Theory of elastic constants of athermal amorphous solids with internal stresses
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arXiv
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| Format: | Preprint |
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2019
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| author | Cui, Bingyu Ruocco, Giancarlo Zaccone, Alessio |
| author_facet | Cui, Bingyu Ruocco, Giancarlo Zaccone, Alessio |
| contents | 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, $μ\sim (z-2d)$, 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, $μ\sim (z-2df)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_09582 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Theory of elastic constants of athermal amorphous solids with internal stresses Cui, Bingyu Ruocco, Giancarlo Zaccone, Alessio Soft Condensed Matter Disordered Systems and Neural Networks Materials Science Other Condensed Matter Statistical Mechanics 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, $μ\sim (z-2d)$, 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, $μ\sim (z-2df)$ 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. |
| title | Theory of elastic constants of athermal amorphous solids with internal stresses |
| topic | Soft Condensed Matter Disordered Systems and Neural Networks Materials Science Other Condensed Matter Statistical Mechanics |
| url | https://arxiv.org/abs/1901.09582 |