Cold Regions Science and Technology 94 (2013) 53–60
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Cold Regions Science and Technology
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Model scale ice — Part B: Numerical model
Rüdiger von Bock und Polach a,b,⁎, Sören Ehlers b
a Aalto University, School of Science and Technology, Department of Applied Mechanics, Marine Technology Group, P.O. Box 15300, FI-00076 Aalto, Finlandb Norwegian University of Science and Technology, Faculty of Engineering Science and Technology, Department of Marine Technology, NO-7491 Trondheim, Norway
⁎ Corresponding author at: Aalto University, School opartment of Applied Mechanics, Marine Technology GrAalto, Finland. Tel.: +358 94513478.
Themodel-scale ice of the Aalto University ice tank isfine-grained ice, which is used in experimental research onArctic marine structures. However, the behavior of model-scale ice is not much explored and a deeper under-standing of the model-scale ice mechanics is required to appraise the validity of full-scale and model-scale iceproperty tests. Therefore, this paper presents a numerical model that accounts for the model-scale icemicro-structure and a variety of the physical effects under load. The numerical model accounts for thenon-homogeneity of the model-scale ice and is based on damage mechanics. The basic mechanical properties,i.e. grain size, elastic strain-modulus, Poisson's ratio, compressive strength and tensile strength, are determinedexperimentally, and the free parameters of the damage model are determined based on experimental evidenceand by reverse engineering until the compliance of the numerical and experimental response is achieved.Furthermore, the response sensitivity on variations of the Poisson's ratio and the non-homogeneities is analyzed.Ultimately, the failure stresses of the numerical model are compared with the experimentally-determinedstrength values. As a result, an experimentally-calibrated numerical model of the model-scale ice is obtained.In the future, this numerical model can be used to evaluate marine designs prior to testing, thus allowing forinexpensive simulation-based design of arctic marine structures.
The increasingmarine activities in Arctic waters involve growing in-terest in experimental and numerical performance predictionmethods.Scaled ice model tests are the state of the art prediction methodassessing the design of ice going ships (see von Bock und Polach et al.,2012). Semi-empirical formulations for ice resistance and ice loads areoften limited to certain ship or structure types, such as the formulationof Lindqvist (1989), which is validated against ice breakers. In von Bockund Polach (2010), it is indicated that the formulation of Lindqvist(1989) does not necessarily apply for tankers or other ships, eventhough it is often used for those. This highlights the need for generallyvalid formulations and methods that allow the analysis of any ice capa-ble design. This is to be achieved by models that respect the physics ofice adequately and the related failure processes bymeans of universallyapplicable prediction methods. The here presented numerical model isdedicated to future simulations in model-scale ice simulation, due toits high importance in the design process and better accessibility totest data.
Already Varsta (1983) used finite elements to determine ice loadson ship hull sections and a comparison with full-scale measurementson board an icebreaker delivered that the calculationmethod predicted
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ock und Polach).
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the right load level. The numerical method by Valanto (2001) is basedon potential flow theory where the ice sheet is modeled as a boundarycondition following plate theory. However, themodel of Valanto (2001)accounts solely for the elastic strain-modulus and the Poisson's ratio asmechanical properties of the ice sheet. The low yield or creep strengthof the model-scale ice (see von Bock und Polach et al., in press) makesthis approach unfit for model-scale ice.
Another approach, the cohesive elementmethod, defines a cohesivezone between the elements, which vanishes upon a certain energylevel. Gürtner (2010) carried out model tests on a Shoulder Ice Barrierat theHSVA to calibrate the cohesive elementmodel. However, such en-ergy based fracture criteria require that the energy release rate is wellapproximated. Hilding et al. (2011) showed that cohesive elementsare capable of simulating full-scale ice structure interaction, however,on a more qualitative scale, which again reflects the difficulty of deter-mining the special input parameters a priori.
A model closer to actual ice physics is presented by Derradji-Aouat(2003). This is a multi-surface failure criterion based on experimentsfor sea ice and accounts for changes in temperature. Wang andDerradji-Aouat (2009) implemented this model in LS-DYNA, butobtained differences between the results of the tests and the simulation.AlsoMartonen et al. (2003)modeled the ice structure interaction basedon a multi-surface failure criterion of Derradji-Aouat (2003) withANSYS. The presented forces in Martonen et al. (2003) have been com-pared with full-scale measurements and show in some cases good cor-relation. To consider the mesh size sensitivity of the failure process,Kolari et al. (2009) introduced a mesh updating method to predict the
Table 1Compilation of parameters required for the damage model.
Parameter Symbol Origin
Elastic modulus E MeasurementYield strength Y MeasurementHardening modulus H Slope of load–displacement curve (measurement)Material constant t Very brittle or more ductile failure (measurement)Material constant S Damage evolution/point of failure (measurement)Critical damage dc Point of failure (measurement)
54 R. von Bock und Polach, S. Ehlers / Cold Regions Science and Technology 94 (2013) 53–60
failure progression in ice. Furthermore, hismesh updating technique al-lows the conservation of mass since no elements are deleted. This tech-nique does not use a prior given crack path, but is updating and refiningthe mesh of the model, thus leading to unpredictable simulation times.Among others (e.g. Duddu and Waisman, 2012; Pralong and Funk,2005;Wells et al., 2011) also Kolari (2007) based the ice failure processon damage mechanics. Jordaan et al. (2005) reported that damagemechanics can also well represent softening of ice that is, for instance,related to recrystallization.
To reproduce the failure of ice observed in measurements Guptaand Bergström (2002) used a shear faulting model. In the numericalmodel, the elements represented grain boundary elements andwere aligned with the grain size. Sinha (1978) developed a viscoelas-tic constitutive model for uniaxial compression. This model reflectsthe total strain as a superposition of the elastic strain, the delayed elasticstrain (due to grain boundary sliding) and the viscous (transient) strain.Karr and Choi (1989) used themodel of Sinha (1978) and added the ef-fect of damage to the delayed elastic and viscous strain components.The damage represents the micro-cracks.
The constitution and failure mechanisms of model-scale ice mightdiffer significantly from S2 sea ice or other ice types. However, someideas from other ice research might be transferable, within limits, tomodel-scale ice in order to describe its material behavior. The motiva-tion for developing a numerical model of model-scale ice is thatmodel-scale ice testing is the state of the art performance predictionmethod. Additionally, model-scale ice reflects ice in an ideal constitu-tion (nearly isotropic in the undeformed state) and the growth andcooling history is known. Most parameters of interest for building anumerical model can be measured in-situ (see von Bock und Polachet al., in press). Measurements in sea ice are disadvantageous, be-cause an ideal constitution of sea ice cannot be defined, as tested icesamples may vary in their grain structure, undergo several coolingand thawing cycles or might have been rafted in their history.Among others, these boundary conditions affect the results of struc-tural tests, but it is not fully understood to what extent.
The numerical model presented in this paper accounts for a variety ofmicro-structural phenomena. Additionally, the model is calibrated with aseries of ice property tests (von Bock und Polach et al., in press). Unlike allknownmethods, this approach is focusing onmodel-scale ice, since mostboundary conditions of the model-scale ice are known while it can bereproduced with reasonable accuracy. The presented material model isbased on damage mechanics and accounts for all observations madeand parameters measured by von Bock und Polach et al. (in press). Thematerial model is readily implemented in LS-DYNA to allow a large num-ber of people to benefit from this work.
2. Modeling of the material constitution
The mechanical model to be used for numerical simulationsaccounts for all findings and measured parameters presented by vonBock und Polach et al. (in press). Herein it must be distinguished be-tween the material model for model ice and modeling of the materialconstitution, i.e. including voids acting as stress concentrators, whichis described in Section 3.
The model-scale ice is a fine grained polycrystalline material withrandomly orientated crystals. Following Karr and Choi (1989) themate-rial is considered isotropic in its undeformed state. The material isnon-homogeneous and contains 1% of air and 4.5% of water (see vonBock und Polach et al., in press). The relatively low yield strengthmarks the early beginning of not-recoverable strains. Therefore a soften-ing modulus must be incorporated due to the relative weakness of thegrain boundaries and their loss of strength with increasing load (i.e. ad-ditional softening). These requirements indicate that damagemechanicsfollowing Lemaitre and Desmorat (1992) are well suited.
Themodel does neither regard viscous effects nor temperature effects.The tests conducted by von Bock und Polach et al. (in press) were neither
suitable to reveal viscous behavior nor considered by the authors of highsignificance for the majority of the ice model-scale tests. A temperaturedependent material behavior, as in Derradji-Aouat (2003), for sea ice isnot considered. This is due to the fact that each model-scale ice sheet isa discrete model of a certain combination of thickness and bendingstrength. Eq. (1) states the damage progression, _D, according toLemaitre and Desmorat (1992) as a function of yield strength, themateri-al constants S and t and the plastic strain �p:
_D ¼ YS
� �t_�p: ð1Þ
Table 1 summarizes the parameters required for the damagemodel and their origin. The reference measurement refers to theconducted experiments and derived parameters in von Bock undPolach et al. (in press).
All parameters in Table 1 can be uniquely determined from themeasurements except S and dc, which are dependent on each other.Lemaitre and Desmorat (2001) give material parameters for steel,ceramics and concrete, and, based on this, 0.2 is used for dc as astarting value for the upper critical damage in compression. There-with, both values are identified to match the final response curve.
3. Numerical modeling and implementation
The numerical model is built using ANSYS and LSPrepost as pre- andpost-processors together with the commercial solver LS-Dyna. Beyondthe elastic regime strain depends on the element size and a reference el-ement sizemust be introduced. In order to align the FEmodel with theice physics, the element size is selected equal to the grain size of0.68 mm, which is determined by von Bock und Polach et al.(in press). The material model is the Lemaitre damage model(* 153 _ MAT _ DAMAGE3), where the damage represents the pro-gressive weakening of the grain boundaries. Therefore, the under-standing of the model is that each finite element represents 1/6 ofsix grains including the grain boundaries they share with each other.
The existing voids of air and water are incorporated with a ran-dom algorithm that defines 1% of the elements as air and 4.5% aswater, following the measured percentages presented by von Bockund Polach et al. (in press). The air elements are deleted and thewater elements are defined as * 001 _ Fluid − Elastic _ Fluid.
The dimensions of the numerical specimens are equal to theexperiments, with the random void distribution depending on thenumber of elements i.e. specimen size.
Fig. 1 displays the numerical model for compression. The upperlayer represents the above water area, where no water voids occur.The area below is a compound of ice (blue) and water voids(brown). Air voids are represented by removing the elements.
The procedure to determine the input parameters for the FE modelis based on reverse engineering. In the Lemaitre damage model thehardening modulus affects the slope of the load–displacement curvebeyond the linear-elastic section, and the critical damage parameterat which the material fails. The material constants S and t controlthe damage progression and hence the rate of weakening withincreasing loading. Those parameters are all unknown and are itera-tively determined. The parameters are considered as determined
Fig. 1. Composition of the numerical model. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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when the simulated load–displacement curves match the measure-ments in terms of load increase and peak force. The material param-eters S and t are considered as global parameters and the hardeningmodulus and critical damage are considered as specimen-individualparameters.
In the FE model, the load is applied with a rigid wall that repre-sents the impact plate or impact cylinder and the determined reactionforce which represents the signal of the load-cell. In the numericalmodel elements with the damage level ≥ dc are deleted. This is aviolation of the principle of mass conservation. However, in realitythe failed grains might change their physical condition to a fluiddue to pressure melting (see also Wang and Derradji-Aouat, 2009).Additionally, as the relative number of deleted elements is small, sois their mass.
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Fig. 2. Comparison of simulation and measurement for (a) compression and (b) tension.
4. Results
4.1. Results of compressive simulation
The damage material model of Lemaitre represents the load–displacement progression in all tensile experiments and most of thecompressive experiments as well (see Fig. 2). Fig. 3 shows one of twomeasurements where the progression of the measured curve couldnot exactly be followed, but the load–displacement progression is stillapproximated in a satisfactory manner.
The critical damage parameter, dc, is determined based on thepoint of failure in the measurements and the hardening modulus isbased on the slope of the curve. The elastic strain-modulus has aminor impact due to the low yield strength. The resulting data of allexperiments are compiled in Table 2. The dc values, which governthe point of failure, follow a log-normal distribution as the peak forcesof the measurements (see von Bock und Polach et al., in press). Fig. 4displays the log-normal fit for the compressive dc values and Fig. 5compiles the resulting combinations of hardening moduli, H, and dc.
The stresses in the numerical specimen are compared with theengineering approach from the measurements, where the stress isdetermined by maximum force divided by the cross-sectional area.The cross sectional area of the numerical specimens is directlymapped from the experiments, which is 652 mm2 ± 6% for the com-pressive specimens and 672 mm2 ± 8% for the tensile specimens. Inthe compressive simulations, the compressive stresses clearly differfrom the analytically determined stresses. However, for the tensile
simulations, the tensile stresses comply with the engineering stresseswith an accuracy of 1%–14%.
5. Sensitivity analysis
5.1. Sensitivity on void distribution
The sensitivity of the specimen's response on the void distribution isinvestigated by simulating the loading of seven numerical specimens
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Fig. 3. Comparison of simulation and measurement visualizing minor deviation in thehigh loading range.
Fig. 4. Log-normal fit to the compressive dc values.
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with different void distributions for both, tension and compressioncases. The distribution of the voids is random (MATLAB random func-tion), but the void ratio in all numerical specimens is equal, henceonly the positioning of the voids differs. Fig. 6 displays the simulationaligned to the measurements (see Fig. 2) together with six other ran-dom void distributions. In compression the variation of the peak forcevaries by ±1% and in tension by ±6%. Figs. 7 and 8 display breakingpatterns of the simulations shown in Fig. 6 and reflect that the failurepattern strongly depends on the void distribution (A, B, C, D) unlikethe maximum force level.
5.2. Sensitivity on the number of voids
The number of voids in the numerical specimens is based on satis-fying the overall density of the model-scale ice (see von Bock undPolach et al., in press). This estimation is affected with uncertainty,because several proportions of air, water and ice are possible. Thesensitivity of the load–displacement history on the number of voidsis presented for a high number of voids (1.45% air and 10% water),no voids and the standard amount (1% air and 4.5% water) used inall other calculations. Fig. 9 shows that with an increasing amountof voids the specimen response becomes weaker and the specimenfails earlier. In the case of the compressive specimens (Fig. 9b) the in-creasing amount of voids reduces the specimen stiffness. The maxi-mum force without voids is higher (15.6% for compression, 28% fortension) than for the standard case and lower (13.6% for compression,9% for tension) for the high void density.
5.3. Sensitivity on elastic modulus and Poisson's ratio
The Poisson's ratio of model-scale ice is selected based on com-mon practice at Aalto Ice Tank and literature (see von Bock undPolach et al., in press; Timco, 1980). Nevertheless as it is currentlynot measurable for model ice, the value is affected with uncertainty.Fig. 10 shows the variation of the displacement–load history in de-pendence of the Poisson's ratio variation and the related elastic
Table 2Compilation of parameters required for the damage model.
Parameter Value
Compression Tension
Elastic modulus [MPa] 148Yield strength [kPa] 0.45Material constant t 2Material constant S 25Number of measurements 9 7Critical damage dc 0.005–0.236 5.0 × 10−6–3.4 × 10−2
Log-normal median dc 0.036 0.001Hardening modulus H [MPa] 0.95–1.3 2.8–3.3Average H [MPa] 1.14 3.07
strain-modulus for tension and compression. Fig. 11 displays thesame load case as Fig. 10, but only the Poisson's ratio is varied andthe elastic strain-modulus is kept constant using the value for ν =0.3.
Table 3 displays the impact of the variation of the Poisson's ratioand of the elastic strain-modulus. The results are expressed in rela-tion to the material response for ν = 0.3 and the related elasticstrain-modulus. Table 3 also states that the elastic strain-modulusaffects the maximum force response of the material. Furthermore, intension, the sensitivity towards variations in Poisson's ratio and elas-tic strain-modulus is higher than in compression.
6. Discussion
The paper presents an approach to simulate the failure process ofmodel-scale ice in compression and tension numerically. The numer-ical model accounts for several parameters representing the physicalconstitution of the material. Furthermore, the dependence of the loadresponse on the micro-structure is analyzed.
In addition to the good functionality of the isotropic materialmodel of Lemaitre and Desmorat (2001), it requires solely fourinput parameters to be determined, which all may be determinedbased on physical evidence from the experiments, presented by vonBock und Polach et al. (in press). The applied von Mises yield surfaceand the related yield stress have a significant impact on the model, asthe yield stress affects the damage progression and the determinationof the related parameters (see Eq. (1)). The von Mises yield surface ismainly used for homogeneous materials such as steel and metals,whereas the yield surface for ice can be significantly different (seeDerradji-Aouat, 2003). However, the model-scale ice of Aalto Ice Tankis considered as quite homogeneous and the von Mises yield criterionis considered the best standard as long as the actual yield surface is
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Fig. 5. Varying parameters of damage model.
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Fig. 6. Void distribution variation for (a) tensile specimen and (b) compressive specimen.
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unknown. The determinedmaterial parameters (see Fig. 5) are found tobe different in compression and tension.
The number of voids has a significant impact on the material re-sponse in terms of stiffness and maximum strength. Furthermore,Fig. 9 displays that a reduced amount of voids enforces a more brittlefailure, which indicates that the determined amount of voids (1% airand 4.5% water) is a good estimate for the current set of measure-ments. Fig. 9 indicates that local variations in void amount cause astrong variation of the maximum forces. In tension, the range ofvariation is up to 40% and in compression to 30%. This indicates thatthe local void distributions in the model-scale ice, which can liebetween 0% and 11.45% void ratio, may contribute significantly tothe specimen response variation.
In von Bock und Polach et al. (in press) it has been shown that crackpropagations and crack origin can vary significantly. The samephenom-enon is reflected in the numerical simulations when the local distribu-tion of the air and water voids is changed. The impact of the void
Fig. 7. Variation in the compressive failure pattern for
distribution on the failure load is small (max. 6%), but it strongly affectsthe failure pattern, due to local stress concentrations and related failureinitiation (see Figs. 8 and 7).
In the sensitivity analysis on the Poisson's ratio variation the dam-age parameters for all runs are taken from the simulation case withν = 0.3. Fig. 10 reflects that a lower Poisson's ratio leads to an earlierfailure, which is triggered by the related lower elastic strain-modulus(see Figs. 11 and 10 for comparison). Once the elastic strain-modulusis kept constant (see Fig. 10) the impact of the Poisson's ratio on themaximum response load is notably reduced. In the applied Lemaitremodel (see Lemaitre and Desmorat, 1992) the damage progressionis a function of the yield strength, which however does not varydepending on the Poisson's ratio (see von Bock und Polach et al., inpress). This highlights the significance of the correct elasticstrain-modulus for the simulation.
The determined material parameter dc is found to follow alog-normal distribution (see Fig. 4). Based on this, a representativedc value and a representative H value are determined (see Table 2).Two numerical specimens are set up: two for each, compression andtension. One has a high void density (see Section 5.2) and the smallestmeasured cross-sectional area found in the experiments and theother has the largest cross-sectional area found and no voids. Fig. 9compiles the results of the two simulations together with all mea-surements. The two simulated cases are considered to reflect theweakest and strongest extreme cases.
Fig. 12 shows thatmost of themeasurements are gathered betweenthe two extreme cases, but not all of them. Fig. 9 reflects that an increas-ing void density causes a significant loss in stiffness (especially in com-pression) and resistance. However, the slope of the measured force–displacement curves is quite constant, which in consequence meansthat the variation in the peak force may not be related to the void den-sity. The variation in the void distribution can affect the peak force byabout 6% (see Section 5.1 and Fig. 6), but it does not explain such alarge spread of the critical damage parameters. Furthermore, an in-creasing amount of voids makes the failure less brittle, which impliesthat the variation in the dc values is related to an effect other than thevoids. This effect is however not yet known. The critical damage param-eter appears however well suited to describe the failure process. Fig. 12reflects that the application of one representative set ofmaterial param-eters with a variation of the voids cannot represent the material re-sponse completely. This might suffice for design purposes, butdetailed analyses will require a probabilistic approach where differentdc values are assigned to the different elements.
The calculated stresses from the experiments (von Bock undPolach et al., in press) are compared with the average stresses ofthe ice elements around the failure location of the numerical speci-men. The determined tensile stresses of the numerical model havean average deviation of 7.1% from the stresses determined from theexperiments with the engineering approach (see Section 4.1). This in-dicates that the numerical model is functioning well and that theconducted specimen tests are mainly taking tensile stresses as
(a) void distribution A and (b) void distribution B.
Fig. 8. Variation in the tensile failure pattern for (a) void distribution C and (b) void distribution D.
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planned. However, this does not apply to the compressive specimenexperiments, where already the breaking pattern from the experi-ments indicates that the specimen fails in shear and not in compres-sion. A review of the average compressive stresses in the contactsurface with the impact plate, IP, does not even approach the stressesthat are determined experimentally by dividing the maximum forceby the cross-sectional area. This indicates that the total load is trans-ferred to stresses other than compressive ones and that the currentstandard experimental method for calculating the compressivestrength may not be valid. The sample size and respectively theboundary conditions are affecting the measured compressivestrength (see Timco and Weeks, 2010).
The material model does not account for visco-elastic, visco-plasticor temperature dependent effects, as those effects are not found in theexperiments of von Bock und Polach and Ehlers (in press). However,
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Fig. 9. Numerical specimens with void density variation for (a) tensile specimen and(b) compressive specimen.
the experiments were neither suitable nor targeted for detectingany viscous effects. The presented model will be subjected to refine-ment and additional research to add viscous effects in future.Derradji-Aouat (1992) stated the significance of temperature effectson the material behavior of sea ice. Past measurements, however,showed that the temperature gradient over the model-scale ice thick-ness is very small and does not differ significantly, despite the factthat the ice properties i.e. bending strength and thickness are different.This is due to the cooling history and the embossed properties. In con-sequence, the presented material's parameters are only valid for thisspecific bending strength/thickness combination. Therefore, the herepresented modeling procedure is to be applied further to ice sheetswith different properties in order to develop a material model for allstrength/thickness combinations (similar to the multi-yield surface ofDerradji-Aouat, 2003). The material parameters presented here hence
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Fig. 10. Variation of the Poisson's ratio and the related elastic strain-modulus for(a) tensile test and (b) compressive test.
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Fig. 11. Variation of the Poisson's ratio with constant elastic strain-modulus for (a) ten-sile test and (b) compressive test.
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Fig. 12. Comparison of the measurements together with the upper and lower extremecases for (a) compression, with H = 1.14 MPa and dc = 0.036 and (b) tension, withH = 3.07 MPa and dc = 0.001 (values according to Table 2).
59R. von Bock und Polach, S. Ehlers / Cold Regions Science and Technology 94 (2013) 53–60
reflect only one discrete step in the overall futurematerial model. Addi-tionally, future work will be dedicated to increase the element size forenhancing the practical use of the presented numerical material model.
7. Conclusion
The damage model of Lemaitre appears to be well suited to modelthe behavior of model-scale ice. It allows modeling of the failure pro-cess in compression and tension. In addition to representing the load–displacement curves correctly, the material model is able to reflectthe failure mode sufficiently. The obtained failure patterns of the sim-ulations reveal that those are significantly affected by the local voiddistributions. The variations in the failure patterns of the simulationscomply with the observations from the experiments. The sensitivityanalysis and the simulation of extreme cases additionally indicatedthat the void distribution is not the only parameter that is affectingthe obtained spread in the critical damage parameters. This must bebased on effects that are not known yet, which are however well
Table 3Variation of the maximum force in relation to the simulation with ν = 0.3 and E =148 MPa.
Test Poisson's ratio Elastic modulus [MPa] Max force variation %
represented by the applied material model. For obtaining a globalpicture of ice–structure interactions, simulations are recommendedto be probability based by varying the dc values. Solely, a variationof the void distributions or void amounts cannot explain the spreadof the experimental results.
As with other materials to which damage mechanics are applied,the model ice has a lower tolerance for damage in tension than incompression. The numerical model shows that the tensile tests fromvon Bock und Polach et al. (in press) are well suited for determiningthe tensile strength of the model ice, but in contrary, the compressivetests are not suitable to determine the actual compressive strength ofthe ice, as the failure process is very complex and not purelycompressive.
Acknowledgment
We would like to thank the researchers from SINTEF Trondheim,Erling Østby, Mario Polanco-Loria, Magnus Eriksson, Odd-Geir Lademoand Berstad Torodd for their interest and time spent in advising ourwork. Furthermore, we want to express our gratitude to Pentti Kujalafor his comments and Hannah Ploskonka for her language review.
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