László Mester

The new physical-mechanical theory of

granular materials

2009

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Contents

Introduction ………………………………………………………………………….3

Granular material as a distinct state of matter …….…………………………………4

Physical properties of the granular material in relation to

the different states of matter …………………………………………….…..6

Granular material as a state of matter …….………..…………………………10

Physical-mechanical basic laws of the non-cohesive granular materials …………..14

Law I ………………………………………………………………………….15

Law II ………………………………………………………………………...15

Law III ………………………………………………………………………..17

Law IV ………………………………………………………………………..24

Stresses in the non-cohesive granular materials ……………………………………26

Active stress state …………………………………………………………………..31

Development of the active stress state ………………………………………..31

Pressures acting on the vertical retaining wall ……………………………….36

Arch formation of granular materials …………………………………………...…39

Condition of the arch formation ……………………………………………...39

Character of the discharge ……………………………………………………44

Mechanism of the arch formation …………………………………………….46

Geometric equation of the arch ………………………………………………47

Principle of the hopper design ………………………………………………..50

Experimental results ………………………………………………………….53

Stresses in cohesive granular materials …………………………………………….55

Lateral pressure ………………………………………………………………55

Inclination angle of the free slope ……………………………………………59

Active stress state …………………………………………………………….64

Summary …………………………………………………………………………...69

Bibliography ……………………………………………………………………….73

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Introduction

Following Coulomb’s and later Rankine’s work the physical-mechanical theoretical

research of granular materials has been characterised by the use of stress analysis

deduced for solids since the 18th century. Others seem to detect the characteristic

features of viscous liquids in granular materials, therefore they describe the physical

behaviour of granular materials using the laws pertinent to viscous fluids. In my

opinion most of the theorems, which were put for cotinuums, cannot be applied to

the aggregation of separate, solid granules. Only those natural laws can be

considered as the starting point of examination, which are also valid for the universal

material.

This work – breaking away from the previous tradition – would like to approach the

physical mechanical properties of granular materials from a new point of view. As a

result, the critical analysis of theories formulated earlier in this research area is not

the objective of this paper, since the new principles were laid down irrespective of

those hypotheses. Contrary to the continuum theory, by examining the equilibrium

and kinetic state of individual granular particles this new thesis is based on simple

experiments, on the Newtonian laws, and on an empirical law, the law of friction.

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Granular material as a distinct state of matter

The physical appearance of materials found in nature is quite varied. The most

substantial part of the Earth’s surface is covered by oceans, seas, lakes, that is to say,

covered by water. The dry land is more diverse: one can find rocky mountain ridges,

surfaces covered by gentle slopes and deserts with sand dunes. In places the earth is

covered by snow or ice in winter. Above the surface level the wind is blowing, or we

can experience a period of calm, that is to say we can feel the air. The sun is shining

above us and we know that inside the sun one would find another state of matter.

The outward form of the water that covers substantial part of the Earth is in itself

diverse. At normal temperature and pressure water is liquid, but with the increase of

temperature it evaporates more and more quickly, and it turns into water vapour. Fog

or clouds form. When water vapour freezes, and precipitates in cold, then snow falls,

and it condenses into a granular material. When snow melts, the result is liquid,

which in turn becomes solid when it freezes. That is to say water can exist in liquid,

vapour (gas), snow (granular) and ice (solid) states. In each of its phases water has

different physical properties, and behaves conforming to different laws.

Physics differentiates among the most prominent forms of appearance of material by

classifying them into the different states of matter: plasma, gas, liquid and solid

Some material cannot be put strictly under one category, because they bear the

physical properties of two or more states of matter. These materials, however, can be

described by applying the laws pertaining to materials in a similar state of matter.

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Granular material cannot be put under any one of the above mentioned four

categories. Furthermore the physical-mechanical properties of granular material do

not make it possible to describe its behaviour successfully using the physical laws of

one or more phases.

A granular material is a conglomeration of large number of solid particles related to

one another, where the granules – as the constituent of the aggregate – in spite of the

affecting forces retain their form, and the incidentally arising cohesive force between

the granules is substantially smaller than the inner cohesion of the individual

granules.

The overall, coherent physical system of granular material has not been set up yet,

scientific analysis is available only for a few prominent, primarily soil mechanical

problems. Several theories have been applied for the handling of these problems,

which speculations, however led to contradictory results. Furthermore no connection

resting on firm, uniform physical foundations exists between the theories, or if there

is a relation, it is disputable. The majority of mechanical theories dealing with

granular materials apply the method of stress analysis deduced for solids, which

procedure presupposes, that the granular material is a solid phase continuum.

There are theories, according to which granular materials can be approached with

the laws pertaining to viscous liquids, since granular material exhibits viscoelastic

and viscoplastic properties. Although, the difficult theoretical notions provide an

approximate solution to individual mechanical problems, they cannot be applied to

an overall, reliable description of granular material behaviour.

The opinions concerning the state of matter of granular materials are not unanimous.

This is reflected in the fact that granular material does not have a single uniform

name, for example the following designations: scattering, powderlike, loose,

granulated, grainy, particulate, granular and gritty are all used.

The physical behaviour and properties of granular materials exhibit substantial

qualitative difference from the materials in other states of matter, and should

therefore be considered an additional state of matter in its own right

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The idealised notion of granular material makes the simplified explanation of its

physical behaviour, properties and origin possible, similarly to the hypotheses

applied to perfect gases, ideal liquids and crystalline solids. The ideal granular

material is a conglomeration of large number of solids (granules) where the mass of

the solid particles is small compared to the mass of the material, in this aggregate

attractive force does not operate between the particles, Coulomb friction rule

governs them.

Physical properties of the granular material in relation to the different

states of matter

The basis for classification according to states of matter depend on the question

whether the material can hold its own shape and volume or not. The basic criteria of

classification of the three classical states of matter are the following:

- gases: have no definite shape or volume;

- liquids: have no definite shape, but have definite volume

- solids: have definite shape and volume.

Regarding the question of definite shape and volume it is the characteristic of the

granular material that:

- In part it has definite shape, the granular aggregate holds its shape in the angle

of repose, but under this angle it takes the shape of its container. This attributive

places the material between liquids and solids.

- In part it has definite volume, but it can be compressed to a limited extent. The

compressibility of granular materials stands between the compressibility of gases

and solids.

The researches on substance structure concerning the states of matter found that the

determining factors in the question whether a material holds a definite shape or

volume lie in the physical properties of its constituents and the nature of interaction

between these particles. For this reason modern physics studies the kinetic state of

the smallest particles attributed to the material, their relative position and the particle

interaction when defining the different states of matter. This made it possible that in

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natural science, in addition to the three classical states of matter, a fourth state of

matter, was accepted, the plasma state.

It seems to be necessary to emphasise the expression smallest constituent

characteristic of the material, since it is of primary importance in the definition of

the different states of matter. Consequently

- A material in plasma state consists of the disintegrated parts of molecules or

atoms, the molecule ions or atomic ions. The plasma state is characterized by the

interactions of atomic or molecule ions and electrons, and not by the other parts of

the atom or molecule.

- The physical properties of a material in the gas state are determined by the

interactions of gas molecules – in case of noble gases the interactions of atoms.

- Regarding liquids the determining factors are again the movements of the

atoms or molecules, and the nature of relation between them. In the case of water it

is the interactions of H

O molecules and not the hydrogen or oxygen atoms, or the

water drop, which characterize the liquid.

- In the case of crystalline solids the physical properties of a material in the solid

state can be explained with the nature of interaction of the atoms, molecules or ions

positioned in the lattice nodes of a crystal structure, and cannot be described for

example with the interactions of elemental crystals and crystallites, or with the

individual properties of the atoms and its parts, which constitute the molecules

positioned in the lattice points.

- In the case of granular material the smallest constituent characteristic of the

material is the granule. The atomic particles, the atoms and molecules that constitute

the granules, are not direct characteristics – at least in physics they cannot be

regarded as significant physical properties– of a material, just like as in the case of

gases and liquids, where material is not characterised by the atomic particles, which

constitute the atom or molecules, or by the individual physical properties and

interactions of atoms either.

The most significant characteristics of the three classical states of matter can be

summarised in the following way:

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Gaseous state: The molecules of gases – in case of noble gases the gas atoms –

move freely in the space available for them, they collide elastically with one another

in a random motion. The average distance between the constituting particles of gases

is relatively big in proportion to their size, the intermolecular forces between the

particles are very weak. In the case of ideal gases intermolecular force can be

disregarded. Molecules move with a translational, rotational and vibrational motion.

Gases evenly fill the space available for them, that is to say they have no definite

shape or volume.

Liquid state: The intermolecular forces between the smallest constituents

characteristic of liquids, between the atoms or molecules are strong enough to

prevent the particles moving away from each other as a consequence of thermal

motion, but not strong enough to prevent their change of position. Compared to

gases, the translational motion of the molecules are smaller, while they also carry

out rotational and vibrational movement. Due to their motion and proximity the

constituting particles collide elastically with one another all the time, thus touch one

another, therefore liquids have a definite volume. The force of attraction between the

particles is so small compared to the Earth’s gravitational force that it is not enough

for individual shape formation, as a result liquids have no definite shape.

Solid state: The smallest constituting particles, characteristic of solids are the atoms,

molecules or ions. Their position is fixed and geometrically determined in a

crystalline structure, particles carry out only vibrational motion. The intermolecular

forces are strong, which prevent their permanent displacement from their state of

equilibrium. As a result solids have definite shape and volume.

Granular material exhibits significant differences from the aggregational properties

of the three classical states of matter. The constituents of an ideal granular material,

the granules are at a relative rest. There are no forces of attraction between the

particles, the material is kept in an aggregate by the compressive forces originating

from the gravitational force, by the shear forces arising on the surface of the

granules, and by the static friction force. Due to these forces the ideal granular

material remains stable until the angle of repose is reach, thus it has only partly a

definite shape. The constituting particles are in constant contact, therefore in

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quiescent state granular material has definite volume. Under pressure the material is

compressed, the granules take up a more efficient space filling position. The

compressibility of granular material is small compared to gases, but it is big in

comparison to solids.

The physical properties of ideal granular material exhibit the following significant

differences in characteristics compared to the features of other states of matter:

- In contrast to gaseous state: the constituting particles are in constant contact

with each other, and it has definite volume;

- In contrast to liquid state: granular material has in part a definite shape;

- In contrast to gaseous and liquid state: The constituting particles are in a

relative collision free, quiescent state and static friction force – shear force – arise in

them;

- Solid state: there is no attractive force between the constituting particles,

therefore granular material has only in part a definite shape.

Granular material exhibits such qualitative differences concerning the most

substantive characteristics of the different states of matter that its definition as a

separate state of matter in its own right becomes justified.

The brief, straight to the point definition – with no pretence to completeness – of the

idealised case of the states of matter is the following.

Perfect gas: disordered aggregate of molecules (in case of noble gases atoms) with

no intermolecular forces, where the molecules move far apart from each other,

undergoing random elastic collisions.

Ideal liquid: Aggregate of molecules moving close to each other, undergoing

constant elastic collisions.

Crystalline solid: The ordered aggregate of vibrating atoms, molecules or ions,

which are fixed in their structure with great force.

Ideal granular materials: the aggregate of relatively static particles, which are in

constant contact with each other, in this assembly the force between the constituting

particles is composed of the compressive force arising from the gravitational force

and of the friction force, which is proportional to it, there is no cohesion force

between the particles.

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The notion of granular material, as a separate state of matter is primarily important

from a mechanical viewpoint. (The basis for categorization into different states of

matter has a mechanical origin: the reason for having a definite shape or volume can

be deduced from the intermolecular forces between the constituting particles.) The

mechanical properties of the different states of matter show distinctive, substantial

differences:

- gases respond to an increase in pressure with the significant reduction of their

volume (at constant temperature the multiplication product of volume and pressure

is constant), therefore they can withstand compressive stress only in part, at the

expense of volume change. In perfect gases there is no attractive force between the

constituting particles, therefore no tensile stress can arise in the material. In static –

not flowing – gases no shear stress arises.

- liquids has small compressibility, from a mechanical point of view they can be

regarded incompressible, therefore they can withstand compressive stress. The

intermolecular forces are strong enough to prevent the constantly colliding

molecules from moving far away from one another. The state of equilibrium or

stability can only be attained under a given outside pressure, that is to say, from a

mechanical standpoint a liquid cannot withstand tensile stresses. (When the pressure

is around p=0 the liquid breaks up, its molecules fly apart and turn into gaseous

state) There is no friction in ideal liquids, in real liquids static friction does not arise

either.

- solids can be regarded as incompressible, the constituting particles join

together with great force, therefore they can withstand tensile, compressive and

shear stress.

- ideal granular material has small compressibility, therefore it can withstand

compressive stress. In the non-cohesive granular materials only shear stress arises in

addition to compressive stress, no tensile stress manifests itself.

Granular material as a state of matter

The notion that granular material must be regarded as a separate state of matter can

be justified not only because its distinct physical properties, which differentiate it

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from other states of matter, but also because granular material is one of the existing

outward forms of raw material, that is to say it is one of defined states of matter.

Granular material – as the conglomeration of large number of solids, where the

constituting solids are small in proportion to the total mass of the material –

generally comes into being when large-sized solids are mechanically cut up, or when

the solids themselves break up into smaller pieces. Its formation, that is to say, the

bringing of the material into a granular state can be achieved not only in a

mechanical way, as it is also true for the granular materials in nature, which were

formed not exclusively by mechanical disintegration either. Granular material can be

produced via a thermodynamic method.

It is known, that if the kinetic energy of the molecules of a liquid exceeds a

threshold it changes into gaseous state and if it goes below another threshold the

liquid turns into solid and the process of crystallization begins. The threshold values

of the thermodynamic state parameters characteristic of the different states of matter

can be illustrated in a p – t (pressure-temperature) diagram. In Figure 1 the p – t

diagram of H

O can be seen.

Figure 1. p-t diagram of H

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The curve that joins the triple point and the origin of the p – t diagram is called the

sublimation curve. The material passes from the solid ice phase into the vapour

phase of the gas state by for example pressure reduction and by the crossing of the

sublimation curve. Crossing the sublimation curve backwards, from vapour phase

granular snow is formed not ice. The granular material – in case of water, the snow –

comes into being as a result of crystallization in the local clusters. The process can

be called local crystallization, the physical explanation of which lies in the

phenomenon that the molecules which move with slow translational motion (under

low temperatures) cannot leave the attraction field of the van der Waals type forces –

for example as a result of heat loss – therefore the translational motion of the

molecules ceases.

The molecule pair - bonding the new molecules, which collide into them - form a

crystal lattice, the growing crystals then bring about the granules. The density of

molecules in the gaseous state is very low in comparison to the molecular density of

the solid state, therefore the local crystallization processes, which are relatively far

from one another bring about the multitude of separate granules, which after having

precipitated form a granular conglomeration.

Under constant temperature the process of getting from gas phase to granular phase

is accompanied by heat loss, which is the sum of the melting and the evaporation

heat.

The states of matter change at the phase boundaries of the p – t diagram. If the

matter crosses the sublimation curve from the solid phase toward the gas phase, we

will get a gas, however changing the direction crossing the curve from the gas phase

we will obtain a granular material. Thus the states of matter, in compliance with the

direction of crossing the phase boundaries are the following:

gas → local crystallization → granular → melting → liquid → evaporation →gas;

nevertheless, from the other direction:

gas → condensation → liquid → freezing → solid → sublimation → gas.

From granular phase to solid phase we can get by crossing the same phase boundary

twice:

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granular → melting → liquid → freezing → solid.

Granular material can be produced directly from liquid phase if we place a multitude

of crystal nuclei - which are approximately at an equal distance from one another -

into a supercooled liquid at the same time. The crystal growth is hindered by the

neighbouring crystals, whose geometric crystal position is not symmetrical or

congruent, therefore no or only occasional lattice forces develop between the

crystals, the inner cohesive force of the individual granules are substantially greater

than the incidental cohesive forces acting between the granules.

The substantial physical properties of the granular material differ significantly from

the characteristics of those materials whose chemical properties are identical,

nevertheless belong to the solid, liquid or gaseous state. Its volume weight, its

refractional, thermodynamical, acoustic, electric and mechanical properties and

behaviour, and the fact that most material can be brought into granular state justifies

the classification of the granular material as a distinct state of matter by its own

right.

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Basic physical-mechanical laws of the non-

cohesive granular materials

I. In the non-cohesive granular materials only compressive and shear stresses

can develop.

II. In the non-cohesive granular materials at a quiescent state the stresses

developed by the vertical-direction compressive stresses act downwards in the





zone measured from the vertical direction. (



is the angle of friction of

the material.)

III. The value of the lateral pressure rising from the self-weight of the non-

cohesive granular material is (



), the half of the product of the depth (h) and

volume weight (γ), its direction deviates from the horizontal downwards with the

angle of friction developed in the material, if the surface is horizontal and over the

given depth the material fills the space evenly closing an angle



with the

horizontal.

IV. The non-cohesive granular materials conform to the physical-mechanical

laws characteristic of them until their constituting elements, the grains keep their

relative quiescent state. When the grains go into motion – collide with each other -,

the granular materials behave according to the physical-mechanical laws of the

liquids.

The physical-mechanical laws of the non-cohesive granular materials prevail with a

statistical character, because the material itself consists of a multitude of different

grains.

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Law I

Law I is the physical-mechanical definition of the non-cohesive granular materials.

In ideal liquids only compressive stresses develop, the non-cohesive granular

materials are capable of withstanding compressive and shear stresses, while solids

are capable of bearing compressive, shear and tensile stresses. The non-cohesive

granular materials differs from the solids in the respect that they are not capable of

withstanding tensile stress, and they are distinct from the ideal liquids because shear

stresses also develop in them. At the same time the components of the liquids are in

constant relative motion – collide with each other -, while the components of the

granular materials, the grains are in a relative quiescent state.

Law II

Law II formulates the direction of the spreading of the vertical compressive stresses.

The natural stability of the free slope provides its experimental proof (Figure 2).

Figure 2. The boundary equilibrium position of the grain located

on the side of the slope

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In the conglomeration of grains the self-weight of the grains produces the vertical

compressive stress. If a stress vector acted on the grain marked A located on the side

of the slope– in boundary equilibrium position - inclined at an angle of more than





from the vertical then the grain would loose its equilibrium position and

slide down.

Figure 3. The stresses developed by the vertical compressive stresses

incline at an angle bigger than



to the horizontal.

Further experiments prove the correctness of the Law II (Figure 3). If a compressive

stress making a smaller than



angle with the horizontal acted on the grain marked

A, the angle of inclination angle



of the natural slope would be smaller than



. If

the vertical compressive stress induced, for example, a horizontal stress, it would

thrust down the grains located on the side of the slope. The material would spread

and would take a kind of shape that is illustrated in Figure 4. However, it does not

exist.

Figure 4. If the compressive stresses induced horizontal stresses, this

would be the position of the granular material.

- 17 -

Law III

In the cases described in the Law III the lateral pressure is



, and its direction

inclines from the horizontal downward with the angle of friction rising in the inside

of the material. Its proof is as follows:

Figure 5. Infinite Quadrant of the Horizontal Terrain

Figure 5 shows that part of the non-cohesive granular material aggregate of infinite-

expansion and horizontal-terrain, which is cut out theoretically by two vertical

planes perpendicular to each other, (consequently, it shows an infinite quadrant of

the horizontal terrain,) which makes the planar execution of the mechanical tests

possible. According to the Law II, from the material part under the section AB only

reaction stresses produced by the material part over the plane AB can act on the

plane OA. If we took off the granular material located in the triangle OAB, then the

material would remain stable in the natural angle of repose AB inclining at an angle



to the horizontal. On the plane OA, stresses can only rise from the self-weight of

the granular material located in the space part OAB. On the plane AB an equilibrium

boundary position exists; the material having a friction coefficient of



tg

does

not slide as yet on the slope with an inclination of angle



. If we increased the slope

angle with





of a very low value, then the material above it could slide down with

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constant acceleration on the slope with the inclination angle





, i.e. it would

exert a force in direct proportion to its mass and acceleration on the plane OA. Due

to the physical properties of the granular materials there are infinitely many slopes

with an inclination angle





, above which angle, on the slopes increased by the

angle





, the weight of the materials exerts slope-direction stresses on the plane

AO. According to Figure 5, on the slope with an inclination angle of





produced as a result of the depth increased by Δh, the granular material ADC weighs

heavily on the slope with its self-weight (ΔG) and with the weight of the material

located above it (G). The material amount ADC is supported on section Δh. The

projection of the surface section Δh, perpendicular to the slope is

)cos(



 hF

. Considering that the Δh is very little, therefore, the stress

distribution can be considered even, so it can be stated for the slope-direction stress

developed there:

)cos)(sin(













In the unit-length space part





ctg

G 

and

,cos

sin2





 h

that is,





ctgh

G 

furthermore,

).cos(



 hF

The (



cossin 

) in the equation can also be expressed:

)cos()sin(



 tg

),sinsincos(cos

cos

sin

sincoscossin









,sin

cos

sin

sincos







- 19 -

)sin(cos

cos

sin











cos

sin







Substituting the values of the G, ΔG, ΔF and



cossin 

into the relation

written for the





)cos(

cos

s in

cos





































ctg

)cos(

cos

sin

)(



















hhctg

)sinsincos(cos

cos

sin

)(

sin

cos



















s in

cos

coss in

































hhh

sin

cossin





















hhh

but the



tg

can be expressed from the triangle ACE of Figure 5:

sin

cos









sin

cossin













therefore,

sin

cossin





















hhh

sinsin















hhh

- 20 -

2 h

hhh 













hh 



if the Δh is very little, that is Δh → 0, then α →



, consequently





→





, i.e. the

direction of the stress inclines at an angle



to the horizontal. Consequently,

lim











that is,









As a result of the deduction it can be established that the stress distribution is linear

against the depth.

However, in case of granular materials one cannot speak of a stress in the classical

sense, since the force effects are transmitted at the contact points of the granules, i.e.

from one point to another, not on a surface perpendicular to the given direction. Not

on a surface, because the material is a discontinuum and the grains touch each other

only at points. Therefore, the meaning of the stress can be interpreted as the average

force imparted to one surface and these average forces are transferred from one

granule to another. The direction and size of these forces manifest themselves as a

statistical average on a given surface.

Figure 6. Division of the average force acting on the grain located

next to the wall into a horizontal and a vertical component

- 21 -

Let us examine what the magnitude of the force is, which the above deduced stress





, inclining at an angle



to the horizontal, exerts on a vertical wall. Through the

contact points force arrives from the neighbouring granules at the grain –which is

supported by a frictionless vertical wall – shown in Figure 6. The resultant of this

force effect should be equal with the product of the stress





calculated for the

surface

F

, i.e. considering its magnitude and direction the compression force

acting on this grain corresponds to the statistical average of the force exerted at this

depth in the given granular material. This granule presses the vertical frictionless

wall with a horizontal





force on the wall section

F

at the contact point. The

vertical-direction force





is received by the grain or grains located under it.

The surface of sections

F

and

F

are equal surfaces on statistical average,

because, considering their shape and position, the grains are spheres on statistical

average; the projections of the spheres from any direction are of equal surface area.

(If a granular material – for example, rice – consists of oblong grains; considering

the random arrangement of the grains the average of their projection taken in any

direction is a circle, i.e. the grains must be considered as spheres on statistical

average.)

Consequently, it can be written for the vector triangle of the Figure 6





cos



but since

FF 

therefore





cos

It comes from the result of the above consideration that in granular material the

stresses – the average forces calculated for a given surface – can be decomposed or

added as vectors. From the





cos

equation the factor of static pressure is

received after the







substitutions, that is











cos



cos







- 22 -

Up to this point, only the shear stresses generated by the stresses acting

perpendicularly to the direction inclining at an angle



to the horizontal and rising

from the self-weight were taken into account at the deduction of the static pressure.

The shear stresses



, produced by the horizontal stress components -







cos



- reduce the vertical stresses hγ with







 cos

that is

cos

sin

cos















sin



Considering that the stresses





act in pairs on the theoretic plane OA assumed

inside the granular material (Figure 5), therefore, the vertical-direction stresses hγ

are reduced by 2



, consequently



2

that is

,sin









and

)sin1(



 h

Consequently, the figure of the stresses acting inside the non-cohesive granular

materials at quiescent state can be constructed (Figure 7).

Figure 7. Stresses acting at a quiescent state

- 23 -

If the plane OA according to Figure 5. is a theoretic plane assumed inside the

granular material, then the angle of friction is



there. In this case a compressive

stress of





, inclined at an angle



to the horizontal acts on both sides of the plane

OA (Figure 7.). On the plane OA the





’s can be divided into horizontal and

vertical components (Figure 8.). The horizontal components have the value







cos



they are perpendicular to lane OA and satisfy the action-reaction law. The vertical

stress components of



complement the vertical stresses to hγ symmetrically to

plane OA in a reciprocal way.

Figure 8. The horizontal and vertical stress component at a quiescent state

If the plane OA according to Figure 5 is a frictionless wall, then that is capable of

taking only horizontal stress, i.e. the horizontal component of the





, which is







cos



At the same time the vertical component of the





complements the vertical stress

component of the material OAB to hγ.

If the plane OA is an actual rough, rigid wall, which serves to support the OAB

material amount, then the development of the static pressure can be interpreted as

follows.

- 24 -

After filling up the OAB material amount behind the OA wall the value of the







is not reached immediately, since due to the inclined stress effect and

because of the friction rising on the wall a weight force intake is realised on the

plane OA. As a result of the weight-force intake the force acting on the plane AB is

reduced; so it is not capable of producing stresses with the value of



and in the

direction AB. The freshly filled-in material comes to a standstill by finishing its

consolidation motion. As a result of the quiescent state the friction rising on the wall

is reduced to zero, consequently the stresses





are decomposed into their

horizontal and vertical components. In the OAB material the vertical components

complement the vertical stress components to hγ, and supplement the weight of the

OAB material to





ctg

. At this time the horizontal component of stress





acts

on the OA wall. Consequently the horizontal component of the static pressure is:

,cos









and, therefore the factor of static pressure is (λ):

cos







Law IV

Law IV can be proved experimentally.

As a result of the experiment demonstrated in Figure 9, due to the collision of the

grains, the granular material behaves according to the laws of the communicating

vessels.

- 25 -

Figure 9. a) state of rest b) state formed as a result of vibration

In Figure 10 as a result of the vibration, and due to the collision of the grains the

body with bigger volume weight γ

and the body with smaller volume weight γ

, -

which were place into the granular material with volume weight γ - sinks to the

bottom of the vessel, or floats on the surface of the granular material respectively;

consequently the law of Archimedes prevails.

Figure 10. a) state of rest b) state developed as a result of the vibration

As a result of the vibration the components of the granular material, the grains

collides into each other, and due to this effect the pressure changes to hγ in every

direction.

- 26 -

Stresses in the non-cohesive granular materials

We performed the examination of the horizontal plane quadrant cut out by the

vertical plane in the proving procedure of the Law III. If this plane inclines towards

the material compared to the vertical, – closing an angle β with the horizontal, - and

the terrain is horizontal, then, generalizing the former deduction the magnitude of

lateral pressure of this granular assembly can be determined in the plane of angle β.

Figure 11. Infinite quadrant of the horizontal terrain confined

with an inclined plane

Using the markings of Figure 1, it can be said that the ADC granular material weighs

on the slope with an inclination angle





, produced as the result of the Δh

- 27 -

depth increase(as it is marked on Figure 11) with its self-weight (ΔG) and with the

weight of the material above it (G). Line segment Δh supports the ADC material

amount in order to prevent its sliding down. Since Δh is very little, therefore, the

stress distribution can be considered even in the area, thus the following can be

formulated for the slope-direction stress rising there:

)cos)(sin(













where the ΔF is the projection of the surface segment Δh perpendicular to the slope.

Figure 12.

h

part of Figure 11. Figure 13.

F

part of figure 11.

The G given in the equation can be expressed for the unit-long space parts ΔG and

ΔF from Figures 11, 12 and13:

),(





ctgctg

G 

and







sin2

where m can be expressed with the help of Figure 11:

)sin(



 zm

, and

)90cos(









thus

),sin(

)90cos(











- 28 -

that is

),sin(

sin











Replacing the value of the m into the relation written for the ΔG:

sinsin

)sin(















The value of ΔF can be expressed with the help of Figure 13:

 

,)(sin



 zF

 

.)(sin

sin











The equality





cos

sin

cossin





was deduced during the proof of the Law II.

Replacing the value of G, ΔG, ΔF and



cossin 

into the relation written for

the σ

 

cos

sin

)(sin

sin

sins in

)sin(

)(



































ctgctg

 

)(sin

sinsin

)sin(

)(

cos

sinsin































hctgctgh

sin)cos(cos)sin(

sin

sincoscossin

)cossin(

cos

sin





























hctgh

)cos()sin(

)cossin()cossin(

cos























ctg

ctghctgh

)cos()sin(

cossin

cos2

























ctg

hhh

but



ctg

can be expressed from the triangle ACE of Figure 11 by employing the

triangle ADF of Figure 12:

AFCD

ctg







where:



sin

CD 

and

)cos(



 zAF

- 29 -

but



sin

)90cos(











therefore

)cos(

sin











)cos(

sinsin

ctg















since

),sin(

sin











)sin(

sin

)cos(

sinsin























ctg

)sin(sin

)cos(sinsin













ctg

);(

)sin(sin

sin













 ctg

ctg

replacing the value of



ctg

into the relation of the





)cos(

)sin(

)sin()cos(

)sin(sin

cossin

cos2







































ctg

hhh

)cos()cos(

sin

cossin

cos2































ctg

hhh









cossin

sin)cossin)((

2 





ctghhh

),1(





tgctg







if Δh is very little, that is Δh → 0, then α →



, consequently





→





, i.e. the

direction of the stress closes an angle



with the horizontal.

Consequently

),1(

lim









tgctg





- 30 -

that is























tgh

The horizontal component of the





is:

)

sin

(cos











Examining the three special values of the slope angle β of the plane, (which is the

angle at which the plane inclines to the horizontal), it can be established that if





, then

0





, i.e. the non-cohesive granular material will stop in the free

slope without support.

90



, then the static pressure acting on this plane is:









, and







cos









, then the static pressure acting on this plane is:









sin1

2 



, and



















- 31 -

Active stress state

Development of the active stress state

The small-size horizontal-direction displacement – tilt – of the vertical wall

supporting the non-cohesive granular material being in a state of rest causes

expansion in the material. The motion of the material follows the displacement of

the retaining wall into the horizontal direction loosened up, which appears as a

relatively two-direction displacement from a given point of the interior of the

material. As a result of the displacement following the expansion, the effect of the

shear stresses mobilised by the horizontal stress components of the static pressure

ceases (breaks up). The relatively two-direction displacement inside the material

terminates the vertical-direction shear stresses in pairs, therefore the vertical stress

increases to hγ. At the same time the material begins to carry out a consolidation

motion.

Figure 14. Stress model in quiescent a position

- 32 -

The vertical stress of Figure 7 and 8 (state of rest) can be divided into two stresses of

the same size and direction (Fig. 14). The development of the active stress state can

be explained in the horizontal-surface non-cohesive granular materials by the change

of the so obtained stress model. The shear stresses, which were terminated due to the

effect of the expansion, change the stress model of Figure 14 to that of Figure 15.

Figure 15. Change of the stress model in an active state

The consolidation motion occurs in the direction of the biggest stresses, i.e. in the

direction of the stress resultants. The directions of the resultants of the stress pairs –

which can be read from Figure 15 – incline at an angle





to the horizontal

and at an angle





to each other. Considering the acting (resultant) stress

directions this stress condition consequently, corresponds to the Rankine active

stress state. Stress

starts the consolidation motion. This motion is reduced by the

multiplication product of



and stress

- a stress perpendicular to the stress

mobilises shear stress. The magnitude of stress K inclining at an angle





to the horizontal, consequently is



tgRRK 

Stress

consists of two stresses.

and

can be expressed from the illustrations

of the stress vectors in Figure 16.

Stress

presents itself as the sum of two stresses; the sum of its stress components

taken for this direction (

) develops the shear stress.

- 33 -

Figure 16. The motion started by the resultant stress mobilises shear tress