DETERMINATION OF HIGHER ORDER COEFFICIENTS FOR A CRACK PARALLEL TO AN INTERFACE

Transcription

1 Clemso Uiversity TigerPrits All Theses Theses 8- DETERMINATION OF HIGHER ORDER COEFFICIENTS FOR A CRACK PARALLEL TO AN INTERFACE Vekata lakshma Kolluru Clemso Uiversity, lakshma.kvl@gmail.com Follow this ad additioal works at: Part of the Egieerig Mechaics Commos Recommeded Citatio Kolluru, Vekata lakshma, "DETERMINATION OF HIGHER ORDER COEFFICIENTS FOR A CRACK PARALLEL TO AN INTERFACE" (). All Theses This Thesis is brought to you for free ad ope access by the Theses at TigerPrits. It has bee accepted for iclusio i All Theses by a authorized admiistrator of TigerPrits. For more iformatio, please cotact kokeefe@clemso.edu.

2 DETERMINATION OF HIGHER ORDER COEFFICIENTS FOR A CRACK PARALLEL TO AN INTERFACE A Thesis Preseted to the Graduate School of Clemso Uiversity I Partial Fulfillmet of the Requiremets for the Degree Master of Sciece Mechaical Egieerig by Vekata Lakshma Kolluru August Accepted by: Dr. Paul F. Joseph, Committee Chair Dr. Loy L. Thompso Dr. Gag Li

3 ABSTRACT A geeral method based o the sigular itegral equatios is developed to computatioally determie the higher order coefficiets i mixed mode fracture mechaics. These k ad T coefficiets are defied with respect to a polar coordiate system cetered at a crack tip, ad give asymptotic expressios for stresses ad displacemets accordig to the William s eigefuctio expasios, (r, ) ij (r) k I f Ik ij (, ) k f II IIk ij (, ) I IT I IT II IIT ij ij ij T f (, ) ( r) T f (, ) T f (, ), i r, ; j r,, I the above expressio the = terms correspod to the modes I ad II stress itesity factors ad the so called, T-stress. From a method poit of view, the higher order k-coefficiets are easily obtaied, while the T-coefficiets require sigificat postprocessig of the sigular itegral equatio solutio. A plaar crack parallel to a iterface betwee two elastic materials ad subjected to far-field tesio is cosidered as a example ad extesive results are preseted. This example is chose due to the aomalous behavior of a closig crack tip as the crack approaches the iterface for certai material combiatios. Such Comiou cotact zoes occur eve i a tesile field whe the crack is withi a critical distace from the iterface. Numerous results are provided that compare the asymptotic solutios with that of the full-field. It is show that up to four k-coefficiets ad may T-coefficiets ca be determied for h/a =., where h is the distace of the crack from the iterface ad a is the half-crack legth. While the applicatio of the method to the case of a crack parallel ad very close to a iterface focuses o the aomaly of a closig crack tip, i geeral the ability to determie ii

4 higher order coefficiets ca be used to quatify the size of the zoe i which liear fracture mechaics is valid. iii

5 DEDICATION To my parets Mr. Kolluru Sri Rama Sayasi Setty ad Mrs. Kolluru Vekata Rata Kumari. iv

6 ACKNOWLEDGMENTS I would like to thak my advisor, Dr. Paul F. Joseph, for his guidace ad support throughout the thesis. I also thak my committee members Dr. Loy L. Thompso ad Dr. Gag Li, for their patiece ad uderstadig. v

7 TABLE OF CONTENTS Page TITLE PAGE... i ABSTRACT... ii DEDICATION... iv ACKNOWLEDGMENTS... v LIST OF TABLES... viii LIST OF FIGURES... ix CHAPTER I. INTRODUCTION... II. FREDHOLM KERNELS AND EXPRESSIONS FOR STRESS AROUND CRACK TIP Fredholm Kerels Expressios for stresses i all three regios aroud crack tip... 5 III. IV. KERNELS FOR A CRACK PARALLEL TO AN INTERFACE AND THE COMPUTATIONAL DETERMINATION OF HIGHER ORDER TERMS... 3 RESULTS: STRESS INTENSITY FACTORS, T-STRESS COEFFICIENTS AND THE STRESS FIELD AROUND THE TIP OF A CRACK THAT IS PARALLEL TO AN INTERFACE Covergece Study Asymptotic coefficiets as a fuctio of the bi-material costats Crack Opeig Displacemet ad Comiou Cotact Zoes Stresses alog the lie of the crack Full-field polar plots for the stresses aroud the crack tip Compariso of full-field ad asymptotic solutio aroud the crack tip vi

8 Table of Cotets (Cotiued) V. SUMMARY AND CONCLUSION... APPENDICES... 4 A: Appedix A... 5 B: Appedix B... 9 REFERENCES... vii

9 LIST OF TABLES No table of figures etries foud. Table Page 4. Coverged asymptotic coefficiets defied by Equatios (3-4, 3-43, 3-56, 3-65) for a/h =., =.48, = Covergece study of asymptotic coefficiets with respect to N defied i Equatio (3-9) for a/h =., =.98, = Same as Table 4. for a/h = Covergece of xx i frot of the crack tip with respect to the umber of terms as defied i (4-) for a/h =., =.98, =.495. The percet error measure, E M, is defied i (4-) Covergece of xx o the upper crack surface with respect to the umber of terms as defied i (4-3) for a/h =., =.98, =.495. The percet error measure, E M, is defied i (4-) viii

10 LIST OF FIGURES Figure Page. Two boded dissimilar materials with a crack parallel to the iterface of two materials Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= Cotour plot for Mode-I T coefficiet whe = ad h/a= Cotour plot for Mode-II T coefficiet whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= Cotour plot for Mode-I T coefficiet whe = ad h/a= Cotour plot for Mode-II T coefficiet whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= Cotour plot for Mode-I T coefficiet whe = ad h/a= Cotour plot for Mode-II T coefficiet whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= Cotour plot for Mode-I T coefficiet whe = ad h/a= Cotour plot for Mode-II T coefficiet whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= ix

11 List of Figures (Cotiued) Figure Page 4.9 Cotour plot for Mode-I T coefficiet whe = ad h/a= Cotour plot for Mode-II T coefficiet whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= Cotour plot for Mode-I (T coefficiet)/ whe = ad h/a= Cotour plot for Mode-II (T coefficiet)/ whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Cotour plot for Mode-II k coefficiet whe = ad h/a= Cotour plot for Mode-I T coefficiet whe = ad h/a= Cotour plot for Mode-II T coefficiet whe = ad h/a= Cotour plot for Mode-I (k coefficiet)/ whe = ad h/a= Cotour plot for Mode-II (k coefficiet)/ whe = ad h/a= Cotour plot for Mode-I (T coefficiet)/ whe = ad h/a= Cotour plot for Mode-II (T coefficiet)/ whe = ad h/a= Cotour plot for Mode-I k coefficiet whe = ad h/a= Close up view of the left side of Figure Close up view of the right side of Figure 4.33 which shows a regio i the upper right had corer were the mode I stress itesity factor is zero Cotour plot for Mode-II k coefficiet whe = ad h/a= Close up view of the left side of Figure x

12 List of Figures (Cotiued) Figure Page 4.38 Close up view of the right side of Figure Cotour plot for Mode-I T coefficiet whe = ad h/a= Close up view of the left side of Figure Close up view of the right side of Figure Cotour plot for Mode-II T coefficiet whe = ad h/a= Close up view of the left side of Figure Close up view of the right side of Figure Comiou cotact zoe at the tip of a iterface crack A Comiou cotact zoe of legth, ad its value as a fuctio of the Dudurs parameter, Comiou cotact zoes for a crack parallel to a iterface Normalized crack opeig displacemet for a rage of h/a for the material pair, =.98, = A close-up view of the ormalized crack opeig displacemet showig that for the material pair, =.98, =.495, the case of h * from Figure 4.47 has bee obtaied Aother close-up view of the ormalized crack opeig displacemet without the square root weight fuctio that shows more clearly that the case of h * from Figure 4.47 has bee obtaied xi

13 List of Figures (Cotiued) Figure 4.5 Normalized σ xx stress o upper surface of the crack for differet h/a Page ad a material pair correspodig to α=.98, β=.495. The figure o the right is a close-up view of the crack tip Same as Figure 4.5 for the lower crack surface Normalized σ xx stress i frot of the crack for differet h/a ad a material pair correspodig to α=.98, β=.495. The figure o the right is a close-up view of the crack tip Same as 4.53 for the shear stress, τ xy Same as 4.53 for yy Full field solutio for σ xx o the upper surface of the crack compared to the asymptotic solutios for h/a =.,.,. ad Same as Figure 4.56 for the lower crack surface Full field solutio for σ xx i frot of the crack compared to the asymptotic solutios for h/a =.,.,. ad Full field solutio for xy i frot of the crack compared to the asymptotic solutios for h/a =.,.,. ad Full field solutio for yy i frot of the crack compared to the asymptotic solutios for h/a =.,.,. ad xii

14 List of Figures (Cotiued) Figure 4.6 Plot for ormalized stress σ rr / σ for h/a =. ad r/a=. Page ad α=.98, β= Plot for ormalized stress τ rθ / σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress σ θθ / σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress σ e /σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress σ rr /σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress τ rθ / σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress θθ /σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress σ e /σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress σ rr / σ for h/a =. ad r/a=. ad α=.98, β= Plot for ormalized stress τ rθ / σ for h/a =. ad r/a=. ad α=.98, β= xiii

15 List of Figures (Cotiued) Figure 4.7 Plot for ormalized stress σ θθ / σ for h/a =. ad r/a=. Page ad α=.98, β= Plot for ormalized stress σ e /σ for h/a =. ad r/a=. ad α=.98, β= Plot comparig the stress σ e /σ with the asymptotic solutio for h/a =. ad r/h=.3 ad α=.98, β= Plot comparig the stress σ e /σ with the asymptotic solutio for h/a =. ad r/h=.3 ad α=.98, β= Plot comparig the stress σ e /σ with the asymptotic solutio for h/a =. ad r/h=.3 ad α=.98, β= xiv

16 CHAPTER I INTRODUCTION A crack parallelig a iterface of two boded dissimilar liearly elastic ad isotropic materials is cosidered for discussio i this thesis. Cosider the followig figure. which defies the geometry. Figure. Two boded dissimilar materials with a crack parallel to the iterface of two materials I the above figure, a is the half-crack legth,, ad, are the material properties of the material- ad material- respectively. The material which is above the iterface ad has the crack is material-. The oe below the iterface is material-. The iterface lies parallel to the crack ad the crack is alog the X-axis. The ceter of the crack is cosidered as the origi. h is the distace of the crack from the iterface ad is

17 very small compared to the crack legth a. A tesile load of is applied ormal to the iterface. Whe the tesile load is applied, shear may develop at the iterface. To avoid the developmet of shear at the iterface, loads ad are applied parallel to the iterface which compesates ay shear occurred. A geeral ad accurate method is used to calculate the coefficiets of asymptotic expasio of stresses. Sigular itegral equatio approach is used to calculate the full field stresses ad they are compared. Stress itesity factor, K derived usig Liear Elastic Fracture Mechaics (LEFM), ca be used to characterize the crack tip coditios. Fracture ca be predicted by usig stress itesity factor ad ca be cosidered a material costat. If ay loadig ad geometry gives the same K, the material will respod i the same way with respect to crack growth. But there are may cases, where the stress itesity factor does ot accout for differet geometric o-liearities ad materials. The plastic zoe ear the crack tip is very small whe compared to other dimesios like crack legth (a). Liear Elastic Fracture Mechaics caot be applied whe the plastic zoe is more widespread. Elastic-Plastic Fracture Mechaics (EPFM) is applied i such a case which uses J-Itegral approach to defie the stress field at the crack tip. To accout for differet materials ad geometric o-liearity, the above two approaches might ot be sufficiet sometimes. I such a case higher order terms ca be used.

18 I the polar co-ordiate system, the stresses ad displacemets ca be expressed as asymptotic series for small distaces from the crack tip. The Eige fuctio expasio ear the crack tip accordig to Williams [] ca be expressed as (r, ) ij (r) k I f Ik ij (, ) k f II IIk ij (, ) I IT I IT II IIT ij ij ij T f (, ) ( r) T f (, ) T f (, ), i r, ; j r,, (.) I Ik II IIk I IT r r r r u ( r, ) ( r) k g (, ) k g (, ) T r g (, ) ( ) r T I IT (, ) gr T gr (, ) E cos F si, I Ik II IIk I IT u ( r, ) ( r) k g (, ) k g (, ) T r g (, ) I IT II IIT ( r) T g (, ) T g (, ) Gr E si F cos. 8 (.) (.3) where, r is the small distace from the crack tip, k ad T are coefficiets that deped o geometry ad loadig ad they are costat. Eve though oly a tesile load is applied, the problem has a mixed mode ature iduced because of the two materials beig boded. This behaviour ca be see from equatios (.), (.) ad (.3). f ad g are agular fuctios. The expasio of the agular fuctios for stress ad displacemets for mode I ad mode II are detailed i Appedix A. If LEFM is applied to a problem, it has two legth scales amely physical legth scale (r p ) ad mathematical legth scale(r m ). The physical legth scale defies the zoe i 3

19 which LEFM does ot cosider the pheomea. While mathematical scale ivolves the trucatio of the above series for acceptable level of error. A sigular itergratio equatio approach has bee adopted for the first time for a mixed mode problem to determie the higher order coefficiets although it has bee applied for mode I ad mode II cases separately. The amplitude of crack tip sigularity is defied by stress itesity factor. If the stress field aroud the tip is completely characterized by stress itesity factor, it is called sigularity domiated zoe. Literature Review: Larsso ad Carlsso [] first came up with the sigificace of T term i William s expasio. They studied plastic zoe ahead of crack tip usig fiite elemet aalysis for commoly used fracture specimes. They determied that T-stress is the differece betwee x solutios usig fiite elemets for differet specimes ad for boudary layer for elemets alog the crack surface. Leevers ad Rado [3] came up with a ratio of stress itesity factor ad T-stress as biaxiality ratio B, which was odimesioalised by geometric parameter like crack legth a. They came up with the importace of T-stress as a secodary fracture parameter whe two specimes are subjected to same stress itesity. Betego ad Hacock [4] used the modified boudary layer approach ad provided a elastic plastic fiite elemet solutio. They showed that a egative T-stress ear the crack tip reduces the stresses idepedet of radial distace from crack tip. Kfouri [5] used Eshelby s method to evaluate elastic T-term. This ivolves determiig 4

20 the T-terms usig cotour J-itegrals alog paths close to the crack tip for three differet geometries. This method is also suited for fiite elemet aalysis. Sham [6] developed higher order weight fuctios for calculatig power expasio coefficiets of a regular elastic field i a D body without body forces for both iterior poits ad crack tips. Sham [7] determied the elastic T-term usig higher order weight fuctios. Sham preseted values of T-term for sigle otched specimes subjected tesio loadig, pure bedig ad three-poit bed. The method to determie T-terms was based o fiite elemet methods. Chidgzey ad Deeks [8] determied the coefficiets usig scaled boudary fiite elemet method. Accordig to Chidgzey ad Deeks, if the scalig ceter is at the crack tip, the scale boudary fiite elemet solutio gives the Williams [] expasio ad so stress itesity factor ad T-stress ca be determied easily. Xiao ad Karihaloo [9] used hybrid crack elemet which allows the calculatio of higher order terms directly. Seed ad Nowell [] determied the T-stress usig distributed dislocatios method. The sigular itegral equatio approach is used to calculate the stress itesity factors ad T-stress by takig a example i which the crack is ormal ad iclied to a free surface of a half plae loaded by a far field tesio. Broberg [] determied the T- stress usig dislocatio arrays which gave more accurate results tha the fiite elemets. Che [] et al. followed crack frot positio ad crack back positio techiques to calculate the T-stress at crack tip usig complex variable fuctio. Che [3] et al. i aother study used the perturbatio method for a slightly curved crack. All the above work has bee doe to calculate higher order terms usig differet methods. 5

21 Erdoga [4] was the first to cosider a crack parallel to a iterface ad calculate stress itesity factor based o a set of itegral equatios. Equatios were solved for three adjoiig sets of materials with symmetric ad ati-symmetric uiform tractios o the crack surfaces. Hutchiso et al. [5] derived coditios for a crack to propagate parallel to a iterface betwee two boded dissimilar materials. The work shows how the stress itesity factors for such a problem ca be calculated if the loadig ad geometry are kow. Eglad [6] cosidered a crack alog the iterface of two materials. Whe equal ad opposite ormal pressure are applied o a crack there is aomalous behavior at crack tips. Accordig to Eglad, the upper ad lower surfaces of the crack wrikle ad overlap ear the crack tip which is ot physically possible. Comiou ad Dudurs [7] came up with a mathematical solutio which is a closed crack tip with a small cotact zoe. Gautese ad Dudurs [8] came up with a solutio to how to solve the itegral equatio exactly. They came up with simple formulae which calculate the legth of the cotact zoes ad Mode II stress itesity factor. Rice ad Sih[9] foud a method to determie Goursat fuctios for a iterface crack problem. It ivolves eigefuctio expasio ad complex fuctio theory. 6

22 CHAPTER FREDHOLM KERNELS AND EXPRESSIONS FOR STRESS AROUND THE CRACK TIP. Fredholm Kerels: I this chapter the Fredholm kerels for stresses i differet material zoes are derived. The eight ukows i Navier s equatios of elasticity, are reduced to two equatios with oly two ukows, horizotal ad vertical displacemets, u(x,y) ad v(x,y), respectively. The reduced equatios are d du dv u, dx dx dy (.) d du dv v. dy dx dy (.) where, 3 4 for plae strai ad 3 / for plae stress. Fourier trasforms are applied o equatios (.) ad (.).Whe the trasform is i x, the expoetial Fourier trasform is give by ix ix f (, y) f ( x, y) e dx, f ( x, y) f (, y) e d (.3) ad whe the Fourier trasform is i y, ix ix f ( x, ) f ( x, y) e dy, f ( x, y) f ( x, ) e d (.4) I this problem, the trasform is i y ad, Fourier trasforms become 7

23 ix ix u(, y) u( x, y) e dx, u( x, y) u(, y) e d, (.5) ix ix v(, y) v( x, y) e dx, v( x, y) v(, y) e d (.6) Applyig (.5) ad (.6) to (.) ad (.) coverts the partial differetial equatios to a costat coefficiet ordiary differetial equatio, ad that gives y u(, y) A ( ) ya ( ) e A ( ) ya ( ) e, (.7) 3 4 y y v (, y) i A ( ) y A ( ) e i A3 ( ) y A4 ( ) e (.8) The boudary coditios ivolve the stresses applied to the surface of the layer, so the Fourier trasforms are applied to the stress ad strai relatios 3 v u xx( xy, ), (.9) y x v 3 u yy ( xy, ), (.) y x (, ) u v xy xy y x. (.) Whe (.3) is applied to (.9),(.),(.), 3 y 3 xx, y i A A y e A3 A4 y e y. y (.) 8

24 y yy, y i A A y e A3 A4 y e y y, 3 4 xy y A A y e A A y e (.3) (.4) Whe the material properties for the three regios are applied, equatios (.7), (.8), ad (.), (.3) ad (.4) ca be writte as Regio : 3 y 3 xx, y i A A y e A3 A4 y e y, 3 4 xy y A A y e A A y e y yy, y i A A y e A3 A4 y e y, u y A ya e A ya e y y (.5) y (.6) y (.7) y 3 4 (.8) y y v, y i A y A e A3 y A4 e (.9) 9

25 Regio : 3 y xx, y i A5 y A6 e (.), y xy y A5 y A6 e (.) y yy, y i A5 y A6 e (.), u y A ya e y 5 6 (.3) y v, y i A5 y A6 e (.4) Regio 3: 3 y xx, y i A7 y A8 e (.5), y xy y A7 y A8 e (.6) y yy, y i A7 y A8 e (.7), u y A ya e y 7 8 (.8) y v, y i A7 y A8 e (.9)

26 The followig boudary coditios are applied to the correspodig equatios above to get the arbitrary itegral fuctios A through A 8 which are cumbersome ad are ot show here. u3 u, y v3 v, y, 3 yy yy y, 3 xy xy y xy xy, y h yy yy, y h u u, y h v v, y h (.3) (.3) (.3) (.33) (.34) (.35) (.36) (.37) Equatios for ( xy, ) ad ( xy, ) ca ow be writte as yy y y i x SS e SS e e d px i lim y xy (.38) y y i x SS e SS e e d qx i lim 3 4 y Where, (.39) SS A A y (.4)

27 SS A3 A4 y (.4) SS3 A A y (.4) SS4 A A y ad, a 3 4 (.43) i t, g t u3 u Ux g t e dt a a i t, f t v3 v Vx f t e dt a t (.44) t (.45) p x ( ) lim yy ( x, y) y q x ( ) lim xy ( x, y) y (.46) (.47) Solvig the equatios (.38) ad (.39) ad evaluatig them at y= ad makig use of the followig itegrals, si( ( )) t x e y t x d, (.48) y ( t x) y y( t x) e si( ( t x)) d, (.49) y ( t x) ( t x) 3 y ( t x) y e si( ( t x)) d (.5) y ( t x) y y e cos( ( t x)) d, (.5) y ( t x)

28 y y ( t x) e cos( ( t x)) d, (.5) y ( t x) y y 3( t x) y e cos( ( t x)) d, (.53) y ( t x) gives the followig sigular itegral equatios, b b b f() t ( ) (, ) ( ) (, ) ( ) t x a a a dt f t K x t dt g t K x t dt p x a x b b b b gt () ( ) (, ) ( ) (, ) ( ) t x a a a dt f t K x t dt g t K x t dt q x a x b (.54) (.55) Ad the kerels are, 4a h t x h ( t x) 8a h t x t x K x t a 4h t x 4h t x 4h t x a (, ) 3 4a h t x h ( t x) 8a h t x t x K x t a 4h t x 4h t x 4h t x a (, ) 3 (.56) 8a h 4h 3( t x) a h K x t K x t a 4h t x 3 (, ) (, ) 3 4h t x (.57) (.58) 3

29 Where, a, a (.59) A similar method as above is followed to derive 3 ( xy, ) ad the correspodig kerels. 3 Equatio for xy, xx ca be writte as, xx i y i x 3 lim 5 lim xx, y y SS e e d x y (.6) where, 3 SS5 A7 A8 y (.6) Equatio (.6) solved as above to get the followig sigular equatio, b b b f() t 3 dt f ( t) L( x, t) dt g( t) L( x, t) dt g( x) xx( x, ) t x a a a (.6) Ad the kerels are, 4a h t x h ( t x) 8a h t x 3 t x L x t a a 4h t x 4h t x 4h t x (, ) 3 8a h 4h 3( t x) 4a h 4 h ( t x) 3 h L x t a a 4h t x 4h t x 3 (, ) 3 4h t x (.63) (.64) 4

30 To derive the kerels of u 3 ( x, y) i y i x 3 lim 6 lim, y y SS e e d u x y (.65) where, (.66) SS6 A7 A8 y Takig a derivative with respect to x, ad evaluatig at y=, (.65) ca be writte as b b b 3 g( t) 4 v ( x, ) dt f ( t) M( x, t) dt g( t) M ( x, t) dt f ( x) t x a a a x (.67) where the kerels are 3 6ha 4h 3 t x 4h t x h (, ) 3 M x t ha 4h t x 4h t x 4 h t x M h t x 8h a t x 8h a t x t x ( x, t) 3 4h t x 4h t x 4h t x (.68) (.69). Expressios for stresses i all the three regios aroud the crack tip The equatio set from (.5) through (.9) are used to derive the kerels for differet stresses i differet regios. The boudary coditios (.3) to (.37) are applied to get the itegral fuctios as doe earlier. 5

31 Regio : Equatios for ( xy, ), ( xy, ) ad ( xy, ) ca be writte as, xx yy xy i y y i x { SS9e SS e } e d xx( x, y) (.7) i y y i x { SS7e SS8 e } e d yy ( x, y) (.7) y y i x { SS4e SS5 e } e d xy ( x, y) (.7) Where, 3 SS9 A A y 3 SS A3 A4 y SS7 A A y SS8 A3 A4 y (.73) (.74) (.75) (.76) SS4 A A y (.77) SS5 A A y (.78) 3 4 Solvig equatios (.7), (.7) ad (.7) ad makig use of the itegrals from (.48) to (.53), we get the followig itegral equatios 6

32 b a b ( ) f (, ) ( ) g (, ) xx(, ) a f t XX x t dt g t XX x t dt x y (.79) b a b ( ) f (, ) ( ) g (, ) yy (, ) a f t YY x t dt g t YY x t dt x y (.8) b a b ( ) f (, ) ( ) g (, ) xy (, ) a f t XY x t dt g t XY x t dt x y (.8) ad the kerels for regio are a3( t x) a4( h y)( h y)( t x) XX f ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) XX YY YY ( t x) y ( t x) a y ( t x) ( y ( t x) ) 4 a4h( h y) 3( h y) ( t x) ( t x) (.8) a (4 h y) ( h y) ( t x) 4 a h( h y)( h y) ( h y) 3( t x) a5 ( h y) 4 4 g ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) y y( y ( t x) ) a y ( t x) ( y ( t x) ) (.83) a6( t x) a4( h y) ( t x) 4 a4h( h y) 3( h y) ( t x) ( t x) f ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) ( t x) y ( t x) a y ( t x) ( y ( t x) ) (.84) a y ( h y) ( t x) 4 a h( h y)( h y) ( h y) 3( t x) a7 ( h y) 4 4 g ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) a y t x y( y ( t x) ) ( ( ) ) (.85) 7

33 XY XY a7 ( h y) 4 4 f ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) a y t x y( y ( t x) ) ( ( ) ) a y ( h y) ( t x) 4 a h( h y)( h y) ( h y) 3( t x) (.86) a6( t x) a4( h y) ( t x) 4 a4h( h y) 3( h y) ( t x) ( t x) g ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) ( t x) y ( t x) a y ( t x) ( y ( t x) ) (.87) The above equatios ca be ormalized usig b a b a, b a b t r x s a (.88) f ( t) f ( r), g( t) g( r) (.89) i b a i XX j( s, r) XX j( x, t) (.9) i b a i YYj( s, r) YY j( x, t) (.9) i b a i XYj( s, r) XYj( x, t) (.9) Regio 3: The method i the derivatio of kerels i regio is followed here. Equatios for 3 ( xy, ), 3 ( xy, ) ad 3 ( xy, ) ca be writte as, xx yy xy i y i x 3 SS6 e e d xx( x, y) (.93) 8

34 i y i x 3 SS7 e e d yy ( x, y) (.94) y i x 3 SS8 e e d xy ( x, y) (.95) Where, 3 SS6 A7( ) A8( ) y (.96) SS7 A7( ) A8( ) y (.97) 7 8 SS8 A ( ) A ( ) y (.98) Solvig the equatios (.4), (.5) ad (.6) ad usig the itegrals from (.48) to (.53), we get the followig itegral equatios b a b ( ) (, ) ( ) (, ) (, ) f g xx a f t XX x t dt g t XX x t dt x y (.99) b a b ( ) f (, ) ( ) g (, ) yy (, ) a f t YY x t dt g t YY x t dt x y (.) b a b ( ) f (, ) ( ) g (, ) xy (, ) a f t XY x t dt g t XY x t dt x y (.) ad the kerels for regio 3 are 3 a8 ( t x) a4( h y)( h y)( t x) XX f ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) a9 ( t x) a y ( t x) a y ( t x) ( y ( t x) ) 4 a4h( h y) 3( h y) ( t x) ( t x) (.) 9

35 3 a ( h y) a (4 h y) ( h y) ( t x) 4 a h( h y)( h y) ( h y) 3( t x) XX g ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) YY YY 4 4 a y a y( y ( t x) ) a y q ( y ( t x) ) (.3) 3 a ( t x) a4( h y) ( t x) 4 a4h( h y) 3( h y) ( t x) ( t x) f ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) a ( t x) a y ( t x) a y ( t x) ( y ( t x) ) (.4) a y ( h y) ( t x) 4 a h( h y)( h y) ( h y) 3( t x) 3 a3( h y) 4 4 g ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) XY XY a y y t x a ( y ( t x) ) ( ( ) ) (.5) a y ( h y) ( t x) 4 a h( h y)( h y) ( h y) 3( t x) 3 a3( h y) 4 4 f ( x, t) 3 a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) a y y t x a ( y ( t x) ) ( ( ) ) (.6) 3 a ( t x) a4( h y) ( t x) 4 a4h( h y) 3( h y) ( t x) ( t x) g ( x, t) 3 Regio : a ( h y) ( t x) ( h y) ( t x) ( h y) ( t x) a ( t x) a y ( t x) a y ( t x) ( y ( t x) ) Equatios for ( xy, ), ( xy, ) ad ( xy, ) ca be writte as, xx yy xy (.7) i y i x SS e e d xx( x, y) (.8)

36 i y i x SS e e d yy ( x, y) (.9) y i x SS3 e e d xy ( x, y) (.) Where, 3 SS A5( ) A6( ) y (.) SS A5( ) A6( ) y (.) 5 6 SS3 A ( ) A ( ) y (.3) Solvig the equatios (.8), (.9) ad (.3) ad usig the itegrals from (.48) to (.53), we get the followig itegral equatios b a b ( ) f (, ) ( ) g (, ) xx(, ) a f t XX x t dt g t XX x t dt x y (.4) b a b ( ) f (, ) ( ) g (, ) yy (, ) a f t YY x t dt g t YY x t dt x y (.5) b a b ( ) f (, ) ( ) g (, ) xy (, ) a f t XY x t dt g t XY x t dt x y (.6) ad the kerels for regio are XX a ( t x) ( a h a y) y( t x) ( x, t) a y ( t x) y ( t x) f 4 (.7)

37 XX YY YY XY XY a y ( a6h a7 y) y ( t x) ( x, t) a y ( t x) y ( t x) 8 g f 4 a ( t x) ( a h a y) y( t x) ( x, t) a y ( t x) y ( t x) a y ( a6h a7 y) y ( t x) ( x, t) a y ( t x) y ( t x) g 4 f 4 a y ( a6h a7 y) y ( t x) ( x, t) a y ( t x) y ( t x) g 4 a ( t x) ( a h a y) y( t x) ( x, t) a y ( t x) y ( t x) (.8) (.9) (.) (.) (.) The costats from a to a are listed i the Appedix B.

38 CHAPTER 3 KERNELS FOR A CRACK PARALLEL TO AN INTERFACE AND THE COMPUTATIONAL DETERMINATION OF HIGHER ORDER TERMS Followig Erdoga (97) ad Achebach (98) the itegral equatios for a crack parallel to a iterface are: b b b f() t dt f ( t) K( x, t) dt g( t) K( x, t) dt p( x) a x b, t x a a a (3-) b b b gt () dt f ( t) K( x, t) dt g( t) K( x, t) dt q( x) a x b, t x a a a (3-) where: d dv f ( t) v ( t, ) v ( t, ), dt dt 3 (3-3) d du g( t) u ( t, ) u ( t, ), dt dt p x 3 (3-4) ( ) lim yy ( x, y) y (3-5) q x ( ) lim xy ( x, y) y. (3-6) The kerels i (3-) ad (3-) are: 4a h t x h ( t x) 8a h t x t x K x t a 4h t x 4h t x 4h t x a (, ) 3 4a h t x h ( t x) 8a h t x t x K x t a 4h t x 4h t x 4h t x a (, ) 3. 3

39 8a h 4h 3( t x) a h K x t K x t a 4h t x 3 (, ) (, ) 3 4h t x (3-7) The costats are give by:, a / a, (3-8) / where the Dudurs (969) parameters are: a a 4aa a a,. a a a a (3-9) The solutio of these itegral equatios allows for the determiatio of the K coefficiets usig: I k COD uy uy u u b x ( ),, (3-) II k CSD ux ux ur ur b x ( ),, (3-) where the polar coordiate variable i (3-) ad (3-), = b x for the right crack tip at x = b. Followig the Appedix A, the mode I T coefficiets are determied usig either or I I rr, (3-), k ( ) T ( ), x b I II rr (3-3), T ( ) k ( ), b x The required expressio to make use of (3-) or (3-3) is 4

40 b b b f() t i dt f ( t) L( x, t) dt g( t) L( x, t) dt g( x) xx( x, ) t x a a a, (3-4) where the kerels are: 4a h t x h ( t x) 8a h t x 3 t x L x t a a 4h t x 4h t x 4h t x (, ) 3 8a h 4h 3( t x) 4a h 4 h ( t x) 3 h L x t a a 4h t x 4h t x 3 (, ) 3 4h t x (3-5) ad the plus sig for the delta fuctio term is for the upper crack surface (i = 3) while the egative sig is for the lower crack surface (i = ). The mode II T coefficiets are determied usig either, or u (,) ( ) ( ) 4 II II k T, (3-6) u ( ) ( ) (, ) I II k T 4. (3-7) The required expressio i this case is b b b 3 g( t) 4 v ( x, ) dt f ( t) M( x, t) dt g( t) M ( x, t) dt f ( x) t x a a a x, (3-8) where the kerels are 5

41 3 6ha 4h 3 t x 4h t x h (, ) 3 M x t ha 4h t x 4h t x 4 M h t x h t x 8h a t x 8h a t x t x ( x, t) 3 4h t x 4h t x 4h t x (3-9) These fuctios caot be expressed i terms of a ad a aloe, although as will be show umerically, the mode II T-coefficiets determied usig (3-8) are oly fuctios of these two costats. Next cosider the umerical work. Numerical solutio. Equatios (3-) ad (3-) are ormalized usig b a b a, b a b t r x s a, (3-) f ( t) f ( r), g( t) g( r), (3-) b a Kij ( s, r) Kij ( x, t) (3-) which gives. f ( r) p( s) dr f ( r) K ( s, r) dr g( r) K ( s, r) dr s, (3-3) r s g( r) q( s) dr f ( r) K ( s, r) dr g( r) K ( s, r) dr s. (3-4) r s Defiig the o-dimesioal legth parameter, h b a, (3-5) which is simply h divided by the half-crack legth, the ormalized kerels (3-) become 6

42 b a K( s, r) K( x, t) 4a r s ( r s) 8a r s r s 4 r s 4 r s 4 rs a 3 a b a K( s, r) K( x, t), (3-6) 4a r s ( r s) 8a r s r s 4 r s 4 r s 4 rs a 3 a, (3-7) 8a 4 3( r s) a K s r K s r a 4 3 (, ) (, ) 3 rs 4 rs. (3-8) Takig ito accout that both crack tips are closed ad the stress is sigular at both tips, the solutio of (3-3) ad (3-4) is obtaied by usig f() r N a T ( r) a T ( r), r r i i i i i i N gr () N bt ( r) bt ( r), (3-9) r r i i i i i i N where the T i fuctios are Chebyshev polyomials of the first kid. The expressios to the right take advatage of symmetry. Usig (3-) ad (3-), (3-9) ca be itegrated to obtai, b a a b a a V ( s) U ( s) s U ( s) s N N i i i i i i i i 7

43 b a b b a b U( s) U ( s) s U ( s) s N N i i i i i i i i. (3-3) A useful expressio to ormalize displacemet i terms of material istead of material, which ca be used to correspod to a egative h/a, is (3-3) Startig the sum at oe i (3-3) istead of zero automatically satisfies the requiremets, d f ( t) dt v3( t, ) v ( t, ) dt v3( t, ) v( t, ), dt d g( t) dt u3( t, ) u( t, ) dt u3( t, ) u( t, ), dt (3-3) due to the orthogoality coditio,, i j Ti( t) Tj( t) dt /, i j t. (3-33), i j Substitutio of (3-9) ito (3-3) ad (3-4) leads to the followig system of liear equatios: T() r ( ) N N i ps ai K( s, r) dr b ( ) i Ti r K( s, r) dr i r s r i, (3-34) T() r ( ) N N i qs ai Ti ( r) K( s, r) dr bi K( s, r) dr i i r s r. (3-35) The sigular itegral is give by 8

44 i dr U i() s r s r, (3-36) T() r where U i- are the Chebyshev polyomials of the secod kid. Determiatio of the K coefficiets. Give the umerical solutio of (3-9) usig (3-34) ad (3-35), (3-3) ad (3-4) together with (3-) ad (3-) ca be used to determie the K-coefficiets. First (3-9) is itegrated to obtai, aiti() s d d i f ( x) v3( x, ) v( x, ) v3( x, ) v( x, ) dx b a ds s N N s Ti( q) ai v3( x, ) v ( x, ) ai dq s U i ( s) b a i q i i N (3-37) Followig Aathasayaam, et. al (7), (3-37) ca be expressed mathematically i terms of ( - s) as follows, where b N i v3( x, ) v( x, ) s ai d( s) a, (3-38) i i i i d d, d i. (3-39) () Takig ito accout that the radial coordiate, at the right crack tip (x = b) i (3-) is b a b x s, (3-4) 9

45 the left side of (3-38) is ow expressed usig (3-) as follows: k b a b a I uy u y b a s ( ) ( ). (3-4) Equatig (3-38) ad (3-4) term by term gives the ormalized mode I k coefficiets as follows: k ( ) ( ) ( ) N I i ai d s b a s i b a I N I k () b b a i k ( b) ( ) aid, b a i I N I k ( a) b a i i ( ) ( ) ( ) i b a i k a a d. (3-4) It is observed that for = the ormalized stress itesity factor is obtaied. I the same maer the mode II coeficiets at both crack tips are II N II k () b b a i k ( b) ( ) bi d, b a i II N II k ( a) b a i i ( ) ( ) ( ) i b a i k a b d. (3-43) I order to uderstad how the asymptotic coefficiets ca represet the actual stress ad displacmet fields aroud the crack tip, it is ecessary to develop expressios for the stresses i from of the crack tip. The logical ad simple startig poit are the stresses 3

46 which are easily obtaied from (3-) ad (3-), or more covieetly, (3-3) ad (3-4). These expressios give, yy ( x) f( r) dr f ( r ) K ( s, r ) dr g ( r ) K ( s, r ) dr, (3-44) r s ( ) xy x gr dr f ( r ) K ( s, r ) dr g ( r ) K ( s, r ) dr. (3-45) r s It is oted that the above two expressios are valid o ad off the crack surfaces, although whe o the crack this is simply a expressio of the boudary coditios. Determiatio of the T coefficiets. The ext step is to obtai a expressio for the T-coefficiets, which must iclude the radial stress compoet from (3-4), which is repeated below, b b b 3 f() t xx t x a a a. (3-46) ( x, ) dt f ( t) L ( x, t) dt g( t) L ( x, t) dt g( x) The first step is to express (3-46) i terms of ormalized quatities usig (3-) ad (3- ) as follows: 3 xx( x, ) f( r) dr f ( r ) L ( s, r ) dr g ( r ) L ( s, r ) dr g ( s ), (3-47) r s where (3-47) is valid for all s ad 4a r s ( r s) 8a r s 3 r s L s r a a 4 r s 4 r s 4 rs (, ) 3, 3

47 8a 4 3( r s) 4a 4 ( r s) 3 L s r a a (, ) 3 r s r s 4 rs (3-48) Aathasayaam et. al (7) obtaied the T-coefficiets were obtaied very accurately by combiig (3-46) with yy, the key poit beig the elimiatio of the sigular itegral. Followig this approach, (3-44) ad (3-47) give: 3 yy ( x) xx ( x, ). f ( r) K ( s, r) L ( s, r) dr g( r) K ( s, r) L ( s, r) dr g( s) (3-49) Off of the crack, the right side of (3-49) ca be expaded i terms of small values of (s - ) usig the otatio, M 3 I M yy ( x) xx( x, ) h ( s ) O( s ), (3-5) where the h costats are obtaied from itegrals of kow fuctios usig (3-49). This same combiatio of stresses o the upper crack surface ca be expressed asymptotically as follows: I r, r, T ( r). (3-5) rr Comparig (3-5) ad (3-5) gives the ormalized coefficiets, T b a I I h, (3-5) 3

48 Now cosider the case whe x is o the crack ad the last term of (3-47) plays a role. First from (3-37 to 3-39), Ti () s s i d ( s). (3-53) Whe x is o the crack, the right side of (3-49) ca be expaded i terms of small values of ( - s) usig the otatio, M N 3 yy ( x, ) xx( x, ) h ( s) O( s) bi d ( s) i M i, (3-54) where the h costats are obtaied from itegrals of kow fuctios usig (3-49). This same combiatio of stresses o the upper crack surface ca be expressed asymptotically as follows: I II rr k T,, ( ) ( ). (3-55) Comparig (3-54) ad (3-55) gives the ormalized T coefficiets as follows T I h, I T b a ( ) h, > (3-56) ad the idetical result (3-4) for the K coefficiets. I order to determie the T- coefficiets for mode II, (3-8) is used i the ormalized form, 33

49 3 4 v ( x, ) g( r) dr x r s f ( r) M( s, r) dr g( r) M ( s, r) dr f ( s), (3-57) where b a Mi( s, r) Mi( x, t). b r s 3 6 a 4 3 r s 4 r s (, ) a 3 M s r M b 4 r s 4 r s 4 rs 8 a r s 8 a r s b r s ( s, r) 3 4 r s 4 r s 4 r, s b. (3-58) Oce agai elimiatig the sigular itegral by combiig (3-54) with (3-45) as follows: xy x v x x 3 ( ) 4 (, ) f ( r) K( s, r) M( s, r) dr g( r) K( s, r) M ( s, r) dr f ( s). (3-59) Whe s is off the crack, the f-bar fuctio is zero ad the right side of (3-59) ca be expaded i terms of small values of (s - ) usig the otatio, 34

50 xy ( x) h ( s ) O( s ) 3 M 4 v ( x, ) 3 M, (3-6) x where the h costats are obtaied from itegrals of kow fuctios usig (3-59). This same combiatio of quatities i frot of the crack tip ca be expressed asymptotically as follows: 4 u (,) II r, T ( ) G. (3-6) Comparig (3-6) ad (3-59) gives the ormalized T-coefficiets, II 3 T b a h,, (3-6) II where for = the rigid body rotatio term, G T. Whe s is o the crack, the right side of (3-59) ca be expaded i terms of small values of ( - s) usig the otatio, xy ( x) 3 4 v ( x, ) x M N 4 M i h ( s) O( s) ai d ( s), (3-63) i where the h costats are obtaied from itegrals of kow fuctios usig (3-59). This same combiatio of stresses o the upper crack surface ca be expressed asymptotically as follows: r 4 u, 35

51 4 4 I II k( ) T ( ) G. (3-64) Comparig (3-6) ad (3-6) gives the ormalized T coefficiets as follows II 4 T b a ( ) h, where for = the rigid body rotatio term, G T, (3-65) II, ad the idetical result (3-4) for the k coefficiets. A very iterestig poit is that the T coefficiets for mode II should oly be depedet o the bi-material costats itroduced i (3-8). We have ot bee able to write the kerels i (3-9) as a fuctio of these costats, ad therefore the expressios (3-59), (3-6) ad (3-63) must be such that whe the evaluatio i (3-6) ad (3-65) is made, all depedece o costats other tha (3-8) disappears. A critical expressio used to obtai the T coefficiets is obtaied by usig partial fractios ad complex variables, ( s) Im Im r s s r i, r i 4 ( ) ( ) si ( ) ( s), ta ( r) 4. (3-66) r This gives c s, (3-67) 4 ( rs) where 36

52 c si ( ) ( r) 4. (3-68) The other required expressios are 4 ( rs) 4 ( rs) 3 c, s c ci ci, (3-69) i 3 c, 3 s c ci ci. (3-7) i Usig (3-49) ad (3-5) for the mode I T coefficiets ad (3-59) ad (3-6) for mode II, requires that the followig be expressed i terms of small distaces from the right crack tip, i.e., s ear : ( r s)( a a) (, ) (, ) 3 K s r L s r 8 a ( r s) ( r s), (3-7) 4 ( rs) 4 ( rs) 4a 4 ( r s) 6a 4 3( r s), 4 ( rs) 4 ( r s) 4 ( r s) 3 ( a a) (, ) (, ) 3 K s r L s r a a b K ( s, r) M( s, r) 4 ( rs) (3-7) 3 a ( ) 4 ( r s) 8 a ( ) 4 3( r s) 3 ( ) 4 ( r s) ( ) 4 ( r s) (3-73) ( r s)( a a b ) K ( s, r) M ( s, r) 4 ( rs) 37

53 4 a ( r s)( ) ( r s) 3 ( ) 4 ( r s) (3-74) The ext step is to write the above four expressios i terms of oe series as follows: I K ( s, r) L ( s, r) hf ( s), I hf ( a a) ( r) c c a ( r) ( r) c 3( r) c 3( r) c c3 (3-75) I K ( s, r) L ( s, r) hg ( s), hg ( a a ) c 4a 4 ( r) c ( r) c c I a 4 3( r) c 6( r) c 3c (3-76) II K ( s, r) M( s, r) hf ( s), hf ( a a b ) c a 4 ( r) c ( r) c c II 8a 4 3( r) c 6( r) c 3c (3-77) II K ( s, r) M ( s, r) hg ( s), II hg ( a a b ) ( r ) c c 38

54 4 a ( r) ( r) c 3( r) c 3( r) c c I the above it is uderstood that j c (3-78),. (3-79) Comparig (3-54) ad (3-56), ad usig (3-75) ad (3-76) for the case o the crack, T b a T hf f r hg g r dr, (3-8) I I I I ( ) ( ) ( ) ad similarly for mode II, usig (3-63) leads to T b a T hf f r hg g r dr. (3-8) II II II II ( ) ( ) ( ) I order to compare full field results to the asymptotic expressios obtaied usig the stress itesity factors ad T-stress coefficiets, the followig results apply off the crack: r, I xx I I I k ( r) T ( r) T ( s ) ( s ) r (3-8), ( s ) yy I I k( r) k (3-83) r xy II II k ( r) k ( s ), ( s ) (3-84) ( s ) k 4 vr (,) II II k ( r) T ( r) r k ( ) (3-85) II II T s ( s ) 39

55 Similarly, alog the crack flaks where, the pertiet expressios are: r, T ( r) k ( r) xx I II II k ( s) I T s (3-86) 4 vr (, ) I II k( r) T ( r) r k ( s) I II T ( s). (3-87) 4

56 CHAPTER 4 STRESS INTENSITY FACTORS, T-STRESS COEFFICIENTS AND THE STRESS FIELD AROUND THE TIP OF A CRACK THAT IS PARALLEL TO AN INTERFACE I this Chapter all the results are preseted startig with a covergece study for the modes I ad II, k- ad T-coefficiets. This is followed by a set of cotour plots of the coefficiets for the full rage of ad. After this the focus is o details of the stress field for the case whe the crack tip closes for =.98 ad =.495. Several plots of displacemet ad stress alog the lie of the crack ad for the stress field aroud the crack tip are preseted. 4. Covergece Study The umerical approach detailed i the previous chapter is used to determie the K ad T-coefficiets usig a double precisio computer program writte i Fortra. Ufortuately there are o results for validatio from the literature for the higher order coefficiets for a crack parallel to a iterface. However, the approach has bee validated for the pure mode I case by Aathasayaam, et. al (7) ad for the pure mode II case by Aathasayaam (8). Furthermore, as will be see throughout this Chapter, the asymptotic results are cosistet with the full field stress field. I Table 4., for a/h =., coverged values of the four differet coefficiets are preseted to show how covergece behaves with respect to the coefficiet order,. This 4

57 is a difficult case for covergece sice the legth parameter, h/a, is small. As show by Aathasayaam, et. al (7), it is easier to obtai the T-coefficiets tha the K- coefficiets, the reaso beig that the former are the result of a itegratio of a fuctio, while the latter are obtaied by evaluatig the derivative of the fuctio at the edpoit, s = -. Table 4.: Coverged asymptotic coefficiets defied by Equatios (3-4, 3-43, 3-56, 3-65) for a/h =., =.48, =.495 ( a) I II I II ka k a a a T T ( a) e E E E E E E E E E E E E+8.648E+8.75E E E E E E E E E E+5 6.9E+7.55E E E+8 7??.33479E E E E+3 I Table 4. covergece of the 4 th K- ad th T-coefficiets, with respect to the parameter, N defied i (3-9), are preseted for h/a =., =.98 ad =.495. Covergece data for the case of h/a =. are preseted i Table 3, which shows a icreasig level of difficulty as the crack gets closer to the iterfaced, i.e., as the legth parameter, h/a, becomes smaller. 4

58 Table 4.: Covergece study of asymptotic coefficiets with respect to N defied i Equatio (3-9) for a/h =., =.98, =.495. N I 3 ka 3 II 3 I II k3 a T( a) T ( a) ( a) ( a) From (4) From (4) From (54) From (6) E E E E E E E E E E E E E E E E E E E E E E E E E+8.644E E E E E E E E E E E+3 Table 4.3: Same as Table 4. for a/h =.. I 3 II 3 I II N ka 3 k3 a T( a) T ( a) ( a) ( a) From (4) From (4) From (54) From (6) E E E E E E E E E E E E E E E E E E E E+9 I the last table of this sectio it is demostrated that as the umber of terms is icreased for small r/a, the asymptotic solutios coverge to the full-field solutio. To do this the followig percet error defiitio is made for the xx stress i frot of the crack tip: 43

59 E M r, xx M xx, (4-) xx r, A where from (3-8) the asymptotic series ivolves the mode I k- ad T-coefficiets as follows, r, M I k Axx T ( s ). (4-) ( s ) xx M I Values of the error measure from (4-) are preseted i Table 4.4. Table 4.4: Covergece of xx i frot of the crack tip with respect to the umber of terms as defied i (4-) for a/h =., =.98, =.495. The percet error measure, E M, is defied i (4-). r/ a xx r, E E E E E+.44E+.368E E+ -.55E+..36E+.5E+.9E+ -.8E- -.77E-3..3E+.553E+.56E- -.5E E E+.563E-.45E E-7 -.E- I Table 4.5 the error of the xx compoet of stress o the upper crack surface is preseted usig the followig asymptotic expressio from (3-86), 44

60 M II r, k Axx T s ( s), (4-3) xx M I where ow the mode II k-coefficiets are ivolved. Table 4.5: Covergece of xx o the upper crack surface with respect to the umber of terms as defied i (4-3) for a/h =., =.98, =.495. The percet error measure, E M, is defied i (4-). r/ a xx r, E E E E 3..76E- -.65E E+3.538E+.9E+3..9E+ -.44E+ -.49E+.E-.E-..7E E+ -.44E-.33E-4.7E E E E-4.39E-7.9E- Two thigs are clear from these tables: ) the value of r/a must be small eough ad ) if it is, as the umber of terms icreases, the error drops, i.e., the asymptotic solutio quatified by a few costats that ca be determied from a aalysis such as this, defie the coditios of stress aroud the crack tip. I the ext sectio a study of the effect of the bi-material costats, ad, is made by usig cotour plots. 4.. Asymptotic coefficiets as a fuctio of the bi-material costats The stress itesity factor results have bee checked with values reported i the literature; however, there are o reported values for the higher order terms. Oe of the 45

61 most covicig argumets that the results are correct is that the mode II, T-coefficiets are oly a fuctio of ad. This was ot prove aalytically; rather the umerical solutios preseted i this sectio cofirm it. For a give set of ad, multiple values of,,, ad that correspod to these values give the same results. I the previous sectio it was demostrated that several higher order coefficiets ca be determied very accurately usig the sigular itegral equatio approach. I this sectio the versatility of the formulatio is used to geerate cotour plots for the full rage of material pair possibilities ad for o-dimesioal crack legths of a/h =,.,.,. ad.. Results are preseted for the first eight coefficiets, which correspods to the first two i each of the four categories: modes I ad II, K- ad T- coefficiets. It is recalled from the discussio ivolvig Equatios (3-6) through (3-65) that the first mode II T-coefficiet is actually the rigid body rotatio term, G, first itroduced i Equatio (A-). For each crack legth through h/a =. there are a series of eight plots i the order: I II I II I II I II k k T T G k a k a T a T a,,,,,,, a a a a. (4-) For the case of h/a =., oly the first coefficiets i (4-) are preseted due to umerical difficulty. There are a total of forty-four figures that follow i this sectio. The first eight figures, Figures , are for a/h =. The remaiig figures correspod to a/h =.,.,. ad., respectively. 46

62 .5 k Figure 4.. Cotour plot for Mode-I k coefficiet whe = ad h/a=.5.5 k Figure 4.. Cotour plot for Mode-II k coefficiet whe = ad h/a=

63 .5 T Figure 4.3 Cotour plot for Mode-I T coefficiet whe = ad h/a=.5.5 T Figure 4.4. Cotour plot for Mode-II T coefficiet whe = ad h/a=.5 48

64 k Figure 4.5 Cotour plot for Mode-I k coefficiet whe = ad h/a= k Figure 4.6 Cotour plot for Mode-II k coefficiet whe = ad h/a=.5 49

65 .5 T Figure 4.7 Cotour plot for Mode-I T coefficiet whe = ad h/a= T Figure 4.8 Cotour plot for Mode-II T coefficiet whe = ad h/a=

66 k Figure 4.9 Cotour plot for Mode-I k coefficiet whe = ad h/a=..5 k Figure 4. Cotour plot for Mode-II k coefficiet whe = ad h/a=. 5

67 .5 T Figure 4. Cotour plot for Mode-I T coefficiet whe = ad h/a=..5 T Figure 4. Cotour plot for Mode-II T coefficiet whe = ad h/a=

68 .5 k Figure 4.3 Cotour plot for Mode-I k coefficiet whe = ad h/a=. k Figure 4.4 Cotour plot for Mode-II k coefficiet whe = ad h/a=. 53

69 T Figure 4.5 Cotour plot for Mode-I T coefficiet whe = ad h/a= T Figure 4.6 Cotour plot for Mode-II T coefficiet whe = ad h/a=. 54

70 .4.5 k Figure 4.7 Cotour plot for Mode-I k coefficiet whe = ad h/a=..5 k Figure 4.8 Cotour plot for Mode-II k coefficiet whe = ad h/a=. 55

71 T Figure 4.9 Cotour plot for Mode-I T coefficiet whe = ad h/a=..5 T Figure 4. Cotour plot for Mode-II T coefficiet whe = ad h/a=. 56