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Potential fracture configurations of a cracked solid under mixed mode-I/III loading

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Abstract

The potential fracture configurations of a cracked solid under mixed Mode-I/III loading are investigated based on a three-dimensional energy-based model, which can effectively capture the physical process of energy release induced by multiple cracks initiation from a crack tip. In order to maximize the energy release rate, four integral subintervals for Ji-integral around a crack tip have been suggested, based on which the four concerned energy-based driving forces have been identified. In this regard, when the driving forces reach the critical value, the concerned local boundaries around a crack tip will fracture, respectively, in a form of either wing cracks or crack extension. A series of potential fracture configurations for cracked solids under the mixed Mode-I/III loading, such as the crack tri-branching, symmetrical branching, side-branching, kinking and extension, can be theoretically predicted from the combination of the triggered new cracks. Some understanding on the fracture mechanism in engineering and experimental research should be refreshed based on the present theoretical investigations. Typical fracture configurations and concerned K-based effective fracture toughness predicted by the present modelling and Griffith’s criterion agree well with the experimental results in the available literature.

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Abbreviations

\(\alpha_{i}\), \(\alpha_{ic}\) :

Angle between the shifting direction of the local boundary and \(x_{i}\)-axis and its critical value

\(\alpha\) :

Inclined angle of the crack plane relative to the loading roller

\(r, \, \theta\) :

Polar coordinates

\(\varphi\) :

Normalized parameter

\(\mu\) :

Poisson’s ratio

\(\sigma_{ij}\) :

Stress components

\(A,A_{i}\), \(s,s_{i}\) :

Boundary surfaces or integration paths around the crack tip

E :

Young's modulus

G :

Energy release rate for boundary shifting

\(G^{{{\text{tri}} - b}}\), \(G_{\max }^{tri - b}\), \(G_{C}^{{{\text{tri}} - b}}\) :

Energy release rate, its maximum and critical value for crack tri-branching

\(G_{\max }^{{{\text{side}} - b}}\), \(G_{\max }^{{{\text{sym}} - b}}\), \(G_{\max }^{{{\text{kinking}}}}\) :

Energy-based configuration driving forces for crack side-branching, symmetrical-branching and kinking, respectively

\(G_{\max }^{{A - {\text{ext}}}}\) :

Energy-based driving force for boundary A moving in x1-direction in a form of crack extension

\(G_{\max }^{{A - {\text{wing}}}}\) :

Energy-based driving force for boundary A moving as a wing crack

\(J_{i}\) :

Conservation integrals

\(K_{{{\text{eff}} - C}}^{{{\text{tri}} - b}}\), \(K_{{{\text{eff - }}C}}^{{{\text{side}} - b}}\), \(K_{{{\text{eff}} - C}}^{{{\text{sym}} - b}}\) :

K-based effective fracture toughness for crack tri-branching, side-branching and symmetrical-branching, respectively

\(K_{{{\text{eff}} - C}}^{{A - {\text{wing}}}}\), \(K_{{{\text{eff}} - C}}^{{A - {\text{ext}}}}\) :

K-based effective fracture toughness for local boundary A moving as a wing crack or crack extension

\(K_{{{\text{IC}}}}\) :

Fracture toughness for pure Mode-I crack extension

\(K_{I}\), \(K_{III}\) :

Stress intensity factors for the Mode-I or Mode-III deformations

P :

Load

\(r_{o}\) :

Polar coordinate of the boundary surface around crack tip

\(T_{i}\) :

Stress vector acting on the integration surface or path

\(u_{i}\) :

Displacement components

\(v_{i}\) :

Direction of the local boundary \(A_{i}\) shifting

w :

Strain energy density

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos.: 50771052, 50971068 and 11272141) and Natural Science Foundation of Liaoning (Grant Nos.: LS2010100 and 20102129).

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Authors and Affiliations

  1. College of Petroleum and Natural Gas Engineering, Liaoning Petrochemical University, Fushun, 113001, LN, People’s Republic of China

    L. Wang, Y. J. Xie & H. Yuan

  2. College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao, 266000, SD, People’s Republic of China

    L. Wang

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Correspondence to Y. J. Xie.

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Wang, L., Xie, Y.J. & Yuan, H. Potential fracture configurations of a cracked solid under mixed mode-I/III loading. Arch Appl Mech 93, 2033–2049 (2023). https://doi.org/10.1007/s00419-023-02371-x

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