Forhelmetapplications Shearstrainratedependencyofexpandedpolystyrenefoam R I T

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IN

DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2019,

Shear strain rate

dependency of expanded

polystyrene foam

For helmet applications

JONATHAN BERGSTRÖM

CHRISTOFFER ÅHMAN

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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R ^OYAL I NSTITUTE OF T ^ECHNOLOGY

M

ASTER

’

S

T

HESIS

Shear strain rate dependency of

expanded polystyrene foam

For helmet applications

Authors:

Jonathan Bergström Christoffer Åhman

Supervisors:

Dr. Peter Halldin Prof. T.Christian Gasser

A thesis submitted in fulfillment of the requirements for the degree of Master in engineering

in the

Department of Solid Mechanics

May 27, 2019

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iii

Abstract

Shear properties of expanded polystyrene foam are investigated, with special emphasis on strain rate dependency. A method to measure simple shear at low and high strain rates is developed by designing a simple shear fixture. The fixture is installed in an Instron ElectroPuls E3000 and in a drop tower system to capture the range of strain rates 0.01 to 150 s⁻¹. Beyond this, the shear data is complemented with compression experiments executed in the Instron machine for strain rates up to 1 s⁻¹. These experiments are conducted on five different densities of expanded polystyrene foam.

The experimentally acquired shear and compression data is used for material mod- elling in the commercial finite element software LS-DYNA, with the material model denoted Modified honeycomb. To investigate the interaction between shear and compression loading in expanded polystyrene foam, another method is developed.

A fixture is designed to subject foam samples to a load case similar to that in oblique helmet impact experiments, and is used in the drop tower system. This method is used to calibrate the material model by comparing experimental and simulated data and adjusting material model parameters.

Further, the produced material model is validated through comparison of oblique impact simulations and laboratory experiments on a bicycle helmet. The compared variables are the translational acceleration, rotational acceleration and rotational velocity, which in the laboratory experiment are measured in a head form placed in the helmet.

It is concluded that expanded polystyrene foam stiffens with increased strain rate, in both shear and compression. Beyond strain rate dependency, a linearly stiffening with density is observed. It is also concluded that finite element simulations using the produced material model tend to over predict translational acceleration and under predict rotational acceleration and rotational velocity of the head in oblique helmet impact laboratory experiments.

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v

Acknowledgements

Thank you Dr. Peter Halldin for your guidance and investment in this project. It has been an honor to work with you. Your energy and knowledge have motivated and inspired us.

We are grateful for your help Peter Arfert, for the time spent building the fixtures and cutting all of our test samples. Your experience has helped us tremendously in the choice of design and technical solutions.

Thank you Marcus Arnesen and Daniel Lanner for your help along the way. Your support in hard times have made it easier to handle challenges.

It has been a pleasure to get to know you Dr. Madelen Fahlstedt. Thank you for supplying us with testing equipment and assistance.

Thank you Martin Öberg for you experienced guidance on testing and your gen- erosity in lending experimental equipment. We have always had a great time visit- ing you.

A special thanks to our supervisor Prof. Thomas Christian Gasser for your avail- ability and guidance in hard times. Your feedback has improved our work.

Thank you DYNAmore Nordic AB for the free LS-DYNA licence and great technical support.

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vii

List of Figures

1.1 Linear and oblique impact testing of bicycle helmets. . . 1

2.1 Impact rig with an angled anvil used for oblique impact testing of helmets. . . 3

2.2 Three types of oblique helmet impact tests performed by MIPS, representing rotation around three axes. . . 4

2.3 Non-commercial helmet used for research and development purposes by MIPS. . . 4

2.4 Experimentally measured and simulated parameters for a backward impact of the bicycle helmet. . . 5

2.5 Typical stress versus strain behavior of expanded polystyrene foam in compression. . . 6

2.6 Typical strain rate dependency of expanded polystyrene foam in compression. . . 7

2.7 Drop weight combined shear and compression test rig. . . 8

2.8 Shear test fixture used by Jungstedt and Arnesen . . . 8

2.9 Combined shear and compression test rig used by Mills and Gilchrist . 9 2.10 Input stress versus strain curves in MIPS’s parameter set for Fu Chang material model, for different strain rates. . . 10

3.1 Schematic illustration of the simple shear fixture which can be installed in the Instron machine or the drop tower. . . 14

3.2 Schematic illustration of the combined shear and compression fixture which can be installed in the drop tower. . . 14

3.3 FEM mesh used in simulations of the simple shear fixture installed in the drop tower. . . 15

3.4 Drop tower used for simple shear and combined shear and compression experiments on expanded polystyrene foam. . . 16

3.5 Instruments used in drop tower experiments. . . 17

3.6 Side view of the simple shear fixture. . . 18

3.7 Prepared sample for use in the simple shear fixture. . . 18

3.8 Gluing station for preparation of simple shear samples. . . 19

3.9 Proxxon Thermocut used to reduce load carrying area of shear samples of density 100 kgm⁻³. . . 20

3.10 Simple shear fixture installed in the Instron machine. . . 20

3.11 Simple shear fixture installed in the drop tower. . . 22

3.12 Compression test setup in the Instron machine. . . 23

3.13 Sample for use in the combined shear and compression fixture. . . 24

3.14 Combined shear and compression fixture installed in the drop tower. . 24

3.15 FEM mesh of the combined shear and compression experiments. . . . 26

3.16 FEM mesh used for simulations of a backward impact with the bicycle helmet. . . 26

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x

4.1 Velocity versus displacement of T-part in the simple shear fixture simulation. . . 27 4.2 Vertical force versus displacement of T-part in the simple shear fixture

simulation. . . 28 4.3 Averaged engineering shear stress versus strain at strain rate 0.01 s⁻¹. 30 4.4 Typical force versus time curve for a high rate shear experiment in the

drop tower. . . 31 4.5 Scale factor for strain rate sensitivity of EPS in shear. . . 31 4.6 Averaged engineering compressive stress versus strain at strain rate

0.01 s⁻¹. . . 32 4.7 Scale factor for strain rate sensitivity of EPS in compression. . . 32 4.8 Force versus time for the combined shear and compression experi-

ment on EPS samples of density 25 kgm⁻³. . . 33 4.9 Density dependency of EPS for compression and shear at strain rate

0.01 s⁻¹. . . 33 4.10 Averaged engineering compressive stress versus strain at strain rates

0.01, 0.1 and 1 s⁻¹and the Nagy extrapolated curve at rate 150 s⁻¹. . . 34 4.11 Scale factor for strain rate sensitivity of EPS in compression, experi-

mental and Nagy extrapolation data. . . 35 4.12 Compression stress versus strain for material model denoted Modi-

fied honeycomb. . . 36 4.13 Shear stress versus strain for material model denoted Modified hon-

eycomb. . . 37 4.14 Force versus time for the combined shear and compression experi-

ment and simulation. . . 37 4.15 Combined shear and compression experiment and simulation side by

side. . . 38 4.16 Motion of the head during a backward impact test of the bicycle helmet. 39 4.17 Motion of the head during a lateral impact test of the bicycle helmet. . 40 4.18 Motion of the head during a pitched impact test of the bicycle helmet. 41

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xi

List of Tables

2.1 Required parameter inputs for material model denoted Fu Chang. . . . 10 2.2 Material parameters defined for used cards in the material model Mod-

ified honeycomb. . . 11 2.3 Input parameters for Modified honeycomb. . . 12 3.1 Drop heights for shear tests in the drop tower for each density. . . 21 4.1 Mean and standard deviations of densities and dimensions of samples

used for simple shear experiments. . . 29 4.2 Mean and standard deviations of densities and dimensions of samples

used for compression experiments. . . 29 4.3 Mean and standard deviations of densities and dimensions of samples

used for the combined shear and compression experiments. . . 29 4.4 Drop heights for shear tests in the drop tower for each density, with

corresponding engineering strain rates. . . 30 4.5 Two calibrated sets of parameters representing the experimental data

acquired in this thesis, used as input in the LS-DYNA material model denoted Modified honeycomb. . . 36

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1

1 Introduction

Bicycle helmet test standards validate protective properties through linear impact experiments, according to test standards EN1078 and CPSC [15, 16]. It is proposed that the test standards are complemented with oblique impact tests to account for the helmet’s ability to absorb rotational energy in the brain [3]. Figure 1.1 shows linear and oblique impact tests. For both tests a helmet is dropped on an anvil, with the difference that the anvil is flat in linear impact tests and angled in oblique impact tests. During linear impact tests the translational acceleration of the head is measured, while oblique impact tests also include rotational acceleration and rotational velocity measurements of the head.

FIGURE1.1: Linear and oblique impact testing of bicycle helmets.

Most bicycle helmets are made of expanded polystyrene (EPS) foam. To predict the translational acceleration in an experiment, simulations in finite element method (FEM) software are made. Knowledge about EPS in uni-axial compression is sufficient for predicting translational acceleration in linear impact experiments.

For oblique impact tests knowledge about uni-axial compression is not sufficient.

Current FEM simulations fail to predict all three parameters simultaneously, a choice between predicting translational or rotational motion must be made. Beyond knowledge of uni-axial compression, shear properties are also required for oblique impact test simulations [6]. The knowledge of EPS shear properties is limited.

EPS is strain rate dependent in compression and potentially also in shear. Several test methods to investigate high shear strain rate behavior exists [8]. No test method allows to test strain rates up to 150 s⁻¹, without the involvement of compression or torsion. A new method is developed to examine simple shear behavior of EPS below strain rate 150 s⁻¹. The method includes a fixture which is suited for placement in a conventional tensile testing machine (Instron ElectroPuls E3000) and in a drop tower to examine strain rates up to 150 s⁻¹.

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2 Chapter 1. Introduction

1.1 Objective

The objective is to characterize material behavior of EPS with a special emphasis on shear properties, such as strain rate dependency. Experimental data is implemented in a FEM material model toward improving the predictability. It should predict the translational acceleration, rotational acceleration and rotational velocity of the head in an oblique impact experiment.

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3

2 Background

The company MIPS AB performs oblique impact experiments. Figure 2.1 shows an angled anvil test rig. The rig simulates oblique impacts by guiding a helmet with an instrumented Hybrid III head form to a 45 degree angled impact anvil. The head is equipped with nine accelerometers.

FIGURE2.1: Impact rig with an angled anvil used for oblique impact testing of helmets.

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4 Chapter 2. Background

Figure 2.2 shows three oblique impact test cases, which represent rotation around three different axes. The cases are specified as backward, lateral and pitched impact. The variables acquired from all types of oblique impact testing of helmets are translational acceleration, angular acceleration and angular velocity of the head.

FIGURE2.2: Three types of oblique helmet impact tests performed by MIPS, representing rotation around three axes.

Figure 2.3 shows a non-commercial bicycle helmet used for research and development purposes by MIPS. This helmet is used in oblique impact experiments and FEM simulations in the commercial FEM software LS-DYNA (LSTC). Figure 2.4A and 2.4B shows the translational acceleration and rotational velocity of the head, from an experiment and corresponding simulation for a backward impact test of the bicycle helmet. These graphs exemplify the problem with predicting all three variables simultaneously with a single FEM simulation.

FIGURE2.3: Non-commercial helmet used for research and development purposes by MIPS.

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Chapter 2. Background 5

(A) Translational acceleration of the head.

(B) Rotational velocity of the head.

FIGURE2.4: Experimentally measured and simulated parameters for a backward impact test of the bicycle helmet. The graphs exemplify the problem with predicting all three variables simultaneously with a

single FEM simulation.

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2.1 Expanded polystyrene foam

EPS is a closed cell foam with properties such as light weight, good thermal insula- tion, moisture resistance, durability, acoustic absorption, low thermal conductivity and energy absorbing [1, 4]. EPS is commonly used in helmet liner due to its high energy absorption capability and the cost to benefit ratio [9].

Figure 2.5 shows the typical compressive material response of EPS. It is divided into three regions. The first region is linear elastic and is controlled by cell edge bending, cell wall stretching and compression of gas trapped in cells. The second region is a wide collapse plateau of ductile failure, where strains are no longer recoverable. This happens when cell walls cannot withstand the acting bending moment. This causes plastic stretching and plastic hinges to occur, together with increased fluid pressure inside cells. In the third region the material reaches densification and stiffens [9].

FIGURE2.5: Typical stress versus strain behavior of EPS in compression. The three zones are highlighted: (I) linear elastic, (II) collapse

plateau and (III) densification [9].

EPS is isotropic and shows a clear strain rate dependency in compression load [13, 7, 1]. Figure 2.6 shows the typical strain rate dependency of EPS. The velocity 0.1 and 20 ms⁻¹ corresponds to an engineering strain rate of 2.7 and 533 s⁻¹, respectively.

The elastic modulus increases as well as the stress level of the plateau region, with an increasing strain rate.

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2.2. Dynamic shear testing methods 7

FIGURE2.6: Typical strain rate dependency of EPS in compression.

(B) is an enlargement of the area marked in (A). The velocity 0.1 and 20 ms⁻¹represents engineering strain rates of 2.7 and 533 s⁻¹, respec-

tively [1].

Whilst the experimental data is limited, it is shown that EPS has a stronger rate dependency when loaded in tension than in compression [1]. From dynamic drop tests combining compression and shear loading it is concluded that the impact angle tends to affect the level of compressive and shear yield stress [10]. Examination of helmet liner loaded biaxially shows that the shear response of EPS is a function of axial compression [6]. No experimental data of EPS in high rate simple shear is available in the open literature.

2.2 Dynamic shear testing methods

The yield behavior can be predicted from compression and tension data for small strains using theories as Tresca, Von Mises, Nadai-Tresca and Nadai-von Mises. For high strain regions this is not possible. Therefore shear experiments are needed to accurately predict mechanical behavior [8].

Several methods for high shear strain rate testing exist. Depending on what strain rate is aimed for, different testing machines are used. With torsional testing, strain rates up to 10⁴s⁻¹can be reached. Some testing machines used for torsional testing

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are high speed hydraulic torsional machines, impact torsional testing machines and torsional Kolsky (Split-Hopkinson) bar machines. Methods as double-notch shear testing and punch loading achieve rates between 10³-10⁴s⁻¹[8].

Figure 2.7 shows a drop weight combined shear and compression test rig. A is a frame, B is a falling weight, C is a tup with force measurement, D is an anvil, E is a specimen, F are stopping devices and α is an angle of inclination [8]. The test can reach strain rates up to 200 s⁻¹. A combined shear and compression load on the specimen is achieved by having an inclination along the loading axis up to 10 degrees.

FIGURE2.7: Drop weight combined shear and compression test rig [8].

Figure 2.8 shows a shear fixture installed in an Instron ElektroPuls E3000, used by Jungstedt and Arnesen. The shear fixture is restricted to actuator velocity of the Instron machine [5].

FIGURE2.8: Shear test fixture used by Jungstedt and Arnesen [5].

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2.3. Material models 9

Figure 2.9 shows an oblique impact test rig used by Mills and Gilchrist. A v-shaped aluminum block is dropped vertically to impact two foam samples. The test is executed from different drop heights and with different impact angles (15ô, 30ô and 45ô). The top of the foam samples are displaced vertically and the shear and normal forces are measured with a tri-axial load cell [12].

FIGURE2.9: Combined shear and compression test rig used by Mills and Gilchrist [12].

2.3 Material models

Several material models in LS-DYNA are available to model EPS. Depending on application some may be more suited than others. For oblique impact testing of helmets two important parameters are shear and strain rate behavior. The chosen material model is Modified honeycomb which is suitable to model different strain rates for shear and compression separately. Material model behavior is presented without numeric values of variables, since these are results.

For in house simulations at MIPS, the material model denoted Fu Chang is used.

The variable set used for the in house simulations is presented together with the material models behavior.

2.3.1 Fu Chang

LS-DYNA material model denoted Fu Chang log log interpolation, is used to model strain rate effects in low and medium density foams. The material model requires density ρ, Young’s modulus E, viscous coefficient µ to model damping effects, hys- teretic unloading factor H_{u f}, shape factor for unloading S_{u f} that increases energy dissipation and exponent for unloading n. At a certain tension cut-off stress σ_cofail- ure occurs and the behavior is perfectly plastic. Table 2.1 shows in house parameter set.

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TABLE 2.1: Required parameter inputs for material model denoted Fu Chang.

Density ρ 1·10⁻⁷ kg·mm⁻³

Young’s modulus E 1·10⁻³ GPa

Viscous coefficient µ 0.3 −

Hysteretic unloading factor H_{u f} 0.5 −

Shape unloading factor S_{u f} 5 −

Unloading exponent n 1 −

Tension cut-off stress σ_co 1·10¹¹ GPa

The material behavior is defined by engineering stress versus engineering strain curves. Compression is defined as positive strains and tension as negative strains.

Figure 2.10 shows the stress versus strain curves defined for engineering strain rates from quasi static to 10 s⁻¹ used in the in house parameter set at MIPS. The strain rates are evaluated in the first principal direction. Strain rate effects are accounted for in the defined span and are not extrapolated.

FIGURE2.10: Input stress versus strain curves in MIPS’s parameter set for Fu Chang material model, for different strain rates.

The constitutive equation for the model is,

σ(t) =σ^∗[E^N(t), ˙E^N(t), S(t)], (2.1) where σ^∗is the engineering stress, E is the engineering strain and S is a state variable [11].

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2.3. Material models 11

2.3.2 Modified honeycomb

LS-DYNA material model denoted Modified honeycomb, is made for use in aluminum honeycomb crushable foam materials with anisotropic behavior. Table 2.2 defines the four cards used in the material model that define behavior, there are in total seven cards and the used ones are number 1, 2, 3 and 7.

TABLE2.2: Material parameters defined for used cards in the material model Modified honeycomb.

Card # Defined parameters

Card 1 Fully compacted elastic modulus, density, Poisson’s ratio and material viscosity coefficient.

Card 2

Engineering stress σ_ij versus engineering strain e_ij curves in all six normal and shear directions, if element type is not 0 or 9 logarithmic strain is instead expected.

Card 3 Initial elastic modulus, E_ij, and initial shear modulus, G_ij, in the six directions.

Card 7 Scale factor λ_ij versus natural logarithm of the absolute value of deviatoric strain rate, ˙e_ij⁰, for all six directions.

The stresses are uncoupled and the updated stresses are defined as, σⁿ⁺¹

trial

ij =σ_ijⁿ+E_ij∆eij, (2.2)

when i= j and

σⁿ⁺¹

trial

ij =σ_ijⁿ+2G_ij∆eij, (2.3)

when i6= j. The trial stresses are used and for the next time step each component is checked to ensure it does not exceed the curves defined in Card 2,

|σⁿ⁺¹

trial

ij | <λ_ijσ_ij(e_ij). (2.4) If Equation 2.4 does not hold, ideal plasticity is used to return towards the load curve in the next time step [11]. The symbol, description and unit of all required parameters are shown in Table 2.3.

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TABLE2.3: Input parameters for Modified honeycomb.

Symbol Description Unit

RO Material density [kg·mm⁻³]

E Young’s modulus for compacted material [GPa]

PR Poisson’s ratio for compacted material [-]

SIGY Yield stress for fully compacted material [GPa]

VF Relative volume at full compaction [-]

MU Material viscosity [-]

BULK Bulk viscosity flag [-]

LCA Load curve ID for normal stress versus strain [-]

LCS Load curve ID for shear stress versus strain [-]

LCSR Load curve ID for strain rate effects [-]

ExxU⁽¹⁾ Elastic modulus in uncompressed configuration [GPa]

GxxU⁽²⁾ Shear modulus in uncompressed configuration [GPa]

LCSRx⁽³⁾ Load curve ID for strain rate effect (normal) [-]

LCSRxx⁽⁴⁾ Load curve ID for strain rate effect (shear) [-]

(1)One value is needed for each direction xx = [AA, BB, CC].

(2)One value is needed for each direction xx = [AB, BC, CA].

(3)One curve is needed for each direction x = [A, B, C].

(4)One curve is needed for each direction xx = [AB, BC, CA].

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13

3 Method

To characterize EPS behavior, both shear and compression tests are made. In total five densities are examined: 25, 40, 50, 75 and 100 kgm⁻³. To capture shear behavior of EPS a test fixture is designed to be placed in an Instron ElectroPuls E3000 or a drop tower. Figure 3.1 shows a schematic illustration of such a fixture.

The installation of the shear fixture in the Instron machine or a drop tower to examine strain rates between 0.01 and 0.1 s⁻¹or between 30 and 150 s⁻¹, respectively.

The compression tests are made in the Instron machine and are examined for strain rates of 0.01, 0.1 and 1 s⁻¹, further explanation in Section 3.2.3. In Section 3.2 more details about the drop tower and experimental setups are reported.

Experimental shear and compression data is implemented in LS-DYNA’s material model denoted Modified honeycomb. The calibration of the material model, is based on a built fixture combining shear and compression. Figure 3.2 shows a schematic illustration of the combined shear and compression fixture. The design of the device is inspired by the fixture proposed by Mills and Gilchrist, described in Section 2.2.

Section 3.5 contains further explanation on how to calibrate the material model.

The calibrated material model is used in oblique impact helmet simulations of the helmet shown in Figure 2.3. The translational acceleration, rotational acceleration and rotational velocity of the head obtained in the simulation are compared with experimental data.

3.1 Fixture design

Two test fixtures are designed, a simple shear fixture and a combined shear and compression fixture. A foam sample in the simple shear fixture is simulated in LS- DYNA, for the case when it is installed in the drop tower. Potential problems with oscillations of the fixture and the ability to keep the strain rate constant during a test.

The simple shear fixture is designed to be placed either in the drop tower or in the Instron machine. The fixture aims to acquire experimental data from foam samples of width, w^s, height, h^s, and depth, d^s. Figure 3.1 shows a schematic illustration of the simple shear fixture with sample sizes, where depth is in the direction perpendicular to the xy-plane. It is symmetric in order to avoid bending. Foam samples are attached between T-part and Sides via Glue plates. Sides are mounted to Bottom, which is mounted on the load cell. As T-part translates vertically with a velocity of v^sand shears the foam samples. As T-part is displaced∆^s, the load cell measures the force F^sin the y-direction.

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14 Chapter 3. Method

FIGURE3.1: Schematic illustration of the simple shear fixture which can be installed in the Instron machine or the drop tower.

The combined shear and compression fixture is designed to be placed in a drop tower. It aims to acquire experimental data from foam samples of width, w^sc, height, h^sc, and depth, d^sc. Figure 3.2 shows a schematic illustration of the fixture with the sample sizes, where depth is in the direction perpendicular to the xy-plane. Foam samples are attached to Plates. The load cell is mounted between Plate and Leg.

Legs are mounted in Base. As Impactor vertically displaces the foam samples upper surfaces,∆^sc, the load cell measures normal force, F_y^sc, and tangential force, F_x^sc.

FIGURE3.2: Schematic illustration of the combined shear and compression fixture which can be installed in the drop tower.

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3.1. Fixture design 15

3.1.1 Simple shear fixture simulation

The simple shear fixture installed in a drop tower is simulated in LS-DYNA. Two parameters are investigated: (i) ability to remain a constant strain rate through an entire test and (ii) influence of inertia. Figure 3.3 shows the geometry to represent the simple shear fixture and mesh used in simulations. In the simulation the parameters w^s, h^sand d^sare 15, 75 and 20 mm, respectively. The finite element formulation 1 in LS-DYNA is used. On top of T-part an Impactor is attached, using matching nodes and the bottom of the load cell is fixed. For Impactor, T-part, Glue plates, Sides, Bot- tom and Mounting a rigid material model is used. The load cell is modelled as steel using an elastic material model. For foam samples the material model Fu Chang is used.

FIGURE3.3: FEM mesh used in simulations of the simple shear fixture installed in the drop tower.

The mass of Impactor is analyzed to determine the reduction of velocity when an initial velocity corresponding to a free fall from 1.8 m is set to Impactor. A velocity reduction less than 10 % is desired over a 15 mm displacement of T-part. Analyzed masses are 5, 10, 20 and 30 kg.

The combined mass of Sides, Bottom and Mounting is analyzed to determine the influence of inertia when a constant prescribed velocity is set for Impactor, also representing a free fall from 1.8 m. The combined mass of the three parts is modified by changing their density, while dimensions remain the same. The analyzed combined masses are 0.01, 0.1, 1, 10 and 100 kg.

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3.2 Laboratory experiments

The compressive and shear properties of the EPSs are tested separately. The compressive tests are executed in the Instron machine and require only foam samples.

The low strain rate shear experiments are executed in the Instron machine and the high strain rate shear experiments are made in the drop tower, they require the simple shear fixture. Combined shear and compressive tests are executed in the drop tower and require the combined shear and compression fixture.

Additional equipment is needed for tests in the drop tower. Figure 3.4 shows the drop tower where the base and pillars are made out of aluminum. The drop tower’s pillars are vertically aligned by placing the drop tower on a base of concrete with re- inforcement bars, surrounded by a stabilizing mold of wood. To avoid oscillations a 5 mm thick rubber mat is placed between the base and drop tower bottom. Impactor height is adjusted with a wire connected to a winch.

FIGURE3.4: Drop tower used for simple shear and combined shear and compression experiments on EPS.

Figure 3.5 shows the instruments used for drop tower testing. Drop activator is a bolt that carries the mass of Impactor and Drop trigger is a rope. To initiate the test Drop activator is released by pulling it out from its equilibrium with Drop trigger.

Impactor can only move vertically. A MotionBLITZ EoSens mini2 Mikrotron high- speed camera is used to record the impact to ensure that the samples displacements are symmetric. High-speed cameras require high frequency light, it is provided by an Aputure Light Storm LS 1S. The recording starts when Camera trigger passes by Camera activator.

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3.2. Laboratory experiments 17

FIGURE3.5: Instruments used in drop tower experiments.

To avoid oscillations, fixtures are placed on a heavy base of steel and a 5 mm thick rubber mat. Force is measured with a 9067C Kistler load cell and acquired with a LabAmp Type 5165A Kistler acquisition system. To calibrate the load cell a 5995A Kistler one-channel hand held charge amplifier is used.

In the case of high strain rate shear experiments industrial dampers stops the impactor after the test. Also, a M3L/100 MEL laser is used to record the displacement.

3.2.1 Foam material

Experiments are made with EPS samples of five densities, ρ, namely 25, 40, 50, 75 and 100 kgm⁻³. EPS is purchased from two manufacturers. EPS of densities 25 and 40 kgm⁻³ are purchased from Jackon AB in sheets of thickness 13 and 38 mm. EPS of densities 50, 75 and 100 kgm⁻³ are purchased from EON Helmets in sheets of thickness 13 mm.

EPS’s density and thickness in a sheet tend to vary more than other foam materials such as expanded polypropylene and low density polyethylene [2]. Therefore, each sample is weighed using a scale with 0.01 g accuracy and the dimensions are measured using a caliper with 0.05 mm accuracy.

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3.2.2 Shear

Figure 3.6 shows the simple shear fixture. For tests of strain rates lower than 1 s⁻¹, it is installed in the Instron machine, whereas for tests of strain rates higher than 30 s⁻¹, it is installed in the drop tower.

FIGURE3.6: Side view of the simple shear fixture.

The foam samples are glued with Araldite 2015 onto Glue plates in batches of 20.

To avoid dis-symmetry two similar samples in the batch are used for a test. For density 25 and 40 kgm⁻³, samples with similar width, w^s, are run together. For density 50, 75 and 100 kgm⁻³samples with similar contact area, h^stimes w^s, are run together. Figure 3.7 shows a prepared sample where dimensions height h^s, width w^sand depth d^sare illustrated. It is important that the screw holes in the two Glue plates holding the same sample are aligned and that the sample is placed in the middle of both Glue plates.

FIGURE3.7: Prepared sample for use in the simple shear fixture.

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Figure 3.8 shows a developed gluing station, used to reduce the gluing process’s affect on test results. To align screw holes, steel rods steers Glue plates. To make the sample end up in the middle, nuts are used as spacers.

FIGURE3.8: Gluing station for preparation of simple shear samples.

Low rate protocol

Figure 3.10 shows the simple shear fixture installed in the Instron machine. It aims to acquire experimental data from foam samples of width, w^s, height, h^s, and depth, d^s. Samples of dimension 13x20x75 mm are used. An exception is for samples of density 100 kgm⁻³, which are modified because the glue is to weak to withstand the shearing load. Figure 3.9A shows a Proxxon Thermocut which is used to reduce the depth, ¯d^s, by ten percent of the density 100 kgm⁻³ samples. The filament is highlighted with a red line. Figure 3.9B shows the removed area from the samples of density 100 kgm⁻³.

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(A)

(B)

FIGURE3.9: (A) shows a Proxxon Thermocut used to reduce the load carrying area of shear samples of density 100 kgm⁻³. (B) highlights

the removed area.

Shear experiments of strain rates lower than 1 s⁻¹ are executed. The fixture is mounted on the lower plate with double sided tape. The actuator moves the upper plate at a set displacement rate. The vertical force F^s is measured in a load cell beneath the lower plate. The speed is set such that the engineering strain rates are 0.01 and 0.1 s⁻¹. For density 25 kgm⁻³three tests are made for each strain rate. For the densities 40, 50, 75 and 100 kgm⁻³five tests are made for each strain rate.

FIGURE3.10: Simple shear fixture installed in the Instron machine.

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From the tests of strain rate 0.01 s⁻¹ a stress versus strain curve is used as input in Card 2, described in Section 2.3.2. The force F^s extracted from the load cell is transformed to engineering stress as

τ= ^F

s

h^sd^s (3.1)

and the displacement∆^sis transformed to engineering strain as γ= ^∆

s

w^s. (3.2)

The average shear stress versus strain ¯τ_0.01(γ)for strain rate 0.01 s⁻¹is calculated as,

¯τ0.01(γ) = ¹ n

∑

n k=1

(τ_0.01,k(γ)), (3.3)

where n is the number of experiments executed. The average peak stress ¯τ_peak,mfor each strain rate m is calculated as,

¯τ_peak,m = ¹ n

∑

n k=1

max(τ_m,k(γ))_, _(3.4) where n is the number of experiments executed for strain rate m. The ratio between average peak stresses for different strain rates are calculated as,

λ^s_m = ^¯τ^peak,m

¯τ_peak,0.01, (3.5)

where λ^s_m represents the shear scale factors used as input for λ_ij when i 6= j, described in Section 2.3.2.

High rate protocol

Shear experiments at strain rates higher than 30 s⁻¹ are executed with the simple shear fixture installed in the drop tower. It aims to acquire experimental data from foam samples of dimension 13x20x75 mm. The mass of Impactor is 33 kg and it is dropped onto the fixture from different heights. Table 3.1 presents the drop heights used for each density. For each density and drop height H_low and H_high, five tests are executed.

TABLE 3.1: Drop heights for shear tests in the drop tower for each density.

Density ρ kgm⁻³ 25 40 50 75 100

Low drop height H_low mm 31 46 60 30 60

High drop height H_high mm 200 160 200 200 200

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Figure 3.11 shows the fixture placement on the drop tower’s bottom and between the two drop tower pillars.

FIGURE3.11: Simple shear fixture installed in the drop tower.

The stress is calculated using Equation 3.1 and the average peak stress ¯τ_peak,m for each strain rate is calculated as,

¯τ_peak,m= ¹ n

∑

n k=1

max(τ_m,k(t)), (3.6)

where m is strain rate and n is the number of experiments executed for strain rate m.

The shear scale factors are calculated using Equation 3.5.

Extracted data from the laser is position of Laser trigger versus time. The data is numerically derived to obtain the velocity of Laser trigger. The velocity at the impact moment v_imp is used to calculate the strain rate as

˙e= ^v^imp

w^s , (3.7)

where w^sis the width of the foam sample.

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3.2.3 Compression

Figure 3.12 shows the Instron machine with a compression sample. The upper plate is displaced such that the specimen is compressed up to 90 percent strain. The displacement rate is set such that the strain rate is 0.01, 0.1 and 1 s⁻¹. The load is measured in the load cell beneath the lower plate.

FIGURE3.12: Compression test setup in the Instron machine.

The sample dimensions are width, w^c, and depth, d^c, of 30 mm and height, h^c, of 13 mm. The height is in the compression direction. The width and depth spans the area A^c, perpendicular to the compressive direction. For all densities two tests are made for the strain rate 0.01 s⁻¹and four tests are made for the strain rates 0.1 and 1 s⁻¹. From the tests with strain rate 0.01 s⁻¹, a stress versus strain curve is obtained. It is used as input in Card 2, described in Section 2.3.2. The force F^cextracted from the load cell is transformed to engineering stress as

σ= ^F

c

A^c (3.8)

and the displacement∆^c is transformed to engineering strain as e= ^∆

c

h^c. (3.9)

The average stress versus strain ¯σ_m(e)for each strain rate m is calculated as,

¯σm(e) = ¹ n

∑

n k=1

σ_m,k(e), (3.10)

where n is the number of experiments executed for strain rate m. The ratio between the averaged stress versus strain curves at each strain for different strain rates are calculated as,

λ^c_m(e) = ^¯σ^m(e)

¯σ0.01(e)^. ^(3.11)

The mean value of the ratios is calculated as,

λ^c_mean,m=mean(λ^c_m(e)), (3.12) where λ^c_mean,m represents the compression scale factors used as input for λ_ij when i= j, described in Section 2.3.2.

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3.2.4 Combined shear and compression

Figure 3.14 shows the combined shear and compression fixture. It aims to acquire experimental data from foam samples with dimensions w^sc = 50 mm, d^sc = 50 mm and h^sc= 38 mm. Figure 3.13 shows the dimensions of the samples and the grid. The grid on the sample allows capturing the deformation with the high speed camera and comparison with the calibration simulations, described in Section 3.5. The combined shear and compression experiments are made on foam samples of density 25 kgm⁻³. The mass of Impactor is 8 kg and it is dropped from a height of 46 cm.

FIGURE3.13: Sample for use in the combined shear and compression fixture.

Figure 3.14 shows the position of the fixture on the drop tower’s base, between the two pillars. A 30 degree angled impactor is dropped on the foam samples. On top of the foam samples double sided tape is attached to only displace the foam samples vertically.

FIGURE3.14: Combined shear and compression fixture installed in the drop tower.

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3.3. Density dependency 25

3.3 Density dependency

The density dependencies of EPS in shear and compression are investigated. The density dependence in shear is illustrated by the ratio between peak stresses at strain rate 0.01 s⁻¹calculated as,

λ^s_d= ^¯τ^peak,d

¯τpeak,25

, (3.13)

where d is density and the index 25 denotes samples of density 25 kgm⁻³. However, for compression the density dependency is investigated over the whole strain spec- trum and not only at the peak values. The ratio between the averaged stress versus strain curves at strain rate 0.01 s⁻¹for all densities are calculated as,

λ^c_d(e) = ^¯σ^d(e)

¯σ25(e) ^(3.14)

and mean values of the curves are calculated as,

λ^c_mean,d =mean(λ^c_d(e)). (3.15) The ratios λ^s_dand λ^c_mean,d are used to investigate the density dependency.

3.4 Compression data extrapolation

Compression tests are only executed up to strain rate 1 s⁻¹. For higher strain rates the data is complemented with Nagy’s extrapolation theory. The measured data from compression experiments is extended into higher strain rates by extrapolation according to,

σ(e) =σ₀(e) ^˙e

˙e₀

n(e)

, (3.16)

where the exponent n(e)is defined as,

n(e) =a+be. (3.17)

In Equation 3.16, σ₀(e)and ˙e₀are the reference stress and corresponding strain rate.

The extrapolation works within the limits of strain rates 10⁻³< ˙e < 10³[14].

3.5 Calibration

Given the experimental data for densities 25 and 75 kgm⁻³material models are produced using Modified honeycomb. Figure 3.15 shows the mesh used to simulate the combined shear and compression experiment in LS-DYNA. The material model for density 25 kgm⁻³ is calibrated, such that the combined shear and compression simulation matches the normal and tangential forces of the experiments. Figure 3.15 shows the finite element mesh and where the foam samples are fixed. The v-shaped part is representing the impactor which is rigid and only allowed to move in the vertical direction with a mass of 8 kg. The initial velocity of the impactor is set to correspond to a free fall from 46 cm. The forces F_N and F_T are measured in the bottom of one foam sample. The factor used to scale the shear curve in the material model for density 25 kgm⁻³in order to gain coherence between simulation and experiment, is used to scale the shear curve for material model of density 75 kgm⁻³.

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FIGURE3.15: FEM mesh of the combined shear and compression experiments.

3.6 Validation

Figure 3.16 shows the mesh of the FEM model used to simulate a backward impact test, of the bicycle helmet shown in Figure 2.3. The head used in simulations is a model of a Hybrid III head form, developed by DYNAmore Nordic AB. The head and helmet translates vertically at a velocity of magnitude 6.2 ms⁻¹until it impacts a rigid surface, representing an angled impact anvil. During impact the helmet is deformed. The velocity direction of both the head and helmet are changed and they start rotating. Translational acceleration, rotational acceleration and rotational velocity are compared when the EPS in the helmet is modelled with the Fu Chang and Modified honeycomb models, described in Section 2.3. To simulate lateral and pitched impact tests the LS-DYNA model used for the backward impact test is ro- tated.

FIGURE3.16: FEM mesh used for simulations of a backward impact with the bicycle helmet.

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27

4 Results

The investigations of masses in the simple shear fixture simulation are presented followed by the results from low and high strain rate shear, compression and combined shear and compression experiments. Thereafter the density dependency investigations are presented. Later the extrapolation of compression strain rate behavior is presented followed by the input parameters for the calibrated material models.

Lastly results from the three types of oblique impact helmet validation tests are presented.

4.1 Simple shear fixture simulation

Figure 4.1 shows the velocity of T-part versus displacement in the simulation, for different masses of Impactor. The initial velocity of Impactor is set to 5.94 ms⁻¹, representing a free fall from 1.8 m. A Impactor mass of 30 kg is needed to achieve a velocity reduction of less than 10 percent.

FIGURE 4.1: Velocity versus displacement of T-part in the simple shear fixture simulation. The initial velocity is 5.94 ms⁻¹, representing a free fall from 1.8 m. The different curves represent different

masses of Impactor.

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28 Chapter 4. Results

Figure 4.2 shows the vertical force versus displacement of T-part in the simulation, for different combined mass of Sides, Bottom and Mounting. In the simulation a constant prescribed velocity of 5.94 ms⁻¹is set for Impactor, representing a free fall from 1.8 m. A mass of approximately 1 kg is acceptable.

FIGURE4.2: Vertical force versus displacement of T-part in the simple shear fixture simulation. Impactor has a constant prescribed velocity of 5.94 ms⁻¹, representing a free fall from 1.8 m. The different curves represent different combined masses of Sides, Bottom and Mounting.

4.2 Laboratory experiments

The data from the low and high strain rate shear, compression and combined shear and compression experiments are presented.

4.2.1 Foam sample measurements

Table 4.1 presents the mean values and standard deviations of densities and dimensions of the simple shear samples.

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TABLE 4.1: Mean and standard deviations of densities and dimensions of samples used for simple shear experiments.

Reference density ρ kgm⁻³ 25 40 50 75 100

Mean density ¯ρ^s kgm⁻³ 25.08 41.50 52.54 77.80 100.39 Standard deviation of density σ_ρ^s kgm⁻³ 1.28 0.90 2.34 2.50 3.21

Mean width w¯^s mm 12.89 12.58 19.91 20.14 20.10

Standard deviation of width σ_w^s mm 0.11 0.06 0.08 0.06 0.08

Mean depth d¯^s mm 20.15 20.27 13.60 13.25 11.45

Standard deviation of depth σ_d^s mm 0.12 0.07 0.47 0.80 2.47

Mean height ¯h^s mm 74.97 75.09 74.87 75.10 75.07

Standard deviation of height σ_h^s mm 0.09 0.05 0.12 0.12 0.14 Table 4.2 presents the mean values and standard deviations of densities and dimensions of the compression samples.

TABLE 4.2: Mean and standard deviations of densities and dimensions of samples used for compression experiments.

Reference density ρ kgm⁻³ 25 40 50 75 100

Mean density ¯ρ^s kgm⁻³ 24.29 41.00 52.48 79.84 100.35 Standard deviation of density σ_ρ^s kgm⁻³ 0.91 0.61 1.37 2.30 3.61

Mean width w¯^s mm 30.15 30.15 30.11 30.15 30.21

Standard deviation of width σ_w^s mm 0.05 0.02 0.06 0.07 0.06

Mean depth d¯^s mm 30.12 30.13 30.05 30.17 30.22

Standard deviation of depth σ_d^s mm 0.12 0.04 0.09 0.03 0.04

Mean height ¯h^s mm 12.79 12.52 13.50 13.97 14.04

Standard deviation of height σ_h^s mm 0.03 0.04 0.24 0.45 0.51 Table 4.3 presents the mean values and standard deviations of densities and dimensions of the combined shear and compression samples.

TABLE 4.3: Mean and standard deviations of densities and dimensions of samples used for the combined shear and compression ex-

periments.

Reference density ρ kgm⁻³ 25

Mean density ¯ρ^s kgm⁻³ 24.68

Standard deviation of density σ_ρ^s kgm⁻³ 0.34

Mean width w¯^s mm 50.07

Standard deviation of width σw^s mm 0.16

Mean depth d¯^s mm 50.10

Standard deviation of depth σ_d^s mm 0.07

Mean height ¯h^s mm 37.57

Standard deviation of height σ_h^s mm 0.01

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4.2.2 Shear

Figure 4.3 shows the averaged engineering shear stress ¯τ_0.01versus engineering shear strain γ for each density at strain rate 0.01 s⁻¹.

FIGURE4.3: Averaged engineering shear stress versus strain at strain rate 0.01 s⁻¹. The experiments are done with the simple shear fixture in the Instron machine. The different curves represent different

densities of EPS.

Table 4.4 presents engineering strain rates corresponding to the low and high drop heights, for each density.

TABLE 4.4: Drop heights for shear tests in the drop tower for each density, with corresponding engineering strain rates.

Density ρ kgm⁻³ 25 40 50 75 100

Low tests Drop height H_low mm 31 46 60 30 60 Strain rate ˙e_low s⁻¹ 54 69 73 34 68 High tests Drop height H_high mm 200 160 200 200 200

Strain rate ˙e_high s⁻¹ 143 125 140 134 135 Figure 4.4 shows the typical force versus time curve for a high shear strain rate experiment. At (1) the impactor is released by pulling the sprint out, causing small vibrations in the tower. At (2) the impactor hits the T-part. At (3) the peak force is reached. At (4) the impactor hits the dampers. The peak force is used to calculate

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the shear scale factors as they are transformed into stresses and divided by the peak stress of rate 0.01 s⁻¹.

FIGURE4.4: Typical force versus time curve for a high rate shear experiment in the drop tower. This test is for density 25 kgm⁻³. Arrows

indicate characteristic periods in the test cycle.

Figure 4.5 shows the shear scale factors λ^s_mversus the engineering strain rate for all densities, where m is the strain rate.

FIGURE4.5: Scale factor for strain rate sensitivity of EPS in shear.

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4.2.3 Compression

Figure 4.6 shows the averaged engineering compressive stress ¯σ_0.01versus engineer- ing compressive strain e for each density at strain rate 0.01 s⁻¹.

FIGURE4.6: Averaged engineering compressive stress versus strain at strain rate 0.01 s⁻¹. The experiments are done in the Instron ma-

chine. The different curves represent different densities of EPS.

Figure 4.7 shows the compression scale factors λ^c_mversus the engineering strain rate for all densities, where m is strain rate.

FIGURE4.7: Scale factor for strain rate sensitivity of EPS in compression.

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4.3. Density dependency 33

4.2.4 Combined shear and compression

Figure 4.8 shows force versus time for the combined shear and compression experiment.

FIGURE4.8: Force versus time for the combined shear and compression experiment on EPS samples of density 25 kgm⁻³.

4.3 Density dependency

Figure 4.9 shows the density dependency scale factors of EPS in compression, λ^c_mean,d, and shear, λ^s_d, versus density. The graph is complemented with linear least square fits of the curves.

FIGURE4.9: Density dependency of EPS for compression and shear at strain rate 0.01 s⁻¹.

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4.4 Compression data extrapolation

Figure 4.10 shows the averaged engineering compressive stress σ versus engineer- ing strain e for density 25 kgm⁻³ together with the Nagy extrapolated curve. The strain rates are 0.01, 0.1 and 1 s⁻¹ for the experimental curves and 150 s⁻¹ for the extrapolated curve. The stress level shown in Figure 4.10A is between 0.24 and 0.42 MPa with a scale factor between the highest and lowest curve of 1.33.

(A)

(B)

FIGURE4.10: Averaged engineering compressive stress versus strain of EPS at strain rates 0.01, 0.1 and 1 s⁻¹ and the Nagy extrapolated curve at rate 150 s⁻¹. (B) is an enlargement of the area marked in (A).

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4.5. Calibration 35

Figure 4.11 shows the scale factors λ^c_m versus engineering strain rate. Both experimental and Nagy extrapolation data are shown.

FIGURE4.11: Scale factor for strain rate sensitivity of EPS in compression, experimental and Nagy extrapolation data.

4.5 Calibration

Table 4.5 presents two calibrated sets of parameters for material model Modified honeycomb. Figure 4.12 shows the compression engineering stress versus engineering strain curves used as input in LS-DYNA. Figure 4.13 shows the shear engineering stress versus engineering strain curves used as input in LS-DYNA. The curves in Figure 4.3 are scaled with a factor 0.3 until the peak, followed by constant stress to avoid numerical issues.

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TABLE4.5: Two calibrated sets of parameters representing the experimental data acquired in this thesis, used as input in the LS-DYNA

material model denoted Modified honeycomb.

Symbol Unit Value Value

RO [kg·mm⁻³] 25·10⁻⁹ 75·10⁻⁹ E [GPa] 26.3·10⁻³ 9.1·10⁻³

PR [-] 0.01 0.01

SIGY [GPa] 0.1 0.1

VF [-] 0.1 0.1

MU [-] 0.01 0.01

BULK [-] 0.0 0.0

LCA [-] 251⁽¹⁾ 751⁽¹⁾

LCS [-] 252⁽²⁾ 752⁽²⁾

LCSR [-] -1⁽³⁾ -1⁽³⁾

ExxU [GPa] 26.3·10⁻³ 16.4·10⁻³ GxxU [GPa] 2.6·10⁻³ 11.8·10⁻³ LCSRx [-] 253⁽⁴⁾ 753⁽⁴⁾ LCSRxx [-] 254⁽⁵⁾ 754⁽⁵⁾

(₁)Load curves from Figure 4.12.

(2)Load curves from Figure 4.13.

(3)The value -1 flags use of individual strain rate effects in each direction.

(₄) Scale factor curve for compression from Figure 4.7, 253 and 753 are curves for density 25 and 75 kgm⁻³respectively.

(5)Scale factor curve for shear from Figure 4.5, 254 and 754 are curves for density 25 and 75 kgm⁻³respectively.

FIGURE 4.12: Engineering stress versus engineering strain in compression for use in material model denoted Modified honeycomb.

The curves represent densities of 25 and 75 kgm⁻³.

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4.5. Calibration 37

FIGURE4.13: Engineering stress versus engineering strain in shear for use in material model denoted Modified honeycomb. The curves

represent densities of 25 and 75 kgm⁻³.

Figure 4.14 shows the simulation of the combined shear and compression experiment and the experimental data, of EPS samples with density 25 kgm⁻³. Impactor weights 8 kg and is dropped from 46 cm. The data beyond approximately 20 ms of the simulation is of no relevance to this thesis.

FIGURE4.14: Force versus time for the combined shear and compression experiment and simulation with samples of density 25 kgm⁻³.

Impactor weights 8 kg and is dropped from 46 cm.

Forhelmetapplications Shearstrainratedependencyofexpandedpolystyrenefoam R I T

Shear strain rate

dependency of expanded

polystyrene foam

For helmet applications

JONATHAN BERGSTRÖM

CHRISTOFFER ÅHMAN

R OYAL I NSTITUTE OF T ECHNOLOGY

M

’

T

Shear strain rate dependency of

expanded polystyrene foam

For helmet applications

Abstract

Acknowledgements

Contents

List of Figures

List of Tables

1 Introduction

1.1 Objective

2 Background

2.1 Expanded polystyrene foam

2.2 Dynamic shear testing methods

2.3 Material models

3 Method

3.1 Fixture design

3.2 Laboratory experiments

∑

∑

∑

∑

3.3 Density dependency

3.4 Compression data extrapolation

3.5 Calibration

3.6 Validation

4 Results

4.1 Simple shear fixture simulation

4.2 Laboratory experiments

4.3 Density dependency

4.4 Compression data extrapolation

4.5 Calibration

R ^OYAL I NSTITUTE OF T ^ECHNOLOGY