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Characteristic Tensor for Evaluation of Singular Stress Field Under Mixed-Mode Loadings

Kei Saito1, 2, *, Tei Hirashima1, Ninshu Ma2, *, Hidekazu Murakawa2
1 JSOL Corporation, 2-5-24 Harumi, Chuo-ku, Tokyo, 104-0053, Japan.
2 Osaka University, 11-1 Mihogaoka, Ibaraki, Osaka, 567-0047, Japan.
* Corresponding Authors: Kei Saito. Email: ;
   Ninshu Ma. Email:

Computer Modeling in Engineering & Sciences 2020, 122(2), 415-432.

Received 17 October 2019; Accepted 25 December 2019; Issue published 01 February 2020


A characteristic tensor is defined using stress tensor averaged in a small circular domain at the crack tip and multiplied by the root of domain radius. It possesses the original stress tensor characteristics and has a simple relationship with conventional fracture-mechanics parameters. Therefore, it can be used to estimate stress intensity factors (SIFs) for cracks of arbitrary shape subjected to multiaxial stress loads. A characteristic tensor can also be used to estimate SIFs for kinked cracks. This study examines the relation between a characteristic tensor and SIFs to demonstrate the correlation between the characteristic tensor and fracture-mechanics parameters. Consequently, a single straight crack and a kinked crack of finite length existing in a twodimensional, infinite isotropic elastic body in a plane stress state, were considered to investigate the properties of the characteristic tensor under mixed-mode loadings. To demonstrate the practical utility of the characteristic tensor, the stress distribution obtained through finite element analysis (FEA) was used to estimate mixed-mode SIFs, and the values of estimated SIFs were compared with those obtained using an analytical solution. Results demonstrate that SIFs estimated under mixed-mode loadings exhibit a good agreement with the analytical values. This indicates that the proposed characteristictensor-based approach is effective in extracting features of singular stress fields at crack tips, and can be employed to estimate values of fracture-mechanics parameters, such as SIFs. Owing to its simplicity, the proposed approach can be easily incorporated in commercial FE codes for practical applications to simulate the crack-growth problem under both static and dynamic loading scenarios. The excellent applicability of the characteristic tensor greatly contributes to efficiency of the design process in industries.


Characteristic tensor, stress-intensity factor, mixed-mode loading, kinked crack.

Cite This Article

Saito, K., Hirashima, T., Ma, N., Murakawa, H. (2020). Characteristic Tensor for Evaluation of Singular Stress Field Under Mixed-Mode Loadings. CMES-Computer Modeling in Engineering & Sciences, 122(2), 415–432.


This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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