Analysis of inertia effect on a three point bend specimen loaded by the
Hopkinson pressure bar
Xiaoxin Zhang1,2, Gonzalo Ruiz2 and Rena C. Yu2
College of Mechanical and Electrical Engineering, Harbin Engineering University, 150001Harbin, China.
2
ETSI Caminos, C. y P., Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain.
1
ABSTRACT
In this study, the inertia effect was investigated in detail using experimental-numerical method on the three-point bending
specimen loaded by a Hopkinson pressure bar. The results show that the nominal load (external load) does not represent the
“true load” (effective load) of the specimen due to inertia effect during the dynamic fracture process. Most of the nominal load
is used to maintain the balance with the inertia force during loading process, only a very small portion of the nominal load is
used to generate deformation and fracture the specimen. Moreover, the strain and strain rate distribution are non-uniform
around the crack tip, the rate-dependent material model should be adopted in numerical simulation.
Introduction
The determination of the dynamic behavior of materials under impact loading continues to attract considerable attention from
both theoretical and experimental viewpoints [1-10].
It is widely understood that, under dynamic loading conditions, loads are added to the specimen quickly and the specimen is
rapidly increased to high velocity, this enables the specimen to produce considerable acceleration. For example, in drop
weight test, Banthia et al. [11] pointed out, the force required to produce the acceleration is defined as the inertia force, the
nominal load (external load) is not the true load required to deform and fracture the specimen, the true load maybe only a
fraction of the nominal load. In order to get the true load on the specimen in Charpy impact test, Kobayashi [12] proposed the
method of moving averages to eliminate the inertia effect; Liu and Zhou [13] did the same by smoothing the peaks and troughs
on the load-time curves, and found that the peak load could be used to calculate the fracture toughness.
Another important feature of a dynamic process is the near-tip strain rate sensitivity. Lei et al. [14] studied this aspect for a
Charpy V-notch test specimen using ADINA. They showed that the strain rate field is not uniform, and this rate sensitivity
means that not only the fracture toughness and the yield strength but also the fracture mechanism, the ductile-brittle transition,
vary with the strain rate.
However, the researches mentioned above were concentrated on the Charpy or drop weight impact test, the inertia effect and
strain rate sensitivity are not that clear for the impact tests using a Hopkinson pressure bar, whose resulted loading rate is
often one or two order higher. This is our motivation to analyze in detail the three-point bending specimen loaded by a
Hopkinson pressure bar.
The structure of the paper is as follows: the experiment setup, the material characterization and the specimen geometry are
given in section 2. The numerical simulation is described in section 3. The results and discussion are presented in section 4,
and some conclusions are extracted in section 5.
Experimental procedure
(1) Experimental setup
The schematic diagram of the experimental apparatus is shown in Fig. 1. The apparatus consists mainly of an air gun, a
projectile, a Hopkinson pressure bar, a speed-measuring device and recording equipment. The Hopkinson pressure bar (800
mm long) and the projectile (290 mm long), both with a diameter of 14.5 mm are made of a high-strength steel. The
dimensions of the Hopkinson pressure bar satisfy the one-dimensional stress wave propagation theory and are appropriate for
the size of the specimen. In addition, the length of the bar is sufficient to identify the incident wave and the reflected wave.
The bend specimen is placed on an anvil (geometrically comparable to the one in a Charpy pendulum), and rapidly loaded
until fracture occurs by a compressive wave generated in the Hopkinson pressure bar due to axial impact with the projectile
that was launched from the air gun. The amplitude of the compressive pulse can be controlled by the projectile velocity. At the
interface between the specimen and the Hopkinson bar, part of the incident compressive pulse
ε i (t )
is transmitted through
the specimen, causing its fracture, while the other part is reflected back to the Hopkinson pressure bar as a tensile
pulse ε r (t ) . The incident pulse ε i (t ) and the reflected pulse
ε r (t )
were recorded by a strain gauge mounted at the center
of the Hopkinson pressure bar. The nominal dynamic load P(t) and the loading point displacement
the one-dimensional stress wave theory as [3],
∆(t )
are determined by
t
P(t ) = EA[ε i (t ) + ε r (t )] ; ∆ (t ) = c [ε i (t ) − ε r (t )]dt
∫
(1)
0
where E is the Young’s modulus and A is the cross-sectional area of the Hopkinson pressure bar, c is the stress wave speed
of the bar (5189 m/s).
Figure 1. Experimental apparatus and its recording system
(2) Material characterization and specimen geometry
The material used is a shipbuilding steel, whose nominal composition is Fe-0.061C-0.91Mn-0.6Si-0.014P-0.005S-0.72Cr0.651Ni-0.48Cu (wt. %). The properties of the steel are given in Table 1. The dimensions of the specimens are 10 mm × 10
mm × 55 mm (width × thickness × length). All of the specimens were notched and fatigue precracked to a crack-length to
specimen-width ratio (a/w) of around 0.5 following ASTM E399 [15]. A fixed support span of 40 mm was used throughout this
investigation.
TABLE 1. Properties of the steel
Yield strength
(MPa)
480
Ultimate
strength
(MPa)
600
Area
reduction
Elongation
Poisson’s
ratio
Density
. -3
(kg m )
Modulus
(GPa)
77%
30%
0.3
7800
210
Taking into account the rate effect, the relationship between the dynamic yield strength
σ yd of
the material and the strain
rate ε& is expressed in terms of the PERZYNA model, which is shown as:
σ yd = 480[1 + (
ε&
64560.7
) 0.275 ]
(2)
Where 480 is the measured static yield strength in MPa; the exponent 0.275, representing the strain rate hardening parameter
-1
and the denominator 64560.7, standing for the material viscosity in s were fitted from the experimental data [16].
Numerical simulation
(1) Determination of the effective load and displacement
A numerical simulation method is adopted to determinate the effective load and displacement, the governing equation used is
as follows:
[M ] {u&&}+ [K ] {u} = {R}
(3)
Where [K] is the assembled stiffness matrix, [M] is the assembled mass matrix, { u&& } and {u} are the nodal acceleration vector
and the nodal displacement vector, respectively, and {R} is the nodal load vector. Eq. (3) was solved by the so-called
Newmark-β method [17]. A time increment of 0.5µs was taken for the integration process, when α and β were chosen 0.5 and
0.25, respectively, in this case the numerical computation is unconditionally stable.
(a)
(b)
Figure 2. (a) Finite element mesh. (b) The location of the analytical points, O is the crack tip.
From Eq. (3), the nominal load {R} is decomposed into two parts, the inertia force [M ] {u&&} and the effective force [K ] {u} .
Since the direct extracting of the effective term is not possible in ANSYS, we first input the nominal load P(t) determined by the
one-dimensional stress wave theory (Eq. (1)) to get the effective displacement of the specimen loading point. Then,
introducing this effective displacement into Eq. (3) and setting the mass matrix null, the effective load applied to the specimen
can be determined as a function of time.
The effect of the pressure bar can be represented by the nominal load P(t) only in case there is no loss of contact between the
bar and the specimen. Since we are interested in the deformation behavior before crack initiation, this is precisely the case.
This is based on the phenomenon observed in the literature [9, 10]: the specimen actually fractures prior to “take off”.
One half of the specimen and one support (anvil) were modeled due to symmetry as shown in Fig. 2(a). The material of the
anvil is a high strength steel, the yield strength, the elastic modulus, the Poisson’s ratio and the density are 1650 MPa, 210
3
GPa, 0.3 and 7800 kg/m , respectively. The mesh of the specimen and support consisted of 1127 nodes and 392 four-node
iso-parametric elements in total. Taking into account the stress and strain concentration at the crack tip, the mesh was refined
in that zone, with the smallest element size being 0.1 mm.
The specimen is assumed to be under plane-strain condition, a static elastic-plastic material model and the strain rate effect of
the yield strength (Eq. (2)) are fed into ANSYS as material model, and it is assumed to obey the Von Mises yield criterion and
isotropic hardening. Moreover, linear elastic behavior is adopted for the support material.
Using the contact wizard we conducted a contact analysis between the specimen and the support; the contact element type
was selected as Surface-to-Surface; the initial penetration was included and other parameters were determined as default.
The zone near the contact point was also refined, as shown in Fig. 2(a). Consequently, the possible loss of contact between
the specimen and the supports is properly taken into consideration by this analysis.
(2) Determination of the strain rate field around the crack tip
Eleven analytical points in all are selected along X and Y axis as shown in Fig.2 (b), five points for each direction, to study the
strain rate distribution around the crack tip, the point “O” is in correspondence with the crack tip, the direction of the positive X
axis is defined along the crack line. The distance between the neighbor points is 0.1 mm.
In ANSYS post-processing, the equivalent strain of each analytical point was extracted, then the (equivalent) corresponding
strain rate was obtained by differentiation.
Results and discussion
A small strain gauge attached near the crack tip was used to detect the crack initiation, its measuring principle can be seen
elsewhere [3, 4]. For the specimen we are going to model, the crack initiation time is measured as 25.90 µs . It should be
noted that the numerical simulation is not correct any more after this point, as the crack propagation was not modeled during
the finite element analysis.
(1) Effective load
Fig. 3 shows a comparison between the numerical effective displacement and the experimental loading point displacement,
obtained from Eq.(1). The numerical results are slightly smaller than the experimental ones owing to the incomplete contact
between the pressure bar and the specimen, nevertheless, similar trend is found and the agreement is reasonable. Similar
observation is also given in [18] for a 40Cr steel specimen.
0.45
Displacement, ∆ (mm)
0.40
Numerical result
Experimental result
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
10
20
30
40
Time, t (µs)
Figure 3. Comparison of the numerical and experimental displacement at the specimen loading point
Fig. 4 shows the comparison of the nominal load measured by Eq. (1) and the effective load calculated with respect to time.
We have to point out that, as mentioned before, the effective load is only valid before 25.9 µs , when the crack initiated. The
crack initiation is in correspondence with the post-peak for the nominal load, while it occurs at the maximum of the effective
load. Moreover, the nominal force shows some oscillation due to the inertia effect, while the effective one turns out to be quite
smooth. It is also evident that the effective load is much less than the nominal load. This indicates that most of the nominal
load is used to maintain the balance with the inertial force distributed over the specimen. For instance, at 16.50 µs , this leaves
only 13.28% of the nominal load (19.05 kN) to work as the effective one (2.53 kN).
20
Nominal load
Effective load
Load, P (kN)
15
10
5
0
0
10
20
30
40
Time, t (µs)
Figure 4. Comparison of the nominal load and the effective load
(2) Strain rate distribution
Fig. 5 shows the strain rate versus loading time at constant distance away from the crack tip in both X and Y direction. It can
be seen that the strain rate fluctuates with loading time and reaches a maximum before the crack initiation. With increasing
distance away from the crack tip, the fluctuation and the maximum of the strain rate decreases.
600
10000
9000
.
1
2
3
4
5
7000
-1
400
0
1
2
3
4
5
8000
strain rate, ε(s )
-1
strain rate, ε (s )
500
6000
.
300
200
5000
4000
3000
2000
100
1000
0
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
0
2
4
6
8
Time, t (µs)
10
12
14
16
18
20
22
24
26
Time, t (µs)
(a)
(b)
Figure 5. Strain rate distribution versus time at constant distances away from crack tip (a) in X direction (b) in Y direction
It can be seen that the strain rate in Y direction is around one order greater than that in X direction. This non-uniform influence
of the inertia requires an explicit rate consideration in numerical simulations, otherwise the material deformation would not be
estimated correctly.
Furthermore, in Fig. 6, we compare the strain rate at the crack tip using a rate-independent material model and a ratedependent one (the yield strength follows Eq. (2)). It is noted that the maximum strain rate for the former is about four times of
that for the latter. This again confirms the necessity for rate consideration.
25000
Static material model
Dynamic material model
-1
strain rate, ε(s )
20000
.
15000
10000
5000
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
Time, t (µs)
Figure 6. Comparison of the strain rate at the crack tip using the two material models
Fig.7 shows the strain rate versus time after eliminating the inertia effect. The curves show less oscillation in both directions
and the magnitude of the strain rate is comparable to that of Fig. 5. In X direction, for points 1, 2 and 3, the strain rate first
increases, then decreases slightly after reaching a maximum before the crack initiation; for points 4 and 5, it increases all the
time until the crack initiation is reached. For all points in Y direction, including the crack tip, the strain rate increases
monotonically before the crack initiation.
600
10000
1
2
3
4
5
.
-1
6000
.
300
200
4000
2000
100
0
0
2
4
6
0
1
2
3
4
5
8000
strain rate, ε (s )
400
-1
strain rate, ε(s )
500
8
10
12
14
16
18
20
22
24
0
26
0
2
4
6
8
Time, t (µs)
10
12
14
16
18
20
22
24
26
Time, t (µs)
(a)
(b)
Figure 7. Strain rate distribution after eliminating inertia effect at constant distances away from crack tip (a) in X direction (b) in
Y direction
The distinct influence of the inertia at both directions can be further revealed, taking the example of the crack tip. As shown in
Fig. 8, after eliminating the inertia effect, the curve goes through the average of the peaks and the troughs of the original one
in Y direction. But it is not the case in X direction.
10000
Not eliminate inertia effect
Eliminate inertia effect
-1
strain rate, ε (s )
8000
.
6000
4000
2000
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
Time, t (µs)
Figure 8. Comparison of the strain rate of the crack tip
Fig.9 shows the equivalent strain before and after eliminating the inertia effect. In X direction, the inertia intends to slow down
the corresponding movement, and leads to lower strain. On the contrary, in Y direction, it enhances the movement, and results
in higher strain. Precisely because of this mechanism, the inertia effect influences the strain rate distribution differently in two
directions.
It should be pointed out that, although the inertia holds a large portion of the nominal load, it contributes to only around 10% of
the total strain. Therefore as a first approximation, we can say that the deformation and fracture come only from the effective
load.
0.06
0.006
0.005
0.04
Strain, ε
Strain, ε
0.004
Not eliminate inertia effect
Eliminate inertia effect
Point 1 in Y direction
0.05
Not eliminate inertia effect
Eliminate inertia effect
Point 1 in X direction
0.003
0.002
0.03
0.02
0.01
0.001
0.00
0.000
0
2
4
6
8
10
12
14
16
18
20
22
24
0
26
2
4
6
8
10
12
14
16
18
20
22
24
26
Time, t (µs)
Time, t (µs)
(a)
(b)
Figure 9. Comparison of the equivalent strain in (a) X direction and (b) Y direction
Conclusion
We analyzed in detail the inertia effect for the three-point bending specimen loaded by the Hopkinson pressure bar. The inertia
leads to oscillations both on the force and strain rate versus time curves.
On the one hand, around the crack tip, it influences the strain and strain rate distribution non-uniformly, this calls for an explicit
rate consideration of the material model. On the other hand, even though the inertia force counterbalances a large portion of
the nominal load, it contributes to only around 10% of the total strain. It is the effective load that actually deforms and fractures
the specimen.
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