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


Department of Civil and Environmental Engineering,

University of California at Berkeley

Collaborators l.jpg

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

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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

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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


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


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)



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Meshfree Simulation of Strain Localization



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



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.

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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

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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]


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

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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]


Zhou-Rosakis Model ([1996])



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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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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


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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

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Macroscale Picture of Shear Bands in Rosakis Model Speed



With Heat Conduction

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Comparison with Experiment at High Impact Speed Speed


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) :



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





200 m

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2.5mm Speed


DT (K)




Field of View

1.1 x 1.1 mm




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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)

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Mesh Dependency of Band Width (Temperature)for adiabatic shear band

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Inside of the Shear Band (Temperature)


(mesh dependency)

With Heat Conduction

(resolved band width)

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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.

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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

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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 ?

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Formulations mesh-dependent

  • Finite deformation thermo-mechanical system

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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.

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Example I: mesh-dependent Gurson-Tvergaard-Needleman model

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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

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Size-effects mesh-dependent

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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

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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:

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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

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Crack Morphology - 3D mesh-dependent


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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

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Line Intersect Plane mesh-dependent

Solve for l

Plane intersection if

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Parametric Visibility Algorithm mesh-dependent

Pick l, Define:




Convergence when

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3D Results (cont) mesh-dependent

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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

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V mesh-dependent I. Simulation of Impact and penetration

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Impact and Penetration mesh-dependent



Brittle Failure Shear Localization

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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)


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


Effect of void volume fraction






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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




Details of the Projectile



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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