1. Introduction
The accurate characterization of a material’s mechanical properties is mandatory to successfully determine its strength and, consequently, its suitability for specific engineering projects and applications. This requirement not only applies to manufactured materials employed in industrial products, but also to naturally generated ones such as rock materials, which are involved in engineering projects related to underground work, mining or civil topics. Consequently, a precise characterization is usually required to determine the capabilities of a given material subjected to various stress conditions. To achieve this, its tensile, compressive and triaxial strengths at different stress ratios must be obtained.
In the case of ductile materials, it is common to subject a specimen to direct tensile tests. However, for brittle materials, any limitations to achieving the desired dog-bone shape during the specimen preparation process or the possibility of inducing cracks when fixing the specimen to the testing device mean that an indirect tensile test [
1,
2,
3,
4,
5,
6,
7,
8], also known as the Brazilian test [
9], is still preferred. Nevertheless, new procedures to determine the uniaxial tensile strength (UTS) are increasing in popularity [
10,
11], as they provide a true uniaxial stress state (UTS) rather than a combined biaxial one [
12,
13].
The determination of the uniaxial compressive strength (UCS) is typically performed using the standardized uniaxial compression test (UCT), employing prismatic or cylindrical geometries for most materials [
14,
15,
16,
17,
18,
19,
20]. Nevertheless, indirect methods, such as the point load test [
21,
22,
23,
24] or the Schmidt–Hammer test [
25,
26], are becoming increasingly popular, as no particular standardized geometry shape must be reproduced; this is extremely useful in those cases where the available samples can be limited or difficult to obtain due to location.
Although the UCT has been meticulously standardized by several societies and institutions to ensure its trustworthiness and reliability, discrepancies between its theoretical foundations and the expected experimental results have been documented. From an elastic perspective, the material considered is subject to a uniform compression stress along the contact between the specimen and the platens, which induces a homogeneous compressive stress field inside the specimen. However, it is widely accepted that in the vicinity of the contact, a non-homogeneous triaxial stress field is generated due to shear stresses caused by the different stiffness values between the materials of the platen and specimen. Notably, these shear stresses may evolve into friction when the applied load allows slippery contact conditions to be induced. Several theoretical formulations seek to capture this end-effect phenomenon [
27,
28,
29]; however, the influence of constrained displacement at end points is still under discussion among the scientific community. In this sense, different slenderness ratios have been proposed to limit its influence [
30,
31,
32]. Nevertheless, consensus on the relative stiffness of post-peak behavior exists, which is considered to be dependent on the friction between the platen and the specimen, instead of being an intrinsic property of the tested material [
31,
33].
All of these discrepancies may explain the variability in the accepted failure patterns related to the UCT [
34,
35], although the specific testing conditions responsible for them remain uncertain. It is worth highlighting here that some discrepancies may also exist in the characterization of elastic constants depending on the measuring technique employed, although novel research directions currently allow the results between procedures to be corrected [
36,
37,
38].
The material strength for confined stress states is addressed by the triaxial compression test. It is conducted using a Hoek cell; not only are stresses on the ends of a cylindrical specimen applied, but a uniform fluid pressure is also applied to its lateral surface [
39,
40]. Thus, by selecting adequate ratios between the axial and confining stresses, the material’s behavior can be predicted. Nevertheless, this test implies that the minor (
) and intermediate (
) principal stresses are equal; thus, deviations from the triaxial test results must be expected when this requirement is not fulfilled [
41,
42].
To determine a material’s strength under real-world conditions, failure envelopes must be obtained by combining the results from a Brazilian test, a uniaxial compression test and a triaxial compression test. The failure envelopes of brittle materials are usually established by means of the Mohr–Coulomb or Hoek–Brown criterion [
39,
43]. The former considers a linear relationship between the major (
) and minimum (
) principal stresses, whereas the second establishes a non-linear envelope, as shown in Equations (1) and (2), where
is the friction coefficient and
and
are material constants, setting
for the case of an intact rock.
It is worth highlighting that the sign criterion used relates to compressive and tensile stresses as positive and negative stress values, respectively, as is commonly used in the field of rock mechanics. Nevertheless, it is pertinent to emphasize that the Mohr–Coulomb criterion is mainly formulated to address the compressive area (
) of the
chart (
Figure 1). Hence, its straightforward lengthening into the tensile area (
) would lead to excessively imprecise results. To soften this limitation, a truncated Mohr–Coulomb (TMC) criterion is used instead (
Figure 1), which restricts the allowable tensile states (
) to those under the maximum threshold value set by the indirect tensile strength obtained in the Brazilian test (
). Therefore, the compressive and tensile values are encompassed in a more consistent failure envelope.
This approach is conservative for bearing purposes, as elastic theory guarantees higher tensile values before reaching failure. In this sense, the Hoek–Brown criterion is even more conservative, as it equates the indirect tensile strength with the uniaxial one, thereby transforming the point () into ().
Due to the relevance of accurately determining UCS for engineering purposes, the influence of relative stiffness in the induced stress field inside the specimen is investigated to ascertain UCT suitability to generate homogeneous compression fields. For this purpose, numerical UCT simulations considering ratios between the platen () and specimen () stiffnesses of 3, 1 and 0.05 were performed. Subsequently, and based on the induced stress fields obtained, the location of the failure initiation point for brittle materials and its correlation with main failure patterns registered in the current literature are discussed. The results highlight that uniaxial compression stress states at the failure initiation point are only guaranteed for platens with similar stiffness to the tested specimen, and that brittle failure patterns depend on relative stiffness.
2. Numerical Model
To address the influence of relative stiffness between the platen and the specimen in the output results obtained from the standardized UCT, numerical simulations were performed using the commercial software FLAC 7 3D version 7.00, which is widely used in the field of rock mechanics modeling [
44]. Therefore, the elastic behavior was predicted using a finite difference method (FDM) [
45,
46]. The latter is based on substituting the partial derivatives of a desired function by finite differences on a given domain [
44]. Hence, the continuity of this method’s results is one of its essential characteristics, but it also reveals one of its major drawbacks, which is that it does not adequately address fractures in the material. The materials in our proposed analysis were limited to the elastic range, with no damage evolution, as only the failure initiation point was determined; FDM therefore remains a suitable choice to model the UCT for our stated purposes.
Standards suggest different acceptable length ()-to-diameter () ratios for the tested specimens. Nevertheless, the obtained results can be considered acceptable if the aspect ratio () is between 2 and 3. Indeed, a value of is simultaneously proposed by main rock mechanics standards. For this reason, the specimen used in the numerical model is 50 mm in diameter and 125 mm in length. It can be argued that such dimensions do not strictly fulfill the minimum 54 mm diameter standard. Nevertheless, this recommendation is due to the intrinsic heterogeneity of most rock materials, in which grain sizes of several millimeters may behave as large inclusions, and thus cause the test results to deviate.
Regarding the platen dimensions, they were all 50 mm in diameter and 40 mm in thickness to ensure that most standards were simultaneously satisfied.
As the geometries used in the numerical model are theoretically perfect—their shape exactly matches that of ideal cylinders—no further considerations regarding geometric deviations were made. However, this limits the results to the case in which specimen ends can be considered perfectly flat for practical purposes.
All the specimens and platens were considered to be homogeneous and isotropic materials. Furthermore, as the current research addresses the failure of brittle materials, their plastic behavior was considered neglectable; therefore, the specimens had perfectly elastic behavior up to the initiation of failure.
On each mesh element, the stress state can be defined by their maximum (), intermediate () and minimum () principal stresses and their directions. However, as only and are considered in most applied failure criteria for brittle materials, the values will be neglected in subsequent sections of this article.
To accurately reproduce the contact phenomenon reported during test execution, we only imposed displacements as boundary conditions. Hence, instead of distributing stresses along the contact surfaces in an arbitrary manner, the top of the upper plate ( mm) was displaced towards the specimen with a velocity of m/s, in order for equilibrium conditions to be fulfilled between each calculation step. No other restrictions or assumptions were made on the shear stresses or horizontal displacements at the surface of the top platen. Under this approach, stresses are induced inside the specimen due to the compression caused by the platen during its displacement. Although the use of an interphase is widely accepted as the means to solve the problem under certain contact restrictions, the authors consider that it may cause biased results, as many assumptions may not necessarily hold true during real test executions. For this reason, the simulation was performed without the use of interphase between the elastic bodies. Contrarily, the model was meshed, assigning a common node for the specimen and the platen points in contact. Consequently, the results are limited to those cases where neither slippery nor relative horizontal displacement between contact surfaces is allowed. Hence, this analysis cannot be extended to the case of friction or the use of effective lubricants.
The addressed problem is elastically three-dimensional; however, considering that platens and the specimen are perfect solids of revolution and that contact conditions have radial symmetry, the problem can be treated as an axisymmetric one. Therefore, the stress state on each element is independent of its angular position; thus, only 45° of the cylinders were used as geometric input to the model (
Figure 2).
The mechanical properties of the assigned materials, as well as the ratio that relates the platen (
) and specimen (
) Young’s modulus and Poisson’s ratio (
), are shown in
Table 1. The M2 material properties were elected to match representative values of rock materials [
40], whereas M1 and M3 were aimed at representing a much stiffer material and an extensively softer one, respectively. For this purpose, mechanical properties similar to those of steel and polymethyl methacrylate were selected [
47].
Additionally, to address the influence of platen stiffness on the failure initiation point, as well as in the associated failure pattern, three different limestone lithotypes were used [
48]. These limestones were selected to cover a wide spectrum of different
,
and
values, as shown in
Table 2.
A new stress defined as , equal to 70% of the limestone lithotype 2 UCS value, was selected to normalize all of the stress states and simultaneously ensure that the results were limited to the elastic range. Once the value was reached in the points of the outer circumference, the platen stopped its movement.
4. Platen Stiffness Influence on Failure Criterion
The UCT is aimed at determining the UCS of a given material. Within this context, it may be argued that only failure criteria on the compressive area of the
chart would be sufficient to address the failure initiation point in standardized UCT specimens. However, based on the results shown in
Section 3, it becomes imperative to consider potential tensile failure in the analysis of the failure initiation point. Thus, even if the aim of this test is to provoke failure of the specimen in a point with coordinates
, meaning that in the ideal scenario no stresses other than compressive stress will be induced, it is essential to verify this scenario for all feasible combinations of relative stiffness. The tensile strength of most brittle materials is not determined by the direct tensile test, but by the Brazilian test. Therefore, the stress state in the failure initiation point is no longer uniaxial; in fact, it is subjected to biaxial stress where
. It is worth mentioning here that this last statement is widely accepted among the scientific community when failure is initiated at the centre of the specimen, which is only the case for certain contact conditions [
12,
49]. Furthermore, even in those cases where the failure initiation point is located in the centre of the specimen, deviations from the accepted ratio
arise as a function of the contact length between the specimen and the jaw during test execution [
7].
The failure envelope of each material in the tensile area (
) was defined by a linear criterion that connects the points
and
of the material. Consequently, the stress state regarding the Brazilian test is placed in the
chart, considering that failure is initiated in the centre of the specimen, and that the influence of the contact length is negligible. Hence, the failure envelopes for the compressive and tensile areas are defined by Equation (1)—the Mohr–Coulomb criterion—and Equation (4), respectively.
To understand how failure is initiated under varied contact conditions, the interactions between the induced stress field and the failure criteria for each case must be carefully analysed. Firstly, interactions between stress states and the failure envelope in the tensile area were addressed (
Figure 8). In the case of stiffer platens (
), its characteristic tensile bulb suggests that failure is likely reached in the tensile area rather than in the compressive one (
). Furthermore, the failure initiation point correlates with significantly high negative
values located far from the desired vicinity of the
axis; this yields results that diverge significantly from the intended uniaxial stress state. For all considered limestone specimens, the failure initiation point is located at the vertical axis and at a distance of approximately 16.7% of the total length of the specimen from both ends, with a ratio
. This result applies independently of the failure criterion used, so even when the Mohr–Coulomb criterion is lengthened from the compressive to the tensile area, the predicted failure initiation remains unchanged and fails due to the existence of a
stress component that deviates testing results from the originally desired uniaxial stress state.
In contrast, the ratio induced a stress state in the failure initiation point that can be assumed to be uniaxial compression for practical purposes. Additionally, it is located in the compression area, thereby eliminating the influence of tensile stresses on the obtained strength and establishing a stronger relation with the theoretical foundations of the test. Nevertheless, it is worth reiterating that the load ratio highly encourages one to take these results as the real UCS value for any further engineering applications. As the failure is always initiated in the compression area of the plot, no tensile failure envelope needs to be considered.
Independently of the failure criterion chosen, all the materials reached failure by the influence of tensile stresses in the case of softer platens (). Even if only the Mohr–Coulomb criterion is used for the compressive region, but lengthened to the tensile area, the specimen reaches failure in the tensile area of the chart. In that case, the stress ratio at the failure initiation point is , indicating that tensile stress surpasses compressive stress significantly, exceeding reasonable limits to correlate the result with the uniaxial compressive strength of the material. Consequently, it is not recommended to use platens that are significantly less stiff than the tested specimen. It is worth mentioning that softer platens would induce failure in all considered materials, whereas the rest of the platens would not induce failure in any of the specimens.
Figure 9 simultaneously shows the failure envelope for the tensile and compressive areas for all considered rock materials. In addition, it not only includes the stress states previously used in
Section 3 and up to this point in
Section 4, but also those incremented by a factor of 5, the approximate moment in which failure is reached by compression in the case of softer (
) platens (
Figure 9). Stress fields prove that failure is caused by the existence of a tensile stress component
in stiffer (
) and softer (
) platens.
Correlating previous initiation failure points within the
chart with their actual coordinates (
) inside the specimen provides meaningful insights into the predominant failure patterns usually documented as the test output. In the case of stiffer platens (
), the generated failure pattern exhibits a characteristic cone shape (
Figure 10a). This result applies independently of the failure criterion used (tensile or compressive), so even when the Mohr–Coulomb criterion is lengthened from the compressive to the tensile area, the predicted failure initiation point remains unchanged and fails due to the existence of a
stress component that deviates testing results from the uniaxial stress state originally desired.
Platens with the same stiffness as the tested material (
) induced a stress field inside the specimen that initiates failure at the outer circumference of both ends, generating a failure pattern defined by an oblique plane that connects them (
Figure 10b). Therefore, all the similar failure patterns strongly suggest that failure is reached in the compression area and in a uniaxial stress state. This evidence facilitates a preliminary assessment of test validity results without needing any additional tools.
In
Figure 10c, the location of the failure initiation point for the case of softer platens (
) is depicted. This point is situated along the longitudinal axis of the specimen, suggesting a failure plane that contains this axis and splits the specimen into two symmetrical halves. Consequently, the induced stress state highly correlates with the phenomenon of axial splitting during the execution of the test.
The findings presented in this section apply for the materials described in
Table 2. Although they cover a wide spectrum of material properties, a straightforward extrapolation of these results to all materials cannot be made, as there is no reason to refuse the possibility of slight differences in the stress field that lead to slightly different failure initiation points in the case of stiffer or softer platens, where points located closer or further to the ends would be acceptable. Nevertheless, it can be ensured that at least three different failure patterns exist, and that they depend on the relative stiffness for specimens with length-to-diameter ratios of 2.5. However, this affirmation can reasonably be extended to those cases where a specimen’s slenderness is sufficiently elevated to guarantee an area of a nearly uniform compressive stress field at the mid height of the specimen, while simultaneously being low enough to preclude buckling effects.
Although the preceding results were derived from numerical simulations, experimental evidence of this phenomena can be found in the current literature.
Figure 11, extracted from reference [
33], shows the standardized UCT performed on cylindrical and prismatic specimens after modifying the contact conditions. In particular, four different cases were addressed. The reference cases in
Figure 11 were executed subject to the recommendations set by the standards, whereas glued, grease, Teflon and brush plates were used to vary the boundary conditions at specimen ends, from impeded horizontal displacement to the complete absence of friction between platen and specimen. In both the cylindrical and prismatic geometries, the experimental results are in obvious agreement with the predictions made by the numerical simulations. The reference and glued sheets show a failure pattern defined by the formation of two cones due to the significantly higher stiffness of the platens and the impeded movement sets by glue in the ends of the specimen. After reducing the influence of friction by using grease, the failure pattern obtained experimentally is an oblique plane joining both top and bottom ends, exactly as in the numerical simulations. It is worth noting that the same material and the greased material used in reference [
33] was used to reduce the influence of friction and shearing stresses along the boundaries; consequently, both achieved similar results. Eventually, platens with significantly lower stiffnesses led to axial splitting or failure planes parallel to the longitudinal axis of the specimen, as previously predicted by simulations.
It is important to emphasize that different failure patterns are caused even if the specimen’s material is considered to be homogeneous and isotropic. Hence, the influence of platen stiffness in more complex real scenarios may be boosted by additional external factors not considered in this analysis, such as high temperature and melting and freezing cycles.
5. Conclusions
Numerical simulations were performed to determine the influence of platen stiffness on the induced stress field inside specimens subjected to the standardized uniaxial compression test (UCT). Subsequently, the locations of the expected failure initiation points and their dependence on the induced stress field and their related failure patterns were addressed. The results show that when the relative stiffness between the platen () and the specimen () is , a cone-shaped failure pattern caused by tensile failure is produced. Additionally, a compression bulb in the vicinity of the contact due to the impeded horizontal movement in the top and bottom ends of the specimen is generated for this stiffness ratio. Conversely, for , the induced tensile stresses are concentrated in a narrow band in the vicinity of the contact, leading to failure initiation points located along the longitudinal axis of the specimen; therefore, a failure pattern characteristic of axial splitting was generated. However, when the platen and specimen stiffnesses are equal (), a uniform and homogeneous stress field is induced inside the specimen, where maximum stresses can be considered constant and can be neglected for engineering applications. In this latter case, the failure pattern is formed by an oblique plane joining the outer circumference of both ends of the specimen. Therefore, platen material should not be established independently from the tested material. In fact, platen stiffness should be determined considering the material to be characterized, aiming to ensure a stiffness value as close as possible to that of the specimen’s. Furthermore, these results allow one to establish the validity of testing results by a simple visual inspection of the generated failure pattern. Finally, those cases where slippery occurred between platen and specimen points along the contact cannot be addressed, considering the findings exposed in this study, as no relative displacement between points was allowed in the numerical model used. Contrary to this, when lubricants are employed to reduce the influence of shearing stresses at the contact, the results may correspond to the case of equal stiffness (), as no horizontal displacement at the specimen ends is constrained; however, further research on the possible deviations from the treated problem must be carried out to ensure definitive conclusions.