Meshfree methods and simulations of material failures
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Meshfree Methods and Simulations of Material Failures Shaofan Li Department of Civil and Environmental Engineering, University of California at Berkeley Collaborators Dr. Bo C Simonsen, Technical University of Denmark; Dr. Daniel C. Simkins,

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Meshfree methods and simulations of material failures l.jpg

Meshfree Methods and Simulations of Material Failures

ShaofanLi

Department of Civil and Environmental Engineering,

University of California at Berkeley


Collaborators l.jpg
Collaborators

Dr. Bo C Simonsen,

Technical University of Denmark;

Dr. Daniel C. Simkins,

University of South Florida;

Dr. Sergey N. Medyanik,

Northwestern University


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Table of Contents

  • Introduction: What is Meshfree Method

  • Simulations of Large Deformations

  • Simulations of Strain Localizations

  • Simulations of Dynamics Shear Band Propagations

  • Simulations of Ductile Fracture

    6. Simulations of Impact and penetrations

    7. Conclusions



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Meshfree Methods

  • Smooth Particle Hydrodynamics (SPH)

  • Element Free Galerkin (EFG)

  • Reproducing Kernel Particle Method (RKPM)

    1. Unknown is represented by convolving a smooth kernel with dependent variable

    2. Discretize by evaluating integral via nodal integration



Rkpm kernel l.jpg
RKPM Kernel

  • f(x-y) is a smooth compactly-supported function, e.g. cubic spline

  • PT(x) = [1 x y z xy xz …] is vector of monomial terms

  • b(x) (called the normalizer) is used to regain discrete partition of unity


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Moment Equation

Note and P(0) = (1, 0, 0, …, 0) then



Compare fem shape function with meshfree shape function l.jpg
Compare FEM shape function with Meshfree shape function

(a) FEM ; (b) Meshfree


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Compare FEM shape function with Meshfree shape function

(a) FEM ; (b) Meshfree


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Meshfree discretization and Meshfree shape functions


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(a) Shape function

(b) Derivative

(c) Derivative

(d) Derivative

“The Cloud”: 3-D Meshfree shape function and its first derivatives generated

by the tri-linear polynomial basis, P(X) = (1, x1 ,x2 , x3 , x1x2 ,x2x3 , x3x1 , x1x2x3 )


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1.2 A Few Virtues of Meshfree Methods

Convergence Property

as as

Reproducing Property

For That is

Non-local Interpolation (Discrete Convolution)


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* Large Support Size

It enables meshfree discretization/interpolation to endure large mesh distortion and

sustain computation without remeshing;

is deformation map



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Example 2.1 : Compression of A Rubber Cylinder

FEMMeshfree

50% Compression

65% Compression

85% Compression

90% Compression



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Example 2.2 A pinched cylindrical shell

Material Properties

Computational Parameters

Number of Particles: 30300

Time Step:



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Example 2.3 Hemispheric Shell with Pinched Load

Summary:

Material: Elastic-plastic material;

Geometry: Hemisphere shell with radius of 1 inch, thickness of 0.04 inch.

Particles: 12,300


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(c) t = 3.010-3s

(b) t = 1.510-3s

(a) t = 0.510-3s

(f) t = 7.510-3s

(d) t = 4.510-3s

(e) t = 6.010-3s


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Example 2.4 The snap-through of a conic shell

Material Properties

Computational Parameters

Number of Particles: 12300

Time Step:


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The snap-through of a 3D conic shell

(a) (b) (c)

(c) (d) (e)


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Excessive mesh distortion (hourglassing)

ABAQUS/Explicit

RKPM/Explicit



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Meshfree Simulation of Strain Localization

Hardening

Softening

From Reid, Gilbert, and Hahn [1966]



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Meshfree Methods: Element-free Galerkin (EFG)

Fleming and Belytechko [1997]


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Shearband Path for a Plate with 31 Holes (FEM vs. Meshfree Methods)

(b) 60  90 mesh

(c) 90  60 mesh

(a) 60  60 mesh

(d) 60  90 particles

(e) 90  60 particles

(c) 60  60 particles




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FEM Methods)Meshfree

(a) 20  20

(b) 30  20

(c) 40  20

(d) 50  20


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4. Meshfree Multiscale Computations Methods)

= Scale 0 + Scale 1 + Scale 2 + …...

1-D Meshfree Multiscale Basis Functions for Polynomial Basis P = ( 1, x, x2, x3 )



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[0,0] [1,0] [0,1]

[2,0] [1,1] [0,2]

Hierarchical Partition of Unity of Quadratic Basis



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Multiscale Analysis [1,0] [0,1]

Spectral Meshfree Adaptive Simulations


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Multiscale Analysis [1,0] [0,1]

(a) Total Scale(b) Low Scale(c ) High Scale(d) Adaptive Pattern

Total Particles: 3321

Particles used for adaptive calculation: 361


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Example 4.2 Multiresolution Analysis [1,0] [0,1]

Total Scale

Low Scale

High Scale


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Tension of A Bar with A Hole [1,0] [0,1]

Meshfree (particles)

Zoom in

Kinematically

Admissible

Mode I

4272 particles/layer

with 3 layers in

thickness direction

Mesh-based (element distortion)


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Symmetry slip line solution [1,0] [0,1]

Shear Plane Development for a bar with A Hole in Tension (I)


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Simulation of anti-symmetric [1,0] [0,1]slip line solution



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How to simulate curve shearband [1,0] [0,1]

Curved Shear Band Formation in Double-Notched Bar in Tension

13364 particles

Reference 1: Ewing, D.J.F. and Hill, R. J. Mech. Phys. Solids, 15, 115 (1967)



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Remark: [1,0] [0,1]FEM’s regularized strong discontinuous element technique may be able to simulate a shearband, only if one knows the shearband path a priori. If the shearband path has unknown and evolving curvature, it will be almost impossible to employ such technique.


3d simulation capacity l.jpg
3D Simulation capacity [1,0] [0,1]

(a) (b) (c)

(d) (e) (f)



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Necking of a thin plate [1,0] [0,1]

0.0 s 1.010- 5 s 2.010- 5 s

3.010- 5 s 4.010- 5 s 5.010- 5 s


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Experimental Results on Multiple Shear Band [1,0] [0,1](Anand and Spitzig, 1982)

250 m

0.1 mm

1 mm

Computational Results by Meshfree Method


Iv dynamic shear band simulations l.jpg
IV: Dynamic Shear-Band Simulations [1,0] [0,1]

Recently, we have done some massive

parallel computations on simulations of

stress collapse inside the adiabatic shear band.


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(a) [1,0] [0,1]

(b)

Morphology of adiabatic shear band: (a) scanning electron micrograph of a

Shear band; (b) Details of a shear band (After Leech, P. W, (1985)).


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Meshfree Simulation of Dynamic Shear Band Propagation [1,0] [0,1]

Experiment on impact-loaded plate

Kalthoff & Winkler [1987], Mason, Rosakis and Ravichandran [1994], and Zhou, Rosakis and Ravichandran [1996a,b]

Failure Mode Transition


Kalthoff problem l.jpg
Kalthoff Problem [1,0] [0,1]

(a) V < Vc

(b) V > Vc



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Freund-Hutchinson Model ([1 [1,0] [0,1]985]

and

Zhou-Rosakis Model ([1996])

and

where


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Propagation of Shear Band and Crack at Intermediate Impact Speed

Experimental observation (Rosakis et al., 1999), Impact speed 23.4m/s, C-300 steel

Computation by meshfree method (Contour of effective stress)


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HIGH-SPEED INFRARED IMAGING SYSTEM Speed

Image Acquisiton Rate: 1 MHz (1 million frames per second)

Response Time: 750 ns

Noise: 2 mV or ~2o temperature resolution

1.1 mm

Complete System

Chip


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Numerical Techniques Simulations Speed

of Fracture and Crack Growth

  • Erosion Algorithm (Rashid [1968])

  • 2. Traditional FEM remeshing (Wawrzynek and Ingraffea [1987]);

  • 3. Cohesive FEM (Xu-Needleman [1994], and

  • Camacho and Ortiz [1997]; )

  • 4. Meshfree Methods (Belytschko et al [1994]);

  • 5. X-FEM and Level set method (Mose et al [1999],

  • and Belytschko et al [2001]).


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Ductile-to-Brittle Transition Speed

(a) Computational Configuration; (b) “Mesh” or Particle Distribution


Macroscale picture of shear bands in rosakis model l.jpg
Macroscale Picture of Shear Bands in Rosakis Model Speed

Adiabatic

Experimental

With Heat Conduction


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Comparison with Experiment at High Impact Speed Speed

ExperimentComputation

Impact speed 29.4 m/s, C-300 steel

(Computation V = 37 m/s)


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Dynamic Shear Band Propagation Speed with Heat Conduction

  • Motivation

  • Heat Conduction may help to regularize the solution

Formulas for shear band width estimation:

Bai(1986) :

Wright(1995):

Olson(1981),Merzer(1982):

Where k is thermal conductivity, ac(~10-3/K), m (~10-2) are softening and rate sensitivity

parameters, =1-10, and =105 – 106/sec are strain and strain rate inside band

  • Look inside the shear band, see its microstructure (“hot spots” were found in experiments)

  • 3) Measure parameters of shear band.


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Detailed Shear Band Structure ( temperature signature) Speed

Using Reproducing Kernel Particle Method(RKPM), an numerical meshfree method, we were able to predict for the first time: the ductile-to-brittle failure mode transition; the character change of the curved shear band with the projectile impact speed; an intense, high strain rate region in front of the shear band tip, which we believe to cause a stress collapse that drives the formation and propagation of a dynamic shear band.

Multiple Scale Analysis of Shear Band: comparison with experiments

17:1

1:1

250:1

250:1

200 m


Slide76 l.jpg

2.5mm Speed

TRANSITION OF CRACK TIP PLASTIC ZONE INTO A SHEAR BAND

DT (K)

t=18ms

t=20ms

t=19ms

Field of View

1.1 x 1.1 mm

t=21ms

t=23ms

t=22ms


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Comparison between Experiment and Computation Results (Temperature)

A. (t =12 s) B. (t = 32 s) C. (t = 72 s)

D. (t =12 s) E. (t = 36 s) F. (t = 72 s)

Experiment (A, B, C), Computation (D,E,F)


Mesh dependency of band width for adiabatic shear band l.jpg
Mesh Dependency of Band Width (Temperature)for adiabatic shear band


Inside of the shear band l.jpg
Inside of the Shear Band (Temperature)

Adiabatic

(mesh dependency)

With Heat Conduction

(resolved band width)


Hot spots in a small model l.jpg
Hot Spots in a Small Model (Temperature)

With Heat Conduction solution is regularized, band width is about 20 microns.

Instability of viscous flow occurs inside the shear band.



Similar instabilities l.jpg
Similar Instabilities (Temperature)

1) Instability of plane Couette flow (viscous fluid)

  • Instability in a layer of thermo-viscoplastic material

  • “Structures in shear zones due to thermal effects”

  • (1991, Molinary and Leroy)


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Shear-Band Bifurcation (Temperature)


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Experimental Results. (Temperature)

by Wittman et. al. (1990)

Numerical Simulation


V simulation of ductile fracture l.jpg
V (Temperature). Simulation of ductile fracture




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X-FEM mesh-dependent


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Difficulties and Challenges: mesh-dependent

  • No remeshing and no adaptive discretization is allowed;

  • 2.Fracture criteria;

  • 3.Prediction and simulation main features of ductile failures;


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Objectives: mesh-dependent

To Seek a simple, viable,

and accurate ductile crack growth numerical algorithm that can be used in both engineering computations and scientific research.











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Why is shape function discontinuous mesh-dependent

At the crack tip ?


Formulations l.jpg
Formulations mesh-dependent

  • Finite deformation thermo-mechanical system


Weak form l.jpg
Weak form mesh-dependent


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Solution Methodology mesh-dependent

  • Lagrangian Finite Deformation

  • Explicit time integration

  • Operator split for heat conduction


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Constitutive Model mesh-dependent

  • Additive decomposition in rate of deformation

  • Use the Jaumann rate to satisfy objectivity


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Numerical Simulations mesh-dependent

2D Plane strain

uniaxial tension test:

A plate with pre-notched

center crack subjected

prescribed velocity.


Example i gurson tvergaard needleman model l.jpg
Example I: mesh-dependent Gurson-Tvergaard-Needleman model



Crack surface configuration i l.jpg
Crack Surface Configuration (I) mesh-dependent

(a) t=6.0µs, (b)=14.0µs


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Crack Surface Configuration (II) mesh-dependent

(a) t = 18.0 µs (b) t=22.0µs


Size effects l.jpg
Size-effects mesh-dependent


Example ii thermal viscoplastic model l.jpg
Example II: mesh-dependent Thermal-viscoplastic model


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Crack Propagation in an Adiabatic Shearband mesh-dependent

Thermo-mechanical coupling

Thermo-elasto-viscoplastic material constitutive


Example iii johnson cook l.jpg
Example III: Johnson-Cook mesh-dependent

  • elasto-visco-plastic


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Damage Model mesh-dependent

  • Johnson-Cook

Di are parameters

D>1 implies fracture


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Operator Split for Energy Equation mesh-dependent

Adiabatic heating, solved in constitutive update:

Heat conduction, solved with momentum:


2d results cont l.jpg
2D Results (cont) mesh-dependent


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Crack Modeling-3D mesh-dependent

  • Still through cracks

  • Crack tip fiber - generalized crack tip

  • Cracks propagate from fiber to fiber

  • Damage averaged over nodes on fiber

  • Crack panels - natural extension of 2D

    • not necessarily planar!

  • Visualization by brute force 3D meshing


Crack morphology 3d l.jpg
Crack Morphology - 3D mesh-dependent

Fiber


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Visibility-3D (cont) mesh-dependent

  • Compute intersection of lines with panels

  • Two cases:

    • If panel is parallel to z-axis or x-y plane, treat as a planar quadrangle

    • otherwise, use non-linear iteration to find intersection


Line intersect plane l.jpg
Line Intersect Plane mesh-dependent

Solve for l

Plane intersection if



Parametric visibility algorithm l.jpg
Parametric Visibility Algorithm mesh-dependent

Pick l, Define:

Geometry:

Residual:

Newton-Raphson:

Convergence when


3d results cont l.jpg
3D Results (cont) mesh-dependent


3d results con t l.jpg
3D Results (con mesh-dependent ’t)



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Front Half mesh-dependent


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Inside View mesh-dependent


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Explosion of a Cylinder mesh-dependent



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Simulation of a thin cylinderical shell mesh-dependent

under thermal-mechanical loading


V i simulation of impact and penetration l.jpg
V mesh-dependent I. Simulation of Impact and penetration


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Impact and Penetration mesh-dependent

KPa

KPa

Brittle Failure Shear Localization



Slide157 l.jpg

mesh-dependent A multiple-scale constitutive law with length scale dependent cohesive model

where is the flow stress. are material constants.

 In the multi-scale formulation, the flow stress is proposed as

 Inclusion of strain rate and temperature effect (The modified Johnson-Cook model)

with

the multi-scale length scale dependent cohesive model is embedded in 0 .

 = 0.005 , T0 = 293K ,  = 0.01 , 0 = 0.002, 0.002 /sec

Localization-induced Cohesive Law


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A Demonstration of the Model mesh-dependent

Strain Rate Effect Temperature Effect


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Comparison of yield surface between proposed smooth cap model and Drucker-Prager model

Effect of compression strength

Drucker-Prager

Effect of void volume fraction

f=0.0

f=0.1

f=0.2

f=0.3

Drucker-Prager


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Problem Statement model and Drucker-Prager model

Area of interest

Geometric Parameters:

W = 50.8 cm T = 15.24 cm L =50.8 cm

d = 1.09 cm h = 1.35 cm

Material Properties:

Projectile = 8770 Kg/m3 , Target = 2300 Kg/m3

ETarget = 2058 Mpa mtarget = 90.456 Kg mprojectile = 0.011 Kg

Target = 0.3

The Projectile is modeled as a rigid body in the simulation

v = 1666 m/sec

Time step dt = 3e-7 sec

Total steps 17400

T

W

L

Details of the Projectile

d

h


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Single Projectile Movie - Stress model and Drucker-Prager model


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Single Projectile Movie - Damage model and Drucker-Prager model


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Damage Evolution model and Drucker-Prager model


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Multiple Projectiles - Stress model and Drucker-Prager model


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Conclusions model and Drucker-Prager model

  • Thank You !


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Bifurcated Energy Release Solutions model and Drucker-Prager model


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Meshfree Methods model and Drucker-Prager model


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