HUGUES BERGER-PELLETIER

MODELLING OF SPLIT HOPKINSON PRESSURE BARS

Adaptation of a compression apparatus into tension

Mémoire présenté

à la Faculté des études supérieures et postdoctorales de l’Université Laval

dans le cadre du programme de maîtrise en Génie Mécanique

pour l’obtention du grade de Maître ès sciences (M.Sc.)

DÉPARTEMENT DE GÉNIE MÉCANIQUE

FACULTÉ DES SCIENCES ET DE GÉNIE

UNIVERSTÉ LAVAL

QUÉBEC

2013

© Hugues Berger-Pelletier, 2013

II

Résumé Les Barres d’Hopkinson sont couramment utilisées pour tester les matériaux à des hauts

taux de déformations. Souvent, différents systèmes de barres sont utilisés pour tester les

matériaux en tension ou en compression.

Par contre, il serait pratique d’utiliser un seul système, pour prendre des mesures en

tension et en compression. Des études ont été faites pour convertir le système de

compression existant du centre de Recherche et Développement pour la Défense Canada

(RDDC) de ValCartier.

Un concept a été choisi parmi 6 systèmes de tension déjà existants. Le choix a été validé

avec un modèle d’éléments finis fait sur LS-Dyna. Le modèle a été calibré sur des

résultats de compression fournis par le RDDC. Il fut ensuite modifié pour intégrer le

nouveau concept.

À cause d’un manque de ressources, les résultats de simulation sur LS-Dyna n’ont pu être

comparés avec des résultats expérimentaux, puisqu’un premier prototype n’a pu être

fabriqué.

III

Abstract The Split Hopkinson Pressure Bars (SHPB) is a common method used to characterize

materials at high rates of strain. First used to experiment on materials in compression, the

method was adapted to do tests in tension and torsion. The compression apparatus

consists of a specimen sandwiched between 2 pressure bars, called the input bar and the

output bar. A third bar, the striker, is launched at the input bar. Upon impact, a

compressive pulse traveling toward the specimen is generated. This load is partially

transmitted into the specimen and the output bar, the rest of it being reflected back into

the input bar. Using measurements of the input, transmitted and reflected pulse, it is

possible to develop the stress-strain response of the material deforming at high strain

rates. This is achieved using strain gages adequately placed on both pressure bars.

Many researchers use a different SHPB system when it comes to tension tests. Many

methods exist, but all of them are based on compressive experiments. It would therefore

be convenient to only have one system, which is capable of taking measurements both in

compression and tension. Based on the compressive SHPB apparatus used by the

Defense, Research and Development Canada (DRDC) center in ValCartier, studies were

made to convert the compressive system into a tensile setup.

The goal was to modify it with minimum changes possible, in order to easily go back and

forth between the two configurations. A design choice was made, considering 6 existing

tension systems. To validate the decision, a finite element model was created using LS-

Dyna. The modal was first aligned with the compression results provided and then

modified to implement the selected design. Because of a lack of available resources, LS-

Dyna simulation results were not compared with experimental data, as it was not possible

to create a first prototype.

IV

Acknowledgements This research project would not have been possible without the support of many people. I

wish to express my gratitude to Professor Augustin Gakwaya, who offered great help and

whose assistance was always crucial.

I also wish to express my sincere thanks to Réjean Arsenault, who was a mentor through

this project.

Moreover, I am deeply appreciative of Manon Bolduc’s support and supervision; that

made it possible to complete this research.

Special thanks to Dennis Nandlall that allowed me to work with a team of highly

qualified professionals.

V

Table of contents Résumé ................................................................................................................................ II

Abstract ............................................................................................................................. III

Acknowledgements ........................................................................................................... IV

Table of contents ................................................................................................................ V

List of tables ...................................................................................................................... IX

List of figures ..................................................................................................................... X

Chapter 1: Introduction to Split Hopkinson Pressure Bars ................................................ 1

1.1 Uses of SHPB ....................................................................................................... 2

1.2 SHPB fundamentals .................................................................................................. 2

1.3 Research objective .................................................................................................... 5

1.4 Thesis methodology .................................................................................................. 5

Chapter 2: Historical overview and literature review ......................................................... 7

2.1 Introduction ............................................................................................................... 8

2.2 Historical overview ................................................................................................... 8

2.3 Recent developments ................................................................................................ 9

2.4 Tension SPHB ........................................................................................................... 9

2.4.1 Hollow striker .................................................................................................... 9

2.4.2 Collar/Ring ....................................................................................................... 10

2.4.3 Grooved fixture ................................................................................................ 11

2.4.4 Side bars ........................................................................................................... 13

2.4.5 Hat specimen .................................................................................................... 14

2.4.6 Input tube ......................................................................................................... 15

2.5 Comparison of various tension methods ................................................................. 16

2.5.1 Advantages and disadvantages ........................................................................ 16

VI

2.5.2 Scoring system ................................................................................................. 17

2.6 Concept selection and conclusion ........................................................................... 19

Chapter 3: Split Hopkinson bars theory ............................................................................ 21

3.1 Introduction ............................................................................................................. 22

3.2 Wave propagation in Hopkinson pressure bars ...................................................... 22

3.2.1 Motion equations ............................................................................................. 22

3.2.2 Pressure wave motion in a medium ................................................................. 24

3.2.3 Reflection and Transmission............................................................................ 25

3.2.4 Application to Hopkinson pressure bars .......................................................... 27

3.2.5 Effective length of a specimen ......................................................................... 29

3.3 Lagrangian diagram ................................................................................................ 30

3.3.1 Visiting the wave motion equations ................................................................. 30

3.3.2 Graphical representation .................................................................................. 30

3.3.3 Compression versus Tension ........................................................................... 32

Chapter 4: Simulation considerations ............................................................................... 40

4.1 Introduction ............................................................................................................. 41

4.2 Geometrical considerations ..................................................................................... 41

4.3 Mesh considerations................................................................................................ 41

4.4 Material considerations ........................................................................................... 42

4.4.1 Elastic (Hooke) law.......................................................................................... 43

4.4.2 Plastic-kinematic law ....................................................................................... 44

4.4.3 Johnson-Cook material model with a damage law .......................................... 44

4.5 Conclusion .............................................................................................................. 45

Chapter 5: Compression apparatus model generation ...................................................... 46

5.1 Introduction ............................................................................................................. 47

VII

5.2 Model creation in LS-Dyna ..................................................................................... 47

5.3 Mesh creation .......................................................................................................... 48

5.4 LS-Dyna boundary conditions................................................................................. 49

5.5 Model presentation.................................................................................................. 50

5.6 Results ..................................................................................................................... 50

Chapter 6: Creation of the 3D tension SHPB model ........................................................ 54

6.1 Introduction ............................................................................................................. 55

6.2 Specimen geometry ................................................................................................. 55

6.2.1 Size considerations........................................................................................... 56

6.2.2 Threaded ends and simulation considerations ................................................. 57

6.3 Collar geometry ...................................................................................................... 58

6.4 Model summary ...................................................................................................... 58

6.5 Aluminum specimen simulations ............................................................................ 59

6.5.1 Variable striker speed ...................................................................................... 60

6.5.2 Variable striker length...................................................................................... 62

6.5.3 Variable specimen diameter ............................................................................. 63

6.5.4 Collar/specimen only ....................................................................................... 65

6.5.5 Specimen material ............................................................................................ 68

6.6 Conclusion .............................................................................................................. 71

Chapter 7: Conclusion....................................................................................................... 72

7.1 Thesis retrospective ................................................................................................ 73

7.2 Discussions and recommendations ......................................................................... 74

7.2.1 Machining feasibility ....................................................................................... 74

7.2.2 Specimen adjustment ....................................................................................... 74

7.2.3 Importance of experimental validation ............................................................ 75

VIII

7.3 Conclusion .............................................................................................................. 76

References ......................................................................................................................... 77

Appendix A: Compressive SHPB LS-DYNA model ....................................................... 81

Appendix B: Tensile SHPB LS-DYNA model (example) ............................................... 86

Appendix C: Transmission and reflection coefficients derivation ................................... 91

IX

List of tables Table 1: Methods advantages and disadvantages ............................................................. 16

Table 2: Scores for each advantages and disadvantages ................................................... 18

Table 3: Tension methods total scores .............................................................................. 19

Table 4: Aluminum and Steel properties .......................................................................... 47

Table 5: Main dimensions of the SHPB setup .................................................................. 48

Table 6 : Comparison between theoretical and simulated stress values ........................... 52

Table 7 : Summary of the model properties and characteristics ....................................... 58

Table 8: Johnson-Cook parameters for aluminum 6061-T6 ............................................. 59

Table 9 : Johnson-Cook parameters for Titanium and Weldox steel ................................ 68

X

List of figures Figure 1: Simple sketch representing a standard SHPB system ......................................... 3

Figure 2: Example of stress versus strain curve obtained via SHPB testing [1] ................. 4

Figure 3: Shu et al. tensile SHPB ..................................................................................... 10

Figure 4: Lee et al. collar/specimen assembly .................................................................. 11

Figure 5: Haugou et al. sleeve and specimen assembly .................................................... 12

Figure 6: Haugou et al. full fixture ................................................................................... 12

Figure 7: Fixtures after tensile loading. ............................................................................ 13

Figure 8: Side bars apparatus ............................................................................................ 13

Figure 9 : Specimen used by Eskandari and Nemes ......................................................... 14

Figure 10 : Fixture and specimen assembly ...................................................................... 14

Figure 11 : Hat specimen apparatus .................................................................................. 14

Figure 12: Harding et al. assembly ................................................................................... 15

Figure 13: Pressure wave traveling in a medium .............................................................. 24

Figure 14: Wave propagation from one bar to another ..................................................... 26

Figure 15: Typical specimen geometry............................................................................. 28

Figure 16: General pulse shape ......................................................................................... 31

Figure 17: Wave front traveling through a bar, shown in a Lagrange diagram ................ 31

Figure 18: Collar type of tension SHPB ........................................................................... 32

Figure 19: Lagrange diagrams for tension and compression SHPB ................................. 33

Figure 20: Symmetry of a Lagrange diagram ................................................................... 33

Figure 21: Lagrange diagram showing the spurious wave ............................................... 34

Figure 22: Lagrange diagram for a non-symmetrical SHPB apparatus ............................ 36

Figure 23 : Full Lagrange diagram for the proposed setup ............................................... 39

Figure 24 : Typical SHPB setup ....................................................................................... 42

Figure 25: Nodes correspondence at bars extremities ...................................................... 42

Figure 26: Linear kinematic hardening representation ..................................................... 44

Figure 27 : Finer mesh at the end of the bars .................................................................... 48

Figure 28 : Typical mesh of the cross section ................................................................... 49

Figure 29 : The full model of the compression apparatus ................................................ 50

Figure 30 : Initial gap between the striker and the input bar ............................................ 50

XI

Figure 31 : Pulse going through the input bar for a striker speed of 17.3 m/s .................. 51

Figure 32 : Results, at different striker speeds, presented in A. Bouamoul's report [19] . 52

Figure 33 : Results of the 3D model created for this thesis .............................................. 53

Figure 34 : Standard specimen taken from ASTM volume 3.01 section E8M [22] ......... 55

Figure 35 : Standard specimen shapes, with threaded ends, taken from ASTM volume

3.01 section E8M [22]....................................................................................................... 56

Figure 36: Gap between the collar and the specimen ....................................................... 56

Figure 37 : Representation of the specimen with short ends and no threads .................... 57

Figure 38 : Strains measured at the input bar strain gauges for each speeds .................... 60

Figure 39: Strains measured at the output bar strain gauge, for each speed ..................... 61

Figure 40: Rise time for a 175 mm long striker ................................................................ 62

Figure 41 : Rise time for a 125 mm long striker ............................................................... 63

Figure 42 : Specimen stresses ........................................................................................... 64

Figure 43 : Input bar stress when only the collar is installed............................................ 65

Figure 44: Stress in the output bar when only the collar is installed ................................ 66

Figure 45: Stress in the input bar when only the specimen is installed ............................ 67

Figure 46: Stress probed in the output bar ........................................................................ 69

Figure 47: Stress probed in the input bar .......................................................................... 70

1

Chapter 1:

Introduction to Split Hopkinson Pressure Bars

2

1.1 Uses of SHPB Engineers, material specialists and scientists often use material properties in their

calculations. For structural resistance, the elastic modulus, yield strength, ultimate

strength, Poisson coefficient, etc. are important parameters required to determine any

structure strength.

It is possible to find these parameters in handbooks and scientific papers, where their

values are often based on experimental characterization methods. The most common

experimental method used is the classical traction test, which gives a lot of information

on a material, including the elastic modulus.

However, this test is performed at very low strain rates (in the order of 0.025 mm-s/mm

[1]) and the material is therefore deformed very slowly. Most of the time, material

behavior is different at higher strain rates of deformation. Consequently, when a design

must take into account high rates of strain, it is preferable to use relevant material data.

For example, a car crash simulation should be done using material properties at high

strain rates. This data can be found experimentally using the Split Hopkinson Pressure

Bars or SHPB, which is probably the most common way of finding material properties at

high strain rates, including the widely used strains versus stress curve.

1.2 SHPB fundamentals The standard SHPB (compression) setup includes 3 bars: the striker, the input bar and the

output bar. Normally, these 3 bars are made of the same material. The striker is launched

at the input bar at a specific speed. Upon impact, a stress pulse traveling toward the

specimen is generated in the input bar. That pulse is then partially transmitted into the

specimen and the output bar. The rest is reflected back into the input bar.

3

Figure 1 is a simple sketch representing a standard compressive SHPB system. The figure

also shows 2 strain gages, one on each pressure bar. These gauges record the strains

during the test. The duration depends mostly on the pressure bars materials and lengths.

Figure 1: Simple sketch representing a standard SHPB system

Often, the three bars are made of steel, with high yield strength. It is important that only

the specimen deforms beyond its elastic limit. The following equations, calculating the

specimen behavior in terms of the pressure bars strain history, are valid only if both bars

are deformed within their elastic limit.

Based on wave propagation in solids, the stress in the specimen is given by [2]:

𝝈𝒔(𝒕) = 𝑬𝑨𝟎𝑨𝜺𝑻(𝒕) (1.1)

Where 𝜎𝑠(𝑡) is the stress in the specimen as a function of time, 𝐸 and 𝐴0 are the pressure

bars elastic modulus and section respectively, 𝐴 is the specimen section and 𝜀𝑇(𝑡) is the

transmitted strain history (measured in the output bar).

The specimen strain history is found using the reflected pulse, measured in the input bar:

𝒅𝜺𝒔(𝒕)𝒅𝒕

= −𝟐𝑪𝟎𝑳

𝜺𝑹(𝒕)

(1.2)

4

With this equation, the specimen strain rate 𝑑𝜀𝑠(𝑡)𝑑𝑡

is found, using the input bar length 𝐿,

the wave speed of the wave in the bar 𝐶0 = �𝐸𝜌 and the reflected pulse 𝜀𝑅(𝑡).

The strain is then calculated by integrating the previous equation:

𝜺𝒔(𝒕) = −

𝟐𝑪𝟎𝑳

� 𝜺𝑹(𝒕)𝒅𝒕𝒕

𝟎 (1.3)

Figures of stress or strain versus time are usually not relevant in engineering. Both curves

are often merged together, in order to obtain the commonly used “Stress versus Strain”

curve. An example of such a curve is presented in figure 2.

Figure 2: Example of stress versus strain curve obtained via SHPB testing [1]

The main problem encountered with SHPB is called the “spurious wave reflection”. This

problem appears when the strain gauges are not properly placed on the pressure bars (see

section 3.3.3). For example, if the input bar strain gauge is too close from the specimen,

the readings of the gauge would include both the incident pulse and the reflected pulse.

Such a situation is undesirable and therefore, preliminary calculations must be made in

order to find the proper positioning of the strain gauges on the pressure bars.

5

For tensile SHPB setups, preventing the spurious waves reflection problem requires even

more care in placing the gauges. Depending on what type of tensile apparatus is used, it is

possible that the pressure bars of a system are not necessarily of the same length.

1.3 Research objective The main goal of this research is to find a way to convert a compression SHPB setup into

a tension SHPB setup. There exist many methods that can be used to perform tensile tests

on materials, using split Hopkinson pressure bars. The difficulty is to find the method that

requires the least possible modifications to the existing apparatus, in order to easily return

to the compression configuration. It would therefore be possible to take measurements on

materials both in compression and tension, using only one experimental testing

equipment.

To achieve this objective, the following process was followed:

• Conduit a literature review of SHPB tension test methods

• Conceptual design analysis and concept selection

• 3D LS-Dyna compression model alignment

• 3D LS-Dyna tension model creation

Upon completion of every sub-objective listed above, a valid LS-Dyna model analysis of

the chosen concept will be used to complete the remaining tasks such as:

• Simulation of stress – strain curves for a specific material, in the event that it is

not possible or required to perform extensive experiments.

• Finding material parameters by comparing and adjusting simulated results with

experimental results, in the event that they are unknown.

• Predicting experimental results.

1.4 Thesis methodology Chapter two contains a brief historical overview of the SHPB method. This overview

shows how the method was first created and how it evolved throughout the years. It

finishes with general information on recent developments. The chapter also presents a

literature review of the different tension tests most commonly used. These methods are

6

then compared to find which one is best suited for converting the compression SHPB

apparatus.

Chapter three describes the governing equations of wave motions in solids. These

equations, applied to split Hopkinson pressure bars, give the relation between the

pressure bars strain measurements and the specimen behavior. While the basic

mathematical development is presented for compressive tests, it is also valid for tension.

Therefore, the difference between tension tests and compression tests is presented

qualitatively. Finally, section 3 explains the Lagrangian diagram technique, which plots

the wave front position on the setup in terms of elapsed time.

Chapter four presents issues that have to be considered when creating the Finite Element

Analysis (FEA) model. For instance, this is the time where mesh considerations are

described, as well as constitutive laws to be used in the different models is presented.

Chapter five is dedicated to the creation of the compression SHPB model. It contains

information on how the compression modal was created and aligned.

This is then followed by the simulation of the tension assembly, in chapter six. Both the

creation of the model and the results are described in this chapter.

The thesis concludes with chapter seven, which contains a discussion of the results

presented in the previous chapters. Moreover, recommendations are given regarding

potential future experiments.

7

Chapter 2:

Historical overview and literature review

8

2.1 Introduction Split Hopkinson Pressure Bars testing has been used for a number of years now. This is

one of the reasons why a lot of research has been done on SHPB testing, in order to use it

for different applications. Many areas of engineering rely on SHPB tests to settle issues

such as material resistance, vibration and wave motion and dynamic data recording.

The current chapter is an overview of the historical evolution of SHPB testing. It shows

its first applications and the advancements that were then made to improve it. The most

recent developments are described in section 2.3.

Section 2.4 is a literature review of the tension testing methods. The most common ways

of acquiring tension data with split Hopkinson pressure bars are described and compared.

Taking into account certain criteria, one method is selected for implementation into the

compression DRDC test setup.

2.2 Historical overview In 1913, Bertram Hopkinson developed a test [1] that allows plotting of pressure as a

function of time during impact experiments (or explosions). The test consists of a bar that

is struck at one end, having a small spherical projectile fixed with grease at the other end.

When the pulse reaches the spherical projectile, it is launched into the air and a ballistic

pendulum measures the momentum. Doing a sufficient number of tests, with different

projectiles, it was possible to find the maximum pressure in the bar and the impact

duration. However, it was still difficult to plot a precise relation between pressure and

time.

During the 1940s, Dennison Bancroft solved Pochhammer and Love equations in order to

find mathematical expressions for longitudinal wave speed in a cylindrical bar. The

usefulness of the equations, with regard to split Hopkinson pressure bars testing, was not

found until later, when the arrival of computers made data processing easier. Still in the

40s, Davies found a method of measuring pressure in a bar with condensers. Pressure had

always been measured using a ballistic pendulum until then. Finally, at the end of the 40s,

Kolsky adds a second pressure bar (the output bar). Kolsky’s setup is the one mostly used

today, with two pressure bars, as described in section 1.2. Kolsky also found the relations

9

between the pressure bars stress/strain history to the specimen behavior. Note that Kolsky

was still using the condensers method developed by Davies.

In the 1960s, with the arrival of strain gauges, many improvements were made. Krafft, et

al. was the first to implement these strain gauges into a SHPB test. They were then

followed by other researchers, notably Hauser, et al. and Lindholm and Yeakly.

2.3 Recent developments Since the introduction of strain gauges, most developments were made based on the

arrival of computers and data recording devices, such as oscilloscopes. Apart from new

data recording methods, Kaiser [1] elaborated in 1998 a method to correct for data

dispersion. He also developed a way of merging the stress versus time curve with the

strain versus time curve, hence plotting the standard stress versus strain curve. In 2002,

Tasneem [3] studied techniques to shape the strain pulse that travels through the pressure

bars, using finite element analysis tools.

Many researchers also found new applications of the SHPB testing assembly. For

example, Al-Mousawi, et al [4], Bateman, et al [5], and Chen, et al [6] developed a

method to test viscoelastic materials, including rubbers, which had been considered too

“soft” for SHPB experiments.

2.4 Tension SPHB The aim of this section is to show the main methods used today and which one could best

be adapted to an already existing compressive SHPB apparatus.

The names given to the different methods of this section were chosen arbitrarily. Most

methods found in the literature don’t have a name and therefore, to easily identify each of

them, the principal characteristic of each setup is used for the name.

2.4.1 Hollow striker

This is the main method where the tensile loading is “direct”. As explained previously,

the initial pulse is normally a compressive one. Most tensile testing setups bypass this

pulse, only to have the reflected tensile wave affect the specimen (indirect loading). But,

10

with the method explained here, a hollow striker generates a tensile pulse at the

beginning of the testing, requiring no bypass.

Shu et al. [7] used this kind of experimental assembly to obtain mechanical properties of

AM50A alloy. Their equipment is shown in figure 3:

Figure 3: Shu et al. tensile SHPB

From figure 3, it is seen that Shu et al. added components to the assembly in order to

improve the results. The hollow striker slides on the input bar, which is connected to the

anvil. When the gun barrel releases the striker, it hits the anvil bar, generating two waves.

One is tensile and travels along the input bar toward the right. The other is compressive

and travels toward the left of the anvil bar. This pulse is then absorbed (by the absorber

bar), since it is of no interest. The tensile wave, upon arrival at the specimen/input bar

interface, is partially reflected and partially transmitted, through the specimen. At this

point, it is basically what was explained in section 1.2, but one should remember that at

this point, the pulse is tensile.

2.4.2 Collar/Ring

In this method (illustrated in figure 3), the loading is indirect and the compressive wave

bypass is achieved by a collar. The collar is inserted between the input and output bar. It

is important not to “fix” the collar to the pressure bars. The whole point of the method is

11

to have the compressive wave completely transmitted to the output bar via the collar.

Because it is not fixed to the pressure bars, the collar can only transmit compressive

loads. Therefore, when the transmitted wave (which should be the entire compressive

wave) is going to reach the free end of the output bar and return as a tensile load, it

should only be “supported” by the specimen. Lee et al. [8] used this system to

characterize different materials in tension.

Figure 4: Lee et al. collar/specimen assembly

Figure 4 shows that the specimen is threaded in order to fix it to the pressure bars. It is

surrounded by the split ring and collar assembly. It is obvious that a fit of the collar as

perfect as possible is required else the specimen will become a transmitter of the

compressive wave.

2.4.3 Grooved fixture

When doing tests on SHPB setups, it is important to have the best possible impedance

match. The impedance of a bar is given by the following expression:

𝑰 = 𝑪𝟎𝝆𝑺 (2.1)

Where 𝐶0 = �𝐸𝜌 is the longitudinal wave velocity in a given material, 𝜌 is the density

and S is the cross section area.

It can be seen that difference in cross section, such as the one at the pressure bar -

specimen interface, creates a mismatch. In order to minimize that mismatch, Haugou et

al. [9] designed a fixture that has approximately the same cross section as the pressure

12

bars (see figure 5). This method works the same way as the collar method, but here the

specimens are still accessible.

Figure 5: Haugou et al. sleeve and specimen assembly

The fixture consists of two grooved maraging steel sleeves. Figure 4 shows only one of

the sleeves. Four specimens are glued with epoxy adhesive in the grooves. The choice of

4 specimens is based on the necessity of having a significant amplitude of the transmitted

pulse. The second sleeve is inserted so that no gap exists between the sleeves. Like the

collar method, a gap between the two sleeves would cause the specimens to be the

transmitter of the compressive pulse.

Figure 6: Haugou et al. full fixture

The centering sleeve shown in figure 5 is present to hold the fixture while the epoxy

adhesive polymerizes. Both ends of the fixture are threaded (not shown in fig. 6), for

assembly with the pressure bars.

Figure 6 presents the full assembly after 2 tests, showing the separation of the two sleeves

that occurred after passage of the tensile wave.

13

Figure 7: Fixtures after tensile loading.

2.4.4 Side bars

The following method, used by Eskandari and Nemes [10] to test composite laminates,

involves a structure of side bars surrounding the pressure bars. This structure is what is

used to bypass the compressive wave. Eskandari and Nemes compared this setup to the

one from section 2.1, meaning it is considered as a direct loading apparatus.

Figure 8: Side bars apparatus

The striker doesn’t, however, hit directly the input bar. It hits the side bars connector, as

can be seen on Figure 8. The compressive wave then propagates along the side bars, until

it reaches the other end connector. Since this connector is fixed to the input bar, the

reflected tensile pulse can be transmitted to the SHPB system itself. Hence, the first pulse

reaching the specimen is a tensile one, which is why the method is considered as “direct”.

14

Eskandari and Nemes used a fixture made of cylinders to hold the specimen (figure 8).

The cylinders are threaded at one extremity to install them on the pressure bars. On the

other extremities, a slot is machined to insert the specimen. In order to fix the specimen

in the slotted cylinders, holes are drilled in the fixture to allow the injection of epoxy

(figure 9).

Figure 9 : Specimen used by Eskandari and Nemes

Figure 10 : Fixture and specimen assembly

Of course, it is noted that this kind of fixture could also be used in other methods, such as

the one explained in section 2.4.2.

2.4.5 Hat specimen

Lindholm and Yeakley [11] proposed a method where a hat-shaped specimen is used.

The apparatus is almost identical to a compressive test, except that the specimen has an

unusual shape.

Figure 11 : Hat specimen apparatus

15

Clearly, when the incident bar strikes the bottom of the specimen, a tensile deformation

occurs in the surrounding surface of the hat (see figure 10). This technique is not accurate

for determining the elastic modulus. In addition, Lindholm et al. [11] found discrepancies

between round and hat-shaped specimen with the strength of the hat specimens being

lower than the round ones. This difference was thoroughly studied and the only error

source found was the geometry of the specimens.

2.4.6 Input tube

This method, developed by Harding et al [12], is made of a hollow input bar (tube) in

which the specimen assembly is installed. To load the specimen, the striker is launched at

the hollow input bar. The compressive pulse then travels along the tube until it reaches

the yoke, then creating tension in the specimen, as shown in the following sketch (figure

11):

Figure 12: Harding et al. assembly

More specifically, the yoke, upon arrival of the compressive pulse, pulls the lower end of

the specimen. The difference of velocity between the two ends loads the specimen in

tension.

16

Harding and Welsh [13] later improved the method by adding an instrumented bar, in

front of the inertia bar, to test fiber-reinforced composites.

2.5 Comparison of various tension methods In order to compare the different setups described in the previous sections, focus was put

on the final objective: adapting an already existing compressive SHPB apparatus. A list

(table 1) was created showing advantages and disadvantages for each experimental

method. Of course, more elements could be added, as this table is subjective. In this list,

letters were attributed for further reference. For example: M1 for “Method 1” or M1A2

for “Method 1 Advantage 2”.

2.5.1 Advantages and disadvantages Table 1: Methods advantages and disadvantages

Method Advantages Disadvantages

M1 : Hollow striker • M1A1: Direct tensile

loading of the specimen

• M1A2: High strain rates

achieved

• M1D1: High amount of

changes required to adapt

a compressive apparatus

M2 : Collar • M2A1: Low amount of

changes required to adapt

a compressive apparatus

• M2A2: Very high strain

rates achieved

• M2D1: Specimen is

hardly reachable when

setup is completely

assembled

M3 : Grooved

sleeves

• M3A1: Moderate amount

of changes required to

adapt a compressive

apparatus

• M3A2: Good impedance

match

• M3A3: High strain rates

achieved

• M3D1: Hard to machine

sleeves for smaller setups

• M3D2: Mainly for sheet

specimen

17

M4 : Side bars • M4D1: High amount of

changes required to adapt

a compressive apparatus

• M4D2: Medium to low

strain rates achieved

• M4D3: Pulse goes

through many

components before

reaching specimen

M5 : Hat specimen • M5A1: Medium amount

of changes required to

adapt a compressive

apparatus

• M5D1: The specimen

geometry affects the

result

M6 : Input tube • M6D1: High amount of

changes required to adapt

a compressive apparatus

• M6D2: Pulse goes

through many

components before

reaching the specimen

2.5.2 Scoring system

The characteristics listed in table 1 above could be attributed a score (from -5 to +5), in

order to improve the comparison. For example, keeping in mind that the goal is to adapt a

compressive SHPB setup, the disadvantage of the hollow striker method could be

considered “worse” than the one from the collar method. Indeed, having a lot of changes

to implement seems “worse” than having almost no access to the specimen. Again, scores

presented in the table are debatable.

Referring to the identifications created in the previous table, the scores are presented in

table 2 are as follows:

18

Table 2: Scores for each advantages and disadvantages

Method Advantages Disadvantages

M1 • M1A1: 5

• M1A2: 3

• M1D1: -5

M2 : Collar • M2A1: 5

• M2A2: 4

• M2D1: -2

M3 : Grooved sleeves • M3A1: 3

• M3A2: 3

• M3A3: 4

• M3D1: -1

• M3D2: -3

M4 : Side bars • M4D1: -4

• M4D2: -3

• M4D3: -3

M5 : Hat specimen • M5A1: 3 • M5D1: -4

M6 : Input tube • M6D1: -4

• M6D2: -3

Of course, the score given to each advantage and disadvantage is subjective. Therefore,

the final choice that will be recommended in this report is still an open issue.

19

The total scores for each method are then shown in table 3:

Table 3: Tension methods total scores

Method Score

M1 3

M2 7

M3 6

M4 -10

M5 1

M6 -7

However, even if this table suggests considering the collar method, it would be wiser to

analyze more thoroughly two methods. In fact, only the range of the score is significant

with the number itself being of little interest. From the previous discussion, it is clear that

M2 and M3 showed the methods that are potentially feasible. However, readers could

disagree with this conclusion and more elements could be added to the comparison.

2.6 Concept selection and conclusion Section 2.4 and 2.5 presented different methods used to test materials, in tension, at high

strain rates, using split Hopkinson pressure bars. Six methods were shown and while

there are probably others, these six were the ones most often found in the literature.

Most setups consist of indirect loading of the specimen, using different methods to

bypass the initial compressive pulse. Two methods, however, are considered as “direct”

and the one using a hollow striker is widely used.

With regard to the 2 first sub-objectives of section 1.3, advantages and disadvantages

were found for each tensile method. A score was then given, allowing a better

comparison since every advantage and disadvantage has a different importance.

20

After discussing the results and for the purpose of this thesis work, it was chosen to focus

the analysis on the collar/ring method. It is the one that shows to have the best potential

for easy implementation on a compressive SHPB apparatus.

21

Chapter 3:

Split Hopkinson bars theory

22

3.1 Introduction This chapter is dedicated to the development of the equations governing the split

Hopkinson pressure bars experiments. Most of the development is derived from wave

propagation in solids.

In the first section, the general theory behind wave propagation in solids is explained. It

is followed by its application to SHPB method and how it is possible to extract indirectly

the specimen behavior.

Another section describes the Lagrangian diagram method, primarily used to find the best

positioning of each strain gauge on the pressure bars. In this diagram, one can see the

position of the wave front in the assembly, in function of time. For example, it is possible

to determine where and when there is a superposition between the reflected and the

transmitted pulse, therefore rejecting this location as a good one for strain gages.

Finally, simulating the whole assembly using finite element analysis requires knowledge

in continuum mechanics. More specifically, explanations on material behavior are

presented in a third section. While modeling the pressure bars is fairly easy, since they

are limited to elastic behavior, it is harder to model the specimen which deforms beyond

its elastic limit.

3.2 Wave propagation in Hopkinson pressure bars Split Hopkinson pressure bars are usually made of steel, because of its very high elastic

limit. Experiments are always deforming the pressure bars elastically. Therefore,

considering the pressure bars are made of steel, linear elastic behavior can be considered

in the mathematical development that will follow. Moreover, the test consists of a stress

pulse traveling longitudinally through a cylindrical bar. The following equations can then

be regarded for the longitudinal wave propagation case.

3.2.1 Motion equations

The local force equations of motion, in indicial notion, are stated as:

𝝈𝒊𝒋,𝒋 + 𝝆𝒃𝒊 = 𝝆𝒂𝒊 (3.1)

23

With 𝜎𝑖𝑗 being the stress tensor, 𝑏𝑖 body forces, 𝑎𝑖 acceleration and 𝜌 material density

(considered constant). In the case of split Hopkinson pressure bars, body forces can be

neglected and only the stress gradient is considered, giving:

𝝏𝝈𝒊𝒋𝝏𝒙𝒋

= 𝝆𝒂𝒊 (3.2)

Since 𝑎𝑖 = 𝜕2𝑢𝑖𝜕𝑡2

, with 𝑢𝑖 being the displacements tensor, then:

𝝏𝝈𝒊𝒋𝝏𝒙𝒋

= 𝝆𝝏𝟐𝒖𝒊𝝏𝒕𝟐

(3.3)

All this can be rewritten considering uniaxial deformation as:

𝝏𝝈𝟏𝟏𝝏𝒙𝟏

= 𝝆𝝏𝟐𝒖𝟏𝝏𝒕𝟐

(3.4)

From Hooke’s law, uniaxial stress-strain relation (with the hypothesis of small

displacements) is:

𝝈𝟏𝟏 = 𝑬𝜺𝟏𝟏 = 𝑬

𝝏𝒖𝟏𝝏𝒙𝟏

(3.5)

Where E is the material elastic modulus. Using equations (3.4) and (3.5) together leads to

the uniaxial wave motion equation, with 𝐶0 = �𝐸𝜌

𝑪𝟎𝟐

𝝏𝟐𝒖𝟏𝝏𝒙𝟏𝟐

=𝝏𝟐𝒖𝟏𝝏𝒕𝟐

(3.6)

24

3.2.2 Pressure wave motion in a medium

In the case of a pressure wave traveling in a medium, as shown in the following figure

13, the solution [2] to equation 3.6 is of type:

𝒖(𝒙, 𝒕) = 𝒇 �𝒕 −𝒙𝑪𝟎� + 𝒈�𝒕 +

𝒙𝑪𝟎� (3.7)

Figure 13: Pressure wave traveling in a medium

In equation 3.7, f and g are two arbitrary functions. It is obvious that, since the pulse is

traveling in the positive x direction, function 𝑔 �𝑡 + 𝑥𝐶0� could be discarded. The

displacement is given by equation 3.7 and depends on the fact that the boundary

conditions are:

𝒖 = �̇� = 𝟎 𝒇𝒐𝒓 𝒕 = 𝟎,𝒂𝒕 𝒙 > 𝟎 (3.8)

Using these boundary conditions, Achenbach [15] showed that

𝒖(𝒙, 𝒕) = 𝒇 �𝒕 −𝒙𝑪𝟎� − 𝑨 𝒇𝒐𝒓 𝒕 >

𝒙𝑪𝟎

𝒖(𝒙, 𝒕) = 𝟎 𝒇𝒐𝒓 𝒕 <𝒙𝑪𝟎

(3.9)

25

Therefore, a particle located at 𝑥 = 𝑥1 remains at rest until time 𝑡 = 𝑡1 = 𝑥1𝐶0

, showing

that the wave travels at a speed 𝐶0 in the medium. Note that A is a constant, required to

satisfy the boundary conditions defined by equation 3.8.

Again, using the boundary conditions and mathematical integrations shown in [2], the

stress in the medium is given:

𝝈𝒙(𝒙, 𝒕) = −𝒑(𝒕 −𝒙𝑪𝟎

) (3.10)

3.2.3 Reflection and Transmission

When a propagating pulse meets an interface, that may be either a change of material or a

free end, it is either partially transmitted with a portion being reflected or fully reflected,

as in the case of a free end. The nature of the interface then determines how much the

transmitted or reflected pulse is dispersed or disturbed.

For an incident stress wave of shape:

(𝝈𝒙)𝒊 = 𝒇 �𝒕 −𝒙𝑪𝟎� (3.11)

the reflected wave, traveling in the other direction, would be of shape:

(𝝈𝒙)𝒓 = 𝒈�𝒕 +𝒙𝑪𝟎� (3.12)

with the total stress being:

𝝈𝒙 = (𝝈𝒙)𝒓 + (𝝈𝒙)𝒊 (3.13)

The case of the free end boundary condition is simpler with the pulse being totally

reflected and of the opposite sign, as shown by Graff [16]. This means that, a

26

compression wave will reflect as a tension wave and vice versa, at a free end.

Mathematically, using equations 3.11to 3.13 and considering that the stress must vanish

at the free end:

𝝈𝒙 = (𝝈𝒙)𝒓 + (𝝈𝒙)𝒊 = 𝟎

𝒈�𝒕 +𝒙𝑪𝟎� = −𝒇�𝒕 −

𝒙𝑪𝟎�

(3.14)

This clearly shows that reflected wave is of the opposite sign to the incident pulse.

For the case of transmission into another rod, the impedance ratio between the two bars is

what determines how well the wave is transmitted. The lower the impedance mismatch,

the better is the transmission.

Considering the case of the figure 14, taken from [14]:

Figure 14: Wave propagation from one bar to another

At the interface, the following principle must hold : “At each of the two pressure bar –

specimen interfaces, the velocity of each material just to the left and right of the interface

must be equal, since they are in intimate contact at all times. The forces just to the left

and right of each interface must balance one another to satisfy equilibrium. For the case

of figure 14, equilibrium of forces and continuity of velocities thus require satisfaction of

the following:

𝑨𝟏(𝝈𝒓 + 𝝈𝒊) = 𝑨𝟐𝝈𝒕

𝒗𝒊 + 𝒗𝒓 = 𝒗𝒕 (3.15)

These equilibrium conditions lead to the transmission and reflection equations:

𝜎𝑖

𝜎𝑟 𝜎𝑡 A1, ρ1, C1

A2, ρ2, C2

27

𝝈𝒕 =𝟐𝑨𝟏𝝆𝟐𝑪𝟐

𝑨𝟏𝝆𝟏𝑪𝟏 + 𝑨𝟐𝝆𝟐𝑪𝟐𝝈𝒊 (3.16)

𝝈𝒓 =𝑨𝟐𝝆𝟐𝑪𝟐 − 𝑨𝟏𝝆𝟏𝑪𝟏𝑨𝟏𝝆𝟏𝑪𝟏 + 𝑨𝟐𝝆𝟐𝑪𝟐

𝝈𝒊 (3.17)

The transmission coefficient 𝛼 could be expressed as:

𝜶 =𝟐𝑨𝟏𝝆𝟐𝑪𝟐

𝑨𝟏𝝆𝟏𝑪𝟏 + 𝑨𝟐𝝆𝟐𝑪𝟐 (3.18)

The reflection coefficient 𝛽 is therefore:

𝜷 = 𝟏 − 𝜶 (3.19)

Considering the previous development, one can see that by varying 𝜌 and E (therefore C),

different impedance matches can be achieved. Users of SHPB setups often have two sets

of pressure bars, one in steel and another one in aluminum, for example. Recalling that

the impedance definition, as shown in equation 2.1, is 𝐼 = 𝜌𝑆𝐶0, aluminum has an

impedance 3 times smaller than that of steel (for the same surface). Hence, when testing

specimen made of low density materials, one could use aluminum pressure bars.

3.2.4 Application to Hopkinson pressure bars

Wave propagation relations can be applied specifically for SHPB apparatus, in order to

link specimen strain and stress to measurements made on the two pressure bars. When

experimenting with split Hopkinson pressure bars, no strain gages are normally installed

on the specimen and strain readings from the pressure bars are what is used to plot the

specimen behavior.

From the previous work, we can say that, for the incident and reflected pulse:

𝒖(𝒙, 𝒕) = 𝒇(𝒙 − 𝑪𝟎𝒕) + 𝒈(𝒙 + 𝑪𝟎𝒕) = 𝒖𝒊 + 𝒖𝒓 (3.20)

𝝐 = 𝒇′ + 𝒈′ = 𝜺𝒊 + 𝜺𝒓 (3.21)

28

�̇� = 𝑪𝟎(−𝒇′ + 𝒈′) = 𝑪𝟎(−𝜺𝒊 + 𝜺𝒓) (3.22)

Remember that the incident, reflected and transmitted strains 𝜀𝑖, 𝜀𝑟 and 𝜀𝑡 are known

through strain gauges placed on the pressure bars. Also, the displacement 𝑢1 is given by,

considering equation 3.20:

𝒖𝟏 = 𝑪𝟎 � (−𝜺𝒊 + 𝜺𝒓)

𝒕

𝟎𝒅𝒕 (3.23)

In order to have the expression of 𝑢2, we must rewrite equations 3.20 to 3.23 for the

output bar:

𝒖(𝒙, 𝒕) = 𝒉(𝒙 − 𝑪𝟎𝒕) = 𝒖𝒕 (3.24)

𝝐 = 𝒉′ = 𝜺𝒕 (3.25)

�̇� = −𝑪𝟎𝒉′ = −𝑪𝟎𝜺𝒕 (3.26)

𝒖𝟐 = −𝑪𝟎 � 𝜺𝒕

𝒕

𝟎𝒅𝒕 (3.27)

With the displacements at the specimen ends, it is possible to find its elongation and its

average strain:

𝜺𝒔 =

𝒖𝟐 − 𝒖𝟏𝒍𝒔

=𝑪𝟎𝒍𝒔� (−𝜺𝒕 + 𝜺𝒊 − 𝜺𝒓)𝒕

𝟎𝒅𝒕 (3.28)

where 𝑙𝑠 is the initial length of the specimen. Note that for a specimen that doesn’t have a

uniform shape, such as the one shown in figure 15 below, it is better to use the effective

length 𝑙𝑒𝑓𝑓. A short explanation of the matter is presented in section 3.2.5.

Figure 15: Typical specimen geometry

29

The forces, acting on each faces of the specimen, are simply:

𝑷𝟏 = 𝑨𝟏𝑬(𝜺𝒊 + 𝜺𝒓) (3.29)

𝑷𝟐 = 𝑨𝟐𝑬𝜺𝒕 (3.30)

The rest of the development is based on the hypothesis that the specimen is deformed

uniformly, which means, that the forces on each sides are equal, giving 𝑃1 = 𝑃2 and

therefore, considering equations 3.29 and 3.30: (𝜀𝑖 + 𝜀𝑟) = 𝜀𝑡. Using this assumption

with equation 3.28 and effective length 𝑙𝑒𝑓𝑓:

𝜺𝒔 =

−𝟐𝑪𝟎𝒍𝒆𝒇𝒇

� 𝜺𝒓𝒕

𝟎𝒅𝒕 (3.31)

The average stress in the specimen is then given by:

𝝈𝒔 =

𝑷𝟏𝑨𝒔

=𝑷𝟐𝑨𝒔

= 𝑬𝑫𝟎𝟐

𝑫𝒔𝟐 𝜺𝒕 (3.32)

where 𝐴𝑠 is the specimen cross-section area and 𝐷0 and 𝐷𝑠 are the diameter of the

pressure bar and of the specimen respectively.

3.2.5 Effective length of a specimen

As mentioned by Elwood et Al. [15], the strain in the uniform narrow section is more

important than the one at the extremities. Because of its smaller cross-section, the

narrower part of the specimen will deform more at the passage of a stress wave. Also,

Elwood et Al. [15] found that the effective length 𝑙𝑒𝑓𝑓 is basically independent of strain

rate.

Normally, the effective length is found experimentally. Using a small strain gauge placed

on the narrow section of the specimen shown in figure 15, results of equation 3.31 can be

compared with actual measurements of the strains in the specimen. In this work,

simulations results are going to be used instead of experiments results.

30

3.3 Lagrangian diagram When designing a split Hopkinson pressure bars setup, it is important to carefully choose

the location of the 2 strain gauges. A compromise must be made so that the gauges are as

close as possible to the specimen, without being too close so that there is a superposition

of the incident and reflected waves [16].

A useful tool to determine the best location for the strain gauges is the Lagrange diagram.

This diagram shows the position of the wave front on the bars, with respect to time. This

diagram is actually based on the governing equations of wave propagation in a rod.

Therefore, the next sections are going to use both the Lagrange diagram and the

unidirectional wave propagation equations to find the best location for strain gauges in a

compression and tension SHPB apparatus.

3.3.1 Visiting the wave motion equations

Section 3.2.2 showed that, for a pressure wave traveling through a medium, a solution for

the wave motion is given by equation 3.7:

𝒖(𝒙, 𝒕) = 𝒇 �𝒕 −𝒙𝑪𝟎� + 𝒈�𝒕 +

𝒙𝑪𝟎� (3.33)

Being more interested in strains, we can write this solution as:

𝜺(𝒙, 𝒕) = 𝜺𝟏 �𝒕 −𝒙𝒄� + 𝜺𝟐 �𝒕 +

𝒙𝒄� (3.34)

3.3.2 Graphical representation

In a standard compressive SHPB test, the wave that is generated in the input bar is

usually a square pulse, of duration 𝑡0 = 2𝐿0𝐶0� , where 𝐿0 and 𝐶0 are the length of the

striker and its wave speed respectively, as shown in figure 16. Note that in most SHPB

setups, the material of the striker is the same as the one of the 2 pressure bars.

31

Figure 16: General pulse shape

The Lagrange diagram of a single cylindrical bar could be as follows (figure 17 [17]),

where a represents the position of the gauge:

Figure 17: Wave front traveling through a bar, shown in a Lagrange diagram

The line shows how much time it takes the wave front to travel to a certain point x on the

bar. What the graphic doesn’t show is the actual duration of the wave. This means that,

knowing the wave velocity 𝑐 = 𝐶0 and its travel duration t0, it is possible to determine the

best location for the gauges. If the strain gauge is too close to the free end, where the

wave is being reflected, both the incident wave and the reflected wave could be measured

at the same time, hence giving mixed results.

32

One must place the strain gauge far enough from the free end to have a fully reflected

wave. It takes 𝑡𝑎 = 𝑎𝑐 seconds to have a first reading on the gauge (the time it takes the

incident wave to reach the strain gage). Note that this reading is going to last t0 seconds.

A second reading is going to appear 𝑡𝑏 = 2(𝐿−𝑎)𝑐

seconds after the first one. If tb is shorter

than t0, there will be superposition of the incident and the reflected waves. Therefore, it is

required to have 𝐿0 < (𝐿 − 𝑎).

3.3.3 Compression versus Tension

It is important to remember that the concept chosen to adapt the compressive setup is the

one suggested by Theodore Nicholas [18]. In order to get a tensile wave in the specimen,

a collar is used to have the first compressive pulse completely transferred to the output

bar, without affecting the specimen. The effect of the reflected tensile wave is what is

going to be measured.

Figure 18: Collar type of tension SHPB

Since this setup is really similar to a compressive setup, it is possible to compare the

Lagrange diagrams of both methods:

33

Figure 19: Lagrange diagrams for tension and compression SHPB

The left diagram of figure 19, picturing a compressive test (figure 19 a), shows that at the

specimen/input bar interface, part of the pulse is reflected because of the impedance

mismatch between the two mediums. The second reflection, at the end of the output bar,

is of no interest since it does not provide any useful data that can be used to calculate the

behavior of the specimen.

On the other hand, during a tensile test (figure 19 b), the reflection at the end of the

output bar is important because it is actually the beginning of the test. The first part of the

test, where the compressive wave is transmitted to the output bar, could be discarded,

since no measurement is recorded during that time.

With respect to the tensile test (figure 19 b), consider only the part after the reflection of

the compressive pulse, at the end of the output bar. It can be seen that the Lagrange

diagram is then symmetric to that of the compressive test (figure 20).

Figure 20: Symmetry of a Lagrange diagram

The output bar of the compressive setup basically becomes the input bar of the tensile

setup.

34

Moreover, the method proposed by Theodore Nicholas [18] allows use of the same

pressure bars for both types of test. This means that, using the same materials, the wave

speed c of the pressure bars does not change. Therefore, the pulse duration remains as

t0=2L0/c0. This is why the development mentioned in section 2.2 is still valid and the

gauges do not need to be placed differently, depending on the type of experiments. Of

course, this requires the apparatus to be symmetric. The two pressure bars must then be

of same length and the gauges placed symmetrically on each side of the specimen.

Figure 19 b shows that there is a first reflection at the input bar/collar interface since the

impedance between both bars does not match perfectly. It is important to make sure that

there won’t be any superposition between this spurious reflected wave and the

transmitted wave. This is easily verified using the development proposed in section 2.2.

Figure 21 shows the same diagram as Figure 18, but this time the spurious wave is

represented with a dash line and it is shown for a longer time.

Figure 21: Lagrange diagram showing the spurious wave

The spurious wave goes through the input bar strain gauge twice. There is a first reading

after its reflection from the input bar/collar interface. Upon arrival at the free end of the

input bar, the spurious wave is reflected a second time toward the 1st strain gauge.

The following provides the full development of the readings sequencing on both pressure

bars after the impact. Times are calculated under the assumption that the setup is

symmetric and neglecting the time it takes for the wave to travel along the specimen:

35

1. First reading on input bar: compressive initial pulse, at 𝑇0 = 𝑡𝑎 = 𝑎𝑐

2. Second reading on input bar: spurious reflected tensile pulse at

𝑇1 = 𝑡𝑎 + 2𝑡𝑏 = 𝑎𝑐

+ 2(𝐿−𝑎)𝑐

3. Third reading on input bar: spurious reflected compressive pulse at

𝑇2 = 𝑡𝑎 + 2𝑡𝑏 + 2𝑡𝑎 = 3𝑎𝑐

+ 2(𝐿−𝑎)𝑐

4. Fourth reading (recorded) on input bar: transmitted tensile pulse and reflected

tensile spurious wave, at 𝑇3 = 3𝑡𝑎 + 4(𝐿−𝑎)𝑐

5. First reading on output bar: compressive transmitted pulse, at

𝑇4 = 𝑡𝑎 + 2(𝐿−𝑎)𝑐

= 𝑇1

6. Second reading (recorded) on output bar: reflected tensile pulse, at

𝑇5 = 3𝑡𝑎 + 2(𝐿−𝑎)𝑐

7. Third reading (recorded) on output bar: reflected compressive pulse and

transmitted spurious compressive wave, at 𝑇6 = 3𝑡𝑎 + 4(𝐿−𝑎)𝑐

It is important to note that what needs to be recorded, in a tensile test, are points 4, 6 and

7. The other readings appear because the tensile load is indirect, with the first pulse being

compressive and is therefore bypassed.

Considering the test is compressive, one can notice that the results come from the logic

that no superposition is achieved if:

(𝑻𝟏 − 𝑻𝟎) ≥ 𝒕𝟎 = 𝟐𝒕𝒃 ≥ 𝒕𝟎 (3.35)

If the same logic is applied for the spurious reflected waves in a tensile test, then there

exists also 2 conditions:

(𝑻𝟑 − 𝑻𝟏) ≥ 𝒕𝟎 = (𝟐𝒕𝒂 + 𝟐𝒕𝒃) ≥ 𝒕𝟎 (3.36)

(𝑻𝟑 − 𝑻𝟐) ≥ 𝒕𝟎 = 𝟐𝒕𝒃 ≥ 𝒕𝟎 (3.37)

36

Therefore, if 𝑡𝑏 ≥ 𝑡0 was respected for the compressive test, then it is necessary that

(2𝑡𝑎 + 2𝑡𝑏) ≥ 𝑡0 is respected for the tensile test, since ta is non-negative. The second

condition of the tensile test is the same as the one for the compressive apparatus.

A problem of superposition is present at points 4 and 6. In figure 21, the dash lines,

showing spurious waves, are merging with the full lines, showing the recorded pulse.

This is caused by the setup symmetry and the most obvious way to solve this problem is

to “remove” that symmetry, having bars of different lengths.

Figure 22 shows a new Lagrange diagram [8] where the pressure bars have different

lengths. Even if figure 21 shows a longer input bar, this is not taken into consideration in

the following equations.

Figure 22: Lagrange diagram for a non-symmetrical SHPB apparatus

The process described in section 3.2, can be written with no symmetry.

1. First reading on input bar: compressive initial pulse, at 𝑇0 = 𝑡𝑎𝑖𝑛 = 𝑎𝑖𝑛𝑐

2. Second reading on input bar: spurious reflected tensile pulse at

𝑇1 = 𝑡𝑎𝑖𝑛 + 2𝑡𝑏𝑖𝑛 = 𝑎𝑖𝑛𝑐

+ 2(𝐿𝑖𝑛−𝑎𝑖𝑛)𝑐

3. Third reading (recorded) on input bar: transmitted tensile pulse at

𝑇2 = 𝑡𝑎𝑖𝑛 + 2𝑡𝑏𝑖𝑛 + 2𝑡𝑏𝑜𝑢𝑡 + 2𝑡𝑎𝑜𝑢𝑡 = 𝑎𝑖𝑛𝑐

+ 2(𝐿𝑖𝑛−𝑎𝑖𝑛)𝑐

+ 2(𝐿𝑜𝑢𝑡−𝑎𝑜𝑢𝑡)𝑐

+ 2𝑎𝑜𝑢𝑡𝑐

37

4. Fourth reading on input bar: reflected compressive spurious wave, at

𝑇3 = 3𝑡𝑎𝑖𝑛 + 2𝑡𝑏𝑖𝑛

5. First reading on output bar: compressive transmitted pulse, at

𝑇4 = 𝑡𝑎𝑖𝑛 + 𝑡𝑏𝑖𝑛 + 𝑡𝑏𝑜𝑢𝑡

6. Second reading (recorded) on output bar: reflected tensile pulse, at

𝑇5 = 𝑡𝑎𝑖𝑛 + 𝑡𝑏𝑖𝑛 + 𝑡𝑏𝑜𝑢𝑡 + 2𝑡𝑎𝑜𝑢𝑡

7. Third reading (recorded) on output bar: reflected compressive pulse, at

𝑇6 = 𝑡𝑎𝑖𝑛 + 𝑡𝑏𝑖𝑛 + 3𝑡𝑏𝑜𝑢𝑡 + 2𝑡𝑎𝑜𝑢𝑡

8. Fourth reading on input bar: transmitted compressive spurious wave, at

𝑇7 = 3𝑡𝑎𝑖𝑛 + 3𝑡𝑏𝑖𝑛 + 𝑡𝑏𝑜𝑢𝑡

Where:

𝑎𝑖𝑛 is the distance of the 1st strain gauge from the input bar free end

𝐿𝑖𝑛 is the length of the input bar

𝑎𝑜𝑢𝑡 is the distance of the 2nd strain gauge from the output bar free end

𝐿𝑜𝑢𝑡 is the length of the output bar

Most of the time in SHPB setups, the strain gauges are at the same distance from the

specimen, therefore fixing (𝐿𝑖𝑛 − 𝑎𝑖𝑛) = (𝐿𝑜𝑢𝑡 − 𝑎𝑜𝑢𝑡) = 𝑏 and 𝑡𝑏𝑖𝑛 = 𝑡𝑏𝑜𝑢𝑡 = 𝑡𝑏. Here

are all the limitations for a tensile setup to obtain any superposition of the signals:

• (𝑇1 − 𝑇0) > 𝑡0 = 2𝑡𝑏 ≥ 𝑡0

• (𝑇2 − 𝑇1) > 𝑡0 = �𝑡𝑎𝑖𝑛 + 4𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡� − �𝑡𝑎𝑖𝑛 + 2𝑡𝑏� ≥ 𝑡0 = 2𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡 ≥ 𝑡0

• (𝑇3 − 𝑇2) > 𝑡0 = �3𝑡𝑎𝑖𝑛 + 2𝑡𝑏� − �𝑡𝑎𝑖𝑛 + 4𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡� ≥ 𝑡0 = 2𝑡𝑎𝑖𝑛 − 2𝑡𝑏 −

2𝑡𝑎𝑜𝑢𝑡 ≥ 𝑡0

• (𝑇5 − 𝑇4) > 𝑡0 = �𝑡𝑎𝑖𝑛 + 2𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡� − �𝑡𝑎𝑖𝑛 + 2𝑡𝑏� ≥ 𝑡0 = 2𝑡𝑎𝑜𝑢𝑡 ≥ 𝑡0

• (𝑇6 − 𝑇5) > 𝑡0 = �𝑡𝑎𝑖𝑛 + 4𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡� − �𝑡𝑎𝑖𝑛 + 2𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡� ≥ 𝑡0 = 2𝑡𝑏 ≥ 𝑡0

• (𝑇7 − 𝑇6) > 𝑡0 = �3𝑡𝑎𝑖𝑛 + 4𝑡𝑏� − �𝑡𝑎𝑖𝑛 + 4𝑡𝑏 + 2𝑡𝑎𝑜𝑢𝑡� ≥ 𝑡0 = 2𝑡𝑎𝑖𝑛 −

2𝑡𝑎𝑜𝑢𝑡 ≥ 𝑡0

38

From the 4th condition, we can immediately find that:

𝒂𝒐𝒖𝒕 ≥ 𝑳𝟎 (3.38)

Entering this new value in the 1st condition, we have

𝑳𝒐𝒖𝒕 ≥ 𝟐𝑳𝟎 (3.39)

From the 3rd condition, we find that:

𝒂𝒊𝒏 ≥ 𝟑𝑳𝟎 (3.40)

And again, including the last result in the 1st condition we conclude that:

𝑳𝒊𝒏 ≥ 𝟒𝑳𝟎 (3.41)

In other words, a tensile setup like this one requires that the input bar be twice as long as

the output bar, which is twice as long as the longest striker used. This means that the

input bar must be at least 4 times the length of the striker. Nicholas et al. [18] also arrived

at the same conclusion, using the same kind of experimental.

In the current compressive setup, both pressure bars are 800 mm long and the longest

striker used has a length of 400 mm. On the other hand, supplied maraging steel bars are

only available in sections of 914 mm, which leaves approximately 910 mm after

machining. Considering this would be the longest pressure bar available, the longest

striker that could be used would have to be 227.5 mm long.

A Lagrange diagram of this situation is presented in the figure 23. For the sake of

simplicity, the input bar is 900 mm instead of 910. As expected, the use of a 225 mm (4

times shorter than 900 mm) striker represents the limit of the system. Indeed, there are

39

locations on the diagram where the pulses “almost” intersect. These locations are circled

in red on the figure.

In order to keep a safety margin, most simulations in this report are made with a 175 mm

long striker.

Figure 23 : Full Lagrange diagram for the proposed setup

40

Chapter 4:

Simulation considerations

41

4.1 Introduction The main software used to simulate SHPB tests, for this thesis work, is LS-Dyna. This

section does not constitute a tutorial for LS-Dyna, but describes major elements that were

taken into consideration to conduct the simulations.

When creating finite element models of parts interacting with each other, contact

definition is crucial because it is known that the quality of a simulated contact is closely

related to the quality of the mesh composing the parts. LS-Dyna’s default pre-processor,

LS-Prepost, is not efficient enough regarding mesh controls and thus pre-processors are

needed to create geometries and thus meshes.

Also, the first step in simulating the tensile results is to create a good compressive model.

Previous work [19] shows the creation of a 2 dimension model for use with LS-Dyna. The

methodology used was based on a 2D model and simulations were compared to

experimental results.

4.2 Geometrical considerations The parts of the existing compressive SHPB are as follows:

• 2 maraging steel pressure bars: dia 14.5 mm X 800 mm length

• 1 maraging steel striker bar: dia 14.3 mm X 200 mm length

• 1 aluminium 6061-T6 specimen: dia 10.5 mm X 5 mm length

In reality, the end of the striker bar has a 254 mm radius instead of being flat. This

improves the contact between the striker and the input bar. For the simulations, because

of the perfectly ortogonal contact, the striker is flat at both ends. Moreover, the striker

diameter is slightly smaller than 14.5 mm to fit the air gun bore. For the simulation,

diameter was to be exactly 14.5 mm, again given that the contact is orthogonal when the

striker hits the incident bar.

4.3 Mesh considerations The model consists of 4 basic components in the case of compression SHPB and 5 in the

case of the tension SHPB, because of the added collar. The parts are:

42

• Specimen

• Input bar

• Output bar

• Striker

• Collar (in the case of the tension system)

Figure 24 : Typical SHPB setup

All the parts listed are in contact at some point during experiment and, therefore, mesh

definition is extremely crucial. In FEA analysis, the quality of the contact between two

parts is closely related to the mesh. It is important that parts that are in contact have

coincident nodes at contact interface. Figure 25 shows the elements at the extremities of

the input and striker bars, with the nodes being coincident.

Figure 25: Nodes correspondence at bars extremities

4.4 Material considerations The material constitutive relations that are used for both compression and tension models

are:

• Elastic (for the bars)

• Plastic-kinematic (for the sample)

43

• Johnson-Cook with damage (for the sample in tension)

The plastic-kinematic constitutive model of LS-Dyna (see section 4.4.2) is used in

compression case in order to compare with a previously done study conducted by A.

Bouamoul study [19]. The Johnson-Cook model with damage law is used to simulate the

deformation of the specimen until failure.

4.4.1 Elastic (Hooke) law

The Hooke’s law is used as a constitutive model in the simulation of the pressure bars,

the striker and the collar behavior. Since all of those components are not to be deformed

plastically, this simple law is sufficient for the model.

As presented by Mase [20], the general elastic relation is

𝝈𝒊𝒋 = 𝑪𝒊𝒋𝒌𝒎𝜺𝒌𝒎 (4.1)

This relation links strains to stresses and the fourth-order tensor of the previous equation

is given by:

𝑪𝒊𝒋𝒌𝒎 = 𝝀𝜹𝒊𝒋𝜹𝒌𝒎 + 𝝁�𝜹𝒊𝒌𝜹𝒋𝒎 + 𝜹𝒊𝒎𝜹𝒋𝒌�

+ 𝜷�𝜹𝒊𝒌𝜹𝒋𝒎 − 𝜹𝒊𝒎𝜹𝒋𝒌� (4.2)

For an isotropic material, equations 4.1 and 4.2 reduce to

𝝈𝒊𝒋 = 𝝀𝜹𝒊𝒋𝜺𝒌𝒌 + 𝟐𝝁𝜺𝒊𝒋 (4.3)

where 𝜆 and 𝜇 are Lamé constants. The well-known Young modulus and Poisson ratio

can both be expressed as a function of in term of Lamé constants:

𝑬 =

𝝁(𝟑𝝀+ 𝟐𝝁)𝝀 + 𝝁

(4.4)

𝝂 =

𝝀𝟐(𝝀 + 𝝁) (4.5)

44

4.4.2 Plastic-kinematic law

This constitutive model is suited for isotropic material with kinematic hardening. Strain

rate dependency is not taken into account here as this as little effects on the aluminum

response. Moreover, to consider only isotropic hardening, parameter 𝛽 must be set to 1 in

LS-Dyna.

In figure 26, hardening is modeled simply as linear isotropic hardening:

Figure 26: Linear kinematic hardening representation

To account for plasticity, a tangent modulus is required, as well as the yield stress. It is

not necessary to model kinematic hardening as it is mostly suited for cyclic loading.

Therefore, one could think of this law as linear until the yield stress is reached, followed

by another linear law until failure has occured.

4.4.3 Johnson-Cook material model with a damage law

As mentioned earlier, this law is often used for ductile materials when strain rate

dependency and damage are to be considered. This means that, unlike the plastic

45

kinematic law shown above, here strain rate effects and damage evolution until failure,

instead of linear strain increase until the limit is reached.

As presented by LS-Dyna Keyword User Manual [21], Johnson-Cook expresses the flow

stress as follows:

𝝈𝒚 = �𝑨 + 𝑩𝜺𝒑�𝒏�(𝟏 + 𝑪 𝐥𝐧 �̇�∗)(𝟏 − 𝑻∗𝒎) (4.6)

Where A, B, C, m and n are material constants.

𝜀�̅� is the effective plastic strain

𝜀̇∗ is the effective total strain rate normalized to quasi-static threshold rate (the quasi-

static threshold rate is the strain rate under which no rate effects are apparent).

and 𝑇∗ = 𝑇−𝑇𝑟𝑜𝑜𝑚𝑇𝑚𝑒𝑙𝑡−𝑇𝑟𝑜𝑜𝑚

For damage, the total strain at fracture is given by:

𝜺𝒇 = �𝑫𝟏 + 𝑫𝟐𝒆𝑫𝟑𝝈∗�(𝟏 + 𝑫𝟒 𝐥𝐧 �̇�∗)(𝟏 + 𝑫𝟓𝑻∗) (4.7)

Where D1 to D5 are material damage parameters.

Fracture occurs when the overall cumulated damage parameter reaches the value 1.

𝑫 = �

∆𝜺𝒑�

𝜺𝒇 (4.8)

4.5 Conclusion The main considerations taken into account when creating the finite elements model are:

• Using the same geometry as the one currently in place at DRDC laboratories.

• Having a good quality mesh, especially at contact zone interfaces

For tensile simulations, the Johnson-Cook with damage law is what is commonly used

when modeling metals under high rates of strain.

46

Chapter 5:

Compression apparatus model generation

47

5.1 Introduction In order to create a tensile Hopkinson pressure bar apparatus, based on a compressive

setup, a 3 dimensions simulation model needs to be created. With this model, it is

possible to predict what the results of the tensile experiments are going to be and the

design can therefore be adjusted as a consequence.

The first step in simulating the tensile results is to create a good compressive model. A

previous report [19] shows the creation of a two dimensional model on LS-Dyna.

The current paper shows the simulation, in three dimensions, of the compressive

apparatus. The methodology used was based on the 2D model and other experimental

results. This was done in order to have a good starting point for the 3D tension model.

It was decided to have the tension model in 3D, as it is more versatile.

5.2 Model creation in LS-Dyna The materials constitutive laws used in the 3D model are the same as those used in the

2D model [19] (see table 4):

Table 4: Aluminum and Steel properties

Maraging Steel Aluminum 6061-T6

Constitutive law = Elastic Constitutive law = Plastic kinematic

Young Modulus = 182.1 GPa Young Modulus = 73.08 GPa

Density = 8064 kg/m3 Density = 2690 kg/m3

Poisson ratio = 0.3 Poisson ratio = 0.33

Elastic limit = 335 MPa

Tangent modulus = 645.7 MPa

48

In accordance with the 2D compressive model, the dimensions for the 3D model are

shown in Table 5:

Table 5: Main dimensions of the SHPB setup

Pressure bars Diameter: 14.5 mm Length: 800 mm

Striker Diameter: 14.5 mm Length: 200 mm

Specimen Diameter: 10.5 mm Length: 5.5 mm

5.3 Mesh creation Figure 27 shows a pressure bar, where the mesh is finer at both ends (only one end is

shown for clarity). At a certain distance from the contact interfaces (20 mm in this case),

it was decided to “stretch” the elements, to save on computation time. Because of the

small diameter/length ratio of the pressure bars, it is not possible to respect an aspect ratio

(length/width) of 2:1 for longer elements. It is safe to assume that this will not affect the

results.

Figure 27 : Finer mesh at the end of the bars

Figure 28 shows a cross-section view of the specimen interface

49

Figure 28 : Typical mesh of the cross section

5.4 LS-Dyna boundary conditions The boundary conditions are created using LS-Prepost are:

• Contact definition

• Velocity generation

All contacts are the same in the model and automatic-surface-to-surface default

configuration was used.

The system modeled contains 3 contacts interfaces:

1. Striker/Input bar interface

2. Input bar/specimen interface

3. Specimen/Output bar interface

The striker is given an initial velocity, which changes with every test, ranging from 17.3

m/s to 37.2 m/s.

50

5.5 Model presentation The full 3D model, is presented in figure 29:

Figure 29 : The full model of the compression apparatus

In the model, there is a small gap between the striker and the input bar. This gap helps

surfaces detection and therefore, proper contact between the two parts. Because of this

gap, there will be a delay between the theoretical wave front position and the simulated

results. Data manipulation might be required when comparing theoretical results and

simulated results in order to bring the curves in phase.

Figure 30 : Initial gap between the striker and the input bar

5.6 Results Two impacting rods produce a pulse of amplitude:

𝜎 =

𝑉𝜌𝐶2

(5.1)

51

With V being the striker velocity, 𝜌 the density.

Looking at the simulated results shown in figure 30, the pulse amplitude is as expected;

331 MPa for a striker impact velocity of 17.3 m/s.

Figure 31 : Pulse going through the input bar for a striker speed of 17.3 m/s

In section 3.3.2, it was mentioned that the pulse duration is 𝑡0 = 2𝐿0𝐶0� . The simulated

results are in agreement with this equation, as with the calculated theoretical duration of

84 μs.

Moreover, given that the wave velocity C is 4750.7 m/s for maraging steel, the

compressive pulse should take 168 μs to go back and forth in the input bar. As noted in

figure 30, there is indeed a delay of approximately 170 μs between the incident

compressive pulse and its reflection.

Table 6 shows a comparison between the theoretical values and the simulated results:

-500,00

-400,00

-300,00

-200,00

-100,00

0,00

100,00

200,00

300,00

0,00 100,00 200,00 300,00 400,00 500,00

Stre

ss (M

Pa)

Time (μs)

17.3 m/s

52

Table 6 : Comparison between theoretical and simulated stress values

Theoretical Simulated Error (%)

Pulse amplitude 331 MPa 333 MPa 0.60

Pulse duration 84 μs 88 μs 4.78

Pulse speed 4750.7 m/s 4751.4 m/s 0.01

Regarding the difference between simulated and theoretical pulse duration, the error is

related to the fact that the theoretical values consider a square pulse.

Bouamoul [19] presents results for multiple striker speeds, using a specimen that has a

diameter of 10.5 mm and a length of 5 mm. Figure 32 was taken from Bouamoul [19]

calculation report. Figure 33 shows the results from the 3D model created for this thesis

work. It can be seen that both figures are almost identical.

Figure 32 : Results, at different striker speeds, presented in A. Bouamoul's report [19]

53

Figure 33 : Results of the 3D model created for this thesis

The comparison shows that the method used to create the 3D model of the split

Hopkinson pressure bars is correct, as the results seem to agree well with those obtained

from the 2D model, which were validated with experimentations.

54

Chapter 6:

Creation of the 3D tension SHPB model

55

6.1 Introduction Using the same methodology as the one developed for the compression model, the

tension SHPB apparatus is created in LS-Dyna. The main differences between

compression and tension models are:

• Contact definition between the specimen and the pressure bars is now tied-

surface-to-surface. This contact prevents separation of the bodies and therefore

represents the fixation of the specimen to the pressure bars via threads.

• Specimen geometry is in accordance with ASTM volume 3.01 section E8M. This

is done in order to have a standard specimen shape.

• Specimen material follows Johnson-Cook material model.

• A collar made of the same material as the pressure bars is added.

6.2 Specimen geometry The ASTM volume 3.01 section E8M defines the standard tests methods for tension

testing of metallic materials (using metric dimensions), as shown in figure 34.

Figure 34 : Standard specimen taken from ASTM volume 3.01 section E8M [22]

As there is no restriction regarding the extremities of the specimen, the gauge section has

standard dimensions. One should refer to the ASTM volume [22] to have information

regarding other dimensions, as the specimen primarily used in the current study is of the

dog bone shape shown in figure 34 and 35.

56

Figure 35 : Standard specimen shapes, with threaded ends, taken from ASTM volume 3.01 section E8M [22]

6.2.1 Size considerations

A compromise need to be made between the size of the specimen and the size of the

collar. The bigger the collar is when compared to the specimen; the lower the

compressive initial pulse is in the specimen.

But, considering the pressure bars have a fixed diameter of 14.5 mm, there is a combined

maximum size for the collar and the specimen assembly. That is, a bigger collar requires

a smaller specimen. However, a smaller specimen is easier to damage with the initial

compressive pulse. Moreover, the size of the collar is directly linked to the size of the

specimen ends as shown in figure 36.

Figure 36: Gap between the collar and the specimen

Gap between collar and

specimen

57

Section 6.5.3 presents multiple simulations with specimens of different sizes, with all the

threaded ends having the same diameter of 7 mm. It is important to note that once the

pressure bars are machined with threaded holes, only one size of specimen threaded ends

can be used. Thus, it was decided to fix the end diameter at 7 mm. Moreover, 7 mm is a

reasonable value, as it corresponds to less than half the cross-section surface of the

pressure bar, the rest being available for the collar. As for the diameter of the gauge

section, it is fixed at 6 mm.

6.2.2 Threaded ends and simulation considerations

As the thread on the specimen will not be included in the model, the ends are shorter in

the model than what they will be in the experimental setup. Because it is difficult to

model the male and female threads of the specimen and pressure bars, a tied contact is

used and the threaded section of each part is discarded. The tied contact prevents

separation of the bodies, allowing the tensile pulse to be transmitted through the

specimen and pressure bars.

Figure 37 : Representation of the specimen with short ends and no threads

This simple approach was chosen because taking into account the threaded fixation of the

specimen into the pressure bars would have greatly increased the modeling complexities

and the computation. While this would be more realistic, it was decided to take a simpler

approach at this time because adding the complexities would not offer much more value

for a conceptual stand point. A discussion on this issue is presented in section 7.1 and

recommendations are also suggested.

58

6.3 Collar geometry As mentioned in section 6.2, the collar geometry is directly related to the specimen size.

It is better to have as small a gap as possible between the collar and the specimen, since

the thinner the collar is, the less efficient it will be.

A radial gap of 1 mm was chosen, as this seemed to be an appropriate value. For

example, a radial adjustment with tolerances h5/D10 [23], which is considered a

clearance fit, has an approximate maximum gap of 0.1 mm between the shaft and the

bore [23]. For the case discussed in this work, no adjustment at all is required, as no

friction/contact is allowed between the specimen and the collar. A tight adjustment

creates friction and introduces errors in the results. Moreover, it is important to properly

lubricate the interior of the collar, to minimize the friction between it and the specimen.

The exterior diameter of the collar is equal to the diameter of the pressure bars, in order

to minimise the impedance mismatch.

6.4 Model summary Table 7 summarizes the principal characteristics of the model

Table 7 : Summary of the model properties and characteristics

Input bar length 910 mm

Input bar diameter 14.5 mm

Output bar length 455 mm

Output bar diameter 14.5 mm

Specimen length (gauge section) 20 mm

Specimen diameter (gauge section) 6 mm

Striker length 175 mm

Striker diameter 14.5 mm

59

Collar external diameter 14.5 mm

Collar internal diameter 8.5 mm

Collar length 24 mm

Contact type specimen/pressure bars

interfaces

Tied surface to surface

Contact type striker/input bar interface Automatic surface to surface

6.5 Aluminum specimen simulations As it was the case in compression simulations, aluminum 6061-T6 is used as a “base” to

calibrate the tension model, because of its insensitivity to strain rate effects.

The parameters used in the Johnson-Cook formulation, as presented in section 4.4.3, are

taken from Banerjee’s study [24]. They also include the parameters for the Mie-

Gruneisen equation of state.

Table 8: Johnson-Cook parameters for aluminum 6061-T6

A 324 MPa

B 114 MPa

C 0.002

n 0.42

m 1.34

D1 -0.77

D2 1.45

D3 -0.47

D4 0

60

D5 1.6

Mie-Gruneisen C0 5386 m/s

Mie-Gruneisen Γ0 1.99

Mie-Gruneisen Sα 1.339

6.5.1 Variable striker speed

For the specimen geometry described in section 6.4 summary, different striker speeds

(20, 25, 30, 35 m/s) were used to conduct the numerical experiment.

Using fixed specimen geometry, different striker velocities will show different

stress/strain amplitudes. This should give an idea of how fast the striker can be launched

without plastically deforming the specimen on the first compression pass. Figure 38

shows the model results for different the striker velocities chosen.

Figure 38 : Strains measured at the input bar strain gauges for each speeds

-0,50

-0,40

-0,30

-0,20

-0,10

0,00

0,10

0,20

0,00 100,00 200,00 300,00 400,00 500,00 600,00 700,00

Stra

in (%

)

Time (µs)

Input strains (20 m/s)

Input strains (25 m/s)

Input strains (30 m/s)

Input strains (35 m/s)

Transmitted pulse

61

One should remember that, unlike the compression model, the input bar in the tension

model shows the transmitted pulse. The first measurement, at approximately 160 µs, is

the compressive wave. The second measurement is at approximately 490 µs and is the

transmitted tensile wave (note the positive sign of the strains). Contrary to the

compression SHPB, the transmitted pulse is measured on the input bar.

One noticeable fact in figure 38 is that the transmitted pulse is mostly constant with

respect to the striker initial velocity. There is however a small difference between each

transmitted pulse and this could be explained by the fact that the impedance mismatch

between the pressure bars and the specimen is important. Therefore, it could be that the

pulse is “filtered” by all those passages through the specimen region. This would mean

that the impedance mismatch has a higher effect on the transmitted pulse than the initial

pulse.

The variation of strain in the specimen, with regard to the striker speed, is shown in

figure 39, which presents the strains measured in the output bar.

Figure 39: Strains measured at the output bar strain gauge, for each speed

-0,50

-0,40

-0,30

-0,20

-0,10

0,00

0,10

0,20

0,30

0,40

0,50

0,00 200,00 400,00 600,00 800,00

Stra

ins (

%)

Time (µs)

Output strains (20 m/s)

Output strains (25 m/s)

Output strains (30 m/s)

Output strains (35 m/s)

Reflected pulse

Incident tensile pulse

Initial compressive pulse

62

Figure 39 shows that the striker velocity has a significant influence on the specimen

strain, as can be seen in the amplitude change of the reflected pulse measured on the

output bar.

6.5.2 Variable striker length

The effect of having a shorter/longer striker is to have a pulse of different wave lengths.

This is particularly helpful to prevent wave superposition problems, as discussed in

section 3.3.

Figures 40 and 41 show the rise time for both strikers of 175mm and 125mm in length.

Figure 40: Rise time for a 175 mm long striker

-0,300

-0,250

-0,200

-0,150

-0,100

-0,050

0,000

0,050

0,100

100,000 150,000 200,000 250,000 300,000

Stra

ins (

%)

Time (µs)

Striker 175

63

Figure 41 : Rise time for a 125 mm long striker

The vertical bars that were inserted in figures 40 and 41 are there in order to show the

theoretical rise time calculated with the formula presented in section 3.3.2. Because the

pulse is not perfectly square, it is obvious that there will be some difference between the

theory and the simulations.

Nevertheless, one can say that the simulated results closely reproduce and are in good

agreement with the theoretical results that were presented in chapter 3 (section 3.3.2).

6.5.3 Variable specimen diameter

Specimen diameter is critical because, the smaller is the specimen, the easier it is to

deform it. On the other hand, this creates a high impedance mismatch. Moreover, chances

of plastically deforming the specimen with the initial compressive pulse are higher with a

smaller specimen.

Figure 42 shows the stress that was calculated for specimens of different diameters.

-0,300

-0,250

-0,200

-0,150

-0,100

-0,050

0,000

0,050

0,100

100,000 150,000 200,000 250,000 300,000

Stra

ins (

%)

Time (µs)

Striker 125

64

Figure 42 : Specimen stresses

Figure 42 shows that the initial compressive pulse is partially reflected back into the

specimen (approximately at 300 µs). The smaller the specimen, the higher is the

reflection, and this is in accordance with the theory presented in section 3.2.3 (equations

3.18 and 3.19).

We can also notice that the specimens do not break with the first tensile pulse. The 3mm

and 4mm diameters break at the second pulse, while the 5mm specimen breaks right at

the beginning of the 3 pulse. As for the 6 mm specimen, its dimensions are so large that it

does not break.

Because of the impedance mismatch, there is always a reflection back into the specimen

when the pulse reaches a pressure bar/specimen interface. At approximately 950 µs, the

compressive reflection in the 6mm specimen creates a large compressive stress. This is

because the 3 tensile loadings stretched the specimen and created constriction of the

cross-sectional area, hence increasing the residual stress.

-500,00

-400,00

-300,00

-200,00

-100,00

0,00

100,00

200,00

300,00

400,00

500,00

600,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00Stre

ss (M

Pa)

Time (μs)

3 mm diameter

4 mm diameter

5 mm diameter

6 mm diameter

65

6.5.4 Collar/specimen only

It is useful to study what happens when the model only runs with the collar or the

specimen. Of course, in this case, this will not be a tension test, but a standard

compression one.

From equations 3.18 and 3.19, it is found that the transmission coefficient of the collar is

approximately 0.93, meaning that a very small part of the initial wave will be reflected

back in the input bar. This is shown in figure 43:

Figure 43 : Input bar stress when only the collar is installed

The measurements of the output bar clearly shows that indeed, most of the pulse is

transmitted through the bars, as presented in figure 44:

-600,00

-500,00

-400,00

-300,00

-200,00

-100,00

0,00

100,00

200,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00

Stre

ss (M

Pa)

Time (µs)

Input bar stress

66

Figure 44: Stress in the output bar when only the collar is installed

Note that the multiple reflections are present because after the first impact, the collar falls

from the setup, creating a gap between both pressure bars.

Again, from equations 3.18 and 3.19, it is found that the transmission coefficient of the

specimen is approximately 0.33, meaning that most of the pulse should be reflected back

into the input bar.

Figure 45 shows the stress calculated in the input bar when only the specimen is installed

(without the collar).

-500,00

-400,00

-300,00

-200,00

-100,00

0,00

100,00

200,00

300,00

400,00

500,00

600,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00Stre

ss (M

Pa)

Time (µs)

Output bar stress

67

Figure 45: Stress in the input bar when only the specimen is installed

Remembering that the specimen is fixed to the pressure bars, it is then possible to deduce

why multiple transmissions after the first one can be observed (and not only the

reflections as seen in figure 44). Considering that the initial pulse introduced by the

striker is approximately 380 MPa, figure 45 shows that more than 66 % is reflected back

into the input bar. This could be due to the fact that the specimen is not a perfect cylinder;

its shape being that of a standard tension test specimen. Nonetheless, the fact that most of

the pulse is reflected back into the input bar when only the specimen is installed is in

accordance with the theory presented in chapter 3.

It is a difficult task to predict, when both the collar and the specimen are assembled, how

much stress goes through each of them. Using the theory of springs in parallel, an attempt

was made to derive an equation that would give the value of how much of the initial

pulse goes through the specimen. Unfortunately, the behaviour of the collar/specimen

assembly is not one that could be modelled using two springs in parallel.

-600,00

-500,00

-400,00

-300,00

-200,00

-100,00

0,00

100,00

200,00

300,00

400,00

500,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00

Stre

ss (M

pa)

Time (µs)

Input bar stress

68

6.5.5 Specimen material

The final simulations presented in this section are for different specimen materials. So

far, only aluminum 6061-T6 was used. The collar material is kept as maraging steel.

The two other materials that were tested are:

• Titanium Ti-6Al-4V, whose properties were taken from Donald R. Lesuer and

Mary M. Leblanc study [25].

• Weldox 460 E steel, whose properties were taken from S. Dey et al study [26].

The choice of the steel and titanium was based on information availability. Studies on

high strain rates behaviour were found for these two types of material and research works

were also done using Johnson-Cook constitutive relations. That is why the two materials

listed above were chosen.

Table 9 : Johnson-Cook parameters for Titanium and Weldox steel

Titanium Ti-6Al-4V Weldox 460 E steel

A 862 MPa 499 MPa

B 331 MPa 382 MPa

C 0.012 0.008

n 0.34 0.458

m 0.8 0.893

D1 -0.09 0.636

D2 0.25 1.936

D3 -0.5 -2.969

D4 0.014 -0.014

D5 3.87 1.014

69

Figure 46 shows the stress measured in the output bar:

Figure 46: Stress probed in the output bar

The reflected wave is higher for the aluminum than it is for both steel and titanium. But,

steel and titanium reflected waves have similar amplitudes.

There is a significant difference between aluminum density and steel density. Therefore,

even if aluminum has a slightly higher wave speed C0, it does not compensate for the

effect of such a low density in the impedance calculation. The higher impedance

mismatch leads to higher reflection in the output bar. As for titanium, its density is lower

than that of steel, but higher than aluminum, thus its wave speed C0 is also higher than

that of steel, which helps to achieve a lower mismatch.

As for the input bar, figure 47 shows the following results:

-600,00

-400,00

-200,00

0,00

200,00

400,00

600,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00

Stre

ss (M

Pa)

Time (µs)

Aluminum

Steel

Titanium

70

Figure 47: Stress probed in the input bar

Of all three materials, considering the specimen geometry does not change, aluminum is

the one with the lowest impedance, followed by titanium and then steel, which is the

material with the highest impedance.

Surprisingly, results show that the transmitted pulse is higher for titanium than for steel.

Since the pressure bars are also in steel, a higher transmission coefficient is expected for

steel. The fact that the pressure bars are from maraging steel and the specimen in Weldox

460 does not affect their impedances difference, because both materials have similar

densities and Young modulus.

The results described could be explained using the used Johnson-Cook formulation.

Some parameters are approximate since they were taken from literature and no

experimental characterisation was done within the scope of this thesis work.

-600,00

-500,00

-400,00

-300,00

-200,00

-100,00

0,00

100,00

200,00

300,00

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00

Stre

ss (M

Pa)

Time (µs)

Steel

Titanium

Aluminum

71

6.6 Conclusion The foregoing analysis shows that the developed and used model is in agreement with

theory. The most important issue is that it confirms that the collar allows most of the

compressive pulse to bypass the specimen, which is what is required for the system to

work. Simulations showed that the results of the output bar strain gauges were of similar

amplitude as those taken from the input bar strain gauge.

The analysis also showed that specimen geometry is critical. Specimen diameter must be

selected carefully. On one hand, in order to prevent plastic deformation of the specimen

by the initial compressive pulse, the diameter must be large enough while on the other

hand, it must be small enough so that the tensile pulse causes the specimen to deform

plastically.

Finally, calibration of the model with experimental results is required. While the

preliminary results presented in this chapter indicate that the collar concept is a viable

method of transforming a compressive SHPB setup into a tension test apparatus, it needs

to be validated experimentally.

72

Chapter 7:

Conclusion

73

7.1 Thesis retrospective The basic objective of this thesis was to find a way to convert a compressive Split

Hopkinson Pressure Bars (SHPB) apparatus into a tension apparatus.

Chapter 1 introduced the general concept of SHPB tests.

Chapter 2 presented a brief history of SHPB evolution and also presented a relatively

restrained literature review of what is commonly used to perform tensile testing using

Split Hopkinson Pressure Bars. In this chapter, it was shown that the best concept (with

respect to the selection criteria) that can be used to convert the compression setup into a

tension setup was the “collar” one. The critical part of this chapter was to properly define

the selection criteria. For the case discussed in this paper, the whole concept selection

work was presented to the DRDC subject matter experts for approval before conducting

the study.

Chapter 3 presented the basic mathematical theory of SHPB calculation. Wave

propagation theory is briefly explained and the chapter concludes with the presentation of

the Lagrange diagrams. The Lagrange diagram showed that, when using a collar to

perform tension tests with SHPBs, both the input and the output bars must be of different

lengths. More specifically, it was shown that the input bar must be twice the length of the

output bar, making it difficult to manufacture. Due to the desire to reuse as many parts as

possible from the compression setup, the input bar would have to be almost 1000 mm

long, with a diameter of 14.5 mm. However, it is difficult to machine such a bar and keep

it “straight”.

In chapter 4, aspects that were considered important to create an appropriate finite

element model of the SHPB apparatus were explained. An adequate meshing of the

geometries, especially at the contact interfaces, was required in order to facilitate the

contact detection and therefore minimising numerical errors. The constitutive laws used

in the model were presented, with a list of all the material parameters, along with a list of

references from the literature. One should remember that using material parameters found

in the literature was a necessary step, since the developed model constitutes a preliminary

evaluation of the selected concept and no in-house experimental data was available.

74

Chapter 5 presents the method used to create the 3D model. Before creating the finite

element model of the tension apparatus, a 3D model of the compression setup was

created, based on a previous existing 2D model of the said setup. This is because the 2D

model in question had been calculated with experimental results. Hence using that 3D

compression model as a starting point to create the tension model was deemed a useful

approach instead than starting from scratch.

The results of the tension model can be found in chapter 6. One important issue that was

presented is that the collar allows the propagation of the compressive pulse from the input

bar to the output with a transmission coefficient of almost 100%. Also, chapter 6 showed

that dimensioning of the specimen is critical. While the specimen must be large enough

in order to sustain the initial compressive wave, it must be however small enough to

allow occurrence of its plastic deformation. By the time being, it would be difficult to fix

a rule about the value of the diameter the specimen should have based only on simulation

results, because the tension model has not been yet compared with experimental results.

(only the compression model has been).

7.2 Discussions and recommendations

7.2.1 Machining feasibility

First of all, it is important to discuss the feasibility of building the tension system

pressure bars. With results and analysis presented, the required length of the input bar, in

order to prevent spurious reflected waves, has to be approximately 900 mm. When

manufacturing such a rod, shape tolerances are required. While the values of such

tolerances are not discussed here, they have to be taken into consideration because a rod

which is long (900 mm) and small (14 mm of diameter) is difficult to keep aligned.

Moreover, a surface finish of quality Ra10 must be machined on the ends of the bars, to

have the best contact surfaces as possible.

7.2.2 Specimen adjustment

The specimen must be well fitted between the two pressure bars. A method that is used to

have the fit as ok as possible, when screwing the specimen in the pressure bars, is the use

of Teflon tape. Such tape minimizes the gap between the female/male threads.

75

The specimen length must be such that it is neither too long, nor too short. A short

specimen is difficult to deform and therefore the strains would be ow. A long specimen

creates reflection problems and there are difficulties in bringing the specimen into

equilibrium [27].

The specimen diameter is another parameter that requires fine tuning. A specimen too

small will plastically deform under the compressive wave. A specimen that is too large

might not deform enough so that a full stress-strain curve can be created.

7.2.3 Importance of experimental validation

Experiments would allow one not only the fine tuning of the specimen geometry, but also

the understanding of specific features inherent to the selected concept.

For instance, the FEA model that was created does not include thread effects. It is

possible that, when taking real measurements, noise induced by the threaded fixation

makes it difficult to obtain good data.

In fact, validation of the model needs to be done experimentally before embarking on

developing such a system and this should be the next step in continuing this development.

76

7.3 Conclusion The main objective of the current thesis was to perform a feasibility study of converting a

compression split Hopkinson pressure bars setup into a tension split Hopkinson pressure

bars setup. To complete this study, the following steps were accomplished:

1. Literature review of existing tension split Hopkinson pressure bars concepts;

2. Concept selection;

3. Compression FEA model alignment and validation;

4. Tension FEA model creation, based on the compression aligned model.

Completion of those 4 steps showed that it is possible to have a convertible

compression/tension apparatus setup using the collar method. FEA simulation also

showed results that were mostly in agreement with the theory.

Because it was not possible, as part of this thesis, to present experimental results, it was

also impossible to align the tension model with such data. Therefore, the model presented

in this thesis should be used as a basis in order to create a more accurate one, based on

experimentations.

Future work should be focused on creating the tension split Hopkinson pressure bars

apparatus presented in this work, so that the FEA model itself can be calibrated. By

jointly using the experimental setup and the calibrated model, it would then be possible to

characterize materials at high rates of strain.

77

References

78

[1] Michael Adam KAISER, Advancements in the Split Hopkinson Bar Test, Faculty of

the Virginia Polytechnic Institute and State University, 1998, 94 pages.

[2] J. D. ADENBACH, Wave Propagation in elastic solids, North Holland, 1999, 425

pages.

[3] N. TASNEEM, Study of wave shaping techniques of Split Hopkinson Pressure Bars

using finite element analysis, 2002, 93 pages

[4] AL-MOUSAWI et al, The use of Split Hopkinson Pressure Bars Technique in High

Strain Rate Material Testing, Proceedings Institution of Mechanical Engineers, 1996, vol.

211, part C, 20 pages.

[5] BATEMAN et al, A study of Shock Mitigating Materials in a Split Hopkinson Bars

configuration Phase II, Sandia National Laboratories, 1998

[6] CHEN et al, A Split Hopkinson Bars Technique for Low Temperature Impedance

Materials, Experimental Mechanics, 1998, page 1-5

[7] SHU et al, Tensile Mechanical Properties of AM50A Alloy by Hopkinson Bar, Key

Engineering Materials, 2007, vol. 340-341, 247-254

[8] LEE et al, Dynamic material property characterization by using Split Hopkinson

Pressure Bars technique, Nuclear Engineering and Design 226, 2003, 119-125

[9] HAUGOU et al, On the use of the non-direct tensile loading on a classical split

Hopkinson bar apparatus dedicated to sheet metal specimen characterization,

International Journal of Impact Engineering 32, 2006, 778-798

[10] ESKANDARI et al, Dynamic Testing of Composite Laminates with a Tensile Split

Hopkinson Bar, Journal of Composite Materials 34, 2000, 260-273

[11] LINDHOLM et al, Effect of strain rate on yield strength, tensile strength and

elongation of three aluminium alloys, Journal of Materials Engineering and Performance

4, 1995, 216-220

79

[12] HARDING et al, Tensile testing of materials at impact rates of strain, Journal of

Mech. Eng. Sc. 2, 1960, 88-96

[13] HARDING et al, A tensile testing technique for fiber-reinforced composites at

impact rates of strain, Journal of Materials Science 18, 1983, 1810-1826

[14] Karl F. GRAFF, Wave Motion in Elastic Solids, Dover, 1991, 649 pages

[15] ELWOOD et al, A tensile technique for materials testing at high strain rates, Journal

of Physics E: Scientific Instruments, 1982, vol 15.

[16] C.H. NGUYEN, H.J. SCHINDER, On Spurious Reflection Waves in Hopkinson

Bars Tensile Tests using a collar, J. Phys IV, France 7, 1997

[17] S.W. PARK, M. ZHOU, Separation of Elastic Wave in Split Hopkinson Bars using

one-point Strain Measurements, Experimental Mechanics, 1999

[18] T. NICHOLAS, Tensile Testing of Materials at High Rates of Strain, Experimental

Mechanics, 1980

[19] A. BOUAMOUL, 2D Hopkinson Bars Simulation Analysis, Defense R&D Canada,

2006

[20] G.Thomas MASE, George E. MASE, Continuum Mechanics for Engineers, CRC,

1999, 377 pages

[21] LS-DYNA Keyword User’s Manual, Livermore Software Technology Corporation

(LSTC), 2007, Volume 1, Version 971

[22] ASTM 2004, volume 3.01, section E8M

[23] Jean-Louis FANCHON, Guide des Sciences et Technologies Industrielles,

NATHAN, 2001, pages 91-92

[24] B. BANERJEE, MPM Validation : Sphere-Cylinder impact tests: Energy Balance,

University of Utah, 2004

80

[25] D.R. LESNER, M.M. LEBLANC, Modeling Large Strain, High Rate Deformation

in metals, Supporting Technologies, Engineering Research Development and Technology

[26] S. DEY et al, On the Influence of Constitutive Relation in Projectile Impact of Steel

Plate, Internal Journal of Impact Engineering, 2005

[27] D.J. PARRY et al, Stress Equilibrium Effects Within Hopkinson Bar Specimens,

Journal de Physique IV, 1994

81

Appendix A:

Compressive SHPB LS-DYNA model

82

$# LS-DYNA Keyword file created by LS-PREPOST 2.4 - 27Nov2009(14:51) $# Created on Dec-17-2009 (15:10:45) *KEYWORD *TITLE $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title *CONTROL_TERMINATION $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $ endtim endcyc dtmin endeng endmas $# endtim endcyc dtmin endeng endmas 4.5000E-4 0 0.000 0.000 0.000 *DATABASE_ELOUT $# dt binary lcur ioopt 1.0000E-6 0 0 1 *DATABASE_GLSTAT $# dt binary lcur ioopt 1.0000E-6 0 0 1 *DATABASE_MATSUM $# dt binary lcur ioopt 1.0000E-6 0 0 1 *DATABASE_BINARY_D3PLOT $# dt lcdt beam npltc psetid 1.0000E-6 0 0 0 0 $# ioopt 0 *DATABASE_EXTENT_BINARY $# neiph neips maxint strflg sigflg epsflg rltflg engflg 0 0 3 1 1 1 1 1 $# cmpflg ieverp beamip dcomp shge stssz n3thdt ialemat 0 0 0 1 1 1 2 1 $# nintsld pkp_sen sclp unused msscl therm iniout iniout 0 0 1.000000 0 0 0STRESS STRESS *DATABASE_HISTORY_SOLID $# id1 id2 id3 id4 id5 id6 id7 id8 5017 258601 0 0 0 0 0 0 *BOUNDARY_SPC_SET $# nsid cid dofx dofy dofz dofrx dofry dofrz

83

4 0 1 1 0 0 0 0 *SET_NODE_LIST_TITLE Input_spc $# sid da1 da2 da3 da4 4 0.000 0.000 0.000 0.000 $# nid1 nid2 nid3 nid4 nid5 nid6 nid7 nid8 99 54 0 0 0 0 0 0 *BOUNDARY_SPC_SET $# nsid cid dofx dofy dofz dofrx dofry dofrz 3 0 1 1 0 0 0 0 *SET_NODE_LIST_TITLE Specimen_spc $# sid da1 da2 da3 da4 3 0.000 0.000 0.000 0.000 $# nid1 nid2 nid3 nid4 nid5 nid6 nid7 nid8 199394 199396 0 0 0 0 0 0 *BOUNDARY_SPC_SET $# nsid cid dofx dofy dofz dofrx dofry dofrz 2 0 1 1 0 0 0 0 *SET_NODE_LIST_TITLE Output_spc $# sid da1 da2 da3 da4 2 0.000 0.000 0.000 0.000 $# nid1 nid2 nid3 nid4 nid5 nid6 nid7 nid8 262820 262775 0 0 0 0 0 0 *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE_ID $# cid title 1Striker_input contact $# ssid msid sstyp mstyp sboxid mboxid spr mpr 1 2 0 0 0 0 0 0 $# fs fd dc vc vdc penchk bt dt 0.000 0.000 0.000 0.000 0.000 0 0.0001.0000E+20 $# sfs sfm sst mst sfst sfmt fsf vsf 1.000000 1.000000 0.000 0.000 1.000000 1.000000 1.000000 1.000000

84

... MESH…

*PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Input bar $# pid secid mid eosid hgid grav adpopt tmid 1 1 1 0 0 0 0 0 *SECTION_SOLID_TITLE Input bar $# secid elform aet 1 1 0 *MAT_ELASTIC_TITLE Maraging Steel $# mid ro e pr da db not used 1 8064.00001.8210E+11 0.300000 0.000 0.000 0 *PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Specimen $# pid secid mid eosid hgid grav adpopt tmid 2 2 2 0 0 0 0 0 *SECTION_SOLID_TITLE Specimen $# secid elform aet 2 1 0 *MAT_PLASTIC_KINEMATIC_TITLE Aluminium 6061-T6 $# mid ro e pr sigy etan beta 2 2690.00007.3080E+10 0.330000 3.3500E+8 6.4570E+8 0.000 $# src srp fs vp 0.000 0.000 0.540000 0.000 *PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Striker $# pid secid mid eosid hgid grav adpopt tmid 3 3 1 0 0 0 0 0

85

*SECTION_SOLID_TITLE Striker $# secid elform aet 3 1 0 *PART $# title Output bar $# pid secid mid eosid hgid grav adpopt tmid 4 4 1 0 0 0 0 0 *SECTION_SOLID_TITLE Output bar $# secid elform aet 4 1 0 *INITIAL_VELOCITY_GENERATION $#nsid/pid styp omega vx vy vz ivatn 1 3 0.000 0.000 0.000 18.200001 0 $# xc yc zc nx ny nz phase iridid 0.000 0.000 0.000 0.000 0.000 0.000 0 0

86

Appendix B:

Tensile SHPB LS-DYNA model (example)

87

$# LS-DYNA Keyword file created by LS-PREPOST 2.4 - 27Nov2009(14:51) $# Created on Mar-23-2010 (13:43:46) *KEYWORD *TITLE $# title LS-DYNA keyword deck by LS-PrePost *CONTROL_TERMINATION $# endtim endcyc dtmin endeng endmas 0.001000 0 0.000 0.000 0.000 *DATABASE_ELOUT $# dt binary lcur ioopt 1.0000E-6 0 0 1 *DATABASE_GLSTAT $# dt binary lcur ioopt 1.0000E-6 0 0 1 *DATABASE_MATSUM $# dt binary lcur ioopt 1.0000E-6 0 0 1 *DATABASE_BINARY_D3PLOT $# dt lcdt beam npltc psetid 1.0000E-6 0 0 0 0 $# ioopt 0 *DATABASE_EXTENT_BINARY $# neiph neips maxint strflg sigflg epsflg rltflg engflg 0 0 3 1 1 1 1 1 $# cmpflg ieverp beamip dcomp shge stssz n3thdt ialemat 0 0 0 1 1 1 2 1 $# nintsld pkp_sen sclp unused msscl therm iniout iniout 0 0 1.000000 0 0 0STRESS STRESS *DATABASE_HISTORY_SOLID $# id1 id2 id3 id4 id5 id6 id7 id8 1845 58657 208825 313481 0 0 0 0 *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE_ID $# cid title 1Striker input contact $# ssid msid sstyp mstyp sboxid mboxid spr mpr 1 2 0 0 0 0 0 0 $# fs fd dc vc vdc penchk bt dt

88

0.000 0.000 0.000 0.000 0.000 0 0.0001.0000E+20 $# sfs sfm sst mst sfst sfmt fsf vsf 1.000000 1.000000 0.000 0.000 1.000000 1.000000 1.000000 1.000000

… MESH…

*PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Specimen $# pid secid mid eosid hgid grav adpopt tmid 1 1 2 1 0 0 0 0 *SECTION_SOLID_TITLE Specimen section $# secid elform aet 1 1 0 *MAT_JOHNSON_COOK_TITLE Aluminium 6061-T6 $# mid ro g e pr dtf vp 2 2703.00002.6000E+107.0000E+10 0.330000 0.000 0.000 $# a b n c m tm tr epso 2.9000E+8 2.0500E+8 0.350000 0.011000 1.340000 925.00000 293.00000 1.000000 $# cp pc spall it d1 d2 d3 d4 896.00000 0.000 2.000000 0.000 -0.770000 1.450000 -0.470000 0.000 $# d5 1.600000 *EOS_GRUNEISEN_TITLE Aluminium EoS $# eosid c s1 s2 s3 gamao a e0 1 5386.0000 1.339000 0.000 0.000 1.990000 0.480000 0.000 $# v0 0.000 *PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Input bar

89

$# pid secid mid eosid hgid grav adpopt tmid 2 2 1 0 0 0 0 0 *SECTION_SOLID_TITLE Input bar section $# secid elform aet 2 1 0 *MAT_ELASTIC_TITLE Maraging Steel $# mid ro e pr da db not used 1 8064.00001.8210E+11 0.300000 0.000 0.000 0 *PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Output bar $# pid secid mid eosid hgid grav adpopt tmid 3 3 1 0 0 0 0 0 *SECTION_SOLID_TITLE Output bar section $# secid elform aet 3 1 0 *PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Striker $# pid secid mid eosid hgid grav adpopt tmid 4 4 1 0 0 0 0 0 *SECTION_SOLID_TITLE Striker section $# secid elform aet 4 1 0 *PART $---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8 $# title Collar $# pid secid mid eosid hgid grav adpopt tmid 5 5 1 0 0 0 0 0 *SECTION_SOLID_TITLE Collar section $# secid elform aet

90

5 1 0 *INITIAL_VELOCITY_GENERATION $#nsid/pid styp omega vx vy vz ivatn 1 3 0.000 0.000 0.000 25.000000 0 $# xc yc zc nx ny nz phase iridid 0.000 0.000 0.000 0.000 0.000 0.000 0 0

91

Appendix C:

Transmission and reflection coefficients derivation

92

Force balance :

𝐴1(𝜎𝑖 + 𝜎𝑟) = 𝐴2𝜎𝑡

Velocity continuity :

𝜎𝑖𝜌1𝐶1

−𝜎𝑟𝜌1𝐶1

=𝜎𝑡𝜌2𝐶2

Transmission coefficient :

𝛼 =𝜎𝑡𝜎𝑖

Equation derivation :

𝜎𝑖 −𝜌1𝐶1𝜌2𝐶2

𝜎𝑡 = 𝜎𝑟

𝐴1 �𝜎𝑖 + 𝜎𝑖 −𝜌1𝐶1𝜌2𝐶2

𝜎𝑡� = 𝐴2𝜎𝑡

2𝐴1𝜎𝑖 = 𝐴2𝜎𝑡 + 𝐴1𝜌1𝐶1𝜌2𝐶2

𝜎𝑡

2𝐴1𝜌2𝐶2𝜎𝑖 = 𝐴2𝜌2𝐶2𝜎𝑡 + 𝐴1𝜌1𝐶1𝜎𝑡

𝛼 =2𝐴1𝜌2𝐶2

𝐴2𝜌2𝐶2 + 𝐴1𝜌1𝐶1

𝜎𝑖

𝜎𝑟 𝜎𝑡 A1, ρ1, C1

A2, ρ2, C2