[Paper Review] Atomistic theory of the shear band direction in amorphous solids

J. Ashwin, Oleg Gendelman+2 more|arXiv (Cornell University)|Apr 15, 2013

Metallic Glasses and Amorphous Alloys3 citations

TL;DR

This paper develops an atomistic theory explaining why shear bands in amorphous solids form at angles other than 45° to the principal stress axis under non-volume-conserving loading. Using Eshelby inclusion models for plastic instabilities, it derives an analytic formula for the shear band angle based on the ratio of eigenstrain components ζₙ/ζₖ, showing that only pure shear (volume-preserving) yields exactly 45°, while other loading protocols produce angles between 30° and 60°, in excellent agreement with simulations and experiments.

ABSTRACT

One of the major theoretical riddles in shear banding instabilities is the angle that the shear band chooses spontaneously with respect to the principal stress axis. Here we employ our recent atomistic theory to compute analytically the angle in terms of the characteristics of the Eshelby inclusion that models faithfully the eigenfunction of the Hessian matrix that goes soft at the plastic instability. We show that loading protocols that do not conserve volume result in shear bands at angles different from 45$^o$ to the strain axis; only when the external strains preserve volume like in pure shear, the shear bands align precisely at 45$^o$ to the strain axis. We compute an analytic formula for the angle of the shear band in terms of the characteristics of the loading protocol; quantitative agreement with computer simulations is demonstrated.

Motivation & Objective

• To resolve the long-standing theoretical puzzle of why shear bands in amorphous solids form at angles other than 45° relative to the principal stress axis under various loading protocols.
• To develop a microscopic, atomistic explanation for the asymmetry in shear band orientation observed in uniaxial compression versus extension.
• To derive an analytic formula for the shear band angle based on the eigenstrain characteristics of Eshelby inclusions representing fundamental plastic instabilities.
• To demonstrate quantitative agreement between the theoretical prediction and computer simulations of shear band angles under different loading conditions.

Proposed method

• Model the fundamental plastic instability as an Eshelby inclusion with anisotropic eigenstrain ε*ₐᵦ = ζₙ nₐ nᵦ + ζₖ kₐ kᵦ, where n and k are orthogonal directions along the principal stress and its perpendicular.
• Use linear elasticity theory to compute the displacement field uᶜ(X) generated by the Eshelby inclusion, derived from the Hessian matrix's soft eigenmode associated with the plastic instability.
• Minimize the total energy of a line of such inclusions to determine the preferred orientation of the shear band, leading to a condition for the angle θ with respect to the principal stress axis.
• Derive the key formula: θ = cos⁻¹√[1/2 − (ζₙ + ζₖ)/(4(ζₙ − ζₖ))], which determines the shear band angle from the loading-dependent ratio |ζₙ/ζₖ|.
• Validate the theory by computing ζₙ/ζₖ from simulation data via averaging non-affine displacement vector lengths in the core of plastic events.
• Compare theoretical predictions with observed angles in uniaxial compression (46°) and extension (54°), finding excellent agreement.

Experimental results

Research questions

• RQ1 Why do shear bands in amorphous solids form at angles other than 45° to the principal stress axis under non-volume-conserving loading?
• RQ2 How does the loading protocol influence the orientation of shear bands in amorphous solids at the atomistic level?
• RQ3 What is the microscopic origin of the observed asymmetry in shear band angles between uniaxial compression and extension?
• RQ4 Can the angle of shear band formation be predicted analytically from the characteristics of the plastic instability?

Key findings

• The shear band angle is predicted by the analytic formula θ = cos⁻¹√[1/2 − (ζₙ + ζₖ)/(4(ζₙ − ζₖ))], which depends solely on the ratio of eigenstrain components ζₙ and ζₖ.
• Only under volume-preserving (pure shear) conditions, where ζₙ = −ζₖ, does the theory predict a precise 45° shear band angle.
• For uniaxial compression, the computed ratio |ζₙ/ζₖ| ≈ 1.15 yields a predicted angle of θ ≈ 46°, matching the observed 46° ± 1° in simulations.
• For uniaxial extension, the higher ratio |ζₙ/ζₖ| ≈ 4.05 predicts θ ≈ 54°, in excellent agreement with the observed 54° ± 1°.
• The theory predicts that all shear band angles must lie between 30° and 60°, with 30° and 60° as universal limits for extreme loading conditions.
• The theory explains the observed asymmetry in shear band angles between compression and extension as arising from the anisotropic response of the non-affine displacement field to different loading protocols.