Plastic material parameters and plastic anisotropy of tungsten single crystal: a spherical micro-indentation study

Abstract

Enhancement of toughness is currently a critical engineering issue in tungsten metallurgy. The inherent toughness of tungsten single crystals is closely related to the capacity for local plastic slip. In this study we have investigated the plastic behavior of tungsten single crystals by means of micro-indentation experiments performed on specimens exposing (100), (110), and (111) surfaces. In parallel, FEM simulations were carried out with the Peirce–Asaro–Needleman crystal plasticity model considering both {110} 〈111〉 and {112} 〈111〉 slip systems. Plastic material parameters were identified by comparing the measured and predicted load–displacement curves as well as pile-up profiles. It is found that both measured and simulated plastic pile-up patterns on the indented surfaces exhibit significant anisotropy and orientation dependence, although the measured and simulated load–displacement curves manifest no such orientation dependence. The height and extension of pile-ups differ strongly as a function of surface orientation. The FEM simulations are able to reproduce the observed features of spherical indentation both qualitatively and quantitatively.

Appendix

Method of least squares used for numerical fitting

In this work the reverse fitting scheme is formulated as a minimization problem with respect to the input parameters, where the objective function quantifies the total discrepancy between the experimental results and the corresponding numerical data in load–displacement curves and the selected surface pile-up profiles of imprints.

The objective function is formulated as

$$ \omega (\vec{p}) = \frac{{\int_{0}^{{h_{\hbox{max} } }} {\left| {L_{\exp } - L_{\text{simu}} } \right|{\text{d}}h} }}{{\int_{0}^{{h_{\hbox{max} } }} {L_{\exp } {\text{d}}h} }} + \frac{{\int_{0}^{{\chi_{\hbox{max} } }} {\left| {U_{\exp 1} - U_{{{\text{simu}}1}} } \right|{\text{d}}\chi } }}{{\int_{0}^{{\chi_{\hbox{max} } }} {\left| {U_{\exp 1} } \right|{\text{d}}\chi } }} + \frac{{\int_{0}^{{\psi_{\hbox{max} } }} {\left| {U_{\exp 2} - U_{\text{simu2}} } \right|{\text{d}}\psi } }}{{\int_{0}^{{\psi_{\hbox{max} } }} {\left| {U_{\exp 2} } \right|{\text{d}}\psi } }}\;, $$

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where $\vec{p}$ represents the vector collecting the input parameters to identify; L _exp and $L_{\text{simu}}$ denote the experimental and numerical load–displacement curves, respectively; U _exp1, $U_{{{\text{simu}}1}}$, U _exp2, and $U_{{{\text{simu}}2}}$ are section profiles from experiment and simulation and the subscripts 1 and 2 indicate different paths, i.e., crystallographic direction [001] and [011] in plane (100), along which the cross-section are taken; χ _max and ψ _max are distances along the selected paths. Obviously, the dependence of the calculated quantities $L_{\text{simu}}$ and $U_{\text{simu}}$ on the input parameters in vector $\vec{p}$ is implicit through the FEM modeling. This indicates the objective function ω is non-explicit and generally non-convex and non-smooth function of $\vec{p}$.

The minimization of the above objective function $\omega (\vec{p})$ starts from a suitably chosen set of initial input parameters, usually those expected on the basis of available literature data or previous experience.