Campulcrs & Slrucruns. Vol. 13, pp. 595-599. 1981 Printed in Great Briiin.

CWJ949/81/050595-05102.0010 Pergamon Press Lld.

THREE DIMENSIONAL SINGULAR ELEMENT

M. A. HIJSSAIN, L. F. COFFIN and K. A. ZALESKT

Corporate Research and Development, General Electric Company, Schenectady, New York

(Received 22 November 1980: received for publication 10 December 1980)

Abstract-The finite element method has become a powerful tool for computation of stress intensity factors in fracture mechanics. The simulation of singular behavior in the stress field is accomplished using “quarter points,” following the methods of Barsoum[l] and Henshell-Shaw[2]. The analysis has also been extended to cubic elements[3] and transition elements[4]. However, these concepts cannot be easily extended to three dimensional cases without additional conditions. Progress has been hampered firstly due to a variety of possible shapes the element may possess near the singular edge of the crack, and secondly due to the complexity of algebraic expressions that have to be manipulated.

In the present investigation we extensively used MACSYMA[5], a large symbolic manipulation program at MIT, thereby alleviating some of these difficulties. A simple condition between mid-side nodes has been derived which simulates the proper singular behavior along the crack.

In the investigation we first study a simple collapsed brick element. This is then generalized to a curved crack front. A few results are derived which can be used to compute the stress intensity factors. The concept of the transitional element has also been outlined. The stability of singular element has been discussed. Some of these ideas have been applied to a specific problem with unusual crack geometry. The analysis was carried out using ADINA on VAX machine. ADINA was implemented on VAX by W. E. I_orensen.

STRMGHT CRACK FRONT

Shape functions for a quadratic generic brick element shown in Fig. l(a) can be written in a condensed form

Ni(Z9 79 5) =:(I+ &)(l + rlT]i)(l + l&)(&i t 7%

t iii - 2M?izrli2l?

tf(l-P)(lt~rli)(ltC(;)(1-&2)

+$(I -V2)(1t55i)(1 + &)Cl-TiY

+f(1-12)(1+ 5ti)(1t9S)(1-li7~

i=1,...20. (1)

where (&vi, 6) are the coordinates of the nodes. The transformation to the physical coordinates are

X = 2 NiXi, y =

cl

k i-l Niy, z = 2 NiZi. (2)

i=1 i-1

Without loss of generality consider the collapsed element with singular edge along y-axis, and the quarter points as shown in Fig. l(b). Using eqns (1) and (2) and the coordinates of Fig. I(b). and using MACSYMA we have

X=~(ft1)2{(~tl)cOS(I+(l-~)}

y=iLg

L z=g(~t1)2sina. (3)

From the above, the Jacobian determinant is given by

J(x,y,z) L” * J = a(5, = jj sm (Y (5 t 1)‘. (4)

Note the simplicity of these expressions, which check

very well with their counter-parts in the two dimensional case. It can be seen from eqns (3) and (4) that x and z have a double zero and the Jacobian has a triple zero at 6 = - 1. As will be seen, this leads to square-root sin- gularities in the strain component. It can further be proved that the quarter-point is the only location which gives the coalescence of roots at 6 = - 1 in eqns (3) and (4).

a

CRACK FRONT

Fig. I(a). Twenty node brick element showing the numbering sequence. (b) The collapsed brick element with straight crack

front, with x 18 = l/2 (xI+x6) etc.

595

5% M. A. HUSSAIN et al

For the isoparametric elements the Cartesian com- As can be seen in Fig. 2, [, 6 = - 1 is the crack front. portents U, u and w are given by, Some of the coordinates are, x,, = x5 =x,, yr5 = y5 = y,,

xl6 = xl21 yib = Yl2, x8 = x20 = x4, Y, = Y, = Y4r x6 = xl6 = 20

u = ,za Niui etc. (5) X2, Y6= YlS= Y2, zb= h, ft8= hf2, X9=X19=x3, Y9=

y19 = Y3. z7 = hi 219 = N2i Xl3 =x9, Y13 = y9, xi5 = Xllr

1

The strain can be obtained from y1s= YII. ZI3=;iZb, 215=jz7. x9=x1 +$x*-x,), Xl1 =

aw 1 ay aw ax aw a~ - ----- aZ = a(x,y,rf a( a[ all as a5 I ( 1

a!5 It. 5) ay aw ax a~ ax ay aw ax aw ax +- --_-- +_ ----- ( a7 al ~6 at x i ( at atl al a!: a7 )I . (6)

i (xj f 3x4), etc. Our aim is to study the behavior of the

mapping, eqn (21, near the crack front. For this purpose let

I=-f+S (9)

After an extensive manipulation it can be shown that, J=-1+s

aw A, +&+A, From eqns (2) and (9) and using the coordinates cor-

;i-;=m ([cl) (7) respo~ing to Fig. 2 we have

where A,, A2, A, are very extensive expressions given in terms of nodal displacement components and [ and n. If we tie the collapsed nodes together (i.e. w1 = wit’= w5 =

= wi6= wq= wz,, =ws=O) we have A,=O, A,#O. z’view of eqn (31, using polar coordinates (x = r cos &, z = t sin et we see that the strain component given by eqn (7) has the required inverse square-root singularity at the crack tip. The stronger Ilr singularity can be eli- minated by tieing the nodes together.

All the components of disptacements (u, u, w), on the plane of the crack can be simplified as

u=-~(~+f){(2u,o_sr,-u~)~2+(+~u,,+h~ +u3-u2)q -4u,l-2u,o-4u9+2u~+2u?}

-$w 021(2U,, -2U9- Ujt U&j

+ (2U, g-t- .?Us - U3 - t&t}. (81

In view of eqn (3), the expression given above for displacement can be used directly for computation of stress intensity factors for all modes, i.e. the coefficient of ([t 1) in eqn (8) gives the P2 term in Westergaard soltion.

CURY~ CR&X FROWF

As seen in the previous section, the three dimensional collapsed, straight crack-front element is simply an extension of the two dimensional element. Now consider the general case with a curved crack front. For simplicity the plane of the crack is taken at z = 0 with points 1,2,3, 4, 10, 12, f4 arbitrary and 9, Ii, 13, I5 as quarter points.

I92 Fig. 2. Collapsed element with curved crack front.

x = {n(n - 11x1-t n(n + IIxr + 2(1- r121x*J

-++ l,{x, - x2 - x3 + x4 + 2(x*0 - x121)6

-~{11-rl)(xl-X2)f(?+l)(x,-x,l

f 2(n2 - lHX14 - Xl&j2,

*A~3

8 * (10)

The expression for y is similar to that of x. The first term corresponds to the location of the crack. From the discussion in the previous section it is ctear that we need. a second order zero in 6 for simuiation of square root singularity. From eqn (IO) this is possible if,

(&o-x,2~=~(x +X4)-(X2+X3)1. (11)

Hence the curved crack front, in general, will have the necessity singularities provided the difference in the coordinates of the mid-side nodes along the crack front satisfy the averaging condition of the form (11).

SPECUL CASES

In this section we consider special cases which may be directiy applicable for computation of stress intensity factors. Consider a crack in the pfane z = 0 with y-axis tangent as shown in Fig. 3.

The pertinent coordinates are

Xl = Yl = 0, xt = L cos 8, y2 = L sin 8, 1 1 X9” - 4 X2. Y9 = iY2. (12)

Fig. 3. The crack front tangent to y-axis.

Three dimensional singular element 591

The mapping becomes (q = f = - I),

x=~(f+l)2cos6

L y = - (6 t 1)2 sin S 4

iz=x2ty2=(Zt 1)4$

and the displacement become

(13)

-;(2s-ul-U,)(s+ 1)2. (14)

Similar expressions are valid for other displacement components and hence eqn (14) together with eqn (13) can be directly used for stress intensity factors for all modes.

Consider a more general case shown in Fig. 4. The pertinent coordinates are

XI =x17 = xg = y, = y,, = y5 = .?, = 217 = Ig = 22 = 29 = 0

x2 = L cos S, y2 = L sin S, x9 = $ x2, x9 = f y2,

X6 = L cos W, y6 = L sin U’, 26 = h, Xi3 = $X6, y13 = fy6,

213 = $ z6, Xl8 =; (X6 + X2),

yl8=~(y,+y2)r~l8=;(Z6+Z2).

The mapping then becomes, for v= - 1,

x=-~(ttl)2L{cos6(C-l)-cosw(~tl)}

Y =-~(~+l)2L{sin6(~-l)-sinw(~tl)}

z=+ l)‘(l-t 1)h

hence

x2+ z2 = 9 =$(ft l)‘{[cos S(l- 1)

-cos w([t l)]‘L2t({t l)2h2}.

(15)

(16)

(17)

From above (5 t 1) can be solved for r and used for a stress intensity computation from the displacement given below.

u=~{(U,+u;-2ul~)~2+(-4u9-us+U2+4u,3)j

t4u9-2u6-2u2+2u18 t4u13} t (I+ i)2

x~{(2u9tu6-u2-2u13)~-2u9tu6tu2-2u1)}.

(18)

In eqn (18) the crack front displacement has been taken as zero.

!JTABILITY OF COLLLAF’SED ELEMENT

It is known[6] that the singularity in the collapsed element is quite sensitive to a perturbation of mid-side node opposite to the crack front. This phenomenon also occurs in three dimensional case. In this section we outline the disappearance of the singularity using per- turbation of nodes 18 and 19 of the straight crack front, as shown in Fig. 5. Denoting the perturbed quantities with an asterisk we have

X:8=X,8tt;Z:*=z,*tE xt9 = x19 + E, zf9 = ZIP + t!. (19)

The mapping becomes

z*=~(*t1)2(~tl)sina-~(f+lWS’-l)~~ (20)

It is sufficient to consider a special ray x* = z* and hence

(21)

The Jacobian determinant becomes (neglecting e2 term)

J L3 . = 3 sm a(6 t 1)3

t~(~t1)2(3~tl)(~tl)(ltsina-cosa)a. (22)

Along the ray x* = z* we can solve for (5 t 1) from (20) in the form (I cos 6 =x* = y*)

([tl)=&{[r, rCOse+C22~2]“2+C2C}. (23)

Fig. 4. A more general case with crack front tangent to y-axis. Fig. 5. Perturbation of nodes 18 and 19.

598 M. A. HUSSAIN et al.

As seen from previous discussions, see for example eqn (7), (t+ 1) is no longer vanishes as r+O unless e+O. This illustrates the disappearance of the singularity at the crack front.

In actual practice this perturbation may occur for several reasons, e.g. automatic mesh generation. This may lead to shifting of singularities in the Jacobian determinant especially for higher integration order. We found it practical to continue the computation even though the Jacobian was close to zero or negative since ultimately the “singular” contribution to the stiffness matrix has to be eliminated due to boundary conditions of the crack. (The stiffness matrix has a logarithmic singularity for the collapsed elements.)

TRANSITlON ELEMENT The three dimensional element can also be used as

transition element. These elements are located in the immediate vicinity of the singular elements with the nodes so adjusted as to reflect or extrapolate the inverse square root singularity in the stresses and strains along the crack front. These were first studied by Lynn and Ingraffea, Ref.[4].

Without loss of generality consider the element shown in Fig. 6. Some of the pertinent coordinates are

x,=I,y,=-LI2.~,=o,x~=pL,yg=-L/2,z~=o, x2 = L, y2 = - Ll?. I2 = 0, X6 = L cos a, y, = - L/2, z6 = L sin 0, x5 = cos (a), y, = - L/2, z5 = sin a,. . . (24)

Again it can be shown that the coalescence of the roots at x = 0 together with the condition BL > I gives

@L=$L+2dL+l). (25)

With the above value of location of nodes we have the mapping

x=@(L)+V(L)-~+I)~{cosrr(~+I)t(l-C)}

z=~(~V(L)+%‘(L)-ftl)2(lt[)sina

y=+L (26)

Fig. 6. Transient element.

and the Jacobian determinant becomes

I=-$$#=k(t/(L)-I)sina([V(L)+V(L)-5 , 9

t I)‘.

As can be seen the x, y have a second order zero and the Jacobian has a third order zero at, I= q(L)+ l/ti(L)- I, the edge of the crack. Hence the brick element can be used as transition element. However, as opposed to singular element, the unwanted I/r type of singularity may hamper the use of transition element.

APPLtCATtON In fatigue experiments, to establish the threshold con-

dition for fatigue crack growth, it was necessary to compute the stress intensity factor distribution for the crack front shown in Fig. 7. This unusual crack front was obtained as a result of the applicationof a progressively decreasing cyclic loading. The test specimen employed was a uniaxial-loaded round bar with a sharp chordal notch, as shown. The material was a high strength nickel- base superalloy. Sortie of the methods discussed were used to compute the stress intensity factors. A preli- minary set of results is given in Table 1. No attempt was

Fig. 7. The cross section of the geometry analyzed.

Table 1.

K I/u K No K lb

I II III

0.0585 0.0704 0.1115 (ti=oY (8=0”) (B=o”)

0.0548 0.0669 0.1024 (0 = 24.2”) ( 6’ = 24.3”) ( t9 = 24.2”)

0.0559 0.0675 0.0897 (8=51.2”) (S=51.0”) (8=51.1”)

0.0636 0.0780 0.0973 (B = 63.8”) (e = 63.8”) (e = 63.8”)

0.0745 0.0895 0.1088 (e = 76.9”) (e = 76.9”) (e = 76.9”)

0.0914 0.1172 0.1374 (e=90°) (e = 90”) (S=W)

Preliminary results of &,. stress intensity factors. along the crack front.

Three dimensional singular element 599

made for computation of stress intensity factors at 4. crackfree surface intersections. At 8 = 90” the crack tip is known to have higher order singularities, Ref.[7], and this is seen in the trend of stress intensity computations. 5.

Rt?FgRgNCEs

1. Roshdy S. Barsoum. On the use of isoparametric finite ele- 6. ments in linear fracture mechanics, Int. 1. Num. Met/r. Engng 10.25-76 (1976).

2. R. D. Henshell and i(. G. Shaw. Crack tip elements are unnecessary. Int. 1. Num. Meth. Engng 9,495-550 (1975).

3. S. L. Pu. M. A. Hussain and W. E. Lorensen, The collapsed 7. cubic isoparametric element as a singular element. Int. J. Num. Meth. Engng 12, 172742 (1978).

P. P. Lynn and A. R. Ingraffea. Transition elements to be used with quarter-point crack tip elements. Inf. J. Num. Meth. Engng 12C. 1031-36 (1978). MACSYMA is a large program for symbolic manipulation at MIT, see MACSYMA Reference Manual, The Mathlab Group, Laboratory for Computer Science, MIT, Cambridge, Mass. (Dec. 1977). W. E. Lorensen and M. A. Hussain, The stability of the isoparametric triangular element as a singular element for crack problems. Developments in Mechanics. Proc. 15th Midwestern Mech. Conf. (Edited by T. C. T. Tiny), Vol. 8 (1977). M. A, Hussain, S. L. Pu and B. Noble, Stress singularity at the vertex of a flat wedge-shaped crack by variational method. Int. J. Engng. Sci. (to be published).