# Signature of local stress states in the deformation behavior of metallic glasses

Sep 11, 2020

### Tensile properties

Typical tensile stress–strain curves of Zr64.13Cu15.75Ni10.12Al10 MG at different temperatures are shown in Fig. 1a. Most monolithic MGs tensile tested at room temperature are prone to catastrophic fracture with zero plastic yielding after elastic deformation1,2. However, here, discernible strain to failure can be detected with decreasing temperature. Note that strain softening takes place once the maximum strength is reached, implying that inhomogeneous flow is the dominant method at cryogenic temperature32. To better visualize this behavior, Fig. 1b shows a magnified view of the stress–strain curves indicated by the dashed rectangle in Fig. 1a, which clearly indicates that both the tensile strength and sustained strain to failure increase with decreasing temperature, as shown in the inset of Fig. 1b. The method for determining the yield strength and strain to failure is given in Supplementary Fig. 1. The tensile strength shows a linear relationship with temperature, and it increases by ~350 MPa when the temperature decreases from 293 to 77 K. The strain to failure also increases with decreasing temperature, although it is improved marginally. Thus, it is obvious that decreasing temperature can yield higher strength and larger strain to failure in monolithic MGs.

### Structure evolution with decreasing temperature

Usually, enhanced strength and ductility are attributed to structural changes33,34. However, the structural evolution due to changes in the environmental temperature is very insignificant. Here, high-energy synchrotron XRD investigations provide insights into the structural changes of MG with temperature. The structure factors in reciprocal space, S(q), at different temperatures are plotted in Supplementary Fig. 2. The first maximum of S(q) reflects the structural evolution in the MRO. The peak position of the first maximum shifts to a higher q value with decreasing temperature, indicating a decrease in the mean atomic bond length. This corresponds to a volume contraction, which can influence the mechanical properties of the MG35. For example, the elastic modulus, E, of the MG monotonically increases with decreasing temperature (Supplementary Fig. 3), indicating continuous stiffening of the MG with decreasing temperature35. In addition, cooling from 293 to 79 K induces thermal stress, which is evaluated to be ~10% of the macroscopic yield stress3. However, the evaluated thermal stress is much lower. The increase in the elastic modulus at 79 K is only 3% (Supplementary Fig. 3), which is not sufficient to explain the ~23% increase in yield strength, implying that the yield strength increase not only depends on the affine contraction of the MRO (macroscopic contraction) underlying the cooling process but may also be linked to the changes in the SRO (degree of heterogeneity).

For the Zr-based MG, the first maximum in the reduced PDF, G(r), is a superposition of ten pair correlations (see Supplementary Table 1). Each partial pair correlation changes with temperature under the anharmonic and asymmetric nature of the interatomic interaction potential, leading to skewness of the non-Gaussian profile in the first maximum of G(r)36,37,38, as shown in Fig. 2a. Therefore, caution should be taken in defining the position of the maximum for such an asymmetric and broad maximum. However, after some reasonable approximations, only three effective atomic pairs, i.e., Zr–Zr, Zr–Cu, and Zr–Ni, are considered39,40, as dominant contributions to G(r) (see Fig. 2a). To explore the structural evolution of the SRO (the first coordination shell) and MRO (beyond the first coordination shell) in real space, the peak center-of-mass, rcom, of each coordination shell in G(r) is determined by:

$$r_{com} = frac{{{int}_{r_{min }}^{r_{max }} {rG(r)} }}{{{int}_{r_{min }}^{r_{max }} {G(r)} }},$$

(1)

where rmin and rmax are the intersection coordinates of G(r) with the line G(r) = 0. Based on the method developed by Poulsen et al.41 and further outlined by Shahabi et al.40, we can estimate the mean strain, (varepsilon _T^i), from the variations in the G(r) peak position for the different coordination shells according to:

$$varepsilon _T^i = frac{{r_T^i – r_{293mathrm{K}}^i}}{{r_{293mathrm{K}}^i}},$$

(2)

where (r_{293,mathrm{K}}^i) and (r_T^i) represent the center-of-mass of the i-th shell in G(r) at 293 K and T, respectively. Similarly, the mean strain can also be calculated by using the peak position of S(q) in reciprocal space. From both real- and reciprocal-space analyses, the variations in the averaged strains for SRO and MRO as a function of temperature are shown in Fig. 2b, which clearly demonstrates that cooling contraction takes place for all coordination shells with decreasing temperature. However, this cooling contraction is nonuniform in different structure motifs (i.e., SRO and MRO), and the MRO contracts more, which is manifested by the wider difference in the mean strain (highlighted by arrow in Fig. 2b). The difference in the MRO on different length scales, such as the second and third coordination shells, becomes slightly stronger only at very low temperatures, indicating that nonaffine thermal strain primarily takes place in the MRO and SRO (Fig. 2b). That is, the SRO may experience cooling expansion due to local rejuvenation in addition to macroscopic cooling contraction when the temperature is down to the cryogenic level, which probably results from chemical SRO variations (changes in the Zr–Zr, Zr–(Cu, Ni) atomic pairs) in the first coordination shell during cooling34,42. Note that the averaged strains estimated in Fig. 2b are inherently related to the structural density within and between the STZs. However, we must emphasize that the actual local variations may indeed be more pronounced than the magnitudes estimated here because the diffraction patterns are recorded over the entire interaction volume11,23.

### Fracture angle and SB dynamics

As demonstrated in Fig. 2b, a decrease in temperature leads to a nonaffine thermal strain in different structure motifs, rendering enhanced structural, and elastic heterogeneities9,10,11,12. Although we cannot clearly distinguish between the local structural excitations and local elastic fluctuations, it is generally accepted that a higher degree of heterogeneity would change the characteristics of the STZ activation process and consequently affect the SB behavior, i.e., SB propagation direction and rate.

The SB propagation direction can be reflected by the shear fracture angle. The fracture angle, θ (or shear-banding angle), between the loading axis and the shear plane measured at different temperatures is shown in Fig. 3 (also see Supplementary Fig. 4). When the temperature decreases from 293 to 77 K, the θ value increases from 54 to 60°, which is in excellent agreement with the theoretical prediction that all experimental θ values should fall in the range of 30–60°43. Moreover, the θ value becomes larger with decreasing temperature, implying that the normal stress has a significant effect on the fracture angle43,44. This can be easily detected from the inset of Fig. 3, which clearly shows that the normal stress increases with decreasing temperature, while the shear stress remains almost constant. The fracture mode factor, ω, i.e., ω = −cot(2θ)43, characterizes the competition between the shear deformation and normal fracture under a uniaxial tension condition. The dependence of θ at different temperatures on ω is also plotted in Fig. 3, showing that the lower the temperature is, the larger ω, which is consistent with the above analysis.

Based on the stick-slip shear dynamics and the atomic-scale cooperative shearing model of STZs, the SB (inserting) sliding velocity, vs, can be expressed as45:

$$v_s = v_0exp left[ {frac{{4Rxi G_{0T}gamma _C^2{Omega}}}{{k_BT}}left( {1 – frac{sigma }{{sigma _0}}} right)^{3/2} + frac{1}{chi }} right],$$

(3)

where v0 is the external loading velocity, kB is the Boltzmann constant, T is the temperature, G0T is the shear modulus at different temperatures, γC is the critical yield shear strain (γC ≈ 0.027), Ω is the volume of the STZ, R ≈ 1/4, ξ~2 – 446, and σ and σ0 are the yield strengths at temperatures T and 0 K, respectively. χ is the effective temperature characterizing the state of the configuration disorder and the density or the total number of STZs47. The effective temperature is a dynamic quantity that depends on the strain rate and the temperature48 and evolves towards its kinetically ideal value, (hat chi). Here, (hat chi = chi _w/ln left( {q_c/dot gamma t_0} right))49, where qc is the normalized strain rate at which the effective temperature diverges, χw is the normalized energy barrier that determines the effective temperature rate dependence, (dot gamma) is the inelastic strain rate, and t0 is the STZ time scale. An approximate analytical solution for the temperature dependence of χ is47:

$$frac{1}{chi } = left{ {begin{array}{*{20}{c}} {frac{1}{{hat chi }}} & {T le T_C} \ {frac{1}{{hat chi }} – frac{Q}{{k_Bbeta }}left( {frac{1}{T} – frac{1}{{T_C}}} right)} & {T ge T_C} end{array}} right.,$$

(4)

where Q is the activation energy for relaxation events, β is the energy scale for the creation of relaxation events, and TC is the crossover temperature (173 K).

Substituting all the parameters into Eq. (3) allows evaluation of the SB sliding velocity, vs, as a function of inverse temperature, as shown in Fig. 4. Apparently, the sliding velocity is much slower at cryogenic temperatures than at high temperatures and increases gradually below 173 K, as indicated by the blue arrow in Fig. 4. When the temperature is higher than 173 K, the value of vs increases dramatically. It must be noted that the vs value below 173 K is comparable to that at room temperature under the compression mode50,51. Moreover, the vs value at room temperature can approach 1200 m/s, which is much faster than the speed of sound but slower than the Rayleigh wave speed, vR [vR ≈ 0.9225cs, where cs is the shear-wave speed (~2100 m/s)]35,50. The vs value dramatically drops by approximately three orders of magnitude when the temperature is decreased from room temperature to the cryogenic level. Unlike the case of compression deformation, tensile deformation is usually dominated by a primary SB, which can have catastrophic consequences, resembling the fracture of brittle MGs with a high crack speed52.

With decreasing temperature, due to the cooling contraction, the overall density increases, while at the atomic scale, the degree of heterogeneity also increases due to the nonaffine thermal contraction. A global increase in density is equivalent to larger barriers for STZ activation, and consequently, reaching the macroscopic yielding point definitely requires more energy (higher loading stress), explaining the observed higher yield stress at low temperature (Fig. 1b). Moreover, microscopically, during plastic deformation, the shear-banding process is controlled by STZ percolation. While STZ activation requires higher energies (higher stresses), the percolation processes correlate to a high degree of heterogeneity due to the nonaffine thermal contraction in the SRO and MRO. In this case, the density of potential sites for SB nucleation should be enhanced in principle, and therefore, multiple SBs should form12,33. However, accommodating the imposed strain does not necessarily require activation of multiple SBs with decreasing temperature32. This is because the activation of a single SB (Supplementary Fig. 4) can also accommodate the imposed strain via strain softening behavior32, giving rise to a higher sustained strain to failure (see Fig. 1b). In addition, large variations in the structural/elastic heterogeneities perturb the percolation process, so the activation of a neighbor STZ becomes more difficult, which, consequently, leads to a slower STZ percolation rate and thus slower SB sliding velocity, and ultimately, runaway instability is retarded53. In other words, 10% of the increase in the yield strength relates to the affine contraction of the MRO, while the rest (13%) could be related to the enhanced degree of heterogeneity, which slows percolation of the STZs and delays SB formation and propagation. Therefore, these results suggest that the increased nonaffine thermal strain in the SRO and MRO may play a paramount important role in the STZ percolation path and influence the formation, morphology, and dynamics of the SB.