A unified theory for brittle and ductile shear mode fracture

ABSTRACT

A unified theory captures both brittle and ductile fracture. The fracture toughness is proportional to the applied stress squared and the length of the crack. For purely brittle solids, this criterion is equivalent to Griffith's theory. In other cases, it provides a theoretical basis for the Irwin-Orowan formula. For purely ductile solids, the theory makes direct contact with the Bilby-Cottrell-Swinden model. The toughness is highest in ductile materials because the shielding dislocations in the plastic zone provide additional resistance to crack growth. This resistance is the force opposing dislocation motion, and the Peach-Koehler force overcomes it. A dislocation-free zone separates the plastic zone from and the tip of the crack. The dislocation-free zone is finite because molecular forces responsible for the cohesion of the surfaces near the crack tip are not negligible. At the point of crack growth, the length of the dislocation-free zone is constant and the shielding dislocations advance in concert. As in Griffith's theory, the crack is in unstable equilibrium. The theory shows that a dimensionless variable controls the elastoplastic behaviour. A relationship for the size of the dislocation-free zone is derived in terms of the macroscopic and microscopic parameters that govern the fracture.

KEYWORDS:

• Mechanical properties
• fracture toughness
• cracks
• Griffith's theory
• Bilby-Cottrell-Swinden model
• dislocation-free zone

Acknowledgments

I am indebted to Professor Adrian P Sutton FRS and Dr Daniel S Balint of Imperial College London and Professor Anthony T Paxton of King's College London for helpful discussion.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Luca Cimbaro http://orcid.org/0000-0002-6295-0744

Additional information

Funding

This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training on Theory and Simulation of Materials [grant no. EP/L015579/1]; and Rolls-Royce plc.