Engineering Fracrute Mechanics
Printed in Great Britain.
Vol. 28, No.
1, pp. 55-65,
1987
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1987 Pergamon
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THE USE OF CONFORMAL MAPPING FOR
CREATING SINGULAR ELEMENTS
Department
G. TSAMASPHYROS? and A. E. GIANNAKOPOULOSS
of Theoretical and Applied Mechanics, The National Technical
Athens, Athens GR-157 73, Greece
University
of
Abstract-In the present work, conformal mapping is used to present a novel eight-noded
crack-tip element for linear elastic plane problems. In the quadratic isoparametric element the
correct strain field singularity l/Jr is developed. The method can be generalized for creating
singular elements for notches and other stress singularities.
INTRODUCTION
ONE OF the major
considerations of finite elements in fracture predictions is the efficient
singularity modelling. Conventional constant stress elements require extremely fine
mesh subdivision
in the vicinity of the crack tip in order to represent the stress and strain field
singularities[ 1J. The FEM is usually based upon assumptions for displacements and/or stresses
which are defined in terms of polynomial interpolations over elements of finite size. The standard finite elements appear to behave excessively stiffly and a very large number of degrees of
freedom are needed to obtain reasonable results. The estimated values of stress intensity factors
depend considerably on which node is taken for computation. Therefore, it is difficult to obtain
exact representation in the region of the singularity.
Special crack tip elements[2] have the disadvantage that they lack the constant strain and
the rigid body motion modes. Therefore, they do not pass the patch test[6] and the necessary
requirements for convergence are not present. As a result, these elements create problems when
they are used in thermal stress analyses.
One way of representing singularities with isoparametric elements is to generalize the
element formulation so that any singularity may be treated by including the proper near-field
terms incorporating the global-local finite element concept[9]. In this way the displacements are
‘enriched’ with terms that give the proper singularity. However, such elements produce, slight
incompatibility between the adjoining element nodes. Even when this compatibility is removed,
a high order (7 x 7) Gaussian quadrature must be used.
A technique which is based on the use of elements defined by numerically integrated shape
functions was used for generating two-dimensional shape functions for generating two dimensional conforming singularity elements for standard conforming elements[3]. Special quadrature
rules were suggested for triangular elements.
A compatible interpolation field is obtained only if the singularity point is surrounded with
the special elements and if that assemblance is in turn surrounded with standard compatible
elements. The interpretation of the results by this method is complicated.
It has been shown that a singularity occurs in isoparametric finite elements if the mid-side
nodes are moved from their normal position to the ‘l/4’ position[7]. By choosing the mid-side
node positions on standard isoparametric elements, the singularity occurs exactly at a corner of
the element. The solutions of ref. [7] are not as accurate as those of crack-tip hybrid elements,
but hybrid elements are more difficult to use. An energy technique was used in ref. [7] for
determination of stress intensity values and difficulties occurred where two fracture modes were
present in the solution.
Strain singularities have been produced through special geometric configurations of &node
quadrilateral elements[4,7].
However, the strain energy (and hence the stiffness) of such
elements is unbounded[8] (of logarithmic order). Other multi-noded quadrilateral isoparametric
crack-tip
tAssistant Professor.
ZPost Graduate Student.
55
56
G. TSAMASPHYROS
and A. E. GIANNAKOPOULOS
elements
have been reported
in the literature[
lo]. These elements
are complicated
and give
unstable results in stress intensity factors.
A new method involving
an eight-node
isoparametric
element can be established
using
conformal
mapping for a novel crack-tip
element. The procedure
of the present work can be
extended
for other plane singular
elements
in linear elasticity
problems
with notches.
An
interpretation
of the finite element results can come using the values of computed displacements
from the crack tip element.
CRACK
given
The transformations
by
TIP
SINGULARITY
that give the geometry
J’=
f
MODELLING
of an 8-noded
are interpolated
isoparametric
I
element
are
(1)
Nit53T)yi .
i=l
The displacements
plane
by
u=
C Nit57q)ui
i=l
(2)
where Ni are the shape (base) functions
corresponding
to the node i, whose co-ordinates
are
(xi, yr) and the displacements
(Ui, ni) in the x-y space system, with 5-n to be the isoparametric
coordinates.
The strains are given by
(3)
with
dNi
ax0
LB1=
The stress are given
_
O?
-dNi
t3Ni
ay
ax
by
(4 = [DI{4
where
[D] is the stress-strain
matrix.
The element
stiffness
(3)
[K] is given
[Kl = 1’ 1’ WITPIIBl~JId5 drl
-1 -1
by
(5)
where [.I] is the Jacobian matrix of the transformation.
In order to obtain a singular element to be
used at the crack tip for linear fracture problems, the strains and therefore the stresses must be
singular.
57
Conformal mapping for singular elements
Y
tc
H.+l
(a)
ICI
(bl
Fig. 1. The transformations
for creating the crack tip element.
The shape functions for a standard 8-noded isoparametric
element (Fig. la) are given by
(6)
i =
1,3,5,7
for corner nodes,
N.=~(l+~~i)(l-112)+~(~+~qi)(l-52)
i = 2,4,6,8
(7)
for midside nodes.
with &, vi the isoparametric coordinates of the ith node of the element.
Consider the mapping from isoparametric coordinates (5, 7) to an auxillary coordinate
system of coordinated (8, H) as in Fig. 1b.
5=2z-1
(8)
77=2H-1
and by inverting (8)
5++
H=$+l)
1)
I
.
Consider also the transformation of the auxiliary coordinate
coordinates (x, y), through a conformal mapping (Fig. lc).
Z=H+iH
(9)
I
f(Z) = x + iy = z2 .
(8, H) to the final physical
(10)
From (10) we get the one to one mapping
(11)
Equations (11) represent a parabolic coordinate system (x, y are the physical coordinates of the
nodes).
The element (Fig. lc) that is formed by the sequence of transformations (9) and (I 1) is a
crack-tip element with the crack-tip on node 1 and the upper lip of the crack be the side l-7 (or
l-3). A similar crack-tip element can be constructed symmetrically to this one with respect to
58
G. TSAMASPHYROS and A. E. G~ANNAKOPOUSOS
x-axis so that the whole tip to be surrounded by 2 singular elements. Note that the mid-side
nodes 2 and 8 are displaced from their nominal position at the quarter distance.
Combining transformations (11) and (9) we get
(13
The isoparametric element proposed here contain a singuIarity that satisfy the necessary
conditions for convergence. Since the same shape functions interpolate both the coordinates and
the displacements of the element, they satisfy inter-element compatibility and continuity[6]. In
addition, since Ii Ni = 1 the element satisfy the constant strain and rigid body motion
conditions[6].
Take lines that emanate from node 1
H=iYZ
(131
with (Yappropriate given constant 0 ~5(Y< 1.
Then from (11) and (12)
Note that
l-CX*
x=xY
(15)
that is, straight lines remain straight in the crack-tip element.
If
and
For cy = 1 or /3 = I
(18)
The line 1-3 ahead of the crack is given by
x ++
y=o
1)2ZO
1.
(191
59
Conformal mapping for singular elements
Along
this line:
x 20.
5=2&-l,
(20)
SO
at
-=-
1
(21)
ax A'
The Jacobian
matrix
is given
by
_-
5+1
2
=
J] =
-n+1
2
--
nfl
2
Along
the line
l-3
the reduced
Jacobian
(22)
-I$+1
2
is
ax
I
-=$[+l)=&.
(23)
at
Therefore,
the Jacobian matrix
The displacement
variation
is singular at node 1 where x = 0.
along l-3 is interpolated
as
(24)
Substituting
(20) into (24) we get
(25)
u=(-3&+2x+l)u,+(‘t&-4x)u2+(-&+2x)u3.
The strain
in the x-direction
Exx
is given
directly
by
au agau
=-=_-_=
(2-~-&,+(2~-4)~~+(-~~+2)~~
ax ax a[
(26)
with E,, - O([- 1). It can be seen that the strain component
required by the Westergaard
solution.
More generally,
the strains for plane problems are given
au
Exx = ax?
and by the chain
au
Eyy
=6’
a II&
singularity
as
by
au av
=-+-
yxy
ay
ax
(264
rule
au au
ax ax
at
a24 au
a77 --
[LI-I[
From
E,., exhibits
the displacement
ax ax
at at
au
au = ag a7.--au
aY
aY
interpolation
ijay
a7
au
aq
Mb)
60
we
G. TSAMASPHYROS
and A. E. GIANNAKOPOULOS
have
(26~)
where aNi/ag and aNi/an are polynomials of second order of 5 and 7.
Inverting the relations (12) we have
5(x, y) =[2~+2(~~+4y~)"*]"~-
4Y
(+ 1
77(x, y) = --
1
1
(264
J
and therefore
a6
=-_ 4~
(t+ 1)2 ax
4 (
-- Y 86
=_._
l+[+lay.
5+1
)J
It is obvious that the radial strain component E,.,along all rays that emanate from node 1 possesses a
l/h singularity. Note that this element does not require a special treatment. A standard
isoparametric element subroutine can be used for the integration throughout the mesh.
The line l-7 along the upper lip is given by
1
x=-&+l)2so
y = 0.
(274
Along this line
77=2&-l,
x 5 0.
(27b)
SO
a77 --=
ax
1
(28)
J-x
and
ax
-=--
w
The displacements
;(,+I)=-;-*.
-
of the upper crack lip (crack profile) are interpolated
u=
N,ul+N7u~+A’~u~=
= (-2x - ,C&
+ (4J-x
v(l+ 77)
2
u7+(1-
+ 4X)&.
by
$)ux
(30)
Conformal
mapping for singular
61
elements
In any singular problem the strain energy density W becomes unbounded. However, in order
that the problem remains physically reasonable, the strain energy E must remain finite. On any ray
r where
H = CE (c = constant) we have
dx dy = (J( dt dn
(31)
with
IJI=$(*+
V+(n+
1)2}.
(32)
Note that whenever the determinant IJ1 is zero the stresses and strains become singular. The
= 0
strain is singular at x = 0 (5 = - 1, n = -1) along any ray from the crack-tip, since IJ(
X=0
y=o
1Jl - W*)
(33)
2
w-o
N>I
41
r
=
Wdxdy=
E=
O(
5-2)
(34)
W-*)IJ( d5d.q
(35)
dt dq.
(36)
which gives
O([-'+*)
Equation (36) proves that E is bounded. Therefore this singular element yields a bounded strain
energy and hence a bounded element stiffness.
In order to examine the instability of this element we would perturbate node 4 by the small
quantities E, E’
x; = 314 + E
y4*=1+e’
I.
A general point (x, y) will be displayed at (x*, y*) according
(37)
to
(38)
Along r) = 0 and with y* =
r
sin 0
4rsin or0
1+1$=---=
1+2&
(39)
with E > -0.5.
(1 + 5) is not a common factor in displacement components. Therefore the singularity
required for the crack problem does not disappear and the crack-tip element is stable.
62
G. TSAMASPHYROS
INTERPRETATION
and A. E. GIANNAKOPOULOS
OF THE RESULTS
FOR MODE I CRACKING
Since the required crack-tip singularity has been established, it remains to evaluate the
stress intensity factors. Classical solutions for the stress and displacement fields around a crack
tip, predict the asymptotic behavior of the singularity along any radial line from the tip of the
crack. It is important in many engineering situations to know the numerical value of the stress
intensity factors, since their critical values determine whether or not the crack will propagate.
Most reliable results can be obtained by using the displacements at points along the free surface
of the crack. The analytical expressions for the vertical displacement variation along rays
emanating from the crack tip in the vicinity of the singularity is given for mode plane, linear
elastic crack problems by
-
u=4%
J
2$
I
(2k+l)sini--sinrj.
38
(40)
For 8 = 7r we recover the crack profile
(41)
where Kr is the stress intensity factor for the opening mode, G = E/2(1 + V) is the shear modulus,
v is the Poisson ratio and
k = (3 - v)l( 1 + V) for plane stress,
for plane strain.
( 3-4v
Since equations (40) and (41) are asymptotic, the stress intensity factor can be evaluated by
equating the coefhcients of Jr from eqs (41) and (30) (-x = r in this case). Then
(in this case -x7= 1). For the more general case (Kr, Ku) see [i3], eq. (23).
NUMERICAL
EXAMPLES
The specimens that were used for our numerical examples were rectangular and were
loaded with a uniform stress co along the edges parallei to the crack as indicated by Fig. 2. Plane
stress conditions were assumed to all cases. Because of the double symmetry of the specimens
only one quarter of them was modeled. Eight-noded elements were used, conforming with the
proposed singular element.
Figure 3 shows the distribution of stresses (a, - pz)/vO along the x-axis (of the crack) for a
mesh shown in Fig. 2 (a,, c~ principal stresses). This distribution was computed from the stresses
at the nodes on the x-axis. The components of the stresses at the nodes were computed simply by
averaging the stresses of the nearest quadrature points of the elements surrounding these nodes.
One can observe the strong peak at the crack-tip for this rather coarse mesh. Note that, in this
case, the singular element is of a size typical of the overall element scheme.
Using the interpolation form (42) we can compute the stress intensity factor for various
geometries and loads. The results for a central crack and for plane stress are given in Table 1.
The results for an edge crack and for piane stress are given in Table 2. For both cases an optimal
mesh [5] with 159 nodes was used. The results are in very good agreement with the theoretical
ones.
63
Conformal mapping for singular elements
Fig. 2. Mesh with crack-tip element.
Fig. 3. Normalized stress distribution ((T, - uJ/aO along the crack line.
Table 1. Stress intensity factor l&/K, for a centrally cracked plate
subject to a uniform load o0 in plane stress (K,, = a0d/7ra).
a/b
Isida[ll]
0.2
0.3
0.4
0.5
1.055
1.123
1.216
1.334
Normalized S.I.F. K,/&
Crack-tip element
% difference
1.049
1.118
1.211
1.329
-0.57
-0.45
-0.41
-0.37
64
G. TSAMASPHYROS
Table 2. Stress intensity
subject to a uniform
alb
0.2
0.3
0.4
0.5
and A. E. GIANNAKOPOULOS
factor ICI/K0 for an edge crack in a plate
load v0 in plane stress (K, = aod&i).
Normalized
S.I.F. K,/K,,
Crack-tip
element
Bowie[l?]
1.38
1.65
2.10
2.85
1.403
1.673
2.107
2.870
EXTENSION
% difference
1.83
1.39
0.33
0.70
TO NOTCH-TIP
ELEMENTS
The above ideas can be extended for creating elements that capture the singularity of a
sharp notch (Fig. 4a). We can create a notch-element using the mapping
with
2
E + iH + llzll e”
=
(44)
((z(I= (E2 + H2)“*.
The singularity is defined by the asymptotic form of the Airy stress function for the notch
F =
r*+lf( 0).
(45)
Then
g = (x2 + y2)‘/’
ei’P = (E2
+ H2)A8/2
eios.
(46)
With this transformation we map the auxiliary plane onto the upper half of the notch (Fig.
4b). A similar element can be described for the lower part of the notch. Along l-3 line, H = 0,
8 = 0 and cp= 0. Then
X=0
‘lAS
(47)
lb)
Fig. 4. Notch
element.
Conformal
mapping
for singular
elements
65
and
(48)
a=
-_=-_x
a:*8
1
(l/AS)-1
.
(49)
Note that for every line emanating from the notch-tip we have the correct singularity.
CONCLUSIONS
Conformal mapping proves to be an efficient way of creating a non-pathological crack-tip
element as well as other notch elements. Special finite elements are not necessary for plane
problems and the whole structure can be analysed using only standard eight-noded elements.
The ideas developed in the present work can be extended to nine-noded and higher order
types of elements as well as for mode II interpolations and to axisymmetric crack-tip analysis.
Furthermore, the proposed crack-tip element can be used for plasticity problems, where the
material is described by a piecewise linear uniaxial stress-strain curve and total deformation
theory is used. In such cases a stress singularity I/&, similar to the elastic solution, can be found
to dominate the asymptotic analysis of the crack-tip[ 141.
Acknowledgements-The
results presented in this paper belong to a research project supported
by the Hellenic
of Research
and Technology.
The financial support of this Ministry is most gradefully
acknowledged.
Ministry
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[l]
(Received
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