Mechanical properties of shear creep and model of mud stone damage in an open coal mine

The classical Nishihara model can better describe the attenuation of the creep process in the rocks and the deformation characteristics of the creep stage at constant speed. However, since the components used in the Nishihara model are ideal linear components, it is difficult to describe the law of nonlinear rock deformation at the accelerated creep stage.19,20,21. However, creep failure at the accelerated creep stage may better characterize the entire rock creep process.22,23,24. Therefore, based on the Nishihara model shown in FIG. 6the author introduced a nonlinear elastomer and an injury variable D and developed a new model of nonlinear shear shear damage as shown in FIG. 7.

Fig. 6
Figure 6
Figure 7
Figure 7

Model of nonlinear shear creep damage.

Elastomers

Liu et al.25 developed an elastomer creep equation,

$$ \ varepsilon_ {1} = \ frac {\ tau} {{k_ {1}}} $$

(1)

where τ is the shear stress of the elastomer, k1 is the modulus of elasticity of the elastomer and ε1 is the deformation of the elastomer.

Nonlinear elastomer

Wang et al.26 defined a new nonlinear function as follows:

$$ f

(2)

where you is an independent variable f

(3)

where k20 is the initial value of the modulus of elasticity of the nonlinear elastomer, k2 is the modulus of elasticity of a nonlinear elastomer.

The constitutional equation for a nonlinear elastomer is

$$ \ varepsilon_ {2} = \ frac {\ tau} {{k_ {2}}} $$

(4)

where τ is the shear stress of the nonlinear elastomer, k2 is the modulus of elasticity of the nonlinear elastomer and ε2 is the deformation of a nonlinear elastomer. The creep equation for a nonlinear elastomer is determined as follows:

$$ \ varepsilon_ {2} = \ frac {\ tau} {{k_ {2}}} = \ frac {\ tau f

(5)

Viscoelastic housing of various parameters

Zhang et al.28 the viscosity coefficient is assumed to be a function of power over time; function η1

(6)

where η10 is the initial coefficient of viscosity of the variable and λ is a constant.

According to the constitutional equation of Kelvin’s material, the creep equation of a variable-parameter viscoelastic body is determined as follows:27:

$$ \ tau = k_ {3} \ varepsilon_ {3} + \ eta_ {1}

(7)

where τ is the shear stress of a variable parameter viscoelastic body, ε3 is the variable parameter viscoelastic body deformation, and k3 is the modulus of elasticity.

The creep equation of a variable parameter viscoelastic body is determined by integrating the equation. (7) as follows:

$$ \ varepsilon_ {3} = \ frac {\ tau} {{k_ {3}}} \ left[ {1 – \exp ( – \frac{{k_{3} }}{{\eta_{{{10}}} \lambda }}t^{\lambda } )} \right]$$

(8)

Damaged all-plastic housing

The author presents the damage variable Dto characterize the degradation of the viscosity coefficient of the creep damage and to construct a viscoplastic housing with respect to the damage. Its constitutional equation can be written as follows29:

$$ \ tau = \ eta_ {2}

(9)

where τ is the shear stress of the damaged viscoplastic body,\ (\ dot {\ varepsilon} _ {4} \) is the rate of deformation of the damaged viscoplastic body, and η2(you ) is a function of the viscosity factor versus time.

Based on the results of many rock creep damage tests, the damage variable Dtook the form of a negative exponential function with respect to time during rock creep30,31,32,33,34,35. In this study, the injury variable is expressed by the equation. (10).

$$ D = 1 – \ exp (- \ alpha t), (0

(10)

where αis a coefficient related to the properties of the rock materials.

Based on \ (\ eta_ {2}

(11)

where ε4 is a deformation of the damaged viscoplastic body, and η2 is the initial viscosity coefficient of the damaged viscoplastic housing.

Development of a nonlinear shear creep damage model

Depending on the nature of the series of components, such a connection can be obtained36,37:

$$ \ varepsilon = \ varepsilon_ {1} + \ varepsilon_ {2} + \ varepsilon_ {3} + \ varepsilon_ {4} $$

(12)

According to Eq. (1), Lyg. (5), Lyg. (8), Lyg. (11) and Eq. (12), the creep equation of the nonlinear shear creep failure model can be obtained as

$$ \ left \ {{\ begin {array} {* {20} l} {\ varepsilon = \ frac {\ tau} {{k_ {1}}} + \ frac {{\ tau bt ^ {c}} } {{k_ {20} (1 + bt ^ {c})}} + \ frac {\ tau} {{k_ {3}}} \ left[ {1 – \exp ( – \frac{{k_{3} }}{{\eta_{{{10}}} \lambda }}t^{\lambda } )} \right]} \ hfill & {\ tau <\ tau_ {s}} \ hfill \\ {\ varepsilon = \ frac {\ tau} {{k_ {1}}} + \ frac {{\ tau bt ^ {c}}} {{k_ {20} (1 + bt ^ {c})}} + \ frac {\ tau} {{k_ {3}}} \ left[ {1 - \exp ( - \frac{{k_{3} }}{{\eta_{{{10}}} \lambda }}t^{\lambda } )} \right] + \ frac {{\ tau - \ tau_ {s}}} {{\ eta_ {2} \ alpha}} \ exp (\ alpha t)} \ hfill & {\ tau \ ge \ tau_ {s}} \ hfill \\ \ end {array}} \ right. $$

(13)

where τs is the yield strength of the rock.

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