Importance and methods to control, vary, and characterize mud strength for studying locomotion

1. Introduction

• Because flowable substrates yield and flow when exerted force exceeds a critical threshold, locomotor performance depends on yield strength, which should be controlled, characterized, and varied.The introduction notes that substrates may also re-solidify after yielding.Source: §1 · pp. 1–2
• Sandy substrates are dominated by coarse grains with frictional, repulsive contacts, whereas fine clay suspended in water can create stronger cohesion in muddy substrates.The introduction characterizes coarse grains as about 1 mm and fine clay as up to about 10 μm.Source: §1 · pp. 2–3
• Despite 17 muddy-locomotion studies, equivalent understanding remains lacking; few varied wetness, and detailed methods for preparing and characterizing controlled, variable mud were unreported.Mud wetness can also change through water evaporation, further motivating methods to track its properties.Source: §1 · pp. 5–6
• Some animal “mud” models were viscous fluids or viscoelastic solids unlikely to undergo solid–fluid transitions; tested fine-particle mixtures were likely below or near the settling limit.The cited mixtures used ϕ = 15% by volume; reported maximum viscosities were 1000 mPa·s for Poly-Bore and 40 mPa·s for methyl cellulose, versus 2090 mPa·s for clay mud at ϕ = 27%.Source: §1 · p. 6

• Most prior substrate-mechanics studies used monolithic gelatin, which cannot undergo mechanically driven solid–fluid transitions.Source: §1 · p. 6
• Table 1 reviews methods for controlling, varying, and characterizing flowable substrates and defines fluid, semi-fluid, semi-solid, and solid wetness levels.Its caption describes fluid as flowing similarly to mud below the settling limit; semi-fluid and semi-solid denote transition regimes with relatively more water or solid particles, respectively.Source: Table 1 · p. 7

Introduction

• Methods for controlling dry and wet sand cannot prepare muddy substrates, especially sticky mud.The passage notes limitations of air-fluidization and sieve-deposition approaches for mud.Source: §1 · p. 13
• Evaporation raises solid volume fraction and reduces wetness in mud and unsaturated wet sand, potentially shifting substrate regime, so prepared mud should be sealed and its yield-strength drift tracked.Source: §1 · p. 13
• The authors speculated that mud is weaker than other flowable substrates such as dry sand and noted that animals and robots can easily get bogged down.Source: §1 · p. 13
• Commercial penetrometers used for sand and soil are insufficiently sensitive to muddy-substrate yield strength except near or above the fracture limit, motivating new tools.The authors report this limitation from preliminary experiments and call for characterization across naturally occurring conditions.Source: §1 · p. 13
• The authors developed low-cost methods to control, systematically vary, characterize, and track yield-strength drift in intermediate-wetness mud that can undergo solid–fluid transitions.They expect the methods to apply to natural muddy substrates in the solid–fluid transition regime.Source: §1 · pp. 13–14
• For a given fine clay, clay-mud yield strength increases monotonically with solid volume fraction by up to several orders of magnitude, and its transition-regime rheology resembles natural mud.Clay mud was selected as a relatively simple test substrate; natural mud can contain both fine clay and coarse grains.Source: §1 · pp. 13–14

2.1. Clay selection

• Kaolin was selected for its low cost and prior use in mud studies, enabling direct comparison with previous results.The reported powder cost was approximately $0.4–4 per kg.Source: §2.1 · p. 14
• After Edgar Plastic Kaolin became unavailable, Georgia Kaolin was used in subsequent experiments; the clays behaved qualitatively similarly but produced different quantitative results.The authors recommend using the same clay type whenever possible.Source: §2.1 · p. 14

2.2. Calculation of clay-to-water ratio for a given wetness

• Solid volume fraction ϕ was defined as solid-particle volume divided by total solid–water mixture volume and decreases as water concentration increases.Using Kaolin and water densities, the authors converted a desired ϕ into the clay and water weights needed for preparation.Source: §2.2 · pp. 14–15

2.3. Automated mud mixer

• The low-cost automated mixer addressed poor uniformity in hand- or drill-mixed mud by feeding clay into water at a constant, slow rate.Preliminary experiments found poor uniformity even in small batches; the system was developed to prepare large quantities uniformly in relatively little time.Source: §2.3 · p. 15
• The mixer dispensed small amounts of clay every 0.5 seconds because larger pours tended to clump, while the mixer ran at 88–100 RPM for uniform mixing.The dipper and rotating cam repeatedly shook powder into the water bowl.Source: §2.3 · p. 16
• A batch containing 1.5 kg of water took 10–20 minutes to prepare, depending on ϕ, and the mixer cost approximately $166.The mixer used an electric stand mixer, a custom dipper, and a rotating cam.Source: §2.3 · p. 16
• Filling a 1.02 m × 0.51 m × 0.16 m testbed with 49.3 kg of mud at ϕ = 34% required 14 batches of 3.5 kg, taking 5 hours over approximately 2.5 days.The batches were combined with existing mud in the testbed.Source: §2.3 · p. 16
• After transfer, each batch was manually mixed with the existing mud for at least 10 minutes, then the mud was sealed immediately to minimize evaporation and drift in ϕ.A metal spatula with beveled edges was used for the manual mixing.Source: §2.3 · p. 16

2.4. Safety during mud preparation

• A filtered ventilation system and custom duct above the mixer remove airborne particles, while a plastic tablecloth contains remaining dust near the mixer.The vent was turned on before preparation and left on for 2–3 hours afterward.Source: §2.4 · p. 18
• Users should wear N95 masks and long-sleeve gloves when handling clay powder, then wash the clothes worn during mixing with water.Source: §2.4 · p. 18
• Personal monitoring during and after mixing found crystalline silica <6.9 μg/m3 and respirable dust at 0.047 mg/m3.The monitoring covered about 5 hours during mixing and 1 hour afterward.Source: §2.4 · p. 18
• Room particle monitoring more than 21 days after mixing found respirable-dust exposure well below OSHA’s 5 mg/m3 limit; similar tests are recommended at other setups.The test lasted about 6 hours.Source: §2.4 · p. 18

2.5. Determining wetness range of solid–fluid transition regime

• The settling limit was approximated by increasing solid fraction in 1–2% increments, mixing and settling samples, then checking whether bottom mud contained visibly more solids than the fluid above.The fraction at which no obvious settling remained after 5 minutes provided an approximate lower bound.Source: §2.5 · pp. 18–19
• More accurate settling-limit measurement can use a 24-hour wait, while more accurate fracture-limit measurement can use a 1-hour wait.Source: §2.5 · p. 19
• The fracturing limit was tested by increasing ϕ in 3–5% increments and checking after each increment; it was exceeded when intrusion and/or dragging caused cracks that did not close or reset after 2 minutes.Source: §2.5 · p. 19

2.6. Clay and solid volume fraction used for each test

• Edgar Plastic Kaolin mud at ϕ = 14%, 27%, 34%, and 39% was used for uniformity, short-term drift, and boundary tests, while Georgia Kaolin mud at 27%, 34%, 39%, and 42% was used for wetness and long-term drift tests.The Edgar Plastic Kaolin boundary-characterization tests used ϕ = 34% and 39%.Source: §2.6 · pp. 19–20
• Most tested solid volume fractions fell within the solid–fluid transition between settling and fracture limits; ϕ = 14% was near settling, and Georgia Kaolin mud at 42% exceeded the fracture limit.The Edgar Plastic Kaolin values were discussed relative to the limits in Fig. 1C, and Georgia Kaolin values relative to Fig. 1D.Source: §2.6 · pp. 19–20

2.7. Mud container sealing

• To limit evaporation and drift in mud properties when not in use, containers were sealed with an airtight lid, plastic wrap over the mud surface, rubber strips, and latching straps.The sealing setup was evaluated by tracking mud-strength drift over multiple days.Source: §2.7 · p. 20

2.8. Mud container size to minimize boundary effect

• No significant boundary effect was found for Edgar Plastic Kaolin mud at ϕ = 34% or 39% when fill height exceeded 10 cm, whereas an effect appeared at about 3 cm.The result is shown in Fig. S1B.Source: §2.8 · pp. 20–21

2.9. Custom-made penetrometers to characterize mud strength

• Commercial penetrometers were generally too insensitive for mud across most of the solid–fluid transition regime and could not measure force changes with depth during intrusion and extraction.Only the most expensive tested device measured the 0.025 m disc probe in stronger Edgar Plastic Kaolin mud at ϕ = 39%; commercial devices target substrates about 1–2 orders of magnitude stronger than most mud in the transition regime.Source: §2.9 · p. 21
• Two custom penetrometers were developed to sensitively measure substrate force across probe depth in mud within the solid–fluid transition regime.The devices were intended to characterize the yield strength of muddy substrates in this regime.Source: §2.9 · p. 21
• The stationary automatic penetrometer controls penetration speed and prescribed depth while measuring vertical force; the portable manual device reaches different locations but has less precise speed and depth control.The automatic device is heavy and table-fixed, whereas the manual device is small and lightweight.Source: §2.9 · p. 22

2.10. Stationary automatic penetrometer

• The penetrometer measures forces up to 49 N, with 0.0011 mm position resolution and 10 μN force resolution; it cost about $300.It weighs 1.61 kg and is 70 cm tall, 5.5 cm wide, and 13 cm thick.Source: §2.10 · p. 22
• A custom translation system drives an intruder disc into the substrate at constant speed while force and displacement sensors record resistance as a function of depth.Motor and load-cell readings were collected using MATLAB.Source: §2.10 · pp. 22–23
• In experiments, a 0.5 cm-thick disc moved up to 4.96 cm into and out of mud at 0.31 cm/s.The force sensor was zeroed at the start, and the disc was cleaned before each use to prevent residue from biasing measurements.Source: §2.10 · p. 23
• Disc radius was reduced from 3 cm to 1 cm when expected forces exceeded the sensor limit, and smaller-disc measurements were scaled by area to 3 cm-disc equivalents.This adjustment was used for higher-ϕ mud and dry mud conditions.Source: §2.10 · p. 23

2.11. Data processing for stationary automatic penetrometer

• Representative tests covered solid–fluid-transition mud and fractured-solid Georgia Kaolin mud, plus solid–fluid-transition Edgar Plastic Kaolin mud.Georgia Kaolin transition-regime samples had ϕ = 20%, 25%, 27%, 34%, and 39%; fractured-solid samples had ϕ = 41% and 42%; Edgar Plastic Kaolin transition-regime samples had ϕ = 14%, 27%, 34%, and 39%.Source: §2.11 · p. 23
• Position and force records were synchronized by timestamps, position was interpolated to the force sampling frequency, and cyclic load-cell noise was removed with a 0.05–15 Hz band-stop filter.The load-cell noise arose from small cyclic movement along the threaded rod.Source: §2.11 · p. 23
• For each trial, depth was zeroed at first contact, detected above 0.03 N; force–depth plots were visually checked and manually re-offset when threshold-based alignment was incorrect.The synchronized force and depth data were plotted as force versus depth.Source: §2.11 · pp. 23–24

2.12. Portable manual penetrometer

• At 0.545 kg and about $60, the portable penetrometer can be moved to characterize substrate strength on a large animal or robot testbed.It is 44.5 cm high, 26 cm wide, and 10.5 cm thick; it moves the intruder up to 14 cm, measures forces up to 49 N, and has 0.11 mm position and 10 μN force resolution.Source: §2.12 · p. 24
• A handle and LEGO rack-and-gear mechanism convert manual rotation into linear intruder motion, while a force sensor and Hall-effect encoder record force and displacement.Rapid penetration, especially in lower-ϕ mud, could spin the encoder magnet too quickly to detect; the prototype added resistance using a powered servo motor over the gear rack.Source: §2.12 · p. 24
• A custom guide rail supported precise measurements at different locations over multiple tests and days, while keeping the probe away from container walls and bottom to avoid boundary effects.A weighted 3-D-printed frame helped keep the disc perpendicular to the mud surface, but these additions were not strictly necessary for general use.Source: §2.12 · p. 25

2.13. Data processing for portable manual penetrometer

• For representative Georgia Kaolin mud samples across solid–fluid transition and fractured-solid regimes, initial contact was identified when force exceeded 0.3 N, then displacement and force were cropped and plotted as force versus depth.The sampled volume fractions were 20%, 25%, 27%, 34%, 39%, 41%, and 42%.Source: §2.13 · p. 25
• For long-term drift comparisons, force records were divided into penetration and extraction phases, cropped to 3 cm penetration depth, and interpolated to average trials across locations, trials, and days.Phase boundaries were estimated from displacement and time data.Source: §2.13 · p. 25
• The portable instrument’s force–depth profile accuracy was tested and verified against the stationary automatic penetrometer for mud of different strengths.The comparison used mud characterization from both instruments (Fig. 5A, B).Source: §2.13 · p. 26

2.14. Experiments to characterize spatial uniformity and short-term temporal drift of mud

• Spatial uniformity was assessed from vertical force measurements at four locations, with three trials at each location.The study used Edgar Plastic Kaolin mud at ϕ = 39% and a fixed-depth commercial pocket penetrometer.Source: §2.14 · p. 26
• Short-term drift was assessed by repeating measurements every 10 minutes for one hour, averaging the three trials and then the four locations at each interval.The hour matched the typical duration of animal experiments in a day.Source: §2.14 · p. 26
• To compare spatial variation and temporal drift with the strength range attainable by varying wetness, the ϕ = 39% force was averaged across space and time and compared with forces for ϕ = 14%, 27%, 34%, and 39%.The wetness-comparison forces came from stationary automatic penetrometer data at 0.0064 m depth, scaled to the 0.025 m commercial probe disc area because vertical penetration force scales with intruder area.Source: §2.14 · p. 26

2.15. Experiments to characterize dependence on mud strength wetness

• Vertical force–depth data characterized Georgia Kaolin mud at ϕ = 20%, 25%, 27%, 34%, and 39% in the solid–fluid transition regime, and at ϕ = 41% and 42% in the fractured solid regime.Measurements used portable manual and stationary automatic penetrometers.Source: §2.15 · p. 27
• Penetration was limited to 4 cm because greater depths would exceed sensor force ranges; this maximum was well beyond fish sinkage of about 0.3 cm (Ramesh et al., submitted).Source: §2.15 · p. 27
• Three trials were collected for each mud fraction, with the mud manually re-mixed and flattened after each trial to reset the disturbed substrate.Data were processed following the procedures in Sec. 2.11 and Sec. 2.13.Source: §2.15 · p. 27

2.16. Experiments to characterize long-term drift of mud strength

• Penetration tests tracked Georgia Kaolin mud with ϕ = 27%, 34%, and 39% for 27, 114, and 106 days, and ϕ = 42% for 24 days.The 27%, 34%, and 39% muds were in the solid–fluid transition regime, while 42% mud was in the fractured solid mud regime.Source: §2.16 · p. 27
• Mud was quite uniform spatially, so measurements on subsequent days were reduced to four locations.On Day 0, tests covered 18 locations; some measurements continued at 18 locations to check repeatability after the penetrometer was improved.Source: §2.16 · pp. 27–28
• To quantify long-term temporal drift for each mud strength, researchers averaged penetration-phase vertical force over 0–1 cm across trials and locations, then compared each day’s force at 1 cm across days.Source: §2.16 · p. 28

3.1. Vertical force vs. depth in mud is complex

• During intrusion, upward force increased nonlinearly and plateaued with depth; during extraction, cohesive mud sticking to the intruder produced downward force that could persist after lift-off.In the solid–fluid transition regime, the downward force could grow in magnitude as the intruder became shallower; for fractured Georgia Kaolin at ϕ = 42%, it first grew and then diminished.Source: §3.1 · pp. 28–29

3.2. Mud yield strength increases exponentially with solid volume fraction

• Strength measured by vertical force increased exponentially by about 80 for Georgia Kaolin mud (ϕ = 20–42%) and about 950 for Edgar Plastic Kaolin mud (ϕ = 14–39%).The fixed-depth penetration force was used as a measure of mud strength across wetness levels.Source: §3.2 · p. 29

3.3. Controlled mud is spatially uniform

• For Edgar Plastic Kaolin mud at ϕ = 39%, mean penetration force across four locations was 14.3 ± 0.7 N, with location-to-location trial coefficients of variation of 2–9%.The four location means were 14.1 ± 0.6 N, 14.9 ± 0.3 N, 13.4 ± 0.3 N, and 14.7 ± 1.3 N.Source: §3.3 · p. 29

3.4. Controlled mud strength drifts little within a short time

• For mud at a given ϕ, the reported value increased by 8%, from 14.3 ± 0.9 N to 15.4 ± 1.1 N.Yield-strength variation for mud of a given ϕ was much smaller than the 13.6 N range achievable by varying ϕ (±1.3 N).Source: §3.4 · p. 31

3.5. Strength drifts more over a long time but is much slower than without sealing

• Sealed mud strength changed by an average of ~14% within a day but increased over multiple days, while mud left open increased 133 times in 7 days.Sealed samples showed gradual increases of 31% over 27 days at ϕ = 27%, 184% over 114 days at ϕ = 34%, and 138% over 106 days at ϕ = 39%.Source: §3.5 · pp. 31–32, Figure 4 · p. 32

3.6. Mixer and sealing method apply to muddy substrates with both clay and grains

• The automated mixer can be adapted to feed clay and coarse grains into water at prescribed ratios, and sealing natural mud or wet sand can limit evaporation during storage.The authors state the methods can be applied to or modified for natural muddy substrates in the solid–fluid transition regime.Source: §3.6 · p. 33

3.7. Mud is more challenging for locomotion than other flowable substrates

• Mud strength varied by 3 orders of magnitude with wetness, was smaller than dry sand (even loosely packed) and snow, and two orders of magnitude smaller than wet sand and snow.Source: §3.7 · p. 33
• Mud can bog down animals, robots, or vehicles more easily than other flowable substrates, and its downward extraction force yields about the same mechanical energy loss as sand.Sand has negligible downward extraction force, whereas mud's large downward force contributes to the energy lost during extraction.Source: §3.7 · p. 35

3.8. Penetrometers are useful for other flowable substrates

• Custom penetrometers measure force by depth during intrusion and extraction, spanning weak transition-regime clay mud through drier mud and serving other flowable substrates.Commercial penetrometers are insufficiently sensitive for weaker mud and do not provide the useful force-versus-depth measurements described here.Source: §3.8 · pp. 35–36

3.9. Penetration test to characterize mud yield strength is more nuanced

• Mud penetration force rises nonlinearly with depth, and this relationship differs across ϕ, making fixed-depth commercial tests inadequate.For the tested clay mud, strength-versus-ϕ dependence was basically the same at different depths, while greater depth improved signal-to-noise ratio.Source: §3.9 · p. 36

3.10. Characterization in the deep penetration limit is needed

• Strongly cohesive clay mud may lack hydrostatic-like pressure, so its observed depth dependence may reflect a shallow surface-boundary effect that requires testing in the deep-penetration limit.The proposed test is whether force becomes independent of depth far beyond intruder size; spanning shallow and deep limits would inform systems moving on or burrowing into substrates.Source: §3.10 · pp. 36–37

3.11. Characterizing non-vertical intrusion, stickiness, rate-dependence, and hysteresis is needed

• Vertical disc intrusion and extraction capture only part of mud's locomotor challenges; broader tests should characterize drag, shear, stickiness, rate dependence, and hysteresis.The study tested localized intrusion at a fixed low speed (~0.3 cm/s) and one intrusion-extraction cycle, motivating characterization across speeds and histories.Source: §3.11 · pp. 37–38
• A companion-animal study found mud stickiness strongly affected locomotion and varied complexly with wetness; at ϕ = 27%, disc-measured stickiness fell, yet greater sinking increased mud on the animal, producing the largest downward and backward pull.Source: §3.11 · pp. 37–38

3.12. Summary

• The developed methods prepare uniform mud of different wetness, minimize evaporation during storage, and use custom stationary and mobile penetrometers to characterize it.Yield-strength drift was ~8% over ~1 hour, while evaporation made mud about two times stronger over many weeks.Source: §3.12 · pp. 38–39
• Mud, except when dried out, is 1–2 orders of magnitude weaker than other flowable substrates and sticks much more during extraction, so animals and robots can more easily get bogged down.Source: §3.12 · p. 39

• Codes for real-time control and data acquisition for the portable manual and stationary automatic penetrometers are available at https://doi.org/10.6084/m9.figshare.30984856.Source: Data availability · p. 40

Supplementary figures

• In smaller-height mud, force rose quickly near the end of penetration, unlike in larger-height mud, for ϕ = 34% and 39%.The comparison used mud depths of 3 cm and 10 cm.Source: Figure S1 · p. 49
• The processing figures show how displacement was offset to determine depth at initial contact, using force thresholds of 0.03 N and 0.3 N for automatic and manual penetrometers.They plot displacement and depth over time, force over time, and force against depth for Georgia Kaolin clay mud at ϕ = 42%.Source: Figure S2 · p. 50
• The final stationary automatic penetrometer used a longer threaded rod, stabilizing 3D-printed supports, and a shaft coupler to improve reach and reduce wobble and alignment difficulty.These changes followed a first version with a shorter rod and a 3D-printed flange.Source: Figure S3 · p. 51, Supplementary figures · p. 52
• Portable manual penetrometer revisions addressed rack bending, insufficient resistance at lower ϕ, side-to-side movement, and a frame that wore easily.The final version used a metal gear rack and gear, a linear motion shaft, and added weights to keep the intruder disc perpendicular to the mud surface.Source: Figure S3 · p. 51, Supplementary figures · p. 52
• The final portable manual penetrometer used a metal gear and set screw to secure the linear motion shaft, with weights helping keep the intruder disc perpendicular to the mud surface.The shaft was attached to a 3D-printed part holding the magnet.Source: Figure S3 · p. 51, Supplementary figures · p. 52
• The Georgia Kaolin clay tests compare downward penetration force across ϕ at different depths, normalizing each depth’s average force to its value at ϕ = 42%.Error bars show ± s.d. across 3 trials for each treatment.Source: Figure S4 · p. 53, Supplementary figures · p. 53