Elastoplastic Modelling of Cyclic Shear Deformation of Amorphous Solids

TL;DR

The paper develops an energy landscape–based elastoplastic model for amorphous solids under both uniform and cyclic shear, representing the material as a lattice of mesoblocks each carrying a mesostate with energy $E_0$, stability range $\gamma$, and a DOS $\Omega(E_0)$. Elastic interactions are treated nonlocally via the finite element method, allowing realistic Eshelby-like stress redistribution, while plastic events move the system between mesostates according to a prescribed protocol for updating $E_0$ and $\gamma_0$. The work reproduces known phenomena such as brittle-to-ductile crossover under annealing and the Bauschinger effect, and qualitatively captures the cyclic-yielding diagram, threshold energies, and failure-time divergences, while revealing novel features like trenching and intermediate regimes near yield whose presence depends on model choices (notably the DOS mean and plastic increment rule). The results emphasize the sensitivity of cyclic-deformation physics to the mesoscale energy landscape and the importance of nonlocal elasticity, offering a framework to study annealing, fatigue, and memory effects in amorphous solids and guiding future three-dimensional and parameter-calibrated extensions. Overall, the approach provides a quantitatively interpretable link between microscopic landscape properties and macroscopic yielding and fatigue phenomena in glasses.

Abstract

We develop an energy-landscape based elasto-plastic model to understand the behaviour of amorphous solids under uniform and cyclic shear. Amorphous solids are modeled as being composed of mesoscopic sub-volumes, each of which may occupy states - termed mesostates -- drawn from a specified distribution. The energies of the mesostates under stress free conditions determine their stability range with respect to applied strain, and their plastic strain, at which they are stress free, forms an important additional property. Under applied global strain, mesostates that reach their stability limits transition to other permissible mesostates. Barring such transitions, which encompass plastic deformations that the solid may undergo, mesostates are treated as exhibiting linear elastic behavior, and the interactions between mesoscopic blocks are treated using the finite element method. The model reproduces known phenomena under uniform and cyclic shear, such as the brittle-to-ductile crossover with annealing and the Bauschinger effect for uniform shear, qualitative features of the yielding diagram under cyclic shear including the change in yielding behaviour with the degree of annealing, across a `threshold level', and dynamic phenomena such as the divergence of failure times on approach to the yield point and the non-monotonic evolution of the local yield rate. In addition to these results, we discuss the dependence of the observed behaviour on model choices, and open questions highlighted by our work.

Figures (12)

• Figure 1: (a) A single mesostate with energy $E(\gamma) = E_0 + \frac{\mu}{2}(\gamma - \gamma_0)^2$ is stable over a finite range in strain, $\gamma \in (\gamma_0 - \sqrt{-E_0},\gamma_0 + \sqrt{-E_0})$. (b) Three choices of Gaussian density of states are shown, with the standard deviation fixed at $0.1$. (c,d,e) Plastic strain increment rules. For the results in the main text the model choices are: $\mu_{DOS}=-0.5$ and maximal plastic increment.
• Figure 2: (a) The stress $\sigma$ is plotted for strain-controlled uniform deformation for various annealing levels as indicated by the parent temperature $T_p$. (Inset) Energy $E_0$ evolution is plotted. (b) Distribution of local $E_0$ in the steady state is plotted with blue points and dashed orange line denotes a Gaussian distribution with mean $-0.525$ and standard deviation $0.098$. The density of states is plotted for comparison. (c,d) The change in the plastic strain field between configurations at $\gamma = 2.0$ and $\gamma = 4.0$ is plotted for (c) poorly annealed ($T_p = 3.16$) and (d) well-annealed sample ($T_p = 0.04$. The plastic activity is diffuse for PA samples while it is strongly localized inside a band for WA samples. (inset) Row-averaged $E_0$ profile is plotted with the innermost blue curve at $\gamma = 4.0$ till the outermost red curve at $\gamma = 20.0$ in steps of $\gamma = 4.0$. (e) Bauschinger Effect: The forward and reverse strain response is plotted for a fresh sample (dashed and dotted lines respectively) and a pre-sheared sample (solid lines). Significant anisotropy in forward and reverse response can be observed for the pre-sheared sample. The inset shows the initial loading-unloading curves. (f) The distribution of local distances to strain thresholds is plotted in either direction for the pre-sheared and fresh sample. System size is $L=512$.
• Figure 3: (a) Per site elastic energy ($E$) evolution is plotted for (a) poorly annealed sample ($T_p = 3.16$) and (b) well annealed sample ($T_p = 0.06$) for a range of $\gamma_{max}$ values below and above the yield point. Color-maps of $E_0$ field for configurations taken at points numbered $1-4$ are shown to the right of the respective panel.
• Figure 4: (a) Steady state energies are plotted at various shear amplitudes for various degrees of annealing (b) Evolution of number of plastic events per site per cycle - the yield rate - is plotted for a poorly annealed sample ($T_p = 3.16$) above and at the yield point $\gamma_{max}^{yield} = 0.43$. (c) Number of cycles to reach an absorbing state ($\tau_{abs}$) and to fail ($\tau_f$) are plotted for a poorly annealed sample. Dashed lines are power-law fits with exponents $-3.2$ ($\tau_{abs}$, green) and $-1.02$ ($\tau_f$, orange); vertical line denotes the yield amplitude in the main panel and the inset. (Inset) $\tau_f$ is plotted for a well-annealed sample. Dashed line shows a power-law fit with exponent $-2.2$. (d) Energy profiles averaged along the direction of the shear band are shown (centered by hand at zero, only the right half is plotted) for a poorly annealed sample ($T_p = 3.16$). Dashed lines are fits to a flat top Gaussian profile $c_1 + c_2 e^{-(x/c_3)^6}$. (Inset) The shear band width is plotted as function of the driving amplitude. Dotted red line is a power law fit $w/L = w_0/L + A(\gamma_{max} - \gamma_{max}^{yield})^{0.73}$. Blue dashed line denotes the fit, $w/L = A(\gamma_{max} - \gamma_y)^{1/2}$, where $\gamma_y$ is a free parameter, with the fit value being $\gamma_y = 0.373$.
• Figure S1: (Left) The shear strain field in response to a single mesoblock in the centre having a non-zero stress-free plastic strain $\gamma_0^{xy} = \gamma_0^{yx} = 1.0; \gamma_0^{xx,yy} = 0.0$. (Right) The analytically derived kernel from Liu et al. and the FEM kernel show a $r^{-2}$ decay in the long field. The source value for the analytically derived kernel is $0.52$ and for the FEM derived kernel it is $0.47$.