Indentation and grain orientation: why hardness and modulus change from grain to grain

Indentation modulus of a crystal

For an isotropic solid the unloading stiffness is S = 2 M a with M = E/(1 − ν²). Vlassak and Nix (1993, 1994) found M for any anisotropic half-space from the surface displacement under a point load. That displacement is w = h(θ)/r: it falls off as 1/r like the isotropic case, but its strength h depends on the direction θ in the surface. Replacing h(θ) by its average over θ gives the "equivalent isotropic" modulus, M = 1/(π ⟨h⟩). The page computes ⟨h⟩ from the Barnett-Lothe tensor: for each in-plane direction m, L = (1/π)∫₀^π [(mm) − (mn)(nn)⁻¹(nm)] dω, where (ab)_jk = a_iC_ijklb_l and m, n turn by ω in the plane that holds the surface normal. Then M = 2 / ⟨n·L⁻¹·n⟩, averaged over m. For an isotropic solid this returns E/(1 − ν²) exactly.

Check of the calculation

For diamond (C₁₁ = 1079, C₁₂ = 124, C₄₄ = 578 GPa) the code gives 1126 GPa on the cube plane and 1165 GPa on the octahedral plane, the values of Vlassak et al. (2003). For copper, Vlassak and Nix (1994) measured the (111) indentation modulus about 10% above the {100} one; with the Simmons and Wang constants this calculation gives 13% (134, 147 and 152 GPa on 001, 101 and 111). Delafargue and Ulm (2004) give closed-form approximations of the same kind for orthotropic solids.

Young's modulus along the normal

1/E = S₁₁ − 2(S₁₁ − S₁₂ − S₄₄/2)(l²m² + m²n² + n²l²), with l, m, n the direction cosines. It is what a tensile test along the normal would give; it is not what an indenter measures. The polycrystal M uses the Hill average of E and ν.

GPa, room temperature. A = 2C₄₄/(C₁₁ − C₁₂), the Zener ratio (1 for an isotropic crystal).

Hardness and pile-up

Under uniaxial compression a crystal yields when the resolved shear stress on its best slip system reaches a critical value, so yield scales as 1/m, with m the largest Schmid factor on the {111}<110> systems. That gives 0.408 on (001) and (011) and 0.272 on (111). Under an indenter, material flows in all directions and many slip systems work at once, so hardness is closer to an average over systems (like a Taylor factor) and changes much less. In copper, Vlassak and Nix (1994) measured only about 6% change in hardness with orientation, less than the change in modulus. The measured ratios in the chart are from 301LN austenitic steel grains (Roa et al. 2015: 3.13, 3.36 and 3.56 GPa on (001), (101) and (111)); that steel can also transform under the tip, so treat them as one example.

The pile-up picture is a sketch, not a simulation. Wang et al. (2004) found, with a conical tip on copper single crystals, in experiments and crystal plasticity simulations, lobes along the lines where the slip planes meet the surface: four-fold on (001), two-fold on (011) and six-fold on (111). The page weights each {111} plane trace by (1 − (p·n)²) Σ (b·n)² (steep planes whose slip directions point out of the surface lift the most material), which gives those three patterns. The crystal itself has three-fold symmetry about [111]; a Berkovich tip adds its own three-fold shape to the pattern.

Size of the plastic zone

Johnson's expanding cavity model for a cone of face angle β to the surface: (c/a)³ = E tanβ / (6Y(1 − ν)) + 2(1 − 2ν)/(3(1 − ν)). A Berkovich tip acts like a cone of half-angle 70.3° (β = 19.7°) with contact radius a = h tan 70.3° = 2.79 h (area 24.5 h²). The page uses copper's polycrystal E = 127 GPa and ν = 0.345 (Hill average). At Y = 200 MPa, c = 3.9 a, about 11 h. Pile-up and sink-in change a, and the elastic field that sets the stiffness reaches further than c, so a boundary can change M even when it is outside the plastic zone. Grains are 3D Voronoi cells of mean spacing d; the count in the zone comes from points spread through the half-sphere.