The program for direct shear testing was selected based on the results of the Proctor tests and in an effort to assess the effects of void ratio, water content, and CaL content. A number of different water contents were selected including optimum, dry, and wet optimum. For each individual CaL content value, five different sample configurations as shown in Figure 2 were tested. Three configurations are at the optimum water content, and they have the relative compactions of 100% (point A), 95% (point E), and 90% (point C). Two additional configurations are at the relative compaction of 95% (point D and point B). As shown in Figure 2, point D is located dry of optimum, while point B is located wet of optimum. These configurations were tested at CaL concentrations of 2%, 4%, 6%, 9%, and 14%. In addition, dry sand was tested at relative compactions of 100%, 95%, and 90%. Furthermore, each single configuration included five direct shear tests at the following normal stresses: 62 kPa, 92.9 kPa, 123.9 kPa, 185.9 kPa, and 247.8 kPa.

All direct shear tests were tested under displacement control whereby a slow displacement rate was selected. Effective friction angle and cohesion values were deduced from the direct shear test program. They are listed for all tested sample configurations in .

Table 1 Values of effective cohesion and friction angle for all test points.

The cohesion itself reached a maximum value of 18.3 kPa at *χ*_{l}=6% (). A plot of peak shear stress versus normal stress for *χ*_{l}=6% is shown in Figure 4.

Figure 4 Peak shear stress versus normal stress for *χ*_{l}=6% (A, E, C, D, and B).

Shear force and horizontal and vertical displacements were continuously recorded during the shear phase of direct shear tests by using a data acquisition system. Failure was defined by the first attainment of the maximum shear stress. Shear stress versus horizontal displacement responses for configuration of 6% (E) at different normal stresses is depicted in Figure 5.

Figure 5 Shear stress versus horizontal displacement for *χ*_{l}=6% (E).

Figure 6 shows vertical displacement versus horizontal displacement for different confining stress levels for configuration of 6% (E). It should be noted that an increase in the sample thickness indicates dilation, which is negative herein.

Figure 6 Change in thickness versus horizontal displacement, *χ*_{l}=6% (E).

Addition of CaL decreases dilatancy whereby the highest rate of dilation occurs usually at point A, which is followed by B, E, D, and C. Among D, E, and B, the configurations which have equal initial void ratios, the material at point B is often the most dilatant, and it produces the highest shear stress at failure.

In order to achieve better understanding of the interactions between sand and bonding material, a further analysis is necessary. To this end, additional equations have been derived starting from the basic definitions of phase relationships given in Eqs. (1)–(7). The derived equations provide the basis for determination of the portion of the total cross-sectional area, which is occupied by CaL-water paste. A schematic of the load-bearing cross-sectional area, as well as the entire cross-sectional area, is depicted in Figure 7.

Figure 7 Schematic of load-bearing cross-sectional area.

It follows from Eq. (6) that

$$\frac{{A}_{w}}{A}=\frac{{G}_{s}w}{1+e}\text{\hspace{1em}(8)}$$(8)

Similarly for CaL area ratio, the following is obtained:

$$\frac{{A}_{l}}{A}=\frac{{G}_{s}{\chi}_{l}}{{G}_{l}\mathrm{(}1+e\mathrm{)}}\text{\hspace{1em}(9)}$$(9)

And adding Eqs. (8) and (9) results in the following:

$$\frac{{A}_{w+l}}{A}=\frac{{G}_{s}}{1+e}\mathrm{(}w+\frac{{\chi}_{l}}{{G}_{l}}\mathrm{)}\text{\hspace{1em}(10)}$$(10)

which gives a normalized area ratio as follows:

$$\frac{{A}_{w+l}}{A{\chi}_{l}}=\frac{{G}_{s}}{1+e}\mathrm{(}{w}_{w/l}+\frac{l}{{G}_{l}}\mathrm{)}\text{\hspace{1em}(11)}$$(11)

Equation (11) shows that water content, gravimetric lignin content, and void ratio affect the normalized area ratio. The latter is a ratio of the cross-sectional areas occupied by CaL and water paste and a total cross-sectional area, which is further divided by gravimetric lignin content. Thus, there are two different constituents carrying a load: CaL-water paste and solid skeleton. Upon mixing CaL and water, they form a paste, which acts as a bonding agent. This concept is illustrated in Figure 7.

From the equilibrium of forces depicted in Figure 7, it follows that the external or total normal force that is carried by these two constituents is given by the following:

$$F={F}_{c}+{F}_{l+w}={\sigma}_{c}{A}_{c}+{\sigma}_{l+w}{A}_{l+w}\text{\hspace{1em}(12)}$$(12)

where *σ*_{c} is the interparticle contact stress and *σ*_{l+w} denotes a stress in the CaL-water paste.

By dividing Eq. (12) by the entire cross-sectional area, *A*, the following is obtained:

$$\sigma ={\sigma}_{c}\frac{{A}_{c}}{A}+{\sigma}_{l+w}\frac{{A}_{l+w}}{A}={\sigma}^{\prime}+{\sigma}_{l+w}\frac{{A}_{l+w}}{A}\text{\hspace{1em}(13)}$$(13)

Thus, there is a portion of the external or total stress that is carried by the contacts of sand particles, which is also known as the effective stress *σ*′. Adding the CaL-water bonding agent leads to an increase in the load-bearing cross-sectional area, as indicated in Figure 7. Thus, the external load is now also carried by the bonding agent, within which it induces a stress denoted as *σ*_{l+w}.

By applying the Mohr-Coulomb criterion, which holds based on the experimental results obtained in this study, the following is obtained:

$$\tau ={\sigma}^{\prime}\mathrm{tan}{\varphi}^{\prime}\text{\hspace{1em}(14)}$$(14)

where *ϕ* is the friction angle of the S-CaL-W mix in terms of total stress. Next, Eq. (14) is combined with Eq. (13) resulting in

$$\tau ={\sigma}^{\prime}\mathrm{tan}{\varphi}^{\prime}+{{\sigma}^{\prime}}_{l+w}\frac{{A}_{l+w}}{A}\mathrm{tan}{\varphi}^{\prime}={\sigma}^{\prime}\mathrm{tan}{\varphi}^{\prime}+{c}^{\prime}\text{\hspace{1em}(15)}$$(15)

Setting *c*′=0 implies that Φ=Φ′. Thus, the friction angle in terms of total stress is equal to the effective friction angle. Also, it follows from Eq. (15) that the effective cohesion (*c*′) is given by

$${c}^{\prime}={{\sigma}^{\prime}}_{l+w}\frac{{A}_{l+w}}{A}\mathrm{tan}{\varphi}^{\prime}=\frac{{F}_{l+w}}{A}\mathrm{tan}{\varphi}^{\prime}\text{\hspace{1em}(16)}$$(16)

Thus, the cohesion is provided by the bonding agent. According to Eq. (16), the amount of cohesion depends on the ratio of the portion of a total cross-sectional area that is occupied by CaL-W mix, and the total cross-sectional area. The cohesion also depends on the stress in the CaL-W mix (*σ*_{l+w}) and the effective friction angle of the S-CaL-W mix (Φ′). In addition, Eq. (15) shows that S-CaL-W mix can sustain a shear stress in the absence of normal stress. This further implies that it also possesses a tensile strength, which is clearly derived from the presence of the bonding agent.

Eq. (16) also gives the following expression for a stress in the CaL-W paste:

$${\sigma}_{l+w}=\frac{{c}^{\prime}}{\frac{{A}_{l+w}}{A}\mathrm{tan}{\varphi}^{\prime}}\text{\hspace{1em}(17)}$$(17)

Next, the expression for the limiting normal stress (*σ*_{l}), which is the maximum normal stress that can be applied to the S-CaL-W mix while still producing a larger peak shear stress at failure as compared to dry sand, is given as follows:

$${\sigma}_{l}=\frac{{c}^{\prime}}{\mathrm{tan}{{\phi}^{\prime}}_{{{\rm X}}_{l}=0\%}\text{-}\mathrm{tan}{\phi}^{\prime}}\text{\hspace{1em}(18)}$$(18)

For normal stresses larger than the limiting normal stress, the S-CaL-W mix is superseded in ranking of peak shear stress by the dry sand. The limiting normal stress provided by each percent of CaL content decreases with the increasing CaL content (Figure 8). It remains the largest in configuration B, followed by C, A, E, and D. Configuration C is positioned high because of the lowest value of the friction angle in dry sand at relative compaction of 90%. In addition, the range of normalized limiting stress for a given *χ*_{l} decreases with increasing *χ*_{l}.

Figure 8 Normalized limiting stress versus gravimetric lignin content.

Figure 9 shows an increase in a normalized cohesion with an increase in the normalized area ratio. All sample configurations except A, C, D, and E at 2% of CaL appear to follow a unique trend.

Figure 9 Normalized cohesion versus normalized area ratio (all points).

Figure 10 shows an increase in the normalized cohesion with an increase in the normalized area ratio for a smaller number of sample configurations than those included in Figure 9.

Figure 10 Normalized cohesion versus normalized area ratio (selected points).

Sample configurations in Figure 10 were selected by excluding 2A, 2C, 2D, 2E, 9B, and all configurations at *χ*_{l}=14%. This improved the regression coefficient significantly (from 0.47 to 0.89). Furthermore, the depicted graph shows that the peak experimentally obtained value of a normalized cohesion is about 4 (kPa/%). To provide a rationale for excluding the aforementioned sample configurations from Figure 10, a mass of CaL-water mix is divided by the total mass of the mix. This provides the mass of the bonding agent as a percentage of the total mass of the mix. It is because the bonding agent consists of extremely small CaL particles that the percentage of CaL-W paste resembles the percentage of fines contained in the coarse-grained soils, which is evaluated during the soil classification procedure. The results are shown in .

Table 2 Ratio of a mass of S-CaL-W paste and a total mass of the S-CaL-W mix.

They indicate that there is simply not enough of CaL-W paste at very low water and gravimetric CaL contents corresponding to sample configurations 2A, 2C, 2D, and 2E because these configurations result in the percentage of fines smaller than 5%. This is based on the fact that sands containing <5% of fines are named clean sands according to the Unified Soil Classification System (USCS) [15]. On the other end of the spectrum sands with more than 12% fines contain a significant amount of fines, thus losing the gradation characteristics in their name and ending up only as simply clayey and/or silty sands. In analogy with this USCS [15] procedure, the sample configurations that contain more than 12% fines are also excluded in Figure 10 as they are likely to form a different material. Furthermore, the results depicted in Figure 9 clearly indicate that there is an overabundance of a bonding agent for sample configurations containing more than 12% fines. Thus, the increase in cohesion becomes not only significantly smaller but also the difference in normalized cohesion values among the sample configurations A, B, C, D, and E becomes significantly smaller at *χ*_{l}=14%. This also indicates that the effect of void ratio and water content becomes less significant at 14% of CaL. In addition, Figure 9 shows that sample configurations 2A, 2B, 2C, and 2D are all located away from all remaining sample configurations. In summary, the results shown in Figure 10 and indicate that the mixes containing anywhere from 4% to 9% of CaL (including 2B and excluding 9B) exhibit a common behavior, which differs from the remaining samples (Bold values in ). This behavior corresponds to the range of an effective contribution of the CaL-water paste. Specifically, the lower bound of the range indicates that a sufficient amount of CaL-water is necessary, and the upper bound indicates that too much of CaL-water paste begins to weaken the S-CaL-W mix. The latter probably happens because of the lower strength of CaL-W mix as compared to the sand skeleton and also because of the possibility that sand particles might be pulled away from each other due to the presence of too much CaL-water paste. This can also be observed in Figure 11, which shows optical microscope scans of S-CaL-W mixes at 0, 6, and 14% of CaL. The balance of CaL, water, and sand appears to be nearly optimal in Figure 11B because CaL-W paste appears to coat the sand particles evenly without the presence of too much paste. On the other hand, at 14% of CaL, which is depicted in Figure 11C, the overabundance of lignin is evident, thus pushing the sand particles away from each other.

Figure 11 Magnified image of (A) *χ*_{l}=0% (dry sand), (B) S-CaL-W mix for *χ*_{l}=6%, and (C) S-CaL-W mix for *χ*_{l}=14%.

Figure 10 can serve as a design chart, which is based on the experimental data from this study. Specifically, by first selecting the water content, void ratio, and CaL content the material design curve depicted in Figure 10 provides the normalized cohesion, which can be obtained for the selected parameters.

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