On Probabilistic Distribution of Forces in Granular Materials: A Statistical Mechanics Approach
- PDF / 117,956 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 27 Downloads / 163 Views
MM2.2.1
On Probabilistic Distribution of Forces in Granular Materials: A Statistical Mechanics Approach Alfonso H.W. Ngan Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. CHINA. ABSTRACT The contact force distribution in stressed grain piles under simple compaction or gravity has been studied intensively in recent years. In the present investigation, by performing discrete element simulations, it is found that the contact force distribution in a stressed granular packing can be described by a generalized form the Second Law of Thermodynamics involving minimization of a free energy functional containing an energy and an entropy component. The relative importance of energy and entropy is controlled by a parameter known as the “mechanical temperature”. INTRODUCTION The distribution of contact forces in random granular packings has been the subject of intensive investigations in the past decade [1-12]. Because of structural disorderness, the internal force distribution in random packings due to external loadings will not be uniform. In the rigid grain regime, it has generally been agreed that the large force distribution follows an exponential decay, while the small force distribution is more debatable [1-3]. In the compressible regime, the force distribution undergoes a gradual transition from the Maxwell-Boltzmann (strictly exponential) behaviour at small applied pressures to a Gaussian form at larger pressures [6]. This work is based on the belief that a satisfactory description of forces or indeed any other quantities in random granular packings should involve concepts from statistical mechanics [13]. In a stressed granular packing, there is a strain energy functional due to the applied loading but at the same time, there is also an entropy functional, due to the random nature of the system. We first postulate that mechanical equilibrium is reached when the energy contribution balances the entropy contribution. This condition is then used to derive the allowable forms of the contact force distribution in a stressed granular packing. The results are then compared with discrete element simulations.
GENERALISED CONCEPTS OF STATISTICAL MECHANICS As Bagi [4] and Evesque [5] have pointed out, a force distribution P(f ) enables the definition of an entropy in the statistical sense ∞
S = − ∫ P( f ) ln[ P( f )]df . 0
(1)
MM2.2.2
Bagi and Evesque both argued that in a structurally random packing, the entropy in eqn. (1) should attain maximum value. This criterion always yields the exponential Maxwell-Boltzmann distribution which, in general, does not match the usually peaked experimental or simulated force distributions reported in the literature. If only entropy is maximized, the contribution of energy is ignored and this is against the spirit of the Second Law of Thermodynamics, one version of which states that when equilibrium is reached, the free energy defined by F = E − θS
(2)
should reach a minimum, rather than the energy E alone should reach a minimum or the en
Data Loading...
