- Email: [email protected]
Mechanical stress reduction in a pressurized 2D-FGM thick hollow cylinder with finite length
Accepted Manuscript Mechanical stress reduction in a pressurized 2D- FGM thick hollow cylinder with finite length Amir Najibi, Assistant Professor PII:
S0308-0161(15)30126-5
DOI:
10.1016/j.ijpvp.2017.05.007
Reference:
IPVP 3616
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 29 November 2015 Revised Date:
15 January 2017
Accepted Date: 9 May 2017
Please cite this article as: Najibi A, Mechanical stress reduction in a pressurized 2D- FGM thick hollow cylinder with finite length, International Journal of Pressure Vessels and Piping (2017), doi: 10.1016/ j.ijpvp.2017.05.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
ACCEPTED MANUSCRIPT
Mechanical Stress Reduction in a Pressurized 2D- FGM Thick Hollow Cylinder with Finite Length 1. Amir Najibi. (Corresponding author).
Assistant Professor of Mechanical Engineering.
RI PT
Faculty of Mechanical Engineering, Semnan University, Semnan, Iran.
Postal Address: Faculty of Mechanical Engineering, Semnan University, Campus 1, Semnan, Iran, P.O. Box: 35131-19111.
AC C
EP
TE D
M AN U
SC
Tel: +98 23 3338 3345. Fax: +98 21 88013029. E-mail: [email protected]
1
ACCEPTED MANUSCRIPT
Mechanical Stress Reduction in a Pressurized 2D- FGM Thick Hollow Cylinder with Finite Length Abstract
RI PT
The material composition evaluation in functionally graded materials is the main concern of a designer to enhance the appropriate functionality of these materials to adopt them in their applications. In this study, new 2D-FGM
material model based on the Mori-Tanaka scheme and third order transition
SC
function was developed for a thick hollow cylinder with finite length. The cylinder is subjected to the internal or external non-uniform pressure and also
the finite element method was performed to analyse axisymmetric elastic stress.
M AN U
It has been demonstrated that the values of the stresses evaluated from external pressure loading were higher than the internal pressure loading ones. Subsequently, the values of Normalized Effective Stress (NES) vs. metallic volume fraction along the horizontal centre line have not been changed significantly by the variation of nz. In addition, the values of the NES which
TE D
evaluated on the centre point of cylinder wall have been increased by increasing of the nr. Finally, it has been shown in details that the lowest value of the maximum NES was related to the nr=20 and nz=0.1 for both internal and external pressure loading. It means that the ceramic 2 (Si3N4) rich cylinder wall
EP
has the lowest value of maximum NES. Keywords: 2D- FGM; Thick hollow cylinder; Mori-Tanaka material model;
AC C
Material distribution optimization.
Nomenclature a, b
Inner and outer radii
c1, c2
First and second ceramic
Ftrans.(Vi)
Transition function
l
Cylinder length
m1, m2
First metal and second metal
2
nr, nz
Radial and axial power law exponents
P(x,y)
General material properties
Vc
Volume fractions of ceramic phase
Vm
Volume fractions of metallic phase
Vi
Volume fractions of basic materials
Vib
constant value
ρ r. z
Mass density of functionally graded material
K
SC
Bulk module Shear module
δ
Constant value
δij
Unit matrix
γ (Vi)
Value of transition function
.
Radial, axial, shear and Hoop stresses Principle stresses
TE D
,
M AN U
G
σ .σ .σ .σ
Effective stress
D
u. w P.P B x
S
N
Ω
The yields stresses for each constituent
EP
σ
The effective yield stress
Matrix of elasticity coefficients
AC C
σ
RI PT
ACCEPTED MANUSCRIPT
Radial and axial displacement
Internal and external pressure
Nodal interpolation matrix Nodal displacement matrix Linear derivative matrix Element shape functions matrix Element volume domain 3
ACCEPTED MANUSCRIPT Element boundary domain
MUMPS
MUlti-frontal Massively Parallel sparse direct Solver
HCL
Horizontal Centre Line
CP
Centre point
NES
Normalized Effective Stress
RI PT
Γ
Introduction
SC
In recent years, the idea of functionally graded material (FGM) has been commenced as a thermal barrier material. The space shuttles, combustion engines, nuclear plants and ovens,
M AN U
require effective high temperature resistant materials to improve the strength of such machine elements. The continuity of the material properties along a direction, prevents cracking, stress singularities and separations of the interface which are the weaknesses of the multi-layers composite plates that have been commonly used in the machines and equipment elements
TE D
(Choules and Kokini, 1996; Noda, 1999). The variation of thermal and mechanical material properties beside core ductility and smooth stress distribution have enabled the functionally graded materials to bear up severe thermal and mechanical shocks (Noda, 1991; Reddy and
EP
Chin, 1998; Tanigawa, 1995).
The exact prediction of the effective material properties of the FGMs according to the
AC C
simple rule of mixture is applicable only for determining the effective mass density of them (Nemat-Alla, 2003). Accordingly, the simple rule-of-mixtures is generally inappropriate for the determination of the effective thermal and mechanical properties. So, several models have been developed over the years to infer the effective properties of the FGMs. Among these models, Mori–Tanaka (Cheng and Batra, 2000; Ferreira et al., 2005) and the self-consistent models (Qian et al., 2004; Reiter and Dvorak, 1998; Reiter et al., 1997; Vel and Batra, 2002, 2003) were more popular. The Mori–Tanaka scheme is applicable to regions of the graded
4
ACCEPTED MANUSCRIPT microstructure that have a well-defined continuous matrix and a discontinuous particulate phase. Abudi and Pindera have introduced the concept of two- dimensional FGM which material properties were varying in two directions (Aboudi et al., 1996). Cho and Ha
RI PT
characterized thermoelastic performance of heat resisting FGMs, under arbitrary thermal and boundary conditions (Cho and Ha, 2002). Besides, the reductions of thermal stresses by developing and composition optimization of two-dimensional functionally graded materials
SC
were investigated by Nemat-Alla (Nemat-Alla, 2009). Nemat-Alla et al. studied the elasticplastic behavior of two-dimensional functionally graded materials under thermal loading
M AN U
(Nemat-Alla et al., 2009). Asgari et al. studied the dynamic behavior of a 2D-FG thick hollow cylinder with finite length under impact loading (Asgari et al., 2009). A new material model has been developed according to Mori-Tanaka scheme for 2D-FGM (Shojaeefard and Najibi, 2014; Najibi and Talebitooti, 2017).
TE D
Li and Peng have investigated the elastic analysis of a pressurized FG tube (Li and Peng, 2009). In that study, the obtained equation has been solved approximately by expanding the solution as series of the Legendre polynomials. Although an elastic solution
EP
for a thick cylinder made of FGM has been presented by Chen and Lin (Chen and Lin, 2008),
AC C
but Tutunku utilized infinitesimal theory of elasticity for the solution of stresses and displacements in the FG cylindrical vessels subjected to internal pressure (Tutuncu, 2007). Besides, Ghannad et al. demonstrated an analytical solution for homogeneous and isotropic truncated thick conical shell (Ghannad et al., 2009). Nie and Batra represented the analytical solutions for the plane strain static deformations of an FG hollow circular cylinder with radial dependent Young’s modulus and Poisson’s ratio (Nie and Batra, 2010). The exact solution of steady-state two-dimensional axisymmetric mechanical and thermal stresses for a short hollow cylinder composed of FGM have been developed by Jabbari et al. (Jabbari et al.,
5
ACCEPTED MANUSCRIPT 2009). Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties has been performed according to generalized theory of thermoelasticity by Abbas (Abbas, 2014). The material properties have been taken as a function of temperature and graded in the radial. The effects of temperature-dependent
RI PT
properties, volume fraction parameter and the thermal relaxation time on the physical quantities behaviour have been calculated by using finite element method. Asgari presented an optimization of volume fraction distribution in a thick hollow cylinder with finite length
SC
made of two-dimensional functionally graded material which subjected to steady state thermal and mechanical loadings (Asgari, 2015). Finite element and Genetic Algorithm
M AN U
jointed with interior penalty-function method have been employed to find the global solution of the numerical and optimization problem. Simple rule of mixture and temperature independent material properties have been utilized to simulate the material properties. The results indicate that the uniform distribution of safety factor as design criteria instead of
TE D
minimizing peak effective stress affects the optimum material distribution. Elastic mechanical stress analysis has been performed by utilizing the finite element method for a new 2D-FGM material model based on Mori-Tanaka scheme and third-order transition
EP
function (Najibi and Shojaeefard, 2016).
AC C
The previous studies about FG cylinders have been performed mostly on the evaluation of the thermal stresses as well as the material properties distribution were following predetermined (exponential or power) functions of the special direction. Accordingly, a few numbers of studies have been conducted for pressurized 2D-FG thick cylinders with finite length. Furthermore, the material distribution is the simple rule of mixtures which is not demonstrating the actual material properties of the FGM structures and linear triangle elements will not be able to demonstrate the actual stress distribution because of material variation along two directions.
6
ACCEPTED MANUSCRIPT In this paper, the mechanical stresses in a cylinder made of 2D-functionally graded materials exposed to a non-uniform axisymmetric internal or external pressure loading have been investigated. Then, the responses of structure like stresses under mechanical loading for various values of volume fraction have been studied. In addition, the finite element method
RI PT
(FEM) is utilized for quasi-static elastic analysis of the 2D-FGM thick hollow cylinder with finite length. The elastic finite element equations are developed and numerically solved in both radial and axial directions. The nonlinear properties of constituent materials in FG
SC
cylinder are considered to the following new developed Mori-Tanaka scheme which introduced during current study of 2D-FGM material distribution.
M AN U
Finally, the maximum values of normalized effective stresses for different values of nr and nz in both internal and external pressure loading have been evaluated and plotted in different contours to achieve the minimum value of normalized effective stresses through the
Problem formulation
TE D
cylinder wall.
In this section the governing equations for a 2D-FGM thick hollow cylinder of internal radius
EP
“a”, external radius “b”, and finite length “l”. will be derived. The volume fraction distributions in two radial and axial directions are introduced and graded finite element and
AC C
Mori- Tanaka material model will be implemented to develop the material distribution through the cylinder wall.
Equilibrium equations
Without body force and quasi-static assumption, the equation of Equilibrium in the axisymmetric cylinder coordinates is demonstrated as: %&'' %(
+
%&'* %+
+
&'' ,&-(
7
=0
(1a)
ACCEPTED MANUSCRIPT +
0123 0
0133 0
+
123
= 0
1b
The stress–strain relations from the Hook’s law in matrix form are: 2
Where D is the matrix of elasticity coefficients as:
9:
8
, :
0 0 B 0 A , :A @
(3)
SC
D =
1−υ υ υ = υ 1−υ υ < υ υ 1−υ < 0 0 ; 0
RI PT
σ = D ε
coordinates.
M AN U
E and C are the Young’s modulus and Poisson’s ratio that depend on r and z The linear strain–displacement relations for this study are: , ε
0D 0
= . ε D
=
0F 0
TE D
ε =
. ε
= G0 + 0D
0F 0
H
4
Where, u and w are the displacement components along the r and z directions respectively.
EP
The upper surface of cylinder is roller constrained to prevent from axial displacement and the lower surface is constrained along the radial direction. Additionally, the inner or outer
AC C
of cylinder is subjected to non-uniform pressures pi and po respectively as given: P a. z = 150 sin G P H MPa and P b. z = 0
5a
P b. z = 150 sin G P H MPa and P a. z = 0
5b
O
O
8
ACCEPTED MANUSCRIPT Volume fraction in 2D-FGM cylinder Consider the volume fractions of 2D-FGM at any arbitrary point in the 2D-FGM axisymmetric cylinder shown in Figure 1. In the present cylinder, the inner surface and outer
M AN U
SC
RI PT
surface is respectively made of two distinct ceramics and two metals.
Figure 1. 2D-FGM and material distribution of the axisymmetric cylinder with predetermined
TE D
point and line (C1=ZrO2, C2=Si3N4 / m1=Ti-6Al-4V, m2= SUS304). The volume fraction distributions of the materials can be expressed as (Asgari et al., 2009): (,V X'
AC C
EP
ST = U1 − GW,VH Y
(6a)
,\ ^2
V[ = UG],\H Y
V = U1 − G
,\ ^2
(6b) ^3
H Y U1 − G H Y
],\
,\ ^2
P
^3
(7b)
^3
(7c)
V = U1 − G],\H Y UG P H Y ,\ ^2
V = UG],\H Y U1 − G P H Y
9
(7a)
ACCEPTED MANUSCRIPT V_ = UG
,\ ^2
^3
H Y UG H Y
],\
P
(7d)
where subscripts 1, 2, 3 and 4 denote the first and second ceramic and the first and
second metal respectively. Also a( and a+ are parameters that represent the basic constituent materials makes 1, i.e. (Mori and Tanaka, 1973),
(8)
SC
∑c V =1
RI PT
distributions in r and z-directions. The sum of the volume fractions of all the constituent
Rules of mixture of FGM
M AN U
As it is mentioned in the previous literatures, the Mori–Tanaka scheme for estimating the effective material properties is appropriate to the regions of graded microstructure which have a well-defined continuous matrix and a discontinuous particulate phase. It is assumed that the matrix phase, indicated by the subscript 1, is reinforced by spherical particles of a
TE D
particulate phase, indicated by the subscript 2. The following relations give the rules of mixture for different thermal and mechanical properties (Benveniste, 1987; Shen, 2009). As presented in Reiter et al. and Reiter and Dvorak, the Mori–Tanaka model was
EP
shown to yield an accurate prediction of the properties with a well-defined continuous matrix
AC C
and discontinuous inclusions; therefore, the matrix and the inclusion will change by moving from the ceramic rich surface to a metal one. The axisymmetric cylinder region divided into 3 zones as ceramic rich, the transition and metal rich zone which 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 (Reiter and Dvorak, 1998; Reiter et al., 1997).
10
ACCEPTED MANUSCRIPT
dg ,df
=
9 ,hg
je ,jf
jg ,jf
=
f =
hg
dg ,df / df ,_jf hg
9 ,hg
(9a)
kg lkf kf mef
(9b)
jf odf 9 pjf q df 9 jf
Transition function
(9c)
RI PT
de ,df
SC
To accommodate the discontinuities in the homogenized material properties anticipated at the boundaries of different zones, the following transition functions were defined. Let Pγ # Pβ
M AN U
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: (Reiter and Dvorak, 1998). Pt
TE D
P = γ V Ps + 1 − γ V
(10)
Defining the value of material parameter within a small interval Vib±δ/2, such that,
EP
u Sv = 1 wxy Sv < SvW − {/2
AC C
u Sv = |}(VX~. Sv wxy SvW − {/2 < Sv < SvW + {/2 u Sv = 0 wxy Sv > SvW + {/2
(11a) (11b) (11c)
While the third-order transfer function was taken as, |}(VX~. Sv = 2 Sv − SvW /{ − 3 Sv − SvW / 2{ + ½
(12)
Rules of mixture of 2D-FGM The rules of mixture for the 2D-FGM can be developed with some mathematical 11
ACCEPTED MANUSCRIPT manipulation. For any point on the 2D-FGM axisymmetric cylinder region with volume fractions S ,S ,S and S_ as shown in Figure 1, the rules of mixture for the different mechanical properties may be obtained as: •‚ = |}(VX~. ƒS‚ „ × •‚T + 1 − |}(VX~. ƒS‚ „ × •‚† ‡Š ,‡‰
‡ˆŠ ,‡Š ‡‰ ,‡Š
=
=
‹Š
9 ,‹Š
‡Š ,‡‰ / ‡‰ ,_Œ‰
9 ,‹‰
‡‰ ,‡Š / d• ,_ŒŠ
9 ,‹f
‡g ,‡f / ‡f ,_Œf
‹f ‡f ,‡g
‡f ,‡g / ‡g ,_Œg
‹Ž ‡Ž ,‡•
‡Ž ,‡• / ‡• ,_Œ•
‹• ‡• ,‡Ž
TE D
•† =
9 ,‹Ž 9 ,‹•
× •T
SC
‹g ‡g ,‡f
+•
+•
•† = |}(VX~. S† × •† + 1 − |}(VX~. S† •† =
(13c)
‡• ,‡Ž / ‡Ž ,_ŒŽ
(13e)
(13f)
× •†
+•
(13d)
M AN U
•T =
9 ,‹g
(13b)
‹‰
•T = |}(VX~. ST × •T + 1 − |}(VX~. ST •T =
RI PT
‡ˆ‰ ,‡‰
(13a)
(13g) (13h)
+ •_
(13i)
EP
The procedure applied for obtaining bulk module •‚ will be applicable for the •‚ . The yield stress across the cylinder wall can be obtained from below equation (Nemat-Alla et
AC C
al., 2009):
where
‘‚
σ = σ V + σ V + σ V + σ _ V_
is the effective yield stress and
‘v are
14
the yields stresses for each constitute
respectively (Nemat-Alla et al., 2009). In contrast to the simple rule of mixture, the material properties for this model is highly position dependent and also have interconnection with each other. “f” index shows the effective material properties in equations 13. 12
ACCEPTED MANUSCRIPT Material model verification To display the feasibility of the proposed model, the modeling predictions have been compared with the experimental data in which the elastic modulus of the Al/Al2O3 composites as determined from the resonance frequency of bend bars was determined for Al
RI PT
volume fractions between 2.5 and 28% (Neubrand et al., 2002). Al/Al2O3 FGMs were prepared by an adaptation of the gradient materials by foam compaction (GMFC) replication process. The samples had a plate geometry (35 × 33 × 3 mm3) with a non-homogeneous Al
SC
content along the longest axis of the plate. Half of the plate always consisted of Al/Al2O3 composite with an aluminum volume fraction of 2.5%. The other half of the plate (a region of
M AN U
length of17.5 mm) contained a composition gradient. The material properties of Al and Al2O3 are: EAl = 69 GPa; ʋAl = 0.33; αAl = 23.1×10-6 C-1; EAl2O3 = 390 GPa; ʋAl2O3 = 0.2; and
αAl2O3 = 7.7×10-6 C-1[28]. In figure 2, the Voigt model estimates the upper bound for the experimental data. The proposed model was found to fit well with the experimental results,
TE D
which Young’s modulus of the composite with its interpenetrating microstructure could not be described by Voigt rule-of-mixture.
The Elastic modulus contour for nr=nz=2 according to the new developed Mori-
AC C
EP
Tanaka material model has been illustrated in the figure 3.
13
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 2. Young’s modulus as a function of Al volume fraction for Al/Al2O3 composites. Graded finite element modelling
TE D
Graded elements can be compared with conventional homogeneous elements such as those used in traditional layered analysis of FGMs. It would be noted that the graded element incorporates the material property gradient at the size scale of the element, while the
EP
homogeneous element produces a stepwise constant approximation to a continuous material
AC C
property field. The special effects of these discontinuities will be more considerable in the 2D-FGMs because of its 2D non-homogeneity. Then, the graded elements are applied by means of direct sampling properties at the Gauss points of the element (Kim and Paulino, 2002). Based on these facts, the graded finite element is strongly preferable for modelling of the present problem.
14
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 3. The elastic modulus contour for nr=nz=2.
TE D
For implementing the finite element method to solution of the elastic problem, an axisymmetric ring element with rectangular cross-section with third order (cubic) sixteen
EP
nodes interpolation with Lagrange shape function is used to discrete the domain.
Displacement approximation
AC C
A finite element approximation for displacements is given by: “ ≈ ∑– •
–
x
–
= • “
(15)
where, N is the element shape functions matrix and x (u and w) is the nodal
displacements matrix and the sum ranges over the number of nodes associated with an element.
15
ACCEPTED MANUSCRIPT Strain-displacement equations Using equations 4 and 17, the strain-displacement equations are given by: — = ˜ “ ≈ ∑– ˜ •
–
x
–
=
∑– ™
–
x
–
= ™ “
(16)
RI PT
In general, for the axisymmetric problem, the strain matrix at each node of an element is given by: %(
0
0
%œ %+
œ (
0
%œ
%+ • %œ %(
SC
=›
%œ
(17)
M AN U
™
š
Weak form
The weak form for the governing equations is written for the problem domain and also written as a sum over the element domains. Thus, the weak form the equilibrium equation, in
TE D
a single element becomes as (Zienkiewicz and Taylor, 2000): – {ž–Ÿ. ={ “
š
U
¨¦
™
š
¡¢– −
¥¦§
•
š
£ ¡¤– Y
(18)
EP
The Galerkin method of solution was obtained from this form using approximations to the dependent variables and their virtual forms.
AC C
Performing the sum over all elements and noting that the virtual parameters { “ are
arbitrary, a semi-discrete problem given by the set of ordinary differential equations will be attained:
q σ
= g
(19)
=∑ q
(20)
where q σ
16
ACCEPTED MANUSCRIPT g =∑ g
=
¬-
B
«
g where p = p or p
σ dΩ = =
¯-°
N
¬«
B
«
p dΓ
and dΩ = 2²y¡y¡³
D B dΩ
(22)
(23)
RI PT
q
(21)
The FGM cylinder in this study has the inner radius of a=0.1m and the outer
SC
radius b= 0.15m and the length of l= 0.2m. The axisymmetric finite element model, used in the current study of the elastic problem, contains 4,800 sixteen-nodes quadrilateral elements.
M AN U
This number of elements results from uniform dividing of the cylinder wall into 120 elements through the axial direction and 40 elements through the radial directions. The numerical solutions of the present investigation were carried out for (ZrO2/Si3N4/Ti-6Al-4V/SUS304), 2D-FGM cylinder of different composition and the material properties for each constituent
1998).
TE D
have been taken from the Reddy and Chen study in the room temperature (Reddy and Chin,
The numerical solutions of the present investigation were conducted for 2D-FGM
EP
cylinder of different material distribution compositions. Then, the resulting displacements
AC C
were determined by the solution of the axisymmetric elastic problem under the determined pressure distribution as described above using the parallel sparse direct linear solver MUMPS (MUlti-frontal Massively Parallel sparse direct Solver) that works on general systems of form Ax = b. Finally, the stresses on each node were averaged according to the stresses on its associated elements and shape function.
Finite element verification For verification and implementation, because similar work is not almost performed in the
17
ACCEPTED MANUSCRIPT previous literatures, the numerical FEM codes were simplified to examine a pressurized hollow thick FG cylinder made of constituents with constant Poisson’s ratio of 0.3 and that the spatial variation of Young’s modulus follows a power law function on the nondimensional radial coordinate "r" by: X
RI PT
´ y̅ = ´¶ y̅
(24)
where, y · = y/y¸ . E0 is a constant that includes the same dimensions as E and also n is
SC
a dimensionless constant, which dictates the profile of the material gradation across the thickness. The closed-form solution of a pressurized hollow FG cylinder can be found in
M AN U
Horgan and Chan (Ching and Yen, 2005; Horgan and Chan, 1999). In that study, a thick hollow cylinder with the inner radius ra=5mm and outer radius rb=10mm has been considered. The hollow cylinder had been subjected to uniform inner pressures of pi on the inner surface of it. Due to symmetries about the horizontal and the vertical centroid axes,
TE D
only a quarter of the cylinder discretized with 6560 third order16 nodes quadrilateral Lagrange shape function elements.
Stress components are normalized to the applied pressure, and it is assumed that a
EP
plane strain state of deformation governs in the cylinder. Accordingly, Figures 4 and 5 display the variation of the normalized radial stress
((
and hoop stress
¹¹
along the
AC C
normalized radial direction when the cylinder is under the internal pressure only, respectively. For value of n=3, the solution obtained by the FE method is found to very good agreement with its corresponding exact solution.
18
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
Figure 4. Comparison of normalized radial stress between FEM and exact solution.
Figure 5. Comparison of normalized hoop stress between FEM and exact solution. Results and discussion 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
19
ACCEPTED MANUSCRIPT material for failure (Reddy and Chin, 1998). The effective stress (von Mises) is defined as where
,
and
are the principal stresses. –‚‚
=º
−
+
−
+
−
»
/
(25)
RI PT
Figure 6 shows the effective stress distribution through the cylinder wall for nr=nz=2 for both internal and external pressure loadings. It is obvious that the effective stress pertained to external pressure loading is higher than internal one especially on the region near the inner
AC C
EP
TE D
M AN U
SC
surface of cylinder wall.
Figure 6. Comparison of effective stress distribution between internal and external pressure loading for nr=nz=2. 20
ACCEPTED MANUSCRIPT Figures (7-10) show radial, axial, hoop and shear stresses vs. metallic volume fraction (Vm) for different values of the nr and nz=2 along the horizontal center line (HCL) respectively. The radial stress related to the internal pressure will decrease when the values of nr
RI PT
and Vm increase but the radial stress pertained to external pressure increases, by increasing the values of nr and Vm (figure 7).
The radial stresses for internal and external pressure loading coincide with each
SC
counterpart in the same value of -73.3 MPa when Vm= 0.05, 0.13 and 0.36 for nr= 3, 2 and 1
AC C
EP
TE D
M AN U
respectively.
Figure 7. Radial stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure. Figure 8 demonstrates that the axial stress vs. Vm along the HCL for internal and external pressure have been reached zero when Vm= 0.11, 0.23 and 0.49 for nr=3, 2 and 1, respectively. 21
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 8. Axial stress vs. metallic volume fraction for different values of nr and nz=2 with
TE D
internal or external pressure.
When Vm and nr increase, the hoop stress decreases rarely for both internal and
AC C
EP
external pressure loading as illustrated in figure 9.
22
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 9. Hoop stress vs. metallic volume fraction for different values of nr and nz=2 with
TE D
internal or external pressure.
Figure 10 shows that by increasing the value of nr, the peak shear stresses occur sooner in the lower values of Vm. It is obvious that for the internal and external pressure
AC C
volume fraction.
EP
loading the peak value of shear stresses for each nr, would happen at the same metallic
23
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 10. Shear stress vs. metallic volume fraction for different values of nr and nz=2 with
TE D
internal or external pressure.
Figures 11 demonstrates the Normalized Effective Stress (NES) vs. the normalized r along the HCL for different values of nr and nz=3, when the internal loadings have been
AC C
EP
applied. The higher values of NES have come about in the inner surface of cylinder.
24
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 11. Normalized effective stress vs. normalized r for different values of nr and nz=3
TE D
with internal pressure.
Figures 12 demonstrates NES vs. Vm along the HCL for different values of nz and nr=2. It can be concluded from these figures that by increasing of the nz, the NES increases
EP
until metallic volume fraction of 0.42. For the values of Vm greater than 0.42, the increasing
AC C
of the nz leads to no significant reduction of the NES.
25
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 12. Normalized effective stress vs. Vm for different values of nz and nr=2 with
TE D
external pressure.
The comparison of NES vs. Vm along the HCL for internal and external pressure loading with nz= 0.5, 2, 20 and nr=2 has been illustrated in figure 13. The NES is reducing by
EP
increasing of the metallic volume fraction for internal pressure loading significantly for Vm’s less than 0.42, external pressure loading leads to higher NES (maximum of 23%) than
AC C
internal one. There is no significant difference between resultant NES pertained to internal and external pressure loading for Vm’s more than 0.42.
26
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 13. Normalized effective stress vs. Vm for different values of nz and nr=2 with internal
TE D
and external pressure.
Figure 14 depicts the comparison of NES resultant from internal and external pressure loadings for nr=1,2,3 and nz=2. External pressure loading for each material distribution
EP
causes more NES than pertained internal pressure loading, which demonstrates almost the same maximum difference about 24% equally. It is obvious that by increasing of nr and Vm,
AC C
the value of NES extracted from both of the internal and external pressure loading have been decreased.
27
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 14. Normalized effective stress vs. Vm for different values of nr and nz=2 with internal and external pressure.
TE D
Figure 15 illustrates NES vs. nz on the center point of cylinder wall (CP) for different values of nr. For the same value of nz, by increasing the value of nr, NES would increase. nz’s
AC C
each value of nz.
EP
less than 2 predict NES<1 for each value of nr and nr’s less than 3 demonstrates NES<1 for
28
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 15. Normalized effective stress vs. nz for different values of nr and nz with internal
TE D
pressure.
The comparison of NES for internal and external pressure loading on CP for nr=1,2,3 and different values of nz is depicted in figure 16. Internal pressure loading estimates lower
AC C
EP
NES than external one, almost maximum of 24% for each curves of the nr.
29
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 16. Normalized effective stress vs. nz for different values of nr and nz with internal and
TE D
external pressure.
The material composition of the presented 2D-FGM cylinder is optimized by minimizing the maximum of the normalized effective stresses through the cylinder wall for
EP
each kind of material distribution and presented boundary conditions in such a way that some optimum non-homogenous parameters, nr and nz can be obtained.
AC C
The design variables for the current problem are the non-homogenous parameter nr
and nz which determine the composition variations through the 2D-FGM cylinder. The following conditions for the design variables are introduced as (Shen, 2009): 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 value nr and nz= 0.1will 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 in the variations of the composition according to Noda (Noda, 1999). 30
ACCEPTED MANUSCRIPT The maximum value of normalized effective stress through the cylinder wall for each value of nr and nz has been presented in the figure 17, when the internal pressure loading has been applied. This contour shows that the minimum values of NES belong to the value of nr=20 which demonstrates that the ceramic rich cylinder wall would depict lower maximum
RI PT
NES’s. It is obvious that the highest value of NES belongs to nr= 0.1 and nz= 20. It means that the metal 1 (Ti-6Al-4V) rich cylinder wall will have the worst strength to withstand this internal pressure loading. When nz= 0.1 for nr‘s> 1 and when nz= 0.5 for nr‘s> 2 the values of
SC
NES are less than 1 which means that for these material distributions the structure will be able to withstand the internal pressure loading, in addition while the value of nz increases the
M AN U
maximum NES increases and NES=0.67 related to nr= 20 and nz= 0.1 is the smallest value of the normalized effective stresses. The above expressions show that the nz lower than 0.5 almost for each value of nr will demonstrate superior characteristic to the other material distribution to tolerate higher internal pressure.
TE D
Similarly, based on the figure 17, the maximum value of normalized effective stress through the cylinder wall for each value of nr and nz when the external pressure has been
AC C
EP
applied, was plotted in the figure 18.
31
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
TE D
Figure 17. Maximum normalized effective stress vs. nr and nz for internal pressure. The characteristics of the NES for each value of the nr and nz are the same as NES one when the internal pressure has been applied. When nz= 0.1 for nr‘s> 2 the values of NES are
EP
less than 1 which means that for these material distributions the structure will be able to withstand the external pressure loading. In addition, while the value of nz increases the
AC C
maximum NES increases and the lowest value of NES is 0.82 for nr= 20 and nz= 0.1. It is noteworthy to mention that for both kinds of loading the lowest value of NES were related to the same composition of nr= 20 and nz= 0.1; however, the value of NES is greater for external pressure loading than internal one.
32
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
TE D
Figure 18. Maximum normalized effective stress vs. nr and nz for external pressure. Figure 19 shows the volume fractions distribution of 4 constituents through the cylinder wall for the lowest value of NES (optimum) pertained to the internal or external
AC C
EP
pressure loading.
33
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Conclusion
TE D
Figure 19. Volume fraction distribution of 4 constituents for optimum material composition.
Numerical axisymmetric elastic stress analysis was performed for a thick hollow cylinder
EP
with finite length, made of 2D-FGM material, in which the material distributions have been estimated based on the new developed material distribution according to the Mori- Tanaka
AC C
scheme and third order transition function. The results of FEM were verified with exact solution from the previous literature. The corresponding material, stresses and normalized effective stress distribution were assessed for different values of nr and nz with internal or external pressure loading. It is shown that the values of the stresses extracted from external pressure loading were higher than the internal pressure loading ones. Also the values of NES vs. metallic volume fraction along the HCL have not been changed significantly by the
34
ACCEPTED MANUSCRIPT variation of nz and nr=2. In addition, the values of the NES evaluated on the centre point of cylinder wall have been increased by increasing of the nr. Finally, the effects of different material properties distribution on the maximum normalized effective stresses have been investigated for both internal and external pressure loading. It has
RI PT
been demonstrated that the lowest value of the maximum NES was related to the nr=20 and nz=0.1 for both internal and external pressure loading. In other word, the ceramic 2 (Si3N4) rich cylinder wall has been the lowest value of the maximum NES, because of its higher
SC
tensile yield than other materials in this cylinder wall.
M AN U
References:
Abbas I. A., 2014. Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties. Meccanica 49(7), 1697-1708. Aboudi, J., Pindera, M.-J., Arnold, S.M., 1996. Thermoelastic theory for the response of materials functionally graded in two directions. International Journal of Solids and Structures 33, 931-966.
TE D
Asgari, M., Akhlaghi, M., Hosseini, S.M., 2009. Dynamic analysis of two-dimensional functionally graded thick hollow cylinder with finite length under impact loading. Acta mechanica 208, 163-180.
EP
Asgari M, 2015. Material distribution optimization of 2D heterogeneous cylinder under thermo-mechanical loading. Structural Engineering and Mechanics 53(4), 703-723.
AC C
Benveniste, Y., 1987. A new approach to the application of Mori-Tanaka's theory in composite materials. Mechanics of materials 6, 147-157. Chen, Y., Lin, X., 2008. Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials. Computational Materials Science 44, 581-587. Cheng, Z.-Q., Batra, R., 2000. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Composites Part B: Engineering 31, 97-106. Ching, H., Yen, S., 2005. Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads. Composites Part B: Engineering 36, 223240. Cho, J., Ha, D., 2002. Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM. Computer methods in applied mechanics and engineering 191, 3195-3211.
35
ACCEPTED MANUSCRIPT Choules, B., Kokini, K., 1996. Architecture of functionally graded ceramic coatings against surface thermal fracture. Journal of engineering materials and technology 118, 522-528. Ferreira, A., Batra, R., Roque, C., Qian, L., Martins, P., 2005. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Composite Structures 69, 449-457.
RI PT
Ghannad, M., Nejad, M.Z., Rahimi, G., 2009. Elastic solution of axisymmetric thick truncated conical shells based on first-order shear deformation theory. Mechanika 5, 13-20. Horgan, C., Chan, A., 1999. The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. Journal of Elasticity 55, 43-59.
SC
Jabbari, M., Bahtui, A., Eslami, M., 2009. Axisymmetric mechanical and thermal stresses in thick short length FGM cylinders. International Journal of Pressure Vessels and Piping 86, 296-306.
M AN U
Kim, J.-H., Paulino, G., 2002. Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. Journal of Applied Mechanics 69, 502-514. Li, X.-F., Peng, X.-L., 2009. A pressurized functionally graded hollow cylinder with arbitrarily varying material properties. Journal of Elasticity 96, 81-95. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica 21, 571-574.
TE D
Najibi, A. and Shojaeefard, M.H., 2016. Elastic Mechanical Stress Analysis in a 2D-FGM Thick Finite Length Hollow Cylinder with Newly Developed Material Model. Acta Mechanica Solida Sinica, 29(2), pp.178-191. Najibi, A. and Talebitooti, R., 2017. Nonlinear transient thermo-elastic analysis of a 2D-FGM thick hollow finite length cylinder. Composites Part B: Engineering, 111, pp.211-227.
EP
Nemat-Alla, M., 2003. Reduction of thermal stresses by developing two-dimensional functionally graded materials. International Journal of Solids and Structures 40, 7339-7356.
AC C
Nemat-Alla, M., 2009. Reduction of thermal stresses by composition optimization of twodimensional functionally graded materials. Acta mechanica 208, 147-161. Nemat-Alla, M., Ahmed, K.I., Hassab-Allah, I., 2009. Elastic–plastic analysis of twodimensional functionally graded materials under thermal loading. International Journal of Solids and Structures 46, 2774-2786. Neubrand, A., Chung, T.-J., Rödel, J., Steffler, E.D., Fett, T., 2002. Residual stresses in functionally graded plates. Journal of Materials Research 17, 2912-2920. Nie, G., Batra, R., 2010. Exact solutions and material tailoring for functionally graded hollow circular cylinders. Journal of Elasticity 99, 179-201. Noda, N., 1991. Thermal stresses in materials with temperature-dependent properties. Applied Mechanics Reviews 44, 383-397.
36
ACCEPTED MANUSCRIPT Noda, N., 1999. Thermal stresses in functionally graded materials. Journal of Thermal Stresses 22, 477-512. Qian, L., Batra, R., Chen, L., 2004. Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method. Composites Part B: Engineering 35, 685-697.
RI PT
Reddy, J., Chin, C., 1998. Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses 21, 593-626. Reiter, T., Dvorak, G.J., 1998. Micromechanical models for graded composite materials: II. Thermomechanical loading. Journal of the Mechanics and Physics of Solids 46, 1655-1673.
SC
Reiter, T., Dvorak, G.J., Tvergaard, V., 1997. Micromechanical models for graded composite materials. Journal of the Mechanics and Physics of Solids 45, 1281-1302. Shen, H.-S., 2009. Functionally graded materials: nonlinear analysis of plates and shells. CRC press.
M AN U
Shojaeefard, M.H. and Najibi, A., 2014. Nonlinear Transient Heat Conduction Analysis for a Thick Hollow 2D-FGM Cylinder with Finite Length. Arabian Journal for Science and Engineering, 39(12), pp.9001-9014. Tanigawa, Y., 1995. Some basic thermoelastic problems for nonhomogeneous structural materials. Applied Mechanics Reviews 48, 287-300.
TE D
Tutuncu, N., 2007. Stresses in thick-walled FGM cylinders with exponentially-varying properties. Engineering Structures 29, 2032-2035. Vel, S.S., Batra, R., 2002. Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA journal 40, 1421-1433.
EP
Vel, S.S., Batra, R., 2003. Three-dimensional analysis of transient thermal stresses in functionally graded plates. International Journal of Solids and Structures 40, 7181-7196.
AC C
Zienkiewicz, O., Taylor, R., 2000. The finite element method, 5 ed.
37
ACCEPTED MANUSCRIPT Figure Lists Figure 1. 2D-FGM and material distribution of the axisymmetric cylinder with predetermined point and line (C1=ZrO2, C2=Si3N4 / m1=Ti-6Al-4V, m2= SUS304). Figure 2. Young’s modulus as a function of Al volume fraction for Al/Al2O3 composites. Figure 3. The module of elasticity contour for nr=nz=2.
RI PT
Figure 4. Comparison of normalized radial stress between FEM and exact solution. Figure 5. Comparison of normalized hoop stress between FEM and exact solution.
Figure 6. Comparison of effective stress distribution between internal and external pressure loading for nr=nz=2.
SC
Figure 7. Radial stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure.
internal or external pressure.
M AN U
Figure 8. Axial stress vs. metallic volume fraction for different values of nr and nz=2 with Figure 9. Hoop stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure.
Figure 10. Shear stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure.
TE D
Figure 11. Normalized effective stress vs. normalized r for different values of nr and nz=3 with internal pressure.
Figure 12. Normalized effective stress vs. Vm for different values of nz and nr=2 with external pressure.
EP
Figure 13. Normalized effective stress vs. Vm for different values of nz and nr=2 with internal and external pressure.
AC C
Figure 14. Normalized effective stress vs. Vm for different values of nr and nz=2 with internal and external pressure.
Figure 15. Normalized effective stress vs. nz for different values of nr and nz with internal pressure.
Figure 16. Normalized effective stress vs. nz for different values of nr and nz with internal and external pressure. Figure 17. Maximum normalized effective stress vs. nr and nz for internal pressure. Figure 18. Maximum normalized effective stress vs. nr and nz for external pressure. Figure 19. Volume fraction distribution of 4 constituents for optimum material composition.
38
ACCEPTED MANUSCRIPT Highlights The new material distributions have been developed based on the Mori- Tanaka scheme and third order transition function.
than the internal pressure loading ones.
RI PT
The values of the stresses extracted from external pressure loading were higher
The values of NES vs. metallic volume fraction along the HCL have not been changed significantly by the variation of nz.
increased by increasing of the nr.
SC
The values of the NES evaluated on the centre point of cylinder wall have been
M AN U
The lowest value of the maximum NES was related to the nr=20 and nz=0.1 for
AC C
EP
TE D
both internal and external pressure loading.
S0308-0161(15)30126-5
DOI:
10.1016/j.ijpvp.2017.05.007
Reference:
IPVP 3616
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 29 November 2015 Revised Date:
15 January 2017
Accepted Date: 9 May 2017
Please cite this article as: Najibi A, Mechanical stress reduction in a pressurized 2D- FGM thick hollow cylinder with finite length, International Journal of Pressure Vessels and Piping (2017), doi: 10.1016/ j.ijpvp.2017.05.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
AC C
EP
TE D
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
ACCEPTED MANUSCRIPT
Mechanical Stress Reduction in a Pressurized 2D- FGM Thick Hollow Cylinder with Finite Length 1. Amir Najibi. (Corresponding author).
Assistant Professor of Mechanical Engineering.
RI PT
Faculty of Mechanical Engineering, Semnan University, Semnan, Iran.
Postal Address: Faculty of Mechanical Engineering, Semnan University, Campus 1, Semnan, Iran, P.O. Box: 35131-19111.
AC C
EP
TE D
M AN U
SC
Tel: +98 23 3338 3345. Fax: +98 21 88013029. E-mail: [email protected]
1
ACCEPTED MANUSCRIPT
Mechanical Stress Reduction in a Pressurized 2D- FGM Thick Hollow Cylinder with Finite Length Abstract
RI PT
The material composition evaluation in functionally graded materials is the main concern of a designer to enhance the appropriate functionality of these materials to adopt them in their applications. In this study, new 2D-FGM
material model based on the Mori-Tanaka scheme and third order transition
SC
function was developed for a thick hollow cylinder with finite length. The cylinder is subjected to the internal or external non-uniform pressure and also
the finite element method was performed to analyse axisymmetric elastic stress.
M AN U
It has been demonstrated that the values of the stresses evaluated from external pressure loading were higher than the internal pressure loading ones. Subsequently, the values of Normalized Effective Stress (NES) vs. metallic volume fraction along the horizontal centre line have not been changed significantly by the variation of nz. In addition, the values of the NES which
TE D
evaluated on the centre point of cylinder wall have been increased by increasing of the nr. Finally, it has been shown in details that the lowest value of the maximum NES was related to the nr=20 and nz=0.1 for both internal and external pressure loading. It means that the ceramic 2 (Si3N4) rich cylinder wall
EP
has the lowest value of maximum NES. Keywords: 2D- FGM; Thick hollow cylinder; Mori-Tanaka material model;
AC C
Material distribution optimization.
Nomenclature a, b
Inner and outer radii
c1, c2
First and second ceramic
Ftrans.(Vi)
Transition function
l
Cylinder length
m1, m2
First metal and second metal
2
nr, nz
Radial and axial power law exponents
P(x,y)
General material properties
Vc
Volume fractions of ceramic phase
Vm
Volume fractions of metallic phase
Vi
Volume fractions of basic materials
Vib
constant value
ρ r. z
Mass density of functionally graded material
K
SC
Bulk module Shear module
δ
Constant value
δij
Unit matrix
γ (Vi)
Value of transition function
.
Radial, axial, shear and Hoop stresses Principle stresses
TE D
,
M AN U
G
σ .σ .σ .σ
Effective stress
D
u. w P.P B x
S
N
Ω
The yields stresses for each constituent
EP
σ
The effective yield stress
Matrix of elasticity coefficients
AC C
σ
RI PT
ACCEPTED MANUSCRIPT
Radial and axial displacement
Internal and external pressure
Nodal interpolation matrix Nodal displacement matrix Linear derivative matrix Element shape functions matrix Element volume domain 3
ACCEPTED MANUSCRIPT Element boundary domain
MUMPS
MUlti-frontal Massively Parallel sparse direct Solver
HCL
Horizontal Centre Line
CP
Centre point
NES
Normalized Effective Stress
RI PT
Γ
Introduction
SC
In recent years, the idea of functionally graded material (FGM) has been commenced as a thermal barrier material. The space shuttles, combustion engines, nuclear plants and ovens,
M AN U
require effective high temperature resistant materials to improve the strength of such machine elements. The continuity of the material properties along a direction, prevents cracking, stress singularities and separations of the interface which are the weaknesses of the multi-layers composite plates that have been commonly used in the machines and equipment elements
TE D
(Choules and Kokini, 1996; Noda, 1999). The variation of thermal and mechanical material properties beside core ductility and smooth stress distribution have enabled the functionally graded materials to bear up severe thermal and mechanical shocks (Noda, 1991; Reddy and
EP
Chin, 1998; Tanigawa, 1995).
The exact prediction of the effective material properties of the FGMs according to the
AC C
simple rule of mixture is applicable only for determining the effective mass density of them (Nemat-Alla, 2003). Accordingly, the simple rule-of-mixtures is generally inappropriate for the determination of the effective thermal and mechanical properties. So, several models have been developed over the years to infer the effective properties of the FGMs. Among these models, Mori–Tanaka (Cheng and Batra, 2000; Ferreira et al., 2005) and the self-consistent models (Qian et al., 2004; Reiter and Dvorak, 1998; Reiter et al., 1997; Vel and Batra, 2002, 2003) were more popular. The Mori–Tanaka scheme is applicable to regions of the graded
4
ACCEPTED MANUSCRIPT microstructure that have a well-defined continuous matrix and a discontinuous particulate phase. Abudi and Pindera have introduced the concept of two- dimensional FGM which material properties were varying in two directions (Aboudi et al., 1996). Cho and Ha
RI PT
characterized thermoelastic performance of heat resisting FGMs, under arbitrary thermal and boundary conditions (Cho and Ha, 2002). Besides, the reductions of thermal stresses by developing and composition optimization of two-dimensional functionally graded materials
SC
were investigated by Nemat-Alla (Nemat-Alla, 2009). Nemat-Alla et al. studied the elasticplastic behavior of two-dimensional functionally graded materials under thermal loading
M AN U
(Nemat-Alla et al., 2009). Asgari et al. studied the dynamic behavior of a 2D-FG thick hollow cylinder with finite length under impact loading (Asgari et al., 2009). A new material model has been developed according to Mori-Tanaka scheme for 2D-FGM (Shojaeefard and Najibi, 2014; Najibi and Talebitooti, 2017).
TE D
Li and Peng have investigated the elastic analysis of a pressurized FG tube (Li and Peng, 2009). In that study, the obtained equation has been solved approximately by expanding the solution as series of the Legendre polynomials. Although an elastic solution
EP
for a thick cylinder made of FGM has been presented by Chen and Lin (Chen and Lin, 2008),
AC C
but Tutunku utilized infinitesimal theory of elasticity for the solution of stresses and displacements in the FG cylindrical vessels subjected to internal pressure (Tutuncu, 2007). Besides, Ghannad et al. demonstrated an analytical solution for homogeneous and isotropic truncated thick conical shell (Ghannad et al., 2009). Nie and Batra represented the analytical solutions for the plane strain static deformations of an FG hollow circular cylinder with radial dependent Young’s modulus and Poisson’s ratio (Nie and Batra, 2010). The exact solution of steady-state two-dimensional axisymmetric mechanical and thermal stresses for a short hollow cylinder composed of FGM have been developed by Jabbari et al. (Jabbari et al.,
5
ACCEPTED MANUSCRIPT 2009). Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties has been performed according to generalized theory of thermoelasticity by Abbas (Abbas, 2014). The material properties have been taken as a function of temperature and graded in the radial. The effects of temperature-dependent
RI PT
properties, volume fraction parameter and the thermal relaxation time on the physical quantities behaviour have been calculated by using finite element method. Asgari presented an optimization of volume fraction distribution in a thick hollow cylinder with finite length
SC
made of two-dimensional functionally graded material which subjected to steady state thermal and mechanical loadings (Asgari, 2015). Finite element and Genetic Algorithm
M AN U
jointed with interior penalty-function method have been employed to find the global solution of the numerical and optimization problem. Simple rule of mixture and temperature independent material properties have been utilized to simulate the material properties. The results indicate that the uniform distribution of safety factor as design criteria instead of
TE D
minimizing peak effective stress affects the optimum material distribution. Elastic mechanical stress analysis has been performed by utilizing the finite element method for a new 2D-FGM material model based on Mori-Tanaka scheme and third-order transition
EP
function (Najibi and Shojaeefard, 2016).
AC C
The previous studies about FG cylinders have been performed mostly on the evaluation of the thermal stresses as well as the material properties distribution were following predetermined (exponential or power) functions of the special direction. Accordingly, a few numbers of studies have been conducted for pressurized 2D-FG thick cylinders with finite length. Furthermore, the material distribution is the simple rule of mixtures which is not demonstrating the actual material properties of the FGM structures and linear triangle elements will not be able to demonstrate the actual stress distribution because of material variation along two directions.
6
ACCEPTED MANUSCRIPT In this paper, the mechanical stresses in a cylinder made of 2D-functionally graded materials exposed to a non-uniform axisymmetric internal or external pressure loading have been investigated. Then, the responses of structure like stresses under mechanical loading for various values of volume fraction have been studied. In addition, the finite element method
RI PT
(FEM) is utilized for quasi-static elastic analysis of the 2D-FGM thick hollow cylinder with finite length. The elastic finite element equations are developed and numerically solved in both radial and axial directions. The nonlinear properties of constituent materials in FG
SC
cylinder are considered to the following new developed Mori-Tanaka scheme which introduced during current study of 2D-FGM material distribution.
M AN U
Finally, the maximum values of normalized effective stresses for different values of nr and nz in both internal and external pressure loading have been evaluated and plotted in different contours to achieve the minimum value of normalized effective stresses through the
Problem formulation
TE D
cylinder wall.
In this section the governing equations for a 2D-FGM thick hollow cylinder of internal radius
EP
“a”, external radius “b”, and finite length “l”. will be derived. The volume fraction distributions in two radial and axial directions are introduced and graded finite element and
AC C
Mori- Tanaka material model will be implemented to develop the material distribution through the cylinder wall.
Equilibrium equations
Without body force and quasi-static assumption, the equation of Equilibrium in the axisymmetric cylinder coordinates is demonstrated as: %&'' %(
+
%&'* %+
+
&'' ,&-(
7
=0
(1a)
ACCEPTED MANUSCRIPT +
0123 0
0133 0
+
123
= 0
1b
The stress–strain relations from the Hook’s law in matrix form are: 2
Where D is the matrix of elasticity coefficients as:
9:
8
, :
0 0 B 0 A , :A @
(3)
SC
D =
1−υ υ υ = υ 1−υ υ < υ υ 1−υ < 0 0 ; 0
RI PT
σ = D ε
coordinates.
M AN U
E and C are the Young’s modulus and Poisson’s ratio that depend on r and z The linear strain–displacement relations for this study are: , ε
0D 0
= . ε D
=
0F 0
TE D
ε =
. ε
= G0 + 0D
0F 0
H
4
Where, u and w are the displacement components along the r and z directions respectively.
EP
The upper surface of cylinder is roller constrained to prevent from axial displacement and the lower surface is constrained along the radial direction. Additionally, the inner or outer
AC C
of cylinder is subjected to non-uniform pressures pi and po respectively as given: P a. z = 150 sin G P H MPa and P b. z = 0
5a
P b. z = 150 sin G P H MPa and P a. z = 0
5b
O
O
8
ACCEPTED MANUSCRIPT Volume fraction in 2D-FGM cylinder Consider the volume fractions of 2D-FGM at any arbitrary point in the 2D-FGM axisymmetric cylinder shown in Figure 1. In the present cylinder, the inner surface and outer
M AN U
SC
RI PT
surface is respectively made of two distinct ceramics and two metals.
Figure 1. 2D-FGM and material distribution of the axisymmetric cylinder with predetermined
TE D
point and line (C1=ZrO2, C2=Si3N4 / m1=Ti-6Al-4V, m2= SUS304). The volume fraction distributions of the materials can be expressed as (Asgari et al., 2009): (,V X'
AC C
EP
ST = U1 − GW,VH Y
(6a)
,\ ^2
V[ = UG],\H Y
V = U1 − G
,\ ^2
(6b) ^3
H Y U1 − G H Y
],\
,\ ^2
P
^3
(7b)
^3
(7c)
V = U1 − G],\H Y UG P H Y ,\ ^2
V = UG],\H Y U1 − G P H Y
9
(7a)
ACCEPTED MANUSCRIPT V_ = UG
,\ ^2
^3
H Y UG H Y
],\
P
(7d)
where subscripts 1, 2, 3 and 4 denote the first and second ceramic and the first and
second metal respectively. Also a( and a+ are parameters that represent the basic constituent materials makes 1, i.e. (Mori and Tanaka, 1973),
(8)
SC
∑c V =1
RI PT
distributions in r and z-directions. The sum of the volume fractions of all the constituent
Rules of mixture of FGM
M AN U
As it is mentioned in the previous literatures, the Mori–Tanaka scheme for estimating the effective material properties is appropriate to the regions of graded microstructure which have a well-defined continuous matrix and a discontinuous particulate phase. It is assumed that the matrix phase, indicated by the subscript 1, is reinforced by spherical particles of a
TE D
particulate phase, indicated by the subscript 2. The following relations give the rules of mixture for different thermal and mechanical properties (Benveniste, 1987; Shen, 2009). As presented in Reiter et al. and Reiter and Dvorak, the Mori–Tanaka model was
EP
shown to yield an accurate prediction of the properties with a well-defined continuous matrix
AC C
and discontinuous inclusions; therefore, the matrix and the inclusion will change by moving from the ceramic rich surface to a metal one. The axisymmetric cylinder region divided into 3 zones as ceramic rich, the transition and metal rich zone which 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 (Reiter and Dvorak, 1998; Reiter et al., 1997).
10
ACCEPTED MANUSCRIPT
dg ,df
=
9 ,hg
je ,jf
jg ,jf
=
f =
hg
dg ,df / df ,_jf hg
9 ,hg
(9a)
kg lkf kf mef
(9b)
jf odf 9 pjf q df 9 jf
Transition function
(9c)
RI PT
de ,df
SC
To accommodate the discontinuities in the homogenized material properties anticipated at the boundaries of different zones, the following transition functions were defined. Let Pγ # Pβ
M AN U
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: (Reiter and Dvorak, 1998). Pt
TE D
P = γ V Ps + 1 − γ V
(10)
Defining the value of material parameter within a small interval Vib±δ/2, such that,
EP
u Sv = 1 wxy Sv < SvW − {/2
AC C
u Sv = |}(VX~. Sv wxy SvW − {/2 < Sv < SvW + {/2 u Sv = 0 wxy Sv > SvW + {/2
(11a) (11b) (11c)
While the third-order transfer function was taken as, |}(VX~. Sv = 2 Sv − SvW /{ − 3 Sv − SvW / 2{ + ½
(12)
Rules of mixture of 2D-FGM The rules of mixture for the 2D-FGM can be developed with some mathematical 11
ACCEPTED MANUSCRIPT manipulation. For any point on the 2D-FGM axisymmetric cylinder region with volume fractions S ,S ,S and S_ as shown in Figure 1, the rules of mixture for the different mechanical properties may be obtained as: •‚ = |}(VX~. ƒS‚ „ × •‚T + 1 − |}(VX~. ƒS‚ „ × •‚† ‡Š ,‡‰
‡ˆŠ ,‡Š ‡‰ ,‡Š
=
=
‹Š
9 ,‹Š
‡Š ,‡‰ / ‡‰ ,_Œ‰
9 ,‹‰
‡‰ ,‡Š / d• ,_ŒŠ
9 ,‹f
‡g ,‡f / ‡f ,_Œf
‹f ‡f ,‡g
‡f ,‡g / ‡g ,_Œg
‹Ž ‡Ž ,‡•
‡Ž ,‡• / ‡• ,_Œ•
‹• ‡• ,‡Ž
TE D
•† =
9 ,‹Ž 9 ,‹•
× •T
SC
‹g ‡g ,‡f
+•
+•
•† = |}(VX~. S† × •† + 1 − |}(VX~. S† •† =
(13c)
‡• ,‡Ž / ‡Ž ,_ŒŽ
(13e)
(13f)
× •†
+•
(13d)
M AN U
•T =
9 ,‹g
(13b)
‹‰
•T = |}(VX~. ST × •T + 1 − |}(VX~. ST •T =
RI PT
‡ˆ‰ ,‡‰
(13a)
(13g) (13h)
+ •_
(13i)
EP
The procedure applied for obtaining bulk module •‚ will be applicable for the •‚ . The yield stress across the cylinder wall can be obtained from below equation (Nemat-Alla et
AC C
al., 2009):
where
‘‚
σ = σ V + σ V + σ V + σ _ V_
is the effective yield stress and
‘v are
14
the yields stresses for each constitute
respectively (Nemat-Alla et al., 2009). In contrast to the simple rule of mixture, the material properties for this model is highly position dependent and also have interconnection with each other. “f” index shows the effective material properties in equations 13. 12
ACCEPTED MANUSCRIPT Material model verification To display the feasibility of the proposed model, the modeling predictions have been compared with the experimental data in which the elastic modulus of the Al/Al2O3 composites as determined from the resonance frequency of bend bars was determined for Al
RI PT
volume fractions between 2.5 and 28% (Neubrand et al., 2002). Al/Al2O3 FGMs were prepared by an adaptation of the gradient materials by foam compaction (GMFC) replication process. The samples had a plate geometry (35 × 33 × 3 mm3) with a non-homogeneous Al
SC
content along the longest axis of the plate. Half of the plate always consisted of Al/Al2O3 composite with an aluminum volume fraction of 2.5%. The other half of the plate (a region of
M AN U
length of17.5 mm) contained a composition gradient. The material properties of Al and Al2O3 are: EAl = 69 GPa; ʋAl = 0.33; αAl = 23.1×10-6 C-1; EAl2O3 = 390 GPa; ʋAl2O3 = 0.2; and
αAl2O3 = 7.7×10-6 C-1[28]. In figure 2, the Voigt model estimates the upper bound for the experimental data. The proposed model was found to fit well with the experimental results,
TE D
which Young’s modulus of the composite with its interpenetrating microstructure could not be described by Voigt rule-of-mixture.
The Elastic modulus contour for nr=nz=2 according to the new developed Mori-
AC C
EP
Tanaka material model has been illustrated in the figure 3.
13
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 2. Young’s modulus as a function of Al volume fraction for Al/Al2O3 composites. Graded finite element modelling
TE D
Graded elements can be compared with conventional homogeneous elements such as those used in traditional layered analysis of FGMs. It would be noted that the graded element incorporates the material property gradient at the size scale of the element, while the
EP
homogeneous element produces a stepwise constant approximation to a continuous material
AC C
property field. The special effects of these discontinuities will be more considerable in the 2D-FGMs because of its 2D non-homogeneity. Then, the graded elements are applied by means of direct sampling properties at the Gauss points of the element (Kim and Paulino, 2002). Based on these facts, the graded finite element is strongly preferable for modelling of the present problem.
14
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 3. The elastic modulus contour for nr=nz=2.
TE D
For implementing the finite element method to solution of the elastic problem, an axisymmetric ring element with rectangular cross-section with third order (cubic) sixteen
EP
nodes interpolation with Lagrange shape function is used to discrete the domain.
Displacement approximation
AC C
A finite element approximation for displacements is given by: “ ≈ ∑– •
–
x
–
= • “
(15)
where, N is the element shape functions matrix and x (u and w) is the nodal
displacements matrix and the sum ranges over the number of nodes associated with an element.
15
ACCEPTED MANUSCRIPT Strain-displacement equations Using equations 4 and 17, the strain-displacement equations are given by: — = ˜ “ ≈ ∑– ˜ •
–
x
–
=
∑– ™
–
x
–
= ™ “
(16)
RI PT
In general, for the axisymmetric problem, the strain matrix at each node of an element is given by: %(
0
0
%œ %+
œ (
0
%œ
%+ • %œ %(
SC
=›
%œ
(17)
M AN U
™
š
Weak form
The weak form for the governing equations is written for the problem domain and also written as a sum over the element domains. Thus, the weak form the equilibrium equation, in
TE D
a single element becomes as (Zienkiewicz and Taylor, 2000): – {ž–Ÿ. ={ “
š
U
¨¦
™
š
¡¢– −
¥¦§
•
š
£ ¡¤– Y
(18)
EP
The Galerkin method of solution was obtained from this form using approximations to the dependent variables and their virtual forms.
AC C
Performing the sum over all elements and noting that the virtual parameters { “ are
arbitrary, a semi-discrete problem given by the set of ordinary differential equations will be attained:
q σ
= g
(19)
=∑ q
(20)
where q σ
16
ACCEPTED MANUSCRIPT g =∑ g
=
¬-
B
«
g where p = p or p
σ dΩ = =
¯-°
N
¬«
B
«
p dΓ
and dΩ = 2²y¡y¡³
D B dΩ
(22)
(23)
RI PT
q
(21)
The FGM cylinder in this study has the inner radius of a=0.1m and the outer
SC
radius b= 0.15m and the length of l= 0.2m. The axisymmetric finite element model, used in the current study of the elastic problem, contains 4,800 sixteen-nodes quadrilateral elements.
M AN U
This number of elements results from uniform dividing of the cylinder wall into 120 elements through the axial direction and 40 elements through the radial directions. The numerical solutions of the present investigation were carried out for (ZrO2/Si3N4/Ti-6Al-4V/SUS304), 2D-FGM cylinder of different composition and the material properties for each constituent
1998).
TE D
have been taken from the Reddy and Chen study in the room temperature (Reddy and Chin,
The numerical solutions of the present investigation were conducted for 2D-FGM
EP
cylinder of different material distribution compositions. Then, the resulting displacements
AC C
were determined by the solution of the axisymmetric elastic problem under the determined pressure distribution as described above using the parallel sparse direct linear solver MUMPS (MUlti-frontal Massively Parallel sparse direct Solver) that works on general systems of form Ax = b. Finally, the stresses on each node were averaged according to the stresses on its associated elements and shape function.
Finite element verification For verification and implementation, because similar work is not almost performed in the
17
ACCEPTED MANUSCRIPT previous literatures, the numerical FEM codes were simplified to examine a pressurized hollow thick FG cylinder made of constituents with constant Poisson’s ratio of 0.3 and that the spatial variation of Young’s modulus follows a power law function on the nondimensional radial coordinate "r" by: X
RI PT
´ y̅ = ´¶ y̅
(24)
where, y · = y/y¸ . E0 is a constant that includes the same dimensions as E and also n is
SC
a dimensionless constant, which dictates the profile of the material gradation across the thickness. The closed-form solution of a pressurized hollow FG cylinder can be found in
M AN U
Horgan and Chan (Ching and Yen, 2005; Horgan and Chan, 1999). In that study, a thick hollow cylinder with the inner radius ra=5mm and outer radius rb=10mm has been considered. The hollow cylinder had been subjected to uniform inner pressures of pi on the inner surface of it. Due to symmetries about the horizontal and the vertical centroid axes,
TE D
only a quarter of the cylinder discretized with 6560 third order16 nodes quadrilateral Lagrange shape function elements.
Stress components are normalized to the applied pressure, and it is assumed that a
EP
plane strain state of deformation governs in the cylinder. Accordingly, Figures 4 and 5 display the variation of the normalized radial stress
((
and hoop stress
¹¹
along the
AC C
normalized radial direction when the cylinder is under the internal pressure only, respectively. For value of n=3, the solution obtained by the FE method is found to very good agreement with its corresponding exact solution.
18
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
AC C
EP
TE D
Figure 4. Comparison of normalized radial stress between FEM and exact solution.
Figure 5. Comparison of normalized hoop stress between FEM and exact solution. Results and discussion 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
19
ACCEPTED MANUSCRIPT material for failure (Reddy and Chin, 1998). The effective stress (von Mises) is defined as where
,
and
are the principal stresses. –‚‚
=º
−
+
−
+
−
»
/
(25)
RI PT
Figure 6 shows the effective stress distribution through the cylinder wall for nr=nz=2 for both internal and external pressure loadings. It is obvious that the effective stress pertained to external pressure loading is higher than internal one especially on the region near the inner
AC C
EP
TE D
M AN U
SC
surface of cylinder wall.
Figure 6. Comparison of effective stress distribution between internal and external pressure loading for nr=nz=2. 20
ACCEPTED MANUSCRIPT Figures (7-10) show radial, axial, hoop and shear stresses vs. metallic volume fraction (Vm) for different values of the nr and nz=2 along the horizontal center line (HCL) respectively. The radial stress related to the internal pressure will decrease when the values of nr
RI PT
and Vm increase but the radial stress pertained to external pressure increases, by increasing the values of nr and Vm (figure 7).
The radial stresses for internal and external pressure loading coincide with each
SC
counterpart in the same value of -73.3 MPa when Vm= 0.05, 0.13 and 0.36 for nr= 3, 2 and 1
AC C
EP
TE D
M AN U
respectively.
Figure 7. Radial stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure. Figure 8 demonstrates that the axial stress vs. Vm along the HCL for internal and external pressure have been reached zero when Vm= 0.11, 0.23 and 0.49 for nr=3, 2 and 1, respectively. 21
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 8. Axial stress vs. metallic volume fraction for different values of nr and nz=2 with
TE D
internal or external pressure.
When Vm and nr increase, the hoop stress decreases rarely for both internal and
AC C
EP
external pressure loading as illustrated in figure 9.
22
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 9. Hoop stress vs. metallic volume fraction for different values of nr and nz=2 with
TE D
internal or external pressure.
Figure 10 shows that by increasing the value of nr, the peak shear stresses occur sooner in the lower values of Vm. It is obvious that for the internal and external pressure
AC C
volume fraction.
EP
loading the peak value of shear stresses for each nr, would happen at the same metallic
23
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 10. Shear stress vs. metallic volume fraction for different values of nr and nz=2 with
TE D
internal or external pressure.
Figures 11 demonstrates the Normalized Effective Stress (NES) vs. the normalized r along the HCL for different values of nr and nz=3, when the internal loadings have been
AC C
EP
applied. The higher values of NES have come about in the inner surface of cylinder.
24
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 11. Normalized effective stress vs. normalized r for different values of nr and nz=3
TE D
with internal pressure.
Figures 12 demonstrates NES vs. Vm along the HCL for different values of nz and nr=2. It can be concluded from these figures that by increasing of the nz, the NES increases
EP
until metallic volume fraction of 0.42. For the values of Vm greater than 0.42, the increasing
AC C
of the nz leads to no significant reduction of the NES.
25
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 12. Normalized effective stress vs. Vm for different values of nz and nr=2 with
TE D
external pressure.
The comparison of NES vs. Vm along the HCL for internal and external pressure loading with nz= 0.5, 2, 20 and nr=2 has been illustrated in figure 13. The NES is reducing by
EP
increasing of the metallic volume fraction for internal pressure loading significantly for Vm’s less than 0.42, external pressure loading leads to higher NES (maximum of 23%) than
AC C
internal one. There is no significant difference between resultant NES pertained to internal and external pressure loading for Vm’s more than 0.42.
26
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 13. Normalized effective stress vs. Vm for different values of nz and nr=2 with internal
TE D
and external pressure.
Figure 14 depicts the comparison of NES resultant from internal and external pressure loadings for nr=1,2,3 and nz=2. External pressure loading for each material distribution
EP
causes more NES than pertained internal pressure loading, which demonstrates almost the same maximum difference about 24% equally. It is obvious that by increasing of nr and Vm,
AC C
the value of NES extracted from both of the internal and external pressure loading have been decreased.
27
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 14. Normalized effective stress vs. Vm for different values of nr and nz=2 with internal and external pressure.
TE D
Figure 15 illustrates NES vs. nz on the center point of cylinder wall (CP) for different values of nr. For the same value of nz, by increasing the value of nr, NES would increase. nz’s
AC C
each value of nz.
EP
less than 2 predict NES<1 for each value of nr and nr’s less than 3 demonstrates NES<1 for
28
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 15. Normalized effective stress vs. nz for different values of nr and nz with internal
TE D
pressure.
The comparison of NES for internal and external pressure loading on CP for nr=1,2,3 and different values of nz is depicted in figure 16. Internal pressure loading estimates lower
AC C
EP
NES than external one, almost maximum of 24% for each curves of the nr.
29
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Figure 16. Normalized effective stress vs. nz for different values of nr and nz with internal and
TE D
external pressure.
The material composition of the presented 2D-FGM cylinder is optimized by minimizing the maximum of the normalized effective stresses through the cylinder wall for
EP
each kind of material distribution and presented boundary conditions in such a way that some optimum non-homogenous parameters, nr and nz can be obtained.
AC C
The design variables for the current problem are the non-homogenous parameter nr
and nz which determine the composition variations through the 2D-FGM cylinder. The following conditions for the design variables are introduced as (Shen, 2009): 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 value nr and nz= 0.1will 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 in the variations of the composition according to Noda (Noda, 1999). 30
ACCEPTED MANUSCRIPT The maximum value of normalized effective stress through the cylinder wall for each value of nr and nz has been presented in the figure 17, when the internal pressure loading has been applied. This contour shows that the minimum values of NES belong to the value of nr=20 which demonstrates that the ceramic rich cylinder wall would depict lower maximum
RI PT
NES’s. It is obvious that the highest value of NES belongs to nr= 0.1 and nz= 20. It means that the metal 1 (Ti-6Al-4V) rich cylinder wall will have the worst strength to withstand this internal pressure loading. When nz= 0.1 for nr‘s> 1 and when nz= 0.5 for nr‘s> 2 the values of
SC
NES are less than 1 which means that for these material distributions the structure will be able to withstand the internal pressure loading, in addition while the value of nz increases the
M AN U
maximum NES increases and NES=0.67 related to nr= 20 and nz= 0.1 is the smallest value of the normalized effective stresses. The above expressions show that the nz lower than 0.5 almost for each value of nr will demonstrate superior characteristic to the other material distribution to tolerate higher internal pressure.
TE D
Similarly, based on the figure 17, the maximum value of normalized effective stress through the cylinder wall for each value of nr and nz when the external pressure has been
AC C
EP
applied, was plotted in the figure 18.
31
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
TE D
Figure 17. Maximum normalized effective stress vs. nr and nz for internal pressure. The characteristics of the NES for each value of the nr and nz are the same as NES one when the internal pressure has been applied. When nz= 0.1 for nr‘s> 2 the values of NES are
EP
less than 1 which means that for these material distributions the structure will be able to withstand the external pressure loading. In addition, while the value of nz increases the
AC C
maximum NES increases and the lowest value of NES is 0.82 for nr= 20 and nz= 0.1. It is noteworthy to mention that for both kinds of loading the lowest value of NES were related to the same composition of nr= 20 and nz= 0.1; however, the value of NES is greater for external pressure loading than internal one.
32
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
TE D
Figure 18. Maximum normalized effective stress vs. nr and nz for external pressure. Figure 19 shows the volume fractions distribution of 4 constituents through the cylinder wall for the lowest value of NES (optimum) pertained to the internal or external
AC C
EP
pressure loading.
33
M AN U
SC
RI PT
ACCEPTED MANUSCRIPT
Conclusion
TE D
Figure 19. Volume fraction distribution of 4 constituents for optimum material composition.
Numerical axisymmetric elastic stress analysis was performed for a thick hollow cylinder
EP
with finite length, made of 2D-FGM material, in which the material distributions have been estimated based on the new developed material distribution according to the Mori- Tanaka
AC C
scheme and third order transition function. The results of FEM were verified with exact solution from the previous literature. The corresponding material, stresses and normalized effective stress distribution were assessed for different values of nr and nz with internal or external pressure loading. It is shown that the values of the stresses extracted from external pressure loading were higher than the internal pressure loading ones. Also the values of NES vs. metallic volume fraction along the HCL have not been changed significantly by the
34
ACCEPTED MANUSCRIPT variation of nz and nr=2. In addition, the values of the NES evaluated on the centre point of cylinder wall have been increased by increasing of the nr. Finally, the effects of different material properties distribution on the maximum normalized effective stresses have been investigated for both internal and external pressure loading. It has
RI PT
been demonstrated that the lowest value of the maximum NES was related to the nr=20 and nz=0.1 for both internal and external pressure loading. In other word, the ceramic 2 (Si3N4) rich cylinder wall has been the lowest value of the maximum NES, because of its higher
SC
tensile yield than other materials in this cylinder wall.
M AN U
References:
Abbas I. A., 2014. Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties. Meccanica 49(7), 1697-1708. Aboudi, J., Pindera, M.-J., Arnold, S.M., 1996. Thermoelastic theory for the response of materials functionally graded in two directions. International Journal of Solids and Structures 33, 931-966.
TE D
Asgari, M., Akhlaghi, M., Hosseini, S.M., 2009. Dynamic analysis of two-dimensional functionally graded thick hollow cylinder with finite length under impact loading. Acta mechanica 208, 163-180.
EP
Asgari M, 2015. Material distribution optimization of 2D heterogeneous cylinder under thermo-mechanical loading. Structural Engineering and Mechanics 53(4), 703-723.
AC C
Benveniste, Y., 1987. A new approach to the application of Mori-Tanaka's theory in composite materials. Mechanics of materials 6, 147-157. Chen, Y., Lin, X., 2008. Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials. Computational Materials Science 44, 581-587. Cheng, Z.-Q., Batra, R., 2000. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Composites Part B: Engineering 31, 97-106. Ching, H., Yen, S., 2005. Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads. Composites Part B: Engineering 36, 223240. Cho, J., Ha, D., 2002. Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM. Computer methods in applied mechanics and engineering 191, 3195-3211.
35
ACCEPTED MANUSCRIPT Choules, B., Kokini, K., 1996. Architecture of functionally graded ceramic coatings against surface thermal fracture. Journal of engineering materials and technology 118, 522-528. Ferreira, A., Batra, R., Roque, C., Qian, L., Martins, P., 2005. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Composite Structures 69, 449-457.
RI PT
Ghannad, M., Nejad, M.Z., Rahimi, G., 2009. Elastic solution of axisymmetric thick truncated conical shells based on first-order shear deformation theory. Mechanika 5, 13-20. Horgan, C., Chan, A., 1999. The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. Journal of Elasticity 55, 43-59.
SC
Jabbari, M., Bahtui, A., Eslami, M., 2009. Axisymmetric mechanical and thermal stresses in thick short length FGM cylinders. International Journal of Pressure Vessels and Piping 86, 296-306.
M AN U
Kim, J.-H., Paulino, G., 2002. Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials. Journal of Applied Mechanics 69, 502-514. Li, X.-F., Peng, X.-L., 2009. A pressurized functionally graded hollow cylinder with arbitrarily varying material properties. Journal of Elasticity 96, 81-95. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica 21, 571-574.
TE D
Najibi, A. and Shojaeefard, M.H., 2016. Elastic Mechanical Stress Analysis in a 2D-FGM Thick Finite Length Hollow Cylinder with Newly Developed Material Model. Acta Mechanica Solida Sinica, 29(2), pp.178-191. Najibi, A. and Talebitooti, R., 2017. Nonlinear transient thermo-elastic analysis of a 2D-FGM thick hollow finite length cylinder. Composites Part B: Engineering, 111, pp.211-227.
EP
Nemat-Alla, M., 2003. Reduction of thermal stresses by developing two-dimensional functionally graded materials. International Journal of Solids and Structures 40, 7339-7356.
AC C
Nemat-Alla, M., 2009. Reduction of thermal stresses by composition optimization of twodimensional functionally graded materials. Acta mechanica 208, 147-161. Nemat-Alla, M., Ahmed, K.I., Hassab-Allah, I., 2009. Elastic–plastic analysis of twodimensional functionally graded materials under thermal loading. International Journal of Solids and Structures 46, 2774-2786. Neubrand, A., Chung, T.-J., Rödel, J., Steffler, E.D., Fett, T., 2002. Residual stresses in functionally graded plates. Journal of Materials Research 17, 2912-2920. Nie, G., Batra, R., 2010. Exact solutions and material tailoring for functionally graded hollow circular cylinders. Journal of Elasticity 99, 179-201. Noda, N., 1991. Thermal stresses in materials with temperature-dependent properties. Applied Mechanics Reviews 44, 383-397.
36
ACCEPTED MANUSCRIPT Noda, N., 1999. Thermal stresses in functionally graded materials. Journal of Thermal Stresses 22, 477-512. Qian, L., Batra, R., Chen, L., 2004. Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method. Composites Part B: Engineering 35, 685-697.
RI PT
Reddy, J., Chin, C., 1998. Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses 21, 593-626. Reiter, T., Dvorak, G.J., 1998. Micromechanical models for graded composite materials: II. Thermomechanical loading. Journal of the Mechanics and Physics of Solids 46, 1655-1673.
SC
Reiter, T., Dvorak, G.J., Tvergaard, V., 1997. Micromechanical models for graded composite materials. Journal of the Mechanics and Physics of Solids 45, 1281-1302. Shen, H.-S., 2009. Functionally graded materials: nonlinear analysis of plates and shells. CRC press.
M AN U
Shojaeefard, M.H. and Najibi, A., 2014. Nonlinear Transient Heat Conduction Analysis for a Thick Hollow 2D-FGM Cylinder with Finite Length. Arabian Journal for Science and Engineering, 39(12), pp.9001-9014. Tanigawa, Y., 1995. Some basic thermoelastic problems for nonhomogeneous structural materials. Applied Mechanics Reviews 48, 287-300.
TE D
Tutuncu, N., 2007. Stresses in thick-walled FGM cylinders with exponentially-varying properties. Engineering Structures 29, 2032-2035. Vel, S.S., Batra, R., 2002. Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA journal 40, 1421-1433.
EP
Vel, S.S., Batra, R., 2003. Three-dimensional analysis of transient thermal stresses in functionally graded plates. International Journal of Solids and Structures 40, 7181-7196.
AC C
Zienkiewicz, O., Taylor, R., 2000. The finite element method, 5 ed.
37
ACCEPTED MANUSCRIPT Figure Lists Figure 1. 2D-FGM and material distribution of the axisymmetric cylinder with predetermined point and line (C1=ZrO2, C2=Si3N4 / m1=Ti-6Al-4V, m2= SUS304). Figure 2. Young’s modulus as a function of Al volume fraction for Al/Al2O3 composites. Figure 3. The module of elasticity contour for nr=nz=2.
RI PT
Figure 4. Comparison of normalized radial stress between FEM and exact solution. Figure 5. Comparison of normalized hoop stress between FEM and exact solution.
Figure 6. Comparison of effective stress distribution between internal and external pressure loading for nr=nz=2.
SC
Figure 7. Radial stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure.
internal or external pressure.
M AN U
Figure 8. Axial stress vs. metallic volume fraction for different values of nr and nz=2 with Figure 9. Hoop stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure.
Figure 10. Shear stress vs. metallic volume fraction for different values of nr and nz=2 with internal or external pressure.
TE D
Figure 11. Normalized effective stress vs. normalized r for different values of nr and nz=3 with internal pressure.
Figure 12. Normalized effective stress vs. Vm for different values of nz and nr=2 with external pressure.
EP
Figure 13. Normalized effective stress vs. Vm for different values of nz and nr=2 with internal and external pressure.
AC C
Figure 14. Normalized effective stress vs. Vm for different values of nr and nz=2 with internal and external pressure.
Figure 15. Normalized effective stress vs. nz for different values of nr and nz with internal pressure.
Figure 16. Normalized effective stress vs. nz for different values of nr and nz with internal and external pressure. Figure 17. Maximum normalized effective stress vs. nr and nz for internal pressure. Figure 18. Maximum normalized effective stress vs. nr and nz for external pressure. Figure 19. Volume fraction distribution of 4 constituents for optimum material composition.
38
ACCEPTED MANUSCRIPT Highlights The new material distributions have been developed based on the Mori- Tanaka scheme and third order transition function.
than the internal pressure loading ones.
RI PT
The values of the stresses extracted from external pressure loading were higher
The values of NES vs. metallic volume fraction along the HCL have not been changed significantly by the variation of nz.
increased by increasing of the nr.
SC
The values of the NES evaluated on the centre point of cylinder wall have been
M AN U
The lowest value of the maximum NES was related to the nr=20 and nz=0.1 for
AC C
EP
TE D
both internal and external pressure loading.