In. --'-\. .7 :--..o"\ '. i ,-o ‘. 2' ".177 ’° ' . \ :- 0 ~ ”‘ .---~ “.1 I do ‘ . . I . a- . .. - -- V‘ g a “ C a -u- ‘. ’V«“- II -flv. - . . a ‘ ' ’ - .. In. ‘~'§‘ ‘~-‘:" . . n...-~ .5. b. O 3“ - A I -, ‘ ‘ . . :. um... “ ha IV ""‘°' ‘ a.’ 'vnuovi a a .2 ...-ESE E;-'&°.'.3.'.S -8 .... n I:.‘- n..." '.. Q. ’- M“ u...-. mnsg“ Q‘- . 13‘: a‘ .U 0 F~.s-v-n ': “i U‘ bun-dn-U‘S -1 '._:‘~ e ‘5 "Q0 ,: ‘ ‘ us. “gt v. ‘.Q V ‘0 .." q ‘ us. ‘ Q .‘ a ‘ m“.‘ ‘. a! Q“ a "‘ ‘I-V‘n .. 'v‘; "u . ‘*Qv..S. .' . ‘oce “Se A; ‘L- . . ' V‘ 5..-: 4 ..‘ ‘ ‘ .‘ V ‘I..‘ ‘I'. “R a. 0 Q ‘ ‘ - Q. kuree‘" .- ~-3::;‘:'-s: s isvi. are A c ‘1 ‘-~ 01:3 IIL:~!. u“‘. e ‘ F ~ a: ‘ —. \- S ._,: €: ‘ *er: K. “Q? :3‘“ O.\: . =5;.--; - ca, h ~a: La“ ‘ to 3 CA..’. V K : .3 L“; u u. :O‘c‘. t ABSTRACT SOLUTION OF SOME MIXED BOUNDARY VALUE PROBLEMS OF THREE-DIMENSIONAL ELASTICITY BY THE METHOD OF LINES BY John Paul Gyekenyesi A semi-numerical method is developed for solving a set of coupled partial differential equations subject to mixed and possi- bly coupled boundary conditions. The application of this method to these equations leads to coupled sets of simultaneous ordi- nary differential equations. Their solutions are obtained along sets of continuous lines in a discretized region. When de-cou~ pling of the equations and their boundary conditions is not possible, the use of a successive approximation method permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by presenting pre- viously unavailable solutions for a number of mixed boundary value problems in three-dimensional elasticity. Stress and displacement distributions are obtained for two finite geometry, rectangular bars which are loaded by a uniform surface stress distribution. The first bar contains a through-thickness central crack while the second bar has double edge cracks. Stress intensity factors KI for both configurations are presented. l .- . <.-r. O. .' u’p'l e.- ..... -..---_.._._.. - -0 o ’ ‘ o . "'J" .- p -A _—_ . \ § - - . ‘-‘*--'- -u .g~v ---v-— —.l . >‘- 0.- A. ,-- ‘ t. - ‘ ~‘ 0“. . ' ' -v-‘--‘ -d‘ .--.. .. _ ‘¢«. . '~ . . - . ‘ a... c -- "‘v.‘ . -. s ‘_ - .“‘- “t“ s ‘V‘. - John Paul Gyekenyesi An independent treatment of problems in cylindrical coor- dinates is also included. Stress and diSplacement distributions are calculated for finite circular bars each with a penny shaped crack. Approximate results for an annular plate containing internal surface cracks are also presented. DiSplacement distributions for each problem are calculated by applying the method of lines to the Navier-Cauchy equations. By comparing them.to known solutions, the results of the circular bar are used to examine the rate of convergence and the accuracy of‘this method. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional elasticity problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results.are obtained even from the use of a relatively coarse grid. Ill ~v-90\fi ‘ ‘ vv-v‘ov' v. . '— -~--— ‘.v.."l-. .- O h a.» I - q 'l A”.-. .0 -"'. .‘ I I .N v ‘Q‘ (a- . 3 ~92, . . huge“ e. “ SOLUTION OF SOME MIXED BOUNDARY VALUE PROBLEMS OF THREE-DIMENSIONAL ELASTICITY BY THE METHOD OF LINES BY John Paul Gyekenyesi A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1972 {xii}? To my Mother and Father, Katalin and Gyorgy Laszlo ii .- . I . . sc‘uap e.— “ u- - "'e :‘OIIUO ..;Iae u-.. 2. - acme-‘ - .n- “VII. 5": A. '1- \v‘.; .Q’ .’ :3 l o a \ Q “Ray. ‘? r~ c.‘ ~. . w. . ‘g.:'s \ ‘ d ere “a. . qr-r.er\. -. ‘~Q‘ Na 0“-‘ ‘~.:‘ 'v v . w- - ‘4 ,:: . O ‘H u.‘~:a\ ‘1 “A tie. Baa “.: ‘r‘: ... I. a-uE: ACKNOWLEDGMENTS The author wishes to express his deepest appreciation to Dr. A. Mendelson, the writer's thesis advisor, for his assist- ance during the course of this research and for his encouragement and continuing guidance throughout the past years. Thanks are also due to the other members of the graduate committee, Professors R. W. Little, W. N. Sharpe, W. A. Bradley and N. L. Hills for reviewing this manuscript. Very special thanks are due my wife, Erika, for her help during my doctoral studies. The help provided by Mr. J. Kring of the Lewis Research Center in setting up and running the computer programs will always be appreciated. Finally, I wish to thank Lewis Research Center of the National Aeronautics and Space Administration for providing generous support for this project. iii TITLE . . . . . DEDICATION. . . ACKNOJLEDGEMENT LIST OF TABLES. LIST OF FIGURES LIST OF SYMBOLS Chapter 1. INTRODUCTION. . 2. SOLUTION OF THE NAVIER-CAUCHY EQUATIONS IN RECTANGULAR CARTESIAN COORDINATES BY THE METHOD OF LINES . TABLE OF CONTENTS 2.1 Governing Equations. . . . . . 2.2 Method of Lines. . 2.2.1 2.2.2 2.2.3 2.3 Solution of Simultaneous Differential Reduction of the First Navier-Cauchy Kquation and Associated Boundary Conditions. Reduction of the Second Navier-Cauchy Equation and Associated Boundary Conditions. Reduction of the Third Navier-Cauchy Equations and Associated Boundary cmditima. C U C C O 0 Equations With Constant Coefficients . . 2.3.1 2 3 2 Evaluation of Matrix Functions. Evaluation of the Particular Integral . iv 0 Page ii iii Vii xiv 10 IO 13 16 27 35 42 ”6 SO |IH _ n. v. v. v. ‘ p _. 2. z. .u .. o. .u— o—- _ .L —.o ._ . 1. . . . a ‘- I ‘ .n‘ ... * Q A.“ .7 a c n a s a 5 “no . v a . ae- -.\v v-II- F- duo‘s-5 I‘Oe 0.. be. . .-- A-.- a - .- e‘- .. 2. ‘5- .Rv . «3 eat 4. Page 2.” Application to Specific Geometries . . . . . . . 5H 2.u.1 Bar With Through-Thickness Central Crack. 55 2.”.2 Bar With Through-Thickness Double Edge Cracks. . . . . . . . . . . . 66 SOLUTION OF THE NAVIER-CAUCHY EQUATIONS IN CYLINDRICAL COORDINATES BY THE METHOD OF LINES. . . . 71 1 Governing Equations. . . . . . . . . . . . . . . 71 2 Ordinary Differential Equations and Boundary Conditions in the Radial Direction. . . 72 3.3 Ordinary Differential Equations and Boundary Conditions in the Circumferential Direction. . . 82 3.8 Ordinary Differential Equations and Boundary Conditions in the Axial Direction . . . 90 3.5 Solution of Simultaneous Differential Equations With Variable Coefficients . . . . . . 97 3. 3. 3.5.1 Evaluation of the Radial Particular Integral . . . . . . . . . . . 105 3.6 Application to Specific Geometries - Annular Plate With Internal Surface Cracks . . . 108 3.7 Axisymmetric Problems. . . . . . . . . . . . . . 117 3.7.1 Ordinary Differential Equations and Boundary Conditions in the Radial Direction. . . . . . . . . . . . . 118 3.7.2 Ordinary Differential Equations and Boundary Conditions in the Axial Direction . . . . . . . . . . . . . 122 3.7.3 Solution of Differential Equations for the Axisymmetric Case . . . . . . . . 126 3.8 Application to Specific Geometries . . . . . . . 130 3.8.1 Solid Cylindrical Bar With a Penny Shaped Crack. . . . . . . . . . . 131 3.8.2 Hollow Cylindrical Bar With a Penny Shaped Crack. . . . . . . . . . . 1H0 3.9 Stress Intensity Factor. . . . . . . . . . . . . 1&3 RESULTS AND DISCUSSION. 0 O C O C C O O O O O O O O O 1‘48 cs. ’ l . e. . " - D 'e- -u--— ‘ - e -'o ; I‘m... b - ‘ x .. . . ~- . 'ce ...--.G ‘ -o' a 5%... I . C . I h . ‘ av ..~ . . I ‘ q‘p ”-0.. ~ - ‘ I. -_. -_ ‘ HUI-'- .\o. -. ~ .._.. I.‘ I‘ ~—-.‘. ,. -‘ .- . -.‘--h‘e-—.. ’ 1... ~ - 'e -——0 7"10..,-.-" .. ‘n I ‘ . s-UCI.-" -. :0.—.'...e..—-. . q n .g -C~e| "‘th- W. ‘ --““f~ .. . ' ~ . e CO. V. -~' I! \~... Jr "" .OA-. 5. 'w 7"... no”.. fl . v .. ‘ A 'g. \ eA .. v-“‘ v- _ - H “nee". u - .d-‘. I A J A " ne vd "" an. A ‘I-‘-v. . -. K a 'o-—., ~ 5. H' - ’1 e i. . ‘ v .e..,,‘-:l _ A ‘ I -- u.‘ N:, ." ~u. fl .- - 5 o 5.. ‘ is.“ 5. Solid Cylindrical Bar With a Penny Shaped Crack . . . . . Hollow Cylindrical Bar With a Penny Shaped Crack . . . . . . . . . Annular Plate With Internal Surface Cracks . Bar With Through- -Thickness Central Crack . Bar With Through-Thickness Double Edge Cracks. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . LIST OF REFERENCES 0 C O 0 O O 0 O O O O O O O O O O O O O APPENDICES A. B. C. EVALUATION OF THE COEFFICIENT MATRIX EIGENVALUES AND EIGENVECTORS. . . . . . . . . . . FOURTH-ORDER RUNGE-KUTTA INTEGRATION FORMULAS . COMPUTER LISTINGS O O O O I O O O O O O O O O O O O C.l DOD 014:0) Solid Cylindrical Bar With a Penny Shaped Crack . . . . . . . . . . . . Hollow Cylindrical Bar With a Penny Shaped Crack . . . . . . . . . . . . . Annular Plate With Internal Surface Cracks . . Bar With Through-Thickness Central Crack . Bar With Through-Thickness Double Edge Cracks. vi Page 199 163 167 176 I92 206 209 213 225 227 227 282 255 279 298 at. ...L o u no-..- ‘V‘n h-._a- »..S ‘ -.,°- -‘ 'la ps- .. w. .6 ... p.” .u C :. J. .. S “v E Z O. .. A. ... I. an. a. o A» ... .t C u... .8 .u .1. a “a. .F. .v t. C. (x .a. e .-b .3 .2 be . ¢ uh. A. Q. :e. - . . v. . . g 2:. J» 2‘ 2W 2‘ . q .C . .@ .2“ .Q 3. an “a. ~.. —. ..~ z.“ —D Q. ~. ‘3 a A... . . a. C :. r. 2. C u. .C S t r 3 ~‘ 3 c. 3 a. h . .~.& .....:.. .l..r; ea C file... 9 . L . . g . ‘Q ~ (\ ‘ ‘. . e 3‘ m» S :u C Ca a»: a“ I.\. Q» C “A. a. wk. 3. ~‘ u. A. u» s ‘ . ‘ Cs . c .. . a I. n : a. . u... .. .. . . . . u... .... r a a. .. .2 a. ... E C x. c at .3 S 3. 3. Us I r ...:. I .C L .2 T. r a... .L r .w ... E .L s Q :. .a\ “o X e~§ u.. 0 us. 4 \ c ' A.) a “\U a fi; 1“ c. ‘ v .1 X C a a... ...a .C F. C; .u.‘ r .. r... . a c I. C. . a. a. u .‘ :a ac .u S» B». .. y. . \. As he u . a .. 2. . S .7. = . . a :. 2 i . . . S e .u e «.v . . a u . w u . . ‘n ~\ ~. he a se u.... n. An .| Puma um I”. .\ a —‘~ U .| NV ‘\5 a J. “ ‘fi. 1. Ab he an 3‘ V4- . \. \. .y u \. . e a. v ‘n . \ ~ Ce as AV \~ st “\ In. so :h. u v I .u. ~ 1 PF» LIST OF TABLES Table Page 1. Non-dimensionalized radial displacements §—-%~ for a solid cylindrical bar with a penny ° shaped crack under uniform normal tension. a = 1.0, B - 1.77, t = 1.68 (16 axial and 13 radial lines) . . 155 2. Non-dimensionalized axial displacements~§—-§- for a solid cylindrical bar with a penny shap8d crack under uniform normal tension. a = 1.0, B = 1.77, L = 1.68 (16 axial and 13 radial lines) . . . . . . . 156 3. Non-dimensionalized radial stress ggé' for a solid cylindrical bar with a penny shaped crack under uniform normal tension. 5 = 1.0, B = 1.77, L = 1.68 (16 axial and 13 radial lines)° . . . . . . . . . . . 157 u. Non-dimensionalized circumferential stress ggi- for a solid cylindrical bar with a penny shaped ocrack under uniform normal tension. 3 = 1.0, B a 1.77, i = 1.68 (16 axial and 13 radial lines) . . . . . . . 158 5. Non-dimensionalized axial stress 332- for a solid cylindrical bar with a penny shapegocrack under uniform normal tension. a = 1.0, B = 1.77, L = 1.68 (16 axial and 13 radial lines) . . . . . . . 159 6. Dimensionless radial displacements E E- for an annular plate with internal'surface ° cracks under uniform radial tension on the outside surface . 168 7. Dimensionless circumferential displacements E_.X. . . O b for an annular plate with internal surface 0 . cracks under uniform radial tension on the outside surface . . . . . . . . . . . . . . . . . . . 169 8. Dimensionless axial displacements B E- for an annular plate with internal surface 0 cracks under uniform radial tension on the outside surface . 170 vii s e .n- --~-.= -- _.a--Van- o~r 10. ll. 12. 13. 19. 15. 16. Dimensionless radial stress distribution gE-for an annular plate with internal surface 0 cracks under uniform radial tension on the outside surface . . . . . . . . . . . . . . . . Dimensionless circumferential stress distribution 39, for an annular plate with internal surface 00 cracks under uniform radial tension on the outside surface . . . . . . . . . . . . . . . . . . Oz Dimensionless axial stress distribution --for an annular plate with internal surface 0 cracks under uniform radial tension on the outside surface . . . . . . . . . . . . . . . . . . . . . . Dimensionless x-directional displacements %—-g- for a rectangular bar under uniform tensiono containing a through-thickness central crack. a = 1.0, B = 2.0, L = 1.75, f = 1.5 (88-96-128 x-y-z directional lines respectively) . . . . . . . Dimensionless y-directional displacements %—-E- for a rectangular bar under uniform tension ° containing a through-thickness central crack. 5 I 1.0, b = 2.0, L = 1.75, t = 1.5 (98-96-128 x-y-z directional lines reSpectively) . . . . . . . W a 0.1.. Dimensionless z-directional displacements for a rectangular bar under uniform tension containing a through-thickness central crack. a = 1.0, B = 2.0, t = 1.75, E = 1.5 (48—26-128 x-y-z directional lines respectively) . . . . . . . v1 ml: Dimensionless x-directional displacements 3- for a rectangular bar under uniform tension ° containing through-thickness double edge cracks. a = 1.0, B = 2.0, t = 1.75, E = 1.5 (#8-96-128 x-y-z directional lines respectively) . . . . . . . Dimensionless y-directional displacements §_.X. . . o a for a rectangular bar under unlform tenSlon containing through-thickness double edge cracks. 3 = 1.0, B = 2.0, L 3 1.75, t = 1.5 (98-96-128 x-y-z directional lines respectively) . . . . . . . viii Page 171 172 173 183 184 185 199 200 .- .neoae 00"‘t--:.*" .3. . ,..- 0". G of... I-AOC-q' ,- - V‘lh‘ulfiol‘ ‘ I ‘ o .1 - " .I-. U 17. Page Dimensionless z-directional displacements %—-E- for a rectangular bar under uniform tension 0 containing through-thickness double edge cracks. a = 1.0, b = 2.0, L = 1.75, t = 1.5 (98-96-128 x-y-z directional lines respectively) . . . . . . . . 201 ix a. . . . .. 9.. v. .3 on an» 7. .1 a . .. ‘ Z. .3 v. 2. .u a. A. “we... . . . . .c . ... p. . T“. 8. ... v. v. m c. 2. :0 . . e A» Eu :— a» .v o. In “a s. we a g F. N :‘ s?” .y n“ a) .m .1‘ a» .s "w u. ... .. to .2. .. T. H.‘ w... . t a. . . H. .u .1" I... ”a ... L.» .C p.. I... mm 2. a“ . . i... s: :H .nae «a. .~‘ Figure 1. ll. 12. 13. LIST OF FIGURES Sets of lines parallel to Cartesian coordinates . Rectangular bar with through-thickness central crack under uniform tension . . . . . . . . Rectangular bar with through-thickness double edge cracks under uniform tension. . . . . . Sets of lines in the direction of cylindrical coordinates . . . . . . . . . . . . . . Annular plate with internal surface cracks under uniform external tension. . . . . . . . . . . Part of annular plate with internal surface cracks. Sets of parallel lines for axisymmetric problems. . Solid cylindrical bar with a penny shaped crack . . Hollow cylindrical bar with a penny shaped crack. . Dimensionless axial displacement distribution for a solid cylindrical bar with a penny shaped crack. . . . . . . . . . . . . . . . . . . . Dimensionless radial displacement distribution for a solid cylindrical bar with a penny shaped crack. . . . . . . . . . . . . . . . . . . . Dimensionless axial stress distribution for a solid cylindrical bar with a penny shaped crack at 2 = O . . . . . . . . . . . . . . . Calculation of the stress intensity factor KI for a solid cylindrical bar with a penny shaped crack. . . . . . . . . . . . . . . . . . . . Page 15 56 67 73 110 111 119 132 191 151 152 153 153 A ‘i '4' c A.) ~ -- - ‘t; "" "=-'lo-'~ . e , .,‘ .- ‘K' a- . \ -0 -a ."‘O uv--u ' - p'-“" A‘ 0.v;.. e '00:- .= U. . . ‘ -..- F. -- -- p \ \‘ ---....‘--.'- v- --. -. . ‘ - _ _ ‘23:: A- 9-: .. --.. e--‘IO-~‘—~ ‘ en a . A.. G ..~.---~ _ ‘ -- C a ~ ‘ . - .’ -ee. - ..~--v‘ »: --..‘ _ - :e 9 = M. ‘ -. . . .. ~ ‘ . - q . '- ----_-a--;~—U ._ a? ‘ ‘-~ ‘ u... -VQ a ..--‘v‘ -.- ‘ \ “Ck: _ A _ -..--e- ”a?” ' W 01 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. Dimensionless crack opening displacements for solid cylindrical bars with penny shaped cracks of various lengths and radii . . . . . . . . . Dimensionless axial displacement distribution for a hollow cylindrical bar with a penny shaped crack. . Dimensionless radial displacement distribution for a hollow cylindrical bar with a penny shaped crack. . Dimensionless axial stress distribution for a hollow cylindrical bar with a penny shaped crack at 2 = O. O C O O C O 0 9 0 t‘ C O I O O O 0 O 0 e 0 Calculation of the stress intensity factor K for a hollow cylindrical bar with a penny shaped crack. . . . . . . . . . . . . . . . . . . . . Dimensionless crack opening displacement for a rectangular bar under uniform tension containing a through-thickness central crack . . . . . . . . . . Dimensionless bar end extension for a rectangular bar under uniform tension containing a through-thickness central crack . . . . . . . . . . Dimensionless normal displacement distribution in the crack plane for a rectangular bar under uniform tension containing a through-thickness central crack. Dimensionless y-directional normal stress distribution in the crack plane for a rectangular bar under uniform tension containing a through-thickness central crack . . . . . . . . . . . Dimensionless z-directional normal stress distribution in the crack plane for a rectangular bar under uniform tension containing a through-thickness central crack . . . . . . . . . . . Dimensionless x-directional normal stress distribution in the crack plane for a rectangular bar under uniform tension containing a through-thickness central crack . . . . . . . . . . . xi Page 154 164 164 165 165 177 178 178 179 180 181 '- -- I h ‘ b. ' -3----G--... ~- . -o -, ‘ -." - v:- - ., -'. a .-—~—- 5 ‘-'o- - ‘e- :- uva..b-..-..6 a .. .. »-';--_.-. A- - . ‘ - ~.a "‘-“Ob'a. H- ‘ II -. -’ o.— — "\\ a -.- -b..-- .g- ‘ “-' ‘.-- --‘... *.-.‘V"-. o-p— 9—--._ *[v .o-e~-5. ._.-- ‘ .- ‘ ~ - _‘ * ~'\ v- 6 __ . “' "'---..--:: --_ ~ VS -v‘ ‘1' u“ ..'§‘-s‘-_. -“ -a- a -n' -v.“-“'-.‘5 ~ 0 2 - . - .‘A'-.‘ 4—- 5.' - M \ ' 5‘- ‘~‘—‘-'--OI"‘-: .. ‘ _ ‘ .T‘av - . ~‘. 1“. '-- ."- ' I— u.. -§-. -._'_ ”-5- -..- ‘- ‘0 . ~ ‘u. ‘e . ‘I C'e . 3“. “. “~~-: , -v..‘c“‘ v. ‘ vtk .‘ - ~. “ 5".I “. “ h - ‘ 1- ~:. ‘. SF ‘_ -“. . '- “‘p. ‘- s..--"‘c‘_fi “. '-e. c‘ - ‘w J “‘ .1 ‘ ‘ -" ‘ -~ . u‘: . - ~ P- e ‘u‘ . “ c‘ F. A ‘m‘ N' " ‘V‘. “F a :G-,\- A ~»~:. . 9‘ . ‘. -1‘~Q'v-. “‘ ‘fl “'5 ‘ . .s '. a ‘ ' I qr‘a‘: ‘ A7. “ p "“c ~--e 1 ‘1. an. “ h V “‘ ‘‘o:... 7-- -~ 5- “ ‘ “is .2” c.‘ ._ 25. 26. 27. 28. 29. 30. 31. 32. 33. 340 Calculation of the stress intensity factors KI for a rectangular bar under uniform tension containing a through-thickness central crack . Variation of the stress intensity factor K across the thickness for a rectangular bar under uniform tension containing a through-thickness central crack. . . . . . . . Dimensionless crack opening displacement for a rectangular bar under uniform tension containing through-thickness double edge cracks. . Dimensionless bar end extension for a rectangular bar under uniform tension containing through-thickness double edge cracks . . . . . Dimensionless normal displacement distribution in the crack plane for a rectangular bar under uniform tension containing through-thickness double edge cracks . . . . . . . . . . . . . . Dimensionless y-directional normal stress dis- tribution in the crack plane for a rectangular bar under uniform tension containing through- thickness double edge cracks . . . . . . . . . Dimensionless z-directional normal stress dis- tribution in the crack p;:ne for a rectangular bar under uniform tension containing through- thickness double edge cracks . . . . . . . . . Dimensionless x-directional normal stress dis- tribution in the crack plane for a rectangular bar under uniform tension containing through- thickness double edge cracks . . . . . . . . . Calculation of the stress intensity factors KI for a rectangular bar under uniform tension containing through~thickness double-edge cracks. Variation of the stress intensity factor K across the thickness for a rectangular bar under uniform tension containing through-thickness double edge cracks . . . . . . . . . . . e . . xii Page 182 182 193 194 194 195 196 197 198 198 - D \-- --o.,. ' .‘nc--‘-.. . . Cl—pa -— -'---—.-..C- O - “ H'-- ~V'_ ( ""'“ van.“ . ~" --.- - V v-U-_,‘_-_ . . ‘ a .. . " "' '0 -" 0"." -... - . . \‘hg-‘-‘“ ~‘N‘b --:.- r . ' . ‘v- '- o-‘ on..-’ O 3‘ w-n‘ ’3? -, "'1': v- . -. hr. 0 35. 36. 37. Page Schematic representation of the solid cylindrical bar with a penny shaped crack computer program . . . . . . . . . . . . . . . . 228 Schematic representation of the annular plate with internal surface cracks computer program . . . . . . . . . . . . . . . . . . . 256 Schematic representation of the bar with through-thickness central crack computer program . . . . . . . . . . . . . . . . . . . 280 xiii . In. [A(KX,X)]9 [ACKr,r)] [D11 [D] [D- LIST OF SYMBOLS half crack width or crack width in rectangular coordinates, crack radius in cylindrical coordinates coefficient matrices of first order differential equations partitioned submatrices of [A] for i = l, 2 and j = l, 2 constants in difference equation solution half bar width in rectangular coordinates, outside radius in cylindrical coordinates particular integrals of first order differential equations for i = l, 2, 3, u. constants in difference equation solution constants of integration for modified Bessel's equation surface crack length 6 7“ ti~~ from internal hole elements of coefficient matrix [ch], i: 1’2, 9 o 0’ 3NR-‘4 depth of surface crack along internal hole surface diagonal matrix for transforming [Kl] into a symmetric matrix diagonal matrix similar matrix functions to [A-.] associated with (Ky,y) for i = l, 2, j = 11 2 xiv h h h x’ y' 2 hr’ he, hz i. is j. k Io, Il partitioned matrix functions products of partitioned matrix functions Naperian or natural logarithm base dilatation in rectangular, cylindrical and axisymmetric cylindrical coordinate problems Young's modulus of elasticity error matrix for accuracy check of matrix functions functions of given variables initial value vectors for the first order differential equations i = l, 2, 3, u partitioned subvectors of the coupling vectors {r}, {s} and {t} shear modulus of elasticity increments along Cartesian coordinate axes increments along cylindrical coordinate axes square root of minus one identity matrices of different order, i = l, 2 integers modified Bessel functions of the first kind modified Bessel functions of the second kind stress intensity factor for opening mode elements of coefficient matrices i = l, 2’ I O 0. 18 coefficient matrices of second order differential equations in Cartesian coordinates xv‘ - e. ‘1 '1 (It [Kr] 9 [K9], l:KZC [KPSJ’ [Kzsalz [KxiJ’ [Kyigé [K '1. [K '3’ r1 61EK 2 L m n NX, NY, NZ, NR, Ne NOC NIC [P], [P1], [P2]. [P3] r.0 r, e, z {r} R {F} J zsh:I zi] zci:I coefficient matrices of second order differential equations in cylindrical coordinates coefficient matrices of second order differential equations for axisymmetric cylindrical coordinates tri-diagonal component matrices component matrices of [KX], [Ky] and [K2] component matrices of [KrJ’ [K6] and [ch] number of lines in x or radial directions half bar length in Cartesian coordinates number of lines in y or circumferential directions number of lines in z or axial directions number of lines in a given plane number of normal lines outside crack surface number of normal lines inside crack surface modal matrices of [Kx], LKlJ, LKQJ and [Krs] internal hole radius cylindrical coordinate directions coupling vector for x-directional or radial second order differential equations distance from crack edge coupling vector for x-directional or radial first order differential equations xvi. en‘ L")! E ,"J 1.- {s} {E} {t} {f} cI x,y,z {x1}, {x2}. {x3} coupling vector for y-directional or circum- ferential second order differential equations coupling vector for y-directional or circum- ferential first order differential equations half bar or plate thickness coupling vector for z-directional or axial second order differential equations coupling vector for z-directional or axial first order differential equations components of displacement along the coordinates x-directional or radial diSplacement x-directional or radial dependent variable in the first order differential equations y-directional or circumferential displacement y-directional or circumferential dependent variable in the first order differential equations z-directional or axial displacement z-directional or axial dependent variable in the first order differential equations rectangular Cartesian coordinate directions eigenvectors of [K1], [K2] and [Kx] respectively solutions of second order difference equation Kroenecker delta eigenvalues of [K2] components of the strain tensor Lame ' a constant eigenvalue! of [Xx] and [Krs] respectively xvii v [QP(A)] O [aij] a. O 1] ax, 0y, oz, axy, ozx, ayz or: Get Oz: are' arz’ 0&2 “rs: 093. azs’ arzs eigenvalues of [K1] hyperbolic trigonometric function variable transformed radial displacements variable of integration Poisson's ratio matrizant of [A] partitioned submatrices of the matrizant of [A]i=l,2,j=l,2 variable of integration i = l, 2, 3 . . . coefficient in difference equation direction angle in complex plane of “i applied uniform surface stress components of the stress tensor components of the stress tensor in Cartesian coordinates components of the stress tensor in cylindrical coordinates stress components for axisymmetric cylindrical coordinate prrb‘ams circumferential length of annular plate cutout Laplacian in Cartesian, cylindrical and axisymmetric cylindrical coordinates respectively diagonal matrix having eigenvalues of [Kx] for its elements diagonal matrix having eigenvalues of [K , rs:I for its elements diagonal matrices with eigenvalues as their elements xviii , A ‘ ~:.,....I --v-e.r.-| ‘III‘ It. ‘A ~r- I g d a. 'v..:.“\ ' “ls-S: [A11]. [A12] Subscripts: Xe Yez r, 6, zc o, B 1.5.k Superscripts: ~ diagonalized matrix functions [All] and [A12] refer to rectangular Cartesian coordinates refer to cylindrical coordinates refer to partitioning of variables for treating mixed boundary conditions integers refers to axisymmetric cylindrical problems refers to cylindrical coordinates denotes lines along which coupling variables are written non-dimensionalized variables derivative of displacements with respect to independent variable in corresponding direction transpose of a matrix denotes symmetric xix —. _ . ' “‘n A. .“e U- U. u .- ‘ G .'e 4.:‘ ~ .‘h. QI. ‘-"-“ » - b O \.-., _'.‘A- ‘~ 0 ova. ..4..“ -3 ‘- ‘e._ 1 .- “3‘4 ‘.:‘v~-- .o is 5--.. “ . O D . u n H ~| Nvlehfip ~.._ 1.. ' -\ ~ . ‘ U. F‘-‘:A“ e b .5“ ‘v— e I. e: ‘h‘ . "‘;4*. Vs ~\e u C { s“ . s: . p “~.“ : a 1“ i_ \. xe‘L . mar- eg~fl ~S . C u ‘. sia‘ . . ma S ‘ CHAPTEva. INTRODUCTION The object of a problem in elasticity is usually to calcu- late the displacement and stress distributions in an elastic body which is subject to given body forces or surface conditions. These distributions are the solutions of the applicable field equations which mathematically describe the behavior of engineer- ing materials. The solution of this general system of equations is, however, usually too difficult to evaluate. For many problems of practical interest, some simplifying assumptions can be made regarding the displacement or stress distributions which then make the solution of these equations relatively simple. These assumptions are generally based on the geometry and loading of the problem at hand. Plane stress or plane strain solutions are two examples that result from these simplifying assumptions. Since all real bodies are three-dimensional in nature, situations arise when these simplifying assumptions cannot be validly accepted.' One important class of elasticity problems falling in this category is found in the theory of fracture mechanics. It is essential, therefore, that a method be developed that makes the solution of the general field equations possible. At present, only few analytical solutions of three- 1 . . a ‘ I » n . ........._.vn '- ac- 'I e~-0«~-- II n - .‘u . ‘Iol a ' .- ..... . 2'.” 4» r . ’5’: .- .. ~n...vll§" .-. ‘ .H.-.-‘ ‘0‘ . " .- ‘ a. U I-I-.v.‘ " . I . .e-.._‘ .' l U \ ’ .......-..__~’ ~--. 0 P‘. . . - ‘Q-. ‘ ' ‘ “4‘ .v .. h... -. v . an. R. 9-.“ -. ...n.'_6” i... . .n.‘ I - a» ... u... A ~‘0':'§'Am3 5“ e- ‘-h‘.." by . e..- a... .I'. u."~ neg” ‘ 9'. as. ‘V..~‘- .! . ~'l- -.;.\ A, A.~~ _ ' . \- a I ~ - -.‘- - \ .. “ I '..V:\:‘V u _ . up" “ - :‘-- ‘ e a in." e :t‘. s_".~ § . ‘ue " n“‘~ "‘»., Q‘. - . "y -9... o l A 5" ‘ 2 dimensional problems exist,and even these solutions are fre- quently based on some symmetry condition required to simplify the governing equations. Recently, with the availability of large digital computers, the use of a number of approximate methods was attempted,but these methods yielded only partial results for these problems. Among these approximate methods are the finite difference, direct potential, finite element, eigenfunction expansion and the line method of analysis. Of all these solution techniques, the line method of analysis is probably the least known and least used method in three-dimensional elasticity. Although the concept of this method for solving partial differen- tial equations is not new, its useful application in the past has been limited to simple examples. Because of their practical importance and inherent singularities, the work in this disserta- tion will concentrate on three-dimensional bodies containing flaws or cracks. Assuming that the method of lines can be successfully applied to these solids which contain large stress and strain gradients, its use for less complicated, three-dimen- sional, elasticity problems should present little difficulty. The phenomenon of structural failure by catastrophic crack propagation at average stresses well below the yield strength has been known for many years. Large scale failures have occurred in such diverse structures as ships, storage tanks, aircraft and rocket motors. These brittle failures have occurred with 22's ;:.::ease. . . 3,4 5:. ..,' 1;; s. ’ D ‘92-uo-o- .,~ . ‘ . “In“.‘n- .. g o ‘ I . :.... -_: bee..- .._ :u.._,”‘_ - _ ‘..‘:: 6.“ eeeb Ino:-. .- O : _ ‘_ ‘3'.Vh :0 1.96-. :- V a. _ “ ‘ «€.€E;:_g_ as 1»- G . ‘ . _ Q. ‘. .A . . . at-.. 'V ~£ a.aTe I‘: ~ a. . ' Ub-QPI Strebé _ ‘e. a.. . I I. ' ~‘.!‘ .L; ‘5 R. C b. i‘ ‘:" ‘v- \ e‘c“.e. . e “9 “ “‘k F I ..‘ .V‘ A e “‘a~K ¢ ' ' a. Ls :r‘fl I H A‘ : ‘ '6 IA Mtueren‘ to ‘ -q L ~€;;.:_ ‘ e Que:. r n e{ ‘ u M. . NJQe-4:~ ‘.|“e:: ~ ‘1‘: 3 increasing frequency as the strength and size of our structures have increased (1). Recent military and aerospace requirements for very high strength, light weight hardware have given added importance to the problem of brittle fracture. Brittle fracture, it should be noted, refers to a material failure with negligible plastic deformation while materials with ratios of yield strength to Young's modulus greater than leO'3 are regarded as high strength materials (2). In general, when high strength materials contain small cracks or flaws they are found to behave in a brittle manner and fail prematurely at low design stress levels. The main goal of fracture mechanics is the prediction of the load at which a structure weakened by a crack will fail. Knowledge of the stress and displacement distributions near the crack tip is of fundamental importance in evaluating this load at failure. Ludwig (3) has pointed out that the state of stress near the crack front is triaxial in nature which today is recog- nized as the main cause of brittle behavior in some materials. It is important, therefore, that for structures with cracks or inherent flaws, a three-dimensional solution of the stresses be obtained. Although certain analytic methods for the solution of three-dimensional elasticity problems have been developed (U), this is a very difficult problem. This difficulty arises because existing mathematical techniques are not suitable for solving the o I ,- ---Av-- a- .. ‘ fl ' -- :.o;.-v.uu v- e o .._. .-.. ..; :—: -an'--'-~e0 " N ' .'. - b - '0 .II. : -‘d'IOU ......9; Q“ uo.:...v\. -. o -‘u- . - o . u..- a- 'V' v. .y‘ H ---. It .-. g . -~ cc- ---. - “' -““ ~;;.:‘ .. ca... ';:5 ‘-- v. .II. .-~‘-.: I- ‘1 v--- k... “v: § QC ..‘ ‘ "O._ - =.'"¢»Q__ “a ‘ in; b. . .- u' . . .‘ ‘ ‘ h “Q” e A...‘ ‘ G‘ ‘fi- ” ‘ asee a" ‘ :Vi .. . In, a. ‘."_‘:I ~ \._ «Lav-g . h‘b 5 ‘Neea’ 4: ~t ‘Q‘. 5" 0“- “s3\c,- u‘b I H... ~‘IOV‘ weze Sq‘lav’ m‘.:‘ ‘. ""tEy-I a- .‘a‘i ’:—. 5 ‘. ..~‘ A i ‘3. ev'a\ '» CH « § . “New.“ .t 5- ~— “~eav .- u equations of elasticity when solids with geometric singularities are involved. The lack of a systematic approach to the solution of three-dimensional crack problems has limited the previously obtained solutions to a particular crack geometry under simple types of loading. In the past, most of the stress analysis of cracked bodies has been based on the plane theory of elasticity. A good summary of the results of this work has been presented by Paris and Sih (5). Most of these two-dimensional results come from the application of eigenfunction expansions, the complex variable formulation and boundary point collocation. It must be recog- nized that these solutions are all based on linear elasticity even though small amounts of plasticity and other non-linear effects arise near the crack tip. These non-linearities arise because linear elastic analysis of crack problems will always pre- dict stresses near the crack tip which approach infinity as the inverse square root of the distance from the crack tip. Real materials cannot possibly sustain such a s+rnes state and hence must develop a small plastic zone in which the linear elastic solution is not valid. Irwin (6), however, showed that even though linear elastic analysis does not admit these non-lineari- ties, the elastic solution of the gross displacement and stress fields has practical importance in estimating the onset of fracture. e- . - .:p It 0"; . at...) cue. ~ rax’ :‘~' A‘ O ' eue'fiitl '0 . . .,.. ., .- u. . I l "“‘ C“ ‘4 .00.. . . “-I.-- q I . i w ‘ IO—l‘.-.fl.... .h 'ey . ;‘_ t O . on c a F r .-" ‘ol _ . . =':‘-~ 7.“- .-.v~evu. . '- .' lh‘- '- .':FL . “"\ ‘.‘ M \ . “ eyed" 2“;- h..‘ I I \fi‘._; ‘ I v‘-.=‘Y' t.e 0. 3‘ C . $.65. 1p: v..‘.8 : . F . "e A. ~.r"-.‘ ». v- e 3;. “a “ (- '«f'. \ 2 . ‘v &~‘ V mi“ Crafl, Q 5‘ ‘ .- 'I'r 9.5 w 5 Early three-dimensional solutions of crack problems involved an extension of the so-called Griffith crack, that is a central crack in an infinite plate under uniform tension, to three- dimensions. These analytical solutions usually described the stresses near circular or ellipsoidal cavities enclosed in infinitely large solids. In 19H6,Sack (7) presented the solution for a flat circular crack in an infinite solid under uniform tension. Following Sack, other solutions for circular and ellipsoidal cavities were obtained by Sneddon (8), Sternberg and Sadowsky (9), Green and Sneddon (10). In reference (8), Sneddon applied Hankel transform methods to Love's biharmonic strain function and reduced the mixed boundary value problem of the axisymmetric half-space to a set of dual integral equations. More recently, Smith (ll) solved the problem of a semi-circular edge crack in a semi-infinite body. In his solution, Smith used super— position methods in conjunction with an iteration technique to satisfy the stress free boundary conditions of the crack and side faces. Using Smith's method, Alavi (12) obtained the solution for a circular crack embedded in a semi-infinite solid. Kassir and Sih (13, I“) solved the more general problem of an embedded elliptical crack subject to prescribed shear and linearly varying normal pressure on the crack surface. Solutions of three-dimen- sional crack problems with quadratic and higher order loadings were presented by Segedin (15). Shah (16) solved the problems of ;—-;.‘ ,3- “H—‘z’- can-nun». °" ". ' e .. .--\ ~‘ "- e D --o ...e \- ’,..e' I . ~:'S U .' .---.‘ .U u... . -| . ..-~“- flan . unoueo- 5'--— n -A» 9*? ~v~‘-“ 5' :‘ ouh” A; - neon-QU'°: '0 ~. e - ' I . u , f. 5 ‘ ,.. “.1 .-.‘--e\. “a. .. fl ‘ D“ ' ‘ ve ~e55< “’A- 1 :oQVU" . AA. . -\ .uQ. F' ‘ ...-..c ‘ ’5 .c. e ‘Ae ~:‘ av. . . ‘R ‘ in- wahbt\ §ee a . e ~ In. - ‘ 1., 00‘ e ~- ' -e— §ae .a .i I. . ?“ M‘s-(- a vari: 3.32335“ ~‘ .- d 1:1,:Qess be”: :1 wk- ex:~- -' e ‘S~~:s (A A v. 7-) «no 1 ray 5:. .v ‘ “=S‘:~: “ ‘ ~ ‘5 if“ ‘ . \Ie ‘ u .uQ" 5‘ b“ ‘vfl (A & . e“ . ‘g“ in eye.“ ‘s Va. ‘ I ‘e- ‘4 ~‘ . v ‘afh ‘ ‘ ‘ V- “‘A ~ £110: -§ 6 embedded hyperbolic and parabolic cracks. In 1968, Hartranft and Sih (17) extended the eigenfunction expansion method of Williams to three dimensions and showed its application to an infinite solid weakened by a plane of discontinuity. Common to all the work cited above is that they represent solutions of crack problems in infinite or semi-infinite solids. Only limited work has been done on three-dimensional solutions of crack problems in finite geometry solids. Irwin (18) in 1962 estimated the stress intensity factor for a part through ellipti- cal crack in a flat plate. Some experimental investigation of fracture in thin sections containing through and part-through cracks was performed by Orange, Sullivan and Calfo (19) and Kuhn (20). After making certain assumptions on the nature of the thickness variation of the stresses, Hartranft and Sih were successful in obtaining partial results for cracks in finite thickness bodies using variational methods (21) and eigenfunction expansions (22). Studies at NASA, Lewis Research Center, are currently being conducted using three-dimensional scattered light photoelasticity to evaluate this solution. In 1966,8ih, Williams and Swedlow (23) attempted the use of Galerkin's biharmonic stress functions together with the eigenfunction expansion of Williams in obtaining the solution for a cracked plate with finite thick- ness. However, they were unable to obtain complete results. Among the application of numerical methods to the solution . . .o.- ,. " "we. a e. an.» 'O‘IV-Ifl . I 'P ”A w e. .' -v >v-!§ ... .'" I - - q-. "’9" as “at. n-ve -huu--. ‘ V “'5" “-Oeo..- Mu... .. {Vere-n 5.- ‘ e 0:. e .A A... . .- u-I-o .' in.._ V: ‘go..“ .. no‘. .' . ...,_, a; ., IA I ‘A-"PH . . Ce .'-‘b‘ V'- .. I.— . l .D .- . " ...:Se 0 e;_a ‘°—.‘. . ’u . U C A v‘.§‘.y ‘ '0': ‘u.- -. v..- ' ~ g ‘ en .:;es :v 4 . “6-. l .. r h . - "~w€ 8.2.. . .r I K. | I k ’ ‘lu‘ ‘ \e 8:. I H. . it.“ ; .m‘e“ a AA“! . k- . § y .W 3: S" a “£2.11 . . “.§ 5" .’: V-.‘. ‘l .. .“ e.. ”‘5‘: "y g,;- \Qg‘nr‘, v.‘ OOer PC‘ ~ . ‘D "it: (a. ' 4:). .'. A. A‘ ~ '5 been 5 ‘ ex. pm “337:" A“C:al "r y. 7 of three-dimensional elasticity problems, Walker (2H) attempted to solve the problem of a rectangular bar with a central crack. After a successful plane elasticity solution, be extended the direct potential method of Rizzo to three dimensions but was unable to obtain meaningful results. In using the direct poten- tial method, Walker obtained singular integral equations in terms of boundary tractions and displacements. The numerical solution of these equations was then used in an analogous manner to Green's boundary formula in potential theory to express the interior stresses and displacements. In 1970, Cruse and Van Buren (25) were more successful in applying this method to a finite rectan- gular bar with a single edge crack. Recently, Ayres (26) pre- sented a complete finite difference formulation for the computa» tion of stresses and deformations in a three—dimensional elastic— plastic solid. However, the inherent inaccuracies in a complete finite difference solution of the Navier-Cauchy equations of three* dimensional elasticity are well known, especially when mixed boundary conditions are involved. Other possible numerical solution of three-dimensional crack problems may involve the use of three-dimensional finite elements such as the tetrahedron (27) or the isoparametric hexahedron ele- ments (28). At present, however, the finite element method has not been extensively investigated for the solution of three- dimensional crack problems. 8 Another method of solution may be obtained by applying the method of lines (29) to the partial differential equations of eqtilibrium. Faddeva (30) discusses the application of this method for the solution of a single elliptic, hyperbolic or parabolic partial differential equation. In a similar manner, Henrici' (31) discusses the solution of the one-dimensional con- duction equation obtained through the use of this technique. For three-dimensional elasticity problems, the essence of this semi- analytical procedure is to reduce the three Navier-Cauchy equan tions to three sets of simultaneous ordinary differential euqations, whose solutions can then be obtained in closed form. Since the dependent variables in the resulting equations and their boundary conditions are coupled, the use of a successive approximation procedure becomes necessary. In addition, the closed form solution of the resulting ordinary differential equations may, in some cases, be difficult to evaluate. However, the usefulness of this method in three-dimensional elasticity has been demonstrated by Irobe (32). It is the primary objective of this dissertation to present a simple and systematic approach to the solution of three—dimen- sional elasticity problems with mixed boundary conditions. These problems are more complex then those in (32), since the geometric singularities involved require a large number of lines. Hence, an extension of the solution methods presented in (32) becomes necessary. .’ -— -9 "~- 3 - o.~-v-—J - . up- a‘...... n ‘ edl: cu: Vite-.0- . .. ._ 0' .""';V"C v. Ueu...‘.‘l --. :9. Os-ae ’——---_ 0'. .‘v .c-.&- as a..- _._- .'._ \ 'e—‘\ ‘ ’ ""'-'~ -‘—..-~- 0:---g. ”V" . j d -u...‘- -.=-e .n . “'c~~-- -...' 2. \\~ h""" -- a- — u.-. _‘ "' Os ooh. a-..‘ a 'h 0 ‘- " $.53: n: - ~--. ”" u- Q ~._‘ ' , “"‘ - m----=‘. -T‘ I . -._ ' -~-. A - ‘G ‘q.‘ — _ -..__ 9 Problems in both rectangular Cartesian and cylindrical coordinates are considered since they result in different types of ordinary differential equations. Detailed results are given for two rectangular bars which are loaded by a uniform surface stress distribution. The first bar contains a through-thickness central crack while the second bar has double edge cracks. Stresses and displacements are also listed for a cylindrical solid with a penny shaped crack. Similar numerical results for the case of a central hole along the cylinder axis are also included. In addition, approximate results for an annular plate containing internal surface cracks are presented. -...‘l‘ u. -‘ --‘_._ . _ fll" G- ‘ . .‘b. 'v ' I I... A- .3 -;-~ r.-.” - -V““ ..~h '-‘A' ‘ “ hQWU-‘Vce. . v. ‘ C .. A11.'~ ‘-“ A .v--vfi-..6 :;Q‘ - . = a a “I ...‘ M .. . f‘ ‘- “w~. Q . ‘ I. _ 1 de 1".“ ‘ A e‘ . 5| a...‘ J R . I <3". ”‘0 a ‘ Q | n Luca -A._°M: byv~ l.‘. ‘ ‘ ‘V‘q_- .."~ as. ‘5“. . s‘,‘ ' e- ‘ ‘ .“ ~ ‘ “‘ t a “\ a s‘ ‘ . q " 4 5"- g- “ -“= e “ "- ~.,:. 5“ , . ‘~-_~‘ 5 a “V, ‘ ~‘ ~ ~ '0 a...~ . .‘Va‘r CHAPTER 2 SOLUTION OF THE NAVIER-CAUCHY EQUATIONS IN RECTANGULAR CARTESIAN COORDINATES BY THE METHOD OF LINES Let us consider a finite solid with rectangular boundaries which is loaded by a given boundary stress or displacement dis- tribution. For all the problems discussed in this thesis, the following assumptions shall apply: a. The deformations are infinitesimal, that is products of displacement gradients can be neglected. b. All deformations are elastic. c. All materials are homogeneous and isotropic. d. Body forces will be neglected. 2.1 Governing Equations Problems satisfying the first three of the above assumptions fall in the general class of linearized elasticity, for which the field equations neglecting body forces are listed below. Using the standard summation convention, the equilibrium equations are 0 i, j = l, 2, 3 (2.1) °ji.i Hooke's law is, 10 - c A ' .41... ‘-':--- \. 'v.‘-C u-— - a . D .‘ -9 - he. -. x h" "-.‘-\Ae a... .o .1.“ un'-" -’ a..- -V‘J' _ .‘— ' Q ‘ ’F fl‘ .‘ _ ,. ‘-- U. _.3 --= v." G . :.Ir ’4 ‘ W-e x.-- -C_.-¢ ' C : “" H'- A‘ - :ve .-\rn. "r- ‘ ; . . -fi .“_1 ‘A - ‘1‘ "U .‘- .- i. ‘ .1- o. b-‘ :1 ‘ ‘e ‘u- v. v. “.‘c r-” ‘v‘, s.‘:‘: ""‘ "-‘o (‘ \ e P. ‘ . -‘_ a ‘ 11 t O" = A61 l3 'ekk + 2Gei. (2.2) 3 J Strain-diSplacement relations are l _ 613- = -2—(u. . + u. .) (2.3) The solution must satisfy these equations at all interior points of the body and, in addition, prescribed conditions on stress and/or displacements must be met on the bounding surfaces. For mixed boundary value problems, displacements are prescribed over a portion of the boundary while stresses are prescribed over the remaining part. The above three sets of equations may be com» bined to form three partial differential equations in terms of displacements by substituting the strain-displacement relations (2.3) into Hooke's law (2.2) and the result in turn being substi- tuted into (2.1). The resulting equations Gui,jj + (A + G)uj,ji = 0 1,3 = l, 2, 3 (2.4) are called the Navier-Cauchy equations of elasticity. For problems formulated in rectangular Cartesian coordinates, these equations can be written as 3er 2 0+ G)-3-;‘—+GVu 0 (2.5) Be (A+G)-—£+Gv2v 3y 3e 2 (A+G)_.£+GVw 32 I O (2.6) (2.7) ll 0 v 9" ..:.'3 ...e (0.. I r. q a; e‘~ ‘— e... . U. an ab = 2X av 12 where the dilatation, e is given by r! e = __.+ ——-+ ——- (2.8) and the Laplacian, V2, is 2 2 2 2 3x2 ay2 32 The stress-displacement relations, which are needed in satisfying the boundary stress distributions and in expressing the interior stresses, can be obtained from substituting the strain-displace- ment relations into Hooke's law. The following equations, listed in the form used for Cartesian coordinate problems, are obtained: B an ‘3 3w (l-v) -+ v(v +—) (2.10) x (l+v)(1-2v) [ 3): FY— 32 :I 0 ll E 5v 3w Bu y (l+v)(l-2v) [ .ay az 3* - E ‘3" (3U 3v) 0‘ "' (134)) "—+ V —+ I—-— (2012) z (l+v)(l-2v) [ az ax 3y] - 5 av an] O _ + (2.13) xy 2(l_+v) [5; 3? cxzx = J— 33.» 2‘1 (2.1M 2(l+v) 32 8x B aw 3v 6 :: .__...__... ...... + _.._ (2.15) yz 2(l+v) [33’ 32] Solution of these equations will be obtained by applying the 13 method of lines, which is described in the following section, to equations (2.5) through (2.7) and by satisfying all relevant boundary conditions. 2.2 Method of Lines Approximate solutions for second order, elliptic and linear partial differential equations are frequently obtained by the finite difference technique (33). For three-dimensional elas— ticity problems, this may involve the solution of an enormous number of algebraic equations in order that reasonable accuracy may be attained. This will be particularly true for problems involving steep stress and strain gradients which arise at geometric singularities and thus require close grid Spacing in these regions. An approximate solution with greater accuracy and much fewer equations to solve can be constructed, however, by applying the method of lines to these equations. The line method lies midway between completely analytical and grid methods. The basis of the method is substitution of finite differences for the derivatives with respect to all the independent variables except one, with respect to which the derivatives are retained. This approach re- places a given partial differential equation with a system of simultaneous ordinary differential equations. These equations describe the dependent variable along lines which are parallel to the coordinate in whose direction the derivatives were retained. - ”'1'--"'=r - —-5--uo U.-— . -_ -- -y- : c a: c: v. 0d- U. -. ‘ Q '- 5 §. _ .— ““~-~m.-a- , _ . I- .a- ;._‘_g i ~.—.--._° “. _ n .‘ --.~ I h - ‘ "¢ ‘— vu'bn-'..~ .. -- _. . \‘-~° ‘cu ~-.,-~ ‘ V . ‘ “a “v~-. - --~.-h -. .- :- ‘- ‘.‘A. ‘ e - ‘ 'h *l . --Q:- ‘ --€:~ ~ ~Qh.. _ ‘ . ..-Q ~ \-.. .- ‘2. ' . “‘e e, ‘ -' c: a” . .J'h z\1~ ‘ ""“ ' '5 5.. ‘ -§ ‘ \a. \‘_ ‘c‘e “.fi‘ ‘~“. \\ v..‘.’ d I ‘1. “~.: \ w“ ‘ iqr~E:‘ ‘ . N \ es \“ \nee \‘\~ ‘ r» s ‘ §\ 11+ It can be noted that this method can be applied to a higher-order linear (or nonlinear) partial differential equation or a system of partial differential equations. Application of the line method is most useful when the resulting ordinary differential equations are linear and have constant coefficients. Equations (2.5) through (2.7) lead to linear differential equations with constant coefficients. Since in three-dimensional elasticity problems solutions of three partial differential equations are desired, three sets of parallel lines must be constructed. An arbitrary rectangular grid consisting of these three sets of parallel lines is shown in Figure 1(a). It is assumed, of course, that a grid of this type will sufficiently describe the geometry of the problem in ques- tion. The lines parallel to the x axis are numbered as l, 2, 3---NY, NY+l---2NY, 2NY+l---3NY, 3NY+l---£. The lines parallel to the y axis are numbered 1, 2, 3---NZ, NZ+l---2NZ, 2NZ+l--- 3NZ, 3NZ+l---m. Finally, lines parallel to the z axis are numbered as l, 2, 3---NX, NX+l---2Nx, 2Nle~~»‘”V, 3NX+l---n. This numbering system is chosen so that the resulting variables in the computer listings are identified through double subscripts only. The first subscript identifies the line along which the variables are calculated while the second subscript indicates the position along that line. For convenience, the lines are evenly spaced with hx, h and hz each equal to some different Y 15 ~ 5 ‘§ -\‘ ~~ ~ “ ". 3 n - NXXNY NZ .. ZNZ l NYX NZ 7 m - NZxNX hx z (a) Three sets of lines parallel to x-y-z coordinates. (b) Set of interior lines parallel to x-coordinate. Figure 1. - Sets of lines parallel to Cartesian coordinates. ,_.---¢-o - 9 ~ ~ 6-- I...- .‘Is’ - .- 0--— go ..... -— - u— .-L'. U‘ can... u d . . -. . -. - , - ..'I _ g, .. v1...“ we... 0 ‘ _ ‘ ’ < . A - .v- .7." 'Q Iev _b.-I~' - v . un— crc- - .l-,.-. VI. - - ‘ -- 0' I \ n .0... :3 E -‘~-- ' h..“'~ ‘ . -- 5v.-- .5 a -‘ a -. D. A, V. ...e ‘I‘. ’II V. a. N.- U ‘ l ‘1'“. 3 v1] L \,_ l‘. I ha, ‘ N “’5‘: _’ 5‘ ' g- '3 .- ~.Ia'o ‘ - y. me .,‘ 1:)- Q A a ‘ h 5 Q , ‘;"\._ . s \‘$ _ §“ 16 constant, although this is not absolutely necessary. The advan« tage of uniform line spacing is that the resulting ordinary dif- ferential equations can be solved more easily than those that are derived from non-uniform line spacing. Inherent to coupled systems of partial differential equations, such as the Navier-Cauchy equations, is that solutions for the de- pendent variables are possible only at points where the three sets of parallel lines intersect. These points are usually called nodes in a discretized body. This limitation is the result of coupling among the equations, which makes the particular solution of the ordinary differential equations valid only at the nodes. 2.2.1 Reduction of the First Navier-Cauchy Equation and Associated Boundary Conditions. For the solution of equation (2.5), the lines parallel to the x-axis in Figure 1(a) are considered. The x-directional diSplace- ments of points along these lines will be denoted as ul, u2, . . ., ufi. We define til, i 2, 613, . . ., 6|, as the derim vatives of the y-directional displacements of the same points on these lines with respect to y and éll, wl2, él3, . . ., élfi as the derivatives of the z-directional displacements of the same points on these lines with respect to 2. These displacements and derivatives can then be regarded as functions of x only since they are variables upon lines which are parallel to the x~axis. Substituting equations (2.8) and (2.9) into equation (2.5) and l7 expressing this equation along a general, x-directional line, denoted as (ij) in Figure l(b) gives 2 2 2 .1. 33 + §l+ .311 + (l—2v) 3w;- + 134-14} = o (2.16) ax ax By 62 1 3x ‘ az’" By the above given definitions, we write (2.16) as 2 d2u-~ 2 2 d u-- , l u l] + i. vl + £1... Wlij + (1-2v)[ 21- + (2...; + ii ]: O dx2 dx ij dx dx ayL 822 (2.17) where an eXpression for the last term in (2.17) is still needed. Introducing finite difference calculus (33), the partial deriva— tives of u along the x-directional line (ij) of Figure l(b) can be written as follows: 3 u l .._—.. z ____ . . -. 2 . (av?) h? (“1+l.3 “In ”1-1,J) (2 l8) ‘ lg] y 2 3 u ~ 1 (322) - it? (U1, 1+1- ?ul] + 1.11 j-l) (2.1.9) Using equation8(2.l8) and (2.19) in equation (2.l7), the general equation along interior lines is obtained. Thus, 2 . . d U' ‘ .- U'+l I + Ill-l _ 13 + (l 2v) [7 ( 2 + 2 ) uij + 1 ,3 .J ax? 2(1—vl h; hi hg u. . l + U0 °_l f..(X) 1.3+ 1:3 + ll___. = o (2 20) h2 2(l-v) Z -'r-\-v-- F- Nov‘.--..5 - \ v I I . . e .1 . a r. .. 2‘ .. .. - . .. N. z. .n A.» A,— ._. .v JV .‘ . u. . .—~ .\~ . .% an Q r. . . . a. C a r JJ . .c _ .. D . c. s .2 . . . S $¢ . . a» C . n. .w. .q. we. a» . .u un- ~J «.v AV :‘ .. . . . . .. . e C h. .: 3 .1 5 a\ u.‘ I. . a . . Lee 2. .2 u a» L e . . ._ . a... :k A e .. a: a: . I E was A. a. .Q J. .» a. .M w T. .u v a . l. n. . u v. .n. x. .. .2. s L. : .... .. . : F. l . . VA .. ~ Sh s . in L ~ .3 .3 a. .e nu. .: . . .. D . 18 ._ _ d9 dw where fij(x) - a; ij + d; ij (2.21) Similar differential equations are obtained for the displacements uij of the points on the other x-directional lines. Since each equation has the terms of the displacements of the points on the surrounding lines, these equations constitute a system of ordi- nary differential equations for the displacements ul, u2, . . ., uE. Equations (2.20) and (2.21) are applicable to interior lines only, since for bounding surface lines the central difference expressions for the required second derivatives involve imaginary lines. In order that central difference derivative approximations may be used, expressions for these fictitious line displacements must be found which are independent of the other differential equations. Since three-dimensional elasticity problems have three boundary conditions at every point of the bounding surface and a second order ordinary differential equation can satisfy only a total of two conditions, some of the boundary data can be used to find expressions for these imaginary displacements. Hence, condi- tions of normal stress and displacement will be enforced through the constants of the homogeneous solutions while shear stress boundary data will be incorporated into the differential equations of the surface lines. As an example of this procedure, let us take the first x- directional line which is formed by the intersection of the x-z . e g e \ F u . -vn. . l9 and x—y coordinate planes of a solid. For shear stress free surfaces, the boundary data gives . = O 2.22 ozx x—y coordinate plane ( ) cyx x-z coordinate plane = O (2.23) Using equation (2.14) in equation (2.22) we obtain along line 1 that (2:) = m 1 In terms of the discretized displacements shown in Figure 1(a), this equation gives _ dw In a similar manner, equation (2.23) leads to u = u + 2h dV (2 25) lYe 2 y a;'l ” Note that the x-directional fictitious lines are numbered lYn and lYe with n and e indicating adjacent north and east line lSt x-directional line. Substitution of equa- positions to the tions (2.2”) and (2.25) into the general equation (2.20), leads to the following equation along line 1: PC ..A ~ .1- Q be- ~uV 1.. I. 2O 2 dx by h by hz ?i(x) + __ = o (2.26) 2(1-v) where e e d :10.) = 91' + £131 . (l-2v) 3—1 + 3—91 (2.27) dxl Xm hydl hzdxl Equations (2.26) and (2.27) are typical of the corner line equa- tions with the exception of some sign changes in (2.27) at the other corners. It will be convenient to non-dimensionalize the above equaa tions with respect to some characteristic dimension of the prob- lem at hand. For the numerical examples presented, the following variables were introduced: h u = B- ; = 31 f1 = l- 1 a a X a " V " 2 ii El ) v = _. = = '2.28 a y a Y a k ~ ~ . h w = y. z = -z— h a .2. a a z a where a is the crack dimension. Equations (2.26) and (2.27) can then be written as (A. ;-. - - - - ‘ - O I - - - —— ‘ A -o -.v\.. a <- ~..‘: 2,— s a x s y}, -- _ ~69- ‘. b 0“. ‘0 'C“-- Cy. ,. . ~§ ‘.. .I .r ,. .. ,. O s s dx 2(1‘V) 71(2) ETIflg) - (2.29) 715.) = 9;] +2?— + (l-2v) 3.9.1”; + 9—5391 1(230) dx 1 dx 1 by dx 1 hz x 1 Introducing matrix notation, the differential equations along the x-directional lines can be expressed in the form 2 §:?. {6} = [xx] {5} + {r(§)} (2.31) x _ . 2x1 2x2 2x1 lxl where the non-dimensionalized coefficient matrix [KX] and the column vectors {u} and {r(x)} are given below. [le] 2EKXZJ o o o NYXNY NYxNY [Kx2] [leJ l [Kx23 0 0 Nnmy Nnmy i Nnmy \\ \ \ \" K = o \x o (2.32) [ x] \\ \\ \\ 2x2 0 0 [KX2] [le] [KXQJ NYxNY NYxNY NYxNY o o o 2[Kx2] [le] NYxNY NYxNY __ _J In 2.); ‘-‘ -- — -..- [Kx 2x1 dzfi 2 ~ -——%-+ SEZZXI. - :2—'+ :2— ul +«;3- G + d; 2(l-v) h h 2 ~ .——~ u ~2 2 2 NY+l ?1(§) "_m__ z (2.29) 2(l-V) 7162) = d3] +9-3— + (l-M 3—93 + 3‘2: 10.30) Introducing matrix notation, the differential equations along the x-directional lines can be expressed in the form 2 1:7 {a} = £le {a} + {rm} dx , 2x1 (2.31) 2x2 2x1 2x1 where the non-dimensionalized coefficient matrix column vectors [Kx] and the {u} and {r(x)} are given below. NYXNY NYXNY [Kx2] [leJ i [Kx2] O O NYxNY NYxNY . l NYxNY \\ \x \\ ‘2m-—i. T 1 z o \ \ \ o (2.32) \ \ \ 0 0 [KXQJ [le] [Kx2] NYxNY NYxNY NYXNY O 0 O 2EKX2] [le] NYxNY NYxNY .\‘-' e 1““; o 22 where the submatrices [le] and [Kx2] are kl -2k2 O O 0 -k2 kl -k2 O 0 \\ \ \ [K ] - 0 \ \ 0 x1 \\ \\ \\ NYxNY o : 0 -k2 kl -k2 l k (l-2v) 2 + 2 l 2(1-v) g2 52' y > z I, .. (l—MUF s ‘l ) l_v; ~’) 1 .rl J [13 o o o o \ [Kx2] - o o \ o o ‘\ NYxNY o o 0 -ka 0 0 O i 0 0 -k3 nu- »-l |4l|lllll lllll,\ : r l/\|r|li up: G~ . . .... .3 .. ex . . A: . . Fe . .A .2 .L .. u .I k : Lil-2.2.2- —~J—-- 3 2(l-v) g: Note that k1 = 2(k2 + k3). K ~ ) f ~ {u}l {f(x)}l {a}, {2%)}, {a} = ( I ) {pm} :i 2x1 . 2x1 {E}NZ-l {f(x)}NZ_l where the partitioned column vectors are r ~ W ~ ”1 (. uNY+1 U2 uNY+2 {a}, = ( . ) {a}, = NYxl ~ NYxl ~ uNY-l u2NY-1 L ”NY .‘J L u2NY > (2.33) J id}l .‘i‘fxl ('- “z-2NY+1 “L-2NY+2 {a} =1 I' NZ-l l , NYxl . “L-NY-l “L-NY \ . U(x))l = ”.1...— 2(1-v) NYxl ) J l 24 r‘ ' “n-Nyuw uL-NY+2 a = . \ {u}NZ < I ( NYxl ~ “2-1 \ uz ' d6 dé 2 de~ 2 an + + ._. 31-1 3351+(1-2v)[5;35' Ed] 1 dé d6 2 de 'l' + (l-2V) '1- aiiz '35 2 [:32'dnJ2 8. & dv d " a: + 3% + (l-2v)[§—-%;-] Nx'l NY'l ’2 My.; d6| d6 ‘2 d6 2 + + (1-2v) ~- -= afi NY . 5: NY [: '_ d“ + §;'d {H30}2 = - NYxl 2(l-v) 93 d2 d6 kdx + l dx + —: NY+2 d“ +—:.- do «fie-'2‘ 2NY ..dx 2S 2 .c, l +(l-2V)[h_d—:] NY+l y X NY+1 do NY+2 d% dx 2NY-l -2 dv NY + (l-2v) [E‘- :17] 2 - Y X 2NY J 26 f . . a?) + di‘i + (1 2 )[2 d6] - _l' d” 2-2mm 3's? z-zlml v B; d" 2-2NY+1 d6 at": a! 2-2NY+2 + a 2'2“” min) = - l < . - NZ-l 2(l-v) e e x. .3 NYxl d" + d" 3'” 2,-NY-l 3’ l-NY-l 46' all 2 dv — + — + (1-2 ) - — — L d” z-NY d“ l-NY v [ 13y (ML-NY J F - .: £4 +dw +(12v)2_d‘\7_2dfi 2—NY+1 '35 £~NY+1 5y 3' 5: a? ' ‘ 2 a 3;) + ‘31-;- + (l-2v) - fi— 2"] Luau-2 2.-NY+ z 4" LNY+2 1 ' ° ' {Am} = - - ' ' NZ 2(l-v) < . . - 2 NYXl 332‘- + g;- 1' (l-ZV) [" *- 3:] 2—1 2—1 ‘z x 2—1 d5 2 d9 2 an a} + a}; - (l-2v ‘8" a'!‘ "’ 3‘ a: k i z " -:.. z. . a,“ ;....E 1.6 e- , u... ' ... , “‘~ .~ .,.;- gr \ - ~.-...s--a. . -’ o ‘0 - 5;- ‘ I. ‘A "““-.~... A . ‘ ..... . A g” . - - \Ii ...‘"" ON --v... » X"'Vfl~.-',., . ‘ . , icehue.V..a‘ -. , _ A i , "FAN,“ ' 5-.- 5|.,‘ ‘:‘ A V c A. ‘L y“ 0"“ ‘ a ‘u \ by» “‘-.es ‘Q .‘h. ‘g‘.~4.:‘:" ‘u‘ges h. ‘C (OH ‘I‘, . ~J are , ‘r' ‘ l \ ‘J. ‘ '1' 31% ,, ”S w‘ ‘1‘: ‘~‘ “ . ‘a‘CEr‘. . 27 Since the elements of the coefficient matrix [Kx] are all con- stants, equations (2.31) are differential equations with constant coefficients. The coupling terms from the second and third Navier— Cauchy equations appear only in the vector {r(x)} which makes the particular solution of (2.31) a function of this coupling. Assuming that {r(i)} is known, solutions of (2.31) can be ob- tained in closed form when boundary data at the end points of the x-directional lines are given. 2.2.2 Reduction of the Second Navier-Cauchy Equation and Associated Boundary Conditions For the solution of equation (2.6), the lines parallel to the y-axis in Figure 1(a) must be utilized. The y-directional displacements of points along these lines will be denoted as v1, v2, V3, . . ., vm. We define fill, ol2, pl . ., him as the 3, . derivatives of the x-directional displacements of the same points on these lines with respect to x and all, «[2, WI3, . . ., elm as the derivatives of the z-directional displacements of the same points on these lines with respect to 2. These diaplacements and derivatives can then be regarded as functions of y only since they are variables upon lines which are parallel to the y axis. By substituting equations (2.8) and (2.9) into equation (2.6) and analogously expressing this equation along a general y-directional line (ij) to equation (2.20), the following equation will be obtained: 2 2 2 3 Bu 3v 3w _ a v a v a v _ 3Y[3X + B), + 32] + (l 2V) [TX "' 53,-?- + 322] -- 0 (2.314) ij 1] 2 2 d V:L + du + dw + (l 2 )[d v1] + (32v 32v) ]_ O __. __. _ v A - dy2 dy lj dy i] A 8x2 3Z2 l] (2.35) In a similar manner to equations (2.18) and (2.19), we have 2 9 v 1 (8x2) 2 1'12 (vi+l’j " 2V1]. + vi-l’j) (2°36) ij x 2 3 v l I (322) - h? (vi.j+l ’ 2vij + Vi,j-l) (2.37) ij z Combining equations (2.36), (2.37) and (2.35), the general equa- tion along y-directional, interior lines is obtained. Thus, we have 2 d Vij (1-2v) _ 2 + 2 v-. + Vi+1,j + vlmlsj 2 + 2(l-v) h? 7g2' 13 h2 dy X z v. . + v. .- f..(Y) + 133+:2 123 1],, 271—711“) = 0 (2.38) db 3;. (2.39) where f..(y) = 1] 3§'4° ij 1] Similar differential equations are obtained for the displace- ments along the other y-directional lines. Since each equation contains displacements of the surrounding lines, these equations l I l I . .. .. . l :0 3' :4“ . . . .u 3x ‘u .. J. . 1" as. .- . ,, . 1.. ‘0 ‘ ‘1 ‘. nu ~ I . + -, ¢ ~ 1.0 . A ' a.‘ “4.:_' .v I ‘. ‘0 . ~' I. u.:\ -' :V‘ ~<'., ““"“U. “.5. ;~.. I I ‘— .. ‘ § -X . l e- - ‘2 q . V ‘ A - a: :2 ' o ‘\ o .V: u ”M a quatl ‘ A o ' | I ‘V‘,‘ ‘k‘cv’ a o J‘- . 1» ~ ‘ Q ‘L ("ere P {4(4) ._ : c S:‘: w \l,“‘ar A ‘ § '- 3“- s .3“ ”“3 v.“ ‘V,‘~ ‘. “it; 2 2 2 .3..[au «9- av + aw] + (1-2v)[——3V + L ‘2’+ ———a V] = 0 (2.3M) 3y 5')? '53? 32 ax2 By 322 ij 1) d2v 2 l +du +3.3 +(l2)[dvij+(a2v+32v) ]_ O _ _ - v — -——- - dy2 dy l] dy ij dy2 3X2 322 ij (2.35) In a similar manner to equations (2.18) and (2.19), we have 2 3 v 1 (3x2) 2 7-h2 (vi-+1.3. - 2V1]. + vi-l’j) (2.36) ij x 2 3 v 1 (822) — h2 (visj+l - 2vij + vi,j-l) (2.37) ' z Combining equations (2.36), (2.37) and (2.35), the general equa— tion along y-directional, interior lines is obtained. Thus, we have 2 d v13 (1-2V) _ 2 + 2 v + vi+1,j + V1-1,j + 2 l-v h'? ' 1,2" 13 h2 dy2 x z x Vi 3+1 + Vi j-l *ij(y) + ’ h2 ’ + 2 1..“ =3 0 (2c38) z where fi.(y) = %E- dw (2.39) 3 y ij Y ij Similar differential equations are obtained for the displace- ments along the other y-directional lines. Since each equation contains diaplacements of the surrounding lines, these equations Q - a ‘IAO.,‘ ._ ; ""~---5 ...4.. ~ \ C . . _ ..' ‘1' oi. ~A.\.. -v. -‘ as-.- .. :_ _ ......3 , a H in nun—fl '. .~-..‘ - .v. - bUhc—.-.',.: ‘l. "v .. . -‘ ‘H-A-‘l "‘ -‘vxpo '.”.' ---e -Q. ~ . l A. .h“ to ~ ‘a V. en: . . -' ~ : - 9 run-3; o.— a- .“ ‘\ "\ b : . ~~i‘.‘ec ‘n I et ‘0 . . \Y |"* ‘ ey‘e tn“ ..a k s \‘. 12“ ‘\ ‘A .Q-‘e‘ ‘ ‘ o .‘ * 'I ‘*"A e. :-|~L VH‘. 29 form a set of ordinary differential equations for v1, v2, . . . 9 v . m Noting that equations (2.38) and (2.39) are defined fully for interior lines only, expressions for the imaginary diSplace- ments lew and len must be obtained from the shear stress conditions in case of the first y-directional boundary line. Since the first y-directional line is formed by the intersection of the y-z and y-x coordinate planes, shear stresses on these planes in the y direction are utilized. For shear stress free surfaces, the boundary data gives a = O (2.HO) z y y-x coordinate plane oxy = O (2.41) y-z coordinate plane Using equations (2.13) and (2.15) the following equations will be obtained in terms of the discretized displacements shown in Figure 1(a): du lew = VNZ+1 + 2hx 537 1 (2.42) _ dw Vlzn ' "2 + 2hz g; 1 (2W3) Note that subscripts w and n again describe west and north adjacent line positions with respect to the first y-directional line. Substitution of equations (2.42) and (2.u3) into the general equation (2.38) results in the following equation along ..‘ I Ouch .l a .II I 'u I, q \ ‘ \- hi, —_'+ “u? — . ‘, L'\--|'; ... i! a - l 0 f I ‘\ . - ‘2-- .I' ?‘ I fi.. I 5 Va ‘..i‘ o . . . a., bl ‘ I Q ‘~' 4‘ a I a '- _ ., 'v“ I...e::: ‘ "‘ ~-v.,=‘- - ‘Pm-‘-.,_" “4». in ""Ms Ta“ it I. . g l ”a “ ‘. " .- UH; §ue ‘;. I . F \. ~ . ‘ ,.," ‘~" *'1 “.1 Hi o - “2E A C“ ~CE.: ‘ I .A‘ .5. 7“ ~ r I ‘ .\‘ ' U Vfi til." r, L-\ '. - \'-3 HL' P. I .§\ ~ \ H C \ / 30 line 1: d2v 2 2v 1 + (l-2v) [_ (h2 + 2 + V2 + NZ+l 2 2(l-v) 2 2 )V1 2 2 dy z hx hz hx {l(y) + m) = 0 (2.1)”) ‘ ° 2 du 2 d 1' (y) = 951 + 3"- + (1-2v) —--—+ —l (2.L+s) l dY 1 dy l hx dy hz dy l Non-dimensionalizing these equations according to (2.28) and introducing matrix notation, the ordinary differential equations along the y-directional lines can be written as 2 i— {3} = [K J {G} + {sum (me) g2 Y (1. mxl mxm mxl mxl The coefficient matrix [Ky] is given by [Kyl] 2[Ky2] o o o NZxNZ NZxNZ [KY2] [Kyl] [KY2] 0 . 0 NZxNZ NZxNZ NZxNZ = \ ‘ \ [Ky] 0 \ \ \ o (2. Ln) mxm o o [KY2] [1(le [Ky2] NZxNZ NZxNZ NZxNZ o o o 2[K,,2] [1(le NZxNZ NZxNZ 9- - ._ _. ."m a t -. .......- 1"" (a! 31 where the submatrices [Kyl] and [KY2] are — kn -2k3 o o o ‘7 = \\ ‘\ ‘\ [KY1] o \ \ \ o NZxNZ o D -k3 k, —k3 o o o -2k3 kn k = (1-2.) ' 1 3 271-») LEI-2 Z k _ (l-2v) ' 2 + 2 u ‘ "‘“"‘ 2 '2' 2(1.\’) LHZ fix Ivote that k” = 2(k3 + k5). 0 -k5 o o o _ \\ [KY2] - o o \\ o o szm o ' o o -«5 o o o o o -x 5 L -l k = (1-2v) _£_ 5 2(1-v) {,3 The vectors {6} and {s(§)} can be written as ":Fa 9-. 1 -_; IUOV“ il“ U‘.»~I‘ , . a v Q.“ . "'0‘ "‘<¢. a u 5-5:. K . U i r!- 1“ v' 5‘ . ' ‘ He (9“,. w.vrs 31 where the submatrices [Kyl] and [KY2] are — ‘ z \\ ‘~ ‘\ [1(le 0 \ \ \ 0 NZxNZ o 0 -k3 k” 5 -k3 o o o ~2k3 ku k = (1‘2 V) l 3 2(‘1'-"v) "5"2 Z k _ (1-2v) 2 + 2 u ‘ """' 2 '7?’ 2(l—V) Hz HX Iqote that k” = 2(k3 + RS)- 0 -k5 o o o - \\ [Ky2] - o o \\ o o szm o ' o o ~«5 o o o o o -x 5 L ' dl k : —-—-——-—( 1‘2V) ._]_'._ 5 2(1-V) 33 The vectors {6} and {s(§)} can be written as , . a") \‘l . ..'., . lufl‘e .— g " 5;. a?‘ 3" ‘ ~ “1 {v}l {912 ' ) {V}NX-l {a} K NX J 32 r {s<§>} = < mxl L where the partitioned column vectors are {31. = ) 1 hflle {V}NX-l=< Nle vm-2Nz+1 Vm-2Nz+2 vm-Nz-1 vm-NZ \ 2 r M2 = ( Nle r {;}NX = < Nle \ {;<§)}l {;<§>}2 O O {;<§)}Nx_l {f(§)} ? NX J vNZ+1 vNz+2 3 P V2Nz-1 ~ V7N7 J 1 ~ vm—NZ+l Gm-NZ+2 : ? G m-l (2.98) {*(?)}1 = 2(11 ) ‘V Nle {;(y)}2 = .;;JL.. 2(1-v) Nle 35 O I W an d” 2 an 2 32 21's? *2? ”1*?“ firm; 1 l X y z 1 gg_ da «<49» 9.1} . w[ ( 2.9.2 d? d + l-2v) Ex d? NZ-l NZ-l z-1 3g dé ( 2 .13 d? d~ + l-ZV) Hz d- NZ+1 NZ+1 NZ+1 dfi d6 37 + 3% NZ 2 NZ+2 d5! 3% ~ +-1: dy dy 2NZ-l 2NZ-l dél 3% + (1 [q 2 33] “L- + '7 _2v) " F”: dY d h d 2NZ y 2NZ z y 2Nz J _. ,_. w W” . . .4 .23.“ ..- .m. .m m. + + ‘ l a V W“ Int I r 1» .. V . .c...... .c..... on ... 11.1. ( l— _ o . mu 2 W?” q. ?. “—7sz _ _ :(J _.L. PIIL . _ - 2.,” U HV F".\. 5.. w... _ _ .1], 1* ..J v (\ (\ _ +. + (s. {. all 0.. _ + «H + n.“ ”U nu Hm Mn . _ ,1 _ _ m w.- m m 1 r t . I (1).]. I ['11. o. f... m- 0‘: 7... a~v.n .1. c u .7... u 1. J L :. . . I...Il ,“c ,.-U .~. .1. .- f... + .4 . + + 2 +. . l m; _ + 2 . 1 v; v: . .7. {.4 r a \ Y.\ l.‘ '1 \ . I... \r‘ ‘1 _ _. m. .- m. m.- w.1 m“ . 11.3.. 1.1....) .m. TV. 1...?) . . 1+. 1.- ; dd 6.; / {ik )\ \nl, U 11 _ ll _ /I\ 9.. : 11+ "r—v {:5(' .3?)} - r i\ .2X‘. ‘1 ‘ I ’2 a“ P‘ ‘1‘:qu ".e “‘ :Voo-sb a ‘ 33 2 A“: (Is. “8 “av.. tL-SSAA‘ - «c-: '“F 9L IV. “.8 C h the z-aXis :_ -5. N. 1 5.:P¢a:e:e:‘.t S ~25" ‘ a f‘dyemen. A: in 2 On;‘. J V‘ V ‘. ‘3 "m 5‘6. V‘ ea. ‘u . F A «V: 35 Equations (2.46) form a set of equations that are similar to equations (2.31). Assuming that {3(9)} is known, solutions of (2.96) can be obtained in closed form when boundary data at the end points of the y-directional lines are given. 2.2.3 Reduction of the Third Navier-Cauchy Equations and Associated Boundary Conditions For the solution of equation (2.7), the lines parallel to the z-axis in Figure 1(a) must be employed. The z-directional displacements of points along these lines will be denoted as "19 W2, W3, . . ., wh. We define fill, al2, ul3, . . ., aln as the derivatives of the x-directional displacements of the same points on these lines with respect to x and v[l, 6|2, v[3, . . ., 61“ as the derivatives of the y-directional displacements of the same points on these lines with respect to y. These displacements and derivatives can then be regarded as functions of z only since they are variables upon lines which are parallel to the z axis. Following the same procedure as in the deriva- tion of equations (2.20) and (2.38), the differential equation for a general interior line (ij) along the z-direction in Figure 1(a) is 2 d w. 0 0 o 0 11 + (1-2v) [_( 2 + 2 ) "13° + W1+1,3 + w1--1,3 h 2(1-v) 2 2 dz2 x hy hx w. + . . ..(z) + 121*1 2 "lb-1‘1] 'r&._.§L_._.JL = o (2.1-£9) h 2(l-v) Y where ~: :‘ '3”: ' 1.1.1?“ 2 iirezticml 1L7 pacer-ts 5'; of mm) 51 . . H (in, NC iefined full) fictitious dis the shear stre beef-Hy line. the inter-secti- messes on tr 5th Stress 1 A ‘V J~l TN .1 \L‘VV 36 where iij(z) = 92. +%Y. (2.50) dz ij Zij Similar differential equations are obtained for the other 2- directional lines. Since each equation has the terms of the dis- placements "i of the points on the surrounding lines, a system of ordinary differential equations is obtained for wl, w2, w3, . . ., "n- Noting again that equations (2.u9) and (2.50) are defined fully for interior lines only, expressions for the fictitious displacements "lxw and "lxs must be obtained from the shear stress conditions in case of the first z-directional boundary line. Since line 1 in the z-direction is formed by the intersection of the z-x and z-y coordinate planes, shear stresses on these planes in the z-direction are utilized. For shear etreee free surfaces we obtain _ du Wlxw - W2 + 2hX E;- (2.51) l _ dv (:le - mm“ + 2hy 'd'z' 1 (2.52) Substituting equations (2.51) and (2.52) back into the general equation (2.89) and non-dimensionalizing the result according to (2.28), leads to the following equation along line 1: 2~ d w 1 (l-2v) _ 2 2 ~ 2 ~ 2 F*m[(§3—*T')W1+TW2*VT“NX+1] z x by x 21(2) + 2(I-v) ll 0 '1 ([0 - ‘ I 0“ - . - .. . . . ’ lor‘- 9.... .- ..... v»...._5 m‘ .. ‘ ' A. _ ‘ - . v- A. p- a e. an. .. ..'H.- "'~ ~ :. ‘Q I ' ~‘ -..: .. x._ - “\L" = p 5. \- a. i _ r E s N‘ , ‘1 1 .1 , " u ‘- ‘\ \ “a, “‘e s- 37 l 32 ‘h A N V I 1 Introducing matrix notation, the differential equations along the z-directional lines can be written as [K2] {3} + {t(2)} (2.53) nxl nxn nxl nxl where the coefficient matrix [K2] and the column vectors {a} and {t(2)} are given below. [K21] 2[K22] o o o NXXNX NXXNX [K22] [K21] [Kz2] O O NXXNX NXXNX NXXNX \\ ‘\ K o \\ “ ‘\ o 2.5a [ z] \ \ \ ( ) nxn O O [Kz2] [Azlil LK22] NXXNX NXXNX NXXNX o o o 2[1<223 [K21] NXXNX NXXNX where the submatrices [K21] and [K22] are \. ‘Hs V'Q S y ‘1 [K211 NXXNX [Kz2] NXXNX Note that k6 38 2(k2 + k5). k6 -2k5 O 0 O \\_ \\ \\ 0 \\ \\ ‘\ 0 0 0 -k5 k6 -k5 o o o -2k5 k6 = (1-2v)[ 1 ] ~2 2(1-v) hX = (1-2v) 2 + 2 2(1—3) 52 ii? x y P, _ o ~k2 o o o 0 ‘\ O \\ O 0 o o o 0 -k2 _ ._L 22.22. 1. 2(l-v) 52 Y r \ {V4}l {a}2 {w} = ( . P nxl {W}NY-1 k {w}NY 2 r ~ 5 W1 1'}2 {ml = < NXxl ~ wNx—1 Q \ ”X J K 1 wn-2Nx+1 a3n-2NX+2 {§}NY-1 = i ° NXxl ~ wn-NX-l an-NX 39 {t(2)} = nxl {Q}NY ‘ NXxl f - {f(z)}l {#2)}2 4 E {muml N QNx+1 WNX+2 w2Nx-1 *2Nx J wn-NX+l wn-NX+2 (2.55) {£(2)} 1 NXxl 4O A : 2(l-v) + 32 32 dz ux+2 NX+2 22. 3% d2 +-.-.- 2NX-1 dz 2NX-l + (l-2v) [- 33 Ldz 36 32 2NX + 2NX r . . x 33 + 35 (1 2 ) 2 33 ’t- ,1: + - V =_-—=- dz dz h dz n-2NX+l n-2NX+l L. X _2NX+1 dfi 36 "r + 37 dz n-2NX+2 z n_2NX+2 {25(2)} - 1 J . . ”H m . . > NXxl d5 3% .T "3 dz n-NX-l dz n‘NX-l “—5- _: - v - 3:.— T dz - dz ‘ h dz k n NX n NX x n-Nx J r W 9-1-1- +9«‘=’- +(l‘2v)!—?¢—9-g--2:—i§ dz n-NX+l dz n-NX+l th dz hy dz n-NX+1 3.121 g“: + (l-2v) [- '13?- 1%] ‘ 2 z -Nx+2 n Nx+ n y n-NX+2 ” ‘ l 4 . . E {£(z)}NY - 2(1-v) . o . NXxl 9E. + §§_ + (1 2 ) F 3_.92. d2 n‘l d2 “-1 \J L By d2 “-1 d6 d6 2 dfi 2 d0 .. -. -..-.-.[.... 1% k dz n dz hx dz by dz n 41 9’. " anfld.‘ " , ME .ue'-vo3-} - l - --~.u- 6‘. :6 Hula--us .u- . ‘ I D 1 “. \ put-.' A 4.. .‘E ...a...&._ .. . ~ ' “‘v #0.»... . . L -o-e~.-w.G- ‘4 ’I-I I I .‘., - qu . “nee .' ....-(. 9 ‘ ‘ P ‘ u ‘ " \F‘~'O.p' ‘.' iv.‘."“ v ’ I ”,e-;.r‘ ’ ‘V .0.\.-e_ “ \ . \f‘ "§~ A“ ~ 0"“..V‘. Ex:e.si'.'ely at“ §:.“\. '0'. ‘3 SLX;‘Q g 9 F c“ ‘rr‘b’mi t1: EquatiCTS. we S‘fi‘uqti‘. eofi“s e eqla‘:hh Du ..VHS‘ :- t .‘ in Me: \ 4e expre< $1: “as re‘; A t M ".Vd 1‘ I fades and e‘ in t‘. R ‘ a ‘e arF~Lc ‘ V‘; "‘ue‘ IA 'e‘went ‘ An Nit-'- C .alt‘cn cf 42 Equations (2.53) are a set of equations that are analogous to the previously obtained two sets of differential equationso Assuming that l{t(z)} is known, closed form solutions of (2.53) can be obtained when boundary conditions at the end points of the z-directional lines are given. Finally, it can be noted that the three coefficient matrices obtained are all singular. 2.3 Solution of Simultaneous Differential Equations With Constant Coefficients Solution methods for a system of ordinary differential equations with constant coefficients have been investigated extensively and are available in the literature (34). One rela- tively simple method is that of the power series solution (3M). In applying this method to a set of second order differential equations, we must, by suitable transformations, reduce the given simultaneous equations to a new set of first order differential equations. The solution of this set of first order equations can then be expressed in terms of a power series. The advantage of this method is that it avoids the problem of finding the eigen- values and eigenvectors of a given coefficient matrix. However, in the application of this method serious convergence difficulties may arise in evaluating the power series at large values of the independent variable. An entirely different method of solution involves the trans- formation of the original coordinates into a new set of . O "N’vV‘"'-. A sv- vvvo dong-ea " . l . F op~ 4- ,! a? ‘ .11 ”to: Me" 59. . 15:25, me first. I e :3. QO~APS VOFA; V‘UQ5.VtD ..ev~ 0"" A-A“ cut ._,.5 , UK) c...“ . 1 "‘ Iv °»w." _e v —} CST. .. 'vn- I F‘-$. “:39. .q ~-CT. - 236$, .19;- ~ . v . ,‘- ‘ -eu _: '1 J‘. : . i' V‘. . — “h. - v v. ‘H t h. F ‘lu A‘- V. A \- ~ ,.' \- a"; 25re .- #3 coordinates in which the differential equations become uncoupled. In this new set of coordinates, usually called principal coordi- nates, the matrix equations are reduced to a set of scalar equations whose solutions can be easily evaluated. The required transformation, however, is only possible when the matrix of eigenvectors, usually denoted as the modal matrix, can be accurately constructed. The solution method employed in this dissertation is a combination of the above described two proce- dures. Since equations (2.31), (2.46) and (2.53) are all of the same type it will be sufficient to discuss equations (2.31) only. Hence, let us define the following new variables: ~ ~ Ul = El 02 = 32 . . . u, = u2 - dfil - duZ - dfifl. (2.56) U1+3. " a3:— Un+2 " 3;:— ' ° ' U29. ‘ 3;? In terms of these variables, equations (2.31) can be written as 9.3m = [A] {u} + {5(2)} (2.57) dx 22x1 22x22 22x1 21x1 where l l [0] [I] 2x1 Ext [A] = (2.58) [Kx] [0] 22x22 le lxl' h. .1 n ‘ . n ‘R we. _ me >v-‘b-x‘ I;- .3. I: m) ""/~LL‘ .. A); .... Av huge A: .‘ V ‘ ‘ a N {3(2)} [aflnn an '{O} ixl {5(2)} = (2.59) 22x1 {r(x)} 2x1 The solution of equation (2.57) is well known and can be written as (an) ~ X {use} = eEAJ" {U(o)} + emi f e-Wn {?(n)}dn (2.60) o 22x1 22x22 22x1 22x21 22x22 22x1 where {0(0)} is a vector which consists of the boundary values dfi 32' by the following equation: ~ 4. i e o e e of u and at x = 0 and eEA] is a matrix series given ~ 2~2 3~3 eEAJX = [1]+L%B-+£§§]_;i_+£ig7§_+... (2.61) Using equation (2.58), the powers of matrix [A] can be expressed as , [KxJn [0] 1 [A]2n = n = o, 1, 2, 3, . . . n L [0] [Kx] J P [o] [len - [A]2“*l n = o, 1, 2, . . . L [lenfl [0] - I ' .--o~o--O-F‘ '. ‘ s a..- bg~u~-.'. ... .- foe-u; I p " ~ 1‘1 :' ‘~ - . - .“\ athX 5 ~ - . U A ' A ‘LXLL .I \. h- "me ~--- _ 2. . S“ \- d.“ .1, » ‘— mg" f ‘w- (A . won‘t. A“ -. ‘\ us Substituting the powers of matrix [A] into equation (2.61) yields P fi 2:) 22m “1 ” ~2m+1 m [K 3 ii) [K (2m)' X (2m+l)! X m=0 m=0 em]: ___ .. .. ~2m+l ~2m 13x 2 m—x w ,m Qmfl)’ X 2m! X Lmzo m=0 .J p 1'! [All(Kx’i)] [Al2(KX’i)] ~ 2x2 2x2 eEAJx = (2.62) 22 22 [A21(Kx,x)] [A22(Kx,x)] x 2 1+ 6 Since coshy = l + X_.+ I----+ I—-+ . . . . 2: u! 6! 3 5 7 sinhy = y+L+L+l.-+ . . . . 3! 5! '7! the submatrices [Aij] i,j = 1,2 can be expressed as the matrix functions shown below. l/2~ = 2.63 [All] [A22] cosh [Kx] x ( ) 1x2 2x9. 1/2 ’1 . 1/2~ [A12] = ([Kx] ) Slnh [Kx] x (2.6%) 2x2 a - . huni" s- n .. V- ‘ n we? 1,\ M... . “""wv i:::l 3“-..:.J ’ . . -— 03...‘ ‘“ “ “"‘“b -u g- - ‘ A 2 ' . .. ""I- . . .- -.‘ ““hh ‘A :bghsgvl, uI‘ . P ‘1,” so -..-G'°“- v. u.,. .- V. _. ' o .-;-.p....:-‘ " bvavO -r..v.. “; Q s' Q - “AA 9.... new,“ .4 _‘. “' "'e «9.. L- - “‘:b' we Sara? .S to use L #- t“5~ at $377.:- 46 [A21] 3 [KXJEA12] (2.65) _ where [KXJI/2 is a matrix whose square is equal to [Kx]° A good summary of matrix function definitions and theory can be found in (35). 2.3.1 Evaluation of Matrix Functions As can be noted from equation (2.60), the accuracy of the solution for a given problem will depend largely on the number of terms retained in (2.61). In the course of this work, a number of different methods were employed in evaluating (2.61). A short description of each technique is given below. The accuracy of each method can be checked by substituting equation (2.58) into the identity of eEAJR . e-[AJR = [I] (2.66) In terms of the submatrices [Aij] this equation yields [A11(Kx,x)]2 - [xxJEA12(KX,2>12 = [I] (2.67) 1x2 1x2 2x2 ix. The most straightforward calculation procedure for evaluating the submatrices [Aij] for each value of the independent variable 2 is to use the matrix series definitions (2.62) and truncate them at some given values. However, for increasing values of a, a large number of terms must be taken and if matrices with orders of 20 or larger are involved, the numerical procedure becomes 1+7 inefficient. In order to avoid the computation of a large number of terms, additive formulas for these matrix functions may be obtained from using the identity of [A121 [A122 [A](%1+§2) e ° e e In terms of the submatrices [Aij] this identity yields the following two equations: \ [All(Kx’§l+i2)] = [All(KX’§l)][All(Kx’§2)] + [KXJEA12(KX,§l)][A12(KX,§2)J > (2.68) [A12(Kx,§1+§2)] = [A12(Kx,§l)][All(Kx,§2)] + ["‘11(Kx:5‘1)3E""12(1 (2.99) 2mxl {v(o)} {Pu} {Fuo} mxl mxl NICxl (F,B} Values of {Fun} and {F38} are given by equations (2.91) mcfi‘n ‘Je (my 29‘ :1|\ DI: 60 respectively. An analogous solution to equations (2.86) along the y-directional lines can be written as W F ' (e(§)} [911(Ky,y)][012(xy,9)] {7(0)} {B3(y)} (. M {9(9)}J [021(Ky,9)][D22(Ky,§)] (9(0))J kfe.(y)} (2.99) where [Dij] are similar matrix functions to [Aij] and {B3} {Bu} are analogous particular integrals to (2.8a) and (2.85). From equation (2.99), we can express {6(L)} in a partitioned form consistent with (2.93) as. , 3 {Va} FiDzlall [Dglagj {r30} {83“} NICxl NICXNIC. NICxNOC NICxl NICxl . = + {98} [D2131] [D2182] {F33} {B3B} NOCxl - - NOCxNIC NOCxNOCJ . - NOCxl NOCxl - - y=L b y=L y=L r '7 [922.11 [922.21 ma] (3..) + NICxNIC iNICxNOC NICxl NICxl [D2281] [D2282] {PUB} {B48} NOCxNIC NOCxNOC , ., NOCxl NOCx .. L. -- y=L y=L (2.95) Equation (2.95) leads to two matrix equations involving the two unknown vectors {F36} and {F98}' Solution of these equations yields 61 -l 1 {FMS} : [D8] {V8}9:E NOCxl NOCxNOC NOCxl -1 z - D D ~ ~ D ~ V ~_~ [ a1 [ 218113,:L [ 21.13y=n ( ,}y_L NOCXNOC NOCxNIC NICxNIC NICxl -l - [Ba] [Db] (338)§:fl NOCxNOC NOCxNOC NOCxl -l - [Da] [0.] ((P,,) + {39.}9=t) NOCXNOC NOCxNIC NICXl NICXl - {BHB}§=E (2.96) NOCxl where -1 [Da] = ([92282] ‘ [D2181] [D2191] [D2292])§=£ NOCXNOC NOCxNOC NOCxNIC NICxNIC NICxNOC -l [Db] = ([D2lB2] ' [D2181] [D2lal] [D21a2])§=i NOCxNOC NOCXNOC NOCxNIC NICxNIC NICXNOC -1 [DC] = ([D22Bl] ‘ [D2181] [DQlal] [D2261])9=E NOCxNIC NOCxNIC NOCxNIC NICxNIC NICxNIC I An“. u‘q“ 62 -1 9 ~ ~ ~ ~ {F ED21011§=E {va}y=L - {83a}y=L NICxl NICxNIC NICxl NICxl -1 ‘ [D21Gl]§=£ ED21623§=£ {338}§=L NICxNIC NICxNOC NOCxl -1 - ED.1.119=£ £922.1J~.g ((9... + {B..}9.z) NICxNIC NICxNIC NICxl NICxl .1 ~ ~ ~ " [D2161]§=£ [D2262]y=L ({Fue} + weaker.) (2°97) NICxNIC NICxNOC NOCxl NOCxl where {Fae} is given by (2.96). Note that although the full matrix [D2l]§=£ is singular, the partitioned submatrices are not. Equations (2.96) and (2.97) together with the given boundary data completely specify the initial value vector needed for the solution of equations (2.96). Certain conclusions using the shear stress conditions can also be noted for problems involving zero or uniform normal dis- placements in a given plane. Using symmetrv conditions (2.88), for example, we have dfi d~ fligo 8 tylfi=o = O (2698) plane plane The zero shear stress conditions on axy and 8x2, together with equation (2.98), will lead to . n. .fid flu .nu u VA- 1 ‘ 3'5 »-\Jo. so“ i V e a . a ‘ re .. h e 5 C s a: awn Mar. 3. C 9 C S 3 u n e T. S T. .2 S O .t .e 9 O r ~ .. l I .1 it v(. .2 C C P. a. .. . I § 6 a s 0. km mm a F. A,» '7' a S “a a.“ :u D. .9. so Q. Hi. : . . . n- Go a .. n. 6,. 63 BV _ 3% _ 322:0 “ fiat-so - 0 (2.99) plane plane Since 52.: §=.2;. and g¥-= £7-3¥, we can conclude at once that x dy 3x dx dz 3x {r(i)}§:o = 0 (2.100) 2x1 Equation (2.100) shows that all the elements of vector {2(2)} in the plane of zero x—directional displacements are zero. Simi- lar conclusions can also be obtained for the elements of the vectors {9(9)} and {t(2)}. mxl nxl There are problems when it is more convenient to apply a given boundary diSplacement, {9(6)}, rather than a uniform normal stress, {00(3)}. Although we shall not present detailed results for this case, the analogous boundary equations to equations (2.96) and (2.97) are listed below. - " -1 ~ 1 . {F99} - [Da] (v, M NOCxl NOCxNOC NOCxl -1 ~ ~ “ [D3] [DllBl]§=t [Dllal]§=£ {va}y= NOCxNOC NOCxNIC NICxNIC NICxl - [‘D' 1"1 a NOCxNOC NOCxNOC NOCxl NOCxNOC [Eb] NOCxNOC [5'] c NOCxNIC 93.} NICxl 6H -1 r ~ [0a] (BC) ((3,0) + (Buaiygi) NOCxNOC NOCxNIC NICxl NICxl {Bu8}~_ L (2.101) NOCxl -1 (ED12823 ' [D1181] [D1101] [D12a23)§=g NOCxNOC NOCxNIC NICxNIC NICxNOC -1 (ED11821 ' [D1181] [Dlldl] ED11823)§=1 NOCxNOC NOCxNIC NICxNIC NICxNOC -1 ([91282] ‘ [D1181] [D1101] [912613)§=L NOCxNIC NOCxNIC NICxNIC NICxNIC -1 E W11613§ {vo}§=fl {Baa}§=£ NICxNIC NICxl NICxl [D 3i~ [D 1* ~ (8 }~ ~ 1161 y=L 11.2 y=L 38 y=L NICxNIC NICxNOC NOCxl [D11613yiL [D1201]§=i({F40} * {Baa §=i) NICxNIC NICxNIC NICxl NICxl [D1131];:L [01292]§=£(iFuB} + {Bu8}~=i) (2,102) NICxNIC NICxNOC NOCxl NOCxl nth tnese .0 a-» A; ‘eV~Vr U. .0 A. F... ~ ,. . .-D:4-a~.e:.€ ”in-2e 65 With these equations and the given boundary data, the initial vector of equations (2.u6) can be evaluated for the case of a displacement loaded rectangular bar containing a central crack. Once the displacement field in the bar has been calculated and the successive approximation procedure has converged, the normal stress distributions along the sets of parallel lines can be obtained from the following equations: {a } x 1x1 {0 } mxl {oz} nxl E(l-v) (1+v)(l-2v) vB (1+v)(l-2v) E(l-v) (1+v)(1-2v) vE (l+v)(l-2v) E(l-v) (1+v)(l-2V) v3 (l+v)(l-2V) {U}along x lines 2x1 ({V} + {w}) along x lines (2.103) 2x1 2x1 {v}along y lines mxl ({u} + {%}> (2.1ou) along y lines mxl mxl {W} along 2 lines nxl ({é} + {6}) along 2 lines (2°105) nxl nxl Note that the above equations involve only derivatives that can . . ‘ " P'H-o... ue.-vq__, E 66 be evaluated in closed form. Hence, we expect that the normal stress boundary conditions will be accurately enforced? The shear stresses at each node can be obtained from applying equa« tions (2.13) through (2.15). These equations, however, involve derivatives that can only be evaluated through the use of finite difference calculus. In general, this presents no impor- tant loss of accuracy since values of the shear stresses are an order of magnitude smaller than the normal stresses (25), This same conclusion was obtained when the shear stresses in our examples were investigated. Numerical results for the problem of Figure 2(a) are listed and discussed in Chapter 4. 2.4.2 Bar With Through-Thickness Double Edge Cracks A problem closely related to the previously described central crack solution is that of a rectangular bar with through- thickness double edge cracks. The configuration and applied loading of this problem are shown in Figure 3a For the geometry shown, similar conclusions can be drawn about symmetry conditions and validity of previous solutions as for the central crack problem. The non-dimensionalized variables of the bar in Figure 3(a) are made identical to those of Figure 2(a) so that comparison between the two solutions will be possible° Most of the equations and their boundary conditions for this problem are identical to those of the central crack configura- tiona The only difference is in the mixed boundary conditions 3. E FlSUr ~ » LDouble edge cracks X (b) Discretized region of rectangular bar with double edge cracks. Figure 3. - Rectangular bar with th rough-thickness double edge cracks under uniforn. tension. ‘ e on )' Q A :‘KU I ’ - ‘- .\ \ “A-“ ‘1. ‘ ""~- gnu _ 68 of the crack plane. Equations for the initial value vector {V(o)}, in the partitioned form of (2.93), are listed below. {F3a} - O NOCxl V ~ ~ {Fug} = - ——- ({u} + {w})-_0 (2.106) l-v y" over crack NICxl NICxl NICxl Note that these given boundary conditions are the same as those in (2.91) except that the partitioning subscripts a and B have been interchanged. Using the condition of a = c we y §=i 0 obtain == -1 : ~ {rug} [ma] {Va}§=L NOCxl NOCxNOC NOCxl := -1 z - [0,] [D21a2]“=i ED21823§=g {Ve}§=i NOCxNOC NOCxNIC NICxNIC NICxl - LfigJ‘i [0;] {Baai§=i NOCxNOC NOCxNOC NOCxl :: -l ‘= - - "' [D3] [DC] ({Fqs} 4’ {B48}y=L) NOCxNOC NOCxNIC NICxl NICxl {a (2 107) 4a §=i NOCxl Sher i" ~ 3 ‘ :CL3~e e V . . the” Ce: ‘. sefi 69 where .. _ -1 - - [Da] - (EDQQGlJ - [D21a23 [D2132] [D2281])y=L NOCxNOC NOCxNOC NOCxNIC NICxNIC NICxNOC :: -1 ~ ~ [Db] = (EDQIGlJ ‘ ED2100] [D2182] [D2lBl])y=L NOCxNOC NOCxNOC NOCxNIC NICxNIC NICxNOC =: _ -l [Dc] ' (“322001 " ED2102] [D2182] [D2282])§=f, NOCxNIC NOCxNIC NOCxNIC NICxNIC NICxNIC _ ‘1 : ~ - {r33} - [021821§=i {v8}y=L NICxl NICxNIC NICxl _l ~~ ‘ ED21823§=L E”218135;:13 {Baa}§=£ ‘ {B38}y=L NICxNIC NICxNOC NOCxl NICxl :l~ ~ ~ ~ ~ - [D2182 JyzL [D2281]y:L ({Fua} + {Baa}y=L) NICXNIC NICXNOC NOCxl NOCXl -1 - [02182];=£ [D2282]§=i ((Fgai + {B48}§=i) (2.108) NICxNIC NICxNIC NICxl NICxl Similar equations can be derived for a diSplacement loaded double edge crack bar by using the given diaplacements rather than their derivatives at the § = i plane. The details of that prob- lem are not considered in this paper. Numerical results for the err}. {a 4U . «a 'H u ‘ H. -' 5-»; - V— . . fl 7O problem of Figure 3(a) are presented and discussed in Chapter IV along with the solutions of the other examples. ‘ B ‘I '4 II (D ’1 9 th: CHAPTER 3 SOLUTION OF THE NAVIER-CAUCHY EQUATIONS IN CYLINDRICAL COORDINATES BY THE METHOD OF LINES 3.1 Governing Equations The discretization technique presented in the previous chapter cannot be readily extended to problems having circular boundaries. For problems of this geometry5it is more convenient to formulate the field equations and associated boundary condi- tions in cylindrical coordinates. In this coordinate system, the Navier-Cauchy equations (2.”) can be written as follows: 8e __9.+1-2 v2-.1— -2.3.1’. =0 3.1 3p ( V)[(c :2 u :736 ( ) 8e 1 c 2 L 2 an _ :3-6—1- (1 2v) [(Vc r2)v+;-2-fi] - 0 (3.2) Be 2 __E+(1-2v)Vw = 0 (3.3) 32 c where the dilatation, ec, is given by Bu 1 av u 8w = — —-- - -- 3.0 ec 3r + r as + r + 82 ( ) and the Laplacian, V2 is c. 72 2 2 2 v2 = 8 +.-l‘_%_+_l__.a___+.a_._ (3.5) C 3P2 r P P2 392 322 The stress-displacement relations, obtained by substituting the strain-displacement relations into Hooke's law, can be written in the following form: or = kec+2G§% (3.6) 06 = Aec+2s (.Eggn‘ii) (3.7) oz = Aec+2c§zi (3.8) Ore .—. G(%%§‘Y;*§%) (3.9) on = c(-§-¥+§-‘z1) (3,10) 082 = G(%+%§i) (3.11) Solution of these equations can again be obtained by using the line method together with the successive approximation procedure and the applicable boundary conditions. 3.2 Ordinary Differential Equations and so. , . ;;nditions in the Radial Direction Following the line method as discussed in the previous chapter, we construct three sets of lines in the direction of the cylindrical coordinate axes. An arbitrary cylindrical grid consisting of these three sets of lines is shown in Figure u. The numbering of the lines is analogous to that shown in Figure i 75 my 6‘ fi ,1 l ' 4 I I IZN I ,+ Axial lines-/ Radial llnes-/ X,f l-NBxNZ m -NZxNR n-NRxNB HCircumterential Figure 4. - Sets of lines in the direction of cylindrical coordinates. C «an r. —.e ‘ _ flu... ‘. pc. “#5335 ~¢g~~ 7n 1. For convenience, the lines are evenly spaced with hr’ he and hz each equal to some given constant. The advantage of a more easily calculated solution with even line spacing is not obtained in fully three-dimensional cylindrical coordinates because some of the resulting differential equations have variable coefficients. In addition, a closed form solution of the eigen- value problem for the remaining equations is also impractical. The previously discussed limitations on the validity of particu« lar solutions also apply to problems in cylindrical coordinates. For the solution of equation (3.1), the radial lines of Figure u must be utilized. The radial displacement of points along these lines will be denoted as ul, u2, . . ., ufi. We define ill, v 2, . . ., 6|, now as the derivatives of the cir- cumferential displacements of the same points on these lines with respect to e and wll, wl2, . . ., w], as the derivatives of the axial displacements of the same points on these lines with respect to z° These displacements and derivatives can then be regarded as functions of the radius only. From equations (3 l). (3.4) and (3.5), the following equation is obtained along the first radial line: ..-e v, “For: 2 2 d . d u1 1 d6 1 ' 1 u1 u1 dw d ul 2 +____ -_._vl+..__d -_+._.+(l-2v) 2 dr r dr 1 r2 r r r2 r dr 2 2 du 3 u 3 u u +1_.l.+l_ l + 1-4-2.44] . o r dr P2 362 322 r2 2 l (3.12) where, by using finite difference calculus, we have 32ul l 2 = Tm, -2ul+ulee) (3.13) 36 ‘6 32ul 1 2 = T (UNe+l - 21.11 + Ulen) (3011+) 3z hz The use of zero shear stress boundary conditions in the radial direction on the r-z and r-e coordinate planes gives respectively _ dv “lee - u2 + 2her 3:11 - 2hevll (3.15) _ dw Substituting these equations into equation (3.12) leads to the following ordinary differential equation: 76 2 d u1 + 1 du1 _ u1 + (1-2v) _ w 2 + 2 u dr2 r dr ,2 2(l—v) r7h§ hg l + 2u2 + 2UN6+1 {l(r) (3 17) r2h§ h; 2(l-V) ' where u -3) . l dv do r l r r l r l 2 2 dw + (l-2v)['%'§%- 2 v+h_-d— (3.18) r 3 r he 2 r 1 Similar differential equations are obtained for the diSplace- ments ui of the points on the other radial lines. Since each equation contains the displacements of the surrounding lines, these equations constitute a system of ordinary differential equations. Noting that equation (3.17) was derived for a corner line, the form of the equations for interior lines and surface lines will differ according to the application of known shear stress conditions. It will be convenient to non-dimensionalize equations (3.17) and (3.18) with respect to some characteristic dimension. For the penny shaped crack problems, which are to be discussed later in this report, the same variables can be used as in (2.28) with the following modifications: (DI mld (ll lv-s (3.19) Introducing matrix notation, the differential equations along the radial lines can be expressed in the form shown below. 2 (gt-'+ -'-'- dr 1 3 2x1 2x2 1x1 7) {a} = [Kr('r)] {a} + mm 2X1 (3.20) where the coefficient matrix [Kr(r)] and the column vectors {3} and {r(r)} [xrwn 2x1 are given below. "F’ [Krl] 2EKP2] 0 0 0 NGXNB NGxNe [Kr2] [Krl] [Kr2] O O NexNG NGXNG NexNe \\ \\ \\ 0 \\ ‘\ \\ O \ \ 0 O [Kr2] [Krl] [:Kr2:I NexNe NBXNB NexNe 0 O O 2EKr2] [Krl] NexNe NexNe _ (3.21) a. 78 where the submatrices [Krl] and [Kr2] are r _ \\ \. ‘\ [Kr1(r)] = 0 \ \ \ 0 \. \. \. NexNe 0 0 'k8 k7 -k8 i" 42 (l-2v) 2 2 k = -——-- + -—- 7 2(l-v) 132326 3% k8 = (IL-2V2 l 2(l-v) g2fi% P k 0 O 0 0T ' 3 0 -}<3 0 O O \\ \~ \\ [K ] - 0 \\ \\ \~ 0 r2 \\ ‘\ \\ NexNe O 0 0 -ka 0 O O O 0 -k3 2% r a A “I 1 79 Note that the coefficient matrix [Kr(r)] is a function of the radius and since the sum of all the elements in any given row is zero, it is also singular. The column vectors are written as r r {all 7 {mail ) {a}2 {#3)}, {a} ={ I > {rm} =$ I > (3.22) 1x1 . 2x1 . {fiiNZ_l {;(3)}NZ_1 m mm} L. u NZ ,4 L NZ where the partitioned column vectors of {a} are the same as those in the previous chapter except that they are of order Nexl. The partitioned vectors {KN}i i = l, 2, . . ., NZ, are given by 80 —--——-—(LN;3) 3| + %§~3— + 3:: + (l-2v)[—~r~12 9: ~:~ v + g:- d .r.‘ l r l r 1 rhe dr be u. - ~ 6 f: 2 do (1’23) ’ + i 9-.— d- + (l-2v)[:fi-— T] r 2 1‘ dr 2 d 2 z r 2 _ 1 ' {HM} = --—-—-- é . 1 2(l-v) . Nexl . (Liv-3) t, + 1 at} + as + (1 W) 2 an] m 1"": “'3.- - '— ~ ;~2 Ne-l 1” d!“ N6 1 d? N601 hz d" NG-l Sixégl-QI + %,E; dw + (l-2v) 212-9; + :V + “ N9 P d!“ N8 dr Ne rhe dr ; E q 6 r . . (LN-3) - 1 dv do 2 do “‘2‘"- + $3: + '5':- + (l-2v) 75—5? - i Ne+l T N6+1 r Ne+l r e r (“v-3) : 1 d6 dé -2 v + “:3: + '8':- r Ne+2 r F Ne+2 F N8+2 who}, = ‘ 1 { . . 2(1"V) . e NGXl e ' (LN-3) l + 133 + g3 ;2 2Ne—1 r dr 2NB-l dr 2Ne-1 u - 6 a - o 2“ (”23)v| .33.- ; ”1-2.0 .39.”;— ‘9: .0. 'S_.‘ (“mud . Nexl {2min Nexl 2(1 1 -v) 1 2(1 -v) ( ( L (av-3) 5| (43-3) 6| r (“v-3) 2 v 1‘ (Uv~3) :. P (“v-3) 5| ---- v 1 d9 * 2-2N8t1 F a? 8]. £-2N6+l do * at I + (l 2 ) ['2 92'- l-2N6+l v :3; a + 1 d + 33 —:.- l-2N6+2 F. r l-2N6+2 d? £-2N6+2 1 d 36 + r'ar a: L-Ne-l ’ ’ l-Ne-l ’ 2-Ne-1 1 d6 d6 - av +31"; 3" +(1-2v) ——-. -.-+ L-NO z-Ne r -N6 rh dr 2 6 + %-%5 + g;‘ + (l-2v) =3— l-N6+l £~N3+l £-N8+1 h 1 36' d6 * (1 2 )[ + "- + '7 - v - z-Ne+2 * d? z-Ne+2 d” n-Ne+2 1 36 l + (1 2 ) [’ 2 la ] -v — "'1" 1-1 1"-3511 a? 5;"? 2_ 1 d db 2 - 2 do +33% +3“; +(l-2v) -2 V‘W‘T- l L r fie rhe dr 2 3] ~2~ ’ he 1-Ne J W 9—3- T’ (“I-Li] dr-7‘ fidr e ’ he 2 L-Ne+l g_ldo h, E? l-N6+2 1 2 d9 §;'3§ 01: W 2 ~~ rhe V ]£-2N6+l IA A, (.1..;.. Q ap~.- ‘c‘ s m-“ ~S “A“; ‘~‘O~ r— i 82 Assuming that {r(r)} is known, solutions of equations (3.20) can be obtained in closed form using the given boundary data at the end points of the radial lines. 3.3 Ordinary Differential Equations and Boundary Conditions in the Circumferential Direction For the solution of equation (3.2), ordinary differential equations are developed along the circumferential lines of Figure 4. The displacements along these lines will be denoted as v1, v2, . . ., Vm- We define ull, fil2, . . ., film as the derivatives of the radial displacements of points on these lines with respect to r, and wll, &|2, . . ., aim as the derivatives of the axial displacements of the same points on these lines with respect to z. u and W are defined, of course, as the radial and axial displacements respectively. These displacen ments and derivatives can then be regarded as functions of 3 only, since they are variables along circumferential lines. Following a similar procedure to that used in the previous section, the set of differential equations obtained along these lines is listed below. d2 ~ "“ {9} = [K J {v} + {3(6)} (3.23) dB? 6 mxl mxm mxl mxl We note that for equations (3.23), the shear stress boundary conditions in the 3 direction were utilized. An inspection of 83 Figure 4 shows that for any given subset of NZ circumferential lines, the radius is a constant. In order that [K6] and {5(3)} be expressed in a form similar to (3.21) and (3.22), we define the radius for each subset as ri i = l, 2, 3, . . ., NR. Then the coefficient matrix [K3] and the vectors {0} and {s(§)} can be written as follows: [Kel(rl)1 [Keu(?1)3 o o o C” NZXNZ NZXNZ [K65(ri)] [K62(ei)] [x66(2i)] o o NZXNZ NZXNZ NZXNZ \ \ \ [K6] = 0 ~\H\\ \\,\\ ‘~ ‘\ 0 mm o o [K95(ri)] [K92(ri)] [K96(ri)] NZXNZ NZXNZ NZXNZ 0 0 o [Keu}NR_l {£(5)} L NR J where the partitioned column vectors are ”(5”1 Nle {£(e)}2 = 2(l-V) 2(l-v) 88 F - z z 2? - ~ 2~ - (3-uv)934 '9’ $11.11!- + $1912., + (l-2V) #%E-'g-g+=*i 2': d6 1 d9 1 de 1 h, e hz de 1 L. . _ (‘2. do 2 do - do 1 do do (3-l-N) (TB—[2 + rl gal? + r1 3.8—2 + (l-2v) EXT-“é- - “'6- 2 L (3 u ) an} + g at - 3: + (1 2v) r‘1 do dfi .- V "" - "'" " -.:-——-"- d5 NZ-l 1 d5 NZ-l d5 NZ-l hr 59 d9 NZ-l ( 33! + i, ail + ; dél + (1 2V) 2~1 dfi do 2’1 do — ‘T - f— - —-—— F‘— 3 av) 35 NZ 1 3“5.142 1 d9 NZ hr 3? d5 hz d5 NZ L i - z : 2} (3-lw) 93 + 22 ~31 + r 1‘3- + (l-2v) 7-2-99- d6 d3 2 a h d9 NZ+1 NZ+1 NZ+1 z NZ+1 do - 33 33 (3-uv) + r + $2 35-1192 2 35mm ETA—NZ+1, do 33 - dd (3-uv) ——- + ?2 -—+ r -—- d9 2uz-1 d9 2NZ-l 2 d9 2uz-1 - : : -2I~ (3-uu) S%- + r2 95- 22 ggi + (l-2v) - 2 2% K 2uz d 2N2 2N2 hz d 2NZ - - 1 (f(o))NR_1 I 573:3? Nle - 1 (6 -——-—- {i )}"R ' 2(1-v) Nle J r (3-4v) (3-uv) (S-Uv) (3-uv) (3-uv) (3-uv) (S-Nv) (3-u ) £39 D ‘ 2; 33} + i 39’ + P 93} + (l-2v) _.§§Ll 45 ”R'1 d5 "R'1 as E m-2Nz+1 m-2Nz+l m-2NZ+1 z cal . do , ab! 33{ ' rNR-J. 3? NR-l 3§[ m-2NZ+2 m-2NZ+2 m-2NZv2 an d6 d6 :5! * ad * m-NZ- m-NZ- m-NZ-l an ab dé '2’NR 1 dD + + 1-2 ) ' 35 7‘1411.1 35 ’ ’Nn-l it! ( V t SE m-NZ m- m- m-NZ -29 do ~ db dbl NR do an .~_ + r ‘__ o I __ ¢ - .__ - as NR d6 ua dB ‘1 V) T, as at ’ m-NZfil m-NZ+1 n-NZ+1 an ad 46 ’2’NR do do _.. ... o - - dbl * NR dbl ’NR 33'} * ‘1 2") “R", 3‘5 3'6 m-Nz+2 m-NZ+2 m-NZ+2 d d6 d6 '2iun d d Q ~ u u .. + i —— r ——- + (1-2 ) - 49' NR d5 NR an " "h" 35 a: m— m-l m-l P m 1 I U -2; 2* do an . da NR do an NR d9 -- +5 - + -- +(l-2v) -—-—-—-—--‘——— d5! NR d6 “NR d! S d5 db h’ as In M m 1 EB d6 m-2NZ+1 2’NR du “§;' 38 m-NZol m-NZ+2 VAN Afr-A, avvv'h .npl‘ u..v& u ‘ .ne ..* Ce ‘ Q ‘\‘ 90 Assuming that {5(5)} is known, solutions of equations (3.23) can be obtained in closed form when boundary data at the end points of the circumferential lines are specified. Note that even though the elements of [K6] are constants, a closed form solution of the associated eigenvalue problem in accordance with the decomposition methods of Appendix A is not possible. In addition, we find that [K9] is also singular ale though this is not as evident from its elements as in the case of the other coefficient matrices. 3.u Ordinary Differential Equations and Boundary Conditions in the Axial Direction Application of the line method to equation (3.3) will result in a set of ordinary differential equations along the axial lines of Figure 4. The displacements along these lines will be denoted as wl, w2, . . ., wn. We define ull, ul2, . . a, film as the derivatives of the radial displacements of points on these lines with respect to r and G[l, v12, . . ., elm as the derivatives of the circumferential displacements of the same points on these lines with respect to 6. These displacements and derivatives are then functions of 2 only, since they are variables along axial lines. Using z-directional shear stress boundary conditions for the corner and surface lines, the simultaneous differential equam tions along the axial lines can be written as follows: h‘ .. cl u¢¢ ‘1 91 r‘H 22 H4 II [ch] {w} + {tc(2)} (3.26) nxl nxn nxl nxl From Figure 4, one can note that the radius varies with the line position within each subset of NR axial lines. Using the nota- tion of the previous section, these radii are denoted as ri i = l, 2, 3, . . ., NR. The coefficient matrix [ch] is then given by -TK J 2EK J o o o 7, zcl zc2 NRXNR NRxNR [KZCQJ [ch1] [ch2] 0 0 NRxNR NRxNR NRxNR \\ \\\\ \t \. [K J = o \\ ‘x o (3.27) zc \\ ‘\ \t ' \\- ‘\ nxn 0 O [ch2] I:chl [ch2:I NRxNR NRxNR NRxNR 0 0 0 2D} + [n:(A)] £321 {r(n)}dn (3.33) 21x1 22x22 21x1 22x21 0 21x22 22x1 where {U(6)} is a vector which consists of the boundary values of (Hz) and [i-g-gfififl at i": O. The matrizant of [A] is an infinite matrix integral series given by lOO - r a [3%)] = [I] +f [Molndol +f [A(92)]d02 O O O °2 ‘33 f [Malfldpl + [A(p3)]d93 [A(p2)]d92 O O O 2 'kp [A(pl)]dol + . . . (3.3“) 0 Substituting equation (3.31) into equation (3.3”) yields ,1 [I] o [0] pllZI] [Q (A)] = + dpl ° 0 [I] o Lug] £0] 91 " [0] p [I] o [0] p [I] +f 1 2 C192 f 2 1 1 C191 0 ----[1133 + -1— [K (p )Jdo pl r l l 93 r 3 3 O O 3 92 l O O l 3 92 [922.] = [I] + f é; [Kr(02)]dp2 f DlEIJdOl + O 0 2x2. Differentiating these integral series with reSpect to r gives d a? [all] = iEIlfrg-I [K (01)]dpl + . - 102 i d ~ 3 l . I I __ K d dr 12 PE 3+ PE 1 I p2E1.(132)]£>2 f almdo1+~ EKJ+£EKJfEIdeo r f r 2 2 O «I r—I K) 1.4 II In. [-1 K) M H l._l II fi‘hd ‘UIA [.3 [Kr(pl)]d°l + . . f [KP] J‘plEIJdpl + . o . 0 dm] d5 22 ulna Inspection of these equations shows that the following relation- ships exist among these four submatrices: 1 d (3.35) 3.3-51321] = [521111133 1 d :35 [912] - [922] (3.36) d 2351:3221 = EQlQJEKr] 102 ii d ~ e l I + [I __. K ) d dr 12 rE] r JIDQEMOQJW ' f plEIJdpl + o £31321] = .1 [Kr] + .331er f [1132332 0 . l f 53L.[Krmlndpl + . . . l“ d _ l O .I r—1 :3 l—J II '1‘] Inspection of these equations shows that the following relation- ships exist among these four submatrices: 1 d g3; [“11] “ [921] (3.35) .. d 1 d W :E‘fi—[Qm] - [922] ) (3.36) d 351322] - EQlQJEKr] J y- b e L ' . :0. ”e :8. once that t: 4“ us: (:41: d--. Eran»: ._ ‘iAg‘l — d. .- sEPJJat iVe 103 From the definition of these matrix series we can conclude at once that their initial values are [3111,14, = [I] [912Jr=o = [0] (3.37) [922]%=o = [I] Since the values of the submatrices [Qij] are difficult to obtain from their series definitions, the above simultaneous matrix equations may be evaluated by a suitable numerical tech- nique such as the Runge-Kutta method. The necessary initial con- ditions for this numerical solution are listed in (3.37). In using a numerical method for the solution of a given differential equation, it is usually necessary to solve for the derivative of the dependent variable at the initial point. If the region of interest includes the point a = 0, equation (3.35) shows that §§.[921] at 3 = O is not finite. This problem can be avoided by defining a new variable [9:1] as [9* J = i3[n J (3 38) 21 21 ° Using equation (3.38) in (3.35), the following simultaneous matrix differential equations will be obtained: T.‘ LX r , . \ov-V x vu-v- «.9. ~o- If. a» k»... m pie “A“ t e Xe?- 10” d 1 :'.° -=-[9 J = ---[Q J; E9 J _ = [I] dr ll £2 21 ll r-0 (3. 39) d :‘c 3 9: . 7': __ By applying L'Hospital's rule, all the necessary derivatives in equations (3.36) and (3.39) can now be evaluated. Note that the matrix [K;] is obtained from multiplying the coefficient matrix [Kr] by 32 such that no element of [R}] contains an r in its denominator. If one were to use the series definitions for evaluating the matrix functions [Qij], similar difficulties would be encountered since at 2 = 0 some of the integrals would diverge. ‘This divergence is the result of trying to evaluate improper integrals of the second kind. At this time, the analogy between solutions (3.33) and (2.60) may be noted. Reference (34) shows in detail that the matrizant of a matrix of constants is identically equal to the exponential matrix series (2.61). However, indirect metnous such as additive formulas and accuracy checks such as (2.87) for evaluating and checking these matrix functions are not available when variable coefficient differential equations are treated. It is evident from the above discussion that for specific examples a closed form solution of equations (3.20) is not pos- sible. However, the advantage of the line method over complete . V ’C q +1- \A3 9 w" .2 E}... + as '..¢~ p‘-o. V: .a A; U‘ 44 w . :u 105 finite difference solutions is still obtained in that higher order numerical solutions are utilized for the computations. Additional comments on the selection of the matrizant method over a direct numerical solution of equations (3.20) can now be made. The advantage of using equation (3.33) is in its ability to express two point differential equation solutions directly in terms of given boundary data. Direct numerical techniques, such as the single-step Runge-Kutte method or the multi-step predictor- corrector methods (33) usually require that all initial point data be known. Since in a two point boundary value problem some of the initial data is unknown, indirect methods such as "shooting" and successive approximations must be employed (33). This, of course, leads to an inefficient use of the computer and should be avoided whenever possible. In addition, it is well known that direct numerical solutions for a system of two point differential equations are not available. 3.5.1 Evaluation of the Particular Integral for the Radial Differential Equations In a similar manner to equation (2.80) we represent the particular integral in partitioned form as \ {31(3)} 13 2x1 -l = [3"] {F(n)}dn (3.33) {32(3)} ° 0 22x22. 22x1 2x1 J IL' ‘ lens i Hewever, {911) E tions ‘2‘?“ C s 33 106 ~ W ~ r- 1-l W {Bl(r)} r [011(Kr,n)] [012(Kr,n)] {O} : ?dn {B2(f))J O L_[Q21(Kr’n)] [022(Kr,n)1‘ {r(n))) It was found that equations (2.83) also apply to the matrix func- tions [Qij] and thus the above integrals may be written as r {31(2)} [912(Kr,n)] {r(n)}dn (3.ul) 03: {32(3)} [911<11 {r(n)}dn (3.1.2) 0 However, the simple relationships of (2.63) between matrices [all] and [922] and of (2.65) between matrices [021] and [912] in an analogous manner to [Aij] are not valid. Note that the particular integrals in the circumferential and axial direc~ tions can also be expressed in a similar manner to equations (2.80) and (2.85). Since {r(n)} in the above integrals is unknown, we start the solution of the problem by assuming zero values for the required quantities. Using the partitioned form of the matrices, the solution of equations (3.30) can be written as 107 ~ 1 ~ ~ {u(r)} = §'[911(Kr:r)]({Fi} + {Bl(r)}) £xl 2x2 2x1 2x1 1 + g-[nl21({r2} + {32(3)}) (3.33) 1x2 2x1 2x1 where F = ~~}~_~ { l} {ru r‘rinitial l . ~ {F2} = {791-333)} = {33.3.}... r dr E. . . r r=rinitial initial . ~ 1 ~ {3(3)} = ([321] - ;§-[nll}> 2x1 + ([022(Kr,f)] - %§-[nlzcxr,r)1)({r2} + {32(3)}) 1" (3.uu) Equations (3.”3) and (3.4M) give us the first estimate for the vectors {3(3)}(1) and {6(2)}(1). It is assumed of course that the boundary vectors {Fl} and {F2} ‘are known. Using the calculated values of {0}(l) and {0}(l) we can evaluate the vector {8(0)}(1). An analogous equation to (2.86) will then give us the first value of {9(5)}(1) and {3(8)}(1). Using the first solutions along theradia1.and circumferential directions in the vector {tc(2)}, the axial solution is obtained from a similar equation to (2.86). If with the repetition of this 108 procedure convergence of the calculated variables occurs, an approximate solution of a given problem can be determined. Regarding the convergence of this process, the comments and error checks of the previous chapter apply only to the axial and circumferential equations. Since equation (2.87) is not applicable to the matrix functions in the radial direction, their accuracy can only be checked by varying the Runge-Kutta integration increment. Since the homogeneous solutions of equa- tions (3.30) are independent of the other two sets of differential equations, this integration step can be arbitrarily small. The values of the coupling terms in {2(2)}. {8(0)} and {tc(2)} can only be determined by the use of finite difference calculus, since they involve displacements and derivatives that are defined only at the nodes. Similar approximations of derivatives near boundaries must be made as in the case of rectangular coordinate problems. 3.6 Application to Specific Geometries - Annular Plate With Internal Surface Cracks A problem of some practical importance is that of an annular plate containing part-through cracks on the inside surface and which is loaded by a uniform radial stress of. °o on the outside surface. In order to minimize the numerical computations, we have assumed four internal cracks located symmetrically at ninety degrees to each other. The closed form solution of this problem 109 is extremely difficult to evaluate because the stress and dis- placements fields of a circular hole interact with the singular stress fields of the cracks. Figure 5 shows the geometry and loading of the problem under investigation. Because of the symmetric geometry and loading, only one-sixteenth of the origi- nal plate has to be discretized. Figure 6 shows this region of interest and the assumed crack geometry. The displacement fields in the plate are described by the solutions of the three sets of simultaneous ordinary differential equations. Inspection of Figure 6 shows that non-dimensionalization with reSpect to the outside radius is more convenient in this case since the crack has two characteristic dimensions. Values of the non-dimension- alized variables a, b, E, d, t and 00 used for this problem are also shown in Figure 6. At this time we return to solutions (3.u3) and (3.44) and note that the initial value vectors {F1} and {F2} are unknown. Using equation (3.6), the given normal stresses on the inside and outside surfaces of the plate can be written in discretized form as follows: {{1} = - A ,1. {6} + 34.1.33}- .. (3.1.3) o A+2G f0 Po r=ro 110 Q 0 \\\\\\\\\ \ \-Four equally spaced internal surface cracks \\\\°°\\\\\ 2 Figure 5. - Annular plate with internal surface cracks under uniform external tension. 111 z v-HB Figure 6. - Part of annular plate with internal surface cracks. 111 Nf v-U3 Flgu re 6. - Part of annular plate with internal surface cracks. 112 L - l A l ' 1. ~ it) Using equation (3.u5) in the definition of {F2} and combining this with the definition of {F1}, the following equation will be obtained: V ' 26 {F2} = -J—-— -.-.—-.. ,_ - A {mic-1. +—-——-——— {F1} (3.47) A+2G rO r=r 1+2G o (1+26)~2 0 r0 In the partitioned matrix form, solution (3.33) at r = b can be written as {Mg} LE921(KP.B)] [922(Kr,b)1j £ng + {32(3)} (3...L+8) From equation (3.u8) two matrix equations can be constructed which when combined with equation (3.u6) lead to a similar equation to (3.47) relating the vectors {Pi} and {F2}. The result of this manipulation is 113 - A -1 §_ 3 - 00 {F1} ' A+2G ma:1 ({B}+{w}) {:25 - [0a] {>323} -l -1 - {31(3)} - [pa] [abjir2i- [933 [9b]{32(5)} (3.u9) where [e ] = ——39-—=—-[e (K ,B)J - [921(K ,B)] a ()+2c)b2 11 P r [ab] = ——39———5 [012(Kr,b)] - [022(K ,B)] (A+QG)B r Simultaneous solution of equations (3.07) and (3.09) gives us the following results: A -1 _1 {7 2. F } [0 [0 J - + { 1 M28 C] a ({5} 6}) 11:13 A -1 -1 i_ z + i+2c [9c] ma] [913] ({go} + {"D f~=i4 O O _ [QCJ‘ltnaJ‘l {17233} - [ncl'l{Bl(B)} - [cci‘lteaitpb](32(3)} (3.50) where 2G -1 [QC] = [I] + ~ 2 [0a] [9b] (A+2G)ro 11% 2(3). -1 -1 A {7 2. {F} = [0] [0 1 [£2 1 - [I] 7-}1- w} 2 ((109230? c a b “2G )({“o { >55 0 2G -1 - [0 J lEQa 3 :} ().-i'2(3)ro 2 C {1:2G 261-1 3 ~;7-EQC 1 lEQa 1 {:g :}+ {:é:£) (A+2G)2r ( 11:13 2G -1 ~ -1 -1 ~ (we); 2 ([30] (31(3)) + rec] ma] [3b1(32(3)}) O + (3.51) Equations (3.50) and (3.51) define the necessary initial vectors for the case of zero inside surface radial stress and for a given radial stress 0 on the outside. 0 For the problem shown in Figure 6, the boundary conditions in the z direction are analogous to those developed in Chapter 2 for the x—directional boundary conditions. As a result of using equation (3.8), the analogous equation to (2.90), of course, will contain additional terms because of the cylindrical coordinates. Along the circumferential direction, the following boundary conditions are enforced in the crack plane through the use of vectors {FBC} and {Fae} which are defined analogously to those in equation (2.93): {3(3)} = {O} (3.52) outside crack (I: U) m (Ll (n 115 n. _ ~ A ~:. ~:. {V(O)}OVeI‘ - " {U10ver " m ({I‘U} ‘l' {PW})over (3.53) crack crack crack and from the symmetry of the problem {3(90)} = {0} (3.5a) A similar partitioning technique to that described in Chapter 2 must be followed to correctly evaluate the elements of {F3c} and {Fuc}' Care must be exercised in performing this operation since for surface cracks the elements of the associated matrices must be reordered to arrive at the desired form of (2.93). Investigation of the shear stress boundary conditions involving planes of zero or uniform normal diSplacements will lead to similar conclusions about the coupling vectors {r(r)}, {3(5)} and {tc(§)} as in rectangular coordinate problems. As an example, we consider the symmetry condition (3.5”). The zero shear stress conditions in that plane are H 0 OP 0 6 (3.55) I O 082! 8=6 0 Using equations (3.9) and (3.11) we obtain 135 (7 a?) '2"? = '4' “—2- r 38 ~ ~ r ~ 6 3r ~ ~ 8:80 8: o 8:80 {3.56) is»? - -251 1:35.” 32.... 6:80 6:80 . 317 a?) . ~ ~ Since a, -—- and -—= are all zero in the plane 8 = 6 , we 3} 32 0 find from equation (3.25) that {s(8)}5=5 = O. 0 Once the successive approximation procedure has converged and the displacement fields in the plate have been determined, the normal stress distributions along the sets of parallel lines can be calculated from the following equations: {or} = (ment) + A {2} EL ((3) + (in) . f (3.57) r 2 along radial lines {06} = (M26) {3,- ‘17} +{3}> + 1({0} + (53)) (3.58) r r along circum. lines _ ,5 -21 1 {h 3 {Oz} - ()(+2G){w} + 1612. v ‘L%‘}+{F})along axial (3.59) Noting the terms in the above equations, we expect to satisfy normal stress boundary conditions again with good accuracy. The shear stresses at each node can be obtained from using equations (3.9) through (3.11). These equations, however, can be evaluated only through the use of finite differences. Computed stress and displacement results for the problem of Figure 6 are tabulated 117 in the following chapter. 3.7 Axisymmetric Problems The field equations (3.1) through (3.3) and the stress- displacements relations (3.6) through (3.11) are greatly simpli- fied for problems that have circular symmetry. It is known that for problems of this geometry, the circumferential displacement is inherently zero at every point and all the remaining variables are independent of the circumferential variable 8. Equations (3.1) through (3.11) are then reduced to the following form: where the dilatation, e Bes 2 l ...._ + (1.2\)) (V - _) u = 0 (3.60) 3r [: 8 r2:] 36 2 S + (l-2V) V w = O (3.61) 32 S s9 and the Laplacian, V2, are given by an u éw_ e = -—- — (3062) 3 3r r 32 2 2 \72 = L.» 1L+§__ (3.63) 3 3r2 r 8r 322 The stresses in terms of displacement variables are - .23 u 3w are - (A+2G) 8r + A(;.+ 3;) (3.6M) o = (kl-2(3) 3+).3‘i+3-‘i) (3.65) as r 3r 32 - 23, 3u u 628 - (Ni-26) az + AGE-tn?) (3.66) 118 0 = G 33-4439- (3.67) rzs 3r Dz The other two shear stresses are zero at every point in the body. Solution of these equations can again be obtained by the method of lines together with the successive approximation procedure and the applicable boundary conditions. 3.7.1 Ordinary Differential Equations and Boundary Conditions in the Radial Direction Since at this time the solution of only two partial differential equations is desired, two sets of parallel lines are constructed along the axes of the independent variables. Figure 7 shows an arbitrary axisymmetric cylindrical body with the necessary discretization. For convenience the lines are evenly spaced with hr and hz each equal to some given constant. The radius of the solid is assumed to be uniformso that the length of all radial and axial lines is the same. The uniform spacing of lines again provides an advantage for more easily evaluating the resulting differential equations. This --l_cws from the axisymmetric condition since for these problems both sets of ordinary differential equations have constant coefficients. However, the particular solutions for these problems are limited in the same manner as those for the previous cases. For the solution of equation (3.60), the ordinary differen- tial equations are developed along the radial lines in Figure 7. 119 / \ // \ Figure 7. - Sets of parallel llnes for axisymmetric problems. 120 The radial displacements along these lines will be denoted as ul, u2, . . ., uNR” We define wll, w 2, . . ., QINR as the derivatives of the axial diSplacements of the same points on these lines with reSpect to 2. Using equation (3.62) and (3.63) in equation (3.60), the following equation is obtained along the first radial line of Figure 7: d2 du 32u ° “1 + 1.1.. .31 + (1‘2“ 1 + l 933. = o (3.68) 2 r dr 2 2(1-v) 8 2 2(l-v) dr dr r z 1 where 2 a U1 - U2 " 2111 + L113 2 .. 2 (3.69) 32 hz The zero shear stress condition on the plane 2 = 0, gives dw : 2 ——- .7 uls hz dr 1 + u2 (3 O) Combining equations (3.68), (3.69) and (3.70) yields 2 dul+ldul Bi +2(l~2v) u _2(l-»2v) u 13:1”) - o dr2 r a?" 1.2 _2(1-v-)h2 2 2 (.2 1 2711177 2 -. (3.71) where d” 2 d {l(r) = J’- + (l-2v) -—)-‘i (3.72) dr 1 hz d? 1 Similar differential equations are obtained for the other radial lines. Noting that equation (3.71) was developed for a 121 surface line, the form of the interior line equations will differ according to the application of known shear stress conditions. The above equation can again be non-dimensionalized with respect to a crack dimension "a" in accordance with equations (2.28) and (3.19). Using matrix notation, the system of differential equations along the radial lines can be written as 2 d 1 d 1 ~ (3+ :5? .. 33.) {u} = [Krs] {£1} + {rs(r)} (3.73) NRxl NRxNR NRxl NRxl where the coefficient matrix [Krs] and the column vectors {u} and {rS(r)} are given below. F - 2k3 -2k3 o o o [Krs] = o “ \. \‘-\. ‘\ \‘ o (3.7a) \ \ \ NRxNR o 0 -k3 2k3 -k3 o o o -2k3 2k3 ‘- .J k = (l-2v) .1_ 3 2(l-v) 52 Z r, W u‘1 u2 m4 . NRxl ~ uNR-l E K NRJ fi~(§)} = S NRxl 2(1-v) 9;, L.d'r NR 1 ( 2 )2 (ml 1' v '3'“ h d? 1 d6 3%. > (3.75) 1’. £1 d? NR-l -(1-2v)§_.‘1§ hz dI‘NR J In contrast to the general radial coefficient matrix (3.21), the matrix (3.7”) has constants for its elements. also singular. Note that it is Although the above matrices can also be obtained by reducing the general cylindrical coordinate problem of Section 3.2, this reduction is not so evident. Assuming that {rs(r)} is known, closed form solutions of equations (3.73) can be obtained. 3.7.2 Ordinary Differential Equations and Boundary Conditions in the Axial Direction Solution of equation (3.61) is obtained by constructing a set of ordinary differential equations along the axial lines of Figure 7. as Wl. W2, 0 o a, wNze We define (1'1, 11,2. 0 The axial displacements along these lines are denoted o. {JINZ as the derivatives of the radial displacements of the same points on these lines with respect to r. Using equation (3.62) and 123 and (3.63) in equation (3.61) the following equation is developed along the first axial line of Figure 7: . 2 2 2 du 1 du d w1 3 w1 1 aw1 d ”1 dz 1 r1 2 dz2 8r2 1‘1 3r dz2 (3.76) Using equation (3.67) and the known zero shear stress condition along the axis of the cylinder, we have L333. = -Li! (3.77) Pl 3P Pl dz 1 Symmetry of the problem also leads to 32 "1 = .1._.(w2 - 2wl + w2) (3.78) 3r2 h2 1‘ Since r1 = 0 along the axis of the cylinder, the terms contain- ing r1 in the denominator must be investigated. For a uniformly loaded axisymmetric problem, the radial displacements along the cylinder axis are inherently zero. This implies that the term 1.9.2 d is indeterminate and L'Hospital's rule can be used to rlz 1 find its limit. Thus, we have 2.. 32+ . lim l.du = 3r dz 1 = ‘ég rail "'3— r + r1 dz (3.79) 1 Substituting equations (3.77), (3.78) and (3.79) into equation (3.76) gives 124 2 d w _ I (z) 21 _ 2(1—2v)2 “1 + 2(1 2v)2 w2 + l = 0 (3.80) dz 2(l-v)hr 2(1-v)hr 2(l-V) where f ( ) - (1+2 )-—-d6 (3 81) 1 z - v dz 1 ' Similar differential equations are obtained for the other axial lines. Since equation (3.80) was developed for the special case of a line along the cylinder axis, the form of the equations along the interior and surface lines will differ according to the position of the line. It will be convenient to non-dimension- alize these equations according to the method of the previous section. Using matrix notation, the system of differential equations along the axial lines of Figure 7 can be written as 2 53:— {ca} = [K28] {a} + use» (3.82) dz2 Nle NZxNZ Nle Nle From Figure 7 it can be nored that for any given axial line the radius is a constant. We define the radius at each axial line as ii, i = l, 2, . . .NZ, where E1 = O and ENZ = b. For a uniform radial increment hr, the radius at any point is (i-l)hr. Using this notation, the coefficient matrix [Kzs] can be written as follows: 12S , fi 2 -2 0 0 O 0 l 3 2 2' o [K J - :12 \ (3 83) ZS - 2 21"3 21-1 . ° ° " at? \ ‘ 27—2- 0 NZxNZ 1‘ 1‘ \. 0 O 0 \\ 2 \\\\ 0 0 0 0 -2 2 L. .4 k 8 (1-2V) g- l 2(l-v) h2 r i = 2. 3’ ‘0 o o, NZ-l . k . . 21-1 18 The upper diagonal elements of [Kzs] are given by - 21:2'-2—' -_ k while its lower diagonal elements are given by -(_._—§: 3)....158 for i = 2, 3, . . ., NZ-l. Note that the elements of [Kzs] are all constants and since the sum of the elements in any given row is zero, the coefficient matrix is also singular. The column vectors in equation (3.82) are listed below. ~ T ~ ~ ~ ~ {w} = l wl W2 W3 0 o o wNZ Nle f : du (1+2v) a; 99 . 9.99 d9 2 hr d2 2 99 __1 9 dz 3 2hr dz 3 1 d‘ 1 d” {t (2)} = ' -§- . -Ji S 2(l-v) dz 1 (1-l)fir dz 1 Nle . . 99 . l 99 di NZ_1 (NZ-2mr d2 NZ—l d6 1 dfi -2 an 1 dfi dz NZ 1.5-dz 11::d2 5-; L b i = 2, 3, . Assuming that {ts(2)} is known, the solution of equations (3.82) can be obtained by the previously discussed matrix methods. 0 ’ NZ‘l 3.7.3 Solutions of the Ordinary Differential Equations for Axisymmetric Problems Although the methods discussed previously can be employed to solve equations (3.73) and (3.82), there are certain simpli- fications that are possible. First of all we note that the 127 coefficient matrix [Krs] is a tri—diagonal matrix having the desired form of (A.l). Thus, a closed form solution of the eigenvalues and eigenvectors of [Krs] is possible and from equations (A.16) and (A.18) these eigenvalues and eigenvectors are respectively ._ i—1 . Xi - 2k3 [l - COS (3.95) o {fi(o)} {F 483 NICxl 45 {9,88} NOCxl J k where we adopted the partioning scheme of (2.93) in order that the mixed boundary conditions in the crack plane can be con- veniently handled. We follow an analogous development as in Chapter 2 from equations (2.93) through (2.97). The results of this development completely define the initial vector (3.95) and they are listed below. {F383} = {O}E=o NOCxl out81de crack (3.96) {FHSo} = ”A (6} + .3 A+2G r =o NICxl over crack Equations (3.96) are specified boundary conditions in the crack plane. We also have 136 _ l -1 -1 a fig {Fuse} - H26 [Dal {coal->103] ((1181 + {5})241 NOCxl NOCxNOC NOCxl NOCxNOC NOCxl NOCxl [n 3‘1 [D J [n 1‘1 { } A+2G a 2181 21a1 °oa NOCxNOC NOCxNIC NICxNIC NICxl . 3 " “DarJL [92161:I [D2lGlJ-l ({fio}*{§£})~ .. z=L NOCxNOC NOCxNIC NICxNIC NICxl NICxl - [Dal‘l [Db] (8358(i)}-{B.SB(L)} NOCxNOC NOCxNOC NOCxl NOCxl + rig-[v.34 [v.1 ((51.1%) + G r 2:0 NOCxNOC NOCxNIC NICxl NICxl - [Dal'l [Dc]. {8 (£)} (3.97) NOCxNOC NOCxNIC NOCxl USO Matrices [D3], [Db], [Dc] and the partitioned submatrices of [D13] are the same as those in equations (2.96) and (2.97) except that in this case they are functions of (K28, i=6). Note that the particular integrals, the applied stress vector, the radial displacements and its derivatives are also partitioned according to their location with respect to the crack. The crack opening {Fssa NICxl 137 displacement is given by l [D 1‘1 i } AED “1 {= is A+2G 21.1 coo ‘ mod1 “9} + ; 2:6 NICxNIC NICxl NICxNIC NICxl NICxl ~ ‘1 .. {3383(L)} - [D2lal] [D2162] {B388(L)} NICxl NICxNIC NICxNOC NOCxl -1 - [021,13 [622,11 {Busa(L)} NOCxNIC NICxNIC NICxl A -1 3 ~ A+2G [D2181] [1322“] (Na) +{%})§=o NICxNIC NICxNIC NICxl NICxl -1 2161] [D22Q2] {F488} NICxNIC NICxNOC NOCxl [D -1 ~ [DQlal] [922a,] {3486(L)} (3.98) NICxNIC NICXNOC NOCxl where {F483} is given by equation (3.97). Although the matrix [D21(Kzs ,i)] is singular, the partitioned matrices are not. In calculating the above equations, the indeterminate terms at E = 0 must be carefully considered. Similar conclusions regarding the elements of the coupling 138 vectors {rS(r)} and {ts(2)} can also be noted from the zero shear stress conditions along lines of uniform normal disc placements as in equation (2.100). Equation (3.67), when applied to the coupling vectors, gives {rs(r)}s = {0} r=o NRxl (3.99) {tS(z)};go = {0} NOCxl out81de crack Once the diSplacement field in the bar has been calculated and the successive approximation procedure has converged, the normal stress distributions along the radial lines can be ob» tained from {ops} = (A+26){t°1} + 1({3} + {{3} )for 2 a! o 2‘ (3.100) {ops};=o = 2(1+e)(fi};=o + A{%}%=o for E = o {06 } = (Ki-2G) {3} + 1({51} + {$}T) for i a! o S r (3.101) {698};=0 = 2(1+6){i1}i=0 + {w}§=o for 9 = o 139 ().-l-2G){w}T + A {u} +{g- })for (A+2G){w}}=0 + 2A{u}}=o for r = O *3! ‘Nu O {0.3} (3.102) } {023 For convenience, all the normal stresses are expressed along radial lines in the above euqations. Note that at r = 0, the radial and circumferential stresses are equal. The shear stress at each node can be calculated from {ans} = 2_(_—1-v) ({g— :+} {53;- (3.103) along radial lines where for the required derivatives, finite difference calculus must be used. In using the given stress equations in cylindrical coordi- nates, the relationships between Lame's constants, Young's modulus and Poisson's ratio are needed, since the non-dimension- alized diSplacements and their derivatives are expressed in terms O’ of some constant times E2» The needed identities are (1+26) = 5(1'V) (3.104) (1+v)(l-2v) A = “5 (3.105) (l+v)(l-2v) (1+6) = E (3.106) 2(l+v)(l-2\)) 140 Numerical values of the stress and displacement distribu- tions for the problem of Figure 8 were calculated for a number of different geometry cylinders. These results are given and discussed in the following chapter. 3.8.2 Hollow Cylindrical Bar Withemfenny'Shaped Crack A problem very similar to that described in the previous section is that shown in Figure 9. As shown in this figure, an internal hole is assumed through the center of the penny shaped crack, which introduces stress free surfaces on both sides of the crack edge. For convenience in the manipulations, the radius of this hole is taken as hr’ that is, the radial incre- ment. The differential equations in the radial direction are identical to equations (3.73) and only their boundary conditions need be modified. Since the radius at any point in this problem can be expressed as ri = ioh i = l, 2, 3, . . ., NZ, the r form of the coefficient matrix (3.83) will be different than for the solid cylinder. The coupling vector {tS(E)} must also be modified accordingly. The results of these modifications are shown below. r d3 + 1 d0 ( (2 1 ) <1er _=- f- — + 1-2 1.- - T:— T dz 1 r1 d2 V) hr I‘l dz 1 ~ - 1 d3 1 dfl % t z = + ——-+ __. 3.107) { sh( )} M1404 3% 1 $1 02 i Nle . i ' o it; +19-B-(l—2v) 3—1-1 5-1-3- kdz NZ 5 dz hr E d2. NZJ 141 L y Penny shaped crack radius - a ll v-1I3 1}. Figure 9. - Hollow cylindrical bar with a penny shaped crack. 192 We also have _, 2 -2 o 0 0 0 1 3 5 __. 2 .__ 1+ u 0 0 0 \~ \\ \\ 0 ‘\ ‘\ \( O O k \ \ \ [K J = 18 - \ . (3.108) zsh -—- 21-1 2 +1 2 0 0 ‘( , ) \ «(1. ) 0 21 \ 21 NZxNZ \ \ 0 0 o \ 2 \ \ o 0 o o 12 2 where i = 2, 3, . . ., NZ-l for both (3.107) and (3.108). Solutions of the radial differential equations for this problem can be obtained from similar equations to (3.43) and (3.uu). The initial condition vectors {Flsh} and {F2sh} are obtained from the zero radial stress conditions on the inside and outside surfaces. The results can be derived by setting {3} and {00} equal to zero and setting r0 = Er in equations (3.50) and (3.51). Solutions of the axial differential equations are expressed by a similar equation to (2.94). The initial condition vectors needed in these solutions are identical to those listed in equations (3.96), (3.97) and (3.98). The normal stress distri- butions are found by using equations (3.100) through (3.102) where the equations for E = 0 are not applicable. 143 Equations similar to those described by (3.97) and (3.98) can also be developed for a diSplacement loaded cylinder. An applied external radial surface load could also be easily in- corporated into the modified form of equations (3.50) and (3.51). Numerical results for the problem of Figure 9 are presented in the next chapter. The case of an additional external load of do on the cylinder of Figure 9 was also investigated. Detailed results for this problem are of no specific interest but some general conclusions about the effect of this radial load will be reported in Section u.2. 3.9 Stress Intensity Factor Most of the early analytical work in fracture mechanics was based on the plane theory of elasticity. Consequently, the elastic stress field equations usually employed in the engineering analysis of real problems are based on the plane elasticity assumptions. It is customary in fracture mechanics to describe the crack opening displacement as a superposition of three basic deformation modes (5). The first mode, mode I defines an opening mode where the crack surfaces are displaced normal to the crack plane. The second mode, mode II, is described by displacements in which the crack surfaces slide over one another perpendicular to the leading edge of the crack. The third mode, mode III, defines a tearing displacement where the crack surfaces slide with respect to one another parallel to the 14L; leading edge of the crack. Superposition of these three modes is sufficient to describe the most general case of crack tip stress and displacement fields. The stress intensity factcrs K and K for these three modes are des1gnated K II III 1’ respectively. Since the examples discussed in this dissertaw tion have geometric symmetry and are symmetrically loaded, only the opening mode of crack displacement will be obtained. Values of KI, as subsequently defined in this section, are calculated from the obtained crack opening displacements. Knowledge of this factor for a given geometry and loading can then be possibly used to predict failure of a structural component. Since three-dimensional problems are neither in a state of plane strain or plane stress, the definition of a stress intensity factor for these problems must be first established. The first problem to be considered in detail is Sneddon's penny shaped crack solution. Reference (#2) gives the crack opening displacement as . 400(1-v2) /""‘T' Va ~ . (3.109) wl_ = Z- O "E which for small values of R, where R = a - r, becomes 2 uc°(l-v ) wl = 2aR (3.110) 3‘0 nB Paris and Sih (5) list the stress intensity factor for this problem as follows: lLiS KI =.\/":12.T: OM (3.111) In terms of this stress intensity factor, the crack opening displacement (3.110) becomes (3.112) “I = K 2(l-v2)V2R 2:0 I EN/;_. Rearranging this equation in terms of the known dimensionless O E w - O displacements -—-—JZ-0 gives 00 a B c K o (3.113) where u(1—v2) B 2na (3.11u) Then a plot of equation (3.113) as the 4/1?»- 0 gives us the desired value of §"CIKI from which an equivalent stress intensity factor for finite geometry cylinders can be calculated. For the rectangular Cartesian coordinate problems, the crack opening displacement near the crack tip is given by (5) 2(l-v) R w — . 3. vly=o G KI A/ 2“ plane strain ( 115) 2 A/ R (3 ) v| KI 2" plane stress .116 y=o (l+v)G lu6 Note that by definition the plane strain and plane stress stress intensity factors are equal while the above displacements are approximately 12.5% different for v = 1/3. Since the results, to be discussed later, indicate that most of the bar in the thickness direction is approximately in a state of plane strain, equation (3.115) is selected to calculate the stress intensity factor. Rearranging equation (3.115) so that the dimensionless crack opening displacements can be utilized leads to illym 00 a (3.117) Q 0| ('3 H X H II a a where CI is given by (3.114). A plot of the above equation as A//§:+ 0 can then be used to calculate KI. Since the crack opening displacement is a function of the thickness variable, the stress intensity factor obtained above will vary in the z direction. However, if we were to account for the non-plane strain conditions near the surface by using equation (3.116) or a corrected equation (3.115) for the definition of KI, this variation in the z direction would be minimized and the stress intensity factor would become a constant across the thickness of the bar by definition. This approach would essentially result in a continuously varying definition of KI across the thickness. is It must be noted that the above description of KI completely arbitrary and it is questionable if it has any 1”? real significance in three—dimensional elasticity problems. However, values of K based on equation (3.117), are still I) presented so that comparison between the calculated results and the published plane strain solutions (5) will be possible. CHAPTER M RESULTS AND DISCUSSION A computer program has been written for each of the numeri- cal examples using the solution techniques of the preceeding chapters to calculate the stress and displacement fields in each case. With the exception of the annular plate with internal surface cracks example, all numerical computations were perm formed for two arbitrary grid increments so that the convergence of the finite difference approximations could be checked. In a given direction, uniform line spacing was used in all calcula= tions with no other restriction being placed on the selection of the grid size. In general, an attempt was made to use a finer grid along the direction of largest variable change. Previously given numerical differentiation formulas show that their truncan tion errors are of Oh2 with the exception of some boundary terms in the coupling vectors {r}, {s} and {t} where 0h derivative approximations were used. However, the use of parabolic differentiation formulas in these isolated cases led to essentially identical results. Since numerical differentia- tion is inherently an inaccurate procedure, analytic differentia- tion is used wherever possible such as in evaluating the normal 1H8 1&9 stress distributions. An inspection of the ordinary differential equations and their boundary conditions shows that decoupling of the dependent variables is impossible and the previously discussed successive approximation procedure must be employed. Since the derivatives {u}, {v} and {w} depend on similar matrix functions, parti- cular integrals and initial value vectors as the corresponding displacement vectors, convergence of the displacements will assure convergence of their corresponding derivatives. This conclusion has been confirmed by the results of the numerical examples presented below. The computations for all the examples were performed on an IBM-360 time sharing digital computer using double precision arithmetic. Since the storage capacity of these computers is essentially unlimited, no attempt was made to optimize the use of the involved arrays. u.1 Solid Cylindrical Bar With a Penny Shaped Crack In order to establish the validity of the line method, detailed results will be first presented for the solid cylindri- cal bar containing a penny shaped crack and loaded normal to the crack plane. A convenient combination of the non- dimensionalized variables for which the evaluation of the required matrix functions presents no difficulty is as follows: 150 a = 1 Hz = ,1u ~ ~ 2 L = 1.68 h = ——- = .1176 4.1 r 17 ( ) B = 1.77 The selection of these dimensions results in the construction of 16 axial and 13 radial lines in Figure 8. According to the above choice of hr, the number of lines inside the crack surface must be 9. The computations for this problem involved 22 iterations in the successive approximation procedure using an arbitrary con« vergence criterion of 10's. This convergence criterion is defined as the maximum difference in the absolute values between successively calculated displacements at any point. The largest element of the error matrix (2.87) was .0001 at a = 1.68. The Runge-Kutta integration increment in evaluating the diagonalized matrix functions [Qij] was .001. The approximate execution time for a problem of this size using the above given data is 3 minutes on an IBM-360 computer. The results of these computations are presented in Figures 10 through 1n and Tables 1 through 5. For easy comparison of data, some of these figures include Sneddon's results for an infinite solid. Figures 10(b) and 11(b) show displacement dis: tributions for an identical bar which were calculated from a grid having only 9 radial and 9 axial lines. Figure 1a contains 151 ~ S-LO v-U3 2.5—— 1 1331.77 ’ii,-o.1176 1.68 ". ~ . 2.0_ L 1.68 hz 0.140 l 1» 3| 2° (a) Dimensionless axial displacement distribution (16x13 grid). ll ~ 3 - 1.0 v - 113 2. 5,—— z B’ - 1. 77 ii, - 0. 222 2.0 ”’3 L- 1.68 Hz - 0.210 7 (b) Dimensionless axial displacement distribution i9x9 grid). Figure 10. - Dimensionless axial displacement distribution for a solid cylindrical bar with a penny shaped crack. 152 .2357 L O }/ N i—il- L; 3| 2° (a) Dimensionless radial displacement distribution (16x13 ' grid). 1 =10 v=U3 .70 -1.77 2,-0.222 -1.68 nz - 0.210 .60 . 50 .40 .30 .20 (b) Dimensionless radial displacement distribution (9x9 grid). Figure 11. - Dimensionless radial displacement distribution for a solid cylindrical bar with a penny shaped crack. 153 \'.. \ l(wa) I: '5=1.0 hr=0.1176 :l E=1.77 'Fiz=0.140 3 ||l I = 1. 68 (16x13 grid) Crack I l v = 1/3 edge I l ? locations! \\ ' I ,rgnetlglon s sol. l | | Figure 12. - Dimensionless axial stress distribution for a solid cylindrical bar with a penny shaped crack at ’i’ - 0. 5-1.0 5,-0.1176 A 3' 1. 77 hZ - 0.140 L - 1. 68 (16x13 grid) 3 ___ v - 113 ---- Extrapolated CI=M1-v% Intercept - 2.1 W755 H 22:“ KI=1.48500\/a- xH lntercept- 1.6 o 7... 10° inegvdonM 1“ (b-L-OO) KI=l.l3oo\/5 l l J I l 1 0 .20 .40 .60 .80 1.0 Rla Figure 13. - Calculation of the stress intensity factor KI for a solid cylindrical bar with a penny shaped crack. 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Figure 13 shows the method for calcu- lating an equivalent plain strain stress intensity factor KI for three-dimensional problems from knowing the crack opening displacement. Inspection of Figures 10(a) and 10(b) clearly shows the advantage of the line method over other numerical solutions. A relatively coarse grid of 9 axial and 9 radial lines gave almost identical results to that obtained by using a 16 by 13 grid. Note that an approximate 100% change in fir and a 50% change in hz resulted in about a 2% change in the axial displacements. Since the bar is of finite size, the crack opening diSplacement is expected to be higher than Sneddon's solution. Consistency of the results with this conclusion is obvious from Figure 14. Figures 10 also show that the axial displacement is highest at the end of the bar above the center of the crack. If the bar were of sufficient length, the highest displacement curve would be a straight line with a constant axial displacement. Note that the crack opening diSplacement curve is assumed to be zero at the crack edge rather than at the first adjacent node outside the crack plane. Figures ll(a) and ll(b) Show the radial contraction of the bar as a function of the length and radius. The contraction is the largest, as one would expect it, at the outside surface in 161 the plane of the crack. Note that symmetry of the problem requires zero slopes along the displacement axes in both Figures 10 and 11. In the plane of the crack, a sudden in- crease in contraction takes place as can be seen from the curves shown. This increase is due to the constraint free crack surface which permits the higher contraction rate of the bar outside the crack radius to exert a strong influence on the material inside the crack. Similar arguments about the agree- ment of the radial displacements in Figures ll(a) and ll(b) for corresponding positions can also be made as for Figure 10. Figure 12 shows the stress distribution normal to the crack plane as a function of the distance from the crack edge. It is known that for linear elastic problems, this stress dis- tribution approaches infinity near the crack tip as the inverse square root of the distance from the crack edge. Establishment of this type of singularity is, however, difficult when numerical methods are used because values of the normal stress are needed within a distance of .05a or less of the crack edge. With the equal spacing of lines used throughout the examples, the minimum node location for these examples is about .06a. For the range of i shown in Figure 12, this inverse square root singularity is not valid. However, for the range shown, the obtained stress curve closely resembles Sneddon's solution as can be noted. Obviously, the absolute value of this stress is 162 greater for a finite size bar than for Sneddon's infinite solid. Note that the stress near the outside radius rapidly approaches the value of the applied dimensionless stress of unity. Figure 13 shows the calculation of the stress intensity factor KI according to equation (3.113). The intercept of Sneddon's solution is 1.6 which gives a stress intensity factor of 1.13 003VKE: 'The stress intensity factor for an infinite cylindrical solid containing a penny shaped crack is given by equation (3.111) as 1.13 ooqugl Thus, the validity of using the method of Section 3.9 for calculating these values of KI is established. The stress intensity factor obtained from the 2.1 intercept is 1.485 oo'N/E: Hence, the finite bar discussed in Figure 13 has an approximately 31% higher stress intensity factor than the infinite solid. Figure 14 shows the crack opening displacement for several cylindrical bars with different lengths and radii. The obtained results clearly show that as the length and diameter of the bars are increased, Sneddon's solution for an irfinite bar is rapidly approached. For a bar with b = i = 3.43, the maximum difference is only about 7%. Tables 1 through 5 show selected results from the computer listings. The accuracy of the normal stress and displacement boundary conditions can easily be noted from the numerical data listed. 163 4.2 Hollow Cylindrical Bar With a Penny Shaped Crack In this example the dimensions of the bar were intention- ally increased along with the inclusion of a central hole through the cylinder axis. The expected result of these changes from the problem of (4.1) would be the minimization of the crack influence upon the calculated displacement and stress fields. Selected results of the computations for this problem are shown in Figures 15 through 18. The physical dimensions of the prob— lem are also listed in these figures. Figure 15 shows the dimensionless axial displacement dis- tribution as a function of the radius and axial position. Note that the maximum crack opening displacement is less than that shown in Figure 14 for solid cylinders. The reason for this is that there is no load applied over the central hole surface and the effect of this is to offset the weakening influence of the hole. As expected, the displacement curves are essentially con- stant once the results are plotted beyond the vicinity of the crack. Figure 16 shows the radial contraction of the bar. This contraction is maximum at the outside surface and its variation along the z direction is only about 9%. For a bar without a crack, this outside contraction would be a constant along the z direction. Comparison of the curves in Figure 16 to those in Figures 11 clearly shows that the crack effect on the radial a = 1.0 L - 2. 67 i) B’= 2.67 v- 113 ,2. 3.0 __ r0 = 0.222 2.677 2.5 — 2.0— 1. 777 3 "o ““ ° ”F M 1.0—— ,50— .2227 . .___ I i l l | g 0 .50 1 0 1.~5 2.0 2 5 3.0 r Figure 15. - Dimensionless axial displacement distribution for a hollow cylindrical bar with a penny shaped crack. ll N N a - 1.0 L - 2.67 B’- 2.67 v =- 113 1.2—- ~ ~ r0 - 0.222 r 2.677 .80— i" ii: ' 40x 1. 337 \1\ 0.2227 1 I l 1 l I 1 1 L l; .50 1 0 ~ 1 5 2.0 2 5 z ~ 164 Figure 16. - Dimensionless radial displacement distribution for a hollow cylindri cal bar with a penny shaped crack. 165 (L ,r-C rack edge 2' 8 i location l l l l; 0 1. 0 1.50 2. 0 2.50 3. 0 0? Flgu re 17. - Dimensionless axial stress distribution for a hollow cylindrical bar with a penny shaped crack at z - 0. 1 3-10 E-ze7 3” b - 2.67 v- 10 "r'o- 0.222 .— 2intercept» 1.7 2 3‘ ~“~ C _ 4(1" 1) ) 1.— ‘— KI - 1.20 oo./5 l l l l I; 0 20 .40 .60 80 l 0 Rla Figure 18. - Calculation of the stress lntensltylactor KI for a hollow cylindrical bar with a penny shaped crack. 166 contraction is more pronounced for bars with smaller overall dimensions relative to the crack radius. Note the sudden in- crease in contraction near the crack plane along the inside hole radius. This increase is caused again by the crack plane which through its constraint free surface permits a contraction rate approaching the contraction near the crack edge. The dimensionless axial stress distribution in the crack plane is shown in Figure 17. The shape of this curve is similar to those shown in Figure 12. Note the relatively constant stress beyond a radius of 2 which shows that the region of largest stress variation is between i = 1 and r = 2. Similar com- ments about the axial stress singularity can also be made in this case as for the solid cylindrical bar problem. The calculation of the stress intensity factor KI using the crack opening displacement is shown in Figure 18. Note that even though the crack opening of Figure 15 is smaller than Sneddonis solution in Figure 14, the calculated stress intensity factor of 1.20 oO‘U/E- is somewhat greater than the 1.13 Cofiv/g- obtained in Figure 13. This is due to the more negative slopes obtained for finite sized bars in Figures 13 and 18. However, the stress intensity factor of Figure 18 is considerably less than that calculated for problem (4.1) in Figure 13. The application of a uniform radial stress, 00, to the outside surface of this hollow cylinder was found to have no 167 effect on the crack opening diSplacement. However, the axial displacements beyond the crack plane were considerably lowered. The radial displacements for this problem were all in the out- ward direction rather than inward as shown in Figure 16. The results of Figure 17 were also found to be independent of this radial surface load. As expected, the main effect of a radial surface load was observed to be in the calculated circumferential and radial stress fields. 4.3 Annular Plate With Internal Surface Cracks As a first attempt at the solution of a general three- dimensional problem in cylindrical coordinates, the problem of Figure 6 was solved using a grid of 16 lines in all directions. Because of the relatively coarse grid involved, the calculated data is listed in Tables 6 through 11. The construction of meaningful figures from these results is obviously difficult. It must be noted, however, that due to the unknown nature of the resulting solutions, the use of a coarse grid is always recom- mended in generating the first set of displacements. This practice may greatly complicate the programming of the necessary equations,but it has the advantage of providing results quickly from which the numerical limitations can be immediately recog- nized. Since the construction of a general computer program for this problem requires a great amount of effort, the listings of Appendix B-3 apply only to the specific case when NR = N8 = NZ 2 u 168 . . . . Eu Table 6. - DimenSionless Radial Displacements 3.5. for an o Annular Plate with Internal Surface Cracks Under Uniform Radial Tension on the Outside Surface 5 ~ .25 .50 .75 1.00 6 0° .435 .671 .786 .930 15° .940 .890 .896 .998 T 2 = .00 30° 1.168 .992 .968 1.058 1 45° 1.240 1.026 .994 1.081 0° I .506 .680 .789 .930 15° .907 .860 .886 .994 1 2 = .10 30° 1.092 .953 .957 1.055 45° 1.152 .986 .984 1.078 l 0° .724 .656 .779 .927 T 15° .795 .761 .856 .985 2 = .20 30° .898 .856 .932 1.048 45° ‘ .942 .892 .961 1.073 l 0° .716 .659 .769 .915 1 15° .752 .727 .845 .978 2 = .30 30° .815 .809 .926 1.049 45° .845 .843 .957 1.077 1 169 Table 7. — Dimensionless Circumferential Displacements "1‘5 L d V °l CD for an Annular Plate with Internal Surface Cracks Under Uniform Radial Tension on the Outside Suriace 8 00 1.0 30° 45° 1 .00 .718 .566 .302 .000 . l l i .10 .592 .2150 .238 .000 T ‘ '1; I" = .25 i .20 .000 .073 .063 000 l ‘ .30 .000 .009 .012 .000 l l .' .00 .425 .285 .138 .000 T I ; .10 .350 .232 .115 .000 2 = .50 .20 .000 .075 .057 .000 g .30 .000 .023 .026 .000 l i l , .00 .000 .037 .029 .000 i .10 .000 .033 .026 .000 T ‘ t r : .75 . i .20 .000 .023 .020 .000 s l l .30 .000 .014 .014 .000 1 l i i I 000 .000 .006 .00” J T ‘ 1 .10 .000 .007 .005 .000 E : r = 1.0 3 l .20 .000 .009 .007 .000 g l i .30 .000 .010 .007 .000 i 170 e I 0 0 EW Table 8. — DimenSionless Ax1al Displacements 3—5' for an Annular 0 Plate with Internal Surface Cracks Under Uniform Radial Tension on the Outside Surface 2 .00 .10 .20 .30 F .25 .000 -.218 -.342 -.416 .50 .000 -.108 -.217 -.322 T : 0° .75 .000 -.089 -.175 -.258 l 1.00 .000 -.076 -.152 -.226 .25 .000 -.098 -.215 -.338 T .50 .000 -.049 -.131 -.232 = 15° .75 .000 -.075 -.15” -.237 1.00 .000 -.074 -.139 -.226 .25 .000 -.068 -.167 -.281 T .50 .000 -.OHO -.106 -.195 2= 30° .75 .000 «.067 -.1Ul -.222 1.00 .000 -.O75 ‘.152 -.231 .25 .000 ".061 -0153 “oLCL .50 0000 -0039 'olOl -6186 = 45° .75 .000 -.055 -.l37 -.218 1.00 .000 -.075 -.153 4.233 l 171 0 Table 9. - Dimensionless Radial Stress Distribution 35- for an o Annular Plate with Internal Surface Cracks Under Uniform Radial Tension on the Outside Surface f. l ~ .25 .50 .75 1.00 8 0° .000 .020 1.062 1.00 T 15° .000 .346 .837 1.00 ‘ 2 = .00 , 30° .000 .293 ..748 1.00 45° .000 .282 .726 1.00 1 0° .000 .027 1.048 1.00 ‘ 15° ‘ .000 .405 .872 1.00 p 2 = .10 30° : .000 .424 .801 1.00 45° 5 .000 .424 .783 1.00 1 0° .000 1.305 1.026 1.00 1 15° .000 .968 .980 1.00 2 = .20 30° .000 .840 .932 1.00 45° .000 .801 .915 1.00 ‘ 0° .000 .872 1.005 1.00 . 15° ? .000 .950 1.004 1.00 2 = .30 30° .000 .966 .986 1.00 45° .000 .956 .975 1.00 i 172 . . 09 Table 10. - Dimensionless Circumferential Stress 6—- o Annular Plate with Internal Surface Cracks Under for an Uniform Radial Tension on the Outside Surface 2 ~ .25 .50 .75 1.00 8 0° .00 ’.00 1.702 1.251 A 15° .308 .803 1.542 1.339 _ 2 = .00 30° .209 1.043 1.460 1.416 45° .240 1.145 1.441 1.443 $ 0° .00 .00 1.659 1.248 15° .703 .957 1.534 1.331 T 2 = .10 30° .803 1.196 1.472 1.410 45° .863 1.270 1.456 1.439 1 0° 4.569 2.821 1.506 1.236 150 3.609 2.011 1.526 1.313 T 2 = 020 30° 2.971 1.706 1.506 1.400 45° 2.748 1.629 1.498 1.432 0° 3.050 1.858 1.442 _ 1 1 15° 3.110 1.879 1.500 1.301 2 = .30 30° 3.171 1.825 1.523 1.397 45° 3.167 1.775 1.524 1.432 1 173 0 Table 11. - Dimensionless Axial Stress 33-for an Annular Plate 0 with Internal Surface Cracks Under Uniform Radial Tension on the Outside Surface f . ~ .25 .50 .75 1.00 8 . 00 ’1097’4 -l.O‘-|>9 .030 . -.Olu~ 15° - .939 " 0126 00,43 .040 1 2 = .00 300 - .631 .041 .064 .060 ; , 45° - .544 .082 .070 .062 ’ 0° -1.597 x -1.073 .028 -.010 15° - .867 - .210 .032 .031 | ~ z = .10 30° - .579 .006 .053 .046 .450 -' 0'48? ‘ .055 0059 00,48 1 0° 0500 0313 0002 -0004 ‘ 15° -3.135 -1.996 -1.504 -1.310 2 = .20 300 - .077 .070 .039 .019 45° - .097 .073 .042 .020 ¢ 0° .000 .000 .000 .000 7 15° .000 .000 .000 .000 ’2 = .30 30° .000 .000 .000 .000 ‘ 45° .000 .000 .000 .000 ‘ 174 in Figure 6. Although the results in Tables 6 through 13 are preliminary, they still demonstrate that the previously presented solution techniques permit the computation of the required dis- placement and stress fields. As a result of having 16 lines in all directions, the ~ coordinate increments used in this problem were hr = .25, 32 = .10 and he = 15°. Using a convergence criterion of 10"6, the number of successive calculations in the iteration procedure was 41. The Runge—Kutta increment in evaluating the matrix functions Qij was .001 while the largest error matrix element in both the circumferential and axial directions was less than 10"5 at all points. Table 6 shows the dimensionless radial displacement dis“ tribution as a function all three coordinates. It can be noted that the outward radial displacement increases as a func- tion of angular position while changes in the z direction are most pronounced at the inside radius. Table 7 lists the calcue lated circumferential displacements. These results show that below the crack plane, the circumferential displacements are esSentially zero while the maximum crack opening is at 8 = E = 0 and at 5 = .25 as expected. Note that the crack opening decreases with an increase in radius and thickness. Table 8 displays similar results for the calculated axial dis- placements. While both radial and circumferential displacements 175 are extensional, the axial displacements are negative indicating a contraction in that direction. This contraction is a maximum at the plate surface and in the crack plane decreases with an increase in radius. Displacement Tables 6, 7 and 8 clearly show the accuracy of the enforced displacement boundary conditions and that the number of normal lines inside the crack plane for this example is 4. Table 9 contains the dimensionless radial stress distribu- tion as a function of the coordinates. The zero normal stress boundary condition on the inside surface and the applied radial stress of unity on the outside surface are evident in the results of this table. Note that in the plane of the crack, that is for 0 < 2 < .15, the radial stress varies gradually between 0 and 1 while below the crack plane it is close to unity everywhere except on the inside surface. The maximum value of this stress occurs in the crack plane at r = .5 just outside the crack edge. As expected, the radial stress is tensile or zero every- where in the body. The dimensionless circumferential stress distribution is shown in Table 10. It is expected that this Stress be the maximum tensile stress in the body since the crack plane is normal to the circumferential axis. Because the interaction between the hole and crack stress fields is most pronounced at the inside radius, this stress should approach infinity most 176 rapidly at the inside boundary just below the crack edge. The results of Table 10 seem to confirm this observation, although we must also remember that the nodes are closer to the crack boundary below the crack surface than along the radius. Note that at E = 1, the stresses are essentially constant along the z direction. Table 11 contains the dimensionless axial stress distribu~ tion as function of the coordinates. The zero normal stress boundary condition at 2 = .3 is obvious from the results in this table. Note that the axial stress is either zero or come pressive in most of the region except just below the crack edge where a tensile axial stress seems to be generated. Because of the coarse grid used, the numerical results presented for this example are probably somewhat inaccurate in magnitude, but they do indicate some previously unknown variations in the stress and displacement fields for this problem. This conclusion is possible in that the line method does not usually require a fine grid for good results as was shown in Se'tien 4.1. 4.4 Bar With Through-Thickness Central Crack The solution of the problem shown in Figure 2 was obtained by using two different sets of lines along the coordinate axes. Selected results of these computations are shown in Figures 19 through 26 and Tables 12 through 14. The computation time for the example containing a 35 x 70 x 98 line grid was about 30 177 rPlane stress ll ’7’ solution ~ Ea 1'0 B . U3 / z b . 2,0 nx - 0.1333 2.40 [1.5 /.9' i_'- 1.75 fiy- 0.250 2.m_ I t'l.5 hZ‘O.30 ~ ‘ ‘ ‘ ‘. \ ‘ 1’ ”Na 1. 60— 0 1.20—- ---Plane strain solution .80— .40— 0 1 1 l 1 g 2" «no (a) Dimensionless crack opening displacement (48x96x128 5— 0 grid). 2.40“.— E 3-1.0 v- 113 b- 2.0 iix- 0.1538 2.00 L- 1.75 E), - 0.2917 1.60_ 't"-1.5 32 = 0.375 1. 20 —- .80- .40— l l l ; 0 20 40 ~ .60 80 l 0 x (D) Dimensionless crack opening displacement (35x70x98 grid). Figure 19. - Dimensionless crack opening displacement for a rectangular bar under uniform tension containing a through-thickness central crack. 178 t—l '3 iii: ii-2.o fix-0.1333 m ~ ~ 1.__ L- 1.75 ny - 0.250 ‘t"-1.5 ill-0.30 1 I l 1 l 1 0 .40 .80 ~ 1.2 1.6 2.0 X Figure 20. - Dimensionless bar and extension for a rectangular bar under unlform tension containing a through-thickness central crack. ll 3? 2.4— 0. .5327 5.1.0 p.113 1.60— b-2.0 fix-o.1333 c t- 1.75 ii -o.250 #‘I? ‘i’ ”y w .932,» # -1.5 112-0.30 .304} l l l l o .40 .go 1.2 1.6 2 Figure 21. - Dimensionless normal displacement distribution in the crack plane for a rectangular bar under uniform tension contain- ing a through-thickness central crack. 179 rCrack edge location (a) Dimensionless y-directional normal stress as a function of 32'. ll 6*— .5 x 1.0667 4__ \ 1.207 1.337 \ 21-! J \ l l l l l l; 0 .10 .80 1,2,2 1. 6 2. 0 2. 4 2 lb) Dimensionless y-directional normal stress as a function of 2. Figure 22. - Dimensionless y-directional normal stress distribution in the crack plane for a rectangular bar under uniform tension containing a through-thickness central crack. 180 ii . 3.20_ : Crack edge location I 3-10 v=U3 | B’= 2.0 iix - 0.1333 2.40— I L- 1.75 'riy= 0. 250 I i=1.5 ~ 160— : I .80— i l =2 . . L J r” 1 13+. 0 V 1.0 1.2 1;.4 1.6 1.8 2.0 o X ~ (a) Dimensionless z-directional normal stress as a function of x. l | l : 0 .40 .80 1.2 1.6 2.0 2.4 E (b) Dimensionless z-directional normal stress as a function of '2. Figure 23. - Dimensionless z-directional normal stress distribution in the crack plane for a rectangular bar under uniform tension containing a through-thickness central crack. “n. 181 l 2AA l 3-1.0 v-ll3 : 5-2.0 fix-0.1333 1.80— I 31.75 27'0'250 : t-1.5 112-0.30 | 1.20— l I l .6Gl- | I l u° 0 V 1.0 1.2 14 1.6 1.8 2.0 l l l 1 l l 0 .40 .80 1.2 1.6 2.0 2.4' lb) Dimensionless x-directlonal normal stress as a function of "2'. Figure 24. - Dimensionless x-dlrectlonal normal stress distribution in the crack plane for a rectangular bar under unlionn tension containing a th rough-thickness central crack. 182 5?— ----- Extrapolated 0 .41-04 1 E753 4— E {'2' 1.5 0 ~ K -2.57a rY'OJ mib°3_ I ow ’4’ KI.2‘4900~‘/i [2'Q9 K -2.530 KIu2.3Taow\ 2 l 1.. O .20 .40 .60 80 1.0 Rla Figure 25. - Calculation ofthe stress intensity factors KI for a rectangular bar under unlform tension containing a th rough-thickness central crack. 3.50L 34.0 i‘-1.5 5-10 v-U3 'i.’ - 1. 75 H 3‘ a— 8 _ ‘3, [Based on plane strain definition 2. 50- ' x-slad-O'f'ilfinnffilo-US'EFOFNW plane strain definition 2.. ' I 1. 0 .40 .go 1. 2 1.6 Figure 26. - Variation of the stress intensity factor K across thethlckness for a rec- tangufar’ r under unlform tension con- taining’ a fh rough-thickness central crack. 2185 Table ML - Dimensionless x-Directional Displacements §””§ for a Rectangular Bar Under 0 Uniform Tension Containing a Through-Thickness Central Crack. 5 = 1.0, b = 2.0, i = 1.75, t = 1.5 (48-96-128 xey-z Directional Lines Respectively) ' m i 0.0 .40 .80 1.20 1.60 2.00 y 0.0 .000 -.474 -.828 -1.026 -1.095 -1.192 .50 .000 -.143 -.275 - .541 - .783 - .920 T i = .00 1.0 .000 -.025 -.092 - .242 - .420 - .565 $ 1.75 .000 .169 .257 .238 .148 .029 0.0 .000 -.479 -.846 -1.00 -1.030 -1.105 .50 .000 -.119 -.244 - .498 - .742 - .888 t i = .90 1.0 .000 -.015 -.076 - .220 - .393 - .540 1.75 .000 .180 .276 .260 .163 .042 l 0.0 .000 -.395 -.713 - .890 - .991 -1.091 .50 .000 -.062 -.156 - .422 - .699 - .862 T i = 1.5 1.0 .000 .009 -.035 l - .182 - .367 - .520 1.75 .000 .179 .278 .263 .168 .050 184 Table 13. - Dimensionless y-Directional Displacements £23. for a Rectangular Bar Under Uni- 0 form Tension Containing a Through-Thickness Central Crack. 5 a 1.0, S a 2.0, i a 1.75, t = 1.5 (48-96-128 x-y-z Directional Lines Respectively) t .00 .25 .50 1.00 1.50 1.75 .00 2.112 2.227 2.315 2.578 2.893 3.048 T .60 2.197 2.322 2.415 2.638 2.952 3.107 2 = .00 .90 2.218 2.334 2.420 2.638 2.956 3.117 1.50 2.249 2.323 2.402 2.631 2.945 3.125 .00 1.252 1.354 1.552 2.014 2.453 2.660 .60 1.287 1.424 1.622 2.072 2.512 2.716 ~ T x = .80 .90 1.297 1.431 1.627 2.074 2.515 2.721 1.50 1.313 1.411 1.591 2.029 2.471 2.686 .00 .000 .270 .572 1.173 1.713 1.966 .60 .000 .288 .599 1.202 1.736 1.986 T 2 s 1.60 .90 .000 .291 .604 1.206 1.739 1.986 1.50 .000 .290 .589 1.179 1.716 1.965 .00 .000 .139 .351 .868 1.372 1.620 .60 .000 .174 .384 .883 1.385 1.630 a = 2.0 .90 .000 .176 .386 .890 1.390 1.638 1 1.50 .000 .167 .377 .881 1.380 1.626 185 Table 14. - Dimensionless z-Dirootional Displacements 5-8-3- for a Rectangular Ber Under 6 Uniform Tension Containing a Through-Thickness Central Crack. I I 1.0, B I 2.0, L a 1.75, t I 1.5 (48-96-128 x-y-z Directional Linea Reepecttvely) I .00 .30 .60 .90 1.20 1.50 I .00 .000 -.000 -.034 -.053 -.041 .025 T 080 .000 -I05“ '0128 ‘0205 .6266 '6267 9 I 0.0 1:50 0000 '0105 .0223 -.355 -.u92 ‘e628 l 2.00 .000 -.J.J.9 -.237 -.350 -.465 -.554 J .00 .000 -.040 -.094 -.134 -.167 -.194 T i .80 .ooo -.078 -.156 -.284 -.309 -.877 , 9 I .50 105° 000° “0110 .6223 -.340 «458 -0657 2000 .000 'all? .0231 -.333 'IHSl -.5‘+9 .00 0°00 “0080 -.160 ‘02“; '03?“ -.H08 .eo .ooo -.096 -.185 -.278 -.874 -.472 T . 1- - 9 I 1.0 l 1.80 .000 -.100 -.218 -.320 -.436 -.543 2000 .000 .011“ .0226 .0335 -.440 -.544 l .00 0000 .0117 .022“ -.334 MUG}. .961“ T 080 .000 '0116 -.218 'a 319 -.'$29 .6558 1 - 9 I 1.75 1.60 .000 «.113 -.2.19 M321 -.422 -.526 2.00 .000 .1116 .022“ .0327 -.427 .6526 186 nfixnxtes while for the #8 x 96 x l28 grid it was about 50 minutes. The runnber of iterations for these problems was 28 and 29 rem spectively° The largest element of the error matrix was “CCOl iJI'the y direction at the end of the bar when the matrix func~ tions associated with the larger grid were calculated. A cona vergence criterion of 10"6 was applied to all three displacements in calculating the successive approximations. The dimensionless crack opening displacements are shown in Figure 19. Inspection of Figures 19(a) and 19(b) shows that the finite difference approximations have been sufficiently refined when the results of the H8 x 96 x 128 grid were calcu« lated. An approximate 15% change in hx, 17% change in hy and a 25% in 52 resulted in a maximum of l“5% change in the crack opening displacement. Figure 19(a) also shows the results of the plane elasticity solutions obtained by Mendelson (43) for the problem shown. The plane stress solution gives the highest crack opening diaplacement while the plane strain solution is very close to the curve obtained at the center of the bare Values ~ at z = .9 and 2 = 1.5 are between the two plane solutions as can be expected. The shapes of the obtained curves are all elliptical which can be easily shown by plotting the equation of an ellipse with the coordinate intercepts as the major and minor axes. Note that the maximum change in the crack opening is about 6.5% across the given thicknesso 187 A phi of the bar end extension is shown in Figure 20. This disphumment is maximum at the center of the bar with the varia— timlin the 2 direction being less pronounced than in the crack pflane. Because the center of the bar is more constrainea, the ywfirectkmal displacements are maximum on the surface of the bar in bofiifigures l9 and 20. Figure 21 shows the variation of the crack opening disPlacements across the thickness of the bar. Near the crack edge, that is at a = .932, the displacement at x O the changes in the curve is almost a constant while 2 direction are most rapid for O 5 z 5 .5. Similar conclusion is possible from Figure 19(a) where the curves at 2 = .9 and 2 = 1.5 are very close to each other. Figure 22 contains a plot of the stress distribution normal to the crack plane as a function of both bar width and thickness. Figure 22(a) shows the same general stress distribution as Figure 12, indicating that the normal stress is singular, as expected, at the crack edge. However, the resolution of our grid :near the crack is not sufficient to establish an inverse square root singularity. Inspection of these curves shows that the stress is highest near the center of the bar and it rapidly approaches the applied stress for values of 52 > 1.5. Figure 22(b) shows that the variation in the 2 direction is largest near the crack edge and as 51 increases, the stress curves become more constant. Note that the stress at SE = 1.066 I. 188 indicates a central region of uniform stress for O < 2 < 1.1 a. and a boundary region of 1.1 < z < 1.5 where the stress drops significantly to the surface value. Similar results were ob» tained by Cruse and Van Buren (25) for a single edge crack bar Specimen. Figures 23 describe the oz stress distribution in the crack plane. The results of Figure 23(a) indicate that this normal stress is also singular near the crack edge and its value approaches zero with increasing 2. This is expected since at the surface is free of this normal stress. Note that 2 = 1.5, = 1.6 indicating for 2 = 1.2, the stress becomes zero at a that this normal stress also vanishes on part of the 2 = 2.0 face. This result seems consistent since the stress field is expected to be three-dimensional in the vicinity of the crack and approach the crack-free bar solution at locations far from normal stress mainly as the crack. Figure 23(b) shows this oz a function of the thickness variable, 2. An unexpected increase in this stress near the center of the bar can be noticed from the results shown. This sudden increase in oz appears to be the result of the way this stress increases as it approaches the (a = 1.0). Note that the variation across the thick- is more continuous than in Figure 22(b) with no However, the curve crack edge ness in ‘02 noticeable central or boundary region shown. at 5-2 = 1.066 seems to indicate that as the crack edge is being 189 approached, most of the variation will occur near the surface of the bar. Figures 24 show the Ox normal stress distribution in the crack plane as a function of both 52 and 2. Figure 2L+(a) indicates that this stress is also singular at the crack edge and itis maximum near the center of the bar. Results in this figure also show that Ox is zero only on the face i. = 2.0 and has a given, non-zero value everywhere on the face 2 = 1.5. This is contrary to the results shown in Figure 23 for oz and as a consequence the sudden increase in <3x near the center of the bar has not been obtained in Figure 21+(b). Figure 21+(b) shows a central region of uniform stress and a boundary layer through which the stress decreases to the surface values. Note that the variation in Ox across the thickness decreases as x increases and approaches a constant value of zero at x = 2.0. Figure 2u(a) also indicates that the value of Ox remains essentially constant for 1.2 < 32 < 1.35. This constant c:x stress is different, of course, as 2 increases from zero to 1.5. The results in Figures 19 through 21+ cannot be checked against known data but the crack opening diaplacements of Figure 19 indicate reliability of the reported stress and dis- placement distributions. Limited results reported in (25) agree with some of our conclusions regarding the normal stress 190 distrflnnfions but a detailed comparison between results is not possible because of the problems treated. Figure 25 shows the calculation of the opening mode stress intensity factor from the plane strain crack opening displace» In terms of the ordinate intercept, the stress intensity = INTERCBPT . (.704) oo\/a—. Using the extrapolated values in Figure 25, the stress intensity factors ment. factor is given by KI KI for selected values of 2 were calculated. A plot of these stress intensity factors as a function of 2 is shown in Figure 26. Since the stress intensity factor is proportional to the crack opening displacement, the value of KI from the plane strain definition is maximum at the surface and minimum near the center. Similar results were obtained in reference (25) for a single edge crack specimen where it was shown that as plane stress conditions are approached, the stress intensity factor increases. Srawley and Brown (5) have also found that the plane strain fracture toughness, which is directly related to the stress intensity factor, is considerably less than the plane stress fracture toughness. Figure 19(a) shows that at the center of the bar plane strain conditions exist while at the surface 2 = 1.5, the crack opening is about half-way between the plane strain and plane stress solutions. Since the plane stress crack opening is about 12.596 higher than the plane strain crack displacement, it 191 is pnnxmed that equation (5.115) be revised as follows: - 1.0625 [511310. KI [2%.] (4.2) V .. y=0 z=1.5a Equathx1(4.2) will now be used to predict the corrected value of &[ at z = 1.5. A similar corrected plane strain crack openhngdisplacement equation may be written for each position along the ‘Z axis and a continuously corrected value of KI can be calculated. Figure 26 also shows a plot of this correc- ted stress intensity factor. The value of this corrected stress intensity factor is a constant across the thickness which then agrees with the fact that plane stress and plane strain stress intensity factors are equal. Figure 26 shows that this cor- rected value of KI is 2.37 oo./E. In reference (2), Brown and Srawley report a plane elasticity stress intensity factor of 2.11 oo./E for a bar having an identical width and half crack length. Note that the result of Figure 26 is about 12 per- cent higher than this value which must be attributed to the fi- :nite length.and thickness of the bar in Figure 2. The solution i112meference (2), of course, is for a bar with infinite length and thickness. Unables 12 through 14 show selected results from the dis— placements obtained in the computations. The y-directional dis- placements are all extensional while in the z-direction only 1’ ‘1 192 contraction is possible. A somewhat unexpected result is ob~=~ taineiin Table 12, which indicates that parts of the bar con— tract While other parts expand in the x-direction. ihfi Bar With Through-Thickness Double Edge Cracks Selected results for the problem of Figure 3 are presented in Figures 27 through SH and Tables 15 through 17. These res suits were obtained with two different sets of lines along the coordinate axes which were identical to the two sets used for the problem of Section u.u. The convergence criteria and error matrix elements were also the same as those for the central crack problem. However, the number of required iterations in- creased greatly with the fine grid using 60 iterations while the more coarse grid requiring 48 successive calculations. As a result, the execution time for the finer, #8 x 96 x 128 line grid problem was about 80 minutes. The dimensionless crack Opening diSplacements are plotted in Figure 27. Comparison of Figures 27(a) and 27(b) shows that the finite difference approximations have sufficiently converged when the #8 x 96 x 128 line grid was used in the final calculae tions. Contrary to the central crack problem, the crack opening displacements in Figure 27 are independent of the 2 coordia nate. Similar results were obtained by Cruse and Van Buren (25) for the single edge crack bar Specimen. One must also note that for this problem the crack opening curve is no longer elliptical 193 ii '52 1.0 vs 113 6B b-2.0 Fx=o.1333 'E-1.75 Fly-0.250 i=1.5 fiz=o.3oo 4— \‘¥ For all values of '7: 2— 0 l l l l I l ( 3—ao 4r— t-l.5 112-0.375 a) Dimensionless crack opening displacement (48x96x128 grid). 6i— z=- 1.0 v= 113 b-2.o Ex-o.1538 LL75 fiy-o.2917 ‘\- For all values of “z’ , l l l l l o 1. 2 1. 4 1:6 1. 8 2. o X (b) Dimensionless crack opening displacement (35x70x98 grid). Figure 27. - Dimensionless crack opening displacement for a rectangular bar under uniform tension containing th rough- thickness double edge cracks. 4' 194- (1|— :a:=1.0 v=113 ? b=2.0 Ex=o.1333 ,r-LS £4.75 fiy=0250 ' 4__ t=1.5 nz=o.3oo l l l l l _ 1 0 .40 .80 ~ 1.2 1.6 2.0 x Figure 28. - Dimensionless bar end extension for a rectan- gular bar under uniform tension containing through- thickness double edge cracks. a-LO v-U3 5-2.0 fix-o.1333 ii I- 1. 75 fiy- 0.250 e— t-1.5 hz'0.3m 2’ 2.007 41.—— c 1.607 25+”o 7:: 1.1.1 2l-— l.%667 l l L I , o .40 .go 1.2 16 2 Figure 29;. - Dimensionless normal displacement distribwlion in the crack plane for a rectangu- lar bar under uniform tension containing th rough-thickness double edge cracks. 195 ll 8" ~ I a=LO v-U3 ’5- 2.0 Ex - 0.1333 : 6_ ‘i-1.75 fiy-0.250 | t - 1.5 nZ - 0.300 2 | 0‘\ I 4"" .9« : | 1.5 | 2-— , Crack edge location/1| l l l l l g 0 .20 .40 .60 .80 1.0 (a) Dimensionless y-directional normal stress as a function Hi i ° - ., .L x 0.9327 6— \ 4— .800] \ 2_ .001 I l | I +7 0 40 .80 12 16 “z’ (b) Dimensionless y-directional normal stress as a function of 2. Figure 30. - Dimensionless y-directional nonnal stress dis- tribution in the crack plane for a rectangular bar under uniform tension containing th rough-thickness double edge cracks. H 196 4+- __ 5- 1.0 v - 113 I B’- 2.0 iix- 0.1333 I 3i- i- 1. 75 iiy- 0.250 I __ 't"-1.5 E, - 0.300 | l 2—- 2. i "’ 0\ I .9 ‘x l 1" I Crack | "' edge 1 1.2 l 1 | :15 l locat'0'1‘1 ° 0 .20 .40 .60 .80 1.0 a: 2' D (a) Dimensionless z-dlrectional normal stress as a function of x. g 32’ 3I- 0. 932] 2r- 1r- L l 0 40 80 1.2 1 6 2 0 NZ (b) Dimensionless z-directional normal stress as a function of 2. Figure 31. - Dimensionless z-directional normal stress dis- tribution in the crack plane for a rectangular bar under uniform tension containing th rough-thickness double edge cracks. 0 197 t“ 34.0 v=113 B’-—-2.0 fix=0.1333 £4.75 5-0250 — t=1.5 hz'0.300 I Crack edge location4 I l I I I I I I I I I I I I I I I .20 .40 ~ .60 .80 1.0 (a) Dimensionless x-directional normal stress as a function of if. I — II (b) Dimensionless 5-directional normal stress as a function of 2. Figure 32. - Dimensionless x-directional normal stress distri- bution in the crack plane for a rectangular bar under uni- form tension containing th rough -thickness double edge cracks. 4". 198 7“. c . 411 - 021 I EVZna ’5’ = 1.0 ---- Extrapolated B- 2.0 ... 6— I: - 1. 75 :H 1‘ Intercept = 5.75 t-1.5 Lu|°° \\ V ' U3 \ 5"“ 7" For all values of 2’ KI g 4.04 Oofi Ill __ 4 I | l I l g 0 .20 .40 .60 .80 1.0 Rla Figure 33. - Calculation of the stress intensity factors KI for a rectangular bar under uniform tension containing through- thickness double edge cracks. '5 = 1.0 II ’5 = 2.0 5" i’ - 1. 75 Based on plane strain 7". 1,5 definition-I to i V = U3 :21 o 4;: L O I I I I l 1.2 1.6 ONO b O on C Figure 34. - Variation of the stress intensity factor K across the thickness for a rec- tangular ar under uniform tension con- taining th rough -thickness double edge cracks. 199 Table 15. - Dimensionless x-Directionei Displacements §—~3 for a Rectangular Bar Under Uniform Tension Containing Through-Thickness Double Edge Cracks. 5 . 2.0, i- 1.75, i: - 1.5 O I O (uB-QB-lQB x-y-z Directional Lines Respectively) s 0.0 .00 .00 1.20 1.60 2.00 9 0.0 .000 .000 .017 1.002 1.210 1.268 T 35° 0000 .05“ ' oOlB - .128 - .082 - .05“ 2 - .00 1000 .000 .022“ - eu78 ' e700 - .852 - .9H3 1.75 .000 -.0uo -1.231 -l.688 -1.000 -2.101 I 0.0 .000 .300 .70u 1.070 1.152 1.100 050 .000 0030 - 0°60 ' el87 ' .158 ' .133 t - l I .90 1.00 .000 -.250 . .507 . .700 - .020 - .000 1.75 .000 -.000 -1.205 -1.700 -2.000 -2.201 0.0 .000 .292 .599 .031 .011 .eul .so .000 -.o20 - .190 _'- .37u - .euu' h; .310 2 - 1.5 1000 .000 ‘0293 . 0616 ' 0892 '10038 “10113 1.75 .000 -.003 -1.017 -1.001 -2.100 -2.201 200 Ev Table 16. - Dimensionless y-Directional Displacements -—1; for a Rectangular Bar Under b . 2.0, i . 1.75, E = 1.5 Go Uniform Tension Containing Through-Thickness Double Edge Cracks. = 1.0, (08-96-128 x-y-z Directional Lines Respectively) 9 .00 .25 .50 1.00 1.50 1.75 2 .00 .000 .250 .500 1.230 1.070 2.260 T .60 .000 .222 .003 1.106 1.056 2.200 .00 .000 .220 .503 1.150 1.060 2.257 - 0.0 1.50 .000 .200 .529 1.102 1.007 2.277 l .00 .000 .500 1.001 1.016 2.070 2.050 .60 .000 .500 1.011 1.025 2.521 2.070 I .00 .000 .520 1.025 1.000 2.536 2.001 , .80 1.50 .000 .601 1.006 1.070 2.507 2.031 I .00 3.370 3.032 3.500 3.705 0.036 0.305 .60 3.360 3.393 3.002 3.600 5.131 0.307 I a 1.60 .90 3.370 3.300 3.000 3.700 0.130 0.003 1.50 3.012 3.060 3.007 3.721 0.161 0.020 I .00 0.507 0.536 0.512 0.600 0.790 5.060 .60 0.515 0.090 0.063 0.506 0.030 5.106 I .00 0.512 0.001 0.053 0.530 0.001 5.100 a 2.0 1.50 0.572 0.536 0.075 0.537 0.030 5.196 I 201 Table 17. - Dimensionless z-Directional Displacements EEE- for a Rectangular Bar Under 0 Uniform Tension Containing Through-Thickness Double Edge Cracks. 6 = 1.0, B a 2.0, i = 1.75, t = 1.5 (00-96-120 x-y-z Directional Lines Respectively) 2 .00 .30 .60 .90 1.20 1.50 R .00 .000 -.220 -.036 -.607 -.071 -1.120 I .80 .000 -.201 -.3QH -.609 -.888 -l.288 9 = 0.0 1060 0000 -0100 -0179 -0238 '02?“ " .293 2.00 .000 -.059 -.102 -.l23 -.123 - .119 .00 .000 -.189 -.379 -.566 -.798 - .928 .80 .000 -.167 -.337 -.513 -.702 - .888 T 9 = .50 1.60 .000 -.090 -.l72 -.299 -.317 - .375 2.00 .000 -.052 -.110 -.169 -.205 - .219 .00 .000 -.130 -.270 -.900 -.519 - .625 .80 .000 -.118 -.201 -.363 -.079 - .58“ T 9 = 1.0 1.60 .000 --057 -.l37 -.222 -.301 - .370 2.00 .000 -.033 -.097 -.165 —.231 - .287 .00 .000 -.095 -.188 -.267 -.315 - .308 .80 .000 -0097 -e186 “.26” -.323 - .3103 1 9 = 1.75 1.60 .000 -0081 -elug -0221 -.297 III .37“ 2.00 .000 -.067 -.123 -.l90 -.270 - .361 202 and for 1.3 < a < 2.0 the displacement curve is essentially a straight line. These results appear to be consistent since at the bar edge no constraints exist against this mode of deformation. Figure 28 shows the dimensionless bar end extension. This extension is maximum, as expected, at the edge of the bar and it is also somewhat dependent on the 2 coordinate. The small variation in the 2 direction is the result of the uneven con- straints developed in the central region of the bar, which in planes far from the crack spread over the entire cross section of the problem. As can be noted, this end extension is somewhat higher near the surface than at the center of the bar. Figure 29 shows the variation of the crack opening displacements across the thickness of the bar. As discussed previously, these curves also show constant diSplacements along the 2 axis. Figure 30 contains a plot of the stress distribution normal to the crack plane. Inspection of these figures shows that this stress is maximum at the center of the bar and is singular near the crack edge. As previously mentioned, the type of singularity is difficult to establish but the shape of these curves is similar to that obtained in the other examples. The minimum value of this stress occurs at i = 0 but even at this point the stress is at least 00% higher than the applied stress. Figure 30(b) shows that the variation in the 2 direction is 203 largest near the crack edge and becomes more gradual with decreasing values of 2. Note that the stress near the crack edge again indicates a central region of approximately uniform stress and a boundary region beyond 2 = 1.1 where the stress drops significantly to the surface values. This agrees with the results obtained for the central crack problem. Figures 31 describe the dimensionless oz stress distribu- tion in the crack plane. The results of Figure 31 are very similar to those shown in Figure 23 for the central crack prob- lem with one major difference. Figure 31(a) indicates that for all values of i this stress has a given, non-zero value and it vanishes only on the surface 2 = 1.5. This result, of course, follows from the relationship between the crack and bar widths for the problem described in Figure 3. The singular nature of this stress near the crack edge is evident from Figure 31(a). The variation of oz across the thickness is shown in Figure 31(b). Note that the curve near the crack edge, that is at i = .932, begins to display an internal region of uniform stress and a boundary region with significant variation. The variation across the thickness becomes more gradual as the value of 2 decreases. Figures 32 contain the °x normal stress distribution in the crack plane as a function of both bar width and thickness. Figure 32(a) shows that this stress is also maximum at the center 201+ of the bar and is singular near the crack edge. The constant value of this stress just outside the crack edge vicinity is similar to the results obtained for the central crack problem. Figures 32 also indicate that this normal stress is greater than the applied stress at all points in the central portion of the problem. Figure 32(b) displays the ox stress distribution across the thickness of the bar. Note the central region of uniform stress and a boundary region with significant stress variation. As the value of 2 decreases, the stress distribu- tion in the 2 direction becomes a constant. This is expected since far from the crack edge, the stress field should approach a one-dimensional state of stress. Figure 33 shows the calculation of the opening mode stress intensity factor from the plane strain crack opening diSplacement. Since the obtained displacements are independent of the 2 coordinate, the stress intensity factor shown is a constant across the thickness of the bar. In addition, note that the continuous correction for the changing conditions from plane strain to plane stress, as applied to the central crack problem, has no meaning for the crack opening diSplacement of Figure 27. As a consequence, Figures 33 and 30 each contain only a single curve. The results of Figure 33Ilead to a stress intensity factor of 0.00 oo’\/g: Brown and Srawley in reference (2) report a plane elasticity stress intensity factor of 2.05 OoflV/;_ 205 which is lower than that reported for the central crack problem. In view of the fact that the crack opening diSplacements and normal stress distributions at correSponding locations are considerably higher for the double edge crack problem than for the central crack problem, the value of 4.00 OOAv/g- seems to be the more realistic solution for this stress intensity factor. Tables 15 through 17 show selected values of the diSplace- ments obtained in the computations. Table 15 shows that in the crack plane, the x-directional displacements are outward while in the other planes along the y-axis they are inward. Table 16 shows that all the y-directional displacements are extensional and that the crack opening displacement is essentially constant across the thickness. Table 17 shows that in the z-direction only contraction is possible which is maximum on the surface of the bar. CHAPTER 5 SUMMARY AND CONCLUSIONS The line method of analysis was investigated for the solu- tion of coupled partial differential equations which were sub- ject to coupled and mixed boundary conditions. The use of this method was illustrated by solving the Navier-Cauchy equations of elastic equilibrium for a number of mixed boundary value problems in three-dimensional elasticity. Problems in both rectangular Cartesian and cylindrical coordinates were investigated. The application of the line method to the Navier-Cauchy equations in Cartesian coordinates led' to coupled sets of ordinary differential equations with constant coefficients. In cylindrical coordinates, this same solution technique results in coupled sets of ordinary differential equations, some of which have variable coefficients. Analytical methods, in conjunction with a successive approximation procedure, were used to obtain the solution of these resulting ordinary differential equations. One advantage of solving directly for displacements in solids containing geometric singularities is that the displace- ments are not singular. In addition, stresses are expressed in terms of first order partial derivatives only, which minimizes 206 207 inherent inaccuracy in higher order numerical differentiation. It is for this reason, that numerical solution of displacement potentials or the Galerkin vector should be avoided since the stresses are expressed as second and third derivatives of these quantities respectively. The advantage of the line method over other numerical solutions is that it minimizes the required numerical differentiations and thus, it may be considered as a semi-analytical approach to the solution of a problem. Stress and displacement distributions were calculated in two rectangular bars, one of which contained a through-thickness central crack while the other had double edge cracks. The need for these specific solutions has existed for a number of years in fracture toughness testing. As expected, the results of the central crack problem indicate that near the center of the given geometry solid, the conditions are approximately in a state of plane strain. As one proceeds from the center of the bar to its surface in the thickness direction, plane stress conditions are approached. Hence, displacements are maximum near the surface of the bar while normal stresses are maximum near its center. The singular nature of the normal stresses near the crack edge was established and three-dimensional stress and diSplacement distri- butions were successfully calculated. An equivalent plane strain stress intensity factor for three—dimensional problems was intro- duced. Similar results are reported for a double edge crack bar 208 which indicate that changes along the thickness direction in the displacements parallel to the applied load are less significant. The calculated normal stress distributions, however, led to identical conclusions in this case as for the central crack problem. Solutions in cylindrical coordinates were obtained for an annular plate containing internal surface cracks. The axisym- metric problem of a solid cylinder with a penny shaped crack was used to check the convergence and accuracy of this method. Results with good accuracy were obtained even from the use of a relatively coarse grid. The stress and displacement solutions of the above examples show that the method of lines provides a simple and systematic approach to the solution of some three- dimensional, mixed boundary value, elasticity problems. At this time, some improvement in the solution techniques and the use of the computer for Cartesian coordinate problems may be indicated. Since the resulting ordinary differential equations are readily solved by the normal mode method, the numerical computations may be minimized by performing the succes- sive approximation calculations in principal coordinates. Mani- pulations of diagonalized matrices should minimize both the round-off and inherent error which necessarily arise in all numerical computations. In addition, considerable savings in the cost of the required computer time will be possible. LIST OF REFERENCES 10. LIST OF REFERENCES Srawley, J. E. and Esgar, J. 8.: Investigation of Hydrotest Failure of Thiokol Chemical Corporation 260 Inch Diameter SL-l Motor Case, Technical Memorandum, NASA TMX-ll94, January 1966. Brown, W. F., Jr. and Srawley, J. E.: Plane Strain Crack Toughness Testing of High Strength Metallic Materials, ASTM Special Technical Publication No. 410, 1966. Ludwig, P. and Scheu, R.: Stahl u. Eisen. 43, 999, 1923. . Lur'e, A. I.: Three-Dimensional Problems of the Theory of Elasticity. Interscience Publishers, 1964. Paris, P. C. and Sih, G. C.: Stress Analysis of Cracks, Fracture Toughness Testing and Its Applications. STP No. 381, ASTM, pp. 30-81, 1965. . Irwin, G. R.: Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate. J. of App. Mech., 1957. Sack, R. A.: Extension of Griffith's Theory of Rupture to Three Dimensions. Proc. Phys. Soc. (London), 58, pp. 729-736, 1996. Sneddon, I. N.: The Distribution of Stress in the Neighbor- hood of a Crack in an Elastic Solid. Proc. Roy. Soc. A-187, pp. 229—260, 1006. Sadowsky, M. A. and Sternberg, E.: Three Dimensional Solution for a Stress Concentration Around a Circular Hole in a Plate of Arbitrary Thickness, J. of Appl. Mech., 16, 149, 1949. Green, A. E. and Sneddon, I. N.: The Distribution of Stress in the Neighborhood of a Flat Elliptical Crack in an Elastic Solid. Proc. Cambridge Phil. Soc., 46, 159, 1950. 209 ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 210 Smith, F. W.: Stresses Near a Semi-Circular Edge Crack. Ph.D. Dissertation, University of Washington, 1966. Alavi, M. J.: Stresses Near a Circular Crack in a Half Space. Ph.D. Dissertation, Colorado State University, 1968. Kassir, M. K. and Sih, G. C.: Three Dimensional Stress Distribution Around an Elliptical Crack Under Arbitrary Loadings. J. of Appl. Mech., Transactions of ASME, Sept. 1966. Kassir, M. K. and Sih, G. C.: Geometric Discontinuities in Elastostatics. J. of Math., Vol. 16, No. 9, 1967. Segedin, C. N.: Some Three—Dimensional Mixed Boundary Value Problems in Elasticity. Department of Aeronautics and Astronautics, University of Washington, June 1967. Shah, R. C.: Stresses and Stress Intensity Factor for Embedded Hyperbolic and Parabolic Cracks. Ph.D. Thesis, University of Washington, 1966. Hartranft, R. J. and Sih, G. C.: The Use of Eigenfunction Expansions in the General Solution of Three Dimensional Crack Problems. J. Math. Mech., 19, 123, 1969. Irwin, G. R.: Crack Extension Force for a Part Through Crack in a Plate. J. of Appl. Mech., December 1962. Orange, T. W., Sullivan, T. L. and Calfo, D. P.: Fracture of Thin Sections Containing Through and Part-Through Cracks. NASA TM x-52700, 1070. Kuhn, P.: Residual Tensile Strength in the Presence of Through Cracks or Surface Cracks. NASA TN D-5432, 1970. Hartranft, R. J. and Sih, G. C.: An Approximate Three- Dimensional Theory of Plates with Application to Crack Problems. Technical Report No. 7, Lehigh University Institute of Research, May 1968. Sih, G. C. and Hartranft, R. J.: Variations of Strain Energy Release Rate with Plate Thickness. NASA Grant NGR-39~OO7-025, to be published in Int. Journ. of Fract. Mech., 1972. 23. 25. 26. 27. 28. 29. 30. 31. 211 Sih, G. H., Williams, M. L. and Swedlow, J. L.: Three- Dimensional Stress Distribution Near a Sharp Crack in a Plate of Finite Thickness. California Institute of Technology, Technical Report AFML-TR-66-242, November 1966. . Walker, Jr., G. E.: A Study of the Applicability of the Method of Potential to Inclusions of Various Shapes in Two and Three-Dimensional Elastic and Thermo-Elastic Stress Fields. Ph.D. Dissertation, University of Washington, 1969. Cruse, T. A., Van Buren, W.: Three-Dimensional Elastic Stress Analysis of a Fracture Specimen with an Edge Crack. Carnegie-Mellon University, Department of Mechanical Engineering, Report SM-2l, January 1970. AyreS, D. J.: A Numerical Procedure for Calculating Stress and Deformation Near a Slit in a Three-Dimensional Elastic-Plastic Solid. NASA TM X-52440, June 1968. Gallagher, R. H., Padlog, J. and Bijlaard, P. P.: Stress Analysis of Heated Complex Shapes. J. Am. Rocket Soc., 32, pp. 700-707, 1962. Zienkiewicz, O. C., Irons, B. M., Ergatoudis, J., Ahmad, S. and Scott, F. C.: Iso-Parametric and Associated Element Families for Two- and Three-Dimensional Analysis, Finite Element Methods in Stress Analysis. Tapir, Technical University of Norway, Trondheim, 1969. Mikhlin, S. G. and Smolitskiy, K. L.: Approximate Methods for Solution of Differential and Integral Equations. American Elsevier Publishing Company Inc., New York, 1967. Faddeva, V. N.: The Method of Lines Applicable to Some Boundary Problems. Akademia Nauk SSR. Matematicheski Institut im V. A. Sleklova, V. 28, 1949. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York, 1962. Irobe, M.: Method of Numerical Analysis for Three-Dimensional Elastic Problems. Japan National Congress of Applied Mechanics, 16th, University of Tokyo, October 19, 1966, Proceedings. Central Scientific Publishers, Tokyo, pp. 1-7, 1968. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 00. 45. ”60 212 Carnahan, B., Luther, H. A. and Wilkes, J. 0.: Applied Numerical Methods. John Wiley and Sons, Inc., 1969. Frazer, R. A., Duncan, W. J. and Collar, A. R.: Elementary Matrices and Some Applications to Dynamics and Differenm tial Equations. Cambridge at the University Press, 1963. Frame, J. 8.: Matrix Functions and Applications. IEEE Spectrum, March-July 1964. Frame, J. S. and Needler, M. A.: State and Covariance Extrapolation from the State Transition Matrix. Proc. of the IEEE, Vol. 59, No. 2, February 1971. Halmos, P. R.: Finite Dimensional Vector Spaces. Van Nostrand, Princeton, 1958. Rutishauser, H.: Solution of Eigenvalue Problems with the LR-Transformation. National Bureau of Standards, Appl. Math. Series, 49, pp. 47-81, 1958. Doust, A. and Price, V. E.: The Latent Roots and Vectors of a Singular Matrix. The Computer Journal, A Publicam tion of the British Computer Society, Vol. 7, No. 3, pp. 222—227, 1964. Wylie, C. R. Jr.: Advanced Engineering Mathematics. McGraw- Hill Book Company, Inc., Second Edition, 1960. Hildebrand, F. B.: Advanced Calculus for Engineers. Prentice-Hall, Inc., Eighth Printing, 1958. Sneddon, I. N.: Crack Problems in the Mathematical Theory of Elasticity. North Carolina State College, Department of Mathematics and Engineering Research, Raleigh, 1961. Mendelson, A., Gross, B. and Srawley, J. E.: Evaluation of the Use of a Singularity Element in Finite-Element Analysis of Center«Cracked Plates. NASA Technical Note E-6680, 1972. Lancaster, P.: Theory of Matrices. Academic Press, 1969. Scheid, F.: Theory and Problems of Numerical Analysis. Schaum's Outline Series, McGraw-Hill Book Company, 1968. Harris, D. L.: Numerical Methods Using FORTRAN. Charles E. Merrill Books, Inc., Columbus, Ohio, 1960. APPENDIX A EVALUATION OF THE COEFFICIENT MATRIX EIGENVALUES AND EIGENVECTORS A close investigation of equation (2.32) shows that the coefficient matrix [KX] can be decomposed into component 2X2 matrices having the following tri-diagonal format: I“ _ 2 -2 -1 2 -l -l 2 -1 \ \ \\ \\ \\ A.1 \\ \\ -1 2 —1 2 2 I. I I 1 MxM It is a Simple matter then to find the eigenvalues and eigene vectors of this type of matrix. Noting that in equation (2.32) we have NZ rows of submatrices each of order NY, we express the coefficient matrix as 213 214 “8.3 = k, [Ill (9 [K11 + 031K221 ® 1121 (02) 2X2 NZxNZ NYXNY NZxNZ NYxNY where <:) denotes the Kroenecker product of two matrices (37). Matrices [Kl] and [K2] have the desired form of (A.l) but are of different order. Associated with the matrices [K1] and [K2] are the following two eigenvalue problems: [Kl] {X1} = utXl} (A 3) NYxNY NYxl NYxl [K2] {X2} = 6{X2} (A.0) NZxNZ Nle Nle where uj, j and 6i, i = l, 2, . . ., NZ represent the eigenvalues of [K2]. 1, 2, . . ., NY denote the eigenvalues of [K1] The original eigenvalue problem associated with the coefficient matrix [Kx] can be written as [xx] 1x3} = 11x3} (A.5) 2x1 2x1 2x1 After some matrix manipulations involving Kroenecker products (37), it can be shown that the eigenvalues le and the corre- sponding matrix of eigenvectors {X3113 can be expressed in terms of the component matrix eigenvalues and eigenvectors. The 215 results of these manipulations are A1] = kadi + k2uJ (A.6) {x3113 * = {x211 69 {x113 (A.7) 1x2 NZxNZ NYXNY where i = 1, 2, . . ., NZ and j = 1, 2, . . ., NY. Equations (A.6) and (A.7) reduce the problem of (A.5) to that of finding the eigenvalues and eigenvectors of (A.1). The eigenvalues of the tri-diagonal matrix (A.l) can be obtained by using difference equation theory. Let us consider th the case when the eigenvalues are denoted as uj. Then the j difference equation can be written as Xj+l + (u-2)xj + xj-l = O (A.8) We define 2p 2 u—2, where by Gersgorin theorem on bounds of eigenvalues (04), we must have [pl : 1. Equation (A.8) can now be written as a linear, second order difference equation with constant coefficients, that is x + 2px. + x. = O (A.9) j-l 3 3+1 Following standard solution techniques (45), we assume that x = aj (A.10) 216 Substituting equation (A.10) into equation (A.9) and noting that [pl 2 l we find that the values of a are given by 01 = cos 0 + i Sin 0 = el¢ - v- - _ ~10 02 - cos 0 - 1 Sln ¢ - e where cos 0 = - p H‘ P U) P :3 '9- | I H" ”O I [.1 'T = . /-1 The solution of (A.9) can now be written as “j = Ate”)j + 002‘”)3 (A.11) or x:j = Acosjp-I-E'i'sin jo Constants A and B can be evaluated from the following boundary conditions: X6 = x2 (A.12) XM-l = Xn+1 where for the matrix [Kl], M 2 NY. Applying equations (A.12) to equations (A.1l) gives 217 [(e10)2 - 11A + [(e‘i‘i)2 - 110 ll 0 (A.13) (61¢1M’l[2 - 110 I O For a non-trivial solution of equations (A.13), the determinant of the coefficients must vanish. This condition leads to the following characteristic equation: e = . 2nn Sin -——- M-1 01 Since equations (A.13) are linearly dependent, we may select a convenient value for A1 [022] = [f2(rn + g, [allin [023] = [f2Crn + 23 [allJn [QQQJ = [f2(rn + h, [allln In the above formulas, increment and n path of integration“ + 2£Qll]’ [QQlJn + g£QQl])] h h + hElel. [QQlJn + th231>J h h + §{Qll], [921]“ + 3£Q21J>J h + §{Q22])] h + Eiqizl’ [Q213n + thl3J, [agljn + th233>J (B. (B. h is the arbitrary integration denotes the instantaneous position along the C10) ell) 012) 13) la) APPENDIX C Appendix C includes a copy of each of the computer programs prepared for the five numerical examples presented, Since the work was performed on an IBM-360 digital computer, the enclosed listings use Fortran IV algebraic type language: A brief de~ scription of the required inputs, subroutine functions and a short flow diagram for each problem is also given; C.l Solid Cylindrical Bar With a Penny Shaped Crack A schematic representation of the main subroutines of the computer program is shown in Figure 350 The program is divided into five parts four of which are called from the main program” The main program is denoted as 85 while the subroutines are designated as 31, 82, 83 and Set Lubroutine Sl — LKCS uses the Runge~Kutta algorithm to calculate the diagonaiired radial matrix functions. Subroutine 33 - PCT generates the needed first deri- vatives tor the kunge—Kutta solution“ Subroutine 82 « MATle uses the Gauss-Jordan maximum pivot strategy (33) to numerically generate all the required matrix inversesv Subroutine 84 - MAPOWR evaluates the required matrix series for the solution of the constant coeificient differential equations. The itqnit for each {multine it; typed ifltit the beflrHthg of 228 Main program - SS Calls other parts of program Evaluates eigenvalues, modal matrix stores all required matrix functions Performs successive approxi- mation calculations, evaluates coupling vectors, initial value vectors, particular integrals Generates and prints all output Subroutine Sl - RKGS Subroutine $3 - FCT Calculates diagonalized radial Generates required first matrix functions derivatives Subroutine SZ - MATINV Generates matrix inverses as needed Subroutine S4 - MAPOWR Entry MASER Calculates axial matrix functions Entry MXTRA Figure 35. - Schematic representation of the solid cylindrical bar with a penny shaped crack computer program. 229 each part using the word DATA for identification. The eigen- values oi the radial and the elements of the axial coefficient matrices are given in SS and SH respectively. The enclosed program is generalized in that the grid size can be modified by suitable changes in the DATA, COMMON and DIMENSION statements only. The successive approximation procedure is performed in the main program 85 and the symbol KTR is used to follow the number of repeated calculations. Information about matrix function error checks, singularity of matrices and number of iterations is printed at the terminal while the bulk of the output is being stored in the computer for a more efficient printing operation. The output for this problem includes the displacements {u} and {w}, their derivatives {u} and {Q} and the stresses {°r8}’ {deg}, {028} and {orzs}. 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The input and output data are handled identically to those for the solid cylindrical bar and only the initial value vectors that are derived from given boundary conditions are differentc This program, in addition to the axial load C2, includes the possibi~ lity of having an outside surface load of CS0 243 taAnfi N 00 OOo0n4 z.AuA 0 DO AfioAOXHAfiaA4> I.AnA A DO fiuA54x I.Aufi A DO Alzuzz III": onoAMx \w.MIOO.A\MaDD4.h4wO 4h4O AMA.MA.3 .AMADz .AMA.MA4> .Azz:.22z0x ZDAMZMIAO h4m0.a.»Mmh.4.3.>.x ZDAMAOw¢a w4mDDO szx.x. >2Ah4z szhaommaM 02m zmahmm MOZAHZDD 000.0\A0400#I+A34NDuAfl4ND OOOoo\AMOMo§I+AO0ADuAO0AD A00¢D+AGDODuAfiAOD A04M0+A00mDuAO0MD dz.Aufi M 00 A4mA.Ox0 h0m 4440 II+Oxu0x A04¢D§I+AO0NDnAfiva A04M0§I+AO0ADuA00hD A04¢D%OOO.N+A040DNAMOOD A34MD*OOO.N+Afi.mDuAfi.mD «Z.Aufl 4 DO A4mA.Ox. hum 4440 A34¢D§000.N+AODODuAOOQD A34M04000oN+A00mDuA14MD A00¢D§II+A50NDHAODMD A04M0*II+A1.ADqu.hD mz.Auw M DO A4mA.Ox. h0m 4440 II+0xqu A1440§II+A04NDuAODMD AODMD*II+A04ADuAfi.hD Afi.¢DuA5.0D AMOMDuAfiva mz.Aufi N 00 A4mA.0x0 h0u 4440 AMDNDuAWDMD AfivADuAfiDND dz.Anfi A DO 30.ANA 0 DO OOO.N\Iqu wx\aIuI AMA4ND.AMA4AD.AMADOD.AMA4MD ZDAMZMIAO AMA4mD.AMADND.AMA4¢DaAMA4Mo \m020m\ ZDIZDO \AO¢.0.¢O.NIOMMMMMMMMMMMMMMM.0.MAx fix.fifi.mx.mz 4h4O ANID.II44 0444mm hAOA4QIA A4uA.0x.ND.AD. 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The schematic representation of these four parts for the bar with a through-thickness central crack computer program is shown in Figure 37. The main program is denoted as CCRAG while the subroutines are designated as CBITT, CNINV and CVECT. Sub- routine CBITT-EIGEN calculates the eigenvalues, eigenvectors, modal matrices and the matrix functions required in the x-y-z directional differential equation solutions. Subroutine CMINV- MATINV uses the Gauss-Jordan maximum pivot strategy to numeric- ally generate all the required matrix inverses. Subroutine CVECT—VECTOR is used to evaluate the coupling vectors in the x-y-z directions respectively. The input for each routine is typed in at the beginning of each part using the DATA statements. Note that the subroutines CBITT-EIGEN and CVECT-VECTOR are each called three times, with the arguments of the call vector defining +he coordinate direction along which the variables are evaluated. The program is com~ pletely general in that the dimensions of the bar or the selected grid increments can be changed by suitable changes in the DIMENSION, DATA and COMMON statements only. The program includes a large number of PRINT statements which are used to follow the execution of the program at the conversation terminal. Note that 280 Main program - CCRAG Calls other parts of program Stores all required matrix functions Performs successive approximation calculations, evaluates initial value vectors, particular integrals Generates and prints all output Subroutine CBITT - EIGEN Calculates eigenvalues, eigenvectors, modal matrices and required matrix functions in x-y-z directions Subroutine CMINV - MATINV Generates matrix inverses as needed Subroutine CVECT - VECTOR Evaluates coupling vectors in x-y-z directions Figure 37. - Schematic representation of the bar with th mug-thickness central crack computer program. 281 most matrix multiplications in this program are performed in a column by column operation which is done solely to minimize the paging in the IBM-360 computer. The output for this problem includes the displacements after each iteration and their derivatives {u}, {0}, {w}, and the normal stresses {ox}, {0y} and {oz} after convergence of the successive approximation procedure has been obtained. Information about matrix function error checks, singularity of matrices and number of iterations is also displayed at the terminal. 282 x20—Ifi 000 00 00000000 55—00 x20—n— 000 00 00000000 55—00 X0\——.fi—¢I——.5—>—a 000 000N0000 55—00 xz.—nfi 000 00 000—0000 55—00 x20—I— 000 00 00000000 55—00 —.00 0004 00 020 .0542100 —00N 00000000 55—00 —00~ 52—ma 00000000 55—00 .00000000—00000000—52205250a 00 00050000 55—00 —+5Zfiu523 00000000 55—00 ——l20540400*~0u00 00000000 55—00 >20—uz 00 00 00000000 55—00 ——lt—540000§—0u00 00000000 55—00 ~20—ut 00 00 000N0000 55—00 0u523 000—0000 55—00 —.521—40*0000~051000I.5zx—40 00000000 55—00 —.~000000l0000—.§N0+———0.0000|0000——*—0I—5zx—40 00000000 55—00 —+5zxu5zx 00000000 55—00 ——Ifi—540400§>0u~0 00050000 55—00 >20—Ifi 00 00 00000000 55—00 ——I——540400*N00—0 00000000 55—00 N20—u— 00 00 00000000 55—00 ou5zx 00000000 55—00 000— I——.—.—0—4x 5 000N0000 55—00 xt.—u— 5 00 000—0000 55—00 0000 I——0—.1004x 0 00000000 55—00 xt.—ufi 0 00 0000N000 55—00 xt0—I— 0 00 0000~000 55—00 0000 u——.—.w.~4x 0 0005~000 55—00 xto—Ifi 0 00 0000~000 55—00 x10—u— 0 00 0000~000 55—00 0000 u4—0—050—4x N 0000~000 55—00 xt0—Ifi N 00 0000~000 55—00 xx.—I— N 00 00000000 55—00 ——t~20540000\—auN0 000—~000 55—00 ——z>z0540400\—nu>0 0000~000 55—00 —~8~2§—t>20540400nx0 0000—000 55—00 —INzu—th 0000—000 55—00 —I>Zu—t>z 0005—000 55—00 —Ixzu—xxz 0000—000 55—00 N4u~0 0000—000 55—00 —4u—0 0000—000 55—00 II“: 0000—000 55—00 0fiux: 000N—000 55—00 00u~z 000——000 55—00 Nfiu>z 0000—000 55—00 —finxz 00000000 55—00 \00005000000N00—0—00\ —Q 4540 00000000 55—00 .0000005N00—0000005—0 20—020x—0 00050000 55—00 .0000000500—030030—x 20—020t—0 00000000 55—00 .000400.00040..00.00.05Q..00.00.N5m0.000000—5Q..00.00.>—a N 00000000 55—00 I0—00000000.00000..000000—00000..00.00.00.—00.000~00.000000—0 — 00000000 55—00 I..—fi.01.0fi.04x..—fi.0fi.0fi0N4x.——fi00fi.0fi0—4x.—000000xx0 20—0202—0 00000000 55—00 .NI001i40 004400 5—0—aax— 000N0000 55—00 .40—003.000N00—fi.~40—4.II.—x.04x.~4x.—4x0200—0 02—5000000 000—0000 55—00 Qflhfifi—lfifldmmmfihfifiao 283 .—.00§—00—0>—0u—00—.—50 xx0—u— 00 00 x20—u1 00 00 —.N0 0000 00 020 ..5n—fi000 N0 05 00 .0.00\0t05u—0000 00 05 00 .—0Ow0fi. u— atm5i—5040u.fi.00 —.1.40_IZ—00u0205 .—fl.00.10000u—0000 xtu—ufi N0 00 .3.40#>u—7040 x10—nfi —0 00 ————5 —txz0—u—— 0— 00 ..N—0 0004 00 020 ..5<2000 000w 52—00 0000N*——Qt.7ax—>—0u——at0fiat0>—a fi+>2nfiax >xto—ufi N—0 00 ~+>Zfl~0t >xt0—u— N—0 00 >ZINIXZH>XI 0000N\——02000x0>—au——0t05020>—a fi+zufiat >20—ufi ——0 00 —+In—Qx >20—u— ——0 00 0000~\.—01030>—0u.—0I.fi.>—a >20—u7 0—0 00 —+Zu—at >20—u— 0—0 00 0000N\.—00010>—0I——00020>—0 0+Iufia! 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