Finite compressibility and strain hardening in elasto-plastic models of amorphous matter

Abstract

We study a mesoscopic elasto-plastic model of amorphous matter with varying dimensionless compression modulus, $K/μ$, where $K$ and $μ$ are the compression and shear moduli. We study both cyclic shear with amplitude $Γ$ and forward steady shear. In cyclic shear, the terminal behavior is, in order of increasing $Γ$: i) trivially elastic, ii) hysteretic but with microscopically reversible limit cycles, iii) diffusive with no return to previously visited configurations. We show that the transition between i) and ii) at the onset point $Γ_0$ is determined by the Eshelby back stress, $σ_0$, which depends on the Poisson ratio. Systems with smaller $K/μ$ (more compressible) are effectively harder with a higher $Γ_0$ and a correspondingly larger purely elastic regime in cyclic loading. In forward shear, $σ_0$ plays a similar role where lower $K/μ$ results in a higher steady state flow stress, $σ_y$. We show that increasing $K/μ$ increases the amplitude of stress redistribution after a local yielding event without changing the net stress relaxation and relate this to the assumptions in mean-field descriptions of amorphous solids. A striking feature of the model is the emergence of a complex hardening behavior in the absence of any ad-hoc hardening parameters: a transition between a kinematic and an isotropic hardening behavior precisely at $Γ_0$ associated with the hysteresis transition. The enhanced plastic response for incompressible systems is also seen in amorphous alloys where it is usually attributed to excess free volume, while in the present model, it arises from the dependence of the Eshelby backstress on the Poisson ratio. Our results should have important implications for amorphous metallic alloys or other glassy systems where $K/μ$ can vary with composition, age, quench procedure, or mechanical processing history.

Figures (14)

• Figure 1: The real space $G_{11}$ (left) and $G_{22}$ (right) near the origin for a system of size $L=128$ and $\lambda/\mu=4$. Other than at the origin, $G_{11}=-G_{22}$ apart from a correction term which vanishes as $1/L^2$.
• Figure 2: The constrained stress $\sigma_c$ in mode-2 vs. $\lambda/\mu$. The squares are the discretized solution for a system size $L=128$. The solid line is the continuum solution (multiplied by $4A[0,0]=1.2732$) for a circular inclusion in an infinite medium.
• Figure 3: A schematic of a single elementary shear transformation in the automaton at a site $[0,0]$, and the resulting changes in local strain and energy in a few nearby sites on the grid after re-equilibration. The parabolas show the local energy, $U$, of the sites as a function of the total strain, $\gamma$. The center of the $n$-th parabola is at $\gamma=2n$, and the transition between basin $n$ and basin $n\pm 1$ occurs at $\gamma=2n\pm 1$. The stress, $\sigma$, as a function of total strain, $\gamma$, is then $\sigma=\gamma-2F[\frac{\gamma}{2}+\frac{1}{2}]$ where $F$ is the floor function defined in the text. The four different identical landscapes correspond to the four different tiles shown. The red and green dots indicate local strain values before and after a shear transformation at the site labeled $[0,0]$. After the transformation, sites $[1,0]$ and $[0,1]$ are at a higher energy and stress, site $[1,1]$ is at a lower energy and stress, while the transforming site, site $[0,0]$, is at a much lower -- but non-zero -- energy and at a negative stress.
• Figure 4: Top row: plastic strain $\epsilon$ (ensemble average) in the limit cycles vs. strain $\gamma$ for several different strain amplitudes $\Gamma$ and four different values of Lame parameter $\lambda$. Middle row: the strain rate $|\frac{d\epsilon}{d\gamma}|$ vs $\gamma$ for the curves in the top row. Bottom row: the strain rate $|\frac{d\epsilon}{d\gamma}|$ vs $\gamma-\gamma_*$ for the curves in the top row where $\gamma_*$ is the plasticity onset strain. The solid line in the bottom row is at $|\frac{d\epsilon}{d\gamma}|=0.2$
• Figure 5: Top row: The (negative) onset strain, $\pm\gamma_*^{+(-)}$, in the forward (and reverse) direction for different values of the first Lame parameter $\lambda$. The solid lines are $\gamma_*=\Gamma_0-(\Gamma-\Gamma_0)$ with $\Gamma_0$ determined by taking the average $(\Gamma+\gamma_*)/2$ for a given $\lambda$. Middle row: The (negative) stress at onset, $\pm\sigma_*^{+(-)}$, in the forward (and reverse) direction. The dashed line is a guide to eye with a slope = -1/2. Bottom row: The (negative) plateau strain, $\pm\epsilon_p^{+(-)}$, in the forward and (reverse) direction. The dashed line is a guide to eye with a slope = 1/2. The vertical lines are at $\Gamma_0$ for the different $\lambda$ values. In all cases, the crosses and pluses are for the forward and reverse directions, respectively. The dots in b) and c) mark the averages.