W13-79M/831030!47-!?%Oi0010
Perpamon Prw Ltd
ANALYTICAL SOLUTION FOR EMBEDDED ELLIPTICAL
CRACKS, AND FINITE ELEMENT ALTERNATING
METHOD FOR ELLIPTICAL SURFACE CRACKS,
SUBJECTED TO ARBITRARY LOADINGS
Center for the Advancement
T. NISHIOKA and S. N. ATLURI
of Computational Mechanics, Georgia Institute of Technology, School of Civil
Engineering, Atlanta, GA 30332, U.S.A.
Abstract-The
complete solution for an embedded elliptical crack in an infinitesolid and subjected to
arbitrary tractions on the crack surface is rederived from Vijayakumar and Atluri’s general solution
procedure. The general procedure for evaluating the necessary elliptic integrals in the generalized solution
for elliptical crack is also derived in this paper. The generalized solution is employed in the Schwartz
alternating technique in conjunction with the finite element method. This finite element-alternating method
gives an inexpensive way to evaluate accurate stress intensity factors for embedded or elliptical cracks in
engineering structural components.
1. INTRODUCTION
THEPROBLEMS
of embedded and surface elliptical cracks have received much attention due to the fact that
the actual flaw from which fracture is initiated in a structural component, can be approximated, often by
an ellipse or a part of an ellipse. S&e analytical solutions to these problems have been limited to the
case of an embedded crack in an infinite solid with a relatively simple loads, many numerical methods
have been developed to analyze fracture susceptibility of embedded or surface flawed, three-dimensional
engineering structures. However the agreement on the stress intensity factor solution at the deepest
point of semi-elliptical surface flaw was extremely poor. The discrepancies obtained by early methods
were sometimes 250%.
Recently, a critical evaluation of numerical solutions to the benchmark surface flaw problem has
been made in Ref. [I]. In this study, three typical numerical methods such as the Schwartz alternating
method, the finite element method and the boundary integral equation method were compared in terms
of the stress intensity factors along flaw border. It was concluded in Ref. [ 11that the alternating method
was very inexpensive, but it gave about 210% precision which was rather poor comparing with 25%
accuracy of the 3-D hybrid crack elementf2-41 and “boundary integral equation” methods[5].
In the alternating method[6,7] the analytical solution, which is a basic solution required in the
alternating technique, has been limited to a cubic pressure variation on the crack surface[8]. This
limitation is thought to be one of the major reasons for the inaccurate solution through the alternating
method. Since 1971, no work has appeared in literature to generalize the solution in Ref.[8] to an
arbitrary pressure variation on the crack surface, due to the seemingly insurmountable mathematical and
algebraic difficulties.
Recently, Vijayakumar and Atluri[9] have derived a general solution procedure for an embedded
elliptical crack, subject to arbitrary crack-face tractions, in an infinite solid. The first part of the present
paper, a more detailed solution, as well as a general procedure for the evaluation of the required elliptic
integrals, is derived. The solutions in Ref. [9] and the present paper represent a generalization, hitherto
thought to be unachievable, of the potential representation of Segedin [ lo] and Shah and Kobayashi [S] to
solve both the problems of arbitrary normal and shear tractions on the crack face.
The major objective of the latter part of the present paper is to show that when this generalized
solution is implemented in the alternating method in conjunction with the finite element method, the
alternating method becomes a very inexpensive procedure for routine evaluation of accurate stress
intensity factors for elliptical cracks in various structural components. In order to save computational
time, several special techniques are developed in the present paper. Results obtained by the present finite
element alternating method are compared with the benchmark estimate[l] and other results available in
literature.
241
248
T. NISHIOKA
and S. N. ATLURI
2.POTENTIALFUNCTIONSFORANELLIPTICALCRACK
IN AN INFINITE
SOLIDWITH ARBITRARY
CRACK-FACETRACTIONS
Using the well-known Trefftz’s formulation[ll], the problem is reduced to finding the appropriate
potential functions [9]. Suppose that x1 and x2 are Cartesian coordinates in the plane of the elliptical crack
and x1 is normal to the crack-plane as shown in Fig. 1, such that:
(x,/a,)z+ (Xl/tlJ” = 1, a, > a?
(1)
describes the border of the elliptical crack of aspect ratio (a,/~). The ellipsoidal coordinates &(a. = 1.23)
are defined as the roots of the cubic equation
+-(g&-(&)-($=0
(21
where we write eqn (2) in an alternate form:
w(s) = P(s)lQ(s)
(3)
P(s) = (s - 5Xs - MS - 5J; Q(s) = s(s + a$(s+ a:)
(4)
and
where
The elliptic boundary (1) in the plane x3 = 0 corresponds to the curve & = & = 0. The crack surface itself,
namely, the region inside ellipse (1) in the plane x3 = 0 is given in a simple manner by the surface & = 0.
The three potential functions f,(a = 1,2,3) of the Trefftz’s formulation[9,11] are taken to be:
(5)
where
k+l+l
[w(s)1
ds
-\/acs>’
(61
In ew (5) Cu,k,lare undetermined coefficients and the commas are introduced for convenience only.
Fig. I. Elliptical crack in an infinite solid.
249
Solution for embedded elliptical cracks
Denoting by fu,a the partial derivative of f, with respect to x,(/3 = 1,2, 3) we write for the first, second,
and third partial derivatives of f, as:
The components of displacement ui and stress “ii in terms of J”~(CX
= 1,2,3) are given by
UI= (1 - 2Wl.j +A,,) - (3 - 4V)fl.j -t x3(0,,
(loa)
&?= (I - 2yN2.3 + f3.2)- (3 - 4v)f*,3+ %(V.AQ
(lob)
(1Oc)
and
(IN
A
@_
wf3.22+ 2vf3.,,
- 2f2.32
- 2vf1,31+
X3tV.fM
(Ilb)
2/N - 2v)f3,,2
- u - VW,,*3
+.fxJ f x3tmu1
(llc)
72 =
~I-2 =
(Ild)
c33 = 2/-d-f3.33+ x3(vh,331
@3, = 2Y[
- (1 -
VIf1.33
+ dfl,ll
+f2,2,1
f .@a31
UW
C3?= b.4 - (1 - v)f2,33+ d_fr,,*i- f2.22)-I-xm,231
(1W
and
w
v.f^ = f,,, +f*,*+f,,rc
where Jo and v are the shear modulus and Poission’s ratio.
By successive differentiation, it can be shown from eqn (6) that, since ~(5~) = 0,
Fir,=
I
cc 8k+‘mk+‘+’
Ei
axfad
-= ds
--
dQW
= ak,al,am,gk+i+l
f c3 1 ’ 3
ds
~Q(s)
(13)
wherein
k, = k; 1, = 1; m, = 0.
(14)
In eqn (13), we have used the additional notation that ah implies the jth partial derivatives with respect to x,.
Similarly, the first order partial derivatives of Fk, with respect to xa (/3 = 1,2, 3) can be expressed by
I
Fkl.0= =
where
(1.9
Ez
k, = k + S,,; I, = I+ &;
m, = S38;
where &,a etc., are the well-known Kronecker deltas.
In the case of the second and third order partial derivatives, we derive:
(16)
T.NISHIOKAandS.N. ATLURI
250
where
F;lPy = (k + I + l)!
(18)
(19)
a%
Pn=z=-2/(a;+s)
(20)
(a=1,2,3)
in which a3 = 0, and
(21)
where
GO =
(k+ I+ M'QW
p;,p~p3”,x~,x~x,“,
ko(ko-21)
(53 - 5A53 - 52)
I
lo(lo-;)
2PlXl
I
mo(mo-1)
2P2X2
.\=t3
ko=k+8,p+8,y;10=i+S~p+S2y;mO=S38+S3~
k, = k,+ S,,; I, = I,+ Szs; m, = m,+ i&
.
(22)
.
The partial derivatives of F’& in eqn (21) are given in Appendix. It is noted that the above derivatives
are needed (i) in satisfying the boundary conditions on the crack-face and (ii) in evaluating the far-field
stresses in the solid containing the elliptical crack which is subject to arbitrary tractions.
It is now seen from eqns (13) to (21) that we need to evaluate a generic integral of the type:
11
a,kl a,a;lh
k+l+l
ds
ec’
To accomplish this, we expand Ok+“’ in terms of x2 and carry out term by term differentiations. Then
we get
2q-zr-I,
x,2pmZq-k,
(r)! (2p-2q-k,)!(2q*22r-I,)!(2~1m,)!
. (2p - 2q)! (2q - 2r)! (2r)!
(p-q)!
(q-r)!
2r-m,
where (.) denotes a multiplication, and
Thus one of the key algebraic steps in the successful application of the analytical work in Ref.[9], in
conjunction with an alternating method, is the evaluation of generic elliptic integrals in eqn (25).
3. SYSTEMATIC PROCEDURE OF EVALUATION
OF THE ELLIPTIC INTEGRALS -Ip-+q_,,,
In general, the integrals of eqn (25), for a given set of parameters p, q, and r, can be evaluated in
terms of incomplete elliptic integrals of the first and second kinds, and Jacobian elliptic functions[l2].
Although the closed-form expressions for lower order components of .l,_,, q-r,r were given in Ref. [7,8],
as pointed out in Ref. [7,8], deriving the closed-form expressions of the elliptic integrals involves
251
Solution for embedded elliptical cracks
exorbitant, if not unpleasant, algebraic work. Therefore, it is important to develop a systematic generic
procedure to evaluate these integrals for arbitrary values of p, 4 and r.
Thus, we rewrite eqn (25) using Jacobian efliptic functions[l2] as:
Jp-q,q-r.r= F
= ;y
MI
(sn”Pu)(nd2q-“u)(nc”U) du
I0
2
2
Lp.q-r.r
(26)
where
SFI’Uj
= L&f+
53).
(27)
The following identities for Jacobian elliptic functions are used.
sn’u + cn’u = 1; k2sn2u + dn’u = 1;
dn’u - k2cn2u = k12; ki2sn2u i- cn2u = dn’u
1
1
ndu = dnlr_ncu=G:sdu=e
(28)
dnu
where
k? = (0: - a;)/& k’?= 1 - k*.
(29)
We may derive by using integration by parts in eqn (26) that:
L,, 4-r, I = C2r_Ll)k,2 {(sn2p+‘u)(nc2r-i u)(nd2q-“-‘u)l$
+ [2(-p f r - 1)+ 2(p - q - r + 2)kZIL,,_,
r-j
t k’(-2p + 2q - 3)L,,q-,,-,I.
(30)
Thus, we need the starting values of L,,,_,.,_, and Lp,y-r,r_2 to evaluate Lp,U_r.r.The lowest order
starting values are:
&.,-I
L, q,
-2
=
=
U’(sPr2Pu)(nd2qu)(nc-2u)du
I0
u’(sn2PU)(nd’4U)(nc-4U)du..
(314
@lb)
The above integrals can be reduced as follows:
j+Y+2/p(2-Y~p
r.,.,.-2=&x&
c2 ((p- 1)- jf!j!(2j_o
y_o
y-J
y)!~!~~(q_j-y~
(3%)
where
I,, =
1?m+2
=
u’nd%
du
2m(2- k’)J,, + (1 - 2n1)1~,,,_~-kZmulcnulnd2mi’ul
(2m
+ l)kf2
(33)
(34)
T. NISHIOKA and S. N. ATLURI
-
Pfesent
qo,o
Closed-Form
Systematic
Procedure
Expressions
17,81
Fig. i. Elliptic integrals obtained by the present systematic procedure.
For 2(p - j - y) < 0 in eqns (32a) and (32b), we find I_*, = Gzm,where
G2mct
_
-
ulsnu,cnu,
k2dnZm-’
-I
(1 - 2m)k”G~~_~+ 2mC2 - k21Gz,
(2m + 1)
(35)
Thus, finally we see that one needs the following starting values for evaluating the general terms of
I 2m+2Jand G2m+2:
I, = G,, = F(u,) = u1
I,=~IE(u,)-kZsnu,cdu,J
Gz = E(u,)
(36)
where Ffu,) and E(u,) are incomplete elliptic integrals of the first and second kinds respectively,
The procedure shown in eqns (2&o@ is implemented in a computer program to evaluate the elliptic
integrals numerically. The variation of numerical values of the elliptic integrals JI,, with the parameters
1, m, and n is shown in Fig. 2. To check the validity of the present procedure and computer program, the
integral values are compared with those obtained by the closed-form expressions in Ref.[7,8]. For the
lower order components of J,,,,, the present integral values are exactly the same as those from the
closed-form expressions. For the higher order components of Jr,,,,, the integral values change monotonically with the parameters.
4. RELATION BETWEEN CRACK-FACE TRACTIONS
AND POTENTIAL, FUNCTIONS
Let the tractions along the crack surface be expressed in the form
(37)
so that the values of (i, j) specify the symmetries of the load with respect to the axes of the ellipse. The
solution in terms of the potential function is assumed in the form
(38)
253
Solution for embedded elliptical cracks
The potential functions expressed by eqn (5) are to be understood as that k and I replace (2k - 21+ i)
and (21-t j), respectively. In the Trefftz formulation[ll] the boundary conditions can be expressed in
terms of the potential functions:
t39d
in which the boundary condition for fj is uncoupled from f, and fi. However, if eqns (39a) and (39b) are
directly used as in Ref. [9], there will be singular terms in the equations relating the coefficients C of eqn
(37) to coefficients A of eqn (38).
To overcome this difficulty, we use alternative forms for the boundary conditions, Since f,(a = I, 2,3)
are harmonic functions,
f&33= -L, LI - f,.zz(o = 1,273).
(40)
Then, eqns (39a) and (39b) can be rewritten as follows:
Substituting eqns (37) and (38) into eqns (41a, b), we obtain the follo~~ing linear algebraic equations
upon comparing coefficients of like powers in the polynomial series. For the Mode I problem,
m=O,l ,...,
M
n=O, l,...,
m.
WW
For the mixed problem of Modes II and III,
n=O,l,...,m
(4%)
where
L,=(m-n+k-I+i),L,=nt/+j,
I”.j~=(2k+i+jt1)!(2L,+2S,,i!(2L,?+26,”)!(lY_1
a
(k-m)!
(IL, + S,,)!
z”.”
I2
=
(2ki-itj-tl)!
(k-m+i+j-l)!
(43)
2)
(Lz + 8&J!
(2L,+2-2i)!(2L,t2-2j)!
(L,tl-i)!
(&+1-j)!
’
(44
(45)
and the elliptic integrals J,,,,, LZ,0(O),
etc are generally defined in eqn (25). It should be noted that the elliptic
integrals Jt_,+$.L,.O(O),etc. in eqns (42a) and (42b) involve no singularities.
2%
T. NISHIOKA and S. N. ATLURI
The relation between the parameters A and parameters C in eqn (42) can be summarized in a matrir
form:
14 = IN
Nxl
ICI
(46
NxN Nxl
where N is the total number of coefficients A or C. For a complete polynomial expressed by eqn (37)
the maximum degree of the polynomial M, and the number of coefficients N can be expressed
respectively, by M, = 2M + 1 and N = (&f + 1)(2&f+ 31x3. For an incomplete polynomial, the maximum
degree of poIynomia1 and the number of coefficients depend on not only the parameter M but alsc
parameters i and j in eqn (37).
5. STRESS INTENSITY FACTORS
Once the coefficients C are determined by solving eqn (46) for given loadings on the crack surface,
the stress intensity factors corresponding to these loads are computed from the following equationr9].
For the Mode I problem,
AlI4
2i
5
i
i=Q
k=()
,=,)
j=”
(-2)2k+i+i(2k + i + j + I)!
(471
where 6 is the elliptic angle and
A = a: sin2 $ + ai cos2 8.
(48)
For the mixed mode problem of Modes II and III,
in which
H,
=:
i
232
(_2)2k+i+j
(2k
+
i
+
j
.+
I)!
i=Oj=Ok=Ol=O
H,
=
(~)2k-2’+~(~)2’+i
Cy;iLl.,
f.sl)
‘.
i
1:
$
i
ix0
j-0
k=O
I=”
(_2)2kih-J(2k
+ 3 _
i
_
j)!(~)*k~*‘+‘-i(~)zi+‘~i
c~!;“:,;J)
($2)
6. FINITE ELE~~T
~TE~AT~G
METHODS
In the three-dimensional Schwartz alternating method for elliptical crack problem, which was originally
developed by Kobayashi et a/.[& 71, two analytical solutions were required as follows:
The elliptical crack subjected to cubic normal loading on the crack surface, in an infinite solid. This
limitation was imposed due to the non-availabihty, until the present work, of a general solution for a general
polynomial type normal loading on the crack face.
I’A similar method can be found in Ref.[19].However, in Ref. [19], the analytical solution for the elliptical crack subjected to cubic
normal loading[7,81 was used, while the generalized solution described in the present paper is employed in the present method.
Additional information on finite element alternating method may be found in Ref. [l9J.
255
Solution for embedded elliptical cracks
Solution 2
A semi-infinite
body subjected to uniform normal and shear stress over a rectangular portion of the
surface.
However, as mentioned earlier, Solution 1 has been limited only to cubic polynomial variation along the
crack surface. This is one of the reasons for causing error in the classical alternating method[6,7].
Moreover, the use of Solution 2 has restricted the application of the classical alternating method to only
media bounded by straight surfaces.
To overcome these difficulties, Solutions 1 and 2 can be replaced by
Solution 1
The elliptical crack subjected to arbitrary normal and shear loading on the crack surface, in an
infinite solid, as presently discussed;
Solution
2
A general numerical solution technique such as the finite element method or boundary element
method.
Solution 1 is explained in the earlier part of the present paper. In the present paper the finite element
method is used to generate Solution 2 because of its simplicity.
The present finite element alternating method requires the following steps as shown in Table 1.
(1) Solve the untracked body under the given external loads by using finite element method. The
untracked body has the same geometry with the given problem except the crack. To save computation
time in solving the finite element equations, a special solution technique is implemented. This will be
explained later.
(2) Using finite element solution, we compute the stresses at the location of the original crack.
(3) Compare the residual stresses calculated in Step (2) with a permissible stress magnitude. In the
present study one percent of the maximum external applied stress is used for the permissible stress
magnitude.
(4) To satisfy the stress boundary condition, reverse the residual stresses. Then determine
Table 1. Flow chart for finite element-alternating
technique
Solve the untracked body under external
by using finite
loads
element method (FEM)
Using FEMsolutions
compute stresses
t
fitting
crack face
at
NO
stresses
in step
(2)
I
t
step 5
Step 6
step 7
,crac!i.
Determine coefficients
C in
the potential
functions
c
Calculate the k-factors
current iteration
for
‘+
Calculate residual stresses an external
surfaces of the body due to the loaded
Reverse them and calculate
equivalent nodal farces.
i
step 8
the
Consider
nodal forces in step (7) as external
applied theloads
acting on the untracked body
I
Add the k-factor
solutions
iterations
of
I
1
256
T. NISHIOKA and S. N. ATLURI
coefficients A of eqn (37) by using the following least squares method:
I,=
f s.
(&-
&)2dS
(a = I, 2,3)
(53)
where &a is the reversed residual stresses calculated by the finite element solution, ai:! is defined by
eqn (37), S, is the region of the crack, and I, are the functionals to be minimized.
Rewriting eqn (37) in a matrix form:
and substituting eqn (54) into eqn (53), we obtain the relation between the coefficients A and the residual
stresses:
where
(5) Determine the coefficients C of eqn (38) in the potential functions by solving eqn (46)
= [K’U]).
(6) Calculate the stress intensity factors for the current iteration by substituting coe~cients C in eqns
(472-02).
(7) Calculate the residual stresses on external surfaces of the body due to the loads in Step (4).
To satisfy the stress condition, reverse the residual stresses and calculate equivalent nodal forces. These
nodal forces {Q} can be expressed in terms of coefficients C:
(0
lQh,, = -IGlmCCl
(58)
[Gl, = j- I~lT~~lbI dS
(59)
and
S “8
where m denotes the number for finite element, [IV] is the matrix of isoparametric element shape
functions, [n] is the matrix of the normal direction cosines and [p] is the basis function matrix for
stresses and can be derived from eqn (11). In order to save computational time, the matrix [G], is
calculated prior to the start of iteration shown in Table 1.
(8) Consider the nodal forces in Step (7) as externai applied loads acting on the untracked body.
Repeat a11steps in the iteration process until the residual stresses on the crack surface become
negligible (Step 3). To obtain the final solution, add the stress intensity factors of all iterations.
As shown in the flow chart for the finite element alternating method, we need to solve the following
type of finite element equations:
[Us”,
q’, . . . t q’7 = lQ”, Q’, . . . , Q”1
(60)
and
Qi = Qi(qi-‘): i = I, 2,. . . , n
(61)
in which the superscript denotes the cycle of iteration, [K] is the global (assembled~ stiffness matrix of
Solution
257
for embeddedellipticalcracks
body and remains the same during the iteration process, and q’ is the nodal displacement
vector for ith iteration. Q’ is the nodal force vector for ith iteration and depends on the solution for the
previous iteration qi-’ as expressed by eqn (61).
An efficient equation solver OPTBLOK developed by Mondkar and Powell[l31 is used to save
computational
time in solving eqn (60). The solution algorithm is devided into three parts, i.e. (i)
reduction of stiffness matrix, (ii) reduction of load vector, and (iii) back substitution. In OPTBLOK, the
reduction of stiffness matrix is done only once, although the reduction of load vector and back
substitution may be repeated for any number of load cases. Thus, denoting CPU time for each part by
T,, T2, and T3 respectively. The total CPU time T in solving eqn (60) using OPTBLOK can be expressed
the untracked
by
T=T,+(nt1)(TztT3)=(T,tTztT7)tn(TZtT-J
(62)
where n is the total number of iterations. Since T, is much larger than (T2 t T3), a great amount of
computational time can be expected to be saved by comparing with the case in which eqn (60) is solved
for each iteration (T* = (n t l)(T, t Tz t TJ). To illustrate this situation, we consider the example given
in Ref.[13]. For a set of linear equations with the number of equations of 1960, and half band width of
200, the CPU time for reduction of load vector and back substitution was about 5.6% of the total CPU
time (T, t T3 = 0.056T). Since for a typical problem the present alternating method needs 3 iterations
(n = 3), the additional cost in this case is only about 16.8%, which is considerably smaller than 300% in
the case when eqn (60) is solved for each iteration.
7. ANALYSES OF EMBEDDED CRACKS
All numerical analyses given in the present paper concern the Mode I crack problem. A linear
elastic material with Poisson’s ratio 0.3 is used for all analyses. All calculations were made by using a
CDC CYBER 74.
7.1 Circular crack in an infinite solid
We consider an infinite solid containing
a penny-shaped
crack, the faces of which are subject to
normal stresses of the type (i) ai? = -pox: t pox:, and (ii) o-1-2= -3p,x:x2 t pox:. The known analytical
solution@] for these problems are (i) KI = 16p, cos e/(&r),
and (ii) KI = 32p, sin 38/35&)
respectively. The analytical solution for the circular crack (~/a, = 1) for arbitrary loadings can be rederived
from the solution presented in the present paper by taking the proper limits as k + 0 and rearranging the
equations extensively. In the present analysis for convenience, the above crack was modeled as an
ellipse, with (&/a~) = 0.982. It is seen from the results presented in Figs. 3(a, b) that the present
10,
=33
= -3P,x,
2
x* + p,x:
0: 5..
a0
\I
Y
oc ,-
-0
5
(a)
Fig. 3. Stress intensity
(b)
factor for a circular crack in an infinite solid and subjected
CT’:,’
= - P,xT t P,,x;, and (b) CT\!)= - 3P,,x:x2 t P,,x:.
to pressure:
(a)
758
T. NISHfOKA and S. N. ATLURI
analytical solution for the nearly circular crack with (ar/a,) = 0.982 is in excellent accord with the
analytical solution[8] for the circular crack (aJar = 1). Especially, both the solutions are practically the
same around 8 = 0”.
7.2 Circular crack in a finite
~0~~~ bar
Consider a round bar with an embedded circular crack and subjected to pure tension or pure
bending at the ends of the bar. The crack is located at the center of the bar as shown in Fig. 4. To satisfy
the boundary conditions on the external surfaces of the bar, the finite element alternating method was
used with the present analytical solution for the nearly circular crack (u2/uI = 0.982). Figure 5 shows the
finite element breakdown for the untracked round bar. Only one-eighth of the round bar is modeled due
either to symmetry or to antisymmetry. The prescribed displacements imposed on the finite element
model for the untracked bar are also shown in Fig. 5. The analytical solution used as Solution 1 in the
present alternating method was that corresponding to eqns (37) and (38) with (A4 = 2, i = 0, j = 0) for the
tension problem and with (M = 2, i = 1, j = 0) for the bending problem. The matrix [G] given in eqns (58)
and (59) is calculated on the surfaces of r = R and x3 = L, prior to the start of iteration as shown in Table
1.
Normalized stress intensity factors for various geometries under pure tension as well as pure
bending are summarized in Table 2. For the tension problems, the present results differ less than 0.6%
from other results obtained by an axisymmetric finite element method[14], and integral equation
method [15].
For the bending problem, the present result differs by - 1.l% from an approximate solution obtained
by Benthem and Koiter[tft]. Since Benthem’s approximate solution differs by 0.8% from Sneddon’s
solution, the difference between the present result and true solution may be expected to be much
smaller.
Figure 6 shows the variation of residual stress on the crack surface with each cycle of iteration in
the alternating technique (see Step 2 in flow chart). It is seen that the residual stress decreases rapidly
and monotonically with the number of iterations.
I
0982R
Fig. 4.
Fig. 5.
Fig. 4. Circular crack in a round bar subjected to remote tension or remote bending.
Fig. 5. Finite element breakdown of an untracked round bar
Solution for embedded elliptical cracks
Table ?. Comparison of stress intensitv factors for a circular crack in a round bar
/
I
/ LOAD
I
a/R
Others
i----
I
ITfXlSiOtl
o.5
i
1.114
(Yamamoto)
1.402
(Yamamoto)
1.072
1.081
(f0.8%)
(Sneddon)
(Benthem)
0.6801
(Benthem)
I L/R=-
L/R=-
Tension
a/R=05
L/R=1 0
I,0
09
0
I23
4
Cycle of lterotlons
Fig. 7.
Fig. 6
Fig. 6. Variation of residual stress on the crack surface.
Fig. 7. Elliptical crack in a finite-thickness plate.
7.3 Elliptical crack in a finite-thickness plate
The geometry of a finite-thickness plate with an elliptical crack, subjected to tension loading at the
end of the plate is shown in Fig. 7. Shah and Kobayashi[17] have solved a similar problem, in which the
plate has the same crack lengths and thickness as here, but has an infinite length and width. Figure 8
shows the finite element breakdown for the untracked plate. Due to the symmetries of the plate, only
one-eighth of the plate was model by finite elements. The matrix [G] given in eqns (58) and (59) is
calculated on the surfaces of xl = W, x2 = h, and x3 = L, prior to the start of iteration as shown in Table
1.
The variation of the stress intensity factor obtained by the present method is shown in Fig. 9 and
compared with the result obtained by Shah and Kobayashi[l7]. The stress intensity factors are
normalized by that of the crack in an infinite solid. The present result agrees excellently with Shah and
Kobayashi’s result especially near the surface of the plate (6 = 90”) where there is more of the effect of
finite plate thickness. It is noted that in the present finite element alternating method six terms (1, x:, x:, XT,
x:x:, x$ for &)(A4 = 2, i = 0, j = 0) were used to satisfy the boundary condition of the crack, while in the
classical alternating method [ 171only three terms (1, xi, xi) for a$: were used.
The CPU time for the analysis of an embedded crack in a finite body was about 800 sec. with a CDC
CY BER 74.
T. NISHIOKA and S. N. ATLURI
260
30
Elliptml
60
72
Angle 0 (Degree)
Fig. 9
Fig. 8.
Fig. 8. Finite element breakdown for an untracked finite-thickness plate.
Fig. 9. Stress intensity factor for an elliptical crack in a finite-thickness plate; a:/a~ = 0.4. az/h = 0.X
8. ANALYSES
OF SURFACE
CRACKS
Next, we consider a semi-elliptical surface flaw in a plate under remote tension and bending as
shown in Figs. lO(a, b). This problem is one of the “benchmark” problems proposed by the Three
Dimensional Fracture Analysis Workshop at Battelle Columbus Laboratories in 1976[18]. A critical
evaluation of various numerical solutions to the benchmark problem has been recently made in Ref. [l],
to establish a “best estimate” of the stress intensity factor variation along the flaw border. The derived
best estimate curve for the stress intensity factor is believed within F 3% of the actual value along the
crack front.
fb)
Fig. 10. Benchmark semi-elliptical surface flaw problem; (a) remote tension and lb) remote bending;
a2i2al = 0.25, H/ W B 2.0, a!/ W S 0.2.
261
Solution for embedded elliptical cracks
x3t -
r
H/w=20
!3Jw =02
a,/20,:025
0,/t
=o 25
” =03
80
459
elements
nodes
-
Fig. 1I. Finite element breakdown for an untracked plate.
The results of the present analyses are compared with the best estimate of magnification factors
defined by the following equations: for the tension problem,
_
F, =
E&
I-J
We)
(63)
a: sin” 0 t a: cos2 0)“4
for the bending problem,
where E(k) is the complete elliptic integral of the second kind, and stresses u,, and C* are defined in Figs.
IO(a, b) respectiveiy.
The typical finite element model used for the untracked plate is shown in Fig. 11, which consists of
1377 total degrees of freedom (before imposition of boundary condition) and total number of 80 finite
elements. The matrix [G] in eqn (58) is calculated on the surfaces of x1 = W, x2 = 0, x2 = t, and x1 = H,
prior to the start of iteration process.
8.1 Residual stress distribution on semi-ellipticul crack
In fitting polynomials in a bounded region using the least squares method, it is well known that
accuracy of fitting in the fitting region can be increased with the increasing number of polynomial terms;
however, in the region outside of the fitting region the fitted curve may change drastically. In the present
alternating method, as explained earlier, the solution for the entire elliptical crack in an infinite solid is
impIemented as Solution 1.
262
T. NISHIOKA and S. N. ATLURI
x,=0
TYPE C
1._
r____.._
ji___^^__
02
0
0,
x2
08
i
orA
‘I
50
Elliptical
Fig. 12.
Ankle
B
(Degreel
Fig. 13
Fig. 12. Distributions of Residual stress over the entire elliptical crack surface.
Fig. 13. Magnification factors with various prescribed fictitious residual stress distributions; 42~1, = 0.25,
a>/t = 0.25, remote tension.
For these reasons, in Step 4 of the alternating method, it is necessary to define stress over the entire
crack plane including the portion of the crack which lies outside of the finite body.
Numerical experimentation is done for arriving at an optimum pressure distribution on the crack
surface extended into the fictitious region. Three types of fictitious stress distributions are prescribed as
shown in Fig. 12. The results obtained by the present alternating method are compared in terms of the
magnification factors as shown in Fig. 13 for the tension problem and Fig. 14 for the bending problem.
Twelve terms of the fifth order polynomia1 (M = 2, i = 0, j = 0,l) in eqn (37) were used for c$. All the
present results agree excellently with the benchmark estimates [ I]. In fact, the differences of the present
results from the benchmark estimates are within the error band of the benchmark estimate as shown in
Figs. 13 and 14. The variations of residual stress on the crack surface with each cycle of iteration in
these analyses are compared in Fig. 15.
It is seen from the figure that Type A gives unstable convergency while Types B and C give
monotonic convergencies. Moreover, Type C is the easiest way to prescribe the ~ctitious stress distribution
especially for a part-elliptical crack or corner quadrant-elliptical crack. Therefore Type C is used for the
following analyses.
To check the effect of the degree of polynomials for the applied stress a\? in eqn (37), the stress
intensity factors are also obtained by using the cubic order polynomial (M = 1, i = 0, j = 0, 1). The
results are compared with those obtained by the fifth order polynomial in Fig. 16 for the tension problem
and in Fig. 17 for the bending problem. It is clearly seen that the stress intensity factors become closer to
the benchmark estimate [ll when the higher order polynomial is used for the fitting of the residual stress
in Step (4), (see Table 1).
8.2 Analysis of the “benchmark” problem
For the geometry of the plate containing a semi-elliptical crack as shown in Fig. 10, further analyses
are made only changing the ratio a,/t from 0.25 to 0.5 and 0.75. The variation of the ma~ification factor
263
Solution for embedded elliptical cracks
TYPE
A
BendIng
-
Present
‘--K
Bwchmarh Estimate
(1pRo, =0.25
0,/t =0.25
o,/w=0.2
H/W=20
with Error Ba7d (I )
14
13
12
I I
I
TYPE
B
L
‘...
..,,_
IO
09
08
07
-..
..,,
.‘..
‘....,:‘y...
.._....y- ...._..___
__
-L -‘-..__.__ -.---.._._.._......
0
30
60
Elliptical
90
Angle B (Degree)
Fig. 14. Magnification factors with various prescribed fictitious residual stress distributions;
az/t = il.25, remote bending.
from
IO
0,6
08
o TensIon;
d”‘=~
A
&“‘=a,
Bendlng;
m
0,7
0.6
0.6
TYPE
B
o.6
TYPE
05
00,
000
a2/2a, = 0.25,
L
012345
0
Cycle
Fig. 15. Comparision
I
of
2
3
4
Iterations
of residual stress variations.
01234
C
264
T. NISBIOKA and S. N. ATLURI
IS
a,&=025
t
a,/t=0.25
“--DOP=3
-
Present
0
Benchmark
o<o+
30
Elliptical
60
Angle
f If
Estimate
90
8 (Degree)
Fig. 16.
Fig. 16. Magnification factors
Elliptical
Angle @ (Degree)
Fig. 17.
with various degree of polynomials in the fitting of the residual stress;
az/2al = 0.25, a& = 0.25, remote tension.
Fig. 17. Magnification factors with various degree of polynomials in the fitting of the residuai stress;
a2/2al = 0.25, aa/: = 0.25, remote bending.
for az/t = 0.5 is shown in Fig. 18 for the tension problem and in Fig. 19 for the bending problem. For the
tension problem the present result agrees well with the benchmark estimate [ I] at the deepest point of the
crack (0 = 90”),while the present result is higher than the upper error bound of the benchmark estimate at the
surface of the plate (8 = 0)o. For the bending problem the present result is close to the upper error bound of
the benchmark estimate at both the points of B = 0” and t?= 90”.
The variation of magni~cation factor for a,/r = 0.75 is shown in Fig. 20 for the tension problem and
in Fig. 21 for the bending problem. As seen from the figures all the present results agree well with the
benchmark estimate except near the plate surface.
Figure 22 shows the variation of the magnification factor F,,, for the tension probtem with the
fractional crack depth. At the deepest point of the crack (e = 90”) the present results agree excellently
with the benchmark estimate[l] while at the surface of the plate (0 = 0“) the present results are on the
upper error bound of the benchmark estimate.
Figure 23 shows the variation of the magnification factor Fh for the bending problem with the
fractional crack depth. As seen from the figure the present results agree well with the benchmark
estimate[l]
at the deepest point (6 = 900). At the plate surface (0 = 00) the present results give higher
values of Fb for a&t = 0.5 and 0.75 than the upper limit of error band shown in Ref. [l].
It is noted that the scatter bounds of the numerical solutions compared in Ref. [ 11were several times
bigger than the estimation error bands. Therefore it may be concluded that the present finite element
alternating method gives a comparable precission to the 3-D hybrid crack element[2,41 and the boundary
integral equation method[5].
The CPU time for the analysis of surface crack was about 850 sec. with a CDC CYBER 74.
9. CONCLUSION
The complete solution for an embedded efliptical crack in an infinite elastic solid and subjected to
arbitrary tractions on the crack surface is rederived from Ref.[9]. While the solution can be reduced to a
1‘0
”
0
.
30
Angle
60
90
)
e (Degree)
Benchmark
Estimate
with
Error Band ( I
Present
Ellipttcol
.. .. . ..
...0
..I
-
_...
-
.I
---
-
-
-.
Fig. 18. Magnificationfactor for a surface flaw in a plate: a2/2al = 0.25,
az/t = OS, remote tension.
‘;;
.-s
t
2
b
Lz
a,/t=O.5
-
I
30
Angle
f?encl#nark
with Error
Resent
Elliptical
--o--
-
/t =0.5
60
)
90
e (Degree)
Estimate
Band ( i
_..
-
-
Fig. 19. Magnification factor for a surface flaw in a plate; 0212~
= 0.25, azlt = 0.5, remote bending.
0
I.5
a2
Elliptical
30
with
60
90
B (Degree)
Band (I 1
Angle
Error
Benchmark Estimate
Present
a2/2al = 0.25, al/t = 0.75, remote tension.
Fig. 20. Magnification factor for a surface flaw in a plate;
0
-
:
:
30
Angle
Present
Elliptical
:
-
=0,75
8 (Degree)
60
90
Fig. 21. Magnification
factor for a surface flaw in a plate:
a2/2al = 0.25, ap/t = 0.75, remote bending.
0
’
I
0.0
15
0,/t
001
0.0
.’
1.0’.
=
U
04
Depth
Crack
Error
02
with
Benchmark
Present
a,/t
06
0.8
(I)
Estimate
Band
Fig. 22. Variation of magnification
factor with the fractional
az/t remote tension, u2/2a, = 0.25.
0”
I
c
;Ff
5_
E
b
0
2
LLY
1.5
2.0
crack depth
IO
Depth
0.4
:
factor
a,/t
Q6
with
i
+
I.0
the fractional
0.8
)
depth a:/t; remote bending, aJ2a1 = 0.25.
of magnification
Crack
:
0.2
Fig. 23. Variation
Preseni
Benchmark Estmate
..._... with Error Band ( I
00
t
0
001 :
’5
i
crack
Solution for embedded elliptical cracks
267
closed-form solution for a relatively simple loading such as constant or linear variation of the tractions,
for a high order polynomial variation of the tractions the solution procedure requires a digital computer.
The genera] evaluation procedure for the necessary elliptic integrals is also derived. The integral
values obtained by the present procedure agree exactly with those obtained by the closed-form
expressionj7,Sl.
The alternating method in conjunction with the presently derived analytical solution and the finite
element method is developed, to analyse an embedded or surface elliptical crack in an arbitrary-shaped
finite solid. The present finite element aIternating method Leads to accurate evaluation of stress intensity
factors and is about one order of magnitude inexpensive in computing costs as compared to those with
the hybrid finite element method [2,4].
It is also demonstrated that the stress intensity factors obtained by the alternating method can be
improved when the degree of polynomials in the applied stress for the analytical solution is increased,
However, the advantage of using the presently developed higher order solutions can be more dramatically demonstrated in problems wherein the stress solution for the untracked solid, at the location of the
crack, is more complex than for problems solved here. Such problems, as for instance of pressurized
cylinders with internal or external surface cracks, or of cracks at pressure-vessei-nozzle junctions, will
be discussed in a forthcoming companion paper.
Acknowledgements-The authors gratefully acknowledge the financial supportfar this work provided by U.S.A.F.O.S.R. under grant
81-0057to Georgia Institute of Technology. They appreciate the timely encouragement of Dr. Anthony Amos, the responsible AFOSR
program official. They thank Ms. Margarete Eiteman for her care and patience in preparing this manuscript.
REFERENCES
[I] J. J. McGowan (Ed.), A critical evaluation of numerical solutions to the “benchmark” surface problem. SESA Monogruph (1980).
121S. N. Atluri and K. Kathiresan, Stress analysis of typical flaws in aerospace structural components using three-dimensional hybrid
displacement finite element methods. AIAA Paper78-513, Proc. A~AA-ASPS 19th SMD Conf.Bethesda MDpp.340-350.(Aug.
1978).
131S. N. Atluri and K. Kathiresan, Three dimensional analysis of surface flaws in thick walled reactor pressure vessels using
displacement-hybrid finite element methods. N~cfear ~~gi~g~e~ig~51,163-176(1979).
[4] S. N. Atluri and K. Kathiresan, Stress intensity factor solutions for arbitrarily shaped surface flaws in reactor pressure vessel
nozzle corners. Itlt. J. Pressure Vessels Piping 8, 3 13-332(1980).
[S] J. Heliot, R. Labbens, A. Peiiissier-Tanon, Benchmark problem no. l-semi-elliptical surface crack-results of computation. Znt.J.
Fracture 15, R197-R202(1979).
161A. S. Kobayashi. A. N. Enetanyaa and R. C. Shah, Stress intensity factors of ellipticai cracks. P~spec~s oi Fracture ~er~~u~ics
(Edited by G. C. Sih, H. C. Van Elst and D. Broek), pp. 525-544.Noordhoff International, Leyden (1975).
[7] R. C. Shah and A. S. Kobayashi, On the surface flaw problem. The Surface Crack: Physical Probfetns and ~um~utati(~~uf
Solutions (Edited by .I. L. Swedlow), pp. 79-124. The American Society of Mechanical Engineers (1972).
181R. C. Shah and A. S. Kobayashi, Stress intensity factor for an elliptical crack under arbitrary normal loading. Engng Fracture
Mech. 3, 71-96 (1971).
[9] K. Vijayakumar and S. N. Atluri. An embedded elliptical flaw in an infinite solid, subject to arbitrary crack-face tractions. J. Appl.
Mech. 48, 88-96 (1981).
IlO] C. M. Segedin. Some three-dimensional mixed boundary-value problems in elasticity. Report 67-3. Department of Aeronautics
and Astronautics College of Engineering, University of Washington, June 1967.
(111E. Trefftz, Kandbueh der Pk~sik, Vol. 6, u. 92. Snrin~er-Verbal. Rerlin (192%.
i 12j R. F. Byrd and M. D. Friedman, Handbook of Eifipts Infegr& for En&&s and Scientists, Springer-Verlag, Berlin (1971).
(131D. P. Mondkar and G. H. Powell, Large capacity equation solver for structural analysis. C~~pff~.Structures 4,699-728 (1974).
[ 141 Y. Yamamoto and Y. Sumi. Stress intensity Factors in a cracked axisymmetric body calculated by the finite element methcd J.
Socl Naval Architerts of Japan 133, 179-187(1973).
[W I. N. Sneddon and R. J. Ta& The effect of a penny-shaped crack on the dist~bution of stress in a long circular cylinder. 18~ J.
Enninn Sri. I, 391-409f 1963).
[16] J. P. Benthem and W. T. Koiter, Asymptotic approximations to crack problems. ~e~~o~~ of A~afysis und Sof~fiuns of Cmk
P~/)b~e~~(Edited by G. C. Sib), pp. 131-178.Noordhoff International, Leyden (1973)
[ 171 R. C. Shah, and A. S. Kobayashi, Ellipticalcrack in a finite-thickness plate subjected to tensile and bending loading. L Pressure Vessel
rerk~ofl~~~: pp. 47-54 (Feb. 1974).
[IS] L. E. Hulbert. Benchmark problems for three-dimensional fracture analysis. lnt. J; Fracture 13, 87-91 (1977).
[I91 F. W. Smith and T. E. Kullgren, Theoretical and experimental analysis of surface cracks emanating from fastener holes
.~FFDI_-TR-7~1~~Air Force Flight Dynamics Laboratory (Feb. 1977).
(R&oed 2 &ember
1981;receiaed far ~ubl~&otion t February 1982)
APPENDIX
&
~~
fi
To evaluate the rtress components at a given point (xl, x2, x2) by using a computer, a general evaluation procedure for obtaining
the partial derivatives of F&, is also one of the key algebraic steps in the successful application of the present analytical solution. We
Spstematic procedure of evaluation of the partial d&&es
268
T. NISHIOKA and S. N. ATLURI
rewrite eqn (18) in the following form:
F&, = (k + 1+ I)! T, Tz
Ts
iAl!
where
7-c= x$“‘:Tc= p&);
The parameter k,, I, and m, are given by eqn
(1%
T; = p:‘(&); TX= p?‘(<,); Ts = d/Q!&).
(A?)
Then, using the chain rule of differentiation, the derivatives of F&,, can be evaluated by
aFirar_
--,k+l+l)!“&.R.
ax6
t ,431
where
R, = T, Tz.. T,_, T,+,
Denoting 2
6
.Tc,.
IA4r
by T,J and 5dx by l.s
where the required partial derivatives & are given by
2xaQM
Q’(s) and P’(s) indicate, respectively, the derivative of Q(s) and P(s) expressed by eqn (4) with respect to s.
b.44)