Ecole Sigle

Doctorale

Renée El

Melhem Bat

Blaise Pascal

Insa : R Gourdon

M Stéphane

M Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

Isabelle Mme

Von Bueltzingloewen

Barres

d'intensité des contraintes dynamique, glace

Keywords: crack propagation, dynamic stress intensity factor, ice, split Hopkinson pressure bar, extended finite element method propagation de fissure, facteur

The phenomena relating to the fracture behaviour of ice under impact loading are common in civil engineering, for offshore structures, and de-ice processes. To reduce the damage caused by ice impact and to optimize the design of structures or machines, the investigation on the dynamic fracture behaviour of ice under impact loading is needed.

This work focuses on the dynamic crack propagation in ice under impact loading. A series of impact experiments is conducted with the Split Hopkinson Pressure Bar. The temperature is controlled by a cooling chamber. The dynamic process of the ice fracture is recorded with a high speed camera and then analysed by image methods. The extended finite element method is complementary to evaluate dynamic fracture toughness at the onset and during the propagation.

The dynamic behaviour of ice under impact loading is firstly investigated with cylindrical specimen in order to obtain the dynamic stress-strain relation which will be used in later simulation.

We observed multiple cracks in the experiments on the cylindrical specimens but their study is too complicated. To better understand the crack propagation in ice, a rectangular specimen with a pre-crack is employed. By controlling the impact velocity, the specimen fractures with a main crack starting from the pre-crack. The crack propagation history and velocity are evaluated by image analysis based on grey-scale and digital image correlation. The main crack propagation velocity is identified in the range of 534 ± 83m/s. It slightly varies during the propagation, first increases and keeps constant and then decreases.

The experimentally obtained parameters, such as impact velocity and crack propagation velocity, are used for simulations with the extended finite element method. The dynamic crack initiation toughness and dynamic crack growth toughness are determined when the simulation fits the experiments. The results indicate that the dynamic crack growth toughness is linearly associated with crack propagation velocity and seems temperature independent in the range -15 ∼ -1 • C.

Acknowledgment/Remerciements

Pursuing a Ph. D likes walking in the desert. You do not know how far you have come and where you are going although you have the destination in the beginning. But in this journey, I am not alone, many people inspired, helped and accompanied me. Before I finish my thesis, I am delighted to give my heartfelt acknowledgements.

First of all, I acknowledge the help, support and guidance of my supervisor, NELIAS Daniel, with the utmost gratitude. I appreciate that he gave me the opportunity to study in LaMCoS as a Ph.D student. It's a new world for me. Living in a different culture, studying with knowledgeable teachers, working with excellent colleagues, all of these, make me different.

Meanwhile, I am grateful to my co-supervisor MAIGRE Hubert. He taught me the knowledge on fracture mechanics and shared his experiences on SHPB experiments. Most of all, I want to thank him for persistent supporting me. Thanks to his guidance, encouragement and patience, when I failed in experiments, made no progress in simulations, I never give it up. Without his help, I couldn't finish my thesis.

Appreciation should be also going to CHAUDET Philippe, GUILHERMOND Jean-Pascal, VALVERDE Paul for their help on my experimental studies such as specimen machinery, apparatus installation.

I appreciate the support from my colleagues. ROUX Louis and LION Nicolas exchanged the SHPB experimental experience to me. HABOUSSA David, CHEMIN Alexandre and JAN Yannic shared the knowledge of the software EUROPLEXUSE with me. The discussion with ZHAO Lv about the problem on dynamic fracture mechanics inspired me.

I also appreciate LACROIX Samy, BOUCLLIER Robin, ZHANG Yancheng, XIE Jing, ROUX Clément, AZEGGAGH Nacer, LU Ye, ZHU Wenqi, FEKAK Fatima-Ezzahra etc. I spent joyful days with them in LaMCoS.

I acknowledge the financial support provided by China Scholarship Council (CSC). I am eternally indebted to my parents for their infinite love since I was born. And I am also indebted to my husband, for his company and support.

Résumé

Les phénomènes liés au comportement à la rupture de la glace sous impact sont fréquents dans le génie civil, pour les structures offshore, et les processus de dégivrage. Pour réduire les dommages causés par l'impact de la glace et optimiser la conception des structures ou des machines, l'étude sur le comportement à la rupture dynamique de la glace sous impact est nécessaire.

Ces travaux de thèse portent donc sur la propagation dynamique des fissures dans la glace sous impact. Une série d'expériences d'impact est réalisée avec un dispositif de barres de Hopkinson. La température est contrôlée par une chambre de refroidissement. Le processus dynamique de la rupture de la glace est enregistré avec une caméra à grande vitesse et ensuite analysé par des méthodes d'analyse d'images. La méthode des éléments finis étendus complète cette analyse pour évaluer la ténacité dynamique.

Au premier abord, le comportement dynamique de la glace sous impact est étudié avec des échantillons cylindriques afin d'établir la relation contrainte-déformation dynamique qui sera utilisée dans les simulations numériques plus tard.

Nous avons observé de multi-fissuration dans les expériences sur les échantillons cylindriques mais son étude est trop difficile à mener. Pour mieux comprendre la propagation des fissures dans la glace, des échantillons rectangulaires avec une pré-fissure sont employés. En ajustant la vitesse d'impact on aboutit à la rupture des spécimens avec une fissure principale à partir de la pré-fissure. L'histoire de la propagation de fissure et de sa vitesse sont évaluées par analyse d'images basée sur les niveaux de gris et par corrélation d'images. La vitesse de propagation de la fissure principale est identifiée dans la plage de 534 ± 83m/s. Elle varie légèrement au cours de la propagation, dans un premier temps elle augmente et se maintient constante ensuite et diminue à la fin.

Les paramètres obtenus expérimentalement, tels que la vitesse d'impact et la vitesse de propagation de fissure, sont utilisés pour la simulation avec la méthode des éléments finis étendus. La ténacité d'initiation dynamique et la ténacité dynamique en propagation de fissure sont déterminées lorsque la simulation correspond aux expériences. Les résultats indiquent que la ténacité dynamique en propagation de fissure est linéaire vis à vis de la vitesse de propagation et semble indépendante de la température dans l'intervalle -15 ∼ -1 • C. [Schulson, 2001] . . . . . . . . . . . . . . . . . . . . 14 1.5 Uni-axial failure stress versus strain rate [Schulson, 2001] . . . . . . . . . . . . . 14 1.6 Tensile and compressive strength of ice as a function of temperature [Petrovic, 2003] 15 1.7 Dynamic stress versus strain curves at cryogenic temperatures [START_REF] Wu | Dynamic compressive behavior of ice at cryogenic temperatures[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

List of Figures

1.8 Fracture toughness of ice over loading rate [START_REF] Bentley | Fracture toughness of columnar freshwater ice from large scale dcb tests[END_REF]] . . . . . . . . . 17

1.9 Fracture toughness of ice over ice grain size [Petrovic, 2003] . . . . . . . . . . . 17 1.11 DCB specimens with wedge-loading configuration [START_REF] Bentley | Fracture toughness of columnar freshwater ice from large scale dcb tests[END_REF]] . . . . . 19

1.12 Modified ring-test specimens [START_REF] Fischer | Fracture toughness of ice and firn determined from the modified ring test[END_REF]

General introduction

The phenomena relating to the fracture behaviour of ice under impact loading are common in civil engineering, for offshore structures, and de-ice processes. To reduce the damage caused by ice impact and to optimize the design of structures or machines, the investigation on the dynamic fracture behaviour of ice under impact loading is needed.

The different failure modes such as fragment, spalling, crushing were observed and studied. Since the mechanism of these failures is complicated, it is tough to study the fracture behaviour of ice with multiple cracks.

The main objective of this work is to study the crack propagation in ice under impact loading. To better understand the fracture behaviour of ice under impact loading, the simplest case is chosen, i.e., a main crack propagating along a straight line. A main crack propagation in ice under impact is a fundamental study for other multiple fractures.

A series of impact experiments is conducted with the Split Hopkinson Pressure Bar. The temperature is controlled by a cooling chamber. The dynamic process of the ice fracture is recorded with a high speed camera and then analysed by image methods. The extended finite element method is complementary to evaluate dynamic fracture toughness.

The thesis is structured in four main parts : I-The first chapter is devoted to a general literature review regarding the various field considered in this work : first, we recall the major concepts in linear elastic fracture mechanics and dynamic fracture mechanics. Next, the mechanical properties of ice and the influence factors are addressed. Finally, the experimental and numerical approaches commonly used in this field are pointed up.

II-The second chapter describes the experimental procedure used in this work in detail and presents the main results. The impact experiments is conducted by the Split Hopkinson Pressure Bar with different dimensions of specimens. The temperature is controlled by a cooling chamber. The dynamic process of the ice fracture is recorded with a high speed camera. The stress-strain curves from the SHPB measurement identify the dynamic behaviour of ice under impact loading. The images captured by the high speed camera shows the crack propagation path.

III-The third chapter presents the image analysis methods used in this work and describes the crack propagation in ice under impact loading more accurately than naked-eyes. The concepts of grey scale and digital image correlation are introduced first. Then two methods are used to locate the crack tip, observe the crack path and measure the crack propagation velocity. Next, we try to use the displacement fields extracted by DIC to calculate the stress intensity factor. Due to the bad image quality, the results are not accurate enough.

IV-The last chapter is dedicated to the numerical method used in this work. Due to bad conditions of illumination, it is hard to get high resolution images for DIC to estimate SIFs. The eXtended Finite Element Method(XFEM) is used to assess dynamic crack propagation criteria in this work. When the simulation fits the experiment, the crack initiation toughness and the crack growth toughness can be evaluated from the simulation results. The factors that influence the fracture toughness are found, and the relationship between the crack growth toughness and crack propagation velocity is established.

The thesis ends with a summary of the main results obtained and an overview of some perspectives.

Chapter 1

State of the art

In this first chapter, we address a brief bibliographic review regarding the various fields considered in this work. First, the major concepts of fracture mechanics used in this work are reviewed. Then, the mechanical properties of ice especially the fracture behaviour are recalled. Finally, the experimental and the numerical method used to study the fracture behaviour of ice are introduced. Fracture mechanics originates from the investigation on the failure under a stress lower than the strength of a material at the beginning of the 20th century.

The symbolic work is contributed by Inglis [Inglis, 1913] who studied the stress distribution in an infinite plate with a crack simplified as an elliptic hole under tension, and determined stress concentration around the hole. The stresses tend toward infinity when distance from the crack tip tends toward zero. It explains the puzzled for years problem that the actual strength of the material is much lower than the theoretical strength. It is regarded as the firstly presentation of fracture mechanics.

In order to evaluate the crack propagation, the fracture criteria are developed base on energy, stress, displacement and new parameters.

Griffith [Griffith, 1921] studied crack propagation in glass ceramics and other brittle materials, and described the failure with an energy release rate criterion rather than a maximum stress for brittle material crack propagation. And then Irwin [Irwin, 1948] and Orowan [Orowan, 1949] extended the energy theory to a crack tip with a small scale yield in metallic material independently. Dugdale [Dugdale, 1960] and Barenblatt [Barenblatt, 1962] studied the plastic zone close to the crack tip and developed D-B model which is an energy approach considering non-linear aspects.

Irwin [Irwin, 1957] studied stress field near the crack tip, proposed a new fracture parameter, stress intensity factor, to evaluate the intensity of the singularity in the vicinity of a crack tip, and established the stress intensity factor criterion. And the critical value is called fracture toughness which represents the potential ability of a material to withstand a given stress field in the vicinity of a crack tip.

Wells [Wells, 1965] proposed crack opening displacement criterion to determine crack initiation under elastoplastic condition on the basis of a large number of experiments and engineering experiences. But it is not a strict characterization parameters for elastoplastic crack tip stress and strain field.

Rice [Rice, 1968] proposed J integral which using the path-independent integral loop around crack tip as an average parameter to express strain concentration characteristic near crack tip avoids calculating elastoplastic stress-strain field near the crack tip directly.

Then fracture mechanics is expanded to dynamic conditions. Mott [Mott, 1948] conducted a quantitative calculation of rapid crack propagation velocity and introduced kinetic energy to Griffith energy criterion. Yoffe [Yoffe, 1951] proposed a model of a crack that propagates from a fixed length at a constant velocity. Broberg [Broberg, 1960] considered a crack that propagates from zero initial length at constant velocity. Bluhm [Bluhm, 1969] proposed an important crack arrest issue which is identified through the crack tip equation of motion. Kostrov [Kostrov, 1966] found the first exact stress intensity factor solution for transient crack growth under mode III crack propagation.

Here, we only give a brief review and some major concepts of fracture mechanics for good understanding of the following.

Fracture mode

Consider a three-dimensional medium containing a crack plane (O, x, z), the crack front will be the line (O, z). The fracture can be divided into three elementary types according to the stress systems in the vicinity of the crack and the associated displacements shown in FIG. 1.1.

Mode I -Opening mode (a tensile stress normal to the plane of the crack).

Mode II -Sliding mode also called in-plane shear mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front).

Mode III -Tearing mode also called out-of-plane shear mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front).

Asymptotic solutions

For a homogeneous, isotropic and linearly elastic solid, the magnitude of the stress field in the vicinity of the crack can be described by a parameter K i , the subscript correspond to each elementary mode.

The stress and displacement expressions are known as Westergaard's asymptotic solutions [Westergaard, 1939]. The crack tip coordinate system is illustrated in FIG. 1.2, the center O is the crack tip, X-axis is the crack propagate direction, Y -axis is normal to the crack plane. The

The asymptotic solution in Mode I

The stress field is given by :

                                       σ xx (r, θ) = K I 1 √ 2πr cos( θ 2 )(1 -sin( θ 2 )sin( 3θ 2
))

σ yy (r, θ) = K I 1 √ 2πr cos( θ 2 )(1 + sin( θ 2 )sin( 3θ 2
))

σ xy (r, θ) = K I 1 √ 2πr cos( θ 2 )sin( θ 2 )sin( 3θ 2 ) σ xz (r, θ) = 0 σ yz (r, θ) = 0 σ zz (r, θ) = ν(σ xx + σ yy ) in plane strain σ zz (r, θ) = 0 in plane stress (1.1)
The displacement field is given by :

               u x (r, θ) = K I 2µ r 2π cos( θ 2 )(κ -cos(θ)) u y (r, θ) = K I 2µ r 2π sin( θ 2 )(κ -cos(θ)) u z (r, θ) = 0 in plane strain (1.2) 1.1.3.

The asymptotic solution in Mode II

The stress field is given by :

                                       σ xx (r, θ) = -K II 1 √ 2πr sin( θ 2 )(2 + cos( θ 2 )cos( 3θ 2
))

σ xy (r, θ) = K II 1 √ 2πr sin( θ 2 )cos( θ 2 )cos( 3θ 2
)

σ yy (r, θ) = K II 1 √ 2πr cos( θ 2 )(1 -sin( θ 2 )sin( 3θ 2
))

σ xz (r, θ) = 0 σ yz (r, θ) = 0 σ zz (r, θ) = ν(σ xx + σ yy ) in plane strain σ zz (r, θ) = 0 in plane stress (1.3)
The displacement field is given by :

               u x (r, θ) = K II 2µ r 2π sin( θ 2 )(2 + κ -cos(θ)) u y (r, θ) = K II 2µ r 2π cos( θ 2 )(2 -κ -cos(θ)) u z (r, θ) = 0 in plane strain
(1.4)

The asymptotic solution in Mode III

The stress field is given by :

                                   σ xx (r, θ) = 0 σ xy (r, θ) = 0 σ xx (r, θ) = 0 σ xz (r, θ) = -K III 1 √ 2πr sin( θ 2 ) σ yz (r, θ) = 0 σ zz (r, θ) = 0 σ zz (r, θ) = 0 (1.5)
The displacement field is given by :

           u x (r, θ) = 0 u y (r, θ) = 0 u z (r, θ) = 2K III µ r 2π sin( θ 2
)
(1.6)

Fracture Criterion

In order to evaluate the crack propagation, the fracture criteria are developed based on energy, stress, displacement, J-integral and other new parameters. Here, we only introduce the fracture criterion based on the stress intensity factor and J-integral which will be used in the following section.

Stress intensity factor

The stress fields in the vicinity of the crack tip has the singularity of type 1 √ r . The stresses tend toward infinity when r tends toward zero. It means that the stress can be no longer used as a criterion in crack propagation. Since K i can be used to describe the magnitude of the stress field in crack tip, it is called stress intensity factor which is an extremely important fracture mechanics parameter.

The stress intensity factor expressions can be described by the stress and displacement fields as Eq.1.7 and Eq.1.8.

           K I = lim r→0 √ 2πrσ y (r, 0) K II = lim r→0 √ 2πrτ xy (r, 0) K III = lim r→0 √ 2πrτ yz (r, 0) (1.7)                  K I = 2µ κ + 1 √ 2π lim r→0 u y (r, π) √ r K II = 2µ κ + 1 √ 2π lim r→0 u x (r, π) √ r K III = 2µ 2 √ 2π lim r→0 u z (r, π) √ r (1.8)
The fracture criterion is established by Irwin [Irwin, 1957] based on the stress intensity factor expressed as Eq.1.9. The crack will grow when the stress intensify factor has increased to a critical value which is called fracture toughness commonly denoted by K IC for mode I. The mode II and mode III have the same expressions. It represents the potential ability of a material to withstand a given stress field in the vicinity of a crack tip.

State of the art

K I ≥ K IC (1.9) 1.1.4.2 J-integral
The concept of J-integral was developed by Cherepanov [Cherepanov, 1967] and Jim Rice [Rice, 1968] independently. It is a way to calculate the strain energy release rateor work (energy) per unit fracture surface area in a material [START_REF] Meyers | Mechanical behavior of materials[END_REF] and characterize the stressstrain field at a crack tip by an independent integration path around the crack as shown in FIG. 1.3. FIGURE 1.3: J-integral path around a crack tip Consider a crack in a plate subjected to an arbitrary remote loading with an arbitrary closed contour Γ around the crack tip. Then the J-integral is defined in the absence of any body forces as

J x k = Γ Q k dΓ (1.10)
where k = 1, 2 and

Q k = W n k -σ i j ∂u i ∂x k n j (1.11)
W is the total strain-energy density defined as

W = ε i j 0 σ i j dε i j (1.12)
The critical J-integral denoted as J IC in Mode I is reached when the crack extension is happened. J-integral is a widely used criterion to assess quantitatively the toughness of a material.

1.1.5 Dynamic fracture mechanics 1.1.5.1 Inertial effect Dynamic fracture mechanics focuses on the fracture processes in which the material inertia effects cannot be ignored. The inertial effect can be caused by high rate loading or high speed crack propagation. The inertial effects depend on the loading conditions, the material properties and the geometrical configuration.

In our work, the impact is at a high rate loading, the rate of the dynamic stress intensity is in the magnitude of 10 5 ∼ 10 6 kPa • √ m/s and the crack propagation velocity is also high, about 500m/s. Both aspects cannot be ignored. Therefore the crack propagation in ice under the impact should be taken as a dynamic fracture mechanics problem.

Dynamic fracture toughness

In analogy with the static fracture mechanics, the crack tip stress field is described by dynamic stress intensity factor in dynamic fracture mechanics. The dynamic fracture toughness is a function of temperature, loading rate and crack propagation velocity [Freund, 1998] expressed in Eq.1.13.

σ i j = K √ r f i j (v, θ, T, KI ) (1.13)
Where, v is the crack propagation velocity, r is the distance, θ is the angle, T is temperature, KI is the rate of the dynamic stress intensity factor used to present the rate of loading.

The dynamic fracture criterion is separated in crack initiation, growth and arrest. The dynamic fracture toughness are determined through experiments, unfortunately until now there is no standard for dynamic fracture tests. The experiments should be conducted at interested range of temperatures and rates of loading.

The instantaneous stress intensity factor is denoted as K I (t). The dynamic stress intensity factor when the crack initiates is identified as the dynamic crack initiation toughness, denoted as K initiation ID . If t f is the crack initiation time, the dynamic crack initiation criterion is described as :

K I (t f ) = K initiation ID (T, KI ) (1.14)
Once a crack has initiated, the crack propagation velocity must be considered. The dynamic crack growth criterion is expressed as :

K I (t, v) = K growth ID (v, T, KI ) (1.15)
A commonly accepted view is that the dynamic crack growth toughness has a relationship with the crack propagation velocity. Based on linear elastodynamic theory, the crack propagation velocity is limited only by the rate of the stored elastic energy transferred to the crack tip, the terminal value is taken as the Rayleigh wave velocity along a free fracture surface [Freund, 1998].

The experimental values measured by investigators in different materials are much lower than the Rayleigh wave velocity. Many scholars focus on finding the relationship between the instantaneous stress intensity factor and the crack propagation velocity [Freund, 1998] [Grégoire et al., 2007] [Ravi-Chandar, 2004a] [Petrenko and Gluschenkov, 1996]. In this work, we pay much attention on this issue.

It needs to be stressed that the dynamic crack initiation toughness does not belong to the function of the crack growth toughness. It means K growth ID (v → 0, T, KI ) = K initiation ID (T, KI ). This should be possibly due to the bluntness of the initial crack, the intrinsic rate dependence of the material, or inertial effects [START_REF] Ravi-Chandar | Dynamic fracture[END_REF].

The dynamic crack arrest toughness is identified as the minimum dynamic stress intensity factor for which a crack keeps on propagating. The criterion is defined as :

K I (t) < K arrest ID (T )
(1.16)

Mechanical properties of ice

The mechanical properties of ice have been investigated for decades and applied extensively to various domains, such as civil engineering, offshore structures, de-ice processes and aerospace Engineering.

In the last century, various experiments with different ice types, dimensions, conditions have been conducted. The mechanical properties of ice in the quasi-static loading are investigated extensively and deeply.

Since the crash of the Space Shuttle Columbia in 2003 may be attributed to the potential ice impact, the dynamic behaviour of ice, especially under impact, draws the attention of scholars. The ice behaviour at high strain rate, be impacted or impact on structures are investigated. With the development in experimental and numerical methods, different experiments are conducted and simulation models are established. The failure modes such as fragment, spalling, crushing after impact were observed and studied. The investigations trend to lower temperatures and high loading rates.

Types of ice

Ice possesses 12 crystal structures, as well as 2 amorphous states [Schulson, 1999]. At ordinary pressures, the stable phase is termed ice I which has two closely related variants Ih and Ic. Ice Ih is hexagonal and is obtained by freezing water ; Ice Ic is cubic and is formed by vapour deposition at low temperatures.

In our work, the ice specimen is laboratory-grown ice, belong to Ice Ih also called ordinary ice, which is the common terrestrial form.

Elastic modulus

The elastic behaviour of ice is characterized by its structure and is influenced by temperature. At temperatures near the melting point, Young's modulus of single crystals varies from 12GPa along the least compliant direction to 8.6GPa along the most compliant direction. Along directions within the basal plane Young's modulus is 10GPa [Fletcher, 2009]. For polycrystalline ice, at a temperature of -10 • C Young's modulus of ice is reported in the range of 9.7 ∼ 11.2GPa and Poisson's ratio is 0.29 ∼ 0.32 [Petrovic, 2003]. At a temperature of -5 • C, typical values of Young's modulus and Poission's ratio are reported 9.0MPa and 0.33 [Fletcher, 2009].

The scattered values make difficult to determinate the parameters for the simulation. We need to measure the elastic modulus ourselves in the same condition as for the crack propagation test.

Brittle behaviour

Ice exhibits two kinds of inelastic behaviour under compression. At low strain rates, the material is ductile. Its stress-strain curve is characterized by ascending and descending branches. At higher strain rates ice exhibits brittle behaviour as shown in Fig. 1.4. Its stress-strain curve is characterized by an ascending branch only, and failure occurs after inelastic strains < 0.003 [Schulson, 2001]. Fig. 1.5 illustrates the effect of strain rate on the uni-axial compressive strength (C, upper curve) of ice, the tensile strength(T) is shown for comparison. The peak in the compressive strength versus strain rate marks the macroscopic ductile-to-brittle transition which is defined in terms of a specific strain rate ≈ 10 -4 ∼ 10 -3 s -1 in small test specimens. And the transition increases with decreasing grain size, increasing salinity, confinement and pre-deformation [Schulson, 2001].

The interested strain rate of our work is in the range 10 1 ∼ 10 2 s -1 and in which ice shows a brittle behaviour.

Tensile and compressive strength

When we talk about the strength of ice, two important parameters must be described : the temperature and the strain rate.

Generally, the strength of ice increases with decreasing temperature in both tension and compression as reported by Haynes [Haynes, 1978] and summeried by Petrovic at the temperature FIGURE 1.5: Uni-axial failure stress versus strain rate [Schulson, 2001] range of 0 ∼ -40 • C [Petrovic, 2003] as shown in Fig. 1.6. And recently, Wu [START_REF] Wu | Dynamic compressive behavior of ice at cryogenic temperatures[END_REF] conducted the experiments in much lower temperature and observed the same tendency. When the temperatures decrease from -15 • C to -125 • C the compression strength increases from 32MPa to 112MPa, and keeps nearly constant in the range 112 ∼ 120MPa as the temperatures decreases from -125 • C to -173 • C the corresponding strain rates varies from 272s -1 to 678s -1 . At the temperature ranges between -10 ∼ -20 • C, the tensile strength of ice is in a wide range of 0.7 ∼ 3.1MPa [Petrovic, 2003]. FIGURE 1.6: Tensile and compressive strength of ice as a function of temperature [Petrovic, 2003] The tensile strength is essentially independent of strain rate, and is only slightly dependent upon temperature, increasing by < 25% upon cooling from -50 • C to -15 • C, and is heavily affected by grain size [START_REF] Schulson | A brittle to ductile transition in ice under tension[END_REF]. While the compressive strength is strain rate sensitive, it increases from 2MPa to 10MPa as the strain rate increase from 10 -7 s -1 to 10 -2 s -1 at the temperature ranges between -10 ∼ -20 • C [Schulson, 2001].

Equiaxed and randomly oriented aggregates of fresh-water ice fail under uni-axial tension via transgranular cleavage at applied stresses around 1MPa [START_REF] Schulson | Fractography of ice[END_REF]].

As stated above, the strength of ice is sensitive to the temperature, strain rate which reminded us to carefully control when we conduct the experiments.

Fracture behaviour

The fracture behaviour of ice can be characterized by fractured size as sliding, fragment, spalling and crushing.

The fracture toughness of ice is reported in a large range, from 50kPa • √ m to 400kPa • √ m [START_REF] Liu | Fracture toughness of fresh-water ice[END_REF]] [Bentley et al., 1989] [Dempsey, 1991] [Petrovic, 2003]. It depends on the variables of temperature, loading rate and ice grain size.

FIGURE 1.7: Dynamic stress versus strain curves at cryogenic temperatures [START_REF] Wu | Dynamic compressive behavior of ice at cryogenic temperatures[END_REF] Generally, the fracture toughness of ice increases with decreasing temperature in the range -1 • C to -46 • C [START_REF] Liu | Fracture toughness of fresh-water ice[END_REF].

The fracture toughness of ice increases with the loading rate in the range 10 -1 kPa • √ m/s to 10 4 kPa • √ m/s reported by Bentley [START_REF] Bentley | Fracture toughness of columnar freshwater ice from large scale dcb tests[END_REF] , 1988]. The fracture toughness exhibits a decrease with increasing grain size as shown in Fig. 1.9 [Petrovic, 2003].

1.3 Experimental study on ice

1.3.1 Quasi-static experiment
The conventional specimens in the fracture experiments are CT, CCP, Single Edge Notched Plate (SENT), DECP, 3PT and 4PT whose geometries are illustrated in Fig. 1.10. In which, a is the pre-crack length, W is the specimen width and 2H is the specimen length [START_REF] Standard | Annual book of astm standards[END_REF]. These specimens are widely used in metallic materials, ceramics, polymers and composite materials. FIGURE 1.8: Fracture toughness of ice over loading rate [START_REF] Bentley | Fracture toughness of columnar freshwater ice from large scale dcb tests[END_REF] FIGURE 1.9: Fracture toughness of ice over ice grain size [Petrovic, 2003] But for ice, the standard for fracture toughness measurement are not yet established. With more restrictions of keeping the geometries, three-point-bending Fig. 1.10e and four-point-bending Fig. 1.10f loading configurations are widely used, and other geometries are also developed. Since the measured values can be compared only with the same standard geometry [Liu, 2005]. The obtained values must be introduced with the geometries.

Bentley [START_REF] Bentley | Fracture toughness of columnar freshwater ice from large scale dcb tests[END_REF]] used large scale double cantilever beam (DCB) specimens with wedge-loading configuration to investigate the fracture toughness of ice. Fischer [START_REF] Fischer | Fracture toughness of ice and firn determined from the modified ring test[END_REF] used the modified ring test to study the fracture toughness of ice.

Dynamic experiment

The phenomena relating to the fracture behaviour of ice under impact are common in various domains, such as ice formed in the high sky and potential impacts on air-plane, hail storm impacts FIGURE 1.11: DCB specimens with wedge-loading configuration [START_REF] Bentley | Fracture toughness of columnar freshwater ice from large scale dcb tests[END_REF] (a) (b) FIGURE 1.12: Modified ring-test specimens [START_REF] Fischer | Fracture toughness of ice and firn determined from the modified ring test[END_REF] on auto-mobiles and buildings, icebreakers open fairways and machines de-ice on highways as shown in Fig. 1.13. To avoid or reduce the damage caused by ice impact, the force generated by ice impact should be estimated. To optimize design of machines used in ice area, the dynamic fracture behaviour of ice under impact loading is studied. A great deal of work has been done by NASA [START_REF] Pereira | Forces generated by high velocity impact of ice on a rigid structure[END_REF]] [Melis et al., 2009] [Fasanella et al., 2006] [Chamis et al., 1994] [Reddy et al., 1991] [Lange and Ahrens, 1982]. Two main kinds of experiments are conducted to study the dynamic fracture behaviour of ice : impact and be impacted. The first type of the experiment is to measure the forces on rigid structures upon high speed impact of ice. Combescure [START_REF] Combescure | Experimental study of high-velocity impact and fracture of ice[END_REF]] study the high-velocity ice fracture mechanisms with the experiments of ice impacting on rigid supports. The fragmentation modes and the deformation are observed and strain response on the impacted plate are measured during the impact. The failure process of the ice specimen is shown in Fig. 1.14. The measured crack propagation velocity is about 500m/s which can be used to verify our work.

Tippmann [START_REF] Tippmann | Experimentally validated strain rate dependent material model for spherical ice impact simulation[END_REF] observed the failure progression of an ice sphere during impact with a high speed video and measured the spherical ice impact forces.

Be impacted experiment conducted by Split Hopkinson bars

The second type is a rigid structure impacting on ice at different strain rates. Usually Split Hopkinson Pressure Bars (SHPB) are used in this case.

Dutta [START_REF] Dutta | A fracture study of ice under high strain rate loading[END_REF] conducted experiments with S2 freshwater ice at -10 • C and -40 • C and a strain rate in the range of 0 to 20s -1 and indicated the specimens failed progressively in a brittle manner by axial splitting.

Shazly [START_REF] Shazly | High strain-rate behavior of ice under uniaxial compression[END_REF] study the dynamic response of ice at a higher strain rates from 60s -1 to 1400s -1 at the initial test temperatures of -10 • C and -30 • C and indicated the positive strain-rate sensitivity and ice behaving stronger at the lower test temperature over the interested range of strain rates.

Kim [START_REF] Kim | Compressive strength of ice at impact strain rates[END_REF] measured the compressive strength of ice at high strain rates range of 400 ∼ 2600s -1 and reported a trend of continuously increasing strength for strain rates beyond 10 1 s -1 can be observed that is to say ice is a rate sensitive material . FIGURE 1.14: Failure history of a conic and a cylindrical specimen impacting on a rigid plate [START_REF] Combescure | Experimental study of high-velocity impact and fracture of ice[END_REF] The range of temperature is extended by Wu [START_REF] Wu | Dynamic compressive behavior of ice at cryogenic temperatures[END_REF] who studied the dynamic behaviour of distilled water ice at temperatures from -15 • C to -173 • C and at strain rates in the range of 100s -1 to 1350s -1 . He reported the peak compressive strength increase from 32MPa to 112MPa as the test temperatures are decreased from -15 • C to -125 • C and remain nearly constant in the range of 110 ∼ 120MPa in test temperature from -125 • C to -173 • C .

From above dynamic experiments, we have got the force generated by the ice impact, the approximate crack propagation velocity, the mean strain-stress curves, the fracture process. But we do not know the fracture mechanism of ice under impact loading, such as the critical conditions of the fracture appearance, the intensity factor of the crack initiation and propagation. The multiple cracks make difficult to study the crack tip local information. To better understand the fracture FIGURE 1.15: Failure history of a spherical specimen impacting on a rigid plate [START_REF] Tippmann | Experimentally validated strain rate dependent material model for spherical ice impact simulation[END_REF] (a) [START_REF] Dutta | A fracture study of ice under high strain rate loading[END_REF] (b) [START_REF] Shazly | High strain-rate behavior of ice under uniaxial compression[END_REF] FIGURE 1.16: Multiple fracture specimen after impacted by SHPB behaviour of ice under impact loading, the simplest case is chosen in this work, i.e., a main crack propagating along a straight line. A main crack propagation in ice under impact is a fundamental study for other multiple fractures.

Numerical method for dynamic fracture problem

The numerical method is a complementation of the experimental method. Because of the advantage of high efficiency and low cost, it is widely used and intensively studied. Here, we introduce the significant development in numerical method for dynamic fracture problem briefly.

Finite element method

The finite element method has been widely used for fracture problems for decades. New ideas and techniques are developed to complement the disadvantages of meshing during crack propagation.

Swenson presented a finite element model of mixed-mode dynamic crack propagation in which the discrete cracks are allowed to propagate through the mesh in arbitrary directions [START_REF] Swenson | Modeling mixed-mode dynamic crack propagation nsing finite elements : theory and applications[END_REF]. Nishioka incorporated the Delaunay automatic triangulation into an automatic moving finite element method which are used to simulate mixed-mode impact fracture tests with fracture-path prediction mode [START_REF] Nishioka | Dynamic fracturepath prediction in impact fracture phenomena using moving finite element method based on delaunay automatic mesh generation[END_REF]. Réthoré presented a stable numerical scheme for the finite element simulation of dynamic crack propagation with remeshing using an energy approach [START_REF] Rethore | A stable numerical scheme for the finite element simulation of dynamic crack propagation with remeshing[END_REF].

Meshless methods

The meshless methods have two main advantages in fracture problem that it is possible to model arbitrary and complex propagation of cracks without remeshing and adaptive refinement at the crack tip is easily accomplished [START_REF] Fleming | Enriched element-free galerkin methods for crack tip fields[END_REF].

Because it does not rely on a mesh but rather on an aggregation of points. Each point has a domain of influence on which interpolation functions are built. These functions are null on the boundary and outside the domain of influence. This is why the influence domain is generally called the support. Enrichment functions can be inserted on these supports and used to model a crack tip [START_REF] Fleming | Enriched element-free galerkin methods for crack tip fields[END_REF]] [Belytschko et al., 1996].

Element-free-Galerkin(EFG) methods is one of the meshless methods which are based on moving least-square(MLS) interpolates. It is applied to study static and dynamic fracture, wave propagation and the mixed-mode dynamic crack propagation in concrete [START_REF] Belytschko | Elementfree galerkin methods for static and dynamic fracture[END_REF]] [Belytschko et al., 1995a] [Belytschko and Tabbara, 1996] [Belytschko et al., 2000][Lu et al., 1995]. Then it is developed to 3-D problem [START_REF] Krysl | The element free galerkin method for dynamic propagation of arbitrary 3-d cracks[END_REF].

The partition of unity

The partition of unity method (PUM) is developed for the solution of problems where the classical FEM approaches fail or are prohibitively expensive. One of problems is the crack propagation. It allows for the construction of conforming ansatz spaces with local properties determined by the user [START_REF] Babuška | The partition of unity method[END_REF].

Duarte presented a partition of unity method for the simulation of three-dimensional dynamic crack propagation in which the approximation spaces are constructed using a partition of unity and local enrichment functions. It allows for modelling of arbitrary dynamic crack propagation

Extended finite element method

The extended finite element method (XFEM) is a versatile tool for the crack propagation problems which has enormous advantages than the other numerical methods mentioned above. First, the crack surface and crack front are completely independent of the mesh. When the crack propagates, remeshing is not needed. Second, the crack surface is constructed in terms of nodal values by additional basis functions especially combined with level sets. The the crack surface do not must to be placed on element edges. It can be put inside an element and cut the element. It simplifies the model. Furthermore, the singular stress field in the vicinity of crack tip is solved with enrichment functions, the requirement of refinement is reduced [START_REF] Belytschko | A review of extended/generalized finite element methods for material modeling[END_REF].

Belytschko presented a new enrichment technique with loss of hyperbolicity to avoid the difficulties encountered with the original XFEM in time-dependent problems and enable both the crack speed and crack direction to be determined for a given material model [START_REF] Belytschko | Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment[END_REF].

Réthoré proposed an energy-conserving scheme for dynamic crack growth using the extended finite element method [START_REF] Réthoré | An energyconserving scheme for dynamic crack growth using the extended finite element method[END_REF].

Menouillard introduced an explicit formulation of the extended finite element method with a lumped mass matrix for enriched elements. They also reported the critical time step of the extended element is of the same order as the corresponding element without extended degrees of freedom [START_REF] Menouillard | Efficient explicit time stepping for the extended finite element method (x-fem)[END_REF].

Grégoire simulated a mixed-mode crack propagation with extended finite element method which consisted with a dynamic crack initiation toughness, a crack orientation along the maximum principal stress and a simple equation for the calculation of the crack speed [START_REF] Grégoire | Dynamic crack propagation under mixed-mode loading-comparison between experiments and x-fem simulations[END_REF].

Bui [START_REF] Bui | Analysis of generalized dynamic intensity factors of cracked magnetoelectroelastic solids by x-fem[END_REF] used the extended finite element method integrated with Newmark time integration scheme and the level set method to accurately capture the crack geometry to study the generalized dynamic intensity factors of cracked homogeneous and linear magnetoelectroelastic solids considering the effects of dynamic impact loads, time-step sizes, material polarization directions and interaction cracks.

Conclusion and recommendations

The objective of this work is to study the crack propagation under impact loading and to evaluate the fracture criteria. According the above literature review, we designed the work implementation process with matters needing attention. I-The dynamic Young's modulus of ice is scattered reported in the literature. It is depended by the strain rate and the experimental condition. It is difficult to chose the suitable value. Therefore, we need to measure the elastic modulus ourselves in the same condition with the crack propagation test.

II-The dynamic behaviour of ice is complicated. We try to study the crack propagation in ice under impact loading with the simplest case, a main crack propagating along a straight line.

A main crack propagation in ice under impact is a fundamental study for other multiple fractures.

III-The behaviour of ice is sensitive to the ice type, the specimen geometry, the test temperature and strain rate which reminded us to control them carefully when we conduct the experiments.

IV-Compared to the other criteria, the stress intensity factor is accustomed used. We also chose this parameter to evaluate the crack propagation.

V-The fracture happens very quickly, we need to observe the process with the help of a high speed camera and image analysis methods.

VI-Compared to the other numerical methods, the extended finite element method has enormous advantages. It is chosen to simulate the experimental process.

Chapter 2

Experimental study on dynamic behaviour of ice

Introduction

The mechanical properties of ice are sensitive to the structure of ice, the configuration of specimen, the experimental condition. The strength and Young's modulus of ice were reported in a large range in different literature. Precise values were needed for the following simulation in evaluating the fracture toughness. In order to obtain the strength and dynamic Young's modulus in the same condition with the crack propagation experiment, three groups of experiments were designed firstly :

1) A group of experiments with the cylindrical specimen of a diameter of 50mm was conducted with different impact velocity to obtain the range of strength and Young's modulus.

2) A group of experiments with the cylindrical specimen of a diameter of 50mm with confinement was conducted to investigate the confinement effect. Since the ice specimen in the crack propagation experiment was in an intermediate state between with and without confinement.

3) A group of experiments with the cylindrical specimen of a diameter of 20mm was conducted with a critical impact velocity to obtain the critical strength and Young's modulus of specimen fracture.

Then, with the purpose of investigating the crack propagation in ice under impact loading, two groups of experiments were designed : 1) A group of experiments with the rectangular specimen with a wedge and a pre-crack was conducted with different impact velocity to obtain a main crack propagation.

2) A group of experiments with the rectangular specimen with a hold and a pre-crack was conducted with different impact velocity to obtain a main crack propagation.

For the specimen with wedge, the waves transferred and reflected complicatedly on interface of the wedge and the specimen. It was hard to do signal analysis. Such a problem could be avoided with the rectangular specimens with hole. In this type specimen, the input bar impact on the specimen directly. But the stress distribution was more complex near the hole. So both types of specimens were used.

SHPB experiment 2.2.1 Test apparatus

The Split Hopkinson Pressure Bar is a mechanical testing apparatus designed specifically to study dynamic mechanical properties of materials at high strain rates from 10 1 to 10 4 s -1 . The apparatus has been designed and developed a century ago. Hopkinson [Hopkinson, 1914] firstly measured the pulse induced by an impact on a long bar. Then Davies [Davies, 1948] done a critical work and established the measurement technique. After, Kolsky [Kolsky, 1949] developed the apparatus into two long bars and a short specimen between them. Since then, SHPB are widely used to investigate the dynamic behaviour of materials.

In our work, the impact tests were performed with the Split Hopkinson Pressure Bar which comprised a striker bar, a long input bar and an output bar with a short specimen between them. The pressure was supplied by an air pump. When the striker bar sustained the pressure and went through the tube, the optical sensors glued on the tube recorded signals which were used to estimate the velocity of the striker bar. A longitudinal compressive stress wave ε i (t) was produced into the input bar by the striker bar impacting on. Once the wave arrived at the interface of the input bar and the specimen, a portion was reflected as the reflected wave ε r (t), the other went through the specimen and transfers into the output bar called the transmitted wave ε t (t). The strains caused by the stress wave were measured with gauges on the input and output bar respectively as voltage signals.

Since the mechanical properties of ice are sensitive to the temperature, especially under dynamic loading. A cooling chamber with a liquid Nitrogen tank was used to control the temperature of the specimen.

The experimental process was recorded with a high speed camera connected to a computer and enlightened with two high brightness low heat lamps. The signals from the optical sensors and the strain gauges were collected by an acquisition card connected to the computer. A Multi-Channel Data Link (MCDL) box was used to establish the synchronization between the SHPB signals and high speed camera videos. A schematic view of the experimental system is shown in FIG. 2.1 .

FIGURE 2.1: System of impact experiment for ice

Since the high speed photography method was chosen to study the crack propagation, obtaining high quality images became an important issue. FIG. 2.2 is a picture of the cooling chamber. Even equipped with the high brightness lamps, it was still dark inside. Despite the window was made of three-layer glasses, the formation of fog between the glasses and the reflective action affected the image quality seriously. Only few tests supplied clear enough images to be analysed correctly.

SHPB measurement

The SHPB measurement technique is based on three basic assumptions. First, the wave propagates in the bars can be described by one-dimensional wave propagation theory. Second, the stress and strain fields in the specimen are homogeneous in its axial direction. Third, the specimen inertia effect and the friction effect in the compression test are negligible [Zhao, 2003].

FIGURE 2.3: SHPB measuring technique

The incident, reflected and transmitted waves were shifted to the bar-specimen interfaces as shown in FIG. 2.3. In a first approximation, one-dimensional elastic wave propagation theory and superposition principle were used to estimate forces and velocities on both bar-specimen interfaces following Eqs.2.1.

F input = S B E(ε i (t) + ε r (t)) F out put = S B Eε t (t) and V input = C 0 (ε i (t) -ε r (t)) V out put = C 0 ε t (t) (2.1)
where F input , F out put , V input , V out put are the forces and velocities on the bar-specimen interfaces. S B is the cross-section of the bars, C 0 is the longitudinal wave speed in the bars.

With an assumption of homogeneous stress and strain fields, the mean strain, mean stress and strain rate of the specimen were obtained by Eqs.2.2 and then converted to Eqs.2.3 by substituting the stress waves into forces and velocities.

σ S = F input + F out put 2S S εS = V out put -V input L S ε S = t 0 V out put (τ) -V input (τ) L S dτ (2.2) σ S = ES B 2S S [ε t (t) + ε r (t) + ε r (i)] εS = C 0 L S [ε i (t) -ε r (t) -ε t (t)] ε S = t 0 [ε t (t) + ε r (t) -ε i (t)]C 0 L S dτ (2.3)
In equation 2.2 and 2.3, ν is the Poisson's ratio and E the Young's modulus of nylon bars. S S is the cross-section and L S is the length of the specimen.

Experimental conditions

As known difficulties in dynamic experiments, especially for ice, most of the experiments fails for various reasons, such as multiple cracks occurrence, bad image quality, incident signals overlap with reflected signals, or the presence of unavoidable defects in the tested specimen. Only a few experiments can be used to supply useful information. The experimental conditions will be presented in details in the following sections. The dynamic fracture behaviour depends on the test temperature and the loading rate. Well controlled experimental conditions are essential to study it.

2.2.3.1 Temperature [-15 • C, -1 • C]
The temperature range investigated in our work was from -15 • C to 0 • C. Although the melting point of ice is one of the most interesting temperature, it was hard to maintain a speckle pattern on the surface of the specimens at 0 • C. At first, the experiments were conducted at indoor temperature without the cooling chamber. The temperature of the specimen during the experiment needed to be measured. Before the ex-periment, the specimen was stored in refrigerator at -15 • C. After the specimen was brought out, the variations in temperature inside and outside the cylindrical specimen of diameter 50mm were measured with a temperature sensor. The temperature versus time is shown in FIG. 2.4. The experiment would be conducted in 2 minutes. Therefore the temperature when the ice specimen fails should be about -14 • C. Then, we used a cooling chamber to control the temperature accurately. After the specimen was brought out of the refrigerator, the specimen was stored in the cooling chamber for 10 minutes until the temperature reached the required value. The experiments were conducted at the temperatures of -15 • C, -7 • C and -1 • C.

Impact velocity [1m/s, 5m/s]

The impact velocity was another key factor for the experiments. Two kinds of impact velocity were measured. One velocity was the striker bar impacting on the input bar measured by two optical sensors glued on the tube. The distance between the two sensors was 50mm, the time was determined by signal1 and signal3. Therefore, this velocity was determined. Another velocity was the input bar impacting the specimen determined by the incident signal with Eq.2.4. This kind of impact velocity is more meaningful. It is the velocity at which the input bar impacts the specimen.

To distinguish them, the first kind is called the striker velocity, and the second one the impact velocity in the following section.

V impact = C 0 ε i (t) (2.4)
From experiments, the impact velocity that guarantees a main crack generation was in the range 1 ∼ 5m/s. If the impact velocity was less than 1m/s, no crack initiated. If the velocity was larger than 5m/s, multiple fracture happened. From a large number of experiments, the critical impact velocity of the specimen fracture was about 2m/s.

2.2.3.3 Strain rate [10s -1 , 100s -1 ]
The strain rate is depending on the properties of the bars and the impact velocity. At the beginning, the input and output bars made of nylon with a diameter of 50mm were used and which crushed the ice specimens completely. Then, the diameter of the bar was reduced to 20mm in order to obtain one main crack. With the purpose of controlling the impact velocity and duration precisely, striker bars made of PVC, PMMA and nylon with different lengths and diameters were tried. The best results were obtained with a PMMA bar of diameter 12mm and length 470mm.

Under the previous impact velocity, the strain rate was in the range of 10-100s -1 . And the experiments indicated the strain rate increased with the length of specimen decrease.

Selection of bars 2.2.4.1 Selection of the input and output bars

In order to guarantee a good wave transmission on the bar-specimen interfaces, the impedances of the bars and the specimen should be taken into account. Nylon was used for the bars because of the similar impedance with ice. The impedance Z is described by Eq.2.5, S is the section, ρ is the density, E is the Young's modulus, and the subscripts B and S present the bar and the specimen, respectively. The mechanical properties of ice and nylon are listed in TAB. For experiments with cylinders, the diameter of the specimen was the same than for bars. For rectangular specimens, the width of the specimen was the same the one of bars, and the section were designed with a restrictive condition of Z B ∼ Z S . At first, the bars with diameter of 50mm were used. After an impact, the ice specimen was crushed totally even controlling the pressure to the minimum value. Then the diameter of bars were changed to 20mm. By controlling the impact velocity, the cylindrical specimens did not break and the rectangular specimens fractured with a main crack.

Selection of the striker bar

To perform the SHPB measurement, the first thing is extracting the incident, reflected and transmission signals. When the stress waves go through the strain gauge on the bar, the voltage signal sharply increases. The pulse is taken as the impact signal. The first pulse in the input bar is the incident signal, and the second is the reflected signal in opposite direction. The first pulse in the output bar is the transmission signal.

But sometimes the end of incident signal overlaps the begin of the reflected signal as shown in Fig. 2.5a which makes separation the incident and reflected signals impossible. A good input signal should be as shown in Fig. 2.5b, a period equals to zero exists between the incident and reflected signals in which the input bar at the strain gauge location does not sustain stress wave. The end of the incident signal and the begin of the reflected signal can be clearly identified. To achieve it, the impact pulse should be short and the input bar should be long enough. The length of the input bar is restricted to the size of the laboratory. The shape of the impact pulse is determined by the striker bar. Therefore, selection a suitable striker bar is a solution easier to implement which guarantees the separation of the incident signal and the reflected signal. The theoretical duration of the impact pulse is given by the length of the striker bar. With rings, the mass increases and the duration also increases. And the mass of the foam rings can be ignored, but the nylon rings can not.

Number

Material The ice type, the specimen geometry and its preparation have all some effects on the dynamic fracture behaviour. In our experiments, laboratory-grown specimens was ordinary ice, which were prepared by running water with few bubbles. Deionized water and boiled water were also tried, they showed same results. The density of the ice specimens was about 907Kg/m 3 . It is much easier and cheaper to prepare such ice than trying to reproduce a particular one knowing that impact experiments failed very often.

Two main kinds of specimens were tested. First cylindrical specimens were used for observing the failure mode of the ice under impact loading and studying the constitutive laws of the material at high strain rates and then extracting the parameters for the following simulations. The cylindrical specimens were also tested with a tube to study the confinement influence on the dynamic behaviour of ice. At the beginning, the cylindrical specimens had the same diameter than the bars of 50mm and the length were 100mm. Then the diameter of specimens was changed to the one of the bars, i.e. 20mm and the length set to 45mm.

Second rectangular specimens were used for observing the crack propagation in ice under impact and investigating the dynamic fracture criteria. Three different geometries were used : i)rectangular specimen of 230mm long with a hole and a pre-crack, ii) rectangular specimen with a hole, but of half-length to get higher strain rate, and iii) rectangular specimen of length 230mm with a notch and a pre-crack. The rectangular specimen geometry is shown in Fig. 2.12.

For the rectangular specimen with notch, the waves transferred and reflected in a complex manner at the interface between the wedge and the specimen. It is hard to do signal analysis. Such a problem can be avoided with the rectangular specimen with hole. But the stress distribution is more complex near the hole. So both of specimens were used.

Fabrications of ice specimen 2.3.2.1 Cylindrical ice specimens

Cylindrical specimens of diameter 50mm and length 100mm were prepared in a metallic tube with two metallic caps. The caps could slide in the tube and well sealed with the tube. The inside water did not leak.

Cylindrical specimens of diameter 20mm and length 45mm were prepared in a metallic tube with two rubber caps. The rubber cap at the bottom of the tube could hold the water and prevent Then the specimen with the tube and caps was put into a fridge for 20 hours at -15 • C.

Afterwards, two ends were polished to ensure a good contact with the bars.

Rectangular ice specimens

The fabrication of the rectangular specimens was complicated due to the wedge or hole and the pre-crack. They were prepared with the following procedure :

1. The rough specimens were obtained by pouring water into a cuboid box whose inner surfaces were covered with foam to provide space for volume expansion during freezing. And then the box was put into a fridge for 20 hours at -15 • C.

2. The specimen was polished until the bubbles at the bottom of the specimen were scarce and both short ends were smooth to ensure a good contact with the bars.

3. The triangular notch was made by putting a metallic cube on a short side of the specimen until the local ice melted and formed a notch. The hole was made by a thread drill. The pre-crack was made by a saw blade.

4. After processing, the specimen was put back into the fridge until the water surface was refrozen.

5. Random speckles on the specimen surface for image analysis were prepared by sprinkling toner uniformly. Attention must be paid to avoid internal cracks and prevent ice forming again in the pre-crack.

It is important to note that during the preparation, melting is unavoidable, especially at the pre-crack tip. Therefore, the slight changed dimensions of the specimens cause a difficulty of guaranteeing uniform size and stable quality. In which, Signal1 and Signal3 were collected from the optical sensors glued on the tube, Signal5 and Signal7 were collected from the strain gauges glued on the input bar and output bar.

In order to demonstrate clearly, signal5 and signal7 are magnified 100 times.

Then the meaningful part of the signal used for measure the strain was zoomed in Fig. 2

.14a

The correction of the offset voltage was done by vertical shifting the signals to zero as shown in Fig. 2.14.

Then the incident, reflected and transmission signals were extracted from the input and output signals and plotted after horizontal shifting as shown in Fig. 2.15.

Signal filtering

Since the signal was noisy, a filter was designed to improve the quality of the signal by reducing the frequencies of the noise. An example of the filtering process with the incident signal is shown in Fig. 2.16.

First, the signal expressed by the time domain was converted into the frequency domain by As shown in Fig. 2.16b, the imaginary components before being filtered is expressed in red. The noise distributed in the high frequency part, of which the frequency is large than 10 4 Hz. After, a low-pass filter was designed as shown in the Fig. 2.16d to attenuate the the high frequency and kept the low frequency part. The blue curve expresses the imaginary components after being filtered. The real component is filtered in the same way as shown in Fig. 2.16c. Finally, the complex signal was rebuilt in time domain shown in Fig. 2.16a. The curve in blue is the filtered signal, it keeps the same shape with the original signal which validates the design of the filter. If the shape of signal changed, the filter should be changed.

Signal conversion

The signals collected by strain gauges were voltage signals. They needed to be converted into strain signals before analysis by Eq.2.6 as shown in Fig. 2.18. In equation 2.6, V measure is the voltage signal from gauges, and V supply is the supply voltage. κ is the strain gauge sensitivity factor, ν is the Poison's ratio of the bar. In our work, V supply = 12V , κ = 2.11, and ν = 0.41.

V measure V supply = 1 2 (ν +
All the signal processing was performed with MATLAB.

Experiment results

Cylindrical ice specimens 2.5.1.1 Cylindrical ice specimens diameter of 50mm

The specimen in this group had a diameter of 50mm and a length of 95mm, approximately. The experiment was conducted with the 50mm-diameter nylon bars at indoor temperature. As discussed before the temperature when the ice specimen failed was about -14 • C.

The experimental parameters and results for this group of experiments are listed in appendix A, Tab.A.1 and Tab.A.2,Tab.A.3 and Tab.A.4, respectively.

The impact velocity is a key factor to classify the fracture behaviour.

When the input velocity is larger than 5m/s, the specimen crushed once it was impacted by the input bar. The fracture process of a typical experiment is shown in Fig. 2.19, and the parameters of the experiment are listed in Tab.2.7. 2013030505 384×288 20 6.9 50000 TABLE 2.7: Image recording parameters for an experiment with a cylindrical ice specimen of diameter of 50mm several axial cracks generated once the input bar impacted on the specimen. The cracks took about 200µs to propagate from the left to the right side, the crack propagation velocity was about 500m/s. In this process, the ice specimen still kept as a whole cylinder but cracks went through inside. Since the stress wave arrived at the interface of the specimen and the output bar, the reflection stress wave generated. Several axial cracks propagated from the right side and reached the middle of the specimen at about 300µs. Then, radial cracks appeared in the middle. The fracture mode could be attributed to the brittle cleavage.

Several stress-strain curves belonging to this kind of experiments are plotted in Fig. 2.21. The stress increases quickly in the beginning, then reaches the peak stress, after it decreases slowly which indicates the fragmented ice still sustains the stress. The residual strength can be attributed to the friction between ice fragments. For this kind of experiments, the dynamic Young's modulus is in the range 4.12 ± 1.72GPa, the peak stress is in the range of 3.57 ± 1.05MPa .

The peak stress values were lower than the result reported by Shazly [START_REF] Shazly | High strain-rate behavior of ice under uniaxial compression[END_REF]. The peak stress in his work was reported in the range of 10 ∼ 60MPa at a high strain rate of 90 ∼ 1111s -1 at -10 • C. The lower compressive strength can be attributed to unavoidable defects in the laboratory-grown ice specimen. Besides, in our experiment the compressive strength of ice under impact shows a linear association with impact velocity as shown in Fig. 2 observed that the strength increases with the strain rate. But in our experiments, this positive sensitivity was not observed. The main reason may be the size effect of the brittle material and the inertial effect. The unapparent sensitivity between the peak stress and strain rate can be attributed to the scattered strain rates due to the bad propagation into the fractured specimen. Meanwhile, the strain rate is determined by the impact velocity and the length of specimen. In our experiment, the length is almost uniform. Therefore, the impact velocity is linear associate with the strain rate. So basically, the linear association between the compressive strength and the strain rate or impact velocity has the same sense in our case. Both the lower impact velocity and the lower strain rate can explain the lower compressive strength.

Cylindrical ice specimens diameter of 50mm with confinement

If the confinement effect would change the strength of ice a lot that is an interesting issue. Furthermore, according to the boundary condition the ice specimen in the crack propagation experiment is in an intermediate state between with and without confinement. Therefore, a group of experiments was designed for investigating the confinement effect on the strength of ice and validating the simulation results.

The confinement was supplied by a rigid metallic tube with a thickness of 8mm. The specimen in this group had a diameter of 50mm and a length of 90mm approximately. The experiment was conducted with 50mm-diameter nylon bars at indoor temperature. The experimental parameters and results for this group of experiments are listed in appendix A Tab.A.5 and Tab.A.6, respectively.

Several typical stress-strain curves are plotted in Fig. 2.24. After the peak stress, the stress decreases slightly and then keeps at a plateau value which indicates the fragmented ice still sustains the stress. Then the specimen is compressed. The residual strength can be attributed to the friction of the ice fragments and the radial stress supplied by the confinement. After the impact, most of the specimens fracture but still keep the shape.

The peak stress is in the range of 1.28 ∼ 5.12MPa with the impact velocity in the range of 1.08 ∼ 5.19. The peak stress increases with the impact velocity. It is close to the results without confinement.

Cylindrical ice specimens diameter of 20mm

The specimens with the diameter of 50mm fracture with multiple cracks under the impact even if the impact velocity is very low. The critical condition of the fracture happening is interesting but can not be obtained. In order to reduce the impact given to the specimen, the diameter of bars are changed to 20mm. At the same time, the diameter of the specimen is also changed to 20mm. The smaller volume has two main influence on the experiment : first, the specimen contains fewer defects which may increase the strength of the specimen. Second, the temperature of the specimen increase quickly after being brought out of the refrigerator. To avoid it, a cooling chamber was used From the video, we can observe the fracture behaviour of 20mm-diameter specimen is different than the one of 50mm-diameter specimen at a lower impact velocity. The axial crack does not happen at the first impact from the input bar. Until the reflected stress generated by the contact with the output bar, the axial cracks initiated from the output bar to input bar. Then the specimen fractures. A typical process is shown in Fig. 2.25, and the parameters of the experiment are listed in Tab.2.9.

Number

Resolution Interval (µs) Exposure (µs) Frame rate ( f ps)

384×192

12.5 11.9 80000 TABLE 2.9: Image recording parameters for an experiment with a cylindrical ice specimen of diameter of 20mm The Young's modulus of ice is dispersive in a large range in different condition. Roland summarized the range of Young's Modulus and maximum stress values for different strain rates [START_REF] Ortiz | Assessment of damage model and strain rate effects on the fragile stress/strain response of ice material[END_REF], shown in Fig. 2.27. The results indicated from 10 ∼ 800s -1 , the Young's modulus increased with the strain rate increasing.

In our work, the strain rate was in the range of 10 ∼ 200s -1 . The range of Young's modulus FIGURE 2.27: Range of Young's Modulus and maximum stress values of ice for different strain rates [START_REF] Ortiz | Assessment of damage model and strain rate effects on the fragile stress/strain response of ice material[END_REF] could be estimated as 2 ∼ 8GPa according to Fig. 2.27 which confirmed the previous range we obtained 2.4Gpa ∼ 7.9Gpa. But it was still a large range.

The peak stress in our experiment was lower than Roland summarized. This may be caused by unavoidable defects in laboratory-grown specimens, such as small bubbles or micro-cracks.

According to the different fracture types, the previous experiments were divided into three groups : the specimen crushed from the input bar side once being impacted, several axial crack initiated from the input bar side once being impacted, several axial crack initiated from the output bar side after the stress wave arriving at the interface of the specimen and the output bar. The range of the impact velocity, the strain rate, the peak stress and Young's modulus are summarized in Tab.2.10. For a main crack propagation problem, the fracture type of specimen is in an intermediate state between crack initiating from the input bar side and the crack initiating from the output bar side. Therefore, the dynamic Young's modulus is determined about 4GPa, this value would be used in the following simulation.

From Tab.2.10, the strain rate is a dispersive feature to classify the experiments. Because for SHPB experiment, the strain rate is decided by impact loading and the length of specimen as expressed in Eq.2.7. Even with the same loading, the different length changes the strain rate. And the peak stress not always appears at the maximum strain rate. For the following rectangular specimen, the stress distribution is not uniform, it is hard to use the strain rate to classify experiments.

εS = C 0 L S [ε i (t) -ε r (t) -ε t (t)] (2.7)
While, the impact velocity is clear to classify the experiments. For the following experiment, a main crack propagation, the impact velocity should be similar with the type of several axial crack initiating from the input bar side, but a smaller range than 1.6 ∼ 5.0m/s. The impact velocity is also an easier value to control during the SHPB experiment than the strain rate.

Rectangular ice specimens

The cylindrical specimens were used to investigate the dynamic fracture behaviour of ice under impact loading. The rectangular specimens were used to investigate the crack propagation in ice under impact loading.

Rectangular ice specimens with a wedge and pre-crack

First, a rectangular specimen with a wedge and pre-crack was employed to conduct the experiment. The dimensions and fabrication have been introduced before.

A carbon powder was used to plot the random speckles on the surface of the specimen for the image analysis. The temperature needs to be well controlled to avoid melting of the surface of the specimen to maintain the speckles. Another reason is that the dynamic behaviour of ice is sensitive to the temperature.

One key factor for type of specimen is to choose the appropriate wedge to assure an as good as possible contact with the specimen. The refraction and reflection of the stress wave on the interfaces of the wedge and specimen are difficult to analyse. The only useful information from the signals is the impact velocity measured from the incident signal which is not affected by the refraction and reflection.

In order to reduce the refraction and reflection of the stress wave, the same material as the one used for bars, i.e. nylon, was first chosen to make a wedge. The experimental parameters are listed in appendix A, Tab.A.9. Before the impact, the specimen was put into the chamber for 15 minutes until the temperature reached the assigned value. The resolution of the high speed camera was reduced to get the higher interval. Because the crack propagates along the pre-crack, the height of the image could be cut into a narrow band.

After the impact, the ice specimen fractures into several pieces. The crack paths are shown in Fig. 2.28. The crack path in experiment No.2014022803 is smooth and follows a straight line, the angle between the pre-crack and the crack propagation direction is up to about 7 • from the horizontal axis. The others break along lines, and change directions obviously. Especially in experiment No.2014022804, the crack path changes direction more than 60 • . The difference may be caused by the impact velocity. The experiment No.2014022803 has the minimum value in the five experiments. The angle between the pre-crack and the crack propagation direction can be attributed to the bluntness of the pre-crack and the unbalanced forces on the two wedge surfaces due to the bad contact between the specimen and the wedge. The major drawback of this group of experiments is lacking calibration. In the experiment with the cylindrical specimen, the whole length of the specimen is captured by the high speed camera, and the length is pre-measured. But in this group, in order to increase the interval, the rectangular specimen is only captured a narrow band after the pre-crack and the begin and the end part are not captured. Therefore, the calibration can not be done and the crack path in the image can not be evaluated.

In the second group of the experiment conducted by the specimen with a notch, the wedged is 57 made by PVC. The purpose of this group is to find the appropriate impact velocity to get a main crack propagation. The experimental parameters are listed in appendix A, Tab.A.11.

After the impact, the ice specimen fractures into several pieces. The crack paths are shown in Fig. 2.29. A ruler close to the specimen is also captured in the image used for calibration in image analysis. Multiple cracks happen in this group, and are difficult to analyse. In order to get a main crack propagation, the impact velocity still needs to be reduced.

The major drawback of this group is that the process of the first two experiments are not recorded due to the inappropriate set up trig point. In the third group, a wood wedge was used to transfer the impact to the specimen. The impact velocity was in the range of 1.87 ∼ 4.03m/s. The experimental parameters are listed in appendix A, Tab.A.13. A main crack is obtained as shown in Fig. 2.30.

Rectangular ice specimens with a hole and pre-crack

For the rectangular specimens with notch, the waves transferred and reflected complicatedly on interface of the wedge and the specimen. It was hard to do signal analysis. Such a problem could be avoided with the rectangular specimens with hole. But the stress distribution was more complex near the hole where the compression was converted to the traction to make the crack Through the previous practice, the impact velocity was well controlled. 2m/s is the critical impact velocity for a main crack initiation and propagation in ice. The small scatter may be due to the micro-structure of the specimen and the bluntness of the pre-crack tip.

The crack propagates along the direction of the pre-crack in five experiments. In the others, the crack propagation direction changes without regularity. For the specimen with a hole, the compression is converted into the traction. During the process, the defects in the specimen affect the stress distribution more strongly than for the specimen with notch.

The drawback of this group experiment is the quality of the images which is too poor to distinguish the crack initiation. The fog which formed between the three-layer glass window reduced the quality of the images dramatically.

Conclusion and recommendations

This chapter presented a series of experiments for investigating the dynamic behaviour of ice under impact loading using the Split Pressure Hopkinson bars. Even if the experiments were difficult to perform : in the preparation of specimens, in the control of impact loading to get a unique crack propagation, in the capture of high quality images. We managed to find preliminary results about the dynamic behaviour of ice.

I-With the SHPB signals the dynamic stress-strain behaviour has been obtained. The peak stress is in the range of 1.5MPa ∼ 8.0M pa and increases with the impact velocity in the range of 0.5 ∼ 6.0m/s. When the impact velocity is controlled at the critical value the fracture appears, the dynamic Young's modulus is 4GPa which is prepared for the following simulation.

II-The cylindrical specimen fracture mode with several axial cracks is observed with the help of the high speed camera. With the impact velocity larger than 2m/s, the crack initiates once it is impacted by the input bar and propagates from the input bar side, then the opposite direction cracks appear. Finally, the radial cracks may happen. With the impact velocity near to 2m/s, the crack initiates until the stress wave reach the output bar and reflects and propagates from the output bar side. The crack propagation velocity is about 500m/s. III-In the well controlled experiments with the rectangular specimen, a main crack propagates in Mode I along the pre-crack in the horizontal direction, and finally the specimen separates into two main parts. The crack initiation is hard to distinguish by naked-eyes, therefore the image analysis methods will be introduced.

Despite our efforts, most experiments failed due first to bad image quality and also to irregular crack paths.

With the purpose of increasing the image quality, one solution is to employ a larger chamber which can contain the lamps inside it to improve the lightening. The other solution is to design vacuum interlayer window to achieve insulation and prevent frost patterned on the glass when the temperature decreases. The preparation of the speckles used for DIC needs also be improved.

The irregular crack paths may be due to the specimen geometry and its preparation, the contact between bar and specimen. The small change in the specimen geometry and the partial poor contact conditions can unbalance the stress distribution, which may explain why the crack is not propagating along the pre-crack direction. The presence of defects within the ice may also affect the crack propagation.

Chapter 3

Evaluation of crack generation and propagation by image analysis

This chapter will introduce image analysis methods in our investigation. First, the digital image is analysed based on the grey scale. The crack tip position is located, the crack path is observed and the crack propagation velocity is measured. Then the image correlation method is used to extract the displacement and strain fields although it lags behind the grey scale method in evaluating the crack. According to the DIC results, we try to use the J-integrals to calculate the stress intensity factor. However, due to the bad image quality the displacement field is not accurate enough to evaluate the stress intensity factor when the crack is propagating.

Introduction

All information from the SHPB measurement technique is based on the assumption of homogeneous stress and strain fields. Unfortunately, for the crack propagation problem, the area of interest is the vicinity of the crack tip where the stress field has singularity. Local displacement and strain fields measurement are needed. The high-speed photography technique as a non-contact method was used to capture the crack propagation process.

From the images, it was hard to observe the crack initiation and propagation by naked eyes. Then, the image analysis method was used to locate the crack tip position and evaluate the crack propagation. The crack path was enhanced by subtracting an image from the previous one, and the crack tip was located and crack propagation velocity was measured.

The digital image correlation was used to extract the displacement and strain field. After, the displacement field was used to estimate the stress intensity factor which is the most important feature to evaluate the crack propagation.

Digital images

All information from the SHPB measurement technique is based on the assumption of homogeneous stress and strain fields. Unfortunately, for the crack propagation problem, the area of interest is in the vicinity of the crack tip where the stress field has singularity. Local displacement and strain fields measurement are needed. The high-speed photography technique as a non-contact method is used to locate the crack tip position and evaluate the crack propagation.

Image recording parameters

The high speed camera used in our experiment to record the experimental process is a Phantom v710 camera as shown in Fig. 3.1. It can deliver 7350 frames per second (fps) in the resolution of 1280 × 800. A high frame rate is essential for dynamic experiment. It can be obtained by reducing the resolution to a lower value. In our experiment, the crack propagates along the pre-crack in the horizontal direction. Therefore, the height of the area of interest was reduced for obtaining a high frame rate. The trigger signal can be an external TTL pulse (hardware produced). A Multi-Channel Data Link (MCDL) box was used to supply the pulse and then the synchronization between the SHPB signals and camera were established.

Experimental digital images

A typical experiment is presented below. The image recording parameters are listed in Tab.3.1. The high speed camera focused on the specimen and recorded transformations of the specimen during impact. An example is given in Fig. 3.2 where 9 consecutive images show the crack propagation within the ice specimen. The duration of the crack propagation is about 216µs. The main crack propagates in Mode I along the pre-crack in the horizontal direction, and finally the specimen separates into two main parts.

At 216µs, the crack has already propagated to the end of the area of interest, but the crack path is hard to observe with naked-eyes. Therefore, the image analysis are needed to locate the crack tip.

Grey scale 3.3.1 Principle of grey scale

The image recorded by high speed camera is a grey-scale digital image. It contains a fixed number of rows and columns of pixels. Pixels are the smallest individual element in an image, hold quantized values that represent only intensity information at any specific point varying from black at the weakest intensity to white at the strongest [Johnson, 2006]. Therefore the images can be taken as grey scale matrices carrying out mathematical calculations. The images as grey-scale matrices were used to evaluate the crack tip position and the crack path. The image contrast was enhanced by subtracting the gray-scale matrix of an image from the previous one, if the crack propagated during the interval time of the two images, because of a low level of deformation, a difference appeared and could be taken as a crack path. The results are shown in Fig. 3.3. In the first image, nothing is visible. From the second to the eighth image, obvious differences appear which are considered as a crack propagation. The crack tip locations are estimated for each image. The length of the crack propagation at each interval of 27µs is measured, thereby the crack propagation velocity is evaluated. The whole crack path can be obtained by subtracting the first image from the last one as shown in the 9 th image.

Digital image correlation 3.4.1 Principle of digital image correlation

The digital image correlation method was also used to analyse the images. As a non-contact measurement technique to obtain the displacement and strain fields in an interesting area, it was [START_REF] Peters | Digital imaging techniques in experimental stress analysis[END_REF].

Before the experiment, a random speckle pattern on the surface of the specimen was prepared.

During the experiment, an image was captured before loading as the reference image and subsequent images were captured as the deformed images corresponding to the loading process. A series of images is shown in Fig. 3.4.

Before analysis, an area was chosen as the interesting area, and a small square subset was defined. The size of the subset was determined by the quality of the image and the speckle pattern on the surface of the specimen. As a general rule, larger subset sizes increase the accuracy whereas a smaller subset increase the resolution.

During the analysis, each subset of the reference image was sought in the deformed images based on the matching the pixel gray scale values [START_REF] Roux | Digital image correlation and fracture : an advanced technique for estimating stress intensity factors of 2d and 3d cracks[END_REF]. Once the same location of a subset in the deformed image was identified, the local displacements could be quantified.

Consider a series of images is recorded by high speed camera. The intensity patterns f (x, y) and f * (x * , y * ) correspond to the reference image and the deformed image respectively as described in Eq. 3.1. They are assumed to be one-to-one correspondence in the deformed image. Where u(x, y) and v(x, y) represent the displacement components of the reference subset center in an x and y coordinate system. Assume that point is G. The matching procedure is to find the corresponding subset centered at P * in the deformed image in the x * and y * coordinate system as shown in Fig. 3.5. The criterion is the subset in deformed image has the maximum similarity with the reference image. Consequently, two images are corresponding by the two central points G and G * . The point G * is expressed by Eq.3.2.

x * = x + u(x, y) y * = y + v(x, y) (3.2)
The displacement fields can be approximated using the Taylor series around a point G(x 0 , y 0 )

with Eq.3.3.

x * = x + u + ∂u ∂x ∆x + ∂u ∂y ∆y + 1 2 ∂ 2 u ∂x 2 ∆x 2 + 1 2 ∂ 2 u ∂y 2 ∆y 2 + ∂ 2 u ∂x∂y ∆x∆y y * = y + v + ∂v ∂x ∆x + ∂v ∂y ∆y + 1 2 ∂ 2 v ∂x 2 ∆x 2 + 1 2 ∂ 2 v ∂y 2 ∆y 2 + ∂ 2 v ∂x∂y ∆x∆y (3.3)
where v and u are the vertical and horizontal displacements, ∆x = xx 0 and ∆y = yy 0 .

To evaluate the similarity between the reference and the deformed images, a correlation coefficient should be predefined and computed. Here, a correlation coefficient is introduced in Eq.3.4.

C = ∑ S [ f (x, y) -f * (x * , y * )] 2 ∑ S f 2 (x, y) (3.4)
Where S is the subset used for calculation, f and f * are the grey scale light intensities corresponding to all the points in the subset of the reference image and deformed images respectively.

In order to improve the displacement fields, the correlation coefficient is minimized. The optimal displacement field consists with the minimum of the correlation coefficient.

More details about the principle of DIC can be found in the references [START_REF] Chu | Applications of digital-imagecorrelation techniques to experimental mechanics[END_REF]] [Pan et al., 2009].

Evaluation crack generation and propagation

The displacement fields from the same experiment were estimated by image correlation as shown in Fig. 3.6 and Fig. 3.7.

The displacement fields in X direction do not supply useful information. The displacement fields in Y direction shows the specimen is separated into two main parts following the impact loading. The top part is with positive displacement and the bottom part with negative displacement.

Then the strain fields were extracted from the displacement fields. The strain fields ε xx and ε yy during the crack propagation are shown in Fig. 3.8 and Fig. 3.9, respectively.

The displacement field along X illustrates that no sliding happens between the top and bottom parts, since no discontinuity can be observed. One may conclude that the main crack does not propagate as Mode II.

The crack path can be observed obviously in the strain filed ε yy . It demonstrates that the crack propagates in Mode I.

U x 0µs U x 81µs U x 162µs U x 27µs U x 108µs U x 189µs U x 54µs U x 135µs U x 216µs FIGURE 3.6: Horizontal displacement U x measured by DIC (mm)
It is difficult to identify the crack path from strain field measurement. One possibility consists in plotting strain field discontinuities. Another one is to plot the maximum values, as shown in Fig. 3.10, where the crack path is clearly identified.

There the crack path and crack tip can not be evaluated in the first four images. In the image at 27µs and 54µs, it could be supposed there is not deformation on the surface of specimen, but only internal crack. In image after 81µs the dots did not merge into a line, it could be supposed the deformation occurred but hard to identify a recognized crack. After 108µs, a recognized crack appeared, the precise location of the crack tip is still difficult to identify. The most part of crack path can be recognized at 216µs, and confirms that there is only one main crack.

By DIC, the most part of crack path is distinguished, but crack tip location is hard to define at different times. The possible reason is that the speckles on the surface of the specimen was not good enough which makes the displacement field discontinuity.

All the DIC processing was performed with VIC 2D. 3.4.3 Evaluation SIFs by the J-integrals contour Γ around the crack tip. Then the J-integral is defined in the absence of any body forces as

J k = Γ Q k dΓ (3.5)
where k = 1, 2 and

Q k = W n k -σ i j ∂u i ∂x k n j (3.6) ε xx 0µs ε xx 81µs ε xx 162µs ε xx 27µs ε xx 108µs ε xx 189µs ε xx 54µs ε xx 435µs ε xx 216µs FIGURE 3.8: Strain ε xx measured by DIC
W is the total strain-energy density defined as

W = ε i j 0 σ i j dε i j (3.7)
An arbitrary but continuous function S(x 1 , x 2 ) is defined with the property

S(x 1 , x 2 ) = 1 (3.8) S(x 1 , x 2 )
is the coordinate of the node.

on the node inside. S(x 1 , x 2 ) = 0 (3.9) on the node outside.

(J k ) domain = - A W ∂S ∂x k -σ i j ∂u i ∂x k ∂S ∂x j dA (3.10)
Where A is the J-integral domain.

Rewritten in a form convenient to numerical computation as 3.11) represents the energy change per unit crack extension.

(J k ) domain = - A W ∂S ∂x k -u x k T [σ] S dA (
where

u x k T = ∂u 1 ∂x k ∂u 2 ∂x k ( 3
S T = ∂S ∂x 1 ∂S ∂x 2 (3.14)
The strain-energy distribution W is

W = 1 2 [σ 11 ε 11 + σ 22 ε 22 + 2σ 12 ε 12 ] (3.15)
The domain integral is

(J k ) domain = ∑ i J x k i (3.16)
where J k i is the area integral over the ith element.

In the isoperimetrical representation, the displacements within the element are defined by the shape functions N j and the nodal displacements

u α = N j (u α ) j (3.17)
where α = 1, 2, N j = N j (ξ, η), ξ , η are the parent coordinates and j = 1 to the number of nodes on the element.

(J k i ) domain = - 1 -1 1 -1 W ∂S ∂x k -u x k T [σ] S (det [J ])dξdη (3.18)
The area integral J x k i of the ith element can be computed using Gaussian quadrature as

(J k i ) domain = - M G ∑ m=1 M G ∑ n=1 W ∂S ∂x k -u x k T [σ] S w m w n (det [J ]) (3.19)
where M G is the number of Gaussian quadrature points used in each direction ξ and η ,w m and w n are the Gaussian weights and det[J] is the determinant of the Jacobian matrix [J].

S function is defined using the element shape functions

S (ξ, η) = N j S j (3.20)
In our case, the (x, y, O) coordinate is established according to the image. The crack tip is signed with a red point located by grey scale or image correlation, respectively as shown in Fig. 3.11. x e and y e are the Cartesian coordinates of each element with 4-nodes. u e and v e are the X and the Y displacements. All these values are from digital image correlation results. S e is S function described as before, at the node (the blue point) inside the integral zone, S e = 1, at the node (the grey point) outside the integral zone, S e = 0 as shown in Fig. 3.12. FIGURE 3.12: 6 pixel × 6 pixel J-integral zone centred with crack tip

x e =            x e 1 x e 2 x e 3 x e 4            , y e =            y e 1 y e 2 y e 3 y e 4            , u e =            u e 1 u e 2 u e 3 u e 4            , v e =            v e 1 v e 2 v e 3 v e 4            (3.21)
according to the element position to the integral domain, S e is different. The shape function is defined as

N 1 (r, s) = 1 4 (1 -r)(1 -s) N 2 (r, s) = 1 4 (1 + r)(1 -s) N 3 (r, s) = 1 4 (1 + r)(1 + s) N 4 (r, s) = 1 4 (1 -r)(1 + s) (3.23)
Partial derivatives of shape function

N 1, r = - 1 4 (1 -s) N 1,s = - 1 4 (1 -r) N 2, r = 1 4 (1 -s) N 2,s = - 1 4 (1 + r) N 3, r = 1 4 (1 + s) N 3,s = 1 4 (1 + r) N 4, r = - 1 4 (1 + s) N 4,s = 1 4 (1 -r) (3.24)
x r = N r x e x s = N s x e y r = N r y e y s = N s y e (3.25)

Jacobian matrix [J ] defined by

J e = x r x s y r y s (3.26) N x = y s N r -y r N s det(J e ) (3.27)
N y = - x s N r + x r N s det(J e ) (3.28)
Partial derivatives of S function ∂q ∂x = N x q e ∂q ∂y = N y q e (3.29)

Thus the strain is

ε xx = ∂u ∂x = N x u e ε yy = ∂u ∂y = N x v e γ xy = ∂v ∂x + ∂u ∂y = N x v e + N y u e (3.30)
The stress is σ = Cε (3.31) where

σ =      σ xx σ yy τ xy      , ε =      ε xx ε yy γ xy      (3.32)
for linear elastic plane stress

C = E 1 -ν 2    1 ν 0 ν 1 0 0 0 1-ν 2    (3.33)
The strain-energy density is

w = 1 2 σε T (3.34)
The J integral in an element can be rewritten as

J = A σ xx ∂u ∂x + τ xy ∂v ∂x -w ∂S ∂x + τ xy ∂u ∂x + σ yy ∂v ∂x ∂S ∂y dA (3.35)
Derivation process :

(J k ) domain = - A W ∂S ∂x k -u x k T [σ] S dA (3.36)
J 1 is usually termed the J-integral, J 2 is called the product integral. So J-integral only compute The J integral in an element is computed with the equation below

J 1 . When k = 1 (J x 1 ) domain = - A W ∂S ∂x -u x T [σ] S dA = - A W ∂S ∂x -
J = 1 -1 1 -1 Q(r, s)drds (3.39) Q(r, s) = σ xx ∂u ∂x + τ xy ∂v ∂x -w ∂S ∂x + τ xy ∂u ∂x + σ yy ∂v ∂x ∂S ∂y det(J e ) (3.40)
J is computed using Gaussian quadrature as

J ≈ Q(r 1 , s 1 ) + Q(r 2 , s 2 ) + Q(r 3 , s 3 ) + Q(r 4 , s 4 ) (3.41) Gaussian quadrature 1 -1 f (x)dx = n ∑ i=1 w i f (x i ) (3.42) Gauss -Legendre quadrature Number of points n = 2, Points x i = ± 1 √ 3 , weight w i = 1 where, r = - 1 √ 3 1 √ 3 1 √ 3 - 1 √ 3 s = - 1 √ 3 - 1 √ 3 1 √ 3 1 √ 3 (3.43)
Since the stress intensity factor K I is a very well known quantity, a relation has been established to convert the volume integral J I to it with Eq.3.44.

K I = √ E * • J (3.44)
Where,

   E * = E plane stress E * = E 1 -ν 2 plane strain (3.45)
The present problem can be considered a plane stress problem.

The stress intensity factors in a narrow band at different times of the typical experiment No.2014030603 are plotted in Fig. 3.13.

Due to the difficulty of maintaining speckles on the surface of a specimen and lightening issues in the cooling chamber, the displacement field established by digital image correlation was not accurate enough to evaluate the stress intensity factors when the crack is propagating. The values in different images scattered a lot and were meaningless.

The SIF is estimated by MATLAB with the displacement from VIC 2D . 3.5 Results

Comparison between the two methods

First, two methods are compared in distinguishing the crack tip. The area near to the pre-crack is defined as the area of interest as shown in Fig. 3.14. At 54µs, it is hard to observe the change by naked eyes. Then the maximum ε yy strain component obtained by DIC method and the subtracted image are shown in Fig. 3.15. The crack tip can be defined in the subtracted image at 54µs, but until 108µs, the similar path appears in the DIC results.

Then, two methods are compared in distinguishing the crack path as shown in Fig. 3.16. At The DIC results lag behind the directly subtracted images. It may be attributed to two main reasons.

First, the ice specimen is about 30mm in thickness. In the very beginning, the crack initiates internally and does not reach into the surface, no deformation appears or the deformation is too small to be detected. Therefore, the subtracted image can show the difference inside the specimen. But the DIC method is based on the deformation of the surface and it can not give useful results.

Second, due to the bad lightening, the images is not precise enough for distinguish the crack tip by DIC method.

Since the precise location of the crack tip is difficult to identify by DIC method. The attempt to evaluate the stress intensity factor by DIC is irrealizable for the crack propagation in ice under impact loading.

The crack tip can be more easily detected by Image grayscale subtraction.

Crack propagation velocity

Crack propagation velocities measured by image subtraction are shown in Fig. 3.17. The experimental conditions are listed in Tab. 3.2,Tab.3.3 and Tab.3.4. The crack propagation velocity varies with the crack length. It firstly increases and then keeps at a constant value. Afterwards, it decreases. It reaches 534 ± 83m/s, which is about 25% of the Rayleigh wave velocity in ice (2000m/s). This velocity is the crack propagation velocity in the area of interest not the whole specimen. Usually the area is chosen from the end of the pre-crack to the end of the specimen. The interaction between the specimen and the output bar cannot be ignored. Therefore the crack propagation velocity at the end of the specimen is not considered. Meanwhile the crack bifurcation decreases the velocity, the value after the crack bifurcation is also not analyzed.

Petrenko [START_REF] Petrenko | Crack velocities in freshwater and saline ice[END_REF] gave the average crack propagation velocity in laboratory-grown ice in the range of 100 ∼ 1320m/s with a strain rate of 10 -3 s -1 . He explained that the large scatter was due to the grain boundaries and crystallographic orientation in ice and Due to the quality of the images, the crack propagation length in each interval is not very accurate, however it is sufficient enough to demonstrate the tendency over the crack length.

Combescure [START_REF] Combescure | Experimental study of high-velocity impact and fracture of ice[END_REF] reported the crack propagation velocity in cylindrical ice projectile slowed down between 300 ± 75m/s with a mean strain rate around 30s -1 and indicated the deceleration was due to the bifurcation. Comparing with our experiments, the mean strain rate is in the same magnitude (10 ∼ 40s -1 ), the higher crack propagation velocity can be caused by keeping only one main crack propagating and avoiding the bifurcation.

Gagnon [Gagnon, 2003] measured the crack propagation velocity in the 0 ∼ 550m/s range in a thick-edge-loaded freshwater ice sheet and indicated the crack jump velocities were generally the highest at the beginning of the test. Due to the fact that our dimensions are much smaller, our value can be taken as the beginning one. The crack propagation velocity in ice measured from our experiments is close to the value reported by Gagnon.

Conclusion and recommendations

This chapter introduced two image analysis methods used in our investigation to evaluate the crack propagation, the grey-scale concept and digital image correlation. The crack propagation in ice under impact loading was described by image analysis. I-After the impact, the ice specimen is separated into two main parts. The crack propagates in Mode I. The crack tip position is located, the crack path is observed and the crack propagation velocity is measured. The crack propagation velocity firstly increases and then keeps at a constant value. Afterwards, it decreases. It reaches 534 ± 83m/s, which is about 25% of the Rayleigh wave velocity in ice (2000m/s).

II-The digital image correlation lags behind the subtracting grey scale directly in evaluating the crack propagation.

We have tried to use the displacement field extracted by DIC to calculate the J-integrals. However, due to the bad image quality the displacement field is not accurate enough to evaluate the stress intensity factor when the crack is propagating. A numerical method is needed to evaluate the crack propagation and will be introduced in the next chapter.

Chapter 4

Numerical study on crack propagation

This chapter introduces the numerical method to assess dynamic crack propagation criteria. The eXtended Finite Element Method(XFEM) is used to simulate the fracture processes. In the simulation, a crack is represented by two level sets, SIFs are evaluated by volume integrals. When the simulation fits the experiment, the crack initiation toughness and the crack growth toughness can be evaluated from the simulation results. The factors that influence the fracture toughness are found, and the relationship between the crack growth toughness and crack propagation velocity is established.

Introduction

As the digital image correlation was not able to evaluate the stress intensity factor due to the quality of the image, the speckles on the specimen and the illumination condition. Numerical simulations were used to access the dynamic crack criteria of propagation as a complement.

Because of the advantages on analysing the crack propagation problem, the extended finite element method was chosen to simulate the crack propagation.

The simulation was conducted by CAST3M software with a 2D model. The specimen with the input and output bars of SHPB set-up was simulated. But only the specimen was meshed.

The stress intensity factor of ice under impact loading was estimated when the simulation fitted the experiment. The crack initiation and growth toughness were evaluated according to the synchronization of the signals and images from experiments and the results from simulation. The influence factors such as crack propagation velocity, the rate of increase of stress intensity factor were discussed.

Extended finite element method 4.2.1 Advantages of XFEM in crack problem

The extended finite element method(XFEM) was chosen to simulate the test process because it has a lot of advantages on analysing crack propagation problem compared to the other numerical methods. First, the crack surface and crack front are completely independent of the mesh and therefore remeshing is not needed when the crack propagates. Second, the crack surface is constructed in terms of nodal values by additional basis functions especially combined with level sets, instead of being located on element edges, which simplifies the model. Furthermore, the singular stress field in the vicinity of crack tip is solved with enrichment functions, and the requirement of refinement is reduced [START_REF] Belytschko | A review of extended/generalized finite element methods for material modeling[END_REF].

Basic XFEM approximation

Considering a finite element model of a homogeneous, elastic cracked body is shown in Fig. 4.1. The set of all nodes in the classical finite element mesh is denoted by S, the set of nodes of elements around the crack tip is denoted by S C and indicated by the circled nodes, the set of nodes of elements cut by the crack but except S C is denoted by S H and indicated by the squares. Nodes in S C and S H served as tip enriched and step enriched nodes, respectively [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF].

The approximation of the XFEM displacement field at time t is expressed in Eq.4.1. The first item is a classical finite element approximation, the others are enrichment approximation. N i (x) is the shape function associated with node i. b j (t) are the nodal degrees of freedom corresponding to the step function H(ψ(x,t)). B l (r, θ) is used to define the span fields of the near-tip asymptotic and the a l k (t) are corresponding nodal degrees of freedom. The crack is represented by two level sets, and ψ(x,t) has opposite signs on the two sides of the crack [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]][Stolarska et al., 2001].

h (x,t) = ∑ i∈I u i N i (x) + ∑ j∈S H b j (t)N j (x)H(ψ(x,t)) + ∑ j ∑ k∈S C N k (x)( 4 ∑ l=1 a l k (t)B l (r, θ)) (4.1)
where H() is the Heaviside step function given by Eq. 4.2. The discontinuity is defined as a crack.

H( f ) = 1, f > 0 0, otherwise (4.2)
The branch function is defined by Eq. 4.3

B l (r, θ) = √ rsin θ 2 , √ rcos θ 2 , √ rsin θ 2 sinθ, √ rcos θ 2 sinθ (4.3)
where(r, θ) is a polar co-ordinate system with its origin at the crack tip and θ = 0 tangent to the crack at its tip. The above functions span the asymptotic crack tip solution of elasto-statics, and √ rsing θ 2 takes into account the discontinuity across the crack face.

The direction in which the crack will propagate from its current tip, θ c , is obtained using the maximum hoop stress criteria [START_REF] Dolbow | A finite element method for crack growth without remeshing[END_REF]. The angle θ c depends on the stress intensity factors, K I and K II , and is given by

θ c = 2 arctan 1 4   K I K II ± K I K II 2 + 8   , -π < θ c < π (4.4)

Description of the crack by level set

The level set method (LSM) is a numerical technique developed by Osher and Sethian [START_REF] Osher | Fronts propagating with curvaturedependent speed : algorithms based on hamilton-jacobi formulations[END_REF] to track the motion of interfaces. In this method, an interface is modelled by a continuous function defined for each point of the space which gives the distance from the considered point to this interface. The interface is directed and distance will be counted positively or negatively according to which side of the interface lies to the point of interest [START_REF] Pommier | Extended finite element method for crack propagation[END_REF].

The level set method coupled with the extended finite element method to model crack growth is reported by Stolarska for the first time in two-dimensions [START_REF] Stolarska | Modelling crack growth by level sets in the extended finite element method[END_REF]. Then Moës developed it into three-dimensions [START_REF] Moës | Non-planar 3d crack growth by the extended finite element and level sets-part i : Mechanical model[END_REF].

A level set separates a space into two zones, a positive and a negative one. For describing a crack, two level sets are needed as shown in Fig. 4.2. One level set denoted as ψ (the green solid line) separates the space into two zones, could be taken as a crack path and its expansion along tangent direction. The other level set denoted as φ (the blue dotted line) defines the crack front surface which is perpendicular to the first one. The intersection of two level sets is the crack tip (the red point).

Model

A numerical simulation was established as a 2D model shown in Fig. 4.4. The specimen with the input and output bars of SHPB setup were simulated. But only the specimen was meshed. The bars were simulated with impedance boundary conditions to reduce the computational time. At each node on the specimen-bar interface, the stress and velocity were assumed by Eq.4.5. This was working only at the beginning before the wave reflected at the ends of the bars. The validity of the impedance condition has been checked by Grégoire et al [START_REF] Grégoire | Dynamic crack propagation under mixed-mode loading-comparison between experiments and x-fem simulations[END_REF].

The impact velocity was obtained from experiments by Eq. 4.6, and simplified as shown in Fig. 4.5.

     σ • n = -z(v -v impact • n)n input bar σ • n = -z(v • n)n out put bar z = ρ bar c bar (4.5) V impact = C 0 ε i (t) ( 4

Algorithm

The discrete equation is as below.

M u n+1 -2u n + u n-1 ∆t 2 + K u n+1 + 2u n + u n-1 4 = F n (4.7)
Where M and K are the mass and the stiffness matrices. u n is the displacement at time step t n . F n is the vector of external forces at time step t n . Then the displacement at time step t n+1 is calculated.

( M ∆t 2 + K 4 )u n+1 = F n + 2( M ∆t 2 - K 4 )u n -( M ∆t 2 + K 4 )u n-1 (4.8)
Node j is one of the nodes on the interface between input bar and the specimen. And the external force on node j is as below :

σ j = ρCv 0 -ρCΣv j ϕ j (4.9)
where, ϕ j is the shape function. Then, external forces can be expressed by all the nodes on the interface.

F i = i σϕ j = i ρC(v 0 -∑ v j ϕ j )ϕ i = ρCv 0 i ϕ j -∑ v j ρC i ϕ j ϕ i (4.10)
Φ i is defined as

Φ i = ρCv 0 i ϕ i (4.11)
K imped is defined as

K imped = ρC i ϕ j ϕ i (4.12) v j is expressed by v j = u n+1 -u n+1 2∆t (4.13)
Then the displacement at time step t n+1 is converted into Finally , the synchronization among the signal, image, and numerical results were established. The crack initiation toughness was the intersection point of the stress intensity factor history and the crack initiation time. The Young's modulus in the simulation was 4Gpa according to the experimental results. 3GPa and 5GPa was also checked. The stress intensity factor history at the pre-crack tip position is shown in Fig. with the dynamic Young's modulus 3GPa, 4GPa, 5GPa. The dispersion could be accepted even there was an experimental error in measuring the dynamic Young's modulus.

Evaluation of crack growth toughness

Once a crack initiated in ice under impact, to estimate the crack growth toughness the crack propagation velocity must be taken into account.

In simulations, once the stress intensity factor at the pre-crack tip was larger than crack initia- The crack length of each section was measured from the image based on the experiment as shown in Fig. 4.11a, and also obtained the velocity by previous image analysis. The time step used in the simulation was not the same with the interval time. So the crack propagation velocity used in the simulation is an equivalent value as shown in Fig. 4.11b. The crack growth path was pre-defined and divided into sections as shown in Fig. 4.11c.

Then we calculated the stress intensity factors at the end of each section in the interval considered as the crack growth toughness.

One typical stress intensity factor history is shown in Fig. 4.12.

Numerical results

Crack initiation toughness

The crack initiation toughness is estimated to be 204 ± 13kPa • √ m. The related experimental conditions are listed in Tab.4.2. The dynamic value is in the range of 50 ∼ 400kPa • √ m as reported under quasi-static loading [START_REF] Liu | Fracture toughness of fresh-water ice[END_REF]][Bentley et al., 1989] [Dempsey, 1991] [Petrovic, 2003]. As noted before, the crack initiation time is estimated by the crack length and the crack propagation velocity in the onset interval. The onset crack length can be measured from images, but the onset crack propagation velocity can be identified only roughly. It could caused the crack initiation toughness scattered across the interval. It is why the toughness identified from the previous image and the next image is also plotted in FIG. The crack initiates when the loading still increases before it reaches plateau. Therefore, the loading rate is taken into account when we assess the fracture behaviour. The loading rate is represented by the increase rate of the initial dynamic stress intensity factor KI also reported in the legend of Fig.

Crack growth toughness and propagation velocity

When the crack grows, the crack propagation velocity should be taken into account. The crack growth toughness versus the crack propagation velocity is plotted in Fig. 4.15. From this data, it is possible to obtain the fitted curves, as reported by Freund et al [Freund, 1998], showing that the crack growth fracture toughness is linearly associated with the crack propagation velocity. Despite experiment 2014053006, the fitted curves belonging to each group are paralleled to a first approximation.

Conclusion and recommendations

The extended finite element method was employed to assess the dynamic crack propagation criteria in this chapter. When the simulation fitted the experiment, the crack initiation toughness and the crack growth toughness were evaluated from the simulation results. The relationship between the crack growth toughness and crack propagation velocity was established. II-With the crack propagation velocity in the previous chapter, the crack path was pre-defined and divided into sections. Then the crack growth toughness in each section is determined to III-The crack growth toughness is sensitive to the loading rate represented by the rate of increase of the initial dynamic stress intensity factor and seems independent to the temperature in the range -15 ∼ -1 • C.

In this chapter, we only introduced the crack criteria obtained by pre-defining the crack history in the simulation. We also took much time and effort to define a universal relation between the crack propagation velocity and the fracture toughness and tried to use the crack criteria to predict the crack propagation. But we did not get meaningful results consistent well with the experimental results. Of all the possible reasons one might think it is due to the defects in the specimen, it can not be taken as a homogeneous material. The growth fracture toughness is scattered along the fitted curve as shown in FIG. 4.15. We can eliminate the defect effect by curve fitting to assess the crack propagation, but we can not forecast the crack propagation ignoring the defects with the criteria defined with the fitting curve.

General conclusions and prospects

Conclusions

Experimental and numerical works have been done in this study with the objective of investigating the crack propagation in ice under impact loading. A series of impact experiments with different ice specimens has been conducted using Split Pressure Hopkinson bars. The image analysis method has been employed to describe the crack propagation history. The extended finite element method has been used to simulate the experimental process. We managed to establish the methodology of investigating crack propagation in ice and find preliminary results about the dynamic fracture behaviour of ice. I-The methodology of investigating dynamic crack propagation in ice is established by SHPB experiments, image analysis and XFEM simulation.

II-The dynamic stress-strain behaviour has been obtained from experiments with cylindrical specimens. The measured dynamic Young's modulus is about 4GPa which is used in the numerical simulation. The peak stress is in the range of 1.5MPa ∼ 5.5MPa.

III-The rectangular specimen is separated into two main parts after the impact. V-The crack growth toughness is sensitive to the loading rate represented by the rate of increase of the initial dynamic stress intensity factor. The dynamic behaviour of ice under impact loading seems temperature independent in the range -15 ∼ -1 • C.

Prospects

The results obtained by experiments and simulations are only a first approximation of the dynamic fracture behaviour of the ice under impact loading. I-The experimental result in this work was dispersive, the may reason was that the quality of the specimen was not uniform. The laboratory-grown ice had avoidable defects which had influences on the results. Since the methodology was established, the dynamic crack propagation in particular ice can be investigated in the same way.

II-The future work will focus on evaluating the fracture toughness with the digital image correlation method. The results can be taken as the verification and validation of the simulation values. To attain it, two aspects need to be improved, the quality of images and the precision of the DIC algorithm.

III-The fracture toughness seems independent to the temperature in the investigated range of temperature. In order to investigate the sensitivity of the dynamic fracture behaviour of ice, the temperature needs to a larger range. Meanwhile, the experiments need to be conducted under a larger range of impact loadings.

IV-Regarding the numerical aspect, more effort should be devoted also to define a universal relation between the crack propagation velocity and the fracture toughness and also to predict the crack propagation. This is to be continued.

A.

1. 10

 10 Conventional fracture specimen geometries . . . . . . . . . . . . . . . . . . . . 18

  concepts of Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 A Brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Fracture mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Asymptotic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 The asymptotic solution in Mode I . . . . . . . . . . . . . . . . 1.1.3.2 The asymptotic solution in Mode II . . . . . . . . . . . . . . . . 1.1.3.3 The asymptotic solution in Mode III . . . . . . . . . . . . . . . 1.1.4 Fracture Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.1 Stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . 1.1.4.2 J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Dynamic fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.1 Inertial effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5.2 Dynamic fracture toughness . . . . . . . . . . . . . . . . . . . . 1.2 Mechanical properties of ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Types of ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Elastic modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Brittle behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Tensile and compressive strength . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Fracture behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Experimental study on ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Quasi-static experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Dynamic experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.1 Impact directly experiments . . . . . . . . . . . . . . . . . . . . 1.3.2.2 Be impacted experiment conducted by Split Hopkinson bars . . . 1.4 Numerical method for dynamic fracture problem . . . . . . . . . . . . . . . . 1.4.1 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Meshless methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 The partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Extended finite element method . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusion and recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Major concepts of Fracture mechanics 1.1.1 A Brief history

FIGURE 1 . 1 :

 11 FIGURE 1.1: Three modes of crack stress and displacements

FIGURE 1 . 2 :

 12 FIGURE 1.2: Crack tip coordinate system

  FIGURE 1.10: Conventional fracture specimen geometries

FIGURE 1 .

 1 FIGURE 1.13: Phenomena relating to the fracture behaviour of ice

FIGURE 2 .

 2 FIGURE 2.2: Cooling chamber

FIGURE 2 . 4 :

 24 FIGURE 2.4: Internal and external temperature of the specimen vs. time

  FIGURE 2.5: Distinguish the incident and reflected signal

FIGURE 2

 2 FIGURE 2.6: Nylon, PMMA, PVC striker bars

FIGURE 2 .

 2 FIGURE 2.9: Input signals of PMMA striker bars with different lengths

FIGURE 2 .

 2 FIGURE 2.11: Input signals of different striker bars with foam rings

FIGURE 2 .

 2 FIGURE 2.12: Dimensions of two types of rectangular specimens

2. 4 FIGURE 2 .

 42 FIGURE 2.13: Signals from a typical experiment

  FIGURE 2.14: Offset voltage correction

FIGURE 2 .

 2 FIGURE 2.15: Incident, Reflected and Transmission voltage signals

  FIGURE 2.16: Signal filtered with a low-pass filer

FIGURE 2 .

 2 FIGURE 2.17: Signals before and after filter

FIGURE 2 .

 2 FIGURE 2.18: Strain signals after processing

FIGURE 2 .

 2 FIGURE 2.19: Cylindrical ice specimen of diameter of 50mm under impact

FIGURE 2 .

 2 FIGURE 2.20: Stress-strain curves and the evaluated dynamic Young's modulus

FIGURE 2 .

 2 FIGURE 2.22: Impact velocity vs. peak stress

FIGURE 2 .

 2 FIGURE 2.24: Stress strain curves for cylindrical ice specimen with confinement

FIGURE 2 .

 2 FIGURE 2.25: Cylindrical ice specimen of a diameter of 20mm under impact

FIGURE 2 .

 2 FIGURE 2.26: Mean values for cylindrical ice specimen of diameter 20mm

  FIGURE 2.28: Crack path in rectangular ice specimen with a wedge and pre-crack Group 1

  FIGURE 2.29: Crack path in rectangular ice specimen with a wedge and pre-crack Group 2

  FIGURE 2.30: Crack path in rectangular ice specimen with a wedge and pre-crack Group 3

  FIGURE 2.31: Crack path in rectangular ice specimen with a notch and pre-crack

Contents 3 .

 3 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Digital images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1 Image recording parameters . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.2 Experimental digital images . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Grey scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

FIGURE 3

 3 FIGURE 3.1: High speed camera phantom v710

FIGURE 3 . 2 :FIGURE 3 . 3 :

 3233 FIGURE 3.2: Crack propagation images from a rectangular ice specimen

FIGURE 3 . 4 :

 34 FIGURE 3.4: Reference and deformed images

  FIGURE 3.7: Vertical displacement U y measured by DIC (mm)

FIGURE 3 . 9 :

 39 FIGURE 3.9: Strain ε yy measured by DIC

  FIGURE 3.10: Maximum strain ε yy measured by DIC

FIGURE 3 .

 3 FIGURE 3.11: Coordinate system established by images for J-integral

FIGURE 3 .

 3 FIGURE 3.13: SIFs evaluated by J-integral, (Pa • √ m)

FIGURE 3 .

 3 FIGURE 3.14: Area of interest near to pre-crack

FIGURE 4 .

 4 FIGURE 4.1: Enrichment to the classical finite element approximation

  u

FIGURE 4 . 2 :

 42 FIGURE 4.2: Crack path presented by two level sets

  The simulation was conducted by CAST3M software. The level set was realized by PSI-PHI function which describes the local frame of the crack. The volume integral was provided by GT HETA function. The procedure is illustrated in Fig.4.3. An example will be given to illustrate the simulation process.

FIGURE 4 . 3 :

 43 FIGURE 4.3: Methodology to access crack propagation

FIGURE 4 . 4 :

 44 FIGURE 4.4: Two types of XFEM numerical simulation models

FIGURE 4 . 5 :

 45 FIGURE 4.5: Impact velocity used in the simulation evaluated by experimental incident signal

  FIGURE 4.6: Numerical stress intensity factor at pre-crack tip without propagation

FIGURE 4 .

 4 FIGURE 4.10: Crack initiation toughness estimated with different Young's modulus

FIGURE 4 .

 4 FIGURE 4.11: Crack path divided according to the experimental results

FIGURE 4 .

 4 FIGURE 4.13: Dispersive range of crack initiation toughness

I

  -The crack initiation time was estimated by the synchronization between SHPB signals and images. Then the time corresponded with the simulation. The corresponding initiation toughness in the simulation is determined to be 204 ± 13kPa • √ m.

FIGURE 4 .

 4 FIGURE 4.15: Crack growth toughness history versus crack propagation velocity

  1.1 Three modes of crack stress and displacements . . . . . . . . . . . . . . . . . .

6 1.2 Crack tip coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 J-integral path around a crack tip . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Ductile to brittle transition

  [START_REF] Combescure | Experimental study of high-velocity impact and fracture of ice[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.15 Failure history of a spherical specimen impacting on a rigid plate[START_REF] Tippmann | Experimentally validated strain rate dependent material model for spherical ice impact simulation[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.16 Multiple fracture specimen after impacted by SHPB . . . . . . . . . . . . . . . . 22 2.1 System of impact experiment for ice . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Cooling chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 SHPB measuring technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Horizontal displacement U x measured by DIC (mm) . . . . . . . . . . . . . . . 71 3.7 Vertical displacement U y measured by DIC (mm) . . . . . . . . . . . . . . . . . 72 3.8 Strain ε xx measured by DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.9 Strain ε yy measured by DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.10 Maximum strain ε yy measured by DIC . . . . . . . . . . . . . . . . . . . . . . . 75 3.11 Coordinate system established by images for J-integral . . . . . . . . . . . . . . 76 3.12 6 pixel × 6 pixel J-integral zone centred with crack tip . . . . . . . . . . . . . . 77 3.13 SIFs evaluated by J-integral, (Pa • √ m) . . . . . . . . . . . . . . . . . . . . . . . 81 3.14 Area of interest near to pre-crack . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.15 Evaluation crack tip by strain from DIC and difference from subtracting images . 82 3.16 Evaluation crack path by original image, difference image and maximum strain field 82 Crack initiation toughness estimated with different Young's modulus . . . . . . . 98 4.11 Crack path divided according to the experimental results . . . . . . . . . . . . . 99 4.12 Stress intensity factor history from numerical simulation . . . . . . . . . . . . . 99 4.13 Dispersive range of crack initiation toughness . . . . . . . . . . . . . . . . . . . 100 4.14 Fracture toughness history of ice in simulation depending on increase rate of initial dynamic stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.15 Crack growth toughness history versus crack propagation velocity . . . . . . . . 102 vii List of Tables 2.1 The mechanical properties of Nylon bars compare to ice . . . . . . . . . . . . . 34 2.2 Data for tests conducted by nylon striker bars with different lengths . . . . . . . 36 2.3 Data for tests conducted by PVC pipe striker bars with different lengths . . . . . 37 2.4 Data for tests conducted by PMMA striker bars with different lengths . . . . . . 37 Data for impact tests with rectangular specimens with a notch . . . . . . . . . . . 83 3.3 Data for impact tests with long rectangular specimens with a hole . . . . . . . . 84 3.4 Data for impact tests with short rectangular specimens with a hole . . . . . . . . 84 4.1 Mechanical properties of ice and Nylon used in the simulation . . . . . . . . . . 95 4.2 Crack initiation toughness of ice . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.1 Experimental parameters of cylindrical ice specimen 50mm in diameter group 1 . 106

. . . . . . . . . . . . . . . . 19 1.13 Phenomena relating to the fracture behaviour of ice . . . . . . . . . . . . . . . . 20 1.14 Failure history of a conic and a cylindrical specimen impacting on a rigid plate [2.4 Internal and external temperature of the specimen vs. time . . . . . . . . . . . . 33 2.5 Distinguish the incident and reflected signal . . . . . . . . . . . . . . . . . . . . 35 v 3.6 3.17 Crack propagation velocity vs. crack length . . . . . . . . . . . . . . . . . . . . 84 4.1 Enrichment to the classical finite element approximation . . . . . . . . . . . . . 90 4.2 Crack path presented by two level sets . . . . . . . . . . . . . . . . . . . . . . . 91 4.3 Methodology to access crack propagation . . . . . . . . . . . . . . . . . . . . . 92 4.4 Two types of XFEM numerical simulation models . . . . . . . . . . . . . . . . . 93 4.5 Impact velocity used in the simulation evaluated by experimental incident signal . 93 4.6 Numerical stress intensity factor at pre-crack tip without propagation . . . . . . . 96 4.7 Position of incident signal and impact loading . . . . . . . . . . . . . . . . . . . 96 4.8 Translation between incident signal in experiment and impact velocity in simulation, ∆t1 = L B 2C 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.9 Crack initiation toughness estimated by the synchronizing experimental signals, images and numerical results, ∆t2 crack initiation time . . . . . . . . . . . . . . 97 4.10 2.5 Data for tests conducted by PVC pipe striker bars with different rings . . . . . . 38 2.6 Data for tests conducted with different striker bars with foam rings . . . . . . . . 39 2.7 Image recording parameters for an experiment with a cylindrical ice specimen of diameter of 50mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.8 Image recording parameters for an experiment with a cylindrical ice specimen of diameter of 50mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.9 Image recording parameters for an experiment with a cylindrical ice specimen of diameter of 20mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.10 Dynamic Young's modulus, peak stress and impact velocity of ice in different fracture type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 Image recording parameters for a typical experiment No.2014030603 . . . . . . 66 3.2 A.2 Experimental results of cylindrical ice specimen 50mm in diameter group 1 . . . 107 A.3 Experimental parameters of cylindrical ice specimen 50mm in diameter group 2 . 108 ix

TABLE 2 .

 2 2.1. Some values are collected from literatures, and others are measured by experiments. Z B = S B ρ B E B and Z S = S S ρ S E S 1: The mechanical properties of Nylon bars compare to ice

	(2.5)

TABLE 2 .

 2 Input signal of nylon striker bars with different lengthsThen, PVC pipes with a cross section, 12mm in diameter with different length were tried. The detailed data on the tests are listed in TAB.2.3. As shown in FIG.2.8, the reflected signal can not be distinguished either. It was concluded that strike bars made of PVC are not suitable to conduct the test.

		0.02
											Length 400mm
											Length 500mm
		0.01									Length 800mm
		0
		-0.01
	Voltage (V)	-0.02
		-0.03
		-0.04
		-0.05
		0 -0.06	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01
							time (s)
		FIGURE 2.7:

2: Data for tests conducted by nylon striker bars with different lengths

TABLE 2 .

 2 3: Data for tests conducted by PVC pipe striker bars with different lengths Input signals of PVC pipe striker bars with different lengths Third, PMMA bars 12mm in diameter with different lengths were tried. The detailed data on the tests are listed in TAB.2.4. As shown in FIG.2.9, the striker bar made of PMMA with the lengths of 470mm and 650mm both avoid the superposition of the incident signal and reflected signal. In addition shorter is the bar, better is the signal.

		0.01
											Length 400mm
											Length 800mm
		0.005
		0
		-0.005
	Voltage (V)	-0.01
		-0.015
		-0.02
		-0.025
		0 -0.03	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01
						time (s)
		FIGURE 2.8: Number	Material Diameter Length Pressure Ring
						(mm)		(mm)	(kPa)
		2014092601 PMMA 12		470	60		Foam
		2014092608 PMMA 12		650	56		Foam

TABLE 2 .

 2

		0.04
											Length 470mm
		0.03									Length 650mm
		0.02
		0.01
	Voltage (V)	-0.01 0
		-0.02
		-0.03
		-0.04
		0 -0.05	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01
							time (s)

4: Data for tests conducted by PMMA striker bars with different lengths

TABLE 2 .

 2 5: Data for tests conducted by PVC pipe striker bars with different ringsIn order to sustain the striker bar well, two foam rings were used. Three materials of striker bars with two foam rings were tried. The detailed data on the tests are listed in TAB.2.6. As shown in FIG.2.11, the striker bar made of PMMA with two foam rings provides the best signals.

		Diameter Length Pressure Ring
		(mm)	(mm)	(kPa)
	2014092309 PVC	12	400	80	2 Nylon
	2014092312 PVC	12	400	70	1 Nylon
	2014092315 PVC	12	400	85	1 Foam

TABLE 2

 2 .22. Whereas, Shazly

		5					4.54m/s No.2013020803
		4.5					2.22m/s No.2013020806 3.28m/s No.2013030307
		4
		3.5
	stress (MPa)	2 2.5 3
		1.5
		1
		0.5
		0 0	0.01	0.02	0.03	0.04	0.05	0.06
					strain
	FIGURE 2.21: Stress-strain curves of cylindrical ice specimen diameter of 50mm
	Number	Resolution Interval (µs) Exposure (µs) Frame rate ( f ps)
	2013070405 320×160	10,41	5,81		96045

.8: Image recording parameters for an experiment with a cylindrical ice specimen of diameter of 50mm

TABLE 2 .

 2

	Type of fracture	Impact velocity Strain rate	Peak stress Young's modulus
		(m/s)	(s -1 )	(MPa)	(GPa)
	Crush once being impacted 5.0 ∼ 6.0	66.63 ± 17.42 7.35 ± 0.41 7.07 ± 0.80
	Crack initiating input side	1.6 ∼ 5.0	51.83 ± 47.42 3.57 ± 1.05 4.12 ± 1.72
	Crack initiating output side 0.5 ∼ 1.5	39.25 ± 22.40 0.85 ± 0.38 3.82 ± 0.55

10: Dynamic Young's modulus, peak stress and impact velocity of ice in different fracture type

  The crack propagates in Mode I. With the help of image analysis, the crack propagation history and the crack propagation velocity are obtained. The crack propagation velocity firstly increases and then keeps at a constant value. Afterwards, it decreases. It reaches 534 ± 83m/s, which is about 25% of the Rayleigh wave velocity in ice (2000m/s).

	IV-The crack initiation time is estimated by the synchronization between SHPB signals and images. The corresponding initiation toughness in the simulation is 204 ± 13kPa • √ m. The
	crack path is pre-defined and divided into sections according to the crack propagation ve-locity. Then the crack growth toughness is determined to be 208 ± 48kPa • √ m which is
	linearly associated with the crack propagation velocity.

  2 Cylindrical ice specimens A.2.1 Cylindrical ice specimens of diameter of 50mm

	Number	Diameter Length Pressure Striker velocity
		(mm)	(mm)	(KPa)	(m/s)
	2013020801 49,1	31,9	122	15,87
	2013020802 49,0	43,0	122	14,05
	2013020803 48,9	44,0	122	11,82
	2013020804 49,0	104,5	122	14,06
	2013020805 49,2	116,9	122	16,23
	2013020806 48,0	34,0	82	5,93
	2013020807 48,0	47,0	81	5,94
	2013020808 48,0	47,0	81	16,67
	2013020809 49,0	105,0	81	21
	2013030505 48,5	104,1	80	22,11
	2013030506 48,1	108,5	83	23,81

TABLE A

 A Experimental results of cylindrical ice specimen 20mm in diameterA.3 Rectangular ice specimens with a wedge and pre-crack

	Number A.2.3 Cylindrical ice specimens of diameter of 20mm Weight Diameter Length Density	Pressure Striker velocity
	(g)	(mm)	(mm)	(g/mm 3 ) (KPa)	(m/s)
	2013070401 159.77 49.00	93.55	0.906	75	3.27
	2013070402 160.27 49.02 Number Temperature Pressure Striker velocity Impact velocity 95.74 0.887 76 3.60 2013070403 165.62 50.04 95.11 0.886 82 4.81 Number Temperature Pressure Striker velocity Impact velocity ( • C) (kPa) (m/s) (m/s) ( • C) (kPa) (m/s) (m/s)
	.1: Experimental parameters of cylindrical ice specimen 50mm in diameter group 1 Number Impact velocity Peak stress Strain Strain rate Strain rate at peak stress at peak stress maximum (m/s) (MPa) (s -1 ) (s -1 ) 2013020801 5,25 7.24 0.0027 52.267 52.65 2013020802 2013020803 4.54 4.48 0.0061 131.35 188.87 2013020804 5.49 5.59 0.0032 64.38 94.14 2013020805 6.41 2013020806 2.22 2.48 0.0032 66.64 95.4 2013020807 3.28 3.64 0.0052 92.33 127.45 2013020808 3.89 4.62 0.0044 105.77 114.51 2013020809 5.35 7.21 0.0024 51.93 53.44 2013030505 5.98 7.01 0.0035 72.66 73.07 2013030506 5.68 7.95 0.0025 57.21 59.86 TABLE A.2: Experimental results of cylindrical ice specimen 50mm in diameter group 1 2013070404 173.59 49.59 97.46 0.923 89 2014030601 -1 60 6.02 1.87 5.48 2013070405 171.30 49.31 97.36 0.922 99 6.61 2013070406 126.81 49.39 73.76 0.898 85 6.69 2013070407 148.10 48.21 88.41 0.918 78 4.81 2013070408 167.69 48.88 96.50 0.927 80 5.28 2013070409 148.36 48.36 93.15 0.868 92 4.17 2013070410 172.73 49.59 96.55 0.927 99 5.24 2013070411 162.36 49.03 95.05 0.905 100 7.32 2013070413 171.31 49.62 97.21 0.912 94 6.44 2013070414 162.96 49.47 93.57 0.907 93 5.00 2013070415 166.50 49.16 94.56 0.928 94 6.05 Number Impact velocity Peak stress Strain Strain rate Number Impact velocity Peak stress Strain Strain rate 2014030501 -7 60 6.50 4.19 Strain rate 2014030602 -1 60 7.34 2.82 Strain rate at peak stress at peak stress maximum (m/s) (MPa) (s -1 ) (s -1 ) 1.67 1.62 0.0068 12.50 18.73 1.91 2.29 0.0063 8.07 16.25 2.68 2.73 0.0015 16.76 29.21 3.12 3.42 0.0098 15.80 31.52 3.92 4.05 0.0168 20.77 39.48 3.69 2.93 0.0011 12.65 29.00 3.06 2.62 0.0017 20.82 38.47 2.30 2.05 0.0113 18.60 26.69 3.05 3.47 0.0091 13.09 23.38 4.33 5.40 0.0095 12.45 28.57 3.84 4.69 0.0010 8.46 36.83 3.81 4.81 0.0098 9.61 28.00 2.82 2.74 0.0122 20.02 31.33 3.54 4.13 0.0113 14.12 25.19 TABLE A.4: Experimental results of cylindrical ice specimen 50mm in diameter group 2 Number Weight Diameter Length Density Pressure Striker velocity (g) (mm) (mm) (g/mm 3 ) (KPa) (m/s) 2013071901 161.49 49.51 92.91 0.903 81 4.72 2013071902 154.89 48.97 89.98 0.914 99 6.14 2013071903 153.63 48.95 89.92 0.908 78 3.94 2013071904 150.52 49.22 89.25 0.887 108 7.28 2013071905 161.79 49.46 92.52 0.911 91 5.47 2013071906 162.88 49.56 93.48 0.904 72 1.66 2013071907 151.67 49.40 87.10 0.909 75 2.42 2013071908 165.97 49.42 94.18 0.919 84 4.90 2013071909 157.22 49.33 89.93 0.915 75 3.07 2013071910 155.93 49.13 90.07 0.914 120 8.79 2013071911 165.84 48.92 96.69 0.913 121 8.36 2013071912 161.88 49.28 93.58 0.907 89 6.25 2013071913 159.70 49.34 91.61 0.912 80 4.90 2013071914 160.76 49.32 92.15 0.914 83 3.40 2013071915 160.24 49.39 91.34 0.916 85 3.97 TABLE A.5: Experimental parameters of cylindrical ice specimen 50mm in diameter with confine-ment at peak stress at peak stress maximum (m/s) (MPa) (s -1 ) (s -1 ) 2.59 1.40 0.0133 35.27 40.70 3.51 3.58 0.0192 20.18 50.93 2.10 1.76 0.0047 18.50 23.51 4.25 4.70 0.0216 23.07 65.93 3.12 1.47 0.0299 40.28 50.06 1.08 1.28 0.0063 19.10 21.09 2.75 2.55 0.0134 20.50 30.21 1.56 1.67 0.0039 9.26 15.30 5.19 4.82 0.0319 36.64 81.56 4.89 4.13 0.0308 39.99 74.47 2.73 2.67 0.0051 18.25 28.16 1.73 2.11 0.0174 24.68 33.03 2.13 2.01 0.0079 16.99 26.33 2.68 2.66 0.0170 17.63 39.70 TABLE A.6: Experimental results of cylindrical ice specimen 50mm in diameter with confinement Number Temperature Diameter Length Pressure Striker velocity • C (mm) (mm) (kPa) (m/s) 2014053012 -1 20 40 100 6.90 2014053013 -1 20 50 95 2.94 2014053014 -1 20 51 95 3.45 2014053015 -1 20 45 95 3.11 TABLE A.7: Experimental parameter of cylindrical ice specimen 20mm in diameter Number Impact velocity Peak stress Strain Strain rate Strain rate at peak stress at peak stress maximum (m/s) (MPa) (s -1 ) (s -1 ) 2014053012 1.43 1.34 0.0015 28.40 62.75 2014053013 0.56 0.64 0.0064 16.85 19.90 2014053014 0.52 0.94 0.0054 6.90 15.79 2014053015 0.60 0.48 0.0060 17.27 18.53 Temperature Pressure Striker velocity ( • C) (kPa) (m/s) 2014022801 -7 70 9.96 2014022802 -7 60 7.42 2014022803 -7 60 7.62 2014022804 -1 60 6.45 2014022805 -1 60 6.50 TABLE A.9: Experimental parameters of rectangular ice specimen with a wedge and pre-crack Group 1 Number Resolution Exposure(µs) Interval (µs) Frame rate 2014022801 640*400 29.53 38.46 26003 2014022802 512*320 14.41 25.63 39013 2014022803 512*320 14.41 25.63 39013 2014022805 512*256 14.74 20.40 49026 TABLE A.10: Image parameters of rectangular ice specimen with a wedge and pre-crack Group 1 2014030502 -7 60 7.41 3.85 2014030503 -7 60 7.34 3.92 2014030504 -7 60 8.36 4.91 2014030505 -7 50 5.71 1.88 TABLE A.11: Experimental parameters of rectangular ice specimen with a wedge and pre-crack Group 2 Number Resolution Exposure(µs) Interval (µs) Frame rate 2014030501 512*344 14.56 27.01 37016 2014030502 512*344 14.56 27.01 37016 2014030503 512*344 14.56 27.01 37016 2014030504 512*344 14.56 27.01 37016 TABLE A.12: Image parameters of rectangular ice specimen with a wedge and pre-crack Group 2 2014030603 -1 60 8.06 2.27 2014030604 -1 60 6.90 1.89 2014030605 -1 60 5.98 1.51 2014030606 -1 60 8.77 4.03 TABLE A.13: Experimental parameters of rectangular ice specimen with a wedge and pre-crack Group 3 Number Resolution Exposure(µs) Interval (µs) Frame rate 2014030601 512*344 14.56 27.01 37016 2014030602 512*344 14.56 27.01 37016 2014030603 512*344 14.56 27.01 37016 2014030604 512*344 14.56 27.01 37016 2014030605 512*344 14.56 27.01 37016 TABLE A.8: Number 2014022804 512*256 14.74 20.40 2014030505 512*344 14.56 27.01 37016 49026 2014030606 512*344 14.56 27.01 37016
	106 110					107 109 111

TABLE A .

 A 14: Image parameters of rectangular ice specimen with a wedge and pre-crack Group 3A.4 Rectangular ice specimens with a hole and pre-crack A.4.1 Rectangular ice specimens with a hole and pre-crack of length of 230mm

	Number Number	Temperature Pressure Striker velocity Impact velocity Dimensions Resolution Exposure(µs) Interval (µs) Frame rate State
	( • C) 2014052701 512*344	(kPa) 25.68	(m/s)	27.01	(m/s)	(mm 3 ) break
	-8 2014052702 512*344	60	25.68	5.68	27.01	3.66	225*90*35 break
	-1 2014052703 512*192	60	18.66	5.06	20.00	3.49	223*90*30 not
	-1 2014053003 384*192	60	14.56	7.08	12.50	5.38	178*90*28 break
	-1 2014053004 384*192	100 14.56	4.63	12.50	1.40	123*90*29 break
	-1 2014053005 384*192	120 14.56	8.53	12.50	2.68	123*90*29 break
	-1 2014053006 384*192	120 14.56	8.99	12.50	2.86	118*90*27 break
	-1 2014053007 384*192	120 14.56	8.65	12.50	2.87	129*90*29 break
	-1 2014053008 384*192	110 14.56	6.62	12.50	2.04	168*90*24 break
	-1 2014053009 384*192	110 14.56	6.68	12.50	2.10	160*90*40 not
	-1 2014053010 384*192	120 14.56	8.90	12.50	2.79	230*90*40 break
	-1 2014053011 384*192	130 14.56	10.57	12.50	3.29	230*90*40 break
		-1	140	10.80		2.34	230*90*40
	TABLE A.15: Experimental parameters of rectangular ice specimen with a hole and pre-crack
	Group 1

TABLE A .

 A 16: Image parameters of rectangular ice specimen with a hole and pre-crack Group 1 A.4.2 Rectangular ice specimens with a hole and pre-crack of length of 120mm

	Number	Temperature Pressure Striker velocity Impact velocity
		• C	(kPa)	(m/s)	(m/s)
	2014102001 -5	100	5.38
	2014102002 -5	90	5.32	2.74
	2014102003 -5	70	2.81	1.24
	2014102004 -10	80	4.27	1.95
	2014102005 -10	75	3.42	1.58
	2014102006 -10	80	4.20	1.99
	2014102007 -10	85	4.42	2.07
	2014102008 -10	95		2.50
	TABLE A.17: Experimental parameters of rectangular ice specimen with a hole and pre-crack
	Group 2
	Number	Resolution Exposure(µs) Interval (µs) Frame rate State
	2014102001 512*320	4.53	25.63	39013	break
	2014102002 512*320	4.53	25.63	39013	break
	2014102003 464*280	8.94	25.10	45003	not
	2014102004 464*280	8.94	25.10	45003	break
	2014102005 368*280	8.57	18.18	55016	not
	2014102006 368*280	8.57	18.18	55016	not
	2014102007 368*280	8.57	18.18	55016	not
	2014102008 368*280	8.57	18.18	55016	break

TABLE A .

 A 18: Image parameters of rectangular ice specimen with a hole and pre-crack Group 2

	A.
		Number Number	Temperature Pressure Striker velocity Impact velocity Resolution Exposure(µs) Interval (µs) Frame rate State
		• C 2014102101 512*320	(kPa) 4.53	(m/s) 25.63	(m/s)	break
		2014102101 -5 2014102102 432*320	85 8.94	4.81 25.10	2.40	not
		2014102102 -15 Number Temperature Pressure Striker velocity Impact velocity 85 4.81 2.11 2014102103 432*320 8.94 25.10 not
		2014102103 -15 • C 2014102104 432*320	90 (kPa) 8.94	4.81 (m/s) 25.10	2.11 (m/s)	break
		2014102104 -15 2014102121 -15	95 100	5.10 7.46	2.13 3.77
		2014102105 -15 2014102122 -15 2014102105 656*320	95 100 8.50	5.38 6.02 39.31	2.21 1.98	not
		2014102106 -15 2014102123 -15 2014102106 656*320	100 100 8.50	5.49 6.25 39.31	2.60 2.01	break
		2014102107 -15 2014102124 -15 2014102107 656*320	100 100 8.50	5.88 6.17 39.31	2.06 2.03	break
		2014102108 -15 2014102125 -15 2014102108 512*256	100 100 8.56	6.10 6.17 20.40	2.01 2.06	break
		2014102109 -15 2014102126 -15 2014102109 512*256	100 110 8.56	5.49 7.35 20.40	1.98 1.98	not
		2014102110 -10 2014102127 -15 2014102110 512*256	110 100 8.56	5.49 6.17 20.40	2.01 2.01	break
		2014102111 -15 2014102128 -15 2014102111 512*256	100 100 8.56	6.25 6.41 20.40	1.97 2.00	break
		2014102112 -15 2014102129 -15 2014102112 512*256	100 105 8.56	5.68 8.06 20.40	2.00 1.95	break
		2014102113 -15 2014102130 -15 2014102113 512*256	95 100 8.56	7.04 6.10 20.40	1.95 2.00	not
		2014102114 -15 2014102131 -15 2014102114 512*256	100 105 8.56	6.17 6.25 20.40	1.95 2.10	break
		2014102115 -15 2014102132 -15 2014102115 512*256	100 105 8.56	5.95 7.25 20.40	1.96 1.95	break
		2014102116 -15 2014102133 -15 2014102116 512*256	100 105 8.56	5.95 6.67 20.40	2.03 1.98	break
		2014102117 -15 2014102134 -15 2014102117 512*256	100 105 8.56	6.10 6.67 20.40	2.01	break
	2014102118 -15 TABLE A.20: Experimental parameters of rectangular ice specimen with a hole and pre-crack 100 6.02 2.06 2014102118 512*256 8.56 20.40 break
	Group 4	2014102119 -15 2014102119 512*256	100 8.56	5.88 20.40	2.09	break
		2014102120 -15	100	6.17	2.06
		2014102120 512*256	8.56	20.40		not
	TABLE A.19: Experimental parameters of rectangular ice specimen with a hole and pre-crack
	Group 3
	120						119

TABLE A .

 A 21: Image parameters of rectangular ice specimen with a hole and pre-crack Group 3 121

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2016LYSEI043/these.pdf © [L. Yao],[2016], INSA Lyon, tous droits réservés

The stress intensity factor is the most important feature to evaluate the crack propagation. In our work, we use the fracture criterion based on the stress intensity factor which was calculated from the displacement field extracted by digital image correlation by the J-integrals using an equivalent domain integral method[START_REF] Raju | An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems[END_REF].Consider a crack in a plate subjected to an arbitrary remote loading with an arbitrary closed

)u n-1 (4.14)

Determination of numerical parameters

To perform the numerical simulation of the experiments, we need to know the mechanical behaviour of ice. As it behaves like a very brittle material we limit our investigation to the Young's modulus under impact. Mechanical properties used in simulations are listed in Tab.4.1. Some values are collected from literatures, and others are measured experimentally as discussed previously.

Mechanical properties Nylon Ice

Density (kg/m 3 ) 1100 900 Dynamic Young's modulus (GPa)

3.6 [START_REF] Karimzada | Modelisation of dynamic crack propagation criteria[END_REF] 4.0 Poisson's ratio 0.41 [START_REF] Karimzada | Modelisation of dynamic crack propagation criteria[END_REF] 0.3 [Petrovic, 2003] Velocity of longitudinal waves (m/s) 1600

Evaluation of crack initiation toughness

The crack initiation toughness of ice under impact was estimated when the simulation fitted the experiment. From experiments, the impact velocity was measured from the gauges, and the crack initiation time was estimated from image analysis.

First, assuming the crack did not propagate during the loading process. A stress intensity factor history at the pre-crack tip was calculated as show in Fig. 4.11.

After, the stress intensity factor curve needed to be connected with the experimental signal by making the impact velocity in the simulation coincide with the incident signal from the experiment. The beginning time of impact velocity in the input signal was determined noted as black line.

Since the incident signal was collected from the gauge glued in the middle of the input bar, whereas the impact velocity acted on the specimen directly in the simulation as shown in Fig. 4.7.

The wave needed to propagate half length of the input bar from the gauge to the specimen. The impact velocity needed to be shifted. The synchronization between the signal, and numerical results was established with translating impact velocity and SIF history by time ∆t1 = L B 2C 0 , where C 0 is the longitudinal wave velocity in Nylon and L B is the length of the input bar as shown in Fig. 4.8.

After, the crack initiation time was estimated from image analysis. The crack initiation time

Appendix A A.1 Introduction

The experimental parameters and results are summarised here briefly.

The mechanical properties of ice are sensitive to the structure of ice, the configuration of specimen, the experimental condition. The strength and Young's modulus of ice were reported in a large range in different literature. Precise values were needed for the following simulation in evaluating the fracture toughness. In order to obtain the strength and dynamic Young's modulus in the same condition with the crack propagation experiment, three groups of experiments were designed firstly :

1) A group of experiments with the cylindrical specimen of a diameter of 50mm was conducted with different impact velocity to obtain the range of strength and Young's modulus.

2) A group of experiments with the cylindrical specimen of a diameter of 50mm with confinement was conducted to investigate the confinement effect. Since the ice specimen in the crack propagation experiment was in an intermediate state between with and without confinement.

3) A group of experiments with the cylindrical specimen of a diameter of 20mm was conducted with a critical impact velocity to obtain the critical strength and Young's modulus of specimen fracture.

Then, with the purpose of investigating the crack propagation in ice under impact loading, two groups of experiments were designed :

1) A group of experiments with the rectangular specimen with a wedge and a pre-crack was conducted with different impact velocity to obtain a main crack propagation.

2) A group of experiments with the rectangular specimen with a hold and a pre-crack was conducted with different impact velocity to obtain a main crack propagation.

For the specimen with wedge, the waves transferred and reflected complicatedly on interface of the wedge and the specimen. It was hard to do signal analysis. Such a problem could be avoided with the rectangular specimens with hole. In this type specimen, the input bar impact on the specimen directly. But the stress distribution was more complex near the hole. So both types of specimens were used. The phenomena relating to the fracture behaviour of ice under impact loading are common in civil engineering, for offshore structures, and de-ice processes. To reduce the damage caused by ice impact and to optimize the design of structures or machines, the investigation on the dynamic fracture behaviour of ice under impact loading is needed.

This work focuses on the dynamic crack propagation in ice under impact loading. A series of impact experiments is conducted with the Split Hopkinson Pressure Bar. The temperature is controlled by a cooling chamber. The dynamic process of the ice fracture is recorded with a high speed camera and then analysed by image methods. The extended finite element method is complementary to evaluate dynamic fracture toughness at the onset and during the propagation.

The dynamic behaviour of ice under impact loading is firstly investigated with cylindrical specimen in order to obtain the dynamic stress-strain relation which will be used in later simulation.

We observed multiple cracks in the experiments on the cylindrical specimens but their study is too complicated. To better understand the crack propagation in ice, a rectangular specimen with a pre-crack is employed. By controlling the impact velocity, the specimen fractures with a main crack starting from the pre-crack. The crack propagation history and velocity are evaluated by image analysis based on grey-scale and digital image correlation. The main crack propagation velocity is identified in the range of 534±83m/s which confirms the previous results. It slightly varies during the propagation, first increases and keeps constant and then decreases.

The experimentally obtained parameters, such as impact velocity and crack propagation velocity, are used for simulations with the extended finite element method. The dynamic crack initiation toughness and dynamic crack growth toughness are determined when the simulation fits the experiments. The results indicate that the dynamic crack growth toughness is linearly associated with crack propagation velocity and seems temperature independent in the range -15̴ -1 °C.