Finite compressibility and strain hardening in elasto-plastic models of amorphous matter
We study a mesoscopic elasto-plastic model of amorphous matter with varying dimensionless compression modulus, $K/\mu$, where $K$ and $\mu$ are the compression and shear moduli. We study both cyclic shear with amplitude $\Gamma$ and forward steady shear. In cyclic shear, the terminal behavior is, in order of increasing $\Gamma$: 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 $\Gamma_0$ is determined by the Eshelby back stress, $\sigma_0$, which depends on the Poisson ratio. Systems with smaller $K/\mu$ (more compressible) are effectively harder with a higher $\Gamma_0$ and a correspondingly larger purely elastic regime in cyclic loading. In forward shear, $\sigma_0$ plays a similar role where lower $K/\mu$ results in a higher steady state flow stress, $\sigma_y$. We show that increasing $K/\mu$ 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 $\Gamma_0$ 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 $K/\mu$ can vary with composition, age, quench procedure, or mechanical processing history.