Ratul Dasgupta, H. George E. Hentschel, Itamar Procaccia
In recent research it was found that the fundamental shear-localizing instability of amorphous solids under external strain, which eventually results in a shear band and failure, consists of a highly correlated array of Eshelby quadrupoles all having the same orientation and some density $\rho$. In this paper we calculate analytically the energy $E(\rho,\gamma)$ associated with such highly correlated structures as a function of the density $\rho$ and the external strain $\gamma$. We show that when the strain $\gamma$ is smaller than some characteristic $\gamma_{_{\rm Y}}$ the minimum energy solution is attained for $\rho=0$ (i.e. isolated localized plastic events). For $\gamma \ge \gamma_{_{\rm Y}}$ there is a bifurcation allowing a finite density of quadrupoles. The paper culminates with an analytic estimate of $\gamma_{_{\rm Y}}$, without any free parameters, in terms of material characteristic constants.
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http://arxiv.org/abs/1208.3333
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