Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: Ideal plastic behavior

Abstract

Assuming ideal plastic behavior for an isotropic matrix containing a misfitting spherical precipitate, the total amount of work expended during elasto-plastic deformation is calculated and compared with the total strain energy in the corresponding pure elastic state. For precipitates larger than one micron (µm), the effective yield stress is taken as the macroscopic yield stress while for smaller precipitates, size-dependent yield stresses are obtained from the Ashby-Johnson model. In the case of coherent submicron precipitates, the effective yield stress becomes the theoretical yield strength and thus plastic relaxation is not possible unless the transformation stress is extremely large. For incoherent submicron precipitates, the effective yield stress is approximately inversely proportional to the precipitate radius,r. Hence plastic relaxation again is not possible whenr < 10 nm, but whenr ≃100 nm the strain energy can decrease by 10 ∼ 40 pct at a misfit of 3 pct. For supra-micron particles, however, the ratio of the effective yield stress to the shear modulus becomes 10⁻³ or less, and plastic relaxation can reduce the strain energy by factors of 3 to 15 at misfits of 1 to 3 pct. Under this circumstance, the plastic zone becomes wide, its radius ranging from 3 to 5r.