panels and shells of revolution and illustrated the influence of the parameters of the power-law distribution on the mechanical behavior of shell structures.And furthermore the study concerned six types of shell structures that illustrate the influence of the power-law exponent, of the power-law distribution and of the choice of the four parameters on the mechanical behavior of shell structures considered [11].
Using the first-order shear deformation theory, Thai [12] had studied the problem of bending and free vibration of functionally graded materials.Liew [13] studied the vibration response of functionally graded material plates under the coupling of mechanical load and electromagnetic load based on the first-order shear deformation theory using the finite element method.Zenkour [14] used first-order shear deformation theory and classical elastic theory to study the structural bending deformation of the sandwich structure under mechanical loading.Ferreira [15] used threeorder shear deformation theory and mesh-less method to study the structure response of functional gradient materials under static load.Reddy [16] studied the deformation of functional gradient rectangular plate based on the threeorder shear deformation theory.Brischetto [17] studied the analytic solution of the structural displacement response of functionally graded material plates under transverse loading.Ray [18] used the finite element method to study the structural response of the functional gradient fiber piezoelectric materials in exponential distribution.Kulikov [19] adopted surface sampling method to solve the problem of threedimensional thermal stress of functional gradient materials.
Due to physical performance parameters of functionally gradient materials with the coordinate changing continuously, the constant coefficient differential equation turns into the variable coefficient differential equation while the theory of elastic body deformation is derived; it is difficult to solve such problem by analytical method.The numerical method is used to solve complex problem while the large computation is in process.The analytical solution is very difficult to obtain, while the precision of the solution will be declined for numerical method for FGM structure, and it also requires a lot of computational memories while using numerical method.In recent years, the researchers have proposed a seminumerical and semianalytic method, that is called the strip element method (SEM) [20], which combined the accuracy of the analytical method with the efficiency of the numerical method.
This study considered the boundary and initially applied strip element method to functionally graded materials structure.Meanwhile this paper firstly considered the displacement response of FGM structure with variable gradient parameters.The paper obtained the displacement response of functionally graded material structure under mechanical loading and further investigated the relationship between the gradient parameters and displacement response of functionally graded material structure.It provides guidance for the application of FGM.

Strip Element Method Theory
The strip element method is a series of strip elements separated by the solution domain as shown in Figure 1.The displacement function is the continuous function of  direction, and the displacement of the element is obtained through the line displacement interpolation.
Based on the finite element method in  direction, it is assumed that the displacement functions of functionally graded materials are where () represents matrix of shape functions for displacement interpolation; () represents continuous function of  direction;  represents a complex unit;  represents circle frequency;  represents the computing time.
The shape functions can be derived from the displacement function of a single element.The matrix of shape functions () [20] is where ℎ represents element thickness; I represents the second order identity matrix.
According to Kausel's equation, the dynamic equilibrium equation of the system is where  represents displacement vector;  represent differential operator matrix;  represents stress vector and  = [      ]  ; superscript  denotes transpose, where the expression for  is It can be seen that the internal displacement of the element is obtained by the interpolation of the nodal displacement from (1).The residual value must be generated in the interpolation calculation, and the residual value is expressed as follows: Since the internal displacement function of the element is interpolated by the nodal displacement, the residual value must exist in the element; then  ̸ = 0.

Mathematical Problems in Engineering 3
According to the principle of virtual work, for a deformed body in equilibrium state, the total virtual work done by the external force is equal to the sum of the work done by the internal force and the work done by the residual value, as follows: where   is tiny virtual displacement,  represents external mechanical load, and  represents nodal stress vector.Given an external load as where  represents external load amplitude, the element stress is where stress   is where The displacement function is substituted into the virtual work equation based on the theory of thin plate deformation and we obtain where  0 ,  1 , and  2 represent coefficient matrix. represents mass matrix.
It can be seen from ( 10) that the governing equation obtained by element discretization is the ordinary differential equation of , which reduces the difficulty of the solution; it also reflects the idea of  direction discretization.
The coefficient matrix is where The expression for the mass matrix is Assembling all elements, (10) becomes as where  0 ,  1 , and  2 are the coefficients matrix for the element assembly in the solving domain;   is the mass assembly matrix;   is the external mechanical load;   () is the displacement function in the solving domain.Assume that the displacement function form is Assume that external mechanical load form is Substituting ( 17) and ( 18) into ( 16), we obtain represents the load acting on the nodal line, while   = 0 (19) is transformed into a homogeneous equation and the eigenvalue equation for the k is obtained.
Equation ( 20) can be written The 2M (M = 6N − 2(N − 1) = 4N − 2) eigenvalue is obtained by solving (21), N is the element number, and the eigenvectors corresponding to the  eigenvalues are represented by By means of the modal superposition method, the solution of the equation can be expressed as where Equation ( 23) is a fundamental solution of the equation.
There are 2M constants  in this fundamental solution, so it is necessary to determine the unknown coefficients by the boundary conditions.Since (23) is the fundamental solution in the problem domain, the displacement at any point satisfies (23) and the displacement is satisfied at the right boundary of the solution.
That can be written as In the same way, on the left side Equations ( 27) and (28) are assembled and sorted, and the constant  expression can be obtained where   is the displacement at the boundary By (24), (  ) can be expressed as The superscript  represents the right boundary.
Similarly, the left boundary condition has the same form as the displacement matrix at the right boundary.The difference between the two forms is that the  coordinate value at the right boundary is changed into the  coordinate value at the left boundary.

Application of the Boundary Conditions
The stress vector of the internal element can be expressed as Substituting ( 1) into (34), we have where  1 and  2 are element coefficient matrices.
Calculate  1 and  2 for all elements, and obtain the overall stress vector   as follows: where  1 and  2 are the assemble matrix of the element coefficient matrix.The displacement is simplified by using the left and right boundaries The stress vector of the inner nodal line is equal to the average value of the stress vector of the adjacent left and right elements.We obtain In (40),   = [      ] is an external load acting on the left and right boundaries, where the stiffness matrix  is

Establishment of Gradient Parameter Model
The functional gradient materials studied in this paper and the model is composite by metal and ceramic materials.The metal material is 1Cr 18 Ni 11 Nb and the ceramic material is Si 3 N 4 .The physical properties of the two materials are shown in Table 1.
Because functionally graded materials are continuously changed by different materials according to the design requirements, the physical performance parameters of the materials are expressed as a function of the volume fraction of the material, the physical properties of the materials, and the content of components; it denotes where (), ](), and () are the elastic modulus, Poisson's ratio, and density of FGM plates, respectively. C , ] C , and  C are the elastic modulus, Poisson's ratio, and density of ceramic materials, respectively. M , ] M , and  M are the elastic modulus, Poisson's ratio, and density of metal materials, respectively. C is the volume fraction of the ceramic.
To ensure the continuity of the material, the sum of the metal volume fraction and the ceramic volume fraction at any location of the material is 1.
where  M is the volume fraction of metals.The volume fraction change function of the functionally graded metal is where  is the vertical position and 0 ≤  ≤ ;  is FGM plate thickness;  ≥ 0 is the gradient parameters.By (44), it can be seen that when  = 0,  M = 1 and  C = 0. Substituting that into (42) we have () =  M , When  = 1 and  = 0 are functionally graded materials, the volume fraction of metals varies as shown in Figure 2. Figure 2 can be shown: when  = 0, metal volume fraction is 0 and this position is pure ceramic material; when  = , metal volume fraction is equal to 1 and this position is the pure metal homogeneous material.So when  = 1 and  = 0, the transition form of materials is a continuous transition of ceramic to metal.When the gradient parameter  = 1, the change of the volume fraction of the functionally graded material is continuous linear change.When the gradient parameter is lesser than 1, the change of the metal volume fraction decreases.When the gradient parameter is greater than 1, the metal volume fraction of the functional gradient material is slower in the early and then the volume fraction is accelerated with the thickness increased.

Validation of Strip Element Method.
Based on the finite element (FEM) theory, we used ANSYS to verify the results of SEM solution.The model of Figure 3 is solved by using the strip element method and finite element method.In Figure 3,  = 100 mm,  = 100 mm, the lower boundary is fixed,  = 100 N/mm, and uniform load is applied on the upper boundary.The solution domain is divided into 20 elements and the material properties in each element are metal material.The metallic material property is shown in Table 1.The displacement of -direction and -direction at  = 10 mm,  = 50 mm, and  = 90 mm is shown in Figures 4 and 5 by strip element method and finite element method (FEM).
From Figures 4 and 5 we can see that the displacement obtained by strip element method and finite element method in -direction and -direction at different position is almost  the same.From  = 10 mm and  = 50 mm to  = 90 mm, the displacement in -direction is increased.The accuracy of the strip element method is verified.

The Influence of Element Number for the Results.
In the calculation process, because of the discrete of the solving domain, the size of the element has an influence on the accuracy of the displacement solution.Figures 6 and 7 are the -direction displacement and the -direction displacement at  = 10 mm using 10 elements, 20 elements, and 50 elements, respectively.It can be known that displacement results have small difference when the element numbers are 10 and 20, while when the element numbers are 20 and 50, the result difference is not obvious, so in this calculation the element number is set to 20 for ensuring that precision and calculation speed.are shown in Figure 8 at  = 10 mm in the  direction.It can be seen from Figure 8, with the increase of gradient coefficient, that the displacement response decreases, and this is due to the fact that metal volume fraction decreases in the functionally graded materials and ceramic volume fraction increases, causing the structure stiffness to increase.In addition, the displacement in the  direction at  = 10 mm has nonlinear characteristics.The displacements with different gradient parameters are shown in Figure 9 at  = 10 mm in the  direction.It can be seen from Figure 9 that the displacement in  direction at  = 10 mm with the increase of  coordinate has the characteristics of first increasing, then decreasing, and then increasing.As the gradient parameter increases, the and as the change of spatial coordinates, the curve is approximatively changed to straight with the increase of gradient coefficient, structural stiffness is gradually increased and the displacement response under external loading is reduced.

Conclusion
This paper presents the variable gradient parameters model of the functional gradient materials.The displacement response of considering different gradient parameters model is analyzed in mechanics loading using the strip element method; there are some useful results.(1) The displacement response result is almost the same by using finite element method and strip element method, so as to verify the validation of the strip element method.
(2) With the ceramic volume fraction increasing, the metal volume fraction decreases in the functionally graded materials, the structure stiffness increases, and displacement response decreases.
(3) As the gradient coefficient increases, the influence of the gradient coefficient on the structure displacement response decreases gradually.

Figure 1 :
Figure 1: Strip element method for solving model.

Figure 2 :
Figure 2: Law of change of volume fraction of metal with thickness under different gradient parameters at  = 1 and  = 0.

Figure 3 :
Figure 3: Physical model of plane problem.

Figure 4 :
Figure 4: The displacement in -direction in different position by FEM and SEM.

Figure 5 :Figure 6 :
Figure 5: The displacement in -direction in different position by FEM and SEM.

Figure 13 :
Figure 13:  direction displacement for different gradient parameters at  = 90 mm.

Table 1 :
Physical properties of 1Cr 18 Ni 11 Nb and Si 3 N 4 materials.  and   have the same expressions as   and   .The difference is that   and   in    and    are replaced by    =   exp(    ) and    =   exp(    ), respectively.