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Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 1/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
6446cc51e042
SSLV154 – Circular crack in mixed mode
Summary:
The purpose of this test is to validate the calculation of the stress intensity factors (SIFs) along the bottom of acrack 3D in mixed mode, within the framework of linear elasticity.
This test brings into play a cube with a central circular crack planes, subjected to a force inclined compared tothe plan of the crack.
This test contains 2 modelings:• Modeling a: the crack is with a grid (FEM);• Modeling b: the crack is not with a grid, it is represented by level-sets (X-FEM)• Modeling C: the crack is with a grid (FEM) for the incompressible elements (_INCO_UPG, _INCO_UP);
For each modeling, SIFs are evaluated by the orders POST_K1_K2_K3 and CALC_G.The digital values are compared with the analytical values.
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Code_Aster Versiondefault
Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 2/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
6446cc51e042
1 Problem of reference
1.1 Geometry
The crack is to circulairE (penny shaped ace) of ray a=2m in the plan OXY . The with dimensionsone of the cube is length L=10a . Thus, it is considered that the crack is in an infinite medium.
1.2 Material properties
The material is elastic isotropic whose properties are:E=200000MPa
=0,3
1.3 Boundary conditions and loadings
Taking into account symmetries of the structure, crack and loadings, only half of the structure aremodelled: half space such as Y0 . Conditions of symmetry are thus applied to the face in Y=0 : onthis face, following displacement Y is blocked.
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Figure 1.1-1: geometry of the fissured cube
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Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 3/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
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The structure is subjected to a constraint xx=1MPa . The local reference mark x , y , z is
obtained by rotation of an angle =/4 around the axis OY .
Thus, we have:
on the higher face:• ZZ= sin 2
• ZX= cos sin
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Figure 1.3-2: Constraints in the total reference mark
Figure 1.3-3: Localreference mark
Figure 1.3-1: Condition of symmetry
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Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 4/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
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on the lower face:
• ZZ=− sin2
• ZX=− sin cos
on the side face of right-hand side:• XX= cos2
• XZ= cos sin
on side face of left:• XX=− cos2
• XZ= cos sin
In order to block the rigid modes of body, the point D1 L/2,0 ,0 is blocked according to X and Zand the point D2 −L /2,0 ,0 is blocked according to Z .
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Figure 1.3-4: Blocking of the rigid modes
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Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 5/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
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2 Reference solution
2.1 Method of calculating used for the reference solution
The reference solution is an analytical solution resulting from (1) , for a circular crack of ray a in aninfinite medium, subjected to a force of uniform surface inclined of an angle with the plan of thecrack, stress intensity factors for a point A placed on the face of crack are worth:
K I=2
sin2 a
K II=4
2−sin coscosa
K III=4 1−
2−sin cossin a
being the angle characterizing the position of the point A on the circular bottom (see Figure 1.3-1).
2.2 Results of reference
For the loading considered and a=2m , it Table 2.2-1 give the analytical values of SIFs along half ofthe bottom of crack, for understood enters 0 ° and 180 ° . These values are also presented onFigure 2.2-1.
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Figure 2.2-1: Values of reference of SIFs
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Angle (°) K I Pa .M K II Pa .M K III Pa.M
0 7,978E+05 9,387E+05 0,0E+05
5 7,978E+05 9,351E+05 5,727E+04
10 7,978E+05 9,244E+05 1,141E+05
15 7,978E+05 9,067E+05 1,701E+05
20 7,978E+05 8,821E+05 2,247E+05
25 7,978E+05 8,507E+05 2,777E+05
30 7,978E+05 8,129E+05 3,285E+05
35 7,978E+05 7,689E+05 3,769E+05
40 7,978E+05 7,191E+05 4,224E+05
45 7,978E+05 6,638E+05 4,646E+05
50 7,978E+05 6,034E+05 5,034E+05
55 7,978E+05 5,384E+05 5,382E+05
60 7,978E+05 4,693E+05 5,690E+05
65 7,978E+05 3,967E+05 5,955E+05
70 7,978E+05 3,211E+05 6,175E+05
75 7,978E+05 2,430E+05 6,347E+05
80 7,978E+05 1,630E+05 6,471E+05
85 7,978E+05 8,181E+04 6,546E+05
90 7,978E+05 5,750E-11 6,571E+05
100 7,978E+05 -1,630E+05 6,471E+05
110 7,978E+05 -3,211E+05 6,175E+05
120 7,978E+05 -4,693E+05 5,690E+05
130 7,978E+05 -6,034E+05 5,034E+05
140 7,978E+05 -7,191E+05 4,224E+05
150 7,978E+05 -8,129E+05 3,285E+05
160 7,978E+05 -8,821E+05 2,247E+05
170 7,978E+05 -9,244E+05 1,141E+05
180 7,978E+05 -9,387E+05 8,050E-11
Table 2.2-1: Values of reference
2.3 Bibliographical references
(1) TADA H., PARIS P., IRWIND G.: The stress analysis of aces handbook, 3rd éd., 2000
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Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 7/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
6446cc51e042
3 Modeling A
3.1 Characteristics of modeling
In this modeling, the crack is with a grid (case FEM). The grid comprises a torus surrounding thebottom of crack.
3.2 Characteristics of the grid
Many nodes: 9967Number of meshs and type: 864 PENTA15 and 1568 HEXA20The length characteristic of an element close to the bottom to crack is of 0,12m .
The nodes mediums of the edges of the elements touching the bottom of crack are moved with thequarter of these edges.
3.3 Sizes tested and results
Crowns of integration of the field theta for the order CALC_G are:RINF=0,1m and RSUP=0,5m .
A smoothing of the type is chosen LEGENDRE.The parameter ABS_CURV_MAXI of the operator POST_K1_K2_K3 is selected so as to retain 5 nodeson the segment of extrapolation.
To test the value of K I for all the points of the bottom of crack, one tests it min and it max valuesalong the bottom.K II is tested only at the point such as =0° (where K II is normally maximum).
K III is tested only at the point such as =90° (where K III is normally maximum).
Theoretically, it would be necessary to test the absolute value of K II and K III because the sign isarbitrary.
3.3.1 Values resulting from CALC_G
The values are in Pa .M .
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Figure 3.1-2: grid of the structure
Figure 3.1-1: radiant grid
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Identification Type of reference Value of reference % Tolerance
max K I ‘ANALYTICAL’ 7.978 105 5.5%
min K I ‘ANALYTICAL’ 7.978 105 4%
K II in =0° ‘ANALYTICAL’ 9.386 105 9%
K III in =90° ‘ANALYTICAL’ 6.570 10 5 3%
3.3.2 Values resulting from POST_K1_K2_K3
The values are in Pa .M .
Identification Type of reference Value of reference % Tolerance
max K I ‘ANALYTICAL’ 7.978 105 3%
min K I ‘ANALYTICAL’ 7.978 105 2%
K II in =0° ‘ANALYTICAL’ 9.386 105 2%
K III in =90° ‘ANALYTICAL’ 6.570 10 5 2%
3.4 Remarks
Figure 3.4-1, Figure 3.4-2, Figure 3.4-3 and Figure 3.4-4 show the evolution of G , K I K II and K III
along the bottom of crack. It is noted that K3 is not correct. That is due to inaccuracies of the localbase at the ends, which are propagated on all the face because of polynomials of LEGENDRE.
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Figure 3.4-1: G along the bottom of crack
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6446cc51e042
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Figure 3.4-2: K1 along the bottom of crack
Figure 3.4-3: K2 along the bottom of crack
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For information, Figure 3.4-5 present a comparison of SIFs along the bottom of crack with asmoothing of LAGRANGE. It is noted that there is a more important error on the points at the ends of
the bottom for K II and K III resulting from CALC_G. That is due to a particular treatment of the pointsat the ends, which does not function although if the solution is constant along the bottom. Thatexplains why the values of K I are better. This particular treatment is carried out only for the results (
G , K I K II and K III ) resulting from the option CALC_K_G, smoothing LAGRANGE.
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Figure 3.4-4: K3 along the bottom of crack
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Figure 3.4-5: SIFs along the bottom of crack (FEM)
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4 Modeling B
4.1 Characteristics of modeling
In this modeling, the crack is not with a grid (case X-FEM).In order to obtain better a precision on the results, the free initial grid was refined on the level of thebottom of crack using the order MACR_ADAP_MAIL.
4.2 Characteristics of the grid
Many nodes: 1146Number of meshs and type: 64573 TETRA4The length characteristic of an element close to the bottom to crack is of 0,07m .
4.3 Sizes tested and results
The choice of the digital parameters for the postprocessing of SIFs is identical to that done formodeling A.
Moreover, in order to smooth the results of CALC_G (inevitable on a free grid), it is necessary post-totreat SIFs only in one limited number of points, distributed uniformly along the bottom of crack. 21points of postprocessing here are chosen (initially, there are 289 points along the bottom of crack).That also makes it possible to reduce time CPU of postprocessing.
One chooses same manner 21 points of postprocessing for POST_K1_K2_K3.
4.3.1 Values resulting from CALC_G
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Figure 4.1-1: grid of the structure refined
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The values are in Pa . M .
Identification Type of reference Value of reference % Tolerance
max K I ‘ANALYTICAL’ 7.978 105 6%
min K I ‘ANALYTICAL’ 7.978 105 1%
K II in =0° ‘ANALYTICAL’ 9.386 105 14%
K III in =90° ‘ANALYTICAL’ 6.570 10 5 10%
4.3.2 Values resulting from POST_K1_K2_K3
The values are in Pa . M .
Identification Type of reference Value of reference % Tolerance
max K I ‘ANALYTICAL’ 7.978 105 2%
min K I ‘ANALYTICAL’ 7.978 105 11%
K II in =0° ‘ANALYTICAL’ 9.386 105 7%
K III in =90° ‘ANALYTICAL’ 6.570 10 5 2%
4.4 Remarks
For information, Figure 4.4-1 present a comparison of SIFs along the bottom of crack. One notes thesame type of error as that underlined in the paragraph §3.4.
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Figure 4.4-1: SIFs along the bottom of crack (FEM)
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Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
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Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 16/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
6446cc51e042
5 Modeling C
5.1 Characteristics of modeling
Identical to modeling A except for the elements 3D_INCO_UPG and 3D_INCO_UP
5.2 Characteristics of the grid
Many nodes: 9967Number of meshs and type: 864 PENTA15 and 1568 HEXA20The length characteristic of an element close to the bottom to crack is of 0,06m .
The nodes mediums of the edges of the elements touching the bottom of crack are moved with thequarter of these edges.
5.3 Sizes tested and results
Crowns of integration of the field theta for the order CALC_G are:RINF=0,1m and RSUP=0,5m .
A smoothing of the type is chosen LAGRANGE.The parameter ABS_CURV_MAXI of the operator POST_K1_K2_K3 is selected so as to retain 5 nodeson the segment of extrapolation.
To test the value of K I for all the points of the bottom of crack, one tests it min and it max valuesalong the bottom.K II is tested only at the point such as =0° (where K II is normally maximum).
K III is tested only at the point such as =90° (where K III is normally maximum).
Theoretically, it would be necessary to test the absolute value of K II and K III because the sign isarbitrary.
5.3.1 Values resulting from CALC_G
The values are in Pa .M obtained starting from the case test SSLV154A.
Identification Type of reference Value of reference % Tolerance
max K I ‘ANALYTICAL’ 8.361 105 3%
min K I ‘ANALYTICAL’ 8.361 105 1%
K II in =0 ° ‘ANALYTICAL’ 8.644 105 2%
K III in =90 ° ‘ANALYTICAL’ 6.747 10 5 3%
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Code_Aster Versiondefault
Titre : SSLV154 - Fissure circulaire en mode mixte Date : 16/07/2015 Page : 17/17Responsable : GÉNIAUT Samuel Clé : V3.04.154 Révision :
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6 Summary of the results
This CAS-test validates the calculation of the stress intensity factors of a crack 3D in mixed mode.One notes for the order CALC_G (option CALC_K_G) important errors at the points ends if it K is notconstant along the bottom of crack.
This problem does not appear for the order POST_K1_K2_K3.
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
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