Formulation of Critical Questions for Understanding Brittle Fracture

1
Formulation of Critical Questions for
Understanding Brittle Fracture

• J. J. Mecholsky, Jr.
• Materials Science Engineering
• University of Florida
• Gainesville, FL 32611

MS T 2007
2
Bond Breaking Leads to Characteristic Features
Log v
U
s
Kc
e
r
Log K Log (Ys c ½)
3
Fracture In Materials
There are several fundamental questions that need
to be answered
How do bonds break?
Once a bond breaks, how do the ensembles of
broken bonds propagate?
Is there a mathematical formulation which
permits insight into the fracture process at
all scales?
4
Hypothesis Fracture is a fractal process.
Implications ? Bond breaking is the
primary factor controlling


the formation of fracture surface
structure at the macroscopic length scale.
? Fracture topography is statistically
self-similar (or self-affine).
? Fracture topography scales from atomic to
macroscopic
lengths.
5
Questions guide presentation

• What is fractal geometry?
• How is fractal geometry related to fracture?
• What do we know about fractal fracture?
• What do we need to do in the future to determine
  the extent of the effect of fractal fracture?

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What Is A Fractal?
Benoit Mandelbrot A rough or fragmented
geometric shape that can be subdivided in parts
that are similar in shape (self-similarity), each
of which is (at least approximately) a
reduced-size copy of the whole shape (scale
invariance).
z z2 C
Defined by a Fractional Dimension (i.e. 1.3, 2.4,
etc. where D 0.3,0.4,etc.)
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Fractal Geometry A Renewed Math

• Characteristics
• Non-differentiable
• Defined by a Fractional Dimension (i.e. 1.5, 2.4,
  etc.)
• Self-similar features
• Scale invariant features

• http//math.rice.edu/lanius/frac/

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Characteristic Markings Are Observed on the
Fracture Surface
Hackle
Mist
Mirror
Fracture origin
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The Mirror Region Is Not Smooth
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Mist and Hackle Appear Similar in Shape
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Mist and Hackle Appear Similar in Shape
mist
mist
hackle
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Many Observations Lead To The Conclusion Of
Fractal Fracture
Scaling Behavior Self-Similarity In A
Plane Scale Invariance
13
FRACTAL DIMENSION IS MEASURED ALONG CONTOUR
Log Length (A-B)
Slope -D
A
Log (Ruler Length)
B
A-B Slit Island Contour
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Fractal Dimension Varies For Different Materials
J. J. Mecholsky, Jr., Fractography, Fracture
Mechanics and Fractal Geometry An Integration,
Ceram. Trans. 64, in Fractography of Glasses and
Ceramics III, eds. J. P. Varner, V.D. Frechette,
G. D. Quinn, Am. Ceram. Soc. (1996).
16
Mandelbrot, Passoja Paullay first applied
fractal geometry to fracture

Fractal character of fracture surfaces of metals
Benoit B. Mandelbrot Dann. E. Passoja Alvin J.
Paullay Nature 308, 721 - 722 (1984).
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There are over 10,000 references for fractal
fracture!

• Fractal fracture mechanicsA review Engineering
  Fracture Mechanics V. 51, Issue 6, Aug 1995, pp
  997-1033
• Evidence of fractal fracture. T Chelidze, Y
  Guguen, International Journal of Rock Mechanics
  and Mining Sciences 2733, 223-225, Pergamon
  Press, 1990.
• On the Interpretation of the Fractal Character of
  Fracture Surfaces , Dauskardt, R H Haubensak, F
  Ritchie, R O Acta Metall. Mater. V. 38, no. 2,
  pp. 143-159. Feb. 1990
• Surfaces Formed by Subcritical Crack Growth in
  Silicate Glasses, Guin and Wiederhorn and,
  Inter. J. of Fracture 140(1-4)15-26 (2006)

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Many others observed fractal fracture
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Bond Breaking Leads to Characteristic Features
Log v
U
s
Kc
e
r
Log K Log (Ys c ½)
20
Characteristic Markings Are Observed on the
Fracture Surface
s M / r1/2
Hackle
Mist
Mirror
Fracture origin
s KC/ (Y c1/2)
21
Fracture Mechanics Fractography Provide A
Framework for Quantitative Analysis
KIC Y s c 1/2 Crack Boundary KB1 Y1 s
r11/2 Mirror-Mist Boundary KB2 Y2 s
r21/2 Mist-Hackle Boundary KB3 Y3 s
r31/2 Crack Branching Boundary
c/rj constant
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D Is Related to Mirror-to-Flaw Size Ratio
Mecholsky Freiman J ACerS 74123136 (1991)
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D Is Related To Flaw-to-Mirror Size Ratio
KB1 Y s r11/2
KC D1/2 E a01/2
Mecholsky Freiman J ACerS 74123136 (1991)
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Toughness Increases With Fractal Dimension
Kc E a01/2 D1/2
g ½ a0 ED
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Fractal Geometry Has Been Used In Failure Analysis

• Silicon J. Mater Res 6,1248 91
• Ocala Chert J Mater Sci Ltrs 7, 1145 88
• Intermetallics J. Mater. Sci.6,1856 91
• Si3N4 J. Mater. Sci. 32 6317 97
• Dental Glass Cer. JACerS 78 3045 95
• Epoxy Resin Scanning 20 99 98

2 g a0 ED Kc a01/2 ED 1/2 D
c/r1
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Bond Breaking Leads to Characteristic Features
Log v
U
s
Kc
e
r
Log K Log (Ys c ½)
27
Bell Dean Model Used for MO Calculations
cf. Varshneya, Fundamentals of Inorganic
Glasses (After Bell and Dean, Nature 212, 1354
1966)
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cf. Varshneya, Fundamentals of Inorganic Glasses
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Groups of atoms at crack tip behave in a similar
manner along crack front
s22
Crack tip
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MO Simulates Bond Breaking At The Crack Tip
2
1
Simulated SiO2 D displ. 1 Å
31
2
1
32
2
2
1
1
33

2

1
34
2
1
35
2
1
36
2
1
37
2
1
38
2
1
39
2
1
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Strain Can Be Measured In Model
a0 a / e c a / c-c
a
a
c
c'
J. Non-Crystalline Solids 260 (1999) 99-108.
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Energy Geometry Are Related In The Fracture
Process
g fracture energy E Elastic Modulus D
Fractal Dimensional Increment a0 Characteristic
Structural Parameter KC Fracture Toughness
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Modeling and Experimental Results Agree
a0 2g/ (ED)
J. K. West, J. J. Mecholsky, Jr, and L. L.
Hench, The Quantum and Fractal Geometry of
Brittle Fracture, J. Non-Crystalline Solids 260
(1999) 99-108.
43
Notice Similarity of Fracture Energy Equation To
Freiman et al.

• g ½ D E a0 fractal approach
• g k E d 0 classical
  approach

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a0 Is Related To Structure
a0 2 g / ( ED)
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Characteristic Markings Are Observed on the
Fracture Surface
Hackle
c / r constant
Mist
Mirror
Fracture origin
c
r
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Fracture Is A Fractal Process

• Fracture transcends many length scales
  Self-similar (or self-affine), scale invariant
  characterized by D.

• Hypothesis 2g a0 E D

• Observations seem to support the
  hypothesis. c/r D g ? ED
• (D is a geometric energy scaling factor.)
• (a0 is a fracture surface structural element)

• MD MO modeling provide framework for
  understanding macroscopic observations

48
Atomic Scale Structure Expands To Fracture Surface
2 g a0 ED
Log v
U
s
Kc
e
r
Log K Ys c 1/2
49
Implications of fractal fracture need to be
considered

• If Fracture is a fractal process
• Geometry at atomic scale related to macroscopic
  scale
• An algoritm can be found to recreate a fracture
  surface
• Relatively simple rules should govern the
  fracture process
• Quantum mechanics should be related to LEFM

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CONCLUSIONS

• The atomic basis of fracture is a series of
  bond reconfigurations resulting in the
  production of more space.

• Fractal fracture implies that the same fracture
  process should be able to be observed at all
  length scales.

• Experimental data acquisition analytical
  modeling have to be interactive to be
  successful.

• Analytical models have to explain fractal
  nature of fracture, mirror, mist hackle and
  crack branching.

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What should be done in the future?

• Raman spectroscopy should be related to atomic
  scale modeling as a check
• Models of fracture should include quantum
  mechanics
• Continuum vs. discrete phenomena should be
  investigated (fracture is a discrete event and
  yet can be modeled using a continuum approach)
• There has to be an interactive cooperation
  between analysis (modeling) and experimentation.

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Many Have Contributed To This Work

• Dr. Dann Passoja Dr. Alvaro Della Bona
• Dr. Jon West Dr. Tom Hill
• Dr. Steve Freiman Jacques Cuneo
• Dr. Joe Simmons Prof . Jeff Thompson
• Dr. Jim Runt Dr. Ananth Naman
• Dr. Larry Hench Jim Plaia
• Prof. Tom Mackin Dr. Zheng Chen
• Karen Feinberg-Ringel
• Former PSU Students Dr. Ben Tsai
• Colleagues Prof. Jason Griggs
• Dr. Luis Carney
• Former UF Students

Partial Support ONR United Technologies
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University of Florida, Gainesville FL

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Technical Approach
Modeling e.g., Molecular Orbital (MO)
Modeling of Fracture - determines a0. a0 is a
characteristic structural parameter that measures
strained bonds.
Experimental determination and comparison of
parameters obtained in modeling, i.e., a0 and D
D is the fractal dimensional increment
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The Mirror/Mist Region Is Not Smooth
57
Mandelbrot, Passoja Paullay first applied
fractal geometry to fracture

Fractal character of fracture surfaces of metals
Benoit B. Mandelbrot Dann. E. Passoja Alvin J.
Paullay Nature 308, 721 - 722 (1984).
58
Stresses at Crack Tip Are Near Theoretical Values
s22
Crack tip
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Theoretical Strength Can Be Obtained Using
Fractal Geometry

• (1a) sij K / (2 p r)1/2 f(q) LEFM
• (1b) s22 Kc / (2 p r)1/2
• (2) gc g0 ½ a0 ED
  Fractal Geometry
• (3a) st E a0 ½ D ½ / (2 p a0)1/2 combine
  (1) (2)
• let r a0
• (3b) st E D ½ / (2 p )1/2 No
  potential energy expression needed!

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Theoretical Strength Predicted Between E/10 and
E/p
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CONCLUSIONS

• Fractal fracture implies that the same fracture
  process should be able to be observed at all
  length scales.

• Experimental data analytical modeling
  
  have to be interactive to
  be successful.

• All models should be compared to (real)
  experimental data.

• Analytical models have to explain fractal
  nature of fracture, mirror, mist hackle and
  crack branching.

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Critical Questions Need To Be Asked

• What are the energetic geometric steps to
  fracture?

• Is a flat fracture (of primary bonds) possible
  above absolute zero? What is bond rupture?

• Is roughness a meaningful parameter in
  fracture?

• How does energy scale?

• How does a crack propagate at all length
  scales?

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Relationship Holds For Large Size Stress Range
s r1/2 constant
J.J. Mecholsky, Jr., Fractography of Optical
Fibers, in ASM Engineered Materials Handbook, 4,
Ceramics and Glasses, Section 9 Failure
Analysis, (1992).
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FSA Can Be Applied To Single Crystals
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Slate Fracture Surface Shows Characteristic
Features
Courtesy of Prof. Yet-Ming Chiang
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Fracture Markings Can Last 4000 Years
Titis Sarcophagus - Egypt c. 2500 BC Prof.
Greenhut
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Many Tools Are Needed for Unified Fracture Theory
Model Scale Experiment
MO sub-atomic fractoemission Ab initio
atomic AFM Quantum nano
STM Mechanics Raman Fluorescence

Fractal Geometry
MD micro crack velocity Monte
Carlo SEM Finite diff.
Meso AFM
FEM Macro fractography
fracture mechanics

Fracture Process
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Epoxy Fracture With Glass Fiber As Origin
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Glass Fiber Fractures Within Epoxy Matrix
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Fracture In Steel Shows Characteristic Features
AISI 4340
s 790MPa
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Strain Can Be Measured In Model
a0 a / e c a / c-c
a
a
c
c'
J. Non-Crystalline Solids 260 (1999) 99-108.
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Brittle Fracture Is Observed At Many Length Scales
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Brittle Fracture Can Be Observed At Many Length
Scales
Courtesy of Dr. Darryl Butt UF
75
Characteristic Markings Are Observed on the
Fracture Surface
Hackle
Mist
Mirror
Fracture origin
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Replicas Provide Multiple Slit Islands for
Analysis
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Energy Geometry Are Related In The Fracture
Process
g fracture energy E Elastic Modulus D
Fractal Dimensional Increment a0 Characteristic
Structural Parameter KC Fracture Toughness
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Fracture Behavior Appears Related To Material
Class
a0 2g/ (ED)
81

Fracture Surfaces Polished Surfaces
3BaO5SiO2

Glass Kc 0.7 I D 0.1
82
Fractal Geometry Has Been Used In Failure Analysis

• Silicon J. Mater Res 6,1248 91
• Ocala Chert J Mater Sci Ltrs 7, 1145 88
• Intermetallics J. Mater. Sci.6,1856 91
• Si3N4 J. Mater. Sci. 32 6317 97
• Dental Glass Cer. JACerS 78 3045 95
• Epoxy Resin Scanning 20 99 98

2 g a0 ED Kc a01/2 ED 1/2 D
c/r1
83
FSA Can Be Applied To Single Crystals
Single Crystal Silicon
84

Simulated Fracture Can Form Fracture Surface

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Simulated Fracture Can Form Fracture Surface

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Simulated Fracture Can Form Fracture Surface

87

Simulated Fracture Can Form Fracture Surface

88

Surface Can Be Created From MD Simulation

89

Slit Island Contour Can Be Made From 3-D Map

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FRACTAL DIMENSION IS MEASURED ALONG CONTOUR
Log Length (A-B)
Slope -D
A
Log (Ruler Length)
B
A-B Slit Island Contour
91
Summary

• At the atomic level, quantum mechanics describes
  the fracture process as a ring contraction
  process dictated by minimum energy and
  availability of free volume.

On the molecular scale, MD modeling describes
creation of the fracture surface.

• On the macroscopic scale, mirror, mist hackle
  form c/r D

• At all length scales, 2g a0 E D

92

Potentials Depend on Bond Behavior
If r lt a, then Stillinger-Weber potential is
used. If r gt a, then Born-Mayer potential is
used.

Following bond breakage at r a, each Si atom
will gain one positive charge for each lost
neighbor.
93

Standard MD Techniques Are Used

94

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Fractal Dimension Varies For Different Materials
J. J. Mecholsky, Jr., Fractography, Fracture
Mechanics and Fractal Geometry An Integration,
Ceram. Trans. 64, in Fractography of Glasses and
Ceramics III, eds. J. P. Varner, V.D. Frechette,
G. D. Quinn, Am. Ceram. Soc. (1996).
97
MD Simulations Experimental Results Agree
Y. L. Tsai, T. P. Swiler , J. H. Simmons and J.
J. Mecholsky, Jr., in Computational Modelling of
Materials and Processing, J. H. Simmons, et al.
(eds) The Am. Cer. Society, Ceram. Trans. 69
(1997) 217.
98
Outline
Fundamental Questions About Fracture -
How does a crack propagate at all length
scales?

• Analytical Tools - Quantum Mechanics (QM),
• Molecular Dynamics (MD), ab initio, Monte Carlo,
  FEM, FD

Experimental Tools - Fractography (FSA) ,
Fracture Mechanics (FM) Fractal Analysis (FA)
Fractoemission (FE), Crack Velocity
Measurements

• Conclusions - FSA, FM, FA, MD QM combine to
  form model of the scaled fracture process.

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Characteristic Features Aid Failure Analysis
KC Y s (c)1/2 KBj Y s (rj)1/2 c
(a b )1/2 rj / c constant
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Modeling Has to Explain Several Observations

• Scaling
• topography (mirror, mist, etc.)
• self similarity (self affinity)
• Fracto-emission
• Chaotic (Velocity) Behavior