INFLUENCE OF STRENGTH HETEROGENEITY FACTOR ON CRACK SHAPE IN LAMINAR ROCK-LIKE MATERIALS

Influence of Strength Heterogeneity Factor on Influence of Strength Heterogeneity Factor on Crack Shape in Laminar RockCrack Shape in Laminar Rock--Like MaterialsLike Materials

Jerzy Podgórski*

Józef Jonak**

* * FacultyFaculty ofof CivilCivil EngineeringEngineering**** FacultyFaculty ofof MechanicalMechanical EngEng..

Lublin Lublin UniversityUniversity ofof TechnologyTechnologyPolandPoland

IUTAM Symposium, Kazimierz Dolny, 23-27 May 2005 2

IntroductionIntroduction The cracking of laminar elastic-brittle materials is the subject of testing in a number

of branches of today’s engineering. It is connected with the fact that these materials are commonly used for various kinds of technical ceramics, composites or in manufacturing technologies. A separate group of issues concerns structural mechanics including underground engineering structures.

The influence of strength heterogeneity factor hf = f1 / f2 (strength of the basic material to strength of the weaker layer) in the crack propagation problem for laminar materials has not been fully understood so far. And for example it has been found in experiments carried out only recently that for the same rock categories (e.g. limestone) the change of hf ratio defining the asymmetry degree of their strength has a decisive importance for the generation of the load exerted on heading machine cutters

The Finite Elements Method has been applied to this analysis (for the analysis of stresses) as well as the “lost elements” method being applied to the crack propagation analysis

PJ failure criterion proposed by Podgórski (1985) the practical application of which has been described in paper (Podgórski 2002) has also been used to analyze the issue discussed in this paper and therefore it does not need to be discussed in further detail.


PJPJ failurefailure criterioncriterion

- material constants

02020100 CJPCC

JJP arccoscos 31

2/32

3

233

JJJ

131

0 I 232

0 J

210 ,,,, CCC

- stress tensor invariants


PJPJ failurefailure criterioncriterion Classical failure criteria like Huber-Mises, Tresca, Drucker-Prager, Coulomb-

Mohr as well as some new proposed by Lade, Matsuoka, Ottosen are some particular cases of the general form PJ criterion.

Material constants can be evaluated on the basis of some simple material test results like:

fc - failure stress in uniaxial compression,

ft - failure stress in uniaxial tension,

fcc - failure stress in biaxial compression at 1/2 = 1,

f0c - failure stress in biaxial compression at 1/2 = 2,

fv - failure stress in triaxial tension at 1/2/3 = 1/1/1,

For concrete or rock-like materials some simplifications can be taken on the basis of test results in biaxial stress state and R. M. Haythornthwaite“tension cutoff” hypotesis:

fcc =1.1 fc , f0c =1.25 fc , fv = ft .


Limit Limit curvescurves inin 11 –– 22 planeplane

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Compression

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Com

pres

sion PJ criterion

Huber-Mises criterion

Drucker-Prager criterion

fcc

foc

fcft

2 / f

c

1 / fc


PJPJ andand DruckerDrucker--PragerPrager criteriacriteria inin tt00 –– ss0 0 planeplane

-0.2 0.0 0.2 0.4 0.6 0.8 1.0Mean stress (compression) -

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Oct

ahed

ral s

hear

str

ess f t

f c

PJ criterion

Drucker-Prager criterion

Tension meridian

Compression meridian

o / fc

o / f

c


PJPJ andand DruckerDrucker--PragerPrager limit limit surfacesurface


Geometry Geometry ofof analysedanalysed problemproblem

T x y R x y z

T x y R x y z

T x y R x y z

T x y R x y z

T x y R x y z

T x y R x y z

T x y R x y z

T x y R x y z

P

Ph=11

mm

b=4m

ma=

12m

m

88m m114m m

28m

m

128m

m


Finite element modelFinite element model

88m m114m m

28m

m

11m

m

128m

mz

y

P

4275 4275 nodesnodes


MaterialMaterial constantsconstants

For basic material of the „strongest layer” strength in uniaxial compression: fc1=20.0 MPa strength in biaxial compression: fcc =22.0 MPa, strength in biaxial compression: f0c =25.0 MPa strength in uniaxial tension: ft1=2.0 MPa Young modulus: E=20.0 GPa Poisson ratio: v=0.2

For the material of the „weaker layer” Case #2 : hf=2 strength in uniaxial compression: fc2=10.0 MPa strength in biaxial compression: fcc=11.0 MPa, strength in biaxial compression: f0c=12.5 MPa strength in uniaxial tension: ft2=1.0 MPa Young modulus: E=10.0 GPa Poisson ratio: v=0.2

fc1 / fc2 =2 ft1 / ft2 =2

Case #4 : hf=4 : fc2=5.0 MPa : fcc=5.5 MPa, : f0c=6.25 MPa : ft2=0.5 MPa : E=10.0 GPa : v=0.2

fc1 / fc2 =4 ft1 / ft2 =4

Case #1 : hf=1 : fc2=20.0 MPa : fcc=22.0 MPa, : f0c=25.0 MPa : ft2=2.0 MPa : E=10.0 GPa : v=0.2

fc1 / fc2 =1 ft1 / ft2 =1


CalculationCalculation procedureprocedure

1. Stress calculation for initial value of the force P,2. Search for the element with maximal value of reduced stress correspond

to considered failure criterion,3. Evaluation of the force P=Pcr for which the element with maximum stress

is in the critical state,4. Removing the chosen element from the analysed FEM mesh or changing

its stiffness,5. Start next step of crack propagation process.

repeat• Calculate stresses caused by unit load p. • if calculation failed then Model_destroyed else

• Select of a model element in which the effort value according to the assumed JP criterion achieves the highest value.• Determine of the value of critical force Pcr at which the effort in the selected element achieves the critical value• Remove of the selected element from FEA mesh

• increase Step_Nountil Model_destroyed or (Step_No > Step_Max)


CrackCrack propagationpropagation –– resultsresults hhff =1=1

Displacements

0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Uy [mm]

10.000

15.000

20.000

25.000

30.000

35.000

40.000

P cr [

N]

1

4

710

13 16

19 22

25

28

31

3437

40

43

4649

52 55 58 61 64 67



Displacements

0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Uy [mm]

12.000

16.000

20.000

24.000

28.000

32.000

P cr [

N]

1

4

7

10

13

1619 22

25

28

31

34

37

4043

46

49

52

55

58

61

64

67

70

73

76

79

82

85



Displacements

0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Uy [mm]

5.000

10.000

15.000

20.000

25.000

30.000

P cr [

N]

1

47

10

13

16 19 2225

28

31

34

3740

43

46

49

52

5558

61

64

6770

73

7679

82

85

88



Displacements

0.0012 0.0014 0.0016 0.0018 0.0020 0.0022 0.0024

Uy [mm]

5.000

10.000

15.000

20.000

25.000

30.000

P cr [

N]

2

5

8

11

14

17

20 23

26

29

3235

38

414447

5053

56

59

6265

68

71

7477

80



Displacements

0.0012 0.0014 0.0016 0.0018 0.0020 0.0022 0.0024

Uy [mm]

8.000

12.000

16.000

20.000

24.000

28.000

P cr [

N]

2

5

8

11

141720

23

2629

32

35

3841

44

47

50

53

56

59

62

65

68

71

74

77

80

83

86


ConclusionsConclusions

The Finite Elements Method analysis showed explicitly that in the case of laminar brittle materials the mechanics of the crack development process depends strictly on the value of strength heterogeneity factor hf = f1 / f2 .

Greater strength heterogeneity causes the material in its “weaker” layer to be more easily destroyed. As a result of this the whole laminar material is delaminated faster.

Three different cases of crack shape was found for value of hf factor:- hf < 4- hf = 4 (delamination)- hf > 4 (shearing)

In order to understand the whole destruction cycle of the material including the “exit” of the crack onto the surface, further simulations are required in which the finite elements mesh area will be enlarged.


ReferencesReferences

Podgórski J.(1985), General Failure Criterion for Isotropic Media. Journal of Engineering Mechanics ASCE, 111 2, 188-201.

Podgórski J. (2002), Influence Exerted by Strength Criterion on Direction of Crack Propagation in the Elastic-Brittle Material. Journal of Mining Science 38 (4); 374-380, July- August, Kluwer Academic/Plenum Publishers.

Podgórski J., Jonak J., Jaremek P. The Strength Asymmetry Effect in Laminar Rock-Like Materials on Crack Propagation, MPES Proceedings, Wrocław 1.09-3.09. 2004, Balkema


Acknowledgment

This paper has been prepared within the framework of a Polish Scientific Research Committee grants No 5 T12A 015 23 and 8 T12A 064 21.

INFLUENCE OF STRENGTH HETEROGENEITY FACTOR ON CRACK SHAPE IN LAMINAR ROCK-LIKE MATERIALS

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