Stress-rate dependency of uniaxial compressive strength of hard rock with regard to test procedure standards

The uniaxial compressive strength test of hard rock is one of the most worldwide applied tests for characterization of hard rock in rock engineering and engineering geology. The uniaxial compressive strength as the results of this test is a basic parameter, used, for example, for the design of rock engineering structures. In the commonly applied standards, stress-controlled test procedures using constant stress rates are recommended by a wide range of stress rates varying between 2 and 60 MPa/min and/or a specific minimum test time. Though the effect of stress-rate dependency of hard rock is generally known, most investigation is focused on dynamic action behavior using high stress rates and/or fast actions. Strain-controlled test procedures are often used as well. For stress-controlled test procedures within the recommended range of stress-rates, the data base is rather poor, the effects to the results are reported by only very few researchers. The research described in this paper aims to close this gap and focuses on the stress-rate dependency of uniaxial compressive strength of hard rock. Seven different stress rates varying between 1 and 100 MPa/min on five different types of hard rock (quartzite, granodiorite, gabbro, sandstone, basalt) using five to fifteen single tests per stress rate have been executed. A significant increase of uniaxial compressive strength by increasing stress rates has been stated; the increase may not be ignored in assessing rock strength in rock engineering projects. The effect has to be considered especially when different parties are involved in the site investigation programs.


Introduction
The uniaxial compressive strength test of hard rock is one of the most worldwide applied tests for characterization of hard rock in rock mechanics and engineering geology. The uniaxial compressive strength as the results of this test is a basic parameter used, for example, for the design of rock engineering structures. A number of standards for the test procedure of testing the uniaxial strength are available; among them, the recommendation for determining the compressive strength and deformability, issued by the International Society for Rock Mechanics (ISRM), and the similar standard, issued by the American Society for Testing and Materials (ASTM) D 7012-14, is commonly applied. In Germany, German standard DIN 18141-1 or the recommendation of the German Geotechnical Society (Deutsche Gesellschaft für Geotechnik, DGGT ) is used normally. In France, the "Norme Francaise" (French Norm) P 94-420 is used. An overview of selected standards is displayed in Fig. 1.
All of the above quoted test standards recommend stresscontrolled test procedures by static or static-like stress rates (either in the units MPa/s or MPa/min) and/or minimum test durations. An overview of the different recommended stress rates is shown in Fig. 1 as well. ISRM and ASTM standards recommend static or static-like stress rates between 0.5 and 1.0 MPa/s (equal to 30 to 60 MPa/min) but recommend different minimum test time (ISRM: 5 to 10 min, ASTM 2 to 15 min). German standard DIN 18141-1 or the recommendation of the German Geotechnical Society ("Deutsche Gesellschaft für Geotechnik" (DGGT )) recommend static or static-like stress rates of 0.033 to 0.1666 MPa/s (equal to 2 to 10 MPa/min) with a minimum test time of 5 min. The recommendation of the German Geotechnical Society additionally recommends the use of a stress rate of 1.0 MPa/s (equal 1 3 4 Page 2 of 13 to 60 MPa/min) for chisel use for TBM tunneling purposes. A further German standard DIN EN 1926 is used when hard rocks are applied as construction material; this standard specifies a stress rate of 1.0 MPa/s (equal to 60 MPa/min), but without a minimum test duration.
The stress-rate dependency of the uniaxial compressive strength of hard rock is generally known: the uniaxial compressive strength increases by increasing stress rates; however, research focused on stress-controlled test procedures by static or static-like stress rates as recommended in the above quoted standards is astonishingly rare. Only three studies are covering the complete range of stress rates given in those standards (Fig. 2); all of them have a weak statistical base due to the few numbers of tests per stress rate (Table 1). The data base allowing to quantify the effect to the test results is therefore very unsatisfactory. A systematic research of the stress-rate dependency of the strength of hard rock in the range of the stress rates recommended in the standards is missing indeed. The results presented in this paper aim to contribute to close this data and knowledge gap.
Many research works are undertaken using much higher stress rates respectively dynamic loading far beyond stress rates of 1.0 MPa/s. In those studies, it was observed that the compressive strength of hard rock increases exponentially by increasing stress rates beyond ca. 1·10 3 MPa/s (equal to 60.000 MPa/min) with a factor of 5 to 6 compared to the strength observed at static or static-like stress rates. Such research is reported, for example, in Zhao et al. (1999), Zhao and Li (2000), and Frew et al. (2001). A good overview on that topic can be found in Zhang (2014).
It can be seen that the stress rates recommended in the standards are only partially covered by the available studies. Only Stowe (1969), John (1972), and Efimov (2007) tested the strength in this range of stress rates. John (1972), Efimov (2007), and Komurlu (2018) compared different types of hard rock. Obert et al. (1946), Stowe (1969), and Zhu et al. (2019) have tested only one type of hard rock using different stress rates. Moreover, the number of tests per stress rate is in many cases limited to three to five tests. The conclusions are therefore in need of improvement from a statistical point of view. The results of those research are quoted in Table 1. When the authors used a range of stress rates larger than the range recommended in the standards, only these results have been selected and quoted. Efimov (2007) provided results not by tabled numbers but by stress rate strength plots only, where the results cannot be reported and displayed exactly. Moreover, Efimov used rectangular specimens where all other authors used cylindrical ones.
This research reveals that there is an increase in strength by increasing stress rates in a range between 4 and 41% compared to the strength of the lowest stress rate. It is assumable that this large variation in differences might by caused partly by the small number of single tests, different geometry of

Rock types and test program
For this test program, cylindrical specimens from five rock types have been tested, those are the following: 1. Quartzite (noted as Q(A)) 2. Granodiorite (noted as GrD(A)) 3. Gabbro (noted as Gbr(A)) 4. Sandstone (noted as Sst(A)) 5. Basalt (noted as B(A)) Samples of these rocks have been collected as rock blocks from either construction sites or quarries. The size of the samples ranges between 20 and 50 cm. Cores for processing cylindrical specimens for testing have been bored by laboratory core drilling (Figs. 3 and 4).
The basalt and the quartzite are fine to very fine-grained rocks; the gabbro and the granodiorite are fine-grained to medium-grained rocks. The particle size of the sandstone is generally fine to medium sand grains ( Table 2). The rock matrix of all rock types is homogenous and isotropic. The sandstone has sedimentation planes where the direction of core drilling was rectangular in order to assure the comparability of the results. The cementation of the crystals of the quartzite, granodiorite, gabbro, and basalt is generally very good; the sand grains of the sandstone show in contrast a weak cementation.
The diameter of all cores was between 44 and 45 mm. All specimens have been prepared with a ratio length (L) to diameter (D) L/D between 2.00 and 2.08. Twelve specimens of the quartzite had ratios length to diameter L/D somewhat smaller (up to L/D = 1.88) resp. larger (up to L/D = 2.13). The processed ratios L/D are shown in Fig. 5.
The end planes of the specimens have been prepared according to the recommendation of the German Geotechnical Society with a roughness below ± 0.1 mm and been paralleled by less than ± 3'. The specimens have then been oven-dried (105 °C) to 0% water content and then cooled down in an exsiccator. The geometry has been determined and each specimen was photographed. After, they were stored in dry conditions until their testing. The dry densities of the specimens have been determined; the results are shown in Table 2 and Fig. 6.
All specimens have been inspected with regard to inhomogeneities such as microcracks, etc. Specimens with such inhomogeneities were separated and not been tested for this program or have been excluded in some cases after testing if inspection of the failure planes revealed hidden inhomogeneities, which is the reason for the differences in numbers of tests per test parameter (Table 3). In Fig. 7, photos of selected specimens are displayed. The last photo shows a   (Table 3). A number of five to fifteen single uniaxial compression tests for each stress rate have been executed, a number of 249 single tests in total. After evaluating the first results, the test program has been focused more on the stress rates 1 MPa/min, 10 MPa/min, and 100 MPa/min. The test program on the gabbro and the basalt has been merged completely to testing these three stress rates only.
All tests have been executed using a rigid testing machine according to DIN EN ISO 7500-1 standard. The strength was defined as the peak strength of the specimens. Photos of selected specimens before and after the test are shown in Fig. 8. The granodiorite, the gabbro, and the sandstone showed shear failure behavior (Fig. 8). Both the quartzite and the basalt showed vertical splitting and very brittle behavior with explosive-like failure. After explosive-like failure of those rocks, only more or less small fragments of these specimens could have been collected (Fig. 8).

Results
The results are given in Table 4 and are displayed in Fig. 9a to f as box plot diagrams where the results are plotted against the tested stress rates in semi-logarithmic scale. The plots contain the median and the mean values as well as the regression curves.
It can be seen that the uniaxial compressive strength values increase by increasing stress rates. Regression analyses show that the overall best regression of the increase fits by a logarithmic regression curve approach (which is a linear curve in a semi-logarithmic scale plot). These regression curves are plotted in those diagrams as well as the mathematical description and the coefficient of correlation R 2 .
The levels of the uniaxial compression strengths range from ca. 250 MPa (basalt), ca. 200 MPa (quartzite), ca. 160 MPa (gabbro), ca. 100 MPa (granodiorite), to ca. 45 MPa (sandstone), and comprises therewith a large, representative variation of strengths of hard rocks from medium strong to extremely strong rock (terms according to EN ISO 14689-1).

Stress-rate dependency of the uniaxial compression strength
As mentioned before, the uniaxial compressions strength increases by increasing stress rates. Regression analyses show that the overall best regression fits by a logarithmic regression curve approach. This approach is used for a comparison of the increase in strength of the different rock types.
The different levels of strength can be compared by normalization of the results to the strength stated at the lowest stress rate of 0.01666 MPa/s (1 MPa/min). These normalized mean strength values are plotted against the stress rate (Fig. 10).
This analysis reveals that the strength increases by a factor from ca. 1.20 (equal to ca. 20% in increase) for  The increase in strength within the ranges of stress rates recommended in the standards is about 7 to 11%, resp. 3% for the sandstone when the German standard DIN 18141-1 or the recommendation of the German Geotechnical Society with stress rates of 0.033 to 0.1666 MPa/s (equal to 2 to 10 MPa/min) is applied. The differences when the recommendation of the ISRM or the ASTM standard D 7012-14 is applied are in an order from 3 to 4% resp. 1% for the tested sandstone.
The plot in Fig. 10 displays the increase of the mean statistical values. An analysis of the strengths of different specimens coming from one single rock block sample shows the same effect. This is shown in Figs. 11 and 12 where the single results of tests on specimens coming from sample block no. 05 (nine single tests) and block 11 (twelve single tests) of the sandstone are displayed. The uniaxial compression strength of each stress rate is more or less constant while the plots reveal the increase of strength within the same block sample.
These single results of the sandstone are comparatively homogenous. An analysis on 16 tests on specimens from five block samples of the more heterogenous granodiorite shows qualitatively an increase of strength, too. In Fig. 13, these test results of 16 single tests of specimens coming from these five rock block samples of the granodiorite are plotted. The overall variation in increase of uniaxial compressive strength for this rock type is displayed in Fig. 14 as normalized uniaxial compressive strength plot. The increase of strength is in an order of more than 50% in the range of stress rates between 1 and 100 MPa/min, while the average statistical increase is 22%.
It can be assumed that the strength of a single block sample is more or less homogenous as it is proven by the results of block sample 28 on tests where two specimens at a stress rate of 2 MPa/min have been tested. The tested strength is 113.5 MPa, resp. 107.0 MPa at this stress rate, which can more or less considered the same level. The strength tested at higher stress rates of 10 MPa/min is 118.9 MPa and 132.4 MPa at 50 MPa/min (Fig. 13) which is equal to an increase of ca. 8% (10 MPa/min) resp. ca. 20% (50 MPa/ min).

Conclusions
The presented research proves by a broad statistical base that there is a dependency of the uniaxial compressive strength to the stress rates used in uniaxial compression strength tests even when static or static-like stress rates as recommended in different test standards are applied. This effect can be deduced from both the overall increase of strength when the results of all tests are analyzed (Fig. 9a to e) and from the analysis of the results of tests on specimens coming from single rock blocks (exemplarily shown in Figs. 11 and 12 for sandstone and Figs. 13 and 14 for granodiorite). That means that the expression that a specific rock type has a specific strength (tested at static or static-like stress rates) has to be treated with caution.
The average increase in strength ranges from ca. 20% to ca. 27% for the tested stress rates from 1 MPa/min to 100 MPa/min for the quartzite, the granodiorite, the gabbro, and the basalt. The sandstone shows an increase of only ca. 7%. It is assumable that the reason for this is the cementation of the crystals/the grains. The cementation of the sandstone grains is much weaker than the cementation of the other rock types.
The results prove the relevance of the stress rate to the results and to characteristic values deducted from uniaxial compressive tests. In specific rock types, as the example of the granodiorite proves, slight differences in stress rate can cause great quantitative differences in results. It must therefore be differentiated between the overall statistical difference in strength at different stress rates and single or few results of strength testing. The latter point has to be considered in ground investigation projects in rock engineering when/where in many cases only few tests are undertaken. Single results might show much higher strengths at higher stress rates than the average increase of strength suggests. This might have major effect to the design of the construction process, e.g., the design/choice of rock cutting tools, bits for rock drilling, and consumption of explosives.
This effect has also to be considered for the determination of a characteristic value of a specific rock type either if a "cautious estimate" or a statistically based analysis-as suggested, for example, in the Eurocodes-is used for it. Given that the coefficient of variation is for all stress rates at the same level, the determination of a characteristic value would lead to different strength values.
The increase in strength within the ranges of stress rates recommended in the standards is about 7 to 11%, resp. 3% for the sandstone when the German standard DIN 18141-1 or the recommendation of the German Geotechnical Society with stress rates of 2 to 10 MPa/min is applied. The differences when the recommendation of the ISRM or the ASTM standard D 7012-14 is applied are in an order from 3 to 4% resp. 1% for the tested sandstone. It is recommended to consider this effect in future development of the standards. The increase of strength by increasing stress rates at static or static-like stress rates given as recommendations in the ISRM and ASTM standards may be in an average order of up to 4% between 30 to 60 MPa/min and for the German standards up to 11% for stress rates of 2 to 10 MPa/min. The increase, especially for the German standards, is significant and may not be ignored. It is recommended by the author to consider the stress rate effect in all upcoming revisions of the standards. The applied stress rate should be clearer displayed in test protocols and reports. It is suggested to add an index to the parameter term UCS (for example, UCS 0.5 as a parameter term when the stress rate is 0.5 MPa/s or UCS 1.0 as parameter term when the stress rate is 1.0 MPa/s). It is recommended that a specific level of stress rate has to be defined for strength testing of rocks in the upcoming revisions of the standards, e.g., the Eurocode 7, which are currently being prepared.
In site investigation of rock engineering projects, the test standard and the type of test (strain rate-controlled vs. stress rate-controlled) have to be defined as well as the stress rates (or strain-rates) to be applied. Only keeping those test parameters constant over the duration of such projects from the first site investigations to the tests during the construction phase guarantees comparable results for both reliable engineering and for the contract management. Contractually have the chosen parameters to be fixed in the project contracts-especially when different laboratories in a project are involved. Tender documents and contracts defining properties of hard rock by its uniaxial compressive strength must be clear on this point. Using different stress rates for testing the same geotechnical unit/rock would lead to different characteristic values, i.e., to misleading design of the structure or to the design of the construction process.
It is assumable that the reason for the increase is the time-dependent effect of the internal stress rearrangement in the rock matrix resp. the crystal/sand grains. This point should be subjected to further research by investigating the dependency of the strength vs. stress rate with regard to the material composition. This could be undertaken by microcomputer tomography (micro-CT) scans during the testing, for example. Mineralogical behavior under different stress rates must then be investigated in order to deduce material laws. Further testing of strength of rock types with weak cementation of the grains/minerals is required in order verify this assumption.
Further research on this topic must be undertaken in order to enlarge the database and to quantify the range of increase in strength as well as its dependency on different rock types. Since there are also large numbers of research on this topic undertaken by using strain-controlled test procedures, a research study should also investigate comparisons of stresscontrolled to strain-controlled tests using a large number of specimens of the same origin.