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A procedure to determine the tangential true stress-strain behavior of pipes
International Journal of Pressure Vessels and Piping 128 (2015) 59e68
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International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
A procedure to determine the tangential true stress-strain behavior of pipes I. Barsoum a, b, *, K.F. Al Ali a a
Department of Mechanical Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates Department of Aeronautical & Vehicle Engineering, Division of Lightweight Structures, Royal Institute of Technology e KTH, Teknikringen 8, 100 44 Stockholm, Sweden b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 March 2014 Received in revised form 27 November 2014 Accepted 28 November 2014 Available online 19 February 2015
Determining the tangential mechanical properties of a tube is essential for simulation of various manufacturing processes that involve the use of a tubular geometry. The aim of this study is to develop a procedure to determine the tangential true stress-strain behavior of pipes. For this purpose a modified ring test setup is proposed consisting of a ring specimen loaded with two separate D-blocks. Using a finite element model, an optimized ring specimen geometry is obtained. The optimized ring specimen exhibits uniform tangential distribution in the gauge region of the specimen and necking occurs consistently at the center of the gauge length. It is found that friction has a substantial effect on the mechanical response of the ring test for which two different setups to reduce friction are proposed. One using lubricated D-blocks (DB) and one using lubricated D-blocks with needle roller bearing (RB). Assisted by the FE model, the friction during the experiment is account for and a data analysis procedure to determine the tangential stress-strain curve of the pipe is proposed. It is found that the results using this procedure show very good agreement with previously published results. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Aluminum Pipes Stress-strain curve Tangential mechanical properties Experiments Finite element modeling Oil and gas
1. Introduction Various manufacturing processes require detailed knowledge of the tangential mechanical behavior of pipes to obtain proper designs and evaluations of the final product. One such process commonly employed in the petroleum industry is the Solid Expandable Tubular Technology (SETT) [1e3]. It is a cold working process in which a tubular casing is expanded radially by pushing a conical mandrel axially into the tube to obtain a desired radial permanent expansion. The technology enables mono-diameter casing, which is being used to replace the conventional multidiameter telescopic casing configuration used in oil wells. This is achieved by expanding multiple tubes of a certain diameter in the well resulting in a rather uniform diameter throughout the well hole providing a larger downhole diameter than the conventional telescopic configuration, hence resulting in an increased rate of oil production. However, to assess the mechanical integrity of the tube being expanded such that to select an appropriate tubular steel
* Corresponding author. Department of Mechanical Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates. E-mail address: [email protected] (I. Barsoum). http://dx.doi.org/10.1016/j.ijpvp.2014.11.002 0308-0161/© 2015 Elsevier Ltd. All rights reserved.
grade, it is of paramount importance to know the tangential stressstrain behavior of the tube. According to ASTM A370 [4], the tensile properties of pipes is to be determined from a specimen cut in the axial direction. Alternatively, ASTM A370 specifies a method to determine the transverse properties of pipes by cutting a circumferential strip from the pipe, flatten it and tensile test it. The work hardening from the flattening steps alters the transverse behavior considerably and the result will not represent the tangential true stress-strain curve of the pipe. Crone et al. [5] conducted a study to determine the effect of flattening the specimen on the tested mechanical properties of the pipe and concluded that flattening has a large effect on the experimental results. The material testing standard ASTM D2290 [6] specifies the split-disc test method to determine the tangential yield and tensile strength of thermoplastic pipes. The specimen is a ring cut from the pipe and is loaded with two half discs, which are inserted internally in the ring specimen and are pulled apart with a tensile testing machine. Kaynak et al. [7] used this method to study the effect of the filament winding process parameters on the tangential tensile strength of continuous fiber reinforced epoxy composite tubes, in Ref. [8] the influence of defects on the tensile strength of glass fiber reinforced composite pipes was studied with this testing method
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and in Ref. [9] the effect of aging of HDPE plastic pipes on the failure strain was investigated. Though the method is found efficient in determining the tangential strength and failure strain of the pipes, it does not provide a way to determine the full tangential stressstrain curve nor does it quantify the effect of friction between the disks and the ring specimen. Few researchers [10e12] have addressed this issue in the past and made attempts to experimentally measure the tensile transverse properties of pipes. Arsene et al. [11] addressed the importance of the ring specimen geometry in view of the effect of bending on the test results. They performed finite element analysis and used a notched ring specimen with three die inserts to load the ring specimen to overcome the issue with bending involved in the classical ring test, as recommended in the international standard ISO 8495-8496 [13]. Wang et al. [10] conducted a study to determine the transverse properties of tubular products and proposed a modified split-disc experimental setup, which uses D-blocks. The contact between the D-blocks and the internal surface of the ring specimen will mimic the internal pressure experienced in a pipe. However, the reason to why the selected ring specimen geometry is used is not given. Friction between the D-blocks and the specimen is considered negligible, which is justified by the use of lubricated thin Teflon sheets in-between the D-blocks and the specimen. However, this assumption was not confirmed and no attempts were undertaken to quantify its effect on the mechanical response of the ring test. Most recently Dick and Korkolis [14] conducted a thorough investigation on the mechanics of the ring hoop tension test (RHTT) and found that the friction can have a large effect on the uniaxial stress state in the specimen. This study addresses some of the aforementioned shortcomings of previous ring test setups and presents a novel testing and evaluation method for determining the tangential stress-strain behavior of pipes. The first part of the work presents an optimization study to find an optimum ring specimen geometry aided by the use of the finite element method (FEM). The second part presents the experimental setup and the means used to minimize the friction effects involved in the experimental setup and the third part provides a data analysis procedure and the determination of the tangential true stress-strain curve of a pipe.
Fig. 1. Failed ring specimens using RHTT (Ring Hoop Tension Test) test method outlined in Ref. [10].
through the D-blocks mimics the internal pressure experienced in pipes. The objective of the optimization study is to obtain a specimen configuration for which necking appears within the gauge length of the specimen and for which a uniform tangential stress state is achieved within the gauge length while accounting for the effect of friction between the D-blocks and specimen. In Fig. 2 the geometry of the specimen is shown, which has an initial specimen thickness t0 ¼ 3 mm, specimen outer diameter D ¼ 60 mm, initial width of reduced gauge section w0 ¼ 3 mm, fillet radius r ¼ 6 mm and a specimen width of 15 mm. The three optimization parameters to be addressed are the gauge length as given by angle a,
2. Optimization of the ring specimen Most of the attempts [6,10,11,15,16] to determine the tangential stress-strain behavior of pipes have used a similar specimen configuration consisting of a ring cut from the pipe with a gauge section manufactured to localize the necking location. Preliminary experiments were conducted following the experimental setup, specimen geometry and testing method in Ref. [10]. The difference was in the choice of material. In Ref. [10] a low-carbon hot-rolled steel tube with outer diameter 95.25 mm and thickness 2.2 mm was used whereas in this study an aluminum alloy Al 6063-T5 tube was used. The Al 6063-T5 material was delivered in 200 mm long seamless tubes with outer diameter of 60 mm and wall thickness of 3 mm. For all tests necking occurred in the transition between the large radius and the gauge length, as shown in Fig. 1, rather than at center of the gauge length. This indicates that the specimen could be subjected to non-uniform tangential stresses, bending and frictional forces between the D-block and the ring specimen. In order to develop a validated method to determine the tangential stress-strain behavior of pipes, an optimum ring specimen configuration need to be chosen. As there are no standards outlining the specimen dimensions for such tests a study to optimize the ring specimen configuration is motivated. The specimen selected for the optimization is a ring specimen, as shown in Fig. 2, for which the application of a tensile load
Fig. 2. (a) Test setup and (b) configuration of the ring specimen.
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specimen orientation defined as the angle q between the center of the specimen gauge length and tensile loading axis and the friction coefficient m between the outer surfaces of the D-blocks and internal surface of the ring specimen. The requirement of a uniform tangential stress distribution in the gauge length is controlled by a and q. This requirement is imposed as it is important to eliminate or minimize any other stress components, such as radial and axial, to ensure uniaxial tangential stresses in the gauge length. A non-uniform tangential stress distribution in the gauge length would also indicate that the specimen is subjected to bending, hence altering the location of necking. Having a necking location outside the measurement gauge length would invalidate the experiment. Furthermore, friction is expected to be present during the test due to the contact between the Dblocks and the ring specimen, which will influence the mechanical response of the test considerably. By varying m the effect of friction can be quantified. The optimization process was undertaken with aid of a finite element model of the test setup. Finite element analysis of the test setup with different combinations of gauge angles a, specimen orientation q and frictions coefficient m were simulated with Abaqus 6.11 [17]. The details of the finite element model are outlined in the next section. 2.1. The finite element model The main purpose of the finite element model is to simulate the mechanical response of the test setup shown in Fig. 2, taking into consideration all the geometric details of the specimen and the D-blocks and the contact interaction between them. This will guide in the selection of the optimum ring specimen geometry. The FE model consists of two D-blocks and a ring specimen, as shown in Fig. 2, where the ring specimen is inserted onto the Dblocks through which the tensile force is transmitted to the inner surface specimen mimicking an internal pressure condition. A master-slave contact interaction is assigned between the outer surfaces of the D-blocks and the inner surface of the ring specimen, with an isotropic friction behavior with a coefficient of friction ranging from 0 m 0.5. A reference node is defined at the center of each pin hole in the D-blocks, to which the internal surfaces of the pin holes are kinematically coupled. The lower reference node is fixed whereas the upper reference node is given a displacement boundary condition D in the y-direction to resemble a tensile force F as shown in Fig. 3. Due to reflective symmetry in load and geometry about the xy-plane, only half the geometry is modeled. The ring specimen is made of aluminium alloy Al 6063-T5 for which the uniaxial stress-strain curve is given by Eqn. (1) and the first row of Table 1. The D-blocks are made AISI 1080 carbon steel, for which the Young's modulus is 210 GPa, Poisson's ratio 0.3 and yield strength 375 MPa (0.2% proof strength). The D-blocks have, in comparison to the ring specimen, about three time higher stiffness and strength. During the ring test, steel is expected to only deform within the elastic regime whereas the ring specimen will undergo elastic and plastic deformation. Hence, the ring specimen material is modeled as elasticeplastic with isotropic hardening according to Eqn. (1) and the D-blocks are modeled as linear-elastic allowing for large deformation theory. The ring specimen and the D-blocks were meshed with three dimensional 20-node hexahedral continuum elements (C3D20R) with reduced integration. The mesh density was refined in the gauge region of the specimen for which an element size of 0.5 mm was used whereas the rest of the model was meshed with an element size of 4 mm. A total of about 6500 elements where used in the half-symmetry model in Fig. 3.
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Fig. 3. Half-symmetric model of the ring test.
2.2. Optimization procedure and results As outlined earlier, the three parameters to be varied in the optimization study are the gauge angle a, the specimen orientation q and the friction coefficient m. In order to cover a large interval, several values were considered for each parameter: a ¼ 9 , 14 and 24 ; q ¼ 0 , 30 and 60 ; m ¼ 0.0, 0.10, 0.25 and 0.50. The effect of the gauge angle a on the location of necking was studied by varying a while keeping the specimen orientation fixed at q ¼ 0 . Fig. 4(a)e(c) are contour plots showing the resulting neck formation for the three gauge angles and m ¼ 0.0. For a ¼ 9 and 14 in Fig. 4(a) and (b), respectively, the neck forms at the centre of the gauge length while for a ¼ 24 in Fig. 4(c) two necks are formed away from the centre. For increasing a > 24 the two necks tend to form closer to the transition region between the notch and the gauge length (see Fig. 1). In order to focus the strain measurements in the gauge length during, this deformation mode is not favourable. It is desired to have a single neck forming in the centre of the gauge length. Hence specimens with a 24 are excluded. Next the effect of friction on the necking formation is studied and a ¼ 9 and 14 are considered. For each specimen gauge angle four different friction coefficients were analysed m ¼ 0.0, 0.10, 0.25 and 0.50. As shown in Fig. 4(d)e(e), the friction coefficient has a strong influence on the necking behaviour. For a gauge angle of a ¼ 14 and friction coefficient m 0.25 two necks are formed whereas for a specimen with a ¼ 9 one centre neck was always formed regardless of the friction coefficient. During the ring test the amount of friction will depend on the surface characteristics and level of lubrication of the contacting surfaces between the D-blocks and the ring specimen. Hence, a specimen with a gauge angle a ¼ 9 is chosen as this proved to form a neck at the centre consistently for all friction levels. The friction has also a large effect of on the mechanical response of the test, as shown in Fig. 5 where the axial load F (kN) versus axial displacement D (mm) of the D-block obtained from FEM are plotted for four different friction coefficients and a Table 1 Material parameters showing the mechanical properties of Al 6063-T5. Al 6063-T5
E (GPa)
so (MPa)
n
Axial (tensile test) Tangential (ring test, DB) Tangential (ring test, RB)
64
142
0.123 0.128 0.122
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Fig. 4. Mises stress contours showing the effect of specimen gauge length and friction coefficient on neck formation.
specimen with a ¼ 9 and q ¼ 0 . The mechanical response has three distinct regions of deformation, where the first corresponds to a gradual reduction of the initial clearance between the ring specimen and the D-block, the second region corresponds to elastic deformation and the third region to plastic deformation. The instance of onset of necking (dF/dD ¼ 0) is also noticeable from the force displacement curve as indicated by the hollow markers. It is also observed that increase in friction coefficient will shift the loadedisplacement curve upward and delay onset of necking while maintaining the trend of the curve. Hence, friction present during the test will have significant effect on the mechanical response and measures must be taken to minimize then friction such that the mechanical response of the material itself is obtained.
Fig. 5. Load F (kN) versus displacement D (mm) of the D-block for Al 6063-T5 ring specimen obtained from FEM at four different friction coefficients.
Next the effect of varying the specimen orientation angle q on the mechanical response (F vs. D) and on tangential stress distribution (sf). Three different specimen orientation angles are considered q ¼ 0 , 30 and 60 . As shown in Fig. 6(a)e(c), corresponding to a ¼ 9 and m ¼ 0.10, the specimen orientation angle q is found to have a negligible effect on the loadedisplacement curve. In order to assure a homogenous tangential stress distribution and low radial stresses in gauge length, the effect of specimen orientation angle q on the tangential and radial stress distribution in the gauge length is analyzed. Fig. 6 shows the tangential stress distribution sf evaluated on a path at the outer surface of the gauge region (r ¼ D0/2 and 1 f/a 1 in Fig. 2) at four different increments as indicated with hollow markers (inc. 1e4), where increment 4 corresponds to a post-necking stage. For specimen orientation angles q ¼ 0 and 30 the tangential stress distribution was found to be fairly uniform along the gauge length throughout the entire on-loading stage. However, for specimen orientation angle q ¼ 60 the stress distribution as shown in Fig. 6(c) is not as uniform, especially toward f/a > 0, which indicates that bending stresses will be influencing the uniformity of the tangential stresses. In Fig. 7(a) the ratio between the radial stress and tangential stress (sr/sf) is plotted for a path at the center of the ring specimen (r ¼ (D e t0)/2 and 1 f/a 1 in Fig. 2), which shows a negligible radial stress. Fig. 7(b) shows a contour plot of the tangential stress distribution. Hence, in this study the specimen gauge angle selected is a ¼ 9 , which will be tested at a specimen orientation angle q ¼ 30 . This optimum specimen geometry does not need any tapering of the gauge region in order to localize necking, which is the case in Refs. [10,11,15]. The value q ¼ 0 is not chosen in this study since it is impractical when conducting the strain measurements with the video extensometer during the ring test. At this juncture it should be mentioned that in Ref. [10] a specimen with a gauge angle of a ¼ 30 is used. Such specimen would, in the view of the optimization results obtained herein, lead to a neck formation at the transition between the notch and the gauge length for all specimen orientation angles q and friction coefficient m. This is also verified experimentally as shown in Fig. 1 for which a ¼ 30 .
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Fig. 6. Force versus displacement (F vs. D) and tangential stress distribution with a ¼ 9, m ¼ 0.10 and specimen angles (a) q ¼ 0 , (b) q ¼ 30 , and (c) q ¼ 60 .
3. Experiments In this section the experimental setup and procedure of the ring test is outlined. As was concluded from the optimization study, a ring specimen with gauge angle of a ¼ 9 and a specimen orientation angle q ¼ 30 will be chosen for the ring test setup. One of the objectives of the experimental investigation is to develop a better understanding of the effect of friction during the ring test and measures to minimize it, especially as this has not been addressed nor quantified previously [6,10,11,15]. 3.1. Axial true stress-strain curve As a preliminary experiment, the true axial stress-strain behavior of the tube material is to be established. Microscopic examination on axial and tangential cuts of the aluminum
Al6063-T5 tubes revealed a similar microstructure suggesting a rather isotropic mechanical response in the two directions. Hence, the axial true stress-strain curve of the tube material will serve as a curve of reference for the tangential true stress-strain curve determined with the ring specimen developed in this study. Tensile axial samples were prepared by cutting out smooth round bar specimens with 2 mm diameter from the thickness of aluminum tubes with their tensile axis in the axial direction of the tube. The axial stress-strain relations were established by conducting uniaxial tests on these smooth round bar specimens with a tensile testing machine [1]. The strain was measured using a Tinius & Olsen video extensometer. In Fig. 8 the true stress versus true strain from the uniaxial tests are shown. Using the measured cross sectional diameter and curvature of the neck, the tensile test data were Bridgman corrected [18] to account for the triaxial stress state arising due to the formation of the neck.
Fig. 7. Showing (a) radial to tangential stress ratio (sr/sf) and (b) contour plot for the tangential stress distribution sf for a ¼ 9 , q ¼ 30 and m ¼ 0.10.
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the ring specimen two strain gauges are glued at each end of the gauge region as shown in Fig. 9(b)e(c). It was found that the strain gauges gave consistent readings which agreed with the readings from the video extensometer and indicated uniform loading through the gauge section. 3.2.1. D-block ring test setup e (DB) All the internal edges of the ring specimens are smoothened with sandpaper so that no edge contact between the D-block is present during the test. In order to reduce friction between the Dblocks and the inner surface of the ring specimen two different testing setups were considered. In the first setup, shown in Fig. 9(b), the internal surface of the ring specimen and D-blocks are lubricated with Amberglide PTFE lubricant, which is a multipurpose lubricant containing Teflon micro-particles known to produce smooth contact with a low friction. This test setup is referred to as the DB-setup.
Fig. 8. True axial stress-strain curve for Al 6063-T5 tube. The solid line correspond to experimental data, hollow circles show the Bridgman corrected data and dash line represents the curve fit according to Eqn. (1).
Experimental data obtained from uniaxial tests were curve fitted to obtain the following exponential hardening stressestrain relationship as shown in Eqn. (1)
s¼
8 > <
Eε n ε > : so ε0
ε εo ε > εo
(1)
where E is the modulus of elasticity, so denotes the initial yield stress, n is the strain hardening exponent and εo¼so/E¼0.0022. The material and model parameters are listed in Table 1. 3.2. Ring test setup The testing setup consists of an upper and lower fixture, Dblocks and pins which are mounted to an MTS Alliance RF/150 tensile testing machine. The fixtures were manufactured using a numerically controlled wire cut electric discharge machine with cut tolerances of 1 mm. This is to assure precise alignment once mounted to the MTS machine. The ring specimens were manufactured from Al 6063-T5 pipes with outer nominal diameter 60 mm and nominal thickness of 3 mm. The ring test experimental setup is shown in Fig. 9(a). The D-blocks were aligned and fixed to the fixture by set-screws, which prevents them from rotating during the test. Markings were introduced on the D-blocks to mark the specimen orientation angle q ¼ 30 such that the ring specimen could be oriented accordingly. The deformation and strain will be measured primarily by a Tinius & Olsen video extensometer which is mased on digital image correlation technique (DIC). The video extensometer system consists of a high resolution camera, as shown in Fig. 9(a), and a DIC post-processing software. The outer surface of the gauge region is sprayed with a matt white color and a matt black color to create an irregular speckled pattern as shown in Fig. 9(d). Target points along with their boxed area range are assigned, which are used to track the movement of the speckles within each box and correlate it the original undeformed pattern. Here four target points are assigned. The strain is measured between target point 1 and 2, and the instantaneous width of the gauge section is measured between target point 3 and 4. In order to assure uniform loading without any bending of the gauge region of
3.2.2. Roller needle bearing ring test setup e (RB) In the second setup, shown in Fig. 9(c), the size of the D-blocks is reduced and a half needle roller bearing is placed between each D-block and the ring specimen. An SKF needle roller bearing of model K 45 563 28 with a total of 34 rollers is used for this propose. The bearing is cut in two halves, lubricated with Amberglide PTFE lubricant before used in the test. The added flexilibty of the rollers underneath the specimen in addition to the application of the lubricant will further reduce the friction during the ring test. This test setup is referred to as the RB-setup. A finite element model of each of the two setups DB and RD is modelled adequately including the details of the setup as shown in Fig. 10(b)e(c). 4. Data analysis and results In this section expressions for the true tangential stress and strain are derived and the analysis and post-processing of the experimental results are presented. As indicated by Fig. 9(d) the video extensometer will measure the strain between target point 1 and 2 equivalent to the change of length L, which has an initial value of L0 ¼ 2w0 ¼ 6 mm. However, the tangential (hoop) engineering strain on the outer surface of the gauge section is related to the change of arc length Lf, shown in Fig. 10(a), by the following expression
ef ¼
Lf 1 Lf0
(2)
where the initial arc length is given by
Lf0 ¼ D0 sin1
L0 D0
(3)
and the current arc length is given by
Lf ¼ D0 ð1 þ er Þsin1
L0 ð1 þ em Þ D0 ð1 þ er Þ
(4)
where em is the experimentally measured linear engineering strain between target point 1 and 2, and er is the radial engineering strain, e.g. change in thickness. It is assumed that plastic deformation in the gauge region is incompressible. Hence, the plastic incompressibility requirement is given by
ef þ ez þ er ¼ 0
(5)
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Fig. 9. (a) The general ring test experimental setup, (b) D-block test setup (DB), (c) needle-roller bearing test setup (RB), and (d) speckled ring specimen gauge region with corresponding target points for digital image correlation.
where ez is the axial engineering strain, e.g. change in width of the gauge section of the ring specimen, which is measured experimentally as the strain between target point 3 and 4 in Fig. 9(d). It is rather difficult to measure the change in thickness, therefore er needs to be estimated. The finite element model revealed that the rate of change of thickness and rate of change of width within the gauge section of the ring specimen are equal, as shown in Fig. 11 where er and ez are evaluated at the center and off-center and plotted versus axial displacement D. A similar finding on a rectangular specimen was reported in Ref. [19]. Thus, er¼ez, which with Eqn. (5) gives
er ¼ ef 2
(6)
Now, using Eqns. (3), (4) and (6) in Eqn. (2) gives an implicit expression for the tangential (hoop) engineering strain as
2 ef sin1 ef ¼
2sin1
! 2L0 ð1þem Þ D0 ð2ef Þ
L0 D0
1
(7)
In order to determine ef the non-linear equation Eqn. (7) needs to be solved for iteratively for each increment of measured strain em. However, for large outer pipe diameter D0 pipes and a relatively small gauge length L0 Eqn. (7) reduces to
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Fig. 10. (a) Gauge region of ring specimen, (b) FEM model for DB-setup and (c) FEM model for RB-setup.
Fig. 11. Engineering strain versus axial displacement obtained from FEM.
ef ¼ em
(8)
The true tangential strain is now given by
εf ¼ lnð1 þ em Þ
(9)
In order to obtain the transverse stress in the ring specimen the transverse force needs to be estimated. Based on the free body cut shown in Fig. 10(a)e(b) and assumption of thin-walled cylinder the tangential load is based on force equilibrium related to the axial force as Ff¼F/2. This assumption is confirmed and validated with the finite element model. In Fig. 12 the tangential force Ff is plotted versus the engineering tangential strain ef for obtained experimentally using the two test setup DB (D-block) and RB (roller needle bearing) and from FEM with different friction coefficients. For the DB-setup the FEM model shown in Fig. 10(b) is utilized whereas for the RB-setup the model in Fig. 10(c) is utilized. It should be mentioned here that the engineering strain ef obtained from FEM is determined with exactly the same procedure as in the experiments outlined above. As shown, for the DB-setup, the FEM results with a friction coefficient of m¼0.25 matches the DB experimental results closely, whereas for the RB-setup the FEM results with a friction coefficient of m¼0.13 matches the RB experimental results markedly well. The FEM results pertaining to frictionless conditions m¼0.0 are also plotted in Fig. 12. In view of the results in Fig. 12, it can be concluded that each of the test setups possesses different levels of friction, where the DBsetup has the higher level and the RB-setup the lower level of which none give frictionless conditions. The fact that a difference in
mechanical response between the two setups and frictionless conditions exists shows that the measures used, e.g. application of PTFE lubricant and use of needle roller bearings, did reduce the friction but did not eliminate it completely as the lowest friction that can be achieved is m¼0.13 with the RB-setup. In previous studies [10,11], the friction effect was disregarded since it was
Fig. 12. Tangential force Ff versus engineering tangential strain ef for DB-setup and RB-setup.
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assumed to be negligible. Evident by Fig. 12, the friction effect is dominant and needs to be accounted for. Here the friction effect is accounted for through a post-processing procedure of the experimental and FEM result for each test setup. The vertical shift in force introduced in the load vs. strain curve due to friction is estimated by comparing the FEM results of DB (m¼0.25) and RB (m¼0.13) with the frictionless curve (m¼0.0), respectively. The difference in force is due to the friction associated with the test setup and the corrected experimental tangential force is obtained by subtracting this difference from the experimentally obtained data. Hence, the corrected tangential force FfDB and FfDB for each test setup, respectively, is given by DBðExpÞ
DBðm¼0:25Þ DBðm¼0:0Þ Ff Ff
(10)
RBðExpÞ
RBðm¼0:13Þ RBðm¼0:0Þ Ff Ff
(11)
FfDB ¼ Ff FfRB ¼ Ff
The true tangential (hoop) stress is now given by the corrected tangential force and the instantaneous cross section area A of the gauge length as DB sDB f ¼ Ff
. A
and
RB sRB f ¼ Ff
. A
(12)
The instantaneous cross section area is given by Refs. A¼w0t0(1þez)(1þer), which with the findings er¼ ez in Fig. 11 gives A¼w0t0(1þez)2, where ez is the change in width and is measured experimentally between target point 3 and 4. Alternatively, the instantaneous cross section area can be given by Ref. A¼w0t0/(1þef). Comparing the two expressions yields (1þef)(1þez)2¼1, which confirms the incompressibility requirement in the gauge section. Fig. 13 shows the tangential (hoop) true stress obtained from Eqns. 10e12 versus the true tangential (hoop) strain obtained from Eqn. (9) for both the DB and RB setup for the Al 6063-T5 ring specimen. The axial true stress strain for Al 6063-T5 obtained from Refs. [1e3] in Fig. 1 is also plotted for reference. As shown, the tangential and axial true stress-strain curves agree markedly well indicating that the Al 6063-T5 pipes have isotropic mechanical behavior. Moreover, the developed experimental procedure with its post-processing step, which corrects for the friction, is an accurate method for determining the tangential true stress-strain behavior of pipes. This is particularly confirmed by the fact that the procedure gives consistent results for both the test setups DB
Fig. 13. Tangential and axial true stress-strain curves for Al 6063-T5 pipes.
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and RD, with very good agreement with the axial stress-strain curve. Using Eqn. (1) with Fig. 13 gives a strain hardening exponent n ¼ 0.128 for the DB-setup and n ¼ 0.122 for the RB-setup, which matches the value obtained from the axial test (n ¼ 0.123) reported in Table 1. 5. Conclusions In this study a method to determine the tangential true stressstrain behavior of pipes is developed. An optimized ring specimen is proposed for the purpose, which showed to have uniform stress distribution in the gauge region. The optimized ring specimen give a neck formation at the center of the gauge region consistently, without having to taper the gauge region. It is also found that friction is present and has a substantial effect on the mechanical response of the ring test, which was neglected in previous studies. Two test setups are proposed, one using lubricated Dblocks (DB) and one using lubricated D-blocks with needle roller bearing (RB). By use of FEM it was found that the RB-setup has the lowest friction with m¼0.13, whereas the DB-setup has a friction coefficient of m¼0.25. The friction effect is accounted for by postprocessing procedure of the experimental results. With aid of the FEM results, the shift in force associated friction is subtracted from the experimental results. Relations are derived for determining the true tangential strain and the true tangential stress, which gave very reliable results agreeing well with axial true stress-strain curve. The experimental setups along with the data analysis procedure makes this method very useful for assess ductility of pipes and for determining the ductile failure locus of pipes. It can also be applied for structural integrity analysis of pipes containing longitudinal defect. Acknowledgement The authors would like to acknowledge the financial support of this work provided by Abu Dhabi National Oil Company (ADNOC) and the Petroleum Institute under the grant RAGS-14326-2014. Mr. Arman Molki and Mr. Shrinivas Bojanampati are acknowledged for their assistance in the lab. References [1] Seibi AC, Barsoum I, Molki A. Experimental and numerical study of expanded aluminum and steel tubes. Eng Procedia 2011;10:3049e55. [2] Barsoum I, Khan F, Molki A, Seibi A. Ductile failure modelling of expanded aluminium tubes with embedded circular holes. In: ASME pressure vessels & piping conference. Paris, France; 2013. [3] Barsoum I, Khan F, Molki A, Seibi A. Modeling of ductile crack propagation in expanded thin-walled 6063-T5 aluminum tubes. Int J Mech Sci 2014;80: 160e8. [4] ASTM. Standard test methods and definitions for mechanical testing of steel products. A 370e07a. 2007. [5] Crone DG, Collins LE, Bian YK, Weber P. The effect of sample flattening on yield strength measurement in line pipe. In: Proceedings of the Asme International pipeline conference 2010, Vol 2; 2010. p. 477e82. [6] ASTM. Standard test method for apparent hoop tensile strength of plastic or reinforced plastic pipe by split disk method. D 2290-08. 2008. [7] Kaynak C, Erdiller ES, Parnas L, Senel F. Use of split-disk tests for the process parameters of filament wound epoxy composite tubes. Polym Test 2005;24: 648e55. [8] Buarque EN, d'Almeida JRM. The effect of cylindrical defects on the tensile strength of glass fiber/vinyl-ester matrix reinforced composite pipes. Compos Struct 2007;79:270e9. [9] Laiarinandrasana L, Devilliers C, Oberti S, Gaudichet E, Fayolle B, Lucatelli JM. Ring tests on high density polyethylene: full investigation assisted by finite element modeling. Int J Pres Ves Pip 2011;88:1e10. [10] Wang H, Bouchard R, Eagleson R, Martin P, Tyson WR. Ring hoop tension test (RHTT): a test for transverse tensile properties of tubular materials. J Test Eval 2002;30:382e91. [11] Arsene S, Bai J. A new approach to measuring transverse properties of structural tubing by a ring test. J Test Eval 1996;24:386e91.
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[12] García JJ, Rangel C, Ghavami K. Experiments with rings to determine the anisotropic elastic constants of bamboo. Constr Build Mater 2012;31: 52e7. [13] MEtallic materials tube ring expanding test, International Standard ISO 8495e8496. [14] Dick CP, Korkolis YP. Mechanics and full-field deformation study of the Ring Hoop Tension Test. Int J Solids Struct 2014;51:3042e57. ne S, Bai J. A new approach to measuring transverse properties of [15] Arse structural tubing by a ring test - experimental investigation. J Test Eval 1998;26:26e30.
[16] Bae B-K, Cho S-K, Seok C-S. A study on ring tensile specimens. Mater Sci Eng A 2008;483e484:248e50. [17] ABAQUS. Analysis User's Manual, Version 6.11. Providence, RI, USA: Dassault mes Simulia Corp.; 2011. Syste [18] Bridgman PW. Studies in large plastic flow and fracture. Cambridge, MA: Harvard University Press; 1964. [19] Choung JM, Cho SR. Study on true stress correction from tensile tests. J Mech Sci Technol 2008;22:1039e51.
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
A procedure to determine the tangential true stress-strain behavior of pipes I. Barsoum a, b, *, K.F. Al Ali a a
Department of Mechanical Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates Department of Aeronautical & Vehicle Engineering, Division of Lightweight Structures, Royal Institute of Technology e KTH, Teknikringen 8, 100 44 Stockholm, Sweden b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 March 2014 Received in revised form 27 November 2014 Accepted 28 November 2014 Available online 19 February 2015
Determining the tangential mechanical properties of a tube is essential for simulation of various manufacturing processes that involve the use of a tubular geometry. The aim of this study is to develop a procedure to determine the tangential true stress-strain behavior of pipes. For this purpose a modified ring test setup is proposed consisting of a ring specimen loaded with two separate D-blocks. Using a finite element model, an optimized ring specimen geometry is obtained. The optimized ring specimen exhibits uniform tangential distribution in the gauge region of the specimen and necking occurs consistently at the center of the gauge length. It is found that friction has a substantial effect on the mechanical response of the ring test for which two different setups to reduce friction are proposed. One using lubricated D-blocks (DB) and one using lubricated D-blocks with needle roller bearing (RB). Assisted by the FE model, the friction during the experiment is account for and a data analysis procedure to determine the tangential stress-strain curve of the pipe is proposed. It is found that the results using this procedure show very good agreement with previously published results. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Aluminum Pipes Stress-strain curve Tangential mechanical properties Experiments Finite element modeling Oil and gas
1. Introduction Various manufacturing processes require detailed knowledge of the tangential mechanical behavior of pipes to obtain proper designs and evaluations of the final product. One such process commonly employed in the petroleum industry is the Solid Expandable Tubular Technology (SETT) [1e3]. It is a cold working process in which a tubular casing is expanded radially by pushing a conical mandrel axially into the tube to obtain a desired radial permanent expansion. The technology enables mono-diameter casing, which is being used to replace the conventional multidiameter telescopic casing configuration used in oil wells. This is achieved by expanding multiple tubes of a certain diameter in the well resulting in a rather uniform diameter throughout the well hole providing a larger downhole diameter than the conventional telescopic configuration, hence resulting in an increased rate of oil production. However, to assess the mechanical integrity of the tube being expanded such that to select an appropriate tubular steel
* Corresponding author. Department of Mechanical Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates. E-mail address: [email protected] (I. Barsoum). http://dx.doi.org/10.1016/j.ijpvp.2014.11.002 0308-0161/© 2015 Elsevier Ltd. All rights reserved.
grade, it is of paramount importance to know the tangential stressstrain behavior of the tube. According to ASTM A370 [4], the tensile properties of pipes is to be determined from a specimen cut in the axial direction. Alternatively, ASTM A370 specifies a method to determine the transverse properties of pipes by cutting a circumferential strip from the pipe, flatten it and tensile test it. The work hardening from the flattening steps alters the transverse behavior considerably and the result will not represent the tangential true stress-strain curve of the pipe. Crone et al. [5] conducted a study to determine the effect of flattening the specimen on the tested mechanical properties of the pipe and concluded that flattening has a large effect on the experimental results. The material testing standard ASTM D2290 [6] specifies the split-disc test method to determine the tangential yield and tensile strength of thermoplastic pipes. The specimen is a ring cut from the pipe and is loaded with two half discs, which are inserted internally in the ring specimen and are pulled apart with a tensile testing machine. Kaynak et al. [7] used this method to study the effect of the filament winding process parameters on the tangential tensile strength of continuous fiber reinforced epoxy composite tubes, in Ref. [8] the influence of defects on the tensile strength of glass fiber reinforced composite pipes was studied with this testing method
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and in Ref. [9] the effect of aging of HDPE plastic pipes on the failure strain was investigated. Though the method is found efficient in determining the tangential strength and failure strain of the pipes, it does not provide a way to determine the full tangential stressstrain curve nor does it quantify the effect of friction between the disks and the ring specimen. Few researchers [10e12] have addressed this issue in the past and made attempts to experimentally measure the tensile transverse properties of pipes. Arsene et al. [11] addressed the importance of the ring specimen geometry in view of the effect of bending on the test results. They performed finite element analysis and used a notched ring specimen with three die inserts to load the ring specimen to overcome the issue with bending involved in the classical ring test, as recommended in the international standard ISO 8495-8496 [13]. Wang et al. [10] conducted a study to determine the transverse properties of tubular products and proposed a modified split-disc experimental setup, which uses D-blocks. The contact between the D-blocks and the internal surface of the ring specimen will mimic the internal pressure experienced in a pipe. However, the reason to why the selected ring specimen geometry is used is not given. Friction between the D-blocks and the specimen is considered negligible, which is justified by the use of lubricated thin Teflon sheets in-between the D-blocks and the specimen. However, this assumption was not confirmed and no attempts were undertaken to quantify its effect on the mechanical response of the ring test. Most recently Dick and Korkolis [14] conducted a thorough investigation on the mechanics of the ring hoop tension test (RHTT) and found that the friction can have a large effect on the uniaxial stress state in the specimen. This study addresses some of the aforementioned shortcomings of previous ring test setups and presents a novel testing and evaluation method for determining the tangential stress-strain behavior of pipes. The first part of the work presents an optimization study to find an optimum ring specimen geometry aided by the use of the finite element method (FEM). The second part presents the experimental setup and the means used to minimize the friction effects involved in the experimental setup and the third part provides a data analysis procedure and the determination of the tangential true stress-strain curve of a pipe.
Fig. 1. Failed ring specimens using RHTT (Ring Hoop Tension Test) test method outlined in Ref. [10].
through the D-blocks mimics the internal pressure experienced in pipes. The objective of the optimization study is to obtain a specimen configuration for which necking appears within the gauge length of the specimen and for which a uniform tangential stress state is achieved within the gauge length while accounting for the effect of friction between the D-blocks and specimen. In Fig. 2 the geometry of the specimen is shown, which has an initial specimen thickness t0 ¼ 3 mm, specimen outer diameter D ¼ 60 mm, initial width of reduced gauge section w0 ¼ 3 mm, fillet radius r ¼ 6 mm and a specimen width of 15 mm. The three optimization parameters to be addressed are the gauge length as given by angle a,
2. Optimization of the ring specimen Most of the attempts [6,10,11,15,16] to determine the tangential stress-strain behavior of pipes have used a similar specimen configuration consisting of a ring cut from the pipe with a gauge section manufactured to localize the necking location. Preliminary experiments were conducted following the experimental setup, specimen geometry and testing method in Ref. [10]. The difference was in the choice of material. In Ref. [10] a low-carbon hot-rolled steel tube with outer diameter 95.25 mm and thickness 2.2 mm was used whereas in this study an aluminum alloy Al 6063-T5 tube was used. The Al 6063-T5 material was delivered in 200 mm long seamless tubes with outer diameter of 60 mm and wall thickness of 3 mm. For all tests necking occurred in the transition between the large radius and the gauge length, as shown in Fig. 1, rather than at center of the gauge length. This indicates that the specimen could be subjected to non-uniform tangential stresses, bending and frictional forces between the D-block and the ring specimen. In order to develop a validated method to determine the tangential stress-strain behavior of pipes, an optimum ring specimen configuration need to be chosen. As there are no standards outlining the specimen dimensions for such tests a study to optimize the ring specimen configuration is motivated. The specimen selected for the optimization is a ring specimen, as shown in Fig. 2, for which the application of a tensile load
Fig. 2. (a) Test setup and (b) configuration of the ring specimen.
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specimen orientation defined as the angle q between the center of the specimen gauge length and tensile loading axis and the friction coefficient m between the outer surfaces of the D-blocks and internal surface of the ring specimen. The requirement of a uniform tangential stress distribution in the gauge length is controlled by a and q. This requirement is imposed as it is important to eliminate or minimize any other stress components, such as radial and axial, to ensure uniaxial tangential stresses in the gauge length. A non-uniform tangential stress distribution in the gauge length would also indicate that the specimen is subjected to bending, hence altering the location of necking. Having a necking location outside the measurement gauge length would invalidate the experiment. Furthermore, friction is expected to be present during the test due to the contact between the Dblocks and the ring specimen, which will influence the mechanical response of the test considerably. By varying m the effect of friction can be quantified. The optimization process was undertaken with aid of a finite element model of the test setup. Finite element analysis of the test setup with different combinations of gauge angles a, specimen orientation q and frictions coefficient m were simulated with Abaqus 6.11 [17]. The details of the finite element model are outlined in the next section. 2.1. The finite element model The main purpose of the finite element model is to simulate the mechanical response of the test setup shown in Fig. 2, taking into consideration all the geometric details of the specimen and the D-blocks and the contact interaction between them. This will guide in the selection of the optimum ring specimen geometry. The FE model consists of two D-blocks and a ring specimen, as shown in Fig. 2, where the ring specimen is inserted onto the Dblocks through which the tensile force is transmitted to the inner surface specimen mimicking an internal pressure condition. A master-slave contact interaction is assigned between the outer surfaces of the D-blocks and the inner surface of the ring specimen, with an isotropic friction behavior with a coefficient of friction ranging from 0 m 0.5. A reference node is defined at the center of each pin hole in the D-blocks, to which the internal surfaces of the pin holes are kinematically coupled. The lower reference node is fixed whereas the upper reference node is given a displacement boundary condition D in the y-direction to resemble a tensile force F as shown in Fig. 3. Due to reflective symmetry in load and geometry about the xy-plane, only half the geometry is modeled. The ring specimen is made of aluminium alloy Al 6063-T5 for which the uniaxial stress-strain curve is given by Eqn. (1) and the first row of Table 1. The D-blocks are made AISI 1080 carbon steel, for which the Young's modulus is 210 GPa, Poisson's ratio 0.3 and yield strength 375 MPa (0.2% proof strength). The D-blocks have, in comparison to the ring specimen, about three time higher stiffness and strength. During the ring test, steel is expected to only deform within the elastic regime whereas the ring specimen will undergo elastic and plastic deformation. Hence, the ring specimen material is modeled as elasticeplastic with isotropic hardening according to Eqn. (1) and the D-blocks are modeled as linear-elastic allowing for large deformation theory. The ring specimen and the D-blocks were meshed with three dimensional 20-node hexahedral continuum elements (C3D20R) with reduced integration. The mesh density was refined in the gauge region of the specimen for which an element size of 0.5 mm was used whereas the rest of the model was meshed with an element size of 4 mm. A total of about 6500 elements where used in the half-symmetry model in Fig. 3.
61
Fig. 3. Half-symmetric model of the ring test.
2.2. Optimization procedure and results As outlined earlier, the three parameters to be varied in the optimization study are the gauge angle a, the specimen orientation q and the friction coefficient m. In order to cover a large interval, several values were considered for each parameter: a ¼ 9 , 14 and 24 ; q ¼ 0 , 30 and 60 ; m ¼ 0.0, 0.10, 0.25 and 0.50. The effect of the gauge angle a on the location of necking was studied by varying a while keeping the specimen orientation fixed at q ¼ 0 . Fig. 4(a)e(c) are contour plots showing the resulting neck formation for the three gauge angles and m ¼ 0.0. For a ¼ 9 and 14 in Fig. 4(a) and (b), respectively, the neck forms at the centre of the gauge length while for a ¼ 24 in Fig. 4(c) two necks are formed away from the centre. For increasing a > 24 the two necks tend to form closer to the transition region between the notch and the gauge length (see Fig. 1). In order to focus the strain measurements in the gauge length during, this deformation mode is not favourable. It is desired to have a single neck forming in the centre of the gauge length. Hence specimens with a 24 are excluded. Next the effect of friction on the necking formation is studied and a ¼ 9 and 14 are considered. For each specimen gauge angle four different friction coefficients were analysed m ¼ 0.0, 0.10, 0.25 and 0.50. As shown in Fig. 4(d)e(e), the friction coefficient has a strong influence on the necking behaviour. For a gauge angle of a ¼ 14 and friction coefficient m 0.25 two necks are formed whereas for a specimen with a ¼ 9 one centre neck was always formed regardless of the friction coefficient. During the ring test the amount of friction will depend on the surface characteristics and level of lubrication of the contacting surfaces between the D-blocks and the ring specimen. Hence, a specimen with a gauge angle a ¼ 9 is chosen as this proved to form a neck at the centre consistently for all friction levels. The friction has also a large effect of on the mechanical response of the test, as shown in Fig. 5 where the axial load F (kN) versus axial displacement D (mm) of the D-block obtained from FEM are plotted for four different friction coefficients and a Table 1 Material parameters showing the mechanical properties of Al 6063-T5. Al 6063-T5
E (GPa)
so (MPa)
n
Axial (tensile test) Tangential (ring test, DB) Tangential (ring test, RB)
64
142
0.123 0.128 0.122
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Fig. 4. Mises stress contours showing the effect of specimen gauge length and friction coefficient on neck formation.
specimen with a ¼ 9 and q ¼ 0 . The mechanical response has three distinct regions of deformation, where the first corresponds to a gradual reduction of the initial clearance between the ring specimen and the D-block, the second region corresponds to elastic deformation and the third region to plastic deformation. The instance of onset of necking (dF/dD ¼ 0) is also noticeable from the force displacement curve as indicated by the hollow markers. It is also observed that increase in friction coefficient will shift the loadedisplacement curve upward and delay onset of necking while maintaining the trend of the curve. Hence, friction present during the test will have significant effect on the mechanical response and measures must be taken to minimize then friction such that the mechanical response of the material itself is obtained.
Fig. 5. Load F (kN) versus displacement D (mm) of the D-block for Al 6063-T5 ring specimen obtained from FEM at four different friction coefficients.
Next the effect of varying the specimen orientation angle q on the mechanical response (F vs. D) and on tangential stress distribution (sf). Three different specimen orientation angles are considered q ¼ 0 , 30 and 60 . As shown in Fig. 6(a)e(c), corresponding to a ¼ 9 and m ¼ 0.10, the specimen orientation angle q is found to have a negligible effect on the loadedisplacement curve. In order to assure a homogenous tangential stress distribution and low radial stresses in gauge length, the effect of specimen orientation angle q on the tangential and radial stress distribution in the gauge length is analyzed. Fig. 6 shows the tangential stress distribution sf evaluated on a path at the outer surface of the gauge region (r ¼ D0/2 and 1 f/a 1 in Fig. 2) at four different increments as indicated with hollow markers (inc. 1e4), where increment 4 corresponds to a post-necking stage. For specimen orientation angles q ¼ 0 and 30 the tangential stress distribution was found to be fairly uniform along the gauge length throughout the entire on-loading stage. However, for specimen orientation angle q ¼ 60 the stress distribution as shown in Fig. 6(c) is not as uniform, especially toward f/a > 0, which indicates that bending stresses will be influencing the uniformity of the tangential stresses. In Fig. 7(a) the ratio between the radial stress and tangential stress (sr/sf) is plotted for a path at the center of the ring specimen (r ¼ (D e t0)/2 and 1 f/a 1 in Fig. 2), which shows a negligible radial stress. Fig. 7(b) shows a contour plot of the tangential stress distribution. Hence, in this study the specimen gauge angle selected is a ¼ 9 , which will be tested at a specimen orientation angle q ¼ 30 . This optimum specimen geometry does not need any tapering of the gauge region in order to localize necking, which is the case in Refs. [10,11,15]. The value q ¼ 0 is not chosen in this study since it is impractical when conducting the strain measurements with the video extensometer during the ring test. At this juncture it should be mentioned that in Ref. [10] a specimen with a gauge angle of a ¼ 30 is used. Such specimen would, in the view of the optimization results obtained herein, lead to a neck formation at the transition between the notch and the gauge length for all specimen orientation angles q and friction coefficient m. This is also verified experimentally as shown in Fig. 1 for which a ¼ 30 .
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63
Fig. 6. Force versus displacement (F vs. D) and tangential stress distribution with a ¼ 9, m ¼ 0.10 and specimen angles (a) q ¼ 0 , (b) q ¼ 30 , and (c) q ¼ 60 .
3. Experiments In this section the experimental setup and procedure of the ring test is outlined. As was concluded from the optimization study, a ring specimen with gauge angle of a ¼ 9 and a specimen orientation angle q ¼ 30 will be chosen for the ring test setup. One of the objectives of the experimental investigation is to develop a better understanding of the effect of friction during the ring test and measures to minimize it, especially as this has not been addressed nor quantified previously [6,10,11,15]. 3.1. Axial true stress-strain curve As a preliminary experiment, the true axial stress-strain behavior of the tube material is to be established. Microscopic examination on axial and tangential cuts of the aluminum
Al6063-T5 tubes revealed a similar microstructure suggesting a rather isotropic mechanical response in the two directions. Hence, the axial true stress-strain curve of the tube material will serve as a curve of reference for the tangential true stress-strain curve determined with the ring specimen developed in this study. Tensile axial samples were prepared by cutting out smooth round bar specimens with 2 mm diameter from the thickness of aluminum tubes with their tensile axis in the axial direction of the tube. The axial stress-strain relations were established by conducting uniaxial tests on these smooth round bar specimens with a tensile testing machine [1]. The strain was measured using a Tinius & Olsen video extensometer. In Fig. 8 the true stress versus true strain from the uniaxial tests are shown. Using the measured cross sectional diameter and curvature of the neck, the tensile test data were Bridgman corrected [18] to account for the triaxial stress state arising due to the formation of the neck.
Fig. 7. Showing (a) radial to tangential stress ratio (sr/sf) and (b) contour plot for the tangential stress distribution sf for a ¼ 9 , q ¼ 30 and m ¼ 0.10.
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the ring specimen two strain gauges are glued at each end of the gauge region as shown in Fig. 9(b)e(c). It was found that the strain gauges gave consistent readings which agreed with the readings from the video extensometer and indicated uniform loading through the gauge section. 3.2.1. D-block ring test setup e (DB) All the internal edges of the ring specimens are smoothened with sandpaper so that no edge contact between the D-block is present during the test. In order to reduce friction between the Dblocks and the inner surface of the ring specimen two different testing setups were considered. In the first setup, shown in Fig. 9(b), the internal surface of the ring specimen and D-blocks are lubricated with Amberglide PTFE lubricant, which is a multipurpose lubricant containing Teflon micro-particles known to produce smooth contact with a low friction. This test setup is referred to as the DB-setup.
Fig. 8. True axial stress-strain curve for Al 6063-T5 tube. The solid line correspond to experimental data, hollow circles show the Bridgman corrected data and dash line represents the curve fit according to Eqn. (1).
Experimental data obtained from uniaxial tests were curve fitted to obtain the following exponential hardening stressestrain relationship as shown in Eqn. (1)
s¼
8 > <
Eε n ε > : so ε0
ε εo ε > εo
(1)
where E is the modulus of elasticity, so denotes the initial yield stress, n is the strain hardening exponent and εo¼so/E¼0.0022. The material and model parameters are listed in Table 1. 3.2. Ring test setup The testing setup consists of an upper and lower fixture, Dblocks and pins which are mounted to an MTS Alliance RF/150 tensile testing machine. The fixtures were manufactured using a numerically controlled wire cut electric discharge machine with cut tolerances of 1 mm. This is to assure precise alignment once mounted to the MTS machine. The ring specimens were manufactured from Al 6063-T5 pipes with outer nominal diameter 60 mm and nominal thickness of 3 mm. The ring test experimental setup is shown in Fig. 9(a). The D-blocks were aligned and fixed to the fixture by set-screws, which prevents them from rotating during the test. Markings were introduced on the D-blocks to mark the specimen orientation angle q ¼ 30 such that the ring specimen could be oriented accordingly. The deformation and strain will be measured primarily by a Tinius & Olsen video extensometer which is mased on digital image correlation technique (DIC). The video extensometer system consists of a high resolution camera, as shown in Fig. 9(a), and a DIC post-processing software. The outer surface of the gauge region is sprayed with a matt white color and a matt black color to create an irregular speckled pattern as shown in Fig. 9(d). Target points along with their boxed area range are assigned, which are used to track the movement of the speckles within each box and correlate it the original undeformed pattern. Here four target points are assigned. The strain is measured between target point 1 and 2, and the instantaneous width of the gauge section is measured between target point 3 and 4. In order to assure uniform loading without any bending of the gauge region of
3.2.2. Roller needle bearing ring test setup e (RB) In the second setup, shown in Fig. 9(c), the size of the D-blocks is reduced and a half needle roller bearing is placed between each D-block and the ring specimen. An SKF needle roller bearing of model K 45 563 28 with a total of 34 rollers is used for this propose. The bearing is cut in two halves, lubricated with Amberglide PTFE lubricant before used in the test. The added flexilibty of the rollers underneath the specimen in addition to the application of the lubricant will further reduce the friction during the ring test. This test setup is referred to as the RB-setup. A finite element model of each of the two setups DB and RD is modelled adequately including the details of the setup as shown in Fig. 10(b)e(c). 4. Data analysis and results In this section expressions for the true tangential stress and strain are derived and the analysis and post-processing of the experimental results are presented. As indicated by Fig. 9(d) the video extensometer will measure the strain between target point 1 and 2 equivalent to the change of length L, which has an initial value of L0 ¼ 2w0 ¼ 6 mm. However, the tangential (hoop) engineering strain on the outer surface of the gauge section is related to the change of arc length Lf, shown in Fig. 10(a), by the following expression
ef ¼
Lf 1 Lf0
(2)
where the initial arc length is given by
Lf0 ¼ D0 sin1
L0 D0
(3)
and the current arc length is given by
Lf ¼ D0 ð1 þ er Þsin1
L0 ð1 þ em Þ D0 ð1 þ er Þ
(4)
where em is the experimentally measured linear engineering strain between target point 1 and 2, and er is the radial engineering strain, e.g. change in thickness. It is assumed that plastic deformation in the gauge region is incompressible. Hence, the plastic incompressibility requirement is given by
ef þ ez þ er ¼ 0
(5)
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Fig. 9. (a) The general ring test experimental setup, (b) D-block test setup (DB), (c) needle-roller bearing test setup (RB), and (d) speckled ring specimen gauge region with corresponding target points for digital image correlation.
where ez is the axial engineering strain, e.g. change in width of the gauge section of the ring specimen, which is measured experimentally as the strain between target point 3 and 4 in Fig. 9(d). It is rather difficult to measure the change in thickness, therefore er needs to be estimated. The finite element model revealed that the rate of change of thickness and rate of change of width within the gauge section of the ring specimen are equal, as shown in Fig. 11 where er and ez are evaluated at the center and off-center and plotted versus axial displacement D. A similar finding on a rectangular specimen was reported in Ref. [19]. Thus, er¼ez, which with Eqn. (5) gives
er ¼ ef 2
(6)
Now, using Eqns. (3), (4) and (6) in Eqn. (2) gives an implicit expression for the tangential (hoop) engineering strain as
2 ef sin1 ef ¼
2sin1
! 2L0 ð1þem Þ D0 ð2ef Þ
L0 D0
1
(7)
In order to determine ef the non-linear equation Eqn. (7) needs to be solved for iteratively for each increment of measured strain em. However, for large outer pipe diameter D0 pipes and a relatively small gauge length L0 Eqn. (7) reduces to
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Fig. 10. (a) Gauge region of ring specimen, (b) FEM model for DB-setup and (c) FEM model for RB-setup.
Fig. 11. Engineering strain versus axial displacement obtained from FEM.
ef ¼ em
(8)
The true tangential strain is now given by
εf ¼ lnð1 þ em Þ
(9)
In order to obtain the transverse stress in the ring specimen the transverse force needs to be estimated. Based on the free body cut shown in Fig. 10(a)e(b) and assumption of thin-walled cylinder the tangential load is based on force equilibrium related to the axial force as Ff¼F/2. This assumption is confirmed and validated with the finite element model. In Fig. 12 the tangential force Ff is plotted versus the engineering tangential strain ef for obtained experimentally using the two test setup DB (D-block) and RB (roller needle bearing) and from FEM with different friction coefficients. For the DB-setup the FEM model shown in Fig. 10(b) is utilized whereas for the RB-setup the model in Fig. 10(c) is utilized. It should be mentioned here that the engineering strain ef obtained from FEM is determined with exactly the same procedure as in the experiments outlined above. As shown, for the DB-setup, the FEM results with a friction coefficient of m¼0.25 matches the DB experimental results closely, whereas for the RB-setup the FEM results with a friction coefficient of m¼0.13 matches the RB experimental results markedly well. The FEM results pertaining to frictionless conditions m¼0.0 are also plotted in Fig. 12. In view of the results in Fig. 12, it can be concluded that each of the test setups possesses different levels of friction, where the DBsetup has the higher level and the RB-setup the lower level of which none give frictionless conditions. The fact that a difference in
mechanical response between the two setups and frictionless conditions exists shows that the measures used, e.g. application of PTFE lubricant and use of needle roller bearings, did reduce the friction but did not eliminate it completely as the lowest friction that can be achieved is m¼0.13 with the RB-setup. In previous studies [10,11], the friction effect was disregarded since it was
Fig. 12. Tangential force Ff versus engineering tangential strain ef for DB-setup and RB-setup.
I. Barsoum, K.F. Al Ali / International Journal of Pressure Vessels and Piping 128 (2015) 59e68
assumed to be negligible. Evident by Fig. 12, the friction effect is dominant and needs to be accounted for. Here the friction effect is accounted for through a post-processing procedure of the experimental and FEM result for each test setup. The vertical shift in force introduced in the load vs. strain curve due to friction is estimated by comparing the FEM results of DB (m¼0.25) and RB (m¼0.13) with the frictionless curve (m¼0.0), respectively. The difference in force is due to the friction associated with the test setup and the corrected experimental tangential force is obtained by subtracting this difference from the experimentally obtained data. Hence, the corrected tangential force FfDB and FfDB for each test setup, respectively, is given by DBðExpÞ
DBðm¼0:25Þ DBðm¼0:0Þ Ff Ff
(10)
RBðExpÞ
RBðm¼0:13Þ RBðm¼0:0Þ Ff Ff
(11)
FfDB ¼ Ff FfRB ¼ Ff
The true tangential (hoop) stress is now given by the corrected tangential force and the instantaneous cross section area A of the gauge length as DB sDB f ¼ Ff
. A
and
RB sRB f ¼ Ff
. A
(12)
The instantaneous cross section area is given by Refs. A¼w0t0(1þez)(1þer), which with the findings er¼ ez in Fig. 11 gives A¼w0t0(1þez)2, where ez is the change in width and is measured experimentally between target point 3 and 4. Alternatively, the instantaneous cross section area can be given by Ref. A¼w0t0/(1þef). Comparing the two expressions yields (1þef)(1þez)2¼1, which confirms the incompressibility requirement in the gauge section. Fig. 13 shows the tangential (hoop) true stress obtained from Eqns. 10e12 versus the true tangential (hoop) strain obtained from Eqn. (9) for both the DB and RB setup for the Al 6063-T5 ring specimen. The axial true stress strain for Al 6063-T5 obtained from Refs. [1e3] in Fig. 1 is also plotted for reference. As shown, the tangential and axial true stress-strain curves agree markedly well indicating that the Al 6063-T5 pipes have isotropic mechanical behavior. Moreover, the developed experimental procedure with its post-processing step, which corrects for the friction, is an accurate method for determining the tangential true stress-strain behavior of pipes. This is particularly confirmed by the fact that the procedure gives consistent results for both the test setups DB
Fig. 13. Tangential and axial true stress-strain curves for Al 6063-T5 pipes.
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and RD, with very good agreement with the axial stress-strain curve. Using Eqn. (1) with Fig. 13 gives a strain hardening exponent n ¼ 0.128 for the DB-setup and n ¼ 0.122 for the RB-setup, which matches the value obtained from the axial test (n ¼ 0.123) reported in Table 1. 5. Conclusions In this study a method to determine the tangential true stressstrain behavior of pipes is developed. An optimized ring specimen is proposed for the purpose, which showed to have uniform stress distribution in the gauge region. The optimized ring specimen give a neck formation at the center of the gauge region consistently, without having to taper the gauge region. It is also found that friction is present and has a substantial effect on the mechanical response of the ring test, which was neglected in previous studies. Two test setups are proposed, one using lubricated Dblocks (DB) and one using lubricated D-blocks with needle roller bearing (RB). By use of FEM it was found that the RB-setup has the lowest friction with m¼0.13, whereas the DB-setup has a friction coefficient of m¼0.25. The friction effect is accounted for by postprocessing procedure of the experimental results. With aid of the FEM results, the shift in force associated friction is subtracted from the experimental results. Relations are derived for determining the true tangential strain and the true tangential stress, which gave very reliable results agreeing well with axial true stress-strain curve. The experimental setups along with the data analysis procedure makes this method very useful for assess ductility of pipes and for determining the ductile failure locus of pipes. It can also be applied for structural integrity analysis of pipes containing longitudinal defect. Acknowledgement The authors would like to acknowledge the financial support of this work provided by Abu Dhabi National Oil Company (ADNOC) and the Petroleum Institute under the grant RAGS-14326-2014. Mr. Arman Molki and Mr. Shrinivas Bojanampati are acknowledged for their assistance in the lab. References [1] Seibi AC, Barsoum I, Molki A. Experimental and numerical study of expanded aluminum and steel tubes. Eng Procedia 2011;10:3049e55. [2] Barsoum I, Khan F, Molki A, Seibi A. Ductile failure modelling of expanded aluminium tubes with embedded circular holes. In: ASME pressure vessels & piping conference. Paris, France; 2013. [3] Barsoum I, Khan F, Molki A, Seibi A. Modeling of ductile crack propagation in expanded thin-walled 6063-T5 aluminum tubes. Int J Mech Sci 2014;80: 160e8. [4] ASTM. Standard test methods and definitions for mechanical testing of steel products. A 370e07a. 2007. [5] Crone DG, Collins LE, Bian YK, Weber P. The effect of sample flattening on yield strength measurement in line pipe. In: Proceedings of the Asme International pipeline conference 2010, Vol 2; 2010. p. 477e82. [6] ASTM. Standard test method for apparent hoop tensile strength of plastic or reinforced plastic pipe by split disk method. D 2290-08. 2008. [7] Kaynak C, Erdiller ES, Parnas L, Senel F. Use of split-disk tests for the process parameters of filament wound epoxy composite tubes. Polym Test 2005;24: 648e55. [8] Buarque EN, d'Almeida JRM. The effect of cylindrical defects on the tensile strength of glass fiber/vinyl-ester matrix reinforced composite pipes. Compos Struct 2007;79:270e9. [9] Laiarinandrasana L, Devilliers C, Oberti S, Gaudichet E, Fayolle B, Lucatelli JM. Ring tests on high density polyethylene: full investigation assisted by finite element modeling. Int J Pres Ves Pip 2011;88:1e10. [10] Wang H, Bouchard R, Eagleson R, Martin P, Tyson WR. Ring hoop tension test (RHTT): a test for transverse tensile properties of tubular materials. J Test Eval 2002;30:382e91. [11] Arsene S, Bai J. A new approach to measuring transverse properties of structural tubing by a ring test. J Test Eval 1996;24:386e91.
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[12] García JJ, Rangel C, Ghavami K. Experiments with rings to determine the anisotropic elastic constants of bamboo. Constr Build Mater 2012;31: 52e7. [13] MEtallic materials tube ring expanding test, International Standard ISO 8495e8496. [14] Dick CP, Korkolis YP. Mechanics and full-field deformation study of the Ring Hoop Tension Test. Int J Solids Struct 2014;51:3042e57. ne S, Bai J. A new approach to measuring transverse properties of [15] Arse structural tubing by a ring test - experimental investigation. J Test Eval 1998;26:26e30.
[16] Bae B-K, Cho S-K, Seok C-S. A study on ring tensile specimens. Mater Sci Eng A 2008;483e484:248e50. [17] ABAQUS. Analysis User's Manual, Version 6.11. Providence, RI, USA: Dassault mes Simulia Corp.; 2011. Syste [18] Bridgman PW. Studies in large plastic flow and fracture. Cambridge, MA: Harvard University Press; 1964. [19] Choung JM, Cho SR. Study on true stress correction from tensile tests. J Mech Sci Technol 2008;22:1039e51.