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Residual stress analyses of re-autofrettaged thick-walled tubes
International Journal of Pressure Vessels and Piping 98 (2012) 57e64
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Residual stress analyses of re-autofrettaged thick-walled tubes G.H. Farrahi a, *, George Z. Voyiadjis b, S.H. Hoseini a, E. Hosseinian c a
School of Mechanical Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA c Aerospace Engineering Department, Shahid Sattari Aeronautical University of Science and Technology, Tehran, Iran b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 February 2012 Received in revised form 9 July 2012 Accepted 10 July 2012
In this paper the effect of the re-autofrettage process on the residual stress distribution at the wall of a thick-walled tube is considered. For accurate material behavior modeling, it is assumed that the yield surface is a function of all the stress invariants. Also for estimating the behavior of the material under loadingeunloading process, a modified Chaboche’s hardening model is applied. For evaluation of this unloading behavior model a series of loadingeunloading tests are conducted on specimens that are made of the high strength steel, DIN1.6959. In addition the finite element simulations are implemented to simulate the re-autofrettage process and to estimate the residual stresses. The numerical results show that the re-autofrettage process without heat treatment only improves the residual stress distribution in high autofrettage percentage and for low autofrettage percentage this method is not beneficial. However, the re-autofrettage process with ‘‘heat soak’’ treatment generally improves the residual stress distribution. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Re-autofrettage process Nonlinear kinematic hardening First stress invariant Lode angle parameter Residual stress distribution
1. Introduction Autofrettage process is applied to introduce the compressive residual stresses at the bore of thick-walled tubes. The compressive residual stresses enhance the fatigue life time of the thick-walled tubes. The thick-walled tubes are typically constructed from high strength steels. Some of these materials have nearly elasticperfectly plastic behavior in the loading phase of a tensioncompression test, but in the unloading phase they show a strong nonlinear hardening behavior. Milligan et al. [1] showed that the Bauschinger effect [2] significantly decreases the yield strength of high strength steels in compression as a result of prior tensile plastic overload. The Bauschinger effect also reduces the compressive residual stresses at the bore of the thick-walled tubes [3]. For decreasing the influence of the Bauschinger effect on residual stresses distribution at the bore of thick-walled tubes, the reapplication of the autofrettage procedure was proposed [4]. In swage autofrettage which is carried out by inserting an oversized mandrel into the bore of thick-walled tubes, Iremonger and Kalsi [5] showed that a more uniform stress is obtained by inserting the mandrel two times such that in each time from different end. But their results showed that such process does not increase the compressive residual stresses. For hydraulic autofrettage process, * Corresponding author. Tel.: þ98 21 66165533; fax: þ98 21 66000021. E-mail address: [email protected] (G.H. Farrahi). 0308-0161/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2012.07.007
Parker [6] proposed a manufacturing procedure for enhancing residual stresses. The procedure involves initial autofrettage with one or more ‘‘heat soak plus autofrettage’’ sequences. He showed that the Bauschinger effect is significantly reduced with this procedure. Troiano et al. [7] also show the possibility of mitigating the Bauschinger effect via an intermediate heat soak between loading and unloading phase of a tension-compression test. Jahed et al. [8] showed that with applying different pressures in first and second autofrettage processes, higher compressive stress can be obtained. Parker and Huang [9] also showed that the reautofrettage process can be beneficial for spherical vessels. Parker et al. [10,11] showed that the hydraulic re-autofrettage of a swageautofrettaged tube with a low temperature post-autofrettage thermal treatment, results in higher compressive residual stresses. The important aspect of residual stresses estimation in reautofrettage process is the accurate modeling of loadingeunloading behavior. There are many researchers who considered the loadingeunloading behavior of high strength steels. Troiano et al. [12] evaluated the uniaxial Bauschinger effect in several high strength steels and showed that the reduction of the Young’s modulus and yield strength occurs after unloading. Parker et al. [13] proposed a nonlinear kinematic hardening model for predicting the material behavior during initial load reversal. More recent works such as Jahed et al. [14], Hojjati and Hassani [15] and Farrahi et al. [16,17] introduced methods to predict the accurate unloading behavior and precise residual stress estimation.
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All research mentioned above applied the experimental results of the uniaxial tension-compression tests or reversed torsion tests to describe the material behavior under reversed loading. However, recent research studies have shown that the stress state should be incorporated into the constitutive description of the plastic deformation. Spitzig et al. [18e20] based on experimental study, demonstrated the effect of hydrostatic pressure on yielding of metals. Brunig [21] similar to the DruckerePrager yield condition in soil mechanics proposed a yield criterion as a function of the first stress invariant to describe the effect of the hydrostatic pressure on metal plasticity. Later Brunig et al. [22] added the third deviatoric stress invariant in the yield criterion to describe the deformation of metals. Bai and Wierzbicki [23] introduced a yield criterion that is a function of pressure and Lode angle and compared their model with experimental tests conducted on aluminum 2024-T351. Mirone and Corallo [24] showed that, for the metals they tested, the hydrostatic stress has a significant effect on fracture and an insignificant effect on the stress-plastic strain relationship, while the Lode angle has an opposite role. Gao et al. [25] introduced a stressstate dependent plasticity model using the non-associated flow rule and Voyiadjis et al. [26] also introduced a yield criterion based on the DruckerePrager yield condition with incorporating the effect of the Lode angle. In this paper for estimating the residual stresses at the bore of the thick-walled tubes, the plasticity model proposed by Voyiadjis et al. [26] is considered. This model was used by Farrahi et al. [27] to analyze the residual stresses which are introduced into the bore of thick-walled tubes under single autofrettage process. In this plasticity model, the yield criterion and the hardening parameters are affected by stress invariants. In addition the effect of the previous plastic deformation history is incorporated into the hardening parameters. For evaluating the loadingeunloading behavior a series of tensionecompression tests are conducted on specimens which are made of a high strength steel, DIN1.6959. A finite element procedure is also implemented to assess the mentioned loadingeunloading model. Good agreement is obtained between the experimental loadingeunloading results and the simulations from the numerical modeling. Using this model and numerical methods, the re-autofrettage process is simulated in a thick-walled tube and the effect of the “heat soak” process is also considered.
The elastoeplastic behavior of the material considered in this section is assumed to be rate independent and elastically isotropic. Since the strains are infinitesimal, the total strain rate is decomposed into the elastic and plastic components:
¼ 3_ eij þ 3_ pij
(1)
where the superscripts e and p designate the elastic and plastic _ is components, respectively. The accumulated plastic strain rate, p, expressed as:
qffiffiffiffiffiffiffiffiffi p p p_ ¼ m 3_ ij 3_ ij
(2)
where m is the inverse magnitude of the normal vector to the yield surface. The Cauchy stress tensor is defined as:
~ sij ¼ D ijkl
3 kl
p
3 kl
(3)
~ is the fourth-order where s is the Cauchy stress tensor, and D isotropic elastic stiffness tensor that is affected by the damage through the plastic deformation. Isotropic elastic stiffness tensor is defined as:
(4)
~ are Lame constants that are defined as: l and D where ~
~ l ¼
~ vE ð1 þ vÞð1 2vÞ
~ ¼ G
~ E 2ð1 þ vÞ
(5)
~ is the Young’s modulus in the damaged configuration and where E is defined as:
~ ¼ Eð1 fÞ E
(6)
where E is the initial Young’s modulus and f is a scalar damage function which is defined as a function of the accumulated plastic strain. This damage function has no effect on the yield surface and is determined experimentally. In this study, the Poisson’s ratio is assumed to remain unchanged through the plastic deformation. The yield function, f, is defined as follows:
f ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðs XÞ : ðs XÞ Y 0 2
(7)
where X is the kinematic hardening back stress tensor that describes the movement of the yield surface in the deviatoric space and Y is the radius of the yield surface. For accurate estimation of the yield surface movement, the back stress tensor can be divided into finite components, where each component is evaluated independently. Kinematic hardening back stress according to the additive decomposition method proposed by Chaboche and Rousselier [28,29] can be expressed as follows:
Xij ¼
M X k¼1
ðkÞ
Xij
(8)
ðkÞ
where Xij is the k-th component of the back stress tensor. The radius of the yield surface, Y, can be defined as:
Y ¼ sy þ R
2. Formulation of the plasticity model
3_ ij
~ ~ ~ D ijkl ¼ ldij dkl þ G dik djl þ dil djk
(9)
where R is the isotropic hardening force and describes the change in the size of the yield surface. In this work the plastic strains are evaluated using the normality rule 3_
p ij
vF ¼ l_ vsij
(10)
where l_ is the multiplier of time-independent plasticity which will be determined using the consistency condition. In this research, the general formulation of the yield condition and hardening parameters proposed by Voyiadjis et al. [26] are considered. This model modifies the DruckerePrager yield condition with incorporating the Lode angle parameter. Therefore the linear effect of the first stress invariant and nonlinear effect of the Lode angle on the yield condition are assumed. The yield condition is defined as follows:
sy ¼ s0 a1 I1 a2 q
(11)
where s0, a1 and a2 are the material constants, I1 is the first Cauchy stress tensor invariant and q is the Lode angle parameter [26]. The Lode angle parameter is defined as follows:
G.H. Farrahi et al. / International Journal of Pressure Vessels and Piping 98 (2012) 57e64
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 27 J32 q ¼ sinð3qÞ ¼ t1 4 J23
3. Experimental procedures
(12)
where J2 and J3 are the second and the third deviatoric stress invariants. The range of the Lode angle parameter varies between zero and unity. The coefficient of the Lode angle parameter must be limited to a special interval so that the convexity of the yield surface is satisfied. The isotropic hardening force is defined as a function of the first stress invariant, accumulated plastic strain and the previous plastic deformation history. The isotropic hardening force is defined as [26]:
R ¼
Q1 þ Q2 eb1 I1
1 eb2 kp
(13)
where Q1, Q2, b1 and b2 are the material constants and k is an exponential decay function of the previous accumulated plastic strain, p. Eq. (13) shows that the compression accelerates the increase of the yield surface radius whereas tension does not significantly change it. The previous plastic deformation is related to the earlier plastic deformation, after which an elastic unloading occurs. Similar to the isotropic hardening, the previous plastic deformation affects the kinematic hardening as well. This effect can be represented as an exponential decay function multiplied by the kinematic hardening constants. The following formulation is proposed for each kinematic hardening component:
2 ðkÞ ðkÞ ðkÞ X_ ¼ C ðkÞ z 3_ p gðkÞ z X ðkÞ p_ 3
(14)
where z is an exponential decay function of the previous accumulated plastic strain, p. This correction means that with increasing the effect of the previous accumulated plastic strain, the translation rate of the yield surface in the stress space at the current plastic deformation is decreased. To evaluate the plastic strain tensor, the yield function should be differentiated with respect to the stress tensor. Therefore vf =vsij should be calculated first. By differentiation of Eq. (7) with respect to the stress tensor, results in the following equations: 3_
p
b 3ðs XÞ Y ¼ p_ 2Y
(15)
b ¼ 2Y vY Y vsij
(16)
By differentiation of the first Cauchy stress tensor invariant and the Lode angle parameter the following equations are obtained [26]:
vI1 ¼ dij vsij
(17)
vq 27J3 3sij J3 1 1 2 ¼ 2 adjðsÞij þ I1 sij þ dij J2 dij I12 vsij J2 3 3 9 8qJ 3
(18)
2
where dij is the Kronecker delta and ‘adj’ is the adjoint matrix of the Cauchy stress tensor. Using Eq. (17) and Eq. (18), vY=vsij can be calculated as follows:
vY ¼ vsij
b1 dij Q2 eb1 I1 Q2 þ Q2 eb1 I1
R a1 dij
1 1 2 þ I1 sij þ dij J2 dij I12 3 3 9
59
27a2 J3
!
8qJ23
3sij J3 2 adjðsÞji J2
To evaluate the loadingeunloading behavior, a series of tensioncompression tests are carried out on smooth round bars. The diameter of the smooth round bar specimens is 8 mm. The diameter in the specimen shoulder is equal to 12 mm, and the length of the gauge section for these specimens is 12.5 mm. The configuration of this specimen is shown in Fig. 1. For evaluating the loadingeunloading behavior, four loading reversals are applied on test specimens. According to this loading pattern each specimen is loaded in tension up to the prescribed plastic strain level, and then unloading and reversed loading is conducted. After reversed loading the specimen is reloaded in tension up to another plastic strain level and after that, once again the unloading and reversed loading is carried out. The selected material is DIN1.6959 steel which is a high strength, quenched, tempered, forged alloy steel used for construction of pressurized thick-walled tubes. The general chemical and mechanical specifications of this steel were introduced by Farrahi et al. [17]. All experiments are conducted using a 30-ton capacity Instron servohydraulic machine and all of the loading patterns are carried out under displacement control. The experimental results are simulated computationally using the finite element method. Finite element model of the specimen is built in Abaqus/Explicit, by means of the user material subroutine VUMAT. The value of the plasticity model parameters which are calibrated from experimental tests for DIN1.6959 are presented in Table 1 [26,27]. As mentioned previously, in high strength steels the Young’s modulus during the plastic deformation changes because of the damage and the dislocation movement that occurs in the material [3,30]. In the tested material, the Young’s modulus changes significantly through the plastic deformation. The Young’s modulus decreases sharply in the beginning of the plastic deformation but the rate of this change decreases, with further plastic deformation [27]. Another aspect that was observed in the experimental tests on DIN1.6959 is the recovery of the Young’s modulus after reverse loading. Reloading after the tension-compression process recovers some amount of the loss in the Young’s modulus [26]. Such behavior, the reduction in Young’s modulus during the tensile unloading and the recovery of Young’s modulus during the compressive unloading, is reported by Troiano et al. [3] for a series of high strength steels with similar behavior as DIN1.6959. According to the experimental tests which were carried out on DIN1.6959 [26], the following expression is proposed for the change in Young’s modulus due to plastic deformation as an isotropic damage:
f ¼ 0:41e9:2pc
0:6
1 e5:5pt
0:6
where pt is related to the accumulated plastic strain in tension and similarly pc is related to the accumulated plastic strain in compression. In this study, the Poisson’s ratio is assumed to remain unchanged through the plastic deformation. Fig. 2 displays the forceedisplacement response of the test specimen simulation along with the experimental results under the first three reversals of the loading pattern that earlier were defined. In Fig. 2 a and b, each specimen is loaded in tension up to
(19)
Eq. (15) is very difficult to solve analytically and therefore a numerical method should be implemented to solve this equation [26].
(20)
Fig. 1. The smooth round bar specimen configuration.
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Table 1 Loadingeunloading behavior parameters calibrated from experimental tests for DIN1.6959. E ¼ 210 GPa
Q1 ¼ 160 MPa
C1 ¼ 1840 GPa
g3 ¼ 1800
v ¼ 0.3 s0 ¼ 610 MPa a1 ¼ 0.01 a2 ¼ 40
Q2 ¼ 20 MPa b1 ¼ 0.001 b2 ¼ 18 k ¼ 0:9e62p þ 0:1
C2 ¼ 2760 GPa C3 ¼ 260 GPa g1 ¼ 10 000 g2 ¼ 10 000
z1 ¼ 0:98e445p þ 0:02 z2 ¼ 0:99e550p þ 0:01 z1 ¼ 1.0
a prescribed plastic strain level, then unloading and reverse loading is conducted up to a different compression plastic strain level. After the compression loading, the specimen is reloaded in tension. As seen in Fig. 2, the loadingeunloading model can accurately predict the material behavior under the reversed loading, also the change of the Young’s modulus through the plastic deformation is captured by this model. These experimental results show the significant influence of the Bauschinger effect on the yield strength and plastic deformation through the compression part of loading. The comparison between the numerical and experimental results in Fig. 2 shows that the Bauschinger effect is precisely simulated by this model. Fig. 3 displays the forceedisplacement response of the smooth round bar simulation along with the experimental results under the loading pattern that earlier is defined. In Fig. 3 each case shows the results for two tests and each test has the same loadingeunloading path but reloading and second unloading are carried out in different paths. As seen in Fig. 3, there is a good agreement between the proposed loadingeunloading model and the experimental results. In each reversal of the loading pattern the
Fig. 2. The forceedisplacement response of the smooth round bar in the first three reversals of the defined loading pattern for different pre-strain levels.
change of Young’s modulus and the plastic deformation are accurately simulated by this model. 4. Re-autofrettage simulation Autofrettage is a technique used on thick-walled tubes to introduce advantageous residual stresses and to enhance fatigue life time. In hydraulic autofrettage process a pressure is applied into the bore of the thick-walled tube to plastically deform it. This pressure is not so high to burst the tube. When the pressure is removed, the elastic recovery of the outer wall puts the inner wall into compression, providing a residual compressive stress. These compressive residual stresses increase the fatigue life time and load carrying capacity of the tube. The thick-walled tubes are
Fig. 3. The forceedisplacement response of the smooth round bar in two loadingeunloading processes for different pre-strain levels.
G.H. Farrahi et al. / International Journal of Pressure Vessels and Piping 98 (2012) 57e64
typically constructed from high strength steels. These materials have nearly elastic-perfectly plastic behavior in the loading phase of a tension-compression test, but in the unloading phase the Bauschinger effect significantly decreases the yield strength of high strength steels in compression as a result of prior tensile plastic overload [3]. This decrease of the compressive yield strength also decreases the advantageous residual stresses at the thick-walled tubes. Therefore eliminating the Bauschinger effect can increase the compressive residual stresses after autofrettage process. Reapplication of the autofrettage process is proposed by some researchers for this purpose [6e11]. The re-autofrettage process can be carried out with and without heat treatment. In heat treatment procedures, the ‘‘heat soak’’ process is applied on all or part of the thick-walled tube by applying a slowly heating and cooling process [6]. The ‘‘heat soak’’ process reduces the microscopic dislocations, but this process does not significantly change the residual stresses introduced by the previous plastic deformation. It is assumed that after the ‘‘heat soak’’ process, the material behavior is similar to the virgin material containing a residual stress field when the next load is applied [6]. For evaluating the residual stresses at the bore of a thick-walled tube under the re-autofrettage process the proposed loadingeunloading behavior is implemented in Abaqus/Explicit, by means of the user material subroutine VUMAT. An axisymmetric FEM model of a thick-wall tube with inner radius a and outer radius b is constructed in which the outer radius is twice as big as the inner radius. The magnitude of a and b are selected equal to 50 mm and 100 mm, respectively. It is assumed that the thick-walled tube has the open ends and the length of the tube is selected in a manner that the effect of the end conditions on the residual stresses distribution is negligible. Therefore, in the FEM model the length of the tube is chosen equal to twice of the outer radius. In reautofrettage simulations the process is considered with and without heat treatment. As mentioned above, the material behavior after ‘‘heat soak’’ process is considered as a virgin material containing a residual stress field when the next autofrettage pressure is applied. Fig. 4 shows the relation between the applied pressure into the thick-walled tube and the autofrettage percentage during the reautofrettage process without heat treatment. The autofrettage percentage is the percent of the tube thickness that yielded during the re-autofrettage process. It is assumed that the yield initiation begins when the accumulated plastic strain reaches a specific value, 0.0001. As seen in Fig. 4, the re-autofrettage process needs less pressure in comparison with the single autofrettage process for producing the same autofrettage percentage. This means that the
Fig. 4. The relation between the autofrettage pressure vs. autofrettage percentage for re-autofrettage process without heat treatment.
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load carrying capacity during the re-autofrettage process is smaller than the single autofrettage process for the same autofrettage percentage. According to this fact, the autofrettage process without heat treatment can produce the compressive residual stresses but it may decrease the load carrying capacity of a thick-walled tube. The re-autofrettage percentage is calculated using the total accumulated plastic strain that is estimated in the loading phase of the prior autofrettage and re-autofrttage processes. For estimating the effect of the re-autofrettage process without heat treatment on residual stress distribution, different prior autofrettge and re-autofrettage procedures are considered. In prior autofrettage phase, the twenty, forty and sixty autofrettage percentages are applied and for re-autofrettage phase the sixty and eighty autofrettage percentages are implemented. The results of these simulations are shown in Fig. 5. In Fig. 5 (a and c) the hoop residual stress distribution and in Fig. 5 (b and d) the axial residual stress distributions are presented. As seen in Fig. 5, for reautofrettage process that is less than sixty percent, no residual stress improvement is observed in the thick-walled tube. For the case of the forty percent prior autofrettage and sixty percent reautofrettage processes, the residual stress distribution at the bore of the thick-walled tube improves less than two percent as shown in Fig. 5 (a and b). For higher re-autofrettage percentage the residual stress distributions improvement are noticeable. As shown in Fig. 5 (c and d) for the case of the sixty percent prior autofrettage and eighty percent re-autofrettage processes, the hoop residual stress and axial residual stress distribution at the bore of the thickwalled tube improves up to ten and twenty percent, respectively. It should be mentioned that this residual stress distributions are obtained at the high autofrettage percentage and according to Fig. 4 such re-autofrettage process without heat treatment can decrease the load carrying capacity of the thick-walled tube. Fig. 6 shows the relation between the applied pressure in the thick-walled tube and the autofrettage percentage during the reautofrettage process with ‘‘heat soak’’ process. In this procedure it is assumed that after ‘‘heat soak’’ process the history of the plastic deformation is eliminated and the elastic and plastic behavior of the material returns to the original state. As seen in Fig. 6, the reautofrettage process required further pressure in comparison with the single autofrettage process for producing the autofrettage percentage which is less than prior autofrettage percentage. For reautofrettage percentage that is bigger than prior autofrettage percentage, the applied pressure is same for prior autofrettage and re-autofrettage processes. This means that the load carrying capacity during the re-autofrettage process with ‘‘heat soak’’ process becomes bigger than the single autofrettage process. According to this fact, the autofrettage process with ‘‘heat soak’’ process can enhance the compressive residual stresses, as shown in the following figures, as well as the load carrying capacity of a thick-walled tube. In Fig. 6, the re-autofrettage percentage is calculated using the accumulated plastic strain which is produced only by re-autofrettage process. For evaluating the effect of the re-autofrettage procedure with ‘‘heat soak’’ treatment, different prior autofrettge and reautofrettage processes are considered. The percentage of prior autofrettage and re-autofrettage processes is varied between twenty and eighty percent. The results of these simulations are shown in Fig. 7. In Fig. 7 (a) and 7 (b) the hoop residual stress and axial residual stress distributions are presented as a 3D surface, respectively. As seen in Fig. 7, this re-autofrettage process generally improves the residual stress distribution at the thick-walled tube. This improvement in comparison with the re-autofrettage process without heat treatment is remarkable. For the case of the sixty percent prior autofrettage and sixty percent re-autofrettage processes, the residual stress distribution at the bore of the thick-
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Fig. 5. The hoop (a and c) and axial (b and d) residual stress distribution at the thick-walled tube under re-autofrettage process without heat treatment at different autofrettage percentages.
walled tube improves up to thirty percent in comparison with single autofrettage process. For higher re-autofrettage percentage the residual stress distributions improvement are more noticeable. For practical autofrettage process, the ASME Code [31] limits the autofrettage process up to forty percent. According to this maximum autofrettage percentage for applying in the prior autofrettage and re-autofrettage processes, the hoop and axial residual stress distribution at the bore of the thick-walled tube improves up to twenty four and fifty percent in comparison with single autofrettage process, respectively. Parker [6] estimated the 28 percent increase in residual hoop stresses for a tube which re-autofrettaged to the same level following intervening heat treatment. That tube
Fig. 6. The relation between the autofrettage pressure vs. autofrettage percentage for re-autofrettage with ‘‘heat soak’’ process.
was made of A723 steel and initially seventy percent overstrained. In comparison with this work the proposed model estimated the 33 percent increase in residual hoop stresses for a similar tube made of DIN1.6959 steel.
Fig. 7. 3D representation of the hoop (a) and axial (b) residual stress distribution of the thick-walled tube under re-autofrettage process with ‘‘heat soak’’ treatment at different autofrettage percentages.
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63
Fig. 8. Comparison between ideal and real hoop (a and c) and axial (b and d) residual stress distributions of the thick-walled tube under re-autofrettage process with ‘‘heat soak’’ treatment.
Fig. 8 compares the residual stress distributions produced by reautofrettage process with ‘‘heat soak’’ treatment and the ideal autofrettage process. The ideal autofrettage indicates the single autofrettage simulation that neglects the Bauschinger effect. On the other hand, the damage and the hardening corrections according to Eq. (6), Eq. (13) and Eq. (14) are not incorporated into the constitutive equation. In Fig. 8 (a and c) the hoop residual stress distribution and in Fig. 8 (b and d) the axial residual stress distributions are presented. As seen in Fig. 8 (a and c), the re-autofrettage process with ‘‘heat soak’’ treatment increases the compressive hoop residual stresses in comparison with single autofrettage process. This process cannot completely eliminate the Bauschinger effect but as mentioned by Parker [6] significantly decreases its influence. For axial residual stresses as shown in Fig. 8 (b and d) the compressive residual stresses are increased by re-autofrettage process even more than ideal autofrettage process. Although the role of hoop residual stresses in fatigue life time is more significant than axial residual stresses but this improvement of axial residual stresses is noticeable.
5. Conclusions In this paper, for simulating the re-autofrettage process a loadingeunloading behavior based on a plasticity model which depends on stress state is considered. According to this model, the yield criterion is defined as a function of the first stress invariant and Lode angle parameter. In addition the unloading behavior is assumed to depend on the damage, first stress invariant and the previous plastic deformation history. To evaluate the proposed loadingeunloading behavior, a series of tension-compression tests are carried out on smooth round bar specimens made of
high strength steel DIN1.6959. Subsequently, the proposed loadingeunloading is simulated by the finite element method in the Abaqus/Explicit, by means of the user material subroutine VUMAT. The numerical results show that the proposed loadingeunloading model can accurately predict the material behavior through the tension-compression tests. This loadingeunloading model is applied to simulate the reautofrettage process in a thick-walled tube by FEM. The numerical results show that the re-autofrettage process without heat treatment only improves the residual stress distribution in high autofrettage percentage and for low autofrettage percentage this method is not beneficial. Also the re-autofrettage processes without heat treatment may decrease the load carrying capacity of the thick-walled tube. In spite of this method the re-autofrettage process with ‘‘heat soak’’ treatment generally improves the residual stress distribution and also increases the load carrying capacity of the thick-walled tube. For low autofrettage percentage which is used in practical applications according to the ASME Code, the second re-autofrettage method is very beneficial. It is essential to mention that for practical applications, the re-autofrettage process with ‘‘heat soak’’ treatment cannot retain all the residual stresses created by the prior autofrettage process. However, if it is assumed that the real behavior of the material is expected to be between the ideal re-autofrettage with and without heat treatment it can be also concluded that the re-autofrettage with heat treatment can improve the fatigue life time and load carrying capacity of the thick-walled tube. References [1] Milligan RV, Koo WH, Davidson TE. The Bauschinger effect in a high-strength steel. Journal of Basic Engineering 1966:480e8.
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[2] Bauschinger J. Ueber die Veranderung der Elasticitatagrenze und dea Elasticitatamoduls Verschiadener Metalle. Zivilingenieur 1881;27:289e348. [3] Troiano E, Parker AP, Underwood JH, Mossey C. Experimental data, numerical fit and fatigue life calculations relating to Bauschinger effect in high strength armament steels. ASME Journal of Pressure Vessel Technology 2003;125: 330e4. [4] Walkup WN. What about Re-autofrettage. In: Antalffy Les, editor. Proceedings of PVP conference in Atlanta, Georgia, high pressure technology at the dawn of the new millennium, Vol. 418. New York: ASME; 2001. p. 13e8. [5] Iremonger MJ, Kalsi SK. A numerical study of swage autofrettage. ASME Journal of Pressure Vessel Technology 2003;122:347e51. [6] Parker AP. A re-autofrettage procedure for mitigation of Bauschinger effect in thick cylinder. ASME Journal of Pressure Vessel Technology. 2004;126: 451e545. [7] Troiano E, Underwood JH, Parker AP. Finite element investigation of Bauschinger effect in high-strength A723 pressure vessel steel. ASME Journal of Pressure Vessel Technology 2006;128:185e9. [8] Jahed H, Ahmadi B, Shambouli M. Re-autofrettage. ASME Journal of Pressure Vessel Technology 2006;128:223e6. [9] Parker AP, Huang X. Autofrettage and reautofrettage of a spherical pressure vessel. ASME Journal of Pressure Vessel Technology 2007;129:83e8. [10] Parker AP, Underwood JH, Troiano E. Hydraulic Re-Autofrettage of a Swage Autofrettaged Tube. Proceedings of ASME Pressure Vessels and Piping Division Conference 2009, PVP2009e77213, Prague, Czech Republic. [11] Parker AP, Underwood JH, Troiano E. Impact of Intensity of Residual Stress Field Upon Re- Yielding and Re-Autofrettage of an Autofrettaged Thick Cylinder. Proceedings of ASME Pressure Vessels and Piping Division Conference 2010, PVP2010-25020, Bellevue, WA, USA. [12] Troiano E, Parker AP, Underwood JH. Mechanisms and modeling comparing HB7 and A723 high strength pressure vessel steels. ASME Journal of Pressure Vessel Technology 2004;126:473e7. [13] Parker AP, Troiano E, Underwood JH, Mossey C. Characterization of steels using a revised kinematic hardening model incorporating Bauschinger effect. ASME Journal of Pressure Vessel Technology 2003;125:277e81. [14] Jahed H, Farshi B, Hosseini M. Fatigue life prediction of autofrettage tubes using actual material behavior. International Journal of Pressure Vessels and Piping 2006;83(10):749e55. [15] Hojjati MH, Hassani A. Theoretical and finite-element modeling of autofrettage process in strain-hardening thick-walled cylinders. International Journal of Pressure Vessels and Piping 2007;84(5):310e9. [16] Farrahi GH, Hosseinian E, Assempour A. On the material modeling of the autofrettaged pressure vessel steels. ASME Journal of Pressure Vessel Technology 2009;131(5):051403.
[17] Farrahi GH, Faghidian SA, Smith DJ. Reconstruction of residual stresses in autofrettaged thick-walled tubes from limited measurements. International Journal of Pressure Vessels and Piping 2009;86(11):777e84. [18] Spitzig WA, Sober RJ, Richmond O. Pressure dependence of yielding and associated volume expansion in tempered martensite. Acta Metallurgica 1975;23:885e93. [19] Spitzig WA, Sober RJ, Richmond O. The effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implications for plasticity theory. Metallurgical and Materials Transactions A 1976;7:1703e10. [20] Spitzig WA, Richmond O. The effect of pressure on the flow stress of metals. Acta Metallurgica 1984;32:457e63. [21] Brunig M. Numerical simulation of the large elasticeplastic deformation behavior of hydrostatic stress-sensitive solids. International Journal of Plasticity 1999;15:1237e64. [22] Brunig M, Berger S, Obrecht H. Numerical simulation of the localization behavior of hydrostatic-stress-sensitive metals. International Journal of Mechanical Sciences 2000;42:2147e66. [23] Bai Y, Wierzbicki T. A new model of metal plasticity and fracture with pressure and lode dependence. International Journal of Plasticity 2008;24:1071e96. [24] Mirone G, Corallo D. A Local viewpoint for evaluating the influence of stress triaxiality and lode angle on ductile failure and hardening. International Journal of Plasticity 2010;26:348e71. [25] Gao X, Zhang T, Zhou J, Graham SM, Hayden M, Roe C. On stress-state dependent plasticity modeling: significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule. International Journal of Plasticity 2011;27:217e31. [26] Voyiadjis GZ, Hoseini SH, Farrahi GH. A plasticity model for metals with stress invariants and plastic deformation history dependency. International Journal of Plasticity 2012 [accepted Manuscript]. [27] Farrahi GH, Voyiadjis GZ, Hoseini SH, Hosseinian E. Residual stress analysis of the autofrettaged thick-walled tube using nonlinear kinematic hardening. ASME Journal of Pressure Vessel Technology 2012 [accepted Manuscript]. [28] Chaboche JL, Rousselier G. On the plastic and viscoplastic constitutive equations based on the internal variables concept. In: SMIRT-6 post conf. Paris, T.P: ONERA; 1981. 8e11. [29] Chaboche JL, Rousselier G. On the plastic and viscoplastic constitutive equations, part I: rules developed with internal variable concept. part II: application of internal variable concepts to the 316 stainless steel. ASME Journal of Pressure Vessel Technology 1983;105:153e64. [30] Puskar A. A correlation among elastic modulus defect, plastic strain and fatigue life of metals. Materials Science Forum 1993;119e121:455e60. [31] ASME Pressure Vessel and Piping Design Code, Design Using Autofrettage. Division 3, Section 8, Article KD-5 1997; 71e3.
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International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Residual stress analyses of re-autofrettaged thick-walled tubes G.H. Farrahi a, *, George Z. Voyiadjis b, S.H. Hoseini a, E. Hosseinian c a
School of Mechanical Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA c Aerospace Engineering Department, Shahid Sattari Aeronautical University of Science and Technology, Tehran, Iran b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 February 2012 Received in revised form 9 July 2012 Accepted 10 July 2012
In this paper the effect of the re-autofrettage process on the residual stress distribution at the wall of a thick-walled tube is considered. For accurate material behavior modeling, it is assumed that the yield surface is a function of all the stress invariants. Also for estimating the behavior of the material under loadingeunloading process, a modified Chaboche’s hardening model is applied. For evaluation of this unloading behavior model a series of loadingeunloading tests are conducted on specimens that are made of the high strength steel, DIN1.6959. In addition the finite element simulations are implemented to simulate the re-autofrettage process and to estimate the residual stresses. The numerical results show that the re-autofrettage process without heat treatment only improves the residual stress distribution in high autofrettage percentage and for low autofrettage percentage this method is not beneficial. However, the re-autofrettage process with ‘‘heat soak’’ treatment generally improves the residual stress distribution. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Re-autofrettage process Nonlinear kinematic hardening First stress invariant Lode angle parameter Residual stress distribution
1. Introduction Autofrettage process is applied to introduce the compressive residual stresses at the bore of thick-walled tubes. The compressive residual stresses enhance the fatigue life time of the thick-walled tubes. The thick-walled tubes are typically constructed from high strength steels. Some of these materials have nearly elasticperfectly plastic behavior in the loading phase of a tensioncompression test, but in the unloading phase they show a strong nonlinear hardening behavior. Milligan et al. [1] showed that the Bauschinger effect [2] significantly decreases the yield strength of high strength steels in compression as a result of prior tensile plastic overload. The Bauschinger effect also reduces the compressive residual stresses at the bore of the thick-walled tubes [3]. For decreasing the influence of the Bauschinger effect on residual stresses distribution at the bore of thick-walled tubes, the reapplication of the autofrettage procedure was proposed [4]. In swage autofrettage which is carried out by inserting an oversized mandrel into the bore of thick-walled tubes, Iremonger and Kalsi [5] showed that a more uniform stress is obtained by inserting the mandrel two times such that in each time from different end. But their results showed that such process does not increase the compressive residual stresses. For hydraulic autofrettage process, * Corresponding author. Tel.: þ98 21 66165533; fax: þ98 21 66000021. E-mail address: [email protected] (G.H. Farrahi). 0308-0161/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2012.07.007
Parker [6] proposed a manufacturing procedure for enhancing residual stresses. The procedure involves initial autofrettage with one or more ‘‘heat soak plus autofrettage’’ sequences. He showed that the Bauschinger effect is significantly reduced with this procedure. Troiano et al. [7] also show the possibility of mitigating the Bauschinger effect via an intermediate heat soak between loading and unloading phase of a tension-compression test. Jahed et al. [8] showed that with applying different pressures in first and second autofrettage processes, higher compressive stress can be obtained. Parker and Huang [9] also showed that the reautofrettage process can be beneficial for spherical vessels. Parker et al. [10,11] showed that the hydraulic re-autofrettage of a swageautofrettaged tube with a low temperature post-autofrettage thermal treatment, results in higher compressive residual stresses. The important aspect of residual stresses estimation in reautofrettage process is the accurate modeling of loadingeunloading behavior. There are many researchers who considered the loadingeunloading behavior of high strength steels. Troiano et al. [12] evaluated the uniaxial Bauschinger effect in several high strength steels and showed that the reduction of the Young’s modulus and yield strength occurs after unloading. Parker et al. [13] proposed a nonlinear kinematic hardening model for predicting the material behavior during initial load reversal. More recent works such as Jahed et al. [14], Hojjati and Hassani [15] and Farrahi et al. [16,17] introduced methods to predict the accurate unloading behavior and precise residual stress estimation.
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All research mentioned above applied the experimental results of the uniaxial tension-compression tests or reversed torsion tests to describe the material behavior under reversed loading. However, recent research studies have shown that the stress state should be incorporated into the constitutive description of the plastic deformation. Spitzig et al. [18e20] based on experimental study, demonstrated the effect of hydrostatic pressure on yielding of metals. Brunig [21] similar to the DruckerePrager yield condition in soil mechanics proposed a yield criterion as a function of the first stress invariant to describe the effect of the hydrostatic pressure on metal plasticity. Later Brunig et al. [22] added the third deviatoric stress invariant in the yield criterion to describe the deformation of metals. Bai and Wierzbicki [23] introduced a yield criterion that is a function of pressure and Lode angle and compared their model with experimental tests conducted on aluminum 2024-T351. Mirone and Corallo [24] showed that, for the metals they tested, the hydrostatic stress has a significant effect on fracture and an insignificant effect on the stress-plastic strain relationship, while the Lode angle has an opposite role. Gao et al. [25] introduced a stressstate dependent plasticity model using the non-associated flow rule and Voyiadjis et al. [26] also introduced a yield criterion based on the DruckerePrager yield condition with incorporating the effect of the Lode angle. In this paper for estimating the residual stresses at the bore of the thick-walled tubes, the plasticity model proposed by Voyiadjis et al. [26] is considered. This model was used by Farrahi et al. [27] to analyze the residual stresses which are introduced into the bore of thick-walled tubes under single autofrettage process. In this plasticity model, the yield criterion and the hardening parameters are affected by stress invariants. In addition the effect of the previous plastic deformation history is incorporated into the hardening parameters. For evaluating the loadingeunloading behavior a series of tensionecompression tests are conducted on specimens which are made of a high strength steel, DIN1.6959. A finite element procedure is also implemented to assess the mentioned loadingeunloading model. Good agreement is obtained between the experimental loadingeunloading results and the simulations from the numerical modeling. Using this model and numerical methods, the re-autofrettage process is simulated in a thick-walled tube and the effect of the “heat soak” process is also considered.
The elastoeplastic behavior of the material considered in this section is assumed to be rate independent and elastically isotropic. Since the strains are infinitesimal, the total strain rate is decomposed into the elastic and plastic components:
¼ 3_ eij þ 3_ pij
(1)
where the superscripts e and p designate the elastic and plastic _ is components, respectively. The accumulated plastic strain rate, p, expressed as:
qffiffiffiffiffiffiffiffiffi p p p_ ¼ m 3_ ij 3_ ij
(2)
where m is the inverse magnitude of the normal vector to the yield surface. The Cauchy stress tensor is defined as:
~ sij ¼ D ijkl
3 kl
p
3 kl
(3)
~ is the fourth-order where s is the Cauchy stress tensor, and D isotropic elastic stiffness tensor that is affected by the damage through the plastic deformation. Isotropic elastic stiffness tensor is defined as:
(4)
~ are Lame constants that are defined as: l and D where ~
~ l ¼
~ vE ð1 þ vÞð1 2vÞ
~ ¼ G
~ E 2ð1 þ vÞ
(5)
~ is the Young’s modulus in the damaged configuration and where E is defined as:
~ ¼ Eð1 fÞ E
(6)
where E is the initial Young’s modulus and f is a scalar damage function which is defined as a function of the accumulated plastic strain. This damage function has no effect on the yield surface and is determined experimentally. In this study, the Poisson’s ratio is assumed to remain unchanged through the plastic deformation. The yield function, f, is defined as follows:
f ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðs XÞ : ðs XÞ Y 0 2
(7)
where X is the kinematic hardening back stress tensor that describes the movement of the yield surface in the deviatoric space and Y is the radius of the yield surface. For accurate estimation of the yield surface movement, the back stress tensor can be divided into finite components, where each component is evaluated independently. Kinematic hardening back stress according to the additive decomposition method proposed by Chaboche and Rousselier [28,29] can be expressed as follows:
Xij ¼
M X k¼1
ðkÞ
Xij
(8)
ðkÞ
where Xij is the k-th component of the back stress tensor. The radius of the yield surface, Y, can be defined as:
Y ¼ sy þ R
2. Formulation of the plasticity model
3_ ij
~ ~ ~ D ijkl ¼ ldij dkl þ G dik djl þ dil djk
(9)
where R is the isotropic hardening force and describes the change in the size of the yield surface. In this work the plastic strains are evaluated using the normality rule 3_
p ij
vF ¼ l_ vsij
(10)
where l_ is the multiplier of time-independent plasticity which will be determined using the consistency condition. In this research, the general formulation of the yield condition and hardening parameters proposed by Voyiadjis et al. [26] are considered. This model modifies the DruckerePrager yield condition with incorporating the Lode angle parameter. Therefore the linear effect of the first stress invariant and nonlinear effect of the Lode angle on the yield condition are assumed. The yield condition is defined as follows:
sy ¼ s0 a1 I1 a2 q
(11)
where s0, a1 and a2 are the material constants, I1 is the first Cauchy stress tensor invariant and q is the Lode angle parameter [26]. The Lode angle parameter is defined as follows:
G.H. Farrahi et al. / International Journal of Pressure Vessels and Piping 98 (2012) 57e64
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 27 J32 q ¼ sinð3qÞ ¼ t1 4 J23
3. Experimental procedures
(12)
where J2 and J3 are the second and the third deviatoric stress invariants. The range of the Lode angle parameter varies between zero and unity. The coefficient of the Lode angle parameter must be limited to a special interval so that the convexity of the yield surface is satisfied. The isotropic hardening force is defined as a function of the first stress invariant, accumulated plastic strain and the previous plastic deformation history. The isotropic hardening force is defined as [26]:
R ¼
Q1 þ Q2 eb1 I1
1 eb2 kp
(13)
where Q1, Q2, b1 and b2 are the material constants and k is an exponential decay function of the previous accumulated plastic strain, p. Eq. (13) shows that the compression accelerates the increase of the yield surface radius whereas tension does not significantly change it. The previous plastic deformation is related to the earlier plastic deformation, after which an elastic unloading occurs. Similar to the isotropic hardening, the previous plastic deformation affects the kinematic hardening as well. This effect can be represented as an exponential decay function multiplied by the kinematic hardening constants. The following formulation is proposed for each kinematic hardening component:
2 ðkÞ ðkÞ ðkÞ X_ ¼ C ðkÞ z 3_ p gðkÞ z X ðkÞ p_ 3
(14)
where z is an exponential decay function of the previous accumulated plastic strain, p. This correction means that with increasing the effect of the previous accumulated plastic strain, the translation rate of the yield surface in the stress space at the current plastic deformation is decreased. To evaluate the plastic strain tensor, the yield function should be differentiated with respect to the stress tensor. Therefore vf =vsij should be calculated first. By differentiation of Eq. (7) with respect to the stress tensor, results in the following equations: 3_
p
b 3ðs XÞ Y ¼ p_ 2Y
(15)
b ¼ 2Y vY Y vsij
(16)
By differentiation of the first Cauchy stress tensor invariant and the Lode angle parameter the following equations are obtained [26]:
vI1 ¼ dij vsij
(17)
vq 27J3 3sij J3 1 1 2 ¼ 2 adjðsÞij þ I1 sij þ dij J2 dij I12 vsij J2 3 3 9 8qJ 3
(18)
2
where dij is the Kronecker delta and ‘adj’ is the adjoint matrix of the Cauchy stress tensor. Using Eq. (17) and Eq. (18), vY=vsij can be calculated as follows:
vY ¼ vsij
b1 dij Q2 eb1 I1 Q2 þ Q2 eb1 I1
R a1 dij
1 1 2 þ I1 sij þ dij J2 dij I12 3 3 9
59
27a2 J3
!
8qJ23
3sij J3 2 adjðsÞji J2
To evaluate the loadingeunloading behavior, a series of tensioncompression tests are carried out on smooth round bars. The diameter of the smooth round bar specimens is 8 mm. The diameter in the specimen shoulder is equal to 12 mm, and the length of the gauge section for these specimens is 12.5 mm. The configuration of this specimen is shown in Fig. 1. For evaluating the loadingeunloading behavior, four loading reversals are applied on test specimens. According to this loading pattern each specimen is loaded in tension up to the prescribed plastic strain level, and then unloading and reversed loading is conducted. After reversed loading the specimen is reloaded in tension up to another plastic strain level and after that, once again the unloading and reversed loading is carried out. The selected material is DIN1.6959 steel which is a high strength, quenched, tempered, forged alloy steel used for construction of pressurized thick-walled tubes. The general chemical and mechanical specifications of this steel were introduced by Farrahi et al. [17]. All experiments are conducted using a 30-ton capacity Instron servohydraulic machine and all of the loading patterns are carried out under displacement control. The experimental results are simulated computationally using the finite element method. Finite element model of the specimen is built in Abaqus/Explicit, by means of the user material subroutine VUMAT. The value of the plasticity model parameters which are calibrated from experimental tests for DIN1.6959 are presented in Table 1 [26,27]. As mentioned previously, in high strength steels the Young’s modulus during the plastic deformation changes because of the damage and the dislocation movement that occurs in the material [3,30]. In the tested material, the Young’s modulus changes significantly through the plastic deformation. The Young’s modulus decreases sharply in the beginning of the plastic deformation but the rate of this change decreases, with further plastic deformation [27]. Another aspect that was observed in the experimental tests on DIN1.6959 is the recovery of the Young’s modulus after reverse loading. Reloading after the tension-compression process recovers some amount of the loss in the Young’s modulus [26]. Such behavior, the reduction in Young’s modulus during the tensile unloading and the recovery of Young’s modulus during the compressive unloading, is reported by Troiano et al. [3] for a series of high strength steels with similar behavior as DIN1.6959. According to the experimental tests which were carried out on DIN1.6959 [26], the following expression is proposed for the change in Young’s modulus due to plastic deformation as an isotropic damage:
f ¼ 0:41e9:2pc
0:6
1 e5:5pt
0:6
where pt is related to the accumulated plastic strain in tension and similarly pc is related to the accumulated plastic strain in compression. In this study, the Poisson’s ratio is assumed to remain unchanged through the plastic deformation. Fig. 2 displays the forceedisplacement response of the test specimen simulation along with the experimental results under the first three reversals of the loading pattern that earlier were defined. In Fig. 2 a and b, each specimen is loaded in tension up to
(19)
Eq. (15) is very difficult to solve analytically and therefore a numerical method should be implemented to solve this equation [26].
(20)
Fig. 1. The smooth round bar specimen configuration.
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Table 1 Loadingeunloading behavior parameters calibrated from experimental tests for DIN1.6959. E ¼ 210 GPa
Q1 ¼ 160 MPa
C1 ¼ 1840 GPa
g3 ¼ 1800
v ¼ 0.3 s0 ¼ 610 MPa a1 ¼ 0.01 a2 ¼ 40
Q2 ¼ 20 MPa b1 ¼ 0.001 b2 ¼ 18 k ¼ 0:9e62p þ 0:1
C2 ¼ 2760 GPa C3 ¼ 260 GPa g1 ¼ 10 000 g2 ¼ 10 000
z1 ¼ 0:98e445p þ 0:02 z2 ¼ 0:99e550p þ 0:01 z1 ¼ 1.0
a prescribed plastic strain level, then unloading and reverse loading is conducted up to a different compression plastic strain level. After the compression loading, the specimen is reloaded in tension. As seen in Fig. 2, the loadingeunloading model can accurately predict the material behavior under the reversed loading, also the change of the Young’s modulus through the plastic deformation is captured by this model. These experimental results show the significant influence of the Bauschinger effect on the yield strength and plastic deformation through the compression part of loading. The comparison between the numerical and experimental results in Fig. 2 shows that the Bauschinger effect is precisely simulated by this model. Fig. 3 displays the forceedisplacement response of the smooth round bar simulation along with the experimental results under the loading pattern that earlier is defined. In Fig. 3 each case shows the results for two tests and each test has the same loadingeunloading path but reloading and second unloading are carried out in different paths. As seen in Fig. 3, there is a good agreement between the proposed loadingeunloading model and the experimental results. In each reversal of the loading pattern the
Fig. 2. The forceedisplacement response of the smooth round bar in the first three reversals of the defined loading pattern for different pre-strain levels.
change of Young’s modulus and the plastic deformation are accurately simulated by this model. 4. Re-autofrettage simulation Autofrettage is a technique used on thick-walled tubes to introduce advantageous residual stresses and to enhance fatigue life time. In hydraulic autofrettage process a pressure is applied into the bore of the thick-walled tube to plastically deform it. This pressure is not so high to burst the tube. When the pressure is removed, the elastic recovery of the outer wall puts the inner wall into compression, providing a residual compressive stress. These compressive residual stresses increase the fatigue life time and load carrying capacity of the tube. The thick-walled tubes are
Fig. 3. The forceedisplacement response of the smooth round bar in two loadingeunloading processes for different pre-strain levels.
G.H. Farrahi et al. / International Journal of Pressure Vessels and Piping 98 (2012) 57e64
typically constructed from high strength steels. These materials have nearly elastic-perfectly plastic behavior in the loading phase of a tension-compression test, but in the unloading phase the Bauschinger effect significantly decreases the yield strength of high strength steels in compression as a result of prior tensile plastic overload [3]. This decrease of the compressive yield strength also decreases the advantageous residual stresses at the thick-walled tubes. Therefore eliminating the Bauschinger effect can increase the compressive residual stresses after autofrettage process. Reapplication of the autofrettage process is proposed by some researchers for this purpose [6e11]. The re-autofrettage process can be carried out with and without heat treatment. In heat treatment procedures, the ‘‘heat soak’’ process is applied on all or part of the thick-walled tube by applying a slowly heating and cooling process [6]. The ‘‘heat soak’’ process reduces the microscopic dislocations, but this process does not significantly change the residual stresses introduced by the previous plastic deformation. It is assumed that after the ‘‘heat soak’’ process, the material behavior is similar to the virgin material containing a residual stress field when the next load is applied [6]. For evaluating the residual stresses at the bore of a thick-walled tube under the re-autofrettage process the proposed loadingeunloading behavior is implemented in Abaqus/Explicit, by means of the user material subroutine VUMAT. An axisymmetric FEM model of a thick-wall tube with inner radius a and outer radius b is constructed in which the outer radius is twice as big as the inner radius. The magnitude of a and b are selected equal to 50 mm and 100 mm, respectively. It is assumed that the thick-walled tube has the open ends and the length of the tube is selected in a manner that the effect of the end conditions on the residual stresses distribution is negligible. Therefore, in the FEM model the length of the tube is chosen equal to twice of the outer radius. In reautofrettage simulations the process is considered with and without heat treatment. As mentioned above, the material behavior after ‘‘heat soak’’ process is considered as a virgin material containing a residual stress field when the next autofrettage pressure is applied. Fig. 4 shows the relation between the applied pressure into the thick-walled tube and the autofrettage percentage during the reautofrettage process without heat treatment. The autofrettage percentage is the percent of the tube thickness that yielded during the re-autofrettage process. It is assumed that the yield initiation begins when the accumulated plastic strain reaches a specific value, 0.0001. As seen in Fig. 4, the re-autofrettage process needs less pressure in comparison with the single autofrettage process for producing the same autofrettage percentage. This means that the
Fig. 4. The relation between the autofrettage pressure vs. autofrettage percentage for re-autofrettage process without heat treatment.
61
load carrying capacity during the re-autofrettage process is smaller than the single autofrettage process for the same autofrettage percentage. According to this fact, the autofrettage process without heat treatment can produce the compressive residual stresses but it may decrease the load carrying capacity of a thick-walled tube. The re-autofrettage percentage is calculated using the total accumulated plastic strain that is estimated in the loading phase of the prior autofrettage and re-autofrttage processes. For estimating the effect of the re-autofrettage process without heat treatment on residual stress distribution, different prior autofrettge and re-autofrettage procedures are considered. In prior autofrettage phase, the twenty, forty and sixty autofrettage percentages are applied and for re-autofrettage phase the sixty and eighty autofrettage percentages are implemented. The results of these simulations are shown in Fig. 5. In Fig. 5 (a and c) the hoop residual stress distribution and in Fig. 5 (b and d) the axial residual stress distributions are presented. As seen in Fig. 5, for reautofrettage process that is less than sixty percent, no residual stress improvement is observed in the thick-walled tube. For the case of the forty percent prior autofrettage and sixty percent reautofrettage processes, the residual stress distribution at the bore of the thick-walled tube improves less than two percent as shown in Fig. 5 (a and b). For higher re-autofrettage percentage the residual stress distributions improvement are noticeable. As shown in Fig. 5 (c and d) for the case of the sixty percent prior autofrettage and eighty percent re-autofrettage processes, the hoop residual stress and axial residual stress distribution at the bore of the thickwalled tube improves up to ten and twenty percent, respectively. It should be mentioned that this residual stress distributions are obtained at the high autofrettage percentage and according to Fig. 4 such re-autofrettage process without heat treatment can decrease the load carrying capacity of the thick-walled tube. Fig. 6 shows the relation between the applied pressure in the thick-walled tube and the autofrettage percentage during the reautofrettage process with ‘‘heat soak’’ process. In this procedure it is assumed that after ‘‘heat soak’’ process the history of the plastic deformation is eliminated and the elastic and plastic behavior of the material returns to the original state. As seen in Fig. 6, the reautofrettage process required further pressure in comparison with the single autofrettage process for producing the autofrettage percentage which is less than prior autofrettage percentage. For reautofrettage percentage that is bigger than prior autofrettage percentage, the applied pressure is same for prior autofrettage and re-autofrettage processes. This means that the load carrying capacity during the re-autofrettage process with ‘‘heat soak’’ process becomes bigger than the single autofrettage process. According to this fact, the autofrettage process with ‘‘heat soak’’ process can enhance the compressive residual stresses, as shown in the following figures, as well as the load carrying capacity of a thick-walled tube. In Fig. 6, the re-autofrettage percentage is calculated using the accumulated plastic strain which is produced only by re-autofrettage process. For evaluating the effect of the re-autofrettage procedure with ‘‘heat soak’’ treatment, different prior autofrettge and reautofrettage processes are considered. The percentage of prior autofrettage and re-autofrettage processes is varied between twenty and eighty percent. The results of these simulations are shown in Fig. 7. In Fig. 7 (a) and 7 (b) the hoop residual stress and axial residual stress distributions are presented as a 3D surface, respectively. As seen in Fig. 7, this re-autofrettage process generally improves the residual stress distribution at the thick-walled tube. This improvement in comparison with the re-autofrettage process without heat treatment is remarkable. For the case of the sixty percent prior autofrettage and sixty percent re-autofrettage processes, the residual stress distribution at the bore of the thick-
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Fig. 5. The hoop (a and c) and axial (b and d) residual stress distribution at the thick-walled tube under re-autofrettage process without heat treatment at different autofrettage percentages.
walled tube improves up to thirty percent in comparison with single autofrettage process. For higher re-autofrettage percentage the residual stress distributions improvement are more noticeable. For practical autofrettage process, the ASME Code [31] limits the autofrettage process up to forty percent. According to this maximum autofrettage percentage for applying in the prior autofrettage and re-autofrettage processes, the hoop and axial residual stress distribution at the bore of the thick-walled tube improves up to twenty four and fifty percent in comparison with single autofrettage process, respectively. Parker [6] estimated the 28 percent increase in residual hoop stresses for a tube which re-autofrettaged to the same level following intervening heat treatment. That tube
Fig. 6. The relation between the autofrettage pressure vs. autofrettage percentage for re-autofrettage with ‘‘heat soak’’ process.
was made of A723 steel and initially seventy percent overstrained. In comparison with this work the proposed model estimated the 33 percent increase in residual hoop stresses for a similar tube made of DIN1.6959 steel.
Fig. 7. 3D representation of the hoop (a) and axial (b) residual stress distribution of the thick-walled tube under re-autofrettage process with ‘‘heat soak’’ treatment at different autofrettage percentages.
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Fig. 8. Comparison between ideal and real hoop (a and c) and axial (b and d) residual stress distributions of the thick-walled tube under re-autofrettage process with ‘‘heat soak’’ treatment.
Fig. 8 compares the residual stress distributions produced by reautofrettage process with ‘‘heat soak’’ treatment and the ideal autofrettage process. The ideal autofrettage indicates the single autofrettage simulation that neglects the Bauschinger effect. On the other hand, the damage and the hardening corrections according to Eq. (6), Eq. (13) and Eq. (14) are not incorporated into the constitutive equation. In Fig. 8 (a and c) the hoop residual stress distribution and in Fig. 8 (b and d) the axial residual stress distributions are presented. As seen in Fig. 8 (a and c), the re-autofrettage process with ‘‘heat soak’’ treatment increases the compressive hoop residual stresses in comparison with single autofrettage process. This process cannot completely eliminate the Bauschinger effect but as mentioned by Parker [6] significantly decreases its influence. For axial residual stresses as shown in Fig. 8 (b and d) the compressive residual stresses are increased by re-autofrettage process even more than ideal autofrettage process. Although the role of hoop residual stresses in fatigue life time is more significant than axial residual stresses but this improvement of axial residual stresses is noticeable.
5. Conclusions In this paper, for simulating the re-autofrettage process a loadingeunloading behavior based on a plasticity model which depends on stress state is considered. According to this model, the yield criterion is defined as a function of the first stress invariant and Lode angle parameter. In addition the unloading behavior is assumed to depend on the damage, first stress invariant and the previous plastic deformation history. To evaluate the proposed loadingeunloading behavior, a series of tension-compression tests are carried out on smooth round bar specimens made of
high strength steel DIN1.6959. Subsequently, the proposed loadingeunloading is simulated by the finite element method in the Abaqus/Explicit, by means of the user material subroutine VUMAT. The numerical results show that the proposed loadingeunloading model can accurately predict the material behavior through the tension-compression tests. This loadingeunloading model is applied to simulate the reautofrettage process in a thick-walled tube by FEM. The numerical results show that the re-autofrettage process without heat treatment only improves the residual stress distribution in high autofrettage percentage and for low autofrettage percentage this method is not beneficial. Also the re-autofrettage processes without heat treatment may decrease the load carrying capacity of the thick-walled tube. In spite of this method the re-autofrettage process with ‘‘heat soak’’ treatment generally improves the residual stress distribution and also increases the load carrying capacity of the thick-walled tube. For low autofrettage percentage which is used in practical applications according to the ASME Code, the second re-autofrettage method is very beneficial. It is essential to mention that for practical applications, the re-autofrettage process with ‘‘heat soak’’ treatment cannot retain all the residual stresses created by the prior autofrettage process. However, if it is assumed that the real behavior of the material is expected to be between the ideal re-autofrettage with and without heat treatment it can be also concluded that the re-autofrettage with heat treatment can improve the fatigue life time and load carrying capacity of the thick-walled tube. References [1] Milligan RV, Koo WH, Davidson TE. The Bauschinger effect in a high-strength steel. Journal of Basic Engineering 1966:480e8.
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[2] Bauschinger J. Ueber die Veranderung der Elasticitatagrenze und dea Elasticitatamoduls Verschiadener Metalle. Zivilingenieur 1881;27:289e348. [3] Troiano E, Parker AP, Underwood JH, Mossey C. Experimental data, numerical fit and fatigue life calculations relating to Bauschinger effect in high strength armament steels. ASME Journal of Pressure Vessel Technology 2003;125: 330e4. [4] Walkup WN. What about Re-autofrettage. In: Antalffy Les, editor. Proceedings of PVP conference in Atlanta, Georgia, high pressure technology at the dawn of the new millennium, Vol. 418. New York: ASME; 2001. p. 13e8. [5] Iremonger MJ, Kalsi SK. A numerical study of swage autofrettage. ASME Journal of Pressure Vessel Technology 2003;122:347e51. [6] Parker AP. A re-autofrettage procedure for mitigation of Bauschinger effect in thick cylinder. ASME Journal of Pressure Vessel Technology. 2004;126: 451e545. [7] Troiano E, Underwood JH, Parker AP. Finite element investigation of Bauschinger effect in high-strength A723 pressure vessel steel. ASME Journal of Pressure Vessel Technology 2006;128:185e9. [8] Jahed H, Ahmadi B, Shambouli M. Re-autofrettage. ASME Journal of Pressure Vessel Technology 2006;128:223e6. [9] Parker AP, Huang X. Autofrettage and reautofrettage of a spherical pressure vessel. ASME Journal of Pressure Vessel Technology 2007;129:83e8. [10] Parker AP, Underwood JH, Troiano E. Hydraulic Re-Autofrettage of a Swage Autofrettaged Tube. Proceedings of ASME Pressure Vessels and Piping Division Conference 2009, PVP2009e77213, Prague, Czech Republic. [11] Parker AP, Underwood JH, Troiano E. Impact of Intensity of Residual Stress Field Upon Re- Yielding and Re-Autofrettage of an Autofrettaged Thick Cylinder. Proceedings of ASME Pressure Vessels and Piping Division Conference 2010, PVP2010-25020, Bellevue, WA, USA. [12] Troiano E, Parker AP, Underwood JH. Mechanisms and modeling comparing HB7 and A723 high strength pressure vessel steels. ASME Journal of Pressure Vessel Technology 2004;126:473e7. [13] Parker AP, Troiano E, Underwood JH, Mossey C. Characterization of steels using a revised kinematic hardening model incorporating Bauschinger effect. ASME Journal of Pressure Vessel Technology 2003;125:277e81. [14] Jahed H, Farshi B, Hosseini M. Fatigue life prediction of autofrettage tubes using actual material behavior. International Journal of Pressure Vessels and Piping 2006;83(10):749e55. [15] Hojjati MH, Hassani A. Theoretical and finite-element modeling of autofrettage process in strain-hardening thick-walled cylinders. International Journal of Pressure Vessels and Piping 2007;84(5):310e9. [16] Farrahi GH, Hosseinian E, Assempour A. On the material modeling of the autofrettaged pressure vessel steels. ASME Journal of Pressure Vessel Technology 2009;131(5):051403.
[17] Farrahi GH, Faghidian SA, Smith DJ. Reconstruction of residual stresses in autofrettaged thick-walled tubes from limited measurements. International Journal of Pressure Vessels and Piping 2009;86(11):777e84. [18] Spitzig WA, Sober RJ, Richmond O. Pressure dependence of yielding and associated volume expansion in tempered martensite. Acta Metallurgica 1975;23:885e93. [19] Spitzig WA, Sober RJ, Richmond O. The effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implications for plasticity theory. Metallurgical and Materials Transactions A 1976;7:1703e10. [20] Spitzig WA, Richmond O. The effect of pressure on the flow stress of metals. Acta Metallurgica 1984;32:457e63. [21] Brunig M. Numerical simulation of the large elasticeplastic deformation behavior of hydrostatic stress-sensitive solids. International Journal of Plasticity 1999;15:1237e64. [22] Brunig M, Berger S, Obrecht H. Numerical simulation of the localization behavior of hydrostatic-stress-sensitive metals. International Journal of Mechanical Sciences 2000;42:2147e66. [23] Bai Y, Wierzbicki T. A new model of metal plasticity and fracture with pressure and lode dependence. International Journal of Plasticity 2008;24:1071e96. [24] Mirone G, Corallo D. A Local viewpoint for evaluating the influence of stress triaxiality and lode angle on ductile failure and hardening. International Journal of Plasticity 2010;26:348e71. [25] Gao X, Zhang T, Zhou J, Graham SM, Hayden M, Roe C. On stress-state dependent plasticity modeling: significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule. International Journal of Plasticity 2011;27:217e31. [26] Voyiadjis GZ, Hoseini SH, Farrahi GH. A plasticity model for metals with stress invariants and plastic deformation history dependency. International Journal of Plasticity 2012 [accepted Manuscript]. [27] Farrahi GH, Voyiadjis GZ, Hoseini SH, Hosseinian E. Residual stress analysis of the autofrettaged thick-walled tube using nonlinear kinematic hardening. ASME Journal of Pressure Vessel Technology 2012 [accepted Manuscript]. [28] Chaboche JL, Rousselier G. On the plastic and viscoplastic constitutive equations based on the internal variables concept. In: SMIRT-6 post conf. Paris, T.P: ONERA; 1981. 8e11. [29] Chaboche JL, Rousselier G. On the plastic and viscoplastic constitutive equations, part I: rules developed with internal variable concept. part II: application of internal variable concepts to the 316 stainless steel. ASME Journal of Pressure Vessel Technology 1983;105:153e64. [30] Puskar A. A correlation among elastic modulus defect, plastic strain and fatigue life of metals. Materials Science Forum 1993;119e121:455e60. [31] ASME Pressure Vessel and Piping Design Code, Design Using Autofrettage. Division 3, Section 8, Article KD-5 1997; 71e3.