1 Introduction

Creep property, a time-dependent behaviour, is the inherent attributes of rocks, soft mudstone in particular (Sun 2007). In deep underground engineering, the creep deformations of rocks become more common due to the high stress (He et al. 2005). The time-dependent deformations of rocks negatively affect the mining safety (Li et al. 2017; Mishra and Verrna 2015; Sainoki and Mitri 2017; Zhou et al. 2019), backfill mining (Guo et al. 2019; Sun et al. 2013), coalbed methane production (Danesh et al. 2016, 2017; Guo et al. 2018; Kang et al. 2015), and geo-sequestration (Liu et al. 2015; Sone and Zoback 2014). Consequently, to enhance the understanding of this phenomenon is of importance to guarantee the stability of underground engineering. Macro-scale laboratory creep experimental methods, such as uniaxial and triaxial compression tests, have been broadly used to investigate the rheological deformations of rocks which threw light on their creep characteristics (Sha et al. 2018; Xiong et al. 2019; Yang et al. 2015a, b). However, these experiments were too complicated. On the one hand, the load-holding time of some triaxial compressive tests was about several days (Sone and Zoback 2014; Xu et al. 2014) and that for uniaxial compression tests was above 6 h (Jia et al. 2018; Yang et al. 2015a, b). On the other hand, the specimens used in macro-scale laboratory creep tests needed to meet strict size requirements: the typical cylindrical specimen size was Φ50 mm × 100 mm (Mishra and Verrna 2015) and the size of the smaller prismatic samples was 25 mm × 25 mm × 50 mm (Lu and Wang 2017). Besides, centimetre-scale laboratory experiments regard rocks consisting of various constituents (Guery et al. 2008) as homogeneous bodies, where the underlying mechanism of large creep deformation of soft mudstone yet remains to be relatively insufficiently understood. Adopting a new technique to identify various mineralogical constituents and investigate their creep characteristics is very meaningful since the individual behaviour of each constituent dictates the overall behaviour of macro-scale mudstone.

Nanoindentation technique, also called the depth-sensing indentation, was first proposed and used by Kalei in 1968 in Russia (Kalei 1968). This technique has proven to be an effective and convenient method for determining the elastoplastic mechanical properties of solids based on a small rock sample, most notably elastic modulus, hardness (Oliver and Pharr 1992) and fracture toughness (Zeng et al. 2019). Merged with CT (Zhang et al. 2018a, b, c), SEM–EDS (Slim et al. 2019; Sun et al. 2020a, b), XRD (Zhao et al. 2019), and 3D printing (Kong et al. 2019) observing the microstructures of rock samples, nanoindentation technique was useful to investigate the mechanical behaviors of mineralogical components of shale, granite, coal or mudstone samples. The influence of clay minerals and kerogen on both the hardness and elastic modulus of organic-rich and clay-bearing shale was investigated (Abedi et al. 2016; Alstadt et al. 2015). Zhang Fan et al. obtained the hardness and elastic modulus of compositions in granite (Fan et al. 2017). Taking Chinese sub-bituminous coal as an example, the relationship between its morphologies and its nano-mechanical properties was investigated (Zhang et al. 2018a, b, c); Meng et al. and Sun et al. calculated the mean values of the nanoindentation mechanical parameters to represent that of standard coal samples (Meng et al. 2020; Sun et al. 2020a, b) and discussed the influence of quartz and kaolinite on the mechanical properties of coal matrix (Sun et al. 2020a, b). For mudstone samples, the nanoindentation technique can test the elastic and plastic mechanics of its minerals (Magnenet et al. 2011) and reveal its heterogeneity (Auvray et al. 2017; Zhang et al. 2018a, b, c). However, all geomaterials also exhibit a viscoelastic response, which has encouraged researchers to improve the utilization of this technique on capturing and modelling the creep characteristics of geomaterials. Slim et al. covered the role of the organic matter on creep rates of gas shale by using the nanoindentation technique and obtained that organic matter drives its creep rates (Slim et al. 2019). Liu et al. used nanoindentation technique to examine the creep behaviour of multi-phase oil shale by applying Burgers models and the logarithmic function to quantify creep curves of various phases in oil shale (Liu et al. 2017, 2018 ). Nevertheless, rock creep was usually observed as an increase in depth during the load-holding period at maximum load (Slim et al. 2019), and there is no work on how the creep strain–time data can be used to model the nanoindentation creep response. Besides, widely used grid nanoindentation method with hundreds of indentations (Liu et al. 2017, 2018 ) complicate the data analysis.

In this context, the nanoindentation technique coupled with XRD, SEM, and EDS, were used to inspect the creep behaviours of mineralogical constituents in mudstone. This paper is structured as follows: The studied two-constituent mudstone was first presented together with the nanoindentation experimental protocol and experimental method to illustrate the heterogeneous properties of mudstone and the necessity of the targeting nanoindentation technique (Sect. 2). A theoretical analysis of nanoindentation creep constitutive models and the Oliver-Pharr method followed; the creep models are three viscoelastic models used to fit the nanoindentation strain–time curves (Sect. 3). The calculating and fitting results of nanoindentation experimental data were then given to distinguish the mechanical properties of the two constituents in mudstone as well as to determine and verify suitable nanoindentation creep model (Sect. 4).

2 Material and methodology

2.1 Material

A broken soft mudstone block obtained from a deep underground coal mine was researched in this study. As shown in Fig. 1a, the coal mine named Tongting is located in Anhui province, China. As well, the sampling layer is above the 7# coal strata deposited at the Permian layer, where the stratigraphic column is presented in Fig. 1b.

Fig. 1
figure 1

Sampling area and layer in Anhui Province, China

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To quantify the mineralogical constituents of this mudstone sample, the XRD experiment was carried out based on the standards No. SY/T 5163–2018; D8 ADVANCE (Bruker, Germany) was used to conduct this test at the range of 2θ (2-theta) from 10° to 65° with the scanning speed of 2°/min. The powder diffraction spectrum is given in Fig. 2a, which shows that the mudstone consisting of 62.57% kaolinite and 37.43% quartz is a two-constituent geomaterial.

Fig. 2
figure 2

Diffraction spectrum of XRD and images of SEM–EDS

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Then, the images of the polished surface of the sample got from SEM–EDS experiments, Fig. 2b to observe and determine the distribution of both constituents in the mudstone sample. The left-top one is the SEM image, in which there are both smooth and unsmooth surfaces; the smooth surfaces are particle-shaped and embedded in the rest unsmooth surfaces regarded as a matrix. The area of unsmooth surfaces is much larger than that of smooth particles, which may represent kaolinite in mudstone and the particles could be quartz. To verify this hypothesis, the right-top EDS image shows the distribution of kalium, silicon, aluminium, and oxygen, and the left-bottom and right-bottom images give the distributions of a single atom of silicon and aluminium, respectively. Since the chemical composition of kaolinite is Al2Si2O5(OH)4 and that of quartz is SiO2, the distributions of atoms can help determine the types of minerals in mudstone. The smooth surfaces are the enrichment areas of silicon atom and an oxygen atom, and its enrichment degree of the silicon atom is much higher than the rest areas. Also, the region does not contain aluminium atoms. EDS images verify that on the surface of this mudstone sample, the smooth particles are quartz and the unsmooth surfaces are kaolinite.

The mudstone is a typical heterogeneous material composed of clay minerals(kaolinite) as matrix and silt inclusions(quartz) embedded in it. The mechanical properties of mudstone are determined by its various constituents (Li et al. 2019b), which encourage us to investigate the mechanical performances of its constituents at the meso-scale. Besides, at the beginning of creep experiments, some standard cylindrical samples (100 mm × Φ50 mm) were prepared to research the mechanical properties of soft mudstone. However, this experimental plan was dropped due to the influence of macro-scale cracks and pores in the mudstone block on the obtaining of reliable mechanical parameters. Therefore, a new experimental method was used in this study, nanoindentation technique, based on small intact samples.

2.2 Methodology

2.2.1 Nanoindentation creep experimental protocol

All nanoindentation creep experiments were conducted using an Anton Paar TTX-NHT3 nano-indentor system equipped with a Berkovich tip. Loads of this system range from 0.1 to 500 mN; the resolutions of load and depth are 20 mN and 0.04 nm, respectively. And the thermal drift of it is less than 0.01 nm/s after allowing thermal equilibrium to be reached for more than 1 h at room temperature (21 °C), which was automatically compensated in this study. As shown in Fig. 3a, a nanoindentation load-time curve is composed of the loading stage(o-a), load-holding stage(a-b) and unloading stage(b-c), where the 180 s load-holding stage at peak load of 50 mN will be used to analyse creep characteristics (Liu et al. 2018). And both the loading and unloading rates are 100 mN/min.

Fig. 3
figure 3

Nanoindentation creep experimental protocol illustrated by the nanoindentation load-time curve and nanoindentation creep deformation-time curve

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After the nanoindentation test, there is a nanoindentation creep deformation-time curve (Fig. 3b). Compared with the deformations during the loading and unloading stages, creep deformation is too small to be observed. Thus, the creep deformation starting from 0 was analysed separately. During the load-holding period, the indenter is displaced further into mudstone. This curve illustrates the portion of displacement that was creep controlled and will be used to analyse the creep characteristics of constituents in mudstone. However, this curve reflects the relationship between creep deformation and time, not that between the typical creep strain and time.

2.2.2 The targeting nanoindentation technique

To make indentations cover representative areas of the material surface and obtain the mechanical properties of all constituents in materials (Slim et al. 2019), the grid nanoindentation method was generally adopted to conduct statistical analysis (Liu et al. 2018). As the name implies, this nanoindentation method has hundreds of indents with uniform intervals on the surfaces of materials of interest to form a grid, which can be used to detect the heterogeneous characteristics of complex materials and perform statistical analysis (Constantinides et al. 2006). Notwithstanding, mechanical parameters obtained from the grid nanoindentation method hinge on the selection of indented surface and the number of indentations (Deirieh et al. 2012; Liu et al. 2018). This experimental method was not of high-efficiency and accuracy. For example, the junction area between mineralogical constituents would affect the accuracy of experimental results.

The whole experimental process used in this study is to replace the grid nanoindentation method, and named as the targeting nanoindentation technique (TNIT) is graphically illustrated in Fig. 4. An intact small mudstone block from a standardized cylindrical sample was cut into a very small volume of mudstone sample (10 mm × 10 mm × 10 mm). The experimental surface fixed with epoxy resin was then polished and exposed. The polishing protocol detailed in (Sun et al. 2020a, b) was followed. This protocol includes a coarse polishing on 3000 grit emery papers and a fine polishing using oil-based diamond suspension with abrasive grain sizes ranging between 3 and 0.5 μm. SEM image shows that there are many meso-scale cracks on the polishing surface. Thus, to measure their creep properties of both constituents in mudstone and avoid the influence of cracks, the intact surfaces including smooth quartz and unsmooth kaolinite were chosen for probing indenter. Images of EDS and SEM illustrate the types and contents atoms to identify the types of mineralogical ingredients in the meso-scale zone. Finally, nanoindentation experiments will be conducted on both elements in a targeting manner, and the mechanical properties of both constituents in mudstone can be obtained easily.

Fig. 4
figure 4

Process of the targeting nanoindentation technique

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The TNIT is a coupled experimental method, where the nanoindentation method is joined with other microstructure/constituent characterization technique. The constituents of mudstone are required to be observed and identified before conducting nanoindentation experiments. During the detecting process, not only the mineralogical constituents are determined, but also the scale separation (hmax < D/10, where hmax is the maximum depth of indenter and D is the size of quartz) is satisfied. The size of quartz is larger than 25 μm, and that of clay matrix (kaolinite) is larger than quartz in Figs. 2b and 4, while the depth of indents on quartz and kaolinite are around 0.7 and 2.5 μm. Consequently, the technique can be readily incorporated into an inner computer program of the nanoindentation system to improve the experimental efficiency of the nanoindentation technique on heterogeneous materials, geomaterials in particular.

3 Theories

3.1 Nanoindentation creep model

3.1.1 Nanoindentation strain–time curve

For nanoindentation creep experiments, the stress and strain rate can be written as (Hackney et al. 2012; Mayo and Nix 1988):

$$\sigma = \frac{\beta F}{{h^{2} \left( t \right)}}$$
(1)
$$\dot{\varepsilon } = \frac{\text{d}h\left( t \right)}{{\text{d}t}}\frac{1}{h\left( t \right)}$$
(2)

And then the strain can be obtained by integrating Eq. (2):

$$\varepsilon = \ln \frac{h\left( t \right)}{{h\left( 0 \right)}}$$
(3)

where, \(\sigma\) is the stress, MPa; \(\dot{\varepsilon }\) is the strain rate, s−1; \(\varepsilon\) is the strain; \(h\left( t \right)\) is the indentation depth, nm; \(t\) is the creep time, s; \(h\left( 0 \right)\) is the indentation depth when creep time is 0, nm; and \(\beta\) is an indenter parameter(\(\beta\) = 0.0407 for Berkovich indenter (Hackney et al. 2012)).

Based on the nanoindentation creep data obtained from stage a-b in Fig. 3, and Eq. (3), four nanoindentation curves of mudstone are shown in Fig. 5. Two curves labeled 1-1 and 1-2 are got from indentations on kaolinite and the other two curves named 2-1 and 2-2 are indented on quartz in mudstone.

Fig. 5
figure 5

Nanoindentation strain–time curves of kaolinite (1-1 and 1-2) and quartz (2-1 and 2-2) in mudstone

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In Fig. 5a, the whole strain–time curves with a load-holding time of 180 s are delivered; and their first stages of 15 s are enlarged in Fig. 5b to help observe the initial part of the load-holding period. The patterns of nanoindentation strain–time curves of both constituents in mudstone are all remarkably similar in form. These curves can be divided into (i) transient creep stage and (ii) steady creep stage. During the period of the transient creep stage, the strain rates are very considerable and decrease sharply with time. After recording the steady stage, the strain develops at almost a constant rate. Also, the load-holding time of 5 s is the dividing point between two stages. Note that there is no tertiary (accelerating) creep stage and initial instantaneous strain in nanoindentation creep-time curves. Consequently, nanoindentation creep curves can be expressed by viscoelastic creep models that do not include elastic elements in series and plastic elements.

3.1.2 Nanoindentation viscoelastic creep models

Here, based on the patterns of nanoindentation strain–time curves, three nanoindentation viscoelastic creep models were proposed in Fig. 6.

Fig. 6
figure 6

Three nanoindentation creep models. Notes a The Kelvin model, with one elastic element and one viscous element in parallel; b the standard viscoelastic model, with one viscous element and one Kelvin model in series; and c the extended viscoelastic model, with one viscous element and two Kelvin models in series

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For the elastic element in Fig. 6, the stress (\(\sigma\)) is proportional to the strain (\(\varepsilon\)), while for the viscous element, it is proportional to the strain rate \((\dot{\varepsilon }\;{\text{or}}\;\text{d}\varepsilon /\text{d}t)\). The relationship between stress and strain can be written as Eq. (4) for the elastic element; and that between stress and strain rate, for the viscous element is given in Eq. (5).

$$\sigma = E\varepsilon$$
(4)
$$\sigma = \eta \dot{\varepsilon } = \eta \frac{\text{d}\varepsilon }{{\text{d}t}}$$
(5)

where, \(E\) is the modulus, GPa, and \(\eta\) is the viscosity, GPa s.

According to the stress and strain relationship of the parallel and the series structures in Fig. 6, the creep function of the Kelvin model can be summarized as:

$$\varepsilon = \frac{\sigma }{{E_{1} }}\left( {1 - e^{{\frac{{E_{1} }}{{\eta_{1} }}t}} } \right)$$
(6)

The standard viscoelastic model and the extended viscoelastic model can be collected by:

$$\varepsilon = \sum\limits_{i = 1}^{n} {\frac{\sigma }{{E_{i} }}\left( {1 - e^{{\frac{{E_{i} }}{{\eta_{i} }}t}} } \right)} + \frac{\sigma }{{E_{n + 1} }}t$$
(7)

where, \(n\) is the number of the Kelvin model in series in a creep viscoelastic model. If \(n = 1\), Eq. (7) is the standard viscoelastic model; and if \(n \ge 2\), it is the extended viscoelastic model. For fitting the nanoindentation creep strain–time curves, Eq. (7) is further simplified as:

$$\varepsilon = \sum\limits_{i = 1}^{n} {A_{i} (1 - e^{{B_{i} t}} )} + Ct$$
(8)

where, \(A_{i}\), \(B_{i}\) and \(C\) are fitting constants, and \(A_{i} = \sigma /E_{i}\), \(B_{i} = E_{i} /\eta_{i}\) and \(C = \sigma /E_{n + 1}\).

3.2 Oliver-Pharr method

The Oliver-Pharr method gives the hardness and elastic modulus of materials by accounting for the curvature in the unloading stage of the load-depth curve and indentation profile (Oliver and Pharr 1992, 2004). As shown in Fig. 7a, a typical load-depth curve can be obtained as the indenter enters into and exits from the surface of mudstone. The curve consists of a loading segment (o-a), load-holding segment (a-b), and unloading segment (b-c); also, segment b-d is the curve tangent of unloading segment at its initial portion. In Fig. 7b, a typical Berkovich indentation is composited by two surface profiles under load (before and after the load-holding segment or creep) and the residual surface profile after load removal.

Fig. 7
figure 7

Nanoindentation calculating schematic diagram of Oliver-Pharr method

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It is noted that the load-holding method is mandatory to derive stable unloading data for the calculation of hardness and elastic modulus (Hu and Li 2015; Shi et al. 2019; Zeng et al. 2019) and the holding time is 180 s in this study(Liu et al. 2018). The stable unloading curve is usually well described by a simple power relation (Oliver and Pharr 1992):

$$F = \alpha (h - h_{p} )^{m}$$
(9)

where, \(\alpha\) and \(m\) are the power law fitting constants and determined by fitting procedure. Then the contact stiffness is calculated by analytically differentiating this expression at the peak load or the maximum depth, i.e.,

$$S = \left( {\frac{dF}{{dh}}} \right)_{{F = F_{m} /h = h_{m} }} = \alpha m(h_{m} - h_{p} )^{m - 1}$$
(10)

The contact depth hc and the projected area of contact between the indenter and indented constituents Ac are measured by:

$$h_{\text{c}} = h_{m} - \varepsilon \left( {h_{m} - h_{r} } \right),$$
(11)
$$A_{\text{c}} = 24.5h_{\text{c}}^{2} + C_{1} h_{\text{c}}^{1} + C_{2} h_{\text{c}}^{1/2} + \cdots + C_{8} h_{\text{c}}^{1/128}$$
(12)

where, \(C_{1}\),…, \(C_{{8}}\) are the fitting constants; \(\varepsilon\) is the geometric constant for Berkovich indenter (ε = 0.75 (Oliver and Pharr 1992)); Ac is the projected contact area at peak load, nm2. Note that the lead term of Eq. (12) describes Berkovich indenter perfectly.

As a result, the hardness is the average load under the indenter tip and reduced modulus can be calculated from BASh formula (Bulychev et al. 1975).

$$H_{\text{IT}} = \frac{{F_{m} }}{{A_{\text{c}} }}$$
(13)
$$E_{\text{r}} = \frac{S\sqrt \pi }{{2\eta \sqrt {A_{\text{c}} } }}$$
(14)

where, HIT is the hardness, MPa; Er is the reduced modulus, GPa; η is the geometric constant for Berkovich indenter (η = 1.034 (Oliver and Pharr 2004)). When the effects of non-rigid indenters on the measurement of elastic modulus of mudstone are ignored, reduced modulus can be used to represent it. Otherwise, it can be calculated by

$$\frac{{1 - \nu_{\text{IT}}^{2} }}{{E_{\text{IT}} }} = \frac{1}{{E_{\text{r}} }} - \frac{{1 - \nu_{\text{i}}^{2} }}{{E_{\text{i}} }}$$
(15)

where, \(\nu_{\text{IT}}\) is the general Poisson’s ratio of constituents in mudstone and \(\nu_{\text{IT}} = 0.3\) in this study; Ei and vi are the elastic modulus and the Poisson’s ratio of the diamond indenter and Ei = 1141 GPa and vi = 0.07 (Oliver and Pharr 1992, 2004).

4 Results and discussions

4.1 Hardness and elastic modulus of kaolinite and quartz

Two nanoindentation tests were made on two constituents (kaolinite and quartz) in mudstone to calculate their hardness and elastic modulus. In Fig. 8, the nanoindentation load-time curves and indentations on kaolinite are labelled 1-1 and 1-2; and the results of quartz are named 2-1 and 2-2.

Fig. 8
figure 8

Load-depth curves of two constituents in mudstone

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The load-depth curves on different constituents have the same pattern: During the loading segment, the depths increase sharply with the load at first and then slightly; there is a small creep deformation during the load-holding segment; and after unloading, elastic deformations recover while there are unrecoverable plastic deformations. However, for different constituents, the magnitudes of depths are not the same. The responses associated with the indentations on kaolinite (1-1 and 1-2) give deeper indentation maximum depth (Approx. 2500 nm); while the responses associated with an indentation in quartz have lower indentation maximum depths (Approx. 700 nm).

Based on the typical Oliver-Pharr method, the hardness and elastic modulus were calculated to distinguish the mineralogical constituents. In Table 1, the mean values of hardness and elastic modulus of kaolinite are 355.75 MPa and 17.79 GPa, respectively. And that of quartz in the mudstone are much greater than them, which are 9055.60 MPa and 66.54 GPa. According to prior studying results, the median values for hardness and elastic modulus of sand grain are 10.74 GPa and 66.9 GPa, respectively (Daphalapurkar et al. 2010); and that of Cluster 1 (organic matter or clays) of shales labeled Sample 2, Sample 3 and Sample 4 are from 16.36 to 18.57 GPa (Li et al. 2019a), which verifies the accuracy obtained in this study. The mechanical performances of the kaolinite matrix are much smaller than that of quartz in mudstone, which explains why the argillaceous rocks are soft. For one constituent, the calculated mechanical parameters are close, meaning the targeting nanoindentation technique can be used to obtain the mechanical parameters of various mineralogical compositions in this mudstone.

Table 1 Hardness and elastic modulus of kaolinite and quartz in mudstone
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4.2 Nanoindentation creep parameters

In this section, the proposed nanoindentation creep viscoelastic models in Fig. 6, will be used to fit the nanoindentation experimental data on kaolinite and quartz in mudstone. Figure 9 gives the fitting results of kaolinite by using the Kelvin model, the standard viscoelastic model, and the extended viscoelastic model, respectively. Figure 10 shows that of quartz in mudstone.

Fig. 9
figure 9

Fitting results of kaolinite by using the Kelvin model, the standard viscoelastic model, and the extended viscoelastic model

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Fig. 10
figure 10

Fitting results of quartz by using the Kelvin model, the standard viscoelastic model, and the extended viscoelastic model

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Fitting curves by using the Kelvin model did not generate desirable results. As shown in Figs. 9a and 10a, the Kelvin model cannot fit the transient creep stage, where both kaolinite and quarte in mudstone have high strain rates. When describing the final part of the steady creep stage by the Kelvin model, the strain rate is zero, which brings a decided contrarian view to the steady creep stage of nanoindentation creep strain–time curves. Consequently, the Kelvin model cannot be used to describe the creep properties of constituents in mudstone. The values of R2 less than 0.94 also show that the fitting goodness of the Kelvin model is much weaker than the other two viscoelastic models.

For the viscoelastic models consisting of a viscous element and one or several Kelvin models, fitting results are very good. This result shows that the creep model presented by Eq. (8) can be used to describe the nanoindentation creep behaviors of both constituents in mudstone and the more of the number of the Kelvin models, the better of the fitting results. However, with the increase of the number of the Kelvin models, the fitting constants are probably fallacious. For example, as shown in Fig. 10c, the coefficient of the first-order term of 2-2 is  -3.24 × 10–4, which should be a positive value in reality; and the fitting value of \(E_{2} /\eta_{2}\) for experiment labeled 2-1 is much larger than that of the experiment 2-2. Consequently, for calculating the creep parameters, the standard viscoelastic model is the most useful model among them.

According to the fitting results by using the standard viscoelastic model, the creep parameters of both kaolinite and quartz in mudstone can be obtained, as shown in Table 2.

Table 2 Creep parameters of kaolinite and quartz in mudstone
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Note that the values of h0 was got from the mean values during the load-holding period to minimize the influence of creep deformations on calculated results. And concurring with the above-calculated hardness and elastic modulus, the creep parameters of different constituents in the mudstone are various, too. For example, the mean value of the first viscous element of kaolinite is 111.52 GPa s, while this value of quartz is much larger, which is 3066.21 GPa s. This comparison means that the soft clay minerals in mudstone have large creep deformation, which reveals the mechanism of large creep deformation of argillaceous engineering.

4.3 Verification of nanoindentation creep model

To verify that the viscoelastic creep models (Eq. (8)) can perfectly fit the nanoindentation strain–time curves and that the calculated creep parameters may be unreliable when n ≥ 2, four nanoindentation experiments were made on the quartz in a sandstone obtain from Suntuan Coal Mine, Anhui, China. Figure 11 gives the load-depth curves and indentations of them. The load-depth curves were used to calculate the elastic modulus and hardness and analyse the creep characteristic of quartz in sandstone; the indentations illustrate the application of the targeting nanoindentation technique.

Fig. 11
figure 11

Four load-depth curves and indentations on quartz in sandstone

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Using the standard viscoelastic model and the extended viscoelastic model to fit the four nanoindentation strain–time curves of quartz in sandstone, fitting results are shown in Fig. 12. And the fitting constants are given in Table 3.

Fig. 12
figure 12

Fitting results of strain–time curves of quartz in sandstone by using the standard viscoelastic model and the extended viscoelastic model

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Table 3 Mechanical parameters of quartz in sandstone
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According to the fitting results in Fig. 12, both the standard viscoelastic model and the extended viscoelastic model can perfectly fit the nanoindentation data, with goodness above 0.99. However, like the obtained results in Sect. 4.2, fitting constants derived from the extended viscoelastic model tend to be mutually contradictory in Table 3, which are bold in this table.

As a result, the standard viscoelastic model was verified to be a suitable function to fit nanoindentation creep strain–time curves and calculate the creep parameters. The obtained hardness, elastic modulus, and creep parameters are shown in Table 4. It also shows that the mechanical properties of quartz in sandstone are slightly larger than that of quartz in mudstone. For example, the mean elastic modulus of quartz in sandstone is 85.81 GPa, and that in mudstone is 66.54 GPa. The reason for the smaller mechanical properties of quartz in mudstone is probably due to the influence of soft kaolinite as a matrix in mudstone or the crystal orientation since quartz is highly anisotropic (Eliyahu et al. 2015; Timms et al. 2010).

Table 4 Mechanical parameters of quartz in sandstone
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5 Conclusions

In this paper, the viscoelastic characteristics of kaolinite and quartz in mudstone were investigated by using the targeting nanoindentation technique. Conclusions are as follows.

  1. (1)

    The examined soft mudstone sample in this study is composed of kaolinite and quartz, in which kaolinite is soft, as well as a matrix with hard quartz embedded in it.

  2. (2)

    For broken soft rocks that cannot provide intact standardized samples, the nanoindentation method can be used to investigate their mechanical properties by yielding stable mechanical parameters. Compared with the grid nanoindentation technique, the TNIT is efficient. This experimental technique can be realized by coupling the nanoindentation technique and observing experimental methods, like XRD, SEM, and EDS. Using this experimental method, the mechanical properties of constituents in geomaterials can be quickly obtained.

  3. (3)

    The nanoindentation creep strain–time curve reflects the viscoelastic properties of mudstone. This curve can be divided into two stages: a transient stage and a steady stage. The standard viscoelastic model consisting of a viscous element where a Kelvin model in series can perfectly fit the nanoindentation creep strain–time curves of mudstone and provide reasonable rheological parameters. And the proposed model was also used to fit creep curves of quartz in sandstone with high goodness, which verifies the nanoindentation creep model. Even though the extended viscoelastic can better fitting nanoindentation creep curves, it could yield unreasonable creep parameters.

  4. (4)

    The mechanical properties of quartz in mudstone and sandstone are slightly different, which may be due to the influence of soft clay minerals in mudstone that softens the mechanical performance of quartz in it or the crystal orientation of quartz.