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(v a “:2 -. ll. 2 I a Illllllllllllllllllllllllllllllllllllllllllllllllllllll -1‘-"¥5¢ 1293 00781 6733 This is to certify that the dissertation entitled Optimization of Geometry and Strain Gage Techniques in Fatigue Testing of Hourglass Profile Sheet Steel Specimens presented by Tadeusz Tomasz Stawiarski has been accepted towards fulfillment of the requirements for Mechanics Ph ' D ° degree in Q/s’fm Major professor 1114/ /d’ 9 MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771 LIBRARY Michigan State University PLACE IN RETURN BOX to remove We checkout from your record. TO AVOID FINES return on or betore due due. - 9 DATE DUE DATE DUE DATE DUE fin: _I — .— MSU le An Afflrmdlve AdlorVEquel Opportunity 1m oWM! OPTIMIZATION OF GEOHETEX'ARD STRAIN GAGE TECHNIQUES IN EKTIGUE TESTING OF HOURGLASS PROFILE SHEET STEEL SPECIMENS BY Tadeusz Tomasz Stawiarski A DISSERTATION Submitted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1989 9043574 ABSTRACT OPTIHIZATION OF GEOHETRY AND STRAIN GAGE TECHNIQUES IN FATIGUE TESTING OF HOURGIASS PROFILE SHEET STEEL SPECIMENS BY Tadeusz Tomasz Stawiarski Specimen buckling in the push-pull fatigue mode is the main problem encountered when testing thin sheet steels. The selection of an optimal specimen grip, specimen geometry, and strain measurement technique is the subject of numerical analysis. Analyzing the deformation of the specimen finite-element model, with ANSYS 4.3, in three modes of loading, elastic, plastic, and cyclic, leads to the conclusion that a thin short-gage-section specimen can produce data having acceptable accuracy. For the type of material and geometry considered, the gage section length equal to 3/16 of an inch and hourglass profile radius .7 inches was found to be optimal. ACKNOWLEDGMENTS First of all, I would like to express my appreciation to my advisor Dr. John F. Martin for his guidance, inspiration and valuable assistance through the whole period of my research and studies. I would like to thank all those who have helped and contributed to this work. This includes Dr. Kali Mukhergee who gave me a great support and advice. Also, I want to say "thank you" to Dr. Ivona Jasiiflc for her help, advice, and valuable discussions, to Dr. Nicholas J. Altiero, Dr. Larry Segerlind and Dr. Peter Schroeder'nmumoers of my guidance committee for a great support and encouragement. A separate note of appreciation goes to my best friemullDiane for all the patience, trust, understanding, and help in preparing the manuscript of this work. iii NOMENCLATURE axial load area of minimum cross section average axial stress total axial strain elastic component of axial strain plastic component of axial strain total transverse strain elastic transverse strain plastic transverse strain elastic Poisson's ratio plastic Poisson’s ratio stress components in plane stress components of the flow velocity thickness of the specimen maximum shear stress or radius of the Mohr's circle iv TABLE OF CONTENTS PAGE LIST OF TABLES ..................................................... vii LIST OF FIGURES .................................................... viii CHAPTER 1 INTRODUCTION AND BACKGROUND ........................... l 1.1. Introduction ........................................ l 1.2. Development of the Testing Method ................... 2 1.3. Load Control Methodology ............................ 7 1.4. Room Temperature Testing Technique .................. 9 1.5. Testing Technique Used in Elevated Temperature ...... 10 1.6. Definition of Strain Uniformity Error and Relative Error ............................................... 12 CHAPTER 2 TECHNICAL APPROACH IN RESEARCH AND ANALYSIS ........... 16 2.1. Research Objective .................................. 16 2.2. Technical Approach .................................. 19 2.2. Estimation of the Strain Uniformity Error by Analytical Methods .................................. 20 2.2. Numerical Approach to Strain Uniformity Error Estimation Analysis ................................. 26 2.3. Work Schedule in Preliminary Stage of the Research.. 31 CHAPTER 3 ANALYSIS IN THE ELASTIC RANGE ......................... 34 3.1. Technical Approach in the Analysis .................. 34 3.2. Stress and Strain Fields Analysis in the Elastic Range ............................................... 72 3.2. Stress Fields Analysis of the Straight-Profile Specimen ............................................ 72 3.2. Stress Fields Analysis of the Full-Hourglass Profile Specimen ................................... 86 CHAPTER 4 ANALYSIS IN THE LARGE SCALE PLASTIC DEFORMATION RANGE. 92 4.1. Technical Approach in the Analysis .................. 92 4.2. Large Scale Plastic Deformation of the Thin Sheet Steel Specimen ...................................... 96 4.2. Simulation Results of the Tensile Test with the Straight Profile Specimen ........................... 96 4.2. Simulation Results of the Tensile Test with the Full-Hourglass Profile Specimen .................... 121 4.3. Analysis of Stress and Strain Distributions in the Large Plastic Deformation Range .................... 129 4.3. Stress and Strain Fields Analysis in a Straight- Profile Short-Gage-Section Specimen ................ 129 4.3. Stress and Strain Fields Analysis in a Full- Hourglass Profile Specimen ......................... 142 CHAPTER 5 ANALYSIS OF CYCLIC PLASTICITY IN A THIN SHEET METAL SPECIMEN......... .................................... 154 5.1. Technical Approach in the Analysis ................. 154 5.2. Analytical Results - Cyclic Loading ................ 157 5.2. Simulation Results of Cyclic Loading Test Utilizing the Straight-Profile Short Gage-Section Specimen... 157 5.2. Simulation Results of the Cyclic Loading Test Utilizing the Full-Hourglass Profile Specimen ...... 174 5.3. Analysis of Stress and Strain Distribution in the Cyclic Loading ..................................... 186 5.3. Stress and Strain Fields Analysis in a Straight- Profile Short Gage-Section Specimen ................ 186 5.3. Stress and Strain Fields Analysis in a Full- Hourglass Profile Specimen ......................... 197 CHAPTER 6 EFFECT OF HOURGLASS PROFILE CURVATURE IN EXPERIMENTAL ERROR EXAMINATION .................................... 207 6.1. Technical Approach in the Analysis ................. 207 6.2. Simulation Results of Tests Conducted in the Elastic Range Using an Hourglass Profile Specimen 211 6.3. Simulation Results of Tests Conducted with an Hourglass Profile Specimen in Plastic Range and Cyclic Mode of Loading ............................. 216 6.4. Development of Plastic Deformation in an Hourglass- Profile Specimen ................................... 240 CHAPTER 7 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ............ 249 7.1. Summary.and Recommendations .......................... 249 7.2. Conclusions .......................................... 257 LIST OF REFERENCES ................................................. 259 vi LIST OF TABLES TABLE PAGE 3.1 Geometry of the finite-element models ....................... 38 3.2 Results of the analysis in the elastic range of loading ..... 48 4.1 Results of the analysis in the plastic range of loading ..... 98 5.1 Results of the analysis in the cyclic mode of loading ....... 158 6.1 Analysis of hourglass profile curvature effect in the » elastic range of loading; relative error in percentages ..... 212 6.2 Analysis of hourglass profile curvature effect in the plastic range and cyclic mode of loading; relative error in percentages .............................................. 213 vii FIGURE 1. l. l. l 2 3 .10 .11 .12 .13 .14 .15 .16 LIST OF FIGURES Physical arrangement for the fatigue test conducted with a thin sheet metal specimen [4] ................................ Specimens used in the test conducted in room temperature a) static tension test specimen b) fatigue specimen [3] ...... Selected geometry of specimens; design of a) static tension specimen b) fatigue specimen [4] ............................. model of the sheet metal specimen; a) referrential model of specimen in uniaxial loading; b) model of the specimen fixed in the grip .................................................. Model of the hourglass profile specimen fixed in the grip used in the study of hourglass profile curvature effects ..... A quadrant of the specimen finite-element model .............. a) Nodal point numbering; b) Element mesh numbering for the straight-profile specimen finite-element idealization ........ Finite element mesh for the full-hourglass profile specimen idealization ................................................. Loading program for the static tensile test simulation conducted in the elastic range ............................... Measurement of the axial strain on the center specimen plane surface; laser based technique (ISG) ......................... Measurement of the average transverse strain conducted a) across the minimum specimen width; b) across the specimen thickness .................................................... Measurement of the average axial strain conducted throughout the entire length of the gage section ........................ Measurement of the average axial strain conducted through the 2/3 of gage section length ............................... Stress-strain response in the simulated static tensile test . conducted in the elastic range according to the laser based method 1 (ISG), and specimen model A ......................... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the laser based method 1 (ISG), and specimen model B ......................... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the laser based method 1 (ISG), and specimen model C ......................... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the laser based method 1 (ISG), and specimen model D ......................... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 2 and the specimen model D .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 3 and the specimen model D .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 4 and the specimen model D .......... Stress-strain response in the simulated static tensile test viii PAGE 5 5 13 21 22 35 37 39 41 42 43 45 46 49 50 51 52 53 54 55 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 conducted in the elastic range according to the axial measurement based method 5 and the specimen model D .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 4 and the specimen model A .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 4 and the specimen model B .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 4 and the specimen model C .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 5 and the specimen model A .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 5 and the specimen model B .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial measurement based method 5 and the specimen model C .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 2 and the specimen model A .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 2 and the specimen model B .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 2 and the specimen model C .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 3 and the specimen model A .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 3 and the specimen model B .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 3 and the specimen model C .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 2 and the specimen model E .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the transverse measurement based method 3 and the specimen model E .......... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial strain measurement method 1 and specimen model E .................... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial strain measurement method 4 and specimen model E .................... Stress-strain response in the simulated static tensile test conducted in the elastic range according to the axial strain measurement method 5 and specimen model E .................... Contour plot of the axial stress in step 7 of the loading the ix 56 58 59 60 61 62 63 65 66 67 68 69 70 71 73 74 75 76 .35 .36 .37 .38 .39 .40 .41 .42 .43 NH .10 .ll .12 straight-profile 3/16 in. gage-section specimen in the elastic range ............................................... 78 Contour plot of the axial stress in step 11 of loading the specimen of .96 in. gage section ............................. 79 Contour plot of the transverse stress in step 7 of the loading the straight-profile 3/16 in. gage-section specimen in the elastic range ......................................... 81 Contour plot of the shear stress in step 7 of the loading the straight-profile 3/16 in. gage-section specimen in the elastic range ................................................ 82 Contour plot of the equivalent stress in step 7 of the loading the straight-profile 3/16 in. gage-section specimen in the elastic range ......................................... 83 Plastic zone in model A of specimen in two different steps of loading a) 14,000 psi; b) 20,000 psi ......................... 85 Contour plot of the equivalent stress in step 7 of loading the full-hourglass profile specimen .......................... 87 Contour plot of the axial stress in step 7 of loading the full-hourglass profile specimen .............................. 88 Contour plot of the transverse stress in step 7 of loading the full—hourglass profile specimen .......................... 89 Contour plot of the shear strain in step 7 of loading the full-hourglass profile specimen .............................. 90 Schematic of analog computer circuit ......................... 93 Loading program for the tensile test simulation conducted in the plastic range ............................................ 97 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the laser based method 1 (ISG) and the specimen model A ...................... 99 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the laser based method 1 (ISG) and the specimen model B ...................... 100 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the laser based method 1 (ISG) and the specimen model C ...................... 101 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the laser based method 1 (ISG) and the specimen model D ...................... 103 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 2 and the specimen model D... 104 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 3 and the specimen model D... 105 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 4 and the specimen model D .......... 106 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 5 and the specimen model D... 107 Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 4 and the specimen model A .......... 108 Stress-strain response in the simulated static tensile test .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 conducted in the plastic range according to the axial strain measurement based method 4 and the specimen model B .......... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 4 and the specimen model C .......... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 5 and the specimen model A... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 5 and the specimen model B... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 5 and the specimen model C... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 2 and the specimen model A... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 2 and the specimen model B... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 2 and the specimen model C... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 3 and the specimen model A... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 3 and the specimen model B... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 3 and the specimen model B... Four stages of plastic zone development at various steps of loading: a) 20,000 psi; b) 24,000 psi; c) 28,000 psi; d) 32,000 psi ................................................ Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 1 and the specimen model E .......... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 4 and the specimen model E .......... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the axial strain measurement based method 5 and the specimen model E... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 2 and the specimen model E... Stress-strain response in the simulated static tensile test conducted in the plastic range according to the transverse strain measurement based method 3 and the specimen model E... Contour plot of the axial stress in step 21 of loading the straight-profile 3/16 in. gage section specimen .............. Contour plot of the equivalent stress in step 21 of loading the straight-profile 3/16 in. gage section specimen .......... 109 110 111 112 113 114 115 116 118 119 120 122 123 124 125 126 127 130 132 .31 .32 .33 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 Contour plot of the axial strain in step 21 of loading the straight-profile 3/16 in. gage section specimen .............. 133 Contour plot of the transverse strain in step 21 of loading the straight-profile 3/16 in. gage section specimen .......... 134 Contour plot of the transverse strain in step 21 of loading the straight-profile 3/16 in. gage section specimen .......... 135 Path plot of the total axial strain along the specimen central axis ................................................ 137 Path plot of the total axial strain across the specimen width at the central section ....................................... 138 Path plot of the total axial strain across the specimen width at the line of the grip ...................................... 139 Path plot of the total transverse strain along the specimen width at the central section; strain in the direction of the specimen thickness 6T2 ....................................... 140 Path plot of the total transverse strain across the specimen width at the central section; total strain in the direction of the specimen width ETx .................................... 141 Contour plot of the axial stress in step 21 of loading the full-hourglass profile specimen .............................. 143 Contour plot of the equivalent stress in step 21 of loading the full-hourglass profile specimen .......................... 145 Contour plot of the axial strain in step 21 of loading the full-hourglass profile specimen .............................. 146 Path plot of the total axial strain across the specimen width at the central section ....................................... 147 Contour plot of the transverse strain in step 21 of loading the full-hourglass profile specimen .......................... 149 Path plot of the total transverse strain across the specimen width at the central section; total strain in the direction of the specimen width eTx .................................... 150 Path plot of the total axial strain along the specimen central axis ......................................................... 151 Path plot of the total axial strain across the specimen width at the line of the grip ...................................... 153 Loading program for the cyclic loading test simulation conducted in the push-pull mode .............................. 155 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG) and the specimen model A ...................... 160 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG) and the specimen model B ...................... 161 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG) and the specimen model C ...................... 162 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG) and the specimen model D ...................... 163 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 2 and the specimen model D... 164 5. 5. 5. 7 8 9 .10 .11 .12 .13 .14 .15 .16 .17 .18. .19 .20 .21 .22 .23 .24 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 3 and the specimen model D... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 4 and the specimen model D .......... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 5 and the specimen model D... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 4 and the specimen model A... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 4 and the specimen model B... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 4 and the specimen model C... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 5 and the specimen model A... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 5 and the specimen model B... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 5 and the specimen model C... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 2 and the specimen model B... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 2 and the specimen model C... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 3 and the specimen model B... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 3 and the specimen model C... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 2 and the specimen model A... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 3 and the specimen model A... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 4 and the specimen model B... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the axial strain measurement based method 5 and the specimen model E... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG) and the specimen model E ...................... xiii 165 166 167 168 169 170 171 172 173 175 176 177 178 179 180 182 183 184 O‘C‘ LAN .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 .40 .41 .42 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 2 and the specimen model E... 185 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the transverse strain measurement based method 3 and the specimen model E... 187 Contour plot of the axial stress in step 7 of cyclic loading the straight-profile 3/16 in. gage section specimen .......... 188 Contour plot of the axial stress in step 21 of cyclic loading the straight-profile 3/16 in. gage section specimen .......... 189 Contour plot of the equivalent stress in step 7 of the cyclic loading the straight-profile 3/16 in. gage-section specimen.. 191 Contour plot of the equivalent stress in step 21 of the cyclic loading the straight-profile 3/16 in. gage-section specimen.. 192 Contour plot of the axial stress in step 8 of the cyclic loading the straight-profile 3/16 in. gage-section specimen.. 193 Contour plot of the axial stress in step 15 of the cyclic loading the straight-profile 3/16 in. gage section specimen.. 194 Contour plot of the axial strain in step 7 of the cyclic loading the straight-profile 3/16 in. gage section specimen.. 195 Contour plot of the axial strain in step 21 of the cyclic loading the straight-profile 3/16 in. gage section specimen.. 196 Contour plot of the transverse strain in step 7 of the cyclic loading the straight-profile 3/16 in. gage-section specimen.. 199 Contour plot of the transverse strain in step 21 of the cyclic loading the straight-profile 3/16 in. gage-section specimen ..................................................... 200 Contour plot of the axial stress in step 7 of loading the full-hourglass profile specimen .............................. 201 Contour plot of the axial stress in step 21 of loading the full-hourglass profile specimen .............................. 202 Contour plot of the axial strain in step 7 of loading the full-hourglass profile specimen .............................. 203 Contour plot of the axial strain in step 21 of loading the full-hourglass profile specimen .............................. 204 Contour plot of the transverse strain in step 7 of loading the full-hourglass profile specimen .......................... 205 Contour plot of the transverse strain in step 21 of loading the full-hourglass profile specimen .......................... 206 Hourglass profile sheet metal short gage section specimen fixed in the grip ............................................ 208 A quadrant of the hourglass profile specimen model ........... 210 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 4; specimen profile radius equal to 1 in.. 218 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 4; specimen profile radius equal to .5 in. 219 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 4; specimen profile radius equal to .75 in 220 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 4; specimen profile radius equal to .25 in 221 6. 6. 6. 7 8 9 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 5; specimen profile radius equal to .25 in Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 5; specimen profile radius equal to .75 in Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 5; specimen profile radius equal to .5 in. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 5; specimen profile radius equal to .25 in Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG); specimen profile radius equal to 1 in ........ Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG); specimen profile radius equal to .75 inches.. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG); specimen profile radius equal to .5 inches... Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the laser based method 1 (ISG); specimen profile radius equal to .25 inches.. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 2; specimen profile radius equal to l in.. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 2; specimen profile radius equal to .75 in Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 2; specimen profile radius equal to .5 in. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 2; specimen profile radius equal to .25 in Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 3; specimen profile radius equal to 1 in.. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 3; specimen profile radius equal to .75 in Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 3; specimen profile radius equal to .5 in. Stress-strain response in the simulated cyclic loading test conducted in the push-pull mode according to the strain gage measurement method 3; specimen profile radius equal to .25 in Influence of the profile curvature on the plastic zone development a) radius 1 in. b) radius .25 in ................. Contour plot of the equivalent stress in the specimen of 3/16 in. gage-section and the profile radius equal to .4 in ....... Contour plot of the equivalent stress in the specimen of 3/16 in. gage-section and the profile radius equal to .3 in ....... 222 223 224 225 227 228 229 230 231 233 234 235 236 237 238 239 242 245 246 6. 6 26 .27 Contour plot of the equivalent stress in the specimen of 3/16 in. gage-section and the profile radius equal to .25 in ...... 247 Contour plot of the equivalent stress in the specimen of 5/16 in. gage-section and the profile radius equal to .5 in ....... 248 CHAPTERI INTRODUCTION AND BACKGROUND 1.1. Introduction In the last decade there has been a significant increase in the production and design of structural and mechanical components having smaller cross-sectional areas and lower weight. Therefore, thin sheet steel parts are more frequently used as load carrying members. This situation.creates a demand for new testing techniques and modification of traditional methods. During the manufacturing,processes of thin Sheet steel products, the mechanical properties will change in accordance to the degree of applied plastic deformation. As a consequence, the material properties are very much affected by the conditions of the manufacturing process. In particular, material anisotropy and fatigue properties can alter radically during the manufacturing process. For this reason accurate static and fatigue property evaluations obtained from thin longitudinal specimens are essential. Data acquired from dependable and practical testing methods are necessary to better understand long term mechanical behavior of thin structural components, especially subjected to cyclic loading at temperatures within the creep range. Presently, there is a significant 1 demand for reliable, low cycle fatigue data for use in the design and manufacture of thin structural components. 1.2. Development of the Testing Method A.survey of technical literature indicates the existence of a considerable amount of fatigue data which has been successfully generated on thin sheet metal in room temperature. For plate thicknesses greater than 0.055 inches, standard uniaxial specimens with straight gage sections having approximately 0.3 inches length are most extensively employed. These specimens, however, are not adequate for the test conditions involving large strain ranges over 0.0075, small thickness below 0.10 inches (2.54 mm), and elevated temperatures up to 1300°F (700°C). Buckling, of course, is the main problem encountered when test conditions involve compressive loads. On the other hand, inherent disadvantages exist when conducting fatigue tests which are restricted to tensile loadings only. This introduces the mean stress effects, and renders invalid data. Thin specimens are more apt to buckling. Their resistance to buckling further drops during the testing conducted.imieelevated temperatures. A technique involving buckling guides has been tested. However, it has been discarded because of detrimental effects. The testing methods reported by Marsh et a1 [1], who initially machined their specimens with cylindrical gage sections from steel sheets of 1015 (UNS G10150) and varying in thicknesSes from 0.10 in. to 0.20 in. (2.54 mm to 5.08 mm) were successful. These methods failed, however, when testing 1020 sheet steel, which is 30% thinner than those studied by Marsh et a1. Elevated temperature fatigue data are frequently generated on hourglass shaped cylindrical specimens. Research has proven that this particular hourglass profile configuration inhibits buckling, and in addition has many other advantages which far outweigh existing disadvantages. A report published by Slot at al [2] describes a testing method using fatigue specimens possessing a unique shape. This technique allows strain amplitudes as high as 0.03 and strain rates from 1/100 to 1/100 000 per second. Other parameters include hold times from 0 to 3 hours, and temperatures ranging from 77°F to 1800°F (25°C to 1000°C). The technique involves experimental procedures which are executed on custom designed servo-controlled equipment permitting push-pull programming. To acquire the above described.performance and secure reliable data, the technique requires implementation of cylindrical specimens with an hourglass shaped profile. The geometry of cylindrical specimens with an hourglass profile is preferential to the use of straight-profile specimens for the following advantages: 1. Strain can be measured and controlled at the cross section where fracture is most likely to occur. 2. Relatively large compressive strain can successfully be produced in the material without buckling the specimen. 3. Localized heating of the specimen is readily accomplished at the minimum diameter cross-section. 4” A diametral displacement sensor can be easily attached matjm specimen for accurate measurement read out. 5. Internal and external flaws are less likely to disturb the outcome of the test since only a slightly localized portion of the specimen was exposed to maximum strain. The following disadvantages of the hourglass specimen exist: 1. Strain measurements are more precise on straight profile specimens because transverse changes are much smaller than axial measurements. 2. .Axial strain, which is the test parameter of primary interest, is not measured directly but rather indirectly via both transverse strain and load measurements. For this reason, an additional conversion circuit is needed in order to program the electronic control system. .After most other fatigue testing methods failed, it became apparent, that in order to solve the problem of buckling the thin sheet metal during the fatigue testing, one viable technique remained. It is similar to the one described by Slot et a1, and has been modified for use on thin sheet metal. In order to minimize the possibility of buckling in push-pull loading nmuha, a sheet metal specimen with an hourglass shaped profile should be utilized and a special custom made fixture with the induction coil unit allowing localized specimen heating. The hourglass profile specimens have been used successfully in an attempt to prevent buckling at strains up to 0.02 at room temperature. This technique has been described by Martin [3]. Figure 1.1 shows the physical arrangement for such a test [4]. Figure 1.2 shows two types of specimens used by Martin [3]. Both specimens are machined from sheets having a nominal thickness of 0.07 in. (1.78 mm). The material used is a spheroid shaped annealed cold rolled 1020 steel. Fatigue tests have been initially attempted on specimens with straight rectangular gage sections (Figure 1.2 a). lfiuaxvidth of the SPEC'MEN GRIP weoce TRANSVERSE EXTENSOMETER Figure 1.1. Physical arrangement for the fatigue test conducted with a thin sheet metal specimen [4] 0.0|" Deep Cuts Ml“ bl- I) I ‘76 'is Pigure1.2. Specimens used in the test conducted in roan temperature a) static tension test specimen b) fatigue specimen [3] straight gage-section varies from 0.07 inches to 0.4 inches (1 78 mm to 12.7 mm). None of these configurations consistently prevents buckling for reversed strains limits of $0.005. Consequently a specimen configuration of the type shown in Figure 1.2 (b) and a loading fixture that would minimize the possibility of buckling have been used. A load cell mounted in series with the specimen, and the displacement sensor positioned at the minimum cross section in direction of specimen thickness furnish electrical analogs of the measured stress and transverse strain. The thickness displacement sensor is in contact with the specimen on both side surfaces providing the point contact on one side and flat surface contact on the other. The total axial strain eTy is than computed from the average stress a and transverse strain 6 as: av Ttr (Ty -f(oav,eTtr); (l 1) The average axial stress is obtained by dividing the load by minimum cross-sectional area. Then, the computed total axial strain is used to control the entire test. The load cell continuously provides the analog signal of the applied load. aav - F/A; (1.2) eey - Gav/E - F/AE; (1.3) - e + e ; (1.4) u - - e / e ; (1.5) - - ° 1.6 "p Eptr/ epy’ ( ) cetr — - eey we - - aavue/ E; (1.7) 6ptr - 6Ttr ' eetra eTtr ' ( ' aavue/E) . 6Ttr + aavve/E; (1'8) so that: In - 6 ; 1.9 py ptr / "p ( ) then: aav 1 aav ye e - - ( e + ); (1.10) Ty E -;— Ttr _——E____ Gav 2 eTy - E - 26Ttr - ; (1.11) The computed axial strain is used as the controlling parameter for all CEStS . 1.3. Load Control Methodology Fatigue testing load control methods are discussed in the research described by Slot et a1 [2] . Test parameters which can be programmed are: stress, transverse strain, axial strain, plastic strain and actuator displacement. The choice of loading the specimen by cyclic load at a constant rate between limits on the displacement of either the mechanical or ‘hydraulic machine results in a technique that does not assure a constant strain amplitude and strain rate. In addition, with this unreliable control system, any distribution of elastic and plastic strain that occurs due to geometry, cyclic strain hardening, softening, relaxation during hold periods or changes in the mean strain will result in a deviation from the constant strain amplitude. This problem is encountered once again, and creates a difficulty in maintaining a constant stress amplitude in the stress controlled tests. In particular, when the slightest permanent deformation occurs, the return of the actuator to the initial position is not equivalent to specimen unloading. Even though the actuator moves at a constant rate, the strain rate in the specimen varies when the specimen deforms plastically. The modeling of periodic variations in the strain is very difficult because periodic variation of the actuator motion will not produce the same variation of strain. A considerable improvement is gained if the strain measuring device is uwunted on the gage section of the specimen. Then, a choice can be made by the researcher to control.euufl1 loading reversal determined by limits preset on the output of the strain measuring device. This technique will maintain the amplitude of strain averaged over the length of the gage section. The intracycle variation of strain will not be controlled, because a constant velocity of the actuator does not yield a constant rate due to plastic deformation. The most efficient and precise method involves the use of servo- controlled testing machines operated in a closed loop control. It requires so-called feedback signal which is an analog representative of the test parameter (stress in stress controlled test or strain in strain controlled test). This signal is continuously compared to a programmed "demand signal" to provide an instantaneous correction to applied load, so that a controlled parameter stays in compliance with the program of load and deformation for the test. The control is continuous, so that both amplitude and rate can be programmed. When the fatigue is to be investigated in the low cycle range, the plastic deformation in each cycle complicates the relationship lnetweerl stress and strain. 1.4. Room Temperature Testing Technique For reasons stated above, the axial strain was selected as a control parameter in the testing procedure described by Slot et a1 [2] and implemented also in fatigue testing conducted by Martin on the thin sheet metal in room temperature. During each test, the voltage analog of axial strain given by equation (1.11) was continuously calculated by ) analog computer. Tfiuaload.(P) and negative transverse strain ( 6tr were the inputs to the circuit. Careful wiring of the transverse strain gage would assure that expansion of the contact points would produce a negative voltage output, and upon contraction, would produce a positive output. This would enable direct transverse strain control, if desired“ The output was recorded and used as the control parameter in the test. A testing program was successfully completed without buckling failures by using thin sheet metal hourglass profile specimens. The data obtained by using both types of specimen were very similar when 10 obtained in static tensile loading. This produced confidence in the testing technique utilizing the hourglass profile specimen. As a result of the testing, it was learned that the acquired data aligned very closely to typical patterns obtained for other mild steels. Specimens of both types produced nearly identical initial stress- strain curves. The major difference which occurred was found in dissimilarity of finite fracture properties. lfixrthermore, the hourglass specimen exhibited stronger characteristics with higher ultimate strength and lower ductility. This difference was attributed to uneven stress distribution produced by the curvature over the test section” and.departing from conventional uniaxial stress and strain conditions due to the short gage-section of an hourglass specimen (Saint Venant's Law). Computation of engineering stress as input into the conversion circuit is considerably easier than working with the true stress and true straixL ‘The differences are negligible as long as the strain is small, say less than 0.05 1.5. Testing Technique Used in Elevated Temperature There are no apparent reasons why the above descrdluui technique, which has been successful in room temperature, should not work in the temperature ranges from 900°F to 1,300°F. An experimental sethiq) for such a testing has been prepared. The conditions for testing were designed to obtained standard strain-life data for thin components of 0.07 inches (1.8 mm) thickness in elevated temperatures up to 1,300°F 11 Appropriate test conditions and required apparatus are described by Martin and Waterbury [4]. A detailed description of the applied technique and test results have been reported by Martin [5]. An aluminum clad sheet steel and a stainless steel were the materials tested. The fixture was positioned in such a way so as to allow a 3/16 inches portion of the gage section to remain exposed for heating purposes and free from the grip. When designing the grip, the intention was specifically to allow the grip itself to be heated by the induction coil. Hence, it has been considered that it could have to withstand the elevated temperatures. The test parameters measured were the transverse strain and the axial load. These parameters were then converted to axial strain by the conversion circuit in the beforehand described way. The axial strain was then used as the control parameter. To insure that the exposed sides of the clad sheet steel specimens would not oxidize, which would become the initiation of the crack growth, these portions of the specimen were placed in an inner argon atmosphere. The strain gage technique applied across the thickness of the specimens was successful in room temperature, however in elevated temperatures an alternative method was needed to prevent disturbances introduced by the noises from the induction coil. To assure the stronger signal, it was decided to apply the strain gage which measured the transverse strain across the minimum width of the specimen. A third alternative method using a laser based computer-controlled interferometric extensometer, was also considered. This device is usually referred to as an "interferometric strain gage" (ISG) [6][7][8]. This precise technique measures strain over a very small 12 gage length, exhibited on minute gage length usually 100 pm. The actual gage consists of shallow reflecting indentations on the surface of the specimen. Because of this, the strain measurement is considered to be a non-contacting technique. A Vicker's hardness tester is used, which makes pyramidally shaped indentations into the material, with a diamond impress. The indentations are illuminated with the laser beam, and this produces two sets of interference fringe patterns. Strain can then be measured in real time, by monitoring the visible patterns with a sensing device connected to a computer. The ISC technique was used in the project, to be able to evaluate constants needed for the conversion circuit and also to compare data obtained from axial specimens when different measuring techniques were used. Figure 1.3 illustrates the specimen designs which were selected for (a) static tension testing and (b) fatigue testing. All tests were successful. Data generated on fatigue specimens had been derived from machined clad sheet steel. 1.6. Definition of Strain Uniformity Error and Relative Error As an independent part of the project, an analysis was conducted to estimate the error produced by departure from uniaxial stress conditions, to perform specimen design optimization, asxmflJ.as to evaluate the accuracy of results obtained by various strain gage techniques. It must be emphasized, that short gage sections (approx. 3/16 inches) are required to assure the prevention of buckling during fatigue testing in the push-pull mode of loading, especially when thin 13 8’ 7:»: 3.700 >1 —\ /__ 43 fl .75 R b) a 3.700 " 1 -——I.503 —-1\y|:'/-33° *H-‘INS n .750 R Figure 1.3. Selected geanetry of specimens: design of a) static tension specimen; b) fatigue specimen [4] 14 sheet specimens are used. Direct axial strain measurements are impractical or extremely difficult in conditions introduced by such a short gage-section. The tested gage section is directly affected by inherent fixture conditions. This results in a departure from uniaxial stress conditions. The fatigue specimen has an hourglass shaped profile which additionally distorts the stress field. In these conditions, estimation of readings accuracy is essential for reliable test data. Uniaxial stress conditions are optimal for testing, but impossible to assure especially when thin sheet metal is tested at elevated temperatures. Uneven stress and strain fields result inaniLumven specimen deformation. Consequently, different readings are produced depending on the type of strain gage technique which is applied, and the selected direction of strain measurements. Hence, the applied strain measurement method directly affects testing accuracy. Readings which are obtained from this type of testing, differ from the true material stress-strain response taken in uniaxial stress conditions. It is considered that this difference is equal to a specific error which arises. It may be a result of specimen fixing conditions, profile geometry, the type of strain gage techniques which are used, as well as the range and mode of loading. Because this error is produced directly by uneven strain distribution, for the purpose of this research, the above mentioned error is referred to as strain uniformity error. The Initio of strain uniformity error to the true material stress- strain response is defined as a relative error and expressed in percentage. 15 These definitions emphasize that departure from uniform strain distribution results in readings which are different than true uniaxial stress-strain response. Using a relative error is more meaningful for use in comparison studies than using strain uniformity error, especially in wide range of loading. The positive \ufihie of the relative error implies that the particular measurement technique overestimates strain readings, whereas the negative value of this error implies that the readings are underestimated. CHAPTER 2 TECHNICAL.APPROACH IN RESEARCH AND ANALYSIS 2.1. Research Objective Difficulty has long been recognized in producing a perfectly uniform, uniaxial stress state for the measurement of mechanical properties. It is difficult to have a truly uniaxial test specimen. Some irregularity in the stress state is inevitable in the transitional region. Saint Venant’s principle dictates the traditional use of the long- gage-length tensile specimen in order to reduce the effects of discontinuity in the cross section, and to compensate for fixture cunn- ditions in the grip. The area of the specimen removed from the transition radii or the grip section, in terms of distance, equal ap- proximately one width of the specimen, has over the entire cross section a fairly uniform axial stress. According to the Annual Book of Standards [9], the gage length of the rectangular tension test specimen should be four times larger than the specimen width to assure truly uniaxial stress conditions. When various conditions are imposed, such as elevated temperature, cyclic loading, large strains, small thicknesses of sheet metal, then 16 17 test-specimen geometries should be modified. There are some par- ticularly attractive features of using the hourglass profile specimen (also known as the zero-gage-length specimen) in fatigue testing of thin sheet metal. These include resistance to buckling when loaded in compression, ease of heating, since only a small part of the specimen needs to be at uniform temperature, and simplified data acquisition by using transverse measurements for calculations of both transverse as well as axial strains. For a given specimen width, the gage section of the hourglass profile specimen is much shorter than the gage section of the static tensile test specimen. Axial strain is assumed to be computed from the transverse strain and axial load (or average axial stress). There is one major concern in using the hourglass profile thin sheet metal specimen and can be found in the uneven stress distribution due to departure from conditions of Saint Venant's Law. Also, the effect of an uneven stress distribution due to the hourglass shape of the specimen profile must be examined. As a greater number of high- temperature mechanical properties data is generated and compared, the question arises as to what causes the apparent differences in properties reported for the same material, yet extracted from different specimens. Stress and strain distributions in hourglass profile fatigue specimens with differing geometries and gage section lengths must be critically examined. A determination of internal stress and strain distributions creates the possibility of assessing accuracy of mechani- cal properties exhibited by tests on a particular specimen. Extensometry correction factors can be calculated and applied in test- ing procedures for selected cases in order to optimize accuracy of 18 mechanical properties indicated by tests on a given specimen. The ac- quired data helps to determine the choice of profile, gage secthni length, and extensometry techniques before fatigue testing of the thin sheet steel specimens is initiated. The objective of this portion of the research is to develop a reliable and reproductable method for analyzing strain uniformity error during the fatigue test of an hourglass profile sheet steel specimen. Particular interest should be focused on the length of the short gage section and the various methods of fixing the specimen in the grip due to the influence which positioning has on strain uniformity error cal- culations. Furthermore, plastic zone initiation and development is examined in the particular specimen undergoing cyclic loading. Preliminary testing indicates a direct correlation between the specimen deformation under cyclic loading and the geometry of position- ing the specimen to the test fixture. The object of this analysis is to determine how these dependent variables interact. Traditionally, the axial strain value has been calculated by dividing the amount of change in extensometer reading by the initial gage length. In view of the uneven stress distribution within the gage sectixni, an uneven strain distribution is also exhibited. Further in- vestigation into this pattern is needed. The deviation from the ideal strain distribution along the axial direction is significant. For this reason much attentixni is given to the influence of axial gage length on the strain uniformity error. The objective of this work is to conduct the above described analysis in the elastic strain range, large plastic strain range, and cyclic mode of loading. 19 Extensometry techniques restrict measurements to the surface of the specimen. fhu>techniques of transverse strain measurement have been applied to fatigue testing at both room, and elevated tempera- tures. First, strain gage measurements in the direction of specimen width, and secondly, measurement across specimen thickness. Accuracy of these techniques is evaluated by estimation of the strain uniformity error involved. These results can be compared with the results of similar analysis involving strain gage techniques of measurement in the axial direction. It is an objective of this work to consider different strain gage lengths when analyzing strain measurement in the axial direction, and to compare the results of this analysis with results obtained from the analysis of transverse strain measurements. The final objective of this work is to analyze the effect of the hourglass profile curvature and its influence on the strain uniformity error. It is also necessary to examine the effects of the hourglass profile curvature upon the development of cyclic plasticity. 20 2.2. Technical Approach Complex conditions of fixing the specimen in a grip are ap- proximated for an analysis using the simple idea of a thin plate element in a plane stress state. Tensile and compressive loading is accomplished on the straight line across the width of the specimen, (Figure 2.1). This model has a more conservative fixing condition than the specimen in the real grip. The calculated absolute value of strain uniformity error, when using the model, is this way not smaller than absolute value of the real error. Also, the amount of plasticity produced is not less than in the specimen under testing. It is an intention to analyze the geometry and different tech- niques of strain gage measurement in such a way that the appropriate choice of the gage section length and specimen profile is possible. In the optimization process buckling of the specimen in compression is considered to be a major limiting factor. To avoid this obstacle, at- tention is given to keep the gage section length minimal. When analyzing the relative error a referential model is repre- sented by the thin element of the same geometry but under uniform loading, (Figure 2.1). . Estimation of the strain uniformity error produced by time intro- duction of the hourglass profile specimen for testing,a thin sheet steel in room and elevated temperatures has developed in two direc- tions. A literature search was undertaken to find a theoretical solution to the developed model. Secondly, a thorough study was con- ducted on a general purpose analytical program ANSYS, version 4.3, for a finite element analysis. 21 11 1 1111. Y Fixing Ax=0 _X b) 1% .{k / / /% I Figure 2.1. Nbdel of the sheet metal specimen; a) referential nodal of specimen in uniaxial loading; b) model of the specimen fixed in the grip 22 Y Fixing [x 0.. / / / /z I Figure 2.2. Model of the hourglass profile specimen fixed in the grip used in the study of hourglass profile curvature effects. 23 2.2.1. Estimation of the Strain Uniformity Error by Analytical Methods In the search for a theoretical solution, certain assumptions have been made. The first assumption, being that the gage section of a flat specimen was short enough to prevent buckling. Secondly, it had either a straight or an hourglass profile as shown in Figure 2.2. In addi- tion, the ends were assumed to be clamped with the rigid grip and loaded by the uniform displacement or pressure. This loading assured zero displacement in the transverse direction and a coupled displace- ment 1J1 the axial direction of all points on the line of loading (line of contact with the grip). At first glance, this problem appears to have a simple theoretical solution. This is very misleading. The literature review shows only solutions to the purely elastic problem. These solutions consider a material which has only linear elastic stress-strain characteristic and they involve the geometrical singularity at the corners of Una specimen. The attempt to find the solution for this particular elastic problem was initiated by Knein in 1926 [10] at University of Aachen. In his approach, Knein assumed that the material remains elastic in all ranges of loading. Consequently, he concentrated on analysis of geometric singularity using the Airy stress function for the rectan- gular wedge. Later, Williams considered a general angle wedge [11]. Ruth (1953), Williams (1956), and England (1971) [12],[13],[l4] subsequently rephrased the problem for analysis in terms of complex variables. Most recently, the contact problem for an elastic rectangle has been analyzed by Prasad et al in 1970's [15],[l6], and [17]. All these 24 works, however emphasize how difficult the problenligs‘to solve while using the usual methods of elasticity only. It must be emphasized that the singular elasticity solutions presented in these works cannot accurately describe stress and strain distributions in real materials, which sustain limited elastic deforma- tion, and deform plastically after exceeding elastic limit. Nevertheless, the singular elasticity solutions can furnish useful information on the location of the stress maximum. Plastic deformation in the rectangular specimen is initiated at the corner points of the rectangle. This indicates that in a straight gage-section specimen, material having immediate contact with the grip is first to yield and subsequently plastic deformation propagates to the other regions. Results of finite-element analysis prove this thesis. Development of ‘plastic deformation in this fashion is a disadvantegous feature of the rectangular specimen. The obtained graphs, which are presented in the following chapters, describe how the plastic deformation develops in this type of strain gage section. In conclusion, the results of infinitesimal linear elastic theory indicate a predominant maximum stress at the grip site. From an ex- perimental stand point, this is an undesirable condition. The above can be considered as an another reason for using an hour glass-profile specimen, which concentrates deformation in the minimum cross section of its central part, while testing the sheet steel materials. Next, the attention has been directed towards the methods used.ir1 the theory'of plasticity in search for, at least, an approximate solu- tion of plastic flow and stress and strain distribution in the plastic 25 range of loading. The most common method used is the method of charac- teristics or otherwise defined as the method of slip lines. ILt considers the state of plane plastic strain which yields a zero transverse strain where this control parameter is actually measured. Hence, this method cannot be applied. The plane stress solution in the theory of plasticity considers a rigid-perfectly plastic model of the solid. Until now only axisymmetric problems are known to have been considered and have included strain hardening effects. The solution of the plane stress problem utilizing the von Mises yield criterion, ac- cording to Szczepinski [l8], considers three stress components ax, 0y and fxy at every point of the body; two components of the flow velocity v and v ; and a thickness h. x Y A set of six equations is given by the equations of equilibrium: 6 a ___ (ha ) + ___ (hr ) - 0; (2.1) 6x x 6y xy 6 6 ___ (ho ) + ___ (hr ) - 0; (2.2) . ay y ax XV Condition of incompressibility: _EE + _i_ ( hvx) + _f_ (hvy) - o; (2.3) at 8x 6y Selected yield criterion and the associated plastic flow rule: a) For von Mises yield criterion: 2 2 2 a - a a + a + 31 - 3k ; (2.4) x x y y xy 8v 6v 6v 8v 1 X _ 1 Y - 1 X + y ; (2.5) 20 - a 6x 20 - 0 By 67 By dx x y y x xy b) For Tresca yield criterion: 26 2 4 2 2 (ox- 0y) + fxy - 4k , (2.6) 6v av av 6v 1 x _ l y _ 1 x + y ; (2.7) a - a 6x a - 0 8y 4r . 6y 6x x y y x xy The state of stress is coupled with the state of strain, in this set, because in the equations of equilibrium appears an h, which is a function of time. The solution of this problem is not known yet. Sokolowski [19] has solved its simplified version assuming h - const. everywhere and through the entire process. Hill [20] solved the plane stress plastic problem assuming that thickness is known everywhere, h - h (x,y), and is constant throughout the entire process. In the above mentioned solutions the condition of incompressibility is not satisfied and the thickness is considered to be constant during the entire deformation. This is the main reason why these solutions cannot be used in the analysis of the considered problem. For the above stated reasons, the main attention in the discussed research has concentrated on the numerical analysis of the problem. It has been assumed that the determination of the internal stress and strain distributions would allow the possibility of assessing the ac- curacy of the mechanical properties as indicated by tests on a particular specimen. Furthermore, the data obtained by this, would guide the geometric choice of an hourglass profile gage section specimen. The geometry would be defined by the ratio of minimum width of the specimen to the height of its testing section ’as well as to the radius of its profile. 27 2.2.2. Numerical Approach to Strain Uniformity Error Estimation The development of numerical approach to strain uniiknnnity error estimation is based upon similar work conducted by Dewey [21]. The method used by Dewey considered an application of the nonlinear finite- element code CREEP-PLAST developed by Oak Ridge National Laboratory. CREEP-PLAST code was designed primary for high-temperature design of axisymmetric and plane structures. Among the features of CREEP- PLAST utilized were an incremental-plasticity solution, with a bilinear representation of the stress-strain curve, an equation-of-state creep formulation, ability to specify increments in loading or displacement on selected groups of nodal points, and a restart procedure that per- mits running a large number of loading steps Vdflfll intermediate examinations of the solution. An analysis of stress and strain distributions was conducted by Dewey at anmdent and elevated temperatures using both geometric varia- tions; an hourglass and cylindrically shaped specimens. Both types of these specimens were examined in terms of strain state uniformity and extensometry accuracy for application in high temperature mechanical properties studies. Data was collected in this analysis, and subsequently used in the determination of shape to be used in creep, plasticity, and fatigue ap— plications. Stress and strain fields as well as displacements in these two specimens have been studied in detail. Analysis revealed that specimen filets and fixing conditions affected strain uniformity in both specimens. A slight tapering was required to assure that the rupture occurred in the region of midspecimen. This created an additional distortion on 28 existing stress and strain uniformity. In the hourglass-profile specimen, the curvature of the hourglass profile distorted the stress and strain fields even more. The displacements of the ruxial point of finite-element model at which the extensometer was fastened was used to compute the apparent strain from the extensometer reading. Corresponding to these nodal-point displacements, the mean axial strain has been computed at the center of the gage section. Collected data made it possible to assess the accuracy of the mechanical properties as indicated by tests on both analyzed specimens. Assuming a similar approach for a thin sheet metal, in the first step, the choice of the finite element program package was made and its performance examined on sample problems, for which the results were easy to verify analytically. It was decided to use ANSYS version 4.3, a general purpose finite element program developed by Swanson Analysis Systems, Inc. This choice was made for the following reasons: a. The program has been generally known and used since 1970. Modifications that have been made by Swanson Analysis Systems every year for the last 17 years are based on a vast response from the users, which allowed instant verification of the results. (Certain con- fidence, which has been built concerning accuracy of the results obtained while using the program, has resulted.) Thanks to the above mentioned reasons, ANSYS has become a trustworthy tool in solving ad- ‘vanced engineering problems. This has been a great advantage especially for the situations where the verification of the results is difficult [22]. 29 b. The program can handle problems involving plastic deformation. When solving a problem of this kind, the solution is obtained by a spe- cial iterative technique called the initial stiffness method. In this method the proper solution is obtained by iterations (series of linear solutions). The convergence is obtained when the ratio of the con- tribution to the total plastic strain in the n'th iteration, to the elastic strain is less than the criterion value. c. The program contains a library of finite elements which are readily suitable for this particular application. For the considered project the choice of 2-D Isoparametric Solid Element seems to give the best results in the shortest time. Four nodal points, which have two degrees of freedom at each node define creep, swelling, stress stiff- ening, and large rotation capabilities. ANSYS gives the option of including or excluding modified extra displacement shape functions. A decision was made to run the analysis with the extra shape functions, which would permit the element to acquire a deformation not otherwise possible. Inclusion of the extra shapes allows higher order displace- ment effects to be characterized with fewer elements [23]. d. In the modeling of the nonlinear properties of the material a bilinear representation of the stress-strain curve and an equation—of- state creep formulation has been utilized. The program also has the ability to specify increments in loading or displacement on selected groups of nodal points that permits running a large number of loadings steps with intermediate examinations of the solution [24]. e. Von Mises yield criterion and the associative flow rule with kinematic workhardening and the Bauschinger effects have been used to model the plastic flow. Using the kinematic hardening rule is an ap- proximation which will introduce some error especially in reversed 30 loading. This will produce certain overestimation of the plastic deformation. Suppose that the specimen is cyclically loaded with a controlled average stress between a and a . . Then, rule of kinematic workhar- max min dening predicts that a steady state involving alternating plastic strain will set in after the first cycle of loading. The rule of isotropic workhardening implies that the specimen will shake down to an elastic state. It. has been observed in tests, however, that a steady cycle of alternating plastic flow is reached after a certain number of cycles or asymptotically. This implies anisotropic workhardening model like one described by Mroz [25] and [26]. Mréz suggested modeling methods, which were more descriptive of the deformation and workhardening process, especially in the reversed loading regime and the result of that may give more accurate data. It could be applied in the future studies with a properly modified finite element program. It is important to note here, though, that kinematic workhardening model is a good approximation of the real material be- havior. This is a kind of anisotropic workhardening model that has been successfully used in analyzing variety of engineering problems. It gave good results. For this reason it has been expected to perform well in the analysis. f. Finallgr the program provides the user with a very versatile graphics capability for displaying the results by means of color maps and contour graphs. It has to be noted that this recent development in finite-element programs makes the whole analysis possible. 'Phe amount of produced data in elastic, plastic and cyclic mode of loading, with 31 many intermediate steps cannot be examined, reviewed and properly in- terpreted without the graphical display capabilities. This capability also enables the control and elimination of all possible errors in the modeling technique, choice of loading increment and the application of proper boundary conditions. g. The last feature of the program, which has been successfully used, was its mesh generator. After selecting the proper element and material properties, the geometry of the object was defined in the preprocessor solid modeling subprogram. Having the geometry defined, meshing routine was activated. The meshing routine automatically divides the whole object into finite elements according to instructions which concentrate or expand the mesh in certain regions of interest. This can increase accuracy in some places with large stress gradient and on the other hand can also save computation time in regions where the state of stress and deformation is almost uniform. Also, this fea- ture is especially suitable because it enables fast modification of the model and rerun of the program with only minor changes in the code, keeping the same number of elements. Consequently, solutions for dif- ferent profiles of the model, but with the comparable accuracy, can.be obtained in a fast and efficient way. This has a very big importance for the conducted studies [27]. All of the above given reasons explain why the ANSYS finite ele- ment program has been selected for the discussed research. 32 2.3. Work Schedule in Preliminary Stage of the Research The following is a record of all works-done in the preliminary stage of the research and in the sequence how they developed: 1. ANSYS code was used to design the program simulating loading the straight gage section specimen by displacement, applied in such a way, so that stress and deformation would not be uniformly distributed. The contour plots have been recorded in each increment of loading in the elastic range, plastic range, and after the load has been released and reversed, and then again released and reversed, until the whole cycle has been completed. The analysis of the results of this test has shown that after the entire cycle the distribution of all stresses and strains matched with a good accuracy. Hence, it has been concluded that the program can be successfully applied to cyclic loading. 2. In the next step the specimen with an hourglass shape profile has ‘been modeled and the mesh of elements generated. Two models have been created and used. The first model has a crude mesh of 54 elements while the second had 150 elements. The reason of building the first, smaller model was to test the program code and simultaneously save com- puter and user time. Its performance proved that it can provide one with the necessary information about the range of loading and deforma- tion, and prepare the necessary data for building,the final model, which uses the fine mesh, whole capabilities of the computer, and gives accurate results. The developed model has been loaded by cyclic dis- placement or cyclic pressure on the line of grip limiting .96 inches height of the gage section. Both methods of loading produced results which almost perfectly matched for the same values of loading in the first and the second cycle. Even residual stresses and elastic strains 33 in the unloaded specimen matched after the whole loading cycle. There is a great practical use of these calculations. The model loaded by applied pressure in the stress controlled cycle may give time solution which require verification. Sometimes the solution does not converge. Then, the calculated displacements may be used in the alternative mode of loading, i.e. the displacement loading. In such a case it is much easier to obtain convergence and convergence criterion may be stronger. This gives more accurate results. 3. The last task in the preliminary stage was to create the code, which would give the possibility of integrating the strains along the chosen paths through the selected nodes. In the result of dividing the integrals by the appropriate dimensions, the mean total strains were obtained as well as their elastic and plastic components. CHAPTER3 ANALYSIS IN THE EIASTIC RANGE 3.1. Technical Approach in the Analysis Elastic range analysis is based upon the assumption that specimen deformation is elastic when the average axial stress is less then the elastic limit of the material under test. In this case, the average axial strain is directly related to transverse strain as defined by elastic Poisson's ratio. Primary, the analysis examines the effect of the fixing conditions on stress and strain distribution. In order to eliminate the effect introduced by the hourglass curvature of the fatigue specimen, this curvature is considered to be equal to zero. In this sense, the model is reduced to that of the thin rectangular plate in the plane stress state. A stress analysis was performed using a finite-element idealization of the specimen quadrant such as one shown in Figure 3.1. Furthermore, the loading is imposed by uniform pressure applied to the elements along the line where the specimen is in direct contact with a grip. The imposed boundary conditions restrict displacement of all nodal points positioned on the line of loading. For all nodal points, the 34 7// 066W? Fixing 1111111 3/16" Figure gaaaaaee 3.1. A quadrant of the specimen finite-element model. 36 displacement in the axial direction is coupled and equal to zero in the transverse direction, (Figure 3.2). The conditions described above represent a simplification of specimen fixing techniques, contact, and loading methods. Four rectangular FEM models have been developed to satisfy Una requirements of this detailed analysis. Standard dimensions used on all specimens represent a thickness of .075 inches and a width of .33 inches. Table 3.1 shows gage section lengths, and ratios of length to width of the four rectangular specimens. In addition a fifth model treats the full hourglass profile with the exact boundary conditions as those used for model D which has the longest gage section of .96 inches, (Figure 3.3). Since the gage section is this long, and the Inininuun width of the specimen is only .33 inches, the influence of the boundary conditions at the grip site upon the stress and strain distribution at the nddspecimen should be minimal as it is determined by Saint Venant's Law. IPlanning the geometry of the models in the method described above allows an examination of the effect of the curvature apart from the effects introduced by the boundary conditions. It is essential to separate the effects of these two variables in order to obtain reliable and comprehensive data. Successful experiments have been conducted on full-hourglass profile specimens as represented by model E. Preliminary testing revealed that a profile radius equal to .75 inches was optimal for cyclic loading. Tested specimens having either a smaller or larger radius buckled or ruptured at the grip site. Considering the strain uniformity error involved, readings obtained from these tests must be examined and appropriate correction factors should be applied where 37 ii 9111:1111;iisiéziéziééiiziiii 94S 96 97 98 99 100 101 108 103 104 18 $43,111.11: 1°11: (.a'1 (b) Figure 3.2 a) Nodal point numbering. b) Elenent mesh numbering for the straight-profile specinen finite-element idealization. 38 Table 3.1 Geometry of the finite—element models MODEL GAGE SECTION RATIO LENGTH LENGTH/WIDTH A 3/ 16 in. .57 B 4/ 16 in. -76 C 5/ 16 in. .9 5 D .96 in. 2.9 E .96 in. 2.9 39 Figure 3.3 Finite element nesh for the full—hanglass—grofile specimen idealization. 40 needed” Ifiualninimum width of .33 inches for the full-hourglass model is standardized as in all rectangular models. ffiua length of the gage section for this specimen is considered to be twice the maximum width of .48 inches. 'Taking advantage of the symmetry, a stress analysis of the full- hourglass specimen has been conducted on a specimen model quadrant. All models are subjected to the same regime of loading, which is static tension beginning at zero with increments of 2,000 psi up to the material elastic limit of 20,000 psi, (Figure 3.4). The Young's modulus assumed is 30,000 ksi, elastic Poisson's ratio .33, and plastic Poisson's ratio is .5. Displacements oftuxkfl.points in the finite-element model where the extensometers are fastened can be used to compute the apparent strains from extensometers readings. Corresponding to these nodal- point displacements, the axial strain is computed at the center of the gage section using the transverse strain and the axial strain measured through three various gage lengths. lfimeassumed elastic limit corresponds to the elastic engineering strain of .0007. Five different average strain measurement techniques are analyzed: 1. Measurement of the axial strain at the center of the specimen plane surface (XY plane), (Figure 3.5). It is referred to as the interferometric strain gage technique (ISG). The use of this technique is simulated by reading the strain at the center node of the finite element mesh. In this case the gage length is considered to be equal zero. 2. Measurement of the average transverse strain is made across specinuni thickness, (Figure 3.6). In the finite-element solution, the technique is simulated by calculation of the transverse strain in the 41 .mmqmu 03mm? 05 5 6396.50 sausage? ummu maflmcmu canon—m 93 new EmHmOHQ 58H .v.m gown :wno. mmofieowfi E omega Eda: m N K. .0 b V m. 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Techniques utilizing measurements in the axial direction through either a part or the entire gage section exhibit a noticeable relative error, even in the case of the long gage- section model. When the gage section length of a specimen with the straight profile is decreased, the relative error further decreases. Furthermore, when all three analyzed axial strain.uwasurement techniques are compared, it can be observed that the relative error is smaller for shorter gage lengths. It can be observed that in the elastic range, short gage-section- length models A, B, and C indicate relatively small relative error when the axial measurement techniques are used. The maximum relative error for these models is -5%, and is produced by the shortest model, model A with the technique based on method 4, (Figure 3.17). TUnis technique and model are most affected by the uneven axial strain distribution. Figure 3.18 and Figure 3.19 show a decrease of the relative error in the technique based on method 4, when the length of the gage section is increased to 4/16 inches, and to 5/16 inches as well. The decrease in the gage length also has a positive affect on the relative error. Figures 3.20, 3.21, and 3.22 show a relative error for models having a 3/16 inches, 4/16 inches, and 5/16 inches gage-section length. However, the gage lengths in these cases are equal only to 2/3 of the gage section lengths, (method 5). Similarly as in method 4, while using method 5, the relative error decreases, when the gage section length is increased. 58 .< .3005 «6.53on 9.3 and v 0059. comma ungwuumMm—u H308 05 On 93.3004“. won—mu 03mme 93 5 00.55% ummu 33:3 033m “Human—elm m5 5 wmsoawwu fimfimlmumuum .54.” an {Oxbow $0.00 :00 \m. 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Figures 3.23, 3.24, and 3.25 illustrate that the relative error ixnnxbved in method 2 decreases with an increased gage-section length. The axial strain is calculated directly, dividing transverse strain by the negative value of'the elastic Poisson's ratio. Hence, the relative error of the transverse strain measurement and calculated axial strain are equal. Figures 3.26 through 3.28 illustrate results of a simulated tension test utilizing transverse strain measurement technique according to method 3. Similar as by method 2, the relative error decreases here with an increasing gage-section length. A maximum relative error value of -29% is found when model A is used having a 3/16 inches gage-section length. In this technique the measured transverse strain is underestimated, whereas in the technique based on, method 2, tflne strain is overestimated. Both transverse strain measurement methods yield similar errors having comparable magnitude but opposite sign. 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Figure 3.34-illustrates a typical contour plot of axial stress in step 7 of loading a straight-profile 3/16 inches gage-section specimen model. The average axial stress in this step of loading is 12,000 psi. In the midsection of the model under examination, results of analysis Idisclosed.a.variation in axial stress. A value of 9,600 psi was found along the edges and 12,500 psi at the center. The substantially large 'portion of the model sustains an axial stress near the value of 12,000 psi. IkIaddition, there is no great variation in stress along the axial direction. However, when the corner points of thexmxkfl.are approached, the stress gradient sharply increases. The numerical value of this maximum axial stress depends on the size of the finite element. This is typical for an element with geometric singularity at the corner node. The estimated maximum axial stress for the given size of this finite-element mesh is 20,760 psi. When the average axial stress approaches an elastic limit of 20,000 psi, a noticeably large plastic zone is observable. The plastic zone expands from the corner point of the quadrant under analysis, until it extends over the entire model. An analysis of the axial stress contour plot conducted for a large gage-section length specimen model D reveals, that the influence of the grip site boundary conditions on the stress distribution at the midspecimen is negligible, (Figure 3.35). The stress field is distorted in the neighborhood of the specimen model corner and along the line of loading. 78 STRAIGHT-PROFILE SHORT GAGE-SECTION SPECIMEN TENSILE LOADING OF 12,000 PSI ANSYS 4.3 F=13123 .AXIAL STRESS G=13711 =7 H=14299 ITER=30 I=14887 DIRECTION=Y J=15475 K=16063 MX=20764 L=16651 MN=9599 M=17239 A=10183 N=17827 B=10771 0:18415 C=11359 P=19003 D=11947 Q=19591 E=12535 R=20179 Figme 3.34. Contour plot of the axial stress in step 7 of the loading the straight-profile 3/16" gage-section specimen in the elastic range. 79 STRAIGHT—PROFILE LONG GAGE-SECTION SPECIMEN, .96 IN. 'I‘ENSIIE LOADING OF 20,000 PSI ANSYS 4.3 F=20433 AXIAL STRESS G=20815 STEP=11 H=21197 ITER=30 I=21579 DIRECTION=Y J=21961 K=22343 MX=25394 L=22725 MN=18146 =23107 A=18523 N=23489 B=18905 0:23871 C=19287 P=24253 D=19669 Q=24635 E=20051 R=25017 the specimen of .96 in. gage-section. Figure 3. 35 . Contour plot of the axial stress in step 11 of loading 80 An analysis of both the transverse stress field (Figure 3.36) and shear stress field (Figure 3.37) must be completed in orrkn: to better understand the mechanism of departure from the uniaxial stress state. The transverse stress field minimum value is zero and is located on the edge of the model. This value changes slightly with a small gradient along the transverse axis reaching 2,000 psi at the center of the model. It increases also along the axial direction towards the line of loading. Maximum transverse stress occurs in model corners and reaches a value of 5,000 psi. A similar analysis of shear stress indicates a variation with a steadily increasing gradient. The value of this stress is increasing from zero at the center of the model up to a maximum in the corner points. This maximum value is 4,210 psi. To cunnplete the stress distribution field analysis, one must also consider the equivalent stress calculated using the von Mises formulaxjxni. When the axial loading is slowly increasing, the equivalent stress (08) calculated in the rectangle corner, exceeds the elastic limit much before the average axial stress reaches the value of that limit, (Figure 3.38). This stress changes with a relatively small gradient in the transverse direction and an even smaller change in the axial direction. When approaching the models quadrant corner, the stress gradient sharply increases. It is here, in the corner, where the equivalent stress reaches its maximum value. When the average axial stress is only 12,000 psi,tim: mathmi estimated equivalent stress in the corner finite-element is 20,062 psi. This indicates the initiation of plastic deformation within this region. The amount of plasticity at the rectangle corner is small, because the stress gradient is fairly large in this region. 81 STRAIGHT-PROFILE SHORT GAGE-SECTION SPECIMEN TENSILE LOADING OF 12,000 PSI ANSYS 4.3 F=1690 TRANSVERSE STRESS G=1967 STEP=7 H=2244 ITER=30 I=2521 DIRECTICN=X J=2798 K=3075 Mx=5286 L=3352 MN=33.3 M:3629 A=305 N=3906 B=582 0:4183 C=859 P=4460 D=1 1 36 Q=4737 E=1 41 3 R=5014 Figure 3.36. Contour plot of the transverse stress in step 7 of the loading the straight-profile 3/16" gage-section specimen in the elastic range . 82 STRAIGHT-PROFILE SHORT GAGE-SECTION SPECIMEN TENSILE LOADING OF 12,000 PSI ANSYS 4.3 F=1333 SHEAR STRESS G=1555 STEP=7 H=1777 ITER=3O I=1999 DIRECTION=XY J=2221 K=2443 MX=421O L=2665 MN=9.62 M=2887 A=223 N=3109 8:445 0:3331 C=667 P=3553 D=889 Q=3775 E=1111 R=3997 Figure 3. 37. Contour plot of the shear stress in step 7 of the loading the straight-profile 3/ 16" gage—section specimen in the elastic range. 83 S'I'RAIGHT—PROFIIE SHORT GAGE-8mm SPECIMEN TENSILE LOADING OF 12,000 PSI ANSYS 4 . 3 F=12890 W STRESS G=1 3442 ' STEP=7 H=1 3994 ITER=30 I=1 4546 VW MISES J=1 5098 K=1 5650 =20062 L=1 6202 MN=9582 M=1 6754 A=101 30 N=17306 B=10682 0:1 7858 C=11234 P=18410 D=1 1 786 Q=1 8962 E=12338 R=19514 Figure 3. 38. Contour plot of the equivalent stress in step 7 of the loading the straight-profile 3/16" gage—section specimen in the elastic range . 84 Figure 3.39 presents a plastic zones in model A of specimen in two steps of loading; (a) step 8, when an average axial stress is equal to 14,000 psi; (b) step 11, when this stress is equal to 20,000 psi. The amount of plasticity developed in step 8 of loading is very small but considerable. However, in step ll, when the loading is equal to the elastic limit of the examined material, the size of the plastic zone is not larger than 5%. It can be observed that this small amount of plasticity influences the strain uniformity error. In the presence of a small portion of plasticity, the amount of measured strain increases when compared to the referential model under the same average axial stress. The manner in which the above described influences the strain uniformity error depends largely upon the condition if the measuring technique overestimates or underestimates the axial strain. Since the majority of the discussed axial measurement techniques underestimate the measured average axial strain, the presence of a small plastic zone found in the elastic range slightly decreases the relative error close to and at the elastic limit. Similarly, when the technique based on measurements of the transverse strain across the specimen width is applied, it can be observed that the relative error decreases at the elastic limit because this method underestimates the strain readings. 0n the other hand though, the technique based on measurements of the transverse strain across the specimen thickness overestimates strain readings. For that reason, the increase in the strain read by the strain gage, in the presence of a small amount of plasticity, slightly decreases the accuracy of readings. This difference, though, is practically negligible for models A and E (prime models of interest). The 85 Plastic zone Plastic zone Figure 3.39. Plastic zone in model A of specimen in two different steps of loading a) 14,000 psi; b) 20,000 psi. 86 application of the strain measurement technique across the specimen thickness (Hi the full-hourglass profile specimen as well as the short gage-section length specimen (3/16 inches) yields very stable results in spite of small plasticity amount produced below the elastic limit. In conclusion, from the stress fields analysis stand point, in the straight-profile specimens, the best conditions for the control parameter measurement exist along the central axial direction, transverse central direction, or at the model surface center point. Ax.these locations and directions, stress changes are small, shear stress is equal to zero, and transverse stress is small, as well as having a small gradient. 3.2.2. Stress Fields Analysis of the Full-Hourglass Profile Specimen Similar analysis has been conducted with the full-hourglass profile model. As more and more load is applied, the plastic zone begins to develop at the specimen minimum cross section, which is found at the bottom of the hourglass profile. Figure 3.40 through 3.éfi3 illustrate stress distribution fields in the full hourglass profile specimen representin equivalent stress (a ) lobal axial stress (a ) lobal 8 e 8 :f transverse stress (ax) and shear stress (7 ). XY In the full-hourglass profile model, the average axial stress III tine region of midsection is much larger than in the region adjacent to the line of loading, (Figure 3.41). For this reason, the amount of plasticity found in the corner section of the model is very small and negligible. With an assumed finite element mesh size, the maximum 87 :13 H: in b B \o ANSYS 4.3 4 STRESS STEP=7 ITER=3O VG‘I MISES ux=13759 MN=4031 N A=4534 B=5047 =5560 D=6073 ._. E=6586 1L F=7099 .14 G=7612 =8125 I=8638 =9151 K=9664 L=10177 M=10690 N=11203 o=1 171 6 p=12229 Q=1Z742 L R=13255 All-'7': ”/14: 17‘ K I l I L I I [III I 'C M H x EVIL-WWW TENSILE LOADING (1“ 12,000 PSI Figure 3. 40. Contour plot of the equivalent stress in step 7 of loading the full-hourglass profile specimen. 88 “ML J \{ an)? - é \Fk‘ir .. N W_ I I 4 _.I 44 “1h M f X FULLPHOURGLASS PROFILE SPECIMEN TENSILE IQADII‘B OF 12,000 PSI Figure 3. 41 . Contour plot of the axial stress in step 7. of loading the full-hourglass profile specimen. ANSYS 4.3 STRESS STEP=2 ITER=3O DIRECTION=Y =13827 =3847 A=4366 B=4892 C=5418 D=5944 =6470 =6996 G=7522 H=8048 I=8574 J=9100 K=9626 L=10152 M=10678 N=11204 0:11730 P=12256 Q=12782 R=13308 89 ANSYS 4.3 STRESS STEP:7 ITER=30 DIRECTION=X MX=3167 MN=-371 A=-191 =-4.48 C=183 D=370 E=557 F=744 G=931 H=1118 I=1305 J=1492 K=1679 L=1866 M=2053 N=2240 0=2427 P=2614 Q=2801 R=2988 FULL-HOURGLASS PROFILE SPECIMEN TENSILE LOADING OF 12,000 PSI Figure 3.42. Contour plot of the transverse stress in step 7. of loading the full-hourglass profile specimen. 90 FULL—HQIRGLASS PROFILE SPECIMEN TENSILE LOADING OF 12,000 PSI Figure 3.43. Contour plot of the shear strain in step 7 of loading the full—hourglass profile specimen. 91 equivalent stress and plasticity has not been detected in that region, (Figure 3.40). The plastic deformation is concentrated at the minimum specimen transverse cross section. At the elastic limit of 20,000 psi, however, it spreads over a.lindted region at the bottom of the hourglass profile. Forlxnfln the equivalent stress and axial stress, (Figures 3,40 and 3.41) , the maximum stress is indicated at the bottom of the hourglass profile. The maximum value for the transverse stress is indicated at the corner of the model quadrant, (Figure 3.42). Close to the elastic limit, in all specimens, some plastic deformation is involved either at the corner point of the straight- profile specimen or in the bottom of the hourglass profile of the full- hourglass profile specimen. An equivalent stress obtained through the calculation utilizing the von Mises formulation indicates a maximum value equal only to 13,760 psi in the full—hourglass profile model. One can conclude then, that much higher values of the average axial stress are needed to initiate plastic deformation in the case of full-hourglass profile specimen than in the case of short-gage-section specimen. However, a larger stress gradient along both, the central axial direction and central transverse direction, reveal that with full- hourglass profile specimens a greater strain uniformity error can be involved when taking measurements along these directions. CHAPTER 4 ANALYSIS IN THE LARGE SCALE PLASTIC DEFORMATION RANGE 4.1. Technical Approach in the Analysis Plastic range analysis focuses on large scale plastic deformation, which occurs in a specimen much above the elastic limit. In the case, when axial strain is the control parameter for the test, but the measurement technique is applied in the transversesciirection, a procedural difficulty exists. The Poisson's ratio has different value for the plastic strain than for elastic. For this reason one must find the means of separating the elastic and plastic part of the average transverse strain. This can be accomplished by calculatjxnn according to fornuflxi (1.11). The axial strain is measured indirectly via total transvermesnmain and the transverse elastic strain, which is calculaxmuitising the average axial stress, as described in Chapter 1. The schematic on Figure 4.1 represents an analog computer strain circuit which was used for calculating axial strain by Martin [3]. The two inputs needed for the circuit are first, the negative value of the total transverse strain (-6 ), and secondly, the total load (P). The Ttr transverse strain gage is wired in such a way that if an1 expansixni of the cxnitact points occurs, a negative voltage output is produced, and 92 93 C 2 0' V 8r ___ av _ 28TH av e y E E 2 (yavve 8py — 28TH + O E ' Gav ' 86y = E O ' eTtr 10K 20K -—AAA~—~ P 49.9K V1 V + 1 vi— 20K 20K 20K D Figure 4.1. Schanatic of analog canputer circuit [3] 94 likewise, a contraction of the contact points produces a positive output. After the specimen is fixed in the grip and the extensometer mounted, time variable potentiometers V1 and V are set. A dummy load 2 voltage is applied to the input of Amplifier 1. V1 is adjusted for the appropriate voltage, which can be calculated using the elasth: modulus E, and the area of the specimen cross section. The ‘potentiometerg \QZ is adjusted for zero output at Amplifier 2, during specimen cycling within the elastic range under load control. Elastic and plastic strains are summed by Amplifier 3. Then, the output from this amplifier is recorded and used as controlling parameter throughout the entire test. Since the difference between total measured transverse strain and the calculated average transverse elastic strain is always treated as the transverse plastic strain, the additional error affects tflua calculation of the axial strain.when using the above described technflan The difference between the measured average transverse strain and the calculated transverse elastic strain does not pertain only to the plastic strain but also to the uneven strai11<1istribution caused by the boundary conditions and.the geometry of the specimen. Fbr this reason the existing difference is not equal to the plastic component of the average transverse strain alone. Hence, the method of the axial strain calculation by itself introduces an error that has to be accounted for. An axial strain can be calculated using elastic and plastic components of the average total transverse strain and appropriate Poisson's ratios. When the conversion circuit is incorporatmxi, it is 95 assumed that the elastic strain field in the spechmuinddsection is uniform which in the case of short—gage-section specimens is not true. For the purpose of this research, the occurring difference in the strain reading iscfififixmd.as an axial strain conversion error. The estimation of this error is possible thanks to the method of integrating the average transverse plastic strain along the given path across time SpeClflKHlidldCh. The plastic strain component can also be calculated across the thickness in specimen center point. The axial strain conversion error will affect the readings, especially hntjmaelastic region and close above the elastic limit, where the plastic component of average transverse strain is much smaller than elastic. .As the plastic defornmtion.increases in the specimen, this error decreases. iflunllarge plastic strains are produced, the above described error becomes negligible. The question arises as to how much this effect influences the strairlinniformity error near the elastic limit and within the elastic range. This could, in particular affect readings (hiring cyclic loading. Analysis in the plastic range is performed with the same models as in the elastic range. To simulate plastic properties of the material, the stress-strain curve is fitted with a bilinear curve of the elastic modulus E1 = 30,000 ksi and the plastic modulus E - 7,500 ksi. In the 2 elastic range the specimen is loaded with an average axial stress of 12,000 psi, followed by two loading steps with stress increment of 4,000 psi up to the elastic limit of 20,000 psi, and finalljrivith the 96 loading steps and with the same loading incrementtq3tx>the plastic range of 88,000 psi, (Figure 4.2). When the specimen is loaded with the maximum applied axial stress, the total average axial strain produced is in the range of .01. 4.2. Large Scale Plastic Deformation of the Thin Sheet Steel Specimen 4.2.1. Simulation Results of the Tensile Test with the Straight Profile Specimen Analysis of large scale plastic deformation in the straight and full-hourglass profile specimens, is based upon static tensile tests simulations. It is conducted by means of finite-elenunit models undergoing large plastic deformation. It is assumed that the reference model undergoes a deformation which follows the pattern of the bilinear curve. iNnis curve is fitted to a stress-strain curve obtained from a static tensile test, which had been performed with the identical material like that used for the analysis. Table 4.1 displays the relative error obtained from each analyzed strain measurement technique. This error is calculated with respect to the assumed reference model. Subsequently, figures show the stress- strain response in simulated static tensile tests. In axial measuremen techniques, the relative error increases by 50-100% with respect to the error obtained in the elastic range. Figures 4.3 through 4.6 display the results of a simulated large- ‘plastic strain tensile-test“ using the laser based strain gage technique. The relative error here is very similar to the errrn: found in the elastic range, quite small and almost negligible. 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IEJIIIIzII‘” PIIII ML-HGJRGIASS PROFILE SPECIMEN TENSILE LOADING OF 88,000 PSI Figure 4.43. Contour plot of the transverse strain in step 21 of loading the full-hourglass profile specinen. 150 FULL-HOURGLASS PROFILE SPECIMEN TENSILE LOADING OF 88,000 PSI (10' ) -2.oo_. -3.00._ -3.20 -3.40 —3.60 -3.80 -4.00 —4.20 -4.40 --004‘§'2 —4.60 -4.80 I I I I I I | l‘eTXI I x O I .040 I .080 I .120 I .160 I .200 .020 .060 .100 .140 .180 Figure 4.44. Pfilth plot of the total transverse strain across the specimen width at the central section; total strain in the direction of the specimen width eTx. 151 FULLPHOURGLASS PROFILE SPECIMEN TENSILE LOADING OF 88,000 PSI (10'3 8.80 ) — for the best readings accuracy .00973 8.40 8.00 7.60 7.20 6.80 6.40 6.00 5.60 5.20 4.80 l J, I I I Y 0 .100 .200 .300 .400 .500 .050 .150 .250 .350 - .450 Figure 4.45. Path plot of the total axial strain along the specimen central axis. 152 is the axial strain value indicated by the referential model in the same step of loading. The path plot curve displays thattflw axial strain value sharply drops when approaching the grip site. This explains why the best accuracy can be accomplished only with a very short gage length technique, in particular the zero gage length of the laser based method 1. In Figure 4.46 a total axial strain distribution across the line of loading is presented in the form of a path plot. This plot displays even.distrflnnfion.of strain across a large portion of the specimen model width. It changes, however. when approaching the model corner region. The sharp rise of the calculated strain is ixniicated.i11 this region. 1S3 FULLPHOURGLASS PROFILE SPECIMEN TENSILE LOADING OF 88,000 PSI (10- 8.40 ) 8.00 _ for the best readings accuracy .00973 7.60 7.20 6.80 6.40 6.00 5.60 5.20 4.80 4.40 l l l l l l l J l IY Figure 4.46. Path plot of the total axial strain across the specimen width at the line of the grip. CHAPTHIS ANALYSIS OF CYCLIC PLASTICITY IN A THIN SHEET METAL SPECIMEN 5.1. Technical Approach in the Analysis Two types of fatigue specimen models, representing both the straight profile and full-hourglass profile specimen, are subject to analysis of deformation under cyclic loading. The specimen undergoes a sequence of cyclic loading. Uniform pressure is applied along the line of loading where the grip is secured across the specimen width. In this type of analysis, the specimen is initially loaded in the elastic range, iJI tensirnl, up to the elastic limit of 20,000 psi. Above that limit, the specimen is considered to be in plastic range of loading. The axial pressure, applied on the line of loading is increased with an increment of 4,000 psi up to 32,000 psi which is the maximum average axial stress. After this stress value is reached, the load is released to zero. Then, the direction of loading is reversed in order to load the specimen in compression. In compression, the specimen is loaded with an increment of -4,000 psi up to the maximum compressive loading of -32,000 psi. Subsequently, the load is released and the specimen is loaded again in tension with an increment of 4,000 psi up no a maximum tensile loading of 32,000 psi. 154 155 .mp9: HgaIzma 93 5 pouospcoo Sflumaafim ummu ashamed 0326 93 new Emumoua 9.3 In gown :mno. 33535 NM.— Room 3.5a: xN 0N 0.x mx Ax @x Wx Vx Max Nx xx Ox 0. m. R. Q m. xv m. N. 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MN00.0| \ 000 .0I \ m 0 00 00 .0: \s 000.0I \ K0 . m 0 0000 0.. . 00 0 \t q \n\ 0 . 000.0 ..\. \ . 0000.0 \ 000.0 «N10 m.N00.0 m 10611.31 am [0 573 01.3.0448 703m afizuarm 9.00.0000 0000 0:0 \0 0\N 0:000 2.6.de ._<_X< m0< 173 060.: Salsma m5 5 69546600 umwu mega 0.326 @333m 93 5 wmconmmu fimflmlmauum .3...“ an .0 360.: 5.500% m5— ccm m BBQ: comma ucmehsmmmE fimbm 1.68 Bu on mcflcuooom 63300.... 00.50 :0\\m. D «”530 9.6ka fldwnnd wmdfimz‘d «wfifidgodkK» 06$380~ Eokwgea \m E «N \\ RN \2\ xx“ \Yx“ k wah \ i; \ m \\ «NS . \ {Owwomm $0.50 33 KO m.\N 0:008 27‘um ._<_X< m0< U WN00 .0l N00 .0l Wu00.0| s 00 .0... W000 .0... W000 .0 x00.0 ”000.0 N000 wN00.0 106113) 31;; [a 9/3 ‘uymns 703x12 360.1.amo 174 The relative error is increasing when.the gage section is decreasing and basically has the same value as obtaimmxi in the large scale plastic range. Consequently, the strain measurement techniques are examined which utilize methods of transverse strain measurement. In the case of models B and C it can be observed that the transverse strain measurement gives basically the same relative error in cyclic loading as in the large scale plastic range. The method of measurement across the specimen thickness overestimates (Figures 5.16 and 5.17), and the method of measurement across the specimen width underestimates values of strain readings, (Figures 5.18 and 5.19). In the case of model A, the readings accuracy is better in cyclic loading than in the large scale plastic range because the error increases with the increasing plastic deformation, (Figure 5.20). Relatively smaller accuracy is indicated by model A when the measurement is conducted across the specimen width, (Figure 5.21). In this method, the relative error is increasing very fast with the decreasing gage section length. It more than.doubles when the gage section length is decreased from 4/16 inches to 3/16 inches. 5.2.2 Simulation Results of the Cyclic Loading Test Utilizing the Full- Hourglass Profile Specimen Inna last column of Table 5.1 shows the results of relative error analysis in the cyclic loading test utilizing the full-hourglass profile specimen. 175 .m H009: 558% may 98 m @059: comma ”_chng 593m omugmqmfl 9.3 on mango .. woos :3:sz m5 5 E05080 ummu @568.“ 0:28 “swung—5m m5 5 wmcoamwu fimnumnmmouum 67m EOE gowuomm which :0\\V D 0:39.50» EOHHFCS D 3R0 «3.93% 08....86 $088038 «$82.8 3904.5 0 0? 0m. 0 0Nl 0Vl \K .. 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D 083.0800 E0533... 0 3.890 880.58 gdwg 000.8090 «83:830.,‘50 . 0m. 0 0Nl 0VI \ ‘5 \8\I \ 8 \8 E xx 8 :5 0‘ xx 303008 00.00 0d» \0 0\N 0.2.3.0 Z.<~_._.m ._<_X< m0< WN00 .0l N00 .0l ”800.0... x 00 .0I W000 .0l 0000.0. x00.0 m4 00 .0 N000 WN00.0 mfiuaz at” [0 9/3 ‘ujmus 700x!) 3604mm 224 25 m. 8 H8860 03088 0:00.00 8.50008 mm 0050... 0008080880... 0080 58.30 0.3 00 05000008 000:. Sauna 05 5 00000050 080» 050.83 0300 0008.350 05 5 0880008 585000000 .06 030E .88.... w. 83.0.8.0... D 0633.000 E05383. D 389» 880.38 0.00.8.0 00.0.8090 s83§8830£50 0V 0N 0 0N| 0Vl WN00.0I V 8 0N p .. - N88 .8: \\ 88 88 .8: X\ \8. \ . . 0.000 .0l 05 bx W000.0 - .08 \\\.a H... .880 \\ «~78; 1061131 am f0 {.73 100.148 201sz afizmamo N00.0 m.N00 .0 2.0.3008 00.00 08.3 \0 m.\N 0:000 Z_< 225 2:0 mm. 00 08:00 050080 0000000 00.50000 «m 00:00:. 05.500880... 0080 808000 05 00 000000008 0005 0000:0080 0:0 :0 000050000 0000 0000800 000000 00080300 0.00 :0 00000000 008000-000000 .36 000500 .20. ”N. 830.00.... D 02.30.00» 62.80.3003. U 3.886 880.88 0.00.8.0 000L090 «80308304.: 0% 0m. 0 0Nl QVI wwoodl \ ‘\ ”NI P1. N00.0l man 00 .0l 6\ \\ x 00 .0! XRN 0000.0! \t . 0: 8 8888.8 80 R “88.8 \h . \va 8 8088.8 \ N88 .8 mfiua; am [a 9/3 ‘uwfis 703m) amuarm ahpl 0N00.0 2.00.0008 00.00 0...; \0 0\N 0.2.00.0 Z_< 226 Finite-element analysis yields results which indicate an increase of relative error'tx) -6% when the profile radius is equal to 1 inch, (Figure 6.11). Ifmroduction of the hourglass profile with a radius equal to .75 inches increases the error to -7%, (Figure 6.12). A decrease in the profile radius to .5 inches yields a relative error with a value of -9%, (Figure 6.13). Finally, a radius equal to .25 inches yields a relative error with a value of -lh%, (Figure 6.14). Relative error studies conducted on specimens undergoing cyclic loading reveal that the introduction of the hourglass profile is increasing the negative value of that error. In the plastic range of cyclic loading,tumwmical value of that error are approximately two times larger then in.the elastic range. Strain measurements conducted in the axial direction detect a smaller axial deformation.tiun1 in the referential model in both elastic and plastic ranges of loading. This is primarily due to an increase in the transverse cross-sectional area. along the axial direction introduced by the hourglass profile, which as a consequence, increases specimen axial stiffness in this direction. Studies undertaken on transverse measurement techniques and cyclic loading simulation also reveal some worthwhile results. Similarly like in elastic range, an introduction of the hourglass profile to specimen model A does not decrease the accuracy of readings in method based on measurements taken across specimen thickness. Actually, detailed finite-element analysis indicates an improvement in readings accuracy. Figure 6.15 illustrates the stress-strain response of an hourglass profile model with the profile radius equal to 1 inch, vdmich yields a relative error equal to 21%. That is less than what has been found in the elastic range for this model, and less than what was indicated by the straight profile model A, (Figure 6.15). When the 227 .5 P 8 1.58 88m.” 338a 28% :8: F 858. e88 .892 m5 8 9888a moo:— Ssalsma $5 5 6955950 umwu 58a 0393 non—maze? m5 5 $9.09me cwgmlmmmuum .56 ”Hawk 6...... a 3.538... D 0.63380» E05353 0 ER» humus» ~85 $588098 «haugdgofikx . . 0». ON 6 ONI OVI 360 6: Vtao- & woo .on c \ we oo o: \ R x 00 .0I \ R m m. wooo on 2. 3 m ‘5 Q m- \t \m\ 2 S m . 2 S a \l moooo m. u good at m. m N . waooo on .o \ V Nooo \N. \ $0.. WNOQS 362$".an as \0 $32.00 was» 98 880k. UGO.)— 0_._0>0 I Z_O I Z_O I 2.6.um ._<_X< 230 00:05 mm. 00 H0000 03.000 030000 20000 000:. Sauna 05 5 000000000 “0000 000003 03.06 0000.15.00 05 :0 00000000 5050:0830 .36 0005.....— .2.0 MN. 03030... 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Figures 6.17 and 6.18 illustrate the stress-strain response for an hourglass-profile specimen model having a radius of .5 inches and .25 inches. These indicate a further decrease of relative error to 18% and 14%, respectively. Method 2 of strain measurement yields larger readings than readings obtained from the referential model undergoing an identical program of loading. By all appearances, this is attributed to an uneven strain distribution, which indicates a transverse strain maximum in the central specimen region. A detailed analysis of the measurements according to method 3 reveals that this method is most sensitive for the introduction of the hourglass profile. In plastic range of cyclic loading, strain measurements conducted across the width of model A indicate a relative error equal to -26%. This error value increases to -29% when the radius of the introduced hourglass profile is equal to 1 inch, (Figure 6.19). Figures 6.20 and.6.21.illustrate a stress-strain response of the models which have profile radii relatively equal to .75 inches and .5 inches. The model with the profile equal to .75 inches indicates.a relative error of -30%, whereas the model with a radius equal to .5 indicates that this error is equal to -33%. Finally, the introduction of an hourglass profile having a radius equal to .25 inches yields a large relative error of -39%, (Figure 6.22). In the plastic range of cyclic loading, the results obtained by means of method 3 to be more accurate than results obtained with this method in elastic range. Nevertheless, all the numerical values 233 25 ms. 00 00000 000000 0000000 00.50000 «N 00:00:. 0000000000.: 0000 000000 0.00 00 000000000 0005 300:2qu 0:0 5 000000000 0000 0000000 0326 0000.350 000 5 00000000 0000001000000 .006 0.00.0: .20 man. 03.030... D 02.0300» E08303. U «0000 000.80 »0......00 00.0.0090 «032.000.000B0 . 0». ON 0 QNI 00.! 000 .0l \ \ m N00 .0! 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