Jamming is a phenomenon occurring in systems as diverse as traffic, colloidal suspensions and granular materials. A theory on the reversible elastic deformation of jammed states is presented.First, an explicit granular stress-strain relation is derived that captures many relevant features of sand, including especially the Coulomb yield surface and a third-order jamming transition. Then this approach is generalized, and employed to consider jammed magneto-and electro-rheological fluids, again producing results that compare well to experiments and simulations.

A complete continuum mechanical theory for granular media, including explicit expressions for the energy current and the entropy production, is derived and explained. Its underlying notion is: granular media are elastic when at rest, but turn transiently elastic when the grains are agitatedsuch as by tapping or shearing. The theory includes the true temperature as a variable, and employs in addition a granular temperature to quantify the extent of agitation. A free energy expression is provided that contains the full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state deformations, it is equivalent to hypoplasticity, a state-of-the-art engineering model.

The complete magnetodissipative structure of ferrofluid dynamics is derived from general principles, without reference to the angular momentum of the ferromagnetic grains. The results are independent of most microscopic details, and easily interpret two previous experiments. Both the Debye theory and the effective-field theory by Shliomis are shown to be special cases of the new set of equations.

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1. EM A ,U ni versi t at der B undesw ehr H am burg,H ol stenhofw eg 85,22043 H am burg 2. M ax-Pl anck-Insti tut f ur Pol ym erforschung,Postfach 3148,55021 M ai nz 3. Insti tut f ur T heoreti sche Physi k,U ni versi t at H annover,30167 H annover 4. T heoreti sche Physi k III,U ni versi t at B ayreuth,95440 B ayreuth,G erm any (PR L 84,3223,(2000)) In the l i m i t ofi n ni te yi el d ti m e for stresses,the hydrodynam i c equati ons for vi scoel asti c,N on-N ew toni an l i qui ds such as pol ym er m el ts m ust reduce to that for sol i ds. T hi s pi ece of i nform ati on su ces to uni quel y determ i ne the nonl i near convecti ve deri vati ve, an ongoi ng poi nt of contenti on i n the rheol ogy l i terature. A l lvi scoel asti c non-N ew toni an ui ds behave as N ewtoni an onesatl ow frequenci es,and assol i dsathi gherfrequenci es. A consi stent hydrodynam i c descri pti on needs to re ect thi s fact and m ust therefore contai n, as speci al cases, both the hydrodynam i c theory for i sotropi c l i qui ds and sol i ds. T he l i qui d l i m i t i s wel lheeded i n the pol ym erl i teratureand uni versal l y correctl y i m pl em ented [ 6,7] . T he sol i d l i m i t i s probl em ati c,as we shal lsee,and com pati bi l i ty especi al l y i n the nonl i near regi m e ofl arge di spl acem ents and rotati ons has so far proven el usi ve. T he reason behi nd i t i s probabl y the l ack ofa consi stent hydrodynam i c theory for sol i ds.T hel astsentencem ay com easa surpri se,butthepoi nt we are m aki ng here i s:A l though both the nonl i nearel asti ci ty theory [ 6] and the l i neari zed hydrodynam i cs for crystal s [ 1,8,9]are wel lknow n and establ i shed,a consi stent hydrodynam i c theory that i ncl udes both nonl i near and i rreversi bl e term s i s not { i n spi te ofsom e i nsi ghtful papers [ 10] . O ne of the obstacl es i s that such a theory necessari l y em pl oysa strai n tensordi erentfrom the one custom ari l y used [ 8] . T he usualstrai n tensor i s of the Lagrange type,deri ved from equati ons ofm oti on for m ass poi nts,w hi l e a fram ework to set up hydrodynam i c equati ons i ncl udi ng di ssi pati ve term s onl y exi sts i n the Eul eri an descri pti on { w hi ch consi ders evol uti on of el d vari abl es at spati al poi nts. C onsi stency forbi ds a m i xi ng ofboth descri pti ons and requi res an Eul eri an strai n tensor [ 11] . (W e note that the l i near hydrodynam i c theory m ay m i x both descri pti ons,as the sm al l ness of the di spl acem entsensuresthatthe di screpancy i snegl i gi bl e. )T he presentati on ofthe nonl i near hydrodynam i c theory for sol i ds i s w hat we shal ldo rst. T hen these equati onsare general i zed fornon-N ew toni an ui ds by addi ng rel axati on-type term s to account for a ni te yi el d ti m e ofthe stresses,such that i n the hi gh frequency l i m i t the theory i s unchanged,but i n the l ow frequency l i m i t onl y the term s ofthe i sotropi c l i qui d hydrodynam i cs rem ai n. So, by ensuri ng the val i d l i qui d and sol i d l i m i t...