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