Post-peak Stress-strain Relationship of Rock Mass Based on howk brown

Procedia Earth and Planetary Science 5 (2012) 289 – 293

1878-5220 © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, Environment and Engineering.

doi:10.1016/j.proeps.2012.01.049

Procedia Earth and Planetary

Science

Procedia Earth and Planetary Science 00 (2011) 000–000

www.elsevier.com/locate/procedia

2012 International Conference on Structural Computation and Geotechnical Mechanics

Post-peak Stress-strain Relationship of Rock Mass Based on Hoek-Brown Strength Criterion

HAN Jianxin

^a

, LI Shucai

^b

, LI Shuchen

^b

, WANG Lei

^b

, a*

a School of Statistics and Mathematics,Shandong Finance University,Jinan,Shandong 250014,China

bGeotechnical and Structural Engineering Research Center,Shandong University,Jinan,Shandong 250061,China

Abstract

The post-peak mechanical characteristic of rock mass plays a leading role in stability of geotechnical engineering.

Based on Hoek-Brown strength criterion and evolutional behavior of strength parameters, choosing major principal strain



₁ as strain softening parameter, this paper presents the method of solving post-peak stress-strain relationship.

In numerical case, the post-peak stress-strain curves of some rock masses are obtained. The effect of evolutional law of strength parameters



_c,

m

^,

s

^and

a

to post-peak stress-strain curve is discussed, and it’s concluded that the greater the residual values of



_c,

m

^,

s

are and the smaller the residual value of

a

is; the greater the residual strength is and the post-peak stress-strain curve falls more gently.

© 2011 Published by Elsevier Ltd.

Keywords: Hoek-Brown strength criterion;strength parameter evolution;strain-softening ;post-peak;stress-strain relationship; rock mass

1. Introduction

In the exploitation of underground resource, the stability control of surrounding rock mass is an issue which must be considered. The post-peak mechanical characteristic of rock mass plays a leading role in stability control of surrounding rock mass. Rock mass is heterogeneous brittle material and its deformation characteristic has marked difference from metallic materials. Especially in the post-peak deformation phase, the deformation has strain softening character and the mechanical behaviour is

* Corresponding author. Tel.: 15953197370.

E-mail address: [email protected].

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, Environment and Engineering. Open access under CC BY-NC-ND license.

Open access under CC BY-NC-ND license.

comparative complex. It is difficult to explain by the classical strength theory, so it has important theoretical and engineering meaning to study the post-peak deformation.

In regard to studying post-peak mechanical behaviour of rock, a lot of research work has been done from different aspects by researchers[1-15]. Initially, to simply the problem, the classical strength theory, that is, elastic perfectly plastic behaviour model is often adopted in studying post-peak mechanical behaviour. But Hoek and Brown were among the first authors to reject the elastic perfectly plastic assumption as inappropriate for rock masses of average or high geotechnical quality and three typical models were presented in according with geological strength index(GSI).The three typical models are respectively elastic perfectly plastic behaviour model, elastic brittle behaviour model and strain softening model. Strain softening model are difficult to characterize, since both residual failure criterion and peak failure criterion along with other post-failure parameters need to be estimated. Estimating the residual failure criterion is no easy task, and a number of researchers have take on the challenge of deepening our knowledge of rock mass stress-strain behaviour[1-10]. But there is a trend that the model becomes more and more complex which makes it difficult to engineering application. Additionally, the paper on directly constructing the post-peak stress-strain relationship based on Hoek-Brown strength criterion and strength parameters evolutional behaviour is seldom. In this paper, based on Hoek-Brown strength criterion and strength parameters evolutional behaviour, selecting principal strain



₁ as strain softening parameter, on the assumption that the strength parameters is a piecewise linear function of



₁, the post-peak stress-strain relationship are presented and the effect of strength parameters evolutional law to post-peak stress-strain curves is discussed.

2. Strength criterion, strain softening parameter and evolutional law of strength parameters

In this work, the basic thinking and method of establishing the stress-strain relationship is described as follows: First, this paper selects strength criterion to establish the connection of stress with strength parameters. Then, determines the relationship of strength parameters and strain softening parameter according to experimental method et al. Finally, the relationship of stress and strain can be obtained by means of strength parameters which are as intermediate variables. Below we describe the strength parameters, strain softening parameter and evolutional law of strength parameters which will be adopted in this paper.

In respect of studying the strength characteristic of rock mass, the common criterions include Mohr— Coulomb strength criterion, Hoek-Brown strength criterion and Drucker-prager strength criterion et al. In this paper, we select Hoek-Brown strength criterion to study post-peak strength characteristic of rock mass. The express of Hoek-Brown strength criterion is

) ( 3 3

1

( )

) ) ( ( ) (



 



 









a

c

m

c

s  

 







(1) where



is strain softening parameter,



_c,

m

^,

s

^and

a

are strength parameters. In the post-peak strain softening phase, these strength parameters are variable with the variation of strain softening parameter



.

When a strain-softening behaviour model is considered, the softening parameter



controls the strength capacity between the peak and the residual failure criterion. The parameter



can be defined in different ways, but so far there has no been a wide support among researchers for any of its possible forms[10] . There are two different common ways of defining this parameter. First it is defined as a function of internal variables,Second it is defined in an incremental way. The first method is adopted and the major principal strain



₁ will be chosen for the development of the studies presented in this paper,

In respect of evolution of strength parameters, generally it is assumed that the strength parameters can be described by bilinear functions of the softening parameter



[8, 10]. In this paper, Hoek-Brown

criterion is selected as strength criterion and the major principal strain



₁ is selected as softening parameter. It is assumed that strength parameters



_c,

m

^,

s

^and

a

can be described by piecewise linear function of



₁ as shown in express (2) ,



 











 





r r

r p

p p p

r p r

















 











1 1 1 1

,

, ) ) (

(

(2)

where



represent one of



_c,

m

^,

s

^and

a

, The subscripts ‘p’ and ‘r’ denote the peak and residual values respectively.



_r is the value of



₁ from which the residual behaviour starts.Substituting (2) into (1), the post-peak stress-strain relationship can be determined. Because the expression is more verbose, we don’t list it here.

3. Analysis of some examples

Below we explain the specific method to solve the post-peak stress-strain relationship of rock mass through an example. Through data arrangement of rock mass in literature [8], the parameters and data of the rock mass can be obtained as follows:

E  5700

MPa,

v  0 . 30

,



_cp

 30

MPa,



_cr

 25

MPa,

m

_p

 1 . 70

^,

m

_r

 0 . 85

^,

s

_p

 0 . 0039

^,

s

_r

 0 . 0019

^,

a

_p

 0 . 55

^,

a

_r

 0 . 60

^,



_p

 0 . 0059

,

0139 . 0

r





,



₃

 15

MPa. In order to compare with other rock masses we set number of this rock mass to1. Substituting the above data into formulae (2) and (1) , the stress-strain relationship can be obtained. But it didn’t be listed as it is too verbose. The corresponding stress-strain curve is shown in Fig.1.

Fig.1 stress-strain curve of rock mass

In the same time, in order to study the effect of strength parameters to post-peak stress-strain curve, we set other four different rock masses which parameters are listed in table 1. The four rock masses stress-strain curves are also presented as shown in Fig.2-5. From the Fig.2-5 the following conclusion can be drawn: When the confining pressure is

15

MPa, the greater the residual value of



_c,

m

,

s

are and the smaller of the residual value of

a

is, the greater the residual strength is and the post-peak stress-strain curve falls more gently. The effect of



_cand

m

to the post-peak stress-strain curve is comparatively marked, whereas the effect of

s

and

a

to the post-peak stress-strain curve is smaller.

Table 1 Data of rock mass

Number



^cp

/Mpa



cr

/Mpa

m

^p

m

_r

s

p

s

_r

a

p

a

_r



p



_r

1 30.00 25.00 1.70 0.85 0.0039 0.0019 0.55 0.60 0.0059 0.0139

2 30.00 10.00 1.70 0.85 0.0039 0.0019 0.55 0.60 0.0059 0.0139

3 30.00 25.00 1.70 0.50 0.0039 0.0019 0.55 0.60 0.0059 0.0139

4 30.00 25.00 1.70 0.85 0.0039 0.0010 0.55 0.60 0.0059 0.0139

5 30.00 25.00 1.70 0.85 0.0039 0.0019 0.55 0.65 0.0059 0.0139

Fig.2 the effect of



_cr to post-peak stress-strain curve (left) Fig.3 the effect of

m

_r ^to post-peak stress-strain curve (right)

Fig.4 the effect of

s

_r to post-peak stress-strain curve (left)

Fig.5 the effect of

a

_r to post-peak stress-strain Curve (right)

4. Conclusion

In this work, based on the evolutional behaviour of the strength parameters, a general method of establishing the post-peak stress-strain relationship is introduced. First, the paper determines the strength criterion and strain softening parameters, then determines the relationship between the strength

parameters and the softening parameter, finally, connects the stress with strain through the strength parameters which are as intermediate variable and we can obtain the post-peak stress-strain relationship.

Based on the Hoek—Brown strength criterion, selecting the principal strain



₁ as softening parameter, we obtained the post-peak stress-strain relationship.

The effect of strength parameters



_c,

m

,

s

and

a

to the post-peak stress-strain curve can be drawn as follows: the greater the residual values of



_c,

m

,

s

are and the smaller the residual value of

a

is, the greater the residual strength is and the post-peak stress-strain curve falls more gently.

Acknowledgements

The study is supported by the National Key Basic Research and Development Program (973 Program NO.2010CB732002), the National Natural Science Foundation of China (NO.51179098) and the Natural Science Foundation of Shandong Province (NO. 2009ZRB02285)

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