Mathematical model for the plastic flow and ductile fracture of polycrystalline solids

It is mathematically shown that ductile fracture after finite plastic strain is a necessary consequence of the polycrystalline nature of the materials. A closed-form equation for the plastic strain to fracture of a fine-grained polycrystal with no voids is derived. The mathematical model for the plastic deformation is grounded on the physical hypothesis that adjacent grains slide with a relative velocity proportional to the local shear stress resolved in the plane of the shared grain boundary, when exceeds a finite threshold. Hence plastic flow is governed predominantly by the in-plane shear forces making grain boundaries to slide, and the induced local forces responsible for the continuous grain reshaping are much weaker. The process is shown to produce a monotonic hydrostatic pressure variation with strain that precludes a stationary flow. The hydrostatic pressure dependence on strain has two solutions. One of them leads to superplasticity, the other one is shown to diverge logarithmically at a finite fracture strain and then represents ductile behaviour. Emphasis is done in the mathematical aspects of the deformation of the polycrystal up to the initiation of fracture. Although theoretical predictions agree well with mechanical tests of commercial alloys, technical issues like the effects of the presence and evolution of porosity and other imperfections, or how fracture evolves after initiation are left for a more specific communication.