Introduction

The depletion of shallow resources due to rapid economic and social development has driven a global shift toward deep resource extraction1,2. However, current mining engineering practices have outpaced fundamental mechanical theory, leading to inefficiencies and a lack of direction in deep mining3. This has prompted extensive research into rock mass deformation and failure under high‒stress conditions4,5,6. The second law of thermodynamics suggests that energy instability is the fundamental characteristic of solid material deformation and failure. Thus, understanding the evolution of strain energy during rock loading is crucial for predicting geological hazards and effectively managing underground resources.

The thermodynamic approach is a key method in rock mechanics for studying rock damage and failure. Xie et al.7,8 proposed that rock deformation and failure result from the interplay between released ue and ud, provided methods to calculate these energies during uniaxial compression tests. Peng et al.9 explored the relationship between energy conversion and damage in coal rocks under conventional triaxial compression via similar methods, whereas Zhao et al.10 studied the energy and damage evolution in granites with varying brittleness under true triaxial compression.

In research on rock energy evolution under compression, the method of calculating ud and ue is central. Although elastic theory‒based methods have been effective, they often assume a linear stress–strain relationship during unloading11,12. However, factors such as confining pressure, rock anisotropy, strain rate, size, and time effects cause deviations from this ideal model13,14,15,16,17. This raises doubts about the accuracy of traditional calculation methods. Previous studies have shown that during cyclic loading‒unloading, the area enclosed by the unloading curve and the strain axis represents stored ue, which can be calculated via curve integration. Many scholars have investigated the evolution of ud and ue during cyclic tests15,18,19,20. For example, Zhang et al.21 analyzed the energy evolution in red sandstone, whereas Xie et al.22 studied the damage mechanisms in water‒bearing coal samples. Extending research on cyclic loading‒unloading, scholars have sought to refine the method of calculating ud and ue under uniaxial compression by integrating unloading curves. Gong et al.23,24,25,26 conducted single‒cycle loading–unloading tests on various rocks, revealing linear energy storage and dissipation laws. They addressed the challenge of calculating energy at peak stress, which is critical issue in uniaxial compression tests. Liu et al.27 further demonstrated that the ue values obtained from cyclic tests align closely with those from single‒cycle tests, suggesting a new method for calculating energy under uniaxial compression. Compared to uniaxial tests, true triaxial tests better simulate the complex stress states in real engineering scenarios. The aforementioned studies proposed a method for calculating ue by integrating unloading stress‒strain curves. However, this approach has not yet been applied to ud and ue analysis in true triaxial tests.

To elucidate the problems associated with the use of integration methods for calculating ud and ue in rocks under true triaxial compression, we described by a combination of figures and text is employed. Figure 1a shows the stress‒strain curve of true triaxial compression, Fig. 1b shows the stress‒strain curve for the σ1 loading stage of true triaxial compression, and Fig. 1c shows the stress‒strain curve for σ1 unloading in true triaxial compression.

Fig. 1
figure 1

Stress‒strain curve of true triaxial tests.

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Figure 1a shows that by integrating the loading stress‒strain curve, one can calculate the u in all three directions during the simultaneous loading phase of the three principal stresses under true triaxial compression. During the σ1 loading phase, integrating curve D2D in Fig. 1b as σ1 increases yields of u in the σ1 direction (area enclosed by points DD1D3D2). In Fig. 1b, the changes in u in the σ2 and σ3 directions are represented by the areas enclosed by points BB1B3B2 and AA1A3A2, respectively. However, the amount of ue and ud contained in the areas enclosed by BB1B3B2 and AA1A3A2 cannot be determined. Upon unloading of σ1, integrating curve DD5 in Fig. 1c yields the ue released in that direction (area enclosed by points DD1D4D5). In Fig. 1c, the changes in u in the σ2 and σ3 directions are represented by the areas enclosed by points B5B4B3B2 and A5A4A3A2, respectively. However, the ue and ud components within these areas remain indeterminable.

To address these challenges, we designed a true triaxial stress‒balanced unloading test (ensuring simultaneous unloading of three principal stresses to zero). On the basis of the true triaxial stress‒balanced unloading tests, we calculated the evolution laws of ud and ue in the σ2 and σ3 directions induced by σ1 loading. Building on these findings, we investigated the relationship between true triaxial ue and u, revealing that the evolution of rock ud and ue under true triaxial compression also adheres to the linear energy storage law. Leveraging the true triaxial stress‒balanced unloading test and the linear energy storage law, we propose a novel method for calculating rock ud and ue under true triaxial compression. Furthermore, the influence of the evolution laws of ud and ue under true triaxial compression on the discrimination of rock bursts is discussed.

Test equipment, sample and true triaxial compression curve

Test equipment

To investigate the calculation of ud and ue for rocks under true triaxial compression, we conducted true triaxial tests via the true triaxial disturbed unloading rock mechanics test system. A diagram of the true triaxial disturbed unloading rock mechanics test system diagram is shown in Fig. 2.

Fig. 2
figure 2

True triaxial disturbed unloading rock mechanics test system.

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The system employs a 3-platen arrangement, where each of the three orthogonal axes is independently controlled to apply different principal stresses (σ1, σ2 and σ3) on the sandstone samples.The loading system is equipped with high‒precision load cells and displacement transducers to measure the applied forces and deformations accurately. To minimize the friction effects between the platens and the sample, we applied white Vaseline to the sample. White Vaseline significantly reduces frictional resistance, ensuring uniform stress distribution.

Sandstone samples

The sandstone used in this study originates from a single source, and the samples were processed into cubes with a side lengths of 100 mm sides. The uniaxial compressive strength is approximately 45 MPa. Figure 3 displays the sandstone samples used in this study.

Fig. 3
figure 3

Sandstone samples and petrography.

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A detailed petrographic analysis of the sandstone samples was conducted to characterize their mineralogical composition. The key findings are as follows: the sandstone is predominantly composed of quartz (approximately 85%), feldspar (approximately 10%), and minor amounts of mica and lithic fragments (approximately 5%). grain size and sorting: the grains are generally well‒sorted, with an average sizes ranging from 0.2 to 0.5 mm; the texture is medium‒grained, indicative of a relatively high degree of uniformity in sedimentation; cementation: the sandstone exhibits silica cementation, with some areas showing partial clay cement. The sandstone samples used in this study were obtained from the Pz (Paleozoic erathem) Formation, which is located in the Luling Coal Mine in Anhui Province.

TTC test scheme and test curve analysis

Detailed procedure for the true triaxial compression test:

Step 1::

Load the sandstone sample in the σ1, σ2, and σ3 directions at loading rates of 0.2, 0.15, and 0.1 MPa/s until it reaches 40, 30 and 20 MPa, respectively.

Step 2::

The values of σ2 and σ3 should remain constant while loading σ1 at a rate of 0.2 MPa/s until rock failure.

The data presented in Fig. 4 indicate that the σ1 of sandstone failure under true triaxial loading (σ3 = 20 MPa, σ2 = 30 MPa) is 191.5 MPa.

Fig. 4
figure 4

True triaxial compression stress‒strain curves.

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The data presented in Fig. 4 indicate that during the true triaxial loading process, the sandstone also exhibited clear stages of compaction, elasticity, plasticity, and postpeak failure. After the principal stresses in all three directions reach the initial stress level simultaneously, as σ1 increases, the strains in the σ2 and σ3 directions gradually decrease. In the study of rock triaxial strain under true triaxial compression via elastic theory, the σ1 generally increases, whereas the elastic strain decreases in the directions of the σ2 and σ3 without a change in residual strain. However, as a natural geological material, rock contains many inherent flaws and is not completely an elastic body. Therefore, the use of elasticity theory to analyze the mechanical properties of rocks is not consistent with the nature of the rocks themselves. Therefore, the accuracy of the analysis results is questionable when elasticity theory is used to analyze the mechanical properties of rocks.

True triaxial stress‒balanced unloading test and analysis

Method for calculating strain energy via elastic theory

The strain energy density analysis in this study is based on two principles: first, that the experimental process complies with the first law of thermodynamics, i.e., different forms of energy follow the law of conservation during transfer and conversion; second, the system is closed, i.e., there is no heat exchange between the material and the outside world, namely, the system is closed9. The strain energy density produced by the external force acting on the rock is u, part of which is ud, and the other part is stored inside the rock under certain conditions in the form of a reversible ue, which can be described via the following equation:

$$u = u^{d} + u^{e}$$
(1)

The equation for calculating the value of u for rocks under true triaxial compression can be described by the following equation24:

$$u = \int\limits_{0}^{{\varepsilon_{1} }} {\sigma_{1} } d\varepsilon_{1} + \int\limits_{0}^{{\varepsilon_{2} }} {\sigma_{2} } d\varepsilon_{2} + \int\limits_{0}^{{\varepsilon_{3} }} {\sigma_{3} } d\varepsilon_{3}$$
(2)

where ε1,ε2 and ε3 represent the strains in the σ1, σ2, and σ3 directions, respectively.

Under complex stress states (true triaxial stress state, conventional triaxial stress state), the method for calculating ud and ue based on elastic theory is usually used for calculation, and the equation are as follows:

$$\left\{ {\begin{array}{*{20}c} {u_{1}^{e} = \frac{{\sigma_{1}^{2} - \mu \left( {\sigma_{1} \sigma_{2} + \sigma_{1} \sigma_{3} } \right)}}{2E}} \\ {u_{2}^{e} = \frac{{\sigma_{2}^{2} - \mu \left( {\sigma_{1} \sigma_{2} + \sigma_{2} \sigma_{3} } \right)}}{2E}} \\ {u_{3}^{e} = \frac{{\sigma_{3}^{2} - \mu \left( {\sigma_{1} \sigma_{3} + \sigma_{2} \sigma_{3} } \right)}}{2E}} \\ {u^{e} = \frac{{\left[ {\sigma_{1}^{2} + \sigma_{2}^{2} + \sigma_{3}^{2} - 2\mu \left( {\sigma_{1} \sigma_{2} + \sigma_{2} \sigma_{3} + \sigma_{1} \sigma_{3} } \right)} \right]}}{2E}} \\ \end{array} } \right.$$
(3)

In the study of the mechanical properties of rock under cyclic loading–unloading, many scholars have considered factors such as different confining pressures, rock anisotropy, strain rates, sizes, and time effects. It is reported that the complex internal structure of rocks and the microcracks within rocks cause their responses to change in the stress state to deviate from the ideal elastic model, casting doubt on the accuracy of the method for calculating ud and ue based on elastic theory13,14,15,16,17.

Supplemental test of true triaxial compression strain energy density analysis and scheme

To precisely calculate the strain energy evolution of rocks under true triaxial compression, we designed a true triaxial stress‒balanced unloading test. The stress diagram depicted in Fig. 5 shows a graphical illustration of the true triaxial stress‒balanced unloading test.

Fig. 5
figure 5

True triaxial stress-balanced unloading test.

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A true triaxial stress-balanced unloading test involves the concurrent reduction of stress in the σ1, σ2 and σ3 directions, with the specific requirement that σ1, σ2 and σ3 simultaneously decrease to zero. True triaxial stress-balanced unloading tests help in preventing internal damage due to stress concentration or redistribution. Therefore, true triaxial stress-balanced unloading tests do not damage the rock further; thus, the stability of the internal rock structure is maintained.

The true triaxial compression test in this paper can be divided into two phases. In Phase 1 the simultaneous loading of principal stresses in three directions occurs. In Phase 2, σ1 loading occurs. When stage 2 is performed, the increase in σ1 causes a change in the ud and ue of σ2 and σ3. The true triaxial stress-balanced unloading tests can be categorized as follows: Supplemental Test 1 (Phase 1 of the true triaxial stress-balanced unloading test) and Supplemental Test 2 (Phase 2 of the true triaxial stress-balanced unloading test).

The detailed procedure for Supplemental Test 1 is as follows:

Step 1::

Load the sandstone sample in the σ1, σ2, and σ3 directions at loading rates of 0.2, 0.15, and 0.1 MPa/s until loads 40, 30, and 20 MPa, respectively, are reached.

Step 2::

Unloading in the σ1, σ2, and σ3 directions at unloading rates of 0.2, 0.15, and 0.1 MPa/s until loads of 0, 0, and 0, respectively, are reached.

For the true triaxial stress-balanced unloading test, to ensure that all three principal stresses reach 0 simultaneously, the unloading rates for the other two directions must be designed on the basis of σ1. The specific rates are presented in Table 1.

Table 1 Unloading rate of the supplemental test.
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The detailed procedure for Supplemental Test 2 (Stage 2 true triaxial stress-balanced unloading test) is as follows:

Step 1::

Load the sandstone sample in the σ1, σ2, and σ3 directions at loading rates of 0.2 MPa/s, 0.15 MPa/s, and 0.1 MPa/s until loads of 40 MPa, 30 MPa, and 20 MPa, respectively, are reached.

Step 2::

For four separate samples, load σ1 at rates of 0.2 MPa/s to 60 MPa, 76.5 MPa (stress ratio of 0.4), 115 MPa (stress ratio of 0.6), and 153 MPa (stress ratio of 0.8), respectively.

Step 3::

Simultaneously, unload all three principal stresses to 0.

Method for calculating the strain energy density under true triaxial compression

Evolution law of strain energy density in true triaxial compression Phase 1

A schematic diagram of ue during Phase 1 of the true triaxial stress-balanced unloading test is depicted in Fig. 6.

Fig. 6
figure 6

Schematic stress–strain diagram for Supplemental Test 1.

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Figure 6 shows that in Phase 1, ue in σ3, σ2, and σ1 directions of sandstone are represented by the areas enclosed by points AA1A31, BB1B31, and CC1C31, respectively.

Phase 1 u can be expressed as follows:

$$\left\{ {\begin{array}{*{20}c} {u_{11}^{{}} = \int\limits_{0}^{{\varepsilon_{10} }} {\sigma_{1} d\varepsilon_{1} \begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} & {} \\ \end{array} } & {} & {} \\ \end{array} } } \\ {u_{21}^{{}} = \int\limits_{0}^{{\varepsilon_{20} }} {\sigma_{2} d\varepsilon_{2} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u_{31}^{{}} = \int\limits_{0}^{{\varepsilon_{30} }} {\sigma_{3} d\varepsilon_{3} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u_{{}}^{1} = \int\limits_{0}^{{\varepsilon_{10} }} {\sigma_{1} d\varepsilon_{1} + \int\limits_{0}^{{\varepsilon_{20} }} {\sigma_{2} d\varepsilon_{2} + \int\limits_{0}^{{\varepsilon_{30} }} {\sigma_{3} d\varepsilon_{3} } } } } \\ \end{array} } \right.$$
(4)

In Eq. (4) \(u_{11}^{{}}\), \(u_{21}^{{}}\), \(u_{31}^{{}}\), and \(u_{{}}^{1}\) represent σ1, σ2, σ3 and the overall u in Phase 1, respectively.

The ue of Phase 1 can be expressed by the following equation:

$$\left\{ {\begin{array}{*{20}c} {u_{11}^{e} = \int\limits_{{\varepsilon_{11} }}^{{\varepsilon_{10} }} {\sigma_{1} d\varepsilon_{1} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u_{21}^{e} = \int\limits_{{\varepsilon_{21} }}^{{\varepsilon_{20} }} {\sigma_{2} d\varepsilon_{2} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u_{31}^{e} = \int\limits_{{\varepsilon_{31} }}^{{\varepsilon_{30} }} {\sigma_{3} d\varepsilon_{3} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u_{e}^{1} = \int\limits_{{\varepsilon_{11} }}^{{\varepsilon_{10} }} {\sigma_{1} d\varepsilon_{1} + \int\limits_{{\varepsilon_{21} }}^{{\varepsilon_{20} }} {\sigma_{2} d\varepsilon_{2} + \int\limits_{{\varepsilon_{31} }}^{{\varepsilon_{30} }} {\sigma_{3} d\varepsilon_{3} } } } } \\ \end{array} } \right.$$
(5)

In Eq. (5) \(u_{11}^{e}\), \(u_{21}^{e}\), \(u_{31}^{e}\), and \(u_{e}^{1}\) represent σ1, σ2, σ3 and the overall ue in Phase 1, respectively.

The ud in Phase 1 can be expressed by the equation:

$$\left\{ {\begin{array}{*{20}c} {u_{11}^{d} = u_{11} - u_{11}^{e} \begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} & {} \\ \end{array} } & {} & {} \\ \end{array} } \\ {u_{21}^{d} = u_{21} - u_{21}^{e} \begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } \\ {u_{31}^{d} = u_{31} - u_{31}^{e} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } \\ {u_{d}^{1} = u^{1} - u_{e}^{1} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } \\ \end{array} } \right.$$
(6)

In Eq. (6) \(u_{11}^{d}\), \(u_{21}^{d}\), \(u_{31}^{d}\) and \(u_{d}^{1}\) represent σ1, σ2, σ3 and the overall ud in Phase 1, respectively.

Method for calculating the strain energy density for true triaxial compression tests

A schematic diagram of the stress–strain for Supplemental Test 2 is depicted in Fig. 7. In Fig. 7, the ue of the sandstone in the σ3, σ2, and σ1 directions is represented by the areas enclosed by points A2A3A32, B2B3B4, and DD1D3, respectively. The strain at point D is ε12, that at point D3 is ε13, that strain at point B2 is ε22, that at point B4 is ε23, that at point A2 is ε12, and that at point A32 is ε13.

Fig. 7
figure 7

Schematic stress–strain diagram for Supplemental Test 2.

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The value of u for rock during true triaxial compression can be calculated via Supplemental Test 2:

$$\left\{ {\begin{array}{*{20}c} {u_{1}^{{}} = \int\limits_{0}^{{\varepsilon_{12} }} {\sigma_{1} d\varepsilon_{1} \begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} & {} \\ \end{array} } & {} & {} \\ \end{array} } } \\ {u_{2}^{{}} = \int\limits_{0}^{{\varepsilon_{22} }} {\sigma_{2} d\varepsilon_{2} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u_{3}^{{}} = \int\limits_{0}^{{\varepsilon_{32} }} {\sigma_{3} d\varepsilon_{3} \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } } \\ {u = \int\limits_{0}^{{\varepsilon_{10} }} {\sigma_{1} d\varepsilon_{1} + \int\limits_{0}^{{\varepsilon_{20} }} {\sigma_{2} d\varepsilon_{2} + \int\limits_{0}^{{\varepsilon_{30} }} {\sigma_{3} d\varepsilon_{3} } } } } \\ \end{array} } \right.$$
(7)

In Eq. (7), u1, u2, and u3 and u represent σ1, σ2, σ3 and the overall u of true triaxial compression, respectively.

The ue during true triaxial compression can be expressed as follows:

$$\left\{ {\begin{array}{*{20}c} {u_{1}^{e} = \int\limits_{{\varepsilon_{13} }}^{{\varepsilon_{12} }} {\sigma_{1} d\varepsilon_{1} } } \\ {u_{2}^{e} = \int\limits_{{\varepsilon_{23} }}^{{\varepsilon_{22} }} {\sigma_{2} d\varepsilon_{2} } } \\ \begin{gathered} u_{3}^{e} = \int\limits_{{\varepsilon_{33} }}^{{\varepsilon_{32} }} {\sigma_{3} d\varepsilon_{3} } \hfill \\ u^{\prime e} = \int\limits_{{\varepsilon_{13} }}^{{\varepsilon_{12} }} {\sigma_{1} d\varepsilon_{1} + \int\limits_{{\varepsilon_{23} }}^{{\varepsilon_{22} }} {\sigma_{2} d\varepsilon_{2} + \int\limits_{{\varepsilon_{33} }}^{{\varepsilon_{32} }} {\sigma_{3} d\varepsilon_{3} } } } \hfill \\ \end{gathered} \\ \end{array} } \right.$$
(8)

In Eq. (8) \(u_{1}^{e}\), \(u_{2}^{e}\), and \(u_{3}^{e}\) and u’e represent σ1, σ2, σ3 and the overall ue of true triaxial compression, respectively.

The ud during true triaxial compression can be expressed as:

$$\left\{ {\begin{array}{*{20}c} {u_{1}^{d} = u_{1} - u_{1}^{e} } \\ {u_{2}^{d} = u_{2} - u_{2}^{e} } \\ {u_{3}^{d} = u_{3} - u_{3}^{e} } \\ {u^{d} = u - u^{e} } \\ \end{array} } \right.$$
(9)

In Eq. (9) \(u_{1}^{d}\), \(u_{2}^{d}\), and \(u_{3}^{d}\), and ud represent σ1, σ2, σ3 and the overall ud of true triaxial compression, respectively.

Strain energy density calculation method of true triaxial compression test phase 2

The value u in phase 2 of the true triaxial compression test can be described as:

$$\left\{ {\begin{array}{*{20}c} {u_{{}}^{2} = u - u_{{}}^{1} } \\ {u_{12}^{{}} = u_{1}^{{}} - u_{11}^{{}} } \\ {u_{22}^{{}} = u_{2}^{{}} - u_{21}^{{}} } \\ {u_{32}^{{}} = u_{3}^{{}} - u_{31}^{{}} } \\ \end{array} } \right.$$
(10)

In Eq. (10), u12, u22, u32, and u2 represent σ1, σ2, σ3, and the overall u of rock in true triaxial compression test Phase 2, respectively.

The ue in Phase 2 of the true triaxial compression test can be expressed as follows:

$$\left\{ {\begin{array}{*{20}c} {u_{e}^{2} = u^{e} - u_{e}^{1} } \\ {u_{12}^{e} = u_{1}^{e} - u_{11}^{e} } \\ {u_{22}^{e} = u_{2}^{e} - u_{21}^{e} } \\ {u_{32}^{e} = u_{3}^{e} - u_{31}^{e} } \\ \end{array} } \right.$$
(11)

In Eq. (11), \(u_{12}^{e}\), \(u_{22}^{e}\), \(u_{32}^{e}\) and \(u_{e}^{2}\) represent σ1, σ2, σ3 and the overall ue of rock in true triaxial test Phase 2, respectively.

The ud in Phase 2 of the true triaxial compression test can be expressed as follows:

$$\left\{ {\begin{array}{*{20}c} {u_{d}^{2} = u^{2} - u_{e}^{2} } \\ {u_{12}^{d} = u_{12}^{{}} - u_{12}^{e} } \\ {u_{22}^{d} = u_{22}^{{}} - u_{22}^{e} } \\ {u_{32}^{d} = u_{32}^{{}} - u_{32}^{e} } \\ \end{array} } \right.$$
(12)

In Eq. (12), \(u_{12}^{d}\), \(u_{22}^{d}\), \(u_{32}^{d}\), and \(u_{d}^{2}\) represent σ1, σ2, σ3 and the overall ud of rock in true triaxial test Phase 2, respectively.

Analysis of the true triaxial stress-balanced unloading test

The true triaxial stress-balanced unloading test is illustrated in Fig. 8. As shown in Fig. 8, during the true triaxial loading process, when σ2 and σ3 reach the initial stress level (σ1 = 40 MPa, σ2 = 30 MPa, and σ3 = 20 MPa), with increasing σ1, both σ3 and σ2 exhibit a decreasing strain trend. The strain variations in the true triaxial compression test were determined on the true triaxial stress-balanced unloading test, and the results are shown in Table 2.

Fig. 8
figure 8

Stress–strain curves from the true triaxial stress-balanced unloading tests.

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Table 2 Strain of sandstone under true triaxial compression.
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The data presented in Fig. 9a indicates that under true triaxial compression, an increase in σ1 increases the elastic strain of σ1, whereas the elastic strains of σ2 and σ3 decrease. As shown in Fig. 9b, with the increasing σ1, the residual deformation in all the σ1, σ2 and σ3 directions tended to increase. Combining the aforementioned analysis, it can be inferred that in true triaxial compression experiments, an increase in σ1 results in an increase in residual strain in the σ2 and σ3 directions, while simultaneously causing a decrease in elastic strain in the σ2 and σ3 directions. Furthermore, the magnitude of the decrease in elastic strain in the σ2 and σ3 directions exceeds the magnitude of the increase in residual strain.

Fig. 9
figure 9

Strain evolution of rock under different stress levels.

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Strain energy density evolution and damage analysis under true triaxial compression

Evolution analysis of the strain energy density of rock under true triaxial compression

On the basis of the true triaxial stress-balanced unloading test and the true triaxial ud and ue calculation methods proposed in this paper, the energy density of sandstone under true triaxial compression is shown in Table 3.

Table 3 Evolution analysis of strain energy density of rock under true triaxial compression.
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Figure 10 illustrates the evolution of ud and ue of σ1, σ2 and σ3 in sandstone under true triaxial compression. With increasing σ1, u and ud and ue in the σ1 direction progressively increase, whereas in the σ2 and σ3 directions, u and ue gradually decrease as ud increases. Moreover, during σ1 loading, the decrease in ue in the σ2 and σ3 directions exceeds the ud generated in these directions. This suggested that during σ1 loading, the ue in σ2 and σ3 was progressively released or converted into ud, with the remaining portion continuing to be stored within the rock.

Fig. 10
figure 10

Evolution of the strain energy density of rock under true triaxial compression.

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Damage analysis under true triaxial compression

While the true triaxial stress-balanced unloading test can calculate ud and ue at specific stress levels, the ud and ue for arbitrary stress states in true triaxial compression tests cannot be calculated. To study rock peak energy evolution patterns, some researchers have employed exponential polynomials for fitting. While this approach can calculate peak strain energy, minor parameter variations during polynomial parameter identification often result in significant errors in energy storage limits. On the basis of the energy dissipation characteristics throughout the uniaxial loading process of rocks, Gong et al.23,24,25,26 proposed linear energy storage and dissipation laws, revealing a linear relationship between u and ue. To investigate the energy storage limits of rocks under true triaxial compression, an analysis of the input and dissipated strain energy densities along the three principal stress directions was conducted, revealing that the ud and ue evolution in sandstone under true triaxial compression also adheres to the linear energy storage law. The results of the fitting are shown in Table 4.

Table 4 Fitting equation for the strain energy density of rock under true triaxial compression.
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The ud of rocks during loading reflects the damage degree of damage to the rock, and is expressed as follows27:

$$D = \frac{{u^{d} }}{{u_{p}^{d} }}$$
(13)

where \(u_{p}^{d}\) represents ud at the peak load and D is the damage variable.

The change laws of ud and ue of rock under true triaxial compression can be obtained via calculation, as shown in Fig. 11. The ue and ud of sandstone subjected to complex stress conditions can be quantified through a combination of true triaxial stress-balanced unloading tests and the application of a linear energy storage law. As evident from Fig. 11, during the elastic stage, the rock’s u, ue, and ud demonstrate approximately linear changes. In the plastic stage, the growth rates of ud and stored ue became nonlinear, with the growth rate of the rock energy density reaching its maximum as failure approached. Figure 11 shows the damage‒stress evolution curve for rock under true triaxial compression, where, during the initial loading stage, the rock’s internal damage exhibits an approximately linear increase with increasing stress; as the stress exceeds 140 MPa, the rock transitions into the plastic stage, with the rate of internal damage growth progressively accelerating, reaching its peak rate of damage growth as failure approaches.

Fig. 11
figure 11

Strain energy density and damage evolution of rock.

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Verification of the strain energy density calculation method

In the field of solid material damage evolution, numerous researchers posit that acoustic emission (AE) counts serve as indicators of material deterioration. These counts exhibit heightened activity when new cracks form and expand within the substance28,29,30.

Investigations into the uniaxial cyclic loading‒unloading of rock samples have revealed that the unloading phase does not induce rock damage. Notably, AE counts enter an “intermittent period” during the unloading stage of these tests31. These findings suggest that AE activity is minimal or absent in rocks when no damage occurs.

Building upon previous AE research, we conducted AE monitoring during true triaxial stress‒balanced unloading tests. Our objective was to analyze the frequency of AE counts throughout the stress reduction phase, thereby assessing the occurrence and extent of potential rock damage. The results of our experimental observations are visually presented in Fig. 12, which provides a comprehensive illustration of the AE counts activity patterns during the unloading process.

Fig. 12
figure 12

True triaxial stress‒balanced unloading test AE counts.

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Figure 12 shows it can be seen that during the true triaxial loading stage, the intensity of the AE counts in rock gradually increases with increasing stress. As the stress level intensified during the loading stage, the frequency of AE counts in the rock sample exhibited a corresponding increase. Conversely, upon transitioning to the unloading phase of the true triaxial stress‒balanced test, a marked shift in the AE behavior was observed. The previously active AE counts entered a quiescent period characterized by a significant reduction in detectable emissions. This abrupt cessation of AE activity strongly implies the absence of new crack formation or propagation within the rock structure during unloading. The observed acoustic silence during stress reduction supports the hypothesis that the unloading process in our proposed true triaxial stress‒balanced unloading test primarily involves the release of ue. Importantly, this phenomenon occurs without inducing additional damage to the rock sample. These findings provide substantial validation for the ud and ue calculation methodology introduced in this study.

Influence of the unloading stress path on the strain energy density analysis under true triaxial compression

In the above study a true triaxial stress‒balanced unloading test was used to analyze the evolution of ud and ue in true triaxial compression tests. The question is whether the unloaded stress path of σ1 in the true triaxial test affects the evolution of the three principal stress directions of ud and ue.

To investigate the effect of the unloading path of σ1 on the three principal stress directions for ud and ue under true triaxial compression, we design the true triaxial loading‒unloading test, with the following procedural steps:

Step 1::

Load the sandstone sample along the σ1, σ2, and σ3 directions at rates of 0.2 MPa/s, 0.15 MPa/s, and 0.1 MPa/s respectively, until 40 MPa, 30 MPa, and 20 MPa are reached.

Step 2::

For the four samples, increase σ1 at rates of 0.2 MPa/s to 60 MPa, 76.5 MPa (stress ratio 0.4), 115 MPa (stress ratio of 0.6), and 153 MPa (stress ratio of 0.8).

Step 3::

Decrease σ1 to 40 MPa at a rate of 0.2 MPa/s.

Step 4::

Simultaneously, reduce σ1, σ2, and σ3 to 0 at rates of 0.2 MPa/s, 0.15 MPa/s, and 0.1 MPa/s respectively.

The stress path is illustrated in Fig. 13.

Fig. 13
figure 13

True triaxial loading‒unloading test.

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The stress‒strain curve of a true triaxial loading‒unloading test is shown in Fig. 14.

Fig. 14
figure 14

Stress–strain curves from the true triaxial loading‒unloading test.

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As depicted in Fig. 14, the true triaxial loading‒unloading test can provide results of ue in the σ1 direction. During the loading process of σ1 from 40 to 60 MPa, 76.5 MPa, 115 MPa, and 153 MPa, the changes in ud and ue in the σ2 and σ3 directions can be obtained, but the changes in the values of u within these variations cannot be determined. On the basis of generalized Hooke’s law, it can be assumed that the reduction of u in the σ2 and σ3 directions induced by the loading of σ1 is entirely due to ue, and similarly, the increase in u in the σ2 and σ3 directions induced by the unloading of σ1 is entirely due to ue.

The ud and ue of a true triaxial loading‒unloading test are shown in Table 5.

Table 5 Strain energy density of a true triaxial loading‒unloading test.
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A comparison of the ue of the three principal stress directions and the overall ue of the sandstone under the two unloading stress paths is shown in Fig. 15.

Fig. 15
figure 15

Patterns of ue evolution under various unloading stress paths under true triaxial compression.

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To further demonstrate the advantages of the true triaxial stress‒balanced unloading test proposed in this study, a comparison was made between the elastic strain energy densities in the three principal stress directions calculated from the true triaxial single loading‒unloading test and the actual ue value under true triaxial compression. A comparison revealed that in the direction of σ1, the ue calculated from the true triaxial loading‒unloading test was lower than the actual ue under true triaxial compression. The difference between the two increased with the increasing σ1, reaching 0.05 mJ·mm−3 at 153 MPa. Conversely, the ue in the σ3 and σ3 directions calculated from the true triaxial loading‒unloading test exceeded the actual ue of the rock under true triaxial compression. The disparity between the two increases with the increasing in σ1. At 153 MPa, the differences in ue between the true triaxial loading‒unloading test results calculations and the true triaxial compression test results are 0.0301 and 0.0204 mJ·mm−3 in the σ2 and σ3 directions, respectively.

Discussion

A rockburst is a phenomenon in which coal ejection occurs during deep coal seam mining, releasing kinetic energy along with noise, vibrations, air blasts, or shock waves. In severe cases, it can cause damage to the roof and floor and even destroy tunnels, leading to casualties, and is considered a typical geological hazard in deep mining engineering32. Many researchers have proposed various coal burst tendency indices, such as the \(A^{\prime}_{CF}\)33, WT34, CEF35, peak energy impact index36, and KET37.

In the abovementioned research on rockburst determination indices, most of the indices are closely related to the evolution of ud and ue in uniaxial compression tests. However, this study revealed that there are significant differences in the calculation of ud and ue between true triaxial compression and uniaxial compression of rock. Under true triaxial compression conditions, a change in the principal stress in one direction usually induces changes in triaxial strain energy storage. This complexity in the stress‒strain relationship may limit the applicability of determination indices derived from traditional uniaxial compression tests in practical applications. In addition, this study explores the effects of different σ1 unloading stress paths on the ud and ue of sandstone. Research has shown that different unloading paths significantly influence the evolution patterns of ud and ue. This finding indicates that in practical engineering, the complex stress paths experienced during deep rock mass excavation need to be incorporated into the assessment of rockburst tendencies.

Given the complexity of ud and ue calculations under true triaxial compression conditions, multiaxial multi‒axial stress state tests and stress path analyses that more closely resemble actual working conditions in the assessment of rockburst tendencies are recommended. This approach can not only increase the accuracy of determination but also provide a more reliable theoretical foundation for disaster prevention and mitigation measures in deep coal seam mining.

Conclusion

This study analyzed the characteristics of the changes in the σ2 and σ3 strain energy density induced by increasing σ1 under true triaxial compression and investigated the influence of the σ1 unloading stress path on sandstone energy. The research findings are as follows:

  1. 1.

    On the basis of the true triaxial stress‒balanced experiment, a new method for the analysis of the strain energy density of sandstone under true triaxial compression is proposed, and the correctness of the analysis method is verified by analyzing the acoustic emission monitoring results of the true triaxial stress‒balanced unloading test.

  2. 2.

    As σ1 increases, both the elastic strain and residual strain in the σ1 direction progressively increase, whereas the elastic strain in the σ2 and σ3 directions decreases, accompanied by a gradual increase in residual strain. Notably, the reduction in elastic strain exceeds the increase in residual strain.

  3. 3.

    During σ1 loading, ue and ud in the σ1 direction exhibit nonlinear growth trends. Conversely, the ue in the σ2 and σ3 directions decreases, while the ud increases. The magnitude of the decrease in ue surpasses the increase in ud.

  4. 4.

    Under true triaxial compression, the evolution of strain energy density in the three principal stress directions of sandstone follows the linear energy storage law. Under under actual triaxial compression conditions, the unloading process of σ1 represents an accumulation of ue in the σ2 and σ3 directions.