ICE CRUSHING

Failure mechanisms

THOMAS OLOFSSON AND LENNART FRANSSON

Division of Structural Engineering

JIM SANDKVIST

SSPA Maritime Consultants AB

ABSTRACT

The ice failure mechanism close to the contact has been studied in a series of small scale crushing tests at relatively high indentation velocities. An iterative numerical model based on successive shear failure was used to predict the crushing behaviour. The model parameters and the initiation of vertical cracks has been evaluated using fracture mechanics.

From observations and recorded data it is concluded that the contact consist of a narrow line where the melting is attained at high indentation rates.

The theoretical analysis and simulation of experiments indicates that the ratio of shear strength to contact pressure controls the fractal behaviour of ice crushing. The contact pressure and shear strength are related to the indentation velocity. As the indentation rate increases the ratio of shear strength to contact pressure decreases causing finer fragmentation of the crushed ice.

At present the 2-D model predicts an upper limit of the ice load similar to the Morgan-Nuttall formula for shear failure of an ice sheet with a friction-less indentor.

PREFACE

The work presented in this report has been carried out within a joint project between SSPA Maritime Consulting AB and div. of Structural Engineering, University of Luleå, with the title "Dynamisk respons av interaktion vid iskrossning". The financial support has been granted by SIND and COLDTECH. Their support is greatly acknowledged.

The experimental program has been carried out at the div. of Structural Engineering research laboratory, TEST-Lab. The following persons are greatly acknowledged for their contributions in the test program: Lars Åström, Georg Danielsson TEST-Lab. Per Gren Experimental Mechanics, University of Luleå. Björn Forsman and Sven Westling from SSPA Maritime Consultants.

Finally the authors wants to thanks Carina Hannu, div of Structural Engineering, who did the final preparation of the manuscript.

Luleå in November 1991

Thomas Olofsson Lennart Fransson Jim Sandkvist

NOMENCLATURE

Greek letters

-- angle between ice edge and indentor

-- a coefficient, (1+)

-- viscous strain rate

-- friction angle in Mohr-Coloumb formula

-- Poisson's ratio

-- flake angle

-- stress, compressive or tensile

-- uniaxial compressive strength, defined positive

-- melting pressure/stress, compressive defined positive

-- maximum contact pressure

-- normal stress acting on flake/crack surface, compressive defined positive

-- nominal contact pressure

-- maximum viscous stress level, compressive defined positive

-- shear stress

-- shear strength

Latin letters

Acontact -- contact area

Aflake -- flake area

a -- contact width

a split -- the splitting contact width, vertical cracks

b -- indentor width

C -- a constant

c -- "cohesion" constant in Mohr-Coloumb formula

D -- plate stiffness

E -- elastic modulus

E I -- beam stiffness

F -- force, contact force.

Fmax -- maximum contact force

Fsplit -- the force required to propagate a vertical crack

G, GIc -- strain energy release rate, critical strain energy release rate mode I

h -- ice thickness

K -- stress intensity factor

KI, KII, KIII -- stress intensity factor for mode I, II and III.

KIc,KIIc -- fracture toughness for mode I and II

k -- ratio of shear strength to maximum contact pressure

Q -- geometric shape factor

R -- universal gas constant

r -- crack radius

T -- temperature

U -- activation energy

u -- deflection of indentor plate

INTRODUCTION

Sanderson (1988) classified the principal failure mechanisms observed during indentation experiments in:

* creep

* radial cracking

* buckling

* flaking or spalling

* crushing

The governing parameters are aspect ratio and indentation rate. At high indentation rates and small aspects ratios the most commonly observed failure mechanisms is flaking and/or crushing.

The flaking mechanism is described as in plane horizontal cracks that bend upwards and downwards creating fragments of ice that break away. This mechanism have so far been little investigated. At higher rates still, the ice is pulverised and extruded upwards and downwards. The ice - structure contact has often in the past been modelled as a viscous layer but recent research has shown that the contact consist of a narrow line with contact pressures reaching the melting point.

The scope of the present study is to investigate the ice failure mechanisms close to the contact between a moving icefield and a structure at relatively high indentation rates. A series of small scale indentation experiments has been conducted with the purpose to observe the contact mechanism between the ice edge and the moving indentor.

Chapter 2 describes the experimental program and chapter 3 presents the measured results and observed behaviour.

In chapter 4 a theoretical model is presented based on an iterative numerical procedure using the Mohr-Coloumb criteria. The parameters are evaluated using fracture mechanics and the model is compared to experimental data for one series of tests. Finally in chapter 5 the results from the experimental program and the theoretical analysis are reviewed.

EXPERIMENTAL PROGRAM

Test configuration

Small specimens of natural sea ice were compressed by a 270 kN servo-hydraulic MTS cylinder at constant indentation rate. All tests were performed in a cold room at -10°C . The load-cell was detached from the cylinder and placed underneath the specimen to reduce the noise induced by the moving cross-head. The displacement was registered by the stroke gauge inside the cylinder. The load and displacement signals was conditioned, filtered and converted by separate 16 bits A/D converters with a sampling frequency of 2 kHz. The indentor was fitted with a 12 mm Lexan window for observations of the contact between the ice specimen and the indentor. In some tests, the contact was filmed with a high speed camera via a mirror attached to the indentor. Both conventional high speed film and high speed b/w video was used to study the contact. Fig 2.1 shows the test configuration.

Test method

Two types of specimen geometry and indentor plates was used in the experiments, see fig 2.2 and fig 2.3

In the first two series of indentation experiments, A and B, the specimen had a pyramid shaped top to provide a stable on-going crushing behaviour. These specimens were loaded by a 12 mm Lexan plate supported on four sides. The third series of experiments, series C, was conducted on specimens with a wedge shape top, loaded with a two side supported 12 mm Lexan plate to obtain a two dimensional configuration.

In test series A and B the specimen was loaded in stroke control with a constant indentor speed of 100 mm/s. In series C three different indentor speeds were utilized: 1 mm/s,10 mm/s and 100 mm/s.

Specimens

Test series A was conducted on naturally grown freshwater columnar lake ice in Luleå. The ice thickness was typically 16 cm with large crystals in the order of 100 mm. Flooding occurred shortly before the ice was sampled. The ice blocks were stored some days at -10°C before the tests was performed.

Test series B was on ice from the same lake as series A but it had grown 26 days longer in mild temperatures. The ice consisted of 5 cm porous snow ice and 28 cm columnar ice. One ice block was photographed in vertical light with black background, appendix 1. Different types of vertical bubble-bands can be seen as well as horizontal planes of bubbles. The horizontal bubble-planes was located at the ice bottom when the ice surface was flooded.

Test series C was collected from the Luleå archipelago and consisted of brackish sea ice with a salinity of 0.1 ppt. The used bottom half of the collected ice specimens was columnar with an average crystal diameter of 50 mm.

The specimen was cut with a band-saw into two different geometries with different angles to the growing directions, see fig 2.4.

Test series

Table 2.1 summarizes the test data, specimen geometry and orientation with respect to growth direction for all tests. The specimen B3 had snow-ice on top of the specimen. B4 was cut 180° to B3 resulting in a bottom layer of snow ice. The snow ice layer was measured to approximately 50 mm.

Table 2.1. Test series A, B and C.

------------------------------------------------------------------------------ Test No Specimen geometry Orientation Indentor Speed series fig 2.2 fig 2.4 fig 2.3 [mm/s] [degrees] ------------------------------------------------------------------------------ A 1-12 Pyramid 90 Four side 100 B 1-2 Pyramid 90 Four side 100 3-4 Pyramid 0 Four side 100 5-6 Pyramid 45 Four side 100 C 11, 21, 31, Wedge 90 Two side 100 41 51 Wedge 0 Two side 100 12, 22, 32, Wedge 90 Two side 10 42 52 Wedge 0 Two side 10 13, 23, 33, Wedge 90 Two side 1 43 53 Wedge 0 Two side 1 ------------------------------------------------------------------------------

RESULTS

Load measurements

Recorded load-displacement curves for all tests are shown in appendix 2. Different sampling frequency was used in test series A-C. In test series A the sampling frequency was 2 kHz and in series B, 1 kHz. In series C three different sampling frequencies was utilized, 500 Hz for 100 mm/s, 160 Hz for10 mm/s and 20 Hz for tests with indentation rate of 1 mm/s.

Fig 3.1 shows the recorded peak loads, mean values, versus specimen orientation. A tendency for higher ice-load was noted when the specimens were oriented at 0°-45° to the growing direction, test B3-B6 and C51.

Fig 3.2 shows typical recordings of load versus indentor movement for series A-C with indentation rate of 100 mm/s. Although different type of ice, specimen shape and indentors was used in series A-C the resulting curves are similar. Note the difference in sampling frequency.

Fig 3.3 and 3.4 shows the result from test series C. Fig 3.3 shows the maximum load versus indentation velocity. Fig 3.4 shows the recorded load - indentor movement curves for test C11, C12 and C13 with indentation rate 100, 10 and 1 mm/s respectively.

From the recorded data and the observations made it is clear that the indentation rate effects the crushing behaviour in two ways:

* The peak load increases with decreasing velocity, especially between 1 and 10 mm/s.

* For the 1 mm/s indentation rate the load-displacement curve shows a marked saw-tooth pattern where the loading cycle is somewhat curved indicating creep at the contact. Also the fragments consisted of mainly larger pieces, flakes. At higher indentation rates smaller fragments breaks off resulting in a chaotic load-displacement behaviour.

Fracture toughness measurements

Fracture toughness measurements was made on specimens taken from the same iceblock as test series C. A field instrument based on chevron notched specimens developed by Stehn (1990), was used to determine the value of the apparent fracture toughness, KQ. (Since no standard exist for fracture toughness measurement on ice, the critical stress intensity factor, KIc, has been denoted apparent fracture toughness, KQ.) Table 3.1 shows measured KQ values along with density and salinity measures on four specimens. In specimen C1 and C2 the crack plane was made parallel to the growth direction. Specimens C3 and C4 was tested with the crack perpendicular to the direction of growth.

Table 3.1. Apparent fracture toughness, KQ, density and salinity s for test series C.

------------------------------------------------------------------------------ Specimen Density Salinity s Apparent fracture [kg/m3] [o/oo] toughness KQ [kPa(m)1/2] ------------------------------------------------------------------------------ C1 884 0.05 127 C2 887 0.11 111 C3 893 0.04 166 C4 894 0.20 72 Mean: 890 4.8 0.1 0.07 119 39 ------------------------------------------------------------------------------

The result shows a tendency for lower fracture toughness values as the salinity and thus the brine volume increases. These results coincide with the observations made by Stehn (1990).

Observations

The ice crushing can be characterized as fractal flaking until the specimen broke into two halves. A dark jiggling line was observed at the centre of the contact area. The observed line was about 1-2 mm thick and became stable parallel to the growing direction in test series A and B if the specimen were cut at 90° to the growing direction. If the specimens were parallel or at 45° direction to the growing direction the line sometimes rotated and became more like a cross or a triple-junction. In test series C the contact line was stable in the direction of the wedge, probably due to the deflection of the 2 side supported indentor. We felt that the dark area shown on the video recordings was a function of the measured load. Typical contact lines are shown in fig 3.5. Video print-outs of the contact line are shown in appendix 1.

Initially only small flakes formed and was transported almost perpendicular to the indentor direction. After approximately 20-30 mm, larger pieces of ice broke off and was transported in the in the crack surfaces direction. On several specimens a vertical crack was observed during and after the test was performed. Some test ended when the crack went through the specimen and split it in two halves, but usually the vertical crack stopped at a distance 5-10 cm from the top. In appendix 1 the picture from test serie A specimens shows vertical cracks on several specimens. Also the picture of test C33 in fig 3.6 shows the developing of a vertical and an inclined crack during indentation.

The maximum load occurred immediately before a large piece broke off. This was also used to synchronize the video recordings with the load measurements. The load dropped to less than half after a major crack been formed.

Considering the rough sample preparation technique and difference in crystal orientation and size the apperance of the load-displacement curves were surprisingly similar.

Pressure melting

One of the objectives of the experiments was to find evidence of pressure melting. Even though the contact area is difficult to measure, an attempt was made to estimate the dark area from the video-recorded experiments in series A.

Table 3.2 shows the estimated contact areas and contact pressures for two pressure distributions for test A1-A6. The assumed pressure distribution are evenly distributed and triangular distributed with the maximum pressure in the middle of the contact.

Table 3.2. Contact area and contact pressure estimates form series A experiments.

------------------------------------------------------------------------------ Specimen Maximum Indentatio Nominal Contact Nominal Contact load [kN] n [mm] area [mm2] area [mm2] pressure pressure [MPa] [MPa] ------------------------------------------------------------------------------ 1 4.08 25 832 71 4.9 58 (116)* 2 3.32 18 431 44 7.7 75 (150) 3 5.52 32 1380 80 4.0 69 (138) 4 3.48 27 967 50 3.6 70 (140) 5 3.34 33 1452 53 2.3 63 (126) 6 2.68 30 1218 57 2.2 47 (94) ------------------------------------------------------------------------------

* The values within paranthesis is the estimated contact pressure if a triangular distribution is assumed.

The estimated contact stresses is in the same order as the melting pressure at -10°C. Nordell (1989) studied the melting pressure of ice and gave a value of 117.8 MPa at -10°C for ice with a density of 943 kg/m³.

Although the estimated contact stress can only give the order of magnitude other evidence of pressure melting was noted:

* A sharp central wedge and a band of ice powder always stuck to the indentor after the test was finished and the indentor was removed from the specimen.

* In some experiments were the indentor was stopped and kept in a fixed position the load increased in time, probably due to volume expansion during refreezing of melted ice at the contact.

The last observation was studied in more detail in series C were the indentor was held in a fixed position a couple of seconds after the indentation test was completed. Fig 3.7 shows the load and displacement versus time at the end of the indentation test for three tests with identation rates of 1, 10 and 100 mm/s. Note that the load decreases for 1 mm/s while it increases for 10 and 100 mm/s. This result indicates that pressure melting is a function of indentation rate and below certain levels no pressure melting occurs.

ANALYSIS

Background

Mechanical models of the ice crushing process are often based on the assumption of a crushed layer between the intact ice sheet and the structure, Kurdyumov and Kheisin (1976), Kärnä and Turunen (1989), Singh et al (1990), Marcellus et al (1990). The intact ice is pulverized and this produces a zone of crushed debris which is extruded upwards and downwards. The extrusion process has been modelled by hydrodynamic lubrication theory, Jordaan and Timco (1988), Jordaan et al (1989) or as a Mohr-Coulomb friction material, Hallam and Pickering (1988).

In Joensuu and Riska (1989), Fransson et al (1990), a contact line was observed in laboratory ice crushing tests were the crushed material was free to extrude.The same contact line have been observed at a ship in the Baltic sea equipped with window in a study by Riska et al 1990. Based on these observations Joensuu and Riska proposed the contact line theory. Fig 4.1 shows the contact according to layer and line theory.

Daley (1990) proposed an iterative failure model based on the contact line. As the moving indentor hits the ice the material in direct contact is crushed giving a constant pressure over the contact area equivalent to the ice crushing "strength". The shear stress generated in the intact ice is compared with the shear strength for different angles of a shear plane running from the middle of the contact to the edges of the intact ice. If the ice fails the flake is removed and the geometry is updated before the next increment is applied. Daley used in his model Tresca and Mohr-Coulomb failure criteria. Despite the simplicity in the assumption made by Daley, the model produces complex load - displacement curves apparently similar to those found in laboratory indentation experiments.

In a recent paper by Jordaan et al (1990) the contact zone is divided into three zones; a zero pressure zone, a extrusion zone and zones of high pressures. In the zero pressure zone, also called spall area, where the pressure is negligible compared to the high pressure zones, local spalling can take place. In the "low" pressure zone ice particles are extruded between the intact ice and the structure. The high pressure zones are a collection of areas where the local pressure is very high, high enough for pressure melting to occur. In simulation of indentation experiments they found a parabolic pressure-area relationship where the peak pressures reached the level of pressure melting.

Contact model

In this study the result from Daley's numerical contact model are compared with the experimental result. A number of minor modifications in the interpretation of the "crushing strength" have been made. Daley's model has also been generalized to incorporate arbitary two dimensional polygon ice blocks hit by an indentor whose normal is inclined to the velocity vector.

Consider the moving ice-sheet in fig 4.2. A triangular pressure will be generated at the contact with a maximum pressure equal to and limited by the melting pressure, . A flake will form from the centre of contact to the edge if the driving shear force is greater than the shear resistance. Using Trescas simple shear failure criteria,

(1) = c

the flake angle can then be calculated as the angle of maximum shear stress, .

Maximizing the shear stress over the angle gives the flake angel as:

(2) Equation 2

The numerical procedure can be summarized in the following steps:

i) Move the indentor (or the polygonal ice block) a small distance. If indentor is in contact with the ice-front proceed to the next step.

ii) Remove "crushed" portion, (update the ice geometry) and determine the contact width (a) and calculate the total acting force as:

(3) Equation 3

where Acontact is the contact area ie, contact width (a) times width of the indentor (b).

iii) For each line i bounding polygonal ice block calculate the total gross shear stress for a possible flake with angle =/2 as:

(4) Equation 4

where Aflake is the flake area ie, flake length times the width of the indentor (b). The flake length can be calculated from the intersection between line i and the flake. If the flake starting at the centre of contact and running with an angle does not intersect line i, flaking on that edge is not possible.

iv) If the calculated shear stress exceeds the shear strength remove the flake and update the geometry. Then proceed to step i).

The numerical procedure is naturally dependent on the step size, ie, the distance the indentor moves for each step. To get a procedure as independent as possibly with regard to step size the following actions has been taken:

* The calculated load at a flake failure has been back-calculated from the failure condition, ie the maximum shear stress allowed is .

* If the maximum shear stress is reached in several possible flakes during one step the failure is taken at the flake with the highest shear stress.

However due to the simple algorithm the best way to ensure good numerical result is to set the step size to a small value.

Daley analyzed the numerical procedure above using chaos theory. The chaotic behaviour is controlled by the ratio of the shear strength to the maximum pressure:

(5) Equation 5

For k>0.5 no flaking is possible and the maximum force is governed by the crushing strength, ie the force determined by eq (3) with the contact width equal to the ice thickness. For 0.2071<k<0.5 flakes will form at 45°. The next level of flakes will form at angles 22.5° if 0.0995<k<0.2071 and so on. Fig 4.3 shows the contact model for different six different values of k , (after Daley 1990).

The Mohr-Coloumb failure criteria

The Mohr-Coloumb failure condition is often used for geological materials . In its simplest form the shear strength is dependent on a constant (cohesion) and on the internal friction times the normal compressive stress on the failure surface:

(6) Equation 6

The flake angle in this case can be determined by:

(7) Equation 7

With the Mohr-Coloumb criteria we can also in some respect control the formation of the flake angles. The Mohr-Coloumb criteria is often used in combination with a tension limiting condition, the so called tension cut-off limit. The normal compressive stress, , is calculated after the same principles as for the shear stress, ie

(8) Equation 8

Fracture mechanics consideration

The Tresca and Mohr-Coloumb formulas are both macroscopical failure criteria. The term cohesion and friction angle is often used in connection with Mohr-Coloumb material. The physical explanation of cohesion and friction angle is related to the internal glue and friction between grains in granular material like soil, sand etc. In recent years microscopical models has grown in importance. Fracture mechanics has been used successfully to explain crack propagation in brittle material such as glass, ceramics but also in plastic materials such as metals. Generally the stress intensity factor for an embedded crack in an infinite medium subjected to a uniform tensile stress,

, is given by:

(9) Equation 9

The term Q is a geometric factor dependent on the shape of the initial crack. For a penny shaped crack with diameter of 2 r, the shape factor is Q = 2/. Melville (1977) showed that the stress intensity factor for a multi-axial compressive state of stress can be expressed as:

(10) Equation 10

Replacing KII with KIIc and rewriting eq (10) gives:

(11) Equation 11

The "cohesion" c in eq (6) can be expressed in terms of fracture toughness as:

(12) Equation 12

The cohesion is a function of initial crack length and since the magnitude decreases with increasing crack length the maximum stress required to propagate the crack is at the onset of crack propagation. As the crack starts to propagate it will go right through the specimen if the load is held constant. If the displacement is held constant the load will drop until the stress intensity KII<KIIc.

Many experiments end when the specimen splits in half. Kendall (1978) studied this problem for glassy material. He concluded that the force required to propagate a vertical crack is given by:

(13) Equation 13

where b is the indentor width, h is the ice thickness, E the youngs modulus and G the critical energy release rate. In linear fracture mechanics giving:

(14) Equation 14

Using eq (3) the corresponding value of the splitting contact width can be evaluated as:

(15) Equation 15

Design load

The upper theoretical limit of the contact force is obtained for a (1:st level) crack initiated at the middle of a ice sheet and running to the edge at an angle of

. The maximum force can be calculated as:

(16) Equation 16

where h is the ice thickness, b is the width of the indentor and is the friction angle. If the flake angle =(90+ )/2, then eq(16) reduces to:

(17) Equation 17

Croasdale et al (1977) proposed a wedge shear failure criteria based on plastic limit theory and a Tresca material with the shear strength c = 0.5 of the uniaxial strength in crushing, :

(18) Equation 18

where is the nominal pressure Fmax /h b. The first term in Croasdales formula is equal to eq(16) if c is replaced with 0.5 . The second term is the contributions from the wedge edge surfaces.

If eq(17) is rewritten in terms of nominal pressure and uniaxial strength equal to 0.5c we get:

(19) Equation 19

For a friction angel = 0 the result is unity and adding the contribution by the wedge end surfaces the equation becomes:

(20) Equation 20

This equation is referred to as the Morgan-Nuttall formula, Ashton (1986).

Assessment of material parameters

In order to use the governing equations the following parameters has to be evaluated:

* Fracture toughness, KIc and KIIc

* Maximum contact pressure,

* The shape and size of flaws, Q and r

* The internal friction

In mixed mode linear fracture mechanics analysis the mode I condition KIc=KI is replaced by:

(21) Equation 21

for crack propagation, where = 1+in plane stress condition. The two equations for flaking and splitting involves single mode fracture mechanics, mode I and mode II respectively. Using eq (21) the fracture toughness for mode I, KIc, can be used for KIIc in eq (12).

An approximate value of the melting pressure as a function of temperature is obtained by the equation:

(22) Equation 22

A temperature of T= -10° C gives a melting pressure of 100 MPa. However, due to creep and micro-cracking in the vicinity of the contact the maximum pressure may not reach the melting point, especially when the indentation is low. The experiments with 1 mm/s indentation rate showed no signs of pressure melting.

Time-dependent deformations are generally categorized by a delayed elastic part and viscous creep. Based on data from creep test in the range from 0.1 to 10 MPa, Sanderson (1988) used the power law, Glen (1955), Michel (1977), to fit the viscous creep rate to different types of ice:

(23) Equation 23

where C is a constant, U is the activation energy, R is the universal gas constant and T is the temperature in degrees Kelvin. is the stress and the corresponding strain rate in the viscous creep process. Making the crude assumption that all deformation is viscous creep the maximum stress at a certain strain rate can be calculated as:

(24) Equation 24

The minimum value of eq(22) and eq (24) gives the maximum contact pressure. In fig 4.4 the contact pressure is plotted versus strain rate for different temperatures with the melting pressure as limiting condition. The values used in eq (24) are those which apply for columnar ice, Sanderson (1988).

For the tests conducted in this study, the calculated maximum contact pressure are:

* = 21 Mpa for indentation rate 1 mm/s

* = 46 Mpa for indentation rate 10 mm/s

* = 99 MPa for indentation rate 100 mm/s

The calculated strain rates are defined by indentation rate / specimen length.

Although the use of eq (24) is questionable, it gives reasonable pressure levels despite the fact that the power law been extrapolated to higher strain rates and stress levels than those to which it is strictly applicable.

The denominator in eq (12), which involves the shape and size of the "initial" flaw, can be thought of as a damage coefficient that express inhomogenities such as:

* air bubbles

* grain boundaries

* existing microcracks in ice crystals

It is also plausible that the high contact pressure will generate a zone of nucleated microcracks in the vicinity of the contact and thus increasing the "effective" flaw size. Based on observation from Cole (1986), Sanderson suggested a length of nucleated cracks of 0.65 times the grain diameter. The observation was based on polycrystalline ice with mean grain diameter ranging from 2 to 6 mm. For the specimens in series C where the grain diameter was considerably larger, approximately 50 mm, the nucleated crack diameter would be in the order of 30 mm.

The internal friction angle has been evaluated from direct shear test by Roggensack (1975), in Ashby (1986). The friction angle was determined to approximatly 25°. Daley (1990) used the value =11° which he evaluated from multi-axial experiments made on ice by Riska and Frederking.

Comparison with experiments in series C

Three tests in series C were used for a qualitative comparison with simulated results. The main objective here was to determine the qualitative nature of:

* the force - displacement/time curves

* the influence of indentation velocity

Table 4.1 shows the model parameters for the simulations of test C41, C12 and C33. The values for the cohesion, c in the Mohr - Coulomb criteria was selected to match the maximum peak in the corresponding test. The value on was choosen to match the time/displacement history.

The step-size used in the simulation is equal to the sampling rate used in the experiments, except for test C41. With the given values of the parameters it was found that the load- displacement curve changed significantly when the displacement step size used in the simulation was reduced. The simulated curve therefore for test C41 shows mean values based on 4 points from a simulation with 2 kHz sampling rate.

Table 4.1. Model simulation parameters

------------------------------------------------------------------------------
 Simulated    Indentation   Cohesion c    Friction      Maximum       Ratio         
    test      rate (mm/s)   (MPa)         angle         contact       k=
                                         (degrees)      pressure     2c/mcp*
                                                        (MPa)         
------------------------------------------------------------------------------
    C41       100           0.55          11            60            0.018         
    C12       10            0.625         11            30            0.042         
    C33       1             1.3           11            17            0.153         
------------------------------------------------------------------------------

* mcp=

The simulation of the three tests are compared with experiments in fig 4.5, 4.6 and 4.7. Note that after approximatly 30 mm test C12 and C33 split in two halves.

For a friction angle of 11° the critical values of k governing the possible failure angles , flake levels Daley (1990), are:

* Level 1: k<0.4122, = 50.5

* Level 2: k<0.1508, = 30.75

* Level 3: k<0.0586, = 20.875

* Level 4: k<0.0212, = 15.938

The flake levels for the simulated tests are level 1 (close to 2) for test C41, up to level 3 for test C12 and above 4 levels for test C33.

In table 4.2 and 4.3 the simulation parameters are compared with the suggested values from chapter 4.3. The values for the flaw diameter are obtained from eq(12) using KQ for KIIc with two shape factors, through crack and penny-shaped. The force and contact width necessary to initiate and propagate the vertical crack have been evaluated using eq(13) and eq(15).

Table 4.2. Simulation parameters and evaluated crack size, split width and split force.

------------------------------------------------------------------------------
Indentation     Maximum         Cohesion    Crack diameter     Split     Split     
   rate         contact         c  (MPa)       (mm)            width     force     
  (mm/s)        pressure                    Q=1   Q=2/pi       asplit    Fsplit    
                (MPa)                                          (mm)      (kN)     
------------------------------------------------------------------------------
    1           17               1.3         5    12           3.17      1.83        
    10          30               0.625       23   57           1.76      1.83        
   100          60               0.55        30   74           0.87      1.83        
------------------------------------------------------------------------------

The maximum pressure used in the simulation is lower than the values predicted by eq(24), especially for the 100 mm/s indentation rate, (99 MPa to 60 MPa). However, the general trend is the same. If the crack diameter is assumed to be 0.65 of the average grain diameter the cohesion evaluated from eq(12) is 0.55 MPa for a through crack shape and 0.86 MPa for a penny shaped crack.

The sampling rate in the 100 mm/s experiments was set too low to pick-up the high frequency crushing. Also the load-cell and the conditioning unit filters the signal so the highest peaks might be obscurred. In the fitting procedure, direct graphically comparison was used. The result are not complete since only one value of the friction angle was used.

Influence of flexible indentor plate

The flexible indentor plate (12 mm Lexan) used in these experiments did not remain flat during loading. Based on thin plate theory the central deflections for the four side supported indentor plate can be evaluated as:

(25) Equation 25

where F is the point-load acting in the middle of the plate, b is the length of the sides and D the plate stiffness.

The two side supported plate is loaded with a line load and the central deflection is given by simple beam theory as:

(26) Equation 26

where F is the total load in N and b the span between the support and EI the beam stiffness.

The calculated deflection for the Lexan plate is 0.3 mm/kN for the 4-sided supported and 0.6 mm/kN for the 2-sided supported indentor. A typical load of 5 kN would give a deflection of 3 mm in the case of the two sided supported Lexan plate.

It can be assumed that the concavity of the indentor during load was a pre-requisite condition for continuously crushing without causing the specimen to slip off the indentor. In later test we observed that the shape of the contact line was similar to yield lines of a deformed plate. In test series C with the two-sided supported indentor plate, the contact line was straight and parallel to the supported sides. The conclusion is that the flexibility of the indentor will indeed effect the contact line pattern and thus the crushing behaviour.

DISCUSSION AND CONCLUSIONS

The contact model proposed by Daley have serveral merits. It is simpel to use. The model input can be physically linked to ice properties. The model output can be tested against recorded data and observations. Furthermore the recognition of the ratio of pressure build-up to ice strength provides new insights to the complex behaviour of ice crushing.

Although only simple linear fracture mechanics been used and crude estimates of the variations in maximum pressures with loading rate, the results from the comparisons are in reasonably agreement with respect to the general outline of the load-displacement curves as wells as predicted crack length and maximum pressures. The role of microcracking in the contact zone is ignored in the sense that no explicit formulation of damage has been included in the model. We could not determine the size and the location of microcracks in our present experimental set-up. Therefore the effect on the pressure level was modelled as viscous creep. When new observations on the microfracturing process in the vicinity of the contact zone has been made a less rigid model can be formulated. The distinction between microcracking, flaking and pressure melting is an area of future research.

The predicted force for vertical crack propagation, eq (14), may at first appear to be low. However, in many experiments a vertical crack was observed, but propably due to end friction the crack was restrained at the end of the specimen. Kendall predicted only a vertical crack propagation with a constant velocity. Vertical cracks has been observed in field and laboratory indentations experiments. It appears that the vertical crack does not set the upper limit for the load a moving ice sheet can generate, but it may be a pre-requisite for the flaking mechanism to occur.

To be of use for design purposes the model must be extended to include:

* flaking in three dimensions

* pressure distribution along the contact line

* extrusion

For now the 2-D contact model gives an upper limit equal to the Croasdale / Morgan-Nuttall formulas.

Finally, the use of fracture mechanics may prove to be fruitful. Other conditions has to be investigated in relation with ice crushing. Ashby and Hallam (1986) investigated crack propagation for inclined cracks in compression and gave formulas based on mode I fracture toughness. The nucleation of cracks is another equally important issue to determine, since most propagation formulas include the crack sizes and distribution. The linkage to fracture mechanics gives the possibility to make simple field tests to determine design loads on structures.

REFERENCES

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APPENDIX 1 : PHOTOGRAPHS

APPENDIX 2 : TEST RESULTS, SERIES A-C