Energy exchanges during indentation tests. in fresh-water ice

Transcription

1 Annals of Glaciology < nternational Glaciological Society nergy exchanges dring indentation tests in fresh-water ice DVNDR S. SODH U.S. Army Cold Regions Research and ngineering Laboratory, 7 Lyme Road, Hanover, NH 3755, U.S.A. ABSTRACT. The data from a small-scale experimental stdy on ice-strctre interaction are sed to compte the energy exchanges that take place dring creep deformation and intermittent and continos crshing of ice. The energy spplied by the carriage is partiy stored in the strctral spring, partly converted to kinetic energy, partly dissipated in deforming and extrding the ice and partly dissipated as heat in the damping mechanisms of the strctre. xcept for the heat dissipation, all other forms of energy were compted from the experimental data, and the heat dissipation was compted from the energy balance sing the first law of thermodynamics. Plots of all forms of energy are shown in graphical form, in which their relative magnitdes, times of occrrence and interplay can be seen. The main reslt of this stdy is the thesis that intermittent crshing or ice-indced vibration takes place whenever there is an imbalance between the rates of work done by the carriage and the indentor and that there are no vibrations when these rates of work are eqal. NTRODUCTON Crshing is a common mode of ice failre dring interaction between a moving ice sheet and a strctre. Under certain conditions, the interaction leads to strctral vibrations. Peyton (1968) and Blenkarn (197) reported on ice-indced vibration of steel strctres in Cook nlet, Alaska. Recently, Jefferies and Wright (1988) reported on the vibration of Molikpaq, a 11 m wide strctre, as a reslt of interaction with a mlti-year seaice floe. Many small-scale experimental stdies have been condcted to increase nderstanding of the manner in which sch an interaction takes place. For extended discssions on this sbject, the reader is referred to review papers by Sodhi (1988) and, for a slightly different point of view, MaaWinen (1988). n a recent paper, Sodhi (in press) reported on indentation tests that were condcted in a laboratory test basin by pshing vertical, flat indentors into the edges of floating, fresh-water ice sheets to stdy dynamic icestrctre interaction dring crshing failre of ice. The experimental data of that stdy are sed here to compte the energy exchanges taking place between the strctre and the ice. A brief description of the experimental set-p and the main reslts of the stdy are discssed first, followed by the reslts of the present analysis of the data. XPRMNTAL ST-UP The strctre sed in the stdy consisted of a mass and a spring (Fig. la) to simlate a single degree-of-freedom system. Two plates were attached to each other with for hinged links and two brackets with a spring between them. The top plate was monted nder the main carriage that spans the test basin. On the bottom plate were monted the spports for the indentor plate and abot kg mass of lead weights. The spring consisted of two alminm bars attached to each other by a nmber of rods. By removing or installing the rods in the middle of the spring, it was possible to change the nspported span of the alminm bars, ths changing the stiffness of the spring. ndentor plates of different widths were spported on three load cells so that the tre interaction force cold be measred, as opposed to interpretation of the strctral response as that force. Besides measrement of the interaction forces, the displacements of the carriage and of the indentor were measred separately with respect to a fixed datm. Frther, the relative displacement of the indentor with respect to the carriage was measred independently. Hereafter, the relative displacement will be designated as strctre displacement becase it is eqal to the deformation of the spring. An accelerometer was also placed on the lower plate to measre its acceleration dring the interaction. These measrements provided a complete set of data on the interaction forces and the response of the strctre. All experiments were done with fresh-water, colmnar ice, which was grown in the basin after seeding with a fine mist of water sprayed in the air. ce sheets of different thicknesses ( mm) were grown. For a particlar vale of spring stiffness, experiments were done by moving 7

2 Displacement Transdcer Main Carriage ndentor Carriage ce Sheet / j--r---1- Fresh Water ndentor Spports Load Cell a ndentor Plate Test Basi n System Bondary r , ndentor Carriage :;:.,.,.,J-oo--- -;.; m.. X.- b X r ---- L J Fig.. Schematic sketch of the apparats: (a) experimental set-p, (b) system and its elements. The explanation for the different symbols is given in the text. the carriage at three different velocities. For tests condcted at high velocities, the carriage velocity was changed in discrete steps withot stopping an experiment. Dring the corse of the experiments, the following parameters were varied: ice thickness, indentor width, carriage velocity and spring stiffness. Frther details on these tests are discssed by Sodhi (in press). RSULTS Three modes of ice behavior were observed dring the stdy: creep deformation at low indentation velocities, intermittent crshing at intermediate indentation velo- 8 cities and continos crshing at high indentation velocities. At times, the strctre vibrated at its natral freqency, a sitation referred to as resonance. Typical records of the experimental data for the for different interactions are shown in Figres -6. Figre shows part of the data in Figre 3 at an expanded time scale. ach figre presents time records of interaction force (F i ), displacement of the carriage (xc) and the indentor (xi)' and strctre displacement (x r ) of the indentor relative to the carriage. t can be seen that the compatibility condition of xr = Xc - Xi is satisfied, within the error bonds of the displacement trandcers, at all times. The force records can be interpreted in terms of

3 1 o 5 Test No Max. Force = kn C 3 n. Cl 1.!? >- Cl Qj c -1 LJ.J - z Velocity =.6 mm/s Fig.. Reslts of a test in which creep deformation of ice took place. effective pressre by dividing the force scale by the contact area, and the scale on the righthandside reflects the pressre scale for a particlar test. The maximm effective pressre was in the range 8-13 MPa dring the creep deformation and intermittent crshing modes, whereas it was very low, between 1 and MPa, dring the continos crshing mode. To obtain force-displacement graphs, the corresponding vales of force and displacement are paired for each scan of the data. On the bottom of each figre, three forcedisplacement plots are shown for the carriage, the indentor and the strctre. The force vales in these three plots at a given instant are assmed to be the same becase the carriage, the indentor and the strctral spring experience more or less the same force, as discssed later. The areas nder these three plots represent the cmlative energy transferred or stored; these are also shown in Figres -6 with respect to time. The comptational procedre for obtaining these plots is discssed below. The apparats (Fig. b) is considered as a thermodynamic system. Any heat transferred to the system is considered positive and any work done by the system is considered positive. Dring a process between two states, 8 6 t a.. :::l :: > ""B W 7 z ::s e 1 Test No. 3 M ax. F orce = 8 kn C Cl 85 n J 6 >- e> c LJ.J z ::s - -, l Vel. = 19. mm/s e 1 8 :::l... Cl a.. a.. > T W o Fig. 3. Reslts of a test in which ict-strctre interaction reslted in intermittent crshing. n the plots of displacement time-histories, the relative displacement of the strctre with respect to the carriage has been denoted as strctre displacement and plotted with respect to the origin in each part of the figre. Similarly, in the force-displacement plots, displacements of the carriage and the indentor have been translated to the origin shown in the bottom part of the figre. the energy exchange satisfies the following eqation according to the first law of thermodynamics Q. = (P) + (K) - Wc + Wi' (1) where Q. is the incremental heat transferred to the system, :!. (P) the incremental change in its potential energy, (K) the incremental change in kinetic energy of the system, and :!.Wc and :!.Wi the incremental work done by the carriage and the indentor, respectively. The comptation of Wc and W i between any two instants of time can be accomplished by mltiplying the average force by the corresponding incremental displacements of the carriage and the indentor, respectively. The incremental change in kinetic energy is compted from :!. (O.S m ; i ), where m is the mass of the vibrating system, and Xi the indentor velocity obtained by differentiating the displacement record. The incremental potential energy stored in the strctre is given by :!. (O.S Kx r } where K is 9

4 3 Test No Max. Foree = 11.3 kn Test No. 11 & 1 o (.) co Ci CS i;; o-o '--t'----io Vel. = 17. mm/s OJ iii:i:i::::q_----t tij O o 3 Fig.. Detailed plot of reslts shown in Figre 3. n the plots of displacement time-histories, the relative displacement of the strctre with respect to the carriage has been denoted as strctre displacement and plotted with respect to the origin of the figre. Similarly, n the force-displacement plots, displacements of the carriage and the indentor have been translated to the origin shown in the bottom part of the figre s 81 c 785 (.) : : t Vel. = 11. mm/s 76=======t====== CS "f.!j.wc O.-:...! =_t.:;...-_=+...::...L o Fig. 5. Reslts of a test in which the strctre vibrated at its natral freqency. n the plots of displacement time-histories, the relative displacement of the strctre with respect to the carriage has been denoted as strctre displacement and plotted with respect to the origin of the figre. Similarly, in the force-displacement plots, displacement of the carriage and the indentor have been translated to the origin shown in the bottom part of the figre. the strctral stiffness. The stiffness was determined from the slope of the graph between the interaction force and the strctral displacement, dring the phase when the interaction force increased and the vibrations were minimal. When there is no vibration taking place dring a process, we have A(P) -A Wc + A W i =, wpich means that the incremental increase in the potential energy of the system is eqal to the incremental work done on the system, or vice versa. This eqation can be rewritten for a conservative system as shown below: where Fc and F i, respectively, are the forces exerted by the carriage and the indentor. Dring continos crshing and steady-state creep deformation of ice, the strctre () displacement xr is constant (i.e. AX r = ), and we get Fc = Fi becase the rates of the carriage and indentor displacement are eqal. Dring the initial stages of creep deformation (e.g. Fig. for t<5s), we get Fe = KX r and Fc = Fi by sing the compatibility condition (i.e. Axc = AXi + Ax r ) and the condition that two of three displacements are independent of each other. Dring a process when there are strctral vibrations taking place, qation does not hold, and AQ. can be compted from qation as given below: qation 3 gives an estimate of the energy dissipation in the frictiomil and damping mechanisms of the strctre. n qation 3, all the variables were measred dring the experiments except the force, F c ' exerted by the carriage. 5

5 Z :::s 3 Test No. 6 Max. Force = 6.1 kn C1l a... a... > 1 8 C 7 () 6 C1l a. (/) J 6 >- c w Vel. = 177. mm/s P " UQ z :::s 1 :::J (/) (/) U W o o Displacement Fig. 6. Reslts of a test which continos ice crshing took place. n the plots of displacement time-histories, the relative displacement of the strctre with respect to the carriage has been denoted as strctre displacement and plotted with respect to the origin ofthefigre. Similarly, in theforce-displacement plots, displacements of the carriage and the indentor have been translated to the origin shown in the bottom part of the figre. nvoking the eqilibrim of the indentor and the carriage, according to the free-body diagrams shown in Figre 1 b, we get Fc = Fi + mxi. By sbstitting Fc in qation 3, LlQ. was compted sing the acceleration of the indentor measred dring the experiments. Plots of the cmlative work done by the carriage (l Wc) and the indentor (l Wi), the potential energy stored in the strctral spring (P), the kinetic energy (K) of the system and the energy dissipated in the form of heat (l are shown in Figres -6. DSCUSSON n the plots of cmlative energy shown in Figres -6, it can be seen that most of the energy spplied by the carriage is dissipated by the indentor into the deformation, breakage and extrsion of ice in front of the indentor. Different types of ictrctre interaction and energy exchanges occr at different indentation velocities. t is sefl to introdce a concept of generalized impedance of the ice at the indentor interface as the ability of the ice to move at a velocity nder the action ofa force. n this way, we can look at the transmission of energy from the carriage to the ice throgh the strctre. Whenever the velocity of the indentor is eqal to that of the carriage (e.g. steady-state creep or continos crshing), this sitation is characterized as impedance matching becase all the energy is transmitted withot any change in the stored potential energy of the strctre. When the indentor velocity is different from that of the carriage, all the energy is not transmitted from the carriage to the ice, and part of it is stored in the strctre and released later when the indentor is able to move throgh the ice after ice failre. The sitation described here is similar to that in testing machines (Sinha and Frederking, 1979), where the strain rate measred by an extensometer placed on an ice specimen is less than that calclated from the cross-head speed. The stiffness of a testing machine plays an important part in the rate of loading a specimen. Dring creep deformation of ice at low indentation velocities «1 mm S-), the interaction force increased first and then decreased to a constant vale (Fig. ). n this case, the kinetic energy of the system was negligible de to small magnitdes of velocity. Dring the initial interaction with ndamaged ice, the indentor lagged behind the carriage, cased extensive micro-cracking in the ice, and then attained a constant displacement rate behind the carriage. The potential energy was stored in the strctre dring the initial interaction period, and it was later released ntil a steady-state condition of constant energy dissipation by the indentor, eqal to that spplied by the carriage, was reached. Dring intermittent crshing at the intermediate indentation rate (Figs. 3-5), the interaction takes place in a cyclical manner, which is also discssed by Sodhi and Nakazawa (199). n each cycle, the interaction force increases, casing a relative displacement between the carriage and the indentor and a storage of potential energy in the strctre. When the force reaches a certain vale, the ice fails, reslting in a sdden forward motion of the indentor. The stored strctral energy is sed to move the indentor forward and to extrde the crshed ice ot of the path of the indentor. Dring this time, as shown by the plots of energy in Figre, there is an exchange of energy between the stored potential energy, the kinetic energy of the system and the work done by the indentor, whereas the work done by the carriage Ll Wc dring that time interval is small. Sometimes, the indentor travels backward. Dring these periods, the interaction force is zero and, hence, the work done by the indentor is also zero. A new cycle begins with increase of interaction force and storage of energy in the strctre (as shown in Figs. 3 and ). From the plots of cmlative energy, the portions of energy transformed or exchanged into different forms can be established at any instant in time. For example, the energy plots in Figre indicate that jst before the ice failre, 78% and % of the energy spplied by the carriage is stored in the spring and dissipated by the indentor, respectively. At the end of an intermittent crshing event (Fig. ), the portions of energy spplied by the carriage are 95% dissipated into the ice by the indentor and 5% dissipated in the damping mechanisms of the strctres. 5 1

6 Sodhi: ndentation tests in fresh-water ice When the strctre vibrates at its natral freqency (Fig. 5), the increase and decrease in the potential energy and the kinetic energy of the strctre approximately follow a sinsoidal form. The overall effect is that the major portion of the energy spplied by the carriage is dissipated in the ice. Lastly, the balance of energy (Ld is the heat dissipated in overcoming the frictional and damping force of the strctre, and its plot increases with time when there are strctral vibrations. From data given in Figre 5, the damping factor (defined as the ratio of eqivalent viscos damping coefficient to the critical damping coefficient) can be calclated from the estimation of energy loss (LdQJ per cycle. The damping factor was fond to be.1, which is close to the vale of.1 obtained from the ratio of any two consective amplitdes of the system when it was displaced from its static eqilibrim position and released with zero initial velocity. At high indentation velocities, the interaction switches sddenly from intermittent crshing to continos crshing, as shown in Figre 6. The stored potential energy in the strctre remains constant, whereas the rate of energy spplied by the carriage eqals that dissipated by the indentor. The srprising reslt of continos crshing is the low effective pressre generated dring the interaction. There is a need to investigate the reason for this low vale of effective pressre. Among the plots of energy shown in Figres -6, the rates of energy transfer or exchange are highly variable becase of different rates and modes of ice failre. Similar energy comptations have been given by Nakazawa and Sodhi (199), bt these were done only dring the period of increasing interaction forces. J ordaan and Timco (1988) estimated the different energy components dring the interaction. Becase the displacements of the strctre and the indentor were not measred independently, they dedced the strctre's displacement from its stiffness. They limited their analysis to one single failre event. Moreover they theorized that the increase in interaction force takes place dring the extrsion process, which was not fond to be the case in the observations made dring the experimental stdy described above. n Figres -6, the cmlative work done by the indentor (Ldwi) was fond to be a non-decreasing fnction. This signifies that there is always energy dissipation into the ice and no transfer of energy from the ice to the strctre. This rles ot any possibility of ice-indced vibration as a reslt of negative damping, as proposed by Blenkarn (197) and Maattanen (1988). The above discssion pertains to the reslts of experiments done in this stdy, in which the ratio of the strctral stiffness to the effective stiffness of ice was small (.5 to.1). n the case of rigid strctres, the stored potential energy in the ice sheet may be greater than that in the strctre. New sets of experiments are needed to stcly this type of ice-strctre interaction. CONCLUSON nergy exchanges dring ice-strctre interaction are compted sing the data obtained from a small-scale experimental stdy. On the whole, most of the energy 5 spplied by the carriage was dissipated by the indentor in deforming and/or extrding the ice, bt the rate of energy inpt to the strctral system does not always eqal that of the otpt. Whenever the ice is not able to move ot of the way of the indentor at the same rate as that of the carriage, there will be relative displacement of the indentor with respect to the carriage, reslting in storage of energy in the strctre. The stored energy is later released after ice failre, when the ice is able to move nder redced levels of force, and exchange into kinetic energy and work done in deforming and extrding ice. This imbalance between the rates of work done by the carriage and the indentor reslts in ice-indced vibrations. Dring steady-state processes of continos creep and crshing, the rate of the energy spplied by the carriage eqals that dissipated by the indentor, reslting in no strctral vibrations. RFRNCS Blenkarn, K.A Measrement and analysis of ice forces on Cook nlet strctres. Second Annal Offshore Technology Conference, 197 Proceedings. Hoston, Texas. Vol.. Hoston, TX, Offshore Technology Conference, (OTC 161). Jefferies, M.G. and W.H. Wright Dynamic response of "Molikpaq" to ice-strctre interaction. n Sodhi, D.S., C.H. Lk, and N.K. Sin ha, eds. OMA 1988 Hoston. Proceedings of the Seventh nternational Conference on Offshore Mechanics and Arctic ngineering presented at... Hoston, Texas, Febrary 7-1, Vol.. Arctic engineering and technology. New York, American Society of Mechanical ngineers, 1-. Jordaan,.]. and G.W. Timco Dynamics of the icecrshing process. J. Glaciol., 3( 118), Maattanen, M ce-indced vibrations of strctres - self-excitation. n Sacki, H. and K.-i. Hirayama, eds. Proceedings, the 9th nternational Symposim on ce, 3-7 Agst 1988, Sapporo, Japan. Vol.. Delft, nternational Association for Hydralic Research. Committee on ce Problems, Nakazawa, N. and D.S. Sodhi ce forces on flat, vertical indentors pshed throgh floating ice sheets. CRRL Spec. Rep Peyton, H.R Sea ice forces. ce pressres against strctres. Ottawa, National Research Concil, (Technical Memorandm 9.) Sinha, N.K. and R.M.W. Frederking ffect of test system stiffness on strength of ice. POAC 79. The Fifth nternational Conference on Port and Ocean ngineering Under Arctic Conditions at the Norwegian nstitte of Technology, Agst 13-18, Proceedings. Vol. 1. Trondheim, University of Trondheim. Norwegian nstitte of Technology, Sodhi, D.S ce-indced vibrations of strctres. n Sacki, H. and K.-i. Hirayama, eds. Proceedings, the 9th nternational Symposim on ce, 3-7 Agst 1988, Sapporo, Japan. Vol.. Delft, nternational Association for Hydralic Research. Committee on ce Problems, Sodhi, D.S. n press. ce-strctre interaction dring indentation tests. n Jones, S.]., R. McKenna, ]. Tillotson, and.]. Jordaan, eds. UTAM/AHR Sympo-

7 sim on ce- nteraction, St. John's, Newfondland, Canada, Agst, 1-17, Berlin, etc., Springer-Verlag. Sodhi, D.S. and N. Nakazawa Freqency of intermittent ice crshing dring indentation tests. AHR 9. Proceedings of the 1th nternational Symposim on ce, Agst -3, 199, spoo, Finland. Vol. 3. spoo, Helsinki University of Technology, The accracy of references in the text and in this list is the responsibility of the athor, to whom qeries shold be addressed. 53