Rates of Stress in Dense Unbonded Frictional Materials During Slow Loading

MATTHEW R. KUHN

University of Portland, 5000 N. Willamette Blvd, Portland, OR 97203 U.S.A. Email: kuhn@up.edu

1 Introduction

This chapter concerns the transmission and evolution of stress within granular materials during slow, quasi-static deformation. Stress is a continuum concept, and its application to assemblies of discrete grains requires an appreciation of the marked nonuniformity of stress when measured at the scale of individual grains or grain clusters. As an example, numerous experiments and simulations have demonstrated that externally applied forces are borne disproportionately by certain grains that are arranged in irregular and ever-changing networks of force chains. Although much attention has recently been given to the transmission of force at low strains, the current work focuses on the transmission of stress within granular materials at both small and large strains.

When a densely packed assembly of unbonded particles is loaded in either triaxial compression or shear, the behaviour at small strains is nearly elastic, and the volume is slightly reduced by the initial loading (an initial Poisson ratio less than 0.5, Figure 1). Plastic deformation ensues at moderate strains, at which an initially dense material becomes dilatant, and this trend of increasing volume continues during strain hardening, at the peak strength, and during strain softening. At very large strains, the material reaches a steady condition of flow, referred to as the "critical state" in geotechnical engineering practice, in which the material flows at a constant, albeit expanded, volume while sustaining a constant shearing or compressive effort. Besides studying behaviour at the initial and peak states, we will also consider experimental results at this steady state condition and the manner in which the inter-granular forces are distributed and changed during steady state flow. These conditions are investigated with numerical simulations of an idealized assembly of circular disks. We will explore mechanisms that underlie the changing stress by separating the stress rate into various constituents and then study their relative influences during simulated loading.

Notation

In this chapter, vectors and tensors are represented in both indexed and unindexed forms with the use of upper and lower case glyphs: A or Aij for tensors, and a or ai for vectors. Inner products are computed as

a · b = aibi, A · B = AijBij, (1)

and tensors are often represented as the dyadic products of vectors:

a [cross product] b = aibj (2)

A juxtaposed tensor and vector will represent the conventional product

Ab = Aijbj. (3)

No contraction is implied with superscripts (e.g., acbc). The trace of a tensor is defined as

trace (A) = A11 + A22 + A33. (4)

Tensile stresses and extension strains are positive, although the pressure p will be positive for compressive conditions.

2 Partitioning the Stress Rate

The average Cauchy stress [??] within a granular assembly can be computed as a weighted average of the contact forces between grains:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where summation is applied to the set of M contacts within the assembly, and each contact c represents an ordered pair of contacting particles p and q, c = (p, q). The sum is of the dyadic products fc [cross product] Ic, where fc is the contact force exerted by particle q upon particle p, and branch vector VIc connects a reference (material) point on particle p to a reference point on particle q (Figure 2). For the numerical simulations in this study, these reference points are placed at the centres of circular disks. The current volume (area) of the three-dimensional (two-dimensional) region is represented by V. Equation 5 applies, of course, only under ideal conditions. The absence of kinetic terms limits Equation 5 to slow, quasi-static deformation, and the lack of body force terms implies a zero-gravity condition. Equation 5 also avoids the complexities that are associated with peripheral particles in a finite region and with contacts that can transmit couples between particle pairs. Although these complexities can be difficult to evaluate in physical experiments, they can be circumvented altogether in numerical simulations that exclude both gravity forces and contact moments, and in which the boundaries are periodic.

The formulation (5) for average stress has been employed in a number of ways. Its use in numerical simulations allows the direct calculation of stress from the inter-particle contact forces, but without the supplemental operation of identifying boundary particles and computing the external forces on those particles, an advantage that is particularly appropriate when the boundaries are periodic. Equation 5 has also been the starting point for estimating a macro-scale stiffness from the micro-scale behaviour at particle contacts, and modest successes have been reported at small strains." The equation has also led to important insights into the nature of stiffness and strength in granular materials. These insights have been primarily gained by partitioning the average stress [??] into various contributions that arise from the distributions and directions of the contact forces fc. Several studies have partitioned the stress into two contributions: one from the normal components of the contact forces, and the other from the tangential components. These studies have shown...