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A nonlinear analysis of thermal stresses in an incompressible functionally graded hollow cylinder with temperature-dependent material properties
European Journal of Mechanics A/Solids 55 (2016) 212e220
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A nonlinear analysis of thermal stresses in an incompressible functionally graded hollow cylinder with temperature-dependent material properties Amin Moosaie* Department of Mechanical Engineering, Yasouj University, 75914-353 Yasouj, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2014 Accepted 19 September 2015 Available online xxx
A nonlinear thermoelastic analysis of a thick-walled cylinder made of functionally graded material is performed. The dependence of material properties on temperature is taken into account. This makes the governing equations nonlinear. Thus, the perturbation technique is employed to solve the nonlinear heat conduction equation analytically. The so-obtained temperature field is then supplied to elasticity equations which are solved exactly for the case of incompressible elastic material to get displacement and stress distributions. Finally, the temperature field, material properties and radial stress versus the radial direction are plotted and discussed. © 2015 Elsevier Masson SAS. All rights reserved.
Keywords: Thick-walled cylinder Temperature-dependent material properties Perturbation method
1. Introduction Functionally graded materials (FGM) are a class of advanced composites whose physical (mechanical, thermal, etc.) properties vary smoothly and continuously in space. That is the material properties are continuous functions of spatial coordinates. These materials are useful in applications when different requirements are to be met by the material. If these requirements are diverse, which can arise in many engineering applications, e.g. in aerospace and biomedical applications, then it might occur that one single material fails to satisfy all requirements. Fiber-reinforced and laminated composites are one solution for this ongoing and everincreasing demand. Despite their usefulness in many areas, they are prone to stress concentration due to material discontinuities as well as damages like delamination. For example, in aerospace applications, more specifically during atmospheric reentry of a space vehicle, the vehicle structure is subjected to mechanical as well as thermal loads. Thus, the material needs to withstands both kinds of loads. Metals like steel and aluminum possess good strength under mechanical loads. However, when it comes to thermal loads at very high temperatures, they are prone to creep and even melting. On the other hand, ceramics are generally good in resisting thermal
* Tel.: þ98 743 322 1711. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.euromechsol.2015.09.005 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.
loadings, but they are often fragile under mechanical loads. So, the structure can be made of a synthetic material which is a blending of both steel and ceramics. That is, the outer surface subjected to aerodynamic heating is made of ceramics and when moving through the thickness towards the inner layers of the structure, the material gradually and continuously changes to steel which is very good at withstanding mechanical loads. This is only one example, but such demands constantly arise in advanced technologies nowadays. Since FGM's are often to be used under thermal loadings, the thermoelastic analysis of structures made of FGM's is of paramount importance. Many authors have addressed such a thermal analysis and their works can be browsed in the literature. Using perturbation technique, Obato and Noda (1994) analyzed steady onedimensional thermal stresses in hollow cylindrical and spherical objects. Ootao et al. (1995) made a transient analysis of thermal stresses in a functionally graded hollow cylinder due to a moving heat source in the axial direction. Lutz and Zimmerman (1996), Zimmerman and Lutz (1999) gave the analytical solution for the thermal stresses in thick cylindrical and spherical shells made of FGM's graded in the radial direction. Tanigawa et al. (1999) analytically investigated thermal stresses in a FG semi-infinite body graded with a power function in the thickness direction. Jabbari et al. (2002, 2003) considered an FG hollow cylinder graded by a power function in the radial direction and developed analytical solutions for one- and two-dimensional thermal stresses. Liew et al.
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
(2003) presented an analysis of thermoelastic problem in an FG hollow cylinder. They proposed a novel technique by which they derived the solution for an inhomogeneous material from the solution for a homogeneous material. You et al. (2005) gave an analytical solution for elastic stresses in thick-walled spherical vessels under internal pressure. They considered the shell to be made of a functionally graded material confined between two inner and outer homogeneous layers. Chen and Lin (2008) performed an elastic analysis of a thick-walled FGM cylindrical shell graded exponentially in the radial direction. Peng and Li (2010) proposed a method to solve steady thermal stresses in a hollow cylinder made of functionally graded materials with physical properties varying in the radial direction. The aforementioned works are just few examples from the large body of literature addressing mechanical and thermal stresses in thick-walled cylindrical and spherical shells. However, all these works address functionally graded materials with temperatureindependent material properties. However, the physical reality for both homogeneous and functionally graded materials is the dependence of material properties on temperature. This is more pronounced especially when the large temperature differences are involved in the problem. That is, the dependence of physical properties on temperature becomes negligible at lower temperature differences. Nevertheless, there are certain applications in which one cannot neglect this fact and has to take temperaturedependent material properties into account. But, this makes the analysis very complicated as the governing differential equations become nonlinear. On the author's knowledge, so far only numerical solutions are used to perform such a full nonlinear analysis. Awaji and Sivakumar (2001) studied the transient thermal stresses of a FGM hollow circular cylinder cooled by surrounding medium using the finite difference method. Azadi and Azadi (2009) presented a transfinite element method for transient analysis of thermal stresses in a functionally graded hollow cylinder with temperature-dependent material properties. Analytical solution of thermal stresses with temperaturedependent material properties has been always an interesting research topic and a number of works have been published on this matter addressing homogeneous materials, e.g. (Trostel, 1958). In this paper, an analytical solution of the nonlinear thermal and thermoelastic problem for an FGM thick-walled cylindrical shell with temperature-dependent material properties is presented. For this purpose, the temperature field is obtained using perturbation technique. Then, the thermoelastic problem is solved exactly for the case of n ¼ 0:5 with n being the Poisson ratio. On the author's knowledge, it is for the first time that such a nonlinear analysis for FGM shells is presented using analytical tools. The remainder of this paper is organized as follows. Section 2 presents the nonlinear governing equations along with the required boundary conditions. The analytical solution of heat conduction and thermoelastic problems are discussed in Section 3. Section 4 contains the results and discussions and finally the paper is concluded in Section 5. 2. Governing equations and boundary conditions In this section, the governing differential equations of temperature field and thermal stresses in a hollow cylinder made of a functionally graded material with temperature-dependent properties are presented and discussed. First we shall introduce the FG model which is used to describe the spatial variation of material properties within the cylinder. For a cylinder which is functionally graded in the radial direction r, the temperature-dependent material properties can be written as E ¼ Eðw; rÞ, at ¼ at ðw; rÞ and l ¼ lðw; rÞ with w and r being the
213
temperature field and the radial direction, respectively. In this paper, we propose a separable model in which E ¼ g1 ðwÞf1 ðrÞ, at ¼ g2 ðwÞf2 ðrÞ and l ¼ g3 ðwÞf3 ðrÞ. In the literature for homogeneous materials such as steel, the functions g1 and g2 are taken to be linear functions of temperature whereas g3 is considered both linear (Nowinski, 1959) and quadratic (Stanisic and McKinley, 1962) functions of w.
E ¼ Eðw; rÞ ¼ E0 E1 w E2 w2 f1 ðrÞ;
(1)
at ¼ at ðw; rÞ ¼ ðat0 þ at1 wÞf2 ðrÞ;
(2)
l ¼ lðw; rÞ ¼ ðl0 l1 wÞf3 ðrÞ;
(3)
in which E, at and l are respectively the elasticity modulus, thermal expansion coefficient and heat conductivity, and fi ðrÞ is the grading function. E0 , E1 , E2 , at0 , at1 , l0 and l1 are material constants. The Poisson ratio n is assumed to be constant for FG materials. In the present study, we consider an incompressible elastic material with n ¼ 0:5. This assumption is necessary to obtain an exact solution of thermoelasticity equations. In a future investigation, we will address thermal stresses in an FG material with an arbitrary Poisson ratio by using a perturbation solution of thermoelasticity equations. In this work, we consider an FGM with a grading function of the form ðr=Ro Þmi . Thus, the above equations can be rewritten as
E ¼ Eðw; rÞ ¼ E0 E1 w E2 w2 ðr=Ro Þm1 ;
(4)
at ¼ at ðw; rÞ ¼ ðat0 þ at1 wÞðr=Ro Þm2 ;
(5)
l ¼ lðw; rÞ ¼ ðl0 l1 wÞðr=Ro Þm3 :
(6)
2.1. Heat conduction In this section, the governing equations of steady heat conduction in a functionally graded hollow cylinder with temperaturedependent heat conductivity are presented. These equations can be obtained by considering the energy conservation law, the Fourier constitutive equation for heat conduction and the dependence of heat conductivity on spatial coordinates and temperature as pointed out in Equation (6). Thus, we start with the steady energy conservation law without heat generation:
V$q ¼ 0;
(7)
where V and q are the nabla operator and heat flux vector, respectively. The Fourier constitutive law along with the energy conservation equation (7) gives the following field equation for the temperature:
V$ðlVwÞ ¼ 0:
(8)
This equation for an infinitely long cylinder with axisymmetric temperature field reads
1 d dw rlðw; rÞ ¼ 0: r dr dr
(9)
In order to be solved, this equation needs two boundary conditions in r direction. In this paper, we consider prescribed temperatures at inner and outer surfaces of the cylinder:
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wðr ¼ Ri Þ ¼ wi ;
wðr ¼ Ro Þ ¼ wo ;
(10)
3. Analytical solution In order to obtain the thermal stresses in the cylinder, first the heat conduction problem is to be solved in order to obtain the temperature field. Once the temperature field is known, we proceed to solve the thermoelasticity problem. In this section, analytical solution of these problems are presented. In what follows, we first present the solution of the heat conduction problem and then the thermoelastic solution is given.
where Ri and Ro are the inner and outer radii of the hollow cylinder, respectively. However, the method presented in this paper is not restricted to these boundary conditions and other types of boundary conditions can be treated as well. When a homogeneous material is considered, l is assumed to be constant whereas for a functionally graded material, l depends on the spatial coordinates. Both cases lead to linear differential equations, but the former yields a linear differential equation with variable coefficients. However, experimental observations show that the properties of real materials depend on temperature. This results in governing equations becoming nonlinear. Thus, the analysis of the problem gets more difficult. Due to this inherent difficulty, it is often tried to refrain from such a full nonlinear description. However, the advancement of technology pushes us towards a more accurate description for the behavior of engineering materials.
The heat conduction problem is solved using the perturbation method. For this purpose, a small parameter ε is to appear in the nonlinear differential Equation (9). For most of engineering materials, one can postulate that the value of l1 is much smaller than the value of l0 . But, in order to make the comparison reasonable, we first have to non-dimensionalize Equation (3):
2.2. Thermoelasticity
lwref ¼ wref εw f3 ðrÞ; l0
For the thermoelastic analysis, we use the Navier equations for a thick-walled cylindrical shell. For this purpose, we start with the differential form of the equilibrium condition which in axisymmetric cylindrical coordinates reads
dsr 1 þ sr s4 ¼ 0: r dr
(11)
In the above equations sr and s4 are the radial and circumferential or hoop stresses, respectively. This axisymmetric problem has only two displacement components in radial (r) and axial (z) directions denoted by u and w, respectively. The strain-displacement relations then read
εr ¼
du ; dr
u ε4 ¼ ; r
εz ¼
dw : dz
(12)
The generalized Hooke's law of thermoelasticity for an FG material with temperature-dependent properties and n ¼ 0:5 is
1 1 sr s4 þ sz þ εr ¼ Eðw; rÞ 2
wðrÞ Z
at ðT; rÞdT;
wðrÞ Z 1 1 s4 ðsr þ sz Þ þ at ðT; rÞdT; Eðw; rÞ 2
(14)
(15)
The thermoelastic problem is subjected to the following boundary conditions. The cylinder is loaded with internal and external pressures pi and po , respectively:
sr ðr ¼ Ro Þ ¼ po :
(17)
in which wref is a reference temperature. We can now postulate that ε≪1 for most of solid materials in the range of temperatures normally present in engineering problems. The rationale for this assumption is as follows. The heat conductivity at a reference temperature w0 is l0 f3 ðrÞ and by varying the temperature, it changes to a new function ðl0 l1 wÞf3 ðrÞ. However, as long as the temperature variation from the reference value is not very intense, one expects that the change in heat conductivity is small. Thus, l1 has to be small compared to l0 . Therefore, we can define the small parameter ε as
ε¼
wref l1 ≪1: l0
(18)
In the sequel, wref is taken to be unity, i.e. wref ¼ 1, for the sake of simplicity. Thus, it does not appear anymore in the subsequent equations. Equation (9) can be casted as
d dw dw dl dw d2 w rl ¼l þr þ rl 2 ¼ 0; dr dr dr dr dr dr
(19)
m3 dl d r ¼ ðl0 l1 wÞ dr dr Ro dðl0 l1 wÞ r m3 d r m3 þ ðl0 l1 wÞ : ¼ dr Ro dr Ro
(20)
The use of the chain rule of differentiation yields
T¼0
sr ðr ¼ Ri Þ ¼ pi ;
wref l1 ; l0
in which
T¼0 wðrÞ Z 1 1 sz sr þ s4 þ εz ¼ at ðT; rÞdT: Eðw; rÞ 2
ε¼
(13)
T¼0
ε4 ¼
3.1. Solution of heat conduction problem
(16)
As a special case, when pi ¼ po ¼ 0 then the stresses in the cylinder are solely due to thermal effects. On the other hand, when the temperature is zero everywhere within the cylinder, then the stresses are purely mechanical due to internal and external pressures.
dðl0 l1 wÞ dðl0 l1 wÞ dw dw ¼ ¼ l1 : dr dw dr dr
(21)
Substituting Equation (21) into Equation (19) reveals
dw dw d2 w þ rðl0 l1 wÞ 2 ¼ 0: ð1 þ m3 Þðl0 l1 wÞ rl1 dr dr dr
(22)
By dividing both sides of Equation (22) by the non-zero constant l0 we have
dw dw d2 w þ rð1 εwÞ 2 ¼ 0: ð1 þ m3 Þð1 εwÞ εr dr dr dr
(23)
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Equation (23) is a nonlinear differential equation and we employ the regular perturbation method (Poincare's technique) to solve it. For this purpose, we expand the temperature w as a power series of ε:
w ¼ w0 þ εw1 þ ε2 w2 þ / ¼
∞ X
εn wn :
(24)
n¼0
This expansion can be truncated at any order to give an approximate temperature field, e.g.
w0 ¼ w0 ;
(25)
w1 ¼ w0 þ εw1 ;
(26)
w2 ¼ w0 þ εw1 þ ε2 w2 :
2 2 d w0 dw0 d w1 dw 1 þ ε r ε r þ ð1 þ m Þ þ ð1 þ m3 Þ 1 3 dr dr dr2 dr2 2 2 d w0 dw0 dw0 dw0 d w2 r þ ε2 r w0 r þ ð1 þ m3 Þ dr dr dr dr2 dr 2 2 dw d w1 dw þ ð1 þ m3 Þ 2 w0 r þ ð1 þ m3 Þ 1 dr dr dr2 2 d w0 dw0 þ O ε3 ¼ 0: w1 r þ ð1 þ m3 Þ dr dr2
(28)
(29)
(30)
This general solution is subjected to the following boundary conditions:
w0 ðr ¼ Ri Þ ¼ wi ;
w0 ðr ¼ Ro Þ ¼ wo :
(31)
Applying these boundary conditions to the general solution (30) gives the following relations for the integration constants C1;0 and C2;0 :
C1;0
C2;0 ¼ wo m3 ; Ro
C2;0 ¼
wo wi 3 3 Rm Rm o i
(32)
:
(33)
The Oðε1 Þ problem is obtained by setting the bracket multiplied by ε1 in Equation (28) to zero:
d2 w1 dw O ε1 : r þ ð1 þ m3 Þ 1 dr dr2 2 d w0 dw0 dw dw0 þr 0 : þ ð1 þ m Þ ¼ w0 r 3 dr dr dr dr2
which is an inhomogeneous Cauchy-Euler differential equation whose solution is composed of two parts, i.e. the homogeneous solution and a particular solution. The homogeneous solution of Equation (35) is the same as the solution of Equation (29). A particular solution of Equation (34) can be easily calculated. The general solution of Equation (34) reads 2 C2;0 C2;1 þ 2m ; m r 3 2r 3
(36)
which is to be subjected to the following boundary conditions:
w1 ðr ¼ Ri Þ ¼ 0;
w1 ðr ¼ Ro Þ ¼ 0:
(37)
2 C2;0 C2;1 C1;1 ¼ m3 2m ; Ro 2Ro 3
(38)
2 3 3 C2;0 R2m R2m o i C2;1 ¼ m3 : 3 2 Rm R o i
(39)
Finally, we come to the Oðε2 Þ problem. In order to obtain the Oðε2 Þ problem we need to set the bracket multiplied by ε2 in Equation (28) to zero:
2 d2 w2 dw d w1 dw O ε2 : r þ ð1 þ m3 Þ 2 ¼w0 r þ ð1 þ m3 Þ 1 2 2 dr dr dr dr 2 d w0 dw0 : þ w1 r þ ð1 þ m Þ 3 dr dr2 (40)
which can also be obtained by setting ε ¼ 0 in Equation (28). This is a Cauchy-Euler differential equation whose general solution reads
C2;0 : r m3
(35)
Application of these boundary conditions to the general solution (36) yields the integration constants:
This series can be continued and more terms could be obtained. But in this paper, we restrict ourselves to the above order of approximation and do not go any further. Since Equation (28) is to be valid for arbitrary values of ε, the bracketed terms are to vanish. This results in a series of problems which we solve in the sequel. First, the Oðε0 Þ is to be solved:
w0 ðrÞ ¼ C1;0 þ
d2 w1 dw dw dw0 ; O ε1 : r þ ð1 þ m3 Þ 1 ¼ r 0 dr dr dr dr2
(27)
0
d2 w0 dw O ε0 : r þ ð1 þ m3 Þ 0 ¼ 0; dr dr2
The first term in the right-hand side of Equation (34) vanishes due to Equation (29). Thus, the Oðε1 Þ problem can be written as
w1 ðrÞ ¼ C1;1 þ
Substituting expansion (24) in Equation (23) and grouping all terms with the same power of ε we get
215
(34)
This equation can be rewritten in the following form by taking into account Equations (29) and (35):
d2 w2 dw dw dw0 : O ε2 : r þ ð1 þ m3 Þ 2 ¼ rw0 0 dr dr dr dr2
(41)
This is again an inhomogeneous Cauchy-Euler differential equation whose general solution is written as the sum of the homogeneous solution and a particular solution:
w2 ðrÞ ¼ C1;2 þ
2 3 C2;0 C2;2 C1;0 C2;0 þ þ : m r 3 2r 2m3 6r 3m3
(42)
The boundary conditions for the Oðε2 Þ problem are
w2 ðr ¼ Ri Þ ¼ 0;
w2 ðr ¼ Ro Þ ¼ 0;
(43)
which give the following values for the integration constants C1;2 and C2;2 : 2 3 C2;0 C2;2 C1;0 C2;0 C1;2 ¼ m3 ; 3 3 Ro 2R2m 6R3m o o
C2;2
2 3 3 3 3 3 R2m R3m C1;0 C2;0 R2m C2;0 R3m o o i i : þ ¼ m3 m3 3 3 Rm R 2 Rm 6 R o o i i
(44)
(45)
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A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Now, the approximate temperature fields obtained by the perturbation method read
C2;0 þ O ε1 ; w0 ðrÞ ¼ ε0 w0 ¼ ε0 C1;0 þ m r 3
(46)
w1 ðrÞ ¼ w0 þ ε1 w1 ! 2 C2;0 C2;0 C2;1 þ ε1 C1;1 þ m þ 2m þ O ε2 ; ¼ ε0 C1;0 þ m 3 3 3 r r 2r (47) w2 ðrÞ ¼ w0 þ ε1 w1 þ ε2 w2 ! 2 C2;0 C2;0 C2;1 0 1 ¼ ε C1;0 þ m3 þ ε C1;1 þ m3 þ 2m r r 2r 3 ! 2 3 C2;0 C2;2 C1;0 C2;0 þ þ O ε3 : þε2 C1;2 þ m3 þ r 2r 2m3 6r 3m3
wo wi ln RRoi
(49)
;
(50)
C2;0 ¼ wo C1;0 lnRo :
(51)
2 C1;0
2
ln2 r þ C1;1 lnr þ C2;1 ;
ln Ro ln Ri
2
ln RRoi
C1;1 ¼
2 C1;0
C2;1 ¼
w ¼ A1 z þ A2 ;
2
εz ¼
dw ¼ A1 : dz
(59)
Constants A1 and A2 are to be determined from the constraints. If we assume that wðz ¼ 0Þ ¼ 0, then we have A2 ¼ 0. We will discuss later about how to determine A1 . With this, Equation (58) can be rewritten as
εr þ ε4 þ εz ¼
(52)
in which the integration constants C1;1 and C2;1 are obtained by applying the boundary conditions: 2
(58)
in which kinematic relations (12) are employed. In this problem, the axial strain εz does not depend on the axial coordinate z and hence the axial displacement w is a linear function of z:
εr þ ε4 þ εz ¼
2 C1;0
du u dw þ þ ; dr r dz
du u 1 d þ þ A1 ¼ ðruÞ þ A1 : dr r r dr
(60)
By using strain-stress relations (13), (14) and (15), the sum of normal strains reads
The first-order solution can also be obtained as
w1 ðrÞ ¼
In this subsection, an exact solution of thermoelastic problem for the case of n ¼ 0:5 is presented and the displacement and stress fields are calculated. A solution technique for the case with arbitrary n is in progress and will be addressed in a future work. The analytical solution starts with the sum of normal strains
εr þ ε4 þ εz ¼
where the integration constants C1;0 and C2;0 are given by
C1;0 ¼
3.2. Solution of thermoelastic problem
(48)
The above solution is valid for the case when m3 s0. The case m3 ¼ 0 which describes a homogeneous material with temperature-dependent material properties needs special care. in this case, the zeroth-order solution reads
w0 ðrÞ ¼ C1;0 lnr þ C2;0 ;
Now, the approximate temperature fields can be obtained using Equations (25)e(27). In the following subsection, these approximate solutions are examined for their accuracy and then they will be used to solve the thermoelastic problem. The latter leads to displacement and stress distributions within the cylinder thickness.
2
;
at ðT; rÞdT:
(61)
This differential equation has the following general solution for the radial displacement uðrÞ:
3 uðrÞ ¼ r (54)
wðrÞ Z
T¼0
(53)
ln2 Ro C1;1 lnRo :
1 d ðruÞ þ A1 ¼ 3 r dr
Z
0 B r@
wðrÞ Z
1 A C C at ðT; rÞdT Adr 1 r þ 1 : 2 r
(62)
T¼0
Considering Equation (2), it shall be noted that
Finally, the second-order solution is as follows:
w2 ðrÞ ¼
3 C1;0
6
ln3 r þ
2 C2;0 C1;0
2
wðrÞ Z
ln2 r þ C1;2 lnr þ C2;2 :
(55) T¼0
The homogeneous boundary conditions yield the following formulae for the integration constants C1;2 and C2;2 : 3 C1;0 6
C1;2 ¼
3 C2;0 C 2 ln Ro ln3 Ri þ 2 1;0 ln2 Ro ln2 Ri ln RRoi
;
C2;1 ¼
6
ln3 Ro
2 C2;0 C1;0
2
ln2 Ro C1;2 lnRo :
1 ðat0 þ at1 TÞf ðrÞdT ¼ at0 w þ at1 w2 f ðrÞ: 2
T¼0
(63) From Equation (15) the following relation for axial stress sz is obtained:
(56) sz ¼
3 C1;0
wðrÞ Z
at ðT; rÞdT ¼
(57)
1 sr þ s4 þ A1 Eðw; rÞ Eðw; rÞ 2
wðrÞ Z
at ðT; rÞdT: T¼0
Now, using Equations (13), (14), (62) and (64) we have
(64)
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
wðrÞ Z
sr s4 ¼2Eðw;rÞ
at ðT;rÞdT T¼0
4 Eðw;rÞ 2Eðw;rÞ C1 2 ¼ 3 r r2
Z
4Eðw;rÞ r2
Z
0 B r@
wðrÞ Z
1 C at ðT;rÞdT Adr
dsr Eðw; rÞ 4 C 2 ¼ 3 1 dr r3
0
T¼0
sr ¼
r 2 at ðw; rÞ
dw dr : dr
(66)
Z Eðw; rÞ 4 dw 2 C dr dr þ C2 : 2 r a ðw; rÞ t 1 3 dr r3
4 po pi 3 D2 þ ; 3 D1 2 D1
(67)
(68)
Z Z Eðw; rÞ dw 2 dr dr r a ðw; rÞ C2 ¼ pi þ 2 t dr r3 r¼Ri Z 4 Eðw; rÞ C1 dr ; 3 r3 r¼Ri
D1 ¼
ZRo
Eðw; rÞ dr; r3
D2 ¼
Ri
r 2 at ðw; rÞ
dw 4 Eðw; rÞ dr þ C1 2 : dr 3 r
ZRo D3 ¼
EðwðrÞ; rÞ2prdr:
(75)
Ri
In this section, some selected results obtained from the proposed analytical solution are presented. The results presented in this section are obtained using the following numerical values of parameters:
E0 ¼ 2:1 105 MPa;
E1 ¼ 27:5 MPa= C;
E2 ¼ 0:141 MPa= C2 ;
(76)
at1 ¼ 1 108 C2 ;
(77)
l1 ¼ 0:0293 W=m C2 :
(78)
(70)
These are realistic values for engineering materials. However, they are used here just for the demonstration of the results and any other set of parameters can be used as well. Because, it is immaterial for the analytical solution which parameters set is used. First, temperature profiles are shown. In order to check the validity of the presented approximate solution based on the perturbation method, a numerical solution of the full nonlinear heat conduction problem is supplied for comparison. Fig. 1 shows the temperature profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation. The solution of full nonlinear governing equations obtained by an accurate numerical solution based on the finite difference method is also plotted for comparison. The convergence of perturbation series from the zeroth-order solution (i.e., linear problem) towards the “exact” solution by increasing the order of approximation is demonstrated
(71)
The axial stress sz is calculated from Equation (64) once sr and s4 are known. The remaining quantity to be determined now is A1 which depends on boundary condition in the axial direction z. Two cases for this boundary condition can be considered. First, is a plane strain state in which
εz ¼ A1 ¼ 0:
where
l0 ¼ 50:16 W=m C;
Z Eðw; rÞ dw 2 dr dr: r a ðw; rÞ t dr r3
Z
(74)
T¼0
(69)
Ri
2Eðw; rÞ r2
3
7 C at ðT; rÞdT A2prdr 5;
The circumferential stress s4 is obtained using Equation (65):
s4 ¼ sr
1
at0 ¼ 1:2 105 C1 ;
in which
ZRo
wðrÞ Z
4. Results and discussions
The integration constants C1 and C2 are determined from boundary conditions (16):
C1 ¼
B @
Z
The radial stress sr is obtained by integrating once the above differential equation:
Z
2 Z 1 6 A1 ¼ 4P þ p po R2o pi R2i þ EðwðrÞ; rÞ D3
dw 4 Eðw;rÞ r 2 at ðw;rÞ dr C1 2 : dr 3 r (65)
Thus, the equilibrium condition (11) can be written as follows:
217
(72)
This happens, for example, when the cylinder is clamped between two rigid walls and its elongation is not allowed. The second case is when an external axial force P is exerted on the cylinder. In this case, the integration of axial stress sz over any cross section must equals P in order for the axial equilibrium to hold, i.e.
ZRo sz ðrÞ2prdr ¼ P:
(73)
Ri
Inserting the axial stress sz from Equation (64) into Equation (73) and performing the integration yields
Fig. 1. Temperature profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different approximations and compared with a numerical solution of the full nonlinear equation.
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A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Fig. 2. Temperature profiles for wi ¼ 0 C, wo ¼ 500 C and m3 ¼ 2 obtained by different approximations and compared with a numerical solution of the full nonlinear equation.
Fig. 3. Temperature profiles for wi ¼ 0 C, wo ¼ 1000 C and for different values of m3 obtained by the second-order approximation. From top to the bottom, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
in this figure. In order to show the effect of temperature difference between the inner and outer surfaces of the cylinder, temperature profiles for wi ¼ 0+ C, wo ¼ 500+ C and m3 ¼ 2 are depicted in Fig. 2. We see that both w1 and w2 almost coincide with the “exact” solution for this case. It means that the higher the temperature difference the more terms in the perturbation series are required in order to get an accurate solution.
Fig. 4. Elasticity modulus profiles for wi ¼ 0 C, wo ¼ 1000 C and different values of m1 . From bottom to the top, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
Fig. 5. Thermal expansion coefficient profiles for wi ¼ 0 C, wo ¼ 1000 C and different values of m2 . From bottom to the top, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
Fig. 6. Heat conductivity profiles for wi ¼ 0 C, wo ¼ 1000 C and different values of m3 . From bottom to the top, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
The results presented in the remainder of this paper are obtained by setting wi ¼ 0+ C and wo ¼ 1000+ C. Fig. 3 shows how the value of m3 affects the temperature profile. Larger values of m3 lead to higher temperature levels and vice versa. Also, the curvature of temperature curves changes with the values of m3 . This means that for large positive values of m3 the temperature gradient near to the inner surface is sharp and this sharp gradient weakens by approaching the outer surface. However, for negative values of m3 the behavior is converse. This trend can be explained as follows. For
Fig. 7. Elasticity modulus profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Fig. 8. Thermal expansion coefficient profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
Fig. 9. Heat conductivity profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
positive m3 , the heat conductivity assumes its minimum value at the inner surface and it increases by approaching the outer surface. Moreover, the radial heat flux is constant in this one-dimensional problem. Thus, by increasing l in the radial direction, the temperature gradient vw=vr has to decrease in order to have a constant heat flux. A converse statement holds for the negative values of m3 . Once the temperature distribution is obtained, one could proceed to compute the profiles of material properties in the radial direction r. The profiles of elasticity modulus E as functions of r for different values of m1 are depicted in Fig. 4. It is observed that the
Fig. 10. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 3 obtained by different orders of approximation.
219
Fig. 11. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
Fig. 12. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 1 obtained by different orders of approximation.
elasticity modulus generally increases by decreasing m1 from positive values towards negative values. A similar behavior is observed for thermal expansion coefficient at and heat conductivity l as seen in Figs. 5 and 6, respectively. Figs. 7e9 respectively show the material properties E, at and l as functions of r for m ¼ m1 ¼ m2 ¼ m3 ¼ 2 obtained by different orders of approximation. Each figure shows the respective material property as predicted by zeroth-, first- and second-order approximate solutions. It shall be noted here that the zeroth-order solution corresponds to the linear solution where temperature dependence
Fig. 13. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 0 obtained by different orders of approximation.
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A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
It is now worthwhile to look at the distribution of radial stress component within the cylinder. Radial stress profiles for m ¼ m1 ¼ m2 ¼ m3 ¼ 3; …; 3 are depicted in Figs. 10e16, respectively. Two general trends can be observed in these figures. First, the peak of the radial stress for m ¼ 3 occurs near the inner surface and by gradually increasing the value of m up to 3, the peak location moves towards the outer surface. Second, for negative values of m, the zeroth-order solution overestimates the radial stress whereas for positive values of m, the zeroth-order solution underestimates the radial stress, as compared to the second-order solution. For example, the second-order solution predicts the peak of radial stress about 25% higher than the zeroth-order solution for m ¼ 3. 5. Conclusions Fig. 14. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 1 obtained by different orders of approximation.
In this paper, an analytical solution of thermal stresses in a hollow cylinder made of FGM with temperature-dependent material properties is presented. After introducing the governing equations and boundary conditions of the problem, the heat conduction problem is analytically solved using perturbation technique. This leads to an approximate solution, but the order of accuracy can be increased in a systematic manner. Then, an exact solution for the elasticity equations is derived for the case of n ¼ 0:5. Finally, the temperature field, material properties and radial stress component obtained by different orders of approximation are plotted versus the radial direction. References
Fig. 15. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
Fig. 16. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 3 obtained by different orders of approximation.
is disregarded. The first-order solution is a considerable improvement over the zeroth-order solution whereas the further improvement of the second-order solution is modest. It is noticeable that the difference between different orders of solution for the thermal expansion coefficient is small as compared to other material properties shown here. The peak of elasticity modulus profile shown in Fig. 7 moves towards the outer surface by increasing the order of approximation.
Awaji, H., Sivakumar, R., 2001. Temperature and stress distributions in a hollow cylinder of functionally graded material: the case of temperature-dependent material properties. J. Am. Ceram. Soc. 84, 1059e1065. Azadi, M., Azadi, M., 2009. Nonlinear transient heat transfer and thermoelastic analysis of thick-walled FGM cylinder with temperature-dependent material properties using Hermitian transfinite element. J. Mech. Sci. Tech. 23, 2635e2644. Chen, Y.Z., Lin, X.Y., 2008. Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials. Comput. Mater. Sci. 44, 581e587. Jabbari, M., Sohrabpour, S., Eslami, M.R., 2002. Mechanical and thermal stresses in functionally graded hollow cylinder due to radially symmetric loads. Int. J. Press. Vessels Pip. 79, 493e497. Jabbari, M., Sohrabpour, S., Eslami, M.R., 2003. General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to nonaxisymmetric steady-state loads. ASME J. Appl. Mech. 70, 111e118. Liew, K.M., Kitipornchai, S., Zhang, X.Z., Lim, C.W., 2003. Analysis of the thermal stress behaviour of functionally graded hollow circular cylinders. Int. J. Solids Struct. 40, 2355e2380. Lutz, M.P., Zimmerman, R.W., 1996. Thermal stresses and effective thermal expansion coefficient of a functionally graded sphere. J. Therm. Stress. 19, 39e54. Nowinski, J., 1959. Thermoelastic problem for an isotropic sphere with temperature dependent properties. ZAMP 10, 565e575. Obato, Y., Noda, N., 1994. Thermal stress in hollow circular cylinder and a hollow sphere of a functionally graded material. J. Therm. Stress. 14, 471e487. Ootao, Y., Akai, T., Tanigawa, Y., 1995. Three-dimensional transient thermal stress analysis of a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction. J. Therm. Stress. 18, 497e512. Peng, X.L., Li, X.F., 2010. Thermo-elastic analysis of a cylindrical vessel of functionally graded materials. Int. J. Press. Vessels Pip. 87, 203e210. Stanisic, M.M., McKinley, R.M., 1962. A note on thermal stresses in hollow cylinders. Ing. Arch. 27, 227e241. Tanigawa, Y., Morishita, H., Ogaki, S., 1999. Derivation of system of fundamental equations for a three-dimensional thermoelastic field with nonhomogeneous material properties and its application to a semi-infinite body. J. Therm. Stress. 22, 689e711. €rmespannungen in Hohlzylindern mit temperatur-abh€ Trostel, R., 1958. Wa angigen Stoffwerten. Ing. Arch. 26, 134e142. You, L.H., Zhang, J.J., You, X.Y., 2005. Elastic analysis of internally pressurized thickwalled spherical pressure vessel of functionally graded materials. Int. J. Press. Vessels Pip. 82, 347e354. Zimmerman, R.W., Lutz, M.P., 1999. Thermal stress and effective thermal expansion in a uniformly heated functionally graded cylinder. J. Therm. Stress. 22, 177e188.
Contents lists available at ScienceDirect
European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
A nonlinear analysis of thermal stresses in an incompressible functionally graded hollow cylinder with temperature-dependent material properties Amin Moosaie* Department of Mechanical Engineering, Yasouj University, 75914-353 Yasouj, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2014 Accepted 19 September 2015 Available online xxx
A nonlinear thermoelastic analysis of a thick-walled cylinder made of functionally graded material is performed. The dependence of material properties on temperature is taken into account. This makes the governing equations nonlinear. Thus, the perturbation technique is employed to solve the nonlinear heat conduction equation analytically. The so-obtained temperature field is then supplied to elasticity equations which are solved exactly for the case of incompressible elastic material to get displacement and stress distributions. Finally, the temperature field, material properties and radial stress versus the radial direction are plotted and discussed. © 2015 Elsevier Masson SAS. All rights reserved.
Keywords: Thick-walled cylinder Temperature-dependent material properties Perturbation method
1. Introduction Functionally graded materials (FGM) are a class of advanced composites whose physical (mechanical, thermal, etc.) properties vary smoothly and continuously in space. That is the material properties are continuous functions of spatial coordinates. These materials are useful in applications when different requirements are to be met by the material. If these requirements are diverse, which can arise in many engineering applications, e.g. in aerospace and biomedical applications, then it might occur that one single material fails to satisfy all requirements. Fiber-reinforced and laminated composites are one solution for this ongoing and everincreasing demand. Despite their usefulness in many areas, they are prone to stress concentration due to material discontinuities as well as damages like delamination. For example, in aerospace applications, more specifically during atmospheric reentry of a space vehicle, the vehicle structure is subjected to mechanical as well as thermal loads. Thus, the material needs to withstands both kinds of loads. Metals like steel and aluminum possess good strength under mechanical loads. However, when it comes to thermal loads at very high temperatures, they are prone to creep and even melting. On the other hand, ceramics are generally good in resisting thermal
* Tel.: þ98 743 322 1711. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.euromechsol.2015.09.005 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.
loadings, but they are often fragile under mechanical loads. So, the structure can be made of a synthetic material which is a blending of both steel and ceramics. That is, the outer surface subjected to aerodynamic heating is made of ceramics and when moving through the thickness towards the inner layers of the structure, the material gradually and continuously changes to steel which is very good at withstanding mechanical loads. This is only one example, but such demands constantly arise in advanced technologies nowadays. Since FGM's are often to be used under thermal loadings, the thermoelastic analysis of structures made of FGM's is of paramount importance. Many authors have addressed such a thermal analysis and their works can be browsed in the literature. Using perturbation technique, Obato and Noda (1994) analyzed steady onedimensional thermal stresses in hollow cylindrical and spherical objects. Ootao et al. (1995) made a transient analysis of thermal stresses in a functionally graded hollow cylinder due to a moving heat source in the axial direction. Lutz and Zimmerman (1996), Zimmerman and Lutz (1999) gave the analytical solution for the thermal stresses in thick cylindrical and spherical shells made of FGM's graded in the radial direction. Tanigawa et al. (1999) analytically investigated thermal stresses in a FG semi-infinite body graded with a power function in the thickness direction. Jabbari et al. (2002, 2003) considered an FG hollow cylinder graded by a power function in the radial direction and developed analytical solutions for one- and two-dimensional thermal stresses. Liew et al.
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
(2003) presented an analysis of thermoelastic problem in an FG hollow cylinder. They proposed a novel technique by which they derived the solution for an inhomogeneous material from the solution for a homogeneous material. You et al. (2005) gave an analytical solution for elastic stresses in thick-walled spherical vessels under internal pressure. They considered the shell to be made of a functionally graded material confined between two inner and outer homogeneous layers. Chen and Lin (2008) performed an elastic analysis of a thick-walled FGM cylindrical shell graded exponentially in the radial direction. Peng and Li (2010) proposed a method to solve steady thermal stresses in a hollow cylinder made of functionally graded materials with physical properties varying in the radial direction. The aforementioned works are just few examples from the large body of literature addressing mechanical and thermal stresses in thick-walled cylindrical and spherical shells. However, all these works address functionally graded materials with temperatureindependent material properties. However, the physical reality for both homogeneous and functionally graded materials is the dependence of material properties on temperature. This is more pronounced especially when the large temperature differences are involved in the problem. That is, the dependence of physical properties on temperature becomes negligible at lower temperature differences. Nevertheless, there are certain applications in which one cannot neglect this fact and has to take temperaturedependent material properties into account. But, this makes the analysis very complicated as the governing differential equations become nonlinear. On the author's knowledge, so far only numerical solutions are used to perform such a full nonlinear analysis. Awaji and Sivakumar (2001) studied the transient thermal stresses of a FGM hollow circular cylinder cooled by surrounding medium using the finite difference method. Azadi and Azadi (2009) presented a transfinite element method for transient analysis of thermal stresses in a functionally graded hollow cylinder with temperature-dependent material properties. Analytical solution of thermal stresses with temperaturedependent material properties has been always an interesting research topic and a number of works have been published on this matter addressing homogeneous materials, e.g. (Trostel, 1958). In this paper, an analytical solution of the nonlinear thermal and thermoelastic problem for an FGM thick-walled cylindrical shell with temperature-dependent material properties is presented. For this purpose, the temperature field is obtained using perturbation technique. Then, the thermoelastic problem is solved exactly for the case of n ¼ 0:5 with n being the Poisson ratio. On the author's knowledge, it is for the first time that such a nonlinear analysis for FGM shells is presented using analytical tools. The remainder of this paper is organized as follows. Section 2 presents the nonlinear governing equations along with the required boundary conditions. The analytical solution of heat conduction and thermoelastic problems are discussed in Section 3. Section 4 contains the results and discussions and finally the paper is concluded in Section 5. 2. Governing equations and boundary conditions In this section, the governing differential equations of temperature field and thermal stresses in a hollow cylinder made of a functionally graded material with temperature-dependent properties are presented and discussed. First we shall introduce the FG model which is used to describe the spatial variation of material properties within the cylinder. For a cylinder which is functionally graded in the radial direction r, the temperature-dependent material properties can be written as E ¼ Eðw; rÞ, at ¼ at ðw; rÞ and l ¼ lðw; rÞ with w and r being the
213
temperature field and the radial direction, respectively. In this paper, we propose a separable model in which E ¼ g1 ðwÞf1 ðrÞ, at ¼ g2 ðwÞf2 ðrÞ and l ¼ g3 ðwÞf3 ðrÞ. In the literature for homogeneous materials such as steel, the functions g1 and g2 are taken to be linear functions of temperature whereas g3 is considered both linear (Nowinski, 1959) and quadratic (Stanisic and McKinley, 1962) functions of w.
E ¼ Eðw; rÞ ¼ E0 E1 w E2 w2 f1 ðrÞ;
(1)
at ¼ at ðw; rÞ ¼ ðat0 þ at1 wÞf2 ðrÞ;
(2)
l ¼ lðw; rÞ ¼ ðl0 l1 wÞf3 ðrÞ;
(3)
in which E, at and l are respectively the elasticity modulus, thermal expansion coefficient and heat conductivity, and fi ðrÞ is the grading function. E0 , E1 , E2 , at0 , at1 , l0 and l1 are material constants. The Poisson ratio n is assumed to be constant for FG materials. In the present study, we consider an incompressible elastic material with n ¼ 0:5. This assumption is necessary to obtain an exact solution of thermoelasticity equations. In a future investigation, we will address thermal stresses in an FG material with an arbitrary Poisson ratio by using a perturbation solution of thermoelasticity equations. In this work, we consider an FGM with a grading function of the form ðr=Ro Þmi . Thus, the above equations can be rewritten as
E ¼ Eðw; rÞ ¼ E0 E1 w E2 w2 ðr=Ro Þm1 ;
(4)
at ¼ at ðw; rÞ ¼ ðat0 þ at1 wÞðr=Ro Þm2 ;
(5)
l ¼ lðw; rÞ ¼ ðl0 l1 wÞðr=Ro Þm3 :
(6)
2.1. Heat conduction In this section, the governing equations of steady heat conduction in a functionally graded hollow cylinder with temperaturedependent heat conductivity are presented. These equations can be obtained by considering the energy conservation law, the Fourier constitutive equation for heat conduction and the dependence of heat conductivity on spatial coordinates and temperature as pointed out in Equation (6). Thus, we start with the steady energy conservation law without heat generation:
V$q ¼ 0;
(7)
where V and q are the nabla operator and heat flux vector, respectively. The Fourier constitutive law along with the energy conservation equation (7) gives the following field equation for the temperature:
V$ðlVwÞ ¼ 0:
(8)
This equation for an infinitely long cylinder with axisymmetric temperature field reads
1 d dw rlðw; rÞ ¼ 0: r dr dr
(9)
In order to be solved, this equation needs two boundary conditions in r direction. In this paper, we consider prescribed temperatures at inner and outer surfaces of the cylinder:
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wðr ¼ Ri Þ ¼ wi ;
wðr ¼ Ro Þ ¼ wo ;
(10)
3. Analytical solution In order to obtain the thermal stresses in the cylinder, first the heat conduction problem is to be solved in order to obtain the temperature field. Once the temperature field is known, we proceed to solve the thermoelasticity problem. In this section, analytical solution of these problems are presented. In what follows, we first present the solution of the heat conduction problem and then the thermoelastic solution is given.
where Ri and Ro are the inner and outer radii of the hollow cylinder, respectively. However, the method presented in this paper is not restricted to these boundary conditions and other types of boundary conditions can be treated as well. When a homogeneous material is considered, l is assumed to be constant whereas for a functionally graded material, l depends on the spatial coordinates. Both cases lead to linear differential equations, but the former yields a linear differential equation with variable coefficients. However, experimental observations show that the properties of real materials depend on temperature. This results in governing equations becoming nonlinear. Thus, the analysis of the problem gets more difficult. Due to this inherent difficulty, it is often tried to refrain from such a full nonlinear description. However, the advancement of technology pushes us towards a more accurate description for the behavior of engineering materials.
The heat conduction problem is solved using the perturbation method. For this purpose, a small parameter ε is to appear in the nonlinear differential Equation (9). For most of engineering materials, one can postulate that the value of l1 is much smaller than the value of l0 . But, in order to make the comparison reasonable, we first have to non-dimensionalize Equation (3):
2.2. Thermoelasticity
lwref ¼ wref εw f3 ðrÞ; l0
For the thermoelastic analysis, we use the Navier equations for a thick-walled cylindrical shell. For this purpose, we start with the differential form of the equilibrium condition which in axisymmetric cylindrical coordinates reads
dsr 1 þ sr s4 ¼ 0: r dr
(11)
In the above equations sr and s4 are the radial and circumferential or hoop stresses, respectively. This axisymmetric problem has only two displacement components in radial (r) and axial (z) directions denoted by u and w, respectively. The strain-displacement relations then read
εr ¼
du ; dr
u ε4 ¼ ; r
εz ¼
dw : dz
(12)
The generalized Hooke's law of thermoelasticity for an FG material with temperature-dependent properties and n ¼ 0:5 is
1 1 sr s4 þ sz þ εr ¼ Eðw; rÞ 2
wðrÞ Z
at ðT; rÞdT;
wðrÞ Z 1 1 s4 ðsr þ sz Þ þ at ðT; rÞdT; Eðw; rÞ 2
(14)
(15)
The thermoelastic problem is subjected to the following boundary conditions. The cylinder is loaded with internal and external pressures pi and po , respectively:
sr ðr ¼ Ro Þ ¼ po :
(17)
in which wref is a reference temperature. We can now postulate that ε≪1 for most of solid materials in the range of temperatures normally present in engineering problems. The rationale for this assumption is as follows. The heat conductivity at a reference temperature w0 is l0 f3 ðrÞ and by varying the temperature, it changes to a new function ðl0 l1 wÞf3 ðrÞ. However, as long as the temperature variation from the reference value is not very intense, one expects that the change in heat conductivity is small. Thus, l1 has to be small compared to l0 . Therefore, we can define the small parameter ε as
ε¼
wref l1 ≪1: l0
(18)
In the sequel, wref is taken to be unity, i.e. wref ¼ 1, for the sake of simplicity. Thus, it does not appear anymore in the subsequent equations. Equation (9) can be casted as
d dw dw dl dw d2 w rl ¼l þr þ rl 2 ¼ 0; dr dr dr dr dr dr
(19)
m3 dl d r ¼ ðl0 l1 wÞ dr dr Ro dðl0 l1 wÞ r m3 d r m3 þ ðl0 l1 wÞ : ¼ dr Ro dr Ro
(20)
The use of the chain rule of differentiation yields
T¼0
sr ðr ¼ Ri Þ ¼ pi ;
wref l1 ; l0
in which
T¼0 wðrÞ Z 1 1 sz sr þ s4 þ εz ¼ at ðT; rÞdT: Eðw; rÞ 2
ε¼
(13)
T¼0
ε4 ¼
3.1. Solution of heat conduction problem
(16)
As a special case, when pi ¼ po ¼ 0 then the stresses in the cylinder are solely due to thermal effects. On the other hand, when the temperature is zero everywhere within the cylinder, then the stresses are purely mechanical due to internal and external pressures.
dðl0 l1 wÞ dðl0 l1 wÞ dw dw ¼ ¼ l1 : dr dw dr dr
(21)
Substituting Equation (21) into Equation (19) reveals
dw dw d2 w þ rðl0 l1 wÞ 2 ¼ 0: ð1 þ m3 Þðl0 l1 wÞ rl1 dr dr dr
(22)
By dividing both sides of Equation (22) by the non-zero constant l0 we have
dw dw d2 w þ rð1 εwÞ 2 ¼ 0: ð1 þ m3 Þð1 εwÞ εr dr dr dr
(23)
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Equation (23) is a nonlinear differential equation and we employ the regular perturbation method (Poincare's technique) to solve it. For this purpose, we expand the temperature w as a power series of ε:
w ¼ w0 þ εw1 þ ε2 w2 þ / ¼
∞ X
εn wn :
(24)
n¼0
This expansion can be truncated at any order to give an approximate temperature field, e.g.
w0 ¼ w0 ;
(25)
w1 ¼ w0 þ εw1 ;
(26)
w2 ¼ w0 þ εw1 þ ε2 w2 :
2 2 d w0 dw0 d w1 dw 1 þ ε r ε r þ ð1 þ m Þ þ ð1 þ m3 Þ 1 3 dr dr dr2 dr2 2 2 d w0 dw0 dw0 dw0 d w2 r þ ε2 r w0 r þ ð1 þ m3 Þ dr dr dr dr2 dr 2 2 dw d w1 dw þ ð1 þ m3 Þ 2 w0 r þ ð1 þ m3 Þ 1 dr dr dr2 2 d w0 dw0 þ O ε3 ¼ 0: w1 r þ ð1 þ m3 Þ dr dr2
(28)
(29)
(30)
This general solution is subjected to the following boundary conditions:
w0 ðr ¼ Ri Þ ¼ wi ;
w0 ðr ¼ Ro Þ ¼ wo :
(31)
Applying these boundary conditions to the general solution (30) gives the following relations for the integration constants C1;0 and C2;0 :
C1;0
C2;0 ¼ wo m3 ; Ro
C2;0 ¼
wo wi 3 3 Rm Rm o i
(32)
:
(33)
The Oðε1 Þ problem is obtained by setting the bracket multiplied by ε1 in Equation (28) to zero:
d2 w1 dw O ε1 : r þ ð1 þ m3 Þ 1 dr dr2 2 d w0 dw0 dw dw0 þr 0 : þ ð1 þ m Þ ¼ w0 r 3 dr dr dr dr2
which is an inhomogeneous Cauchy-Euler differential equation whose solution is composed of two parts, i.e. the homogeneous solution and a particular solution. The homogeneous solution of Equation (35) is the same as the solution of Equation (29). A particular solution of Equation (34) can be easily calculated. The general solution of Equation (34) reads 2 C2;0 C2;1 þ 2m ; m r 3 2r 3
(36)
which is to be subjected to the following boundary conditions:
w1 ðr ¼ Ri Þ ¼ 0;
w1 ðr ¼ Ro Þ ¼ 0:
(37)
2 C2;0 C2;1 C1;1 ¼ m3 2m ; Ro 2Ro 3
(38)
2 3 3 C2;0 R2m R2m o i C2;1 ¼ m3 : 3 2 Rm R o i
(39)
Finally, we come to the Oðε2 Þ problem. In order to obtain the Oðε2 Þ problem we need to set the bracket multiplied by ε2 in Equation (28) to zero:
2 d2 w2 dw d w1 dw O ε2 : r þ ð1 þ m3 Þ 2 ¼w0 r þ ð1 þ m3 Þ 1 2 2 dr dr dr dr 2 d w0 dw0 : þ w1 r þ ð1 þ m Þ 3 dr dr2 (40)
which can also be obtained by setting ε ¼ 0 in Equation (28). This is a Cauchy-Euler differential equation whose general solution reads
C2;0 : r m3
(35)
Application of these boundary conditions to the general solution (36) yields the integration constants:
This series can be continued and more terms could be obtained. But in this paper, we restrict ourselves to the above order of approximation and do not go any further. Since Equation (28) is to be valid for arbitrary values of ε, the bracketed terms are to vanish. This results in a series of problems which we solve in the sequel. First, the Oðε0 Þ is to be solved:
w0 ðrÞ ¼ C1;0 þ
d2 w1 dw dw dw0 ; O ε1 : r þ ð1 þ m3 Þ 1 ¼ r 0 dr dr dr dr2
(27)
0
d2 w0 dw O ε0 : r þ ð1 þ m3 Þ 0 ¼ 0; dr dr2
The first term in the right-hand side of Equation (34) vanishes due to Equation (29). Thus, the Oðε1 Þ problem can be written as
w1 ðrÞ ¼ C1;1 þ
Substituting expansion (24) in Equation (23) and grouping all terms with the same power of ε we get
215
(34)
This equation can be rewritten in the following form by taking into account Equations (29) and (35):
d2 w2 dw dw dw0 : O ε2 : r þ ð1 þ m3 Þ 2 ¼ rw0 0 dr dr dr dr2
(41)
This is again an inhomogeneous Cauchy-Euler differential equation whose general solution is written as the sum of the homogeneous solution and a particular solution:
w2 ðrÞ ¼ C1;2 þ
2 3 C2;0 C2;2 C1;0 C2;0 þ þ : m r 3 2r 2m3 6r 3m3
(42)
The boundary conditions for the Oðε2 Þ problem are
w2 ðr ¼ Ri Þ ¼ 0;
w2 ðr ¼ Ro Þ ¼ 0;
(43)
which give the following values for the integration constants C1;2 and C2;2 : 2 3 C2;0 C2;2 C1;0 C2;0 C1;2 ¼ m3 ; 3 3 Ro 2R2m 6R3m o o
C2;2
2 3 3 3 3 3 R2m R3m C1;0 C2;0 R2m C2;0 R3m o o i i : þ ¼ m3 m3 3 3 Rm R 2 Rm 6 R o o i i
(44)
(45)
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Now, the approximate temperature fields obtained by the perturbation method read
C2;0 þ O ε1 ; w0 ðrÞ ¼ ε0 w0 ¼ ε0 C1;0 þ m r 3
(46)
w1 ðrÞ ¼ w0 þ ε1 w1 ! 2 C2;0 C2;0 C2;1 þ ε1 C1;1 þ m þ 2m þ O ε2 ; ¼ ε0 C1;0 þ m 3 3 3 r r 2r (47) w2 ðrÞ ¼ w0 þ ε1 w1 þ ε2 w2 ! 2 C2;0 C2;0 C2;1 0 1 ¼ ε C1;0 þ m3 þ ε C1;1 þ m3 þ 2m r r 2r 3 ! 2 3 C2;0 C2;2 C1;0 C2;0 þ þ O ε3 : þε2 C1;2 þ m3 þ r 2r 2m3 6r 3m3
wo wi ln RRoi
(49)
;
(50)
C2;0 ¼ wo C1;0 lnRo :
(51)
2 C1;0
2
ln2 r þ C1;1 lnr þ C2;1 ;
ln Ro ln Ri
2
ln RRoi
C1;1 ¼
2 C1;0
C2;1 ¼
w ¼ A1 z þ A2 ;
2
εz ¼
dw ¼ A1 : dz
(59)
Constants A1 and A2 are to be determined from the constraints. If we assume that wðz ¼ 0Þ ¼ 0, then we have A2 ¼ 0. We will discuss later about how to determine A1 . With this, Equation (58) can be rewritten as
εr þ ε4 þ εz ¼
(52)
in which the integration constants C1;1 and C2;1 are obtained by applying the boundary conditions: 2
(58)
in which kinematic relations (12) are employed. In this problem, the axial strain εz does not depend on the axial coordinate z and hence the axial displacement w is a linear function of z:
εr þ ε4 þ εz ¼
2 C1;0
du u dw þ þ ; dr r dz
du u 1 d þ þ A1 ¼ ðruÞ þ A1 : dr r r dr
(60)
By using strain-stress relations (13), (14) and (15), the sum of normal strains reads
The first-order solution can also be obtained as
w1 ðrÞ ¼
In this subsection, an exact solution of thermoelastic problem for the case of n ¼ 0:5 is presented and the displacement and stress fields are calculated. A solution technique for the case with arbitrary n is in progress and will be addressed in a future work. The analytical solution starts with the sum of normal strains
εr þ ε4 þ εz ¼
where the integration constants C1;0 and C2;0 are given by
C1;0 ¼
3.2. Solution of thermoelastic problem
(48)
The above solution is valid for the case when m3 s0. The case m3 ¼ 0 which describes a homogeneous material with temperature-dependent material properties needs special care. in this case, the zeroth-order solution reads
w0 ðrÞ ¼ C1;0 lnr þ C2;0 ;
Now, the approximate temperature fields can be obtained using Equations (25)e(27). In the following subsection, these approximate solutions are examined for their accuracy and then they will be used to solve the thermoelastic problem. The latter leads to displacement and stress distributions within the cylinder thickness.
2
;
at ðT; rÞdT:
(61)
This differential equation has the following general solution for the radial displacement uðrÞ:
3 uðrÞ ¼ r (54)
wðrÞ Z
T¼0
(53)
ln2 Ro C1;1 lnRo :
1 d ðruÞ þ A1 ¼ 3 r dr
Z
0 B r@
wðrÞ Z
1 A C C at ðT; rÞdT Adr 1 r þ 1 : 2 r
(62)
T¼0
Considering Equation (2), it shall be noted that
Finally, the second-order solution is as follows:
w2 ðrÞ ¼
3 C1;0
6
ln3 r þ
2 C2;0 C1;0
2
wðrÞ Z
ln2 r þ C1;2 lnr þ C2;2 :
(55) T¼0
The homogeneous boundary conditions yield the following formulae for the integration constants C1;2 and C2;2 : 3 C1;0 6
C1;2 ¼
3 C2;0 C 2 ln Ro ln3 Ri þ 2 1;0 ln2 Ro ln2 Ri ln RRoi
;
C2;1 ¼
6
ln3 Ro
2 C2;0 C1;0
2
ln2 Ro C1;2 lnRo :
1 ðat0 þ at1 TÞf ðrÞdT ¼ at0 w þ at1 w2 f ðrÞ: 2
T¼0
(63) From Equation (15) the following relation for axial stress sz is obtained:
(56) sz ¼
3 C1;0
wðrÞ Z
at ðT; rÞdT ¼
(57)
1 sr þ s4 þ A1 Eðw; rÞ Eðw; rÞ 2
wðrÞ Z
at ðT; rÞdT: T¼0
Now, using Equations (13), (14), (62) and (64) we have
(64)
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
wðrÞ Z
sr s4 ¼2Eðw;rÞ
at ðT;rÞdT T¼0
4 Eðw;rÞ 2Eðw;rÞ C1 2 ¼ 3 r r2
Z
4Eðw;rÞ r2
Z
0 B r@
wðrÞ Z
1 C at ðT;rÞdT Adr
dsr Eðw; rÞ 4 C 2 ¼ 3 1 dr r3
0
T¼0
sr ¼
r 2 at ðw; rÞ
dw dr : dr
(66)
Z Eðw; rÞ 4 dw 2 C dr dr þ C2 : 2 r a ðw; rÞ t 1 3 dr r3
4 po pi 3 D2 þ ; 3 D1 2 D1
(67)
(68)
Z Z Eðw; rÞ dw 2 dr dr r a ðw; rÞ C2 ¼ pi þ 2 t dr r3 r¼Ri Z 4 Eðw; rÞ C1 dr ; 3 r3 r¼Ri
D1 ¼
ZRo
Eðw; rÞ dr; r3
D2 ¼
Ri
r 2 at ðw; rÞ
dw 4 Eðw; rÞ dr þ C1 2 : dr 3 r
ZRo D3 ¼
EðwðrÞ; rÞ2prdr:
(75)
Ri
In this section, some selected results obtained from the proposed analytical solution are presented. The results presented in this section are obtained using the following numerical values of parameters:
E0 ¼ 2:1 105 MPa;
E1 ¼ 27:5 MPa= C;
E2 ¼ 0:141 MPa= C2 ;
(76)
at1 ¼ 1 108 C2 ;
(77)
l1 ¼ 0:0293 W=m C2 :
(78)
(70)
These are realistic values for engineering materials. However, they are used here just for the demonstration of the results and any other set of parameters can be used as well. Because, it is immaterial for the analytical solution which parameters set is used. First, temperature profiles are shown. In order to check the validity of the presented approximate solution based on the perturbation method, a numerical solution of the full nonlinear heat conduction problem is supplied for comparison. Fig. 1 shows the temperature profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation. The solution of full nonlinear governing equations obtained by an accurate numerical solution based on the finite difference method is also plotted for comparison. The convergence of perturbation series from the zeroth-order solution (i.e., linear problem) towards the “exact” solution by increasing the order of approximation is demonstrated
(71)
The axial stress sz is calculated from Equation (64) once sr and s4 are known. The remaining quantity to be determined now is A1 which depends on boundary condition in the axial direction z. Two cases for this boundary condition can be considered. First, is a plane strain state in which
εz ¼ A1 ¼ 0:
where
l0 ¼ 50:16 W=m C;
Z Eðw; rÞ dw 2 dr dr: r a ðw; rÞ t dr r3
Z
(74)
T¼0
(69)
Ri
2Eðw; rÞ r2
3
7 C at ðT; rÞdT A2prdr 5;
The circumferential stress s4 is obtained using Equation (65):
s4 ¼ sr
1
at0 ¼ 1:2 105 C1 ;
in which
ZRo
wðrÞ Z
4. Results and discussions
The integration constants C1 and C2 are determined from boundary conditions (16):
C1 ¼
B @
Z
The radial stress sr is obtained by integrating once the above differential equation:
Z
2 Z 1 6 A1 ¼ 4P þ p po R2o pi R2i þ EðwðrÞ; rÞ D3
dw 4 Eðw;rÞ r 2 at ðw;rÞ dr C1 2 : dr 3 r (65)
Thus, the equilibrium condition (11) can be written as follows:
217
(72)
This happens, for example, when the cylinder is clamped between two rigid walls and its elongation is not allowed. The second case is when an external axial force P is exerted on the cylinder. In this case, the integration of axial stress sz over any cross section must equals P in order for the axial equilibrium to hold, i.e.
ZRo sz ðrÞ2prdr ¼ P:
(73)
Ri
Inserting the axial stress sz from Equation (64) into Equation (73) and performing the integration yields
Fig. 1. Temperature profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different approximations and compared with a numerical solution of the full nonlinear equation.
218
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Fig. 2. Temperature profiles for wi ¼ 0 C, wo ¼ 500 C and m3 ¼ 2 obtained by different approximations and compared with a numerical solution of the full nonlinear equation.
Fig. 3. Temperature profiles for wi ¼ 0 C, wo ¼ 1000 C and for different values of m3 obtained by the second-order approximation. From top to the bottom, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
in this figure. In order to show the effect of temperature difference between the inner and outer surfaces of the cylinder, temperature profiles for wi ¼ 0+ C, wo ¼ 500+ C and m3 ¼ 2 are depicted in Fig. 2. We see that both w1 and w2 almost coincide with the “exact” solution for this case. It means that the higher the temperature difference the more terms in the perturbation series are required in order to get an accurate solution.
Fig. 4. Elasticity modulus profiles for wi ¼ 0 C, wo ¼ 1000 C and different values of m1 . From bottom to the top, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
Fig. 5. Thermal expansion coefficient profiles for wi ¼ 0 C, wo ¼ 1000 C and different values of m2 . From bottom to the top, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
Fig. 6. Heat conductivity profiles for wi ¼ 0 C, wo ¼ 1000 C and different values of m3 . From bottom to the top, temperature profiles belong to m3 ¼ 3; 2; 1; 0; 1; 2 and 3.
The results presented in the remainder of this paper are obtained by setting wi ¼ 0+ C and wo ¼ 1000+ C. Fig. 3 shows how the value of m3 affects the temperature profile. Larger values of m3 lead to higher temperature levels and vice versa. Also, the curvature of temperature curves changes with the values of m3 . This means that for large positive values of m3 the temperature gradient near to the inner surface is sharp and this sharp gradient weakens by approaching the outer surface. However, for negative values of m3 the behavior is converse. This trend can be explained as follows. For
Fig. 7. Elasticity modulus profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
Fig. 8. Thermal expansion coefficient profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
Fig. 9. Heat conductivity profiles for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
positive m3 , the heat conductivity assumes its minimum value at the inner surface and it increases by approaching the outer surface. Moreover, the radial heat flux is constant in this one-dimensional problem. Thus, by increasing l in the radial direction, the temperature gradient vw=vr has to decrease in order to have a constant heat flux. A converse statement holds for the negative values of m3 . Once the temperature distribution is obtained, one could proceed to compute the profiles of material properties in the radial direction r. The profiles of elasticity modulus E as functions of r for different values of m1 are depicted in Fig. 4. It is observed that the
Fig. 10. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 3 obtained by different orders of approximation.
219
Fig. 11. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
Fig. 12. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 1 obtained by different orders of approximation.
elasticity modulus generally increases by decreasing m1 from positive values towards negative values. A similar behavior is observed for thermal expansion coefficient at and heat conductivity l as seen in Figs. 5 and 6, respectively. Figs. 7e9 respectively show the material properties E, at and l as functions of r for m ¼ m1 ¼ m2 ¼ m3 ¼ 2 obtained by different orders of approximation. Each figure shows the respective material property as predicted by zeroth-, first- and second-order approximate solutions. It shall be noted here that the zeroth-order solution corresponds to the linear solution where temperature dependence
Fig. 13. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 0 obtained by different orders of approximation.
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A. Moosaie / European Journal of Mechanics A/Solids 55 (2016) 212e220
It is now worthwhile to look at the distribution of radial stress component within the cylinder. Radial stress profiles for m ¼ m1 ¼ m2 ¼ m3 ¼ 3; …; 3 are depicted in Figs. 10e16, respectively. Two general trends can be observed in these figures. First, the peak of the radial stress for m ¼ 3 occurs near the inner surface and by gradually increasing the value of m up to 3, the peak location moves towards the outer surface. Second, for negative values of m, the zeroth-order solution overestimates the radial stress whereas for positive values of m, the zeroth-order solution underestimates the radial stress, as compared to the second-order solution. For example, the second-order solution predicts the peak of radial stress about 25% higher than the zeroth-order solution for m ¼ 3. 5. Conclusions Fig. 14. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 1 obtained by different orders of approximation.
In this paper, an analytical solution of thermal stresses in a hollow cylinder made of FGM with temperature-dependent material properties is presented. After introducing the governing equations and boundary conditions of the problem, the heat conduction problem is analytically solved using perturbation technique. This leads to an approximate solution, but the order of accuracy can be increased in a systematic manner. Then, an exact solution for the elasticity equations is derived for the case of n ¼ 0:5. Finally, the temperature field, material properties and radial stress component obtained by different orders of approximation are plotted versus the radial direction. References
Fig. 15. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 2 obtained by different orders of approximation.
Fig. 16. Radial stress for wi ¼ 0 C, wo ¼ 1000 C and m3 ¼ 3 obtained by different orders of approximation.
is disregarded. The first-order solution is a considerable improvement over the zeroth-order solution whereas the further improvement of the second-order solution is modest. It is noticeable that the difference between different orders of solution for the thermal expansion coefficient is small as compared to other material properties shown here. The peak of elasticity modulus profile shown in Fig. 7 moves towards the outer surface by increasing the order of approximation.
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