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Coupled thermoelasticity and second sound in finite length functionally graded thick hollow cylinders (without energy dissipation)
Materials and Design 30 (2009) 2011–2023
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Coupled thermoelasticity and second sound in finite length functionally graded thick hollow cylinders (without energy dissipation) Seyed Mahmoud Hosseini * Department of Mechanical Engineering, Khorasan Research Institute of Sciences and Food Technology, P.O. Box 91735-139, Mashhad, Iran Department of Industrial Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 13 May 2008 Accepted 30 August 2008 Available online 17 September 2008 Keywords: Coupled thermoelasticity Functionally graded materials Finite length cylinders Thermal shock
a b s t r a c t In this article, the coupled thermoelasticity behavior of functionally graded thick hollow cylinders is studied. The governing coupled thermoelasticity and the energy equations are solved for a finite length functionally graded cylinder subjected to thermal shock load. The coupled thermoelastic equations are considered based on Green–Naghdi theory. The mechanical properties of cylinder are graded across the thickness as a power law function of radius. The cylinder is assumed to be made of many isotropic sub-cylinders (layers) across the thickness. Functionally graded properties are created by suitable arrangement of layers and governing equations are expanded in longitudinal direction by means of trigonometric function expansion. The Galerkin Finite Element and Newmark Methods are used to analyze the cylinder. The dynamic behavior of temperature distribution, mechanical displacement and thermal stresses is obtained and discussed. The second sound and elastic wave propagation are determined for various kinds of variation in the mechanical properties. The comparison of present results with published data shows the excellent agreement. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Functionally graded materials (FGMs) are two-phase particulate composites in which material composition and microstructure vary continuously from one surface to the other especially from ceramic to metal. The volume fractions of the FGM constituents are gradually varied by power law or other functions of position. In the recent years, there are some researches on dynamic analysis, transient thermoelasticity and wave propagation of functionally graded structures. A number of these works have been carried out in the steady-state and transient thermoelasticity [1–3]. The theoretical treatment of the steady-state thermoelastic problem of a functionally graded cylindrical panel due to non-uniform heat supply in the circumferential direction was carried out by Ootao and Tanigawa [4]. They obtained the exact solution for the two-dimensional temperature change in a steady-state, and thermal stresses of a simple supported cylindrical panel under the state of plane strain. Hosseini et al. [5] presented an analytical method to determine the transient temperature distribution of functionally graded thick hollow cylinders. To obtain the analytical solution, they used the composition of Bessel functions. Han and Liu [6] presented a computational method to investigate SH wave in functionally graded material plates.
* Address: Department of Mechanical Engineering, Khorasan Research Institute of Sciences and Food Technology, P.O. Box 91735-139, Mashhad, Iran. Tel.: +98 912 190 5942; fax: +98 511 5003150. E-mail address: [email protected] 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.08.048
The material properties were assumed to be a quadratic function in the thickness direction. A novel spectral element was presented to study the wave propagation behavior in FGM beams subjected to high frequency impact loads by Chakraborty and Gopalakrishnan [7]. The propagation of stress waves in functionally graded materials was studied numerically by means of the theory of laminated composite wave-propagation algorithm by Berezovski et al. [8]. There are some works on wave propagation problems by using multilayer modeling of FGM. The better rule-of-mixture called as SBS method for FGMs has been used in FG plates [9], FG cylinders [10,11] and functionally graded piezoelectric material (FGPM) cylinders [12]. A two-dimensional numerical simulation model for the elastodynamic wave propagation in two linear elastic, isotropic, joint half-spaces was presented by Vollmann et al. [13]. They considered that the border between the two half-spaced was graded in a way, which the values of elastic properties and the densities vary smoothly from one half-space to the other. The vibration and dynamic analysis of functionally graded cylinders was studied by Hosseini et al. [14]. The mean velocity of radial stresses wave propagation, natural frequency and dynamic behavior of cylinder were presented in their work. The modified smoothed particle hydrodynamics (MSPH) method to study the propagation of elastic waves in functionally graded materials was studied by Zhang and Batra [15]. In their work, an artificial viscosity is added to the hydrostatic pressure to control oscillations in the shock wave. Du et al. [16] presented an exact approach to investigate Love waves in FGPM layer bonded to a semi-infinite homogeneous solid. The coupled
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
thermoelastic and the energy equations were simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load by Bahtui and Eslami [17]. They used classic theory of coupled thermoelasticity and Galerkin Finite Element Method to solve the equations. In this article, coupled thermoelasticity analysis of finite FGMcylinder is presented by using Green–Naghdi (GN) theory without energy dissipation. The gradation of properties through the thickness is assumed to be of power law type. The time history of radial displacement, temperature, radial and hoop stresses distribution across thickness of cylinder is obtained for various kinds of power law exponents in three sections across the length. The second sounds and elastic wave propagation are illustrated for two examples. Finally, the results are compared with the published data and there is a good agreement between them. 2. Governing equations The mechanical properties of a material cylinder such as Young’s modulus ‘‘E”, mass density ‘‘q”, thermal expansion coefficient ‘‘a”, thermal conductivity ‘‘K” and specific heat conduction ‘‘C” are graded across the thickness of a cylinder. The Poisson’s ratio is considered a constant value (m = 0.3). The elastic constants such as Young’s modulus ‘‘E”, mass density ‘‘q”, thermal expansion coefficient ‘‘a”, thermal conductivity ‘‘K” and specific heat conduction ‘‘C” of a functionally graded material can be shown by
P ¼ ðP out Pin Þ
r a n ba
þ Pin ;
ð1Þ
where ‘‘P” is material property, ‘‘n” is a non-negative volume fraction exponent and subscripts ‘‘in” and ‘‘out” stand for inner and outer surface mechanical properties (elastic constants). The inner and outer radii of the cylinder are ‘‘a” and ‘‘b”, and ‘‘r” is the radius of cylinder. The FGM cylinder is divided into ‘‘m” sub-cylinder (m layers) and each sub-cylinder is assumed to be isotropic. The number of layers depends directly to size of thickness and the maximum value of deference between elastic constants of each two layers that can be defined by researcher. The CPU time should be considered to determine the number of layers or the maximum value of deference between elastic constants of each two layers. The material properties of the Jth layer are found by
n ðJ 1Þh P ¼ ðP out Pin Þ þ P in ; ba
ð2Þ
the most important benefit of using power function with nonlinear volume fraction to show the variation of elastic constants in mathematical form. The FG cylinder with finite length ‘‘L”, inner radius ‘‘a” and outer radius ‘‘b” has been considered to analyze the problem in this paper. The inner and outer surfaces of cylinder can be assumed to be ceramic or metal which elastic constants vary continuously from inner to outer surface as a power function of radius with nonlinear volume fraction. The schematic of cylinder and functionally graded properties are shown in Fig. 1. The composition of ceramic and metal depends directly to volume fraction and exponent of it ‘‘n”. The governing equations of motion and heat conduction based on GN Theory of thermoelasticity, without energy dissipation at each isotropic sub-cylinder, are given as [18]
€ lr2 U þ ðk þ lÞrdiv U crT þ qQ ¼ qU; € ¼ qg_ þ k r2 T; cT€ þ cT 0 div U
ð5Þ
where ‘‘U” is the displacement vector, ‘‘T” is the temperature change above the uniform reference temperature ‘‘T0”, ‘‘Q” is the external _ is the external rate of supply of heat (both ‘‘Q” and force and ‘‘g” _ are not considered in this work). ‘‘q” is the mass density, ‘‘c” ‘‘g” is the specific heat, ‘‘k” and ‘‘l” are the Lame constants and
c ¼ ð3k þ 2lÞb ;
ð6Þ
*
*
where ‘‘b ” is the coefficient of volume expansion and ‘‘k ” is a material constant characteristic of the mentioned theory. To analyze the problem, the non-dimensional parameters are used as follows:
r z v ¼ 1 ðk1 þ 2l1 Þ U; r ¼ ; z ¼ ; t ¼ t; U l l l l c1 T 0 T r r r h r ¼ h ¼ T¼ ; r ; r ; T0 c1 T 0 c1 T 0
r z ¼
rz s rz ¼ rz ; ; s c1 T 0 c1 T 0
ð7Þ
where ‘‘l” is a standard length and ‘‘v” is a standard speed. The ‘‘k1”, ‘‘l1” and ‘‘c1” stand for first layer mechanical properties (mechanical properties of inner surface). The governing equation and heat transfer can be rewritten by using non-dimensional parameters as following:
€ þ ðC 2 C 2 Þrdiv U C 2 DT ¼ U; C 2s r2 U p s p € C 2 r2 T ¼ T€ þ e div U;
ð8Þ ð9Þ
T
where
where h is
ba h¼ : m
ð4Þ
ð3Þ
In the cylindrical structures such as tubes and cylindrical shells, the main heat transfer and heat conduction is usually across thickness (radial direction) of cylinder. In the high temperature applications, FGMs with radial variation in elastic constants are used to create thermal resistance ability across thickness of cylinders. There are some researches in heat transfer and thermoelasticity of FG structures which elastic constants vary in radial direction such as [2,4,17]. To state the variation of elastic constants in mathematical form, there are some models which the best of them is shown in Eqs. (1) and (2). This power function with nonlinear volume fraction has been used in the most of research in the thermoelasticity and dynamic analysis of FGMs field. In this model, the variations in elastic constants from ceramic to metal are modeled using various values of exponent ‘‘n”. For fixed ceramic and metal materials (for example alumina and aluminum), there are infinite kinds of variations (variation rates) from ceramic (alumina) to metal (aluminum) because of power function exponent ‘‘n” can vary from zero to infinity (in mathematical view). This phenomenon is
k þ 2l k ; C 2T ¼ ; q1 v2 q1 c1 v2 cT 0 c1 ¼ : q1 c1 ðk1 þ 2l1 Þ
C 2p ¼
e
C 2s ¼
l ; q1 v2 ð10Þ
Z b a
Outer surface material L
Inner surface material R
R
Fig. 1. The schematic of FG cylinder and variation of materials from inner to outer surfaces.
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The equations for axisymmetry and plane strain conditions can be obtained as follows:
The stresses and heat flux rate continuity conditions in terms of displacement and temperature are
u o2 u 1 ou o2 u þ C 2p þ C 2s 2 C 2p C 2s 2 2 r or or oz r 2 2 2 o w 2 oT € ; þ Cp Cs ¼u Cp or oroz " # 2 2 1 ou o2 u 2o w 2 1 ow 2o w 2 2 þ C p 2 þ ðC p C s Þ þ Cs 2 þ Cs r or or oz or oz r oz
ðk þ 2lÞk u ou kk ow kk c þ þ k T or ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1 ðk þ 2lÞkþ1 u c ou kkþ1 ow kkþ1 ¼ þ þ kþ1 T; or ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1
C 2p
oT € C 2p ¼ w; oz " # " # 2 € ow € u € 1 oT o2 T 2 o T € ou : ¼T þe þ þ þ þ CT or 2 r or oz2 or r oz
ð11Þ
lk
ou lk þ ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ ð12Þ
L ; l
ð14Þ
z ” is axial stress. The continuity conditions to be enforced where ‘‘r at any interface between two layers are written as
Þk ¼ ðu Þkþ1 ; ðu
ð15Þ
er
ou ¼ ; or
eh
u ¼ ; r
ez
ow ¼ ; oz
ou ow crz ¼ þ : oz or
ð16Þ ð17Þ ð18Þ ð19Þ ð20Þ
The non-dimensional stresses and heat flux rate in terms of non-dimensional displacement and temperature can be written as
ðr; z; tÞ ¼ u r ; z; tÞ ¼ wð r; z; tÞ ¼ Tð
ðk þ 2lÞ ow k ou k c u þ þ T; oz ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ or ðk1 þ 2l1 Þ c1
ð22Þ
ð23Þ
crz ¼ crz c1 T l ou l ow þ : ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ or
ðcp
ð30Þ
hn ðr ; tÞ sin bz;
ð31Þ
ð35Þ
c1 q1 m
0.8 n = 0.05 n = 0.2 n=1 n=5 t (nondimensional time) = 0.3 z/L = 0.3
0.6
0.5
0.4
0.3
0.2
0.1
The heat flux rate in Green–Naghdi Coupled Thermoelasticity Theory is
-0.1
oT : or
wn ðr; tÞ cos bz;
oF n oF n F n ðkkþ1 kk Þ ðkkþ1 kk Þ ¼ bwn Þ ðcp Þ r or k or kþ1 q1 v2 q1 v2 ðc ck Þðc1 þ 2l1 Þ kþ1 hn ; 2
0
q_ ¼ C 2T
n¼1 1 X
ð29Þ
The stresses and heat flux rate continuity conditions at any interface between two layers in the new form are
ð24Þ
¼
F n ðr ; tÞ sin bz;
n¼1 1 X
o2 F F oW oh € 2 1 oF 2 2 2 2 þ C b F ðC C Þ b C C 2p ¼ F; ð32Þ p s p s r or r 2 or 2 or or o2 w 1 ow oF F € C 2p bh ¼ w; ð33Þ C 2s 2 þ C 2s C 2s b2 w ðC 2p C 2s Þb þ r or or or r " # " # 2 € F€ 1 oh 2 o h 2 oF € € CT þ ð34Þ b h ¼hþe þ bw : or 2 r or or r
ð21Þ
rz z ¼r c1 T 0 ¼
ð28Þ
n¼1
Nondimensional Temperature
ðk þ 2lÞ ou k ow k c u þ þ T; r ðk1 þ 2l1 Þ or ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1
1 X
0.7
rh h ¼r c1 T 0 ¼
ð27Þ
The solutions which satisfy the boundary conditions (14) are
rr r ¼r c1 T 0 ðk þ 2lÞ ou k ow k c u ¼ þ þ T; or ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1
k
oT oT oT ¼ C 2T ! C 2T C 2T ¼ 0: or kþ1 or k or kþ1
ow ; or
C 2p
_ are r ”, ‘‘s rz ”, ‘‘q” The subscript ‘‘k” stands for kth layer and ‘‘r non-dimensional radial stress, shear stress, heat flux rate, respectively. The Hook’s law and strain–displacement relations are as following:
rr ¼ ðk þ 2lÞer þ keh þ kez ð3k þ 2lÞaT; rz ¼ ðk þ 2lÞez þ keh þ ker ð3k þ 2lÞaT; rh ¼ ðk þ 2lÞeh þ ker þ kez ð3k þ 2lÞaT; srz ¼ lcrz ;
lkþ1 ou þ oz ðk1 þ 2l1 Þ
where b ¼ np Ll After substituting Eqs. (29), (31) into Eqs. (11)–(13), the governing equations can be written in new form as
r Þk ¼ ðr r Þkþ1 ðr
k ¼ ðwÞ kþ1 ; ðs rz Þk ¼ ðs rz Þkþ1 ; ðwÞ _ k ¼ ðqÞ _ kþ1 : ðTÞk ¼ ðTÞkþ1 ; ðqÞ
oT C 2T or
ð13Þ
The parameters ‘‘u” and ‘‘w” are radial and axial displacements. The simply supported boundary conditions are taken as
¼r z ¼ 0 at z ¼ 0; u
lkþ1 ow ¼ or ðk1 þ 2l1 Þ
ð26Þ
ð25Þ
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Nondimensional Radial Distance
Fig. 2. Temperature distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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ow ow c2s c2s ¼ c2s F n k c2s F n kþ1 ; or kþ1 or k oh oh 2 oh 2 oh ðcT Þk ¼ ðcT Þkþ1 ! ðc2T Þk ðc2T Þkþ1 ¼ 0: or or or or
ð36Þ ð37Þ
each element by using the Galerkin Finite Element Method. The following dynamic finite element governing equation for each element can be obtained by using linear shape function as following:
F n ¼ N1 F in þ N2 F iþ1 n ; 3. Finite element modeling To solve the governing equations for functionally graded cylinder, each isotropic sub-cylinder is divided into many elements in radial direction. The mass and stiffness matrices are obtained for
wn ¼ N1 win þ N2 wiþ1 n ;
ð38Þ
hn ¼ N1 hin þ N2 hniþ1 ; € þ ½K / ¼ ½f ½M /
ð39Þ
e
e
e
with
0.8 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0.7 n=1 z/L = 0.3
0.6
0.04
0.035 n=1 z/L = 0.3
Nondimensional Radial Displacement
0.03 Nondimensional Temperature
t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0.5
0.4
0.3
0.2
0.1
0.025
0.02
0.015
0.01
0.005 0 0 -0.1
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5 -0.005
Fig. 3. Temperature distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
-4
x 10
0.5 n = 0.05 n = 0.2 n=1 n=5
14 t = 0.3 z/L = 0.3
x 10
n = 0.05 n = 0.2 n=1 n=5
0
Nondimensional Axial Displacement
12 Nondimensional Radial Displacement
1.05
Fig. 5. Radial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
-3
16
1
10
8
6
4
t = 0.3 z/L = 0.3
-0.5
-1
-1.5
-2
2 -2.5 0
-2
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 4. Radial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
-3
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 6. Axial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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h / ¼ F in
win
hin
F iþ1 n
wiþ1 n
iT hiþ1 ; n
ð40Þ
where ‘‘e” and ‘‘i” stand for eth element and ith node. The elements of force, mass and stiffness matrices are given in Appendix. The global dynamic equilibrium equations for FGM cylinders become
€ þ ½Kf/g ¼ ½f : ½Mf/g
ð41Þ
Once the finite element equilibrium equation is established, different numerical methods can be employed to solve them in space and time domains. The Newmark method with suitable time step is used and the equilibrium equation is solved.
4. Numerical examples and discussion The standard length ‘‘l” is determined equal to inner radius of FG cylinder ‘‘a” in the presented cases and the number of layers is 50 in non-dimensional value. The first numerical examples are carried out under the following conditions: Inner surface (aluminum),
am ¼ 23 106 1= C; cm ¼ 0:896 kJ=kg C;
r ¼ 1;
qm ¼ 2707 kg=m3 ; Em ¼ 70 Gpa and outer surface (alumina),
-4
2
x 10
0.1 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0
0
-0.1 Nondimensional Radial stress
Nondimensional Axial Displacement
-2
-4 n=1 z/L = 0.3 -6
-8
n=1 z/L = 0.3
-0.2
-0.3
-0.4 -10
-0.5
-12
-14
-0.6 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1
1.5
Fig. 7. Axial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
0.2 n = 0.05 n = 0.2 n=1 n=5
t = 0.3 z/L = 0.3
0.2
n = 0.05 n = 0.2 n=1 n=5
0
-0.2 Nondimensional Hoop stress
0
-0.2
-0.4
-0.6
-0.8
-0.8
-1
-1.2 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 8. Radial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
t = 0.3 z/L = 0.3
-0.4
-0.6
-1
1.05
Fig. 9. Radial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
0.4
Nondimensional Radial stress
t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 10. Hoop stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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r ¼ 1:5;
ac ¼ 7:4 106 1= C; cc ¼ 0:76 kJ kg C; 3
qc ¼ 3800 kg=m ; Ec ¼ 380 Gpa The boundary conditions for the first example are assumed as
q_ ¼ HðtÞ;
¼ 0; u
oT ¼ 0 ðisolatedÞ; or
¼ 0 at inner surface; w ¼ 0; u
¼ 0at outer surface; w
where H(t) is Heaviside function and q_ is the heat flux, q_ ¼ C 2T oT . or Fig. 2 shows the temperature distributions across the thickness of cylinder in t ¼ 0:3 for various values of power law exponents
‘‘n”. It can be seen that the temperature waves for big values of ‘‘n”, propagate faster than small values of ‘‘n”, because the cylinder shows further ceramic behavior for small values of ‘‘n”. There is a thermal resistance in ceramic materials. The peak point of diagram in Fig. 2 (inner surface) is decreased if the value of ‘‘n” is increased. The temperature of inner surface is increased when the material properties of FG cylinder tend to ceramic properties. It is also concluded that the thermal wave for big values of ‘‘n” is propagated faster than small values of ‘‘n”. The time history of temperature wave propagation for n = 1 is illustrated in Fig. 3. Fig. 4 shows the radial displacement waves across the thickness of cylinder in
0.1 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0.1 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0
-0.1
Nondimensional Axial stress
Nondimensional Hoop stress
-0.1
0
-0.2 n=1 z/L = 0.3
-0.3
-0.4
-0.2 n=1 z/L = 0.3
-0.3
-0.4
-0.5
-0.5 -0.6 -0.6 -0.7 -0.7 -0.8
1
1.05
1.1
-0.8 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
Fig. 11. Hoop stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Nondimensional Radial Distance
1.5
Fig. 13. Axial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
-3
2
0.2 n = 0.05 n = 0.2 n=1 n=5
0
x 10
n = 0.05 n = 0.2 n=1 n=5
1.5 t = 0.3 z/L = 0.3 1
Nondimensional Shear stress
Nondimensional Axial stress
-0.2
t = 0.3 z/L = 0.3
-0.4
-0.6
0.5
0
-0.5
-1
-0.8 -1.5 -1 -2
-1.2
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 12. Axial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
-2.5
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 14. Shear stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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t ¼ 0:3 for various values of ‘‘n”. The radial displacement wave propagation for small value of ‘‘n” is slower than big values of ‘‘n” because the ceramic properties for small values of ‘‘n” are further than big values of ‘‘n”. The radial displacement wave propagations for n = 1 are shown in Fig. 5. The behaviors of axial displacement across thickness of cylinder for various ‘‘n” are shown in Fig. 6 and the time history of axial displacement can be seen in Fig. 7. Figs. 8–15 show the dynamic behaviors of radial, hoop, axial and shear stresses for various values of ‘‘n” and various instants of time. The wave propagations can be seen for all stresses. The velocity of wave propagation depends on mechanical properties and especially on the exponent of mechanical properties
power function ‘‘n”. Figs. 10–13 show the same behaviors for hoop and axial stresses. The reason of similarity can be found according to Eqs. (22) and (23) and also Figs. 5 and 7. The slopes of diagrams ou in Fig. 5 for most points across the thickness are bigger than or the values of parameters ur and oowz . In Eqs. (22) and (23), the coefficients of parameter oour are the same. The effects of parameters ur and ow on the values of hoop and axial stresses are very small against oz parameter oour . Therefore the values of hoop and axial stresses depend on parameter oour and their coefficients, approximately. Figs. 2–15 are valid for section z/L = 0.3 of the cylinder. The presented method and results in this article could be validated with the published results reported by Taheri et al. [19]. To
-3
-3
x 10
3
10 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
2
Taheri et al Present 8 t = 0.4 Nondimensional Radial Displacement
Nondimensional Shear stress
1
0 n=1 z/L = 0.3
-1
x 10
-2
-3
z/L = 0.5 n=0
6
4
t = 0.2 2
-4 0 -5
-6
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
-2
1.5
Fig. 15. Shear stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1
1.05
1.1
1.15
1.2 1.25 1.3 Nondimensional Radius
1.35
1.4
1.45
1.5
Fig. 17. The comparison of radial displacement distribution for various ‘‘t” in z/L = 0.5 and n = 0 with Ref. [19].
0.4 Taheri et al Present 0.35
0
Present Taheri el al t = 0.2
-0.05
0.25
Nondimensional Radial Stress
0.3
Nondimensional Temperature
z/L = 0.5 n=0
z/L = 0.5 n=0
t = 0.4
0.2
0.15 t = 0.2
0.1
-0.1
t = 0.4
-0.15
-0.2 0.05
0
-0.05
-0.25
1
1.05
1.1
1.15
1.2 1.25 1.3 1.35 Nondimensional Radius
1.4
1.45
1.5
Fig. 16. The comparison of temperature distribution for various ‘‘t” in z/L = 0.5 and n = 0 with Ref. [19].
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 18. The comparison of radial stress distribution for various ‘‘t” in z/L = 0.5 and n = 0 with Ref. [19].
2018
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
verify the results, the power law exponent ‘‘n” is considered to be zero in the first example and the material of inner surface is assumed to be the same material as in Ref. [19]. Figs. 16–19 show the temperature, radial displacement, radial and hoop stresses wave propagation at the section z/L = 0.5 of cylinder for the present work and the published data of Ref. [19]. In the section z/L = 0.5 of the cylinder (middle section of the cylinder across the length), the plane strain conditions are valid. There is a good agreement between the present results and those obtained in Ref. [19]. For second example, the following boundary conditions are assumed for the cylinder:
¼ 0; w ¼ 0 at inner surface; T ¼ 100ð1 e10;000t Þ; u oT ¼ 0; w ¼ 0 at outer surface ¼ 0 ðisolatedÞ; u or and the inner surface is considered to be made of ceramic (alumina) and outer surface of metal (aluminum). In this example, the results are presented in the section z/L = 0.7 of the cylinder. Fig. 20 shows the non-dimensional temperature distribution across the thickness of the cylinder in t ¼ 0:1 for various values of power law exponents. For small values of ‘‘n”, the wave is propagated faster than big values of ‘‘n” because the material properties of the cylinder converge
0.05
100
Present z/L = 0.5 n=0
0
t = 0.05 t = 0.1 t = 0.15 t = 0.2
Taheri el al n=1 z/L = 0.7
80 -0.05 60
Nondimensional Temperature
Nondimensional Hoop Stress
t = 0.2 -0.1 t = 0.4 -0.15
-0.2
40
20
-0.25 0 -0.3
-0.35 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 19. The comparison of hoop stress distribution for various ‘‘t” in z/L = 0.7 and n = 0 with Ref. [19].
-20 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 21. Temperature distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
100
5 n = 0.05 n = 0.2 n=1 n=5
80
n = 0.05 n = 0.2 n=1 n=5
4
Nondimensional Radial Displacement
Nondimensional Temperature
60
40 t = 0.1 z/L = 0.7 20
0
t = 0.1 z/L = 0.7 2
1
0
-20
-40 0.75
3
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 20. Temperature distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
-1 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 22. Radial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
2019
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
to ceramic properties for big values of ‘‘n”. The temperature wave propagation for n = 1 can be seen in Fig. 21. The velocity of wave is different in each point of thickness because the material properties are different in these points. The behavior of radial displacement for various values of ‘‘n” and the time history of radial displacement across the thickness of the FG cylinder can be seen in Figs. 22 and 23 for second example. The radial displacement wave propagation is illustrated in Fig. 23. The radial displacement wave propagation velocities for small values of ‘‘n” are faster than those of big values of ‘‘n” be-
cause the mechanical properties of FG cylinder are close to mechanical properties of ceramic for big values of ‘‘n”. The time history of axial displacement and the behavior of axial displacement across the thickness of cylinder for various ‘‘n” are observed in Figs. 24 and 25. The axial displacement wave propagation can be seen in Fig. 25. Figs. 26–33 illustrate the variation of radial, hoop, axial and shear stresses across the thickness of cylinder for various ‘‘n” and various instants of time. The stresses wave propagations and the effect of exponent ‘‘n” on the stresses behavior are shown in these
4
0.05 t = 0.05 t = 0.1 t = 0.15
t = 0.05 t = 0.1 t = 0.15
n=1 z/L = 0.7
3 Nondimensional Axial Displacement
Nondimensional Radial Displacement
0.04
n=1 z/L = 0.7 2
1
0.03
0.02
0.01
0
0 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
-0.01 0.75
1
Fig. 23. Radial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
0.8
0.95
1
Fig. 25. Axial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
0.08
150 n = 0.05 n = 0.2 n=1 n=5
t = 0.1 z/L = 0.7
0.07
n = 0.05 n = 0.2 n=1 n=5
100 t = 0.1 z/L = 0.7 Nondimensional Radial Stress
0.06
Nondimensional Axial Displacement
0.85 0.9 Nondimensional Radial Distance
0.05
0.04
0.03
0.02
50
0
-50
0.01 -100 0
-0.01 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 24. Axial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
-150 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 26. Radial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
2020
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023 20
20 t = 0.05 t = 0.1 t = 0.15 t = 0.2
n=1 z/L = 0.7 0
t = 0.05 t = 0.1 t = 0.15 t = 0.2
0
Nondimensional Hoop Stress
Nondimensional Radial Stress
-20 -20
-40
-60
-40
n=1 z/L = 0.7
-60
-80
-80
-100 0.75
-100
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 27. Radial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
-120 0.75
0.95
1
20 n = 0.05 n = 0.2 n=1 n=5
0
n = 0.05 n = 0.2 n=1 n=5
0
-20 Nondimensional Axial Stress
-20 Nondimensional Hoop Stress
0.85 0.9 Nondimensional Radial Distance
Fig. 29. Hoop stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
20
-40 t = 0.1 z/L = 0.7
-60
-80
-40 t = 0.1 z/L = 0.7
-60
-80
-100
-100
-120
-120
-140 0.75
0.8
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 28. Hoop stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
-140 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 30. Axial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
5. Conclusions
be multilayer cylinder across the thickness. Material properties in each layer are assumed to be constant and functionally graded properties are resulted by suitable arrangements of the layers in a multilayer cylinder. The results of this procedure can be outlined as:
Coupled thermoelasticity without energy dissipation of finite length functionally graded thick hollow cylinders is studied by using Green–Naghdi theory. The calculations of the mechanical displacements, temperature distribution and thermal stresses have been provided by a hybrid numerical solution (Galerkin and Newmark methods). To analyze the problem, the cylinder is assumed to
1. The coupled thermoelasticity without energy dissipation in functionally graded thick hollow cylinder is studied and developed by using Green–Naghdi theory. The wave propagations are analyzed and the time history of temperature, mechanical displacement and thermal stresses are discussed for various values of power law exponents.
figures. The behaviors of hoop and axial stresses are similar and their reason was described in the first example.
2021
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023 20
0.2 t = 0.05 t = 0.1 t = 0.15 t = 0.2
0
0.1
0
-40
Nondimensional Shear Stress
Nondimensional Axial Stress
-20
n=1 z/L = 0.7
-60
-0.1
-0.3
-100
-0.4
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
0.2
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 33. Shear stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
The components of mass matrices for each element
m11 ¼ a5 ;
m12 ¼ 0;
m13 ¼ 0;
m14 ¼ a6 ;
m22 ¼ a5 ;
m23 ¼ 0;
m24 ¼ 0;
m15 ¼ 0;
m16 ¼ 0;
-0.2
Nondimensional Shear Stress
0.8
Appendix
n = 0.05 n = 0.2 n=1 n=5
0
-0.5 0.75
1
Fig. 31. Axial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
n=1 z/L = 0.7
-0.2
-80
-120 0.75
t = 0.05 t = 0.1 t = 0.15
-0.4
m21 ¼ 0;
t = 0.1 z/L = 0.7
m25 ¼ a6 ;
m26 ¼ 0;
-0.6
m31 ¼ -0.8
e a13 R
m34 ¼
-1
e a13 R
m41 ¼ a6 ;
e R 3
m32 ¼ e ba5 ;
;
e a8 ;
m33 ¼ a5 ;
m35 ¼ e ba6 ;
m36 ¼ a6 ;
m42 ¼ 0;
m43 ¼ 0;
m44 ¼ a11 ;
m52 ¼ a6 ;
m53 ¼ 0;
m54 ¼ 0;
m45 ¼ 0;
m46 ¼ 0; -1.2
m51 ¼ 0; -1.4 0.75
0.8
0.85
0.9
0.95
1
Nondimensional Radial Distance Fig. 32. Shear stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
m61 ¼
e a14
m64 ¼ 2. The phenomenon of stress wave front is observed due to the theory of elasticity and the governing equations are solved by general elasticity solutions. 3. Results for axisymmetric thermal shock loads applied to the inner surface are presented for finite length functionally graded thick hollow cylinder with simply supported edges. The behaviors of functionally graded thick hollow cylinders are determined and discussed for inner surface temperature shock and Heaviside heat flux of inner surface. 4. The employed method in this paper based on the achieved results can be used in analyzing of vibration, dynamic behavior, coupled and uncoupled thermoelasticity of functionally graded materials.
m55 ¼ a11 ;
m56 ¼ 0;
R
e a8 ;
e a14 R
e R 3
m62 ¼ e ba6 ; ;
m63 ¼ a6 ;
m65 ¼ e ba11 ;
m66 ¼ a11 ;
where R = r{i+1} ri. For simplicity, we assumed that r ¼ r in the Appendix. The components of stiffness matrices for each element
c2p ri c2p c2s b2 a5 c2p c2s a1 ; R 2 c2p 2 2 ba13 ¼ cp cs ; k13 ¼ a13 ; R R
k11 ¼ k12
c2p r i c2p þ c2s b2 a6 c2p c2s a2 ; R 2 ba c2p 13 2 ¼ cp c2s ; k16 ¼ a13 ; R R
k14 ¼ k15
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
b bR k21 ¼ c2p c2s a13 þ c2p c2s ; R 3
k22 ¼
k24 k26
c2s r i R
c2s 2
a6 ¼
k23 ¼ a5 c2p b;
b ¼ c2p c2s a13 þ c2p c2s ba8 ; R ¼ a6 c2p b;
k25
c2 r i c2 ¼ s þ s c2p b2 a6 ; R 2
a7 ¼
k32 ¼ 0;
k34 ¼ 0;
k41 k42
k35 ¼ 0;
k33 ¼ k36
N1 N 2 r dr
1 r iþ1 r i R ðr iþ1 þ r i Þ 3 ðr iþ1 þ ri Þ þ r iþ1 r 3i r 4iþ1 r4i ; 2 3 4 R 1
2
Z
r iþ1
ri
a8 ¼ k31 ¼ 0;
r iþ1
ri
¼ c2p b2 a5 ;
Z
Z
r iþ1
N1 N 2 dr ¼
ri
c2T r i c2T c2T b2 a5 ; R 2
a9 ¼
c2 r i c2 ¼ T þ T c2T b2 a6 ; R 2
Z
r iþ1
ri
c2p r iþ1 c2p ¼ c2s b2 a6 c2p c2s a2 ; R 2 c2p 2 2 ba14 ¼ cp cs ; k43 ¼ a14 ; R R
a10 ¼
Z
riþ1
ri
c2p r iþ1 c2p þ c2s b2 a11 c2p c2s a9 ; R 2 ba c2p 14 ¼ c2p c2s ; k46 ¼ a14 ; R R
¼
k44 ¼
a12 ¼
b k51 ¼ c2p c2s a14 þ c2p c2s ba8 ; R c2s r iþ1 c2s k52 ¼ c2p b2 a6 ; k53 ¼ a6 c2p b; R 2
k55
a13 ¼
2 1 ri R 2r i 3 ðr iþ1 þ ri Þ r iþ1 r 3i þ r 4iþ1 r4i ; 4 3 2 R 2
Z
riþ1
k62 ¼ 0;
k64 ¼ 0;
k65 ¼ 0;
k63 k66
a14 ¼
Z
Z
Z
r iþ1
r iþ1
ri
Z
r iþ1
ri
a5 ¼
Z ri
¼
1 R2
r iþ1
riþ1
ri
ri
a4 ¼
N1 r dr ¼
1 R 1 r iþ1 ðr i þ r iþ1 Þ r 3iþ1 r 3i ; R 2 3
N1 r dr ¼
1 R 1 r i ðr i þ r iþ1 Þ þ r3iþ1 r 3i : R 2 3
riþ1
Z
riþ1
Force matrices for each element
oF ; or r¼ri oF ¼ c2p r i or
c2 r iþ1 c2T ¼ T c2T b2 a6 ; R 2
f11 ¼ c2p ri
c2 r iþ1 c2T ¼ T þ c2T b2 a11 ; R 2
f 41
f 21 ¼ c2s r i
r¼riþ1
;
f 51
ow or
; r¼r i
f 31 ¼ c2T r i
ow ¼ c2s r i ; or r¼riþ1
oh or
;
r¼r i
f 61 ¼ c2T r i
oh : or r¼riþ1
References
alpha1 ¼
a3 ¼
R ; 3
ri
where
a2 ¼
Z
N22 dr ¼
ri
b ¼ c2p c2s a14 þ c2p c2s bR=3; R c2s r iþ1 c2s ¼ þ c2p b2 a11 ; k56 ¼ a11 c2p b; R 2
k61 ¼ 0;
N22 r dr
1
ri
k54
N22 1 r iþ1 ri dr ¼ R 2r ln þ R i r2 ri r iþ1 R2
riþ1
Z
R 1 3 2 3 ð þ r Þ r r R ; r r r iþ1 i i iþ1 iþ1 i 3 R2 2 1
N 22 1 r iþ1 R 2r i R þ ðr iþ1 þ r i Þ ; dr ¼ 2 r2i ln 2 r ri R
ri
a11 ¼
k45
R ; 3
N21 dr ¼
N21 1 r iþ1 R 2riþ1 R þ ðriþ1 þ r i Þ ; dr ¼ 2 r 2iþ1 ln 2 r ri R
N1 N2 1 r iþ1 R þ ðriþ1 þ r i Þ ; dr ¼ 2 r iþ1 ri ln 2 r ri R
N21 1 r iþ1 r iþ1 ; dr ¼ 2r ln þ R þ R iþ1 r2 ri ri R2
N1 N2 1 r iþ1 dr ¼ ð r þ r Þ ln 2R iþ1 i r2 ri R2 N21 rdr
1 r2iþ1 R 2r iþ1 3 ðr iþ1 þ ri Þ r iþ1 r3i þ r4iþ1 r 4i 4 3 2
[1] Fukui Y, Yamanaka N. Elastic analysis for thick-walled tubes of functionally graded material subjected to internal pressure. JSME Int J, Series I 1992;35(4). [2] Praveen GN, Reddy JN. Nonlinear transient thermoelastic analysis of functionally graded ceramic–metal plates. Int J Solids Struct 1998;35: 4457–76. [3] Zimmerman RW, Lutz MP. Thermal stress and thermal expansion in a uniformly heated functionally graded cylinder. J Thermal Stress 1999;22:88–177. [4] Ootao Y, Tanigawa Y. Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply. Int J Mech Sci 2005;47:1769–88. [5] Hosseini SM, Akhlaghi M, Shakeri M. Transient heat conduction in functionally graded thick hollow cylinder (by analytical method). Heat Mass Transfer 2007;3:669–75. [6] Han x, Liu GR. Effects of SH waves in a functionally graded plate. Mech Res Commun 2002;29:327–38. [7] Chakraborty A, Gopalakrishnan S. A spectrally formulated finite element for wave propagation analysis in functionally graded beams. Int J Solids Struct 2003;40:2421–48. [8] Berezovski A, Engelbrecht J, Maugin GA. Numerical simulation of twodimensional wave propagation in functionally graded materials. Eur J Mechan A/Solids 2003;22:257–65. [9] Han X, Liu GR, Lam KY. Transient waves in plates of functionally graded material. Int J Numer Methods Eng 2001;52:851–65.
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023 [10] Han X, Liu GR, Xi ZC, Lam KY. Transient waves in a functionally graded cylinder. Int J Solids Struct 2001;38:3021–37. [11] Han X, Liu GR, Xi ZC, Lam KY. Characteristics of waves in a functionally graded cylinder. Int J Numer Methods Eng 2002;53(3):653–76. [12] Han X, Liu GR. Elastic waves propagation in a functionally graded piezoelectric cylinder. Smart Mater Struct 2003;12(6):962–71. [13] Vollmann J, Profunser DM, Bryner J, Dual J. Elastodynamic wave propagation in graded materials: simulations, experiments, phenomena, and applications. Ultrasonics 2006;44:e1215–21. [14] Hosseini SM, Akhlaghi M, Shakeri M. Dynamic response and radial wave propagation velocity in thick hollow cylinder made of functionally graded materials. Eng Comput 2007;24:288–303.
2023
[15] Zhang GM, Batra Rc. Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method. J Comput Phys 2007;222:374–90. [16] Du J, Jin X, Wang J, Xian K. Love wave propagation in functionally garded piezoelectric material layer. Ultrasonics 2007;46:13–22. [17] Bahtui A, Eslami MR. Coupled thermoelasticity of functionally graded cylindrical shells. Mech Res Commun 2007;34:1–18. [18] Green AE, Naghdi PM. Thermoelasticity without Energy Dissipation. J Elasticity 1993;31:189–208. [19] Taheri H, Fariborz S, Eslami MR. Thermoelastic analysis of an annulus using the Green–Naghdi model. J Thermal Stresses 2005;28(9):911–27.
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Materials and Design journal homepage: www.elsevier.com/locate/matdes
Coupled thermoelasticity and second sound in finite length functionally graded thick hollow cylinders (without energy dissipation) Seyed Mahmoud Hosseini * Department of Mechanical Engineering, Khorasan Research Institute of Sciences and Food Technology, P.O. Box 91735-139, Mashhad, Iran Department of Industrial Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 13 May 2008 Accepted 30 August 2008 Available online 17 September 2008 Keywords: Coupled thermoelasticity Functionally graded materials Finite length cylinders Thermal shock
a b s t r a c t In this article, the coupled thermoelasticity behavior of functionally graded thick hollow cylinders is studied. The governing coupled thermoelasticity and the energy equations are solved for a finite length functionally graded cylinder subjected to thermal shock load. The coupled thermoelastic equations are considered based on Green–Naghdi theory. The mechanical properties of cylinder are graded across the thickness as a power law function of radius. The cylinder is assumed to be made of many isotropic sub-cylinders (layers) across the thickness. Functionally graded properties are created by suitable arrangement of layers and governing equations are expanded in longitudinal direction by means of trigonometric function expansion. The Galerkin Finite Element and Newmark Methods are used to analyze the cylinder. The dynamic behavior of temperature distribution, mechanical displacement and thermal stresses is obtained and discussed. The second sound and elastic wave propagation are determined for various kinds of variation in the mechanical properties. The comparison of present results with published data shows the excellent agreement. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Functionally graded materials (FGMs) are two-phase particulate composites in which material composition and microstructure vary continuously from one surface to the other especially from ceramic to metal. The volume fractions of the FGM constituents are gradually varied by power law or other functions of position. In the recent years, there are some researches on dynamic analysis, transient thermoelasticity and wave propagation of functionally graded structures. A number of these works have been carried out in the steady-state and transient thermoelasticity [1–3]. The theoretical treatment of the steady-state thermoelastic problem of a functionally graded cylindrical panel due to non-uniform heat supply in the circumferential direction was carried out by Ootao and Tanigawa [4]. They obtained the exact solution for the two-dimensional temperature change in a steady-state, and thermal stresses of a simple supported cylindrical panel under the state of plane strain. Hosseini et al. [5] presented an analytical method to determine the transient temperature distribution of functionally graded thick hollow cylinders. To obtain the analytical solution, they used the composition of Bessel functions. Han and Liu [6] presented a computational method to investigate SH wave in functionally graded material plates.
* Address: Department of Mechanical Engineering, Khorasan Research Institute of Sciences and Food Technology, P.O. Box 91735-139, Mashhad, Iran. Tel.: +98 912 190 5942; fax: +98 511 5003150. E-mail address: [email protected] 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.08.048
The material properties were assumed to be a quadratic function in the thickness direction. A novel spectral element was presented to study the wave propagation behavior in FGM beams subjected to high frequency impact loads by Chakraborty and Gopalakrishnan [7]. The propagation of stress waves in functionally graded materials was studied numerically by means of the theory of laminated composite wave-propagation algorithm by Berezovski et al. [8]. There are some works on wave propagation problems by using multilayer modeling of FGM. The better rule-of-mixture called as SBS method for FGMs has been used in FG plates [9], FG cylinders [10,11] and functionally graded piezoelectric material (FGPM) cylinders [12]. A two-dimensional numerical simulation model for the elastodynamic wave propagation in two linear elastic, isotropic, joint half-spaces was presented by Vollmann et al. [13]. They considered that the border between the two half-spaced was graded in a way, which the values of elastic properties and the densities vary smoothly from one half-space to the other. The vibration and dynamic analysis of functionally graded cylinders was studied by Hosseini et al. [14]. The mean velocity of radial stresses wave propagation, natural frequency and dynamic behavior of cylinder were presented in their work. The modified smoothed particle hydrodynamics (MSPH) method to study the propagation of elastic waves in functionally graded materials was studied by Zhang and Batra [15]. In their work, an artificial viscosity is added to the hydrostatic pressure to control oscillations in the shock wave. Du et al. [16] presented an exact approach to investigate Love waves in FGPM layer bonded to a semi-infinite homogeneous solid. The coupled
2012
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
thermoelastic and the energy equations were simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load by Bahtui and Eslami [17]. They used classic theory of coupled thermoelasticity and Galerkin Finite Element Method to solve the equations. In this article, coupled thermoelasticity analysis of finite FGMcylinder is presented by using Green–Naghdi (GN) theory without energy dissipation. The gradation of properties through the thickness is assumed to be of power law type. The time history of radial displacement, temperature, radial and hoop stresses distribution across thickness of cylinder is obtained for various kinds of power law exponents in three sections across the length. The second sounds and elastic wave propagation are illustrated for two examples. Finally, the results are compared with the published data and there is a good agreement between them. 2. Governing equations The mechanical properties of a material cylinder such as Young’s modulus ‘‘E”, mass density ‘‘q”, thermal expansion coefficient ‘‘a”, thermal conductivity ‘‘K” and specific heat conduction ‘‘C” are graded across the thickness of a cylinder. The Poisson’s ratio is considered a constant value (m = 0.3). The elastic constants such as Young’s modulus ‘‘E”, mass density ‘‘q”, thermal expansion coefficient ‘‘a”, thermal conductivity ‘‘K” and specific heat conduction ‘‘C” of a functionally graded material can be shown by
P ¼ ðP out Pin Þ
r a n ba
þ Pin ;
ð1Þ
where ‘‘P” is material property, ‘‘n” is a non-negative volume fraction exponent and subscripts ‘‘in” and ‘‘out” stand for inner and outer surface mechanical properties (elastic constants). The inner and outer radii of the cylinder are ‘‘a” and ‘‘b”, and ‘‘r” is the radius of cylinder. The FGM cylinder is divided into ‘‘m” sub-cylinder (m layers) and each sub-cylinder is assumed to be isotropic. The number of layers depends directly to size of thickness and the maximum value of deference between elastic constants of each two layers that can be defined by researcher. The CPU time should be considered to determine the number of layers or the maximum value of deference between elastic constants of each two layers. The material properties of the Jth layer are found by
n ðJ 1Þh P ¼ ðP out Pin Þ þ P in ; ba
ð2Þ
the most important benefit of using power function with nonlinear volume fraction to show the variation of elastic constants in mathematical form. The FG cylinder with finite length ‘‘L”, inner radius ‘‘a” and outer radius ‘‘b” has been considered to analyze the problem in this paper. The inner and outer surfaces of cylinder can be assumed to be ceramic or metal which elastic constants vary continuously from inner to outer surface as a power function of radius with nonlinear volume fraction. The schematic of cylinder and functionally graded properties are shown in Fig. 1. The composition of ceramic and metal depends directly to volume fraction and exponent of it ‘‘n”. The governing equations of motion and heat conduction based on GN Theory of thermoelasticity, without energy dissipation at each isotropic sub-cylinder, are given as [18]
€ lr2 U þ ðk þ lÞrdiv U crT þ qQ ¼ qU; € ¼ qg_ þ k r2 T; cT€ þ cT 0 div U
ð5Þ
where ‘‘U” is the displacement vector, ‘‘T” is the temperature change above the uniform reference temperature ‘‘T0”, ‘‘Q” is the external _ is the external rate of supply of heat (both ‘‘Q” and force and ‘‘g” _ are not considered in this work). ‘‘q” is the mass density, ‘‘c” ‘‘g” is the specific heat, ‘‘k” and ‘‘l” are the Lame constants and
c ¼ ð3k þ 2lÞb ;
ð6Þ
*
*
where ‘‘b ” is the coefficient of volume expansion and ‘‘k ” is a material constant characteristic of the mentioned theory. To analyze the problem, the non-dimensional parameters are used as follows:
r z v ¼ 1 ðk1 þ 2l1 Þ U; r ¼ ; z ¼ ; t ¼ t; U l l l l c1 T 0 T r r r h r ¼ h ¼ T¼ ; r ; r ; T0 c1 T 0 c1 T 0
r z ¼
rz s rz ¼ rz ; ; s c1 T 0 c1 T 0
ð7Þ
where ‘‘l” is a standard length and ‘‘v” is a standard speed. The ‘‘k1”, ‘‘l1” and ‘‘c1” stand for first layer mechanical properties (mechanical properties of inner surface). The governing equation and heat transfer can be rewritten by using non-dimensional parameters as following:
€ þ ðC 2 C 2 Þrdiv U C 2 DT ¼ U; C 2s r2 U p s p € C 2 r2 T ¼ T€ þ e div U;
ð8Þ ð9Þ
T
where
where h is
ba h¼ : m
ð4Þ
ð3Þ
In the cylindrical structures such as tubes and cylindrical shells, the main heat transfer and heat conduction is usually across thickness (radial direction) of cylinder. In the high temperature applications, FGMs with radial variation in elastic constants are used to create thermal resistance ability across thickness of cylinders. There are some researches in heat transfer and thermoelasticity of FG structures which elastic constants vary in radial direction such as [2,4,17]. To state the variation of elastic constants in mathematical form, there are some models which the best of them is shown in Eqs. (1) and (2). This power function with nonlinear volume fraction has been used in the most of research in the thermoelasticity and dynamic analysis of FGMs field. In this model, the variations in elastic constants from ceramic to metal are modeled using various values of exponent ‘‘n”. For fixed ceramic and metal materials (for example alumina and aluminum), there are infinite kinds of variations (variation rates) from ceramic (alumina) to metal (aluminum) because of power function exponent ‘‘n” can vary from zero to infinity (in mathematical view). This phenomenon is
k þ 2l k ; C 2T ¼ ; q1 v2 q1 c1 v2 cT 0 c1 ¼ : q1 c1 ðk1 þ 2l1 Þ
C 2p ¼
e
C 2s ¼
l ; q1 v2 ð10Þ
Z b a
Outer surface material L
Inner surface material R
R
Fig. 1. The schematic of FG cylinder and variation of materials from inner to outer surfaces.
2013
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
The equations for axisymmetry and plane strain conditions can be obtained as follows:
The stresses and heat flux rate continuity conditions in terms of displacement and temperature are
u o2 u 1 ou o2 u þ C 2p þ C 2s 2 C 2p C 2s 2 2 r or or oz r 2 2 2 o w 2 oT € ; þ Cp Cs ¼u Cp or oroz " # 2 2 1 ou o2 u 2o w 2 1 ow 2o w 2 2 þ C p 2 þ ðC p C s Þ þ Cs 2 þ Cs r or or oz or oz r oz
ðk þ 2lÞk u ou kk ow kk c þ þ k T or ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1 ðk þ 2lÞkþ1 u c ou kkþ1 ow kkþ1 ¼ þ þ kþ1 T; or ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1
C 2p
oT € C 2p ¼ w; oz " # " # 2 € ow € u € 1 oT o2 T 2 o T € ou : ¼T þe þ þ þ þ CT or 2 r or oz2 or r oz
ð11Þ
lk
ou lk þ ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ ð12Þ
L ; l
ð14Þ
z ” is axial stress. The continuity conditions to be enforced where ‘‘r at any interface between two layers are written as
Þk ¼ ðu Þkþ1 ; ðu
ð15Þ
er
ou ¼ ; or
eh
u ¼ ; r
ez
ow ¼ ; oz
ou ow crz ¼ þ : oz or
ð16Þ ð17Þ ð18Þ ð19Þ ð20Þ
The non-dimensional stresses and heat flux rate in terms of non-dimensional displacement and temperature can be written as
ðr; z; tÞ ¼ u r ; z; tÞ ¼ wð r; z; tÞ ¼ Tð
ðk þ 2lÞ ow k ou k c u þ þ T; oz ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ or ðk1 þ 2l1 Þ c1
ð22Þ
ð23Þ
crz ¼ crz c1 T l ou l ow þ : ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ or
ðcp
ð30Þ
hn ðr ; tÞ sin bz;
ð31Þ
ð35Þ
c1 q1 m
0.8 n = 0.05 n = 0.2 n=1 n=5 t (nondimensional time) = 0.3 z/L = 0.3
0.6
0.5
0.4
0.3
0.2
0.1
The heat flux rate in Green–Naghdi Coupled Thermoelasticity Theory is
-0.1
oT : or
wn ðr; tÞ cos bz;
oF n oF n F n ðkkþ1 kk Þ ðkkþ1 kk Þ ¼ bwn Þ ðcp Þ r or k or kþ1 q1 v2 q1 v2 ðc ck Þðc1 þ 2l1 Þ kþ1 hn ; 2
0
q_ ¼ C 2T
n¼1 1 X
ð29Þ
The stresses and heat flux rate continuity conditions at any interface between two layers in the new form are
ð24Þ
¼
F n ðr ; tÞ sin bz;
n¼1 1 X
o2 F F oW oh € 2 1 oF 2 2 2 2 þ C b F ðC C Þ b C C 2p ¼ F; ð32Þ p s p s r or r 2 or 2 or or o2 w 1 ow oF F € C 2p bh ¼ w; ð33Þ C 2s 2 þ C 2s C 2s b2 w ðC 2p C 2s Þb þ r or or or r " # " # 2 € F€ 1 oh 2 o h 2 oF € € CT þ ð34Þ b h ¼hþe þ bw : or 2 r or or r
ð21Þ
rz z ¼r c1 T 0 ¼
ð28Þ
n¼1
Nondimensional Temperature
ðk þ 2lÞ ou k ow k c u þ þ T; r ðk1 þ 2l1 Þ or ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1
1 X
0.7
rh h ¼r c1 T 0 ¼
ð27Þ
The solutions which satisfy the boundary conditions (14) are
rr r ¼r c1 T 0 ðk þ 2lÞ ou k ow k c u ¼ þ þ T; or ðk1 þ 2l1 Þ r ðk1 þ 2l1 Þ oz ðk1 þ 2l1 Þ c1
k
oT oT oT ¼ C 2T ! C 2T C 2T ¼ 0: or kþ1 or k or kþ1
ow ; or
C 2p
_ are r ”, ‘‘s rz ”, ‘‘q” The subscript ‘‘k” stands for kth layer and ‘‘r non-dimensional radial stress, shear stress, heat flux rate, respectively. The Hook’s law and strain–displacement relations are as following:
rr ¼ ðk þ 2lÞer þ keh þ kez ð3k þ 2lÞaT; rz ¼ ðk þ 2lÞez þ keh þ ker ð3k þ 2lÞaT; rh ¼ ðk þ 2lÞeh þ ker þ kez ð3k þ 2lÞaT; srz ¼ lcrz ;
lkþ1 ou þ oz ðk1 þ 2l1 Þ
where b ¼ np Ll After substituting Eqs. (29), (31) into Eqs. (11)–(13), the governing equations can be written in new form as
r Þk ¼ ðr r Þkþ1 ðr
k ¼ ðwÞ kþ1 ; ðs rz Þk ¼ ðs rz Þkþ1 ; ðwÞ _ k ¼ ðqÞ _ kþ1 : ðTÞk ¼ ðTÞkþ1 ; ðqÞ
oT C 2T or
ð13Þ
The parameters ‘‘u” and ‘‘w” are radial and axial displacements. The simply supported boundary conditions are taken as
¼r z ¼ 0 at z ¼ 0; u
lkþ1 ow ¼ or ðk1 þ 2l1 Þ
ð26Þ
ð25Þ
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Nondimensional Radial Distance
Fig. 2. Temperature distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
ow ow c2s c2s ¼ c2s F n k c2s F n kþ1 ; or kþ1 or k oh oh 2 oh 2 oh ðcT Þk ¼ ðcT Þkþ1 ! ðc2T Þk ðc2T Þkþ1 ¼ 0: or or or or
ð36Þ ð37Þ
each element by using the Galerkin Finite Element Method. The following dynamic finite element governing equation for each element can be obtained by using linear shape function as following:
F n ¼ N1 F in þ N2 F iþ1 n ; 3. Finite element modeling To solve the governing equations for functionally graded cylinder, each isotropic sub-cylinder is divided into many elements in radial direction. The mass and stiffness matrices are obtained for
wn ¼ N1 win þ N2 wiþ1 n ;
ð38Þ
hn ¼ N1 hin þ N2 hniþ1 ; € þ ½K / ¼ ½f ½M /
ð39Þ
e
e
e
with
0.8 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0.7 n=1 z/L = 0.3
0.6
0.04
0.035 n=1 z/L = 0.3
Nondimensional Radial Displacement
0.03 Nondimensional Temperature
t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0.5
0.4
0.3
0.2
0.1
0.025
0.02
0.015
0.01
0.005 0 0 -0.1
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5 -0.005
Fig. 3. Temperature distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
-4
x 10
0.5 n = 0.05 n = 0.2 n=1 n=5
14 t = 0.3 z/L = 0.3
x 10
n = 0.05 n = 0.2 n=1 n=5
0
Nondimensional Axial Displacement
12 Nondimensional Radial Displacement
1.05
Fig. 5. Radial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
-3
16
1
10
8
6
4
t = 0.3 z/L = 0.3
-0.5
-1
-1.5
-2
2 -2.5 0
-2
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 4. Radial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
-3
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 6. Axial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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h / ¼ F in
win
hin
F iþ1 n
wiþ1 n
iT hiþ1 ; n
ð40Þ
where ‘‘e” and ‘‘i” stand for eth element and ith node. The elements of force, mass and stiffness matrices are given in Appendix. The global dynamic equilibrium equations for FGM cylinders become
€ þ ½Kf/g ¼ ½f : ½Mf/g
ð41Þ
Once the finite element equilibrium equation is established, different numerical methods can be employed to solve them in space and time domains. The Newmark method with suitable time step is used and the equilibrium equation is solved.
4. Numerical examples and discussion The standard length ‘‘l” is determined equal to inner radius of FG cylinder ‘‘a” in the presented cases and the number of layers is 50 in non-dimensional value. The first numerical examples are carried out under the following conditions: Inner surface (aluminum),
am ¼ 23 106 1= C; cm ¼ 0:896 kJ=kg C;
r ¼ 1;
qm ¼ 2707 kg=m3 ; Em ¼ 70 Gpa and outer surface (alumina),
-4
2
x 10
0.1 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0
0
-0.1 Nondimensional Radial stress
Nondimensional Axial Displacement
-2
-4 n=1 z/L = 0.3 -6
-8
n=1 z/L = 0.3
-0.2
-0.3
-0.4 -10
-0.5
-12
-14
-0.6 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1
1.5
Fig. 7. Axial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
0.2 n = 0.05 n = 0.2 n=1 n=5
t = 0.3 z/L = 0.3
0.2
n = 0.05 n = 0.2 n=1 n=5
0
-0.2 Nondimensional Hoop stress
0
-0.2
-0.4
-0.6
-0.8
-0.8
-1
-1.2 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 8. Radial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
t = 0.3 z/L = 0.3
-0.4
-0.6
-1
1.05
Fig. 9. Radial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
0.4
Nondimensional Radial stress
t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 10. Hoop stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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r ¼ 1:5;
ac ¼ 7:4 106 1= C; cc ¼ 0:76 kJ kg C; 3
qc ¼ 3800 kg=m ; Ec ¼ 380 Gpa The boundary conditions for the first example are assumed as
q_ ¼ HðtÞ;
¼ 0; u
oT ¼ 0 ðisolatedÞ; or
¼ 0 at inner surface; w ¼ 0; u
¼ 0at outer surface; w
where H(t) is Heaviside function and q_ is the heat flux, q_ ¼ C 2T oT . or Fig. 2 shows the temperature distributions across the thickness of cylinder in t ¼ 0:3 for various values of power law exponents
‘‘n”. It can be seen that the temperature waves for big values of ‘‘n”, propagate faster than small values of ‘‘n”, because the cylinder shows further ceramic behavior for small values of ‘‘n”. There is a thermal resistance in ceramic materials. The peak point of diagram in Fig. 2 (inner surface) is decreased if the value of ‘‘n” is increased. The temperature of inner surface is increased when the material properties of FG cylinder tend to ceramic properties. It is also concluded that the thermal wave for big values of ‘‘n” is propagated faster than small values of ‘‘n”. The time history of temperature wave propagation for n = 1 is illustrated in Fig. 3. Fig. 4 shows the radial displacement waves across the thickness of cylinder in
0.1 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0.1 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
0
-0.1
Nondimensional Axial stress
Nondimensional Hoop stress
-0.1
0
-0.2 n=1 z/L = 0.3
-0.3
-0.4
-0.2 n=1 z/L = 0.3
-0.3
-0.4
-0.5
-0.5 -0.6 -0.6 -0.7 -0.7 -0.8
1
1.05
1.1
-0.8 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
Fig. 11. Hoop stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Nondimensional Radial Distance
1.5
Fig. 13. Axial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
-3
2
0.2 n = 0.05 n = 0.2 n=1 n=5
0
x 10
n = 0.05 n = 0.2 n=1 n=5
1.5 t = 0.3 z/L = 0.3 1
Nondimensional Shear stress
Nondimensional Axial stress
-0.2
t = 0.3 z/L = 0.3
-0.4
-0.6
0.5
0
-0.5
-1
-0.8 -1.5 -1 -2
-1.2
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 12. Axial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
-2.5
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 14. Shear stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.3 and certain ‘‘t” (first example).
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
t ¼ 0:3 for various values of ‘‘n”. The radial displacement wave propagation for small value of ‘‘n” is slower than big values of ‘‘n” because the ceramic properties for small values of ‘‘n” are further than big values of ‘‘n”. The radial displacement wave propagations for n = 1 are shown in Fig. 5. The behaviors of axial displacement across thickness of cylinder for various ‘‘n” are shown in Fig. 6 and the time history of axial displacement can be seen in Fig. 7. Figs. 8–15 show the dynamic behaviors of radial, hoop, axial and shear stresses for various values of ‘‘n” and various instants of time. The wave propagations can be seen for all stresses. The velocity of wave propagation depends on mechanical properties and especially on the exponent of mechanical properties
power function ‘‘n”. Figs. 10–13 show the same behaviors for hoop and axial stresses. The reason of similarity can be found according to Eqs. (22) and (23) and also Figs. 5 and 7. The slopes of diagrams ou in Fig. 5 for most points across the thickness are bigger than or the values of parameters ur and oowz . In Eqs. (22) and (23), the coefficients of parameter oour are the same. The effects of parameters ur and ow on the values of hoop and axial stresses are very small against oz parameter oour . Therefore the values of hoop and axial stresses depend on parameter oour and their coefficients, approximately. Figs. 2–15 are valid for section z/L = 0.3 of the cylinder. The presented method and results in this article could be validated with the published results reported by Taheri et al. [19]. To
-3
-3
x 10
3
10 t = 0.1 t = 0.2 t = 0.3 t = 0.4 t = 0.5
2
Taheri et al Present 8 t = 0.4 Nondimensional Radial Displacement
Nondimensional Shear stress
1
0 n=1 z/L = 0.3
-1
x 10
-2
-3
z/L = 0.5 n=0
6
4
t = 0.2 2
-4 0 -5
-6
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
-2
1.5
Fig. 15. Shear stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.3 and certain ‘‘n” (first example).
1
1.05
1.1
1.15
1.2 1.25 1.3 Nondimensional Radius
1.35
1.4
1.45
1.5
Fig. 17. The comparison of radial displacement distribution for various ‘‘t” in z/L = 0.5 and n = 0 with Ref. [19].
0.4 Taheri et al Present 0.35
0
Present Taheri el al t = 0.2
-0.05
0.25
Nondimensional Radial Stress
0.3
Nondimensional Temperature
z/L = 0.5 n=0
z/L = 0.5 n=0
t = 0.4
0.2
0.15 t = 0.2
0.1
-0.1
t = 0.4
-0.15
-0.2 0.05
0
-0.05
-0.25
1
1.05
1.1
1.15
1.2 1.25 1.3 1.35 Nondimensional Radius
1.4
1.45
1.5
Fig. 16. The comparison of temperature distribution for various ‘‘t” in z/L = 0.5 and n = 0 with Ref. [19].
1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 18. The comparison of radial stress distribution for various ‘‘t” in z/L = 0.5 and n = 0 with Ref. [19].
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
verify the results, the power law exponent ‘‘n” is considered to be zero in the first example and the material of inner surface is assumed to be the same material as in Ref. [19]. Figs. 16–19 show the temperature, radial displacement, radial and hoop stresses wave propagation at the section z/L = 0.5 of cylinder for the present work and the published data of Ref. [19]. In the section z/L = 0.5 of the cylinder (middle section of the cylinder across the length), the plane strain conditions are valid. There is a good agreement between the present results and those obtained in Ref. [19]. For second example, the following boundary conditions are assumed for the cylinder:
¼ 0; w ¼ 0 at inner surface; T ¼ 100ð1 e10;000t Þ; u oT ¼ 0; w ¼ 0 at outer surface ¼ 0 ðisolatedÞ; u or and the inner surface is considered to be made of ceramic (alumina) and outer surface of metal (aluminum). In this example, the results are presented in the section z/L = 0.7 of the cylinder. Fig. 20 shows the non-dimensional temperature distribution across the thickness of the cylinder in t ¼ 0:1 for various values of power law exponents. For small values of ‘‘n”, the wave is propagated faster than big values of ‘‘n” because the material properties of the cylinder converge
0.05
100
Present z/L = 0.5 n=0
0
t = 0.05 t = 0.1 t = 0.15 t = 0.2
Taheri el al n=1 z/L = 0.7
80 -0.05 60
Nondimensional Temperature
Nondimensional Hoop Stress
t = 0.2 -0.1 t = 0.4 -0.15
-0.2
40
20
-0.25 0 -0.3
-0.35 1
1.05
1.1
1.15 1.2 1.25 1.3 1.35 Nondimensional Radial Distance
1.4
1.45
1.5
Fig. 19. The comparison of hoop stress distribution for various ‘‘t” in z/L = 0.7 and n = 0 with Ref. [19].
-20 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 21. Temperature distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
100
5 n = 0.05 n = 0.2 n=1 n=5
80
n = 0.05 n = 0.2 n=1 n=5
4
Nondimensional Radial Displacement
Nondimensional Temperature
60
40 t = 0.1 z/L = 0.7 20
0
t = 0.1 z/L = 0.7 2
1
0
-20
-40 0.75
3
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 20. Temperature distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
-1 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 22. Radial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
to ceramic properties for big values of ‘‘n”. The temperature wave propagation for n = 1 can be seen in Fig. 21. The velocity of wave is different in each point of thickness because the material properties are different in these points. The behavior of radial displacement for various values of ‘‘n” and the time history of radial displacement across the thickness of the FG cylinder can be seen in Figs. 22 and 23 for second example. The radial displacement wave propagation is illustrated in Fig. 23. The radial displacement wave propagation velocities for small values of ‘‘n” are faster than those of big values of ‘‘n” be-
cause the mechanical properties of FG cylinder are close to mechanical properties of ceramic for big values of ‘‘n”. The time history of axial displacement and the behavior of axial displacement across the thickness of cylinder for various ‘‘n” are observed in Figs. 24 and 25. The axial displacement wave propagation can be seen in Fig. 25. Figs. 26–33 illustrate the variation of radial, hoop, axial and shear stresses across the thickness of cylinder for various ‘‘n” and various instants of time. The stresses wave propagations and the effect of exponent ‘‘n” on the stresses behavior are shown in these
4
0.05 t = 0.05 t = 0.1 t = 0.15
t = 0.05 t = 0.1 t = 0.15
n=1 z/L = 0.7
3 Nondimensional Axial Displacement
Nondimensional Radial Displacement
0.04
n=1 z/L = 0.7 2
1
0.03
0.02
0.01
0
0 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
-0.01 0.75
1
Fig. 23. Radial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
0.8
0.95
1
Fig. 25. Axial displacement distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
0.08
150 n = 0.05 n = 0.2 n=1 n=5
t = 0.1 z/L = 0.7
0.07
n = 0.05 n = 0.2 n=1 n=5
100 t = 0.1 z/L = 0.7 Nondimensional Radial Stress
0.06
Nondimensional Axial Displacement
0.85 0.9 Nondimensional Radial Distance
0.05
0.04
0.03
0.02
50
0
-50
0.01 -100 0
-0.01 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 24. Axial displacement distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
-150 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 26. Radial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
2020
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023 20
20 t = 0.05 t = 0.1 t = 0.15 t = 0.2
n=1 z/L = 0.7 0
t = 0.05 t = 0.1 t = 0.15 t = 0.2
0
Nondimensional Hoop Stress
Nondimensional Radial Stress
-20 -20
-40
-60
-40
n=1 z/L = 0.7
-60
-80
-80
-100 0.75
-100
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 27. Radial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
-120 0.75
0.95
1
20 n = 0.05 n = 0.2 n=1 n=5
0
n = 0.05 n = 0.2 n=1 n=5
0
-20 Nondimensional Axial Stress
-20 Nondimensional Hoop Stress
0.85 0.9 Nondimensional Radial Distance
Fig. 29. Hoop stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
20
-40 t = 0.1 z/L = 0.7
-60
-80
-40 t = 0.1 z/L = 0.7
-60
-80
-100
-100
-120
-120
-140 0.75
0.8
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 28. Hoop stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
-140 0.75
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 30. Axial stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
5. Conclusions
be multilayer cylinder across the thickness. Material properties in each layer are assumed to be constant and functionally graded properties are resulted by suitable arrangements of the layers in a multilayer cylinder. The results of this procedure can be outlined as:
Coupled thermoelasticity without energy dissipation of finite length functionally graded thick hollow cylinders is studied by using Green–Naghdi theory. The calculations of the mechanical displacements, temperature distribution and thermal stresses have been provided by a hybrid numerical solution (Galerkin and Newmark methods). To analyze the problem, the cylinder is assumed to
1. The coupled thermoelasticity without energy dissipation in functionally graded thick hollow cylinder is studied and developed by using Green–Naghdi theory. The wave propagations are analyzed and the time history of temperature, mechanical displacement and thermal stresses are discussed for various values of power law exponents.
figures. The behaviors of hoop and axial stresses are similar and their reason was described in the first example.
2021
S.M. Hosseini / Materials and Design 30 (2009) 2011–2023 20
0.2 t = 0.05 t = 0.1 t = 0.15 t = 0.2
0
0.1
0
-40
Nondimensional Shear Stress
Nondimensional Axial Stress
-20
n=1 z/L = 0.7
-60
-0.1
-0.3
-100
-0.4
0.8
0.85 0.9 Nondimensional Radial Distance
0.95
0.2
0.85 0.9 Nondimensional Radial Distance
0.95
1
Fig. 33. Shear stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
The components of mass matrices for each element
m11 ¼ a5 ;
m12 ¼ 0;
m13 ¼ 0;
m14 ¼ a6 ;
m22 ¼ a5 ;
m23 ¼ 0;
m24 ¼ 0;
m15 ¼ 0;
m16 ¼ 0;
-0.2
Nondimensional Shear Stress
0.8
Appendix
n = 0.05 n = 0.2 n=1 n=5
0
-0.5 0.75
1
Fig. 31. Axial stress distribution across thickness of cylinder for various ‘‘t” in z/L = 0.7 and certain ‘‘n” (second example).
n=1 z/L = 0.7
-0.2
-80
-120 0.75
t = 0.05 t = 0.1 t = 0.15
-0.4
m21 ¼ 0;
t = 0.1 z/L = 0.7
m25 ¼ a6 ;
m26 ¼ 0;
-0.6
m31 ¼ -0.8
e a13 R
m34 ¼
-1
e a13 R
m41 ¼ a6 ;
e R 3
m32 ¼ e ba5 ;
;
e a8 ;
m33 ¼ a5 ;
m35 ¼ e ba6 ;
m36 ¼ a6 ;
m42 ¼ 0;
m43 ¼ 0;
m44 ¼ a11 ;
m52 ¼ a6 ;
m53 ¼ 0;
m54 ¼ 0;
m45 ¼ 0;
m46 ¼ 0; -1.2
m51 ¼ 0; -1.4 0.75
0.8
0.85
0.9
0.95
1
Nondimensional Radial Distance Fig. 32. Shear stress distribution across thickness of cylinder for various ‘‘n” in z/L = 0.7 and certain ‘‘t” (second example).
m61 ¼
e a14
m64 ¼ 2. The phenomenon of stress wave front is observed due to the theory of elasticity and the governing equations are solved by general elasticity solutions. 3. Results for axisymmetric thermal shock loads applied to the inner surface are presented for finite length functionally graded thick hollow cylinder with simply supported edges. The behaviors of functionally graded thick hollow cylinders are determined and discussed for inner surface temperature shock and Heaviside heat flux of inner surface. 4. The employed method in this paper based on the achieved results can be used in analyzing of vibration, dynamic behavior, coupled and uncoupled thermoelasticity of functionally graded materials.
m55 ¼ a11 ;
m56 ¼ 0;
R
e a8 ;
e a14 R
e R 3
m62 ¼ e ba6 ; ;
m63 ¼ a6 ;
m65 ¼ e ba11 ;
m66 ¼ a11 ;
where R = r{i+1} ri. For simplicity, we assumed that r ¼ r in the Appendix. The components of stiffness matrices for each element
c2p ri c2p c2s b2 a5 c2p c2s a1 ; R 2 c2p 2 2 ba13 ¼ cp cs ; k13 ¼ a13 ; R R
k11 ¼ k12
c2p r i c2p þ c2s b2 a6 c2p c2s a2 ; R 2 ba c2p 13 2 ¼ cp c2s ; k16 ¼ a13 ; R R
k14 ¼ k15
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S.M. Hosseini / Materials and Design 30 (2009) 2011–2023
b bR k21 ¼ c2p c2s a13 þ c2p c2s ; R 3
k22 ¼
k24 k26
c2s r i R
c2s 2
a6 ¼
k23 ¼ a5 c2p b;
b ¼ c2p c2s a13 þ c2p c2s ba8 ; R ¼ a6 c2p b;
k25
c2 r i c2 ¼ s þ s c2p b2 a6 ; R 2
a7 ¼
k32 ¼ 0;
k34 ¼ 0;
k41 k42
k35 ¼ 0;
k33 ¼ k36
N1 N 2 r dr
1 r iþ1 r i R ðr iþ1 þ r i Þ 3 ðr iþ1 þ ri Þ þ r iþ1 r 3i r 4iþ1 r4i ; 2 3 4 R 1
2
Z
r iþ1
ri
a8 ¼ k31 ¼ 0;
r iþ1
ri
¼ c2p b2 a5 ;
Z
Z
r iþ1
N1 N 2 dr ¼
ri
c2T r i c2T c2T b2 a5 ; R 2
a9 ¼
c2 r i c2 ¼ T þ T c2T b2 a6 ; R 2
Z
r iþ1
ri
c2p r iþ1 c2p ¼ c2s b2 a6 c2p c2s a2 ; R 2 c2p 2 2 ba14 ¼ cp cs ; k43 ¼ a14 ; R R
a10 ¼
Z
riþ1
ri
c2p r iþ1 c2p þ c2s b2 a11 c2p c2s a9 ; R 2 ba c2p 14 ¼ c2p c2s ; k46 ¼ a14 ; R R
¼
k44 ¼
a12 ¼
b k51 ¼ c2p c2s a14 þ c2p c2s ba8 ; R c2s r iþ1 c2s k52 ¼ c2p b2 a6 ; k53 ¼ a6 c2p b; R 2
k55
a13 ¼
2 1 ri R 2r i 3 ðr iþ1 þ ri Þ r iþ1 r 3i þ r 4iþ1 r4i ; 4 3 2 R 2
Z
riþ1
k62 ¼ 0;
k64 ¼ 0;
k65 ¼ 0;
k63 k66
a14 ¼
Z
Z
Z
r iþ1
r iþ1
ri
Z
r iþ1
ri
a5 ¼
Z ri
¼
1 R2
r iþ1
riþ1
ri
ri
a4 ¼
N1 r dr ¼
1 R 1 r iþ1 ðr i þ r iþ1 Þ r 3iþ1 r 3i ; R 2 3
N1 r dr ¼
1 R 1 r i ðr i þ r iþ1 Þ þ r3iþ1 r 3i : R 2 3
riþ1
Z
riþ1
Force matrices for each element
oF ; or r¼ri oF ¼ c2p r i or
c2 r iþ1 c2T ¼ T c2T b2 a6 ; R 2
f11 ¼ c2p ri
c2 r iþ1 c2T ¼ T þ c2T b2 a11 ; R 2
f 41
f 21 ¼ c2s r i
r¼riþ1
;
f 51
ow or
; r¼r i
f 31 ¼ c2T r i
ow ¼ c2s r i ; or r¼riþ1
oh or
;
r¼r i
f 61 ¼ c2T r i
oh : or r¼riþ1
References
alpha1 ¼
a3 ¼
R ; 3
ri
where
a2 ¼
Z
N22 dr ¼
ri
b ¼ c2p c2s a14 þ c2p c2s bR=3; R c2s r iþ1 c2s ¼ þ c2p b2 a11 ; k56 ¼ a11 c2p b; R 2
k61 ¼ 0;
N22 r dr
1
ri
k54
N22 1 r iþ1 ri dr ¼ R 2r ln þ R i r2 ri r iþ1 R2
riþ1
Z
R 1 3 2 3 ð þ r Þ r r R ; r r r iþ1 i i iþ1 iþ1 i 3 R2 2 1
N 22 1 r iþ1 R 2r i R þ ðr iþ1 þ r i Þ ; dr ¼ 2 r2i ln 2 r ri R
ri
a11 ¼
k45
R ; 3
N21 dr ¼
N21 1 r iþ1 R 2riþ1 R þ ðriþ1 þ r i Þ ; dr ¼ 2 r 2iþ1 ln 2 r ri R
N1 N2 1 r iþ1 R þ ðriþ1 þ r i Þ ; dr ¼ 2 r iþ1 ri ln 2 r ri R
N21 1 r iþ1 r iþ1 ; dr ¼ 2r ln þ R þ R iþ1 r2 ri ri R2
N1 N2 1 r iþ1 dr ¼ ð r þ r Þ ln 2R iþ1 i r2 ri R2 N21 rdr
1 r2iþ1 R 2r iþ1 3 ðr iþ1 þ ri Þ r iþ1 r3i þ r4iþ1 r 4i 4 3 2
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