multiscale modeling fracture peridynamics
TRANSCRIPT
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Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energy under contract DE-AC04-94AL85000.
Stewart Silling
Multiphysics Simulation Technologies Department
Sandia National Laboratories
Albuquerque, New Mexico, USA
Seminar given at
University of Texas, San Antonio
February 16, 2012
SAND2012-1141C
Multiscale Modeling of Fracture with
Peridynamics
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First: a puzzle
• Consider a 1D lattice with linear elastic force interactions.
– No transverse loads.
– Prescribed axial displacements at ends.
– The equilibrium displacement field contains transverse deflection as shown.
• What’s going on? – What sort of material would do this?
– Is this even legit mechanically?
Deformed
Undeformed
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Outline
• Peridynamics basics
• Coarse graining
• Multiscale fracture
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Purpose of peridynamics
• To unify the mechanics of continuous and discontinuous media within a single,
consistent set of equations.
Continuous body Continuous body
with a defect Discrete particles
• Why do this?
• Avoid coupling dissimilar mathematical systems (A to C).
• Model complex fracture patterns.
• Communicate across length scales.
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Why this is important
• Cracks:
• Standard approaches implement a fracture model after numerical
discretization.
• Particles:
• Standard approaches require a separate coupling method to relate
particles to continuum.
Complex crack path in a composite
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Peridynamics basics:
Horizon and family
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Peridynamics basics:
Bonds and bond forces
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Peridynamics basics:
Material modeling
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Kinematics:
Deformation state
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Material modeling:
Bonds and states
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Peridynamic vs. standard equations
Kinematics
Constitutive model
Linear momentum
balance
Angular momentum
balance
Peridynamic theory Standard theory Relation
Elasticity
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Any material model in the classical theory
can be used in peridynamics…
• Example: Large-deformation, strain-hardening, rate-dependent material model.
– Material model implementation by John Foster.
0% strain 100% strain
Necking of a bar under tension
Taylor impact test
Test
Emu
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…and there are peridynamic materials that
cannot be represented in the standard theory
• Examples
• Bond-pair materials: resist angles changes between opposite bonds.
• Discrete particles: any multibody potential can be represented with peridynamic states.
Multibody potential
Bond-pair
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Convergence of peridynamics
to the standard theory
In this sense, the standard theory is a subset of peridynamics.
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How damage and fracture are modeled
Bond elongation
Bond force density Bond breakage
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Bond breakage forms cracks “autonomously”
Broken bond
Crack path
• When a bond breaks, its load is shifted to its neighbors, leading to progressive failure.
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Energy balance for an advancing crack
There is also a version of the J-integral that applies in this theory.
Crack
Bond elongation
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EMU (and LAMMPS) numerical method
• Integral is replaced by a finite sum: resulting method is meshless and Lagrangian.
Method is also available in Sierra (D. Littlewood)
Discretized model in the
reference configuration
• Looks a lot like MD!
• LAMMPS implementation by M. Parks & P. Seleson
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Predicted crack growth direction depends
continuously on loading direction
• Plate with a pre-existing defect is subjected to
prescribed boundary velocities.
• These BC correspond to mostly Mode-I loading with a
little Mode-II.
Contours of vertical displacement Contours of damage
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Effect of rotating the grid
in the “mostly Mode-I” problem
Damage Damage, rotated grid
Damage Displacement
Network of identical bonds in many
directions allows cracks to grow in
any direction.
Original grid direction
30deg
Rotated grid direction
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Fragmentation example:
Same problem with 4 different grid spacings
Dx = 3.33 mm
Dx = 2.00 mm
Dx = 1.43 mm
Dx = 1.00 mm
Brittle ring with
initial radial velocity
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Dynamic fracture in a hard steel plate
• Dynamic fracture in maraging steel (Kalthoff & Winkler, 1988)
• Mode-II loading at notch tips results in mode-I cracks at 70deg angle.
• 3D EMU model reproduces the crack angle.
EMU*
Experiment
S. A. Silling, Dynamic fracture modeling with a meshfree peridynamic code, in Computational Fluid and Solid Mechanics 2003, K.J. Bathe, ed., Elsevier, pp. 641-644.
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Example of long-range forces:
Nanofiber network
Nanofiber membrane (F. Bobaru, Univ. of Nebraska)
Nanofiber interactions due to van der Waals forces
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Peridynamic dislocation model
Example: Dislocation segment in a square with free edges
100 x 100 EMU grid
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Coarse-graining:
Reducing the degrees of freedom
• Start with a detailed description (level 0).
• Choose a coarsened subset (level 1).
• Model the system using only the coarsened DOFs…
• Forces on the coarsened DOFs depend only on their own displacements.
• These forces should be the same as you would get from the full detailed model.
• After coarsening, the level 0 DOFs no longer are modeled explicitly.
Small-scale MD model
Blue: Level 0
Red: Level 1
Even cats find this interesting
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Linearized peridynamics
SS, Linearized theory of peridynamic states, J. Elast. (2010)
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Coarse-graining:
Reduce the number of degrees of freedom
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Each level’s displacements are determined
by the next higher level
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Each level has the same mathematical structure
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Refinement
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Example: Coarse graining
of an elastic block (answer to puzzle)
Level 0
Level 1
Level 2
• A homogeneous, isotropic rectangle is
stretched from its lower corners.
• Problem is modeled at three levels:
• Level 0 (purple)
• Level 1 (green)
• Level 2 (red)
• Displacements and forces on bc nodes
agree between all three calculations.
Curvature in level 2 results from tensor
nature of the micromoduli.
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Level 0 composite model
Fiber (red)
Matrix (purple)
2D fiber-reinforced composite: fibers are much stiffer than the matrix.
Stiff bond
Soft bond
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3 levels of coarsening
0
2
1
3
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Coarsened composite micromodulus
Level 0 Level 1
Level 2 Level 3
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Composite bar stretch: displacements
Level 0 Level 1
Level 2 Level 3
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Level 0 composite model with defect
Fiber (red)
Matrix (purple)
Crack (green)
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Composite bar with defect:
displacements
Level 0 Level 1
Level 2 Level 3
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Coarse grained model of Brazilian test
(static crack)
Level 0: 6646 nodes
Level 6357: 289 nodes
Coarsen
Solve on small grid
Refine if needed
Solve on large grid
Load = 5.455
Load = 5.432 Load = 5.441
Almost identical displacements
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Issues with this coarse graining method
Level 0 Level 1
• Similar to static condensation.
• The number of bonds connected to each node grows with each coarsening level.
• Large memory requirements.
• Experience shows:
• Can be used for coarse-graining relatively small regions or subregions.
• Not suitable for large-scale material mechanics modeling
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Multiscale approach for growing cracks:
Multiple horizons
Each successive level has a larger
length scale (horizon). Crack process zone
The details of damage evolution
are always modeled at level 0.
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Multiscale approach for growing cracks:
Concurrent solution strategy
Crack
Level n
Level 2
Level 1
Level 0:
Within distance d of ongoing damage
Level 0
Level 1
Level 2
Level n
Time
Refin
e
Coa
rsen
Solve (fine)
Solve (coarse) R
efin
e
Coa
rsen
Concurrent solution strategy Level 0 region follows the crack tip
• Refinement:
• Level 1 acts as a boundary condition on level 0.
• Coarsening:
• Level 0 supplies material properties (e.g., damage) to higher levels.
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Crack growth in a brittle plate
Level 2
Level 1 Level 0
Damage process zone
Initial damage
v v
Crack
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Brittle crack growth:
Bond strain near crack tip
Colors show the largest strain among all bonds connected to each node.
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Brittle crack growth:
Damage progression and velocity
Damage process zone
v1 velocity
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Brittle crack under shear loading
Bond strain Damage process zone
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Crack growth in a heterogeneous medium
• Crack grows between randomly placed hard inclusions.
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Dynamic brittle crack:
Branching
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Fracture due to indentation
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Discussion
• Coarse graining method*:
• Exact but expensive.
• Linear only.
• No adjustable parameters.
• Two-way coupling (coarsening + refinement): consistent multiscale method.
• Multiscale damage method:
• Non-linear, dynamic.
• Low memory requirements.
• Small time step applied only in level 0.
• With big computers, appears to offer the potential to model the details of
heterogeneous material failure at all physically relevant length scales.
*SS, A coasening method for linear peridynamics, Int. J. Multiscale Computational Engineering (2011).
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Extra slides
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Nonlocality as a result of homogenization
• Homogenization, neglecting the natural length scales of a system, often doesn’t give good answers.
Indentor Real
Homogenized, local Stress
Claim: Nonlocality is an essential feature of a realistic homogenized model of a heterogeneous material.
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Proposed experimental method for measuring the peridynamic horizon
• Measure how much a step wave spreads as it goes through a sample.
• Fit the horizon in a 1D peridynamic model to match the observed spread.
Time
Free surface velocity
Peridynamic 1D
Visar
Spread
Projectile Sample
Visar
Laser
Local model would predict zero spread.
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Material modeling: Composites
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Splitting and fracture mode change in composites
• Distribution of fiber directions between plies strongly influences the way cracks grow.
Typical crack growth in a notched laminate
(photo courtesy Boeing) EMU simulations for different layups
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Polycrystals: Mesoscale model*
• Vary the failure stretch of interface bonds relative to that of bonds within a grain.
• Define the interface strength coefficient by
Large favors trans-granular fracture.
*
*
g
i
s
sβ
= 1 = 4 = 0.25
• What is the effect of grain boundaries on the fracture of a polycrystal?
Bond strain
Bond force
*is
*gs
Bond within a grain
Interface bond
* Work by F. Bobaru & students
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Dynamic fracture in PMMA: Damage features
Microbranching
Mirror-mist-hackle transition*
* J. Fineberg & M. Marder, Physics Reports 313 (1999) 1-108
EMU crack surfaces EMU damage
Smooth
Initial defect
Microcracks
Surface roughness
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Dynamic fracture in PMMA: Crack tip velocity
• Crack velocity increases to a critical value, then oscillates.
Time (ms)
Cra
ck tip
ve
locity (
m/s
)
EMU Experiment*
* J. Fineberg & M. Marder, Physics Reports 313 (1999) 1-108