A new method to solve crack problems based on G2 theory Elementary Fracture Mechanics
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Transcript of A new method to solve crack problems based on G2 theory Elementary Fracture Mechanics
21st ALERT Doctoral School Friday 8th October 2010George Exadaktylos Slide 1 of 43
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A new method to solve crack problems based on G2 theory
Elementary Fracture Mechanics
George ExadaktylosTechnical University of Crete
http://minelab.mred.tuc.g
Dedicated to the memory of Prof. Ioannis Vardoulakis
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The presence of crack-like voids is known to have a profound effect on the strength and mechanical properties of materials. Griffith (1921) first showed that the low tensile strength of glass could be explained by the presence of slit like cracks. Irwin (1957) extended Griffith's ideas by introducing the concept that a Critical Energy Release Rate (ERR) governs fracture, while Barrenblatt (1962) developed the concepts of modes I, II, and III crack tip Stress Intensity Factors (SIF’s) as governing extension, shearing, and tearing modes of deformation. Irwin (1957) showed the equivalence of the ERR and the crack tip SIF’s, and crack tip SIF’s for a wide variety of crack geometries have long before been compiled (e.g. Sih (1973)).
Grittith A. A. The phenomenon of rupture and flow in solids. Phil. Trans. R. Soc. Lond, 221A, 163-198 (1921).Irwin G. R. Analysis of stresses and strains near the ends of a crack traversing a plate. J. Appl. Mech. 24, 361-364 (1957).Barrenblatt G. I. Mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, Vol. 7. Academic Press, New York (1962).Sih G. C. Handbook of Stress Intensity Factors. Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem (1973).
Historical notes
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Modes of fracture (Ch 1 CWS)
Mode II and III cracks bear a certain analogy to edge and screw dislocations, respectively, in the sense that displacement discontinuities exist along the crack surface behind the crack tips.
The superposition of the three basic modes is sufficient to describe the most general plane case of local crack tip stress and deformation fields.
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Remarks:
1. The theoretical foundation of Fracture Mechanics may be soundly basedupon the Linear Small Strain Theory of Elasticity, regardless of thephenomena occuring within the plastic zone at the crack tip which precipitatesfracture.2. The failure criterion may be based upon a limiting intensity of the local elasticstress field in the neighborhood of the crack tip. This limit may be specified interms of a single parameter. This parameter was initially recognized historically to be the so called Strain Energy Release Rate (SERR) and was later shown to be related to the SIF (fracture toughness).3. The correct value of the SIF to be applied in design should be based uponappropriate experimental data in combination with the appropriate theoretical Stress analysis (the latter is also our topic for discussion!!).4. Cracks are taken to be branch cuts i.e. lines or surfaces of displacement discontinuity in 2D or 3D.
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Sixth framework program of the European unionStresses & displacements in cracked bodies (Ch 2 CWS)
Ixy
IIyy
ZRey
ZImyZRe
Step 1: Choose the following Westergaard function
2
1I
)]az)(bz[(
)z(gZ
Step 2: Moving the origin to the crack tip αζ z
2/1
2
1
])[(
)(
f
ab
agZ I
Step 3: Expand f(ζ) in MacLaurin series (analytic): ....aa2
K)(f 2
21I
iyxz
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Sixth framework program of the European unionState of affairs at the crack tip
2
3cos
2sin
2cos
2
2
3sin
2sin1
2cos
2
2
3sin
2sin1
2cos
2
2/1
2/1
2/1
r
Kr
K
r
K
Ixy
Iy
Ix
0
2cos)1(2
2sin
2
2sin21
2cos
2
22/1
22/1
z
Iy
Ix
u
r
G
Ku
r
G
Ku
Plane strain solution:
yxz
II ZK 2/1
0)2(lim
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)y,x(uu
0u
0u
zz
y
x
2sin
r2
G
Ku
2cos
)r2(
K
2sin
)r2(
K
2/1III
z
2/1III
yz
2/1III
xz
2
KZ II1
0III
Kinematicsof mode IIIcrack
Westergaard
Stress functionSolution
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Sixth framework program of the European unionDetermination of Stress Intensity Factors (Ch 3 CWS)
1. Crack tip solutions using the Westergaard Stress Function (Westergaard, 1936)
IIxx ZyZ ImRe IIyy ZyZ ImRe
Ixy Zy Re
2/122 )( az
CzZI
Choose:
2/322
2
2/322
2
2/122 )( az
Ca
az
Cz
az
C
dz
dZZ I
I
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212/12
1
lim iCCC
z
a
CzZ zI
0lim2/322
2
az
CaZ zI
0;; 11 zxyzyyzxx CC
The value of constant C may be determined from the BC’s away from the crack and the resulting stresses must satisfy BC’s at the crack surface.At infinity:
But the stresses
II ZK 2/1
0)2(lim
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Sixth framework program of the European union2a. Muskhelishvili’s approach using complex potential functions &
conformal mapping
2/11
1
lim22 zziKKKzz
III
The concept of complex SIF
Consider the mapping function: )(z
2/1121 )(22
1
LimiKKK
1
2z
12 K
For the right-hand crack tip
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Sixth framework program of the European unionExample: Straight crack subjected to a concentrated force
00
0
00
0
02/122
loglog1
1
111
4 b
F
iQPF where:
0 corresponds to z=b
43
This problem is of particular interest as it may be used as a Green’s Influence Function to form the solution to other problems.
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Another efficient method for solving plane elasticity crack problems and estimating the SIFs at crack tips is the method which reduces the problem to a Cauchy type singular integral equation by considering a curvilinear crack composed of a series of edge dislocations.
2b. Muskhelishvili’s approach using Cauchy type SIEs
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,)(
)(
,)(
)(
dz
zdzz
dz
zdzz
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Sixth framework program of the European union3. Finite element methods
yr
rK 0
2/11 lim2
2/1
201
2
14lim
r
EuK
y
r
Discretization techniques such as FEM are not basically well suited to problems containing singularities. In order to deal with this difficulty, two basic approaches have been developed:
a) Non-singular crack-tip modelsb) Singular crack-tip elements
a)Non-Singular Models
By using a very high density of elements near the tips, SIFs were derived from near tip Eqs. for stresses or displacements (latter more accurate than the former)
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It has been briefly shown that even though much achievement has been made in crack modeling techniques (analytical and numerical), a simple and practical crack modeling technique is still needed, in particular for complex multiple crack growth problems (in Structural Geology, Rock Mechanics, Structural Engineering, Petroleum Engineering, Biomechanics etc).
Motivation
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Constant normal and shear displacement discontinuities
The Displacement Discontinuity (DD) method, as originally presented by Crouch [1976,1990], is based on a solution of the Neuber-Papkovitch displacement functions, which expresses the stresses and displacements at a point due to a Constant Displacement Discontinuity (CDD) (dislocation) over a finite line segment (i.e. a branch cut).
sss
nnn
uuD
uuD
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2/121)1(2
)0,()0,()(ˆ xpG
xuxuxu yyy
Analytical solution::
Assumption: line segments are small enough so that the DD along Oy-axis can be taken as constant over each segment
Numerical approximation: NjD jy ,...,1,
Fundamental sol.: On each segment-j with a CDD along its entire length the stress is:
1
1
)1()0,(
2
xD
Gx yyy
The uniformly pressurized crack problem
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Sixth framework program of the European unionIf this constant DD occurs at the j-th element of the crack then eq. (3) takes the form
22)(
1
)1()0,(
jj
yj
yy
axx
DGa
x j
The stress at the midpoint of the i-th segment due to a DD at the j-th segment is found by setting ixx
,
22)(
1
)1()0,(
jji
yj
iyyiyy
axx
DGa
x j
Superposition: ji y
N
j
ijyyiyy DAx
1
)0,(
22)()1(
jji
jij
axx
aGA
Influence coefficients for the special case of y=0
Unknown (?)Unknown (?)
It is noteworthy that the central point used by Crouch in his DD element formulation represents the first Gauss-Chebyshev integration point.
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Eq. (9) is an approximation of the following Singular Integral Equation of the 1st kind
p
x
dDG
i
y
1
1
21
Hadamard finite-part
integral
0,10
0,1
0
yxu
yxp
x
y
yy
xy
BC’s of the uniformly pressurized crack problem
NipDA jy
N
j
ij ,...,1,
1
CDD approximate solution
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Overestimation of the relative displacements between the crack surfaces in theCrack tip region is a consequence of assuming that the normal stress at the mid-point of the i-th line segment represents the average stress over the segment.
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Sixth framework program of the European unionConsider the extreme case in which the crack is modeled by one constant
DD element of width 2
pDG
yyy )1()0,0(
G
pDy
)1(
This solution for the opening of the crack may be compared with the exact solution
pG
Dy)1(2
The numerical solution overestimates the maximum opening of the crack in this case by π/2
.
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rDr
GK
rDr
GK
rDr
GK
zr
III
xr
II
yr
I
2lim
4
,2
lim14
,2
lim14
0
0
0
Ιn turn, overestimation of relative displacements of crack lips means overestimation of SIF’s!!!....see Eqs below
1. Avoid more elaborate elements with more than one collocation points.2. Simply you need a better measure of the average stress at each discontinuity location than the simple midpoint value used.
Perspectives
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The impact of non-linearly varying local stress fields on the constitutive law for the stress
dxfL
y
L
L
2/
2/
1
2)()( oxfxfxf
yy
2)(2
1)()( xfxfxfxf
ydx
dLy
dx
ydLyy
x
22
2
22
24
11
24
1
Field theories, which are based on averaging rules that include the effect of higher gradients, are called higher gradient theories. In particular above rule (B.5) represents a 2nd gradient rule, and can be readily generalized in 2D and 3D by introducing the Laplacian operator instead of the second derivative.
For linearly varying fields:For linearly varying fields:
For quadratically varying fields:For quadratically varying fields:
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Sixth framework program of the European unionEducational example: Prescribed profile of a Mode III crack
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)(2
),(2
22
22
yzyzyz
xzxzxz
G
G
y
w
x
wyzxz
2
1,
2
1
0),1(1)0,( 2 c xHxDxwc
z
0,,02
2
yx
y
w
.;)(11
)(),(
,;)(,
,;)(,
,;)()(),(
22
22
/1222
22
2
/1
xeB
eAFyxy
w
xeAGFyx
xeAGFyx
xeBeAFyxw
y
yc
ycyz
ysxz
yyc
Fourier integral transform:
Prescribed profile of a Mode III crack
0
cos)(2
);(
dxxfxxfFc
Extra BC
Simple G2 theory
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.0,
21
;21
;23
,22
21
;21
;21
,1212
20,2
122
5
2
212
2
1
cc
xcF
c
xcFDc
G
x
cc
zcyz
For c=0:
Solution of the single straight Mode III crack problem
)()1()0,( ctxHDxw z
2
122
2;
2
1;2,
2
32
1
110,xF
xD
G
xz
yz
0
2/1222/1 cos112
20,
dxJDcG
xc
cz
cyz
xxHxDFAc
zc
;11 2
0,,02
2
yx
y
wBC #2.BC #2.
0,
)( A
aB
BC #1.BC #1.
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Sixth framework program of the European union1, ii
2
122
2;
2
1;2,
2
32
1
110,xF
xD
G
xz
yz
21st ALERT Doctoral School Friday 8th October 2010George Exadaktylos Slide 31 of 43
Sixth framework program of the European unionIn the sequel we seek that value of the length scale that gives the exact agreement
of the mid-point displacement of the uniformly pressurized CDD with the analytical solution for the uniformly pressurized crack,assuming that the latter is discretized
with only one element.
22
21210,0
G
TDT
GDT z
zyz
The above value of DD should be equal to the mid-point opening of the Mode-I crack as it given from Eq. (2)
G
T
G
T2
21 2
22
1 Solution!..
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Sixth framework program of the European unionConstruction of the new G2CDD element
(G2 stands for grade-2 or 2nd gradient theory)
ji zijyziyz DAx
0,
2
22
22
2
2
22
3
211
jji
jij
jjji
jij
axx
xxa
aaxx
GaA
Creation of new Influence Functions !!!....
New G2-term
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Comparison of G2CDD with CDD & LDD: 1st case ofMode III crack problem under uniform shear
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Sixth framework program of the European unionThe case of three co-linear straight cracks
kK
kE
bc
acc
K
K
K
kK
kE
ab
ac
bc
abb
K
K
K
kK
kE
ab
aca
K
K
K
lCIII
II
I
lBIII
II
I
lAIII
II
I
1
,1
,
22
22
22
22
22
22
22
22
Analytical solution:
22222 acbck
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Generalization for multiple cracks of any shape
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Validation of G2CDD against known results
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Interaction of multiple cracks with a free surface
This solution is constructed by superposition from the infinite body results presented in Part I by using the classical method of images (Hirth and Lothe, 1982).
yxjiSijIijAijij ,.)()(
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Sixth framework program of the European unionValidation of G2CDD against known results
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Sixth framework program of the European union Conclusions
• A brief account of existing analytical methods for attacking LEFM problems has been made.
• A new CDD element was presented for the numerical solution of Mode I, II and III crack problems, based on the strain
gradient elasticity theory in its simplest possible Grade-2 (second gradient of strain or G2 theory) variant.
• It has the advantage of simplicity, yet it has rather good accuracy appropriate for the fast solution of multiple crack
problems.