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Progressive failure constitutive model for softening behavior of rocks based on maximum entropy theory

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Abstract

Considering that rock failure is a gradual process when subjected to triaxial stress conditions, a new statistical damage constitutive model is proposed to describe the progressive failure of rocks. The model is based on continuous damage mechanics and maximum entropy theory while the commonly used statistical damage model is based on continuous damage mechanics and the conventional Weibull distribution, which is used to describe the strength of mesoscopic rock elements. Weibull distribution is a distribution function with a specific assumption that the nth central moment and the geometric mean of the statistical variable are constant. The maximum entropy distribution is the only unbiased distribution and the Weibull distribution is a special case of the maximum entropy distribution. According to the maximum entropy theory, the damage variable is defined without any prior assumptions of the theoretical distributions. The rock is hypothesized to be divided into two parts: the damaged portion and undamaged portion. The bearing capacity of the damaged part is also considered in the new model so that it is more in accordance with the actual situation. The mesoscopic rock elemental strength is calculated based on energy release rate principles to avoid the deficiencies in using the conventional stress or strain criteria approaches, and the effect of rock initial fissures is emphasized. A new method is presented to determine the unknown parameters in the constitutive equations. The applicability of the new statistical damage constitutive model is verified by experimental data. It is shown that the theoretical model is in good agreement with the test data trend and can simulate the softening behavior of rock well. Admittedly, the proposed model proposed in this paper is a basic model without considering some important aspects of rock deformation mechanics, such as the absences of the complex stress conditions. The purpose of this paper was to illustrate that the constitutive model can be established in the framework of continuous damage mechanics and maximum entropy theory.

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Abbreviations

A :

Total area of the representative surface

A u :

Undamaged area of the representative surface

A d :

Damaged area of the representative surface

C :

Positive constant

c :

Cohesion of rock

D :

Damage variable

F :

Resultant acting on A

E :

Modulus of intact rock

E u :

Modulus of undamaged rock

E i :

Unloading modulus

F u :

Resultant acting on A u

F d :

Resultant acting on A d

F(x):

Distribution function of rock elements strength

f(x):

Probability density function

f p (x):

Most likely distribution function for x

G(X):

Lagrange function

G c :

Critical value of energy release rate

G i :

Energy release rate in i direction

G max :

Maximum energy release rate

H(X):

Information entropy

H(X)max :

Maximum of information entropy

i, j :

Subscript symbols (i, j = 1, 2, 3)

K :

Elemental strength parameter

K i :

Material constant

k :

Order of moment function

m :

Total number of discrete statistical variables

m k :

Calculated value from available data

N :

Total number of rock elements

N d :

Number of failed rock elements

n :

Number of discrete statistical variables

n j :

Normal to the representative surface

P n :

Probability of discrete statistical variables

P np :

Most likely probability for x n

R :

Sliding interval of x

U d :

Dissipated energy

U e :

Releasable energy

\(u_{k} \left( x \right){\text{ or}}\;u_{\alpha } \left( x \right)\) :

Moment function

X :

Statistical variable

x :

Value of continuous statistical variable

x n :

Value of discrete statistical variable

α :

Number of Lagrange parameters

β :

Serial number of experimental data

γ :

Numbers of data points

\(\varepsilon_{i} , \, \varepsilon_{i}^{\text{u}} ,\;{\text{and}}\;\varepsilon_{i}^{\text{d}}\) :

Principal apparent strain, latent strain of damaged part and undamaged part

λ α :

Lagrange parameters

\(\sigma_{c}\) :

Uniaxial compressive strength

\(\sigma_{ij} , \, \sigma_{ij}^{\text{u}} ,\;{\text{and }}\sigma_{ij}^{\text{d}}\) :

Stress tensor acting on A, A u, and A d, respectively

σ 1, σ 2, and σ 3 :

Principal stresses acting on A

\(\sigma_{1}^{\text{u}} , \, \sigma_{2}^{\text{u}} ,{\text{ and}}\;\sigma_{3}^{\text{u}}\) :

Principal stresses acting on A u

\(\sigma_{1}^{\text{d}} , \, \sigma_{2}^{\text{d}} ,{\text{ and }}\sigma_{3}^{\text{d}}\) :

Principal stresses acting on A d

υ :

Poisson’s ratio

φ :

Internal friction angle

\(\phi\) :

Value of a specific angle

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Acknowledgments

This work was financially supported by the Major State Basic Research Project (2011CB201201), Provincial Science and Technology Support Project (2012FZ0124), and the International Science and Technology Cooperation Program of China (2012DFA60760).

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Authors and Affiliations

  1. Key Laboratory of Energy Engineering Safety and Disaster Mechanics, Ministry of Education, School of Architecture and Environment, Sichuan University, Chengdu, 610065, China

    C. B. Li, L. Ren & J. Wang

  2. Key Laboratory of Energy Engineering Safety and Disaster Mechanics, Ministry of Education, Institute of New Energy and Low-Carbon Technology, Sichuan University, Chengdu, 610065, China

    L. Z. Xie

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  1. C. B. Li
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Correspondence to L. Z. Xie.

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Li, C.B., Xie, L.Z., Ren, L. et al. Progressive failure constitutive model for softening behavior of rocks based on maximum entropy theory. Environ Earth Sci 73, 5905–5915 (2015). https://doi.org/10.1007/s12665-015-4228-7

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