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Fractal Geometry Applied To FractureJ. J. Mecholsky, Jr.Materials Science & Engineering DepartmentUniversity of FloridaGainesville, FL 32611-6400
[email protected] Tutorial Series: prepared for and produced by theInternational Material Institute for New Functionality in GlassAn NSF sponsored program – material herein not for sale Available at www.lehigh.edu/imi
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σ
ε
Log v
Log K = Log (Yσ c ½)
U
rKc
Bond Breaking Leads to Characteristic Features
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Outline•Experimental Observations - One View of Fracture
•Fundamental Questions About Fracture -How does a crack propagate at all length scales?
• Experimental Tools - Fractography (FSA) , Fracture Mechanics (FM) & Fractal Analysis (FA) Fractoemission (FE), Crack Velocity Measurements
•Analytical Tools - Quantum Mechanics (QM),Molecular Dynamics (MD), ab initio, Monte Carlo, FEM, FD
•Conclusions - FSA, FM, FA, MD & QM combine to form model of the scaled fracture process.
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Mirror
Mist
Hackle
Fracture origin
Characteristic Markings Are Observed
on the Fracture Surface
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J.J. Mecholsky, Jr., Fractography of Optical Fibers, in ASM Engineered Materials Handbook, 4, Ceramics and Glasses, Section 9: FailureAnalysis, (1992).
Relationship Holds For Large Size & Stress Range
σ r1/2 = constant
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Energy & Geometry Are Related
In The Fracture Process
2 γ = a0 [ED*]
KC = Yσ (c)1/2 = (2 E γ)1/2
γ = fracture energy
E = Elastic Modulus
Y = Geometry & Loading Constant
σ = Fracture Stress (Strength)
C = Critical Crack Size
KC = Fracture Toughness
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Characteristic Features Aid Failure Analysis
KC = Y σ (c)1/2 KBj = Y σ (rj)1/2
c = (a b )1/2 rj / c = constant
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Fracture Mechanics & Fractography Provide A Framework for Quantitative Analysis
KIC = Y σ c 1/2 Crack Boundary
KB1 = Y1 σ r11/2 Mirror-Mist Boundary
KB2 = Y2 σ r21/2 Mist-Hackle Boundary
KB3 = Y3 σ r31/2 Crack Branching Boundary
[c/rj = constant]
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FSA Can Be Applied To Single Crystals
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Slate Fracture Surface On Window Sill at Kimbull Union Academy
Courtesy of Prof. Yet-Ming Chiang
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Fracture Markings Can Last 4000 Years
Titi’s Sarcophagus - Egypt c. 2500 BC [Prof. Greenhut]
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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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AISI 4340
Fracture In Steel Shows Characteristic Features
σ =790MPa
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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]
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Mirror
Mist
Hackle
Fracture origin
Characteristic Markings Are Observed
on the Fracture Surface
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Mist and Hackle Appear Similar in Shape
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Many Observations Lead To The Conclusion Of Fractal Fracture
• Scaling Behavior
• Self-Similarity In A Plane
• Scale Invariance
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Fractal Geometry –A Renewed Math• Characteristics:
• Non-differentiable• Defined by a Fractional
Dimension (i.e. 1.3, 2.4, etc.)• Self-similar features• Scale invariant features
–http://math.rice.edu/~lanius/frac/
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SCALING CAN BE DESCRIBED BY FRACTAL GEOMETRY Fractal Geometry is - a non-Euclidean geometry that exhibits - self-similarity (or self-affinity) & - scale invariance and is characterized by the
- fractal dimension, D.
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Replicas Provide Multiple Slit Islands for Analysis
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Log (Ruler Length)
Slope = -D*Log Length (A-B)
B
A
FRACTAL DIMENSION IS MEASURED ALONG CONTOUR
A-B = Slit Island Contour
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Material Class D* (dimensionless) Single Crystals Glasses Glass Ceramics Polycrystalline Ceramics Polymers Metals
0.07-0.12 0.07-0.1 0.06-0.3 0.06-0.35 0.2-0.29 0.06-2.5
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).
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(D*) 1/2
K IC
PolycrystallineMaterials
Glass-Ceramics
Single Crystals
0.2 0.3 0.4 0.5 0.60
1
2
3
4
5
(MPa-m 1/2 )
Toughness Increases With Fractal Dimension
Kc = E a01/2 D*1/2
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D* Is Related To Flaw-to-Mirror Size Ratio
KIC = E (a01/2) D*1/2 = Y σa (c)1/2
KB1= E (b0
1/2) = Y1σa (r1)1/2 [if a0 = b0 and Y = Y1]
then, D* = c/r1
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D* Is Related To Flaw-to-Mirror Size Ratio
Mecholsky & Freiman J ACerS 74[12]3136 (1991)
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D* Is Related to Mirror-to-Flaw Size Ratio
Mecholsky & Freiman J ACerS 74[12]3136 (1991)
05
1015
0 5 10 151/D*
r1/cGlass
PyroceramZnS
SiPZTAl2O3
(Solid Line is 1:1)
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3BaO•5SiO2
D*=0.13
Kc=1.0 I
AR= 1.4
D*=0.16
Kc=1.3 I
AR= 3.6
D*=0.26
Kc=2.2 I
AR = 8
I =MPam1/2
Fracture Surfaces Polished Surfaces
Glass:
Kc = 0.7 I
D* = 0.1
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Energy & Geometry Are Related
In The Fracture Process
2 γ = a0 [ED*]
KC = (Ea01/2) D*1/2 = (2 E γ)1/2
γ = fracture energy
E = Elastic Modulus
D* = Fractal Dimensional Increment
a0 = Characteristic Structural Parameter
KC = Fracture Toughness
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020406080
100
0 50 100 150
ED* (GPa)
2γ (J/m2)
Glass Ceramics
Glasses
Single Crystals
Fracture Behavior Appears Related To Material Class
a0 = 2γ/ (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 γ = a0 [ED*] ; Kc = a01/2 ED* 1/2 ; D* = c/r1
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Energy & Geometry Are Related
In The Fracture Process
2 γ = a0 [ED*]
KC = (Ea01/2) D*1/2 = (2 E γ)1/2
γ = fracture energy
E = Elastic Modulus
D* = Fractal Dimensional Increment
a0 = Characteristic Structural Parameter
KC = Fracture Toughness
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Modeling Has to Explain Several Observations
• Scaling– topography (mirror, mist, etc.)– self similarity (self affinity)
• Fracto-emission• Velocity (Chaotic) Behavior
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Much evidence exists for chaotic and fractal scaling behavior
Fracto-emission : light, particles, molecules, etc. [e.g., cf. Langford, et. al., J.Mat. Res. 4, 1272 (1989)] Dynamic instability : chaotic crack velocities at branching [e.g., cf. Fineberg, et. al., Phys.Rev.Ltrs 67,4(1991)] Self-similar fracture surfaces : crack branching [e.g., Ravi-Chandar & Knauss, Inter.J.Fracture 26,65-80(1984) Kulawansa et al., J. Mater. Res.9,2476 (1994) Mecholsky et al., Adv. In Cer.22 ACerS (1988); J. Materials Res.13 ,11 (1998).] Scaling : Energy (γc) & Geometry (a0, D*) [e.g., cf. Passoja, Adv. In Cer. 22, 101 (1988) ACerS; Mecholsky, Cer.Trans. 64 385-93, ACerS (1996); West et al., J. Non-crystalline Solids, 260 (1999) 99-108; Y. Fahmy, J. C. Russ and C. C. Koch, J. Mater. Sci. 6, 1856-1861 (1991).]
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Fracture In Materials
• Is there a mathematical formulation which permits insight into the fracture process at all scales?
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?