Yielding behaviour of glasses under shear deformation at constant pressure

TL;DR

The paper addresses how yielding in glasses under shear is affected when external pressure is held constant rather than global volume, focusing on dilatancy and shear-band formation. It employs athermal quasistatic shear simulations of the Kob–Andersen binary Lennard-Jones glass at multiple pressures, examining both uniform and cyclic shear for well- and poorly annealed histories. The main finding is that qualitative yielding behavior remains similar to constant-volume results, with the peak stress scaling linearly with external pressure and stable, pressure-dependent-width shear bands forming under cyclic loading while maintaining lower density inside the bands. Stress–strain curves collapse across pressures when scaled by the peak stress, and the shear-band width grows with cycle number following a sub-cubic law before saturating; these results underscore the role of dilatancy and volumetric effects in yielding and suggest directions for future work on volumetric plasticity and composition segregation, including material-specific cases like silica.

Abstract

Computer simulations of yielding of glasses under shear have typically been performed under constant volume, strain controlled protocols. However, volumetric effects, such as the dilatancy associated with plastic rearrangements, and the observed reduction of density in shear bands, make it interesting to consider constant pressure shear protocols. We present a computational investigation on the nature of yielding of glasses under constant-pressure conditions, for different pressures. For uniform shear, the stress-strain curves at different pressures differ only by the stress scale. We find stable shear bands under cyclic shear whose steady-state width increases with an increase in external pressure, with density within shear bands being lower compared to the average values reached. Cyclically sheared well annealed glasses yield with a discontinuous dilation at the yield point, whereas the poorly annealed glasses undergo compaction before yielding accompanied by dilation. The external pressure influences the quantitative mechanical response of the glasses, but the qualitative behaviour is similar at different pressures, and remains the same as that of yielding at the constant-volume strain-controlled conditions. We discuss directions along with further investigations may be pursued, based on the results presented.

Figures (4)

• Figure 1: Variation of pressure for constant volume shear: a) Pressure decreases for a poorly annealed sample till the yield shear amplitude is reached. For a well annealed sample, it varies negligibly till the system yields at a larger strain amplitude, beyond which the pressures are the same as for the poorly annealed glass. b) Pressure is linear with energy but increases with a larger slope beyond the yielding transition.
• Figure 2: Density, Energy and Stress across the yielding transition ($N = 4000$, $e_{IS} = -7.05$): Uniform shear- a) Evolution of number density with shear strain for a well-annealed glass at different external pressures under uniform shear. The density decreases both below and above the yield strain ($\gamma^Y \approx 0.1$). b) Energy density increases with strain, with the strain dependence changing at $\gamma^Y$. c) Stress $\sigma$ as a function of strain. The peak stress ($\sigma_P$) before failure increases linearly with an increase in external pressure (shown in bottom left inset). All the stress-strain curves collapse on top of each other when stress values are rescaled with its peak value ($\sigma / \sigma_P$, top right inset). Cyclic shear- d) Evolution of number density with shear cycles at stroboscopic (at $\gamma=0$ after each cycle) configurations for strain amplitude $\gamma_{max} = 1.2$ for which failure happens in the first cycle. The system undergoes dilation and eventually reaches a steady state. The steady state number density ($\rho_{ss}$) increases linearly with increase in pressure (shown in the inset). The solid lines are fits to the data points used to obtain steady state values. e) The energy density $U/V$ of the system increases with cycles, and decreases with an increase in external pressure. The steady state value is linearly dependent on external pressure (as shown in the inset). f) The steady state value of stress at $\gamma = \gamma_{max}$ denoted by $\sigma_{max}$, increases with shear amplitude up to the yield point($\gamma_{max}^Y=0.10$). A sharp drop is observed for shear amplitudes greater than $\gamma_{max}^Y$ at all values of external pressures considered. The solid lines are guides to the eye. The peak value of $\sigma_{max}$ increases linearly with external pressure (bottom left inset), and the $\sigma_{max}/\sigma_P$ are independent of constant volume or constant pressure cyclic shear conditions (top right inset).
• Figure 3: Yielding diagrams at constant pressure: The steady state values of number density $\rho$ and energy density $U/V$ are shown for poorly annealed ($e_{IS}=-6.92$, marked by open squares) and well annealed ($e_{IS}=-7.05$, marked by open circles) glasses. a) Number density: The poorly annealed glass undergoes compaction for small shear amplitudes followed by dilation after the yield point for all the values of pressure considered. The results for P=2 are shown in black, for P=$10^{-3}$ in red, and for P=$-2$ in green colors. The magenta and blue solid lines are the yield strain amplitudes for glasses with $e_{IS}=-6.92$ and $e_{IS}=-7.05$, respectively. b) Energy density ($U/V$) vs. shear amplitude at different external pressures. The monotonic change in density dominates in the product of $\rho \times U/N = U/V$, resulting in a monotonic variation of $U/V$ with pressure. The energy density reflects the changes with annealing and strain amplitude seen in the number density and energy per particle. c) Energy per particle ($U/N$): The poorly annealed glass undergoes annealing before the yield strain amplitude, followed by a discontinuous jump and subsequent increase with $\gamma_{max}$. The well annealed glass shows negligible annealing before yielding, and a larger discontinuous change at yielding. $U/N$ shows non-monotonic variation with pressure.
• Figure 4: Stability of shear bands at constant pressure (N=64000, $e_{IS}=-7.05$ and $\gamma_{max}=0.10$): a) Slab-wise cycle-to-cycle mean squared displacement profile ($MSD_{Slab}$) plotted as a function of the coordinate perpendicular to the shear band plane. We consider the glass to have yielded when the peak of $MSD_{Slab}$ is greater than $0.8$ (as a convenient but arbitrary choice), and denote the corresponding cycle by $n_f$. The $n_f$ value for the data shown is $4$. The shear band width $\sigma$ is obtained from a Gaussian fit to $MSD_{Slab}$, and dividing by the box length $L$ gives the fraction of the box length occupied by the shear band($\sigma / L$). b) Growth of the shear band with shear cycles after the formation of the shear band. The exponent of the power law growth is $\approx 0.2$, increasing mildly with pressure, and not close to $1/3$ as proposed in LiuJCP2022. For reference, power laws with exponent $1/5$ and $1/3$ are shown with magenta and blue dashed lines respectively. c) The shear band width increases and approaches a steady state value, being larger for larger pressure, as shown in the inset. The solid lines are fits to the binned data points used to obtain the steady state value. d) Mean squared displacement between subsequent stroboscopic configurations after the formation of the shear band ($MSD_{System}$) showing a high value at the formation of shear band, ultimately reaching a slightly lower steady state value. The steady state $MSD_{System}$ is higher for higher pressure. e) Variation of the number density with shear cycles inside (shown by open circle) and outside(shown by open squares) the shear band, and the system average value(shown by solid lines) at $P = 2$, $10^{-3}$ and $-2$ shown in black, red and green colors. The number density is lower inside the shear band compared to the average and the values outside. f) Steady state value of number density inside and outside of shear band as well as of the total number density of the system increases with pressure.