Tuning Johnson-Cook material model parameters for …cj82pb67g/... · me through out my life. I am unable to ﬁnd the words to express my appreciation of their support. I am unable

Tuning Johnson-Cook Material Model Parameters for Impact of High

Velocity, Micron Scale Aluminum Particles

A Thesis Presented

by

Arash Alizadeh Dehkharghani

to

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in

Mechanical Engineering

Northeastern University

Boston, Massachusetts

August 2016

NORTHEASTERN UNIVERSITYGraduate School of Engineering

Thesis Signature Page

Thesis Title: Tuning Johnson-Cook Material Model Parameters for Impact of

High Velocity, Micron Scale Aluminum Particles

Author: Arash Alizadeh Dehkharghani NUID: 001102511

Department: Mechanical and Industrial Engineering

Approved for Thesis Requirements of the Master of Science Degree

Thesis Advisor

Dr. Sinan MuftuSignature Date

Department Chair

Dr. Hanchen HuangSignature Date

Associate Dean of Graduate School:

Dr. Sara Wadia-FascettiSignature Date

To my family

ii

Contents

List of Figures v

List of Tables vii

Acknowledgments viii

Abstract of the Thesis ix

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective and Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Arrangement of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature Review 52.1 Cold Spray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Particle Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 High Strain Rate Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Split-Hopkinson Pressure Bar Test . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Dynamic tests in tensile and torsional loadings . . . . . . . . . . . . . . . 12

2.4 Ultra High Rate of Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Modeling and Finite Element Simulation of High Velocity Impact for a Single Micron-Scale Particle 163.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 FEM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Modeling and analysis technique . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Element Type and Failure Criterion . . . . . . . . . . . . . . . . . . . . . 183.2.3 Interaction and Contact Properties . . . . . . . . . . . . . . . . . . . . . . 203.2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.5 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.6 Adiabatic Shear Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.7 Effect of Using Temperature Dependent Material Properties . . . . . . . . 24

3.3 Mesh Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Laser Induced Single Particle Impact Experiments . . . . . . . . . . . . . . . . . . 28

iii

3.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Material Model for High Strain Rate Deformation of Metals 324.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Johnson-Cook Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.2 Zerilli-Armstrong Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.3 Voyiadjis-Abed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.4 Preston-Tonk-Wallace Model . . . . . . . . . . . . . . . . . . . . . . . . 354.2.5 Khan-Huang-Liang Model . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.6 Gao-Zhang Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.7 Summary of All Constitutive Models . . . . . . . . . . . . . . . . . . . . 38

4.3 Bilinear Johnson-Cook Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Optimization of Bilinear Johnson-Cook parameters 415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Optimization Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5 Estimated Particle Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.6 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Conclusion 586.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 61

A Simulation Results Using the Classic and Bilinear Johnson-Cook Material Models 70A.1 Simulation Results Using the Classic JC Model . . . . . . . . . . . . . . . . . . . 70A.2 Simulation Results Using the Bilinear JC Model . . . . . . . . . . . . . . . . . . . 75

B Table of Temperature Dependent Material Properties for Aluminum 6061 79

iv

List of Figures

1.1 Comparison of gas temperature and particle velocities in thermal spray processes andcold spray process. VPS is vacuum plasma spray and LPPS is low pressure plasmaspray [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Schematic diagram of a cold spray gun [1]. . . . . . . . . . . . . . . . . . . . . . 2

2.1 Schematic plot of correlation between particle velocity and deposition efficiency [2]. 62.2 Particle impact on a solid surface: effect of impact velocity and particle size on

features of the interaction. Regions characteristic of certain impact phenomena areshown [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Strain rate regimes and associated instrument and experimental conditions [4]. . . . 92.4 Schematic view of split-hopkinson pressure bar [4]. . . . . . . . . . . . . . . . . . 102.5 A compression split-Hopkinson pressure bar facility at Los Alamos National Lab

(Figure from [4]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Experimental method for high strain rate testing [4]. . . . . . . . . . . . . . . . . . 122.7 Schematics of dynamic tension loading [5]. . . . . . . . . . . . . . . . . . . . . . 132.8 (a) Schematics of Split Hopkinson Pressure Bar for dynamic punch shear tests, and

(b) sample holder in SHPB [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Isometric view of the developed 3D model. . . . . . . . . . . . . . . . . . . . . . 173.2 Shear stress versus nominal shear strain for a typical work-hardening material during

a torsion experiment [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 rebound velocity as a function of particle impact velocity for different failure strains.

This analysis conducted by using Copper as a material properties for both particleand substrate [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Traction separation response of interacting surfaces with a linear failure mechanism. 213.5 Temperature dependent material properties for Aluminum 6061 [9]. Properties

include a) Elastic Modulus; b) Poisson’s Ratio; c) Thermal Expansion; d) ThermalConductivity; and e) Specific Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Deformed shapes of particle using bilinear JC material model for 19.75 µm particleand impact velocity of 663 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

v

3.7 Time history analysis of particle for a) average strain; b) average strain rate; c)average temperature; d) average yield stress. Note that the particle diameter was19.75 µm and impact velocity was 663 m/s. Average values are the mean of allintegration point at each increment time. . . . . . . . . . . . . . . . . . . . . . . 27

3.8 Diagram of the optical setup for LIPIT [10]. . . . . . . . . . . . . . . . . . . . . . 293.9 particle velocity and excitation pulse energy relation [10]. . . . . . . . . . . . . . . 303.10 multiple exposure photograph of a particle with constant ∆t [11]. . . . . . . . . . 30

4.1 Comparison between the stress-strain rate behavior predicted by the mechanism-based material model and experimental data for aluminum alloy 6061-T6 [12]. . . . 39

4.2 flow stress in classic JC (left) and Bilinear JC (right) . . . . . . . . . . . . . . . . 40

5.1 Stress-Strain rate behavior in a semi log plot [13]. . . . . . . . . . . . . . . . . . . 435.2 SEM images on which the parameter fitting is based [11] and the definition of the

dimensions D1 and D2 used in calculating the ellipticity ratio Re. . . . . . . . . . 455.3 Optimization process with modified steepest descent method. . . . . . . . . . . . . 465.4 Comparison of the deformed particle by using the initial JC parameters. . . . . . . 495.5 Comparison of the deformed particles by using the optimized JC parameters. . . . 505.6 Time history of the particle’s volume in case-4. Note that the small increase in

volume is due to thermal expansion. . . . . . . . . . . . . . . . . . . . . . . . . . 525.7 Comparison of the experimental particle contours with those that are computed by

using the adjusted particle diameter DSEM and the modified JC parameters reportedin Table 5.10. Note that the JC parameters for each impact speed were different asreported in Table 5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.8 Comparison of the experimental particle contours with those that are computed byusing the adjusted particle diameter DSEM and the average value of the modified JCparameters reported in Table 5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.9 Comparison of the experimental coefficient of restitution with simulated results.Lines represents the simulation data and dots are the experiments. . . . . . . . . . 57

A.1 Comparison of the deformed shape of particle between experiment (a, b) and simula-tions (c, d, e, f) where the impact velocity. is 556 m/s . . . . . . . . . . . . . . . . 72



A.4 Strain rate contour of iso-surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.5 Comparison of the deformed shape of particle between experiment (a, b) and simula-

tions (c, d) where the impact velocity. is 556 m/s . . . . . . . . . . . . . . . . . . 76A.6 Comparison of the deformed shape of particle between experiment (a, b) and simula-

tions (c, d) where the impact velocity. is 691 m/s . . . . . . . . . . . . . . . . . . 77A.7 Comparison of the deformed shape of particle between experiment (a, b) and simula-

tions (c, d) where the impact velocity. is 859 m/s . . . . . . . . . . . . . . . . . . 78

vi

List of Tables

3.1 Temperature dependent material properties of Sapphire [14]. . . . . . . . . . . . . 243.2 material properties for Aluminum 6061 at room temperature. . . . . . . . . . . . . 253.3 Mesh convergence study for single particle impact simulation. . . . . . . . . . . . 28

4.1 Bilinear Johnson-Cook material properties for Al-6061 . . . . . . . . . . . . . . . 39

5.1 Material properties of Al-6061 [15, 9]. . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Temperature dependent material properties of Sapphire [14]. . . . . . . . . . . . . 425.3 Bilinear Johnson-Cook material properties for Al-6061 obtained by SHPB test. . . 435.4 Bilinear Johnson-Cook parameters at each attempt of the optimization process for

the case-1, where Dp = 20.75 µm, and Vp = 175 m/s. . . . . . . . . . . . . . . . . 475.5 Bilinear Johnson-Cook parameters at each attempt of the optimization process for

the case-2, where Dp = 24.4 µm, and Vp = 286 m/s. . . . . . . . . . . . . . . . . . 475.6 Bilinear Johnson-Cook parameters at each attempt of the optimization process for



the case-5, where Dp = 19.75 µm, and Vp = 663 m/s. . . . . . . . . . . . . . . . . 485.9 Bilinear Johnson-Cook parameters at each attempt of the optimization process for

the case-6, where Dp = 23.4 µm, and Vp = 699 m/s. . . . . . . . . . . . . . . . . . 485.10 Summary of the optimized values of the parameters for the bilinear JC equation . . 515.11 Diameter and volume estimates of the particles. Dr is the reported particle diameter

[11]. Vr is the particle volume based on Dr. VA is the particle volume computed byAbaqus based on Dr after meshing. VSEM is the particle volume estimated based onthe post-impact SEM image of the particle. DSEM is the particle diameter based onVSEM . DBF is the particle diameter that best fits the experimental results obtainedby using different sizes in Abaqus. E −DSEM and E −DBF are the %-errors inparticle diameters with respect to Dr. . . . . . . . . . . . . . . . . . . . . . . . . 53

A.1 Classic Johnson-Cook material properties for Al-6061 . . . . . . . . . . . . . . . . 71

vii

Acknowledgments

The current thesis is the product of contribution of many people who have been helpingme through out my life. I am unable to find the words to express my appreciation of their support.

I am deeply thankful to my supervisor, Prof. Sinan Muftu for his guidance and infinitesupport in every single step of this research. His intelligence, knowledge, and wise comments havealways lit up new ways and ideas.

I would like to express my great appreciation of the inspiring work of Prof. Jae-HwangLee and Ms. Wanting Xie from University of Massachusetts, Amherst, on the single particle impacttests. This data had a remarkable influence on my entire research.

My sincere thanks to Prof. Andrew Gouldstone, and Prof. Teiichi Ando for their de-tailed review, constructive suggestions and excellent advice during the preparation of this thesis.

The financial support of US Army (ARL W911NF-15-2-0026) which gave me the opportu-nity and encouragement to expand my research, is gratefully acknowledged.

I would like to thank my classmates in 244H and 225B Forsyth Building for all the help,guidance and friendship they have given me.

Last but not least, I wish I was able to thank my lovely family. I am grateful to my parentswho loved me with infinite generosity and provided me with the opportunity to be where I am.Without them, none of this would have been possible.

viii

Abstract of the Thesis

Tuning Johnson-Cook Material Model Parameters for Impact of High

Velocity, Micron Scale Aluminum Particles

by

Arash Alizadeh Dehkharghani

Master of Science in Mechanical and Industrial Engineering

Northeastern University, August 2016

Dr. Sinan Muftu, Adviser

Cold Spray (CS) is an additive manufacturing process which uses the extreme plasticdeformation of micron scale particles to repair surface defects. This process requires acceleratingparticles to very high velocities (200-1000 m/s) by a supersonic compressed gas jet at temperatureswell below particle’s melting point. Although many metals and alloys have been successfullyprocessed using the CS techniques, the accurate dynamic responses of individual metallic particlesrelated to the deformation characteristics are still largely unknown. Therefore the main objectiveof this research is to investigate the mechanics of single particle impact. The outcome of this studycan be used to study multi-particle impact and ultimately study the mechanics of 3D-printed metalsusing CS technology.

Numerical simulation has been used to produce the particle impact results. Simulationsshow that the material experiences very high strain rates (107-108 s−1) causing severe plasticdeformation. To conduct an accurate analysis in the simulations, the flow stress of the material shouldpredict appropriate metal behavior at that range of strain rate. In this study the Bilinear Johnson-Cookmaterial model has been used to predict the flow stress and Aluminum-6061 was chosen as theparticle’s material property. Simulations included the effects of high strain-rate (HSR) plasticity, heatgeneration and dissipation, material damage, and surface interactions in three dimensions.

High strain rate experimental results are usually done by using Split Hopkinson PressureBar (SHPB). Since these experiments have limitation on the maximum strain rate applied to thesample, there is no accurate data for the flow stress of the material at the high rates encountered inCS. Therefore, in this thesis a computational material model calibration has been performed for theBilinear JC model for the HSR applications. The optimization process uses the method of steepestdescent to find the best constants in the Bilinear JC constitutive law. The difference between ellipticity

ix

ratio of the deformed particle in simulation and experiments is used as the objective function, and theparameters of the Bilinear JC equation are modified until the objective function is satisfied.

The optimized bilinear Johnson-Cook model was used to simulate the deformed shapeof particles. The results show a very good agreement between the simulations and single particleimpact experiments. The optimized bilinear JC-model was further verified by comparing simulationresults of the particle rebound velocity and the coefficient of restitution (COR) to experimental data.The methodology developed in this thesis can be used to develop the model parameters for differentmaterials and other HSR material models.

x

Chapter 1

Introduction

1.1 Background

In recent years, research on is conducted to find materials with superior mechanical

properties to satisfy the requirements for different designs imposed by many fields and applications in

order to obtain competitive products. Some of the major modes of failure in mechanical engineering

and structural applications, such as; fatigue, fracture, wear, aging and corrosion, are very sensitive to

the properties of surfaces. These defects usually originate from outer layers and propagate inward.

This makes the surface improvement an effective approach, in increasing the durability and life of

the components. In the following, two distinct methods of coating are discussed.

In last decade, thermal spray technologies have evolved from fairly crude processes that

were relatively difficult to control into increasingly precise tools, by taking into account the properties

of the deposited material and coating requirements. The limitation of some coating types such as;

High Velocity Oxygen Fuel (HVOF) process, Plasma Spray Systems, and thermal spray coating

technology have been overcome by the cold gas spray coating technology. Figure 1.1 shows a

comparison of different coating methods by considering the temperature and velocity of the sprayed

particles [1]. In this diagram, we see that the cold spray requires much lower process temperature

with respect to other coating types. The cold spray coating technology was originally developed

in the mid-1980s at the Theoretical and Applied Mechanics division of the Russian Academy of

Sciences Institute in Novosibirisk by Dr. Anatolii Papyrin and his colleagues [16]. They successfully

deposited a wide range of pure metals, metal alloys, and composites onto a variety of substrate

materials, and demonstrated the feasibility of cold spray process for various applications.

The coating process known as Cold Spray is a material deposition process in which

1

CHAPTER 1. INTRODUCTION

Figure 1.1: Comparison of gas temperature and particle velocities in thermal spray processes and

cold spray process. VPS is vacuum plasma spray and LPPS is low pressure plasma spray [1].

relatively small particles (ranging in size from approximately 5 µm to 100 µm in diameter) in solid

state are accelerated to high velocities (typically 200-1000 m/s), and subsequently form a deposit on

a metallic substrate. Currently, Cold Spray is an advanced surface repair process, for metal surfaces

that does not induce thermal stresses in the deposited material. This makes CS uniquely compatible

with many aerospace materials. Unlike other repair processes that use heat to fuse materials together,

Cold Spray uses a compressed gas to accelerate metal powders to supersonic speeds and plastic

deformation to attach to the substrate. The resulting surface is comparable to the original substrate in

terms of the material strength.

Figure 1.2: Schematic diagram of a cold spray gun [1].

Bonding of the particles in this process is presumed to be the result of extensive plastic

deformation due to the high kinetic energy upon impact; therefore, the velocity of the particles play

the most important role in material deposition. During the process, powders are accelerated by

injection into a high velocity stream of gas which is generated through a converging-diverging nozzle.

The schematic diagram of the cold spray equipment is shown in Figure 1.2. As the process continues,

2


the particles impact the substrate and form bonds with it, resulting in a uniform, almost porosity-free

coating with high bonding strength. Note that the particles remain in the solid state and are relatively

cold, so the cohesion of the deposited material, is accomplished in solid state.

The applications of cold spray coating is mostly on the surface treatment. Wear resistance

is one of these applications. Wolfe et al. [17] used cold spray coating successfully in order to

coat 4140 alloy for wear resistance applications. The obtained results are presented in detail in [1].

Other papers [18, 19, 20] also show the potential of using cold spray coating in wear resistance

applications. Corrosion protection is also another application of Cold Spray. The advantages of

cold spray technology for deposition of corrosion-resisting layers are evident, however, a major

consideration in the selection of cold spray processing for a given application is its cost relative to

alternatives [1]. Another application of Cold Spray is to repair the damaged component. The US

Army Research Laboratory (ARL) Center for Cold Spray Technology [21] has developed a cold spray

process to reclaim magnesium components that show significant improvement over existing methods

and is in the process of qualification for use on rotor-craft. The cold spray repair has been shown to

have superior performance in the tests conducted up to date. It can be incorporated into production,

and has been modified for field repair [1]. The feasibility of using the cold spray process to repair

non-structural magnesium aircraft components, has demonstrated satisfactory results obtained from

adhesion, corrosion testing, and micro-structural analysis. In these tests, cold spray coating was not

pulled off the substrate and the coating did not fail cohesively [1]. It should be noted that cold spray

coating can be considered a young technology with respect to other types of thermal coating. The

applications of this technique are gradually increasing in the surface treatment industry. This section

has only mentioned some of the important current applications of cold spray technology, which is

used all over the world.

1.2 Objective and Problem Description

Not only many constitutive relations are not capable of predicting the behavior of material

at ultra high strain rates, but also there are no reliable experimental data to fit them onto a material

model. Therefore, in this study the main focus is to achieve a material model that can anticipate the

mechanical behavior of material in very high strain rate applications such as Cold Spray. In the Cold

Spray process the gas carries micron-scale particles with a supersonic velocity and they impact the

substrate. The process includes very large strain rates, since the duration of the process is in the order

of nano-seconds and the deformation is really large. This constitutive model, needs to be precise in a

3


wide range of impact velocities (200 m/s - 1000 m/s) and particle diameters (15 µm - 50 µm).

1.3 Arrangement of the Thesis

Material presented in this thesis is divided into six chapters. This chapter (Chapter 1)

provided a brief background on the Cold Spray technique and its applications followed by the

motivations and objectives of this research. In Chapter 2, a survey of literature of the cold spray and

the high strain rate experiments are presented. In Chapter 3, numerical simulation of single particle

impact is explained along with a comprehensive study of the techniques used to model the impact

process in Abaqus finite element software. The effect of using temperature dependent material

properties in simulations is also discussed. In Chapter 4, simulation results using classic Johnson-

Cook material model and Bilinear Johnson-Cook material model are illustrated. In Chapter 5, the

constants for Bilinear JC model is optimized to find the exact deformation of single particle impact.

In this chapter, the results of optimized bilinear JC model is also compared to the experimental data.

Finally, this thesis finishes by explaining the summary, conclusion and future works in Chapter 6.

4

Chapter 2

Literature Review

2.1 Cold Spray

Cold spray is a technique that accelerates powder particles towards a substrate using a

supersonic compressed gas jet, as mentioned in Chapter 1. Upon impact the particles experience

extreme and rapid plastic deformation. They stack up to form a solid state deposit that is well bonded

to the substrate. Particles adhere to the substrate only if their impact velocity is above a critical

value. This value varies between 500 and 900 m/s, depending on the material and the particle size.

Therefore the particle velocity prior to impact [1, 22] is the main parameter that determines what

phenomenon occurs upon the impact of spray particles, whether it would be the deposition of the

particle or the erosion of the substrate. Figure 2.1 is a schematic plot which shows the correlation

between particle velocity and deposition efficiency. Critical velocity for a given powder is defined as

the velocity that an individual particle must obtain in order to deposit after impacting the substrate [1].

In other words, for a given material, the critical velocity is the velocity at which the transition from

erosion of the substrate to deposition of the particle takes place. Experimental investigations also

reveal that successful bonding is achieved only above this velocity. However, this value is associated

with the temperature and thermo-mechanical properties of the sprayed material [22, 16], as well as

the characteristics of the substrate [23, 24]. Temperature of the particles have also an effect on both

critical velocity and deposition efficiency [1].

The coating process, depends on using high kinetic energy, provided by the supersonic

gas flow. The high velocity of the particles is obtained in a divergent-convergent nozzle. The gas

temperature along with many other process parameters, play considerable role in coating quality

from increasing deposition efficiency to the porosity control of the final deposition.

5

CHAPTER 2. LITERATURE REVIEW

be explained by the occurrence of local shear instabilities at par-ticle-substrate and particle-particle interfaces due to thermalsoftening, as first shown by Assadi et al. (Ref 18). Later workwas aimed at refining the procedures for calculation and its ap-plications to different spray materials (Ref 19, 20). Based on theconcept of bonding by shear instabilities and by combining theresults from modeling and experimental investigations, analyti-cal expressions were recently developed to predict the ranges ofoptimum spray conditions with respect to the mechanical prop-erties of spray materials, spray particle sizes, and particle tem-peratures (Ref 21).

Another focus in cold spraying is the exploration of the fea-sibility of different feedstock materials. As shown in the latestproceedings of the International Thermal Spray Conference(ITSC) conferences, quite a number of investigations concernprocess development for standard engineering materials anddeal with different aspects to customize cold spraying for vari-ous spray powders envisaging short-term applications. To dem-onstrate the versatility of the process, a couple of such examplesare also given in the present article. With respect to powder de-velopments, significant attention is being paid to new materialtypes. It has been shown that cold spraying is suitable to processcoatings on the basis of nanocrystalline alloys (Ref 22),nanocrystalline composites (Ref 23, 24), or bulk metallic glasses(Ref 25), just to mention a couple.

This article reports on the current “state of the art” in cold-spray process development. Taking the example of copper as thespray material, the influences of optimized spray conditions oncoating quality are reported. With respect to the above-mentioned understanding of the process, special emphasis isgiven to current investigations to tune cold-spray parameters todifferent applications. The given examples of coating micro-structures of various materials should demonstrate the versatilityof the process.

2. Procedures

2.1 Determination of Critical Velocities

For a variety of spray materials, the critical velocity for suc-cessful impacts was determined by correlating the deposition

efficiency with the respective spray particle size distributions. Arange of various deposition efficiencies were obtained by usingdifferent nozzle geometries (Ref 7) or by varying process param-eters such as gas temperature and pressure (Ref 12). The presentinvestigation is based on a method by which the effects of gastemperature are minimized by using the same parameter settingsand different nozzle types (Ref 7). For each set of process con-ditions, particle velocity and temperature were calculated by us-ing CFD. This approach assumes that the mechanical propertiesand the extent of oxidation are similar and independent of par-ticle size for the particular feedstock powder.

2.2 Investigations of Coating Microstructures andBond Strength

To evaluate coating quality with respect to porosity and theamount of well-bonded particle-particle interfaces, cross sec-tions were subsequently prepared and investigated by opticalmicroscopy. The microstructural features of embedded copperparticles and particle-particle interfaces were revealed bychemical etching with a solution of 25 mL of H2O, 25 mL ofammonia (25% in H2O), and 5 ml of hydrogen peroxide (3% inH2O). A selection of coating cross sections was further investi-gated by scanning electron microscopy.

Bond strength measurements were performed in a tensilemode according to the European standard EN582 (comparableto ASTM standard C 633-01; Ref 26). Before cold spraying,steel substrates were degreased and grit blasted. To avoid thethermal influences by preparation, the gluing of as-sprayedbond-strength samples to counterbodies was performed at 60 °Cby using adhesive DP 490 (3M, St. Paul, MN). The oxygen con-tents of the initial feedstock and of the coatings were measuredby using a commercial analyzer of the model TC300 type manu-factured by LECO (St. Joseph, MI). The coatings were mechani-cally detached from the substrate by bending the substrate sideof the flat coupons over a sharp edge with an radius of 5 mm,using a mechanical testing machine and a procedure similar tothat in 3-point bending tests.

2.3 Nozzle Developments

By launching cold spraying, nozzle designs of a slightlytrumpet-like shape were originally propagated (Ref 3, 5). Refer-ring to radial symmetry, the smallest cross section of such a“standard” nozzle of a slightly trumpet-like shape has a width of2.7 mm, a length of the diverging section of 70 mm, and an ex-pansion ratio of 8.8. Using the numerical codes of fluid dynam-ics, gas and particle flow for certain process conditions werecalculated (Ref 7). The numerical codes of fluid dynamics, inparticular the method of characteristics (MOC), were also usedto develop new nozzle designs that allow a more homogeneousparticle acceleration (Ref 27).

3. Results and Discussion

3.1 Evaluation of Critical Velocities

In the high-velocity gas stream, smaller particles are acceler-ated to significantly higher velocities than large particles (Ref 7,14). Only very small particles are subject to deceleration in the

Fig. 3 Schematic of the correlation between particle velocity and de-position efficiency. The transition between abrasion in the low-velocityrange and deposition for high velocities defines the critical velocity.Vcrit, critical velocity.

Journal of Thermal Spray Technology Volume 15(2) June 2006—225

Peer

Review

ed

Figure 2.1: Schematic plot of correlation between particle velocity and deposition efficiency [2].

2.2 Particle Impact

Figure 2.2 shows the classification of impact phenomenon with respect to particle diameter

and impact velocity [3]. Particles impacting onto the surface of a solid body can rebound from

the surface, stick to the surface or penetrate into the bulk. Often, the impact of a particle on a

surface causes a deformation or destruction of both the particle and the substrate. In the following,

characteristic of each region is discussed briefly.

At low velocity impacts (<100 m/s), small particles (<1.0 µm) can stick to the substrate

after impact. In these cases (number 6 and 7 in Figure 2.2), adhesion is governed by the van der

Waals and electrostatic forces, [25]. Sticking happens when the velocity of impacted particle is

between two critical velocities. If the impact velocity is below the first critical velocity, particles

will attach to the surface. On the other hand, if the impact velocity is above the second critical

velocity, impacted particles will rebound. In the intermediate range there is a probability of sticking

of particles, which depends on the impact velocity.

In region number 8 (Figure 2.2) adhesion can also happen during a low velocity impact

of larger particles (10 < dp < 100 µm) [26]. The rebound velocity of such particles decreases with

decreasing particle size.

Larger particles (1–10 mm) impacting with velocities up to about 40 m/s on a plate typically

rebound from the surface without breaking but they leave plastic deformation on both particle and

impacted surface. The size of the crater increases with increasing impact velocity. In order to predict

velocities and angles of rebound as well as the size of the plastic prints, Hertzian theory can be

6


S.V. Klinkov et al. / Aerospace Science and Technology 9 (2005) 582–591 583

Nomenclature

c speed of sounddp particle diameterFa particle-surface adhesion force acting on a particle during contactf ballistic limit (cf. Eq. (3))f0 coefficient in the relation between adhesive force and contact radius (an adhesion force per unit length, cf. Eq. (1))ks coefficient in the relation between the impact pressure and the shock velocity (cf. Eq. (2))pshock initial pressure in shocked region at particle-surface contact area near the axisrc radius of the contact area between particle and surfacetc contact timevp impact velocity of a particleρp density of particles

the erosion of optical surfaces and domes by particle im-pacts is of a great concern. Furthermore, disregarding thecold spray phenomenon can result in deviations of the dragforce and heat exchange and, consequently, failure. Here,characteristic impact velocities are similar to those in coldsprays, or smaller.

In spite of the fact that impact phenomena have beenstudied for a long time, the discovery of the cold spray phe-nomenon was unexpected and there is as yet no completetheory explaining the phenomenon of cold spray deposition.It is therefore time to consider more closely a circle of phe-nomena accompanying impact interactions of bodies and toshow how the main features depend on particular parametersof an impact process. In the following different impact phe-nomena are reviewed in the context of their relationship tothe cold spray phenomenon in order to gain a better under-standing of the latter. Thereafter, characteristics of impactsoccurring under conditions of cold sprays, are discussed indetail.

2. Phenomena characteristic of the impact of particlesonto plane solid surfaces

Particles impinging onto the surface of a solid body canbe reflected off the surface, stick to the surface or pen-etrate into the bulk. Often, the impact of a particle on asurface causes a deformation or destruction of both, the par-ticle and the solid body. In the following, different resultsof particle impacts on plane surfaces will be classified. Ifnot noted otherwise spherical particles and normal impactsare considered. As will be explained later scaling is difficult.Therefore, the classification is based on two important di-mensional parameters describing processes occurring duringimpacts: impact velocityvp and diameterdp of impingingparticles (the diameter will also be called the size of a par-ticle). Regions characteristic of certain impact phenomenahave been identified by studying results reported in the lit-erature. These regions are displayed in Fig. 1 and will bediscussed in the following sections.

Fig. 1. Particle impact on a solid surface: Influence of impact velocity andparticle size on features of the interaction. Regions characteristic of certainimpact phenomena are shown. For details see text.

2.1. Low-velocity impacts of particles

At low impact velocities (vp ≈ 1–100 m/s) small parti-cles (dp ≈ 0.1–1.0 µm) can stick to the surface after impact.Adhesion is governed by van der Waals and electrostaticforces, e.g. [24,50]. This is typical of filters. Correspondingregions that have been studied in the literature are denotedby the numbers 6 and 7 in Fig. 1.

Sticking is characterized by two critical velocities. If theimpact velocity is below the first critical velocity, particlesare collected by the surface. If the impact velocity is abovethe second critical velocity, impinging particles rebound. Inthe intermediate range there is a probability of sticking (orrebound) of particles, which depends on the impact velocity.

The probability of sticking has been determined exper-imentally, for example, for silica spheres (dp = O(1 µm))impinging on a quartz crystal (cf. Fig. 2, after [43]). In thisfigure the two critical velocities are of the order of 1 m/s andabout 10 m/s, respectively. The two lines represent an upperand lower limit (i.e. standard deviation) of sticking probabil-ities. The true value of the probability lies between the twocurves.

Figure 2.2: Particle impact on a solid surface: effect of impact velocity and particle size on features

of the interaction. Regions characteristic of certain impact phenomena are shown [3].

applied [27].

The impact of large particles (1–10 mm) at higher velocities (50–3000 m/s) illustrated in

Figure 2.2 as ”Ballistic”. These particles have high kinetic energy, therefore the amount of plastic

strain and strain rate considerably increases during the impact. As a result of that, both material

hardening and heat generation happen in the material. Heating consequently results in a thermal

softening. Thus, in a correct simulation it is necessary to take into account a competition between

processes of dynamic hardening and thermal softening.

At impact velocities greater than 2000–3000 m/s stresses arising in bodies on impact,

considerably exceed the yield point of materials. Under these conditions phase transformation

happens and solids behave like liquids. Locally, the flow velocities are comparable to the sound

speed. This is connected with the appearance of strong shock waves. Hypervelocity impacts of

particles on semi-infinite bodies [28, 29], and on plates [30, 31] have been studied extensively. The

corresponding regions have been depicted by number 2 in Figure 2.2. Hypervelocity impacts of

particles of intermediate size [32] and of micro-particles [33] are labeled with number 3,4 and 5.

Experiments done by Klinkov et al., [3] shows some penetrations when particles accelerated

to velocities of about 1–3 km/s by using hollow-charge explosions. They observed that some particles,

7


in the range of 100 µm penetrate far into the body. Penetration distances were equal to 103–104 times

greater than diameters of the particle. This effect is called super-deep penetration which is indicated

by SDP in Figure 2.2. After impact, the material of the body collapses and therefore contains a

significant level of plastic deformation [34]. Note that in the case of 1–10 mm particles impacting a

surface at similar velocities, the depth of penetration is typically about 4–5 times the particle diameter

[35].

At moderate and low impact velocities (5–300 m/s) a repeated impact of intermediate-sized

particles (30–500 µm) on the same part of a surface, results in an erosion and destruction of the

surface. In brittle materials erosion happens when the particle is impacting on normal direction

[36, 37], however, in ductile material, the highest rate of erosion observed when the impact happens

on oblique angle [38, 39]. Klinkov et al., [3] explained that ductile materials is worn out by processes

of ductile cutting and plowing. This results in the formation of raised lips of material that are

vulnerable to removal by subsequent impacts. Nevertheless, brittle materials are worn out by a

process of fracture that is more intensive when the impact is close to normal.

Finally, impacts of micron scale particles on surfaces at velocities of approximately

200–1200 m/s, correspond to the region denoted as cold spray in Figure 2.2. Under impact conditions

corresponding with the cold spray region, particles can form strong bonding with the surface after

impact. The cold spray phenomenon is used for coating surfaces. In these applications, particles

are accelerated by a supersonic gas flow. Hence, in this process the impact velocity depends on the

temperature and pressure of the gas and also the gas type in the reservoir ahead of the nozzle.

2.3 High Strain Rate Experiments

When it comes to simulating a physical phenomenon, mechanical and thermal behavior

of a material plays a significant role, and should be determined as accurately as possible. For

example, in a creep test the variation of temperature in a process is really important or, in a crash

test the hardening and rate of applying the load are substantial information. In this section, the

dynamic behavior of a material and how this behavior can be distinguished from its static behavior

is discussed. First, the history of Split-Hopkinson Pressure Bar (SHPB) and the new developments

in this experiment are explained. In the second part, the SHPB apparatus and determination of the

dynamic material behavior are illustrated. In the next part, tensile and torsional dynamic loading

experiment based on the concept of SHPB is described. Finally, other types of dynamic experiment

and limitation of these kind of tests are briefly discussed.

8


Introduction to High Strain Rate TestingSia Nemat-Nasser, University of California, San Diego

HIGH STRAIN RATE TESTING is impor-tant for many engineering structural applica-tions and metalworking operations. In struc-tural applications, various components must bedesigned to function over a broad range ofstrain rates and temperatures. In metalworkingoperations, materials undergo large amounts ofstrains at various temperatures and strain rates.The constituent materials must, therefore, becharacterized at the strain rates and tempera-tures of the intended application. Conventionalservohydraulic machines are generally used fortesting at quasi-static strain rates of 1 s– 1 or less(Fig 1). With special design, it is possible to at-tain greater strain rates, up to about 100 s– 1,with conventional load frames. For higherstrain rates, other test methods are required. Ta-ble 1 summarizes various methods in terms ofthe ranges of the strain rates that they canachieve.The articles in this Section describe various

methods for high strain rate testing. Severalmethods have been developed, starting with thepioneering work of John Hopkinson (Ref 1)

and his son, Bertram Hopkinson (Ref 2, 3).Based on these contributions and also on an im-portant paper by Davies (Ref 4), Kolsky (Ref 5)invented the split-Hopkinson pressure bar,which allows the deformation of a sample of aductile material at a high strain rate, whilemaintaining a uniform uniaxial state of stresswithin the sample. The basic concept of theKolsky apparatus involves a test samplesandwiched between an input and output bar, asdescribed in detail in the article “ ClassicSplit-Hopkinson Pressure Bar Testing” in thisSection. This technique provides a capability tomeasure the stress-strain response of ductilematerials at a high strain rate, usually betweenapproximately 50 s– 1 and 104 s– 1, depending onthe sample size, over the entire stress-straincurve. Strains exceeding 100% can be achievedwith the Hopkinson bar method. The maximumstrain rate that can be attained in a Hopkinsonbar varies inversely with the length of the testspecimen. The maximum strain rate is also lim-ited by the elastic limit of the Hopkinson barsthat are used to transmit the stress pulse to the

test sample. These basic factors of theHopkinson bar method and some specializedcompression and tension tests are discussed inthe article “ High Strain Rate Tension and Com-pression Tests” in this Section. An overview ofshear test methods (other than the torsionalKolsky bar method) is also provided in the arti-cle “ High Strain Rate Shear Testing.”The most important characteristic of Kolsky’s

split-Hopkinson compression apparatus is thatit allows high strain rate deformation while thesample is, in fact, in dynamic equilibrium, thatis, the stress gradient is essentially zero alongthe sample. It is thus possible to develop theuniaxial stress-strain response of many materi-als at a variety of strain rates. Because the re-sponse of most materials depends on both thestrain rate and the temperature, the techniqueallows developing constitutive relations thatexpress the uniaxial stress to the correspondingstrain rate and temperature. From such results,one is able to produce experimentally based,three-dimensional constitutive models for nu-merous materials.The series of articles on split-Hopkinson com-

pression testing begins with the article “ Classic

Table 1 Experimental methods for highstrain rate testing

Applicable strain rate, s–1 Testing technique

Compression tests<0.1 Conventional load frames0.1– 100 Special servohydraulic frames0.1– 500 Cam plastometer and drop test200– 104 Hopkinson (Kolsky) bar in

compression103– 105 Taylor impact test

Tension tests<0.1 Conventional load frames0.1– 100 Special servohydraulic frames100– 103 Hopkinson (Kolsky) bar in tension104 Expanding ring>105 Flyer plate

Shear andmultiaxial tests<0.1 Conventional shear tests0.1– 100 Special servohydraulic frames10– 103 Torsional impact100– 104 Hopkinson (Kolsky) bar in torsion103– 104 Double-notch shear and punch104– 107 Pressure-shear plate impact

Intermediatestrain rates

0

Increasing stress levels

Uniaxial strain &simple shear

Uniaxial & shear stress

Hopkinsontechniques

Light gas gunor

explosivelydriven

plate impact

Servohydraulic &screw machines

Very highstrain rates

106

10–6

10–8

10–4

10–2

100

102

104

Creep Quasi-staticHigh strain

rates

Constant loador

stress machine

Specialservohydraulic

machines

Shockloading

Dynamicconsiderations

in testing

Strain versus timeor

creep raterecorded

Constantstrain rate

tests

Constantstrain rate

tests

Uniaxialstress &torsion

tests

Uniaxialstrain &

shear tests

Inertia forces neglected

Isothermal

Inertia forces important

Adiabatic/quasi-isothermal

Strain rate (s–1)

Fig. 1 Strain rate regimes and associated instruments and experimental conditions

ASM Handbook, Volume 8: Mechanical Testing and EvaluationH. Kuhn, D. Medlin, editors, p427–428DOI: 10.1361/asmhba0003293

Copyright © 2000 ASM International®All rights reserved.

www.asminternational.org

Figure 2.3: Strain rate regimes and associated instrument and experimental conditions [4].

The evolution of current Split-Hopkinson pressure bar test for conducting high strain

rate testing of materials, goes back to 1872 where John Hopkinson was investigating the effect

of impulsive forces on the iron wires [40, 41]. Three decades later, his son, Bertram Hopkinson

suggested the Hopkinson bar experiment while he was investigating the stress pulse propagation

in metal bars [42]. In these experiments he determined how to measure the pressure produced by

an explosive or impact of bullets. Of course the limitation of equipments on that era affected the

accuracy of experiments. Davis [43] in 1948 used an electronic device to record wave propagation

and improved the measurement technique. He also was the first one to find a way to measure the axial

and radial strain in the Hopkinson pressure bar experiments. One year later, Kolsky [44] modified

the mechanism of the experiment. He divided the bar into two (incident bar and transmitted bar) and

placed the specimen between these two bars. That’s why the name of the experiment changed to the

Split Hopkinson Pressure Bar (SHPB). He also obtain the stress-strain relations of materials such as

rubbers, plastics, polythene, Polymethyl methacrylate (PMMA), copper, and lead under dynamic

loading conditions.

After that, the concept of this dynamic test did not change and scientists improved the

accuracy of the experiments by minimizing the friction and radial inertia effects to produce a uniform

stress in the specimen. They also expanded the experiment to different materials and various kind

9


of loading types such as tension, torsion, compression followed by torsion, torsion followed by

compression, tri-axial loading, and dynamic indentation. These new developments allow us to reach

more than 100% of total strain in SHPB test [45]. However, the maximum strain rate that can be

found in a test varies inversely with the length of the specimen and can go up to 104s−1 strain rate

by using SHPB test. Figure 2.3 illustrates an appropriate technique and experiment for each range of

strain rate.

2.3.1 Split-Hopkinson Pressure Bar Test

A modern SHPB apparatus consists of two symmetrical bars (transmitted bar and incident

bar), bearing and fixtures for alignments, a device to accelerate the striker bar, strain gauges to

measure the stress wave propagation and a computer to record and analyze the data (Figure 2.4).

strength (Ref 25– 29) or the fracture toughness(Ref 30) of an impact-loaded material. The ba-sic theory of how to reduce the pressure bardata based upon one-dimensional stress waveanalysis, as presented in the theory of the split-Hopkinson pressure bar section, is common toall three loading stress states. Of the differentHopkinson bar techniques (i.e., compression,tension, and torsion) the compression bar re-mains the most readily analyzed and least com-plex method to achieve a uniform high-ratestress state. In addition, the compression baruses simple right-regular solid samples. Detailsof the dynamic loading of materials in tensionusing either the tensile split-Hopkinson pressurebar or expanding ring test are compared andcontrasted with the compression Hopkinson bartechnique discussed in the section “ Stress-StateEquilibrium during Split-Hopkinson PressureBar Testing” in this article.

An alternate method of probing the mechani-cal behavior of materials at high strain rates, ofthe order of 103 s−1, is the Taylor rod impacttest. This technique, named after G.I. Taylor(Ref 31), who developed the test, entails firinga solid cylinder of the material of interestagainst a rigid target. The deformation inducedin the rod due to the impact in the Taylor testshortens the rod as radial flow occurs at the im-pact surface. The fractional change in the rodlength can then, by assuming one-dimensionalrigid-plastic analysis, be related to the dynamicyield strength. By measuring the overall lengthof the impacted cylinder and the length of theundeformed (rear) section of the projectile, thedynamic yield stress of the material can be cal-culated (Ref 31). The Taylor test technique of-fers an apparently simplistic method to ascertaininformation concerning the dynamic strengthproperties of a material. However, this test rep-resents an integrated test rather than a uniqueexperiment with a uniform stress state or strainrate, as does the split-Hopkinson pressure bar.Accordingly, the Taylor test has been usedmost prevalently as a validation experiment inconcert with two-dimensional finite- elementcalculations.

This article describes the techniques involvedin measuring the high-strain-rate stress-strain

response of materials using a split-Hopkinsonpressure bar, hereafter abbreviated as SHPB(Ref 18). The focus of this article is on the gen-eralized techniques applicable to all SHPBs,whether compressive, tensile, or torsion. Em-phasis is given to the methods of collecting andanalyzing compressive high-rate mechanical prop-erty data and a discussion of the critical experi-mental variables that must be controlled toyield valid and reproducible high-strain-ratestress-strain data. Comparisons and contrasts tothe differences invoked when using a tensileHopkinson bar in terms of loading technique,sample design, and stress-state stability also arediscussed.

Principles of theSplit-Hopkinson Pressure Bar

While there is no universal standard designfor SHPB test apparatus, all facilities sharecommon design elements. A compressionHopkinson bar test apparatus consists of thefollowing:

• Two long, symmetrical bars• Bearing and alignment fixtures to allow the

bars and striking projectile to move freelywhile retaining precise axial alignment

• Compressed gas launcher/gun tube or alter-nate propulsion device for accelerating aprojectile, termed the striker bar, to producea controlled compressive pulse in the inci-dent bar

• Strain gages mounted on both bars to mea-sure the stress-wave propagation in the bars

• Associated instrumentation and data acqui-sition system to control, record, and analyzethe stress-wave data in the bars (Ref 18)

In a compression split-Hopkinson pressurebar, a sample is sandwiched between an elas-tic incident and a transmitted bar (Fig. 1). Theterms incident/input and transmitted/outputare used interchangeably throughout this ar-ticle to describe the two pressure bars used inthe SHPB. The elastic displacements mea-

sured in these bars are in turn used to determinethe stress-strain conditions at each end of thesample.

The bars used in a split-Hopkinson bar setupare traditionally constructed from a high-strengthstructural metal, AISI-SAE 4340 steel, maragingsteel, or a nickel alloy such as Inconel. Suchconstruction is used because the yield strengthof the selected pressure bar material deter-mines the maximum stress attainable within thedeforming specimen given that the pressurebars must remain elastic. Inconel bars havebeen previously used for elevated-temperatureHopkinson bar testing because this alloy’s elas-tic properties are essentially invariant up to800 °C (Ref 3). Because a lower-modulus ma-terial increases the signal-to-noise level, the se-lection of a bar material with lower strengthand lower elastic modulus material for the barsis sometimes desirable to facilitate high-resolu-tion dynamic testing of low-strength materialssuch as polymers or foams. Researchers haveselected bar materials possessing a range ofelastic stiffnesses from maraging steel (210GPa) to titanium (110 GPa) to aluminum (90GPa) to magnesium (45 GPa) (Ref 5, 32), andfinally, to polymer bars (<20 GPa) (Ref 17,33– 35). Alternately, the signal-to-noise of aHopkinson bar used to test polymeric materialscan be increased using a hollow tubular trans-mitted pressure bar (Ref 36). While this tech-nique can yield increased transmitted wavemeasurement sensitivity, the absolute resolu-tion of the sample stress-strain data for poly-meric materials must still address the elasticwave dispersion in the tubular-transmitted bar(Ref 37, 38).

The length, l, and diameter, d, of the pressurebars are chosen to meet a number of criteria fortest validity as well as the maximum strain rateand strain level desired in the sample. Thelength of the pressure bars must first ensureone-dimensional wave propagation for a givenpulse length; for experimental measurementson most engineering materials, this propagationrequires approximately 10 bar diameters. Toreadily allow separation of the incident and re-flected waves for data reduction, each barshould exceed a length-to-diameter (L/D) ratioof ~20. In addition, the maximum strain ratedesired will influence the selection of the bardiameter because the highest strain-rate testsrequire the smallest diameter pressure bars (thisaspect of bar design is discussed in a later sec-tion). The third consideration affecting the se-lection of the bar length is the amount of totalstrain desired to be imparted into the specimen;the absolute magnitude of this strain is relatedto the length of the incident wave. The pressurebar must be at least twice as long as the incidentwave if the incident and reflected waves are tobe recorded without interference. In addition,because the bars must remain elastic during thetest, the displacement and velocity of the barinterface between the sample and the bar can beaccurately determined. Depending on the sam-ple size, for strains >30% it may be necessaryfor the split-Hopkinson bars to have an L/D

Classic Split-Hopkinson Pressure Bar Testing / 463

Air

Fig. 1 Schematic of a compression split-Hopkinson pressure bar

Figure 2.4: Schematic view of split-hopkinson pressure bar [4].

In a compression test, samples are placed in between the incident bar and the transmitted

bar. The elastic displacements are measured in these two bars to determine the stress-strain relation

in the sample. These bars are constructed from a steel which has a high yield strength. The maximum

attainable pressure depends on the yield strength of material. The experiments are only valid if the

deformation of these two bars are elastic. It is really important to choose an appropriate length and

diameter for the pressure bars to ensure the validity of the test, and to reach the maximum amount of

desirable strain rate and also find the maximum strain in the sample. To do that, first, the stress wave

10


should propagate in just one-dimension and to make that happen the length to diameter ratio should

be greater than 20. Second, to reach higher strain rates, the diameter of the pressure bars should be

small enough. Finally, to retain the desirable amount of total strain the length of the bars should be

long. For instance to reach 30% of total strain, the length over diameter ratio need to be 100 or more

[4].

464 / High Strain Rate Testing

ratio of 100 or more (Ref 3). There are similarrequirements for bar L/D ratios to allow waveseparation for compression, tensile, and torsionHopkinson bars.

For proper Hopkinson bar operation, the barsmust be physically straight, free to move with-out binding, and carefully mounted to ensureoptimal axial alignment. Precision bar align-ment is required for both uniform and one-dimensional wave propagation within the pres-sure bars as well as for uniaxial compression

within the specimen during loading. Bar align-ment cannot be forced by overconstraining orforceful clamping of curved pressure bars in anattempt to straighten them because this clamp-ing violates the boundary conditions for one-di-mensional wave propagation in an infinite cy-lindrical solid. Lack of free movement of thebars will lead to additional noise on the waveforms measured on the pressure bars. Bar mo-tion must not be impeded by the mountingbushings but rather must remain free to readily

move along the bar axis. Accordingly, it is essen-tial to apply precise dimensional specificationsduring construction and assembly. Pressurebars are often centerless ground along theirlength to achieve the uniform diameter andstraightness required. In typical bar installa-tions, as schematically shown in Fig. 1, the pres-sure bars are mounted to a common rigid baseto provide a rigid and straight mounting plat-form. Construction of the Hopkinson pressurebar facility, compression, tension, or torsion,on an optical rail beam rigidly attached to anI-beam, can be used to facilitate reproduciblealignment. Figure 2 shows one of the compres-sion Hopkinson bar facilities at Los Alamos Na-tional Laboratory, where an optical rail is usedto maintain accurate pressure bar alignment. In-dividual mounting brackets or stanchions withslip bearings through which the bars pass aretypically spaced every 200 to 300 mm (8 to 12in.), depending on the bar diameter and stiff-ness. Mounting brackets are generally designedso that they can be individually translated toadjust bar alignment within each stanchion.

The most common method of generating anincident wave in the input bar is to propel astriker bar to impact the end of the incidentbar. The striker bar is normally fabricatedfrom the same material and is of the same di-ameter as the pressure bars. The length and ve-locity of the striker bar are chosen to producethe desired total strain and strain rate withinthe test specimen. While elastic waves canalso be generated in an incident bar throughthe adjacent detonation of explosives at thefree end of the incident bar, as Hopkinson did(Ref 7), it is more difficult to ensure a one-di-mensional excitation within the incident bar bydirect explosive loading.

The impact of a striker bar on the free end ofthe incident bar develops a longitudinal com-pressive incident wave in this bar, designatedεi, as denoted in Fig. 3. Once this wave reachesthe bar-specimen interface, a part of the pulse,designated εr, is reflected while the remainderof the stress pulse passes through the specimenand, upon entering the output bar, is termed thetransmitted wave, εt. The time of passage andmagnitude of these three elastic pulses throughthe incident and transmitted bars are recordedby strain gages normally cemented at the mid-point positions along the length of the two pres-sure bars. Figure 3 shows an illustration of thestrain-gage data measured as a function of timefor the three wave signals during the testing ofa 304L stainless steel sample using maragingsteel pressure bars. The incident and transmittedwave signals represent compressive loadingpulses, while the reflected wave is a tensile wave.

Using the wave signals from the gages on theincident and transmitted bars as a function oftime, the forces and velocities at the two inter-faces between the pressure bars and the speci-men can be determined. When the specimen isdeforming uniformly, the strain rate within thespecimen is directly proportional to the ampli-tude of the reflected wave. Similarly, the stresswithin the sample is directly proportional to the

Gas Gun

Incident Bar

Transmitted Bar

Bar Stop

Induction HeatingCoil

Optical Rail

BarMover

Fig. 2 A compression split-Hopkinson pressure bar facility at Los Alamos National Laboratory

−1.5

−1.0

−0.5

0

0.5

1.0

1.5

2.0

2.5

0 100 200Time, μs

Str

ain

gage

out

put,

V

300 400 500

Incident wave

Transmitted wave

Reflected wave

Incident wave

Transmitted wave

Reflected wave

Fig. 3 Strain-gage data, after signal conditioning and amplification, from a compression split-Hopkinson pressurebar test of a 304 stainless steel sample showing the three stress waves measured as a function of time. Note

that the wave positions in time are arbitrarily superimposed due to the time delays used during data acquisitions.

LIVE GRAPHClick here to view

ratio of 100 or more (Ref 3). There are similarrequirements for bar L/D ratios to allow waveqseparation for compression, tensile, and torsionpHopkinson bars.

Figure 2.5: A compression split-Hopkinson pressure bar facility at Los Alamos National Lab (Figure

from [4]).

Pressure bars are usually attached to a rigid body, for a better alignment and providing a

straight platform (for example the I-beam in figure 2.5). Figure 2.5 shows one of the compression

Hopkinson bar facilities at Los Alamos National Laboratory, where an optical rail is used to maintain

accurate pressure bar alignment. Impact of the striker bar to incident bar happens due to propulsion

of the striker. This impact, develops a horizontal compressive wave in the bar. Of course, a portion

of this wave reflects while the remaining reaches the specimen which they call it transmitted wave.

The time of the passage and magnitude of these elastic pulses are recorded by strain gauges normally

cemented at the midpoint positions along the length of the two pressure bars. These wave signals from

the gauges on the incident and transmitted bars, can be plotted as a function of time. By integrating

11


the mentioned plot we can determine the forces and velocities at the two interfaces between the

pressure bars and the specimen. Because the specimen deformed uniformly, the strain rate within the

specimen is directly proportional to the amplitude of the reflected wave. Similarly, the stress within

the sample is directly proportional to the amplitude of the transmitted wave. The reflected wave is

also integrated to obtain strain and is plotted against stress to give the dynamic stress-strain curve for

the specimen.

In recent years, a number of new improvement have been designed to improve the accuracy

of the experiment. Optical measurement is a new technique to find the radial strain in a sample using

linear laser line-measuring technique. In this method it is not necessary to attach a strain gauge to

the sample [46, 47] and this provides a significant advantage.

Introduction to High Strain Rate TestingSia Nemat-Nasser, University of California, San Diego

HIGH STRAIN RATE TESTING is impor-tant for many engineering structural applica-tions and metalworking operations. In struc-tural applications, various components must bedesigned to function over a broad range ofstrain rates and temperatures. In metalworkingoperations, materials undergo large amounts ofstrains at various temperatures and strain rates.The constituent materials must, therefore, becharacterized at the strain rates and tempera-tures of the intended application. Conventionalservohydraulic machines are generally used fortesting at quasi-static strain rates of 1 s– 1 or less(Fig 1). With special design, it is possible to at-tain greater strain rates, up to about 100 s– 1,with conventional load frames. For higherstrain rates, other test methods are required. Ta-ble 1 summarizes various methods in terms ofthe ranges of the strain rates that they canachieve.The articles in this Section describe various

methods for high strain rate testing. Severalmethods have been developed, starting with thepioneering work of John Hopkinson (Ref 1)

and his son, Bertram Hopkinson (Ref 2, 3).Based on these contributions and also on an im-portant paper by Davies (Ref 4), Kolsky (Ref 5)invented the split-Hopkinson pressure bar,which allows the deformation of a sample of aductile material at a high strain rate, whilemaintaining a uniform uniaxial state of stresswithin the sample. The basic concept of theKolsky apparatus involves a test samplesandwiched between an input and output bar, asdescribed in detail in the article “ ClassicSplit-Hopkinson Pressure Bar Testing” in thisSection. This technique provides a capability tomeasure the stress-strain response of ductilematerials at a high strain rate, usually betweenapproximately 50 s– 1 and 104 s– 1, depending onthe sample size, over the entire stress-straincurve. Strains exceeding 100% can be achievedwith the Hopkinson bar method. The maximumstrain rate that can be attained in a Hopkinsonbar varies inversely with the length of the testspecimen. The maximum strain rate is also lim-ited by the elastic limit of the Hopkinson barsthat are used to transmit the stress pulse to the

test sample. These basic factors of theHopkinson bar method and some specializedcompression and tension tests are discussed inthe article “ High Strain Rate Tension and Com-pression Tests” in this Section. An overview ofshear test methods (other than the torsionalKolsky bar method) is also provided in the arti-cle “ High Strain Rate Shear Testing.”The most important characteristic of Kolsky’s

split-Hopkinson compression apparatus is thatit allows high strain rate deformation while thesample is, in fact, in dynamic equilibrium, thatis, the stress gradient is essentially zero alongthe sample. It is thus possible to develop theuniaxial stress-strain response of many materi-als at a variety of strain rates. Because the re-sponse of most materials depends on both thestrain rate and the temperature, the techniqueallows developing constitutive relations thatexpress the uniaxial stress to the correspondingstrain rate and temperature. From such results,one is able to produce experimentally based,three-dimensional constitutive models for nu-merous materials.The series of articles on split-Hopkinson com-

pression testing begins with the article “ Classic

Table 1 Experimental methods for highstrain rate testing

Applicable strain rate, s–1 Testing technique

Compression tests<0.1 Conventional load frames0.1– 100 Special servohydraulic frames0.1– 500 Cam plastometer and drop test200– 104 Hopkinson (Kolsky) bar in

compression103– 105 Taylor impact test

Tension tests<0.1 Conventional load frames0.1– 100 Special servohydraulic frames100– 103 Hopkinson (Kolsky) bar in tension104 Expanding ring>105 Flyer plate

Shear andmultiaxial tests<0.1 Conventional shear tests0.1– 100 Special servohydraulic frames10– 103 Torsional impact100– 104 Hopkinson (Kolsky) bar in torsion103– 104 Double-notch shear and punch104– 107 Pressure-shear plate impact

Intermediatestrain rates

0

Increasing stress levels

Uniaxial strain &simple shear

Uniaxial & shear stress

Hopkinsontechniques

Light gas gunor

explosivelydriven

plate impact

Servohydraulic &screw machines

Very highstrain rates

106

10–6

10–8

10–4

10–2

100

102

104

Creep Quasi-staticHigh strain

rates

Constant loador

stress machine

Specialservohydraulic

machines

Shockloading

Dynamicconsiderations

in testing

Strain versus timeor

creep raterecorded

Constantstrain rate

tests

Constantstrain rate

tests

Uniaxialstress &torsion

tests

Uniaxialstrain &

shear tests

Inertia forces neglected

Isothermal

Inertia forces important

Adiabatic/quasi-isothermal

Strain rate (s–1)

Fig. 1 Strain rate regimes and associated instruments and experimental conditions

ASM Handbook, Volume 8: Mechanical Testing and EvaluationH. Kuhn, D. Medlin, editors, p427–428DOI: 10.1361/asmhba0003293

Copyright © 2000 ASM International®All rights reserved.

www.asminternational.org

Figure 2.6: Experimental method for high strain rate testing [4].

2.3.2 Dynamic tests in tensile and torsional loadings

As mentioned earlier, the Hopkinson experiment is also used to determine the shear and

tensile strength of a material in a dynamic loading test. In addition to the compression version of

Kolsky SHPB, the tensile version of this experiment for tension which is Split Hopkinson Tension

12


Bar (SHTB) was also developed to obtain the characteristics of materials under dynamic tensile

loading. The initial design of dynamic tension apparatus was by Harding. In this experiment there

was a single elastic bar and a specimen, attached inside a hollow tube [48]. Later, the design was

modified and the entire system was established inside the tube [49, 50]. The specimen was also

modifed to satisfy the tensile test criteria. Lindholm [51] proposed the top-hat specimen and Nicholas

[5] introduced the threaded specimen with a rigid collar. A tensile pulse is generated in the incident

bar by loading the end of the incident bar (Figure 2.7) through direct impact of a mass with a flange

at the end of the incident bar [48]. Another loading method is to release a tensile pulse which is

stored in the incident bar by using a clamping fixture [52].

in Fig. 21 show the quite different damage levels, although thesamples in these three typical tests were approximately subjectedto the same incident wave. With bar/sample interfaces fully lubri-cated, the samples are completely fragmented into small pieces,featuring a typical splitting failure mode. This failure mode con-firms 1D stress states during the dynamic tests. In contrast, withfriction constrain at the boundary interfaces, the splitting is con-strained significantly and the recovered samples feature a shearcone. It can be concluded that without proper lubrication, themeasured strength values will be over-estimated and the failuremode will also be quite different.

5.2. Dynamic tension test

For rock materials, dynamic tension tests can be approximatelycategorized into two approaches: direct tension (Howe et al., 1974;Goldsmith et al., 1976; Huang et al., 2010a) and indirect tension(Klepaczko and Brara, 2001; Wu et al., 2005; Schuler et al., 2006;Dai et al., 2008; Kubota et al., 2008; Wang et al., 2009; Zhouet al., 2012). For dynamic direct tension method, the specimen issubjected to dynamic tensile load which is directly generated bySHTB. The dynamic indirect tension method includes Brazilian disc(BD) method (Zhou et al., 2012), semi-circular bend (SCB) method(Dai et al., 2008) and spalling method (Klepaczko and Brara, 2001;Wu et al., 2005; Kubota et al., 2008; Erzar and Forquin, 2010).

5.2.1. Dynamic direct tension methodIn order to generate directly dynamic tensile loading in spec-

imen, the tensile versions of Kolsky bar system have beendesigned. The early design of dynamic tension experiments was ahollow tube, inside which incident and transmitted bars with aspecimen sandwiched in between are placed (Hauser, 1966;Harding and Welsh, 1983) (Fig. 22a). The principle of this device

is to convert the external axial compression impact in externaltube into tension wave in incident bar via the external tube wherethe Kolsky bar system is placed. This approach was just anextension of a compression version of Kolsky bar system anddirectly utilized the launching system of a compression bar sys-tem. However, since the whole setup was inside a solid tube, it isinconvenient for instrumentation (e.g. strain gauge) and directobservation of the specimen deformation. Then, a new configu-ration with top-hat specimen was proposed by Lindholm andYeakley (1968), where a top-hat specimen is placed between thesolid incident bar and the hollow transmission tube (Fig. 22b).When the compression stress wave in the incident bar arrives atthe specimen, a tensile load is produced on the specimen gaugesection and a compression stress wave is transmitted into thetransmitted tube. The advantage of this design is that the spec-imen does not need to be attached to the bar ends. Nicholas (1981)introduced a way to use only the compression bar system toachieve tensile experiments (Fig. 22c). The modifications include(i) the specimen which is threaded into the incident bar andtransmitted bar, and (ii) a rigid collar placed over the specimen.The function of rigid collar allows the initial compression wave topass through the collar without virtually touching the specimenand most of the initial compression energy is transferred into thetransmitted bar. The transmitted compression wave is reflected tobe a tensile wave, and propagates back to load the specimen onlybecause the collar cannot support tensile load. However, in thisdesign, the specimen with the collar will inevitably deform whensubjected to the initial compression wave.

The most efficient loading method for Kolsky tension bar isdirect tension. The first way is to store elastic energy by stretching asection of incident bar in tension (Staab and Gilat, 1991; Cadoniet al., 2009). A clamp is used to divide the pre-stressed and stressfree section in the incident bar. Suddenly releasing or breaking the

Fig. 22. Schematics of four types of dynamic direct tension methods.

K. Xia, W. Yao / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 27e5940

Figure 2.7: Schematics of dynamic tension loading [5].

On the other hand, Goldsmith [53] was the first one who was after shear strength of

the material in a dynamic test. He used solid cylindrical Barre granite specimen in torsional split

Hopkinson bar test. Then, the torsional split Hopkinson bar with thin-walled tubes was utilized to

obtain the pure shear strength by Lipkin et al., [54, 55]. Gilat, explained more about the details of

torsional split Hopkinson bar in the ASM handbook [6]. According to Gilat, the friction and lateral

inertia effects is negligible in the torsional split-Hopkinson bar, however, it is difficult to prepare

the thin-walled specimens and mount them on the bars. Therefore, several other methods had been

developed to perform the dynamic shear tests, such as, compression shear method [56], punch shear

method [57, 58] and a split Hopkinson pressure shear bar (SHPSB) [59]. The compression shear

method was used for large strain test in SHPB. The SHPSB system consists of a wedge-shaped end

incident bar and two transmitted bars, using an optical system to obtain shear strength and shear

strain.

13


punch tests were also used to investigate the mechanical properties of metals [16, 17]. Motivated by the needs from engineering applications and the existing studies, the objective of this paper is to develop a dynamic punch shear device to quantify the dynamic shear properties of brittles solids.

Punch

head

Sample

Holder

Punch

head

Sample

Holder

a)

b)

Loading

Loading

Fig. 1. Schematics of punch shear devices used by: a) Mazanti and Sowers [7] and b) Stacey [5].

GENERIC TESTING PRINCIPLES A 25 mm diameter SHPB system is utilized to apply the dynamic load for punch shear tests (Fig. 2). The SHPB consists of a striker bar, an incident bar and a transmitted bar. The length of the striker bar is 200 mm. The incident bar is 1500 mm long and the strain gauge station is 787 mm from the specimen. The transmitted bar is 1000 mm long and the stain gauge station is 522 mm away from the specimen. The bars are made from Maraging steel, with a yielding strength of 2.5 GPa, density 8100 kg m-3, Young’s modulus 200 GPa and one dimensional stress wave velocity 4970 m/s. A gas gun lunches the striker bar to impact on the incident bar and generates an elastic compressive wave toward the sample. At the sample assembly, the incident wave will be separated into two waves: an elastic tensile wave reflected back into the incident bar and a compressive wave transmitted into the transmitted bar. The incident wave i , reflection wave r and transmitted wave t are measured by strain gauges mounted on the incident bar and the transmitter bar, respectively.

Striker Incident bar Transmitted bar

Strain gauge

holderSample Fig. 2. Schematics of Split Hopkinson Pressure Bar for dynamic punch shear tests.

Using these three waves, the dynamic forces P1 and P2 on both ends (Fig. 3) of the sample assembly can be calculated [18]:

)]()([)(1 ttEAtP ri (1)

)()(2 tEAtP t (2)

158

(a)

where E and A are Young’s modulus and cross-sectional area of the bars, respectively.

Incident bar

Frontcover

Rear supporter

Transmitted bar

P1 P2

Fig. 3. Schematics of the sample holder in SHPB.

When the test is under dynamic force equilibrium condition (i.e. P1 = P2), the inertial effect in the dynamic test can be ignored [14, 19]. In this case, the punch shear stress in the sample can be calculated using the following equation:

DBP

(3)

where is the punch shear stress; P = P1 = P2 is the loading force; D and B are the diameter of incident bar and the thickness of the disc specimen, respectively. It is noted here that we divide the load by the total shear area to obtain the shear stress, in a similar way to most other static punch shear studies. The maximum value of is considered as the punch shear strength 0 of the sample tested. A special holder is designed to support and protect the sample during dynamic punch tests. Conventional punch shear systems for static tests usually have two kinds of punch heads: cylindrical punch head and block punch head (Fig. 1). For dynamic punch tests using SHPB, an annular holder is usually adopted [15-17]. In this paper, the stainless steel holder consists of a front cover and a rear supporter, which are jointed by screw to hold the sample as shown in Fig. 3. The purpose of the front cover is to reduce the bending force during tests and additional damage on samples during and after the tests. The inner diameter of the rear supporter is 25.4 mm, 0.4 mm larger than the diameter of the incident bar to accommodate shear deformation. The incident bar serves as the punch head and the rear supporter is attached to the transmitted bar. The outer diameter of the entire holder is 57 mm. APPLICATIONS TO LONGYOU SANDSTONE Sample preparation

Fig. 4. a) Typical virgin and tested samples; b) The rock ring and rock plug produced in a typical dynamic punch shear test

(the unit in the picture is centimeter).

159

(b)

Figure 2.8: (a) Schematics of Split Hopkinson Pressure Bar for dynamic punch shear tests, and (b)

sample holder in SHPB [6].

Dynamic punch shear method is widely used to measure dynamic shear strength of materi-

als. Zhao et al. [57] measured the shear strength of Bukit Timah granite by using the punch shear

tests in the range of 101 - 104s−1 using a pneumatic hydraulic machine, which can neglect the wave

propagation effects. Huang et al. [58] created a dynamic punch shear method in SHPB to measure

the dynamic shear strength of rocks (Figure 2.8a). In the dynamic design, the sample assembly is

composed of a front cover, a disc sample and a rear supporter (Figure 2.8b). The purpose of the front

cover is to reduce the bending force during test and to prevent additional damage to the specimen

after test. The inner diameter of holder is 0.4 mm larger than that of bars. A Teflon adaptor is used to

connect the rear holder to the transmitted bar [58]. The hole in the rear supporter recovers the sample

after the shear test.

2.4 Ultra High Rate of Loading

In very high strain rates ( > 105s−1), experiments have been done in the atomic scale

to determine the strength of materials. For example, Bringa et al. [60] utilized the large-scale

non-equilibrium molecular dynamics method to obtain the shear strength of copper at strain rates

higher than 109s−1. They validated the MD simulations with dynamic experimental data at the same

spatial and temporal scales, both at the lattice level and globally. Murphy et al. [61] used uniaxial

shock compression test at 100 GPa to measure the shear stress and shear strain in a single crytal

copper at 1010s−1 strain rate. They took the advantage of X-ray diffraction method and VISAR

(velocity interferometer system for any reflector) measurement to study structure of the material

and find the information about the strength of the material directly. X-ray diffraction is a powerful

experimental technique for characterizing the state of shock-compressed material, because it gives

14


direct experimental information on crystal strain as a function of applied loading (pressure), from

which strength can be inferred [62]. In addition to experimental procedures, theoretical methods

and non-equilibrium molecular dynamic (NEMD) simulations have also been developed to calculate

the materials behavior at high strain rates. According to these studies, the strength of the material

can increase dramatically (to ∼ 1 GPa) for extreme strain rates. Comley et al. [62] also used X-ray

diffraction to measure the strain state of the compressed crystal and reached up to 35 GPa of shear

strength at a shock pressure of 181 GPa.

15

Chapter 3

Modeling and Finite Element Simulation

of High Velocity Impact for a Single

Micron-Scale Particle

3.1 Introduction

In this chapter, the characteristic of finite element model, used to analyze the single

particle impact is expressed in detail. First, the creation of geometry by using Python scripts and the

advantages of explicit time integration in simulating the dynamic impact are explained. Then, the

element type used in simulations and features associated with this type of elements are discussed.

In this section, the criterion and circumstances of element failure are also expressed. Next, the

properties of contact between the particle and substrate are defined. Then, the boundary conditions

and the use of symmetry in the problem are presented. In the Section 3.2.7 temperature dependent

material properties and its effects in simulation are covered. Study on the mesh independence in

single particle, dynamic impact simulations is also conducted. Finally, the chapter is finished by

explaining the single particle impact experiments, preformed at UMass Amherst University under

supervision of Dr. Jae-Hwang Lee [10].

16

CHAPTER 3. MODELING AND SIMULATION

3.2 FEM Model

Supersonic impact of a spherical particles on a flat substrate has been simulated by using

commercial finite element analysis software, ABAQUS/Explicit 6.13-2 with Lagrangian formula-

tion. Fully coupled thermal-stress analysis was performed with the brick elements, that have both

displacement and temperature degrees of freedom. The interactions between the particle and the

substrate is defined by using the general contact algorithm. The properties of this interaction includes

the definition of friction in tangential behavior, hard contact in normal direction, heat generation due

to mechanical interaction and cohesive behavior. In the following sections, each feature is explained

in detail.

3.2.1 Modeling and analysis technique

The whole process of modeling an impact simulation is controlled by a Python script. The

script creates the non-dimensional 3D model, assigns material property, defines contact, meshes the

parts, launches the analysis, and gets all the necessary results from output files. In this study only a

quarter of the whole shape is simulated due to the symmetry. The particle is assumed to be a perfect

sphere and the substrate a perfect cylinder. In the Python script, all dimensions are with respect to the

particle diameter. For example the height and the radius of the cylinder is 12.5 times larger than the

particle diameter, and the mesh size is 25 times smaller than particle diameter. Schematic isometric

view of the assembled model can be find in Figure 3.1.

rp

25 r p

Figure 3.1: Isometric view of the developed 3D model.

17


The explicit dynamics analysis procedure uses the central difference integration rule to

integrate the equations of motion for the body, so that’s why it is conditionally stable without

adding any kind of damping to the system. An explicit dynamic analysis is computationally efficient

for the analysis of large models with relatively short dynamic response times. It is also stable in

complicated contact problems, large deformation and inelastic dissipation (to generate heat from

plastic deformation) with respect to implicit dynamic analysis which uses the direct-integration

algorithm. However, the accuracy of the results needed to be checked due to some simplification

in formulation of dynamic explicit algorithms. In the case of impact simulation, it is reasonable to

use the explicit package because this phenomena happens in a relatively short time, it involves large

deformation due to high velocity impact and it is a complicated contact problem.

3.2.2 Element Type and Failure Criterion

The element type used in this work is C3D8RT. This is a first order, 3D continuum element

with 8 nodes and one integration point. The displacement and temperature variation is linear along

each direction. The elements with reduced integration are referred to uniform strain or centroid strain

elements. Reduced integration elements decrease running time of simulations, especially in three

dimensional problems. However, the elements can be distorted in such a way that the strains at the

integration point become zero, which in turn, leads to uncontrolled distortion of the mesh. This

problem is known as Hourglassing [63]. In Abaqus/Explicit, first-order, reduced-integration elements

have the capability to control this issue. Hourglass control, attempts to minimize the excessive

distortion without introducing extra constraints on the element’s physical response.

It was mentioned that particle impact at supersonic velocities, involves large deformation,

therefore, it is necessary to add a damage control mechanism for the model. The progressive damage

and failure for ductile metals involves damage initiation and damage evolution. Damage initiation can

be detected by using shear strain, forming limit diagram (FLD) [64], or Marciniak-Kuczynski (M-K)

[65] criteria. After damage initiation, the material stiffness is degraded progressively according

to the specified damage evolution response. The progressive damage models allow for a smooth

degradation of the material stiffness, which makes them suitable for dynamic situations. In this

study shear criterion is utilized to determine the onset of damage due to shear band localization. The

criterion for damage initiation is met when w =∑(

∆εpεp f

)is equal to one. ∆εp is an increment of

the equivalent plastic strain (PEEQ), εp f is the failure shear strain and w is the damage parameter

which increases monotonically at each increment during the analysis.

18


Observations of material failure during explosive and impact problems, often shows

localized deformation that may either dominate or contribute to failure. This mechanism, which

is called adiabatic shear instability, has been known and studied for many years [7]. Figure 3.2

illustrates the typical response of a work-hardened material. In slow loading, (no adiabatic heating)

the material may show continuous hardening out to large shear strains, as indicated by the upper curve

in Figure 3.2. However, plastic work involves heat generation and as a result of that the temperature

of the material increases. Since metals tend to be softened by increasing temperature, eventually the

flow stress will reach a maximum at γmax, followed by strain softening, as indicated schematically

by the middle curve. In a perfect material with perfectly uniform distributions of stress, strain, and

temperature, softening may continue indefinitely. In a work hardened material, the adiabatic stress,

reaches a maximum at some temperature and plastic strain.

provide the macroscopic linkage between the ‘‘softening’’ or ‘‘damaging’’ material and the ultimate loss of

stress carrying capability that arises in the context of the technologically important class of problems that

involve strong discontinuities and/or much weaker velocity gradients.

The current review of our model proceeds with a summary for the criterion of stress collapse due toadiabatic shear. The problem of simple shearing is reviewed and the details of ultimate stress collapse are

described as a function of both the driving imperfections and the homogeneous response of a work-

hardening, strain-rate hardening and thermally softening material. We will describe a simple generalization

of this scheme to formulate a macroscopic failure model for damage due to shear localization and the

incorporation of this more general model within the context of finite element analysis techniques. We will

exercise the model by way of example problems designed to highlight strengths and weaknesses in the

current formulation and to understand its appropriateness for describing failure for a certain class of

materials and driving conditions. Finally we will summarize our model and describe some of its currentassumptions as well as general directions for further development.

2. Summary of a criterion for adiabatic shear

The ideas, upon which the numerical model for initiation of an adiabatic shear band in this paper is

based, were developed in two papers some eight to ten years ago, (Wright, 1992, 1994) and have also been

discussed in a recent comprehensive summary (see Wright, 2002). However, we only recently recognizedthat the earlier papers actually contained the beginnings of an idea that could be developed into a com-

putational failure model. These ideas will be outlined in this section; for further details, the reader may refer

to the original papers.

Consider the dynamic response of a thin-walled tube in torsion, as experienced by a specimen in a

Kolsky bar experiment. Fig. 1 illustrates the typical response of a work-hardening material. In slow

loading, or if adiabatic heating is somehow suppressed, the material may show continuous hardening out to

large shear strains, as indicated by the upper curve in the figure. However, plastic working will heat the

material, and because metals tend to soften with increasing temperature, eventually the flow stress willreach a maximum at cmax stress, followed by strain softening, as indicated schematically by the middle curve.

In a perfect material with perfectly uniform distributions of stress, strain, and temperature, softening may

continue indefinitely.

Fig. 1. Shear stress versus nominal shear strain for a typical work-hardening material during a torsion experiment.

3024 S.E. Schoenfeld, T.W. Wright / International Journal of Solids and Structures 40 (2003) 3021–3037

Figure 3.2: Shear stress versus nominal shear strain for a typical work-hardening material during a

torsion experiment [7].

The considered material failure in these simulations depend on the local equivalent plastic

strain. A constant failure strain, εf , is used as the criterion for material failure. Different values

of εf were tested by Yildirim et al., [8] to find an appropriate one which prevents excessive mesh

distortion but does not alter impact behavior significantly. They observed that (Figure 3.3) when the

failure shear strain is in the interval of [2,∞) impact behavior is not significantly altered and mesh

distortion is greatly reduced.

Element removal is also activated in the numerical simulation when the shear failure

criterion is met at an integration point. As a result of failure, all the stress components will be set to

19


156

Figure 6.2: Deformed mesh structure of the particle and the substrate (a), and rebound velocity as a function of particle impact velocity (b) for failure strains εf = 1, εf = 2, and no failure condition (εf = ∞). Particle and substrate are both copper.

Investigated parameters and material combinations are provided in Table 6.2. Oxygen free,

high conductivity (OFHC) copper, aluminum 1100-H12, 316L steel, commercially pure (CP)

titanium and molybdenum were considered in this work as particle and substrate materials. These

materials were selected as they differ in key properties, as follows: Copper is a low strength and

high density material; aluminum is a low strength and low density material; steel and

molybdenum are high strength and high density material; and titanium is a high strength and low

Figure 3.3: rebound velocity as a function of particle impact velocity for different failure strains.

This analysis conducted by using Copper as a material properties for both particle and substrate [8].

zero and that integration point fails. In FEA, when all of the integration points at any one section of

an element fail, the element is removed from the mesh. In the case of first-order reduced-integration

solid element, removal of the element takes place as soon as its only integration point fails.

There is also another method to control the excessive distortion in simulation which is

mostly utilized in metal forming simulations and it is called arbitrary lagrangian eulerian (ALE).

Yildirim et al., [8] and Assadi et al., [22] reported that, adaptive remeshing, causes unphysical shape

of the out-flowing jet of material at the interface which is far from reality. Therefore, in this study

adaptive remeshing is ignored.

3.2.3 Interaction and Contact Properties

The general contact algorithm is used for modeling the contact and interaction between the

particle and substrate. For the normal behavior of interaction properties, hard contact is specified as

a pressure overclosure relationship, which minimizes the penetration of the particle surface into the

substrate surface at the constraint locations and does not allow the transfer of tensile stress across

the interface. When the impact happens and surfaces are in contact, the contact pressure can be

transmitted between surfaces. The surfaces can also separate if the contact pressure reduces to zero.

Separated surfaces could come into contact again if the clearance between them reduces to zero.

In the tangential direction, the basic form of the Coulomb friction model is utilized for the

friction. In this model, two contacting surfaces can carry shear stresses up to a certain magnitude

20


across their interface before they start sliding relative to one another. It is also assumed that the

friction coefficient is the same in all directions (isotropic friction).

In fully coupled temperature-displacement simulations, heat can be generated due to the

dissipation of energy created by the mechanical interaction of contacting surfaces. The source of heat

in these kind of analyses is frictional sliding and it is distributed between the interacting surfaces of

particle and substrate equally.

The definition of cohesive behavior is utilized to study the bonding/rebound behavior of

particles impacting a substrate. This approach introduces an interfacial cohesive strength parameter

(σc) into the finite element model in order to model the bonding upon impact. Traction-separation

relationship is assumed for the behavior of surfaces in contact. This theory assumes a linear elastic

behavior followed by the initiation and evolution of damage in the interface. The initiation occurs

at (δc, Tc) and the evolution can be predicted by different kinds of relations, however, in this study

it is assumed that this relationship is linear until separation reaches its ultimate value δf . Figure

3.4 shows a typical traction-separation response with a linear failure mechanism. In the case of

impact simulations, deboning happens when the kinetic energy of the particle, overcomes the fracture

energy or energy dissipated due to failure Gc. Although, in most sections of this study the physical

bonding has not been taken into the account and the particles are rebounding after impact, this option

is available in the Python script and can be switched on or off depending on the application.

11

Traction

Separation

Tc

δc δf

Gc

Figure 3.4: Traction separation response of interacting surfaces with a linear failure mechanism.

3.2.4 Boundary Conditions

Based on the symmetry in the problem, only a quarter of the complete shape is modeled.

Therefore symmetry constraints are applied on each side of the sectioned body. The bottom part of

21


the substrate is also set to be fixed. Both particle and substrate are initially at room temperature. The

particle has an initial velocity which is the same as impact speed in vertical direction. In this study

the gravitational force acting on the particle is ignored.

One of the demanding aspects of high velocity impact simulations is the possibility of

wave reflection from the boundaries and its effect on the solution. It has been also mentioned by

Assadi et al., [22] that the waves induced by the particle impact could have a notable effect on the

results. In our simulations the substrate needs to be large enough that the reflected waves do not

return the interface of substrate and particle during impact. In addition, the duration of the numerical

analysis needs to be set long enough to make sure that the particle will detach completely and the

total energy in the system will become stable. Thus, by taking both of these points into account,

the radius and the height of the cylindrical substrate are set to be 25 times larger than the radius of

particle, with this size, the simulation time can be up to 200 ns [8]. With such large substrate the

reflected waves reach the impact zone only after the rebound of particle.

3.2.5 Material Properties

The properties of material used in the simulations are provided in this section. All material

properties used in the finite element simulations were taken from material databases and the literature

[9, 15, 14]. Manes et al., [13, 66] calculated Bilinear Johnson-Cook parameters for Aluminum 6061

using the experimental and theoretical data. This is done by fitting Bilinear Johnson-Cook model

constants (A, B, n, C1, C2 and m) to the experimental data and using nonlinear multiple regression.

All material properties are temperature dependent. Figure 3.5 shows the variation of each property

with respect to temperature for Aluminum 6061. The respective table for these properties can be find

in Appendix B. Table 3.1 depicts the material properties of Sapphire as a function of temperature.

Note that the density is the only material property that is not defined as a function of temperature due

to some limitation with the explicit package of ABAQUS software.

3.2.6 Adiabatic Shear Instability

Studies show that thermal softening cause shear instability at the interface of contact. This

could be the reason for bonding between the particle and substrate [22]. According to Bever et al.[67]

a majority of the plastic work at the tip of a propagating crack is converted to heat while a small

percentage is actually stored in the material due to dislocation interaction [67]. Numerous studies

have been conducted to measure the amount of heat generated due to plasticity. For instance, Rosakis

22


17 | P a g e

AppendixAThe following five figures represent the temperature dependent material properties for Al-6061, extracted from MPDB database [9] and implemented in Abaqus simulations.

a) b)

c) d)

e)

Figure 3.5: Temperature dependent material properties for Aluminum 6061 [9]. Properties include

a) Elastic Modulus; b) Poisson’s Ratio; c) Thermal Expansion; d) Thermal Conductivity; and e)

Specific Heat.

23


Table 3.1: Temperature dependent material properties of Sapphire [14].

20◦C 500◦C 1000◦C

Density, kg/m3 3980

Elastic Modulus,GPa 416 390 364

Poisson′s ratio 0.231 0.237 0.244

Thermal conductivity, W/mK 33 11.4 7.22

Specific heat, J/kgK 755 1165 1255

Thermal Expansion coefficient, 10−6K−1 4.6 7.1 8.1

et al. [68] used the Split Hopkins Pressure Bar in conjunction with an infrared detector to investigate

the fraction of plastic work converted to heat. In their experiments, the SHPB loads the specimen

dynamically with a controllable strain rate. Loading time is very short, so it can be assumed that the

impact is totally adiabatic. Since there is no time for conductive, radiative or convective heat loss, it

is expected that the dynamic loading apparatus will provide highly accurate measurements of the

converted plastic work fraction.

Assuming adiabatic conditions the converted plastic work fraction, β, can be defined as

follows [68],

ρcpT = βWp (3.1)

The left hand side is the amount of heat flux added to the system, ρ is the density, cp is the specific

heat capacity, T is the temperature, ( ˙ ) represents the time derivative and Wp is the plastic dissipated

energy which can be defined as,

Wp =

∫ t

0σεpdt (3.2)

Based on the literature [68, 69, 70], this inelastic heat fraction is in the range of 85% to

95% depending on the material and it is mostly constant in high strain rate applications.

3.2.7 Effect of Using Temperature Dependent Material Properties

In finite element simulations the obtained results rely on the accuracy of the material model

and material properties. One of the key factors that alters the behavior of material response is heating.

Therefore, the temperature dependence of material properties should be considered in the simulations.

24


The material model that is used for simulating the particle impact is the Bilinear Johnson-Cook

model. As presented in Chapter 4, temperature is one of the main factors that helps softening of

the stress strain relationship in this material model. In the following, the role of using temperature

dependent material properties is discussed.

Figure 3.5 shows the variation of each material property (elastic modulus, Poisson’s ratio,

thermal expansion, thermal conductivity and specific heat) with respect to temperature for Aluminum

6061. On the other hand, Table 3.2 depicts the material properties of Al-6061 at room temperature.

The following is an analysis of the temperature dependent material response for Al-6061. Other

simulation properties such as substrate material properties, particle diameter, impact velocity, and

inelastic heat fraction are kept constant in order to focus on the effects of temperature dependent

material properties in simulations.

Table 3.2: material properties for Aluminum 6061 at room temperature.

Density, kg/m3 2700

Elastic Modulus,GPa 68.9

Poisson′s ratio 0.33

Thermal conductivity, W/mK 220

Specific heat, J/kgK 904

Thermal Expansion coefficient, 10−6K−1 23.6

The simulated deformed shape of the particle after impact is illustrated in Figure 3.6. In

these simulations, the diameter of the particle was 19.75 µm and impact velocity was 663 m/s. The

particle was made of Aluminum and substrate from Sapphire. In this picture the red curve represents

the deformed shape of particle based on constant material properties and the blue curve represents

the same shape but temperature dependent material properties. Both shapes are nearly identical,

however, the red curve deformed a bit more. It was observed taht as the impact velocity increases,

the difference between two curves becomes more and more evident.Here in Figure 3.6 it is seen that

the red curve (temperature independent material properties) spreads more in the vertical direction

and compressed more in the longitudinal direction. This kind of deformation was not expected from

a material using temperature dependent material properties (blue curve), since their strength should

be degraded due to temperature softening which causes more deformation.

In order to answer this apparent dilemma an in-depth analysis has been carried out. Figure

3.7 is the time history analysis of an impact simulation. Blue and red plots represent data for the

25


−10 −5 0 5 10

−5

0

5

10

15

X

Y

bilinear temp dep

bilinear temp indep

Figure 3.6: Deformed shapes of particle using bilinear JC material model for 19.75 µm particle and

impact velocity of 663 m/s.

temperature dependent and temperature independent material properties, respectively. Based on the

time history plots the average values for strain and strain rate of temperature independent material

properties simulation are slightly higher (Figure 3.7 a, b). On the other hand, the temperature plot

shows that average temperature increases slower with temperature independent material properties

(Figure 3.7 c).

Considering the Bilinear Johnson-Cook relationship explained in Section 4.3, higher strain

and strain rate can help the hardening of material resulting in a higher yield stress. However,

increasing the temperature of material results in softening and decreasing the yield stress of material.

Depending on the intensity of temperature effect or the strain and strain rate effect, the final result

of material strength could be harder, neutral or softer as compared to the simulation with constant

material properties. In the case of Aluminum 6061, the combination of strain hardening, strain rate

hardening and temperature softening, causes the temperature dependent material properties to behave

slightly harder. This idea can also be shown by considering the average yield stress of the material

(Figure 3.7 d). The average yield stress is slightly higher for the case of temperature dependent

material properties. That’s why particle using temperature dependent material properties deformed

slightly less in Figure 3.6.

Previously, it was mentioned that the particle’s temperature increases slowly in the case of

using temperature dependent material properties. This is due to the effect of specific heat. Based

on Figure 3.5 e the specific heat increases with temperature. This implies that as the temperature

26


increases, the material needs more heat energy to increase 1 unit of temperature. Therefore, for a

given amount of plastic work hardening, as converted to heat, the increase of temperature is relatively

lower with respect to the case of temperature independent material.

a) b)

c) d)

Figure 3.7: Time history analysis of particle for a) average strain; b) average strain rate; c) average

temperature; d) average yield stress. Note that the particle diameter was 19.75 µm and impact

velocity was 663 m/s. Average values are the mean of all integration point at each increment time.

27


3.3 Mesh Convergence Study

It is well-known that element size strongly affects the results of numerical simulations.

Therefore, in order to find an appropriate mesh size, a mesh convergence study was carried out for

single particle impact. A very dense mesh has been generated on the impact area and the element

sizes have been increased getting far from the impact zone. ”C3D8RT” linear hexahedron, 8 node,

reduced integration elements have been used to mesh the particle and the substrate. The size of the

elements in the impact zone area and particle have been chosen as functions of the particle diameter

which are dp/10, dp/15, dp/20, dp/25, and dp/30. The results are summarized in Table 3.3.

Table 3.3: Mesh convergence study for single particle impact simulation.

Mesh Total no. Simulation Space Vr

size of Nodes time,mins required,GB m/s

dp/10 36,074 2 0.47 20.6

dp/15 120,191 5 1.56 23.62

dp/20 283,083 14 3.68 19.7

dp/25 535,311 35 6.95 19.3

dp/30 928,170 81 12.05 19.0

It was observed that the energy dissipated by damage and removing the failed elements are

relatively high due to large element sizes in first and second cases. As the element size decreases this

energy becomes more stable. In addition, Table 3.3 shows that the rebound velocity results converged

in the last three cases. Also, by considering the computational time and disk space required for a

single analysis, the average element size near the impact zone in the substrate and for the particle

was selected as dp/25. Moreover, the experimental results also reveal that the rebound velocity for

this case is 19.2 m/s which has a good agreement with the selected mesh density case.

3.4 Laser Induced Single Particle Impact Experiments

Laser-induced projectile impact test (LIPIT) [10] is illustrated in Figure 3.8, used to

accelerate a micron scale particle , to study the high strain rate phenomenon in particle impacts. In

this experimental setup an excitation laser pulse is generated by a pulsed laser system that has an

optical parametric oscillator. The pulse, results in ablation and expansion of the gold film in the form

28


of gas. The polymer layer, which is between the gold layer and a particle expands and ejects the

particle. Those particles can accelerate up to 4 km/s. Figure 3.9 illustrates the relation between the

pulse energy and particle velocity [10].

Figure 3.8: Diagram of the optical setup for LIPIT [10].

Figure 3.10 shows images of a particle moving toward a substrate. The velocity of particles

are measured by capturing their positions with two pulses that have a specific phase delay.

The excitation laser pulse generates a supersonic gas plume surrounded by a shock front

trailed by the particle. Deceleration of the particles by air is negligible until the particles overtake

the shock front. Therefore, by placing a target sample at a position where the shock front arrives

before the particle, the flight speed can be used as the impact speed. As demonstrated in Figure 3.9

the kinetic energy of the particles is linearly proportional to the input energy after subtracting the

energy used for ablation. More information on the experiment can be found in Lee et al. [10, 71].

29


A B

d

cv = 2.3 (E –0.08)0.5

Speed of sound

Secondarydelay

Beamsplitter

Polarizer

Mach cone

Pgroup

S

3.4 km s–1

Primary delay(optical fiber)

Target

To microscope

Excitation pulse� = 800 nm

Probe pulse� = 400 nm

Spe

ed o

f bea

ds, v

(km

s–1

)�/2

Plate

Excitation pulse energy, E (mj)0.0

0

1

2

0.5 1.0

Figure 8 | Laser-induced projectile impact test. (a) Diagram of the optical setup for LIPIT. (b) The speed of the m-projectiles is proportional to the square

root of the excitation pulse energy for a given thickness of the absorbing polymer layer. The error bar represents the maximum and minimum speeds for

each median value and the green line is a fitting curve. (c) Double exposure photograph with a 22-ns interval at 250 ns after the excitation laser pulse. The

scale bar is 200mm. (d) Expanded section of (c). Here, each m-projectile appears twice; for example, the m-projectile marked as ‘A’ is the same m-projectilemarked as ‘B’ and the corresponding speed is 3.4 km s� 1. Only the m-projectiles outside of the shock front present a Mach cone, appearing when an

object’s speed exceeds the speed of sound. The sizes of the m-projectiles appear differently depending on their distance to the microscope objective lens.

The scale bar is 100mm.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2166 ARTICLE

NATURE COMMUNICATIONS | 3:1164 | DOI: 10.1038/ncomms2166 | www.nature.com/naturecommunications 7

& 2012 Macmillan Publishers Limited. All rights reserved.

Figure 3.9: particle velocity and excitation pulse energy relation [10].

Capability of the UMass micro-ballistic system

Δx=47.3 μm Δt=100 ns v=473 m/s

Δx=45.9 μm Δt=100 ns v=459 m/s

Trajectory: 0.51 mm

12-exposure micrograph

Figure 3.10: multiple exposure photograph of a particle with constant ∆t [11].

30


3.5 Summary and Conclusions

A 3D finite element analysis has been developed in order to predict the deformed shape

of a particle impacting a substrate. Abaqus/Explicit 6.13-2 has been used to perform the numerical

analysis. Python scripts are used to create the input file, post-process the Abaqus outputs and preform

a series of numerical tests to accurately detect the outer shape of particle. In this chapter, it is found

that temperature dependent material properties for Aluminum 6061 does not change the deformed

shape of particle significantly. Furthermore, the mesh convergence study revealed that reasonable

convergence can be obtained by using an element size of dp/25 for the particle and in the impact

zone of the substrate.

31

Chapter 4

Material Model for High Strain Rate

Deformation of Metals

4.1 Introduction

This chapter gives a brief overview of different constitutive laws which can predict the

behavior of material at very high strain rates.

4.2 Background

One of the most important requirements to mimic a real world application in a numerical

analysis is to have an accurate constitutive model. The goal of this study is to find a proper

material model which can model material behavior in cold spray application. Particle spraying is a

dynamic phenomena which involves large and nonlinear deformation, high strain rate plasticity, and

temperature spike at the impact interface which could cause local melting and bonding between a

particle and substrate. An appropriate constitutive law should consider all of these effects at the same

time. There are a few material models that can predict the behavior of material in such conditions,

including Johnson-Cook (JC) [72, 73], Zerilli and Armstrong (ZA) [74, 75], Voyiadjis and Abed

(VA) [76], Preston-Tonk-Wallace (PTW) [77], Khan-Huang-Liang (KHL) [78, 79, 80], Gao and

Zhang (GZ) [81]. In the following, a brief introduction to each material model is given.

32

CHAPTER 4. MATERIAL MODEL FOR HIGH STRAIN RATE DEFORMATION OF METALS

4.2.1 Johnson-Cook Model

Johnson and Cook [72, 73] proposed this material model in 1983. This is a phenomeno-

logical model which is widely used in most computer codes for static and dynamic analysis, such

as; metal cutting [82, 83], cold spray [8], crash tests [84] and etc. It is relatively easy to calibrate in

the experiments and can predict the behavior of the material flow stress at different strain rates and

temperatures.

Johnson-Cook model was the first model that defined the flow stress as a function of plastic

strain, strain rate, and temperature. Equation 4.1 is the definition of flow stress based on Johnson and

Cook. Here A, B, n, C, and m are material constants, εp is the equivalent plastic strain, εp is the

plastic strain rate, ε0 is the strain rate at which the material constants are obtained and it is known as

reference strain rate. T ∗ is the homologous temperature and defined in equation 4.2, where, Tm is

the melting temperature, Tr is the reference temperature, and T is the absolute temperature.

σ =(A+Bεnp

)(1 + C ln

(εpε0

))(1− T ∗m

)(4.1)

T ∗ =

0 T ≤ TrT−TrTm−Tr Tr < T < Tm

1 T ≥ Tm

(4.2)

This model has been used for simulations to predict the dynamic behavior of materials in the literature

regularly. The advantage of the JC model is that the strain, strain rate and temperature are considered

at the same time and they are directly proportional to the flow stress predicted by this model. However,

it has been indicated that the model may not be accurate when the strain rate of the system is over

104s−1 [81, 85, 78]. This inaccuracy is somewhat related to experimental limitations. For example

in Hopkinson pressure bar test, the strain guages cannot capture the rate if it is greater than 104s−1.

Therefore these JC material constants calibrated to predict the behavior of materials in a specific

range. In addition, the dependency of work hardening on the logarithm of strain rate is linear in this

model which is a simplified relation. For these reasons, this model is not usually able to accurately

predict the material behavior at very high strain rates.

33


4.2.2 Zerilli-Armstrong Model

The initial Zerilli-Armstrong model is a physically based model proposed for different

metals, that are sensitive to strain rate and temperature [74, 75]. The ZA model has different forms

of constitutive relations for body-centered-cubic (BCC) materials and face-centered-cubic (FCC)

materials. The constitutive equation for BCC materials such as tantalum is expressed as follows,

σ = C0 + C1exp(− C3T + C4T ln εp

)+ C5ε

np (4.3)

where, C0, C1, C3, C4, C5, and n are material constants, εp is the equivalent plastic strain, εp is the

plastic strain rate, and T is the temperature. The first term C0 is related to Hall–Petch relation and

equals to σ0 + kd−1/2 , where d is the grain size of the material and k is the micro-structural stress

intensity. In this model, it is presumed that the strain hardening is independent of temperature and

strain rate. The exponential term in equation 4.3 is used to describe the thermal stress component

based on experimental observation. This definition is inappropriate as the thermal stress component

goes to zero only when the temperature tends to infinity. It is noted that, the thermal stress component

must disappear at the melting point of material. In order to resolve this issue, the model was modified

by Abed et al. [86].

σ = C2εnp

(1−X1/2 −X +X3/2

)+ C6 (4.4)

X = C4T ln

(ε0

εp

)(4.5)

where, C2, C4, C6 are material constants, εp is the plastic strain rate, and ε0 is the reference strain

rate.

The modified ZA model generally improves the results at temperatures above 300 K.

However, the work hardening is expressed independent of temperature and strain rate just like the

initial ZA model. This is why these two models are not appropriate to model the materials whose

responses are strongly depended on temperature and strain rate.

34


4.2.3 Voyiadjis-Abed Model

To improve the prediction capability of flow stress at high strain rates and temperatures,

Voyiadjis and Abed [76] modified the ZA model as follows,

σ = Y

[1− (β1T − β2T ln εp)

1/q

]1/p

+Bεnp + Ya (4.6)

where, Y , β1, β2, Ya, B, p, q and n are material constants, εp is the plastic strain rate, ε0 is

the reference strain rate and T is the temperature. The last two terms are athermal components

of flow stress and they are the same forms as the ZA model in equation 4.3. The first term is

thermal flow stress and it is related to the strain rate and temperature. It was modified from the

ZA model and derived by using the concept of thermal activation energy as well as the dislocation

interaction mechanism where, the mobile dislocation density evolution was also taken into account.

By modifying the thermal component of flow stress, the prediction capability at high strain rates and

temperatures is improved as they reported in [76].

4.2.4 Preston-Tonk-Wallace Model

This model was developed by Preston et al. [77] to describe the behavior of materials at

very high strain rates. It is a complex constitutive model proposed based on the dislocation motion

during plastic deformation. According to the theory, thermal activation mechanism of dislocations

has a significant influence on the deformation by weak shocks that cause strain rates up to 105s−1.

However, the strain rate in explosively driven deformations or in high-velocity impacts is sometimes

much higher than 105s−1, and thus the plastic constitutive model based on only the thermal activation

mechanism can result in a significant error. In order to model the material behavior accurately at a

strain rate up to 1012s−1, Preston et al. proposed a plastic constitutive model considering nonlinear

dislocation drag effects that are predominant in a strong shock regime. The model is given by

equation 4.7.

τ = τs +1

p

(s0 − τy

)ln

1−[1− exp

(− p τs − τy

s0 − τy

)]exp

− pθψ

(s0 − τy)[exp(pτs−τys0−τy

)]

(4.7)

where, τ is a normalized flow stress and equals to τ/G ( τ is the shear stress and G is the shear

35


modulus ). τs and τy are the normalized work hardening saturation stress and normalized yield stress,

respectively. The variables, p, q, and s0 are dimensionless material constants. Following equations

define τs and τy.

τs = max

{s0 −

(s0 − s∞

)erf

[κ T ln

(γζ/ε

)], s0

(ε/γζ

)β }(4.8)

τy = max

{y0 −

(y0 − y∞

)erf

[κ T ln

(γζ/ε

)], min

[y1

(ε/γζ

)y2, s0

(ε/γζ

)β }(4.9)

where the material constants s0 and s∞ are the values that τs takes at zero temperature and very

high temperature, respectively. y0 and y∞ have analogous interpretations. κ and γ are dimensionless

material constants. Scaled temperature T is also defined by TTm

, where the absolute temperature is T

and the melting temperature is Tm. The parameter ζ is defined by

ζ =1

2

(4πρ

3M

)1/4(Gρ

)1/2

(4.10)

where ρ is the density and M is the atomic mass. ζ−1 has the meaning of the time required for

a transverse wave to cross an atom. Shear modulus G is taken to be a function of density and

temperature; G(ρ, T ) = G0(ρ)(1− αT ), where α is the G dependency on scaled temperature and

G0 is the shear modulus at absolute temperature. Constants y0, y1, y2, y∞, s0, s1, s∞, κ, γ, p, and β

are fitted from experiments. In 2009 Kim and Shin [85] modified the existing PTW to be applicable

in wide ranges of strain, strain rate, and temperature.

4.2.5 Khan-Huang-Liang Model

Khan et al. [79, 80] introduced this model based on the JC model. They considered the

work hardening as a coupled function of strain and strain rate. Later, Huh et al. [87, 88] presented

the modified version of KHL model. This model correlated better to the experimental results in

compression with the KHL model. Equation 4.11 represents the flow stress in the modified KHL

model.

σ =

(A+B

(1− ln ε∗

lnDp0

)n1

εn0

)(1 + C(ln ε∗)p

)(1− T ∗m) (4.11)

36


where ε∗ = ε/εref , n1 = a (1− T ∗b), m = m1 +m2.lnεp and Dp0 is chosen to be 106 per second.

The material constants of the proposed model have some physical meaning. A is the yield stress at

the quasi static state. B and n0 represent the work hardening at the quasi static strain rate, C and

p are the coefficients of the strain rate hardening. a and b represent the change of work hardening

rate with respect to the temperature and finally, m1 and m2 are the strain rate dependent thermal

softening parameters, respectively.

4.2.6 Gao-Zhang Model

This physically based model was proposed by Gao and Zhang in 2011 [81] for deformation

of material response at strain rates higher than 104 s−1. As mentioned above, some of the previous

models do not have the ability to predict the correct material behavior (e.g. copper) at strain rates

above 104 s−1. Unlike the other models, the density of dislocations is not assumed to be constant in

this model. In fact, it is defined as a function of equivalent plastic strain, strain rate and temperature.

The flow stress in this model is given as follows,

σ = σth + σath (4.12)

σth = C

√√√√{1 + tanh

[c0 log

(ε

εs0

)]}(ε

εs0

)c1T {1− exp

[−k0

(ε

εs0

)−c1Tε

]}{

1−[−c2 T ln

(ε

ε0

)]1/q}1/p

(4.13)

σath = σG +B[1− exp (−ka0 ε)

]1/2(4.14)

where σth and σath are thermal and non-thermal stress components, respectively. C is reference

thermal stress. k0, c0, c1, c2, p and q are all material constants for thermal stress component. ε0 and

εs0 are reference strain rate and saturated strain rate, respectively. σG is the stress due to initial

defects, B and ka0 are material constants for non-thermal stress component.

37


4.2.7 Summary of All Constitutive Models

There are numerous of material models which might predict the behavior of material in

high strain rate applications. Among all, only six of them are described in this section. Some of

them are simple models, while others require many material constants. Determining these constants

are quite complicated and requires a large number of experiments. Rahmati et al. [89] compared

all six different material models mentioned above in his paper. They utilized cold spray experiment

data, and built six different user subroutines for each material model for Abaqus software. Pure

Copper material properties was used for both particle and substrate in simulations. According to this

paper, jetting phenomenon can only be estimated by the JC and the PTW models. VA, modified KHL

and modified ZA models did not predict the flow stress over a wide range of strains and strain rates.

Thus, they cannot predict the cold spray process accurately. GZ model overestimates the flow stress

for copper and cannot anticipate the deformed shapes of particles. This model was also unable to

predict the critical velocity of particles attaching to the substrate. JC model is the simplest model

among all six material models, and it is very popular in high strain rate applications. Therefore,

for almost all metals, material constants have been reported in the literature. However, this model

underestimates the flow stress at very high strain rates. Therefore, for impact simulations using this

criteria overpredict the deformation. In addition, in this model the strain rate and temperature effects

are uncoupled which implies that the strain rate sensitivity is independent of temperature.

4.3 Bilinear Johnson-Cook Model

As mentioned in the previous section, the classic JC material model only works fine in the

low or intermediate strain rates regimes. the bilinear JC model compensates this issue by adding an

extra condition for high strain rates. is really high. Equations 4.15 and 4.16 explain more about this

material model.

σ =(A+Bεnp

)(1 + C ln

(εpε0

))(1− T ∗m

)(4.15)

C =

C1 εp < εc

C2 εp ≥ εc(4.16)

The proportionality constants C1, C2 and the reference strain rate ε0 are determined

experimentally. The coefficient C and the reference strain rate ε0 control the strain-rate dependent

38


hardening behavior. Experiments show that as the plastic strain rate increases, the relation between

stress and log of strain rate is not linear anymore (Figure 4.1).

Manes et al. presented these coefficients as shown in Table 4.1. They used the experimental

results [90, 66] and optimized the parameters for a ballistic impact application [13].

Supporting Technologies

data for the Ti-6Al-4V alloy. The figures also showthe regions of the stress-strain curves that aredominated by discrete-obstacle plasticity and bydrag-controlled plasticity. For both alloys, the agree-ment between the model predictions and experimen-tal data is excellent.

Gurson Void Growth Model

Observations have been made that ductile frac-ture in metals is related to the nucleation andgrowth of voids. Conventional plasticity models, forexample, von Mises, are based on the assumptionof plastic incompressibility and can not predict thegrowth of voids during yielding. Studies have indi-cated18–20 that void growth during tensile loadingis related to the hydrostatic component of stress,and that this porosity increase directly affectsmaterial yielding.

In these observations it was assumed that thematerial surrounding a void was incompressible.Gurson8 proposed a pressure-sensitive macroscopicyield surface that relates void growth to the evolu-tion of microscopic (pointwise physical quantities ofthe matrix material) and macroscopic quantities toaccount for the behavior of void-containing solids.Here, macroscopic refers to the average values ofphysical quantities, which represent the materialaggregate behavior. As defined by Gurson, the yieldsurface for a ductile material is:

(12)where σo is the tensile flow stress of the micro-scopic matrix material, q and p are the equivalent

Φ =

+

+

=q

q fp

q fσ σ0

2

10

222

32

1 0cosh – ,

stress and hydrostatic stresses of the macroscopicmaterial, and f is the current void volume fractionwhich is a function of the initial porosity, the voidgrowth, and nucleation during yielding. The materialparameters q1, q2 are defined by Gurson.

The Gurson model was added to NIKE2D byB. Engelmann. For the current study, a version ofthe NIKE2D Gurson model was modified to correctlyaccount for the evolution of plastic strain in themicro (matrix) material and to account for strainrate sensitivity. The model was added to DYNA3D.

The response of a notched bar under uniaxialtensile loading was simulated to demonstrate theDYNA3D application of the Gurson model.Substantial hydrostatic tension is created in thenotched regions of the bar for this type of loading.This hydrostatic stress accelerates void growth andleads to the eventual coalescence of voids andductile failure of the bar. Failure was assumed tocorrespond to the loss of load-carrying capability inthis displacement-controlled simulation.

The bar was assumed to have the following mater-ial properties: E = 20.7 GPa, υ = 0.3, yield stress =690 MPa, with a linear hardening modulus of 1,540MPa. The initial void fraction was assumed to beequal to 0.050.

The initial and deformed shapes of the tensilespecimen are shown in Fig. 11, which also depictsthe regions of predicted high void growth. The effectof rate-dependence is shown in Fig. 12, where anincreased loading rate resulted in an increasednormalized axial load (actual axial load/initialyield strength), with softening similar to the rate-independent Gurson model results.

Also shown in Fig. 12 is the conventional plas-ticity solution, which does not exhibit the

Engineering Research Development and Technology6-12

●

10–4 10–2 100 102 104

Strain rate (s–1)

300

350

400

450

500St

ress

(M

Pa)

Drag-control

PredictionData

Discrete-obstacle-controlled

●

●

●

●

●

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●●

● ●●●●

Figure 9. Comparison between the stress-strain rate behav-ior predicted by the mechanism-based material model andexperimental data for aluminum alloy 6061-T6. Regions ofthe stress-strain rate curve that are dominated by discrete-obstacle plasticity and drag-controlled plasticity are shown.

10–310–4 10–110–2 101100 103 104102

Strain rate (s–1)

1500

1400

1300

1200

1100

1000

900

Stre

ss (

MPa

)

Drag-controlled

PredictionData

Discrete-obstacle-controlled

●

●

●

●●

●

●●

Figure 10.Comparison between the stress-strain rate behaviorpredicted by the mechanism-based material model andexperimental data for titanium alloy Ti-6Al-4V. Regions ofthe stress-strain rate curve that are dominated by discrete-obstacle plasticity and drag-controlled plasticity are shown.

TA 610 Lesuer_qk 7/27/99 9:29 AM Page 6-12

Figure 4.1: Comparison between the stress-strain rate behavior predicted by the mechanism-based

material model and experimental data for aluminum alloy 6061-T6 [12].

Lesuer et al. [12] explained that the deformation in terms of strain rate is controlled by two

sequential processes; the cutting (or bypassing) of obstacles by dislocations, and the drag on moving

dislocations by phonons or electrons. They believe that different rates represent the deformation

kinetics associated with discrete obstacles or drag. The Figure 4.1 shows the regions of the stress

versus strain rate curve that are dominated by discrete-obstacle plasticity or by drag-controlled

plasticity.

Table 4.1: Bilinear Johnson-Cook material properties for Al-6061

A,MPa B,MPa n C1 C2 m Tm ε0 εc

JC-4 [66] 270 154.3 0.2215 0.002 0.1301 1.34 925 1 597.2

JC-5 [13] 270 138.2 0.1792 0.002 0.1301 1.34 925 1 597.2

Figure 4.2 shows the flow stress as two functions of strain and strain rate. The definitions

for JC-1 to JC-5 can be found in Tables A.1 and 4.1. Left picture is the classic JC plotted for a

39


constant temperature and the right picture is the bilinear JC. The transition from low strain rate to

high strain rate is clear in the right picture. The jump in this figure can be interpreted as an extra

hardening for the material.

2 | P a g e

fitted line is 0.1301. This stress-strain rate relation consist of two linear fitted line, that’s why they call it Bilinear Johnson-Cook equation and can be summarized as follows:

1 0

2 0

and 1 if and if

p c

c p c

CC

C

A, MPa B, MPa C2 , K

JC-4 [4] 270 154.3 0.2215 0.002 0.1301 1.34 925 597.2

JC-5 [3] 270 138.2 0.1792 0.002 0.1301 1.34 925 597.2

Table 1: constants for bilinear JC flow stress

Figure 2 represents the flow stress as two functions of strain and strain rate. Left picture is the classic JC plotted in three different temperatures and the right picture is the bilinear JC. The transition from low strain rate to high strain rate is clear in the right picture. The jump in this figure can be interpreted as an extra hardening for the material.

Figure 2: flow stress in classic JC (left) and Bilinear JC (right)

If we consider the simulation one more time, we will understand that in the first 25 ns of the impact (before rebounding happens) the strain rate of the particle is in its maximum amount.

Figure 4.2: flow stress in classic JC (left) and Bilinear JC (right)

4.4 Conclusion

In this chapter various material models used to represent the material behavior of high

strain rates. The introduced Bilinear JC model is going to be used in the next chapter to simulate

impact of Al-6061 particles on Sapphire substrate.

40

Chapter 5

Optimization of Bilinear Johnson-Cook

parameters

5.1 Introduction

The goal of this chapter is to calibrate the material constants in the bilinear Johnson-Cook

(JC) model for Al-6061 particle impact simulations. To this end, a multi-variable parameter fit

procedure was performed by using the results of the impact experiments performed by the UMass-

Amherst group [11] and finite element simulations.

In particular, a modified version of the method of steepest descent was used to find the

values of the six JC-model parameters by fitting the aspect ratio of the experimentally obtained

particle shapes to the simulated ones (Section 5.4). Results were further improved by considering

the possibility of size variability of the original particles (Section5.5). The average JC-parameters

obtained by the parameter fit were shown to be very effective in predicting the deformed particle

shapes for a wide range of 175 – 699 m/s impact velocities (Section 5.6).

The computational process for this section was done in Python and Abaqus. The analysis

is set to be dynamic explicit, and 3D elements are utilized to mesh the particle and substrate. Each

element has 8 nodes and 1 integration point, and all material properties are defined temperature

dependent except the density.

41

CHAPTER 5. OPTIMIZATION OF BILINEAR JOHNSON-COOK PARAMETERS

5.2 Background

In the optimization procedure, temperature dependent material properties are obtained

from the MPDB [9] database software, presented in Chapter 3 were used as mentioned earlier. Tables

5.1 and 5.2 depict other properties used for Aluminum-6061 and Sapphire.

Table 5.1: Material properties of Al-6061 [15, 9].

Density, kg/m3 2700

Reference Temperature, K 293

Melting Temperature, K 925

Plastic heat conversion ratio 0.9

Table 5.2: Temperature dependent material properties of Sapphire [14].

20◦C 500◦C 1000◦C

Density, kg/m3 3980

Elastic Modulus,GPa 416 390 364

Poisson′s ratio 0.231 0.237 0.244

Thermal conductivity, W/mK 33 11.4 7.22

Specific heat, J/kgK 755 1165 1255

Thermal Expansion coefficient, 10−6K−1 4.6 7.1 8.1

The Johnson-Cook material model is used to characterize the dependence of the yield

stress σ on the plastic strain εp, rate of plastic strain εp and temperature T , as follows,

σ =(A+Bεnp

)(1 + C ln

(εpε0

))(1−

(T − TrTm − Tr

)m)(5.1)

whereA is the static yield stress of material, parametersB, and n are the strain-hardening parameters,

C controls the strain rate hardening, ε0 is the reference strain rate, Tr is a reference temperature

and Tm is the melting temperature of the material, and m is the temperature exponent. The classic

JC-model has 5 experimentally determined constants A, B, n, C, m. Recent split-Hopkinson bar

experiments for Al-6061 show that strain-rate hardening increases dramatically at high strain rates

42


[12]. Manes et al. [66, 13] summarized this observation in the context of Johnson-Cook model [72]

by modifying the strain rate coefficient C as follows,A. Manes et al. / Procedia Engineering 10 (2011) 3477–3482 3481

10-4

10-2

100

102

104

0.9

1

1.1

1.2

1.3

1.4

1.5

Strain rate (s-1)

σ/σ st

(-)

Experimental data Lesure et al.J-C [5]

10-4

10-2

100

102

104

0.9

1

1.1

1.2

1.3

1.4

1.5

Strain rate (s-1)

σ/σ st

(-)

Experimental data Lesure et al.Optimized model

Fig. 3. Stress-strain rate behavior in the semi-log plane: C=0.002 and 0ε =1 s-1 (left); stress-strain rate behavior in the semi-log

plane: C=0.1301 and 0ε =597.2 s-1 (right). In both cases the solid line is saturated at the value of 1 because in LS-DYNA 0εrepresents the threshold under which the strain rate effects are negligible (the second term of the equation 2 is equal to the unity).

0 0.1 0.2 0.3 0.4 0.50

200

400

600

800

1000

Engineering strain (-)

Eng

inee

ring

stre

ss (

MP

a)

ExperimentalNumerical

278.20.0

Fig. 4. Comparison between experimental end numerical stress-strain curves (up to 50% of deformation): ~4600 s-1, ~3000 s-1, ~1900 s-1, static (left); temperature increment distribution (K) in the specimen at ~4100 s-1 and 5 mm of stroke.

For what concerns the strain rate sensitivity identification, usually, only the parameter C is considered as an optimization variable, while 0ε is set equal to 1 s-1. In Figure 3 there is the comparison between the

experimental stress-strain rate results and the J-C model obtained with the strain rate sensitivity coefficients taken from [5]. The experimental data are shifted according to the requirement of 0ε =1 s-1

and normalized respect to the stress obtained in the test at the lowest strain rate.

s--1Stress-strain rate behavior in the semi-log plane: C=0.002 and ε =1 (left); stress-strain rate behavior in the semi-logε0εs--1plane: C=0.1301 and ε =597.2 (right). ε0ε Figure 5.1: Stress-Strain rate behavior in a semi log plot [13].

C =

C1 εp < εc

C2 εp ≥ εc(5.2)

C1 and C2 are two experimentally determined coefficients that show the additional increase in the

yield stress when the plastic strain rate εp is greater than an experimentally determined critical plastic

strain rate εc . In the small strain rate region (εp < εc), the yield stress is a weak function of strain

rate with less than 2% effect on the yield stress. On the other hand, the yield stress can increase

more than 200% in the high strain rate region. Thus the strain rate effects in the bilinear JC-model

are characterized by C1, C2 and εc. The bilinear Johnson-Cook material parameters for Al-6061,

obtained by split-Hopkinson bar test are given in Table 5.3.

Table 5.3: Bilinear Johnson-Cook material properties for Al-6061 obtained by SHPB test.

A,MPa B,MPa n C1 C2 m Tm ε0 εc

270 154.3 0.2215 0.002 0.1301 1.34 925 1 597.2

43


5.3 Optimization Process

Express the bilinear-JC model parameters in the following vector,

x ={A, B, C2, m, n, εc

}(5.3)

Define the ellipticity ratio R for the deformed particle as follows,

R =D1

D2(5.4)

where the dimensions D1 and D2 are demonstrated on a deformed particle in Figure 5.2. The

ellipticity ratio can be defined for the experimental measurements (Re) and for the simulations

(Rs). Note that ellipticity ratio of the simulations Rs depends on the chosen JC parameters, that is

Rs = Rs(x). The following objective function (normalized least square error) is defined to measure

the error between the two ellipticity ratios,

E(x) =(Re −Rs)2

R2e

× 100 (5.5)

For a given deformed particle (e.g. Figure 5.2) the objective (error) function is minimized by using a

modified version of the method of steepest decent [91, 92]. The x vector which minimizes the error

(Equation 5.5) is found iteratively. The values given in Table 5.3 are used as initial conditions. The

gradient of the objective function is determined numerically with respect to each parameter xi of the

x vector. The parameter xi which should be optimized is identified by comparing all the gradients.

The one with the maximum absolute value is modified as follows,

x(k+1)i = x

(k)i − ni x

(k)i (5.6)

where the correction coefficient is determined from,

n =1√∑N

i=1

(∂E∂xi

)2

(∂E

∂xi

)(5.7)

44


and N is the length of the x vector. The procedure outlined by equations 5.6 and 5.7 is repeated

until the error is minimized for each element of the x vector. Figure 5.3 illustrates the optimization

process.

5 | P a g e

Figure 1 : SEM images on which the parameter fitting is based [4], and the definition of the dimensions D1 and D2 used in calculating the ellipticity ratio Re.

D1

D2

Figure 5.2: SEM images on which the parameter fitting is based [11] and the definition of the

dimensions D1 and D2 used in calculating the ellipticity ratio Re.

45


‐ Slightly Modified Method of Steepest descent is used for optimization process

Initial Values for JC parameters

Find critical parameter: Max{ }

Update the critical parameter

, ∑

Calculate , ∈ , , , , , ,

Is a local minimum?

Remove this parameter from x vector

Y

N

END

All parameters removed from x?

Y

N

Figure 5.3: Optimization process with modified steepest descent method.

5.4 Optimization Results

In this section, the optimized parameters for the bilinear-JC model are presented. UMass-

Amherst experiments that we considered (Figure 5.2) consist of six different particles impacting a

sapphire substrate at six different velocities. The reported particle diameters Dr were determined

from SEM images before impact to be 20.75, 24.4, 24.3, 22.7, 19.75 and 23.4 µm. The impact

velocities of these particles were measured to be 175, 286 416, 530, 663 and 699 m/s, respectively.

Particles were collected post impact and their morphology was imaged by SEM . We digitized these

images by using Matlab. The ellipticity ratio of Re was determined from these images. Details of the

optimization procedure outlined in the previous section are given below for each case.

Figure 5.4 shows the computed and measured particle shapes by using the bilinear-JC

model parameters reported by Manes et al. [66] (Table 5.3). The detailed progression of the

optimization procedure is outlined in Table 5.4 to Table 5.9. The columns marked in green, indicate

the parameter on which optimization is performed. For example in Table 5.9, attempt-1 identified C2

as the critical parameter, followed by n in attempt-2. All parameters converged to a very tight error

tolerance value after attempt-3.

46


Table 5.4: Bilinear Johnson-Cook parameters at each attempt of the optimization process for the

case-1, where Dp = 20.75 µm, and Vp = 175 m/s.

A,MPa B,MPa n C2 m εc E(%)

initial 270 154.3 0.2215 0.1301 1.34 597.2 0.1482

Attempt1 270 154.3 0.2215 0.027 1.34 597.2 1.47E-04

Attempt3 270 154.3 0.2215 0.027 1.74 597.2 1.90E-06




initial 270 154.3 0.2215 0.1301 1.34 597.2 1.536

Attempt1 270 154.3 0.2215 0.011 1.34 597.2 1.21E-04

Attempt3 270 154.3 0.2215 0.011 1.44 597.2 2.02E-06




initial 270 154.3 0.2215 0.1301 1.34 597.2 3.122

Attempt1 270 154.3 0.2215 0.027 1.34 597.2 1.61E-03

Attempt3 270 154.3 0.24 0.027 1.34 597.2 5.50E-04




initial 270 154.3 0.2215 0.1301 1.34 597.2 2.472

Attempt3 270 154.3 0.2215 0.055 1.34 597.2 2.43E-04

47





initial 270 154.3 0.2215 0.1301 1.34 597.2 15.103

Attempt1 270 154.3 0.2215 0.03 1.34 597.2 0.0759

Attempt2 270 154.3 0.2215 0.027 1.34 597.2 0.0034

Attempt3 270 154.3 0.26 0.027 1.34 597.2 3.47E-05




initial 270 154.3 0.2215 0.1301 1.34 597.2 17.736

Attempt1 270 154.3 0.2215 0.03 1.34 597.2 0.0728

Attempt2 270 154.3 0.2215 0.027 1.34 597.2 0.0197

Attempt3 270 154.3 0.27 0.027 1.34 597.2 7.16E-09

The particle shapes computed by using the optimized bilinear JC-parameters are reported

in Figure 5.5. These shapes show a remarkable resemblance to the overall shape of the actual,

deformed particles. Nevertheless, there is a difference in the “volumes” of the two shapes that cannot

be reconciled by parameter fitting. This point will be addressed in more detail below.

Table 5.10 gives a summary of the optimized bilinear JC parameters for all six cases

considered in this work. The C2 parameter is the most sensitive parameter that was adjusted by the

optimization algorithm. The algorithm did not findA,B and εc to have any effect on the minimization

of the objective function. The C1 value reported in Table 5.10 was not part of the optimization

process and was not included in the x vector which is mentioned earlier. This parameter corresponds

to the low strain rate regime in which we can barely find a material point in our simulations. The

average values of all of the parameters are reported in Table 5.10.

48


a) Dp = 20.75 m, and Vp = 175 m/s b) Dp = 24.4 m, and Vp = 286 m/s

c) Dp = 24.3 m, and Vp = 416 m/s d) Dp = 22.7 m, and Vp = 530 m/s

e) Dp = 19.75 m, and Vp = 663 m/s f) Dp = 23.4 m, and Vp = 699 m/

Figure 1 : Comparison of the deformed particle by using the initial JC parameters.

Figure 5.4: Comparison of the deformed particle by using the initial JC parameters.

49


a) Dp = 20.75 m, and Vp = 175 m/s b) Dp = 24.4 m, and Vp = 286 m/s

c) Dp = 24.3 m, and Vp = 416 m/s d) Dp = 22.7 m, and Vp = 530 m/s

e) Dp = 19.75 m, and Vp = 663 m/s f) Dp = 23.4 m, and Vp = 699 m/s

Figure 2 : Comparison of the deformed particles by using the optimized JC parameters.

Figure 5.5: Comparison of the deformed particles by using the optimized JC parameters.

50


By using the modified steepest descent method, we were able to reduce the objective

function to a negligible amount. However, the match between the deformed shapes of particles in the

experiments and the optimized simulations is not yet exact. Error introduced while measuring the

size of the particle from an oblique SEM image could be one of the reasons for this mismatch. In

addition, there could be some error in the initial volume and/or diameter of the particles. In the next

section we investigate the effect of volume and particle size.

Table 5.10: Summary of the optimized values of the parameters for the bilinear JC equation

A,MPa B,MPa n C1 C2 m εc

Manes et al. [66] 270 154.3 0.2215 0.002 0.1301 1.34 597.2

Case1 : Dp = 20.75µm,Vp = 175m/s 270 154.3 0.2215 —– 0.027 1.74 597.2

Case2 : Dp = 24.40µm,Vp = 286m/s 270 154.3 0.2215 —– 0.011 1.44 597.2

Case3 : Dp = 24.34µm,Vp = 416m/s 270 154.3 0.24 —– 0.027 1.34 597.2

Case4 : Dp = 22.74µm,Vp = 530m/s 270 154.3 0.2215 —– 0.055 1.34 597.2

Case5 : Dp = 19.75µm,Vp = 663m/s 270 154.3 0.26 —– 0.027 1.34 597.2

Case6 : Dp = 23.40µm,Vp = 699m/s 270 154.3 0.27 —– 0.027 1.34 597.2

average 270 154.3 0.239 0.002 0.029 1.42 597.2

Standard Deviation 0 0 0.0215 0 0.0143 0.1602 0

5.5 Estimated Particle Diameters

In the previous section it was mentioned that the volume of the particles needed to be

adjusted. First, we need to prove that the volume of the particle does not change after the impact.

Then, by calculating the volume based on the deformed shape, the initial diameter of the particle can

be computed.[93]

It is known that, the volume of a metal undergoing plastic deformation remains unchanged

during deformation, and the volume that is experiencing elastic deformation will change as follows,

∆V

V0= trace(ε) (5.8)

However, this elastic volume dilatation is recoverable. Moreover, according to the molecu-

lar dynamic (MD) simulations insignificant amount of the FCC structures of the Aluminum convert

51


to BCC during deformation. Therefore, we expect the volume of the particle to remain unchanged.

Figure 5.6 shows time history of the particle volume computed for case-4. The change in the com-

puted particle volume in the first 100 ns is insignificant. Note that the increase in volume is due to

thermal expansion which is expected to diminish at steady state. Therefore, we conclude and expect

that the initial and post-impact particle volumes to remain the same.

11 | P a g e

Table 10 : Summary of the optimized values of the parameters for the bilinear JC equation.

A B C1 C2 m n c

Manes et al. [1] 270 154.3 0.002 0.1301 1.34 0.2215 597.2Case 1: Dp = 23.40 m, Vp = 699 m/s 270 154.3 ---- 0.027 1.34 0.27 597.2Case 2: Dp = 19.75 m, Vp = 663 m/s 270 154.3 ---- 0.027 1.34 0.26 597.2Case 3: Dp = 22.74 m, Vp = 530 m/s 270 154.3 ---- 0.055 1.34 0.2215 597.2Case 4: Dp = 24.34 m, Vp = 416 m/s 270 154.3 ---- 0.027 1.34 0.24 597.2Case 5: Dp = 24.40 m, Vp= 286 m/s 270 154.3 ---- 0.011 1.44 0.2215 597.2Case 6: Dp = 20.75 m, Vp = 175 m/s 270 154.3 ---- 0.027 1.74 0.2215 597.2

average 270 154.3 0.002 0.029 1.42 0.239 597.2standard deviation 0 0 0 0.0143 0.1602 0.0215 0

Table 11 : Diameter and volume estimates of the particles. Dr is the reported particle diameter [4]. Vr is the particle volume based on Dr. VA is the particle volume computed by Abaqus based on Dr after meshing. VSEM is the particle volume estimated based on the post-impact SEM image of the particle. DSEM is the particle diameter based on VSEM. DBFis the particle diameter that best fits the experimental results obtained by using different sizes in Abaqus. E-DSEM and E-DBF are the %-errors in particle diameters with respect to Dr.

Case Dr Vr VA VSEM DSEM E-DSEM DBF E-DBF

6 20.75 4677.92 4658.17 3646.23 19.05 8.2 19.15 7.75 24.40 7606.21 7491.36 6085.58 22.60 7.4 22.20 9.04 24.30 7513.07 7491.36 5098.80 21.30 12.3 21.10 13.23 22.70 6124.58 6104.18 3841.69 19.39 14.6 19.10 15.92 19.75 4033.67 4016.63 3261.21 18.36 7.0 19.00 3.81 23.40 6708.82 6686.48 4463.24 20.38 12.9 21.20 9.4

Figure 5 : Time history of the particle’s volume in case-3. Note that the small increase in volume is due to thermal expansion.

6.00E+03

6.10E+03

6.20E+03

6.30E+03

6.40E+03

6.50E+03

6.60E+03

0 20 40 60 80 100 120

Volu

me,

μm

3

Time, ns

Case 3, Vi=530 m/s

Figure 5.6: Time history of the particle’s volume in case-4. Note that the small increase in volume is

due to thermal expansion.

By taking into consideration that the volume of a particle should not change, we can

estimate an initial diameter for the particle, based on its final volume. To this end, first, an image

processing technique is used to measure the post-impact volume of the particles. The border of a

particle is extracted as a 2D plot. Then, the approximate volume (VSEM ) is calculated by rotating

this 2D picture about its middle axis. Note that this is an approximation of the volume, because, it

is assumed that the particle is perfectly symmetrical. Nevertheless, a particle diameter (DSEM ) is

computed based on this volume. The error (E −DSEM ) between this diameter and the reported

diameter (Dr) is on the order of 10% as shown in Table 5.11. It is reasonable to state that this error

has contribution from both the initial measurements and post-impact volume assessment procedure.

Second, the original particle diameter is estimated by making it as a fitting parameter.

The simulations were run with different initial particle diameters until the error with respect to the

post-impact SEM images is minimized. This is done by visual inspection. The particle diameter that

52


gives the smallest error is deemed the best fit diameter DBF . Table 5.11 shows that in general the

error (E −DBF ) is also on the order of 10%. More interesting is the fact that the estimated particle

diameters DBF and DSEM , which are obtained by completely different approaches, are very close

to one another.

Figure 5.7 shows the simulation results, based on estimated DSEM particle diameters. In

each case the red curves represent computed particle shape of the optimized bilinear JC parameters

reported in Table 5.10. The blue curves represent the particle shapes obtained from the SEM images.

The volumetric error mentioned in the previous section is reduced remarkably well.

Table 5.11: Diameter and volume estimates of the particles. Dr is the reported particle diameter [11].

Vr is the particle volume based on Dr. VA is the particle volume computed by Abaqus based on Dr

after meshing. VSEM is the particle volume estimated based on the post-impact SEM image of the

particle. DSEM is the particle diameter based on VSEM . DBF is the particle diameter that best fits

the experimental results obtained by using different sizes in Abaqus. E −DSEM and E −DBF are

the %-errors in particle diameters with respect to Dr.

Case Dr Vr VA VSEM DSEM E−DSEM DBF E−DBF

1 20.75 4677.92 4658.17 3646.23 19.05 8.2 19.15 7.7

2 24.40 7606.21 7491.36 6085.58 22.60 7.4 22.20 9.0

3 24.30 7513.07 7491.36 5098.80 21.30 12.3 21.10 13.2

4 22.70 6124.58 6104.18 3841.69 19.39 14.6 19.10 15.9

5 19.75 4033.67 4016.63 3261.21 18.36 7.0 19.00 3.8

6 23.40 6708.82 6686.48 4463.24 20.38 12.9 21.20 9.4

53


a) Vp = 175 m/s b) Vp = 286 m/s

c) Vp = 416 m/s d) Vp = 530 m/s

e) Vp = 663 m/s f) Vp = 699 m/s

Figure 3 : Comparison of the experimental particle contours with those that are computed by using the adjusted particle diameter DSEM and the modified JC parameters reported in Table 10. Note that the JC parameters for each impact speed were different as reported in Table 10.

Figure 5.7: Comparison of the experimental particle contours with those that are computed by using

the adjusted particle diameter DSEM and the modified JC parameters reported in Table 5.10. Note

that the JC parameters for each impact speed were different as reported in Table 5.10.

54


5.6 Verification

Next, the average values of the optimized bilinear JC parameters reported in Table 5.10 are

used to simulate the impact process. DSEM values reported above were used as the diameters of the

undeformed particles. Figure 5.8 shows the comparison of these simulations with the experimentally

obtained contours. The match is generally very reasonable.

Further verification is obtained by comparing the coefficient or restitution (COR) as a

function of impact velocity in experiment and simulation (Figure 5.9). In these simulation the average

values of the modified bilinear JC parameters reported in Table 5.10 utilized in conjunction with the

adjusted particle diameter DSEM available in Table 5.11. The figure is plotted in the logarithmic

scale. Experimental data is illustrated by colored dots, and each color represents a specific particle

diameter. Simulations are depicted by two lines. The lines are the upper limit and lower limit of the

particle sizes available in this figure. Both experiment and simlation show that the results are not a

function of particle diameter. Results show a good resemblance between simulation and experiment.

5.7 Conclusion

In this chapter a combined experimental and simulation approach was presented in order to

find the parameters of a bilinear-JC flow stress relationship for Al-6061. It was shown that by using

the modified method of steepest decent and by allowing the particle volume to be a fit parameter,

remarkably good match between experimental and theoretical results can be obtained.

55


a) Vp = 175 m/s b) Vp = 286 m/s

c) Vp = 416 m/s d) Vp = 530 m/s

e) Vp = 663 m/s f) Vp = 699 m/s

Figure 4 : Comparison of the experimental particle contours with those that are computed by using the adjusted particle diameter DSEM and the average value of the modified JC parameters reported in Table 10.

Figure 5.8: Comparison of the experimental particle contours with those that are computed by using

the adjusted particle diameter DSEM and the average value of the modified JC parameters reported

in Table 5.10.

56


Coefficient of restitutionImpact speeds 50 950 m/sMeasurements: Particle diameters 15 – 25 m (UMass Amherst, May 2016)Simulations: 15 and 25 m particle diameter (Calibrated, bilinear JC model)

Dp = 15 μmDp = 25 μm

Figure 5.9: Comparison of the experimental coefficient of restitution with simulated results. Lines

represents the simulation data and dots are the experiments.

57

Chapter 6

Conclusion

6.1 Summary and Conclusion

In this thesis, an investigation of micron-scale particles, impacted on substrate at supersonic

velocities is presented. The major goal of this work was to study the results of single particle impact

in both experiments and computer simulations to find a material model that can predict the behavior

of material under very high strain rates, based on those observations. To this end, the finite element

method has been used to simulate this physical phenomenon. In these 3D analyses, high strain

rate plasticity, temperature dependent material properties, heat generation and contact algorithm

have been used. Since, impact velocities vary from 50 to 1000 m/s and diameter of particles are in

the range of 20 to 50 micro-meters, simulating all combinations of particle diameters and impact

velocities are very time consuming. Therefore, Python scripts were used to automate all the process

of creating a geometry, running the simulation to creating the outputs and updating input file for the

later simulations.

The main focus of this thesis was to find a material flow model that is suitable to simulate

mechanics of Al-6061 particles on a sapphire substrate. The classic Johnson-Cook and the bilinear

Johnson-Cook material models were considered to this end. The equation and constants of classic

JC constitutive law, were obtained using Split Hopkinson Pressure Bar. In these experiments, the

equation was optimized to give an appropriate results in the range of 10−4 to 104 s−1. The bilinear

JC model, was proposed to predict the material behavior under ultra high strain rates. In the high

velocity impact simulation, it was observed that the strain rate go beyond 108 in certain regions

near the impact interface. Therefore, the classic Johnson-Cook model was not able to predict the

deformed shape of particles after the impact. As the classic-JC model does not account for high

58

CHAPTER 6. CONCLUSION

strain rate strengthening, it is not suitable for particle impact simulations.

Since, there is no large scale experiment equipment to measure the yield stress of material

as a function of strain rate, this relationship should be obtained by using an optimization technique.

The bilinear JC model has six different constants. By using the method of steepest descent, and

considering the least square error of the ellipticity ratio of deformed particles as an objective function,

we were able to optimize the bilinear JC model for the high strain rate applications.

6.2 Future Work

In this work, impact of a single particle onto a substrate has been simulated under various

scenarios by using the finite element method. The material parameters of the bilinear JC model are

optimized, leading to very good agreement between the experiments and simulations. This work can

be extended in a number of ways. The following is some recommendations for future work, some of

which are already under progress:

I. In this study, impact of single Aluminum particle to Sapphire substrate was studied. The results

of the bilinear JC model can be used to model the single Aluminum particle to Aluminum

substrate. The cohesive zone model (CZM) can also be applied to these simulation for

studying the bonding/rebound situation. The cohesive stress in these cases can be a function of

temperature, pressure or both. The resultant simulations can be compared to experimental data

for validations and further investigations such as interface strength.

II. Although mechanics of single particle impact, reveals valuable information about the particle

deformation, multi-particle impact simulations could be a reasonable goal to achieve. By

simulating the multi-particle impact, a lot of new areas of studies will be opened. For instance,

a metal plate with a surface defect can be simulated. Then the cold spray coating can be

applied on top. Finally, the strength of the new plate can be measured based on the tensional

force, torsional force and fatigue loading. These results can be compared to the results of a

plate with defects to reveal new ideas and improve the conditions of coating and spraying in

real world applications.

III. In section 4.2, six different constitutive laws were explained. All of the models are capable

of predicting the behavior of material at very high strain rates. One of the interesting future

studies, could be the modeling of all constitutive laws and trying to compare the result of all

59

CHAPTER 6. CONCLUSION

simulation results. Most probably some of them need to be optimized to give proper results. In

that case, the same method of optimization, explained in chapter 5 can be applied.

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69

Appendix A

Simulation Results Using the Classic and

Bilinear Johnson-Cook Material Models

A.1 Simulation Results Using the Classic JC Model

In this section, the deformed shapes of the particles in the experiments are compared

to simulations. The impact experiments were performed in UMass at Amherst [10]. In these

experiments [11], Aluminum 6061 particles and a sapphire substrate were used. The idea behind

these experiments was to study only the deformation of the particles. The diameters of the particles in

these experiments were in the range of 19 to 24 micrometers and the impact velocities were between

200 and 1100 m/s.

Simulations were conducted by using Abaqus/explicit 6.13-2. All experimental conditions

were simulated to duplicate these experiments as closely as possible. Dynamic explicit was used

in simulate that includes coupled temperature and displacement degrees of freedom. In these

simulations, Three different configurations for the particle and the substrate were compared:

I. Deformable particle and substrate (Young’s modulus of substrate is set to 1000 times greater

than the particle’s).

II. Deformable particle and rigid substrate.

III. Deformable elastic-plastic particle and perfectly elastic substrate.

These modelling approaches are referred to as the type-I, type-II and type-III models, here after.

Elastic plastic behavior with exponential isotropic hardening is utilized for the mechanical properties

70

APPENDIX A. SIMULATION RESULTS USING THE CLASSIC AND BILINEAR JC MODEL

of the particle in these simulations. There are three sets of Johnson-Cook parameters that can be

found in the literature for material behavior of Al-6061. The information about these three can be

seen in Table A.1. Other material properties are reported in Chapter 3.

Table A.1: Classic Johnson-Cook material properties for Al-6061

A,MPa B,MPa n C m Tm ε0

JC-1 [12] 324 114 0.42 0.002 1.34 925 1

JC-2 [94] 200 203.4 0.35 0.011 1.34 925 1

JC-3 [13] 270 154.3 0.2215 0.002 1.34 925 1

Deformed shapes of particles are presented in Figures A.1, A.2 and A.3, using three

different modeling approaches and the three JC parameters (From Table A.1). All simulations results

compared to the single particle impact experiments [10]. In each figure, (a) and (b) show the scanning

electron microscope (SEM) image of the particle after impact. Figure (c) represents the type-II

analysis using JC-1 parameters (Table A.1) as a material model. In each figure, (d), (e) and (f)

display the results for type-III modeling approach using JC-1, JC-2, and JC-3 as a material models,

respectively.

we found that the shock waves produced by the impact is very large for the type-I analysis,

where the particle and substrate are deformable but the Young’s modulus of the substrate is much

greater than the particle. This requires very small time increments putting a large computational

cost on simulations in terms of time and capacity. For instance, a typical analysis by using type-III

approach, JC-3 material model and taking the advantage of multiprocessing completes in 15 to 20

minutes, however, type-I analyses takes 5 to 6 hours. Therefore, type-I was not pursued any further.

Although simulations reveal reasonable match between the experiment and analysis at

slow impacts, as the impact velocity increases, the finite element predictions become less accurate.

In general, simulations predicts more deformation than what is observed in the experiments. It can

be concluded that the flow stress predicting by the classic Johnson-Cook equation needs to be more

sensitive to the strain rate. Therefore, to compensate the excessive deformation, the flow stress should

provide higher yield stress. In other words, the amount of hardening predicted by JC due to plastic

deformation is not sufficient to reproduce the exact deformations as experiments.

71


dp p = 556 m/s, Mp = 9.8x10-12 kg, KEp = 2.8x10-6 J

a) Front view of deformed particle b) Top view of deformed particle

c) Type II with JC-1 d) Type III with JC-1

e) Type III with JC-2 f) Type III with JC-3

10 m 10 m

30.4

9.1

31.4

9

31

8.8

30.6

8.8

Figure A.1: Comparison of the deformed shape of particle between experiment (a, b) and simulations

(c, d, e, f) where the impact velocity. is 556 m/s

72


dp p = 691m/s, Mp = 7.6x10-12 kg, KEp = 3.6x10-6 J

a) Front view of deformed particleb) Top view of deformed particle



10 m10 m

35.1

6

35.9

5.6

34.8

5.6

35.9

5.5



73


dp p = 859m/s, Mp = 1.1x10-11 kg, KEp = 8.3x10-6 J

a) Front view of deformed particle b) Top view of deformed particle



10 m

10 m

60

3.7

57.1

3.6

55.3

3.8

58.4

3.4



74


The simulation results show the limitations of the Johnson-Cook model at high velocities.

According to Johnson and Cook papers [72, 73], this material model is calibrated for application

in the range of 10−4 − 104s−1. However, simulation results reveals that due to very high shear

deformation, the strain rate usually goes up to 108s−1 in the contact interface regions. Figure A.4 is

a transparent contour of strain rate at a specific increment time. This plot shows that almost all of the

particle elements are in the second regime of stress-strain rate plot, where εp > εc.

3 | P a g e

Figure 3 is a transparent contour of strain rate at a specific increment time. This plot shows that almost all of the particle elements are in the second regime of stress-strain rate plot. It means that the new bilinear Johnson-Cook flow stress can provide more hardening for the material model. This extra hardening will prevent the particle from severe deformation and it helps to match the deformed shapes of particles in simulations and experiments. This could be one of the solutions which could help to anticipate the behavior of the dynamic impact phenomenon precisely.

Figure 3: Strain rate contour of iso-surfaces

2. Results

Phase-I experiments of UMass Amherst University [6] is used as a reference to compare the simulation results with experiments. In these experiments Aluminum 6061 and sapphire is used for the particle and substrate material respectively. A summary of material characteristics are available in table 2.

Table 2: Experiments material characteristics

Figure A.4: Strain rate contour of iso-surfaces

Therefore, a modified material model is needed to predict the precise deformation in high

strain rates. Manes et al. [13, 66] suggested a Bilinear Johnson-Cook material model for Aluminum

6061 and they calibrated the constants for their specific application with the method of optimization.

In the next section, the simulations results using the bilinear Johnson-Cook are compared to the

experiments.

A.2 Simulation Results Using the Bilinear JC Model

The deformed shapes of particles are presented in Figures A.5, A.6 and A.7, using two

bilinear JC parameters (Table 4.1). All simulations results compared to the single particle impact

experiments conducted at UMass-Amherst by Professor J. H. Lee [11]. In each Figure, (a) and (b)

refer to the SEM images of the particle after the impact, (c) represents the analysis using a set of

JC parameters that Manes et al. [66] found for Aluminum 6061 material and using Split Hopkinson

pressure bar test. In the same picture, (d) displays the results for another set of JC parameters which

is optimized by Gilioli et al. [90] to find the exact deformation shape of a bullet impact on the

helicopter tail rotor transmission.

Schematically, it is clear that the particles simulated by using the bilinear JC material

model, deformed less than simulations performed by using the classic JC material model. Therefore,

75


we conclude that the extra term which is added to the classic JC equation ”C2” can play a significant

role in high strain rate applications and it can compensate for the underestimation of classic JC

material model.

To compare classic the JC results with bilinear JC model, a new parameter is defined. The

Ellipticity ratio is defined as the maximum vertical distance to the maximum horizontal distance

in the front view of deformed particle shape. This study shows that in three different cases that

comparison have been made, the bilinear JC model improved the ellipticity ratio by 89%, 138% and

246% for the cases where the impact velocity was 556 m/s, 691 m/s and 859 m/s, respectively.

dp = 18.5μm, Vp = 556 m/s, Mp = 9.8x10-12 kg, KEp = 2.8x10-6 J

a) Front view of deformed particle

b) Top view of deformed particle

c) Type III with JC-4

10μm 10μm

22.8

12.4

d) Type III with JC-5

22.9

12.4


(c, d) where the impact velocity. is 556 m/s

76


dp = 17.5μm, Vp = 691m/s, Mp = 7.6x10-12 kg, KEp = 3.6x10-6 J



c) Type III with JC-4

10μm

10μm

26.6

9.7

d) Type III with JC-5

26.8

9.7



77


dp = 20.0μm, Vp = 859m/s, Mp = 1.1x10-11 kg, KEp = 8.3x10-6 J



c) Type III with JC-4 d) Type III with JC-5

10μm

10μm

37.8

7.6

38.1

7.5



78

Appendix B

Table of Temperature Dependent

Material Properties for Aluminum 6061

Data is taken from MPDB material database software [9].

Temperature Elastic Modulus Poisson’s

Ratio

Thermal

Expansion

Thermal

Conductivity

Specific Heat

K GPa 10−6/K W/m K J/kg K

10 76.5951 0.3239 0.2525 14.3824 1.5609

20 76.5621 0.3240 0.5045 28.4645 8.8829

30 76.4967 0.3242 0.9477 41.0270 33.4402

40 76.4011 0.3243 1.7028 52.2049 81.9529

50 76.2777 0.3245 2.8316 62.1332 149.1508

60 76.1285 0.3248 4.3375 70.9469 223.9242

70 75.9555 0.3250 6.1652 78.7811 297.9033

80 75.7608 0.3253 8.2007 85.7707 368.3660

90 75.5463 0.3255 10.2179 92.0509 433.5356

100 75.3137 0.3258 11.8261 97.7565 492.5810

110 75.0647 0.3262 13.1962 103.0226 545.6162

120 74.8011 0.3265 14.3611 107.9842 593.7009

130 74.5243 0.3268 15.3513 112.7788 637.9319

140 74.2358 0.3271 16.1946 116.8674 677.6165

79

APPENDIX B. TEMPERATURE DEPENDENT MATERIAL PROPERTIES FOR AL-6061

150 73.9370 0.3274 16.9163 120.6998 712.0432

160 73.6291 0.3278 17.5390 124.2883 742.6290

170 73.3136 0.3281 18.0826 127.6453 771.5068

180 72.9913 0.3284 18.5646 130.7830 798.7493

190 72.6634 0.3287 18.9998 133.7137 824.4276

200 72.3309 0.3290 19.4002 136.4498 848.6113

210 71.9946 0.3293 19.7756 139.0036 871.3683

220 71.6554 0.3296 20.1328 141.3873 892.7655

230 71.3138 0.3298 20.4761 143.6133 912.8678

240 70.9707 0.3301 20.8074 145.6938 931.7389

250 70.6264 0.3303 21.1257 147.6413 949.4410

260 70.2816 0.3306 21.4275 149.4679 966.0347

270 69.9364 0.3308 21.7067 151.1861 981.5790

280 69.5914 0.3310 21.9547 152.8080 996.1318

290 69.2465 0.3312 22.1677 154.3460 1009.7491

300 68.9021 0.3314 22.3724 155.8133 1022.4857

310 68.5580 0.3315 22.5752 157.5203 1034.3947

320 68.2143 0.3317 22.7762 159.1733 1045.5277

330 67.8709 0.3319 22.9753 160.7722 1055.9351

340 67.5275 0.3320 23.1726 162.3171 1065.6656

350 67.1838 0.3321 23.3680 163.8080 1074.7662

360 66.8395 0.3323 23.5616 165.2448 1083.2828

370 66.4941 0.3324 23.7533 166.6276 1091.2597

380 66.1470 0.3325 23.9431 167.9564 1098.7395

390 65.7977 0.3327 24.1311 169.2311 1105.7634

400 65.4454 0.3328 24.3173 170.4518 1112.3714

410 65.0892 0.3329 24.5016 171.6185 1118.6016

420 64.7284 0.3331 24.6840 172.7311 1124.4907

430 64.3620 0.3333 24.8646 173.7898 1130.0742

440 63.9889 0.3335 25.0433 174.7943 1135.3858

450 63.6079 0.3337 25.2202 175.7449 1140.4577

460 63.2179 0.3339 25.3952 176.6414 1145.3208

80


470 62.8176 0.3342 25.5684 177.4839 1150.0045

480 62.4055 0.3345 25.7397 178.2723 1154.5365

490 61.9803 0.3349 25.9091 179.0067 1158.9432

500 61.5403 0.3353 26.0767 179.6871 1163.2495

510 61.0839 0.3357 26.2425 180.3134 1167.4786

520 60.6094 0.3362 26.4064 180.8858 1171.6525

530 60.1150 0.3368 26.5684 181.4040 1175.7914

540 59.5987 0.3374 26.7286 181.8683 1179.9144

550 59.0588 0.3382 26.8869 182.2785 1184.0388

560 58.4929 0.3390 27.0434 182.6347 1188.1804

570 57.8992 0.3399 27.1980 182.9368 1192.3538

580 57.2752 0.3409 27.3570 183.1850 1196.5717

590 56.6187 0.3420 27.5120 183.3790 1200.8456

600 55.9274 0.3433 27.6650 183.5191 1205.1855

610 55.1986 0.3446 27.8170 183.6051 1209.5998

620 54.4300 0.3461 27.9680 183.6371 1214.0954

630 53.6187 0.3478 28.1180 183.6151 1218.6778

640 52.7622 0.3496 28.2660 183.5390 1223.3509

650 51.8576 0.3515 28.4130 183.4089 1228.1173

660 50.9019 0.3537 28.5590 183.2247 1232.9778

670 49.8923 0.3560 28.7040 182.9865 1237.9321

680 48.8256 0.3585 28.8480 182.6943 1242.9780

690 47.6988 0.3613 28.9900 182.3481 1248.1122

700 46.5085 0.3642 29.1310 181.9478 1253.3295

710 45.2514 0.3674 29.2710 181.4935 1258.6236

720 43.9243 0.3708 29.4100 180.9852 1263.9864

730 42.5235 0.3745 29.5480 180.4228 1269.4085

740 41.0455 0.3785 29.6840 179.8064 1274.8790

750 39.4867 0.3828 29.8200 179.1359 1280.3854

760 37.8433 0.3873 29.9550 178.4115 1285.9137

770 36.1116 0.3922 30.0880 177.6330 1291.4486

780 34.2720 0.3974 30.2210 176.8004 1296.9731

81


790 32.3480 0.4030 30.3520 175.9139 1302.4689

800 30.3240 0.4089 30.4830 174.9732 1307.9159

810 28.1950 0.4152 30.6120 173.9786 1313.2929

820 25.9560 0.4218 30.7410 172.7800 1315.3000

830 23.6040 0.4289 30.8690 171.3400 1319.2000

840 21.1330 0.4365 30.9950 169.6600 1322.9000

850 18.5400 0.4444 31.1210 167.6900 1326.1000

82

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