# Simulation of fracture of elastoplastic materials in mode III: from brittle to ductile

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## Abstract

The initiation of a longitudinal shear crack (mode III fracture) in elastoplastic materials with the limiting strain is considered. The crack propagation criterion is formulated using a modified Leonov–Panasyuk–Dugdale model using an additional parameter—the width of the plasticity zone (the width of the pre-fracture zone). Conditions of a small-scale yielding in the presence of a stress field singularity in the vicinity of the crack tip are considered. A two-parameter criterion for quasi-brittle fracture of an elastoplastic material is formulated for mode III cracks. The proposed fracture criterion includes the deformation-based criterion, which is formulated at the crack tip, as well as the force-based criterion, formulated at the tip of a model crack. The lengths of the initial and model cracks differ by the length of the pre-fracture zone. The application of the proposed strength criterion to the determination of fracture loads for bodies containing longitudinal shear cracks is demonstrated. The diagrams of quasi-brittle fracture are constructed for a strip of finite width with an edge crack in case of the out-of-plane deformation. It is proposed to determine the model parameters using an idealized simple shear diagram and the critical stress intensity factor. For quasi-ductile and ductile types of fracture, the limiting loads are determined numerically. The propagation of plastic zones in the vicinity of the crack tip under quasistatic loading is successively described by the nonlinear finite element method. It is shown that the shapes of the numerically simulated plastic zones differ significantly from the known classical concepts.

## Keywords

Brittle Quasi-brittle Quasi-ductile and ductile fracture Elastoplastic materials Limiting strain## List of symbols

- \( a \)
Width of the pre-fracture zone

- \( a_{f} \)
Width of the plasticity zone for full-scale yielding

- \( K_{\text{III}}^{{}} \)
Stress intensity factor (SIF)

- \( K_{{{\text{III}}\infty }} \)
SIF generated by stresses \( \tau_{\infty } \) applied to distant boundaries

- \( K_{{{\text{III}}\Delta }} \)
SIF generated by constant stresses \( - \tau_{Y} \) acting in the pre-fracture zone

- \( K_{\text{III}}^{0} \)
Critical SIF according to the necessary criterion

- \( K_{\text{III}}^{*} \)
Critical SIF according to the sufficient criterion

- \( K_{\text{IIIc}}^{{}} \)
Experimentally identified critical SIF

- \( l_{0} \)
Initial crack length

- \( \bar{l}_{0} = l_{0} /r_{0} \)
Dimensionless length of the initial crack

- \( l = l_{0} + \Delta \)
Model crack length

- \( l^{*} = l_{0} + \Delta^{*} \)
Critical length of the model crack

- \( \bar{l}^{*} = l^{*} /r^{*} \)
Dimensionless critical length of the model crack

- \( l^{f} = l_{0} + \Delta^{f} \)
Critical crack length for full-scale yielding

- \( L \)
Strip width

- \( \bar{L}_{0} = L/r_{0} ,\;\bar{L}^{ * } = L/r^{ * } \)
Dimensionless strip widths

- \( r_{0} \)
Effective diameter of fracture structures for brittle materials

- \( r^{*} \)
Effective diameter of fracture structures for quasi-brittle materials

- \( \gamma_{0} \)
Maximum elastic shear strain

- \( \gamma_{1} \)
Maximum shear strain

- \( \Delta \)
Length of pre-fracture zone

- \( \Delta^{*} \)
Critical length of the pre-fracture zone

- \( \Delta^{f} \)
Critical length of the plasticity zone for full-scale yielding

- \( \delta^{*} \)
Critical crack opening displacement

- \( \tau_{nom} \)
Nominal shear stresses

- \( \tau_{Y} \)
Shear yeild strength

- \( \bar{\tau }_{\infty } = \tau_{\infty } /\tau_{Y} \)
Dimensionless remotely applied shear stresses

- \( \bar{\tau }_{\infty }^{0} = \tau_{\infty }^{0} /\tau_{Y} \)
Dimensionless critical load according to the necessary criterion

- \( \bar{\tau }_{\infty }^{*} = \tau_{\infty }^{*} /\tau_{Y} \)
Dimensionless critical load according to the sufficient fracture criterion

- \( \bar{\tau }_{\infty }^{f} = \tau_{\infty }^{f} /\tau_{Y} \)
Dimensionless critical load for full-scale yielding

## Notes

### Acknowledgements

This work was supported by the Russian Science Foundation (Grant No. 19-19-00126).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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