Stress between Two Spherical Particles with Nonzero Contact Angle

A micromechanical theoretical development for evaluating interparticle forces and suction stress in contacting, monosized, unsaturated spherical particles with a constant contact angle equal to zero was presented in Chapter 4. This section takes the analysis several steps further by considering contact angle as a nonzero material variable. As before, simple cubic (SC) packing and tetrahedral (TH) packing are considered to represent end members in granular soil fabric. It is presumed that the range in material properties and water

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Simple cubic: radius R Coordination number = 6 Layer spacing = 2R Unit volume = 8R³ Void ratio = 0.91 Porosity = 47.6%

Tetrahedral: radius R Coordination number = 12 Layer spacing = 2R(2/3)^0.5 Unit volume = 4(2R³)^0.5 Void ratio = 0.34 Porosity = 26.0%

(a)

(b)

Figure 5.1 Uniform spheres in (a) simple cubic packing order and (b) tetrahedral close packing order.

retention and suction stress behavior of real soil, particularly coarse-grained material such as silt or sand, falls somewhere between these two idealized scenarios. Figures 5.1a and 5.1b illustrate geometries for uniformly sized spheres coordinated under SC packing and TH packing, respectively. Unit volumes for SC and TH packing have void ratios of 0.91 and 0.34, respec-tively, corresponding to porosities of 47.6 and 26.0%.

Two quantities are required to analyze suction stress and its dependency on water content in such particle arrangements: the capillary force between the particles and the water content of the particle / pore water system. The capillary force between two contacting spherical particles for a toroidal me-niscus geometry and zero contact angle (Fig. 4.20) was derived earlier as

2 2 2

F_{sum ⫽} ua_␲R _⫺ ua_␲r2 _⫺T 2s _␲r2_⫹ uw_␲r2 (5.2) Figure 5.2 shows a more general system geometry for describing the water content between two particles with a nonzero and variable contact angle _␣.

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O

Figure 5.2 Geometrical constraints for defining the water meniscus between con-tacting spheres with consideration for a variable contact angle: (a) system radii and angles and (b) two-dimensional surface boundaries of water lens.

Here, the water lens represented by radii r₁and r₂can be written in terms of filling angle_␪, the common particle radius R, and contact angle_␣ as

1 _⫺cos _␪

r_1⫽ R (5.3)

cos(_{␪ ⫹ ␣})

sin_␣

r²_⫽ R tan_{␪ ⫺}r¹冉1_⫺ ^cos␪冊 (5.4) When contact angle is equal to zero, eqs. (5.3) and (5.4) reduce to those proposed by Dallavalle (1943) [eq. (3.38)] as presented in Chapter 3.

The water content of the system may be evaluated by considering the volume of the water lens. In a two-dimensional projection, the water lens is bounded by the three hatched surfaces shown in Fig. 5.2b. The rectangle BFGH, which is a cylinder in three dimensions having radius R sin _␪ and height R, bounds the bottom half of the symmetrical water lens. The partial circle of radius R defined by FBEOI bounds the water lens on the top half.

The partial circle defined by IKJ bounds the water lens on both sides. Rotated in three dimensions, these surfaces become volumes that can be used to define the total volume of the water lens. The volume of the rotated area BFGH for one unit particle is

3 2

Vc_⫽2_␲R sin _␪ (5.5)

The volume of the rotated area FBEOI for one unit particle is

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2_␲

3 2 3 2

Vs_⫽ 2_␲R sin _␪cos _{␪ ⫹} R (1_⫺ cos_␪) (2_⫹ cos_␪) (5.6) 3

The volume of the rotated area IJK for one unit particle is 2 r cos (₁ 3 _{␪ ⫹ ␣})

V^r_⫽ 2_␲冋r² _⫹r¹_⫺ ^{3 (␲/ 2)⫺ (␪ ⫹ ␣) ⫺sin(␪ ⫹ ␣) cos(␪ ⫹ ␣)}册

1 ₂

r [_{1␲ ⫺}2(_{␪ ⫹ ␣}) _⫺sin 2(_{␪ ⫹ ␣})]

2

(5.7) Accordingly, the total volume of the water lens Vlis

Vl_⫽ Vc_⫺Vs_⫺ Vr

3 2 3 2

⫽ 2_␲R sin _{␪ ⫺} 2_␲R sin _␪cos _␪

2_{␲ 3 2}

⫺ R (1_⫺ cos _␪) (2_⫹ cos _␪) _⫺ Vr (5.8) 3

Determining gravimetric water content for each unit cell of particles in SC packing requires summation of three orthogonal water lens volumes and can be expressed as

3Vl

w_SC⫽ (5.9)

V_sphereGs

where V_sphereis the volume of one soil particle (i.e., V_{sphere⫽ ␲}–⁴₃ R³) and Gsis the specific gravity of the soil solids. It follows that water content can be written in terms of the angles_␪and _␣ as

9 ₂ 9 ₂

w_SC⫽ sin _{␪ ⫺} sin _␪cos_␪

2Gs 2Gs

3 2 9Vr

⫺ (1 _⫺cos _␪) (2_⫹cos _␪)_{⫺ 3} (5.10)

2Gs 4Gs_␲R

In TH packing, gravimetric water content is simply twice that of SC pack-ing for the same fillpack-ing angle:

w_{TH ⫽}2 w_SC (5.11)

For zero contact angle, the limits of the pendular water regime in SC and TH packing are 0.063 g / g gravimetric water content and 0.032 g / g,

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tively. These values represent water contents where the individual water lenses between neighboring particles begin to touch each other and the meniscus geometry idealized in Fig. 5.2 is no longer valid.

Equations (5.10) and (5.11) are plotted in Fig. 5.3a to show relationships between filling angle_␪and gravimetric water content for contact angle equal to zero (R _⫽ 1 mm, Gs⫽ 2.65). Similar analytical solutions for R _⫽ 1 mm, Gs ⫽ 2.65, and _{␣ ⫽} 0_⬚ developed previously by Dallavalle (1943) and Cho and Santamarina (2001) are included for comparison. Figure 5.3b shows w_SC and w_TH as functions of _␪for contact angle equal to 0_⬚, 20_⬚, and 40_⬚. It can be seen here that the increase in contact angle has a significant effect on the volume of the pore water lens and the corresponding water content of the two-particle system. Larger contact angles, which may be considered to co-incide with a wetting process, result in higher water contents for a given filling angle _␪. Zero contact angles, which might correspond to a drying process, result in relatively low water contents. This observation forms the basis for an analysis of contact angle hysteresis presented in Section 5.2.

As introduced in Chapter 4, effective stress resulting from suction stress can be evaluated by dividing the interparticle capillary force, that is, eq. (5.2), by the area over which it acts. Taking the cross-sectional area of one particle (_␲R²) as an elementary area, and employing eq. (4.50) to describe surface tension Tsin terms of the spherical radii r₁and r₂, eq. (5.2) can be written in terms of a stress contribution due to capillarity _␴was

2 2

r₂ 2r r_{2 1}

␴ ⫽w ua_⫺ R2(ua_⫺ u )w _⫺ R (r2 _2⫺ r )₁ (ua _⫺ u )w

2 2

r₂ 2r r_{2 1}

⫽ u^a_⫺ 冋^R2_⫹ ^{R (r2 2⫺ r )1} 册(u^a_⫺ u )^w r r2_{2 2⫹} r₁

⫽ ua_⫺ R r2 _2⫺ r₁(ua_⫺ u )w (5.12)

and the effective stress under an external total stress _␴is r r2_{2 2 ⫹} r₁

␴⬘ ⫽ ␴ ⫺ ␴ ⫽ ␴ ⫺w ua_⫹ R r2 _{2 ⫺} r₁(ua_⫺ u )w (5.13)

which is in the same form as Bishop’s (1959) single-valued effective stress equation for unsaturated soil, that is, eq. (5.1). Equating the two leads to

r r2_{2 2 ⫹}r₁

␴ ⫺u_{a⫹ ␹}(u_a⫺ u )_{w ⫽ ␴ ⫺}u_{a ⫹ 2} (u_{a ⫺}u )_w (5.14) R r_{2 ⫺}r₁

where

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0.00

Cho and Santamarina (2001)

R = 1 mm

Figure 5.3 Relationship between filling angle and gravimetric water content for 1-mm spheres in simple cubic (SC) and tetrahedral (TH) close packing: (a) for_{␣ ⫽}0_⬚ and (b) for_{␣ ⫽}0_⬚,_{␣ ⫽}20_⬚, and_{␣ ⫽} 40_⬚.

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r r2_{2 2 ⫹} r₁

␹ ⫽ ₂ (5.15)

R r_{2 ⫺} r₁

Equation (5.15) can now be used in conjunction with eqs. (5.3) and (5.4) to write the effective stress parameter _␹ as a function of filling angle _␪and contact angle_␣ as

1 _⫺cos _␪ cos _{␪ ⫺}sin_␣ 2

␹ ⫽冋tan_{␪ ⫺}^{cos(␪ ⫹ ␣) cos ␪} 册

tan_{␪ ⫹} (sin_␣/ cos _␪)(1_⫺ cos _␪) / cos(_{␪ ⫹ ␣})

(5.16) tan_{␪ ⫺}(2 _⫺sin_␣/ cos_␪)(1_⫺ cos_␪) / cos(_{␪ ⫹ ␣})

The above equation can be used to investigate the dependency of_␹ on water content and contact angle, as shown subsequently.