=Paper=
{{Paper
|id=Vol-3179/Paper_18.pdf
|storemode=property
|title=Models and information Technology Implementation of Contact Interaction of Theedge of a Rock Cutting Tool with Granular Rocks Modeling
|pdfUrl=https://ceur-ws.org/Vol-3179/Paper_18.pdf
|volume=Vol-3179
|authors=Yaroslav Petrivskyi,Mykhailo Tymchuk,Volodymyr Petrivskyi,Viktor Shevchenko
|dblpUrl=https://dblp.org/rec/conf/iti2/PetrivskyiTPS21
}}
==Models and information Technology Implementation of Contact Interaction of Theedge of a Rock Cutting Tool with Granular Rocks Modeling==
https://ceur-ws.org/Vol-3179/Paper_18.pdf
Models and Information Technology Implementation of
Contact Interaction of the Edge of a Rock Cutting Tool With
Granular Rocks Modeling
Yaroslav Petrivskyia, Mykhailo Tymchuka, Volodymyr Petrivskyib and Viktor Shevchenkob
a
Rivne State Humanitarian University, St. Bandery str. 12, Rivne, 33000, Ukraine
b
Taras Shevchenko National University of Kyiv, Bohdan Hawrylyshyn str. 24, Kyiv, 01001, Ukraine
Abstract
In the article a mathematical model of the destruction of strong rocks cracked normal
separation for the developed technology of directional splitting of the geological environment
with a rock cutting tool a cyclic load is presented. Indicated that quality model indicators are
the values of the physicomechanical structure geomaterial, a criterion for the steady growth of
a normal separation crack, stress intensity factor of the first kind, the length of the initial crack
deepening, angle and type of sharpening the edge of the rock cutting tool. The solution to the
problem is based on a numerical model of a plane boundary value problem theory of elasticity.
The condition for the opening of microcracks in the contact area is established. cutting part of
rock cutting tool with granular rock structures, as well as an indicator of the initial crack output
to the surface for model example.
Keywords 1
Anti-filter screen, rock cutting tool, mathematical model, crack, area of weakened connections,
discontinuous displacement method, dummy force method.
1. Introduction
The last decades have been marked by a huge increase of energy and mineral resources
consumption. This fact creates a huge amount of industrial waste, which significantly affects
the environmental status of the territories. Modern landfills for solid wastes are engineering
specialized structures, where organized controlled storage of solid household wastes in
compliance with technical and sanitary norms, reducing the negative impact of waste on the
atmospheric air, soil, water pool to the normative level.
One of the effective ways to prevent the negative effects of industrial and domestic waste
accumulation on the environment is the construction of various types of protective anti-filtering
structures. In particular, this problem can be solved by creating in the underlying array rocks
antifiltration collector screen, which has the necessary strength characteristics and allows
directional collection of filtrates. The arrangement of the screen involves the creation in the
array of technological cavity rocks and filling it with hardening barrier material. In the case of
reinforcing the cutting edge of the rock-cutting tool, its immersion in the array occurs from the
initial moment of impact - the arrival of the initial wave of stress. Under the edge, in the zone
of greatest stresses, the core of the seal of the damaged rock begins to grow, due to which a
smoother redistribution and transfer of the impact pulse to the array occur. In articles [1,2]
theoretical and applied bases of technology of infiltration leaching of uranium from
technogenic deposits of complex form underground have been developed and substantiated. A
Information Technology and Implementation (IT&I-2021), December 01β03, 2021, Kyiv, Ukraine
EMAIL: prorectorsgu@ukr.net (A. 1); vsirf.17@gmail.com (A. 2); vovapetrivskyi@gmail.com (A. 3); gii2014@ukr.net (A. 4)
ORCID: 0000-0001-9749-8244 (A. 1); 0000-0003-1793-5999 (A. 2); 0000-0001-9298-8244 (A. 3) ; 0000-0002-9457-7454 (A. 4)
Β©οΈ 20222 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
194
key element of the leaching process scheme is the anti-filtration manifold screen, which allows
direct filtrate collection. In papers [1, 2, 3] a method of creating an anti-filtration manifold
screen in the underlying bedrock, which ensures the filtrate is collected and brought to the
surface is proposed. The method involves arranging the screen by creating an inclined process
cavity in the underlying rocks and then filling it with a curing barrier material. Also, the method
and a destructive device for mining a mountain range and forming a technological cavity is
proposed [2, 3].
A review of the problems of the stress-strain state of the region under the action of special
loads, rock masses in the vicinity of technological workings and mathematical methods for
solving them is given quite fully in the works [1, 4-12]. At the same time, the well-known
mathematical methods and research methods of such problems give only a general approach to
solving a particular problem, and therefore, for each specific case, in order to obtain reliable
results, detailed elaboration is necessary taking into account the specifics of the technology.
Thus, the main goal and objective of the study is to create a mathematical model of the
destruction of strong rocks by normal separation cracks for directional cracking of the
geological environment by the proposed rock cutting tool in the process of shock-rotational
cyclic loading. Schematically, the process of destruction can be described as follows.
2. Main part
The problem of implementing new technologies for mining rock mass when creating
technological workings by an underground method is a complex task of computer technologies
in the geomechanical theory of stability, the integral components of which are: establishing
patterns of formation of stable and unstable cracks; creation of a numerical model of the
destruction of granular material; development of algorithms for the implementation of
numerical calculations, allowing to establish the patterns of formation and development of
cracks in discrete media loaded with various rock cutting tools.
The formation of structural damage occurs due to the violation of bonds between the
elements of the microstructure of the material, which always has some heterogeneity. So,
metals are characterized by grains and intercrystalline matter, rocks by microheterogeneity of
their tectonic formation, concretes and other composite materials by the composition of the
matrix and filler, their ratio, filler orientation. With the opening of microcracks in such
materials, local microstresses increase, caused by the action of the applied tensile stress, which
contributes to the rate of accumulation of local stresses. Consequently, in addition to the
existing internal tensile stress, microstress forces begin to act, depending on the morphological
features of the environment.
Currently, there are enough models and experimental data that allow us to study the process
of formation and development of microcracks in materials. However, not all existing models
take into account the structural features of the studied material samples. So, in the process of
numerical simulation of fracture when the fracture criterion is fulfilled in a certain
computational cell, all the material in this cell is destroyed and is considered a set of small
particles. Further, for this cell, the mechanical properties of the material change: the ability to
resist stretching is lost, its elastic moduli decrease, and the stress deviator is reset. In this case,
not all cracks in the fracture patterns intersect or close, forming contoured particles, which is
caused by both certain computational difficulties and physical reasons. The former are
associated with large distortions of the computational grid when describing fracture, which is
why the calculation cannot be carried out far enough in terms of deformations. On the other
hand, the deformation conditions simulated in the calculations inhibit the growth of cracks.
Such circumstances complicate the automatic processing of such fracture patterns; therefore,
special methods were needed to refine the images obtained in the calculations.
195
For our case, modeling the destruction of a rock mass by a rock-breaking tool, according to
the technological scheme [1,2], has the following distinctive and characteristic features. When
the lateral boundaries are far removed from the impact site, plastic deformation develops deep
into the massif. The emergence of deformation localization areas on the free surface with close
lateral boundaries leads to gouging out of large pieces of geomaterial. In a brittle material, from
the sides of the affected area, where tensile stresses arise, the formation of a destroyed area is
observed. Repeated exposure, which is characteristic of periods of failure, can lead to the
propagation of cracks into the interior of the specimen. A zone of the most loaded and deformed
material is formed near the tip of the cutting edge of the rock cutting tool. It is in it that the
main part of the destroyed particles is formed. The stress state field has a complex structure
and most closely corresponds to a combination of compression and shear. To determine the
fractional composition of the spall, it is necessary to calculate the behavior of the mesovolume
of the rock under the given impact conditions. For the numerical implementation of the fracture
process, further in the main part of the article, a technique is proposed using the node separation
algorithm and an explicit description of crack opening, which takes into account the complexity
of the interaction of destroyed particles during multiple cracking under conditions of
simultaneous compression and shear.
When simulating deformation and fracture processes, which is determined by the process
conditions, a dynamic approach is used that includes equations describing the motion of a
continuous medium, defining relations that concretize the behavior of a particular material
within the framework of the selected medium model. The main system of equations expresses
the laws of conservation of mass, momentum and energy, as well as geometric relationships
connecting, for example, the components of the strain rate tensor with the components of the
displacement velocity vector. For example, nonlinear parabolic equations as a general system
of equations for the dynamics of elastoplastic media as kinetic equations, when it corresponds
to the physics of the process (similar equations determine the nucleation and propagation of
damage and inelastic deformation fronts in a loaded nonlinear medium, which propagate at
velocities significantly lower than the propagation velocity elastic disturbances).
A distinctive feature of the choice of a mathematical model for the propagation of an impact
crack with repeated exposure to a rock cutting tool [1, 2] is the formation of new cracks and
areas of destruction with each new repeated exposure. For this, it is proposed to calculate the
stress-strain state near a single crack, describe its behavior, and perform using a stepwise
calculation procedure or the stationary grid method. The problem is realized by solving the
partial differential equations:
π 2 π’π₯ π 2 π’π₯ 1 π 2 π’π₯ π 2 π’π¦
+ + ( + ) = 0,
ππ₯ 2 ππ¦ 2 1 β 2 β π ππ₯ 2 ππ¦ππ₯
π 2 π’π¦ π 2 π’π¦ 1 π 2 π’π₯ π 2 π’π¦
+ + ( + ) = 0,
ππ₯ 2 ππ¦ 2 1 β 2 β π ππ₯ππ¦ ππ¦ 2
where π’π₯ β axial ΠΠ₯ offset; π’π¦ β axial ΠY offset; π β Poisson's ratio.
In an array, the volume of compression increases with the constant gradient of compression
to the core of the compression. The growth of stress occurs as a result of the immersion of the
cutting edge and the growth of the core of the seal, and as a result of the internal reflection of
the wave energy emitted by the edge of the deep into the array. When the stresses reach the
boundary value for a given breed, values in the zone bordering the core of the seal begin to
give rise to microcracks, which release part of the elastic energy and cause the splitting off of
the volume of compression of a particular layer. Part of this layer passes into the core of the
seal (adhering to the cutting edge), and the other part forms the region of weakened ligaments
around the nucleus. In the future, this process is repeated. In the next, adjacent nucleus, the
196
layer of stress reaches the limit value, the origin of microcracks, their growth and the increase
of the area of weakened bonds occurs. At the next time there is an increase of the nucleus again,
and so long as the resistance of the breed to the immersion of the edge does not equal the force
of the impulse or until the blow ends. This complex spatial problem can be reduced to a plane
problem of the theory of elasticity, if we divide it into n elementary problems, as follows. The
cutting edge of the rock-cutting tool interacts with the array along the circle arc. It is natural to
assume that if a crack occurs and reaches the surface of the face in the furthest point of the arc,
the same processes will occur in all other points of the interaction arc. Thus, we arrive at the
problem of the propagation of a crack in a half-infinite plate with a damped velocity (Fig. 1).
Figure 1: Scheme of propagation of the crevice.
The peculiarity of the problem is the proximity of the crack to the edge. Because of this
proximity, the crack will not develop straightforwardly, but will tend to a surface free from
stress (surface of the face). This hypothesis is in good agreement with the main provisions of
the theory of Bussinesca.
Also, it should be noted that the process of growth of a spall crack also depends on the
physicomechanical structure of the geomaterial, the criterion for the stable growth of a normal
detachment crack, the stress intensity factor of the first kind, as well as the length of the initial
deepening crack, the angle and type of edge sharpening of the rock cutting tool.
In the proposed model, the geological environment (rock), according to [11, 12], has a
granular structure, consists of a set of grains that are bonded to each other with cement having
microcracks of different order of length. Microcracks differ in aggregate composition, width
and shear resistance. It is assumed that adhesion along the length of the crack is absent or an
order of magnitude smaller than along cracks of shorter length and width. The angle of friction
for smaller cracks is several degrees larger than for long ones. Upon destruction, the rocks
exhibit the property of a discrete medium consisting of a set of individual particles that, due to
the load, are displaced relative to one another.
Thus, a granular geomedium is modeled by a grid of hexogonal, much smaller compared to
body size, homogeneous and isotropic grains with a constant averaged grain diameter of the
order of 10-1 mm and a mathematical section between grains as microcracks. Between the grain
space (microcracks) are filled with a linearly elastic material with properties different from
those of the grain (Young's modulus E=10n n=2,3,4 MPa, Poisson's ratio Ξ½=0,3, intergranular
πΈ
width 0,01-0,-01 mm, shear modulus πΊ = 2β(1+Ξ½)).
197
The boundary conditions at the contact boundary of the rock cutting tool were modeled by
the case of the action of the wedge-shaped edge of the tool on the rock of the above structure.
The brittle properties of a material, regardless of its strength and the method of load application,
determine the mechanism of interaction with the working tool. A design scheme for the
interaction of a wedge-shaped edge of a rock cutting tool with an array of brittle geological
material under impact is presented on the Figure 2. A force π is applied to the tool, under the
action of which, overcoming resistance to penetration, the tool is immersed in the array. Let us
assume that the width of the tool blade and the width of the interaction area are equal:
π = π, (1)
where π β interaction area width, m; π β tool blade width, m.
The material has isotropic properties. Also, during the simulation, the following
assumptions were made:
οΎ The width of the tool blade is equal to the width of the impact area;
οΎ Material properties are isotropic.
P
πΌ π½
A
b
h
ππ1
ππ
A
ππ2
Figure 2: Scheme of the interaction of the wedge-shaped edge of the rock cutting tool with
an array of brittle geological material.
t the first stage, a linear contact of the working tool with a step occurs, stresses arise in the
material that exceed its contact strength, local destruction of the material is observed and the
tool moves deep into the massif. The process is symmetrical, so one half of it is considered. As
the tool advances, the contact area of its lateral surface with the material appears, on which a
normal force occurs:
π
ππ = 2βsin πΌ, (2)
where Ξ± β half the angle of the rock cutting tool.
At the same time, the reaction acts from the side of the mass of material:
ππ = πΡππππ β π, (3)
where ΟΡompr β material compressive strength, Pa;
contact area of the lateral surface of the tool with the material can be calculated with the
next formula use:
198
πββ
π = cos πΌ, (4)
where h β immersion depth of the tool, m.
Equating expressions for ππ and substituting the value of S, we find the depth of immersion
of the edge of the rock cutting tool in the form:
πβcos πΌ
β = 2βsin πΌβπ . (5)
ΡΠΆΠ°Ρ βπ
The normal force ππ can be decomposed into two components ππ1 and ππ2 , which are
directed towards the open surface and the depth of the massif, respectively. Of interest is ππ1 ,
under the action of which shear stresses in the plane A-A, emerging on the free surface:
π
πΠβΠ = π π1 , (6)
π΄βπ΄
where SA-A β cross-sectional area of the material in the plane A-A, m2.
The component ππ1 is equal:
ππ1 = π β sin π½, (7)
where Ξ² β angle between the lateral surface of the tool and the plane Π-Π.
Cross-sectional area of the material in the plane Π-Π is equal to:
πββ
ππ΄βπ΄ = cos(πΌ+π½). (8)
Normal force ππ can also be decomposed into components ππ3 and ππ4 , as shown on the
figureg 2. The component ππ4 acts at an angle Ξ³ and creates shear stresses in the plane A-B,
which emerges on the side surface of the step. Using the previous reasoning, the dependencies
describing the process of spallation of an array of material can be represented as follows:
π
ππ΄βπ΅ = π π4 , (9)
π΄βπ΅
ππ4 = ππ β cos(πΌ β πΎ), (10)
πβπ
ππ΄βπ΅ = cos πΎ . (11)
The presented dependences make it possible to determine the angles Ξ² and Ξ³, at which the
material is βpuncturedβ and βchippedβ. The above dependences are valid for a limited volume
of rock - oversized. According to the design scheme shown in Fig. 2, with the impact interaction
of the wedge-shaped edge of the rock cutting tool with the rock area, the normal components
of the impact force Ο_n create tensile stresses in the A-A plane, in which a split can occur. In
addition, shear stresses appear, acting in the planes A-B and A-C, along which, most likely,
chipping or gouging occurs, respectively. Depending on the ratio of the energy of a single
impact, the strength of the material, the size of the interaction area, as well as the point of
impact, the destruction of the rock will occur in one of the indicated planes. Taking into account
the previous studies, the model characterizing the boundary conditions for the problem of
impact interaction of the wedge-shaped edge of a rock-cutting tool with an array of brittle
geological material can be described by the formulas:
πβπ£ 2
β = 2βπΉ , (12)
π£ = β2 β π β π», (13)
πΉ = 2 β π β sin πΌ, (14)
ππ = πΡππππ β π , (15)
199
2βπββ
π = cos πΌ, (16)
πβπ£ 2
π= 2 , (17)
π1 = π β (π β β), (18)
π
ππ1 = ππ β cos πΌ, (19)
1
πΌ
πΎ=2 , (20)
ππ2 = ππ β cos(πΌ β πΎ), (21)
πβπ
π2 = , (22)
cos πΎ
π
π2 = ππ2 , (23)
2
ππ3 = ππ sin π½, (24)
πββ
π3 = cos(πΌ+π½), (25)
π
π3 = ππ3 , (26)
3
where: h β immersion depth of the tool, m; m β impact mass, kg; v β impact velocity, m/s; P β
impact force, Π; g β acceleration of gravity, m/s2; h β lifting height of the striking part, m; ππ
β normal component of impact force, Π; Ξ± β half the angle of the tool tip, rad; ΟΡompr β material
compressive strength, MPa; S β the area of contact of the lateral faces of the tool with the rock,
m2; a β interaction area width, m; T β the kinetic energy of the instrument at the moment of
impact, J; S1 β split surface area, m2; c β oversized height, m; Οn1 β normal stresses in the split
plane, MPa; Ξ΄ β the angle of inclination of the cleavage surface to the horizon, rad; Οn2 β
component of the impact force in the cleavage plane, Π; S2 β cleavage surface area, m2; Ο2 β
shear stresses in the cleavage plane, MPa; Ξ² β the angle between the split surface and the side
edge of the tool, rad; Οn3 β impact force component in the split plane, Π; S3 β gouge surface
area, m2 ; Ο3 β shear stresses in the puncture plane, MPa.
The developed mechanism of the initial stage of impact fracture makes it possible to
establish the influence of the energy of a single impact, the strength of the material and the
shape of the tool, as well as the point of impact on the main parameters of the fracture process.
To increase the stresses in the interaction area, the energy of a single impact should be increased
and the angle of sharpening of the working tool should be reduced. The developed mechanism
for the destruction of oversized objects makes it possible to establish the dependence of the
energy of a single blow, necessary for destruction, on the size, physical and mechanical
properties of the rock and the angle of the tool tip.
When modeling the process of destruction of a granular geostructure, which occurs
sequentially in two stages [13]: delocalized accumulation of microcracks and their combination
into large cracks up to the moment of the formation of main cracks, leading to macrofracture
and separation of the body into parts, according to [6], the criterion of critical disclosure of
microcracks along the grain boundaries of the material:
4βΠ2
πΏΠΊΡ = πβΠβπ1 , (27)
ΠΊΡΡ
where Π1 β stress intensity factor of the 1st kind, πΠΊΡ β material strength limit.
An important component in modeling the process of formation and further propagation of
an impact crack is the problem of calculating the stress-strain state in the area of fracture and
the formation of new cracks. This requires an algorithm that implements the physical essence
of the occurrence of discontinuities. In this case, the calculation of the stress-strain state near a
single crack, the description of its behavior, it is advisable to carry out using the well-known
methods of fracture mechanics, concerning the description of the crack surfaces, using the
200
stepwise calculation procedure or the stationary mesh method. The implementation of the
highest quality calculation of the stress-strain state of an arbitrary fracture region guarantees a
more accurate identification of the fracture surfaces, although, according to the principle of the
adequacy of the construction of mathematical models, sufficient information content of
arbitrary fracture is achieved without a precise description of their surfaces [14, 15].
Thus, it is proposed to explicitly model the occurrence of cracks in the process of
mining a rock mass with a rock-cutting tool of percussion action with the creation of new free
surfaces. The application of the Lagrange approach to the description of the motion of a
medium allows one to interpret the behavior of points and their boundaries as elements of the
material structure. We accept that the processes of destruction of geomaterial are localized
along the boundaries of the cells. In this case, damage to such non-integral regions leads to the
formation of fracture crack surfaces, for which the same number of nodes remained on each of
the fracture edges. This procedure is carried out through the division of the nodes of the
computational grid, when new boundaries are formed along the boundaries of the
computational cells. Such an algorithm of fracture along the boundaries of cells guarantees a
better picture of the direction and shape of cracks together with the field of the stress-strain
state in the region of the apex. In addition, the inconsistency of the boundaries of the
computational cells guarantees the fulfillment of the law of conservation of their mass and
nodes, as well as the correspondence of the boundaries of the computational cells during
discretization, in the case of interfaces in heterogeneous rock massifs. In the standard method,
each of the nodes simultaneously belongs to its four neighboring cells, which determine its
mass as the average value of four surrounding cells. For the case of inconsistency of the
boundaries of the computational cells, it is observed that it is combined into one node out of
four before the destruction of the rock mass begins. At the onset of fracture, which is regulated
by the appropriate criterion for the critical opening of microcracks along the grain boundaries
of the geomaterial, the group of combined cells disintegrates with the formation of not one, but
two three, four nodes in accordance with the shape and direction of the cracks and the area of
destruction. The scheme for calculating the parameters of the stress-strain state does not change
in this case, since the calculations are performed in the grid cells. After the separation of the
group, its constituent nodes belong to the newly formed surfaces of the crack edges. In the
further calculation of the crack growth, the corresponding conditions are specified at all newly
formed boundaries, which, when opened, are equivalent to the formation of a free surface. As
mentioned, when the fracture criterion is met, the crack opens along the boundaries of the
corresponding design cells. Since the calculation uses a rectangular mesh, each newly formed
crack can be directed in two orthogonal directions, which determines both the fracture area and
the shape together with the direction of the cracks. The choice of the partitioning scheme for
each node is made by checking the state of each of the cell boundaries, since this determines
the configuration of the destroyed area.
The solution of the problem of opening and developing a crack under the influence of
the edge of a rock-cutting tool is made on the basis of a numerical model of a flat boundary
value problem of the theory of elasticity by methods of discontinuous displacements and
fictitious loads. The methods are based on the analytical solution of the problem of the infinite
Π₯ΠY plane, the displacements in which suffer a constant in magnitude discontinuity within the
finite segment. The solution is considered as a special module of the boundary-element
program of numerical solution. The physical discontinuity displacement is modeled as a linear
crack, the sides of which are displaced relative to each other. In this case, the surfaces are
shifted by a constant along the entire crack. In the general case, an arbitrary distribution of the
relative displacement is considered. The discontinuous displacement method is based on the
notion that displacement discontinuities continuously distributed along a crack are replaced by
a discrete approximation by breaking the crack into N boundary elements and, within each
201
element, displacement discontinuities are assumed to be constant. Knowing the analytical
solution for one constant discontinuity displacement and summing up the effects of all N
elements, we find a numerical solution to the problem. For the case when the distribution of
displacement discontinuities along the crack is unknown, for the correct formulation of the
problem, one needs to know the distribution of the forces applied on the crack contour. In this
case, the values of the elementary discontinuity displacements for each of the elements that are
necessary to cause such efforts are determined. This is achieved by solving the system of
corresponding algebraic equations. The boundary integral equations of this method are
equivalent to the following system of linear equations of the form [5, 16, 17]:
ππ π ππ π
ππ π = βπ
π=1 (π΄π π ππ + π΄π π ππ )
{ , (28)
ππ π ππ π
πππ = βπ
π=1 (π΄ π
ππ π + π΄ π
ππ π )
Μ
Μ
Μ
Μ
Μ
π = 1, π,
where
π
ππ = ππ π
{ π ,
ππ = πππ
1 β€ π β€ π,
π
ππ = π·π π
{ π ,
ππ = π·ππ
π + 1 β€ π β€ π,
j
Asi s , β boundary stress influence factors for the problem under consideration. Coefficient
j
Asi n gives the actual tangential stress ο³ si in the center of the i segment caused by a constant
unit normal load Pni ο½ 1 attached to j segment.
The sum of the initial and additional stresses expresses the total stresses in the i element is
equal:
(ππ π )π = (ππ π )0 + ππ π
{ , (29)
(πππ )π = (πππ )0 + πππ
Equations (29) were used to construct an algebraic system for finding unknown quantities
X s and X nj (i = 1, ..., N). Boundary elements ( 1 ο£ j ο£ M ) are characterized by zero-equal full
j
stresses ο¨ο³ si ο©ΠΏ , ο¨ο³ ni ο©ΠΏ as a result 2Π equations of the system (28) has next form:
π π π π
β(ππ π )0 = βπ π π
π=1 (π΄π π ππ + π΄π π ππ )
{ , (30)
ππ π ππ π
β(πππ )0 = βπ
π=1 (π΄ π
ππ π + π΄ π
ππ π )
1 β€ π β€ π.
Another 2(N-M) equations obtained by combining system (28) and the condition
ο³ n ο½ ο K n ο Dn , ο³ si ο½ ο K s ο Ds , where K n and K s - normal and tangential stiffness
i
characterizing the elastic contact element:
π ππ π ππ π
0 = πΎπ π ππ + βπ
π=1 (π΄π π ππ + π΄π π ππ )
{ , (31)
π ππ π ππ π
0 = πΎππ ππ + βππ=1 (π΄ π
ππ π + π΄ π
ππ π )
π + 1 β€ π β€ π.
Systems (30) and (31) were solved by the standard numerical method.
202
On Figures 3 and 4 for a fixed shock pulse, the dependence of the crack length on the radius
of the rock-cutting tool r (at fixed ο³ 0 ) and the tensile strength ο³ 0 (at fixed r ) is shown.
Figure 3: Dependence of the crack length on the radius of the tool:1 - argillites, 2 β sandstones, 3 -
granite.
Figure 4: Dependence of the length of the crack on the strength of the rock on the gap: 1.r = 5, 2.r
= 15, 3. r = 25, 4. r = 35.
3. Conclusions
1. The formation of microcracks in granular geomaterial occurs along the grain boundaries
of the rock in the direction of action of the cutting edge of the rock cutting tool.
2. During the calculations using the boundary element method, it is found that if the initial
crack length is much smaller than the distance to the free surface, then the tear gap develops
without bending until the crackβs length l and its distance D to the surface the values are
203
ο²l
approximately in proportion ο£ ο°ο¬ο΄ . If this correlation with the growth of the crack ceases to
D
be fulfilled, then its trajectory gradually curves up until the crack reaches the surface.
3. The opening of microcracks in the contact area of the cutting part of the rock cutting tool
with rock of a granular structure occurs under the condition of a higher value of tangential
stresses of the value of normal stresses for cutting edge sharpening angles οͺ ο£ 37 0 . In the case
ππ = 0 and ππ = 0 full cracking occurs along the boundaries of the structural grains of the
geomaterial.
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