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Elastic Mechanical Stress Analysis in a 2D-FGM Thick Finite Length Hollow Cylinder with Newly Developed Material Model
Acta Mechanica Solida Sinica, Vol. 29, No. 2, April, 2016 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
Elastic Mechanical Stress Analysis in a 2D-FGM Thick Finite Length Hollow Cylinder with Newly Developed Material Model Amir Najibi1⋆
Mohammad Hassan Shojaeefard2
1
( Faculty of Mechanical Engineering, Semnan University, Semnan, Iran) (2 School of Mechanical Engineering, Iran University of Science & Technology, Iran) Received 4 April 2014, revision received 8 November 2014
ABSTRACT In this paper a new 2D-FGM material model based on Mori-Tanaka scheme and third-order transition function has been developed for a thick hollow cylinder of finite length. Elastic mechanical stress analysis is performed by utilizing the finite element method. The corresponding material, displacement and stress distributions are evaluated for different values of nr and nz . Moreover, the effects of different material property distributions on the effective stress with respect to the metallic phase volume fraction are investigated. It is demonstrated that the increase in nr and Vm leads to a significant reduction in the effective stress. Finally, it is shown that the ceramic phase rich cylinder wall has lower maximum effective stresses of which the lowest value of effective stress has been evaluated for nr = 20 and nz = 5. This minimum value is about half the maximum effective stress which has been evaluated for the non-FGM cylinder case (nr = nz = 0.1).
KEY WORDS 2D-FGM, elastic analysis, thick hollow cylinder, Mori-Tanaka material model
Nomenclature a, b c1 , c2 Ftrans (Vi ) m1 , m2 nr , nz P (x, y) pmax Vc Vm V1 , V2 Vco Vci Vib K G δ
⋆
inner and outer radii first ceramic and second ceramic transition function first metal and second metal radial and axial power law exponents general material properties pressure amplitude volume fractions of ceramic phase volume fractions of metallic phase volume fractions of basic materials the volume fractions of the ceramic phase on the outer surfaces the volume fractions of the ceramic phase on the inner surfaces constant value bulk module shear module constant value
Corresponding author. E-mail: [email protected]
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δij γ(Vi )
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unit matrix value of transition function
I. Introduction With the development of new industries and modern processes, a new class of composite materials called functionally graded materials (FGMs) able to tolerate severe thermal loadings are described[1] . In fact, FGMs are composite materials that are formed of two or more constituent phases with a composition that is continuously variable[2] . With the materials used in FGMs proposed first by Japanese scientists in the mid-1980s, FGMs have found numerous applications in different related fields; e.g. FGM sensors (Muller et al. 2003) and actuators, FGM metal/ceramic armor, FGM photo detectors, and FGM dental implant[3–7] . Birman and Byrd presented a review of principal developments in FGMs involving heat transfer issues, stress, stability and dynamic analyses, testing, manufacturing and design, applications, and fracture[8] . FGMs have many applications in those industries using drastically different temperatures in their operating environment such as aerospace structural applications and fusion reactors[9, 10]. Zimmerman and Lutz studied the thermal stress and thermal expansion in a uniformly heated functionally graded cylinder. They proposed an analytical solution for solving the thermoelasticity equations in the steady state. In an analytical solution of FGM problems, exponential functions for continuous gradation of material properties were considered under thermal and mechanical loads[11] . Additionally, Tutuncu and Ozturk proposed an exact solution for stresses in functionally graded pressure vessels[12] . Mechanical and thermal stresses in a functionally graded hollow cylinder due to a radially symmetric load were studied by Jabbari et al.. This study also proposed a general solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to a non-axisymmetric steady state load. They considered thermal loading in the steady state and axisymmetric as well as non axisymmetric[13, 14] . There are several studies that have focused on two-dimensional FGMs. However, exponential functions have been used in all of these studies for continuous gradation of the material properties, and the employment of these functions for material properties commonly leads to enhanced analytical solution with special functions[15–19] . The element-free Galerkin method was used to analyze the two-dimensional quasi-static heat conduction and thermoelastic problems[20] . Spatial distribution of ceramic volume fraction was obtained by piecewise bi-cubic interpolation of volume fractions defined at a finite number of grid points. Additionally, Nemat-Alla proposed the idea of adding a third material to the conventional FGM constituents in order to withstand serious thermal stresses. The FGM defined was considered as a 2DFGM, the study of which showing that it has a higher capability to relax thermal stress than conventional FGM[21] . In studying hollow thick cylinders, although the FGM is supposed to be graded in most cases, it is assumed that each of the layers is homogeneous. An analysis of the thermal stress behaviour of functionally graded hollow cylinders was performed by Liew and Kitipornchai, who concluded that thermal stress is bound to occur in the FGM cylinder, except in the insignificant case of zero temperature[22] . Furthermore, Chen and Tong studied thermo-mechanically coupled sensitivity analysis and design optimization of functionally graded circular cylinders under thermal and mechanical loading in the static and non- static state[23] . Besides, mechanical and thermal stresses of a functionally graded circular hollow cylinder of finite length were studied by Shao, who presented the solutions of temperature, displacements, and thermal/mechanical stresses in a functionally graded circular hollow cylinder[24] . In addition, Hongjun et al. proposed the elastic solutions of heterogeneous elastic hollow cylinders. They analyzed the problem with two different methods, one was a cylinder with multi-layers and the other a cylinder with continuously graded properties[25] . An analytical solution of 2D non-axisymmetric elasticity and thermo-elasticity problems for a radially inhomogeneous hollow cylinder has been proposed by Tokovoyy et al. According to this study, a solution of the equation for the governing stress in the form of Fourier series was presented[26] . Hosseini Kordkheili et al. have presented an analytical thermo-elasticity solution for hollow finite-length cylinders made of functionally graded materials subjected to thermal loads, internal pressure and axial loadings. They concluded that stress and displacement components behave nonlinearly, especially for regions near the longitudinal ends of functionally graded cylinders[27] . Asgari et al. have investigated the dynamic analysis of a two-dimensional functionally graded thick
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hollow cylinder of finite length under impact of loading[28] . Asgari and Akhlaghi have studied transient heat conduction in a two-dimensional functionally graded hollow cylinder of finite length. They have implemented the Crank-Nicolson finite difference method to solve time dependence equations of the heat transfer problem. Their results show that using 2D-FGM leads to a more flexible design[29] . Asemi et al. investigated an elastic solution of a two-dimensional functionally graded thick truncated cone of finite length under hydrostatic combined loads[30] . A nonlinear transient stress and wave propagation analysis of FGM thick cylinders has been made according to a unified generalized thermoelasticity theory by Shariyat. He performed nonlinear generalized and classical thermoelasticity analyses for functionally graded thick cylinders with temperature-dependent material properties subjected to various thermomechanical shocks at their inner and outer surfaces using Hermitian elements. He employed the Mori-Tanaka homogenization model of variation of the material properties instead of the simple rule of mixtures[31] . Previous studies on FGM cylinders devoted more to thermal stresses and the variation of material properties just considered the radial direction. Furthermore, the material distribution estimation as the simple rule of mixtures is not displaying the actual material properties of FGMs. In this paper, a cylinder exposed to a non-uniform axisymmetric pressure loading made of 2Dfunctionally graded materials is investigated. The finite element method (FEM) is used for static analysis of the 2D-FGM thick hollow cylinder of a finite length. The elasticity equations are derived and numerically solved in both radial and axial directions. The nonlinear properties of constituent materials of the FG cylinder are considered to satisfy the newly developed Mori-Tanaka scheme. Moreover, the responses of structure-like displacements and stresses, under mechanical loading for various values of volume fraction have been studied and discussed in detail.
II. Volume Fraction in 2D-FGM Cylinder Consider the volume fractions of a 2D-FGM at an arbitrary point in the 2D-FGM axisymmetric cylinder shown in Fig.1. In the present cylinder, the inner and outer surfaces are made of two distinct ceramics and metals, respectively. The volume fraction distributions of the materials can be expressed as[29]
n r−a r b−a n r−a r Vm = b−a
Vc = 1 −
(1a) (1b)
Fig. 1. 2D-FGM and material distribution of the axisymmetric cylinder (c1 =ZrO2 , c2 =Si3 N4 / m1 =Ti-6Al-4V, m2 = SUS304).
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V1 V2 V3 V4
nr h z nz i 1− = 1− L n r − a r z nz = 1− b−a L nr h z nz i r−a = 1− b−a L nr r−a z nz = b−a L
r−a b−a
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(2a) (2b) (2c) (2d)
where Vc and Vm are ceramic and metallic volume fraction and V1 , V2 , V3 and V4 denote the first and second ceramic and the first and second metal volume fraction, respectively. a, b and L are the inner, outer radii and length of cylinder wall, respectively. Also nr and nz are parameters that represent the basic constituent distributions in r- and z-directions, respectively. The sum of the volume fractions of all the constituent materials makes 1, i.e. X Vi = 1 (3) i=1
III. Rules of Mixture of FGM The Mori-Tanaka scheme for estimating the effective material properties is applicable to regions of the graded microstructure with a well-defined continuous matrix as well as a discontinuous particulate phase. It is assumed that the matrix phase (denoted by subscript 1) is reinforced by spherical particles of a particulate phase (denoted by subscript 2). The following relations give the rules of mixtures for different thermal and mechanical properties[32, 33]. According to Reiter et al. as well as Reiter and Dvorak, the Mori-Tanaka model was shown to yield accurate prediction of the properties with a well-defined continuous matrix and discontinuous inclusions. Therefore, the matrix and inclusion will vary by moving from the ceramic rich surface to a metal one. The axisymmetric cylinder region is divided into 3 zones, the ceramic rich, the transition, and metal rich zone, while in the ceramic zone the matrix is ceramic and the metal is inclusion. The transition zone, in which skeletal microstructures characterized by a wide transition zone between the regions with predominance of one of the constituent phases, is defined with an appropriate transition function and in the metal rich zone the metal is matrix and the ceramic is inclusion[34, 35] . Kf − K1 V2 = K2 − K1 1 + (1 − V2 ) [3 (K2 − K1 ) / (3K1 − 4G1 )]
(4a)
Gf − G1 V2 = G2 − G1 1 + (1 − V2 ) (G2 − G1 )/(G1 + f1 )
(4b)
f1 =
G1 (9K1 + 8G1 ) 6 (K1 + 2G1 )
(4c)
where K1 , K2 and G1 , G2 are Bulk and Shear module for matrix and inclusion, respectively and V1 and V2 are the corresponding volume fractions. The subscript f represents effective material properties. 3.1. Transition function In order to accommodate the discontinuities in the homogenized material properties which are predicted at the boundaries of different zones, the following transition functions are defined by Reiter and Dvorak[35] . Accordingly, Pγ #Pβ denote the magnitudes of a certain effective material parameter predicted by two different averaging methods γ and β at a given volume fraction Vi = Vib of materials ci and mi as inclusion in a matrix ci or mi at such a region boundary. Moreover let[35] : P = γ (Vi ) Pγ + [1 − γ (Vi )] Pβ
(5)
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Defining the value of material parameter within a small interval Vib ± δ/2, such that, γ (Vi ) = 1
for Vi < Vib − δ/2
γ (Vi ) = Ftrans (Vi ) γ (Vi ) = 0
(6a)
for Vib − δ/2 < Vi < Vib + δ/2
for Vi > Vib + δ/2
(6b) (6c)
While the third-order transfer function is taken as,
Ftrans (Vi ) = 2
Vi − Vib δ
3
−
3 (Vi − Vib ) 1 + 2δ 2
(7)
where Vi is volume fraction and Vib ± δ/2 determines the boundaries of transition function. Ftrans (Vi ) is the governing transition function for the transition zone and γ (Vi ) is the effective material property in the transition zone[35] .
3.2. Material model validation To demonstrate the ability of the proposed model to estimate the effective material properties of FGM, the proposed model (M-T) is compared with the experimental data in which the Young’s modulus of the Al/Al2 O3 composites is determined for Al volume fractions between 2.5 and 28%[36] . The samples have plate geometry of 35 × 33 × 3 mm3 , with a non-homogeneous Al content along the longest axis of the plate. Half the plate consists of Al/Al2 O3 composite with an aluminum volume fraction of 2.5%. The other half of the plate (a region of length 17.5 mm) contains a composition gradient. The material properties of Al and Al2 O3 are: EAl = 69 GPa; νAl = 0.33; EAl2O3 = 390 GPa; νAl2O3 = 0.2[36] . In Fig.2, the results of the proposed model have been found in good agreement with experimental ones. The elastic modulus contour for nr = 1 and nz = 0.5 according to the newly developed Mori-Tanaka material model is illustrated in Fig.3.
Fig. 2. Young’s modulus as a function of Al volume fraction for Al/Al2 O3 composite. Fig. 3. The elastic modulus distribution contour for nr = 1 and nz = 0.5.
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IV. Governing Equations The governing equations for the linear strain-displacement relation and the quasi-static case can be written as σij,j = 0
(8a)
σij,j = λεkk δij + 2µεij
(8b)
εij =
1 (ui,j + uj,i ) 2
(8c)
where the Lam´e coefficients λ and µ are the function of radial and axial position according to the newly developed Mori-Tanaka model.
V. Finite Element Modelling In order to solve the governing equations, the finite element method has been utilized and significant work has been done in previous investigations to perform the continuous variation of constituent functionally graded materials properties in finite element modelling. It is shown that use of the continuously varying properties of non-homogeneous materials in the iso-parametric finite element formulation does not pose a computational problem from Li et al. . This formulation provides enormous flexibility in meshing. Further, this method can be extended straightforwardly to three dimensional problems[37] . Finite element modelling of FGM requires creating very fine mesh of elements even for relatively small sized bodies, particularly in nonlinear analyses. In general, there are two procedures that can be selected to be considered for the material properties in the FE formulation. They include the properties assignment for each element individually or dividing the entire structure into numerous regions then assigning properties to each region[37–39] . A formulation of a finite element with a spatially varying material property field for solving boundary value problems involving continuously non-homogeneous materials has been presented by Santare and Lambros. The specific element studied was a 2D-plane stress element with linear interpolation and an exponential material property gradient[40]. Graded finite elements within the framework of a generalized iso-parametric formulation for materials within an internal property gradient were proposed by Kim and Paulino. In their study, the mean properties computed by integration within each element were employed for stiffness matrix. Generally, the graded finite elements give more precise local stress than conventional homogeneous elements[41] . A novel method including the definition of the properties as functions of the temperature has been presented by Rousseau et al. Then, the properties have been estimated in solving the functions[42] . In the current investigation, mechanical properties of materials are taken into consideration through finite element analysis. When the inner ceramic surface of the FGM cylinder is going to be subjected to the mechanical load, the stresses will develop inside the cylinder. The mechanical properties are calculated based on the volume fractions and the rules of mixture for FGMs are found in the preceding section. The FGM cylinders investigated have the inner and outer radius of a = 0.1 m and b = 0.15 m, respectively and a length of 0.2 m. The axisymmetric finite element model used in this study of the elastic problem contains 4,000 sixteen-noded rectangular elements. This number of elements results from uniform division of the cylinder wall into 100 elements through the axial direction and 40 elements through the radial direction. The mechanical boundary conditions used in the finite element model are as follows: all nodes at the top and bottom of the cylinder wall are roller supported in order to prevent their movement in the z-direction. Similarly, the outer surface of cylinder wall (r = 0.15 m) is stress free and the inner surface of cylinder wall is subjected to the following internal pressure and pmax = 300 MPa: πz pr = pmax sin MPa (9) L The numerical solutions of the present investigation are found for (ZrO2 /Si3 N4 /Ti-6Al-4V/SUS304), 2D-FGM cylinder of different composition and the material properties for each of the constituents at room temperature are taken from the research by Reddy and Chen[2] . Then the resulting displacements were determined by the solution of the axisymmetric elastic problem under the pressure distribution
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determined as described in Eq.(9). Finally, the stresses on eachnode are averaged according to those on its associated elements and shape function.
VI. Finite Element Verification In order to validate the proposed methodology and approach, the numerical FEM codes are simplified for examination of a FG cylinder made of a two-phase ceramic/metal composite which is analyzed by meshless local Petrov-Galerkin method[43] . In their study a thick hollow cylinder with an inner radius ra = 5 mm and outer radius rb = 10 mm is considered. The hollow cylinder is subjected to uniform pressures on its inner surface. Because of symmetries about the horizontal and the vertical centroidal axes, only a quarter of the cylinder is discretized with 6560 quadrilateral elements with third-order Lagrange shape functions. The stress components are normalized to the applied pressure. Besides, it is presumed that a plane strain state of deformation is dominant in the cylinder. The material constituents are Al for the metal phase and SiC for ceramic phase with material properties of EAl = 70 GPa and ESiC = 427 GPa; νAl = 0.3 and νSiC = 0.17. The volume fraction of the ceramic phase is assumed to have radial dependence by a power law function as[43] n r − ra i o i Vc = Vc + (Vc − Vc ) (10) rb − ra where Vco and Vci denote the volume fractions of the ceramic phase on the outer and the inner surfaces of the cylinder, n is a power law index that dictates non-homogeneity of material constituents through the thickness and ra and rb are inner and outer radii, respectively. The effective material properties at a point in the FG cylinder are determined by using the Mori-Tanaka method. Figure 4 depicts a comparison of the normalized values of σrr and σθθ between MLPG and FEM methods, along the radial direction for value of n = 2. According to Fig.4, the variation of the normalized radial stress obtained by the MLPG and FEM method agrees with each other.
Fig. 4. Comparison of normalized radial stress between FEM and MLPG methods. Fig. 5. The normalized radial stress contour through the cylinder wall for nr = 1 and nz = 2.
VII. Results and Discussion For parametric study of the effect of static internal pressure and also material distribution compositions on the elastic behaviour of a 2D-FGM cylinder, the stresses have been normalized according to the pmax .
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Figure 5 shows the normalized radial stress distribution for nr = 1 and nz = 2 through the cylinder wall. The radial stress on the inner cylinder surface follows the loading pattern, so it achieves maximum normalized compression stress of 1 on the internal surface. Normalized axial stress for nr = 1 and nz = 2 is illustrated in Fig.6. The stress contour shows maximum value of about 600 MPa on the top internal surface of the cylinder that is about twice the maximum internal pressure loading. Figure 7 represents the normalized hoop (tangential) stress distribution through the cylinder wall. The material properties exponents are nr = 1 and nz = 2 and the maximum value of normalized hoop stress is about 2.5 times the maximum internal pressure. This value has been attained on top of internal cylinder surface, in which this value is even higher than that of axial stress at this region.
Fig. 6. Normalized axial stress contour for nr = 1 and nz = 2.
Fig. 7. Normalized hoop stress contour for nr = 1 and nz = 2.
The last stress contour is shear stress for nr = 1 and nz = 2. The maximum compression stress, greater than 80 MPa which is one-third less than the maximum internal pressure, achieved within half the top of cylinder wall, as shown in Fig.8.
Fig. 8. Normalized shear stress contour for nr = 1 and nz = 2.
Fig. 9. Radial displacement contour through the cylinder wall for nr = 1 and nz = 2.
The radial and axial displacement contours through the cylinder wall have been demonstrated in Figs.9 and 10 for nr = 1 and nz = 2. The effective stress can be evaluated from the principal stresses using the distortion energy theory or von Mises-Hencky theory and can be compared with the yield strength of the material at fault[44] .
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Fig. 10. Axial displacement contour through the cylinder wall for nr = 1 and nz = 2. Fig. 11. Normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 1 and 5.
The effective stress (von Mises) is defined as follows: σeff
h i1/2 1 2 2 2 = (σ1 − σ2 ) + (σ2 − σ3 ) + (σ3 − σ1 ) 2
(11)
where σeff is von Mises stress and σ1 , σ2 and σ3 are the principal stresses. Figure 11 shows the normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 1 and 5. When nz = 1, the maximum normalized von-Mises stress is about 3 while for nz = 5 the value of normalized von Mises stress is about 2.7, which means by increasing the value of nz the the value of von Mises stress will decrease. Figure 12 shows normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 5 and vice versa. When nz = 5, the stress contour gets sooner its higher value of the stress along the radial direction. Radial displacement with respect to the metallic phase volume fraction (Vm ) along the horizontal centre line (HCL) for nz = 1 and different values of nr is illustrated in Fig.13. According to the figure, when the value of nr increases, the radial displacement decreases. Moreover, when the metallic phase volume fraction increases, the radial displacement decreases excessively and this reduction depends on the value of the nr . Accordingly, for nr ’s higher than 1, significant reduction occurs at lower values of Vm while this phenomenon occurs for nr ’s lower than 1 with a smooth reduction in the higher values of Vm , on the radial displacement curves. Additionally, by considering Fig.13, it can be concluded that the increase in ceramic inclusions will result in the decrease in radial displacement. In addition to this, Fig.14 illustrates the radial displacement along HCL with respect to the metallic phase volume fraction for nr = 1 and different values of the nz . By increasing the variation of material along the axial direction, the radial displacement increases consistently. This figure also demonstrates that the increase in ceramic inclusions leads to the reduction in the radial displacement. Moreover, a comparison of the effective stress (von-Mises) along the HCL with the metallic volume fraction with nz = 2 and different values of nr shows that although by increasing the metallic volume fraction, the effective stress decreases severely, this reduction occurs for lower values of Vm for nr ’s higher than 5 as depicted in Fig.15. In the same metallic volume fraction, by increasing the nr , the
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Fig. 13. Radial displacement vs. metallic phase volume fraction along the HCL for nz = 1 and different values of nr .
Fig. 12. Normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 5 and vice versa.
Fig. 14. Radial displacement vs. metallic phase volume fraction along the HCL for nr = 1 and different values of nz .
Fig. 15. Effective stress vs. metallic phase volume fraction along the HCL for nz = 2 and different values of nr .
effective stress decreases severely while for smaller values of Vm these differences become greater and greater. Figure 16 shows a comparison of effective stress along the HCL with the metallic volume fraction with nr = 2 and different values of the nz . In the same metallic volume fraction there is no distinguishable change in different values of nz on the effective stress curves. Accordingly, by increasing the metallic volume fraction, the effective stresses will decrease significantly. The composition of a 2D-FGM cylinder is optimized by minimizing the stresses, namely, the effective stress. In the context of this study, this effective stress is minimized in such a way that an optimum non-homogeneous parameters, nr and nz can be obtained. The design variables for the current problem are the non-homogeneous parameters nr and nz , which control the composition variations through the 2D-FGM cylinder. The following conditions for the design variables are introduced:
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Fig. 16. Effective stress vs. metallic phase volume fraction along the HCL for nr = 2 and different values of nz .
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Fig. 17. Variation of effective stresses on the centre point (CP) of the cylinder wall vs. nz .
The values of the non-homogeneous parameter, nr and nz vary in the range 0 ≤ nr and nz ≤ ∞. The zero value is represented by 0.1 to obtain a 2D-FGM and ∞ is represented by 20. The values nr and nz = 0.1 will demonstrate 2D-FGM instead of nr and nz = 0.0 which indicates conventional FGM. Also, nr and nz = ∞ is represented by 20, whereas higher values have negligible effect on the variations of the composition according to Noda[45] . Also, nr and nz = ∞ represents the conventional FGM[46] . In Fig.17 variations of effective stresses on the centre point (CP) of the cylinder wall with respect to the nz values have been plotted. Each line has been dedicated to a value of nr . The lowest value of effective stress on the centre point of cylinder wall is 417 MPa for nr = 1 and nz = 0.3. This value is 107 MPa lower than the maximum value of effective stress in which nr = 5 and nz = 0.5. The right centre point of the cylinder wall (Right CP) is chosen for evaluation of the variation of effective stress with respect to nz values. Each line has its special value of nr as shown in Fig.18. By increasing the value of nr , the effective stress decreases when nz is more than 2. For each value of nr , the lowest value of effective stress is obtained for nz = 0.1 and the maximum value of effective stress is achieved for nz = 1. It is obvious from this figure that the lowest value of effective stress is for nr = 20 and nz = 0.1, which means that there is no material properties variation along the axial direction and the cylinder wall is ceramic 2 (Si3 N4 ) rich. The maximum value of normalized effective stress through the cylinder wall for each value of nr and nz has been evaluated and plotted in Fig.19. This contour shows that the minimum value of normalized effective stresses belongs to the value of nr = 20 which demonstrates that the ceramic rich cylinder
Fig. 18. Variation of effective stresses on the right centre point of the cylinder wall vs. nz .
Fig. 19. The maximum normalized value of effective stress through the cylinder wall for each value of nr and nz .
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wall would depict lower maximum effective stresses. It is obvious from this contour that the highest value of normalized effective stress belongs to nr = nz = 0.1 which means that there is no material properties variation along r- and z-direction. When there is no material properties variation along z-direction (nz = 0.1), the cylinder wall will experience higher maximum effective stress. As the value of nz increases, the maximum normalized effective stress decreases, but it has a minimum value of 735 MPa for nr = 20 and nz = 5 which is 2.45 times higher than maximum internal pressure. In fact, the material composition of ceramics 1 and 2 need to be compromised to achieve the minimum value of maximum effective stresses. The above expressions show that the ceramic rich cylinder wall variation of material in the z-direction exhibits a superior characteristic over the conventional FGM’s (1D-FGM’s) characteristic in tolerating higher internal pressure.
VIII. Conclusions A new 2D-FGM material model based on the Mori-Tanaka scheme and third-order transition function was developed for a thick hollow cylinder with finite length in this study. Elastic axisymmetric mechanical stress analysis was performed by utilizing the finite element method. The results of FEM were verified to be in good agreement with those using the MLPG method from previous literature. Material, displacement and stresses distribution were evaluated for different values of nr and nz . The effects of different material properties distribution on the effective stress with respect to the metallic phase volume fraction are investigated. Accordingly, increasing nr and Vm will lead to a significant reduction in effective stress, whereas variation of nz can not have any important change in the effective stress. Finally, it is shown that the ceramic phase rich cylinder wall in addition to its variation of material properties along the z-direction has lower maximum normalized effective stresses of which the lowest value of stress was evaluated for nr = 20 and nz = 5. It is worth notice that the effective stress for nr = nz = 0.1 is about twice the value for nr = 20 and nz = 5.
References [1] Shen,Hui-Shen, Functionally Graded Materials: Nonlinear Analysis of Plates and Shells. CRC Press, Taylor & Francis Group, 2009. [2] Reddy,J.N. and Chin,C.D., Thermo-elastical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses, 1998, 21: 593-626. [3] Muller,E., Drasar,C., Schilz,J. and Kaysser,W.A., Functionally graded materials for sensor and energy applications. Materials Science and Engineering, 2003, A362: 17-39. [4] Qiu,J., Tani,J., Ueno,T., Morita,T., Takahashi,H. and Du,H., Fabrication and high durability of functionally graded piezoelectric bending actuators. Smart Materials and Structures, 2003, 12: 115-121. [5] Liu,L.S., Zhang,Q.J. and Zhai,P.C., The optimization design of metal/ceramic FGM armor with neural net and conjugate gradient method. Materials Science Forum, 2003, 423-425: 791-796. [6] Paszkiewicz,B., Paszkiewicz,R., Wosko,M., Radziewicz,D., Sciana,B., Szyska,A., Macherzynski,W. and Tlaczala,M., Functionally graded semiconductor layers for devices application. Vacuum, 2008, 82: 389-394. [7] Watari,F., Yokoyama,A., Omori,M., Hirari,T., Kondo,H., Uo,M. and Kawasaki,T., Biocompatibility of materials and development to functionally graded implant for bio-medical application. Composites Science and Technology, 2004, 64: 893-908. [8] Birman,V. and Byrd,L.W., Modelling and analysis of functionally graded materials and structures. Applies Mechanics Review, 2007, 60: 195-216. [9] Nemat-Alla,M., Ahmed,Kh.I.E. and Hassab-Allah,I., Elastic-plastic analysis of two-dimensional functionally graded materials under thermal loading. International Journal of Solids and Structures, 2009, 46: 2774-2786 [10] Asgari,M. and Akhlaghi,M., Transient thermal stresses in two-dimensional functionally graded thick hollow cylinder with finite length. Archive of Applied Mechanics, 2010, 80: 353-376. [11] Zimmerman,R.W. and Lutz,M.P., thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. Journal of Thermal Stresses, 1999, 22(2): 177-188. [12] Tutuncu,N. and Ozturk,M., Exact solution for stress in functionally graded pressure vessels. Composite Part: B, 2002, 32: 683-686. [13] Jabbari,M., Sohrabpour,S. and Eslami,M.R., Mechanical and thermal stress in a functionally graded hollow cylinder due to radially symmetric loads. International Journal of Pressure Vessels and Piping, 2002, 79: 493-497.
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[14] Jabbari,M., Sohrabpour,S. and Eslami,M.R., General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to non-axisymmetric steady state loads. ASME Journal of Applied Mechanics, 2003, 70: 111-118. [15] Clements,D.L., Kusuma,J. and Ang,W.T., A note on antiplane deformations of inhomogeneous materials. International Journal of Engineering Science, 1997, 35: 593-601. [16] Nemat-Alla,M. and Noda,N., Thermal stress intensity factor for functionally gradient half space with an edge crack under thermal load. Applied Mechanics, 1996a, 66(8): 569-580. [17] Nemat-Alla,M. and Noda,N., Study of an edge crack problem in a semi-infinite functionally graded medium with two dimensionally non-homogenous coefficient of thermal expansion under thermal load. Journal of Therm. Stresses, 1996b, 19: 863-888. [18] Nemat-Alla,M. and Noda,N., Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. Acta Mechanica, 2000, 144(2-3): 211-229. [19] Nemat-Alla,M., Noda,N. and Hassab-Allah,I., Analysis and investigation of thermal stress intensity factor for edge cracked FGM plates. Bulletin of the Faculty of Engineering, 2001, 29: 89-102. [20] Goupee,A.J. and Vel,S.S., Two-dimensional optimization of material composition of functionally graded materials using meshless analyses and a genetic algorithm. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 5926-5948. [21] Nemat-Alla,M., Reduction of thermal stresses by developing two-dimensional functionally graded materials. International Journal of Solids and Structures, 2003, 40: 7339-7356. [22] Liw,K.M., Kitipornchai,S., Zhang,X.Z. and Lim,C.W., Analysis of thermal stress behaviour of functionally graded hollow circular cylinder. International Journal of Solids and Structures, 2003, 40: 2355-2380. [23] Chen,B. and Tong,L., Termomechanically coupled sensitivity analysis and design optimization of functionally graded materials. Computer Methods in Applied Mechanics and Engineering, 2005, 194(18-20): 1891-1911. [24] Shao,Z.S., Mechanical and thermal stresses of a functionally graded circular hollow cylinder with finite length. International Journal of Pressure Vessels and piping, 2005, 82: 155-163. [25] Hongjun,X., Zhifei,S. and Taotao,Z. Elastic analyses of heterogeneous hollow cylinders. Mechanics Research Communications, 2006, 33: 681-691. [26] Tokovyy,Yu.V. and Ma,Ch.Ch., Analysis of 2d non-axisymmetric elasticity and thermoelasticity problems for radially inhomogeneous hollow cylinders. Journal of Engineering Mathematics, 2008, 61(2-4): 171-184. [27] Kordkheili,S.A. and Hosseini Naghdabadi,R., Thermoelastic analysis of functionally graded cylinders under axial loading. Journal of Thermal Stresses, 2008, 31(1): 1-17. [28] Asgari,M. Akhlaghi,M. and Hosseini,S.M., Dynamic analysis of two-dimensional functionally graded thick hollow cylinder with finite length under impact loading. Acta Mechechanica, 2009, 208: 163-180. [29] Asgari,M. and Akhlaghi,M., Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length. Heat and Mass Transfer, 2009, 45: 1383-1392. [30] Asemi,K., Salehi,M. and Akhlaghi,M., Elastic solution of a two-dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads. Acta Mechanica, 2001, 217: 119-134. [31] M. Shariyat, Nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory. International Journal of Mechanical Science, 2012, 65(1): 24-37. [32] Mori,T. and Tanaka,K., Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 1973, 21: 571-574. [33] Benveniste,Y., A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials, 1987, 6: 147-157. [34] Reiter,T., Dvorak,G.J. and Tvergaard,V., Micromechanical models for graded composite materials. Journal of the Mechanics and Physics of Solids, 1997, 45: 1281-1302. [35] Reiter,T. and Dvorak,G.J., Micromechanical models for graded composite Materials: II. Thermomechanical loading. Journal of Mechanics and Physics of Solids, 1998, 46: 1655-1673. [36] Neubrand,A., Chung,T.J., Rodel,Steffler,E.D. and Fett,T., Residual stress in functionally graded plates. Journal of Materials Research, 2002, 17(11): 2912-2920. [37] Li,C., Zou,Z. and Duan,Z., Multiple iso-parametric finite element method for non-homogeneous media. Mechanics Research Communications, 2000, 27(2): 137-142. [38] Bleek,O., Munz,D., Schaller,W. and Yang,Y.Y., Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading. Engineering Fracture Mechanics, 1998, 60(5-6): 615-623. [39] Bao,G. and Wang,L., Multiple cracking in functionally graded ceramic/metal coatings. International Journal of Solids and Structures, 1995, 32(19): 2853-2871.
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[40] Santare,M.H. and Lambros,J., Use of graded finite elements to model the behaviour of non-homogeneous materials. ASME Journal of Applied Mechanics, 2000, 67: 819-822. [41] Kim,J.H. and Paulino,G.H., Iso-parametric graded finite elements for non-homogeneous isotropic and orthotropic materials. ASME Journal of Applied Mechanics, 2002, 69(4): 502-514. [42] Rousseau,C.E. and Tippur,H.V., Compositionally graded materials with cracks normal to the elastic gradient. Acta Materialia, 2000, 48: 4021-4033. [43] Ching,H.K. and Yen,S.C., Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads. Composites: Part B, 2005, 36: 223-240. [44] Obata,Y. and Noda,N., Transient thermal stress in a plate of functionally gradient materials. Ceramic Trans., Functionally Gradient Materials, 1993, 34: 403-410. [45] Noda,N., Thermal stresses in functionally graded materials. Journal of Thermal Stresses, 1999, 22: 477-512. [46] Nemat-Alla,M., Reduction of thermal stresses by composition optimization of two-dimensional functionally graded materials. Acta Mechanica, 2009, 208: 147-161.
ISSN 0894-9166
Elastic Mechanical Stress Analysis in a 2D-FGM Thick Finite Length Hollow Cylinder with Newly Developed Material Model Amir Najibi1⋆
Mohammad Hassan Shojaeefard2
1
( Faculty of Mechanical Engineering, Semnan University, Semnan, Iran) (2 School of Mechanical Engineering, Iran University of Science & Technology, Iran) Received 4 April 2014, revision received 8 November 2014
ABSTRACT In this paper a new 2D-FGM material model based on Mori-Tanaka scheme and third-order transition function has been developed for a thick hollow cylinder of finite length. Elastic mechanical stress analysis is performed by utilizing the finite element method. The corresponding material, displacement and stress distributions are evaluated for different values of nr and nz . Moreover, the effects of different material property distributions on the effective stress with respect to the metallic phase volume fraction are investigated. It is demonstrated that the increase in nr and Vm leads to a significant reduction in the effective stress. Finally, it is shown that the ceramic phase rich cylinder wall has lower maximum effective stresses of which the lowest value of effective stress has been evaluated for nr = 20 and nz = 5. This minimum value is about half the maximum effective stress which has been evaluated for the non-FGM cylinder case (nr = nz = 0.1).
KEY WORDS 2D-FGM, elastic analysis, thick hollow cylinder, Mori-Tanaka material model
Nomenclature a, b c1 , c2 Ftrans (Vi ) m1 , m2 nr , nz P (x, y) pmax Vc Vm V1 , V2 Vco Vci Vib K G δ
⋆
inner and outer radii first ceramic and second ceramic transition function first metal and second metal radial and axial power law exponents general material properties pressure amplitude volume fractions of ceramic phase volume fractions of metallic phase volume fractions of basic materials the volume fractions of the ceramic phase on the outer surfaces the volume fractions of the ceramic phase on the inner surfaces constant value bulk module shear module constant value
Corresponding author. E-mail: [email protected]
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δij γ(Vi )
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unit matrix value of transition function
I. Introduction With the development of new industries and modern processes, a new class of composite materials called functionally graded materials (FGMs) able to tolerate severe thermal loadings are described[1] . In fact, FGMs are composite materials that are formed of two or more constituent phases with a composition that is continuously variable[2] . With the materials used in FGMs proposed first by Japanese scientists in the mid-1980s, FGMs have found numerous applications in different related fields; e.g. FGM sensors (Muller et al. 2003) and actuators, FGM metal/ceramic armor, FGM photo detectors, and FGM dental implant[3–7] . Birman and Byrd presented a review of principal developments in FGMs involving heat transfer issues, stress, stability and dynamic analyses, testing, manufacturing and design, applications, and fracture[8] . FGMs have many applications in those industries using drastically different temperatures in their operating environment such as aerospace structural applications and fusion reactors[9, 10]. Zimmerman and Lutz studied the thermal stress and thermal expansion in a uniformly heated functionally graded cylinder. They proposed an analytical solution for solving the thermoelasticity equations in the steady state. In an analytical solution of FGM problems, exponential functions for continuous gradation of material properties were considered under thermal and mechanical loads[11] . Additionally, Tutuncu and Ozturk proposed an exact solution for stresses in functionally graded pressure vessels[12] . Mechanical and thermal stresses in a functionally graded hollow cylinder due to a radially symmetric load were studied by Jabbari et al.. This study also proposed a general solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to a non-axisymmetric steady state load. They considered thermal loading in the steady state and axisymmetric as well as non axisymmetric[13, 14] . There are several studies that have focused on two-dimensional FGMs. However, exponential functions have been used in all of these studies for continuous gradation of the material properties, and the employment of these functions for material properties commonly leads to enhanced analytical solution with special functions[15–19] . The element-free Galerkin method was used to analyze the two-dimensional quasi-static heat conduction and thermoelastic problems[20] . Spatial distribution of ceramic volume fraction was obtained by piecewise bi-cubic interpolation of volume fractions defined at a finite number of grid points. Additionally, Nemat-Alla proposed the idea of adding a third material to the conventional FGM constituents in order to withstand serious thermal stresses. The FGM defined was considered as a 2DFGM, the study of which showing that it has a higher capability to relax thermal stress than conventional FGM[21] . In studying hollow thick cylinders, although the FGM is supposed to be graded in most cases, it is assumed that each of the layers is homogeneous. An analysis of the thermal stress behaviour of functionally graded hollow cylinders was performed by Liew and Kitipornchai, who concluded that thermal stress is bound to occur in the FGM cylinder, except in the insignificant case of zero temperature[22] . Furthermore, Chen and Tong studied thermo-mechanically coupled sensitivity analysis and design optimization of functionally graded circular cylinders under thermal and mechanical loading in the static and non- static state[23] . Besides, mechanical and thermal stresses of a functionally graded circular hollow cylinder of finite length were studied by Shao, who presented the solutions of temperature, displacements, and thermal/mechanical stresses in a functionally graded circular hollow cylinder[24] . In addition, Hongjun et al. proposed the elastic solutions of heterogeneous elastic hollow cylinders. They analyzed the problem with two different methods, one was a cylinder with multi-layers and the other a cylinder with continuously graded properties[25] . An analytical solution of 2D non-axisymmetric elasticity and thermo-elasticity problems for a radially inhomogeneous hollow cylinder has been proposed by Tokovoyy et al. According to this study, a solution of the equation for the governing stress in the form of Fourier series was presented[26] . Hosseini Kordkheili et al. have presented an analytical thermo-elasticity solution for hollow finite-length cylinders made of functionally graded materials subjected to thermal loads, internal pressure and axial loadings. They concluded that stress and displacement components behave nonlinearly, especially for regions near the longitudinal ends of functionally graded cylinders[27] . Asgari et al. have investigated the dynamic analysis of a two-dimensional functionally graded thick
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hollow cylinder of finite length under impact of loading[28] . Asgari and Akhlaghi have studied transient heat conduction in a two-dimensional functionally graded hollow cylinder of finite length. They have implemented the Crank-Nicolson finite difference method to solve time dependence equations of the heat transfer problem. Their results show that using 2D-FGM leads to a more flexible design[29] . Asemi et al. investigated an elastic solution of a two-dimensional functionally graded thick truncated cone of finite length under hydrostatic combined loads[30] . A nonlinear transient stress and wave propagation analysis of FGM thick cylinders has been made according to a unified generalized thermoelasticity theory by Shariyat. He performed nonlinear generalized and classical thermoelasticity analyses for functionally graded thick cylinders with temperature-dependent material properties subjected to various thermomechanical shocks at their inner and outer surfaces using Hermitian elements. He employed the Mori-Tanaka homogenization model of variation of the material properties instead of the simple rule of mixtures[31] . Previous studies on FGM cylinders devoted more to thermal stresses and the variation of material properties just considered the radial direction. Furthermore, the material distribution estimation as the simple rule of mixtures is not displaying the actual material properties of FGMs. In this paper, a cylinder exposed to a non-uniform axisymmetric pressure loading made of 2Dfunctionally graded materials is investigated. The finite element method (FEM) is used for static analysis of the 2D-FGM thick hollow cylinder of a finite length. The elasticity equations are derived and numerically solved in both radial and axial directions. The nonlinear properties of constituent materials of the FG cylinder are considered to satisfy the newly developed Mori-Tanaka scheme. Moreover, the responses of structure-like displacements and stresses, under mechanical loading for various values of volume fraction have been studied and discussed in detail.
II. Volume Fraction in 2D-FGM Cylinder Consider the volume fractions of a 2D-FGM at an arbitrary point in the 2D-FGM axisymmetric cylinder shown in Fig.1. In the present cylinder, the inner and outer surfaces are made of two distinct ceramics and metals, respectively. The volume fraction distributions of the materials can be expressed as[29]
n r−a r b−a n r−a r Vm = b−a
Vc = 1 −
(1a) (1b)
Fig. 1. 2D-FGM and material distribution of the axisymmetric cylinder (c1 =ZrO2 , c2 =Si3 N4 / m1 =Ti-6Al-4V, m2 = SUS304).
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V1 V2 V3 V4
nr h z nz i 1− = 1− L n r − a r z nz = 1− b−a L nr h z nz i r−a = 1− b−a L nr r−a z nz = b−a L
r−a b−a
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(2a) (2b) (2c) (2d)
where Vc and Vm are ceramic and metallic volume fraction and V1 , V2 , V3 and V4 denote the first and second ceramic and the first and second metal volume fraction, respectively. a, b and L are the inner, outer radii and length of cylinder wall, respectively. Also nr and nz are parameters that represent the basic constituent distributions in r- and z-directions, respectively. The sum of the volume fractions of all the constituent materials makes 1, i.e. X Vi = 1 (3) i=1
III. Rules of Mixture of FGM The Mori-Tanaka scheme for estimating the effective material properties is applicable to regions of the graded microstructure with a well-defined continuous matrix as well as a discontinuous particulate phase. It is assumed that the matrix phase (denoted by subscript 1) is reinforced by spherical particles of a particulate phase (denoted by subscript 2). The following relations give the rules of mixtures for different thermal and mechanical properties[32, 33]. According to Reiter et al. as well as Reiter and Dvorak, the Mori-Tanaka model was shown to yield accurate prediction of the properties with a well-defined continuous matrix and discontinuous inclusions. Therefore, the matrix and inclusion will vary by moving from the ceramic rich surface to a metal one. The axisymmetric cylinder region is divided into 3 zones, the ceramic rich, the transition, and metal rich zone, while in the ceramic zone the matrix is ceramic and the metal is inclusion. The transition zone, in which skeletal microstructures characterized by a wide transition zone between the regions with predominance of one of the constituent phases, is defined with an appropriate transition function and in the metal rich zone the metal is matrix and the ceramic is inclusion[34, 35] . Kf − K1 V2 = K2 − K1 1 + (1 − V2 ) [3 (K2 − K1 ) / (3K1 − 4G1 )]
(4a)
Gf − G1 V2 = G2 − G1 1 + (1 − V2 ) (G2 − G1 )/(G1 + f1 )
(4b)
f1 =
G1 (9K1 + 8G1 ) 6 (K1 + 2G1 )
(4c)
where K1 , K2 and G1 , G2 are Bulk and Shear module for matrix and inclusion, respectively and V1 and V2 are the corresponding volume fractions. The subscript f represents effective material properties. 3.1. Transition function In order to accommodate the discontinuities in the homogenized material properties which are predicted at the boundaries of different zones, the following transition functions are defined by Reiter and Dvorak[35] . Accordingly, Pγ #Pβ denote the magnitudes of a certain effective material parameter predicted by two different averaging methods γ and β at a given volume fraction Vi = Vib of materials ci and mi as inclusion in a matrix ci or mi at such a region boundary. Moreover let[35] : P = γ (Vi ) Pγ + [1 − γ (Vi )] Pβ
(5)
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Defining the value of material parameter within a small interval Vib ± δ/2, such that, γ (Vi ) = 1
for Vi < Vib − δ/2
γ (Vi ) = Ftrans (Vi ) γ (Vi ) = 0
(6a)
for Vib − δ/2 < Vi < Vib + δ/2
for Vi > Vib + δ/2
(6b) (6c)
While the third-order transfer function is taken as,
Ftrans (Vi ) = 2
Vi − Vib δ
3
−
3 (Vi − Vib ) 1 + 2δ 2
(7)
where Vi is volume fraction and Vib ± δ/2 determines the boundaries of transition function. Ftrans (Vi ) is the governing transition function for the transition zone and γ (Vi ) is the effective material property in the transition zone[35] .
3.2. Material model validation To demonstrate the ability of the proposed model to estimate the effective material properties of FGM, the proposed model (M-T) is compared with the experimental data in which the Young’s modulus of the Al/Al2 O3 composites is determined for Al volume fractions between 2.5 and 28%[36] . The samples have plate geometry of 35 × 33 × 3 mm3 , with a non-homogeneous Al content along the longest axis of the plate. Half the plate consists of Al/Al2 O3 composite with an aluminum volume fraction of 2.5%. The other half of the plate (a region of length 17.5 mm) contains a composition gradient. The material properties of Al and Al2 O3 are: EAl = 69 GPa; νAl = 0.33; EAl2O3 = 390 GPa; νAl2O3 = 0.2[36] . In Fig.2, the results of the proposed model have been found in good agreement with experimental ones. The elastic modulus contour for nr = 1 and nz = 0.5 according to the newly developed Mori-Tanaka material model is illustrated in Fig.3.
Fig. 2. Young’s modulus as a function of Al volume fraction for Al/Al2 O3 composite. Fig. 3. The elastic modulus distribution contour for nr = 1 and nz = 0.5.
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IV. Governing Equations The governing equations for the linear strain-displacement relation and the quasi-static case can be written as σij,j = 0
(8a)
σij,j = λεkk δij + 2µεij
(8b)
εij =
1 (ui,j + uj,i ) 2
(8c)
where the Lam´e coefficients λ and µ are the function of radial and axial position according to the newly developed Mori-Tanaka model.
V. Finite Element Modelling In order to solve the governing equations, the finite element method has been utilized and significant work has been done in previous investigations to perform the continuous variation of constituent functionally graded materials properties in finite element modelling. It is shown that use of the continuously varying properties of non-homogeneous materials in the iso-parametric finite element formulation does not pose a computational problem from Li et al. . This formulation provides enormous flexibility in meshing. Further, this method can be extended straightforwardly to three dimensional problems[37] . Finite element modelling of FGM requires creating very fine mesh of elements even for relatively small sized bodies, particularly in nonlinear analyses. In general, there are two procedures that can be selected to be considered for the material properties in the FE formulation. They include the properties assignment for each element individually or dividing the entire structure into numerous regions then assigning properties to each region[37–39] . A formulation of a finite element with a spatially varying material property field for solving boundary value problems involving continuously non-homogeneous materials has been presented by Santare and Lambros. The specific element studied was a 2D-plane stress element with linear interpolation and an exponential material property gradient[40]. Graded finite elements within the framework of a generalized iso-parametric formulation for materials within an internal property gradient were proposed by Kim and Paulino. In their study, the mean properties computed by integration within each element were employed for stiffness matrix. Generally, the graded finite elements give more precise local stress than conventional homogeneous elements[41] . A novel method including the definition of the properties as functions of the temperature has been presented by Rousseau et al. Then, the properties have been estimated in solving the functions[42] . In the current investigation, mechanical properties of materials are taken into consideration through finite element analysis. When the inner ceramic surface of the FGM cylinder is going to be subjected to the mechanical load, the stresses will develop inside the cylinder. The mechanical properties are calculated based on the volume fractions and the rules of mixture for FGMs are found in the preceding section. The FGM cylinders investigated have the inner and outer radius of a = 0.1 m and b = 0.15 m, respectively and a length of 0.2 m. The axisymmetric finite element model used in this study of the elastic problem contains 4,000 sixteen-noded rectangular elements. This number of elements results from uniform division of the cylinder wall into 100 elements through the axial direction and 40 elements through the radial direction. The mechanical boundary conditions used in the finite element model are as follows: all nodes at the top and bottom of the cylinder wall are roller supported in order to prevent their movement in the z-direction. Similarly, the outer surface of cylinder wall (r = 0.15 m) is stress free and the inner surface of cylinder wall is subjected to the following internal pressure and pmax = 300 MPa: πz pr = pmax sin MPa (9) L The numerical solutions of the present investigation are found for (ZrO2 /Si3 N4 /Ti-6Al-4V/SUS304), 2D-FGM cylinder of different composition and the material properties for each of the constituents at room temperature are taken from the research by Reddy and Chen[2] . Then the resulting displacements were determined by the solution of the axisymmetric elastic problem under the pressure distribution
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determined as described in Eq.(9). Finally, the stresses on eachnode are averaged according to those on its associated elements and shape function.
VI. Finite Element Verification In order to validate the proposed methodology and approach, the numerical FEM codes are simplified for examination of a FG cylinder made of a two-phase ceramic/metal composite which is analyzed by meshless local Petrov-Galerkin method[43] . In their study a thick hollow cylinder with an inner radius ra = 5 mm and outer radius rb = 10 mm is considered. The hollow cylinder is subjected to uniform pressures on its inner surface. Because of symmetries about the horizontal and the vertical centroidal axes, only a quarter of the cylinder is discretized with 6560 quadrilateral elements with third-order Lagrange shape functions. The stress components are normalized to the applied pressure. Besides, it is presumed that a plane strain state of deformation is dominant in the cylinder. The material constituents are Al for the metal phase and SiC for ceramic phase with material properties of EAl = 70 GPa and ESiC = 427 GPa; νAl = 0.3 and νSiC = 0.17. The volume fraction of the ceramic phase is assumed to have radial dependence by a power law function as[43] n r − ra i o i Vc = Vc + (Vc − Vc ) (10) rb − ra where Vco and Vci denote the volume fractions of the ceramic phase on the outer and the inner surfaces of the cylinder, n is a power law index that dictates non-homogeneity of material constituents through the thickness and ra and rb are inner and outer radii, respectively. The effective material properties at a point in the FG cylinder are determined by using the Mori-Tanaka method. Figure 4 depicts a comparison of the normalized values of σrr and σθθ between MLPG and FEM methods, along the radial direction for value of n = 2. According to Fig.4, the variation of the normalized radial stress obtained by the MLPG and FEM method agrees with each other.
Fig. 4. Comparison of normalized radial stress between FEM and MLPG methods. Fig. 5. The normalized radial stress contour through the cylinder wall for nr = 1 and nz = 2.
VII. Results and Discussion For parametric study of the effect of static internal pressure and also material distribution compositions on the elastic behaviour of a 2D-FGM cylinder, the stresses have been normalized according to the pmax .
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Figure 5 shows the normalized radial stress distribution for nr = 1 and nz = 2 through the cylinder wall. The radial stress on the inner cylinder surface follows the loading pattern, so it achieves maximum normalized compression stress of 1 on the internal surface. Normalized axial stress for nr = 1 and nz = 2 is illustrated in Fig.6. The stress contour shows maximum value of about 600 MPa on the top internal surface of the cylinder that is about twice the maximum internal pressure loading. Figure 7 represents the normalized hoop (tangential) stress distribution through the cylinder wall. The material properties exponents are nr = 1 and nz = 2 and the maximum value of normalized hoop stress is about 2.5 times the maximum internal pressure. This value has been attained on top of internal cylinder surface, in which this value is even higher than that of axial stress at this region.
Fig. 6. Normalized axial stress contour for nr = 1 and nz = 2.
Fig. 7. Normalized hoop stress contour for nr = 1 and nz = 2.
The last stress contour is shear stress for nr = 1 and nz = 2. The maximum compression stress, greater than 80 MPa which is one-third less than the maximum internal pressure, achieved within half the top of cylinder wall, as shown in Fig.8.
Fig. 8. Normalized shear stress contour for nr = 1 and nz = 2.
Fig. 9. Radial displacement contour through the cylinder wall for nr = 1 and nz = 2.
The radial and axial displacement contours through the cylinder wall have been demonstrated in Figs.9 and 10 for nr = 1 and nz = 2. The effective stress can be evaluated from the principal stresses using the distortion energy theory or von Mises-Hencky theory and can be compared with the yield strength of the material at fault[44] .
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Fig. 10. Axial displacement contour through the cylinder wall for nr = 1 and nz = 2. Fig. 11. Normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 1 and 5.
The effective stress (von Mises) is defined as follows: σeff
h i1/2 1 2 2 2 = (σ1 − σ2 ) + (σ2 − σ3 ) + (σ3 − σ1 ) 2
(11)
where σeff is von Mises stress and σ1 , σ2 and σ3 are the principal stresses. Figure 11 shows the normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 1 and 5. When nz = 1, the maximum normalized von-Mises stress is about 3 while for nz = 5 the value of normalized von Mises stress is about 2.7, which means by increasing the value of nz the the value of von Mises stress will decrease. Figure 12 shows normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 5 and vice versa. When nz = 5, the stress contour gets sooner its higher value of the stress along the radial direction. Radial displacement with respect to the metallic phase volume fraction (Vm ) along the horizontal centre line (HCL) for nz = 1 and different values of nr is illustrated in Fig.13. According to the figure, when the value of nr increases, the radial displacement decreases. Moreover, when the metallic phase volume fraction increases, the radial displacement decreases excessively and this reduction depends on the value of the nr . Accordingly, for nr ’s higher than 1, significant reduction occurs at lower values of Vm while this phenomenon occurs for nr ’s lower than 1 with a smooth reduction in the higher values of Vm , on the radial displacement curves. Additionally, by considering Fig.13, it can be concluded that the increase in ceramic inclusions will result in the decrease in radial displacement. In addition to this, Fig.14 illustrates the radial displacement along HCL with respect to the metallic phase volume fraction for nr = 1 and different values of the nz . By increasing the variation of material along the axial direction, the radial displacement increases consistently. This figure also demonstrates that the increase in ceramic inclusions leads to the reduction in the radial displacement. Moreover, a comparison of the effective stress (von-Mises) along the HCL with the metallic volume fraction with nz = 2 and different values of nr shows that although by increasing the metallic volume fraction, the effective stress decreases severely, this reduction occurs for lower values of Vm for nr ’s higher than 5 as depicted in Fig.15. In the same metallic volume fraction, by increasing the nr , the
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Fig. 13. Radial displacement vs. metallic phase volume fraction along the HCL for nz = 1 and different values of nr .
Fig. 12. Normalized von Mises stress distribution through the cylinder wall for nr = 1 and nz = 5 and vice versa.
Fig. 14. Radial displacement vs. metallic phase volume fraction along the HCL for nr = 1 and different values of nz .
Fig. 15. Effective stress vs. metallic phase volume fraction along the HCL for nz = 2 and different values of nr .
effective stress decreases severely while for smaller values of Vm these differences become greater and greater. Figure 16 shows a comparison of effective stress along the HCL with the metallic volume fraction with nr = 2 and different values of the nz . In the same metallic volume fraction there is no distinguishable change in different values of nz on the effective stress curves. Accordingly, by increasing the metallic volume fraction, the effective stresses will decrease significantly. The composition of a 2D-FGM cylinder is optimized by minimizing the stresses, namely, the effective stress. In the context of this study, this effective stress is minimized in such a way that an optimum non-homogeneous parameters, nr and nz can be obtained. The design variables for the current problem are the non-homogeneous parameters nr and nz , which control the composition variations through the 2D-FGM cylinder. The following conditions for the design variables are introduced:
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Fig. 16. Effective stress vs. metallic phase volume fraction along the HCL for nr = 2 and different values of nz .
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Fig. 17. Variation of effective stresses on the centre point (CP) of the cylinder wall vs. nz .
The values of the non-homogeneous parameter, nr and nz vary in the range 0 ≤ nr and nz ≤ ∞. The zero value is represented by 0.1 to obtain a 2D-FGM and ∞ is represented by 20. The values nr and nz = 0.1 will demonstrate 2D-FGM instead of nr and nz = 0.0 which indicates conventional FGM. Also, nr and nz = ∞ is represented by 20, whereas higher values have negligible effect on the variations of the composition according to Noda[45] . Also, nr and nz = ∞ represents the conventional FGM[46] . In Fig.17 variations of effective stresses on the centre point (CP) of the cylinder wall with respect to the nz values have been plotted. Each line has been dedicated to a value of nr . The lowest value of effective stress on the centre point of cylinder wall is 417 MPa for nr = 1 and nz = 0.3. This value is 107 MPa lower than the maximum value of effective stress in which nr = 5 and nz = 0.5. The right centre point of the cylinder wall (Right CP) is chosen for evaluation of the variation of effective stress with respect to nz values. Each line has its special value of nr as shown in Fig.18. By increasing the value of nr , the effective stress decreases when nz is more than 2. For each value of nr , the lowest value of effective stress is obtained for nz = 0.1 and the maximum value of effective stress is achieved for nz = 1. It is obvious from this figure that the lowest value of effective stress is for nr = 20 and nz = 0.1, which means that there is no material properties variation along the axial direction and the cylinder wall is ceramic 2 (Si3 N4 ) rich. The maximum value of normalized effective stress through the cylinder wall for each value of nr and nz has been evaluated and plotted in Fig.19. This contour shows that the minimum value of normalized effective stresses belongs to the value of nr = 20 which demonstrates that the ceramic rich cylinder
Fig. 18. Variation of effective stresses on the right centre point of the cylinder wall vs. nz .
Fig. 19. The maximum normalized value of effective stress through the cylinder wall for each value of nr and nz .
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wall would depict lower maximum effective stresses. It is obvious from this contour that the highest value of normalized effective stress belongs to nr = nz = 0.1 which means that there is no material properties variation along r- and z-direction. When there is no material properties variation along z-direction (nz = 0.1), the cylinder wall will experience higher maximum effective stress. As the value of nz increases, the maximum normalized effective stress decreases, but it has a minimum value of 735 MPa for nr = 20 and nz = 5 which is 2.45 times higher than maximum internal pressure. In fact, the material composition of ceramics 1 and 2 need to be compromised to achieve the minimum value of maximum effective stresses. The above expressions show that the ceramic rich cylinder wall variation of material in the z-direction exhibits a superior characteristic over the conventional FGM’s (1D-FGM’s) characteristic in tolerating higher internal pressure.
VIII. Conclusions A new 2D-FGM material model based on the Mori-Tanaka scheme and third-order transition function was developed for a thick hollow cylinder with finite length in this study. Elastic axisymmetric mechanical stress analysis was performed by utilizing the finite element method. The results of FEM were verified to be in good agreement with those using the MLPG method from previous literature. Material, displacement and stresses distribution were evaluated for different values of nr and nz . The effects of different material properties distribution on the effective stress with respect to the metallic phase volume fraction are investigated. Accordingly, increasing nr and Vm will lead to a significant reduction in effective stress, whereas variation of nz can not have any important change in the effective stress. Finally, it is shown that the ceramic phase rich cylinder wall in addition to its variation of material properties along the z-direction has lower maximum normalized effective stresses of which the lowest value of stress was evaluated for nr = 20 and nz = 5. It is worth notice that the effective stress for nr = nz = 0.1 is about twice the value for nr = 20 and nz = 5.
References [1] Shen,Hui-Shen, Functionally Graded Materials: Nonlinear Analysis of Plates and Shells. CRC Press, Taylor & Francis Group, 2009. [2] Reddy,J.N. and Chin,C.D., Thermo-elastical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses, 1998, 21: 593-626. [3] Muller,E., Drasar,C., Schilz,J. and Kaysser,W.A., Functionally graded materials for sensor and energy applications. Materials Science and Engineering, 2003, A362: 17-39. [4] Qiu,J., Tani,J., Ueno,T., Morita,T., Takahashi,H. and Du,H., Fabrication and high durability of functionally graded piezoelectric bending actuators. Smart Materials and Structures, 2003, 12: 115-121. [5] Liu,L.S., Zhang,Q.J. and Zhai,P.C., The optimization design of metal/ceramic FGM armor with neural net and conjugate gradient method. Materials Science Forum, 2003, 423-425: 791-796. [6] Paszkiewicz,B., Paszkiewicz,R., Wosko,M., Radziewicz,D., Sciana,B., Szyska,A., Macherzynski,W. and Tlaczala,M., Functionally graded semiconductor layers for devices application. Vacuum, 2008, 82: 389-394. [7] Watari,F., Yokoyama,A., Omori,M., Hirari,T., Kondo,H., Uo,M. and Kawasaki,T., Biocompatibility of materials and development to functionally graded implant for bio-medical application. Composites Science and Technology, 2004, 64: 893-908. [8] Birman,V. and Byrd,L.W., Modelling and analysis of functionally graded materials and structures. Applies Mechanics Review, 2007, 60: 195-216. [9] Nemat-Alla,M., Ahmed,Kh.I.E. and Hassab-Allah,I., Elastic-plastic analysis of two-dimensional functionally graded materials under thermal loading. International Journal of Solids and Structures, 2009, 46: 2774-2786 [10] Asgari,M. and Akhlaghi,M., Transient thermal stresses in two-dimensional functionally graded thick hollow cylinder with finite length. Archive of Applied Mechanics, 2010, 80: 353-376. [11] Zimmerman,R.W. and Lutz,M.P., thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. Journal of Thermal Stresses, 1999, 22(2): 177-188. [12] Tutuncu,N. and Ozturk,M., Exact solution for stress in functionally graded pressure vessels. Composite Part: B, 2002, 32: 683-686. [13] Jabbari,M., Sohrabpour,S. and Eslami,M.R., Mechanical and thermal stress in a functionally graded hollow cylinder due to radially symmetric loads. International Journal of Pressure Vessels and Piping, 2002, 79: 493-497.
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2016
[14] Jabbari,M., Sohrabpour,S. and Eslami,M.R., General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to non-axisymmetric steady state loads. ASME Journal of Applied Mechanics, 2003, 70: 111-118. [15] Clements,D.L., Kusuma,J. and Ang,W.T., A note on antiplane deformations of inhomogeneous materials. International Journal of Engineering Science, 1997, 35: 593-601. [16] Nemat-Alla,M. and Noda,N., Thermal stress intensity factor for functionally gradient half space with an edge crack under thermal load. Applied Mechanics, 1996a, 66(8): 569-580. [17] Nemat-Alla,M. and Noda,N., Study of an edge crack problem in a semi-infinite functionally graded medium with two dimensionally non-homogenous coefficient of thermal expansion under thermal load. Journal of Therm. Stresses, 1996b, 19: 863-888. [18] Nemat-Alla,M. and Noda,N., Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. Acta Mechanica, 2000, 144(2-3): 211-229. [19] Nemat-Alla,M., Noda,N. and Hassab-Allah,I., Analysis and investigation of thermal stress intensity factor for edge cracked FGM plates. Bulletin of the Faculty of Engineering, 2001, 29: 89-102. [20] Goupee,A.J. and Vel,S.S., Two-dimensional optimization of material composition of functionally graded materials using meshless analyses and a genetic algorithm. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 5926-5948. [21] Nemat-Alla,M., Reduction of thermal stresses by developing two-dimensional functionally graded materials. International Journal of Solids and Structures, 2003, 40: 7339-7356. [22] Liw,K.M., Kitipornchai,S., Zhang,X.Z. and Lim,C.W., Analysis of thermal stress behaviour of functionally graded hollow circular cylinder. International Journal of Solids and Structures, 2003, 40: 2355-2380. [23] Chen,B. and Tong,L., Termomechanically coupled sensitivity analysis and design optimization of functionally graded materials. Computer Methods in Applied Mechanics and Engineering, 2005, 194(18-20): 1891-1911. [24] Shao,Z.S., Mechanical and thermal stresses of a functionally graded circular hollow cylinder with finite length. International Journal of Pressure Vessels and piping, 2005, 82: 155-163. [25] Hongjun,X., Zhifei,S. and Taotao,Z. Elastic analyses of heterogeneous hollow cylinders. Mechanics Research Communications, 2006, 33: 681-691. [26] Tokovyy,Yu.V. and Ma,Ch.Ch., Analysis of 2d non-axisymmetric elasticity and thermoelasticity problems for radially inhomogeneous hollow cylinders. Journal of Engineering Mathematics, 2008, 61(2-4): 171-184. [27] Kordkheili,S.A. and Hosseini Naghdabadi,R., Thermoelastic analysis of functionally graded cylinders under axial loading. Journal of Thermal Stresses, 2008, 31(1): 1-17. [28] Asgari,M. Akhlaghi,M. and Hosseini,S.M., Dynamic analysis of two-dimensional functionally graded thick hollow cylinder with finite length under impact loading. Acta Mechechanica, 2009, 208: 163-180. [29] Asgari,M. and Akhlaghi,M., Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length. Heat and Mass Transfer, 2009, 45: 1383-1392. [30] Asemi,K., Salehi,M. and Akhlaghi,M., Elastic solution of a two-dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads. Acta Mechanica, 2001, 217: 119-134. [31] M. Shariyat, Nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory. International Journal of Mechanical Science, 2012, 65(1): 24-37. [32] Mori,T. and Tanaka,K., Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 1973, 21: 571-574. [33] Benveniste,Y., A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials, 1987, 6: 147-157. [34] Reiter,T., Dvorak,G.J. and Tvergaard,V., Micromechanical models for graded composite materials. Journal of the Mechanics and Physics of Solids, 1997, 45: 1281-1302. [35] Reiter,T. and Dvorak,G.J., Micromechanical models for graded composite Materials: II. Thermomechanical loading. Journal of Mechanics and Physics of Solids, 1998, 46: 1655-1673. [36] Neubrand,A., Chung,T.J., Rodel,Steffler,E.D. and Fett,T., Residual stress in functionally graded plates. Journal of Materials Research, 2002, 17(11): 2912-2920. [37] Li,C., Zou,Z. and Duan,Z., Multiple iso-parametric finite element method for non-homogeneous media. Mechanics Research Communications, 2000, 27(2): 137-142. [38] Bleek,O., Munz,D., Schaller,W. and Yang,Y.Y., Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading. Engineering Fracture Mechanics, 1998, 60(5-6): 615-623. [39] Bao,G. and Wang,L., Multiple cracking in functionally graded ceramic/metal coatings. International Journal of Solids and Structures, 1995, 32(19): 2853-2871.
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Amir Najibi et al.: Elastic Mechanical Stress Analysis in a Hollow Cylinder
· 191 ·
[40] Santare,M.H. and Lambros,J., Use of graded finite elements to model the behaviour of non-homogeneous materials. ASME Journal of Applied Mechanics, 2000, 67: 819-822. [41] Kim,J.H. and Paulino,G.H., Iso-parametric graded finite elements for non-homogeneous isotropic and orthotropic materials. ASME Journal of Applied Mechanics, 2002, 69(4): 502-514. [42] Rousseau,C.E. and Tippur,H.V., Compositionally graded materials with cracks normal to the elastic gradient. Acta Materialia, 2000, 48: 4021-4033. [43] Ching,H.K. and Yen,S.C., Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads. Composites: Part B, 2005, 36: 223-240. [44] Obata,Y. and Noda,N., Transient thermal stress in a plate of functionally gradient materials. Ceramic Trans., Functionally Gradient Materials, 1993, 34: 403-410. [45] Noda,N., Thermal stresses in functionally graded materials. Journal of Thermal Stresses, 1999, 22: 477-512. [46] Nemat-Alla,M., Reduction of thermal stresses by composition optimization of two-dimensional functionally graded materials. Acta Mechanica, 2009, 208: 147-161.