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in this paper , we consider the relation between jamming ( a kinetic process ) and the laws of static mechanical equilibrium of particulate media . we describe first a simple model of jamming in a colloid , sheared between parallel plates . we assume that force chains ( linear arrays of rigid particles in contact ) develop along the major stress compression axis and span the sample . the resulting jammed state can support a shear stress indefinitely , but does this by virtue of a self - organized internal structure ( the force chain array ) which has developed in direct response to the applied load itself . if a different load is now applied ( e.g. if the material is sheared in some different direction ) , the force chains can not sustain the load but must flow and rejam in a different configuration . the model thus provides a simple example of a  fragile " material  one which can not support certain types of infinitesimal stress increment without plastic reorganisation . the continuum mechanics of fragile materials is very different from conventional elastic or elastoplastic theory . we propose that fragility may be generic in jammed systems , and argue further that a pile of cohesionless poured sand can be viewed as jammed in the required sense : its mechanical integrity results purely from the applied load ( gravity ) . some simple models that we have recently developed for poured sand @xcite , such as the fixed principal axes ( fpa ) model , indeed display fragile behaviour ; in fact they correspond to assuming a particular simplified geometry for the granular skeleton of force chains . fragile models require that the behaviour of a pile of cohesionless grains ( fragile ) is quite different from a hypothetical pile where each grain is firmly glued to its neighbours on first coming to rest ( elastic ) . this is plausible because for random packings of glued grains , a finite fraction of the interparticle forces will be under tension ( forbidden in the cohesionless case ) and will remain so until the load is large enough to cause appreciable elastic deformation of individual grains . for simple fragile models such as fpa , the same criterion defines a crossover between fragile behaviour and a form of ( anisotropic ) elastoplasticity . a fuller discussion of these ideas is given in @xcite , where further references may be found . consider a concentrated colloidal suspension of hard particles , confined between parallel plates at fixed separation , to which a shear stress is applied . above a certain threshold of stress , this system exhibits enters a regime of strong shear thickening ; see , e.g. @xcite . the effect can be observed in the kitchen , by stirring a concentrated suspension of corn - starch with a spoon . in fact , computer simulations suggest that , at least under certain idealized conditions , the material will jam completely and cease to flow , no matter how long the stress is maintained @xcite . in these simulations , jamming apparently occurs because the particles form  force chains " @xcite along the compressional direction ( fig . [ fig : jammed ]  ( a ) ) . even for spherical particles the lubrication films can not prevent direct interparticle contacts ; once these arise , an array or network of force chains can indeed support the shear stress indefinitely . ( we ignore brownian motion , here and below , as do the simulations ; this could cause the jammed state to have finite lifetime . )    to model the jammed state , we start from a simple idealization of a force chain : a linear string of at least three rigid particles in point contact . crucially , this chain can only support loads _ along its own axis _ ( fig.[fig : paths ]  ( a ) ) : successive contacts must be collinear , with the forces along the line of contacts , to prevent torques on particles within the chain @xcite . note that neither friction at the contacts , nor particle aspherity , can change this longitudinal force " rule .    as a minimal model of the jammed colloid , we take an assembly of such force chains , characterized by a unique director @xmath0 , in a sea of  spectator " particles , and incompressible solvent . this is obviously oversimplified , for we ignore completely any collisions between chains , the deflections caused by weak interactions with the spectator particles , and the fact that there must be some spread in the orientation of the chains themselves @xcite . with these assumptions , in static equilibrium , with no body forces acting , the pressure tensor @xmath1 ( defined as @xmath2 , with @xmath3 the usual stress tensor ) must obey @xmath4 here @xmath5 is an isotropic fluid pressure , and @xmath6 ( @xmath7 ) a compressive stress carried by the force chains . ( [ nn ] ) defines a material that is mechanically very unusual . it permits static equilibrium only so long as the applied compression is along @xmath0 ; while this remains true , incremental loads ( increase or decrease in stresses at fixed major compression of the stress tensor ) can be accommodated reversibly , by what is ( at the particle contact scale ) an elastic mechanism . but the material is certainly not an ordinary elastic body , for if instead one tries to shear the sample in a slightly different direction ( causing a rotation of the principal stress axes ) static equilibrium can not be maintained without changing the director @xmath0 . now , @xmath0 describes the orientation of a set of force chains that pick their ways through a dense sea of spectator particles . accordingly @xmath0 can not simply rotate ; instead , the existing force chains must be abandoned and new ones created with a slightly different orientation . this entails dissipative , plastic , reorganization , as the particles start to move but then re - jam in a configuration that can support the new load . our model jammed colloid is an idealized example of  fragile matter " : it can statically support applied shear stresses ( within some range ) , but only by virtue of a self - organized internal structure , whose mechanical properties have evolved directly in response to the load itself . its incremental response can be elastic only to _ compatible _ loads ; _ incompatible _ loads ( in this case , those of a different compression axis ) , even if small , will cause finite , plastic reorganizations . the inability to elastically support _ some _ infinitesimal loads is our proposed definition of the term  fragile " ( which , up to now , has not been given a precise technical meaning in this context ) . we argue that jamming may lead _ generically _ to mechanical fragility , at least in systems with overdamped internal dynamics . such a system is likely to arrests as soon as it can support the external load ; since the load is only just supported , one expects the state to be only marginally stable . any incompatible perturbations then force rearrangement ; this will leave the system in a newly jammed but ( by the same argument ) equally fragile state . this scenario is related , but not identical , to several other ideas in the literature @xcite . these include the emergence of rigidity by successive buckling of force chains in glasses and granular matter @xcite ; the concepts of self - organized criticality ( soc ) @xcite ( see also @xcite ) , and those of mechanical percolation which underly recent  hypoplastic " models of granular matter @xcite . fragility is also connected with recent ideas concerning isostaticity and marginal mechanics in ( frictionless ) sphere packings @xcite ( see section [ marginal ] ) . our ideas are , in addition , reminiscent of the ( much older ) concept of a self - selecting critical state in theories of soil mechanics @xcite . however the latter is primarily concerned with _ dilatancy _ : the tendency of dense particulate media to expand upon shearing . jamming can be viewed as the constant - volume counterpart of this process : if expansion is prevented , jamming results . consider the idealized jammed colloid of ( fig .  [ fig : jammed ]  ( a ) ) . so far we allowed for an external stress field ( imposed a the plates ) but no body forces . what body forces can it now support _ without _ plastic rotation of the director ? various models are possible . one is to assume that eq . ( [ nn ] ) continues to apply , with @xmath8 and @xmath9 now varying in space . if @xmath5 is a simple fluid pressure , a localized body force can be supported only if it acts along @xmath0 . thus ( as in a bulk fluid ) no static green function exists for a general body force . ( note that , since eq . ( [ nn ] ) is already written as a continuum equation , such a green function would describes the response to a load that is localized in space but nonetheless acts on many particles in some mesoscopic neighbourhood . )    for example , if the particles in fig . [ fig : jammed ]  ( a ) were to become subject to a gravitational force along @xmath10 , then the existing force chains could not sustain this but would reorganize . applying the longitudinal force rule , the new shape is easily found to be a catenary , as realized by hooke @xcite , and emphasized by edwards @xcite . on the other hand , a general body force can be supported , in three dimensions , if there are several different orientations of force chain , possibly forming a network or granular skeleton " @xcite . a minimal model for this is : @xmath11 with @xmath12 directors along three nonparallel populations of force chains ; the @xmath6 s are compressive pressures acting along these . body forces cause @xmath13 to vary in space . we can thus distinguish two levels of fragility , according to whether incompatible loads include localized body forces ( _ bulk _ fragility , _ e.g. _ eq .  ( [ nn ] ) ) , or are limited to forces acting at the boundary ( _ boundary _ fragility , _ e.g. _ eq .  ( [ osl ] ) ) . in disordered systems one should also distinguish between macro - fragile responses involving changes in the _ mean _ orientation of force chains , and the micro - fragile responses of individual contacts . below we focus on macro - fragility , but in practice the distinction may become blurred . in any case , these various types of fragility should not be associated too strongly with minimal models such as eqs . ( [ nn],[osl ] ) . it is clear that many granular skeletons have a complex network structure where many more than three directions of force chains exist . such a network may nonetheless be fragile ; see section [ marginal ] below . returning to the simple model of eq . ( [ osl ] ) , the chosen values of the three directors ( two in 2-d ) clearly should depend on how the system came to be jammed ( its  construction history " ) . if it jammed in response to a constant external stress , switched on suddenly at some earlier time , one can argue that the history is specified purely by the stress tensor itself . in this case , if one director points along the major compression axis then by symmetry any others should lie at rightangles to it ( fig . [ fig : jammed ]  ( b ) ) . applying a similar argument to the intermediate axis leads to the ansatz that all three directors lie along principal stress axes ; this is perhaps the simplest model in 3-d . one version of this argument links force chains with the fabric tensor @xcite , which is then taken coaxial with the stress @xcite .    with the ansatz of perpendicular directors as just described , eq . ( [ osl ] ) becomes a fixed principle axes " ( fpa ) model @xcite . although grossly oversimplified , this leads to nontrivial predictions for the jammed state in the colloidal problem , such as a constant ratio of the shear and normal stresses when these are varied in the jammed regime . such constancy is indeed reported by laun @xcite in  the regime of strong shear thickening " ; see @xcite . we now turn from colloids to granular materials . although the formation of dry granular aggregates under gravity is not normally described in terms of jamming , it is a closely related process . indeed , the filling of silos and the motion of a piston in a cylinder of grains both exhibit jamming and stick - slip phenomena associated with force chains ; see @xcite . and , just as in a jammed colloid , the mechanical integrity of a sandpile disappears as soon as the load ( in this case gravity ) is removed .    in the granular context , a model like eq . ( [ osl ] ) is interpreted by saying that a fragile granular skeleton of force chains is laid down at the time when particles are first buried at the free surface ; so long as subsequent loadings are compatible with this structure , the skeleton will remain intact . if in addition the skeleton is rectilinear ( perpendicular directors ) this forces the principal axes to maintain forever the orientation they had close to the free surface ( fpa model ) . however , we do not insist on this last property and other models , based on an oblique family of directors , have also been developed @xcite . ( the construction history of a sandpile allows this since the orientation of the free surface and/or gravity provides a reference direction in addition to that given by the stress tensor at the onset of jamming @xcite . ) note also that , for a conical sandpile we require in addition to eq . ( [ osl ] ) one further relation among stresses , found for example by assuming that @xmath14 everywhere ; see @xcite . it turns out that the fpa models account quite well for the forces measured experimentally beneath conical piles of sand , constructed by pouring cohesionless grains from a point source onto a rough rigid support @xcite . ( note that the two dimensional case  a wedge  may have special features and the effectiveness of fpa for this is much less clear @xcite . ) as mentioned previously , fragile models such as this show very different mechanics from conventional forms of elasticity or elastoplasticity . for example , in 2-d , when combined with stress continuity ( @xmath15 for sand under gravity ) , eq . ( [ osl ] ) gives differential equations for the stress tensor which are hyperbolic @xcite . with a zero - force boundary condition at the upper surface of a pile , this gives a well - posed problem : the forces acting at the base follow uniquely from the body forces by integration . ( analogous remarks apply to eq . ( [ osl ] ) in 3-d . ) if _ different _ forces are now imposed at the base , rearrangement is inevitable @xcite . ( this is boundary - fragile behaviour . ) the same does not hold @xcite within a traditional elastoplastic modelling approach @xcite whose equations are elliptic in elastic zones . in such models of the sandpile the forces acting at the base can not be found without specifying a displacement field there . to define this displacement , one would normally invoke as _ reference state _ the one in which the load ( gravity ) is removed . for cohesionless poured sand , this state is undefined @xcite , just as it is for a jammed colloid which , in the absence of the applied shear stress , is simply a fluid . one route to an elastic reference state is to consider a hypothetical sandpile where each grain becomes firmly  glued " to its neighbours ( or the base ) , upon first coming to rest . the resulting medium is surely elastic , and must therefore be governed by elliptic equations . this does not mean that it is a conventional homogeneous elastic continuum ( for which the states of zero strain and zero stress coincide ) . indeed , a glued pile built under gravity will certainly have nonzero stresses if gravity is later removed @xcite . more importantly , for a typical disordered packing of near - rigid , glued grains , there will arise many _ tensile contacts _ even under a purely compressive external load . thus the problem of glued and unglued piles might , in practice , be extremely dissimilar . the problem of cohesionless granular media is a highly nonlinear one , because ( a ) grains are typically very rigid ( on a scale set by the stresses that arise ) , and ( b ) there can be no tensile forces between any grains in the entire system . the first issue is addressed in @xcite where we show that , for some fragile models , there is in fact a smooth crossover to more conventional elastoplastic physics when grains are deformable , with fragility emerging as the limiting behaviour for rigid particles .    in this rigid particle limit , where the longitudinal force rule ( lfr ) becomes valid , fragility will be recovered in a granular skeleton of force chains whose coordination number @xmath16 with @xmath17 dimension of space ( e.g. fig . [ fig : jammed ]  ( b ) in two dimensions ) . this is the same rule as applies for packings of _ frictionless _ hard spheres , which also obey the lfr  not because of force chains , but because there is no friction . indeed , regular packings of frictionless spheres , which show  marginal " or isostatic " mechanics have been studied in detail recently @xcite . note that although our idea of fragility is closely related to the concept of isostaticity ( see e.g. @xcite ) , it is apparently not identical ; in our models of fragile materials the isostatic condition applies only to a loadbearing substructure ( the force skeleton ) and not the whole packing . it would be interesting to test this idea numerically by measuring the mean coordination number among the subset of particles that carry strong forces . interestingly , moukarzel has recently argued @xcite that random packings of frictionless spheres _ generically _ become isostatic ( @xmath16 ) in the rigid particle limit . his arguments appear to depend only on the lfr and the absence of tensile forces , so they should , if correct , equally apply to any granular skeleton that is made of force chains of rigid particles . it remains to be seen whether moukarzel s ( somewhat unintuitive ) arguments can be made rigorous . pending this , we prefer for the moment our own qualitative reasoning , which generically links fragility to jamming , as the motivation for setting @xmath16 in our simplified models ( such as fpa ) of the granular skeleton . such models , in which the skeleton is represented as a rectilinear or oblique array with no disorder , are obviously convenient for calculation . however , disorder will not remove the fragility ( though it will change eqs . ( [ nn],[osl ] ) ) unless it causes the mean coordination number of the skeleton to increase . the jammed state of colloids , if it indeed exists in the laboratory , has not yet been fully elucidated by experiment . it is interesting the even very simple models such as eq . ( [ nn ] ) can lead to nontrivial and testable predictions ( such as the constancy of certain measured stress ratios ) . such models suggest an appealing conceptual link between jamming , force chains , and fragile matter @xcite . however , further experiments are needed to establish the degree to which they are useful in describing real colloids .    for granular media , the existence of tenuous force - chain skeletons is clear @xcite ; the question is whether such skeletons are fragile . several theoretical arguments have been given , above and elsewhere , to suggest that this may be the case , at least in the limit of rigid particles . moreover , simulations show strong rearrangement under small changes of compression axis ; the skeleton is indeed  self - organized " @xcite . experiments also suggest cascades of rearrangement @xcite in response to small disturbances . these findings are consistent with the fragile picture . the mechanics of fragile models such as eqs . ( [ nn],[osl ] ) differ strongly from those of conventional elasticity . for example , if an infinitesimal point force is exerted on top of a layer of sand , we expect the resulting pressure distribution at the base to form an annulus @xcite ; an elastic model would predict the maximum pressure to be directly beneath the applied force . it could be very fruitful to perform new ( but quite simple ) experiments of this kind . we thank r. ball , e.  clement , s.  edwards , m. evans , p.  evesque , p .- g . de gennes , g.  gudehus , j.  goddard , j. jenkins , d. levine , j. melrose , s. nagel , j. socolar , c.  thornton , l.  vanel and t.  a. witten for discussions . work funded in part by epsrc ( uk ) gr / k56223 and gr / k76733 . bouchaud , m.  e.  cates , and p.  claudin , j. physique i , * 5 * 639 ( 1995 ) ; j .- bouchaud , p.  claudin , m.  e.  cates , j.  p.  wittmer , in _ physics of dry granular media _ , h.j . herrmann , j.p . hovi and s. luding , eds . , nato asi 1997 ( cond - mat/9711135 ) . r.  s.  farr , j.  r.  melrose , and r. c. ball , phys . e , _ 55 _ , 7203 ( 1997 ) ; r.  c.  ball , j.  r.  melrose , adv . colloid interface sci . , _ 59 _ , 19 ( 1995 ) ; j.  r.  melrose , r.  c.  ball , europhys . lett . , _ 32 _ , 535 ( 1995 ) . s. alexander , phys . reports * 296 * , 65 ( 1998 ) ; p. s. cundall and o. d. l. strack , in _ mechanics of granular materials : new models and constitutive relations _ , j .. t. jenkins and m. satake , eds , elsevier 1983 . r. hooke , _ a description of helioscopes , and some other instruments _ john martin , london ( 1676 ) . see anagram on title page concerning  the true mathematical and mechanical form of all manner of arches . " ; reproduced as fig.2.2 of @xcite . see , e.g. , _ proc . 3rd int . conf . on powders and grains , durham nc 18 - 23 may 1997 _ , r. p. behringer and j.  t. jenkins , eds . balkema , rotterdam 1997 , especially f.  cantelaube and j.  d. goddard , pp . 185 - 194 .
we present simple models of particulate materials whose mechanical integrity arises from a jamming process . 
 we argue that such media are generically  fragile " , that is , they are unable to support certain types of incremental loading without plastic rearrangement . in such models , 
 fragility is naturally linked to the marginal stability of force chain networks ( granular skeletons ) within the material . 
 fragile matter exhibits novel mechanical responses that may be relevant to both jammed colloids and cohesionless assemblies of poured , rigid grains . 
 [ multiblock footnote omitted ]