sslv154 - circular crack in mixed mode...sslv154 – circular crack in mixed mode summary: the...

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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 :

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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)



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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.


Figure 1.1-1: geometry of the fissured cube



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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


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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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 .


Figure 1.3-4: Blocking of the rigid modes



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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.


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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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 .


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 .


max K I ‘ANALYTICAL’ 7.978 105 3%




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.


Figure 3.4-1: G along the bottom of crack



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Figure 3.4-2: K1 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 3.4-5: SIFs along the bottom of crack (FEM)



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4 Modeling B


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.


Many nodes: 1146Number of meshs and type: 64573 TETRA4The length characteristic of an element close to the bottom to crack is of 0,07m .


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.



Figure 4.1-1: grid of the structure refined



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The values are in Pa . M .






4.3.2 Values resulting from POST_K1_K2_K3

The values are in Pa . M .






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.


Figure 4.4-1: SIFs along the bottom of crack (FEM)



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5 Modeling C


Identical to modeling A except for the elements 3D_INCO_UPG and 3D_INCO_UP


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.


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.


The values are in Pa .M obtained starting from the case test SSLV154A.




K II in =0 ° ‘ANALYTICAL’ 8.644 105 2%

K III in =90 ° ‘ANALYTICAL’ 6.747 10 5 3%




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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.


sslv154 - circular crack in mixed mode...sslv154 – circular crack in mixed mode summary: the...

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