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Experimental study and mathematical modelling of the interaction between the hemispherical bottom of a container with cohesive soil

https://doi.org/10.38013/2542-0542-2020-4-87-94

Abstract

The object of this study was a product (container) for air defence systems. Experimental and computational studies of the processes of force interaction of a solid body, modelling the bottom of a container, with silt-loam soils of different moisture were carried out in order to assess the parameters of the interaction between the container and the soil. It is shown that the developed empirical model of soil penetration, which assumes constant stress on the indenter-soil contact surface, is suitable for describing the obtained experimental data.

For citation:


Shamgunov R.F., Igumnov L.A., Zhegalova K.P., Kotov V.L., Metrikin V.S., Kazakov D.A. Experimental study and mathematical modelling of the interaction between the hemispherical bottom of a container with cohesive soil. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(4):87-94. https://doi.org/10.38013/2542-0542-2020-4-87-94

Introduction

Air defence products of S-400 type may employ long- and short-range munitions installed in transporter-launcher containers (TLC) of the same launcher. Short-range missiles are launched one after another with small intervals (less than 3 s). However, at each missile launch the launcher and its container make longitudinal and lateral oscillations that exceed this interval, which may create unfavourable conditions for the next missile launch. Therefore such oscillations have to be dampened. To determine damping parameters and develop technical means for suppressing such oscillations, it is necessary to develop mathematical models simulating interaction between solid bodies (TLC, launcher) and soils of different type. Obtaining sufficiently accurate results for real products requires carrying out of computational and experimental studies on physical models, which is actually the subject of this paper.

Modelling of solid body – soil interaction is also important for damping oscillations of rotating radars mounted on movable or stationary jacked-up platforms, since heavy rocking may dramatically distort the pattern of illumination and reception, as well as reduce service life of rotary support devices.

For estimation of forces acting on air defence product (container) during its force interaction with soil, it is necessary to have experimental data of soil strength or adhesion. To determine the magnitude of adhesion forces in dispersed cohesive soils and sticky formations, with consideration of their changing over time, the ball-indenter method of N. A. Tsytovich is applied [1]. During testing performed using a special instrument or a special installation at the site ball indenter depressions are measured under a certain constant load. In so doing, load magnitude should not be too small, so that elastic deformations of soil could be neglected. Used as estimated relationships is the A. Yu. Ishlinsky’s solution [2], derived for a model of ideal rigidplastic medium. As demonstrated by respective studies, adhesion determined by the ballindenter method should be regarded as a certain complex characteristic, enabling to estimate not only adhesion as such, but, in case of low plasticity soils, to some extent also internal friction, which can be used, e.g., in computation of the ultimate loading of clay soils as per the formulas for ideally coherent bodies (without consideration of friction, which is automatically accounted for by soil adhesion value) [1]. The mean contact pressure is defined as the relationship between normal pressure on the sphere and indentation projection area on the deformed half-space surface, or Meyer hardness [3].

Another model for assessment of resistance to quasistatic indentation of body into soil, one that has found wide practical use [4][5], is based on presumption of constant character of stress acting along the normal to the indenter surface. Considering dispersion of properties of real soils and uncertainty of their composition, such model may prove more preferable, given proper calibration on the basis of experimental data [6].

This paper offers the results of experimental and computational studies of the processes of force interaction of a solid body, modelling the bottom of a container, with silt-loam soils of different moisture content carried out in order to assess the parameters of interaction between container and soil. It is shown that the developed empirical model of soil penetration, which assumes constant stress on the indenter – soil contact surface, is suitable for describing the obtained
experimental data.

Experimental procedure

The performed tests involved penetration of a steel indenter into specially prepared soil contained in a steel ring, under static loading mode. The tests were performed on two lots of soil with different moisture content, contained in a metal ring. The material used was silt-loam soil with 25 % moisture content for lot 1 and 20 % for lot 2. The soil was specially prepared and compacted in the metal rings. The tests partly were held with lubrication of indenter surface with mineral oil.

To conduct the tests, a special rigging was fabricated. The indenters with diameter of 5, 10, and 13.5 mm were made of steel 40Kh. The metal ring was manufactured from an electrically welded metal line-weld tube as per [7]. The outside diameter of the ring is 88 mm, inside diameter – 81.4 mm, ring height – 60 mm. The dimensions of the indenters and the rings were selected in accordance with the minimum sample sizes, depending on indenter diameter as provided in [8].

The experiments were conducted on a Z100 ZWICK/ROELL testing machine, which enables to run tensile and compression tests on solid and tubular samples, perform complex loading of tubular samples with simultaneous time-synchronised setting of parameters: longitudinal force (displacement) changing rate – torque (torsional angle) changing rate – internal pressure changing rate. The limiting values of testing equipment parameters by force: ±100 kN. The measuring equipment of the machine includes a force transducer for 0–250 kN, accuracy class 1 as per ISO 7500-1. The indentation depth was measured by the ‘indirect’ method using a standard displacement measurement device, connected to the moving raverse of Z100 testing machine, with the accuracy of the order of 1 µm of the measured magnitude (axial displacements of the tested sample rigidly attached to the moving traverse of the machine are sensed by a digital displacement transducer, which is integrated in the installation and ensures traverse positioning in a required point with the error of max. 1 µm). Control over the experiment, reading off and processing the data obtained during the test is provided by means of an electronic system comprising a personal computer with software package TestXpert2, adapted for the given testing machine. The testXpert software is tuned for carrying out tests of various types and offers a possibility to obtain the necessary experimental parameter values of the process being investigated, in the parametric form or in other forms, with their subsequent processing. Metrological support of the experiments described here was provided by equipment having Certificates of Verification issued by the Nizhny Novgorod Centre for Standardization, Metrology and Certification.


Test results

The performed tests involved penetration of a steel indenter into specially prepared soil contained in a steel ring, under static loading mode. The tests were performed on two lots of soil with different moisture content.

Lot 1. Indenter motion speed (testing machine moving traverse rate) was selected in accordance with the testing machine capacity: V1 = 10 mm/min, V2 = 8 mm/s. It was shown that change of the loading rate within this range had no effect on the results. Therefore, for testing the next lot of samples, the selected speed was V1 = 10 mm/min.

Lot 2. The selected indenter motion speed was V1 = 10 mm/min. For samples Nos. 2.2, 2.3, 2.4, 2.6, 2.9, 2.10, indenter was lubricated with mineral oil before test (in order to reduce friction forces on the indenter side surface when indentation depth was greater than the indenter radius). It was shown that indenter lubrication had no effect on the obtained results.

Recalculation of the machine plot “force ~ indentation depth” into plot “stress ~ deformation” was performed by the following formulas:

The obtained plots are shown in Figs. 3, 4.

Model of solid sphere – soil quasistatic interaction

In the mechanics of soil, for approximation of normal pressure on the surface of body interacting with soil, commonly used [4][5] relationship is as follows:

σn = αV2 + γ,

where α – soil density factor, considered in more detail in [9][10]; V – normal component of the body motion speed vector; γ – constant factor, equal to soil compression strength.

Shear stress on the surface of body moving in the medium is defined in accordance with the Coulomb friction model

στ = kf σn,

where kf – constant skin friction coefficient.

The resistance force equation can be represented as:

where φ – angle reckoned from the sphere top towards the free surface.

After integration we have:

The mean pressure is defined as

where r – contact area radius.

(at φ = 0 → r = R, with further penetration of the indenter, the contact area does not increase).

Applying linear interpolation for the mean pressure, depending on the penetration depth, we approximately determine resistance force as follows: 

where

In derivation, relationship r2 = 2Rw – w2 was also used.

Let us consider the case of small depths of sphere penetration. Assuming  , and neglecting the terms of higher order of vanishing 0(w), we have a linear relationship of indentation resistance force vs. penetration depth:

where  denotation – dimensionless penetration depth.

In case of low-rate penetration (quasistatic indentation), we have  and

Transforming the expression for resistance force with account of equation r2 = 2Rw – w2, we have

Hence, the relationship between quasistatic indentation resistance force and current middle area (maximum cross-section area) of the indenter surface determines the value of soil strength. Curve deflection from the horizontal straight line is an indication of friction input into the resultant resistance force.

To determine the values of constant factors, we make use of the experimental data.

Results of experiments and computations

For validation of the obtained relationships, the results of experiments with an indenter of 10 mm diameter were taken.

Given in Fig. 5 are the indentation resistance forces obtained through experiment (dark markers – no lubrication, light markers – with lubrication) and empirical relationships in the presence of friction (solid line) and with no friction (dashed line) under parameter values γ = 0.215 МPа, kf = 1/3 (the values were determined by the least-squares method).


Fig. 5
. Indentation resistance force: experiment and computation

Fig. 6 shows the indentation resistance forces referenced to the current middle area, obtained through experiment (dark markers – no lubrication) and on the base of theoretical dependence, in the presence of friction (solid line) and with no friction (dashed line).


Fig. 6
. Soil strength: experiment and computation

It can be noticed that under parameter values γ = 0.215 МPа, kf = 1/3 the computation results quite well correspond to the obtained experimental results.

For soil of the other experimental lot, similar comparisons were drawn. Summing up all of the above, the undertaken computational and experimental studies have demonstrated practical applicability of the empirical model and its sufficient sensitivity to changes in soil moisture content.

Conclusions

The undertaken experimental and computational studies of the processes of force interaction between a solid body, modelling the bottom of a container, and soil conducted with the use of advanced experimentation technology and state-ofthe art equipment have enabled to obtain assessments of the investigated process characteristics depending on the indenter penetration rate, soil strength, indenter geometry, as well as assess the magnitude of occurring stresses.

The tests with penetrating a steel indenter into specially prepared soil contained in a steel ring, under static loading mode, have demonstrated that change of the loading rate within a given range (minimum rate – 10 mm/min, maximum – 8 mm/s) and indenter lubrication have no significant effect on the characteristics of indenter penetration process.

The indentation plots and experimental data provided enable to obtain reference material on the magnitude of solid body (container bottom) penetration depending on its weight, geometry, and soil strength.

The applied state-of-the-art measuring equipment of the complex, traverse displacement high-precision transducers, and the testXpert software made it possible to represent the necessary experimental parameter values of the investigated process in a parametric form, convenient for practical use.

The conducted tests with steel sphere indentation into silt-loam soil showed that with moisture content change from 25 to 20 % there was a nearly two-fold increase in strength. To obtain a quantitative relationship between soil strength and moisture content and/or other parameters, additional experimental studies are required.

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About the Authors

R. F. Shamgunov
70th anniversary of Victory Nizhny Novgorod Plant
Russian Federation

Shamgunov Rif Fayzelkhakovich - Cand. Sci. (Engineering), Head of the Bureau of Hydraulic Systems. Research interests: hydromechanics.

Nizhny Novgorod


L. A. Igumnov
Lobachevsky State University of Nizhny Novgorod (UNN)
Russian Federation

Igumnov Leonid Aleksandrovich - Dr. Sci. (Phys.-Math)., Prof., Lobachevsky State University of Nizhny Novgorod (UNN); Chief Researcher, Research Institute of Mechanics, Research interests: theoretical and experimental mechanics.

Nizhny Novgorod


K. P. Zhegalova
Lobachevsky State University of Nizhny Novgorod (UNN)
Russian Federation

Zhegalova Ksenia Petrovna - Postgraduate student, Research interests: mechanical testing.

Nizhny Novgorod


V. L. Kotov
Lobachevsky State University of Nizhny Novgorod (UNN)
Russian Federation

Kotov Vasily Leonidovich - Dr. Sci. (Phys.-Math.), Prof., Lobachevsky State University of Nizhny Novgorod (UNN); Chief Researcher, Research Institute of Mechanics, Research interests: computational mathematics and mechanics of geomaterials and porous media.

Nizhny Novgorod


V. S. Metrikin
Lobachevsky State University of Nizhny Novgorod (UNN)
Russian Federation

Metrikin Vladimir Semenovich - Cand. Sci. (Engineering), Assoc. Prof., Research interests: mechanics (strength, dynamics, resource).

Nizhny Novgorod


D. A. Kazakov
Lobachevsky State University of Nizhny Novgorod (UNN)
Russian Federation

Kazakov Dmitry Alexandrovich - Cand. Sci. (Engineering), Laboratory Head, Research Institute of Mechanics, Research interests: physical and mechanical testing of materials.

Nizhny Novgorod


For citation:


Shamgunov R.F., Igumnov L.A., Zhegalova K.P., Kotov V.L., Metrikin V.S., Kazakov D.A. Experimental study and mathematical modelling of the interaction between the hemispherical bottom of a container with cohesive soil. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(4):87-94. https://doi.org/10.38013/2542-0542-2020-4-87-94

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