Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 1/8

Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.02 booklet: Breaking process
R7.02.08 document

Calculation of the factors of intensity of the constraints
by extrapolation of the field of displacements

Summary:

One describes here a method of calculation of K1, K2 and K3 in 2D (plane and axisymmetric) and 3D by extrapolation of
jumps of displacements on the lips of the plane fissure. This method is applicable only to the case of the fissures
plane, in homogeneous and isotropic materials. It is usable using the command
POST_K1_K2_K3.
The method used is less precise than calculation starting from the bilinear form of the rate of refund of
the energy and of singular displacements [R7.02.05] (option CALC_K_G of operator CALC_G_THETA_T in
2D). It however makes it possible to easily obtain approximate values of the factors of intensity of
constraints, in particular in the case 3D, for which the method of singular displacements is not
operational that with linear elements (option CALC_K_G of operator CALC_G_LOCAL_T).
One can have an idea of the precision of the results by recomputing G by the formula of Irwin, from
values of K1, K2 and K3, and by comparing this value with that obtained by the energy method (method
“THETA”). The precision of the results is clearly improved if the elements touching the bottom of fissure
(quadratic elements) nodes mediums located at the quarter of the edges have (elements known as of “Barsoum”).

Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 2/8

1
Position of the problem

The method recommended for calculation of the factors of intensity of the constraints is the use of
bilinear form of the rate of refund of energy and singular displacements (option CALC_K_G
CALC_G_THETA_T [R7.02.05]). This method is operational and precise, but usable
only in 2D (forced plane and plane deformations) and in 3D for linear elements.
This method is not available into axisymmetric.

Indeed, in the general case in 3D and axisymmetric, one does not experience the development
asymptotic of the stress field, and thus not the form of the singularity. K1 and K2 (and K3 in 3D)
then do not have clear significance.

One can however give a significance to K1 and K2 (and K3) under certain assumptions ([bib1], [bib2]):

1) the fissure is plane
2) the behavior is elastic, linear, isotropic and homogeneous
3) one places oneself in a normal plan at the bottom of fissure

Under these assumptions, one can show that in the normal plan at the bottom of fissure in a point M, them
expressions of the stress fields (or displacements) are identical to the case of
plane deformations; in this case, one can thus give a precise significance to K1, K2 and K3. It is
the same definition as in plane deformations, and moreover, the formula of Irwin who allows to calculate G with
to start from K1, K2 and K3 remains valid. This situation is easily transposable with the axisymmetric case, if
the fissure is plane.

These expressions are, in 3D, for a plane fissure, in a point M:


lim
E
2

K (M) =

U

1
R 0
m

(
2
8 1 -) [
] R

lim
E
2

K (M) =

U

2

R 0
N

(
2
8 1 -) [
] R
lim
E
2
K (M)


3
=
[C]
R 0 (
8 1 +)
R

with:

T, N in the plan of the fissure
T tangent vector at the bottom of fissure in M,
N normal vector at the bottom of fissure in M,
m normal vector in the plan of the fissure
[U] jump of displacement enters the lips of fissure,
R = MP where P is a point of the normal plan at the bottom of fissure in Mr.

Note:

· One can note that the signs of K2 and K3 depend on the orientation on T and N. This is not
not too awkward insofar as the criteria of rupture or fatigue use only them
absolute values of K2 and K3.
· One can also give expressions according to the stress fields, but them
values of the vectors forced on the lips of the fissure are less precise than them
displacements (bus exits of a transport of the points of Gauss to the nodes).
· To have an idea of the precision of the calculation of K1, K2 and K3 by extrapolation of the jumps of
displacements, it is enough to recompute G by the formula of Irwin starting from the factors of intensity of
constraints and to compare this value with that obtained with G_THETA.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 3/8

2
Implementation of the methods of extrapolation:

The methods of extrapolation of displacements are implemented in POST_K1_K2_K3, afterwards
extraction of the fields of displacements on the two lips of the fissure.

They are tested with or without a grid of the type “Barsoum” (the nodes not nodes on
sides of the quadratic elements concerning the bottom of fissure are moved with the quarter of with dimensions).

Three methods are programmed:

·
Method 1: one calculates the jump of the field of displacements squared and one divides it by R.
Various values of K ² are obtained (except for a multiplicative factor) by extrapolation in
R =0 of the segments of straight lines thus obtained. If the solution were perfect (asymptotic field
analytical everywhere), one should obtain a line. Actually, one obtains almost a line
with a grid of the type “Barsoum”, and a nonright curve if not:



Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 4/8

·
Method 2: one traces the jump of the field of displacements squared according to R. Les
approximations of K are (always except for a multiplicative factor) equal to the root of
slope of the segments connecting the origin to the various points of the curve.



·
Method 3: one identifies the stress intensity factor K starting from the jump of displacement
[U] by a method of least squares. Retiming is done on a segment length
dmax, where dmax is the parameter fixed in operand ABSC_CURV_MAXI of the operator
POST_K1_K2_K3:
dmax
1
2
K minimizes J (K) =
([U (R)]- K R) Dr.
2 0
That is to say thus the formula clarifies to calculate K:
2 dmax
1 nbno 1
-
K =
[U (R)]

rdr =
(R - R) ([U] R - U R
2
2
I 1
+
I
I 1
+
I 1
+
[])
I
I
R
R
m
0
m
i=0

where nbno is the number of nodes on the segment of retiming [0, dmax]. It is noticed that
in this expression K is, for a fixed dmax, the linear shape of the field of
displacement.

These three methods were validated on tests SSLP313 (fissure inclined in an infinite medium) and
SSLV134 (plane fissure, in the shape of disc, in a space infinite 3D). The reference [bib2] recommends
the use of method 1, with grid of the type “Barsoum”.

Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 5/8

3
Validity of the methods suggested

One evaluates the precision of the methods of extrapolation on tests whose analytical solutions are
known. One compares the results with the more precise methods founded on calculation of the rate of
restitution of energy and on the singular functions.

For highlighting well the inaccuracy of the methods of extrapolation, POST_K1_K2_K3 provides
systematically for the first two methods the values maximum and the values minimum
(on the whole of the calculated points) of the factors of intensity of the constraints, as well as the value of G
recomputed by the formula of Irwin. One will thus qualify precise method which provides values
maximum and close relations minimum, and to which the rate of refund of energy is close to that calculated
by method “THETA”.

Method 3 provides as for it only one value for each stress intensity factor
and for G. the precision of the method will thus be estimated only by comparison between the value
G recomputed by the formula of Irwin and the rate of refund of energy calculated by the method
“THETA”. Method 3 can be regarded as a weighted average of the rates of refund
of energy extrapolated in each nodes. It will thus be checked that the error obtained with this method is
lower to the maximum of error of the two preceding methods.

One has here briefly the results obtained on a test 2D and a test 3D.

3.1
Test SSLP313: 2D C_PLAN

It is about a fissure inclined in an infinite medium subjected to a uniform stress field in
a direction (analytical reference solution in plane constraints). The fissure opens in mode
mixed (K1 and K2).

For the Aster test, the fissure is with a grid in a rather large plate. The grid is very fine.
results are as follows:

Reference solution (CALC_G_THETA_T)

K1 aster
K2 aster
Gtheta
3.6037E+06 2.7000E+06 1.0013E+02

POST_K1_K2_K3: grid without nodes of edges to the quarter

K1_max method
K1_min
K2_max
K2_min G_max G_min difference
difference
between Gmax between Gmin
and Gtheta
and Gtheta
1 3.54E+06
3.19E+06
2.63E+06 1.92E+06 9.73E+01 6.94E+01 ­ 3,33% ­ 30,70%
2 3.51E+06
3.33E+06
2.61E+06 2.25E+06 9.57E+01 8.08E+01 ­ 4,50% ­ 19,32%
3 3.50E+06
2.59E+06
9.47E+01 - 5,47%

POST_K1_K2_K3: grid with nodes of edges to the quarter

K1_max method
K1_min
K2_max
K2_min G_max G_min difference
difference
between Gmax and Gmin and
Gtheta
Gtheta
1 3.61E+06
3.60E+06
2.70E+06 2.69E+06 1.01E+02 1.01E+02 1,29% 1,07%
2 3.60E+06
3.53E+06
2.69E+06 2.65E+06 1.01E+02 9.75E+01 1,02% ­ 2,67%
3 3.56E+06 2.66E+06 9.88E+01 - 1,42%

Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 6/8

On this test one notes that the grid of the type “Barsoum” is essential if results are wanted
precis. With “Barsoum” method 1 is more stable. It provides values of G (starting from K1 and
K2) to approximately 1% of solution G_THETA. Methods 2 and 3 lead to errors from 1 to 2,5%.

On the other hand, with a normal grid, the results vary much (between ­ 3% and - 30% of the solution).
It is the same with linear elements. In the case of a grid without elements of
“Barsoum”, method 3 is most precise.

3.2
Test SSLV134: 3D

It is about a plane fissure in the shape of disc in an infinite medium 3D subjected to a field of
constraints uniform in a direction (known analytical reference solution under the name of
“penny shape ace”). The fissure opens in mode 1 (K1 alone) [V3.04.134].

For this test, the fissure is with a grid in a block parallelepiped. The grid is relatively coarse.
One compares the results with those of method “THETA”.

Reference solution:

K1 G
room
1,6.106 11,59

The results are as follows:

POST_K1_K2_K3: grid without nodes of edges to the quarter

K1_max method
K1_min
G_max
G_min
difference
difference
between Gmax and Gmin and
Gtheta
Gtheta
1 1.56E+06
1.45E+06
1.11E+01 9.63E+00 ­ 4,32%
­ 16,91%
2 1.53E+06
1.49E+06
1.06E+01 1.01E+01 ­ 8,35%
­ 13,08%
3 1.52E+06 1.05E+01 ­ 9,51%

POST_K1_K2_K3: grid with nodes of edges to the quarter

K1_max method
K1_min
G_max
G_min
difference
difference
between Gmax and Gmin and
Gtheta
Gtheta
1
1.61E+06 1.59E+06 1.18E+01 1.16E+01 1,32%
­ 0,06%
2
1.59E+06 1.53E+06 1.15E+01 1.07E+01 ­ 0,42%
­ 7,87%
3 1.55E+06 1.10E+01 ­ 5,16%

On this test one still notes that the grid of the type “Barsoum” is essential if one wants
precise results. With “Barsoum” method 1 is most stable. It provides values of G (to
to start from K1 and K2) to 1% solution G_THETA.

3.3 Conclusion

It should well be noticed that the asymptotic expression of displacements is valid only for R
tending towards 0. This is why the grids of the type “Barsoum” are definitely preferable (and
almost obligatory). It is also necessary to take care not to choose a field of extrapolation too much
large (distance dmax of operator POST_K1_K2_K3 of about 4 to 5 elements).

With Barsoum, method 1 gives the most precise results and most stable, that it is in
2D or in 3D. If the grid does not comprise elements of Barsoum, one then advises to use
results of method 3.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 7/8

4 Bibliography

[1]
H.D. BUI “Mécanique of brittle fracture” Masson (1978)
[2]
J. Lemaître, J.L.Chaboche “Mécanique of the solid materials” Dunod 1996

Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of the factors of intensity of the constraints


Date:
26/05/05
Author (S):
E. Key CRYSTAL, J.M. PROIX
:
R7.02.08-B Page
: 8/8

Intentionally white left page.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

Outline document