A Split Hopkinson Pressure Bar Apparatus for High Strain Rate Testing of 
Interpenetrating Phase Composites (IPC): Measurements and Modeling  
 
by 
 
Chandru Periasamy 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
May 14, 2010 
 
 
 
 
Keywords: structural foams, dynamic failure, interpenetrating phase composites, 
syntactic foam, split Hopkinson pressure bar, finite element analysis 
 
 
 
 
 
Approved by 
 
Hareesh V Tippur, Chair, Professor of Mechanical Engineering 
Jeffrey C Suhling, Quina Distinguished Professor of Mechanical Engineering 
James S Davidson, Associate Professor of Civil Engineering 
 
 
 
 
 
 
Abstract 
 
 
 Novel materials with enhanced mechanical performance and multifunctionality 
are of interest to automotive, aerospace and marine industries alike. Designing materials 
with multiple thermo-mechanical attributes while satisfying lightweight constraint is 
rather challenging. This challenge becomes even greater when materials are to withstand 
elevated rates of loading. In this thesis, (i) a 3D Interpenetrating Phase Composite (IPC) 
made of Syntactic Foam (SF) in an open-cell aluminum scaffold is proposed for high-
strain rate applications and (ii) a split-Hopkinson pressure bar apparatus is developed for 
characterizing low-impedance materials such as the SF and IPC materials at high-strain 
rates. The SF used for making IPC is prepared by dispersing hollow glass microballoons 
into an epoxy matrix. The resulting SF is a two phase composite on a microscopic scale 
and a lightweight homogeneous, isotropic material on a macro scale. The infusion of SF 
in its uncured state into the aluminum scaffold results in an IPC with a continuous 3D 
interpenetrating network throughout the material volume. 
A split Hopkinson pressure bar has been developed and calibrated for carrying out 
high-strain rate stress-strain response measurements on low impedance materials. 
Dynamic compression characteristics, including strength and energy absorption features 
of IPC made of four different volume fractions of microballoons in SF are measured 
using this apparatus and compared. Measurements on pure SF counterparts are also 
carried out for evaluation relative to IPC. The results show that in general, the IPC foams 
 ii
 iii
outperform the SF samples in terms of compressive strength and energy absorption per 
unit volume. The underlying deformation processes are evaluated optically using real-
time high-speed photography as well as post-mortem analysis of deformed samples using 
scanning electron microscopy. The former includes the development of a novel grid-
based method for measuring surface displacements and strains.  
An idealized elasto-plastic unit cell finite element (FE) model based on explicit 
dynamics is proposed for studying IPC foams. The 3D geometry of the aluminum 
component in the cubic unit cell is modeled as a tetrakaidecahedron, a 14 sided 
polyhedron, also called the Kelvin cell. The SF constituent of the IPC is modeled as an 
occupant of the rest of the unit cell. The computational model incorporates infinite 
elements to represent the far-field regions surrounding the unit cell. The infinite elements 
help simulate stress wave dynamics of a sample undergoing deformation in a split 
Hopkinson pressure bar. The computations are successfully compared with measurements 
for all the four cases. The contour plots of von-Mises stress, equivalent plastic strain and 
displacements in the loading direction are analyzed at different strain levels. In addition, 
the kinetic and strain energy histories absorbed by SF and aluminum ligaments in the IPC 
are quantified. 
 
 
 
 
 
 
Acknowledgments 
 
 
 I thank my advisor, Dr. Hareesh V. Tippur, for his financial support and technical 
guidance without which my work would not have been a success. I consider the time I 
spent in the ?Failure characterization and optical techniques? laboratory at Auburn 
University, USA under Dr. Tippur?s supervision to be of immense value in advancing my 
technical education. I thank Dr. Jeffrey Suhling and Dr. James Davidson for their 
willingness to be on my MS committee and for reviewing this thesis. I thank the U.S. 
Army Research Office (ARO) for funding this research through grants W911NF-08-1-
0285 and W911NF-06-1-0126 (DURIP). Also, this work was made possible in part by a 
grant of high performance computing resources and technical support from the Alabama 
Supercomputer Authority. 
It would be unfair on my part not to acknowledge and thank Dr. Robert Jackson, 
as without his support and help my induction into Auburn University would not have 
been smooth. It was the opportunity to work on his team that introduced me to technical 
research and helped me develop my modeling skills. Dr. Dongyeon Lee, Kailash Jajam, 
Vinod Kushvaha, Allen Craven and Robert Bedsole deserve mention here as my good 
colleagues who made my experience inside and outside the laboratory stimulating, 
pleasant and eventful. I thank Rahul Jhaver for sharing his experimental knowledge and 
results. My parents, brother and grandmother have been my greatest believers and 
supporters in all of my aspirations. I dedicate this thesis to them. 
 iv
 
 
 
 
 
Table of Contents 
 
 
Abstract ............................................................................................................................ ii 
Acknowledgements ......................................................................................................... iv 
List of Tables ................................................................................................................ viii 
List of Figures ................................................................................................................. ix 
1. Introduction ...................................................................................................................1 
 1.1 Motivation ..............................................................................................................1 
 1.2 The syntactic foam / aluminum foam based IPC ...................................................3 
1.3 Objectives ...............................................................................................................5 
1.4 Organization of the thesis .......................................................................................6 
2. Literature review ...........................................................................................................8 
2.1 Split Hopkinson Pressure Bar (SHPB) ...................................................................8 
 2.2 Previous works on IPC .........................................................................................10 
3. Material preparation and sample dimensions .............................................................13 
3.1 Material preparation .............................................................................................13 
3.2 Sample dimensions ...............................................................................................16 
4. Experimental procedure ..............................................................................................18 
4.1 Experimental setup ...............................................................................................19 
 4.2 Location of strain gages .......................................................................................22 
4.3 Measurement of strain ..........................................................................................23 
 v
4.4 Stress, strain and strain rate ..................................................................................24 
5. Evaluation of the setup ................................................................................................27 
5.1 Dynamic equilibrium and constant strain rate ......................................................27 
 5.2 Effect of pulse shapers and striker length ............................................................31 
5.3 SHPB calibration ..................................................................................................33 
6. Experimental results ....................................................................................................34 
6.1 Quasi-static compressive stress-strain response ...................................................34 
 6.1.1 Syntactic foam ...........................................................................................34 
6.1.2 IPC .............................................................................................................37 
6.2 Dynamic compressive stress-strain response .......................................................39 
 6.2.1 Syntactic foam ...........................................................................................40 
6.2.2 IPC .............................................................................................................43 
6.3 Energy absorption .................................................................................................46 
6.3.1 Quasi-static ................................................................................................46 
6.3.2 Dynamic .....................................................................................................48 
6.3.3 Dynamic energy absorption history ...........................................................50 
6.4 Failure progression ...............................................................................................51 
7. Finite element modeling and analysis of IPC foam ....................................................56 
7.1 Geometric modeling .............................................................................................56 
 7.2 Computational model ...........................................................................................58 
7.2.1 Material properties .....................................................................................58 
7.2.2 Yielding and hardening ..............................................................................59 
7.2.3 Explicit integration scheme in ABAQUS ..................................................63 
 vi
 vii
7.3 Discretization ........................................................................................................64 
7.3.1 Infinite elements ? Background .................................................................67 
7.4 Boundary conditions and loading .........................................................................68 
7.5 Other analysis parameters ....................................................................................70 
7.6 Results from finite element analysis ....................................................................70 
7.6.1 Stress-strain response .................................................................................70 
7.6.2 Comparison between experimental and finite element results ..................71 
7.6.3 Analysis of contour plots ...........................................................................73 
7.6.4 Energy absorption ......................................................................................78 
8. Summary .....................................................................................................................81 
References   .....................................................................................................................85 
Appendix A   ...................................................................................................................92 
 
 
 
 
 
 
 
 
 
 
List of Tables 
 
 
Table 6.1 Quasi-static properties of syntactic foam and IPC foam samples ..................39 
Table 6.2 Dynamic properties of syntactic foam and IPC foam samples .......................42 
Table 7.1 Elastic properties of SF and aluminum used in the finite element model ......59 
Table 7.2 Inelastic properties of Al 6101 used in the finite element model ...................59 
viii 
 
 
 
 
 
 
 
List of Figures 
 
 
Figure 1.1 Cross section view of a human bone ...............................................................2 
Figure 1.2 Syntactic foam used in submersible vehicles in marine environments ...........4 
Figure 1.3 (a) Open-cell metal foams made of different materials, (b) Closed-cell 
aluminum alloy foam ......................................................................................4 
Figure 1.4 Schematic of a 3D interpenetrating network ...................................................5 
Figure 2.1 Hopkinson?s setup ...........................................................................................9 
Figure 3.1 (a) Rubber mold with uncured SF (b) Al foam inserted into mold 
containing uncured SF (c) Cast and machined SF and IPC samples  ...........14 
Figure 3.2 (a) Cross-section of IPC foam with 30% volume fraction of SF, (b) SEM 
micrograph of IPC.........................................................................................15 
Figure 3.3 (a) Representation of pores and cells, (b) Sample dimensions ......................16 
Figure 4.1 Photographs of the actual SHPB setup used for dynamic compression 
experiments ...................................................................................................18 
Figure 4.2 Split Hopkinson pressure bar setup ...............................................................20 
Figure 4.3 Schematic representation of the locations of incident, reflected and 
transmitted waves in a SHPB setup ..............................................................23 
Figure 4.4 Schematic illustration of the SHPB ...............................................................24 
 
 ix
Figure 5.1 Representative SHPB strain histories for (a) Syntactic foam sample with 
30% microballoon volume fraction, (b) IPC foam sample with 30% 
microballoon volume fraction .......................................................................28 
Figure 5.2 Demonstration of constant strain-rate for (a) SF-30 and (b) IPC-10 and 
dynamic stress equilibrium for (c) SF-30 and (d) IPC-10  ...........................30 
Figure 5.3 Effect of pulse shapers and repeatability on SF-30 samples .........................31 
Figure 5.4 Effect of striker length on SF-30 samples .....................................................32 
Figure 5.5 True stress-strain response for cast acrylic at a strain rate of ~ 1250 / sec ...33 
Figure 6.1 Effect of microballoon volume fraction on quasi-static compression 
response of syntactic foam (SF) samples ......................................................35 
Figure 6.2 SEM images of 30% V
f
 syntactic foam at (a) 14% strain and (b) 58% 
strain showing crushing of microballoons at various stages of static 
deformation ...................................................................................................36 
Figure 6.3 Effect of microballoon volume fraction on quasi-static compression 
response of IPC foam samples ......................................................................37 
Figure 6.4 SEM image of statically deformed IPC foam at 58% strain .........................38 
Figure 6.5 Effect of microballoon volume fraction on dynamic stress-strain response 
of SF samples ................................................................................................41 
Figure 6.6 (a) Side view of a dynamically deformed syntactic foam sample revealing 
a network of shear bands, (b) SEM image of cross section of dynamically 
deformed syntactic foam ...............................................................................43 
Figure 6.7 Effect of microballoon volume fraction on dynamic stress-strain response 
of IPC foam samples .....................................................................................44 
 x
Figure 6.8 (a) Side view of a dynamically deformed IPC foam sample revealing 
multiple shear bands and (b) SEM image of cross section of dynamically 
deformed IPC foam .......................................................................................45 
Figure 6.9 Energy absorbed (a) per unit volume and (b) per unit mass by SF and IPC 
foam samples under quasi-static loading up to 22% true strain ....................47 
Figure 6.10 Energy absorbed (a) per unit volume and (b) per unit mass by SF and 
IPC foam under dynamic loading up to 22% true strain .............................49 
Figure 6.11 Strain energy absorption history of (a) SF and (b) IPC with increasing 
strain in SHPB tests ......................................................................................51 
Figure 6.12 Schematic of the experimental setup used for high-speed photography to 
study real-time failure progression in SF and IPC samples in the SHPB 
set up ............................................................................................................52 
Figure 6.13 Stages of dynamic deformation of SF-30 ....................................................53 
Figure 6.14 Stages of dynamic deformation of IPC-30 ..................................................55 
Figure 7.1 Schematics representing the building blocks of an idealized Kelvin cell 
based IPC ......................................................................................................57 
Figure 7.2 Check for appropriateness of discretization ..................................................65 
Figure 7.3 Illustration showing (a) the infinite element layer in the FE model and 
stress wave behavior in the (b) absence and (c) presence of infinite 
element layer .................................................................................................66 
Figure 7.4 Load and boundary conditions ......................................................................69 
Figure 7.5 True stress?true strain responses of different IPC obtained from finite 
element analysis  ...........................................................................................71 
 xi
 xii
Figure 7.6 Comparison of experimental and FEA results of (a) IPC-10, (b) IPC-20, 
(c) IPC-30 and (d) IPC-40.............................................................................72 
Figure 7.7 Mises stress contour plots from dynamic FE model for IPC-30 ...................74 
Figure 7.8 Equivalent plastic strain contour plots from dynamic FE model for 
IPC-30 ...........................................................................................................75 
Figure 7.9 x-displacement contour plots from dynamic FE model for IPC-30 ..............77 
Figure 7.10 History of (a) kinetic energy and (b) strain energy absorption by 
aluminum and SF in unit-cell IPC-30 ..........................................................78 
Figure 7.11 History of percentage of (a) kinetic energy and (b) strain energy 
absorption by SF and aluminum in IPC-30 .................................................79 
Figure A1 Quasi-static and dynamic compressive responses of neat epoxy (?Epo-
Thin?) used in SF and IPC foams .................................................................92 
 
 
 
1 INTRODUCTION 
 
1.1 Motivation 
The continued human quest for making lives more comfortable has increased 
demands on engineering and technology. Lately, researchers have been looking towards 
nature for inspiration to meet some of these demands such as those involving materials 
and structures. The field of science and engineering which focuses on studying and 
researching bio-inspired materials and structures is called biomimetics [1]. Naturally 
occurring structural materials have undergone millions of years of evolutionary processes 
to adapt to the changing needs of their environment and functionality. So, it is only wise 
to adopt some of these examples in the process of developing new structural materials. 
Recent advances in commercial and defense automotives, as well as marine 
construction, require innovative lightweight materials that could serve in the design of 
stiff and tough structures capable of absorbing impact energy. This concurrence of 
material properties cannot be easily formulated into a single material system. The 
microstructure of some naturally occurring materials such as bone (Fig 1.1) and nacre  
have a 3D interpenetrating structure [2], often made up of two or more biological phases. 
Thinking along these lines, and considering the combined requirement of structural 
integrity and energy absorption, an Interpenetrating Phase Composite (IPC) [3] made of 
two phases ? one, brittle Syntactic Foam (SF) and the other, ductile aluminum (Al) foam 
1?
?
is being proposed in this thesis work. In general, IPCs are multiphase materials in which 
each constituent forms a continuous 3D network within the material volume. Thus, each 
phase in its stand alone state can be imagined to have an open-cellular microstructure.  
 
 
Figure 1.1: Cross section view of a human bone 
(From www.health.state.ny.us/diseases/conditions/osteoporosis/needknow.htm) 
 
Even though IPCs are heterogeneous on a micro/meso scale, the macro scale 
response is often isotropic. Each phase offers its own unique property to the IPC as a 
whole. For example, one phase might offer good toughness and thermal conductivity, 
while the other might enhance stiffness and dielectric properties. This co-existence of 
desirable properties without significant directional dependency or distinct weak planes 
(as opposed to conventional layered composites) makes an IPC attractive for structural 
applications. That is, each phase of an IPC contributes its unique property to the overall 
structural response synergistically. The above mentioned possibilities invite an indepth 
study on the IPC.  
 
 
2?
?
1.2 The syntactic foam / aluminum foam based IPC 
In this work, the IPC is made of two phases ? a polymer based Syntactic Foam 
(SF) and an open-cell aluminum (Al) foam. SF is a low density composite comprised of 
epoxy (or, any polymer) and hollow glass (or, any ceramic) microballoons. The 
microballoons are microscopic hollow spheres, usually a few micrometers to several 
hundred microns in diameter, and wall thickness ranging from fraction of a micron to a 
few microns. When these two materials are mechanically blended and cured, they form a 
two phase composite on a micro scale. Considering the size of the microballoons and 
their uniform distribution within the epoxy matrix, the resulting composite can be 
regarded as an isotropic material on a macro scale. The microballoons are hollow (with a 
partial pressure) which makes the syntactic foam lighter as well as buoyant. This unique 
combination of low density and buoyancy makes SF an ideal choice for marine 
applications (Fig. 1.2). Metal foams, on the other hand, are broadly classified into open-
cell (Fig. 1.3(a)) and closed-cell (Fig. 1.3(b)) foams. They are typically used for energy 
absorption and heat dissipation applications. The closed-cell foams in particular are used 
as layers in sandwiched sheets for energy absorption. The open-cell foams, on the other 
hand, are also used for other applications [2] such as cryogenics, filteration, missile 
baffles, etc.  Due to their relatively high cost, metal foams are only used in areas of 
advanced technology, such as aerospace structures. 
 
3?
?
 
Figure 1.2: Syntactic foam used in submersible vehicles in marine environments 
(From www.naval-technology.com) 
 
4?
?
between the two phases is occupied by one of the two phases.) It can be seen that each 
  
(a) (b) 
Figure 1.3: (a) Open-cell metal foams made of different materials (from 
www.ergaerospace.com); (b) Closed-cell aluminum alloy foam (from Reference [4]) 
 
The IPC used in this work is made by infusing the SF into the open-cell aluminum 
foam. The term ?Interpenetrating? signifies that each phase of the composite is 
continuous and forms a network of its own. Figure 1.4 shows a schematic of an 
interpenetrating network. The region shown in red (gray) represents one phase of the 
network, and the region shown in white is the second phase. (In reality, the empty space 
phase is three dimensionally continuous and also forms an interpenetrating network. A 
material of this type can prove to be advantageous in many applications as it can offer the 
combined properties of the two different materials synergistically. For example, in the SF 
/ aluminum foam IPC, the aluminum foam might add to the stiffness of the whole 
structure, whereas the aluminum foam by itself does not have appreciable stiffness.  
 
 
Figure 1.4: Schematic of a 3D interpenetrating network 
1.3 Objectives 
wing are the primary objectives of this work: 
? Dev up for characterizing the 
h different volume fractions (V
f
) of microballoons in 
dynamic compression properties, viz., stress-strain response, compressive 
strength and energy absorption of SF and IPC foams using the SHPB.  
 
The follo
elop a split Hopkinson pressure bar (SHPB) set 
compression behavior of low impedance materials and calibrate it for accuracy and 
repeatability of measurements.  
? Prepare SF and IPC samples wit
SF. 
? Test 
5?
?
? Comparatively study the dynamic compression properties of SF and IPC and the 
effect of microballoon volume fraction V
f 
in the SF on the overall performance of 
erparts.  
erstand the deformation processes in SF and IPC materials.  
This thesis has been structured in such a way that the experimental aspects are 
putational work is detailed later on. The next chapter 
present
IPC. 
? Evaluate the high-strain rate performance of SF and IPC relative to the quasi-static 
count
? Perform high-speed photography and microstructural investigation of the deformed 
samples to und
? Develop a finite element model of the IPC and simulate dynamic compression 
response using the measured mechanical responses of its constituents. 
? Validate the computational model by comparing the computed high-strain rate IPC 
responses with the measurements. 
 
1.4 Organization of the thesis 
presented first and then the com
s a literature review on (a) the Split Hopkinson Pressure Bar (SHPB) setup used 
for high-strain rate experimental characterization of materials and (b) the previous works 
on IPC. Chapter 3 describes material preparation methods for both SF and IPC foams. 
The experimental procedure and the evaluation of the SHPB apparatus, respectively, are 
presented in Chapters 4 and 5. The results of the experiments are detailed and analyzed in 
Chapter 6. For the purpose of comparison of the strain rate effects, the appropriated 
quasi-static results from a previous work [2] are also presented intermittently throughout 
this chapter. Chapter 7 presents the finite element procedure, including details on 
6?
?
7?
?
geometric modeling, relevant material properties, model discretization and application of 
boundary conditions. A comparison of the experimental and computational results is also 
presented in this chapter. Finally, Chapter 8 summarizes the results of this work. 
 
 
 
2 LITERATURE REVIEW 
 
2.1 Split Hopkinson Pressure Bar (SHPB) 
A split Hopkinson pressure bar is a device used to study the mechanical behavior 
of a material at high-strain rates. The principle behind the SHPB was first introduced by 
Bertram Hopkinson [5] in 1914. Hopkinson used the principle to measure the pressure 
developed in a bar when a blast occurs at one end of it. In his set up (Fig. 2.1), a bar was 
suspended with one end attached to a detonation mechanism and the other end in 
magnetic contact with a time piece. When a blast occurs at one end, a pressure pulse 
travels through the suspended bar and impacts the time-piece. At the free end of the time 
piece, the pressure pulse reflects back as a tensile pulse, and when it reaches the time 
piece ? bar interface, the time piece separates from the bar and flies into a momentum 
trapping mechanism. The energy recorded by the momentum trap corresponds to the 
energy of twice the wave length of the pressure pulse in the time piece.  
Subsequently, Hopkinson?s principle was used by Robertson [6] in 1921 and by 
Landon and Quinney [7] in 1923. The technique was then modified in 1948 by Davies [8] 
who used condensers for measuring displacements in the pressure bar. In 1949, Kolsky 
[9] modified the setup further by adding a second pressure bar and placed the specimen 
between the two bars. He used a cylindrical condenser microphone to measure the 
amplitude of the pressure pulse produced by firing a detonator at the free end of the 
8?
?
incident bar. At the free end of the second bar, he used a parallel plate condenser 
microphone to measure displacement. He used the signals from the two condensers to 
measure the stress-strain response of the specimen. Most recently, researchers have 
replaced the detonator with a striker impacting the incident bar (at a known velocity) and 
the condensers with strain gages pasted on the pressure bars.  
 
 
Momentum Trap 
Gun Cotton Cylinder Bar Time Piece 
Recorder 
Suspension Rails 
Figure 2.1: Hopkinson?s Setup 
(Reproduced from reference [5]) 
 
The invention of the split Hopkinson bar has helped researchers to dynamically 
test a variety of solids including metals, polymers, ceramics and composites [10]. 
Modifications to the conventional Kolsky bar have been made by many researchers to 
9?
?
test brittle materials [11, 12], low-impedance or soft materials [13, 14, 15] and structural 
foams [16]. To test low-impedance materials at high strain rates, Chen et al. [13] used a 
hollow high-strength aluminum alloy for the bar material in place of the conventionally 
used solid circular cross sectioned steel bars. It has been reported in the same work that 
the reduced cross sectional area of the hollow aluminum bars increased the amplitude of 
the transmitted strain signal in the bars by an order of magnitude. Song and Chen [16] 
obtained the dynamic compressive response of epoxy syntactic foam using a SHPB setup 
in which they used an annealed copper pulse shaper at the striker-incident bar interface. 
In their work, it has been demonstrated that the specimens were under dynamic stress 
equilibrium during loading due to the effect of the pulse shaper. Recently, Subhash and 
Liu [17] tested an epoxy-based polymeric foam for its crushability under rigid lateral 
confinement. In their work, they used magnesium bars to test samples under low stress, 
and steel bars for samples under high stress. Thus, it is evident that material properties 
(density in particular) play a significant role in the stress wave history in the bars. 
Likewise, Bouix, et al. [18] used nylon bars in their SHPB setup to dynamically 
characterize the compression properties of polypropylene foams. It is to be noted that 
nylon which is visco-elastic, was the choice material for testing a low impedance rubbery 
material. 
 
2.2 Previous works on IPC 
The primary material system under investigation in this work is an IPC, which is a 
two phase composite. The following is a review of some of the works on different IPC 
systems available in the literature. Among the existing works on IPCs, Prielipp et al. [19] 
10?
?
studied the mechanical properties of aluminum/alumina interpenetrating composite and 
measured fracture strength and fracture toughness of the composite as a function of 
ligament diameter and volume fraction of the metal reinforcement. The metal reinforced 
interpenetrating composites consistently had higher fracture strength. Breslin et al. [20] 
characterized an aluminum/alumina IPC obtained using the liquid phase displacement 
reaction method. The resulting IPC showed enhanced density, thermal conductivity and 
CTE characteristics, without compromising stiffness or fracture toughness. Travitzky et 
al. [21] showed that the residual stresses developed in silicon during solidification of 
molten Si within an Al
2
O
3
 matrix results in the greater strength and fracture toughness of 
the resulting Al
2
O
3
/Si interpenetrating system. Wegner and Gibson [22] developed finite 
element models to predict the elastic, strength and thermal expansion properties of two 
phase IPCs. They attributed the enhancement in the thermo-mechanical properties to the 
contiguity of the phase with the most desirable property. In a later work [23], the same 
authors studied the mechanical behavior of resin-impregnated porous stainless steel. The 
yield strength, ultimate strength, elongation at failure and the elastic modulus were all 
found to increase with an increasing volume fraction of steel. Skirl et al. [24] examined 
the thermal expansion behavior of alumina/aluminum IPC. A pressure infiltration 
technique was used to introduce aluminum into slip cast and then sintered alumina. The 
tensile and compressive residual stresses in alumina and aluminum phases were found to 
enhance the overall thermal coefficient of expansion. An increase in failure strain with 
increased metal content was also reported. Mayer and Papakyriacou [25] studied the 
fatigue behavior of graphite/aluminum IPC. The lightweight metals, such as aluminum, 
were infiltrated into polycrystalline graphite to improve the fracture toughness of 
11?
?
12?
?
polycrystalline graphite. A 30% increase in the cyclic strength and a 10% increase in the 
endurance limit were also reported. Tilbrook et al. [26] measured the effective 
mechanical properties of alumina-epoxy interpenetrating composites and reported strong 
dependence of properties on the composition and the processing of the material itself. 
Etter et al. [27] examined the flexural strength and fracture toughness of 
graphite/aluminum IPC at room temperature and at 300
o
C. A 200% improvement in both 
these characteristics for IPC over un-infiltrated material at room temperature was 
reported. Additionally, no significant drop in properties was seen at elevated 
temperatures. Jhaver and Tippur [28] demonstrated the feasibility of a lightweight IPC 
foam by infiltrating a syntactic foam (SF) [29-31] into an open-cell aluminum network 
and examining quasi-static uniaxial compression response experimentally and 
numerically. They showed that the compressive strengths and energy absorption 
capacities of IPC samples were greater than those of the corresponding syntactic foam 
samples. 
As noted in the above reports, the enhancements in the effective mechanical 
properties of IPC are dictated by the contiguity of the phase with the most desirable 
properties. An IPC foam formed by a continuous interpenetrating 3D network of 
syntactic and aluminum foams has the potential for improving structural integrity with 
good energy absorption characteristics, thus making it a worthwhile material system for 
further study. This work builds on the feasibility study of the IPC foam first reported in 
Ref. [28].   
 
 
 
3 MATERIAL PREPARATION AND SAMPLE DIMENSIONS 
 
 This chapter details the preparation of the IPC foam material and the choice of 
dimensions of the samples. In the first section, the sequential procedure adopted for the 
preparation of the SF samples is described, followed by a description of the preparation 
of the IPC samples. Later, a microscopic view of an IPC sample cross section is 
discussed with regard to the bonding between SF and aluminum and the randomness of 
the microballoon distribution. Lastly, a discussion on the sample dimensions and the 
reason for choosing those dimensions will be presented. 
 
3.1 Material preparation 
 The constituents used for preparing the SF foam were a low viscosity epoxy (Epo-
Thin
TM
 from Beuhler, Inc. USA, mass density of the resin ~1100 kg/m
3
) and hollow glass 
microballoons (K-1
TM
 microballoons from 3M Corp., bulk density 125 kg/m
3
) of average 
diameter of ~60 ?m and wall thickness of ~0.6. Syntactic foam samples were prepared by 
mixing the desired quantity of microballoons into epoxy resin. The uncured syntactic 
foam was then transferred into a silicone rubber mold (Fig. 3.1 (a)) after vacuuming (at 
approximately -75 kPa gage pressure) to remove any trapped air bubbles in the mixture. 
The mixture was then allowed to cure for at least seven days before being removed from 
the mold and machined to size as shown in Fig. 3.1 (c). 
13?
?
     
Al Foam
Uncured SF 
Rubber mold
(a) (b)
 
 
Cured SF 
Cured IPC 
Machined SF sample
Machined IPC sample 
Machined IPC sample 
 
(c)
Figure 3.1: (a) Rubber mold with uncured SF, (b) Aluminum foam inserted 
into mold containing uncured SF, (c) Cast and machined SF and IPC samples 
 
 The metal foam used in the IPC was a commercially available open-cell Duocel
? 
aluminum (Al 6101-T6) preform obtained from ERG Aerospace Corp., with a pore 
density of 40 pores per inch (~8% relative density). The metal foam was cleaned with 
14?
?
acetone and then coated with silane, ?-aminopropyltrimethoxysilane 
(H
2
NC
2
H
4
NHC
3
H
6
Si(OCH
3
)
3
), to enhance the bond strength between aluminum 
ligaments and syntactic foam. To prepare the IPC, the degassed, uncured SF was 
prepared the same way as before. Then, the silane coated aluminum foam was slowly 
inserted into the mold (as shown in Fig. 3.1 (b)) containing uncured syntactic foam so 
that the syntactic foam filled the open-pores of the aluminum network. After curing for at 
least seven days, the unfinished sample was removed from the mold for machining (Fig. 
3.1 (c)). 
  
Figure 3.2: (a) Cross-section of IPC foam with 30% volume fraction of SF,  
(b) SEM micrograph of IPC; (From Ref. [28])  
 A photograph of the cross-section of the cured IPC foam is shown in Fig. 3.2(a).  
The shiny metallic ligaments of the aluminum preform interspersed within the syntactic 
foam can be readily observed.  Figure 3.2(b) shows a micrograph of the cross section of 
the IPC with 30% microballoon volume fraction in SF. The interface between aluminum 
ligaments and syntactic foam is well defined without voids or cracks suggesting a good 
bond between the two phases. The microballoons also appear to be uniformly distributed 
within the epoxy matrix. 
15?
?
3.2 Sample dimensions 
 The cured SF and IPC samples were removed from the mold and rested for about 
a week before machining. The sample dimensions were determined based on a previous 
work by Gibson [32]. Gibson has analyzed size effects on foam test specimens, and 
proposed a specimen thickness (length) to cell size ratio (t/d) of 8 or above, so that the 
specimen better represents the bulk. By size effects, Gibson means the effect on the 
measured material properties when the macroscopic sample dimensions are of the order 
of the pore size. The aluminum foam used in this thesis work has a pore density of 40 
pores per inch with 22 ? 26 cells per inch according to the manufacturer. Therefore, it can 
be assumed that there are approximately 24 cells per inch on an average (Figure 3.3(a) 
shows the relative sizes of pores and cells). Thus, the individual cell size is approximately 
1.06 mm (0.042 in). A t/d ratio of 9 was chosen for the SHPB tests. Accordingly, the 
samples used in the dynamic testing had a length of 9.5 mm (0.375 in) and a diameter of 
12.7 mm (0.5 in) as shown in Fig. 3.3(b). 
 
    
12.7 mm
  9.5 mm 
 
Figure 3.3: (a) Representation of pores and cells (From www.ergaerospace.com); 
(b) Sample dimensions  
 
(a) (b) 
16?
?
17?
?
 From the dimensions shown in Fig. 3.3(b), the thickness (t) to diameter (D) ratio 
(or, slenderness ratio S
0
) is 0.75. This S
0
 was chosen with reference to the work published 
by Malinowski and Klepczko [33] in which they suggested an optimum value
 
for use in 
SHPB tests for different material categories based on the axial inertia and interfacial 
friction of the sample material. According to their work, samples made of low flow stress 
materials such as gold and uranium should have 0.1 ? S
0
 ? 0.5. For samples made of 
moderate density metals like aluminum and copper they suggest an optimum S
0
 range of 
0.5 to 1.0, and for hard materials like high strength steels, an optimum S
0
 range of 1.0 to 
1.5. The IPC used in this work consists of aluminum and SF, which are made of 
moderately dense and brittle, but not hard materials. Logically, the integrated effect of 
combining the two materials in the IPC suggests a S
0
 in the 0.5 to 1.0 range for the SHPB 
test samples. 
 
 
 
4 EXPERIMENTAL PROCEDURE 
 
The dynamic experiments were performed using a Split Hopkinson Pressure Bar 
[9] (SHPB) apparatus (Fig. 4.1) developed for this study. This chapter provides a detailed 
description of the experimental setup, the working principle and the theoretical basis for 
obtaining the dynamic stress-strain response of SF and IPC samples. 
(a) 
 
 
 
 
Pressure 
cylinder 
Pressure 
gage 
Air inlet
I-beam 
Bronze 
bearing 
(b) 
Gun 
barrel 
Solenoid 
valve 
Striker 
Incident 
bar 
18?
?
 
 
(c) 
 
Oscilloscope 
Signal conditioning unit
Location of 
strain gage 
Figure 4.1: Photographs of the actual SHPB setup used for dynamic 
compression experiments 
 
4.1 Experimental setup 
Figures 4.1 and 4.2 show photographs and a schematic of the SHPB setup. It 
consists of an air pressure cylinder with a barrel, a striker (25.4 mm diameter and 406 
mm long), a pulse shaper, an incident bar (25.4 mm diameter and 2438 mm long), a 
transmitter bar (25.4 mm diameter and 2438 mm long), a stopper, two strain gages (one 
on each bar), a signal conditioning unit (Ectron
TM
 Wheatstone bridge with signal 
conditioning amplifiers) and a data acquisition system (LeCroy
TM
 high-speed digital 
oscilloscope).  
The cylinder, barrel and the bars were all mounted on two steel I-beams supported 
by adjustable legs. The adjustable legs were used for the leveling and aligning the bars. A 
solenoid valve was connected the cylinder and the barrel was used to launch the striker. 
The solenoid valve was encased within a ?-metal sheet to attenuate the magnetic flux 
field that would otherwise generate electrical noise in the output signals and trigger the 
Incident bar 
Transmitter bar 
19?
?
data acquisition circuit prematurely. A series of through holes were drilled into the end of 
the gun barrel so that when the striker was launched, the air ahead of the striker escaped 
through the holes laterally and the air resistance on the striker was minimized. To reduce 
friction, the incident and transmitter bars were supported on a series of bronze bearings 
mounted on the I-beams. Both the striker and the two bars were made of Al-7075 of the 
same diameter in order to eliminate the impedance mismatch between them. The 7075 
aluminum alloy was chosen for its strain rate independent characteristics. 
The sample was held snugly in between the incident and the transmitter bars prior 
to loading. The sample ends were coated with a thin layer of grease to position the 
sample and minimize friction at the interfaces. A 6.4 mm diameter disc made of a stack 
of four sheets of adhesively backed copper tapes each 0.1 mm thick was used at the front 
end of the incident bar as the pulse shaper. The interface between the pulse shaper and 
the incident bar was also coated with grease for tackiness. The use of pulse shaper helped 
to achieve dynamic stress equilibrium (to be described later) in the sample for the loading 
duration. The pulse shaper, the incident bar, the sample and the transmitter bar were all 
pushed against each other so that there were no gaps at any of the interfaces.  
Transmitter bar 
Strain gage 
Sample
Strain gage 
Incident bar 
Striker 
 
Figure 4.2: Split Hopkinson pressure bar setup 
Data acquisition 
system
Signal Conditioner 
Stopper
Cylinder 
Pulse shaper
20?
?
The strain histories in the bars were captured by two strain gages fixed one on 
each bar. The strain gages (CEA-13-062UW-350) had a grid resistance of 350? and a 
gage length of 1.6 mm (0.062 in) and were purchased from Vishay
?
 Micro-
Measurements
?
. One practical problem while conducting experiments was that the lead 
wires from the strain gages were breaking off at the terminals on the gages due to the 
intense stress waves. To overcome this, one long piece of masking tape was used to first 
wind the lead wires together. Then, the wire pair was attached to the bar using the 
remaining free end of the tape which was wrapped around the bar. In addition to the 
gages on the bars, two external dummy gages of the same specifications were mounted 
on an unstressed bar placed in the same atmospheric conditions as the incident and the 
transmitter bars. The leads from the strain gages on the bars and the external dummy 
gages were connected to the signal conditioning unit. The output from the signal 
conditioning unit was connected to the digital oscilloscope. The strain gage signals were 
filtered using a built in (?Eres? or Enhanced resolution filter) filter available in the 
oscilloscope. A resolution enhancement of 1.5 bits was chosen. The filtered data was then 
used to calculate the actual strain in both the bars as a function of time.  
The striker was pushed all the way to the back of the gun barrel, and the pulse 
shaper was fixed on the incident bar. The sample was then held between the bars and they 
were all pushed against the stopper at the end of the transmitter bar. Next, the air cylinder 
was pressurized and the striker was launched towards the incident bar by opening the 
solenoid valve using an electrical switch. When the striker impacted the incident bar, a 
compressive stress wave (incident wave) originated at the front end of the bar and 
traveled towards the sample. A part of the stress wave (transmitted wave) traveled into 
21?
?
the sample and subsequently into the transmitter bar. Simultaneously, a portion of the 
stress wave was reflected (reflected wave) at the incident bar-sample interface and 
traveled back into the incident bar as a tensile stress wave. All three pulses were captured 
by the two strain gages mounted on the incident and the transmitter bars, and recorded as 
voltage histories in the data acquisition system after being processed by a signal 
conditioning unit
1
. The oscilloscope acquired the strain signals as voltage histories at a 
rate of 2.5 x 10
6
 samples per second. These voltage histories were then used to calculate 
the strain histories of the bars.  
 
4.2 Location of Strain Gages 
The location of the strain gages on the incident bar was chosen based on the 
elastic wave speed (C
0
) in the bar, the time period of the wave and the total length of the 
bar (L
bar
) to make sure that there was no overlapping of the incident and the reflected 
stress wave signals. The bar wave speed nullnull null
null
null
null  in the Al 7075 bars is 5123 m/s [16], 
the striker length (L
striker
) was 406.4 mm (16 in) and the length of the bars (L
bar
) were  
2438.4 mm (96 in). From the above data, the time period (T) [34] of the stress wave pulse 
was calculated as 
ker
0
2
stri
L
C
= 160 ?s. Then, the time taken by the wave to traverse the 
incident bar once was calculated as 
0
bar
L
C
= 475 ?s. 
????????????????????????????????????????????????????????????
1
?To ensure that the circuitry and output voltage values from the signal conditioning unit were correct, a 
strain gage of the same specifications was mounted on a cantilever beam and the strains for various loads 
were measured. These measured strains were compared against theoretically calculated strains based on the 
one-dimensional beam equations, and the error factor was determined. This error factor was multiplied to 
the output voltages corresponding to the gages on the incident and the transmitter bars to ensure accuracy. 
22?
?
 
Figure 4.3: Schematic representation of the locations of incident, reflected and 
transmitted waves in a SHPB setup 
 
From the above calculations, it is evident that the time period of the wave 
corresponds to twice the length of the striker. So, the strain gage on the incident bar was 
located at approximately 1200 mm from the incident bar?specimen interface which is 
more than twice the length of the striker. The schematic representation of the above 
discussion is shown in Fig. 4.3 using the so-called x-t diagram. Accordingly, the incident 
and the reflected wave signals will not overlap during a typical test.  
 
4.3 Measurement of strain 
The strain gages on the bars were separately connected to Wheatstone bridge 
circuits in the signal conditioning unit. A half-bridge circuit was used in this study. The 
external dummy gage of the half bridge circuit was mounted on a rod of the same 
material as the bars so that the temperature effects on the working gage would be 
nullified. The rod with the dummy gage was placed in the vicinity of the bars so as to 
Specimen Strain gage 
Transmitter bar 
Strain gage 
Incident bar 
Striker 
Distance (x)
Incident wave 
 
 
T 
Reflected wave
 i 
Transmitted wave 
m 
(t) 
e  
23?
?
maintain the same atmospheric conditions. The strain in the bars is recorded as voltages, 
V
0
 using the oscilloscope. If S
g
 is the gage factor of the strain gage, ? is the strain in the 
bar and R
g 
is the resistance of the strain gage, then change in resistance ?R
g
 in the strain 
gage is given by, 
?null
null
null
null
null null
null
null   (4.1)
 voltage for an initially balanced bridge V
0
 when the gain or amplification is G, 
null
null
 .                                                           
The output
is given by 
null null 
null
null
null
 
?null
null
null
null
                        (4.2)   
where V
e
 is the exc n voltage fo  From the last two equations, the s
null
null
null
null
null
itatio r the bridge. train 
can be expressed as, 
nullnull 
nullnull
null
 .                      (4.3) 
 
.4 Stress, strain and strain rate 4
The stress ()t? , strain rate ()t?&   and strain ()t?  histories of the sample were 
calcula e sp  ba
Figure 4.4: Schematic illustration of the SHPB 
 
ted using th lit Hopkinson r equations. The details of the derivation [34] of 
the equations are provided as follows. 
 
Transmitter bar (A
0
, ?
0
) Incident bar (A
0
, ?
0
)Striker 
1    2
v
1          
 v
2
 
 
 
24?
?
The strain rate or the rate of change of linear dimension in the specimen is given by 
nullnull
nullnull null 
nullnull
null 
null
null
nullnullnullnullnull
null
nullnullnull
null
 .         (4.4)  
As seen in Fig. 4.4, null
null
 and null
null
are interface velocities at 1 and 2 and L is the specimen 
=
                     (4.5)  
where C
0 
and  are the wave speeds n the bars and the specimen respec
null
null
     (4.6) 
From Eqs. (4.4), (4.5)
 
length. At the interfaces, at t  0,  
null
null
null null
null
null
null
 , null
null
null null
null
null
null
     
 C  i , tively, and the 
subscripts I, R (and T) denote incident, reflected and transmitted, respectively. At t > 0, 
null
null
 
is reduced because of the reflected wave. That is,  
null null
null
nullnull
null
null null
null
null .     
 and (4.6), the strain-rate expression can be re-written as,  
nullnullnullnullnull null 
null
0
nullnull
null
nullnull
null
nullnull null
0
null
null
 
null
  . 
 The strain in the specimen can be obtained by integrating the strain rate: 
 null nullnull
null
nullnullnull null null
null
nullnullnullnull null
null
nullnullnullnullnullnull
null
null
.        nullnullnullnull null 
null
null
null
     (4.7) 
By assuming stress equilibrium in the specime l times from 0 to t, t  stres
 
null
null
nullnullnullnull null
null
nullnullnull
2null
n at al he s in the 
sample can be written as the average of the stresses on the specimen at the two interfaces 
as, 
nullnull  
where, 
null
null
null
nullnull
null
nullnull
null
null , 
null
null
nullnullnull null null
null
null
null
null
null
 . 
null
null
nullnullnull null null
25?
?
26?
?
Here, A
0
 and A are cross sectional areas of the bars and specimen, respectively, and E
0 
is 
the elastic modulus of the bars. Thus, 
nullnull 
null
null
null
null
nullnull
nullnull
null
nullnullnull null null
null
nullnullnullnull null
null
nullnullnullnull . 
For dynamic equilibrium, null
null
nullnullnull nullnull
null
nullnullnull d null
null
nullnull
null
null null
null
 . Hence, we get, 
null
 an
nullnullnullnull null null
null
null
null
null
null
nullnullnull ,          (4.8) 
nullnullnullnullnull null null 
nullnull
null
null
null
null
nullnullnullnull null null
nullnull
null
null
 ,          (4.9) 
null
null
null
null
The last three equatio
SHPB and were used to calculate the stress, strain and strain rate in the specimen as a 
function of time, and subsequently, to obtain stress-strain curves. Once the engineering 
stresses (? )? and strains (? ) were obtained, constant volume approximation was 
invoked to obtain the corresponding true stress (? ) and true strain (? ) values using: 
nullnull 
) 
null  nullnull.         (4.10) 
 
ns (4.8), (4.9) and (4.10) are the governing equations of the 
eng eng
true true
null
nullnullnullnull
nullnull null1nullnull
nullnullnull
null ,              (4.11) 
null
nullnullnullnull
nullnull
nullnullnull
null1 nullnull
nullnullnull
null .         (4.12
 
 
 
5 EVALUATION OF THE SETUP 
 
The SHPB setup developed in the laboratory was tested for its ability to produce a 
constant strain rate, dynamic stress equilibrium and experimental repeatability. Strikers of 
different lengths, ranging from 8 inches to 16 inches, were used to check the appropriate 
striker lengths for ensuring that the samples would undergo an appreciable 
strain/deformation within one stress pulse. In this chapter, the above issues are presented 
in addition to the calibration tests of the setup. 
 
5.1 Dynamic equilibrium and constant strain rate 
The standard SHPB equations for strain rate, strain and stress in the specimen 
assume that the specimen is under dynamic stress equilibrium and experiencing a 
constant strain rate during deformation. The dynamic stress equilibrium essentially means 
that the instantaneous forces at the front and back ends of the specimen are equal. That is, 
the sum of the strains or output voltages corresponding to the incident and reflected stress 
waves should equal the strain or output voltage corresponding to the transmitted wave. In 
real situations, however, this does not occur unless the rate of loading is reduced to allow 
sufficient time for the specimen to reach dynamic equilibrium. This is often achieved by 
the use of so-called pulse shapers. A pulse shaper essentially moderates the initial impact 
allowing sufficient time for the specimen to attain stress equilibrium before undergoing 
27?
?
large deformation. So, different types of pulse shapers were used in this work at the 
striker / incident bar interface to attain the best possible dynamic equilibrium in the 
sample.  
 
Time (x 10
-6
) sec
0 200 400 600 800 1000
St
r
a
i
n
 (
x 1
0
-6
) 
 
-2000
-1500
-1000
-500
0
500
1000
1500
2000
Incident wave
Reflected wave
Transmitted wave
Strain rate ~ 1500 /sec
?
(a) 
Time (x 10
-6
) sec
0 200 400 600 800 1000
Str
a
in
 (x 1
0
-6
)
-2000
-1500
-1000
-500
0
500
1000
1500
2000
Incident wave
Reflected wave
Transmitted wave
 
(b) 
Figure 5.1: Representative SHPB strain histories for (a) Syntactic foam 
sample with 30% microballoon volume fraction, (b) IPC foam sample 
with 30% microballoon volume fraction 
28?
?
The pulse shapers made of a stack of copper tapes that are more compliant than 
aluminum resulted in reducing the rate of loading and smoothing the instantaneous input 
strain wave profile as seen in Fig. 5.1. The representative strain histories for SF-30 and 
IPC-30 samples obtained by using the above mentioned pulse shaper that resulted in good 
dynamic stress equilibrium are shown in Fig. 5.1. In Figs. 5.1(a) and (b), the solid line 
corresponds to the strain history in the incident bar, and the dotted line corresponds to the 
one in the transmitter bar. As can be seen from the strain histories, the initial compressive 
wave ramps up for about 20 microseconds to attain the maximum amplitude instead of an 
instantaneous jump. Subsequently, the amplitude remains constant for about 125 ?s 
before starting to ramp down. The tensile reflected wave from the incident bar / specimen 
interface reaches the strain gage on the incident bar at approximately 250 ?s after the end 
of the incident wave. The transmitted wave is seen as a compressive signal in Fig. 5.1, 
and it lasts for almost 125 ?s before its amplitude starts to ramp down. 
The condition that the sample should be under a constant strain-rate during the 
deformation was verified by plotting the sample strain-rate histories (Figs. 5.2(a) and 
(b)). The equations [34] for SHPB reveal that the strain rate in the specimen is 
proportional to the reflected wave induced strain in the incident bar. The nearly constant 
strain region in the reflected wave (Fig. 5.1) suggests that the specimen was under a 
constant strain rate for about 125 ?s during the dynamic loading event. The strain-rate 
histories shown in Figs. 5.2 (a) and (b) for the representative SF-30 and IPC-10 samples 
confirm this. As was the case for the incident (as well as reflected) strain waves, the 
strain-rate histories also follow the ramp-up and ramp-down patterns during the initial 
29?
?
and final stages of the pulses, respectively. Also, it is evident that the strain-rates remain 
nearly constant for about 125 ?s for both SF and IPC cases.  
 
Time (microseconds)
0 50 100 150 200 250
S
t
rain rat
e (
s
-1
)
0
500
1000
1500
2000
2500
3000
?
Time (microseconds)
0 50 100 150 200 250
S
t
rai
n
 ra
t
e
 
(s
-1
)
0
500
1000
1500
2000
2500
3000
 
(a) (b) 
Time (microseconds)
0 50 100 150 200 250
E
rror %
-100
-80
-60
-40
-20
0
20
40
60
80
100
??
Time (microseconds)
0 50 100 150 200 250
Er
ro
r
 %
-100
-80
-60
-40
-20
0
20
40
60
80
100
?
(c) (d) 
Figure 5.2: Demonstration of constant strain-rate for (a) SF-30 and (b) IPC-10, 
and dynamic stress equilibrium for (c) SF-30 and (d) IPC-10 
 
The dynamic stress equilibrium condition was verified by plotting the percentage 
difference between the incident wave and the (reflected + transmitted) wave with respect 
to the incident wave. From Figs. 5.2(c) and (d), it can be seen that the percentage 
difference is about 9% or less for most part of the loading pulse duration and that the 
dynamic equilibrium condition is satisfied quite well given the highly transient nature of 
30?
?
the dynamic event. Among the various shapers that were tried (Fig. 5.3), a stack of four 
0.25 inch diameter copper tapes resulted in good equilibrium and was used in all 
experiments unless stated otherwise. 
 
5.2 Effect of pulse shapers and striker length 
As noted earlier, different types of pulse shapers were used at the striker / incident 
bar interface to attain the best possible dynamic equilibrium in the sample. The stress-
strain curves corresponding to the different pulse shapers are shown in Fig. 5.3. From the 
figure, it can be seen that there is no significant difference in the stress-strain curve, even 
when different pulse shapers were used. The plots also confirm the repeatability of the 
experimental setup. Nevertheless, the pulse shaper that gave the best dynamic 
equilibrium of the test sample was used for all other experiments.  
 
Strain
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
St
r
e
s
s
 (
M
Pa
)
0
20
40
60
80
100
No pulse shaper 
0.04" thick, 1/8" dia Cu 
0.04" thick, 1/4" dia annealed Cu 
4 stacks, 1/4" dia Cu tapes 
3 stacks, 1/4" dia Cu tapes 
 
Figure 5.3: Effect of pulse shapers and repeatability on SF-30 samples  
(strain-rate ~ 1500/s) 
 
31?
?
As mentioned at the beginning of this chapter, strikers of different lengths were 
used to test samples to select the most appropriate and practical striker length. From the 
discussion in Section 4.2 it is clear that the loading pulse duration is directly proportional 
to the striker?s length when the striker and the bars are made of the same material. It 
should also be noted that the longer the loading pulse, the larger the sample deformation 
within the first pulse. This is evident from Fig. 5.4. The line-dot-dot patterned line 
corresponds to an 8 in. long striker. The maximum strain attained by the sample by using 
this striker was ~12%. Longer strikers of lengths 12 in. and 16 in. yielded higher strains 
of ~16% and ~20%, respectively. In Fig. 5.4, the line-line patterned line corresponds to 
the 12 in. striker and the line-dot-line patterned line corresponds to the 16 in. striker. 
Strain
0.00 0.05 0.10 0.15 0.20 0.25
S
t
ress (M
P
a)
0
20
40
60
80
100
203 mm (8") striker
305 mm (12") striker 
406 mm (16") striker 
?
Figure 5.4: Effect of striker length on SF-30 samples (strain-rate ~ 1500/s) 
 
 
32?
?
5.3 SHPB calibration 
Since the primary motivation for this study is to study low impedance materials, 
the SHPB apparatus was also calibrated relative to the published results [35] in the 
literature for a widely studied low impedance material.  The material tested was a 
commercially produced cast acrylic rod stock of 12.7 mm dia.  Cylindrical samples of 
length 6.35 mm were machined from the stock and tested under conditions similar to 
those used for SF and IPC samples.  The resulting true stress-strain diagram for a strain 
rate of approximately 1250 /sec is shown in Fig. 5.5. The results are compared with those 
reported in Ref. [35] for specimens made from cast acrylic sheet stock of identical 
specimen dimensions. 
True strain
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
True st
ress
 (M
P
a
)
0
50
100
150
200
250
300
350
Current work 
Ref. 35 
 
Figure 5.5: True stress-strain response for cast acrylic at a strain rate of ~ 1250 / sec  
 
The overall similarity in the stress-strain response with the reported ones is clearly 
evident. The minor deviations are attributed to differences in the specimen preparation, 
stock material (sheet vs. rod) and strain rates (~ 1500 / sec in Ref.  [35]). 
33?
?
 
 
 
6 EXPERIMENTAL RESULTS 
 
This chapter presents the dynamic compression responses of SF and IPC obtained 
from the experiments. To appreciate the strain rate effects on these two material systems, 
quasi-static [2, 28] stress-strain responses of SF and IPC previously reported by Jhaver 
and Tippur [28] are presented first. Later, the dynamic stress-strain responses obtained 
from the current work are presented along with a comparative analysis of quasi-static and 
dynamic responses. Finally, a summary of the energy absorption characteristics of SF and 
IPC under quasi-static conditions [2, 28], which are also later compared against the 
corresponding results under dynamic conditions, is also presented.  
 
6.1 Quasi-static compressive stress-strain response 
6.1.1 Syntactic foam 
The static compressive true stress vs. true strain responses for four types of 
syntactic foams (SF) (10, 20, 30 and 40% V
f
) are shown in Fig. 6.1. These responses 
show three distinct stages of deformation typical of structural foams. An initial linear 
elastic region is followed by a distinct drop in stress with increasing strain after yielding. 
(This softening response is attributed to the one seen for neat epoxy sample tested under 
identical static loading conditions, and is presented in Appendix A.) With increasing 
strain, the softening response is followed by a region of nearly constant stress. In this 
34?
?
plateau region, microballoons within the SF fail progressively. Further increase in load 
causes densification (signified by an increasing stress) as microballoons collapse 
completely. As seen in Fig. 6.1, the stress-strain responses of syntactic foam samples of 
all four volume fractions follow a similar trend. It is also evident that the compressive 
strength decreases with increasing volume fraction of microballoons in the SF. The 10% 
V
f
 samples (SF-10) have a compressive strength of 63.2 MPa, whereas the SF-40 samples 
have 34.4 MPa, or a 45% decrease. (The elastic modulus values also show a decreasing 
trend with increasing microballoon volume fraction. The SF-10 samples have an elastic 
modulus of ~1600 MPa and the SF-40 samples, ~1260 MPa, which is a 21% decrease.) 
The true strains corresponding to the initial maximum stress are 4.5%, 4.0%, 3.3% and 
3.1% for the SF-10, SF-20, SF-30 and SF-40 samples, respectively.  
True strain
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
True st
ress (M
Pa)
0
20
40
60
80
100
SF-10 
SF-20 
SF-30 
SF-40 
 
Figure 6.1: Effect of microballoon volume fraction on quasi-static 
compression response of syntactic foam (SF) samples [2] 
35?
?
The plateau stress values of SF samples in the order of increasing volume 
fractions are approximately 40 MPa, 33.5 MPa, 26 MPa and 20.6 MPa. This trend of 
decreasing plateau stress with increasing microballoon volume fraction is also consistent 
with those predicted by Kim and Plubrai [36]. Furthermore, as the volume fraction of 
microballoons increase in SF, the densification strain also increases monotonically, from 
~50% true strain for 10% V
f
 to ~75% true strain for 40% V
f
. The deformed samples were 
sectioned and examined in a scanning electron microscope. Figure 6.2 shows 
micrographs of deformed SF-30 samples corresponding to 14% (Fig. 6.2(a)) and 58% 
(Fig. 6.2(b)) strains. 
  
          
Figure 6.2: SEM images of 30% V
f
 syntactic foam at (a) 14% strain and (b) 58% 
strain showing the crushing of microballoons at various stages of static 
deformation. (The loading was in the vertical direction) [2] 
 
The onset of microballoon failure is evident in Fig. 6.2(a). Figure 6.2(b) shows 
the microballoons to be almost completely crushed and flattened, which explains the 
densification or the increase in stress with increase in strain after ~50% strain. 
Furthermore, compression of the matrix and crushing of microballoons is uniform in the 
36?
?
entire field of view.  The direction of load in the micrographs is along the vertical axis, 
but a skew in the direction of the fractured microballoons is seen in Fig. 6.2(a) suggesting 
a tendency for shear failure.  
 
6.1.2 IPC 
The quasi-static results for four different IPC samples with different volume 
fraction of microballoons in SF (10%, 20%, 30% and 40%) are shown in Fig. 6.3. The 
IPC foams respond in a way similar to SF samples (Fig. 6.1). There is an initial linear 
elastic region, followed by yielding and softening signifying the onset of nonlinearity. 
The softening is then followed by regions of plateau stress and densification.  
True strain
0.00.20.40.60.81.01.21.4
True stress 
(M
Pa)
0
20
40
60
80
100
IPC-10 
IPC-20 
IPC-30 
IPC-40 
 
Figure 6.3: Effect of microballoon volume fraction on quasi-static compression 
response of IPC foam samples [2] 
 
37?
?
In the case of IPC, however, the onset of nonlinearity is also affected by 
progressive bending of the aluminum ligaments, which in turn causes crushing of 
microballoons in between the ligaments (Fig. 6.4). The plateau region for IPC foam is 
relatively narrow when compared to the one for the corresponding syntactic foam. 
That is, densification starts at a relatively lower strain in IPC. The elastic modulus for 
10%, 20%, 30% and 40% volume fraction IPC foam samples are 2354 MPa, 2123 
MPa, 1852 MPa and 1702 MPa, respectively. The compressive strengths of the IPC 
samples in increasing order of volume fraction are 81.0 MPa, 63.1 MPa, 51.5 MPa 
and 44.0 MPa. The plateau stresses for the samples show a similar trend as the 
compressive strengths. That is, samples with a lower V
f 
of microballoons have higher 
plateau stresses. The true plateau stress values for the 10%, 20%, 30% and 40% V
f
 
samples are approximately 59 MPa, 50 MPa, 40 MPa and 31 MPa, respectively, 
which is a ~10 MPa increase in plateau stress value for a 10% decrease in V
f
 of 
microballoons.  
 
 
Figure 6.4: SEM image of statically deformed IPC foam at 58% strain. 
(The loading was along the vertical direction) [2] 
38?
?
After yielding, the onset of densification also follows a uniform trend, wherein 
lower V
f
 samples have lower densification strain. The approximate true strain values 
which mark the onset of densification are in the 45%-55% range. The quasi-static 
compression properties are consolidated in Table 6.1. 
 
Designation 
Microballoon 
volume fraction (%)
Density 
(kg/m
3
) 
Compressive 
strength 
(MPa) 
Elastic 
modulus (MPa)
SF-10 10 995  63 ? 1.6 1600 ? 25  
SF-20 20 870  52 ? 2.2 1530 ? 35 
SF-30 30 796  42 ? 1.4 1448 ? 28 
SF-40 40 696  34 ? 1.8 1261 ? 42 
IPC-10 10 1169  81 ? 1.6 2354 ? 26 
IPC-20 20 1044  63 ? 2.3 2123 ? 32 
IPC-30 30 986  52 ? 3.6 1852 ? 27 
IPC-40 40 862  44 ? 1.9 1702 ? 26 
 
Table 6.1: Quasi-static properties of syntactic foam and IPC foam samples 
 
6.2 Dynamic compressive stress-strain response 
SF and IPC samples with 10%, 20%, 30% and 40% microballoon volume 
fractions were tested at a strain rate of ~1500 per second using the split Hopkinson 
pressure bar described in the previous chapter. The maximum strain attained by a sample 
depends on the time period of the incident stress wave. This in turn depends on the length 
39?
?
of the striker used [34]. Several striker lengths, 203 mm (8 in.) ? 406 mm (16 in.), were 
considered. In view of a relatively large expected strain to failure, the longest feasible 
striker length was considered. The incident stress pulse generated by the 406 mm long 
striker (impact velocity ~15 m/s) had a total time period of ~200 ?s with a constant strain 
lasting over 100 ?s as shown in the representative strain histories in Fig. 5.1 for SF and 
IPC samples with 30% volume fraction of microballoons. The maximum engineering 
strain experienced by the sample for this pulse was a little over 25%. So, conservatively, 
all analyses were based on a maximum specimen strain value of 25% and the stress-strain 
data once the stress pulse started dropping off was not considered in the analyses. 
 
6.2.1 Syntactic foam 
The dynamic true stress - true strain responses of SF with four different volume 
fractions of microballoons obtained using a 406 mm striker at a strain rate of ~1500 per 
second are shown in Fig. 6.5. The response of syntactic foam has two distinct regions. An 
initial linear elastic response is followed by a monotonically decreasing stress region with 
increasing strain. As in the quasi-static cases, the compressive strengths of the samples 
decrease with increasing microballoon volume fraction. The compressive strengths of the 
10%, 20%, 30% and 40% volume fraction samples were approximately 104 MPa, 80 
MPa, 62 MPa and 50 MPa, respectively. The relative decrease in the compressive 
strengths for every 10% increase in the microballoon volume fraction is ~20%. Among 
the notable differences, dynamic loading clearly suppresses the distinct softening seen in 
quasi-static cases after the initial peak stress in all volume fractions, as reported in Ref. 
[28]. Instead, a more gradual softening is evident in Fig. 6.5. The tendency for the SF 
40?
?
with lower volume fraction of microballoons (10% and 20%) to soften after attaining the 
maximum stress is somewhat more distinct than for those with higher volume fractions of 
microballoons (30% and 40%). These can be further attributed to the differences in neat 
epoxy response under static and dynamic loading conditions as described in Appendix A. 
After yielding, the stresses for SF with lower V
f
 of microballoons remain consistently 
higher than that for SF with higher V
f
. The difference in stress values after yielding 
between specimens with different volume fractions of microballoons is approximately 
constant at all strains (within the observation window). The elastic responses under static 
and dynamic conditions also show differences.  The measured elastic modulus for the 
four cases of syntactic foams is shown in Table 6.2.  
True Strain
0.00 0.05 0.10 0.15 0.20 0.25
Tr
ue Stres
s
 
(
M
Pa)
0
20
40
60
80
100
120
140
SF-10 
SF-20 
SF-30 
SF-40 
 
Figure 6.5: Effect of microballoon volume fraction on dynamic stress-
strain response of SF samples (Strain rate ~1500/sec) 
 
41?
?
Designation 
Microballoon 
volume fraction 
(%) 
Density 
(kg/m
3
) 
Compressive 
strength 
(MPa) 
Elastic 
modulus 
(MPa)  
SF-10 10 995  104 ? 4 3460 
SF-20 20 870  80 ? 3 3100 
SF-30 30 796  62 ? 5 2700 
SF-40 40 696  50 ? 3 2400 
IPC-10 10 1169  120 ? 4 - 
IPC-20 20 1044  100 ? 5 - 
IPC-30 30 986  80 ? 6 - 
IPC-40 40 862  60 ? 3 - 
 
Table 6.2: Dynamic properties of syntactic foam and IPC foam samples 
 
It should be noted that the dynamic elastic modulus values shown in Table 6.2 
were obtained using ultrasonic pulse-echo measurements [37]. (This method was 
preferred over direct measurement from the stress-strain response due to weak stress 
equilibrium in the specimen during the initial stages (up to about 30 ?s) of stress wave 
propagation in the specimen. The elastic moduli of the IPC are not reported here as the 
pulse-echo transducer could not be used effectively to determine wave speeds in IPC due 
to wave dispersion problems). The results indicate a higher elastic modulus under 
dynamic conditions when compared to the respective quasi-static [28] counterparts. A 
backlit photograph of a deformed SF sample with 30% V
f
 of microballoons is shown in 
42?
?
Fig. 6.6(a).  The sample in this case was loaded along the vertical direction. A network of 
shear bands crisscrosses the entire sample and the bands are oriented at approximately 
?45
o
 to the loading direction.   
        
Figure 6.6: (a) Side view of a dynamically deformed syntactic foam sample revealing 
a network of shear bands, (b) SEM image of cross section of dynamically deformed 
syntactic foam; Careful observation reveals crisscrossing shear failure planes.  
(The loading was along the vertical direction)  
 
A micrograph of a 30% V
f
 specimen is shown in Fig. 6.6(b). Again, the 
direction of loading was along the vertical direction of the image. The cracks that 
appear in the image are skewed at an angle of approximately 45
o
 to the direction of 
application of the load suggesting failure due to shear localization. In regions away 
from the crack, the microballoon footprints are essentially circular suggesting very little 
deformation away from the shear bands. 
 
6.2.2 IPC 
The IPC samples of four microballoon volume fractions viz. 10% (IPC-10), 20% 
(IPC-20), 30% (IPC-30) and 40% (IPC-40) were tested and the results are shown in Fig. 
43?
?
6.7. The dynamic compression response of IPC foams followed trends similar to that of 
the corresponding syntactic foams. The response shows a linear region in the beginning, 
followed by a modest nonlinear response until a maximum stress is reached.  
Subsequently a monotonic reduction of stress with increasing strain is seen in the 
observation window up to 22% true strain. As in the case of SF samples, the yield 
strengths of IPC samples decrease with increasing volume fraction of microballoons. 
True strain
0.00 0.05 0.10 0.15 0.20 0.25
True st
ress (M
P
a
)
0
20
40
60
80
100
120
140
IPC-10 
IPC-20 
IPC-30 
IPC-40 
 
Figure 6.7: Effect of microballoon volume fraction on dynamic stress-
strain response of IPC foam samples (Strain rate ~1500/sec) 
 
In the order of increasing microballoon V
f
, the maximum stress values attained 
are approximately 120 MPa, 100 MPa, 80 MPa and 60 MPa. The percentage decrease in 
the compressive strength for the IPC-20 with respect to that of IPC-10 is 17%, and that of 
IPC-30 with respect to that of IPC-20 is 20%. The IPC-40 has a 25% decrease in 
44?
?
compressive strength with respect to the IPC-30. Figure 6.8 shows a photograph and a 
micrograph of a deformed 40% V
f
 IPC.  As in the SF samples, the photograph of the 
deformed sample in Fig. 6.8(a) clearly shows a well-defined network of shear bands 
throughout the specimen (the shear bands are slightly more pronounced and father apart 
relative to the SF counterparts due to the presence of the aluminum network.). The SEM 
image in Fig. 6.8(b) shows underlying microscopic failure mechanisms in the IPC. Two 
types of failures can be identified from this image. One is the formation of cracks in the 
syntactic foam (similar to the one seen for SF in Fig. 6.6(b)). The crack in Fig. 6.8(b) is 
again inclined at ?45? to the direction of the load along the vertical direction of the 
image, which suggests that the failure occurred due to shear localization. However, shear 
bands here are interrupted by the metallic ligaments of the IPC.  The other type of failure 
in IPC, which is absent in pure SF samples, is the debonding of the interfaces between the 
syntactic foam and the aluminum ligaments.   
 
      
Figure 6.8: (a) Side view of a dynamically deformed IPC foam sample revealing 
multiple shear bands and (b) SEM image of the cross section of dynamically 
deformed IPC foam. (The loading was along the vertical direction) 
 
45?
?
6.3 Energy absorption 
An aspect of foams that is attractive for structural applications is energy 
absorption. A comparison of energy absorbed per unit volume and per unit mass by the 
SF and the corresponding IPC samples under dynamic loading conditions was made. The 
energy absorbed per unit volume (U) was evaluated by calculating the area under the true 
stress ? true strain curve as, 
7
L ? ? : ? ; @?

4
                                 (6.1) 
where ()? ? is the stress as a function of strain. The energy absorbed was evaluated up to a 
strain value of 22% as U
0.22
. The energy absorption per unit mass was also evaluated by 
dividing the energy absorbed per unit volume by the density of the sample. Again, to 
appreciate the effect of strain rate on energy absorption, the quasi-static results on energy 
absorption [2, 28], are presented before the dynamic counterparts. 
 
6.3.1 Quasi-static 
The absorbed energy (U
0.22
)
 
evaluated from true stress-true strain graphs [2, 28] is 
plotted as histograms for all the cases in Fig. 6.9. As was the case for the compressive 
strength and the elastic modulus, the absorbed energies also show a decreasing trend with 
increasing volume fraction of microballoons. The energy absorbed per unit volume (Fig. 
6.9 (a)) by the SF samples of 10%, 20%, 30% and 40% microballoon V
f
 are 10.4 MJ/m
3
, 
8.5 MJ/m
3
, 6.8 MJ/m
3
 and 5.4 MJ/m
3
, respectively. The relative decrease in U
0.22 
for 
every 10% increase in microballoon V
f
 is 18-20%. The energy absorbed by the IPC 
samples is higher than that by the corresponding syntactic foam samples by 34%-37%. 
The actual values are approximately 14.0 MJ/m
3
, 11.5 MJ/m
3
, 9.2 MJ/m
3
 and 7.4 MJ/m
3
 
46?
?
for 10%, 20%, 30% and 40% cases, respectively. The percentage decreases in U
0.22 
for 
the IPC-20 / IPC-10 pair, IPC-30 / IPC-20 pair and the IPC-40 / IPC-30 pair are 
approximately 20% in each case.  
 
Microballoon volume fraction
10% 20% 30% 40%
E
ner
gy
 
abso
r
ptio
n p
er 
unit v
olume
 (M
J/
m
3
)
0
5
10
15
20
25
30
SF 
IPC 
(a) 
?
Microballoon volume fraction
10% 20% 30% 40%
E
ner
gy
 ab
sor
p
ti
on pe
r u
n
it mas
s (
kJ/kg)
0
5
10
15
20
25
SF 
IPC 
(b) 
 
Figure 6.9: Energy absorbed (a) per unit volume and (b) per unit mass by SF and 
IPC foam samples under quasi-static loading up to 22% true strain 
 
47?
?
The energy absorption per unit mass are plotted in Fig. 6.9(b) and they also 
show a similar trend. The IPC samples have absorbed 12.0 kJ/kg, 11.0 kJ/kg, 9.4 
kJ/kg and 8.6 kJ/kg which are higher when compared to the syntactic foams of 
corresponding volume fraction of microballoons, respectively. The percentage 
increase in the energy absorption per unit mass in the IPC foams with respect to the 
syntactic foams is approximately 10%-14%. 
 
 
6.3.2 Dynamic 
The data on energy absorption, U
0.22
,
 
under dynamic loading conditions is 
shown in Fig. 6.10. The absorbed energies of SF and IPC foam samples under 
dynamic loading conditions are clearly higher than those observed under quasi-static 
loading conditions (Fig. 6.9). Energy absorbed per unit volume by SF-10, SF-20, SF-
30 and SF-40 samples are 21.9 MJ/ m
3
, 19.6 MJ/m
3
, 16.4 MJ/m
3
 and 12.1 MJ/m
3
, 
respectively and that by IPC-10, IPC-20, IPC-30 and IPC-40 samples are 24.5 MJ/m
3
, 
22.0 MJ/m
3
, 18.3 MJ/m
3 
and 12.7 MJ/m
3
,
 
respectively. The percentage reduction in 
U
0.22
 per unit volume for the SF-20 / SF-10 pair, SF-30 / SF-20 pair and the SF-40 / 
SF-30 pair are approximately 11%, 16% and 26% respectively. The percentage 
reduction in U
0.22
 per unit volume for the IPC-20 / IPC-10, IPC-30 / IPC-20 pair and 
the IPC-40 / IPC-30 pair are approximately 10%, 17% and 30%, respectively. The 
trend of the percentage reduction in the energy absorption per unit volume suggests 
that the rate of reduction in U
0.22
 would be greater than the rate of increase in the 
microballoon volume fraction for dynamic loading.  
 
48?
?
 
Microballoon volume fraction
10% 20% 30% 40%
E
n
e
r
g
y
 abso
r
ption
 pe
r unit v
o
l
u
m
e
 (M
J/m
3
)
0
5
10
15
20
25
30
SF 
IPC 
?
Microballoon volume fraction
10% 20% 30% 40%
E
n
ergy
 
absor
p
tion per unit 
mass (kJ/kg)
0
5
10
15
20
25
SF 
IPC 
(a) 
(b) 
 
Figure 6.10: Energy absorbed (a) per unit volume and (b) per unit mass by SF and 
IPC foam under dynamic loading up to 22% true strain 
 
However, in the case of energy absorbed per unit mass, the syntactic foam 
appears to perform better than the corresponding IPC foam under dynamic loading 
conditions. The energies absorbed per unit mass by the SF-10, SF-20, SF-30 and SF-
49?
?
40 samples were 22.0 kJ/kg, 22.5 kJ/kg, 20.6 kJ/kg and 17.4 kJ/kg, respectively and 
that absorbed by IPC-10, IPC-20, IPC-30 and IPC-40 samples were 20.9 kJ/kg, 21.0 
kJ/kg, 18.5 kJ/kg and 14.8 kJ/kg, respectively. Again, these are nearly twice the 
respective quasi-static values. When compared with the SF samples, the IPC samples 
had lower energy absorption per unit mass by 5%, 6.7%, 10.2% and 14.5%, 
respectively. (It should be noted that the differences between 10% and 20% SF and 
between the SF and IPC of the two volume fractions are relatively small with 
overlapping error bars. Hence, no claim regarding the trend between these two 
volume fractions is made.) 
Accordingly, if energy absorption at a slight increase (~10%) in weight is 
acceptable for an application, IPC still outperforms SF even under dynamic 
conditions. Another interesting observation is that the increasing trend of the energy 
absorption per unit mass with decreasing microballoon volume fraction appears to be 
reversed at a microballoon volume fraction of approximately 10%. Therefore, the 10% 
microballoon volume fraction could be the threshold level in terms of absorbed energy 
per unit mass. 
 
6.3.3 Dynamic energy absorption history 
Plots of cumulative absorbed energy at various strain values were also plotted to 
evaluate the history of energy absorption during dynamic compression. The variations in 
energy absorption per unit volume corresponding to the dynamically loaded SF and IPC 
samples are shown in Fig. 6.11. The cumulative energies absorbed per unit volume for 
both the SF and IPC samples follow a nearly linear trend beyond yielding. Also, as seen 
50?
?
from the graphs, the energies absorbed by lower V
f
 samples are consistently higher than 
that absorbed by higher V
f
  samples throughout the loading history. 
 
Engineering strain
0.00 0.05 0.10 0.15 0.20 0.25
S
t
rai
n
 energy
 (MJ/
m
3
)
0
5
10
15
20
25
SF-10 
SF-20 
SF-30 
SF-40 
Engineering strain
0.00 0.05 0.10 0.15 0.20 0.25
S
t
rai
n
 en
ergy
 (M
J/m
3
)
0
5
10
15
20
25
IPC-10 
IPC-20 
IPC-30 
IPC-40 
(a) (b) 
?
Figure 6.11: Strain energy absorption history of (a) SF and (b) IPC 
with increasing strain in SHPB tests 
 
6.4 Failure progression 
Another interesting outcome under dynamic loading conditions is strain recovery 
in SF and IPC foam samples.  The final lengths of the deformed samples were measured 
after experiments on both SF and IPC. Interestingly, for SF samples, the final measured 
lengths were more than that predicted by the SHPB equations after imposing a 25% 
engineering strain. This suggests that the SF samples had partially recovered (sprung-
back) after the loading pulse had ended. Although the source/s of spring-back in SF is/are 
unclear at this point and would be interesting to study in the future, this phenomenon was 
negligible in the case of IPC samples. Whatever caused the spring-back to occur in the 
SF, is probably overcome in the IPC by the aluminum ligaments. Once aluminum 
ligaments undergo plastic deformation, they prevent the SF from springing-back. This 
51?
?
aspect of SF relative to IPC needs to be considered in an application if spring-back is a 
design factor.  
This issue was further examined optically by tracking the progression of 
deformation under static and dynamic loading conditions for both SF and IPC foams. To 
understand the failure of SF and IPC foams including the above noted spring-back 
behavior in SF, real-time photography of progression of deformation during loading was 
carried out. The schematic of the experimental setup is shown in Fig. 6.12. This was also 
particularly important under dynamic loading conditions to assess progressive (failure 
originating at the impact end and propagating along the length of the bar) vs. steady-state 
deformation as stress waves propagate and deform the sample. 
Grating lines on flattened 
sample surface 
Incident bar Transmitter bar 
Lamp Lamp 
Hi speed 
camera
 
Figure 6.12: Schematic of the experimental setup used for high-speed photography to 
study real-time failure progression in SF and IPC samples in the SHPB set up 
52?
?
 
Two narrow (3 mm wide) diametrically opposite flat surfaces were machined on 
the foam samples and were printed with a line grating (pitch = 0.4 mm). The grating lines 
were first drafted using Solid Edge
TM
 software and then printed on a decal paper. The 
printed grating lines on the decal paper were then transferred to the specimen surface. A 
few selected photographs of the specimen with gratings recorded using digital high-speed 
photography (186,000 frames/sec) are shown in Fig. 6.13.   
 
 
Figure 6.13: Stages of dynamic deformation of SF-30.  Images (a)-(c) correspond to 
he first loading pulse and (d)-(f) correspond to later time instants showing presence ot f
?
 
sp  ring-back.  The pitch of the coarse grating lines is 0.4 mm. The sample was loaded
in the horizontal direction; t = 0 corresponds to the time when deformation begins. 
?
53?
?
The image in Fig. 6.13(a) corresponds to the undeformed sample just before the 
onset of the loading pulse in SHPB. The subsequent images show the deformed samples 
at different time instants (t). It is evident from the deformed length (l) that there was 
continuous decrease in the sample length up to 135 ?s.  This time instant is also when the 
loading pulse starts to decay due to the finite striker length. That is, t = 135 ?s marks the 
end of the constant strain rate region of the loading pulse. The approximate strain at 135 
?s calculated directly from the image is ~23% which is close to the obtained engineering 
strain of 22% using SHPB equations. Around 195 ?s the sample starts to spring-back. At 
the 850
th
 ?s the sample has regained about 0.7 mm in length which is about 9.5% of the 
deformed length at the end of the loading pulse at 135 ?s. Figures 6.13(d), (e) and (f) 
correspond to time instants after the end of the first incident stress wave, and before the 
second incident stress wave.   
From the images shown in Fig. 6.13, due to the coarseness of the gratings used, 
shear bands seen on the specimen surface in Fig. 6.6(a) are not readily evident although a 
hint of these bands can be visualized in Fig. 6.13(c) and beyond. Much higher gratings 
densities and magnification are needed to achieve the necessary spatial and temporal 
resolutions in this regard. To contrast the response of SF with that of IPC, a similar 
optical observation using high-speed photography was also carried out for an IPC sample.  
A selection of corresponding real-time images is shown in Fig. 6.14. Noticeably, the 
quality of the images in these experiments is not as good as the ones in Fig. 6.13 due to 
the presence of shiny aluminum ligaments in the IPC which perturb the overall light field 
making it difficult to image the specimen.  Nevertheless, qualitative observations 
regarding the nature of deformation progression can be made. The spring-back 
54?
?
phenomenon seen in SF is nearly absent in IPC samples under stress wave loading as 
evident from Figs. 6.14 (d), (e) and (f).  
 
Figure 6.14: Stages of dynamic deformation of IPC-30.  Images (a)-(c) 
correspond to the first loading pulse and (d)-(f) correspond to later time 
instants showing absence of spring-back.  The pitch of the coarse grating lines 
mm. The sample was loaded in the horizontal direction; t=0 corresp
to the time when deformation be
is 0.4 onds 
gins.
 
55?
?
 
 
 
7 FINITE ELEMENT MODELING AND ANALYSIS OF IPC FOAM 
 
In view of the geometric complexity of modeling the IPC foam computationally, a 
unit-cell analysis was undertaken. A single cell of the IPC foam was used to predict the 
dynamic compressive response. The geometry was modeled using solid modeling 
software Solid Edge and imported into the finite element analysis software 
ABAQUS/Explicit. In ABAQUS, the dynamic/explicit algorithm was chosen to solve the 
temporal deformation process numerically. The following details the methodology 
adopted for geometric modeling and finite element analysis. 
 
7.1 Geometric modeling 
In 2005, Gong et. al., [38] studied the compressive response of unfilled open-cell 
foams. In their work, they reported that the number of sides of a cell range from 9 to 17, 
the average being 13.7 (see, insert in Fig. 3.1(b)). Accordingly, they chose a 14 sided 
polyhedron (tetrakaidecahedron) [39] called a Kelvin cell to represent a single cell of the 
open-cell foam in their analysis. In this research, the same Kelvin cell was adopted as an 
idealized representation of a unit-cell of the aluminum open-cell foam. The polyhedron 
consists of 6 squares and 8 hexagons, with all the edges having the same length l, as 
shown in Fig. 7.1. The height, h of the Kelvin cell is nullnull2?2null [38]. The length of the 
ligaments in the actual aluminum foam was ~1.5 mm and the same dimension was 
56?
?
chosen for the ligaments in the idealized model. The resulting wire mesh of the Kelvin 
cell model had a height, h ~ 4.24 mm. Further, the Kelvin cell was modeled as a solid 
with triangular cross section ligaments based on the actual cross sectional shape of the 
aluminum ligaments, increasing the total height of the model to ~ 4.69 mm. The cross 
section of the ligaments was chosen to be an equilateral triangle with ~ 0.78 mm sides for 
reasons to be discussed shortly. In reality, the cross section along the actual ligaments 
was not a constant, but considering the complexity of geometric modeling, the ligaments 
were modeled as having a constant cross sectional dimension.  
 
 
4.69 mm
 
1.5 mm 
3D array of Kelvin cells
Unit cell IPC Ligament 
 
 
 
Side view of unit cell IPC
Figure 7.1: Schematics representing the building blocks of an idealized 
Kelvin cell based IPC 
57?
?
Though there are some non conformances of the model geometry to the actual 
geometry, the aluminum volume fraction in the model matches the volume fraction range 
(8 -10% as claimed by the manufacturer of the aluminum foam) of the aluminum in the 
actual test sample. Thus, the resulting IPC model had an overall dimension of ~ 4.69 mm 
x 4.69 mm x 4.69 mm. The region not occupied by aluminum was ?filled? with the SF 
material. In Fig. 7.1, the periodic arrangement of the Kelvin cells in 3D space resulted in 
a regular, well organized idealized open-cell foam whereas in actual aluminum foam, the 
cells are not periodic. Nevertheless, the randomness of the actual aluminum foam cells 
were assumed to be uniform on the scale of the test sample dimensions. 
 
7.2 Computational model 
7.2.1 Material properties 
The material definitions for the SF region were assigned the compressive material 
properties of the respective microballoon volume fraction SF obtained from the SHPB 
experiments (Chapter 6). The elastic moduli were obtained by computing the slopes of 
the stress-strain curves obtained from the SHPB experiments. The Poisson?s ratios and 
the density of the SF are shown in Table 7.1. The aluminum region was assigned a 
bilinear stress strain property corresponding to Al 6101 based on Alcoa Inc. datasheet
1
. 
The elastic regime for aluminum was assigned a Young?s modulus of 69 GPa and a 
Poisson?s ratio of 0.35. The plastic regime was assigned an initial yield stress of 172 MPa 
at a plastic strain of 0, and a stress of 200 MPa at a plastic strain of 0.148, for Al 6101 as 
shown in Table 7.2. 
????????????????????????????????????????????????????????????
1
?The metallic phase of the IPC was assumed to be strain rate independent. 
58?
?
Material Elastic Modulus 
(MPa) 
Poisson ratio Density 
(kg/m
3
) 
Al 6101 69000  0.35 2700 
SF-10 5122  0.34 995 
SF-20 4529  0.34 870 
SF-30 4459  0.34 796 
SF-40 3835  0.34 696 
 
Table 7.1: Elastic properties of SF and aluminum used in the finite element model 
 
Plastic strain Stress (MPa) 
0 172  
0.148 200  
 
Table 7.2: Inelastic properties of Al 6101 used in the finite element model 
 
7.2.2 Yielding and hardening 
The von-Mises yield criterion was used to define material yielding. Further, 
yielding was assumed to be isotropic. This criterion uses the equivalent uniaxial plastic 
strain and the corresponding uniaxial stress input to define the yield surface [40, 41]. The 
material property input for the plastic regime was supplied in the form of stress as a 
tabular function [40] of plastic strain. In ABAQUS, the stress at any given state within 
the maximum plastic strain defined will be interpolated [40] based on the supplied tabular 
59?
?
data. Beyond the maximum plastic strain defined, the stress remains constant and equal to 
the stress value in the previous step in the data [40]. The hardening and softening 
behaviors are also assumed to be isotropic. This means that the yield surface is the same 
[40] in all directions based on increasing or decreasing stress as plastic straining occurs. 
In the current material system, aluminum could be undergoing strain hardening, while the 
SF could be softening. The yield function [41] for isotropic hardening is expressed as, 
nullnullnullnull null null
null
nullnull
nullnull
,nullnull
nullnullnull
nullnull
nullnull
nullnull
nullnull
nullnullnullnullnullnull
 
where ?
0
 is the equivalent uniaxial stress, ?
pl
 is the equivalent plastic strain and ? is  
temperature. Reference [41] details the algorithm that ABAQUS uses to solve the 
nonlinear material model based on the von-Mises yield criterion. Some of the details are 
as follows: 
The total strain can be partitioned into elastic and plastic strains as, 
            (7.1) 
where ? is the total strain and ?
el
 and ?
pl
 are the elastic and plastic strain components, 
respectively. For a given stress state, the strain states could be different depending on the 
loading path. Due to this, increments of plastic strain (nullnull ) need to be computed based 
on the loading path before obtaining the total strain for a particular state of stress [42]. 
The Prandtl-Reuss plasticity relations state that the increments of plastic strain depend 
only on the instantaneous deviatoric stresses [42]. Hence, the deviatoric stress matrix, S is 
obtained using 
                              (7.2) 
where ? is the stress matrix, I is the identity matrix and p is the volumetric stress given 
by 
60?
?
nullnullnull
null
null
nullnullnullnullnullnullnullnull.   
nullnullnullnullnull
nullnullnull
null2nullnull
nullnull
nullnull
nullnull
nullnull
nullnullnullnull
nullnull
null ;                 null null
null
null
                                       (7.3)  
Further, p and S are related to strains under elastic conditions as  
  and  null  ,         (7.4) 
where, K and G denote the bulk modulus and shear modulus, respectively, ?
vol
 is the 
elastic volumetric strain, and null  is the deviatoric elastic strain. It is to be noted that bold 
symbols ?, S and I represent (3x3) matrices. Now, writing the incremental plastic strain 
as a product of its scalar value and its direction,  
 ,   null null null
null
null
null:null
null
null
                    (7.5) 
where, e
pl
 is the deviatoric plastic strain, null  is the scalar deviatoric plastic strain, n is the 
direction vector and q is the estimated equivalent uniaxial stress. Equation (7.5) is called 
the integrated flow rule.  
null
nullnull
pl
nullnull nullnull
nullnull
step. It is understood that nullnull corresponding to a particular nullnull
nullnull
 is obtained from the 
material definition input as stated at the beginning of this section. Therefore, the 
condition to be verified is, 
nullnullnullnullnullnullnull
nullnull
null.                         (7.6) 
To check this condition, the expression for q would first be obtained as follows: 
The above equations are then solved for the state of the material at the end of 
every increment. The state of the material refers to the estimated equivalent uniaxial 
stress (q) as a function of deviatoric equivalent plastic strain (e ). The state of the 
material will be verified by checking if the estimated stress (q) equals the scalar 
equivalent stress ( ) corresponding to the scalar deviatoric plastic strain ( ) of that load 
61?
?
The change in the deviatoric plastic strain (?e
pl
) between the current and the previous 
increment is computed as 
?null
nullnull
null?nullnull
nullnull
null
nullnull2nullnullnull
nullnull
|
null
null?nullnull?nullnull
nullnull
nullnull
?null
null1null
nullnull
null
 .           (7.7)  
Using Eqs. (7.1) and (7.7), Eq. (7.2) is modified as 
          (7.8) 
where,  is the change in the total strain. In the above equation, the term in the 
parentheses is the estimated elastic strain value at the end of the current iteration. The 
subscript t represents the previous time step. Now, substituting for n in the above 
equation from Eq. (7.5), 
?nullnull
nullnull
nullnullnull2nullnullnull ;           nullnull null nullnull
nullnull
|
null
null?nullnull
nullnull3null?nullnull
nullnull
null3nullnull? ;                  null? null null
null
null
.                 (7.9) 
The inner product of the above equation with itself gives 
nullnull:nullnull .       (7.10) 
As q must now satisfy Eq. (7.6), the above equation is modified as 
3nullnullnull? null?nullnull
nullnull
nullnullnullnullnull0
The above equation is then solved using Newton?s method to get, 
null
nullnull
null
nullnullnullnull?null?nullnull
nullnull
nullnullnullnull
nullnullnullnull
.                     (7.11) 
 ;                    null null
nullnullnull
nullnullnull
nullnull
 ,                   (7.12) 
where, null
nullnull
 is the error in the plastic strain increment in case Eq. (7.6) is not satisfied. H is 
the strain hardening function, which is nothing but the relationship between the 
respective stresses and plastic strains given as the material definition input (in this work, 
the input is given in a tabular format). Now, the new ?nullnull  or the new plastic strain 
increment value for the next iteration for the same load step is obtained as 
?nullnull null?nullnull nullnull .           (7.13)  
nullnull
nullnull nullnull nullnull
62?
?
The iterati
zero. A
, the deviatoric and volumetric stresses 
corresp
.2.3 Explicit integration scheme in ABAQUS 
motion
, the explicit central difference operator 
satisfie
nullnull
nullnull
nullnullnull
nullnull
null
null                                                                                           (7.14) 
where nullnull
nullnullnull
 represents th
em
?null
on is then carried out until ?nullnull
nullnull
 converges, or until null
nullnull
 approaches 
t this stage, the solution for that increment is fully defined. The converged change 
in scalar equivalent plastic strain (?nullnull
nullnull
) is used in Eq. (7.9) to obtain the deviatoric stress 
(S) that corresponds to the equilibrium state for that particular increment. The volumetric 
stress (p) is also now computed using Eq. (7.3). 
In essence, for each increment of load
onding to the equilibrium states are computed based on the converged value of 
change in the scalar equivalent plastic strain at every integration point in the model.  
 
7
The explicit central difference integration rule is used to solve the equations of 
 governing a dynamic event in ABAQUS. The implementation of this rule in 
ABAQUS [40] is summarized in the following: 
 At the beginning of every increment, t
s the dynamic equilibrium equations; the result of which is the acceleration at that 
time step, 
nullnull
nullnullnull
null nullnullnull
e acceleration with null as the degree of freedom, the subscript i is 
the incr ent in an explicit dynamic step, null is the mass matrix, null is the applied load 
vector and null is the internal force vector. The velocity for the time nullnull
null
 is computed 
using the acceleration at time t. Then, using this computed velocity and the displacement 
at time t, the displacement at time nullnull?null is computed as, [40]: 
nullnull
nullnullnull
null
null
null
nullnullnull
nullnullnull
null
null
null
null
?null
nullnullnullnullnull
null?null
nullnullnull
null
63?
?
nullnull
nullnullnull
         (7.15) 
and 
null
nullnullnullnullnull
nullnull
nullnullnull
null?null
nullnullnullnullnull
nullnull
nullnullnull
null
null
null
 .         (7.16)  
At time t = 0, the initial v
fied otherwise, and the mid-increments nullnull
nullnull
null
null
alues of velocity and acceleration are set to zero, unless 
speci
null
 and nullnull
nullnull
null
null
null
 are given as 
nullnull
nullnull
null
null
null
nullnullnull
nullnullnull
null
?null
nullnullnull
null
nullnull
nullnullnull
 
and 
nullnull
null
         (7.17) 
 nullnull null
null
null
nullnullnull
nullnullnull
null
?null
nullnullnull
nullnull
nullnullnull
,           (7.18) 
respectively. 
The displacements will then be used to obtain the state of the material as 
described above. Unlike the implicit scheme, iterations are not used here to achieve 
convergence. Instead, the stability of the solution is governed by the appropriate time step 
ch is dictated by the time required by the stress wave to travel the smallest 
elemen
US and discretized using four 
noded linear tetrahedron (C3D4) [40] elements. To check for the appropriateness of the 
odel meshed with elements having 0.2 mm and 0.1 mm edge length 
were simulated and the stress vs. strain results corresponding to the two mesh sizes were 
size (?t), whi
t and the maximum wave speed in the model defined [40]. The time step size 
should be chosen to be sufficiently small, typically a third or lower than the time taken by 
the stress wave to travel the smallest element in the mesh. 
 
7.3 Discretization 
The unit cell model of the IPC was used to predict the overall response of the test 
sample. The geometric model was imported into ABAQ
mesh refinement, a m
64?
?
compared. The comparison (Fig. 8.2) shows that the results nearly overlap on each other. 
So, for all further analyses, the model was meshed with elements having an approximate 
element edge of length 0.2 mm.  
True strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
True st
ress
 (M
P
)
a
40
60
80
100
IPC-30 SHPB 
0
20
IPC-30 FEA 0.2 
IPC-30 FEA 0.1
 
Figure 7.2: Check for appropriateness of discretization 
 
In order that the model better represents the sample, the unit cell is imagined to be 
located deep within a large material volume. The Solid Edge model of the unit cell has 
exposed faces, and does not represent a cell within the material volume. This is because, 
in a dynamically ire length of the 
sample before reflecting at the end of the sample, whereas in the model the stress waves 
would 
 loaded sample, the stress waves travel through the ent
get reflected at the boundaries of the cell itself. To overcome this shortcoming, a 
skin / layer of SF and aluminum material was modeled adjacent to the corresponding 
material at the boundary of the cell. This skin was meshed using the so-called infinite 
elements [43].  
 
65?
?
 
 
 
Syntactic foam 
Infinite element skin 
Aluminum 
(a) 
 
 
 
 
 
 
(c) 
 
Figure 7.3: Illustration showing (a) the infinite element layer in the FE model and stress 
wave behavior in the (b) absence and (c) prese ment layer 
Reflected wave 
Incident wave 
Free surface 
(solid line)
(broken line)
(b) 
 
 
 
T
Incident wave 
Infinite elem
layer 
ent 
ransmitted wave 
nce of infinite ele
 
 
66?
?
7.3.1 Infinite elements ? Background 
 
small c ly the 
most important work on infinite elements was carried out by Bettess [43] in 1977. He 
suggested Lagrange polynomial type shape functions that include an exponential decay 
ts should satisfy two main requirements, one 
that the
null nullnull
null
null
nullnull null
nullnull
The infinite elements are used in situations where the region of analysis is very
ompared to the region surrounding it [40]. One of the earliest and probab
term. Bettess states that the infinite elemen
 shape functions should be realistic and the other that it should lead to integrations 
over the finite element domain. Accordingly, Bettess proposed shape functions for one 
dimensional domain presented as, 
null
null null?
null
null
nullnull
null
null
nullnull
null
null
nullnullnull
 
i
varying from 1 to n-1 and j ? i. Among all the points in the domain, the n  point is 
infinitely distant and all other points have finite x-coordinates. As reported by Bettess, in 
the above expression, L is an arbitrary distance and serves as the measure of severity of 
the exponential decay.  
il regions and plane frames. Two types of infinite elements were 
develop
nullnullnull
nullnullnull
where, n is the number of points in the domain, x is the coordinate and  N  is defined for i 
th
Since its inception, infinite elements have been developed and used by several 
researchers. In 1987, Haggblad and Nordgren [44] used infinite elements to model the far 
field regions to study a nonlinear soil-structure interaction effect. Viladkar et al. [45] also 
used infinite elements coupled with finite elements to idealize and study interactions 
between the far field so
ed and used by Park et al. [46] to analyze the hydrodynamic forces on offshore 
structures. One type was used to model radiation in the fluid at infinity and the other type 
67?
?
used to model the fictitious bottom boundary of the deep water. Later in 1993, Zhao and 
Valliappan [47] developed a 3D dynamic infinite element that satisfies four different 
requirements all of which are very useful in representing practical problems much more 
efficiently. Two of those requirements used in the FE model of the current work are, 
interfacial displacement compatibility between finite and infinite elements and 
displacement continuity at the common boundary between infinite elements possessing 
different material properties.  
In ABAQUS, as infinite elements are associated with the far field regions, their 
behavior is assumed to be linear and the material response isotropic [40]. The particular 
infinite element used in this analysis is a 3D, 8 node, one-way infinite brick (CIN3D8) 
[40]. The infinite elements (CIN3D8) transmit any stress wave incident on them to a 
hypothetically infinite distance in one direction and thus prevent any reflection of stress 
waves 
side. On the side opposite to that of the support, and on the surface of the unit cell which 
at the boundary of the unit cell. This phenomenon made possible by the 
implementation of a layer of infinite elements in the finite element model enforces the 
condition that the unit cell is located within a larger volume of IPC material. Those 
infinite elements adjacent to aluminum were assigned the properties of aluminum, and 
those that are adjacent to SF were assigned the SF properties as seen in Fig. 7.4. Even 
though all the sides of the IPC have infinite element layers, for clarity purposes, the 
infinite element layer on only one side is shown in Fig. 7.4. 
 
7.4 Boundary conditions and loading 
The unit cell is supported on the free surface of the infinite element layer on one 
68?
?
marks the boundary between the unit cell IPC and the infinite element layer, a velocity 
pulse of magnitude 10250 mm/s for a time period of 250 microseconds is applied. This 
velocity corresponds to the approximate particle velocity in the incident bar at the 
cle velocity (V
PI
) in the incident bar is given by, 
null
nullnull
nullnull
 
 
 
 
Figure 7.4: Load and boundary conditions 
 
 
incident bar / sample interface. The parti
nullnullnull
nullnull
nullnullnull
null 5.123null 10
null
nullnull null? null0.002 null 10250nullnull null? , 
where C
bar
 is the incident bar wave speed and null
nullnullnull
 (~ 0.002) is the maximum plateau 
strain in the incident bar caused by the stress wave. The time period of 250 microseconds 
was chosen as it is the approximate duration for which the actual test samples were 
loaded in the SHPB experiments.  
 
 
   
x 
y 
z 
Infinite element layer with 
aluminum properties 
 lay
rtie
Infinite element er with 
SF prope s 
Uniform velocity pulse 
v(t) = 10250 mm/s 
69?
?
7.5 Other analysis parameters 
The analysis was set up to simulate the dynamic problem for 250 microseconds, 
same as the loading pulse duration. The load steps were automatically selected by the 
ABAQUS software based on the fastest wave speed and the smallest element size in the 
model [40]. The load step period is chosen to be less than the time taken by the stress 
wave to traverse the smallest element in the model. As a verification procedure, the 
software checks if the formation speed of the material is less than one-third the wave
speed in the material at each load step. As rge deformations were expected during the 
deformation, geometri ded in the analysis 
procedure. 
 
analysis 
7.6.1 
 de  
la
cal nonlinearity effects were also inclu
7.6 Results from finite element 
Stress-strain response 
The stress and strain histories of all the elements (aluminum and SF) on the 
surface of the cell on the side directly opposite to the loaded surface were obtained. From 
the data obtained, the average stress and strain histories were obtained, which were used 
to plot the dynamic stress-strain curves. The same procedure was followed for all the 
numerical IPC models corresponding to 10%, 20%, 30% and 40% microballoon volume 
fractions in SF, and the results are shown in Fig. 7.5. The trends are similar to those 
observed from the experimental results discussed in Ch. 6, Fig. 6.7. 
70?
?
True strain
0.00 0.05 0.10 0.15 0.20 0.25
True st
0
20
40
60
80
140
100
120
ress
 (M
P
a
)
IPC-10 FEA 
IPC-20 FEA
IPC-30 FEA
IPC-40 FEA 
 
Figure 7.5: True stress ? true strain responses of different IPC obtained 
from finite element analysis 
 
7.6.2 Comparison between experimental and finite element results 
The graphs in Fig. 7.6 show the dynamic stress-strain responses of all volume 
fraction IPC models obtained from the finite element analysis. In general, a good 
agreement between the measured results and finite element results is observed. The 
differences between the experimental results and the simulations are attributed to the 
following
 
? In the actual specimens micro cracking occurs whereas the numerical model does not 
 issues:  
allow any debonding to occur at material interfaces. 
71?
?
? The material property of aluminum used in the model corresponds to the quasi-static 
compressive response. This assumption was used considering the fact that aluminum 
is r in rate independent. 
lections back into the sample at 
elatively stra
? The infinite elements transmit all the stress waves that are incident on them. In the 
real situation however, there will be stress wave ref
the sample / transmitter bar interface within the duration of 250 microseconds.  
 
True strain
0.
?
True strain
000 0.05 0.10 0.15 0.20 0.25
True s
t
ress
 
(M
P
a
)
0
20
100
140
0
20
100
140
40
60
80
120
IPC-10 EXP 
IPC-10 FEA 
.00 0.05 0.10 0.15 0.20 0.25
Tru
e
 stre
ss
 
(M
Pa)
40
60
80
120
IPC-20 EXP 
IPC-20 FEA 
?
?
True strain
0.00 0.05 0.10 0.15 0.20 0.25
T
r
ue st
ress
 
(M
P
a)
0
20
40
60
80
100
120
140
IPC-30 EXP
IPC-30 FEA
?
True strain
0.00 0.05 0.10 0.15 0.20 0.25
T
r
ue
 str
e
ss 
(M
P
a
)
0
20
40
60
80
100
120
140
IPC-40 EXP 
IPC-40 FEA 
 
Figure 7.6: Comparison of experimental and FEA results of (a) IPC-10, 
(b) IPC-20, (c) IPC-30 and (d) IPC-40 
 
(b)
(c) (d)
(a) 
72?
?
7.6.3 alysis of contour plots 
From the finite element analysis, the von-Mises stresses, the equivalent plastic 
strains, displacements along the loading direction, the kinetic energy and the total strain 
energy of the whole model for up to 22% true strain were obtained. To appreciate the 
spatial and temporal distribution of these quantities in a dynamically deforming IPC, the 
FE results (contour plots) corresponding to the IPC-30 model is presented. In these 
contour p he IPC. 
The right hand side image  section made at exactly 
half the length of the unit cell along the z-direction. From the stress contour plots (Fig. 
t minum ligaments experience the maximum stress, 
approx
 
An
lots, the left hand side images show contours on the outer surface of t
s show contours on the x-y plane of the
7.7), i  can be seen that the alu
imately 200 MPa, equal to the yield stress of aluminum. The maximum stress 
attained by the SF however is approximately 70 MPa, which is close to the yield stress of 
SF-30 obtained from the experiments. From the figures, it is also evident that the stress 
concentrations occur in the vicinity of the ligament joints. This is expected considering 
the geometry and the material discontinuity present in the model.  
 
 
 
 
 
 
 
73?
?
 
   
 
 
 
                                               
Figure 7.7: Mises stress contour plots from dynamic FE model for IPC-30  
Left side images: whole IPC, Right side images: sectioned IPC 
(Loading is along the x-direction) 
 
                                              
 
(b) 12% strain 
(c) 20% strain 
(a) 5% strain 
A A
B 
C 
B 
C 
A 
B 
C 
A
B 
C 
B 
C 
C 
A
 
A 
B
74?
?
 
                         
 
             
 
   
strain contour plots from ic FE model for IPC-30 
side images: whole IPC, Rig ectioned IPC 
(Loading is along the x-direction) 
                    
                      
           
Figure 7.8: Equivalent plastic 
ft 
 dynam
 sLe ht side images:
(a) 5% strain 
(b) 12% strain 
(c) 20% strain 
A 
B 
C 
A
B 
A 
B 
C 
A
B 
A 
B 
C 
C 
B 
A
C 
C 
75?
?
The contour plots of the equivalent plastic strain ar  the 
contour plots, it is obvious that the SF regions have experienced larger pla c strains 
compared to that in aluminum. A much more interesting observation from the figures is 
that the locations of localized maximum plastic strain are not too close to the ligament 
joints. This suggests that very close to the ligament joints, the stiffer aluminum regions 
ic strain have restrained the SF from ng as much as the SF regions 
at are further away. The contours also help in appreciating the percentage of strain 
energy absorbed by the aluminum and the SF regions. 
Unlike the stress contours and equivalent plastic strain contours, the disp ent 
contours are rather continuous (Fig. 7.9). This is expected as debonding between the 
aluminum and SF regions are not allowed to occur in the model. That is, the material 
interfaces are modeled as perfectly bonded. The contours on the cross-section view reveal 
that up  the length of the unit cell fr  end the displacem  at th  
ce s 
reversed. Th d gradually 
decreas
e shown in Fig. 7.8. From
sti
with low plast deformi
th
lacem
 to half om the loading eent
nter is more than that at the periphery. Beyond the mid-section of the cell, this trend i
at is, the displacement is a maximum at the periphery an
es towards the center symmetrically. Near the mid-section, the displacement is 
nearly constant along the cross section.  
 
 
 
 
 
 
76?
?
 
              
 
   
.9: x-displacement contour plo odel for IPC-  
Left side images: whole IPC, Right side images: sectioned IPC 
(Loading is along the x-direction) 
                           
 
           
Figure 7 ts from dynamic FE m 30
(a) 5% strain 
(c) 20% strain 
A 
A
B 
B 
A 
(b) 12% strain 
C 
C 
C 
C 
C 
C 
B 
B 
B 
B 
A
A 
A
77?
?
7.6.4 Energy absorption 
It was shown earlier (see Section 6.2.2) that the compressive strengths and the 
stresses beyond yield point in IPC are higher than their corresponding pure syntactic 
foam. In other words, the area under the stress-strain curve (or the energy absorbed per 
unit volume) for a dynamically deforming IPC is greater than that for a SF. The 
introduction of aluminum into the SF has helped enhance this property among others 
such as the compressive strength. At this stage it was thought to be a worthwhile effort to 
analyze the proportion of various energies absorbed by the aluminum and SF phases in 
the IPC. The kinetic energy (KE) trends and the total strain energy (SE) trends are of 
particular interest. Previously, the equivalent plastic strain contours gave a hint regarding 
the proportion of strain energy absorbed by the two phases of the composite foam. To 
quantify the proportions shared between the SF and the aluminum phases, the KE history 
and the SE history for a duration of 125 microseconds, which approximately corresponds 
to the 22% strain, are plotted below.  
Engineering strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Kin
e
tic e
nergy
 (mJ)
0
1
3
5
(a) 
2.0
(b) 
2
4
SF-30
AL
IPC-30 
Engineering strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
S
tra
in ene
rgy
 (J)
0.0
1.2
0.4
0.8
1.6
SF-30
AL
IPC-30
 
Figure 7.10: History of (a) kinetic energy and (b) strain energy 
absorption by aluminum and SF in unit-cell IPC-30 
78?
?
Figures 7.10(a) and (b) show the amount of KE and SE absorption by SF and 
aluminum per unit cell of IPC. To quantify the proportion of these two energies shared by 
SF and aluminum, the percentage of KE and SE absorbed by SF and aluminum per cell is 
plotted in Figs., 7.11(a) and (b), respectively. From the Fig. 7.11(a), it can clearly be seen 
that the KE absorbed by the aluminum region accounts for more than 20% of the total KE 
absorbed by the IPC throughout the loading history. It should be realized that the volume 
fraction of aluminum in the model IPC is ~8 o, 8% of the IPC material absorbs at 
least 20% of the total KE.  
%. S
Engineering strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
20
40
60
80
100
20
40
60
80
100
Kine
t
i
c en
er
gy
 %
SF-30
AL
Engineering strain
S
t
r
ain e
nergy
 
%
SF-30
AL
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
 
E >> KE) is to be noted here. Although, the strain energy 
absorbed per unit cell (Fig. 7.11 (b)) by the aluminum region is considerably lower with 
(a) (b) 
Figure 7.11: History of percentage of (a) kinetic energy and (b) strain 
energy absorption by SF and aluminum in IPC-30 
 
From Fig. 7.11(b), it is seen that about 11% of the total SE is absorbed by 
aluminum during most of the loading history. Again, comparing the proportion of the SE 
absorbed by aluminum (~11%) to that of the volume fraction of aluminum in the model 
IPC (~8%), there is an encouraging, though not very prominent trend. The difference in 
KE and SE magnitudes (S
79?
?
80?
?
res  to that by the SF, the fact that aluminum prises only ~ 8% of the total volume 
of the IPC cell makes the energy absorbed by the aluminum phase more significant. This 
suggests that if the aluminum phase is replaced with a more compliant material, the 
energy absorption by the resulting IPC might be higher than that observed in the 
aluminum foam / SF IPC. It should also be noted that using a more compliant material 
might prove to be detrimental to the overall structural integrity of the IPC. Nevertheless, 
the proposed FE m
characteristics of di
pect  com
odel could be effectively used to predict the energy absorption 
fferent material combinations. 
 
 
 
8 SUMMARY 
 
A split Hopkinson pressure bar was developed for characterizing the compression 
response of low impedance materials. The apparatus was successfully calibrated against 
compression data in the literature for acrylics at strain rates of 1250 /sec. The 
repeatability of stress-strain responses was demonstrated for different stress pulse 
durations as well as modest variations in pulse shapers. 
Using this SHPB apparatus, the compression characteristics of Interpenetrating 
Phase Composite (IPC) foams were studied. The IPC foams were prepared by infusing 
uncured epoxy-based syntactic foam (SF) into open-cell aluminum preforms/scaffolds. 
Curing of SF resulted in an IPC structural foam with a 3D interconnectivity. The dynamic 
compression responses of IPC foams made of SF with 10% - 40% V
f
 of hollow glass 
microballoons were studied at strain rates of ~1500/sec.  The responses of IPC foams 
were also evaluated relative to their pure SF counterparts. The failure progression in both 
types of samples was recorded using high-speed photography to examine the real-time 
deformation and failure processes. The micrographs were also used to explain the 
underlying failure mechanisms. 
A unit-cell based finite element model of an IPC was developed using a Kelvin 
cell representation of aluminum ligaments of the open-cell scaffold. The dynamic 
compression response was simulated using ABAQUS structural analysis software. The 
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?
3D geometries for the SF and aluminum regions of the unit cell were modeled using 
Solid Edge. Elasto-plastic experimental measurements were used as the material property 
inputs for the SF region in the finite element IPC model. Infinite elements were used to 
model the external surface boundary of the unit-cell to idealize its presence within a large 
volume of the material. 
  The major conclusions of this work are as follows: 
? The V
f
 of microballoons in SF plays a dominant role in the overall response of the SF 
and IPC foams.  A monotonic increase in elastic modulus, yield stress, and plateau 
stress are evident as V
f
 of microballoons decreases.  The IPC foams consistently have 
higher values for each of these characteristics relative to the corresponding SF.  
? Unlike static results, under dynamic conditions SF and IPC samples show a stress-
strain response that has only two dominant regimes ? an initial proportional loading 
zone up to a maximum stress, and a subsequent monotonically softening zone.  The 
foams with lower V
f
 of microballoons tend to soften more rapidly than the ones with 
higher V
f
. The elastic modulus and maximum stress increase with decreasing V
f
 of 
microballoons under high-strain rate loading.  The values for IPC are again higher 
than those for SF under dynamic conditions.  The dynamic maximum stress values 
are also higher for both SF and IPC relative to the quasi-static ones. 
? The dynamic energy absorption values (U
0.22
) up to 22% strain are nearly twice the 
corresponding static values for both IPC and SF samples.  The energy absorbed per 
unit volume by the IPC under dynamic conditions is about ~10% higher than the SF 
counterparts.  The energy absorbed per unit mass, however, favors SF under dynamic 
conditions. This is unlike the observation under static conditions. 
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?
? The energy absorbed per unit mass for both the SF and IPC foams under dynamic 
loading follow a linear trend with respect to the applied strain.    
? The failure of SF and IPC under dynamic conditions is dominated by the formation of 
an extensive network of shear bands in SF.  Also, under dynamic conditions the 
microballoons are not as uniformly crushed as in static cases.  The IPC samples show 
debonding of aluminum ligaments from the surrounding SF as well. 
? Unlike static results, the high-speed optical recordings of the deformation process 
reveal significant spring-back in SF whereas it is negligible in case of IPC. Also, the 
failure process is rather uniform in both SF and IPC and no evidence of spatially 
progressive failure (crushing) is seen at the loading rates of 1500/sec used in this 
study.  
? A unit-cell model of the IPC using a space-filling polyhedron ? tetrakaidecahedron or 
a Kelvin cell was successfully developed using ABAQUS. Infinite elements were also 
used to model the external surfaces of the unit-cell.  
? The stress-strain responses of the model IPC?s with 10% - 40% V
f
 of microballoons in 
SF were compared to those from experiments. A good agreement between the 
simulated and the experimental results was observed. 
? Contour plots of von-Mises stress, equivalent plastic strain and displacement in the 
loading direction were analyzed. It was observed that maximum stress concentrations 
occurred at the vicinity of the aluminum ligament joints and the SF region 
experienced higher plastic strains than the aluminum region. The proportions of 
kinetic and strain energies absorbed by aluminum and SF regions were also analyzed. 
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84?
?
? Citing the conformance of the results of the unit-cell model to those of the 
experimental results, the proposed method of finite element analysis could be used 
effectively to predict the energy absorption characteristics of different material 
combinations. 
 
 
 
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APPENDIX A 
The epoxy resin used for preparing SF and IPC foam samples was characterized 
under both static and dynamic loading conditions. For completeness, the corresponding 
data is reported in Fig. A1. The quasi-static response of neat epoxy shows three distinct 
regions: initial linear elastic region, softening region following yielding, and a plateau 
stress region. A yield stress of approximately 80?2 MPa is followed by a plateau stress of 
~ 48 MPa up to 22% true strain.  At higher strains (not shown), there is a densification of 
stress. (The sample continues to densify up to 70% strain). 
True strain
0.00 0.05 0.10 0.15 0.20 0.25
True st
ress 
(M
Pa)
0
25
50
75
100
125
150
175
200
Dynamic 
Static 
Strain rate ~ 1200 / sec
Strain rate ~ 0.001 / sec
?
Figure A1: Quasi-static and dynamic compressive responses of neat epoxy 
(?Epo-Thin?) used in SF and IPC foams 
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?
The dynamic response, on the other hand, has only two distinct regions. 
Following an initial linear response with a higher elastic modulus than the quasi-static 
counterpart, a maximum yield stress of ~185?5 MPa is attained by the neat epoxy 
sample. Subsequently, the sample shows a monotonic softening response until the stress 
pulse ends.