Rock Mechanics and Rock Engineering

, Volume 49, Issue 6, pp 2043–2054 | Cite as

Effects of Thermal Damage and Confining Pressure on the Mechanical Properties of Coarse Marble

Original Paper


Heating treatment generally causes thermal damage inside rocks, and the influence of thermal damage on mechanical properties of rocks is an important topic in rock mechanics. The coarse marble specimens drilled out from a rock block were first heated to a specific temperature level of 200, 400 and 600 °C except the control group left at 20 °C. A series of triaxial compression tests subjected to the confining pressure of 0, 5, 10, 15, 20, 25, 30, 35 and 40 MPa were conducted. Coupling effects of thermal damage and confining pressure on the mechanical properties of marbles including post-peak behaviors and failure modes, strength and deformation parameters, characteristic stresses in the progressive failure process had been investigated. Meanwhile, accompanied tests of physical properties were carried out to study the effect of thermal damage on microstructure, porosity and P-wave velocity. Finally, the degradation parameter was defined and a strength-degradation model to describe the peak strength was proposed. Physical investigations show that porosity increases slowly and P-wave velocity reduces dramatically, which could be re-demonstrated by the microscopy results. As for the post-peak behaviors and the failure modes, there is a brittle to ductile transition trend with increasing confining pressure and thermal effect reinforces the ductility to some degree. The comparative study on strength and deformation parameters concludes that heating causes damage and confining pressure inhibits the damage to develop. Furthermore, crack damage stress and crack initiation stress increase, while the ratios of crack damage stress to peak strength and crack initiation stress to peak strength show a decreasing trend with the increase of confining pressure; the magnitude of crack damage stress or crack initiation stress shows a tendency of decrease with the increasing heating temperature and the tendency vanishes subjected to high confinement.


Coarse marbles Thermal damage Crack damage stress Crack initiation stress Peak strength model 

1 Introduction

In numerous deep underground rock engineerings, such as oil or gas exploitation and geothermal energy extraction, high temperature is one of the important factors affecting transport characteristics or stability of surrounding rocks. For instance, in a nuclear waste disposal project, high level radioactive waste decays in a long term and releases a large quantity of heat, which thus causes temperature rise of surrounding rocks. The microcracks induced by thermal stress may break integrity of surrounding rocks and even provide potential channels of nuclear leakage. In addition, the mechanical properties of rocks suffering from high temperature was one of the major themes of the German Continental Deep Drilling Program (KTB), of which the main superdeep borehole reached a final depth of 9101 m and a temperature of 265 °C (Emmermann and Lauterjung 1997). Consequently, research into thermal effect on the physical and mechanical properties of rocks provides a crucial way to improve the safety of corresponding projects.

Until recently, massive laboratory researches have been carried out on rock specimens under high temperatures to investigate temperature influences on physical and mechanical properties. Those rock properties studied consist of the strength and deformation properties (compressive or tensile strength, dynamic impact strength, elastic modulus, post-peak characteristics, creep deformation, etc.) (Yavuz et al. 2006; Chen et al. 2012; Sun et al. 2013; Liu and Xu 2014; Tian et al. 2014), acoustic properties (ultrasonic wave velocity and acoustic emission) (Cantisani et al. 2009; Gónzalez-Gómez et al. 2015; Inserra et al. 2013; Yavuz et al. 2010), transmission properties (permeability, thermal or electrical conductivity) (Rosengren and Jaeger 1968; Liu et al. 2011). In general, with the increase of temperature, strength properties, acoustic and thermal conductivity decreased gradually, while permeability and electric conductivity enhanced instead. On one hand, temperature changes could induce microcracks due to the mismatch in the thermal properties of the different mineral compositions (intergranular) or within grains (intragranular). On the other hand, heavy thermal treatment could cause complicated chemical changes, such as the phase transformation of the quartz from α phase to β phase, thus causing dramatical variation of properties. However, every coin had two sides, and a threshold temperature existed for strength properties of high porosity sandstones. Below this threshold temperature, the Young’s modulus and the UCS could increase to some extents, whose reason was that thermal stress played a role in accommodating deformation and closing cracks. Although numerous experiments have been conducted, the mechanism of thermal effect on rock properties is still not completely understood. Furthermore, most of the published researches paid attention to uniaxial compression properties of thermal damaged rocks. Therefore, it is necessary to systematically study the physical and mechanical properties of marble specimens under both effect of confining pressure and high temperature damage.

Experimental observations indicated that based on crack initiation, propagation and coalescence, the progressive failure of brittle rocks could be divided into several stages and related stress thresholds or characteristic stresses, referred to as the crack closure stress σcc, the crack initiation stress σci, and the crack damage stress σcd (Brace et al. 1966; Martin and Chandler 1994; Cai et al. 2004; Xue et al. 2014; Zhao et al. 2015). Among the characteristic stresses, σci and σcd, equal to about 30–50 and 70–85 % of the peak strength, are scale-independent parameters of rocks, which have practical meaning for the stability assessment of surrounding rocks. It was widely accepted that σcd represented the long-term strength of rocks. Martin and Chandler (1994) suggested that the increase in load above σcd was a temporary strain-hardening effect. Moreover, σci was recognized as the lower limit of the long-term strength, and σcd was for the upper limit of the long-term strength. Martin and Christiansson (2009) suggested σci was considered to be an important parameter for estimating field spalling strength. Many researches had been made to investigate influence factors on σci and σcd (Martin and Stimpson 1994; Zhao et al. 2013; Nicksiar and Martin 2013). Martin and Chandler (1994) concluded that σcd was dependent on accumulated damage and σci independent by damage-controlled testing of Lac du Bonnet granite. The results of a laboratory investigation into the effect of grain size published by Eberhardt et al. (1999b) indicated that σcd was seriously affected by grain size and σci remained unaffected. Pestman and Van Munster (1996) conducted a series of acoustic emission experiment on sandstone and found that there was a positively linear correlation between σci and confining pressure. Peng et al. (2013) study experimentally the influence of water pressure and considered that σci had an increase trend while σcd and σpeak decreased with increasing water pressure. As previously mentioned, various rock engineerings were relevant to high temperature condition, and thus thermal cracking undoubtedly had certain influence on the progressive failure of rocks. Nevertheless, the thermal damage effect on characteristic stresses has seldom been reported. To study thermal effect is instrumental in mastering the mechanism of crack development inside rocks under high temperature environment and understanding deeply the progressive failure of rocks concerning with high temperature.

This study systematically investigates coupling effects of thermal damage and confining pressure on the mechanical properties of marbles. Thermal treatment was firstly implemented by heating specimens up to 600 °C for a period of 4 h. Then, a series of triaxial compression tests for marble specimens treated by different temperatures, accompanied with optical and acoustic tests, were carried out. Experimental results including the physical properties, the strength and deformation properties, especially variation of σci and σcd, are described in detail. Considering strongly nonlinear of the peak strength, an empirical model of the negative exponential type is proposed to describe the strength well.

2 Experimental Design

2.1 Description of Rock Samples

Tested rock was mined from Henan Province, China. A massive intact marble block with uniform texture, no visible defects and a rough dimension of 60 × 50 × 15 cm3 was collected in situ. Based on X-ray diffraction analysis for powered samples (Fig. 1), the main mineral was calcite, accounted for over 95 % of the total volume. In order to ensure homogeneity of the tested specimens, all the rock specimens were drilled out of the marble block with the drilling proceeding along the short dimension using a diamond coring bit with an inner diameter of 50 mm. The marble specimens were then cut to 100 mm height, and finally the ends of them were finely grinded and polished to meet the specifications recommended by ISRM (Ulusay and Hudson 2007). P-wave velocity of every rock specimen that had experienced 2-week air drying was measured at room conditions. The average P-wave velocity of the specimens before heating is 5814 m/s and the coefficient of variation is 1.84 %. It could be seen from the petrographical study (Fig. 2a) that the marble is massive structure and coarse granular crystalloblastic texture with the grain size about 2 mm. In addition, density and porosity were determined as recommended by ISRM (Ulusay and Hudson 2007) and described in Table 1.
Fig. 1

X-ray diffraction pattern of marble sample

Fig. 2

Microphotographs of coarse marbles by polarization microscopy under crossed polars: a 20 °C, b 200 °C, c 400 °C, d 600 °C (The marks in the photographs point out different cracking: encircled one grain boundary microcracks, encircled two trans-granular microcracks)

Table 1

Physical properties of marble specimens before thermal treatment

Rock name

Grain density (g/cm3)

Bulk density (g/cm3)

Saturation density (g/cm3)

Total porosity (%)

Effective porosity (%)

P-wave velocity (m/s)








2.2 Testing Facilities and Procedures

The heating device in this study is a box-type resistance furnace (SX3-10-12). The maximum operating temperature is 1200 °C, the rated power is 10 kW and the temperature control precision is 1 °C. The marble specimens were divided into four groups including a control group, which wasn’t subjected to thermal treatment. Each heating group had its corresponding temperature, 200, 400 and 600 °C, respectively. The rock specimens were firstly heated to the target temperature in the furnace at the heating rate of 10 °C/min. The heating rate in previous researches varied from 1 to 30 °C/min. Once the target temperature was reached, temperature was kept constant for 4 h to ensure specimens heated sufficiently. Finally the specimens were left in the furnace to cool down to the room temperature at a lower rate than heating rate.

The uniaxial or triaxial compression tests were performed using a hydraulic servo-controlled compression system (TAW-3000). The test apparatus consists of a loading frame, a pressure chamber, and the measurement and control electronics. The maximum axial load is 3000 kN, the maximum confining pressure is 100 MPa, and the measurement ranges for the axial and lateral extensometers are 8 and 4 mm, respectively. The non-linearity of the two extensometers is less than 0.01 % of the full scale measuring range. The controller (DOLI EDC-220), a significant part of the electronics, could measure, process and control the load, the strain and other parameters, and then enable people to implement test steps.

After thermal treatment and before mechanical tests, representative specimens were chosen for the physical properties tests, including polarizing microscopy test, density and porosity test and ultrasonic velocity test. In every test, each group had no less than 3 tested samples. As ultrasonic pulse transmission technique was non-destructive, P-wave velocity of every specimen experienced thermal treatment was measured. In the following mechanical tests, each of the four specimen groups were subjected to the confining pressure of 0, 5, 10, 15, 20, 25, 30, 35 and 40 MPa, respectively. Firstly, both axial and confining pressures were loaded with a loading rate of 0.25 MPa/s until the target confining pressure was reached. Then, the loading mode control was switched to axial-deformation control. While the confining pressure was kept constant, the axial load continued to be applied at a constant rate of 0.075 mm/min. Test ended until a stable residual strength was reached. The axial strain, lateral strain, and axial stress were recorded during the loading process.

3 Experimental Results

3.1 Physical Properties and Microstructures

One of the intuitive changes for specimens through heating is a significant color change. The color change is attributed to chemical reactions caused by high temperature, such as the oxidation of transition metals or the dehydration of organic materials even in the presence of minor amounts (Gónzalez-Gómez et al. 2015). As seen from the Fig. 5, the color of the specimen without heating is milk-white, slightly with pearly luster. The specimen experienced 200 or 400 °C treatment turn to show cream yellow. With further increase of the temperature to 600 °C, the color of the specimen is red-gray while pearly luster fades out.

Thermal treatment could cause the changes in the microstructure of rocks, inducing cracks initiation. Orthogonal polarizing microscopy tests of thin sections, which were prepared from the specimens experienced different heating temperatures, were conducted using a polarizing optical microscope (Leica DMLA CW4000). In specimens without heating (Fig. 2a), rock grains are cemented closely and almost no microcracks could be observed. Microcracks, especially the grain boundary microcracks, begin to appear and gradually increase when applied temperatures increase from 200 to 400 °C (Fig. 2b, c). There are not only a large number of grain boundary microcracks but also several non-negligible trans-granular microcracks inside the specimen experienced 600 °C treatment (Fig. 2d).

Porosity and P-wave velocity could provide quantitative indicators for assessing the degree of thermal damage. Porosity is related to aperture of cracks, while P-wave velocity is closely sensitive to both morphology and amount of cracks. The effective porosity of the rock specimen is determined using the Saturation and Caliper Techniques recommended by ISRM (Ulusay and Hudson 2007). The effective porosity of the rock specimen experienced different temperatures treatment is calculated from
$$n_{\text{eff}} = \frac{{100\left( {M_{\text{sat}} - M_{\text{s}} } \right)}}{{V\rho_{\text{w}} }}\%$$
where Msat and Ms are respectively saturated-surface-dry mass and grain mass, V is bulk volume, ρw is water density.
The gain density of the specimen is determined following the Pulverization Method by filling pycnometer with rock powder, as recommended by ISRM (Ulusay and Hudson 2007). Then, the total porosity of the specimen is calculated from
$$n_{t} = 100\left( {1 - \frac{\rho }{{\rho_{\text{g}} }}} \right)\%$$
where ρ and ρg are bulk density and grain density, respectively.
P-wave velocity tests were carried out using an ultrasonic pulse analyzer (NM-4A), the transducer frequency of which was set at 50 kHz. Before thermal treatment, P-wave velocity of every rock specimen was measured at room conditions. Then, P-wave velocity of the specimen after thermal treatment was measured again at room conditions. As shown in Fig. 3, the average P-wave velocity of the specimens before heating is 5814 m/s. When the specimens have been treated at 200, 400 and 600 °C, the average P-wave velocity reduces dramatically to about 36, 20 and 13 % of its original value, separately. As for the porosity in Fig. 3, with an increase of the treatment temperature, the effective porosity and the total porosity increase slowly and synchronously. The results redemonstrate a gradual rise in quantity and aperture with an increase of the treatment temperature.
Fig. 3

Porosity and P-wave velocity of marble specimens after heating to different temperatures

3.2 Stress–Strain Relations and Failure Modes

The triaxial stress–strain curves for rock specimens without heating or experienced 200–600 °C treatment are presented in Fig. 4a–d. In the initial deformation stage, i.e., the crack closure stage, the stress–strain curve is up concave due to the closure of existing microcracks. In the low confining pressures (i.e., 0–5 MPa), the crack closure stage of the stress–strain curve for the specimen with higher treatment temperature shows to be longer and more remarkable. Whereas, the non-linearity in the initial deformation stage gradually diminishes with increasing confining pressure (i.e., above 5 MPa). This is due to the fact that the microcracks in the rock specimen are partly closed by the applied confining pressure prior to the application of the axial stress.
Fig. 4

Triaxial stress–strain curves for rock specimens: a 20 °C, b 200 °C, c 400 °C, d 600 °C

It could be seen that the post-peak behaviors for all four kinds of specimens show a similar brittle to ductile transition trend with increasing confining pressure. As for the specimen without heating (Fig. 4a), the post-peak behavior in the uniaxial compression shows a typical brittle manner, i.e., the axial stress drops immediately to zero once reaching its peak. With the increase of the confining pressure (namely, 5–25 MPa), the post-peak behavior shows a strain-softening manner, i.e., the curve remains at peak for a while and then drops to its residual strength. Under high confining pressures (namely, 30–40 MPa), the curve drops relatively slowly and gets a comparatively high residual strength, showing a transition to an ideal plastic manner. As for the specimen experienced heating (Fig. 4b–d), the ductility of the stress–strain curves is reinforced. Subjected to the same confining pressure, the post-peak behavior of a high-temperature treated specimen shows more ductile than that of a low-temperature treated one.

The failure modes for rock specimens without heating or experienced 200–600 °C treatment are presented in Fig. 5a–d. On one hand, the failure mode varies with increase of confining pressure. Under the uniaxial compression, the rock specimen fails in a splitting way. In moderate confining pressures, the rock specimen shows a single shearing failure, which occurs along a clearly defined plane. A conjugate shearing or ductile failure can be observed when the specimen is tested under high confining pressures. On the other hand, the thermal heating could enhance the ductility of rock specimens to a large degree, which has been demonstrated by the post-peak behavior. With the increase of temperature, the transitional confining pressure between two neighboring failure modes decreases (Fig. 5).
Fig. 5

Failure modes of marble specimens after heating to different temperatures: a 20 °C, b 200 °C, c 400 °C, d 600 °C (Photographs under different confining pressures, i.e., 0, 5, 10, 15, 20, 25, 30, 35 and 40 MPa, are from left to right respectively. The marks in the photographs point out different failure modes: encircled one longitudinal splitting, encircled two single shear; encircled three conjugate shear; encircled four ductile)

As for post-peak behaviors and failure modes mentioned previously, confining-pressure effect makes it present a brittle to ductile transition feature and thermal effect reinforces its ductility. The phenomena can be explained as follows: (1) Different parts inside a rock specimen, such as minerals, bounding matrix, cracks and pores, have different strength. When confining pressure is low, low-strength materials inside the specimen yield in the first place. High-strength materials, which haven’t failed in the pre-peak region, begin to unload with the loss of axial loading. Meanwhile, plastic deformation is concentrated in low-strength materials such as no-persistently joints, and the whole specimen shows a brittle manner. However, when confining pressure is high, high-strength materials gradually yield due to the increase of axial loading capacity. Plastic deformation tends to be more evenly distributed within the whole specimen, so the plastic or ductile manner appears. (2) Thermal treatment induces microcracks inside the specimen, which improves the proportion of low-strength materials. Whereas, axial loading capacity of the heated specimen in confined conditions is almost equal to that of the unheated one. That is to say, high-strength materials, the proportion of which is less than before, should provide the same loading capacity as before. As a consequence, the yield of high-strength materials and the homogenization of plastic deformation would occur in advance, and the transitional confining pressure between two neighboring deformation behaviors decreases.

3.3 Deformation and Strength Behavior

As could be seen in Fig. 6, under the same confining pressure, the Young’s modulus of the specimen reduces as temperature is increased. This phenomenon is due to the different levels of fragmentation inside specimens after heating treatment. However, when results on the same temperature conditions are analyzed, it could be seen that there is an increasing trend of the Young’s modulus with an increase of confining pressure.
Fig. 6

Relationship between elastic modulus and confining pressure

As shown in Fig. 7, under the same confining pressure, the peak strain (namely, axial strain at peak strength) of a specimen at high temperature is obviously higher than that at low temperature or without heating, except for few points. The reason is that compaction of more microcracks inside a high-temperature treated specimen produces larger strain value. However, on the same temperature condition, there is a positive linear correlation between the peak strain and confining pressure. The following linear function is used to express the relationship:
$$\varepsilon_{1} = A\sigma_{3} + B$$
where A and B are model constants. According to the fitted curves (Fig. 7), the slopes of four curves nearly parallel to each other, which indicates that the peak strain of the different-temperature treated specimens has the same sensitivity to confining pressure.
Fig. 7

Relationship between peak axial strain and confining pressure

According to the peak strength data in Fig. 8, under the same confining pressure, the peak strength decreases with an increase of the treatment temperature. It is crystal clear that heating treatment induces damage inside rock and thus causes the strength degradation. When the same temperature condition is considered, the peak strength increases with an increase of confining pressure. The peak strength of the high-temperature treated specimen is more sensitive to confining pressure than that of the low-temperature one. That is to say, the higher the treatment temperature is, the higher the increase rate of the confined strength is. There is a large strength difference in the low confining pressure due to thermal induced damage. With increasing confining pressure, the strength difference gradually decreases as the initial microcracks begin to close owing to application of the confining pressure. In other words, it is the initial cracks that mainly control the peak strength under the low confining pressure, while the high confinement is the key factor that greatly affected the peak strength.
Fig. 8

M–C model to describe relationship between peak strength and confining pressure

3.4 Characteristic Stress

For the sake of determining characteristic stresses, various stress–strain methods (denoted as “SS”) and acoustic emission methods (denoted as “AE”) have been presented by researchers. Those methods have different application conditions, difficult degrees and subjectivity. The crack damage stress was at turning point where reversal of volumetric strain-axial strain curve occurred, and thus could be identified by the volumetric strain method without difficulty. The crack volumetric strain method, in which the crack volumetric strain was calculated by subtracting the elastic volumetric strain from the total volumetric strain, was proposed by Martin and Chandler (1994a) to determine σci based on the curve of calculated crack volumetric strain versus axial strain. Unfortunately, it was difficult to use this method in practice to determine σci when there were a large number of microcracks prior to testing, due to its strong dependence on the elastic constants (Eberhardt et al. 1998). In terms of the general knowledge that lateral strain could define the onset of cracking more clearly, the lateral strain response (LSR) method was developed by Nicksiar and Martin (2012) for determining σci objectively. Although both strain response methods were theoretically approximate, the good agreement among the results suggested LSR method valid. The AE method could be used to determining σcd and σci as proposed by Eberhardt et al. (1998). Since it was difficult to differentiate between the background noise and the cracking-source acoustic events, the AE method was still deemed to be a phenomenological method. In the present study, the volumetric strain method is used to determine σcd while LSR is for σci.

The variation of σcd or σci with confining pressure is shown in Fig. 9. At the same treatment temperature, σcd and σci increase with increasing confining pressure except for few data points. The increase rate of both stress thresholds is relatively high on condition of low confining pressures, and slows down subjected to high confining pressure. The ratios of characteristic stresses for each specimen to its peak strength are also calculated and plotted in Fig. 9. For the same temperature specimens, there is a trend of decreasing ratios as confining pressure increases. Besides, the ratios for heated specimens reduce comparatively more than those for unheated specimens.
Fig. 9

Relationship between characteristic stress or its normalized data and confining pressure: a 20 °C, b 200 °C, c 400 °C, d 600 °C

Confining-pressure effect on σcd or σci, which causes an increase of the magnitude and a decrease tendency of the ratio, can be explained as follows: As confining pressure increases, the loading capacity of the specimen increases, and thus characteristic stresses increase accordingly. On the other hand, with increasing confining pressures, rocks’ behavior shows a transition from brittle to ductile, and thus the magnitude of the incremental strength above σcd, recognized as a temporary strain-hardening effect, increases. As the degree of brittleness reduces, the ratio decreases.

As seen in Fig. 10, the magnitude of σcd and σci shows a trend of decrease with the increasing heating temperature, and σcd is more sensitive to the treatment temperature than σci. However, this decrease trend gradually weakens as the confining pressure increases. Suffering from comparatively high confining pressure, for example 20–40 MPa, there is no obvious correlation between the characteristic stresses and the variation of temperature any more. Similar to peak strength, the magnitude of σcd and σci is principally controlled by the thermal effect under the low confining pressure, while the governing factor changes to the confining-pressure effect subjected to high confinement.
Fig. 10

Relationship between characteristic stress and heating temperature: aσcd,bσci

Thermal effect on σcd or σci is a decrease in the magnitude and thermal damage has a greater impact on σcd. Since thermal treatment causes a rise in amount and aperture of microcracks, the stages of crack initiation, propagation and coalescence occur more easily. To some extent, the result is consistent with the research by Eberhardt et al. (1999a), which concluded that both σcd and σci dramatically decreased with increased sampling damage.

4 Proposed Strength-Degradation Model

The Mohr–Coulomb failure criterion, a linear function of the confining pressure, is utilized to express the strength of the specimens according to
$$\sigma_{1} = \frac{2c\cos \phi }{1 - \sin \phi } + \frac{1 + \sin \phi }{1 - \sin \phi }\sigma_{3}$$
where σ1 and σ3 are the major and minor principal stresses, respectively; c and ϕ are the cohesive strength and the frictional angle of the rock mass, respectively.
The fitted model parameters and the squared correlation coefficients are summarized in Table 2. It could be seen from the fitting results (Fig. 8) that c decreases with the increase of the treated temperature and ϕ increases instead. Furthermore, although the squared correlation coefficients for fitting the Mohr–Coulomb model are relatively high without exception, in this case, Mohr–Coulomb failure criterion does not always match the test data well and overestimates the UCS. It is the strongly nonlinearity of the peak strength that causes the overestimation (Peng et al. 2014; Bahrani and Kaiser 2013).
Table 2

Mohr–Coulomb model coefficients





















In order to further investigate the influence of confinement on the strength degradation of thermal damaged rock specimens, the degradation parameter D is defined according to
$$D = \frac{{\sigma_{{{\text{peak}}(20)}} - \sigma_{\text{peak}} }}{{\sigma_{{{\text{peak}}(20)}} }}$$
where σpeak (20) and σpeak are the peak strength under the same confining pressure of the unheated specimen and the heated specimen, respectively.
A negative exponential function model is used to describe the variation of the degradation parameter D affected by confining pressures, i.e.,
$$D = D_{0} \exp \left( { - n\sigma_{3} } \right)$$
where D0 is the initial value of degradation parameters, according to the condition of uniaxial compression, n is the model parameter mainly affecting curvature of the curve.
The best-fit curves are obtained using the least squares method, plotted in Fig. 11. It could be easily concluded that with the increase of heating temperature, fitted parameter D0 increases and n decreases. Furthermore, D0 and n could be expressed as a function of the thermal temperature, separately.
Fig. 11

Degradation parameter D

By substituting Eq. (5) into Eq. (6), the strength-degradation model to describe the variation of the peak strength for thermal damaged rocks is obtained as
$$\sigma_{\text{p}} = \left( {1 - D} \right)\sigma_{{{\text{p}}(20)}} = \left\{ {1 - D_{0} (T)\exp \left[ { - n(T)\sigma_{3} } \right]} \right\}\sigma_{{{\text{p}}(20)}}$$
The variation of the peak strength for unheated rocks is described by Mohr–Coulomb model, and then, that for heated rocks is described by the strength-degradation model, in which both the confining pressure and the temperature are taken into account. In comparison of Figs. 8 and 12, it could be seen that the new model described better than the M–C model, avoiding the overestimation of the strength under the low confining pressure.
Fig. 12

Strength degradation model to describe peak strength of thermal damaged rocks

5 Conclusions

A series of triaxial compression tests for marble specimens treated by four kinds of temperatures, accompanied with optical and acoustic tests, were carried out. The effect of thermal damage on the physical and mechanical properties e.g. porosity, P-wave velocity, strength and deformation parameters and characteristic stresses, were studied. The following conclusions can be drawn:
  1. 1.

    Orthogonal polarizing microscopy test shows a gradual rise of microcracks, i.e., grain boundary microcracks first grow extremely and then trans-granular microcracks occur. With an increase of the treatment temperature, the effective porosity and the total porosity slowly increase, while the P-wave velocity reduces dramatically, which indirectly reflecting thermal cracking. Besides that, significant color changes of specimens are observed.

  2. 2.

    According to the post-peak behaviors and the failure modes of thermal damaged rocks on the triaxial tests, there is a brittle to ductile transition trend with increasing confining pressure and heating enhances the ductility to some degree.

  3. 3.

    The compared study on strength and deformation parameters concludes that heating causes damage and confining pressure inhibits the damage to develop. Under the same confining pressure, with higher treatment temperature, Young’s modulus and the peak strength are lower and axial strain at peak strength is higher. As for the confined effect, at the same treatment temperature, Young’s modulus and axial strain at peak strength increase, the peak strength increases at a decreased rate and the strength difference diminishes, with the increasing confining pressure.

  4. 4.

    Concerning the variation of characteristic stresses, σcd and σci increase, while the ratios σcdpeak or σcipeak show a decreasing trend with the increase of confining pressure. The magnitude of σcd or σci shows a trend of decrease with the increasing heating temperature, and this decrease trend gradually weakens as the confining pressure increases.

  5. 5.

    Finally, an empirical model of the negative exponential type is proposed, which described the peak strength well.




The research work presented in this paper is supported by the National Natural Science Foundation of China (Grant No. 51579189), the National Basic Research Program of China (‘‘973’’ Program, Grant No. 2011CB013501), and the Fundamental Research Funds for the Central Universities. The authors are grateful for these financial supports.


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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  • Mengdi Yao
    • 1
    • 2
  • Guan Rong
    • 1
    • 2
  • Chuangbing Zhou
    • 1
    • 3
  • Jun Peng
    • 1
    • 2
  1. 1.State Key Laboratory of Water Resources and Hydropower Engineering ScienceWuhan UniversityWuhanChina
  2. 2.Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of EducationWuhan UniversityWuhanChina
  3. 3.School of Civil Engineering and ArchitectureNanchang UniversityNanchangChina

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