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.
IUTAM Symposium, Kazimierz Dolny, 23-27 May 2005 3
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
IUTAM Symposium, Kazimierz Dolny, 23-27 May 2005 4
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 .
IUTAM Symposium, Kazimierz Dolny, 23-27 May 2005 5
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
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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
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PJPJ andand DruckerDrucker--PragerPrager limit limit surfacesurface
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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
IUTAM Symposium, Kazimierz Dolny, 23-27 May 2005 9
Finite element modelFinite element model
88m m114m m
28m
m
11m
m
128m
mz
y
P
4275 4275 nodesnodes
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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
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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)
IUTAM Symposium, Kazimierz Dolny, 23-27 May 2005 12
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
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CrackCrack propagationpropagation –– resultsresults hhff =2=2
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
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CrackCrack propagationpropagation –– resultsresults hhff =3=3
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
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CrackCrack propagationpropagation –– resultsresults hhff =4=4
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
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CrackCrack propagationpropagation –– resultsresults hhff =5=5
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
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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.
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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