Critical state mechanics applied to filtered iron tailings for safe dry-stacking design

Abstract

Recent tailings dam failures have prompted the adoption of dry stacking, where filtered tailings are compacted into stable piles. Although filtration reduces water content, residual pore water under high stress can still trigger liquefaction, raising stability concerns. This study compiles parameters of critical state soil mechanics from existing literature on filtered iron ore tailings from the Iron Quadrangle region, Minas Gerais, Brazil. The analysis reveals that the critical state parameters M_tc, λ, and Γ exhibit nearly normal distributions, influenced by mineralogy and grain shape. However, fines content, ranging from 33.3% to 98.0%, may cause slight deviations. Specific gravity and fine content distributions are bimodal, possibly reflecting processing efficiency. Correlation analysis shows a moderate positive relationship between λ and Γ (Pearson’s R = 0.720) and a minimal effect of fines content on density (Pearson’s R = 0.091). Representative values of M_tc = 1.41, λ = 0.037, and Γ = 0.933 were obtained, with M_tc and Γ showing low coefficients of variation, while λ showed a high one.

Keywords:
: filtered iron tailings; dry stacking; constitutive model; critical state.

1. Introduction

The disposal of mining tailings in dams has been a common practice for decades, permitting their consolidation and alteration of geotechnical properties, such as increase of shear strength and reduction of permeability and compressibility along time. However, with technological advancements and the growth of mining activities, waste generation has significantly increased (Nery, 2013; Villar, 2002), making it essential to seek safer and more sustainable destination methods.

Disasters, such as the failures of the Mariana and Córrego do Feijão tailings dams, highlighted the need for a better understanding of the geomechanical behavior of tailings and the environmental, social, and economic impacts associated with structural failures, and resulted in stricter regulations. Also, mining activities should conform to the United Nations Sustainable Development Goals (SDGs) (ONU, 2015).

As a safer alternative, dry stacking has been gaining global traction. Although this technique significantly reduces the water content of the tailings, they do not become completely dry and may still pose liquefaction risks in high stacking piles due to high stress conditions and the low water retention capacity of these materials (Consoli et al., 2022).

For stability analysis of tailings dams and, more recently, dry stacking of filtered tailings, specific stress-strain models have been developed. Following geotechnical evolution, one of the first design approaches was to combine the linear elastic model for displacements with Mohr-Coulomb criterium for failure. After experimental observations, a hyperbolic curve was proposed for the stress-strain behavior of soils. Also, for tailings the hyperbolic behavior has been suggested. For example, Adajar and Zarco (2021) noticed that mine tailings showed brittle failure in dry conditions and ductile failure when saturated and proposed a modified hyperbolic model that accurately predicts stress-strain responses for ductile behavior but not for brittle failure.

More recent and advanced constitutive models, such as Cam Clay (Schofield and Wroth, 1968), NorSand (Jefferies, 1993), SaniSand (Dafalias and Manzari, 2004), and Casm (Yu, 1998) enable the simulation of various field conditions.

This study analyzes the geomechanical behavior of iron ore tailings from a Brazilian mining region in light of critical state mechanics, using a database of laboratory test results available in the literature.

2. Methodology

Firstly, a brief overview of the critical state model is presented. Next, a literature review shows the geomechanical characterization of iron ore tailings from the Iron Quadrangle, Minas Gerais, Brazil. This region, with approximately 7,000 square kilometers, is responsible for 51% of the national iron ore production, Brazil being the second largest world producer (ANM, 2023).

The dataset was primarily sourced from academic theses (77%) focused on the iron ore tailings from the Iron Quadrangle, with additional information obtained from two reports concerning tailings dam failures, also in this region.

Statistical analyses were performed through the Jamovi software (R CORE TEAM, 2021; Jamovi project, 2022). First, normality hypothesis was tested by Shapiro and Wilk (1965) test with confidence interval of 95% (p >0.05). For the parameters with normal distributions, average, standard deviation, median, and variation coefficient were determined. Also, correlations among the physical indices and the critical state parameters of the materials were studied.

2.1 Introduction to critical state soil mechanics (CSSM)

There are several methods to estimate the deformations resulting from the application of stresses on the soil. The linear elastic model is the simplest, establishing a relationship based on a constant proportionality, which is characteristic of the material (Hooke’s law). Since the linear elastic model does not predict failure, it must be linked to a plasticity failure criterion, such as Tresca, Von-Mises, Mohr-Coulomb or Drucker-Prager. Among these, the Mohr-Coulomb failure criterium is considered the best fit for soils, and it estimates the major principal stress (σ₁) by the minor principal stress (σ₃) - neglecting the secondary stress (σ₂) - according to Eq.1 (Nader, 2015).

(1) $${\sigma }_1=kp{\sigma }_3+2c\sqrt{kp}$$

With: • c: soil cohesion;

• kp: passive thrust coefficient calculated by the internal friction angle (ϕ) of the soil (Eq. 2)

(2) $$Kp=\frac{1+\mathrm{sen}(\phi )}{1-\mathrm{sen}(\phi )}={\tan }^2(45+\frac{\phi }{2})$$

The main unfavorable points of this model are:

• The Mohr Coulomb failure criterion defines strength limits but does not, by itself, provide deformation predictions. Therefore, it is generally combined with the linear elastic model, generating an elastic-plastic model;

• It does not consider σ₂;

• It indicates infinite strength for stresses close to the hydrostatic stress state.

Critical state is the scenario in which the soil deforms without changing the void ratio (dilatancy and increase of dilatancy null) and at constant stress.

Independently of the initial void ratio (either maximum, e_max, or minimum, e_min) and the stress path developed under different solicitations, a soil sample under determined octaedric stress (σ₁ = σ₂ = σ₃) will reach the same void ratio, and this final void ratio is called critical (e_c).

For example, in Figure 1 samples of the same soil with different void ratios subjected to triaxial compression tests with the same confining stress will reach at the end the same void ratio, i.e., for a given soil and a constant confining stress, the critical void ratio at the end of a triaxial compression test is the same, regardless of the initial relative density. The behavior of dense soil after reaching the peak strength is characterized by residual strength and massive deformation.

Figure 1
Stress-strain behavior leading to e_c, despite compacity (Kramer, 1996).

A loose sand under shearing suffers volume decreases while it deforms, and the shear stress increases asymptotically. When a dense sand is subjected to loading, to allow deformation, the grains need to roll over each other, leading to a volume increase, generating a structure with more voids; the shear stress reaches a peak strength and then tends asymptotically to a residual value (Kramer, 1996).

The relationship between e_c and stress state is called critical state line (CSL) or steady state line (SSL) depending on how the stress state is represented. The SSL presents the critical void ratio (in this case, called steady state ratio - e_s) as a function of confining stress (Figure 2) (Schofield and Wroth, 1968; Kramer, 1996), and the CSL, as a function of p’ (Eq. 4), as shown in Figure 3.

Figure 2
SSL (Kramer, 1996).

Figure 3
CSL (Schofield and Wroth, 1968).

As presented in Figure 3, for each confining stress there is a critical void ratio, which decreases as the confining stress increases. The CSL may be presented as Eq. 3 (Schofield and Wroth, 1968).

(3) $$\mathrm{e}=\mathrm{\Gamma }-\lambda \ln p^{\prime }$$

With: Γ is the critical void ratio for 1 kPa; λ represents the slope of the critical state; p’: octaedric stress, calculated by Eq. 4:

(4) $$p=\frac{{\sigma }_1^{\prime }+{\sigma }_2^{\prime }+{\sigma }_3^{\prime }}{3}$$

When the state parameter Ψ (the difference between a given void ratio and the critical or steady state void ratio for the same p’) is positive, it indicates a soil in a loose state, and therefore, with contractile behavior and a greater probability of liquefaction when under undrained conditions.

The interpretation of the normally consolidated line (NCL) relative to the CSL varies among constitutive models. The Cam-Clay model, as its name suggests, was primarily developed for clays and assumes that the critical state lies parallel to the NCL, offset by the value (λ-κ)ln(2), as shown in Figure 4. The parameters λ and κ are the slopes, respectively, of the normally consolidated line (NCL) and the recompression line (RL).

Figure 4
CSL and NCL for Cam Clay (Schofield and Wroth, 1968).

In contrast, the NorSand model introduces a more flexible framework, proposing that multiple NCLs may exist depending on the initial density and relative compaction of the material (Figure 5).

Figure 5
CSL and NCL for NorSand (Jefferies and Been, 2016).

By testing samples on different compacity it is possible to define their D_mín and M_tc, as seen in Figure 6.

Figure 6
η_máx = f(D_mín) (Jefferies and Been, 2016).

With: M_tc: slope of the CSL obtained by compression for null dilatancy; N: slope of η_máx = f(D_mín), represents work increment which didn’t dissipate with distortional strain; D_mín: minimum dilation, thus maximum compacity.

The NorSand model establishes a relationship between stress and void index, making it particularly suitable for sandy soils (Robertson et al., 2019). Its formulation captures the influence of initial density on mechanical behavior more explicitly than traditional models. This characteristic is especially advantageous for representing the behavior of filtered iron ore tailings, which often display stress-dependent dilatancy and contractive tendencies similar to loose sands. As such, NorSand (Jefferies, 1993) provides a more realistic framework for modeling the response of these materials under various loading conditions.

Other models developed to simulate the mechanical behavior of mining tailings can be cited, such as Sanisand (Dafalias and Manzari, 2004), developed for sands with complex loading histories, and includes features, such as rotational hardening and fabric evolution, which improve the prediction of cyclic and monotonic responses. The CASM (Clay and Sand Model) (Yu, 1998) framework unifies the modeling of both coarseand fine-grained materials and introduces a generalized critical state formulation that is also applicable to tailings. Other models, such as UBCSAND (Beaty and Byrne, 2011) and PM4Sand (Boulanger and Ziotopoulou, 2017), have been calibrated for liquefaction-prone materials and are promising in simulating the seismic response of tailings deposits. Despite these advancements, accurate model calibration remains a challenge due to limited availability of well-defined parameters.

2.2 Geotechnical data

Considering the destructive impact of liquefaction of tailings dams, several authors tested and modeled iron tailings. The data compiled in Table 1 refer to iron tailings from the Iron Quadrangle, Brazil.

Table 1
Summary of properties.

Reference	GS	FC	Mtc	λ	Γ	Type of test	Confining stress (kPa)	Compaction
Pereira (2005)	3.400	61.0	1.570	-	-	CIU	50, 100, and 200	8 mm rod (numbers of blows necessary to achieve maximum dry density)
	3.340	34.0	1.410	-	-
	3.950	74.0	1.580	-	-
	4.800	24.0	1.300	-	-
	3.630	24.0	1.630	-	-
	4.150	50.0	1.270	-	-
	3.370	47.0	1.330	-	-
	2.920	47.0	1.070	-	-
Motta (2008)	3.220	89.0	1.450	-	-	CIU and CID	100, 200, 300 and 400	Hydraulic press equipped with a mold featuring a dual loading system for applying compaction force in two stages/ directions
Rezende (2013)	2.970	56.0	1.290	-	-	CID	75, 150, 300 and 550	None (slime)
	2.840	40.0	1.400	-	-
	2.940	70.0	1.320	-	-
	3.000	34.0	1.310	-	-
Morgenstern et al. (2016)	3.900	98.0	1.330	0.024	0.865	CIU and CID	50, 100, 200, 300, 400 and 600	None (slime)
Eloi (2021)	4.462	92.0	1.397	0.037	1.037	CIU	100, 200 and 400	Moist tamping
	4.550	54.0	1.449	0.069	1.250
Telles (2017)	2.800	39.0	1.360	0.048	1.017	CIU and CID	100, 150, 300, 400 and 500	Double static compaction
Robertson et al. (2019)	4.500	-	1.380	0.039	1.120	CIU and CID	100, 300 and 800	Proctor (standard)
	4.500	-	1.380	0.039	1.040
	4.500	-	1.380	0.039	1.020
Carneiro (2021)	3.209	91.1	1.384	0.041	0.800	CIU and CID	100, 2000, 400, 800 and 1600	Proctor (standard))
Nóbrega (2021)	4.994	35.0	1.365	-	-	CID	50, 100, 200, 400 and 1600	Moist tamping
Balbino (2022)	4.231	93.0	1.443	0.015	0.584	CIU and CID	50, 100, 200 and 400	Proctor (standard)
	4.230	93.0	1.247	0.057	0.866
Faria (2022)	4.548	91.9	1.400	0.037	1.040	CID	100, 200 and 400	Moist tamping
Oliveira (2022)	3.050	51.7	1.400	0.047	0.768	CIU and CID	200, 400, 800 and 1600	Moist tamping
	3.050	51.7	1.400	0.009	0.768
	3.050	51.7	1.400	0.010	0.768
	2.973	33.3	1.350	0.044	0.855
	2.973	33.3	1.350	0.017	0.855
	2.973	33.3	1.350	0.012	0.855
Silva et al. (2022)	2.960	80.0	1.350	0.083	1.280	CIU and CID	50, 100, 150, 200, 300, 400 and 600	Harvard miniature

From the literature reviewed, studies involving non-plastic fines were selected, resulting in the exclusion of only one study (Balbino, 2022). Oliveira (2022) varied the initial void ratio of the samples.

The uncertainties in the compiled dataset derived from different materials and different testing methods should be acknowledged. Factors, such as fines and iron contents, sample preparation, boundary conditions, and stress path control, can influence the reported values of critical state parameters. Therefore, while the statistical analysis of the compilation offers valuable insights, caution is advised when applying these parameters directly to site-specific designs without additional validation.

3. Results and discussion

Table 1 presents the compiled data and Table 2 shows results of the statistical analysis of the previously reported values. Density and histograms plots for each parameter are shown from Figure 7 to Figure 11.

Table 2
Descriptive analyses.

	GS	FC	Mtc	λ	Γ
Quantity	32	29	32	18	18
Mean	3.63	54.0	1.38	0.037	0.933
95% Confidence interval mean lower bound	3.37	45.5	1.34	0.027	0.843
95% Confidence interval mean upper bound	3.88	62.5	1.41	0.047	1.02
Median	3.35	50.0	1.38	0.039	0.865
Mode	3.01 4.56	35.1	1.41	0.039	0.768
		53.6
		94.3
Standard deviation	0.71	22.30	0.10	0.02	0.02
Minimum	2.80	24.00	1.07	0.01	0.58
Maximum	4.99	98.0	1.63	0.08	1.28
Variation coefficient	0.20	0.41	0.07	0.54	0.02
Skewness	0.476	0.694	-0.027	0.485	0.284
Std. error skewness	0.414	0.434	0.414	0.536	0.536
Kurtosis	-1.39	-0.73	3.09	0.14	-0.13
Std. error kurtosis	0.809	0.845	0.809	1.040	1.040
Shapiro-Wilk W	0.860	0.898	0.904	0.936	0.947
Shapiro-Wilk p	< .001	0.009	0.008	0.247	0.384

Figure 7
Density distribution of G_S values.

Figure 8
Density distribution of FC values.

Table 2 shows that only λ and Γ have normal distributions. Therefore, mean, median, standard deviation, and variation coefficient only have meaning for these parameters. For the other parameters, the modes were adopted as more representative values instead of the mean. By selecting the modes, the analysis captures the dominant trends more accurately within each population subset.

For G_S, there are two distinct data groupings, and the mean falls in a low-density region between the peaks, misrepresenting the actual material behavior. For FC, three modes can be seen in Figure 8. It is expected that the particle size distribution and the specific gravity of solids together provide an indication of the efficiency of the ore processing. Considering that iron oxides have specific gravities varying between 4.9 and 5.3, and quartz, 2.6, the higher the specific gravity of the tailing’s solids, the higher the quantity of lost iron in the tailings, and therefore, the worse the processing. The bimodality of G_S and the three modes of FC may be related to the type or efficiency of the processing. However, there is no information in the reviewed articles to support this assumption.

M_tc (Figure 9 density distribution of M_tc values Figure 9) has a well-defined peak, but the distribution cannot be considered normal. M_tc, λ and Γ (Figure 9 to Figure 11) are parameters primarily influenced by mineralogy, grain shape, and structure. Since the tailings studied originate from the same region, it is assumed that their mineralogical composition is similar. Additionally, the grain shape is generally angular due to the beneficiation processes. However, the fines content ranged from 33.3% to 98.0%, which may account for the distribution of M_tc being not perfectly normal. The relatively small sample size for key parameters, such as λ and Γ (n = 18), constrains the statistical power of the analysis. Larger datasets may better capture material variability and improve confidence in these estimates.

Figure 9
density distribution of M_tc values.

Figure 10
Density distribution of λ values.

Figure 11
Density distribution of Γ values.

Figure 12
Relation between parameters. (*** strong relation).

Table 3 presents the representative values of the critical parameters. Since the density curve of parameter M_tc was not normal, the mode was considered more representative than the average (mean). For λ and Γ, as the distribution is normal, it is considered the mean.

Table 3
Representative values of the critical state parameters.

	Mtc	λ	Γ
Mean/mode	1.41	0.037	0.933
Standard deviation	0.10	0.02	0.02

Figure 12 was developed with the software Jamovi and shows the linear correlation (Pearson’s R) between the parameters. The Figure is read as a table:

• Pearson’s R is shown by crossing the column of the first one with the line of the second one, above the diagonal. For example, the correlation of the FC with G_S is presented on the second column, first line, with the value of 0.091. A strong relation is represented by Pearson’s R close 1 or -1, so that the software marks strong relations with “***". The closest relationship was found between the parameters of the CSL, λ and Γ, with Pearson’s R of 0.720;

• In the diagonal, the density curves of each parameter are shown;

• Beneath the diagonal is shown the graphs which relate each parameter.

The collected data indicates a weak relation between FC and G_S. Therefore, the effect of residual iron is negligible on the specific gravity.

4. Study limitations

It is important to acknowledge the inherent uncertainty in the compiled dataset due to material variability and differences in data quality and testing methodologies across the original studies. Factors, such as sample preparation, fines content, iron content, boundary conditions, and stress path control can significantly influence the reported values of critical state parameters. Furthermore, the relatively small sample size for key parameters-particularly λ and Γ, with ν = 18, limit the statistical robustness and generalizability of the findings. The absence of complete metadata in some sources, e.g. mineralogical composition, further constrains the interpretation. While the compilation provides meaningful insights into the behavior of filtered iron ore tailings, caution is advised when applying these parameters to site-specific designs without further validation or supporting laboratory testing.

5. Conclusion

This study compiled and analyzed a database of geomechanical parameters from laboratory tests on filtered iron ore tailings, originating from the Iron Quadrangle, Brazil, within the framework of Critical State Soil Mechanics (CSSM). Representative values were identified for key parameters: M_tc = 1.41, λ = 0.037, and Γ = 0.933. Among these, Γ showed low variability, while λ was more sensitive to material variability.

The findings support the suitability of using critical state parameters (M_tc, λ, and Γ) to describe the behavior of filtered tailings for dry-stacking design, due to the observed distributions approaching normality, which suggest relatively homogeneous material behavior. While certain variables, such as fines content and specific gravity, displayed complex (non-normal) distributions, the core critical state parameters showed sufficient statistical consistency to inform preliminary modeling. However, due to inherent data limitations and potential site-specific differences, these parameters should not be applied directly without further calibration and validation through targeted laboratory testing.

Funding information

National Council for Scientific and Technological Development - CNPq, Research fellowship, Process 310978/2025-4.

Data availability

The authors state that all data supporting the findings of this study are contained within the article itself, and no additional datasets were generated or analyzed during the current study.

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Conflict of interests

The authors declare that there is no conflict of interest.

Associate Editor

Jório Coelho

Publication Dates

• Publication in this collection
  28 Sept 2026
• Date of issue
  2026

History

• Received
  07 Aug 2025
• Accepted
  18 May 2026