Modeling the Influence of Non-Constant Poisson’s Ratio on Crack Formation Under Uniaxial Compression of Rocks and Concrete

Abstract

This article considers the effect of constant and variable Poisson’s ratio on cracking in concrete and rock specimens under uniaxial compression using mechanical systems modeling methods. The article presents an analysis of the data confirming the increase in Poisson’s ratio under specimen loading. A system of equations for modeling the effect of Poisson’s ratio on cracking under uniaxial compression is proposed. The comparison showed that the model with a constant Poisson’s ratio predicts a thickness of the surface layer with cracks that is underestimated by approximately 10%. In practice, this means that the model with a constant Poisson’s ratio underestimates the risk of failure. A technique for analyzing random deviations of Poisson’s ratio from the variable mathematical expectation is proposed. The comparison showed that the model with a variable Poisson’s ratio leads to results that are more cautious, i.e., it does not potentially overestimate the safety factor. The model predicts an increase in uniaxial compression strength when using external reinforcement. An equation is proposed for determining the required wall thickness of a conditional reinforcement shell depending on the axial compressive stress. The study contributes to understanding the potential vulnerability of load-bearing structures and makes a certain contribution to increasing their reliability.

1. Introduction

Poisson’s ratio, compared to other key rock properties such as elastic modulus, compressive strength, tensile strength, and shear strength, is not often at the forefront of engineering calculations [1]. The use of Poisson’s ratio allows us to simplify indirect measurements, since it is used in relationships that link difficult-to-measure properties with more easily measurable ones. For example, the brittleness of a material increases if Poisson’s ratio decreases [2]. With increasing load, the values of Poisson’s ratio increases and at the boundary of brittleness and plasticity, it can reach a value of 0.5 [3]. Standard tests of granite and other rocks under uniaxial compression have shown [1] that Poisson’s ratio is stress-dependent, which contradicts its mechanical interpretation as an elastic constant [4]. There is also a certain relationship between the confining pressure and the Poisson’s ratio of the intact rock [3]. The value of Poisson’s ratio can significantly influence the boundaries between the stages of the fracture process, since the crack propagation threshold can change by more than 40% with a change in Poisson’s ratio by ±0.05 [5,6]. It is therefore important to determine the effect of changes in Poisson’s ratio on the bearing capacity of, for example, supports in mine workings.

Geotechnical calculations are typically based on the assumption that Poisson’s ratio remains constant, which does not always align with the behavior of rocks at different deformation stages, as evidenced by the analysis of uniaxial compression test results. Indeed, it is widely recognized [7,8,9] that during the initial stage of axial compression, the applied energy is primarily consumed by the closure of pre-existing cracks, resulting in relatively small lateral strains. This means that Poisson’s ratio, defined as $={\displaystyle \frac{\left|{\epsilon }_{lateral}\right|}{\left|{\epsilon }_{axial}\right|}}$, approaches zero in the first stage. In the second stage, once all pre-existing cracks are closed, the rock behaves as an elastic material up to 0.4–0.6 of its peak stress [7]. The second stage establishes conditions for stable crack growth, though it does not exclude the possibility that the weakest mineral grains or their aggregates may fail during this phase.

The results of standard tests of granite for uniaxial compression [7,8,9,10] known from the literature show that the crack initiation stress is from 0.3 to 0.5 of the ultimate strength for uniaxial compression. The formation of new cracks at loads below 0.4 of the peak stress is further confirmed by the positive correlation between acoustic emission intensity and loading rates (from 0.0005 mm/s to 0.01 mm/s) in uniaxial compression tests on cylindrical granite specimens [11].

A sign of the third stage is a steady growth of cracks up to a load of 0.7–0.9% of the peak value. The fourth stage involves unstable crack propagation until the load reaches its maximum (${\sigma }_{peak}$) [7]. In the post-peak (fifth stage), the specimen undergoes failure. Our analysis focuses solely on axial and lateral deformations during the pre-peak loading stages.

To illustrate this, we performed an analysis using high-resolution images from [11], which enabled the determination of axial and lateral strain values for granite. Based on these data, we calculated Poisson’s ratio at different deformation stages, particularly at the boundary points of the third stage (Figure 1).

The cited study [11] presents experimental stress–strain curves from tests on cylindrical granite specimens (D = 50 mm, H = 100 mm) at loading rates of 0.0001 mm/s (Figure 1), 0.001 mm/s, and 0.01 mm/s. By processing the diagrams from [11] for loading rates of 0.001 mm/s and 0.01 mm/s using the same methodology as in Figure 1, we obtained the corresponding Poisson’s ratio values at boundary points of the third stage, showing their dependence on loading rate (

Table 1. Values of Poisson’s ratio at points on the boundary of the third stage according to Figure 1.

Loading Rate mm/s	Poisson’s Ratio, Point a	Poisson’s Ratio, Point b
0.0001	0.07	0.14
0.001	0.07	0.15
0.01	0.08	0.14

).

The obtained data demonstrate that for the studied granite, Poisson’s ratio values increase with loading but remain virtually independent of loading rate within the range of 0.0001 to 0.01 mm/s. This finding naturally raises the question: how does Poisson’s ratio behave during unloading and reloading cycles? Experimental data addressing this question are presented in study [12], which revealed that granite’s Poisson’s ratio increases from near-zero values to 0.5 with increasing load–unload cycles across a temperature range of 25 °C to 800 °C.

Poisson’s ratio values depend not only on loading conditions but also on factors such as structural heterogeneity, rock mineralogical composition [1,13], moisture content, temperature, and other parameters [14,15,16,17,18], in addition to the aforementioned effects. For instance, undrained rock specimens exhibit higher Poisson’s ratio values compared to drained ones [19].

In studying the effect of Poisson’s ratio on rock fracturing, special attention should be paid to crack patterns and their localization. Observations of crack development in cylindrical soil–rock mixture specimens under uniaxial deformation using X-ray computed tomography [20] revealed that tensile cracks initiate in the specimen’s surface layer. Both experimental and theoretical studies of spalling yield similar conclusions, as does damage pattern analysis in uniaxial compression tests of similar materials [21,22,23,24].

These experimental findings led to the hypothesis of a surface layer of thickness t where tensile stresses and cracks develop when the tensile stress reaches its maximum. The mathematical formulation of this hypothesis, incorporating equilibrium conditions along with Dinnik’s hypothesis on vertical-to-horizontal stress ratio [25,26], resulted in [27] in a theoretical model expressed by the following two equations for estimating circumferential tensile stress (${\sigma }_t$) as a function of Poisson’s ratio (ν) and axial compressive stress (${\sigma }_c$) in the cross-section at mid-height of a cylindrical specimen with radius $R$:

$${\sigma }_t=n{\sigma }_c\left({\displaystyle \frac{R}{t}}-1\right),$$

(1)

$$n={\displaystyle \frac{\nu }{1-\nu }}.$$

(2)

These experimental findings led to the hypothesis of a surface layer of thickness t where tensile stresses and cracks develop when the tensile stress reaches its maximum [27]. It is important to note that the tensile strength of rocks is lower than their shear strength, which is why tensile cracks appear before shear cracks. Once tensile cracks form, the growth of tensile stresses ceases, while shear stresses continue to increase until shear cracks develop. The failure pattern may vary depending on the specific material properties [28,29] and loading conditions [30,31]. Let us consider the “hourglass” failure mode, for which Equation (1) is applicable. This mode is typical for rock pillar damage [30].

From a physical perspective, the compressive stress ${\sigma }_c$ is external to the specimen and determines its strength (we account for the distinction between specimen strength and material strength [32,33]). Increasing compressive stress leads to simultaneous growth of both tensile and shear stresses. The growth of σ_t stops when it reaches its peak value (${\sigma }_{t,peak}$), resulting in tensile crack formation and spalling of part of the specimen. This reduces the transverse dimension (effective radius) and cross-sectional area at the specimen’s mid-height.

Figure 2 schematically shows the origin of tensile stresses (1), adapted from [16]. The model assumes that rock, as a heterogeneous material, consists of particles bonded at their contact points. Each contact is assumed to act as a pair of elastic springs allowing normal and shear displacement. If the tensile or shear force reaches its limit, the corresponding bond breaks. Particles with broken tensile bonds that remain in contact with others can still provide shear resistance through friction under compression [16]. Figure 2 demonstrates that insufficient strength of the aforementioned particle bonds in the surface layer—in the absence of confining (here, horizontal) pressure—leads to particle detachment and spalling. Conversely, confining pressure counteracts transverse deformations, increasing the allowable vertical load. In these scenarios, Poisson’s ratio plays a critical role, yet its influence remains insufficiently studied in this context.

As Equation (1) shows, decreasing the transverse dimension R reduces circumferential tensile stress, ultimately driving the formation of new tensile cracks toward zero (if any new tensile cracks appear, they only occur in the weakest grains or at grain boundaries in grain conglomerates, if present). However, shear stresses increase not only due to a growing load but also because of the reduced area of the inclined section where these shear stresses act.

Thus, the influence of circumferential tensile stresses weakens, consequently reducing energy expenditure on tensile crack initiation and propagation, while increasing energy consumption for shear crack formation and growth. Therefore, specimen loading and “hourglass” failure begins with tensile crack growth and reduction in the cross-sectional area at mid-height, which diminishes the role of tensile cracks while simultaneously enhancing the significance of shear stresses, shear cracks, and shear failure. These failure stages are illustrated in Figure 3a–c using garnet amphibolite as an example [27]; Figure 3d [34] shows vertical cracks in the surface layer of a concrete specimen at the first failure stage. Figure 4 schematically shows the geometric aspects of the failure mechanism depicted in Figure 3.

The reduction in the specimen’s transverse dimension at mid-height (Figure 4) decreases both the cross-sectional area and the inclined sectional area where shear stresses act. These stresses depend on compressive stress (${\sigma }_c$), internal friction coefficient $\mu =\mathit{tan}\phi$ (where $\phi$ is the friction angle), and cohesion (${\tau }_0$), which equals the yield strength at zero normal stress [7].

The combined effect of these factors on the specimen’s load-bearing capacity (i.e., the peak compressive stress) results in nonlinear equations, whose solution methods are widely covered in the literature. However, certain research gaps persist and require further investigation [30,34].

A literature review revealed that a model predicting load-bearing capacity—defined as the compressive stress causing granite (and similar materials) to crack in construction structures while accounting for Poisson’s ratio variability—would be of significant practical interest.

The innovative potential of such models lies in their usefulness as auxiliary tools for assessing the technical condition of both surface and underground structures, both modern and historical. Analysis of the application of models will expand our understanding of ways to improve the reliability and stability of these structures.

The purpose of this work is to develop a model for analyzing the effect of variable Poisson’s ratio on crack formation under uniaxial compression of rocks and concrete as an artificial analog of rocks.

2. Methodology and Results

2.1. Introductory Remarks on Research Methodology

In this section, we model the process of change in the relative thickness of the cracked surface layer (t/R) as a function of applied load. The t/R ratio is a relative estimate of the cracked layer thickness, so this ratio is one of the possible variants of the damage variable. The t/R ratio reflects not only a quantitative assessment but also the aforementioned localization of damage in the form of tensile cracks within the surface layer. These cracks will develop when the tensile stress (1) reaches its peak value ${\sigma }_{t,peak}$.

Using the condition ${{\sigma }_t=\sigma }_{t,peak}$, we derive from Equation (1) the relative thickness of the damaged surface layer (3) [27]:

$${\displaystyle \frac{t}{R}}={\displaystyle \frac{1}{1+\frac{{\sigma }_{t,peak}}{n{\sigma }_c}}},$$

(3)

The Poisson’s ratio for granite may range between 0.1 and 0.33 within the theoretically possible values from 0 to 0.5 [1]. For instance, $\nu =0.24$ was reported in study [35], where ${\sigma }_{t,peak}=2.69$ MPa, and ${\sigma }_{c,peak}=$41.5 MPa.

2.2. Verification Case: Comparison with a Known Solution

Let us consider a verification case. As noted above, experimental data [7] indicate that tensile cracks initiate when the axial load exceeds a threshold value ${\overline{\sigma }}_c$, corresponding to 0.4–0.6 of the peak stress [7]. Taking these data into account, we rewrite Equation (3) as Equation (4):

$${\displaystyle \frac{t}{R}}={\displaystyle \frac{1}{1+\frac{{\sigma }_{t,peak}}{n({\sigma }_c-{\overline{\sigma }}_c)}}}.$$

(4)

Figure 5 presents the computational results for the relative thickness of the cracked surface layer (4) at $\nu =0.24$ and ${\sigma }_{t,peak}=2.69$ MPa for three threshold values of ${\overline{\sigma }}_c$,: 16.60 MPa, 20.75 MPa, and 24.90 MPa.

Accordingly, three limiting values of t/R were obtained: 0.745, 0.709, and 0.661. These values are consistent with the known damage estimate of 0.6916 for a granite specimen under uniaxial compression, obtained using an alternative method [35].

2.3. Effect of Poisson’s Ratio Variations on Cracked Layer Thickness

The Equations (4) and (5) above model the state of a rock specimen under axial compressive load assuming a constant Poisson’s ratio. However, rock testing [1] and experimental data processing, including the results shown in Figure 1, demonstrate that Poisson’s ratio changes under load. A recent study [4], which proposed an innovative method for determining the stress dependence of Poisson’s ratio of granitic rocks, showed that Poisson’s ratio is not a constant for rigid rocks, since in experiments its values depend nonlinearly on stress. To approximate the Poisson’s ratio of granitic rocks from stress, a nonlinear equation with two material constants was proposed and experimentally verified (using standard uniaxial compression tests). The results of study [4] create new opportunities for continuing studies of the effect of Poisson’s ratio on the mechanical behavior of brittle materials, including the topic of our work. However, article [4] appeared when the main part of our work was completed. Nevertheless, we took into account the conceptual aspects of theory [4], albeit in an approximate interpretation. Of the two conceptual aspects studied in [4], our work takes into account only the increase in the Poisson ratio with increasing load, but according to a linear law, which limits the scope of application to a range of compressive stresses from 20% to 80% relative to the peak stress under uniaxial compression. Thus, the following Equation (5) models the dependence of Poisson’s ratio from ${\nu }_0$ to ${\nu }_n$ with increasing load:

$$\nu ={\nu }_0+{\displaystyle \frac{{\nu }_n-{\nu }_0}{{\sigma }_{c,peak}}}{\sigma }_c.$$

(5)

Supplementing Equations (4) and (2) with Equation (5), we write the system of equations as:

$${\displaystyle \frac{t}{R}}={\displaystyle \frac{1}{1+\frac{{\sigma }_{t,peak}}{n({\sigma }_c-{\overline{\sigma }}_c)}}},$$

(6)

$$n={\displaystyle \frac{\nu }{1-\nu }},$$

(7)

$$\nu ={\nu }_0+{\displaystyle \frac{{\nu }_n-{\nu }_0}{{\sigma }_{c,peak}}}{\sigma }_c.$$

(8)

To account for random deviations in Poisson’s ratio (e.g., within ±10% of the value predicted by Equation (1)), the algorithm should generate a normally distributed random number for each simulated specimen state, with the expected value from (8) and an appropriate standard deviation (see Section 2.8 for details).

2.4. Constant Poisson’s Ratio: Comparison with a Known Solution

We apply Equations (7)–(9) to model the influence of Poisson’s ratio on granite behavior using reference data from [31]: ${\sigma }_{c,peak}=246$ MPa, ${\sigma }_{t,peak}=14.9$ MPa, and $\nu$ = 0.225 (constant).

Figure 6 presents the computed relative thicknesses of the cracked surface layer (6) for three threshold values of ${\overline{\sigma }}_c$: 98.4 MPa, 123.0 MPa, and 147.6 MPa.

Accordingly, three limiting values of t/R were obtained: 0.736; 0.700; and 0.651. These values do not contradict the known assessment of damage to a granite specimen under uniaxial compression, obtained in another way and equal to 0.6916 [35].

2.5. Example: Non-Constant Poisson Ratio

Let us consider the application of Equations (6)–(8) to modeling the influence of Poisson’s ratio on the behavior of granite, using the initial data from [36]: ${\sigma }_{c,peak}=246$ MPa, ${\sigma }_{t,peak}=14.9$ MPa, ${\nu }_0$ = 0.16, ${\nu }_n$ = 0.29, and mean value ${\nu }_{mean}$ = 0.225. The value of ν is determined by Equation (8).

Figure 7 illustrates a comparison of the computational results for variable and constant Poisson’s ratios at three threshold values of ${\overline{\sigma }}_c$: 98.4 MPa, 123.0 MPa, and 147.6 MPa. Accordingly, the three critical values for damage assessment t/R are 0.802, 0.771, and 0.730.

The estimates of the relative thickness of the surface cracked layer presented in Figure 7, as well as those given above, do not contradict the known relative assessment of damage in a granite specimen under uniaxial compression, obtained by a different method and equal to 0.6916 [35].

2.6. Comparison of Models: Conclusions for Practice

Table 2. Dependence of the parameter ${\left(t/R\right)}_{max}$ on the crack initiation threshold ${\overline{\sigma }}_c$.

Model	Poisson’s Ratio	${(\mathit{t}/\mathit{R})}_{\mathit{m}\mathit{a}\mathit{x}}$
		${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.4{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.5{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.6{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$
1	Constant, $\nu =$ 0.22	0.742 (100%)	0.706 (100%)	0.657 (100%)
2	Non-constant, $\nu ={\nu }_0+{\displaystyle \frac{{\nu }_n-{\nu }_0}{{\sigma }_{c,peak}}}{\sigma }_c$	0.802 (108%)	0.771 (110%)	0.730 (112%)

presents a comparison of the calculation results using the models from Section 2.4 and Section 2.5.

The comparison of simulation results (Table 2) shows that the model using the commonly adopted constant Poisson’s ratio underestimates the thickness of the surface cracked layer by approximately 10%. In practice, this means that a model with a constant Poisson’s ratio tends to underestimate the risk of failure.

2.7. Example: Random Deviations of Poisson’s Ratio from a Constant Mean Value

This section uses initial data from [31]: ${\sigma }_{c,peak}=246$ MPa, ${\sigma }_{t,peak}=14.9$ MPa, ${\nu }_0=0.16$, ${\nu }_n=0.29$, and mean value ${\nu }_{mean}=0.225$.

The values of ν lie within the interval 0.225 $\pm$ 0.0650 or, by the three-sigma rule, $0.225\pm 3{\nu }_{sdv}$ with a probability of 0.9973 [37]. Thus, the standard deviation is ${\nu }_{sdv}=0.065/3=0.022$, which corresponds to $0.022/0.225=0.098$ (or 9.8%) of the mean value.

To generate random Poisson’s ratio values, a standard normal-distribution number generator was used, with a mean (expected value) of 0.225 and a standard deviation of 0.022. Figure 8 illustrates the simulation results for three values of crack initiation threshold.

The solid lines in Figure 8 are analogous to those in Figure 6. Table 2 above contains the corresponding numerical estimates.

2.8. Example: Random Deviations of Poisson’s Ratio from a Non-Constant Mean Value

As in the previous section, this section uses the initial data from [36]: ${\sigma }_{c,peak}=246$, MPa, ${\sigma }_{t,peak}=14.9$ MPa, ${\nu }_0=0.16$, ${\nu }_n=0.29$, and ${\nu }_{mean}=0.225$. The values of ν lie within the interval $0.225\pm 0.065$ or, by the three-sigma rule $0.225\pm 3{\nu }_{sdv}$ with a probability of 0.9973 [37]. The standard deviation is ${\nu }_{sdv}=0.022,$ which corresponds to $0.022/0.225=0.098$ (or 9.8%) of the mean value.

In the considered example, ν increases linearly with the applied load and is determined by Equation (8), while the standard deviation also varies linearly as 0.098·ν. In accordance with the discrete approach mentioned earlier (Section 2.1), at each point of the loading process, a standard normally distributed random number generator was used to produce Poisson’s ratio values, with the expected value ν (8) and a standard deviation of 0.098·ν. Figure 9 shows the simulation results for three values of the crack initiation threshold $\overline{\sigma }$.

Table 2 contains additional quantitative estimates for Figure 9. These estimates represent the mean values of the dependent and independent variables. Since Poisson’s ratio in this case exhibits random deviations from its mean values, the relative thickness of the cracked layer will also have random deviations from its expected value. The variability is measured by the standard deviation (root mean square deviation), which quantifies the degree of dispersion or scatter of the data relative to the mean (expected) value.

2.9. Root Mean Square Deviation of the Relative Cracked Layer Thickness

The root mean square deviation of the relative cracked layer thickness is determined by the following Formula (9):

$${{\left({\displaystyle \frac{t}{R}}\right)}_{sdv}={\displaystyle \frac{d\left(\frac{t}{R}\right)}{d\nu }}\nu }_{sdv}.$$

(9)

Equations (6)–(8) are sufficient to calculate the derivative (10):

$${\displaystyle \frac{d\left(\frac{t}{R}\right)}{d\nu }}={\sigma }_{t,peak}\left({\sigma }_c-{\overline{\sigma }}_c\right){\left(\left({\sigma }_c-{\overline{\sigma }}_c-{\sigma }_{t,peak}\right)\nu +{\sigma }_{t,peak}\right)}^{-2}.$$

(10)

Figure 10 illustrates the change in standard deviation for the examples discussed in Section 2.4 and Section 2.5.

Figure 10 shows that the standard deviation estimates for the model with a non-constant Poisson ratio are lower than those for the model with a constant ratio. The upper estimates of variability in the cracked layer thickness (14%) exceed the variability estimate of Poisson’s ratio in the initial data (9.8%, see Section 2.8 above).

2.10. Lateral Pressure Effect

In previous sections, we focused on uniaxial compression, although underground rocks in natural conditions experience stresses in all directions. Therefore, tests conducted without lateral pressure do not reflect the true stress state of rock under natural conditions. However, uniaxial compression test results are widely used in various rock classification systems [1,7,38] due to technical feasibility and economic considerations. Uniaxial compression tests are relatively simple, and many researchers have extensively studied methods for analyzing test results and modeling, though new challenges have emerged [2,39,40]. For instance, pillars in room-and-pillar mining systems operate under uniaxial compression conditions [41,42,43]. However, in some cases, external reinforcement is advisable to create a limiting effect, reduce crack formation, and increase the load-bearing capacity of columns. The problem of analyzing parameters of artificial lateral pressure created by external reinforcement remains relevant today [41,42].

Section 2.10.1 and Section 2.10.2 analyze the effect of lateral pressure ($p=s·{\sigma }_c$) on cracked layer thickness in axially compressed granite specimens.

2.10.1. Model with Constant Poisson’s Ratio

Building upon and extending the results of [27], we can show that the relative cracked layer thickness is determined by the system of Equations (11) and (12):

$${\displaystyle \frac{t}{R}}={\displaystyle \frac{1-\frac{s}{n}}{1+\frac{{\sigma }_{t,peak}}{n({\sigma }_c-{\overline{\sigma }}_c)}}},$$

(11)

$$n={\displaystyle \frac{\nu }{1-\nu }}.$$

(12)

The physical meaning of the problem corresponds to computational results obtained when ${\sigma }_c\ge {\overline{\sigma }}_c$ and $s\le n$.

Using Equations (11) and (12), let us consider an example with initial data from [36]: ${\sigma }_{c,peak}=246$ MPa, ${\sigma }_{t,peak}=14.9$ MPa, and $\nu =0.225$. For the model with constant Poisson’s ratio, Figure 11 shows the influence of proportional lateral pressure $p=s·{\sigma }_c$ on the relative thickness of the cracked surface layer t/R as a function of axial stress ${\sigma }_c$ for three values of crack initiation threshold ${\overline{\sigma }}_c$.

As expected, lateral pressure reduces surface layer cracking. The following section presents the corresponding quantitative estimates.

2.10.2. Model with Variable Poisson’s Ratio

If Poisson’s ratio depends linearly on the axial stress, as shown in [1], then, expanding the results from [27], it can be shown that the basis of the model of the influence of the proportional lateral pressure $p=s·{\sigma }_c$ on the relative thickness of the destroyed surface layer $t/R$ are Equations (13)–(15):

$${\displaystyle \frac{t}{R}}={\displaystyle \frac{1-\frac{s}{n}}{1+\frac{{\sigma }_{t,peak}}{n({\sigma }_c-{\overline{\sigma }}_c)}}}$$

(13)

$$n={\displaystyle \frac{\nu }{1-\nu }}$$

(14)

$$\nu ={\nu }_0+{\displaystyle \frac{{\nu }_n-{\nu }_0}{{\sigma }_{c,peak}}}{\sigma }_c.$$

(15)

Equation (13) shows that $t/R$ depends linearly on lateral pressure when other conditions remain constant. Consider an example using initial data from [36]: ${\sigma }_{c,peak}=246$ MPa, ${\sigma }_{t,peak}=14.9\mathrm{M}\mathrm{P}\mathrm{a}$, ${\nu }_0=0.16$, and ${\nu }_n=0.29$. Figure 12 illustrates the simulation results for three values of the crack initiation threshold.

Table 3. The influence of lateral pressure on the thickness parameter of the cracked layer ${\left(t/R\right)}_{max}$.

Model	Poisson’s Ratio	s	${(\mathit{t}/\mathit{R})}_{\mathit{m}\mathit{a}\mathit{x}}$
			${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.4{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.5{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.6{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$
1	Constant, $\nu =$ 0.22	0	0.742 (100%)	0.706 (100%)	0.657 (100%)
		0.05	0.614 (83%)	0.584 (83%)	0.544 (83%)
		0.10	0.486 (65%)	0.463 (66%)	0.431 (66%)
2	Non-constant, $\nu ={\nu }_0+{\displaystyle \frac{{\nu }_n-{\nu }_0}{{\sigma }_{c,peak}}}{\sigma }_c$	0	0.802 (108%)	0.771 (109%)	0.730 (111%)
		0.05	0.704 (95%)	0.677 (96%)	0.640 (97%)
		0.10	0.605 (82%)	0.583 (83%)	0.551 (84%)

contains numerical estimates for comparison between the constant and variable Poisson’s ratio models.

The simulation results demonstrate that when ${\overline{\sigma }}_c=0.4{\sigma }_{c,peak}$:

▪

With lateral pressure equal to 5% of axial compressive stress, Models 1 and 2 predict a reduction in fractured layer thickness from 100% to 83% (17% decrease) and from 108% to 95% (13% decrease), respectively;

▪

Doubling the lateral pressure reduces the layer thickness from 100% to 65% (35% decrease) and from 108% to 82% (26% decrease), respectively.

Similar trends are observed for ${\overline{\sigma }}_c=0.5{\sigma }_{c,peak}$ and ${\overline{\sigma }}_c=0.6{\sigma }_{c,peak}$. The minor deviation from the linear $t/R$ versus lateral pressure relationship noted in Section 2.10.2 is attributed to numerical rounding effects.

3. Discussion

The simulation results (Table 3) demonstrate that when ${\overline{\sigma }}_c=0.4{\sigma }_{c,peak}$:

▪

At lateral pressure equal to 5% of axial compressive stress, Models 1 and 2 predict reductions in fractured layer thickness by 17% and 13%, respectively.

▪

Doubling the lateral pressure reduces the thickness by 35% and 26%, respectively.

Similar trends were observed for ${\overline{\sigma }}_c=0.5{\sigma }_{c,peak}$ and ${\overline{\sigma }}_c=0.6{\sigma }_{c,peak}$. The minor deviation from the linear $t/R$ versus lateral pressure relationship noted in Section 2.10.2 can be attributed to numerical rounding effects.

These findings are based on laboratory tests of high-compressive-strength granite specimens (${\sigma }_{c,peak}=246$ MPa) [36], as previously discussed (Section 2.8).

For comparative analysis, we examined practical case studies from the literature [43,44]. The cited study analyzed nine rock pillars, five of which had failed. Using data from [44], we reapplied our analysis to two pillars using the model from Section 2.10.1. Computational results obtained from Equations (11) and (12), with ${\sigma }_{t,peak}={\sigma }_{c,peak}/12$ and $\nu =0.24$, are presented in

Table 4. Effect of lateral pressure p on maximum fractured layer thickness parameter ${\left(t/R\right)}_{max}$.

Pillar	${\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$, MPa	s	$\mathit{p}=\mathit{s}·{\mathit{\sigma }}_{\mathit{c}}$	${\overline{\mathit{\sigma }}}_{\mathit{c}}=0.4{\mathit{\sigma }}_{\mathit{c},\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	${(\mathit{t}/\mathit{R})}_{\mathit{m}\mathit{a}\mathit{x}}$
Failed Pillar	22	0	0	8.8	0.695
		0.05	1.1		0.585
		0.10	2.2		0.475
Unbroken Pillar	17	0	0	6.8	0.695
		0.05	0.85		0.585
		0.10	1.7		0.475

.

From a practical standpoint, it is important to note that the relative estimates of fractured layer thickness ${\left(t/R\right)}_{max}$ were identical for both failed and intact pillars (Table 4). Consequently, this parameter cannot serve as a predictive indicator for comparative safety assessment. However, alternative models may prove more effective for evaluating damaged layer thickness.

The presented case studies demonstrate that lateral pressure reduces crack formation and enhances the load-bearing capacity of pillars. Practical implementation of this approach involves external reinforcement techniques such as shotcrete, wire mesh, and steel cables [39]. Reinforcement is effective when the tensile strength of reinforcing elements exceeds that of the host rock [45]. The literature reports indicate that pillar reinforcement is not always successful [43,46], highlighting the need for further research in this area. A key consideration is the determination of tensile stress in pillar reinforcement materials [44,47].

To determine tensile stresses in external reinforcement materials, we propose a mechanical model representing the reinforcement as a hypothetical thin-walled cylindrical shell subjected to internal pressure. We suggest considering the internal pressure on this hypothetical shell as equal to the radial stress, which we determine using Dinnik’s hypothesis [25,26], following the approach of work [27]:${\sigma }_R=n{\sigma }_c$, where $n=\nu /(1-\nu )$. For a shell with radius $R$, wall thickness $\overline{t}$, and material tensile strength ${\overline{\sigma }}_t$, we derive:

$${\displaystyle \frac{\overline{t}}{R}}={\displaystyle \frac{n{\sigma }_c}{{\overline{\sigma }}_t}}.$$

(16)

From Equation (16), we obtain the required shell thickness as a function of axial compressive stress ${\sigma }_c$:

$$\overline{t}={\displaystyle \frac{n{\sigma }_{c}R}{{\overline{\sigma }}_t}}.$$

(17)

Equation (17) shows that reducing the radius or increasing the tensile strength of the reinforcement material decreases the required shell thickness. However, radius reduction also decreases the cross-sectional area of the pillar, leading to increased shear stresses and potential pillar collapse. This represents a separate research problem beyond the scope of the current study. Within our framework, studies such as [48,49,50,51,52,53,54] are of particular scientific, methodological, and practical interest.

4. Conclusions

In this work, we apply mechanical system modeling techniques to analyze the influence of constant and variable Poisson’s ratios on brittle material specimens, with granite as a case study. The initial data for the simulation examples are taken from known experimental data available in the literature. The study yielded the following results.

The data analysis confirmed the variability of Poisson’s ratio under load. Lateral deformation grows faster compared to axial deformation, so the Poisson’s ratio $\nu =\left|{\epsilon }_{lateral}\right|/{\epsilon }_{axial}$ increases with increasing load (Figure 1).

A system of Equations (6)–(8) is proposed to model the influence of Poisson’s ratio variability on crack formation in the specimen’s surface layer under uniaxial compression. A comparison of simulation results (Table 2) showed that a model with a constant Poisson’s ratio underestimates the thickness of the cracked surface layer by approximately 10%. In practical terms, this means that a model with a constant Poisson’s ratio underestimates the risk of failure.

Using the three-sigma rule, a methodology is proposed for analyzing random deviations of Poisson’s ratio from a non-constant mathematical expectation (Section 2.8).

The influence of lateral pressure on the thickness of the cracked surface layer was investigated. A comparison of simulation results with constant and non-constant Poisson’s ratio (Table 3) showed that increasing lateral pressure linearly reduces the thickness of the fractured layer—by 35% and 26% in one of the examples, respectively. For practical applications, this means that a model with a variable Poisson’s ratio leads to more conservative results, i.e., it does not overestimate the safety factor.

The discussed examples confirm the increased strength of columns due to external reinforcement. To determine tensile stress in the external reinforcement material, Equation (17) is proposed for iterative determination of the required thickness of a conditional reinforcing shell depending on axial compressive stress.

Although the proposed models match experimental data, it is important to note that they remain approximate. Therefore, further research is required to evaluate the influence of Poisson’s ratio on the strength and crack formation in rocks and concrete (as an artificial rock analog), particularly focusing on nonlinear models where Poisson’s ratio varies with load.