Analytical predictions of yield stress of a strain gradient plasticity material reinforced by small elastic particles
Jonas Faleskog, Peter Gudmundson
Abstract
Theories describing the important role of small particles for strengthening of metals have evolved since the pioneering work of Orowan. Here, this problem is analysed by a strain gradient plasticity (SGP) theory. The structure of the governing equations on non-dimensional form reveals that the plastic strain in the matrix material is to zeroth order approximation constant for a sufficiently small particle size a in comparison to material length scale ℓ. Based on this observation, a perturbation solution has been developed by expansions of all field variables in terms of a/ℓ and the volume fraction of particles f. The simple structure of the plastic strain field is also exploited to derive an upper bound solution from the principles of virtual work and maximum plastic dissipation. These analytical solutions are then used to derive expressions for the yield stress taking into account a random distribution of particles of various size and shape with elastic constants that differ from the matrix. The accuracy and range of validity of these solutions are demonstrated by comprehensive 2D and 3D finite element analyses of material volumes containing realistic distributions of particles of spherical and spheroidal shape of various elastic modulus. The results show that significant strengthening will arise provided that the representative particle size is smaller than the material length scale ℓ of the SGP material.