Micromechanical bounds for the effective elastic moduli of granular materials
Introduction
In many industrial, geotechnical and geophysical applications dealing with granular materials, knowledge of the mechanical behaviour of granular materials is important. This knowledge is expressed by a constitutive relation, which usually is based on continuum mechanics, and involves heuristic assumptions. Contrary to this approach is the micromechanical approach, in which a granular material is modelled as an assembly of particles that interact at contacts. The micromechanical approach therefore incorporates the discrete nature of granular materials. An objective of this approach is to derive macroscopic characteristics from microscopic characteristics, such as contact geometry and contact constitutive relation.
The relatively simple case that is considered here is that of the effective elastic behaviour of two-dimensional assemblies of non-rotating particles with bonded contacts. It is expected that many salient features of this simple system will hold, at least qualitatively, for more complicated systems. Some applications of the current model are the initial elastic deformation of dense or cemented granular materials and certain fibrous media.
The outline of this paper is as follows. In Section 2 some micromechanical quantities and the contact constitutive relation that is considered here are described. Then minimum potential energy and minimum complementary energy principles are recapitulated in Section 3 that were derived by Kruyt and Rothenburg (1998). To use these principles to obtain rigorous bounds for the moduli, compatible fields for the relative displacements at the contacts and equilibrated fields for the forces at the contacts must be constructed. The construction of compatible fields is fairly trivial, while a general construction of equilibrated fields of contact forces (Satake, 1992) is described in Section 4. In Section 5, a compatible field that is close to force equilibrium is determined based on local analyses, which in turn gives an upper bound for the moduli. Similarly, based on local analyses, an equilibrated field that is close to compatibility is determined, which yields a lower bound for the moduli. In Section 6 the results of discrete element simulations (Cundall and Strack, 1979) are used to compare the obtained bounds with the actual moduli, and to analyse the approximate fields for the relative displacements and forces at the contacts. Finally, findings from this study are discussed.
The usual sign convention from continuum mechanics is employed for stress and strain, according to which tensile stresses and strains are considered positive. The summation convention is adopted, implying a summation over repeated subscripts.
Section snippets
Micromechanics
Branch vectors lpqi are defined as the vectors that connect the centres of particles p and q that are in contact. These branch vectors form closed loops, or polygons, as depicted in Fig. 1. For future reference the polygon vector hRSj (Rothenburg, 1980; Kruyt and Rothenburg, 1996) is also defined in Fig. 1: it is the vector that is obtained by counter-clockwise rotation over 90° of the rotated polygon vector gRSi that connects the centres of adjacent polygons R and S.
Contacts can be identified
Variational principles
Two variational principles were derived by Kruyt and Rothenburg (1998) for assemblies of non-rotating particles with the linear elastic contact constitutive relation (4). These principles are discrete analogues of the classical minimum potential energy and minimum complementary energy principles in continuum elasticity (see for example Washizu, 1968).
Compatible and equilibrated solutions
To use these variational principles, general expressions for compatible relative displacement fields and equilibrated force fields must be constructed. The first is trivial, while the second is more involved. Both will be discussed here to show the analogy between the two approaches.
Local adjustment fields
To obtain an upper bound for the moduli from the minimum potential energy principle (7), a compatible relative displacement field will be constructed which leads to approximate equilibrium. To obtain a lower bound for the moduli from the minimum complementary energy principle (8), an equilibrated force field will be constructed which leads to approximate compatibility. Both approaches involve a local adjustment to a uniform field that uses only information on neighbours (particles or polygons).
Comparison with discrete element simulations
Discrete element simulations, as proposed by Cundall and Strack (1979), were performed with large assemblies of 50,000 disks from a fairly wide lognormal particle size distribution for nine different isotropic assemblies with average coordination numbers Γ in the range 4–6, and for each average coordination number for nine different stiffness ratios λ in the range 0.05–1.0. Two loading paths are sufficient for isotropic assemblies, compressive and shearing loading. The simulations are fully
Discussion
A micromechanical study was made of the elastic behaviour of two-dimensional assemblies of non-rotating particles with bonded contacts. Based on the results of discrete element simulations, it was shown that the case of non-rotating particles is the limit case of couple stiffness kω→∞. In the absence of body forces, the family of equilibrated contact forces is described, using force potentials (Satake, 1992). Applying minimum potential energy and minimum complementary energy principles,
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