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

Abstract

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: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H 0 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 H 0. 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: ( σ σ 0 ) 2 = 1 + l ̂ χ , where σ is the effective flow stress in the presence of a gradient, σ 0 is the flow stress in the absence of a gradient, χ is the effective strain gradient and l ̂ a characteristic material length scale, which is, in turn, related to the flow stress of the material in the absence of a strain gradient, l ̂ ≈ b( μ σ 0 ) 2 . For materials characterized by the power law σ 0 = σ ref ε 1 n , the above law can be recast in a form with a strain-independent material length scale l. ( builtσ σ ref ) 2 = ε 2 n + l χ l = b( μ σ ref ) 2 = l ̂ ( σ 0 σ ref ) 2 . This law resembles the phenomenological law developed by Fleck and Hutchinson, with their phenomenological length scale interpreted in terms of measurable material parametersbl].