__, 9: I at th- ‘A-"fl‘ " . . 4.. J. I ‘1 t 3 '3, t.....—.-— h i f '5 O 2 . ,7 (”I ' I ‘. ' l I” .- " Q; .o "u t ‘- ‘J J at! .u' 0- f» 2 8 .°— 0 tiff-v3.1?” "'1! ',' . E _. r. ‘11? can b 4" ‘- *' ‘-’ ‘ This is to certify that the dissertation entitled A Numerical Method for the Treatment of Kinked Cracks in Finite BodieS’ presented by Ukhwan Sur has been accepted towards fulfillment of the requirements for Ph. D. degreein Mechanics; flab ocean/.5 CUIHOO ' Mavfessor 3/5/53 7 Date / v MS U in an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LlBRARlES .J—n. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. A.NUHERICAL.HETHOD FOR.THE TREAIHENT 0P KINKED CRACKS IN FINITE BODIES By Ukhwan Sur A DISSERIAIION Submitted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR.OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1987 ABSTRACT ANUHERICALHETBODFORTHETREAMT OF KINKED CRACKS IN FINITE BODIES By Ukhwan Sur A displacement discontinuity method is developed for modelling cracks in linear-elastic two-dimensional infinite domains. This method is then coupled to the standard boundary-element method for the treatment of cracks in finite two-dimensional regions. The hybrid approach has been implemented on the computer and representative results are presented. Problems studied include a variety of crack shapes, including cracks with kinks. The hybrid method is shown to be an effective technique for the study of cracks of arbitrary shape in finite bodies. The ability of the method to handle cracks with kinks is a distinct advantage over other known approaches. Furthermore, the application of this method to crack propagation problems possesses none of the computational problems associated with other approaches. AGKNWLEDGWS I would like to express appreciation to those who have helped and contributed to making this work a reality. This includes my advisor, Dr. Nicholas J. Altiero, for his support, inspiration and assistance through this period and Drs. John F. Martin, Larry J. Segerlind and David Yen, members of my guidance committee. A final note of appreciation must go to my wife, Oksoon, for her tolerance and encouragement . ii TABLE OF CONTENTS PAGE LIST OF TABLES .................................................... iv LIST OF FIGURES .................................................... v CHAPTER 1 INTRODUCTION AND BACKGROUND ............................ 1 CHAPTER 2 BOUNDARY ELEMENT FORMULATION ........................... 8 2.1 THE BOUNDARY INTEGRAL EQUATION METHOD .............. 8 2.2 NUMERICAL TREATMENT ................................ 12 CHAPTER 3 CRACK PROBLEMS: INFINITE DOMAIN ........................ 24 3.1 THE DISPLACEMENT DISCONTINUITY METHOD .............. 24 3.2 NUMERICAL TREATMENT ................................ 30 CHAPTER 4 CRACK PROBLEMS: FINITE DOMAIN .......................... 35 4.1 COUPLING OF BOUNDARY ELEMENT METHOD AND DISPLACEMENT DISCONTINUITY METHOD .................. 35 CHAPTER 5 EXAMPLES AND DISCUSSION ................................ 40 5.1 EXAMPLES OF INFINITE DOMAIN CRACK PROBLEMS ......... 40 5.1.1 STRAIGHT CRACK .................................. 40 5.1.2 CIRCULAR ARC CRACK .............................. 46 5.1.3 KINKED CRACK .................................... 49 5.2 EXAMPLES OF FINITE DOMAIN PROBLEMS ................. 57 5.2.1 STRAIGHT CRACK .................................. 57 5.2.2 KINKED CRACK .................................... 66 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS ........................ 71 BIBLIOGRAPHY ....................................................... 73 iii LIST OF TABLES TABLE Page 2.1 Singular integrals ............................................ 19 5.1 Displacement discontinuity along left half of a straight crack in an infinite domain ................................... 44 5.2 Stress intensity factors at tip A of a kinked crack (a1 - 45°) .................................................... 50 5.3 Stress intensity factors for an off-center straight crack ..... 62 5.4 Stress intensity factors for an angled crack in a large plate.62 5.5 Stress intensity factors for an angled crack in a finite plate ......................................................... 65 iv LIST OF FIGURES FIGURE Page 1.1 Crack propagation in an arbitrarily shaped body ............... 7 2.1 Description of region of interest ............................. 9 2.2 Discretized model of region R ................................. 13 2.3 Definition of vectors a and b ................................. 17 3.1 Source point and field point in an infinite plane ............. 24 3.2 Line defining a crack in the infinite plane ................... 26 3.3 Equal and opposite crack surface tractions .................... 29 3.4 Descretized crack surface ..................................... 30 4.1 Center cracked plate with tensile load ........................ 36 4.2 Linear superposition of the boundary element and dispacement discontinuity models .......................................... 37 5.1 Straight crack in an infinite domain .......................... 40 5.2 Dependence of cn on number of nodes ........................... 41 5.3 Dependence of en on location of transition node ............... 43 5.4 Dependence of CD on number of nodes when transition nodes are used ...................................................... 45 5.5 Circular arc crack ............................................ 46 5.6 SIFs for a circular arc crack under biaxial stress ............ 47 5.7 SIFs for a circular arc crack under pure shear ................ 48 5.8 Geometry of a kinked crack in an infinite plane ............... 49 5.9 Mode I SIFs for an asymmetric kinked crack under uniaxial stress ........................................................ S3 FIGURE Page 5.10 Mode II SIFs for an asymmetric kinked crack under uniaxial stress ........................................................ 54 5.11 SIFs for an asymmetric kinked crack under various biaxial conditions .................................................... 55 5.12 SIFs for an anti-symmetric kinked crack in an infinite plate..56 5.13 Center-cracked test specimen .................................. 58 5.14 SIFs vs. a/b for center-cracked test specimen ................. 59 5.15 An off-center straight crack .................................. 60 5.16 Angled crack in a large plate ................................. 61 5.17 Angled crack in a finite plate ................................ 64 5.18 Asymmetric kinked crack in a finite plate .................... 67 5.19 Anti-symmetric kinked crack in a finite plate ................. 68 5.20 SIFs for an asymmetric kinked crack in a finite plate ......... 69 5.21 SIFs for an anti-symmetric kinked crack in a finite plate ..... 70 vi INTRODUCTION AND BACKGROUND The presence of cracks in a structure as shown in Figure 1.1 generally reduces the fatigue and static strength of the structure because the stresses and strains are highly magnified at the crack tip. It has been established that parameters deduced from linear elastic fracture mechanics can be used to determine the stress and strain magnification at the crack tip. These parameters, the stress intensity factors (SIF), incorporate applied load levels, geometry, and crack size in a systematic manner and may be evaluated from elastic stress analysis using the finite element method (FEM) or the boundary element method (BEM). The BEM is the numerical form of the "boundary integral equation method” (DIEM) . The FEM has been used extensively to solve elastic plastic problems in fracture mechanics. The FEM is very effective for solving problems with material and geometric nonlinearities, dynamic effects and inhomogeneities (e.g. boundaries, inclusions, interfaces). However, FEM is not the most effective method for problems involving singularities (e.g. mathematically sharp cracks) or localized events such as point sources and sinks. For the analysis of a sharp crack, a fine field discretization is needed near the crack tip to capture the rapidly varying stress and displacement fields. Furthermore, analysis of crack propagation necessitates constant re-defining of the finite element topology which may present severe computational difficulties. The main disadvantage of the FEM is that a domain discretization is required to perform the analysis. The BEM involves discretization only of the boundary of the structure and the governing differential equation is solved exactly in the interior, leading to greater accuracy . The boundary integral equation method has also been developed and extensively used for the analysis of problems in continuum mechanics. This method reduces the order of the problem, basically by using the divergence theorem of Gauss. However, the system matrix is full and unsymmetric so the reduction in order may be achieved only superficially. It is particularly effective for problems with singularities and dominantly linear response, and for modeling infinite domains. The method has also proven effective for the treatment of material nonlinearities. Geometric nonlinearities are still under s tudy . Several strategies have been proposed for the analysis of crack problems using the boundary integral equation method. These methods include representing the crack as a notch, symmetric crack modelling, use of special Green's functions, and flat crack modelling. Representing the crack as a notch or replacing the crack plane with symmetric boundary conditions, i.e. symmetric crack modelling, removes the singularity in the algebraic system of equations which is obtained when the upper and lower crack surfaces are modelled in the same plane [1]. However, representation of the crack as a notch increases the modelling error due to the notch opening and symmetric modelling is limited to symmetric problems. The special Green's function approach [2-4,5] possesses the advantage that crack geometry and crack tip singularities are fully embedded in the boundary equations, i.e. no modelling of the crack surface is required. The disadvantage is that some two-dimensional and all three-dimentional problems can not be formulated using the special Green's function approach [4] . Also, complex arithmetic is required for the problems for which it does apply . Flat crack modelling represents the displacements along the crack surface as the relative displacement between the two crack surfaces. This scheme has two critical deficiencies as a mathematical model for crack geometries, as pointed out by Cruse [4]. First, if there are no tractions on the outer boundary and only crack surface loading, a non-unique boundary integral equation is generated. Second, two unknown displacement variables, i.e. the relative and total displacements, exist along the crack. A possible solution for these problems is also given by Cruse [4]. An extensive discussion of the special Green's function and flat crack modelling techniques is given by Cruse [4] . The success of the BIEM in linear elasticity has motivated many attempts at application of this method to bodies containing cracks. These efforts have been hampered by certain difficulties associated with treating the two crack surfaces as ”boundary" leading to an underdetermined system of equations . In addition to BIEM, an alternative approach, suggested in [6,7,8], involves combining the boundary-integral equations of the two crack surfaces into a set of equations associated with a single surface integral. These new integral equations are precisely those obtained if one models the crack as a continuous layer of edge dislocations. The fact that a crack is equivalent to a continuous array of edge dislocations has long been known [9] . Gol'dstein and co-workers employed this fact to develop an integral equation model for curvilinear cracks in an infinite plane subjected to arbitrary load. This method has since been developed for treatment of in-plane cracks of arbitrary shape. However, efforts to apply this technique to kinked cracks have not been successful [34] . The more important purpose of this work is to consider the behavior and characteristics of non-linear shaped cracks which are kinked, a phenomenon generally observed in the macroscopic or microscopic crack growth or propagation process. Crack morphology seems to have significant meaning in crack propagation and fatigue crack growth. The brittle crack propagation and fatigue crack growth in the mixed mode loading state has received considerable attention. Generally in these cases a crack does not follow a straight path, but rather a curved or a kinked path. Only a few reliable solutions for the stress intensity factors (SIF) of non-linear shaped cracks have been obtained for special cases. Some solutions for the stress intensity factors of non-linear shaped cracks in the state of longitudinal shear were given by Sih [10], Nakagama [11], and Smith et a1. [12]. However analytic solutions for the stress intensity factors of non-linear shaped cracks in two-dimensional stress states which are more important to crack morphology are difficult to come by. Generally in these cases, both the stress intensity factors of mode 1 and mode 2 appear. For this reason, the analysis is not easy and many interesting problems can be expected to be found . In this work we restrict our discussion to the problem of linear and non-linear shaped cracks in the two-dimensional elastic stress state. Recently, several experimental or numerical analyses for these kinds of problems have been reported [13]. Anderson [14] solves these problems on the basis of Muskhelishvili's method, but the value of his numerical results are questionable [15,16] . Kitagawa [l7] constructed a general analytical method for determination stress intensity factors of non-linear shaped cracks in an infinite isotropic homogeneous plate in the two-dimensional elastic stress state. This method includes a polynomial approximation and truncation procedure [18] of a conformal mapping function. Also Kitagawa [17] obtained numerical results for various cases . When a crack is oriented asymmetrically, the new crack initiates at an angle to the old one. The calculation of stress intensity factors for kinked cracks is difficult and there have been many attempts at their solution [19,20,21-24,25-28]. With most solutions, the analysis is such that the limit for an infinitesimally small kink cannot be obtained readily from the analysis for a finite kink. Recently, Lo [24] has presented a convincing solution that models the crack as a continuous distribution of dislocations, in a manner that can handle both the finite and the infinitesimal kink within the same formulation. A new model, suggested here, has very good features in comparison with the edge dislocation model. This new set of integral equations is precisely that obtained if one models the crack as a line of displacement discontinuity. Knowledge of the edge dislocation distribution leads directly to the relative crack surface displacements and to a complete field solution since the edge dislocation distribution is simply the derivative of the displacement discontinuity along the crack surface. It is, however, more appealing to formulate the equations in terms of the displacement discontinuity itself since the dislocation distribution is singular at tips but the displacement discontinuity is zero. The equations derived on this basis are, however, not integrable, a fact which has discouraged progress in this direction. An effort to develop a displacement discontinuity formulation has been presented by Crouch [29] but his numerical treatment results in modelling the crack as a discrete set of dislocation dipoles, a rather cumbersome variation of the dislocation density approach. Here, the displacement discontinuity is obtained through a single formulation and this method can handle kinked cracks very well. The displacement discontinuity method presented here is based on the analytical solution to the problem of a discontinuity in displacement over a finite line segment in an infinite elastic solid. Physically, one may imagine a displacement discontinuity as a line crack whose opposing surfaces have been displaced relative to one another. This method is based on the notion that one can make a discrete approximation to a continuous displacement discontinuity along a crack. An effective hybrid method has also been developed here to model fracture problems in finite plane domains. This hybridization by (incrementally) linear superposition combines the best features of two component methods. Boundary elements are used to model the finite domain while a continuous distribution of displacement discontinuity (dislocation dipoles in two dimensions) are used to model the crack. This method allows modelling of the crack "independently" of the Boundary Element mesh. x2? original crack surface -- propagation path R boundary traction T crack surface traction 4 w x1 Figure 1.1. Crack propagation in an arbitrarily shaped body. CHAPTER 2 BOUNDARY ELDENT FORMULATION 2.1 W For the plane boundary-value problem of linear elasticity illustrated in Figure 2.1, the displacement at a point x on B is related to the displacements and tractions at all other points on B by Somigliana's identity, i.e. aij(x)uj(x) + JB(uc)1.j(x,x)uj(x)ds(x) - JB(uR)i.j(x,x)tj(x)ds(x) (2.1) where the integral on the left hand side is interpreted in the Cauchy principal-value sense. The function (uc)1.J(x,x) is the displacement in the i direction at x due to a unit displacement discontinuity applied in the j direction at i in the infinite elastic plane and (uR)1.j(x,x) is the displacement in the 1 direction at x due to a unit force applied in the j direction at i in the infinite elastic plane. The coefficients, aij , depend on the character of the boundary at x (e.g. aij - 1/2 613 at a smooth boundary point). As shall be seen, knowledge of a is not required. 13 Figure 2.1 Description of region of interest. 10 At a point x in R, the displacements and stresses can be calculated from the equations ui(x) - JB(uR)i_j(x,i)tJ(i)d§ - JB(uc)i.j(x,x)uj(x)ds (2.2) aik(x) - JB(HR)1k.j(x,x)tj(x)ds - JB(Hc)ik.J(x,x)uj(x)ds (2.3) where the influence functions (IIR)1k J(x,x) and (IIc)1k j(x,x) give the stress components at x due to a unit force applied in the j direction at i , and a unit displacement discontinuity in the j direction at :1, respectively, in the infinite plane. At each point x on B and in each direction, either u (x) or t .l J is known. Therefore, eq. (2.1) can be used to solve for the unknown (X) values of u:] (x) and tj (x), thus giving complete boundary information. The displacements and stresses at any internal point can then be determined by integration using eqs. (2.2) and (2.3). It can be shown that, for plane stress, the influence functions of eqs. (2.1), (2.2) and (2.3) are given by (“R)1.k - [-<3-u>6,klogp + <1+v>qiqk1/(8«c> (no)... - [2(1+u)<fi.q:-fi2q:> + <1-v>fi.q1 + <3+v>fi2q21/(4«p) ll _ 3 - 3 - - (uc)1.2 - [2(1+u)(-n,q,-n,q2) + (l+3u)n2q1 + (3+v)n1q2]/(4Np) (uc)2.1 - [2(1+y)(-fi2q:-fi,q:) + (3+u)fi,q, + (1+3u)fi,q,]/(a«p) - 3 - 3 - - (uc)2.2 - [2(1+u)(-n1q1+n2q2) + (l-V)n2q2 + (3+V)n1q1]/(4xp) s (HR)11.1 [‘2(1+V)Q1 ' (I‘V)Q1]/(43P) (nR>.2.. - [2<1+u)q: - <3+v>q.1/<4«p> (nR>22.. - [2<1+u)q: - <1+3u>q11/(4«p> (HR).... - [2<1+u>q: - (1+3v)q21/(4«p> (nR>.2.2 - [2<1+u>q: - (3+u>q11/<4«p> 3 (HR)22 2 ['2(1+V)Q2 ' (l‘V)Q2]/(4”P) 2 _ 2 _ 2 (Hc)11.1 G(l+u)[(l+4q1-8q: )n, + 2q1q2(l-4q1)n2]/(2xp ) 22_ 2_ 2 (Hc)12.1 C(1+V)[(1'BQIQ2)U2 + 2Q1Q2(1'AQ1)H1]/(2WP ) 22_ 2.. are)...1 C(1+v)[(1-8q1q2)n1 + 2q1q2<1-4q2>n21/<2«p2> 12 (Hc)11.2 ' (Hc)12.1 (H°)12 2 ' (Hc)22.1 (Hc)22.2 - G(l+u)[(l+4q:-8q;)n2 + 2q1q2(l-4q:)nl]/(2np2) (2.4) where /2 _ 2 _ 2 1 P - [(X1'x1) + (Xa'xz) ] QI ' (xi'i1)/P Q2 ' (Xz'i27/P (2.5) and 5,, H2 are the components of the outward-directed unit normal vector at a point i on the boundary, G is the shear modulus and u is I Poisson 5 ratio. 2.2 Numerical treatment Eq. (2.1)cnu1be solved numerically if the boundary B is approximated by N straight segments, as shown in Figure 2.2. For this model, eq. (2.1) can be written as 13 x2 4b Figure 2.2 Discretized model of region R. 14 (n) (n) (n) - - - aij uj +mgl Jm (uc)1 J(x ,x) uj(x) ds (n) - - - - § Jn(“R)1 j(x ,x) tj(x) ds (2.6) m-l (n) where u J n-l, ...... ,N, j-l,2. are the displacements components at node n and The displacements and tractions on each segment m can be approximated using shape functions so that uj(x) - uj(m-1)N1(€) + uj(“"N. - _ (m) tj(x) tj (2.7) where N1(E) - (1-€)/2. N2(€) - (1+€)/2 (m-l) (m) i - N1(£) x + N2(€) x a; - [(sm - sm_1)/2] d5 - (Asm/Z) d5 (2.8) and 6 is a local coordinate for the segment m with value -1 at node m- 1, value 0 at the center of the segment, and value 1 at node m. Note that the order of of t (x) in the interval is less that that J of uj (x). This model allows discontinuities of t (x) on the boundary J and it is consistent within elements, i.e. linear displacements and constant tractions on each element. 15 If eqs. (2.7) and (2.8) are substituted into eq. (2.6) the following is obtained 2 egg) uj(n) + Asn mgl Jm(uc)i.j(x(n),€)N1(€)dfou§m-l) m+1 + As“, E351“, (uc>,.JN.de-u (“0 3; J § - Asm m-lJm (uR)1.J,_1N,<£>de 31.j(mvn) - Astm(uc)i.j(X(n).€)N2(€)d€ (m.n) 01.3 - c Jams)i J,oj(x‘“’.e)de where (uc)ij and (uR)1.1 are given by eq. (2.4). To put (uc)1 j and (uR)1 j in the proper form, we require p,<;p q2,111and112as functions of 5. Employing eq. (2.8), we have 17 (n) x -x - ai - bif where we have defined 31 _ xi(n) _ ;1(m) and (m) ‘ (m) bi- xi - xi I4 Figure 2.3 Definition of vectors a and b. (2.11) 18 Then R - a a - 2a b 5 + b b 52 i i i i i i q; - (al-bl)/p q: - (32'b2)/P (2.12) and n1 - (2/Asm)-b2 n2 - -(2/Asm)ob1. The integrals given by eq. (2.10) can now be computed by Gaussian quadrature except for the following special integrals A1.J(m’“) - Asm J:.(uc>,.1(x‘“’.5)N.(e>de‘ l m-n 1 61.3%“) " G J-l‘m1.1(x(n)’€)d€ ‘ (2.13) 1 81.j(m’n’ - Asm J-. m-n+1 1 c1 j‘m'“) - G J-.(uR),.j 0 n-m-l B1 1(m.n)_B2 (m.n)J n-m A1 1‘(m.n)_.A2 1(m.n)‘ t (1-u)/(2«) n-m-l B1 2(m.n)__Bz 1(m.n)J 20 Eq. (2.9) can also be written in matrix form as [uc]{Gu} - [uR]{F}. (2.14) This is a system of equations relating nodal displacements to resultant segment forces. In order to solve a well-posed elasticity problem, it is necessary to re-pose this system of equations in terms of nodal forces. Therefore, a transformation {Fl - [I‘]{F} (2.15) relating the vector of nodal forces {F} to the vector of segment forces (F) , is introduced into the system (2.15). The simplest physical interpretation of the transformation is to replace the segment forces by nodal forces equal to the average of the segment forces adjacent to each node , or F (n) - 1/2 [F(“) + F(“+1)]. (2.16) i i i The form of [F] for this transformation is II I 0 0 - . 0. 0 I I 0 - - 0 [I‘]- 1/2 0 0 0 I . . 0 o e e o o e e (2.17) [I 0 0 0 Id 21 where I is a 2x2 identity matrix. For an odd number of nodes, the inverse of [F] is [r]' - -1 1 I -I - . - I (2.18) L-I I-I I ---I_ and eq. (2.14) becomes ,1 [uCIIGul - [uRJIP] {F}. (2 19) ILt should.be noted that, for an even number of nodes, [I1 has no inverse. We can obtain the diagonal 2x2 blocks of [uc] through a simple observation. If we apply a rigid body displacement to the body (i.e. 1 2 N 1 2 N u1 - u1 -.... - u1 , u2 - u2 -.... - u2 ), this will generate no stress so that {F} - {0} and it follows that (“°)(2n-1)(2n-1)' g§%‘“°)(2n-1)(2m-1) -§ (“°)(2n-1)(2n )' 33%(“°)(2n-1)(2m ) 22 -E ‘“°’(2n )(2n-1)' g;g‘“°’(2n )(2m-1) (uc)(2n )(2n )- $41ch2n )(2m ). (2.20) .After solving eq. (2.19), stresses and displacements can be calculated anywhere in the body using a discretized form of eqs. (2.2) and (2.3),i.e. u 101) - ’m211m(uC)LJ(x(n),i) uj (i) d; +m§1 Jm(uR)1.J(x(n),x) t (i) a; (2.21) J aik(“> - -mgl Jm(nc)1koj(x(n),i) uj(x) a; +m§1 Jm(nR)1k.j(x(“),i) cj(i) a; (2.22) where i-l,2, k-l,2, u1(n) are the displacements at field point x(n), (n) and 01k are the stresses at x (n). If eqs. (2.7) and (2.8) are substituted into eqs. (2.21) and (2.22) the following is obtained 23 u (m- 1) 1(n) ' - Asmg1Jm(uc)1.j(x(n).€)N1(€)d€'uj As § I (no), JN2d5-tj( m) (2.23) (n) (n) . (m-l) “1k - - Asm m2, Jm(nc),k_j(x .e)N.(£>de uJ - As 5 Jm (nc). kjd£-u‘m) m m-l j (x‘n).e)de-cj‘m’ (2.24) + A8111 mil Jm - [uR*][F]'1{F} - [uc*]{Gul (2.25) (0,, 2x)» - [nR*1[r1"(Fi - [Hc*]{Gu} (2.26) * * All the entries of the matrices [uR ], [uc*], [Hc*] and [HR ] are calculated by numerical integration using Gauss-Legendre quadrature ( Conte and deBoor [36]). CHAPTER 3 CRACK PROBLEMS: INFINITE DOMAIN 3.1 The Displacement Discontinuity Method Consider an infinite elastic plane in which there is a point, i, at which some "source" of stress is located and a field point, x, at which the stresses are to be computed. At each of these points, we will be referring to small integral surfaces as shown in Figure 3.1 described by unit normals fl, and n, respectively. X Figure 3.1 Source point and field point in an infinite plane. 24 25 Let us define functions «1 and a, at x such that 8x1 an, 011 ' 8x2 022 ' 5;: (3.1) an, dsz a - - 12 - 8x1 8x2 If we now introduce displacement discontinuities of unit magnitude in the x1 and x2 directions at i, we can obtain : + _ 3 _ 3 _ _ (“€71.1 ' -§é%;21[2n2q1 + 2n1q2 ' n2q1-3n1q2] + - s _ s _ _ (NC)1 2 ' (NC)2 1 ' 2np [2n1q1 -2n2q2 -n1q1+n2q2] (3.2) G(1+u) - 3 - 3 - - (NC)2 2 ' 2wp [-2n2q1 -2n1q2 +3n2q1+n1q2] where («c)1 1 - 1r1 at x due to a unit displacement discontinuity in the 1 direction, at i in the infinite domain, etc. Next consider a line of length 2, as shown in Figure 3.2, across which the displacement is discontinuous by amounts c1(s) and c2(s). 26 x2 I} Figure 3.2 Line defining a crack in the infinite plane. 27 Then, by superposition, we have at x : 2 - *1 ' Jo[(FC)1.1¢1 + (*C)1.2C2]d5 t2 - Jf [(«c)2.1c1 + (sc)2.2c2]ds - + (3.3) where c1 - ui - ui, i-l,2, and all integrals are interpreted in the Cauchy principal-value sense. Suppose that the line is a crack, the surfaces of which are subjected to equal and opposite tractions t1(s) and t2(s) as shown in Figure 3.3. Then, t1 ' ’[011n1 + 012n2] 6K1 dX2 3'1 dxl ' ' a—x2 E + a, E dxl - ds and similarly dtz t2 - - ds so that «1(3) - -J: t:(§)a§ 1-1,2 0 (3.4) (3.5) (3.6) 28 along the crack, and eqs.(3.3) become J£[(1rc)1 1c1 + (arc)1 2c2]ds - -J: ttds e a o (3.7) J:[(fic)1.2cl + (”C)2.202]dS - -J:ot:d§. Thus, if t:(s) and. t:(s) are given, eqs.(3.7) can be solved for c1(s) and c2(s). Then the displacement discontinuities normal and tangential to the crack surfaces are (3.8) Ct - Cznl ' 61112. Once cn and ct are known, we can readily compute the stress intensity factors as follows E 21 KIfis-O - 8 J c cn(€) E 2i KIlls-o ' 8 J e °t(‘) (3.9) E Z! KIls-2 - 8 J c cn(£-€) K % J %1 ct(£-e) Ills-2 - where 6 << 2. 29 it Figure 3.3 Equal and opposite crack surface tractions. 30 3.2 Numerical Treatment Eq. (3.7) can be solved numerically if the crack surface is approximated by N straight segments, as shown in Figure 3.4 X2“ x1 Figure 3.4 Discretized crack surface. For this model, eq. (3.7) can be written as - - - * mgljm(«c)i.j(x(n),x)cj(x)ds - '"1 (n) (3.10) where x(n)is the mid-point of element n, n - 1,.. ,N, and summation on j is implied. 31 The displscement discontinuities in each segment m can be approximated using shape functions so that (m-l) J (m) c (i) - Nlc + N2cj (3.11) J where N1(€) - (1-€)/2. N2(€) - (1+€)/2 + N,(e) x(m) (3.12) a; - [(sm-sm_1)/2]-d£ - [Asm/2]-d£. If eq. (3.11) and (3.12) are substituted into eq. (3.10) the following is obtained Gm§1[A§?jn)c§m-l) + BiTjn)c§m)] — -n:(n) (3.13) where (m.n) (m.n) Ai.j ' JmN‘Di.j d5 (m.n) 32 93'5“) - [ 00 68833 3686 8%: 12 $6 In 6.39.1 damnvfid mm 00 new on Dr 0 L p H I A . O. rl. sown—.0 uo 02.0035. ll 1 6:32 I a I fimdl 1 __ < A I fiod vun // I a w u. I 7nd 3 _ < 0.. //I’ an 0 0; III 04. 56 0h .0020 37:622.. :0 E xooto next} otumEExmlmco :0 “—0 mhzm Nim mtnmfi 2.33 00 W? L L :nu< -L..IlIl-l.l.LI-._l- -IIIIII Fo.onAoN0\ n I P N.oImoN0\ n mlm F ¢.onnowv\ n «I4 _. m.onnomv\ n 0.6 0N... Ian... 57 5.2 e f n P o ems Here we employ the coupled model developed in chapter 4 to find numerical solutions for some finite domain problems. 5.2.1 Straight—91.8213 (a) Central symmetric crack with tensile load (Mode I) The results for the stress intensity factors (SIFs) of the crack problem shown in Figure 5.13 are plotted in Figure 5.14. The length of the crack is varied, while the boundary element mesh is maintained and the same number nodes are used define the crack. With this coupled model, it can be seen that good agreement with the analytical solution of Isida [32] is obtained up to a ratio of a/b - 0.7. For a/b > 0.7, the SIFa are slightly lower than the analytical values. (b) Central unsymmetric crack with tensile load The SIFs for the problem shown in Figure 5.15 are given in Table 5.3. Results agree well with the analytical solution given by Isida [32]. (c) Angled crack in a large plate The results for an angled crack as shown in Figure 5.16 are shown in Table 5.4. Several angles of inclination are considered. As expected, the results are very close to the analytical solution for an 58 a l l; I :l t [I l 1) | h . I . I: | i : 2a 2: b - 2 I b h - 3b 13 nodes h on crack surface I) I) ; .l'. ”9-, n4" ,4?” L b I b A Figure 5.13 Center-cracked test specimen. 59 .cochomam ummu 09.001.01.3ch Low n\0 .m> mn:m vim 830E £6 0.P 0.0 0.0 #0 N0 0.0 r L Li L! L! 7 b 0.0 SE sooxm II I .532 I 1 1 n 1 a 1 a I 0._. I u a I A. 1 a I 10.N 1 a I I fi 1 n 1 a 0.0 on /\0 /')l 60 Tip 2 Figure 5.15 An off-center straight crack. b - 2 h - 3b b1- 1.5 e - 0.5 13 nodes on crack surface 61 |/>fi 10 2a 0 H v 4: 4* ' [4;7 14;? r¢¥7 riLv r;;7 l; 10 J 13 nodes on crack surface 2a - J2 KI - ojka sinzfi K11 - a/ua sinfl-cosfi Figure 5.16 Angled crack in a large plate. 62 Table 5.3 Stress intensity factors for an off-center straight crack. K x x [COUPL] ___l_ __l_ aJua [COUPL] a/ra [32] K1 [32] Tip 1 1.395 1.405 0.99 Tip 2 1.189 1.200 0.98 lhble SJL Stress intensity factors for an angled crack in a large plate. K K K + K + _7_1_ .___11_ ___1_ ___ll_ ang1e(fi) a «a [COUPL] a/xa [COUPL] a/xa ajxa 90 0.000 0.000 0.000 0.000 60 0.249 0.427 0.250 0.433 45 0.497 0.493 0.500 0.500 30 0.745 0.427 0.750 0.433 0 0.994 0.000 1.000 0.000 + Analytical solution for an angled crack in an isotropic infinite medium is given by K1 2 - ajna sin fl and K II - a/«a sinfl cosfi. 63 crack in an infinite medium, since the plate is very large compared with the size of the crack. (d) Angled crack in a plate Results for the problem shown in Figure 5.17 are given in Table 0 o 5.5. Excellent accuracy has been obtained for 45 and 90 cracks, for which analytical results are available. 64 a r v h ’ 0 D 1 D fl 4} 7 " ”—lr W 2a 3 a - J2 b - 2 ' 0 h - 3b 13 nodes h on crack surface ' 0 : _fl 0 A 0 I777 777'! 7777 L b A b :l Figure 5.17 Angled crack in a finite plate. 65 Table 5.5. Stress intensity factors for an angled crack in a finite plate. K K K K K COUPL K an 1cm) _1_ _I.L __1_ __1_1_ _11 1 g aJna ajxa aJxa aJaa XI [32] KII [COUPL] [COUPL] [32] [32] 60 0.363 0.507 _ _ _ _ 45 0.713 0.570 0.730 0.600 .98 .95 30 1.000 0.467 0 1.449 0.000 1.488 0.000 .97 66 5.2.2 Kinked Crack Because an exact solution does not exist, the kinked crack within a finite body can not provide a simple check on the accuracy of the numerical solution. Nevertheless, let us introduce some of numerical results. Figure 5.20 presents results for an asymmetric kinked crack in a finite body as shown in Figure 5.18 while Figure 5.21 presents results for an anti-symmetric kinked crack as shown in figure 5.19. The tendencies of the numerical results are very similar to those of such cracks in infinite bodies, but have higher values. Computer CPU times for the problems that have been presented here were less than 10 sec for all cases using the Prime 750 computer. 67 0' l L T 3] ' H h 0 4 b1 0 " 1::{ 4 . 2 0 u a D H b - 2 0 h - 3b 13 nodes h on crack surface 0 D L Figure 5.18 Asymmetric kinked crack in a finite plate. 68 a 1i 4 L J :L 0 U h 0 U 0 b1 01 1: ' W P” : 2a 0 b - 2 ’ ” h - 3b 13 nodes h on crack surface ’ » r A?" :I b b ”9» .3. I: 1 Figure 5.19 Anti-symmetric kinked crack in a finite plate. 69 .320 BE: 0 E x0000 Dmex otumEExmo :0 $0 mnzm DNA.“ mtsmE 6020de mm mm mm. mm We 0 6.013me 0.0 mm T. . N.onmomv\rn mlm . - m.onAo~v\rn «In 13.7. . o._quNv\ n I a I Tmndl . __.....< a i 13.0.. .N . . N .... ImNd w . _H< a .. Tmnd a .. I 1mm; 70 .803 BE: 0 E xooto nmxcmx owmeExmlbco co u.0 mnzm —N.m 0.390 3035 Ox. 00 9V on m_ o L p _ L w MN.—.l. SdHAoN