The main steps of using damage theory to study problems are: first, define a suitable state parameter-damage variable, then determine the damage evolution equation and constitutive relation of an object under external load according to the external load, and finally solve the stress-strain field and the overall damage field at a certain point inside the object according to the corresponding initial and boundary conditions. Therefore, the definition of damage variable is the basis. Correct and reasonable damage variables can not only simplify the problem, but also make the damage evolution equation and constitutive equation easy to establish, which has clear physical significance. Therefore, the definition of damage variables plays a fundamental role in the study of impact damage model. The definition methods of damage variables in several common rock impact theoretical models are analyzed and compared below.

Kipp-Grady(K-G) damage model holds that there are a large number of primary cracks in rocks, and the spatial distribution of their length and direction is random. Under the action of external load, some cracks are activated and propagated. The damage variable d is defined as (Yang Jun, 1996).

Rock fracture and damage

Where: Cg is the crack propagation speed, which is set as a constant; K and m are material constants; ε is volume strain; It is time. Meanwhile, if V(t), -V(t) and N(t) are the rock volume affected by a single fracture, the average value of the rock volume affected by a single fracture and the number of fractures contained in the rock, the formula (13- 1) can be expressed as: d (t) =-v (t). K and m are material constants; ε is volume strain; It is time. Meanwhile, if V(t), -V(t) and N(t) are the rock volume affected by a single fracture, the average value of the rock volume affected by a single fracture and the number of fractures contained in the rock, the formula (13- 1) can be expressed as: d (t) =-v (t). K and m are material constants; ε is volume strain; It is time. Meanwhile, if V(t), -V(t) and N(t) are the rock volume affected by a single fracture, the average value of the rock volume affected by a single fracture and the number of fractures contained in the rock, the formula (13- 1) can be expressed as: d (t) =-v (t). K and m are material constants; ε is volume strain; It is time. Meanwhile, if V(t), -V(t) and N(t) are the rock volume affected by a single fracture, the average value of the rock volume affected by a single fracture and the number of fractures contained in the rock, the formula (13- 1) can be expressed as: d (t) =-v (t). K and m are material constants; ε is volume strain; It is time. Meanwhile, if V(t), -V(t) and N(t) are the rock volume affected by a single fracture, the average value of the rock volume affected by a single fracture and the number of fractures contained in the rock, the formula (13- 1) can be expressed as: d (t) =-v (t). K and m are material constants; ε is volume strain; It is time. Meanwhile, if V(t), -V(t) and N(t) are the rock volume affected by a single fracture, the average value of the rock volume affected by a single fracture and the number of fractures contained in the rock, the formula (13- 1) can be expressed as: d (t) =-v (t).

Chen E P and Taylor L M( 1984) introduced the relationship between crack density Cd and effective Poisson's ratio and damage variable D of cracked materials based on the K-G model, and established the TCK damage model (Chen E P et al., O'connell R J et al.,1974; Yang r, 1987), the relationship between damage variable and macro modulus of materials is shown in formula (13-2):

Rock fracture and damage

Budiansky B and O'connell R J obtained the macroscopic equivalent modulus expression of the fractured body with randomly distributed flat cracks by self-consistent method:

Rock fracture and damage

Where, k, e, g and μ are the bulk modulus, elastic modulus, shear modulus and Poisson's ratio of undamaged and damaged rocks respectively; Cd is the crack density; D is the damage variable.

The damage model established by Yang et al. holds that the initiation and propagation of cracks in rocks are determined by plastic strain. When the plastic strain at a certain point in the rock is greater than a certain critical value, the original crack initiates and propagates. Crack propagation leads to rock damage. The damage factor d defined in this model is (O'connell R J et al., 1974): D = 1-exp (S2). The elastic constants before and after rock damage are as follows:

Rock fracture and damage

In the formula, e, g and μ are the elastic modulus, shear modulus and Poisson's ratio of undamaged rock respectively, and the quantities in the damaged rock are marked. In this model, there is an obvious relationship between elastic constants and damage variables before and after rock damage.

The above definition of damage variables is not convenient for engineering application, and some simple and clear definition methods of damage variables are often used in engineering practice.

(1) the damage variable is defined by the damage area: if the cross-sectional area a of the material is reduced to the effective bearing area A* due to the formation and expansion of distributed microcracks and micropores, the damage variable d is defined as (Yu Tianqing et al., 1993).

Rock fracture and damage

(2) Define the damage variable according to the reduction of elastic modulus: let the elastic modulus of the material be E without damage, and the damage variable D be defined as (Dai Jun, 2002) because the effective elastic modulus after damage becomes.

Rock fracture and damage

Because the elastic modulus of rock can be measured directly by experiments, the damage variable is easy to quantify, and this definition has been widely used in practice.

(3) Damage variable defined according to stress wave attenuation: When a material is damaged, the change of its microstructure will cause the change of elastic wave velocity propagating inside the material, so the damage of the material can be defined as

Rock fracture and damage

Where v is the elastic wave velocity before and after material damage.

(4) Damage variable defined by fractal dimension: The research work of Yang Jun et al. shows (Yang Jun et al., 1999) that the distribution of cracks in rocks is a fractal, and the physical meaning of the fractal dimension of cracks can be understood as the parameter of the degree of crack filling space in rocks, and the process of rock damage is also the process of increasing the fractal dimension. The relationship between damage variable and fractal dimension is as follows:

Rock fracture and damage

Where: β is the shape influence factor, 0.