For this state in two space dimensions, a new loop-force formulation confirms the idea of a locally linear constitutive equation of the form
where 
, with the added result that the coefficients 
under a local spatial average.The Marginally Rigid State of Matter, and its Hydrodynamics
R C Ball and R Blumenfeld
Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
Abstract:
A collection of rigid grains will consolidate until the state of Marginal Rigidity is achieved.
For this state in two space dimensions, a new loop-force formulation confirms the idea of a locally linear constitutive equation of the form where , with the added result that the coefficients under a local spatial average.
Equations governing the deformation of such systems at the yield surface give a new interpretation to the coefficients 
, as the contribution to bulk deformation from rolling motion of individual grains. Thus the symmetric part of the strain rate tensor is given by 
where the first term is conventional plasticity associated with slip between grains and the second term is the contribution of grain rolling with local angular velocity 
.
The combination 
can now be interpreted as the force conjugate to the rolling field 
, and the appearance of these in the statics and hydrodynamics distinguishes granular from other states of matter.
The Physics Problem (1):
Let rigid grains settle until they come to rest in a pile
The question:
What are the equations governing the transmission of stress through this system?
Microscopic answer: Macroscopic version:
Balance force on each grain 
?. d eq'ns in d dimensions.
Balance torque on each grain 
?. 
equations
Dilemma:
The macroscopic equations are underdetermined:
further constitutive relations require to be found.
Marginal Rigidity at the grain level
Model grains as rigid, with point contacts.
For grains with static friction, the intergranular forces between contacts amount to
force degrees of freedom per grain
As the pile settles and mean coordination number 
increases,
the intergranular forces can first support force and torque balance when
for grains with friction
for grains without friction (except spheres, which are anomalous)
At 
there are exactly enough contacts for the intergranular forces to stabilise the pile:
we call this the Marginally Rigid State, and under ideal conditions this should be selected dynamically.
At this condition the intergranular forces are uniquely determined by force and torque balance
- so we should seek a corresponding constitutive equation in terms of the stress field alone.
Static Constitutive Equations for the Marginally Rigid State
For grains with friction the following result seems to obtain:
where the coefficients 
are antisymmetric with respect to interchange of 
.
A result of this form was postulated by Cates and collaborators [1],
has been obtained explicitly for simple periodic lattices [2],
and more general discussions in two dimensions are given in [3],[4].
The latest work [4] shows that for grains with friction varies strongly with position,
such that its local spatial average vanishes (CMMP Poster SPpP1.13).
Physics Problem (2):
Load the pile until it yields.
Question:
How does the pile subsequently move?
Answer:
Symmetrised strain rate law:
Conventional plasticity + Grain Rolling
This equation is supplemented by:
Condition for grain sliding 
Yield Locus 
Force balance 
Constitutive Equation
These eq'ns determine: symmetric tensor 
, scalar 
, vector 
, and angular velocity 
The Contribution of Grain Rolling
It is obvious that grains can roll over each other.
For periodic arrays, explicit calculation shows that grain rolling contributes to the shear rate with the same coefficients as appear in the static constitutive equation.
More widely, this is a conjecture - supported by the observation that the dissipation associated with pure rolling is correctly zero, 
.
Global rotational invariance is preserved, because the spatial average of vanishes.
Without grain rolling, equation counting shows the system of equations is not (in general) compatible with the static constitutive equation - so whilst a sample might follow the constitutive equation until yield, it could not follow it thereafter.
Interpretation of the Static Constitutive Equation:
is the generalised force conjugate to grain rolling;
because rolling couples to macroscopic deformation,
the balancing of this force cannot be ignored in the macroscopic transmission of stress.
Acknowledgements
RCB and RB acknowledge fruitful collaboration with SF Edwards and D Grinev, and recent discussions with C Thornton.
This work is supported by EPSRC grant GR/L59757.
Selected References
1. JP Wittmer, P Claudin and ME Cates, J Phys I (France) 7, 39 (1997).
2. RC Ball in Structure and dynamics of materials in the mesoscopic domain, eds. M Lal, RA Mashelkar, BD Kulkani and VM Naik (Imperial College Press, London 1999).
3. SF Edwards and D Grinev, Phys Rev Lett 82, 5397 (1999).
4. R C Ball and R Blumenfeld, Preprint cond-matt 0008127, August 2000.