The influence on macroscopic work hardening of small, spherical, elastic particles dispersedwithin a matrix is studied using an isotropic strain gradient plasticity framework. An analyticalsolution for strain hardening, i.e. the flow stress as a function of plastic strain, based ona recently developed model for initial yield strength is proposed. The model accounts forrandom variations in particle size and elastic properties, and is numerically validated againstFE solutions in 2D/3D unit cell models. Excellent agreement is found as long as the typicalparticle radius is much smaller than the material length scale, given that the particle volumefraction is not too large (< 10%) and that the particle/matrix elastic mismatch is within arealistic range. Finally, the model is augmented to account for strengthening contribution from shearable particles using classic line tension models and successfully calibrated againstexperimental tensile data on an 𝐴𝑙 − 2.8𝑤𝑡%𝑀𝑔 − 0.16𝑤𝑡%𝑆𝑐 alloy.

An analytical flow stress model, based on isotropic strain gradient plasticity theory, for precipitation hardened materials, is proposed and evaluated against tensile data on a 15 wt% Cr - 5 wt% Ni (15-5) PH stainless steel. The 15-5 PH material was aged at 500 °C for 1 h, 2 h, 5 h and 50 h to obtain a wide range of precipitate sizes. Detailed characterisation of precipitates was obtained using atom probe tomography (APT). A second material, a 15-5 stainless steel without added Cu was heat treated to obtain a similar matrix microstructure as in the 15-5 PH, but without Cu precipitates. Tensile testing revealed that the heat treated 15-5 PH material covered the full range from under- to overaged conditions. The analytical model, which accounts for stress reducing effects of plastic relaxation around particles, manages to capture the experimental data in a very satisfying manner using only a total of three tunable parameters. It is believed that the proposed model can offer an alternative to the much more commonly used work hardening models based on the internal variable approach.

An analytical model, based on an isotropic strain gradient plasticity theory, describing work hardening during cyclic straining in a metal reinforced by a dispersion of non shearable particles is presented. The yield criterion is expressed in terms of isotropic and kinematic hardening contributions and the model is validated against full field finite element (FE) solutions on a 2D axi-symmetric unit cell model. Excellent agreement between analytical and FE results is obtained. The theory presented includes mixed energetic/dissipative contributions from higher order stresses in both bulk and at particle/matrix interfaces. In particular, the influence of a quadratic interface free energy that transitions into a linear form at some threshold value of plastic strain is investigated. It is shown that such an energy is capable of capturing the experimentally observed phenomenon of inflections in the reverse stress-strain curve. It is argued, based on the well known phenomenon where particles are shielded by Orowan dislocation loops during reverse strain, that an energetic interface contribution could be physically relevant for low plastic strains.

Room temperature cyclic tensile/compression tests were carried out on a precipitation hardened martensitic 15-5 PH stainless steel in order to validate a previously developed work hardening model by the authors, based on strain gradient plasticity. Identical tests were also made on a 15-5 steel containing no precipitates to serve as model input for the matrix material. The model was calibrated to the experimental data up to a forward plastic strain of 1% and the rest of the cyclic stress strain curves were predicted by the model with generally very good agreement. We believe that the model's capability to predict the cyclic stress strain behaviour of the composite strengthens the role of continuum modelling within material micro mechanics. Despite low plastic strain amplitudes (0.25%, 0.5% and 1%) no signs of inflection on the reverse flow curves were observed, even when tested at a temperature of -50 °C. Moreover, the results suggest that the strain gradient plasticity related higher order stresses that exist in close vicinity to the particles most likely have a dissipative character in the current alloy. However, this does not constitute a general conclusion as it should depend on parameters such as temperature, alloy stacking fault energy, etc.