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International Journal of Advances in Applied Mathematics and Mechanics
Stress analysis in strain hardening two-layer composite tubes subject
to cyclic loading of internal pressure
Research Article
Ahmet N. Eraslana, Tolga Akisb,∗
aDepartment of Engineering Sciences, Middle East Technical University, 06531, Ankara, Turkey bDepartment of Civil Engineering, Atılım University, 06836, Ankara, Turkey
Received 13 August 2015; accepted (in revised version) 11 November 2015
Abstract: Cyclically loaded two-layer composite tubes in the elastic and elastic-plastic stress states are examined in this study. An analytical model based on Tresca’s yield criterion, its associated flow rule, and linear strain hardening is devel-oped to estimate the stresses in the tube assembly with axially constrained ends. Then a comprehensive analysis is performed for tubes having different dimensions under one cycle of loading, unloading, and reloading of internal pressure. This cycle demonstrates an autofrettage process in which a residual stress field is formed inside the tube assembly.
MSC: 74A10• 74C05
Keywords:Elastoplasticity• Composite tubes• Strain hardening•Tresca criterion • Cyclic loading
© 2015 The Author(s). This is an open access article under the CC BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
Thick-walled tubes are commonly used structures that have a wide range of applications in engineering. Chemi-cal and nuclear industry, power plants, pipelines, and military equipment are the major areas of usage of these assem-blies. Due to having wide application areas in different branches of industry, the analysis and design of thick walled tubes, most of which are subjected to internal pressure, are important.
The problem of a single-layered tube under internal pressure was treated in purely elastic stress state by several researchers (Timoshenko [1], Timoshenko and Goodier [2], Ugural and Fenster [3], Boresi, Schmidt and Sidebottom [4]). On the other hand, the fully plastic solution of the problem was reported by Mendelson [5], Nadai [6] and Boresi, Schmidt and Sidebottom [4]. In addition to these studies, Durban and Kubi [7], Lazzarin and Livieri [8], Parker [9], and Perry and Aboudi [10] investigated the internally pressurized tube problem in elastic-plastic stress state. Among them, Lazzarin and Livieri [8], Parker [9], and Perry and Aboudi [10] also studied the autofrettage process of thick-walled tubes. Autofrettage is defined as a process that increases the elastic strength of the tube assembly by initially subjecting the vessel to high internal pressure so that plastic deformations occur in the tube [4]. This process also increases the fatigue lifetime by slowing down the development and propagation of cracks [10].
There also exist studies in which internally pressurized multilayer composite tubes in the elastic and partially plastic states were investigated [11–13]. In recent years, studies focusing on tubes made of functionally graded mate-rials (FGM) were performed by several researchers both in elastic and elastic-plastic stress states. For example, Fukui and Yamanaka [14], Horgan and Chan [15], Tutuncu and Ozturk [16], and Tutuncu [17] treated the internally pressur-ized FGM tube problem in elastic stress state. On the other hand, Eraslan and Akis [18] investigated the elastoplastic response of such tubes and Jahromi et al. [19,20] studied the autofrettage of FGM pressure vessels. Besides these studies, the mechanical and thermal stresses in FGM tubes were studied by Jabbari, Sohrabpour and Eslami [21] and Eraslan [22].
∗
Corresponding author.
Nomenclature
a,b,c Inner, interface, and outer radii of the tube assembly, respectively
Ci Integration constants
E Modulus of elasticity
P Pressure
r,θ,z Cylindrical coordinates
ri Elastic-plastic border radius
u Radial displacement
ǫi Strain components ǫEQ Equivalent plastic strain
η Hardening parameter
ν Poisson’s ratio
σi Stress components
σ0,σy Initial and subsequent yield stress
In the literature, there are also several studies in which thick-layered tubes under combined loading conditions are analyzed. For example, Imaninejad and Subhash [23] investigated thick-walled cylinders subjected to internal pressure and proportional axial loading. Limam, Lee and Kyriakides [24] studied dented tubes under combined bend-ing and internal pressure. Eraslan and Apatay [25] investigated nonlinear strain hardening preheated tubes subjected to internal pressure. Other examples of different types of loading including combined ones on the tubes and other basic structures can be found in [26–31].
In the present work, we extend the internally pressurized two-layer composite tube problem to a cyclically loaded two-layer tube in the elastic and elastic-plastic stress states. A detailed analysis is performed for these tubes under one cycle of loading, unloading, and reloading of internal pressure. This loading cycle under consideration demonstrates an autofrettage process in which a residual stress field is formed inside the tube assembly. In the liter-ature, there are few studies on the cyclic behavior of pressurized tubes. Mahbadi and Eslami [32] studied the cyclic loading behavior of single-layer thick-walled tubes under different types of loading including internal pressure and they used a numerical iterative method in their study. Megahed and Abbas [33] studied the influence of reverse yield-ing on the residual stresses that developed after autofrettage of a syield-ingle-layer tube. Darijani, Kalgarnovin and Naghd-abadi [34] obtained closed-form solutions of the internally pressurized elastoplastic tube problem in which forward and reversed loading is considered. In contrast to those studies, in our work two-layer composite thick-walled tubes are analyzed in which the locations of the yielding along the tube assembly depend on the tube dimensions. The so-lutions obtained in this study may be used in the analysis and design of the two-layer tube assemblies under internal pressure. In addition, the results of this analytical work may be used as benchmark problems for numerical solutions. As mentioned above, the geometry considered here consists of two tightly fitted concentric tubes. A long tube of inner radiusaand outer radiusbis placed in a tube of the same length and of inner radiusband outer radiusc. The assembly is then constrained axially and at this stage it is stress-free. As the internal pressure is applied, stresses are formed in both of the tubes and increase in magnitude with increasing pressures. Depending on the tube dimensions, the yielding commences at the inner surface, at the interface, or at both locations at a critical value of pressureP=Pe. ForP >Pe, the assembly will consist of plastic and elastic regions. When the pressure load is removed, the plastic strains will be frozen and the assembly will have elastic behavior when it is reloaded. In the framework of small deformations, a state of plane strain and Tresca’s yield condition analytical solutions are obtained for the two-layer tubes under the above-mentioned load cycle (loading-unloading-reloading).
2. Formulation
2.1. All elastic
Cylindrical polar coordinates (r,θ,z) are considered. A state of plane strain (ǫz=0) and infinitesimal deforma-tions are presumed. Furthermore, the notation and basic equadeforma-tions of elasticity as given by Timoshenko and Goodier [2] are used. The strain-displacement relations:ǫr =d u/d r,ǫθ=u/r, the equation of equilibrium in the radial direc-tion
dσr
d r +
σr−σθ
and generalized Hooke’s law
ǫr=ǫrp+ 1
E[σr−ν(σθ+σz)] , (2)
ǫθ=ǫp θ+
1
E[σθ−ν(σr+σz)] , (3)
ǫz=ǫpz+1
E[σz−ν(σr+σθ)] , (4)
form the basis for the present analysis and are valid in both elastic (with plastic strainsǫpi =0) and plastic regions. In the equations above,σj represents the normal stress,rthe radial coordinate,ǫj the normal strain,E the modulus of elasticity,νthe Poisson’s ratio, anduthe radial component of the displacement vector. For purely elastic deformations of a tube, the axial stress is obtained from Eq. (4) as
σz=ν(σr+σθ) . (5)
The axial stressσzmay be eliminated from the radial and circumferential strain expressions and results are substituted in the strain-displacement relations to obtain stress-displacement relations. Substituting the stresses in the equation of equilibrium (1), one obtains a differential equation in radial displacementuwith the solution
u(r)=C1
r +C2r, (6)
whereCirepresents an arbitrary integration constant. The stresses are then determined as
σr(r)= E 1+ν
· −C1
r2+
C2 1−2ν
¸
, (7)
σθ(r)= E 1+ν
·
C1
r2+
C2 1−2ν
¸
, (8)
σz(r)= 2νEC2
(1+ν)(1−2ν). (9)
Eqs. (6)-(9) are nothing but the elastic equations of a homogeneous tube with fixed ends. For a two-layer composite tube assembly, the same stress and displacement expressions are valid. However, these expressions contain four un-known integration constants to be determined:C1I andC2Ifor the inner tube andC1I I andC2I I for the outer tube. The elastic solution of composite tubes subjected to internal pressure follows.
2.2. Onset of yield
Using subscript 1 to mark material properties (E,νandσ0(yield stress)) of the inner and 2 for the outer tube and superscriptsIandI I to mark inner and outer tubes, respectively, the boundary conditions for a tube assembly subject to internal pressure and a traction-free outer surface can be stated asσIr(a)= −PandσI Ir (c)=0. In addition, at the interface of the tubes radial stresses and displacements are equal and henceσrI(b)=σrI I(b),uI(b)=uI I(b). Application of these four conditions results in
C1IE1=
a2b2P N1[b2(E1M2−E2M1)+c2(E1N2+E2M1)]
E1(b2−a2)(b2M
2+c2N2)+E2(c2−b2)(a2N1+b2M1), (10)
C2IE1=
a2P M1[b2(E1M2+E2N1)+c2(E1N2−E2N1)]
E1(b2−a2)(b2M
2+c2N2)+E2(c2−b2)(a2N1+b2M1), (11)
C1I I=
a2b2c2P N2(M1+N1)
E1(b2−a2)(b2M
2+c2N2)+E2(c2−b2)(a2N1+b2M1), (12)
C2I I=
a2b2P M2(M1+N1)
E1(b2−a2)(b2M
2+c2N2)+E2(c2−b2)(a2N1+b2M1), (13) where
N1=(1+ν1),M1=N1(1−2ν1),N2=(1+ν2),M2=N2(1−2ν2). (14)
Like in homogeneous ones, the stress state in the two-layer tube assemblies under internal pressure satisfiesσθ> σz>σr throughout. However, unlike the deformation behavior of a single tube, in a two-layer composite tube under internal pressure different modes of plastic flow may take place. According to the Tresca’s yield criterionσθ−σr=σ0, plastic deformation may first begin in the inner tube atr=a, or in the outer tube at the interfacer=b. These two different modes imply the existence of a critical interface radiusb=bCat which yielding commences simultaneously in both tubes at the inner surface (r=a) and at the interface (r=bC). This critical radius may be determined from
σIθ(a)−σI r(a) σ01
=
σI Iθ(b)−σI I r (b)
After some algebraic manipulations, we get
bC=
c
q σ02
£
c2σ02(E
1N2+E2M1)2+4a2E2σ01(E1M2−E2M1) (M1+N1)¤ 2σ02(E1M2−E2M1)
+c 2(E
1N2+E2M1) 2(E2M1−E1M2)
¾1/2
. (16)
The existence of the critical interface radius depends mainly on the material properties of the tubes. This subject is discussed comprehensively in [13]. If the critical interface radius exists, the plastic flow will start at the inner surface for the values ofb≥bCand at the interface of the assembly forb<bC. The nondimensional elastic limit pressure that causes yielding at the inner surfacer=a, which is determined fromσI
θ(a)−σ I
r(a)=σ01, is obtained as
Pe=
Pe σ01
=E1(b
2−a2)(b2M
2+c2N2)+E2(c2−b2)(a2N1+b2M1) 2b2[E1(b2M
2+c2N2)+E2M1(c2−b2)]
. (17)
If the interface radius is less thanbC, yielding begins at the interface according toσI Iθ(b)−σI Ir (b)=σ02. The elastic limit becomes
Pe=
σ02[E1(b2−a2)(b2M2+c2N2)+E2(c2−b2)(a2N1+b2M1)] 2a2c2E
2σ01(M1+N1)
. (18)
3. Elastic-Plastic solution
As stated before, for composite tubes under internal pressure, the stresses satisfyσθ>σz>σr and the yield condition isσy=σθ−σr. The associated flow rule and the equivalence of increment of plastic work giveǫp
θ= −ǫ p r, ǫpz =0 andǫ
p
θ =ǫEQwithǫEQ, being the equivalent plastic strain. On the other hand, for a linear strain material the hardening law can be expressed as
σy=σ0(1+ηǫEQ), (19)
whereηrepresents the hardening parameter. Manipulating strain-displacement relations, Hooke’s law, and plastic strain relations, one may obtain the following stress-displacement relations:
σr= − σ0 2+H(1+ν)
+ E
[2+H(1+ν)](1+ν)(1−2ν) ½
[1+Hν(1+ν)]u
r +[1+H(1−ν
2 )]d u
d r
¾
, (20)
σθ= σ0
2+H(1+ν)
+ E
[2+H(1+ν)](1+ν)(1−2ν) ½
[1+H(1−ν2)]u
r +[1+Hν(1+ν)] d u d r
¾
, (21)
whereH=ησ0/Erepresents the nondimensional hardening parameter. The governing differential equation for this plastic region is obtained by combining Eqs. (20) and (21) and the equation of equilibrium (1). The result is
r2d
2u
d r2+r
d u d r −u=
2σ0(1+ν)(1−2ν)
E[1+H(1−ν2)] r. (22)
The general solution is
u(r)=C3
r +C4r+
σ0(1+ν)(1−2ν)(2 lnr−1)r
2E[1+H(1−ν2)] , (23)
and as a result
σr(r)= − E HC3 [2+H(1+ν)]r2
+ EC4
(1+ν)(1−2ν)
+ (2 lnr −1)σ0
2[1+H(1−ν2)], (24)
σθ(r)= E HC3 [2+H(1+ν)]r2
+ EC4
(1+ν)(1−2ν)
+ (2 lnr +1)σ0
2[1+H(1−ν2)], (25)
σz(r)= 2EνC4 (1+ν)(1−2ν)
+ 2νlnrσ0
1+H(1−ν2), (26)
ǫθp= −ǫpr = 2C3 [2+H(1+ν)]r2
− (1
−ν2)σ0
4. Elastic reloading
As soon as the tube assembly is completely unloaded, the plastic strains are frozen and the assembly behaves purely elastic again. Since the plastic strain distribution is known, the stress distributions upon reloading can be calculated considering the plastic strains on the right-hand side as permanent plastic strains.
The elastic strains are related to the stresses via Hooke’s law,
ǫr=1
E[σr−ν(σθ+σz)]+ǫ
p
r, (28)
ǫθ=1
E[σθ−ν(σr+σz)]+ǫ
p
θ, (29)
ǫz=0= 1
E[σz−ν(σr+σθ)]+ǫ
p
z, (30)
whereǫpi represents the permanent plastic strains after unloading of the internal pressure. Sinceǫpz=0, the axial stress is the same as that given by Eq. (5). The plastic strain expressions in Eq. (27) can be written as
ǫpr = −
A r2+B; ǫ
p θ=
A
r2−B, (31)
where
A= 2C3
2+H(1+ν), (32)
B= (1 −ν2)σ0
E[1+H(1−ν2)]. (33)
The axial stressσzmay be eliminated from the radial and circumferential strain expressions given in Eqs. (28) and (29) and results, and the plastic strains given in Eq. (31) are substituted in the strain-displacement relationsǫr =d u/d r, ǫθ=u/rto obtain the following stress displacement relations:
σr= E
(1+ν)(1−2ν) ·
νu
r +(1−ν) d u d r
¸
+ E
1+ν ·
A r2−B
¸
, (34)
σθ= E
(1+ν)(1−2ν) ·
(1−ν)u
r +ν d u d r
¸
− E
(1+ν) ·
A r2−B
¸
. (35)
The governing differential equation is obtained by substitution of these stress expressions in the equation of equilib-rium (Eq. (1)). The result is
r2d
2u
d r2+r
d u d r −u=
2B(1−2ν)r
1−ν . (36)
The general solution is derived as
u(r)=C5
r +C6r−
B(1−2ν)(1−2 lnr)r
2(1−ν) . (37)
and as a result
σr=
E
1+ν ·
−C5
r2+
C6 1−2ν
¸
+ E
1+ν ·
A r2−
B(1−2 lnr) 2(1−ν)
¸
, (38)
σθ= E 1+ν
·
C5
r2+
C6 1−2ν
¸
− E
1+ν ·
A r2−
B(1+2 lnr) 2(1−ν)
¸
, (39)
σz=
2EνC6 (1+ν)(1−2ν)
+2EνBlnr
1−ν2 . (40)
5. Numerical results
A composite system consisting of steel inner (E=200 GPa,ν=0.3,σ0=430 MPa,H=0.5) and aluminum outer (E =70 GPa,ν=0.35,σ0=100 MPa,H=0.25) tubes is considered. In addition, for the presentation of numerical results the following nondimensional and normalized quantities are used:
r=r
c; σj=
σj σ01
; u= uE1 σ01c
; ǫpj = ǫpjE1
σ01
;P= P σ01
0.0 0.2 0.4 0.6 0.8 1.0 1.2
s
tr
e
s
s
e
s
a
n
d
d
is
p
la
c
e
m
e
n
t
r
σ
σ
θ −
r
σ
σ
θ −θ
σ
θ
σ
z
σ
z
σ
u
ST
AL
r
σ
-0.4 -0.2
0.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate
70
.
0
=
a
=0.886638C
b
Fig. 1.Elastic response of steel-aluminum tube assembly of inner radiusa=0.7 subjected to the elastic limit internal pressure
P=Pe=0.213214 forb=bC=0.886638.
Here we also define the following dimensionless integration constants:
C1=C1/c2 ; C2=C2 ; C3=C3/c2;
C4=C4 ; C5=C5/c2 ; C6=C6. (42)
In order to obtain the unknown integration constants, the following boundary and interface conditions are valid: Boundary conditions: 1.σr(a)= −P; 2.σr(c)=0.
Interface conditions: 1.σIr(b)=σI Ir (b); 2.uI(b)=uI I(b).
In addition, for a tube having elastic and plastic deformations, the following conditions should be taken into consid-eration:
1.σpr(ri)=σer(ri); 2.σp θ(ri)=σ
e
θ(ri); 3.u p(r
i)=ue(ri).
Here,ri is defined as the plastic-elastic border radius at the inner (i =1) or outer tube (i =2) and the superscripts
e andprepresent the plastic and elastic regions, respectively. For example, if the inner tube has plastic and elastic regions, the conditions that should be used are: 1.σrp I(r1)=σe Ir (r1); 2.σ
p I θ (r1)=σ
e I
θ (r1); and 3.u
p I(r1)=ue I(r1). The
superscriptse I andp I are used to indicate the plastic and elastic regions in the inner tube. For the determination of the integration constants of the plastically predeformed tube assembly subjected to the reloading of the internal pres-sure, the general boundary and interface conditions given above are valid. Additionally, at the plastic-elastic interface of the tube, the following conditions should be used: 1.σrr(ri)=σer(ri); 2.ur(ri)=ue(ri). Here, the superscriptr rep-resents the reloading region. Finally, the solution data for which the figures are drawn are given in the Appendix. The values used in the solution and the parameters that are evaluated during calculations are presented for each figure in this part.
First, the inner radius is taken asa=0.7. By virtue of Eqs. (16) and (17), the critical interface radius and the corresponding critical elastic limit internal pressure are calculated asbC =bC/c=0.886638 andPe=0.213214, re-spectively. If these values are substituted in Eqs. (10)-(13), the unknown integration constants can be obtained. (The calculated integration constantsC1I,C
I 2,C
I I 1,andC
I I
2 are given in the Appendix-Fig. 1). Fig. 1shows the consequent stresses and displacement. As seen in this figure,σθ(a)−σr(a)=1 andσθ(bC)−σr(bC)=1, which implies the forma-tion of plastic regions at the inner surface and at the interface of the tube assembly. It should also be noted for this figure that the radial stress and displacement are continuous at the interface satisfying interface conditions.
For the values of pressureP >Pe the composite tube becomes partially plastic, consisting of an inner plastic region ina≤r≤r1, an inner elastic region inr1≤r≤bC, an outer plastic region inbC≤r≤r2, and an outer elastic region inr2≤r≤c. The unknowns to be determined for the solution are:C3I,C
I
4(inner plastic),C I 1,C
I
2(inner elastic),
of the plastic regions is shown inFig. 2(a). As seen in this figure, both plastic regions expand with increasingP, and at
P=P1=0.257815 the outer tube becomes fully plastic sincer2=c. The stresses in partially plastic state for this tube assembly may be observed by considering an internal pressurePinPe<P<P1.
For the tube assembly considered (a=0.7), assigningb=0.90>bCand using Eq. (17) the elastic limit internal pressure is obtained asPe=0.218906. InFig. 2(b), the expansion of plastic regions in the pressurized tube is given. Plastic deformation commences at the inner surface as soon asP=Pe=0.218906. As this plastic region spreads into the assembly asP increases, another critical limit,P=P1=0.226060, is reached at which the outer tube begins to yield atr=bbased onσθ−σr=σ0as well. Thereafter, both plastic regions expand with increasing internal pressure and atP=P2=0.266289 the outer tube becomes fully plastic while the inner tube has plastic and elastic regions.
Assigning the interface radiusb=0.80<bC and using Eq. (18), the elastic limit pressure is obtained asPe= 0.135444.Fig. 2(c) shows the expansion of plastic regions in the tube assembly. The yielding begins at the interface as soon asP=Pe=0.135444 is reached. As this plastic region spreads into the assembly asPincreases, another critical limit,P=P1=0.166937, is reached at which the inner tube begins to yield atr=abased onσθ−σr =σ0as well. Thereafter, both plastic regions expand with increasing internal pressure and atP =P2=0.191958 the outer tube becomes fully plastic while the inner tube has plastic and elastic regions.
In order to show the distributions of stresses, plastic strains, and displacement, the same assembly (a=0.7) withb=bC=0.886638 is considered. ForP=0.24<P1the distributions of stress and plastic strain components, and radial displacement are plotted inFig. 3(a). The corresponding residual stresses and displacement upon complete removal of the loadP1=0.24 are displayed inFig. 3(b). In order to determine the elastic response of the plastically predeformed tube assembly subjected to the reloading of the internal pressure, four regions should be considered: an inner reloading region ina≤r≤r1, an inner elastic region inr1≤r≤bC, an outer reloading region inbC≤r≤r2, and an outer elastic region inr2≤r≤c. Eight unknown integration constants are determined:CI
5andC6I(inner reloading region),C1IandC2I(inner elastic region),C5I IandC6I I(outer reloading region), andC1I I andC2I I (outer elastic region). For this case, the elastic limit reloading pressure is calculated asPer=0.230293 and the corresponding stresses and deformation are plotted inFig. 3(c). As seen in this figure, plastic deformation caused by the reloading of internal pressure commences at the inner tuber=aaccording to the same yield conditionσy=σθ−σr.
0.22 0.23 0.24 0.25 0.26 0.27
p
re
s
s
u
re
ST
AL
1
r
2
r
257815
.
0
1=
P
213214 . 0
=
P
0.20 0.21
0.70 0.75 0.80 0.85 0.90 0.95 1.00
border radius
70
.
0
=
a
213214 . 0
=
e P
886638 . 0
=
C b
0.22 0.23 0.24 0.25 0.26 0.27 0.28
p
re
s
s
u
re
ST
AL
218906 . 0 = e
P
226060
.
0
1
=
P
266289
.
0
2
=
P
0.20 0.21
0.70 0.75 0.80 0.85 0.90 0.95 1.00
border radius
90 . 0 =
b
70
.
0
=
a
(a) (b)
0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21
p
re
s
s
u
re
ST
AL
166937
.
0
1
=
P
191958
.
0
2
=
P
1
r
r
20.12 0.13
0.70 0.75 0.80 0.85 0.90 0.95 1.00
border radius 135444
. 0
=
e P
70
.
0
=
a
b =0.80(c)
Fig. 2.Expansion of plastic regions with increasing pressure in the steel-aluminum tube assembly (H1=0.5,H2=0.25) of inner
0.0 0.2 0.4 0.6 0.8 1.0 1.2 s tr e s s e s , p la s ti c s tr a in s a n d d is p la c e m e n
t
σ
θ−
σ
r rσ
σ
θ−
θσ
θσ
zσ
zσ
u
ST
AL
rσ
1r
r
2p θ
ε
p rε
pε
p θε
-0.4 -0.20.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate p r
ε
70
.
0
=
a
=0.886638C b -0.06 -0.04 -0.02 0.00 0.02 0.04 re s id u a l s tr e s s e s a n d d is p la c e m e n t 0 θ
σ
0 θσ
0 zσ
0 zσ
0 rσ
0u
ST
AL
0 rσ
0u
-0.10 -0.080.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate
70
.
0
=
a
=0.886638C b (a) (b) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 s tr e s s e s a n d d is p la c e m e n t r
σ
σ
θ−
σ
θ−
σ
rθ
σ
θσ
zσ
zσ
u
ST
AL
rσ
-0.4 -0.20.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate
70
.
0
=
a
bC =0.886638(c)
Fig. 3.(a) Partially plastic response of the steel-aluminum tube assembly of inner radiusa=0.7 subjected to the internal pressure
P=0.24 forb=bC=0.886638,H1=0.5, andH2=0.25, (b) residual stresses and displacement upon complete unloading, (c)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 s tr e s s e s , p la s tic s tr a in s a n d d is p la c e m e n t r
σ
σ
θ −r
σ
σ
θ −θ
σ
θσ
zσ
zσ
u
ST
AL
rσ
p θε
p θε
p rε
pε
1r
-0.4 -0.20.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate p r
ε
70
.
0
=
a
=0.886638C b -0.12 -0.08 -0.04 0.00 0.04 0.08 re s id u a l s tr e s s e s a n d d is p la c e m e n t 0 θ
σ
0 θσ
0 zσ
0 zσ
0 rσ
0u
ST
AL
0 rσ
0u
-0.16 -0.120.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate
70
.
0
=
a
=0.886638C b (a) (b) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 s tr e s s e s a n d d is p la c e m e n t r
σ
σ
θ −
σ
θ −σ
rθ
σ
θσ
zσ
zσ
u
ST
AL
rσ
-0.4 -0.20.70 0.75 0.80 0.85 0.90 0.95 1.00
radial coordinate
70
.
0
=
a
=0.886638C
b
(c)
Fig. 4. (a) Partially plastic response of the steel-aluminum tube assembly of inner radiusa=0.7 subjected to the internal pressure
P=P1=0.257815 forb=bC=0.886638,H1=0.5, andH2=0.25, (b) residual stresses and displacement upon complete
6. Concluding remarks
An analytical model, based on Tresca’s yield criterion, its associated flow rule, and linear hardening, is devel-oped to estimate the elastic, elastic-plastic, and elastic reloading behavior of two-layer concentric tubes with axially constrained ends. Analytical expressions for the critical values of pressure leading to onset of yield for both elastic and reloading elastic cases are obtained for different tube dimensions. It is found that the elastic limit for the plastically predeformed tube assembly is higher than that for the undeformed tube assembly.
References
[1] S.P. Timoshenko, Strength of Materials: Part II Advanced Theory and Problems. D. van Nostrand Company 3rd ed., New York, 1956.
[2] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity. McGraw-Hill 3rd ed., New York, 1970.
[3] A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity. Prentice Hall 2nd SI ed., New Jersey, 1987. [4] A.P Boresi, R.J. Schmidt, O.M. Sidebottom, Advanced Mechanics of Materials. Wiley 5th ed., New York, 1993. [5] A. Mendelson, Plasticity: Theory and Application. Macmillan, New York, 1968.
[6] A. Nadai, Plasticity. McGraw-Hill, New York, 1931.
[7] D. Durban, M. Kubi, A general solution for the pressurized elastoplastic tube, J. of Applied Mechanics 59 (1992) 20–26.
[8] P. Lazzarin, P. Livieri, Different solutions for stress and strain fields in autofrettaged thick walled cylinders, Int. J. of Pressure Vessels and Piping 71 (1997) 231–238.
[9] A.P. Parker, Autofrettage of open end tubes-pressures, stresses, strains and code comparisons, J. of Pressure Vessel Technology-Transactions of the ASME 123 (2001) 271–281.
[10] J. Perry, J. Aboudi, Elasto-plastic stresses in thick walled cylinders, J. of Pressure Vessel Technology-Transactions of the ASME 125 (2003) 248–252.
[11] A.N. Eraslan, T. Akis, Deformation analysis of elastic-plastic two layer tubes subjected to pressure: An analytical approach, Turkish J. Eng. Env. Sci. 28 (2004) 261–268.
[12] T. Akis, A.N. Eraslan, Yielding of long concentric tubes under radial pressure based on von Mises criterion, J. Fac. Eng. Arch. Gazi Univ. 20 (2005) 365–372 (in Turkish).
[13] A.N. Eraslan, T. Akis, Yielding of two-layered shrink-fitted composite tubes subjected to radial pressure, Forschung Engineering Research 69 (2005) 187–196.
[14] Y. Fukui, N. Yamanaka, Elastic analysis for thick-walled tubes of functionally graded material subjected to internal pressure, The Japan Society of Mechanical Engineers 35 (1992) 379–385.
[15] C.O. Horgan, A.M. Chen, The pressurized hollow cylinder or disk problem for functionally graded isotropic lin-early elastic materials, J. of Elasticity 55 (1999) 43–59.
[16] N. Tutuncu, M. Ozturk, Exact solutions for stress in functionally graded pressure vessels. Composites: Part B 32 (2001) 683–686.
[17] N. Tutuncu, Stress in thick-walled FGM cylinders with exponentially varying properties, Engineering Structures 29 (2007) 2032–2035.
[18] A.N. Eraslan, T. Akis, Plane strain analytical solutions for a functionally graded elastic-plastic pressurized tube, Int. J. of Pressure Vessels and Piping 83 (2006) 635–644.
[19] B.H. Jahromi, G.H. Farrahi, M. Maleki, H.N. Hashemi, A. Vaziri, Residual stresses in autofrettaged vessel made of functionally graded material, Engineering Structures 31 (2009) 2930–2935.
[20] B.H. Jahromi, A. Ajdari, H.N. Hashemi, A. Vaziri, Autofrettage of layered functionally graded metal-ceramic com-posite vessels, Comcom-posite Structures 92 (2010) 1812–1822.
[21] M. Jabbari, S. Sohrabpour, M.R. Eslami, Mechanical and thermal stress in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. of Pressure Vessels and Piping 79 (2002) 493–497.
[22] A.N. Eraslan, Stresses in FGM pressure tubes under non-uniform temperature distribution, Structural Engineer-ing and Mechanics 26 (2007) 393–408.
[23] M. Imaninejad, G. Subhash, Proportional loading of thick-walled cylinders. Int. J. of Pressure Vessels and Piping 82 (2004) 129–135.
[24] A. Limam, L.H. Lee, S. Kyriakides, On the collapse of dented tubes under combined bending and internal pres-sure, Int. J. of Mechanical Sciences 55 (2012) 1–12.
[25] A.N. Eraslan, T. Apatay, Analytical solution of nonlinear strain hardening preheated pressure tube, Turkish J. Eng. Env. Sci. 32 (2008) 41–50.
[26] A.R. Shahani, S.M. Nabavi, Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading, Applied Mathematical Modeling 31 (2007) 1807–1818. [27] C.S. Oh, Y.J. Kim, K.B. Yoon, Elastic-plastic behaviours of pressurized tubes under cyclic thermal stresses with
[28] X.T. Zheng, F.Z. Xuan, Autofrettage and shakedown analysis of strain hardening cylinders under thermo-mechanical loadings, J. of Strain Analysis for Engineering Design 46 (2011) 45–55.
[29] M. FarhadNia, N. Namdaran, J.E. Jam, M. Zamani, O. Yaghobizadeh, Analysis investigation of composite lattice conical shells as satellite carrier adapter for aerospace applications, Int. J. of Advances in Applied Mathematics and Mechanics 1 (2014) 40–51.
[30] K.R Gaikwad, Mathematical modelling of thermoelastic problem in a circular sector disk subject to heat genera-tion, Int. J. of Advances in Applied Mathematics and Mechanics 2 (2015) 183–195.
[31] C.N. Nath, S.A. Shah, The effect of rigidity on torsional vibrations in a two layered poroelastic cylinder, Int. J. of Advances in Applied Mathematics and Mechanics 3 (2015) 116–121.
[32] H. Mahbadi, M.R. Eslami, Cyclic loading of thick vessels based on the Prager and Armstrong-Frederic kinematic hardening models, Int. J. of Pressure Vessels and Piping 83 (2006) 409–419.
[33] M.M. Megahed, A.T. Abbas, Influence of reverse yielding on residual stresses induced by autofrettage, Int. J. of Mechanical Sciences 33 (2003) 139–150.
[34] H. Darijani, M.H. Kargarnovin, R. Naghdabadi, Design of thick-walled cylindrical vessels under internal pressure based on elasto-plastic approach, Materials and Design 30 (2009) 3537–3544.
Appendix
Solution Data :
Fig. 1: a=0.7,b=0.886638, P =0.213214,CI
1=6.84775×10
−4,CI
2=3.20626×10
−4,CI I
1 =7.58051×10
−4,CI I 2 = 2.27415×10−4.
Fig. 2(a):a=0.7,b=0.886638,H1=0.5,H2=0.25.
Fig. 2(b):a=0.7,b=0.9,H1=0.5,H2=0.25.
Fig. 2(c):a=0.7,b=0.8,H1=0.5,H2=0.25.
Fig. 3a=0.7,b=0.886638,H1=0.5,H2=0.25,P=0.24, (a)CI
3=1.00014×10
−3
,C4I=5.90196×10
−4
,r1=0.749230,
CI1=7.84480×10
−4
,CI2=3.68356×10
−4
,CI I3 =1.08112×10
−3
,CI I4 =2.85347×10
−4
,r2=0.948570,CI I
1 =8.67650×10
−4 ,
CI I2 =2.60295×10
−4
, (b)C1I=7.70802×10
−4
,CI2=3.60906×10
−4
,CI I1 =8.53283×10
−4
,C2I I=2.55985×10
−4
, (c)CI5= 9.68969×10−4,CI6=5.75599×10−4,CI1=7.53305×10−4,C2I=3.53759×10−4,CI I5 =1.04661×10−3,CI I6 =2.74994×10−4,
CI I1 =8.33139×10
−4
,C2I I=2.49942×10
−4 .
Fig. 4: a=0.7,b=0.886638,H1=0.5,H2=0.25,P =0.257815, (a)CI
3=1.11350×10
−3
,C4I=5.92977×10
−4 ,r1= 0.790550,CI1=8.73395×10
−4
,CI2=4.12386×10
−4
,C3I I=1.20155×10
−3
,C4I I=2.89287×10
−4
, (b)CI1=8.28018×10
−4 ,
CI2=3.87696×10
−4
,CI I1 =9.16621×10
−4
,C2I I=2.74986×10
−4
, (c)CI5=1.05452×10
−3
,CI6=5.65360×10
−4
,C1I= 8.14413×10−4,CI
2=3.84769×10
−4
,C5I I=1.13625×10
−3
,CI I6 =2.69698×10
−4 .
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