Finite Element Analysis of Mixed-Mode I/II Fracture in Marble Using SENB Specimens

Authors

  • Yanxiang Chen Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China
  • Enping Guo Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China
  • Linhua Huang Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China
  • Song Li Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China

DOI:

https://doi.org/10.13052/ejcm2642-2085.3521

Keywords:

Mixed-mode fracture, Fracture criterion, Single-Edge Notched Beam (SENB), Brittle rocks, Finite element simulation

Abstract

The fracture behavior of brittle rocks, like fractures in marble, is important for various engineering applications, including mining, underground facilities, hydraulic mining, and tunneling. While most of the prior work has focused on pure Mode I loading, rock structures undergo mixed-Mode I/II loading in which fractures appear from tensile and shear stresses. This research specifically looks at mixed-Mode I/II fractures in which vertical pre-cracked single edge notched beam specimens allow for a variety of Mode mixites without the difficult construction of angled cracks. Finite element simulations in ABAQUS focused on calculating Mode I (KI) as well as II (KII) stress intensity factors (SIF) and T-stress, with mesh convergence ensuring accuracy. To predict fractures in polygons, the maximum tangential stress (MTS) and generalized maximum tangential stress (GMTS) criteria were used. The outcomes display that pure Mode I yields the lowest fracture load while, with Mode II contribution, it increases and reaches the peak at pure Mode II. There is an increase in the crack propagation angle with respect to the increase of Mode II dominance, displaying the effect of negative T-stress on cracking density. The methodology is single-edge notched beam (SENB) samples for the mixed-mode fracture of brittle rocks incorporating numerical simulations with fracture criteria. This gives systems for predictive modelling to work with in experimental design.

Downloads

Download data is not yet available.

Author Biographies

Yanxiang Chen, Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China

Yanxiang Chen, female, born in Yongzhou, Hunan Province, is a member of the Communist Party of China. She holds a doctoral degree and is a senior engineer as well as a National Class-1 Certified Structural Engineer. In recent years, she has published more than 10 academic papers in domestic and international journals. She has been involved in two projects supported by the National Natural Science Foundation of China and two educational reform projects funded by the Provincial Department of Education. Additionally, she has led two municipal-level research projects, guided students in one provincial-level innovative research program, and directed one university-level high-quality course. Her main research focuses on monitoring and early warning of unstable rock masses, rock mechanics, and engineering. She teaches the courses Fundamental Principles of Concrete Structures and Building Information Modeling (BIM).

Enping Guo, Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China

Enping Guo was born in Linfen, Shanxi, China, in 1987. She is an associate professor in Hunan Institute of Science and Engineering. Her master’s degree is from Hebei University of Engineering. Her research interest is structural engineering.

Linhua Huang, Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China

Linhua Huang was born in Yongzhou, Hunan, China, in 1986. He is a lecturer in Hunan University of Science and Engineering. His master’s degree is from Hunan Institute of Science and Engineering. His research interest is tunnel engineering.

Song Li, Hunan Provincial Key Laboratory of Intelligent Protection and Utilization Technology in Masonry Artifacts, Hunan Institute of Science and Engineering, Yongzhou 425199, China, School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China

Song Li was born in Yongzhou, Hunan, China, in 1985. He is an associate professor in Hunan University of Science and Engineering. His master’s and doctor’s degree are from Hunan Institute of Science and Engineering. His research interest is rock mechanics.

References

S. Ghouli, B. Bahrami, M. R. Ayatollahi, T. Driesner, and M. Nejati, “Introduction of a scaling factor for fracture toughness measurement of rocks using the semi-circular bend test,” Rock Mech. Rock Eng., vol. 54, no. 8, pp. 4041–4058, 2021, doi: 10.1007/s00603-021-02468-1.

M. R. Ayatollahi, S. Ghouli, and B. Bahrami, “Experimental and theoretical fracture analyses for three biomaterials with dental applications,” J. Mech. Behav. Biomed. Mater., vol. 103, p. 103612, 2020, doi: 10.1016/j.jmbbm.2019.103612.

L. Liu, H. Li, and X. Li, “A state-of-the-art review of mechanical characteristics and cracking processes of pre-cracked rocks under quasi-static compression,” J. Rock Mech. Geotech. Eng., vol. 14, no. 6, pp. 2034–2057, 2022, doi: 10.1016/j.jrmge.2022.03.013.

M. R. M. Aliha, G. R. Hosseinpour, and M. R. Ayatollahi, “Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials,” Rock Mech. Rock Eng., vol. 46, no. 5, pp. 1023–1034, 2013, doi: 10.1007/s00603-012-0325-z.

N. Razavi, M. R. M. Aliha, and F. Berto, “Application of an average strain energy density criterion to obtain the mixed mode fracture load of granite rock tested with the cracked asymmetric four-point bend specimens,” Theor. Appl. Fract. Mech., vol. 97, pp. 419–425, 2018, doi: 10.1016/j.tafmec.2017.07.004.

A. M. Al-Mukhtar and B. Merkel, “Simulation of the crack propagation in rocks using fracture mechanics approach,” J. Fail. Anal. Prev., vol. 15, no. 1, pp. 90–100, 2015, doi: 10.1007/s11668-014-9907-2.

T. L. Anderson and T. L. Anderson, Fracture Mechanics: Fundamentals and Applications. CRC Press, 2005.

J. Caicedo and A. Portela, “Direct computation of stress intensity factors in finite element method,” Eur. J. Comput. Mech., pp. 309–335, 2017.

M. R. M. Aliha and M. R. Ayatollahi, “Mixed Mode I/II brittle fracture evaluation of marble using SCB specimen,” Procedia Eng., vol. 10, pp. 311–318, 2011, doi: 10.1016/j.proeng.2011.04.054.

J.-P. Liu et al., “Cracking mechanisms in granite rocks subjected to uniaxial compression by moment tensor analysis of acoustic emission,” Theor. Appl. Fract. Mech., vol. 75, pp. 151–159, 2015, doi: 10.1016/j.tafmec.2014.12.006.

A. Mardoukhi, Y. Mardoukhi, M. Hokka, and V.-T. Kuokkala, “Effects of heat shock on the dynamic tensile behavior of granitic rocks,” Rock Mech. Rock Eng., vol. 50, no. 5, pp. 1171–1182, 2017, doi: 10.1007/s00603-017-1168-4.

Y. Wang and X. Hu, “Determination of tensile strength and fracture toughness of granite using notched three-point-bend samples,” Rock Mech. Rock Eng., vol. 50, no. 1, pp. 17–28, 2017, doi: 10.1007/s00603-016-1098-6.

M. R. M. Aliha and M. R. Ayatollahi, “Rock fracture toughness study using cracked chevron notched Brazilian disc specimen under pure modes I and II loading – A statistical approach,” Theor. Appl. Fract. Mech., vol. 69, pp. 17–25, 2014, doi: 10.1016/j.tafmec.2013.1 1.008.

M. R. M. Aliha, M. R. Ayatollahi, and J. Akbardoost, “Typical upper bound-lower bound mixed mode fracture resistance envelopes for rock material,” Rock Mech. Rock Eng., vol. 45, no. 1, pp. 65–74, 2012, doi: 10.1007/s00603-011-0167-0.

G. S. Xeidakis, I. S. Samaras, D. A. Zacharopoulos, and G. E. Papakaliatakis, “Crack growth in a mixed-mode loading on marble beams under three-point bending,” Int. J. Fract., vol. 79, no. 2, pp. 197–208, 1996, doi: 10.1007/BF00032936.

M. R. M. Aliha, R. Pakzad, and M. R. Ayatollhi, “Numerical analyses of a cracked straight-through flattened Brazilian disk specimen under mixed-mode loading,” J. Eng. Mech., vol. 140, no. 1, pp. 219–224, Jan. 2014, doi: 10.1061/(ASCE)EM.1943-7889.0000651.

J. Akbardoost, M. R. Ayatollahi, M. R. M. Aliha, M. J. Pavier, and D. J. Smith, “Size-dependent fracture behavior of Guiting limestone under mixed mode loading,” Int. J. Rock Mech. Min. Sci., vol. 71, pp. 369–380, 2014, doi: 10.1016/j.ijrmms.2014.07.019.

C.-H. Chen, C.-S. Chen, and J.-H. Wu, “Fracture toughness analysis on cracked ring disks of anisotropic rock,” Rock Mech. Rock Eng., vol. 41, no. 4, pp. 539–562, 2008, doi: 10.1007/s00603-007-0152-9.

M. Shimada, “Confirmation of two types of fracture in granite deformed at temperatures to 300∘C,” Tectonophysics, vol. 211, no. 1, pp. 259–268, 1992, doi: 10.1016/0040-1951(92)90063-C.

Q. Guo, H. Su, J. Liu, Q. Yin, H. Jing, and L. Yu, “An experimental study on the fracture behaviors of marble specimens subjected to high temperature treatment,” Eng. Fract. Mech., vol. 225, p. 106862, 2020, doi: 10.1016/j.engfracmech.2019.106862.

W. Zhang, W. Hua, M. Zhou, F. He, Y. Xu, and S. Dong, “Experimental study on the fracture characteristics of sandstone under asymmetric load by using a semi-circular bending specimen,” Theor. Appl. Fract. Mech., vol. 130, p. 104315, 2024, doi: 10.1016/j.tafmec.2024.104315.

Y. Zhao, K. Zheng, and C. Wang, “Mixed-Mode I/III fracture BT – Rock fracture mechanics and fracture criteria,” Y. Zhao, K. Zheng, and C. Wang, Eds., Singapore: Springer Nature Singapore, pp. 89–112, 2024, doi: 10.1007/978-981-97-5822-7_6.

Y. Xu, F. Dai, T. Zhao, N. Xu, and Y. Liu, “Fracture toughness determination of cracked chevron notched Brazilian disc rock specimen via Griffith energy criterion incorporating realistic fracture profiles,” Rock Mech. Rock Eng., vol. 49, no. 8, pp. 3083–3093, 2016, doi: 10.1007/s00603-016-0978-0.

G. Giambanco, M. Puccia, and A. Spada, “Numerical simulation of experimental fracture in BD and CCNBD sandstone specimens using the Advanced Augmented-FEM,” Eng. Fract. Mech., vol. 330, p. 111694, 2025, doi: 10.1016/j.engfracmech.2025.111694.

M. G. Salomon, F. Meftah, and J.-M. Reynouard, “Mixed mode fracture of a higher order beam model in gradient-dependent plasticity,” Eur. J. Comput. Mech., pp. 831–839, 2004.

M. D. Kuruppu and K. P. Chong, “Fracture toughness testing of brittle materials using semi-circular bend (SCB) specimen,” Eng. Fract. Mech., vol. 91, pp. 133–150, 2012, doi: 10.1016/j.engfracmech.2012.01.013.

Y. Obara, K. Nakamura, S. Yoshioka, A. Sainoki, and A. Kasai, “Crack front geometry and stress intensity factor of semi-circular bend specimens with straight through and chevron notches,” Rock Mech. Rock Eng., vol. 53, no. 2, pp. 723–738, 2020, doi: 10.1007/s00603-019-01 930-5.

Q. Lin, X. Bian, P.-Z. Pan, Y. Gao, and Y. Lu, “Criterion of local symmetry visualized in small eccentric single edge notched bend (E-SENB) rock specimens,” Eng. Fract. Mech., vol. 248, p. 107709, 2021, doi: 10.1016/j.engfracmech.2021.107709.

M. R. Ayatollahi, B. Bahrami, A. M. Mirzaei, and M. Yazid Yahya, “Effects of support friction on Mode I stress intensity factor and fracture toughness in SENB testing,” Theor. Appl. Fract. Mech., vol. 103, p. 102288, 2019, doi: 10.1016/j.tafmec.2019.102288.

L. Zhang et al., “Experimental and numerical investigations on the mixed-mode fracture of granite under offset three-point bending loads,” Rock Mech. Rock Eng., 2025, doi: 10.1007/s00603-025-04871-4.

B. Afrasiabian and M. Eftekhari, “Prediction of Mode I fracture toughness of rock using linear multiple regression and gene expression programming,” J. Rock Mech. Geotech. Eng., vol. 14, no. 5, pp. 1421–1432, 2022, doi: 10.1016/j.jrmge.2022.03.008.

B. N. Whittaker, R. N. Singh, and G. Sun, Rock Fracture Mechanics: Principles, Design and Applications. 1992.

A. Jahanbani, S. Pirmohammad, and M. R. Ayatollahi, “Effect of friction at crack surfaces on the fracture behavior of particleboard under compression-shear loading,” Theor. Appl. Fract. Mech., vol. 141, p. 105269, 2026, doi: 10.1016/j.tafmec.2025.105269.

H. M. Westergaard, “Bearing pressures and cracks: bearing pressures through a slightly waved surface or through a nearly flat part of a cylinder, and related problems of cracks,” J. Appl. Mech., vol. 6, no. 2, pp. A49–A53, Mar. 2021, doi: 10.1115/1.4008919.

M. L. Williams, “On the stress distribution at the base of a stationary crack,” J. Appl. Mech., vol. 24, no. 1, pp. 109–114, Jun. 2021, doi: 10.1115/1.4011454.

F. Erdogan and G. C. Sih, “On the crack extension in plates under plane loading and transverse shear,” J. Basic Eng., vol. 85, no. 4, pp. 519–525, Dec. 1963, doi: 10.1115/1.3656897.

M. Cai, “Fracture initiation and propagation in a Brazilian disc with a plane interface: a numerical study,” Rock Mech. Rock Eng., vol. 46, no. 2, pp. 289–302, 2013, doi: 10.1007/s00603-012-0331-1.

D. J. Smith, M. R. Ayatollahi, and M. J. Pavier, “The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading,” Fatigue Fract. Eng. Mater. Struct., vol. 24, no. 2, pp. 137–150, 2001.

R. A. Schmidt, “A microcrack model and its significance to hydraulic fracturing and fracture toughness testing,” in 21st U.S. Symposium on Rock Mechanics (USRMS), Rolla, Missouri, May 1980.

M. R. Ayatollahi and M. R. M. Aliha, “Mixed mode fracture analysis of polycrystalline graphite – A modified MTS criterion,” Carbon N. Y., vol. 46, no. 10, pp. 1302–1308, 2008, doi: 10.1016/j.carbon.2008. 05.008.

Y. Xu, G. Zhao, X. Li, B. Wu, and K. Xia, “Influences of notch width and notch-tip angle on the fracture toughness measurement using the semi-circular bend (SCB) specimen,” Eng. Fract. Mech., vol. 281, p. 109098, 2023, doi: 10.1016/j.engfracmech.2023.109098.

M. Fakhri, E. Haghighat Kharrazi, and M. R. M. Aliha, “Mixed mode tensile – In-plane shear fracture energy determination for hot mix asphalt mixtures under intermediate temperature conditions,” Eng. Fract. Mech., vol. 192, pp. 98–113, 2018, doi: 10.1016/j.engfracmech.2018.02.007.

R. Negru, L. Marsavina, H. Filipescu, and N. Pasca, “Investigation of mixed Mode I/II brittle fracture using ASCB specimen,” Int. J. Fract., vol. 181, no. 1, pp. 155–161, 2013, doi: 10.1007/s10704-013-9 830-7.

Downloads

Published

2026-10-23

How to Cite

Chen, Y. ., Guo, E. ., Huang, L. ., & Li, S. . (2026). Finite Element Analysis of Mixed-Mode I/II Fracture in Marble Using SENB Specimens. European Journal of Computational Mechanics, 35(02), 73–102. https://doi.org/10.13052/ejcm2642-2085.3521

Issue

2026: Vol 35 Iss 2

Section

Original Article