Numerical studies on the inter-particle breakage of a con?ned particle assembly in rock crushing
H.Y.Liu *,S.Q.Kou,P.-A.Lindqvist
Department of Civil and Environmental Engineering,Lulea
?University of Technology,SE-97187Lulea ?,Sweden Received 9June 2003;received in revised form 17August 2004
Abstract
Understanding rock crushing mechanisms may provide an e?cient key to better fragmentation e?ciency.In this
paper,?rstly the fracture processes of a rock specimen under uniaxial and triaxial compressions are simulated using the rock and tool interaction (R–T 2D )code and compared with the results from experimental observations in litera-tures.It is found that,with increasing con?nement,the fracture process is more progressive and the failure mechanism gradually changes from axial splitting to shear fracture.Then the inter-particle breakage process in a particle bed under con?ned conditions is numerically investigated from a mechanics point of view.The results show that when the particle breaks depends on the strength criterion,how it is broken depends on the stress distribution and redistribution,and where it is broken depends on the heterogeneous distribution in the particle.It is found that,irrespective of the particle shape or particle bed arrangement,the fragmentation starts from the particles which are loaded in quasi-uniaxial com-pression.The resulting fragmentation is usually axial splitting between the two highest stressed loading points.After that,the particles which are loaded at ?rst in quasi-triaxial compression,because of the con?nement from the neigh-bouring particles,the loading plate or the container wall,fail progressively.Depending on the location of the loading points,small fragments are torn o?at the loading points with a large piece preserved.In the ?nal stage,the local crush-ing at the highest stressed contact points becomes an important failure mechanism.Through this study,it is concluded that the R–T 2D code can capture the features of the inter-particle breakage process,and a better qualitative understand-ing of the physics and mechanics of deformation and breakage is gained.ó2004Elsevier Ltd.All rights reserved.
Keywords:Particle breakage;Numerical study;Rock crushing;Fracture;Fragmentation;Uniaxial compression;Triaxial compression
1.Introduction
In mining and in the production of ballast materials and pavement aggregates,mechanical
0167-6636/$-see front matter ó2004Elsevier Ltd.All rights reserved.doi:10.1016/j.mechmat.2004.10.002
*
Corresponding author.Tel.:+46920491440;fax:+46920491935.
E-mail address:hong-yuan.liu@ce.luth.se (H.Y.
Liu).
Mechanics of Materials 37(2005)
935–954
crushing is a method widely used to liberate valu-able minerals from ores or to reduce the particle size of rock materials.However,it is energetically very expensive,with a cost close to2%of the world energy production(Tsoungui et al.,1999).Facing this cost,understanding the crushing mechanisms inside rock particles under compression may pro-vide an e?ective key to better fragmentation e?ciency.
On the basis of previous research(Evertsson and Bearman,1997;Tang et al.,2001;Kou et al.,2001),two breakage modes have been iden-ti?ed in mechanical crushing:single-particle and inter-particle breakage.Single-particle breakage occurs when the distance between the chamber walls is equal to or smaller than the particle size, which is relatively simple.Inter-particle breakage occurs when a particle has contact points shared with other surrounding particles,and this is be-lieved to be an important breakage mode in mechanical crushing.It is obvious that the inter-particle breakage process is very complex.To facil-itate an in-depth study of this process,theoretical models have been developed.
An ideal particle bed model is characterized by Scho¨nert(1996)as follows:(1)it possesses a homogeneous structure(stochastic homogeneity);
(2)homogeneous compaction is possible;(3)the volume or mass of the stressed particles is known; and(4)the wall e?ect is negligible in respect of the overall size-reduction e?ect.Previously,the ideal particle bed model was widely used for fundamen-tal research on inter-particle breakage and to deli-ver basic information for comminution(Scho¨nert, 1996;Fandrich et al.,1997).In those researches, comminution within the particle bed is character-ized by the breakage probability and the breakage function.The breakage probability is de?ned as the mass fraction of the progeny smaller than the lower bound of the initial size fraction,and the breakage function describes the particle size distri-bution of the broken progeny.However,most of those researches have not been carried out from the mechanics point of view.It is desirable that, from a mechanics point of view,models developed for understanding the inter-particle breakage mechanisms should take into account the growth and interaction of microcracks,which culminate in the formation of progeny particles under typical loading conditions.This will require the consider-ation of the material properties,particle shape and particle size.With such complex requirements,for many years it has been thought that a general the-oretical approach from a mechanics point of view for calculating the stressing intensity,breakage probability and breakage function seems to be al-most impossible,or at least very di?cult(Liu and Scho¨nert,1996).
During the past few years,with the rapid devel-opment of computing power,interactive computer graphics and topological data structure,the use of computer simulations seems to be the appropriate tool to obtain some clari?cations of the inter-par-ticle breakage process.Most of the numerical models developed to investigate the inter-particle problems inside particle packings from a mechan-ics point of view(Cundall and Strack,1979; Ghaboussi and Barbosa,1990;Rothenburg and Bathurst,1992)are based on the granular material model.In these numerical simulations,the parti-cles are assumed to be completely rigid,and the overall deformation is caused only by relative dis-placements at the contact points(Satake,1992). However,it is not adequate to use the rigid gran-ular material model to study the particle break-age problem in crushing,in which the process of particle breakage is involved.Recently,a granular material model based on the molecular dynamics method with elastic interactions between grains has been implemented by Tsoungui et al.(1999) into a two-dimensional computer simulation code to study the crushing mechanisms of grains inside a granular material under diametric compression. Compared with the previous granular material models,a big step forward has been made,since the model has de?ned well the breakage conditions of a single grain subjected to multiple loads from neighbouring grains and the grains are represented by elastic disks.However,in their model,when a particle ful?ls the fracture criterion,it is replaced with a set of twelve smaller disks of four di?erent sizes,and the particle breakage is not based on mechanical principles.More recently,Kou et al. (2001)used the RFPA(rock failure process analy-sis)model(Tang,1997)to investigate the inter-particle breakage process of a particle assembly
936H.Y.Liu et al./Mechanics of Materials37(2005)935–954
in a container.However,since the post-failure pro-cess is not related to con?ning conditions,they have di?culty in modelling the con?nement from the neighbouring particles and the chamber walls after some particles fail.The present paper is a con-tinued development of Kou et al.?s(2001)research and will mainly concentrate on the inter-particle breakage process under con?ned compression in mechanical crushing.
In the present paper,?rstly the fracture process of a rock specimen under uniaxial and triaxial con-ditions is simulated to investigate the in?uence of con?nement on the fracture process and compared with the experimental studies conducted by Horii and Nemat-Nasser(1985).Then the inter-particle breakage process under con?ned conditions in mechanical crushing is numerically investigated and is discussed in terms of the two loading geo-metries:quasi-uniaxial compression and quasi-tri-axial compression.The work presented herein forms part of an on-going investigation into the fundamental aspects of rock breakage,in order to improve the design of rock fragmentation equipment from the mechanics point of view.
2.Rock and tool interaction code(R–T2D)
In order to improve the understanding of rock behaviour under mechanical tools and then opti-mise the design of fragmentation equipment,the rock and tool interaction(R–T2D)code has been developed on the basis of the rock failure process analysis(RFPA)model(Tang,1997)and the?nite element method(FEM).The main contents of the R–T2D code include the heterogeneous material model(Liu et al.,2002),the Mohr–Coulomb or the double elliptic strength criterion(Liu et al., 2002)and the mesoscopic elemental mechanical model for elastic damage(Liu,2003).Several pub-lished papers(Tang,1997;Tang et al.,2000;Liu et al.,2002;Liu,2003)have introduced the RFPA model and the R–T2D code in detail.Herein just a brief description is given.
In the R–T2D code,the numerical simulation model is constructed on the basis of the heteroge-neous material model with a homogeneous index (m)and elemental seed parameters(the compres-sive strength r0,the elastic modulus E0,etc.). The?nite element method is used to compute the stress and deformation in each element of the built numerical model.The Mohr–Coulomb or the dou-ble elliptic strength criterion is used to examine whether or not the elements undergo a phase tran-sition.In the loading process,an external load is slowly applied on the constructed numerical model step by step.When in a certain step the stresses in some elements satisfy the strength criterion,the elements are damaged and become weak according to the rules speci?ed by the mesoscopic elemental mechanical model for elastic damage(Liu,2003). The stress and deformation distributions through-out the model are then adjusted instantaneously after each rupture to reach the equilibrium state. At positions with an increased stress due to stress redistribution,the stress may exceed the critical value and further ruptures may be caused.The process is repeated until no failure elements are present.The external load is then increased fur-ther.In this way the system develops a macro-scopic fracture.Thus the code links the mesoscopic mechanical model to the continuum damage model and ultimately to the macrostruc-ture failure(Liu et al.,2002).Energy is stored in the element during the loading process and is re-leased as elastic strain energy through the onset of elemental failure.
3.Fracture process of a rock specimen under uniaxial and triaxial compression
Brittle materials are comminuted with grinding media mills and roller mills,in which a particle or a particle assemblage is stressed between two hard surfaces approaching each other.Traditionally, the breakage of material in crushing is regarded as relying upon single-particle breakage without considering the con?nement condition.However, the breakage behaviour of a single-particle without con?nement cannot adequately represent the ef-fects of stressing a large number of particles,which induces much more complicated loading condi-tions for particle surfaces.For this reason,con-?ned conditions are required in simulating particle breakage for the understanding of the
H.Y.Liu et al./Mechanics of Materials37(2005)935–954937
inter-particle breakage behaviour.In this paper,?rstly the fracture of a heterogeneous rock speci-men under uniaxial and triaxial conditions is explored to investigate the in?uence of con?ne-ment on the fracture process and compared with the experimental studies conducted by Horii and Nemat-Nasser (1985).The reason for using a triax-ial test instead of using a single-particle breakage test is that the triaxial test is better documented than the single-particle breakage test on the basis of laboratory experiments.Accordingly,the in-ter-particle fragmentation process of particles subjected to multiple loads from neighbouring par-ticles or machine walls in a rock assembly will be investigated in Section 4.
In the numerical simulation,the triaxial test is simpli?ed as a plane stress problem and a vertical section of the cylindrical sample is considered.The numerical model is constructed following the heterogeneous material model (Liu et al.,2002)with the homogeneous index m =2,which is one of the typical values for the heterogeneous rock.During the loading process,an axial loading dis-placement increment (0.005mm/step)is applied on the loading platens and con?ning pressures of 0and 20MPa are applied on both lateral sides.Fig.1shows the simulated initiation,propaga-tion and coalescence of the fractures in the rock specimen under uniaxial compression.The letters in the ?gure indicate the di?erent loading levels,which are labelled in Fig.2,where the correspond-ing stress–displacement curve and associated fail-
ure event rate are depicted.As shown in Fig.1A,at ?rst there is almost no failure,which corre-sponds to the linear elastic deformation stage (the line before point A in Fig.2).As the axial loading displacement increases,local isolated fail-ures are initiated at a few random sites depending on the heterogeneity of the rock specimen (Fig.1B),which results in the formation of the nonlin-ear deformation stage (curve AB in Fig.2)in the stress–displacement curve.With a small increase in the axial loading displacement after the peak load,the microfractures begin to cluster and be-come clearly localized in Fig.1C,where a macro-scopic crack comes into being.Correspondingly,there is a large stress drop and a big failure event rate (point C in Fig.2).As the loading displace-ment increases,the formed macroscopic crack propagates in a direction sub-parallel to the
maxi-
Fig.1.Simulated fracture process of a rock specimen under uniaxial compression.
938H.Y.Liu et al./Mechanics of Materials 37(2005)935–954
mum compressive axis and therefore a fault plane is formed (Fig.1D),which results in a rapid in-crease in the failure events and a further fall in the stress–strain curve (point D in Fig.2).Hence,mode I cracking is the dominant mechanism.The fault plane is prevented from developing when it comes close to the upper end piece,due to the con-trast between the elastic moduli of the sample and the loading platen.Strain energy is then stored again,and another main macroscopic crack begins to grow (Fig.1E)and a stress drop is induced (point E in Fig.2).Finally,the eventual failure of the specimen is induced by a combination of ax-ial splitting and local shearing (Fig.1F),and the stress–displacement curve attains a residual strength (point F in Fig.2).
Fig.3shows the simulated fracture process of the specimen under triaxial compression,with the corresponding stress–displacement curve and fail-ure event rate illustrated in Fig.4.The con?ning pressure is applied on the rock specimen in the ?rst loading step,and correspondingly an axial loading displacement (0.09mm)is applied on the loading platen to achieve a hydrostatic stress state.Again it can be seen that at ?rst there is almost no failure event (Fig.3A)and the stress–displacement curve has a linear pro?le (the curve before point A in Fig.4).Then the onset of nonlinear deformation (point B in Fig.4)is indicated by the formation of a large number of isolated microfractures (Fig.3B).With the loading displacement increasing,more di?used failed sites develop (Fig.3C)and the stress attains its maximum value (point C in
Fig.4).As the loading displacement increases,in contrast to the uniaxial compression case,where the failure sites propagate parallel to the major principal stress,more and more individual failure sites tend to develop in the con?ned condition,and it is only when the di?used failed sites become dense that extensile cracks begin to propagate from failed sites and link with each other (Fig.3D).Correspondingly,the stress–displacement curve descends rapidly,at the same time as the fail-ure event rate reaches a maximum (point D in Fig.4).A subsequent increase in the loading displace-ment enhances the linkage between the failed sites to form a macroscopic shear fracture plane (Fig.3E),and another big stress drop is induced (point E in Fig.4).The shear fracture plane continues to grow,until ?nally the formation of the macro-scopic through-going shear fracture plane is com-plete and the specimen fails completely on
the
Fig.3.Simulated fracture process of a rock specimen under triaxial compression (the con?ning pressure is 20MPa).
H.Y.Liu et al./Mechanics of Materials 37(2005)935–954939
macroscopic scale with a single shear fracture plane(Fig.3F).
Our numerically simulated results are consistent with previous analytical and experimental research (Nemat-Nasser and Horii,1982;Horii and Nemat-Nasser,1985and Horii and Nemat-Nasser,1986), as well as numerical research(Tang et al.,2000; Fang and Harrison,2002)on rock fracture under uniaxial and triaxial compression.Fig.5compares the fracture patterns obtained in uniaxial and con-?ned compressions using the R–T2D code and the experimental method(Horii and Nemat-Nasser, 1985).It should be noted that in the numerical simulation,the specimen is heterogeneous.In the experimental observations conducted by Horii and Nemat-Nasser(1985),the specimen is homo-geneous but there are macroscopic defects(hetero-geneities),i.e.the multiple pre-existing?aws.The comparison reveals that the R–T2D code captures the micromechanics of rock failure under uniaxial and triaxial compressions.Under uniaxial com-pression,local failure development is mainly man-ifested by the extension of failed sites in the direction of the major principal stress.In other words,axial splitting in the loading direction is the important failure mode.Under triaxial com-pression,the extension of the failed sites is sup-pressed,but the individual failure sites become dense and link with each other to form a shear plane.Moreover,a comparison between the stress–displacement curves reveals that both the peak strength and residual strength noticeably in-crease in triaxial compression,which means that the specimen under triaxial compression is less prone to failure and fails more progressively than does the uncon?ned specimen.
4.Fragmentation process of a rock particle assembly in a container
The calculation of the stress distribution inside particles with multiple contact forces and the pre-diction of the inter-particle fracture process in a particle bed have been the biggest challenges in the mechanical crushing industry.So far a limited amount of research has been carried out in the ?eld of numerical analysis of these problems from the mechanics point of view.The fragmentation process of particles subjected to multiple loads from neighbouring particles or machine walls in a rock particle assembly will be investigated in this section.
4.1.Numerical model
Fandrich et al.(1997)developed the experimen-tal set-up for particle bed breakage shown in
Fig. https://www.doczj.com/doc/6118533506.html,parison between the fracture patterns obtained in(a)uniaxial and(b)con?ned compression using the R–T2D code and the experimental method(Horii and Nemat-Nasser,1985).
940H.Y.Liu et al./Mechanics of Materials37(2005)935–954
6a to imitate the operating principle of mechanical crushing equipment.The pot contains the sample (1),a cylinder (2),a piston (3)and a base (4),with a plate (5)on top to locate the three LVDTs (6)that are ?xed to the cylinder.The entire pot is placed in a loading frame.The load and LVDT signals are logged by data acquisition software running on a PC (7).Evertsson and Bearman (1997)described the sample (1)in more detail to simulate the conditions to which a volume of material is subjected in a real crushing chamber,as shown in Fig.6b.Correspondingly,a similar numerical model,shown in Fig.7a,is constructed to investigate the inter-particle breakage process in a particle assembly.In practice,the shape of the working surface may be plane,cylindrical or spherical.Any combinations of these shapes are possible for investigating a particular con?gura-tion in a mill.Fundamental studies on the break-age behaviour should preferably use two parallel plane surfaces (Scho ¨nert,1996).Therefore,a par-allel plane surface is used in Fig.7a.The numerical test corresponds to the compressing part of the machine cycle of the crusher,when the liners
move
Fig.7.Numerical model and quasi-photoelastic stress fringe pattern:(a)numerical model for a crushing chamber containing 27rock particles with the polydispersed size and (b)quasi-photoelastic stress fringe pattern in a crushing
chamber.
Fig.6.The principle of mechanical crushing:(a)Schematic diagram of the experimental set-up for particle bed breakage (Fandrich et al.,1997)and (b)crushing chamber (Evertsson and Bearman,1997).
H.Y.Liu et al./Mechanics of Materials 37(2005)935–954941
towards each other.The material is then locked between the chamber walls and can only deform elastically or break into smaller particles.Since the maximum radial velocity of the mantle relative to the concave in normal operating conditions is below0.5m/s,it is assumed that the breakage is independent of the strain rate at this level(Kou et al.,2001).In the present work we simply treat the breakage process as quasi-static.
The numerical model(Fig.7a)consists of a crushing chamber and27randomly placed rock particles with radii following the Weibull distribu-tion within the chamber,where the individual particles are subjected to an arbitrary set of con-tact forces.The model consists of a steel con-tainer measuring180mm in width and height. The thickness of the container walls is5mm.A steel platen measuring170mm in width is used as a cover on the top for transferring a compres-sive load down to the rock particles from the ver-tical direction.The chamber contains27particles, which are numbered from1to27for convenience in the following discussion.The particle bed is loaded under form conditions with an assumption of plane strain.In the form-conditioned case,the size reduction and applied force are a function of the displacement.In the simulation,the axial load is increased by moving the upper loading platen downwards step by step in a displacement control fashion.In the model,the walls that have the same modulus as the loading platen impose a horizontal constraint against the particles inside. This provides the necessary con?ned condition for inter-particle breakage.Similar numerical models have also been used by other researchers (Tsoungui et al.,1999;Kou et al.,2001).Com-pared with Tsoungui et al.?s(1999)model,the numerical model in the present paper regards the rock as breakable particles and can deal with irregularly shaped particles,even though circular particles are used here for comparison with Tsoungui et al.?s(1999)experimental results. The in?uence of an irregular shape on the particle breakage process will be discussed in Section5.1. Compared with Kou et al.?s(2001)model,the residual strength of the element after failure relat-ing to the con?nement and the ability of the con-tact point to resist compressive stress but not tensile stress are the main features of the present numerical model.
The R–T2D code randomly generates di?erent circular particles by adjusting the overlapping ele-ments between neighbouring particles to satisfy the de?ned percentage(approximately85%).The di?erent particles consist of di?erent amounts of elements.The elements are de?ned according to the heterogeneous material model(Liu et al., 2002)with the homogeneous index m=2and the following elemental seed parameters:the elastic modulus E0=60GPa,the compressive strength r0=200MPa,etc.The steel container and the steel platen are simulated as homogeneous materi-als whose elastic modulus and strength are?ve times higher than those of rock in order to prevent them from the permanent deformation.
4.2.Quasi-photoelastic stress fringe pattern
In a con?ned particle bed,no particles can es-cape stressing by moving sidewards.The general loading process is that contact forces act on a par-ticle,deform it,and may cause inelastic deforma-tion and breakage.A contact force in general is directed obliquely and generates always a pressure and a shear.A contact can arise between two neighbouring particles or between a hard surface and a particle.Both contact situations cause di?er-ent e?ects on the deformation and the stress distri-bution in the contact volume and thus on the breakage.Therefore,knowledge of the deforma-tion and the stress distribution in the interior of the particle skeleton is helpful in understanding the breakage behaviour.Since stress or strain can-not be measured systematically in situ in the inte-rior of a particle assembly in the container, physical model tests,such as photoelastic tests, are often the only way to investigate a stress?eld under idealized conditions.However,in applying the optical method,some demands have to be met concerning the test technique.One of them is that a transparent and optically sensitive mate-rial such as glass or epoxy resin has to be used (Oda and Iwashita,1999).Therefore,although optical stress measurements in photoelastic materi-als open new perspectives in research on stress ?elds,this method makes it impossible to investi-
942H.Y.Liu et al./Mechanics of Materials37(2005)935–954
gate stresses in materials,such as rocks,which are by no means transparent.
To overcome such a di?culty,the R–T2D code is used to obtain the full?eld stress information for the particles.In order to obtain clear pictures that resemble the photoelastic test,i.e.quasi-photoelastic stress fringe pattern,a numerical model which is the same as that illustrated in Fig.7a is constructed,but the material in this model is considered to be homogeneous.Numeri-cally generated quasi-photoelastic stress fringe patterns in each particle and in the wall of the con-tainer are shown in Fig.7b.This?gure indicates that the overall load produces contact forces be-tween the particles.These contact forces create stress distribution in the particles.The stress distri-bution in the model will be visualized,and the interaction between the particles,as well as be-tween the particles and the container walls,will be examined in more detail in the following.
4.3.Inter-particle breakage process
Fig.8records the total crushing force(F)and the displacement(S)curve obtained during the simulation of the inter-particle breakage process. The equilibrium states labelled by the alphabetical letters A,B,etc.are shown in Figs.9and10in terms of the distributions of the elastic modulus and the major principal stress,respectively.It can be seen that the force–displacement response has the general features common to many brittle materials.Initially,it is relatively sti?and nearly linear(curve AB in Fig.8).Most of particles de-form elastically except for a few failures in the par-ticle bed because of the rock heterogeneity as shown in Figs.9and10A and B.At a load of approximately1364N(point B in Fig.8),the re-sponse begins to soften,mainly due to the break-age of particles1,6,7,12,13,16and24(please refer to Fig.7a for the particle number)as re-corded in Figs.9and10B,and eventually a limit load develops at1858N(point C in Fig.8).The fracture is localized in particles1,4,6,7,12,13, 16,18,19,20,21,23,24and25(Figs.9and10C and D)beyond the limit load(point D in Fig.8). Careful observation reveals that until this stage, the grain fragmentations are mainly located on the grains with smaller sizes.In those grains,the splitting macroscopic cracks are initiated and propagate along the lines between the two highest stressed contact points.The reasons for this frag-mentation are purely geometric.As a matter of fact,with respect to the rest of the packing,the small grains have few contact points with the neighbouring grains or the walls of the crushing chamber,i.e.they are grains under an almost qua-si-uniaxial compression.In Section3,it has been shown that under uniaxial compression the frag-mentation processes develop very quickly and the particle collapses over a very small strain range. In this case,axial splitting between the loading points is the prominent characteristic.For the large grains,the fragmentation is more di?cult, because their large number of surrounding con-tacts create a dominant hydrostatic e?ect around the grains,i.e.quasi-triaxial compression,for example the grains numbered as2,5,8,10,11 and15in Figs.9and10A–D.
As the collapse of the particles with smaller size progresses,the failures spread to the neighbouring large particles.The spreading of the collapse from particle to particle continues,creating an undulat-ing load plateau,as shown in Fig.8after point D.One can observe the fragmentation regime, characterized by the irregular saw-toothed curve (point D,E,F and G in Fig.8)of the load,as a function of the loading displacement.During this stage,although the splitting cracks are still initiated and propagate along the lines between the two most highly stressed contact points(for example the grains numbered as8and15in Figs.9and
H.Y.Liu et al./Mechanics of Materials37(2005)935–954943
10E–G),because the previous failures release the con?nement,grain crushing has also become an important failure mechanism (for example the grains numbered as 2,5,10and 11in Figs.9and 10G).A large number of Hertzian cracks are initi-ated from the highly stressed contact points to form chips.It should be noted that after point F in Fig.8,the force–displacement response begins to climb up to the maximum load (point G in Fig.8),which implies a material-hardening characteristic.After that,the particle assembly exhibits a great many load hills and valleys across the plateau (curve after point G in Fig.8).In this stage,grain crushing around the highest stressed contact points has be-come the dominant failure mechanism,for example the grains numbered as 2,5,10and 11in Figs.9and 10G–I.It is noted that the maximum load (point G in Fig.8)extends much higher than the assembly strength at point C in Fig.8.The average load of the plateau,which will be called the propa-gation load,is about 2000N.
Moreover,during the inter-particle breakage process,the side-walls of the crushing chamber are kept at the same places in the vertical direction.There is no movement in the horizontal direction.The resultant force displacement response on the side-walls shows the similar features as that in the loading platens (the lower curve in Fig.8
).
Fig.9.Simulated inter-particle breakage process (distribution of the elastic modulus).
944H.Y.Liu et al./Mechanics of Materials 37(2005)935–954
The numerically simulated inter-particle break-age process agrees well with the results from the experimental test (Tsoungui et al.,1999),as shown in Fig.11:the grain fragmentation begins ?rst in the small-sized grains,particularly in the grains with uniaxial geometric con?gurations,and the large grains are always di?cult to break because of their hydrostatic environment.4.4.Energy transformation
Energy is one of the important considerations in the breakage process during comminution.The energy is supplied to the particle bed by mov-
ing the loading platen.The force ?ows from the loading platen through the bed towards the bottom platen and the con?nement wall.By inte-gration of the crushing force (F )over the displace-ment of the loading platen (S )in Fig.8,the energy consumption (E )can be obtained E ?
Z
S max
0F d S ?1:003J ;
where S is the displacement of the loading platen,F is the applied force and S max is the maximum displacement of the loading platen when loading the particle
bed.
Fig.10.Simulated inter-particle breakage process (distribution of the major principal stress).
H.Y.Liu et al./Mechanics of Materials 37(2005)935–954945
Figs.12and13show the numerically obtained fracture event rate and the elastic energy release (ENR)inside the particle during the inter-particle breakage process.The failure event rate shows the following expected features:(1)during the ini-tial deformation or linear elastic phase(curve AB in Fig.8),little elastic energy(Fig.13)was re-leased,although some fracture events(Fig.12)oc-curred;(2)an increasing rate of fracture events (Fig.12)accompanied the inelastic phase and the load plateau(point C in Fig.8).This agrees with the understanding that the fracture events are gen-erated by microfractures that result in nonlinear deformation behaviour(Tang et al.,2000).It is important to note that,although nearly60%
(59.2%in fact)of the fracture events(Fig.12)oc-cur before point F in Fig.8,less than35%(34.3% in fact)of the elastic energy is dissipated during this stage,as shown in Fig.13.A comparison be-tween Figs.8,12and13shows a good relationship between the load curve,failure event rate and en-ergy release.Note that most of the large load drop on the load curves shown in Fig.8corresponds to a high failure event rate(Fig.12)and a big energy release(Fig.13).
The applied work corresponds to the energy consumed in breakage of the particles and energy losses.According to the accumulated energy re-lease shown in Fig.13and the applied work shown in Fig.8,about18%of the applied work is con-sumed in breaking https://www.doczj.com/doc/6118533506.html,pared with the single-particle breakage conducted by Tang et al. (2001),inter-particle breakage has a lower energy utilization ratio because of local crushing at con-tact points.The applied energy is mainly con-sumed by(a)acoustic emission,(b)the formation of new surfaces,and(c)local crushing at contact points,etc.
4.5.Fragment size distribution
A quantitative description of the fragment size reduction would be helpful for a better under-standing of the comminution in mechanical crush-ing and especially for modelling the crushing performance.We have studied the distribution of
Fig.11.Experimental inter-particle breakage process(Tsoungui et al.,1999). 946H.Y.Liu et al./Mechanics of Materials37(2005)935–954
fragment size corresponding to the inter-particle breakage process,as shown in Fig.14,in which the letters correspond to those in Fig.9.An image analysis program,Particle2D developed by Wang (1998),has been used to measure the fragment size. At present,the way in which the degree of reduc-tion is de?ned has not been standardized.Here the diameter of the equivalent circle area(DECA) of a particle has been used to measure the size of the particle.In Fig.14,the abscissa is the DECAs of the particles(mm)and the y-coordinate is the cumulated weight distribution(%).As can be seen from the?gure,before crushing(Step A),less than 8%of the DECAs of the particles are smaller than 12mm.With the loading displacement increasing, the size reduction e?ect increases rapidly before Step F(the onset of the material-hardening re-gime)in Fig.14:from less than8%of the DECAs of the particles being smaller than12mm to more than70%of the DECAs of the particles being smaller than12mm,as shown in Fig.14(Step B, C,D and E).After Step F,local crushing at con-tact points becomes the important failure mecha-nism and the size reduction e?ect increases slowly,as shown in Fig.14(Step F,G,H and I). In practice,in order to reduce the?nes or to con-trol the microcracks within the reduced particles,a careful design of the normal stroke is important. With the assumed mechanical properties,and the size and shape of the assembled rock particles,as well as the height of the container,the present sim-ulation indicates that a normal stroke between0.3and0.4mm(point F in Fig.8)may be a good choice to avoid over-breakage(Step F in Fig.
14).After crushing(Step I),more than90%of the DECAs of the particles are below12mm.Be-sides the fragment size distribution also depends on parameters such as the bed height,the stroke and the initial size of the crushed particle.
5.Discussions
In mechanical crushing,the comminution e?ect is achieved by two surfaces gripping the material between them and forcing the material with a com-pressive force imposed by the surfaces.Under-standing the breakage mechanism inside the material is rather important for crusher design. However,so far studies have concentrated on the behaviour of cohesionless particles with particular emphasis on the macroscopic response.Few studies have been conducted to investigate the behaviour of a particle assembly at the levels of inter-particle breakage.Recently,Tsoungui et al.(1999)devel-oped a granular material model based on a mole-cular dynamics method to simulate the quasi-static evolution of a packing during compression and grinding.The particles are modelled as elastic disks.When a particle ful?ls the fracture criterion, it is replaced with a set of twelve small disks of four di?erent sizes,which are?tted into its original vol-ume and are then treated as new independent particles.
H.Y.Liu et al./Mechanics of Materials37(2005)935–954947
In Section4,we have presented the results of numerical simulation of inter-particle breakage in a2D particle assembly resembling real crushing. Compared with the work of Tsoungui et al. (1999),our investigation has taken a step forward in that the particle breakage in our model is com-pletely based on mechanical principles.In the investigation,the individual particle in the particle bed is divided into many mesoscopic elements.The amount of elements in the particle depends on the particle size and the elemental size.The physical–mechanical parameters of elements,i.e.the elastic modulus,the strength,etc.follow the heterogeneous material with the homogeneous index(m=2)and the elemental seed parameters(E0=60GPa, r0=200MPa,etc.).During the inter-particle breakage process,the particle is loaded by the neighbouring particles,the side-walls of the crush-ing chamber or the loading plate through the con-tact points.The?nite element method is used to calculate the stress and deformation of the ele-ments in the particle.Depending on the heteroge-neity and the stress distribution,under a certain load level,the stresses of some elements in the particle may satisfy the double elliptic strength criterion.Those elements will fail and the physi-cal–mechanical properties of the elements will be changed according the mesoscopic mechanical model for elastic damage.Microcracks will then initiate at the locations of failed elements.After each failure,the stress and deformation distribu-tions through the particle assembly are adjusted instantaneously to reach the equilibrium state. Due to stress redistribution,the stress of some ele-ments in the particle may exceed the critical values and further microcracks may initiate.The process is repeated until no further failure occurs in the particle bed under the current load level.The external load is then increased further.Depending on the loading conditions,the microcracks nucle-ate in di?erent modes to form macroscopic cracks. In the case of the quasi-uniaxial compression,the microcracks are?rstly initiated at the centres of particles with the elemental stress satisfying the tensile strength and extend following tortuous path till the two highest stressed contact points to form tensile splitting crack to result in particle breakage depending on the particle heterogeneity.In the case of the quasi-triaxial compression,the microcracks are?rstly initiated around the contact points with the elemental stress satisfying the shear strength.With more elements failing around the highly stressed contact points,the microcracks be-come dense and link with each other to form shear crack to result in grain crushing near the highly stressed contact points.Therefore,in our model, when the particle breaks depends on the strength criterion,how it is broken depends on the stress distribution and redistribution inside the particle, and where it is broken depends on the hetero-geneous distribution in the particle.Moreover,it is seen from Figs.9and10that the shapes of the pregnancies originating from the larger one by breaking are by no means regular.
Besides,the raw materials in the crushing cham-ber are di?erent types of rock materials that are crystalline,heterogeneous and often irregularly shaped.The granular material model has di?cul-ties in simulating the breakage process of real het-erogeneous particles with various irregular shapes, which may cause a completely di?erent particle bed arrangement and con?nement,and then in?u-ence the inter-particle breakage process.In the fol-lowing,the in?uence of the particle shape and the con?nement on the inter-particle breakage process will be discussed.
5.1.In?uence of the particle shape on the
inter-particle breakage
In order to investigate the in?uence of the par-ticle shape on the inter-particle breakage process,a numerical model similar to that illustrated in Fig. 7a is constructed,as shown in Fig.15.The particle bed consists of irregularly shaped particles,which have the same con?guration as those used by Kou et al.(2001).However,in Kou et al.?s (2001)model,since the post-failure process is not related to con?ning conditions,it shows di?culty in modelling the con?nement from the neighbour-ing particles and the chamber walls after some particles fail.If the particles in Fig7a are polydi-spersed,we can regard the particles in Fig.15as monodispersed,which means that the irregularly shaped particles have an approximately equal area.The irregular shape determines the number
948H.Y.Liu et al./Mechanics of Materials37(2005)935–954
of contacts among neighbouring particles and the walls of the crushing chamber.
Fig.15shows the quasi-photoelastic stress fringe patterns of irregularly shaped particles in a steel container.Similarly to Fig.7b,Fig.15indi-cates that the overall load produces contact forces between the particles.These contact forces create stress distribution in the particles.In contrast to the stress ?elds induced in the regular circular disk,there are also stress concentrations around the geometric heterogeneities because of the irregular shapes.
Fig.16visually displays the inter-particle break-age process of irregularly shaped particles.In con-trast to the disk particle breakage process,where the fragmentation starts in the small-sized parti-cles,one can observe that at the beginning of the simulation,the grain fragmentation starts ?rst with a few of the grains contacting the two rigid walls,as shown in the particles numbered as 8and 4(please refer to Fig.15for the particle num-ber)in Fig.16B.The reason for this fragmentation near the walls is also purely geometric.As a matter of fact,with respect to the rest of the packing,the grains in contact with the walls present generally geometric con?gurations close to those of grains submitted to two opposed forces,i.e.quasi-uniax-ial compression,which favours their fragmenta-tion much more.Previously,we have shown in
the polydispersed circular particle bed that the fragmentation starts from the small particle.In fact,irrespective of whether it is a small-sized cir-cular particle or an irregular particle contacting the rigid walls,a particle is selected to break ?rst if two criteria have been ful?lled:(1)the particle is located in such a way that quasi-uniaxial com-pression can be achieved,i.e.in such a way that splitting failure easily occurs;and (2)the stress level of the particle has reached a critical value.After the ?rst few grains have been fractured,one observes that more grains,which are loaded under quasi-uniaxial compression,fragment as shown in the particles numbered as 4,6,8,9,12and 13in Fig.16C–E.At the same time,because of the failure releasing the con?nement,the parti-cles loaded at ?rst under quasi-triaxial compres-sion begin to fail also,as shown in the particles numbered as 1,5and 10in Fig.16C–E.After that,local grain crushing at the contact points becomes an important breakage mechanism,as shown in the particles numbered as 1,4,5,7,8,9,12and 13in Fig.16F–I,although a splitting failure of the quasi-uniaxial compression type (particle 14in Fig.16)still occurs.
Compared with the inter-particle breakage pro-cess investigated by Kou et al.(2001),at the ?rst stage,the fragmentation develops in the same mode.However,after some particles fail,it seems that in the current simulation the particles are more di?cult to fail than those in Kou et al.?s (2001)simulation.It is reasonable since the parti-cle post-failure process in Kou et al.?s (2001)model is not related to con?ning conditions and the con-?nement from neighbouring particles and the chamber wall after some particles fail is di?cult to simulate there.It is because the particle post-failure process in the model is related to the con-?nement that the particles show the trend to be more di?cult to fragment in the current simula-tion.Moreover,it was noted that the particle breakage strength depends on the particle shape.Our results suggest that the more spherical the par-ticle becomes,the higher is the breakage strength that may be expected (particle 7,10,11,14and 15in Fig.16).Presumably,a disk particle has a more regular stress distribution than an irregular particle.In fact,in order to achieve the
same
Fig.15.Quasi-photoelastic stress fringe pattern in particles with irregular shapes.
H.Y.Liu et al./Mechanics of Materials 37(2005)935–954949
propagation load as shown in Fig.8,a higher per-centage (approximately 90%)of irregularly shaped particles (Fig.16)is ?lled into the crushing cham-ber compared with the percentage (approximately 85%)of circular particles (Fig.9).5.2.Two kinds of fracture patterns in the inter-particle breakage process
With a view to achieving a better understanding of comminution,the fracture patterns in the inter-particle breakage process are discussed herein in terms of con?nement from the neighbouring parti-cle,the loading plate or the side-wall.As shown by previous simulations,no particle can escape the stress caused by moving the loading plate in a con-?ned particle bed.The general course of events is that at ?rst the overall load produces contact forces between the particles,and then contact forces act on a particle,deform it,and cause inelastic deformation and breakage.A contact force in general is directed obliquely and generates always a pressure and a shear.A contact can arise between two neighbouring particles or between a hard surface and a particle.Both contact situa-tions cause di?erent con?nements on the deforma-tion and the stress distribution in the contact volume and thus on the
breakage.
Fig.16.Simulated progressive fragmentation process for irregularly shaped rock particles inside a crushing chamber.
950H.Y.Liu et al./Mechanics of Materials 37(2005)935–954
In fact,the fracture patterns of the particles in a con?ned particle bed are rather similar to those of a rock specimen with di?erent con?nement.A careful comparison between the breakage pro-cesses of the particles and the rock specimen in uniaxial and triaxial compression can verify that statement.It is clear that the particles numbered as13,19,20and21(please refer to Fig.7a for the particle number)in Fig.9D and numbered as 4,8,9and12(please refer to Fig.15for the parti-cle number)in Fig.16D are loaded similarly to a rock specimen loaded in uniaxial compression (Fig.1).For example,the particle numbered as8 in Fig.16is loaded by the upper particle numbered as5and the lower particle numbered as12.The con?nement is provided via the container wall from the left and particle number9from the right. However,in this case,con?nement from the lateral sides does not have any important in?uence on the particle breakage process;i.e.the particle is loaded in quasi-uniaxial compression.One can note from Fig.16D that tensile cracks initiated from some-where along the line connecting the two vertical contact points and split the particle into two halves,which is the typical failure pattern in uniax-ial compression.This kind of loading mode also explains why the fragmentation starts from the small particle in the polydispersed disk particle bed(Fig.9),while in the monodispersed irregular particle bed(Fig.16),the fragmentation starts from the particles near the loading plate or the container wall.As a matter of fact,in the poly-dispersed disk particle bed,with respect to the rest of the packing,the small grains have fewer contact points with the neighbouring grains or the walls of the crushing chamber;i.e.they are grains under a quasi-uniaxial compression.Therefore,the frag-mentation processes in the small grains develop very quickly and the particles collapse over a very small loading displacement.In the monodispersed irregular particle bed,the grains in contact with the walls present generally geometric con?gura-tions close to those of grains submitted to two opposed forces,i.e.quasi-uniaxial compression, which favours their fragmentation much more.
The particles numbered as2,5,8,10,11and15 in Fig.9and numbered as1,2,7,10,11and15in Fig.16can be regarded as being loaded in triaxial compression.They do not fail?rst because of con-?nement provided by the neighbouring particle or a hard surface.For example,the particle num-bered as2in Fig.9G is loaded via the relative movement of the upper loading cover and the low-er particle numbered as8.The con?nement is pro-vided via the neighbouring particles numbered1,3 and9.In this case,con?nement from the lateral sides does have an important in?uence on the par-ticle breakage process.In fact,the surrounding contacts create a dominant hydrostatic e?ect around the grain.Therefore,it does not fail?rst. As the loading displacement increases,fragments are torn o?from the main loading points and a large piece is preserved.In this case,grain crushing around the contact points becomes the important fragmentation mechanism.With the loading dis-placement increasing,chips,?nes and crushed zones are formed,particularly in the vicinity of contact locations.
Therefore,according to the above simulations of the inter-particle breakage process,two loading geometries and correspondingly two kinds of fail-ure patterns can be recognized:quasi-uniaxial compression and quasi-triaxial compression.In the quasi-uniaxial compression case a particle is loaded between diametrically opposed surfaces and the resulting stress?eld consists of a zone be-tween the loading points,which is in indirect tension.The resulting fragmentation is usually two pieces of rock approximately half of the origi-nal size of the particle and a collection of?ne rock particles generated in the compression zone.For high reduction ratios,it has been seen that thin particles with one dimension equal to the original particle size often generate from the zone between the loading points(Briggs and Evertsson,1998). Thus,in the rock fragmentation industry,it is sug-gested to arrange the particle bed so that the quasi-uniaxial compression is easily achieved to facilitate its fragmentation.In the case of quasi-triaxial compression,the stress?eld is di?erent.The load-ing is not diametrically opposed,so that a simple major?eld cannot develop.Depending on the location of the loading points,the stress?eld at the points is more direct tension.A large piece is preserved and in addition there are other frag-ments that have been torn o?at the loading points.
H.Y.Liu et al./Mechanics of Materials37(2005)935–954951
The e?ect of tearing o?protruding sections is the physical reason why quasi-triaxial compression improves the shape without destroying the overall size of the rock(Briggs and Evertsson,1998). Thus,in the rock aggregate industry,it is sug-gested to arrange the particle bed so that the qua-si-triaxial compression is easily achieved to obtain the even particle products.Moreover,in this case grain crushing around the loading points becomes an important fragmentation mechanism.Our numerically simulated results are consistent with those observed by Briggs and Evertsson(1998).
6.Conclusions
In this paper,?rstly the fracture processes of a rock specimen under uniaxial and triaxial com-pression are simulated using the rock and tool interaction(R–T2D)code and compared with the experimental observations in literatures to investi-gate the in?uence of con?nement on the fracture process and fracture pattern since in the inter-par-ticle breakage process the con?nement from the neighbouring particle is the mainly loading mode. Then the inter-particle breakage process in a parti-cle bed under con?ned conditions is numerically investigated from a mechanics point of view.Fi-nally,a number of questions related to mechanical crushing are discussed.On the basis of the simu-lated results,the following conclusions can be drawn:
(1)The con?nement has an important in?uence on the fracture process and fracture pattern of rock.Under uniaxial compression,the fracture process develops very quickly,so that the specimen collapses over a very small strain range through the extension of failed sites in the direction of the major principal stress.Under triaxial compression, the extension of the failed sites is suppressed,but the individual failure sites become dense and link with each other to form a shear fracture plane.
(2)A simple description and qualitative model of the inter-particle breakage process in a particle assembly can be summarized as follows.At?rst the overall load produces contact forces between the particles,and then contact forces act on a par-ticle,deform it,and cause inelastic deformation and breakage.In more detail,?rstly the particles deform elastically,which is the elastic deformation https://www.doczj.com/doc/6118533506.html,rge forces are carried by chains of par-ticles that are more or less aligned in the direction of the major compression,depending on the parti-cle bed arrangement.These chains become shorter and more kinked with an increasing load.As the loading displacement increases,the elastic struc-ture gradually fails and the fragmentation regime begins.The fracture is at the beginning initiated in particles located in such a way that the quasi-uniaxial compression is easily achieved,i.e.the connection line of the two highest contact points runs parallel to the major principal stress.With a further increase of the loading displacement,the particles,being loaded at?rst in quasi-triaxial compression because of the con?nement from the neighbouring particles,loading platen or container wall,fail progressively,and therefore the particle assembly becomes less able to carry the load. The crushing propagates in a particle-by-particle fashion,while the average load remains relatively constant.Finally,as the loading displacement in-creases,local grain crushing at the contact points becomes an important mechanism and the densi-?ed assembly recovers a signi?cant sti?ness be-cause of a re-compaction behaviour,which is the assembly hardening regime.
(3)A combination of the numerically obtained force–loading displacement curve,the fragment size distribution and the progressive breakage pro-cess provides a possibility of estimating the neces-sary stroke and input energy for e?ectively breaking particles to a desired extent.With the as-sumed mechanical properties,and the size and shape of the assembled rock particles,as well as the height of the container,the present simulation indicates that a normal stroke between0.3and 0.4mm may be a good choice.
(4)The particle shape has an important in?u-ence on the particle bed arrangement and then in?uences the contact conditions between parti-cles.A comparison between the breakage pro-cesses of particles with di?erent shapes shows that in a particle bed consisting of circular parti-cles,the fragmentation starts?rst with small-sized particles,but in a particle bed consisting of irregu-lar particles,the fragmentation starts?rst with a
952H.Y.Liu et al./Mechanics of Materials37(2005)935–954
few of the grains contacting the two rigid walls. However,in both cases,the fragmentation starts from the particles which are loaded in a quasi-uni-axial compression condition.Besides,in current particle bed arrangements,it seems that quasi-uni-axial compression is more easily achieved in an irregular particle bed than in a circular particle bed.
(5)Two kinds of loading geometries and corre-spondingly two kinds of fracture patterns are recognized:the quasi-uniaxial compressive frac-ture pattern and the quasi-triaxial compressive fracture pattern.In the quasi-uniaxial compressive fracture pattern,a particle is mainly loaded be-tween diametrically opposed points and the result-ing fragmentation is usually two pieces of rock with the main fracture connecting these two op-posed points and are produced by axial splitting, which mainly occurs in the?rst stage of the in-ter-particle breakage process.In the quasi-triaxial compressive fracture pattern,the local crushing at the contact points becomes the important fail-ure mechanism.Depending on the location of the loading points,the stress?eld at the points is more direct tension and small fragments are torn o?at the loading points with a large piece preserved. Thus,in the rock fragmentation industry,it is sug-gested to arrange the particle bed so that the quasi-uniaxial compression is easily achieved to facilitate its fragmentation.In the rock aggregate industry, it is suggested to arrange the particle bed so that the quasi-triaxial compression is easily achieved to obtain the even particle products.
Therefore,it is concluded that the R–T2D code can capture the features of the inter-particle break-age process in the particle bed assembly.Through this study,a better qualitative understanding of the physics and mechanics of deformation and breakage has been achieved,and a great many valuable practical insights have been gained into inter-particle breakages.
Acknowledgments
The?nancial support from LKAB?s Founda-tion for the Promotion of Research and Education at Lulea?University of Technology,Trelleborg AB?s Research and Education Foundation,the Foundation for Technology Transfer,Arne S. Lundberg?s Foundation,and the Knowledge Foundation is greatly appreciated.
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【摘要】黄自先生是我国杰出的音乐家，他以艺术歌曲的创作最为代表。而黄自先生特别强调了钢琴伴奏对于艺术歌曲组成的重要性。本文是以黄自先生创作的具有爱国主义和人道主义的艺术歌曲《天伦歌》为研究对象，通过对作品分析，归纳钢琴伴奏的弹奏方法与特点，并总结黄自先生的艺术成就与贡献。 【关键词】艺术歌曲；和声；伴奏织体；弹奏技巧 一、黄自艺术歌曲《天伦歌》的分析 （一）《天伦歌》的人文及创作背景。黄自的艺术歌曲《天伦歌》是一首具有教育意义和人道主义精神的作品。同时，它也具有民族性的特点。这首作品是根据联华公司的影片《天伦》而创作的主题曲，也是我国近代音乐史上第一首为电影谱写的艺术歌曲。作品创作于我国政治动荡、经济不稳定的30年代，这个时期，这种文化思潮冲击着我国各个领域，连音乐艺术领域也未幸免――以《毛毛雨》为代表的黄色歌曲流传广泛，对人民大众，尤其是青少年的不良影响极其深刻，黄自为此担忧，创作了大量艺术修养和文化水平较高的艺术歌曲。《天伦歌》就是在这样的历史背景下创作的，作品以孤儿失去亲人的苦痛为起点，发展到人民的发愤图强，最后升华到博爱、奋起的民族志向，对青少年的爱国主义教育有着重要的影响。 （二）《天伦歌》曲式与和声。《天伦歌》是并列三部曲式，为a+b+c，最后扩充并达到全曲的高潮。作品中引子和coda所使用的音乐材料相同，前后呼应，合头合尾。这首艺术歌曲结构规整，乐句进行的较为清晰，所使用的节拍韵律符合歌词的特点，如三连音紧密连接，为突出歌词中号召的力量等。 和声上，充分体现了中西方作曲技法融合的创作特性。使用了很多七和弦。其中，一部分是西方的和声，一部分是将我国传统的五声调式中的五个音纵向的结合，构成五声性和弦。与前两首作品相比，《天伦歌》的民族性因素增强，这也与它本身的歌词内容和要弘扬的爱国主义精神相对应。 （三）《天伦歌》的伴奏织体分析。《天伦歌》的前奏使用了a段进唱的旋律发展而来的，具有五声调性特点，增添了民族性的色彩。在作品的第10小节转调入近关系调，调性的转换使歌曲增添抒情的情绪。这时的伴奏加强和弦力度，采用切分节奏，节拍重音突出，与a段形成强弱的明显对比，突出悲壮情绪。 c段的伴奏采用进行曲的风格，右手以和弦为主，表现铿锵有力的进行。右手为上行进行，把全曲推向最高潮。左手仍以柱式和弦为主，保持节奏稳定。在作品的扩展乐段，左手的节拍低音上行与右手的八度和弦与音程对应，推动音乐朝向宏伟、壮丽的方向进行。coda 处，与引子材料相同，首尾呼应。 二、《天伦歌》实践研究 《天伦歌》是具有很强民族性因素的作品。所谓民族性，体现在所使用的五声性和声、传统歌词韵律以及歌曲段落发展等方面上。 作品的整个发展过程可以用伤感――悲壮――兴奋――宏达四个过程来表述。在钢琴伴奏弹奏的时候，要以演唱者的歌唱状态为中心，选择合适的伴奏音量、音色和音质来配合，做到对演唱者的演唱同步，并起到连接、补充、修饰等辅助作用。 作品分为三段，即a+b+c+扩充段落。第一段以五声音阶的进行为主，表现儿童失去父母的悲伤和痛苦，前奏进入时要弹奏的使用稍凄楚的音色，左手低音重复进行，在弹奏完第一个低音后，要迅速的找到下一个跨音区的音符；右手弹奏的要有棱角，在前奏结束的时候第四小节的t方向的延音处，要给演唱者留有准备。演唱者进入后，左手整体的踏板使用的要连贯。随着作品发展，伴奏与旋律声部出现轮唱的形式，要弹奏的流动性强，稍突出一些。后以mf力度出现的具有转调性质的琶音奏法，要弹奏的如流水般连贯。在重复段落，即“小
我国艺术歌曲钢琴伴奏-精 2020-12-12 【关键字】传统、作风、整体、现代、快速、统一、发展、建立、了解、研究、特点、突出、关键、内涵、情绪、力量、地位、需要、氛围、重点、需求、特色、作用、结构、关系、增强、塑造、借鉴、把握、形成、丰富、满足、帮助、发挥、提高、内心 【摘要】艺术歌曲中，伴奏、旋律、诗歌三者是不可分割的重 要因素，它们三个共同构成一个统一体，伴奏声部与声乐演唱处于 同样的重要地位。形成了人声与器乐的巧妙的结合，即钢琴和歌唱 的二重奏。钢琴部分的音乐使歌曲紧密的联系起来，组成形象变化 丰富而且不中断的套曲，把音乐表达的淋漓尽致。 【关键词】艺术歌曲；钢琴伴奏；中国艺术歌曲 艺术歌曲中，钢琴伴奏不是简单、辅助的衬托，而是根据音乐 作品的内容为表现音乐形象的需要来进行创作的重要部分。准确了 解钢琴伴奏与艺术歌曲之间的关系,深层次地了解其钢琴伴奏的风 格特点,能帮助我们更为准确地把握钢琴伴奏在艺术歌曲中的作用 和地位，从而在演奏实践中为歌曲的演唱起到更好的烘托作用。 一、中国艺术歌曲与钢琴伴奏 “中西结合”是中国艺术歌曲中钢琴伴奏的主要特征之一，作 曲家们将西洋作曲技法同中国的传统文化相结合，从开始的借鉴古 典乐派和浪漫主义时期的创作风格，到尝试接近民族乐派及印象主 义乐派的风格，在融入中国风格的钢琴伴奏写作，都是对中国艺术 歌曲中钢琴写作技法的进一步尝试和提高。也为后来的艺术歌曲写 作提供了更多宝贵的经验，在长期发展中，我国艺术歌曲的钢琴伴 奏也逐渐呈现出多姿多彩的音乐风格和特色。中国艺术歌曲的钢琴
写作中，不可忽略的是钢琴伴奏织体的作用，因此作曲家们通常都以丰富的伴奏织体来烘托歌曲的意境，铺垫音乐背景，增强音乐感染力。和声织体，复调织体都在许多作品中使用，较为常见的是综合织体。这些不同的伴奏织体的歌曲，极大限度的发挥了钢琴的艺术表现力，起到了渲染歌曲氛围，揭示内心情感，塑造歌曲背景的重要作用。钢琴伴奏成为整体乐思不可缺少的部分。优秀的钢琴伴奏织体，对发掘歌曲内涵，表现音乐形象，构架诗词与音乐之间的桥梁等方面具有很大的意义。在不断发展和探索中，也将许多伴奏织体使用得非常娴熟精确。 二、青主艺术歌曲《我住长江头》中钢琴伴奏的特点 《我住长江头》原词模仿民歌风格，抒写一个女子怀念其爱人的深情。青主以清新悠远的音乐体现了原词的意境，而又别有寄寓。歌调悠长，但有别于民间的山歌小曲；句尾经常出现下行或向上的拖腔，听起来更接近于吟哦古诗的意味，却又比吟诗更具激情。钢琴伴奏以江水般流动的音型贯穿全曲，衬托着气息宽广的歌唱，象征着绵绵不断的情思。由于运用了自然调式的旋律与和声，显得自由舒畅，富于浪漫气息，并具有民族风味。最有新意的是，歌曲突破了“卜算子”词牌双调上、下两阕一般应取平行反复结构的惯例，而把下阕单独反复了三次，并且一次比一次激动，最后在全曲的高音区以ff结束。这样的处理突出了思念之情的真切和执著，并具有单纯的情歌所没有的昂奋力量。这是因为作者当年是大革命的参加者，正被反动派通缉，才不得不以破格的音乐处理，假借古代的