Triaxial Testing’s 1/6th Rule: Particle-Continuum Analysis of Granular Stability during Compression

The 1/6th Rule of Triaxial Testing says that particles larger than 1/6th the diameter of the specimen will skew test results when not discarded. Although this rule is documented in the procedures of government laboratories, its origin is obscure. In this paper, the rule is derived as a corollary of granular stability. The stability derivation involves particle-continuum analysis. However, instead of a discrete element formulation or Cosserat mechanics, this paper uses the classical solution of Sokolovskii that is the modern basis of Terzaghi bearing capacity theory. Stability of a soil particle is addressed within the continuum by matching leading terms of series expansions, which is also known as the singular perturbation method. This finding is similar to previous derivations involving deformation of geosynthetic reinforced soil.

In Figure 1, the Wu-Pham instability is validated by capacity ratios for unfaced and faced soil structures tested at the US Federal Highway Administration (FHWA) [7].
for K/KA = 0.35 and ϕ = 45°, which are associated with common configurations of geotextile reinforced soil [4,8]. KA = tan 2 (45°-ϕ/2) is the coefficient of active lateral earth pressure for an aggregate. The S/D instability has a companion, the S/B instability, The 1/6 th Rule of Triaxial Testing follows immediately from Equations (2) and (3) because equating them gives  (4) for sand where ϕ = 30°, For gravel, it is more probable that ϕ = 42°. Consequently, procedures of a few organizations also include a 1/10 rule for gravel [3].
This paper now proceeds with derivations that are tailored toward triaxial testing.

Sokolovskii's Solution for a Layer
Geotechnical engineers are familiar with the theory of soil bearing capacity due to Terzaghi [9]. It originated with solutions by Prandtl [10] and Reissner [11] to the equation of Kötter [12] for Mohr-Coulomb materials. The Prandtl-Reissner solutions described failure in metals, clays, and other purely cohesive materials. Terzaghi provided approximations for soil with friction angle ϕ > 0. The current version of that theory has been updated with the solution due to Sokolovskii [13].
Today, Sokolovskii is associated with general slip line theory. When a material fails, it fails along a slip line. This paper examines slip lines in a layer of aggregate compressed between two tensile materials.
For soil in a state of incipient failure, slip lines are inclined at 45° + ϕ/2 to the minor principal direction. Invariants are ln(c + p tanϕ) ± 2θ tanϕ along these lines. In the case c = 0 and ϕ = 45° of an aggregate, these invariants simplify to ln(p) ± 2θ. In Mohr's circle, 2θ represents shear and is the central angle measured from the horizontal. The center of Mohr's circle, p = (σ1 + σ3)/2, represents mean stress. The circle's radius is r = (σ1 -σ3)/2. Figure 2 depicts a layer of soil, and θ is a function of shear stress τ between soil and platen. Shear stress at the top has the opposite sign of shear stress at the bottom. Therefore, near the middle of the layer, shear stress is zero and θ = 0.   (8) where μ = tan ϕ is the soil's coefficient of friction. The equation expresses the ratio of mean pressure at mid-layer (pm) and mean pressure at the platen (p) as a function of change in shear stress (Δθ) at the soil-platen interface at the moment of failure.
The exponential of Equation (8) finds its way into Equations (2) and (3) because W and P are now shown to be related.

Generic Stability without Calculus
There is a calculus-based derivation of the relationship between W and P [4]. Unfortunately, it does not contribute to engineering intuition. A non-calculus-based derivation is provided here.
W is defined in Equation (1) as the ratio of unfaced to faced capacity. By Equation (8), mean stress decreases at the middle of the layer when the shear gradient Δθ > 0. Because this suggests that confining pressure also decreases toward the middle, define Calculation shows that an unfaced failure occurs when (10) so that, for geosynthetics and aggregate where KP = 1/KA, the unfaced capacity q is is the ratio of unfaced to faced capacity; therefore, it appears that and this is confirmed by the validation tests of Figure 1. W's two definitions, Equations (1) and (9) This enables an alternative calculation of P,   (14) because the central angle in Mohr's circle is where  is the shear stress and r is the circle's radius.
Shear stress  near the platen is estimated with the free body diagram, Figure 4, and is the ratio of horizontal to vertical stress.

S/B Continuum Stability
In an unfaced test article, tension T in the platen is zero where physical facing is absent or ϕ3 = 0. Thus, change in tension is equal to the maximum tension. In a test article of breadth B that is symmetric from left to right, tension at the center is a maximum and shear is zero. Maximum shear stress near the platen is approximated as This value represents the maximum possible change in shear stress between Points A and B in Figure 2. Recall that 2r = σ1 -σ3 ≈ (1 -KA) σ1 ; therefore, Equation (14)

S/D Particle-Continuum Stability
Consideration of the Cosserat approach is necessary because some sort of particle-based argument is required. Ultimately, stability calculations benefit in this paper from the brilliant tools of the Prandtl-Sokolovskii legacy. In other words, the author attempted to work upward from particles to continuum, but in the end, works downward from continnum to particles. This paper follows examples from fluid mechanics that go from flow in the far field downward to boundary layers and eddies.

Continuum
Begin with layer stability, Equation (14), that follows from Sokolovskii's continuum solution. It contains the ratio τ/r. In Figure 5, this ratio is examined in the context of the Mohr-Coulomb failure condition.

Particles
With Lref = D, Equation (17) provides a particle-based basis, In order to determine α, it is necessary to examine how zero is approached. The cosine and tangent are replaced by their series expansions,

Synthesis
A purely algebraic comparison of Equations (20) and (21) is provided above. A physical comparison is needed for intuition.
In Equation (20), W ranges between 1 when ϕ = 0 and down near 0 when ϕ = 90°. Failure angle αf = 45° + ϕ/2 can be substituted for ϕ, and then W ranges between 1 when αf = 45° and down near 0 when αf = 90°. When the failure angle is viewed as the angle formed by the base of a soil arch in the layer, Equation (20) says that shallow arches collapse first, that is, at loads associated with small W. This is verified with the finite element analysis (FEA) in Figure 6. for sand where ϕ = 30°. For gravel, it is more probable that ϕ = 42°. Consequently, procedures of a few organizations also include a 1/10 rule for gravel [3]. The method used here is a simple form of singular perturbation used elsewhere in engineering [14].
So far, this derivation ignores pressure in the water jacket surrounding the triaxial specimen. It has been shown elsewhere that this pressure increases the value of K [15,16]. Equation (25) shows that K cancels and has no effect on the 1/6 th Rule.

Conclusion
The 1/6 th Rule for Triaxial Testing is suggested by Equations (2) and (3), the granular stability equations for common configurations of reinforced aggregates, e.g., . In the context of triaxial testing and a range of noncohesive soils, Equations (19) and (24) are derived as replacements for Equations (2) and (3). Nevertheless, the result, suggested by common reinforced soil configurations, is unchanged for triaxial testing.
An intuitive, calculus-free derivation is provided for broader understanding. The meaning of Sokolovskii's solution is explained in detail.
The derivation assumes that friction, however small, always exists between soil and platen. This analysis indicates that common methods of friction-reducing lubrication do not reduce the need for the 1/6 th Rule.
The particle-continuum method used here is called singular perturbation elsewhere in engineering.
Because D/B = KA/2, the rule becomes more restrictive when sand is replaced by gravel in test specimens under compression.