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LIQUEFACTION, CRITICALITY AND COMPLEX SYSTEM Jeen-Shang Lin Department of Civil and Environmental Engineering University of Pittsburgh 4/30/09 NTOU

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9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response f s, (kPa) q c, (Mpa) stress resistance Depth (m) Point Based Criteria

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Motivation Identification of sites that are susceptible to liquefaction triggered by earthquakes is a crucial part of an earthquake hazard mitigation effort. Point failure criteria

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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…..

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Common large scale ground failures: Liquefaction Landslide Ground movement (uplift or subsidence) Debris flow Mud flow 9/6/20075 Kobe-Pitt Symposium on Disaster Risk Reduction and Response

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6Credit: C-Y KU

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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 shouldnt a computation be guided by a proper understanding of physics? 9/6/20077 Kobe-Pitt Symposium on Disaster Risk Reduction and Response What is the deeper question? Reflections on Failure

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Intrinsic failure: Topology of a problem? Key block theory + discrete elements

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53 removable out of 63 47 removable out of 57 Stresses induce fracture and block movements Simplified modeling of flow stress

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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

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

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Evolution of Slope Failure

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Mesh Based Partition of Unity: A new paradigm Problem domain Mathematical Mesh Physical Mesh

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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

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Covers, nodes and Elements Covers are nodes Overlapped covers are elements

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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)

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Formulation Partition of Unity Function: Duarte and Oden (1996) Approximate Function Weak form of the discrete problem with penalty formulation

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Discontinuity Modeling: Add nodes

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Mesh Based Partition of Unity: A new paradigm Problem domain Mathematical Mesh Physical Mesh Failure computation has made substantial Pr

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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

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Covers, nodes and Elements Covers are nodes Overlapped covers are elements

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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)

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Formulation Partition of Unity Function: Duarte and Oden (1996) Approximate Function Weak form of the discrete problem with penalty formulation

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Discontinuity Modeling: Add nodes

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Back to Science Liquefaction : 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.

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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.

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Self-Organized Criticality Castro 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.

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A percolation view? There cant be no flow unless voids are connected into a path through a material domain. 9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response 28

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Failure of a uniform material

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Failure of a heterogeneous material A global failure requires local failure to be connected….

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Failure of a heterogeneous material Failure is not static

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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, p c. One of our central issues is: What portion of a site has to be liquefied before a site is considered liquefied?

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Renormalization Group Method Basic Failure group for Von Mises materials p c is found to be 0.618

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Critical threshold p c is found to be 0.618

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Verification of Percolation failure 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.

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Take a detailed look Cluster size distribution

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Uniaxial experiment: bonded particles (initial void ratio=0.16) Red: location of failed bonds Deviatoric stress vs Axial strain

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Failure Progression Breaking of bonds 0.066% axial strain 2.31% 0.198%0.165%

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same sample L=H (13203 particles) Deviatoric stress vs Axial strain

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Failure Progression Breaking of bonds 0.198% axial strain 0.264%

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using an assembly of particles and conduct numerical experiment for understanding Using an assembly of particles --Looking from a different angle

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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.

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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.

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Cell Traction Force Microscopy Method Jeen-Shang Lin In Collaboration with James H-C. Wang MechanoBiology Laboratory Departments of Orthopaedic Surgery University of Pittsburgh

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

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A B Force-loaded image Null-force image B

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

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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 C

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Selected Features in null force image Matched featured in force loaded image

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A differential based registration Models a mapping function for a given pixel (x,y) from the target I o to source I 1 in the following form m be locally smooth

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A B A

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Vulnerability Assessment of Critical Infrastructure Jeen-Shang Lin Department of Civil and Environmental Engineering University of Pittsburgh

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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?

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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

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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.

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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).

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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,….

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Workshop on Hazard Reduction and Responses in Metropolitan Regions The random rewiring model Watts, 1999

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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

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

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