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Fractional derivative triaxial creep model of Beishan granite considering probabilistic damage evolution

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Abstract

It is helpful to study the creep process of rock by modeling the process of damage evolution in triaxial creep test. Based on the statistical distribution of strength of representative element, a dynamic damage model suitable for the initial loading stage is established. From the perspective of strength deterioration of representative element, the similarity of damage characteristics between characteristic representative element and rock sample during creep is found, and a time-dependent damage model suitable for the constant loading stage is also obtained. And the quantitative identification of parameters in the damage model is realized based on the cumulative acoustic emission (AE) count, which reveals the whole process of damage evolution in triaxial creep test. In addition, the power-law dashpot suitable for the accelerated creep stage is proposed, and the fractional non-Newtonian creep model considering the damage evolution is further established. The parameters of creep model are also identified by the stepwise least square fitting (LSF), and the applicability and superiority of the models are verified.

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All data, models and codes generated or used during the study appear in the published article.

Abbreviations

K :

Strength of representative element (MPa)

f(k):

Probability density function of strength

m :

Shape parameter

λ :

Scale parameter (MPa)

ξ :

Location parameter (MPa)

k′ :

A certain strength (MPa)

N d :

Number of damaged representative elements

N :

Total number of representative elements

F(k):

Probability distribution function of k

D :

Damage variable

I  1 * :

First stress invariant of representative element (MPa)

J 2 * :

Second deviatoric stress invariant of representative element (MPa2)

α :

A constant related to internal friction angle θ

θ :

Internal friction angle (rad)

σ :

Nominal stress matrix

σ * :

Effective stress matrix

C :

Nominal stiffness matrix

C * :

Effective stiffness matrix

ε :

Strain matrix

D 0 :

Damage of rock sample at the end of initial loading stage

m(t):

Shape parameter at creep time t

λ(t):

Scale parameter at creep time t (MPa)

k(t):

Strength of representative element at creep time t (MPa)

D c(t):

Creep damage

\(\overline{k}\) :

Average strength (MPa)

m 0 :

Shape parameter when t = 0

λ 0 :

Scale parameter when t = 0 (MPa)

E e :

Young's modulus of characteristic representative element (MPa)

μ e :

Poisson's ratio of characteristic representative element

ε 1(t):

Axial strain of characteristic representative element at time t

ε c(t):

Creep strain of characteristic representative element at time t

σ 1 :

Axial pressure (MPa)

σ 3 :

Confining pressure (MPa)

σ s :

Long-term strength (MPa)

ε st :

Strain of stable creep

ε is :

Strain of instable creep

a 0 :

Creep constant

a 1 :

Creep constant (h−1)

b 0 :

Creep constant

b 1 :

Creep constant

t c :

Creep failure time (h)

t 0 :

An amount of time slightly larger than tc (h)

β :

Damage parameter

ϕ :

Damage parameter

η γ :

Viscosity coefficient of Abel dashpot (MPa·hγ)

γ :

Order of Abel dashpot

g(t):

A function

a :

Interval of function

b :

Interval of function

a I γt g(t):

Left integral of γ-order Riemann–Liouville type of function g(t)

Γ:

Gamma function

x :

Independent variable of integral

κ :

Consistency coefficient (MPa·hn)

n :

Fluidity index

η a :

Apparent viscosity (MPa·h)

ε e :

Strain of Hooke spring

ε v :

Strain of Abel dashpot

ε vp :

Strain of viscoplastic body

E :

Elastic modulus of Hooke spring (MPa)

δ ij :

Kronecker function

S ij :

Deviatoric stress tensor

e ij :

Deviatoric strain tensor

σ m :

Spherical stress tensor

ε m :

Spherical strain tensor

G 0 :

Shear modulus of elastic body (MPa)

K :

Bulk modulus of elastic body (MPa)

μ :

Poisson's ratio of elastic body

H 1 :

Shear viscosity coefficient of Abel dashpot (MPa·hγ)

F :

Yield function (MPa)

Q :

Plastic potential function (MPa)

F 0 :

Reference value of yield function F

φ(·):

A power function with exponent c

τ :

A independent variable

α′ :

A constant related to dilatancy angle θ’

θ′ :

Dilatancy angle (rad)

σ c :

Initiation stress of micro-crack (MPa)

ε v :

Volumetric strain

C 1 :

Cumulative AE count throughout the initial loading stage

C 2 :

Cumulative AE count throughout the constant loading stage

C 1(σ):

Cumulative AE count in the initial loading stage when σ1-σ3 = σ

R 2 :

Correlation coefficient

C 2(t):

Cumulative AE count in the constant loading stage at time t

T :

Temperature (°C)

ω :

Damage parameter associated with the properties of rock and stress (h−1)

E 0 :

Parameter of one-dimensional fractional Nishihara model (MPa)

E 1 :

Parameter of one-dimensional fractional Nishihara model (MPa)

ηγ 1:

Parameter of one-dimensional fractional Nishihara model (MPa·hγ)

ηγ 2:

Parameter of one-dimensional fractional Nishihara model (MPa·hγ)

G 1 :

Parameter of three-dimensional fractional Nishihara model (MPa·hγ)

H 2 :

Parameter of three-dimensional fractional Nishihara model (MPa·hγ)

ρ :

Value of scale parameter λ0 of sample S0 (MPa)

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Acknowledgements

The present work is supported by the National Natural Science Foundation of China (51827901, 52121003, 52142302), the 111 Project (B14006) and the Yueqi Outstanding Scholar Program of CUMTB (2017A03).

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Authors and Affiliations

  1. School of Energy and Mining Engineering, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Hongwei Zhou

  2. School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Zhimin Jin

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  1. Hongwei Zhou
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  2. Zhimin Jin
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Correspondence to Hongwei Zhou.

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Zhou, H., Jin, Z. Fractional derivative triaxial creep model of Beishan granite considering probabilistic damage evolution. Acta Geotech. 18, 4017–4033 (2023). https://doi.org/10.1007/s11440-023-01848-x

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