A possible link between brittle and ductile failure by viewing fracture as a topological defect
Amit Acharya
(to appear in Comptes Rendus Mécanique)
A continuum model of fracture that describes, in principle, the propagation and interaction of
arbitrary distributions of cracks and voids with evolving topology without a 'fracture criterion'
is developed. It involves a 'law of motion' for crack-tips, primarily as a kinematical consequence
coupled with thermodynamics. Fundamental kinematics endows the crack-tip with a topological
charge. This allows the association of a kinematical conservation law for the charge, resulting
in a fundamental evolution equation for the crack-tip field, and in turn the crack fi eld. The
vectorial crack field degrades the elastic modulus in a physically justi fied anisotropic manner.
The mathematical structure of this conservation law allows an additive 'free' gradient of a scalar
field in the evolution of the crack field. We associate this naturally emerging scalar fi eld with the
porosity that arises in the modeling of ductile failure. Thus, porosity-rate gradients aff ect the
evolution of the crack- field which, then, naturally degrades the elastic modulus, and it is through
this fundamental mechanism that spatial gradients in porosity growth aff ect the strain-energy
density and stress carrying capacity of the material - and, as a dimensional consequence related
to fundamental kinematics, introduces a length-scale in the model. A key result of this work is
that brittle fracture is energy-driven while ductile fracture is stress-driven; under overall shear
loadings where mean stress vanishes or is compressive, shear strain energy can still drive shear
fracture in ductile materials.
