AD-/M>55 93^
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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
AMCMS No. 6in02.H54001
PRON No. lA-7-51701-(02)-M7
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4. TITLE fand SubHde)
COLLAPSED 12-NODE TRIANGULAR ELEMENTS AS CRACK TIP ELEMENTS FOR ELASTIC FRACTURE
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IS. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse aide If neceaeary and Identity by block number)
Fracture Mechanics Finite-Element Method
Isoparametric Elements Singular Elements
Cubic, quadrilateral Elements
Stress intensity Factors
20. 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OD 1 JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE
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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
2. 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
Page
5. (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
6. (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
7. Idealization of a 45-Degree Slant Edge Cracked Panel in Tension. 45
8. (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) BY FOUR-TERM EXPANSION USING APES
K] (EXACT) WITH COLLAPSED TRIANGULAR ELEMENTS 28
11
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. !<! AND K2 FOR 45° EDGE CRACK BY NASTRAN 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
1Swedlow, 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.
4Wilson, 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 expansion-
finite 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
5Tracey, D. M., "Finite Elements for Determination of Crack Tip Elastic Stress Intensity Factors," Engineering Fracture Mechanics, Vol. 3, 1971.
6Blackburn, W. S., "Calculation of Stress Intensity Factors at Crack Tips Using Special Finite Elements," The Mathematics of Finite Elements and Applications, Brunei University, 1973.
7Benzley, 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.
9Barsoum, 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. The 12-node, isoparametric elements have been
implemented in NASTRAN. 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:
10Hussain, 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.
12 u = 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]
Ni(?'^ = 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
(3)
n7,- Zienkiewicz, 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
where [B] is a matrix relating joint displacements to strain field
(4)
[B] - [.. .B.j.... J, [B,] =
9Ni
9x
0
9y
0
9^
9y
9Ni
9x
(5a)
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 (1 - v)/2
(5b)
in which E is Young's modulus and v is Poisson's ratio.
The Jacobian matrix [J] is given by
9Barsoum, R. S., "On the Use of Isoparametric Finite Elements in Linear Fracture Mechanics," International Journal for Numerical Methods in Engineering, Vol. 10, 1976.
10Hussain, 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.
[J] =
— - "- ■■
3X 9x K 9?
9x 9^ 9n 9ri
=
... 9N-J ...
9?
• •• oNi • • •
9n
• •
• • • •
_ _ — __ —
(6)
whenever the determinant of [J] is zero, the stresses and strains become
singular [8-10]. The derivatives of shape functions are
9Ni . l ^ 256 (1 + nni)C-10 + 9(Cf + nfnHOd + m + 27^ + 9qn2)
+ ^^(1 + 9Tin1)(l - n2)(l - nf)
+ ^L(1 +nni)(l - K])^ - 2? - 27^) (7a)
9Ni 9TI
2^6 0 + ^i)[-10 + 9(52 + n^HOn-i + 18n + 27nin2 + 9^)
+ ^^.(1 +95^)0 - ?2)(i - q)
81 256
THE CRACK TIP ELEMENT
(1 + ^i)(l - n12)(9ni - 2n - 27nin
2) (7b)
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
8Henshell, R. D., and Shaw, K. G., "Crack Tip Finite Elements Are Unnecessary," International Journal for Numerical Methods in Engineering, Vol. 9, 1975.
9Barsoum, 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
a0 = IT M + 9a + 93) . ai = IT (-■' " 27a + 27^ ]
\ (10)
a2 = f| (1 - a - 0) , 83 = -yf (1 + 3a - 3B) J
b0 = T6 {-Ul + 9U2 + 9U3 " U4) ' bl = II (U1 " 27U2 + 27u3 " U4)
(11) b2 = ^ (ul " u2 ■ "3 + U4) ' b3 = IT (-Ul + 3u2 " 3u3 + U4)
dx To have singular strain at x = 0 (£; = -1), the reduced Jacobian, ^F »
must vanish at 5 ■ •!. From (8) we have
^-= a1 + 2a2? + 3a3K2 (12)
dx For £; = -IJ HF = 0 leads to the equation dC
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)
Equations (8) and (9) become
x = A (1 + O2 or ir » -1 + zJI" (16)
u = u1 + I (-11^ + 18u2 - 9U3 + 2u4)Jf + j (2u1 - 5u2 + 4U3 - u4) j
H-fC-U! + 3u2 - 3u3 + u4)(^)3/2 (17)
From (17) it is clear ^ has singularity of the order - 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• The Cartesian coordinates of nodal points are
shown in Figure 2 . Using (3),
x = I (1 + 02f(n,a,B)
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
(18)
9x 9^ 9C 9^
[0] = 9x 9y.
_9n 9TI_
1 (1 + C)f(n.a.B) | (1 + Of (n,a,B)
4 (1 + 5)2(cosa - cosp) I (1 + ?)2(sina - sinB) 8 o
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
[0] ■1
'E. in" 9x 9x
K la 9y 9y
2(sina - sing) -2(cosa - cosg) £(1 + ^)sin(a - 3) 5,(1 + ?)sin(a - 3)
-4f'(n»a»3) 4f(rua,3) £(1 + ?)2sin(a - 3) £(1 + C)2sin(a - 3)_
(20)
In terms of polar coordinates (r,0) we have from (18)
1 + C = 2v^ , R = (r/il)cos(e - ^)/cos(^)
n = tan(e - S^/tan^)
(21)
The displacements components u,v at a point (g.n) of the trian-
gular element of Figure 2 are
12 u = 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)
+ 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
The displacement derivatives are
3A 9u_=3u_ci£+3u_3n= 1 -4f'(n,a.B) 0 8x 9C 9x 8n 9x (1 + C)2 Jisinja - 3) 9n
1 SA,
(1 + 5) ilsin(a - 3)
BA, 9u=3u9£+9u9ii_ 1 4f(n,a»g) 0 8y " 85 8y 8n 8y "' (1 + C)2 ^sin(a - g) 3n
1 9A,
: . r ^-{-(cosa - cosB)A1 + 2f(n,a,B)—!-} + (1 + 5) Jlsin(a - B) ' 8n
where
8A0(n.ui)
3n = {(2 + 36n - 54TI2)U1 - (2 - 36n - 54TI2)U1|
(22)
(23)
{(sina - sinB)A1 - 2f,(n,a,B) g-1} + ... (24)
(25)
+ 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. The
order of singularity is (1/r). The leading term vanishes together with
3AO/8TI for all n if
Ul = U10 = U11 = U12 ' Vl = V10 = Vll = V12 (27)
In this case, the order of singularity becomes (l//r). 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] (28)
v = lV ^seF^n.v.,) + F3(n,vi) + 36{F2(r1,vi) - F^n.V^}^
- SeFgCn.v^R] (29)
n
where
F^n,^) = (1 - n)(2u2 - U3) + (1 + n)(2u9 - u8)
F2(n,ui) = (1 - n){-3u2 + 3U3 - u4) + (1 + n)(-3u9 + 3u8 - u7)
FgCn.u^ = 9(1 - n2)[(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^.
(30)
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. The discussion is limited to the
use of nodal displacements. 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.
12
■
2Gu <9) = „.li....<("1)n"lrn"1/2cd2„-lDul<n"e) ' a2n-.Aul("-6)] n n
+ (-1) r [d2nDu2(n,e) + a2nAu2(n,e)]} (31)
n-1 n-1/2 2Gv(e) = I {(-1) r [d D (n.e) + a A (n.e)]
n=l,2,... 2n-l vl 2n-1 vl
n n + (-1) r [d D (n.e) + a A (n.e)]}
2n v2 2n v2 (32)
where
Dul(n,8) = (n - l/2)cos(n - |)e - (< + n - |)cos(n - 1)6 ^
Du2(n,e) = ncos(n - 2)6 - (K + n + 1)cosne
Aul(n,0) = (n - l/2)sin(n - |)0 - (K + n + l/2)sin(n - 1/2)e
Au2(n,e) = nsin(n - 2)9 - (K + n - l)sine (33)
D (n,9) = -(n - l/2)sin(n - |)e - (K - n + |)sin(n - ^)8
D 2(n,e) = -nsin(n - 2)6 - (K - n - l)sin n0
1 5 A (n,6) = (n - 2)cos(n - ^e + (ic - n - 1/2)cos(n - 1/2)6
A „(n,6) = ncos(n - 2)6 + (K - n + 1)cos n( 1v2 in which
(3 - v)/(l + v) for plane stress
3 - 4v for plane strain (34)
13
The coefficients d's and a's are to be determined from the boundary
conditions of a problem. The stress intensity factors K, and K2 are
related to d-j and a-, by
K} = -d^ , K2 = -a1^¥ (35)
In (31) and (32),u and v are displacement components referring to
the local Cartesian coordinates with crack tip as the origin and the
crack on the negative x-axis. A simple transformation can be used to
change the nodal displacements in global coordinates, obtained from
the finite element method, to displacements in local coordinates. For
simplicity, we will assume the local coordinates and the global coor-
dinates are the same.
1. One Term Expansion:
If we retain only the /r term and drop all higher order terms
in the right hand sides of (31) and (32), we shall obtain a set of d-|
and a-i by substituting the displacement components of a nodal point
into the left hand sides of (31) and (32). Numerical results thus
obtained for K-] and l^ vary considerably depending on the locations of
nodal points and the values of displacement components used. From
(28) and (29), the displacement components of a collapsed triangular
1/2 3/2 element are functions of r , r and r . If u* and v* designate the
leading portion of displacements (r ' term only) in (28) and (29),
we have specifically
14
1 1/2 U*(TI = -1) = 2 ir/l) (18u2 - 9U3 + 2u4)
(36) u*(n - 1) = 1 (rA)1/2(18u9 - 9U8 + 2u7)
and similar expressions for v* (replacing u by v). The stress
intensity factors obtained from (35) with d1 and a, computed from
(31) and (32) using u*, v* on the left hand sides and using only the
Jr term in the right hand sides, are independent of r. The use of
the leading portion of displacements u* or v* on the left hand sides
of (31) and (32) is suggested by Tracey [14] and discussed by
Barsoum [15]. For a mode I (or mode II) crack, all a's (or d's) of
(31) and (32) vanish. The stress intensity factor ^ (or K2) can be
obtained by either (31) or (32).
2. Two-Term Expansion: 1 /p
In (31) and (32), if r and r terms are considered, we have
four unknown constants dp a-,, d2, a2 to be determined. Four displace
ment components of any two nodal points should suffice to compute K,
and K2. In practice, we use either the pair of nodes (2,9) or (3,8)
of Figure 2. Two nodes of different r such as (2,3) or (8,9) usually
give poorer results. On the left hand side of (28) and (29), we may
use actual nodal displacements (u2,v2), (iig.Vg) or we may use (u2**,
v2**) and (u9**,v9**) where u2** is the part of u2 corresponding to 1/2
r ' and r terms in (28). For n = ± 1
14Tracey, 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,
15Barsoum, R. S., "Author's Reply to the Discussion by Tracey," Int, Journal for Numerical Methods in Engineering, Vol. 11, 1977.
15
u**(n = -1) = ^ (r/£)1/2(18u2 - 9U3 + 2u4) - f (rA)(5u2 - 4u3 + u4)'
u**(n = 1) = 1 (r/£)1/2(18u9 - 9u8 + 2u7) - f (r/Jl)(5ug - 4u8 + u7)
(37)
For a mode I crack, both a-i and a2 vanish. Only two displacement
components of a node are needed to determine d, and d2. Any of the
following pairs may be used for this purpose: ^.VgK (u3»v3)» ("g'
Vg), (Ug.Vg).
3. Four Term Expansion:
If we take r1/2 up to r2 terms in (31) and (32), the eight
constants di and a-, i = 1, 2, 3, 4 are to be determined from displace-
ment components of four nodal points. We may take two mid-nodes (£/9
and 4J!,/9 from the tip) of the two sides of a collapsed triangular
element as the four nodal points, or we may take four consecutive
nodes of the same radius from the tip (r = Jl/9 or r = 4)1/9). For the
three collapsed triangular elements around a mode I crack shown in
Figure 5(a), displacement components are taken from one of the
following three groups: (i) nodes 11, 12, 13, 14; (ii) nodes 15, 16,
17, 18; (111) nodes 11, 12, 15, 16 of the element (1), nodes 12, 13,
16, 17 of the element (2) and nodes 13, 14, 17, 18 of the element (3).
4. Collocation Method:
Let us, vs be displacements from the asymptotic expansions (31)
and (32) and let u. and vp be displacements from finite elements given
by (28) and (29). For an arbitrarily fixed r, we define
16
■
e = .1 (CusOi) - u^)]2 + [v^e^ - ^(e^]2} (38)
The unknown coefficients d's and a's in (31) and (32) are the set
which minimizes e. In other words, d's and a's are solutions of
3dj
9e-= 0 9aj
^ = 1.2.....n<i ^ 2
where n = 2p, p is the highest exponent of r in the asymptotic
expansion.
NASTRAN IMPLEMENTATION
The NASTRAN implementation of the 12-node quadrilateral follows
that of the 8-node quadrilateral as described in [10]. The dummy
users element facility of NASTRAN is used. This requires coding
routines to calculate element stiffness matrices and stress recovery
computations. Modifications to existing NASTRAN source codes are
made to provide proper output formats for the element. Three-point
Gaussian quadrature is normally used to calculate each partial
integration of the double integral (4). As an option a four-point
Gaussian quadrature may be used instead. All stiffness computations
are performed in double precision while stress recovery is performed
in single precision. Element stiffness matrix computation requires
10 seconds/element on an IBM 360/44. Stress intensity factors are
calculated from nodal displacements by various techniques. Equations
10Hussain, 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.
17
(40) are finally adopted to compute stress intensity factors for
mode I, mode II and mixed mode cracks.
NUMERICAL RESULTS
Computer program APES [16] utilizes the same 12-node quadrilateral
isoparametric elements shown in Figure 1. The high order element
greatly reduces the total number of elements as well as nodal points
needed to model either elasticity or fracture problems. The program
has been designed primarily for user convenience. There are many
convenient features such as the automatic generation of middle nodes
of a side which is a straight line, and the automatic computation of
nodal force for a given distributed and/or concentrated tractions.
However, APES requires the use of special crack tip elements for
fracture problems. There are two different types of crack tip elements
used in APES. One is a small circular core element which completely
encloses a crack tip and which reproduces the singular nature of the
strains there. The other consists of several "enriched" 12-node
elements around a crack tip all having a corner node corresponding to
the tip. In an enriched element, the displacement assumption is
augmented by the leading terms of the singular near field solution
(an extension of the work of Benzley and Beisinger [7]). Both models
lead to about the same high degree of accuracy in stress intensity
factors K-j and Kp.
7Benzley, 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.
^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.
18
In this report, we have used the collapsed triangular elements to
eliminate the use of these special crack tip elements. The same
displacement functions are used for quadrilateral elements and trian-
gular elements, and there is no question regarding the continuity of
displacement between the special crack tip elements and conventional
QUAD-12 elements. There is no need for the use of 8 x 8 Gaussian
quadrature for numerical integration of element stiffness matrix as is
required for the enriched QUAD-12 elements. From the numerical results,
the collapsed triangular elements as singular crack tip elements will
be assessed.
Three mode I crack problems and one mixed mode crack problem are
chosen for numerical computation of stress intensity factors. The
geometries and loads of mode I tension test specimens are given in
Figure 3. The idealization of a half of the single edge crack is
shown in Figure 4. Similar idealization is used also for a quadrant
of a center crack or a double edge crack. Three collapsed triangular
elements surrounding a mode I crack tip are shown in Figure 5(a). The
same idealization is used for NASTRAN and APES (with collapsed trian-
gular elements). For comparison, a circular core with 10-core nodes
around the crack tip is also used in APES and is shown in Figure 5(b).
Around a mixed mode crack tip, six collapsed triangular elements,
shown in Figure 6(a), are used and the corresponding singular 'core'
element surrounded by six QLIAD-12 elements are shown in Figure 6(b).
A 45° slant edge crack, in a rectangular panel under tension, is taken
as an example of mixed mode cracks and is shown in Figure 7. The
19
sixteen-element idealization of the cracked panel is also shown in the
same figure, and the six elements around the crack tip are enlarged
and shown in Figure 6.
Our first goal in the numerical analysis is to find a simple and
accurate way to estimate the stress intensity factor from nodal
displacements obtained from the finite element method. For the
purpose of comparison, stress intensity factors obtained from the
finite element method are normalized by corresponding values which a^e
considered a. exact. Referring to Figure 3, K-j = 1.966 [17] is taken
as exact for the central crack and K-| = 2.00 [18] for the double edge
crack. For the single edge crack ^ = 3.728 for a/b = 0.4, H/b = 4
and K-i = 5.009 for a/b = 0.5, H/b = 3 (first F(a/b) on page 2.11 of
[19]). Normalized stre? . intensity factors for the three mode I crack
specimens are computed ^y APES, using singular 'core' elemest with
10-core nodes. Figure 5(b). Results are tabulated in Table 1 for
h/r = 6 and r0 = 0.01, 0.02 and 0.03. The overall results are better
for r0/a = 0.01. Here and in tables 1 through 8, r0 = 0.01, 0.015,...
should be understood as r0/a = 0.01, 0.015,.., since a = 1 is used for
all mode I crack examples.
17Isida, M., "Analysis of Stress Intensity Factors for the Tension of a Centrally Cracked Strip with Stiffened Edges," Engineering Fracture Mechanics, Vol. 5, 1973.
18Brown, W. F., and Srawley, J. E., "Plane Strain Crack Toughness Testing of High Strength Metallic Materials," ASTM STP-410, 1966.
19Tada, H., Paris, P., and Irwin, G., The Stress Analysis of Cracks Handbook, Del Research Corp., 1973.
20
TABLE 1. RATIOS OF ^ (APES) TO K, (EXACT)
r0 = 0.01 r0 = 0.02 r0 = 0.03
Center crack (exact K-| = 1.966)
Double edge crack (exact K-] = 2.00)
Single edge crack (exact K-, = 5.009)
0.996
1.001
0.974
0.976
0.993
0.964
0.969
0.989
0.966
Replacing the special core element and the regular 12-node, quad-
rilateral elements surrounding the core with collapsed triangular
elements, we can use the APES program for general structures (no cracks)
to obtain displacement components at all nodes. Stress intensity
factors are then obtained from displacements at nodes close to the
crack tip by various techniques mentioned previously. If only one
term expansion is used, the ratios of K-j (Finite Element) to K-. (Exact)
for the three tension test specimens are given in Tables 2-4. The use
of v* (leading term of v) does not always give better results (e.g.
Table 4). Using two term expansion, the same ratios are given in
Tables 5-7. The results from two term expansion show little improve-
ment. Similar results using four term expansion are tabulated in
Table 8. The values in the last column of Table 8 are the linear
average of three values of K-j obtained from three different elements.
Careful study of results tabulated in Tables 2 through 8 reveals that
one term expansion using v at node 14 or 18, Figure 5(a), with r =
1% - 2% of the crack length is the simplest way to estimate K-j and
K-j thus obtained is quite accurate. This technique is adopted in our
NASTRAN to compute K-|.
21
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28
The collocation method requires the computation of displacements
along r=r0 at points between nodes. For the single edge tension
specimen with a/b=0.5, H/b=3.0, the numerical results obtained from
the collocation method are given in Table 9. It can be seen that
values of K-] remain nearly a constant no matter how many terms or how
many mid-points are taken. Because of more computations involved and
since as good or even better results can be obtained by other simpler
methods, the collocation method has not been pursued further.
TABLE 9. ^ (FINITE ELEMENT) FOR SINGLE EDGE CRACK USING
COLLOCATION METHOD. NODAL DISPLACEMENTS OBTAINED FROM APES
WITH 3-COLLAPSED TRIANGULAR ELEMENTS. COLLOCATION POINTS
ARE EQUALLY SPACED ON r=0.01
No. of Mid=Points Between Nodes
— —_
Four-Term Expansion Eight-Term Expansion General Mode I (a'^O) General Mode l(a'=0)
0 2 3 5 9
11 14 19
4.82815 4.82753 4.82743 4.82737 4.82734
4.83417 4.83352 4.83337 4.83322 4.83309
4.90392 4.82446
4.81972
4.85093 4.82966
4.82762
Our second goal in the numerical computation is to assess the
accuracy of NASTRAN in linear fracture, using 12-node, isoparametric
elements with collapsed triangular elements around a crack tip and to
examine the effect of multiple constraints. Normalized stress intensity
factors thus obtained are given in Table 10. It indicates a high degree
29
of accuracy, and the effect of multiple constraints is insignificant.
The multiple constraints tend to increase the stress intensity factors
slightly. In Table 10, by "No" multiple constraint we mean nodes 2 to
10, Figure 5(a), are free to move in both the local x- and y-directions
with respect to node 1, and node 1 has no displacement in the local
y-direction due to the symmetry of the problem (mode I crack). The
column "Yes" gives results obtained by assuming v1 = v2 = ••• = V-JQ = 0
and u-! = ^2 = '" = u-jg-
TABLE 10. ^ (NASTRAN)/^ (EXACT)
■V" 0.01 0.015 0.02
Multiple Constraint No Yes No Yes No Yes
Center Crack a/b=0.4, H/b=4.0 Exact ^=1.966
0.981 1.013 0.982 0.999 0.983 0.994
Double Edge Crack a/b=0.4, H/b=4.0 Exact K-^2.00
1.000 1.021 0.998 1.007 0.999 1.002
Single Edge Crack a/b=0.4, H/b=4.0 Exact ^=3.728
0.980 1.003 0.980 0.991 0.982 0.988
Another goal in the numerical computation is to compare results
using the concept of collapsed triangular elements in APES with results
of APES with singular 'core' element around the crack tip. Ratios of
K-| (APES) to K, (EXACT) are given in Table 11 for the tension test
specimens with r0/a = 0.01. Results in the first column of Table 11
30
are obtained by using three collapsed triangular Elements around the
crack tip where nodes 1 to 10, Figure 5(a), coincide. All vertical
displacements of nodes 1 to 10 are assumed to vanish while they are
free to slide with respect to one another in the x-direction. Results
in the second column are computed by using a singular 'core' element
surrounded by three quadrilateral elements as shown in Figure 5(b)
with h/r0 = 6. Table 11 shows values of stress intensity factors,
obtained by collapsed triangular element, are as accurate as those
obtained by using singular 'core' element.
TABLE 11. ^ (APES)/^ (EXACT)
Center Crack a/b=0.4, H/b=4.0 Exact K-i=1.966
Double Edge Crack a/b=0.4, H/b=4.0 Exact KyZ.OO
Single Edge Crack a/b=0.5, H/b=3.0 Exact ^=5.009
Collapsed Triangular Elements, Fig. 5(a)
0.998
1.005
0.972
Singular "Core" Elements, Fig. 5(b)
0.996
1.011
0.974
In NASTRAN, it is the user's choice to apply the multiple constraint
conditions, equations (27), at the crack tip. But in APES, it is not
yet available for the application of multiple constraints. Since the
effect of multiple constraints is so small. Table 10, that results
from APES with collapsed triangular elements are considered correct
even if the multiple constraint conditions (27) are not satisfied.
31
For the integration of the element stiffness matrices, a 3 x 3
Gaussian quadrature is used in all NASTRAN results in this report while
a 4 x 4 Gaussian quadrature is used in APES.
For a mixed mode crack, six collapsed triangular elements shown
in Figure 6(a) are used. An effective way to estimate the stress
intensity factors is to use v(r0,-Tr) for K-] and u(r0,TT) and u(r0,-TT)
for Kp.
„ , , /H 2G v(ro,TT) u . . -Vgjr 2G v(r0,-7T) 1 /F^ (K+1) ' /r^ (K+1) N
SB 2G u(r0,Tr) -^f 2G u(r0,-TT) I , . K2(TT) = , M-TT) = Q- > (40)
/F^ (K+1) ^ /^(K+1)
K-, = \ (K^TT) + K^-u)) , K2 = i (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
20Bowie, 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 Multiple Constraint Kl K2
1 3x3 No 1.89 0.95 1 3x3 Yes 1.89 0.96 1 4x4 No 1.83 0.92 2 4x4 Yes 1.84 0.93
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 Inter- national 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
3 1 + ? sina
1/2 1 + M + L 4sinesinou '
(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.
10Hussain, 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 Inter-
national 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
32
9
(i-n)[i-5)[-io+9(Cz+nz)]
N2 = jj Ci-n)(i-5z)(i-3Q
N3 = h d-^Ci-^^Cl+SC)
J7 = jj (l+nlCl+^H-lO+QC^+T!2)]
N8 = ^j (i+nHi-S2)(i+3C)
Ng = ^- (l+r1)(l-52)(l-3?)
^4 = 32 a-n)(l+?)[-10+9(C2+Tiz)]
<5 = ^ (iH)(i-n2)(i-3n)
Nio = 32 (i+n)(i-c:i[-io+9C52+nz:']
Nii = 32 (i-CHi-n2)(i+3n)
N = 32- (iH)(i-n2)(i+3n) Ni2 = 32 (i-5)(i-n2)(i-3n)
Figure 1, Shape Functions and Numbering Sequence For a 12-Node Quadrilateral Element.
39
10 9 8 7
II' V
— 1 I2<
• I 9
i
0
6
'5
4
NODE x/l y/H
1 0 . o
2 cos 6/9 sinB/9
3 4cosB/9 4sin3/9
4 cosg sing
5 [2cos3+cosc0/3 (2sin3+sina)/3
6 (cos3+2coscO/3 (sin3+2sina)/3
7 cosa sina
8 4cosa/9 4sina/9
9 cosa/9 sina/9
10 0 0
11 0 0
12 0 0
— X
Figure 2. A Normalized Square in (Cn) Plane Mapped Into a Collapsed Triangular Element in (x,/) Plane with the side % - -1 Degenerated into a Point at the Crack Tip.
40
>D
_JL_
_L
10 T
(a) Center Crack
10
II
10
1
10
(b) Double-Edge Crack
\D
H _l_
(c) Single-Edge Crack
H
Figure 3. Three Tension Test Specimens.
41
SEE FIG. 5
Figure 4. Idealization of a Half of the Single-Edge Cracked Tension Specimen.
42
(a)
25 24 23 22
t—IO
ib)
Figure 5(a). Three Collapsed Triangular Elements Surrounding a Mode I Crack Tip.
(b). Special Core Element and Three Quadrilateral Elements Surrounding a Mode I Crack Tip.
43
(a)
(W
Figure 6(a). Six Collapsed Triangular Elements Surrounding a Mixed Mode Crack Tip.
(b). 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
K> C^
^ LO f\i
s a. T3 C)
o cd u u 0) bO
C cd |H
u be » a
i i-0
o c o
•H
cd
cd o
o
45
(Q)
Figure 8(a). Node 5 Perturbed to 5*. (b). Nodes 20, 21, 23, 24, 26, 27 Perturbed From
Their Nominal Positions.
46
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