Enhancing Geotechnical Engineering Education Through Case-Based Innovation: A Predictive Modeling Framework for Cemented Sand in Strength Theory Teaching

Abstract

The shear strength–suction (here suction induced from unsaturation) relationship is inherently challenging, yet this difficulty is compounded for cemented crushable sands, whose behavior fundamentally diverges from classical clay-centric theories. This paper presents an innovative teaching case study focusing on colloidal-silica-cemented calcareous sand, based on direct shear tests across a full saturation range (0–100%). Our experimental findings reveal two unconventional characteristics that challenge textbook models: (1) suction strength exhibits a positive dependency on normal stress—an inverse trend to conventional expectations; and (2) strength near desiccation drops below saturated values, contradicting the monotonic increasing function typically observed in clays. A review of 20 existing models confirms that none of them can simultaneously capture both features, highlighting a clear gap in both theory and instruction. To address this gap pedagogically, the core novelty of this work lies in the development of a classroom-friendly predictive model that introduces two physical innovations: first, it incorporates normal-stress-dependent suction strength by modifying capillary condensation probability—departing from constant-angle assumptions; second, it accounts for desiccation-induced strength deterioration through a gel crack size effect, which is absent in conventional unsaturated strength formulations. The model retains clear physical interpretability and demonstrates strong agreement with experimental data. By integrating unconventional behavior, model limitations, and novel physically inspired formulations into a coherent case study, this work equips students not only to recognize deviations from classical unsaturated strength theory but also to construct their own mechanistic models in geotechnical engineering education.

1. Introduction: Why This Teaching Case Study Was Chosen

In unsaturated soil mechanics courses, the relationship between suction and shear strength is a core topic. Classical theory typically presents a monotonically increasing curve, i.e., the higher the suction, the higher the strength. This conclusion is documented in several mainstream textbooks and can easily lead students to develop a fixed mindset. However, for cemented materials, the drying process may induce gel shrinkage and micro-crack development, thereby causing strength deterioration and exhibiting a counterintuitive phenomenon of “weaker when drier.” This counterexample serves as an excellent teaching material to guide students toward a deeper understanding of the microscopic mechanisms of unsaturated strength.

Colloidal silica can be used to stabilize marine calcareous sand, thereby improving the stability of calcareous sand beneath foundations. Due to its low viscosity, colloidal silica can seep through sand over long distances and then turn into silica gel for cementing sand. A notable engineering application is the coastal Fukuoka International Airport, where colloidal silica was injected beneath the runway to stabilize the sand [1]. Extensive studies have been performed on the mechanical properties of colloidal-silica-cemented sand [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. However, the existing literature focuses primarily on the saturated state of colloidal-silica-cemented sand, with very few studies on the unsaturated state. Previous studies analyzed the strength characteristics of cemented soils [22]. To date, only [23] have analyzed the permeability of unsaturated colloidal-silica-cemented sand, while its unsaturated shear strength remains unexplored. In practical engineering, colloidal silica-cemented calcareous sands can be applied in coastal reclamation projects, subgrade construction and shore protection works. These exposed fill materials are frequently subjected to complex natural environments, including seasonal drought, intense solar evaporation, fluctuating groundwater tables, and tidal variations. Such conditions easily cause water loss and desiccation inside the treated sand mass, leading to continuous changes in matric suction. Accordingly, investigating the relationship between suction and unsaturated strength is of great necessity for the long-term stability evaluation of colloidal silica-cemented calcareous sands.

For different types of unsaturated soils, extensive experimental studies have been conducted on stress–strain characteristics [24,25,26,27,28,29,30,31,32,33,34,35], and various formulas for unsaturated strength have been proposed [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54]. Therefore, it is necessary to examine whether these existing models can be used to predict the shear strength of unsaturated colloidal-silica-cemented calcareous sand.

The teaching objective of this paper is to help students break their fixed mindset regarding the monotonic increase of unsaturated strength through a real counterintuitive experimental case study and to guide them in understanding how to develop physically inspired strength models from microscopic mechanisms. To this end, direct shear tests were performed on unsaturated colloidal-silica-cemented calcareous sand with degrees of saturation ranging from 0% to 100%. The experimental results reveal two characteristics: (1) the suction-induced component of shear strength is normal-stress-dependent; and (2) a new type of strength–suction relationship is observed, in which the strength near desiccation is lower than that at the saturated state. A review of 20 existing models shows that none of them can simulate the above two characteristics. Therefore, this paper establishes a new model capable of reflecting both characteristics: on the one hand, by modifying the capillary condensation probability, the effect of normal stress on the suction-induced strength component is incorporated; on the other hand, by introducing the crack size effect, the micro-mechanism of shrinkage-induced crack deterioration is accounted for, thereby enabling the prediction of this new type of strength–suction relationship. The following sections are organized according to the pedagogical logic of “experimental observations → model limitations → new model development → classroom application.”

2. Teaching-Oriented Experimental Design

2.1. Materials

The calcareous sand used in this teaching case was sourced from the Philippines, and its particle size was controlled within the range of 0.25–0.5 mm after sieving. Figure 1 shows a photograph of the sand sample and its gradation curve, and

Table 1. Properties of calcareous sand.

Specific Gravity Gs	Max. Void Ratio edmax	Min. Void Ratio edmin	Particle Size d (mm)
2.65	1.15	0.81	0.25–0.5

lists the basic physical properties of the sand. Instructors can guide students to calculate fundamental indices such as the relative density of this sand during classroom exercises. The dominant mineral component is aragonite, followed by magnesian calcite with a lower content. Both are typical minerals of marine calcareous sand, as shown in Figure 1c.

Colloidal silica was purchased from Qingdao Ronghai Jute Fine Chemical Co., Ltd. (Qingdao, Shandong, China). This material is a liquid with fluidity close to that of water. In colloidal silica, nano-silica particles exhibit a stable suspension due to repulsive forces between particles. Such repulsive forces result from the alkaline environment created by the addition of sodium hydroxide. When an acid or salt is added [2,4,5,7,9,11,12,13,15,16,17,19,20,21], the repulsive forces disappear, and the nano-silica particles connect to form a solid silica gel, which can cement sand particles. The concentration of colloidal silica used in this teaching case is 30%, the pH value ranges from 8.5 to 10.0, and the particle size of the nano-silica particles is 10–20 nm. Classroom demonstration experiment: after adding a sodium chloride solution (e.g., 1% NaCl) to the colloidal silica, the fluidity of the material decreases significantly after 3 h (see Figure 2a), and it transforms into a solid gel after 24 h (see Figure 2b). This transformation process is highly suitable for a live demonstration in the classroom, helping students intuitively understand the formation mechanism of cementitious material.

2.2. Specimen Preparation Method for Teaching (Operational Steps for Student Laboratory Work)

The specimen size of the colloidal-silica-cemented sand is 61.8 mm in diameter × 20 mm in height. The specimen preparation steps are as follows, which instructors can use as operational guidelines for laboratory sessions:

Step 1: Mix 10% NaCl solution and 30% colloidal silica together in a ratio of 1:9, and pour this mixture into the mold.

Step 2: Use the pluviation method [55] to deposit the sand into the mold.

Step 3: Wrap the entire mold with plastic film to prevent moisture evaporation, and place the mold in a curing box for 3 days to obtain saturated specimens of colloidal-silica-cemented sand.

Figure 3 shows the stabilized specimen after curing, and Figure 4 shows the specimen after shearing.

For unsaturated specimen preparation, the saturated specimens are placed in an oven for different heating durations to control the degree of saturation and then wrapped in plastic film for 48 h to ensure uniformity of moisture distribution. The specimens in this study are only 20 mm thick, which allows uniform heat distribution during drying. Meanwhile, the oven temperature was set at 50 degrees Celsius, a moderate value consistent with the actual ambient temperature of calcareous sand in tropical island environments. Under such conditions, severe moisture gradients and premature microcracks caused by thermal shock can be effectively avoided. In addition, the oven drying method enables precise control of water content at designated levels (95%, 85%, 65%, 50%, 40%, 30%, 25%, and 15%). By contrast, the conventional suction equilibration via salt solution can cover this full range of water contents, but it fails to strictly match our pre-defined saturation values. Therefore, we adopted the accelerated drying approach for specimen preparation. Ref. [56] sealed the sample in a bag for a 48 h period to allow for suction equilibration. The underlying principle is that moisture migrates spontaneously from regions of higher water content to regions of lower water content within the closed system, driven by the water potential gradient, ultimately achieving uniform distribution. Therefore, this study followed this well-established protocol for reaching a uniform moisture distribution after the 48 h wrapping period. These unsaturated specimens in Figure 3 are used to determine the shear strength at different degrees of saturation.

2.3. Suction Measurement Method (Teaching Demonstration of the Filter Paper Method)

The filter paper method was used to determine the soil–water characteristic curve (SWCC). Ref. [57] demonstrated that the filter paper method can reliably measure suctions above 1000 kPa. For suctions up to approximately 100,000 kPa, the method remains applicable. In this study, the matric suction under the fully dried state (oven dryness) was not measured directly using the filter paper method. Instead, it was assigned a value of 10⁶ kPa based on established theoretical findings in the literature. Ref. [58] demonstrated, through thermodynamic considerations, that oven dryness corresponds to a matric suction of 10⁶ kPa. This theoretical value has been widely adopted in subsequent studies. For example, refs. [27,37] used a matric suction of approximately 10⁶ kPa to represent oven-dry conditions in their analyses of unsaturated soil behavior. Ref. [57] also confirmed that a matric suction approaching 10⁶ kPa is thermodynamically sound and consistent with experimental observations. Therefore, although the filter paper method in this study was not employed to measure suctions in the fully dried state, the assigned value of 10⁶ kPa is well supported by existing thermodynamic theory and established practice in unsaturated soil mechanics. The specimen of colloidal-silica-cemented sand and the filter paper were sealed together in an aluminum can. When the specimen and the filter paper reached moisture equilibrium, the water content of the filter paper was measured to determine the matric suction. In this study, only matric suction was measured and discussed, while osmotic suction was not taken into account throughout the tests and data analysis. Our research focuses on the matric suction characteristics of colloidal silica-cemented calcareous sand. Accordingly, we adopted the original calibration equations provided by the manufacturer for Shuangquan filter paper. In this teaching case, Shuangquan No. 203 filter paper (Cytiva, Hangzhou, China) was used, and the following two equations provided by the manufacturer were used to relate water content to matric suction:

$$\log \psi =5.493-0.0767\omega , \omega \le 47\%$$

(1)

$$\log \psi =2.470-0.0120\omega , \omega >47\%$$

(2)

where ψ is the suction, and ω is the water content of the filter paper. Instructors can guide students through this measurement process to understand the fundamental principles of suction measurement in unsaturated soils.

2.4. Direct Shear Test Plan

The direct shear apparatus was provided by Nanjing Ningxi Soil Instrument Factory (Nanjing, China), and a shearing rate of 0.8 mm/min was used. Direct shear tests were performed on colloidal-silica-cemented sand under four different vertical stresses (100 kPa, 200 kPa, 300 kPa, and 400 kPa), with degrees of saturation ranging from 0% to 100%. The detailed test plan is shown in

Table 2. Plan of direct shear tests (for student group work reference).

Material	Degree of Saturation, S (%)	Normal Stress, σv (kPa)
colloidal-silica-cemented sand	100, 95, 85, 65, 50, 40, 30, 25, 15, 0	100, 200, 300, and 400

. Instructors can use this as a design template for student group experiments.

2.5. Experimental Results and Their Pedagogical Interpretation

2.5.1. Teaching Analysis of the Soil–Water Characteristic Curve (SWCC)

Figure 5 shows the soil–water characteristic curve (SWCC) of colloidal-silica-cemented sand. Several fitting formulas for the SWCC have been developed in the literature, among which the [59] formula is now widely used. Furthermore, based on microscale mechanisms, different types of SWCC models have been developed by distinguishing between the adsorption and capillary components of saturation [50,60,61,62,63,64]. Figure 6 shows a schematic view of obtaining the air-entry value ψ_a and the residual suction value ψ_r from the SWCC. For the colloidal-silica-cemented sand in this teaching case, ψ_a = 31 kPa and ψ_r = 691 kPa, and the degree of saturation corresponding to ψ_r is S_r = 18.6%. Instructors can guide students to read these key parameters from the SWCC curve and discuss their influence on unsaturated strength.

2.5.2. Two Major Teaching Highlights of Strength Characteristics

Figure 7 shows the curves of shear stress versus horizontal displacement at different degrees of saturation and under different vertical stresses. It can be observed that as the vertical stress increases, the shear stress increases; as the degree of saturation decreases, the shear stress first increases and then decreases. Figure 7 also shows the vertical displacement behavior: when the degree of saturation S > 0%, the specimens first contract and then dilate; however, when S = 0%, only contraction is observed. Instructors can guide students to discuss the mechanisms underlying this phenomenon.

For the core teaching discovery, Figure 8a shows the curves of shear strength versus degree of saturation, and Figure 8b shows the curves of shear strength versus suction. As the degree of saturation decreases (Figure 8a) or as suction increases (Figure 8b), the shear strength first increases and then decreases. It is particularly noteworthy that at degrees of saturation of 15% and 0%, the shear strength is lower than that at the saturated state, a phenomenon that has not been reported in previous literature. In other words, assuming Δτ = shear strength near desiccation minus saturated strength, Δτ is always greater than 0 for unsaturated soils in the previous literature; however, for colloidal-silica-cemented sand, Δτ < 0 at degrees of saturation of 15% and 0%. The underlying mechanism is the shrinkage and cracking of the silica gel between sand particles near desiccation, as discussed in Section 4.2. This counterintuitive phenomenon represents the core value of this teaching case. Instructors can set up a classroom discussion question here: Why does drying lead to a reduction in strength?

Suction strength is defined as the increment in shear strength from the saturated state to the unsaturated state. Figure 9 shows that the experimentally measured suction strength is normal-stress-dependent, meaning that the suction strength increases with increasing normal stress. This characteristic has not been fully captured by traditional unsaturated strength theories.

For unsaturated soils, ref. [53] summarized three types of shear strength versus suction curves for different soil types (see Figure 10). All of these curves show that the unsaturated strength is always greater than the saturated strength (i.e., Δτ > 0, see Figure 10). However, the colloidal-silica-cemented sand in this teaching case exhibits a fourth distinct type of shear strength versus suction (or degree of saturation) relationship. The next section attempts to use existing unsaturated models to explain this phenomenon and reveal the limitations of these models, thereby leading to the necessity of developing a new model.

3. Simulation by Previous Models: A Classroom Critical Evaluation

Ref. [65] provided a general expression for the strength of unsaturated soils, which is widely used in textbooks and classroom instruction, as shown in Equation (2):

$$\tau _f=c^{\prime }+\left({\sigma }_v-u_a\right)\tan {\varphi }^{\prime }+\left(u_a-u_w\right)\tan {\varphi }^b$$

(3)

where τ_f is the shear strength, c′ is the effective cohesion at saturation, σ_v is the normal stress, u_a is the pore air pressure, u_w is the pore water pressure, ϕ′ is the effective friction angle at saturation, and ϕ^b is the angle of shear strength with respect to suction.

The matric suction ψ and suction strength τ_us are shown in Equations (3) and (4), respectively:

$$\psi =u_a-u_w$$

(4)

$$\tau _{us}=\psi \cdot \tan {\varphi }^b$$

(5)

Substituting Equations (3) and (4) into Equation (2) yields the following:

$$\tau _f=c^{\prime }+\left({\sigma }_v-u_a\right)\tan {\varphi }^{\prime }+\tau _{us}$$

(6)

where c′ + (σ_n − u_a)tan ϕ′ denotes the saturated component of strength, and τ_us denotes the suction strength.

3.1. Summary of Existing Suction Strength Formulas (For Classroom Reference)

We summarized 20 formulas of suction strength τ_us from the existing literature [36,37,38,39,40,41,42,43,44,45,47,48,49,50,52,53,54], as presented in

Table 3. Existing models for predicting shear strength of unsaturated soil.

Type of the Equation	Authors	Shear Strength with Respect to Suction (τus)	Parameters
Increase monotonically	1. Khalili & Khabbaz (1998) [39]	$\tau _{us}=\psi {\left(\psi /{\psi }_a\right)}^{-0.55}\tan {\varphi }^{\prime }$
	2. Tekinsoy et al. (2004) [42]	$\tau _{us}=\left({\psi }_a+p_{at}\right)\mathrm{In}\left(\psi /{\psi }_a+1\right)\tan {\varphi }^{\prime }$
	3. Xu (2004) [43]	$\tau _{us}={\psi }_a^{1-\varsigma }{\psi }^{\varsigma }\tan {\varphi }^{\prime }$	0 ≤ ζ ≤ 1
	4. Gao et al. (2020) Equation (I) [53]	$\tau _{us}=\psi {\left[\psi /\left(\eta \times {\psi }_a\right)\right]}^{-0.55}\tan {\varphi }^{\prime }$	η
	5. Gao et al. (2020) Equation (II) [53]	$\tau _{us}=\left(\eta {\psi }_a+p_{at}\right)\mathrm{In}\left(\psi /{\psi }_a+1\right)\tan {\varphi }^{\prime }$	η
First increase then decrease	6. Vanapalli et al. (1996) Equation (I) [36]	$\tau _{us}=\psi \cdot S^{\kappa }\tan {\varphi }^{\prime }$	κ
	7. Vanapalli et al. (1996) Equation (II) [36]	$\tau _{us}=\psi \left(S-S_r\right)/\left(1-S_r\right)\tan {\varphi }^{\prime }$
	8. Oberg & Sallfors (1997) [38]	$\tau _{us}=\psi \cdot S\tan {\varphi }^{\prime }$
	9. Fredlund et al. (1996) [37]	$\tau _{us}=\psi {\left({\displaystyle \frac{S-S_r}{1-S_r}}\psi \right)}^{\kappa }\tan {\varphi }^{\prime }$	κ
	10. Sun et al. (2000) [40]	$\tau _{us}={\displaystyle \frac{\psi }{1+\psi /a}}\tan {\varphi }^{\prime }$	a
	11. Miao et al. (2002) [41]	$\tau _{us}={\displaystyle \frac{a\psi }{1+\left(1-a\right)\psi /p_{at}}}$	a
	12. Vilar (2006) [44]	$\tau _{us}={\displaystyle \frac{\psi }{a+b\psi }}$	a, b
	13. Houston et al. (2008) [45]	$\tau _{us}=\psi \cdot \tan \left({\varphi }^{\prime }-{\displaystyle \frac{\psi -{\psi }_a}{a+b\left(\psi -{\psi }_a\right)}}\right)$	a, b
	14. Guan et al. (2010) [47]	$\tau _{us}=\left\{\begin{array}{l}\psi \tan {\varphi }^{\prime },\psi <{\psi }_a\\ \left[{\psi }_a+b\left(\psi -{\psi }_a\right)S^{\kappa }\right]\tan {\varphi }^{\prime },\psi \ge {\psi }_a\end{array}\right.$ where $\kappa ={\left[\mathrm{In}\left(\psi \right)-\mathrm{In}\left({\psi }_a\right)\right]}^y$	b, y
	15. Hossain & Yin (2010) [48]	$\begin{array}{ll}\hfill \tau _{us}=& \left({\sigma }_n-u_a\right)\tan \left({\varphi }^{\prime }+{\psi }_D\right)+\psi \cdot S^{\kappa }\tan \left({\varphi }^{\prime }+{\psi }_D\right)\hfill \\ & -\left({\sigma }_n-u_a\right)\tan \left({\varphi }^{\prime }\right)\hfill \end{array}$ where ψD is the dilation angle	κ
	16. Konrad & Lebeau (2015) [49]	$\tau _{us}=\psi \cdot S_{r,c}\tan \varphi$ where $S_{r,c}=\left\{\begin{array}{ll}1\hfill & \psi <{\psi }_a\hfill \\ 1-{\left(1-e^{-\wedge \psi }\right)}^{\alpha }\hfill & \psi >{\psi }_a\hfill \end{array}\right.$	∧, α
	17. Zhou et al. (2016) [50]	$\tau _{us}\psi S^{cap}\tan {\varphi }^{\prime }$ where $S^{cap}={\displaystyle \frac{C\left(\psi \right)-\alpha C\left(\psi \right)A\left(\psi \right)}{1-\alpha C\left(\psi \right)A\left(\psi \right)}}$ $A\left(\psi \right)=1-\mathrm{In}\left(\psi \right)/\left(1-\mathrm{In}{\psi }_d\right)$ $C\left(\psi \right)=0.5\times \mathrm{erfc}\left(\mathrm{In}\left(\psi /{\psi }_m\right)/\left(\sqrt{2}\zeta \right)\right)$ψd = 106 kPa	α, ζ, ψm
	18. Zhai et al. (2019) [52]	$\tau _{us}=\psi \cdot S\tan {\varphi }^{\prime }+c_s$ where cs can be obtained from the SWCC curve and micro contact angle α
	19. Gao et al. (2020) Equation (III) [53]	$\begin{array}{l}\tau _{us}=\psi S\tan {\varphi }^{\prime },\psi \le {\psi }_{\max }\\ \tau ={\tau }_{\max }+\left(\psi S-{\psi }_{\max }{S}_{\max }\right)\tan {\varphi }^{\prime },\psi >{\psi }_{\max }\end{array}$ where ‘max’ = maximum skeleton stress
	20. Pham & Sutman (2023) [54]	$\tau _{us}=\psi \left[\left(1-S_{e}n\right)S_e\right]\tan {\varphi }^{\prime }$, where n = porosity and $S_e=\left(S^{2/3}-S_r^{2/3}\right)/\left(1-S_r^{2/3}\right)$

Note: ψ = suction, ψ_a = air-entry value, ψ_r = residual suction, S = degree of saturation, S_r = degree of saturation corresponding to residual degree of saturation, P_at = atmospheric pressure.

for classroom discussion.

3.2. Classroom Discussion: Limitations of Existing Models

It should be noted that all formulas in Table 3 do not incorporate the effect of normal stress on the suction-induced component of shear strength. Therefore, they cannot simulate the normal-stress-dependent characteristic of suction strength τ_us (e.g., see Figure 9 in Section 2.5.2). This is a critical observation that students should recognize: most classical models treat suction strength as being independent of vertical stress.

The formulas of suction strength τ_us can be divided into two categories based on their trend with increasing suction (see Table 3) as follows.

Category 1: Suction strength τ_us increases monotonically with matric suction.

Category 2: Suction strength τ_us increases first and then decreases with increasing suction.

Table 4. Common parameters of colloidal-silica-cemented sand.

Previous Models (Can Increase First Then Decrease)	Values of Parameters
Air-entry suction, ψa (kPa)	31
Residual suction, ψr (kPa)	691
Degree of saturation at residual suction, Sr (%)	18.6
Effective friction angle at saturation, ϕ′ (degrees)	50.97
Effective cohesion at saturation, c′ (kPa)	11.2

lists the common parameters of existing models, while several specific parameters of the models are presented in

Table 5. Parameters of previous models.

Previous Models (Can Increase First Then Decrease)	Values of Parameters
Vanapalli et al. (1996) Equation (I) [36]	κ = 3.71
Sun et al. (2000) [40]	a = 18.8
Miao et al. (2002) [41]	a = 0.19
Vilar (2006) [44]	a = 0.016, b = 0.45
Houston et al. (2008) [45]	a = 29.5, b = 1.12
Guan et al. (2010) [47]	b = −0.42, y = 4.44
Hossain & Yin (2010) [48]	κ = 4.4 × 104
Konrad & Lebeau (2015) [49]	∧ = 1.41, α = 1.0
Zhou et al. (2016) [50]	α = 0.386, ζ = 1.03, ψm = 157

.

Since the suction strength of colloidal-silica-cemented sand increases first and then decreases with increasing suction (as shown in Figure 8b), the five formulas in Category 1 that exhibit monotonic increase (i.e., Khalili & Khabbaz 1998 [39]; Tekinsoy et al. 2004 [42]; Xu 2004 [43]; Gao et al. 2020 [53] Equation (I); Gao et al. 2020 Equation [53] (II)) are not considered suitable for simulation here and are thus excluded from the following comparison.

Furthermore, for colloidal-silica-cemented sand, the term (S − Sr)^κ in the formula by [37] yields an imaginary number when S < Sr, making it mathematically invalid. The remaining model [48] was excluded because the calculated shear strength from its equation yields negative values, which are physically meaningless. Therefore, this model is not plotted in the figure. Ref. [52] model requires statistical analysis of micro contact angles in pores, which is beyond the scope of typical classroom exercises. Therefore, in Figure 11, the remaining 12 formulas are plotted for comparison.

3.3. Teaching Analysis: Comparison of Model Simulations

As shown in Figure 11, all models provide reasonably good simulations in the low suction stage (ψ < 100 kPa). This indicates that the fundamental mechanisms of suction strength are well captured by existing theories at low suction ranges. However, as the matric suction increases beyond 100 kPa, the errors become progressively larger, and the following four distinct types of deviation can be observed.

(1)

Underestimation in the high suction stage: Simulations from formulas by [36,40,41,44,47,49] tend to be much smaller than the experimental data in the high suction stage. These models fail to capture the sustained strength contribution at high suctions.

(2)

Unrealistic negative strength predictions: Some models predict that unsaturated strength can be lower than saturated strength, but the strength quickly becomes negative as suction increases, which is physically unrealistic. This occurs in [36,45,54] Vanapalli et al. (1996) Equation (II), Houston et al. (2008), and Pham & Sutman (2023). Students should recognize that negative strength is meaningless in this context.

(3)

Overestimation in the high suction stage: Formulas by [38,53] produce strength values that rise too steeply in the high suction stage. A sub-window in Figure 11 is used to show these deviations clearly.

(4)

Closest but still inadequate: The simulation by [50] is the closest to the experimental data among all models. However, when the degree of saturation approaches zero (complete desiccation), the predicted shear strength is not lower than that at the saturated state. This is inconsistent with the experimental data shown in Figure 8a where Δτ < 0 at S = 15% and 0%.

3.4. Classroom Summary: The Need for a New Model

Let Δτ = strength near desiccation minus saturated strength (see Figure 11). For all 20 existing models reviewed, Δτ ≥ 0, meaning they cannot predict the phenomenon where strength near desiccation falls below the saturated strength. Therefore, in the next section, we aim to modify the [50] model—which provides the closest overall fit—to simulate the case when Δτ < 0, while also incorporating the normal-stress-dependent characteristic of suction strength that is missing from all existing models. This step-by-step modification process is presented as a classroom exercise in physically inspired model development.

4. Model Establishment: A Step-by-Step Classroom Development

In this section, we guide students through two necessary improvements to the [50] model. First, we identify the limitations of the existing model by comparing its predictions with the experimental data. Then, we modify the capillary condensation variable and introduce the crack size effect, enabling the model to capture the unsaturated shear strength behavior of colloidal-silica-cemented sand. This step-by-step approach helps students to understand how physical mechanisms translate into mathematical formulations.

Figure 12 presents a side-by-side comparison of the [50] model simulations with experimental data. This figure serves as a core teaching tool to help students visualize two critical discrepancies as follows.

(1)

Missing normal-stress dependence: The experimental data clearly show that suction strength τ_u increases with increasing normal stress σ_v (i.e., τ_u is normal-stress-dependent). However, as shown in Figure 12, the simulated τ_u from the [50] model does not change with σ_v. This indicates that the model fails to capture the coupling between suction strength and confining stress.

(2)

Inability to predict strength reversal near desiccation: The experimental data show that Δτ < 0 (where Δτ = strength near desiccation minus saturated strength) at degrees of saturation of 15% and 0%. In other words, the sand becomes weaker when completely dry than when saturated. However, the simulated Δτ from the [50] model remains ≥0 (see Figure 12), meaning it cannot predict this strength reversal.

The above two aspects can be improved from the perspectives of capillary condensation variable and crack size effect, as shown in the following sections.

4.1. Improvement 1: From Physical Mechanism (Capillary Condensation) to Model Modifications

There are two types of water between solid particles [66], as shown in Figure 13: (1) one is the adsorbed water that adheres closely to particles—this adsorbed water is very thin and does not affect matric suction; (2) the other is capillary water, which is capable of generating matric suction. But it should be noted that if the capillary water on two particles is not joined together, as shown in Figure 13a, pore pressure and water surface tension cannot work together to generate suction. Only when the capillary water on two particles is connected, does suction exist (see Figure 13b). So, there is a probability variable P_cc (0 ≤ P_cc ≤ 1) introduced by [50]. This P_cc represents the proportion of capillary water which bridges two particles together to form suction. This P_cc can be considered as the capillary condensation probability. For the physical mechanism for the positive correlation between suction resistance and normal stress, with the increase of normal stress, the sand particles are pressed more tightly together, which notably enlarges the effective contact area between adjacent particles. Meanwhile, the geometry of the intergranular pores and water menisci is altered accordingly. Such structural change further affects the probability of capillary condensation within the pore menisci. Consequently, the overall suction resistance rises with increasing normal stress. From the microstructural perspective, the above mechanism accounts well for the increased probability of capillary condensation.

Let S be the degree of saturation, S^cap be the capillary part of S, and let S^ads be the adsorbed part of S—[50] proposed Equation (6) based on the work of [67,68]; they also used Equation (7) from [69] and proposed Equation (8).

$$S=S^{cap}+S^{ads}$$

(7)

$$S^{cap}=\left(1-S^{ads}\right)C\left(\psi \right)$$

(8)

$$S^{ads}=\alpha A\left(\psi \right)\left(1-P_{cc}\right)$$

(9)

where

$$A\left(\psi \right)=1-\mathrm{In}\left(\psi \right)/\left(1-\mathrm{In}{\psi }_d\right)$$

(10)

$$C\left(\psi \right)={\displaystyle \frac{1}{2}}\mathrm{erfc}\left[{\displaystyle \frac{\mathrm{In}\left(\psi /{\psi }_m\right)}{\sqrt{2}\zeta }}\right]$$

(11)

And where α, ζ, ψ_m are three parameters, ψ_d = 10⁶ kPa, erfc() denotes the complementary error function.

In Equations (6)–(8), the three equations have four unknown variables, i.e., S, S^cap, S^ads, and P_cc. It should be noted that the four unknown variables cannot be solved based on the three equations. For simplicity, ref. [50] proposed that P_cc = S^cap, then using Equations (6)–(8), S, S^cap, and S^ads can be solved and plotted in Figure 14. The expression of S^cap in the [50] model is shown in Equation (11). Then this S^cap is used in Equation (12) to calculate the suction strength:

$$S^{cap}={\displaystyle \frac{C\left(\psi \right)-\alpha C\left(\psi \right)A\left(\psi \right)}{1-\alpha C\left(\psi \right)A\left(\psi \right)}}$$

(12)

$$\tau _{us}=\psi S^{cap}\tan {\varphi }^{\prime }$$

(13)

From the above process, it can be seen that the variable P_cc of capillary condensation probability can be modified to improve the simulation of suction strength τ_us. That is, P_cc affects the capillary part of saturation, and this capillary part of saturation affects the suction strength τ_us.

From the perspective of a physical mechanism, the distance between two particles may become smaller with increasing normal stress, which leads to a greater probability that the capillary water on the two particles bridge together to form suction. Therefore, there exists a physical mechanism that the capillary condensation probability, P_cc, can be normal-stress-dependent. But it is difficult to determine P_cc directly from a microscopic statistical perspective. So, we resort to the intermediate variable, the capillary part of saturation, to give the form of P_cc. Inspired by [36] Equation (I) in Table 3, using S to the power of κ can enhance the strength, so here we introduce a variable κ* to modify the capillary part of saturation. Letting the modified capillary part of saturation be S^cap*, we propose Equation (13) to express S^cap*:

$$S^{cap*}={\left(S^{cap}\right)}^{\kappa *}$$

(14)

Substituting S^cap with S^cap* in Equation (7), combined with Equations (8), (11) and (13), P_cc can be expressed by Equation (14):

$$P_{cc}=1-{\displaystyle \frac{1}{\alpha A\left(\psi \right)}}\left[1-{\displaystyle \frac{1}{C\left(\psi \right)}}{\left({\displaystyle \frac{C\left(\psi \right)-\alpha C\left(\psi \right)A\left(\psi \right)}{1-\alpha C\left(\psi \right)A\left(\psi \right)}}\right)}^{\kappa *}\right]$$

(15)

Here, κ* is a normal-stress-dependent variable, which leads to the normal-stress-dependent characteristic of P_cc in Equation (14). Substituting S^cap with S^cap* in Equation (12) yields the modified suction strength τ_us, as shown in Equation (15):

$$\tau _{us}=\psi S^{cap*}\tan {\varphi }^{\prime }=\psi {\left(S^{cap}\right)}^{\kappa *}\tan {\varphi }^{\prime }$$

(16)

Based on Equation (15) and test data, the form of κ* can be proposed, as shown in Equation (16) and Figure 15:

$${\kappa }^{*}={\left({\displaystyle \frac{1}{1+{\sigma }_v/P_{at}}}\right)}^{\beta }$$

(17)

where β is a parameter, σ_v is the normal stress, and P_at is the atmospheric pressure. Here σ_v in Equation (16) makes κ* normal-stress-dependent.

Based on Equations (15) and (16), the prediction of suction strength τ_us is plotted in Figure 16 and it increases with increasing normal stress σ_v, i.e., τ_us is normal-stress-dependent, which is consistent with the test data. The prediction of suction strength τ_us by the [50] model is also plotted in Figure 16, but it does not change with normal stress σ_v. So, the modification by this paper can reflect the normal-stress-dependent characteristic of suction strength τ_us.

Substituting Equation (15) into Equation (5) yields a formula for the shear strength τ_f, as shown in Equation (17):

$${\tau }_f=c^{\prime }+\left({\sigma }_v-u_a\right)\tan {\varphi }^{\prime }+\psi {\left(S^{cap}\right)}^{\kappa *}\tan {\varphi }^{\prime }$$

(18)

Based on Equation (17), the shear strength τ_f is simulated, as shown in Figure 17. When β = 0, Equation (17) is equivalent to the [50] model. Figure 17 shows that, compared to β = 0 ([50] model), β = 0.4 yields a better prediction of the peak value of shear strength τ_f. However, when approaching desiccation (e.g., the degree of saturation S = 15% or 0%), it still cannot simulate Δτ < 0, where Δτ = strength near desiccation minus saturated strength (see Figure 17). So, Equation (17) for the shear strength still needs to be improved.

4.2. Improvement 2: From Physical Mechanism (Shrink and Crack) to Model Modifications

In this section, based on the experimental phenomenon of silica gel cracks between sand particles, the size effect in fracture mechanics is introduced to improve the formula of shear strength.

Figure 18 shows the severe shrinkage of pure silica gel due to desiccation, and this shrinkage also results in fragmentation. Figure 19 shows the cracks of silica gel between particles due to desiccation. So, a schematic view of the cracks in colloidal-silica-cemented sand is plotted in Figure 20. The saturated state of colloidal-silica-cemented sand is shown in Figure 20a. When approaching desiccation, the silica gel between sand particles shrinks and cracks (see Figure 20b). Note that suction has little effect on these large cracks, since there is almost no water in such large cracks (see Figure 19). Now we resort to the size effect of the crack in fracture mechanics to consider the weakening effect of these cracks on the strength from the saturated state to the desiccation state.

Ref. [70] proposed a concise and effective strength formula that reflects the size effect, taking into account the influence of crack length a (see Figure 20b) on strength, as shown in Equation (18):

$${\sigma }_f={\displaystyle \frac{{\sigma }_Y}{\sqrt{1+a/a*}}}$$

(19)

where σ_f is the strength affected by the crack, σ_Y is the strength of the specimen without defects, a is the length of the crack (see Figure 20b), a* is a constant and is called the reference crack length depending on the fracture toughness and σ_Y.

Note that the reference crack length a* is a constant in Equation (18), so when the crack length a = 0, we have σ_f = σ_Y, while when a > 0, we have σ_f < σ_Y, and as the crack length a increases, the strength σ_f decreases. So, the core idea of [70] is the size effect term a/a*. Here, we introduce this size effect term a/a* to Equation (17) to reflect the weakening effect of the desiccation-induced crack on the saturated strength, as shown in Equation (19):

$$\tau _f=c^{\prime }+{\displaystyle \frac{\left({\sigma }_v-u_a\right)\tan {\varphi }^{\prime }}{\sqrt{1+a/a*}}}+\psi {\left(S^{cap}\right)}^{\kappa *}\tan {\varphi }^{\prime }$$

(20)

Since the crack length a is a microscopic statistical variable, it is very difficult to determine the size effect term a/a* in Equation (19) from the micro scale. So, we use a function of the capillary part of saturation to represent a/a*, because when approaching desiccation, the crack length a increases, while correspondingly the capillary part of the saturation decreases. The function proposed is 1 − (S^cap)^κ*×b, where b is a parameter. When at the saturation state, the crack length a = 0, and correspondingly 1 − (S^cap)^κ*×b = 0; while during the transition from saturation to desiccation, the crack length a increases, and correspondingly the function 1 − (S^cap)^κ*×b also increases. So, by replacing a/a* with 1 − (S^cap)^κ*×b in Equation (19), we have the formula for shear strength τ_f, as shown in Equation (20):

$$\tau _f=c^{\prime }+{\displaystyle \frac{\left({\sigma }_v-u_a\right)\tan {\varphi }^{\prime }}{\sqrt{1+\left[1-{\left(S^{cap}\right)}^{\kappa *\times b}\right]}}}+\psi {\left(S^{cap}\right)}^{\kappa *}\tan {\varphi }^{\prime }$$

(21)

where κ* is a normal-stress-dependent variable and includes the parameter β (see Equation (16)), and b is the crack size effect parameter.

Table 6. Parameters for predicting shear strength of unsaturated colloidal-silica-cemented sand.

Parameter Description	Parameter	Unsaturated Colloidal-Silica-Cemented Sand
Soil water characteristic curve (SWCC) [50]	α	0.386
	ζ	1.03
	ψm	157
Capillary condensation	β	0.4
Size effect	b	0.075

shows values of the parameters. Here, α, ζ, and ψ_m were proposed by [50] and can be obtained by fitting the soil water characteristic curve (SWCC). β is the parameter related to capillary condensation, which can be fitted by Equations (15) and (16) and the test data. b can be obtained by substituting the values of the desiccation point into Equation (20).

Figure 21 shows the simulation of shear strength τ_f by Equation (20). The following can be seen: (1) the peak suction strength τ_us (τ_us = the increment of τ_f from the saturated state to a partially saturated state, as shown in Figure 12) is normal-stress-dependent, i.e., suction strength τ_us increases with increasing normal stress σ_v; our simulation can match this characteristic, which is owing to the introduction of a normal-stress-dependent characteristic into the capillary condensation probability; (2) near desiccation (e.g., the degree of saturation is 15% or 0%), the shear strength τ_f is less than that at the saturated state (i.e., Δτ < 0 as shown in Figure 21); our formula can describe this characteristic, due to the addition of the crack size effect in the formula. We adopt two quantitative indicators, namely root mean square error (RMSE) and mean absolute percentage error (MAPE), which are defined in Equations (21) and (22). Smaller values of these two metrics represent superior fitting performance:

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-f_i}{f_i}\right|}$$

(22)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-f_i\right)}^2}}$$

(23)

where f_i denotes the i-th measured data, y_i represents the corresponding predicted value, and n stands for the total number of sample points.

The model proposed by [50] exhibits the optimal simulation performance among existing models. Our approach is developed on the basis of the [50] model, so we only conduct comparative analysis between the two methods. For the [50] model, the MAPE and RMSE reach 0.163 and 64.75, respectively. By contrast, our model yields an MAPE of 0.123 and an RMSE of 42.09. Both metrics are evidently lower, demonstrating that the proposed model achieves more accurate simulation results. The experimental method proposed in this study is currently applicable only to calcareous sand solidified with colloidal silica at a concentration of 30%. If specimens prepared with other cemented sands, varied colloidal silica contents, alternative activators or different curing conditions exhibit comparable strength characteristics, the presented method can also be applied accordingly. Nevertheless, the relevant strength laws under these conditions still need to be further verified through laboratory tests.

Classroom takeaway: By comparing Figure 21 with Figure 12 (which showed the limitations of the existing [50] model), students can clearly see how the two proposed modifications—normal-stress-dependent capillary condensation and crack size effect—jointly enable the model to reproduce the unconventional strength behavior observed in the experiments.

5. Discussion and Parameter Analysis: A Classroom Guide to Model Behavior

In this section, we examine how the two newly introduced parameters, β and b, influence the shear strength predicted by our model. Understanding the role of each parameter is essential for students to grasp the physical meaning behind the mathematical formulation.

5.1. Overview of the Two New Parameters

Our model introduces two parameters into the [50] framework, each targeting a specific strength characteristic observed in colloidal-silica-cemented sand:

β (beta): This parameter accounts for the stress-dependent characteristic of the capillary condensation probability. It captures the experimentally observed phenomenon that suction strength increases with increasing normal stress (see Section 2.5.2, Figure 9).

b (b): This parameter accounts for the size effect of cracks that develop near desiccation. It captures the strength deterioration caused by shrinkage and cracking of the silica gel between sand particles when the material becomes sufficiently dry (see Section 2.5.2, Figure 8a).

5.2. Relationship to the Baseline Model [50]

It is important to note that when β = 0 and b = 0, our model degenerates exactly into the [50] model. This baseline model is a well-established and robust framework, as it predicts unsaturated strength using parameters derived directly from the soil–water characteristic curve (SWCC). Moreover, it has been validated across several major soil types [50]. The applicable scope and limitations are discussed as follows. The proposed model (Equation (20)) is an extension of the theoretical framework developed by [50]. In their work, the original model was validated against multiple independent datasets covering reconstituted silty clay, Madrid gray clay, red silty clay, Madrid clay sand, and Nanyang expansive soil. The present model can be mathematically reduced back to [50]’s. original formulation, so it inherently inherits the validated applicability to all these soils. For cemented geomaterials, however, the proposed model is specifically calibrated and limited to colloidal-silica-cemented calcareous sand. Therefore, our modifications should be understood as targeted extensions to address the specific unconventional behaviors of cemented sand, rather than a complete replacement of the existing theory.

5.3. Parameter Effects: A Visual Classroom Comparison

Taking b = 0.075 and β = 0.4 as the baseline values, we separately vary each of the two parameters to calculate RMSE and MAPE. The resulting variations are summarized in

Table 7. Effects of b and β on the goodness-of-fit metric.

Parameter	Effect of b				Effect of β
	b = 0	b = 0.025	b = 0.05	b = 0.075	β = 0.1	β = 0.2	β = 0.3	β = 0.4
MAPE	0.319	0.163	0.066	0	0.027	0.023	0.014	0
RMSE	98.062	49.656	20.030	0	27.019	20.396	11.741	0

for quantitative evaluation of model parameter sensitivity.

Figure 22 provides a side-by-side visual comparison of how β and b affect the predicted shear strength. This figure is designed to help students connect each parameter to its underlying physical mechanism.

(a)

Effect of β (Figure 22a): As β increases, the predicted shear strength increases. The underlying mechanism is that a larger β increases the capillary condensation probability between the sand particles. A higher condensation probability means more extensive formation of capillary water bridges, which in turn increases the suction-induced contribution to the shear strength. In classroom terms: stronger stress-sensitivity of capillary condensation leads to higher strength.

(b)

Effect of b (Figure 22b): As b increases, the predicted shear strength decreases. The mechanism is that b is related to the crack size of the silica gel between the sand particles. A larger b corresponds to larger cracks (or more severe cracking) near desiccation. As the crack size increases, the cementation bridges between particles become more fragmented, leading to a reduction in shear strength. In classroom terms: more extensive drying-induced cracking leads to lower strength.

5.4. Classroom Summary

This parameter analysis serves as a useful classroom exercise: students can be asked to predict how the shear strength curve would change if the cemented sand had a higher gel concentration (which might increase β) or if it were subjected to faster drying (which might increase b). Such thinking experiments reinforce the connection between physical mechanisms and model parameters.

Table 8. Physical meaning and underlying mechanism of parameters.

Parameter	Physical Meaning	Strength Effect	Underlying Mechanism
β	Stress sensitivity of capillary condensation	Increases	Higher capillary condensation probability enhances suction strength
b	Crack size effect near desiccation	Decreases	Shrink and crack weaken cementation between sand particles

presents the physical meaning and underlying mechanism of the parameters.

6. Conclusions

Through direct shear tests on colloidal-silica-cemented calcareous sand across a full range of saturations (0–100%), this paper identified two unconventional strength characteristics: (1) a new type of shear strength versus suction relationship in which the strength near desiccation is lower than that at the saturated state; and (2) the normal-stress-dependent nature of suction strength, i.e., suction strength increases with increasing normal stress. A critical review of the existing models revealed that none of them can simultaneously simulate these two characteristics. Therefore, based on the [50] model, a new model was developed by modifying the capillary condensation probability variable and introducing the crack size effect. The key conclusions, presented as teaching points, are summarized below.

(1)

Normal-stress-dependent suction strength: Experimental results show that suction strength increases with increasing normal stress. However, previous models do not incorporate the effect of normal stress into the suction-induced component of shear strength. Consequently, they cannot simulate this characteristic. This highlights an important lesson for students: classical unsaturated strength models often assume the suction strength is independent of the confining stress, but this assumption may not hold for cemented materials.

(2)

New type of strength versus suction relationship: Experimental results reveal that near desiccation, the shear strength drops below the saturated strength. This phenomenon was not observed in previous tests on other soil types and cannot be simulated by the existing models. This serves as a powerful classroom demonstration that not all unsaturated soils follow the conventional “drier is stronger” rule.

A new model was established to simulate the above two characteristics of colloidal-silica-cemented calcareous sand. The model modifications are directly linked to physical mechanisms, making them suitable for classroom explanation as follows.

(1)

For normal-stress-dependent suction strength: From the perspective of micro-scale capillary condensation, the variable of capillary condensation probability was modified to incorporate the effect of normal stress. This modification enables the new model to simulate the observed, normal-stress-dependent characteristic. The key insight for students is that capillary condensation between particles is not solely a function of suction—it is also influenced by how tightly particles are pressed together.

(2)

For the new type of strength versus suction relationship: By introducing the size effect of gel cracks between sand particles, the model can simulate the phenomenon where strength near desiccation is lower than strength at the saturated state. This modification teaches students that drying can induce not only beneficial suction but also detrimental cracking, and the net effect on strength depends on which mechanism dominates.

Final classroom takeaway: This case study demonstrates that when experimental evidence contradicts existing theories, new physical mechanisms must be identified and incorporated into models. Students learn that model building requires understanding of the underlying physics, such as how capillary condensation couples with stress and how drying-induced cracking can reverse the expected strength trend.