Size effect and asymptotic matching approximations in strain-gradient theories of micro-scale plasticity

To explain the size effect found in the testing of plastic behavior of metals on the micrometer scale, four theories of strain-gradient plasticity, representing generalizations of the deformation theory of plasticity, have been developed since 1993-the pioneering original theory of Fleck and Hutchinson in two subsequent versions, the mechanism-based strain-gradient (MSG) plasticity of Gao and co-workers (the first theory anchored in the concept of geometrically necessary dislocations), and Gao and Huang's recent update of this theory under the name Taylor-based nonlocal theory. Extending a recent study of Bažant in 2000 focused solely on the MSG theory, the present paper establishes the small-size asymptotic scaling laws and load-deflection diagrams of all the four theories. The scaling of the plastic hardening modulus for the theory of Acharya and Bassani, based on the incremental theory of plasticity, is also determined. Certain problematic asymptotic features of the existing theories are pointed out and some remedies proposed. The advantages of asymptotic matching approximations are emphasized and an approximate formula of the asymptotic matching type is proposed. The formula is shown to provide a good description of the experimental and numerical results for the size range of the existing experiments (0.5-100 μm).