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NEW PERSPECTIVE OF FRACTURE MECHANICS INSPIRED BY NOVEL TEST
WITH CRACK-PARALLEL COMPRESSION
ZDENĚK P. BAŽANT
COLLABORATORS: HOANG NGUYEN, MADURA PATHIRAGE, MASOUD REZAEI, MOHSEN ISSA AND GIANLUCA CUSATIS
NU-TAM WEBINARS ON BREAKTHROUGHS IN MECHANICSNORTHWESTERN UNIVERSITY, JUNE 2, 2020
BASED ON: PNAS—IN PRESS (JUNE). EXTENSION: JAM (JULY 2020) DOWNWLOAD NOW FROM AUTHOR’S WEBSITE 1
Video link: https://youtu.be/UyTeopPModA
Standard Fracture Test Specimens— negligible crack-parallel normal stress
3PB
SENT CNT
CT
DCB DEN-EC
WS
2
Why has the crack-parallel stress been ignored?• All the standard notched fracture specimens
have • Generally considered: line cracks, as in
• A line cut in x-direction in a field of homogeneous uniaxial stress causes no stress change in a continuum model
3
σxx
Why:σxx ≈ 0
σxx
– linear elastic fracture mechanics, LEFM(Griffith 1921)
– cohesive crack model, CCM(Barenblatt 1959)
What matters?
σxxvisible crack
Fracture process, not the visible final crack
LargeSpecimen
Small
Quasibrittle Materials— brittle constituents, but inhomogenity size and the RVE or FPZ are not
Fracture Process Zone (FPZ) Size10 km – Arctic Ocean ice cover as a 2D heterogenous
medium consisting of thick floes of approx. 3 kmsize, embedded in thin ice matrix
5 m – Sea ice pushing on oil platform legs
0.5 m – Normal concrete
2 cm – Textile composites
2 mm – Gas or oil shale
( 10-6m – MEMS, polysilicone, embrittled metals—We omit )
– Quasibrittleness is a relative concept5
Brittle Fracture Mechanics Founding in 1921 and Its Evolution into Ductile, Cohesive and Quasibrittle
A A Griffith 1921
1950 – 1980 1980 –
ε or w
Nonlinear fracture mechanics
or cohesive
or
A A Griffith 1921
FPZ is long, wide, softening and tensorial
Damage mechanics
FPZ = a point also a point large
Brittle (dynamic)
Ductile
6
-0.2 0 0.2 0.4 0.6 0.80
0.5
1
1.5
2
Gf /
Gf,0
M7 + crack band model
Tensile strength
σxxLEFM, CCM, basic XFEM
9 tests
σxx /σc
Data from regression of 3 x 9 = 27 tests
9 tests9 tests
Compression strength
New finding: Effect of crack parallel stress σxxon fracture energy Gf of concrete is strong
3 data points – based on regression of 3 x 9 = 27 experimentsM7 microplane prediction is calibrated only by fc and Gf,0
(line crack models)
7
NORMAL CONCRETE
I. Gap Test:
New, yet simple and unambiguous,
test of fracture at crack-parallel compression
8
Basic Idea of the Gap Test– Achieve superposition in sequence
(in statically determinate way)
R RCC
CC
CC
R R
R R
R+C R+C
+
same=
1) 2)
same KI
And keep constant as KI is applied9
again statically determinate
Novel Experiment: GAP TESTSTAGE 1
STAGE 2
C
F = 2C
CC
F = 2C+2R
C+RFPZ
C
R F = 2C+2R
statically determinate
10
Four advantageous key features:
1. At crack mouth, plastic pads to apply crack-parallel compression C.
2. A gap above end supports gets in contact only after the pads yield, to apply bending moment.
3. This way the test beam passes from one statically determinatesystem to another simple and unambiguous evaluation.
4. The static determinacy and Cconstancy enables the size effect method – a robust and unambiguous way to measure fracture energy Gf
Static system of Stage 2
C
C C
R
GAP
GAPclosed
yieldingpads
yielding
H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press. In brief: PNAS in press
C+R
C
steel
0 0.1 0.20
2
4
6
CTOD (mm)(Crack Tip Opening Displacement)
0 2 40
5
10
15
20
25
Load
(kN
)
Displacement (mm)at Load Point
0.45% riseseating effects
Yield line of pads
Typical Measured Load Evolution
Load
(kN
) Under CTOD control
H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press. In brief: PNAS, in press. 11
Gap
Extensometer Steel
a
Plastic blocks (polypropylene)
c)
F
F
Extensometer
no lateral slip
a
D
2L
S
Steel
Plastic block (polypropylene)
Gap
yx
z
Thickness b
σ pad
(MPa
)
Nominal strain
0.45% rise
0.39% rise
DIC used to verify strain field for calibration
Test Setup
12
Test Setup in More Detail
Plastic pads (polypropylene)
13Gap
Extensometer Steel
a
F, δ
Pads
14
Distinguish:
σpad = normal stress under the plastic pad, as measured
σxx = inferred (by FE or DIC) crack-parallel stress at the notch tip FPZ, which is what matters for damageconstitutive model.
Computed Field of Principal Stress Vectors
At M = Mmax, σxx ≠ 0
At M = 0,σxx ≠ 0
Mmax
Uniform tension field σxx
σpad
STAGE 1
gap
steel strap
glued
F = -2T
T T
pads yield in tension
steelglued
T T
steel
F = 2R-2T
T T
pads yield in tension
steelglued
T T
gapclosed
R R
steel strap
gluedsteel
Gap Testfor Tension
15
again statically determinate
statically determinate
STAGE 2
H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press
Optimum Design of Polypropylene Pads
w
slope HF
F
x
y u
w
incompressible u
3u/2
Possible alternative to plastic pad: Tin (St)H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press
16
II. Size effect method— essential part ofgap test to measure
material fracture energy GfStandard RILEM Recommendation, 1990.
ACI-446 endorsed it and recommended it to ASTM C-0917
For quasibrittle materials, distinguishthe initial and total fracture energy, Gf and GF
18
Classical(Hillerborg)
specimens at Pmax
Gf - most structures at Pmax
Gf initialσ σ σft
GF total (2 to 6 Gf)
δ
σft
f1 δ
GF is important for energy absorption at P 0 in dynamic failures (impact)
We consider only Gf .
GF is to be calculated from the ratio GF /Gfdetermined separately.
For Size Effect Test Method of Gf :Specimens Geometrically Scaled (as 1 : 2 : 4)
Depths:D = 4, 8, 16 in.
19H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press
Required 1st and 2nd Order Small- and Large-Size Asymptotic Properties of Type 2 Size Effect on Fracture
For
1 ...Ns
DD
σ ∝ − −0:D →
:D → ∞
Type 2
01 1 ...2NDDD
σ ∝ − +
For σ w
Asymptotic Matchingyields the Size Effect Law (SEL)(1984, 1990, 2004)
20
ZP Bažant, Z.P. (1984). J. Eng. Mech. 110, 518—535; ZP Bažant, MT Kazemi (1990). IJF, 44, 111--131.In detail: ZP Bažant (2004), PNAS, p. 13400
Asymptotic matching
ln σN
ln D
Two-termasymptotes
ln D0
21
|
RILEM Standard Recommendation 1990ACI-446 Recommendation
= dimensionless energy releasefunction of LEFM
Transformation to linear regression:
21
1A
C
Y
X
Size Effect Method to Measure Fracture Energy Gfand Material Characteristic Length cf
ZP Bažant, MT Kazemi (1990). IJF, 44, 111--131.
10 1 10 2 10 3
0.5
1
1.5
0 100 200 300 400 5000
5
10
1/σ N
2 (M
pa-2
)
σ N (M
Pa)
D (mm)
a) b)
D (mm)
1A
C
12
Y = AD+C
Size effect regression of 9 Gap Teststo get Gf for, e.g., the medium crack-parallel compression
(the means and the scatter width)
H. Nguyen,…. Z.P. Bažant, JAM, in press22
Note the closeness of fit
Classical Work-of-Fracture Method of Measuring Total Fracture Energy GF
23
1. J-integral varies with crack length x2. Ambiguity of tail—ratios GF /Gf , ft /f1 matter but vary3. Tail hard to measure; stable postpeak required
Why was it not used?
P
u
?
P
a0M
x
Energyrelease
Rate
FPZ evolution
at σxx= 0
Nakayama 1965Hillerborg 1976
III.Gap Test Results
and Mesoscale Physical Mechanism
24
Found: Effect of crack parallel stress σxxon fracture energy Gf of concrete is strong
-0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Gf /
Gf,0
M7 + crack band model
Tensile strength
σxxLEFM, CCM, basic XFEM
3 data points – based on regression of 3 x 9 = 27 experimentsσxx / σxx,0
3 data points – based on regression of 3 x 9 = 27 experimentsM7 microplane prediction is calibrated by only fc and Gf,0
Strengthening Gf Weakening Gf
25
9 tests
9 tests
9 tests
σpad measured
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
σxx /σc
σxx
c f/c
f,0
M7 + crack band model
Material characteristic length cfdependence on crack-parallel stress σxx
Note: Approx. cf = 0.4 FPZ length
Data from regression of 3 x 9 = 27 tests
σpad measured
H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press. Prelim.: PNAS 26
9 tests
9 tests
9 tests
0 < σxx < 0.75σc 0.75σc < σxx < σc
1) strengthening Gf 2) weakening GfRegimes of:
Static friction without slip enhanced by
compression
Transverse widening of FPZ caused by inclined
slips and splitting
Meso-Scale Mechanisms of σxx Effect on Gf
σyy
w
Splitting bands
w
σyy
Slip + dilation
widened widenedw
σyy
Friction + Interlock
σxx σxx σxx
27
IV.FE Extrapolations and
Predictions for Fiber Concrete and Shale
28
29
The crack band model M7 with its default parameters was scaled to fit only: 1) uniaxial compression strength of the concrete, and 2) the measured basic fracture energy (at σxx = 0).
The predictions for the 2nd and 3rd data points were acceptable (10% error) and excellent (1% error). Overall 7% error (which is normal for concrete)
So it makes sense to use microplane model M7 to make predictions for other situations and materials
Extrapolations Based on Microplane Model M7 Approximately Validated by Gap Tests
σxx / σccrack–parallel stress
Effect of combined in-plane and out-of-plane stresses σxx and σzz on Gf and cf
Calculated for Gap Test Geometry
H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press 30
Note the non-monotonic effect of σzz
0 0.5 1 1.50
0.5
1
1.5
2
σzz /σc=0.9
σzz /σc=0.4
σzz /σc=0Gf /
Gf0
normalconcrete
0 0.5 1 1.50
0.5
1
1.5
2
σzz /σc=0.9
σzz /σc=0.4σzz /σc=0
c f/c
f0
Fracture Energy Characteristic Length
d)
ξ
8.25E-3
0
4.12E-3
Comparisons with classical tensorial strength or damage models based on invariants
31H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press
0 0.5 10
0.5
1
1.5
2
ξ
Gf /
Gf‘
,0
CDPM2 concrete damage model
(Grassl, Abaqus)
0 0.5 10
0.5
1
1.5
2
Drucker-Prager I1 and J 2 criterion
(Abaqus)
Gf /
Gf‘
,0
Fiber Reinforced Concrete (FRC) with 3% Dramix Steel Fibers vs. Plain Concrete
0 0.5 1 1.50
0.5
1
1.5
2
0 0.5 1 1.50
0.5
1
1.5
2
Gf /
Gf0
c f/c
f0
Gf,0 = 86.7 N/mPlain concrete
FRC FRC
Crack–parallel compression σxx / σxx,c
Characteristic LengthFracture Energy
Calculated with M7 for concrete and M7f for FRC – tests yet to be made!
Gf,0 = 143.1 N/m
cf,0 = 11.2 mmPlain concrete
H. Nguyen, …Z.P. Bažant, JAM, in press; FC Caner, ZP Bažant (2013), Eng. Frac. Mech.105, 41--57 32
Gap Test for Gas Shale Predicted with Spherocylindrical Anisotropic Microplane Model
Fracture Energy Characteristic Length
Crack–parallel compression σxx / σxx,c
NOTE: After the Virus, collaborative tests will resume with LANL (Luke Fray, Hari Viswanathan, Bill Carey and Esteban Rougier)
_______________*Li, Cunbao,...Bažant, JMPS 103, 155-178 (2017)H. Nguyen, M. Pathirage, G. Cusatis, Z.P. Bažant, JAM, in press 33
*
0 0.25 0.5 0.75 10
0.5
1
1.5
2
0 0.25 0.5 0.75 10
0.5
1
1.5
2
Gf /
Gf,0
c f/c
f0
– tests yet to be made!
Strong effect of history (or path) of loading, calculated for concrete
σxx / σc
First σNthen σxx
First σxxthen σN
0.9 0.98
1.81σN*
σN*
0.40
0.62 σN*
Reversed: first
σN ∝ KI Hence, no universal formula for Gf as function of σxx exists!
H. Nguyen,…. Z.P. Bažant, PNAS, in press; JAM July 2020 34
Gap Test
First
V. Alternative Tests
Consideredand Retrospective
35
Tested sample
Stiff steel section
2Lx Symmetrically movable
Steel
Steel
Alternative setup – proportional loadingEvaluating Gf would require optimal FE fitting with crack-band damage model
36
Problem: σxx is not constant
Tschegg’s 1995 wedge-splitting test with crack-parallel compression
— pioneering idea!*Tests showed some effect but were ambiguous because:▪ work-of-fracture method
was used▪ no size effect was tested▪ non-uniform stress field▪ bending and friction from
weight of hydraulic jacks
37
_________*Ignored by fracture community due to lack of simplicity and high scatter
• Elasto-plastic metals: Stress triaxiality in an annulus around crack tip, led to extra parameter Q accounting for compressive T-stress (= σxx), and an increase of J-integral based on HRR field for ductile fracture (Shih, Hancock, Tvergaard,… 1990s).
• Crack path deflection due to T-stress in LEFM was solved by Cotterell and Rice (1980)
But both problems are different.
Retrospective
38
VI. New Perspective of
Fracture Mechanics ofQuasibrittle Materials
39
Formula Approximating Present Gap Tests and FE Simulations
H. Nguyen, …Z.P. Bažant, JAM, in press. Prelim.: PNAS in press.40
Can the existing FE programs for LEFM, CCM, XFEMand phase-field model be used? Hardly, because:
— Path dependence of damage is even stronger than it is inplasticity
— Different formulae would be needed for different materials,stress ratios σzz / σxx , plane strain, …
1
• To adjust Gf of the cohesive crack model is ambiguous since many different combinations of tensile strength ft and softening slope are possible, and adjusting ratio GF/Gf is unclear.
• Expected: A strong σxx effect in anisotropic fiber-polymer composites – e.g., a pressurized fuselage is under high biaxial tension.
• Delamination fracture is sure to depend on σxx , σzz .
Ambiguity of cohesive law adjustment
to, e.g., halve Gf
41
2
σ
w
ft ??
▪ Plus Vertex Effect:– Tangential stiffness at start of torsion vs. axial strain of compressed cylinder
rotating principal stress axes
Caner-BažantTests 2002
▪ uni-, bi-, tri-axial ▪ splitting strength▪ proportional or nonproportional▪ post-peak softening at FPZ scale
▪ “non-associated” dilatancy of frictional slip (captured by M7)
▪ tension-compression-torsion▪ unloading-reloading ▪ cycles…
microplaneloadingsurface
Experiment: Caner-Bažant 2002
Tang
entia
l She
ar S
tiffn
ess
Initial Compressive Strain
Paramount Problem: Realistic TensoriaI Damage Model
42** FC Caner, ZP Bažant, J Cervenka, J. (2002). JEM, 24-33;ZP Bažant (!982), J Eng Mech ASCE 109, 849--865
Tensorial invariant-based models
* FC Caner, ZP Bažant (2013), JEM, p. 1714
3
test
test
test
for concrete, fit 22 different benchmark triaxial tests (microplane model does)*
Insufficient Verifications by Limited Test DataF
u
A hundred different damage constitutive models for concrete exist !
- All can work for one crack:- All can fit Mode I test of F(u) softening curve, using: • scalar softening
• volumetric softening • deviatoric softening (~von Mises)
But should also fit: • basic material test data and
benchmark triaxial tests (with approx. FPZ size)
• multiple interacting cracks• gap tests
parallelcracks
h
43
4
• Mazars’ one-param. softening, etc.
etc.
Both the tensorial nature of the FPZ and the minimum possible spacing of parallel cracks can be effectively captured by the crack-band model.
Devil’s Postpile, Sierra Nev.
ZP Bažant, H Ohtsubo, Mech. Res. Comm. 4 (5), 353-366 (1977); Int. J. of Fracture 15, 443—456 (1979)
ad b
Eigenstrain σxx xa) b)
Localizing Not
𝜀𝜀0x
44
ad a
5
Devil’s Postpile, Sierra
σxx
https://www.bing.com/images/search?view=detailV2&ccid=CIZe0oXY&id=6F61F2915E261FB25C6A1A0E5E57AA38C8EBF358&thid=OIP.CIZe0oXYk5sBRKQCMi2jygHaE8&mediaurl=http://www.biblelife.org/globalwarming-dry-cracked-mud.jpg&exph=603&expw=903&q=photos+of+drying+cracks+in+mud+flats&simid=608025394424646963&selectedIndex=0https://www.bing.com/images/search?view=detailV2&ccid=CIZe0oXY&id=6F61F2915E261FB25C6A1A0E5E57AA38C8EBF358&thid=OIP.CIZe0oXYk5sBRKQCMi2jygHaE8&mediaurl=http://www.biblelife.org/globalwarming-dry-cracked-mud.jpg&exph=603&expw=903&q=photos+of+drying+cracks+in+mud+flats&simid=608025394424646963&selectedIndex=0
c)
b)a)
d)
45
Simulated vertical hydraulic cracks in shale with preexisting weak layers if Gf is constant— no localization!(Rahimi,…Bažant, PNAS 2019, p. 1532)
Injection into primary crack
Effects of σxx , σzz in fracking of shale – probably great
3 km deep, vertical crack faces are under high tectonic and overburdencrack-parallel stresses σxx ,σzz , combined by body forces applied by diffusion pressure gradients.
S Rahimi-Aghdam,…ZP Bažant, PNAS 116 (5), 1532—1537 (2019)
6
Continue using Mohr failure envelope for large crack-parallel stress σxx? — in doubt
—widely used in geophysics and in fracking studies.Envelopes from gap tests, extended by FE
46σσ
shea
r stre
ss in
tensit
yτ
τ
7
σI = transverse tensile stress in FPZ, via FEσIII = σxx = crack-parallel compressive stress at FPZ H. Nguyen,…. Z.P. Bažant, PNAS, in press 47
For weakenedGf
For strengthenedGf
Gap Test proves that shear fracture of beams and slabs must be analyzed taking into account a large tensorial FPZ
V
KIc = KIc,0 Reinf. bars
47
Despite 40 years of studies, CCM, LEFM never succeeded; KI 0 at Vmax . This was one experience that motivated the gap test.
8 Montreal 2006
Q Yu, … Bažant, Z.P. (2016). Struct. Concr. 17 (5), 778-789; A Donmez, ZP Bažant (2019), Struct. Conc. 20, 1-13
Promise of Multiscale Approach
• Tensorial damage band shrunken into a line (Remmers, de Borst, Needleman, IJF 2013). Yields a cohesive crack with an embedded subscale tensorial damage band.
• But tensorial damage law remains a big question.
• Calibration by data fits not yet demonstrated.
• Conceptual problem: Minimum crack spacing will not be enforced automatically. Parallel cracks?
48
9
Promise of Phase-Field Model– computationally enticing, alternative. But:
• Single parameter damage—inadequate.
• Needs a realistic tensorial model for softening damage, such as microplane, validated and calibrated by existing triaxial material tests of diverse types, esp. of FPZ size
• Transition from distributed damage to isolated bands? Minimum spacing of parallel cracks?
49
10
ONR Size effect tests 1993, on Arctic Ocean (Dempsey, Bažant collab.)
J Dempsey, PI
80x80x1.8 m specimen
Notch 24 m long
In 1980s, forces F measured on legs of oil platforms in Prudhoe Bay were an order-of-magnitude less than FE predictions with buckling.— One (incomplete) explanation: Size effect— Now 2nd explanation: Crack-parallel compression in large FPZ reduced Gf (or KIc) near 0.
In glaciers — σxx in vertical cracks, depth >1 km
Flat jacks
50
Fracture of Sea Ice with σxx
log D
log
σ
11
___________
ZP Bažant (2002), J. Appl. Mech. ASME 69, 19--24.
Bone tests by Bažant, Kim, Yu (IJF 2013)
SEL
Why? Observed size effect implies large FPZ
From size effect: Cohesive softening law of bone has a long tail
Crack opening displacement
Stre
ss
51
ZP Bažant, KT Kim, Q Yu, K. Xu, IJF 181, 2013
Expected: Effect of σxx , σzzon fracture of bone and biomaterials
12
Hip fracture, tooth fracture, … often occurs at high σxx
Tests of PEEK at Northwestern
(Bažant-Kim,…, IJF 1999)
• Kink-bands in fiber composites are a fracture propagationproblem with strong size effect
• σxx effect — unknown but likely.
52ZP Bažant, J J-H Kim, … IJF 95, 103-141 (1999)
Kink band micro-buckling with shear fractures leads to size effect
13
1) Charles-Evans law for static fatigue crack growth in rocks or ceramics, and
1) Paris law for cyclic fatigue crack growth in quasibrittle materials.
Bažant & Xu tests (ACI J 1991), for 3 specimen sizes
ZP Bažant, K. Xu, ACI Materials J 88 (4) 390—399 (1991).ZP Bažant, PC Prat, M Tabbara, ACI Mat J 87, 12—19 (1990).
Large FPZ suggests: effect of crack-parallel σxx ,σzzon subcritical crack growth
Size effect proves large FPZ
14
53
Tensorial Aspect of Shear Fracture
Torsional shear testsBažant, Prat, Tabbara, ACI J. 1990, 12-19
With axial compression
With none
54
In concrete, compression changes a planar shear crack to an inclined conical shear crack.
Inexplicable if constant Gf or Kc govern.
15
• Seismically sliding geological faults are very thin (< 1 mm), but if the FPZ at front is not, then σxx,σzz should matter for their propagation.
• Cracks growing in prestressed concrete have a different Gf.
• The Gf variation is generally not expected in metals, fine-grained ceramics or polysilicon, since their FPZ is of micrometer scale. But it could matter for MEMS, or devices < 10-4 m.
55
16
Conclusions• Gap Test – simple, unambiguous interpretation – one statically
determinate system transits to another – size effect test of Gf possible
• FPZ – Tensorial !Effective simple approach – FE crack band model.
• The CCM and LEFM (and XFEM) are approximations of rather limited applicability.
• LEFM and CCM remain essential for understanding and teachingfracture mechanics, and for providing accurate benchmark solutionsof special cases which tensorial damage mechanics must match.
• The evidence from the gap test looks compelling but so far is scantand deserves broad scrutiny. Extensive testing and calibration will be needed – for most quasibrittle materials.
56
Download from www.civil.northwestern.edu/people/Bažant video link: https://youtu.be/UyTeopPModA612.pdf. H. Nguyen et al., Perspective of Fracture Mechanics Inspired by Gap Test with Crack-Parallel
Compression. Proc. National Academy of Sciences 52 (2020) June, in press.613.pdf H. Nguyen et al., Gap test of crack-parallel stress effect on quasibrittle fracture and its
consequences. ASME J. of Applied Mechanics 87 (2020), 07, pp. 1012-1—11. 57
Afterthought Many hot research subjects become closed
in a few decades. But, like turbulence, fracture mechanics is different.
This formidable subject has been researched for a century, and probably will for another century.
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http://www.civil.northwestern.edu/people/Ba%C5%BEant
NEW PERSPECTIVE OF FRACTURE MECHANICS INSPIRED BY NOVEL TEST WITH CRACK-PARALLEL COMPRESSION���ZDENĚK P. BAŽANTSlide Number 2Why has the crack-parallel stress been ignored?�Slide Number 4Slide Number 5Brittle Fracture Mechanics Founding in 1921 and Its Evolution into Ductile, Cohesive and QuasibrittleNew finding: Effect of crack parallel stress σxx on fracture energy Gf of concrete is strongSlide Number 8Slide Number 9Slide Number 10Slide Number 11Test SetupTest Setup in More DetailSlide Number 14Slide Number 15Optimum Design of Polypropylene Pads Slide Number 17For quasibrittle materials, distinguish�the initial and total fracture energy, Gf and GFFor Size Effect Test Method of Gf :�Specimens Geometrically Scaled (as 1 : 2 : 4)Required 1st and 2nd Order Small- and Large-Size Asymptotic Properties of Type 2 Size Effect on FractureSize Effect Method to Measure Fracture Energy Gf and Material Characteristic Length cfSlide Number 22Classical Work-of-Fracture Method of Measuring Total Fracture Energy GFSlide Number 24Found: Effect of crack parallel stress σxx �on fracture energy Gf of concrete is strongSlide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Fiber Reinforced Concrete (FRC) with 3% Dramix Steel Fibers vs. Plain ConcreteGap Test for Gas Shale Predicted with Spherocylindrical Anisotropic Microplane ModelStrong effect of history (or path) of loading, calculated for concreteSlide Number 35Alternative setup – proportional loadingTschegg’s 1995 wedge-splitting test with crack-parallel compressionSlide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Continue using Mohr failure envelope for large crack-parallel stress σxx? — in doubtSlide Number 47Promise of Multiscale Approach Promise of Phase-Field ModelSlide Number 50Slide Number 51Slide Number 52Slide Number 53Tensorial Aspect of Shear FractureSlide Number 55ConclusionsSlide Number 57E X T R A SSize Effect Ensuing from First Law of ThermodynamicsSlide Number 60Essence of Microplane Model��������Physical Phenomena Mimicked by Microplane Model��������Size Effect Types in Quasibrittle FractureCrucial Features of Gap Test:Why quasibrittle fracture mechanics? What for?Microstructural Cause of Crack-Parallel Stress EffectBasic Idea of Quasibrittle (of Cohesive Softening, or Nonlinear Softening) Slide Number 67Slide Number 68Slide Number 69Slide Number 70Bottom: Premature failure of Drucker-PragerSize Effect Method to Measure Fracture Energy Gf and Material Characteristic Length cfAsymptotic Matching – Type 2 Size Effect�Size Effect of Divinycell H100 FoamAmbiguities of Work-of-Fracture Method needed for�alternative tests Derivations of Size Effect Law, Used in Gap Test Size Effect Law (of Type 2) for FRC�Slide Number 77Spherocylindrical Model by Shale—Uniaxial compression at dip angles 0 to 90o and confining pressures from 10 to 60 MPaSlide Number 79Review 2:�
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