A soil model based on discrete element method for crash analysis

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Ingegneria Aeronautica

A Soil Model Based on Discrete Element

Method for Crash Analysis

Relatore: Prof. Paolo Astori

Relazione della tesi di:

Jacopo Varriale, Matr. 859358 Anno Accademico 2018/2019

Abstract

A statistical study reported in the state of art defines the most critical im-pact scenario in case of aircraft accidents as an imim-pact against a soft soil. A reliable numerical soil model is of fundamental importance in order to predict the high loads acting on the structure af an aircraft, but, unfortunately, the soil modelling in not an easy task. The discrete element method, a technique used by the geotechnical engineering, gives a satisfactory soil representation. Firstly, the experimental tests carried out in Politecnico di Milano are pre-sented. Subsequently, the numerical models of the experiments are exposed. The soil is calibrated and validated. In the end, the experiment of the crash of the glider is simulated and the numerical model is verified. The results are satisfactory and the computational costs acceptable.

Keywords: soil impact, soil modeling, glider crashworthiness, discrete

element method.

Contents

1.2.1 Test on the glider DG-800 . . . 21

1.2.2 Test on a glider made in carbonium and Dyneema fabric 23 1.3 Conclusions . . . 24

2 Experimental Tests 27 2.1 Descriptions of the impacts of the drop mass . . . 27

2.1.1 Experiment set-up . . . 27

2.1.2 Shock absorbing layer . . . 31

2.1.3 Performed impacts . . . 31

2.2 Description of the glider crash experiment . . . 33

2.2.1 Glider construction . . . 34

2.2.2 Building and filling of the container . . . 37

2.2.3 Instrumentations for the data acquisition . . . 40

2.2.4 Glider crash . . . 42

2.3 Filtering . . . 43

2.4 Dynamic of the soil . . . 44

3 Discrete Element Method 47 3.1 Hypothesis and initial considerations . . . 48

3.2 YADE (Yet Another Dynamic Engine) . . . 50

3.2.1 Collision detection . . . 50

3.2.2 Material . . . 51

3.2.3 Contact model . . . 53 5

6 CONTENTS

4 Numerical Model 55

4.1 Simulations Description . . . 55

4.1.1 Deposition of the particles into the test tank . . . 56

4.1.2 Drop test numerical simulation . . . 58

4.1.3 Filling of the container . . . 59

4.1.4 Glider crash test simulation . . . 60

4.2 Computing time . . . 63

4.2.1 Algorithm of the generation of the bodies . . . 63

4.2.2 Duration of the simulations . . . 65

5 Numerical Results 69 5.1 Calibration and validation of the soil model . . . 69

5.1.1 Comparison between the elasto-plastic model and the cohesive model . . . 74

5.2 Glider crash numerical results . . . 79

List of Figures

1.1 Statistical analysis of the accidents . . . 16

1.2 histogram of mortal injuries . . . 17

1.3 Types of impact soil . . . 18

1.4 Impact of type 4 . . . 19

1.5 Glider DG-800 . . . 22

1.6 Glider made in carbonium and Dyneema fabric . . . 24

2.1 The tower and the lifting system of Politecnico di Milano . . . 28

2.2 Test tank . . . 28

2.3 Drop mass . . . 29

2.4 Damped piezoresistive accelerometer . . . 30

2.5 Strain gauge load cells . . . 30

2.6 Installation of the accelerometers in the cavity of the drop mass 31 2.7 Graphs of the physical quantities researched during the exper-imental tests . . . 33

2.8 Lamination phase . . . 35

2.9 The peel-ply method and the assembly of the molds . . . 35

2.10 The designed ballast and the wingbeam . . . 36

2.11 The supports designed at La.S.T. and the rails of its "launch track" . . . 37

2.12 The final assembly of the glider with the additional components 37 2.13 Technical drawings . . . 38

2.14 Manual compaction . . . 39

2.15 Filled container . . . 39

2.16 The container . . . 40

2.17 Detail of the fixing of the sensors and their relative cables . . . 40

2.18 Disposition of the sensors . . . 41

2.19 The longitudinal acceleration of the fuselage . . . 42

2.20 The angular velocity of the glider and its rotation angle . . . . 43

2.21 Types of low pass filter . . . 44 2.22 Configuration of the impacts in the real and laboratory case . 45

8 LIST OF FIGURES

3.1 DEM algorithm . . . 48

3.2 Interactions scheme between the elements . . . 48

3.3 Particles contacts modeling . . . 49

3.4 Yade . . . 50

3.5 two springs in series representing the normal stiffness between bodies in contact . . . 52

4.1 Trend of the energies present during the settling . . . 57

4.2 Slope of the unbalanced Force . . . 58

4.3 Model of the experimental test implemented with Yade . . . . 59

4.4 Real and virtual cohesive container models . . . 60

4.5 Glider models . . . 61

4.6 Comparison between different glider models . . . 62

4.7 Istants of the glider crash simulation . . . 67

5.1 Acceleration vs Time graph for the different falling heights . . 71

5.2 Velocity vs Time graph for the different falling heights . . . . 72

5.3 Displacement vs Time graph for the different falling heights . 73 5.4 Acceleration vs Time graph of the soil models for the different falling heights. . . 76

5.5 Velocity vs Time graph of the soil models for the different falling heights. . . 77

5.6 Displacement vs Time graph of the soil models for the different falling heights. . . 78

5.7 Longitudinal acceleration of the glider . . . 80

5.8 Angular velocity of the glider . . . 81

List of Tables

2.1 Characteristics of the impacts carried out in the series S2 . . . 32 2.2 Summary of the main characteristics of the series S2 . . . 33 2.3 The main characteristics of the glider . . . 36 2.4 Summary of the instrumentations used for the data acquisition 41 2.5 CFC indexes . . . 44 4.1 Comparison between the total duration of the simulations and

the percentage error for different diameters . . . 66 4.2 Numbers of the elements and durations of the simulations . . 66 5.1 The elasto-plastic costitutive parameters set for the reinforced

concrete drop mass, for the soil and for the test tank . . . 70 5.2 The geometry parameters and the number of the elements . . 70 5.3 Percentage errors . . . 74 5.4 Constitutive parameters of the three models of the soil . . . . 75 5.5 Percentage errors of the three models of soil for the different

falling heights . . . 79 5.6 Parameters considered as known . . . 79 5.7 Percentage errors of the peaks of acceleration . . . 82 5.8 Summary of the costitutive parameters and of the results . . . 83

Nomenclature

∆l Variation of the lenght of the spring

µ Friction coefficient

ν Poisson’s coefficient

φ Intergranular friction angle

θb Bending rotation

θt Twisting rotation

an Normal adhesion

at Tangential adhesion

C Contact point

D Diameter of the spheres

d Distance between the contact point and the center of the sphere

E Young’s modulus

F Total force: sum of Fn and Ft

Ft

t Approximated tangential contact force

Fn Normal contact force

Ft Tangential contact force

g Acceleration of gravity

h Falling height

kb Linear bending modulus

12 LIST OF TABLES

Kn Normal spring stiffness

Kt Tangential spring stiffness

kt Linear twisting modulus

l Spring lenght at rest

Mb Bending moment

Mt Twisting moment

n Normal to the surface

T Torque moment

up

n Plastic component of the normal displacement

un Normal displacement

ut Tangential displacement

Introduction

The design philosophies for aircraft and vehicles of the latest generation are increasingly influenced by the introduction of passive safety requirements within the norms. These new requirements have led to the emergence of complications during the design phases. In fact, the impact phenomenon is extremely rapid and leads the structure to have a highly complex behavior, whose prediction implies burdensome calculation methods, both in terms of computing time as well as in terms of the memory of the computer necessary for the processing. However, the reduction of costs and of construction time of the structures are the advantages deriving from the use of numerical models. The use of experimental tests remains necessary in different phases of the project; in fact, the experimental activity assumes the role of verification and control tool.

The behavior of an aeronautical structure during an impact is considered satisfactory only if the passive safety specifications are followed since in the preliminary phase of the project. It is necessary to adopt all those precau-tions which are useful to limit the damages that may arise due to an accident during the flight. The deformability of the structure of the aircraft is of fun-damental importance because it must be able to absorb much of the energy that develops during the impact. Despite the numerous accidents and their seriousness, a project philosophy that takes into consideration the modern technological devices aimed to absorb energy is not found yet. This fact is primarily linked to a critical aspect of the design of an aircraft, which is the maintenance of high aerodynamic performances. In fact, the benefits deriv-ing from the addition of all the devices dedicated to absorb energy durderiv-ing an impact are not considered sufficient with respect to the loss of performance caused by their introduction. A numerical model capable of predicting the behavior of the structure is therefore of fundamental importance in order to analyze new safety devices without additional external costs. It is important to underline that in order to achieve a correct prediction of an event, it is necessary to model correctly the shock absorbing layer and not only the im-pacting object. The realization of models of soil assumes a fundamental role

14 LIST OF TABLES because it implies an appropriate modeling of all the aspects of the prob-lem: geometry simplification, continuous discretization, materials, boundary conditions, contact definition. The study of the mechanisms that govern the phenomenon is of fundamental importance. Most of aircraft crashes occur on soft soil. The structural analysis of a crash landing on a deformable surface may be of interest in aircraft crashworthiness. One could say, in general, that if a structure is designed for crashing properly on a stiff surface, it will per-form even better on a deper-formable surface, because part of the energy will be absorbed by the latter. But crashworthiness is not only a matter of energy absorption: how this energy is absorbed is important to reduce the loads transferred to the cabin, seats and occupants.

This paper is divided in four parts. In the first part, the main reasons of the impact configuration chosen and the state of art are exposed. The descriptions and the results of the experimental tests carried out at La.S.T. (Laboratorio di Sicurezza dei Trasporti) in Politecnico di Milano concerning the mechanical characterization of the soil (drop mass impact test) and the crash of the glider V1/2 Rondine are shown in the second part. The method on which the numerical model is based and how the experiments are simulated are shown in the third part. The comparison between the numerical results and the experimental results, and therefore, the validity of the model are shown in the fourth part.

Chapter 1

Initial considerations

1.1

Statistical analysis

An important part of the preparation of the test is the statistical study con-cerning the glider accidents. The selection of the most frequent types of impact is essential. Among these, the one that could be made survivable if modifications of the structure of the aircraft are applied must be found. It is necessary to understand what are the characteristics of an impact in terms of attitude, type of soil and impact velocity. The statistical analysis is conducted looking the investigation reports that the national flight safety agencies must issue after each accident. These reports are carried out as ac-curately as possible and their purpose is to identify the causes of the accident to prevent it from happening again. A significant number of impacts for the statistical analysis is reached by analyzing the reports of tha ANSV (Agenzia Nazionale per la Sicurezza del Volo) and the reports of the NTSB (National Transportation Safety Board). The total number of the analyzed accidents is 90.

1.1.1

Types of impact

The expert Martin Sperber identifies four categories of types of impact in his studies [9]:

• type 1: slight negative pitch angle of the fuselage attitude. Inclined descending trajectory (45 deg with respect to the soil surface)

• type 2: high negative pitch angle of the fuselage attitude. Inclined descending trajectory (30 deg with respect to the soil surface).

16 CHAPTER 1. INITIAL CONSIDERATIONS • type 3: fuselage attitude aligned with the velocity direction. Inclined descending trajectory (30 deg with respect to the soil surface). Roll angle not null.

• type 4: fuselage attitude aligned with the velocity direction. Inclined descending trajectory (45 deg with respect to the soil surface).

The statistical analysis conducted shows that the most frequent impact is that of type 4 (Fig 1.1). This type of impact is often caused by a loss of control of the aircraft due to screw or stall in an attempt to carry out a off-field landing. The category "Other" in the Fig 1.1 includes the impact against structures like trees, pylons, rocks etc.. In this category are included also the impacts where the reconstruction of the accident is impossible.

Figure 1.1: Statistical analysis of the accidents

The impacts of type 4 are the most serious for the mortality of the event (Fig 1.2). As it is possible to see, the category "Other" has a high probability of death. In fact, an impact against trees, pylons or rocks involves: very high accelerations, fatal injuries caused by external objects that penetrate the aircraft and fatal injuries caused by components of the aircraft that are forced against the body of the passanger.

1.1. STATISTICAL ANALYSIS 17

Figure 1.2: histogram of mortal injuries

The impact velocity cannot be always determined with certainty. The only cases where the velocity is possible to know are the following:

• The pilot is able to give direct testimony.

• The circumstances are such that it is possible to hypothesize a rather precise value.

• If a data logger is present on the aircraft. The data logger is a device that register the velocity, the altitude and the position of the aircraft through a gps system.

The average impact velocity is around 80 km/h. It is important to point out that no passengers survived an impact that occurred at a velocity higher than 70 km/h.

1.1.2

Soil

Another fundamental goal of the statistical analysis is to identify the most frequent type of impact soil. The mechanical characteristics of the soil against which the aircraft collides play an important role in the determination of the

18 CHAPTER 1. INITIAL CONSIDERATIONS dynamics of the impact. A necessary condition for a successful test is to have a clear idea of the type of soil to be reproduced in the laboratory. The categories for the classification of the soils are:

• Typical grass of the track of a flight field or of an airfield. • Rocky soil near mountainous areas.

• Typical grass of an agricultural field. • Airport runway or road.

• Water of rivers or lakes.

• "Other": impact against trees, pylons or constructions of various type. The most frequent soil in case of impact is grassy (Fig 1.3). The ground reproduced in laboratory is that of a generic field of grass (the one that a pilot would chose in an emergency landing).

1.2. STATE OF ART 19

1.1.3

Conclusion

The most frequent type of impact is characterized by:

• Fuselage attitude: aligned with the velocity direction (45 deg with re-spect to the soil).

• Soil: not cultivated grass.

• Impact velocity: 20 m/s. approximatly.

These conditions lead to the death of the pilot on average. The impact velocity is decreased at 18.33 m/s in the examined case in order to make the impact survivable. A schematic draw of the impact of type 4 is shown in the Fig 1.4.

Figure 1.4: Impact of type 4

1.2

State of art

The experimentation is the basis of the validation of the theoretical models. The numerical models have evolved a lot in the recent years. Despite this, it is impossible and unthinkable to be able to perfectly reproduce every sin-gle aspect of a real phenomenon through a model built through algorithms. Moreover, an element that is not considered relevant for the correct numerical reproduction of the event could instead heavily influence the phenomenon, and vice versa. These considerations can be applied to the crash tests of the gliders. It is important to take into account all the experiments already performed in the past in order to avoid the same mistakes and, above all, to take advantages from the positive aspects derived from these tests. In particular, a real "state of art" regarding the experimental tests concerning

20 CHAPTER 1. INITIAL CONSIDERATIONS the passive safety of the light aircraft does not exist. The only existing tests are attributed to Martin Sperber, an expert at TUEV Rheinland which is a company operating in the sector of the evaluation and verification of prod-ucts and management systems. In the next sections some of the tests carried out by Sperber are reported. It is important to underline that a preliminary test is carried out before the real crash of the glider in order to understand the mechanical characteristics of the soil. In fact, the stresses acting on the impact depend on the designed soil model. The preliminary test consists in dropping a mass at different heights on the soil to be studied. The drop mass is characterized by a material having high values of stiffness and resis-tance (generally reinforced concrete). As it is said in [5], the most obvious mechanical features of the examined phenomenon are:

• The high difference in stiffness and resistance of the materials that make up the impact body and the soil. The drop mass is considered infinitely resistant and not deformable because of these differences in the materials and in favor of safety. This leads to the impossibility of damaging the drop mass. Each deformation process is assumed to be concentrated within the deformable layer. The penetration is con-sidered active because the drop mass is moving within the shock ab-sorbing layer and not the vice versa. Large irreversible deformations progressively accumulate within the soil during the penetration of the drop mass. The loads exerted during the phenomenon are impulsive and cause high displacements of large masses of soil. The breaking of the grains of the soil or the possibility of the emergence of generalized breaking mechanisms are not present.

• The impulsiveness of the event. The drop mass has a high kinetic energy at the moment of impact. This energy is almost completely dissipated during the phenomenon. The energy of the drop mass is partly dissipated locally and partly dissipated in the form of elastic waves that propagate in the surrounding environment. The impact velocity depends solely on the falling height and on the acceleration of gravity in the case of large bodies.

• The heterogeneity of the structure of the soil. The mechanical charac-teristics of the shock-absorbing layer are extremely poorer than those of the reinforced concrete drop mass. The thickness of the soil layer must be large enough in order to prevent that the drop mass hits the bottom of the tank containing the granular material. However, the thickness must be such that the force exchanged between the drop mass and the soil is not influenced by the presence of the soil-bottom tank interface.

1.2. STATE OF ART 21 This hypothesis allows to maximize the dissipated energy and to mini-mize the value of the force exchanged between the falling mass and the structure.

A study introduced by Jaeger et al. (1992) ensures that the maximum im-pact force is the sum of three different contributions: one static, one dynamic and one related to the penetration of the drop mass. The static contribution consists in the bearing capacity of the foundation of a circular section equiv-alent to the maximum section of the projectile. This contribution can be assessed directly on site through loading tests on plate. The dynamic contri-bution depends on the penetration velocity of the projectile: the greater the velocity is, the greater the value of the dynamic contribution is. The third contribution is associated with geometric effects of the second order, that is, the greater the sinking of the object is, the greater the force necessary to allow its penetration is.

1.2.1

Test on the glider DG-800

The first test is carried out in 1994. Friedel Weber witnessed the test. He is a fan of the gliders and was lucky enough to speak directly with Sperber. Weber wrote an article containing the most important considerations concerning the crash. This article is published on the official website of the DG [7]. The experiment involves the crash of a glider against a container which is inclined of about 45 deg with respect to the horizontal. The part behind the wing is not built. The total weight, the center of gravity and the moment of inertia of the fuselage do not reflect the real conditions of the aircraft because the missing parts of it are not restored in any way. The fuselage is in fiberglass and it is made through the molds of the DG-800 that are supplied by the company Glaser-Dirks. The experiment simulates the most serious impact for the structure and therefore for the occupant himself (impact of type 4, see section 1.1.1). This kind of impact is the most frequent among the accidents of gliders. The test is carried out in the laboratory without the presence of wind and with an almost constant temperature. The fuselage is mounted on a sled that ran on rubber wheels. The glider is in solidarity with the guide even at the moment of the impact against the inclined soil. The anthropomorphic dummy adopted is the 50th percentile (Fig 1.5). The glider canopy is much wider than the one adopted for our experiment. The canopy of most of the gliders is much smaller than the one used by Sperber.

A part of the commentary expressed by F. Weber regarding this experi-ment is reported below [2]:

22 CHAPTER 1. INITIAL CONSIDERATIONS

Figure 1.5: Glider DG-800

instrument panel contacts the forward bent head of the crash dummy and pushes it backward. The weights simulating the aircraft mass push the seat back further forward. The crash dummy does not have a compression zone, it is the compression zone!”

The result of this first experiment is disastrous. The fuselage is almost completely destroyed and the anthropomorphic dummy appears to be the "compression zone": This negatively affects the survivability of the pilot and involves a reduction of the habitability of the aircraft, making the escape from the cockpit impossible. However, the test returned fairly good results because the model of the aircraft and the anthropomorphic dummy used are a good representations of the reality. Nevertheless, Sperber did not take into account some errors and inaccuracies:

• The overall weight of the glider used for the crash test was 525 kg which is an excessive weight, especially if the aircraft is not equipped with water ballast positioned in the wings.

1.2. STATE OF ART 23 • The sled through which the aircraft is accelerated is solidly bound to

the glider. This invalidates the results.

• The impact is achieved without the presence of the canopy. The canopy is not a relevant structural element. The error can be neglected. • The test was carried out with a 45 deg inclined container covered with

grass, which dimensions are too small compared to the ones of the nose of the glider. This could cause significant board effects.

• The configuration of the impact considered is not fully representative. In fact, the aircraft impacts with the wing tip in a first moment and with the nose in a second moment.

A good test of general validity is difficult to achieve. Some errors listed above are eliminated during the realization of the subsequent tests.

1.2.2

Test on a glider made in carbonium and Dyneema

fabric

In 1998 the German Ministry of Transport commissioned to Helmut Fendt and Martin Sperber a test for the experimental validation of a new technique of construction adopted for light aircrafts in order to improve their impact resistance and therefore the passive safety. The configuration and the type of impact remains the same as in 1994. The glider is built through a new technique which involves the use of carbon fiber and Dyneema fabric (Fig 1.6). Dyneema fabric is a material very similar to kevlar, which is also used for Formula 1 cars. This new hybrid formulation gives the structure a higher impact resistance than the traditional fuselages. The problem with the new construction method is the weight which is 40 kg higher than required. The compliance with the norms and the flight phase of a heavier object are the main problems of the new construction method.

The results obtained by the experiment are very good, as a journalist underlined in the "Aero Intern" magazine [6]:

“By zero an increasingly loud rolling noise can be heard. The fuselage rolls toward a container filled with dirt and grass. Two meters before contact with the container the drive chain is released. The fuselage with the large steel weights drills itself with a loud bang deeply into the dirt filled container. [...] During that test, a steel bar marked the fuselage after it plunged 130 cm into the earth, otherwise there was no other larger damage to be seen.[...] The middle G force due to the delay was approx. 16.5 G.”

24 CHAPTER 1. INITIAL CONSIDERATIONS

Figure 1.6: Glider made in carbonium and Dyneema fabric

the fuselage remained almost intact. The pilot could survive this type of impact if a good belt system and a durable headrest are used. However, there are some defects also in this experiment:

• The weight is above the norm.

• The rear part of the glider is reproduced even if the weight of the wings are represented by a steel ballast.

• The sled is linked to the glider.

• The container used is too small. In fact, the nose of the glider crossed the rear wall of the container.

• The experiment cannot be considered of general validity.

1.3

Conclusions

The two experiments analyzed can be considered good examples for our study. In fact, the experiment carried out at La.S.T. in Politecnico di Mi-lano tries to correct some of the defects described above. There are no other impact tests similar to the chosen configuration. The examined casuistry is very limited and not sufficient for a depth analysis. The aircraft structure cannot be used for a second test (as most of the objects tested for passive safety purpose). The cost of a test is therefore very high and this is one of the main reasons why there is not a high number of experiments to refer to.

1.3. CONCLUSIONS 25 The possibility to simulate the crash phenomena through accurate numeri-cal models is of fundamental importance because they allow to predict the outcome of the test and, moreover, they drastically reduce both the costs as well as the time of realization of the impacting object.

Chapter 2

Experimental Tests

The two experimental tests carried out at La.S.T. (Laboratorio di Sicurezza dei Trasporti) in Politecnico di Milano are shown in this chapter. The first experimental campaign consists in the soil deposition inside a test tank and subsequently in four impacts of a reinforced concrete mass which fall down by gravity on the soil itself starting from different falling heights. The second test includes three phases: the building of the glider following the directives given by the manufacturer, the building of the container and the deposition of the soil inside of it, and finally, the crash of the glider.

2.1

Descriptions of the impacts of the drop

mass

It is necessary to perform specific experimental tests able to describe qualita-tively and quantitaqualita-tively both the dynamic response of the impacting object as well as the propagation of the dynamic wave in the deformable medium in order to study the dynamic-structural response of the impact mass-soil system.

The tests performed at Politecnico di Milano [5] are shown in the following pages.

2.1.1

Experiment set-up

The laboratory La.S.T. has a tower equipped with a lifting system (Fig 2.1). The soil is deposited inside a test tank which is a reinforced concrete tank buried below the tower (like a pit).

The tower is made of steel and guarantees a useful falling height of almost 20 m from the bottom of the tank. The latter has a circular section with a

28 CHAPTER 2. EXPERIMENTAL TESTS

(a) Tower (b) Lifting system

Figure 2.1: The tower and the lifting system of Politecnico di Milano diameter of 10.7 m and a depth of 2.6 m. The thickness of the bottom of the tank is 50 cm (Fig 2.2). The tank can be considered rigid and fixed in relation to the stresses produced by the impacts. This is due to the higher stiffness, weight and sinking of the tank with respect to the ones of the soil.

Figure 2.2: Test tank

The drop mass used in the tests is a reinforced concrete sphere. The sphere weighs 850 kg and has a diameter of 90 cm (Fig 2.3). The sphere is characterized by a cavity of 20 cm of diameter. A plate is fixed on the bottom of the cavity. This plate is used as the fixing base on which the instrumentations for the data acquisition are placed.

2.1. DESCRIPTIONS OF THE IMPACTS OF THE DROP MASS 29

Figure 2.3: Drop mass

The following measuring instruments are used during the tests:

• Two damped piezoresistive accelerometers ENTRAN EGCS-200 posi-tioned in the center of gravity of the sphere and oriented along the z (vertical) and x directions with a full scale of 200 g (Fig 2.4).

• A damped piezoresistive accelerometer ENTRAN EGCS-50 positioned in the center of gravity of the sphere and oriented along the y direction with a full scale of 50 g.

• Eight strain gauge load cells placed on the bottom of the tank with a full scale of 2000 kg and a sensitivity of 0.05%.

• A sixteen channel programmable acquisition system (Pacific Instru-mentation Model 5400) records the data transmitted by the measuring instruments.

The strain gauge load cells allow to measure the acceleration along three orthogonal axes, one of which coincides with the vertical direction. They are fixed on the bottom of the tank through threaded dowels. The instruments are aligned along the diametrical direction of the tank in order to use the axialsymmetric configuration of the problem (Fig 2.5). The accelerometers are mounted on a steel plate, which is fixed on the bottom of the cavity of the reinforced concrete drop mass through four threaded dowels (Fig 2.6)

30 CHAPTER 2. EXPERIMENTAL TESTS

Figure 2.4: Damped piezoresistive accelerometer

2.1. DESCRIPTIONS OF THE IMPACTS OF THE DROP MASS 31

Figure 2.6: Installation of the accelerometers in the cavity of the drop mass

2.1.2

Shock absorbing layer

The shock-absorbing material consists of quarry inert. The maximum diam-eter of the grains is about one centimdiam-eter. The granular material is deposited inside the test tank without compacting it through a mechanical shovel. The resulting damping layer has a constant thickness of 2 m. The soil is reshaped using an excavator between one impact and the other. This reshaping con-sists in the remotion around the point of impact of almost 1 m of thickness of the layer, and subsequently, the soil is deposited again. The part of the shock absorbing layer near the bottom of the tank is not removed in order to avoid the risk of damaging the load cells. A substantial variation of the density between the superficial and the deep portions of the soil is gradually created due to this procedure.

2.1.3

Performed impacts

The whole experimental campaing performed in Politecnico di Milano under commission of the Structural Engineering Department (Calvetti et al., 2005 [4]) is organized according to four series of impacts:

• Series S1: tests performed on 20 deg inclined soil with respect to the horizontal.

• Series S2: tests performed on horizontal soil.

• Series S3: tests performed on unrestricted horizontal soil.

• Series S4: tests performed on horizontal soil covered with expanded clay.

32 CHAPTER 2. EXPERIMENTAL TESTS Only the impacts of the series S2 are considered for the calibration and the validation of the soil model. The characteristics of this series are summarized in the Tab 2.1.

Table 2.1: Characteristics of the impacts carried out in the series S2 Tests Soil inclination Falling height Impact velocity

(deg) (m) (m s) 1 0 05.00 09.90 2 0 10.00 14.00 3 0 13.70 16.40 4 0 18.45 19.00

The impact force, the acceleration, the velocity and the displacement (penetration) along the vertical axis of the drop mass are the physical quan-tities analyzed during the tests. Their corresponding graphs are reported in the Fig 2.7.

The deceleration of the drop mass is characterized by a particular double peak. This particular trend is due to the presence of a sharp discontinu-ity between the superficial portion of the remodeled soil and the deeper one which has undergone a progressive compaction. The impact phenomenon has a relatively long duration, which is enough to allow the dynamic wave to strike the bottom of the tank and to be reflected. A sort of an additional "lift" on the drop mass is generated due to the collision of the reflected dy-namic wave with the mass itself during its penetration. The stresses in the horizontal direction are negligible. A slight rebound phase is found in all the cases. The rebound phase starts when the velocity becomes negative, that is, around 0.06 s and 0.08 s. The impact force and the penetration of the drop mass increase with the increasing of the falling height. Moreover, the impact assumes a more markedly impulsive character with the increasing of the falling height. The penetration of the drop mass exceeds one meter for relatively low values of the impact velocity. The dimension of the rebound in-creases when the falling height dein-creases. This means that the kinetic energy possessed by the drop mass is not completely dissipated during the impact. The exit velocity is very small and it is insufficient to release the impact mass from its footprint. The denser layer of the soil in the deep part of the tank influences more the value of the peak in the force-time curve when the falling height is higher. The main characteristics of the tests carried out in the series S2 are summarized in the Tab 2.2.

2.2. DESCRIPTION OF THE GLIDER CRASH EXPERIMENT 33

(a) Impact forces (b) Accelerations

(c) Velocities (d) Displacements

Figure 2.7: Graphs of the physical quantities researched during the experi-mental tests

Table 2.2: Summary of the main characteristics of the series S2

Number of impacts Max. falling height (m) Max. impact energy (kJ) Soil material Lower boundary condition 4 18.45 155 sand, loose gravel Rigid underground plate

2.2

Description of the glider crash experiment

The test is carried out in September 2009 at La.S.T. and it is well documented in [1]. The experiment consists in a crash of a glider (V1/2 Rondine) against an inclined container of about 45 deg. The horizontal speed of the glider at the impact moment is 18.33 m/s. The mock-up aircraft used in the test is not provided with the wing and with the tail section, but they are replaced by an appropriate ballast that restore the original values of the mass and of the moments of inertia. The highest attention is given to the repruduction of the

34 CHAPTER 2. EXPERIMENTAL TESTS pitch moment of inertia, while no importance is given to the restoration of the other moments due to the symmetry of the phenomenon and the excellent balance of the aircraft. The glider is mainly made by composite material. It is important to underline that the terrain is to be considered infinitely extended in a real impact, however, the use of a container is necessary to represent the inclined soil profile due to the limited space of the laboratory. The container is considered a great source of intrusiveness. For this reason the container is built using the maximum space available in the laboratory.

2.2.1

Glider construction

The molds of the V1/2 Rondine used to be stored in a deposit at the back of the La.S.T. in via Durando. It was decided to build only the fuselage of the glider because of the impossibility to perform the experimental test with the whole aircraft, which is long almost six meters. The fuselage molds consist of two half-shells joined together by screws. These molds were cut at a distance of about 2300 mm from the nose in order to contain the entire cockpit and half of the attacks of the wings. Subsequently, these two half-shells are separated and cleaned in order to be ready for the lamination process. The laminates used in the project are the following:

• fabric 1717, 160 g m2 arranged at 45 deg. • fabric 1102, 280 g m2 arranged at 45 deg. • fabric 1035, 200 g m2 arranged at 90 deg. • fabric 1102, 280 g m2 arranged at 45 deg. • fabric 1035, 200 g m2 arranged at 90 deg. • fabric 1102, 280 g m2 arranged at 45 deg. • fabric 1102, 280 g m2 arranged at 90 deg. • fabric 1102, 280 g m2 arranged at 45 deg. • fabric 1102, 280 g m2 arranged at 90 deg.

All the fabrics used have an internal distribution of the fibers of 0−90 deg. The distribution shown in the previous list refers to how the fabrics are disposed with respect to the rolling axis of the aircraft.

A resin is passed at the end of the lamination phase. The resin poly-merization was carried out using the technique of empty bag (Fig 2.9). The

2.2. DESCRIPTION OF THE GLIDER CRASH EXPERIMENT 35

Figure 2.8: Lamination phase

half-shells are wrapped in a layer of peel-ply, which is a plastic material that allows the passage of the air and of the excessing resin through vacuum pomps. The characteristics of the finished object are uniform using this tech-nique. These two halves of the fuselage are joined togheter at the end of this process.

(a) peel-ply (b) Assembly of the molds

Figure 2.9: The peel-ply method and the assembly of the molds A ballast to be joined to the back of the fuselage is designed. The struture of the ballast and its particular shape are necessary to re-establish the weight and the moments of inertia of the whole aircraft. The structure is made of iron and consists of four C-shaped beams and two plates placed vertically. Moreover, an additional mass of 32 kg is fixed to the sides of the fuselage (at the same position of the seat approximatly). The wings are restored through an iron beam. The beam has a square profile section of 4 mm of thickness

36 CHAPTER 2. EXPERIMENTAL TESTS and two HEB beams are welded at its ends. It is fixed in correspondence of the wing attacks 2.10.

(a) Ballast (b) Wingbeam

Figure 2.10: The designed ballast and the wingbeam

The wingbeam is necessary because it must hold the aircraft on the sup-ports, which are fixed to the sled from where the "half glider" is "launched". The total weight of the assembled glider is 343 kg. The laboratory has a launch track (rails) long enough to allow the sled, and so the glider, to reach the desired speed of 18.33 m/s (Fig 2.11). The sled used is called "gullwing" due to its lowered shape between the rails. This shape is necessary to give stability to the sled during the arresting phase. The arresting phase occurs through the deformation of iron rods that are mounted in the front part of the sled. The sled flows along two guides that extend for 45 m and it has the capacity to reach the maximum speed of 25 m/s. The main characteristics of the glider are summarized in the Tab 2.3. The final assembly of the glider with the additional components is shown in the Fig 2.12. For more informa-tion about the projects of the supports, of the ballast and of the wingbeam see the reference in bybliography [1].

Table 2.3: The main characteristics of the glider

Total weight C.G. X coord. C.G. Z coord. Moment of inerzia

(from the nose) (from the floor) (Pitch axis)

(kg) (m) (m) (N · m2₎

2.2. DESCRIPTION OF THE GLIDER CRASH EXPERIMENT 37

(a) Designed Support (b) Rails

Figure 2.11: The supports designed at La.S.T. and the rails of its "launch

track"

Figure 2.12: The final assembly of the glider with the additional components

2.2.2

Building and filling of the container

The box containing the soil is a delicate point in the setting up of the test. Its functionality must be analyzed with particular attention in order to know the problems that may arise during the phenomenon. The soil should be considered infinitely extended in a real impact from a statistical point of view. In the case of the crash test carried out at La.S.T., a container had to be used because of the limited space of the laboratory. The presence of this structure is a great source of intrusiveness. The introduced intrusiveness

38 CHAPTER 2. EXPERIMENTAL TESTS consists of two phenomena:

• The rigidity of the soil increases.

• The possibility that the dynamic wave propagates up to the walls, then bounces and returns to the aircraft.

These inconveniences can be avoided in two ways: • The soil is represented by a very large pile of land. • Build a container.

The second option is chosen. The first option could bring better results, but it is discarded for reasons related to the available space of the laboratory.

The container should have the volume of 4.8x4.8x4.8m3 _{to avoid the board}

effects. The board effects are the forces exchanged between the glider and the soil due to the reflection of the dynamic wave against the walls of the container. Such a big container does not fit into the space available in the laboratory, that is the reason why a smaller one is built. The structure is made of 30 mm thick poplar plywood panels in order to avoid a too high increase of the rigidity of the soil. Two C-shaped beams made by steel are welded on the side walls in order to prevent a possible breakage due to a strong impact. A vertical edge is placed in the front part of the box to contain the soil and to prevent it from rolling on the floor. The technical drawings of the container are shown in the Fig 2.13.

(a) Side view (b) Front view

2.2. DESCRIPTION OF THE GLIDER CRASH EXPERIMENT 39 The soil is put in the container once it is built. Layers of ground are inserted in the box, and subsequently, they are compacted using a special compactor in order to realize a profile of the soil inclined of 45 deg. The attendant loaded the soil into the container with a small scraper. He grad-ually compacts the newly-laid soil (Fig 2.16). A problem emerged during this phase: the used scraper was not suitable for the purpose. In fact, the worker had difficulties in climbing and compacting the layers of the soil when the thickness of the ground, which was already inserted, reached about one meter. A manual compaction was tried but with poor results (Fig 2.14).

Figure 2.14: Manual compaction

As a last attempt, the soil was wetted in order to make it more compact, but, also with this solution, both the higher layers of the soil as well as the intermediate ones did not have the adequate degree of compactness. The final result is the very low compactness of the soil and the low increase of the rigidity of the soil (due to the presence of the container). The final configuration of the filled container is shown in the Fig 2.15. For more information about the project of the container see the paper reported in the bybliography [3].

40 CHAPTER 2. EXPERIMENTAL TESTS

(a) Real (b) 3D model

Figure 2.16: The container

2.2.3

Instrumentations for the data acquisition

The glider is instrumented in order to be able to reconstruct precisely its dynamic during the phenomenon. Four accelerometers are mounted: two at the nose and two at the poop. A gyroscope oriented along the pitch axis is mounted in order to capture the rotations of the fuselage during the impact. The accelerometers are of the piezoresistive type with a full scale of 250 g. Their frequency response follows the standard SAE J211 for crash test experiments. The accelerometers record the longitudinal and the vertical accelerations (x and z axis). The transversal accelerations are not recorded because they are not considered relevant due to the symmetry and to the excellent balance of the aircraft. The trends of the velocity and of the displacement are obtained by numerical integration of the acceleration. Other accelerometers and load cells are mounted on the anthropomorphic dummy in order to register the forces acting on it. For more information about the anthropomorphic dummy, please refer to the bibliography [1]. A detail of the fixing of the sensors on the fuselage and their relative cables is shown in the Fig 2.17.

2.2. DESCRIPTION OF THE GLIDER CRASH EXPERIMENT 41 All the signals of the sensors are combined in a single cable, which is connected to a remote station located in an office next to the test area. The cable is laid parallel to the rails in order to dampen the "whip" effect due to the non-negligible mass of the cable itself. An acquisition channel (from 1 to 15) is assigned to each sensor. The sampling frequency of the system is 12500 Hz. The disposition of the sensors is shown in the Fig 2.18, while a summary of the instrumentations used for the data acquisition is reported in the Tab 2.4.

Table 2.4: Summary of the instrumentations used for the data acquisition

Channel number Acquired data Unit lenght Type of sensor Fixing position 1 Strength in the lumbar tract N Load cell Dummy 4 Y-acceleration of the dummy g Accelerometer Dummy 5 Z-acceleration of the dummy g Accelerometer Dummy 6 Pitch angular velocity of the glider m

s Gyroscope Glider

7 X-acceleration of the glider (front) g Accelerometer Glider 8 Z-acceleration of the glider (front) g Accelerometer Glider 10 Z-acceleration of the glider (rear) g Accelerometer Glider 11 Force in the belt N Load cell Belt 14 X-acceleration of the dummy g Accelerometer Dummy 15 X-acceleration of the glider (rear) g Accelerometer Glider

42 CHAPTER 2. EXPERIMENTAL TESTS

2.2.4

Glider crash

The experiment of the crash of the glider can be divided in six phases: • Acceleration of the sled.

• Motion for inertia of the sled which is neither accelerated nor braked. • Impulsive braking of the sled and release of the glider.

• Free flight of the glider. The aircraft evolves according to its own dynamics.

• Impact and penetration of the glider inside the soil. • Secondary impact.

The phases that are taken into consideration in this paper are the im-pact of the glider, its penetration inside the soil and the secondary imim-pact. The accelerometers placed on the front part of the glider are broken during the impact. The acquired data of the load cell placed in the belt cannot be considered valid because of an error in the calibration of the sensor. Both the accelerometers placed in the rear part of the glider as well as the ones connected with the anthropomorphic dummy returned valid results. The longitudinal acceleration of the glider, the angular velocity and the rotation angle along the pitch axis are reported in this paper. The entire collected data of the experiments and their relative graphs are reported in the bybli-ography [1].

The duration of the impact (primary impact, penetration and secondary im-pact) is almost 0.25 s. The trend of the longitudinal acceleration is reported in the Fig 2.19.

2.3. FILTERING 43 The acceleration is characterized by two peaks. The first peak represents the primary impact, while the second is due to the impact of the nose of the glider with the bottom of the container. The maximum acceleration reaches 20 g. The glider performs a moderate positive rotation along the pitch axis (about 1.6 deg). A negative rotation angle is presented during the first instants of the impact. This is due to the free flight phase of the glider, where the gravity "pushes" the nose of the aircraft towards the floor. The slope of the rotation along the pitch axis is obtained by numerical integration of the angular velocity, whose graph is reported in the Fig 2.20.

(a) Angular velocity (b) Fuselage attitude

Figure 2.20: The angular velocity of the glider and its rotation angle

2.3

Filtering

The norm that dictates the basic criteria for the filtering of the data provided by accelerometers and load cells in the aeronautic field is the SAE J211. The SAE J211 imposes that the sampling frequency is approximately 10 times higher than the maximum value of the signal to be detected. The sampling frequency for the test of the glider is set to 12500 Hz in order to satisfy the norm. The others criteria imposed by the SAE J211 concern the types of low pass filters to be used in order to attenuate all the harmonics that are above the frequency of interest (shear frequency and filter order). The various categories of filter are shown in the Fig 2.21.

Moreover, constraints on the calibration of the instruments, on their sen-sitivity and on the environmental conditions are exposed in this norm. Great attention must be given to the value of the Channel Frequency Class (CFC). This value represents the type of the filter. The SAE J211 includes four CFC indices that indicate the value of the shear frequency. The norm provides a

44 CHAPTER 2. EXPERIMENTAL TESTS

(a) Class 600 and 1000 (b) Class 60 and 180

Figure 2.21: Types of low pass filter

table which indicates the CFC to be used for each situation. An extract of the aforementioned table is reported below (Tab 2.5). A MatLab script is necessary to apply the filter on the acquired data. The inputs of the script are the CFC value and the acquired data. The outputs of the script are the filtered data. The data shown in this paper are filtered using a CFC 60 filter.

Table 2.5: CFC indexes

Typical Test Measurements Channel Frequency Class (CFC)

Total vehicle comparison 60

Thorax, sternum accelerations 1000

Lumbar, forces 600

2.4

Dynamic of the soil

The impact point is expected at a relatively low altitude. The soil has an acceptable compaction at the impact point (0.850 m from the floor). Nev-ertheless, the strike suffered a considerable attenuation due to the lack of compactness of the overlying layers which also brought the nose of the glider to impact against the rear wall of the container. The main reasons why the compactness of the earth is not the ideal one are:

• The soil is not well compacted.

2.4. DYNAMIC OF THE SOIL 45 Regarding the second point, even if the earth had been compacted per-fectly, it would have been for horizontal layers and the glider would have impacted with an effective angle of 45 deg to the surface, but with a relative angle between its axis and the compaction direction of 0 deg. The relative angle between the glider axis and the direction of the compaction of the soil is 45 deg in a real case (the soil is horizontal, while the glider is inclined of 45 deg). This difference in the relative angles (Fig 2.22) leads to a dif-ferent impact dynamic between the real case and the case simulated in the laboratory:

• Real case (relative angle 45 deg): the soil responds with greater force. • Laboratory case (relative angle 0 deg): the soil is hit in the worse

di-rection. The exchanged forces are minor.

It is clear that a compacted soil with a relative angle of 45 deg is really complicated to generate. In the conclusion of this experiment it is seen that it is not obvious that a more compacted (and therefore hard) soil produces higher exchanged forces than a softer soil. The direction of compaction is fundamental.

(a) Real case (b) Laboratory case

Chapter 3

Discrete Element Method

The Discrete Element Method (DEM) considers bodies as a group of particles and each particle is considered indipendent from the others. In this way, the real discrete nature of the bodies is taken into account. This method can reproduce an impact simulation properly. Infact, it can evaluate the motion of the particles by computing the dynamic equilibrium equations, while the material structure is free to evolve during the motion of these particles. However, as far as the discrete element method is concerned, the size of the elements is the only difference between the drop mass and the grains of the soil, as well as the material obviously. Moreover, DEM is able to reproduce deformation mechanisms characterized by high displacements and re-adjustments of the structure of the shock-absorbing granular material. The only acting forces during an impact phenomenon are: the weight force, the interactions forces between adjacent particles, the force and inertia moment exchanged at the moment of the impact.

The computational procedure follows a logic similar to that of the finite element method. Each calculation step consists of two phases. In the first phase, the second law of dynamics is applied to each particle; while in the second phase, the force-displacement constitutive bond, which is defined for the existing contacts, is taken into consideration. Through this iterative procedure, it returns in succession for each calculation cycle:

• the motion of the particles caused by the internal forces (weight and in-ertia) and by the external forces (interaction forces with the contacting particles) acting on the particles themselves;

• the contact forces evaluated through the relative displacements between the particles.

Figure 3.1 shows the typical DEM model algorithm, while Figure 3.2 shows the interactions scheme between the elements.

48 CHAPTER 3. DISCRETE ELEMENT METHOD

Figure 3.1: DEM algorithm

Figure 3.2: Interactions scheme between the elements

3.1

Hypothesis and initial considerations

At every calculation step, the simulation of a physical problem using a DEM model involves the updating of the fundamental quantities of the elements (position, speed and acceleration) and of all the interactions existing between them during the contact (the positions and the transmitted forces).

Yade (Yet Another Dynamic Engine) is the software used in the simulations presented below. Yade is an "open source" program useful for the construc-tion of discrete numerical models. The calculaconstruc-tion parts are written in C++.

3.1. HYPOTHESIS AND INITIAL CONSIDERATIONS 49 Moreover, it allows new algorithms implementations and interfaces are pos-sible too. Python is used for the construction of quick and concise scenes, simulation control, post-processing and debugging [8].

The possibility to use spherical bodies makes the model implementation much simpler and more efficient. However, the use of spherical particles is too dras-tic as a simplification of the real shape of the grains characterizing a sandy soil material. In particular, this simplification leads to the impossibility to faithfully reproduce the resistance values (friction angle) commonly observed in a granular material. In fact, the spherical elements can rotate with re-spect to each other, regardless of the surface friction value assigned to it. In order to counter these undesirable effects, all the simulations are performed by inhibiting the particles’ rotation, following a documented methodology in the bibliography (Calvetti et al., 2005).

The discrete element method is based on the modeling of rigid elements with yielding contacts, which means that the trasmission of the contact forces presupposes an overlap between the particles at the contact point. This ap-proach is consistent with the reality because granular materials, such as sand or gravel, are made of higly stiff and resistant particles (relatively to ordinary loads). For this reason, the material deformation is due to the resettling of the particles rather than their significant deformation. In particular, this deformation is limited to the contact points only, where a high concentration of stress exists. The overlap between the DEM model elements represents the real contact points of the deformation of the particles, and must remain contained to maintain the validity of the method. The contact forces are evaluated through the scheme shown in Figure 3.3

Figure 3.3: Particles contacts modeling

The interaction between two elements is decomposed in the normal and tangential directions relative to the contact point. This interaction is

mod-elled through a springs’ system, having a normal stiffness Kn, a tangential

stiffness Kt, and a sliding block ("friction" or "slider") characterized by the

compres-50 CHAPTER 3. DISCRETE ELEMENT METHOD sion, while the tangential contact forces are limited by the Coulomb’s friction law. The corresponding intergranular friction angle is:

φ = arctan µ. (3.1)

For any impact the force exerted between an impacting mass and a ground depends mainly on the rigidity of the considered materials, rather than on the friction angle, as suggested in Calvetti et al. 2005,. For this reason the friction angle values of the glider and of the drop mass can be considered null. Another important parameter in a DEM simulation is the numeric damping. In the case of quasi-static simulations, the energy dissipated by friction within the shock-absorbing layer leads to the origin of parasitic os-cillations around the equilibrium configuration. It is therefore necessary to insert a dimensionless parameter in the equation of motion, in order to dump the aforementioned oscillations. In this way the new equilibrium condition is reached more quickly. It is important to clarify that if "quasi-static" stresses are applied, this parameter does not play a fundamental role, because it limits and regulates only the response of the model and allows the achieve-ment of the equilibrium in a shorter time. However, in the case of dynamic stresses, its influence can not and must not be neglected. Therefore, in all the simulations presented in this paper, the numerical damping is null, as specified in Calvetti et al. 2005 [4].

3.2

YADE (Yet Another Dynamic Engine)

Figure 3.4: Yade

The mathematical model used by the software Yade at each computing cycle is shown in this section.

3.2.1

Collision detection

First of all, the existing interactions within the simulation are identified through a particular algorithm. This must be repeated for each calculation

3.2. YADE (YET ANOTHER DYNAMIC ENGINE) 51 cycle since, at each step, the particles change their mutual position. In Yade there are different contact algorithms and they are subdivided according to classes located within the "IGeom Functor" library. In particular it is necessary to define which kind of geometries will collide during the simulation (e.g. sphere + sphere, sphere + facet, etc . . . ). The inputs of the algorithm are:

• The geometry of the element, identified according to the index "Ig". • The material properties of the contact elements, identified according

to the index "Ip".

3.2.2

Material

Yade contains different material typologies within its code (e.g. elasto-plastic model, visco-elastic model, concrete model, etc. . . ). In the examinated case an elasto-plastic material with the addition of cohesion parameters is set. The parameters that mainly influence the behavior of the simulated body are: • Young’s modulus. • Poisson’s coefficent. • Friction angle. • Density. • Normal cohesion • Shear cohesion

Through these parameters a normal stiffness Knand a tangential stiffness

Kt are defined. The first is directly related to the Young’s module, while

the second depends on the Poisson coefficient, which is defined as the ratio between the two stiffnesses. The algorithm computes the normal stiffness as the stiffness of two springs in series which have a length equal to the radius of the particles taken into consideration. The aformentioned logic is reported in Figure 3.5.

In the case of contact between sphere and facet, the facet is taken into consideration as it has a spring lenght equal to twice the radius lenght of the sphere. Due to this logic, it is possible to assert that in Yade any body can not be considered perfectly rigid. It is important to underline that Yade

52 CHAPTER 3. DISCRETE ELEMENT METHOD

Figure 3.5: two springs in series representing the normal stiffness between bodies in contact

sets the facet element as non-dynamic body by default. In this way the facet elements do not have mass and inertia, and in this case, if a sphere comes into contact with a facet element, it is automatically affected by a force equal and opposite to the one which the sphere itself is subjected.

In the contact case between two spheres of elasto-plastic material, knowing the initial radius of the spheres, the displacements ∆l of the two springs and

the Young’s modulus E of the material, the spring stiffness Kn is calculated

with the following equations:

∆l = ∆l1+ ∆l2. (3.2)

Ki = Ei · ˜Li. (3.3)

Where ˜Li is a characteristic length. It is equal to the spheres diameters

in the illustrated case.

Kn·∆l = F = F1 = F2 =⇒ Kn·∆(l1+ l2) = F (3.4) Kn·( F K1 + F K2 ) = F =⇒ K−1 1 + K −1 2 = K −1 n (3.5) Kn= ( K1· K2 K1+ K2 ) =⇒ Kn = ( E1· ˜L1· E2· ˜L2 E1· ˜L1+ E2· ˜L2 ) (3.6)

3.2. YADE (YET ANOTHER DYNAMIC ENGINE) 53

3.2.3

Contact model

Generally, in DEM models, the contact laws are defined either through strain and deformation or through forces and displacements. The constitutive law of the model under examination was originally proposed by Cundall, and it uses forces and displacements. As explained in the previous paragraph, the forces depend on the material of the particles in contact.

The Cundall & Strack law is non-cohesive elasto-plastic. Two foundamental laws are used in this paper:

• Law2_ScGeom _FrictPhys_CundallStrack.

• Law2_ScGeom6D_CohFrictPhys_CohesionMoment.

the first one is the traditional Cundall’s linear elastic-plastic law. This law has three parameters: tangential stiffness, normal stiffness, and friction coefficient. It modelizes dry contacts (no tensile force is allowed). It is useful for simulating linear compression and for Mohr-Coulomb plasticity surface without cohesion. The second law is an augmented version of the previ0us one with cohesion and torques at contacts. Creep is optional. In particular this law results a good choice for the simulation of linear traction-compression-bending-twisting, with cohesion+friction and Mohr-Coulomb plasticity sur-face. Substantially this law adds adhesion and moments to the previous one. The differences between the two laws can be found in a different way of considering the normal and shear force:

• In the Law2_ScGeom _FrictPhys_CundallStrack the normal force is

(with the convention of positive tensile forces) Fn = min(Kn∗ un,0).

The tangential force is Ft= Kt∗ ut, the plasticity condition defines the

maximum value of the tangential force: Fmax

t = Fn∗tan(φ), with φ the

friction angle and un and ut the normal and tangential displacements

respectively.

• In the Law2_ScGeom6D_CohFrictPhys_CohesionMoment the normal

force is (with the convention of positive tensile forces) Fn= min(Kn∗

(un− upn), an), with an the normal adhesion and upn the plastic part

of normal displacement. The tangential force is Ft = Kt ∗ ut, the

plasticity condition defines the maximum value of the tangential force,

by default Fmax

t = Fn∗ tan(φ) + at, with φ the friction angle and atthe

shear adhesion. If the maximum tensile or maximum tangential force

is reached, the cohesive link is broken, and an, at are set back to zero.

If a tensile force is present, or the contact is lost, the shear strength is

Fmax

54 CHAPTER 3. DISCRETE ELEMENT METHOD

are computed using a linear law with moduli respectively kb and kt, so

that the moments are : Mb = kb∗Θb and Mt = kt∗Θt, with Θb,t the

relative rotations between interacting bodies. The maximum value of moments can be defined and takes the form of rolling friction.

With this premises in mind, it is possible to introduce the computational process performed for the evaluation of the forces at each calculation cycle.

Knowing the normal displacement unand the tangential displacement ut, the

normal and tangential forces are calculated for each new generated contact:

Fn = Kn· un· n. (3.7)

F_tt= Kt· ut. (3.8)

At this point the tangential force is only approximated. In order to obtain the correct value, the following relationship is applied:

Ft =    Ft t · |Fn|·tan φ Ft t se |Fn| > |Fn| ·tan φ Ft t se |Fn| < |Fn| ·tan φ

Finally the overall strength is given by: F = Fn+ Ft and it is applied to

both the spheres in contact. The force acts at the contact point C, which is different from the center of the spheres, then a twisting moment arises. Therefore, the complete system of forces is the following:

             F1+ = F F2+ = −F T1+ = d1·(−n) · F T2+ = d2·(n) · F

Once the exchanged forces are obtained, the displacements generated by them are calculated by numerical integration. These displacements will be used in the next iteration for the calculation of the forces following the same logic explained above.

Chapter 4

Numerical Model

The implemented numerical model of the crash of the glider and of the drop test are shown in this chapter. The phenomena are studied using a Discrete Element Method (DEM). The software used to develop the simulation is Yade.

To be more specific, the following sections are focused on the description of the simulations and their implementational problems.

4.1

Simulations Description

The simulations needed to reproduce the phenomena correctly and coherently are seven:

• Deposition of the particles inside the test tank.

• Four different impact tests at the falling heights 18.45, 13.7, 10 and 5 m.

• Container filling. • Glider crash.

It is important to underline the reason for such a high number of sim-ulations. The first two points are needed to calibrate and validate the soil model that will be used afterwards for the simulated crash of the glider. In fact, as explained in the second chapter, the experimental soil used for the tests of the drop mass impact and for the crash of the glider were the same, with the exception of a little bit of compaction and wetting of the soil of the container. This process is proved to be necessary in order to completely fill the container volume. This implied a modification in the charateristics of the