A J-integral approach in characterizing mechanics of a horizontal crack embedded in a cantilever beam under an end transverse force

Significance Statement

A crack embedded in elastic homogeneous systems is observed to propagate in the direction of maximum energy release rate that is experienced by incipient kink cracks at the tip of the crack. For instance, under mode I crack surface opening conditions, including incipient kink crack directly ahead of the tip of the crack, experiencing maximum energy release rate, therefore, it is the one initiated in the course of crack growth initiation and confines the macro-crack path to the plane of the crack.

Under pure mode II loading conditions, i.e. under relative crack surface sliding conditions, local maximum in the energy release rate is normally experienced by a kink crack at about 70.3° clockwise from the crack plane for a positive mode II stress intensity factor. In an interesting manner, the kink crack path direction estimated implementing a maximum energy release rate criterion is observed to coincide with a path perpendicular to the maximum principal stress near the crack tip.

It has been observed from previous studies that it can be challenging to envision the conditions, under which mode II crack occur, for instance, horizontal crack embedded in a cantilever beam subjected to an end force loading, will initiate and propagate in plane. University of Maryland researchers Dr. Xiaomin Fang and Professor Panos Charalambides employed a J-integral approach for developing estimates of the available elastic energy release rate at crack tips on the right and left. For this reason, they used moment resultants, beam deflections and cross sectional force. Their work is now published in Engineering Fracture Mechanics.

Path independent J-integral equals to the elastic energy release rate which is available to the crack-tip contained within the domain bounded by the integral contour as well as traction free crack surfaces. This is based on linear and elastic isotropic system. Considering the independence of the J-integral, the authors used contours for left and right crack tips. Every contour initiated at the bottom crack surface relative to a system that was positioned at the crack tip, and followed the specimen contour in an anti-clockwise direction and ended at the top surface.

The authors also carried out finite element studies of a cantilever beam with an embedded sharp crack and subjected to an end traverse loading. They systematically placed cracks of different lengths as well as orientation on geometrically admissible locations within the selected beam. They extracted and reported the near-tip energy release rate and mode I and mode II stress intensity factors that dominated the crack tip regions.

Both analytical and numerical results yielded identical energy release rate predictions where the right and left tips experienced identical levels of energy release rate. 2-D finite element models were adopted for parallel studies to extract the energy release rates for a wide range of beam and crack systems. Comparison results indicated significant agreement between finite element and analytical predictions. However, there were slight deviations between two predictions for cracks deeply embedded close to the neutral axis of the beam.

Energy release rates for all systems at the left and right crack tips were observed to be symmetric with respect to the neutral axis of the beam. This suggested that horizontal cracks situated at equal distances below and above the neutral axis of the beam experienced identical energy release rates.

A J-integral approach in characterizing mechanics of a horizontal crack embedded in a cantilever beam under an end transverse force -Advances in Engineering

About The Author

Professor Panos G. Charalambides
Department of Mechanical Engineering
The University of Maryland Baltimore County (UMBC)
1000 Hilltop Circle, Baltimore, MD 21250
Tel. # (410) 455-3346, Fax #  (410) 455-1052, E-mail : [email protected]

Professor Charalambides received his B.Sc. degree in Civil Engineering from Aristotle University of Thessaloniki, Greece in 1981.  He obtained his M.Sc. and Ph.D. degrees in Theoretical and Applied Mechanics at the University of Illinois at Urbana-Champaign in 1983 and 1986 respectively.

During his Post Doctoral studies in the Department of Materials, University of California at Santa Barbara he worked on pioneering studies with several collaborators on the characterization of bimaterial interface fracture.  While at Michigan Technological University, (1989-1993) Professor Charalambides received the prestigious NSF sponsored Presidential Young Investigator award.  He joined the Department of Mechanical Engineering at the University of Maryland Baltimore County in January 1994.

Professor Charalambides served as the Chair of the Department of Mechanical Engineering for a six year period during 2002-2008.  He has advised over 20 PhD and M.S. students and has published extensively in his area of research.

Research Interests
In addition to the  fracture of bimaterial interfaces and delamination in layered systems Dr. Charalambides research contributions include studies on micromechanical modeling, the modeling of unidirectionally reinforced composites, woven ceramic and polymer matrix composites, Fibrous Monolith Composites (FMCs), biomechanics, and optimal fixturing for manufacturing.

Professor Charalambides has technology transfer experience and contributions through the development of the DENDRO Finite Element software and the NIST sponsored OOF/OOF2 Materials Research software.  His most recent research includes modeling the non-linear response of red blood cells as well as predicting model development of tactile MEMS sensors.

Accomplishments & Awards
Professor Charalambides received several Teaching Excellence awards and the NSF sponsored Presidential Young Investigator award.

About The Author

Xiaomin Fang  Ph.D.

Ford Motor Company
15575 Lundy Parkway, iTek W2CK03,Dearborn, MI 48126

(Previously) Department of Mechanical Engineering
The University of Maryland Baltimore County (UMBC)
Tel. #  (410) 858-6599, E-mail : [email protected]

Dr. Fang received his B.Sc. degree in Mechanical Engineering from Shanghai Jiao Tong University, China in 2003.  He obtained his Ph.D. degrees in Mechanical Engineering at the University of Maryland Baltimore County in 2013.

Since joining IT at Ford Motor Company, Dr. Fang has led the enhancement of multiple computer aided engineering (CAE) and knowledge based engineering (KBE) tools for business needs in powertrain design and optimization.  While at University of Maryland Baltimore County (2007-2013), he worked on developing physics based finite element and analytical models enabling damage detection in beam structures.

Research Interests
In addition to his interest on model development for non-destructive damage detection in components and structures, Dr. Fang’s research interests include life prediction modeling for aging infrastructure, automation and physics-based modeling and simulation for the prediction of mechanical system performance, e.g. internal combustion engine system.  His expertise in the development of customized simulation tools enables the utilization of computing resources and enhances the flexibility and scalability to satisfy the evolving research demands.

Accomplishments & Awards
Dr. Fang received  recognition in IT for the quick ramp-up and quality deliverables of the CAE tool enhancement projects.  He also received Dissertation Fellowship Award when he was the Ph.D. candidate at UMBC.


Xiaomin Fang, Panos G. Charalambides. A J-integral approach in characterizing the mechanics of a horizontal crack embedded in a cantilever beam under an end transverse force. Engineering Fracture Mechanics 169 (2017) 35–53.

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