High Stalk Bubble Height Relative to Frost Line in HDPE Drawdown Splitting

Drawn-tube high-stalk extrusion of high-density polyethylene is operated with the frost line deliberately displaced far above the die exit, often by a distance equivalent to 6 to 8 die diameters, so that the molten tube remains an unexpanded or minimally necked stalk through the entire accelerating uniaxial extension zone. In this configuration the bubble height is not a secondary aesthetic variable: it defines the time available for elongational strain accumulation before radial expansion and the corresponding cooling-induced crystallization arrest the molecular network. The relationship between high stalk bubble height and frost line position becomes critical in HDPE drawdown splitting because the solidification front separates two fundamentally different mechanical regimes—molten strain-hardening polymer below the frost line and semicrystalline oriented film above it. A stalk that is too short relative to the frost line produces a partially expanded liquid bubble, which quenches with high transverse orientation and reduced axial stiffness; a stalk that is too tall relative to the frost line generates excessive uniaxial stress at the solidification front, causing longitudinal splitting of the partially solidified web and periodic gauge-band failures. Industrial high-molecular-weight HDPE film grades are typically selected with a melt flow rate between 0.05 and 0.15 g/10 min at 190 °C and 5 kg by ISO 1133-1:2022, and with density between 0.947 and 0.959 g/cm³ by ISO 1183-1:2019, because this combination provides the strain hardening required to sustain a tall stalk without premature rupture. The subsequent sections examine the process windows, equipment limits, crystallization dynamics, orientation balances, and standards-based test correlations that govern the maximum bubble height relative to the frost line before drawdown splitting occurs.

Elongational viscosity of HMW-HDPE in the stalk is not constant; it increases with Hencky strain at high rates due to extensional strain hardening arising from the bimodal molar mass distribution and the presence of high-molar-mass chains that form a transient physical network. The stalk process exploits this nonlinear response because the film bubble is not laterally supported by internal air until the melt reaches the frost line. At the die exit, the shear history in the spiral mandrel imposes a molecular orientation that relaxes only partially; the subsequent uniaxial draw stretches the high-molar-mass fraction and produces a network stress that resists further necking. The draw force required to sustain a tall stalk is transmitted through the partially solidified frost line to the nip. When the frost line is positioned above the expansion zone, the load-bearing section is only a few micrometers thick at the transition, and the local true stress can approach the tensile strength of the semicrystalline web if the stalk height is increased beyond the stable limit. Published data for exact rupture stresses of high-stalk HDPE webs at the frost line under production cooling rates are limited; commercial practice therefore relies on stalk height-to-die diameter ratios as a conservative surrogate.

The production-scale configuration that generates this condition typically consists of a grooved-feed extruder of 75 to 120 mm barrel diameter, a spiral mandrel die of 150 to 350 mm diameter with a die gap between 0.8 and 1.2 mm, a dual-lip air ring, internal bubble cooling, a stabilisation cage, collapsing frames, and a variable-speed nip. In this configuration the high-stalk bubble is normally run with a blow-up ratio of 3:1 to 5:1, and the stalk height is measured from the die lip to the frost line with an infrared pyrometer or manual visual reference. The frost line is not controlled by a single setting; it shifts with extruder output, melt temperature, cooling-air velocity, ambient temperature, internal bubble pressure, and film thickness. When the stalk height is raised without a corresponding increase in melt temperature or reduction in haul-off speed, the tensile force per unit width increases and the web can exhibit longitudinal splitting after leaving the nip or during winding. This limiting behaviour is often encountered on high-output lines producing thin HDPE film below 20 µm, where the drawdown ratio is high and the frost line is held far above the die to force full uniaxial orientation before expansion.

The high-stalk bubble is conventionally operated with a small neck at the die exit where the molten tube is drawn down to a diameter that is slightly smaller than the die diameter before entering the constant-diameter stalk. This neck is the first extensional deformation zone and its shape is determined by the draw-down speed and the die swell of the HDPE melt. A sharp neck concentrates stress and reduces the critical stalk height, while a long gradual neck distributes strain over a greater distance and permits a taller stalk. The neck shape is controlled by the die land length, the die gap, the melt temperature, and the external cooling air directed at the tube. Processors routinely adjust the lower air ring to modify the neck and stalk; if the lower lip is closed too aggressively, the stalk surface is quenched before the interior is oriented, leading to a brittle skin that fails at the frost line.

What Limits High-Stalk Bubble Height Before Melt Splitting Propagates?

The primary limiting condition is the attainment of a critical uniaxial tensile stress at the frost line. The stress arises from the haul-off force, the weight of the molten stalk, the internal bubble pressure, and the viscoelastic stress retained from the die. In a stable high-stalk operation, the force balance at the frost line is dominated by the haul-off force, which is transmitted downward through the quenched film into the partially molten transition zone. The frost line temperature for HDPE is recorded by infrared pyrometers at the outer surface as 110 to 125 °C; the core may remain above 130 °C even when the outer surface appears opaque. This thermal gradient means that the apparent frost line is not a plane of uniform solidification but a steep surface temperature jump. The stalk can sustain the imposed force only if the extensional viscosity of the melt remains higher than the stress divided by the extension rate. At high stalk heights, the extension rate in the lower stalk may be increased beyond the strain-hardening range, and the stalk develops a visible periodic necking that precedes splitting. Draw resonance, bubble breathing, and frost line oscillation are early indicators of this condition. The probability of splitting increases when the stalk height exceeds 8 die diameters for an unmodified bimodal HMW-HDPE at a melt temperature below 220 °C, but the actual limit shifts with die gap, blow-up ratio, and film thickness. Published data for exact splitting thresholds in specific die configurations are limited; the 6 to 8 die diameter range is therefore an empirically accepted operating window rather than a certified universal limit.

Bubble splitting should also be distinguished from melt fracture at the die lip. Melt fracture appears as a rough surface on the stalk below the frost line, and the resulting surface defects act as stress concentrators when the web is quenched. High-stalk splitting, by contrast, initiates at the frost line or just above it as a vertical crack that propagates toward the collapsing frame and the nip. The crack path is aligned with the machine direction because the frozen-in molecular orientation in the stalk produces a transverse tear resistance that is frequently 50 % to 70 % lower than the machine-direction tear resistance. This imbalance is acceptable for many high-density film applications, but it becomes a failure mode when downstream drawdown, printing, or slitting applies transverse tensile forces. The specific equipment type influences the splitting mode: lines with fixed collapsing frames and low-friction slat shields generate lower wrinkle-induced transverse stress than rotating-die lines with non-adjustable frames. The splitting frequency is also sensitive to die-lip cleanliness; degraded particles or carbonised material at the die lip concentrate stress in the stalk and accelerate frost line cracking.

The splitting threshold is also influenced by the physical dimensions and mechanical alignment of the collapsing frame. A collapsing frame angle that is too steep increases the transverse tension applied to the bubble above the frost line, and the resultant splitting can occur even when the stalk height is within the stable range. The fabric slat tension, the coefficient of friction of the film against the slats, and the alignment of the nip rolls all modify the force path from the freeze line to the winding drum. In high-stalk HDPE operations, the collapsing frame should be positioned so that the film enters the nip symmetrically with minimal wrinkle formation; a misaligned frame shifts the frost line laterally and creates a crescent-shaped split that is often incorrectly attributed to drawdown. The exact maximum stalk height before splitting therefore depends not only on the resin and die geometry but also on the downstream conveying forces. Published data for the interaction between collapsing frame geometry and high-stalk splitting are limited; most equipment-specific data are derived from direct line audits rather than peer-reviewed consolidation.

Frost line height on production high-stalk lines is measured by a radiometric infrared pyrometer or a thermographic line scanner directed at the stalk transition, with the target emissivity set to the oxidised polyethylene surface. Manual visual measurement against a calibrated scale is still used for periodic verification, but plant data show operator-to-operator variance of ±25 mm on a 250 mm die can obscure the onset of instability. Internal bubble cooling complicates the measurement because the inner surface of the stalk is cooled more strongly than the outer surface; the visible frost line may therefore lag the internal solidification front by a distance that corresponds to 1 to 2 die diameters. This lag means that a high stalk bubble may appear stable while the inner wall has already begun to crystallise, a condition that reduces transverse stretchability and promotes splitting when blow-up ratio is increased. The frost line position should be recorded together with melt temperature at the die, cooling air temperature, ambient dew point, and line speed to maintain a reproducible process; the data acquisition system should log these variables at a frequency no lower than 1 Hz to detect transient oscillations. Published data for exact control accuracy of closed-loop frost line systems on high-stalk HDPE lines are restricted to proprietary machine-control evaluations; the general industrial practice is to maintain manual or open-loop adjustment of cooling-air flow and internal bubble pressure.

Frost Line Position as a Crystallization Front Marker in HDPE Film Formation

The frost line is best interpreted as a crystallization front marker because the visible opacity of the film coincides with light scattering from growing spherulites or oriented lamellar aggregates when the outer surface drops below the dynamic crystallization temperature of HDPE. In high-stalk extrusion, the cooling rate to the frost line is relatively low compared with low-stalk blown film because the molten stalk has a smaller surface-to-volume ratio and is subjected to lower cooling-air turbulence. The slow cooling and high uniaxial draw before crystallization favour the formation of extended-chain shish structures and row-nucleated lamellae, which produce the characteristic imbalance of mechanical properties in high-density film. The distance between die exit and frost line therefore determines the amount of strain-induced crystallization and the level of residual stress locked into the film. If the frost line is raised by reducing cooling air, the stalk remains molten longer, uniaxial extension increases, and the resulting film develops higher machine-direction tensile strength but lower transverse tear strength. If the frost line is lowered by increasing cooling air, orientation is arrested earlier, and the film retains greater transverse ductility at the expense of axial stiffness. The transition is not linear: at very low frost lines the stalk may widen before the frost line and the process resembles low-stalk tube extrusion, while at very high frost lines the stalk becomes mechanically unstable and splitting initiates at the frost line. Operators therefore treat the frost line position as a proxy for the crystalline orientation state, but the actual orientation must be verified by shrinkage testing or birefringence measurement because local cooling rate and air impingement can decouple the visible frost line from the true solidification front.

The crystallization kinetics of HDPE under extensional flow involve a complex interplay between molecular relaxation time, cooling rate, and nucleation density. At commercial draw rates, the extensional strain rate in the stalk is typically between 0.1 and 1.0 s−1, and the residence time from die to frost line ranges from 1 to 5 s depending on stalk height and line speed. These conditions place the process near the Weissenberg number range where chain stretching enhances nucleation but does not guarantee full crystallization before the frost line. The temperature gradient across the film at the frost line is on the order of 5 to 15 °C under high-stalk conditions, with the inner surface typically warmer than the outer surface when internal bubble cooling is not used. This thermal gradient produces a gradient in lamellar thickness and yield stress across the film, which influences the propagation of tears when the film is split during downstream slitting. The frost line is therefore not a single variable but a thermal boundary condition that interacts with molecular weight distribution, die shear history, and air-ring geometry.

The relationship between frost line height and film crystallinity can be assessed by differential scanning calorimetry on samples taken from the stalk and from the final film. A high-stalk film quenched far above the die may show a broader melting endotherm than a low-stalk film of the same resin, reflecting a broader distribution of lamellar thicknesses. The onset of the melting endotherm is commonly observed near 120 °C and the peak melting temperature near 130 °C for high-density polyethylene. These values shift with comonomer content and cooling rate. In the high-stalk process, the slow cooling of the inner surface produces a transcrystalline layer that may be differentiated from the outer quenched layer by optical microscopy. That layered morphology influences both the dart impact and the splitting behaviour, but it is not easily separated from the effect of orientation in a routine process audit.

Drawdown Ratio, Blow-Up Ratio, and Neck-In Induced Orientation

The drawdown ratio in blown film is defined by the mass balance as the die gap divided by the product of final film thickness and blow-up ratio. For a 1.0 mm die gap, a 4:1 blow-up ratio, and a 25 µm final film, the drawdown ratio is 10:1. High-stalk HDPE lines typically operate at drawdown ratios between 8:1 and 15:1, while the stalk itself may draw uniaxially by a factor of 2 to 4 before radial expansion begins. This dual-stage drawdown is the central processing difference between high-stalk and conventional low-stalk blown film. The first stage, from die exit to frost line, is a uniaxial extensional deformation that aligns polymer chains in the machine direction. The second stage, from frost line to final bubble diameter, is a biaxial deformation that can partially reorient chains in the transverse direction. When the first stage is too aggressive relative to the resin’s extensional strength, the web splits at the frost line; when the second stage is too aggressive, the web can exhibit circumferential cracks or poor dart impact. The blow-up ratio is bounded by the desired film width and the stability of the stalk; at high BUR values above 5:1 the expansion zone above the frost line becomes highly curved, and the transverse stretching can tear the partially crystalline web. At BUR values below 3:1, the final film retains excessive machine-direction orientation and the splitting tendency in downstream drawdown is magnified. These thresholds are not absolute; they depend on die diameter, film gauge, melt temperature, and resin rheology.

Process variableHigh-stalk HDPE rangeEffect on bubble height / frost lineSplitting consequenceReference method
Stalk height to die diameter ratio6:1 to 8:1 typicalSets residence time in uniaxial extensionAbove 8:1 increases frost line tensile stressLine visual or IR pyrometer
Blow-up ratio3:1 to 5:1Controls expansion curvature above frost lineHigh BUR tears partially crystalline webBubble diameter / die diameter
Drawdown ratio8:1 to 15:1Determines final gauge and orientationExcessive DDR increases longitudinal splittingDie gap / film thickness x BUR
Frost line outer surface temperature110 to 125 °CDefines solidification front positionToo low causes premature quench and residual stressIR pyrometer
Melt temperature at die210 to 240 °CControls extensional viscosity and relaxationLow melt temperature promotes frost line crackingMelt thermocouple
Die gap0.8 to 1.2 mmDetermines initial thickness and drawdownSmall gap increases shear stress and melt fractureFeeler gauge

Neck-in induced orientation is distinct from the drawdown orientation created downstream of the frost line. Neck-in occurs when the molten stalk reduces its diameter just above the die, and the corresponding hoop contraction imposes a transverse compressive strain. This deformation is not equivalent to uniaxial extension; it contains a biaxial component that can disturb the subsequent uniaxial orientation. If the neck is too abrupt, the outer surface of the stalk quenches with a wrinkled or irregular circumference, and the frost line is not circular. The asymmetry is visible as a tilted frost line and leads to uneven splitting resistance around the bubble. Using a larger die gap reduces neck-in severity because the thicker melt section resists the radial collapse, but it also lowers the drawdown ratio and increases the final gauge unless the haul-off speed is raised. The optimum die gap for high-stalk HDPE is therefore a compromise between neck-in stability, melt fracture, and the desired drawdown ratio.

Molecular orientation in high-stalk HDPE film is measured by birefringence, thermal shrinkage according to ASTM D2732-20, or wide-angle X-ray scattering because the visible frost line alone does not fully define crystalline orientation. A high frost line may be accompanied by significant thermal relaxation if the melt temperature is high, while a lower frost line with strong cooling can freeze higher surface orientation. Shrinkage testing is used to assess residual stress, and the measured machine-direction shrinkage should be compared to the transverse direction; a large anisotropy indicates an increased tendency for longitudinal splitting during downstream drawdown.

Splitting of high-stalk HDPE film at the collapsing frame is frequently misinterpreted as a raw-material deficiency when the actual cause is a mismatch between frost line height and collapsing angle. With a short collapsing frame and a high frost line, the flattened tube reaches the nip while the inner layers are still warmer than the outer layers, and the resulting differential shrinkage creates local wrinkles that trigger longitudinal tears. The typical solution is to lower the frost line by 1 die diameter or to reduce line speed, but this action also reduces the uniaxial orientation benefit that the high-stalk configuration is intended to create. On rotating-die lines, the helix angle of the die rotation can cause the stalk to twist below the frost line; if the twist is arrested at the collapsing frame, a torsional stress is superimposed on the uniaxial stress and the web splits at the frost line. This torsional splitting mode is specific to rotating-die equipment and is not observed on stationary die lines. Field experience on 90 to 120 mm grooved-feed extruders indicates that high-stalk HDPE splitting is more likely when the stalk height is manually adjusted without compensating the internal bubble cooling; the internal air pressure that stabilises the expanded bubble also provides a small radial force on the stalk, and a sudden change in that pressure produces a frost line position change of several die diameters within seconds. Because the web is still molten below the frost line, such a transient moves the solidification front rapidly upward and the partially solidified film is loaded in tension before it has developed adequate transverse strength.

To reduce splitting risk on high-output lines, the die-lip surface should be inspected for degradation particles at intervals defined by the purging frequency of the extruder. Carbonised polyethylene accumulations at the mandrel lip create localised stress concentrations that follow the stalk and initiate failure at the frost line. The extruder screw and barrel should be maintained within the manufacturer’s permitted wear limits; inadequate solids conveying in a grooved-feed hopper produces throughput fluctuations that move the frost line vertically and trigger periodic splitting. The air-ring lip clearance and internal bubble cooling nozzle alignment should be verified with the film tube stopped, because eccentric cooling produces a tilted frost line and a crescent-shaped gauge profile. Gauge variation below ±5 % of mean thickness is routinely specified for high-stalk HDPE film intended for printing and lamination; higher gauge variation concentrates tension in thin bands and initiates drawdown splitting at the nip. This gauge variation is measured according to ISO 4593:2014 or equivalent capacitance thickness scanning, and the frost line tilt should be less than 10 to 15 mm on a 250 mm die to maintain such uniformity.

When Bimodal HDPE Resins Narrow the High-Stalk Processing Window

Bimodal high-density polyethylene resins are preferred for high-stalk film because the low-molecular-weight fraction improves processability while the high-molecular-weight fraction supplies extensional strain hardening. The processing window narrows when the high-molecular-weight fraction is too low, because the stalk cannot support the gravity load over a tall distance, or too high, because the elastic memory from the die prevents stable necking and produces a rough stalk surface. The rheological fingerprint relevant to bubble height is the elongational relaxation time and the strain-hardening index measured in a filament-stretching or Sentmanat extensional rheometer; resins with a pronounced strain-hardening plateau sustain taller stalks but also retain higher residual stress after quench. The melt flow rate of the base resin is not a sufficient parameter: two HDPE grades with identical 0.10 g/10 min MFR at 190 °C and 5 kg can exhibit different stalk stability because their molar mass distributions and comonomer placements differ. The density and short-chain branch distribution also affect crystallization kinetics at the frost line. Resins with higher alpha-olefin comonomer content crystallise at a lower temperature and require a lower frost line for equivalent quench; if the frost line is held too high for such a resin, the film remains partially molten through the expansion zone and splitting occurs as a ductile rupture rather than a brittle crack. Published data for high-stalk splitting thresholds in specific bimodal resin grades are generally limited to resin supplier technical bulletins and confidential plant trials; the generalised processing range must therefore be verified for each resin lot on the target line.

Long-chain branching is typically low in HDPE but can be introduced through certain catalyst systems or reactive extrusion. A small amount of branching increases strain hardening and can allow a taller stalk without splitting, but it also increases orientation and may reduce transverse tear strength. The resin supplier’s data sheet may report only melt flow rate and density; it does not guarantee high-stalk stability. Laboratory evaluations should include oscillatory shear in the melt at 190 °C across the frequency range 0.01 to 100 rad/s and extensional viscosity growth at Hencky strain rates of 0.1 to 1.0 s−1. Batches that fail these rheological checks often produce a frost line that oscillates vertically by ±50 mm on a 250 mm die even when layflat width and gauge remain within specification. The observed splitting is then traced to batch-to-batch molar mass distribution shifts, not to a change in melt flow rate. For this reason, converters operating high-stalk HDPE lines frequently require the resin supplier to maintain lot-to-lot high-load melt flow rate or elongational viscosity control within a defined range; when this control is absent, the maximum stable bubble height must be reduced by 0.5 to 1.0 die diameters during lot transitions.

Catalyst residues and metal ions in HDPE can influence crystallization and thus the frost line position. Certain high-activity Ziegler or metallocene residues act as nucleating agents, raising the onset temperature of crystallization and moving the visible frost line downward for a given cooling-air setting. This shift is not necessarily detrimental; it can reduce the tensile stress at the frost line by shortening the molten stalk, but it may also reduce the time available for uniaxial extension and alter the orientation balance. The resin producer’s specification seldom reports the nucleation density directly; it is inferred from the thermal analysis of the resin and from the actual frost line position on the line. Converters that switch between catalyst systems may therefore observe a change in stable stalk height of 0.5 to 1 die diameters even when the melt index and density are unchanged. Published data for this specific interaction are sparse; the effect is normally identified by thermal cell measurements and film-test correlation.

Additive packages for high-stalk HDPE film must be selected with attention to their effect on frost line position and splitting tendency. Fluoroelastomer processing aids at concentrations above 1000 ppm can lower the extensional strain at the die lip and destabilise the high stalk because the same slip that reduces melt fracture also lowers the wall adhesion that contributes to stable neck formation. The use of such additives is not universally incompatible; the concentration and the die geometry determine the effect. Antiblock agents with coarse particle size can nucleate cracks at the frost line if they are not well dispersed, and the resulting splitting may be mistaken for a resin failure. Pre-drying of HDPE resin is generally not required at ambient humidity conditions below 60 % RH, but condensation on cold pellets during winter transport can introduce moisture that appears as surface defects and frost line instability. When high levels of recycled in-house edge trim are added to the feed, the lot-to-lot variation in melt flow rate and contamination can move the frost line upward by as much as 1 die diameter; in that case the stable stalk height should be deliberately lowered to prevent splitting. These limitations define the operational boundary of high-stalk bubble height and should be included in standard operating procedures for any line producing HDPE film below 20 µm with a high drawdown ratio.

Measuring Bubble Height, Frost Line, and Splitting Frequency on Production Lines

Quantitative control of high-stalk bubble height requires combining an infrared pyrometer, a visible-light camera or laser profilometer, and a tension transducer at the nip. The infrared pyrometer should be mounted on a vertical linear guide so that the frost line can be tracked automatically and its height stored with time stamps. A visible-light camera with image segmentation can identify bubble diameter transitions and stalk neck-in; these measurements should be calibrated against the actual die diameter and the layflat width. The frost line height measured radiometrically will not perfectly match the visible frost line because the emissivity changes as the polyethylene surface oxidises and crystallises; emissivity settings between 0.90 and 0.95 are typical for opaque HDPE film. Tension at the nip is a better real-time indicator of splitting risk than frost line height alone, because it includes the effects of bubble radius, film thickness, coefficient of friction of the collapsing frame, and winding tension. On a stable high-stalk line, nip tension traces show regular oscillations of ±10 % around a mean value; sustained oscillations above ±20 % or abrupt spikes are correlated with incipient splitting. The splitting frequency is measured as the number of longitudinal tear defects per 1000 m of wound film, and a production standard may allow no more than 1 split per 5000 m for lamination-grade film. These values are not universal and should be defined by the end-use specification.

Thermographic cameras with a frame rate of 30 to 60 Hz can capture transient frost line movement that a single-point pyrometer misses. The temperature data should be synchronised with the nip tension signal and the internal bubble pressure transducer. A rapid increase in nip tension followed by a decrease indicates a partial split that has propagated through the web; the film may still wind without a full break but the roll will contain weak sections. In-line roll inspection systems using line-scan cameras detect these splits as narrow machine-direction lines, but they must be set to distinguish surface scratches from true fracture paths. Scratches generally do not penetrate the film and disappear under reflected-light illumination, while true splits extend through the thickness and show a characteristic bright edge in transmitted light. Calibration of the in-line inspection system should be verified against manual tear testing on retained samples with ASTM D1922-15.

ParameterTest methodRelevance to high-stalk splitting
Melt mass-flow rateISO 1133-1:2022Resin lot consistency
DensityISO 1183-1:2019Crystallinity and frost line temperature
Film thicknessISO 4593:2014Gauge uniformity and drawdown calculation
Tensile propertiesASTM D882-18MD/TD strength balance
Elmendorf tear resistanceASTM D1922-15Transverse tear weakness and split propagation
Dart impactASTM D1709-22Brittle failure under biaxial impact

The measurement records from these methods should be retained with the corresponding extrusion parameters: melt temperature, die temperature, die gap, blow-up ratio, drawdown ratio, stalk height, frost line height, internal bubble cooling pressure, and cooling air velocity. A change in any one of these variables moves the frost line relative to the die and changes the orientation state of the film. Statistical process control limits for frost line position are typically set at ±5 % of the target height for high-stalk HDPE; a change beyond this range requires either a reset of the cooling air or an adjustment of the line speed. Closed-loop control that uses nip tension to adjust internal bubble pressure is available on some film lines, but its effectiveness on high-stalk HDPE depends on the response time of the air supply and the compressibility of the bubble volume. The maximum bubble height that can be maintained without splitting is ultimately defined by the intersection of resin strain hardening, cooling dynamics, and the mechanical strength of the partially solidified frost line.

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