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Based on the assumption that rock strength follows the log normal distribution statistically, this paper establishes a damage constitutive model of rock under uniaxial stress conditions in combination with the Mohr–Coulomb strength criterion and damage mechanical theory. Experiments were carried out to investigate the damage evolution process of rock material, which can be categorized into nondamaging, accelerated growth, constant-speed, similar growth, and speed-reducing growth stages. The evolution process had a good corresponding relationship with the rock stress-strain curves. Based on the statistical damage constitutive model proposed in this paper, a numerical fitting analysis was conducted on the uniaxial compression testing data of laboratory sand rock and on experimental data from other literature, in order to validate the rationality of the constitutive equation and the determination of its parameters and to analyze the effect of internal friction variables on damage variables and compression strength. The research outcomes presented in this paper can provide useful reference for the theory of rock mechanics and for rock engineering.

Experiments have shown that rock is a type of complex medium. Under the application of external loading, rock presents a very complex nonlinear deformation, and the complex deformation results in complexity and uncertainty of rock engineering accordingly. Many investigators have tried to explain the relationship between the deformation of rock and external loading, i.e., the constitutive relationship of rock. The results of early research into rock mechanics include Hooke’s law, Newton’s law of viscosity, and Saint Venant’s ideal plasticity law. The continuous recognition of progress in understanding the mechanical properties of rock has led investigators to propose a variety of constitutive models by means of a subsequent viscoelastic plastic theory of continuum mechanics, such as that put forward by Liu et al. [

Under the long-term application of external load and environmental conditions, many fine cracks and other defects are created inside rock. Owing to the existence of these defects, the mechanical properties of rock differ significantly from metal materials and polymers. The microstructure inside the rock material can cause the deterioration of the mechanical properties of the rock; therefore, the constitutive relationship of the rock material can be investigated from the point of view of damage mechanics. Damage mechanics assumes that the growth and confluence of various defects inside a material can result in the deterioration of its structure and accordingly, affect its mechanical properties.

Lemaitre [

Statistical physics theory has been widely used in the evolution and development of microscopic damage mechanics [

In the constitutive models that consider the statistics of damage evolution, the investigators defined the types of various microelement strength distributions, including Weibull distribution [

It is assumed that

When the internal microelement stress

When loading reaches a certain stress level

From Equations (

The stress level

According to Hooke’s law, the principal strain can be expressed by

We next introduce the strain equivalent hypothesis [

Substituting Equation (

Substituting Equation (

For uniaxial compression experiments, Equation (

According to Equations (

Equation (

Considering the situation of uniaxial stress and substituting Equation (

Assuming that the rock strength follows a log normal distribution, its probability density function is

The two statistical parameters in Equation (

Substituting Equation (

Considering the full stress-strain curve of uniaxial compression and defining

Equation (

To validate the accuracy of the above constitutive model, uniaxial compression tests were conducted in the laboratory for three sand rock specimens as shown in Figure

Specimen of sand rock.

Based on the damage equations presented in this paper, the evolution regularity chart for the three rock specimens in the loading process were prepared as shown in Figures

Evolution of damage variable. (a) Sample no. 1. (b) Sample no. 2. (c) Sample no. 3.

The theoretical and experimental data curves were individually prepared as shown in Figures

Comparison of theoretical results and experimental data. (a) Sample no. 1. (b) Sample no. 2. (c) Sample no. 3.

Ji et al. [

Curves of the theoretical fitting from Ji et al. [

Using the uniaxial compression and shear testing data of sand rock as an example, the related parameters were

Impact of internal friction angle

Impact of internal friction angle

From the viewpoint of the stress-strain relationship, the internal friction angle

From Figure

In this paper, the general form of the rock statistical damage constitutive model is inferred under the uniaxial stress condition, and the model parameters are determined by using uniaxial compression testing. In addition, the reliability of the proposed constitutive model is checked through analysis of testing data and concluded the following:

On the basis of strength theory and the random distribution characteristics of rock, in this research, we established a damage evolution equation based on statistics

Based on the M-C criterion, we established a statistical damage constitutive model, and identified that the model was able to describe the mechanical behaviors of rock under uniaxial compression testing

The variation of the internal friction angle had a certain impact on the uniaxial compression strength and damage evolution and reflected the compression durability of rock

All of the data supporting the conclusions of the study are available in the article and the authors are willing to share the data underlying the findings of the article.

The abstract of this paper had been accepted by the conference of GeoEdmonton 2018, but the full paper has not been submitted to the conference.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper was supported by Priority Academic Program Development of Jiangsu Higher Education Institutions and the Fundamental Research Funds for the Central Universities (2017XKQY044).