ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation
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simulations can serve as a validation and verification tool for complex sample geometries and facilitate data
analysis and constitutive model development for materials exhibiting nonuniform and/or nonconservative
behaviors such as compaction (foams), shear band formation, and fracture.
Split-Hopkinson Pressure Bar Testing of Soft Materials
George T. (Rusty) Gray III and William R. Blumenthal, Los Alamos National Laboratory
Acknowledgments
This work was supported under the auspices of the United States Department of Energy. The authors wish to
acknowledge the assistance of M.F. Lopez for conducting the quasi-static mechanical testing and B. Jacquez,
C.P. Trujillo, and T. Bell for conducting the dynamic mechanical testing. The authors wish to acknowledge
T.A. Mason for critically reviewing this manuscript.
Split-Hopkinson Pressure Bar Testing of Soft Materials
George T. (Rusty) Gray III and William R. Blumenthal, Los Alamos National Laboratory
References
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Hopkinson Bar, Exp. Mech., Vol 38, 1998, p 242–249
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Constitutive Behavior of Teflon and Nylon, Plasticity 99: Constitutive and Damage Modeling of
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(France) IV Colloq., C3 (EURODYMAT 97), Vol 7, 1997, p 91–96
42. G.T. Gray III, W.R. Blumenthal, C.P. Trujillo, and R.W. Carpenter, Influence of Temperature and
Strain Rate on the Mechanical Behavior of Adiprene L-100, J. Phys. (France) IV Colloq., C3
(EURODYMAT 97), 1997, p 523–528
43. G.T. Gray III, High-Strain-Rate Testing of Materials: The Split-Hopkinson Pressure Bar, Methods in
Materials Research, John Wiley Press, 2000
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Compression of Polymers at 300K and 100K, J. Phys. (France) IV Colloq., C3 (DYMAT 91), Vol 1,
1991, p 185–190
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Pressure Bar: Stress State Stability, Exp. Mech., in preparation
Split-Hopkinson Pressure Bar Testing of Ceramics
G. Subhash, Michigan Technological University G. Ravichandran, California Institute of Technology
Introduction
SPLIT-HOPKINSON PRESSURE BAR (SHPB) TESTING, described elsewhere in this Section, was originally
developed by Kolsky (Ref 1) and has been traditionally used for determining the plastic properties of metals
(which are softer than the pressure bar material) at high strain rates in the range of 10
2
to 10
4
s
-1
. Ceramics, on
the other hand, are hard and brittle and typically exhibit higher compressive strength than metals. Because most
of the ceramics reveal only elastic strains to failure, accurate measurement of these strains using the traditional
SHPB technique is often not a trivial task.
In this article, the inherent limitations of the traditional SHPB technique for testing ceramics and the
modifications necessary in design and test procedures are discussed. Before describing the test method for
ceramics, a brief discussion of the operational principle of the traditional SHPB technique and the relevant
assumptions in the derivation of the stress-strain relationship are presented. This is followed by discussion of
the inherent limitations on the validity of these assumptions while testing ceramics and the necessary
modifications in SHPB design and test procedure for high-strength brittle ceramics. Other topics covered in this
article include maximum strain rate that can be obtained in ceramics using an SHPB, the necessity of incident
pulse shaping, specimen design considerations, interpretation of experimental results obtained from SHPB
testing of ceramics, and the effectiveness of the proposed modifications.
Reference cited in this section
1. H. Kolsky, An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading,
Proc. R. Soc. (London) B, Vol 62, 1949, p 676–700
Split-Hopkinson Pressure Bar Testing of Ceramics
G. Subhash, Michigan Technological University G. Ravichandran, California Institute of Technology
Review of Traditional Split-Hopkinson Pressure Bar Operational Principles
A traditional SHPB configuration (Fig. 1) consists of a striker bar, an incident bar (AB), and a transmission bar
(CD). A specimen of suitable dimensions is sandwiched between the incident and transmission bars. The striker
bar is launched from a gas gun at a predetermined velocity towards the incident bar. Upon impact at A, a
compressive pulse is generated in the incident bar and travels towards the specimen. The duration of the pulse
is equal to the round trip travel time of the longitudinal wave in the striker bar. Upon reaching the specimen, a
portion of the incident pulse is transmitted into the transmission bar as a compression pulse, and the remaining
portion is reflected back into the incident bar traveling toward the impact end, A. Strain gages are mounted at
midpoints along the length of the incident and transmission bars to capture the stress pulses as they pass by. In
general, the specimen is chosen to have a lower impedance (product of density, wave velocity, and area of cross
section) than the bar material, and, hence, the reflected pulse is rendered tensile in nature. Based on one-
dimensional analysis, it can be shown that the equations for stress, strain, and strain rate in the specimen are
given by (Ref 1, 2):
(Eq 1)
(Eq 2)
ε
s
(t) =
s
(t)dt
(Eq 3)
where A is cross-sectional area, E is Young's modulus, l is length, σ is stress, ε is strain, is strain rate, and t is
time. The subscripts o, s, T, and R correspond to the bar, specimen, transmitted pulse, and reflected pulse,
respectively; and c
o
is the longitudinal bar wave velocity, which is given by , where ρ
o
is density. In
deriving Eq 1 2 3, the following assumptions are made:
1. The specimen is under a uniform and uniaxial state of stress during deformation.
2. The incident and transmission bars remain elastic at all times during the experiment, and the end
surfaces of the bars in contact with the specimen remain flat and parallel during the deformation of the
specimen.
3. The incident, transmitted, and reflected pulses undergo minimal dispersion as they travel along the
length of the bar.
4. The stress distribution across the cross section of the incident and transmission bars is fairly uniform
(meaning that the measured strains on the surface of the bar using strain gages are representative of the
stress in the elastic bars).
5. The accumulated strain in the specimen (determined from the reflected strain gage signal in Eq 3) is
solely due to a single incident compression pulse reaching the specimen; in other words, the specimen is
subjected to only a single incident stress pulse to cause the observed deformation.
Fig. 1 Schematic of the traditional split-Hopkinson pressure bar text configuration. AB, incident bar;
CD, transmission bar; A, impact location
These assumptions (except the fifth one) are easily satisfied while testing metals where plastic deformation is
desired. Ceramics, on the other hand, are hard, have high strength, and exhibit no more than 1 to 2% strains
before fracture. Therefore, several of the above assumptions are violated while testing ceramics using the
traditional SHPB procedure, and, thus, caution must be exercised while interpreting the experimental data.
Proper modifications must also be made in the testing procedure to ensure the accuracy and reliability of the
stress and strain measurements. Each of the five assumptions (and related issues in SHPB testing of ceramics) is
discussed in the following section.
References cited in this section
1. H. Kolsky, An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading,
Proc. R. Soc. (London) B, Vol 62, 1949, p 676–700
2. H. Kolsky, Stress Waves in Solids, Dover, 1963, p 41–94
Split-Hopkinson Pressure Bar Testing of Ceramics
G. Subhash, Michigan Technological University G. Ravichandran, California Institute of Technology
Limitations of Testing Ceramics by Traditional Split-Hopkinson Pressure Bar Techniques
The first assumption (that the specimen is under a uniform and uniaxial state of stress) ensures that the stress
within the specimen is in equilibrium before the required data (e.g., failure strength) about the ceramic behavior
is extracted from the strain-gage signals. Stress equilibrium in the entire specimen is ensured by choosing the
duration of the incident loading pulse to be sufficiently longer than the travel time of the longitudinal wave
within the specimen. This allows for sufficient wave reflections to occur within the specimen, after which a
state of uniaxial and uniform stress is expected to prevail. While testing metals where plastic properties are of
interest, the specimen undergoes elastic deformation during the initial few reflections of the incident wave
within the specimen, during which a nonuniform state of stress prevails. Therefore, the elastic deformation
obtained for metals using a SHPB test is often neglected, and only plastic strains are plotted.
In the case of ceramics (where only elastic strains form a significant portion of the overall response), specimen
failure can occur before sufficient reflections occur or before stress equilibrium within the specimen has been
established. Thus, the failure strength data obtained under these conditions does not represent the true uniaxial
compressive strength of a ceramic. Ravichandran and Subhash (Ref 3) demonstrated that if the impedance
mismatch between a ceramic specimen and the steel bars is taken into account, it is possible to derive the
magnitude of the stress difference between the two end surfaces of the specimen at any given instant while the
specimen is being loaded by an incident pulse in SHPB testing. Figure 2 illustrates the result of their analysis.
On the y-axis is the normalized parameter R(t) defined as the ratio of stress difference, Δσ(t), between the two
end surfaces of the specimen, B and C, at a given instant to the mean stress, σ
m
(t), in the specimen at that
instant, that is:
(Eq 4)
On the x-axis is the normalized time, where t
s
is the time required for one single transit of the longitudinal wave
within the specimen, that is, t
s
= l
s
/c
s
. The results are plotted as a function of impedance mismatch parameter, r,
between the specimen and the steel bars, which is defined as:
(Eq 5)
Fig. 2 A plot of normalized stress difference between the two ends of a ceramic specimen versus number
of wave reflections within the specimen. See discussion in text. Source: Ref 3
The plot shown in Figure 2 indicates that several reflections are needed before complete stress equilibrium is
achieved within a ceramic specimen. For practical purposes, a stress difference of 5% between the two end
surfaces of the specimen with respect to the mean stress is assumed reasonable. It takes at least four transit
times to reach this value, and, therefore, for the fracture strength data to be valid, the failure of the ceramic
specimen should occur at least after this time has elapsed during loading of the incident pulse. These results
were also confirmed by Chen et al. (Ref 4) using finite element simulations. A typical incident stress pulse
containing equal loading and unloading portions should, therefore, have a total duration, T, of at least eight
transit times (assuming failure occurs only during the loading phase) of the longitudinal wave within a ceramic
specimen, that is:
(Eq 6)
The maximum length, l
s
, of a ceramic specimen for a given incident pulse duration is now limited to:
(Eq 7)
The equation clearly reflects the fact that the time required for stress equilibration increases with specimen
length.
The second assumption (that the incident and transmission bars remain elastic at all times during the
experiment, and the end surfaces of the bars in contact with the specimen remain flat and parallel during
deformation of the specimen) is usually met by making the incident and transmission bars out of a high-strength
material, for example, maraging steel. Notice that the equation for the stress (Eq 1) in the specimen contains the
area of the bar in the numerator and the area of the specimen in the denominator. Therefore, by making the
specimen diameter smaller (or the bar diameter bigger) one can impose a sufficiently high stress level to cause
fracture in a high-strength ceramic specimen while keeping the stress level in the incident and transmission bars
within the elastic limit of the bar material. Because ceramics are harder than steel, however, they can indent
into the steel bars, as illustrated in Fig. 3(a), thus violating the later part of the second assumption (i.e., the end
surfaces of the bars in contact with the specimen remain flat and parallel during the deformation). The
indentation into the steel bars can also cause stress concentration along the circumference of the specimen end
faces and lead to a nonuniform (or nonhomogeneous deformation) and nonuniaxial stress state within the
specimen, thus also violating the first assumption. The stress concentration can lead to premature failure (e.g.,
chipping) of the ceramic by initiating microcracking at these locations. This problem can be reduced by
sandwiching the ceramic specimen between two identical high-strength tungsten-carbide (WC) inserts and
placing this whole assembly in the SHPB testing setup, as shown schematically in Fig. 3(b). The main
requirement in this approach is that the tungsten-carbide inserts should not alter the incident, reflected, and
transmitted stress-wave characteristics as the wave crosses the specimen-bar interfaces to avoid
misinterpretation of strains measured by the strain gages on the bars. This requirement can be achieved by
matching the impedance of the tungsten-carbide inserts to that of the bar material, that is:
(ρcA)
WC
= ρ
o
c
o
A
o
(Eq 8)
This requirement can also be achieved by accordingly choosing the diameter of the inserts. The length of the
inserts is typically chosen to be one fourth of the length of the specimen.
Fig. 3 Problem caused by the hardness of ceramic specimens and techniques developed to solve the
problem. (a) Schematic illustration of indentation of hard ceramic into the bars. (b) Specimen-insert
assembly with impedance matched tungsten carbide inserts. (c) Specimen-insert assembly with matching
diameter conical inserts. (d) Dog-bone shaped specimen with end diameters sized to match the incident
and transmission bars
Alternate insert geometry to reduce the indentation into the bars was proposed by Anderson et al. (Ref 5). In
this design, conical ceramic inserts with matching diameters at both ends, as shown in Fig. 3(c), were used.
Finite element analysis on various specimen-insert configurations (Ref 4) has confirmed that the magnitude of
indentation into the steel bars, as well as the stress concentrations in the specimen, can be eliminated by using
conical inserts. However, the analysis also revealed that conical inserts can alter the reflected and transmitted
wave characteristics considerably due to the impedance mismatch with the steel bars. Other disadvantages of
ceramic conical inserts compared to the constant-diameter impedance-matched tungsten-carbide inserts are
higher manufacturing cost and need to customize the inner diameter of the cone whenever the specimen
dimensions change. A dog-bone shaped ceramic specimen with an end diameter matched to that of the bar,
shown in Fig. 3(d), has also been used in the literature to reduce the stress concentration and avoid indentation
into the bars (Ref 6, 7). Again, this specimen geometry is prohibitively expensive to manufacture from a
ceramic material.
While using inserts to avoid indentation into the bars, it is assumed that the strength of inserts is considerably
greater than the ceramic being tested. However, smaller-diameter high-strength ceramic specimens can cause
stress concentration on the inserts and cause fracture of the inserts before the ceramic specimen. Once the
failure of insert occurs, the ceramic fractures invariably (due to stress concentration caused by the insert
fragments on the ceramic), thus rendering the data on ceramic fracture strength invalid. To eliminate this
possibility, it is recommended that the inserts be laterally confined using steel rings that can provide adjustable
confining pressures, as shown in Fig. 4. To avoid impedance mismatch at the contact surface between the bar
and the insert, the thickness of the steel ring is made slightly smaller than that of the inserts.
Fig. 4 Schematic of the adjustable lateral confinement ring to avoid premature fracture of inserts
The third assumption (that the incident, transmitted, and reflected pulses undergo minimal dispersion as they
travel along the length of the bar) points to the fact that stress-strain response of a ceramic is not measured on
the specimen during its deformation, but at distances farther away from the specimen (and at a time different
from that of specimen deformation) using strain gages on the incident and transmission bar surfaces (Fig. 1).
Under these circumstances, it is imperative that the reflected (which is a measure of strain) and transmitted
(which is a measure of stress) strain-gage signals reflect the true response of the specimen. If dispersion is
present in the propagating pulse, the amplitude and duration of the pulse changes by the time it reaches the
strain gages and results in erroneous measurement of stress and strain within a ceramic. The extent of
dispersion depends on the dominant frequency components present in (or the shape of) a propagating stress
pulse (Ref 3). When the diameter of the bar is comparable to the wavelength of the dominant frequency
component (or when the period of the pulse is comparable to the transit time of the bar wave across the
diameter of the bar), dispersion effects become extremely severe, and the use of surface measurements at
distances farther away from the specimen lead to erroneous measurement of the stress-strain response. It is
suggested that the pulse duration should at least be 10 times larger than the transit time of the longitudinal wave
across the diameter of the bar to minimize the dispersion in a propagating stress pulse (Ref 2, 8). Because the
minimum pulse duration of the incident stress pulse and the maximum obtainable strain within a ceramic are
limited, the maximum strain rate that can be imposed on to a ceramic specimen using SHPB testing is also
limited. This concept is discussed in detail in the section “Limiting (Maximum) Strain Rate” in this article.
One way to avoid the problems associated with the dispersion is to eliminate the use of strain-gage signals from
the bars. Instead, one can mount strain gages directly on the ceramic specimen and record the axial and
transverse strains in situ during the deformation of the specimen. This procedure is also expected to give a more
accurate measure of strain in a ceramic. Because the majority of ceramics undergo only linear elastic response
until failure, the stress to fracture, σ
f
, can be determined from the measured strain to fracture, ε
f
, using Hooke's
law (σ
f
= Eε
f
). However, caution should be exercised while adopting this linear elastic approach on ceramics
that exhibit significant inelastic strain before fracture. For example, zirconia ceramics, such as MgO-PSZ and
Y-TZP, exhibit considerable inelastic strains due to stress-induced transformation and associated microcracking
(Ref 9, 10), as discussed in the section “Pulse Shaping” in this article. In such cases, use of linear elastic
response until fracture can overestimate the failure strength.
The fourth assumption (that the stress distribution across the cross section of the steel bars at a given instant is
uniform) implies that the strain-gage measurements on the surface of the bars are representative of the stress at
any interior point at that cross section of the bar. To further illustrate this concept, consider the stress
distributions in the bar shown in Fig. 5. Obviously, for the stress distribution shown in Fig. 5(a), the measured
strains on the surface of the bars will represent those at any interior point on the bar cross section. For the stress
distributions shown in Fig. 5(b) the strain-gage measurements are not representative of the strains inside the
bars, and, hence, the surface measurements will be meaningless. The nonuniform stress distribution can occur if
either a planar impact is not ensured between the striker and the incident bars or when the duration of the
incident stress pulse is comparable to that of the time required for transit of the longitudinal wave across the
diameter of the bar, which can amplify the dispersion effects (Ref 3).