Granular Masses Research Articles

Experiments on Landquakes Generated by Free‐Falling Granular Masses: Implications for Rockfall Impact Dynamics

AbstractThe properties of rockfalls, such as the volume and geometry of the detached rock mass as well as the number and diameter of rock fragments, greatly affect their propagated behavior, and different strategies are required to mitigate rockfall hazards. Landquakes generated by rockfalls are regarded as good proxies for those properties. To explore the free‐fall impact dynamics of rockfalls, a series of free‐fall experiments on granular masses were designed and conducted with the characteristics of the generated landquakes versus initial variables being analyzed. The results show that the particle diameter is a dominant factor affecting landquakes, with the most significant effect on the mean frequency. The impact area is another key factor, with which both the maximum seismic amplitude and radiated seismic energy have strong positive correlations, while the mean frequency of landquakes shows a weak negative correlation. The maximum seismic amplitude appears to have a poor correlation with the numbers of layers of the free‐falling granular masses, and it was proven that the maximum seismic amplitude is determined by only the lower part of the granular masses. Quantitative relationships between the number of particles and seismic signal characteristics indicate that disintegration needs to be accounted for when landquakes are used to investigate the characteristics of a rockfall. Moreover, the frequency of landquakes decreases with a decrease in the vertical velocity and increase in the horizontal velocity during granular mass collapse.

Investigation of the fabric effect on the arching effect in granular materials

A series of trapdoor experiments and DEM simulations were carried out in this study to examine the arching effect in granular materials, with the focus being on the influence of the direction for particle deposition. In the tests, particles (e.g. rice grains, PVC particles and nylon particles) were dropped as if they were raining at different directions to construct granular material layers with various fabric orientations. It is shown that the mobilization of arching effect depends on the direction of particle deposition, with the arching ratio ρ increasing at first and then decreasing. The arching effect is the weakest at a deposition angle of θ = 30° in obliquely deposited granular masses. It is also found that the vertical displacement of particles in the ultimate phase increases initially; it then decreases with the variation in the orientation of particle deposition, and the largest displacement occurs at θ = 30°.

Bursting Sand Balloons

Using linear elasticity theory, we describe the mechanical response of dry non-cohesive granular masses of Ottawa sand contained by spherical rubber balloons subject to sudden bursting in the earliest instants of the event. Due to the compression imposed by the balloon, the rupture produces a fast radial expansion of the sand front that depends on the initial radius R0, the initial pressure p originated by the balloon, and the effective modulus of compression Ke. The hydrostatic compression approximation allows for the theoretical study of this problem. We found a linear decompression wave that travels into the sand and that induces a radial expansion of the granular front in the opposite direction with similar behavior to the wave but with a slightly lower speed.

Smoothed particle hydrodynamics approach for modelling submerged granular flows and the induced water wave generation

In this study, we develop a Smoothed Particle Hydrodynamics (SPH) 2D-model for simulating fully submerged granular flows and their arising water waves. The granular particles are characterised by a non-Newtonian flow pattern, following a Casson constitutive law, generalised by applying the infinitesimal strain theory to avoid numerical singularities inherited from the original law. The implementation of this rheological model on the weakly compressible viscous Navier-Stokes equations enables the simultaneous modelling of the motion of granular flows and their resulting water waves, establishing a monolithic representation of fluid-structure coupling. The novelty of this model lies in the numerical continuity of the generalised rheological model based mainly on the yield stress criterion, which is computed purely from the mechanical properties of granular materials, including internal friction, cohesion, and viscosity coefficients. The proposed SPH model is validated through two benchmarks available in the literature, representing a submarine landslide along an inclined plane and an immersed granular column collapse. The outcomes of our study illustrate the effectiveness of the proposed model in accurately predicting the motions of submerged granular masses and their resulting water waves, which is crucial for accurately predicting the behaviour of underwater landslides and other natural hazards.

Isolated and combined pneumocystosis of the pulmonary in patients with HIV infection according to the results of pathomorphological study

The most common opportunistic disease in patients with HIV infection is Pneumocystis pneumonia, diagnosis of which is not difficult, and treatment significantly reduces the risk of death. Purpose of the study. To analyze the results of a pathomorphological study of the lungs with isolated pneumocystis and in combination with other lung diseases (bacterial, cytomegalovirus pneumonia and pulmonary tuberculosis) in patients with HIV infection. Materials and methods of research. The results of pathomorphological studies of patients with HIV infection (n=141), who were previously observed in hospitals in the region: patients with pneumocystis pneumonia (n=24), with pneumocystis pneumonia and bacterial pneumonia (n=52), with pneumocystis pneumonia and pulmonary tuberculosis (n=56), with pneumocystis and cytomegalovirus pneumonia (n=9). Results. At the time of death, 72.7% of patients with HIV infection had a viral load level of more than 300000, and 83.7% had a CD4+ lymphocyte level of less than 41 cells/μl. In all patients, treatment with trimethoprim/sulfamethoxazole was started too late. A pathological examination of patients with HIV infection and isolated Pneumocystis pneumonia revealed changes characteristic of this disease. Nosocomial microorganisms were identified in patients with pneumocystosis and bacterial pneumonia. At autopsy in patients with HIV infection with Pneumocystis pneumonia and pulmonary tuberculosis, the picture of tuberculosis was varied, heterogeneous and significantly different from classic granulomatous inflammation. With combined damage to the lungs by pneumocystis and cytomegalovirus, foamy, vacuole-like granular masses (PCS-positive), accumulations of neutrophilic leukocytes were observed in the lumens of the alveoli, and single cells of the “owl’s eye” type were detected. Conclusion. A pathological autopsy confirmed mixed infection in patients with HIV infection.

Scaling behavior of granular columns collapse on erodible-inclined surfaces

This paper explores the effects of slope angle, initial column aspect ratio, and interparticle friction coefficient on the mobility and erodibility of granular columns collapsing on an erodible-inclined layer before spreading and depositing on a nonerodable-horizontal surface using Discrete Element Method. The numerical results showed that the column collapse with higher aspect ratio or smaller friction coefficient induces larger normalized kinetic energy, leading to larger granular masses out the inclined region and longer runout distance. The slope angle, on the other hand, shows trivial impacts on the kinetic energy and apparent friction coefficient. Remarkably, these physical quantities are nontrivially behaved by a new dimensionless number that incorporates the initial column aspect ratio, inclination angle, and interparticle friction coefficient. These findings may significantly complement the better understanding of the behavior of natural disaster events such as landslides, rock and snow avalanches, or pyroclastic flows on complex surfaces.

Effects of brucine on mitochondrial apoptosis and expression of HSP70 in prostate cancer cells.

BackgroundMost prostate cancer patients are already in the middle/advanced stages of the disease at the time of detection. Whether brucine can regulate apoptosis in prostate cancer cells through the expression of heat shock protein 70 (HSP70) has not been reported. We preliminarily investigated the effects of brucine on the mitochondrial apoptosis and HSP70 expression of prostate cancer cells and analyzed brucine’s possible mechanisms in the hope of providing an experimental basis for its clinical application.MethodsThe effect of brucine on the activity of PC-3 cells was determined by the methodology of tetrazolium (MTT) assay method. The effect of brucine on apoptosis was measured by Hoechst 33258 staining and flow cytometry, and western blotting was used to detect the expression levels of the apoptosis-related heat shock protein 70 (HSP70), anti-apoptotic protease activating factor 1 (Apaf-1) and anti-cysteine protease-3 (caspase-3).ResultsAt 24 and 48 h, the activity of human prostate cancer PC-3 cells in the low, medium, and high dose brucine groups was significantly lower than that of the control group, with a dose-dependent difference (P<0.01). The nuclei of the control group fluoresced uniformly with intact nuclei, whereas the nuclei of the brucine group appeared crinkled, and dense granular masses of strong blue fluorescence were visible. The apoptosis rate of human prostate cancer PC-3 cells in the low, medium, and high dose brucine groups was significantly higher than that in the control group (P<0.01) and was dose-dependent. The expression levels of the HSP70 protein in the human prostate cancer PC-3 cells in the low, medium, and high dose brucine groups were significantly lower than those in the control group, while the expression levels of the Apaf-1 and caspase-3 proteins were significantly higher than those in the control group (P<0.01) and showed a dose-dependent relationship.ConclusionsBrucine downregulated HSP70 expression in human prostate cancer PC-3 cells and inhibited the mitochondrial apoptotic signaling pathway, thus acting as an anti-apoptotic agent.

Study on the effect of ZrO2 and SiO2 additions type on the properties of high alumina crucibles for melting heat-resistant alloys and steels

The technology of vibrocasting high-quality refractories from granular masses of different chemical composition, size, configuration and weight of products, in particular crucibles for induction melting of heat-resistant alloys was developed and introduced by JSC “URIR named after A. S. Berezhnoy”. According to the developed technology, mullite-alumina zircon-containing crucibles contain ZrO2 ~ 8 %, zirconia-containing crucibles — ~ 17 %, alumina zircon-containing crucibles — ~ 8 %, ~ 8—10 % and ~ 25 %, have open porosity &lt; 20 %, cold crushing strength depending on the used raw materials and firing temperature within ~ 50—230 N/mm2, thermal shock resistance&gt; 20 thermal cycles (1300 °C — water).&#x0D; The article studies the effect of ZrO2 and SiO2 additives type introduced with zircon or zirconia in combination with a silica-containing additive on the main properties of vibrocasting samples after firing at 1580 °C.&#x0D; As a result of the studies the high alumina samples with 10 % ZrO2, introduced by zircon or zirconia powder, were produced, which are characterized by apparent density of 3.12 and 3.08 g/cm3, open porosity of 14.4 and 15.5 %, cold crushing strength 145 and 210 N/mm2, thermal shock resistance &gt; 20 thermal cycles and 6 thermal cycles (1300 °C — water) respectively. That is, with practically the same chemical composition, samples with zircon have better indicators of apparent density and especially thermal shock resistance than samples with zirconia in combination with silica. The high thermal shock resistance of samples with zircon can be explained by a nature of the pores which long irregular in configuration and connected to each and, as well as microfractured structure due to the decomposition of zircon and the formation of microcrystalline monoclinic zirconia while in the structure of the samples, which included zirconia powder and silica and a mixture of joint grinding of alumina and a silica-containing additive, the pores are predominantly round isolated. Considering also that zircon is a domestic material and is cheaper than imported ZrO2, it is more expedient to use zircon for crucibles.

Dry granular masses impacting on rigid obstacles: numerical analysis and theoretical modelling

The assessment of the time evolution of the impact force exerted by dry flowing masses on rigid obstacles is mandatory for the dynamic design of sheltering structures and the evaluation of the vulnerability of existing structures. In this paper, the results of an extensive numerical campaign performed by employing a discrete element method (DEM) code are presented and the role of different geometrical factors (flow length, height and front inclination) and state parameters (porosity and velocity) on the impact force–time evolution is investigated. The impact process is studied to correlate local information with the macroscopic response and a physically based force–time function, generalising the formula already introduced by the authors for the assessment of maximum impact force, in which each parameter is correlated with the previously mentioned factors, is proposed.

‪Impulse wave generation: a comparison of landslides of block and granular masses by coupled Lagrangian tracking using VOF over a set mesh

AbstractThis paper presents the numerical results of impulsive waves generated by landslides of solid block, granular materials and heavy block sinking. An impulse product parameter P is developed and a wide range of effective parameters are studied. The volume-of-fluid (VOF) and overset mesh methods have been used to study landslide-generated tsunamis. Also, a Lagrangian tracking approach coupled with the VOF to simulate the granular movement was developed. The effect of the water reservoir depth, the landslide height, the landslide density and the geometrical parameters on the wave height (elevation) has been investigated using the open-source OpenFOAM software. The results have been presented for dimensionless distances and the normalized geometry of the landslide in the ranges 5–7, and 1–2, respectively. These numbers have been normalized to the height of the landslide (a). According to the results of simulations, the tsunami formation process is divided into three stages, which were analyzed in detail by considering the interactions between the solid and the water reservoir. The Scott Russell wave has the highest impulse product parameter among the impulse wave mechanisms which is 58.6% of the total impulse production. In addition, the duration of the wave propagation has been computed based on the wave height.

DEM numerical tests on dry granular specimens: the role of strain rate under evolving/unsteady conditions

In this paper, the mechanical behaviour of an ideal dry granular material under both evolving and steady conditions has been studied. Triaxial loading both constant volume and constant pressure Discrete Element (DEM) tests on periodic cells have been performed. The role played by strain rate, void ratio and imposed pressure has been analysed. The aim of the paper is to obtain a numerical data set, at present totally absent in the literature, useful for the definition of constitutive relationships capable of reproducing phase transitions (from solid-to-fluid and vice versa) taking place in granular materials. The novelty of the data set obtained derives also from the type of loading tests performed, referred as heating, cooling and cyclic, in which different strain rate histories are imposed, inspired to what locally occurs in flowing granular masses, accelerating and decelerating according to the slope inclination. The numerical results are interpreted by considering state variables (granular temperature and fabric) and the energies evolution (elastic and kinetic fluctuating) to put in evidence phase transition processes. Again, this interpretation of the results is fundamental to conceive upscaled constitutive relationships based on thermodynamic theories. The numerical results have allowed to highlight the role of the strain acceleration/deceleration, and the evolution of the directional properties of the microstructure even in the collisional regime (never described up to now). Cooling constant volume tests results provide a description, not available in the literature, of the evolving response of the material under pure collisional conditions.

Experimental Insights on the Propagation of Fine‐Grained Geophysical Flows Entering Water

AbstractGranular flows that propagate down a mountainside, then reach the sea, a lake or a river and finally, travel underwater, is a common event on the Earth's surface. To help the description of such events, laboratory experiments on gas‐fluidized granular flows entering water are performed, analyzed, and compared to those propagating in air. The originality of this study lies in the fluidization process, which improves the laboratory modeling of geophysical flows by taking their high mobility into account. Qualitatively, the presence of the water body promotes the generation of a granular jet over the water surface, a leading and largest wave, and a particle‐driven gravity current underwater. Hydrodynamic forces mainly play a dissipative role by slowing and reducing the spreading of the granular mass underwater, but a low amount of grains are still transported by the turbulent fluid as a gravity current far away. The temporal evolution of the granular jet and the particle‐driven gravity current are well described by ballistic motion theory and scaling laws of homogeneous gravity currents, respectively. Most currents propagate with a constant flow‐front velocity along the horizontal bottom, which is controlled by the flow height depending on the water depth. In contrast, the bulk volume concentration of particles in the current is estimated to be nearly constant, interpreted as a critical concentration above which the excess of particles cannot be maintained by the turbulent fluid. This experimental study highlights the complexity of the dynamics and deposits of granular masses when they encounter a water body.

Influence research of an α–form alumina type on the properties of samples from vibrocasting mullite corundum granular masses

The technology of refractories vibrocasting was developed by JSC "URIR named after A. S. Berezhnoy" in particular highly refractory mullite corundum crucibles for melting of heat-resistant alloys in induction furnaces. At manufacturing of highly refractory vibrocasting mullite corundum refractories a-form alumina, as a dispersion binder, which milling to a particles size of less than 10 μm (the content of particles less than 4 μm in such alumina is more than 50 %) is used. For improvement the casting characteristics of vibrocasting masses dispersants are used. At research milling kinetics alumina of all brands was milling in a vibration mill for a specified time. For this research a certain amount of the investigated alumina of each brand was loaded into a vibration mill, and after a specified grinding time (30, 45, 60, 75 and 90 min), a sample of alumina after milling was taken in a specified amount. In this work, the kinetics of changes the dispersion of alumina (S, N and NR brands) during their milling was studied. The dispersion of alumina which were needed (the content of particles less than &lt; 10 μm in such alumina is not less than 90 %, the content of particles less than &lt; 4 μm — not less than 50 %) was achieved in almost the same milling time was found (~ 45 min). With a further increase of milling time to 60, 75 and 90 minutes, the dispersion of alumina N and NR brands increases in comparison with alumina S brand and after 90 minutes of milling, the content of particles &lt; 4 μm in alumina, respectively, is ~ 60, ~ 75 and ~ 80 %. After fired at a temperature of 1580 °C the properties of vibrocasting mullite corundum samples, which contain alumina S, N and NR brands, are characterized by a high level of apparent density (above 3 g/cm3), open porosity (less than 16 %) and cold crushing strength (above 140 N/mm2). The high properties make it possible to use a-form alumina N and NR brands along a-form alumina S brand in the technology of vibrocasting mullite corundum refractories, including the highly refractory mullite corundum crucibles.

Dilatancy in dry granular flows with a compressible μ(I) rheology

Dilatancy plays a key role in mixtures of grains and fluid but is poorly investigated in dry granular flows. These flows may however dilate by more than 10% in granular column collapses. We investigate here dilatancy effects in dry flows with a shallow depth-averaged model designed to be further applied to simulate natural landslides. We use a compressible μ(I), ϕeq(I) rheology with a dilatancy law, where ϕeq(I) is the volume fraction at the equilibrium (i.e. critical) state and I the so-called inertial number. This law is obtained by simply removing the fluid phase in the solid/fluid model of our previous work (Bouchut et al., 2016) [7] and derived from critical state theory. A numerical method is proposed to solve the equations, that have however singularities that are rather difficult to handle.Simulations of granular collapses on horizontal and sloping beds show that the maximum height of the deposits changes as a function of the initial volume fraction with higher (lower) deposits for initially denser (looser) granular masses, as observed with Discrete Element simulations. The front position and the deposit shape behind it are on the contrary poorly affected by the initial volume fraction, as if the flow had almost forgotten its initial state. However subtle effects can be observed with the occurrence of low velocity regimes on steep slopes that strongly depend on the initial volume fraction. Simulations show complex compression/dilation effects during the flow, in particular with front dilation (compression) during the acceleration (deceleration) phases. These effects may dramatically change the effective friction that is observed to decrease at the front in some situations, while the μ(I) rheology without dilatancy would have predicted an increasing friction. The model predicts an increasing dilation of the mass for increasing slopes by up to 10% in the studied configurations, in agreement with laboratory experiments. Our results suggest that this compressible model contains key features to describe granular dilatancy.

Research of vibrocast corundum periclase granular masses structural and rheological properties depending on type and quantity of dispersing additives

For melting of heat­resistant alloys in induction furnaces high refractory vibroсast crucibles are used. Their technology was developed and implemented in JSC “URIR named after A. S. Berezhnoy”. For manufacturing of high refractory corundum periclase crucibles fused corundum and periclase, alumina, and ADS­1 + ADW­1 dispersing additive are used, wherein moisture of vibrocast mass is 4.5—4.6 % and their flowability is 80 %. Along with ADS­1 + ADW­1 additive it is possible to use Castament FS 10 + Castament FW 10 dispersing additive which provides almost the same moisture of masses. Also a new dispersant Castament FS 65 was developed. Therefore researches of Castament FS 65 dispersing additive influence on the structural and rheological properties of vibrocast corundum periclase granular masses in order to establish the possibility of further reducing their moisture is relevant. Influence of dispersing additive Castament FS 65 in the quantity of 0.05; 0.1; 0.2 and 0.3 % on flowability of corundum periclase mass in comparison with influence of optimal quantity (0.2 %) dispersing additive Castament FS 10 + FW 10 (in the ratio 3:1) was studied. Castament FS 65 additive optimal quantity, which significantly increases flowability of mass in comparison to Castament FS 10 + FW 10 additive and can provide decreasing mass moisture from 4.0—4.5 % to 3.5—4.0 % was found as 0.2 %.

A two-layer shallow flow model with two axes of integration, well-balanced discretization and application to submarine avalanches

We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesian or local coordinates for both layers. However, the motion characteristics of the granular layer and the water wave are different: the granular flow velocity is mainly oriented downslope while water motion related to tsunami wave propagation is mostly horizontal. As a result, the shallow approximation and depth-averaging have to be imposed (i) in the direction normal to the topography for the granular flow and (ii) in the vertical direction for the water layer. To deal with this problem, we define a reference plane related to topography variations and use the associated local coordinates to derive the granular layer equations whereas Cartesian coordinates are used for the fluid layer. Depth-averaging is done orthogonally to that reference plane for the granular layer equations and in the vertical direction for the fluid layer equations. Then, a finite volume method is defined based on an extension of the hydrostatic reconstruction. The proposed method is exactly well-balanced for two kinds of stationary solutions: the classical one, when both water and granular masses are at rest; the second one, when only the granular mass is at rest. Several tests are presented to get insight into the sensitivity of the granular flow, deposit and generated water waves to the choice of the coordinate systems. Our results show that even for moderate slopes (up to 30°), strong relative errors on the avalanche dynamics and deposit (up to 60%) and on the generated water waves (up to 120%) are made when using Cartesian coordinates for both layers instead of an appropriate local coordinate system as proposed here.

Effects of granular collisions on the rapid coarse-grained materials flow

A continuum model to predict the mobility of landslides is developed according to the formulation by Savage and Hutter as well as the main ‘microscopic aspects’ affecting the rapid sliding of granular masses. Conservation of mass and momentum equations, describing the geometry and velocity evolution during the motion, are re-written by considering the effects of collisions between solid grains (‘granular temperature’ and ‘dispersive pressures’) typically occurring within a basal thin ‘shear’ layer, causing a ‘fluidification’ effect and corresponding energy dissipations. To this purpose, a ‘frictional−collisional’ rheological law allowing to simulate the energy dissipation related to grain inelastic collisions and friction, simultaneously acting within the basal ‘shear layer’, is developed and applied. The interstitial pressures, and their possible increment along curved paths related to centrifugal force, and the mass variation due to erosion/deposition processes are also considered. The numerical simulation of a documented experimental test is finally carried out.

Multi-phase-flow modeling of underwater landslides on an inclined plane and consequently generated waves

An underwater landslide is a complex multiphase flow phenomenon with potential to cause large waves and destruction. Particle size is an important parameter that affects flow behaviors of multiphase flows. This numerical study examines underwater granular landslides on an inclined plane and their resultant waves with the purpose of exploring how the particle size affects landslide kinematics and wave amplitudes. A rheology-based multi-phase flow model is employed, newly incorporating the Navier-slip condition. Smooth and rough beds that reflect ideal and practical conditions, respectively, are simulated. Five types of particle are used; they are very fine sand, medium sand, coarse sand, granules, and pebbles. The granular masses are initially in the loose-packing condition. Our numerical results demonstrate that a smooth bed results in the faster landslides and the larger impulsive waves than a rough bed. The effects of particle size differ between the two bed conditions. For the rough bed, the relationship between the landslide speed and the particle size is negative, and so is the relationship between the wave amplitude and the particle size. For the smooth bed, the two relationships become concave and non-monotonic.

Mechanical interpretation of dry granular masses impacting on rigid obstacles

The evaluation of impact forces exerted by flowing granular masses on rigid obstacles is of fundamental importance for the assessment of the associated risk and for the design of protection measures. Empirical formulae are available in the literature estimating the maximum impact force; most of them are based on over-simplifying hypotheses about the behaviour of the granular material. For practical applications, formulations based on either hydrodynamic or elastic body models are often employed. These formulations require the use of empirical correcting factors. In this paper, the same DEM method is used to investigate the relationship between the evolution with time of the impact force and the micromechanics of the granular mass. In fact, considering the dynamic nature of impacts, the impulse value is fundamental for the dynamic response of the barrier, and the mere information about the maximum impact force is not sufficient to design protection works, or assess the vulnerability of structures. Information about contact forces and particle velocities will be discussed and critically compared with macroscopic results. In order to progressively introduce the complexity of the impact phenomenon, four geometrical and mechanical conditions are considered: (a) vertical front, confined flow, bonded material; (b) vertical front, confined flow, purely frictional material; (c) vertical front, free surface flow, purely frictional material; (d) inclined front, free surface flow, purely frictional material.

Basal Stress Equations for Granular Debris Masses on Smooth or Discretized Slopes

AbstractKnowledge of basal stresses is essential for analyzing slope stability and modeling the dynamics and erosive potential of debris flows and avalanches. Here we derive and test new algebraic formulas for calculating the shear stress τ and normal stress σ at the base of variable‐thickness granular debris masses in states of static or dynamic equilibrium on slopes. The formulas include a lateral pressure coefficient κ, but use of a fixed value κ = 0.7 yields predictions of σ that on average err by less than 3% and of τ that on average err by less than 13% in matching basal stresses measured in six large‐scale experiments involving wet debris masses with varying geometries and compositions. Much larger prediction errors result from use of infinite‐slope or shallow‐debris approximations. Specialized versions of the new formulas apply if basal topography is discretized and represented by a “staircase” function in a digital elevation model. Use of these formulas to assess static limiting equilibrium conditions shows that the apparent basal Coulomb friction angle ϕtread of debris that engages friction acting on the horizontal surfaces (or “treads”) of a staircase sloping at an angle θ is generally described by tanϕtread = tan (ϕ − θ)+κ tan θ, where ϕ is the true basal friction angle of the same debris in contact with a uniformly sloping bed. Differences between the values of ϕ and ϕtread can greatly influence the results of numerical simulations that use unsmoothed digital elevation model topography to calculate the stability or dynamics of debris masses on slopes.