Delayed elasticity of metallic glasses: Loading time and temperature dependences of the anelastic relaxation

Mehran Nabahat, Narges Amini, Eloi Pineda, Fan Yang, Jichao Qiao, Beatrice Ruta, and Daniel Crespo

Phys. Rev. Materials 6, 125601 – Published 9 December 2022

Abstract

One of the hallmarks of disordered matter is the large amplitude of the anelastic deformation, i.e., the fraction of reversible deformation that is not instantaneously recovered after the release of load but is delayed in time. In this paper, this delayed elasticity is studied for the glass-forming ${Zr}_{46.25} {Ti}_{8.25} {Cu}_{7.5} {Ni}_{10} {Be}_{27.5}$ alloy by means of stress step and recovery experiments. Even at high temperatures, not far from the glass transition, the delayed elasticity can recover an important fraction of the deformation and endure for a long time. Analyzing the effects of loading time and waiting time on the strain evolution, we reveal the presence of an anelastic response with a timescale dependent on loading time and an invariant shape, which indicates the presence of a distribution of reversible relaxation modes following a $τ^{- n}$ law with exponent $n$ between 0.5 and 1. The underlying distribution of energy barriers activated at different temperatures is accordingly shape invariant. Moreover, we found that a distribution of reversible modes corresponding to the high-frequency side of the $α − relaxation$ peak can reproduce the experimental results. The results establish a direct link between the dynamical spectrum and the distribution of activation energies, revealing the origin of the transient creep and anelastic recovery behaviors of metallic glasses.

Received 22 June 2022
Revised 7 October 2022
Accepted 22 November 2022

DOI:https://doi.org/10.1103/PhysRevMaterials.6.125601

Physics Subject Headings (PhySH)

Anelastic deformation Creep Mechanical deformation

Metallic glasses Supercooled liquid

Materials modeling Tension tests

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Mehran Nabahat ¹, Narges Amini ^2,3, Eloi Pineda ^1,*, Fan Yang ³, Jichao Qiao ⁴, Beatrice Ruta ⁵, and Daniel Crespo ¹

¹Department of Physics, Institute of Energy Technologies, Universitat Politècnica de Catalunya—BarcelonaTech, 08019 Barcelona, Spain
²Department of Chemistry, Aarhus University, 8000 Aarhus, Denmark
³Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany
⁴School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072, China
⁵Institut Lumière Matière, Université Claude Bernard—Lyon 1, CNRS, F-69622 Lyon, France

^*eloi.pineda@upc.edu

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Vol. 6, Iss. 12 — December 2022

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Images

Figure 1
Evolution of strain during two consecutive creep-recovery cycles at 580 K. Each creep and recovery segment lasted 600 min, the whole duration of a cycle being 1200 min. To better compare the two curves, the strain and time axes of the second cycle are shifted by setting $ɛ = t = 0$ at the end of the first cycle.
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Figure 2
(a) Creep-retardation $ϕ_{ac} (t) = ɛ_{ac} (t) / ɛ_{ac} (0)$ and (b) recovery-relaxation $ϕ_{ar} (t) = ɛ_{ar} (t) / ɛ_{ar} (0)$ functions obtained at 580 K for the first cycle of creep-recovery tests with different applied stresses. (c) Amplitude of the recovery relaxation and (d) recovery-relaxation functions at 520, 540, 560 and 580 K. All tests were performed with a creep duration of $t^{*} = 600$ min. The dashed line in (c) is the prediction of change of anelastic amplitude calculated by means of Eq. (3).
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Figure 3
(a) Time to reach half recovery after creep segments of duration $t^{*}$ . (b) Recovery-relaxation functions after creep segments of duration $t^{*}$ . (c) Apparent extensional viscosity $η = σ / \dot{ɛ}$ calculated at different times during creep segments initiated at two different waiting times. (d) Creep-retardation and recovery-relaxation functions for two consecutive creep-recovery cycles and a cycle initiated after a waiting time of 1200 min. All tests in this figure were performed at 580 K.
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Figure 4
Symbols: Master curve of the recovery functions obtained in this paper for different values of stress ( $σ = 20 - 400$ MPa), temperature ( $T = 520 - 580$ K), loading time ( $t^{*} = 200 - 800$ min at 580 K) and aging time (3.2–30 h at 580 K). A total number of 30 recovery functions corresponding to different conditions are plotted. Black dashed line: Kohlrausch-Williams-Watts (KWW) function with stretching parameter $β = 0.45$ . Red dashed line: Recovery-relaxation function calculated by Eq. (2) considering $D (τ) \propto τ^{- 0.64}$ and $t^{*} = 1$ .
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Figure 5
(a) Master curve of the loss modulus of Vit4 glass as a function of $ω τ_{α}$ , where $τ_{α}$ is the α relaxation time. See Ref. [3] for details. (b) Distribution of relaxation modes (blue dashed line) considering a distribution of reversible modes $D (τ) \sim τ^{- 0.64}$ (dashed line). The excited modes after loading during $t^{*} = 200$ , 400, 600, and 800 min are shown by the blue, yellow, green, and red areas, respectively. (c) Recovery functions calculated by Eq. (2) with $D (τ) \sim τ^{- 0.64}$ . The legend also shows the calculated times needed to reach half recovery after different excitation times $t^{*}$ , which are in good agreement with the experimental results in Fig. 3.
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Figure 6
(a) Examples of Vit4 stress relaxation curves obtained at different temperatures. (b) Vit4 stress relaxation times at the supercooled liquid (SCL) state (light blue circles) and at the glass state (orange, red, purple, and blue circles). In the glass state, two relaxation times are displayed for each temperature corresponding to the as-quenched state and the most aged sample. The arrows indicate the increase of relaxation time as a function of aging in the glass state. The blueish area in the time-temperature map corresponds to the timescales expected for the viscous relaxation, which depend on the aging state. The higher border of the blueish area is the extrapolated equilibrium Vogel-Fulcher-Tammann (VFT) line $τ_{sr} = τ_{0} \exp (\frac{D T_{0}}{T - T_{0}})$ with $D = 22.7$ and $T_{0} = 372$ K, and the lower border is an Arrhenius line with fictive temperature $T_{f} = 600 K$ and activation energy $E = 270 kJ mo l^{- 1}$ .
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