of elemetary prisms

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Rock Destruction by Shearing on Surface of Elemetary PrismsR.R. Rakhmankulov and A.P. Batalov

National Mineral Resource University, Saint-Petersburg, Russia

Abstract. This work is aimed on consideration of rocks destruction. Today rocks

destruction subject is one of the most challenging due to increasing of

performance on pits, quarries and mines. Research results in field of rocks

destruction gives us a picture of what kind of tool will be used in miningactivities.

Using cutting process as a rocks destruction method, separating from side of

face by cutting tool- creates a complicated area of stressful condition of massif,

and if loading is enough separation of rock occurs by those surfaces, which have

defects, dislocations in structure of internal connections. The more homogenous

structure of material, the more definite loadings, cutting shape and kind of shear.

Loading on a tool depends on material strength, its structure, speed and depth of

cutting, temperature, geometry of tool and many others factors. To mathematize

fact it is necessary to distinguish factors and build the model of process using

them. Majority of scientists accepted as a main factor strength properties of

material- failure stress in compression, displacement, tension (Coulomb, Galileo,

Morse etc.), others - Marriott, Saint-Venant were thinking that destruction occurs

in areas of the largest deformations. I. A. Time proposed very deep theory. Hisresults still allow determining energy, which used for cutting of metal and some of

roach. In our work we would like to combine research in field of destruction of

monolithic, cohesive and discrete, granular material, as by practical observations

destruction of face occurs as secondary process after natural discontinuity and

previous processing of a face, what lead to the formation of defects in surface

area.

Keywords: I.A.Thime, Galileo, material deformation, Marriott, Mohr, rock

cutting, rock destruction.

1 IntroductionSurvey of rock destruction processes by shearing on surfaces of elementary

prisms, research of existing theories of material destruction, process modeling360 R.R. Rakhmankulov and A.P. Batalov

using CAD systems. Researches of rock and ground destruction processes for last

three hundred years gave us general representation about the formation of cracks,

stone chip and others results of rock, ground and metal destruction. A lot of

scientists from around the world made their work for development of this topic.

Among them are Coulomb, Galileo, Mohr etc. By these and many others scientists

have been made a big amount of experiments and theoretical calculations, it was

calculated a large number of formulas. In this way, to date it is possible to allocate

strenght theory of Mohr-Coulomb as a basic tool for strenght definition of

material.

The Mohr-Coulomb strenght theory is widely used in construction and in

mining in relation to a loose incoherent and coherent rocks (grounds), as well as

in relation to a clastic cemented rocks. Incoherent materials occupy an

intermediate position between liquid and solid monolithic rocks. As liquids, theese

rocks occupy volume of any shaped vessel; but loads to a walls and bottom of the

vessel formed considering friction between material and walls and between itsown rock particles. With increasing particles coherence theese materials become

more close to a monolithic materials.

Material destruction by tension is not taken into account. Destruction of

materials only interested in compression and shear.

The strength theory of the Mohr-Coulomb based on the strenght hypothesis of

Mohr. This hypothesis is about the dependence the limiting tangential stress of

average normal stress and the Coulomb hypothesis, which says that dependence is

governed by internal friction in solid body.


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The Mohr-Coulomb strength theory allows to determine the breaking strength

of materials, which having different resistance to tension and compression (brittle

materials) and has significant advantage over the first and second theories.

According to the Mohr strenght theory two stressed conditions equally

dangerous, if for the appropriate main stresses observed ratio of equality.

Factor is the ratio of limiting stresses corresponding to uniaxial tension and

compression.

The Mohr strength hypothesis recommended for brittle materials. For plastic

materials Mohr strength hypothesis is identical to the third hypothesis strength.

2 BodyThe first hypothesis is based on the strength of the assumption that the causes of

the destruction of the material are the largest in absolute value of normal stresses.

Usually, the first hypothesis of strength, as proposed by Galileo, called the

hypothesis of the greatest normal stresses.

The condition on the strength of the first hypothesis (1):

(1)eq 1 p (1)Rock Destruction by Shearing on Surface o f Elemetary Prisms 361If the greatest value will be for the compressive principal stress, a condition of

strength on the strength of the first hypothesis: 3 .

A significant disadvantage of the first hypothesis of strength is that

determination of the equivalent stress completely ignored the other two principal

stresses are affecting the strength of the material (2):

(1)eq 3 compr (2)The first hypothesis is supported by the strength of the experimental data for a

brittle material in tension, when the stress 2 , 3 is much less than 1.

Under uniform compression, for example, cement cube, the first hypothesis of

strength leads to wrong results, as the cube withstand stress is many times greater

than the tensile strength of the uniaxial compression.

At present, the strength of the first hypothesis does not apply, it has only a

historical significance.

Disadvantages of the first hypothesis strength led to the second hypothesis, the

strength offered by Marriott and developed Saint-Venant.

On the second hypothesis, the strength, called the hypothesis of the largest

linear deformation cause of destruction are the greatest linear deformations. The

equivalent stresses are calculated according to the formula (3), where is thePoisson's ratio:(2)eq 1 (2 3 ) (3)


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Considered that for a plastic material is performed Hooke's law up to the yield

point, and for brittle - to a tensile strength that is rough assumption.

The advantage of the strength of the second hypothesis is that when calculating

the equivalent stress it considers all three principal stresses.

With the largest linear deformations hypothesis the destruction of fragile

materials in simple compression can be explained. However, the second

hypothesis is not confirmed by the experiments of strength and is not used for

critical predictions of material behavior.

It should be emphasized that the brittle or plastic state of the material is

determined not only by its properties, but also the views of stressed state,

temperature and rate of loading. Experiments show that plastic materials under

certain loading conditions and temperature behave as brittle and brittle materials

under certain stressful conditions can behave as plastic.

Mohr-Coulomb criterion based on the assumption that the material strength is

generally stressful state depends mainly on the magnitude and sign of the largest

1and smallest 3 main stresses (error due to the fact that not considered 2 ,usually does not exceed 12-15%) . Based on this assumption, any stress can be a

circle of Mora, built on the main stresses 1 and 3 .

If using existing 1 and 3 strength of the material is broken, the circle was

built on these stresses, called the limit. By changing the ratio between 1 and 3 ,362

we obtain for the materia

limit circle ABCDE possithe Mohr circles built for

tensile and compression -

material in compression

When cutting of metal

tilted at an angle of appro

From the work of "So

the destruction of the instr

the sample destruction incFig. 1 Test of ground depend

Using the theory of M

horizontal axis (fig. 2, 3,4

Fig. 2 Work of ground prismR.R. Rakhmankulov and A.P. Batalo

al family of limit circles. The envelope of the family o

ible with sufficient accuracy to replace the line tangent t

r stretching, with a diameter equal to the time v resi

with a diameter equal to the time the resistance of th

compr [1].fracture occurs at the shear plane (Research I.A.Thieme

ximately 1 35o to the direction of the cutter movement.


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oil mechanics. Part 7" [2] seen that in testing ground fo

rument triaxial state of stress, is formed on the surface o

clined to the horizon at an angle

(fig. 1).ding on its initial state [3]

Mohr-Coulomb define the angle of the shear area to th

4).m to form a sliding surface at the time of the limit state [3]ov

of

to

ist

he

e)

or

of

heRock Destruction by ShearinFig. 3 The stress state at theFig. 4 Comparison of cube &

Thereby, we see that aground formed a sliding s

principal normal stresses

surfaces under the surfa

formation of uplift from thFig. 5 Schematic of the forlimit state, with the formationg on Surface of Elemetary Prisms 36

point of ground on the Mohr-Coulomb theory [3]

cylinder crushing strength [4]

at the time of the limit state basis under the stamp on th

surface, directed at an angle to the line of action of th

s. This is perfectly shown in the diagram form slidin

ace of the die at the time of the limit state, with th

he stamp to the surface (fig. 5).rmation sliding surfaces under the base hard foundation at thon of uplift from the foot of the surface [3]63

he

he

ng

hehe364

With the implementatio

which is the protection fo

friction force between the

with stamp and go at the e

prism in the ground. If th

nearby surface is not pre

surface pressure on the gr

into the following layers

secondary and subsequen

sheared at an angle of

arranged at different angle

This paper describes t

therefore we consider tha


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along the corners of repos

does not allow shift prism

of stamp shown on the ro

an array of only one side (Fig. 6 Animation process of

During the work was c

Inventor. From the figur

destruction of rocks by cseen that the destruction

elementary prisms, as cka

the impact of stress on th

stamp.R.R. Rakhmankulov and A.P. Batalo

on a stamp in the soil or rock under it appears prism are

or the surface of stamp, since the ground pressure of th

e surface of stamp and the ground does not give a prism

expense angles of repose formed by introducing the mai

here is more pressure stamp greater flow of ground on

ssurized. In its turn the main prism, like a stamp, has

round, where a similar way the following prisms that ru

of soil and form a whole new prism. Note that all of thnt prism surfaces can slide on the neighboring prisms o

repose, because abutting surfaces of these prisms ar

es to the direction of movement of the stamp.

the process of destruction of rocks, ground are not an

at one side is always open for the movement of prism

se, and the rest are part of the surface of the array, whic

ms inside. As can be seen in figure 6, where the pressur

ock, and the fact that to cut off part of the rock adjoins t

the left).breaking rock by chipping over the surfaces of the prisms

carried out animation process failure to means Autodes

res 7,8,9, which display a sequence of the process o

chipping over the surfaces of elementary prisms can b

n of the rock passes through the sliding surfaces o

azano above, these sliding surfaces formed as a result o

he sample. Platform shift parallel or angled to pressurov

ea,

he

m

in

a

a

un

he

or

rend

ms

ch

re

to

sk

of

be


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of

of

reRock Destruction by Shearing on Surface o f Elemetary Prisms 365Fig. 7 Movement of elementary prisms on the surfaces of chipping. Initial stageFig. 8 Movement of elementary prisms on the surfaces of chipping. Middle stage

During the simulation of the angle at the base of the prism was taken to be 35

degrees to carry out research at various angles up to 45 degrees, and the samestamp - its base must also change their situation - that be different angles to its

direction of movement.366 R.R. Rakhmankulov and A.P. BatalovFig. 9 Movement of elementary prisms on the surfaces of chipping.Final stage. Fulldestruction

3 ConclusionThe properties of such models are far from the natural state of the material, for

example, secondary and subsequent prism (a flat model - triangles), unchanging,

not destroyed, but some extent in good agreement with the phenomenon of

formation of built-up edge on metals, formed by the build-up and the pick cutting

edge in cutting rocks.

The development of this approach to the destruction of connected and discrete

materials may be useful for improving

of elemetary prisms

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