Collapsed 12-Node Triangular Elements as Crack Tip Elements for ...

AD-/M>55 93^ AD TECHNICAL REPORT ARLCB-TR-77047

COLLAPSED 12-NODE TRIANGULAR ELEMENTS AS CRACK TIP ELEMENTS FOR ELASTIC FRACTURE S.L. Pu M.A. Hussain W.E. Lorensen

TECHNICAL LIBRARY

December 1977

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND LARGE CALIBER WEAPON SYSTEMS LABORATORY BENET WEAPONS LABORATORY WATERVLIET, N. Y. 12189

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COLLAPSED 12-NODE TRIANGULAR ELEMENTS AS CRACK TIP ELEMENTS FOR ELASTIC FRACTURE

6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBERfa.)

7. AUTHORfs;

S.L PU

M.A HUSSAIN

W.E. LORENSEN

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Benet Weapons Laboratory Watervliet Arsenal, Watervliet, N.Y. 12189 DRDAR-LCB-RA

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20.

Fracture Mechanics

Finite-Element Method

Isoparametric Elements

Singular Elements

Cubic, quadrilateral Elements Stress intensity Factors

ABSTRACT (Continue on reverse aide II neceaamry and Identity by block number)

For the 12-node bicubic, quadrilateral, isoparametric elements, it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by the simple technique of collapsing the quadrilateral elements into triangular elements around the crack tip and placing the two mid-side nodes of each side of the triangles at 1/9 and 4/9 of the length of the side from the tip. This is analgous to placing the mid-side nodes at quarter points in the (See Other Side) OD 1 JAN 73 1473

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Block No. 20. vicinity of the crack tip for the quadratic, isoparametric element. The advantage of this method are that the displacement compatibility is satisfied throughout the region and that there is no need of special crack tip? elements. The stress intensity factors can be accurately obtained by using general purpose programs having isoparametric elements such as NASTRAN. The use of 12-node isoparametric element program APES may be simplified by eliminating the special crack tip elements.

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TABLE OF CONTENTS Page

INTRODUCTION

1

THE 12-NODE QUADRILATERAL ISOPARAMETRIC ELEMENT

3

THE CRACK TIP ELEMENT

6

DETERMINATION OF STRESS INTENSITY FACTORS

12

1.

One Term Expansion

14

2.

Two-Term Expansion

15

3.

Four Term Expansion

16

4.

Collocation Method

16

NASTRAN IMPLEMENTATION

17

NUMERICAL RESULTS

18

THE STABILITY OF COLLAPSED TRIANGULAR ELEMENTS

33

CONCLUSIONS

35

REFERENCES

36 ILLUSTRATIONS

1.

Shape Functions and Numbering Sequence for a 12-Node Quadrilateral Element.

39

A Normalized Square in (£,n) Plane Mapped Into a Collapsed Triangular Element in (x,y) Plane with the side ? = -1 Degenerated into a Point at the Crack Tip.

40

3.

Three Tension Test Specimens.

41

4.

Idealization of a Half of the Single-Edge Cracked Tension Specimen.

42

2.

Page 5.

6.

7. 8.

(a) Three Collapsed Triangular Elements Surrounding a Mode I Crack Tip.

43

(b) Special Core Element and Three Quadrilateral Elements Surrounding a Mode I Crack Tip.

43

(a) Six Collapsed Triangular Elements Surrounding a Mixed Mode Crack Tip.

44

(b) Special Core Element and Six Quadrilateral Elements Surrounding a Moxed Mode Crack Tip.

44

Idealization of a 45-Degree Slant Edge Cracked Panel in Tension.

45

(a) Node 5 Perturbed to 5*.

46

(b) Nodes 20, 21, 23, 24, 26, 27 Perturbed From Their Nominal Positions.

46

TABLES RATIOS OF K] (APES) TO ^ (EXACT)

21

K, (FINITE ELEMENT)/^ (EXACT) BY ONE-TERM EXPANSION USING APES WITH COLLAPSED TRIANGULAR ELEMENTS

22

K, (FINITE ELEMENT)/^ (EXACT) BY ONE-TERM EXPANSION Utim APES WITH COLLAPSED TRIANGULAR ELEMENTS

23

K, (FINITE ELEMENT)/^ (EXACT) BY ONE-TERM EXPANSION USim APES WITH COLLAPSED TRIANGULAR ELEMENTS

24

Kn (FINITE ELEMENT)/^ (EXACT) BY TWO-TERM EXPANSION mm APES WITH COLLAPSED TRIANGULAR ELEMENTS

25

K-. (FINITE ELEMENT)/^ (EXACT) BY TWO-TERM EXPANSION USim APES WITH COLLAPSED TRIANGULAR ELEMENTS

26

Ki (FINITE ELEMENT)/^ (EXACT) BY TWO-TERM EXPANSION U^ING APES WITH COLLAPSED TRIANGULAR ELEMENTS

27

Kn (FINITE ELEMENT) K] (EXACT)

BY FOUR-TERM EXPANSION USING APES WITH COLLAPSED TRIANGULAR ELEMENTS

11

28

Page 9.

Ki (FINITE ELEMENT) FOR SINGLE EDGE CRACK USING C6LLOCATION METHOD. NODAL DISPLACEMENTS OBTAINED FROM APES WITH 3-COLLAPSED TRIANGULAR ELEMENTS. COLLOCATION POINTS ARE EQUALLY SPACED ON r=0.01.

29

10.

K] (NASTRAN)/Ki (EXACT)

30

11.

K] (APES)/^ (EXACT)

31

12.

!
33

111

INTRODUCTION The direct application of the finite element method to crack problems was studied by a number of investigators [1-3].

No special

attention was given to the singular nature of stress and strain of a crack tip.

Because of the large strain gradients in the vicinity of a

crack tip, it requires the use of an extremely fine element grid near the crack tip.

By comparing the finite element result of displacement

components or stress components at a nodal point with the corresponding asymptotic result of displacement or stress components at that node, the stress intensity factor can be estimated.

The estimated values of a

stress intensity factor vary over a considerable range, depending on which node is taken for computation.

This results in poor estimates if

displacements are taken at nodal points either very close to or far away from the crack tip. An improved finite element technique was developed by Wilson [4]. It combined the asymptotic expansion of displacements in a small circular core region surrounding a crack tip and the finite element approximation outside a polygon approximating the circular arc of the 1

Swedlow, J. L., Williams, M. L., and Yang, W. H., "Elasto-Plastic Stresses and Strains in Cracked Plates," Proceedings First International Conference on Fracture, 1, p. 259, 1966. 2Kobayashi, A. S., Maiden, D. E. and Simon, B. J., "Application of the Method of Finite Element Analysis to Two-Dimensional Problems in Fracture Mechanics," ASME 69-WA/PVP-12 (1969). 3chan, S. K., Tuba, I. S. and Wilson, W. K., "On Finite Element Method in Linear Fracture Mechanics," Engineering Fracture Mechanics, 2, p. 1, 1970. 4 Wilson, W. K., "Combined Mode Fracture Mechanics," Ph.D. Dissertation, University of Pittsburgh, 1969.

core region.

The displacement fields obtained from these two approx-

imations are not, in general, continuous along the asymptotic expansionfinite element interface except at discrete nodal points. An alternative finite element approach to crack problems is the use of special elements in the region of the crack tip, e.g. [5-7]. In [5], Tracey employs quadrilateral isoparametric elements which become triangular around the crack tip.

The displacement functions of

the two types of elements are selected such that displacements are continuous everywhere, and the near tip displacements are proportional to the square root of the distance from the crack tip. Henshell and Shaw [8] and Barsoum [9] showed that special crack tip elements were unnecessary.

For two-dimensional 8-node quadrilateral

elements, the inverse square root singularity of the strain field at the crack tip is obtained by collapsing quadrilateral elements into triangular elements and placing the mid-side nodes at quarter points 5

Tracey, D. M., "Finite Elements for Determination of Crack Tip Elastic Stress Intensity Factors," Engineering Fracture Mechanics, Vol. 3, 1971. 6 Blackburn, W. S., "Calculation of Stress Intensity Factors at Crack Tips Using Special Finite Elements," The Mathematics of Finite Elements and Applications, Brunei University, 1973. 7 Benzley, S. E. and Beisinger, A. E., "Chiles - A Finite Element Computer Program that Calculates the Intensities of Linear Elastic Singularities," Sandia Laboratories, Technical Report SLA-73-0894, 1973. henshell, R. D., and Shaw, K. G., "Crack Tip Finite Elements Are Unnecessary," International Journal for Numerica" Methods in Engineering, Vol. 10, 1975. 9 Barsoum, R. S., "On the Use of Isoparametric FinJte Elements in Linear Fracture Mechanics," International Journal for Numerical Methods in Engineering, Vol. 10, 1976.

from the tip.

The quarter-point quadratic isoparametric elements, as

singular elements for crack problems, have been implemented in NASTRAN by Hussain et al [10]. In order to reduce the computer core requirement and to simplify the modeling of a structure, better known but lower order finite elements have been abandoned in favor of cubic 12-node, isoparametric, quadrilateral elements as described by Zienkiewicz [11].

In this paper,

the concept of quarter-point, quadratic, isoparametric elements is extended to 12-node cubic isoparametric elements.

The correct order of

strain singularity at the crack tip is achieved in a simple manner by collapsing the quadrilateral elements into triangular elements and by placing the two middle nodes of a side at 1/9 and 4/9 of the length of the side from the tip. implemented in NASTRAN.

The 12-node, isoparametric elements have been Both mode I and mixed mode crack problems are

computed by NASTRAN using the collapsed elements to assess the accuracy. The stability of results is discussed when the collapsed triangular elements are used. THE 12-NODE QUADRILATERAL ISOPARAMETRIC ELEMENT A typical 12-node, quadrilateral element in Cartesian coordinates (x,y) which is mapped to a square in the curvilinear space (^,ri) with vertices at (± 1, ± 1) is shown in Figure 1.

The assumption for

displacement components takes the form: 10

Hussain, M. A., Lorensen, W. E., and Pflegl, G., "The Quarter-Point Quadratic Isoparametric Element As a Singular Element for Crack Problems," NASA TM-X-3428, 1976, p. 419. ^Zienkiewicz, 0. 0., The Finite Element Method in Engineering Science, McGraw Hill, London, 1971.

u =

12 I Ni(?,n)ui i=l

12 v = I Ni(?,n)vi 1-1

(1)

where u,v are x,y components of displacement of a point whose natural coordinates are £,n; u.j.v.j are displacement components of node 1 and N.|(£;5n) is the shape function which is given by [11] N (?

i '^

=

256 0

+

5^)0 + miK-W + 9{e + n2)][-lo + 9(q + nf)]

81

+ ^L.(I

+ ^1)(l + 9m.)0 - n2)(l - nj)

81 + ^L(1 +nni)(l + g^Od - ?2)(1 - q)

(2)

for node i whose Cartesian and curvilinear coordinates are (x^.y^) and (^j.ni) respectively.

The details of the shape functions and the

numbering sequence are given in Figure 1. The same shape functions are used for the transformation of coordinates, hence the name isoparametric. 12 i=l

1

12 y = I M^n)y, 1 1=1 1

n7Zienkiewicz, ,-

(3)

0.0., The Finite Element Method in Engineering Science. McGraw Hill, London, 1971.

The element stiffness matrix is found in the usual way and is given by [9,10] [K] = / / [B]I[D][B] det iJld^dn -1 -1

(4)

where [B] is a matrix relating joint displacements to strain field 9Ni

0

9x [B] - [.. .B.j.... J,

[B,] =

0

9^

(5a)

9y 9Ni 9y

9x

and [D] is the material stiffness matrix and is given for the case of plane stress by

[D] =

v

1

v

0

v

1

0

0

0

(5b)

(1 - v)/2

in which E is Young's modulus and v is Poisson's ratio. The Jacobian matrix [J] is given by 9

Barsoum, R. S., "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," International Journal for Numerical Methods in Engineering, Vol. 10, 1976. 10 Hussain, M. A., Lorensen, W. E., and Pflegl, G., "The Quarter-Point Quadratic Isoparametric Element as a Singular Element for Crack Problems," NASA TM-X-3428, 1976, p. 419.

—

3X K

9x 9?

9x

9^

9n

9ri

=

[J] =

-

... 9N-J ... 9?

•

•

• ••

• •

• •

oNi

(6)

• • •

9n _

■■

"-

_

—

__

—

whenever the determinant of [J] is zero, the stresses and strains become singular [8-10].

The derivatives of shape functions are

9Ni . l (1 + nni)C-10 + 9(Cf + nfnHOd + m + 27^ + 9qn2) 256 ^ + ^^(1 + 9Tin1)(l - n2)(l - nf) +

9Ni 9TI

2^6 0

+

^L(1 +nni)(l - K])^ - 2? - 27^)

^i)[-10

+ 9

(52

+

(7a)

n^HOn-i + 18n + 27nin2 + 9^)

+ ^^.(1 +95^)0 - ?2)(i - q) 81 (1 + ^ )(l - n 2)(9n - 2n - 27n n2) i 1 i i 256

(7b)

THE CRACK TIP ELEMENT In an 8-node quadratic isoparametric element, Henshell and Shaw [8] and Barsoum [9] found independently that the strain became singular at the corner node if the mid-side nodes were placed at the quarter points 8

Henshell, R. D., and Shaw, K. G., "Crack Tip Finite Elements Are Unnecessary," International Journal for Numerical Methods in Engineering, Vol. 9, 1975. 9 Barsoum, R. S., "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," International Journal for Numerical Methods in Engineering, Vol. 10, 1976. "■OHussain, M. A., Lorensen, M. E., and Pflegl, G., "The Quarter-Point Quadratic Isoparametric Element as a Singular Element for Crack Problems," NASA TM-X-3428, 1976, p. 419. 6

of the sides from the corner node.

This singularity is achieved in a

similar way for a 12-node isoparametric element by placing the two middle nodes at the 1/9 and 4/9 of the length of the sides from the common node of two sides. For simplicity, let us consider the singularity along the side n = -1 of Figure 1.

In general, the cubic mapping functions are x = a0 + a^ + a2C2 + a^3

(8)

u = b0 + b1? + b2e + b3e

(9)

For £ ■ -1, -1/3, 1/3 and 1, the corresponding values of x and u are x = 0, ai, {&• & u = u-j, u2, u3, u4 The constants a's and b's in terms of these values of x and u are a

+

0 = IT M

9a + 93)

.

a

i

=

IT (-■' "

27a + 27

^

] \ (10)

a2 = f| (1 - a - 0) b

0 = T6 {-Ul

b

2

=

^

+ 9U

2

+ 9U

l " u2 ■ "3

(u

3 " U4)

+ U )

4

, 83 = -yf (1 + 3a - 3B)

'

b

l

=

II

(U

'

b

3

=

IT (-Ul

1 "

27U

2

+ 3u

J

+ 27u

2 "

3 " U4) (11)

3u

3

+ U )

To have singular strain at x = 0 (£; = -1), the reduced Jacobian, must vanish at 5 ■ •!.

4

dx

^F

»

From (8) we have ^-= a1 + 2a2? + 3a3K2

(12)

For £; =

dx

-IJ HF

dC

=

0 leads to the equation

3 = 2a + I

(13)

In order to have the inverse square root singularity for -^ ,

x must be a quadratic function of 5 so that the inverse gives ^ as a function of x'2.

This leads to 83 = 0 or 1 + 3a - 36 = 0

(14)

The solution of (13) and (14) gives a = 1/9

and

3 = 4/9

(15)

ir » -1 + zJI"

(16)

Equations (8) and (9) become x = A (1 + O2

or

u = u1 + I (-11^ + 18u2 - 9U3 + 2u4)Jf + j (2u1 - 5u2 + 4U3 - u4) j H-fC-U! + 3u2 - 3u3 + u4)(^)3/2 From (17) it is clear ^ has singularity of the order

(17) - at x = 0.

The inverse square root singularity at x = 0 along any other ray emanating from node 1 can be achieved by degenerating the quadrilateral element into a triangular element with the side 10, 11, 12, 1 collapsed to a point at the crack tip and placing grid points 2, 9 at 1/9 and 3, 8 at 4£/9 from the tip (Figure 2), where £ is the length of the sides corresponding to n = ± 1• shown in Figure 2 .

The Cartesian coordinates of nodal points are

Using (3),

x = I (1 + 02f(n,a,B)

(18)

y = | (l + U2f'(n.a.g) where f and f are abbreviations f(n.a,B) = (1 - n)cos3 + (1 + n)cosa f(n»a,B) = (1 - n)sin3 + (1 + n)sina The Jacobian matrix is given by

[0] =

9x 9C

9^ 9^

1 (1 + C)f(n.a.B)

| (1 + Of (n,a,B)

9x _9n

9y.

4 (1 + 5)2(cosa - cosp) 8

I (1 + ?)2(sina - sinB) o

9TI_

and the determinant |J| =^(1 + C)3sin(a - 3)

(19)

This shows the strain is singular at x = 0 (^ = -1) along any ray from x = 0 since |J| = 0 at ^ = -1 for all n. For the inverse functions, we have 2(sina - sing) £(1 + ^)sin(a - 3)

'E. in" [0]

■1

9x

9x

(20)

K la 9y

-2(cosa - cosg) 5,(1 + ?)sin(a - 3)

-4f'(n»a»3) £(1 + ?)2sin(a - 3)

9y

4f(rua,3) £(1 + C)2sin(a - 3)_

In terms of polar coordinates (r,0) we have from (18) 1 + C = 2v^

,

R = (r/il)cos(e - ^)/cos(^) (21)

n = tan(e - S^/tan^)

The displacements components u,v at a point (g.n) of the triangular element of Figure 2 are u =

12 I Ni(5,n)ui = Agdi.Ui) + A-jCn.u^O +5)

^

+ A^n.u^d + K)2 + A3(n.u1)(l + K)3 12 v = I Ni(5,T1)vi = AoCn.v^ + A^n.v^d + 5)

(22)

+ A2(T1>vi)(i + d2 + AgCn.v^d + c)3 where

Aodn.^) = {2(-i + 9n2)[(i - n)u1 + (1 + n)u10] + 18(1 - n2)[(l + Sn)^ + (1 - 3n)u12]}/32

(23)

The displacement derivatives are 9u_=3u_ci£+3u_3n= 1 8x 9C 9x 8n 9x (1 + C)2 1 (1 + 5)

ilsin(a - 3)

-4f'(n,a.B) Jisinja - 3)

3A

0

9n

SA, {(sina - sinB)A1 - 2f,(n,a,B) g-1} + ... (24)

BA, 9u=3u9£+9u9ii_ 1 4f(n,a»g) 0 2 8y " 85 8y 8n 8y "' (1 + C) ^sin(a - g) 3n 1 (1 + 5)

9A, : . r ^-{-(cosa - cosB)A1 + 2f(n,a,B)—!-} + Jlsin(a - B) ' 8n

(25)

where 8A0(n.ui) 3n

= {(2 + 36n - 54TI2)U1 - (2 - 36n - 54TI2)U1|

+ 18(3 - 2n - 9T12)U11 - 18(3 + Zr] - 9n2)u12}/32

(26)

Similar expressions for 9v/3x and 9v/9y can be obtained by replacing u-j with v.,-.

10

It can be seen that the strain field is singular at r = 0. order of singularity is (1/r). 3AO/8TI

The

The leading term vanishes together with

for all n if U

l

= U

10

= U

11

= U

12

'

V

l

= V

10

= V

ll

= V

In this case, the order of singularity becomes (l//r).

12

(27)

This is analo-

gous to the constraints given in [12] for quadratic, isoparametric elements.

Hence we have two types of strain singularity at our

disposition.

If nodes 1, 10, 11, 12, which are collapsed into one

point at the crack tip (Fig. 2), are tied together during deformation, the elastic singularity is obtained.

If these nodes are allowed to

slide with respect to each other, then the strain singularity is of the order of (1/r), the perfect plastic singularity. Using the multiple constraint conditions, equations (27), the displacement components at (C,n) relative to the tip may be written in the form

u = yL ,/R[36Fl(n.u1) + Fgdi,^) + 36^2(11,^) - F^n.Uj)}^ - 36F2(n,ui)R] v =

(28)

lV ^seF^n.v.,) + F3(n,vi) + 36{F2(r1,vi) - F^n.V^}^ - SeFgCn.v^R]

n

(29)

where F^n,^) = (1 - n)(2u2 - U3) + (1 + n)(2u9 - u8) F2(n,ui) = (1 - n){-3u2 + 3U3 - u4) + (1 + n)(-3u9 + 3u8 - u7) (30) 2

FgCn.u^ = 9(1 - n )[(l - 3n)u5 + (1 + 3n)u6] - (1 - 9n2)[(l - n)u4 + (1 + n)u7] and Fi(ri,v.) etc. are obtained by replacing u^ with v^. DETERMINATION OF STRESS INTENSITY FACTORS The collapsed, triangular elements around the crack tip have the correct elastic singularity at the tip if all nodes at the tip are tied together during deformation.

Only one set of displacement functions is

used for regular quadrilateral elements and the collapsed triangular elements.

Hence the continuity of displacement components is insured

throughout the region.

The nodal displacements obtained from the finite

element method using higher order polynomials for the displacement field should be quite accurate.

In this section, we will briefly discuss

various techniques to estimate the stress intensity factors from the nodal displacements thus obtained. use of nodal displacements.

The discussion is limited to the

Other techniques, such as J-integral, the

strain energy release rate etc. are not included. The well known classical near crack tip displacements are given by [13] ^Williams, M. L., "On the Stress Distribution at the Base of a Stationary Crack," Journal of Applied Mechanics, Vol. 24, 1957.

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O (n II c ; 3 J3 (40)

/^(K+1) (K2(TT)

+ K2(-7r))

where u(r0,iT) and v(r0,7r) are displacement components of node 26 relative to node 19 in the direction of parallel and normal to the crack face; r is the distance between nodes 26 and 19.

u(r0,-TT) and

v(r0,-Tr) are the same but of node 20 relative to node 1. For a 45° edge crack shown in Figure 7, NASTRAN results for K-| and K2 are tabulated in Table 12 for ro/a=0.01 and for various other conditions.

Again the multiple constraint conditions, namely u-| = u2 =

••• = u-jg and vi = v2 = ••• = v-jg, give little effect on values of K-] and IC.

An obliqued edge crack in a rectangular panel under uniform

tension is solved by Freese using a modified mapping collocation method [20].

K, and K , read from Bowie's graphs (Figure 1.16a and 1.16b of

20

Bowie, 0. L., "Solutions of Plane Crack Problems by Mapping Technique," in Mechanics of Fracture, 1, Edited by G. C. Sih, Noordhoff International Publishing, Leyden, The Netherlands, 1973. 32

[20]), are approximately 1.86 and 0.88 respectively.

Numerical results

of the same problem computed by APES, using special crack tip elements and various idealizations, are given in Figure 12 of [16]. TABLE 12.

K} AND K2 FOR 45° EDGE CRACK BY NASTRAN

B.C.

Integration

1 1 1 2

3x3 3x3 4x4 4x4

Multiple Constraint No Yes No Yes

l

K2

1.89 1.89 1.83 1.84

0.95 0.96 0.92 0.93

K

THE STABILITY OF COLLAPSED TRIANGULAR ELEMENTS In a recent report by Hussain and Lorensen [21], it was found that a slight perturbation in placing the mid-side node opposite to the crack tip for a collapsed 8-node, quadrilateral element led to unstable results in stress intensity factors.

This unstability can be shown in

the collapsed 12-node, quadrilateral element if one or both middle nodes of the side opposite to the crack tip are slightly perturbed from their nominal positions. Let node 5 be perturbed as shown in Figure 8.

Denoting the

perturbed quantities with an asterix, we have '6Gifford, L. N., "Apes - Second Generation Two-Dimensional Fracture Mechanics and Stress Analysis by Finite Elements," Naval Ship Research and Development Center, Report 4799, December 1975. 20 Bowie, 0. L., "Solutions of Plane Crack Problems by Mapping Technique,1 in Mechanics of Fracture, 1, Edited by G. C. Sih, Noordhoff International Publishing, Leyden, The Netherlands, 1973. 21 Hussain, M. A. and Lorensen, W. E., "Isoparametric Elements As Singular Elements for Crack Problems," Watervliet Arsenal Technical Report, to be published.

33

* .„

2 + cosa .

H'1 = —3

+ e

(41) A general point (x,y) given by equation (18) will be displaced at x*M = \ (l+d2[(l-n) + (l+n)cosa] + e ^ (l+0(l-n2)(l-3n)

(42)

y*/£ = 5 (1H)2(1+Tl)sina + e' ^ (l+U(l-ri2)(l-3n)

(43)

Along the line n = -1/3, and replacing y* with r sine in (40), we have 1 + ?

3

1 + M

+

1/2 L 4sinesinou '

sina

(44)

Since (1+?) is a common factor in displacement components, equations (28) and (29), it is seen that the singularity required, for the crack problems disappears along at least the ray n = -1/3 in the collapsed triangular case. As a numerical example, the central crack tension specimen of Figure 3(a) is again used.

If the idealization remained the same as

shown in Figure 4, except that the collapsed elements of Figure 5 were replaced by those of Figure 8(b) (where nodal points 20, 21, 23, 24, 26 and 27 are on a circular arc), the computed stress intensity factor changed from its almost exact value K-| = 1.962 to K-| = 1.421 (nearly 30% error).

If only nodal points 26 and 27 were perturbed to their

new locations in Figure 8(b), the stress intensity factor would become K-] = 1.457 (a 26% error).

34

CONCLUSIONS By a simple manner, the 12-node isoparametric elements can be used to form a singular element for two-dimensional, elastic, fracture mechanics analysis.

The elements have been successfully implemented

in NASTRAN, which can now be more efficiently used for more accurate prediction of stress intensity factors of complicated crack problems. The middle nodes of the side opposite to a crack tip in a collapsed triangular element should be accurately located to avoid unstable results.

The extension of collapsed triangular elements as singular

elements to three-dimensional brick elements can be easily done as in [9,10].

q

^Barsoum, R. S., "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," International Journal for Numerical Methods in Engineering, Vol. 10, 1976. 10 Hussain, M. A., Lorensen, W. E., and Pflegl, G., "The Quarter-Point Quadratic Isoparametric Element As a Singular Element for Crack Problems,"NASA TM-X-3428, 1976, p. 419.

35

REFERENCES 1.

Swedlow, J. L.. Williams, M. L., and Yang, W. H., "Elasto-Plastic Stresses and Strains in Cracked Plates," Proceedings First International Conference on Fracture, 1, p. 259, 1966,

2.

Kobayashi, A. S., Maiden, D. E. and Simon, B. J., "Application of the Method of Finite Element Analysis to Two-Dimensional Problems in Fracture Mechanics," ASME 69-WA/PVP-12 (1969).

3.

Chan, S. K., Tuba, I. S. and Wilson, W. K., "On Finite Element Method in Linear Fracture Mechanics," Engineering Fracture Mechanics, 2. p. 1, 1970.

4.

Wilson, W. K., "Combined Mode Fracture Mechanics," Ph.D. Dissertation, University of Pittsburgh, 1969.

5.

Tracey, D. M., "Finite Elements for Determination of Crack Tip Elastic Stress Intensity Factors," Engineering Fracture Mechanics, Vol. 3, 1971.

6.

Blackburn, W. S., "Calculation of Stress Intensity Factors at Crack Tips Using Special Finite Elements," The Mathematics of Finite Elements and Applications, Brunei University, 1973.

7.

Benzley, S. E. and Beisinger, A. E., "Chiles - A Finite Element Computer Program That Calculates the Intensities of Linear Elastic Singularities," Sandia Laboratories, Technical Report SLA-73-0894, 1973.

8.

Henshell, R. D., and Shaw, K. G., "Crack Tip Finite Elements Are Unnecessary," International Journal for Numerical Methods in Engineering, Vol. 9, 1975.

36

9.

Barsoum, R. S., "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," International Journal for Numerical Methods in Engineering, Vol. 10, 1976.

10.

Hussain, M. A., Lorensen, W. E., and Pflegl, G., "The Quarter-Point Quadratic Isoparametric Element As a Singular Element for Crack Problems," NASA TM-X-3428, 1976, p. 419.

11.

Zienkiewicz, 0. 0., The Finite Element Method in Engineering Science. McGraw Hill, London, 1971.

12.

Barsoum, R. S., "Triangular Quarter-Point Elements As Elastic and Perfectly-Plastic Crack Tip Elements," International Journal for Numerical Methods in Engineering, Vol. 11, 1977.

13.

Williams, M. L., "On the Stress Distribution at the Base of a Stationary Crack," Journal of Applied Mechanics, Vol. 24, 1957.

14.

Tracey, D. M., "Discussion of 'On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics' by R. S. Barsoum", Int. Journal for Numerical Methods in Engineering, Vol. 11, 1977.

15.

Barsoum, R. S., "Author's Reply to the Discussion by Tracey," Int. Journal for Numerical Methods in Engineering, Vol. 11, 1977.

16.

Gifford, L. N., "Apes - Second Generation Two-Dimensional Fracture Mechanics and Stress Analysis by Finite Elements," Naval Ship Research and Development Center, Report 4799, December 1975.

17.

Isida, M., "Analysis of Stress Intensity Factors for the Tension of a Centrally Cracked Strip with Stiffened Edges," Engineering Fracture Mechanics, Vol. 5, 1973.

37

18.

Brown, W. F., and Srawley, J. E., "Plane Strain Crack Toughness Testing of High Strength Metallic Materials," ASTM STP-410, 1966.

19.

Tada, H., Paris, P. and Irwin, G., The Stress Analysis of Cracks Handbook, Del Research Corp., 1973.

20.

Bowie, 0. L., "Solutions of Plane Crack Problems by Mapping Technique," in Mechanics of Fracture, 1, Edited by G. C. Sih, Noordhoff International Publishing, Leyden, The Netherlands, 1973.

21.

Hussain, M. A. and Lorensen, W. E., "Isoparametric Elements As Singular Elements for Crack Problems," Watervliet Arsenal Technical Report, to be published.

38

8J

(i-n)[i-5)[-io+9(Cz+nz)] 32

J7

9

= jj (l+nlCl+^H-lO+QC^+T!2)]

N2

= jj Ci-n)(i-5z)(i-3Q

N

3

=

h d-^Ci-^^Cl+SC)

Ng = ^- (l+r1)(l-52)(l-3?)

^4

=

32 a-n)(l+?)[-10+9(C2+Tiz)]

N

=

32 (i+n)(i-c:i[-io+9C52+nz:']

D

T

10 (a)

10

Center Crack

1 II

10

10

(b) Double-Edge Crack

\D _l_

H

H

(c) Single-Edge Crack

Figure 3.

Three Tension Test Specimens.

41

SEE FIG. 5

Figure 4.

Idealization of a Half of the Single-Edge Cracked Tension Specimen.

42

25

24

23

22

(a)

t—IO

ib)

Figure 5(a). (b).

Three Collapsed Triangular Elements Surrounding a Mode I Crack Tip. Special Core Element and Three Quadrilateral Elements Surrounding a Mode I Crack Tip.

43

(a)

(W

Figure 6(a). (b).

Six Collapsed Triangular Elements Surrounding a Mixed Mode Crack Tip. Special Core Element and Six Quadrilateral Elements Surrounding a Mixed Mode Crack Tip.

44

3^

UM—1

»

•

m I

c o

•H C

H

^

^

II

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a.

T3 C)

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u u 0) bO

LO

f\i

^ C cd |H

u be » a

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o c o

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cd

o

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(Q)

Figure 8(a). (b).

Node 5 Perturbed to 5*. Nodes 20, 21, 23, 24, 26, 27 Perturbed From Their Nominal Positions.

46

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