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I III IIl’IIIIfiIIIIIIIIIII IIIIIII'IIIIII” . III III II IIIIE‘II III III III h I I WI III'IIIII'IIII III III IIIIIIIII III III: I IIIIIIIII II I IIIIIIII II THE RESIDUAL STRAIN DISTRIBUTION AROUND A FASTENER HOLE COLDWORKED WITH A TUBE EXPANDER BY SUSAN EME RY A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SC IENCE Department of Metallurgy, Mechanics, and Material Science 1978 ABSTRACT THE RESIDUAL STRAIN DISTRIBUTION AROUND A FASTENER HOLE COLDWORKED WITH A TUBE EXPANDER BY Susan Emery These experiments were conducted to achieve a uniform radial expansion of a fastener hole. The expansion was done in increments to study the developing pattern for increasing amounts of expansion. A technique was developed to measure the expanded diameter of the hole. After coldworking the hole with a tube expander, residual strain and elastic- plastic boundary location measurements were taken. The results were compared with theoretical predictions. It is evident from the deformation and the residual strain dis- tributions that the expansion was not uniformly radial through the thickness. The residual strain distributions also led to the conclusions that the existing theories aren't complete enough to handle the problem. ACKNOWLE DGEMENTS The author wishes to express her appreciation to Dr. William N- Sharpe, Jr., her major professor, for his consulation and guidance during her course of study and for the opportunity to do this research. Thanks also go to the Office of Scientific Research for providing funds through Grant No. 75-2817 to do this research, and to Mr. Mike Ward for his assistance in preparing the figures. LIST OF TABLES TABLE OF CONTENTS LIST OF FIGURES Chapter 1 2 INTRODUCTION THEORIES 2.1 Material Behavior for the Coldworking Problem 2.2 Nadai Theory 2.3 HsueForman Theory EXPERIMENTAL TECHNIQUES 3.1 3.2 3.3 3.4 3.5 Specimens Expansion Technique Displacement Measurement Technique Strain MeaSurements Thickness Change Measurement Technique RESULTS AND DISCUSSION 4.1 6.35 mm.Thick 7075-T6 Aluminum Specimen 4.1.1 Radial Strains 4.1.2 Tangential Strains 4.1.3 Elastic-Plastic Boundary Location 3.18 mm Thick 7075-T6 Aluminum Specimen 4.2.1 Radial Strains 4.2.2 Tangential Strains 4.2.3 Elastic-Plastic Boundary Location iii Page vi vii 10 13 18 18 24 26 29 36 39 41 44 51 52 S6 69 74 77 4.3 3.18 mm Thick 1100 Aluminum Specimen 77 4-3.1 Radial Strains 84 4.3.2 Tangential Strains 95 4.3.3 Elastic-Plastic Boundary 97 Location 4.4 Discussion 97 5 CONCLUS IONS 1 0 8 REFERENCES 111 APPENDI X , ll 3 iv Table 3.1 3.2 3.3 4.6 LIST OF TABLES Mechanical properties of aluminum.alloys used Initial diameters (in mm) of holes Gage diameters and linear regression results for best straight lines of the data in Figure 3.6 Initial and residual diameter measure- ments (in mm) for the 6.35 mm thick 7075-T6 aluminum specimen Comparison of expanded and residual radial displacements (in mm) for each expansion for the 6.35 mm thick 7075-T6 aluminum specimen Initial and residual diameter measure- ments (in mm) for the 3.18 mm thick 7075-T6 aluminum specimen Comparison of expanded and residual radial displacements (in mm) at the hole edge for each expansion for the 3.18 mm thick 7075-T6 aluminum specimen Comparing two diameter measurements (in mm) obtained by the two techniques used in these experiments Initial and residual diameter measurements (in mm) for the 3.18 mm thick 1100 a1uminum.specimen Comparison of the residual and expanded radial displacements (in mm) at the hole edge for each expansion for the 3.18 mm thick 1100 a1uminum.specimen Page 21 23 29 41 43 56 58 74 83 83 Figure 2.1 2.3 2.4 3.1 3.2 3.3 3.4 3.5 LIST OF FIGURES Page Geometry considered by coldworking theories. 6 Sketches of the stress distributions that 11 occur at different points in the cold- working process: (a) a series of applied loads have caused only elastic stresses in the material, (b) the applied load has caused the material to yield, (c) elastic unloading stresses, (d) stress state that results from.cold- working (sketches (b) and (c) superposed). Residual radial (compressive) strains pre- 15 dicted by the Nadai and Hsu—Forman theories for a 0.1422 mm radial expansion. Residual radial tangential (tensile) strains 16 predicted by the Nadai and Hsu—Forman theories for a 0.1422 mm radial expansion of a 12.7 mm.hole. Stress-strain curve for the 3.18 mm thick 19 1100 aluminum alloy. Stress-strain curves for the 3.18 mm and 20 the 6.35 mm thick 7075-T6 aluminum alloy. Schematic of plate illustrating the thick- 22 ness change measurement directions and the notches used to hold the plate. Photograph of the condenser tube expander 27 illustrating the location of the L1 and L2 measurements used to find the displacement causing the expansion. Longitudinal and cross-sectional drawings 28 of the tube expander (roller unit and pin) and the specimen. vi Figure 3.6 3.7 3.8 3.9 3.10 4.1 4.2 Graph to obtain the best straight line relationship between the two length measurements, L1 and L2, used to deter- mine the expanded diameter. The best straight line coefficients appear in Table 3.3. Each line is for a different diameter gage. Graph relating diameter to dimensions L1 and L2 . Each curve corresponds to a particular L value used in the five best straight line equations in Table 3.3. Photograph of a set of three indentations which forms a gage that measures strains in the radial and tangential directions. Schematic of the indentation pattern applied on four radial lines on both sides of the plate to measure radial and tangential strains. Photograph of the thickness change measure- ment setup, consisting of the LVDT (1), Daytronic amplifier (2), the LVDT holder (3), specimen (4), x-Y recorder (5), X-Y translation stage (6), linear potentiometer (7), and the specimen holder (8). Sketches a-d illustrate the general shape of the deformed material at the hole edge for successively larger expansions of the hole. In each case the hole is to the left of the straight line portion of the sketch. Photograph of the deformed 6.35 mm thick 7075-T6 aluminum specimen. Average residual radial (compressive) strains on the 6.35 mm thick 7075-T6 a1uminum.specimen for a radial expansion of 0.191 mm. Average residual tangential (tensile) strains on the 6.35 mm thick 7075-T6 a1uminum.specimen for a radial expansion of 0.191 mm. Average residual radial (compressive) strains on the 6.35 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.290 mm. - vii Page 30 31 34 35 38 4O 42 45 46 47 Figure 4.6 4.9 4.10 4.12 4.14 Average residual tangential (tensile) strains on the 6.35 mm thick 7075-T6 aluminum specimen for a radial expan- sion of 0.290 mm. Typical profiles obtained on one radial line for the thickness change measure- ment used to locate the elastic-plastic boundary for the two different expansions of the 6.35 mm thick 7075-T6 aluminum specimen. (Scale sensitivity: $0.127 mm.) (1) original profile of the plate (2) profile for the 0.191 mm radial expansion (3) profile for the 0.290 mm radial expansion Comparison of the theoretical and experimental elastic-plastic boundary locations for the 6.35 mm thick 7075-T6 specimen. Photograph of the deformed 3.18 mm thick 7075-T6 aluminum specimen. Average residual radial (compressive) strains on the 3.18 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.030 nmn Average residual tangential (tensile) strains on the 3.18 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.030 mm. Average residual radial (compressive) strains on the 3.18 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.061 mm. Average residual tangential (tensile) strains on the 3.18 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.061 mm. Average residual radial (compressive) strains on the 3.18 mm thick 7075-T6 aluminum Specimen for a radial expansion of 0.142 mm. viii Page 48 53 55 57 60 61 62 63 64 4.18 4.20 4.21 4.22 4.23 4.24 Average residual tangential (tensile) strains on the 3.18 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.142 mm. Average residual radial (compressive) strains on the 3.18 mm thick 7075—T6 aluminum specimen for a radial expansion of 0.276 mm. Average residual tangential (tensile) strains on the 3.18 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.276 mm. Typical profiles obtained on one radial line for the thickness change measure- ment used to locate the elastic-plastic boundary for the four expansions of the 3.18 mm thick 7076-T6 aluminum specimen. (Scale sensitivity: $0.127 mm) (1) original profile on the plate Page 65 66 68 78 (2) profile for the 0.030 mm radial expansion (3) profile for the 0.061 mm radial expansion (4) profile for the 0.142 mm radial expansion (5) profile for the 0.276 mm radial expansion Comparison of the theoretical and experi- mental elastic-plastic boundary locations for the 3.18 mm thick 7075-T6 specimen. Photograph of the deformed 3.18 mm thick 1100 aluminum specimen. Average residual radial (compressive) strains on the 3.18 mm thick 1100 aluminum specimen for a radial expansion of 0.121 mm. Average residual tangential (tensile) strains on the 3.18 mm thick 1100 aluminum specimen for a radial expansion of 0.121 mm. Average residual radial (compressive) strains on the 3.18 mm thick 1100 aluminum specimen for a radial expansion of 0 .178 mm. Average residual tangential (tensile) strains on the 3.18 mm thick 1100 aluminum specimen for a radial expansion of 0.178 mm. ix 80 81 85 86 87 89 Figure 4.25 4.26 Average residual radial (compressive) strains on the 3.18 mm thick 1100 aluminum specimen for a radial expansion of 0.236 mm. Average residual tangential (tensile) strains on the 3.18 mm thick 1100 speci- men for a radial expansion of 0.236 mm. Typical profiles obtained on one radial line for the thickness change measure- ment used to locate the elastic-plastic boundary for the three expansions of the 3.18 mm thick 1100 aluminum specimen. (Scale sensitivity: e0.127 mm) (1) original profile of the plate (2) profile for the 0.121 mm radial expansion (3) profile for the 0.178 mm radial expansion (4) profile for the 0.236 mm radial expansion. Comparison of the theoretical and experimental elastic-plastic boundary locations for the 3.18 mm thick 1100 specimen. Page 90 92 98 100 CHAPTER 1 INTRODUCTION The purpose of this research is to experimentally study the residual strain distribution around a coldworked fastener hole and compare it with theoretical predictions. The expansion (coldworking) technique used is chosen to approximate the boundary conditions of the theoretical analyses. A shape change becomes a stress concentration when the structure is subjected to a stress field. A fastener hole is one example of a stress concentration. The hole edge has a tangential stress approximately three times as great as the applied normal stress when the hole is subjected to a uniform tensile field. Cracks will form at the hole edge if the load produces stresses larger than the yield stress of the material or large enough to cause fatigue damage. A crack in an aircraft grows when it is subjected to the fatigue type of loading that occurs in fueling, takeoff, landing, and general buffeting by air currents. This may lead to premature failure of the aircraft if the cracks get large enough. Two of the techniques used to slow this growth are interference fit fasteners and coldworking. .Studies of the latter process by Sharpe (1) and Poolsuk (2) prompted the present work. ‘The commercial coldworking technique consists of sim- ply pulling an oversized mandrel through a fastener hole. This applies and then removes a radial load at the hole edge that causes yielding of the material and an increase of the hole diameter. Mere significantly however, a com- pressive stress now exists around the hole edge, created by 2 the plastic deformation that occurred. This stress must be overcome before a tensile load will be felt by the fastener hole. In effect then, the applied tensile load is decreased. It is important to determine the exact nature of this stress state because it has been shown that one can calculate the stress intensity factor for radial cracks (3). This allows an estimation of the maximum allowable crack size for safe operation of the aircraft. Sharpe (l) and Poolsuk (2) conducted experiments to discover more exactly what occurs in the coldworking pro- cess. In particular, they looked at the J. 0. King pro- cess of coldworking. The purpose was to determine which of many existing theories best model the situation. Sharpe took strain measurements on three thicknesses of material using an indentation technique, and measured the height of the deformed material. He also did fatigue tests to get data on how the coldworking affects crack growth. Poolsuk (2) used two techniques to measure the location of the elastic-plastic boundary, which can be used to evaluate which theories are useful. Both studies found that the King coldworking process clearly does not give a uniform radial displacement or stress through the thickness of the plate. This is evi- denced by the smaller amount of plastic deformation that occurs on the back side. Sharpe concluded that the process does produce a radially symmetric residual strain field on each side of the specimen. This means that readings may be averaged from several radial lines on a side to get a bet- ter estimate of the actual values. He also indicated that these measurements are reproducible from specimen to speci- men. Clearly, a need exists for an experimental study which carefully generates a uniform radial load at the edge of a circular hole. The specimen must be of such dimensions that one can assume plane stress and an infinite sheet, as 3 do the theories. This will allow one to determine what simplifying assumptions can be made regarding material behavior. The expansion in this study is achieved by means of small revolving rolls surrounding a tapered mandrel. The pressure they exert presumably creates a uniform radial displacement: which, if large enough, results in a radial distribution of stress around the hole in the plate and a uniform distribution of stress through the thickness of the plate. Two theories have been selected (based on results of (l) and (2)) for comparison: one by Nadai (1943) (4) and one by Hsu-Forman (5). These are discussed in greater detail in Chapter 2. Chapter 3 describes the experimental techniques used to expand the hole, and to measure the deformation, the strain, and the elastic-plastic boundary location. Results are presented in Chapter 4 and dis- cussed in Chapter 5. Strain measurement data appears in the Appendix. CHAPTER 2 THEORIES Many theories exist that provide solutions to the coldworking problem. Sharpe (1) considered a number of them as they related to the J. 0. King process for cold- working. Between them, Sharpe (l) and Poolsuk (2) consid- ered eleven theories: Nadai (4), Hsu-Forman (5), Potter- Grandt (6), Adler-Dupree (7), Chang (8), Rich-Impellizzeri (9), Alexander-Ford (10), Swainger (11), Taylor (12), Carter-Hanagud (l3), and Mangasarian (14). When briefly comparing these theories, one finds that two of them.con- sidered plane strain (8,9) and the remainder considered plane stress. Mest of them used the Mises-Hencky yield criterion but some of them used the Tresca yield criterion (12,13). Only three of them accounted for strain harden- ing: Hsu-Forman (5), Adler-Dupree (7), and Alexander-Ford (10). These differing assumptions lead to considerable variation in the predicted elastic-plastic boundaries and residual strains for the coldworking problem (1). The above theories were evaluated (2) by finding which ones most accurately predicted where the elastic-plastic ‘boundary, rp, lies. The position of the elastic-plastic boundary is an important measure of the amount of coldwork- ing. Two experimental techniques were used to determine where rp is located: foil gages and thickness change mea- surements. (The latter is described in more detail in Chapter 3 because it was also used in this study.) Both methods gave very comparable measurements. The results in thinner specimens were acceptably predicted by only two of the theories: Nadai (4) and Hsu-Forman (5). These two theories are presented in the latter part of this Chapter 4 5 after the general nature of the coldworking problem is dis- cussed. 2.l Material Behavior for the Coldworking Problem In the coldworking problem, the geometrical shape under consideration is a flat circular plate. It has a radius of "b" with a circular hole in the center of radius "a", as illustrated in Figure 2.1. The theories use one important boundary condition and two simplifying assumptions for the problem. The boundary condition is that the defor- mation is caused by a uniform positive radial displacement, ua, (or a negative pressure, p) at the hole edge, r = a. From this condition, the assumption follows that the pro- blem is axially symmetric. This means that u6 = 0, 8/39 = 0 and that all of the shear stresses and strains are equal to zero. The second assumption, made by all of the theories, is that the radius of the plate is large enough, compared to the plate thickness and hole diameter, that a state of plane stress exists. (Most of the theories simply assume that the radius is infinitely large.) As a result of these assumptions, the material behavior relationships are simpli- fied considerably. The theories are developed for small deformations, so the 8 << 1. Therefore, engineering strain can be used to get the strain-displacement equations that follow: _ 3n _ u 88 - r (2.2) where u is the radial displacement of the material. (No- tice that there is a tangential strain even though the material displacement is radial. An element has moved out on a radial line and must expand to assume the larger radius.) The equilibrium equations for a plane stress pro- blem with axial symmetry become simply: :pr 353 + -£—E——2-= 0 . (2.3) All three of these relationships hold in both the elastic and the plastic regions. In the elastic region, the boundary conditions for a uniform radial displacement at the hole edge are that the radial stress is zero at the outer edge of the plate and that u = ua at the hole edge, r = a. The result is that the stress in the radial direction is equal to the negative of the tangential stress at the same location. For plane stress e=1(0+o). (2.4) 2 E r 8 Therefore, the strain in the z-direction is zero. By fur- ther manipulation of the stress equations and Equation 2.2, the following expression is found for the stresses: a E u l - or = I—:—:—';§ = - o (2.5) where E and v.are material properties, a is the hole radius, and ua is the displacement at the hole edge. As the radial displacement increases, a load is applied that causes yielding of the material; and a yield criterion must be applied. The two most commonly used criterion are the Mises-Hencky Distortion Energy Theory and the Tresca Maximum Shear Theory. The Mises-Hencky criterion is used in the two theories presented later in this Chapter. However, for simplicity of illustration, in the following discussion the Tresca criterion is used: (2.6) where CI and OIII will be equal to 06 and or, respectively. Then the maximum radial displacement possible that won't cause plastic deformation is do a (1 + v) uaE = 2 E ' (2'7) If the radial displacement becomes larger than this, yield- ing occurs in the region between r = a and the elastic- plastic boundary, rp. As the radial displacement, ua, increases, so does rp. After the desired expansion is accomplished, the load is removed. Thus the resultant stress at the hole edge is zero. Most of the coldworking theories that account for unloading assume that it occurs elastically and with no reverse yielding. So, a tangential stress is removed in addition to the radial stress. The unloading stress dis- tributions are a 2 a 2 Cr = O'm [2:] 1 06 = ”Um [E] (2.8) where am is the magnitude of the radial stress generated at r = a by the loading process. The unloading strain dis- tributions are a 2 a 2 6r = Em [E] p 86 = "Em [E] (2.9) where em = -(1 + v)cm/E. The result of the unloading is that residual stresses and strains remain in the material around the hole. For the plastic region, the boundary conditions are that u = ua at the hole edge and that the stresses, strains, and displacements match the elastic ones at r a r . The stresses at rp are found by using the fact that the radial and tangential stresses are equal in magnitude to each other in the elastic region. And if rp is known, it is possible to calculate values for the elastic region, r 1 rD, with the following equations: r 2 e: = - i (2.10) r ~ 2 c (l + v) r 0 [—2] (2.11) “=7?— r Of course, to get the residual strains one must subtract the unloading distribution from these values. In the plastic region, the strains and equilibrium equation are the same as in the elastic region. See Equa- tions 2.1, 2.2, and 2.4. However, in the plastic region the strain in the z-direction is no longer zero because the radial and tangential stresses are not equal here. In addition, the condition of volume constancy must be used for the strains: 8+8 +8 =0. (2.12) (o - -—) (2.13) where eP/ee varies because of the nature of the stress- strain curve. This is the point at which the various the- ories take different directions. There are several accept- able methods for dealing with this problem, the simplest being to assume that the material is elastic-perfectly plastic. Another method is to assume a modified uniaxial Ramberg-Osgood stress-strain relation. These approaches are used, respectively, by Nadai (4) and Hsu-Forman (5). The stresses that exist in the material at different points in the preceding discussion are illustrated in 10 Figure 2.2. In sketch (a) observe the effect of three expansions in the elastic region. The distributions are caused by the application of uniform radial loads at the hole edge, r = a, to generate a radial displacement. Note that the tangential and radial stresses are symmetric about the r-axis. If the load is removed, the material relaxes back to its original state. In sketch (b) a load has been applied which is much greater than that necessary to cause yielding. Because of the yield criterion (the Tresca cri- Lterion is used in the Figure) and the fact that the radial load is always increasing, then 06 must decrease in the plastic region. The tangential stress begins to relax or flow since the material is assumed to be incompressible. The unloading Stress distribution appears in sketch (c). Notice that the radial and tangential stresses here are opposite in sign to what they are for loading. Sketch (d) is the result of superposing (b) and (c). Notice the large compressive stress that remains at the edge of the hole. This stress is the goal of the coldworking procedure and must be overcome before a tensile load will be felt. Notice also the stresses extending into the elastic region of the material. They are caused by the pressure the plas- tic zone exerts in the elastic region. Observe that these stresses are below the yield point. 2.2 Nadai Theory In 1943 Nadai (4) published a theory about the expan- sion of boiler and condenser tube joints. These joints must remain leak-free at very high temperatures and pres- sures. The tube end is placed in the plate and the tube and plate are plastically deformed to achieve the necessary fit. The expansion is done with a set of small revolving rollers. He considered the plasticity in both the tube and the plate. The present study makes use only of the infor- mation regarding the plate. His assumptions were 1. uniform pressure at the edge of the hole in the plate Figure 2.2 11 Sketches of the stress distributions that occur at different points in the coldworking process: (a) a series of applied loads have caused only elastic stresses in the material, (b) the applied load has caused the material to yield, (c) elastic unloading stresses, (d) stress state that results from coldworking (sketches (b) and (c) superposed). 12 :9 ---dh--‘---------- h- D __it s. N.m musmfim EWOH 8\ .1 h] :9 «x b .1 Q. ---‘------.. -u---- -_-- 4m EWOH QR. $33WOH hr 13 2. a linear approximation to the Mises- Hencky yield criterion 3. perfectly plastic material response In the plastic zone he developes the following equa- tions for the displacement and resulting strains: ‘ u [2:— aE r u = ' P 3 (2.14) [1 +1; 1n[§—]] p u e = e (2.15) r 1 + —- n —— 3 r p [ p (2 r ‘ ue ‘5' 1n [F] " l 8r z p 4 r [1 +.3 1n[£—]1 p 3 r . P J where uaE has already been explained and ue = uaE(rP/a). These strain equations do not include the elastic unloading. To find the residual strains one must add the following: (1 + v) o = + 0 -1 + 2 1n 3— 3 2 (2 16) €r,8 '_ 2 E rp r ’ to Equations 2.15. The residual strains are plotted in Figures 2.3 and 2.4 for a 0.1422 mm radial expansion of a 12.75 mm hole. 2.3 Hsu-Forman Theory In 1975 Hsu and Forman (5) published a theory that was basically the same as Nadai's; but, in addition, it ac- counted for work hardening. Their assumptions are l. uniform pressure at the hole edge 2. Mises-Hencky yield criterion l4 3. a modified uniaxial Ramberg- Osgood representation of the stress-strain curve Specifically, the material behavior is represented by: c e = E for [cl i 00 (2.17) n-l — 2 .2. e - E loo] for lol 1 do For 7075-T6 aluminum, the stress-strain curve is repre- sented by n = 15. The solution is developed in terms of a parameter a which varies between 90° and aa, where aa corresponds to a particular expansion, ua. The stresses, strains, and dis- placements in the plastic region in terms of a for R = l and n = 15 are as follows: c 1 .07895 n 30 = [sin a - .7423 cos a] exp(.10636 (a - 2)) (2'18) 0 [30] 0° — 1 + v 8r 6 = -——§———I[(l - v) cos a + [ ] sin a] (2.19) I /3 . (“—J“ u 0 32 = l +0v [(1 + v) cos aa +[l + v] sin ca] (2.20) 0' |——-—l f3" 3 where for Equation 2.20 v = .5 - (.5 - v’)/(o/co)n-l, v’ is Poisson's ratio of the material, and u0 = 11511:: (see Equa- tion 2.12). The residual strains are graphed in Figures 2.3 and 2.4 for a 0.1422 mm radial expansion of a 12.75 mm hole. The following relationship enables one to express the stresses and strains in terms of r: 15 - - - HSU-FORMAN -— NADAI i- Z LIJ 0 (I LU 0|- Z 4 Ct '— (D C 1 1 1 IO |.5 2.0 2.5 3.0 DISTANCE- '4: Figure 2.3. Residual radial (compressive) strains predicted by the Nadai and Hsu-Forman theories for a 0.1422 mm radial expansion of a 12.7 mm hole. 16 3 ““" HSU‘ FORMAN _ NADA) STRAIN- PERCENT “ ‘- ——l— ___ *fi LO 1.5 2.0 2.5 3.0 DISTANCE— Va Figure 2.4 Residual tangential (tensile) strains predicted by the Nadai and Hsu-Forman theories for a 0.1422 mm radial expansion of a 12.7 mm hole. 17 A .13158 sin a '2 , . a a- .7423 cos a + Sln a exp(.8508(aa-g)) .7423 cos aa+ sin aa mu: sin a One can calculate the location of the elastic-plastic boundary, rp, from this equation by letting a = w/2. (2.23) CHAPTER 3 EXPERIMENTAL TECHNIQUES To evaluate the theories presented in Chapter 2, one must construct a specimen and coldworking process for the experiments that satisfy the boundary conditions. Design- »ing a specimen to satisfy the plane stress and infinite plate criteria is relatively simple. It is difficult, but very important, to find an expansion technique that will give a uniform radial displacement through the thickness at the hole edge. A long tapered mandrel was tried with poor results so another method was selected. A technique was developed to measure the displacement causing the expan- sion. To locate the elastic-plastic boundary, a thickness change measurement developed by Poolsuk (2) was used. This Chapter contains a discussion of the materials, geometries, and experimental techniques used in these experiments. 3.1 Specimens Two different materials were used for this study. One, aluminum type 7075-T6, was an obvious choice. It is a high strength, light-weight alloy used in manufacturing aircraft and space vehicles. Two thicknesses, 6.35 mm and 3.18 mm, of this alloy were used. A softer alloy, aluminum type 1100, was also used to see if it might behave accord- ing to the theories. Its thickness was 3.18 mm. The mechanical properties were obtained by conducting uniaxial tension tests and hardness tests (2). The numerical results appear in Table 3.1 and the stress-strain curves in Figures 3.1 and 3.2. The dimensions of the plate are determined by the assumptions made in the theoretical solutions and by the 18 19 MPO Ksl '0 A v A o (.00 2.00 7 Strains (%) Figure 3.1 Stress-strain curve for the 3.18 mm thick 1100 aluminum alloy. . 20 MPO. r Kai 650- 600.. 550 4" 80 500« 450‘ 400.4 35° #50 «0 8 8 O o l 1 Stress“ 200 q —— 6.35 mm.(l/4in.) _--- 3.18 mm. (vain) ISO IOO - 50- J I. L A L 4 I A L f r 0 ‘ . ' 1'0 . ’ 2:0 Strains ('16) Figure 3.2 Stress-strain curves for the 3.18 mm and the 6.35 mm thick 7075-T6 aluminum alloy. 21 Table 3.1 MeChanical properties of aluminum alloys used ........... Mechanical Alloy 7075—T6 7075-T6 1100 Property Thickness 6.35 mm 3.18 mm 3.18 mm Material . Strength 589 527 79 (Mpa) 0.2 percent Offset Yield 548 503 33 Strength (MP3) Modulus of Elasticity 696 682 675 (x 103 MPa) Poisson's Ratio 0.31 0.31 ' 0.28 Hardness 91RB 90RB 28RB instruments available for conducting the experiments. The 178 mm diameter plate is the largest that would fit in the microscope. The 12.7 mm diameter hole was controlled by the expansion mechanism chosen. The ratio of b to a is equal to 14. A ratio of b/a greater than 10 is acceptable as an infinite plate (9). A schematic of the plate appears in Figure 3.3. The four notches on the circumference are to hold the plate in position for the thickness change mea- surement. It is necessary to construct round, non-tapered holes V due to the nature of the problem. They must be a specific dimension so that the amount of coldworking deformation can be compared from one specimen to the next. The holes were prepared by first drilling them with a 12.7 mm drill and then reaming them to a diameter of 12.75 mm. This produces square edges of the hole and straight sides in the hole. Upon receiving them from the machine shop, the diameters of the holes were measured on a microscope equipped with'an X-Y stage. Measurements taken at various angles around the hole 22 ©——THICKNEss CHANGE MEASUREMENT DIRECTION I2.7 mm. INSIDE DIAMETER I78 mm. OUTSIDE DIAMETER Figure 3.3 Schematic of plate illustrating the thickness change measurement directions and the notches used to hold the plate. showed that the holes were of acceptable roundness and dia- meter. The greatest uncertainty of this measurement is in locating the edge of the hole accurately. The variation in repeated measurements was usually less than 8 microns. To further prepare the specimens, both sides were hand polished to remove the larger surface scratches. insures that one can easily see the indentations that were applied later to measure the strain. First, one sands the plate in running water with four successively smaller grits of sandpaper (240, 320, 400, and 600). Between grit sizes the plate was turned so that the scratches of the succeed- ing grit were at right angles to those of the preceding one. In this manner, one can easily tell when the larger grit's scratches have been removed. The final polish was done with a polishing cloth and a mixture of 1 micron alumina powder and lapping oil. At this step it is important to not polish longer than absolutely necessary. The cloth cathes on the edge of the hole and begins to round it off slightly. In this study the 1100 type aluminum did not receive this pol- ishing (only the sanding) because the indentations were readily visible without it. "After polishing" initial diameter measurements were taken with the microscope. They appear in Table 3.2. The orientation angles refer to diameters that are perpendicu- lar to each other and are approximately along the lines Table 3.2 Initial diameters (in mm) of holes Specimen E(7075-T6) Front 3.18 mm thick Back C(7075-T6) Front 6.35 mm thick Back H (1100) Front 3.18 mm thick Back Orientation 90° 12.769 12.779 12.796 12.774 12.748 12.746 180° 12.754 12.768 12.779 12.795 12.720 12.741 24 where the indentations will be placed. Location of the exact edge of the hole is more difficult at this time on the 7075—T6 specimens because some rounding of the edge has occurred during the final polishing procedure. Even so, the variation in repeated measurements was still usually no greater that 8 microns. 3.2 Expansion Technique Selection of an expansion technique is perhaps the most critical aspect of this project. It is desired to generate a uniform radial deformation at the hole edge since the commercial coldworking process itself does not create a uniform radial expansion through the thickness of the specimen (1,2). One reason may be that the desired expansion is achieved over such a short length of mandrel, about 19 mm. A second reason could be because the mandrel is pulled through the hole, thus creating an axial load. Recall that the theories have assumed this to be zero. Sharpe (1) found that the 1.6 mm thick specimens obviously showed buckling out of the plane. The same phenomena pro- bably occurs in the thicker specimens, though it may not be obvious to the naked eye. Each of these items must be con- sidered when choosing the expansion process, For the first expansion technique used in this study, it was decided to extend the taper over 762 mm. This makes the opposite surfaces of the mandrel more nearly parallel. Consequently, the displacement caused would be more uniform through the thickness. The mandrel would be pushed, a small amount at a time, through the rotating plate; thus minimizing the axial load. The specific amount of expan- sion used with this technique, 0.1254 mm radial expansion of a 6.6 mm diameter hole, matched that used by Sharpe (l), Poolsuk (2), and Adler-Dupree (7). In addition, this amount of expansion is typical of coldwork applications. The mandrel for this expansion technique was made in four sections out of steel drill rod. Diameter measure- ments were taken at 25 mm intervals. It was lubricated 25 with oil and Molycoat and mounted in the tailstock of a lathe. The specimen was mounted in the chuck and rotated as the mandrel was pushed in about 3 mm, withdrawn, and pushed through a little more. After a section of mandrel was used to within 25 mm of its large end, the specimen was removed and strain measurements were taken with a microscope. Two sections of mandrel were used on a 3.18 mm thick 7075-T6 specimen and one section was used on a 6.35 mm thick specimen. This resulted in about 720 strain measurements. At this point the specimens were examined at 7X magnifica- tion in a stereomicroscope. The material at the hole edge was no longer rising sharply out of the plane of the plate as it should be. It had begun to curl out as though mater- ial were being pushed through the hole. Strains could not be measured on this curled part. The side of the plate toward the entering mandrel had a bit more deformation than the other side. Obviously, the displacement was not uni- form through the thickness of the plate. In addition, the plate and mandrel would often gall, leaving bits of aluminum on the mandrel. It was necessary to find another expansion technique. The mechanism subsequently chosen to produce the expan- sion is a condenser tube expander. It is used commercially to expand boiler and condenser tubes in head plates to obtain a leak-free joint as discussed in Nadai (4). During the plastic expansion process, the diameter of the hole in the plate is permanently enlarged a small amount. In using the expander, it is assumed that the stresses created are uniform through the thickness and radially symmetric. Therefore, the flow of the material in the plate should be radial. This complies with the restrictions set by the theories for the problem. The tool consists of three hardened rollers with a slight taper that are mounted symmetrically around a long tapered pin. They are placed at a slight angle to the axis of the pin. This causes the rollers to describe a helix as 26‘ ‘the roller unit moves to increasing diameters on the'pin. It is important to realize that the expansion is produced by increasing the concentrated forces of the three rollers in infinitesimal steps. These steps are minute enough that they have the effect of being a continuous force applied at the circumference of the hole. A succession of the increases generates the desired small uniform radial expan- sion. Figure 3.4 is a photo of the tube expander, and cross-sectional and longitudinal drawings appear in Figure 3.5. Because of the size of the tool, a 0.3048 mm radial expansion of a 12.7 mm diameter hole is necessary. This is approximately twice the size of the expansion used for the first technique; but, since it is of a hole that is approximately twice the size as that for the first tech- nique, it has been assumed that the effect will be the same. To perform the expansion, the plate is placed in the chuck of the lathe to hold it vertical. The chuck is rota- ted by hand so that one can stop the expansion at small intervals. When the pin is restrained from rotating, it is the helical action mentioned above that draws the pin through the rotating plate. After a predetermined length of the pin moves through the hole, a displacement measurement is taken and the expander is removed. The plate is removed from the chuck and final diameter measurements are taken. 3.3 Displacement Measurement Technique To measure the amount of displacement, the tube expan- der was calibrated using a series of five steel ring gages. The diameters of the gage holes were measured with the microscope at ten positions around the hole. The greatest uncertainty in this measurement was in locating the hole edge. As was indicated for the specimen hole diameters, the variation in measurements was usually no more than 8 microns. The average of the readings appears in Table 3.3. The holes were determined to be of acceptable size and roundness. One gage at a time was placed at various Figure 3.4 27 Photograph of the condenser tube expander illus- trating the location of the L and L measure- ments used to find the displacement causing the expansion. ) .cmawowmm on» was fined can pass umaaonv Hmpcmmxm was» may no mmcfi3mup Hmcowuoomlmmouo pcm Hmcwpsuwmcoq m.m wusmflm I I: I I 7 Z 7 \N‘ 29 Table 3.3 Gage diameters and linear regression results for best straight lines of the data in Figure 3.6 Gage No. Diameter (mm) Slope Intercept (cm) 502 12.822 -1.050 7.613 508 12.937 -0.958 6.858 514 13.121 -0.056 6.4165 520 13.290 -0.970 5.725 526 13.429 -l.056 5.091 positions on the roller unit. The distance from the gage to the spacer on the roller unit was measured and recorded as L1' The distance from the roller unit to the handle of the pin was measured and recorded as L2 (see Figure 3.5). These measurements were taken to the nearest 0.5 mm. The graph of these readings appears in Figure 3.6. Each line is for a different diameter gage. A.linear regression was done on the data points for each set to find the best straight line. The slopes and intercepts calculated also.appear in Table 3.3. These were used with particular L1 values to calculate values for the graph in Figure 3.7. In the experiments, L1 and L2 were both measured before and after each expansion. Then the initial and final diameters were read from Figure 3.7 and subtracted to get the diametral expansion. 3.4 Strain Measurements In choosing a strain measurement technique it is impor- tant to consider the deformation around a coldworked hole. Photographs of the deformation for various specimen thick- nesses appear in Sharpe (l) and in Chapter 4 of this work. The most striking feature is the large amount of deformation at the hole edge. Individual grains have rotated and slip lines are visible. From this, one can conclude that the material is not a homogeneous, isotropic medium at the hole edge. The large deformation and the large plastic strains that vary greatly over short distances at the hole edge make the strain measurements difficult. Many excellent strain Lz-ON MANDREL (cm) Figure 3.6 30 ; . i . L.- ON ROLLERS (crn) Graph to obtain the best straight line relation- ship between the two length measurements, L and L , used to determine the expanded diaméter. The bgst straight line coefficients appear in Table 3.3. Each line is for a different dia- meter gage. 31 Figure 3.7 Graph relating diameter to dimensions L and L2. Each curve corresponds to a particUlar L value used in the five best straight line e nations in Table 3.3. DIAMETER (inches) 32 .530” .525- CURVES FROM LEFT To RIGHT CORRESPWD To L522 L.‘2.0 .520 _ LELB LI‘LG L..'I.4 Ll‘l.2 LI'ID L.=O.8 .55“ .50’ .505- 1 l I 1 1 1 Z 3 4 5 6 7 8 L2- LENGTH ON MANDREL (cm) Figure 3.7 33 measuring techniques cannot be used because they miss impor- tant data at the hole edge. Sharpe (l) tried various techniques for measuring strains around the coldworked hole. One which he found to work quite well, and subsequently used for most of his work, is the indentation technique. Two small fiducial marks are put on the specimen to form the gage. Before-and-after length measurements are taken with a microscope using 400x magnification. With this technique, one can get as close as 50 microns to the edge of the hole for the first tangential strain measurement and 150 microns to the edge of the hole for the first radial strain measurement (using the 6.6 mm diameter hole). It is usable in an area of large deforma- tion. The edges of the mark are sometimes distorted because of the large deformation, but the center can be located within acceptable limits of error. This technique is also used in this study. The indentation technique employs the diamond indenter of a Vicker's hardness tester. It is used to apply the pyrimidal indentations to the specimen surface. The dis- tance between the two marks, the gage length, is nominally 200 microns. This distance is measured before and after coldworking. True strain is calculated by taking the natur- al logarithm of the final over the initial length. The lim- iting factor of this measurement is being able to locate the exact center or edge of the indentation. Usually the dis- tance measurement has an uncertainty of 0.1 micron. Since the uncertainties add, when comparing the initial and final measurements, the total uncertainty is 0.4 micron. Then dividing by the gage length gives the uncertainty of the measurement to be 0.2 percent strain. This is acceptable when measuring strains of 2 percent or larger. A set of three indentations forms one gage. A photo— graph of a set appears in Figure 3.8. This configuration allows for both a tangential and radial measurement. A pat- tern of these sets, like the schematic in Figure 3.9, was 3A Figure 3.8 Photograph of a set of three indentations which forms a gage that measures strains in the radial and tangential directions. "' 'v """"""" I00 CI 0 300 CI 700 a 0 EDGE OF 900—- D HOLE I300 0 CI I500 0 I900 D E! (D 2|00-———D 2 g 2500 o o 0 2700 D 2 ’I 200 mia'ons I w 3 3700 U D 3900 I: O I M/ 2 8 5900—— CI CI u. 600—0 8 M E 8I00--- o o I- 8300——— U 22 O W I0300—-—- D I: I0500--—- D Figure 3.9 Schematic of the indentation pattern applied on four radial lines on both sides of the plate to measure radial and tangential strains. 36 applied to each specimen on four radial lines perpendicular to the hole edge. Each pattern is perpendicular to two of the others and on the same diameter as the remaining pat- tern. These lines are located 45° from where the thickness change measurement lines are (see Figure 3.3). They are placed on both sides of the plate. For the 12.75 mm dia- meter hole, the closest tangential strain measurement is 0.1 mm from the hole edge and the closest radial strain is 0.2 mm from the edge. The space between gages is twice that used for the smaller diameter hole used with the first expansion technique. Taking the measurements on the micro- scope and doing the strain calculations is tedious and time consuming. However, this method was selected because it does allow one to obtain reproducible strain measurements close to the hole edge in an area of large deformation. 3.5 Thickness Change Measurement Technique Poolsuk (2) used the thickness change measurement as one method of evaluating the coldworking theories. A detailed description of the technique and apparatus can be found in that work. Briefly, the measurement is based on two conditions that were mentioned in Chapter 2. The first condition is that no thickness change occurs in the elastic region of the plate. This is caused by the fact that in the elastic region the tangential and radial strains are equal in magnitude but opposite in sign (Equation 2.1), so 62 = 0. The second condition is the assumption of volume constancy in the plastic region: When a circular hole in a plate is loaded by a large enough uniform radial displacement, plastic deformation occurs. The material rises sharply out of the plate at the edge of the hole, in the positive and negative z-directions. The slope of the deformation gradually decreases until it gets to the elastic region. Then the slope is zero because in 37 that region there is no deformation in the z-direction, as was stated above. Therefore, by locating where the thick- ness first starts to change, one can find the elastic- plastic boundary. The measurement device used to detect the thickness change is a linear variable differential transformer (LVDT). A photograph of the set-up is in Figure 3.10. This instru- ment is sensitive to a very small thickness change, 0.127 mm for this experiment. To prepare for the measurement, one must first set the contact balls of the LVDT as close as possible to the centerline of the hole. A small weight is placed on the plate edge to balance the specimen weight on the lower ball when the plate is moved horizontally. Then, by lightly holding the balls in contact with the plate, a 30 mm groove is made on the surface of the plate. This length was selected to insure that one is outside the plas- tic region when starting the measurement. Four grooves are made along the lettered directions indicated in Figure 3.3. Slashes are made across the lines to provide reference peaks on the traces to be recorded later. The distance between the slashes is measured on the microscope and recorded. After this, the original thickness traces for each line are graphed using an X-Y recorder. For each line, one must set the null of the LVDT as near as possible to the calibrated null to be able to compare the traces later. Following each expansion a graph is obtained for each of the four directions. The thickness change profiles are super- posed on the original. The point where the superposed curve begins to deviate from the original is the elastic-plastic boundary. Examples of these graphs appear in Chapter 4 for each specimen. 38 Figure 3.10 Photograph of the thickness change measurement setup, consisting of the LVDT (1), Daytronic amplifier (2), the LVDT holder (3), speci- men (4), X-Y recorder (5), X-Y translation stage (6), linear potentiometer (7), and the specimen holder (8). CHAPTER 4 RESULTS AND DISCUSSION In this Chapter the results of the experiments are presented and discussed. Three specimens were tested, two thicknesses of 7075-T6 aluminum and one thickness of 1100 aluminum. All were coldworked using the tube expander described in Chapter 3. In the ensuing discussion the front of the plate refers to the side facing the incoming mandrel. The back is the side away from the incoming mandrel. Two to four expansions were done on the plates. A schematic of typical shapes of the deformation for increasing amounts of expansion appears in Figure 4.1. The residual strains were measured using the indenta- tion technique. Altogether, for the three specimens, about 2500 gage length measurements were taken with the microscope and the strains calculated. The large deformation of large grains causes significant variation of the strains from one radial line to the next, so the strains must be averaged over several positions. In addition, the deformation is not uniform through the specimen thickness. To resolve this dilemma, one must assume, as the theories do, that the material is isotropic and homogeneous. Then the strains measured on the front and back of the plate can be averaged even though they vary considerably. Therefore, the strains that are plotted in this Chapter are the average of eight radial lines, four on the front and four on the back. It is important to note that for all of the tests, strains greater than 0.5 percent strain are certainly significant, while those equal to 0.5 percent strain and less might be ques- tionable, as they are near the range of the error of the 39 Figure 4.1 40 Sketches a—d illustrate the general shape of the deformed material at the hole edge for suc- cessively larger expansions of the hole. In each case the hole is to the left of the straight line portion of the sketch. 41 measurement. The positions of the gages were also averaged because from one radial line to the next, the position of comparable gages may vary as much as 30 microns. The expan- sion displacement and the location of the elastic-plastic boundary were measured as described in Chapter 3. 4.1 6.35 mm Thick 7075-T6 Aluminum Specimen A photograph of the deformed specimen is shown in Figure 4.2. The deformation is uniform around the hole. The thickness change measurement lines are visible, as are many small scratches that show up because of the fine sur- face polish. Two expansions were done on this specimen. The resulting residual strains and elastic-plastic boundary locations were compared to the Nadai and Hsu—Forman theory predictions. The initial and residual diameter measurements of the front and back appear in Table 4.1. Observe that for all of the residual diameter measurements, the front diameters are larger than the back ones by about 0.15 mm. This would imply that the tube expander does not give a radial dis- placement through the thickness of the plate. From calcula- ting the average difference between the front and back dia- meters, and considering the plate thickness, the rollers appear to make a positive angle of 0.6° to 0.7° with a line through the hole from the front edge and perpendicular to the faces of the plate. While this angle is very small, it Table 4.1 Initial and residual diameter measurements (in mm) for the 6.35 mm.thick 7075-T6 aluminum specimen Orientation 90° 180° Initial Front 12.780 12.783 Back 12.777 12.794 lst expansion Front 13.002 13.026 Back 12.879 12.852 2nd expansion Front 13.139 13.128 Back 12.947 12.954 42 Figure 4.2 Photograph of the deformed 6.35 mm thick 7075- T6 aluminum specimen. 43 is apparently large enough to make the expansion nonuniform through the thickness of the plate. The residual radial displacements can be calculated from Table 4.1. The initial diameter for a particular posi- tion is subtracted from.the after-expansion diameter for that position. The two front and two back remainders are then averaged and divided by two to get the residual radial displacement. These are compared below in Table 4.2 to the expanded radial displacement obtained using the L1 and L2 measurements. Observe that the expanded displacement for this specimen is at least 2 1/3 times what the residual dis- placement is. In other words, the specimen relaxes a large amount when the load is removed. It required considerable effort to load and unload this specimen because of the force with which it resisted the displacement and relaxed around the expander. Table 4.2 Comparison of expanded and residual radial dis- placements (in mm) for each expansion for the 6.35 mm thick 7075-T6 aluminum specimen Expansion Displacements Expanded Residual l .191 .077 2 .290 .128 The specimen was observed with a stereomicroscope after each expansion. At the 7X magnification, the indentations were readily visible, as are the thickness change measure- ment lines. After the first expansion, the deformation was uniform on each side of the plate. The deformation on the back was much less than on the front. The back was compara- ble to the first sketch in Figure 4.1, while the front was more like sketch (b) or (c). The edges of the hole were sharply defined and the sides of the hole were smooth. (This last is also observable with the X—Y stage microscope 44 as one looks at the edges.) After the second expansion, similar observations were recorded. The deformation on both sides was greater; and that on the front was still larger than on the back. The surface of the deformation was uneven, at times distorting the indentations: This makes them difficult to measure for the strains. The residual strains for the two expansions are plotted in Figures 4.3-4.6. The first two plots are the radial and tangential strains, respectively, for the first expansion; the second two are the radial and tangential strains for the second expansion. The theoretical residual strain curves plotted are calculated using the measured initial radial displacements. In the succeeding paragraphs, the radial and then the tangential strains will be discussed as to the nature of the results and how they fit the theories. 4.1.1 Radial Strains For both expansions, the radial strains, which are com- pressive, have a standard deviation at the hole edge that is much larger than the measurement uncertainty. This is par? tially due to the fact that the front and back strains are averaged. For example, consider the first expansion and the first two gage positions near the hole edge. The average radial strains on the front are 6.73 percent strain and 7.22 percent strain, respectively; while for comparable positions on the back, the average radial strains are 1.31 percent strain and 2.11 percent strain. (See the Appendix for the strain values.) The deviation is also due to the variablilty of the strains at the highly deformed hole edge. This is caused by large rotations of large grains and is typically 1-2 percent strain higher or lower that the aver- age. At distances of 3 mm and more from the hole edge the standard deviation is on the order of the uncertainty of the measurement. Here the front and back radial strains are quite small and are nearly equal to each other. Observe that for both of the expansions the radial strains at 0.2 mm from the edge of the hole are less by 45 ---- HSU-FORMAN 8 7 6 5 — NADAI 4 STANDARD DEVIATION 3 2 STRAIN - PERCENT o . . [0 LS 2.0 2.5 3.0 DISPLACEMENT- 90 Figure 4.3 Average residual radial (compressive) strains on the 6.35 mm thick 7075—T6 aluminum specimen for a radial expansion of 0.191 mm. 46 - ---- HSU-FORMAN 2 ' — NADAI STANDARD DEVIATION STRAIN - PERCENT 1‘; O, ‘ 1 IO L5 2.0 2.5 3.0 DISPLACEMENT — r/a Figure 4.4 Average residual tangential (tensile) strains on the 6.35 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.191 mm. 47 6': — .— --—-' '—l ---- HSU'FORMAN — NADAI § STANDARD DEVIATION STRAIN - PERCENT LO (.5 ‘20 2.5 3.0 DISPLACEMENT- 90 Figure 4.5 Average residual radial (compressive) strains on the 6.35 mm thick 7075-T6 aluminum specimen for a radial expansion of 0.290 mm. 48 4 - . ---- HSU-FORMAN — NADAI 3‘7 § STANDARD DEVIATION STRAIN - PERCENT DISPLACEMENT- 9,, Figure 4.6 Average residual tangential (tensile) strains on the 6.35 mm thick 7075—T6 aluminum specimen for a radial expansion of 0.290 mm. 49 0.5-1.5 percent strain than those at 0.8 mm from the hole edge. This is typical of both the front and back strains; therefore it is not a result of averaging them. (Refer again to the Appendix for the strain values.) The differ- ence in strain values for these two positions is much less for the first expansion than for the second one. From this, one can infer that for a smaller first expansion the phe- nomena might not occur. The strains might be largest at the hole edge. Also notice that for the second expansion the strain at 1.4 mm from the hole edge is much closer in value to that at 0.8 mm than for the first expansion. This would seem to indicate that if another expansion were done, the highest strain values might be at the 1.4 mm position. This trend in the strain distribution for increasing expansion can be explained by the material movement. The deformed material is forced to rise vertically out of the plate because it can be pushed no further into the plane of the plate. Once out of the plane, this material also receives a radial load from the tube expander. The deformed material is more free to flow in the radial direc- tion than is the material in the plane of the plate. It is not being pushed against the remainder of the plate. There- fore, it does not need to rise as sharply out of the plane it is now in. There is still material flow in the z- direction because of the load being applied to the plate proper. This results in increasing strain at positions near to the hole edge, even though they are not the highest val- ues for the expansion. Remember that upon successive expan- ions (see Figure 4.1) the deformed material flattens out at its peak, which is next to the hole edge. This occurs because the raised part of the deformation is moving in the radial direction. Thus, the position where the highest strain is recorded always shifts in a positive radial direc- tion for successive expansions. For the first expansion, Nadai and Hsu-Forman predict nearly the same curve. It shows about 7.5 percent strain at 50 the hole edge and tapers to almost nothing by 2.5 mm from the hole edge. The experimental data at a distance from.the hole edge follows the curve quite well. However, the pat- tern of the experimental data near the hole edge is enough different from.the theoretical that the applicability of the theories is questionable. The strain at the first gage position is half a percent strain less than that at the second gage position. The strains at the second and third gage positions deviate considerably from.the theoretical strain there. Thus the shape of the distribution is dras— tically different than the theoretical distribution. The strains in this area are all large enough that the variation from.the theory cannot be attributed to measurement error. The theoretical curves fall within the standard deviation of the data. This is not very significant near the hole edge because the standard deviation there is very large. For the second expansion, the Hsu—Forman curve is somewhat higher than Nadai's until about 0.8 mm from the hole edge. Then it is lower by as much as 0.5% strain out to the elastic-plastic boundary. The data follows the Hsu-Forman curve much more closely than for the first expan- sion. (This comment neglects the strain at 0.2 mm from the hole edge.) The strains at 2.6 mm from the hole edge and further out are less than either theory predicts: The Hsu-Forman curve, though it fits better than Nadai's, is still about twice what experimental data is in this part. At about 3.4 mm.from.the hole edge and beyond, the data for this expansion is erratic. Some of the values are positive and a few are a bit larger than the values preceding them on a radial line from the hole edge. The compressive force that the remainder of the plate exerts on the expanding center portion might cause the plate to buckle in this area. This would result in the lower than predicted strains. At the second, third, and fourth gage positions the strains are somewhat larger than those predicted by the theories. The 51 strain at the 0-8 mm position for the second expansion is much closer to the theories than the one for the first expansion. The HsueForman curve lies within most of the standard deviations. Again, this is not terribly significant because the standard deviations are so large near the hole edge. The Nadai curve does not lie within the standard deviations. To summarize, neither theory is very adequate for predicting the residual strains. 4.1.2 Tangential Strains For the tangential strains, which are tensile, the standard deviations are much larger than the measurement uncertainty for strains near the hole. This is due more to the variation of the data from one radial line to the next than it is to averaging the front and back strains. These variations are not quite as large as with the radial strains. .At 1.9 mm from the hole edge and further out, the standard deviations are on the order of the uncertainty of the measurement. The tangential strain distribution is highest near the hole edge. These values drop off rapidly so that beyond a distance of 3 mm from the hole edge the strains are less than 0.1% strain. Referring to the Appendix for the strain values, one can see that two of the radial lines consisten- tently have smaller values at the hole edge, similar to the radial strains. However, these average out with the other values so that this phenomena does not appear on the graph. Also, observe that at distances of 1.3 mm from the hole- edge and further out, the measures strains are frequently negative. Generally, though not always, these average out with the positive values so that expected positive values appear on the graph. The occurrence of these values serves to significantly lower the average strain values in this region. The values here are also erratic in magnitude, with some closer to the hole being smaller than those further 52 away. These also tend to compensate for each other so that the average strains get increasingly smaller as one gets further away from the hole edge. Both of these occurrences are consistent with the idea that buckling occurs in this area. For both expansions the Nadai theory curve is some- what larger than the Hsu-Forman theory curve for the tan- gential strains. Both deviate considerably from the mea- sured strains. Almost none of the data points are near the curves. The curves are within the standard deviations of less than half of the points. Even the shapes of the curves are incorrect. To approximate the experimental data more accurately, the gradient of the curve needs to be much more sharp than it is out to 3 mm from the hole edge. So, one can conclude that for this particular specimen thickness and material, the two chosen theories do not predict the residual tangential strain well. 4.1.3 Elastic-Plastic Boundary Location A sample of the thickness change measurement profiles for this specimen appears in Figure 4.7 for one of two positions around the hole. The lower trace is the original profile, the middle one is after the first expansion, and the third one is after the second expansion. The points where the second and third traces begin to deviate from the original are marked. The elastic-plastic boundary locations are calculated from this information. The two positions are averaged for each expansion and the result plotted in Figure 4.8 with the theoretical predictions. The standard deviations do not appear on the graph because they are small enough that they are within the circle around the plotted point. The experimental results are significantly lower than either of the theories predicts though the Hsu-Forman is closest. This is consistent with the con- clusions drawn from.the strain plots; neither theory is very close but Hsu-Forman is closest. Figure 4.7 53 Typical profiles obtained on one radial line for the thickness change measurement used to locate the elastic-plastic boundary for the two different expansions of the 6.35 mm thick 7075-T6 aluminum specimen. (Scale sensitivity: 20.127 mm) 1) original profile of the plate 2) profile for the 0.191 mm radial expansion 3) profile for the 0.290 mm radial expansion. Figure 4.7 55 2.5 ' 15 :g _ U) . 2 2 . a: B 2.0 - O I . \ 00 2 .. o O 3‘: . Q n: 2! 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