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Designing and analysing Kagome
structures for crash safety applications
M.J.A.M. Walters
MT 05.17
Coach: Dr. Ir. W.J.Witteman
Supervisors: Prof. Dr. Ir. M.G.D. Geers
Prof. Dr. Ir. J.S.H.M. Wismans
Master thesis committee:
Prof. Dr. Ir. M.G.D. Geers
Prof. Dr. Ir. J.S.H.M. Wismans
Dr. Ir. W.J.Witteman
Dr. Ir. F.H.M. Swartjes
Eindhoven University of Technology
Department of Mechanical Engineering
Automotive Engineering
Eindhoven, April 11, 2005
2
Summary
The number of fatal traffic accidents decreases slowly in the Netherlands, but the
number of traffic injuries and casualties is still very high. The same trend is shown in
Europe, where the number of deaths is 42.000 per year. The frontal impact of vehicles
contributes a high percentage to those accidents. Improving the energy-absorption and
deformation of the frontal impact will decrease the injuries and fatalities considerably.
One way of achieving this goal is to improve the crumple zone by using ‘advanced
materials’. Kagome is one of these materials and is chosen for research based on a
preliminary literature study. Fabrication of this truss-structure is done by injection
moulding or laser cutting, CNC folding and than brazing.
The material behaviour of Kagome models are designed, analysed and simulated with
a finite element program Marc-Mentat. Several assumptions have been made and
researched to simplify the models and to implement correctly. Based on validation
with literature results the simulations are reliable. Validation has been done by means
of comparing the static results with literature results. The simulations are executed
quasi-static at compression and shear. Also a dynamic analysis has been executed to
simulate a more realistic impact load. The simulations are the basis for Kagome to be
applied in panel or column configurations. Besides simulations a literature study is
executed on the actuation feature of Kagome where piezoelectric material can be
used. Also the application of the Kagome structure is researched. One type of
application is fully examined, namely the Kagome material applied as a longitudinal
bar in a vehicle. Energy absorption is not sufficient enough for direct application.
However using the material in combination with the common structures can improve
safety. The conclusion is that the models reflect a realistic behaviour of the material
and give bases for further research of application of Kagome.
3
Summary (in Dutch)
Het aantal fatale verkeersongelukken neemt langzaam af in Nederland maar het aantal
gewonden en doden zijn nog steeds erg hoog. Dezelfde trend is te zien in Europa,
waar het aantal verkeersdoden per jaar 42.000 is. De frontale botsingen van
voertuigen leveren hier een groot aandeel in. Het verbeteren van de energie absorptie
en deformatie van frontale botsingen zal de verwondingen en doden aanzienlijk
verlagen. Een manier om dit doel te bereiken is het verbeteren van de kreukelzone
door gebruik te maken van ‘geavanceerde materialen’. Kagome is een van die
materialen die op basis van een voorafgaande literatuurstudie is gekozen voor dit
onderzoek. Fabricage van deze stavenstructuur wordt gedaan door middel van
spuitgieten of laserknippen, CNC vouwproces en lassen. Voor het materiaalonderzoek
van de Kagome zijn modellen gemaakt, geanalyseerd en gesimuleerd in een eindige
elementen programma Marc-Mentat. Meerdere veronderstellingen zijn gedaan en
onderzocht ter vereenvoudiging van de modellen en het verkrijgen van een correcte
implementatie. Uit de validatie met de literatuur blijkt dat de simulaties betrouwbaar
zijn. De validatie is uitgevoerd door middel van de statische resultaten te vergelijken
met resultaten uit de literatuur. De simulaties zijn uitgevoerd in quasi-statische
compressiekracht en dwarskracht. Ook is er een dynamische analyse uitgevoerd om
het gedrag tijdens impact realistischer weer te geven. De simulaties geven de basis
voor de toepassing van Kagome in paneel of kolom configuratie. Naast simulaties is
er ook een literatuurstudie uitgevoerd naar de eigenschap van actueren van de
Kagome waarbij piezoelectrische materialen een rol kunnen spelen. Verder is de
applicatie van de Kagome structuur onderzocht. Eén type toepassing is volledig
onderzocht, namelijk het Kagome materiaal toepassen als langsligger in een voertuig.
Energie absorptie is hierbij niet voldoende voor directe toepassing. Echter het
materiaal gebruiken in combinatie met de gewone constructies kan wel een
verbetering opleveren in veiligheid. De conclusie is dat de modellen een realistisch
beeld weergeven van het materiaalgedrag en als basis kunnen dienen voor verder
onderzoek naar de applicatie van Kagome.
4
Nomenclature
Symbol Explanation Unit
σ Tension N/m2
ε Strain -
ρ Density Kg/m3
E Young’s modulus GPa
σY Yield stress MPa
ν Poisson’s ratio -
σT Tensile stress MPa
N Strain-hardening exponent -
ε& Strain rate s-1
T Temperature oC
εo Strain elastic -
εp Strain plastic -
s Displacement m
A Area m2
vi Initial velocity m/s
F Force N
s Stockiness -
k Radius of gyration for in-plane
bending
-
Lc Length of members in the core mm
Hc The core thickness mm
Rc Core member radius mm
Lf Length of the solid circular face
member
mm
ϕc Angle between truss and plate degrees
tf The thickness of the solid sheet mm
t Axial tension J
Wo Reference energy J
W Work J
W Non-dimensional work -
A,B,m,n Parameters Marc -
n Strain rate sensitivity -
x Dimensionless parameter -
5
Contents
Summary 2
Summary (in Dutch) 3
Nomenclature 4
Contents 5
1 General Introduction 7
1.1. Description of the research issue 9
1.2. Objectives 10
1.3. Strategy of research 11
2 Material Kagome 12
2.1 Introduction 12
2.2 Choice of ‘advanced material’ 12
2.3 Definition Kagome 13
2.4 Fabrication of the Kagome 14
2.4.1 Introduction 14
2.4.2 Rapid prototyping 14
2.4.3 Investment casting 15
3 Model design 16
3.1 Introduction 16
3.2 Core model 16
3.3 Material design 17
3.4 Stress-strain behaviour 18
3.5 Temperature and strain rate effects 19
3.6 Final implementation 20
4 Simulation method for static analyses 21
4.1 Introduction 21
4.2 Compression 21
4.3 Shear 22
4.4 Set-up simulation in Marc 22
4.4.1 Mesh 22
4.4.2 Contact bodies 24
4.4.3 Links 24
4.4.4 Iteration steps 24
4.4.5 Tolerance Marc 24
4.5 Cell design 25
4.5.1 Introduction 25
4.5.2 Area assessment 25
4.5.3 Boundary Conditions 27
4.6 Conclusion 27
5 Static behaviour 28
5.1 Introduction 28
6
5.2 Validation results 28
5.3 Results extended simulations 30
5.4 Optimisation 35
5.4.1 Introduction 35
5.4.2 Thickening 35
5.4.3 Angle 36
5.5 Conclusion 37
6 Dynamic behaviour 38
6.1 Introduction 38
6.2 Set-up in Marc 38
6.3 Results dynamic 39
6.4 Conclusion 42
7 Actuation 43
7.1 General introduction 43
7.2 The design and actuation-types 43
7.3 Piezoelectric materials 46
7.4 Possible application 47
8 Application of the Kagome structure 48
8.1 Introduction 48
8.2 Longitudinal bars 49
8.3 Results 50
8.4 Folding process 51
8.5 Concepts of application 52
Conclusion 54
Suggestions and recommendations 54
Acknowledgement 55
References 56
Appendix A: Deformation Kagome 58
Appendix B: Optimal design truss plates 62
Appendix C: Convert to Power Law 65
Appendix D: Marc scripts 66
Appendix E: Matlab scripts 74
7
1 General Introduction
Mobility is an important part of life in the modern human society. The most common
use of travelling is by vehicle. The trend in the Netherlands over the past decade
clearly shows an increase of car-ownership, namely from 5.30 million in 1992 to 6.88
million in 2003. That’s about 43 percent of the entire population in the Netherlands.
Figure 1 reveals this trend.
Figure 1: Absolute amount of car owners in the period 1992-2003 in the Netherlands, source: CBS
[w9]
In addition, the number of fatal traffic accidents in 2003 has been increased by 2
percent to 1088 fatalities, as table 1 reveals. In spite of the slightly increase, the
Netherlands is together with Sweden and the United Kingdom the top three of traffic
safest countries in the European Union.
Year 1996 1997 1998 1999 2000 2001 2002 2003
Number 1251 1235 1149 1186 1166 1083 1066 1088
Table 1: Traffic fatalities per year in the Netherlands, source: AVV/CBS [w1]
In spite of the improvements in vehicle safety over the past 25 years the ministry of
Transport, Public Works and Water Management in the Netherlands has the policy to
reach a permanent improvement in traffic safety. The target in 2010 is to have less
than 900 fatalities per year and only 17.000 hospital casualties. [9]
8
An even further ambition is to have a maximum of 640 fatalities and 13.500 hospital
casualties in the year 2020. In the near future the emphasis in the traffic safety is in
the field of human behaviour in traffic and vehicle safety. Figure 2 shows the actual
numbers of the traffic fatalities and ambition numbers of the Dutch government. The
government will invest nearly 400 million euro in regional vehicle safety till 2010 and
another 23 million per year in national level to suppress the fatalities and injuries
further. [w2]
Figure 2: Trend of traffic fatalities and ambition mobility memorandum in the Netherlands [9]
Europe has the same kind of aim for reduction as the Netherlands. Besides looking at
the general trend of the traffic fatalities it’s also important to look at the type of
impact since this will assess the emphasis where the improvements should be made in
the near future. The current number of deaths is about 42.000 per year in Europe. In
table 2 the type of accident is put out in the number of these casualties on the road in
Europe. [w7]
Accident Type Percentage [%] No. Casualties
Frontal Impact 28 11.760
Side Impact 25 10.500
Rear Impact 1 420
Rollover 4 1.680
Pedestrians 20 8.400
Motorcycles 16 6.720
Other 6 2.520
Total 100 42.000
Table 2: Number of fatalities at different type of accidents, source ETSC [10]
With 28 percent of all fatalities the frontal impact contributes the most in fatal
injuries. Just beneath this high percentage of the frontal impact, the side impact also
contributes a high amount of fatalities.
9
Passive safety measures have proven the reduction of road accident causalities
considerately. Based on continuing research into passive safety measures a trend to
the future of reducing casualties can be made. [10]
Table 3 shows the potential targets in Europe again categorised by the accident type
by the year 2030. The reduction in casualties has the biggest potential at the front of
the car.
Accident Type No. Casualties Percentage reduction in casualties
by 2030 due to passive safety
measures alone
Frontal Impact 11.760 50
Side Impact 10.500 40
Rear Impact 420 5
Rollover 1.680 10
Pedestrians 8.400 30
Motorcycles 6.720 25
Other 2.520 30
Total 42.000 36
Table 3: Number of fatalities reduction in 2030 at different type of accidents source [10]
The potential numbers of table 3 are only reached due to passive safety measures,
other types of measures that improve the reduction like active measures, collision
avoidance and driver education are not considered here.
1.1. Description of the research issue
A further reduction of fatalities and injuries can be achieved for instance by using new
energy absorbing materials. For great reductions these materials should be
concentrated in the front or side of the vehicle. This thesis concentrates on the frontal
part. Finding the suitable crash (absorbing) design requirements for the frontal of the
vehicle is a complex issue. The optimal safety during a collision means the energy
absorption level must be high but the deceleration level should be low.
The high-energy absorption of a crash safety material is to deal with the entire impact.
This means that the structure must be stiff enough so the mean force is high during
deformation. The occupant compartment won’t or barely deform, which is important
since this is the survival space of the occupant. Serious and sometimes even fatal
injuries of car occupants are caused by intrusion of car parts into the passenger
compartment because stiff structures penetrate in softer parts.
When the vehicle structure is too stiff, than the deceleration level is high and could
lead to internal injuries like rupture of organs or serious concussions. A low
deceleration level is desirable since this will lead to lower injuries of the occupants.
[18] The conclusion is that the structure must be neither too stiff nor too weak. This is
clearly a contradiction and therefore a design dilemma.
10
This dilemma can be dealt with if the optimal deceleration level is known and the new
structure is designed such that the optimum is reached at any given time during
deformation. In figure 3 the optimal deceleration level during a collision with an
impact velocity of 56 km/h (based on EU-regulations) is illustrated.
There is a clear distinction during the deformation, namely three phases with each an
optimal deceleration level. [18]
Figure 3: Deceleration level during an optimal frontal deformation at 56 km/h [18]
1.2. Objectives
The main objective of this thesis is to improve the crashworthiness at the front-end of
the vehicle. Thereby, the amount of energy absorption of safety structures and the
optimal deceleration level during impact are important criteria. One way of improving
the crashworthiness is through applying new safety structures within the crumple zone
of a vehicle. Advanced materials have the potential of complying with the criteria;
they absorb a considerable amount of energy and can be changed in stiffness through
actuation. Since the major variety in frontal impact and different force levels during
collisions (see figure 3) the material must have the feature of variability in the
stiffness of the structure. Therefore an optimal deceleration level may be reached at
any type of collision. By showing the potential of the new material through this thesis
the basis for further research on application of this new safety structure is made.
11
1.3. Strategy of research
A material is chosen for research in this thesis based on a preliminary literature
survey. The criterion of selection is energy absorption and the possibility of actuation.
The new material is generally mapped out and the current status is elaborated. One
aspect of research is the fabrication of the material so the possibility of manufacturing
the material is determined. Chapter 2 discusses this.
Another aspect for research is the geometry of the structure, since this qualifies the
energy absorption and force level as well as the weight of the material.
The parameters of the material behaviour are researched so the characteristic of the
stress-strain behaviour is known. Chapter 3 discusses this.
With quasi-static analyses the material is researched and compared with previous
results. The set-up for the simulation is a compression and shear load test. To validate
the results of the simulation with previous studies the same set of boundary conditions
are used and the chosen structure is compared with a slightly different geometry.
After validation the models are comprehensively tested at larger deformations and
several different configurations for optimisation and a broader view on the material
behaviour. Besides static analyses also dynamic analyses are executed since this
reveals the real behaviour during impact. Chapters 4 through 6 treat these aspects.
Finally the actuation of the material is mapped out as well as the application area for
the new material. This defines the possibilities and limitations of the material and
gives a set-up for concepts of a new design in crash safety. Several applications can
be researched, but the emphasis is on the longitudinal bar. Several concepts for the
application of the material are made. The final two chapters 7 and 8 handle these
subjects.
Suggestions and recommendations conclude the thesis.
12
2 Material Kagome
2.1 Introduction
Improving the energy absorption of the frontal part of a vehicle can be achieved by
using ‘advanced materials’. These lightweight materials use the geometry and
elementary structure to ensure the stiffness and strength of the material. Common
‘advanced materials’ are foam materials. However there are other ‘advanced
materials’ that have a better density-energy absorption relationship. Materials of this
type are honeycomb, trusses or shell elements. Based on the findings in the
preliminary literature survey on ‘advanced materials’ the trusses show large potential
to be used in crash safety applications. [13] Particular two materials are qualified,
namely the Octet-truss en 3D-Kagome-truss structure. The cellular structure of one
cell is illustrated in figure 4.
(a) (b)
Figure 4: (a) Octet cell, (b) Kagome cell
2.2 Choice of ‘advanced material’
Because of the extensiveness of the research a choice has to be made which material
is further researched. One way of comparing both structures is to look at the stress-
strain characteristics during compression. Figure 5 represents the stress-strain
characteristics of both structures.
(a) (b)
Figure 5: (a) Stress-strain characteristics of the 3D-Kagome and Tetrahedral structure [14], (b) Stress-
strain characteristics of the Octet structure [3]
13
However, the two graphs cannot be compared with each other since the conditions of
the measurements are not the same, the octet-specimen is a five layer specimen while
the 3D-Kagome has only one. The material used for the octet structure is an
aluminium alloy while for 3D-Kagome this is a copper/beryllium-casting alloy.
Therefore other criteria are viewed to determine the superior structure. There are three
conclusions based on the preliminary literature survey [13] that are decisive:
• Bedding-in effects [3]
Bedding-in effects occur only for the octet material. This is caused by the structure,
which is made out of two different basic geometries, namely Tetrahedral and
Octahedral, see figure 6. During the initial stages of deformation the pins of the
Tetrahedral core bed into the holes of the triangulated layers. Look at the magnified
picture in figure 5 (b) for the influence on the stress-strain curve. This uncertainty can
lead to undesirable and irregular force levels.
(a) (b)
Figure 6: (a) Octahedral cell, (b) Tetrahedral cell
• Complex structure
The octet structure is much more complex than the Kagome structure. This means that
when simulating the models the octet has a longer simulation time and, when
fabricating the structure, the octet is most likely more expensive.
• Possibility to influence the structure
Recent studies [15,17] have shown the potential to influence and actuate the structure
of a Kagome type and, therefore this structure has the potential to reach or approach
the optimal deceleration level. This feature will be further explained in chapter 8.
Based on these three points the final judgement is clear; the 3D-Kagome structure is
superior compared to the octet structure.
2.3 Definition Kagome
Kagome is a Japanese word. One might guess that this word is the name of a Japanese
scientist. However the word is a translation for bamboo-basket (kago) woven pattern
(me) that is composed of interlaced triangles whose lattice points each have four
neighbouring points. A Japanese scientist Itiro Syôzi studied a honeycomb lattice with
an extra spin at the middle point of each bond to obtain the exact solution for an
antiferromagnet. He found that this honeycomb lattice turns into a new lattice by star-
to-triangle transformation. Therefore, he named this new lattice Kagome. [w5]
The material is also called 3D-Kagome, but this thesis uses the abbreviation
‘Kagome’.
14
2.4 Fabrication of the Kagome
2.4.1 Introduction
The fabrication of a Kagome is complex. One way of fabrication is the use of
injection moulding. A wax or polymer template of the truss core is fabricated as a
sacrificial pattern for investment casting. This however requires the fabrication of a
complex and often expensive die. [2] Furthermore, if a new truss core needs to be
made with a slightly different geometry or proportion an entire new die has to be
made with the additional costs. Another way of fabrication is the procedure laser
cutting, folded by CNC folding and brazing. [2] This is generally also an expensive
procedure.
2.4.2 Rapid prototyping
The rapid prototyping approach offers a better solution, especially in the first stage of
designing. It is less expensive and it has a far more flexible design and manufacturing
option. Rapid prototyping has gained a wide interest in the design community since it
provides a means of: (a) visualizing new product ideas and (b) fabricating parts for
functional testing. The procedure is to fabricate acrylonitrile-butadiene-styrene (ABS)
patterns, which is used for investment casting a beryllium-copper casting alloy (Cu-
2%Be). [2]
The basis of a rapid prototyping technique is a computer generated three-dimensional
solid model of an object, which provides a convenient creation of sandwich panels of
optimal or near optimal designs for performance comparisons. Different panels can be
created easily by varying the geometric parameters of the design.
A slicing algorithm defines tool paths for the outer boundaries of each layer and the
areas in which the material needs to be filled.
An extrusion-based, fused deposition modelling (FDM) process is used. The FDM
process uses acrylonitrile butadiene styrene (ABS) in filamentary form for the
modelling material. The material is fed into a temperature-controlled heater/liquefier
through counter-rotating rollers and is extruded through a fine nozzle in a semi-liquid
state as shown in figure 7.
Figure 7: Schematic view on extrusion process [2]
15
A second liquefier and nozzle system extrudes an easily dissolved material. This
material will support the structure and upon build completion be removed. The
fabricated panel is rotated to an orientation that minimizes the support structure.
After the panel is produced in the rapid prototyping machine it will be placed in a
soluble concentrate and water solution at 70 degrees to dissolve the support structure.
After the part is dry, a coat of clear acrylic spray is applied to fill-in gaps between the
individual layers of the model. [2]
2.4.3 Investment casting
The ABS patterns are being coated with a liquid wax and cleaned with isopropyl
alcohol. Seven wax gates are placed in a staggered arrangement to ensure that quick
molten metal is fed into the two face sheets, see figure 8 (a).
(a) (b)
Figure 8: (a) Attachment of wax runners and burn out vents, (b) Molten metal poured into ceramic shell
after ABS burnout [2]
Casting wax runners are placed connecting the seven gates. Three vent channels allow
gases to escape from the mould during ABS-burn-out, see figure 8 (b) for the final
procedure. The patterns are dipped in ceramic slurry containing colloidal silica and
finally coated with a fine-grained zircon. After drying at room temperature this
procedure was repeated up to 7 times. After placing it in a furnace with a temperature
of about 1066 degrees and held for 1.5 hours the moulds are removed and the vents
are plugged so that only the sandwich panel remains. [2]
16
3 Model design
3.1 Introduction
The designed models in this thesis are originally based on literature. Important is the
type of material, the stress-strain behaviour of that material and the implementation in
a finite element program Marc-Mentat for simulation purposes. By means of
validation the chosen structure for this research, Kagome, is compared with another
truss structure, namely Tetragonal, see figure 6 (b) and figure 5 (b) for the
compressive stress-strain results.
3.2 Core model
The dimensions of the Tetragonal core of the panel (figure 9(a)) are representative to
the near-optimised sandwich panels as Wicks and Hutchinson determined in an earlier
study. [16] The relative density is ρcore ≈ 0.02. The same truss radius and panel height are used for the Kagome core, but to attain the same core density, the truss length is
half that for the tetragonal core. Appendix B explains the principle analytical basis
and formula for the chosen dimensions. [16]
(a)
Figure 9: (a) Panel with Tetrahedral core, (b) Panel with Kagome core
The overall panel size of the Kagome-model (figure 9(b)) is the same as the
Tetrahedral core; therefore comparison with each other is valid. Figure 9 illustrates
the parameters, whereas table 4 shows the values. The nomenclature elaborates the
parameters.
Parameters of
Tetrahedral/Kagome core
Values
Lc 14 mm
Hc 11,4 mm
Rc 1,25/2 mm
Lf (Lf,tetra=½* Lf,Kag) 14,076 mm
tf 1,5 mm
ϕc +/- 550
Table 4: Dimensions of the Tetrahedral core and Kagome core
17
3.3 Material design
The material of the structures in the models is a copper/beryllium alloy (Cu-2%Be).
The reason to use this material is to facilitate the ductility requirements, which
exhibits a strain-to-failure in excess of 20 % in the as-cast state. The
Copper/beryllium is superior to for instance Aluminium, see figure 10 for
clarification. To facilitate a good comparison with numerical and experimental results
of previous studies Cu-Be is also used as basis material in the simulations described in
this thesis. [7,13]
Figure 10: Material behaviour of Ramberg-Osgood curve of Cu-Be Alloy and Al Alloy [7]
To limit the amount of parameters in the system, the plates in which both face sheets
and the core members are constructed have the same material properties. The material
properties of the Alloy Cu-Be are shown in table 5. Composition of the Alloy
C17200: 96,7 % Cu (min), 1,9 % Be, 0,2% Co. [1]
Alloy Density
(ρ) [g/cm3]
Modulus
of
Elasticity
(E) [GPa]
Poisson
ratio (ν) [-] Yield
Strength
(σY) [MPa]
Tensile
Strength
(σT) [MPa]
Coefficient
of Thermal
Exp. (oC)
C17200 ª 8,25 128 0,30 290 415-540 16,7*10-6
ª At room temperature
Table 5: Properties of copper-beryllium alloy [1]
18
3.4 Stress-strain behaviour
For implementing the right behaviour in the models it is important to know the
material behaviour. Therefore the relation between stress and strain for a Cu-Be
casting alloy is examined. The stress-strain [σ(ε)] curve of that material is fit by the Ramberg-Osgood formula, which is shown here: [7]
)()σ/E)(σ(σσ/Eε N
YY 1/+=
The previous study of S. Hyun shows that this fit is representative for the stress-strain
behaviour of Cu-Be casting alloy. The strain-hardening exponent N is 7,4. The other
values for the parameters are shown in table 5. The Ramberg-Osgood fit of the stress-
strain curve is plotted in figure 11.
Figure 11: Stress-strain curve for Cu-Be alloy
To implement the Ramberg-Osgood representation curve in Marc-Mentat, the formula
(1) has to be converted to a representation that is commonly used in Marc-Mentat.
The Power-Law representation is chosen because this has an elastic-plastic isotropic
hardening behaviour, the same behaviour of the applied material. The power law is
written as:
)(εBε)A(εσ nm 20&++=
The coefficients A and B, and exponents m and n are the parameters that are
implemented in Marc-Mentat. The second term of formula 2 is a velocity component.
When the simulation is static this component can be neglected, but in a dynamic
analyses the term ε& is of influence. The strain rate and temperature influence is
discussed in the next section.
19
3.5 Temperature and strain rate effects
The general stress-strain curve for a given material is temperature and strain rate
dependent. The shape and level of the curve is affected in opposite senses. At an
increasing temperature T the stress-strain curve drops while at an increasing strain
rate the stress levels and sometimes work-hardening rate increase. Figure 12 shows
this phenomenon. The sensitivity for these changes varies with the material. Generally
body-centred-cubic metals (e.g. iron, chromium) and ceramic materials are much
more sensitive to T and ε& than face-centred-cubic metals (e.g. aluminium, copper),
while polymeric solids are especially sensitive. [4][5]
Figure 12: (a) Temperature influence on stress, (b) Strain rate influence on stress [5]
The relationship between true stress and strain rate is mentioned before in the second
part of the Power Law formula (2):
)(εBσ n 3&⋅=
Where n = strain rate sensitivity factor
ε& = strain rate
B = material constant
σ = true stress
For most metals n is low and varies between 0.02 and 0.2.
Especially at dynamic analysis this sensitivity is of importance since high strain rates
are achieved. The Cu-Be in this research is compared with the behaviour of Cu
because of the low beryllium. Figure 13 shows the sensitivity of the strain rate versus
the temperature of Al, Ag and Cu. At normal conditions, (T = room temperature) the
strain rate sensitivity is very low (n = 0.01). [5] The contribution of the second part of
the Power Law ( nεB & ) can therefore be neglected. Important: when other materials are
used this term may contribute to the stress-strain relationship.
20
Figure 13: Strain rate sensitivity (n) for several materials at different temperatures [4]
3.6 Final implementation
The complete conversion of the Ramberg-Osgood to the Power-law is shown in
appendix C. The result is:
( )
Nand
σ
EσAtherefore:
)(εεAσεE
σ
σ
Eσσ
/N
Y
Y
m
p
/N
pY
/N
Y
Y
1m
4
1
0
11
=
=
+=⇔
+
=
With the given parameters (table 5) the component A and exponent m can be
obtained. Figure 14 illustrates that the Power Law analytically approaches the
Ramberg-Osgood curve correctly and is therefore a good description for the material
behaviour.
Figure 14: Stress-strain of Ramberg-Osgood versus Power law
21
4 Simulation method for static analyses
4.1 Introduction
The models are implemented in the design-program Unigraphics, with the basic
parameters mentioned in the previous chapter. Figure 15 (a) illustrates the basic
model. After design the structures are meshed with the program GS-Mesher into a
three-dimensional mesh, followed by static or dynamic simulations, which are
executed in the finite element solver Marc-Mentat. Appendix D shows the program-
script used for simulating in Marc. Finally the numerical results are implemented in
Matlab to plot the force against time and to calculate the matching stress and energy
absorption. Appendix E shows the program-script of Matlab. After evaluation of the
results the models are improved if necessary. Figure 15 (b) shows schematically the
course of the research. Two types of static simulations are executed, compression and
shear.
(a) (b)
Figure 15 (a): One cell illustrated in Unigraphics, (b) Course of research
4.2 Compression
Displacements are applied on the models to simulate the compression and shear load.
The assumptions and boundary conditions are based on the simplification and
elementary force applications. Section 4.4 shows the set-up in Marc. For compression,
a displacement ‘s’ in Y-direction is applied to the top plate with area A. This plate is
prohibited from transverse motion and rotation, while the bottom of the cell remains
rigid. Figure 16(a) visualizes this.
(a) (b)
Figure 16: (a) Quasi-static set-up for compression analysis, (b) Quasi-static set-up for shear analysis
22
4.3 Shear
The shear displacements are applied in three principal directions, reflecting the
extremes. The directions are positive (orientation I) and negative (orientation II) along
direction 2 and positive (orientation III) along direction 1, see figure 17. [7]
Displacements are applied horizontally in X-direction, illustrated in figure 16 (b). The
boundary conditions reflect the constraint exerted by the plates. Rotation is
prohibited. The bottom plate is rigid, same as at compression.
Figure 17: Directions displacement at shear analyses
4.4 Set-up simulation in Marc
Before simulations are executed some features of Marc-Mentat need to be further
examined, namely element type, mesh size, contact bodies, links, iteration steps and
tolerance. These features can influence the correctness and simulation-time of the
simulations.
4.4.1 Mesh
The choice of the best mesh element depends on several factors:
• Element accuracy
• The size of the model (number of elements)
• Complexity of the mesh boundary
• Linearity
• Element distortion
The application determines the element choice. Table 6 shows the overview of the
most common elements. The element geometry in the models is volumetric. It’s
important to consider pre-processor software that meshes and creates the models. The
GS-Mesher meshes the volumetric models in 4-node Tetrahedral or 10-node
Tetrahedral elements. The 4-node is favourable with respect to simulation time. [8]
23
However a mesh of 4-node Tetrahedral volume elements is not recommended, since
this leads to irregular smoothness of the model regardless of the size. Therefore this
element is less accurate during simulation. Thereby the 10-node Tetrahedral mesh is
chosen for the models in Marc-Mentat, see table 6.
Stress-Strain assumption Element geometry Common element
Uniaxial stress Line 2-node line
Axisymmetric solid Surface 4-node quadrilateral
Beam bending Line 3-node line
Plane stress Surface 4-node quadrilateral
Plane strain Surface 4-node quadrilateral
Plate bending/ shell Surface 4-node quadrilateral
3-D stress Volume 8-node hexahedral
(10-node Tetrahedral)
Table 6: Common element usage for simulation [8]
The default-mesh size of the GS-Mesher program has a maximum element size of
1.44e-3 m, and a minimum of 1.44e-4 m. The mesh size needs to be small enough to
obtain results that show convergence and are reliable. A check has been performed to
see if the model is meshed properly. The default-settings are compared with a finer
mesh, namely a maximum mesh of 0.85e-3 m. The results are shown in figure 18 and
illustrate that the default mesh is reliable, since a finer mesh barely differs from the
default value and further reduction will only increase the calculation time. The strain
used in the thesis is determined by dividing displacement through core thickness Hc.
Figure 18: Force-strain behaviour for different meshes of the Kagome
24
4.4.2 Contact bodies
The upper and lower plates are assigned as deformable bodies, while the Kagome core
is assigned as second deformable body. Through contacting (gluing) these bodies
interaction between the plates and core is possible.
4.4.3 Links
The boundary conditions are assigned to one node at each plate. After linking the
adjoining nodes that have the same conditions the entire plate is prescribed with the
correct boundaries.
4.4.4 Iteration steps
The iteration steps of the simulations are set high. Thereby small displacement
increments can be implemented. This is to avoid numerical instability, especially near
the yield stress. The iteration steps are set at a default of 40 steps at an overall
displacement of one millimetre. Figure 19 (a) shows the influence at a Tetrahedral
core. To avoid instability the iteration steps are increased to 120 steps. The Kagome
structure is less sensitive to the amount of iteration steps, but at larger deformations
the number of increments also have to be increased to avoid unreliable results.
4.4.5 Tolerance Marc
The tolerance that Marc standard uses is 0.1. The convergence tolerance determinates
the accuracy of the solution and thereby the simulation time. Figure 19 (b) shows that
the default tolerance is inaccurate especially at higher forces. Thereby the
convergence tolerance is set to a minimum of 0.01. This results in smooth and stable
plots as the dashed line illustrates.
(a) (b)
Figure 19:(a) Force-Strain behaviour at 40 iteration steps compare with 120 steps of Tetrahedral
structure, (b) Force-Strain behaviour for convergence tolerance of Marc on Kagome
25
4.5 Cell design
4.5.1 Introduction
Simulating an entire panel of cells costs a considerable amount of time, several days
at least. The models are therefore first simplified to only one cell. The simplified
schemes at figure 16 are used to simulate the behaviour of the Kagome and
Tetrahedral structure at compression and shear.
4.5.2 Area assessment
Boundary conditions are applied to simulate the neighbouring cells at the simulations
so the results are representative for an entire panel, elaborated in section 4.5.3.
The force against strain of one cell without plates is plotted in figure 20 (a). To
simulate a cell with plates, which represents a simplified panel, the area of one plate is
calculated. The corresponding stress-strain curve can be plotted. The force curve is
validated with the stress-strain curve of the literature study of Hyun, [7], see figure 5
(a).
The maximum amount of σcrush +/- 4.9 MPa for Kagome and 4.4 MPa for Tetrahedral are reached at a strain of approximately 0.04. Through validation in Marc the
maximum force is 1.1206e3 N respectively 961 N and is scaled with the maximum
stresses, therefore determining the area of one cell:
24
6
3
Kag m10*2.2410*4.9
10*1.1206
σ
FArea −=== and
If the plates are approached by rectangular with dimensions of 15 by 15 millimetres,
than the associated stress curve for the Kagome in figure 20 (b) is obtained.
⇒
(a) (b)
Figure 20: (a) Force-strain of one Kagome cell, (b) Associated stress-strain curve of one Kagome cell
24
6
3
Tetra m10*2.1810*4.4
10*0.96Area −==
26
Furthermore with the right boundary conditions, further explained in section 5.4.3,
one cell represents the material behaviour of an entire panel since the force scales
with the area of the panel-size at compression. The overall stress shown in the plots is
calculated by dividing the applied force through the associated area.
This is proven in figure 21, by placing two cells next to each other and compares them
with one cell for the same kind of compression and boundary conditions. It makes no
difference in the stress behaviour of the structure as figure 21 (b) illustrates.
(a) (b)
Figure 21 (a): Force-compression curve of one versus two Kagome cells at horizontal configuration, (b)
Stress-strain curve of one versus two Kagome cells
27
4.5.3 Boundary Conditions
To make sure that the boundary conditions, which are set at the end of the plates of
one cell, give a good representation for the adjacent cells, the original set-up in figure
16 is compared with a set up minimum of boundary and therefore simulated as one
cell with no neighbours. Only the z-direction is prescribed, namely z-displacement of
the upper plate while the bottom plate has a fixed z-displacement. Figure 22 shows
the compression of both set-ups. The free boundary set-up has a 10% lower value for
the maximum stress.
More realistic is that the neighbouring cells prohibit the cell to move sideways. There
is always friction between the impact body and the cells while the end of the plate is
fixed to for instance car parts. The real curve lies between the two lines of figure 22,
and more realistic near the fixed boundary curve. It is therefore assumed that the fixed
boundary set-up gives a simplified but reliable approach of the actual experiments,
which is also assumed in literature. [7]
Figure 22: Force-strain curve for difference between fixed and free boundary compression simulation
4.6 Conclusion
With the chosen configurations of Marc-Mentat, the mesh set-up of GS-Mesher and
cell boundary conditions, an correct finite element model is built that can be used for
evaluation purposes of the Kagome structure.
28
5 Static behaviour
5.1 Introduction
There are three categories in the static results determined. First the validation process
of the models is executed; results are compared with literature. Second the extended
static deformation results are shown. Here compression and shear simulations are
executed and the influence of imperfection at the model is checked. Finally the
optimisation is shown of the truss-thickness and truss-angle of the geometry.
5.2 Validation results
Based on literature results small deformations are executed for validating the model
results. Besides Kagome structure also Tetrahedral has been compared for correct
validation. Two previous studies of Hyun [7] respectively Wang [14] are used for
validation. Hyun executed simulations with the finite element program Abaqus, while
Wang executed experiments, which are shown in figure 5 (a).
Figure 23 (a) shows the stress-strain curve of the compression simulation of Kagome,
which can be compared with figure 5 (a). Figure 23 (b) shows the stress-strain curve
illustrated in non-dimensional form. The non-dimensional force is calculated through
dividing the force with πσYRc2. This has been done to compare figures 24 and 25
correctly with the literature. Figure 24 (b) compares the shear behaviour of the model
while figure 25 (b) compares the compression behaviour for Kagome as well as
Tetrahedral.
⇒
(a) (b)
Figure 23: (a) Compression-strain curve of Kagome structure, (b) Non-dimensional form of
compression-strain curve (a)
The maximum stress (σcrush) in figure 23 is the same as the measurement result of figure 5 (a). There is a difference at validation between the elastic behaviour; the
slope of the simulation is slightly higher than the measurement. This difference is also
shown at the simulations executed by Hyun. [7]
29
(a) (b)
Figure 24: (a) Literature result of non-dimensional shear in three directions for Kagome [7], (b)
Simulation result of non-dimensional shear in Marc-Mentat
(a) (b)
Figure 25: (a) Literature result of non-dimensional compression for Kagome and Tetrahedral [7], (b)
Simulation result of non-dimensional compression in Marc-Mentat
The curves of the three different shear directions, figure 24, are nearly the same.
Consequently the Kagome structure is direction-independent, and reveals isotropic
behaviour. [7] Figure 25 shows the Tetrahedral structure has the same elastic
behaviour as the Kagome one. However, it is clear to see that the Kagome has a larger
load carrying capacity after yielding than the Tetrahedral structure. The Kagome
structure is superior to the Tetrahedral structure.
There is a slight difference in both figures between the literature and the own results.
This difference is probably because of different use of simulation program; in the
literature the simulation program Abaqus is used while the own simulations are
executed in Marc-Mentat.
30
5.3 Results extended simulations
The behaviour of the Kagome is examined under extreme conditions and other
configurations so a better judgement can be made for the behaviour of a complete
panel of one layer or column of Kagome cells, see figure 26.
(a) (b)
Figure 26: (a) Panel (of 25 cells) configuration of Kagome structure, (b) Column configuration of
Kagome structure illustrated in Unigraphics
When the Kagome structure is used as a column configuration, therefore having
multiple layers, it is important to look at the change in compression and shear
behaviour at larger deformations. One, two and three layers are compared with each
other, which will predict the behaviour of a Kagome column of several tens of layers.
This saves a considerable amount of simulation time. The panel of one layer
configuration the compression was already discussed in section 4.5.2, now the shear
behaviour of one versus two neighbouring cells are compared. Finally the influence of
imperfections in the Kagome is shown.
Compression
The maximum compressive strain is equal to 75 %. The force level of the different
layers is shown in figure 27. The deformation of the Kagome is illustrated in appendix
A, where at strain steps of 25 % the stress in the structure is shown. During the first
stage of deformation the highest force level is reached followed by softening and
collapse of the Kagome. The force level drops about 550 N.
At about 60 % strain the force increases rapidly; the cell is totally crushed between
the plates and there is no room for further deformation of the Kagome cell. The
remaining material is flattening by the plates.
31
This initiates the start of the second layer at configuration of multi-layers. Figure 27
(b) shows the phase shift in deflection. The high forces are still reached by multi-
layers, namely about 1140 N, but at larger deflections, see the arrow. In the first stage,
all layers contribute to the force till the moment that one cell starts to collapse.
Section 8.4 discusses this process.
(a) (b)
Figure 27: (a) Force of compression simulation for different layers, (b) Enlarged picture of first
compression of (a)
The energy absorption can be found by multiplying the mean force with the
deformable length. The following formulas are used:
[J]DeflectionforceMeanabsorptionEnergy[kN]stepsIteration
forceReactionforceMean ∗=⇒=
The corresponding energy absorption for the different layers of figure 26 (a) is shown
in figure 28. This illustrates a linear increase when multiple layers are applied.
Figure 28: Energy absorption-deflection for different layers
32
Shear
The maximum applied shear strain is equal to 75 % too. The shear has only been
executed in shear I direction since Kagome has isotropic behaviour as mentioned in
section 5.2. The softening process starts at a larger deformation than the compression,
resulting in a constant larger force level at a larger displacement range. Figure 29
shows the shear behaviour of one cell versus two neighbouring cells. The deformation
of one Kagome cell is illustrated in appendix A.
The multi-layer behaviour is shown in figure 30. A shift in displacement is shown
when more layers are used as the arrow shows. The maximum force is not depending
on the number of layers. Softening starts at a displacement of 2e-3 m for one layer,
resulting in a drop of force of about 200 N.
(a) (b)
Figure 29: (a) Force level of shear behaviour of one cell versus two neighbouring cells, (b) Stress level
of shear behaviour of one cell versus two neighbouring cells
Figure 30: Shear behaviour for multi-layer structures
33
Comparison Shear – Compression of one cell
Figure 31 illustrates the difference between compression and shear of the Kagome
structure. The structure can cope with a high stress of σcrush= 5 MPa, when the load direction is in y-direction, see figure 16 (a). When the load direction is in x-direction
(figure 16 (b)) this maximum stress is σcrush= 2.3 MPa, about 2.2 times lower; but the softening process starts at a later stadium. Values that extend the 65 % strain are
unreliable for one cell; since at compression the plates starts to crush the Kagome
completely and at shear extreme necking of the trusses occurs, see appendix A. This
comparison clearly illustrates no collapse occurs when shear is applied and still a
considerable amount of load is dealt with. This is because of its typical geometry of
the Kagome.
Figure 31: Stress-displacement for comparison between shear and compression of Kagome
34
Imperfection of one cell
Another structural material that is used frequently is the honeycomb. This material is
also a high strength lightweight material. However, a big disadvantage is the high
sensitivity of imperfections, which causes the structure to deform at much lower
stresses and even collapse. [12]
Imperfections can be caused during fabrication. To check if the truss material
Kagome has this sensitivity, the perfect structure is compared with a structure with
imperfection in one of the trusses, as shown in the circle of figure 32. The
imperfection results are compared with the compression and shear results of figure 31.
Figure 33 shows the results.
Figure 32: Kagome with imperfection at the end of one truss
(a) (b)
Figure 33: (a) Force-displacement for comparison of imperfections at compression, (b) Force-
displacement for comparison of imperfections at shear
The cell with imperfection has a slightly lower force at compression and shear.
Furthermore at shear the force level drops at a displacement of 2 millimetres. No
collapse occurs during the deformation.
35
5.4 Optimisation
5.4.1 Introduction
The geometry of the Kagome structure is optimal according to appendix B. [16] This
optimum is based on the combination of weight efficiency and strength of the
structure. The weight efficiency is of less importance than strength, since the pass rate
of applying this material in crash safety is energy absorption. By changing the
geometry of the Kagome it is interesting to see the influence on the strength and the
possibility of optimising the materials geometry. There are two ways of changing the
geometry, namely thickening the truss radius Rc and changing the angle ϕ between the trusses and the plate.
5.4.2 Thickening
The default truss diameter is 1.25 mm. Another model has a diameter of 1.3 mm.
Only a small increase of the diameter is researched since the general properties of the
Kagome still are valid, therefore making the structure not solid for instants. A
decrease of the truss diameter will surely decrease the load capacity and is therefore
not considered here. The mean force at compression of the default is 1.04 kN for the
deformation of 1 mm, while the new model has a mean force of 1.15 kN. Figure 34
shows the stress-strain curve difference. This modification results in a capability
increase of 11 %.
Figure 34: Force level at compression for different truss diameters
36
5.4.3 Angle
The default angle is 55 degrees. To illustrate the influence of the angle, it is changed
with 10 degrees in both directions. The compression as well as the shear behaviour is
examined. Figure 35 shows the angle while figure 36 reveals the compression and
shear results at different angles.
Figure 35: Configuration of angle trusses
(a) (b)
Figure 36: (a) Stress-strain behaviour at compression for different angles, (b) Stress-strain behaviour at
shear for different angles
Compression results show that when the angle is 10 degrees increased, the maximum
stress increases with almost 1 MPa. This is an improvement of about 18 %. If the
angle decreases with 10 degrees this stress decreases with 1 MPa. When shear is
applied the opposite would be expected. However at 45 degrees the structure collapse
as figure 36 (b) illustrates. This reveals the critical limit is reached in which the angle
of the trusses can be applied. The stress level decreases with 0.5 MPa for the 65
degrees configuration.
37
5.5 Conclusion
The results of the simulations executed in Marc-Mentat approach the curves of the
simulations executed in Abaqus by Hyun, [7] and approach the measurement results
of Wang. [14] The type of finite element program probably causes the small
difference; in Marc-Mentat the Ramberg-Osgood curve is converted to a Power Law
while the implementation in Abaqus by Hyun is unknown. However, this concludes
the models are simulated correctly and represent a realistic behaviour of the material.
Therefore, further analyses of the models are reliable.
The Kagome has the highest force at compression, but at shear there is still a
considerable amount of force and at a larger deformation period, no collapse occurs.
Therefore, non-perpendicular loads can also be applied. Based on the imperfection
simulation the Kagome has a low sensitivity for imperfections.
By changing the thickness of the trusses with only 0.05 mm the stress behaviour
changes considerably. The same type of behaviour is shown when the angle is
changed. This illustrates the material can be improved, but the optimal weight factor
has not been considered. By knowing the type of application the geometry can be
adapted in such a way that an optimum can be reached for thickness and angle of
trusses.
38
6 Dynamic behaviour
6.1 Introduction
The static analysis represents a good insight in the behaviour of the Kagome structure.
However, the impact of a vehicle does not represent static or quasi-static behaviour
but dynamic. Therefore, a dynamic analysis is executed for the Kagome structure.
The same strategy is used as in the static analysis, namely simulating first one cell
with the same boundary conditions, after which two layers are used to represent the
column configuration. Three situations are researched. First the dynamic response is
compared with the static one. Then the influence of impact velocity is illustrated and
finally the weight factor is shown. The script for this analysis in Marc-Mentat is
shown in appendix D, while the calculation script in Matlab is shown in appendix E.
6.2 Set-up in Marc
To execute a dynamic analysis that represents an impact, a mass is set at an initial
velocity and dropped on the Kagome structure. Figure 37 shows the dynamic set-up, a
block with mass M is set as an impactor. The Kagome structure is fixed at the bottom.
The block’s initial velocity starts when the impactor makes contact with the Kagome
structure.
Figure 37: Dynamic set-up analysis for Kagome
When looking at dynamic analyses strain rate dependency is of great importance.
However, as mentioned before in section 3.5, using Cu/Be this can be neglected in the
simulations. Therefore, no changes are made to the Power Law formula.
Impactor
The mass is chosen such that the Kagome structure is deformed considerably. The
density of the block sets the weight. The mass block is glued to the upper plate of the
Kagome to avoid contact instability and to minimize calculation time.
39
Time
The analysis is set at dynamic transient. The time step is adaptive since in the
beginning of the analysis the force level increases rapidly compared to a later time
period. The duration of the impact is 100 ms since this is globally the duration of a
full frontal impact. [18]
Velocity
The velocity and weight of the impactor determine the amount of impact energy of the
structure. Based on the European regulations for a frontal impact the velocity is set at
56 km/h. This impact energy results in a complete crushed cell. Therefore, lower
velocities are also used.
6.3 Results dynamic
Comparison dynamic – static of one cell
Figure 36 (a) illustrates the difference between the static and dynamic analyses at the
same deflection of the Kagome. The dynamic set-up is an impact velocity of 15,56
m/s and an impact-mass of 0.5 kg. The dynamic deformation of the Kagome is
illustrated in appendix B, which illustrates the stress in the structure at three different
time steps. Those time steps comply with the strains of 25, 50 and 75 %. Figure 38 (b)
shows the associated energy absorption.
(a) (b)
Figure 38: (a) Force-deflection of dynamic versus static simulation of one Kagome cell, (b) Associated
energy absorption-deflection behaviour
Figure 38 shows a small difference between the static and dynamic behaviour at the
same deflection. Since no strain-rate dependence is implemented no large difference
is shown.
40
Comparison one – two layers
Figure 39 (a) shows the difference between in force level for one and two layers. The
settings are: an initial velocity of 10 m/s and an impact-mass of 1 kilogram, while the
duration is 100 milliseconds as mentioned before. Figure 39 (b) shows the energy
absorption against the deflection.
(a) (b)
Figure 39: (a) Force level of one and two layers, (b) Energy absorption against deflection of one and
two layers
At this impact one layer deforms considerably, after reaching maximum deflection in
z-direction the structure starts to fade out. The collapse of the Kagome of one layer
already starts at 2 milliseconds. At the two-layer configuration the structure barely
deforms and therefore the amount of energy is much lower. After reaching maximum
the structure also starts to fade out. In the beginning the force level shows higher
amplitudes but fades out earlier than the one-layer configuration. Since the one layer
is more deformed the fading out part is longer than the two-layer configuration.
41
Weight and velocity influence
At these simulations two layers are used, since one layer simply collapse early and,
thereby, the view of the behaviour is not clear at high velocities and masses. At the
weight simulation the same initial velocity is taken, namely 15,56 m/s. The layers are
compared for 1 kg and 1.5 kg to illustrate the weight factor. Figure 40 shows the force
level and associated energy absorption. Figure 41 shows the force and energy
absorption of the layers for two different initial velocities, namely 10 m/s and 15,56
m/s while the impact-mass is set at 1 kg.
(a) (b)
Figure 40: (a) Force-time behaviour for different weights, (b) Energy absorption-deflection for
different weights
In figure 40 1 kg initiates the deformation of the second layer at a deflection of about
8,55 mm as figure 40 (b) clearly illustrates. The impact-mass of 1,5 kg shows a
complete collapse of the two layers and starts to crush the structure between the
plates, increasing the force extremely. The arrows clearly show this initiation. After
maximum deflection is reached the set-up of 1,5 kg shows higher force amplitudes but
fades out much earlier than the 1 kg set-up. This is because the maximum deflection is
reached earlier when a higher weight is applied.
42
(a) (b)
Figure 41: (a) Force-time behaviour for different velocities, (b) Energy absorption-deflection for
different velocities
Increasing the velocity from 10 m/s to 15,56 m/s with the same impact-mass of 1 kg,
figure 41 (b) illustrates at the arrow the initiation of the second layer before fading
out. The fading out part hardly shows any difference, see figure 41 (a).
6.4 Conclusion
When more layers are applied the deflection decreases. Therefore, higher impact
weights or velocities can be applied before the same deflection is reached. At the
chosen dynamic set-ups the collapse of the Kagome structure starts very early, since
only two layers are used.
43
7 Actuation
7.1 General introduction
The truss structures can also be used as basis for adaptive structures. Individual
actuators that replace some of the members of the truss can satisfy this property.
Altering the length of these actuators changes the macroscopic shape of the structure.
This results in variability in stiffness of an entire structure, which may ultimately
optimise the energy absorption and deceleration level during a collision.
This chapter clarifies the basis in actuation of simple 2D-truss structures and what the
required energy is to actuate a single truss member, in particular how this energy
scales with the stockiness (s) (also know as slenderness) of the members. [15]
7.2 The design and actuation-types
The design of a truss with minimal resistance to actuation can be simplified if the
structure is assumed to be pin-jointed. To optimise the structure the following
properties must apply: the structure should be statically determinate (no self-stress)
and also kinematically determinate (no mechanisms). However, Guest and
Hutchinson [15], [17] have shown that infinite repetitive structures cannot be
simultaneously statically and kinematically determinate. Therefore there is no simple
and obvious solution to the optimal design. However the 2D-Kagome truss has shown
a promising solution [15]. For a clear view on actuation three simple finite structures
with different configurations will be explained with actuation through simplified
analyses, so this is how a structure reacts on a single member actuation. See figure 42,
where the top bar is to be actuated.
(a) (b) (c)
Figure 42: Three structure types for actuation [15]
The three structures are:
(a) Kinematically indeterminate (it has an obvious shear mechanism)
(b) Both statically and kinematically determinate
(c) Statically indeterminate (it has a state of self stress)
44
The following analysis uses standard techniques of structural mechanics and clarifies
the three different structures. For each structure, all bars are assumed to have a cross-
sectional area A, a Young’s modulus E, and a radius of gyration for in-plane bending
k, resulting in a axial stiffness AE and a bending stiffness AEk2. These properties are
also taken for the member that is actuated, although in practice, actuating members
will have different elastic properties, for instance using piezoelectric material. [15]
The structure is not subjected to any external loads and is initially unstressed. The
calculations are done for a small deformation linear-elastic range. They are carried out
for different values of stockiness, where the stockiness (s) for each structure is defined
as s=k/L, where L=length shown in figure 40. The calculation with the actuation is
being produced by extending the length of the actuated bar by a strain εa. Since the structure is not completely flexible, an (negative) axial tension in the bar t arises. The
work done by the extension is W=-tLεa /2. If the structure is completely rigid with t0=-AEεa, the reference energy is W0=(AEL/2)εa
2. [15]
Now the reporting results of the calculations are in the non-dimensional form
0W/WW = .
In the context of designing an easily actuated structure we want W to be small. The
results are shown in figure 43.
Figure 43: Energy required to actuate the three structures at different stockiness [15]
The two structures (a) and (b), which do not have a pin-jointed state of self-stress,
deform primarily in bending state. The non-dimensionalised bending energy scales
with s2, and for fixed pattern of deformation, the dimensionless W scales with s
2.
Structure (c) is different since actuation activates the pin-jointed state of self-stress, so
axial deformation is induced. For fixed pattern of deformation the non-
dimensionalised stretching energy does not vary with s and, therefore, the
dimensionless W is approximately constant.
45
These results reinforce the statement earlier made, namely that an actuated structure
should be both statically and kinematically determinate. However, structure (a) has a
soft shear mode, which has two implications. First the structure cannot efficiently
carry some loads and, secondly, changing the length of any member cannot actuate
the soft mode. Hence for these 2D structures (b) is optimal. This type of configuration
needs to be considered when applying actuation in truss structures. [15]
Figure 44 shows how a Kagome truss structure changes by actuating a single rod
(shown dashed).
Figure 44: Shape of Kagome structure by actuating a single rod [15]
46
7.3 Piezoelectric materials
The actuation of the truss members can be executed with piezoelectric materials.
These materials are so called ‘smart materials’. Piezoelectric materials have sensory
and actuatory properties. When the material is deformed, it gives a small electrical
discharge. This is the sensory property. When the material is subjected to an electrical
current, the material deforms and this is the actuator property. [11][w10]
In this study the actuator property is of interest. This effect is illustrated in figure 45.
The most frequently used piezoelectric material is quartz and Rochelle salt, but more
new materials like the Lead Zirconate Titanate (PZT) offer recently better solutions to
adaptive structural devices. Table 7 reveals important property values of this material.
Figure 45: Influence of electrical current to material [11]
Density 7.7 – 8.1g/cm3
Maximum Energy Density 102 J/m
3
Young’s Modulus 60 – 120 GPa
Tensile Strength 25 MPa (dynamic), 75 MPa (static)
Compressive Strength 520 MPa
Curie Temperature 160 – 350 oC
Operable Temperature Range -273 oC to +/- 80
oC
Inducible strain 1 – 2 µm/m (1 – 2 kV/m) Response Time Very fast (typical kHz, up to GHz)
Table 7: Properties of PZT piezoelectric materials [11]
The temperature has a large influence on the behaviour of the piezoelectric material.
The Curie temperature is the critical value; above this value the material loses its
piezoelectric properties. The operating temperature is limited to 80 oC since
depolarisation starts above this value. [11]
The dynamic behaviour of the piezoelectric materials is very good since they have a
low inertia. It means that they can be actuated more than a thousand times per second.
This makes them very suitable for application in adaptable safety devices where fast
reaction is needed. Another advantage is the resistance to compressive forces,
actuators can withstand several tens of kN’s. However, the material is brittle in
tension and this limits the freedom of application. Recently, piezoelectric composites
are developed, which combine ceramic and polymer materials. These show an
+
-
~
+
-
~
Electrical
Source
Electrical Current Off Electrical Current On
47
improved sensitivity and mechanical performance over the original piezoelectric
ceramics. However, it is a new subject on which few studies have been executed, so
this will not be discussed in more detail. A summary of the advantages and
disadvantages are shown below. It is based on the commonly used piezoelectric
materials like the PZT. [11]
The advantages are:
• Low cost
• Low power requirement during static operation
• High stiffness
• Very high frequencies attainable, thus very fast actuation
• Compact and light
• High position accuracy
• High generation of force per unit of volume
• No maintenance
The disadvantages are:
• Brittleness in tension
• Power consumption increases linearly with frequency and actuator capacitance
• High driving voltage required
• Limited strain
• The possible heath risks of lead in PZT piezoelectric ceramics
7.4 Possible application
Based on the previous two sections a concept can be made on applying actuation and
using piezoelectric material on a Kagome structure. Figure 46 shows this concept. A
Kagome structure of two neighbouring cells is connected with an extra truss member
made out of piezoelectric material. When compression is applied rotation of the centre
nodes occur and the trusses bend as the dashed lines show. By actuating the piezo
material this rotation can be limited and therefore influencing the stiffness as well as
the amount of compression that can be applied. This is merely a suggestion on how to
apply this feature and is not further analysed.
Figure 46: Possible application of actuation in Kagome structure [13]
48
8 Application of the Kagome structure
8.1 Introduction
There are several areas of the vehicle where crash-safety can be improved, for
instance the bumper, longitudinal bar and the hood or side door of the vehicle. Each
area has its own set of requirements with respect to safety. The main requirement is
the force-level in combination with the deceleration level and, therefore, the energy
absorption since this qualifies the scale of injury of the occupant or even the
pedestrian when hit by a vehicle.
As mentioned before the largest contribution of vehicle injuries or fatalities are caused
by frontal impacts. The largest dissipation of the energy is through the longitudinal
bars, about 50 percent of the total energy dissipation of the front-end of a vehicle. See
figure 47 for the general energy distribution.
Figure 47: Estimated energy absorption percentages in the front-end structure of a vehicle [18]
7.5 % 7.5 %
7.5 % 7.5 %
10 % 10 % 20 %
5 % 5 %
5 % 5 %
5 % 5 %
_____________________________________________________________
First half
Second half
longitudinals
engine
front panel
firewall
49
8.2 Longitudinal bars
In this section the common longitudinal bar is replaced with a Kagome bar, see figure
26 (b) shows this bar. To make a good assessment on the replacement or improvement
of the longitudinal bars with the Kagome structure, one has to define a default
longitudinal bar. For the default longitudinal bar the results of the doctoral thesis of
Witteman are used. [18] The longitudinal bar has a square cross-section with a
constant profile thickness of 2 mm, since this gives a stable folding pattern. The
length of the bar is commonly restricted to a deformation length of 60 - 80 cm.
Available space at the front-end of the vehicle may limit this. Figure 48 shows the
default bar. The material that has been used here is the commonly used steel FeP03.
Side viewFront view
Figure 48: Dimensions of default bar
To evaluate the new structure the mean force is of interest. The energy absorption for
the longitudinal bar can be found by multiplying the mean force with the deformable
length as mentioned in section 5.3.
In table 8 the mean force values for 25 different square cross-sections during
deformation of the longitudinal bar are evaluated. This is a quasi-static numerical
calculation. Based on this table the default bar has a mean force of 56 kN.
Thickness
[mm]
Perimeter
[mm]
4 x 25 =100
Mean Force
[KN]
Perimeter
[mm] 4 x 37.5
=150 Mean
Force [KN]
Perimeter
[mm] 4 x 50
=200 Mean
Force [KN]
Perimeter
[mm] 4 x 75
=300 Mean
Force [KN]
Perimeter
[mm] 4 x 100
=400 Mean
Force [KN]
1 12 14 15 17 19
1.5 24 27 30 34 37
2 38 44 49 56 61
3 75 86 96 110 121
4 120 139 154 177 195
Table 8: Quasi-static mean force of 25 different square cross-sections during deformation [18]
50
8.3 Results
Looking at the Kagome structure, based on simplicity and earlier assumptions only
three layers of each one cell is researched to simulate a column. Based on the static
analyses figure 49 (a) shows the corresponding energy plot.
(a) (b)
Figure 49: (a) Energy absorption-deflection of multi layers, (b) Energy comparison of 25 cells versus
36 cells of three layers.
If the default size of the standard longitudinal bar is used, 25 Kagome cells can be
fitted in the default cross-section and with multi-layers a Kagome bar is realised, see
figure 26 (b). The mean force of one Kagome cell is 620 N. Based on figure 21, the
entire panel of 25 cells results in a mean force of 15,5 kN (25 x 620 N). Comparing
this with the mean force of table 8 the structure has a 3.6 times lower mean force
level.
The associated energy absorption for the panel is shown in figure 49 (b). When the
panel is extended with a 6 by 6 cell configuration the energy is improved but still low,
and the panel size is larger, namely 90 x 90 mm. Even if the optimisation at section
5.3 is executed with an overall possible improvement in compression of 29 %, which
results in a mean force of 28.8 kN, the force level is still lower than the default bar.
This concludes that direct replacement of the common longitudinal bars with the
Kagome bars does not fulfil the requirement.
51
8.4 Folding process
Beside the level of energy absorption, it is also very important to have a stable and
regular force level during deformation. An unstable folding process can result in a
high peak force, which could lead to a bending collapse of the bar or other undesired
deformations in the structure (like rupture). During bending the energy absorption is
strongly reduced. Therefore, the standard longitudinal bars use triggering. So
especially the first high peak level is lower and the folding process starts at the front
and proceeds towards the rear end. This gives a more stable folding and, therefore, a
relative constant force level. Figure 50 elaborates this.
Figure 50: folding behaviour with and without triggering [18]
Left is a bar without triggering. When a force is applied at the front, this results in a
high peak force at the beginning of deformation and results finally in buckling of the
bar, as the middle bar shows. With triggering a regular folding pattern and a much
lower first peak load arises, the picture on the right shows the result. [18]
If the Kagome bar is applied, a column of Kagome layers see figure 26 (b), the
folding of the material is now fulfilled through the crush of each layer. The advantage
is that the layers will crush one by one in sequence resulting in a regular deformation.
In appendix B the folding pattern of the Kagome is shown of three layers. In the early
stages of the deformation (at 2,5 %) the cells deform equally. In theory this behaviour
should also occur for further deformations since the cells are uniform isotropic.
However, the simulation shows that at 20 % deformation the middle layer collapses
entirely before the top layer starts (see at 40 %) and finally the lower layer deforms.
This behaviour is caused by the numerical instability of the meshed cells. In reality
this can happen too, imperfections of some cells can cause one layer to collapse
entirely instead of uniform deformation. Therefore the Kagome column needs to be
triggered as well if deformation starts at the front and proceeds towards the rear end.
The front Kagome layer is applied with less Kagome cells, which ensures this layer to
collapse first.
52
8.5 Concepts of application
The concepts mentioned here replace the conventional structures of the front of the
vehicle. The Kagome structure can be applied in combination with the conventional
longitudinal bar as figure 51 illustrates. As mentioned in the preliminary literature
study [13] a numerical study of Santosa and Wierzbicki was executed to research the
energy absorption of filling a column with honeycomb or foam materials. Results
show improvement. Filling the longitudinal bar with the Kagome structure will
improve the energy absorption as well. In the design the folding pattern in
compression needs to be considered. The column and Kagome structures have to fold
stable since the force level needs a constant level as mentioned in the previous section
8.4. Therefore, there is no contact between the Kagome and the inner side of the
column, which may lead to undesirable folding. Since the Kagome attributes to the
force level the thickness of the column wall can be decreased.
Side view
Figure 51: Longitudinal bar filled with Kagome
Another concept is filling the bumper of a vehicle with the Kagome structure.
This application has the same principle as the longitudinal bar, namely filling the
structure with the advanced material, which improves the energy absorption.
However, the force level that needs to be dealt with is much lower than at the
longitudinal bars. The goal is at the first stage of collision that the bumper absorbs a
considerable amount of energy before other safety structures start to work. Figure 52
illustrates this concept.
Top view
Figure 52: Bumper filled with Kagome structure
53
The final concept is making the hood of the vehicle as a sandwich panel with the
Kagome structure as core material instead of a steel plate. Figure 53 illustrates the
cross-section. The hood is thereby lighter than the conventional hood, but the stiffness
stays intact. The hood is becoming a hotter item for vehicle manufactures since the
EuroNCAP tests for pedestrian impact on vehicles. At figure 54 illustrates the test set-
up. With simulations and experiments it is possible to find an optimal geometry for
Kagome for which pedestrians have a low injury scale.
Side view
Figure 53: Sandwich panel of Kagome for hood of vehicle
Figure 54: Test set-up of EuroNCAP for impact pedestrians [w12]
54
Conclusion
The designed and simulated models of the Kagome structure are validated. The use of
the Power Law model for implementation in Marc-Mentat of the material behaviour is
a correct approach. Finding the right set-up in Marc-Mentat so simulations convert
and results are reliable is complex. Since Marc-Mentat uses a considerable amount of
parameters the simulations have sometimes been executed on a ‘trail-and-error’ basis.
Therefore the simulations have taken up a considerable amount of time in this thesis.
Based on the research results several features can be concluded. The Kagome has a
low sensitivity to imperfections and the structure has an isotropic behaviour. The
compression and shear results reveal that Kagome is best applied at compression.
However, the structure is less directional sensitive than for instant honeycomb
structure since no collapse occurs at shear. Small changes in the Kagome geometry
lead to considerable changes in force level, which leaves room for improvement when
looking at the application. When applied as column configuration the folding pattern
sequence is irregular and when applied as panel of one layer, Kagome collapses
entirely at high impact velocities, based on the dynamic analyses. These analyses
show reliable results that reflect an impact situation.
The Kagome should by applied in combination with common safety structures in a
vehicle since force level of Kagome is not sufficient at high impacts. The models with
the complete implementation and program scripts give the bases for further research
on Kagome structures.
Suggestions and recommendations
Based on this master thesis several suggestions and recommendations are made for
future notice:
• Research if other materials, which also have a good ductility like Cu-Be alloy,
improve the energy-absorption.
• Make costs-benefit analyses on the fabrication of Kagome structure when
using it as a safety measurement.
• Use faster calculation program or computer to extend these simulations
without extending the calculation time.
• Extend the models to a complete panel and column and simulate. Then
execute quasi-static experiments with the Kagome structure.
• Extend the dynamic simulations of the Kagome structures and execute a
dynamic experiment by using the crash set-up in the Automotive Lab at the
University.
• Make a complete crash safety structure with Kagome as mentioned in section
8.5 and execute experiments on the crash set-up.
• Research on how to fabricate a Kagome structure with piezoelectric material
and execute simulations and experiments to establish the influence on stiffness
and strength.
55
Acknowledgement
The research of this thesis has been executed at the section Automotive Engineering
of the department Mechanical Engineering at Eindhoven University of Technology.
Thanks to Dr. Ir. W.J. Witteman at the University and Dr. Ir. F.H.M. Swartjes at TNO
in Delft for coaching during the thesis. Also thanks to Dr. Ir. R.H.J. Peerlings for
support of Marc-Mentat. Finally I especially thank my parents for moral support
during my study at the University.
56
References
Books and articles
[1] Callister, W.D. Jr. (2000) Materials Science and Engeneering, An
Introduction. Fifth Edition. University of Utah, U.S.A.
[2] Chiras, S., Mumm, D.R., Evans, A.G., Wicks, N., Hutchinson, J.W.,
Dharmasena, K., Wadley, H.N.G., Fichter, S. (2002) The structural
performance of near-optimized truss core panels. International Journal of
Solids and Structures 39. U.S.A.
[3] Deshpande, V.S., Fleck, N.A., Ashby, M.F. (2001) Effective properties of the
octet-truss lattice material. Journal of the Mechanics and Physics of Solids 49
(8). Cambridge, U.K.
[4] Felbeck, D.K., Atkins, A.G. (1996) Strength and fracture of engineering
solids. Second edition, U.S.A.
[5] Hertzberg, R.W. (1996) Deformation and fracture mechanics of engineering
materials. Fourth Edition, U.S.A.
[6] Hutchinson, R.G., Wicks, N., Evans, A.G., Fleck, N.A., Hutchinson, J.W.
(2003) Kagome plate structures for actuation. International Journal of Solids
and Structures 40. U.K., U.S.A.
[7] Hyun, S., Karlsson, A.M., Torquato, S., Evans, A.G. (2003) Simulated
properties of Kagome and tetragonal truss core panels. International Journal
of Solids and Structures 40. U.S.A.
[8] Lepi, S.M. (1998) Practical guide of finite elements. Michigan U.S.A.
[9] Ministry of Transport, Public Works and Water Management, (2004) Road
safety in The Netherlands, key figures. Netherlands
[10] Oakley, Charles (2004) Roadmap of future automotive passive safety
technology development. Transport Research Laboratory, UK.
[11] Rutten, S.H.L.A. (2003) Smart Materials in Automotive Applications.
MT03.19, Internship Report, Eindhoven University of Technology. The
Netherlands
[12] Wadley, H.N.G., Fleck, N.A., Evans, A.G. (2003) Fabrication and structural
performance of periodic cellular metal sandwich structures. Composites
science and technology. Virginia, U.S.A.
[13] Walters, M.J.A.M. (2004) Literature survey on ‘advanced materials’ in crash
safety applications, MT04.07, Internal Report, Eindhoven University of
Technology, The Netherlands.
57
[14] Wang, J., Evans, A.G., Dharmasena, K., Wadley, H.N.G. (2003) On the
performance of truss panels with Kagomé cores. International Journal of
Solids and Structures 40. Princeton, U.S.A.
[15] Wicks, N., Guest, S.D. (2004) Single member actuation in large repetitive
truss structures. International Journal of Solids and Structures 41. U.S.A.
[16] Wicks, N., Hutchinson, J.W. (2001) Optimal truss plates. International Journal
of Solids and Structures 38. U.S.A.
[17] Wicks, N., Hutchinson, J.W. (-) Sandwich plates actuated by a kagome planar
truss. Cambridge. U.S.A.
[18] Witteman, W.J. (1999) Improved Vehicle Crashworthiness Design by Control
of the Energy Absorption for Different Collision Situations. Ph.D. Thesis,
Eindhoven University of Technology, The Netherlands.
Manual
[19] Schreurs, P, Giesen, H., Vree de, H. (2003) Mentat & Marc handleiding met
Achtergronden en Oefeningen. Eindhoven University of Technology, The
Netherlands.
Websites
[w1] http://www.rws-avv.nl
[w2] http://www.regering.nl/actueel/nieuwsarchief/2004/04April/21/0-42-1_42-
29171.jsp
[w3] http://www.sciencedirect.com
[w4] http://www.swov.nl
[w5] http://www.aip.org/pt/vol-56/iss-2/p12.html
[w6] http://www2.eng.cam.ac.uk/~achl5/more.html
[w7] http://europa.eu.int/scadplus/leg/nl/lvb/l24257.htm
[w8] http://www.passivesafety.com
[w9] http://www.cbs.nl
[w10] http://piezo.com
[w11] http://www.sciencedirect.com
[w12] http://www.euroncap.com
[w13] http://www.minvenw.nl
[w14] http://www-mech.eng.cam.ac.uk/profiles/fleck
[w15] http://www.deas.harvard.edu/hutchinson
58
Appendix A: Deformation Kagome
Deformation of one cell static compression (displacement z=8.55 mm,
Load case = 1 second)
-At 0 % strain- -At 25 % strain-
-At 50 % strain- -At 75 % strain-
59
Deformation of one cell static shear (displacement x = 8.55 mm,
Load case = 1 second)
-At 0 % Strain- -At 25 % Strain-
-At 50 % Strain- -At 75 % Strain-
60
Deformation of one cell dynamic compression (Vi = 15,56 m/s, Mass = 0,5 kg,
Load case = 0.1 second)
- At 0 seconds - - At 0.00052 seconds -
- At 0.00127 seconds - - At 0.00185 seconds -
61
Deformation of three layers static compression (displacement z=22,32 mm,
Load case = 1 second)
-At 0 % Strain- -At 2,5 % Strain-
-At 20 % Strain- -At 40 % Strain-
-At 60 % Strain-
62
Appendix B: Optimal design truss plates
Well-designed structures using truss elements can be highly efficient from a weight
standpoint. With respect to the dimensions of the sandwich structure it’s important to
find an optimal design. This optimal is reached by minimising the weight of the
sandwich and still have a high strength and stiffness. This research has been executed
by N. Wicks and J.W. Hutchinson [16] for an octet truss plate and a sandwich plate
with truss core, see figure 55. The same assumptions and formula’s can be applied to
plates with Kagome structure. [7]
(a) (b)
Figure 55: (a) View of truss core with truss faces, (b) View of truss core with plates
The next sections will explain the realisation of the numerical analyses that gives the
basis for the minimal weight design.
Algorithm at plates subjected to bending and transverse shear
V
Ml:ratio
(2)HL
RL
3
πt2ρW
:facessheetsolidwithareaunitperWeight
(1)L
RL
L
R23ππ2W
:facestrusswithareaunitperWeight
)H3(LL
2
c
2
c
2
cc
f
2
f
2
cc
f
2
f
2
c
2
cf
≡
−+=
+=
−=
63
(5d)buckling)member(core2c4L
4cER3π
cH
cL
fVL
(5c)yielding)member(core2c
πRYσ
cH
cL
fVL
(5b)buckling)member(face4L
ERπ
H
L
3
M
(5a)yielding)member(faceπRσH
L
3
M
:areforcesmembermax.theonsconstraintFour
(4)H
LVLFlyrespective(3)
H
L
3
MF
:membercoreresp.sheetfaceinforceMaximum
2
f
4
f
3
c
f
2
fY
c
f
c
cfc
c
ff
≤
≤
≤
≤
=
=
The optimisation problem
( ) ( )
[ ] (6)/3xxxxx2x3π2ρl
W
:parameterweightessDimensionl
/3xx/lLwhere/lH/l,R/l,L/l,Rx,x,x,xx
2
2
2
4
2
2
2
3
1
2
2
1
2
2
2
4cccff4321
++=
+==≡
−−
r
The optimisation problem requires the minimisation of W/ρl with respect to the four constraints in equation (5), which becomes dimensionless in the form:
( ) (7d)buckling)member(core1/3xxxxxEM
V
π
4
(7c)yielding)member(core1/3xxxxxσ
E
EM
V
π
1
(7b)buckling)member(face1xxxEM
V
3π
4
(7a)yielding)member(face1xxxσ
E
EM
V
3π
1
3/22
2
2
4
1
4
4
32
2
3
2
2
2
4
1
4
2
32
Y
2
1
4
3
2
4
1
2
3
1
42
2
1
Y
2
≤+
≤+
≤
≤
−−
−−
−−
−−
64
There is only one dimensionless material parameter in the problem, σY/E, and only one dimensionless load parameter, V
2/EM.
Using a sequential quadratic programming algorithm these formulas are implemented
and the optimisation is computed. The optimisation is carried out for specified values
of V2/EM. An effective parameter tracking method uses the solution at one value of
V2/EM as an initial guess in the iteration for the solution at a smaller V
2/EM, because
this guess necessarily satisfies all the inequalities in the above equations. See figure
56 for the results. Here the dimensionless weight parameter and the member geometry
parameters are plotted against V/(EM)1/2. The plots are terminated at V/(EM)
1/2=0.002
and σY/E=0.007 because larger values would generate plates that would not be considered as thin.
Figure 56: Normalised weight per unit area of truss plates as a function of the dimensionless load
parameter σY/E=0.007.
65
Appendix C: Convert to Power Law
( )
Nmand
EA
so
AE
E
LawPower
E
E
E
E
E
E
E
EE
E
E
EEE
N
Y
Y
m
p
N
pY
N
Y
Y
N
pY
N
Y
Y
N
pY
Y
Y
N
p
Y
Y
Y
Y
N
p
Y
Y
N
p
YY
p
Y
N
Y
N
Y
Yp
Y
N
Y
Y
pe
1
:
:
11
1
1
/1
0
/1/1
/1/1
/1
/1
/1/1
=
=
+=⇔
+
=
+
=
+=
+=
+=→
+=
+=
−
=
−
+=
+=
σσ
εεσεσ
σσσ
εσ
σσσ
εσ
σσσ
εσ
σσ
σσ
εσ
σσεσσ
σ
εσσ
σ
σσσ
ε
σσσσσ
ε
εεε
66
Appendix D: Marc scripts
The following script is made for the static simulation of one Kagome cell in
compression, named onecellexact.m. This is a general form for all simulations. At
shear simulation for instance only the load direction has to be changed. The meshed
model is imported from the GS-Mesher which has the file extension *.bdf.
Static script
FILES
(INTERFACES) IMPORT NASTRAN (SELECTION) *.bdf (Importing Mesh model) OK
MAIN (MAIN MENU)(PREPROCESSING) MESH GENERATION (srfs) ADD (Adding surfaces to plates) Enter quad points: select corners of top plate,# Enter quad points: select corners of bottom plate,# SWEEP (SWEEP) TOLERANCE
Enter the sweep tolerance: 1E-5 ALL (REMOVE UNUSED) NODES POINTS RETURN CHECK UPSIDE DOWN INSIDE OUT (Answer must be low, 1 or even 0, MAIN else flip element) (MAIN MENU) (PREPROCESSING) MATERIAL PROPERTIES (apply material parameters as NEW mentioned in table 5) ISOTROPIC YOUNG’S MODULUS Enter value for ‘young’s modulus’: 1.3e11 Enter value for ‘poisson’s ratio’: 0.3 Enter value for ‘mass density’: 8250 ELASTIC-PLASTIC (method) POWER LAW COEFFICIENT Enter value for ‘power_law_a’:
290e6*(1.3e11/290e6)^(1/7.4) Enter value for ‘power_law_b’: 1/7.4 OK
OK (elements) ADD Enter add material element list: (all) EXIST. MAIN
67
(MAIN MENU) (PREPROCESSING) GEOMETRIC PROPERTIES (apply geometric parameters) (mechanical elements) 3-D NEW SOLID OK
(elements) ADD Enter add material element list: (all) EXIST. MAIN CONTACT CONTACT BODIES (apply contact bodies and TABLES boundaries) NEW 1 INDEPENDENT VARIABLE TYPE TIME OK FORMULA V1 ENTER SHOW MODEL RETURN
NEW NAME: kagome DEFORMABLE OK (elements) ADD Enter contact body add element list: (select kagome structure)#
NEW NAME: rigidtop RIGID (body control) POSITION PARAMETERS Z: - 0.00855 (apply amount of deformation) TABLE TABLE 1 (time) OK OK
OK (surfaces) ADD
Enter contact body add surface list: (select upper surface)#
NEW
NAME: rigidbodem (fixed) RIGID POSITION OK (elements) ADD Enter contact body add surface list: (select bottom surface)# NEW NAME: toppanel
68
DEFORMABLE OK (elements) ADD Enter contact body add element list: (select upper panel structure)#
NEW NAME: bottompanel DEFORMABLE OK (elements) ADD Enter contact body add element list: (select bottom panel structure)# RETURN
(contact) CONTACT TABLES NEW PROPERTIES (second) G (GLUE PARTS TOGETHER) 1-2,1-3,2-1,2-4,3-1,3-5 (detection method) AUTOMATIC (ANALYSIS) LOADCASES MECHANICAL STATIC (mechanical static properties) CONTACT CONTACT TABLES CTABLE1 OK SOLUTION CONTROL NON-POSITIVE DIFINITE OK CONVERGE TESTING (residual forces) RELATIVE FORCE TOLERANCE: 0.01 OK TOTAL LOADCASE TIME: 1 (fixed) CONSTANT TIME STEP
# STEPS: 360 AUTOMATIC TIME STEP CUT BACK #CUT BACKS ALLOWED: 10 OK MAIN
(ANALYSIS) JOBS (analysis class) MECHANICAL (available) lcase1 (analysis dimension) 3-D ANALYSIS OPTIONS (mechanical analysis options) LARGE DISPLACEMENT (plasticity procedure) LARGE STRAIN ADDITIVE OK JOB RESULTS EQUIVALENT VON MISES STRESS OK ELEMENT TYPES MECHANICAL 3-D SOLID
69
127 OK Enter element list: (all) EXIST. RUN (activate simulation) SUBMIT (1) OK MAIN (POSTPROCESSING) RESULTS (the result is of this script is shown OPEN DEFAULT in appendix A) DEF ONLY CONTOUR BANDS SCALAR: EQUIVALEMENT VON MISES STRESS OK MONITOR HISTORY PLOT
SET NODES Enter history node list: 5# COLLECT DATA Enter first history increment: 0 Enter last history increment: 360 Enter step size: 1 NODES/VARIABLES ADD 1-NODE CURVE (nodes) (SELECT NODE) (global variables) TIME (contact body variables) FORCE RIGIDBODEM RETURN SAVE (selection) onecellexact.txt (graph saving as .txt file)
Dynamic script
This script is made for dynamic simulation; the set-up is based on figure 37.
FILES
(INTERFACES) IMPORT NASTRAN (SELECTION) *.bdf (Importing Kagome model) OK (SELECTION) *.bdf (Importing Block mass model) OK
MAIN (MAIN MENU)(PREPROCESSING) MESH GENERATION SWEEP (SWEEP) TOLERANCE
Enter the sweep tolerance: 1E-5 ALL (REMOVE UNUSED) NODES POINTS RETURN CHECK UPSIDE DOWN
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INSIDE OUT (Answer must be low, 1 or even 0, MAIN else flip element) (MAIN MENU)(PREPROCESSING) BOUNDARY CONDITIONS
NEW NAME: APPLY 1 (BOUNDARY CONDITION CLASS) MECHANICAL (BOUNDARY CONDITION TYPE) FIXED DISPLACEMENT DISPLACEMENT X: 0 DISPLACEMENT Y: 0 OK (NODES) ADD Enter add apply node list: select upper front left corner node, #
NEW
NAME: APPLY 2 (BOUNDARY CONDITION TYPE) FIXED DISPLACEMENT DISPLACEMENT X: 0 DISPLACEMENT Y: 0 DISPLACEMENT Z: 0 OK (NODES) ADD Enter add apply node list: select bottom front left corner node, # MAIN
(MAIN MENU)(PREPROCESSING) INITIAL CONDITIONS (INITIAL CONDITION CLASS) MECHANICAL VELOCITY VELOCITY Z: -15,56 (variable velocity input) OK (NODES) ADD Enter add initial condition node list: select all nodes of block mass (MAIN MENU)(PREPROCESSING) LINKS
(LINKS) NODAL TIES N TO 1 TIES TYPE: 100 (RETAINED) NODE 1: select same node as in apply 1, # ADD TIES: select all node of the block mass RETAINED) NODE 1: select same node as in apply 1, # ADD TIES: select all node of bottom fixed plane MAIN
(MAIN MENU) (PREPROCESSING) MATERIAL PROPERTIES (apply material parameters as NEW mentioned in table 5)
NAME: MAT 1 (kagome structure) ISOTROPIC YOUNG’S MODULUS Enter value for ‘young’s modulus’: 1.3e11 Enter value for ‘poisson’s ratio’: 0.3 Enter value for ‘mass density’: 8250 ELASTIC-PLASTIC (method) POWER LAW COEFFICIENT Enter value for ‘power_law_a’:
290e6*(1.3e11/290e6)^(1/7.4)
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Enter value for ‘power_law_b’: 1/7.4 OK
OK (elements) ADD Enter add material element list: select kagome NEW
NAME: MAT 2 ISOTROPIC YOUNG’S MODULUS Enter value for ‘young’s modulus’: 3e7 Enter value for ‘poisson’s ratio’: 0.33 Enter value for ‘mass density’: 1.48e6 (Variable weight) OK (elements) ADD Enter add material element list: select block mass MAIN (MAIN MENU) (PREPROCESSING) GEOMETRIC PROPERTIES (apply geometric parameters) (mechanical elements) 3-D NEW SOLID 1 OK
(elements) ADD Enter add material element list: select kagome # NEW SOLID 2 OK
(elements) ADD Enter add material element list: select block # MAIN CONTACT CONTACT BODIES (apply contact bodies)
NEW NAME: kagome DEFORMABLE OK (elements) ADD Enter contact body add element list: (select kagome structure)# OK NEW NAME: block mass DEFORMABLE OK (elements) ADD Enter contact body add element list: (select block mass)# RETURN
(contact) CONTACT TABLES NEW PROPERTIES (second) G (GLUE PARTS TOGETHER) 1-2,2-1 (detection method) AUTOMATIC
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(ANALYSIS) LOADCASES MECHANICAL DYNAMIC TRANSIENT (mechanical static properties) LOADS (select loads) APPLY 1,APPLY 2 OK
(mechanical static properties) CONTACT CONTACT TABLES CTABLE1 OK SOLUTION CONTROL NON-POSITIVE DIFINITE OK CONVERGE TESTING (residual forces) RELATIVE FORCE TOLERANCE: 0.01 OK TOTAL LOADCASE TIME: 0.1 (stepping procedure)
(adaptive) MULTI-CRITERIA PARAMETERS INITIAL FRACTION OF LOADCASE TIME: 1E-6 MINIMUM FRACTION OF LOADCASE TIME: 1E-7
MAXIMUM FRACTION OF LOADCASE TIME: 0.01 OK
OK MAIN
(ANALYSIS) JOBS (analysis class) MECHANICAL (available) lcase1 (analysis dimension) 3-D ANALYSIS OPTIONS (mechanical analysis options) LARGE DISPLACEMENT (plasticity procedure) LARGE STRAIN ADDITIVE OK JOB RESULTS EQUIVALENT VON MISES STRESS OK ELEMENT TYPES MECHANICAL 3-D SOLID 127 OK Enter element list: (all) EXIST. RUN (activate simulation) SUBMIT (1) OK MAIN
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(POSTPROCESSING) RESULTS (the result is of this script is shown OPEN DEFAULT in appendix A) DEF ONLY CONTOUR BANDS SCALAR: EQUIVALEMENT VON MISES STRESS OK MONITOR HISTORY PLOT
SET NODES Enter history node list: select bottom node of apply 1# COLLECT DATA Enter first history increment: 0 Enter last history increment: amount of iteration steps used Enter step size: 1 (usually 1) NODES/VARIABLES ADD 1-NODE CURVE (nodes) (SELECT NODE) (global variables) TIME (contact body variables) REACTION FORCE Z RETURN SAVE (selection) dynamic.txt (graph saving as txt file)
HISTORY PLOT
SET NODES Enter history node list: select upper front left corner node, # COLLECT DATA Enter first history increment: 0 Enter last history increment: amount of iteration steps used Enter step size: 1 (usually 1) NODES/VARIABLES ADD 1-NODE CURVE (nodes) (SELECT NODE) (global variables) TIME (contact body variables) DISPLACEMENT Z RETURN SAVE (selection) deflection.txt (graph saving as .txt file)
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Appendix E: Matlab scripts
Static Matlab-script
This Matlab-script is for the static compression analyses run by Marc-Mentat. The
result is written in as a .txt file and Matlab reads this file.
%%%%%%%%%%%%%%%% read file %%%%%%%%%%%%%% [B,K] = textread('onecellexact.txt','%f %f',-1,'headerlines',9); Force1=K; Displ=8.55e-3*B; Rek=Displ/Hc; % calculating strain % stress=Force/Opp; % calculating stress % %%%%%%%%%%%%%%%% parameters %%%%%%%%%%%%% SigmaY=2.9e8; Rc=0.625e-3; Opp=(15e-3)^2; Hc=11.4e-3; %%%%%%%%%%%%%%%% plots %%%%%%%%%%%%%%%%% figure plot(Displ,K) title('Reaction force of one cell') xlabel('deflection') ylabel('Reaction force N') figure plot(Rek,stress) title('stress-strain of one cell') xlabel('strain \epsilon') ylabel('stress \sigma [Pa]') %%%%%%%%%%%%%%%% mean force %%%%%%%%%%%%%% Mean1=[]; p=[1:361]; for n=p; Mforce1=sum(Force1(1:n,:))/n; Mean1=[Mean1,Mforce1]; end %%%%%%%%%%%%%%%% energy %%%%%%%%%%%%%%%% AX1=Mean1'; AY1=Displ1; R1=[]; E1=[]; i=[1:361]; for x=i; Energy1(x)=AX1(x)*AY1(x); R1=[R1,Energy1(x)];
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end R1energy=R1'; %%%%%%%%%%%%%%%% plot %%%%%%%%%%%%%%%%%% figure plot(Displ1,R1energy) title('Energy absorption') xlabel('Deflection [m]') ylabel('Energy absorption [J]') Legend('one')
Dynamic Matlab-script
This script shows the dynamic analyses reading and calculating of the file dynamic.txt
and deflection.txt.
%%%%%%%%%%%%%%% read file %%%%%%%%%%%%%% [B,K] = textread('dynamic.txt','%f %f',-1,'headerlines',9); [C,L] = textread('deflection.txt','%f %f',-1,'headerlines',9); Force1=K; Time=B; Defl=-L; %%%%%%%%%%%%%% parameters %%%%%%%%%%%%%% SigmaY=2.9e8; Rc=0.625e-3; Opp=(15e-3)^2; Hc=11.4e-3; %%%%%%%%%%%%%%%% plot %%%%%%%%%%%%%%%% figure plot(B,K) title('Force dynamic of one cell') xlabel('time [sec]') ylabel('Force [N]') %%%%%%%%%%%%%% Mean Force %%%%%%%%%%%%%% %%%%%%%%%%%%%% Mean Force %%%%%%%%%%%%%% Mean1=[]; p=[1:672]; for n=p; Mforce1=sum(Force1(1:n,:))/n; Mean1=[Mean1,Mforce1]; end %%%%%%%%%%%%%% Energy %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Energy %%%%%%%%%%%%%%%%% AX1=Mean1'; AY1=Defl;
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R1=[]; E1=[]; i=[1:672]; for x=i; Energy1(x)=AX1(x)*AY1(x); R1=[R1,Energy1(x)]; end R1energy=R1'; %%%%%%%%%%%%%%%% plot %%%%%%%%%%%%%%%%% figure plot(Time,R1energy) title('Energy absorption of one cell') xlabel('time [sec]') ylabel('Energy absorption [J]') figure plot(Defl,R1energy) title('Energy absorption of one cell') xlabel('Deflection [m]') ylabel('Energy absorption [J]')
