Theory of elastic constants of athermal amorphous solids with internal stresses

Abstract

A new microscopic derivation of the elastic constants of amorphous solids is presented within the framework of nonaffine lattice dynamics, which makes use of a perturbative form of the low-frequency eigenvectors of the dynamical matrix introduced in [V. Mazzacurati, G. Ruocco, M. Sampoli EPL 34, 681 (1996)]. The theory correctly recovers the shear modulus at jamming, $\mu \sim (z-2d)$, including prefactors in quantitative agreement with simulations. Furthermore, this framework allows us, for the first time, to include the effect of internal stresses. The theory shows that the Maxwell rigidity criterion $z=2d$ is violated with internal stress. In particular, $\mu \sim (z-2df)$ where $f<1$ if the bonds are, on average, stretched, and the solid is thus rigid below the Maxwell isostatic limit, while $f>1$ if the bonds are, on average, compressed. The coefficient $f$ is derived in analytical form and depends only on $d$ and on the average particle displacement from the interaction energy minimum.

arXiv commentSpecial Issue "In Memoriam of Robert P. Behringer"

References in corpus (12)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Jamming, Force Chains and Fragile Matter
• Theory of nonlinear rheology and yielding of dense colloidal suspensions
• Criticality and isostaticity in fiber networks
• Approximate analytical description of the nonaffine response of amorphous solids
• Green's Function Measurements of Force Transmission in 2D Granular Materials
• Local inversion-symmetry breaking controls the boson peak in glasses and crystals
• Residual Stresses in Glasses
• Non-affine response: jammed packings versus spring networks
• Stresses in isostatic granular systems and emergence of force chains
• Elastic Moduli and Vibrational Modes in Jammed Particulate Packings
• Local structure controls shear and bulk moduli in disordered solids

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• Structural and mechanical characteristics of sphere packings near the jamming transition: From fully amorphous to quasi-ordered structures
• Building rigid networks with prestress and selective pruning
• Comparison of the Helmholtz, Gibbs, and Collective-modes methods to obtain nonaffine elastic constants
• Sinusoidal Displacement Describes Disorder in CsPbBr3 Nanocrystal Superlattices
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