1 Introduction

The tension strength of a rock mass is mobilised during excavation when confinement is reduced, and the excavation surface is able to dilate prior to support being installed (Park 2023). At the intact scale, yielding in tension has been identified to commence at 30% of the UCS resulting in the ultimate shear failure of the laboratory sample as individual cracks coalesce under continued loading conditions (Hoek and Martin 2014; Sainsbury and Sainsbury 2017). However, tension strength is seldom directly measured in the laboratory due to limitations associated with project time and costs that affect proper sample preparation and processing (Rabat et al. 2023). As such, the tensile strength of intact geo-material is estimated based on a relationship with UCS or measured via indirect means (Baêsso et al. 2024).

1.1 Empirical Relationship for the Estimation of Tensile Strength

The tensile strength of intact rock is ubiquitously estimated based on a relationship with UCS. This relationship (tension ~ 10% UCS) was derived from a data-set published in Sheorey et al. (1989). The relationship was developed based on a dataset that contains UCS and tension strengths for 23 sedimentary rocks. For the strength properties reported in Sheorey et al. (1989), UCS was measured in only 12 of the 23 cases and Direct Tension (DT) was not measured for any of the samples but estimated based on the results of triaxial testing.

A sub-set of the dataset is presented in Fig. 1 that includes only the datapoints in which UCS was directly measured. The linear trendline of the sub-set datapoints is also provided. The trendline is consistent with the empirical relationship (tension ~ 10%UCS).

Fig. 1
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Relation between unconfined compressive strength and tension based on direct measurement (modified after Sheorey et al. 1989)

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It is known that the tension strengths of geo-materials are influenced by sample mineralogy (Cai 2010; Sainsbury and Kurucuk 2020), and as such, and a ‘one-fit’ empirical relation cannot accurately represent the intact property (Packulak et al. 2022). Previous research conducted on rock samples suggests that the derivation of tension values from “an estimate from UCS gives the most erroneous results” (Perras and Diederichs 2014). Studies completed on a range of rock-types by Perras and Diederichs (2014) suggest tension:UCS ratios of 3–20% that are clearly related to rock-type.

1.2 Indirect Measurements of Tensile Strength

Increased reliability of the tension strength of a geo-material beyond the application of an empirical equation is usually considered through the use of indirect measurements in the field and laboratory. Indirect methods for the quantification of tension strength in the laboratory include flexural (ASTM 2015a; b; Liao et al. 2020) and splitting methods (ISRM 1978; Resan et al. 2020).

Point load testing (PLT) which can also be considered an indirect measurement technique, has been shown to correlate with tension strength and is easily conducted in the field (Butenuth 1997). However, PLT is usually considered unreliable due to errors associated with machine calibration and operator training (Akbay and Altindag 2020). As such PLT it will not be considered in this study. Authors have also determined relationships between Mode 1 fracture toughness and tensile strength (Ameen et al. 2024). This indirect approach is also not considered in this research. Some generalised loading conditions for indirect tension testing are provided in Fig. 2.

Fig. 2
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Direct and indirect methods for the measurement of tension strength in the laboratory (modified after Jones and Sainsbury 2023)

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For the determination of concrete tension strength, both splitting tensile strength (STS) and flexural three-point bending methods (3PB) are common since samples can be batched to fit the specific mould shapes. Forensic strength investigations on concrete usually require core recovery, and, as such, splitting tests are usually performed in these instances. A usual UCS for concrete is in the range 40–60 MPa (Nazmul et al. 2023). For rocks, splitting methods are the most usual for the determination of tension since samples can be obtained directly from geotechnical core. A typical UCS for rock is in the range 100 MPa but can vary from 1.25 MPa to greater than 200 MPa (Hack and Huisman 2002). Indirect tension tests are performed on low strength cemented paste backfill materials (Johnson et al. 2015; Rankine 2004) that have UCS strengths that range from 0.1 to 10 MPa (Sainsbury et al. 2024). The accuracy of each of these most common indirect approaches (STS and 3PB) is considered in this research over the low to typical strength ranges 0.1–100 MPa.

1.3 Direct Measurement of Tensile Strength

Publications that document the DT strength of rock are limited but are summarised in Perras and Diederichs (2014). Due to their delicate nature, even fewer studies have been completed that measure the tensile strength test of low strength geo-materials materials e.g. cemented paste fill (Grabinsky et al. 2022).

Recent direct tension testing of rock has been completed on dumbbell shaped samples (Li et al. 2013; Jones and Sainsbury 2023). The preparation of dumbbell shaped samples is onerous and not commercially viable at the present time. The sample geometry requires the strength of the material to be able to withstand compression during gripping / seating which is not feasible in this study due to the lower strength range of geo-material being considered. A description of the procedure to establish a reliable DT test procedure is provided in the following section.

2 Laboratory Investigation

2.1 Development of a Reliable Test Procedure

To establish a consistent and reliable procedure for the determination of DT over a wide strength range (0.1–100 MPa) a number of experimental set-ups were trialled. The trials are similar to the experiences documented by Zhang and Lu (2018) for DT testing and are described below.

  1. A.

    Traditional 50 mm dia. cylindrical samples were bonded to a steel platen that was loaded (Fairhurst 1961). For this procedure, a bonding agent capable of securing the sample with UCS’s of 0.1–100 MPa was difficult to identify. A range of three typical commercial bonding agents were trialled and are documented in Jones and Sainsbury (2023). Trials resulted in high strength samples de-bonding from the platen prior to failure and lower strength samples failing close to the bond location resulting in invalid experimental results. The experimental set-up trialled is presented in Fig. 3a.

  2. B.

    To increase the bond capacity of the platen to the sample, a screw was be embedded into the sample prior to curing (Alhussainy et al. 2019). During straining of lower strength samples the screw pulled out resulting in invalid experimental results. The experimental set-up trialled is presented in Fig. 3b.

  3. C.

    The sample bond to the platen was increased through a modified dumbbell geometry that increased the grip/bond area (Hoek 1964). Again, during loading, the samples with higher UCS values de-bond from the platen prior to tensile failure. The experimental set-up trialled is presented in Fig. 3c.

  4. D.

    To increase the bonding capacity of the platen to the dumbbell shape sample, a screw was embedded into the sample prior to curing. Again, during testing of low strength samples, the screw pulled out resulting in invalid experimental results. The experimental set-up trialled is presented in Fig. 3d.

  5. E.

    The size of the sample cross sectional area was decreased, and the bond surface area was increased by bonding the sample into endcaps (Hawkes and Mellor 1970). In this case an increased rate of valid tests was achieved but the reliability on the mode of failure was still less than 20% at higher UCS values. The experimental set-up trialled is presented in Fig. 3e. Typical invalid modes of failure presented in Fig. 3f.

Fig. 3
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Traditional modes of direct tension sample assemblage

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Based on the unreliable nature of the traditional techniques for DT testing over the range of UCS values to be tested, the tension response in this study has been established based on a procedure documented by Guo et al. (2022) and Pan and Grabinsky (2021). A bespoke mould and seating arrangement, presented in Fig. 4 has been utilized to ensure only pure tensile strain is applied to the sample eliminating eccentric loading conditions (Zhang et al. 2021).

Fig. 4
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Design of direct tension mould and seating assemblage

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Side moulds have been printed in a Fortus 450mc Production System with front and back pieces cut from 2 mm Perspex to provide transparency during filling and curing (Fig. 5).

Fig. 5
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Design of direct tension sample preparation and testing

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2.2 Sample Preparation

Due to the heterogeneous nature of rock that may provide a significant amount of scatter in results, this investigation has been performed on a series of fine-grain cementitious samples. The samples were prepared from Sika™ 212 High Performance Structural Grout with varying water contents to provide the strength results > 5 MPa and cemented fine-grain tailings at varying binder contents to provide the strength results < 5 MPa. The binder content ranged from 2 to 10% and the water content ranged from 40 to 15% to generate the 0.1 MPa and 5 MPa UCS strengths respectively for the fine-grained cementitious products. Additional details can be found in Sainsbury et al. (2024). The use of a very fine grain material limits heterogeneity within the sample and thus limits the impact of sample scale dependency on the results (Pérez-Rey et al. 2023). As such, cement mortar has been used by many researchers as a proxy for rock (Lee et al. 2013; Duan et al. 2019; Shi et al. 2023) due to its similar UCS, Modulus, Poisson’s ratio, density and brittle failure mode.

Samples were prepared and tested based on the standards and methods provided in Table 1. UCS tests have been performed for each batch of tension results to characterise the compression response.

Table 1 Summary of applied testing and reporting standards
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Samples were mixed in small-scale batches using a commercial 9L Apuro Planetary Mixer. Samples were prepared in three-layers with tamping occurring between each. The samples were individually sealed and cured under zero effective stress at a temperature of 23° and minimum humidity of 50% for 28–29 days prior to testing.

3 Experimental Results

The laboratory tests were completed over the period one-year on three separate Instron Load Frames (10 KN, 50KN and 300 KN capacities) based on the strength requirements. Typical failure modes for a range of strength samples and testing methods are provided in Fig. 6.

Fig. 6
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Typical failure profiles of samples for each stress path over a range of strengths

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Strain rates between 8.33E−05 and 1.67E−04 were applied to the samples (~ extension of 0.25 mm/min). These rates are low enough to ensure that loading induced dynamic effects are not observed in the strength results (Li et al. 2013; Efimov 2020).

3.1 Strength Response

A total of 241 individual tests have been completed and reported on herein. A summary of the results is provided in Fig. 7 along with trendlines for each of the testing method results.

Fig. 7
figure 7

UCS versus tension strength results for direct and indirect methods. DT direct tension, STS splitting tension, 3PB three-point bending

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DT results have been fitted with a Power Law trendline that has a R2 value of 89%. The Equation of the trendline is presented in Eq. 1. The DT results have the least scatter of all the testing procedures which is reflected in the high R2 value.

$$DT=0.25{(UCS)}^{0.75}$$
(1)

where UCS and DT are defined in MPa.

STS results provide a Power Law trendline (Eq. 2) that has a R2 value significantly less than that of the DT dataset (68%). This wide-scatter in results is consistent with other researchers (Perras and Diederichs 2014). The R2 value suggests that using this technique to determine tension may result in values that vary by up to 30% from the real DT value.

$$STS=0.22{(UCS)}^{0.70}$$
(2)

where UCS and STS are defined in MPa.

An equation that can be used to approximate DT from STS is proposed and presented in Eq. 3.

$$DT=1.47\left(STS\right)$$
(3)

The 3PB measured values are consistently higher than the DT results. This outcome is consistent with literature that suggests flexural strength is greater than DT (Chen et al. 2014) and approximately 200% the DT strength (Fernandes Neto and Haach 2023; Sainsbury et al. 2024). The over-prediction in tensile strength is most pronounced at lower strengths but reduces as UCS increases. The 3PB results have been fitted with a Power Law trendline that has a R2 value of 72%. The equation of the trendline is presented in Eq. 4.

$$3PB=0.51{(UCS)}^{0.67}$$
(4)

where UCS and 3PB are defined in MPa.

An equation that can be used to approximate DT from 3PB is proposed and presented in Eq. 5.

$$DT=0.72(3PB)$$
(5)

For each of the test methods the trendlines have been used to compute tension values over the UCS range 0.1–100 MPa. They are presented in Table 2. For comparison purposes it is assumed the DT values are accurate.

Table 2 Direct and Indirect tension values computed for increasing UCS values based on trendlines
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Based on the comparisons in Table 2 it can be concluded that:

  • The 3PB method consistently over-predict DT strength. This over-prediction is more pronounced at lower strengths (e.g., cemented paste backfills).

  • STS methods provide significant variability in results and generally seen to under-predict DT strength. This under-prediction is more pronounced at higher strengths. Based on the tests presented herein, STS are more accurate than 3PB tests to characterise the tensile strength of low strength geo-materials (e.g., cemented paste backfills), however, should be considered to under-estimate DT at higher UCS values (> 25 MPa).

Based on the data-set presented in Fig. 7, empirical relationships to convert STS and 3PB strength to DT strengths based on indicative UCS values are presented in Fig. 8. The relationships are presented in Eqs. (6) and (7) respectively.

Fig. 8
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Relationships to convert indirect tension measurements to direct tension results

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$$DT=STS\left[0.063{\text{ln}}\left(UCS\right)+0.75\right]$$
(6)

This Power Law relationship is significantly different to the linear relationship proposed by Tang et al. (2023). The difference arises due to the increased direct tensile strength range of the materials tested in this study that ranged from (0.035 to 100 MPa) to the material tested by Tang et al. (2023) that ranged from 0.01 to 1 MPa.

$$DT=3PB\left[0.045{\text{ln}}\left(UCS\right)+0.2\right]$$
(7)

A relationship relating 3PB strengths to DT strengths over such a strength range has not been proposed before for geo-materials. Relationships exist for modified concrete (e.g. Alkurdi et al. 2020) however these are limited in their application to rock engineering due to the additives included.

3.2 Failure Response

The inaccuracy of each of the indirect testing methods are further explored through a consideration of example stress–strain curves for a range of the samples UCS strengths (1–80 MPa) that are presented in Fig. 9.

Fig. 9
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Typical stress–strain curves for DT, 3PB, STS tests for a range of indicative UCS strengths 1–80 MPa

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Based on the stress–strain curves it is obvious that the post-peak failure responses of the DT samples are perfectly brittle. This represents a pure tensile failure mechanism. The failure responses of the 3PB tests are more ductile at UCS values of 20 MPa and below. Ductility increases as the sample continues to deform representing the compression that is occurring at the loading location and is more pronounced in lower strength (less stiff) geo-materials. The STS responses are more complex and represent a post peak response possibly representing multiple rupture points that coalesce and/or the effect of anisotropy within the sample (Chen et al. 2024). For each of the example stress–strain curves, the modulus has been computed and is provided in Table 3 along with the comparable compression stress–strain result. The modulus has been computed in each of the cases as a tangent at 50% of the peak strength response.

Table 3 Summary of compression and tension modulus values
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Based on the consideration of the modulus values, the DT and STS are comparable suggesting similar ‘pure’ tension failure modes when the STS tests are performed correctly. It is also observed that when the STS modulus is consistent with the DT modulus, the tension strengths are consistent. This outcome is also observed by Andreev (1991). The modulus results of the 3PB are significantly higher than the DT and STS at UCS values less than 80 MPa. Previous studies have highlighted that if the tension modulus is less than the compression modulus, the tensile strengths derived from indirect techniques are overestimated (Chen and Stimpson 1993). This is certainly the case for the 3PB tests.

3.3 Consideration of the Accuracy of Empirical Relationship to Estimate Tension

The DT results from this study are compared in Fig. 10 with the empirical Sheorey et al. (1989) relationship that is presented as the red trendline and the datapoints used to develop it. Other published datapoints that have previously been referred to in this work are also included for reference.

Fig. 10
figure 10

UCS versus tension strength results

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The current results are consistent with the Sheorey et al. (1989) dataset however, they present less scatter and cover an increased UCS range. Tension results derived from this study and the empirical approach are compared for discrete UCS values and are presented in Table 4.

Table 4 Direct tension empirical versus measured
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The empirical relationship is observed to under-estimate tension below UCS values of 22.5 MPa and over-estimate above this based on a comparison of the trendlines in Fig. 10 and can be observed in the values provided in Table 4. When both the current and Sheorey et al. (1989) data are combined, an updated relationship over the UCS range 0.1–100 MPa is defined (black trendline in Fig. 10) that suggests DT strength is higher than 10% the UCS strength. The relationship can be defined as a Power Law with an R2 value of 73% and is presented in Eq. 8.

$$DT=0.24{(UCS)}^{0.765}$$
(8)

4 Conclusions

A series of 241 unique laboratory tests have been completed on fine grained cementitious material to compare the results of direct and indirect methods for the measurement of tension. Testing has included direct tension, flexure and splitting strengths over the UCS range 0.1–to 100 MPa.

When indirect tensile strength results are compared to the DT results the following conclusions can be made:

  • STS provides an average approximation to DT results over the range 0.1–100 MPa; however, the methods reliability is 30% less that DT and so many more tests must be completed to provide an ‘average’ result. It is most likely that STS results will be more accurate at lower strength and will under-estimate tension up to 30% at values greater than 10 MPa. An empirical correction factor is presented that is based on indicative UCS that relates STS to DT strength.

  • 3PB tests consistently over-predict the tension strength over the UCS range 0.1–100 MPa. This over-prediction is most pronounced at lower strengths and can be related to compressional stresses that build up lower strength samples during straining. An empirical correction factor is presented that is based on the indicative UCS values that relates 3PB strength to DT strength.

The empirical approach to estimate tension based on UCS has been shown to under-estimate tension values below UCS values of 25 MPa and over-estimate tension above this. A new empirical relationship is proposed that includes 241 new datapoint as well as the original 12 datapoints that were measured and documented by Sheorey et al. (1989). The revised empirical relationship has been presented over the UCS strength range 0.1–100 MPa with a R2 value of 89%.