Universal scaling of the stress-strain curve in amorphous solids

Jie Lin and Wen Zheng

Phys. Rev. E 96, 033002 – Published 5 September 2017

Abstract

The yielding transition of amorphous solids is a phase transition with a special type of universality. Critical exponents and scaling relations have been defined and proposed near the yield stress. We show here that, even in the initial stage of shear far below the yield stress, the stress-strain curve of amorphous solids also shows critical scaling with universal exponents. The key point is to remove the elastic part of the strain, and the shear stress exhibits a sublinear scaling with the plastic strain. We show how this critical scaling is related to the finite size effect of the minimum strain to trigger the first plastic avalanche after a quench. We point out that this sublinear scaling between the stress and the plastic strain implies the divergence of a high-order shear modulus. A scaling relation is derived between two exponents characterizing the stress-strain curve and the density distribution of the local stabilities, respectively. We test the critical scaling of the stress-strain curve using both mesoscopic and atomistic simulations and get satisfying agreement in two and three dimensions.

Received 1 May 2017
Revised 30 June 2017

DOI:https://doi.org/10.1103/PhysRevE.96.033002

Physics Subject Headings (PhySH)

Phase transitions Plastic deformation Plasticity

Amorphous materials

Polymers & Soft MatterCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Jie Lin^1,2,* and Wen Zheng^3,†

¹Department of Physics, Center for Soft Matter Research, New York University, New York 10003, USA
²School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
³Department of Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China

^*jielin@g.harvard.edu
^†wenzheng@ustc.edu.cn

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Issue

Vol. 96, Iss. 3 — September 2017

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Images

Figure 1
(a) Before the implementation of shear, one typical way to prepare the amorphous solid is to rapidly quench a fluid at high temperatures to zero temperature. (b) After the quench, one can exert quasistatic shear by the stress-control or strain-control method. The main panel shows the resulting stress- ( $σ$ -) strain ( $γ$ ) curve. The two methods coincide in the thermodynamic limit, and their difference on the microscopic scale is shown in the inset: during the plastic deformation (avalanches), the stress is conserved in the stress-control case (blue) whereas the stress relaxes in the strain-control case (red). The resulting plastic strain increment $δ ε$ and plastic stress drop $δ σ_{p}$ are shown. In the limit $γ \to 0$ (or $σ \to 0$ ), the slope of the stress-strain curve becomes the shear modulus $μ$ . (c) The stress-plastic strain ( $ε$ ) curve after subtracting the elastic strain part. In the low stress limit, the stress has a power-law scaling relation with the plastic strain. The inset shows the difference between the stress-control and the strain-control methods.
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Figure 2
The stress ( $σ$ ) vs plastic strain ( $ε$ ) or plastic stress ( $σ_{p}$ ) for the mesoscopic simulations and atomistic simulations. The black dashed line has a slope of 2/3. The stress is in arbitrary units. For the mesoscopic simulations, the shear modulus is set to be constant. The system sizes are $N = 512^{2}$ in 2D and $N = 64^{3}$ in 3D. The curve is averaged over 100 samples. The initial temperature before the infinite quench is $T = 1$ (2D) and $T = 2$ (3D). For the atomistic simulations, the system sizes are $N = 8192$ (2D) and $N = 4096$ (3D). We first calculate the shear modulus as the average slope of the stress-strain curve in the initial elastic response before the first avalanche. After subtracting the elastic strain according to Eq. (1), we obtain the stress-plastic strain curve for one sample and then average over 100 samples (atomistic). We also compute the plastic strain based on Eq. (11) (plastic strain) and the plastic stress based on Eq. (12) (plastic stress). The curves of the atomistic simulations are shifted arbitrarily on the $y$ axis for clarity. We implement stress-control (strain-control) quasistatic shear to the mescoscopic (atomistic) simulations.
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Figure 3
The distribution of local stabilities $x_{i} = σ^{th} - σ_{i}$ right after the quench $P (x)$ for the mesoscopic model. The black lines have a slope of 0.5 both for 2D and 3D. The insets: the finite size scaling of the minimum strain to trigger the first avalanche right after a quench for the atomistic simulations. The black lines have a slope of $- 2 / 3$ . Using the scaling $γ_{min} \sim N^{- 1 / (1 + θ_{0})}$ , we obtain $θ_{0} = 0.5$ as well.
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