Elastic Crack Growth in Finite Elements With Minimal Remeshing-belytschko and Black-1999

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 45, 601–620 (1999)

ELASTIC CRACK GROWTH IN FINITE ELEMENTS WITH MINIMAL REMESHING T. BELYTSCHKO∗;† AND T. BLACK‡ Departments of Mechanical and Civil Engineering; Northwestern University; 2145 Sheridan Road; Evanston; IL 60208; U.S.A.

SUMMARY A minimal remeshing ÿnite element method for crack growth is presented. Discontinuous enrichment functions are added to the ÿnite element approximation to account for the presence of the crack. This method allows the crack to be arbitrarily aligned within the mesh. For severely curved cracks, remeshing may be needed but only away from the crack tip where remeshing is much easier. Results are presented for a wide range of two-dimensional crack problems showing excellent accuracy. Copyright ? 1999 John Wiley & Sons, Ltd. KEY WORDS:

ÿnite elements; fracture; partition-of-unity

1. INTRODUCTION This paper describes a method whereby cracks and crack growth can be modelled by ÿnite elements with no remeshing. By this method, a crack arbitrarily aligned within the mesh can be represented by means of enrichment functions. Stress intensity factors for the crack are computed with errors less than 1 per cent. The essential idea in this method is to add enrichment functions to the approximation which contains a discontinuous displacement ÿeld. The same span of functions developed in Fleming et al. [4] for the enrichment of the element-free Galerkin method is used. The method exploits the partition of unity property of ÿnite elements which was noted by Melenk and BabuÄska [1], and Duarte and Oden [2], namely that the sum of the shape functions must be unity. This property has long been known, since it corresponds to the ability of the shape functions to reproduce a constant, that is to represent translation, which is crucial for convergence. The method can solve most crack growth problems without any remeshing. However, for severely curved cracks, some remeshing near the crack root may be necessary. Since remeshing near the crack root is much easier than remeshing at the crack tip (it may only need to be

∗ Correspondence

to: T. Belytschko, Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, U.S.A. E-mail: [email protected] † Walter P. Murphy, Professor of Computational Mechanics ‡ Research Assistant, Theoretical and Applied Mechanics Contract=grant sponsor: Oce of Naval Research Contract=grant sponsor: Army Research Oce

CCC 0029–5981/99/170601–20$17.50 Copyright ? 1999 John Wiley & Sons, Ltd.

Received 10 July 1998

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T. BELYTSCHKO AND T. BLACK

done in a few steps of a crack growth simulation), the method can be said to require minimal remeshing. The competition aorded by alternative methods is substantial. These problems can be solved by 1. mesh-free methods such as the element-free Galerkin method [3–5], 2. nite element methods with continual remeshing [6] and 3. boundary element methods. However, this method oers unique advantages: 1. it is a nite element method and can exploit the large body of nite element technology and software, 2. in contrast to boundary elements, it is readily applicable to non-linear problems and 3. in contrast to nite elements with continuous remeshing, it does not require as many projections between dierent meshes. Thus, it is a very attractive method. It is also worthwhile to contrast this with other embedded nite element crack methods such as Oliver [7]. In the aforementioned, the crack is represented as a discontinuity in the displacements within the element. However, crack tips cannot be represented inside an element and the essential singularity of the elds is absent. Incidentally, a combination of these techniques with the present technique may be promising. The paper is organized as follows: Section 2 reviews the governing equations, Section 3 describes the method, examples are presented in Section 4, conclusions and extensions of the method are discussed in Section 5, and the appendix includes a review of the growth law used in this paper and a discussion of interaction integrals for the computation of stress intensity factors.

2. GOVERNING EQUATIONS We consider small displacement elastostatics, which is governed by the equation of equilibrium: ∇·b + b=0

in

(1)

U = ∇s u

(2)

where b = C : U;

In the above equations, ⊂ R2 is the domain of the body, b is the Cauchy stress tensor, U is the small strain tensor, b is the body force, C is the material moduli tensor, u is the displacement, ∇ is the gradient operator, and ∇s is the symmetric gradient operator. The essential and natural boundary conditions are u = u

on

u;

n · b = t

on

t

(3)

where = u ∪ t is the boundary of with unit normal vector n, and u and t are prescribed displacements and tractions, respectively. The crack faces considered are traction-free. Prior to the description of our method, we recall some basics concerning asymptotic elds in fracture mechanics. The computation of stress intensity factors and crack growth laws in two Copyright ? 1999 John Wiley & Sons, Ltd.

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dimensions are described in the appendix. The near tip displacement elds for combined Modes I and II loading are KI p r=2 cos(=2)[ − 1 + 2 sin2 (=2)] u(x; y) = 2 +

KII p r=2 sin(=2)[ + 1 + 2 cos2 (=2)] 2

(4)

KI p r=2 sin(=2)[ + 1 − 2 cos2 (=2)] 2

v (x; y) =

−

KII p r=2 cos(=2)[ − 1 − 2 sin2 (=2)] 2

where  is the Kolosov constant

  3 − 4; = 3−  ; 1+

(5)

plane strain plane stress

where r and  are polar co-ordinates with origin at the crack tip and x1 -axis oriented into the body and parallel to the crack faces. The displacement eld is contained in the span of the following four functions:           √ √ √  √    4 ; r sin r cos ; r sin sin(); r cos sin() (6) { I (r; )}i=1 ≡ 2 2 2 2 These functions will be used to enrich the trial space by explicitly including these functions near the crack tip and along the faces. This is accomplished by using extra degrees of freedom at selected nodes. 3. FORMULATION The method consists of an enrichment of the nite element partition of unity near the crack tip with functions (6) whose span include the two-dimensional plane strain asymptotic crack tip elds. Without enrichment, the FEM requires considerable mesh re nement near the tip and the nite element mesh needs to conform to the geometry of the crack faces. Selected nodes around the crack have extra degrees of freedom associated with the enriching functions. The approximation takes the form of an extrinsic enrichment and can be written as ! nP n E (I ) P h NI (x) uI + ajI j (r; ) (7a) u (x) = h

v (x) =

I =1

j=1

n P

nP E (I )

I =1

NI (x) vI +

j=1

!

bjI j (r; )

(7b)

where (r; ) is a polar co-ordinate system with origin at the crack tip and the x1 -axis parallel to the last segment and N1 (x) are the standard nite element shape functions. The enrichment Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 1. Two enriching strategies near the crack: (a) crack passes through the cells and nodes surrounding the whole crack are enriched; and (b) the mesh conforms to part of the crack and enrichment is only near the crack tip including the last mesh conforming node

coecients aiI and biI are associated with nodes and nE (I ) is the number of coecients for node I ; for this case nE (I ) = 4 at the enriched nodes. In this paper, nE (I ) is set prior to running the program: it is chosen to be 4 for all nodes around the crack tip and zero at all other nodes. The four enrichment functions in equation (6) are used. Thus the enrichment is active only on a set of nodes around the crack. Two enriching methods are represented in Figure 1. The rst method passes the crack through the cells and enriches all nodes in a small region (a tube) surrounding the crack. The enriching Copyright ? 1999 John Wiley & Sons, Ltd.

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functions allow for the proper jump in eld variables along the crack. The second method uses a mesh conforming to a portion of the crack, that is a part of the crack is explicitly modelled by the mesh. Only nodes about the crack tip, including the portion of the crack that is not explicitly modelled, are enriched. Near the crack tip the enriching functions allow for the proper jump in eld variables while away from the tip the mesh is adjusted so that element edges conform to the crack. The enrichment functions (6) are discontinuous along the ray,  = ±, emanating from the crack tip to account for the presence of the crack. We want to show that using functions with a jump discontinuity along the crack produce a solution with zero traction along the crack faces. We consider the general problem and then observe how our discrete approximation satis es the traction condition along the crack. Consider the case where approximations (7) are discontinuous along the curve c . We still insist that our approximation satis es the principle of virtual work with no body force Z Z ∇s v : b d = v · t d ∀v (8)

t

where t is the given traction on t . We use the divergence theorem on the rst integral in (8) to obtain Z Z Z Z v · ∇ · b d − v · ( t − bn) d + v · bn d + v · b n d = 0 ∀v (9) −

−

c

c+

t

If v is chosen to be zero on

t

∪

c−

∪

c−

then

c+

Z

v · ∇ · b d = 0

−

If v is chosen next to be non-zero only on Z

c

t

then

v · ( t − b n) d = 0

t

So we are left with the integrals along the crack faces Z Z v · bn d + v · bn d = 0 c+

(10)

c−

By choosing v to be continuous and non-zero on c+ and zero on Z v · bn d = 0

c−

we obtain from (10)

c+

Consequently, b n = 0 along c+ . Similarly, b n = 0 along c− . Therefore, approximations with a jump discontinuity along the crack satisfy b n = 0 along the crack faces. The numerical approximation using (7) satis es the traction free condition in an averaged sense. Substituting (7) into (10) and using summation notation gives ∀uIi ; aIj Z Z (uIi NI + aIj NI j )ti d + (uIi NI + aIj NI j )ti d = 0 (11) c+

Copyright ? 1999 John Wiley & Sons, Ltd.

c−

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where ti ≡ (b n)i . Therefore, the following conditions holds for each i; j; I : Z Z NI j ti d + NI j ti d = 0 c+

Z

Z

NI ti d + c+

(12)

c−

NI ti d = 0

(13)

c−

The crack is modelled as a series of straight-line segments connecting vertices. The vertices are numbered beginning at the crack tip, which is (x1 ; y1 ) = (xtip ; ytip ). As the crack grows, a new crack segment is added to the model. An initially straight crack may curve and the crack representation develops kinks. The method is able to treat cracks which are not straight by using a procedure similar to [4]. In this procedure, we align the discontinuity in the enriching functions with the crack by a sequence of mappings that rotate each section of the discontinuity onto the crack model. The enrichment functions are constructed through a sequence of maps; one map is used for each straight-line segment of the crack, beginning with the segment at the crack tip. We rst describe the modi cation of the enrichment functions for the segment adjacent to the crack tip segment. The crucial step in the modi cation of the enrichment functions is to change the angle in  y) in terms of equation (6). This is done as follows. Given a point (x; y), we de ne an angle (x; the angle of the segment R (see Figure 2) and the sampling point (x; y) by   Ca ( − R ); ¿R (14)  =  Cb ( − R ); ¡R Ca ≡

=2 3=2 − R

(15)

Cb ≡

=2 R − =2

(16)

Then we map the co-ordinates of the sampling point (x; y) to co-ordinates in the crack tip frame (x∗ ; y∗ ) by  −r sin ())  (x; y) 7→ (x∗ ; y∗ ) ≡ (−l − r cos (); where l≡

q (x2 − xtip ) 2 + (y2 − ytip ) 2

r ≡

p (x2 − x) 2 + (y2 − y) 2

The variables r and  in the enrichment functions (6) are given by p r = (x∗ ) 2 + (y∗ ) 2  ∗ −1 y  = tan x∗ Copyright ? 1999 John Wiley & Sons, Ltd.

(17)

(18) (19)

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Figure 2. Mapping used for kinked cracks: (a) initial geometry for the modi cation of the enrichment functions for the segment adjacent to the crack tip segment; (b) the mapped point (x∗ ; y∗ ) used to determine r and  in the enriching functions (6)

At step n in the mapping sequence the algorithm is as follows (see Figure 3). The angle  y) is computed using (14) where the angles R and  (x; y) are determined from the current (x; con guration. The new enrichment function variables are given by (17) with q l ≡ (xn − xtip ) 2 + (yn − ytip ) 2 r ≡

p (xn − x) 2 + (yn − y) 2

 is less than the length of the next segment in the mapped co-ordinates If r cos () p (xn+1 − xn ) 2 + (yn+1 − yn ) 2 , then the new co-ordinates (17) are used to de ne r and  in the enriching functions i (r; ); i = 1; 4, by (18). Otherwise, the co-ordinates of the remaining crack vertices are rotated using (17) and a new mapping is constructed as just described. Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 3. Mapping used for kinked cracks: (a) the geometric con guration at step n in the mapping sequence (17). The previous crack segments are aligned with the tip segment and a new mapping is needed to compute the enriching functions at (x; y); and (b) the mapped point (x∗ ; y∗ ) used to determine r and  in the enriching functions (6)

The mapping can be imagined as a virtual rotation of the crack to align it with the leading crack segment leaving the eld xed on the perpendicular to the leading crack segment through the rotation point. This sequence of mappings leaves the length of the crack invariant. For longer cracks, a second method is described that will generally change the eective length of the crack in the mapped con guration. For longer cracks it may be necessary to approximate this sequence of maps by a single map. Given the point (x; y), to use mapping (17) it is necessary to nd a point on the discontinuity ray from the tip and the corresponding crack segment to be rotated about this point. The crack segment is the last segment from the tip with negative projected vector xrel on the discontinuity ray (x1 -axis) (see Figure 4). Given this segment, the rotation point (xrot ; yrot ) is the intersection of the extended crack segment with the discontinuity ray. The mapping is constructed using this data as described in the previous paragraphs. The crack length is changed by the projections and rotations onto the discontinuity ray. It is necessary to obtain spatial derivatives of the enriching functions (6) in the original global co-ordinate system. At each stage of the mapping sequence, the derivatives with respect to old and new co-ordinates are related by the chain rule   @   @   − cos  cos  − Ck sin  sin  − sin  cos  + Ck cos  sin  @xold @xnew   @  (20)  @  =   sin  + Ck sin  cos  − sin  sin  − Ck cos  cos  − cos  @yold @ynew Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 4. Geometry for the approximate kink map. The crack is eectively shortened using this method

where Ck = Ca or Cb depending on  and R as in (14). Then the desired derivatives are obtained by multiplying in succession the matrices of each map used in nding the elds.

4. EXAMPLES Several problems are solved to illustrate the eectiveness of the enrichment method in fracture problems. Solutions are given for single- and mixed-mode static problems and for quasi-static crack growth. Basics related to fracture parameters and their computation and crack growth laws in two dimensions are discussed in the appendix. Stress intensity factors are computed with the domain form of interaction integrals. The direction of crack growth is determined from the maximum circumferential (hoop) stress criterion.

4.1. Tensile and shear edge cracks A rectangular plate with an edge crack is shown in Figure 5 and loaded rst in tension and then in shear with plane strain conditions. The geometric parameters are chosen to be a=W = 0·5; h=W = 16·0=7·0 and W = 7·0 in. Four meshes with two dierent enrichment zones near the crack are considered. Two of the four meshes consist of nodes evenly spaced in the x-direction and in the y-direction and are denoted even1 (12 × 24 nodes) and even2 (24 × 48 nodes). The crack passes through the centre of the elements in these two meshes and the enrichment encloses the crack as shown in Figure 1(a). The other two meshes conform to a portion of the crack as is typical in FEM analysis where the mesh conforms to all of the crack. These two meshes are denoted conform1 (280 nodes) and conform2 (1135 nodes). The enrichment zones are near the crack tip only as shown in Figure 1(b). Stress intensity factors are computed on four dierent domains using the domain form of the interaction integrals. The weighting function q(x) is de ned using Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 5. Geometric parameters for edge crack, shown here with shear loading

Table I. Normalized KI values using smaller enrichment zone Domain number dom1 dom2 dom3 dom4

even1 16 enriched

even2 28 enriched

conform1 12 enriched

conform2 21 enriched

0·9768 0·9785 0·9730 0·9729

0·9932 0·9882 0·9883 0·9883

0·9277 0·9869 0·9800 0·9798

0·9953 0·9938 0·9943 0·9942

the nite element structure. Four boxes centred at the crack tip are considered dom1 : 0·5 × 0·5 × 0·5 × 0·5 dom2 : 1·0 × 1·0 × 1·0 × 1·0 dom3 : 2·0 × 2·0 × 2·0 × 2·0 dom4 : 2·5 × 2·5 × 2·5 × 2·5 and for each box q(x) = 1 over the cells completely inside the box and q(x) ramps to 0 over the cells containing the boundary of the box using the nite element shape functions for those cells. 4.1.1. Tensile edge crack. The plate is loaded in tension at the top with  = 1·0 psi and essential boundary conditions are applied to the bottom of the plate. The computed √ mode I stress intensity factors are compared with a nite geometry corrected value KI = C a where the correction is given by Ewalds and Wanhill [8], C = 1·12 − 0·231(a=W ) + 10·55(a=W 2 ) − 21·72(a=W )3 + 30·39(a=W )4 . The normalized KI values for each of the four meshes with the smaller enrichment zone are shown in Table I, and in Table II with the larger enrichment zone. There is excellent agreement with the corrected KI value. Copyright ? 1999 John Wiley & Sons, Ltd.

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ELASTIC CRACK GROWTH IN FINITE ELEMENTS

Table II. Normalized KI values using larger enrichment zone Domain number dom1 dom2 dom3 dom4

even1 54 enriched

even2 90 enriched

conform1 55 enriched

conform2 88 enriched

0·9960 0·9974 0·9958 0·9942

1·0003 0·9986 0·9975 0·9975

0·9332 0·9953 0·9898 0·9899

0·9981 0·9978 0·9988 0·9985

Table III. Normalized KI values using smaller enrichment zone Domain number dom1 dom2 dom3 dom4

even1 16 enriched

even2 28 enriched

conform1 12 enriched

conform2 21 enriched

0·9738 0·9755 0·9690 0·9688

0·9940 0·9886 0·9885 0·9885

0·9231 0·9844 0·9763 0·9759

0·9959 0·9942 0·9945 0·9945

Table IV. Normalized KI values using larger enrichment zone Domain number dom1 dom2 dom3 dom4

even1 54 enriched

even2 90 enriched

conform1 55 enriched

conform2 88 enriched

0·9945 0·9958 0·9943 0·9924

1·0012 0·9995 0·9982 0·9982

0·9291 0·9934 0·9874 0·9873

0·9985 0·9982 0·9993 0·9989

4.1.2. Shear edge crack. The plate is clamped on the bottom and loaded by a shear traction  = 1·0 psi on the top. The reference mixed mode stress intensity factors are [9], √ KI = 34·0 psi in: √ KII = 4·55 psi in: The normalized computed KI values with the smaller zone are shown in Table III, and with in Table IV the larger zone. The normalized KII values with the smaller zone are shown in Table V, and with in Table VI the larger zone on each of the four meshes. Again, the results are in good agreement with the reference solution. 4.2. Centre and centre curved cracks A centre straight crack in a panel is considered to demonstrate how to enrich in multiple regions of the domain and a centre curved crack is considered to show the eectiveness of the kink mapping (17). The parameters for each problem are the elastic modulus E = 3 × 107 psi, Copyright ? 1999 John Wiley & Sons, Ltd.

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Table V. Normalized KII values using smaller enrichment zone Domain number dom1 dom2 dom3 dom4

even1 16 enriched

even2 28 enriched

conform1 12 enriched

conform2 21 enriched

0·9832 0·9968 0·9915 0·9921

1·0009 0·9949 0·9956 0·9957

0·9937 0·9979 0·9888 0·9892

0·9936 0·9929 0·9952 0·9953

Table VI. Normalized KII values using larger enrichment zone Domain number dom1 dom2 dom3 dom4

even1 54 enriched

even2 90 enriched

conform1 55 enriched

conform2 88 enriched

0·9921 0·9974 0·9968 0·9960

0·9981 0·9972 0·9976 0·9977

0·9971 1·0008 0·9927 0·9926

0·9949 0·9947 0·9965 0·9963

Figure 6. Geometric parameters for centre cracks, shown here for the centre curved crack

Poisson’s ratio  = 0·25;  = 1·0 psi, and plane strain conditions are assumed (Figure 6). The crack is situated at y = 0·0 from x = −2·0–2·0. 4.2.1. Centre crack. The purpose of this example is to describe a computational strategy for problems containing multiple crack tips. The full geometry, [−5·0; 5·0] × [−10·0; 10·0], is modelled by meshing the crack from x = −1·0−1·0 and using disjoint enrichment zones at each tip up to and including a small portion of the conforming part of the mesh. Two meshes, mesh1 consists of 402 nodes with 25 enriched nodes at each tip and mesh2 consists of 1606 nodes with 64 enriched nodes at each tip, are used for the computations. The reference solution for this geometry is [8], KI = 2·7641, where a nite geometry correction is used. Copyright ? 1999 John Wiley & Sons, Ltd.

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Table VII. Normalized KI values for Mesh1 SIFs

dom1

dom2

dom3

dom4

Intq Intq2 Jdomq Jdomq2

0·9924 1·0551 0·9894 1·0222

0·9984 0·9946 0·9966 0·9940

1·0053 1·0019 1·0102 1·0007

1·0024 0·9964 0·9980 0·9978

Table VIII. Normalized KI values for Mesh2 SIFs

dom1

dom2

dom3

dom4

Intq Intq2 Jdomq Jdomq2

1·0063 1·0135 1·0045 1·0062

1·0043 1·0029 1·0042 1·0025

1·0051 1·0116 1·0040 1·0070

1·0053 1·0033 1·0039 1·0037

The sensitivity of the interaction domain integral calculations to domain size are checked by using a second weighting function, q2 (x), on four domains centered at the crack tip dom1: 0·5 × 0·5 × 0·5 × 0·5 dom2: 1·0 × 1·0 × 1·0 × 1·0 dom3: 1·5 × 1·5 × 1·5 × 1·5 dom4: 2·0 × 2·0 × 2·0 × 2·0 The weight q2 (x) is 1 at the tip and ramps linearly to 0 at the edges of the boxes. The integration cells align for the q weight function but not for the q2 weight. Therefore, there is larger integration error with the q2 weight function. The idea is that the integration error in the calculation of the integrals is still acceptable since, the weighting function q2 is close to 0 near the edges of the domain box where the integration error is largest. Additionally, KI values are computed with the domain form of the J integral, [10]. The normalized KI values using weight functions q and q2 with domain forms of the interaction and J integrals are shown in Tables VII and VIII. Contours of the normal stress yy are shown in Figure 7. 4.2.2. Centre curved crack. The centre curved crack, shown in Figure 6, is modelled by symmetry with domain [0·0; 10·0] × [−10·0; 10·0]. The stress intensity factors for the in nite body, [11], are given by # " 2 2  1=2 (1 − sin ( =2) cos ( =2)) cos( =2) + cos(3 =2) KI = (R sin( )) 2 1 + sin2 ( =2) # " 2 2  1=2 (1 − sin ( =2) cos ( =2)) sin( =2) + sin(3 =2) KII = (R sin( )) 2 1 + sin2 ( =2) Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 7. Centre crack yy stress contours

Figure 8. Description of the mesh near the curved crack, exaggerated, which is modelled by symmetry. The crack is partially opened along the symmetry boundary to facilitate the application of essential boundary conditions. There the mesh is aligned with the crack opening. The rest of the crack is discretely modelled as an internal boundary de ned by the enriching functions (6)

where R is the radius of the circular arc and 2 is the subtended angle of the arc. Figure 8 shows the discrete model of the crack near the symmetry boundary. The nodes along the symmetry boundary are not enriched to facilitate the application of essential boundary conditions. The enrichment is included along only part of the crack near the crack tip. The crack is partially opened along the remainder of its length as shown. Nodes around the crack tip, including a small part of the crack opening, are enriched with (6). Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 9. Initial geometry for the double cantilever beam specimen

Figure 10. Crack path through the mesh of 1675 nodes for the double cantilever beam specimen with initial perturbation angle of 5·71◦ and step size 0·05 in. The initial specimen is shown followed by the crack path after 12 and then 24 steps

The crack is a circular arc centred at (0·0; 3·75) with radius R = 4·25. The arc extends from (0·0; −0·5) on the symmetry boundary to (2·0; 0·0). The subtended angle is = 28·0725◦ . The stress intensity factors for the in nite body with this crack geometry are KI = 2·0146 and KII = 1·1116. The mesh consists of 1037 nodes with 54 enriched nodes near the crack tip and conforms to the circular crack from (0·0; −0·5) to (0·59300; −0·45842). Three domains are used for the interaction integrals using the weight function q(x) dom1: 0·4 × 0·4 × 0·4 × 0·4 dom2: 0·5 × 0·5 × 0·5 × 0·5 dom3: 0·8 × 0·8 × 0·8 × 0·8 Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 11. Crack paths for double cantilever beam with comparison of dierent initial perturbation angles

Table IX. Normalized KI values SIFs

dom1

dom2

dom3

Kink 11 Kink 21 No map

1·0117 1·0119 0·8240

1·0107 1·0113 0·8225

1·0227 1·0227 0·8356

Table X. Normalized KII values SIFs

dom1

dom2

dom3

Kink 11 Kink 21 No map

1·0056 1·0058 0·8089

1·0141 1·0148 0·8101

1·0567 1·0567 0·8241

The normalized KI and KII values using the kink mapping are compared with results without the mapping. We consider two models of the crack: 10 straight-line segments with 11 vertices (Kink 11) and 20 straight-line segments with 21 vertices (Kink 21), as shown in Tables IX and X. The stress intensity factors exceed the closed-form results for the in nite body. At this level of discretization, there was little, if any, improvement with increased crack discretization. As the domains increase in size, more of the curvature of the crack is involved in the numerical calculation of the stress intensity factors leading to error in the computed values. The mapping of the enrichment functions is de nitely needed to capture the correct behaviour near the crack. Copyright ? 1999 John Wiley & Sons, Ltd.

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4.3. Double cantilever beam In this example, a crack is grown quasi-statically in a double cantilever beam specimen. The geometry is shown in Figure 9, with L = 11·8 in:; h = 3·94 in:; a = 3·94 in: and P = 197 lbs. Plane stress conditions are assumed with elastic modulus E = 3 × 107 psi and Poisson’s ratio  = 0·3. The crack is given a small perturbation at the tip of length 0·3 in with initial angle d of 0·0; 1·43; 2·86, and 5·71◦ as shown in Figure 9. Figure 10 shows the evolution of the crack path through the mesh consisting of 1675 nodes for an initial perturbation angle of 5·71◦ . The crack advances a length 0·05 in: at each step. The stress intensity factors are computed using the domain form of the interaction integrals in a square domain with side lengths 1·2 in: centred at the crack tip. The direction of propagation is determined using equation (23). Figure 11 shows the crack paths for each of the initial perturbation angles and step sizes of 0·1 and 0·05 in: At each step, KII should be monitored. If KII is too large in comparison with KI , then the step size should be shortened and the solution computed again. 5. CONCLUSIONS A method has been presented for enriching nite element approximations so that crack problems can be solved with minimal remeshing. The nite element mesh in most cases need not conform to the shape of the crack. Thus crack growth problems can be solved with no or very little remeshing. The key feature of this method is the use of test and trial functions which are discontinuous. This discontinuity is placed along the path of the crack, so that the crack is not treated by the standard nite element shape functions. The discontinuous functions are added to the test and trial functions by using the original nite elements to generate a partition of unity. The results presented show excellent accuracy for a range of two-dimensional problems. In all cases, stress intensity factors agreed with analytic solutions to about 1 per cent. The extension to problems with material non-linearities should pose only minor diculties. The crack representation is embedded in the approximation, although the square root character of the near tip eld was embedded in the approximation, enrichment with more general elds is certainly conceivable. Extension to three dimensions also seems feasible. There are new dimensions of diculty in enrichment of three-dimensional crack elds in that the enrichment usually depends on a unique de nition of a normal to the crack front, which becomes ambiguous away from the crack front near sharp corners, see [12]. However, with some restrictions these diculties are surmountable. Thus, this is a promising method in a large class of crack growth problems. APPENDIX We brie y review the crack growth law we used to specify the direction of crack growth. Among the criteria for determining the growth direction are: (1) the maximum energy release rate criterion [13] (2) the maximum circumferential (hoop) stress criterion or the maximum principal stress criterion [14], and (3) the minimum strain energy density criterion [15]. In this paper we use the maximum circumferential stress criterion, which is identical to the maximum energy release criterion for problems governed by equation (1). The maximum circumferential stress criterion Copyright ? 1999 John Wiley & Sons, Ltd.

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states that the crack will propagate from its tip in a direction c so that the circumferential stress  is maximum. The circumferential stress in the direction of crack propagation is a principal stress. Therefore, the critical angle c de ning the radial direction of propagation can be determined by setting the shear stress associated with (4) and (5) to zero   1 1 1 (22) cos(=2) KI sin() + KII (3cos () − 1) = 0 r = 2r 2 2 This leads to the equation de ning the angle of crack propagation in the tip co-ordinate system c KI sin(c ) + KII (3cos (c ) − 1) = 0 Solving this equation gives c = 2arctan 14 (KI =KII ±

p (KI =KII ) 2 + 8)

(23)

The stress intensity factors are computed using domain forms of the interaction integrals [9; 16]. For completeness these are discussed here. Co-ordinates are crack tip co-ordinates with the x1 -axis parallel to the crack faces. For general mixed mode problems we have the following relationship between the value of the J integral and the stress intensity factors K2 K2 J = ∗I + ∗II E E

(24)

where E ∗ is de ned in terms of material parameters E (Young’s modulus) and  (poisson’s ratio) as  plane stress  E; ∗ E =  E ; plane strain 1 − 2 Two states of a cracked body are considered. State 1 (ij(1) ; ij(1) ; ui(1) ) corresponds to the actual state and state 2 (ij(2) ; ij(2) ; ui(2) ) is an auxiliary state which will be chosen as the asymptotic elds for modes I and II. The J integral for the sum of the two states is  Z  1 2 1 (1) (2) (1) (2) (1) (2) @(ui + ui ) (1+2) ( + ij )(ij + ij )1j − (ij + ij ) = nj d J 2 ij @x1 Expanding and rearranging terms gives J (1+2) = J (1) + J (2) + M (1; 2) where M (1; 2) is called the interaction integral for states 1 and 2  Z  2 1 (1) @ui (2) @ui (1; 2) (1; 2) = 1j − ij − ij W nj d M @x1 @x1

(25)

(26)

and W (1; 2) is the interaction strain energy W (1; 2) = ij(1) ij(2) = ij(2) ij(1) Copyright ? 1999 John Wiley & Sons, Ltd.

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Figure 12. Derivation of the domain form of the interaction integral used in computation of the stress intensity factors

Writing equation (24) for the combined states gives after rearranging terms J (1+2) = J (1) + J (2) +

2 (K (1) K (2) + KII(1) KII(2) ) E∗ I I

(27)

Equating (25) and (27) leads to the following equation: 2 M (1; 2) = ∗ (KI(1) KI(2) + KII(1) KII(2) ) E

(28)

Choosing state 2 as Mode I asymptotic elds with KI(2) = 1 and KII(2) = 0 gives mode I stress intensity factor for state 1 in terms of the interaction integral KI(1) =

E ∗ (1; Mode M 2

I)

(29)

Then choosing state 2 as mode II asymptotic elds with KI(2) = 0 and KII(2) = 1 gives mode II stress intensity factor for state 1 KII(1) =

E ∗ (1; Mode M 2

II)

(30)

The contour integral de ning M (1; 2) is converted into an area integral by multiplying the integrand by a bounded weighting function q(x) that is 1 on an open set containing the crack tip and vanishes on an outer prescribed contour 0 . Then for each contour (see Figure 12) in this open set where q(x) = 1 and assuming the crack faces are stress free and straight in the interior of the region A bounded by the prescribed contour 0 , the interaction integral may be written as  Z  @u2 @u1 W (1; 2) 1j − ij(1) i − ij(2) i qmj d (31) M (1; 2) = @x1 @x1 C where C = + C+ + C− + 0 and m is the outward unit normal to the contour C. Now using the divergence theorem and passing to the limit as the contour is shrunk to the crack tip, which is Copyright ? 1999 John Wiley & Sons, Ltd.

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justi ed by the Dominated Convergence Theorem, gives the following equation for the interaction integral in domain form:  Z  2 1 @q (1) @ui (2) @ui (1; 2) (1; 2) = − ij −W 1j dA (32) ij M @x @x @x 1 1 j A The condition that the weighting function is 1 on an open set containing the crack tip is easily relaxed to be just = 1 at the tip.

ACKNOWLEDGEMENTS

The support of the Oce of Naval Research and the Army Research Oce is gratefully acknowledged. REFERENCES 1. Melenk JM, Babuska I. The partition of unity nite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996;39:289–314. 2. Durate C, Oden J. hp clouds—a meshless method to solve boundary-value problems. Technical Report, TICAM, 1995. 3. Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. International Journal of Numerical Methods in Engineering 1994;37:229–256. 4. Fleming M, Chu YA, Moran B, Belytschko T. Enriched element-free Galerkin methods for singular elds. International Journal for Numerical Methods in Engineering 1997;40:1483–1504. 5. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996;139:3– 47. 6. Swenson D, Ingraea A. Modeling mixed-mode dynamic crack propagation using nite elements: theory and applications. Computational Mechanics 1988;3:381–397. 7. Oliver J. Continuum modelling of strong discontinuities in solid mechanics using damage models. Computational Mechanics 1995;17:49– 61. 8. Ewalds H, Wanhill R. Fracture Mechanics; Edward Arnold: New York, 1989. 9. Yau J, Wang S, Corten H. A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. Journal of Applied Mechanics 1980;47:335–341. 10. Moran B, Shih C. Crack tip and associated domain integrals from momentum and energy balance. Engineering Fracture Mechanics 1987;127. 11. Gdoutos E. Fracture Mechanics; Kluwer Academics Publishers: Boston, 1993. 12. Black T. Mesh-free applications to fracture mechanics and an analysis of the corrected derivative method. Ph.D. Thesis, Northwestern University, 1998. 13. Nuismer RJ. An energy release rate criterion for mixed mode fracture. International Journal of Fracture 1975;11: 245–250. 14. Erdogan F, Sih G. On the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering 1963;85:519–527. 15. Sih GC. Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture 1974;10:305–321. 16. Shih C, Asaro R. Elastic–plastic analysis of cracks on bimaterial interfaces: Part i-small scale yielding. Journal of Applied Mechanics 1988;55:299–316.

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Int. J. Numer. Meth. Engng. 45, 601–620 (1999)