The design of a formula student front impact attenuator

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1 The design of a formula student front impact attenuator J.M.J. Schormans MT Supervisors: dr. ir. Varvara Kouznetsova Eindhoven University of Technology Department of Mechanical Engineering Computational and Experimental Mechanics Section Mechanics of Materials

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3 Abstract This research is concerned with the development of a front impact attenuator for the University Racing Eindhoven formula student team. Abaqus by Dassault Systèmes S.A. was used to do finite element calculations in order to investigate the deceleration properties and deformations of several front impact attenuator geometries which are made out of Rohacell foam. The results indicate whether or not a foam specimen wil fracture. If a foam specimen will fracture, the precise location of the fracture and global deceleration information can be determined. If a foam specimen does not fracture, deceleration data and changes in foam geometry can be determined accurately. This research shows that buckling in a foam impact attenuator can be prevented and if an anti intrusion plate is used to place the impact attenuator on, it should be placed vertical. This research recommends the implementation of a damage model to increase the precision of the simulation.

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5 Contents 1 Introduction University Racing Eindhoven Problem description Design space Structure of the report The crushable foam material model Elasticity Yield surface Flow potential Hardening Numerical simulations Uniaxial compression test Simple collision The model Rohacell 51WF Rohacell 110IG /2009 crash test Mesh size Yield point prediction New design Design requirements Foam shapes

6 4.2.1 Geometry Anti intrusion plate Conclusion and recommendations Material Geometry Anti intrusion plate Foam geometry Research method Conclusion Recommendations Department of mechanical engineering 2

7 Chapter 1 Introduction 1.1 University Racing Eindhoven University Racing Eindhoven (URE) is a student organized team which has been competing in the Formula Student competition for 5 years. Every year they have built a new small single seated car (figure 1.1) and have competed in races in England, Germany and Italy. During these races the car will be assessed on dynamic and static events. Dynamic events include: acceleration, skid-pad, autocross and endurance. The static events consist of: design cost and presentation judging, technical and safety scrutineering, a tilt, brake and noise test. During the years there has been a lot of technical development within the team and progress is made every year. For the 2009/2010 season the team is focusing on two aspects. First, to improve the previous car and thus fine tuning the design of a new car with a internal combustion engine. Second, the 2008/2009 car will be converted to an electric powered car. Figure 1.1: The 2008/2009 URE formula student car 3

8 1.2 Problem description Part of the safety scrutineering is a Front Impact Attenuator report, in which the team has to prove that the car is equipped with a front impact attenuator which is able to reduce the forces of impact keeping in mind the driver s safety. The impact attenuator has to comply with rules specified by the formula student organization [04]. The most important part of the front impact attenuator is made out of Rohacell foam. This foam absorbs and dissipates the kinetic energy of the car during a crash. Previous designs of the front impact attenuators made by URE are based on very simple and minimalistic calculations concerning the dimensions of the foam which resulted in a too soft crash. The rules specify that the peak deceleration and the average deceleration must me below 40g and 20g respectively. Previous years designs have met these specifications with almost half of the values required. This indicated the design could be optimized with respect to the weight of the design. This leads to the main goals of this thesis: To model the foam material and to make a simulation of the crash test as specified by formula student rules and thus predict its success. To minimize material usage in the front impact attenuator and thus saving weight which is a very important factor in motorsport. To supply the next generation designers of University Racing Eindhoven with a reliable method of modeling to make future front impact attenuator designs less time consuming. To compare numerical simulations with the impact data gathered during the 2008/2009 season crash test. Department of mechanical engineering 4

9 1.2.1 Design space University Racing Eindhoven s resources and time are limited. Therefore it has been chosen to use the mold of the 2008/2009 nose cone again, to make a carbon shell in which the impact attenuator can be placed. This means the space in which the attenuator can be positioned is limited and pre described. The design space is prescribed by the outside dimensions of the carbon shell which can be seen in figure 1.2 and is situated at the front of the car, in front of the anti intrusion plate (indicated in red). This anti intrusion plate is a 1,5 mm steel plate (or equivalent) which is intended to prevent any objects from reaching the driver s feet during a crash. This plate is situated at the red line and covers the cross section of the car. The foam impact attenuator has to be placed inside the carbon shell and on the steel plate. Figure 1.2: Design space Structure of the report This report is organized as follows. In chapter two, the material model which has been implemented will be explained. Following in chapter three, the model will be calibrated using the experiments performed in the paper of Li et al. [02]. Furthermore, the model will be used to do simulations to compare a crash test done in 2008/2009 with numerical simulations. The fourth chapter will deal with the design of the 2009/2010 and designs for the years after that. Finally chapter five will deal with the conclusions and recommendations which can be done based on this research. Department of mechanical engineering 5

10 Chapter 2 The crushable foam material model This chapter will deal with the crushable foam model which is used to model the foam parts of the impact attenuator in Abaqus. 2.1 Elasticity The model can only be used in combination with a simple linear isotropic elastic material model. In Abaqus the required parameters are the Elasticity modulus and the Poisson s ratio. 2.2 Yield surface The yield surface for the plastic part of the behavior is a von Mises circle in the deviatoric plane and an ellipse in the meridional plane. Within the crushable foam model there are two different types of hardening possible: volumetric and isotropic hardening. Within the isotropic hardening model the yield ellipse is centered at the origin and as it evolves it retains its original height to width ratio (Fig. 2.1). This model is based on the model of Deshpande and Fleck [01]. With the volumetric hardening model the point on the yield ellipse which represents hydrostatic loading is fixed and the evolution of the yield surface is driven by the compressive plastic strain (Fig. 2.2). The hardening which occurs under tension is negligible. The parameters in the figures are explained in table

11 Figure 2.1: Isotropic hardening in the meridional plane (p,q) [05] Figure 2.2: Volumetric hardening in the meridional plane (p,q) [05] Department of mechanical engineering 7

12 Table 2.1: Volumetric model parameters p = 1 3 trace σ Pressure stress q = S : S Mises stress 2 3 S = σ + pi Deviatoric stress p c p 0 p t = pc pt Yield stress in hydrostatic compression (p c is always positive) 2 Center of the yield ellipse on the p axis Strength of the material in hydrostatic tension A = pc+pt 2 Size of the horizontal axis B = α A Size of the vertical axis α = A B Shape factor of the yield ellipse As this research is concerned with foams with different yield stresses in tension and compression and the isotropic hardening model assumes the initial yield stress in tension and compression are of equal magnitude, the volumetric model has been chosen. The initial yield surface ellipse is defined by equation (2.1). F = q 2 + α 2 (p p 0 ) 2 B = 0 (2.1) The yield ellipse develops along the line with slope α, which is constant. To define the parameter α there are three material properties required: σ 0 c The initial yield stress in uniaxial compression p 0 c The initial yield stress in hydrostatic compression p t The yield strength in hydrostatic tension Together they are used to calculate α: α = 3k (3kt + k)(3 k) (2.2) With: k = σ0 c p 0 c k t = p t p 0 c (2.3) Department of mechanical engineering 8

13 2.3 Flow potential The volumetric hardening model uses the following relation to define the plastic strain rate: ε pl = ε pl G σ With G being the flow potential, chosen to be: G = (2.4) q p2 (2.5) ε pl is the equivalent plastic strain rate which is defined as: ε pl = σ : εpl G (2.6) The equivalent plastic strain rate is related to the rate of axial plastic strain in uniaxial compression: ε pl = 2 3 εpl axial (2.7) The shape of this flow potential is depicted in Fig Hardening The model assumes that the yield stress in tension remains the same throughout the whole deformation. The yield ellipse intersects the p axis at p t and p c being the yield stress in tension and compression respectively. Contrary to the yield stress under tension, the yield stress under compression evolves as a result of compaction (or dilation) of the material. The change in the yield surface can be expressed as a function of the the size of the yield surface on the hydrostatic stress axis, p c +p t, as a function of the value of volumetric compacting plastic strain. With p t constant, this relation is determined by equation (2.8): p c (ε pl vol ) = σ c(ε pl axial )[σ c(ε pt axial )( α 2 9 ) + pt p t + σc(εpl axial ) 3 3 ] (2.8) in combination with user provided results of an uniaxial compression test, σ c (ε pl axial ), along with the fact that εpl axial = εpl vol in uniaxial compression for the volumetric hardening model. Department of mechanical engineering 9

14 Chapter 3 Numerical simulations This chapter deals with the calibration of the model on a simple compression test performed in the paper of Li et al. [02]. Also in this chapter there will be a comparison between the numerical simulations and a real live crash test performed in Januari 2009 at TV Rheinland TNO. The goal of this chapter is to check the material model implementation in Abaqus and to verify the model against experiments. 3.1 Uniaxial compression test One of the tests performed in the paper of Li et al. [02] is an uniaxial compression test of Rohacell-51WF foam. To replicate this test numerically, a quasi static model has been set up using the Abaqus implicit solver. A rectangular block was modeled with a displacement prescribed in such a way that the final nominal strain would be -75%. An overview of the dimensions of the foam specimen are given in table 3.1 Table 3.1: Foam specimen dimensions length of the foam block 200 [mm] height of the foam block 100 [mm] width of the foam block 200 [mm] In order to fill in the needed hardening data, the results of the uniaxial compression test from the paper of Li et al. [02] were used. To ensure the data was suitable for Abaqus, i.e. providing small enough steps for the iteration process, the hardening curve was fitted with the use of two 10

15 functions. The test was carried out using a single hex element as the stresses and strains are uniform throughout the material for this test. The rest of the model parameters are given in table 3.2. Table 3.2: Material parameters of Rohacell 51WF foam Density 110 [kg/m 3 ] k t [-] k 0.1 [-] Poisson s ratio 0 [-] Young s modulus 22 [MPa] Initial yield stress 0.8 [MPa] A Poisson s ratio of 0 has been taken in accordance with the research of Flores-Johnson et al. [03]. Strain rate effects are assumed to be absent in accordance to the paper of Li et al. [02]. The results of this test can be seen in figure 3.1. Figure 3.1: Uniaxial compression test on Rohacell 51WF The results indicate that the model has been properly implemented as the simulation results match the experimental curve. The material properties used in the uniaxial compression test can now be used to model a simple collision. Department of mechanical engineering 11

16 3.2 Simple collision The model The next step in modeling a crash test is to model a simple collision. The simple collision was modeled using a rectangular block of foam material, two analytical rigid surfaces and a point mass equivalent to the mass required for the formula student regulatory crash test [04]. One of the analytical rigid surfaces represents the wall and is fixed in all degrees of freedom. The second analytical surface represents the anti-intrusion plate, which is at the front of the car. This anti-intrusion plate is fixed in all but the direction perpendicular to the wall. The rectangular foam block is attached to it. The point mass is fixed to the anti intrusion plate which implies it is also fixed in all but the same direction as the anti intrusion plate. The model can be seen in figure 3.2. Figure 3.2: Abaqus simple collision model To execute the finite element calculations, Abaqus explicit was used instead of Abaqus implicit as there are dynamical effects which need to be taken into account. The prescribed nominal strain has been replaced by a velocity of 7 [ m s ] and the point mass was set to be 300 [kg] Rohacell 51WF For the simple collision model, the same Rohacell 51WF parameters as for the uniaxial compression test were used as specified in table 3.2. The deceleration values and dissipation energy were measured to see if there were any irregularities that would indicate an error in the model. The results are plotted in figure 3.3. Department of mechanical engineering 12

17 Figure 3.3: Deceleration (left) and plastic dissipation energy data (right) from the Rohacell 51WF simple collision simulation The results indicate that there was a constant deceleration, which is in accordance to the constant frontal area of the foam. At about 4.5 ms, all the energy is dissipated and the deceleration stops. This indicates that all the model has been implemented properly Rohacell 110IG Model parameters As the 2008/2009 impact attenuator was made of Rohacell 110IG foam, a useful model of Rohacell 110IG foam would be required to simulate the 2008/2009 collision tests. Both Rohacell 110IG and Rohacell 51WF are polymethacrylimide foams. As their number indicates, the density of both foams is different. For the Rohacell 110IG foam a Young s modulus of 160 MPa and a Poisson s ratio of 0 were taken. The hardening curve for Rohacell 110IG (fig: 3.4) was constructed using the compression strength and the maximal compression given in the material data sheet provided by the manufacturer [06]. Department of mechanical engineering 13

18 Design of a formula student impact attenuator Figure 3.4: The hardening curve for Rohacell 110IG However this leaves the compression yield stress ratio k (eq. 2.3) and the hydrostatic yield stress ratio kt (eq. 2.3) to be determined. The hydrostatic yield stress ratio is estimated to be 0.1 as indicated in the Abaqus user manual [05]. However, the compression yield stress ratio is not even approximately known. The simple collision model was used to investigate the influence of the compression yield stress ratio on the deceleration and the plastic dissipation energy, the most important parameters for a crash test simulation. The simulation has been repeated several times using different values for the compression yield stress ratio. The results of the simulations are depicted in figure 3.5 showing the plastic dissipation energy and deceleration. Figure 3.5: Plastic dissipation energy (left) and deceleration (right) at various compression yield stress ratios, other model parameters are given in table: 3.3 As the results indicate there is no difference in energy dissipation and little difference between the deceleration for the different ratios. It has been chosen to model the 2008/2009 crash test with a compression yield strength ratio of because this was the value calculated using the paper of Li [02] for the Rohacell 51WF foam. The material parameters used to model Rohacall 110IG foam are summarized in table 3.3 Department of mechanical engineering 14

19 Table 3.3: Material parameters of Rohacell 110IG foam Density 110 [kg/m 3 ] k t [-] k 0.1 [-] Poisson s ratio 0 [-] Young s modulus 160 [MPa] Initial yield stress 2 [MPa] To examine the difference between the Rohacell 51WF foam and the Rohacell 110WF foam. The dimensions of the simple collision test were changed in such a way that the energy of the collisions would be the same for Rohacell 51WF and for Rohacell 110IG i.e. altering the frontal area of the foam block and thus the mass of the foam block. The dimensions of the Rohacell 110IG block are l b h: [m]. It was chosen to alter the mass by changing the surface and not the length to keep the deceleration distance the same. The results of the plastic dissipation energy and the deceleration values are shown in figure 3.6. Figure 3.6: Comparison between the plastic dissipation energy and deceleration data of the Rohacell 110IG and Rohacell 51WF collisions As expected, the deceleration of the Rohacell 110IG foam is shorter but of greater magnitude. There is a difference in plastic dissipation energy, which can be related to the difference in frontal surfaces. The Rohacell 51WF block s surface is larger so a larger surface will be able to use the elastic area in the stress-strain diagram to dissipate energy. Department of mechanical engineering 15

20 /2009 crash test In preparation for the races in 2009, two crash tests were carried out. There was the URE05 nose cone design, and there was a foam block used. This paragraph wil deal with the crash of the foam specimen. The old URE05 nose cone consisted out of a carbon fiber shell with four foam pads laminated in the sides. However, this nose cone has not been modeled because there is a large role of carbon fiber concerning the evolution of the shape of the cone during the crash and a model of the carbon fiber shell has not been included in the present work. The foam block was modeled using similar analytical rigid planes as used in the model of the simple collision. The Abaqus model for the crash test and the actual test set up are shown in figure 3.7, the dimensions of the model are listed in appendix A. Figure 3.7: The real foam specimen test (left) and the Abaqus model (right) Mesh size In order to provide accurate predictions, the mesh size of the model is very important. The mesh size has to be small enough to accurately model deformation of the material while at the same time, a smaller mesh size will lead to large computation times. For this research, an investigation has been done to find an ideal mesh size to model the foam specimen. Deceleration graphs obtained with models with different mesh sizes have been compared. The comparison of these graphs can be seen in figure 3.8. It can be seen from figure 3.8 that the differences between the graphs decrease along with the mesh size. The mesh sizes of 0.02 and almost do not differ anymore. Taking this comparison into account along with the fact that simulations with an even smaller mesh size will take more time, a mesh size of has been taken to model the foam specimen. Department of mechanical engineering 16

21 Figure 3.8: Deceleration graphs for different mesh sizes Yield point prediction Although the model does not include a damage model, the model is stil valid up to the point of fracture. The point of the fracture can be indicated in the model. To determine the point at which the material cracks, a closer look is taken at the yield surface shown in figure 2.2. The assumption was made that there is negligible hardening in tension. This implies that equation 2.1 can be used in combination with the pressure and von Mises stress to determine if an element is inside the initial yield surface (F < 0), or outside the initial yield surface (F > 0) which implies yielding of the material. As the foam is very brittle in tension, the points at which the pressure stress is positive (left of the q axis) are of most importance in determining yielding which leads to cracks in the foam. Therefore the elements which show a negative pressure for the first time are examined carefully. When done so there are four elements on the top of the block which fit the criterium. The four elements are displayed in figure 3.9. Figure 3.9: Elements having negative pressure (tensile state) in the 2009 model For each of the four elements, F is calculated as a function of the time. The results are shown in figure Department of mechanical engineering 17

22 Figure 3.10: F as a function of time These graphs indicate that the value of F becomes positive at 57 ms. This is the step at which the four elements displayed in figure 3.9 were identified. It is therefore assumed that the foam will start to fracture at the place of the four elements. This is consistent with the footage of the actual crash test done in 2009, which showed that a crack developed through the middle from the front to the back as can be seen in figure Figure 3.11: Crack initiation (right to left) during crash (top view) Department of mechanical engineering 18

23 Finally, the deceleration values of the simulation can be compared with the deceleration values measured during the crash test done in The results are depicted in figure Figure 3.12: Comparison between the deceleration values calculated with Abaqus and the 2009 crash test The two graphs approximately match in the beginning. At the end however, the two graphs start to differ. This is because the Abaqus model is not equipped with a failure model and thus fails to model the crack propagation. It can be concluded that the model is only valid up to the point where the yield strain in tension is reached locally. Department of mechanical engineering 19

24 Chapter 4 New design 4.1 Design requirements The design of the formula student front impact attenuator needs to comply with certain rules. There are rules which specify the dimension of the front impact attenuator and there are the rules specifying the function of the impact attenuator [04]. The most important rules concerning the research are B which states that the attenuator must occupy a space of 200 mm x 200 mm x 100 mm (l x w x h) in front of the car and rule B which states that the attenuator must be able so stop a mass of 300 kg traveling at a velocity of 7 m/s with a maximal deceleration of 40g and a maximal average deceleration of 20g. 4.2 Foam shapes Geometry The first test was done with the minimal dimensions specified by the rules. Again two analytical rigid planes were used. This time however, the analytical rigid plane resembling the front of the car was put at a ten degree angle of vertical to properly model the current design of the front of the car. A point mass was added to resemble the required 300 kg weight. The collision was modeled using the 7 m/s velocity. The model used is depicted on the left in figure

25 Figure 4.1: Numerical model having the minimal dimensions as specified by the rules, initial shape on the left and deformed on the right The problem with the minimal dimensions is that due to the anti intrusion plate which is at an angle, the front impact attenuator is subject to buckling, which is shown on the right in figure 4.1. As Rohacell 110IG foam is very brittle in tension and when the foam specimen is subject to buckling, it would be certain the foam would break and probable that the foam would move beneath the front of the car making it unusable for energy absorption. A solution to this problem would be to change the angles between the anti intrusion plate and the top and bottom of the foam block which can be seen on the left of figure 4.2. Figure 4.2: Foam specimen with adjusted angles, initial shape on the left and deformed on the right Although in the new situation the specimen will yield, the foam block will not buckle. Instead, the foam will collapse in the direction of the movement, shown on the right of figure 4.2. This ensures that the foam will not move out under the anti intrusion plate during the crash. In order to investigate the effect of this geometrical adjustment, angle θ is introduced in figure 4.3. Department of mechanical engineering 21

26 Figure 4.3: Introduction of angle θ to describe the geometry of the new design During the simulations, only this angle will be varied. If angle theta increases with 1 degree the mass of the foam block increases with about 14 gram. As this is less than 3 percent of the total mass of the foam block, this increase in weight is neglected. This means the time at which the yield criterium in tension is reached can be used as a parameter to compare the different models, while at the same time give an indication of the amount of damage a particular foam specimen with the corresponding angle will have. Angle θ has been varied from 0 to 6 degrees and the same damage criterium as in paragraph was used. The results are shown in figure 4.4. Figure 4.4: Time until fracture as a function of angle θ The results shown in figure 4.4 indicate there is a minimum at three and four degrees, which means those foam specimens will fracture more easily with respect to specimens with other angles. One important remark concerning figure 4.4 is that only the specimen with θ equal to zero will buckle. Department of mechanical engineering 22

27 4.2.2 Anti intrusion plate As indicated in paragraph 4.2.1, a foam specimen with θ equal to zero will buckle easily. This is caused by the anti intrusion plate which is placed at an 10 degree angle off vertical. In this paragraph, a simulation will be discussed where the anti intrusion plate is vertical and the foam specimen has the minimal dimensions as explained at the design requirements. In essence, this model is the same as the one used for the simple collision model shown in figure 3.2. If the damage criterium discussed in paragraph is used and the pressure and the von Mises stresses are examined, then the results indicate that there will be no damage in the foam specimen as the pressure does not become negative throughout the simulation. If there is no damage, then the model will be an accurate representative of a real crash test, which implies that the deceleration graph also will be a valid representation of the real deceleration. The deceleration graph of the model is shown in figure 4.5. Figure 4.5: Deceleration for the model with the vertical anti intrusion plate model Figure 4.5 indicates that the deceleration requirements specified by the formula student organization would be easily met. The peak deceleration is below 200 m/s 2 and therefore the average deceleration is also below 200 m/s 2, whilst in order to comply with the rules, the maximal peak deceleration is limited to 40g and the maximal average deceleration is limited to 20g. Department of mechanical engineering 23

28 Chapter 5 Conclusion and recommendations 5.1 Material The Rohacell 110IG foam has proven to be a very brittle foam in tension, which can be used as an energy absorbing material. When evaluating the simulation done with the 2008/2009 foam specimen and the simulation done with the minimal dimensions specified by the formula student rules and the vertical anti intrusion plate, it becomes visible that the attenuator made out of the Rohacell 110IG foam passes the test easily. 5.2 Geometry Anti intrusion plate When considering the tests done, it can be concluded that the anti intrusion plate placed at an angle has a negative effect on the crash in the sense that the foam specimen will fracture. Simulations done with a vertical anti intrusion plate indicate that there would be no fracturing of the foam which leads to an improved accuracy of the results Foam geometry When evaluating the foam specimens intended to fit the design space (with the anti intrusion plate at an angle), it can be concluded that an impact attenuator with a positive angle θ is preferable over a specimen with angle θ equal to zero. When varying angle θ, the time it takes until the specimen 24

29 fractures varies and shows a minimum at 3 and 4 degrees. When evaluating the foam specimen attached to the vertical anti intrusion plate, it can be concluded that there is no damage and that the deceleration would meet formula student standards Research method This research has presented a way to model the continuum behavior of Rohacell 110IG and a way to determine if a foam specimen wil fracture or not. If the foam fractures, the place of the initial fracture can be determined. The yield criterium has proven to work as can be seen with the 2008/2009 simulation. Furthermore the research method used in this report was able to globally reproduce the deceleration graph of the 2008/2009 test. In the case of a vertical anti intrusion plate, the model used in this research indicated that the foam would nog fracture. This would mean that the deceleration graph of this model would be an accurate representation of a real crash test. 5.3 Conclusion With respect to the main goals of this thesis it can be said that the implementation of a material model to investigate changes in a continuum of the foam was a succes. Furthermore the research has used a yield criterium which accurately predicts the point of fracture of a foam specimen, if there is one. This yield criterium was verified with the 2008/2009 crash test. While evaluating different foam geometries, this research has proven that the current design of the anti intrusion plate is not the best one and that the design with the vertical anti intrusion plate is the best one. With respect to the weight optimization, it can be said that the model is optimized in the sense that a minimal angle θ is indicated at which the foam does not buckle. Together with the fact that the angle θ was implemented on the minimal dimensions specified by the rules, the minimal dimensions of a valid and stable front impact attenuator are determined. A point of criticism that can be made is that there was no damage model implemented, and therefore the behavior after fracture is an indication and not 100% reliable. 5.4 Recommendations The recommendations that can be done based on this research are the fact that the anti intrusion plate should be placed vertical when designing a next generation formula student car because this would improve the collapse Department of mechanical engineering 25

30 of the front impact attenuator in the sense that there would not be any buckling. With respect to the material used it can be said that the Rohacell 110IG foam easily met the formula student standards, so research should be done to investigate if lighter materials also would be sufficient. A second recommendation regarding the material used would be to investigate if materials could be used which are not as brittle in tension as Rohacell 110IG foam. The material model used did not include a damage model. To enhance the accuracy of the simulations after the foam fractures, a damage model is necessary. Department of mechanical engineering 26

31 Appendix A: Foam dimensions Figure 5.1: 2008/2009 foam dimensions ([mm]) 27

32 Bibliography [01] Deshpande, V. S., and N. A. Fleck, Isotropic Constitutive Model for Metallic Foams, Journal of the Mechanics and Physics of Solids, vol. 48, pp , [02] Q.M. Li, R.A.W. Mines, R.S. Birch, The crush behavior of Rohacell-51WF structural foam, International Journal of Solids and Structures, vol. 37, pp , [03] E.A. Flores-Johnson, Q.M. Li and R.A.W. Mines, Degradation of Elastic Modulus of Progressively Crushable Foams in Uniaxial Compression, Journal of Cellular Plastics, vol. 44, pp , 2008, DOI: / X [04] 2010 Formula SAE Rules, source: competitions/formulaseries/rules/2010fsaerules.pdf, visited [05] Abaqus user manual version 6.8, Dassault Systémes S.A., 2005 [06] Rohacell data sheet, source: datasheet.aspx?matguid=a d783413fb f115& ckck=1, visited

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