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### Meshfree Methods and Simulations of Material Failures

### Dynamic Shear Band Propagation with Heat Conduction

ShaofanLi

Department of Civil and Environmental Engineering,

University of California at Berkeley

Collaborators

Dr. Bo C Simonsen,

Technical University of Denmark;

Dr. Daniel C. Simkins,

University of South Florida;

Dr. Sergey N. Medyanik,

Northwestern University

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

I. Introduction of meshfree methods

Mesh vs. Meshfree

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

- 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

Note and P(0) = (1, 0, 0, …, 0) then

therefore

Compare FEM shape function with Meshfree shape function

(a) FEM ; (b) Meshfree

Compare FEM shape function with Meshfree shape function

(a) FEM ; (b) Meshfree

(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 )

1.2 A Few Virtues of Meshfree Methods

Convergence Property

as as

Reproducing Property

For That is

Non-local Interpolation (Discrete Convolution)

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

sustain computation without remeshing;

is deformation map

Example 2.1 : Compression of A Rubber Cylinder

FEMMeshfree

50% Compression

65% Compression

85% Compression

90% Compression

Example 2.2 A pinched cylindrical shell

Material Properties

Computational Parameters

Number of Particles: 30300

Time Step:

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

(b) t = 1.510-3s

(a) t = 0.510-3s

(f) t = 7.510-3s

(d) t = 4.510-3s

(e) t = 6.010-3s

Example 2.4 The snap-through of a conic shell

Material Properties

Computational Parameters

Number of Particles: 12300

Time Step:

Meshfree Methods: Element-free Galerkin (EFG)

Fleming and Belytechko [1997]

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

4. Meshfree Multiscale Computations

= Scale 0 + Scale 1 + Scale 2 + …...

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

Multiscale Analysis

Spectral Meshfree Adaptive Simulations

Multiscale Analysis

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

Total Particles: 3321

Particles used for adaptive calculation: 361

Meshfree (particles)

Zoom in

Kinematically

Admissible

Mode I

4272 particles/layer

with 3 layers in

thickness direction

Mesh-based (element distortion)

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

How to simulate curve shearband

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)

Remark: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.

Experimental Results on Multiple Shear Band(Anand and Spitzig, 1982)

250 m

0.1 mm

1 mm

Computational Results by Meshfree Method

IV: Dynamic Shear-Band Simulations

Recently, we have done some massive

parallel computations on simulations of

stress collapse inside the adiabatic shear band.

(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)).

Meshfree Simulation of Dynamic Shear Band Propagation

Experiment on impact-loaded plate

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

Failure Mode Transition

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)

HIGH-SPEED INFRARED IMAGING SYSTEM

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

Numerical Techniques Simulations

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]).

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

Comparison with Experiment at High Impact Speed

ExperimentComputation

Impact speed 29.4 m/s, C-300 steel

(Computation V = 37 m/s)

- 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.

Detailed Shear Band Structure ( temperature signature)

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

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

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)

Hot Spots in a Small Model

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

Instability of viscous flow occurs inside the shear band.

Similar Instabilities

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)

Cohesive FEM: Crack path is pre-determined or it is mesh-dependent

Difficulties and Challenges:

- No remeshing and no adaptive discretization is allowed;
- 2.Fracture criteria;
- 3.Prediction and simulation main features of ductile failures;

Objectives:

To Seek a simple, viable,

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

Why is shape function discontinuous

At the crack tip ?

Formulations

- Finite deformation thermo-mechanical system

Solution Methodology

- Lagrangian Finite Deformation
- Explicit time integration
- Operator split for heat conduction

Constitutive Model

- Additive decomposition in rate of deformation
- Use the Jaumann rate to satisfy objectivity

Numerical Simulations

2D Plane strain

uniaxial tension test:

A plate with pre-notched

center crack subjected

prescribed velocity.

Crack Surface Configuration (I)

(a) t=6.0µs, (b)=14.0µs

Crack Surface Configuration (II)

(a) t = 18.0 µs (b) t=22.0µs

Crack Propagation in an Adiabatic Shearband

Thermo-mechanical coupling

Thermo-elasto-viscoplastic material constitutive

Example III: Johnson-Cook

- elasto-visco-plastic

Operator Split for Energy Equation

Adiabatic heating, solved in constitutive update:

Heat conduction, solved with momentum:

Crack Modeling-3D

- 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

Visibility-3D (cont)

- 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

Simulation of a thin cylinderical shell

under thermal-mechanical loading

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

Strain Rate Effect Temperature Effect

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

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

Conclusions

- Thank You !

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