numerical simulations of quasi-static indentation …
TRANSCRIPT
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NUMERICAL SIMULATIONS OF QUASI-STATIC INDENTATION AND LOW
VELOCITY IMPACT OF ROHACELL 51 WF FOAM
E.A. FLORES-JOHNSON*†, Q. M. LI‡, L. SHEN†
†School of Civil Engineering, The University of Sydney, Building J05, Sydney, NSW 2006, Australia
‡School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Pariser Building,
Manchester M13 9PL, UK
Numerical simulations of quasi-static indentation and low velocity impact of low density
polymethacrylimide (PMI) Rohacell 51WF foam using indenters with different nose shapes (conical,
truncated-conical, hemi-spherical and flat) were carried out using finite element code LS-DYNA. A
2D axisymmetric model was generated. A strain-rate dependent material model and r-adaptive
remeshing was used for low velocity impact simulations. Numerical predictions were compared with
available experimental data with good agreement between numerical simulations and experiments.
The results demonstrated the ability of the model to reproduce the mechanisms of deformation of the
penetration process in quasi-static indentation and low velocity impact. The predicted resistance
force closely matches the empirical results.
Keywords: Finite-element simulation; r-adaptive remeshing; indentation; low velocity impact,
polymer foam.
1. Introduction
Polymeric foams are used in a variety of applications such as automobile industry,
communications, aircraft and aerospace industry, medical equipment, launch vehicles,
rail vehicles, shipbuilding and sports. Polymeric foams are extensively used in energy
absorption applications such as automotive crash safety systems due to their great impact
energy absorbing capability [Gibson and Ashby (1997)]. Strength-to-weight ratio, low
moisture absorption, good fatigue resistance, low heat conductivity, thermoformability
and easy-machining are some of the properties that have made polymeric foams being
increasingly used in structures. One of the main applications of polymeric foams is the
use as core material for sandwich structures, which have been increasingly used in
applications where high stiffness and strength combined with light weight is required
such as in aerospace and marine constructions.
Polymeric foams and polymeric foam cored sandwich structures are prone to impacts
during manufacturing, service life and maintenance operations by a wide range of
* Corresponding author.
Preprint of an article published in [International Journal of Computational Methods 11, 1344004 (2014)] [DOI: http://dx.doi.org/10.1142/S0219876213440040 ] © [copyright World Scientific Publishing Company] http://www.worldscientific.com/worldscinet/ijcm
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projectiles shapes, sizes and velocities. For example, low velocity impact can occur
during manufacturing process or maintenance by tool drops or natural events (e.g.
hailstorms) [Abrate (1998)]. High velocity impacts can occur during aircraft takeoffs and
landings when small foreign objects such as stones or other debris on the runway are
propelled by the tires. Low velocity impacts are considered the most dangerous, because
the damage may escape detection in a routine visual inspection of the impacted surface of
the structure. After an impact, there can be a drastic reduction in compressive strength
due to the delamination of the face sheets and/or core crushing damage [Flores-Johnson
and Li (2011a)]. Foam deformation history during dynamic loading may also have a
critical effect on the integrity of sandwich structures [Daniel and Cho (2011); Daniel et
al. (2013)]. Consequently, a good understanding of indentation and impact response of
foam cores is necessary in order to predict and assess their consequent damages.
Finite element (FE) modeling is a cost-effective tool used to predict behavior of
cellular materials under quasi-static indentation and low velocity. Some of the advantages
of FE simulations are the flexibility to model either a localized region through the
thickness or the entire structure and that geometric and material parameters can be easily
varied [Foo et al. (2006)].
Although there is a large amount of investigations concerning numerical simulations
of the quasi-static and dynamic compression of cellular foams, there is limited amount of
research papers dealing with quasi-static indentation and low velocity impact with
different indenter nose shapes. Some of these investigations include the numerical
simulation of quasi-static indentation by a wide rectangular cuboid into aluminum foams,
which have been studied by Hanssen et al. [2002a] using LS-DYNA. Rizov [2007]
carried out numerical simulation of low velocity impact on PVC foams by cylindrical and
hemi-spherical indenters using ABAQUS/Explicit. Low velocity impact numerical
simulation of polyurethane foams using spherical impactor has been performed by Du
Bois et al. [2004] using LS-DYNA. Lu et al. [2008] studied the quasi-static indentation
of aluminum foams by flat indenter using ABAQUS. Mills and Gilchrist [2000] studied
the quasi-static indentation of polyurethane foams by rectangular blocks and flat indenter
using ABAQUS. Mills and Gilchrist [2000] studied the quasi-static indentation of
polyurethane foams by rectangular blocks and flat indenter using ABAQUS. Xu et al.
[2014] studied the quasi-static indentation of aluminum foams by hemi-spherical and
cylindrical indenters using ABAQUS/Standard.
It is found that most of the indenters and impactors modeled in numerical simulation
of quasi-static indentation and low velocity impact of cellular materials are cylindrical,
rectangular cuboid and hemi-spherical. Therefore, it is necessary to study the nose shape
effects on the indentation and low velocity penetration of polymeric foams.
The present paper presents the results of a numerical study on quasi-static indentation
and low velocity impact on low density PMI Rohacell 51 WF foams using conical,
conical-truncated, spherical and flat indenters/impactors. In this investigation numerical
simulations were carried out using the explicit finite element code LS-DYNA [Hallquist
(2007)]. A 2D axisymmetric model was generated and implicit time integration was used
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for quasi-static indentation. A strain-rate dependent material model and r-adaptive
remeshing was used for low velocity impact simulations.
Available experimental results are also presented to validate the numerical
simulations carried out in this study. Quasi-static indentation and low velocity impact on
Rohacell 51 WF foam are carried out using a wide range of indenter nose shapes. It was
found that the nose shape severely affects the indentation force and the mechanisms
involved that contribute to this force during the penetration.
Good agreement between numerical simulations and experimental results was
observed. The results demonstrated the ability of the model to reproduce the mechanisms
of deformation of the penetration process in low density PMI foam. Low velocity impact
simulations were found to be in good agreement with experiments of Rohacell 51 WF
foam.
2. Experiments
Although experiments of quasi-static indentation and low velocity impact Rohacell 51
WF foam have been presented elsewhere [Flores-Johnson and Li (2010); (2011b)], a
summary of the experimental details and results are presented in this Section as the
numerical predictions in Section 3 are compared with the empirical data.
2.1. Material
The polymeric foam studied in this research is commercially available high performance
Rohacell 51 WF [EVONIK (2008)], which is a closed-cell rigid foam with PMI
(polymethacrylimide) matrix. Rohacell 51 WF has a density of 52 kg/m3. The mechanical
properties of Rohacell 51 WF foam have been reported by Li et al. [2000; 2006], Flores-
Johnson et al. [2008], Kuwabara et al. [2005] and Arezoo et al. [2011].
2.2. Quasi-static indentation tests
A series of quasi-static indentation tests were conducted for Rohacell 51 WF foam using
a range of steel indenters shown in Table 1 [Flores-Johnson and Li (2010)]. The indenters
were fixed to a standard 200 kN INSTRON servo-hydraulic testing machine and the load
was applied at a nominal strain rate of 8.3×10-4
s-1
. The indenters were pushed to 50 mm
maximum depth into the foam specimens. A 100×100×100 mm cube specimen was used.
Typical force-indentation curves for conical indenters #1, #2, #3, truncated indenter
#4, flat indenter #5 and hemi-spherical indenter #6 for Rohacell 51 WF foam are depicted
in Fig. 1. It was observed that the indentation force strongly depends on the indenter nose
shape. For all types of indenters, two distinct regimes were observed, i.e. the initial
immersing regime when the nose of the indenter gradually immerse into the foam and the
plateau-like regime starting from the moment when the indenter nose has completely
immersed into the foam [Flores-Johnson and Li (2010)].
Examinations of the specimens after indentation tests were performed to identify the
different deformation mechanisms involved during the indentation for each particular
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indenter. Figure 2 (Top pictures) shows the cross-sections of the indented specimens for
all types of indenters used in this study. Crushing, tearing and friction forces were clearly
identified.
Table 1. Indenter geometries [Flores-Johnson and Li (2010)].
Indenter # Type Nose Geometry D (mm) l (mm) β ( o )
1
Conical
20 36 74.5
2
20 10 45
3
20 6 31
4 Truncated
20 18 74.5
5 Flat
20 - 90
6 Spherical
20 - -
2.3. Low velocity impact tests
Low velocity impact tests were performed on Rohacell 51 WF foam. Indenters #1-6
(Table 1) were used for impact tests. Indenters with a mass of 0.6 kg were dropped from
4 m heights using a vertical guiding tube with a releasing mechanism [Flores-Johnson
and Li (2011b)]. A piezoelectric accelerometer was attached to the indenters. For impact
tests, same specimen dimensions as those used in quasi-static indentation were used.
Specimens were placed on a steel plate. The displacement and the deceleration of the
projectile during the impact were measured using a non-contacting optical displacement
measurement system (Zimmer 100F) and the attached accelerometer. Based on the
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accelerometer signals, the impact force can be determined by F=mâ where m is the mass
of the impactor and â is the deceleration measured by the accelerometer. The force-time
history from accelerometer signal and the displacement-time history from Zimmer 100F
are used to obtain the force-penetration curve. Typical force-penetration curves of Rohacell 51 WF foam with impactors #1-6 are
depicted in Fig. 3 [Flores-Johnson and Li (2011b)]. The penetration force was found to
be dependent on the impactor nose shape. Two distinct regimes were observed as those
observed in quasi-static indentation tests, i.e. the initial immersing regime when the nose
of the impactor gradually immerse into the foam and the plateau-like regime starting
from the moment when the impactor nose has completely immersed into the foam.
3. Numerical model
Fig. 1. Experimental [Flores-Johnson and Li (2010)] and numerical results of the indentation force for
Rohacell 51 WF.
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6
Numerical simulations of quasi-static indentation of Rohacell 51 WF foams were carried
out with LS-DYNA FE code. The problem was considered to be axisymmetric and four-
node 2D quadrilateral elements were used with the following element formulation:
Axisymmetric solid, symmetry about y-axis, volume weighted with stiffness-based
hourglass control. The material models used in this investigation are material model 63
(MAT_CRUSHABLE_FOAM) for quasi-static indentation and material 163
(MAT_MODIFIED_CRUSHABLE_FOAM) for low velocity impact [Hallquist (2007)].
The materials properties used are listed in Table 2. In material model 163, the stress-
volumetric strain curves at different strain rates are shown in Fig. 4a. The quasi-static
curve was taken from [Flores-Johnson (2008)] and the high strain rate curves were taken
from [Arezoo et al. (2013)]. Figure 4b shows the numerical results of uniaxial
compression of Rohacell 51 WF foam cube specimen at various strain rates.
3.1. Mesh sensitivity study
A mesh sensitivity study was carried out in order to evaluate the mesh density effect. A
Rohacell 51 WF foam specimen (cube with 100 mm edges) was modeled using the
following element sizes, 2 mm x 2mm (1250 elements), 1 mm x 1 mm (5000 elements),
0.5 mm x 0.5 mm (20000 elements) and 0.25 mm x 0.25 mm (80000 elements) (Fig. 5),
respectively. Indenter #6 (hemi-spherical nose) was used in order to apply multiaxial
loading. Contact between the indenter and the foam specimen was modeled using the
CONTACT_2D_AUTOMATIC_SURFACE_TO_SURFACE option. The indenter was
modeled as a rigid body.
Figure 5f shows a comparison between experimental results and simulations using
different mesh densities. It can be seen that numerical results are not sensitive to the
mesh density in the studied range. With the consideration of computation efficiency and
the characteristic cell size (l=300 µm for Rohacell 51 WF), 1 x 1 mm element size was
selected as the optimum element size-computation time for all FE simulations.
Fig. 2 Cross-sectioned indented specimens of Rohacell 51 WF (Top) [Flores-Johnson and Li (2010)] and
numerical simulations (Bottom): a) indenter #1; b) indenter #2; c) indenter #3; d) indenter #4; e) indenter #5
and f) indenter #6.
a)
d) e)
b) c)
f)
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Table 2 Material properties.
Material type Steel (Material 20) Rohacell 51 WF (Materials 63 and 163)
Density ρ (kg/m3) 7850 52
Young's modulus E (MPa) 210000 88
Poisson's ratio ν 0.3 0.01
Tensile stress cutoff (MPa) - 0.2
Damping coefficient - 0.2
Fig. 3 Experimental [Flores-Johnson and Li (2011b)] and numerical results of the impact force for Rohacell 51
WF.
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3.2. Numerical simulation of quasi-static indentation
Indenters #1-6 (Table 1) were modeled as depicted in Fig. 2 (Bottom images). Quasi-
static indentations were simulated at 3 mm/min using implicit time integration. Figure 1
shows the indentation simulations for all indenters at 40 mm of penetration. It should be
mentioned that numerical simulations stopped prematurely before the indentation reached
40 mm for indenter #6 (flat). This occurs because of the very high distortion of the
elements around the edge of the indenter. The elements which are close to the corner
become very slender. Eventually, the elements distort to such an extent that a relatively
small increment of deformation causes a node to pass through the opposite edge. This
element becomes singular. A fillet corner (concave corner) in the edges of the flat
indenter was used without a significant improvement. The problem may be avoided by
using very small incremental steps, which would be computationally expensive when
using implicit method. However, the calculations are considered to be adequate since the
trend of the simulation curves was found to be comparable with experimental results
(Fig. 1). Comparing the top and bottom of Figs. 2(a-f), a fairly good reproduction of the
indentation mechanisms of Rohacell 51 WF foams is observed. Crushed material (denser
mesh in black color) is observed beneath the indenter nose and around the indenter body
as shown in the bisected specimens in Fig. 2 (Top images). Similar observations have
been reported in the numerical simulation of quasi-static indentation in aluminum foams
[Lu et al. (2008)]. Figure 1 compares experimental results with numerical simulations and shows good
agreements for all indenters. When comparing the indentation mechanisms observed for
conical indenters in Figs. 2(a-c), it can be seen that for indenter #1, there is an
accumulation of crushed material around the indenter while for indenter #3 the
accumulation of the crushed material is beneath the indenter. This suggests that there
may be a critical value of the angle β where a transition between deformation
mechanisms occurs. In order to observe this transition, indentation with conical indenters
Fig. 4 a) Stress-volumetric strain curves for various strain rates [Flores-Johnson et al. (2008);
Arezoo et al. (2013)]; b) Numerical results of uniaxial compression at various strain rates.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Str
ess
(MP
a)
Volumetric strain
0.00083 s^-1
9.1 s^-1
2900 s^-1
Strain rate
a) b)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Str
ess
(MP
a)
Volumetric strain
0.0008 s^-1
0.5 s^-1
5 s^-1
50 s^-1
Strain rate
0.4
0.0
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(β=56.3º, 50.2º, 45º, 42º, 40.4º and 38.7º) were simulated (Fig. 6). It can be seen that,
rather than an abrupt transition, the transition of the deformation mechanism happens
gradually. The accumulation of crushed material beneath the indenter increases with the
decrease of the conical angle (Fig. 6).
3.3. Numerical simulation of low velocity impact test
Low velocity impact simulations were carried out at impact velocity of 7.93 m/s. In low
velocity impact simulations, the mesh became severely distorted leading to invalid results
and 2D r-adaptive remeshing for axisymmetric solid elements had to be used in order to
complete the simulation. In r-adaptive method, also referred to as moving grid adaptivity
or rezoning, the number of elements and nodes are fixed, while the nodal positions are
relocated to achieve optimal aspect ratios of the elements within a part at predefined time
intervals [Porcaro et al. (2006)]. When r-adaptivity method is used, a new mesh is
Fig. 5 a) Geometry for mesh sensitivity study; element sizes: b) 2 mm x 2 mm, c) 1 mm x 1 mm, d) 0.5 mm x
0.5 mm and e) 0.25 mm x 0.25 mm; f) Comparison between experiment [Flores-Johnson and Li (2010)] and
simulations for different mesh sizes.
Fig. 6 Numerical simulations of quasi-static indentation with: a) β =56.3º, b) β =50.2º, c) β =45º, d) β =42º,
e) β =40.4º, and f) β =38.7º.
a)
b)
c)
d)
e)
f)
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generated after each remeshing interval which is specified by the user. This can be seen
in Fig. 7 which shows a comparison between meshes for non-adaptive and r-adaptive
methods at 10, 20 and 30 mm of penetration using indenter #6.
This method has been successfully used for large deformations of an axisymmetric
simulation of self-piercing riveting process [Porcaro et al. (2006)] giving accurate
predictions, although the mesh deformation does not visually represent the real process.
Comparison between experiments and numerical simulations for low velocity impact for
all indenters are shown in Fig. 3. A good agreement was observed for indenters #1-3. For
indenters #4-6, a reasonable agreement is observed.
3.4. Discussion on performance of material 63and material 163
Material model 63 is applicable for isotropic foams in applications where cyclic loading
is not dominant [Hallquist (2007)]. For the calibration of model 63, only uniaxial
compressive data is required which make it a good option in comparison with other LS-
DYNA material models where hydrostatic compressive data is needed. Hanssen et al.
[2002b] have reported material model 63 as having a good performance and high
efficiency in computational time in numerical simulations of indentation test for low
density aluminum foams. Material model 63 transforms the stresses into the principal
stress space where the yield surface is defined [Hanssen et al. (2002a)]. The model agrees
with the experimental data on polystyrene foams reported by Shaw and Sata [1966],
which indicates that the compression yielding is governed by the principal compressive
stress. Li et al. [2000] found that Rohacell 51 WF follows the same behavior, which is
classified as the maximum principal stress criterion [Triantafillou et al. (1989)].
Comparison between numerical simulations and experimental results implies that
model 63 is suitable for modeling quasi-static indentation of low density polymeric
foams studied in this research. As shown in Fig. 8, predictions for low velocity impact
test were reasonably good for Rohacell 51 WF using material 163. Figure 8 also shows
that material 63 can be used for the prediction of low velocity impact in Rohacell 51 WF
Fig. 7 Penetration with indenter #6 using non-adaptive method: a) 10 mm, b) 20 mm, c) 30 mm and r-adaptive
method: d) 10 mm, e) 20 mm, f) 30 mm.
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50
Forc
e (k
N)
Penetration (mm)
Experimental results
Simulation: Material 63
Simulation: Material 163
when strain rate curves are not available; however, non-conservative results are obtained,
i.e., a larger penetration is predicted and this should be taken into account.
4. Conclusions
Quasi-static indentation and low velocity impact experiments are reported in this paper
on low density PMI foam (Rohacell 51 WF) for a wide range of indenter nose shapes. It
is found that the nose shape has a large influence on the indentation and impact
resistances. Two indentation regimes, i.e. an immersing regime and a plateau-like regime,
are observed in all indentation and impact tests.
Numerical simulations of quasi-static indentation and low velocity impact were
performed for Rohacell 51 WF using the finite element code LS-DYNA with material
model type 63 and model type 163, respectively. The r-adaptive remeshing method was
used for the low velocity impact simulations. The model agreed very well with
experimental quasi-static indentation data. Good agreement between numerical
simulations and low velocity impact experimental results is also observed.
Acknowledgments
This research was partially funded by the Australian Research Council (ARC) Centre of
Excellence for Design in Light Metals (CE0561574).
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