A smoothed particle hydrodynamics method for virtual drop tests of polycarbonate suitcases
P. Druzhinin1, D. Vandepitte1
1 KU Leuven, Department Mechanical Engineering
Celestijnenlaan 300 B, B-3001, Heverlee, Belgium
e-mail: [email protected]
Abstract Impact is a critical load case for many consumer goods. The ability to withstand such load is also critical
for the design of suitcases. The present work deals with the numerical simulation of a drop test of a
suitcase filled with a sandbag (bag with plastic spheres) in the ABAQUS/Explicit environment. In a
nonlinear dynamic problem as the drop test, proper attention should be paid to the realistic application
of load at impact. A Smoothed Particle Hydrodynamics (SPH) method together with the Mohr-
Coulomb material model is used here to simulate granular behaviour of the plastic spheres. The
influence of the size of the SPH particles on the resulting stress and strain fields’ smoothness has been
investigated. In the final step the calculation with the real suitcase was conducted using the SPH particles.
A good correlation between the simulation and experiment data is obtained.
1 Introduction
For lightweight suitcases the impact of a loaded suitcase on a hard surface may bring about damage to the
case, either to the shell or to the wheels or handles, and also to the luggage that is contained within the
suitcase. The ability to withstand such a load is critical for the design of successful products like suitcases
developed by the Samsonite Europe NV company. The traditional approach to improve the product’s quality
consists of extensive experimental testing in a range of load cases. The load cases imply different values on
drop weight, on suitcase attitude at the time of impact and also on the point of impact. The manufacturer
uses sandbags as a filler of the suitcases to represent the luggage. For each test a number of bags are put
inside the suitcase. Each bag contains a large number of 10mm plastic spheres. These bags represent the
mass and solidity of the typical contents of a suitcase in a realistic way.
Virtual testing through finite element (FE) modelling is a typical solution which reduces the duration of the
development phase and the cost of the product. However, impact is a highly nonlinear dynamic process, and
its simulation is quite delicate. In order to correctly simulate the dynamic behaviour of the case, every aspect
of the experiment, especially the application of load at impact, should be modelled with a high degree of
realism. Consequently, discontinuous behaviour of the plastic spheres should be taken into account by the
use of special computational method like Discrete Element Method, Element-Free Galerkin or Smoothed
Particle Hydrodynamics. Some of them have already found application in commercial FE codes.
Results verification and validation is indeed an important step of a correct analysis. Types of data for
validation depend on experimental conditions. In the terms of the suitcase drop test this can be strain or
acceleration data from gauges or a video of drop and impact recorded on a high-speed camera. In the latter
case the FE model can be verified only by matching deformational patterns of the real suitcase and FE model
captured at the same time points.
The current work focuses on the implementation of the Smoothed Particle Hydrodynamics method (SPH)
for the sandbag modelling, as well as its applicability, and sensitivity in the terms of the drop test simulation.
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2 Samsonite drop test procedure
A conventional drop test assumes dropping a specifically oriented test object from a certain height on a rigid
floor. According to a procedure developed at the Samsonite Europe NV company a series of drops should
be done for a suitcase to test it on each side, some representative wheels and corners for several times. The
test consists of two rotating plates on which the suitcase is placed before the test is initiated (see Figure 1).
These plates rotate around fixed parallel axes and can be locked in the horizontal position, so then to be
released after the suitcase is set in a proper way. Depending on a type of the drop test (on 1 wheel or 4
wheels etc.) the standard drop height can vary from 600 to 900mm, but the table can be lifted up to 1200mm
and higher.
Figure 1: Scheme of the drop test.
For the imitation of baggage the company uses a number of sandbags. Normally they fill the entire volume
of the case to represent maximum loading of the suitcase and also maximum kinetic energy at impact.
Depending on the size of the suitcase, the mass of the sandbag varies from 18 to 23 kg.
The height, width, and depth of the suitcases used in the tests are 690 × 460 × 290 mm.
A high-speed camera located on the floor records the drop and impact of the suitcase. The recorded video
is later used to analyse deformed shape of the case’s elements.
3 Finite element approach
3.1 Impact problem and a choice of a solver
The analysis of a drop test is usually complex and computationally demanding because impact is a highly
nonlinear transient process. It involves dynamics, large deformations, contact interactions, nonlinear
material behaviour, etc. A typical issue for this type of analysis is also a need for a verification of the
calculation results by a real drop test. Compatibility of the test and calculation depends on such factors like
test conditions (drop height, orientation of the object, loading conditions), and FE model parameters (mesh
60
0 m
m
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discretisation, damping, material properties, apposition and nature of load application) which determine all
together its dynamic behaviour. Thus, complexity of such a multifactorial analysis requires the use of a very
efficient computational tool. Different types of dynamic problems, including shock loading and high
velocity impact can be effectively solved by explicit dynamics solvers. There is a number of FE softwares
which implement explicit solutions. LS-DYNA and ABAQUS are commercial codes which are used to
simulate drop tests [1-3]. For the present calculations the ABAQUS/Explicit is used. It is a comprehensive
solver for dynamic problems also suited for drop test simulations.
3.2 FE model
The suitcase consists of several parts, of which two shells and four wheel assemblies are the primary
components for the impact test. The wheel assemblies consist of wheels and brackets, which are used to
connect the assemblies to the shells. Materials are polycarbonate (PC) for the shells, polypropylene (PP) for
the wheels, and polyamide (PA) for the brackets. CAD models of the parts are processed to avoid appearance
of unimportant features on the meshing step. Small geometrical details which do not affect the structural
response are removed to simplify the meshing procedure. S4R (with reduced integration and automatic
hourglass control) shell elements from ABAQUS standard library [4] are used to mesh both shells (see
Figure 2a). The thickness of the shells is set as 2.1mm. For the parts of the wheel assemblies we implemented
C3D4 and C3D8R – tetrahedrons and bricks – solid elements (see Figure 2b). A rigid impact surface is used
as a representation of the floor.
Figure 2: (a) General view of the suitcase FE model and (b) the wheel assembly. An interaction between the model and the floor, and between movable parts of the suitcase is defined by means of the automated general contact with frictionless tangential behaviour. The wheel assemblies are “glued” to the shells in contact areas.
For the purposes of the analysis mechanical behaviour of both materials are taken to be elastic.
This assumption is found to be valid by the observation that no damage is seen after the standard drop
tests. Elastic properties used for the modelling and the parts which they are assigned to are presented in
the Table 1.
Material Young’s modulus
E, MPa Poisson’s ratio Density, kg/m3 Parts
PC 2350 0.37 1200 Shells
PP 1400 0.42 950 Wheels
PA 2800 0.39 1120 Brackets
Table 1: Mechanical properties of the materials.
(b)(a)
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3.3 Methods of load simulation
The sandbag’s mass is the major load in the analysis. The bag itself is also an important structural part of
the model. Specific behaviour of its content makes it impossible to apply common modelling approaches
like the use of Lagrangian solid elements. An implementation of a special method is needed to simulate the
presence of the sandbag and its influence on the dynamic characteristics of the suitcase at impact.
3.3.1 Mass distribution
It is good practice to verify behaviour of an FE model before starting time-consuming high-performance
computations. A series of approximate analysis runs are performed to check contacts and materials
behaviour, and sufficiency density of the mesh discretisation. Due to the fact that these computations are
auxiliary they are preferably completed in a relatively short time. The preferred procedure of load
application would be one that does not increase the number of elements in the model. The most
straightforward procedure of load application would be the distribution of the sandbag mass over the
structural components of the suitcase. The sandbag mass is then modelled as an additional non-structural
mass which is evenly distributed over the shell elements.
However, in spite the simplicity of this approach, its application may lead to unrealistic results. It would
lead to a change of model’s natural frequencies as only the mass of the shell is affected without any changes
to the shell stiffness.
3.3.2 A smoothed Particle Hydrodynamics method for sandbag modelling
In order to reach the highest degree of realism in the simulation, the specific behaviour of a sandbag’s
content should be taken into account. Unlike distributed non-structural mass approach, in this case the
sandbag should be modelled directly by means of some type of finite elements. A major problem with
common Lagrangian methods is that they cause computational errors when the mesh undergoes severe
distortions caused by large deformations. The fact that in some drop tests relatively large deformations are
observed of the suitcase shells and sandbag makes Lagrangian solid elements unfeasible for this simulation.
Fortunately many other comprehensive computational methods have been developed in recent decades to
solve this problem. In the current analysis the sandbag was simulated with the use of a Smoothed Particle
Hydrodynamics method.
The Smoothed Particle Hydrodynamics (SPH) [5] is a Lagrangian mesh-free method. It is based on the idea
of representation of continuous or granular media as a set of discrete particles governed by the appropriate
constitutive model. Basic parameters which are assigned to each particle are its position, velocity, mass,
internal energy, and spatial distance (in terms of ABAQUS modelling procedure is can be also imagined as
a size or diameter of a particle). A kernel function W (see Figure 3) is used to smooth the particle’s properties
over that distance [6].
Figure 3: Kernel function W [6].
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Since the particles are not attached to any grid they do not suffer from mesh distortion. Thus, unlike the
common Lagrangian approach based on a structured mesh, the SPH method can be effectively implemented
to solve problems of metal forming, impact, crack growth and structural disintegration. Further development
of the SPH technology also allows to extend it on geo-mechanical problems. Research has demonstrated
[7-11] that SPH elements provide an adequate representation of soils and granular materials exposed to large
deformations caused by different types of loads.
The method is used in most of explicit solvers. ABAQUS allows users to create domains of the particles
with a predefined diameter. In this way, the internal volume of the suitcase was filled with the SPH elements
by means of ABAQUS preprocessing tools. The cut view of the model is shown in the Figure 4.
Figure 4: Cut view of the model containing the SPH particles (orange spheres).
3.4 Mohr-Coulomb material model for the sandbag
Since the sandbag contains small plastic spheres, which are modelled as discrete particles, it is reasonable
to model their behaviour with a constitutive model of ideal granular material.
One of the most common and simple constitutive models which is being successfully implemented in
different geomechanical applications is the Mohr-Coulomb (MC) model. This model predicts failure
phenomena in brittle materials which are subjected to a multi-axial stress state. It is particularly useful to
simulate frictional materials like soil, sand or gravel. The MC constitutive model describes an elastic-
perfectly plastic behaviour of material.
The MC model expresses that the resistance of a material in shear increases linearly with the magnitude of
compressive stress, as shown by the equation [4] and Figure 5:
.tan c (1)
The principal material parameters which define the failure criteria are the cohesion c – stress, corresponding
to zero plastic strains, the internal friction angle, , and the dilatancy parameter, ψ [12]. The latter parameter
is used to determine the non-associative flow rule. It is needed to model a realistic irreversible change in
volume due to shear. The MC model also supports linear behaviour within the elastic range, so Young’s
modulus, E and Poisson’s ratio, ν should be specified as well.
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Figure 5: the Mohr-Coulomb model sets a limit to shear stress.
The MC model can be used with different computational methods, including the SPH technology [7,10,11].
In the current work the properties of gravel are used as a first approximation to define parameters of the MC
model for the sandbag. The values of the friction angle for many types of gravel and loose sand are normally
close to 30° [12,13]. The cohesion is set as 0, since the plastic spheres were considered as ideal granular
material. The density is calculated on the condition that total mass of the SPH particles should be equal
to 18 kg.
The final values of the parameters are collected in the Table 2.
Young’s
modulus E,
MPa
Poisson’s
ratio
Density,
kg/m3
Friction
angle , ° Dilatancy ψ, °
Cohesion c,
MPa
80 0 0.3 310 30 5
Table 2: Mohr-Coulomb material properties for the sandbag.
3.5 Loading conditions
Different load cases should be defined according to the experimental procedure. Yet it is unnecessary to
reproduce all of the load cases. Only worst case conditions are considered to be the most dangerous and
representative. In this work we consider the impacts on the wheels and corners as the worst cases. These
parts of the suitcase or places of their attachment to the shell are the most vulnerable to getting severely
deformed. The Figure 6 depicts the examples of such drops.
The FE models are rotated to match the orientation of the suitcase in the drops tests which are recorded in
the video images. The initial velocity applied to every node of the models is consistent with the 600mm drop
height and it equals 3.41m/s.
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Figure 6: Images of (a) the suitcase when dropped on its wheels and (b) on the top corners.
4 Investigation of the SPH’s dimension effect
4.1 Problem
An important parameter in the model set-up is the size of the SPH particles. The number of particles in the
sand domain depends on their diameter. The smaller is the diameter of the particles, the higher is the number
of particles which fits in the virtual sandbag. But the large number would increase the number of degrees of
freedom and the calculation time to unrealistic numbers. On the other hand the low number of particles
would cause excessive local deformation of the shell. Thus, the optimal size of the particles have to be
determined considering acceptable accuracy and computational efficiency of the solution.
4.2 FE model
In order to investigate the size effect a simple special purpose FE model is created. It consists of square PC
plate of 300mm edge length and a cylindrical volume of the sand with 150mm diameter falling on the plate.
The plate is meshed by means of S4R shell elements with 1.3mm thickness.
In the analysis a ratio N between the shell element size and the diameter of the SPH particles is the most
important characteristic parameter. A number of models with the constant shell element size (2.5mm) and
different values of the SPH diameters were created. The final N values were following: 0.05, 0.07, 0.1,
0.163, 0.25, and 0.5. While keeping constant the volume and mass of the sand we obtained different numbers
of the SPH particles.
The plate is clamped by the edges. Only gravitational load is applied in this analysis. Examples of described
models are shown in the Figure 7.
(b)(a)
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Figure 7: Examples of the models with (a) 0.1 (2.5/25) and (b) 0.5 (2.5/5) values of the ratio N.
4.3 Comparison of the results
The analysis of the results shows that a change of the particle diameter does not have any influence the total
kinetic energy which is a very important parameter in impact analysis. It remains constant for all models.
At the same time the particle size has a formidable impact on the stress and strain distributions. Two
representative examples of von Mises stress distributions in the plate taken at the same time points are
depicted in the Figure 8. The Figure 8a corresponds to the model with N = 0.1, and the Figure 8b shows the
result for the calculation with N = 0.5. The stress distribution in the second field is more uniform then in the
first one.
Figure 8: Von Mises stresses distribution in the models with (a) 0.1 and (b) 0.5 values of N.
Thus, the stress and strain fields’ smoothness plays the major role in the SPH calculations. For a
numerical comparison we took peak values of von Mises stresses at the same time points. The graph
depicted in the Figure 9 shows the maximum von Mises stress value against the parameter N.
(b) (a)
(b)(a)
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Figure 9: Dependence of von Mises stress in the plate from the parameter N.
Data show that the difference between the last two points is 3.7%. In the present work this value is negligible.
In this way, the following conclusion can be made: the value 0.25 (the diameter of the SPH particles is four
times larger than the shell element size) can be considered as the lowest allowable limit of N.
5 Calculation results and discussion
Both FE models – with distributed mass and SPH particles – were used in calculations. Dropping conditions
on the wheels and top corners are considered to be the most representative load cases. Since the models
should be verified by high-speed video recordings of drop tests, then the most informative results in this
analysis are displacement fields in the models. They are depicted in the Figure 10 and Figure 11 in
comparison with frames from the corresponding drop tests recordings. The points of time 0.01s and 0.015s
respectively were chosen so that the shape changes of the real suitcase would be the most pronounced.
Characteristic deformations of the test suitcase’s shell in the Figure 10c and Figure 11c are outlined in red
for better comparison. Red color on the fields corresponds to maximum displacements.
Visual analysis shows a particularly good correlation between deformed shapes of the model with the SPH
particles (see Figure 10b and Figure 11b) and the real suitcase (see Figure 10c and Figure 11c). At the same
time the result obtained from the model with the distributed mass (see Figure 10a and Figure 11a) does not
prove to be comparable with the experiment data. Zones of the maximum deformations in this case do not
correspond to the ones outlined in the Figure 10c and Figure 11c.
Realistic representation of granular media, the good correlation between the results and experiment data,
simplicity of modelling, and computational efficiency make the SPH method an efficient and reliable
solution for the sandbag simulation. Another method which implements the distributed sandbag’s mass can
be used only for checking the FE model’s consistency and adjusting parameters which are not related to
loading.
40
60
80
100
120
140
160
0 0.1 0.2 0.3 0.4 0.5
Von Mises
stress, MPa
N
STRUCTURAL DYNAMICS: METHODS AND CASE STUDIES 3223
Figure 10: Comparison of deformation patterns of the suitcases with (a) distributed mass and (b) the SPH particles with (c) the real one dropping on the wheels
(the colors represent magnitudes of displacements).
Figure 11: Comparison of deformation patterns of the suitcases with (a) distributed mass and (b) the SPH particles with (c) the real one dropping on the top corners (the colors represent magnitudes of
displacements).
6 Concluding remarks
This paper presents a virtual impact analysis of suitcase which is dropped on a hard floor.
A finite element model of the suitcase is developed in ABAQUS/Explicit. Capabilities of the software allow
to use the model in various assessments simply adjusting material models, contact properties, loading
conditions, and etc.
Simulation results are compared to physical experiments in drop testing. Deformation patterns obtained in
simulations with the use of the SPH method, show a good correlation with the patterns captured in real drop
(b)
(b) (a) (c)
(a) (c)
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tests. The method proves to be effective and reliable approach for the further simulations of sandbag’s
behaviour under impact loading.
The current work presents a qualitative estimation of the results obtained with the use of the SPH method.
A quantitative analysis will be conducted in the next step of the research. It will be done by performing a
new series of drop tests with the use of strain of acceleration gauges.
Acknowledgments
The authors would like to thank Pauline Koslowski and Laurens Van Audenaerde from
Samsonite Europe NV, Belgium for the cooperation and data provision. The funding agency IWT-
Vlaanderen is gratefully acknowledged for their financial support in the programme SIM-NANOFORCE.
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