Large Scale Simulation of Fracture in Disordered Materials

1
Large Scale Simulation of Fracture in Disordered
Materials

• Phani Kumar V.V. Nukala
• Computer Science and Mathematics Division
• Oak Ridge National Laboratory
• Srdjan Simunovic
• Computer Science and Mathematics Division
• Oak Ridge National Laboratory

2
Collaborators

• Stefano Zapperi
• Dipartimento di Fisica
• Universita La Sapienza, Roma, Italy
• Mikko Alava
• Laboratory of Physics,
• Helsinki University of Technology, Finland
• Richard Mills
• Oak Ridge National Lab

3
Acknowledgements

• MICS DOE Office of Science
• INCITE Award Computer resources on BG/L at ANL
• NLCF allocation on Cray-XT3 (Jaguar) at ORNL

• Relevant Journal Publications
• J. Phys. Math. Gen. 36 (2003) 37 (2004) IJNME
  62 (2005)
• European Physical Journal B 37 (2004)
• JSTAT, P08001 (2004) JSTAT (2006)
• Physical Review E 71 (2005a, 2005b, c) 73 (2006a
  2006b)
• Advances in Physics (2006)
• International Journal of Fracture (2006)

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Motivation Size Effect

• Fracture is controlled by
• Stress concentrations
• Material disorder induced fluctuations

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Effect of Disorder

• Weak disorder Stress concentrations dominate the
  disorder effects
• Extreme-value theory
• Size effect L-1/2
• Strong disorder Disorder dominates the stress
  concentration effects
• What is the strength distribution?
• Size effect?

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Motivation Crack Roughness

• Initially, fracture surface roughness and
  toughness were thought to be correlated
• Mandelbrot (1984)
• Experiments on several materials (metals, glass,
  rocks, ceramics)
• Fracture surface is self-affine
• Roughness exponent displays a universal value
  over five decades of length scales (5nm to 0.5
  mm)

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Anisotropic Roughness Scaling
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Anomalous Roughness

• Recent experiments with granite and wood samples
• There exist two roughness exponents (local and
  global)
• Only the local exponent appears to be universal
• Open Question
• How can the fracture surfaces of materials as
  different as metallic alloys and glass, for
  example, be so similar?

• Family-Vicsek scaling
• Anomalous scaling

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Motivation

• What are the size effects and scaling laws of
  fracture of disordered materials?
• What are the signatures of approach to failure?
• What is the relation between toughness and crack
  surface roughness?
• Universality of crack surface roughness?

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Outline

• Random Thresholds Fuse Model
• Numerical Algorithms and HPC
• Numerical Simulation Results (2D and 3D)
• Damage and Percolation
• First- or second-order phase transition?
• Fracture Strength Distributions and Size Effects
• Crack Roughness
• Summary

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Random Thresholds Fuse Model

• Scalar or electrical analogy
• For each bond, assign unit conductance and the
  thresholds are prescribed based on a random
  thresholds distribution
• The bond breaks irreversibly whenever the current
  (stress) in the fuse exceeds the prescribed
  thresholds value
• Currents (stresses) are redistributed
  instantaneously
• The process of breaking one bond at a time is
  repeated until the lattice falls apart

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Algebraic Problem

• Simulation Procedure
• Assemble the initial conductivity matrix of the
  lattice system
• Increase external loads until a bond threshold is
  reached
• Irreversibly break the bond en1 that satisfies
  the failure criterion
• Redistribute forces within the lattice network
• Repeat steps 2-4 until the lattice network is
  disconnected

O(L1.8) in 2D O(L2.8) in 3D
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Sparse Direct Solvers for 2D RFM

• Each An is sparse
• Factorization ( ) can be done
  efficiently
• But, too expensive to factorize every An directly
  (O(L1.8))
• Successive An matrices differ by a rank-p matrix.
• For 2D or 3D fuse and spring models, p 1
• For 2D beam models, p 3
• For 3D beam models, p 6
• Since
• the sparsity of Ln1 is contained in Ln
• for all m lt n, , where is the
  sparsity of Ln
• Multiple rank sparse Cholesky downdate

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Optimal/Block Circulant Preconditioners

• Circulant matrices can be diagonalized by
  discrete Fourier matrices O(N logN)
• Circulant preconditioners can be chosen such that
  the condition number of the system is minimized
• Exhibit favorable clustering of eigenvalues. In
  general, the more clustered the eigenvalues are,
  the faster the convergence rate is
• If A is positive definite, then c(A) is also
  positive definite.

unique optimal circulant preconditioner of A
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Largest Ever System Size Simulations and
Extensive Statistical Sampling

• Large system sizes and extensive sampling
• Largest system sizes ever analyzed
• L 1024 in 2D and L 64 in 3D
• Extensive ensemble sample averaging (2D)
• 50000 samples for system sizes (L 8, 16, 24,
  32, 64)
• 12000 samples for (L 128)
• 1200 samples for (L 256)
• 200 samples for (L 512)
• 10 samples for (L 1024)
• In 3D
• 50000 (L 10), 20000 (L 16), 2512 (L 24)
• 1200 samples for L 32
• 200 samples for L 48
• 12 samples for L 64

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Typical Fracture of a 2D Lattice System

• CPU O(L4.5)
• Capability Issue Previous simulations have been
  limited to a system size of L128.
• Largest 2D lattice system (L1024) analyzed for
  investigating fracture and damage evolution.
• Effective computational gain 80 times

Peak
Failure
J. Phys. A Math. Gen. 36 (2003)
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Fracture of 3D Cubic Lattice System

• CPU O(L6.5)
• Largest cubic lattice system analyzed for
  investigating fracture and damage evolution in 3D
  systems (L64)

peak
Progressive damage accumulation in a 3D cubic
lattice network, L 64. Pre-peak damage
(a)-(c), and (d)-(i) is post-peak damage.
Failure
J. Phys. A Math. Gen. 37 (2004)
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High-Performance Computing

• Parallel computing
• L 64 on 128 processors takes 3 hours
• L 100 on 512 processors takes a day
• L 128 on 1024 processors takes 3 days
• L 200 on 2048 processors takes 20 days!
• Algorithmic development
• Recycling Krylov subspace
• 30 faster

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Scientific Significance

• Damage and percolation
• Is damage in the same universality class as
  percolation?
• Correlated or gradient percolation?
• Is fracture a first-order or second-order phase
  transition?
• Fracture strength distribution and size-effects
• Crack surface roughness
• Anomalous scaling
• Multiscaling
• Origin of universality

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Damage and Percolation
JSTAT, P08001, 2004
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Damage and Percolation
Damage for strong disorder case is it in the
same universality class as that of percolation?
Clearly, percolation scaling is not obeyed
JSTAT P08001 (2004)
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How about Correlated Percolation?
Response flattens for large systems

• There is no correlated percolation at failure

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First or Second-Order Phase Transition?
JSTAT, P08001, 2004 PRE, 2006
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Scaling of Failure Probability Distribution
Cumulative Distribution
Uniform and power law threshold distributions
(p mf)/Df
(p mf)/Df

• Cumulative distribution is universal
• Normal distribution
• No divergent correlation length
• First-order phase transition

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Stiffness (Order Parameter)
Mean field
Peak load
L 1024
L 256
L 512
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Fracture Strength Distribution and Size Effects
EPJB, 37, 2004 PRE, 71, 2005a Advances in
Physics, 2006
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Fracture Strength Distribution Weibull, Gumbel
or Lognormal?

• Lattice topology Triangular and diamond
• Disorder Uniform (D 1), and power law (D
  20)
• Model type Fuse and central-force (spring)
  models

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Size Effects in Notched Specimens
D 1

• Size effect
• Logarithmic in the un-notched case
• L-1/2 scaling when a0/L is large

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Influence of Disorder on Size Effect

• Size effect
• Model recovers strength scaling even when a0 is
  irrelevant, and D is relevant!

• Size effect
• Logarithmic when D 1
• L-1/2 scaling when D is small

a0
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Crack Roughness
PRE, 71, 2005b PRE, 2006 Advances in Physics,
2006 IJF, 2006
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Anomalous Scaling

• Model recovers anomalous scaling as observed in
  experiments
• Local roughness Global roughness

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Anomalous Scaling Power Spectrum Analysis
zloc 0.4
zloc 0.74
z 0.5
z 0.85
3D
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Effect of Notch and Disorder on Roughness

• Anomalous scaling of crack surface
• Disorder
• Notch
• Lattice type

• Universality of roughness.
• Disorder, notch and lattice type are irrelevant
  parameters

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Conclusions and Significant Results

• Percolation vs Localization
• Damage evolution is not in the percolation
  universality class ( ).
• Damage is accumulated in a uniform manner in the
  pre-peak regime, and then suddenly localizes in
  the post-peak regime.
• Damage profiles are uniform until the peak-load,
  and show a peak with exponential tails in the
  post-peak regime.
• Failure Probability Distributions
• Universal and do not depend on lattice topology
  (even 2D and 3D).
• Gaussian distribution indicating that there is no
  divergent correlation length at failure.

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Conclusions and Significant Results

• Fracture Strength
• Lognormal distribution is more adequate than
  Weibull or Gumbel.
• Mean fracture strength decreases logarithmically.
• Weak disorder or large crack size results in
  L-1/2 scaling.
• Crack Roughness
• Exhibits anomalous scaling as recently observed
  in experiments.
• Local roughness is estimated to be 0.72 /- 0.03
• Global roughness is estimated to be 0.83 /- 0.04
• Global width distributions are found to be
  universal with respect to lattice geometry.