# Presentation: 567163

## 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

**Notes:**
The project that I’m going to talk about today is the ...

First of all I’ll talk a little bit about the general concepts, and application of acoustic imaging

Then I’ll go a little into details of mathematics behind those methods.

And then I’ll talk about the details of our new method and it’s contribution to this subject

And finally we’ll see some results from our method and compare it with the ones from previous methods.

## 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

**Notes:**
The project that I’m going to talk about today is the ...

First of all I’ll talk a little bit about the general concepts, and application of acoustic imaging

Then I’ll go a little into details of mathematics behind those methods.

And then I’ll talk about the details of our new method and it’s contribution to this subject

And finally we’ll see some results from our method and compare it with the ones from previous methods.

## 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)

## Motivation: Size Effect

- Fracture is controlled by
- Stress concentrations
- Material *disorder *induced fluctuations

## 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?

## 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)*

## Anisotropic Roughness Scaling

## 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

## 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?

## 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

- 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

- Outline

## 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

- 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

- Random Thresholds Fuse Model

## Algebraic Problem

- Assemble the initial conductivity matrix of the lattice system
- Increase external loads until a bond threshold is reached
- Irreversibly break the bond en+1 that satisfies the failure criterion
- Redistribute forces within the lattice network
- Repeat steps 2-4 until the lattice network is disconnected

**~ O(L****1.8****) in 2D; O(L****2.8****) in 3D**

## 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 Ln+1 is contained in Ln
- for all m < n,            , where      is the sparsity of Ln
- Multiple rank sparse Cholesky downdate:

## Optimal/Block Circulant Preconditioners

- 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

## 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

- 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

- Largest Ever System Size Simulations and Extensive Statistical Sampling

## CPU ~ O(L4.5)

Capability Issue: Previous simulations have been limited to a system size of L=128.

Largest 2D lattice system (L=1024) analyzed for investigating fracture and damage evolution.

Effective computational gain ~ 80 times

- Capability Issue: Previous simulations have been limited to a system size of L=128.
- Largest 2D lattice system (L=1024) analyzed for investigating fracture and damage evolution.
- Effective computational gain ~ 80 times

- Typical Fracture of a 2D Lattice System

- J. Phys. A: Math. Gen. 36 (2003)

- Peak

- Failure

## CPU ~ O(L6.5)

Largest cubic lattice system analyzed for investigating fracture and damage evolution in 3D systems (L=64)

- Progressive damage accumulation in a 3D cubic lattice network, L = 64. Pre-peak damage: (a)-(c), and (d)-(i) is post-peak damage.

- CPU ~ O(L6.5)
- Largest cubic lattice system analyzed for investigating fracture and damage evolution in 3D systems (L=64)

- Fracture of 3D Cubic Lattice System

- J. Phys. A: Math. Gen. 37 (2004)

- peak

- Failure

## 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

## 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

## Damage and Percolation

- JSTAT, P08001, 2004

## Damage and Percolation

- Clearly, percolation
- scaling is not obeyed

- Damage for strong disorder case: is it in the same universality class
- as that of percolation?

- JSTAT P08001 (2004)

## How about Correlated Percolation?

- Response flattens
- for large systems

- There is no correlated percolation at failure

## First or Second-Order Phase Transition?

- JSTAT, P08001, 2004
- PRE, 2006

## Scaling of Failure Probability Distribution

**Cumulative Distribution**

**(p – ********f****)/********f**

**(p – ********f****)/********f**

- Uniform and
- power law
- threshold
- distributions

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

## Stiffness (Order Parameter)

- Mean field

- L = 256

- L = 512

- L = 1024

- Peak load

## Fracture Strength Distribution and Size Effects

- EPJB, 37, 2004
- PRE, 71, 2005a
- Advances in Physics, 2006

## Fracture Strength Distribution: Weibull, Gumbel or Lognormal?

-  Disorder: Uniform (D = 1), and power law (D = 20)
-  Model type: Fuse and central-force (spring) models

## Size Effects in Notched Specimens

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

- D = 1

## Influence of Disorder on Size Effect

- a0

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

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

## Crack Roughness

- PRE, 71, 2005b
- PRE, 2006
- Advances in Physics, 2006
- IJF, 2006

## Model recovers anomalous scaling as observed in experiments
Local roughness: 		Global roughness:

- Model recovers anomalous scaling as observed in experiments
- Local roughness: 		Global roughness:

- Anomalous Scaling

## Anomalous Scaling: Power Spectrum Analysis

- loc = 0.4

-  = 0.5

- loc = 0.74

-  = 0.85

- 3D

## 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

## 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.

- 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.

- 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.

- 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.

- Conclusions and Significant Results