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


Date:
02/05/05
Author (S):
E. Key CRYSTAL
:
R7.02.05-B Page
: 1/14

Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.02 booklet: Breaking process
Document: R7.02.05

Calculation of the coefficients of intensity of constraints
in plane linear thermoelasticity

Summary:

One presents the method of calculation of the coefficients of intensity of constraints KI and KII and in thermoelasticity
linear plane. The formulation regards the rate of refund of energy as a symmetrical bilinear form
field of displacement U and uses the explicit expressions of the fields of singular displacements
known in plane linear elasticity.

Key words:

Breaking process, coefficient of intensity of constraints, thermoelasticity.

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

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


Date:
02/05/05
Author (S):
E. Key CRYSTAL
:
R7.02.05-B Page
: 2/14

Count

matters

1 Expressions of the stress intensity factors KI and KII in linear thermoelasticity 2D .............. 3
1.1 Presentation ..................................................................................................................................... 3
1.2 Formulate IRWIN and rate of refund of energy G ........................................................................ 5
1.3 Decoupling of the modes of rupture I and II ........................................................................................ 6
2 Establishment of KI, KII in linear thermoelasticity 2D in Aster ........................................................ 8
2.1 Types of elements and loadings .............................................................................................. 8
2.2 Environment necessary for the calculation of K, K ......................................................................... 8
I
II
2.3 Bilinear form symmetrical G (. .).................................................................................................. 8
2.3.1 Elementary traditional term ................................................................................................. 8
2.3.2 Term forces voluminal ......................................................................................................... 11
2.3.3 Thermal term .................................................................................................................. 12
2.3.4 Term forces surface ........................................................................................................ 12
2.4 Fields of singular displacements and their derivative ................................................................. 12
2.5 Postprocessing of the results of K and K ....................................................................................... 12
I
II
3 Bibliography ........................................................................................................................................ 14

Handbook of Référence
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Code_Aster ®
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7.4
Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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Author (S):
E. Key CRYSTAL
:
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1
Expressions of the stress intensity factors KI and KII
in linear thermoelasticity 2D

1.1 Presentation
x2

R



0

x1

Are the axes of Cartesian co-ordinates Ox1 in the prolongation of the fissure and Ox2
perpendicular with the fissure. The problem is plane. We will express the Cartesian components
displacements and constraints according to the polar co-ordinates R and.

In linear elasticity, the system of the equilibrium equations, without voluminal force, and the conditions
in extreme cases homogeneous on the fissure, the null constraints ad infinitum, admit a noncommonplace solution
1
-
form U = R G ()
2
I
I
. The constraints are infinite at the bottom of the fissure like R
[bib3].

For an unspecified problem in plane linear elasticity (plane strains or plane stresses),
the field of displacement U can break up into a singular part and a regular part.
singular part, also called singularity, is that clarified above, it contains them
coefficients of constraints. In linear elasticity, the modes of rupture I and II are separate:

U = U + K uI + K uII
R
I
S
II
S

with:


1+ R 1/2


uI

=
cos
K
S1
- cos

E 2
(2) (
)



1+ R 1/2


uI
=
sin
K
S2
- cos

E 2
(2) (
)


1+ R 1/2


uII

=
sin
K
S1
+ cos + 2

E 2
(2) (
)



1+ R 1/2


uII
= -
cos
K
S2
+ cos - 2

E 2
(2) (
)
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Code_Aster ®
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Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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Author (S):
E. Key CRYSTAL
:
R7.02.05-B Page
: 4/14

with:

K = 3 - 4
in plane deformations D_PLAN
K = (3 -)/(1+)
in plane constraints C_PLAN

and:

E modulus YOUNG
Poisson's ratio

The distribution of the singular constraints in the vicinity of the fissure is given by the formulas:

S

= K I + K II
11
I
11
II
11

S

= K I + K II
12
I
12
II
12
S

= K I + K II
22
I
22
II
22

with:


1




3
I

11 = (
cos
1 sin
sin


2 R) 1/2

2 -



2 2


1



3
I

12 = (
cos
sin
cos



2 R) 1/2

2

2

2

1



3
I

22 =
cos
sin
sin


(
1

2 R) 1/2

2 +



2

2


1



3
II

11 = - (
sin
2 cos
cos

2 R) 1/2

2 +



2

2



1



3
II

12 = (
cos
1 sin
sin



2 R) 1/2

2 -



2

2

1



3
II

22 =
sin
cos
cos

(


2 R) 1/2

2

2

2
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Code_Aster ®
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7.4
Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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Author (S):
E. Key CRYSTAL
:
R7.02.05-B Page
: 5/14

1.2
Formulate IRWIN and rate of refund of energy G

In plane linear elasticity, the stress intensity factors are connected to the rate of refund
of energy G by the formula of IRWIN:

2
1
G =
(2 2
K I + KII)
déformatio

in
(DP)

plane

NS
E

1
G =
(2 2
K I + KII)
constraint

in
(CP)

plane

S
E

The demonstration of these formulas can be made starting from the expression of the rate of refund
of energy G established in Code_Aster and known under the name of the method theta [bib5].
Let us recall that G is defined by the opposite of derived from the potential energy compared to the evolution
bottom of fissure.

F

F
F
U


In the Lagrangienne method of derivation of the potential energy, one considers transformations
MR. M
+ (M) of the area of reference 0 in a field which corresponds to
propagations of the fissure. With these families of configuration of reference thus defined
correspond of the families of deformed configurations whose fissure was propagated. Energy
potential definite on is brought back on 0.
One considers the surface F and voluminal forces F respectively applied to F and 0. One
note ((U)) density of free energy, U the field of displacement, T the field of temperature and
the field of vectors describing the direction of transport in = 0, then the general expression of
rate of refund of energy G [bib5] is:
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Code_Aster ®
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Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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Author (S):
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:
R7.02.05-B Page
: 6/14

G =
[(U): (U) - ((U) div] D


Traditional term



-
(T) D


Term
thermics



with

had

T
+
([F) U + F U div] D
Term
voluminal

forces
with

had

F on








+
(F)


U + F
U div - N
D Terme
surface

forces
with

had


S F on






F
F


N
-
(N) (U) D
Term
déplacemen
with

had

on

imposed

ts



U
U

In linear elasticity, G can be regarded as the symmetrical bilinear shape of the field of
displacement U. Density of energy elastic ((U)) is written:

(
1
1
(U)) =
(U): : (U) =
B (U, U)
2
2


while noting:

the tensor of elasticity
B the symmetrical bilinear form defined by: B (U, v) = (U): : (v)

and the bilinear form G (,) associated G is defined by:

1
B
B

G (U, v) =
(v) +
(U
) - B

(U, v) div D
2 U

v




1
+ ([F) v + (F) U + (F v + F U) div] D
U
v
U
v

2

by limiting under the terms traditional and due to the voluminal forces F.

There are G = G (U, U) if U is solution of the elastic problem.

1.3
Decoupling of the modes of rupture I and II

In the method established in Code_Aster, to uncouple the modes from rupture I and II and
to calculate coefficients KI and KII, one uses the bilinear form symmetrical G (,) and
decomposition of the field of displacement U in parts regular and singular [bib7].

G
(
I
I
II
I
I
I
I
II
I
U, custom) = G (U + K U + K
R
I
S
II custom, custom) = G (U R, custom) + K G
I
(custom, custom) + K G
II
(custom, custom)


G

(
II
I
II
II
II
I
II
II
II
U, custom) = G (U + K U + K
R
I
S
II custom, custom) = G (U R, custom) + K G
I
(custom, custom) + K G
II
(custom, custom)
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Code_Aster ®
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Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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Author (S):
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:
R7.02.05-B Page
: 7/14

One shows in plane linear thermoelasticity that uI
II
S and customs are orthogonal for the product
scalar defined by the bilinear form G (,), that terms utilizing the regular part
cancel themselves and finally:

G (
I
I
I
U, custom) = K G
I
(custom, custom)
G (
II
II
II
U, custom) = K G
II
(custom, custom)

Moreover, by writing the rate of refund of energy in the form:

G
G (U, U)
G (U + K I
U + K
II
U, U + K
I
U + K
II
=
=
R
I
S
II
S
R
I
S
II custom)

and like:

G (I
II
II
I
custom, custom) = G (custom, custom) = 0
G (R I
R
II
U, custom) = G (U, custom) = 0

the formula of IRWIN is found:

G (,) = K2 G (I
I
2
II
II
U U
U, custom) + K G
I
S
II
(custom, custom)

with:

1
2
-
G (I I
custom, custom) = G (II II
custom, custom) =

in D_PLAN
E

1
G (I I
custom, custom) = G (II II
custom, custom) =

in C_PLAN
E

Finally:


=
E
K I
G (
I
U U
, S)


1 - 2



=
E
K II
G (
II
U U
, S)
in D_PLAN


1 - 2



K
=
I
E G (
I
U U
, S)


K
=
II
E G (
II
U U
, S)
in C_PLAN

Establishment of the calculation of the coefficients of intensity of constraints in plane linear thermoelasticity
in Code_Aster is realized starting from the expression of the rate of refund of energy G in elasticity
linear 2D, written in symmetrical bilinear form, by introducing the known expressions of
singular displacements, and by using the method theta.
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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Author (S):
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:
R7.02.05-B Page
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2
Establishment of KI, KII in linear thermoelasticity 2D in
Aster

2.1
Types of elements and loadings

To calculate the coefficients of intensity of constraints KI and KII in linear elasticity 2D, it is necessary
to use option CALC_K_G of command CALC_G_THETA. This option is available for all
thermomechanical loading applying to a model of affected two-dimensional continuous medium to
triangles with 3 or 6 nodes, quadrangles with 4, 8 or 9 nodes, and segments with 2 or 3 nodes.
It is valid for a modeling “C_PLAN” or “D_PLAN”.

Note:
One does not take account of the term due to the displacements imposed on U, one thus should not
to impose conditions of DIRICHLET on the lips of the fissure.

2.2
Environment necessary for the calculation of KI, KII

Command CALC_G_THETA makes it possible to recover the model of the problem, the characteristics of
material, the field of displacements and the field theta. For the calculation of the coefficients of intensity of
constraints, it is necessary to add the key word FOND_FISS, which makes it possible to recover a concept of
fond_fiss type where the basic node of fissure and the normal with the fissure are defined.

When that the fissure is laid out along an axis of symmetry, one can also specify
symmetry of the loading by key word SYME_CHAR. By defect one supposes that there is no symmetry.
If one assigns value “SYME” to key word SYME_CHAR, that means that only mode I of rupture acts
(opening of the lips of the fissure) and one automatically assigns the zero value to KII. If one affects
value “ANTI”, then only mode II is active (slip of a lip compared to the other) and KI is
no one.

Let us insist on the need for assigning to all the elements (including those of edges) the values of
YOUNG E and Poisson's ratio moduli, because they are used in the calculation of
singular displacements. These values must be homogeneous on all the support of the field theta.

2.3
Bilinear form symmetrical G (. .)

Note:

Routine GBILIN calculates the rate of refund of energy G in the bilinear form
symmetrical G (U, v) in thermo linear elasticity planes (plane strains or stresses)
for the isoparametric elements 2D.

2.3.1 Elementary traditional term
TCLA = (U): (U) - ((U)) div

Density of energy elastic ((U)) is written in thermo linear elasticity:
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Code_Aster ®
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:
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in D_PLAN:

1 - E
E
((U))
(
)
=
(2 +2
xx
yy)
(
+

2 1+) (1 -
2)
(1+) (1 -
2) xx yy
E
+
2 -
1+ xy
HT

in C_PLAN:
E
E
E
((U)) =
(
2 + 2 +
+
2 -
- 2) (xx
yy) (- 2) xx yy (+) xy HT
2 1
1
1

with HT = Densité of energy due to thermics:

= 3
HT
K (T - D
T F) tr

where:

3
=
E
K
1 -
2

=
thermics

dilation


= tensor
déformatio

of

NS
T
= températur
reference

of

E
ref.

and in a general way, one can write:

2 ((U)) = C (2
2
+
xx
yy) + 2 C
+ 4
2




C - 2
1
2 xx yy
3

xy
HT

with:


(1 -) E

C =
= + µ
C =
E
1 (1+) (1 -)
2
2
1
(1 2
-)



E

E
C =
=
C =

2
(

1+) (1 - 2)
in
D_PLAN


;


in
.
2
C_PLAN
1
2


-

E

E
C =
= µ
C =
3

(
3
2 1+)
(
2 1+)

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Code_Aster ®
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Therefore, while noting ((U)) = (,
U U), one has 2 (U, v) = S1- S1TH with:

U


v
U


v

U


v
U


v

1
X
X
y
y
X
y
y
X
S
=
1
C


+





+ C2

+








X



X

y

y





X
y

y


X




U

U


v

v

X
y
X
y
+
3
C
+


+





y

X

y

X





1
S TH = 3K (U
T - ref.
T) tr (v) + (v
T - ref.
T) tr (U))

where You is the temperature associated with the field with displacement U by the relation:

= ((U) - HT)
where
HT = (T - D
T F)
ij
ij

and the equilibrium equations check.

In the same way, the term (U): (U) can be written:

(U): (U
) = S2 - S2TH

with:

U U
U


X
X
X
X
y
uy uy
U
X
y
y
S2 = C1
+
+






+


X
X X
y X



y X y
y y
U U



X
y
U
X
y
y
uy U
U
X
X
X
y
+ C2
+
+






+


X
X y
y y



y X X
y X
U
U




X
y
U
U
U
X
X
X
y
y
U
X
y
y
+ C3
+


+
+
+




y
X X y
y y
X X
y

X

U U
U v
Terms
X
X
X
X

become in the symmetrical bilinear form
and terms of
X X
X X
U U
1 U v
v U
type
X
y
X
y
X
y

+


become
.
y X
2 there X
y X
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Calculation of the coefficients of intensity of constraints


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TH
v
v
X
X
y
y
v
v
X
y
y
S TH =
1
2
3K (T - T
U
ref.)


+
+
+
X
2
X X
y y
y X
X y
TH
U
U
X
X
y
y
U
U
X
y
y
+
13K (T - T
v
ref.)


+
+
+
X
2
X X
y y
y X
X y
TH1 =
1
in


D_PLAN
where

1 - 2
TH1 =

in C_PLAN

1 -

and finally the traditional term is written:

1
TCLA = (S2 - S2TH) - (S1- S TH
1
) div
2

2.3.2 Term forces voluminal
TFOR = (F) U + F U div

In any rigor, the symmetrical bilinear expression of TFOR is written in (U, v):

1
TFOR (U, v) =
([fu) v + (fv) u+ (fu v +fv U) div]
2

where fu is the voluminal forces associated the field of displacement U for the elastic problem.

but as the expressions which we are brought to calculate are of type TFOR (U, U) and
TFOR (
S
U, U), where U and customs are respectively the field of displacement and the singular field, and
that:

F
S
div
= 0 on
custom
=
((U)

One limits oneself to write:

S
TFOR (
CS
5
.
0
U, v) = CS (
[F.
U) v + F. v div
U
]


=
v = U
with



CS = 1
v = U
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Finally:

F
F

TFOR (U, v) = CS v
X
X

+
+ F
X
div
X X
y y
X



F
F

+ v
y
y

+
+ F
y
div
X X
y y
y


The same remark is valid for the thermal traditional term, the additional term due to
thermics and terms due to the surface forces.

2.3.3 Term
thermics

One makes the assumption that the characteristics of the material (E
,
,) do not depend on
temperature.


1
T
T
TTHE = -
(T) =
3 K tr
+



T
2
X X there y

2.3.4 Term forces surface

In 2D, for the isoparametric elements of edge, one introduced the loadings of the pressure type
shearing and force distributed of real type.

The term forces surface is written in the same way that the voluminal term from:



TSUR = (F) U + F U div - N



.
N

2.4
Fields of singular displacements and their derivative

Singular fields uI
II
S and customs, respectively associated with modes I and II, are known
explicitly like their derivative. They are written according to the polar co-ordinates in
reference mark related to the fissure. The knowledge of the co-ordinates of the basic node of fissure and of its
normal makes it possible to calculate them in the total reference mark 0xy.

The successive introduction of these fields uI
II
S and custom allow, as indicated in [§1], calculation
elementary of the coefficients of intensity of constraints KI and KII.

2.5
Postprocessing of the results of KI and KII

Knowing the values of the coefficients of intensity of constraints KI and KII for a fissure
data, formulas of AMESTOY - BUI and DANG-VAN, allow the calculation of the angle of
propagation of the fissure according to 3 criteria (maximum KI, KII and G maximum) [bib6].


That is to say m a field identical to except that the fissure is prolonged in the direction of angle m
of a segment of straight line length.
Handbook of Référence
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Code_Aster ®
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Titrate:
Calculation of the coefficients of intensity of constraints


Date:
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U

m
F
F
F
m


O
m


Are K (
)
m
I,
, K (
)
m
II,
, G (,)
m stress intensity factors and the rate of refund

of energy of m subjected to the same loading as.

One poses:

K * ()
m
= lim K
I
I (,
)
m
0
K * ()
m
= lim K
II
II (,
)
m
0
G * ()
m
= lim G (,)
m
0

Criteria quoted by AMESTOY - BUI and DANG-VAN [bib6] are:

·
to choose m
*
O such as K (m
I
O) is maximum,
·
to choose m
*
O such as K (m
II
O) is null,
·
to choose m
*
O such as G (Mo) is maximum.

These criteria give very nearby results [bib8].

The results are given in the form of a table of 4 coefficients K11, K21, K12, allowing K22
to calculate K *
*
I and KII in all the cases of loading:

K *
K
K
11
12 K
I
I

=


K *
K
K
21
22 K
II
II
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Calculation of the coefficients of intensity of constraints


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Angle m (°)
K11
K21
K12
K22
0
1
0
0
1
10
0,9886
0,0864
-- 0,2597
0,9764
20
0,9552
0,1680
-- 0,5068
0,9071
30
0,9018
0,2403
-- 0,7298
0,7972
40
0,8314
0,2995
-- 0,9189
0,6540
50
0,7479
0,3431
-- 1,0665
0,4872
60
0,6559
0,3696
-- 1,1681
0,3077
70
0,5598
0,3788
-- 1,2220
0,1266
80
0,4640
0,3718
-- 1,2293
-- 0,0453
90
0,3722
0,3507
-- 1,1936
-- 0,1988

K (-)
m = K ()
m, K (-)
m = - K ()
m, K (-)
m = - K ()
m, K (-)
m = K ()
m
11
11
21
21
12
12
22
22


The search of the angle Mo in CALC_G_THETA is made of 10 degrees in 10 degrees. The angle of
propagation is not calculated and is printed out (in file MESSAGE) only if INFO is worth 2.

3 Bibliography

[1]
H.D. BUI, J.M. PROIX
: “Law of conservation in thermo linear elasticity”
- C.R.
Acad.Sc.Paris, t.298, Série II, n° 8, 1984.
[2]
H.D. BUI: “Associated path independent J-Integrals for separating mixed modes” - J. Mech.
Phys. Solids, Vol. 31, N° 6, pp. 439-448, 1983.
[3]
H.D. BUI: “Breaking process fragile” - Masson, 1977.
[4]
P;. DESTUYNDER, Mr. DJAOUA: “On an interpretation of the integral of Rice in theory of
brittle fracture, Mathematics Methods in the Applied Sciences " - Vol. 3, pp. 70-87, 1981.
[5]
P. MIALON: “Calculation of derived from a size compared to a bottom of fissure by
method théta " - E.D.F. Bulletin of Direction of Etudes and Recherches, Série C, n° 3,
1988, pp. 1-28.
[6]
Mr. AMESTOY, H.D. BUI, Ky DANG-VAN: “Deviation infinistésimale of a fissure in one
arbitrary direction " - C.R. Acad. Sc Paris, t.289 (September 24, 1979), Série B-99.
[7]
E. SCREWS: “Calculation of the stress intensity factors in plane linear elasticity” - Note
intern EDF-DER-MMN, HI-75505D of the 05/07/94.
[8]
P. MIALON: “Study of the rate of refund of energy in a direction marking an angle
with a fissure ", intern E.D.F. HI/4740-07 - 1984 notes.
Handbook of Référence
R7.02 booklet: Breaking process
HT-66/05/002/A

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