Theoretical damage characterisation and damage evolution process of intact rocks based on linear energy dissipation law under uniaxial compression

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Abstract

The investigation of rock damage behaviour is an important requirement for ensuring stability control and safety prediction in rock engineering. In this study, based on the linear energy dissipation law and energy dissipation coefficient, a theoretical method is introduced for characterising the damage of intact rocks under uniaxial compression conditions. The existing uniaxial compression test data of 14 kinds of rock materials (including 6 types of granite, 4 types of sandstone, 2 types of marble, 1 type of slate, and 1 type of limestone) were used to verify the effectiveness of the damage characterization method. The results indicate that the damage evolution process of rocks under uniaxial compression can be fully expressed by the proposed theoretical damage characterisation method, and the damage variable exhibits a quadratic functional relationship with the loading stress level. Moreover, with the linear energy storage law serving as a theoretical basis, the peak dissipated strain energy in a rock damage expression under uniaxial compression can be accurately calculated. The new theoretical damage characterisation method overcomes the shortcomings of conventional damage characterisation methods (e.g. estimating the peak dissipated strain energy through a hypothesis), and provides a new means for analysing rock damage from an energy viewpoint.

Introduction

Rock is a type of geological material with natural defects, such as micro cracks and micro holes. These natural defects gradually expand and penetrate under an external force, resulting in deterioration of the mechanical properties of the rock. The propagation of micro cracks is often considered as process of rock damage accumulation, and ultimately leads to the macroscopic destruction of the rock.1, 2, 3, 4, 5 Investigation of rock damage behaviour is vital for stability control and safety prediction in rock engineering. Furthermore, the mechanical behaviour of rock is dominated by energy, e.g., energy dissipation causes damage to the rock units, and energy release causes failure in the overall structure. The processes of deformation and damage evolution are accompanied by energy dissipation, which has a good correlation with the progressive development of rock damage.6, 7, 8

The dissipated energy is an important parameter for evaluating rock damage, and many scholars have studied the mechanical responses of rocks from the perspective of energy dissipation and obtained corresponding damage relationships.7, 8, 9, 10 Ahmed et al.11 proposed that the energy produced for the propagation of a crack tip is equal to the dissipated energy owing to the formation of the new crack surfaces, leading to the damage of the rock. Based on an energy dissipation principle, Li et al.12 proposed a new damage model for considering the nonlinear behaviours and initial damage of fractured rocks. Cheng et al.13 indicated that the damage evolution behaviours of shale were dependent on structural orientations and confining pressures. Einav et al.14 derived two classes of constitutive models from the perspective of dissipation and free energy potentials. One was a decoupling model for plastic strain and damage, whereas the other was a coupled damage-plasticity yield surface model. Gao et al.15 analysed the linear relationships between damage and equivalent irreversible strains in intact and jointed marble from the perspective of energy dissipation. Jin et al.16 defined a new damage variable for describing the damage of rocks based on the dissipated energy and constitutive energy. After conducting cyclic tests, Xiao et al.17 and Liu et al.18 adopted a ratio of the energy dissipation to the total energy dissipation to describe the fatigue damage evolution of rock. The above-mentioned studies have great significance with regard to determining the energy characteristics and damage of rocks. When energy is used as a damage factor, a widely accepted approach is to use the dissipated energy ratio, which denotes the ratio of the pre-peak dissipated energy to the dissipated energy at the peak strength (known as the ‘peak dissipated energy’). Although numerous approaches have been used to obtain the peak dissipated energy, they cannot have been unable to obtain sufficiently accurate results.

To this end, we proposed a theoretical method for characterising and calculating the damage of rock materials under uniaxial compression (UC) conditions. First, a new representation of a damage variable was established based on a linear energy dissipation law. Then, according to the existing UC and single cyclic loading–unloading uniaxial compression (SCLUC) tests on 14 types of rocks, the damage variable and damage evolution characteristics were analysed. Finally, the proposed method was compared with the conventional method with the data obtained from the UC tests. The findings of this study can provide a good theoretical method for characterising the damage evolution processes of rocks.

Section snippets

Damage variable definition considering energy

Damage refers to the degradation of materials or structures caused by changes in microscopic structural defects (such as micro-cracks, micro-voids) under external influences. A degradation process can be described by a damage variable representing a continuous internal variable. Many factors have been considered for calculating the damage variable, such as the those based on a statistical damage model, or the wave velocity, elastic modulus, or energy dissipation.9,19, 20, 21, 22 Existing

Specimen preparation

As depicted in Fig. 2, to analyse the rock damage process under UC circumstances, 14 types of rock materials were selected as the research objects,30, 32, 34 including six types of granite, four types of sandstone, two types of marble, one type of slate, and one type of limestone. With reference to the standard of International Society for Rock Mechanics (ISRM),38 these rocks were processed into cylindrical specimens, each Φ50 mm × 100 mm. The rock specimens showed no visible defects, with

Stress–strain curves

Certain basic physical and mechanical properties of the rock specimens are listed in Table 1, and their stress–strain curves under UC are depicted in Fig. 5a and b. The representative stress–strain curves of the rocks in the SCLUC tests (designed unloading stress level k = 0.7) are also depicted in Fig. 5c and d.

The characteristics of the stress and strain curves of 14 rock types show significant differences. Yellow rust stone, white marble, and marble exhibit strong plasticity, and their

Discussion

Damage causes deterioration of the internal structures in materials, affecting the overall mechanical properties.7,8 Accurately characterising damage and describing the damage evolution processes of rock materials has great significant for underground engineering construction and evaluation. According to continuum damage mechanics, the deterioration degree of a material can be quantified by a continuous damage variable, and the variable can be characterised based a variety of parameters. For

Conclusions

This study aimed to establish a novel theoretical method for characterising rock damage from the perspective of energy. Considering the linear energy dissipation law in rock materials, a series of UC and SCLUC tests were conducted on 14 types of rocks were conducted, and theoretical analyse and experimental verifications of the new theoretical damage characterisation method were conducted. The following conclusions can be drawn.

Linear energy storage and dissipation laws exist in rock materials,

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 41877272), the Fundamental Research Funds for the Central Universities of Southeast University (Grant No. 2242021R10080) and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2021zzts0867).

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