Shock and Vibration

Volume 2019, Article ID 3291730, 10 pages

https://doi.org/10.1155/2019/3291730

## Numerical Simulation of the Influence of Width of a Prefabricated Crack on the Dimensionless Stress Intensity Factor of Notched Semi-Circular Bend Specimens

^{1}School of Energy Science and Engineering, Henan Polytechnic University, Jiaozuo, Henan 454001, China^{2}Collaborative Innovation Center of Coal Work Safety, Jiaozuo, Henan 454001, China^{3}Institute of Deep Earth Sciences and Green Energy, College of Civil Engineering, Shenzhen University, Shenzhen 518060, China

Correspondence should be addressed to Mingzhong Gao; moc.uhos@0891zmg

Received 12 November 2018; Revised 24 March 2019; Accepted 4 April 2019; Published 5 May 2019

Academic Editor: Salvatore Caddemi

Copyright © 2019 Sheng Zhang et al. 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

To analyze the effect of the width of a prefabricated crack on the dimensionless stress intensity factor of notched semi-circular bend (NSCB) specimens, ABAQUS software was employed to perform numerical calibration of the crack tip stress intensity factor for the width of prefabricated cracks in the range of 0.0∼2.0 mm. The relative errors of the dimensionless stress intensity factor for different widths of prefabricated cracks were analyzed. The results indicate that the dimensionless stress intensity factor shows an approximate linear increase as the width of the prefabricated crack increases. The longer is the length of the prefabricated crack, the “faster” is the increase in speed. The effect of the dimensionless support spacing on the increase in the speed of the dimensionless stress intensity factor due to the increase in crack width is minimal. When the prefabricated crack width is 2.0 mm, the maximum relative error of the dimensionless stress intensity factor is 4.325%. The new formula for the dimensionless stress intensity factor that eliminates the influence of the width of a prefabricated crack is given, which provides a theoretical basis for the more accurate fracture toughness value measured using an NSCB specimen.

#### 1. Introduction

Rock mass is a typical noncontinuous natural medium, whose interior presents many defects, such as joints and cracks [1, 2]. Because rock fractures cannot be simply determined by strength parameter measurement, the existence of internal cracks should be considered. Fracture toughness can characterize the ability of rock fissures to resist crack initiation and propagation [3–5]. Regarding the testing method, the International Society for Rock Mechanics (ISRM) has successively presented chevron notched short rod (SR) specimen, chevron notched three-point bend round bar (CB) specimen, cracked chevron notched Brazilian disc (CCNBD) specimen, and notched semi-circular bend (NSCB) specimen—a total of four types of specimens to test the mode I fracture toughness of rock [6–8]. Besides the four test configurations recommended by the ISRM, some scholars have proposed some specimens of other configurations to test the pure mode I, II, and III or mixed mode fracture toughness of rock materials. Such as diametrally compressed ring (DCR) specimen [9], cracked straight-through flattened Brazilian disk (CSTFBD) [10], edge cracked triangular (ECT) specimen [11], and edge-notched disc bend (ENDB) specimen [12–14]. The NSCB specimen is a new type of specimen employed in static fracture toughness tests, which was recommended by ISRM in 2014. This specimen was proposed by Chong and Kuruppu [15] and has been extensively employed in fracture toughness tests due to its simple structure, ease of specimen preparation, and loading [16–24]. However, in the calculation of fracture toughness by most researchers, the dimensionless stress intensity factor was obtained by substitution in the relevant formula recommended by the ISRM. In the process of deriving this formula, the recommended method applied ABAQUS software to establish a two-dimensional NSCB specimen model that contains line cracks (i.e., zero-width cracks) to calibrate the dimensionless stress intensity factor and then obtain a relevant calculation formula by fitting. During the preparation of a fracture toughness test specimen, which is affected by factors such as preparation methods, an artificial prefabricated crack has a certain width. Its crack tip is not ideally sharp but approximates the arc shape. The arc radius may substantially vary due to different processors, which causes a non-strict correspondence relation between the dimensionless stress intensity factor given by the recommended formula substitution and fracture load obtained by experiments. The obtained fracture toughness exhibits a certain deviation from the true fracture toughness value. Currently, the study which regards the influence of width of the prefabricated crack on the stress intensity factor is primarily focused on a disc-type specimen or three-point bending beams. Zhu and Wang [25] performed detailed experimental research and theoretical analysis of the relationship between the curvature radius of a rock specimen for three-point bending tests and the test value of fracture toughness. Dai and Wang [26] analyzed the influence of crack width on the dimensionless stress intensity factor of CCNBD specimens. Dong et al. [27] investigated the effect of crack form on the stress intensity factor of a cracked straight-through Brazilian disc (CSTBD). Zhang et al. [28] and Zhang and Liang [29] successively investigated the influence of the width of a prefabricated crack on the fracture toughness test of a holed-cracked flattened Brazilian disc (HCFBD) specimen. Zhou et al. [30] compared the fracture behavior and toughness of straight-notch disc specimens with those of central sharp-notch disc specimens. They discovered that the fracture toughness test values of straight-notch specimens were excessive, the crack usually initiated at the corner of the notch, and the test results were more scattered. Cui et al. [31] used a Hopkinson pressure bar to conduct dynamic impact tests of non-ideal crack disc specimens with center incisions. The test results showed that the ideal disc specimen can be replaced with a non-ideal crack disc specimen with a center incision when the width of the prefabricated crack is less than 1 mm. Wang and Luo et al. [32–34] proposed a method that simultaneously determines the fracture toughness and tensile strength of rock using a series of U-shaped notched beams with the same notch depth and different root curvatures. The feasibility of this method was verified by both theoretical analysis and experimentation. Kolhe et al. [35] explored the influence of the crack tip radius on the fracture toughness of CB specimens. Barati and Alizadeh [36] performed research on the ratio of the notch tip curvature to the notch depth and its influence on the plane strain fracture toughness. Cicero et al. [37] examined the influence of the three-point bending beam notch radius on the dimensionless stress intensity factor and fracture toughness by the theory of critical distances (TCD) and explicated the influencing mechanism of the notch tip radius by scanning electron microscopy. Dehghany et al. [38] investigated the effects of the first non-singular stress terms on the fracture assessment of sharp V-notches under mixed mode loading. Hussain and Murthy [39] have proposed a simple, robust, and efficient point substitution type displacement based technique for finite element estimation of the notch stress intensity factors (NSIFs) of sharp V-notched configurations. The technique can acquire accurate NSIFs even in course meshes made of quadratic elements without the use of any special singular elements at the notch tip.

Currently, the influence of the width of a prefabricated crack on the dimensionless stress intensity factor of the NSCB specimen has not been reported. Analysis of the extent to which the width of the prefabricated crack influences the dimensionless stress intensity factor is necessary to improve the method that measures the fracture toughness of NSCB specimens.

For this reason, the dimensionless stress intensity factor of the crack tip was calibrated by ABAQUS software for different widths of the prefabricated cracks of the NSCB specimens in this paper. The influence of the width of a prefabricated crack on the dimensionless stress intensity factor is examined; the relative error of the dimensionless stress intensity factor for different widths of the prefabricated cracks is analyzed; and the corrected formula of the NSCB dimensionless stress intensity factor that eliminates the effect of crack width is obtained. It is worth pointing out that it is not easy to prefabricate a crack on rock, and the crack tip of the manually prepared specimen is not an ideal sharp type, but a U-shaped crack with a certain radian. Therefore, the next numerical research work in this paper is to treat the prefabricated crack into a U-shaped crack, which is consistent with the actual crack form in the specimen used by researchers to test rock fracture toughness.

#### 2. Calibration of Dimensionless Stress Intensity Factor

##### 2.1. Fracture Toughness Principle of NSCB Specimen

Figure 1 shows the loading of an NSCB specimen. The specimen in Figure 1 carries the concentrated load *P*; its radius is *R*; its disc thickness is *B*; the length and width of the artificially prefabricated crack are *a* and 2*b*, respectively; and the support spacing is *S*. Considering the influence of the width of a prefabricated crack, the calculation of the mode I stress intensity factor is shown as follows:where is the concentrated load, is the dimensionless length of the prefabricated crack, is the dimensionless width of the prefabricated crack, is the dimensionless specimen thickness, and is the dimensionless support spacing. The dimensionless stress intensity factor value that is associated with length, width, and thickness of the prefabricated crack and the support spacing of the specimen is calculated by the following formula: