Mathematical Problems in Engineering

Volume 2015, Article ID 960973, 7 pages

http://dx.doi.org/10.1155/2015/960973

## Failure Probability Model considering the Effect of Intermediate Principal Stress on Rock Strength

College of Civil Engineering, Tongji University, Shanghai 200092, China

Received 8 July 2015; Revised 1 November 2015; Accepted 8 November 2015

Academic Editor: Yakov Strelniker

Copyright © 2015 Yonglai Zheng and Shuxin Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A failure probability model is developed to describe the effect of the intermediate principal stress on rock strength. Each shear plane in rock samples is considered as a micro-unit. The strengths of these micro-units are assumed to match Weibull distribution. The macro strength of rock sample is a synthetic consideration of all directions’ probabilities. New model reproduces the typical phenomenon of intermediate principal stress effect that occurs in some true triaxial experiments. Based on the new model, a strength criterion is proposed and it can be regarded as a modified Mohr-Coulomb criterion with a uniformity coefficient. New strength criterion can quantitatively reflect the intermediate principal stress effect on rock strength and matches previously published experimental results better than common strength criteria.

#### 1. Introduction

The stress state in three-dimensional space is defined by three mutually perpendicular stress components (, , ) and it is an important subject in rock mechanics to study the rock failure behavior under complex stress conditions. The traditional considerations suggest that shear failure will occur when the shear stress along some plane in the sample is too large. The extreme values of shear stress in a material are related only to the largest and smallest principal stress (commonly denoted as and ). Based on such considerations, the influence of intermediate principal stress () on the experimental results is not taken into account.

However three-dimensional unequal stress states are very common in engineering practice and it is very important to predict the rock strength with the effect of intermediate principal stress. Many researchers [1–6] have conducted a large number of true triaxial tests on different rock types such as Dunham dolomite, Solnhofen limestone, and granite, to investigate the behavior under triaxial stress conditions. They found that the strength first increases and then reduces with the increase of . At the same time, researchers have developed many numerical models or used commercial software to study the failure processes of rocks under polyaxial stress conditions [7–11]. In these numerical tests, it was clear that the intermediate principal stress has an effect on rock strength and the mechanism of such an effect is discussed. The phenomenon of intermediate principal stress effect seems to be related to the heterogeneity of materials [10, 11]. Without material heterogeneity, local tensile stress cannot be generated in an overall compressive stress environment so that there is no crack initiation and propagation [7]. The intermediate principal stress confines the rock in such a way that fractures can be easier to develop in the direction parallel to and . Fjær and Ruistuen’s numerical results [11] indicate that effect of is related to stress symmetry, rather than the stress level. When was close to or , there were more possible directions for the failure planes with higher stress symmetry. The directions for the failure plane for which the theoretical failure criterion is first fulfilled may not coincide with the directions preferred by the rock’s heterogeneity. The rock will be weaker where the stress state is more symmetric and there are several equivalent directions of the failure plane to choose. These considerations are supported by Chang and Haimson’s observations [1]. They found that stress induced micro cracks in amphibolites were randomly oriented when , while micro cracks became more aligned with the direction of when . Similar phenomenon was also observed by Cai [7] and Pan et al. [10].

According to Fjær and Ruistuen’s considerations [11], affects the failure probabilities in different directions under three-dimensional unequal stress. Which direction the failure plane eventually takes is determined by the heterogeneities of the rock. The heterogeneity is always incorporated by assuming the micro-units’ properties complying with a certain distribution and here Weibull distribution [12] is selected. Weibull distribution is introduced to explain the statistical size effect and later justified theoretically on the basis of some reasonable hypotheses about the statistical distribution and the role of microscopic flaws or micro cracks [13]. Unlike previous Weibull analysis [14–17] with volume consideration in this paper each potential shear failure plane is regarded as a micro-unit and failure probabilities of all directions are calculated. Thus, a new failure probability model considering the failure probabilities in different directions is proposed. By combining the new model with Mohr-Coulomb criterion, a new strength criterion is developed, which can quantitatively describe the effect of intermediate principal stress on rock strength.

#### 2. Modelling

##### 2.1. Failure Probability Model with Volume Considerations

Assume that under the nominal stress the density function of micro flaws per unit volume is . The amount of micro flaws in materials sample with a volume of could be expressed as

The strength of micro-units complies with Weibull distribution [12] and the probability density function is where is Macaulay bracket; when , ; when , ; is stress threshold; is scalar parameter, ; is shape parameter and it can be considered as the uniformity coefficient, .

Stress threshold is often set to zero. In this case, (2) has only two parameters ( and ). The failure probability of materials sample with one flaw can be obtained:

According to weakest link theory, the failure probability of materials sample with a volume of can be represented:

By inserting (1) and (3) into (4), we obtainwhere is considered as a reference volume.

From (5), the average strength can be calculated:

By inserting (5) into (6) and using variable substitution, (6) can be represented:where is gamma function, .

The variance and variation coefficient can be obtained from (5) and (7):

Considering the nonuniform force field, strength of each point in coordinates can be expressed as . Here is the maximum strength and is a dimensionless coordinate function for each point. Thus (5) becomes

Accordingly, (6) becomes

Comparing the average strength in uniform force field and in nonuniform force field, from (6) and (11) we obtain

##### 2.2. Failure Probability Model with Direction Considerations

Shear failure is a basic failure mode of rocks under triaxial compression. Following Fjær and Ruistuen’s considerations [11], affects the failure probabilities in different directions under three-dimensional unequal stress. In order to describe the intermediate principal stress effect, direction considerations instead of volume considerations are made to express failure probabilities in different directions. The shear planes in rock samples are considered as potential failure planes. In order to calculate the probability of each direction, each potential shear failure plane is regarded as a micro-unit. The effect of the intermediate principal stress can quantitatively be estimated by calculating the failure probabilities for all the shear planes and combining these into the total probability for failure.

Figure 1 shows a normal vector (ON) of a shear plane defined by the spherical coordinates and , and the normal and shear stresses are given as