A smoothed particle hydrodynamics method for virtual drop...

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

3216 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

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)

STRUCTURAL DYNAMICS: METHODS AND CASE STUDIES 3217

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].


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.


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.


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)


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)


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


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)


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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