LIQUEFACTION, CRITICALITY AND COMPLEX SYSTEM

Jeen-Shang LinDepartment of Civil and Environmental Engineering

University of Pittsburgh4/30/09 NTOU

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9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response

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Point Based Criteria

MotivationIdentification of sites that are susceptible to liquefaction triggered by earthquakes is a crucial part of an earthquake hazard mitigation effort.

Point failure criteria

•Is the soil susceptible to liquefaction?•If the soil is susceptible, will liquefaction be triggered?•If liquefaction is triggered, will damage occur?

Questions to be answered:

Answers to these questions require tackling some deep questions…..

Common large scale ground failures:

LiquefactionLandslideGround movement (uplift or subsidence)Debris flowMud flow

9/6/2007 5Kobe-Pitt Symposium on Disaster Risk Reduction and Response

6Credit: C-Y KU

Failures often have common fundamental physics and structures: Intrinsic nature, invariant scale measure.

It is important to view them from a unified framework.

Numerical analysis is difficult but not insurmountable.

We are very good at computation. But shouldn’t a computation be guided by a proper understanding of physics?

9/6/2007 7Kobe-Pitt Symposium on Disaster Risk Reduction and Response

What is the deeper question?

Reflections on Failure

Intrinsic failure: Topology of a problem?

Key block theory + discrete elements

53 removable out of 63

47 removable out of 57

Stresses induce fracture and block movements

Simplified modeling of flow stress

Three ways a soil may respond depending on their states (Intrinsic nature?) Castro (1969)• liquefaction failure• dilative response • limited liquefaction failure

Monotonic undrained triaxial tests on Banding Sands

Conventional computational approaches

Constitutive modeling + FEM work quite well,but does not address a more fundamental question.

Evolution of Slope Failure

Mesh Based Partition of Unity: A new paradigm

Problem domain

Mathematical Mesh

Physical Mesh

Same foundation with meshless methods: Partition of unity

• Diffusion element• Element free Galerkin method• Reproducing Kernel Particle Method• h-p cloud method• Partition of unity finite element method• Extended finite element method• Generalized finite element method

Covers, nodes and Elements

Covers are nodesOverlapped covers are elements

The use of finite covers leads to node based computation

Element Free Galerkin method

Nodes lie inside a problem domain.

Boundary, covers and supports (Belytschko et al. 1996.)

Meshed Based Partition of Unity

No restriction on nodes locations

(XFEM)

Formulation

xxxx forforN

0)(;1)(1

0;1)()( 0 SNforC S x

m

kkk

T ap0

)()()( xaxpxu

Partition of Unity Function: Duarte and Oden (1996)

Approximate Function

Nc

i

Nc

iiiii

1 1

)()(ˆ)()( axuxxuh x

ohhhhhhhh VVddd

tu

0)()(:)( vvtvbvuuvu

Weak form of the discrete problem with penalty formulation

Discontinuity Modeling: Add nodes

j

ji

ji

j

ji

jii uuu )()(ˆ)()(ˆ)( xxxxx

Mesh Based Partition of Unity: A new paradigm

Problem domain

Mathematical Mesh

Physical Mesh

Failure computation has made substantial Pr

Same foundation with meshless methods: Partition of unity

• Diffusion element• Element free Galerkin method• Reproducing Kernel Particle Method• h-p cloud method• Partition of unity finite element method• Extended finite element method• Generalized finite element method

Covers, nodes and Elements

Covers are nodesOverlapped covers are elements

The use of finite covers leads to node based computation

Element Free Galerkin method

Nodes lie inside a problem domain.

Boundary, covers and supports (Belytschko et al. 1996.)

Meshed Based Partition of Unity

No restriction on nodes locations

(XFEM)

Formulation

xxxx forforN

0)(;1)(1

0;1)()( 0 SNforC S x

m

kkk

T ap0

)()()( xaxpxu

Partition of Unity Function: Duarte and Oden (1996)

Approximate Function

Nc

i

Nc

iiiii

1 1

)()(ˆ)()( axuxxuh x

ohhhhhhhh VVddd

tu

0)()(:)( vvtvbvuuvu

Weak form of the discrete problem with penalty formulation

Discontinuity Modeling: Add nodes

j

ji

ji

j

ji

jii uuu )()(ˆ)()(ˆ)( xxxxx

Back to ScienceLiquefaction : What happened?

Casagrande had envisioned that a “flow structure” would develop at liquefaction (Castro, 1969): “….during a liquefaction slide the relative position of the grains is constantly changing in a manner which contains a minimum resistance. He explained that the change from a normal structural arrangement of the grains to be flow structure would start almost accidentally in a nucleus and then spread through the mass by a chain reaction, and that such a reaction could explain the spontaneous character of liquefaction slide.”

Liquefaction as a Critical Behavior of a Complex System?

A sand assembly is a complex system, and “critical state”, “phase transition”, “steady state” are also the focus points in the dynamics of various complex systems, including even biological systems (e.g., Kauffman, 1993). On the outset, a steady state in the terminology of nonlinear dynamics is often referred to as a “fixed point” or a “fixed point attractor” (e.g., Williams, 1997), as a system is attracted to a fixed state under sustained loading. The physics governing the process that leads to an attractor generally have something in common. It is believed here that linking liquefaction to the current understanding of nonlinear dynamics may also provide a better understanding for post-liquefaction behavior.

Self-Organized CriticalityCastro and Poulos (1977) have proposed an interpretation for cyclic mobility. They postulated that large strains resulting from cyclic mobility in laboratory on dilative sands are due principally to redistribution of void ratio within the specimen during cyclic loading.

In 1987, Bak, Tang and Wisenfeld proposed the concept of “self-organized criticality” (SOC). They discovered that large, dissipative complex system had the tendency of driving themselves into a critical state with wide range of length and time scale.

A percolation view? There can’t be no flow unless voids are connected into a path through a material domain.

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Failure of a uniform material

Failure of a heterogeneous material

A global failure requires local failure to be connected….

Failure of a heterogeneous material

Failure is not static

Critical Threshold How wide spread needs the liquefaction be within a given site in order for it to fail?

We used percolation theory to address this issue:•A problem domain is first discretized with a regular lattice•Each lattice cell may be vacant or occupied with an occupancy rate, p. We then asked, “What is occupancy level in a lattice at which it is

possible to find a pathway through the unoccupied population of the lattice?”

The critical occupancy rate is called a percolation threshold, or critical threshold, pc.

One of our central issues is: “What portion of a site has to be liquefied before a site is considered liquefied?”

Renormalization Group Method

Basic Failure group for Von Mises materials

2234 )1(2)1(4 ppppppo

pc is found to be 0.618

Critical threshold

22341 )1(2)1(4 ooooo pppppp

pc is found to be 0.618

21

211

31

412 )1(2)1(4 pppppp

2234 )1(2)1(4 ppppppo

Verification of Percolation failure

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Strain (%)

Str

ess

(kP

a)

p=0.1

p=0.3

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•We conduct strain control compression test simulation on Von Mises material using samples consists of 10,000 cells.

•Two strength levels were selected; each cell was assigned one of the two strengths according to the random number generated. If the random number, which lay between 0 and 1, selected was less or equal to an assigned p, the lower strength was given; otherwise the higher strength was used.

Take a detailed look

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Cluster size distribution

Uniaxial experiment: bonded particles(initial void ratio=0.16)

Red: location of failed bonds

Deviatoric stress vs Axial strain

Failure ProgressionBreaking of bonds

0.066% axial strain2.31%

0.198%0.165%

same sample L=H(13203 particles)

Deviatoric stress vs Axial strain

Failure ProgressionBreaking of bonds

0.198% axial strain

0.264%

usingan assembly of particles and conduct numerical experiment for understanding

Using an assembly of particles--Looking from a different angle

A sample triaxial test include unloading reloading

Devistoric stress vs axial strain

Volumetric strain vs axial strain

WE are in the process of tracing the failure location and cluster size, as well as what happen at the largestrain.

Concluding remarks

1. This is a work in progress. We are now using both particle codes and continuum code with bounding surface model with state variables to study the problem involving self –organization after failure.

2. We believe that looking into a scale invariance measure, the phase change from percolation view point can be invaluable.

Cell Traction Force Microscopy Method

Jeen-Shang LinIn Collaboration with

James H-C. WangMechanoBiology Laboratory

Departments of Orthopaedic SurgeryUniversity of Pittsburgh

Cell traction force microscopy (CTFM) is recognized as a simple and effective method for quantifying cell traction forces.

Migration

Traction forces

polyacrylamide gelSubstrate

Cell

A B

“Force-loaded” image “Null-force” image

B

(1) Feature-based registration(2) Non-Feature Based Registration

Image Registration

Phase congruence(1)In the frequency domain, at locations of complex

image intensity variation, the local Fourier components are maximally in phase.

(2)local phase is also invariant to image brightness or contrast.

(3) To determine the phase information spatially, one needs to process the images to simultaneously obtain the spatial and the frequency information.

(4) Use filter bank: Gabor functions

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Feature identification: A Phase Congruence Method

A B

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Selected Features in null force image

Matched featured in force loaded image

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Cell and displacement field (Phase congruence)

A differential based registration

Models a mapping function for a given pixel (x,y) from the target Io to source I1 in the following form

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ym

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Vulnerability Assessment of Critical Infrastructure

Jeen-Shang LinDepartment of Civil and Environmental Engineering

University of Pittsburgh

The deficiency of old mode of thinking

• Designing a functional system is not enough. – Form is as important as function.– Topology robustness.

• We need a new set of tools and metrics.– Shifting from trees to woods mindset– Easy to read a tree, but how do you read the woods?

Conventional Risk Assessment Study

• Identify failure mode of a component

– Develop a fragility of a component

• Identify Hazard

– Develop of a hazard map with various return periods

• Integration and obtain the risk

Small World Network

• A small world is a world of order that embodies a certain degree of disorder. (The world is not ordered, nor is it completely random.)

• It originated from the study of social network: It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances.– Stanley Milgram (1967) the chain is of order six thus six

degree of separation.– Watts and Strogatz, 1998, the small world model

• The small world effect appears to be universal and applies to other networks.

Characteristics of a Small world network

• Sparse: A network has relatively few links. An n-node network of links much less than • Clustered: It is not random

– Links are not uniformly distributed; – Small diameter:

• For a small world network, the largest separation is much less than n, and is closer to ln(n).

22

)1( 2nnn

There are some things about a graph…• Characteristic Path Length, L

– Medium of the averages of shortest path between any two nodes (2.58)• Diameter: (Degree of separation)

– Largest shortest path between any two vertices (6)• Clustering coefficient: How well are your neighbors connected?• Shortcuts, contraction,….

Workshop on Hazard Reduction and Responses in Metropolitan Regions

The random rewiring model

Watts, 1999

Original:

k=3.29, L=4.4,D=8,c=0.5,phi=0.195;psi=0.213

After adding three links:

k=3.5, L=3.72,D=8,c=0.442,phi=0.245;psi=0.331

March 16-18, 2003, J-S Lin Workshop on Hazard Reduction and Responses in Metropolitan Regions