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M, ’ '1‘ 7,‘ ‘ " {ht-35‘” ATE UNIVERSITY U lll'llllll'lllllllillllllllllllllljfllllll 3 1293 00791 4 LIBRARY Michigan State University i.— This is to certify that the thesis entitled ELASTIC-PLASTIC STRAIN AND EFFECT OF NON-SINGULAR STRESS AT THE CRACK TIP IN FINITE THICKNESS PLATE USING VARIOUS EXPERIMENTAL AND ANALYTICAL METHODS presented by Subrato Dhar has been accepted towards fulfillment of the requirements for M.S. . MECHANICS degree in ‘ @fiéw Major professor DateZJ JUL; I771 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution ——.——- a ,7- i _, 7 , 7 , 7 r ,7 a — - --- — —~— —— . —— — -7 * PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU Is An Attirmetive Action/Equal Opportunity institution omen“: ___ ELASTIC-PLASTIC STRAIN AND THE EFFECT OF N ON-SINGULAR STRESS AT THE CRACK TIP IN FIN ITE THICKNESS PLATE USING VARIOUS EXPERIMENTAL AND ANALYTICAL METHODS By SUBRATO DHAR B.Eng.(Metallurgical Engineering), Regional Engineering College, Suratkal, INDIA (1985) A THESIS Submitted to the Department of Materials Science and Mechanics In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanics MICHIGAN STATE UNIVERSITY February, 1992 ELASTIC-PLASTIC STRAIN AND EFFECT OF NON-SINGULAR STRESS AT THE CRACK TIP IN FINITE THICKNESS PLATE USING VARIOUS EXPERIMENTAL AND ANALYTICAL METHODS by Subrato Dhar ABSTRACT It is well known that for most two-dimensional problems, the stress-intensity fac- tor K; can be obtained by analytic techniques or by finite-element analysis. In many instances, the crack length may be of the same order of magnitude as the thickness of the specimen, and the in-plane stress can no longer be assumed to remain con- stant across the thickness direction since the free surface of the specimen can exert appreciable influence on the stress distribution. As a result, the crack driving force will vary along the crack front and the problem becomes three-dimensional. The generalized plane stress solution will not be a limit of the three-dimensional solution. Furthermore, it is incorrect to refer to the stress distribution on the surface layer of a plate with or without a crack as being in a state of generalized plane stress. This problem is rather complicated and may be answered by experimental techniques. Static strain distribution and crack opening displacement near the crack tip in , thick compact tension specimens obtained from multiple embedded grid moire and strain gages were reported by Paleebut and Cloud. Their results are used to calculate the variation in the apparent Mode-l stress intensity for various interior planes and on the surface of the specimen. The stress intensity factor was found to be higher in the midplane than in the surface. The strain energy density concept is then utilized for calculating elastic-plastic strains and stresses near the crack tip. The availability of linear or non-linear elas- tic notch or crack stress-strain solutions enables calculation of strain energy density distribution at and ahead of a notch and crack tip. Subsequently, the strain energy density can be translated into elastic-plastic strains and stresses which actually exist at the notch and crack tip. The stress redistribution caused by the plastic yielding around the crack tip is taken into account so that the value of the strain is improved. The estimated values of crack tip strain based on strain energy density approach are compared with experimental results obtained from embedded grid moire technique and embedded strain gages, and the results are encouraging. From the moire photographs it seems that large scale yielding dominates near the crack tip. In fact, the measured strain is very near the elastic solution, which means, in reality, only small-scale yielding is taking place near the crack tip. The small scale yielding approximation then incorporates the notion that, even though stresses derived from a linear elastic solution are inaccurate within and near a small crack tip yield zone, its dominant singular term governs the deformation state within that zone. This deformation state, well within the yield zone, is expressed solely in , terms of J-integral. Besides this singularity term, there is a nonsingular term acting parallel to the crack tip. There is no non-singular effect on the dominant singularity, although there will, of course, be an effect on the overall shape of the yielded zone. A detailed study using finite element analysis was carried out in the estimation of distribution of stresses and assessing the effects of the non-singular term on plastic zone sizes near the crack tip of a CT specimen. The results are quite comparable with the above two experimental techniques. In the three-dimensional photoelastic investigation, an observation near the crack tip showed that crack blunting is taking place on the surface plane, and the crack tip remained sharp in the interface after a small rupture. In a thick specimen, triaxiality near the crack tip is high and higher crack front singularity increases with distance along the thickness from the surface plane. This causes stress distribution at the midplane near the crack tip to be higher than on the surface plane, and plastic zone size on the surface to be smaller than on the mid-plane. Strain energy density in the surface plane remains elastic and almost flat, whereas strain energy density in the midplane remains fully plastic. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES 1 BACKGROUND 1.1 1.2 1.3 1.4 Purpose and Motivation ......................... Linear Elastic Fracture Mechanics .................... 1.2.1 1.2.2 1.2.3 1.2.4 Inelastic region .......................... Singular region .......................... Non-singular region ........................ Mixed field region ......................... Non-Linear Fracture Mechanics ..................... Experimental Techniques ...... . ................... 2 MATERIALS SPECIFICATION 2.1 Application and General Properties ................... 2.1.1 Material stress and strain curve ................. 2.1.2 Tensile load-elongation curves for CT specimen ........ 3 STRAIN ENERGY DENSITY APPROACH 3.1 The Continuum Mechanics Model .................... 3.1.1 Elastic-plastic stress and strain at the notch tip ........ 3.1.2 Plastic zone size and shape at the notch tip .......... 3.1.3 Increment in the plastic zone size at the notch tip ....... 3.1.4 Elastic-plastic stress and strain at the crack tip ........ 3.1.5 Plastic zone size and shape at the crack tip ........... 3.1.6 Increment in the plastic zone size at the crack tip ....... 3.2 Numerical Method ............................ 3.2.1 Estimation of local stress and strain ............... iii iv 4310me 16 17 18 21 23 23 i 24 26 30 30 33 42 45 48 49 50 51 ‘51 4 NUMERICAL ESTINIATION OF NON-SINGULAR STRESS 59 4.1 Introduction ................................ 59 4.1.1 Modeling of CT Specimen and Boundary Conditions ..... 60 4.1.2 Modeling of Boundary Layer Problem and Boundary Conditions 62 4.2 Estimation of Elastic Non-Singular Stress ................ 65 4.3 Modified Boundary Layer Formulation ................. 73 4.3.1 Ramberg-Osgood Materials ................... 73 4.3.2 Effect of Non-singular Stress ................... 74 4.3.3 Polycarbonate Materials ..................... 83 EXPERINIENTAL TECHNIQUES 93 5.1 Introduction ................................ 93 5.2 Crack Opening Displacement ................. . ..... 94 5.3 Three Dimensional Photoelasticity ................... 107 5.3.1 Fundamentals of the stress freezing method .......... 107 5.3.2 Experimental set up ....................... 108 5.3.3 Slicing the model and interpretation of the resulting fringe pattern109 5.4 Double Embedded Moire Technique ................... 119 5.4.1 Measurement of Three Dimensional Mode-1 SIF ........ 122 5.5 EXperimental Method to Find Size and Shape of the Plastic Zone . . 130 . 5.5.1 Estimation of Increment in Plastic Zone ............ 131 6 DISCUSSION AND CONCLUDING REMARKS 134 6.1 Discussion ................................. 134 6.2 Conclusion ................................. 138 APPENDICES A Fortran code to solve energy density equations 140 B Fortran code for contour mapping 156 C Fortran code containing COD data 163 BIBLIOGRAPHY 169 ii 1.1 2.1 3.1 4.1 4.2 5.1 5.2 LIST OF TABLES Table of Hutchinson, Rosengreen and Rice as presented by Shih [19]. 20 Polycarbonate true stress and strain experimental data ......... 25 Stress concentration factor variation with normalized distance from the crack tip as measured by elastic finite element analysis ......... 38 Average biaxiality ratio (Poisson’s ratio = 0.3). ............ 70 Average biaxiality ratio (Poisson’s ratio = 0.45) ............. 71 Estimated and measured crack tip opening displcement ......... 101 Measured and estimated values 1‘, and Ar,. .............. 132 iii 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2.1 LIST OF FIGURES Deformed regions in front of the crack tip, (a) Four distinct regions, (b) Inelastic region and (c) Singular region ................ Crack opening strain distribution in a CT specimen for a/ W = 0.4966 and 0 = 0° based on equation (1.1) and (1.4) for SN = 4.137 MPa. . . Crack opening strain distribution in a CT specimen for a/W = 0.4966 and 0 = 0° based on equation (1.1) and (1.4) for SN = 17.927 MPa. . Crack opening strain distribution in a CT specimen for a/ W = 0.4966 and 0 = 0° based on equation (1.1) and (1.4) for SN = 31.717 MPa. . Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 60), based on equation (1.1) and (1.4) for SN = 4.137 MPa and p = 0 in plane strain. ....... Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 60), based on equation (1.1) and (1.4) for SN = 4.1371 MPa and p = 0 in plane stress. ...... Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 60), based on equation (1.1) and (1.4) for SN = 4.1371 MPa and p = 0.18 mm in plane strain. . . Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 60), based on equation (1.1) and (1.4) for SN = 4.1371 MPa and p = 0.18 mm in plane stress. Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 60), based on equation (1.1) and (1.4) for a? = 0.1478, 0.6408 and 1.1338 in plane strain ...... Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E co), based on equation (1.1) and (1.4) for 75,5} = 0.1478, 0.6408 and 1.1338 in plane stress ...... True stress-strain curve for polycarbonate. ............... iv 10 11 12 13 14 24 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Experimental notch strains in a sharp notched compact tension speci- men made out of polycarbonate ...................... Tensile load-elongation curve for two different specimens of 2mm thick- ness. The sharply notched fails in a brittle manner and bluntly notched specimen fails in a ductile manner [38]. ................. Optical micrograph near root of the specimen in Figure 2.2: (a) internal plastic zone at point A; (b) internal small crack at the tip of plastic zone at B; (c) surface shear bands at point B; (d) internal plastic zone at point D; (e) internal plastic zone at point E; (f) surface shear bands at point E [38]. .............................. Schematics of notched body extending to a cracked geometry; (a) A smooth sharp notched body and ( b) A sharp pre-cracked compact ten- sion specimen ................................ Stress concentration factor (KT) variation in a CT specimen for a/ W = 0.4966, a/t = 0.9372 and 0 = 0° based on 3D elastic finite element analysis. .................................. Theoretical and experimental notch strains in a sharp notched compact tension specimen made out of polycarbonate ............... Crack opening strain redistribution in a CT specimen throughout the specimen thickness a/ W = 0.4966 and 0 = 0° based on energy density apptoach, loaded to 4.137 MPa. ..................... Crack Opening strain redistribution in a CT specimen throughout the specimen thickness a/ W = 0.4966 and 0 = 0° based on energy density approach, loaded to 17.927 MPa. .................... Crack opening strain redistribution in a CT specimen throughout the specimen thickness a/W = 0.4966 and 9 = 0° based on energy density approach, loaded to 31.716 MPa. ............ - ........ Interpretation of plastic zone size increment. .............. Crack opening strain distribution in a CT specimen throughout the specimen thickness a/ W = 0.4966 and 0 = 0° based on energy density approach, loaded to 4.137 MPa. ..................... Crack opening strain distribution in a CT specimen throughout the specimen'thickness a/W = 0.4966 and 0 = 0° based on energy density approach, loaded to 17.927 MPa. .................... 26 28 28 30 40 42 45 45 46 46 53 3.10 Crack opening strain distribution in a CT specimen throughout the 3.11 3.12 3.13 4.1 14.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 specimen thickness a/W = 0.4966 and 0 = 0° based on energy density approach, loaded to 31.717 MPa. .................... Estimation of plastic zone size based on energy density criterion for mid-plane and surface plane loaded to 4.137 MPa ............ Estimation of plastic zone size based on energy density criterion for mid-plane and surface plane loaded to 17.927 MPa. .......... Estimation of plastic zone size based on energy density criterion for mid-plane and surface plane loaded to 31.717 MPa. .......... CT specimen used in energy based formulation, elastic and elastic- plastic finite element analysis, 3D photoelasticity and multiple embed- ded moire technique, (a) Dimensions according to ASTM E-399 [42] and (b) Loading and boundary conditions applied in FEM analysis. . Finite element modeling of CT specimen with a/ W = (a) 0.3958, (b) 0.4375, (c) 0.4790, (d) 0.5208 and (e) 0.5625 ............... Circular domain inside the CT specimen used for boundary layer and modified boundary layer problem ..................... Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/ra at a/W = 0.3958. ................. Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/ro at a/W = 0.4375. ................. Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/ro at a/W = 0.479 ................... Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/ro at a/W = 0.5208. ................. Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/r, at a/W = 0.5625. ................. Average values of the non-dimensional biaxiality parameter B for CT specimen as a function of r/r, ....................... Comparision of biaxiality values for CT specimen obtained from dis- tributed load at pin center with Leever and Radon [44], Cotterell [45], Kfouri [46] and Larsson and Carlsson [47], with simulation u = 0.3 and 0.45, 3D photoelasticity. ......................... Normalized true stress-strain curve based on empirical equation. . . . Normalized normal stress distribution for 0 = 0°. ........... vi 54 55 56 57 60 62 63 66 66 67 67 68 69 69 74 75 4.13 Normalized crackopening stress distribution in plane strain for 0 = 0°. 76 4.14 Normalized crack opening stress distribution in plane strain for biax- iality ratio B = 0.4890 and for 0 = 0° for modified boundary layer problem ................................... 78 4.15 Normalized crack opening stress distribution in plane strain for biax- iality ratio B = 0.5338 and for 0 = 0° for modified boundary layer problem ................................... 79 4.16 Normalized crack opening stress distribution in plane strain for biax- iality ratio B = 0.5564 and for 0 = 0° for modified boundary layer problem ................................... 80 4.17 Normalized crack opening stress distribution in plane strain for biax- iality ratio B = 0.5703 and for 0 = 0° for modified boundary layer problem ................................... 81 4.18 Comparision of plastic zones between Ramberg-Osgood and Modified Ramberg-Osgood in coordinate system non dimensionalized with re- spect to the characteristic length parameter (J / 00), in plane strain for B = 0 and strain hardening value n = 5 ................ 83 4.19 Comparision of plastic zones between Ramberg-Osgood and Modified Ramberg-Osgood in coordinate system non dimensionalized with re- spect to the characteristic length parameter (J / do), in plane strain for B = 0 and strain hardening value n = 10 ................ 84 4.20 Effect of B = 0.4890 on plastic zones in coordinate system non dimen- sionalized with respect to the characteristic length parameter (J / do), in plane strain for ace/00 = 1.0 and strain hardening value n = 5. . . 85 4.21 Effect of B = 0.4890 on plastic zones in coordinate system non dimen- sionalized with respect to the characteristic length parameter (J / do), in plane strain for deg/0,, = 1.0 and strain hardening value n = 10. . 86 4.22 Effect of B = 0.4890 on plastic zones in coordinate system non dimen- sionalized with respect to the characteristic length parameter (J / 00), in plane strain for deg/0‘0 = 1.1 and strain hardening value n = 5. . . 87 4.23 Effect of B = 0.4890 on plastic zones in coordinate system non dimen- sionalized with respect to the characteristic length parameter (J /a,,), in plane strain for deg/co = 1.1 and strain hardening value n = 10. . 88 4.24 The crack opening stress directly ahead of the crack at a distance 2J/ao and 5.1/0., for all a/ W ratio considered in a CT specimen in modified boundary layer formulation. ....................... 89 _ vii 4.25 Corrected normalized crack opening stress distribution in plane strain for a/W = 0.3958 and 0 = 0° for full scale CT specimen. ....... 4.26 Corrected normalized crack opening stress distribution in plane strain for a/W = 0.5208 and 0 = 0° for full scale CT specimen. ....... 5.1 Normalized crack opening stress distribution throughout the thickness of the specimen based on 3D elastic analytical solution and elastic FEM (NISA). .................................. 5.2 Home made clip gage. .......................... 5.3 Calibration set up for clip gage. ..................... 5.4 Calibration curve for the Clip gage. ................... 5.5 Measurement of clip gage strain after placing in the CT specimen with load varying monotonically. ....................... 5.6 Measurement of back face strain after placing in the CT specimen with load varying monotonically. ....................... 5.7 Relation between clip gage strain and back face strain under monotonic loading. .................................. 5.8 Construction of Load-crack mouth opening displacement (CMOD) di- agram with elastic and plastic components ................ 5.9 The exact profile of the deformed crack front following a parabola with 2p = 30 mm; Z is the direction measuring the thickness of the specimen and X is the distance from the notch to precracked end. ....... 5.10 Crack opening displacement measured behind the crack tip. ..... 5.11 Estimation of Model stress intensity factor for mid plane and surface plane using CTOD technique, (*) indicates remote stress ........ 5.12 Light field isochromatic fringe pattern from disc specimen. ...... 5.13 Dark field isochromatic fringe pattern from disc specimen. ...... 5.14 Fringe order versus stress difference obtained from the disc specimen and superimposed. is the calibration curve. ...... ~ ......... 5.15 Dark field isochromatics fringe pattern in the mid-plane. ....... 5.16 Dark field isochromatics fringe pattern in the surface plane ....... 5.17 Isochromatic fringe pattern in the mid-plane and surface plane repre- senting plastic zone. ........................... 5.18 Estimation of Model stress intensity factor, non-singular stress and biaxiality parameter for CT specimen in mid—plane and surface plane. 5.19 Schematic drawing of the optical processing system ........... viii 90 91 94 96 96 97 97 99 99 100 103 104 105 110 110 112 114 114 117 119 5.20 Schematic of the specimen set up ..................... 5.21 Moire fringe patterns in the quarter plane of a fatigue crack specimen obtained from a submaster having 19.96 1/ mm; (top) patterns parallel to the crack line, (bottom) patterns perpendicular to the crack line; (a, left) not loaded, (b, right) loaded to 31.717 MPa. ......... 5.22 Plot of distance versus fringe order .................... 5.23 Crack opening strain distribution in a CT specimen for a/ W = 0.4966 and 0 = 0° based on embedded grid moire and strain gage technique. 5.24 Transverse strain distribution in a CT specimen for a/ W = 0.4966 and ' 0 = 0° based on embedded grid moire. ................. 5.25 Variation of stress intensity factor throughout the thickness. ..... 5.26 Schematic of deformation near the crack tip. .............. 5.27 Estimation of Mode-l SIF based on COD, 3D stress freezing, Moire and Double embedded moire techniques. . . .i ............. 5.28 Analytical versus experimental results in the non linear zone ...... ix 119 120 122 123 127 127 128 128 CHAPTER 1 BACKGROUND 1.1 Purpose and Motivation This research was started by Paleebut [1] and Cloud [2] in 1980 to develop a new three- dimensional moire method using multiple embedded and surface gratings to measure strain around a coldworked hole through the thickness of a polymeric specimen and to measure the strain components near a crack tip on the surface and in the interior of a thick specimen. One purpose of the present investigation is to further highlight the importance of the above technique in terms of measurement of the strain components near a crack tip on the surface and in the interior of a thick CT specimen. The strain energy density criterion has been adopted to give physical insight in explaining the experimental results. Stress intensity factor was calculated along the crack front by using the moire results previously obtained by Cloud and Paleebut [2]. Apparent Model SIF was then compared with the stress freezing technique of photoelasticity, the results of which are additionally encouraging. Crack opening displacement techniques are used to further explain the degree of strain near a crack tip in the interior and on the surface. The results are compared for amorphous and crystalline polycarbonate to understand the fracture behavior. Measurement of plastic zone size and shape and predictions for any thickness are developed. Numerical analysis was than carried out using finite element packages like NISA [3] and ABAQUS [4]. Elastic analysis was done using NISA, and elastic-plastic analysis was done using ABAQUS. The estimated non-singular stresses from FEM are comparable to those obtained from photoelasticity. The effect of the non-singular stress for a power hardening material is more pronounced than for the materials which show softening. The results of all these approaches leads to understanding of fracture behavior in polycarbonate and, more or less, in any thick finite geometries. 1.2 Linear Elastic Fracture Mechanics In order to predict against failures of engineering structures it is necessary to under- stand fundamental processes causing fracture in solids, and, more important, is the engineering requirement in providing calculable and measureable parameters which can characterize unambiguously such fracture events as the initiation of crack growth at different load levels. This is made possible by testing some suitable specimen in a laboratory. Near the notch and crack tip there exists stress or strain concentration. . The local notch strain can be estimated from the product of nominal stress (strain) and the theoretical stress concentration factor (KT) only when the material at the notch root remains elastic. Several methods and solutions concerning the calculation of elastic stress and strain near notches were given by Howland [5], Peterson [6] and Neuber [7]. The stress and strain field at a crack tip was first found using elastic mechanics by Irwin [8] also then [9] extended stress and strain analysis under narrow- range yield. In the near crack tip, four distinct regions can be identified, as shown in Figure 1.1a. The following comments can be made about stresses in each of the four 201188: 1.2.1 Inelastic region In practice, the region at the crack-tip may not be in the elastic state; and the crack- tip may become blunt due to local plastic yielding. The elastic stress field near. the tip of a blunt crack is given by Creger and Paris [10] in the form: \non singular (a) Figure 1.1. Deformed regions in front of the crack tip, (a) Four distinct regions, (b) Inelastic region and (c) Singular region. J'=‘/-—fi.i(0 )+—— \/'_2r -P-9s(9 ) (H) where f,-,-(0) and g;,-(0) (for i,j = 1,2) are given by (l.2,1.3). r G 0 30 0 l — sinasin— 2 fij = 008- 1+ singsin3—9 (1.2) 2 2 2 o 39 3112-56087 .. 30 —COS7 0 so 9:3 = 008-2- cos-5- (1'3) ‘ i2 . —sm 2 ‘ where (r, 0) are the coordinates of the point of interest, and p is the root radius of the notch as shown in Figure 1.1b. In (1.1), if notch radius p = 0, it turns into (1.4). The result is valid for p less than or equal to 0.18 inch as reported by Zhang and Venugopal [11]. The above equations are independent of material constants and do not differ between plane stress and plane strain in a two dimensional problem. 1.2.2 Singular region In this region, the elastic stress components are given by (1.4 - 1.6), strain components in plane strain and plane stress are given by (1.7 - 1.8) and displacement components in plane strain and plane stress are given by (1.9 - 1.10), in the near vicinity of a sharp crack tip. These components always have a singularity fl and are given in tensorial form (for i,j = 1,2,3) by Paris and Sih [12] in the form: l — singsin3—9 K1 0 aij = mws§ 1 + singsin§29 (l4) dragons-323 0'13 = 0’23 = 0 (1.5) 0 for plane stress 033 = 2ueos 9 K] (16) —7§%}— for plane strain (1 - u) —- (1 + u) sin-gem? .. _ KI 0 - a - so 1 7 6,, —- fleas-2- (1 — u) + (l + u) szn532n7 ( - ) %’-cos% J K 0 (1 - u - 2V2) — (1 + V) singsinéz'i I . . 6;,- = Moos-é (1 — u — 211’) + (l + u) magma? (1'8) 0 u cos?- 1- 21/ + cos"2 1 = 2(1 + 10%!- 51 2( 2) (1.9) u, 1r sing-(2 — 2V - cos’%) a 20 11 K cos— 1— 21/ + cos - ucosd ‘ = 2(1+ of .5"- 2( 2 - .3. . -2 - .— ’ 1 -4 u— _1 " q " 832206411130 .. -6 ”_ v-O.45 ‘ P s.- 0.01267 1 r p-OOmrn . _10 Plil'lLJllLllll111111111LIIIIILLILIIIIIJ. -10 -s -s -4 -2 0 2 4 s s 10 X/(J/E€s) Figure 1.5: Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E so), based on equation (1.1) and (1.4) for Sn = 4.137 MPa and p = 0 in plane strain. 10 10 rI T I I I I I I I IN I I I I I I I II I I I I I I I I I I I I r1 I I r I 1.] 6 E Plane stress 5,, = 4.137 MPa : 6 I — "cos/£0 = 1.0/ / .—- \.\ ': : --e.‘/s. - 1.1 ' ./ \ \ ‘ 1 _ _._.-.,/., .. / \ \ : 4 l- ...... eeq/ee . 1'3‘ / 1’ \\ \. . . - . . ." ‘\ \ \ ° . : '. J, \\ . \ : f} 2 _ {I I‘ \ . " —1 g t '1‘ : ' 2 \ O t i I : a + ' - \ ,_ ”'1 r I . : >‘ -2 ,_ {. ’1 . . _ : : x I I l 1 l- . . \ I .1 -4 '- ! \\\‘ I,’ l ' 5 _ : -\ “-', ./ / / d 6 b \ \\‘-—// . .' : E E 3 3240564 MPa \ vj./ : v - " -3 r e. - 001267 ‘— j . p - 00 mm . -10 .1 l 1 l IJ l J l 1 LI 1 l ILL] l l l l l I l l l I 144 l l l IJ l l 1‘ -10 -s -s -4 -2 0 2 4 s s 10 X/(J/Eeol Figure 1.6: Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (.1 / E 6,), based on equation (1.1) and (1.4) for SN = 4.1371 MPa and p = 0 in plane stress. 11 10 _I I IT I I I I I I I I I I I I I I I I I I I I I I I I I IN I r1 I I I IW-1 6 :_ Plane strain 5a = 4.137 MPa 3 s l —--—e.,/e. = 1.0 -' C — —e /e. - 1% 2 _ ————— e e - . . 4 _. ------- 62/5: 8 1.3 _ 3 1 ’3 2 - _ w : - E ~ I ha 0 :- 1 v .. q \ - .. >5 -2 "' -i -4 L _‘ -s L .2 - r: :- 2 06.4 MPa 3 6 :_ v - 0.45 - ' _ e, - 0.01267 1 . p - 0.16 mm . -10 b1 1 l l l l l l I l l l l; l I I l J l 1 l l I l A L 1L1 l l l l 1J4 l ‘ -10-s -s -4 -2 0 2 4 s s X/(J/Eeo) Figure 1.7: Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 6,), based on equation (1.1) and (1.4) for SN = 4.1371 MPa and p = 0.18 mm in plane strain. p O 12 10 -I I I I I I I 1 I I I I I I I I I I I I I I r] I I I I I I I I I I I I I I d : Plane stress /”..\ 8,,= 4.137 MPa : 8 P- - \ . _ l- q C /-— \ \ a 6 *- — o-e,/e. = 1.0 . - : —--e,/e. - 1.1 .. ' ‘ _ -._.-.,/.. - 1.2 / _ \, \ 3 4 ,_ ----- C . 3 1.3 f " ~“\ \ . "i : \\. \ : A .. ° \t\ / q o 2 '- ° ‘ . . -' U : i ] " i=1 . '. ' I \ O b l I e 'fi - . .1 < ~ :1 1 >\ ‘2 :' 1" l \ :l :- 1 ° 1 -4 :. --',//'/ - J . __ / / / Z _6 __ . s .: - E - 2206.4 MPa \_,./ / - *3 L V - (0351267 /. I .. E n '- .. d - p’- 0.18mm \.../ 2 -10.llllllllllllllllllllLJlllIllllillllJllJ -1o-6 -6 -4 -2 0 2 4 6 6 10 X/(J/Eeo) Figure 1.8: Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 5,), based on equation (1.1) and (1.4) for SN = 4.1371 MPa and p = 0.18 mm in plane stress. 13 10 _I I I [j I T 1 Ti I1 I I I l I I I I I I I I I I I I I If] 1 I I I II I. 8 :_ Plane strain 5.4/5o = 1.0 E 6 7- — —s,./a = 0.1478 3 ~ — —s,./a: - 0.6406 - ; ————— Sn/a. - 1.1338 _, j 4 '— \ —l r— . -1 L d f} 2 _ ) _ h 0 L \ < J v : ./ : \ __ ' ... . >4 -2 ~ ) .- C - 2 -4 '- ./ -I -5 L L - s =- 2206.4 MPa 1 8 '_ v - 0.45 - - _ e. - 0.01267 1 C I _10 1 1 LI 1 1 1 L1 1 1 l 1 1 1 l 1 1 1 I 1_1 1 l 1 1 1 I 1 1 1 l 1 1 1 l 1 1 1 -10-6 -6 -4 -2 0 2 4 6 6 10 X/(J/Eeo) Figure 1.9: Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E ea), based on equation (1.1) and (1.4) for £5 = 0.1476, 0.6408 and 1.1338 in plane strain. l4 10 III1TIIIIII'IIIIIITIIIITII—IIIIlejIIII Plane stress .//;/“\e\/eo= 1.0 ‘::Su/ . - a l \ m III'III G III :5” \ QQQ a "II ‘\—o / /' \o———o y/(J/Eeo) 111l11Ll111l111l111l111l1141111J1_L1l11 III'III'IIIIIIIIIII'III'IITIIIII -2 -4 /n. _6 ‘ E'ZZOBAMPQ. v-O.45 ’3 e.-0.0625 _10 111lJ11l1L1l111l111i11411111111l111l11L -10 -6 -6 -4 -2 0 2 4 6 6 10 X/(J/Eeo) Figure 1.10: Plastic zones in coordinate system, non-dimensionalized with respect to the characteristic length parameter (J / E 60), based on equation (1.1) and (1.4) for £5 = 0.1478, 0.6408 and 1.1336 in plane stress. 15 1.2.3 Non-singular region The general or far-field stress dominates in this region, and the importance of the stress intensity factor in describing singular or non-singular domains was observed by expanding the stress field around a straight, mathematical sharp crack tip by an infinite series given by William [13]. 0’51“ = %+B,,~(0)+C,-j(0)\/F+-~ (1.11) Equation (1.11) applies for a crack in a symmetrically-loaded plane, and crack faces are assumed to be traction free, such that uniform perpendicular and shear compo- nents do not exist. The first term Ag,- in the expansion is singular and dominates very near to the crack tip where linear elastic fracture mechanics is valid. The asymptotic elastic stress field of a symmetrically loaded mode-1 crack is expressed in the form given by (1.12 - 1.14)‘and the corresponding displacement fields in cartesian coordi- nates for plane strain and plane stress are given by (1.15) and (1.16) respectively. 1 — singsing 00, K1 0 i. = - ' Q ' 3_9 1.12 a, '—21rrw82 1+ smzsm 2 + 0 ( ) sin-glcos-i,2 0 0'13 = (723 = 0 (1.13) 0 for plane stress (1 14) 0‘33 = —7§—)-—2ym2¥ K' for plane strain fl 9 29 u 606- 1—2u+cos - 1 2(1 11 K; 21' 2( 2) u; E 1r sin%(2 - 21/ - coszg) ‘same as Irwin [9] . 16 l - V2 c030 +%’r ( ) (1.15) —u(1 + u)sin9 0 29 11 cos- 1— 21/ + cos - ‘ -_- 2(1+ 10%I ‘21“ ’( 2 113 1r sin%(2 - 2v - coszg) ucosH 1 — 112 c039 —e33 + %r ( ) (1.16) usinfl —u(1 + u)sin0 Equation (1.11) gives the exact solution for the region r—yO and can also be used in the area where r is small compared to other geometric dimensions of the problem, such as, for example, crack length “ a”. A requirement of a maximum allowable K1 such that K; < 0.6320'y\/E as in ASTM Standard [14], guarantees that the plastic zone size is much smaller than the crack length. The second term 3;; in the expansion denotes the transverse, nonsingular T or 00,, stressl The third and other higher terms in (1.11) are zero near the crack tip, but are significant at large r. Finally terms A55 and 3,3 are the ones which dominate the crack tip process zone. If we know the value of a“. for a given geometry, than we can perform the same analysis as was done for (1.1) and (1.4). A numerical estimation of the non-singular terms was obtained using FEM and later compared with the photoelastic investigation. 1.2.4 Mixed field region In this region, the stress distribution is affected by the singularity effect of the crack- tip and also by the effect of far-field stress. The use of (1.11) is justified only if a few more higher order terms are taken into consideration along with the first singular terms. '0... was proposed earlier by Irwin [8] and this is measured in stress freezing and in moire technique at 90° to calculate the apparent stress intensity factor. Numerically, T stress is measured at 180°. The T stress acts parallel to the crack flanks, as was later proposed by Rice [15]. 17 1.3 Non-Linear Fracture Mechanics Nonlinear fracture mechanics is largely concerned with inelastic effects. Some in- elasticity is almost always present in the vicinity of a stressed crack tip. Depending on material and conditions, the inelasticity can take various forms, including rate- independent plasticity, creep, and phase change. When the zone of inelasticity is small enough, the solution from linear elasticity can be used to analyze, or, more precisely, to correlate, data from test spcimens. Linear-elastic fracture mechanics has found extensive applications to high-strength, relatively brittle materials. For certain fracture phenomena, such as fatigue crack growth and corrosion cracking, the zone of inelasticity is often small enough to use linear-elastic fracture mechanics. However, the more ductile a material, the more likely it is that the inelastic zone will not be small enough at the point of fracture to justify the use of solutions based on linear elasticity. Under this situation it is essential to use solutions to crack problems based on the theory of plasticity. Nonlinear fracture mechanics encompasses a semiem- pirical approach, which for the most part is an extension of linear-elastic fracture mechanics. The unifying theoretical idea behind non-linear fracture mechanics, for rate independent materials under monotonic loading, is the J-integral by Rice [16]. A small-strain deformation theory of plasticity (i.e., small-strain, nonlinearly elastic material) is assumed as the material model. The strain energy density of the material is W(e) with stress given by (1.17), and the path-independent line integral expression for J is given by (1.18). 6W Ugj = 36—” (1.17) J = [(Wfll - Ugjnju;,1)d3 I (1.18) where u is the displacement vector, 6 is the arc length along the contour, and n is the outward unit normal to any counter I‘ encircling the tip of the crack in a contour clockwise direction. The important role of J is in the measurement of the intensity of the near-tip deformation which has 3: singularity. This suggests an intimate relation between J and the near-tip deformation. A more explicit connection is revealed if a power-law relation between the stress and strain is assumed. A widely used uniaxial 18 stress-strain relation is the Ramberg-Osgood form: :40: %+a(-§;)nforaz do (1.19) where a", is an effective yield stress, 60 is the associated elastic strain with E as the Young’s modulus, and a and n are the parameters chosen to fit the data. Asymp- totically, as the crack tip is approached, the contribution to the strain components that depend linearly on stress components are negligible compared to the power-law terms. The power law relation from (1.19) is f. = (1(1) for a 2 00 (1-20) co 00 If J2 deformation theory [17] is used to extend (1.20) to multiaxial states, then n-l .. i = 1.501(1) fifor a' _>_ 00 (1.21) 60 do O'ij 6,, = «1.55.3ng (1.22) where 5.3 is the stress deviator. For a power-law material, the % singularity in W implies a rfi singularity in the stresses, a 1'53 singularity in the strain, and r31"? variations in the displacements. In the vicinity of the crack tip the elastic strain components are negligible as compared with the plastic strains. The plastic part of the strain dominates the asymptotic solution. The asymptotic crack tip stress, strain and displacement fields are given by (1.23 - 1.25), which were derived by Hutchinson [17], Rice [16] and Rice and Rosengreen [18]. 0'51' = do [w] 055(0,n) (1.23) J :31 - Cij — (160 m] 6.1(0,n) (1.24) 19 J r")? aaocoInr u,- = 0601' [ 11,-(9,n) (1.25) Here the dimensionless constant In and 0-variation of the dimensionless function 02,-, 6’},- and 11',- depend not only on 11, but also on the symmetry of the fields with respect to the crack plane, and whether plane strain or plane stress conditions prevail in the vicinity of the crack tip. These variations are normalized by setting the maximum value of the 0 variations of the effective stress, a"e = [351,53]%, to unity where 53-,- = a},- — (ggifigj. With this normalization, the numerical quantities In assume definite values. All the quantities in (1.23 - 1.25) have been defined by Shih [19] and are listed in Table 1.1 for plane strain only. The contribution of 11 allows for a possible translation of the crack itself. There are two conditions to be met in problems where the J-integral can be used. First, the deformation theory of plasticity must be an adequate model of the small-strain behavior of real elastic-plastic materials under the monotonic loads being considered. Second, the regions in which finite strain effects are important and in which the microscopic processes occur must each be contained well within the region of the small-strain solution dominated by the singularity fields. This latter condition is sometimes called J-dominance and is analogous to the small- scale yielding for the yielding requirements in linear fracture mechanics? It was shown by Hutchinson [17], Walker [20] and Schijve [21] for sharp cracks that, in the case of localized plastic yielding, the energy density distribution in the plastic zone is almost the same as in a linear elastic material. This means that, in the presence of localized small-scale plastic yielding, the gross linear elastic behavior of the material surrounding the notch also controls the deformations in the plastic zone. It follows that the complementary energy density in the plastic zone is equal to that calculated on the basis of the elastic solution. Some investigators, including Molski and Glinka [22] and Glinka [23], have assumed that this conclusion was true for notches and mate- rials exhibiting non-linear stress-strain behavior. The first step involves finding local stress and strain at the notch root, from which the crack initiation can be predicted. The most popular and frequently used formulae in notch analysis are Neuber’s rule [7]; the modified version of Hardrath and Ohman [24]; and Stowell’s [25] approximate formula. Based on energy considerations, elastic-plastic notch strain (stress) analysis *J = 11.3211“? in plane strain and J = fila- in plane stress 20 Table 1.1. Table of Hutchinson, Rosengreen and Rice as presented by Shih [19]. n I In 5e(o=0-) 0;r(a=0°) 039(ozoo) I €;r(0=0’) £561qu 5 5.02 0.4621 1.6836 2.2172 -0.0183 0.0183 10 4.54 0.6691 1.7243 2.4969 -0.0156 0.0156 has been developed [22,23]. It has been shown by Hult and McClintock [26] that the elastic stress distribution near different notches are similar to each other and can be satisfactorily characterized by two parameters: the notch radius and the stress con- centration factor KT. The stress state at the crack tip is reduced to uniaxial stress in the case of plane stress and to biaxial stress in the case of plane strain. The avilability of linear or non-linear elastic notch stress-strain solutions enables us to calculate the energy distribution ahead of a sharp or blunted crack tip. Subsequently, the energy density can be translated into the elastic-plastic strains and stresses which actually exist at the crack tip. Hence it is necessary to know the non-linear stress-strain curve of the analyzed material. The way in which the non-linear stresses and strains are computed is very important. 1.4 Experimental Techniques Experimut methods are available for studing elasto-plastic problems in three dimen- sional cracked structures. Hybrid methods are well known for studying such prob- lems. Three-dimensional photoelasticity data was used by Barispolsky [27] in solving the elasticity problem. Balas, Sladek and Drzik [28] have used holography with the boundary element method. Laser speckle photography was combined with the finite element method to study stresses in a compressed disk by Weathers, Foster, Swinson and Turner [29]. Kannien et.al. [30] and Shih et.al. [31] have used experimentally obtained load-timedisplacement data [30] in a finite element model, and then they simulated the field parameters ahead of a growing crack in a hardening material. The applicabilty of a hybrid scheme for differentiating experimental data for determining full field strain distributions has been demonstrated by Segalman [32]. A review on 21 the progress of the above techniques useful in dynamic fracture study was prepared by Kobayashi [33]. Recently Hareesh and Chiang [34] have demonstrated a hybrid method for studying elastic-plastic displacement and strain fields around a crack tip using moire technique in conjuction with the finite element method. There are many other references in this area which will be mentioned in chapter 5. CHAPTER 2 MATERIALS SPECIFICATION 2.1 Application and General Properties Polycarbonate has been used for the past two decades where reliability and higher performance is needed in adverse conditions. Some critical applications are: aircraft wind shields, pipe lines, pump impellers, cams and gears, truck wheel oil seals, foot- ball helmets etc. The combination of high strength and rigidy with toughness under impact loading or in low temperature environment is important, because this is ob tained without the addition of toughening constituents or impact modifiers. Owing to' increased usage of polymers in structural components, a great deal of attention in recent years has been focused towards modern adhesives. Bonded joints tend to be damage—tolerant because of the high damping behavior of the adhesive layer. Poly- carbonate has time dependent properties and it suffers delayed failure. It shows an elastic-plastic tensile instability point or Luders band formation studied by Brinson [35,36]. For a constant stress input, a variety of stress distributions and interactions will be developed in the adherend and adhesive layer. A simple delayed failure, de- lamination, or rupture at an isolated point will not cause the joint or structure to cease transmitting load from one adherand to the other. But a failure at one point will cause a step increase in stress at the next point with a corresponding change in failure or rupture time. 22 23 2.1.1 Material stress and strain curve Mechanical testing was conducted at room temperature, 25°C, and relative humidity 65 % in a 100kN closed loop servo hydraulic materials testing system. An MTS 1.0 inch extensometer capable of measuring strains up to 15 % was calibrated to 0.1 inch full scale deformation. The sensitivity was better than 21:10 11 {lg—fig. Load was measured with an accuracy of 0.3 % and with a precision of 0.01 lbs. Strain-time and load-time data were digitized and stored on an IBM PC. Data were sampled and stored in the hard disc. Analog signals proportional to load and displacement were recorded using X-Y recorders. The uniaxial tensile tests were conducted in displacement control mode as opposed to strain control because of the high elongation expected. Displacement rate was kept at 5 % per minute. Figure 2.1 represents the true stress-strain curves belonging to polycarbonate. The polycarbonate true stress- strain reponses are in agreement with those reported by an Engineering Handbook [37]. The experimental true stress and strain data are listed in Table 2.1, which will be used later for elastic-plastic analysis at the crack tip using incremental plasticity theory. The fracture behavior can be explained as follows. During the elastic loading, molec- ular bonds along the polymer chain are stretched and secondary bonds may be broken to accomodate the imposed elastic strains. At yielding, however, large scale molecular motion takes place, resulting in permanent deformation. Amorphous polycarbonate (LEXAN) is more suceptible to large scale molecular motion in the form of cavitation because there are a high number of heterogeneous nucleation sites provided by the particles. Crystalline polycarbonate, on the other hand, has no preferred sites for the nucleation of cavities. Thus, for large scale molecular motion to take place, molecular entanglement must be broken, requiring a higher stress for initiating the yield stress. In Figure 2.1 the Lexan showed nonlinear behavior, indicating that it does not obey the power law commonly observed for metals. After the yield, there is a sudden drop in the stress, which is then followed by its steady rise with further straining until fail- ure occurs. Such materials behavior is classified as linear or nonlinear elastic regions followed by strain-softening, yielding and neck formation, propagation and further deformation to failure. The yielded region is represented by a plateau. 0‘ (MP6) 100 I I I j I I 1 I I I I T I I I "—‘x 1 r—WI )- I13 25 d - T‘_ __.. 1 .. h- 24 -1 5 I2 1‘- 24 -o-1 80 P- — ]- (All units are lam) . ’1 - 1- ” .1 60 - - ’I b- A ,I’ . l . I I I 1:. z . , - 1 1 4" .1 1 - I I . , 40 , - ,1 . I . I 20 1 o o 0 Experiment - Merlon Pogcar -bonate andbook [35] ' """" o-e, curve _ o ‘ L 4 l L J l 1 L 1 l 1 L 1 l 0.00 0.24 0.48 0.72 0.96 24 6 (mm/ mm) Figure 2.1: 'D'ue stress-strain curve for polycarbonate. 25 Table 2.1: Polycarbonate true stress and strain experimental data. strain(mm/mm) 0.00000008+00 3.32034003-03 6.64060008-03 9.6590000E-03 1.26770003-02 1.9277000E-02 2.58770008-02 3.07770008-02 3.90770003-02 4.46770008-02 5.1577000E-02 6.87770008-02 7.30770008-02 7.86770008-02 8.5277000E-02 9.18770008-02 9.51770003-02 1.01777008-01 1.24877008-01 1.49877003-01 1.71077008-01 1.90877008-01 stress (MPa) 0.00000008+00 7.32600008+00 1.46520008+01 2.13120008+01 2.79720003+01 3.33000008+01 3.79620003+01 4.1292000E+01 4.66200008+01 4.94830003+01 S.26140008+01 5.5944000E+01 S.5611000£+01 5.19480008+01 4.82850008+01 4.46220008+01 4.32900008+01 4.26240008+01 4.28230008+01 4.31560008+01 4.32900008+01 4.39560003+01 strain (run/mm) 3.42677008-01 3.16277008-01 2.70077008-01 2.43677008-01 3.62477008-01 3.82277008-01 4.48277008-01 5.20877008-01 5.60477003-01 5.80277008-01 6.00077008-01 6.26477008-01 6.52877003-01 6.72677008-01 6.92477008-01 7.12277003-01 7.35377003-01 7.58477008-01 8.50877003-01 9.10277003-01 1.07527708+00 stress (MPa) .52210003+01 .51540008+01 .4622000E+01 .42890008+01 .52230008+01 .5154000E+01 .46220008+01 .39560008+01 4.329000OE+01 4.4622000E+01 4.5954000E+01 4.72860003+01 4.86180003+01 4.9284000E+01 4.99500003+01 5.0949000E+01 5.16150003+01 5.2614000E+01 5.69430008+01 6.06060008+01 7.05090008+01 151515151thth 2.1.2. Tensile load-elongation curves for CT specimen Strain gages were used to measure local strain in CT specimen. Strain gages were bonded very near to the notch tip of each specimen. Two such specimens were then bonded using epoxy resin in such a way that local strain could be measured in the mid-plane as well as in the surface plane. Local stress and strain plots at the notch tip are shown in Figure 2.2. Material true stress -strain curve obtained from tensile testing up to 2 % strain is superimposed. Under slow tensile loading, a sharply notched specimen fails in a brittle manner, and a bluntly notched specimen fails in a ductile manner. Amorphous polycarbonate, therefore, affects the ductile-brittle transition of the fracture mode by varying the notch geometry and also the thickness. More experimental evidence of fracture in amorphous polycarbonate and the limiting condition for the transition of fracture mode from ductile to brittle is shown in Figures 2.3 and 2.4, which were taken from Nisitani [38]. In Figure 2.3 at point A, the shear bands are visible inside the specimen; and the plastic zone is made up of many fine individual shear bands as shown in Figure 2.4 (a). 26 130 I T I I I I 1 T j I I I I I I l I I I 0 r- 0 '1 1 l- 0 -1 0 o .1 104 '- ' o —] >- 0 " 0 r . 4 b o d s 78 L ' - o ' ' 2 U1!- " s I I ‘ Ed _ o' . v' °° 52 - " ° 0 _. O o O ./ 0 0° 0 0 ° o O I 1- 0 q o o a True a-e curve (Fig. 2.1) 23 t. 0 ° ° Mid-plane (SC) _ v v v Surface plane (56) ./ . .4 l l l l l l l J l 1 i 1 l I l 1 J 1 l 00 0.04 0.08 0.12 0.16 0.20 e (mm/mm) Figure 2.2. Experimental notch strains in a sharp notched compact tension specimen made out of polycarbonate. 27 At point B, the plastic zone size expands to about the size of the notch root radius of the specimen, and then a small crack is nucleated at the tip of the plastic zone as shown in Figure 2.4 (b). However, for B, the crack nucleated inside the specimen and shear bands are formed on the surface as shown in Figure 2.4 (c). A bluntly notched specimen failed in a ductile manner with large shear bands. The plastic zone at point D in Figure 2.3 is shown in Figure 2.4 ((1). At E in Figure 2.3, the plastic zone is as large as the size of the notch root radius of the specimen. The process in which the plastic zone expands to the size of the notch root radius is similar to that of the sharply notched specimen. It was also reported by Nisitani [38] that with a notch root radius p = 0.2 mm, all the specimens of 1 mm thickness failed in a ductile manner and all specimens of 2 mm thickness failed in a brittle manner. With p = 0.5 mm, all the specimens of 5 mm thickness failed in a brittle manner, independent of notch depth, c. The general conclusion is that the mode of failure depends on the notch root radius and the thickness in amorphous polycarbonate. 28 2.5 l J . _ F r . 2 mm :- . _ ' p . .13 I . - h . . I F A 2.0 ’ I 2 J \E X 2 .l 0 ' C I 1.5- I a... _‘i \lD G D 3 P . 0.5 l\ " c g 3 1 - Ductile fracture 10}- c , ‘ ,g . a ,. ‘ . 2 Brim. fracture ,g A 0.5 - , . P 8 0.2 . C I 3 . I o l l l J 1 1 0 0.5 L0 1.5 2.0 2.5 3.0 3.5 4.0 Elongation ( mm ) Figure 2.3. Tensile load-elongation curve for two diherent specimens of 2mm thick- ness. The sharply notched fails in a brittle manner and bluntly notched specimen fails in a ductile manner [38]. (e) Figure 2.4. Optical micrograph near root of the specimen in Figure 2.2: (a) internal plastic zone at point A; (b) internal small crack at the tip of plastic zone at B; (c) surface shear bands at point B; (d) internal plastic zone at point D; (e) internal plastic zone at point E; (f) surface shear bands at point E [38]. CHAPTER 3 STRAIN ENERGY DENSITY APPROACH 3.1 The Continuum Mechanics Model The continuum mechanics model for a notched body is based on only equilibrium equations, strain displacement relations and boundary conditions, without using any constitutive equation, as mentioned in the introduction. Consider a notched body of linear or nonlinear elastic material such as in Figure 3.1a, free of body force, but subjected to a slowly increasing surface force on the portion QT of the boundary. Suppose that the maximum applied force on the boundary fly- is denoted by traction T;, the stresses denoted by 0,-1- are in equilibrium within the body, i.e., 0;,- = 0; 0.3 = 01-.- (i, j = 1,2,3), and thus satisfy boundary conditions on (IT and on the notch surface PT. On the boundary surface QT, Uijnj = T.-, and on the notch surface I‘T, 0.373;; = 0, where n, is the unit vector normal to the surface boundary and directed towards the exterior of the body. The infinitesimal strain tensor is defined by q,- = ug'j'j. Once the stress and strain tensors 0;,- and 6;,- are known, the strain energy density per unit volume is given by; 29 30 NOTCH E l ASIlC-PLAST IC ZONE lp (b) Figure 3.1: Schematics of notched body extending to a cracked geometry; (a) A smooth sharp notched body and (b) A sharp pre-cracked compact tension specimen. 31 j, (uses-W = [V (.,,...),dv (3.1) [v (agjegj)dV= [V (angu;+a,-ju,-,j)dV (3.2) [yes-cw = fvwsuswv (3.3) [yoga-adv = [Smuads (3.4) The term 0,-1- J-ugj is zero from the equilibrium condition. Consider now a compact tension specimen of elastic-plastic material with a notch as shown in Figure 3.1b loaded remotely by load P, which causes yielding of the material in a zone defined by R, near the crack tip. Near the crack tip, We have an inelastic region, a singular region, a mixed-mode region, and a non-singular region. In the singular region, stress distribution is best described by (1.4). In the mixed-mode region, stress distribution is affected by both the singularity of the crack tip and the far field stress. In the inelastic region, the crack tip may not be in the elastic state and the tip may become blunt due to local plastic yielding, which invalidates (1.4). In small scale yielding, when the plastic region dimensions R, are negligible compared with specimen geo- metric dimensions, the surrounding elastic singularities set the boundary conditions on the elastic-plastic boundary value problem. That is to say, the plastically yielding material can be described by the surrounding stress field through the inverse square root term in the elastic solution. This stress field is approached by the elastic-plastic solution at a distance which is large compared with the plastic zone size but still small compared with the other geometric dimensions. Thus, the stress field under the small-scale yielding solution for any loading may be obtained by considering a semi— infinite crack, with the assymptotic boundary conditions given by (1.12). If R3 2 R, is satisfied, then from (3.4) traction, T.- = 0.3-n,- = 0.3-(Maj; and displacement vector, u.- = 11“,”). In small scale yielding the total strain energy is the sum of strain en- ergy caused by elastic-plastic strain W(e),, and complementary strain energy density caused by elastic-plastic stresses W(a’)c,. It is given as; 32 ./V (UgjégjdV) = ./V [d(a.-,-e.-,-)] dV (3.5) /V(O','j£.'j) dV = l/‘I(0;jd£;j)dv+£,(£;jdagj) dV . (3.6) [V (W66) dv = We)... + War)... (3.7) Equation (3.7) holds good for proportional loading, where it can be assumed that the plastic flow occurs in the direction of the deviatoric stress minus the back stress for forward loading. Now taking, on, = ‘/l.5D.'jD.'j (3.8) Gutsy 3 em = V 2%‘11 (3.10) 3.1.1 Elastic-plastic stress and strain at the notch tip (3.9) 0;,- = 0:3 - The elastic strain energy density at the blunt notch tip W(C)ne can be calculated; 0'3. 213 (3.11) C.“ 1 l W(e),,e = [o 0.3-deg,- = 5 (amen) = In the case of elastic material behavior, the local stress at the notch tip in plane stress is given by (3.12 - 3.13). 0W 2 One = SnKT ‘ (3.12) 33 axz = 0:2 = 0 (3.13) For plane strain the stress is given by (3.14 - 3.16). From (3.11),(3.12) and (3.14) we obtain (3.17). 0w = one = SnKT (3.14) a“ = 0 (3-15) on = vane (3.16) W(e),,c = (SE—1:31): (3.17) The strain energy density at the notch tip created by nominal stress 5,, is W,, = 33/219. Finally, the relation between the strain energy density at the notch tip caused by one and strain energy density caused by the nominal stress 5,, is W(c),,, = WnKT. It was shown for cracks by Hutchinson [17] and Hult and McClintock [26] that, in the case of localized plastic yielding, the energy density distribution in the plastic zone is almost the same as in the linear elastic material. This means that in the presence of localized small scale plasticyielding, the gross linear elastic behavior of the material surrounding the notch also controls the deformation in the plastic zone. The strain energy in the plastic zone W(e),,,, is given as; W(£),,p = W(€),,e = WnK12~ (3.18) It is known from Glinka [23] that the stress state in the plane stress condition is reduced to uniaxial stress, and that in the plane strain condition the stress state is reduced to biaxial. It is apparent that in the plane stress condition a uniaxial stress state exists at the notch tip, and therefore uniaxial stress-strain relations can be used. A basic assumption made in the development of stress-strain relations for elastic- 34 plastic material is that, for each load increment, the corresponding strain increment can be decomposed into elastic and plastic components. The strain energy density W(c),, is simply given by; W(e).,, = f 0:3 (deifinc) + d5ij(np)) (3-19) W(c)¢, = W(C)ne + W(c),,p (3.20) Now, if the material shows elastic behavior at the crack tip, the elastic strain at the notch tip cm is given by (3.21). em = O’ME (3.21) In actuality, at the crack tip the material shows elastic-plastic behavior inside region Rep, and the corrsponding elastic-plastic strain 5,, is given by Raske and Morrow Dean Jo [39] and ASTM Standard [40] as the exponential rule in (3.22). 1 = a 162) .. a, E + ( K (3.22) Energy density at the notch tip may be calculated on the basis of a non-linear stress- strain curve by the Ramberg—Osgood relationship. ' .. 2 1. _ "’ .. .. .. ”_er; _afL) (”_ev)“ we)”- [0 a,,dc,, _ E + ("H K (3.23) The first term in the right hand side of the above equation shows the elasticity behavior, and the second term shows the plasticity behavior. In fact, when dealing with the plastic zone, assuming small scale yielding such that Re >> Rm it is possible to write from (3.23); 1 W(5)ep = (1_1-II ) aepcep (3.24) 35 W(a¢,) = nW(c)¢,, (3.25) On the boundary of the elastic-plastic regime shown in Figure 3.1b, when R. >> R, La;,-c,-jdV=/Va,,ecncdv (3.26) From (3.20), (3.24) and (3.26) on the boundary R, for small scale yielding, [v [qW(c)., —- W(c),,,] dV = o (3.27) Since the strain hardening exponent n varies from O to 1, then q = 112—"2 for -;- < q < 1. Equation (3.27) is written as; [qW(c)¢,, — W(c),,c] dV = 0 (3.28) In other words, from (3.28); 0.36,,- = amen, ’ (3.29) Equation (3.29) means that the strain energy density for a plastic regime is the same as that of the purely elastic solution under mode-l loading for a piece-wise linear element, i.e. W(c),,, = W(c),,¢. Substituting (3.23) into (3.18) and using (3.17); (3)K%=—+(——>(3)* (a...) where n is the strain hardening exponent, K is the strength coefficient factor, and E is Young’s modulus. The material stress-strain curve showed non-linear behavior below the yield limit, and so the modified energy density due to nominal stress is given by (3.31). It is also assumed that the plastic strain increment is proportional to deviatoric stress at any instant of loading. If the nominal stress 5,, is beyond the proportional limit, then; 36 J. 5?, 5,, 5,, n W" - 272' + (n +1) ('3‘) (3-31) substituting (3.31) into (3.30) and using (3.17) 3:2: Sn Sn I 2 03p 06p ac? % 273 (n+1)(7) ]KT-‘E+(n+1)(7<‘) (332) Therefore, inelastic stress and strain at the blunt notch tip in plane stress can be calculated from (3.33) and (3..34) W(e),, = MK; (3.33) 3., 0,, 3 6,, = ‘E’ + (7?) (3.34) In the plane strain condition, a biaxial state of stress exists and hence thelrelationships given by Dowling et.al [41] for the translation of the uiaxial stress-strain curve aep—ccp into the biaxial plane strain relation 0,,(5) — 6.3,“) have been used. U-flfi e :5 -———— l 3.35 .3320’) 319m ( ) “er a’ = -————- 3.36 6P“) m ( ) ——=—"" (3 37) E: ' y+_E_‘fl. ’1: Substituting (3.35) and (3.36) into (3.34) gives the results for plane strain condition. W(C)ep(b) = W,K% (3.38) 37 I = 0cm) (0mm); 6ep(b) E + K (3.39) and from equation (3.34) 6,, = YE? + 5w = écc + 6,, (3.40) where 6,, is the elastic part of total strain and cm, is the plastic part of total strain. The inelastic stress and strain at the blunt notch tip are calculated from (3.33) and (3.34) for plane stress or else from (3.38) and (3.39) for plane strain, for a given nominal stress 3,, and stress concentration factor KT. It is necessary to know first the variation of stress concentration factor KT ahead of a blunt notch tip (i). A 3D elastic finite element analysis was done using coarse mesh for this geometry using NISA. The radius of this notch is given as 0.1 mm, which yet may be considered as a crack with finite radius. The results are given in Table 3.1. The variation of theoretical stress concentration factor K1 in the mid-plane, quarter-plane and surface plane is shown in Figure 3.2. 38 Table 3.1: Stress concentration factor variation with normalized distance from the crack tip. X/a 0.0000000E+OO 6.00160108-04 1.2003198E-03 1.80047998-03 2.40064008-03 3.00079913-03 3.60096018-03 4.20112018-03 4.80127918-03 5.40143918-03 6.00160018-03 6.60176018-03 7.20192018-03 7.80208118-03 8.40224018-03 9.00239918-03 9.60256012-03 1.02027198-02 1.08028808-02 1.14030398-02 1.20032013-02 1.26033623-02 1.32035218-02 1.38036808-02 1.44038412-02 1.50040003-02 1.56041598-02 1.62043208-02 1.68044798-02 1.74046398-02 1.80048003-02 1.86049618-02 1.92051208-02 1.98052818-02 2.58068808-02 3.18084838-02 3.44510148-02 4.12247058-02 4.79983938-02 5.47720858-02 6.15457738-02 6.83194612-02 7.50931493-02 8.18668448-02 8.86405398-02 9.54142328-02 1.02187923-01 1.08961628-01 1.15735318-01 KT (MPL) 4.14009288+00 3.970267SE+00 3.8197594E+00 3.6851673E+00 3.56386593+00 3.4538043s+00 3.3533514E+00 3.2611802£+00 3.1762137E+00 3.097559SE+00 3.02447512+00 2.95632823+00 2.89259313+00 2.8328097E+00 2.7765844£+00 2.7235801E+00 2.67350138+00 2.6260834E+00 2.58110712+00 2.53836133+00 2.49767283+00 2.45888022+00 2.42183958+00 2.386424BE+00 2.35251813+00 2.32001888+00 2.28883003+00 2.25886688+00 2.230049IE+00 2.20230793+00 2.1755764E+00 2.14979SBE+00 2.1249087E+00 2.10086808+00 1.89815123+00 1.74467663+00 1.62160283+00 1.5159901E+00 1.4174399E+00 1.32547963+00 1.23966888+00 1.15959613+00 1.084877BE+00 1.01515613+00 9.50096618-01 8.89387678-01 8.32738343-01 7.79877078-01 7.30550658-01 KT (QPL) 3.2450964E+00 3.111983BE+00 2.99401213+00 2.88851593+00 2.793437ZE+00 2.70716838+00 2.62843118+00 2.55618528+00 2.48958688+00 2.4279358E+00 2.37065058+00 2.31723SSE+00 2.2672787E+00 2.22041928+00 2.1763484E+00 2.13480238+00 2.09554968+00 2.0583823E+00 2.023129OE+00 1.989623BE+00 1.95773128+00 1.92732488+00 1.89829ISE+00 1.87053268+00 1.84395583+00 1.81848223+00 1.79403583+00 1.77054998+00 1.74796198+00 1.7262179E+00 1.70526508+00 1.685057GE+00 1.66555058+00 1.6467069E+00 1.487813OE+00 1.36751628+00 1.27104823+00 1.18826683+00 1.1110209E+00 1.03894038+00 9.71679938-01 9.08917198-01 8.50351333-01 7.95701988-01 7.44706818-01 6.97121808-01 6.52718788-01 6.11284978-01 5.72621768-01 KT (SPL) 2.03000818+00 1.9467379E+00 1.87293938+00 1.80694508+00 1.7474674E+00 1.69350108+00 1.6442459E+00 1.5990517E+00 1.55739OZE+00 1.5188237E+00 1.48298823+00 1.4495739E+00 1.41832288+00 1.3890092E+00 1.36144033+00 1.33545063+00 1.31089563+00 1.28764SZE+00 1.26559218+00 1.24463ZSE+00 1.2246817E+00 1.20566063+00 1.18749858+00 1.17013363+00 1.15350828+00 1.1375729E+00 1.12228018+00 1.1075883E+00 1.09345818+00 1.07985583+00 1.06674863+00 1.054107SE+00 1.0419047E+00 1.03011688+00 9.30718848-01 8.55465838-01 7.95119113-01 7.43334178-01 6.95012158-01 6.49921308-01 6.07845783-01 5.68583793-01 5.31947268-01 4.97760688-01 4.65860078-01 4.36092738-01 4.08315908-01 3.82396468-01 3.58210278-01 1.22509018-01 1.29282712-01 1.36056398-01 1.42830098-01 1.49603788-01 1.56377488-01 1.6315117E-Ol 1.69924878-01 1.76698558-01 1.8347223E-Ol 1.90245918-01 1.97019608-01 2.03793288-01 2.10566968-01 2.17340648-01 2.24114348-01 2.30888028-01 2.37661703-01 2.44435388-01 2.51209068-01 2.57982778-01 2.64756453-01 2.71530133-01 2.78303818-01 2.85077498-01 2.91851178-01 2.98624853-01 3.05398548-01 3.12172253-01 3.18945938-01 3.25719618-01 6.8452281E-01 6.41572713-01 6.01494858-01 5.64096938-01 5.29199788-01 4.96636218-01 4.66250188-01 4.37896163-01 4.11438148-01 3.86749333-01 3.63711548-01 3.4221417E-Ol 3.2215443E-01 3.03436078-01 2.85969418-01 2.69670758-01 2.54461918-01 2.40270188-01 2.27027428-01 2.14670238-01 2.03139338-01 1.92379478-01 1.82339188-01 1.72970288-01 1.64227872-01 1.56070078-01 1.48457758-01 1.4135450E-01 1.34726248-01 1.28541228-01 1.22769778-01 39 5.36544148-01 5.02878968-01 4.71464998-01 4.42151678-01 4.1479856E-01 3.89274488-01 3.65457248-01 3.432327ZE-Ol 3.22494338-01 3.03142703-01 2.85085178-01 2.68235068-01 2.52511808-01 2.37839943-01 2.2414917E-Ol 2.11373933-01 1.99452898-01 1.88329108-01 1.7794913E-Ol 1.6826329E-01 1.59225128-01 1.50791308-01 1.42921498-01 1.35577953-01 1.28725458-01 1.22331198-01 1.16364483-01 1.10796798-01 1.05601428-01 1.00753468-01 9.62296723-02 3.35641508-01 3.14581813-01 2.94930468-01 2.76593158-01 2.59482098-01 2.4351522E-01 2.2861606E-01 2.14713268-01 2.01740128-01 1.8963447E-01 1.78338363-01 1.67797603-01 1.57961718-01 1.48783568-01 1.40219148-01 1.32227448-01 1.24770103-01 1.17811483-01 1.11318168-01 1.05259078-01 9.96051438-02 9.43292688-02 8.94062153-02 8.48123738-02 8.05257198-02 7.65257118-02 7.27931638-02 6.93102338-02 6.60602088-02 6.30275098-02 6.01975998-02 (*) crack tip is assumed to have a radius (r) of 0.1 mm, and a is the length of the notch. Note: The data depends on the thickness of the specimen used in 3D finite element analysis. There could be a numerical error in data extrapolation, which should be verfied before use for a given problem. 40 3.2 1 _°'—”' Mid plane —] 1. ] °——" Quarter plane P - °°°°°°°° Surface plane \. _. .- .\.\. ; i\\.. 3 151‘ \\ d )- \.\ \ \ d 0.3 L \\ \. \\ : - \ . ‘ b \ \ \ . ..\.'\ .. . \.\.\ .\...\ d \.‘.~:~...\.u . o.opnntnl....1.n.. ...>1‘E‘T Figure 3.2: Stress concentration factor (KT) variation in a CT specimen for a/W = 0.4966, a/ t = 0.9372 and 0 = 0° based on 3D elastic finite element analysis. 41 Monotonically increasing the nominal stress 3,, for a fixed value of K T defined by the normalized distance results in the local stress and strain curve for both plane stress and plane strain as shown in Figure 3.3. Superimposed is the Nueber’s flow rule for comparison sake only. Similarly, varying KT with respect to the normalized distance at a fixed nominal stress 5,, results in the local strain or stress distribution ahead of a notch tip, which will be discussed in the next paragraph. 3.1.2 Plastic zone size and shape at the notch tip In (1.1) it should be noted that the origin of polar coordinates is at a distance of g behind the notch tip. For 1' = g the stress intensity factor can be calculated in terms of the stress concentration factor for deep notches, KI = KTSn % ' (3.41) Substituting (3.41) in (1.1) for the stress intensity factor K1, we get A g , _ P .. 3.5,. s .. aa—Krsnfigfuww «3‘2 () 9,,(9) (3.42) where f,-,- and gij are given in (1.2) and (1.3). The strain distribution obtained from a 2D analytical elastic solution‘for SN = 4.137 MPa, SN = 17.927 MPa and SN = 31.717 MPa were illustrated in Figures 1.3 to 1.5. The effect of finite notch radius increases the strain distribution far from zero notch radius, also strains in plane stress are larger than in plane strain condition. The 2D analytical solution assumes that on the free surface 0,, is zero in plane stress; and also in 2D, K7 is constant. This limitation causes discrepencies in experimental measurements of strain in finite notch or cracked thick geometries? ‘converting (3.42) using stress-strain relation for plane stress and plane strain. lA comparison of three dimensional normalized stress variation with distance between 3D an- alytical elastic solution given by Sih[49] and a finite element solution is given in Figure 5.1. The results are quite comparable. 42 130 I I I I I I I T I I I I I I I y I l I 104 — /. ,x" - 78— - us?- i Z :2 L . 52- - Z I. _ o o 0 True a-e curve (Fig. 2.1) 25 1" --- Surface plane (SED) _ ——-- Mid-plane (SED) ‘ —-—-- Surface plane (Nueber) ‘ _ ------ Mid-plane (Nueber) . o o o Mid-plane (86) v v v Surface plane (56) . O 4 1 J l 1 1 1 l 4 1 1 L 4 4 1 l 1 1 1 0.00 0.04 0.08 0. 1 2 0. 1 6 0.20 e (mm/ mm) Figure 3.3: Theoretical and experimental notch strains in a sharp notched compact tension specimen made out of polycarbonate. 43 The first approximation of plastic zone size 1‘, at the notch tip was derived from the Von-Mises yield criterion on the basis of elastic stress distribution. KTSn P P )3 = — 3.43 2f \10 (:- P) + fl (7‘, ( ) for plane stress; 0 . 0 . 30 0 0 0 30 a = 6 [60353212533123]: + 2c cos-é- + 6 [sinzcos-écos—f]2 (3.44) __ 3 2 39 3- . 2 30 fl — -2- [cos ('3) + -2-sm (~2-)] (3.45) for0=0°,a=2andfl=% MIX/Si JV) +2093 . (3.46) for plane strain; for0=0°,a=(2+8u2_8y) andflzg- ___KTS“ .8. V2— V g i 3 0,... Ni \ (7,) (2+8 8 )+4 (1,) (3.48) In the above equation aya,KT,S,,andp are known and so 1‘, can be found. By 2D elastic solution, the stress distribution is known for 0 varying from 0° to 180° for a given SN and constant KT from which plastic zone sizes are calculated (refer to 44 Figures 1.5 to 1.10. However in 3D case, the situations are different and will be explained in the coming paragraphs. 3.1.3 Increment in the plastic zone size at the notch tip Since the value of r, is known the force can be calculated; (refer to Figures 3.4 to 3.6) F1 = A, aydr - ay(r,)(rp — 0.5p) (3.49) 2 _KTS" w{ \/':” _ \/-;} — a,,(r,,)(r,D — 0. 5p) (3-50) where; W) 1:33. {fififif} (3.51) Because of the plastic yielding at the notch tip, the force E1 cannot be carried through by the material in the plastic zone rp; but, in order to satisfy the equilibrium condi- tions of the notched body, the force has to be carried through by the material beyond the plastic zone Arp. As a result, stress redistribution occurs and the plastic zone 1‘, is increased by an increment Arp. In reality E1 74 F1,E2 95 F2 and E3 sé F3, but it is assumed that E1 = F1,E2 = F; and E3 4—- F3 and F1 > F; > F3. Thus the plastic zone increment Ar, can be calculated, if E1 = F1 = a,(r,,)Ar,. Hence 433:1 1cm . <31 An interpretation of plastic zone size increment is shown in Figure 3.7, and a better Ar, = — 0.5p) (3.52) explanation will be given in Chapter 5. 0.020 . f . . . . . . . S" = 4.137 MPa :1 0.01s { 0.012 —---md plane 5 i . ——°- Quarter plane - W 0.008 :‘ —-—-- Surface plane _: u. . ..\”‘~.. 3 0.004 _. _ \H“~__-——: 0.000 1 4 l 1 1 1 1 l 1 1 J 1 1 I 1 1 1 1 : 0.000 0.004 0.008 0.012 0.016 0.020 x/a Figure 3.4. Crack opening strain re-distribution in a CT specimen throughout the Specimen thickness 8/“, = 0.4966 and 9 = 00 based on ener densit a toach loaded to 4.137 MPa. ‘y Y PP . 0.040 I I fi I I I I I I I I I I I I I I I I I I I I u .1 5,. = 17.927 MPa : 0.032 " E, J '. \ E. : l . 1 0.024 ' F, —--- Mid plane 1 k " . F. ——-- Quarter plane 1 w \ . ‘- F, —-—-- Surface plane . 0.016 - - \ A. -. £0 1 ' \ N. ..\.. : 0.008 \-\4 \QHNM 1 "~~ ........ 21:15..“‘4 0.000 4 . ' - 1 . 1 1 ‘ 0.00 0 04 0.08 O 12 O 16 O 20 Figure 3.5. Crack opening strain re-distribution in a CT specimen throughout the specimen thickness a/W = 0.4966 and 0 = 0° based on energy density approach, loaded to 17.927 MPa. 46 0.080....,...-r.......s.,. ‘ 5,. = 31.717 MPa : 0.064 g 0.048 F, —--- Mid plane -: E F. -—-- Quarter plane - w 1“, -—-—-- Surface plane I 0.032 1 0.016 . k E S - «3 _.;.-..'_i£_——._: ‘--—'““‘= -——-~ 0000, . . . 1 Ti: . l . . . . J.—1.1_.1—1..T.T—.1—-1.—4.‘ 0.00 0.08 0.16 0.24 0.32 0.40 x/a Figure 3.6. Crack opening strain re-distribution in a CT specimen throughout the specimen thickness a/ W = 0.4966 and 0 = 0° based on energy density approach, loaded to 31.716 MPa. ‘ i " " c' -a q Figure 3.7: Interpretation of plastic zone size increment. 47 3.1.4 Elastic-plastic stress and strain at the crack tip The elastic strain energy W,,e in the case of a mathematical sharp notch or crack tip is calculated from the stress intensity factor solution K1, given in an ASTM Standard [42]. The ordinary stress concentration factor is no longer useful. For a CT specimen; K, = 553‘ f (10?) (3.53) 11%) = [2 + 1 3;] [0.886 + 4.64 (%) —- 13.32 (57)? (3.54) where B is the thickness, W is the width, a is the crack length, {:7 is the geometric constant and P is the applied load. The elastic stresses in x and y directions are equal for 0 = 0, and, subsequently at r = a; W... = «lg—7:; (1 g”) (3.55) The stress components ahead of the crack tip are relatively high, and none of them can be neglected. The strain energy density in the plastic zone Whip) is the summation of x and y components. 0’ a 3 ”v as 1'- wnp=fa+~ _ \\.\. '\.\ . \. "J \ \ 4 \ . . ' )- i\\._/ \. .1 . \. 1 . \,\ /- -0.05 *- ‘-' - ' —-—-—Mid plane ‘ - -- Quarter plane .- --- Surface plane 0 o oEmbedded moire ‘ -O.10 1 1 1 1 I 1 1 1 1 L 1 1 1 l 1 1 1 1 -0.10 -0.05 0.00 0.05 0.10 X/ a Figure 3.12: Estimation of plastic zone size based on energy density criterion for mid-plane and surface plane loaded to 17.927 MPa. 57 0.150 I I I I I T I I I 0.075 — m b \ 0.000 > b -0.075 - ’ --—-—Mid plane . -° Quarter plane Surface plane 0 e oEmbedded moire _o.150 1 1 1 1 l 4 1 1 1 1 l l l J -0.150 -0.075 0.000 X/a Figure 3.13: Estimation of plastic zone size based on energy density criterion for mid-plane and surface plane loaded to 31.717 MPa. 0.075 CHAPTER 4 NUMERICAL ESTIMATION OF N ON-SINGULAR STRESS 4. 1 Introduction The purpose of calculating the non-singular stress 00, numerically using finite ele- ments is to compare with the experimental stress freezing technique. Experimentally it is difficult to find the effect of the non-singular stress in elastic-plastic stress analy- sis of the deformed crack tip unless a hybrid technique is employed. The second-term in (1.12) is a constant elastic term and is geometry dependent. Once we know the non-singular constant term, there is no need to carry out real experiments in full scale rather than depend on numerically obtained results? At the same time, compu- tational methods are playing an increasing role in fracture mechanics, but often they lack detailed experimental evidence supporting the inevitable modeling assumptions. Here, the two approaches will be conglomerated to see the best way to understand the effects of non—singular stress. The effect of this non-singular term for different materials will give different results. Three different materials have been analyzed; materials that obey the Ramberg- ‘The analysis for CT specimen was done at MIT by the author using ABAQUS [4]. 58 59 Osgood law, the modifiedIRamberg-Osgood power law, and, finally the experimental stress-strain curve for polycarbonate. The first two are solved based on deformation plasticity, and for polycarbonate the incremental plasticity has been used in plane strain only. There are many methods for evaluating T or 00, stress but only the technique shown by Larsson and Carlson [47] will be used. The total x-direction stress 0,, is determined from the solutions obtained by the elastic finite element method for a given specimen and for the boundary layer problem for elements with one side on the crack surface. T stress for a CT specimen with different crack depths was analyzed by imposing a distributed load around the plane of the loading pins, the results of which are then compared with that of point loading at the pin center and of shear traction applied in the plane of pin center (Leevers and Radon [44]; Cotterell [45]; Kfouri [46] and Larsson and Carlsson [47]). 4.1.1 Modeling of CT Specimen and Boundary Conditions The CT specimen with dimensions from ASTM E—399 [42] is shown in Figure 4.1a. Each of the CT specimens are characterized by dimensionless parameters 5, fl, and u; where a is crack length, W is the width, V is poissons ratio, E is young’s modulus and 0y is yield strength. In the finite element formulation exact notch dimensions are not specified but a simpler one is shown in Figure 4.1b. Because of symmetry only meshes for the upper half of the model are shown. The crack tip was modeled by a semicircular domain on the plane y = 0. There are 32 fans of elements circumferentially, and 32 rings radially. In the plane strain case, a second- order-reduced integration, eight node quadrilateral element-type GPE8R was used. The ratio of radius of the outer boundary to the radius of the first ring elements was of the order of 2.775E+07. The first ring elements were degenerated, so as to collapse a side into a single point at the crack tip. The mesh definition for the remaining part of the geometry is self explanatory, except in the pin loading region. The mesh is sufficiently fine to minimize load distribution irregularities. However, they cannot be completely removed. The semicircular domain has 1040 elements. The mesh is sufficiently fine here. tThe UMAT to be used in ABAQUS was given by Dr. Y. Wang [48]. 60 Yeah». of starter notch plus creel: Sn- 29(21 + a) ] p (L-a 3 1 Surface plane (SPLI x1 a/t- .700 - | 1 ‘ 1.1__,_,.__ Quarter plane (QPL) L X3 1 ::% Hid plane (EFL) ' 1 I 2 1 l 1 1 1 X2 I I ”.1 ' : EF- fio———I-—‘Ju 1i 1 Figure 4.1. CT specimen used in energy based formulation, elastic and elastic-plastic finite element analysis, 3D photoelasticity and multiple embedded moire technique, (a) Dimensions according to ASTM E—399 [42] and (b) Loading and boundary mn- ditions applied in FEM analysis. 61 In Figure 4.2 (a - e) all the nodes starting from 1 to 12801 at an increments of 400 are given 11,, = 0 and at node 12801 both 11, and 11,, are zero, to constrain the rigid body motion. Since degenerated collapsible elements have been used at the crack tip, these nodes are given displacement boundary conditions to match with node 1, using “equation [4] ”. This applies only for the elastic solutiOn. A numerical error was encountered by not imposing this additional boundary condition. The specimen was loaded at the plane of the pin by giving a distributed nodal force obtained by using the shape function of elements. Loading on these types of specimens are complex in nature; but, by Saint-Venants principle, the effect of loading would not be felt near the crack tip. Constraining nodes 3 to 129 similar to node 1 (Bias=%) would result % point concept. A sharp crack tip, and second order isoparametric elements can have such a strain singularity if they are focused at the crack tip. The only advantage obtained is that one can adjust the position of mid nodes by “ Bias [4] ” rather than by “ Singular [4] ”. In the elastic solution it does not matter whether a quarter point or collapsible technique is used, but additional boundary conditions should be applied for the collapsible technique. For elastic-plastic analysis where blunting takes place, these additional boundary conditions do not apply. 4.1.2 Modeling of Boundary Layer Problem and Boundary Conditions Plane-strain crack-tip deformation was modeled by boundary layer formulations using focused meshes of the type shown in Figure 4.3. The mesh typically involved 2048 eight node second order, reduced integration type GPE8R elements. It consists of 64 fans of elements circumfrentially, and 32 rings radially at equal angular intervals 0=5.45°. The radius of the first ring of elements was less than one-hundred thousandth of the radius of the outer ring. However the semicircular domain of radius = 100 in the boundary layer model has the same number of elements, nodes and coordinates as compared with the semicircular domain present in the CT specimen for T stress analysis. Near the crack-tip, degenerated collapsible elements have been used and nodes corresponding to these elements were given displacement boundary conditions to match with node 1. Displacement boundary conditions were imposed on the outer boundary corresponding to displacements associated with the K field. 62 Figure 4.2: Finite element modelling of CT specimen with a/W = (a) 0.3958, (b) 0.4375, (c) 0.4790, (d) 0.5208 and (e) 0.5625. 63 Figure 4.3: Circular domain inside the CT specimen used for boundary layer and modified boundary layer problem. 64 4.2 Estimation of Elastic Non-Singular Stress In the present analysis, the total x-direction stress 0,p is determined from the solutions obtained by the elastic finite element method for the specimen with a given 5 ratio and for the boundary layer problem for elements with one side on the crack surface. Load levels are chosen in such a way that K; < 0150de for all 5. In the first set of experiments, K1 was kept constant for all 5 by lowering the appropriate far field load. The J value, except for the first two rings, are to be constant and should be nearly the same for both the boundary layer problem and the CT specimen. The sensivity of measuring T-stress is extremely delicate. In Larsson and Carlson [47], T stress is measured for a distance 0.02 - 0.24 times the crack length behind the crack tip? Before any further discussion, it is to be noted that T stress is defined by (4.1). T, = 0,(r, 0)g=,r — 03(1', 0)9=« (4.1) ' 61p apecsmen The magnitude of T-stress is defined by Leevers and Radon [44] and is given in (4.2). 7ra B .. rd?! (4.2) In the above equation, B stands for biaxiality ratio. The primary reason for intro- ducing B is as an instrument for mapping the conditions in any of the specimens analysed onto a reproducible condition of crack length and load biaxiality ratio. The crack tip stress field parameter can be regarded as a geometry-independent relative of the load biaxiality ratio. In Appendix D, as an example, the first row indicates the normalized distance 5: where 1‘,- represents the radius of the rings varying from inside to the outside boundary at an increment of 1.53, and r0 = 100 is the outer radius of the semicircular domain. The second row shows the absolute value of transverse stress 0, behind the crack tip of a CT specimen. The third row shows the absolute value of transverse stress behind the crack tip of a boundary layer problem. The fourth row shows the difference in transverse stress between specimen and boundary layer problem. The fifth row shows the biaxiality parameter, and, finally, the last ‘they [47] stated: “The T-stress would have been exact had the average T, been determined from element stresses over a more narrow range close to the crack tip ”. 65 row indicates all the above numerical values corresponding to the node numbers and their position radially in ascending order at an increment of 200. A comparison in the values of T-stress and biaxiality parameter B are also shown for angles varying from _0 = 151° to 180°, for each 5. The results are more consistent for 180°, even though some negative values appear at the nodes near the crack tip, which can be neglected. The numerical values of T-stress corresponding to nodes in each circumferential line are averaged as given by (4.3) and then each of these are again averaged radially to give the net T-stress. T“ = z 3 (4 3) i=1 n m T Tm = ”c (4.4) g m In Figures 4.4 to 4.8 the biaxiality ratio for 0 = 151° to 180° in the range 5: from 0.01 to 0.2 are compared. The results are quite consistent because of the fine mesh employed in the analysis. The net T“ variation with f: is shown in Figure 4.9. The final result in the T stress calculation is shown in Figure 4.10, in which biaxiality ratio B is compared with 5. The results are in agreement with Leevers and Radon [44], Cotterell [45], Kfouri [46] and Larsson and Carlson [47]. The value of B depends on the ratio 5 and value used in this analysis is 0.5. Kfouri [46] had used 0.833 and Larsson and Carlson [45] had used 0.4583. The values of B for u = 0.3 are listed in Table 4.1. For comparison with the photoelastic investigation the values of B for V = 0.45 are listed in Table 4.2. T(1ra)1/2/ K1 1 .00 66 0.60 f- 0.60 :- 0.40 _ 0.20 '- 0.00 l 1 0.00 0.04 0.20 Figure 4.4: Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/ro at a/W = 0.3958. T("a)1/2/K1 1.00 . , . . . . . ‘ ~ r; : :L I : 1; 3 0.60 _- :.:.:.:g : g3; . 3 I -- = 8: : 0.60 :- { “_.m—mu—a-nm-m . 0.40 _b 0.20 :- 0.00 : J 1 1 1 l 1 1 1 1 l 1 1 l 1 I 1 - 0.00 0.04 0.06 0.12 0.16 0.20 Figure 4.5: Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/r, at a/W = 0.4375. 67 1.00..3.r.-3.,-...,....,. 0.60 - :.:.:.:§: 3:: : —— 3 Q 0.60} 'A T‘“—“"_"-"”_"""'"""‘°" g 0.40:- v .- F‘ : 0.20 _- 0.00'111411141111111111111111 0.00 0.04 0.06 0.12 0.16 0.20 r/ro Figure 4.6: Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/r, at a/W = 0.479. 1.00 I T I I I I r I I I j I T I I I I fl I I 0.80 «00" I'IITIIIII I" l 0 I. III' I an. ' II I . ', III' 0 ..l I d :5 ~\ 0.60 > ———-————————e_..e_1_e_em_nm A g 0.40 v [— 0.20 l 1 1 1 l 1 1 1 1 I 1 1 1 1 0.06 0.12 0.16 0.20 r/r° Figure 4.7: Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/r, at a/W = 0.5208. 0.00 1 1 1 1 l 1 0.00 0.04 68 1.00_....,.-..,....,.,.,,,,fi‘ ; IL: :L: : 1; 0.60 _- :.:.:.: : g; . 1 Ed I -3 -3 7-H: : cu\ 0.60 ’- 1 A a g 0.40 - v j 9" : 0.20 1 0.00 4 1 1 LI 1 1 1 1 l 1 1 1 14 1 1 1 1 l 1 L 1 1 : 0.00 0.04 0.06 0.12 0.16 0.20 r/r. Figure 4.8: Variation of non-dimensional biaxiality parameter B for CT specimen as a function of r/r, at a/W = 0.5625. 69 1.00 v v I l 7 ‘ l I ' a : g;;53;3::§§§ E ._ °~°° r -_ ----- 5;: - 3'32: 1 >4 : ' : 3 °~°° g..m_==.==__.__._ ___.__.__. ‘3 ,\ EtiiEE?§§€EEESEEE?E?§§5E???€?§?f§:: g 0.40 _- : V r- :1 E— : : 0.20 _— - 0.00 b 1 l 1 1 1 l 1 1 l 1 4 I 1 1 1 0.00 0.04 0.06 0.12 0.16 0.20 r/ro Figure 4.9: Average values of the non-dimensional biaxiality parameter B for CT specimen as a function of r/ro. 0.8 _ . . 1 , 1 1 ’ O three-dimensional photoelasticity 1 ~ 0.6 l 1 x ’ . 1 N\ I .~ . j > - /,/’/ /’ “even and Radon [48] - A 0.4 - 7’ , —--—-—(a)Calenlatedeelnt - Q " 1’. ' loading at the pin center. - l: ' fl’/ — -— -(b) Calculated m 11.11- 1 V L ,’/ , Man a: the same plane. - ’ ’ . ' ----- Cotterell 4s 1 Ed 0'2 '- I / / —--K!ourf [44] I " ‘ - 0 Lemon and Carmen [4s] 1 P / 5 — flaunt!” (V - 0.3) ' : / ---- 51111111111611 (0 - 0.45) ‘ 0.0 . 1 . . 1 . 1 . ‘ 0.0 0.2 0.4 0.8 0.8 a / w Figure 4.10. Comparision of biaxiality values for CT specimen obtained from dis- tributed load at pin center with Leever and Radon[44], Cotterell[45], Kfouri[46] and Larsson and Carlsson [47], with simulation u = 0.3 and 0.45, 3D photoelasticity. 83 70 Table 4.1: Average biaxiality ratio (Poisson’s ratio = 0.3). B - T*Sqrt(pi*a)/K I. a/H - 0.3958 a/H - 0.4375 a/H - 0.4791 a/w - 0.5208 a/u - 0.5625 1.16778782-04 1.48496718-04 1.80214738-04 2.29011598-04 2.77808472-04 3.52880708-04 4.27952848-04 5.43448498-04 6.58944093-04 8.36629678-04 1.01431538-03 1.28767748-03 1.56103628-03 1.98159493-03 2.40215452-03 3.04917258-03 3.69618168-03 4.69158732-03 5.68698428-03 7.21837722-03 8.74976138-03 1.11057423-02 1.34617402-02 1.70863272-02 2.07109152-02 2.62871648-02 3.18634788-02 4.04423828-02 4.90212853-02 6.22196038-02 7.54179678-02 9.57230408-02 1.16028432-01 1.47267058-01 1.78505672-01 2.26564573-01 5.11893048-01 5.09205928-01 5.06529488-01 5.04365362-01 5.02210723-01 5.00460768-01 4.98719852-01 4.97299338-01 4.95886138-01 4.94724123-01 4.93568922-01 4.92606343-01 .91650888-01 .90837632-01 .90031303-01 .89321832-01 .88618973-01 .87969185-01 .87325738-01 4.86690858-01 4.86061718-01 4.8539380E-01 4.84730788-01 4.83979233-01 4.83230793-01 4.82347902-01 4.81465453-01 4.80430828-01 4.79389112-01 4.78265005-01 4.77122205-01 4.76215833-01 4.75253053-01 4.75467392-01 4.75556418-01 4.79297643-01 .fifififibb 5.58826193-01 5.55892698-01 5.52970868-01 5.50608333-01 5.48256148-01 5.46345738-01 5.44445218-01 5.42894458-01 5.41351683-01 5.40083128-01 5.38822028-01 5.37771188-01 5.36728128-01 5.35840308-01 5.34960053-01 5.34185538-01 5.33418238-01 5.32708878-01 5.32006418-01 5.31313333-01 5.30626502-01 5.29897352-01 5.29173SSE-Ol 5.28353098-01 5.27536038-01 5.26572182-01 5.25608833-01 5.24479343-01 5.23342128-01 5.22114943-01 5.20867368-01 5.19877908-01 5.18826853-01 5.19060848-01 5.19158018-01 5.23242278-01 5.82488938-01 5.79431228-01 5.76385668-01 5.73923098-01 5.71471308-01 5.69480003-01 5.67499008-01 5.65882588-01 5.64274488-01 5.62952218-01 5.61637708-01 5.60542368-01 5.59455148-01 5.58529733-01 5.57612208-01 5.56804898-01 5.56005108-01 5.55265708-01 5.54533503-01 5.5381107E-01 5.53095178-01 5.52335148-01 5.51580698-01 5.5072548E-01 5.4987383E-01 5.48869178-01 5.47865038-01 5.46687702-01 5.45502333-01 5.44223198-01 5.42922793-01 5.41891428-01 5.40795878-01 5.41039763-01 5.41141058-01 5.45398253-01 5.97046243-01 5.93912113-01 5.90790448-01 5.88266333-01 5.85753278-01 5.83712208-01 5.81681698-01 5.80024878-01 5.78376592-01 5.77021278-01 5.75673918-01 5.74551203-01 5.73436803-01 5.72488272-01 5.71547812-01 5.70720323-01 5.69900548-01 5.69142663-01 5.68392168-01 5.67651672-01 5.66917883-01 5.66138863-01 5.65365552-01 5.64488913-01 5.63616043-01 5.62586288-01 5.61557038-01 5.60350293-01 5.59135293-01 5.57824188-01 5.56491288-01 5.55434148-01 5.5431121E-01 5.54561198-01 5.54665023-01 5.59028613-01 6.06311628-01 6.03126666-01 5.99956713-01 5.97395468-01 5.94643396-01 5.92770653-01 5.90706632-01 5.89026103-01 5.67352242-01 5.85975883-01 5.61607622-01 5.63467433-01 5.62335792-01 5.61372546-01 5.00417488-01 5.79577153-01 5.7611465s-01 5.77975013-01 5.77212073-01 5.76460892-01 5.75715712-01 5.74924606-01 5.11139293-01 5.73219116-01 5.72362623-01 5.71316666-01 5.70271678-01 5.69046192-01 5.6761234s-o1 5.66460696-01 5.65127298-01 5.61053766-01 5.62913408-01 5.63167262-01 5.63272703-01 5.6770100s—01 4.89003423-01 5.33837932-01 5.55644253-01 5.70373918-01 5.79234148-01 '* a - crack length, n - width, r* - distance behind the crack tip, K — stress intensity factor, T ' T'SttBBS 71 Table 4.2: Average biaxiality ratio (Poisson’s ratio = 0.45). r. B - T'Sqrt(pi*a)/K a/H - 0.3958 a/u - 0.4375 a/v - 0.4791 a/H - 0.5208 a/u - 0.5625 1.16778788-04 1.48496713-04 1.80214738-04 2.29011593-04 2.77808478-04 3.52880703-04 4.27952842-04 5.43448492-04 6.58944098-04 8.36629673-04 1.01431532-03 1.28767748-03 1.56103623-03 1.98159498-03 2.40215453-03 3.04917252-03 3.69618168-03 4.69158738-03 5.68698428-03 7.21837723-03 8.74976133-03 1.11057422-02 1.34617402-02 1.70863272-02 2.07109153-02 2.62871643-02 3.18634782-02 4.04423823-02 4.90212853-02 6.22196038-02 7.54179678-02 9.57230408-02 1.16028438-01 1.47267052-01 1.78505673-01 2.26564578-01 4.78875943-01 4.76362143-01 4.73858333-01 4.71833798-01 4.69818138-01 4.68181042-01 4.66552428-01 4.65223528-01 4.63901478-01 4.62814418-01 4.61733722-01 4.60833232-01 4.59939408-01 4.59178608-01 4.58424283-01 4.57760578-01 4.57103058-01 4.56495178-01 4.55893228-01 4.55299298-01 4.54710738-01 4.54085908-01 4.53465642-01 4.52762578-01 4.52062408-01 4.51236468-01 4.50410938-01 4.49443038-01 4.48468512-01 4.47416918-01 4.46347828-01 4.45499918-01 4.44599238-01 4.44799748-01 4.44883028-01 4.48382948-01 DO...Dub1bbub-b.fi.hfihhhb‘UU‘UUUUUUMU‘U‘U‘U‘f-flm .22781905-01 .20037618-01 .17304246-01 .15094093-01 .12893628-01 .11106438-01 .09328495-01 .07877768-01 .06434508-01 .05247768-01 .04068008-01 .03081946-01 .02109163-01 .01278608-01 .00455138-01 .99730568-01 .99012758-01 .98349158-01 .97692008-01 .97043628-01 .96401092-01 .95718978-01 .95041868-01 .94274328-01 .93509963-01 .92608278-01 .91707068-01 .90650422-01 .89586558-01 .88438538-01 .87271428-01 .86345788-01 .85362528-01 .85581428-01 .85672328-01 .89493148-01 U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U’U‘U‘U‘U‘U‘U‘U‘U‘U‘U .44918398-01 .42057918-01 .39208788-01 .36905058-01 .34611408-01 .32748548-01 .30895312-01 .29383158-01 .27878788-01 .26641798-01 .25412078-01 .24387388-01 .23370288-01 .22504568-01 .21646215-01 .20890973-01 .20142778-01 .19451068-01 .18766092-01 .18090263-01 .17420538-01 .16709528-01 .16003748-01 .15203698-01 .14406978-01 .13467118-01 .12527743-01 .11426348-01 .10317438-01 .09120793-01 .07904278-01 .0693942E-01 .05914545-01 .06142708-01 .06237458-01 .10220068-01 .58536768-01 .55604788-01 .52684468-01 .50323152-01 .47972185-01 .46062768-01 .44163223-01 .42613273-01 .41071303-01 .39803408-01 .38542943-01 .37492658-01 .36450138-01 .35562788-01 .34682988-01 .33908863-01 .33141968-01 .32432968-01 .31730878-01 .31038143-01 .3035168E-01 .29622902-01 .28899478-01 .28079432-01 .27262813-01 .26299468-01 .25336608-01 .24207708-01 .23071068-01 .21844525-01 .20597593-01 .19608643-01 .18558143-01 .18791998-01 .18889138-01 .22971263-01 U‘U‘U‘U‘U‘U‘U'U‘U‘U‘U‘MU‘U‘U‘U‘U‘U‘U‘U‘U‘UU‘U‘UU'U‘U'U'U‘U‘U‘U‘U'U'M U‘U‘U‘U‘U‘U‘U‘U’U‘U‘U‘UU‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘U‘OO‘U‘U‘U’U‘U'U'U‘U'U‘ .37107318-01 .35445233-01 .33489158-01 .31188448-01 .28676588-01 .26605498-01 .26941618-01 .34837282-01 .63243418-01 .43379078-01 .63652478-01 .45833838-01 .44775138-01 .43874012-01 .42980558-01 .42194428-01 .41415628-01 .40695628-01 .39982648-01 .39279165-01 .38582058-01 .37841963-01 .37107318-01 .36274548-01 .35445238-01 .34466948-01 .33489158-01 .32342713-01 .31188448-01 .29942878-01 .28676588-01 .27672292-01 .26605498-01 .26842978-01 .26941618-01 .31087098-01 4.57462693-01 4. 99405348-01 U‘ .19805198-01 5.33584798-01 5. 41873538-01 '* a - crack length, W - width, r* - distance behind T - T-stress the crack tip, K - stress intensity factor, 72 4.3 Modified Boundary Layer Formulation The modified boundary layer was modeled using focused meshes of the type shown in Figure 4.3. The semicircular domain has an outer radius of the order 106. The mesh typically involved 2048 eight-noded hybrid isoparametric elements consisting of 64 rings radially and 64 fans of elements circumferentially. The ratio of radius of the outer boundary to the radius of the first ring element was on the order of 107. The first ring elements were degenerated so one side collapsed into a single point at the crack tip. The boundary conditions given are the same as for the boundary layer formulation, except for the nodes of the first rings, where blunting is allowed. Dis- placement boundary conditions-were imposed on the outer boundary corresponding to the displacements associated with the K field plus the displacements due to the T or a” stress given by (1.15). The crack-tip fields of the modified boundary layer solution far from the outer bound- ary and outside the crack-tip blunting zone should represent those of any crack with the same values of K1 and T. The formulation does not involve an explicit crack or ligament length, and so a dimensional scale is introduced by the radius at which displacements appropriate to a given K and T field could be applied. The K and T terms were increased in a proportional way by an imposed biaxiality parameter B for all— u,of the CT specimen. The T-stress 1s essentially based on an elastic concept, and it is wcalculated from the remote K field using the biaxiality paramer B. 4.3.1 Ramberg-Osgood Materials The elastic-plastic finite element analysis was done for all“—2 based on deforma- tion plasticity. In deformation plasticity the material response was described by a Ramberg-Osgood power law of the form given in (1.19), with the exponent n = 5 and 10. Poisson’s ratio was set as 0.3, and a = 1, while the ratio of the yield stress 00 to the elastic modulus was 0.0025. A second kind of material response was described by a Modified Ramberg-Osgood power law of the form given in (1. 20), with the same material properties as above, except that material constant a is absent. This equation was made use of by Wang [48] to obtain consistent JIM; as compared to J far for; _<_ 1.0. 73 e f- for a S 00 — = ° (4.5) 6" (ff) for 0' > 00 A comparison between (1.19) and (4.5) is shown in Figure 4.11. The power law part of the modified Ramberg—Osgood for both n = 5 and 10 lie above the Ramberg-Osgood material. This difference is due to the material constant a. If a is sufficiently large, the power law part of the Ramberg-Osgood will lie above the modified one. 4.3.2 Effect of Non-singular Stress For elastic-plastic solution with B = 0, the two power laws gave identical results. The presence of T stress in the elastic-plastic calculations introduced a difference larger by 0.2 % to 0.38 % of K component of the far field. The small-scale yielding solutions are than compared with the HRR singular field. In the vicinity of the crack tip, elastic strains are negligible as compared to plastic strains. The plastic part of the strain dominates the asymptotic solution. The asymptotic crack tip stress, strain and displacement fields were given in ( 1.23 - 1.25). In the present analysis, the value of %9 obtained from finite element analysis of modified boundary layer problem and with the given parameters in Table 1.1 are substituted in (1.23 - 1.25) to give the stress distribution ahead of a crack tip. The stress distribution in z and y directions for B = 0 (0 = 0) for the two different material responses and for n = 5,10 are shown in Figures 4.12 and 4.13. In order to reduce the singularity near the sharp crack tip a circular hole was made. The results for this case are superimposed with the HR solution. The stresses are normalized by the initial yield stress 00, while the radial distances of a point from the crack tip, r, are nondimensionalized by :2. The data will be self similar in the sense that data obtained for a given applied K field falls on the same curve as that for a higher K. The x component stress falls below the HRR field for a distance greater than 24}! for n = 5 and 1+? for n = 10. The y component stress falls below the HRR field for a distance approximately greater than “—599 for n = 5 and 10. 74 2'0 I I I I I I I I I I I I I I I I I I I l I I I I I I I I I I F] I I I II I I 1.8 - .— 1-6 : n-s "',-E 1.4 _- I’o"’ —: r ’1’ CI ‘ n - 1o ‘ - ___________ .13.! 1.2 - ........... _ r- ‘‘‘‘‘‘ d o I " b I- d \ 1.0 - - b _ 1 0.8 F —— lumbar-Osgood: - : E/t. - 0/0, + car/6.)“ : . 4 0.6 - """ Modified lumbar-Osgood [48]: fl : s/c. - 0/0. for a s a. : b c/c. - (rt/0.)“ for c > c. - 0.4 F- - 0.2 - 00 Linjnmll111111iliiiliinliiilrinlniilii1- O 1 2 3 4 5 6 7 8 9 10 é/eo Figure 4.11: Normalized true stress-strain curve based on empirical equation. 0.2/0. 75 I I r l I I I I f I I l T I r r I I I I I I I ' I l I : — - - --Polycarbonate : . ———-Ram berg- Osgood .. ,.. ------- Modified Ramberg -* . ‘ -Osgood .1 ........... I‘IRR .. Boundary layer problem (B = 0) ° 0 ° Notched Boundary layer problem (8 = 0) .0 '0 us a i l’ ".\ - \\, ————— Modified BLP for aa./.1 - 44.38 « . '-\-\ (polycarbonate) . \.'\. l— .\‘.“- _ '- -.\.'.:o§ ....... -.— - LJ J I J 4 LI I J I I I I I I I I I I I I I I I I I 0 1 2 3 4 5 6 7 r/(J/u./J-400.0 ‘ f o----oaa./J - 200.0 +- _G----Oa0./J - 100.0 2 _ ‘----Aac./J-50.0 ------- HRR -. - —-—s-0.n-s ~ ' —--—s-0.n-10 ' 1- _ 0’111l1111141l111[111l111l11L- 0 1 2 3 4 5 6 7 r/(J/Uo) Figure 4.16: Normalized crack opening stress distribution in plane strain for biaxiality ratio B = 0.5564 and for 0 = 0° for modified boundary layer problem. 81 71fTIIIIIIIIIIIIIIIIIII[[IIF ".'. 1 .3 “ Ramberg-Osgood - 6 "ii _, 5 4 O b \ it b 3 ’ o----oa0./J-400.0 ‘ “ G----0a0./J-200.0 " 2 _G----Cla0./J- 100.0 q . t----6a0./J-60.0 ------- m . I- —od-o.n-5 - I. —---B-O.n-IO ‘ 1 *- .. )- . O111l111l11111m1111111111111 O 1 2 3 4 5 6 7 r/(J/¢T..) Figure 4.17: Normalized crack opening stress distribution in plane strain for biaxiality ratio B = 0.5703 and for 0 = 0° for modified boundary layer problem. 82 But at W = 1.2, both the material responses showed the same magnitude and both had maximum at 71°. For n = 10 and B = 0, the material response behavior remains the same until 533,??? = 1.2, but when this value is exceeded the Ramberg- Osgood material response takes over from the Modified ones. The effect of T stress is very significant in the modified boundary layer and these are illustrated in Figures 4.20 to 4.23 for n = 5 and 10 for Ramberg-Osgood material only. The maximum extent of the plastic zone size varies from 71° to 130°. The effect of W conceptually remains the same. There is less T-effect on the dominant singularity (J) and crack opening displacement, although there is an effect on over all shape of the plastic zone. Particular attention is paid to the development of the stress ahead of the crack at a distance r = % and r = % as illustrated in Figure 4.24, for n = 5 and n = 10. Figures 4.25 and 4.26 illustrates the comparison between full field (CT specimen) and the modified boundary layer solutions for 5 0.3958 and 0.5208. They seem to match even at all z 50, which correponds to fully plastic case. 4.3.3 Polycarbonate Materials A detailed analysis for Ramberg-Osgood material was done for various a/ W ratios as was observed earlier. In this section elastic-plastic strain or stresses will be calculated using incremental plasticity theory for a/ W = 0.4966 only. The procedure is the same as before. The semicircular domain consists of 256 elements and 833 nodes. In Figure 4.12 a comparison of normalized normal stress and in Figure 4.13 a comparison of crack opening stress between the boundary layer problem (B = 0) and the modified boundary layer problem (B = 0.5233) for %2 = 44.38 are made. The effect of biaxiality parameter is significant for normalstress, but it is less significant for crack opening stress. 83 100 IIITl—IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII : —Ramber -03‘aod I 80 " ”ca/ac " 1-0 100 90 80 "'Ilgdlfie Rambarg - __ ”.051“; :.:.;?_:;?O. -sgood : : 13.0“" 2 ' ' .- "=: j 60 '- 130" j E 14.0? : 40 E 1§0'_ .2 : 160...? . 3 A 20 l- ; ‘ ‘ _1 ° ’ 110 . ' . e . , _ _ . . . v—; 0 - 180‘ .2- : . . , _ —20 E ' a C 220._ _ : 240 : -60 - .1 h 1 '- .1 3 1 _100 llIIIIIIIIIIJIIILIILIJIIIIIJIIllIlIlllIIIIIJIIILI‘ -100 -60 -60 -40 -20 0 20 4O 60 80 100 X/(J/Uo) Figure 4.18: Comparision of plastic zones between Ramberg-Osgood and Modified Ramberg-Osgood in coordinate system, non dimensionalized with respect to the char- acteristic length parameter (J/ao), in plane strain for B = 0 and strain hardening value 11 = 5 Y/(J/Uo) 100 -100 q _ 60 60 4O 20 IIIT‘IIII'IIII'IIII[IIII'IIII—[IIII'IIIIIIIIT'IrTT IIIIIrIerIIlIIIIWIIIIIIIIIIIIII’IIIITIIII'III —Ramber -Osgood 0' /0. 8 1.0 unloads Ramberg ” 100- - - '9°-~ 80 -Osgood 11.0:“4 :‘ '- "+70. 1201" 2. .,; - -' "“99 1.90:, h. '. . , .. . . . .:_:;_40 so; . 19°-..” . >20 200 210' ' 11141111111111111111111111111IL111I1111IL1111111 I IIIIIIIIIIllllIIJJlIllllIIIIIIIIIIIIIIIIIIIJIIII I -100 -60 -6O -40 -20 0 20 4O 60 60 X/(J/Uo) 100 Figure 4.19: Comparision of plastic zones between Ramberg-Osgood and Modified Ramberg-Osgood in coordinate system, non dimensionalized with respect to the char- acteristic length parameter (J / do), in plane strain for B = 0 and strain hardening value n=10 85 80 IIII'IFIIIIIIIIIIIIIIIIIrIVIIjIlIIIIIIIIIrIIIIIjII Ramberg-Osgood I 0'../0. = 1.0 64 46 32 16 . .. _'“ ’. . ..... l0 - . 25° 260-270-200 290 -—-—-ac./J - 200. 2.131%: : £85" _ao JIIIIIJIIIIIIIIIIIIILIIIIIIIIILIIIILIIIIIIIIIIll -60 -64 -46 -32 -16 0 16 32 46 64 X/(J/Oo) ILLIIIILIIIIILLIIIIIIIIILILLIJJJIIIIIIIIIIIIIILI y/(J/O‘.) O IIIIIIIII'IIII'IIII'IIIIITIII'I—TII'IITI'IIII'ITI - O C Figure 4.20: Effect of B = 0.4890 on plastic zones in coordinate system, non dimen- sionalized with respect to the characteristic length parameter (J / 0,), in plane strain for aa/a. = 1.0 and strain hardening value 11 = 5. 86 100 IIIIIIIIIIIIj'IijTIITWIIIIIIIIIIIIIIIIIIIIIIIFI Ranting-Osgood I 0/0'. = 1.0 8° _.11912‘99"?°i ‘79.. 60 4o " 20 170 . . _,-_' 3.10 y/(J/Uo) . ' '. ’ '. .' ‘ ‘ . v .. . ' _ . ~ ,. . . . a. s a, s - 1 .' 0- . s o c I a .. ' - . .. u . o 1. i '. ‘ o a - I . 's' ' ‘4 - a s I ' n . . . . . ... ‘ \oo ' .. -_ ~ - - ' - ' . ' .~ .- . . 0 .' . . . ..‘. . . . . O .0 _. '1‘ ' , I ' ‘ .- - - o c' - . . -' I ' u I u a ‘ ' .. ' . " u. - ' .. ' '. ' ~ ‘0 g n . ., . -~ - _ ' , ° 0 ‘ . ' 1. . o . s r n a C ' o I . _ ' . ' . . n a . - f a ', u' ' '. ‘ ' ’ a l , . . - - o . a . . - . o - ' i - . - ' _ ' '. ‘ I ‘0 . . s a u- I ' . ' - '- s a; 0 0. _ . a IllllIllllIllllIIIJJIIIIILIJIIIILILIIIIIIIII I—III'IIII'IIII'IIIIIIIIIIIIII'IIII—IIIIr'IIII'IIII .8 IIII -100 41111111I1111I1111I1111I1111I111111111I111111111 -100 -60 -60 -4O -20 0 20 40 6O 60 100 X/(J/Uo) Figure 4.21: Effect of B = 0.4890 on plastic zones in coordinate system, non dimen- sionalized with respect to the characteristic length parameter (J / 0.), in plane strain for aq/a. = 1.0 and strain hardening value 11 = 10. 87 60 l-I I I TTTY I I I I I I I [I I II I I I I ITII I I l I I I I I I I I I I I I—I I l I I I 'j : Ramberg-Osgood _ 48 '- ”/0. = 1.1 100 90 80 all I .130" ‘ ' ' ".59 I 36 L 1.30 " ".69 : 24 E- lsa. ‘ .2 : 16.0... [2.0 : b° I 110‘. .. .10 : l- : 1 . p- . ' d > o - 180 ------ “.30 - V I E f 3 > 12 : 19° ' "-3.60 : E 269" 3.40 1' -24 :— 2'1-9" . ' —: : ml 3 -35 - '- _ " 1 : 78 I 0. 1 .°.--“ J I 400. a -48 L —---ac./J - zoo. - : nan/J - 100. ‘ .. sag/J - 50 I “60 '- llllJlLllllllllllllllJllleLlLlllIlllllllllllllJ‘ -60 -48 -36 -24 -12 O 12 X/(J/Uo) 24 36 48 O 0 Figure 4.22: Effect of B = 0.4890 on plastic zones in coordinate system, non dimen- sionalized with respect to the characteristic length parameter (J /a.), in plane strain for calm, = 1.1 and strain hardening value n = 5. 88 60 _rlrrlrtlrI[In]!Irvltlirlrfi1111lrltilrlITIIlitur- I lumbar-Osgood I 48 '- U/O'. 31.1 100 90 80 q . _uo" f "70. j : .nzo- : .eo : . no; ' " ' 34.0 I . 24— 150. p.739 5 E 10.6., .520 E o ' 170.. . ~10 ‘ b : 3 '. 2 v C ; . p '. _ .' j > 12: 190~""'_ ”j N: --s,so : I zoo '_ aao I -24 f are . . aso - : 22°. . .330 : -36 _- ‘ .310 T. I .300 —B-0. 2 - -----ac./J-400. - -48 l- —°-'I0./J I 200.- : —’° JJ-IOO.: _ acJJ-BO. . _ao '- llllLllllllllllllllllllllUllllliJLlllllllll1111‘ -—60 -4s -3s -24 -12 o 12 24 as 48 x/(J/€T¢.) Figure 4.23. Effect of B = 0.4890 on plastic zones in coordinate system non dimen- sionalized with respect to the characteristic length parameter (J / do), in plane strain for dale, = 1.1 and strain hardening value n = 10. O O 89 5 t I I I I I I I T I I r T Y fit I I I I T l r f MW : fil-HO-w—4-OB‘ ————— Q—-_o_g‘ : 4 ..... ‘ -—-o-c9~-o—eD----o-o-Gb ----- «"0“ + .....ea.———o—o-as~—--o——-o-a-A -——°"‘—’a—°a_ ._oa ——-o-eab —— o—-—o-as O u b ' .. \ I- .. g: : : 2 - - : out-oases ————— alga-5‘ _ ash-0.4376 ——--—61/c,n-6: 1 __ D a/v - 0.000 ------- 21/0, I: - 10_ _ halt-0.5300 -———61/c,n-10. : on-o : O h 1 1 1 1 l 1 1 4 1 l 1 1 1 1 l 1 1 1 Li 1 1 1 4 “ 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4.24: The crack opening stress directly ahead of the crack at a distance 2.1/ac and 5.1/a, for all a/ W ratio considered in a CT specimen in modified boundary layer formulation. s so. gas. ‘1 90 7 IIIIIIIITIITIIIIIIITIII]II .3 Ramberg-Osgood 4 ’ coo ouJJ-400.0 (CT specimen) ’ 111 “Mn-50.0 (CT specimen) b d 11ch - 400.0 (MEL) 2 b ..... “J1 - 50.0 (NHL) ........ m - . —-B - 0. n - 5 ‘ —ooB - O. n I 10 o J41LlljllllljjlllLlllLlLll O 1 2 3 4 5 6 7 r/(J/ : — ------ Surface plane I {'0‘ 0 72 -_ o o o o oExperimental data _' l: ' . \ (Embedded Moire) - V. I .. ‘ b - I \ 0.4a - Q. - e t \. : =3: : ‘K\ ‘.““-. ‘L-‘~."-~.h~ ‘ 0.24 - k. . "\ 5 . ‘T‘t>-. “~~—GL,.._I‘L~~~ cl-~.~.II . : ~11 '\o. \. . _ ‘ -¢..\ °\. \0 ‘ 0.00 1 1 1 1 l 1 1 1 l 1 1 1 fl] 1 1 1 1 1 1 1 1 1 d 0.00 0.08 0.18 0.24 0.32 0 40 r/a Figure 5.25: Variation of stress intensity factor throughout the thickness. Surface Plane Midplane ......... ----_-_---_-_--.._ --... Initial Condition before loading. >----- >11“ After loading to crack 3L5 MPG observed After loading to -------- ----------------- , 3L5 ”Pa for 30 ain. crack Viscoelastic crack tip blunting. combination of extension due to rupture Initial Viscoelastic crack blunting and formation of sharp crack. Figure 5.26: Schematic of deformation near the crack tip. ....j.r............,1...,...°.. 60 _ mm o o 1 : o ' : . O . Q 50 _- 0 '1 .? 40:— 5 A : F . o A A A A A : .T" I ‘ o .. .1 b 30: .+-.+++ +°'+'+ . \ - .+Q++++'++ : ‘3 20 ' w" + + ' 1 M - o o o Moire [10] q I l I o o 0 Embedded Moire 3 10 184 18-6 15 8 A A A Photoelasticity [10] J + + + CTOD ; l .8 - O 1 1 14 l 1 1 1 1 1 1 1 1l1 1 1 1| 1 1 1 1 I 1 1 1 114 1 1 1‘ 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.7 (r/ a) V 2 Figure 5.27: Estimation of Mode-1 SIF based on COD, 3D stress freezing, Moire and Double embedded moire techniques. Y 10.0 - . X o bli. O b _ un IX II \x O O O dill O ‘5’ l 0 - .625 0.0 o ' A S _ A 9 based on "odor tip ..o o o Moire [10] at. - o o 0 Embedded Moire A A A Photoelasticity [10] 0,1 1 1 1 1 1 I 1 J 1 1 1 0.001 0.01 0.1 1.0 10.0 Yaoz/Klz Figure 5.28: Analytical versus experimental results in the non linear zone. 129 5.5 Experimental Method to Find Size and Shape of the Plastic Zone In photoelasticity the appropriate isochromatic lines themselves designate the plastic zones which may satisfy the yield criterion. Now, looking at the moire photograph in Figure 5.21b of the deformed crack tip, one suspects that there is really a plastic yielding. If there is not large scale yielding at the crack tip, then the measured strains should agree with the elastic prediction. The first approximation of plastic zone size rp ahead of a notch tip was derived from the Von-Mises criterion on the basis of elastic stress distribution. The above equation doesn’t describe the shape of the plastic zone. The following methods have been employed to find the shape and extent of the plastic 20116. (a) Experimental determination of constant strain or stress contour in the mid-plane, quarter-plane and surface-plane near the crack tip of the specimen using multiple embedded grid moire technique. (b) Direct measurement of the plastic zone size on the surface plane only. (c) Theoretical prediction of plastic zone size based on (b). By simple geometry the size of the plastic zone indicated by moire photos can be found. The shape can be approximated by an ellipse. Suppose the ellipse’s major half axis is al and minor axis is 01 (= 0111, p is the eccentricity ratio), then the equation for the upper half ellipse is; _ 2 2 (x1 01) + (11.) = 1 (5,35) 01 [101 The equation for the x axis in the 2:1 yl coordinate system is; 311 = tg(—cp) x1 = —:1:1tg

HRITE (26,102) X1,Y1 END IF 102 F0RMAT(1(1PE15 7),2X,1(1PE15 7)) DO 94 IPTI1,N SUM-0.0 DO 93 ICOEFI2,I JCOEFII-ICOEF+2 SUMI(SUM+C(JCOEF))*XX(IPT) 93 CONTINUE SUM-SUM+C(1) BETA-BETA+(YY(IPT)-SUM)**2 94 CONTINUE BETAsBETA/(N-I) C NRITE (6,203)BETA 0203 FORMAT(1H , 10X, 9H BETA IS , F10.5) 95 CONTINUE RETURN END c+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ FUNCTION F(C,X) IMPLICIT REALI8(A-H,O-Z) DIMENSION 0(1) F- C(5)IX**4 + C(4)IX**3 + C(3)*XI*2 + C(2)*x + 0(1) RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE XROOT (C,YO,XMIN,XMAX,RP) IMPLICIT REAL*8(A-H,O-Z) fminIf(c,xnin) fmafo(c,xmax) write(*,*)’y0I ’,y0, ’ XMINI’,xmin,’ XMAXI’,xmax,’ fmin=’,fmin, + ’ fmaxI’,fmax IF((Y0-F(C,XMIN))I(Y0-F(C,XMAX)) .GT. 0 ) GO TO 99 1 XI(XMIN+XMAX)[2.O c URITE (4,4) 'x-’,x IF (ABs(Y0-F(C,X)) .LT. .00001) GO TO 100 IF (Y0-F(C,X) .LT. 0.) THEN XMIN-x 100 99 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 10 15 20 25 30 99 100 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 10 20 + 153 GO TO 1 ELSE XMAXIX GO TO 1 ENDIF RPIX RETURN NRITE(*,*) ’xmin and xmax are wrong’ RP I 99999999.0 RETURN END SUBROUTINE LUDCMQ (A,N,NDIM) IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(NDIM,NDIM) DO 30 II1,N DO 30 JI2,N SUMIO. IF (J .GT. I) GO TO 15 JM1IJ-1 - DO 10 KI1,JM1 SUMISUM+A(I,K)*A(K,J) A(I.J)IA(I,J)-SUM GO TO 30 IM1=I-1 IF(IM1 .E0. 0.) GO TO 25 DO 20 K31,IM1 SUMISUM+A(I,K)*A(K,J) IF (ABS(A(I,I)) .LT. 1.0E-10) GO TO 99 A(I.J)I(A(I.J)-SUM)/A(I.I) CONTINUE RETURN WRITE (6,100) I FORMAT(1H0, 32H REDUCTION NOT COMPLETED BEACAUSE , 38H SMALL VALUE FOUND FOR DIVISOR IN RON , I3) RETURN END SUBROUTINE SOLNO (A,B,N,NDIM) IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(NDIM,NDIM),B(NDIM) BC1)-B(1)/A(1.1) DO 20 II2,N IM1-I-1 SUMIO . DO 10 KIi,IM1 SUMISUM+A(I.K)*B(K) B(I)-(B(I)-SUM)/A(I.I) DO 40 J =2,N NMJP2IN-J+2 NMJPI-N-J+1 30 40 154 SUM-0 . D0 30 KINMJP2,N SUM-SUM+A (NMJP1 ,K) *B (K) B(NMJP1)-B(NMJP1)-SUM RETURN END APPENDICES APPENDIX B Fortran code for contour mapping PROGRAM CRACK C ................................................................... C THIS PROGRAM IS DEVELOPED TO CALCULATE THE PLASTIC C ZONE SIZE AND SHAPE FROM NODAL DATA OBTAINED FROM C ABAOUS. » C ................................................................... C THE FORTRAN CODE Is URITTEN BY SUBRATO DHAR AT MIT C (1990-91) AND REVISED ON 06-04-92. C ................................................................... IMPLICIT REALIBCA-H,O-Z) DIMENSION XX(200),YY(200),C(10) CIIE‘3'...3""...I-C"3..........-n....81 I 4 FORMAT(120) C88888...-88.3.3.888...8.888838838388888888888888883888-8888838888:: OPEN (UNITs11, FILEI’coord’,STATUSI’OLD’) OPEN (UNIT-12, FILEI’stress’,STATUSI’OLD’) OPEN (UNIT-13, FILEI’strain’,STATUSI’OLD’) OPEN (UNIT-14, FILE-’disp’,STATUSI’OLD’)' OPEN (UNITIlS, FILE-’sxx’,STATUSI’UNKNONN’) OPEN (UNIT-16, FILEI’syy’,STATUSI’UNKNOHN’) OPEN (UNIT-17, FILEI’szz’,STATUSI’UNKNOHN’) OPEN (UNIT-18, FILEI’sxy’,STATUSI’UNKNOHN’) OPEN (UNIT-19, FILEI’exx’,STATUSI’UNKNONN’) OPEN (UNITI20, FILEI’eyy’,STATUSI’UNKNONN’) OPEN (UNITI21, FILE=’ezz’,STATUSI’UNKNOUN’) OPEN (UNIT-22, FILEI’exy’,STATUSI’UNKNONN’) OPEN (UNIT-23, FILEI’uxx’,STATUSI’UNKNOWN’) OPEN (UNIT-24, FILEI’uyy’,STATUSI’UNKNOWN’) OPEN (UNIT-25, FILEI’eqstress’,STATUSI’UNKNOWN’) 00000000 000000 155 156 OPEN (UNIT-26, FILEI’conteqs’,STATUSI’UNKNOUN’) c OPEN (UNIT-30, FILEI’meanstrs’,STATUSI’UNKNOWN’) C.‘.....uu---.-.--..ICICIS...-...I...‘....I....-.-I - £222- 2 EI400.0 PNUI0.3 PJI115.8 SIGY-1.0 EPSYI0.OO25 PII3.141569 MI400 C88.."qu.“....u-.I..-.....u..-.-88.B== 3.88:...3381 -— ----- DO 60 y0-1.0,1.5,0.1 DO 20 K=1,129,4 I-o 10 READ(11,I)NODE,X,Y READ(12,I)NODE,INT,311,S22,S33,S44 c READ(13,*)NODE,INT,E11,E22,E33,E44 c READ(14,*)NODE,UX,UY c ------------------------------------------------------------------ NODELI51200+K NODEAI(((NODE-K)/M)IM) ' NODEB-(NODE-K) IF((NODE.EO.(400+K))) THEN DNNTHETA- (atan2 (Y , X) ) ANGLEIDNNTHETA*(18O.dO/3.14159) HRITE(15,102)ANGLE WRITE(16,102)ANGLE WRITE(17,102)ANGLE HRITE(1B,102)ANGLE WRITE(19,102)ANGLE WRITE(20,102)ANGLE HRITE(21,102)ANGLE HRITE(22,102)ANGLE WRITE(23,102)ANGLE HRITE<24,102)ANGLE HRITE(30,102)ANGLE ENDIF 00000000000 IF((NODE.GE.(400+K)) .AND. (NODEA.E0.NODEB))THEN RXYISORT((XIX)+(YIY)) c .................................................................. xx1-(RXYISIGY)/PJ RR1=(S11/SIGY) RR2-(s22/SIGY) RR3I(833/SIGY) RR4-(s44/SIGY) SIGMI(Sli+S22+833)/3.O c ............................................. c xx2-(RXYIEPSYIE)[PJ c EE1I(E11/EPSY) 157 c EE2-(E22/EPSY) c EE3-(E33/EPSY) c EE4I(E44[EPSY) c --------------------------------------------- TEBM1-(s11-522)*(311-522) TERM2I(S22-833)*(822-833) TERM3I(333-811)*(833-811) TERM4I(6.0IS44IS44) TERMSI (TERM1+TERM2+TERM3+TERM4) TERM6ISORT(TERM5) TERM7I0.707106I(TERM6) AMISESITERM7 VON-(AMISE5/SIGY) HBITE(15,101)XX1,RR1 HRITE(16,101)XX1,RR2 HBITE(17,101)XX1,BR3 HBITE<1B,101)XX1,BB4 HBITE<19,101)XX2,EE1 HBITE(20,101)XX2,EE2 HRITE(21,101)XX2,EE3 HBITE(22,101)XX3,EE4 HBITE(23,101)X,UX HBITE(24,101)X,UY HBITE(25,101)XX1,VON HBITE<30,101)XX1,SIGM ENDIF 101 FOBMAT(1(1PE15.7),2X,1(1PE15.7)) c ............................................. I-I+1 IF(NODE .LT. NODEL) GO TO 10 102 FOBMAT(1(1PE15.7)) c .................................................................. CLOSE (25) CALL DATA1 (yo,XXXX,YYYY,XX,YY,BP,C,DNNTHETA) c .................................................................. BEHIND (11) BEHIND (12) BEHIND (13) BEHIND (14) 0 CONTINUE BEHIND (15) BEHIND (16) BEHIND (17) BEHIND (18) BEHIND (19) BEHIND (20) BEHIND (21) BEHIND (22) 0000000000 00 00 00000000I00 158 c BEHIND (23) c BEHIND (24) c BEHIND (30) 50 CONTINUE c .................................................................. STOP END CIW.33."...8..888..“8...--.88.8888838..I:.88838333:=8 3 == SUBROUTINE DATA1 (yO,XXXX,YYYY,XX,YY,RP,C,DNNTHETA) IMPLICIT REALI8(A-H,O-Z) DIMENSION XX(200),YY(200),C(10) c DIMENSION y0(10) C OPEN (UNIT-25, FILEI’eqstress’,STATUSI’OLD’) cum-munuumnmmmmumnsmum.Inna-“.mnsnalunmusnmnsIn. -: 2 DO 999 LIO,6,1 XMINIO.0001*(1O*IL) XMAXIXMINI10.O I I O c ------------------------------------------------------- DO 9999 MI1,1000 READ(25,I,ERRI998)XXXX,YYYY C WRITE(*,*) ’CHECK NUMBER OF DATA POINTS’ IF((XXXX.GT.XMIN).AND.(XXXX.LE.XMAX)) THEN III+1 xx(I)Ixxxx YY(I)IYYYY ENDIF 9999 CONTINUE c ------------------------------------------------------- 998 CONTINUE . write(*,*)’ II’,i,’ XMINI’,xmin,’ XMAXI’,xmax IF(i.gt.0 .and. y0.1t.yy(1) .and. y0.ge.yy(i)) then do 234 jI1,i write(*,I)XX(j),YY(j),’ 1111’ 234 continue CALL FITTING (yO,XX,YY,RP,C,I,DNNTHETA,1,XMIN,XMAX) DNNTHETIDNNTHETA*(180.dO/3.14159) write (I,I)DNNTHET,RP endif BEHIND (25) 999 CONTINUE END C.’.8....---...-u-...--I.3.....--.I.-ICC-..--........“..I..III...8 SUBROUTINE FITTING (yo,XX,YY,RP,C,N,DNNTHETA,INDEX,XMIN,XMAX) IMPLICIT REAL*8(A-H,O-Z) DIMENSION C(10),A(10,10),XN(200),XX(200),YY(200) MSI4 ' MFI4 IF(MF .LE.(N-1)) GO TO 5 MFIN-l 159 c HBITE(6,2oo) ME C200 EOBMAT(1NO, 42HDEGREE OF POLYNOMIAL CANNOT EXCEED N - 1. / C + 1H , 47NBEOUESTED MAXIMUM DECBEE TOO LABCE - BEDUCED TO , 13) 5 MFPI-MF+1 MFP2-HF+2 DO 10 I-1,N 10 XN(I)-1. DO so I-1,MFP1 A(I,1)-O. A(I,MFP2)-o. DO 20 J-I,N A(I.1)-A(I.1)+XN(J) A(I,MFP2)-A(I,MFP2)+YY(J)*XN(J) 20 XN(J)=XN(J)*XXCJ) so CONTINUE DO so I-2,MFP1 A(MFP1,I)-O. DO 40 J-1,N A(MFPI,I)-A(MEP1,I)+XN(J) 4O XN(J)=XN(J)*XX(J) so CONTINUE DO 70 J-2,MFP1 DO so I-1,MF 60 A(I,J)-A(I+1,J-1) 7o CONTINUE C HBITE(6,2OI) ((A(I,J),J-1,MFP2),Is1,HFP1) C201 FORMAT(1HO,9F13.5) CALL LUDCMO (A,MFPl,10) MSPI-Ms+1 DO 95 I-MSPI,MFP1 DO 90 J-1,I - 90 c(J)-A(J,MFP2) . CALL SOLNO (A,C,I,10) IM1-I-1 C HBITE(6,202) IM1,(C(J),J-1,I) C202 EOBMAT (1H0, 14HFOR DECBEE OF , 12, 17H COEFFICIENTS ABE,/ C + 1H , 10x, 11F11.3) c ---------------- ' --------------------------------------------------- IF (INDEx .80. 1) THEN CALL XROOT(C,YO,XHIN,XMAX,RP) X1-RP*COS(DNNTHETA) Y1-RP*SIN(DNNTHETA) HBITE (26,102) X1,Y1 END IF 102 EOBMAT(1(1PE15;7),2x,1(IPE15.7)) c- ------------------------------------------------------------------ BETA-0.0 DO 94 IPT-1,N SUM-=0 . 0 DO 93 ICOEF=2,I JCOEF-I-ICOEF+2 93 94 C C203 95 C---- 160 sun-(SUM+CoCOEF))a-XX(IFT) CONTINUE SUM-SUM+C(1) BETA-BETA+(YY(IPT)-SUM)**2 CONTINUE BETA-BETA/(N-I) HBITE (6,203)BETA FOBMAT<1M , 10x, QH BETA IS , F1o.5) CONTINUE FUNCTION F(C,X) IMPLICIT REAL*8(A-H,O-Z) DIMENSION c(1) F- C(5)*X**4 + C(4)*X**3 + C(3)*X**2 + C(2)*X + c(1) RETURN END 100 99 C---- SUBROUTINE XROOT (C,YO,XMIN,XMAX,RP) IMPLICIT REAL*8(A-H,0-Z) fnin-f(c,xnin) inax-f(c,xmax) write(*,*)’y0- ’,y0, ’ XMIN-’,xmin,’ XMAX-’,xmax,’ fmin=’,fmin, + ’ fnu-’,fnax IF((YO-FCC,XMIN))*(YO-F(C,XMAX)) .GT. 0 ) GO TO 99 X-(XMIN+XMAX)/2.0 HBITE (*,*) ’x-’,x IF (ABSCYO-F(C,X)) .LT. .00001) GO TO 100 IF (YO-F(C,X) .LT. 0;) THEN XHIN'X GO TO 1 ELSE XMAXIX GO TO 1 ENDIF RP-X RETURN HRITE(*,*) ’xnin and xmax are wrong’ RP 3 99999999.0 RETURN SUBROUTINE LUDCHQ (A,N,NDIM) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDIM,NDIM) DO 30 I-1,N DO 30 J-2,N SUM-0. IF (J .GT. I) GO TO 15 JM1-J-1 ' 161 DO 10 x-1,JM1 1o SUM-SUM+A(I,K)*A(K,J) A(I,J)-A(I,J)-SUM GO TO 30 15 IM1-I-1 IF(IM1 .Eq. 0.) CO TO 25 DO 20 K-1,IM1 20 SUM-SUN+A(I,K)*A(K,J) 25 IF (ADS(A(I,I)) .LT. 1.0E-10) GO TO 99 A(I.J)-(A(I,J)-SUM)/A(I,I) 30 CONTINUE BETUBN 99 HBITE (6,100) I 100 FORMAT(1HO, 32H BEDUCTION NOT COMPLETED DEACAUSE , + 38H SMALL VALUE FOUND FOB DIVISOB IN BOH , I3) BETUBN C ------------------------------------------------------ SUBBOUTINE SOLNO (A,E,N,NDIM) IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(NDIM,NDIM),B(NDIM) H(1)-8(1)/A(1.1) DO 20 I-2,N IM1-I-1 SUM-o. DO 10 x-1,IM1 10 SUM-SUM+A(I,K)*B(K) 20 B(I)-(B(I)-SUM)/A(I,I) DO 40 J -2,N NMJP2-N-J+2 NMJFIsN-J+1 SUM-o. DO 30 K-NMJF2,N 30 SUM-SUM+A(NMJP1,K)*B(K) 4o B(NMJPI)-B(NMJPI)-SUM BETUBN c ------------------------------------------------------- APPENDICES APPENDIX C Fortran code containing COD data PROGRAM cod C MEASUREMENT OF CRACK OPENING DISPLACEMENT IN COMPACT C TENSION SPECIMEN MADE OUT OF POLYCARBONATE AND PLEXIGLASS. C THE PROGRAM CONTAINS EXPERIMENTAL DATA AND ANALYTICAL C RESULTS CORRESPONDING TO ELASTIC AND PLASTIC PARTS OF C CRACK OPENING DISPLACEMENT. INTECEB I,J,K,L,M,N DOUBLE PBECISION SLOPE1,X1,SLOPE2,X2,X(10),X(10),Y(10), 1 LOAD1,LOAD2,LOAD(10),LOAD(10) c ---------------------------------------------------------------- OPEN (UNIT-11, FILEs’cod_ca1’, STATUSz’UNKNOHN’) OPEN (UNIT-12, FILE-’lod1_cod’, STATUSs’UNKNONN’) OPEN (UNIT-13, FILEs’lod2_cod’, STATUS=’UNKNOHN’) OPEN (UNIT-14, PILE=’10d1,bfs’, STATUS=’UNKNONN’) OPEN (UNIT-15, FILE-’lod2_bfs’, STATUS=’UNKNOWN’) OPEN (UNIT=16, FILE=’bf91_cod’, STATUS=’UNKNOWN') OPEN (UNIT-17, FILE=’bfs2-cod’, STATUS=’UNKNOHN’) OPEN (UNIT-18, FILE-’lod1_dis’, STATUS=’UNKNOHN’) OPEN (UNIT=19, FILEs’Iod2_dis’, STATUSs’UNKNOHN’) OPEN (UNIT-20, FILEs’poly’, STATUS-’UNKNONN’) C—---CALIBBATION OF CLIP GAGE ------------------------------------ C----x AXIS IS THE DISPLACEMENT OF THE CLIP GAGE, AND Y AXIS IS C----THE CLIP GAGE STBAIN (x 10E-04 nun/mm). SLOPE1810.O DO 100 X1 II 0., 2.4, 0.218 COD-SLOPE1*X1 MRITEC11,1001) X1, COD 1001 FORMAT(1X,4(2X,1PE15.7)) 162 163 100 CONTINUE C----CLIP GAGE FIXED TO THE CT SPECIMEN MADE OUT OF ------------- C----POLYCARBONATE AND PLEXIGLASS ------------------------------- C----X AXIS IS THE CLIP GAGE STRAIN (x 10E-05 mm/mm), AND Y AXIS C----IS THE LOAD (LBS) APPLIED TO THE SPECIMEN.. C----PLEXIGLASS -------------- SLOPE2-10.81 DO 200 X2 I 0., 18.5, 1.6818 LOAD1-SLOPE2*X2 HRITE<12,1002) X2, LOAD1 1002 FORMAT(1X,4(2X,1PE15.7)) 200 CONTINUE C----POLYCARBONATE -------------- SLOPE3-10.0 DO 300 X3 I 0., 84., 7.63636 LOAD2=SLOPE3*X3 HRITEC13,1003) X3, LOAD2 1003 FORMAT(1X,4(2X,1PE15.7)) 300 CONTINUE C ----- DATA FOB LOAD VERSUS BACKPACE STBAIN GAGE IN --------------- C ----- POLYCARBONATE AND PLEXICLASS ------------------------------- C ----- X AXIS IS BACKPACE STBAIN READINGS (x 10-E04 mm/mm) AND Y C ----- AXIS IS THE LOAD (LBS). C ----- POLYCARBONATE ---------- DO 400 I - 1.10.1 x(1) - 0.0 x(2) - 14.0 X(3) - 22.0 x(4) - 30.2 X(5) - 38.4 X(6) - 46.8 X(7) - 55.2 X(8) - 63.6 X(9) - 72.0 X(10) - 75.4 C ....................... 7(1) - 0.0 7(2) - 200.0 7(3) - 300.0 7(4) - 400.0 7(5) - 500.0 Y(6) - 600.0 7(7) - 700.0 Y(8) - 800.0 7(9) - 900.0 Y(10) I 920.0 164 HRITEC14,1004) X(I), Y(I) FORMAT(1X,4(2X,1PE15.7)) CONTINUE 1004 400 C ----- PLEXICLASS ---------- DO 500 x(1) - X(2) x(3) x(4) x(5) x(6) x(7) x(8) 0.0 45.625 91.25 136.875 182.5 190.0 192.0 7(8) - 190.0 HBITE(15,1005) X(J), 7(3) FOBMAT(1X,4(2X,1PE15.7)) CONTINUE 0'4 A .p V IIIIIII 1005 500 C ----- DATA FOR BACK FACE STRAIN GAGE VERSUS CLIP GAGE IN --------- C ----- POLYCARBONATE AND PLEXIGLASS ------------------------------- C ----- X AXIS IS THE CLIP GAGE READINGS (x 10E-05) AND Y AXIS IS C ----- THE BACK FACE STRAIN READINGS (x 10E-05). C ----- POLYCARBONATE ---------- x . 1,6,1 0.0 7(6) 19.0 40.0 62.0 78.5 94.5 0.0 13.0 27.0 43.0 57.5 76.0 HRITE(16,1006) X(K), Y(K) FOBMAT(1X,4(2X,1PE15.7)) CONTINUE 1006 600 165 C ----- PLEXICLASS ---------- DO 700 L - 1,5,1 7(5) - 10.0 HRITE(17,1007) X(L), 7(L) 1007 FORMAT(1X,4(2X,1PE15.7)) 700 CONTINUE C ----- DATA FOR LOAD VERSUS DISPLACEMENT FOR ---------------------- C ----- POLYCARBONATE AND PLEXICLASS ------------------------------- C ----- X AXIS IS THE CRACK OPENING DISPLACEMENT IN mm, AND 7 AXIS C ----- IS THE LOAD (LBs). CONv-4.5 C ----- POL7CABBONATE ---------- x(1) - 0.0 X(2) - 0.2 X(3) . 0.43 x(4) - 0.65 x(s) - 0.88 x(6) - 1.06 x(7) - 0.98 C ....................... 7(1) . 0.0 7(2) - 200 0 7(3) - 410.0 7(4) - 600 0 7(5) - 800.0 7(6) - 920.0 7(7) - 940.0 LOAD(M) - YCM)*CONV HRITE(18,1008) X(M), LOAD(M) 1008 FOBMAT(1X,4(2X,1PE15 7)) 800 CONTINUE C ----- PLEXICLASS ---------- DO 900 N - 1,6,1 X(1) - 0.0 X(2) - 0.02 X(3) - 0.04 x(4) . 0.06 7(5) - 0.08 166 .4 A 00 V III 0) O 0 120.0 7(6) - 180.0 LOAD(N) - Y(N)*CONV HBITE(19,1009) X(N), LOAD(N) 1009 FOBMAT(1X,4(2X,1PE15 7)) 900 CONTINUE '< A 0'1 v I C ------------------------------------------------------ C C ------------------------------------------------------ C C ------------------------------------------------------ C c ----- CALCULATION OF PLASTIC ZONE SIZE ----------------- C C ----- VB IS V(A/H), A IS CRACK LENGTH, H IS THE HIDTM, C ----- V IS TOTAL CRACK MOUTH OPENING DISPLACEMENT, VE IS c ----- ELASTIC PART AND VP IS PLASTIC PART, VEFF IS V(AEFF/H), C ----- AEFF-A+R, R IS THE PLASTIC ZONE RADIUS. V-1.06 VE-0.92 VP-0.14 A-37.49 H-75.45 CONST - (2.163+12.219*(A/W)-20.065*(A/H)*¥2-0.993*(A/H)**3 + 20.609*(A/H)**4-9.931*(A/H)**5) VR - ((1.0+(A/N)**2)/(4.0*(1.0-(A/H)**2)*(A/U)))*CONST VEF- ((A/H)*VR*(VP/VE)) + (A/H)*VR DO 99 B-0.,10.,1. VEFF- (N/(A+R))*VEF HRITE(20,1010) B,VEFF 1010 FOBMAT(1X,4(2X,1PE15.7)) 99 CONTINUE C ----- THE VALUES OF THE VEFF AND R IS KNOWN AND IS EXPRESSED C ----- IN TERMS OF POLYNOMIAL EQUATION USING SUBROUTINE POLDHAR D0 999 030.,10.,1. VEFF1'1.597+0.808*Q-0.166*(Q**2) HRITE(*,*)Q,VEFF1 999 CONTINUE END APPENDICES APPENDIX D 167 For (a/W) = 0.3958 and 0 = 151.88° r' lect) 33(b1p) T 8 MODE 2.77997228-07 1.60019208+02 1.47069092+02 1.69350008-01 1.9999106E+00 309 9.99114493-07 7.1993393E+01 7.1049969E+01 1.03413798-01 1.221247SE+00 909 9.02129643-07 9.04992023*01 9.93636108+01 9.16637905-02 1.002007IE+00 709 1.60913712-06 6.9129610E+01 4.90494693+01 0.01490002-02 9.66699063-01 909 2.06607663-06 3.93490943+01 3.9274799l+01 7.90962903-02 0.06636673-01 1109 2.72301723-06 3.2961001E+01 3.27919003+01 6.94929008-02 0.20660123-01 1309 3.73369303-06 2.0906607E+01 2.90606693+01 6.6042900E-02 7.79917922-01 1909 6.76437133-06 2.692647CEt01 2.60610793+01 6.29907903-02 7.39269923-01 1709 6.29929963-06 2.2140697E+01 2.20005213901 6.01762903-02 7.10661613-01 1909 7.99614003-06 1.9363667E+01 1.9309696E+01 9.77712903-02 6.02239973-01 2109 1.02462033-09 1.73169023001 1.7290912B+01 9.99900008-02 6.61204993-01 2309 1.26304193-09 1.92662908001 1.9212063E+01 9.62190003-02 6.6024302l-01 2909 1.6319624E-09 1.37024903001 1.36496223+01 9.20979008-02 6.26211093-01 2709 1.99900378-09 1.214067SE+01 1.20091718001 9.19037903-02 6.00224973-01 2909 2.96606942-09 1.0922679!+01 1.0972229E+01 9.06462903-02 9.99736603-01 3109 3.13229973-09 9.7070037B+00 9.6976916B+00 6.93921293-02 9.03200003-01 3309 6.00331103-09 0.74690923000 0.6900270!+00 6.09902903-02 9.73640973-01 3909 6.67636993-09 7.7002097E+00 7.76099063000 6.77271293-02 9.63625943-01 3709 6.21669062-09 7.02409SSE+00 6.97703193900 4.70637908-02 9.99791932-01 3909 7.99693623-09 6.26333732+00 6.21693393000 6.6603379E-02 9.67992942-01 4109 9.61620993-09 9.6939409E+00 9.607660£E+00 4.90729003-02 9.61723668-01 6309 1.16770798-04 9.04923362000 6.9990009E+00 6.93491293-02 9.39499713-01 6909 1.60496715-04 4.99694193900 6.9116220E+00 4.69107902-02 9.30660903~01 4709 1.0021673E-06 6.06909293000 0.02699693000 6.66961293-02 9.29669993-01 6909 2.29011992-04 3.67690933000 3.63243603000 4.61929003-02 9.21611603-01 9109 2.77000073-04 3.299097OE+00 3.26120698000 4.30129798-02 9.17600063-01 9309 3.92000703-04 2.9696966l700 2.92994993000 6.39391298-02 9.1612092l-01 9909 6.27992042-06 2.6966969l+00 2.61139943+00 6.32619008-02 9.10909698-01 9709 9.63649693-06 2.6006216B+00 2.3979091l+00 6.30369008-02 9.00232393-01 9909 6.90944095-06 2.14722043+00 2.10660923000 6.20192503-02 9.09619973-01 6109 0.36629675-06 1.94263103900 1.09999903000 6.26320003-02 9.03699913-01 6309 1.01431933-03 1.73099413900 1.69610193000 4.2692629l-02 9.01337212-01 6909 1.20767763-03 1.97372393000 1.9314210E+00 6.23021293-02 4.99999913-01 6709 1.96103623-03 1.60930013900 1.36719293000 6.21996293-02 0.97029063-01 6909 1.90199693-03 1.27647733000 1.23666692+00 6.20307903-02 6.96399192-01 7109 2.60219693-03 1.1439013E+00 1.1020719l+00 6.19097903-02 6.94926223-01 7309 3.06917292-03 1.0369190E+00 9.99116998-01 6.10066292-02 6.93692603-01 7909 3.69610163-03 9.30127932-01 0.00624693-01 6.17020793-02 0.92693163-01 7709 6.69190733-03 9.63023673-01 0.02211913-01 6.16119638-02 6.91606023-01 7909 9.60690623-03 7.97730398-01 7.16216608-01 6.19239123-02 6.90369762-01 0109 7.21037723-03 6.09160163-01 6.66710063-01 6.10621003-02 6.9960399l-01 9309 0.76976132-03 6.10790762-01 9.77399203-01 6.13639373-02 6.00679012-01 0909 1.11097628-02 9.6269039I-01 9.21371632-01 6.12069623-02 6.07971912-01 0709 1.36617603-02 9.06700008-01 6.69607632-01 6.12131793-02 6.96700133-01 0909 1.70963278—02 6.61662913-01 6.20323663-01 6.11393793-02 6.0902060I-01 9109 2.07109198-02 6.16339263-01 3.79272103-01 6.10671623-02 6.06979022-01 9309 2.62071648-02 3.79096103-01 3.36.61.23-01 6.09943623-02 6.96116108-01 9909 3.19636703-02 3.43664638-01 3.02962928-01 6.09221123-02 6.63262093-01 9709 6.06623028-02 3.16066903-01 2.73199093-01 6.00970138-02 6.02903963-01 9909 6.90212993-02 2.06690073-01 2.63909003-01 6.07990793-02 6.91701263-01 10109 6.22196033-02 2.60909313-01 2.20266903-01 6.07660133-02 6.01169098-01 10309 7.96179673-02 2.37333708-01 1.96639123-01 6.06949798-02 6.00979913-01 10909 9.9723060E-02 2.10290168-01 1.77960918001 6.07272901-02 6.00961603-01 10709 1.16029633-01 1.99269973-01 1.90930392-01 6.07306298-02 6.91096013-01 10909 1.67267093-01 1.06065703-01 1.63169462-01 4.09162623-02 6.63193798-01 11109 1.70909678-01 1.60093913-01 1.27006962-01 6.10769703-02 6.99091663-01 11309 2.26964978-01 1.97102203-01 1.19606913-01 6.16992913-02 6.92393603-01 11909 2.76626293-01 1.69272998-01 1.03037022001 6.22367603-02 6.90766663-01 11709 3.66961796-01 1.36909673-01 9.30603713-02 6.39601013-02 9.16276063-01 11909 6.22699303-01 1.27962093-01 0.30600223-02 6.69969693-02 9.30170463-01 12109 9.36249923-01 1.23096993-01 7.90006603-02 6.90961673-02 9.67069398-01 12309 6.90000213-01 1.19372733-01 6.69696263-02 9.16061033-02 6.07060202-01 12909 9.26999923-01 1.1667022E-01 6.06690623-02 9.60003793-02 6.61327163-01 12709 9.99999093-01 1.19019073-01 9.60991193-02 6.17629968-02 7.29379363-01 12909 _l‘l-i. 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