Original Article
Experimental assessment of residual stresses induced by the thermal autofrettage of thick-walled cylinders
J Strain Analysis 2016, Vol. 51(2) 144–160 Ó IMechE 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0309324715616005 sdj.sagepub.com
Seikh Mustafa Kamal1, Arun Chandra Borsaikia2 and Uday Shanker Dixit1
Abstract In this work, an experimental study of the residual stresses present in the thermally autofrettaged thick-walled cylinders is carried out. The idea of thermal autofrettage has been conceived recently, and due to its simplicity, it has the potential to be competitive with the existing hydraulic and swage autofrettage processes. In thermal autofrettage, the beneficial compressive residual stresses at the inner wall are produced by means of thermal gradient across the wall thickness. In this work, the residual stresses setup in the cylinders are determined experimentally using Sachs boring method. The experimental results are compared with the analytically determined residual stresses and found to be in good agreement. In addition to the Sachs boring method, two other methods, namely, microhardness test and the measurement of opening angle as a result of cutting through the wall of the cylinders, are also carried out in order to infer the generation of residual stresses in the autofrettaged cylinders. All three experimental methods advocate the feasibility of thermal autofrettage for producing beneficial compressive residual stresses at the inner wall.
Keywords Thermal autofrettage, residual stresses, Sachs boring, microhardness, opening angle
Date received: 25 August 2015; accepted: 13 October 2015
Introduction Autofrettage is employed to induce beneficial compressive residual stresses at and around the inner wall of a thick-walled cylinder. It is accomplished by the removal of the load that was applied to cause the nonhomogenous elasto-plastic deformation within the wall. The compressive residual stresses reduce the effect of the working stress in the cylinder, when subjected to high pressure. This enhances the pressure-carrying capacity of the cylinder as well as increases its fatigue strength. The hydraulic and swage autofrettage are the traditional methods used for producing compressive residual stresses at the inner wall of pressure vessels. The hydraulic autofrettage involves ultra-high hydraulic pressure and the swage autofrettage is achieved by passing an oversized tapered mandrel through the inner wall of the cylinder. Recently, Kamal and Dixit1 proposed a thermal autofrettage process that utilizes a certain amount of thermal gradient across the wall of the cylinder. The process can potentially produce the beneficial compressive residual stresses at and around the inner wall after the release of the induced thermal gradient.
Due to the influence of the compressive residual stresses on enhancing the pressure-carrying capacity and lifetime, the analysis of the residual stresses in the autofrettaged cylinders has received significant attention of the researchers. Many researchers developed the analytical and numerical models to predict the residual stress distribution in the hydraulically autofrettaged cylinders.2–6 The analytical modeling of the swage autofrettage is not well explored due its transient and localized nature; however, numerous works have been reported on the numerical front using finite element method (FEM).7–10 Kamal and Dixit1 developed an analytical model for the thermal autofrettage process.
1
Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India 2 Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, India Corresponding author: Uday Shanker Dixit, Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India. Emails:
[email protected];
[email protected]
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
Kamal et al.
145
The experimental determination of the residual stress distribution is necessary in order to get confidence in the analytical and numerical model of the process. Researchers have used various well-established techniques for evaluating the residual stresses experimentally such as the Sachs boring method, deep hole drilling method, compliance method, neutron diffraction method and X-ray diffraction (XRD) method. Stacey and Webster11 determined the residual stresses in the hydraulic autofrettage of AISI 4333M4 steel analytically as well as numerically and compared with the experimental measurements made by the Sachs boring and neutron diffraction methods. The influence of work hardening, the Bauschinger effect and the choice of yield criterion were studied. The authors observed that except close to the bore, the experimental data lie between the results predicted by considering von Mises and Tresca criteria irrespective of the material model employed. The determination of the residual stresses in the hydraulically autofrettaged steel cylinders was carried out by Venter et al.12 using neutron diffraction, Sachs boring and the compliance method. They observed a good agreement among the residual stresses measured by all these three techniques. Stacey et al.13 also measured the residual stresses in high-strength low-alloy steel autofrettaged tube using neutron diffraction method and compared the stress distribution with the Sachs boring technique. George and Smith14 measured the residual stresses using the deep-hole technique in a hydraulically autofrettaged steel tube. The residual stresses in swage autofrettaged cylinders were determined by Davidson et al.15 using Sachs boring technique. Zerari et al.16 used XRD method to evaluate residual stresses in cylinders induced by the hydraulic autofrettage. The experimental determination of the residual stresses in the cylinders induced by thermal autofrettage is yet to be carried out. However, preliminary experimental investigation based on the surface microhardness measurement of the thermally autofrettaged cylinder was reported in Kamal and Dixit.1 In this work, the residual stress field in the thermally autofrettaged cylinders is measured experimentally using Sachs boring technique. Furthermore, the measured residual stresses are compared with the analytical model developed by Kamal and Dixit.1 Many researchers showed that the compressive residual stresses increase the surface hardness while the tensile residual stresses decrease the surface hardness.17–19 This work also establishes the generation of residual stresses in the thermally autofrettaged cylinders on the basis of Vickers microhardness test. Parker et al.,20 Parker and Underwood21 reported that the presence of residual stresses can be detected by making a single radial cut in the autofrettaged cylinder and measuring the opening angle due to the released pure bending moment ‘‘locked in’’ by the residual hoop stresses. In this work, the presence of residual stresses is also confirmed by measuring the opening angle after cutting
thermally autofrettaged cylinder along a radius and comparing it with the theoretical opening angle based on the residual hoop stress distribution of Kamal and Dixit.1 Thus, the objective of this article is to carry out an experimental study on the residual stresses of thermally autofrettaged cylinders. Three different experimental techniques have been used and the results have been compared with the theoretical results quantitatively and qualitatively. The article is organized in the following manner. In section ‘‘Thermal autofrettage process,’’ the thermal autofrettage process is briefly described followed by a brief introduction of the Sachs boring method in section ‘‘A brief description of the Sachs boring method.’’ Section ‘‘Results of Sachs boring method along with comparison with the model of Kamal and Dixit’’ presents the determination of the residual stresses by Sachs boring method and its comparison with the theoretical model of Kamal and Dixit.1 In section ‘‘Inference of residual stresses from microhardness test,’’ the residual stresses in the cylinder are inferred on the basis of the Vickers microhardness test. In section ‘‘Demonstration of residual stresses by the measurement of opening angle of radial cut through the wall thickness,’’ the method of the measurement of opening angle is presented in order to verify the presence of residual stresses in the cylinder. Section ‘‘Conclusion’’ concludes the article.
Thermal autofrettage process The thermal autofrettage process utilizes a certain amount of steady-state radial temperature gradient across the wall of a thick cylinder to produce beneficial residual stresses at the inner side. The outer wall is subjected to a higher temperature than the inner wall. The entire wall thickness remains in the elastic state as long as the temperature gradient is not large enough to cause the plastic deformation. When the temperature gradient is gradually increased, the cylinder wall starts yielding first at the inner wall at a certain value of temperature gradient. After crossing this first threshold, the wall of the cylinder undergoes first stage of elasto-plastic deformation, emanating an inner plastic zone propagating from inner radius to some intermediate radius. The outer portion of the wall remains in the elastic state. As the temperature gradient is increased further to reach a second threshold, yielding initiates at the outer wall. The cylinder is subjected to second stage of elastoplastic deformation resulting into two plastic zones, one at the inner and other at the outer separated by an intermediate elastic zone within the wall. On releasing the temperature gradient across the wall, that is, when the cylinder is cooled to room temperature, sufficiently large amount of compressive residual stresses are generated at the inner wall. At the outer wall, smaller tensile residual stresses are generated. The compressive residual stresses at the inner wall offset the tensile stresses
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
146
Journal of Strain Analysis 51(2)
Figure 1. Experimental setup for creating temperature gradient to achieve thermal autofrettage.
produced in the cylinder when it is subjected to high internal pressure. This enhances the pressure-carrying capacity of the cylinder as well as increases the fatigue strength when subjected to cyclic loading. A system for creating desired temperature gradient between the outer and inner walls of the cylinder for achieving thermal autofrettage is designed and fabricated. The fabricated setup is shown in Figure 1. As shown in the figure, a well-insulated ceramic jacketed heater is employed for heating the outer wall of the cylinder to be autofrettaged. The heater is connected through a variac, an ammeter and a digital temperature controller in series to control the outer wall temperature. K-type thermocouples are attached on the outer and inner surfaces and connected to temperature controllers to display the corresponding temperatures. Simultaneously, the inner wall is cooled continuously by flowing cold water through the bore. The cold water is stored in a 200-L capacity tank and is pumped through a poly vinyl chloride (PVC) pipeline using a household 0.5 horsepower (HP) water pump. At the inlet and outlet, the cylinder to be heated is connected to the PVC pipeline using ceramic tube. When the desired temperature is reached, the power is switched off and cold water is forced through the bore till the cylinder cools to the room temperature. This generates beneficial compressive residual stresses at and around the inner wall of the cylinder.
A brief description of the Sachs boring method The Sachs boring method is one of the most popular techniques for measuring the residual stresses. The method was first developed by G Sachs22 in 1927. Traditionally, this method is used for measuring the axisymmetric residual stresses in cylindrical components such as autofrettaged tubes. Researchers have also employed this technique for measuring the residual stress distribution in the cold-worked holes of aluminum alloy plates.23,24 In that case, an axisymmetric
hollow cylindrical specimen was cut from the plate. The specimen contained the cold-worked hole and was subjected to the standard Sachs boring procedure. Garica-Granada et al.25 applied the Sachs boring test for measuring non-axisymmetric residual stresses. In this work, the Sachs boring method is employed for measuring the axisymmetric residual stresses setup in the thermally autofrettaged cylinders. The method of Sachs boring is destructive in nature. The method involves progressively removing incremental layers from the inside of the cylinder. Strain gauges aligned in the hoop and axial directions are pasted on the outer circumference of the cylinder and are used for measuring the strains on the outer radius while removing the material from the inner diameter in a series of increments. The measured strains are then used for analyzing the residual stress distribution in the cylinder using the equations developed by Sachs.22 If a cylinder with inner radius a and outer radius b is considered, the residual hoop (su), axial (sz) and radial (sr) stresses in the removed layer of any radius ri are given by E du (Fb + Fr ) su = (F F ) u ð1Þ b r 1 n2 dFr 2Fr E dL sz = (F F ) L ð2Þ b r 1 n2 dFr E (Fb Fr ) sr = u ð3Þ 1 n2 2Fr where E is the Young’s modulus and n is the Poisson’s ratio of the material. The strain parameters u and L are given by u = eu + nez
ð4Þ
L = ez + neu
ð5Þ
eu and ez being the measured cumulative hoop and axial strains on the outer radius after removing an inner incremental layer, respectively. Fb and Fr are the cross sections of the original cylinder and the bored out cylinder, respectively, given by Fb = pb2
ð6Þ
Fr = pr2i
ð7Þ
Results of Sachs boring method along with comparison with the model of Kamal and Dixit The thermal autofrettage of steel cylinders was carried out using the setup shown in Figure 1.1 The residual thermal stresses generated in the cylinders were evaluated experimentally using standard Sachs boring procedure. The experimental results are compared with the generalized plane strain model developed by Kamal and Dixit.1
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
Kamal et al.
147
Table 1. Material properties of specimens. Material
Yield stress, sY (MPa)
Modulus of elasticity, E (GPa)
Poisson’s ratio, n
Coefficient of thermal expansion, a (/°C)
SS304 Mild steel
215 352
200 219
0.29 0.30
17.8 3 1026 13 3 1026
Table 2. Geometry and autofrettage temperature difference of specimens. Material
Inner radius, a (mm)
Outer radius, b (mm)
Length, L (mm)
Autofrettage temperature difference (Tb 2 Ta)(°C)
SS304 SS304 Mild steel
10 10 12.65
20 25 25.3
60 90 75
130 120 230
Figure 2. Position of the strain gauges on the outer periphery of the cylinder.
Materials The materials of the cylinders subjected to thermal autofrettage were SS304 stainless steel and mild steel. The material properties of SS304 steel and mild steel are summarized in Table 1. The detailed dimensions of the specimen are presented in Table 2.
Measurement of Sachs boring The strain gauges on the outer periphery of the autofrettaged cylinders were pasted such that one strain gauge is aligned in the circumferential direction and the other aligned in the longitudinal direction. The positions of the strain gauges on the outer periphery of the cylinders along with the layers to be removed in each incremental boring are shown schematically in Figure 2. The four-wired strain gages were connected to lead wires. The four-wired system helps in eliminating the temperature effect in the lead wires as well as in eliminating any measurement error due to gauge factor correction and contact resistance. The strain gauges
were sealed using Anabond 666 RTV silicon sealant to prevent the gauges from moisture and any other environmental factor such as liquid coolant during machining. The removal of the layers from the inside diameter of the cylinders was performed in lathe machine. The boring was carried out at a low spindle speed of 145 r/min using a sharp boring tool. The depth of cut was varied from 0.1 to 0.5 mm during the machining at the feed rate of 0.04 mm/rev. The sharp tool and low cutting speed during boring minimizes vibration and reduces the possibility of generating residual stresses in the component due to machining.24 A liquid coolant was employed continuously on the cylindrical specimen during the boring operation to minimize the generation of heat. The released hoop and axial strains, after boring out of each layer of material, measured on the outer radius of the cylinder were recorded and stored using HBM data acquisition system with CATMAN software. During machining, the lead wires from the strain gauges were wound around the circumference of the chuck of the lathe machine and fixed on the circumference with the help of adhesive tape. This enabled the wires to rotate with the chuck during the material removal process. All the strains were set to zero before the boring was carried out. When one layer of the material was removed completely from the cylinder, the machine was stopped and lead wires were taken out from the chuck by removing the adhesive tape. The lead wires were then plugged into the data acquisition system for recording strain data. After recording the strain data, the lead wires were disconnected and again fixed on the circumference of the chuck. Then the machine was started and removal of the next layer of material was carried out. The strain readings were recorded 15 min after boring. This was to allow the specimen to cool to room temperature. The procedure was repeated till about 80% of the thickness got removed from the cylinder.
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
148
Journal of Strain Analysis 51(2)
Table 3. Measured hoop and axial strain in Sachs boring of SS304 with b/a = 2. Radius, ri (mm)
Area, Fr (mm2)
Hoop strain
Axial strain
u
L
10.00 10.30 10.60 10.90 11.20 11.50 11.95 12.45 12.95 13.25 13.75 14.25 14.75 15.25 15.75 16.25
314.16 333.29 352.98 373.25 394.08 415.47 448.63 486.95 526.85 551.54 593.95 673.94 683.49 730.62 779.31 829.58
0 29.59904 3 1026 21.76466 3 1025 22.6142 3 1025 22.51799 3 1025 22.39738 3 1025 22.31968 3 1025 22.0276 3 1025 21.89257 3 1025 21.67629 3 1025 21.38597 3 1025 26.23119 3 1026 22.51949 3 1026 2.1795 3 1026 8.8745 3 1026 1.514 3 1025
0 21.06843 3 1025 21.87362 3 1025 22.8964 3 1025 22.7756 3 1025 22.6735 3 1025 22.4951 3 1025 22.2224 3 1025 22.098 3 1025 21.90245 3 1025 21.53539 3 1025 28.14101 3 1026 23.21732 3 1026 21.187 3 1026 7.1431 3 1026 1.312E 3 1025
0 21.28043 3 1025 22.32675 3 1025 23.48312 3 1025 23.35067 3 1025 23.19943 3 1025 23.06821 3 1025 22.69432 3 1025 22.52197 3 1025 22.24702 3 1025 21.84659 3 1025 28.6735 3 1026 23.48469 3 1026 1.8234 3 1026 1.10174 3 1025 1.9076 3 1025
0 21.3564 3 1025 22.40302 3 1025 23.68066 3 1025 23.53099 3 1025 23.39271 3 1025 23.19101 3 1025 22.83069 3 1025 22.66577 3 1025 22.40534 3 1025 21.95118 3 1025 21.00104 3 1025 23.97317 3 1026 25.33154 3 1027 9.80545 3 1026 1.7662 3 1025
Table 4. Measured hoop and axial strain in Sachs boring of SS304 with b/a = 2.5. Radius, ri (mm)
Area, Fr (mm2)
Hoop strain
Axial strain
u
L
10.00 10.10 10.26 10.45 10.65 10.95 11.40 11.90 12.40 12.90 13.40 13.90 14.40 14.75 15.25 15.70 16.20 16.70 17.20
314.16 320.47 330.71 343.07 356.33 376.68 408.28 444.88 483.05 522.79 564.1 606.99 651.44 683.49 730.62 774.37 824.48 876.16 929.41
0 21.70017 3 1026 24.85328 3 1026 28.11881 3 1026 21.121 3 1025 21.64348 3 1025 22.10368 3 1025 21.91763 3 1025 21.81351 3 1025 21.68411 3 1025 21.41709 3 1025 21.2974 3 1025 21.03193 3 1025 27.98942 3 1026 23.77444 3 1026 21.1402 3 1026 2.2438 3 1026 7.19202 3 1026 1.2517 3 1025
0 22.78439 3 1026 25.54351 3 1026 29.89501 3 1026 21.23175 3 1025 21.7536 3 1025 22.21871 3 1025 22.3162 3 1025 22.012 3 1025 21.83046 3 1025 21.6673 3 1025 21.39649 3 1025 29.81728 3 1026 28.16942 3 1026 26.10308 3 1026 22.4201 3 1026 8.25342 3 1027 4.15729 3 1026 8.64956 3 1026
0 22.53549 3 1026 26.51634 3 1026 21.10873 3 1025 21.49052 3 1025 22.16956 3 1025 22.76929 3 1025 22.61249 3 1025 22.41711 3 1025 22.23324 3 1025 21.91728 3 1025 21.71635 3 1025 21.32645 3 1025 21.04402 3 1025 25.60536 3 1026 21.86623 3 1026 2.4914 3 1026 8.43921 3 1026 1.51119 3 1025
0 23.29444 3 1026 26.99949 3 1026 21.23307 3 1025 21.56805 3 1025 22.24664 3 1025 22.84982 3 1025 22.89149 3 1025 22.55605 3 1025 22.33569 3 1025 22.09243 3 1025 21.78571 3 1025 21.29131 3 1025 21.05663 3 1025 27.23541 3 1026 22.76216 3 1026 1.49848 3 1026 6.31489 3 1026 1.24047 3 1025
The cumulative hoop and axial strains measured in Sachs boring of autofrettaged SS304 and mild steel cylinders are given in Tables 3–5. The strain parameters u and L are evaluated using equations (4) and (5). It is observed that the strains increased from zero to certain value in compression and then gradually decreased. After boring out to certain radial position, the strains changed sign from negative to positive.
Evaluation of residual stresses using Sachs boring data and its comparison with Kamal and Dixit model The residual stresses in the thermally autofrettaged cylinders are evaluated using the strain data recorded in the Sachs boring experiments in equations (1)–(3). The evaluation of residual stresses requires the strain parameters u and L; the original cross section Fb of the
cylinders; the bored out cross sections after each layer of the material removed, Fr, and the differentiation of the strain parameters with respect to the bored out radius, that is, du/dFr and dL/dFr. The evaluated values of u, L and Fr are shown in Tables 3–5 for the respective cylinders. The original cross sections of the SS304 cylinders with b/a = 2 and 2.5 were evaluated to be 1256.64 and 1963.49 mm2, respectively. The original cross section of the mild steel cylinder was 2010.90 mm2. The derivatives du/dFr and dL/dFr are calculated using the central difference method. For example, considering the second row of the values in Table 3, the derivative du/dFr is evaluated to be 25.99369 3 10207. Similarly, the value of dL/dFr is evaluated as 26.19017 3 10207. The residual hoop, axial and radial stresses setup in the SS304 and mild steel cylinders as evaluated from Sachs boring measurement are shown in Figures 3–5 as
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
Kamal et al.
149
Table 5. Measured hoop and axial strain in Sachs boring of mild steel with b/a = 2. Radius, ri (mm)
Area, Fr (mm2)
Hoop strain
Axial strain
u
L
12.65 12.85 13.15 13.40 13.70 14.00 14.45 14.95 15.45 15.95 16.45 16.95 17.45 17.95 18.45 18.95 19.45 19.95
502.72 518.75 543.25 564.1 589.64 615.75 655.97 702.15 749.91 799.23 850.12 902.58 956.62 1012.22 1069.4 1128.15 1188.47 1250.36
0 24.47518 3 1026 21.22299 3 1025 21.3673 3 1025 21.47564 3 1025 21.31324 3 1025 21.01324 3 1025 27.07518 3 1026 25.3174 3 1026 23.28444 3 1026 21.72435 3 1026 1.19935 3 1026 3.90152 3 1026 6.55904 3 1026 1.07984 3 1025 1.38523 3 1025 1.55106 3 1025 1.97969 3 1025
0 29.26904 3 1026 21.51064 3 1025 21.85499 3 1025 22.11941 3 1025 22.52994 3 1025 22.32994 3 1025 22.00752 3 1025 21.71983 3 1025 21.46266 3 1025 21.08474 3 1025 27.03235 3 1026 23.32948 3 1026 1.43336 3 1026 4.34162 3 1026 8.54031 3 1026 1.22906 3 1025 1.66572 3 1025
0 27.25589 3 1026 21.67618 3 1025 21.9238 3 1025 22.11146 3 1025 22.07222 3 1025 21.71222 3 1025 21.30977 3 1025 21.04769 3 1025 27.67242 3 1026 24.97858 3 1026 29.10352 3 1027 2.90267 3 1026 6.98904 3 1026 1.21009 3 1025 1.64144 3 1025 1.91978 3 1025 2.4794 3 1025
0 21.06116 3 1025 21.87754 3 1025 22.26519 3 1025 22.5621 3 1025 22.92391 3 1025 22.63391 3 1025 22.21977 3 1025 21.87935 3 1025 21.56119 3 1025 21.13648 3 1025 26.67254 3 1026 22.15903 3 1026 3.40107 3 1026 7.58113 3 1026 1.2696 3 1025 1.69438 3 1025 2.25962 3 1025
Figure 3. Residual (a) hoop, (b) axial and (c) radial stress distributions in SS304 cylinder (b/a = 2).
a function of bored out radius. The residual stresses are also evaluated using the analytical model of Kamal and Dixit1 based on generalized plane strain and compared with the experimental results. The analytical residual
stress distributions of Kamal and Dixit are given in Appendix 1. It is observed from Figures 3–5 that the experimental Sachs boring residual stresses are in good agreement
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
150
Journal of Strain Analysis 51(2)
Figure 4. Residual (a) hoop, (b) axial and (c) radial stress distributions in SS304 cylinder (b/a = 2.5).
with the residual stresses predicted by Kamal and Dixit1 model. In all the cases, it is observed that very near to the region of inner surface, the experimental compressive residual stresses very closely match with the analytical residual stresses. As the transition of residual stresses from compressive to tensile takes place on the removal of the layer of the material, the difference between the experimental and analytical stresses increases slightly. This may be attributed to the experimental and numerical errors.
Inference of residual stresses from microhardness test The residual stresses influence the surface microhardness. The change in hardness of the surface after any metal working process may be interpreted as one of the qualifying test for the presence of residual stresses. Researchers have been using the measurement of hardness as a means to detect the presence of surface residual stresses. For example, Kokubo17 reported the change in Vickers hardness values in carbon-steel-rolled sheets due to applied stresses under bending load. It was found that the hardness decreases in the presence of tensile stresses while in the presence of compressive
stresses hardness of the surface increases. The influence of residual stresses on the hardness number during shot peening and sheet bending of medium carbon steel was studied by Tosha,18 who observed that hardness numbers increase in the compressive stress field and decrease in the tensile stress field. Simes et al.26 carried out the experimental investigation of the influence of residual stresses on the hardness number obtained in the Vickers indentation test and found that the presence of tensile residual stresses results in a decrease in hardness. In the present analysis, the microhardness on the inner and outer surfaces of the autofrettaged cylinders based on the Vickers diamond indentation test was carried out in order to envisage the generation of thermal residual stresses in the cylinders. To carry out the microhardness test, small test samples were extracted from the cylinders by cutting it radially. The Vickers indentation on the inner and outer surfaces of both the autofrettaged and non-autofrettaged samples was carried out at 500 gf. The measured microhardness values on the inner and outer surfaces of the samples show that the hardness increases on the inner surface of the autofrettaged sample as compared to the non-autofrettaged sample due to the presence of compressive residual stresses. Similarly, due to the presence of tensile residual stresses
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
Kamal et al.
151
Figure 5. Residual (a) hoop, (b) axial and (c) radial stress distributions in mild steel cylinder (b/a = 2).
on the outer surface, a decrease in the measured microhardness was observed. The microhardness test results conducted on the inner and outer surfaces of the SS304 cylinders are presented in Table 6. It is found from Table 6 that in the case of SS304 cylinder with b/a = 2, the average microhardness on the inner surface increases from 256.94 HV with standard deviation of 3.11 to 279.36 HV with standard deviation of 4.61. However, on the outer surface, the average microhardness decreases from 283.18 HV with standard deviation of 3.29 to 224.79 HV with standard deviation of 3.59. In the case of SS304 cylinder with b/a = 2.5, the average microhardness on inner surface increased to 290.95 HV with standard deviation of 3.09 and that on the outer surface decreased to 236.22 HV with standard deviation of 3.27. These results indicate that the inner surface of the cylinder is in compression and the outer surface is in tension. Some methods have been proposed to quantify the residual stresses using indentation.27,28 Furthermore, research is needed in this direction for enhancing the reliability and the ease of experimentation. A suitable calibration may also be required for different materials.
Demonstration of residual stresses by the measurement of opening angle of radial cut through the wall thickness The angle of opening is a measure of the pure bending moment ‘‘locked in’’ to the cylinder during autofrettage. When the autofrettaged cylinders are cut radially, the cylinders open by an angle b as shown in Figure 6. The measurement of this opening angle can be used as a means to demonstrate that the residual stresses are set up in the cylinder.20 In the thermally autofrettaged cylinders, the pure bending moment ‘‘locked in’’ the cylinders is calculated based on the residual stress distribution predicted by Kamal and Dixit.1 In an autofrettaged cylinder of unit length with inner radius a and outer radius b, the total bending moment acting over any radial cut is given by29 Zb M=
su rdr
ð8Þ
a
where su is the net residual hoop stress setup in the cylinder by autofrettage. When the cylinder has
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
152
Journal of Strain Analysis 51(2)
Table 6. Vickers microhardness on the surfaces of SS304 cylinders. Non-autofrettaged cylinder
Autofrettaged cylinder (b/a = 2)
Autofrettaged cylinder (b/a = 2.5)
Hardness on the outer surface (HV)
Hardness on the inner surface (HV)
Hardness on the outer surface (HV)
Hardness on the inner surface (HV)
Hardness on the outer surface (HV)
Hardness on the inner surface (HV)
284.8 287.9 277.5 282.8 282.3 286.8 281.9 278.4 285.8 280.4 288.4 280.4 286.8 284.3 285.8 278.4 279.9 284.8 281.9 284.3
262.3 257.1 261.5 254.1 257.9 254.1 257.5 261.4 255.8 260.5 253.3 252.4 253.5 256.6 253.7 260.1 259.2 253.7 256.2 257.9
223.9 225.4 221.7 229.3 218.7 226.1 219.7 224.8 228.0 220.1 219.7 227.2 227.7 225.6 227.4 229.9 230.3 223.9 221.4 225.1
277.1 278.8 280.1 272.8 276.8 279.2 282.8 280.8 287.7 282.7 271.9 279.3 272.7 276.8 274.9 281.7 288.3 277.1 280.3 285.4
234.1 237.0 230.3 233.2 239.2 239.9 232.4 230.6 238.4 235.2 237.6 233.5 238.9 234.6 240.1 239.6 235.9 241.7 237.5 234.8
286.9 288.3 285.6 288.4 295.1 289.4 295.1 291.4 289.9 297.7 290.4 290.4 294.9 290.1 287.3 293.1 289.9 292.0 292.5 290.7
radial cut when the cylinder has undergone second stage of elasto-plastic deformation is given by Zc M=
su
(plastic zone I)
Zd rdr +
a
su (plastic zone II) rdr
c
Ze +
su
(elastic zone )
Zf rdr + e
d
Zb +
su (plastic zone IV) rdr
su (plastic zone III) rdr
ð10Þ
f
Figure 6. Opening angle in thick-walled autofrettaged cylinder with single radial cut.
undergone first stage of elasto-plastic deformation during thermal autofrettage, the total bending moment released for a cylinder of unit length due to any single radial cut can be given by Zc M=
su
(plastic zone I)
Zd rdr +
a
c
Zb +
In equations (9) and (10), su refers to the residual thermal hoop stresses in the respective zones as given in Appendix 1. The variables c, d, e and f refer to the limits of the various zones. These can be determined by satisfying the continuity of stresses as detailed in Kamal and Dixit.1 The evaluations of the integrals in equations (9) and (10) are provided in Appendix 2. The opening angle b due to the released bending moment M‘‘locked in’’ to the cylinder can be evaluated using the following equation30
su (plastic zone II) rdr ð9Þ
su (elastic zone ) rdr
d
Similarly, the total bending moment released in the thermally autofrettaged cylinder of unit length due to a
b=
4pM 4(b2 a2 ) E N
ð11Þ
where E is the Young’s modulus of elasticity and N is given by 2 b N = (b2 a2 )2 4a2 b2 ln a
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
ð12Þ
Kamal et al.
153
Table 7. Comparison of experimental opening angle with the theoretical opening angle. Sl no.
Material
b/a
(Tb 2 Ta) (°C)
Theoretical opening angle (bth) (°)
Experimental opening angle (bexp) (°)
% Variation
1 2
SS304 SS304
2 2.5
130 120
1.69 0.91
1.98 1.05
14.64 13.33
Here, the thermal autofrettage of SS304 cylinders is considered to measure the opening angles in order to illustrate the existence of residual stresses in the cylinders. The theoretical angle of opening for the cylinders is calculated using equation (11). Equation (11) requires the value of M, which is evaluated either from equation (9) or equation (10) depending on whether the cylinder has undergone first stage or second stage of elastoplastic deformation. For experimental determination of the opening angle, a disk of 15 mm thickness was extracted from each of the autofrettaged cylinders. A radial cut was made in each disk and the opening angle was measured using profile projector. The experimentally measured opening angles in SS304 cylinders are compared with the theoretical opening angles and are presented in Table 7. The comparison of the theoretical and experimental measurement of opening angles in the cylinder shows that the experimentally determined opening angles are slightly greater than the theoretically predicted opening angles. The percentage variation between the experimental and theoretical opening angles is observed to be less than 15%. Thus, it is confirmed that the residual stresses are set up in the thermally autofrettaged cylinders.
Conclusion In this work, an experimental investigation of the thermal autofrettage of thick-walled cylinders is carried out. The thermal autofrettage of SS304 and mild steel with different wall thickness ratios was conducted by creating a thermal gradient between the outer and inner walls of the cylinders. On removal of the thermal gradient, the residual thermal stresses generated in the cylinders were evaluated experimentally using traditional Sachs boring method. The experimentally determined residual stresses are compared with the residual stresses predicted by the generalized plane strain model of Kamal and Dixit.1 The comparison shows that the experimental results are in close agreement with the analytical results. Thus, the model developed by Kamal and Dixit1 can be used to analyze the residual stresses generated in the cylinders during thermal autofrettage. The Vickers microhardness test on the inner and outer surfaces of the autofrettaged cylinders was also conducted to infer the presence of residual stresses. It was found that the microhardness increases on the inner surface due to the residual compressive stresses at the
inner surface and the microhardness decreases on the outer surface due to the presence of tensile residual stresses at the outer surface. There is a scope to develop a model for quantifying the residual stresses based on the measurement of microhardness. To envisage the presence of residual stresses, the method of measuring the opening angle when the thermally autofrettaged cylinders were cut along radial direction to release the ‘‘locked in’’ bending moment was also carried out. It was found that the experimentally measured opening angles are in good agreement with the theoretically predicted opening angles based on the stress analysis of Kamal and Dixit.1 Among the three experimental methods, the method of measuring the opening angle is the easiest to conduct. Although it does not directly provide the distribution of residual stress along the thickness, it can be used for the quick verification of a theoretically predicted distribution. Given a reliable theoretical model, microhardness measurement can be an excellent destructive method for the determination of residual stresses. The overall experimental results reveal that the thermal autofrettage process can be used as a potential manufacturing procedure in the industries to generate the beneficial compressive residual stresses at the inner wall. Due to the simplicity in the procedure of thermal autofrettage, the process has some edge over the existing hydraulic and swage autofrettage processes. A good agreement between the experimental results with the theoretical model of Kamal and Dixit1 indicates that the behavior of thermal autofrettage is easily predictable, a positive feature toward its industrial adaptation. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article. Funding The author(s) received no financial support for the research, authorship and/or publication of this article. References 1. Kamal SM and Dixit US. Feasibility study of thermal autofrettage of thick-walled cylinders. J Press Vess: T ASME 2015; 137(6): 061207-1–061207-18. 2. Thomas DGB. The autofrettage of thick tubes with free ends. J Mech Phys Solids 1953; 1: 124–133.
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
154
Journal of Strain Analysis 51(2)
3. Gao XL. An exact elasto-plastic solution for an openended thick-walled cylinder of a strain-hardening material. Int J Pres Ves Pip 1992; 52: 129–144. 4. Rees DWA. Autofrettage theory and fatigue life of openended cylinders. J Strain Anal Eng 1990; 25: 109–121. 5. Avitzur B. Autofrettage–stress distribution under load and retained stresses after depressurization. Int J Pres Ves Pip 1994; 57: 271–287. 6. Huang XP and Cui WC. Effect of Bauschinger effect and yield criterion on residual stress distribution of autofrettaged tube. J Press Vess: T ASME 2006; 128: 212–216. 7. Iremonger MJ and Kalsi GS. A numerical study of swage autofrettage. J Press Vess: T ASME 2003; 125: 347–351. 8. Chen PCT. Finite element analysis of the swage autofrettage process. Technical report ARCCB-TR-88037, September 1988. Watervliet, NY: US Army Armament Research Development and Engineering Center, Benet Laboratories. 9. Bihamta R, Movahhedy MR and Mashreghi AR. A numerical study of swage autofrettage of thick-walled tubes. Mater Design 2007; 28: 804–815. 10. Perry J and Perl M. A 3-D model for evaluating the residual stress field due to swage autofrettage. J Press Vess: T ASME 2008; 130: 041211(1–6 pages). 11. Stacey A and Webster GA. Determination of residual stress distributions in autofrettaged tubing. Int J Pres Vess Pip 1988; 31: 205–220. 12. Venter AM, de Swardt RR and Kyriacou S. Comparative measurements on autofrettaged cylinders with large Bauschinger reverse yielding zones. J Strain Anal Eng 2000; 35: 459–469. 13. Stacey A, MacGillivary HJ, Webster GA, et al. Measurement of residual stresses by neutron diffraction. J Strain Anal Eng 1985; 20(2): 93–100. 14. George D and Smith DJ. The application of the deep hole technique for measuring residual stresses in an autofrettaged tube. In: Proceedings of PVP 2000, ASME pressure vessel and piping, Seattle, WA, 23–27 July 2000. New York: ASME. 15. Davidson TE, Barton CS, Reiner AN, et al. New approach to the autofrettage of high-strength cylinders. Exp Mech 1962; 2: 33–40. 16. Zerari N, Saidouni T and Benretem A. Determination of residual stresses induced by the autofrettage treatment by the X-rays diffraction method. Mod Mech Eng 2013; 3: 121–126. 17. Kokubo S. Changes in hardness of a plate caused by bending. Sci Rep Tohoku Imperial Univ Jpn (Series I) 1932; 21: 256–267. 18. Tosha K. Influence of residual stresses on the hardness number in the affected layer produced by Shot Peening. In: Proceedings of the second Asia-Pacific forum on precision surface finishing and deburring technology, Seoul, Korea, 22–24 July 2002, pp.48–54. 19. Pharr GM, Tsui TY, Bolshakov A, et al. Effects of residual stress on the measurement of hardness and elastic modulus using nanoindentation. MRS Proc 1994; 338: 127. 20. Parker AP, Underwood JH, Throop JF, et al. Stress intensity and fatigue crack growth in a pressurized, autofrettaged thick cylinder. In: Lewis JC and Sines G (eds)
21.
22. 23.
24.
25.
26.
27.
28.
29.
30.
Fracture mechanics: Fourteenth symposium–Volume I: Theory and analysis (ASTM STP 791). West Conshohocken, PA: American Society for Testing and Materials, 1983, pp.216–237. Parker AP and Underwood JH. Influence of the Bauschinger effect on residual stress and fatigue lifetimes in autofrettaged thick walled cylinders. In: Panontin TL and Sheppard SD (eds) Fatigue and fracture mechanics: 29th volume (ASTM STP1321). West Conshohocken, PA: American Society for Testing and Materials, 1998, pp.565–583. Sachs G. Der Nachweis immerer spannungen in stangen und rohren. Zeit Metall 1927; 19: 352–357. O¨Gzdemir AT and Edwards L. Measurements of the three-dimensional residual stress distribution around split-sleeve cold-expanded holes. J Strain Anal Eng 1996; 31: 413–421. Smith DJ, Pavier MJ and Poussard CP. An assessment of Sachs method for measuring residual stresses in cold worked fastener holes. J Strain Anal Eng 1998; 33: 263– 274. Garcia-Granada AA, Smith DJ and Pavier MJ. A new procedure based on Sachs’ boring for measuring nonaxisymmetric residual stresses. Int J Mech Sci 2000; 42: 1027–1047. Simes TR, Mellor SG and Hills DA. Research note: a note on the influence of residual stress on measured hardness. J Strain Anal Eng 1984; 19(2): 135–137. Abbate A, Frankel J and Scholz W. Measurement and theory of the dependence of hardness on residual stress. Technical report ARCCB-TR-93022, May 1993. Watervliet, NY: US Army Armament Research, Development and Engineering Center, Close Combat Armaments Center, Benet Laboratories. Yonezu A, Kusano R, Hiyoshi T, et al. A method to estimate residual stress in austenitic stainless steel using a microindentation test. J Mater Eng Perform 2015; 24(1): 362–372. Parker AP, Underwood JH, Throop JF, et al. Stress intensity and fatigue crack growth in a pressurized, autofrettaged thick cylinder. Technical report ARLCB-TR82027, September 1982. Watervliet, NY: US Army Armament Research and Development Command, Large Caliber Weapon Systems Laboratory, Benet Weapons Laboratory. Parker AP and Farrow JR. Technical note: on the equivalence of axisymmetric bending, thermal, and autofrettage residual stress fields. J Strain Anal Eng 1980; 15(1): 51–52.
Appendix 1 Residual stress distributions predicted by Kamal and Dixit model Residual stresses in first stage of elasto-plastic deformation. When the cylinders are subjected to first stage of elasto-plastic deformation for achieving thermal autofrettage, the residual stresses in the different zones are given by the following:1
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
Kamal et al.
155
Plastic zone I, a 4 r 4 c r Ea(Tb Ta ) b a2 b2 b 1 + ln ln 1+ 2 a a r b2 a2 2(1 n) ln a r Ea(Tb Ta ) b 2b2 b 1 + 2 ln ln (sz )res = k1 sY (1 + ln r) + C3 + a a b2 a2 2(1 n) ln a Ea(Tb Ta ) r b a2 b2 ln ln 1 2 (sr )res = k1 sY ln r + C3 + a a r b2 a2 2(1 n) ln ba
(su )res = k1 sY (1 + ln r) + C3 +
ð13Þ
ð14Þ
ð15Þ
For Tb . Ta, k1=1, otherwise k1= –1. Plastic zone II, c 4 r 4 d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2(1 n)r 2(1n)1 C6 2(1 n)r 2(1n)1 k1 s Y EaTa Ea(Tb Ta ) r 2 b ln + + + a 2n 1 (2n 1) (2n 1) (2n 1) ln a Ee0 Ea(Tb Ta ) r b a2 b2 1 + ln + ln 1+ 2 a a (2n 1) r b2 a2 2(1 n) ln ba pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2n (sz )res = C5 r1 + 2(1n) + C6 r1 2(1n) + k1 s Y 2n 1 EaTa Ea(Tb Ta ) r 2n + 1 Ee0 b ln + + a 2n 1 (2n 1) (2n 1) (2n 1) ln a 2 Ea(Tb Ta ) r b 2b b 1 + 2 ln + ln 2 a2 a a b 2(1 n) ln a pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi (sr )res = C5 r1 + 2(1n) + C6 r1 2(1n) k1 s Y EaTa Ea(Tb Ta ) r 2n + 1 b ln + + + a 2n 1 (2n 1) (2n 1) (2n 1) ln a Ee0 Ea(Tb Ta ) r b a2 b2 + ln ln 1 2 a a (2n 1) r b2 a2 2(1 n) ln ba
(su )res = C5
ð16Þ
ð17Þ
ð18Þ
Elastic zone, d 4 r 4 b (su )res =
Ea (Tb Ta ) 2(1 n) ln ba b b a2 b2 d2 b d b2 ln ln 1+ 2 (2n 1) n ln 1 + ln a a a a r b2 a2 b2 + d2 (2n 1) r2 d2 b2 d2 b2 + k EaT + 2 s 1 + 1 + 1 Y a b + d2 (2n 1) r2 b2 + d2 (2n 1) r2 2 2 d b 2 Ee0 1 + 2 b + d2 (2n 1) r ð19Þ
(sz )res =
Ea (Tb Ta ) 2(1 n) ln ba r b 2b2 b 2nd2 b d + 2n ln 1 + 2 ln ln n (2n 1) n ln ln 2 2 2 2 a a b a r a a b + d (2n 1) r ln 2 2 2 2 2 2nd d b d b + 2 k1 sY + 2 EaTa 2 Ee0 Ea(Tb Ta ) a 2 2 b b + d (2n 1) b + d (2n 1) b + d2 (2n 1) ln a ð20Þ
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
156 (sr )res =
Journal of Strain Analysis 51(2) Ea (Tb Ta ) 2(1 n) ln ba b b a2 b2 d2 b d b2 2 ln ln 1 2 (2n 1) n ln 1 2 ln a a a a r b2 a2 b + d2 (2n 1) r 2 2 2 2 d b d d b2 + 2 k1 s Y 1 2 + 2 EaTa 2 Ee0 1 2 b + d2 (2n 1) r b + d2 (2n 1) b + d2 (2n 1) r ð21Þ
In the above equations, a is the inner radius, b is the outer radius, c is the interface radius between the plastic zones I and II and d is the interface radius between the plastic zone II and the elastic zone. The constants C3, C5 and C6 are given by ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) pffiffiffiffiffiffiffiffiffiffi 2nb2 1 2(1n) 1 n + n 2(1 n) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d C5 = Q + Ee0 (2n 1)fb2 + d2 (2n 1)g 2n 2(1 n)
ð22Þ
( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 n n 2(1 n) 2nb2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ee0 C6 = P (2n 1)fb2 + d2 (2n 1)g d1 2(1n) 2n 2(1 n)
ð23Þ
where ) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )( 1 n n 2(1 n) Ea(Tb Ta ) d 2n + 1 (1 + n) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P= ln a 2n 1 1 n n 2(1 n) (2n 1) ln ba d1 2(1n) 2n 2(1 n) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 n n 2(1 n) Ea (Tb Ta ) pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 nÞ ln ba d1 2(1n) 2n 2(1 n) b b2 d2 b d ln + 2 (2n 1) n ln ln 2 d a a b + d (2n 1) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 2 1 n n 2(1 n) 2nb pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 sY + 2 2 1 2(1n) (2n 1)fb + d (2n 1)g d 2n 2(1 n) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 n n 2(1 n) 2nb2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EaTa + (2n 1)fb2 + d2 (2n 1)g d1 2(1n) 2n 2(1 n)
ð24Þ
and ( ffi) pffiffiffiffiffiffiffiffiffiffi 1 n + npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(1 n) Ea(Tb Ta ) (1 + n) 2(1n) b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Q = Pd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 2(1n) (2n 1) ln a d 1 n n 2(1 n) 1 n n 2(1 n)
ð25Þ
3 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi n 2(1n) 1 2(1n) 1n + 2nb2 c1 + 2(1n) p ffiffiffiffiffiffiffiffiffiffi d 6 (2n1)fb2 + d2 (2n1)g 7 2n 2(1n) 7Ee0 pffiffiffiffiffiffiffiffiffiffi C3 = R + 6 pffiffiffiffiffiffiffiffi 4 5 1nn 2(1n) 2nb2 c1 2(1n) 1 p ffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffi (2n1)fb2 + d2 (2n1)g 1 2(1n) (2n1) d 2n 2(1n)
ð26Þ
2
2
where R = Qc1 +
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
k1 s Y EaTa + (2n 1) (2n 1) Ea(Tb Ta ) c 2Ea(Tb Ta ) Ea(Tb Ta ) ln + a (2n 1) ln ba (2n 1) ln ba (2n 1)2 ln ba
2(1n)
k1 sY ln c + Pc1
2(1n)
+
The constant axial strain e0 is given by
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
ð27Þ
Kamal et al.
157
k1 sY c2 a2 1 2 n(d2 c2 ) nd2 (b2 d2 ) R c 2 a2 2 + 2 ln c ln a + (c a ) + e0 = + 4 2n 1 b + d2 (2n 1) AE AE 2 2 2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ) pffiffiffiffiffiffiffiffiffiffi ) ( ( pffiffiffiffiffiffiffiffiffiffi Q d1 + 2(1n) c1 + 2(1n) P d1 2(1n) c1 2(1n) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + + AE AE 1 + 2(1 n) 1 2(1 n) " # 2 aTa d2 c2 (b2 d2 ) + A 2(2n 1) 2fb2 + d2 (2n 1)g 2 c c2 1 a(Tb Ta ) d d 6n + 1 2 2 ln (d c ) ln + a 2 a 2 2 2(2n 1) (2n 1) ln ba A 2 3 b d nd2 (b2 d2 ) b 2 ln nd (2n 1) n ln ln 7 d a a b2 + d2 (2n 1) a (Tb Ta ) 6 6 7 b + 6 2 7 2 5 2(1 n) ln a A 4 b b d d 1 ln (b2 d2 ) 2(1 n) ln a 2 a 2 4
ð28Þ
where A is defined as 2
A=
d 2 c2 (b2 d2 ) 2(2n 1) 2fb2 + d2 (2n 1)g pffiffiffiffiffiffiffiffiffiffiffi 8 9 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 c 1 + 2ð1nÞ 2 < 2 1 n + n 2(1 n) c a 2nb 1 = A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ d pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :(2n 1)fb2 + d2 (2n 1)g2n 2(1 n) (2n 1); 2 1 2(1n) dc 1 n n 2(1 n) pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ) ( ( ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi d1 + 2(1n) c1 + 2(1n) 2nb2 1 2(1n) 1 n + n 2(1 n) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi d (2n 1)fb2 + d2 (2n 1)g 1 + 2(1 n) 2n 2(1 n) pffiffiffiffiffiffiffiffiffiffi ) ( pffiffiffiffiffiffiffiffiffiffi ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 n n 2(1 n) d1 2(1n) c1 2(1n) 2nb2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi + ffi (2n 1)fb2 + d2 (2n 1)g d1 2(1n) 2npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2(1 n) 2(1 n)
ð29Þ
Residual stresses in second stage of elasto-plastic deformation. The expressions of the residual stresses in the plastic zones I and II are given by equations (13) to (18). The residual stresses in the elastic zone and two outer plastic zones are given as follows:1 Elastic zone, d 4 r 4 e Ea Ea(Tb Ta ) 1 d 1 e2 1 e2 b a2 b2 Ta + ln su = + 2 + + 2 ln 1+ 2 (2n 1) a 2 2d 2 a 2r r b2 a2 2(1 n) ln ba (2n 1) ð30Þ k1 s Y e2 e2 Ee0 + 1 + 2 2 k1 s Y + 2n 1 2d 2r (1 2n) Ea Ea(Tb Ta ) 2n d 1 e2 b 2b2 Ta + ln sz = + 2 + 1 ln (2n 1) a 2 2d a b2 a2 2(1 n) ln ba (2n 1) ð31Þ 2n e2 Ee0 k1 sY 1 + 2 + + 2n 1 2d (1 2n) Ea Ea(Tb Ta ) 1 d 1 e2 1 e2 b a2 b2 b Ta + ln (sr )res = + 2 + 2 ln 1 2 (2n 1) a 2 2d 2 2r a r b2 a2 2ð1 nÞ ln a (2n 1) k1 s Y e2 e2 Ee0 + 1 + 2 + 2 k1 sY + 2n 1 2d 2r (1 2n) ð32Þ Plastic zone III, f 4 r 4 b r Ea(Tb Ta ) b a2 b2 1 + ln (su )res = k1 sY (1 + ln r) + C7 + ln 1+ 2 a a r b2 a2 2(1 n) ln ba
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
ð33Þ
158
Journal of Strain Analysis 51(2)
r Ea(Tb Ta ) b 2b2 ln (sz )res = k1 sY (1 + ln r) + C7 + 1 + 2 ln a a b2 a2 2(1 n) ln ba Ea(Tb Ta ) r b a2 b2 1 2 ln (sr )res = k1 sY ln r + C7 + ln a a r b2 a2 2(1 n) ln ba
ð34Þ ð35Þ
Plastic zone IV, e 4 r 4 f r Ea Ea(Tb Ta ) 1 d 1 e2 b a2 b2 Ta + ln (su )res = + 2 + 1 + ln ln 1+ 2 (2n 1) a 2 2d e a r b2 a2 2(1 n) ln ba (2n 1) 1 e2 1 r Ee0 + 1 + 2 ln k1 sY + 2n 1 2 e 2d (1 2n) r Ea Ea(Tb Ta ) 2n d 1 e2 b 2b2 Ta + ln (sz )res = + 2 + 2n ln + 1 ln (2n 1) a 2 2d e a b2 a2 2(1 n) ln ba (2n 1) 2n e2 r Ee0 1 + 2 2n ln + k1 s Y + 2n 1 e 2d (1 2n) r Ea Ea(Tb Ta ) 1 d 1 e2 b a2 b2 Ta + ln (sr )res = + 2 + ln ln 1 2 (2n 1) a 2 2d e a r b2 a2 2(1 n) ln ba (2n 1) r 1 e2 1 Ee0 1 + 2 + ln + k1 sY + 2n 1 2 e 2d (1 2n)
ð36Þ
ð37Þ
ð38Þ
In the above equations, e is the interface radius between the elastic zone and plastic zone III and f is the interface radius between the plastic zone III and plastic zone IV. The various constants involved in the above equations are given by1 ) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )( 1 n n 2(1 n) Ea(Tb Ta ) d 2n + 1 (1 + n) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C6 = ln a 2n 1 1 n n 2(1 n) (2n 1) ln ba d1 2(1n) 2n 2(1 n) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 n n 2(1 n) Ea(Tb Ta ) 1 d 1 e2 d 1 e2 b pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln + 2 ln + 2 ð39Þ a 2 2d a 2 2d 2(1 n) ln a d1 2ð1nÞ 2n 2(1 n) (2n 1) ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1 n n 2(1 n) 2n e2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s Y 2 2n 1 2d d1 2(1n) 2n 2(1 n) ( ) p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffi 1 n + n 2(1 n) Ea(Tb Ta ) (1 + n) b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð40Þ C5 = C6 d2 2(1n) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2n 1) ln a d1 + 2(1n) 1 n n 2(1 n) 1 n n 2(1 n) C3 = N +
Ee0 (1 2n)
ð41Þ
where N = C5 c
1 +
pffiffiffiffiffiffiffiffiffiffi 2(1n)
+ C6 c
1
pffiffiffiffiffiffiffiffiffiffi
2(1n)
k1 s Y EaTa Ea(Tb Ta ) c 2n + 1 ln + + + k1 sY ln c a 2n 1 (2n 1) (2n 1) (2n 1) ln ba ð42Þ
C7 = M +
Ee0 (1 2n)
ð43Þ
where e e2 Ea Ea(Tb Ta ) 1 d 1 Ta + ln M= + 2 ln a (2n 1) a 2 2d 2(1 n) ln ba (2n 1) 1 e2 1 f 1 + 2 + ln + + ln f k1 sY 2n 1 2 e 2d
Downloaded from sdj.sagepub.com at INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI on February 17, 2016
ð44Þ
Kamal et al.
C9 =
159
e2 a(Tb Ta ) 2 k s (n 1) e2 (1 n2 ) 1 Y b E 2(1 n) ln a
ð45Þ
f2 a(Tb Ta ) 1 2n f2 2 2 k1 sY 2 C8 = f + C9 (C7 k1 sY ln f) (n 1) + (1 n ) b E E 2 2(1 n) ln a ð46Þ 3 2n 2 1 2 a(Tb Ta ) 2 f 3 f2 f ln f k1 sY f aTa + e0 b 4E 2 a 2 2 2 ln a 2 3 2 2 c a 1 2 n n e2 2 2 2 2 2 (d c ) + 1 + 2 (f d ) 7 ln c ln a + (c a ) + 4 2n 1 2n 1 2 2 2d k1 s Y 6 6 7 e0 = 6 2 2 7 5 BE 4 f 2 1 2 1 b f n ln f (f e2 ) (b2 f2 ) ln b + ln f e 2 4 2 2 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 9 N 2 C5
