László Mester. The new physical-mechanical theory of granular materials

Transcription

1 László Mester Te new pysical-mecanical teory of granular materials 9

2 - - Contents Introduction 3 Granular material as a distinct state of matter 4 Pysical properties of te granular material in relation to te different states of matter 6 Granular material as a state of matter 1 Pysical-mecanical basic laws of te non-coesive granular materials 14 Law I 15 Law II 15 Law III 17 Law IV 4 Stresses in te non-coesive granular materials 6 Active stress state 31 Development of te active stress state 31 Pressures acting on te vertical retaining wall 36 Arc formation of granular materials 39 Condition of te arc formation 39 Caracter of te discarge 44 Mecanism of te arc formation 46 Geometric equation of te arc 47 Principle of te opper design 5 Experimental results 53 Stresses in coesive granular materials 55 Lateral pressure 55 Inclination angle of te free slope 59 Active stress state 64 Summary 69 Bibliograpy 73

3 - 3 - Introduction Following Coulomb s and later Rankine s work te pysical-mecanical teoretical researc of granular materials as been caracterised by te use of stress analysis deduced for solids since te 18t century Oters seem to detect te caracteristic features of viscous liquids in granular materials terefore tey describe te pysical beaviour of granular materials using te laws pertinent to viscous fluids In my opinion most of te teorems wic were put for cotinuums cannot be applied to te aggregation of separate solid granules Only tose natural laws can be considered as te starting point of examination wic are also valid for te universal material Tis work breaking away from te previous tradition would like to approac te pysical mecanical properties of granular materials from a new point of view As a result te critical analysis of teories formulated earlier in tis researc area is not te objective of tis paper since te new principles were laid down irrespective of tose ypoteses Contrary to te continuum teory by examining te equilibrium and kinetic state of individual granular particles tis new tesis is based on simple experiments on te Newtonian laws and on an empirical law te law of friction

4 - 4 - Granular material as a distinct state of matter Te pysical appearance of materials found in nature is quite varied Te most substantial part of te Eart s surface is covered by oceans seas lakes tat is to say covered by water Te dry land is more diverse: one can find rocky mountain ridges surfaces covered by gentle slopes and deserts wit sand dunes In places te eart is covered by snow or ice in winter Above te surface level te wind is blowing or we can experience a period of calm tat is to say we can feel te air Te sun is sining above us and we know tat inside te sun one would find anoter state of matter Te outward form of te water tat covers substantial part of te Eart is in itself diverse At normal temperature and pressure water is liquid but wit te increase of temperature it evaporates more and more quickly and it turns into water vapour Fog or clouds form Wen water vapour freezes and precipitates in cold ten snow falls and it condenses into a granular material Wen snow melts te result is liquid wic in turn becomes solid wen it freezes Tat is to say water can exist in liquid vapour (gas) snow (granular) and ice (solid) states In eac of its pases water as different pysical properties and beaves conforming to different laws Pysics differentiates among te most prominent forms of appearance of material by classifying tem into te different states of matter: plasma gas liquid and solid Some material cannot be put strictly under one category because tey bear te pysical properties of two or more states of matter Tese materials owever can be described by applying te laws pertaining to materials in a similar state of matter

5 - 5 - Granular material cannot be put under any one of te above mentioned four categories Furtermore te pysical-mecanical properties of granular material do not make it possible to describe its beaviour successfully using te pysical laws of one or more pases A granular material is a conglomeration of large number of solid particles related to one anoter were te granules as te constituent of te aggregate in spite of te affecting forces retain teir form and te incidentally arising coesive force between te granules is substantially smaller tan te inner coesion of te individual granules Te overall coerent pysical system of granular material as not been set up yet scientific analysis is available only for a few prominent primarily soil mecanical problems Several teories ave been applied for te andling of tese problems wic speculations owever led to contradictory results Furtermore no connection resting on firm uniform pysical foundations exists between te teories or if tere is a relation it is disputable Te majority of mecanical teories dealing wit granular materials apply te metod of stress analysis deduced for solids wic procedure presupposes tat te granular material is a solid pase continuum Tere are teories according to wic granular materials can be approaced wit te laws pertaining to viscous liquids since granular material exibits viscoelastic and viscoplastic properties Altoug te difficult teoretical notions provide an approximate solution to individual mecanical problems tey cannot be applied to an overall reliable description of granular material beaviour Te opinions concerning te state of matter of granular materials are not unanimous Tis is reflected in te fact tat granular material does not ave a single uniform name for example te following designations: scattering powderlike loose granulated grainy particulate granular and gritty are all used Te pysical beaviour and properties of granular materials exibit substantial qualitative difference from te materials in oter states of matter and sould terefore be considered an additional state of matter in its own rigt

6 - 6 - Te idealised notion of granular material makes te simplified explanation of its pysical beaviour properties and origin possible similarly to te ypoteses applied to perfect gases ideal liquids and crystalline solids Te ideal granular material is a conglomeration of large number of solids (granules) were te mass of te solid particles is small compared to te mass of te material in tis aggregate attractive force does not operate between te particles Coulomb friction rule governs tem Pysical properties of te granular material in relation to te different states of matter Te basis for classification according to states of matter depend on te question weter te material can old its own sape and volume or not Te basic criteria of classification of te tree classical states of matter are te following: - gases: ave no definite sape or volume; - liquids: ave no definite sape but ave definite volume - solids: ave definite sape and volume Regarding te question of definite sape and volume it is te caracteristic of te granular material tat: - In part it as definite sape te granular aggregate olds its sape in te angle of repose but under tis angle it takes te sape of its container Tis attributive places te material between liquids and solids - In part it as definite volume but it can be compressed to a limited extent Te compressibility of granular materials stands between te compressibility of gases and solids Te researces on substance structure concerning te states of matter found tat te determining factors in te question weter a material olds a definite sape or volume lie in te pysical properties of its constituents and te nature of interaction between tese particles For tis reason modern pysics studies te kinetic state of te smallest particles attributed to te material teir relative position and te particle interaction wen defining te different states of matter Tis made it possible tat in

7 - 7 - natural science in addition to te tree classical states of matter a fourt state of matter was accepted te plasma state It seems to be necessary to empasise te expression smallest constituent caracteristic of te material since it is of primary importance in te definition of te different states of matter Consequently - A material in plasma state consists of te disintegrated parts of molecules or atoms te molecule ions or atomic ions Te plasma state is caracterized by te interactions of atomic or molecule ions and electrons and not by te oter parts of te atom or molecule - Te pysical properties of a material in te gas state are determined by te interactions of gas molecules in case of noble gases te interactions of atoms - Regarding liquids te determining factors are again te movements of te atoms or molecules and te nature of relation between tem In te case of water it is te interactions of H O molecules and not te ydrogen or oxygen atoms or te water drop wic caracterize te liquid - In te case of crystalline solids te pysical properties of a material in te solid state can be explained wit te nature of interaction of te atoms molecules or ions positioned in te lattice nodes of a crystal structure and cannot be described for example wit te interactions of elemental crystals and crystallites or wit te individual properties of te atoms and its parts wic constitute te molecules positioned in te lattice points - In te case of granular material te smallest constituent caracteristic of te material is te granule Te atomic particles te atoms and molecules tat constitute te granules are not direct caracteristics at least in pysics tey cannot be regarded as significant pysical properties of a material just like as in te case of gases and liquids were material is not caracterised by te atomic particles wic constitute te atom or molecules or by te individual pysical properties and interactions of atoms eiter Te most significant caracteristics of te tree classical states of matter can be summarised in te following way:

8 - 8 - Gaseous state: Te molecules of gases in case of noble gases te gas atoms move freely in te space available for tem tey collide elastically wit one anoter in a random motion Te average distance between te constituting particles of gases is relatively big in proportion to teir size te intermolecular forces between te particles are very weak In te case of ideal gases intermolecular force can be disregarded Molecules move wit a translational rotational and vibrational motion Gases evenly fill te space available for tem tat is to say tey ave no definite sape or volume Liquid state: Te intermolecular forces between te smallest constituents caracteristic of liquids between te atoms or molecules are strong enoug to prevent te particles moving away from eac oter as a consequence of termal motion but not strong enoug to prevent teir cange of position Compared to gases te translational motion of te molecules are smaller wile tey also carry out rotational and vibrational movement Due to teir motion and proximity te constituting particles collide elastically wit one anoter all te time tus touc one anoter terefore liquids ave a definite volume Te force of attraction between te particles is so small compared to te Eart s gravitational force tat it is not enoug for individual sape formation as a result liquids ave no definite sape Solid state: Te smallest constituting particles caracteristic of solids are te atoms molecules or ions Teir position is fixed and geometrically determined in a crystalline structure particles carry out only vibrational motion Te intermolecular forces are strong wic prevent teir permanent displacement from teir state of equilibrium As a result solids ave definite sape and volume Granular material exibits significant differences from te aggregational properties of te tree classical states of matter Te constituents of an ideal granular material te granules are at a relative rest Tere are no forces of attraction between te particles te material is kept in an aggregate by te compressive forces originating from te gravitational force by te sear forces arising on te surface of te granules and by te static friction force Due to tese forces te ideal granular material remains stable until te angle of repose is reac tus it as only partly a definite sape Te constituting particles are in constant contact terefore in

9 - 9 - quiescent state granular material as definite volume Under pressure te material is compressed te granules take up a more efficient space filling position Te compressibility of granular material is small compared to gases but it is big in comparison to solids Te pysical properties of ideal granular material exibit te following significant differences in caracteristics compared to te features of oter states of matter: - In contrast to gaseous state: te constituting particles are in constant contact wit eac oter and it as definite volume; - In contrast to liquid state: granular material as in part a definite sape; - In contrast to gaseous and liquid state: Te constituting particles are in a relative collision free quiescent state and static friction force sear force arise in tem; - Solid state: tere is no attractive force between te constituting particles terefore granular material as only in part a definite sape Granular material exibits suc qualitative differences concerning te most substantive caracteristics of te different states of matter tat its definition as a separate state of matter in its own rigt becomes justified Te brief straigt to te point definition wit no pretence to completeness of te idealised case of te states of matter is te following Perfect gas: disordered aggregate of molecules (in case of noble gases atoms) wit no intermolecular forces were te molecules move far apart from eac oter undergoing random elastic collisions Ideal liquid: Aggregate of molecules moving close to eac oter undergoing constant elastic collisions Crystalline solid: Te ordered aggregate of vibrating atoms molecules or ions wic are fixed in teir structure wit great force Ideal granular materials: te aggregate of relatively static particles wic are in constant contact wit eac oter in tis assembly te force between te constituting particles is composed of te compressive force arising from te gravitational force and of te friction force wic is proportional to it tere is no coesion force between te particles

10 - 1 - Te notion of granular material as a separate state of matter is primarily important from a mecanical viewpoint (Te basis for categorization into different states of matter as a mecanical origin: te reason for aving a definite sape or volume can be deduced from te intermolecular forces between te constituting particles) Te mecanical properties of te different states of matter sow distinctive substantial differences: - gases respond to an increase in pressure wit te significant reduction of teir volume (at constant temperature te multiplication product of volume and pressure is constant) terefore tey can witstand compressive stress only in part at te expense of volume cange In perfect gases tere is no attractive force between te constituting particles terefore no tensile stress can arise in te material In static not flowing gases no sear stress arises - liquids as small compressibility from a mecanical point of view tey can be regarded incompressible terefore tey can witstand compressive stress Te intermolecular forces are strong enoug to prevent te constantly colliding molecules from moving far away from one anoter Te state of equilibrium or stability can only be attained under a given outside pressure tat is to say from a mecanical standpoint a liquid cannot witstand tensile stresses (Wen te pressure is around p= te liquid breaks up its molecules fly apart and turn into gaseous state) Tere is no friction in ideal liquids in real liquids static friction does not arise eiter - solids can be regarded as incompressible te constituting particles join togeter wit great force terefore tey can witstand tensile compressive and sear stress - ideal granular material as small compressibility terefore it can witstand compressive stress In te non-coesive granular materials only sear stress arises in addition to compressive stress no tensile stress manifests itself Granular material as a state of matter Te notion tat granular material must be regarded as a separate state of matter can be justified not only because its distinct pysical properties wic differentiate it

11 from oter states of matter but also because granular material is one of te existing outward forms of raw material tat is to say it is one of defined states of matter Granular material as te conglomeration of large number of solids were te constituting solids are small in proportion to te total mass of te material generally comes into being wen large-sized solids are mecanically cut up or wen te solids temselves break up into smaller pieces Its formation tat is to say te bringing of te material into a granular state can be acieved not only in a mecanical way as it is also true for te granular materials in nature wic were formed not exclusively by mecanical disintegration eiter Granular material can be produced via a termodynamic metod It is known tat if te kinetic energy of te molecules of a liquid exceeds a tresold it canges into gaseous state and if it goes below anoter tresold te liquid turns into solid and te process of crystallization begins Te tresold values of te termodynamic state parameters caracteristic of te different states of matter can be illustrated in a p t (pressure-temperature) diagram In Figure 1 te p t diagram of H O can be seen Figure 1 p-t diagram of H O

12 - 1 - Te curve tat joins te triple point and te origin of te p t diagram is called te sublimation curve Te material passes from te solid ice pase into te vapour pase of te gas state by for example pressure reduction and by te crossing of te sublimation curve Crossing te sublimation curve backwards from vapour pase granular snow is formed not ice Te granular material in case of water te snow comes into being as a result of crystallization in te local clusters Te process can be called local crystallization te pysical explanation of wic lies in te penomenon tat te molecules wic move wit slow translational motion (under low temperatures) cannot leave te attraction field of te van der Waals type forces for example as a result of eat loss terefore te translational motion of te molecules ceases Te molecule pair - bonding te new molecules wic collide into tem - form a crystal lattice te growing crystals ten bring about te granules Te density of molecules in te gaseous state is very low in comparison to te molecular density of te solid state terefore te local crystallization processes wic are relatively far from one anoter bring about te multitude of separate granules wic after aving precipitated form a granular conglomeration Under constant temperature te process of getting from gas pase to granular pase is accompanied by eat loss wic is te sum of te melting and te evaporation eat Te states of matter cange at te pase boundaries of te p t diagram If te matter crosses te sublimation curve from te solid pase toward te gas pase we will get a gas owever canging te direction crossing te curve from te gas pase we will obtain a granular material Tus te states of matter in compliance wit te direction of crossing te pase boundaries are te following: gas local crystallization granular melting liquid evaporation gas; neverteless from te oter direction: gas condensation liquid freezing solid sublimation gas From granular pase to solid pase we can get by crossing te same pase boundary twice:

13 granular melting liquid freezing solid Granular material can be produced directly from liquid pase if we place a multitude of crystal nuclei - wic are approximately at an equal distance from one anoter - into a supercooled liquid at te same time Te crystal growt is indered by te neigbouring crystals wose geometric crystal position is not symmetrical or congruent terefore no or only occasional lattice forces develop between te crystals te inner coesive force of te individual granules are substantially greater tan te incidental coesive forces acting between te granules Te substantial pysical properties of te granular material differ significantly from te caracteristics of tose materials wose cemical properties are identical neverteless belong to te solid liquid or gaseous state Its volume weigt its refractional termodynamical acoustic electric and mecanical properties and beaviour and te fact tat most material can be brougt into granular state justifies te classification of te granular material as a distinct state of matter by its own rigt

14 Basic pysical-mecanical laws of te noncoesive granular materials I In te non-coesive granular materials only compressive and sear stresses can develop II In te non-coesive granular materials at a quiescent state te stresses developed by te vertical-direction compressive stresses act downwards in te 9 zone measured from te vertical direction ( is te angle of friction of te material) III Te value of te lateral pressure rising from te self-weigt of te non- coesive granular material is ( ) te alf of te product of te dept () and volume weigt (γ) its direction deviates from te orizontal downwards wit te angle of friction developed in te material if te surface is orizontal and over te given dept te material fills te space evenly closing an angle wit te orizontal IV Te non-coesive granular materials conform to te pysical-mecanical laws caracteristic of tem until teir constituting elements te grains keep teir relative quiescent state Wen te grains go into motion collide wit eac oter - te granular materials beave according to te pysical-mecanical laws of te liquids Te pysical-mecanical laws of te non-coesive granular materials prevail wit a statistical caracter because te material itself consists of a multitude of different grains

15 Law I Law I is te pysical-mecanical definition of te non-coesive granular materials In ideal liquids only compressive stresses develop te non-coesive granular materials are capable of witstanding compressive and sear stresses wile solids are capable of bearing compressive sear and tensile stresses Te non-coesive granular materials differs from te solids in te respect tat tey are not capable of witstanding tensile stress and tey are distinct from te ideal liquids because sear stresses also develop in tem At te same time te components of te liquids are in constant relative motion collide wit eac oter - wile te components of te granular materials te grains are in a relative quiescent state Law II Law II formulates te direction of te spreading of te vertical compressive stresses Te natural stability of te free slope provides its experimental proof (Figure ) Figure Te boundary equilibrium position of te grain located on te side of te slope

16 In te conglomeration of grains te self-weigt of te grains produces te vertical compressive stress If a stress vector acted on te grain marked A located on te side of te slope in boundary equilibrium position - inclined at an angle of more tan 9 from te vertical ten te grain would loose its equilibrium position and slide down Figure 3 Te stresses developed by te vertical compressive stresses incline at an angle bigger tan to te orizontal Furter experiments prove te correctness of te Law II (Figure 3) If a compressive stress making a smaller tan angle wit te orizontal acted on te grain marked A te angle of inclination angle of te natural slope would be smaller tan If te vertical compressive stress induced for example a orizontal stress it would trust down te grains located on te side of te slope Te material would spread and would take a kind of sape tat is illustrated in Figure 4 However it does not exist Figure 4 If te compressive stresses induced orizontal stresses tis would be te position of te granular material

17 Law III In te cases described in te Law III te lateral pressure is and its direction inclines from te orizontal downward wit te angle of friction rising in te inside of te material Its proof is as follows: Figure 5 Infinite Quadrant of te Horizontal Terrain Figure 5 sows tat part of te non-coesive granular material aggregate of infiniteexpansion and orizontal-terrain wic is cut out teoretically by two vertical planes perpendicular to eac oter (consequently it sows an infinite quadrant of te orizontal terrain) wic makes te planar execution of te mecanical tests possible According to te Law II from te material part under te section AB only reaction stresses produced by te material part over te plane AB can act on te plane OA If we took off te granular material located in te triangle OAB ten te material would remain stable in te natural angle of repose AB inclining at an angle to te orizontal On te plane OA stresses can only rise from te self-weigt of te granular material located in te space part OAB On te plane AB an equilibrium boundary position exists; te material aving a friction coefficient of tg does not slide as yet on te slope wit an inclination of angle If we increased te slope angle wit of a very low value ten te material above it could slide down wit

18 constant acceleration on te slope wit te inclination angle ie it would exert a force in direct proportion to its mass and acceleration on te plane OA Due to te pysical properties of te granular materials tere are infinitely many slopes wit an inclination angle above wic angle on te slopes increased by te angle te weigt of te materials exerts slope-direction stresses on te plane AO According to Figure 5 on te slope wit an inclination angle of produced as a result of te dept increased by Δ te granular material ADC weigs eavily on te slope wit its self-weigt (ΔG) and wit te weigt of te material located above it (G) Te material amount ADC is supported on section Δ Te projection of te surface section Δ perpendicular to te slope is F cos( ) Considering tat te Δ is very little terefore te stress distribution can be considered even so it can be stated for te slope-direction stress developed tere: In te unit-lengt space part ( G G)(sin cos ) F and tat is furtermore G ctg 1 G cos sin G ctg F cos( ) Te ( sin cos ) in te equation can also be expressed: sin( ) tg cos( ) sin sin cos cos sin (cos cos sin sin ) cos sin cos sin sin cos

19 sin (cos sin ) cos sin cos Substituting te values of te G ΔG ΔF and sin cos into te relation written for te : cos sin ctg sin cos cos( ) sin ctg( ) cos cos( ) cos sin ( ) sin cos (cos cos sin sin ) cos sincos sin sin sin cos sin tg but te tg can be expressed from te triangle ACE of Figure 5: tg tg cos sin sin sin cos sin terefore sin cos sin sin cos sin sin sin

20 - - if te Δ is very little tat is Δ ten α consequently ie te direction of te stress inclines at an angle to te orizontal Consequently tat is lim As a result of te deduction it can be establised tat te stress distribution is linear against te dept However in case of granular materials one cannot speak of a stress in te classical sense since te force effects are transmitted at te contact points of te granules ie from one point to anoter not on a surface perpendicular to te given direction Not on a surface because te material is a discontinuum and te grains touc eac oter only at points Terefore te meaning of te stress can be interpreted as te average force imparted to one surface and tese average forces are transferred from one granule to anoter Te direction and size of tese forces manifest temselves as a statistical average on a given surface Figure 6 Division of te average force acting on te grain located next to te wall into a orizontal and a vertical component

21 - 1 - Let us examine wat te magnitude of te force is wic te above deduced stress inclining at an angle to te orizontal exerts on a vertical wall Troug te contact points force arrives from te neigbouring granules at te grain wic is supported by a frictionless vertical wall sown in Figure 6 Te resultant of tis force effect sould be equal wit te product of te stress calculated for te surface F1 ie considering its magnitude and direction te compression force acting on tis grain corresponds to te statistical average of te force exerted at tis dept in te given granular material Tis granule presses te vertical frictionless wall wit a orizontal F x force on te wall section F at te contact point Te vertical-direction force F y 3 is received by te grain or grains located under it Te surface of sections F1 F and F3 are equal surfaces on statistical average because considering teir sape and position te grains are speres on statistical average; te projections of te speres from any direction are of equal surface area (If a granular material for example rice consists of oblong grains; considering te random arrangement of te grains te average of teir projection taken in any direction is a circle ie te grains must be considered as speres on statistical average) Consequently it can be written for te vector triangle of te Figure 6 x F F 1 cos but since terefore F 1 F cos x It comes from te result of te above consideration tat in granular material te stresses te average forces calculated for a given surface can be decomposed or added as vectors From te cos x equation te factor of static pressure is x received after te substitutions tat is y so y cos x cos

22 - - Up to tis point only te sear stresses generated by te stresses acting perpendicularly to te direction inclining at an angle to te orizontal and rising from te self-weigt were taken into account at te deduction of te static pressure Te sear stresses y produced by te orizontal stress components - x cos - reduce te vertical stresses γ wit y cos tg tat is so Considering tat te stresses y sin cos cos y sin act in pairs on te teoretic plane OA assumed inside te granular material (Figure 5) terefore te vertical-direction stresses γ are reduced by y consequently y y tat is and y sin ( 1 sin) y Consequently te figure of te stresses acting inside te non-coesive granular materials at quiescent state can be constructed (Figure 7) Figure 7 Stresses acting at a quiescent state

23 - 3 - If te plane OA according to Figure 5 is a teoretic plane assumed inside te granular material ten te angle of friction is tere In tis case a compressive stress of inclined at an angle to te orizontal acts on bot sides of te plane OA (Figure 7) On te plane OA te s can be divided into orizontal and vertical components (Figure 8) Te orizontal components ave te value x cos tey are perpendicular to lane OA and satisfy te action-reaction law Te vertical stress components of complement te vertical stresses to γ symmetrically to plane OA in a reciprocal way y Figure 8 Te orizontal and vertical stress component at a quiescent state If te plane OA according to Figure 5 is a frictionless wall ten tat is capable of taking only orizontal stress ie te orizontal component of te wic is x cos At te same time te vertical component of te complements te vertical stress component of te material OAB to γ If te plane OA is an actual roug rigid wall wic serves to support te OAB material amount ten te development of te static pressure can be interpreted as follows

24 - 4 - After filling up te OAB material amount beind te OA wall te value of te is not reaced immediately since due to te inclined stress effect and because of te friction rising on te wall a weigt force intake is realised on te plane OA As a result of te weigt-force intake te force acting on te plane AB is reduced; so it is not capable of producing stresses wit te value of and in te direction AB Te fresly filled-in material comes to a standstill by finising its consolidation motion As a result of te quiescent state te friction rising on te wall is reduced to zero consequently te stresses are decomposed into teir orizontal and vertical components In te OAB material te vertical components complement te vertical stress components to γ and supplement te weigt of te OAB material to ctg At tis time te orizontal component of stress acts on te OA wall Consequently te orizontal component of te static pressure is: x cos and terefore te factor of static pressure is (λ): cos Law IV Law IV can be proved experimentally As a result of te experiment demonstrated in Figure 9 due to te collision of te grains te granular material beaves according to te laws of te communicating vessels

25 - 5 - Figure 9 a) state of rest b) state formed as a result of vibration In Figure 1 as a result of te vibration and due to te collision of te grains te body wit bigger volume weigt γ 1 and te body wit smaller volume weigt γ - wic were place into te granular material wit volume weigt γ - sinks to te bottom of te vessel or floats on te surface of te granular material respectively; consequently te law of Arcimedes prevails Figure 1 a) state of rest b) state developed as a result of te vibration As a result of te vibration te components of te granular material te grains collides into eac oter and due to tis effect te pressure canges to γ in every direction

26 - 6 - Stresses in te non-coesive granular materials We performed te examination of te orizontal plane quadrant cut out by te vertical plane in te proving procedure of te Law III If tis plane inclines towards te material compared to te vertical closing an angle β wit te orizontal - and te terrain is orizontal ten generalizing te former deduction te magnitude of lateral pressure of tis granular assembly can be determined in te plane of angle β Figure 11 Infinite quadrant of te orizontal terrain confined wit an inclined plane Using te markings of Figure 1 it can be said tat te ADC granular material weigs on te slope wit an inclination angle produced as te result of te Δ

27 - 7 - dept increase(as it is marked on Figure 11) wit its self-weigt (ΔG) and wit te weigt of te material above it (G) Line segment Δ supports te ADC material amount in order to prevent its sliding down Since Δ is very little terefore te stress distribution can be considered even in te area tus te following can be formulated for te slope-direction stress rising tere: ( G G)(sin cos ) F were te ΔF is te projection of te surface segment Δ perpendicular to te slope Figure 1 part of Figure 11 Figure 13 F part of figure 11 Te G given in te equation can be expressed for te unit-long space parts ΔG and ΔF from Figures 11 1 and13: G ( ctg ctg ) and m G sin were m can be expressed wit te elp of Figure 11: m z sin( ) and z cos(9 ) tus m sin( cos(9 ) )

28 - 8 - tat is m sin( ) sin Replacing te value of te m into te relation written for te ΔG: sin( ) G sin sin Te value of ΔF can be expressed wit te elp of Figure 13: F z sin ( ) F sin ( ) sin sin Te equality sin cos was deduced during te proof of te Law II cos Replacing te value of G ΔG ΔF and te σ α : sin( ) ( ctg ctg ) sin sin sin cos sin ( ) sin sin cos into te relation written for sin( ) ( ctg ctg ) sin sin sin sin cos sin ( ) sin cos cos sin ( ctg sin cos ) sin sin cos sin( ) cos cos( ) sin 1 ( ctg sin cos ) ( ctg sin cos ) cos ctg sin( ) cos( ) but ctg sin cos cos ctg sin( ) cos( ) ctg can be expressed from te triangle ACE of Figure 11 by employing te triangle ADF of Figure 1: AE CD AF ctg m m were: CD and AF z cos( ) sin

29 - 9 - but z cos(9 ) sin terefore F cos( ) sin so ctg cos( ) sin sin m since m sin( ) sin ctg cos( ) sin sin sin( ) sin replacing te value of sin sin cos( ) ctg sin sin( ) sin ctg ctg ( ); sin sin( ) ctg into te relation of te : ctg sin cos cos sin sin( ) cos( ) sin( ) cos( ) sin sin( ) sin( ) ctg sin cos cos sin cos( ) cos( ) sin ( )( ctg sin cos )sin sin cos (1 ctg tg) if Δ is very little tat is Δ ten α consequently ie te direction of te stress closes an angle wit te orizontal Consequently lim (1 ctg tg)

30 - 3 - tat is Te orizontal component of te is: tg 1 tg sin (cos ) tg Examining te tree special values of te slope angle β of te plane (wic is te angle at wic te plane inclines to te orizontal) it can be establised tat if ten ie te non-coesive granular material will stop in te free slope witout support If 9 ten te static pressure acting on tis plane is: and cos If 45 ten te static pressure acting on tis plane is: 1 and tg45 1 sin

31 Active stress state Development of te active stress state Te small-size orizontal-direction displacement tilt of te vertical wall supporting te non-coesive granular material being in a state of rest causes expansion in te material Te motion of te material follows te displacement of te retaining wall into te orizontal direction loosened up wic appears as a relatively two-direction displacement from a given point of te interior of te material As a result of te displacement following te expansion te effect of te sear stresses mobilised by te orizontal stress components of te static pressure ceases (breaks up) Te relatively two-direction displacement inside te material terminates te vertical-direction sear stresses in pairs terefore te vertical stress increases to γ At te same time te material begins to carry out a consolidation motion Figure 14 Stress model in quiescent a position

32 - 3 - Te vertical stress of Figure 7 and 8 (state of rest) can be divided into two stresses of te same size and direction (Fig 14) Te development of te active stress state can be explained in te orizontal-surface non-coesive granular materials by te cange of te so obtained stress model Te sear stresses wic were terminated due to te effect of te expansion cange te stress model of Figure 14 to tat of Figure 15 Figure 15 Cange of te stress model in an active state Te consolidation motion occurs in te direction of te biggest stresses ie in te direction of te stress resultants Te directions of te resultants of te stress pairs wic can be read from Figure 15 incline at an angle and at an angle 45 to te orizontal 9 to eac oter Considering te acting (resultant) stress directions tis stress condition consequently corresponds to te Rankine active stress state Stress R 1 starts te consolidation motion Tis motion is reduced by te multiplication product of tg and stress R - a stress perpendicular to te stress R1 - R 1 mobilises sear stress Te magnitude of stress K inclining at an angle to te orizontal consequently is K R 1 R tg 45 Stress R 1 consists of two stresses R 1 and R can be expressed from te illustrations of te stress vectors in Figure 16 Stress R 1 presents itself as te sum of two stresses; te sum of its stress components taken for tis direction ( R ) develops te sear stress

33 Figure 16 Te motion started by te resultant stress mobilises sear tress R 1 sin 45 and R cos 45 tus K sin 45 tg cos 45 Te orizontal component of te resultant stress K: K K K cos 45 sin 45 cos 45 tg cos 45 because tat is K 1 1 sin 45 tg cos sin 1 sin cos 45 cos sin

34 cos cos 1 cos 1 1 cos 1 cos 1 cos 1 cos sin 1 cos 4 K K K K sin cos cos cos cos sin cos 1 sin cos 1 sin 1 sin sin sin cos K tg 45 Te vertical component of te resultant stress K is K v K sin 45 K v K v sin 45 tg sin 45 cos and sin 45 cos 45 cos since sin 45 1 sin 1 tus K v 1 sin tg cos K v 1 sin 1 sin cos cos K v

35 Te resultant stress K is K K K v K tg 45 1 sin 45 cos 45 K cos 45 cos 45 K sin 45 cos 45 cos 45 K 1 cos 45 1 K cos 45 Comparing te value of te K wit te value of te wic is - it is conceivable tat te K is bigger Terefore in case of expansion or in case of a more significant displacement of te wall supporting te granular material te motion direction of te material inclines at an angle of 45 to te orizontal Due to te expansion te stress starting te motion can be illustrated according to Figure 17 Figure 17 Motion starting stress in an active state

36 Te motion is realised towards te direction of te resultants of te stresses consequently in te direction inclining at an angle of 45 to te orizontal Te material moves wit its wole material amount ie infinitely many slip planes inclining at an angle of 45 to te orizontal are developed Te orizontal component of te resultant stress K is x ie K cos 45 K sin 45 x sin 45 cos 45 x tg45 x Pressure acting on te vertical retaining wall If te orizontal-terrain non-coesive granular material is supported by a real frictional vertical wall and an expansion occurs in te material due to its displacement ten te stresses acting on te wall can be determined wit te knowledge of te angle (δ) of te friction rising on te wall: A stress wit a magnitude of inclining at an angle δ to te orizontal acts on te wall Te sin -fold amount of tis stress reduces te vertical stress sin Te orizontal stress component: cos is in direct proportion to to

37 te vertical stress just as te ratio of te vertical and te orizontal cos stress components is constant in te stress model acting inside te material Consequently te proportion can be written: Te can be expressed: cos sin cos cos cos sin cos cos sin cos cos (cos sin cos) cos cos cos sin cos Te orizontal component of te is : cos cos cos cos sin cos cos 1 tg cos Te obtained result sows tat if no friction were developed on te retaining wall ten te static pressure would act on it; or looking at it from te oter way round: no friction rises on te retaining wall wen static pressure develops Tis is proved by te evidence of te model experiments If te friction rising on te retaining wall were equal to te friction rising inside te granular material ie ten active pressure would act on te retaining wall wic pressure also prevails inside te material in te active stress state If does not reac te value of te ten an intermediate stage between te static pressure and active pressure emerges near te retaining wall

38 In te past several people performed experimental measurements wit dry sand a non-coesive granular material for te determination of te lateral pressure Taking accuracy and model size into consideration te experiments started by Terzagi in 199 rose above te oter researces In is experiment te retaining wall was a 1-metre-ig and 4-metre-long rigid reinforced concrete structure Te volume of te sandbox was 37 3 m and te displacement of te wall was measured wit an accuracy of 5 mm Te results of te experiment can be summed up as follows: Wile te retaining wall was motionless a orizontal static pressure wit te magnitude of E 4 acted on it At te sligt displacement of te wall te lateral pressure decreased ten due to furter displacement tilt of te retaining wall te orizontal component of te lateral pressure became constant near te value of 9 wile te tangent of te friction developed on te retaining wall moved near te value tg 54 Due to te expansion tat occurred in te sand and as a result of te loosening te surface sank near te displaced wall Te measured values correspond well to te result obtained by means of te previously deduced teoretic formulas: - te static pressure coefficient was 4 cos cos wic is a value caracteristic of te dry sand - te orizontal stress component of te pressure acting on te frictional retaining wall is: cos 1 tg cos te measured value cos 84 and tg ie it is remarkably consistent wit te expected measured value of ca 889 9

39 Arc formation in granular materials Te penomenon of te arc formation is one of te basic questions of te mecanics of te granular materials Te teoretical clarification of arc formation provides solution to suc direct practical problems as te bulk storage of granular materials in silos or teir safe discarge In te oppers of te silos te material often coagulates or an arc is formed wic impedes te gravitational discarge Condition of arc formation In eac case it is always te displacement of a part of te material wic generates arc formation Tis motion can originate from consolidation compaction or for example from te material motion tat follows te opening of te gate located on te bottom of te opper Due to te displacement te stresses in te material are rearranged in a way tat te retaining part of te material tat remains in place takes over also part of te stresses of te moving material part If te stresses wic rose tis way are big enoug and teir direction is adequate an arc will be formed in te material wic will prevent any furter displacement Te arc-forming effect of te displacement prevails wen te material must undergo specific deformation during te displacement ie it must pass troug for example a narrowing crosssection

40 - 4 - On te basis of te aforementioned considerations if we want to follow te process of arc formation an infinitely long symmetric troug wit narrowing cross-section filled wit non-coesive granular material (Fig 18 ) can be cosen as te starting point of te examination Figure 18 Dimensions of te troug Let us assume tat te volume weigt of te material ( ) does not cange as a function of te dept and te material does not compress after te filling A movable bottom-plate closes te b -wide lower opening of te troug wic as a flat and rigid side wall inclining outwards wit an angle β to te vertical Te assumption of an infinite lengt makes te planar examination of te case possible After te removal of te bottom-plate of te troug te material moves off it wants to flow out and undergoes specific deformation as a result of te narrowing cross-section; consequently te model ensures te conditions of arc formation as described before If te granular material is at a quiescent state and te side walls are rigid ten static pressure develops inside te material; consequently te pressure is γ in te vertical and λγ in te orizontal direction were λ is te quotient of te vertical and orizontal pressure ie it is te coefficient of te static pressure

41 Te stresses acting on te side wall inclining outwards at an angle β to te vertical and respectively te resultant force of te stresses (E) can be determined according to magnitude and direction on te basis of Figure 19: Figure 19 Force equilibrium of te troug wit a closed bottom plate filled wit granular material G is te weigt of te material part between te vertical plane and te side wall wic is inclining outwards at an angle β: G tg E is te resultant force of te orizontal static pressure: E Te resultant force from te vector triangle is: E tg For te inclination angle of te resultant force inclined to te orizontal plane it can be written: tat is G tg E tg tg If te side wall can take up te α-direction force and te stresses due to te lateral wall friction ten only vertical stresses wit a magnitude of γ act on te bottom

42 - 4 - plate Tis is only true if were te δ is te angle of te friction on te side wall rising between te wall and te material After removing te orizontal plate closing te discarge opening of te troug sear planes form inside te material due to te fact tat te material can pass troug te narrowing cross-section only by sear If ie only vertical stresses acted previously on te orizontal plate ten te plane of te sear will be vertical Since te material is seared on tat surface to wic te smallest force is needed Te force necessary for searing te non-coesive granular material is expressed by te relation A is te seared surface F A tg using te Coulomb s equation were n n is te perpendicular stress acting on te seared surface is te friction angle of te material Te smallest sear force is necessary for te searing of te vertical plane since te seared surface is te smallest ere and it is also te plane were te orizontal stresses are te smallest (Te orizontal stresses are always smaller tan te vertical or intermediate-direction stresses) Consequently te plane of te sear is vertical From bot of te points B and D of te troug a vertical sear plane is formed if te vertical-direction force rising from te weigt of te material part located above te opening b is equal wit te sear force demand of te two planes: tat is b tg b tg If te size of b is bigger tan tis value ten te material wit a tg widt is torn off in one part and takes wit itself under te effect of te acting sear stresses - te oter material parts as far as te total opening b of te troug If b tg ten only two vertical-direction sear surfaces develops after te removal of te plate closing te discarge opening of te troug At tis time sear stresses arise inside te material on te sear plane wic are produced by te vertical weigt force b Te material part wic is located between te sear

43 plane and te side wall of te troug takes up te vertical sear forces F and transfers tem to te side walls Te side wall can only take up tis vertical-direction plus force entirely if te resultant force ( E ) does not exceeds te angle of te friction arising on te side wall B Figure Force equilibrium after te removal of te bottom plate of te troug It is well discernible on te vector polygon of Figure tat te resultant force E does not exceed te friction angle δ if te inclination angle of te side wall and te friction arising tere is big enoug ie following can be formulated for te limit case F G tg( ) E From te vector diagram te Te sear force F is te alf of te material weigt above te opening since it is divided into two sear planes consequently furtermore b F G tg an E Substituting te values of te F G an E into te relation written for te tg (β+δ) we get: tg b tg ) ( B

44 simplified: consequently if b tg tg( ) b tg ten starting from te lower opening of te troug individual sear planes are formed and te side wall can take up te weigt of te material part located between tem by te rearrangement of te sear stresses if b tg tg( ) In tis case te material cannot flow out and an arc is formed above te opening From te relation relationsip b can be expressed: b tg and te relation obtained for te tg ( ) te b b tg tg( ) tg Tese two equalities and inequalities formulate te condition of te formation of te arc in te non-coesive granular material According to te solution of te arc formation calculated wit te elp of te resultant forces te sear force F diverts te fulcrum of te resultant force E of te quiescent state towards te lower opening of te troug Te intersection point of te line of action of te forces E and F determines te position of E B Te deviation is in direct ratio wit te increase of te angle β Tis deviation can be left out of consideration in case of te practical calculations Since in te event of a significant increase of te angle β te arc already rests directly on te material and not on te side wall (see later) Caracter of te discarge Te caracter of te discarge of te granular material flowing out of te troug can be interpreted wit te previously deduced arc conditions in cases wen any one of te conditions is not fulfilled

45 b a) If ie tg( ) tg ten te side wall cannot take entirely up te resultant force wic canged due to te rearrangement of te sear stresses Te free component of te resultant force E parallel wit te side wall sets off te slide of te material along te side wall Te material part located between te sear plane and te side wall also moves off and mass flow occurs (Figure 1/a) Figure 1 Typical discarges: a) mass flow; b) tunnel flow b b b) If ie tg( ) tg but b tg and tg ten even toug te side wall can take up te resultant force but te weigt force of te material located above te opening is bigger tan te sear force preventing te break away; terefore te material wit a widt of b flows out vertically wile te material part next to it te part between te vertical sear plane and te side wall - remains in its place and flows from above to te vertically moving material part afterwards Tunnel flow develops (Figure 1/b) It is easy to see tat te caracter of te discarge is determined not only by te inclination angle and wall friction of te troug but it is also influenced by te sear resistance friction angle and te pressure conditions of te material It can be observed in silos tat certain granular materials are discarged firstly by mass flow ten later by tunnel flow proving tat te pressure conditions dominating in te opper also influence toug to a smaller extent tan te inclination angle and friction of te opper te caracter of te discarge In te course of our experiments performed wit weat and sand wit te application of oppers wit

46 different inclination angles it could be observed (wen te experiment was carried out in te same opper and wit te same material) tat at first mass flow occurred but wen te discarge was continued te penomenon of tunnel flow appeared Te cange of te caracter of te material discarge occurred at te pre-calculated eigt value Mecanism of te arc formation Consequently te conditions of te arc formation are tat te weigt force of te material above te lower opening of te troug sould be smaller tan ten te sum of te sear forces arising on te sear planes and tat due to teir inclination angle and wall friction te side walls sould be able to take up te forces acting on te tem Te sear stresses developed after te removal of te plate closing te lower opening of te troug are transferred onto te side wall and summed up wit te stresses acting tere Tese resultant stresses form te arc Te arc surface is formed as a result of a second stress rearrangement Figure Te arc is formed as a result of stress rearrangement Between te edges B and D of te troug te distribution of te orizontal components of te resultant stresses is as sown in Figure /a At te places were te orizontal components of te compressive stresses prove to be too small to

47 support te material against te gravitation its sear resistance is smaller tan its weigt force arising from te gravitation tere te dropping of granules Due to te bleed te stresses rearrange presumably according to Figure /b in order to provide enoug compressive stress for te support of te material Te dropping of granules ceases wen identical orizontal stress components of critical value in terms of te dropping of granules on eac point of te arc surface Te fact tat on eac point of te arc te same-value orizontal stress components must act makes te determination of te equation describing te geometric sape of te arc possible Geometric equation of te arc It is known if for te support of an evenly distributed load suc quadratic parabola is used in wose end-point only tangential stress develops ten in eac point of te parabola only a stress wit a parabola-direction and equal-size orizontal component arises Suc a parabola is loaded by no bending moment wic condition is of vital importance Te bending moments would produce tensile and compressive stresses wic a solid body can witstand but te non-coesive granular material is not capable of bearing tensile stresses In te points B and D of te troug an equal-size stress acts in te β+δ direction wile above te arc tere is an almost evenly distributed load Te direction β+δ and te opening widt b definitely determine te parabola Te maximum rise of te parabola tat as te aforementioned caracteristics is: b f tg( ) 4 Wit te coordinate axis y placed in te symmetry plane of te opening of te troug and te axis x leading troug te points B and D (Fig 3) te cuspidal point of te parabola intersects te axis y at te eigt of C

48 Figure 3 Parabola of te arc Te general form of te equation of te quadratic parabola symmetrical to te axis y and running downwards is: y Ax C b since f=c so C tg( ) 4 Te first differential coefficient of te equation of te parabola at te place of b x is y ' tg( ) consequently y Ax C y' Ax and b tg( ) A 1 A can be expressed: A tg( ) b Substituting te values of A and C into te general equation of te parabola: x b y tg( ) tg( ) b 4 b x y 4 b tg( ) we get te geometric equation of te arc From te equation formulating te arc condition te tg ( ) can be substituted: b x b tg y b 4

49 In some cases te arc is lower tan te form given in te above equation It occurs wen te inclination angle of te troug is big In tis case te arc is supported by tat plane of te material wic inclines at an angle ε to te vertical and ε<β In te material te angle of te friction is terefore substitutes te values of β+δ in te equation tat formulates te arc condition and describes te geometric form: b tg( ) tg b x y ( ) 4 tg b Te arc leans directly on te material in te case wen is smaller tan β+δ If from te two arc conditions te equation b tg( ) tg restricts te ratio te limit case can be acieved at te identical ratio of b tg( ) tg tg( ) tg Te equation formulated for te limit case gives a solution for te ε only in case of tg certain values because it is a quadratic equation and its discriminant can be tg negative depending on te ratio of δ and If te discriminant is negative ten te arc will continue to lean on te side wall of te troug If te discriminant is positive ten te equation b x y 4 b determines te geometric form of te arc If from te arc conditions calculated from te following relation: tg( ) b tg restricts te ratio ten te angle ε can be tg tg( ) tg

50 - 5 - Principle of te opper design Te relations formulating te condition of arc formation in granular media makes te design of suc oppers possible from wic te gravitational discarge of te material can be ensured and wic takes up te smallest possible space If none of te arc condition is fulfilled ten te gravitational flow is ensured If but b b tg tg( ) tg ten te gravitation discarge takes place in te form of a tunnel flow If owever b tg( ) tg ten te discarge is of mass flow type At te design of te oppers te goal is in general to ensure te mass flow wit te smallest discarge opening in a way tat te vertical dimension of te opper sould be te smallest one possible Te principal procedure of te design for a opper wit circular cross-section is as follows In te case of oppers wit circular cross-section instead of a widt b opening a radius r discarge opening must be used so instead of te equation evidently is written from wic r b tg r tg r tg

51 Before designing te opper te internal friction angle of te granular material te friction angle developing on te surface of te opper and te volume weigt must be determined Te friction angles can be determined by sear experiments Te normal loads applied to te material wic was filled in te sear box must correspond to te expected pressure values in te opper For te measurement of te friction angle δ it is advisable to make a packing plate of te material of te opper for te sear box Te granular material is filled onto te packing plate placed in te sear plane of te sear box and te angle of te friction is determined by sear experiments If te material stays in te opper for a longer period ten te required reological measurements must also be performed: te searing is carried out by canging te time of te normal loads acting on te sample From te tendency of te curves of te material properties drawn in te logaritm of te time te expectable values of te material caracteristics during a longer storage period can be concluded If te granular material is also coesive or becomes coesive in te course of a longer storage ten for te given normal stress value of te sear te inclination angle to te orizontal axis of te straigt-line drawn from te origin can be taken into account (see later) as te angle of te internal sear resistance (effective friction angle) Figure 4 Hopper-design construction

52 - 5 - Te principal procedure of te opper design is as follows: 1 Let us determine te value of te cos r sin wit te substitution of is: k 1 ; r k1 wit te elp of te tg wic Let us set up te symmetry axis of te opper according to Figure 4 and construct straigt lines wit different r ratios; 3 Let us calculate te opper inclination angle 1 belonging to te straigt line k 1 from te relation k tg( ) tg : 1 k tg arc tg (1 k tg ) tg 4 tg 4k tg ; 4 Let us draw angle 1 to te given point of te straigt line k 1 (On te upper part of te opper te mass flow can be ensured by using te opper inclination angle 1 by using an angle tat is steeper tan tat); 5 Let us calculate te value of angle for a ratio k wic is smaller tan te critical value k 1; 6 Let us construct angle on te intersection point obtained on te straigt line k1and draw its leg as far as te ratio line k ten draw te angle 3 calculated on te basis of te k 3 on te straigt line k ten continuing te construction and calculation we get a opper wit a curve-constituent approaced wit individual line segments; 7 By determining te proportion for te given discarge-opening dimension or for te upper diameter of te opper we obtain a dimension-correct opper sape is obtained from wic te cosen dimensions can be read

53 Consequently te opper profile ensuring a favourable discarge is a curve If tecnological difficulties justify te construction of a opper wit a straigt constituent ten te inclination angle calculated in te dimension of te discarge opening is te decisive factor If we disregard te mass flow te only requirement is tat an angle steeper tan te natural angle of repose must be cosen instead of te ratio k 1 te inclination angle of te opper is indifferent Te great advantage of te opper wit curve constituents is tat its build-in space demand is te smallest possible and it can be inserted as a packing into te existing oppers by wic te discarge difficulties can be efficiently improved Te flow-improving advantages of te yperbolic oppers wit curve constituents are known wic are also proved experimentally by te oppers designed on te basis of te present teory Experimental results After te elaboration of te teory we carried out measurements wit experimental tanks and opper designed for te given material to be stored Tere was a metres ig and 1 metre diameter material column above te opper designed for te material caracteristics and friction parameters of te corn-grits At te straigt conical opper wit an inclination angle β=3º an arc was formed as far as te opper opening wit a diameter of 15 mm wic impeded te gravitational discarge At te same tank but wit a opper wit curve-constituents and wit an opening diameter of 1 mm and wit te construction of a sorter opper obtained a safe discarge (Figure 5)

54 Figure 5 Curve-component opper Te experimental measurements proved our teoretical calculations not only for corn grits but also for wetted sand as a model material te calculations were furtermore verified wit experiments for fertilisers and mixed feeds for extracted soya grits feed lime and alfalfa flour