Indentation size effects in crystalline materials: A law for strain gradient plasticity

We show that the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations. The model leads to the following characteristic form for the depth dependence of the hardness: HH01+h∗h where H is the hardness for a given depth of indentation, h, H0 is the hardness in the limit of infinite depth and h∗ is a characteristic length that depends on the shape of the indenter, the shear modulus and H0. Indentation experiments on annealed (111) copper single crystals and cold worked polycrystalline copper show that this relation is well-obeyed. We also show that this relation describes the indentation size effect observed for single crystals of silver. We use this model to derive the following law for strain gradient plasticity: (σσ0)2 = 1 + l̂χ, where σ is the effective flow stress in the presence of a gradient, σ0 is the flow stress in the absence of a gradient, χ is the effective strain gradient and l̂ a characteristic material length scale, which is, in turn, related to the flow stress of the material in the absence of a strain gradient, l̂ ≈ b(μσ0)2. For materials characterized by the power law σ0 = σrefε1n, the above law can be recast in a form with a strain-independent material length scale l. (builtσσref)2 = ε2n + lχ l = b(μσref)2 = l̂(σ0σref)2. This law resembles the phenomenological law developed by Fleck and Hutchinson, with their phenomenological length scale interpreted in terms of measurable material parametersbl].