Brick 27

Collapsed 27 node Lagrangian element for three-dimensionalstress intensity computations

C.A. Grummitt a, G. Baker b,*

a Cullen, Grummitt and Roe, Consulting Engineers, 126 Wickham Street, Fortitude Valley, Queensland, 4006, Australiab School of Engineering, The University of Warwick, Coventry CV47AL, UK

Abstract

Considered are the details of a collapsed 27 node (singular) Lagrangian element for use in three-dimensional stress

intensity factor calculations. The element is shown to capture both the 1=��rp

singularity near the front as required for

linear elastic fracture mechanics and a 1=r singularity under a relaxed constraint for plastic fracture. It is shown that the

radial and circumferential strains exhibit the singularity, but that direct strain along the front is zero, thus providing

plane strain conditions. Of equal relevance is the quality of mesh at the front. For this we adopt a standard suite of

elements comprising a number of singular `wedge-shaped' elements around the crack tip in the form of a tube, with the

major axis placed along the front. A number of three-dimensional fracture problems are analysed to demonstrate the

accuracy and e�ciency of the collapsed element, even when used with moderate mesh sizes. Ó 1999 Elsevier Science

Ltd. All rights reserved.

1. Introduction

The use of singular elements for stress intensity calculations is now well established, with quarter pointserendipity and collapsed elements available in many commercial codes, at least for two-dimensionalproblems. It is well known that the solution for near-®eld stress around a crack tip in an elastic mediumexhibits a square root singularity, and it is this singularity that singular elements attempt to capture.

Early research [1] into singular elements used shape functions which provided a singularity in the strain®eld, but they su�ered from de®ciencies in basic strain and displacement conditions. It has been shown [2±4]that the required singularity in two-dimensional isoparametric elements can be obtained by simply movingthe mid-side modes to the quarter points. Further, it was shown [5] that triangular elements generated bycollapsing one side of an eight node quarter point element give better results than the quadrilateral parent.Suggested in [6] is that collapsed elements give more accurate results because the quadrilaterals do not have®nite strain energy at the tip, and hence the sti�ness matrix, too, is singular at this point; standard nu-merical integration provides the necessary non-singular sti�ness matrix but accuracy su�ers. Collapsedquarter point elements also exhibit the square root singularity of linear elastic fracture mechanics on all

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Theoretical and Applied Fracture Mechanics 32 (1999) 189±201

* Corresponding author.

E-mail address: [email protected] (G. Baker).

0167-8442/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 6 7 - 8 4 4 2 ( 9 9 ) 0 0 0 3 9 - 7

rays emanating from the tip, whereas quadrilateral singular elements only show the singularity on theboundaries [5].

These concepts can be extended to three-dimensional computations in the sense that a brick element canbe mapped into a prism by collapsing one face into an edge. It was shown in [7] that a collapsed 20 nodebrick element exhibits the required O(1/

��rp

) singularity on any ray from the crack front, on a plane per-pendicular to the front. It appears that the researchers in [8] were the few to consider the Lagrangian family,for both two- and three-dimensional problems. They employed a 27 node Lagrangian brick element forthree dimensional stress intensity computations, and in an e�ort to maximize the region over which thesingularity exists, she found it necessary to move the central and mid-face nodes to the 11/32 points fromthe front, and the mid-side nodes to the quarter points.

In this paper we present the details and validation of a collapsed 27 node Lagrangian brick elementspeci®cally developed for stress intensity calculations along the crack front in three dimensional solids. Themotivation for adopting a high order Lagrangian element is ®rstly that it provides the essential accuracynear a crack front, and secondly that it is fully compatible with ten-node tetrahedra which the authors usefor automatic meshing within the domain of the solid. The motivation for collapsing the 27 node brick is byanalogy with the collapsed serendipity elements [7] used previously. An important contribution of this studyis that the element proposed can retrieve the O�1= ��

rp � singularity for use in linear elastic fracture me-

chanics, or a O�1=r� singularity often required in plastic fracture. Moreover, the singularities can be shownto hold on any ray through the element from the crack tip.

The techniques employed to extract the stress intensity factors from the element response also a�ectaccuracy. The main techniques of displacement extraction, J -integral, virtual crack extension and sti�nessderivative methods were surveyed and shown that the virtual crack extension to be an accurate and stableapproach, albeit more expensive than displacement extraction. Validated in [9] is a number of displacementbased methods, concluding that the quarter point displacement technique (QPDT) was robust and accu-rate. More recent extensions of the in¯uence function technique [10] also hold promise. However, weemphasize that our aim has been to develop an accurate and convenient element on which to base thesetechniques for the analysis of three-dimensional fracture problems.

2. 27 node Lagrangian brick element

The shape functions for the 27 node Lagrangian solid element (shown in Fig. 1) are generated by aproduct of three Lagrange polynomials, each of which takes the general form:

lnk�n� �

�nÿ n0��nÿ n1� . . . �nÿ nkÿ1��nÿ nk�1� . . . �nÿ nn��nk ÿ n0��nk ÿ n1� . . . �nk ÿ nkÿ1��nk ÿ nk�1� . . . �nk ÿ nn�

; �1�

where the polynomial passes through order �n� 1� points, and nk represents the coordinate of the point forwhich the shape function is being determined.

Element shape functions are then formed from the triple product of polynomials in each coordinatedirection:

Ni � NIJK � lnI �n�lm

J �g�lpK�f�; �2�

where n;m; p represent the number of sub-divisions along the coordinate direction, and I ; J ;K refer to thepoint for which the shape function is being evaluated. For the 27 node brick element, there is just one mid-side node along each edge, so that n � m � p � 2. Refer to the explicit form of the 27 quadratic terms inAppendix A, for reference in the following.

190 C.A. Grummitt, G. Baker / Theoretical and Applied Fracture Mechanics 32 (1999) 189±201

3. Collapsed 27 node Lagrangian element

3.1. Shape functions

The singular element is generated by collapsing the face g � 1 of the parent element described above,and by moving all mid-side nodes, mid-face nodes and the central node towards the collapsed edge, asshown in Fig. 2. It has been found in numerical experiments that if the central and mid-face nodes (near thecollapsed edge) are placed at the 11/32 points, as used in [8] for the `un-collapsed' 27 node Lagrangian,then the O�1= ��

rp � singularity does extend over the entire element. On the other hand, we show that if all

Fig. 2. Collapsed 27 node Lagrangian element.

Fig. 1. Twenty seven node Lagrangian element.

C.A. Grummitt, G. Baker / Theoretical and Applied Fracture Mechanics 32 (1999) 189±201 191

mid-side, mid-face and central nodes are placed at the quarter points, then the required singularity doesextend over the whole element. Fig. 3 depicts an arbitrary section through the collapsed element with local`front' coordinate axes �x; y; z� also shown. The de®nition of these coordinates on the reference element isneeded to form the Jacobian for the collapsed element. The parent shape functions (given in Appendix A)can be used to map the collapsed element into normalized space using the geometric transformations:

x �X27

i�1

Ni�n; g; f�xi; y �X27

i�1

Ni�n; g; f�yi; z �X27

i�1

Ni�n; g; f�zi: �3�

The corresponding matrix of local coordinates for the chosen nodal positions is:

x1 x2 . . . x27

y1 y2 . . . y27

z1 z2 . . . z27

26643775

�0 0 ÿ h ÿ h 0 0 ÿ h ÿ h 0 ÿ h

4ÿ h ÿ h

40 ÿ h

4ÿ h ÿ h

40 0 ÿ h ÿ h ÿ h

40 ÿ h

4ÿ h ÿ h

4ÿ h

4ÿ h

4

0 0 d d 0 0 ÿ d ÿ d 0 d4

d d4

0 ÿ d4ÿ d ÿ d

40 0 0 0 0 0 0 0 0 d

4ÿ d

4

L ÿ L ÿ L L L ÿ L ÿ L L 0 ÿ L 0 L 0 ÿ L 0 L L ÿ L ÿ L L 0 0 ÿ L 0 L 0 0

24 35�4�

Substituting these coordinates into the transformations in Eq. (3), and invoking the shape functions in Eqs.(18)±(23) yields the following straightforward result for coordinate transformation:

x � ÿ h4�1ÿ g�2; y � d

4f�1ÿ g�2; z � Ln: �5�

With this result, the Jacobian and hence the global derivatives of the displacement can be formed for use inthe usual ®nite element strain calculations.

That is, we ®nd the Jacobian explicitly:

J �

oxon

oyon

ozon

oxog

oyog

ozog

oxof

oyof

ozof

26643775 �

0 0 Lh2�1ÿ g� ÿ d

2f�1ÿ g� 0

0 d4�1ÿ g�2 0

264375; �6�

and its determinant as:

jJ j� dhL8�1ÿ g�3; �7�

Fig. 3. Cross-section through collapsed element.


from which the singularity of the transformation on the `face' g � 1 is obvious. Finally, the inverse of theJacobian for all points g 6� 1 can be written:

Jÿ1 �0 2

h�1ÿg�4f

h�1ÿg�20 0 4

d�1ÿg�21L 0 0

264375: �8�

This inverse is of course used in the normal way to relate shape function derivatives with respect to ref-erence coordinates �x; y; z� to those with respect to local coordinates �n; g; f�:

ouox

ovox

owox

ouoy

ovoy

owoy

ouoz

ovoz

owoz

264375 � Jÿ1

ouon

ovon

owon

ouog

ovog

owog

ouof

ovof

owof

26643775: �9�

3.2. Order of singularity

Consider that the element exhibits either a O(1/��rp

) singularity or a O�1=r� as required, over the entireelement domain. The point P in Fig. 3 is at a distance r from the crack tip (collapsed edge). From thetransformation in Eq. (5), there results

r ��x2 � y2

p) r � d

4�1ÿ g�2

��hd

� �2

� f2

s: �10�

On a radial line form the crack front, we consider f in e�ect to be constant, and then rearrange Eq. (10) as

�1ÿ g� � C��rp; �11�

where the constant is

C � h4

� �2"

� df4

� �2#ÿ1=4

: �12�

In treating f as constant, we essentially require that all rays from the crack front on the parent element areparallel to the g axis. Given the transformation singularity at the tip, this is a reasonable model. Moreover,if stresses were to be evaluated along a ray towards the tip, i.e., r! 0. then one would normally follow aline of integration points towards the tip. More importantly, the value of f along a given ray is unimportantin the limit because the term f2 in Eq. (12) guarantees that C in Eq. (11) is always ®nite, and as will be seen,the 1/

��rp

singularity therefore dominates as r ! 0.Now, write the usual interpolation of displacements, �u; v;w�, in terms of nodal unknowns:

uvw

8<:9=; �X27

i�1

Ni�n; g; f�ui

vi

wi

8<:9=; �13�

from which we obtain derivatives with respect to the local coordinates. Substitute the shape functionsEqs. (18)±(23) into Eq. (13) noting that n and f are constant on any ray from the tip, and collect like termsin the coordinate values. After some manipulation, we can rearrange the explicit forms for local derivativesas functions of terms �1ÿ g�. For the local derivatives of u, this process reveals:


ouon� a0u�1ÿ g�2 � a1u�1ÿ g� � a2u;

ouog� b1u�1ÿ g� � b2u; �14�

ouof� c0u�1ÿ g�2 � c1u�1ÿ g� � c2u;

where the coe�cients aku; bku; cku are ®nite valued functions of n; f, and the nodal displacements, ui. Similarexpressions for derivatives of v and w can also be obtained.

Finally, substitute the local derivatives into Eq. (9) with the inverse Jacobian from Eq. (8), replacingterms �1ÿ g� using Eq. (11). After collecting together the relevant terms for small deformation de®nition ofstrain, we obtain the strains on any ray from the tip:

ex � �b1u � �cou ��b2u � �c1u��

rp � �c2u

r;

ey � �c0v � �c1v��rp � �c2v

r;

ez � �a0wr � �a1w��rp � �a2w;

cxy � �b1v � �c0u � �c0v ��b2v � �c1u � �c1v��

rp � �c2u � �c2v

r; �15�

cyz � �a0vr � �a1v��rp � �a2v � �c0w � �c1w��

rp � �c2w

r;

cxz � �a0ur � �a1u��rp � �a2u � �b1w � �c0w �

�b2w � �c1w��rp � �c2w

r;

where the coe�cients from Eq. (14) have simply been modi®ed by the constant parts in Eq. (8). For aperfectly elastic crack front, the displacements at nodes 1, 5 and 17 must be identical, as must displacementsat nodes 2, 6 and 18, and 9, 13 and 22, respectively (see Fig. 1). With this constraint, it can be shown thatcoe�cients of 1=r terms reduce to zero, i.e.:

�a2u; �a2v; �a2w; �c2u; �c2v; �c2w � 0: �16�Hence, it is clear that the direct strains ex and ey are dominated by the O�1= ��

rp � singularity; this has also

been con®rmed by numerical calculations on rays from the tip. On the other hand, the direct strain alongthe front, ez, is zero, which indicates that local plane strain conditions are captured along the front itself. Allthree shear strains are also dominated by the singularity. The ®rst corresponds to the classical mode II`shearing' (sliding) intensity, the third to the mode III `tearing' intensity, and the second corresponds to alocal curving of the crack front line.

Finally, observe that if the sets of nodes f1; 5; 17g, f2; 6; 18g and f9; 13; 22g are given the samecoordinates initially, but allowed to remain uncoupled, then Eq. (16) does not hold and the stresses aredominated by a 1=r singularity.

3.3. Crack tip mesh

An integral part of stress intensity calculations in three dimensions is the generation of the mesh near acrack front and the quality of that discretization. Consider the automatic mesh generation in three-dimensional problems in a sequel. It is relevant at this point to describe the form of crack tip mesh.


Given that even modest mesh sizes can produce vast numbers of degrees of freedom in three dimensionalproblems, it is particularly important to achieve accurate results without particularly ®ne tip meshes. Asoutlined brie¯y in [11], a standard mesh of singular elements can be formed in a tube along the front, asillustrated in Fig. 4; the tube consists of a set of eight of the wedge shaped collapsed elements wrappedaround the front. In crack propagation studies, the `fracture tube' moves with the front maintaining thesame quality of discretization, which must be necessarily good near the front. Moreover, given that the tubeconsists of a series of collapsed elements forming the cylinder, the direct stresses obey the 1=

��rp

singularityon every radial line from the tip as required, since this has been shown to occur in each element separately.

Having placed a standard mesh of singular elements around the crack front, the analysis lends itself todisplacement-type extraction techniques to compute the stress intensity factors, and those we use here todemonstrate the utility of the model. Two methods will be presented here: (i) the quarter point displacementtechnique (QPDT); and (ii) the displacement correlation technique (DCT) both of which have been vali-dated in many studies [9]. Both techniques use the di�erence in nodal displacement for nodes behind the tipto evaluate the relative opening, shearing and tearing of the front. The QPDT uses just the di�erence indisplacements between the quarter points nodes, whereas the DCT also extends to nodal displacements atthe extreme of the element. The expressions for stress intensity factors for each method are summarized inAppendix B. Contended in [9,12] is that the QPDT provides more accurate results, and is less sensitive tovariations in the size of quarter point elements, at least in two-dimensional problems. This issue will beassessed for our three-dimensional element.

4. Numerical computations and discussion

Three stress intensity problems are considered, with the objectives being (i) to demonstrate the accuracy ofthe singular elements; (ii) to demonstrate the typical tube meshes employed; and (iii) to consider typical ®niteelement predictions of stress intensity factors along a front. In each case the material properties are the same± E� 200 GPa, v� 0.3 ± and the specimens have a uniaxial tension of 10 MPa applied along the major axis.

4.1. Planar crack in a rectangular prism

Consider a cube with a 100 mm side length under uniaxial tension, with a straight through planar crack50 mm wide embedded in the solid. Fig. 5 shows the crack tip meshes and the surface mesh on the crackfaces inside the solid, where a tube of singular elements is place along each front of the crack.

Fig. 4. Tube of collapsed singular elements along a crack front.


Given the problem symmetry, both mode II and III factors should be zero, and only mode I exists.Table 1 summarizes the values determined from both extraction techniques based on the displacementsolution using the mesh of singular elements shown; the mesh for the solid comprised a similar re®nementof 10-node tetrahedral elements. The theoretical values given in [13] are also shown, and it is clear that theresults are excellent for the mesh used, with the QPDT predictions being consistently better than theDCT.

4.2. Penny shaped crack in a cylinder

For the case of a cylinder in uniaxial extension, the near tip stresses are symmetric around the pennyshaped crack, so that Mode II and III factors should again be zero; no symmetry conditions were imposedon the problem since the goal is to test the element's performance in true three-dimensional situations. Thecrack tip mesh used is shown in Fig. 6: for this example, the radius of the crack tip elements to crack radiusis just 0.1, which is relatively coarse. The ®nite element predictions for stress intensity factors from theQPDT and DCT are shown in Table 2 against the theoretical values [14]. Again both methods are accuratebut with the QPDT giving results very close to the analytical solution.

The comparative accuracy of these results with other studies is also noteworthy. For example, Bank-Sills [8] presented results for a penny-shaped crack in a cylinder with aspect ratio 2:1, but did not reportthe same degree of accuracy obtained here with the collapsed Lagrangian with a similar sized mesh ofeither 20 node isoparametric, or 27 node Lagrangian bricks. Studied in [10] is the case of a small di-ameter penny in a block, as a model of the semi-®nite problem, using 20 node isoparametric elements inABAQUS. They discretized 1/8 of the block (by symmetry) and placed collapsed 20 node isoparametricelements around the front in order to ``optimize'' the solution. Those elements lay within a curvedcylinder consisting of 7 concentric layers and 12 segments around the quarter circle crack. The inner layerof collapsed wedge elements had a diameter of �0.0625 of the block size, which produced a very ®nemesh of more than 44,000 degrees of freedom, including some 3600 nodes on the tube. In a relative errorof 0.6%. This compares with a 1.3% error observed for the collapsed Lagrangian elements. However,only 1 layer of collapsed Lagrangian elements were used around the front (as Fig. 6) with a diameter of

Table 1

Stress intensity factors for a central crack in a rectangular prism

KI KII KIII KI :% error

Theory 1.334 0.000 0.000 ±

QPDT 1.255 0.034 0.001 ÿ5.9

DCT 1.440 0.052 0.001 +7.9

Fig. 5. Crack front mesh for planar crack in a rectangular prism.


0.15 of the block size, requiring just 216 nodes on the tube for the equivalent discretization of 1/8 of theproblem.

4.3. Inclined elliptical crack in a cylinder

For the last example, consider a cylinder in uniaxial extension. This time, however, the crack is ellipticaland inclined to the major axis, as shown in Fig. 7. The main coordinate of interest is the angle, h, measuredfrom the top of the crack on the major axis. Because of the crack's inclination, the Modes II and III factorsare not zero. The Mode II value should be at its maximum at both top and bottom of the `penny'exchanging with Mode III which reaches its peak at the sides, h � 90�.

Fig. 8 shows the calculated variation of stress intensity factor with h for all three modes. Firstly, notethat the Modes II and III stress intensity factors respond as expected, and that the Mode I value increases

Fig. 7. Geometry of the elliptical crack in a cylinder.

Fig. 6. Mesh for penny-shaped crack in cylinder.

Table 2

Stress intensity factors for a penny shaped crack in a cylinder

KI KII KIII KI : %error

Theory 0.6366 0.000 0.000 ±

QPDT 0.6351 0.007 0.000 ÿ0.2

DCT 0.6663 0.010 0.000 +4.6


slightly from its value at the top as h increases. The oscillating response of KI is a feature of all ®nite elementpredictions [8,10], when using calculations at discrete points along the front, such as displacement ex-traction techniques or J -integrals [8,10]. Fig. 8 shows that the stress intensity factors are greater at the inter-element boundaries, where predicted crack mouth openings are larger. This feature can itself be traced tothe C0 continuity formulation of all solid elements, where the lack of rotational compatibility at the inter-element nodes provides relative ¯exibility at the joining nodes and hence oscillating CMOD values; Fig. 9depicts the response based on actual calculations.

It was shown [8] that the KI oscillations in a Lagrangian element are less pronounced than in a 20-nodeisoparametric element, as also noted here in comparison with the results [8] for the penny shaped crack in acylinder. Moreover, an in¯uence function technique such as in [10] may dampen the `wavy response' furthersince the weighted integral along the front `smooths out' the oscillations. Nonetheless, it should be em-phasized again that the objective has been to provide a good singular element, within a consistent meshingalgorithm, on which to base the SIF calculations.

Appendix A. Shape functions for 27 node Lagrangian

In this Appendix A the explicit form of the shape functions for the 27 node Lagrangian solid element arelisted. They are used in detail to demonstrate the order of singularity in the collapsed element.

Fig. 9. Typical crack mouth opening displacements.

Fig. 8. Stress intensity factors for an elliptical crack in a cylinder.


Fig. 1 shows the parent element, measuring 2 units on each side, with the origin of local coordinates�n; g; f� in the centre of the element. Considering ®rst the gf plane of the brick element, the polynomials in nfrom Eq. (1) become

l20�n� �

�nÿ n1��nÿ n2��n0 ÿ n1��n0 ÿ n2�

� 1

2n�nÿ 1�;

l21�n� �

�nÿ n0��nÿ n2��n1 ÿ n0��n1 ÿ n2�

� ÿn�n2 ÿ 1�; �17�

l22�n� �

�nÿ n0��nÿ n1��n2 ÿ n0��n2 ÿ n1�

� 1

2n�n� 1�:

Then, by substituting into Eq. (2), there results the shape functions N1; . . . ;N8 for the corner nodes:

N1 : I � 2; J � 2;K � 2 ) N1 � 1=8 n�n� 1�g�g� 1�f�f� 1�;N2 : I � 0; J � 2;K � 2 ) N2 � 1=8 n�nÿ 1�g�g� 1�f�f� 1�;N3 : I � 0; J � 0;K � 2 ) N3 � 1=8 n�nÿ 1�g�gÿ 1�f�f� 1�;N4 : I � 2; J � 0;K � 2 ) N4 � 1=8 n�n� 1�g�gÿ 1�f�f� 1�;N5 : I � 2; J � 2;K � 0 ) N5 � 1=8 n�n� 1�g�g� 1�f�fÿ 1�; �18�N6 : I � 0; J � 2;K � 0 ) N6 � 1=8 n�nÿ 1�g�g� 1�f�fÿ 1�;N7 : I � 0; J � 0;K � 0 ) N7 � 1=8 n�nÿ 1�g�gÿ 1�f�fÿ 1�;N8 : I � 2; J � 0;K � 0 ) N8 � 1=8 n�n� 1�g�gÿ 1�f�fÿ 1�:

Next, for the mid-side nodes on the face f � 1 there are N9; . . . ;N12:

N9 : I � 1; J � 2;K � 2 ) N9 � ÿ0:25n�n2 ÿ 1�g�g� 1�f�f� 1�;N10 : I � 0; J � 1;K � 2 ) N10 � ÿ0:25n�nÿ 1��g2 ÿ 1�f�f� 1�;N11 : I � 1; J � 0;K � 2 ) N11 � ÿ0:25�n2 ÿ 1�g�gÿ 1�f�f� 1�; �19�N12 : I � 2; J � 1;K � 2 ) N12 � ÿ0:25n�n� 1��g2 ÿ 1�f�f� 1�:

The shape functions for mid-edge nodes on the face f � ÿ1 and the plane f � 0 are respectively:

N13 : I � 1; J � 2;K � 0 ) N13 � ÿ0:25�n2 ÿ 1�g�g� 1�f�f� 1�;N14 : I � 0; J � 1;K � 0 ) N14 � ÿ0:25n�nÿ 1��g2 ÿ 1�f�fÿ 1�;N15 : I � 1; J � 0;K � 0 ) N15 � ÿ0:25�n2 ÿ 1�g�gÿ 1�f�fÿ 1�; �20�N16 : I � 2; J � 1;K � 0 ) N16 � ÿ0:25n�n� 1��g2 ÿ 1�f�fÿ 1�

and

N17 : I � 2; J � 2;K � 1 ) N17 � ÿ0:25n�n� 1�g�g� 1��f2 ÿ 1�;N18 : I � 0; J � 2;K � 1 ) N18 � ÿ0:25n�nÿ 1�g�g� 1��f2 ÿ 1�;N19 : I � 0; J � 0;K � 1 ) N19 � ÿ0:25n�nÿ 1�g�gÿ 1��f2 ÿ 1�; �21�N20 : I � 2; J � 0;K � 1 ) N20 � ÿ0:25n�n� 1�g�gÿ 1��f2 ÿ 1�:

Finally, the shape function for the central node are given by

N21 : I � 1; J � 1;K � 1 ) N21 � ÿ�n2 ÿ 1��g2 ÿ 1��f2 ÿ 1� �22�


and the mid-face nodes:

N22 : I � 1; J � 2;K � 1 ) N18 � 0:5�n2 ÿ 1�g�g� 1��f2 ÿ 1�;N23 : I � 0; J � 1;K � 1 ) N19 � 0:5n�nÿ 1��g2 ÿ 1��f2 ÿ 1�;N24 : I � 1; J � 0;K � 1 ) N20 � 0:5�n2 ÿ 1�g�gÿ 1��f2 ÿ 1�;N25 : I � 2; J � 1;K � 1 ) N17 � 0:5n�n� 1��g2 ÿ 1��f2 ÿ 1�; �23�N26 : I � 1; J � 1;K � 2 ) N18 � 0:5�n2 ÿ 1��g2 ÿ 1�f�f� 1�;N27 : I � 1; J � 1;K � 0 ) N19 � 0:5�n2 ÿ 1��g2 ÿ 1�f�fÿ 1�:

Appendix B. Displacement extraction techniques

Displacement extraction techniques begin with the elastic solution for near tip stresses, then write out thedisplacement at h � �p in the plane perpendicular to the front in terms of stress intensity factor. These arerearranged to give formulae for SIF in terms of nodal displacements on the top and bottom faces of a crack.For this purpose, write the crack opening displacement, crack sliding and crack tearing displacement as

COD � vtop ÿ vbottom;

CSD � utop ÿ ubottom;

CTD � wtop ÿ wbottom:

B.1. QPDT

The QPDT uses di�erences in displacements �u; v;w� at the quarter point nodes, i.e., at r � h=4. For theplane strain case, the stress intensity factors are

KI � E2�1ÿ v2�

��p2h

rCODr�h=4;

KII � E2�1ÿ v2�

��p2h

rCSDr�h=4; �24�

KIII � E2�1ÿ v2�

��p2h

rCTDr�h=4:

B.2. DCT

For this case, crack opening displacements are interpolated through values obtained at the quarterpoints and the element extremes. The stress intensity factors are thus

KI � E4�1ÿ v2�

��p2h

rf4CODr�h=4 ÿ CODr�hg;

KII � E4�1ÿ v2�

��p2h

rf4CSDr�h=4 ÿ CSDr�hg; �25�

KIII � E4�1ÿ v2�

��p2h

rf4CTDr�h=4 ÿ CTDr�hg:


References

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