21st IAHR International Symposium on Ice "Ice Research for a Sustainable Environment", Li and Lu (ed.) Dalian, China, June 11 to 15, 2012 © 2012 Dalian University of Technology Press, Dalian, ISBN 978-7-89437-020-4

Cohesive Zone Method Based Simulations of Ice Wedge Bending: a Comparative Study of Element Erosion, CEM, DEM and XFEM

Wenjun Lu*, Raed Lubbad, Sveinung Løset, and Knut Høyland Sustainable Arctic Marine and Coastal Technology (SAMCoT), Centre for Research-based Innovation (CRI), Norwegian University of Science and Technology, Trondheim, Norway

* wenjun.lu@ntnu.no The ice wedge bending problem is important to study the ice-sloping structure interactions. A lot of experiments, analytical and numerical solutions have been pursuit in the past decades to study ice wedge bending related problems (e.g. beam tests, ice bearing capacity experiments, etc.). Nowadays, due to the advancement in computational mechanics, various numerical methods are at our disposal to simulate this process into detail according to certain material failure theory, e.g. the cohesive zone method. This paper tested four available numerical methods combined with the cohesive zone method in simulating this ice wedge bending scenario. These different numerical approaches include the traditional finite element method with element erosion technique, the cohesive element method (CEM), the discrete element method (DEM) with cohesive contacts and the extended finite element method (XFEM). Based on the simulations, it is found that all methods can reproduce the bending failure mode but the results are mesh-dependent due to the presence of material softening. Further information (e.g. strain rate effects) is needed for a more detailed material constitutive model so as to get a unique solution (i.e. mesh-independence). Apart from that, among all these numerical methods, the element erosion technique turns out to be the most efficient method; CEM is capable of capturing a stabilized ice breaking load but the whole structure appears to be softened with increasing cohesive element density; DEM with cohesive contact alleviate us from the structural softening problem, but it is too computationally expensive that only limited simulations and conclusions are made on it; XFEM is still at its early stage of development, so lots of disadvantages still remains in the market available implementations. However, it is shown in this study, different from those

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discrete fracture approaches (CEM and DEM), that the crack propagation is free from the mesh bounding thus making this method very promising in the future.

1. Introduction When level ice interacting with sloping structures, being either icebreakers or other sloping structures, it is often observed the formation of wedge shaped ice (Kotras, 1983; Lubbad and Løset, 2010; Valanto, 2001). Similarly, when level ice is suffering from a point load, the first formation of radial cracks separate the ice plate into several ice wedges before the final loss of its bearing capacity (Dempsey et al., 1995; Sodhi, 1995; 1996; 1998). Accordingly, it is crucial to study the failure process of a wedge-shaped ice so as to extract the ice breaking load, ice breaking length etc., which are important for the forthcoming interaction process. For the past decades, the ice wedge bending problem has been studied both analytically (see (Lubbad et al., 2008) for a detailed literature review) and numerically (Derradji-Aouat, 1994; McKenna and Spencer, 1993; Sawamura et al., 2008). One similarity of all these methods is the adoption of a continuum approach and focus on the pre-failure process (i.e. when certain failure criterion is reached, the ice wedge is assumed to be failed instantly). Nowadays, the advancement in the modern computation capacity empowered us the possibility to utilize more comprehensive material models to simulate the failure of ice in a progressive failure manner and the material degradation is thus captured. One of such models, the cohesive zone model, stemming from the concept of fracture mechanics (Hillerborg et al., 1976) and being able to simulate the fracture initiation and propagation, is considered as a promising tool in simulating lots of material failure behaviours. This method enables us to simulate the transition from continua to discontinua. The failure of the material becomes a natural output during the simulation. However, this method can be realized by many numerical schemes (e.g. element erosion technique, cohesive element method (CEM), discrete element method (DEM), extended finite element method (XFEM), etc.), each of which is flourishing in their respective academic fields. Konuk et al. (2009) qualitatively reviewed the implementation of the cohesive zone model with several numerical methods. The application of the cohesive zone model in ice structure interaction problems has been realized by CEM (Gürtner, 2009; Gürtner et al., 2010; 2008; Konuk and Yu, 2010) and DEM (Paavilainen et al., 2011). Instead of directly embarking on a global ice structure interaction simulation, the current paper focuses on evaluating these numerical methods with a relatively simple numerical set-up. This is to simulate the ice wedge bending in a progressive failure manner with either ABAQUS-6.11/EXPLICIT or ABAQUS-6.11/STANDARD. The evaluation criteria will be set on the ability of each method to:

• get stable solution in describing bending failure of the ice wedge; • robustly simulate the progressive failure process which is highly nonlinear due to the

material softening; • have satisfying efficiency.

It should be noted that this paper focuses on the evaluation of different numerical methods. The investigation of a suitable ice material model is so far out of the scope of this paper. Relatively

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simple constitutive models (e.g. linear elastic, homogeneous, isotropic, and perfect plastic material model) are utilized in most of the simulations, except otherwise stated. And the cohesive zone method takes care of the ice material's post-failure behaviour. However, convincing, robust and effective constitutive models for different ice features are of great importance. This will be set as a future work combined with one of the numerical method from this paper. The arrangement of this paper is as following: First, the numerical set-up and the cohesive zone model will be briefly introduced. Then each numerical method (i.e. the element erosion technique, CEM, DEM, and XFEM) will be introduced separately together with their bending simulation results. Since material softening is very notorious for mesh objectivity, the effort was later spent on the mesh sensitivity study of each numerical method. In the end, conclusions are made based on the numerical test results.

2. The Numerical Set-up and the Cohesive Zone Model

2.1 The Numerical Set-up The target of this paper is to simulate the bending failure of a wedge shaped ice with different numerical methods. Different from previous simulations, a contact scenario is set up in this study to obtain the boundary condition of the ice wedge in this study. A pressure-over closure relationship is assumed to simulate the possible crushing of the ice before it fails in bending. Similar as in (Lu et al., 2012), the slope of this pressure-over closure relationship is tuned such that the crushing depth is no more than the ice thickness and then kept constant in all numerical tests. The wedge angle in this study is chosen as 45°, similar as in (Lubbad et al., 2008; McKenna and Spencer, 1993). Regarding the length of the ice wedge, according to (Sodhi, 1996), in ice bearing capacity problems, both observation and numerical results show that the radial cracks propagate no more than two times of the characteristic length Lc. While McKenna and Spencer (1993) utilized 4 times Lc as the ice wedge length in their simulation. Derradji-Aouat (1994) and Lubbad et al. (2008) simulated ice wedge with length 96 metres and 300 metres, respectively. In the current study, in order to make a compromise between calculation efficiency and accuracy, a length slightly larger than 4 times the Lc length was chosen for the geometry of the ice wedge. Most geometry parameters and material constants are set similar to (Lubbad et al., 2008) as shown in Table 1. The structure that has contact with the ice wedge is a conical structure with a sloping angle of 45°. This conical structure is assumed to be fixed. The influence from its response on ice is thus neglected. The overall numerical set up is shown in Figure 1. In order to highlight the investigations of different numerical methods, the fluid base in the current study is simply treated as elastic foundations with constant hydrodynamic coefficients as in (Lubbad et al., 2008). A FORTRAN user subroutine was implemented in ABAQUS-6.11/EXPLICIT to simulate the elastic foundation. This user subroutine has been validated against the elastic foundation algorithm within ABAQUS-6.11/STANDARD. Both the commercial elastic foundation code with IMPLICIT solver and self-programmed elastic foundation code with EXPLICIT solver tender almost the same results. For a further detailed treatment of the fluid base including hydrodynamic effects under the same numerical set-up, see (Lu et al., 2012c).

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Table 1. General inputs for the simulations. Young's Modulus E : 3.5 GPa Possion ratio υ : 0.3 Density of ice iρ : 900 3kg/m Density of ice wρ : 1025 3kg/m Ice wedge thickness h : 0.3 m Ice wedge angle: 45° Added mass coefficient for the elastic foundation:

1.23

Hydrodynamic damping coefficient for the elastic foundation:

1.0

Ice wedge length L : is chosen as 40 m since

144 4 30mc

w

DLgρ

× = × =

with 3

212(1 )EhD

υ=

−

Figure 1. The numerical set-up illustrations.

2.2 The Cohesive Zone Model for Ice The cohesive zone model is essentially a conceptually simple way to describe the criteria of fracture initiation and how the crack propagates according to the evolving traction and separation relationship. This method was first introduced by (Hillerborg et al., 1976) to simulate the brittle failure of concrete and termed as the fictitious crack model. The main assumption is the existence of a cohesive zone in front of the crack tip and a cohesive law which is assumed to be a material property that governs crack's initiation and propagation. Experiments are necessary to obtain this cohesive law before any meaningful simulation. With respect to ice, Mulmule and Dempsey(1999; 2000; 1998) back-calculated the cohesive law with a Mode I fracture energy of 15 N/m . For simplicity, as in (Paavilainen et al., 2009; 2011; 2010), a linearly softening law is assumed in the current study together with this reported cohesive fracture energy.

3. Evaluation on Different Numerical Methods Based on the previous established numerical set-up and adopted cohesive zone model, this section tests different numerical methods with a set of evaluation criteria. The basic theory of

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each numerical method will first be presented together with the bending simulation results. Following that, the mesh sensitivity study is presented.

3.1 Numerical Methods Introduction and Bending Failure Reproducibility All the numerical methods that have been tested are conceptually introduced in this section. For detailed formulation of these numerical methods, one is strongly encouraged to the relevant references. After the basic introduction of each method, they are tested to see if the bending failure scenario could be reproduced numerically. In order to capture the bending failure of the ice wedge, the shell element is chosen for discretization. Moreover, in order to more precisely capture the contact algorithm, the continuum shell element with modified Mindlin-Reissner assumption (Belytschko et al., 2000), is adopted in all the simulations except for the XFEM where the enrichment was applied to the first order continuum elements. Before starting with the mesh sensitivity study of each method, the first question to be answered is how many layers of such continuum shell/continuum elements are required to capture the ice wedge's bending failure. Therefore, in the numerical tests, the number of the element layers in the ice thickness direction is first set as a variable. Global horizontal loading histories are extracted for each method so as to identify the required number of element layers to reproduce the stabilized bending failure process.

3.1.1 Element erosion technique The element erosion technique can be viewed as the most 'engineering' and widely utilized approach to simulate the material failure. The concept is rather simple. Before the initiation of damage, the material is modelled as a continuum following certain constitutive models. When a certain failure criterion has been reached, material degradation (softening) occurs. A damage variable is usually assigned to describe the evolution of such material degradation. Usually, when the material is fully damaged (i.e. the damage variable equals to 1), the corresponding element is deleted from the finite element mesh. Thus a crack is explicitly modelled. To sum up, there are three important ingredients in this technique, saying: 1) the constitutive model, which describe the stress-strain relationship, the yield criteria and the flow rule (if with the presence of plasticity); 2) the damage initiation criteria and 3) the damage evolution law. In the current paper, the hyperbolic Drucker-Prager plasticity model is serving as the material constitutive model in the element erosion technique. As shown in Figure 2, the hyperbolic Drucker-Prager model is a modification of the linear Drucker-Prager model by introducing a tension cut-off tp and approximates the original yield function hyperbolically (see the yield functions shown in Figure 2). By assuming the tensile strength and compressive strength of ice in uniaxial test to be 500 kPa and 1000 kPa, respectively, the input parameters for the yield function are as shown in Figure 2. Furthermore, when the yield criterion is reached, the material is assumed to follow a perfect plasticity flow rule before the damage initiation criteria is reached. In analogy to the consideration of bulk dissipation combining with cohesive zone method (Bažant and Planas, 1998), the damage is assumed to initiate when the equivalent plastic strain is 30% of the elastic

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strain. Regarding the damage evolution law, it follows the cohesive zone model, which is essentially a softening curve describing the softening behaviour during material degradation.

Figure 2. The Drucker-Prager plasticity model.

'd equals to 600 kPa; represents the internal cohesion term in the Drucker-Prager criterion;

β equals to 45°; represents the friction angle; tr( ) / 3p = σ is the hydrostatic part of the stress tensor σ ;

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q = S : S is the equivalent Mises stress with S being the deviatoric part of the stress tensor σ ;

tp is assumed to be half the uniaxial tensial strength (250 kPa) since no triaxial tensile strength data for sea ice are available;

Based on the trial simulation, it is found that when more than 4 layers of the continuum shell elements are used, the loading history tends to converge as shown in Figure 3(b). It can also be seen from Figure 3(a) that the crack is formed by eroding the damaged elements from the mesh.

Figure 3. Illustration of the simulation results based on the element erosion technique. (Left: visualization of the ice wedge breaking with damage variables; Right: Bending failure tests with different layers of continuum elements)

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3.1.2 Cohesive element method The cohesive element method is implemented by specially discretizing a continuum into the traditional bulk elements with a stress-strain constitutive relationship and the cohesive element with the constitutive relationship described by the cohesive law. The virtual form of its expression can be found in (Ruiz et al., 2000; Xu and Needleman, 1994). In the finite element framework, if the crack is not known in advance, the cohesive elements are usually inserted along the boundaries of all the bulk elements. Since the cohesive zone model is taken care of by the cohesive elements, the bulk elements can be modelled in a rather simplified fashion. In this model an elastic-perfect plastic model is adopted to model the bulk element. The major challenge of CEM is the convergence issues. The mesh sensitivity of the cohesive element methods with application in ice structure interactions has been studied in (Lu et al., 2012) and will not be repeated here. In the current study, a Matlab code was developed to generate a mesh with an infinitesimally thin layer of cohesive elements along all the boundaries of the bulk elements. However, with such pre-inserted cohesive elements everywhere, it is pointed out by (Klein et al., 2001; Zhou and Molinari, 2004) that the elasticity of the cohesive elements alters the elasticity of the whole structure and a penalty based approach is proposed by (Diehl, 2008a; 2008b) to obtain the stiffness of the cohesive elements. However, since the current paper focuses on evaluating various numerical methods at its very 'raw' form, any remedies to its shortcomings are not further explored here.

Figure 4. Illustration of the CEM simulation results. (Left: visualization of the ice wedge breaking with the CEM simulation; Right: Bending failure tests with different layers of continuum shell elements)

The simulation was first run with different continuum shell element layers. The visualization and simulation results are shown in Figure 4(a) and (b). It can be seen that the CEM method capture the bending failure fairly well when more than 3 layers of continuum shell elements are utilized. The loading histories also tend to converge when more layers of continuum shell elements are used. For the thickness direction, it appears that the mesh dependency does not occur in this numerical test.

3.1.3 Discrete element with cohesive contact Paavilainen et al. (2009; 2011; 2010) applied the concept of combined finite element and discrete element method to simulate the ice fracturing and rubble accumulation. When ice rubbles formed,

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the discrete element method stands for a potential candidate to capture the ice accumulation and clearing process. In their model, the discrete elements are connected with beam elements following a cohesive law during the damage process. Motivated by this methodology, a discrete element method together with a cohesive contact algorithm is applied in this study. Conceptually, it is a rather simple method. Within the system, any two neighbouring discrete elements are following a special contact algorithm. This contact is hard contact under compression, i.e. inter-elements penetration is not allowed in compression. While in tension, loads transferred through the contact are initially elastic. However, when the crack initiation criterion is reached, the contact stiffness is weakened in such a way that the stress and separation follow the cohesive law. Moreover, based on lots of trial simulations and previous discrete methods experiences (Dorival et al., 2008; Schlangen and Garboczi, 1997), a critical damping is introduced in the numerical algorithm. Accordingly, before damage initiation, the tensile contact interactions between any two discrete elements are in analogy to interaction of two bodies tied by springs and dashpots. It should be noted that, different from the traditional discrete element method, the bending deformation is mainly captured by each single element (in the current case, the continuum shell element). Each single element is modelled with a linear elastic constitutive model. A Matlab code was developed to generate these discrete elements and assign each two of them such cohesive contacts as input file for ABAQUS 6.11/EXPLICIT. The challenge with this method is the identification of the contact stiffness (i.e. the relationship between the traction and separation). Based on lots of trial simulations, it is found that the contact stiffness greatly influences the simulation results. Large contact stiffness leads to 'explosive' results. The cohesive contact in the current study is formulated in an uncoupled way as shown in the following equation

n nnn

s ss s

ttt t

t Kt K

Kt

δδδ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥= = =⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

t Kδ [1]

where t is the traction vector in the normal n direction and two orthogonal shear directions ( s and t ). δ is the separation in the corresponding direction. Since the ice material model in the current study is assumed to be isotropic and homogeneous. The contact stiffness tensor reduces to K=K 1 , in which only one parameter is to be identified. Based on energy balance, the following equation in a uniaxial loading case stands:

( ) ( ) ( )A L TAσ ε δ⋅ = ⋅ [2] On the left hand side, Aσ represents the force in the cross section, Lε is the displacement. L is the specimen size. On the right hand side, TA is the traction force while δ is the separation displacement. When in tension, we simply have Tσ σ< >= = which leads to

Lε δ= [3] which corresponds to the definition of an engineering strain. When introducing Young's modulus E and contact stiffness K , the above equation can be reformulated into

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TLE Kσ = [4]

which eventually leads to

EKL

= [5]

It is shown in Eq. [5] that the contact stiffness is related to the specimen size. This definition is similar to the stiffness of the cohesive elements. The choice of L is crucial in this study to adjust the contact stiffness. If L is chosen as the element size, larger contact stiffness is obtained and the simulation becomes 'explosive'. Since in one direction, there will be only one major crack (i.e. the circumferential crack) at concern, it is therefore reasonable to choose L as the size of the ice wedge. Accordingly, based on the loading direction and element position, different contact stiffness is obtained following the above equation and assigned to the corresponding element-to-element contact. By doing so, stabilized solutions are obtained as shown in Figure 5(a). A simulation with different layers of continuum elements to simulate bending failure is also presented in Figure 5(b). It is worth mentioning that this method is very computationally expensive. Three layers of continuum elements overload the cluster with 12 CPUs. However, seen from the current available results, especially with two and three continuum element layers, the loading history shows some resemblance to each other, implying that convergence is possible.

Figure 5. Illustration of the simulation results of DEM with cohesive ties. (Left: visualization of the ice wedge breaking with contact damage variables; Right: Bending failure tests with different layers of continuum shell elements)

3.1.4 Extended finite element method (cohesive segment method) The above introduced CEM and DEM with cohesive ties manage to model the crack in a discrete fashion, i.e. the crack can only exist along discrete boundaries of the bulk/discrete elements. Accordingly, these methods are, in their geometrical representation, mesh dependent. The XFEM method, as a mesh free method, theoretically allows the simulation of arbitrary cracks. A

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Heaviside enrichment term was introduced in the formulation so as to capture the displacement jump across the crack line (ES et al., 1999). As shown in Eq. [6],

,

1( , ) ( ) ( ) ( ) ( ) ( )

d j

m

I jj

X t X t H X X tΓ=

= +∑u N u N a [6]

where u represents the field displacement with the spatial position X at time t . N(X) is the traditional shape function interpolating the nodal displacement uI(t). Comparing with traditional FEM method, the XFEM method has an extra second term on the right hand side of the above equation. A displacement jump is described by introducing the Heaviside function HΓdj(X) at discontinuous boundary Γd. And aj(t) contains all the additional nodal degree of freedom associate with the discontinuity Γd. The XFEM method enables a way to incorporate the discontinuities (cracks) in the finite element formulation. Yet we still need a method to quantify the crack initiation and propagation. The cohesive zone method can then be introduced in the XFEM framework and thus leads to the cohesive segment method (Deborst, 2003; Remmers et al., 2003). Though elegant in theory, implementation of the XFEM with cohesive zone method still poses several challenges.

Figure 6. Illustration of the XFEM simulation results. (Left: visualization of crack propagation inside the damaged element; Right: Bending failure tests with different layers of continuum elements) The cohesive segment method is implemented in ABAQUS 6.11/STANDARD (implicit solver) with only first order continuum elements and second order tetrahedral elements. Therefore even more layers of continuum elements are required to reproduce the bending failure scenario. For calculation considerations, in the current simulation, the ice wedge was separated into three parts, as shown in Figure 6(a). The first part close to the structure is discretized with relatively fine continuum shell elements. The second part in the middle is assumed to be the crack initiation and propagation region. Therefore it is meshed with multi-layers of continuum elements with

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enrichment. The third part is the far end part, which is meshed with continuum shell elements with a relatively coarse mesh. A Matlab code is developed to generate this special mesh in the current study. During the numerical tests, it is found that this method is very sensitive to the tolerance of the crack initiation's numerical error. Since the cohesive segment methodology is selected in the current study, it has been assumed that the crack initiates when traction in any elements reaches a critical threshold (e.g. 500 kPa). However, the incremental nature of the FEM formulation and numerical scheme usually leads to some elements' stresses outcome are way beyond this threshold during some critical calculation steps. Stress inside an element larger than the threshold is physically prohibited. This means that the stress cannot go beyond the fracture surface. Therefore, more numerical iterations are required to bring the calculated stress down to the fracture surface with certain numerical error tolerance (by default 5%). The conflicts here is that, if a high tolerance is used, multiple cracks initiate in the enriched region which leads to numerical difficulties due to a large amount of additional degree of freedoms; if a low tolerance is used, crack can localize, but at the costs of calculation efforts and further numerical convergence difficulties. Therefore, a compromise has been made in the current study. Instead of starting from an initially flawless ice wedge, an infinitesimally shallow (half of the element size) crack is introduced in the upper surface of the ice wedge. The position of the crack is determined such that the ice breaking length equals to the results given previously by the element erosion technique. The introduction of an initial crack localizes the 'initiation' of crack even with a relatively larger numerical error tolerance. The simulation becomes stable and the propagation of crack is not bound to the mesh of the ice wedge, e.g. as shown in Figure 6(a), the crack separate the original intact ice element into two. The same bending simulation tests also show that the global horizontal loading history resembles each other with the tested number of layers as shown in Figure 6(b).

3.2 Mesh Sensitivity Study Based on the criteria set forth previously, it appears that all methods are capable of predicting the bending failure mode when sufficient layers of elements are utilized. The next stage of the investigation is to identify the mesh sensitivity of each method. However, it should be stressed that the simulation results by different numerical methods are not yet comparable since different material models are implemented. For the current study, the focus is on the mesh sensitivity study of each numerical method, rather than on the loading history comparisons among different methods. In the current numerical set-up, as shown in Figure 7, there are three spatial directions where the element size variations are expected. However, when investigating the bending failure reproducibility, the thickness direction mesh sensitivity has already been tested under a different name. In this section, the mesh sensitivity investigation mainly focuses on the mesh size variation in the circumferential and radial directions.

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Figure 7. Mesh sensitivity tests in three directions of the wedge.

3.2.1 Mesh sensitivity tests in the circumferential direction By gradually refining the mesh size in the circumferential direction, the loading histories obtained by each method are illustrated in Figure 8. It can be seen from the figure that with the CEM simulation, the structural stiffness decreases with increasing refinement. This is in agreement with the conclusion that pre-inserted cohesive elements soften the whole structure (Klein et al., 2001; Zhou and Molinari, 2004). Thus some remedy measures must be taken before the CEM simulation could be trusted. Regarding other numerical methods, before breaking, the overall stiffness remains almost the same with different circumferential refinements. In the post-failure regime, the element erosion technique presents results with finer mesh and differs from those with relatively coarse mesh. A scrutinization into the numerical animation shows that the element erosion technique with finer circumferential discretization fails in a different manner. This leads to its change in the post-failure loading curve. The DEM and XFEM offer a better post-failure behaviour in the current study.

Figure 8. Loading history with different circumferential direction refinements. (Element erosion technique (a); CEM simulation (b); DEM with cohesive contact (c); XFEM simulation (d))

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To sum up, except for CEM, all other methods show relatively good mesh objectivity in the circumferential direction before breaking occurs. DEM with cohesive contact and XFEM appear to give satisfactory mesh objectivity in the post-failure loading history.

3.2.2 Mesh sensitivity tests in the radial direction The loading histories based on each numerical method with different radial mesh sizes are shown in Figure 9. It appears that the radial direction mesh size tends to cause the largest discrepancy. This makes sense since the crack is mainly formed circumferentially and therefore influenced by the radial mesh size. For element erosion technique, the ice wedge failure pattern (e.g. number of radial cracks) differs quite a lot with different mesh size, thus leading to large loading history discrepancy. Regarding the CEM simulation, it is again observed the structural stiffness softening with increasing cohesive element density. But one advantage of this method is that the peak ice breaking loads do not deviate too much from each other with radial mesh refinement. In terms of the DEM simulation with cohesive contact, quite large discrepancy is observed under different radial mesh refinement. However, due to computational burden, only a small amount of simulation with rather coarse mesh is conducted here. This method might be able to give relatively better solution if a finer mesh is used. But for the time being, it is not promising due to the burdensome contact algorithm between all neighbouring elements.

Figure 9. Loading history with different radial direction refinements. (Element erosion technique (a); CEM simulation (b); DEM with cohesive contact (c); XFEM simulation (d)) As far as XFEM based simulations are concerned, although theoretically claimed to be a mesh free method, its application to the current ice wedge bending case with the ABAQUS-

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6.11/STANDARD implementation seems to be mesh dependent. This might partly be due to the less suitability of the implicit algorithm comparing with the explicit algorithm to simulate such a transit, contact involved and dynamic governed physical process; and also possibly due to the limitations of XFEM's implementations within ABAQUS-6.11/STANDARD; one more reason is due to the material softening implementation in the simulation. Considering relatively good results were obtained in the thickness and circumferential direction, the first two reasons are supposed to contribute to minor mesh sensitivities. Therefore, the material model with softening emerges as the major culprit for mesh sensitivity. To sum up, all these numerical methods exhibit mesh dependency with varying radial mesh size. The structural stiffness softening is also observed with varying radial mesh size in the CEM based simulation. While the CEM based simulation has an advantage over other methods in the way that the peak ice breaking loads stay rather stable with varying mesh size. Still, further remedies are needed in order to alleviate the mesh sensitivity issue.

3.3 Discussions about the Numerical Tests In the very beginning, three criteria have been established to evaluate these four numerical methods. Regarding bending failure reproduction, all numerical methods have passed with varying degree of soundness. As for mesh sensitivity study, based on the numerical test results with varying mesh size in different direction, it is shown that the radial mesh size causes the most mesh sensitivity for all the numerical methods. This is to say, at least in one case, that all the above numerical simulations are mesh-dependent (including mesh free method). Accordingly, two possible originations of the mesh sensitivity could be thought of. The first possibility is due to the numerical methods that we choose, e.g. discrete cracks can only propagate along the boundaries of the bulk elements in the CEM and DEM methods. To address this type of mesh sensitivity, a vast literature regarding possible measures to alleviate the mesh sensitivity of each numerical method is available, e.g. non-local approach formulation for the element erosion technique; non-intrinsic cohesive elements insertion dynamically for CEM, etc. Another possibility is due to the material model implemented in the simulation. This has been pointed out by (Borst, 2004) that the reason for not converging is not due to traditional numerical discretization, but due to the change of nature of the partial differential equation (PDE) when material softening occurs. The solutions of the PDEs are not unique, thus refining the mesh leads to no-unique results. Therefore, the mesh sensitivity is in its nature not a numerical problem but a mathematical problem due to material softening. Several approaches have been proposed to alleviate the mesh dependency problem with regard to the material softening issue, e.g. the introduction of strain rate consideration. This will be combined with a future study in terms of finding a suitable material constitutive model. In terms of calculation efficiency, it is found out that the element erosion technique is the most effective method; the CEM comes next, following which stands the XFEM. The DEM with

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cohesive contact is the most computationally expensive method among these four. Each method has their own advantages and disadvantages. The application of each method is largely dependent upon the simulation purpose and care should be taken to each numerical methods' pros and cons before implementation.

4. Conclusions In this paper, four different numerical methods are combined with the cohesive zone method to simulate the progressive failure process of an ice wedge bending problem. The ice bending failure scenario is reproduced by all the numerical methods. Following that, the mesh sensitivity of each numerical method is further scrutinized by comparing the loading history with varying mesh size. It is found that all methods are mesh-dependent due to the presence of material softening. Apart from that, advantages and disadvantages of each method are identified and listed below:

• Element erosion technique: This method offers the most computationally effective simulation. However, this method is largely dependent upon the applied constitutive model. Therefore, a constitutive model, which is capable of describing the anisotropic, pressure dependent, and strain rate dependent, etc. natures of ice, is crucial for the correctness of this method. Numerically, the 'creation' of crack is achieved by deleting relevant elements. This brings about a disadvantage of mass imbalance. Especially considering the ice accumulation load covers a major part of the total ice load in terms of ice-sloping structure interactions (LU et al., 2012), necessary remedies are required (e.g. the model update technique implemented in (Kolari et al., 2009))

• CEM and DEM with cohesive contacts: These two methods are geometrically discrete approaches to simulate the fracture initiation and propagation. The potential cracks are predefined to be along the bulk/discrete elements' boundaries. Accordingly the cracks' initiation and propagation are simulated in a 'controlled' manner. Therefore, one advantage of this method is that a largely simplified material constitutive model can be applied to the bulk/discrete elements. By doing so, much less burdensome numerical and experimental tests are required to construct a sound material constitutive law that include most important aspects of the ice material's behaviours. Another advantage is that the crack is formed by deleting the infinitesimally thin cohesive elements, thus no mass imbalance problem involves in these two approaches. However, the existence of cohesive elements alters the structure's overall stiffness. The DEM with cohesive contact, though has no problem with structural softening, is very computationally expensive. Further remedies are required before the implementation of these methods.

• XFEM: This method is very promising in simulating the crack initiation and propagation problems due to its mesh-free nature. However, this method is still at its developing stage. Thus most of its disadvantages still exist in most market available implementations. In the current study, challenges in the localization of fracture initiation are encountered. Further, since only limited element types are equipped with the enrichment application, a

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non-uniform meshing strategy (meshed with both continuum elements and continuum shell elements) is adopted in the current study to capture both cracks and bending failure.

Therefore, each of the above method has its own advantages and disadvantages. Its application mainly is determined by the problem to be solved. Moreover, remedies to the numerical methods are always needed before any convincing results are obtained. Acknowledgements The authors would like to thank the Norwegian Research Council through the project 200618/S60-PetroRisk and the SAMCoT CRI for financial support and all the SAMCoT partners. Further, we acknowledge Prof. O. Hopperstad, Prof. A. Metrikine, Prof. J. Tuhkuri and Mr. T. Nord for valuable discussions, criticism and constructive suggestions. References Bažant, Z. P., Bažant, Zdeněk P., and Planas, J., 1998. Fracture and size effect in concrete and

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