Length-scale dependence of elastic strain from scattering measurements in metallic glasses

Several recent studies have reported that the elastic strain in metallic glasses, as measured from peak shifts in the pair-correlation functions of samples under load, increases with distance from an average atom, approaching the macroscopic strain at large distances. We have verified this behavior using high-energy x-ray scattering on metallic glasses loaded under uniaxial compression, uniaxial tension, and pure shear, and show that the apparent length-scale dependence of elastic strain is not an artifact of the assumption of structural isotropy in the data analysis. Molecular dynamics simulations of a binary Lennard-Jones glass loaded in uniaxial tension reproduce, qualitatively, the behavior observed in the experiments when the elastic strain is calculated from the shifts in the peaks of the pair-correlation function. Under hydrostatic loading, however, the length-scale dependence of elastic strain observed in the simulations is greatly reduced. This suggests that nonaffine atomic displacements, which are smaller under hydrostatic loading than under uniaxial loading, may play a key role in the length-scale dependence of elastic strain. Furthermore, no length-scale dependence is observed in simulations, for either uniaxial or hydrostatic loading, when the elastic strain is calculated from the average local deformation gradient tensor. We explain this apparent contradiction and show that the atomic displacements resulting from elastic loading are largest in the low-density regions between atomic shells around an average atom. Finally, we present an analysis of length-scale dependence of elastic strain calculated from the pair-correlation function for the case of homogeneous deformation, which is in good agreement with the simulations conducted under hydrostatic loading. For uniaxial loading, however, the analysis diverges from both the experimental and simulated results in the first two near-neighbor atomic shells. This suggests, in agreement with our observations from the molecular dynamics simulations, that the observed length-scale dependence of elastic strain from scattering measurements reflects the nature of the nonaffine atomic displacements in the glass.

Figure 1

Experimental setup for the in situ x-ray scattering experiments, using the case of uniaxial tension as an example. $q_{\parallel }$ and $q_{\perp }$ are the scattering vectors approximately parallel and perpendicular to the loading direction, respectively. The azimuthal angle $\psi$ is also shown; note that $q_{\parallel }$ corresponds to $\psi =0^{\circ }$ and $q_{\perp }$ to $\psi ={90}^{\circ }$ on the image plate.

Figure 2

Elastic strain from $g(r,\psi )$ as a function of load and distance from average atom ($r$) for Zr${}_{57}$Ti${}_5$Ni${}_8$Al${}_{10}$Cu${}_{20}$ specimens loaded in (a) uniaxial compression, (b) uniaxial tension, and (c) pure shear. Solid lines represent strain at a given stress during loading, while dashed lines are strain during unloading. (d) Tensile strain at a single load [870 MPa, from part (b)], along with the macroscopic strain recorded by an extensometer attached to the sample (dashed line). With increasing $r$, the elastic strain measured in the x-ray experiment asymptotically approaches the macroscopic strain.

Figure 3

Spherical coordinate convention used in this paper. The loading direction is along the $z$ axis.

Figure 4

First two harmonics of the structure factor, $S_0\left(q\right)$ and $S_2\left(q\right)$, when the sample is (a) unloaded and (b) under 1080 MPa compressive load.

Figure 5

(a) Pair-distribution function harmonics ${\rho }_n\left(r\right)$ obtained from $S_n\left(q\right)$ using Eq. (11). (b) The second harmonic, ${\rho }_2\left(r\right)$, obtained by transformation of $S_2\left(q\right)$ using the spherical Bessel transform, given by Eq. (11), and the sine Fourier transform, given by Eq. (2).

Figure 6

Elastic strain in the loading direction measured from the pair-correlation functions obtained using the isotropic transformation, given by Eq. (2), and the anisotropic transformation, given by Eq. (11). Data are for Zr${}_{57}$Ti${}_5$Ni${}_8$Al${}_{10}$Cu${}_{20}$ loaded in uniaxial compression.

Figure 7

Elastic strain in the model glass subject to uniaxial tension, at $T=0.036$, calculated using three different techniques: shifts in pair-correlation function $g(r,\phi )$ peaks; the deformation gradient tensor, averaging local strains ${\langle F\rangle }_{\epsilon }$; and the deform-ation gradient tensor, averaging atomic positions ${\langle F\rangle }_x$. The imposed strain calculated using Eq. (13) is also shown.

Figure 8

Elastic strain in the model glass subject to uniaxial tension calculated using the ellipsoid-fit method, along with the strain calculated from peak shifts in the pair-correlation function [$g(r,\phi )$], the deformation gradient tensor (${\langle F\rangle }_{\epsilon }$), and the imposed strain.

Figure 9

Elastic strain in the model glass subject to hydrostatic tension calculated using the ellipsoid-fit method, along with the strain calculated from peak shifts in the pair-correlation function [$g(r,\phi )$] and the imposed strain. Note the difference in scale compared to Fig. 8.

Figure 10

Nonaffine displacements in the simulations of elastic deformation. Slices through atomic configurations subjected to (a) uniaxial compression and (b) hydrostatic tension show the nonaffine displacements projected onto the $x$-$y$ plane (perpendicular to the loading axis, $z$). The lengths of the arrows indicating the nonaffine displacements have been multiplied by a factor of 20. In (c) and (d), we show the distributions of the magnitudes and orientations, respectively, of the nonaffine displacements. In (d), the histogram is normalized by a geometrical factor of ${\left(\sin \phi \right)}^{-1}$.

Figure 11

Left-hand and right-hand sides of Eq. (18) obtained from (a) experimental and (b) simulation data in uniaxial tension. Solid curves depict the difference between pair-correlation functions in deformed [$g^{\mathrm{def}}(r,\phi )$] and reference [$g^{\mathrm{ref}}\left(r\right)$] states, i.e., the left-hand term of Eq. (18), of (a) Zr${}_{57}$Ti${}_5$Ni${}_8$Al${}_{10}$Cu${}_{20}$ specimen and (b) model glass. In both cases, deformation is due to a tensile load applied along the $z$ axis; thus, the polar angle $\phi =0^{\circ }$ corresponds to the loading direction. Pair-correlation functions were calculated by averaging over an angular range of $\psi$, or equivalently, $\phi ={2.5}^{\circ }$ in the case of the in situ experiments and $\phi ={18}^{\circ }$ in the case of the simulated system. Dashed curves depict the expected $g^{\mathrm{def}}(r,\phi )-g^{\mathrm{ref}}\left(r\right)$ given by the right-hand side of Eq. (18).

Figure 12

Left-hand and right-hand terms in Eq. (19) are shown here, as solid and dashed curves, respectively, for the simulated system subject to hydrostatic tension. Note that $\epsilon$ here represents $\Delta /3$, where $\Delta$ is the dilatation induced by the imposed stress. Also note that the pair-correlation function in the deformed configuration is isotropic, hence terms involving $\theta$ or $\phi$ do not appear, unlike the case of uniaxial deformation (Fig. 11).

Figure 13

Elastic strain in the loading direction in the model glass subject to uniaxial tension calculated using the ellipsoid-fit method (bottom) along with the pair-correlation function $g\left(r\right)$ (top).