Sensitivity analysis,calibration and validation of a snow indentation model

Quantification of the mechanical behavior of snow in response to loading is of importance in vehicle-terrain interaction studies. Snow, like other engineering materials, may be studied using indentation tests. However, unlike engineered materials with targeted and repeatable material properties, snow is a naturally-occurring, heterogeneous material whose mechanical properties display a statistical distribution. This study accounts for the statistical nature of snow behavior that is calculated from the pressure-sinkage curves from indentation tests. Recent developments in the field of statistics were used in conjunction with experimental results to calibrate, validate, and study the sensitivity of the plasticity-based snow indentation model. It was found that for material properties, in the semi-infinite zone of indentation, the cohesion has the largest influence on indentation pressure, followed by one of the the hardening coefficients. In the finite depth zone, the friction angle has the largest influence on the indentation pressure. A Bayesian metamodel was developed, and model parameters were calibrated by maximizing a Gaussian likelihood function. The calibrated model was validated using three local and global confidence-interval based metrics with good results.     

Predicting terrain parameters for physics-based vehicle mobility models from cone index data

To provide terrain data for the development of physics-based vehicle mobility models, such as the Next Generation NATO Reference Mobility Model, there is a desire to make use of the vast amount of cone index (CI) data available. The challenge is whether the terrain parameters for physics-based vehicle mobility models can be predicted from CI data. An improved model for cone-terrain interaction has been developed that takes into account both normal pressure and shear stress distributions on the cone-terrain interface. A methodology based on Derivative-Free Optimization Algorithms (DFOA) has been developed in combination with the improved model to make use of continuously measured CI vs. sinkage data for predicting the three Bekker pressure-sinkage parameters, k_c, k_ϕ and n, and two cone-terrain shear strength parameters, c_c and ϕ_c. The methodology has been demonstrated on two types of soil, LETE sand and Keweenaw Research Center (KRC) soils, where continuous CI vs. sinkage measurements and continuous plate pressure vs. sinkage measurements are available. The correlations between the predicted pressure-sinkage relationships based on the parameters derived from continuous CI vs. sinkage measurements using the DFOA-based methodology and that measured were generally encouraging.     

Sinkage tests for mobility study, modelling and experimental validation

The sinkage of the bearing tracks or wheels of a vehicle in soil induces a resistance to travel motion. Usually it is determined with methods based on the modelling of soil pressure-sinkage curves. This article presents a new method for modelling soil penetration tests as a result of experimental study of three standard soils. These soils have been chosen to represent the mechanical properties of a range of soils: a sand for frictional soils, a silt for cohesive soils and a silty sand for cohesive frictional soils. The models take into account the mechanical behaviour of soils where a small vertical sinkage can be assumed analogous to elastic behaviour, while for large sinkage, the analogy is with plastic behaviour. A New Model of Mobility (N2M) is proposed. A new equation relating the pressure p and the sinkage z is governed by four parameters which are constant for a specific soil in a given physical state. These parameters can be calculated with two sinkage tests made with two different plate diameters and are particularly stable: a small change of one of them involves a small change of the modelling. They are independent of the size of the sinkage plate and hence could pave the way for the extrapolation to the scale of full size vehicles. For the tested soils, comparison of the model results with experimental tests is very promising.     

On conditions of snow avalanching and landslides

The problem of elastic equilibrium of a wedge-shaped ground or snow massive on a rigid inclined plane under the action of gravity force and a constant force on its outer surface is considered in the three-dimensional statement. An exact solution that allows one to determine the displacements, strains, and stresses at each point of the massive is obtained in final form. An analysis shows that there are several critical relations between the problem input parameters at which the massive equilibrium is impossible. In this case, the earth or snow tears off from the slope and becomes a dangerous avalanche. In particular, this result can be used to predict formation of landslides and snow avalanching in mountains.     

Diffraction of acoustic and elastic waves on a half-plane for boundary conditions of various types

The classical problem of wave diffraction on a half-plane with boundary conditions of different types and its generalizations to elastic media are considered. As a solution method it is proposed to combine the Fourier method of separation of variables and the series summation technique based on the use of integral representations of Bessel functions. The analytic solutions thus obtained are equally efficient in the near- and far-field diffraction regions. The two-term singularity at a corner point (in stresses for elastic media and in the velocity for acoustic media) was discovered for the first time. The knowledge of singularities in the scalar problem allowed one to construct the solution of the vector problem of elastic longitudinal wave diffraction. It is investigated how different types of boundary conditions on both sides of the half-plane affect the solution behavior in the far-field region. Possible physical interpretations of the obtained results are given.     

Sliding a body on snow

Snow is considered as an ideal nonlinear elastoplastic medium. A body performs planeparallel motion on snow. The area of its contact with snow is a part of a rectangular plate. The contact zone changes during the motion of the body. Steady motions are found from the derived equations of motion in the case when the constant external forces and the moment exerted on the body are given. The inverse problem of determining the forces and moments is solved for a given steady motion of a vehicle.     

Motion of a wheel on snow

A model of a snow layer represented by a continuous set of columns whose deformations are described by the nonlinear model of an ideal elastoplastic continuous medium with viscous properties is proposed. Under the action of a rigid wheel on snow, the field of shear stresses is specified by the law of dry friction. Prom the equations of motion describing the plane-parallel motion of the wheel, there are determined a zone of contact of the wheel with snow, the steady motions of the wheel, and a mode of slipping the wheel. The numerical results are given in tables and figures. These results are obtained by solving the nonlinear equations of motion containing definite integrals with variable integration limits.     

Mechanical properties in relation to vehicle mobility of Sepang peat terrain in Malaysia

An analytical framework for determining the mechanical properties of peat and predicting the tractive performance of tracked vehicle is presented. It takes into account the load-sinkage and shearing characteristics of peat as well as all major design parameters of tracked vehicle. An experimental study on the mechanical properties of peat soil was conducted at Sepang area, Selangor, Malaysia. The stiffness values of surface mat and underlying weak peat deposit from load-sinkage test were determined by specially made bearing capacity apparatus. The mean values of surface mat stiffness before and after drainage were found to be 31 and 45.62 kN/m³, respectively and the mean values of underlying peat stiffness before and after drainage were found to be 252 and 380.20 kN/m³, respectively. The mean value of the internal frictional angle, cohesiveness and shear deformation modulus of the peat soil sample were determined using a direct shear box apparatus in the laboratory. The mean values of internal friction angle, cohesiveness and shear deformation modulus before and after drainage were found to be 22.80° and 24.31°, 2.63 and 2.89 kN/m², and 1.21 and 1.37 cm, respectively.     

On vehicle mobility in snow-covered terrain — 1. Problem development and requirements for analysis

The problem of evaluation and prediction of vehicle mobility on snow-covered terrain needs to be studied not on the basis of application of direct technology transfer from vehicle mobility on soil, but on the basis of new perspectives on material (snowpack) properties and response performance. The complexities of snow identification and classification, arising from local environmental control and thermodynamic history, render analogies between snow and soil inapplicable. In addition, it is significant to note that in snow trafficability considerations, the first pass is the worst pass.     

Calibration of a simple cone-penetration model for snow

Numerical studies using the Material Point Method (MPM) have been conducted recently to model snow penetration tests for fine-grained and coarse-grained snows using small cones with diameters ranging from 2.5 mm to 4 mm, and cone half-angles between 15° and 45°. Although numerical studies have gained physical insight of these tests, due to the lengthy computation time needed for the MPM simulations, it is not feasible to use these simulations to develop a stochastic model to assess the large variations of the mechanical properties of snow typically shown in tests. In this paper, we present a simple and efficient physics-based analytical model based on equilibrium and a cavity expansion solution upon which a stochastic model is built to obtain calibrated material parameters for a Drucker–Prager (DP) model such that prediction of the model can be made. Sensitivity analysis of the analytical model indicates that cohesion and interfacial shear (friction) factor contribute significantly to the penetration hardness whereas the friction angle has little contribution. The calibrated material parameters are similar to those estimated via the MPM simulations. The quality of the stochastic model, when compared with test data, was assessed using four interval-based validation metrics with good results.     

The motion of snow avalanches in the conditions of limiting friction

We investigate the equations of motion of large snow avalanches, and in contrast with [1–3] we take into account the fact that the dry friction can reach a critical value above which the snow in the avalanche or the underlaying material cannot sustain the friction. We find asymptotic solutions for long times after the beginning of motion. These solutions describe the avalanche motion in which a part of the snow moves in the conditions of limiting friction over a tilted plane with a uniform layer of snow. The equations which are used to find these asymptotic solutions have the property that for certain depths the flow velocity of small perturbations decreases with increasing depth. This is related to a number of unusual features (from the hydraulic point of view) of the solutions. In particular, on relatively gentle slopes two zones are formed in the avalanche: the forward part, with a large velocity and thickness of the moving layer, and the rear part, which is significantly slower and thinner. The two parts are separated by a narrow region characterized by a sharp decline in velocity and thickness of the moving layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–37, September–October, 1977.     

Expansion of the terrain input base for nepean tracked vehicle performance model, NTVPM, to accept swiss Rammsonde data from deep snow

A study of the correlation between the measured and predicted vehicle performance over undistributed and preconditioned snow using the Nepean Tracked Vehicle Performance Model, NTVPM, has been carried out. It is shown that on undisturbed snow in Fernie, British Columbia, the performance of a BV 206 predicted by NTVPM correlates very well with measured performance obtained in the field. On preconditioned snow, there is also a reasonable correlation between the measured vehicle performance and predicted one using NTVPM. It is found that predictions of vehicle performance made by NTVPM using pressure-sinkage data obtained with the Swiss Rammsonde and with the bevameter are comparable. This indicates that the pressure-sinkage data obtained using the Rammsonde can be used as input to the NTVPM for predicting tracked vehicle performance over snow. It is shown that in comparison with an earlier version, NTVPM-85, the latest version of the Nepean Tracked Vehicle Performance Model, NTVPM-86, which takes into account fully the characteristics of roadwheel suspension systems, provides improved predictions of vehicle performance over snow where track sinkage is significant. It is suggested that the computer simulation model NTVPM, using pressure-sinkage data obtained by the Rammsonde as input, could form a useful interface with cone based models, such as the NATO Reference Mobility Model, to provide them with an additional capability of predicting tracked vehicle performance over snow.     

Computation of flow over a spherically blunted cone for various shock layer and surface blowing conditions

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 100–106, March–April, 1991.     

Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory

Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.     

Finite element modeling of tire/terrain interaction: Application to predicting soil compaction and tire mobility

Tire/terrain interaction has been an important research topic in terramechanics. For off-road vehicle design, good tire mobility and little compaction on terrain are always strongly desired. These two issues were always investigated based on empirical approaches or testing methods. Finite element modeling of tire/terrain interaction seems a good approach, but the capability of the finite element has not well demonstrated. In this paper, the fundamental formulations on modeling soil compaction and tire mobility issues are further introduced. The Drucker-Prager/Cap model implemented in ABAQUS is used to model the soil compaction. A user subroutine for finite strain hyperelasticity model is developed to model nearly incompressible rubber material for tire. In order to predict transient spatial density, large deformation finite element formulation is used to capture the configuration change, which combines with soil elastoplastic model to calculate the transient spatial density due to tire compaction on terrain. Representative simulations are provided to demonstrate how the tire/terrain interaction model can be used to predict soil compaction and tire mobility in the field of terramechanics.     

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated. This paper is partially supported by the Deanship of Scientific Research at King AbdulAziz University (Grant no. 172/427).     

Vibrations of polygonal plates with various boundary conditions

The bending vibrations of polygonal (L-shaped) plates with different shapes and boundary conditions are studied. The natural frequencies are calculated using the inverse-iteration and Kantorovich-Vlasov methods. To take the configuration of the domain into account, the fictitious domain method and an analog of the force method of structural mechanics are used. Different trends in the dependence of the lowest natural frequency of an L-shaped plate on its geometry are illustrated for different boundary conditions.Acorrelation between the extreme values of the bending frequency and some relations for the energy characteristics of the plate is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 63–72, May 2007.